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[ [ "Description of the Operational Mechanics of a Basel Regulated Banking\n System" ], [ "Abstract This paper presents a description of the mechanical operations of banking as used in modern banking systems regulated under the Basel Accords, in order to provide support for a verifiable and complete description of the banking system suitable for computer simulation.", "Feedback is requested on the contents of this document, both with respect to the operations described here, and any known national, regional or local variations in their structure and practice." ], [ "Abstract", "This paper presents a description of the mechanical operations of banking as used in modern banking systems regulated under the Basel Accords, in order to provide support for a verifiable and complete description of the banking system suitable for computer simulation.", "Feedback is requested on the contents of this document, both with respect to the operations described here, and any known national, regional or local variations in their structure and practice." ], [ "Introduction", "There appears to be considerable confusion surrounding the precise operation of the modern banking system, in particular with respect to the regulation of lending and deposit creation, the handling of loan defaults, and the relationships between holdings at the central bank and the bank clearing system and the rest of the system.", "Simulation of the aggregate behaviour of the banking system is well within current computing capabilities, and would be highly beneficial both in exploring the impacts of different regulatory frameworks on the behaviour of the system, and to provide a scientific foundation for economic understanding of the monetary system.", "However for simulation efforts to be successful an accurate description of the mechanical operations used by banks in their day to day operations is required and this does not appear to be currently available either within economic theory, or from the regulatory authorities.", "The descriptions that are currently provided by economic textbooks such as Mankiw[1], and McConnell[2].", "are notably deficient, with important aspects of the system such as the precise handling of loan repayments and loan defaults omitted.", "This paper aims to provide a clear and verifiable description of the fundamental operations of the banking system, which can then be used to build accurate simulations of its behaviour.", "We present these operations following the example of late 19th and early 20th century bookkeeping manuals on banking such as Shand[3], by providing detailed descriptions of the fundamental bookkeeping operations performed by a bank as it processes deposits, lends money, receives repayment on loans, and handles loan defaults in a banking system that consists of two banks, A and B, and a central bank." ], [ "Double Entry Bookkeeping.", "Banking as we understand it today has emerged over several centuries from a set of practices first established in Northern Europe by medieval goldsmiths and traders[4].", "It initially developed as a form of statistical multiplexing whereby access to physical money in the form of gold was managed through day to day bookkeeping practices, operated under the assumption that only a fraction of the underlying liabilities (customer deposits) would be requested at any one time.", "Based on this assumption, goldsmiths would make short term loans of gold to other customers, and as the chits used to represent gold deposits began to be exchanged directly a bank based monetary system developed.", "Over time this system has mutated into today's almost entirely electronic transfer based system, however it still retains the bookkeeping practices of the original system, in particular with respect to the relationship between customer deposits, and interbank liabilities both in the form of reserves at the central bank, and deposits held with other banks in the system.", "The historical antecedents of the system are significant, as several of its current features can probably only be appreciated within that context.", "The mechanical operations used by banks in their day to day processing of money and loans, are in large part a creation of the double entry bookkeeping procedures that evolved to track the customer deposits of physical money, and the associated lending activities of the banks.", "Double entry bookkeeping is based on the principle that a general ledger of assets, liabilities and shareholder equity is constructed from a series of separate accounts or individual ledger books (commonly referred to as T-accounts when presented formally).", "The system of accounts for any bookkeeping entity is deliberately structured so that a separate and opposite entry must be made into two T-account simultaneously for each action that occurs.", "That is for each debit in one T-Account there must be a separate matching credit in a different T-account, and vice versa.", "The practice was developed by the Florentines in the 13th century, initially as an anti-fraud measure, since the separate updates to two separate books could be structured to require different people to maintain the entries in each book.", "In accounting assets are generally the resources owned by a company, and liabilities are resources that the bank owes to another separate entity.", "Customer deposits at a bank for example, are classified as liabilities, but when physical cash is deposited at the bank this is classified as an asset, with the corresponding liability being the customer deposit that was created by the deposit of physical money.", "The terms debit and credit have very specific meaning within bookkeeping that are tied to the type of account being operated on.", "For example, debits to accounts classified as dividends, expenses, assets and losses cause the account's balance to increase, whilst credits to accounts classified as income, revenue, liabilities and stockholder's equity cause these accounts to be increased.", "Debits are listed for all accounts in the left hand column, and credits in the right.", "Table: Example of T-Account Cash HandlingThe balance sheet of assets, versus the liabilities and equity of a bank is built up from the set of individual T-accounts.", "In order to maintain this balance, each T-account is classified as either an asset or a liability.", "Increases to an asset T-account are then recorded on its debit side, and decreases as credits; whilst increases to a T-account classified as a liability are recorded as credits and decreases as debits.", "Table REF shows an example of this when physical cash is deposited at a bank.", "Two entries are made, a debit into the bank's vault cash account which is classified under assets, and a credit into the customer's deposit account as a liability.", "The balance of both T-accounts consequently increases, maintaining equality in the balance book.American and English accounting practices reverse the credit/debit convention, in the English system increases to an asset account are recorded as a credit.", "In this document we follow the American conventions.", "As a consequence the structure surrounding the classification of T-accounts as liabilities or assets can be somewhat unintuitive.", "Revenue and capital for example are typically treated as liabilities, with the justification that capital and profits are 'owed' to the shareholders, although more prosaically this treatment is also required to maintain the balance of bookkeeping operations.", "Similarly the handling of loan defaults by banks uses a 'contra-asset' account, which allows income to be reserved on the Asset side of the ledger against expected losses.", "As a result money is removed from the income accounts, that would otherwise be evaluated to determine profits and paid as dividends to shareholders." ], [ "Bank Model", "Economic models of bank operations are frequently presented at the annual balance sheet level, following the basic accounting identity: $Assets = Liabilities + Stockholder^{\\prime }s\\:Equity$ However, correct analysis of banking behaviour requires a consideration of the details of monetary flows within the banking system in their day to day operations, particularly with respect to the handling of loan defaults, which are hidden by this 'identity'.", "The expanded versions of equation REF : $Assets = Liabilities + Common\\:Stock + Retained\\:Earnings$ and $Assets = Liabilities + Common\\:Stock + (Income - Expenses) - Dividends\\footnote {There arepotential order of evaluation issues with this equation if bracketing is not treatedstrictly.", "It perhaps might also be observed that units are not being correctly treated bythe equality in the equation, and this may cause issues for superficial analyses based onit.", "For example, the majority of assets in the banking context are loans which representcontractually committedflows of money, stock is usually represented in financial instruments that are priced in monetaryunits, whilst income, expenses and dividends typically represent money as it is generallyunderstood.", "}$ show the breakdown within the Stockholder's Equity of the bank's day to day monetary flows and its capital holdings.", "Of particular significance is the definition of expenses, which for a bank includes its provisions for loan write-offs.", "A mistake sometimes found in the economic literature is to simply deduct losses from Stockholder's Equity in the basic accounting equation REF , rather than consider the flow implications of the (Income - Expenses) term in the expanded equation REF , which indicate that banks can write-off loans against income with no effect on stock or capital reserves as long as they remain profitable.", "As stockholder's equity is part of the regulatory capital for a bank, and in part determines its lending limits, this can lead to incorrect assumptions about the system's stability.", "Analysis of the banking system is further complicated by the increasingly abstract nature of money, as the banking system continues its evolution away from physical money to a completely electronic system.", "The system was originally based on empirically derived but known ratios between physical money, the price of precious metals, and the quantity of bank loans made at each local bank, regulated by the requirement that a fixed percentage of reserves against deposits was required to be held at the central bank.The description commonly found in economic textbooks such as Mankiw[1], which appears to have been derived from the 1931 Macmillan Report to the British Parliament[5], probably authored by Keynes[6], incorrectly shows a reserve being withheld from the total customer deposits at the bank, rather than as additional funds owned by the bank and maintained in a fixed relationship to the quantity of money represented as deposits.", "While it is not completely correct to equate bank deposits with physical money, if for no other reason than accounting treatment of the two differs significantly, it is equally invalid to fail to acknowledge the role bank deposits play as the de facto money supply in determining the general price level, and indeed have done for over a century[7].", "The original role of bank deposits as a form of multiplexed access to physical money that is in day to day use remains embedded in the book keeping accounts, and now creates a feature of the system that is generally referred to as liquidity, that is the money available to the bank on the asset side of its balance sheet, to satisfy its day to day demands for cash and transfers within the system through its holdings with the central clearing system and other banks." ], [ "Initial Position", "Examples in this document are based on a banking system consisting of two Banks, A and B, and a simplified Central Bank.", "The general ledgers of the two banks are shown together with their reserve account relationship with the central bank.", "The other holdings of the central bank are not shown.", "The starting position used for the examples in this document is shown in Table REF .", "For the examples shown here, the 2% reserve required of European Banks on accounts with notice periods up to 2 years is used.", "It is assumed all deposit accounts at both banks fall within this classificationSource: European Central Bank, http://www.ecb.int/mopo/implement/mr/html/calc.en.html.", "Reserve accounts held by banks at the central banks are treated as deposit accounts by the central bank, and are consequently classed as liabilities of the central bank.", "A matching amount of central bank assets is shown for completeness.", "Cash, cash equivalents and reserves represent the bank's own money, its 'liquidity'.", "Although originally this would have involved significant holdings of physical cash, today these holdings are predominantly electronic, and their significance derives from their position in the system of ledger books in maintaining receipts as funds flow between banks, rather than directly from customers.", "Money paid into or out of the bank is funnelled through its cash asset journal, with a matching credit or debit in the account the money is processed for.", "In an era of electronic operations, this part of the bank's operation can be classified as a vestigial structure derived from gold standard era operations, but one with significant implications for the behaviour of the larger system." ], [ "Fundamental Operations", "The following list of bookkeeping operations describe the fundamental mechanical actions that any bank must perform to maintain its days to day operations.", "Potentially some of these actions, such as transferring money between accounts can be performed differently when done at the same bank, than when done between banks as opposed to at the same bank, and consequently both possibilities are described.", "Transfer between accounts at different banks, i.e.", "cheque or EFT Transfer between accounts at the same bank.", "Lend money to a customer.", "Lend money to a customer at a different bank.", "Borrow from another bank (or central bank) Payment of interest and capital on a bank loan Write off a loan Increase Capital Holdings Increase Reserve Holdings Central Bank Operations Borrow from the Central Bank (Lender of last resort) Payment of interest on reserves at the Central Bank In the examples below, we first show the set of (credit, debit) tuple operations that are performed using the American convention (increases in assets are debits), and then a worked example following the initial position in Table REF ." ], [ "At the same Bank", "When money is transferred between two accounts at the same bank it is a debit to one account, and a credit to the other, with no change to the aggregate liability for the bank shown on the balance sheet.", "Table: NO_CAPTIONTable: Transfer between accounts at same bankThis is in contrast to the procedure used when money is explicitly transferred between different banks shown in section REF , which could also be applied to a transfer occurring between customers at the same bank.", "While it may seem unlikely that there would be such dramatically different treatment, the potential certainly appears to exist, and this would have systemic implications if allowed.", "It is also not known what if any differences in treatment occur when transfers are performed between branches of the same bank.", "It seems distinctly possible that both forms of accounting could be in use by different institutions within the same banking system.Banks that operate unified bookkeeping across all branches would be able to source larger loans, and could also be expected to cause higher monetary expansion rates as they take advantage of a larger liquidity channel with the central bank's clearing mechanisms." ], [ "Transfer between different Banks", "Transfers between the main commercial banks, (clearing banks in the English system) take place through the central bank's clearing operations.Clearing operations today are usually performed through a real time transaction based system, but historically depended on an end of day exchange and balancing approach[8].", "The exact implementation of the clearing operation, particularly with respect to its tolerance or otherwise for negative balances during the day, may have some systemic implications.", "Smaller banks may use accounts at larger banks, rather than direct access to the central clearing systems.", "In the example below we will show a transfer through the reserve accounts held at the central bank.", "For a transfer from customer A.C1 of 1000 at Bank A, to customer B.C3 at Bank B: Table: NO_CAPTIONThe operations are shown in more detail in Tables REF and REF , which show a transfer of 20 from customer A.C1 at Bank A to customer B.C3 at Bank B.", "Table: Transfer: Step 1: Move money to reservesTable: Transfer: Step 2 transfer money to customer A.C1" ], [ "Lending Money", "Similar issues with liquidity considerations and activity that takes place between banks as opposed to those at the same bank can be seen with bank lending.", "Although banks have to assume that the money they loan may end up on deposit at another bank, and manage their liquidity exposures appropriately, many banks express a clear preference for lending to their own rather than other bank's customers, a preference that is also recommended in early banking literature.", "Both alternatives are detailed below." ], [ "A loan of money to its own customer.", "Manuals on bank bookkeeping from the early 20th century indicate that the practice then was to enter the loan and the deposit simultaneously in the ledger books as shown here and there is no evidence that this practice has ever changed.", "\"If a loan is granted, an entry is made in a Customers' loan register, and passed for entry in the Current Accounts Credit Analysis book.", "Against the credit so placed to his Current Account, the customer draws in the ordinary way.", "Bank Bookkeeping and Accounts, Meelboom (p35-p36)[9].", "For a loan of 500 made by Bank A to its own customer A.C1 the operation proceeds as follows: Table: NO_CAPTIONTable: Loan to Bank's own customerBesides adding to both the loan and customer deposit accounts, the bank may also need to adjust its reserve provisions with respect to the new level of customer deposits.", "In this case 10 is transferred from the bank's cash holdings to the reserve account at the cental bank.", "We assume in this example that the bank is still within its risk weighted capital multiple, and does not need to adjust its capital holdings.", "Banks lend money against their asset holdings, with the total amount they can lend regulated by reserve requirements at the central bank, capital requirements and in the case of loans made to customers of other banks, or directly to other banks(interbank lending), their cash holdings.", "There are restrictions on the total amount of its loans that a bank can maintain.", "Under the Basel accords, it must be within its risk weighted capital restrictions, and it must also be able to meet the reserve requirement on its new level of deposits.", "To lend to another bank's customer, the bank must additionally have available liquidity for the transfer of money for the loan, and in practice since the bank must assume that its funds may be transferred to other banks, these considerations also apply to loans to its own customers." ], [ "Lend to another Bank's Customer", "Lending to a customer at a different bank by contrast requires use of the interbank transfer mechanisms and follows a different sequence of operations, as shown below.", "Table: NO_CAPTIONTable: Loan to another Bank's CustomerTable: Loan to another Bank's customerThis example also illustrates another feature of the system, that the creation of money in the form of customer deposit entries is independent of the money on deposit at the central bank (base money) and within the clearing system unless the system is operating at the limits of its reserve requirements.In any banking system where accounts exist that do not carry reserve requirements (only Net Transaction Accounts require a reserve in the US system, while time deposits of greater than two years do not require reserves in the euro-zone), reserve limits effectively only throttle the system's deposit expansion rate, and do not set absolute limits on expansion." ], [ "Interbank Loan", "A loan to another bank is similar to a loan to a customer at a different bank, with side effects involving liquidity availability.", "It is accounted as a liability at the bank receiving the loan, and as an asset at the bank making it.", "Table: NO_CAPTIONTable: NO_CAPTIONTable: NO_CAPTION" ], [ "Loan Repayment", "Loan repayment is broken into two parts, repayment of the principal outstanding on the loan, and repayment of the interest.", "Repayment of the principal is a balanced operation, with a simple deduction from both sides of the balance sheet in the event that the loan is made to a customer of the same bank which made the loan.", "Interest is received as income by the Bank holding the loan, however its accounting is more complex.", "Strictly, GAAP requirements are that interest is accrued on a daily basis, rather than when it is actually paid, but to simplify the presentation this step is not shown.", "Income is accounted for as a liability, as it nominally represents revenue that will be paid to the shareholders.", "In reality, this is a requirement for the double entry bookkeeping operations surrounding it to remain balanced, which carries implications for the treatment of expenses, and in particular loan defaults or write-offs which are treated as an expense in bank accounting.", "Outside the artificial constraints of bookkeeping practices, income received by the bank has asset like properties, and in particular can be transferred to the asset side when needed to compensate for loan losses.", "This element of bank liquidity is systemically interesting for a number of reasons, in particular loan defaults that can be covered from income do not impact the quantity of bank lending that is regulated by the capital requirement.", "Consequently, when a loan is written off purely against income, the bank is able to extend new loans on its existing capital base, and subject to liquidity availability the total amount of lending it can perform is not affected.", "The example shown in Table REF and REF shows a loan repayment of 100, split into two parts, a 40 principal repayment and a 60 interest payment made by customer A.C1 at Bank A.", "The two payments are processed separately to illustrate the different handling for interest versus capital repayment, and also that while interest repayment is money supply neutral, principal repayment removes the deposit from the system that was originally created by the loan.", "Table: NO_CAPTIONTable: Principal RepaymentTable: Interest Repayment" ], [ "Loan Default", "Losses on loans are initially treated as an expense for banks, and are effectively deducted from income, but there are several stages to this process.", "Additional and potentially systemic complexities can occur if the capital reserve becomes involved.", "In general loan write-offs are a fairly predictable occurrence, several payments have to be missed before a loan can be treated as impaired.", "Banks are required to provision against potential losses on a loan at the same time it is made, and to continuously monitor and adjust loss provisions to match their anticipated losses.", "Banks also have some freedom on how impaired loans are treated, and may elect to write off all or some of the loan, or refinance it.", "If loan losses and other expenses significantly exceed income, then the capital reserve is used to cover the write-off.", "With respect to the Basel Accord Tier 1 and Tier 2 capital lending provisions, the bank may be over capitalised, in which case there is a buffer of capital that can be used for this purpose without any impact on its ability to lend with respect to its risk weighted capital reserve multiplier.", "However, since banks must maintain a limit on their lending that is a multiple of their Tier 1 & 2 reserve funds, if losses are sufficiently high they can push the bank out of regulatory compliance, since it will no longer meet its capital requirement.", "This last situation is rarely recoverable without external intervention.", "The bookkeeping arrangements that are used to represent the first part of this process use a contra-asset account for loan losses, which is then subtracted from gross loans.A contra-asset account is an asset account which has a credit balance, normally asset accounts maintain a debit balance.", "The contra-asset account is itself linked to an asset account, and the book value is the net value of the two accounts.", "It is effectively a way of carrying an offsetting allowance forward for loss provisions or depreciations, linked to specific ledgers for tracking purposes.", "In the example below, we will begin with showing the creation of a loan loss account from income received, and then assume that its contents are sufficient to cover losses from the loan write down of 50 for Bank A.", "The significance of the contra-asset designation of the loss provisions account becomes clearer when the transfer from the interest income account is examined.", "The funds are credited to the loss provisions account, but as it is a contra-asset account they act to reduce the total asset balance.", "When the bank then writes off part of its loan book (50 in the example shown in Table REF , the loss provision account is reduced by the amount of the write-off, as is the loan book.", "Strictly, the loss provision account is debited, reducing its balance, and the loan account is credited, also reducing its balance, since it as an Asset account.", "The net balance of the Assets is unchanged as a result, since the loss provision account has already accounted for the write off[10].", "As a result, the impact of the write-off on the balance sheet usually precedes the actual write-off.", "As long as the loss can be covered from income, then the overall state of the balance sheet (with respect to the reduction in assets caused by the write-off) can be restored by new lending.", "Nor are there any accompanying money supply considerations, since the money removed from the system by the loan write-off is replaced by the new loan.", "Some degree of loan losses can consequently be absorbed by the system with no systemic repercussions for the money supply.", "There may be implications for the larger economic debt supply, depending on the subsequent treatment of the loan, which although removed from the bank's balance sheet, may be sold on for collection.", "Local practices surrounding the handling of bankruptcy and recourse and non-recourse lending will play a part there.", "Table: NO_CAPTIONTable: Loan Writeoff - Initial ConditionsTable: Loan Writeoff - Creation of Loss Reserve AccountTable: Loan Writeoff - Writeoff against loss provisions" ], [ "Increase Capital", "The capital holdings of a bank are initially the shares purchased by its stockholders when the bank is founded.", "The money received by the bank for this purpose becomes its asset cash holdings.", "Although the tradable price of shares varies with stock market conditions, the book value used for common stock held in the bank's capital is the money received by the bank and initially entered into its cash asset ledger.", "Under Basel, capital holdings are divided into two Tiers with regulated definitions for the financial instruments that can be held in the different tiers, and separate ratios for the loans that can be extended against their capital holdings.", "Broadly, Tier 1 consists of common stock and disclosed reserves or retained earnings.", "Tier 2 holds undisclosed reserves, revaluation reserves, additional reserves for loan losses (holdings additional to the loss provisions described above), and subordinated debt.", "(Subordinate debt is money that has been borrowed by the bank, but is subordinate to the claims of the depositors on bank funds.)", "Basel 2 included a Tier 2 category of \"hybrid capital instruments\" which are financial instruments having qualities of both debt and equity.", "The category has proved somewhat controversial, with a number of such instruments being explicitly forbidden by the regulators, and appears to be being removed in Basel 3.", "There appear to be no restrictions or controls on increases to the capital reserve, which can be done from profits, but liquidity would be required for any purchase of financial instruments such as government treasuries.", "Since sales of bank stock add to liquidity, this restriction would not apply to that channel.", "the deposit holder of account A.C1.", "Table: NO_CAPTIONTable: Sale of Stock to increase Capital" ], [ "Borrow from Central Bank", "Central bank operations are in principle no different to other bank operations, but operate from a privileged position in the system with respect to the other banks.", "Table: NO_CAPTIONTable: Loan from Central Bank" ], [ "Payment of interest on reserve holdings by Central Bank.", "Payment of interest on the reserve holdings is a necessary feature of the system, otherwise systemic imbalances would result over time from the asymmetric flow within the system as central bank loans were repaid by the clearing banks.", "In the example in Table REF it is assumed that 10 has been received by the Central Bank as interest payment on its loans, and this is now paid to Bank A as interest on its reserves.", "Table: NO_CAPTIONTable: Initial PositionTable: Payment of Interest on Reserves" ], [ "Reserve Holdings", "Two forms of reserve holdings exert regulatory control within the system.", "The capital reserve regulates the total amount of loans that can be made by the bank, while the reserve held at the central bank, in principle at least regulates the amount of deposits that the bank may hold.", "Potentially, as lending is also linked to deposit creation the central bank reserve can exert some regulation over lending as well.", "For this to occur however, two conditions have to be true.", "One is that the central bank reserve requirement is greater than the capital reserve requirement, otherwise the capital reserve requirement will dominate.", "The other is that the reserve requirement is applied to all deposits accounts without exception.", "Consequently the regulatory effect of the central bank reserve can be diluted in practice.", "A further consideration, with systemic implications, is also the mechanisms by which banks are allowed to increase their central bank reserves.", "In the USA it seems this can be done through the deposit of government treasuries, which in practice would remove systemic control over the quantity of reserves in the system." ] ]

[ [ "The compound class of extended Weibull power series distributions" ], [ "Abstract In this paper, we introduce a new class of distributions which is obtained by compounding the extended Weibull and power series distributions.", "The compounding procedure follows the same set-up carried out by Adamidis and Loukas (1998) and defines at least new 68 sub-models.", "This class includes some well-known mixing distributions, such as the Weibull power series (Morais and Barreto-Souza, 2010) and exponential power series (Chahkandi and Ganjali, 2009) distributions.", "Some mathematical properties of the new class are studied including moments and generating function.", "We provide the density function of the order statistics and obtain their moments.", "The method of maximum likelihood is used for estimating the model parameters and an EM algorithm is proposed for computing the estimates.", "Special distributions are investigated in some detail.", "An application to a real data set is given to show the flexibility and potentiality of the new class of distributions." ], [ "Introduction", "The modeling and analysis of lifetimes is an important aspect of statistical work in a wide variety of scientific and technological fields.", "Several distributions have been proposed in the literature to model lifetime data by compounding some useful lifetime distributions.", "Adamidis and Loukas (1998) introduced a two-parameter exponential-geometric (EG) distribution by compounding an exponential distribution with a geometric distribution.", "In the same way, the exponential Poisson (EP) and exponential logarithmic (EL) distributions were introduced and studied by Kus (2007) and Tahmasbi and Rezaei (2008), respectively.", "Recently, Chahkandi and Ganjali (2009) proposed the exponential power series (EPS) family of distributions, which contains as special cases these distributions.", "Barreto-Souza et al.", "(2010) and Lu and Shi (2011) introduced the Weibull-geometric (WG) and Weibull-Poisson (WP) distributions which naturally extend the EG and EP distributions, respectively.", "In a very recent paper, Morais and Barreto-Souza (2011) defined the Weibull power series (WPS) class of distributions which contains the EPS distributions as sub-models.", "The WPS distributions can have an increasing, decreasing and upside down bathtub failure rate function.", "Now, consider the class of extended Weibull (EW) distributions, as proposed by Gurvich et al.", "(1997), having the cumulative distribution function (cdf) $G(x;\\, \\alpha , \\xi ) = 1 - \\mathrm {e}^{-\\alpha \\, H(x;\\, \\xi )}, \\quad x>0, \\,\\,\\, \\alpha >0,$ where $H(x;\\, \\xi )$ is a non-negative monotonically increasing function which depends on a parameter vector $\\xi $ .", "The corresponding probability density function (pdf) is given by $g(x;\\, \\alpha , \\xi ) = \\alpha \\, h(x;\\, \\xi ) \\,\\mathrm {e}^{-\\alpha \\,H(x;\\,\\xi )}, \\quad x > 0, \\,\\,\\, \\alpha >0,$ where $h(x;\\, \\xi )$ is the derivative of $H(x; \\,\\xi )$ .", "Note that many well-known models are special cases of equation (REF ) such as: (i) $H(x; \\xi ) = x$ gives the exponential distribution; (ii) $H(x; \\xi ) = x^2$ yields the Rayleigh distribution (Burr type-X distribution); (iii) $H(x; \\xi ) = \\log (x/k)$ leads to the Pareto distribution; (iv) $H(x; \\xi ) = \\beta ^{-1}[\\exp (\\beta x)-1]$ gives the Gompertz distribution.", "In this article, we define the extended Weibull power series (EWPS) class of univariate distributions obtained by compounding the extended Weibull and power series distributions.", "The compounding procedure follows the key idea of Adamidis and Loukas (1998) or, more generally, by Chahkandi and Ganjali (2009) and Morais and Barreto-Souza et al.", "(2011).", "The new class of distributions contains as special models the WPS distributions, which in turn extends the EPS distributions and defines at least new 68 (17 $\\times $ 4) sub-models as special cases.", "The hazard function of our class can be decreasing, increasing, bathtub and upside down bathtub.", "We are motivated to introduce the EWPS distributions because of the wide usage of the general class of Weibull distributions and the fact that the current generalization provides means of its continuous extension to still more complex situations.", "This paper is organized as follows.", "In Section 2, we define the EWPS class of distributions and demonstrate that there are many existing models which can be deduced as special cases of the proposed unified model.", "In Section 3, we provide the density, survival and hazard rate functions and derive some useful expansions.", "In Section 4, we obtain its quantiles, ordinary and incomplete moments.", "Further, the order statistics are discussed and their moments are determined.", "Section 5 deals with reliability and average lifetime.", "Estimation of the parameters by maximum likelihood using an EM algorithm and large sample inference are investigated in Section 6.", "In Section 7, we present suitable constraints leading to the maximum entropy characterization of the new class.", "Three special cases of the proposed class are studied in Section 8.", "In Section 9, we provide an application to a real data set.", "The paper is concluded in Section 10." ], [ "The new class", "Our class can be derived as follows.", "Given $N$ , let $X_1, \\ldots , X_N$ be independent and identically distributed (iid) random variables following (REF ).", "Here, $N$ is a discrete random variable following a power series distribution (truncated at zero) with probability mass function $p_n = P(N=n)=\\frac{a_n \\, \\theta ^n}{C(\\theta )}, n=1,2,\\ldots ,\\\\$ where $a_n$ depends only on $n$ , $C(\\theta ) = \\sum _{n=1}^{\\infty }a_n \\,\\theta ^n$ and $\\theta >0$ is such that $C(\\theta )$ is finite.", "Table REF summarizes some power series distributions (truncated at zero) defined according to (REF ) such as the Poisson, logarithmic, geometric and binomial distributions.", "Let $X_{(1)} = \\mbox{min}\\left\\lbrace X_i\\right\\rbrace ^{N}_{i=1}$ .", "The conditional cumulative distribution of $X_{(1)}|N = n$ is given by $G_{X_{(1)}|N=n}(x) = 1-\\mathrm {e}^{-n\\alpha H(x; \\xi )},$ i.e., $X_{(1)}|N = n$ follows a general class of distributions (REF ) with parameters $n\\alpha $ and $\\xi $ based on the same $H(x; \\xi )$ function.", "Hence, we obtain $P(X_{(1)} \\le x, N=n) = \\frac{a_n\\, \\theta ^n}{C(\\theta )}\\left[1-\\mathrm {e}^{-n\\alpha H(x; \\xi )}\\right], \\quad x>0,\\quad n \\ge 1.$ The EWPS class of distributions can then be defined by the marginal cdf of $X_{(1)}$ : $F(x;\\theta ,\\alpha , \\xi ) = 1-\\frac{C(\\theta \\,\\mathrm {e}^{-\\alpha H(x; \\xi )})}{C(\\theta )}, \\quad x>0.$ Table: Useful quantities for some power series distributions.", "[!htbp] Table: NO_CAPTION Special distributions and related $H(x;\\,\\xi )$ and $h(x;\\,\\xi )$ functions.", "The random variable $X$ following (REF ) with parameters $\\theta $ and $\\alpha $ and the vector $\\xi $ of parameters is denoted by $X\\sim \\mbox{EWPS}(\\theta ,\\alpha , \\xi )$ .", "Equation (REF ) extends several distributions which have been studied in the literature.", "The EG distribution (Adamidis and Loukas, 1998) is obtained by taking $H(x;\\,\\xi )=x$ and $C(\\theta ) = \\theta (1-\\theta )^{-1}$ with $\\theta \\in (0,1)$ .", "Further, for $H(x;\\,\\xi )=x$ , we obtain the EP (Kus, 2007) and EL (Tahmasbi and Rezaei, 2008) distributions by taking $C(\\theta ) = \\mathrm {e}^{\\theta }-1, \\theta >0$ , and $C(\\theta ) = -\\log (1-\\theta ), \\theta \\in (0,1)$ , respectively.", "In the same way, for $H(x;\\,\\xi )=x^\\gamma $ , we obtain the WG (Barreto-Souza et al., 2009) and WP (Lu and Shi, 2011) distributions.", "The EPS distributions are obtained from (REF ) by mixing $H(x;\\,\\xi )=x$ with any $C(\\theta )$ listed in Table REF (see Chahkandi and Ganjali, 2009).", "Finally, we obtain the WPS distributions from (REF ) by compounding $H(x;\\,\\xi )=x^\\gamma $ with any $C(\\theta )$ in Table REF (see Morais and Barreto-Souza, 2011).", "Table displays some useful quantities and respective parameter vectors for each particular distribution." ], [ "Density, survival and hazard functions", "The density function associated to (REF ) is given by $f(x;\\theta ,\\alpha , \\xi ) = \\theta \\, \\alpha \\,h(x;\\xi )\\, \\mathrm {e}^{-\\alpha H(x;\\,\\xi )}\\, \\frac{C^{\\prime }(\\theta \\, \\mathrm {e}^{-\\alpha H(x;\\,\\xi )})}{C(\\theta )}, \\quad x>0.$ Proposition 1 The EW class of distributions with parameters $c\\alpha $ and $\\xi $ is a limiting special case of the EWPS class of distributions when $\\theta \\rightarrow 0^+$ , where $c= \\min \\left\\lbrace n \\in \\mathbb {N}: a_n >0\\right\\rbrace $ .", "This proof uses a similar argument to that found in Morais and Barreto-Souza (2011).", "Define $c= \\min \\left\\lbrace n \\in \\mathbb {N}: a_n >0\\right\\rbrace $ .", "We have $\\lim _{\\theta \\rightarrow 0^+} F(x) &= 1 - \\lim _{\\theta \\rightarrow 0^+} \\frac{\\displaystyle {\\sum _{n=c}^\\infty a_n\\left(\\theta \\, \\mathrm {e}^{-\\alpha H(x; \\xi )}\\right)^n}}{\\displaystyle {\\sum _{n=c}^\\infty a_n\\, \\theta ^n}}\\\\&= 1 - \\lim _{\\theta \\rightarrow 0^+} \\frac{\\displaystyle {\\mathrm {e}^{-c \\alpha H(x; \\xi )} + a_c^{-1} \\sum _{n=c+1}^\\infty a_n \\,\\theta ^{n-c}\\mathrm {e}^{-n\\alpha H(x; \\xi )}}}{\\displaystyle {1 + a_c^{-1}\\sum _{n=c+1}^\\infty a_n\\, \\theta ^{n-c}}}\\\\&= 1-\\mathrm {e}^{-c\\alpha H(x; \\xi )},$ for $x>0$ .", "We now provide an interesting expansion for ($\\ref {pdf}$ ).", "We have $C^{\\prime }(\\theta ) = \\sum _{n=1}^{\\infty }n\\, a_n\\, \\theta ^{n-1}$ .", "By using this result in (REF ), it follows that $f(x;\\theta ,\\alpha , \\xi ) = \\sum _{n=1}^{\\infty }p_n\\,g(x;\\, n\\alpha , \\xi ),$ where $g(x;\\, n\\alpha , \\xi )$ is given by (REF ).", "Based on equation (REF ), we obtain $F(x;\\theta ,\\alpha , \\xi ) = 1 - \\sum _{n=1}^\\infty p_n\\,\\mathrm {e}^{-n\\alpha H(x;\\, \\xi )}.$ Hence, the EWPS density function is an infinite mixture of EW densities.", "So, some mathematical quantities (such as ordinary and incomplete moments, generating function and mean deviations) of the EWPS distributions can be obtained by knowing those quantities for the baseline density function $g(x;\\, n\\alpha , \\xi )$ .", "The EWPS survival function is given by $S(x; \\theta , \\alpha , \\xi )= \\frac{C(\\theta \\,\\mathrm {e}^{-\\alpha H(x;\\,\\xi )})}{C(\\theta )}$ and the corresponding hazard rate function becomes $\\tau (x; \\theta , \\alpha , \\xi ) = \\theta \\alpha \\, h(x;\\,\\xi ) \\, \\mathrm {e}^{-n\\alpha H(x; \\xi )}\\,\\frac{C^{\\prime }(\\theta \\, \\mathrm {e}^{-\\alpha H(x;\\,\\xi )})}{C(\\theta \\, \\mathrm {e}^{-\\alpha H(x;\\,\\xi )})}.$" ], [ "Quantiles, moments and order statistics", "The EWPS distributions are easily simulated from (REF ) as follows: if $U$ has a uniform $U(0,1)$ distribution, then the solution of the nonlinear equation $X = H^{-1}\\left\\lbrace -\\frac{1}{\\alpha }\\log \\left[\\frac{C^{-1}(C(\\theta )(1-U))}{\\theta }\\right]\\right\\rbrace $ has the EWPS$(\\theta ,\\alpha , \\xi )$ distribution, where $H^{-1}(\\cdot )$ and $C^{-1}(\\cdot )$ are the inverse functions of $H(\\cdot )$ and $C(\\cdot )$ , respectively.", "To simulate data from this nonlinear equation, we can use the matrix programming language Ox through SolveNLE subroutine (see Doornik, 2007).", "We now derive a general expression for the $r$ th raw moment of $X$ , which may be determined by using (REF ) and the monotone convergence theorem.", "So, for $r \\in \\mathbb {N}$ , we obtain $\\operatorname{E}(X^r) = \\sum _{n=1}^{\\infty }p_n\\, \\operatorname{E}(Z^r),\\\\$ where $Z$ is a random variable with pdf $g(z; n\\alpha ,\\xi )$ .", "The incomplete moments and moment generating function (mgf) follow by using (REF ) and the monotone convergence theorem: $I_X(y) &= \\int ^{y}_{0}x^r\\, f(x)dx = \\sum _{n=1}^{\\infty }p_n\\, I_{Z}(y)\\\\\\multicolumn{2}{l}{\\text{and}}\\\\ M_{X}(t) &=\\sum _{n=1}^{\\infty }p_n\\,\\operatorname{E}\\left(\\mathrm {e}^{tZ}\\right).$ where $Z$ is defined as before.", "Order statistics are among the most fundamental tools in non-parametric statistics and inference.", "They enter in the problems of estimation and hypothesis tests in a variety of ways.", "Therefore, we now discuss some properties of the order statistics for the proposed class of distributions.", "The pdf $f_{i:m}(x)$ of the $i$ th order statistic for a random sample $X_1, \\ldots , X_m$ from the EWPS distribution is given by $f_{i:m}(x) = \\frac{m!}{(i-1)!(m-i)!", "}f(x;\\theta ,\\alpha , \\xi )\\left[1-\\frac{C(\\theta \\, \\mathrm {e}^{-\\alpha H(x;\\,\\xi )})}{C(\\theta )}\\right]^{i-1}\\left[\\frac{C(\\theta \\mathrm {e}^{-\\alpha H(x;\\,\\xi )})}{C(\\theta )}\\right]^{m-i}, \\quad x> 0,$ where $f (x; \\theta , \\alpha , \\xi )$ is the pdf given by (REF ).", "By using the binomial expansion, we can write (REF ) as $f_{i:m}(x) = \\frac{m!}{(i-1)!(m-i)!", "}f(x;\\theta ,\\alpha , \\xi )\\sum _{j=0}^{i-1} (-1)^j \\,\\binom{i-1}{j}\\, S(x;\\theta ,\\alpha ,\\xi )^{m+j-i},$ where $S(x; \\theta , \\alpha , \\xi )$ is given by (REF ).", "The corresponding cumulative function is $F_{i:m}(x) = \\sum _{j=0}^{\\infty }\\sum _{k=i}^{m}(-1)^j\\,\\binom{k}{j}\\,\\binom{m}{k}\\,S(x;\\theta ,\\alpha ,\\xi )^{m+j-k}.$ An alternative form for (REF ) can be obtained from (REF ) as $f_{i:m}(x) = \\frac{m!}{(i-1)!", "(m-i)!}", "\\sum _{n=1}^\\infty \\sum _{j=0}^{i-1} \\omega _j\\, p_n\\, g(x; n\\alpha ,\\xi )S(x;\\theta ,\\alpha , \\xi )^{m+j-1},$ where $\\omega _j = (-1)^j \\binom{i-1}{j}$ .", "So, the $s$ th raw moment $X_{i:m}$ comes immediately from the above equation $\\operatorname{E}\\left(X_{i:m}^s\\right) = \\frac{m!}{(i-1)!", "(m-i)!", "}\\sum _{n=1}^\\infty \\sum _{j=0}^{i-1} \\omega _j\\, p_n\\,\\operatorname{E}\\left[Z^s S(Z)^{m+j-i}\\right],$ where $Z\\sim \\mbox{EW}(n\\alpha , \\xi )$ is defined before." ], [ "Reliability and average lifetime", "In the context of reliability, the stress-strength model describes the life of a component which has a random strength $X$ subjected to a random stress $Y$ .", "The component fails at the instant that the stress applied to it exceeds the strength, and the component will function satisfactorily whenever $X > Y$ .", "Hence, $R = \\operatorname{P}(X > Y)$ is a measure of component reliability.", "It has many applications, especially in engineering concepts.", "The algebraic form for R has been worked out for the majority of the well-known distributions.", "Here, we obtain the form for the reliability $R$ when $X$ and $Y$ are independent random variables having the same EWPS distribution.", "The quantity $R$ can be expressed as $R = \\int _0^{\\infty }f(x; \\theta , \\alpha , \\xi )F(x;\\theta ,\\alpha , \\xi ) dx.$ Substituting (REF ) and (REF ) into equation (REF ), we obtain $R &=& \\int _{0}^{\\infty }\\theta \\, \\alpha \\,h(x;\\xi )\\, \\mathrm {e}^{-\\alpha H(x;\\,\\xi )}\\, \\frac{C^{\\prime }(\\theta \\mathrm {e}^{-\\alpha H(x;\\,\\xi )})}{C(\\theta )}\\left[1-\\frac{C(\\theta \\mathrm {e}^{-\\alpha H(x; \\xi )})}{C(\\theta )}\\right] dx\\\\&=&1 - \\sum _{n=1}^{\\infty }p_n\\int _{0}^{\\infty }g(x;n\\alpha ,\\xi )S(x;\\theta ,\\alpha , \\xi )dx,$ where the integral can be calculated from the baseline EW distribution.", "The average lifetime is given by $t_m =\\sum _{n=1}^{\\infty }p_n\\int \\limits _{0}^{\\infty }\\operatorname{e}^{-n\\alpha H(x;\\, \\xi )}dx.$ Given that there was no failure prior to $x_0$ , the residual life is the period from time $x_0$ until the time of failure.", "The mean residual lifetime can be expressed as $m(x_0;\\theta ,\\alpha ,\\xi ) &=& \\left[\\operatorname{Pr}(X>x_0)\\right]^{-1}\\int \\limits _{0}^{\\infty }y\\,f(x_0+y;\\theta ,\\alpha ,\\xi ) dy\\\\&=&[S(x_0)]^{-1}\\sum _{n=1}^{\\infty }p_n\\int \\limits _{0}^{\\infty }y\\,g(x_0+y;n\\alpha ,\\xi ) dy.$ The last integral can be computed from the baseline EW distribution.", "Furthermore, $m(x_0;\\theta ,\\alpha ,\\xi ) \\rightarrow \\operatorname{E}(X)$ as $x_0 \\rightarrow 0$ ." ], [ "Preliminaries", "Here, we determine the maximum likelihood estimates (MLEs) of the parameters of the EWPS class of distributions from complete samples only.", "Let $X_1, \\ldots , X_n$ be a random sample with observed values $x_1, \\ldots , x_n$ from an EWPS distribution with parameters $\\theta , \\alpha $ and $\\xi $ .", "Let $\\Theta = (\\theta ,\\alpha , \\xi )^\\top $ be the $p \\times 1$ parameter vector.", "The total log-likelihood function is given by $ \\nonumber \\ell _n &=& \\ell _{n}(x; \\Theta ) = n\\left[\\log \\theta + \\log \\alpha - \\log C(\\theta )\\right] - \\alpha \\sum _{i=1}^{n}H(x_i;\\, \\xi ) +\\sum _{i=1}^{n}\\log h(x_i;\\, \\xi )\\\\&+& \\sum _{i=1}^{n}\\log C^{\\prime }(\\theta \\, \\mathrm {e}^{-\\alpha H(x_i;\\, \\xi )}).$ The log-likelihood can be maximized either directly by using the SAS (PROC NLMIXED) or the Ox program (sub-routine MaxBFGS) (see Doornik, 2007) or by solving the nonlinear likelihood equations obtained by differentiating (REF ).", "The components of the score function $U_n(\\Theta ) = \\left(\\partial \\ell _n/\\partial \\theta , \\partial \\ell _n/\\partial \\alpha , \\partial \\ell _n/\\partial \\xi \\right)^\\top $ are $\\frac{\\partial \\ell _n}{\\partial \\alpha } &= \\frac{n}{\\alpha } - \\sum _{i=1}^{n} H(x_i;\\, \\xi ) - \\theta \\sum _{i=1}^n H(x_i;\\, \\xi ) \\mathrm {e}^{-\\alpha H(x_i;\\, \\xi )}\\,\\frac{C^{\\prime \\prime }(\\theta \\, \\mathrm {e}^{-\\alpha H(x_i;\\, \\xi )})}{C^{\\prime }(\\theta \\, \\mathrm {e}^{-\\alpha H(x_i;\\, \\xi )})},\\\\\\frac{\\partial \\ell _n}{\\partial \\theta } &= \\frac{n}{\\theta } - n\\frac{C^{\\prime }(\\theta )}{C(\\theta )} + \\sum _{i=1}^n \\mathrm {e}^{-\\alpha H(x_i;\\, \\xi )}\\,\\frac{C^{\\prime \\prime }(\\theta \\, \\mathrm {e}^{-\\alpha H(x_i;\\, \\xi )})}{C^{\\prime }(\\theta \\, \\mathrm {e}^{-\\alpha H(x_i;\\, \\xi )})}\\\\\\multicolumn{2}{l}{\\text{and}}\\\\ \\frac{\\partial \\ell _n}{\\partial \\xi _k} &=\\sum _{i=1}^{n} \\frac{\\partial \\log h(x_i;\\, \\xi )}{\\partial \\xi _k} - \\alpha \\sum _{i=1}^{n} \\frac{\\partial H(x_i;\\, \\xi )}{\\partial \\xi _k}\\left[1 + \\theta \\mathrm {e}^{-\\alpha H(x_i; \\, \\xi )} \\frac{C^{\\prime \\prime }(\\theta \\, \\mathrm {e}^{-\\alpha H(x_i;\\,\\xi )})}{C^{\\prime }(\\theta \\, \\mathrm {e}^{-\\alpha H(x_i;\\,\\xi )})}\\right].$ For interval estimation on the model parameters, we require the observed information matrix $ J_n(\\Theta )=-\\left( \\begin{array}{cccc}U_{\\theta \\theta } & U_{\\theta \\alpha } & |&U_{\\theta \\xi }^\\top \\\\U_{\\alpha \\theta } & U_{\\alpha \\alpha } & |&U_{\\alpha \\xi }^\\top \\\\-- & --& --& --\\\\U_{\\theta \\xi } & U_{\\alpha \\xi } & | & U_{\\xi \\xi } \\end{array} \\right),$ whose elements are listed in .", "Let $\\widehat{\\Theta }$ be the MLE of $\\Theta $ .", "Under standard regular conditions stated in Cox and Hinkley (1974) that are fulfilled for our model whenever the parameters are in the interior of the parameter space, we have that the asymptotic distribution of $\\sqrt{n}\\left(\\widehat{\\Theta } - \\Theta \\right)$ is multivariate normal $N_p(0, K(\\Theta )^{-1})$ , where $K(\\Theta ) = \\lim _{n \\rightarrow \\infty }J_n(\\Theta )$ is the unit information matrix and $p$ is the number of parameters of the compounded distribution." ], [ "The EM algorithm", "Here, we propose an EM algorithm (Dempster et al., 1977) to estimate $\\Theta $ .", "The EM algorithm is a recurrent method such that each step consists of an estimate of the expected value of a hypothetical random variable and then maximizes the log-likelihood for the complete data.", "Let the complete-data be $X_1, \\ldots , X_n$ with observed values $x_1, \\ldots , x_n$ and the hypothetical random variables $Z_1, \\ldots , Z_n$ .", "The joint probability function is such that the marginal density of $X_1, \\ldots , X_n$ is the likelihood of interest.", "Then, we define a hypothetical complete-data distribution for each $(X_i, Z_i)^\\top , i=1, \\ldots , n$ , with a joint probability function in the form $g(x, z; \\Theta ) = \\frac{\\alpha \\, z\\, a_z\\, \\theta ^z}{C(\\theta )}\\,h(x;\\, \\xi )\\, \\mathrm {e}^{-\\alpha z H(x;\\, \\xi )},$ where $\\theta $ and $\\alpha $ are positive, $x>0$ and $z \\in \\mathbb {N}$ .", "Under this formulation, the E-step of an EM cycle requires the expectation of $Z|X$ ; $\\Theta ^{(r)} = (\\theta ^{(r)}, \\alpha ^{(r)},\\xi ^{(r)})^\\top $ as the current estimate (in the rth iteration) of $\\Theta $ .", "The probability function of $Z$ given $X$ , say $g(z|x)$ , is given by $g(z|x) = \\frac{z\\, a_z\\, \\theta ^{\\theta -1}}{C^{\\prime }(\\theta e^{-\\alpha H(x_i;\\, \\xi )})}\\,\\mathrm {e}^{-\\alpha (z-1) H(x_i;\\, \\xi )}$ and its expected value is $E(Z|X) = 1 + \\theta e^{-\\alpha H(x;\\, \\xi )}\\,\\frac{C^{\\prime \\prime }(\\theta \\, \\mathrm {e}^{-\\alpha H(x;\\, \\xi )})}{C^{\\prime }(\\theta \\, \\mathrm {e}^{-\\alpha H(x;\\, \\xi )})}.$ The EM cycle is completed with the M-step by using the maximum likelihood estimation over $\\Theta $ , where the missing $Z^{\\prime }s$ are replaced by their conditional expectations given before.", "The log-likelihood for the complete-data is $\\textstyle \\ell _n^*(x_1, \\ldots , x_n; \\, z_1, \\ldots , z_n; \\,\\alpha , \\,\\theta , \\,\\xi ) &\\propto n\\log \\alpha + \\log \\theta \\sum _{i=1}^n z_i + \\sum _{i=1}^n \\log h(x_i;\\, \\xi )\\\\&- \\alpha \\sum _{i=1}^n z_i H(x_i;\\, \\xi ) - n\\log C(\\theta ).$ So, the components of the score function $U^*_n(\\Theta ) = \\left(\\partial l^*_n/\\partial \\theta , \\partial l^*_n/\\partial \\alpha , \\partial l^*_n/\\partial \\xi \\right)^\\top $ are $\\frac{\\partial l^*_n}{\\partial \\theta } &= \\frac{n}{\\theta } -\\sum _{i=1}^{n} z_i - n\\frac{C^{\\prime }(\\theta )}{C(\\theta )}, \\quad \\quad \\frac{\\partial l^*_n}{\\partial \\alpha } = \\frac{n}{\\alpha } -\\sum _{i=1}^{n} z_i H(x_i;\\, \\xi ) \\quad \\multicolumn{2}{l}{\\text{and}}\\\\\\frac{\\partial l^*_n}{\\partial \\xi _k} &= \\sum _{i=1}^{n}\\frac{\\partial \\log h(x_i;\\, \\xi )}{\\partial \\xi _k} - \\alpha \\sum _{i=1}^{n} z_i \\frac{\\partial H(x_i;\\,\\xi )}{\\partial \\xi _k}.$ From a nonlinear system of equations $U^*_n(\\widehat{\\Theta }) = 0$ , we obtain the iterative procedure of the EM algorithm $&\\hat{\\alpha }^{(t+1)} = \\frac{n}{\\sum _{i=1}^{n} z_i^{(t)} H(x_i;\\,\\xi ^{(t)})}, \\quad \\quad \\hat{\\theta }^{(t+1)} =\\frac{C(\\hat{\\theta }^{(t+1)})}{C^{\\prime }(\\hat{\\theta }^{(t+1)})}\\frac{1}{n}\\sum _{i=1}^{n}z_i^{(t)} \\multicolumn{2}{l}{\\text{and}}\\\\ &\\sum _{i=1}^{n} \\frac{\\partial \\log h(x_i;\\,\\hat{\\xi }^{(t+1)})}{\\partial \\xi _k} -\\hat{\\alpha }^{(t)}\\sum _{i=1}^{n} z_i^{(t)}\\frac{\\partial H(x_i;\\,\\hat{\\xi }^{(t+1)})}{\\partial \\xi _k} = 0,$ where $\\hat{\\theta }^{(t+1)}, \\hat{\\alpha }^{(t+1)}$ and $\\hat{\\xi }^{(t+1)}$ are obtained numerically.", "Here, for $i=1,\\ldots , n$ , we have $z_i^{(t)} = 1 +\\hat{\\theta }^{(t)} \\mathrm {e}^{-\\hat{\\alpha }^{(t)} H(x_i;\\,\\hat{\\xi }^{(t)})} \\frac{C^{\\prime \\prime }(\\hat{\\theta }^{(t)}\\mathrm {e}^{-\\hat{\\alpha }^{(t)} H(x_i;\\,\\hat{\\xi }^{(t)})})}{C^{\\prime }(\\hat{\\theta }^{(t)}\\mathrm {e}^{-\\hat{\\alpha }^{(t)} H(x_i;\\,\\hat{\\xi }^{(t)})})}.$ Note that, in each step, $\\theta , \\alpha $ and $\\xi $ are estimated independently.", "The EWPS distributions can be very useful in modeling lifetime data and practitioners may be interested in fitting one of our models." ], [ "Maximum entropy identification", "Shannon (1948) introduced the probabilistic definition of entropy which is closely connected with the definition of entropy in statistical mechanics.", "Let $X$ be a random variable of a continuous distribution with density $f$ .", "Then, the Shannon entropy of $X$ is defined by $\\mathbb {H}_{Sh}(f) = - \\int _{\\mathbb {R}}f(x;\\theta ,\\alpha ,\\xi )\\log \\left[f(x;\\theta ,\\alpha , \\xi )\\right] dx.$ Jaynes (1957) introduced one of the most powerful techniques employed in the field of probability and statistics called the maximum entropy method.", "This method is closely related to the Shannon entropy and considers a class of density functions $\\mathbb {F} = \\left\\lbrace f(x;\\theta ,\\alpha , \\xi ):\\operatorname{E}_{f}(T_i(X)) = \\alpha _i,\\, i = 0, \\ldots , m\\right\\rbrace ,$ where $T_i (X), i = 1, \\ldots , m$ , are absolutely integrable functions with respect to $f$ , and $T_0(X) = a_0 = 1$ .", "In the continuous case, the maximum entropy principle suggests deriving the unknown density function of the random variable $X$ by the model that maximizes the Shannon entropy in (REF ), subject to the information constraints defined in the class $\\mathbb {F}$ .", "Shore and Johnson (1980) treated axiomatically the maximum entropy method.", "This method has been successfully applied in a wide variety of fields and has also been used for the characterization of several standard probability distributions; see, for example, Kapur (1989), Soofi (2000) and Zografos and Balakrishnan (2009).", "The maximum entropy distribution is the density of the class F, denoted by $f^{ME}$ , which is obtained as the solution of the optimization problem $f^{ME}(x;\\theta ,\\alpha , \\xi ) = \\arg \\max _{f \\in \\mathbb {F}} \\mathbb {H}_{Sh}.$ Jaynes (1957, p. 623) states that the maximum entropy distribution $f^{ME}$ , obtained by the constrained maximization problem described above, “is the only unbiased assignment we can make; to use any other would amount to arbitrary assumption of information which by hypothesis we do not have\".", "It is the distribution which should not incorporate additional exterior information other than which is specified by the constraints.", "We now derive suitable constraints in order to provide a maximum entropy characterization for our class of distributions defined by (REF ).", "For this purpose, the next result plays an important role.", "Proposition 2 Let X be a random variable with pdf given by (REF ).", "Then, C1.", "$\\operatorname{E}\\left[\\log ( C^{\\prime }(\\theta \\, \\mathrm {e}^{-\\alpha H(X;\\, \\xi )}))\\right] = \\dfrac{\\theta }{C(\\theta )}\\operatorname{E}\\left[C^{\\prime }(\\theta \\, \\mathrm {e}^{-\\alpha H(Y;\\, \\xi )})\\log ( C^{\\prime }(\\theta \\, \\mathrm {e}^{-\\alpha H(Y;\\, \\xi )}))\\right];$ C2.", "$\\operatorname{E}\\left[\\log ( h(X; \\,\\xi ))\\right] = \\dfrac{\\theta }{C(\\theta )}\\operatorname{E}\\left[C^{\\prime }(\\theta \\, \\mathrm {e}^{-\\alpha H(Y;\\, \\xi )})\\log ( h(Y;\\,\\xi ) )\\right];$ C3.", "$\\operatorname{E}\\left[H(X; \\,\\xi )\\right] = \\dfrac{\\theta }{C(\\theta )}\\operatorname{E}\\left[C^{\\prime }(\\theta \\, \\mathrm {e}^{-\\alpha H(Y;\\, \\xi )})H(Y;\\,\\xi )\\right],$ where Y follows the EW distribution with density function (REF ).", "The constraints C1, C2 and C3 are easily obtained and therefore their demonstrations are omitted.", "The next proposition reveals that the EWPS distribution has maximum entropy in the class of all probability distributions specified by the constraints stated in the previous proposition.", "Proposition 3 The pdf f of a random variable X, given by (REF ), is the unique solution of the optimization problem $f(x;\\theta ,\\alpha , \\xi ) = \\arg \\max _{h}\\mathbb {H}_{Sh},$ under the constraints $\\rm {C1}$ , $\\rm {C2}$ and $\\rm {C3}$ presented in the Proposition REF .", "Let $\\tau $ be a pdf which satisfies the constraints C1, C2 and C3.", "The Kullback-Leibler divergence between $\\tau $ and $f$ is $D(\\tau , f) = \\int _{\\mathbb {R}} \\tau (x;\\theta ,\\alpha , \\xi ) \\log \\left(\\frac{\\tau (x;\\theta ,\\alpha , \\xi )}{f(x;\\theta ,\\alpha , \\xi )}\\right) dx.$ Following Cover and Thomas (1991), we obtain $0 \\le D(\\tau , f) &=& \\int _{\\mathbb {R}} \\tau (x;\\theta ,\\alpha , \\xi ) \\log \\left[ \\tau (x;\\theta ,\\alpha , \\xi )\\right] dx - \\int _{\\mathbb {R}} \\tau (x;\\theta ,\\alpha , \\xi ) \\log \\left[f(x;\\theta ,\\alpha , \\xi )\\right] dx\\\\&=& -\\mathbb {H}_{Sh}(\\tau ;\\theta ,\\alpha , \\xi ) - \\int _{\\mathbb {R}} \\tau (x;\\theta ,\\alpha , \\xi ) \\log \\left[f(x;\\theta ,\\alpha , \\xi )\\right]dx.$ From the definition of $f$ and based on the constraints C1, C2 and C3, it follows that $\\hspace{-56.9055pt}\\int _{\\mathbb {R}} \\tau (x) \\log \\left[f(x)\\right] dx &=& \\log (\\theta \\alpha ) + \\frac{\\theta }{C(\\theta )}\\operatorname{E}\\left\\lbrace C^{\\prime }(\\theta \\, \\mathrm {e}^{-\\alpha H(Y;\\, \\xi )})\\log \\left[ h(Y;\\,\\xi ) \\right]\\right\\rbrace - \\log \\left[ C(\\theta )\\right]\\\\&-&\\alpha \\frac{\\theta }{C(\\theta )}\\operatorname{E}\\left[C^{\\prime }(\\theta \\, \\mathrm {e}^{-\\alpha H(Y;\\, \\xi )})H(Y;\\,\\xi )\\right] \\\\&+& \\frac{\\theta }{C(\\theta )}\\operatorname{E}\\left\\lbrace \\log \\left[ C^{\\prime }(\\theta \\, \\mathrm {e}^{-\\alpha H(Y;\\, \\xi )})\\right] C^{\\prime }(\\theta \\, \\mathrm {e}^{-\\alpha H(Y;\\, \\xi )})\\right\\rbrace \\\\&=& \\int _{\\mathbb {R}} f(x;\\theta ,\\alpha , \\xi ) \\log \\left[f(x;\\theta ,\\alpha , \\xi )\\right] dx = - \\mathbb {H}_{Sh}(f),$ where $Y$ is defined as before.", "So, we have $\\mathbb {H}_{Sh}(\\tau ) \\le \\mathbb {H}_{Sh}(f)$ with equality if and only if $\\tau (x;\\theta ,\\alpha , \\xi ) = f(x;\\theta ,\\alpha , \\xi )$ for all $x$ , except for a set of measure 0, thus proving the uniqueness.", "The intermediate steps in the above proof in fact provide the following explicit expression for the Shannon entropy of the EWPS distribution $\\nonumber &&\\mathbb {H}_{Sh}(f) = -\\log (\\theta \\alpha ) - \\frac{\\theta }{C(\\theta )}\\operatorname{E}\\left\\lbrace C^{\\prime }(\\theta \\, \\mathrm {e}^{-\\alpha H(Y;\\, \\xi )})\\log \\left[h(Y;\\,\\xi ) \\right]\\right\\rbrace + \\log \\left[C(\\theta )\\right]\\\\&&+\\alpha \\frac{\\theta }{C(\\theta )}\\operatorname{E}\\left[C^{\\prime }(\\theta \\, \\mathrm {e}^{-\\alpha H(Y;\\, \\xi )})H(Y;\\,\\xi )\\right] - \\frac{\\theta }{C(\\theta )}\\operatorname{E}\\left\\lbrace C^{\\prime }(\\theta \\, \\mathrm {e}^{-\\alpha H(Y;\\, \\xi )})\\log \\left[C^{\\prime }(\\theta \\, \\mathrm {e}^{-\\alpha H(Y;\\, \\xi )})\\right]\\right\\rbrace .$ For some EWPS distributions, the above results can only be obtained numerically." ], [ "Special models", "In this section, we investigate some special cases of the EWPS class of distributions.", "We offer some expressions for moments and moments of the order statistics.", "To illustrate the flexibility of these distributions, we provide plots of the density and hazard rate functions for selected parameter values." ], [ "Modified Weibull geometric distribution", "The modified Weibull geometric (MWG) distribution is defined by the cdf (REF ) with $H(x;\\, \\xi ) = x^\\gamma $ and $C(\\theta ) = \\theta (1-\\theta )^{-1}$ leading to $F(x;\\theta ,\\alpha , \\gamma ,\\lambda ) = 1 -\\frac{(1-\\theta )\\exp \\left(-\\alpha x^\\gamma \\mathrm {e}^{\\lambda x}\\right)}{1 - \\theta \\exp \\left(-\\alpha x^\\gamma \\mathrm {e}^{\\lambda x}\\right)}, \\quad x > 0,$ where $\\theta \\in (0,1)$ .", "The associated pdf and hazard rate function are $f(x;\\theta ,\\alpha , \\gamma ,\\lambda ) &= \\alpha (1-\\theta ) (\\gamma +\\lambda x)\\,x^{\\gamma -1}\\frac{\\exp \\left(\\lambda x - \\alpha x^\\gamma \\mathrm {e}^{\\lambda x}\\right)}{\\left[1-\\theta \\exp \\left(-\\alpha x^\\gamma \\mathrm {e}^{\\lambda x}\\right)\\right]^2} \\multicolumn{2}{l}{\\text{and}}\\\\\\tau (x;\\theta ,\\alpha , \\gamma ,\\lambda ) &= \\alpha (\\gamma +\\lambda x)\\,x^{\\gamma -1}\\frac{\\exp \\left(\\lambda x\\right)}{1 - \\theta \\exp \\left(-\\alpha x^\\gamma \\mathrm {e}^{\\lambda x}\\right)}$ for $x > 0$ , respectively.", "The MWG distribution contains the WG distribution (Barreto-Souza et al.", "(2010)) as the particular choice $\\lambda = 0$ .", "Further, for $\\lambda = 0$ and $\\alpha =1$ , we obtain the EG distribution (Adamidis and Loukas (1998)).", "Figures $\\ref {fig:densityfigewps}$ and $\\ref {fig:hazardfigewps}$ display the density and hazard functions of the MWG distribution for selected parameter values.", "Figure: Plots of the MWG density functions for θ=0.01\\theta = 0.01 (solid line), θ=0.2\\theta = 0.2 (dashedline), θ=0.5\\theta = 0.5 (dotted line) and θ=0.9\\theta = 0.9 (dotdash line).The $r$ th raw moment of the random variable $X$ having the MWG distribution has closed-form.", "It is calculated from (REF ) as $E(X^r) = \\sum _{n=1}^{\\infty } p_n\\, \\mu _r(n),$ where $\\mu _r(n) = \\int _{0}^{\\infty }x^r g(x;\\, n\\alpha , \\gamma , \\lambda )dx$ denotes the $r$ th raw moment of the MW distribution with parameters $n\\alpha , \\gamma $ and $\\lambda $ .", "Here $p_n$ corresponds to the probability function of the geometric distribution.", "Carrasco et al.", "(2008) determined an infinite representation for the $r$ th raw moment of the MW distribution with these parameters expressed as $\\mu _r(n) = \\sum _{i_1, \\ldots , i_r=1}^{\\infty } \\frac{A_{i_1, \\ldots , i_r}\\,\\Gamma (s_r/\\gamma + 1)}{(n\\alpha )^{s_r/\\gamma }},$ where $A_{i_1, \\ldots , i_r} = a_{i_1}, \\ldots , a_{i_r} \\,\\,\\,\\,\\, \\mbox{and} \\,\\,\\,\\,\\, s_r = i_1, \\ldots , i_r,$ and $a_i = \\frac{(-1)^{i+1}i^{i-2}}{(i-1)!", "}\\left(\\frac{\\lambda }{\\gamma }\\right)^{i-1}.$ Hence, the moments of the MWG distribution can be obtained directly from equations (REF ) and (REF ).", "Figure: Plots of the MWG hazard rate functions for θ=0.01\\theta = 0.01 (solid line), θ=0.2\\theta = 0.2 (dashedline), θ=0.5\\theta = 0.5 (dotted line) and θ=0.9\\theta = 0.9 (dotdash line).The density of the $i$ th order statistic $X_{i:m}$ in a random sample of size $m$ from the MWG distribution is given by (for $i = 1, \\ldots , m$ ) $f_{i:m}(x) = \\frac{m!}{(i-1)!", "(m-i)!}", "\\sum _{n=1}^\\infty \\sum _{j=0}^{i-1} \\omega _j\\, p_n\\,\\left[\\frac{(1-\\theta )\\exp \\left(-\\alpha x^\\gamma \\mathrm {e}^{\\lambda x}\\right)}{1 - \\theta \\exp \\left(-\\alpha x^\\gamma \\mathrm {e}^{\\lambda x}\\right)}\\right]^{m+j-i} g(x; n\\alpha , \\gamma , \\lambda ),$ where $g(x; n\\alpha , \\gamma , \\lambda )$ denotes the MW density function with parameters $n\\alpha , \\gamma $ and $\\lambda $ .", "From (REF ), we obtain $\\operatorname{E}\\left(X_{i:m}^s\\right) = \\frac{m!}{(i-1)!", "(m-i)!", "}\\sum _{n=1}^\\infty \\sum _{j=0}^{i-1} \\omega _j\\, p_n\\,\\operatorname{E}\\left\\lbrace X^s \\left[\\frac{(1-\\theta )\\exp \\left(-\\alpha X^\\gamma \\mathrm {e}^{\\lambda X}\\right)}{1 - \\theta \\exp \\left(-\\alpha X^\\gamma \\mathrm {e}^{\\lambda X}\\right)}\\right]^{m+j-i}\\right\\rbrace .$" ], [ "Pareto Poisson distribution", "The Pareto Poisson (PP) distribution is defined by taking $H(x;\\, \\xi ) = \\log (x/k)$ and $C(\\theta ) = \\mathrm {e}^{\\theta }-1$ in (REF ), which yields $F(x; \\theta ,\\alpha , k) = 1 - \\frac{\\exp \\left[\\theta \\left(k/x\\right)^{\\alpha }\\right]-1}{\\mathrm {e}^{\\theta }-1}, \\quad x\\ge k.$ The pdf and hazard functions of the PP distribution are $f(x; \\theta ,\\alpha , k) = \\frac{\\theta \\,\\alpha \\, k^{\\alpha } \\exp \\left[\\theta \\left(k/x\\right)^{\\alpha }\\right]}{(\\mathrm {e}^{\\theta }-1)\\,x^{\\alpha +1}}$ and $\\tau (x; \\theta ,\\alpha , k) = \\frac{\\theta \\,\\alpha \\,k^{\\alpha }\\exp \\left[\\theta \\left(k/x\\right)^{\\alpha }\\right]}{x^{\\alpha +1}\\left\\lbrace \\exp \\left[\\theta \\left(k/x\\right)^{\\alpha }\\right]-1\\right\\rbrace }.$ We obtain the Pareto distribution as a sub-model when $\\theta \\rightarrow 0$ .", "The $r$ th moment of the random variable $X$ following the PP distribution becomes $E(X^r) = \\frac{\\alpha k^r}{(\\mathrm {e}^{\\theta }-1)}\\sum _{n=1}^{\\infty }\\frac{\\theta ^n}{(n-1)!\\,(n\\alpha -r)},\\quad n\\alpha > r.$ In particular, setting $r = 1$ in (REF ), the mean of $X$ reduces to $\\mu = \\frac{\\alpha k}{\\mathrm {e}^{\\theta }-1}\\sum _{n=1}^{\\infty }\\frac{\\theta ^n}{(n-1)!\\,(n\\alpha -1)}, \\quad n\\alpha > 1.$ Figure: Plots of the PP density functions for θ=0.01\\theta = 0.01 (solid line), θ=0.2\\theta = 0.2 (dashed line), θ=0.5\\theta = 0.5 (dotted line) and θ=0.9\\theta = 0.9 (dotdash line).Figure: Plots of the PP hazard functions for θ=0.01\\theta = 0.01 (solid line), θ=0.2\\theta = 0.2 (dashed line), θ=0.5\\theta = 0.5 (dotted line) and θ=0.9\\theta = 0.9 (dotdash line).From equation (REF ), the $s$ th moment of the $i$ th order statistic, for $i = 1, \\ldots , m,$ is given by $\\operatorname{E}\\left(X_{i:m}^s\\right) = \\frac{m!}{(i-1)!", "(m-i)!", "}\\sum _{n=1}^\\infty \\sum _{j=0}^{i-1} \\omega _j\\, p_n\\,\\operatorname{E}\\left[X^s \\left(\\frac{\\exp (\\theta \\left(k/X\\right)^{\\alpha })-1}{\\mathrm {e}^{\\theta }-1}\\right)^{m+j-i}\\right],$ where $p_n$ denotes the Poisson probability function.", "Furthermore, after some algebra, the Shannon entropy for the PP distribution reduces to $\\mathbb {H}_{Sh}(f) = \\log \\left(\\frac{\\mathrm {e}^\\theta -1}{\\theta \\alpha }\\right) - \\frac{\\theta }{\\mathrm {e}^\\theta -1}\\left(\\mu _1 - \\alpha \\mu _2 + \\mu _3\\right),$ where $\\mu _1 &= \\operatorname{E}\\left[\\exp \\left\\lbrace \\theta \\left(\\frac{k}{X}\\right)^\\alpha \\right\\rbrace \\log \\left(\\frac{1}{X}\\right)\\right] = \\frac{1}{2(\\mathrm {e}^\\theta -1)}\\left\\lbrace \\frac{\\mathrm {Chi}(2\\theta )-\\log (2\\theta )+\\mathrm {Shi}(2\\theta )-\\gamma }{\\alpha } - (\\mathrm {e}^{2\\theta }-1)\\log k\\right\\rbrace ,\\\\\\mu _2 &= \\operatorname{E}\\left[\\exp \\left\\lbrace \\theta \\left(\\frac{k}{X}\\right)^\\alpha \\right\\rbrace \\log \\left(\\frac{X}{k}\\right)\\right] = \\frac{\\mathrm {Chi}(2\\theta )-\\log (2\\theta )+\\mathrm {Shi}(2\\theta )-\\gamma }{2\\alpha (\\mathrm {e}^\\theta -1)}\\multicolumn{2}{l}{\\text{and}}\\\\\\mu _3 &= \\operatorname{E}\\left[\\theta \\exp \\left\\lbrace \\theta \\left(\\frac{k}{X}\\right)^\\alpha \\right\\rbrace \\left(\\frac{k}{X}\\right)^\\alpha \\right] = \\frac{\\alpha \\,\\theta \\, k^{2\\alpha }}{4(\\mathrm {e}^{\\theta }-1)}\\left\\lbrace 1- (2\\theta +1)\\mathrm {e}^{2\\theta }\\right\\rbrace ,$ where $\\operatorname{Chi}(z) = \\gamma + \\log z + \\int _{0}^{z}\\frac{\\operatorname{cosh}(t)-1}{t}dt$ is the hyperbolic cosine integral, $\\operatorname{Shi}(z) = \\int _{0}^{z}\\frac{\\operatorname{sinh}(t)-1}{t}dt$ is the hyperbolic sine integral and $\\gamma \\approx 0.577216$ is the Euler-Mascheroni constant." ], [ "Chen logarithmic distribution", "The Chen logarithmic (CL) distribution is defined by the cdf (REF ) with $H(x;\\, \\xi ) = \\exp (x^\\beta )-1$ and $C(\\theta ) = -\\log (1-\\theta )$ , leading to $F(x) = 1 - \\frac{\\log \\left\\lbrace 1-\\theta \\exp \\left[-\\alpha (\\exp (x^\\beta )-1)\\right]\\right\\rbrace }{\\log (1-\\theta )} , \\quad x > 0,$ where $\\theta \\in (0,1)$ .", "The associated pdf and hazard rate function (for $x>0$ ) are $f(x) = \\frac{\\theta \\alpha b x^{b-1} \\exp \\left\\lbrace x^b - \\alpha \\left[\\exp (x^b)-1\\right]\\right\\rbrace }{\\log (1-\\theta )\\left\\lbrace \\theta \\exp \\left[- \\alpha (\\exp (x^b)-1)\\right]-1\\right\\rbrace }$ and $\\tau (x) = \\frac{\\theta \\alpha b x^{b-1}\\exp \\left[x^b - \\alpha (\\exp (x^b)-1)\\right]}{\\left\\lbrace \\theta \\exp \\left[- \\alpha (\\exp (x^b)-1)\\right]-1\\right\\rbrace \\log \\left\\lbrace 1-\\theta \\exp \\left[- \\alpha (\\exp (x^b)-1)\\right]\\right\\rbrace },$ respectively.", "Figure: Plots of the CL density functions for θ=0.01\\theta = 0.01 (solid line), θ=0.2\\theta = 0.2 (dashed line), θ=0.5\\theta = 0.5 (dotted line) and θ=0.9\\theta = 0.9 (dotdash line).As expected by proposition REF , we obtain the Chen distribution as a limiting special case when $\\theta \\rightarrow 0^+$ .", "Figure: Plots of the CL hazard rate functions for θ=0.01\\theta = 0.01 (solid line), θ=0.2\\theta = 0.2 (dashed line), θ=0.5\\theta = 0.5 (dotted line) and θ=0.9\\theta = 0.9 (dotdash line).The density of the $i$ th order statistic $X_{i:m}$ in a random sample of size $m$ from the CL distribution is given by (for $i = 1, \\ldots , m$ ) $f_{i:m}(x) = \\frac{m!}{(i-1)!", "(m-i)!}", "\\sum _{n=1}^\\infty \\sum _{j=0}^{i-1} \\omega _j^*\\, p_n\\, g(x; n\\alpha , b) \\left\\lbrace \\log \\left[1-\\theta \\exp (\\alpha -\\alpha \\operatorname{e}^{x^b})\\right]\\right\\rbrace ^{m+j-1},$ where $g(x; n\\alpha , b)$ is the pdf of the Chen distribution with parameters $n\\alpha $ and $b$ and $p_n$ denotes the logarithmic probability mass function and $\\omega _j^* = (-1)^j \\binom{i-1}{j}\\left[\\frac{1}{\\log (1-\\theta )}\\right]^{m+j-1}.$ In the same way, the $s$ th raw moment of $X_{i:m}$ is obtained directly from $\\operatorname{E}\\left(X_{i:m}^s\\right) = \\frac{m!}{(i-1)!", "(m-i)!", "}\\sum _{n=1}^\\infty \\sum _{j=0}^{i-1} \\omega _j\\, p_n\\,\\operatorname{E}\\left\\lbrace Z^s \\exp \\left[n\\alpha (m+j-1)(1-\\exp (Z^b))\\right]\\right\\rbrace ,$ where $Z\\sim \\mbox{Chen}(n\\alpha , b)$ ." ], [ "Application", "Fonseca and França (2007) studied the soil fertility influence and the characterization of the biologic fixation of $\\mathrm {N}_2$ for the Dimorphandra wilsonii rizz growth.", "For 128 plants, they made measures of the phosphorus concentration in the leaves.", "The data are listed in Table REF .", "We fit the MWG, Gompertz Poisson (GP), PP, Chen Poisson (CP) and CL models to these data.", "We also fit the three-parameter WG distribution introduced by Barreto-Souza et al.", "(2010).", "The required numerical evaluations are implemented using the SAS (PROCNLMIXED) and R softwares.", "Table: Phosphorus concentration in leaves data set.Table: Descriptive statistics.Tables REF and REF display some descriptive statistics and the MLEs (with corresponding standard errors in parentheses) of the model parameters.", "Since the values of the Akaike information criterion (AIC), Bayesian information criterion (BIC) and consistent Akaike information criterion (CAIC) are smaller for the CL distribution compared with those values of the other models, this new distribution seems to be a very competitive model for these data.", "Table: MLEs of the model parameters, the corresponding SEs (given in parentheses) and the statistics AIC, BIC and AICC.Plots of the estimated pdf and cdf of the MWG, WG, GP, PP, CP and CL models fitted to these data are displayed in Figure REF .", "They indicate that the CL distribution is superior to the other distributions in terms of model fitting.", "Figure: Estimated (a) pdf and (b) cdf for the CL, MWG, PP, WG, GP and CP models to the percentage of Phosphorus concentration in leaves data.Table REF lists the values of the Kolmogorov-Smirnov (K-S) statistic and the values of $-2\\ell (\\widehat{\\Theta })$ .", "From these figures, we conclude that the CL distribution provides a better fit to these data than the MWG, WG, GP, PP and CP models.", "Table: K-S statistics and -2ℓ(Θ ^)-2\\ell (\\widehat{\\Theta }) for the exceedances of phosphorus concentration in leaves data set." ], [ "Concluding remarks", "We define a new lifetime class of distributions, called the extended Weibull power series (EWPS), which generalizes the Weibull power series class of distributions proposed by Morais and Barreto-Souza (2011), which in turn extends the exponential power series class of distributions (Chahkandi and Ganjali, 2009).", "We provide a mathematical treatment of the new distribution including expansions for the density function, moments, generating function and incomplete moments.", "Further, explicit expressions for the order statistics and Shannon entropy are derived.", "The EWPS density function can be expressed as a mixture of EW density functions.", "This property is important to obtain several other results.", "Our formulas related with the EWPS model are manageable, and with the use of modern computer resources with analytic and numerical capabilities, they may turn into adequate tools comprising the arsenal of applied statisticians.", "The estimation of the model parameters is approached by the method of maximum likelihood using the EM algorithm.", "The observed information matrix is derived.", "Further, maximum entropy identification for the EWPS distributions was discussed and some special models are studied in some detail.", "Finally, we fit the EWPS model to a real data set to show the usefulness of the proposed class.", "We hope that this generalization may attract wider applications in the literature of the fatigue life distributions." ], [ "Acknowledgements", "tocchapterAcknowledgements We also gratefully acknowledge financial support from CAPES and CNPq." ], [ "tocchapterAppendix A.", "The elements of the $p \\times p$ information matrix $J_n(\\Theta )$ are $J_{\\theta \\theta } &= - \\frac{n}{\\theta ^2} - n\\left[\\frac{C^{\\prime \\prime }(\\theta )}{C(\\theta )} - \\left(\\frac{C^{\\prime }(\\theta )}{C(\\theta )}\\right)^2\\right] + \\theta \\sum _{i=1}^n \\left(\\frac{z_{2i}}{z_{1i}}\\right)^2 H(x_i;\\, \\xi ) \\mathrm {e}^{-2\\alpha H(x_i;\\, \\xi )}\\\\&- \\theta \\sum _{i=1}^n \\frac{z_{3i}}{z_{1i}} H(x_i;\\, \\xi ) \\mathrm {e}^{-2\\alpha H(x_i;\\, \\xi )}\\\\J_{\\alpha \\alpha } &= - \\frac{n}{\\alpha ^2} + \\theta \\sum _{i=1}^n \\frac{z_{2i}}{z_{1i}} H^2(x_i; \\xi ) \\mathrm {e}^{-\\alpha H(x_i;\\, \\xi )} + \\theta ^2 \\sum _{i=1}^n \\frac{(z_{3i}-z_{2i}^2)}{z_{1i}}H^2(x_i; \\xi ) \\mathrm {e}^{-2\\alpha H(x_i;\\, \\xi )}\\\\J_{\\alpha \\theta } &= \\theta \\sum _{i=1}^n \\left[\\left(\\frac{z_{2i}}{z_{1i}}\\right)^2 - \\frac{z_{3i}}{z_{1i}}\\right] H^2(x_i; \\xi ) \\mathrm {e}^{-2\\alpha H(x_i;\\, \\xi )} - \\sum _{i=1}^n \\frac{z_{2i}}{z_{1i}} H^2(x_i; \\xi ) \\mathrm {e}^{-\\alpha H(x_i;\\, \\xi )} \\\\J_{\\alpha \\xi _k} &= - \\sum _{i=1}^n \\frac{\\partial H(x_i;\\, \\xi )}{\\partial \\xi _k} - \\theta \\sum _{i=1}^n \\frac{z_{2i}}{z_{1i}}\\frac{\\partial H(x_i;\\, \\xi )}{\\partial \\xi _k} \\mathrm {e}^{-\\alpha H(x_i;\\, \\xi )} \\left[1 - \\alpha H(x_i;\\, \\xi )\\right] \\\\&+ \\alpha \\theta ^2 \\sum _{i=1}^n \\left[\\frac{z_{3i}}{z_{1i}}-\\left(\\frac{z_{2i}}{z_{1i}}\\right)^2 \\right] \\frac{\\partial H(x_i;\\, \\xi )}{\\partial \\xi _k}H(x_i;\\, \\xi )\\mathrm {e}^{-2\\alpha H(x_i;\\, \\xi )}\\\\J_{\\theta \\xi _k} &= \\theta \\alpha \\sum _{i=1}^n \\left[\\left(\\frac{z_{2i}}{z_{1i}}\\right)^2 - \\frac{z_{3i}}{z_{1i}}\\right] \\frac{\\partial H(x_i;\\, \\xi )}{\\partial \\xi _k} \\mathrm {e}^{-2\\alpha H(x_i;\\, \\xi )} - \\alpha \\sum _{i=1}^n \\frac{z_{2i}}{z_{1i}} \\frac{\\partial H(x_i;\\, \\xi )}{\\partial \\xi _k} \\mathrm {e}^{-\\alpha H(x_i;\\, \\xi )}\\\\J_{\\xi _k \\xi _l} &= -\\alpha \\sum _{i=1}^n \\frac{\\partial ^2 H(x_i;\\, \\xi )}{\\partial \\xi _k \\partial \\xi _l} - \\sum _{i=1}^n \\frac{1}{H(x_i;\\, \\xi )^2}\\frac{\\partial H(x_i;\\, \\xi )}{\\partial \\xi _k}\\frac{\\partial H(x_i;\\, \\xi )}{\\partial \\xi _l} + \\sum _{i=1}^n \\frac{1}{H(x_i;\\, \\xi )}\\frac{\\partial ^2 H(x_i;\\, \\xi )}{\\partial \\xi _k \\partial \\xi _l}\\\\&+ (\\alpha \\theta )^2 \\sum _{i=1}^n \\left[\\left(\\frac{z_{2i}}{z_{1i}}\\right)^2 + \\frac{z_{3i}}{z_{1i}}\\right] \\frac{\\partial H(x_i;\\, \\xi )}{\\partial \\xi _k} \\frac{\\partial H(x_i;\\, \\xi )}{\\partial \\xi _l} \\mathrm {e}^{-2\\alpha H(x_i;\\, \\xi )} \\\\&- \\alpha \\theta \\sum _{i=1}^n \\frac{z_{2i}}{z_{1i}} \\frac{\\partial ^2H(x_i;\\, \\xi )}{\\partial \\xi _k \\partial \\xi _l} \\mathrm {e}^{-\\alpha H(x_i;\\, \\xi )} + \\alpha ^2\\theta \\sum _{i=1}^n \\frac{z_{2i}}{z_{1i}} \\frac{\\partial H(x_i;\\xi )}{\\partial \\xi _k} \\frac{\\partial H(x_i;\\xi )}{\\partial \\xi _l}\\mathrm {e}^{-\\alpha H(x_i;\\xi )}$ where $z_{1i} = C^{\\prime }(\\theta e^{-\\alpha H(x_i;\\, \\xi )}),z_{2i} = C^{\\prime \\prime }(\\theta \\mathrm {e}^{-\\alpha H(x_i;\\, \\xi )})$ and $z_{3i} = C^{\\prime \\prime \\prime }(\\theta \\mathrm {e}^{-\\alpha H(x_i;\\, \\xi )})$ , for $i= 1, \\ldots , n$ ." ] ]

[ [ "On the anomalous t-quark charge asymmetry and noncontractibility of the\n physical space" ], [ "Abstract Heavy flavour production at hadron colliders represents a very promising field to test perturbative QCD.", "The integrated forward-backward asymmetry of the top-antitop quark production is particularly sensitive to any deviation from the standard QCD calculations.", "The two Tevatron collaborations, CDF and D0, reported a much larger t-quark charge asymmetry than predicted by the theory.", "We show that the QCD in noncontractible space, where the minimal distance is fixed by weak interactions, enhances the asymmetry by more than a factor of 3 (5) at the parton level in leading order of the coupling for the Tevatron (LHC) center of mass energies.", "This result should not be a surprise since the asymmetry observable directly explores the far ultraviolet sector of the spacelike domain of the Minkowski spacetime." ], [ "Introduction and motivation", "The missing flavour-mixed light neutrinos, no lepton or baryon number violation and the absence of any candidate for a dark matter particle, call for a substantial improvement of the Standard Model (SM) of electroweak and strong interactions.", "Namely, the dominance of the baryon over antibaryon matter in the Universe suggests that baryon or lepton number be broken in particle physics [1].", "It is well known, for example, that the broken lepton number can induce the breaking of the baryon number [2].", "Any alternative to the presence of the cold dark matter in the Universe supposes drastic changes not only of the General Relativity, but also of the nonrelativistic Newtonian theory of gravity [3].", "However, the direct or indirect detection of the dark matter particle is indispensable.", "The HESS source J1745-290 at the centre of our galaxy (for the most recent anaysis see ref.", "[4]) and the hints from the anomalous positron (or antiproton) abundance from the PAMELA mission [5] (see, for example, the analysis in ref.", "[6]) suggest on the existence of the very heavy dark matter particle when searching for the dark matter annihilation products.", "MOND (with its few relativistic generalizations) represents the alternative to the dark matter paradigm (see a discussion in ref.", "[7]).", "Besides the supersymmetric, grand unified or extra dimensional extensions of the SM, a very conservative alternative to the SM was proposed in [8], called the BY theory, resolving the ultraviolet singularity and the $SU(2)$ global anomaly problems.", "Light and heavy Majorana neutrinos with flavour mixing and lepton CP violation could play a crucial role as hot and cold dark matter particles in the evolution of the expanding and rotating Universe [8], [9].", "The noncontractible space of the BY theory, as an alternative symmetry-breaking mechanism to the Higgs one, introduces into the physical realm a new universal Lorentz and gauge invariant constant (UV cutoff in the spacelike domain of the Minkowski spacetime, see ref.", "[8]) $\\Lambda =\\frac{\\hbar }{c d}=\\frac{2}{g}\\frac{\\pi }{\\sqrt{6}}M_{W} \\simeq 326 GeV$ .", "The enhanced strong coupling at small distances and the absence of the asymptotic freedom in QCD are the immediate consequences of noncontractible space [10].", "We show that electroweak quantum loops with heavy t-quark contributing to the CP violating processes of K and B mesons are affected by the UV cutoff [11].", "The branching fraction for a rare decay $B_{s}\\rightarrow \\mu \\mu $ is lower by more than $30\\%$ in the BY theory compared with the SM owing to the modified short distance part of the amplitude [12].", "ATLAS and CMS experiments at the LHC reported recently [13] a discovery of the 125 GeV resonance.", "It could be the Higgs boson of some theory beyond the SM, but it could be also some pseudoscalar or scalar meson with a substantial component of the pseudoscalar or scalar toponium [14].", "Even the spin 1 boson cannot be excluded as an interpretation of the new 125 GeV resonance [15].", "Anyhow, the Higgs mechanism does not solve the mass problem of particles.", "Eventually, the solutions of the coupled system of nonlinear integral Dyson-Schwinger equations of the UV nonsingular BY theory could resolve the mass problem of elementary particles [16].", "In this paper, we study the implication of the UV cutoff to the leading QCD contribution for the forward-backward asymmetry in the top-antitop production.", "The large discrepancy between the theory and the experiment for this asymmetry observable is reported by the Tevatron collaborations CDF and D0 [17].", "Let us quote the most recent results of the CDF collaboration [18]: parton level asymmetry $A_{FB}(M_{t\\bar{t}} < 450 GeV):\\ Data (\\pm stat\\pm syst)= 0.084 \\pm 0.046 \\pm 0.026\\ vs.\\ SM\\ expectation = 0.047 \\pm 0.014$ ; $A_{FB}(M_{t\\bar{t}} \\ge 450 GeV):\\ Data = 0.295 \\pm 0.058 \\pm 0.031\\ vs.\\ SM\\ $ $expectation = 0.100 \\pm 0.030$ .", "In the next chapter, we present the main ingredients of the calculations while providing more details in the Appendix.", "Results and Conclusions are given in the last chapter." ], [ "Charge asymmetry at the parton level", "Almost invariably, various asymmetry observables of the electroweak or strong interactions are very sensitive to the details of the underlying processes.", "It appears that the t-quark pair charge asymmetry can test QCD loop corrections [19].", "We shall study the dominant quark-antiquark annihilation channel whose structure equals the electron-positron annihilation amplitude modulo coupling and gauge group constant factors [20], [21], [22].", "Let us define the asymmetric part of the differential cross sections [19] $\\frac{d \\sigma ^{q\\bar{q}}_{A}}{d \\cos \\theta } =(\\sigma ^{q\\bar{q}}_{A})^{\\prime }\\equiv \\frac{1}{2}[\\frac{d \\sigma (q\\bar{q} \\rightarrow QX)}{d \\cos \\theta }-\\frac{d \\sigma (q\\bar{q} \\rightarrow \\bar{Q}X)}{d \\cos \\theta }].$ Born cross section (symmetric part of the quark-antiquark annihilation to leading order $\\alpha ^{2}_{s}$ ) is given by [20], [19]: $\\frac{d \\sigma (q\\bar{q}\\rightarrow Q\\bar{Q};Born)}{d \\cos \\theta } =\\alpha ^{2}_{s}\\frac{T_{F}C_{F}}{N_{c}}\\frac{\\pi \\beta }{2 s} (1 + c^{2} +4 m^{2}), \\\\T_{F}=\\frac{1}{2}, C_{F}=\\frac{4}{3}, N_{c}=3,\\beta =\\sqrt{1-4 m^{2}}, m^{2}=\\frac{m^{2}_{Q}}{s}, \\\\s=E_{cm}^{2}, c=\\beta \\cos \\theta ,\\ \\angle (\\vec{p}(q),\\vec{p}(Q))=\\theta .$ The asymmetric part to the leading $\\alpha ^{3}_{s}$ order consists of the virtual, soft and hard gluon emmission differential cross sections [19], [21], [22]: $(\\sigma ^{q\\bar{q}}_{A})^{\\prime } &=& (\\sigma ^{q\\bar{q}}_{A})^{\\prime }(virtual)+ (\\sigma ^{q\\bar{q}}_{A})^{\\prime }(soft) +\\int _{(I)} \\frac{\\partial ^{4} (\\sigma ^{q\\bar{q}}_{A}(hard)-\\sigma ^{q\\bar{q}}_{A}(soft))}{\\partial \\cos \\theta \\partial \\Omega _{\\gamma } \\partial k}d \\Omega _{\\gamma }d k \\nonumber \\\\&+&\\int _{(II)} \\frac{\\partial ^{4} \\sigma ^{q\\bar{q}}_{A}(hard)}{\\partial \\cos \\theta \\partial \\Omega _{\\gamma } \\partial k}d \\Omega _{\\gamma }d k,\\nonumber \\\\(I)&\\ & 0 \\le k \\le k_{1},\\ -1 \\le \\cos \\theta _{\\gamma } \\le 1,\\nonumber \\\\(II)&\\ & k_{1} \\le k \\le k_{2},\\ g_{1}(k,E_{th}) \\le \\cos \\theta _{\\gamma } \\le g_{2}(k,E_{th}).$ The equations for the virtual, hard and soft gluon radiation in the appendix of ref.", "[19] are obtained from the equations in [21], [22] in the limit of the vanishing mass of incoming fermions.", "The QCD in noncontractible space differs from the standard QCD when quantum loops are evaluated with the cutoff in the spacelike domain.", "Thus, one can find two possible sources of deviation from the standard QCD calculation for the asymmetry function $A^{\\infty }(\\cos \\theta ) = \\sigma ^{\\prime }_{A}/\\sigma ^{\\prime }_{Born}$ : (1) calculation of the running coupling $\\alpha ^{\\Lambda }_{s}$ (see ref.", "[10]), (2) box diagram contribution to the virtual correction [19], [21], [22]: $(\\sigma ^{\\Lambda }_{A})^{\\prime }&=&(\\sigma ^{\\Lambda }_{A})^{\\prime }(virtual,\\alpha ^{\\Lambda }_{s})+(\\sigma ^{\\Lambda }_{A})^{\\prime }(soft,\\alpha ^{\\Lambda }_{s})+(\\sigma ^{\\Lambda }_{A})^{\\prime }(difference,\\alpha ^{\\Lambda }_{s})$ $&+&(\\sigma ^{\\Lambda }_{A})^{\\prime }(hard,\\alpha ^{\\Lambda }_{s})=(\\frac{\\alpha ^{\\Lambda }_{s}}{\\alpha ^{\\infty }_{s}})^{3}(\\sigma ^{\\infty }_{A})^{\\prime }(\\alpha ^{\\infty }_{s})+(\\sigma ^{\\Lambda }_{A})^{\\prime }(virtual,\\alpha ^{\\Lambda }_{s})-(\\sigma ^{\\infty }_{A})^{\\prime }(virtual,\\alpha ^{\\Lambda }_{s}),$ $(\\sigma ^{\\Lambda }_{Born})^{\\prime } = (\\frac{\\alpha ^{\\Lambda }_{s}}{\\alpha ^{\\infty }_{s}})^{2}(\\sigma ^{\\infty }_{Born})^{\\prime },$ $A^{\\Lambda } = \\frac{(\\sigma ^{\\Lambda }_{A})^{\\prime }}{(\\sigma ^{\\Lambda }_{Born})^{\\prime }},$ $A^{\\Lambda }(\\cos \\theta )&=&A^{\\infty }+\\delta A^{\\Lambda }_{\\alpha }+\\delta A^{\\Lambda }_{box},\\hspace{142.26378pt} \\\\\\delta A^{\\Lambda }_{\\alpha }&\\equiv & \\frac{\\alpha _{s}^{\\Lambda }-\\alpha _{s}^{\\infty }}{\\alpha _{s}^{\\infty }}A^{\\infty },\\ \\delta A^{\\Lambda }_{box} \\equiv \\frac{(\\sigma ^{\\Lambda }_{A})^{\\prime }(virtual,\\alpha _{s}^{\\Lambda })-(\\sigma ^{\\infty }_{A})^{\\prime }(virtual,\\alpha _{s}^{\\Lambda })}{(\\sigma _{Born}^{\\Lambda })^{\\prime }}, \\nonumber $ $\\Lambda \\ denotes\\ quantity\\ in\\ the\\ BY\\ theory,\\ \\infty \\ denotes\\ quantity\\ in\\ the\\ SM.$ We mean that $(\\sigma ^{\\Lambda }_{A})^{\\prime }(virtual,\\alpha _{s}^{\\Lambda })$ is evaluated with $\\alpha _{s}^{\\Lambda }$ coupling, etc.", "The calculation of the strong interaction running coupling in noncontractible space was performed in the momentum subtraction renormalization scheme to one loop order in ref.", "[10].", "Hard and soft gluon radiations do not contain loop diagrams to leading $\\alpha ^{3}_{s}$ order.", "Our main task should be a reevaluation of the interference term in the cross section containing the box diagram in the virtual correction term.", "To accomplish this in noncontractible space, we have to reduce the amplitude into pieces that are manifestly translationally and Lorentz invariant.", "We render light quark masses nonvanishing as a regulator of the collinear singularity that is canceled away in the asymmetric cross sections.", "Infrared singularity is controlled by the regulator gluon mass and is canceled away in both $A^{\\infty }$ and $\\delta A_{box}^{\\Lambda }$ asymmetry parameters.", "The virtual corrections can be represented with the following expression [22] $\\frac{d \\sigma _{A} (virtual)}{d \\cos \\theta }=\\alpha ^{3}_{s}\\frac{d^{2}_{abc}\\beta _{t}}{32 N_{c}^{2} s}[\\sum _{j=1}^{7}w_{j}I_{j} - (\\theta \\rightarrow \\pi -\\theta )], \\\\d^{2}_{abc}=\\frac{40}{3},\\ \\beta _{t}=\\sqrt{1-4 m_{t}^{2}/s}.\\nonumber $ Definitions are given in the Appendix, as well as the procedure how to evaluate the integrals in noncontractible space to maintain translational and Lorentz invariance.", "Now we can compare the t-quark charge asymmetries to the leading one loop order in the standard QCD and the QCD in noncontractible space.", "The numerics and discussion can be found in the last chapter." ], [ "Results and conclusions", "The difference between the t-quark charge asymmetries of the standard QCD and the QCD in noncontractible space lies in the additional two terms of Eq.", "(2) $\\delta A_{\\alpha }^{\\Lambda }$ and $\\delta A_{box}^{\\Lambda }$ .", "The first additional term $\\delta A_{\\alpha }^{\\Lambda }$ can be evaluated using Table 1 derived from the formulae for $\\alpha _{s}^{\\Lambda }$ in ref.", "[10].", "This correction can enhance the SM asymmetry by up to $47\\%$ for the largest parton $E_{cm}=14 TeV$ .", "The strong coupling $\\alpha _{s}^{\\Lambda }(\\mu )$ is frozen at $\\mu \\simeq 0.5 TeV$ .", "This is not enough to explain the asymmetry observed at the Tevatron [17].", "Fortunately, the second additional term $\\delta A_{box}^{\\Lambda }$ provides the necessary enhancement (see Figure 1 and Table 2).", "We define the integrated charge asymmetry parameter as [19] $A_{int} \\equiv \\frac{\\int _{0}^{1} \\sigma _{A}^{\\prime }d \\cos \\theta }{\\int _{0}^{1} \\sigma _{Born}^{\\prime }d \\cos \\theta }.$ One can conclude that the charge asymmetries at the parton level are enhanced in the BY theory by more than a factor of 3 (5) for Tevatron (LHC) center of mass energies.", "It is evident from Tables 1 and 2 that the deviation from the SM is larger for higher $E_{cm}$ and the virtual correction (box diagram) $\\delta A_{box}^{\\Lambda }$ dominates over the strong coupling correction $\\delta A_{\\alpha }^{\\Lambda }$ .", "It means that the box diagram explores the deep spacelike domain of the Minkowski spacetime to which the asymmetry observable is very sensitive and, in addition, there is no new negative compensation of the real hard and soft contributions (no quantum loops to this order of perturbation) except the new ${\\alpha }^{\\Lambda }_{s}$ factor.", "Table: Running strong couplings at the scale μ=E cm /2\\mu = E_{cm}/2assuming m u =2.5MeVm_{u}=2.5 MeV, m d =5.0MeVm_{d}=5.0 MeV, m s =100MeVm_{s}=100 MeV,m c =1.6GeVm_{c}=1.6 GeV, m b =4.8GeVm_{b}=4.8 GeV, m t =172GeVm_{t}=172 GeV andα s (μ=M Z )=0.12\\alpha _{s}(\\mu =M_{Z})=0.12.Table: Integrated t-quark charge asymmetries for parton E cm E_{cm}evaluated with E th =0.9×E cm /2E_{th}=0.9\\times E_{cm}/2 and m t =172GeVm_{t}=172 GeV.Figure: NO_CAPTION Fig.", "1: Asymmetry parameters $A^{\\infty }$ and $A^{\\Lambda }$ as a function of $x=\\cos \\theta $ ;                              $\\ \\ \\ \\ \\ \\ \\ E_{cm}$ =1.96 TeV, $m_{t}$ =172 GeV, $E_{th}$ =0.9 TeV To find charge asymmetry for hadrons, one has to convolve parton cross sections with parton distributions.", "It is necessary to solve DGLAP and BFKL equations in noncontractible space.", "This work remains for the future.", "It is very unlikely that higher orders of perturbation in the strong coupling or new parton distributions can remove large deviation of the asymmetry from the standard QCD found at the parton level.", "If the LHC confirms the Tevatron results, it will be necessary to investigate the issue to higher perturbative order to reach higher accuracy, because to date, it is the largest discrepancy observed between the standard QCD and the experiment.", "Acknowledgment It is my pleasure to thank the referee for suggestions that improve my presentation in the manuscript.", "Appendix Since the details for the QCD running coupling evaluations can be found in ref.", "[10] and the equations for the SM asymmetries in ref.", "[19], in the Appendix, we outline the equations for the virtual corrections in the SM and the BY theory using notations of refs.", "[21], [22].", "Let us define the energy unit $E=E_{cm}/2$ and the dimensionless mass of the light quark by $\\overline{m}_{u}=m_{u}/E$ and the t-quark by $\\overline{m}_{t}=m_{t}/E$ [22].", "With previously defined $c=\\beta _{t}\\cos \\theta $ the coefficients $w_{j}$ in the sum $\\sum _{j=1}^{7}{w_{j}}I_{j}$ of Eq.", "(3) are as follows [22]: $w_{1}&=&1+c^{2}-2c^{3}+(1-2c)(\\overline{m}_{u}^{2}+\\overline{m}_{t}^{2}),\\ w_{2}=2c(1-c)-\\overline{m}_{u}^{2}-c\\overline{m}_{t}^{2}, \\\\w_{3}&=&2c(1-c)-\\overline{m}_{t}^{2}-c\\overline{m}_{u}^{2},\\ w_{4}=2-c+c^{2}+\\overline{m}_{t}^{2}+\\overline{m}_{u}^{2}, \\\\w_{5}&=&-1-c,\\ w_{6}=1,\\ w_{7}=1-c .$ We need further definitions to describe the process $q(p_{+})+\\bar{q}(p_{-})\\rightarrow t(q_{+})+\\bar{t}(q_{-})$ and its amplitude [21], [22]: $P=\\frac{1}{2}(p_{+}+p_{-}),\\ \\Delta =\\frac{1}{2}(p_{+}-p_{-}),\\ Q=\\frac{1}{2}(q_{+}-q_{-}), \\\\\\overline{\\int }(f(k))\\equiv \\frac{4}{\\pi ^{2}}\\Im \\int \\frac{f(k)d^{4}k}{(\\Delta )(Q)(+)(-)}, \\\\(\\Delta )=k^{2}-2 k\\cdot \\Delta -P^{2}+\\imath \\varepsilon ,\\ (Q)=k^{2}-2 k\\cdot Q-P^{2}+\\imath \\varepsilon , \\\\(\\pm )=k^{2}\\pm 2 k\\cdot P+P^{2}-m_{gluon}^{2}+\\imath \\varepsilon .$ Now we can define dimensionless integrals $I_{j}$ of eq.", "(3): $I_{1}&=&E^{4}\\overline{\\int }(1),\\ I_{2}=E^{2}\\overline{\\int }(k\\cdot \\Delta ),\\ I_{3}=E^{2}\\overline{\\int }(k\\cdot Q),\\ I_{4}=E^{2}\\overline{\\int }(k^{2}), \\\\\\ I_{5}&=&\\overline{\\int }((k\\cdot P)^{2}),\\ I_{6}=\\overline{\\int }((k\\cdot \\Delta )^{2}+(k\\cdot Q)^{2}),\\ I_{7}=\\overline{\\int }((k\\cdot \\Delta )(k\\cdot Q)).$ These integrals can be evaluated by the integrals from ref.", "[21] $[J;J_{\\mu };J_{\\mu \\nu }]=\\int d^{4}k\\frac{[1;k_{\\mu };k_{\\mu }k_{\\nu }]}{(\\Delta )(Q)(+)(-)},$ that are expressed in terms of nine functions: $F,G,F_{\\Delta },F_{Q},G_{\\Delta },G_{Q},H_{P},$ $H_{\\Delta },H_{Q}$ .", "Let us represent seven integrals $I_{j}$ of ref.", "[22] in terms of functions from [21]: $I_{1}&=&\\frac{4}{\\pi ^{2}}\\frac{F+G}{2 P^{2}}E^{4}, \\nonumber \\\\I_{2}&=&\\frac{4}{\\pi ^{2}}(\\Delta ^{2}J_{\\Delta }+\\Delta \\cdot Q J_{Q})E^{2}, \\nonumber \\\\I_{3}&=&\\frac{4}{\\pi ^{2}}(Q^{2}J_{Q}+\\Delta \\cdot Q J_{\\Delta })E^{2}, \\nonumber \\\\I_{4}&=&\\frac{4}{\\pi ^{2}}(4 K_{O}+P^{2}K_{P}+\\Delta ^{2}K_{\\Delta }+Q^{2}K_{Q}+2 \\Delta \\cdot Q K_{X})E^{2}, \\\\I_{5}&=&\\frac{4}{\\pi ^{2}}(K_{O}P^{2}+K_{P}(P^{2})^{2}), \\nonumber \\\\I_{6}&=&\\frac{4}{\\pi ^{2}}(K_{O}(\\Delta ^{2}+Q^{2})+K_{\\Delta }((\\Delta ^{2})^{2}+(\\Delta \\cdot Q)^{2})+K_{Q}((\\Delta \\cdot Q)^{2}+(Q^{2})^{2}) \\nonumber \\\\&+& 2 K_{X}\\Delta \\cdot Q (\\Delta ^{2}+Q^{2})), \\nonumber \\\\I_{7}&=&\\frac{4}{\\pi ^{2}}(\\Delta \\cdot Q K_{O}+\\Delta \\cdot Q \\Delta ^{2} K_{\\Delta }+\\Delta \\cdot Q Q^{2} K_{Q}+ K_{X}(Q^{2}\\Delta ^{2}+(\\Delta \\cdot Q)^{2})), \\nonumber $ where $J_{\\Delta },\\ J_{Q},\\ K_{O},\\ K_{P},\\ K_{\\Delta },\\ K_{Q}\\ and\\ K_{X}$ functions are defined in terms of nine functions $F,...,H_{Q}$ [21].", "We use standard definitions for the scalar two, three and four point functions [23]: $B_{0}(p;m_{1},m_{2})=(\\imath \\pi ^{2})^{-1}\\int d^{4} k [k^{2}-m_{1}^{2}+\\imath \\epsilon ]^{-1}[(k+p)^{2}-m_{1}^{2}+\\imath \\epsilon ]^{-1},\\\\C_{0}(p_{1},p_{2};m_{0},m_{1},m_{2})=(\\imath \\pi ^{2})^{-1}\\int d^{4} k[k^{2}-m_{0}^{2}+\\imath \\epsilon ]^{-1}[(k+p_{1})^{2}-m_{1}^{2}+\\imath \\epsilon ]^{-1} \\\\\\times [(k+p_{2})^{2}-m_{2}^{2}+\\imath \\epsilon ]^{-1},$ $D_{0}(p_{1},p_{2},p_{3};m_{0},m_{1},m_{2},m_{3})=(\\imath \\pi ^{2})^{-1}\\int d^{4} k[k^{2}-m_{0}^{2}+\\imath \\epsilon ]^{-1} \\\\\\times [(k+p_{1})^{2}-m_{1}^{2}+\\imath \\epsilon ]^{-1}[(k+p_{2})^{2}-m_{2}^{2}+\\imath \\epsilon ]^{-1}[(k+p_{3})^{2}-m_{3}^{2}+\\imath \\epsilon ]^{-1}.$ In ref.", "[21] expressions for all nine functions $F,...,H_{Q}$ in the standard QCD can be found.", "The same functions have to be expressed by the previous scalar two, three and four point Green functions in noncontractible space in order to properly restore translational invariance [10], [11], [12].", "Functions $G,F_{\\Delta },F_{Q}$ have already a suitable form of the three point functions [21]: $G &=& \\int d^{4} k (\\Delta )^{-1}(Q)^{-1}(+)^{-1},\\ F_{\\Delta } = \\int d^{4} k(\\Delta )^{-1}(+)^{-1}(-)^{-1},\\ \\\\F_{Q} &=& \\int d^{4} k(Q)^{-1}(+)^{-1}(-)^{-1}.$ Note that all expressions in ref.", "[21] are derived under the assumption of $m_{gluon}\\equiv \\lambda \\ll m_{u},m_{t},E_{cm}$ .", "From their definitions, $G_{\\Delta }$ and $G_{Q}$ can be expressed as: $\\Im G_{Q} = \\frac{1}{\\beta ^{2}_{t}}\\Im F_{Q}+\\frac{2 \\pi ^{2}}{s \\beta ^{2}_{t}}[\\Re B_{0}(-2P;\\lambda ,\\lambda )-\\Re B_{0}(-Q-P;\\lambda ,m_{t})], \\nonumber \\\\\\Im G_{\\Delta } = \\frac{1}{\\beta ^{2}_{u}}\\Im F_{\\Delta }+\\frac{2 \\pi ^{2}}{s \\beta ^{2}_{u}}[\\Re B_{0}(-2P;\\lambda ,\\lambda )-\\Re B_{0}(-\\Delta -P;\\lambda ,m_{u})].$ For functions $F,H_{P},H_{\\Delta },H_{Q}$ , we derive the equations that allow to put these functions in the alternative form expressed only through scalar n-point integrals.", "The linear system for the $F$ function looks as $p_{1}^{2}\\eta _{1}+p_{1}\\cdot p_{2}\\eta _{2}+p_{1}\\cdot p_{3}\\eta _{3}=R_{1}, \\nonumber \\\\p_{1}\\cdot p_{2}\\eta _{1}+p_{2}^{2}\\eta _{2}+p_{2}\\cdot p_{3}\\eta _{3}=R_{2}, \\nonumber \\\\p_{1}\\cdot p_{3}\\eta _{1}+p_{2}\\cdot p_{3}\\eta _{2}+p_{3}^{2}\\eta _{3}=R_{3},$ $R_{1}=\\frac{1}{2}[\\Re C_{0}(p_{2},p_{3};m_{0},m_{2},m_{3})-\\Re C_{0}(p_{2}-p_{1},p_{3}-p_{1};m_{1},m_{2},m_{3}) \\\\-(p_{1}^{2}-m_{1}^{2}+m_{0}^{2})\\Re D_{0}(p_{1},p_{2},p_{3};m_{0},m_{1},m_{2},m_{3})], \\\\R_{2}=\\frac{1}{2}[\\Re C_{0}(p_{1},p_{3};m_{0},m_{1},m_{3})-\\Re C_{0}(p_{2}-p_{1},p_{3}-p_{1};m_{1},m_{2},m_{3}) \\\\-(p_{2}^{2}-m_{2}^{2}+m_{0}^{2})\\Re D_{0}(p_{1},p_{2},p_{3};m_{0},m_{1},m_{2},m_{3})], \\\\R_{3}=\\frac{1}{2}[\\Re C_{0}(p_{1},p_{2};m_{0},m_{1},m_{2})-\\Re C_{0}(p_{2}-p_{1},p_{3}-p_{1};m_{1},m_{2},m_{3}) \\\\-(p_{3}^{2}-m_{3}^{2}+m_{0}^{2})\\Re D_{0}(p_{1},p_{2},p_{3};m_{0},m_{1},m_{2},m_{3})], \\\\p_{1}=2 P,\\ p_{2}=P-\\Delta ,\\ p_{3}=P-Q,\\ m_{0}=m_{1}=\\lambda ,\\ m_{2}=m_{u},\\ m_{3}=m_{t} \\\\\\Rightarrow \\Im F = -\\Im F_{Q}+2\\pi ^{2}(\\Delta ^{2}\\eta _{2}+\\Delta \\cdot Q \\eta _{3}).$ Similarly, we derive the linear system for $H$ functions $p_{1}^{2}\\rho _{1}+p_{1}\\cdot p_{2}\\rho _{2}= M_{1}, \\nonumber \\\\p_{1}\\cdot p_{2}\\rho _{1}+p_{2}^{2}\\rho _{2}= M_{2},$ $M_{1}&=&\\frac{1}{2}[\\Re B_{0}(p_{2};\\lambda ,m_{2})-\\Re B_{0}(p_{2}-p_{1};m_{1},m_{2})+(-\\lambda ^{2}+m_{1}^{2}-p_{1}^{2}) \\\\&\\times & \\Re C_{0}(p_{1},p_{2};\\lambda ,m_{1},m_{2})], \\\\M_{2}&=&\\frac{1}{2}[\\Re B_{0}(p_{1};\\lambda ,m_{1})-\\Re B_{0}(p_{2}-p_{1};m_{1},m_{2})+(-\\lambda ^{2}+m_{2}^{2}-p_{2}^{2}) \\\\&\\times & \\Re C_{0}(p_{1},p_{2};\\lambda ,m_{1},m_{2})], \\\\p_{1}&=&P-\\Delta ,\\ p_{2}=P-Q,\\ m_{1}=m_{u},\\ m_{2}=m_{t}\\\\& & \\Rightarrow \\Im H_{P}=\\Im G+\\pi ^{2}(\\rho _{1}+\\rho _{2}),\\ \\Im H_{\\Delta }=-\\pi ^{2}\\rho _{1},\\ \\Im H_{Q}=-\\pi ^{2}\\rho _{2}.$ The validity of new forms for $F,H_{P},H_{\\Delta }\\ and\\ H_{Q}$ is also checked numerically.", "The virtual corrections can be evaluated by eq.", "(A.1) of ref.", "[22] or by eq.", "(12) of ref.[21].", "We are now prepared for the crucial step to calculate virtual corrections in noncontractible space defining scalar n-point integrals in noncontractible space.", "$B_{0}^{\\Lambda }$ function is outlined in refs.", "[10], [11], [12].", "The similar procedure should be applied to the three point function: $\\Re C_{0}^{\\infty } &=& \\Re C_{0}^{\\Lambda } + \\delta C_{0}^{\\Lambda }(symm), \\\\\\delta C_{0}^{\\Lambda }(symm)&=&\\frac{1}{3}[\\delta C_{0}^{\\Lambda }(p_{1},p_{2};m_{0},m_{1},m_{2})+\\delta C_{0}^{\\Lambda }(-p_{1},p_{2}-p_{1};m_{1},m_{0},m_{2})\\nonumber \\\\& & +\\delta C_{0}^{\\Lambda }(-p_{2},p_{1}-p_{2};m_{2},m_{0},m_{1})], \\nonumber $ $\\delta C_{0}^{\\Lambda }(p_{1},p_{2};m_{0},m_{1},m_{2})=\\pi ^{-2}\\int ^{1/\\Lambda }_{0} dw w^{-5} \\int ^{+1}_{-1}dx\\sqrt{1-x^{2}}\\int ^{+1}_{-1}dy \\\\ \\times \\int ^{2\\pi }_{0}d\\phi [-k^{2}-m_{0}^{2}]^{-1}[-k^{2}+2 (k \\cdot p_{1})+p_{1}^{2}-m_{1}^{2}]^{-1} \\\\ \\times [-k^{2}+2 (k \\cdot p_{2})+p_{2}^{2}-m_{2}^{2}]^{-1}(k=w^{-1}),\\\\where\\ (k \\cdot p_{1})=\\imath kx(p_{1})^{0}-\\vec{k}\\cdot \\vec{p}_{1}, \\\\\\vec{k}=k\\sqrt{1-x^{2}}(\\sqrt{1-y^{2}}\\cos \\phi ,\\ \\sqrt{1-y^{2}}\\sin \\phi ,\\ y).$ All the imaginary parts of the subintegral function in $\\delta C_{0}^{\\Lambda }$ are erased by integration as odd functions in variable x.", "The same decomposition is possible for the four point function, although with four terms necessary for symmetrization in $\\delta D_{0}^{\\Lambda }$ .", "Multidimensional numerical integrations in virtual and real gluon radiations are performed by Suave routine from CUBA library [24] to the relative accuracy of ${\\cal O}(10^{-4})$ with up to 50 million of sampling points per integral." ] ]

[ [ "Long-wavelength limit of gyrokinetics in a turbulent tokamak and its\n intrinsic ambipolarity" ], [ "Abstract Recently, the electrostatic gyrokinetic Hamiltonian and change of coordinates have been computed to order $\\epsilon^2$ in general magnetic geometry.", "Here $\\epsilon$ is the gyrokinetic expansion parameter, the gyroradius over the macroscopic scale length.", "Starting from these results, the long-wavelength limit of the gyrokinetic Fokker-Planck and quasineutrality equations is taken for tokamak geometry.", "Employing the set of equations derived in the present article, it is possible to calculate the long-wavelength components of the distribution functions and of the poloidal electric field to order $\\epsilon^2$.", "These higher-order pieces contain both neoclassical and turbulent contributions, and constitute one of the necessary ingredients (the other is given by the short-wavelength components up to second order) that will eventually enter a complete model for the radial transport of toroidal angular momentum in a tokamak in the low flow ordering.", "Finally, we provide an explicit and detailed proof that the system consisting of second-order gyrokinetic Fokker-Planck and quasineutrality equations leaves the long-wavelength radial electric field undetermined; that is, the turbulent tokamak is intrinsically ambipolar." ], [ "Introduction", "Gyrokinetic theory [1] and gyrokinetic codes [2], [3], [4], [5], [6], [7] are recognized as the fundamental tools for the description of microturbulence in fusion and astrophysical plasmas.", "Gyrokinetic theory consists of the elimination of the degree of freedom associated to the gyration of the charged particle around the magnetic field order by order in an asymptotic expansion in $\\epsilon = \\rho /L \\ll 1$ , where $\\rho $ is the gyroradius and $L$ is the macroscopic scale length of the problem.", "This procedure reduces the phase-space dimension and, more importantly, the degree of freedom averaged out is precisely the one with the shortest time scale.", "The savings in computational time that gyrokinetics has provided have made it possible to simulate kinetic plasma turbulence.", "Derivations of the gyrokinetic equations by iterative methods can be found in references [8], [9], [10], [11], [12], and via Hamiltonian and Lagrangian methods in references [13], [14], [15], [16].", "A recent review of gyrokinetic theory is given in [17].", "The gyrokinetic equations have typically been solved only for the turbulent components of the distribution function and the electrostatic potential (we restrict our discussion to electrostatic gyrokinetics), but in recent years growing supercomputer capabilities have motivated an increasing interest in the extension of gyrokinetic calculations to longer wavelengths and transport time scales.", "However, at least for a tokamak Throughout this paper “tokamak” means “axisymmetric tokamak”., this is a subtle issue, as F. I. Parra and P. J. Catto have discussed in a series of papers [12], [18], [19], [20], [21], [22].", "The main lines of the argument can be stated in a succinct way.", "The perpendicular component of the long-wavelength piece of the plasma velocity depends on the long-wavelength radial electric field through the $\\mathbf {E}\\times \\mathbf {B}$ drift.", "The momentum conservation equation can be used to obtain the three components of the velocity, and from it, derive the radial electric field.", "The plasma velocity is to lowest order parallel to the flux surfaces because the radial particle drift is small.", "Then, the poloidal and toroidal components of the momentum conservation equation are sufficient to calculate the velocity to the order of interest, and by decomposing it in parallel and perpendicular components, the radial electric field can be obtained by making the perpendicular component equal to the $\\mathbf {E}\\times \\mathbf {B}$ drift plus the diamagnetic velocity.", "The poloidal component of the velocity is strongly damped by collisions because the poloidal direction is not a direction of symmetry.", "The poloidal velocity is determined by setting the collisional viscosity in the poloidal direction equal to zero, giving a poloidal velocity proportional to the ion temperature gradient unless collisionality is really small and turbulence can compete with the collisional damping [18], [22].", "Unfortunately, the toroidal component of the momentum equation that would give the toroidal component of the velocity and completely determine the radial electric field is identically satisfied to order $\\epsilon ^2$ by any toroidal velocity [18], [20].", "Since gyrokinetic equations are customarily derived and solved to order $\\epsilon $ , the tokamak long-wavelength radial electric field cannot be correctly obtained from the standard set of gyrokinetic equations available in the literature.", "In the limit in which the velocity is of the order of the diamagnetic velocity, known as low flow limit, the calculation of the radial flux of toroidal angular momentum, which we need to compute the radial electric field, is especially demanding because this flux is smaller than the radial flux of particles and energy in the expansion in $\\epsilon $ .", "The low flow limit is relevant in the study of intrinsic rotation [23], [24], [25].", "In references [22], [24], a method to calculate the toroidal angular momentum conservation equation in the low flow limit to the order in which it is not identically zero is proposed.", "With the toroidal angular momentum equation to this order, it is possible to obtain the toroidal rotation and hence calculate the radial electric field.", "The formula for the radial flux of toroidal angular momentum in [22], [24] is given as a sum of several integrals over the first- and second-order pieces of the distribution functions and the electrostatic potential.", "To avoid calculating these second-order pieces in complete detail, a subsidiary expansion in $B_p/B \\ll 1$ was employed, where $B_p$ is the poloidal magnetic field and $B$ is the total magnetic field.", "With the derivation for the first time of the gyrokinetic equations and change of coordinates in general magnetic geometry up to second order [16], it has become possible to calculate the second-order pieces without resorting to a subsidiary expansion.", "In this article, we present the equations that need to be solved to obtain the long-wavelength second order pieces.", "These equations have not been explicitly written before.", "They contain neoclassical [26], [27] and turbulent contributions.", "The turbulent contributions have never been considered to our knowledge, and the complete neoclassical equations have only been used in the Pfirsch-Schlüter limit in [28].", "Calculations of the neoclassical radial flux of toroidal angular momentum in other collisionality regimes have relied on the $B_p/B \\ll 1$ expansion [29].", "We emphasize that the equations derived here are the first step towards a complete model for the computation of radial transport of toroidal angular momentum in a tokamak.", "The second step, that will be taken in a future publication, includes the derivation of the equations determining the short-wavelength components of the distribution functions and electrostatic potential to second order.", "To ease the reading of the paper, we advance in this introduction which are the equations that we derive, and that will eventually enter the aforementioned complete model for toroidal angular momentum transport in a tokamak.", "They are the long-wavelength Fokker-Planck equations to second order, (REF ) and (REF ), that give the long-wavelength component of the distribution functions; the quasineutrality equation up to second-order (REF ), (REF ), and (REF ), that determines the first and second-order pieces of the long-wavelength poloidal electric field; and the transport equations for density (REF ) and energy (REF ).", "The first-order pieces of the short-wavelength components of the distribution functions and electrostatic potential appear in (REF ), and we give the equations for them in (REF ) and (REF ).", "Carrying the expansion to second order in $\\epsilon $ at long wavelengths also clarifies the issues with the radial electric field raised in references [12], [18], [19], [20], [21], [22], pointed out at the beginning of this introduction.", "Along with the derivation of the equations we give an explicit proof of the indeterminacy of the radial electric field, showing that it cannot be found from the long-wavelength gyrokinetic Fokker-Planck and quasineutrality equations correct to second order.", "This property, known as intrinsic ambipolarity, was first proven for neoclassical transport in [30], [31] and it was shown to hold for turbulent tokamaks in [18] using the identical cancellation of the toroidal angular momentum conservation equation to the order of interest.", "The intrinsic ambipolarity of purely turbulent particle fluxes was shown to hold in [32], even electromagnetically and in general magnetic geometry (that is why the long-wavelength radial electric field in non-quasisymmetric stellarators is determined from neoclassical theory).", "This is, however, the first direct, explicit, and general proof for turbulent tokamaks.", "Instead of resorting to the toroidal angular momentum equation, we write the long-wavelength equations order by order and show that they can be solved for any radial electric field, leaving it undetermined.", "Those readers who are familiar with the Chapman-Enskog results on the derivation of fluid equations from kinetic theory (see the classical monograph [33]) will find that the approach that we adopt at some stages of the proof is very similar.", "The analogy becomes especially clear in Section REF .", "In previous sections the long-wavelength Fokker-Planck and quasineutrality equations have been derived up to second order.", "In Section REF we inspect the second-order piece of the long-wavelength Fokker-Planck equation and learn that it possesses solvability conditions, i.e.", "the existence of solutions of this equation imposes constraints on lowest-order quantities.", "These constraints are transport equations for particle and energy density.", "The way of obtaining them and of showing that we have actually found all the solvability conditions are the aspects particularly reminiscent of the Chapman-Enskog techniques.", "Nevertheless, we have written the paper in a self-contained fashion and no prior knowledge of the Chapman-Enskog theory is assumed.", "The rest of the paper is organized as follows.", "In Section we introduce the gyrokinetic formulation and the essential results and notation from [16] that will be needed here.", "An important element of our derivation is the scale separation between the turbulent short-wavelength fluctuations and the equilibrium long-wavelength profiles.", "In Section 2 we also discuss the implications of this scale separation and formalize the notion of “taking the long-wavelength limit of gyrokinetics”.", "The most laborious part of this work corresponds to explicitly taking the long-wavelength limit of the gyrokinetic system of equations in tokamak geometry by employing the results of [16].", "In Section we do it for the Fokker-Planck equation and in Section for the quasineutrality equation.", "Reaching the final expressions for the long-wavelength limit of the gyrokinetic system to second order involves enormous amounts of algebra, and in order to ease a first reading of the paper the most cumbersome parts of the calculation have been collected in the appendices.", "Using the results of Sections and we prove in Section that the long-wavelength tokamak radial electric field is not determined by second-order Fokker-Planck and quasineutrality equations.", "A complete proof requires computing the solvability conditions imposed by the second-order long-wavelength Fokker-Planck equation, contained in Subsection REF .", "These conditions are transport equations for particle and energy densities, as mentioned above.", "With these transport equations, we show in Section REF that the well-known neoclassical intrinsic ambipolarity property of the tokamak is not broken by the turbulent terms that are specific to gyrokinetics, that is, the radial electric field is left undetermined by a gyrokinetic system of equations correct to second order in $\\epsilon $ .", "Section is devoted to a discussion of the results and the conclusions." ], [ "Second-order electrostatic gyrokinetics", "In this section we state and justify the assumptions of the theory, and we summarize the results from reference [16] that will be needed." ], [ "Kinetic description of a plasma in a static magnetic field", "The kinetic description of a plasma in the electrostatic approximation involves the Fokker-Planck equation for each species ${\\sigma }$ , $\\partial _t f_{\\sigma }+ \\mathbf {v}\\cdot \\nabla _\\mathbf {r}f_{\\sigma }+\\frac{Z_{\\sigma }e}{m_{\\sigma }}\\left(-\\nabla _\\mathbf {r}\\varphi +c^{-1}\\, \\mathbf {v}\\times \\mathbf {B}\\right)\\cdot \\nabla _\\mathbf {v}f_{\\sigma }=\\nonumber \\\\[5pt]\\hspace{28.45274pt}\\sum _{{\\sigma }^{\\prime }}C_{{\\sigma }{\\sigma }^{\\prime }}[f_{\\sigma },f_{{\\sigma }^{\\prime }}](\\mathbf {r},\\mathbf {v}),$ and Poisson's equation, $\\nabla ^2_\\mathbf {r}\\varphi (\\mathbf {r},t)= -4\\pi e\\sum _{\\sigma }Z_{\\sigma }\\int \\,f_{\\sigma }(\\mathbf {r},\\mathbf {v},t)\\mbox{d}^3 v.$ Here $c$ is the speed of light, $e$ the charge of the proton, $\\varphi (\\mathbf {r}, t)$ the electrostatic potential, $\\mathbf {B}(\\mathbf {r})=\\nabla _\\mathbf {r}\\times \\mathbf {A}(\\mathbf {r})$ a time-independent magnetic field, $f_{\\sigma }(\\mathbf {r},\\mathbf {v},t)$ the phase-space probability distribution, and $Z_{\\sigma }e$ and $m_{\\sigma }$ are the charge and the mass of species ${\\sigma }$ .", "We recall that the Landau collision operator between species ${\\sigma }$ and ${\\sigma }^{\\prime }$ reads $C_{{\\sigma }{\\sigma }^{\\prime }}[f_{\\sigma },f_{{\\sigma }^{\\prime }}](\\mathbf {r},\\mathbf {v}) =\\nonumber \\\\[5pt]\\hspace{28.45274pt}\\frac{\\gamma _{{\\sigma }{\\sigma }^{\\prime }}}{m_\\sigma }\\nabla _\\mathbf {v}\\cdot \\int \\leftrightarrow \\over {\\mathbf {W}}(\\mathbf {v}-\\mathbf {v}^{\\prime })\\cdot \\Bigg (\\frac{1}{m_{\\sigma }}f_{{\\sigma }^{\\prime }}(\\mathbf {r},\\mathbf {v}^{\\prime },t)\\nabla _\\mathbf {v}f_{\\sigma }(\\mathbf {r},\\mathbf {v},t)\\nonumber \\\\[5pt]\\hspace{28.45274pt}-\\frac{1}{m_{{\\sigma }^{\\prime }}}f_{{\\sigma }}(\\mathbf {r},\\mathbf {v},t)\\nabla _{\\mathbf {v}^{\\prime }} f_{{\\sigma }^{\\prime }}(\\mathbf {r},\\mathbf {v}^{\\prime },t)\\Bigg )\\mbox{d}^3v^{\\prime },$ where $\\gamma _{{\\sigma }{\\sigma }^{\\prime }}:= 2\\pi Z_{\\sigma }^2 Z_{{\\sigma }^{\\prime }}^2 e^4 \\ln \\Lambda ,$ ${\\bf \\leftrightarrow \\over {\\mathbf {W}}}({w}):= \\frac{|{w}|^2{\\leftrightarrow \\over {\\mathbf {I}}}-{w}{w}}{|{w}|^3},$ $\\ln \\Lambda $ is the Coulomb logarithm, and $\\leftrightarrow \\over {\\mathbf {I}}$ is the identity matrix.", "A direct check shows that the Fokker-Planck equation can also be written as $\\partial _t f_{\\sigma }+ \\lbrace f_{\\sigma },H_{\\sigma }\\rbrace _\\mathbf {X}=\\sum _{{\\sigma }^{\\prime }}C_{{\\sigma }{\\sigma }^{\\prime }}[f_{\\sigma },f_{{\\sigma }^{\\prime }}](\\mathbf {X}),$ where we designate by $\\mathbf {X}\\equiv (\\mathbf {r},\\mathbf {v})$ a set of euclidean coordinates in phase-space, $H_{\\sigma }(\\mathbf {r},\\mathbf {v},t) = \\frac{1}{2}m_{\\sigma }\\mathbf {v}^2 + Z_{\\sigma }e\\varphi (\\mathbf {r},t)$ is the Hamiltonian of species ${\\sigma }$ , and the Poisson bracket of two functions on phase space, $g_1(\\mathbf {r},\\mathbf {v})$ , $g_2(\\mathbf {r},\\mathbf {v})$ , is $\\lbrace g_1,g_2\\rbrace _\\mathbf {X}&= \\frac{1}{m_{\\sigma }}\\left(\\nabla _\\mathbf {r}g_1\\cdot \\nabla _\\mathbf {v}g_2 -\\nabla _\\mathbf {v}g_1\\cdot \\nabla _\\mathbf {r}g_2\\right)\\nonumber \\\\[5pt]&+ \\frac{Z_{\\sigma }e}{m_{\\sigma }^2c} \\mathbf {B}\\cdot (\\nabla _\\mathbf {v}g_1\\times \\nabla _\\mathbf {v}g_2).$" ], [ "Dimensionless variables", "In most of what follows we find it convenient to work with non-dimensionalized variables [16].", "The species-independent normalization $ \\underline{t} = \\frac{c_s t}{L}, \\ \\underline{\\mathbf {r}} =\\frac{\\mathbf {r}}{L}, \\ \\underline{\\mathbf {A}} = \\frac{\\mathbf {A}}{B_0 L}, \\ \\underline{\\varphi } = \\frac{e \\varphi }{\\epsilon _s T_{e0}}, \\ \\nonumber \\\\[5pt]\\underline{H_{\\sigma }} = \\frac{H_{\\sigma }}{T_{e0}}, \\ \\underline{n_{\\sigma }}=\\frac{n_\\sigma }{n_{e0}}, \\ \\underline{T_{\\sigma }}=\\frac{T_\\sigma }{T_{e0}},$ is employed for time, space, vector potential, electrostatic potential, Hamiltonian, particle density, and temperature; and the species-dependent normalization $ \\underline{\\mathbf {v}_{\\sigma }} = \\frac{\\mathbf {v}_{\\sigma }}{v_{t{\\sigma }}}, \\ \\underline{f_{\\sigma }} = \\frac{v_{t{\\sigma }}^3}{n_{e0}} f_{\\sigma },$ for velocities and distribution functions.", "In the previous expressions $L\\sim |\\nabla _\\mathbf {r}\\ln |\\mathbf {B}||^{-1}$ is the typical length of variation of the magnetic field, $B_0$ a typical value of the magnetic field strength, $c_s=\\sqrt{T_{e0}/m_i}$ the sound speed, $T_{e0}$ a typical electron temperature, $n_{e0}$ a typical electron density, and $m_i$ the mass of the dominant ion species, that we assume singly charged.", "Finally, $v_{t{\\sigma }}$ is the thermal speed of species ${\\sigma }$ , $\\epsilon _s=\\rho _s/L$ , where $\\rho _s=c_s/\\Omega _i$ is a characteristic sound gyroradius, and $\\Omega _i={eB_0/(m_ic)}$ is a characteristic ion gyrofrequency.", "We take $v_{t{\\sigma }}=\\sqrt{T_{e0}/m_{\\sigma }}$ as the expression for the typical thermal speed, i.e.", "we assume that $T_{e 0}$ , the characteristic temperature of electrons, is also the characteristic temperature for all species.", "This assumption is justified when the time between collisions is shorter than the transport time scale, leading to thermal equilibration between species.", "The normalization of the electrostatic potential might seem strange at this point but it will be explained in the next subsection.", "The natural, species-independent expansion parameter in gyrokinetic theory is $\\epsilon _s$ .", "Many expressions, however, are more conveniently written in terms of the species-dependent parameter $\\epsilon _{\\sigma }=\\rho _{\\sigma }/L$ , where $\\rho _{\\sigma }=v_{t{\\sigma }}/\\Omega _{\\sigma }$ is a characteristic gyroradius of species ${\\sigma }$ and $\\Omega _{\\sigma }=Z_{\\sigma }e B_0/(m_{\\sigma }c)$ a characteristic gyrofrequency.", "Observe that the relation between $\\epsilon _{\\sigma }$ and $\\epsilon _s$ is $\\epsilon _s=\\lambda _{\\sigma }\\epsilon _{\\sigma }$ , with $\\lambda _{\\sigma }= \\frac{\\rho _s}{\\rho _{\\sigma }} = Z_{\\sigma }\\sqrt{\\frac{m_i}{m_{\\sigma }}}.$ In dimensionless variables, the Fokker-Planck equation (REF ) becomes $\\partial _{\\underline{t}}\\, \\underline{f_{\\sigma }} +\\tau _{\\sigma }\\left\\lbrace \\underline{f_{\\sigma }},\\underline{H_{\\sigma }}\\right\\rbrace _{\\underline{\\mathbf {X}}} = \\tau _{\\sigma }\\sum _{{\\sigma }^{\\prime }}\\underline{C_{{\\sigma }{\\sigma }^{\\prime }}}[\\underline{f_{\\sigma }},\\underline{f_{{\\sigma }^{\\prime }}}](\\underline{\\mathbf {r}},\\underline{\\mathbf {v}}),$ where $\\tau _{\\sigma }= \\frac{v_{t{\\sigma }}}{c_s} =\\sqrt{\\frac{m_i}{m_{\\sigma }}}\\, ,$ and the Poisson bracket of two functions $g_1(\\underline{\\mathbf {r}},\\underline{\\mathbf {v}})$ , $g_2(\\underline{\\mathbf {r}},\\underline{\\mathbf {v}})$ (we no longer write the subindex ${\\sigma }$ in $\\mathbf {v}_{\\sigma }$ ) is defined by $\\lbrace g_1,g_2\\rbrace _{\\underline{\\mathbf {X}}} &=\\left(\\nabla _{\\underline{\\mathbf {r}}}g_1\\cdot \\nabla _{\\underline{\\mathbf {v}}} g_2 - \\nabla _{\\underline{\\mathbf {v}}}g_1\\cdot \\nabla _{\\underline{\\mathbf {r}}} g_2\\right)\\nonumber \\\\[5pt]&+\\frac{1}{\\epsilon _{\\sigma }}\\underline{\\mathbf {B}}\\cdot (\\nabla _{\\underline{\\mathbf {v}}}g_1\\times \\nabla _{\\underline{\\mathbf {v}}} g_2).$ Here $\\underline{\\mathbf {X}}\\equiv (\\underline{\\mathbf {r}},\\underline{\\mathbf {v}})$ are the dimensionless cartesian coordinates.", "The normalized collision operator is $\\underline{C_{{\\sigma }{\\sigma }^{\\prime }}}[\\underline{f_{\\sigma }},\\underline{f_{{\\sigma }^{\\prime }}}](\\underline{\\mathbf {r}},\\underline{\\mathbf {v}}) =\\nonumber \\\\[5pt]\\hspace{28.45274pt}\\underline{\\gamma _{{\\sigma }{\\sigma }^{\\prime }}} \\nabla _{\\underline{\\mathbf {v}}}\\cdot \\int \\leftrightarrow \\over {\\mathbf {W}}\\left(\\tau _{\\sigma }\\underline{\\mathbf {v}} -\\tau _{{\\sigma }^\\prime }\\underline{\\mathbf {v}^{\\prime }}\\right) \\cdot \\Bigg ( \\tau _{\\sigma }\\underline{f_{{\\sigma }^{\\prime }}}(\\underline{\\mathbf {r}},\\underline{\\mathbf {v}^{\\prime }},\\underline{t})\\nabla _{\\underline{\\mathbf {v}}}\\underline{f_{\\sigma }}(\\underline{\\mathbf {r}},\\underline{\\mathbf {v}},\\underline{t})\\nonumber \\\\[5pt]\\hspace{28.45274pt}-\\tau _{{\\sigma }^{\\prime }}\\underline{f_{{\\sigma }}}(\\underline{\\mathbf {r}},\\underline{\\mathbf {v}},\\underline{t})\\nabla _{\\underline{\\mathbf {v}^{\\prime }}} \\underline{f_{{\\sigma }^{\\prime }}}(\\underline{\\mathbf {r}},\\underline{\\mathbf {v}^{\\prime }},\\underline{t}) \\Bigg )\\mbox{d}^3\\underline{v^{\\prime }},\\nonumber \\\\$ with $\\underline{\\gamma _{{\\sigma }{\\sigma }^{\\prime }}}:= \\frac{2 \\pi Z_{\\sigma }^2 Z_{{\\sigma }^{\\prime }}^2 n_{e0} e^4 L}{T_{e0}^2}\\ln \\Lambda .$ Note in passing that $\\underline{\\gamma _{{\\sigma }{\\sigma }^{\\prime }}}$ is the usual collisionality parameter $\\nu _{*{\\sigma }{\\sigma }^\\prime }$ up to a factor of order unity.", "We use the following definition of $\\nu _{*{\\sigma }{\\sigma }^\\prime }$ : $\\nu _{*{\\sigma }{\\sigma }^\\prime }:=L\\nu _{{\\sigma }{\\sigma }^\\prime }/v_{t{\\sigma }},$ where the collision frequency is $\\nu _{{\\sigma }{\\sigma }^\\prime }:= \\frac{4\\sqrt{2\\pi }}{3}\\frac{Z_{\\sigma }^2 Z_{{\\sigma }^{\\prime }}^2 n_{e0} e^4 }{m_{\\sigma }^{1/2} T_{{\\sigma }}^{3/2}}\\ln \\Lambda ,$ which coincides with Braginskii's definition [34] for ${\\sigma }=e$ and ${\\sigma }^{\\prime }=i$ .", "As for equation (REF ), $\\frac{\\epsilon _s \\lambda _{De}^2}{L^2} \\nabla _{\\underline{r}}^2\\,\\underline{\\varphi } (\\underline{\\mathbf {r}}, \\underline{t}) = -\\sum _{\\sigma }Z_{\\sigma }\\int \\underline{f_{\\sigma }}(\\underline{\\mathbf {r}},\\underline{\\mathbf {v}}, \\underline{t}) \\mbox{d}^3 \\underline{v},$ where $\\lambda _{De} = \\sqrt{\\frac{T_{e0}}{4\\pi e^2 n_{e0}}}$ is the electron Debye length.", "We assume that the Debye length is sufficiently small that we can neglect the left-hand side of (REF ), so quasineutrality $\\sum _{\\sigma }Z_{\\sigma }\\int \\underline{f_{\\sigma }}(\\underline{\\mathbf {r}}, \\underline{\\mathbf {v}},\\underline{t}) \\mbox{d}^3 \\underline{v}=0$ holds." ], [ "Gyrokinetic ordering and separation of scales", "In strongly magnetized plasmas a small quantity, $\\epsilon _{\\sigma }=\\rho _{\\sigma }/L\\ll 1$ , naturally arises for each species.", "The smallness of $\\epsilon _{\\sigma }$ implies that two very different length scales exist: the gyroradius scale and the macroscopic scale.", "Also, strong magnetization makes the time scale associated to the gyromotion around a field line, $\\Omega _{\\sigma }^{-1}$ , very small compared to microturbulence time scales.", "It is therefore justified to try to average over the irrelevant gyromotion without losing non-zero gyroradius effects.", "Gyrokinetics is the theory that results from averaging over the gyromotion when the parameter $\\epsilon _{\\sigma }$ (or more precisely $\\epsilon _s$ ) is small.", "We assume that $\\underline{\\gamma _{{\\sigma }{\\sigma }^{\\prime }}}\\sim \\lambda _{\\sigma }\\sim \\tau _{\\sigma }\\sim 1$ for all ${\\sigma },{\\sigma }^{\\prime }$ .", "That is, the only formal expansion parameter is $\\epsilon _s$ .", "This is a maximal expansion in the sense that the different physically reasonable and customary subsidiary expansions (such as expansions in mass ratios) are contained in our results and could be eventually performed in order to simplify the equations.", "As most gyrokinetic derivations, this article relies on a set of ansätze about scale separation and ordering that we proceed to explain.", "First we define a transport or coarse-grain average, that for a given function extracts the axisymmetric component (recall that our aim is to fully work out the axisymmetric case) corresponding to long wavelengths and small frequencies.", "Let $\\lbrace \\psi ,\\Theta ,\\zeta \\rbrace $ be a set of flux coordinates, where $\\psi $ is the poloidal magnetic flux, $\\Theta $ is the poloidal angle, and $\\zeta $ is the toroidal angle.", "A working definition of this averaging operation can be given by $\\langle \\dots \\rangle _{\\rm {T}}= \\frac{1}{2\\pi \\Delta t \\Delta \\psi \\Delta \\Theta }\\int _{\\Delta t}\\mbox{d}t\\int _{\\Delta \\psi }\\mbox{d}\\psi \\int _{\\Delta \\Theta }\\mbox{d}\\Theta \\int _0^{2\\pi }\\mbox{d}\\zeta (\\dots ),$ where $\\epsilon _s\\ll \\Delta \\psi /\\psi \\ll 1$ , $\\epsilon _s\\ll \\Delta \\Theta \\ll 1$ , and $L/c_s\\ll \\Delta t\\ll \\tau _E$ .", "Here $\\tau _E \\sim \\epsilon _s^{-2} L/c_s$ is the transport time scale.", "For any function $g(\\mathbf {r},t)$ , we define $g^{\\rm lw}&:= \\left\\langle g\\right\\rangle _{\\rm {T}}\\nonumber \\\\[5pt]g^{\\rm sw}&:= g - g^{\\rm lw}.$ The following obvious properties will be repeatedly employed: $&&\\left[g^{\\rm lw}\\right]^{\\rm lw}= g^{\\rm lw},\\nonumber \\\\[5pt]&&\\left[g^{\\rm sw}\\right]^{\\rm lw}= 0,\\nonumber \\\\[5pt]&&\\left[g h\\right]^{\\rm lw}= g^{\\rm lw}h^{\\rm lw}+ \\left[g^{\\rm sw}h^{\\rm sw}\\right]^{\\rm lw},$ for any two functions $g(\\mathbf {r},t)$ and $h(\\mathbf {r},t)$ .", "We decompose the fields of our theory using the coarse-grain average: $f_{\\sigma }& = f_{\\sigma }^{\\rm lw}+ f_{\\sigma }^{\\rm sw},\\nonumber \\\\\\varphi & = \\varphi ^{\\rm lw}+ \\varphi ^{\\rm sw}.$ An ansatz is made about the relative size of the long-wavelength and the short-wavelength components.", "The long-wavelength component of the distribution function is assumed to be larger than the short-wavelength piece by a factor of $\\epsilon _s^{-1} \\gg 1$ ; the long-wavelength piece of the potential is itself comparable to the kinetic energy of the particles and its short-wavelength component is also small in $\\epsilon _s$ .", "Summarizing, $ \\frac{v_{t{\\sigma }}^3 f_{\\sigma }^{\\rm sw}}{n_{e0}}\\sim \\frac{Z_{\\sigma }e\\varphi ^{\\rm sw}}{m_{\\sigma }v_{t{\\sigma }}^2} \\sim \\epsilon _s,\\nonumber \\\\[5pt]\\frac{v_{t{\\sigma }}^3 f_{\\sigma }^{\\rm lw}}{n_{e0}}\\sim \\frac{Z_{\\sigma }e \\varphi ^{\\rm lw}}{m_{\\sigma }v_{t{\\sigma }}^2}\\sim 1.$ We need also an ansatz about the size of the space and time derivatives of the long- and short-wavelength components of our fields.", "The long-wavelength components $f_{\\sigma }^{\\rm lw}$ and $\\varphi ^{\\rm lw}$ are characterized by large spatial scales, of the order of the macroscopic scale $L$ , and long time scales, of the order of the transport time scale, $\\tau _E:=L/(c_s\\epsilon _s^2)$ , i.e.", "$\\nabla _\\mathbf {r}\\ln f_{\\sigma }^{\\rm lw}, \\ \\nabla _\\mathbf {r}\\ln \\varphi ^{\\rm lw}\\sim 1/L, \\nonumber \\\\[5pt]\\partial _t \\ln f_{\\sigma }^{\\rm lw},\\ \\partial _t \\ln \\varphi ^{\\rm lw}\\sim \\epsilon _s^2 c_s/L.$ The short-wavelength components $f_{\\sigma }^{\\rm sw}$ and $\\varphi ^{\\rm sw}$ have perpendicular wavelengths of the order of the sound gyroradius, and short time scales, of the order of the turbulent correlation time.", "The parallel correlation length of the short-wavelength component is much longer than its characteristic perpendicular wavelength, and it is comparable to the size of the machine.", "In short, $f_{\\sigma }^{\\rm sw}$ and $\\varphi ^{\\rm sw}$ are characterized by $\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\ln f_{\\sigma }^{\\rm sw},\\ \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\ln \\varphi ^{\\rm sw}\\sim 1/L,\\nonumber \\\\[5pt]\\nabla _{\\mathbf {r}_\\perp } \\ln f_{\\sigma }^{\\rm sw},\\ \\nabla _{\\mathbf {r}_\\perp }\\ln \\varphi ^{\\rm sw}\\sim 1/\\rho _s,\\nonumber \\\\[5pt]\\partial _t \\ln f_{\\sigma }^{\\rm sw}, \\ \\partial _t\\ln \\varphi ^{\\rm sw}\\sim c_s/L.$ The magnetic field only contains long-wavelength components, $\\nabla _\\mathbf {r}\\ln |\\mathbf {B}| \\sim 1/L.$ The above assumptions make the elimination of the gyrophase order by order in $\\epsilon _s$ possible and the resulting equations consistent.", "These assumptions are based on experimental and theoretical evidence.", "In experiments it has been possible to confirm that the characteristic correlation length of the turbulence is of the order of and scales with the ion gyroradius [35].", "The same measurements showed that the size of the turbulent fluctuations scales with the ion gyroradius.", "The characteristic length of the turbulent eddies and the size of the fluctuations are related to each other by the background gradient.", "An eddy of length $\\ell _\\bot \\sim \\rho _s$ mixes the plasma contained within it.", "In the presence of a gradient this eddy will lead to fluctuations on top of the background density of order $\\delta n_e \\sim \\ell _\\bot |\\nabla n_e| \\sim \\epsilon _s n_e\\ll n_e$ .", "In addition to the experimental measurements, there exist strong theoretical arguments in favor of the assumptions above.", "The equations obtained using these assumptions lead to a nonlinear system of gyrokinetic equations for the fluctuations.", "These equations can be implemented in numerical simulations that encompass several ion gyroradii, as is done in [3], [4], [5], [7].", "These simulations converge for numerical domains that are sufficiently large to contain the largest turbulent eddies.", "The model is consistent if the domain size is only several gyroradii across, proving that for sufficiently small gyroradius the turbulence eddies will scale with the gyroradius.", "The flux tube simulations converge, and the characteristic size of the turbulent eddies is indeed of the order of the ion gyroradius.", "In [36] the fluctuation spectrum of the turbulence is studied by varying different parameters.", "The final result is that the spectrum peaks around wavelengths proportional to the ion gyroradius, and although the constant of proportionality depends on the density and temperature gradients and the magnetic field characteristics, it is of order unity.", "The size of the fluctuations is also of order $\\epsilon _s$ .", "Observe that in dimensionless variables the short-wavelength electrostatic potential and distribution functions satisfy $\\underline{\\varphi }^{\\rm sw}(\\underline{\\mathbf {r}}, \\underline{t})\\sim 1,\\nonumber \\\\[5pt]\\underline{f_{\\sigma }^{\\rm sw}} (\\underline{\\mathbf {r}},\\underline{\\mathbf {v}},\\underline{t})\\sim \\epsilon _s,\\nonumber \\\\[5pt]{\\hat{\\mathbf {b}}}(\\underline{\\mathbf {r}}) \\cdot \\nabla _{\\underline{\\mathbf {r}}}\\,\\underline{\\varphi }^{\\rm sw}(\\underline{\\mathbf {r}}, \\underline{t}) \\sim 1,\\nonumber \\\\[5pt]{\\hat{\\mathbf {b}}}(\\underline{\\mathbf {r}}) \\cdot \\nabla _{\\underline{\\mathbf {r}}}\\,\\underline{f_{\\sigma }^{\\rm sw}} (\\underline{\\mathbf {r}},\\underline{\\mathbf {v}},\\underline{t}) \\sim \\epsilon _s,\\nonumber \\\\[5pt]\\nabla _{\\underline{\\mathbf {r}}_{\\perp }}\\, \\underline{\\varphi }^{\\rm sw}(\\underline{\\mathbf {r}},\\underline{t}) \\sim 1/\\epsilon _s,\\nonumber \\\\[5pt]\\nabla _{\\underline{\\mathbf {r}}_{\\perp }}\\, \\underline{f_{\\sigma }^{\\rm sw}}(\\underline{\\mathbf {r}},\\underline{\\mathbf {v}},\\underline{t}) \\sim 1,\\nonumber \\\\[5pt] \\partial _{\\underline{t}} \\underline{\\varphi }^{\\rm sw}(\\underline{\\mathbf {r}}, \\underline{t}) \\sim 1,\\nonumber \\\\[5pt] \\partial _{\\underline{t}} \\underline{f_{\\sigma }^{\\rm sw}} (\\underline{\\mathbf {r}},\\underline{\\mathbf {v}},\\underline{t}) \\sim \\epsilon _s.$ The normalized functions $\\underline{\\varphi }^{\\rm sw}$ and $\\underline{f_{\\sigma }}^{\\rm sw}$ are of different size due to our choice of dimensionless variables, consistent with [16] (see (REF ) and (REF )).", "As for the long-wavelength components, $\\underline{\\varphi }^{\\rm lw}(\\underline{\\mathbf {r}},\\underline{t})\\sim 1/\\epsilon _s,\\nonumber \\\\[5pt]\\underline{f_{\\sigma }^{\\rm lw}}(\\underline{\\mathbf {r}},\\underline{\\mathbf {v}},\\underline{t})\\sim 1,\\nonumber \\\\[5pt]\\nabla _{\\underline{\\mathbf {r}}}\\, \\underline{\\varphi }^{\\rm lw}(\\underline{\\mathbf {r}},\\underline{t}) \\sim 1/\\epsilon _s,\\nonumber \\\\[5pt]\\nabla _{\\underline{\\mathbf {r}}}\\, \\underline{f_{\\sigma }^{\\rm lw}}(\\underline{\\mathbf {r}},\\underline{t}) \\sim 1,\\nonumber \\\\[5pt] \\partial _{\\underline{t}} \\underline{\\varphi }^{\\rm lw}(\\underline{\\mathbf {r}}, \\underline{t}) \\sim \\epsilon _s,\\nonumber \\\\[5pt] \\partial _{\\underline{t}} \\underline{f_{\\sigma }^{\\rm lw}}(\\underline{\\mathbf {r}},\\underline{\\mathbf {v}},\\underline{t}) \\sim \\epsilon _s^2.$ The following convention is adopted when we expand $\\underline{\\varphi }^{\\rm lw}(\\underline{\\mathbf {r}},\\underline{t})$ in powers of $\\epsilon _s$ : $\\underline{\\varphi }^{\\rm lw}(\\underline{\\mathbf {r}},\\underline{t}) :=\\frac{1}{\\epsilon _s}\\underline{\\varphi _0}(\\underline{\\mathbf {r}},\\underline{t}) +\\underline{\\varphi _1}^{\\rm lw}(\\underline{\\mathbf {r}},\\underline{t}) + \\epsilon _s\\underline{\\varphi _2}^{\\rm lw}(\\underline{\\mathbf {r}},\\underline{t}) +O(\\epsilon _s^2).$ Similarly, $\\underline{\\varphi }^{\\rm sw}(\\underline{\\mathbf {r}},\\underline{t}) :=\\underline{\\varphi _1}^{\\rm sw}(\\underline{\\mathbf {r}},\\underline{t}) + \\epsilon _s\\underline{\\varphi _2}^{\\rm sw}(\\underline{\\mathbf {r}},\\underline{t}) +O(\\epsilon _s^2).$ From now on we do not underline variables but assume that we are working with the dimensionless ones unless otherwise stated." ], [ "Gyrokinetic expansion to second order", "The complete calculation of the gyrokinetic system of equations to second order is given for the first time in reference [16] in the phase-space Lagrangian formalism.", "The latter was applied to the problem of guiding-center motion by Littlejohn [37] and has been used extensively in modern formulations of gyrokinetics [15].", "In reference [16] we perform a change of variables in (REF ) and (REF ) that decouples the fast degree of freedom (the gyrophase) from the slow ones in the absence of collisions.", "This decoupling is achieved by eliminating the dependence on the gyration order by order in $\\epsilon _{\\sigma }$ .", "Let us denote the transformation We warn the reader that we call ${\\cal T}_{\\sigma }$ to the transformation that is often called ${\\cal T}_{\\sigma }^{-1}$ in the literature.", "from the new phase-space coordinates $\\mathbf {Z}\\equiv \\lbrace \\mathbf {R},u,\\mu ,\\theta \\rbrace $ to the euclidean ones $\\mathbf {X}\\equiv \\lbrace \\mathbf {r},\\mathbf {v}\\rbrace $ by ${\\cal T}_{\\sigma }$ , $(\\mathbf {r},\\mathbf {v}) = {\\cal T}_{\\sigma }(\\mathbf {R},u,\\mu ,\\theta ,t).$ The transformation is, in general, explicitly time-dependent and is expressed as a power series in $\\epsilon _{\\sigma }$ .", "Here $\\mathbf {R}$ is the position of the gyrocenter, and $u$ , $\\mu $ , and $\\theta $ are deformations of the parallel velocity, magnetic moment, and gyrophase.", "We recall that in [16] the gyrokinetic transformation is written as the composition of two transformations.", "First, the non-perturbative transformation, $(\\mathbf {r}, \\mathbf {v}) = {\\cal T}_{NP, {\\sigma }} (\\mathbf {Z}_g) \\equiv {\\cal T}_{NP,{\\sigma }} (\\mathbf {R}_g, v_{||g}, \\mu _g, \\theta _g)$ , $\\mathbf {r}&=& \\mathbf {R}_g + \\epsilon _{\\sigma }\\mbox{$\\rho $}( \\mathbf {R}_g, \\mu _g, \\theta _g),\\nonumber \\\\[5pt]\\mathbf {v}&=& v_{||g} \\hat{\\mathbf {b}}(\\mathbf {R}_g) + \\mbox{$\\rho $}( \\mathbf {R}_g, \\mu _g, \\theta _g)\\times \\mathbf {B}(\\mathbf {R}_g),$ with the gyroradius vector defined as $\\mbox{$\\rho $}( \\mathbf {R}_g, \\mu _g, \\theta _g ) = - \\sqrt{\\frac{2\\mu _g}{B(\\mathbf {R}_g)}} \\left[ \\sin \\theta _g \\hat{\\mathbf {e}}_1 (\\mathbf {R}_g) - \\cos \\theta _g \\hat{\\mathbf {e}}_2 (\\mathbf {R}_g) \\right].$ The unit vectors $\\hat{\\mathbf {e}}_1 (\\mathbf {r})$ and $\\hat{\\mathbf {e}}_2 (\\mathbf {r})$ are orthogonal to each other and to $\\hat{\\mathbf {b}}= \\mathbf {B}/B$ , and satisfy $\\hat{\\mathbf {e}}_1\\times \\hat{\\mathbf {e}}_2 = \\hat{\\mathbf {b}}$ at every location $\\mathbf {r}$ .", "Second, the perturbative transformation $(\\mathbf {R}_g, v_{||g}, \\mu _g, \\theta _g) = {\\cal T}_{P, {\\sigma }} (\\mathbf {R}, u, \\mu , \\theta , t),$ that we express as $\\mathbf {R}_g &= \\mathbf {R}+ \\sum _{i=1}^n \\epsilon _{\\sigma }^{i+1} {\\mathbf {R}}_{i+1},\\;\\nonumber \\\\v_{||g} &= u + \\sum _{i=1}^n \\epsilon _{\\sigma }^i {u}_i, \\;\\nonumber \\\\\\mu _g &=\\mu + \\sum _{i=1}^n \\epsilon _{\\sigma }^i {\\mu }_i, \\;\\nonumber \\\\\\theta _g &= \\theta + \\sum _{i=1}^n \\epsilon _{\\sigma }^i {\\theta }_i.$ The gyrokinetic transformation is ${\\cal T}_{\\sigma }= {\\cal T}_{NP,{\\sigma }} {\\cal T}_{P,{\\sigma }}.$ At this point we need to mention that the derivation of ${\\cal T}_{\\sigma }$ in [16] assumed that the electrostatic potential had only a short-wavelength component, i.e.", "we assumed $\\varphi =\\varphi ^{\\rm sw}$ and $\\varphi ^{\\rm lw}= 0$ .", "Since $\\varphi ^{\\rm sw}$ is small in $\\epsilon _s$ , this assumption lead to normalizing the electrostatic potential with $\\epsilon _s T_{e0}/e$ .", "It is easy to relax the assumption in [16] that $\\varphi = \\varphi ^{\\rm sw}$ and include $\\varphi ^{\\rm lw}$ .", "In equations (68) and (69) of [16] we display the phase-space Lagrangian after the non-perturbative transformation.", "The Hamiltonian is given by $H = H^{(0)} + \\epsilon _\\sigma H^{(1)},$ with $H^{(0)} = \\frac{1}{2}v_{||g}^2 + \\mu _g B (\\mathbf {R}_g)$ and $H^{(1)} = Z_{\\sigma }\\lambda _{\\sigma }\\varphi ^{\\rm sw}( \\mathbf {R}_g + \\epsilon _{\\sigma }\\mbox{$\\rho $}(\\mathbf {R}_g, \\mu _g, \\theta _g), t).$ Using this expression, it is possible to obtain the perturbative change of variables ${\\cal T}_{P,{\\sigma }}$ by expanding in $\\epsilon _{\\sigma }$ .", "To do so, $H^{(1)}$ must be of order unity, and if instead of $\\varphi = \\varphi ^{\\rm sw}$ we have a long wavelength piece $\\varphi ^{\\rm lw}\\sim 1/\\epsilon _s \\gg 1$ , it would seem that the condition $H^{(1)}\\sim 1$ is not satisfied.", "Fortunately, it is possible to redefine $H^{(0)}$ and $H^{(1)}$ so that the expansion can be performed.", "The new Hamiltonian is given by $H^{(0)} = \\frac{1}{2}v_{||g}^2 + \\mu _g B (\\mathbf {R}_g)+ Z_{\\sigma }\\lambda _{\\sigma }\\epsilon _{\\sigma }\\langle \\phi _\\sigma \\rangle (\\mathbf {R}_g, \\mu _g, t)$ and $H^{(1)} = Z_{\\sigma }\\lambda _{\\sigma }\\tilde{\\phi }_{\\sigma }(\\mathbf {R}_g, \\mu _g, \\theta _g, t),$ where the function $\\phi _{\\sigma }$ is defined as $\\phi _{\\sigma }(\\mathbf {R}_g,\\mu _g,\\theta _g,t) :=\\varphi (\\mathbf {R}_g+\\epsilon _{\\sigma }\\mbox{$\\rho $}(\\mathbf {R}_g,\\mu _g,\\theta _g),t).$ From it we can calculate $\\tilde{\\phi }_{\\sigma }(\\mathbf {R}_g,\\mu _g,\\theta _g,t) :=\\phi _{\\sigma }(\\mathbf {R}_g,\\mu _g,\\theta _g,t)- \\langle \\phi _{\\sigma }\\rangle (\\mathbf {R}_g,\\mu _g,t)$ and $\\langle \\phi _{\\sigma }\\rangle (\\mathbf {R}_g,\\mu _g,t) :=\\frac{1}{2\\pi }\\int _0^{2\\pi }\\phi _{\\sigma }(\\mathbf {R}_g,\\mu _g,\\theta _g,t)\\mbox{d}\\theta _g.$ Here $\\langle \\cdot \\rangle $ stands for the average over the gyrophase.", "We now prove that $H^{(1)}$ is indeed of order unity.", "From the ordering and scale separation assumptions on $\\varphi $ , equations (REF ) and (REF ), we obtain that the turbulent component of $\\phi _{\\sigma }$ is $O(1)$ , i.e.", "$\\phi ^{\\rm sw}_{\\sigma }= \\phi ^{\\rm sw}_{{\\sigma }1} + O(\\epsilon _s),\\nonumber \\\\\\tilde{\\phi }^{\\rm sw}_{\\sigma }= \\tilde{\\phi }^{\\rm sw}_{{\\sigma }1} + O(\\epsilon _s).$ For the long wavelength piece $\\phi ^{\\rm lw}_{\\sigma }$ we use that it is possible to Taylor expand around $\\mathbf {r}= \\mathbf {R}$ to find $\\langle \\phi _{\\sigma }^{\\rm lw}\\rangle (\\mathbf {R}_g,\\mu _g,t) =\\frac{1}{\\epsilon _s}\\varphi _0(\\mathbf {R}_g,t)+\\varphi _1^{\\rm lw}(\\mathbf {R}_g,t)\\nonumber \\\\[5pt]\\hspace{14.22636pt}+\\epsilon _s \\left( \\frac{\\mu _g}{2\\lambda _{\\sigma }^2 B(\\mathbf {R}_g)}(\\leftrightarrow \\over {\\mathbf {I}}-\\hat{\\mathbf {b}}(\\mathbf {R}_g)\\hat{\\mathbf {b}}(\\mathbf {R}_g)):\\nabla _{\\mathbf {R}_g}\\nabla _{\\mathbf {R}_g}\\varphi _0(\\mathbf {R}_g,t)+\\varphi _2^{\\rm lw}(\\mathbf {R}_g,t) \\right)\\nonumber \\\\[5pt]\\hspace{14.22636pt}+O(\\epsilon _s^2)$ and $\\tilde{\\phi }_{\\sigma }^{\\rm lw}(\\mathbf {R}_g,\\mu _g,\\theta _g,t) =\\frac{1}{\\lambda _{\\sigma }} \\mbox{$\\rho $}(\\mathbf {R}_g,\\mu _g,\\theta _g)\\cdot \\nabla _{\\mathbf {R}_g}\\varphi _0(\\mathbf {R}_g,t) + O(\\epsilon _s),$ giving $\\tilde{\\phi }^{\\rm lw}_{\\sigma }= O(1)$ as expected.", "We have expanded up to first order in $\\epsilon _s$ in (REF ) because it will be needed later in this paper.", "Our double-dot convention for arbitrary matrix $\\leftrightarrow \\over {\\mathbf {M}}$ is $\\mathbf {u}\\mathbf {v} :\\leftrightarrow \\over {\\mathbf {M}}= \\mathbf {v} \\cdot \\leftrightarrow \\over {\\mathbf {M}}\\cdot \\mathbf {u}$ .", "The authors of references [38] and [39] already pointed out the usefulness of the separation of the electrostatic potential into a large gyrophase-independent piece and a small gyrophase-dependent one, and exploited it in their derivations.", "We want to write the Fokker-Planck equation in gyrokinetic coordinates.", "Denote by ${\\cal T}^*_{\\sigma }$ the pull-back transformation induced by ${\\cal T}_{\\sigma }$ .", "Acting on a function $g(\\mathbf {X},t)$ , ${\\cal T}_{\\sigma }^* g (\\mathbf {Z},t)$ is simply the function $g$ written in the coordinates $\\mathbf {Z}$ , i.e.", "${\\cal T}_{\\sigma }^* g (\\mathbf {Z},t) = g({\\cal T}_{\\sigma }(\\mathbf {Z},t), t).$ Now, defining $ F_{\\sigma }:={\\cal T}_{\\sigma }^* f_{\\sigma }$ , we transform (REF ) and get: $\\partial _{t} F_{\\sigma }+ \\tau _{\\sigma }\\left\\lbrace F_{\\sigma },\\overline{H}_{\\sigma }\\right\\rbrace _\\mathbf {Z}=\\tau _{\\sigma }\\sum _{{\\sigma }^{\\prime }}{\\cal T}^{*}_{{\\sigma }} C_{{\\sigma }{\\sigma }^{\\prime }} [{\\cal T}^{-1*}_{{\\sigma }} F_{\\sigma },{\\cal T}^{-1*}_{{\\sigma }^{\\prime }}F_{{\\sigma }^{\\prime }}](\\mathbf {Z}, t),$ where ${\\cal T}^{-1*}_{{\\sigma }}$ is the pull-back transformation that corresponds to ${\\cal T}_{\\sigma }^{-1}$ , i.e.", "${\\cal T}_{\\sigma }^{-1*} F_{\\sigma }(\\mathbf {X},t) =F_{\\sigma }({\\cal T}_{\\sigma }^{-1}(\\mathbf {X},t),t)$ , and the Poisson bracket in the new coordinates is expressed as $ \\lbrace G_1,G_2\\rbrace _\\mathbf {Z}&=\\frac{1}{\\epsilon _{\\sigma }}\\left(\\partial _\\mu G_1\\partial _\\theta G_2- \\partial _\\theta G_1\\partial _\\mu G_2\\right)\\nonumber \\\\[5pt]&+\\frac{1}{B_{||,{\\sigma }}^*}{\\bf B}_{\\sigma }^* \\cdot \\left(\\nabla ^*_{\\mathbf {R}}G_1\\partial _u G_2-\\partial _u G_1\\nabla _{\\mathbf {R}}^*G_2\\right) \\nonumber \\\\[5pt]&+ \\frac{\\epsilon _{\\sigma }}{B_{||}^*} \\nabla _{\\mathbf {R}}^*G_1 \\cdot (\\hat{\\mathbf {b}}\\times \\nabla _{\\mathbf {R}}^*G_2 ),$ with $ \\mathbf {B}_{\\sigma }^* (\\mathbf {R}, u, \\mu ) := \\mathbf {B}(\\mathbf {R}) + \\epsilon _{\\sigma }u \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}(\\mathbf {R})- \\epsilon ^2_{\\sigma }\\mu \\nabla _\\mathbf {R}\\times \\mathbf {K}(\\mathbf {R}),$ $ B^*_{||,{\\sigma }} (\\mathbf {R}, u, \\mu ) := \\mathbf {B}_{\\sigma }^* (\\mathbf {R}, u, \\mu ) \\cdot \\hat{\\mathbf {b}}(\\mathbf {R})\\nonumber \\\\[5pt] = B(\\mathbf {R}) + \\epsilon _{\\sigma }u \\hat{\\mathbf {b}}(\\mathbf {R}) \\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}(\\mathbf {R})\\nonumber \\\\[5pt]- \\epsilon ^2_{\\sigma }\\mu \\hat{\\mathbf {b}}(\\mathbf {R}) \\cdot \\nabla _\\mathbf {R}\\times \\mathbf {K}(\\mathbf {R}),$ $\\nabla _{\\mathbf {R}}^* := \\nabla _{\\mathbf {R}} - {\\bf K}(\\mathbf {R})\\partial _\\theta ,$ and $\\mathbf {K}(\\mathbf {R}) = \\frac{1}{2} \\hat{\\mathbf {b}}(\\mathbf {R}) \\hat{\\mathbf {b}}(\\mathbf {R}) \\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}(\\mathbf {R}) - \\nabla _\\mathbf {R}\\hat{\\mathbf {e}}_2 (\\mathbf {R}) \\cdot \\hat{\\mathbf {e}}_1 (\\mathbf {R}).$ The above expression (REF ) for the Poisson bracket is customary in the literature [15].", "The gyrokinetic change of coordinates is not unique, in the sense that there are infinitely many transformations such that the gyromotion is decoupled from the rest of degrees of freedom and such that the coordinate $\\mu $ is an adiabatic invariant.", "To make comparisons with standard references in the literature easy, we have made use of this flexibility by choosing our change of variables so that the Poisson bracket takes the form (REF ).", "The main achievement of [16] was the computation of the gyrokinetic Hamiltonian $\\overline{H}_{\\sigma }=\\sum _{n=0}^\\infty \\epsilon _{\\sigma }^n \\overline{H}_{\\sigma }^{(n)}$ to order $\\epsilon _{\\sigma }^2$ .", "The result is: $\\overline{H}^{(0)}_{\\sigma }= \\frac{1}{2}u^2+\\mu B,$ $\\overline{H}^{(1)}_{\\sigma }= Z_{\\sigma }\\lambda _{\\sigma }\\langle \\phi _{\\sigma }\\rangle ,$ $\\overline{H}^{(2)}_{\\sigma }=Z_{\\sigma }^2 \\lambda _{\\sigma }^2 \\Psi _{\\phi ,{\\sigma }} +Z_{\\sigma }\\lambda _{\\sigma }\\Psi _{\\phi B,{\\sigma }} + \\Psi _{B,{\\sigma }},$ with $\\Psi _{\\phi ,{\\sigma }} &= \\frac{1}{2 B^2} \\left\\langle \\nabla _{(\\mathbf {R}_\\bot /\\epsilon _{\\sigma })} \\widetilde{\\Phi }_{\\sigma }\\cdot \\left( \\hat{\\mathbf {b}}\\times \\nabla _{(\\mathbf {R}_\\bot /\\epsilon _{\\sigma })} \\widetilde{\\phi }_{\\sigma }\\right) \\right\\rangle \\nonumber \\\\ &- \\frac{1}{2 B} \\partial _\\mu \\langle \\widetilde{\\phi }^2_{\\sigma }\\rangle , $ $\\Psi _{\\phi B,{\\sigma }}& = - \\frac{u}{B} \\left\\langle \\left( \\nabla _{(\\mathbf {R}_\\bot /\\epsilon _{\\sigma })} \\widetilde{\\phi }_{\\sigma }\\times \\hat{\\mathbf {b}}\\right) \\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\cdot \\mbox{$\\rho $}\\right\\rangle \\nonumber \\\\ &- \\frac{\\mu }{2 B^2} \\nabla _\\mathbf {R}B \\cdot \\nabla _{(\\mathbf {R}_\\bot /\\epsilon _{\\sigma })} \\langle \\phi _{\\sigma }\\rangle - \\frac{1}{B} \\nabla _\\mathbf {R}B\\cdot \\langle \\widetilde{\\phi }_{\\sigma }\\, \\mbox{$\\rho $}\\rangle \\nonumber \\\\ & - \\frac{1}{4 B} \\left\\langle \\nabla _{(\\mathbf {R}_\\bot /\\epsilon _{\\sigma })} \\widetilde{\\phi }_{\\sigma }\\cdot \\left[\\mbox{$\\rho $}\\mbox{$\\rho $}- ( \\mbox{$\\rho $}\\times \\hat{\\mathbf {b}}) (\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}}) \\right] \\cdot \\nabla _\\mathbf {R}B \\right\\rangle \\nonumber \\\\ &- \\frac{u^2}{B} \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\cdot \\left\\langle \\partial _\\mu \\widetilde{\\phi }_{\\sigma }\\, \\mbox{$\\rho $}\\right\\rangle -\\frac{u^2}{2 \\mu B} \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\cdot \\langle \\widetilde{\\phi }_{\\sigma }\\, \\mbox{$\\rho $}\\rangle \\nonumber \\\\&+\\frac{u}{4} \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}: \\left\\langle \\partial _\\mu \\widetilde{\\phi }_{\\sigma }\\, \\left[ \\mbox{$\\rho $}( \\mbox{$\\rho $}\\times \\hat{\\mathbf {b}}) + (\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}}) \\mbox{$\\rho $}\\right] \\right\\rangle \\nonumber \\\\& +\\frac{u}{4 \\mu } \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}: \\left\\langle \\widetilde{\\phi }_{\\sigma }\\,\\left[ \\mbox{$\\rho $}( \\mbox{$\\rho $}\\times \\hat{\\mathbf {b}}) + (\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}})\\mbox{$\\rho $}\\right] \\right\\rangle $ and $\\Psi _{B,{\\sigma }} &= - \\frac{3u^2 \\mu }{2B^2} \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B\\nonumber \\\\&+ \\frac{\\mu ^2}{4B} (\\leftrightarrow \\over {\\mathbf {I}}- \\hat{\\mathbf {b}}\\hat{\\mathbf {b}}) :\\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\mathbf {B}\\cdot \\hat{\\mathbf {b}}\\nonumber \\\\&- \\frac{3\\mu ^2}{4B^2}|\\nabla _{\\mathbf {R}_\\bot } B|^2+ \\frac{u^2 \\mu }{2B}\\nabla _\\mathbf {R}\\hat{\\mathbf {b}}: \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\nonumber \\\\& + \\left(\\frac{\\mu ^2}{8} -\\frac{u^2 \\mu }{4B}\\right) \\nabla _{\\mathbf {R}_\\perp } \\hat{\\mathbf {b}}: (\\nabla _{\\mathbf {R}_\\perp }\\hat{\\mathbf {b}})^\\mathrm {T}\\nonumber \\\\&- \\left(\\frac{3 u^2 \\mu }{8B} +\\frac{\\mu ^2}{16}\\right) (\\nabla _\\mathbf {R}\\cdot \\hat{\\mathbf {b}})^2 \\nonumber \\\\ & + \\left(\\frac{3 u^2 \\mu }{2B}-\\frac{u^4}{2B^2}\\right) |\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}|^2\\nonumber \\\\&+ \\left(\\frac{ u^2 \\mu }{8B} -\\frac{\\mu ^2}{16}\\right) (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}})^2.$ Here $\\leftrightarrow \\over {\\mathbf {M}}^\\mathrm {T}$ is the transpose of an arbitrary matrix matrix $\\leftrightarrow \\over {\\mathbf {M}}$ , the magnetic field quantities $\\mathbf {B}(\\mathbf {R})$ , $\\hat{\\mathbf {b}}(\\mathbf {R})$ and $B(\\mathbf {R})$ are evaluated at $\\mathbf {R}$ instead of $\\mathbf {r}$ , the functions $\\phi _{{\\sigma }} (\\mathbf {R}, \\mu ,\\theta , t)$ , $\\langle \\phi _{\\sigma }\\rangle (\\mathbf {R}, \\mu , t)$ and $\\tilde{\\phi }_{\\sigma }(\\mathbf {R}, \\mu , \\theta , t)$ are evaluated at $\\mathbf {R}$ , $\\mu $ and $\\theta $ instead of $\\mathbf {R}_g$ , $\\mu _g$ and $\\theta _g$ , and $\\widetilde{\\Phi }_{\\sigma }(\\mathbf {R},\\mu ,\\theta ,t) :=\\int ^\\theta \\widetilde{\\phi }_{\\sigma }(\\mathbf {R},\\mu ,\\theta ^{\\prime },t)\\mbox{d}\\theta ^{\\prime }\\ ,$ where the lower limit of the integral is chosen such that $\\langle \\widetilde{\\Phi }_{\\sigma }\\rangle = 0$ .", "The second-order Hamiltonian is sufficient to obtain the long-wavelength component of the distribution function to order $\\epsilon _{\\sigma }^2$ .", "To check this, the reader can follow the calculation in this article assuming that $\\overline{H}^{(n)}_{\\sigma }$ for $n>2$ are known, and finding that these higher-order terms do not enter the final equations.", "In gyrokinetic variables the quasineutrality equation reads $\\sum _{\\sigma }Z_{\\sigma }\\int |\\det \\left(J_{{\\sigma }}\\right)|F_{\\sigma }\\delta \\Big (\\pi ^{\\mathbf {r}}\\Big ({\\cal T}_{{\\sigma }}(\\mathbf {Z},t)\\Big )-\\mathbf {r}\\Big )\\mbox{d}^6Z = 0,$ with $\\pi ^\\mathbf {r}(\\mathbf {r},\\mathbf {v}):=\\mathbf {r}$ , and the Jacobian of the transformation to order $\\epsilon _{\\sigma }^2$ is $|\\det (J_{{\\sigma }} )| \\equiv B_{||,{\\sigma }}^*.$ The expressions for the corrections $\\mathbf {R}_2$ , $u_1$ , $\\mu _1$ , and $\\theta _1$ found in [16] are $\\mathbf {R}_{{\\sigma },2} &=& - \\frac{2u}{B} \\hat{\\mathbf {b}}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\cdot (\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}})- \\frac{u}{B} \\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\cdot \\mbox{$\\rho $}\\nonumber \\\\&&- \\frac{1}{8} \\hat{\\mathbf {b}}\\left[ \\mbox{$\\rho $}\\mbox{$\\rho $}-(\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}}) (\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}}) \\right]: \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\nonumber \\\\&&- \\frac{1}{2B} \\mbox{$\\rho $}\\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}B- \\frac{Z_{\\sigma }\\lambda _{\\sigma }}{B^2}\\hat{\\mathbf {b}}\\times \\nabla _{(\\mathbf {R}_\\bot /\\epsilon _{\\sigma })} \\widetilde{\\Phi }_{\\sigma },\\\\[5pt]u_{{\\sigma },1} &=& u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\cdot \\mbox{$\\rho $}\\nonumber \\\\&&- \\frac{B}{4}\\left[ \\mbox{$\\rho $}(\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}}) + (\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}}) \\mbox{$\\rho $}\\right] : \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}, \\\\[5pt]\\mu _{{\\sigma },1} &=&- \\frac{Z_{\\sigma }\\lambda _{\\sigma }\\widetilde{\\phi }_{\\sigma }}{B}- \\frac{u^2}{B} \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\cdot \\mbox{$\\rho $}\\nonumber \\\\&&+ \\frac{u}{4} \\left[ \\mbox{$\\rho $}(\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}}) +(\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}}) \\mbox{$\\rho $}\\right]: \\nabla _\\mathbf {R}\\hat{\\mathbf {b}},\\\\[5pt]\\theta _{{\\sigma },1} &=& \\frac{Z_{\\sigma }\\lambda _{\\sigma }}{B}\\partial _\\mu \\widetilde{\\Phi }_{\\sigma }+\\frac{u^2}{2\\mu B} \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\cdot (\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}})\\nonumber \\\\&&+ \\frac{u}{8\\mu } \\left[ \\mbox{$\\rho $}\\mbox{$\\rho $}-(\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}}) (\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}}) \\right]: \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\nonumber \\\\&&+ \\frac{1}{B} (\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}}) \\cdot \\nabla _\\mathbf {R}B.$ These corrections are needed to find the gyrokinetic quasineutrality equation to the order or interest.", "Although it might seem that we also need the next order corrections $\\mathbf {R}_{{\\sigma },3}$ , $u_{{\\sigma },2}$ , $\\mu _{{\\sigma },2}$ , and $\\theta _{{\\sigma },2}$ , it will be shown that they do not contribute in the long-wavelength limit.", "In the following sections we take the long-wavelength limit of (REF ) and (REF ) up to second-order in the expansion parameter." ], [ "Fokker-Planck equation at long wavelengths", "The objective in this section is to take the long-wavelength limit of the gyrokinetic Fokker-Planck equation (REF ) up to second order in tokamak geometry.", "As a preliminary step we must write (REF ) order by order; for this we will expand $F_{\\sigma }$ as $F_{\\sigma }&=\\sum _{n=0}^\\infty \\epsilon _{\\sigma }^n F_{{\\sigma }n} =\\sum _{n=0}^\\infty \\epsilon _{\\sigma }^n F^{\\rm lw}_{{\\sigma }n}+\\sum _{n=1}^\\infty \\epsilon _{\\sigma }^n F^{\\rm sw}_{{\\sigma }n}.$ From the scale separation and ordering assumptions enumerated in Section it follows that $F_{{\\sigma }n}\\sim 1,\\ n\\ge 0, \\nonumber \\\\[5pt]\\hat{\\mathbf {b}}(\\mathbf {R}) \\cdot \\nabla _{\\mathbf {R}} F_{{\\sigma }n}\\sim 1,\\ n\\ge 0.$ Also, the long-wavelength component of every $F_{{\\sigma }n}$ must have perpendicular derivatives of order unity in normalized variables, i.e.", "$\\nabla _{\\mathbf {R}_\\perp }F_{{\\sigma }n}^{\\rm lw}\\sim 1,\\ n\\ge 0.$ Finally, the zeroth-order distribution function must have an identically vanishing short-wavelength component, and the perpendicular gradient of the rest of the short-wavelength components is of order $\\epsilon _{\\sigma }^{-1}$ , $F_{{\\sigma }0}^{\\rm sw}\\equiv 0, \\nonumber \\\\[5pt]\\nabla _{\\mathbf {R}_\\perp }F_{{\\sigma }n}^{\\rm sw}\\sim \\epsilon _{\\sigma }^{-1},\\ n\\ge 1.$ Then, one must use expression (REF ) for the Poisson bracket in gyrokinetic coordinates and the form of $\\overline{H}_{\\sigma }$ given in equations (REF ), (REF ), and (REF ).", "With the help of , writing (REF ) order by order is relatively straightforward.", "In addition to writing the equations order by order, we manipulate them to make them as close as possible to the results obtained in neoclassical theory [26], [27].", "This form of the equations will be useful when we calculate the transport equations for particles and energy in subsection REF .", "Recall that along this paper we assume $\\gamma _{{\\sigma }{\\sigma }^{\\prime }}\\sim \\lambda _{\\sigma }\\sim \\tau _{\\sigma }\\sim 1$ for all ${\\sigma },{\\sigma }^{\\prime }$ , so that the only formal expansion parameter is $\\epsilon _s$ ." ], [ "Long-wavelength Fokker-Planck equation to\n$O(\\epsilon _{\\sigma }^{-1})$", "The coefficient of $\\epsilon _{\\sigma }^{-1}$ in (REF ) simply gives $-\\tau _{\\sigma }B\\partial _\\theta F_{{\\sigma }0} = 0,$ implying that $F_{{\\sigma }0}$ is independent of $\\theta $ ." ], [ "Long-wavelength Fokker-Planck equation to\n$O(\\epsilon _{\\sigma }^0)$", "Equation (REF ) to order $\\epsilon _{\\sigma }^0$ involves the collision operator, which is written in coordinates $\\mathbf {X}\\equiv (\\mathbf {r},\\mathbf {v})$ .", "Therefore, either we transform the collision operator to gyrokinetic coordinates $\\mathbf {Z}\\equiv (\\mathbf {R},u,\\mu ,\\theta )$ or transform the gyrokinetic distribution function to coordinates $\\mathbf {X}$ .", "We choose the second option.", "To write order by order the collision operator we need to obtain certain coefficients of the Taylor expansion of the gyrokinetic change of coordinates ${\\cal T}_{\\sigma }$ and its inverse, ${\\cal T}_{\\sigma }^{-1}$ , $\\mathbf {X}= {\\cal T}_{{\\sigma }}(\\mathbf {Z},t)= {\\cal T}_{{\\sigma },0}(\\mathbf {Z},t) + \\epsilon _{\\sigma }{\\cal T}_{{\\sigma },1}(\\mathbf {Z},t) +O(\\epsilon _{\\sigma }^2),\\\\[5pt]\\mathbf {Z}= {\\cal T}_{{\\sigma }}^{-1}(\\mathbf {X},t) = {\\cal T}_{{\\sigma },0}^{-1}(\\mathbf {X},t)+ \\epsilon _{\\sigma }{\\cal T}_{{\\sigma },1}^{-1}(\\mathbf {X},t)\\nonumber \\\\+\\epsilon _{\\sigma }^2 {\\cal T}_{{\\sigma },2}^{-1}(\\mathbf {X},t)+O(\\epsilon _{\\sigma }^3).$ In the present subsection we need ${\\cal T}_{{\\sigma },0}$ , the transformation ${\\cal T}_{\\sigma }$ , equation (REF ), for $\\epsilon _{\\sigma }= 0$ : ${\\cal T}_{{\\sigma },0}(\\mathbf {R},u,\\mu ,\\theta )=(\\mathbf {R},u\\hat{\\mathbf {b}}(\\mathbf {R})+\\mbox{$\\rho $}(\\mathbf {R},\\mu ,\\theta )\\times \\mathbf {B}(\\mathbf {R})).$ The remaining terms in the Taylor expansions will be employed in subsequent subsections and are computed in the appendices.", "We write the collision operator in the zeroth-order long-wavelength Fokker-Planck equation by employing the pull-back of (REF ) and its inverse, so the equation reads: $\\hspace{42.67912pt} u\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}F_{{\\sigma }0}-\\hat{\\mathbf {b}}\\cdot \\left(\\mu \\nabla _\\mathbf {R}B + Z_{\\sigma }\\nabla _\\mathbf {R}\\varphi _0\\right)\\partial _u F_{{\\sigma }0} - B\\partial _\\theta F_{{\\sigma }1}^{\\rm lw}\\nonumber \\\\[5pt]\\hspace{85.35826pt}=\\sum _{{\\sigma }^{\\prime }}{\\cal T}^{*}_{{\\sigma },0}C_{{\\sigma }{\\sigma }^{\\prime }} [{\\cal T}^{-1*}_{{\\sigma },0} F_{{\\sigma }0},{\\cal T}^{-1*}_{{\\sigma }^{\\prime },0}F_{{\\sigma }^{\\prime } 0}] (\\mathbf {R},u,\\mu ,\\theta ).$ Using that $F_{{\\sigma }0}$ is gyrophase independent and the isotropy property of the collision operator (by which it gives a gyrophase-independent function when acting on a gyrophase-independent function) we immediately deduce that $\\partial _\\theta F_{{\\sigma }1}^{\\rm lw}= 0,$ i.e.", "$F_{{\\sigma }1}^{\\rm lw}$ is gyrophase-independent.", "Actually, it is trivial to prove from the zeroth-order short-wavelength component of equation (REF ) that also $\\partial _\\theta F_{{\\sigma }1}^{\\rm sw}= 0$ , so $\\partial _\\theta F_{{\\sigma }1} = 0.$ We proceed to prove that the solution to (REF ) is a stationary Maxwellian.", "Multiplying (REF ) by $-B\\ln F_{{\\sigma }0}$ and integrating over $u,\\mu $ , and $\\theta $ : $\\hspace{28.45274pt}-\\nabla _\\mathbf {R}\\cdot \\int \\mathbf {B}\\, u( F_{{\\sigma }0}\\ln F_{{\\sigma }0}- F_{{\\sigma }0})\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\nonumber \\\\[5pt]\\hspace{56.9055pt} = -\\int B\\ln F_{{\\sigma }0}\\sum _{{\\sigma }^{\\prime }}{\\cal T}^{*}_{0,{\\sigma }} C_{{\\sigma }{\\sigma }^{\\prime }} [{\\cal T}^{-1*}_{{\\sigma },0}F_{{\\sigma }0},{\\cal T}^{-1*}_{{\\sigma }^{\\prime },0}F_{{\\sigma }^{\\prime } 0}] \\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta .$ Here it is convenient to define the flux-surface average of a function $G(\\psi ,\\Theta ,\\zeta )$ , given by [40] $\\hspace{14.22636pt}\\langle G \\rangle _\\psi :=\\frac{\\int _0^{2\\pi }\\int _0^{2\\pi }\\sqrt{g}\\,G(\\psi ,\\Theta ,\\zeta ) \\mbox{d}\\Theta \\mbox{d}\\zeta }{\\int _0^{2\\pi }\\int _0^{2\\pi }\\sqrt{g}\\,\\mbox{d}\\Theta \\mbox{d}\\zeta }\\, ,$ where $\\sqrt{g}:=\\frac{1}{\\nabla _\\mathbf {R}\\psi \\cdot \\left(\\nabla _\\mathbf {R}\\Theta \\times \\nabla _\\mathbf {R}\\zeta \\right)}$ is the square root of the determinant of the metric tensor in coordinates $\\lbrace \\psi ,\\Theta ,\\zeta \\rbrace $ .", "It will also be useful to define the volume enclosed by the flux surface labeled by $\\psi $ , $V(\\psi ) =\\int _0^\\psi \\mbox{d}\\psi \\int _0^{2\\pi }\\mbox{d}\\Theta \\int _0^{2\\pi }\\mbox{d}\\zeta \\, \\sqrt{g}\\, .$ The flux-surface average of (REF ) yields $\\left\\langle -\\sum _{{\\sigma }^{\\prime }}\\int B \\ln F_{{\\sigma }0}{\\cal T}^{*}_{{\\sigma },0} C_{{\\sigma }{\\sigma }^{\\prime }}[{\\cal T}^{-1*}_{{\\sigma },0} F_{{\\sigma }0},{\\cal T}^{-1*}_{{\\sigma }^{\\prime },0}F_{{\\sigma }^{\\prime } 0}]\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\right\\rangle _\\psi = 0,$ and after multiplying by $\\tau _{\\sigma }$ and adding over $\\sigma $ : $\\left\\langle -\\sum _{\\sigma ,{\\sigma }^{\\prime }}\\tau _{\\sigma }\\int B \\ln F_{{\\sigma }0}{\\cal T}^{*}_{{\\sigma },0} C_{{\\sigma }{\\sigma }^{\\prime }}[{\\cal T}^{-1*}_{{\\sigma },0} F_{{\\sigma }0},{\\cal T}^{-1*}_{{\\sigma }^{\\prime },0}F_{{\\sigma }^{\\prime } 0}]\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\right\\rangle _\\psi = 0.$ Observing that the Jacobian of ${\\cal T}_{{\\sigma },0}$ at the point $(\\mathbf {R},u,\\mu ,\\theta )$ is exactly $B(\\mathbf {R})$ , and using the formula for the change of variables in an integral, we obtain: $\\left\\langle -\\sum _{\\sigma ,{\\sigma }^{\\prime }}\\tau _{\\sigma }\\int \\ln \\left({\\cal T}^{-1*}_{{\\sigma },0}F_{{\\sigma }0}\\right) C_{{\\sigma }{\\sigma }^{\\prime }}[{\\cal T}^{-1*}_{{\\sigma },0} F_{{\\sigma }0},{\\cal T}^{-1*}_{{\\sigma }^{\\prime },0}F_{{\\sigma }^{\\prime } 0}] \\mbox{d}^3v \\right\\rangle _\\psi = 0.$ This equation can be written as $\\Bigg \\langle \\sum _{\\sigma ,{\\sigma }^{\\prime }}\\frac{\\gamma _{{\\sigma }{\\sigma }^{\\prime }}}{2}\\int {\\cal T}^{-1*}_{{\\sigma },0}F_{{\\sigma }0}{\\cal T}^{-1*}_{{\\sigma }^{\\prime },0}F_{{\\sigma }^{\\prime } 0}\\Big (\\tau _{\\sigma }\\nabla _\\mathbf {v}\\ln {\\cal T}^{-1*}_{{\\sigma },0}F_{{\\sigma }0}\\nonumber \\\\[5pt]-\\tau _{{\\sigma }^{\\prime }}\\nabla _{\\mathbf {v}^{\\prime }}\\ln {\\cal T}^{-1*}_{{\\sigma }^{\\prime },0}F_{{\\sigma }^{\\prime } 0}\\Big )\\cdot \\leftrightarrow \\over {\\mathbf {W}}\\cdot \\Big (\\tau _{\\sigma }\\nabla _\\mathbf {v}\\ln {\\cal T}^{-1*}_{{\\sigma },0}F_{{\\sigma }0}\\nonumber \\\\[5pt]-\\tau _{{\\sigma }^{\\prime }}\\nabla _{\\mathbf {v}^{\\prime }}\\ln {\\cal T}^{-1*}_{{\\sigma }^{\\prime },0}F_{{\\sigma }^{\\prime } 0}\\Big )\\mbox{d}^3 v\\mbox{d}^3 v^{\\prime }\\Bigg \\rangle _\\psi = 0,$ which is the entropy production in a flux surface.", "The only solution to this equation is ${\\cal T}^{-1*}_{{\\sigma },0} F_{{\\sigma }0}({\\mathbf {r}},{\\mathbf {v}},t)=\\frac{{n_{{\\sigma }}}({\\mathbf {r}},t)}{(2\\pi {T_{{\\sigma }}}({\\mathbf {r}},t))^{3/2}}\\exp \\left(-\\frac{(\\mathbf {v}-\\tau _{\\sigma }^{-1}\\mathbf {V}(\\mathbf {r},t))^2}{2{T_{{\\sigma }}}({\\mathbf {r}},t)}\\right),$ where the temperature has to be the same for all the species (with the exception of electrons if a subsidiary expansion in the mass ratio is performed, or equivalently, if $\\tau _e\\sim \\lambda _e\\gg 1$ is used).", "That is, in the previous equation, $T_{\\sigma }= T_{{\\sigma }^{\\prime }}$ for every pair ${\\sigma }, {\\sigma }^{\\prime }$ .", "Then, $&& F_{{\\sigma }0}(\\mathbf {R},u,\\mu ,t)=\\nonumber \\\\[5pt]&& \\hspace{28.45274pt}\\frac{n_{{\\sigma }}(\\mathbf {R},t)}{(2\\pi T_{{\\sigma }}(\\mathbf {R},t))^{3/2}}\\exp \\left(-\\frac{\\mu B(\\mathbf {R})+ (u-\\tau _{\\sigma }^{-1}V_{||}(\\mathbf {R},t))^2/2}{T_{{\\sigma }}(\\mathbf {R},t)}\\right).$ In the last expression we have made explicit the fact that the component of $\\mathbf {V}(\\mathbf {R},t)$ perpendicular to the magnetic field has to be zero bacause otherwise $F_{{\\sigma }0}$ would depend on the gyrophase; that is, $\\mathbf {V}(\\mathbf {R},t) = V_{||}(\\mathbf {R},t)\\hat{\\mathbf {b}}(\\mathbf {R},t)$ .", "Now, take the gyroaverage of (REF ) and use (REF ) along with (REF ) to obtain $u\\hat{\\mathbf {b}}\\cdot \\Bigg [ \\frac{1}{n_{{\\sigma }}}\\nabla _\\mathbf {R}n_{{\\sigma }}+\\left(\\frac{\\mu B + (u-\\tau _{\\sigma }^{-1}V_{||})^2/2}{T_{{\\sigma }}} -\\frac{3}{2}\\right)\\frac{1}{T_{{\\sigma }}}\\nabla _\\mathbf {R}T_{{\\sigma }}\\nonumber \\\\[5pt]\\hspace{28.45274pt}+\\frac{1}{T_{\\sigma }}\\left(\\tau _{\\sigma }^{-1}(u-\\tau _{\\sigma }^{-1}V_{||})\\nabla _\\mathbf {R}V_{||}+Z_{\\sigma }\\nabla _\\mathbf {R}\\varphi _0\\right)\\Bigg ]\\nonumber \\\\[5pt]\\hspace{28.45274pt}-\\frac{V_{||}}{\\tau _{\\sigma }T_{\\sigma }}\\hat{\\mathbf {b}}\\cdot (\\mu \\nabla _\\mathbf {R}B+ Z_{\\sigma }\\nabla _\\mathbf {R}\\varphi _0)=0.$ Since this equation has to be satisfied for every $u$ and $\\mu $ , $V_{||}$ must vanish identically and $T_{{\\sigma }}$ must be a flux function.", "Then, from (REF ), we infer that the combination $\\eta _{\\sigma }= n_{{\\sigma }}\\exp \\left(\\frac{Z_{\\sigma }\\varphi _0}{T_{{\\sigma }}}\\right)$ is a function of $\\psi $ and $t$ only, $\\eta _{\\sigma }(\\psi ,t)$ .", "The zeroth-order long-wavelength quasineutrality equation (see (REF ) later on in this paper) gives $\\sum _\\sigma Z_{\\sigma }n_{{\\sigma }} = 0,$ or equivalently, $\\sum _\\sigma Z_{\\sigma }\\eta _{{\\sigma }}\\exp \\left(-\\frac{Z_{\\sigma }\\varphi _0}{T_{{\\sigma }}}\\right) = 0.$ Taking the parallel gradient of this equation, one shows that $\\varphi _0$ and $n_{{\\sigma }}$ are flux functions." ], [ "Long-wavelength Fokker-Planck equation to $O(\\epsilon _{\\sigma })$", "The equation to order $\\epsilon _{\\sigma }$ is fairly more complicated.", "Apart from the material in , we need the long-wavelength limit of the pull-back of $F_{{\\sigma }0}$ by ${\\cal T}_{\\sigma }^{-1}$ to order $\\epsilon _{\\sigma }$ .", "This is computed in (see (REF )).", "The result is $\\left(u\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}- \\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B\\partial _u\\right) F_{{\\sigma }1}^{\\rm lw}- B\\partial _\\theta F_{{\\sigma }2}^{\\rm lw}\\nonumber \\\\[5pt]\\hspace{28.45274pt}+\\left(\\mathbf {v}_\\kappa + \\mathbf {v}_{\\nabla B} + \\mathbf {v}_{E,{\\sigma }}^{(0)}\\right)\\cdot \\nabla _\\mathbf {R}F_{{\\sigma }0}\\nonumber \\\\[5pt]\\hspace{28.45274pt} + \\left[u\\, \\mbox{$\\kappa $}\\cdot \\left(\\mathbf {v}_{\\nabla B} + \\mathbf {v}_{E,{\\sigma }}^{(0)}\\right)-Z_{\\sigma }\\lambda _{\\sigma }\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\varphi _1^{{\\rm lw}}\\right]\\partial _u F_{{\\sigma }0}\\nonumber \\\\[5pt]\\hspace{28.45274pt}= \\sum _{\\sigma ^{\\prime }}{\\cal T}_{{\\sigma },0}^*C_{{\\sigma }{\\sigma }^{\\prime }}\\Bigg [ \\frac{1}{T_{{\\sigma }}} \\left( \\mathbf {v}\\cdot {\\mathbf {V}}^p_{\\sigma }+ \\left( \\frac{v^2}{2T_{{\\sigma }}}-\\frac{5}{2}\\right)\\mathbf {v}\\cdot {\\mathbf {V}}^T_{{\\sigma }}\\right){\\cal T}^{-1*}_{{\\sigma },0}F_{{\\sigma }0}\\nonumber \\\\[5pt]\\hspace{28.45274pt}+ {\\cal T}^{-1*}_{{\\sigma },0}F_{{\\sigma }1}^{\\rm lw},{\\cal T}^{-1*}_{{\\sigma }^{\\prime },0}F_{{\\sigma }^{\\prime } 0} \\Bigg ]+ \\sum _{\\sigma ^{\\prime }}\\frac{\\lambda _{\\sigma }}{\\lambda _{{\\sigma }^{\\prime }}}{\\cal T}_{{\\sigma },0}^*C_{{\\sigma }{\\sigma }^{\\prime }} \\Bigg [ {\\cal T}^{-1*}_{{\\sigma },0}F_{{\\sigma }0}, \\frac{1}{T_{{\\sigma }^{\\prime }}} \\Bigg ( \\mathbf {v}\\cdot {\\mathbf {V}}^p_{{\\sigma }^{\\prime }}\\nonumber \\\\[5pt]\\hspace{28.45274pt}+ \\left(\\frac{v^2}{2T_{{\\sigma }^{\\prime }}}-\\frac{5}{2} \\right)\\mathbf {v}\\cdot {\\mathbf {V}}^T_{{\\sigma }^{\\prime }}\\Bigg ){\\cal T}^{-1*}_{{\\sigma }^{\\prime },0}F_{{\\sigma }^{\\prime } 0} +\\,{\\cal T}^{-1*}_{{\\sigma }^{\\prime },0}F_{{\\sigma }^{\\prime } 1}^{\\rm lw}\\Bigg ],$ where $\\mathbf {v}_\\kappa :=\\frac{u^2}{B}\\hat{\\mathbf {b}}\\times \\mbox{$\\kappa $},\\\\[5pt]\\mathbf {v}_{\\nabla B}:= \\frac{\\mu }{B}\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}B,\\\\[5pt]\\mathbf {v}_{E,{\\sigma }}^{(0)}:=\\frac{Z_{\\sigma }}{B}\\hat{\\mathbf {b}}\\times \\nabla _{\\mathbf {R}}\\varphi _0,$ and $\\mbox{$\\kappa $}:= \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}$ is the magnetic field curvature.", "The velocities ${\\mathbf {V}}^p_{{\\sigma }}$ and ${\\mathbf {V}}^T_{{\\sigma }}$ are defined by ${\\mathbf {V}}^p_{\\sigma }:= \\frac{1}{n_{\\sigma }B}\\hat{\\mathbf {b}}\\times \\nabla p_{\\sigma }, \\quad {\\mathbf {V}}^T_{\\sigma }:= \\frac{1}{B}\\hat{\\mathbf {b}}\\times \\nabla T_{\\sigma }.$ Here, $p_{\\sigma }:= n_{\\sigma }T_{\\sigma }$ is the pressure of species ${\\sigma }$ .", "On the right-hand side of (REF ) we have employed (REF ) to prove that the contribution of $\\varphi _0$ appearing in (REF ) vanishes within the collision operator.", "It is easy to find the equation for the gyrophase-dependent piece of $F_{{\\sigma }2}^{\\rm lw}$ : $- B\\partial _\\theta (F_{{\\sigma }2}^{\\rm lw}-\\langle F_{{\\sigma }2}^{\\rm lw}\\rangle )\\nonumber \\\\[5pt]\\hspace{28.45274pt}=\\sum _{\\sigma ^{\\prime }}{\\cal T}^{*}_{{\\sigma },0}C_{{\\sigma }{\\sigma }^{\\prime }}\\Bigg [ \\frac{1}{T_{{\\sigma }}} \\Bigg ( \\mathbf {v}\\cdot {\\mathbf {V}}^p_{\\sigma }\\nonumber \\\\[5pt]\\hspace{28.45274pt}+ \\Bigg ( \\frac{v^2}{2T_{{\\sigma }}}-\\frac{5}{2}\\Bigg )\\mathbf {v}\\cdot {\\mathbf {V}}^T_{{\\sigma }}\\Bigg ){\\cal T}^{-1*}_{{\\sigma },0}F_{{\\sigma }0} ,{\\cal T}^{-1*}_{{\\sigma }^{\\prime },0}F_{{\\sigma }^{\\prime }0} \\Bigg ]\\nonumber \\\\[5pt]\\hspace{28.45274pt} + \\sum _{\\sigma ^{\\prime }}\\frac{\\lambda _{\\sigma }}{\\lambda _{{\\sigma }^{\\prime }}}{\\cal T}^{*}_{{\\sigma },0}C_{{\\sigma }{\\sigma }^{\\prime }} \\Bigg [ {\\cal T}^{-1*}_{{\\sigma },0}F_{{\\sigma }0}, \\frac{1}{T_{{\\sigma }^{\\prime }}} \\Bigg ( \\mathbf {v}\\cdot {\\mathbf {V}}^p_{{\\sigma }^{\\prime }}\\nonumber \\\\[5pt]\\hspace{28.45274pt}+ \\left(\\frac{v^2}{2T_{{\\sigma }^{\\prime }}}-\\frac{5}{2} \\right)\\mathbf {v}\\cdot {\\mathbf {V}}^T_{{\\sigma }^{\\prime }} \\Bigg ){\\cal T}^{-1*}_{{\\sigma }^{\\prime },0}F_{{\\sigma }^{\\prime } 0}\\Bigg ].$ The gyroaverage of (REF ) yields an equation for $F_{\\sigma 1}^{\\rm lw}$ (recall from Section REF that $F_{{\\sigma }1}$ is gyrophase-independent): $\\left(u\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}- \\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B\\partial _u\\right)F_{{\\sigma }1}^{\\rm lw}\\nonumber \\\\[5pt]\\hspace{28.45274pt}+\\left(\\mathbf {v}_\\kappa + \\mathbf {v}_{\\nabla B} + \\mathbf {v}_{E,{\\sigma }}^{(0)}\\right)\\cdot \\nabla _\\mathbf {R}F_{{\\sigma }0}\\nonumber \\\\[5pt]\\hspace{28.45274pt} + \\left[u\\, \\mbox{$\\kappa $}\\cdot \\left(\\mathbf {v}_{\\nabla B} + \\mathbf {v}_{E,{\\sigma }}^{(0)}\\right)-Z_{\\sigma }\\lambda _{\\sigma }\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\varphi _1^{{\\rm lw}}\\right]\\partial _u F_{{\\sigma }0}\\nonumber \\\\[5pt]\\hspace{28.45274pt}=\\sum _{\\sigma ^{\\prime }}{\\cal T}^{*}_{{\\sigma },0}C_{{\\sigma }{\\sigma }^{\\prime }}\\left[ {\\cal T}^{-1*}_{{\\sigma },0}F_{{\\sigma }1}^{\\rm lw},{\\cal T}^{-1*}_{{\\sigma }^{\\prime },0}F_{{\\sigma }^{\\prime } 0} \\right]\\nonumber \\\\[5pt]\\hspace{28.45274pt}+ \\sum _{\\sigma ^{\\prime }}\\frac{\\lambda _{\\sigma }}{\\lambda _{{\\sigma }^{\\prime }}}{\\cal T}^{*}_{{\\sigma },0}C_{{\\sigma }{\\sigma }^{\\prime }} \\left[ {\\cal T}^{-1*}_{{\\sigma },0}F_{{\\sigma }0},{\\cal T}^{-1*}_{{\\sigma }^{\\prime },0}F_{{\\sigma }^{\\prime } 1}^{\\rm lw}\\right].$ Up to this point our computations are valid for an arbitrary time-independent magnetic field with nested flux surfaces.", "Now, we particularize to the case of an equilibrium tokamak magnetic field: $\\mathbf {B}= I(\\psi )\\nabla _\\mathbf {R}\\zeta + \\nabla _\\mathbf {R}\\zeta \\times \\nabla _\\mathbf {R}\\psi .$ In we show that, in this setting, equation (REF ) can be written as $\\left(u\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}- \\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B\\partial _u\\right)G_{{\\sigma }1}^{\\rm lw}\\nonumber \\\\[5pt]\\hspace{28.45274pt}= \\sum _{\\sigma ^{\\prime }}{\\cal T}_{{\\sigma },0}^*C_{{\\sigma }{\\sigma }^{\\prime }}\\left[ {\\cal T}^{-1*}_{{\\sigma },0}\\left(G_{{\\sigma }1}^{\\rm lw}-\\frac{Iu}{B}\\Upsilon _{\\sigma }F_{{\\sigma }0}\\right),{\\cal T}^{-1*}_{{\\sigma }^{\\prime },0}F_{{\\sigma }^{\\prime } 0} \\right]\\nonumber \\\\[5pt]\\hspace{28.45274pt} + \\sum _{\\sigma ^{\\prime }}\\frac{\\lambda _{\\sigma }}{\\lambda _{{\\sigma }^{\\prime }}}{\\cal T}_{{\\sigma },0}^*C_{{\\sigma }{\\sigma }^{\\prime }} \\Bigg [ {\\cal T}^{-1*}_{0,{\\sigma }}F_{{\\sigma }0},{\\cal T}^{-1*}_{0,{\\sigma }^{\\prime }}\\Bigg (G_{{\\sigma }^{\\prime } 1}^{\\rm lw}- \\frac{Iu}{B}\\Upsilon _{{\\sigma }^{\\prime }}F_{{{\\sigma }^{\\prime }} 0}\\Bigg )\\Bigg ],$ where $G_{{\\sigma }1}^{\\rm lw}&:=& F_{{\\sigma }1}^{\\rm lw}+\\Bigg \\lbrace \\frac{Z_{\\sigma }\\lambda _{\\sigma }}{T_{\\sigma }}\\varphi _1^{\\rm lw}+\\frac{Iu}{B}\\left(\\frac{Z_{\\sigma }}{T_{\\sigma }}\\partial _\\psi \\varphi _0+\\Upsilon _{\\sigma }\\right)\\Bigg \\rbrace F_{{\\sigma }0},$ and $\\Upsilon _{\\sigma }:=\\partial _\\psi \\ln n_{\\sigma }+\\left(\\frac{u^2/2 +\\mu B}{T_{\\sigma }}-\\frac{3}{2}\\right)\\partial _\\psi \\ln T_{\\sigma }\\, .$ It is a remarkable fact that in terms of the functions $G_{{\\sigma }1}^{\\rm lw}$ the first-order Fokker-Planck equations do not involve the electrostatic potential.", "Equation (REF ) is in a form that makes it easy to compare with the results of neoclassical theory [26], [27]." ], [ "Short-wavelength Fokker-Planck and quasineutrality\nequations to $O(\\epsilon _{\\sigma })$", "Here, the equations for $F_{{\\sigma }1}^{\\rm sw}$ and $\\phi _{{\\sigma }1}^{\\rm sw}$ are given because they enter the second-order, long-wavelength piece of the Fokker-Planck equation.", "Before presenting such short-wavelength equations, we need to define a new operator ${\\mathbb {T}_{{\\sigma },0}}$ acting on phase-space functions $F(\\mathbf {R},u,\\mu ,\\theta )$ .", "Namely, ${\\mathbb {T}_{{\\sigma },0}}F(\\mathbf {r},\\mathbf {v}) :=F\\left(\\mathbf {r}-\\epsilon _{\\sigma }{\\cal T}_{{\\sigma },0}^{-1*}\\mbox{$\\rho $}(\\mathbf {r},\\mathbf {v}),\\mathbf {v}\\cdot \\hat{\\mathbf {b}}(\\mathbf {r}), \\frac{v_\\bot ^2}{2B(\\mathbf {r})},\\arctan \\left(\\frac{\\mathbf {v}\\cdot \\hat{\\mathbf {e}}_2(\\mathbf {r})}{\\mathbf {v}\\cdot \\hat{\\mathbf {e}}_1(\\mathbf {r})}\\right)\\right).$ This operator is useful to write some expressions involving the short wavelength pieces of the distribution function and the potential, for which it is not possible to Taylor expand the dependence on $\\mathbf {r}-\\epsilon _{\\sigma }{\\cal T}_{{\\sigma }, 0}^{-1 *} \\mbox{$\\rho $}(\\mathbf {r}, \\mathbf {v})$ around $\\mathbf {r}$ .", "The first-order, short-wavelength terms of (REF ) yield $\\frac{1}{\\tau _{\\sigma }}\\partial _t F_{{\\sigma }1}^{\\rm sw}+\\left(u\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}-\\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B\\partial _u\\right)F_{{\\sigma }1}^{\\rm sw}\\nonumber \\\\[5pt]\\hspace{14.22636pt}+\\left[\\frac{Z_{\\sigma }\\lambda _{\\sigma }}{B}\\left(\\hat{\\mathbf {b}}\\times \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\right)\\cdot \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}F_{{\\sigma }1}^{\\rm sw}\\right]^{\\rm sw}\\nonumber \\\\[5pt]\\hspace{14.22636pt}+\\left(\\frac{u^2}{B}\\hat{\\mathbf {b}}\\times \\mbox{$\\kappa $}+\\frac{\\mu }{B}\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}B+\\frac{Z_{\\sigma }}{B}\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}\\varphi _0\\right)\\cdot \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}F_{{\\sigma }1}^{\\rm sw}\\nonumber \\\\[5pt]\\hspace{14.22636pt}+\\frac{Z_{\\sigma }\\lambda _{\\sigma }}{B}\\left(\\hat{\\mathbf {b}}\\times \\nabla _{\\mathbf {R}_\\bot /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\right)\\cdot \\nabla _\\mathbf {R}F_{{\\sigma }0}\\nonumber \\\\[5pt]\\hspace{14.22636pt}-Z_{\\sigma }\\lambda _{\\sigma }\\Big (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle +\\frac{u}{B}\\hat{\\mathbf {b}}\\times (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}})\\cdot \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\Big )\\partial _u F_{{\\sigma }0}\\nonumber \\\\[5pt]\\hspace{14.22636pt}=\\sum _{{\\sigma }^\\prime } \\left\\langle {\\cal T}_{NP, {\\sigma }}^* C_{\\sigma \\sigma ^\\prime }\\left[{\\mathbb {T}_{{\\sigma },0}}F_{\\sigma 1}^{\\rm sw}-\\frac{Z_{\\sigma }\\lambda _{\\sigma }}{T_{\\sigma }}{\\mathbb {T}_{{\\sigma },0}}\\tilde{\\phi }_{{\\sigma }1}^{\\rm sw}{\\cal T}_{{\\sigma },0}^{-1*} F_{{\\sigma }0}, {\\cal T}_{{\\sigma }^{\\prime },0}^{-1*}F_{{\\sigma }^{\\prime } 0}\\right] \\right\\rangle \\nonumber \\\\[5pt]\\hspace{14.22636pt}+ \\ \\sum _{{\\sigma }^\\prime } \\frac{\\lambda _{\\sigma }}{\\lambda _{{\\sigma }^{\\prime }}}\\left\\langle {\\cal T}_{NP, {\\sigma }}^*C_{\\sigma \\sigma ^\\prime } \\left[ {\\cal T}_{{\\sigma },0}^{-1 *}F_{{\\sigma }0} ,{\\mathbb {T}_{{\\sigma }^{\\prime },0}}F_{\\sigma ^{\\prime } 1}^{\\rm sw}-\\frac{Z_{{\\sigma }^{\\prime }}\\lambda _{{\\sigma }^{\\prime }}}{T_{{\\sigma }^{\\prime }}}{\\mathbb {T}_{{\\sigma }^{\\prime },0}}\\tilde{\\phi }_{{\\sigma }^{\\prime }1}^{\\rm sw}{\\cal T}_{{\\sigma }^{\\prime },0}^{-1 *}F_{{\\sigma }^{\\prime } 0}\\right] \\right\\rangle .$ As for the short-wavelength, first-order quasineutrality equation: $\\sum _{\\sigma }\\frac{Z_{\\sigma }}{\\lambda _{\\sigma }}&\\int B\\Bigg [-Z_{\\sigma }\\lambda _{\\sigma }\\widetilde{\\phi }_{{\\sigma }1}^{\\rm sw}\\left(\\mathbf {r}-\\epsilon _{\\sigma }\\mbox{$\\rho $}(\\mathbf {r},\\mu ,\\theta ),\\mu ,\\theta ,t\\right)\\frac{F_{{\\sigma }0}(\\mathbf {r},u,\\mu ,t)}{T_{{\\sigma }}(\\mathbf {r},t)}\\nonumber \\\\[5pt]&+ F_{{\\sigma }1}^{\\rm sw}\\left(\\mathbf {r}-\\epsilon _{\\sigma }\\mbox{$\\rho $}(\\mathbf {r},\\mu ,\\theta ),u,\\mu ,t\\right)\\Bigg ]\\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta = 0.$" ], [ "Long-wavelength Fokker-Planck equation to $O(\\epsilon _{\\sigma }^2)$", "The second-order contribution to (REF ) is cumbersome.", "In order to avoid lengthy calculations to those readers interested in reaching quickly the final expressions and main results, most of the manipulations in this subsection are deferred to the appendices.", "The pieces of order $\\epsilon _{\\sigma }^2$ in (REF ) yield $ \\left(u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}- \\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B \\, \\partial _u\\right)F^{\\rm lw}_{\\sigma 2}- B \\partial _\\theta F^{\\rm lw}_{\\sigma 3} +\\frac{\\lambda _{\\sigma }^2}{\\tau _{\\sigma }}\\partial _{\\epsilon _s^2 t} F_{{\\sigma }0}\\nonumber \\\\[5pt]\\hspace{14.22636pt} +\\left(\\mathbf {v}_\\kappa + \\mathbf {v}_{\\nabla B} + \\mathbf {v}_{E,{\\sigma }}^{(0)}\\right)\\cdot \\nabla _\\mathbf {R}F^{\\rm lw}_{\\sigma 1} \\nonumber \\\\[5pt]\\hspace{14.22636pt}+\\left[- Z_{\\sigma }\\lambda _{\\sigma }\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\varphi _1^{\\rm lw}+u\\, \\mbox{$\\kappa $}\\cdot \\left(\\mathbf {v}_{\\nabla B} + \\mathbf {v}_{E,{\\sigma }}^{(0)}\\right)\\right]\\partial _u F^{\\rm lw}_{\\sigma 1}\\nonumber \\\\[5pt]\\hspace{14.22636pt} + \\Bigg [\\mathbf {v}_{E,{\\sigma }}^{(1)}-\\frac{u}{B}(\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}})\\left(\\mathbf {v}_\\kappa + \\mathbf {v}_{\\nabla B} + \\mathbf {v}_{E,{\\sigma }}^{(0)}\\right)\\nonumber \\\\[5pt]\\hspace{14.22636pt}- \\frac{u \\mu }{B}(\\nabla _\\mathbf {R}\\times \\mathbf {K})_\\bot + Z_{\\sigma }\\lambda _{\\sigma }\\partial _u\\Psi _{\\phi B,{\\sigma }}^{\\rm lw}\\hat{\\mathbf {b}}+\\partial _u\\Psi _{B,{\\sigma }} \\hat{\\mathbf {b}}\\Bigg ]\\cdot \\nabla _\\mathbf {R}F_{{\\sigma }0}\\nonumber \\\\[5pt]\\hspace{14.22636pt} - \\Bigg \\lbrace Z_{\\sigma }\\lambda _{\\sigma }^2 \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\left[ \\varphi _2^{\\rm lw}+ \\frac{\\mu }{2\\lambda _{\\sigma }^2 B}(\\leftrightarrow \\over {\\mathbf {I}}- \\hat{\\mathbf {b}}\\hat{\\mathbf {b}}) : \\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\varphi _0 \\right]\\nonumber \\\\[5pt]\\hspace{14.22636pt}+\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\Psi _{B,{\\sigma }} + Z_{\\sigma }\\lambda _{\\sigma }\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\Psi _{\\phi B}^{\\rm lw}+ Z_{\\sigma }^2 \\lambda _{\\sigma }^2 \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\Psi _\\phi ^{\\rm lw}\\nonumber \\\\[5pt]\\hspace{14.22636pt}-u\\,\\mbox{$\\kappa $}\\cdot \\mathbf {v}_{E,{\\sigma }}^{(1)}+\\Bigg [\\frac{u^2}{B}\\left(\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}\\right)\\mbox{$\\kappa $}\\nonumber \\\\[5pt]\\hspace{14.22636pt}+ \\mu \\left(\\left(\\nabla _\\mathbf {R}\\times \\mathbf {K}\\right)\\times \\hat{\\mathbf {b}}\\right)\\Bigg ]\\cdot \\left(\\mathbf {v}_{\\nabla B}+\\mathbf {v}_{E,{\\sigma }}^{(0)}\\right)\\Bigg \\rbrace \\partial _u F_{{\\sigma }0}\\nonumber \\\\[5pt]\\hspace{14.22636pt}+\\frac{Z_{\\sigma }\\lambda _{\\sigma }}{B} \\left[\\nabla _\\mathbf {R}\\cdot \\left(\\hat{\\mathbf {b}}\\times \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle F_{{\\sigma }1}^{\\rm sw}\\right)\\right]^{\\rm lw}\\nonumber \\\\[5pt]\\hspace{14.22636pt} - Z_{\\sigma }\\lambda _{\\sigma }\\partial _u \\left[\\left( \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle + \\frac{u}{B}\\left(\\hat{\\mathbf {b}}\\times \\mbox{$\\kappa $}\\right)\\cdot \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\right)F_{{\\sigma }1}^{\\rm sw}\\right]^{\\rm lw}\\nonumber \\\\[5pt]\\hspace{14.22636pt} = \\sum _{{\\sigma }^{\\prime }}\\left[{\\cal T}_{{\\sigma },1}^*C_{{\\sigma }{\\sigma }^{\\prime }}^{(1)}\\right]^{\\rm lw}+\\sum _{{\\sigma }^{\\prime }}{\\cal T}_{{\\sigma },0}^*C_{{\\sigma }{\\sigma }^{\\prime }}^{(2){\\rm lw}}.$ Here, $\\mathbf {v}_{E,{\\sigma }}^{(1)} = \\frac{Z_{\\sigma }\\lambda _{\\sigma }}{B}\\hat{\\mathbf {b}}\\times \\nabla _{\\mathbf {R}_\\perp }\\varphi _1^{\\rm lw},$ $ \\Psi ^{\\rm lw}_{\\phi B} = - \\frac{3 \\mu }{2 \\lambda _{\\sigma }B^2}\\nabla _\\mathbf {R}B \\cdot \\nabla _{\\mathbf {R}} \\varphi _0- \\frac{u^2}{\\lambda _{\\sigma }B^2} (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}})\\cdot \\nabla _{\\mathbf {R}}\\varphi _0,$ and $ \\Psi ^{\\rm lw}_\\phi =- \\frac{1}{2\\lambda _{\\sigma }^2 B^2} |\\nabla _{\\mathbf {R}}\\varphi _0|^2 - \\frac{1}{2B}\\partial _\\mu \\left[\\langle (\\widetilde{\\phi }_{{\\sigma }1}^{\\rm sw})^2 \\rangle \\right]^{\\rm lw}.$ As for the collision operator, $C_{{\\sigma }{\\sigma }^{\\prime }}^{(1)} =C_{{\\sigma }{\\sigma }^{\\prime }}^{(1){\\rm lw}}+ C_{{\\sigma }{\\sigma }^{\\prime }}^{(1){\\rm sw}},$ $\\left[{\\cal T}_{{\\sigma },1}^* C_{{\\sigma }{\\sigma }^{\\prime }}^{(1)}\\right]^{\\rm lw}&=\\left(\\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}+\\hat{u}_1\\partial _u+\\hat{\\mu }_1^{\\rm lw}\\partial _\\mu +\\hat{\\theta }_1^{\\rm lw}\\partial _\\theta \\right){\\cal T}_{{\\sigma },0}^{*}C_{{\\sigma }{\\sigma }^{\\prime }}^{(1){\\rm lw}}\\nonumber \\\\[5pt]&+\\left[{\\cal T}_{{\\sigma },1}^*C_{{\\sigma }{\\sigma }^{\\prime }}^{(1){\\rm sw}}\\right]^{\\rm lw},$ where $ C_{\\sigma \\sigma ^\\prime }^{(1){\\rm lw}}=C_{\\sigma \\sigma ^\\prime } \\Bigg [\\frac{1}{T_{\\sigma }} \\mathbf {v}\\cdot \\left(\\mathbf {V}_{\\sigma }^p+ \\left( \\frac{v^2}{2T_\\sigma } - \\frac{5}{2} \\right)\\mathbf {V}^T_{\\sigma } \\right) {\\cal T}_{{\\sigma },0}^{-1 *}F_{{\\sigma }0} \\nonumber \\\\[5pt]\\hspace{14.22636pt} + {\\cal T}_{{\\sigma },0}^{-1 *} \\Bigg ( G_{{\\sigma }1} - \\frac{Iu}{B}\\Upsilon _{\\sigma }F_{{\\sigma }0}\\Bigg ),{\\cal T}_{{\\sigma }^{\\prime },0}^{-1 *}F_{{\\sigma }^{\\prime } 0} \\Bigg ]\\nonumber \\\\[5pt]\\hspace{14.22636pt} + \\ \\frac{\\lambda _{\\sigma }}{\\lambda _{{\\sigma }^{\\prime }}}C_{\\sigma \\sigma ^\\prime } \\Bigg [ {\\cal T}_{{\\sigma },0}^{-1 *}F_{{\\sigma }0} ,\\frac{1}{T_{{\\sigma }^{\\prime }}} \\mathbf {v}\\cdot \\Bigg (\\mathbf {V}_{\\sigma ^{\\prime }}^p+ \\left( \\frac{v^2}{2T_{\\sigma ^{\\prime }}}- \\frac{5}{2} \\Bigg ) \\mathbf {V}^T_{ \\sigma ^{\\prime }} \\right) {\\cal T}_{{\\sigma }^{\\prime },0}^{-1 *}F_{{\\sigma }^{\\prime } 0}\\nonumber \\\\[5pt]\\hspace{14.22636pt} + {\\cal T}_{{\\sigma }^{\\prime },0}^{-1*}\\left( G_{{\\sigma }^{\\prime } 1} - \\frac{I u}{B} \\Upsilon _{{\\sigma }^{\\prime }} F_{{\\sigma }^{\\prime } 0}\\right)\\Bigg ],$ and $C_{\\sigma \\sigma ^\\prime }^{(1){\\rm sw}} = C_{\\sigma \\sigma ^\\prime }\\left[{\\mathbb {T}_{{\\sigma },0}}F_{\\sigma 1}^{\\rm sw}-\\frac{Z_{\\sigma }\\lambda _{\\sigma }}{T_{\\sigma }}{\\mathbb {T}_{{\\sigma },0}}\\tilde{\\phi }_{{\\sigma }1}^{\\rm sw}{\\cal T}_{{\\sigma },0}^{-1*} F_{{\\sigma }0}, {\\cal T}_{{\\sigma }^{\\prime },0}^{-1*}F_{{\\sigma }^{\\prime } 0}\\right]\\nonumber \\\\[5pt]\\hspace{14.22636pt} + \\ \\frac{\\lambda _{\\sigma }}{\\lambda _{{\\sigma }^{\\prime }}}C_{\\sigma \\sigma ^\\prime } \\left[ {\\cal T}_{{\\sigma },0}^{-1 *}F_{{\\sigma }0} ,{\\mathbb {T}_{{\\sigma }^{\\prime },0}}F_{\\sigma ^{\\prime } 1}^{\\rm sw}-\\frac{Z_{{\\sigma }^{\\prime }}\\lambda _{{\\sigma }^{\\prime }}}{T_{{\\sigma }^{\\prime }}}{\\mathbb {T}_{{\\sigma }^{\\prime },0}}\\tilde{\\phi }_{{\\sigma }^{\\prime }1}^{\\rm sw}{\\cal T}_{{\\sigma }^{\\prime },0}^{-1 *}F_{{\\sigma }^{\\prime } 0}\\right].$ The left-hand side of (REF ) is written by employing again the Poisson brackets obtained in .", "The first-order coordinate transformation that enters explicitly the expression of the collision operator is computed in detail in ; in particular, $\\hat{u}_1,\\hat{\\mu }_1^{\\rm lw},\\hat{\\theta }_1^{\\rm lw}$ are defined in (REF ) and (REF ).", "In we calculate the last term of (REF ), $[{\\cal T}_{{\\sigma }, 1}^{*} C_{{\\sigma }{\\sigma }^\\prime }^{(1){\\rm sw}}]^{\\rm lw}$ , and its gyroaverage.", "Finally, $C_{{\\sigma }{\\sigma }^{\\prime }}^{(2){\\rm lw}}$ is calculated in .", "In we prove that the gyroaverage of (REF ) can be rearranged so that it reads $ \\Big ( u \\hat{\\mathbf {b}}\\cdot & \\nabla _\\mathbf {R}- \\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B\\partial _u \\Big ) G_{{\\sigma }2}^{\\rm lw}+\\frac{\\lambda _{\\sigma }^2}{\\tau _{\\sigma }} \\partial _{\\epsilon _s^2 t} F_{{\\sigma }0}\\nonumber \\\\& - \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\Theta \\partial _\\psi \\Bigg \\lbrace \\frac{Z_{\\sigma }\\lambda _{\\sigma }}{\\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta } \\left[ F_{\\sigma 1}^{\\rm sw}(\\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }} \\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\times \\hat{\\mathbf {b}}) \\cdot \\nabla _\\mathbf {R}\\psi \\right]^{\\rm lw}\\nonumber \\\\ & +\\frac{1}{\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\Theta } \\left\\langle \\left(\\frac{Iu}{B} +\\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}\\psi \\right) \\sum _{\\sigma ^\\prime }{\\cal T}_{{\\sigma },0}^*C_{\\sigma \\sigma ^\\prime }^{(1){\\rm lw}} \\right\\rangle \\Bigg \\rbrace \\nonumber \\\\&- \\partial _u \\Bigg \\lbrace \\Bigg [Z_{\\sigma }\\lambda _{\\sigma }F_{\\sigma 1}^{\\rm sw}\\Big ( \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\nonumber \\\\&+ \\frac{\\mu }{u B} \\partial _\\Theta B(\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}\\Theta ) \\cdot \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\nonumber \\\\&+\\frac{u}{B} [\\hat{\\mathbf {b}}\\times (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}})] \\cdot \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\Big ) \\Bigg ]^{\\rm lw}\\nonumber \\\\& - \\left\\langle \\frac{I}{B}\\left( \\mu \\partial _\\psi B +Z_{\\sigma }\\partial _\\psi \\varphi _0 \\right)\\sum _{\\sigma ^\\prime } {\\cal T}_{{\\sigma },0}^*C_{\\sigma \\sigma ^\\prime }^{(1){\\rm lw}} \\right\\rangle \\Bigg \\rbrace \\nonumber \\\\& + \\partial _\\mu \\Bigg \\langle \\frac{1}{B}\\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}\\psi \\left( \\mu \\partial _\\psi B + Z_{\\sigma }\\partial _\\psi \\varphi _0\\right) \\sum _{\\sigma ^\\prime }{\\cal T}_{{\\sigma },0}^*C_{\\sigma \\sigma ^\\prime }^{(1){\\rm lw}}\\Bigg \\rangle \\nonumber \\\\& = - \\sum _{\\sigma ^\\prime } \\partial _u \\Bigg \\langle \\Bigg [ \\frac{Z_{\\sigma }\\lambda _{\\sigma }\\varphi _1^{\\rm lw}}{u}+\\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}\\nonumber \\\\& - \\frac{1}{u}\\mbox{$\\rho $}\\cdot \\left( \\mu \\partial _\\Theta B \\nabla _{\\mathbf {R}} \\Theta + u^2 \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\right)\\Bigg ] {\\cal T}_{{\\sigma },0}^*C_{\\sigma \\sigma ^\\prime }^{(1){\\rm lw}}\\Bigg \\rangle \\nonumber \\\\&+\\sum _{\\sigma ^\\prime } \\partial _\\mu \\Bigg \\langle \\Bigg [ \\frac{u \\mu }{B} \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}\\nonumber \\\\&- \\frac{1}{B}\\mbox{$\\rho $}\\cdot \\left( \\mu \\partial _\\Theta B\\nabla _{\\mathbf {R}} \\Theta + u^2 \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\right) \\Bigg ] {\\cal T}_{{\\sigma },0}^*C_{\\sigma \\sigma ^\\prime }^{(1){\\rm lw}}\\Bigg \\rangle \\nonumber \\\\&+\\sum _{\\sigma ^\\prime }\\left[ \\left\\langle {\\cal T}_{{\\sigma }, 1}^* C_{\\sigma \\sigma ^\\prime }^{(1){\\rm sw}}\\right\\rangle \\right]^{\\rm lw}+\\sum _{\\sigma ^\\prime } \\left\\langle {\\cal T}_{{\\sigma },0}^*C_{\\sigma \\sigma ^\\prime }^{(2){\\rm lw}} \\right\\rangle ,$ where $G_{{\\sigma }2}^{\\rm lw}$ is defined in (REF ).", "Note that the first-order, short-wavelength pieces of the distribution function and electrostatic potential, $F_{{\\sigma }1}^{\\rm sw}$ and $\\phi _{{\\sigma }1}^{\\rm sw}$ , enter equation (REF ).", "The equations needed to determine them are given in subsection REF .", "Observe also that the time derivative of $F_{{\\sigma }0}$ appears in (REF ), something that has very important consequences.", "We will learn that equation (REF ) has non-trivial solvability conditions that involve the time evolution of certain moments of $F_{{\\sigma }0}$ .", "The function $G_{{\\sigma }2}^{\\rm lw}$ is defined to make comparisons with neoclassical theory easier.", "It is also useful to obtain the solvability conditions in subsection REF with less algebra." ], [ "Long-wavelength quasineutrality equation", "In this section we obtain the quasineutrality equation, (REF ), at long-wavelengths.", "For convenience, we repeat here equation (REF ): $\\sum _{\\sigma }Z_{\\sigma }\\int B_{||,{\\sigma }}^*F_{\\sigma }\\delta \\Big (\\pi ^{\\mathbf {r}}\\Big ({\\cal T}_{{\\sigma }}(\\mathbf {R}, u, \\mu , \\theta ,t)\\Big )-\\mathbf {r}\\Big )\\mbox{d}^3 R\\, \\mbox{d}u\\, \\mbox{d}\\mu \\, \\mbox{d}\\theta = 0,$ with $\\pi ^\\mathbf {r}(\\mathbf {r},\\mathbf {v}):=\\mathbf {r}$ .", "At long wavelengths we can simply expand the argument of the Dirac delta function around $\\mathbf {R}-\\mathbf {r}$ .", "Using that $\\pi ^{\\mathbf {r}}{\\cal T}_{{\\sigma }}(\\mathbf {R}, u, \\mu , \\theta ,t) = \\mathbf {R}+\\epsilon _{\\sigma }\\mbox{$\\rho $}+\\epsilon _{\\sigma }^2\\left(\\mathbf {R}_{{\\sigma },2}+\\mu _{{\\sigma },1}\\partial _\\mu \\mbox{$\\rho $}+\\theta _{{\\sigma },1}\\partial _\\theta \\mbox{$\\rho $}\\right)+ O(\\epsilon _{\\sigma }^3),$ that the first-order term of $B_{||,{\\sigma }}^*$ is odd in $u$ , that $F_{{\\sigma }0}$ is even in $u$ , that $F_{{\\sigma }1}^{\\rm lw}$ does not depend on $\\theta $ , and integrating over $\\mathbf {R}$ , it is straightforward to obtain $\\sum _{\\sigma }Z_{\\sigma }n_{\\sigma }(\\mathbf {r},t) = 0$ to order $\\epsilon _s^0$ , $\\sum _{\\sigma }\\frac{Z_{\\sigma }}{\\lambda _{\\sigma }}\\int B(\\mathbf {r})F_{{\\sigma }1}^{\\rm lw}(\\mathbf {r},u,\\mu ,t)\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta = 0$ to order $\\epsilon _s$ , and $\\sum _{\\sigma }\\frac{Z_{\\sigma }}{\\lambda _{\\sigma }^2}\\Bigg [\\int (BF_{{\\sigma }2}^{\\rm lw}-\\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\times \\mathbf {K}\\, F_{{\\sigma }0}+u\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\times \\hat{\\mathbf {b}}\\, F_{{\\sigma }1}^{\\rm lw})\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\nonumber \\\\[5pt]\\hspace{14.22636pt}-\\nabla _\\mathbf {r}\\cdot \\int (\\mathbf {R}_{{\\sigma },2}^{\\rm lw}+\\mu _{{\\sigma },1}^{\\rm lw}\\partial _\\mu \\mbox{$\\rho $}+\\theta _{{\\sigma },1}^{\\rm lw}\\partial _\\theta \\mbox{$\\rho $})BF_{{\\sigma }0}\\,\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\nonumber \\\\[5pt]\\hspace{14.22636pt}+\\frac{1}{2}\\nabla _\\mathbf {r}\\nabla _\\mathbf {r}:\\int \\mbox{$\\rho $}\\mbox{$\\rho $}BF_{{\\sigma }0}\\, \\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\Bigg ]$ to order $\\epsilon _s^2$ .", "Here everything is evaluated at $\\mathbf {R}=\\mathbf {r}$ .", "In writing the arguments of some functions we have stressed that they are evaluated at $\\mathbf {R}=\\mathbf {r}$ , e.g.", "$n_{\\sigma }(\\mathbf {r})$ , but we should not forget that $n_{\\sigma }$ , for example, depends only on $\\psi $ in flux coordinates.", "Note that to be formally correct we need a unique, species-independent expansion parameter, and we have chosen $\\epsilon _s$ as indicated in Section .", "In we show that (REF ) can be transformed into $\\sum _{\\sigma }\\frac{Z_{\\sigma }}{\\lambda _{\\sigma }^2}\\Bigg [\\int (BF_{{\\sigma }2}^{\\rm lw}+u\\hat{\\mathbf {b}}\\cdot (\\nabla _\\mathbf {r}\\times \\hat{\\mathbf {b}})F_{{\\sigma }1}^{\\rm lw})\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\nonumber \\\\[5pt]\\hspace{28.45274pt}-\\hat{\\mathbf {b}}\\cdot (\\nabla _\\mathbf {r}\\times \\mathbf {K})\\frac{n_{{\\sigma }}T_{{\\sigma }}}{B^2}+\\nabla _\\mathbf {r}\\cdot \\left(\\frac{3}{2}\\frac{\\nabla _{\\mathbf {r}_\\bot } B}{B^3}n_{{\\sigma }}T_{{\\sigma }}\\right)\\nonumber \\\\[5pt]\\hspace{28.45274pt}+\\frac{1}{2}\\nabla _\\mathbf {r}\\nabla _\\mathbf {r}:\\left(\\left({\\leftrightarrow \\over {\\mathbf {I}}}-\\hat{\\mathbf {b}}\\hat{\\mathbf {b}}\\right)\\frac{n_{{\\sigma }}T_{{\\sigma }}}{B^2}\\right)\\nonumber \\\\[5pt]\\hspace{28.45274pt}+\\nabla _\\mathbf {r}\\cdot \\left((\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\hat{\\mathbf {b}})\\frac{n_{{\\sigma }}T_{{\\sigma }}}{B^2}\\right)+\\nabla _\\mathbf {r}\\cdot \\left(\\frac{Z_{\\sigma }n_{{\\sigma }}}{B^2}\\nabla _\\mathbf {r}\\varphi _0\\right)\\Bigg ]=0.$ Observe that the above expressions for the long-wavelength quasineutrality equation are completely general, i.e.", "we have not particularized for tokamak geometry.", "We proceed to do it next by writing (REF ) and (REF ) in terms of the functions $G_{{\\sigma }1}^{\\rm lw}$ and $G_{{\\sigma }2}^{\\rm lw}$ , defined in (REF ) and (REF ).", "This is obvious for the first-order piece of the quasineutrality equation, yielding $\\sum _{\\sigma }\\frac{Z_{\\sigma }}{\\lambda _{\\sigma }} &\\Bigg ( \\int B(\\mathbf {r}) G_{{\\sigma }1}^{\\rm lw}(\\mathbf {r},u,\\mu ,t)\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\nonumber \\\\[5pt]&-\\frac{Z_{\\sigma }\\lambda _{\\sigma }}{T_{\\sigma }}n_{\\sigma }(\\mathbf {r},t) \\varphi _1^{\\rm lw}(\\mathbf {r},t)\\Bigg ) =0,$ and in it is shown that the result for the second-order piece is $\\sum _{\\sigma }&\\frac{Z_{\\sigma }}{\\lambda _{\\sigma }^2}\\int B\\Bigg [G_{{\\sigma }2}^{\\rm lw}+ \\frac{u}{B}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\times \\hat{\\mathbf {b}}\\, G_{{\\sigma }1}^{\\rm lw}\\nonumber \\\\&+\\left(\\frac{Z_{\\sigma }\\lambda _{\\sigma }\\varphi _1^{\\rm lw}}{u} \\partial _u-\\frac{Iu}{B}\\partial _\\psi \\right) G_{\\sigma 1}^{\\rm lw}\\nonumber \\\\&- \\left[ \\frac{Z_{\\sigma }\\lambda _{\\sigma }}{u \\mathbf {B}\\cdot \\nabla _\\mathbf {r}\\Theta } F_{\\sigma 1}^{\\rm sw}(\\hat{\\mathbf {b}}\\times \\nabla _{\\mathbf {r}_\\perp /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle ) \\cdot \\nabla _\\mathbf {r}\\Theta \\right]^{\\rm lw}\\nonumber \\\\& +\\left\\langle \\frac{1}{ u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\Theta } \\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {r}\\Theta \\sum _{\\sigma ^\\prime }{\\cal T}_{{\\sigma },0}^* C_{\\sigma \\sigma ^\\prime }^{(1){\\rm lw}} \\right\\rangle \\nonumber \\\\&+ \\frac{Z_{\\sigma }^2\\lambda _{\\sigma }^2}{2T_{\\sigma }^2}\\left[ \\left\\langle (\\widetilde{\\phi }_{{\\sigma }1}^{\\rm sw})^2 \\right\\rangle \\right]^{\\rm lw}F_{{\\sigma }0}\\Bigg ]\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\nonumber \\\\&+\\sum _{\\sigma }\\frac{Z_{\\sigma }}{\\lambda _{\\sigma }^2}n_{\\sigma }T_{\\sigma }\\Bigg \\lbrace \\frac{Z_{\\sigma }\\lambda _{\\sigma }^2}{T_\\sigma ^2}\\left(\\frac{Z_{\\sigma }}{2 T_\\sigma }(\\varphi _1^{\\rm lw})^2-\\varphi _2^{\\rm lw}\\right)\\nonumber \\\\[5pt]&+\\frac{R^2}{2}\\Bigg [\\left(\\frac{Z_{\\sigma }}{T_\\sigma }\\partial _\\psi \\varphi _0+\\partial _\\psi \\ln n_{\\sigma }\\right)^2\\nonumber \\\\[5pt]&+\\left(\\partial _\\psi \\ln T_{\\sigma }\\right)^2+2\\partial _\\psi \\ln n_{\\sigma }\\partial _\\psi \\ln T_{\\sigma }\\nonumber \\\\[5pt]&+\\partial _\\psi ^2\\ln n_{\\sigma }+ \\frac{Z_{\\sigma }}{T_{\\sigma }}\\partial _\\psi ^2\\varphi _0+\\partial _\\psi ^2\\ln T_{\\sigma }\\Bigg ]\\Bigg \\rbrace =0,$ where $R$ is the major radius coordinate, i.e.", "it is the distance to the axis of symmetry of the tokamak." ], [ "Indeterminacy of the long-wavelength radial electric field", "With the results of Sections and at hand it is reasonably easy to prove that in a tokamak $\\varphi _0(\\psi )$ is not determined by second-order Fokker-Planck and quasineutrality equations.", "In order to be as clear as possible, we divide the argument into three steps.", "In subsection REF we show that the quasineutrality equation gives no information about the radial electric field, even though naively one would have expected to use this equation to solve for it.", "In subsection REF we learn that (REF ) possesses non-trivial solvability conditions and work them out.", "They are transport equations for the lowest order density and temperature functions.", "In subsection REF we prove that these solvability conditions do not yield new equations for the radial electric field.", "The proof amounts to explicitly showing that the turbulent tokamak is intrinsically ambipolar." ], [ "Quasineutrality equation and long-wavelength radial\nelectric field", "It is obvious from equations (REF ) and (REF ) that if $G_{{\\sigma }j}^{\\rm lw}$ , $j=1,2$ are solutions of the first and second-order Fokker-Planck equations, then so are $G_{{\\sigma }j}^{\\rm lw}+ h_{{\\sigma }j}$ , $j=1,2$ , where $h_{{\\sigma }j} &=&\\left[\\frac{n_{{\\sigma }j}}{n_{\\sigma }}+\\left(\\frac{\\mu B+u^2/2}{T_{\\sigma }}-\\frac{3}{2}\\right)\\frac{T_{{\\sigma }j}}{T_{\\sigma }}\\right]F_{{\\sigma }0},$ for an arbitrary set of flux functions $\\lbrace n_{{\\sigma }j}(\\psi ,t), T_{{\\sigma }j}(\\psi ,t)\\rbrace _{\\sigma }$ , with the only restriction $T_{{\\sigma }j}/\\lambda _{\\sigma }^j =T_{{\\sigma }^{\\prime } j}/\\lambda _{{\\sigma }^{\\prime }}^j$ , for all $\\sigma ,\\sigma ^{\\prime }$ (the temperature of the electrons is allowed to be different if we expand in the mass ratio, that is, if we use $\\lambda _e \\sim \\tau _e \\gg 1$ ).", "In other words, the operator acting on $G_{{\\sigma }1}^{\\rm lw}$ in (REF ) and on $G_{{\\sigma }2}^{\\rm lw}$ in (REF ) has a kernel given by (REF ) with an obvious interpretation: it consists of corrections of order $\\epsilon _{\\sigma }^j$ to the zeroth-order particle densities, $n_{\\sigma }$ , and temperatures, $T_{\\sigma }$ .", "Therefore, in order to have a unique solution for the Fokker-Planck equation, one needs to prescribe a condition that eliminates the freedom introduced by the existence of a non-zero kernel.", "An example of such a condition is given by imposing, for $j=1,2$ , $\\left\\langle \\int B G_{\\sigma j}\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\right\\rangle _\\psi = 0, \\mbox{ for every $\\sigma $, and}\\nonumber \\\\\\left\\langle \\sum _\\sigma \\frac{1}{\\lambda _\\sigma ^j} \\int B \\left(u^2/2+\\mu B\\right)G_{\\sigma j}\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\right\\rangle _\\psi = 0.$ Of course, even though this is a natural choice, there are infinitely many different possibilities.", "Assume that $\\textsf {G}_{{\\sigma }j}$ , $j=1,2$ (note the different font) are particular solutions of (REF ) and (REF ) (not necessarily satisfying (REF )).", "Then, any solution of (REF ) and (REF ) is of the form $\\textsf {G}_{{\\sigma }j}^{\\rm lw}+ h_{{\\sigma }j}$ , $j=1,2$ .", "When introduced in (REF ) and (REF ) we find: $\\sum _{\\sigma }\\frac{Z_{\\sigma }}{\\lambda _{\\sigma }} n_{{\\sigma }1}+\\sum _{\\sigma }\\frac{Z_{\\sigma }}{\\lambda _{\\sigma }} \\Bigg ( \\int B(\\mathbf {r}) \\textsf {G}_{{\\sigma }1}^{\\rm lw}(\\mathbf {r},u,\\mu ,\\theta ,t)\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\nonumber \\\\[5pt]\\hspace{28.45274pt}-\\frac{Z_{\\sigma }\\lambda _{\\sigma }}{T_{\\sigma }}n_{\\sigma }(\\mathbf {r}) \\varphi _1^{\\rm lw}(\\mathbf {r},t)\\Bigg ) =0,$ $\\sum _{\\sigma }\\frac{Z_{\\sigma }}{\\lambda _{\\sigma }^2}n_{{\\sigma }2}+\\sum _{\\sigma }\\frac{Z_{\\sigma }}{\\lambda _{\\sigma }^2}\\int B\\Bigg [\\textsf {G}_{{\\sigma }2}^{\\rm lw}+\\frac{u}{B}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\times \\hat{\\mathbf {b}}\\, \\textsf {G}_{{\\sigma }1}^{\\rm lw}\\nonumber \\\\[5pt]\\hspace{28.45274pt}+\\left(\\frac{Z_{\\sigma }\\lambda _{\\sigma }\\varphi _1^{\\rm lw}}{u} \\partial _u-\\frac{Iu}{B}\\partial _\\psi \\right) \\textsf {G}_{\\sigma 1}^{\\rm lw}\\nonumber \\\\[5pt]\\hspace{28.45274pt}- \\left[ \\frac{Z_{\\sigma }\\lambda _{\\sigma }}{u \\mathbf {B}\\cdot \\nabla _\\mathbf {r}\\Theta } F_{\\sigma 1}^{\\rm sw}(\\hat{\\mathbf {b}}\\times \\nabla _{\\mathbf {r}_\\perp /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle ) \\cdot \\nabla _\\mathbf {r}\\Theta \\right]^{\\rm lw}\\nonumber \\\\[5pt]\\hspace{28.45274pt} + \\left\\langle \\frac{1}{ u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\Theta }\\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {r}\\Theta \\sum _{\\sigma ^\\prime } \\mathcal {T}^*_{{\\sigma },0}C_{\\sigma \\sigma ^\\prime }^{(1){\\rm lw}} \\right\\rangle \\nonumber \\\\[5pt]\\hspace{28.45274pt}+ \\frac{Z_{\\sigma }^2\\lambda _{\\sigma }^2}{2T_{\\sigma }^2}\\left[ \\left\\langle (\\widetilde{\\phi }_{{\\sigma }1}^{\\rm sw})^2 \\right\\rangle \\right]^{\\rm lw}F_{{\\sigma }0}\\Bigg ]\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\nonumber \\\\[5pt]\\hspace{28.45274pt}+\\sum _{\\sigma }\\frac{Z_{\\sigma }}{\\lambda _{\\sigma }^2}n_{\\sigma }T_{\\sigma }\\Bigg \\lbrace \\frac{Z_{\\sigma }\\lambda _{\\sigma }^2}{T_\\sigma ^2}\\left(\\frac{Z_{\\sigma }}{2 T_\\sigma }(\\varphi _1^{\\rm lw})^2-\\varphi _2^{\\rm lw}\\right)\\nonumber \\\\[5pt]\\hspace{28.45274pt}+\\frac{R^2}{2}\\Bigg [\\left(\\frac{Z_{\\sigma }}{T_\\sigma }\\partial _\\psi \\varphi _0+\\partial _\\psi \\ln n_{\\sigma }\\right)^2\\nonumber \\\\[5pt]\\hspace{28.45274pt}+\\left(\\partial _\\psi \\ln T_{\\sigma }\\right)^2+2\\partial _\\psi \\ln n_{\\sigma }\\partial _\\psi \\ln T_{\\sigma }\\nonumber \\\\[5pt]\\hspace{28.45274pt}+\\partial _\\psi ^2\\ln n_{\\sigma }+ \\frac{Z_{\\sigma }}{T_{\\sigma }}\\partial _\\psi ^2\\varphi _0+\\partial _\\psi ^2\\ln T_{\\sigma }\\Bigg ]\\Bigg \\rbrace =0.$ Even though $\\varphi _0$ enters this equation to second-order, it cannot be determined.", "The first and second-order pieces of the long-wavelength quasineutrality equation simply give constraints on the corrections $n_{{\\sigma }1}$ and $n_{{\\sigma }2}$ .", "Each function $n_{{\\sigma }j}$ will be determined by a transport equation that appears as a solvability condition for a higher order long-wavelength piece of the Fokker-Planck equation, just as a transport equation for $n_{\\sigma }$ is derived in subsection REF as a solvability condition for equation (REF ).", "Note that we cannot choose the value of $n_{{\\sigma }j}$ ; we can only decide which piece of $G_{{\\sigma }j}$ we call $n_{{\\sigma }j}$ and which piece we leave within $\\textsf {G}_{{\\sigma }j}$ .", "The density corrections $n_{{\\sigma }j}$ cannot be set to zero arbitrarily because we need them to satisfy the solvability conditions of the higher order pieces of the Fokker-Planck equation.Our procedure here has diverged from the canonical Chapman-Enskog approach, where the density of the lowest order Maxwellian is not broken into pieces of different orders.", "Instead, in the Chapman-Enskog theory, the conservation equation for particle density, obtained from the solvability conditions (see subsection REF ), contains terms of different orders.", "Our procedure is different in that the conservation equation will be split into different orders, each giving an equation for a piece $n_{{\\sigma }j}$ .", "In this way we highlight that the quasineutrality equation to $O(\\epsilon _s^2)$ does not allow to solve for $\\varphi _0$ because this would require knowing $n_{{\\sigma }1}$ and $n_{{\\sigma }2}$ , which are determined from higher order pieces of the long-wavelength Fokker-Planck equation.", "We cannot calculate $\\varphi _0$ from the quasineutrality equation to this order, but the first and second-order pieces of the long-wavelength poloidal electric field can be found, respectively, from (REF ) and (REF ).", "This can be viewed by acting on the latter equations with $\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}$ and employing that $n_{{\\sigma }1}$ and $n_{{\\sigma }2}$ are only functions of $\\psi $ .", "Not surprisingly, $\\varphi _1^{\\rm lw}$ and $\\varphi _2^{\\rm lw}$ are determined up to an arbitrary, additive function of $\\psi $ , that can be absorbed by redefining the corrections $n_{{\\sigma }1}$ and $n_{{\\sigma }2}$ .", "Without loss of generality, we fix the ambiguity by taking $\\left\\langle \\varphi _1^{\\rm lw}\\right\\rangle _\\psi = 0$ and $\\left\\langle \\varphi _2^{\\rm lw}\\right\\rangle _\\psi = 0.$ In Subsection REF we explain that the second-order Fokker-Planck equation possesses some solvability conditions.", "We have to show that their fulfillment does not impose any additional conditions that give $\\varphi _0(\\psi ,t)$ , and we do so in subsection REF ." ], [ "Transport equations", "Some of the benefits of writing the Fokker-Planck equation precisely in the form (REF ) will be appreciated in this subsection, where we show that time evolution equations for the lowest order density and temperature functions $n_{\\sigma }$ and $T_{\\sigma }$ are obtained as solvability conditions for the second-order, long-wavelength Fokker-Planck equation.", "That is, we prove that if a solution for $G_{{\\sigma }2}^{\\rm lw}$ (equivalently, for $F_{{\\sigma }2}^{\\rm lw}$ ) exists, then (REF ) imposes certain constraints among lower-order quantities (solvability conditions).", "These conditions turn out to be transport equations for density and energy.", "In we prove that these transport equations are indeed the only solvability conditions obtained from the Fokker-Planck equation up to order $\\epsilon _s^2$ ." ], [ "Transport equation for density.", "Go back to (REF ), multiply by $\\tau _{\\sigma }B/\\lambda _{\\sigma }^2$ , integrate over $u,\\mu ,\\theta $ and take the flux-surface average: $\\partial _{\\epsilon _s^2 t} n_{{\\sigma }}(\\psi ,t)= \\frac{1}{V^{\\prime }(\\psi )}\\partial _\\psi \\Bigg \\langle V^{\\prime }(\\psi )\\int \\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\ \\Bigg \\lbrace \\nonumber \\\\[5pt]\\hspace{28.45274pt}\\left[ F_{\\sigma 1}^{\\rm sw}(\\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\times \\hat{\\mathbf {b}}) \\cdot \\nabla _\\mathbf {R}\\psi \\right]^{\\rm lw}\\nonumber \\\\[5pt]\\hspace{28.45274pt}+\\frac{B}{Z_{\\sigma }\\lambda _{\\sigma }}\\left\\langle \\left(\\frac{Iu}{B} +\\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}\\psi \\right) \\sum _{\\sigma ^\\prime }C_{\\sigma \\sigma ^\\prime }^{(1){\\rm lw}} \\right\\rangle \\Bigg \\rbrace \\Bigg \\rangle _\\psi ,$ where $V^{\\prime }(\\psi )$ is the derivative of the function $V(\\psi )$ , defined in (REF ), which gives the volume enclosed by the flux surface with label $\\psi $ .", "We have also used that for the tokamak the square root of the determinant of the metric tensor (recall (REF )) is $\\sqrt{g} = \\frac{1}{\\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta } \\, .$ Equation (REF ) is a transport equation for each lowest-order particle density function $n_{\\sigma }$ .", "Note that $\\varphi _0$ and $\\varphi _1^{\\rm lw}$ do not appear." ], [ "Transport equation for energy.", "Now, we do something similar for the total energy.", "Multiply (REF ) by $(\\tau _{\\sigma }/\\lambda _{\\sigma }^2)B(u^2/2 + \\mu B)$ , integrate over $u,\\mu ,\\theta $ , and take the flux-surface average.", "Then, $\\partial _{\\epsilon _s^2 t}\\left(\\frac{3}{2}n_{\\sigma }(\\psi ,t) T_{\\sigma }(\\psi ,t) \\right)=\\nonumber \\\\\\hspace{14.22636pt}\\frac{1}{V^\\prime (\\psi )} \\partial _\\psi \\Bigg \\langle V^\\prime (\\psi ) \\int \\left(u^2/2 + \\mu B\\right)\\Bigg \\lbrace \\nonumber \\\\\\hspace{14.22636pt}\\left[ F_{\\sigma 1}^{\\rm sw}(\\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }} \\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\times \\hat{\\mathbf {b}}) \\cdot \\nabla _\\mathbf {R}\\psi \\right]^{\\rm lw}\\nonumber \\\\ \\hspace{14.22636pt}+\\frac{B}{Z_{\\sigma }\\lambda _{\\sigma }} \\left\\langle \\left(\\frac{Iu}{B} +\\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}\\psi \\right) \\sum _{\\sigma ^\\prime }{\\cal T}_{{\\sigma },0}^*C_{\\sigma \\sigma ^\\prime }^{(1){\\rm lw}} \\right\\rangle \\Bigg \\rbrace \\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\Bigg \\rangle _\\psi \\nonumber \\\\\\hspace{14.22636pt} -\\Bigg \\langle \\int B \\Bigg [ F_{\\sigma 1}^{\\rm sw}\\Big ( u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\nonumber \\\\\\hspace{14.22636pt}+ \\frac{\\mu }{B} (\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}B) \\cdot \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\nonumber \\\\\\hspace{14.22636pt} +\\frac{u^2}{B} [\\hat{\\mathbf {b}}\\times (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}})] \\cdot \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\Big ) \\Bigg ]^{\\rm lw}\\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta \\Bigg \\rangle _\\psi \\nonumber \\\\\\hspace{14.22636pt}+\\frac{1}{\\lambda _{\\sigma }}\\partial _\\psi \\varphi _0 \\Bigg \\langle \\int B \\left( \\frac{I u}{B} + \\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}\\psi \\right)\\sum _{\\sigma ^\\prime } {\\cal T}_{{\\sigma },0}^*C_{\\sigma \\sigma ^\\prime }^{(1){\\rm lw}}\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\Bigg \\rangle _\\psi \\nonumber \\\\\\hspace{14.22636pt} +\\Bigg \\langle \\frac{\\tau _{\\sigma }}{\\lambda _{\\sigma }^2}\\int B\\left(u^2/2+\\mu B\\right)\\sum _{\\sigma ^\\prime }\\Bigg [ \\left[\\left\\langle {\\cal T}_{{\\sigma }, 1}^* C_{\\sigma \\sigma ^\\prime }^{(1){\\rm sw}}\\right\\rangle \\right]^{\\rm lw}\\nonumber \\\\\\hspace{14.22636pt} +\\left\\langle {\\cal T}_{{\\sigma },0}^*C_{\\sigma \\sigma ^\\prime }^{(2){\\rm lw}} \\right\\rangle \\Bigg ]\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\Bigg \\rangle _\\psi ,$ which is a transport equation for the energy density of species ${\\sigma }$ .", "The term containing $\\varphi _1^{\\rm lw}$ in (REF ) does not contribute to (REF ) because the collision operator conserves the total number of particles of each species.", "Equation (REF ) gives an equation for the temperature of each species.", "Unless we expand in the mass ratio $\\lambda _e \\sim \\tau _e \\gg 1$ , and allow different temperatures for electrons and ions, this equation still contains the function $G_{{\\sigma }2}^{\\rm lw}$ in $C_{{\\sigma }{\\sigma }^\\prime }^{(2){\\rm lw}}$ and cannot be considered a solvability condition.", "It is possible to prove that for $\\lambda _e\\sim \\tau _e \\gg 1$ , the equations for the electron and ion temperatures do not contain $G_{{\\sigma }2}^{\\rm lw}$ and consequently are independent solvability conditions that determine $T_i$ and $T_e$ .", "However, in general, the only way to eliminate $G_{{\\sigma }2}^{\\rm lw}$ is summing over all species.", "We obtain $\\partial _{\\epsilon _s^2 t} \\left( \\sum _{\\sigma }\\frac{3}{2}n_{\\sigma }(\\psi ,t) T_{\\sigma }(\\psi ,t) \\right)=\\nonumber \\\\\\hspace{14.22636pt}\\frac{1}{V^\\prime (\\psi )} \\partial _\\psi \\Bigg \\langle V^\\prime (\\psi ) \\int \\left(u^2/2 + \\mu B\\right)\\sum _{\\sigma }\\Bigg \\lbrace \\nonumber \\\\\\hspace{14.22636pt}\\left[ F_{\\sigma 1}^{\\rm sw}(\\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }} \\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\times \\hat{\\mathbf {b}}) \\cdot \\nabla _\\mathbf {R}\\psi \\right]^{\\rm lw}\\nonumber \\\\ \\hspace{14.22636pt}+ \\frac{B}{Z_{\\sigma }\\lambda _{\\sigma }} \\left\\langle \\left(\\frac{Iu}{B} +\\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}\\psi \\right) \\sum _{\\sigma ^\\prime }{\\cal T}_{{\\sigma },0}^*C_{\\sigma \\sigma ^\\prime }^{(1){\\rm lw}} \\right\\rangle \\Bigg \\rbrace \\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\Bigg \\rangle _\\psi \\nonumber \\\\\\hspace{14.22636pt} -\\Bigg \\langle \\sum _{\\sigma }\\int B \\Bigg [ F_{\\sigma 1}^{\\rm sw}\\Big ( u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\nonumber \\\\\\hspace{14.22636pt}+ \\frac{\\mu }{B} (\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}B) \\cdot \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\nonumber \\\\\\hspace{14.22636pt} +\\frac{u^2}{B} [\\hat{\\mathbf {b}}\\times (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}})] \\cdot \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\Big ) \\Bigg ]^{\\rm lw}\\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta \\Bigg \\rangle _\\psi \\nonumber \\\\\\hspace{14.22636pt} +\\Bigg \\langle \\sum _{{\\sigma },{\\sigma }^{\\prime }} \\frac{\\tau _{\\sigma }}{\\lambda _{\\sigma }^2}\\int B\\left(u^2/2+\\mu B\\right)\\left[\\left\\langle {\\cal T}_{{\\sigma }, 1}^* C_{\\sigma \\sigma ^\\prime }^{(1){\\rm sw}}\\right\\rangle \\right]^{\\rm lw}\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\Bigg \\rangle _\\psi .$ Here we have used the conservation of momentum and energy by the collision operator to find $ \\sum _{{\\sigma }, {\\sigma }^\\prime } \\Bigg \\langle \\frac{1}{\\lambda _{\\sigma }}\\int B \\left( \\frac{I u}{B} + \\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}\\psi \\right){\\cal T}_{{\\sigma },0}^*C_{\\sigma \\sigma ^\\prime }^{(1){\\rm lw}}\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\Bigg \\rangle _\\psi = 0,\\nonumber \\\\\\sum _{{\\sigma }, {\\sigma }^\\prime } \\Bigg \\langle \\frac{\\tau _{\\sigma }}{\\lambda _{\\sigma }^2}\\int B\\left(u^2/2+\\mu B\\right)\\left\\langle {\\cal T}_{{\\sigma },0}^*C_{\\sigma \\sigma ^\\prime }^{(2){\\rm lw}} \\right\\rangle \\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\Bigg \\rangle _\\psi = 0.$ These expressions are easily deduced from the conservation properties of the collision operator (REF ) by realizing that the summations of collision operators in (REF ) can always be decomposed in binomials of the form $\\int B \\left( \\frac{I u}{B} + \\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}\\psi \\right)( {\\cal T}_{{\\sigma },0}^*C_{\\sigma \\sigma ^\\prime } [ \\lambda _{\\sigma }^{-1} g_{\\sigma }, {\\cal T}_{{\\sigma }^\\prime , 0}^{-1*} F_{{\\sigma }^\\prime 0} ] \\nonumber \\\\+ {\\cal T}_{{\\sigma }^\\prime ,0}^*C_{{\\sigma }^\\prime {\\sigma }} [ {\\cal T}_{{\\sigma }^\\prime , 0}^{-1*} F_{{\\sigma }^\\prime 0}, \\lambda _{{\\sigma }}^{-1} g_{\\sigma }] )\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta = 0,\\nonumber \\\\\\int B \\left( u^2/2 + \\mu B \\right)( \\tau _{\\sigma }{\\cal T}_{{\\sigma },0}^*C_{\\sigma \\sigma ^\\prime } [ \\lambda _{\\sigma }^{-1} g_{\\sigma }, \\lambda _{{\\sigma }^\\prime }^{-1} g_{{\\sigma }^\\prime } ] \\nonumber \\\\+ \\tau _{{\\sigma }^\\prime } {\\cal T}_{{\\sigma }^\\prime ,0}^*C_{{\\sigma }^\\prime {\\sigma }} [ \\lambda _{{\\sigma }^\\prime }^{-1} g_{{\\sigma }^\\prime }, \\lambda _{\\sigma }^{-1} g_{\\sigma }] )\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta = 0,\\nonumber \\\\\\int B \\left( u^2/2 + \\mu B \\right)( \\tau _{\\sigma }{\\cal T}_{{\\sigma },0}^*C_{\\sigma \\sigma ^\\prime } [ \\lambda _{\\sigma }^{-2} g_{\\sigma }, {\\cal T}_{{\\sigma }^\\prime , 0}^{-1*} F_{{\\sigma }^\\prime 0} ] \\nonumber \\\\+ \\tau _{{\\sigma }^\\prime } {\\cal T}_{{\\sigma }^\\prime ,0}^*C_{{\\sigma }^\\prime {\\sigma }} [ {\\cal T}_{{\\sigma }^\\prime , 0}^{-1*} F_{{\\sigma }^\\prime 0}, \\lambda _{{\\sigma }}^{-2} g_{\\sigma }] )\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta = 0,$ where $g_{{\\sigma }} (\\mathbf {r}, \\mathbf {v}, t)$ and $g_{{\\sigma }^\\prime } (\\mathbf {r}, \\mathbf {v},t)$ are just placeholders for the functions that appear in the collision operators.", "We have not written these functions explicitly because their particular form is unimportant for the cancellations.", "To show that the summation of collision operators in (REF ) gives these binomials it is important to keep track of the factors $\\lambda _{{\\sigma }}^{-1}$ that multiply the collision operator arguments.", "Equation (REF ) can be simplified by further cancellations.", "contains the proof that $-\\Bigg \\langle \\sum _{\\sigma }\\int B \\Bigg [ F_{\\sigma 1}^{\\rm sw}\\Big ( u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\nonumber \\\\\\hspace{14.22636pt}+ \\frac{\\mu }{B} (\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}B) \\cdot \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\nonumber \\\\\\hspace{14.22636pt} +\\frac{u^2}{B} [\\hat{\\mathbf {b}}\\times (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}})] \\cdot \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\Big ) \\Bigg ]^{\\rm lw}\\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta \\Bigg \\rangle _\\psi \\nonumber \\\\\\hspace{14.22636pt} +\\Bigg \\langle \\sum _{{\\sigma },{\\sigma }^{\\prime }} \\frac{\\tau _{\\sigma }}{\\lambda _{\\sigma }^2}\\int B\\left(u^2/2+\\mu B\\right)\\left[\\left\\langle {\\cal T}_{{\\sigma }, 1}^* C_{\\sigma \\sigma ^\\prime }^{(1){\\rm sw}}\\right\\rangle \\right]^{\\rm lw}\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\Bigg \\rangle _\\psi \\nonumber \\\\\\hspace{14.22636pt}= O(\\epsilon _s),$ and therefore the final expression for the total energy transport equation is $\\partial _{\\epsilon _s^2 t}\\left( \\sum _{\\sigma }\\frac{3}{2}n_{\\sigma }(\\psi ,t) T_{\\sigma }(\\psi ,t) \\right)=\\nonumber \\\\\\hspace{14.22636pt}\\frac{1}{V^\\prime (\\psi )} \\partial _\\psi \\Bigg \\langle V^\\prime (\\psi ) \\int \\left(u^2/2 + \\mu B\\right)\\sum _{\\sigma }\\Bigg \\lbrace \\nonumber \\\\\\hspace{14.22636pt}\\left[ F_{\\sigma 1}^{\\rm sw}(\\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }} \\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\times \\hat{\\mathbf {b}}) \\cdot \\nabla _\\mathbf {R}\\psi \\right]^{\\rm lw}\\nonumber \\\\ \\hspace{14.22636pt}+\\frac{B}{Z_{\\sigma }\\lambda _{\\sigma }} \\left\\langle \\left(\\frac{Iu}{B} +\\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}\\psi \\right) \\sum _{\\sigma ^\\prime }{\\cal T}_{{\\sigma },0}^*C_{\\sigma \\sigma ^\\prime }^{(1){\\rm lw}} \\right\\rangle \\Bigg \\rbrace \\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\Bigg \\rangle _\\psi .$" ], [ "Time evolution of the lowest-order quasineutrality condition:\nintrinsic ambipolarity of the turbulent tokamak", "The zeroth-order piece of the long-wavelength quasineutrality equation imposes the well-known condition on the lowest order particle densities, equation (REF ): $\\sum _{\\sigma }Z_{\\sigma }n_{{\\sigma }}(\\psi ,t) = 0.$ On the other hand, we have obtained as a solvability condition of the long-wavelength second-order Fokker-Planck equation a time evolution equation for each function $n_{\\sigma }$ , (REF ).", "Thus, we can deduce a time evolution equation for $\\sum _{\\sigma }Z_{\\sigma }n_{\\sigma }$ .", "It is important to find out whether (REF ) is automatically preserved by the time evolution or, on the contrary, its preservation implies additional constraints on low-order quantities.", "In principle, it might have happened that imposing $\\partial _t\\sum _{\\sigma }Z_{\\sigma }n_{\\sigma }= 0$ implied a new equation involving the long-wavelength radial electric field.", "In this subsection we show that this is not the case in a tokamak.", "The contribution of the last term in (REF ) to $\\partial _t \\sum _{\\sigma }Z_{\\sigma }n_{\\sigma }$ vanishes due to the momentum conservation properties of the collision operator (see (REF )).", "As a result, we recover that, in neoclassical theory, $\\partial _t\\sum _{\\sigma }Z_{\\sigma }n_{\\sigma }\\equiv 0$ .", "The contribution of the first term on the right side of (REF ) to $\\partial _t \\sum _{\\sigma }Z_{\\sigma }n_{\\sigma }$ also vanishes, $ \\sum _{\\sigma }Z_{\\sigma }\\Bigg \\langle \\int B\\left[ F_{\\sigma 1}^{\\rm sw}(\\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\times \\hat{\\mathbf {b}}) \\cdot \\nabla _\\mathbf {R}\\psi \\right]^{\\rm lw}\\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta \\Bigg \\rangle _\\psi = 0.$ To prove this property we need the short-wavelength quasineutrality equation to first order in the expansion parameter, given in (REF ).", "We repeat it here for convenience: $\\sum _{\\sigma }\\frac{Z_{\\sigma }}{\\lambda _{\\sigma }}&\\int B\\Bigg [-Z_{\\sigma }\\lambda _{\\sigma }\\widetilde{\\phi }_{{\\sigma }1}^{\\rm sw}\\left(\\mathbf {r}-\\epsilon _{\\sigma }\\mbox{$\\rho $}(\\mathbf {r},\\mu ,\\theta ),\\mu ,\\theta ,t\\right)\\frac{F_{{\\sigma }0}(\\mathbf {r},u,\\mu ,t)}{T_{{\\sigma }}(\\mathbf {r},t)}\\nonumber \\\\[5pt]&+ F_{{\\sigma }1}^{\\rm sw}\\left(\\mathbf {r}-\\epsilon _{\\sigma }\\mbox{$\\rho $}(\\mathbf {r},\\mu ,\\theta ),u,\\mu ,t\\right)\\Bigg ]\\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta = 0.$ Acting on (REF ) with $\\varphi _1^{\\rm sw}(\\mathbf {r},t)\\nabla _{\\mathbf {r}_\\perp /\\epsilon _s}$ , taking the coarse-grain average, and observing that $\\varphi _1^{\\rm sw}(\\mathbf {r},t) =\\phi _{{\\sigma }1}^{\\rm sw}(\\mathbf {r}-\\epsilon _{\\sigma }\\mbox{$\\rho $}(\\mathbf {r},\\mu ,\\theta ),\\mu ,\\theta ,t)+ O(\\epsilon _{\\sigma }),$ we obtain $\\sum _{\\sigma }Z_{\\sigma }\\int B\\left[\\phi _{{\\sigma }1}^{\\rm sw}\\nabla _{\\mathbf {r}_\\perp /\\epsilon _{\\sigma }}\\left(-\\frac{Z_{\\sigma }\\lambda _{\\sigma }\\widetilde{\\phi }_{{\\sigma }1}^{\\rm sw}}{T_{{\\sigma }}} F_{{\\sigma }0} + F_{{\\sigma }1}^{\\rm sw}\\right)\\right]^{\\rm lw}\\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta = O(\\epsilon _{\\sigma }).$ In this expression the functions $\\phi _{{\\sigma }1}^{\\rm sw}$ and $F_{{\\sigma }1}^{\\rm sw}$ are evaluated at $\\mathbf {R}= \\mathbf {r}- \\epsilon _{\\sigma }\\mbox{$\\rho $}(\\mathbf {r}, \\mu , \\theta )$ , but after the coarse grain average we can Taylor expand and, to lowest order, they can be evaluated at $\\mathbf {R}=\\mathbf {r}$ .", "This leads to $-\\sum _{\\sigma }Z_{\\sigma }\\int B\\nabla _{\\mathbf {r}_\\perp /\\epsilon _{\\sigma }}\\left[\\frac{Z_{\\sigma }\\lambda _{\\sigma }(\\widetilde{\\phi }_{{\\sigma }1}^{\\rm sw})^2}{2 T_{{\\sigma }}} F_{{\\sigma }0}\\right]^{\\rm lw}\\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta \\nonumber \\\\[5pt]\\hspace{28.45274pt}+ \\sum _{\\sigma }Z_{\\sigma }\\int B\\left[F_{{\\sigma }1}^{\\rm sw}\\nabla _{\\mathbf {r}_\\perp /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}\\rangle ^{\\rm sw}\\right]^{\\rm lw}\\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta = O(\\epsilon _{\\sigma }).$ The fact that $\\nabla _{\\mathbf {r}_\\perp /\\epsilon _{\\sigma }} g^{\\rm lw}= O(\\epsilon _{\\sigma })$ whenever $g = O(1)$ implies $\\sum _{\\sigma }Z_{\\sigma }\\int B\\left[F_{{\\sigma }1}^{\\rm sw}\\nabla _{\\mathbf {r}_\\perp /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\right]^{\\rm lw}\\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta = O(\\epsilon _{\\sigma }),$ whence we immediately infer equation (REF ), giving $\\partial _t\\sum _{\\sigma }Z_{\\sigma }n_{\\sigma }\\equiv 0,$ identically, at the relevant order.", "Consequently, we have proven that the well-known neoclassical intrinsic ambipolarity property of the tokamak still holds in gyrokinetic theory." ], [ "Discussion of results and conclusions", "At the moment, the problem of extending the standard set of gyrokinetic equations, and therefore computer simulations, to transport time scales is an active research topic.", "Focusing on toroidal angular momentum transport in tokamaks in electrostatic gyrokinetics, the issue has been recently raised by Parra and Catto [12], [18], [19], [20], [21], [22]; they argue that calculating momentum transport correctly requires knowledge of the distribution function and electrostatic potential up to second order in the expansion parameter, the gyroradius over the macroscopic length scale.", "An intimately related result of this series of works is that in a tokamak the system consisting of second-order Fokker-Planck and quasineutrality equations does not determine the long-wavelength radial electric field.", "A method to correctly compute the radial transport of toroidal angular momentum (and therefore the radial electric field) when the second-order pieces of the distribution function and the electrostatic potential are known is given in reference [22], [24].", "Using the recent derivation of the second-order gyrokinetic equations [16] in general magnetic geometry, we have worked out the long-wavelength limit of the Fokker-Planck and quasineutrality equations in a tokamak, a necessary first step towards the formulation of a set of equations to compute the radial transport of toroidal angular momentum without having to resort to subsidiary expansions such as the expansion in $B_p/B\\ll 1$ of references [22], [24].", "Specifically, we have obtained (see the main text for notation and details): The long-wavelength Fokker-Planck equations to second order, (REF ) and (REF ), that give $G_{{\\sigma }1}^{\\rm lw}$ and $G_{{\\sigma }2}^{\\rm lw}$ , and therefore the long-wavelength component of the distribution functions.", "The quasineutrality equation up to second-order (REF ), (REF ), and (REF ), that determines the first and second-order pieces of the long-wavelength poloidal electric field.", "Equivalently, and under conditions (REF ) and (REF ), the quasineutrality equation determines $\\varphi _1^{\\rm lw}$ and $\\varphi _2^{\\rm lw}$ .", "Transport equations for density (REF ) and energy (REF ).", "Equations (REF ) and (REF ), that give the short-wavelength component of the distribution functions, $F_{{\\sigma }1}^{\\rm sw}$ , and electrostatic potential, $\\phi _{{\\sigma }1}^{\\rm sw}$ .", "They are needed because they enter equation (REF ).", "In order to provide a model for toroidal angular momentum transport in tokamaks one still needs to derive explicit equations for the short-wavelength components of the distribution functions and electrostatic potential to second order.", "This will be the subject of a future publication.", "In addition, in this paper, we have given a complete proof that the long-wavelength tokamak radial electric field cannot be determined by simply using Fokker-Planck and quasineutrality equations accurate to second order in the gyrokinetic expansion parameter.", "In other words, we have proven that gyrokinetics does not spoil the well-known neoclassical intrinsic ambipolarity property of the tokamak.", "This research was supported in part by grant ENE2009-07247, Ministerio de Ciencia e Innovación (Spain), and by US DoE grant DE-SC008435." ], [ "Gyrokinetic equations of motion", "Here, the gyrokinetic equations of motion corresponding to the Poisson bracket (REF ) and the gyrokinetic Hamiltonian given in equations (REF ), (REF ), and (REF ), are explicitly written: $ \\frac{\\mbox{d}\\mathbf {R}}{\\mbox{d}t}&=\\lbrace \\mathbf {R},\\overline{H}_{\\sigma }\\rbrace _\\mathbf {Z}=\\nonumber \\\\&\\left( u + Z_{\\sigma }\\lambda _{\\sigma }\\epsilon ^2_{\\sigma }\\partial _u\\Psi _{\\phi B,{\\sigma }} +\\epsilon ^2_{\\sigma }\\partial _u\\Psi _{B,\\phi } \\right)\\frac{\\mathbf {B}_{\\sigma }^*}{B_{||,{\\sigma }}^*}\\nonumber \\\\&+ \\frac{1}{B_{||,{\\sigma }}^*} \\hat{\\mathbf {b}}\\times \\Bigg (\\epsilon _{\\sigma }\\mu \\nabla _\\mathbf {R}B+ Z_{\\sigma }\\lambda _{\\sigma }\\epsilon _{\\sigma }\\nabla _{\\mathbf {R}_\\perp / \\epsilon _{\\sigma }}\\langle \\phi _{\\sigma }\\rangle \\nonumber \\\\&+{Z_{\\sigma }^2\\lambda _{\\sigma }^2 \\epsilon ^2_{\\sigma }}\\nabla _{\\mathbf {R}_\\bot /\\epsilon _{\\sigma }} \\Psi _{\\phi ,{\\sigma }}+ Z_{\\sigma }\\lambda _{\\sigma }\\epsilon ^2_{\\sigma }\\nabla _{\\mathbf {R}_\\bot / \\epsilon _{\\sigma }}\\Psi _{\\phi B,{\\sigma }}\\nonumber \\\\ &+Z_{\\sigma }^2\\lambda _{\\sigma }^2 \\epsilon _{\\sigma }^3 \\nabla _\\mathbf {R}\\Psi _{\\phi ,{\\sigma }}+ Z_{\\sigma }\\lambda _{\\sigma }\\epsilon _{\\sigma }^3 \\nabla _\\mathbf {R}\\Psi _{\\phi B,{\\sigma }}+ \\epsilon _{\\sigma }^3 \\nabla _\\mathbf {R}\\Psi _{B,{\\sigma }} \\Bigg ),$ $ \\frac{\\mbox{d}u}{\\mbox{d}t}&=\\lbrace u ,\\overline{H}_{\\sigma }\\rbrace _\\mathbf {Z}=\\nonumber \\\\&- \\frac{\\mu }{B^*_{||,{\\sigma }}} \\mathbf {B}_{\\sigma }^*\\cdot \\nabla _\\mathbf {R}B - Z_{\\sigma }\\lambda _{\\sigma }\\epsilon _{\\sigma }\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\langle \\phi _{\\sigma }\\rangle \\nonumber \\\\&- Z_{\\sigma }^2\\lambda _{\\sigma }^2 \\epsilon _{\\sigma }^2\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\Psi _{\\phi ,{\\sigma }}- Z_{\\sigma }\\lambda _{\\sigma }\\epsilon _{\\sigma }^2\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\Psi _{\\phi B,_{\\sigma }}\\nonumber \\\\[3pt]&- \\epsilon _{\\sigma }^2 \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\Psi _{B,{\\sigma }}- \\frac{1}{B_{||,{\\sigma }}^*} \\Big [ u \\hat{\\mathbf {b}}\\times (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}})\\nonumber \\\\[3pt]&- \\epsilon _{\\sigma }\\mu (\\nabla _\\mathbf {R}\\times \\mathbf {K})_\\bot \\Big ] \\cdot \\Bigg ( Z_{\\sigma }\\lambda _{\\sigma }\\epsilon _{\\sigma }\\nabla _{\\mathbf {R}_\\bot / \\epsilon _{\\sigma }} \\langle \\phi _{\\sigma }\\rangle \\nonumber \\\\[3pt]& +Z_{\\sigma }^2\\lambda _{\\sigma }^2 \\epsilon _{\\sigma }^2\\nabla _{\\mathbf {R}_\\bot /\\epsilon _{\\sigma }} \\Psi _{\\phi ,{\\sigma }}+ Z_{\\sigma }\\lambda _{\\sigma }\\epsilon _{\\sigma }^2\\nabla _{\\mathbf {R}_\\bot /\\epsilon _{\\sigma }} \\Psi _{\\phi B,_{\\sigma }} \\nonumber \\\\&+Z_{\\sigma }^2\\lambda _{\\sigma }^2 \\epsilon _{\\sigma }^3 \\nabla _\\mathbf {R}\\Psi _{\\phi ,{\\sigma }}+ Z_{\\sigma }\\lambda _{\\sigma }\\epsilon _{\\sigma }^3 \\nabla _\\mathbf {R}\\Psi _{\\phi B,{\\sigma }}+\\epsilon _{\\sigma }^3 \\nabla _\\mathbf {R}\\Psi _{B,{\\sigma }} \\Bigg ) ,$ $ \\frac{\\mbox{d}\\mu }{\\mbox{d}t} = \\lbrace \\mu ,\\overline{H}_{\\sigma }\\rbrace _\\mathbf {Z}= 0,$ $ \\frac{\\mbox{d}\\theta }{\\mbox{d}t}&=\\lbrace \\theta ,\\overline{H}_{\\sigma }\\rbrace _\\mathbf {Z}=\\nonumber \\\\&-\\frac{1}{\\epsilon _{\\sigma }} B - Z_{\\sigma }\\lambda _{\\sigma }\\partial _\\mu \\langle \\phi _{\\sigma }\\rangle - Z_{\\sigma }^2\\lambda _{\\sigma }^2\\epsilon _{\\sigma }\\partial _\\mu \\Psi _{\\phi ,{\\sigma }}\\nonumber \\\\&-Z_{\\sigma }\\lambda _{\\sigma }\\epsilon _{\\sigma }\\partial _\\mu \\Psi _{\\phi B,{\\sigma }}- \\epsilon _{\\sigma }\\partial _\\mu \\Psi _{B,{\\sigma }}\\nonumber \\\\&- \\frac{\\mathbf {B}_{\\sigma }^* \\cdot \\mathbf {K}}{B_{||,{\\sigma }}^*} \\Bigg ( u + Z_{\\sigma }\\lambda _{\\sigma }\\epsilon _{\\sigma }^2\\partial _u\\Psi _{\\phi B,{\\sigma }}+ \\epsilon _{\\sigma }^2 \\partial _u\\Psi _{B,{\\sigma }} \\Bigg )\\nonumber \\\\[3pt]&- \\frac{1}{B_{||,{\\sigma }}^*} (\\mathbf {K}\\times \\hat{\\mathbf {b}}) \\cdot \\Big ( \\epsilon _{\\sigma }\\mu \\nabla _\\mathbf {R}B+Z_{\\sigma }\\lambda _{\\sigma }\\epsilon _{\\sigma }\\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }} \\langle \\phi _{\\sigma }\\rangle \\nonumber \\\\[3pt]&+ Z_{\\sigma }^2\\lambda _{\\sigma }^2 \\epsilon _{\\sigma }^2\\nabla _{\\mathbf {R}_\\bot /\\epsilon _{\\sigma }} \\Psi _{\\phi ,{\\sigma }}+Z_{\\sigma }\\lambda _{\\sigma }\\epsilon _{\\sigma }^2\\nabla _{\\mathbf {R}_\\bot /\\epsilon _{\\sigma }}\\Psi _{\\phi B,{\\sigma }}\\nonumber \\\\[3pt]&+ Z_{\\sigma }^2\\lambda _{\\sigma }^2 \\epsilon _{\\sigma }^3 \\nabla _\\mathbf {R}\\Psi _{\\phi ,{\\sigma }}+ Z_{\\sigma }\\lambda _{\\sigma }\\epsilon _{\\sigma }^3 \\nabla _\\mathbf {R}\\Psi _{\\phi B,{\\sigma }} + \\epsilon _{\\sigma }^3 \\nabla _\\mathbf {R}\\Psi _{B,{\\sigma }} \\Big ).$" ], [ "Some basic properties of the collision operator", "We recall (see, for example, reference [41]) that the collision operator (REF ) satisfies, for every ${\\sigma },{\\sigma }^{\\prime }$ , the conservation properties $\\int C_{{\\sigma }{\\sigma }^{\\prime }}\\mbox{d}^3 v = 0\\nonumber \\\\[5pt]\\int m_{\\sigma }\\mathbf {v}C_{{\\sigma }{\\sigma }^{\\prime }}\\mbox{d}^3 v = -\\int m_{{\\sigma }^{\\prime }}\\mathbf {v}C_{{\\sigma }^{\\prime }{\\sigma }}\\mbox{d}^3 v\\nonumber \\\\[5pt]\\int \\frac{1}{2}m_{\\sigma }\\mathbf {v}^2 C_{{\\sigma }{\\sigma }^{\\prime }}\\mbox{d}^3 v= -\\int \\frac{1}{2} m_{{\\sigma }^{\\prime }}\\mathbf {v}^2 C_{{\\sigma }^{\\prime }{\\sigma }}\\mbox{d}^3 v,$ and the statistical equilibrium condition $C_{{\\sigma }{\\sigma }^{\\prime }}[f_{M{\\sigma }},f_{M{\\sigma }^{\\prime }}] = 0$ when both distribution functions are Maxwellian, $&&{f_{M{\\sigma }}}({\\mathbf {r}},{\\mathbf {v}})=n_{{\\sigma }}(\\mathbf {r})\\left(\\frac{m_{\\sigma }}{2\\pi T_{{\\sigma }}(\\mathbf {r})}\\right)^{3/2}\\exp \\left(-\\frac{m_{\\sigma }{\\mathbf {v}}^2}{2T_{{\\sigma }}(\\mathbf {r})}\\right),\\nonumber \\\\[5pt]&&{f_{M{\\sigma }^{\\prime }}}({\\mathbf {r}},{\\mathbf {v}})=n_{{\\sigma }^{\\prime }}(\\mathbf {r})\\left(\\frac{m_{{\\sigma }^{\\prime }}}{2\\pi T_{{\\sigma }^{\\prime }}(\\mathbf {r})}\\right)^{3/2}\\exp \\left(-\\frac{m_{{\\sigma }^{\\prime }}\\mathbf {v}^2}{2T_{{\\sigma }^{\\prime }}(\\mathbf {r})}\\right),$ with $T_{\\sigma }(\\mathbf {r}) = T_{{\\sigma }^{\\prime }}(\\mathbf {r})$ at every point.", "These are the only solutions to the equations (REF ).", "The easiest way to see this is noting that the entropy production, $-\\sum _{{\\sigma },{\\sigma }^{\\prime }}\\int \\ln f_{\\sigma }C_{{\\sigma }{\\sigma }^{\\prime }}[f_{\\sigma },f_{{\\sigma }^{\\prime }}]\\mbox{d}^3v \\, ,$ is non-negative and vanishes only when $f_{\\sigma }$ and $f_{{\\sigma }^{\\prime }}$ are Maxwellians with the same temperature.", "Another well-known property, derived from (REF ), is $C_{{\\sigma }{\\sigma }^{\\prime }}\\left[\\frac{m_{\\sigma }}{T_{\\sigma }}\\mathbf {v}f_{M\\sigma },f_{M\\sigma ^{\\prime }}\\right]+C_{{\\sigma }{\\sigma }^{\\prime }}\\left[f_{M\\sigma },\\frac{m_{{\\sigma }^{\\prime }}}{T_{{\\sigma }^{\\prime }}}\\mathbf {v}f_{M\\sigma ^{\\prime }}\\right]\\equiv 0.$ This property implies that displacing both Maxwellians by the same average velocity gives another solution of (REF ).", "It is useful to have the explicit translation of these properties into our non-dimensional variables.", "With the definition (REF ) we have $\\int \\underline{C_{{\\sigma }{\\sigma }^{\\prime }}}\\mbox{d}^3 \\underline{v} = 0\\nonumber \\\\[5pt]\\int \\underline{\\mathbf {v}}\\, \\underline{C_{{\\sigma }{\\sigma }^{\\prime }}}\\,\\mbox{d}^3\\underline{v} = -\\int \\underline{\\mathbf {v}} \\,\\underline{C_{{\\sigma }^{\\prime }{\\sigma }}}\\,\\mbox{d}^3 \\underline{v}\\nonumber \\\\[5pt]\\int \\frac{1}{2}\\,\\tau _{\\sigma }\\, \\underline{\\mathbf {v}}^2 \\,\\underline{C_{{\\sigma }{\\sigma }^{\\prime }}} \\, \\mbox{d}^3 \\underline{v} =-\\int \\frac{1}{2}\\, \\tau _{{\\sigma }^{\\prime }} \\, \\underline{\\mathbf {v}}^2 \\,\\underline{C_{{\\sigma }^{\\prime }{\\sigma }}} \\, \\mbox{d}^3 \\underline{v}.$ Also, $\\underline{C_{{\\sigma }{\\sigma }^{\\prime }}} [\\underline{f_{M{\\sigma }}},\\underline{f_{M{\\sigma }^{\\prime }}}]\\equiv 0$ when $&&\\underline{f_{M{\\sigma }}}(\\underline{\\mathbf {r}},\\underline{\\mathbf {v}})=\\frac{\\underline{n_{{\\sigma }}}(\\underline{\\mathbf {r}})}{(2\\pi \\underline{T_{{\\sigma }}}(\\underline{\\mathbf {r}}))^{3/2}}\\exp \\left(-\\frac{\\underline{\\mathbf {v}}^2}{2\\underline{T_{{\\sigma }}}(\\underline{\\mathbf {r}})}\\right),\\nonumber \\\\[5pt]&&\\underline{f_{M{\\sigma }^{\\prime }}}(\\underline{\\mathbf {r}},\\underline{\\mathbf {v}})=\\frac{\\underline{n_{{\\sigma }^{\\prime }}}(\\underline{\\mathbf {r}})}{(2\\pi \\underline{T_{{\\sigma }^{\\prime }}}(\\underline{\\mathbf {r}}))^{3/2}}\\exp \\left(-\\frac{\\underline{\\mathbf {v}}^2}{2\\underline{T_{{\\sigma }^{\\prime }}}(\\underline{\\mathbf {r}})}\\right),$ and $\\underline{T_{{\\sigma }}}(\\underline{\\mathbf {r}})=\\underline{T_{{\\sigma }^{\\prime }}} (\\underline{\\mathbf {r}})$ at every point.", "Finally, $\\underline{C_{{\\sigma }{\\sigma }^{\\prime }}}\\left[\\frac{1}{\\tau _{{\\sigma }}\\underline{T_{\\sigma }}}\\underline{\\mathbf {v}}\\ \\underline{f_{M\\sigma }},\\underline{f_{M\\sigma ^{\\prime }}}\\right]+\\underline{C_{{\\sigma }{\\sigma }^{\\prime }}}\\left[\\underline{f_{M\\sigma }},\\frac{1}{\\tau _{{\\sigma }^{\\prime }}\\underline{T_{{\\sigma }^{\\prime }}}}\\underline{\\mathbf {v}}\\ \\underline{f_{M\\sigma ^{\\prime }}}\\right]\\equiv 0.$" ], [ "Gyrokinetic transformation to first order", "In this appendix we provide explicit expressions for the gyrokinetic transformation $(\\mathbf {r},\\mathbf {v})={\\cal T}_{\\sigma }(\\mathbf {R},u,\\mu ,\\theta ,t)$ to order $\\epsilon _{\\sigma }$ .", "Define $v_{||}&:=\\mathbf {v}\\cdot \\hat{\\mathbf {b}}(\\mathbf {r}),\\\\[5pt]\\mu _0&:=\\frac{(\\mathbf {v}-v_{||}\\hat{\\mathbf {b}}(\\mathbf {r}))^2}{2B(\\mathbf {r})},\\\\[5pt]\\theta _0&:=\\arctan \\left(\\frac{\\mathbf {v}\\cdot \\hat{\\mathbf {e}}_2(\\mathbf {r})}{\\mathbf {v}\\cdot \\hat{\\mathbf {e}}_1(\\mathbf {r})}\\right),$ and let us compute $(\\mathbf {r},v_{||},\\mu _0,\\theta _0)$ as a function of $(\\mathbf {R},u,\\mu ,\\theta )$ to first order in $\\epsilon _{\\sigma }$ .", "From the definition (REF ) we find $(\\mathbf {r},v_{||},\\mu _0,\\theta _0)$ as a function of $(\\mathbf {R}_g,v_{||g},\\mu _g,\\theta _g)$ : $ \\mathbf {r}= \\mathbf {R}_g + \\epsilon _{\\sigma }\\mbox{$\\rho $}_g,\\nonumber \\\\[5pt]v_{||} = v_{||g}+ \\epsilon _{\\sigma }B_g(\\mbox{$\\rho $}_g\\times \\hat{\\mathbf {b}}_g)\\mbox{$\\rho $}_g:\\nabla _{\\mathbf {R}_g}\\hat{\\mathbf {b}}_g+ O(\\epsilon _{\\sigma }^2),\\nonumber \\\\[5pt]\\mu _0 = \\mu _g - \\epsilon _{\\sigma }\\left(\\frac{\\mu _g}{B_g}\\mbox{$\\rho $}_g\\cdot \\nabla _{\\mathbf {R}_g}B_g +v_{||g}(\\mbox{$\\rho $}_g\\times \\hat{\\mathbf {b}}_g)\\mbox{$\\rho $}_g:\\nabla _{\\mathbf {R}_g}\\hat{\\mathbf {b}}_g\\right)\\nonumber \\\\[5pt]\\hspace{28.45274pt}+ O(\\epsilon _{\\sigma }^2),\\nonumber \\\\[5pt]\\theta _0 = \\theta _g +\\epsilon _{\\sigma }\\left(\\mbox{$\\rho $}_g\\cdot \\nabla _{\\mathbf {R}_g}\\hat{\\mathbf {e}}_{2g}\\cdot \\hat{\\mathbf {e}}_{1g}-\\frac{v_{||g}}{2\\mu _g}\\mbox{$\\rho $}_g\\cdot \\nabla _{\\mathbf {R}_g}\\hat{\\mathbf {b}}_g\\cdot \\mbox{$\\rho $}_g\\right)\\nonumber \\\\[5pt]\\hspace{28.45274pt}+ O(\\epsilon _{\\sigma }^2),$ where a subindex $g$ stresses that the quantity is evaluated at $(\\mathbf {R}_g,v_{||g},\\mu _g,\\theta _g)$ .", "Using (REF ), (), (), (), and the identities $(\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}})\\mbox{$\\rho $}:\\nabla _{\\mathbf {R}}\\hat{\\mathbf {b}}= \\mbox{$\\rho $}(\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}}):\\nabla _{\\mathbf {R}}\\hat{\\mathbf {b}}- \\frac{2\\mu }{B}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}},\\nonumber \\\\[5pt]\\mbox{$\\rho $}\\mbox{$\\rho $}+ (\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}})(\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}})= \\frac{2\\mu }{B}(\\leftrightarrow \\over {\\mathbf {I}}-\\hat{\\mathbf {b}}\\hat{\\mathbf {b}}),$ we arrive at $\\mathbf {r}&= \\mathbf {R}+ \\epsilon _{\\sigma }\\mbox{$\\rho $}+ O(\\epsilon _{\\sigma }^2),\\nonumber \\\\[5pt]v_{||} &= u+ \\epsilon _{\\sigma }\\hat{u}_1+ O(\\epsilon _{\\sigma }^2),\\nonumber \\\\[5pt]\\mu _0 &= \\mu + \\epsilon _{\\sigma }\\hat{\\mu }_1+ O(\\epsilon _{\\sigma }^2),\\nonumber \\\\[5pt]\\theta _0 &= \\theta +\\epsilon _{\\sigma }\\hat{\\theta }_1+ O(\\epsilon _{\\sigma }^2),$ where $\\hat{u}_1 &=u\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\cdot \\mbox{$\\rho $}+\\frac{B}{4}[\\mbox{$\\rho $}(\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}})+(\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}})\\mbox{$\\rho $}]:\\nabla _{\\mathbf {R}}\\hat{\\mathbf {b}}\\nonumber \\\\[5pt]&-\\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}},\\nonumber \\\\[5pt]\\hat{\\mu }_1 &=-\\frac{\\mu }{B}\\mbox{$\\rho $}\\cdot \\nabla _{\\mathbf {R}}B-\\frac{u}{4}\\left(\\mbox{$\\rho $}(\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}})+(\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}})\\mbox{$\\rho $}\\right):\\nabla _{\\mathbf {R}}\\hat{\\mathbf {b}}\\nonumber \\\\[5pt]&+\\frac{u\\mu }{B}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}-\\frac{u^2}{B}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\cdot \\mbox{$\\rho $}-\\frac{Z_{\\sigma }\\lambda _{\\sigma }}{B}\\tilde{\\phi }_{{\\sigma }1},\\nonumber \\\\[5pt]\\hat{\\theta }_1&=(\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}})\\cdot \\Bigg (\\nabla _\\mathbf {R}\\ln B + \\frac{u^2}{2\\mu B}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\nonumber \\\\[5pt]&-\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}\\hat{\\mathbf {e}}_2\\cdot \\hat{\\mathbf {e}}_1\\Bigg )-\\frac{u}{8\\mu }\\left(\\mbox{$\\rho $}\\mbox{$\\rho $}- (\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}})(\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}})\\right):\\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\nonumber \\\\[5pt]&+\\frac{u}{2B^2}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B+\\frac{Z_{\\sigma }\\lambda _{\\sigma }}{B}\\partial _\\mu \\tilde{\\Phi }_{{\\sigma }1}.$ It is useful to have the long-wavelength limit of the previous expressions at hand.", "Employing (REF ), and (REF ) we get: $\\hat{u}_1^{\\rm lw}&= \\hat{u}_1\\nonumber \\\\[5pt]\\hat{\\mu }_1^{\\rm lw}&=-\\frac{\\mu }{B}\\mbox{$\\rho $}\\cdot \\nabla _{\\mathbf {R}}B-\\frac{u}{4}\\left(\\mbox{$\\rho $}(\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}})+(\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}})\\mbox{$\\rho $}\\right):\\nabla _{\\mathbf {R}}\\hat{\\mathbf {b}}\\nonumber \\\\[5pt]&+\\frac{u\\mu }{B}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}-\\frac{u^2}{B}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\cdot \\mbox{$\\rho $}-\\frac{Z_{\\sigma }}{B}\\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}\\varphi _0,\\nonumber \\\\[5pt]\\hat{\\theta }_1^{\\rm lw}&=(\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}})\\cdot \\Bigg (\\nabla _\\mathbf {R}\\ln B + \\frac{u^2}{2\\mu B}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\nonumber \\\\[5pt]&-\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}\\hat{\\mathbf {e}}_2\\cdot \\hat{\\mathbf {e}}_1\\Bigg )-\\frac{u}{8\\mu }\\left(\\mbox{$\\rho $}\\mbox{$\\rho $}- (\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}})(\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}})\\right):\\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\nonumber \\\\[5pt]&+\\frac{u}{2B^2}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B+\\frac{Z_{\\sigma }}{2\\mu B}\\left(\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}}\\right)\\cdot \\nabla _\\mathbf {R}\\varphi _0.$ Next, we proceed to calculate the long-wavelength limit of ${\\cal T}_{\\sigma }^{-1*}F_{{\\sigma }0}$ to first order in $\\epsilon _{\\sigma }$ , needed to write (REF ) in Section REF .", "Inverting (REF ) to first order, and recalling (REF ) and the relations $\\partial _u F_{{\\sigma }0 } = -(u/T_{\\sigma }) F_{{\\sigma }0 }$ , $\\partial _\\mu F_{{\\sigma }0 } = -(B/T_{\\sigma }) F_{{\\sigma }0 }$ , one finds that $\\left[{\\cal T}^{-1*}_{\\sigma }F_{{\\sigma }0}\\right]^{\\rm lw}&=& {\\cal T}^{-1*}_{{\\sigma },0}F_{{\\sigma }0}+\\frac{\\epsilon _{\\sigma }}{ T_{{\\sigma }}}\\Bigg [\\mathbf {v}\\cdot {\\mathbf {V}}^p_{\\sigma }+\\left(\\frac{v^2}{2T_{{\\sigma }}}-\\frac{5}{2}\\right)\\mathbf {v}\\cdot {\\mathbf {V}}^T_{\\sigma }\\nonumber \\\\[5pt]&+&\\frac{Z_{\\sigma }}{B}\\mathbf {v}\\cdot (\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {r}\\varphi _0)\\Bigg ]{\\cal T}^{-1*}_{{\\sigma },0}F_{{\\sigma }0},$ with ${\\mathbf {V}}^p_{\\sigma }$ and ${\\mathbf {V}}^T_{\\sigma }$ defined in (REF )." ], [ "Calculations for the Fokker-Planck equation to\n$O(\\epsilon _{\\sigma })$", "In what follows we detail the calculations that recast (REF ) into (REF ) when the magnetic field has the form (REF ).", "First, rewrite (REF ) as $\\left(u\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}- \\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B\\partial _u\\right)F_{{\\sigma }1}^{\\rm lw}\\nonumber \\\\[5pt]\\hspace{28.45274pt}+\\left(\\mathbf {v}_\\kappa + \\mathbf {v}_{\\nabla B} + \\mathbf {v}_{E,{\\sigma }}^{(0)}\\right)\\cdot \\Big (\\nabla _\\mathbf {R}F_{{\\sigma }0} + \\frac{\\mu F_{{\\sigma }0}}{T_{\\sigma }}\\nabla _\\mathbf {R}B\\nonumber \\\\[5pt]\\hspace{28.45274pt}+\\frac{Z_{\\sigma }F_{{\\sigma }0}}{T_{\\sigma }}\\nabla _\\mathbf {R}\\varphi _0\\Big )+\\frac{Z_{\\sigma }\\lambda _{\\sigma }}{T_{\\sigma }} u\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\varphi _1^{\\rm lw}F_{{\\sigma }0}\\nonumber \\\\[5pt]\\hspace{28.45274pt}=\\sum _{\\sigma ^{\\prime }}{\\cal T}^{*}_{{\\sigma },0}C_{{\\sigma }{\\sigma }^{\\prime }}\\left[ {\\cal T}^{-1*}_{{\\sigma },0}F_{{\\sigma }1}^{\\rm lw},{\\cal T}^{-1*}_{{\\sigma }^{\\prime },0}F_{{\\sigma }^{\\prime } 0} \\right]\\nonumber \\\\[5pt]\\hspace{28.45274pt}+ \\sum _{\\sigma ^{\\prime }}\\frac{\\lambda _{\\sigma }}{\\lambda _{{\\sigma }^{\\prime }}}{\\cal T}^{*}_{{\\sigma },0}C_{{\\sigma }{\\sigma }^{\\prime }} \\left[ {\\cal T}^{-1*}_{{\\sigma },0}F_{{\\sigma }0},{\\cal T}^{-1*}_{{\\sigma }^{\\prime },0}F_{{\\sigma }^{\\prime } 1}^{\\rm lw}\\right].$ Denote by $R$ the cylindrical coordinate giving the distance to the axis of the torus, and by $\\hat{\\mbox{$\\zeta $}}$ the unit vector in the toroidal direction.", "The identities $B^2 = \\frac{I^2+|\\nabla _\\mathbf {R}\\psi |^2}{R^2},\\\\[5pt]\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}\\psi = I\\hat{\\mathbf {b}}- RB\\hat{\\mbox{$\\zeta $}},\\\\[5pt]\\nabla _\\mathbf {R}\\cdot \\hat{\\mbox{$\\zeta $}}= 0,\\\\[5pt][\\hat{\\mathbf {b}}\\times (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}})]\\cdot \\nabla _\\mathbf {R}\\psi =\\nonumber \\\\\\hspace{14.22636pt}\\left(\\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}\\right)\\cdot \\nabla _\\mathbf {R}\\psi =\\nabla _\\mathbf {R}\\cdot (I\\hat{\\mathbf {b}}) = \\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\left(\\frac{I}{B}\\right),\\\\[5pt]\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}\\varphi _0 =\\partial _\\psi \\varphi _0(I\\hat{\\mathbf {b}}-RB\\hat{\\mbox{$\\zeta $}}),$ and $(\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}B)\\cdot \\nabla _\\mathbf {R}\\psi = -I\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B$ are useful to write (REF ) as $\\left(u\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}- \\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B\\partial _u\\right) F_{{\\sigma }1}^{\\rm lw}\\nonumber \\\\[5pt]\\hspace{28.45274pt}\\left(u^2\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\left(\\frac{I}{B}\\right)-\\frac{I \\mu }{B}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B\\right)\\Bigg (\\Upsilon _{\\sigma }\\nonumber \\\\[5pt]\\hspace{28.45274pt}+\\frac{Z_{\\sigma }}{T_{\\sigma }}\\partial _\\psi \\varphi _0\\Bigg ) F_{{\\sigma }0}+\\frac{Z_{\\sigma }\\lambda _{\\sigma }u}{T_{\\sigma }}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\varphi _1^{\\rm lw}F_{{\\sigma }0}\\nonumber \\\\[5pt]\\hspace{28.45274pt}=\\sum _{\\sigma ^{\\prime }}{\\cal T}_{{\\sigma },0}^* C_{{\\sigma }{\\sigma }^{\\prime }}\\left[{\\cal T}^{-1*}_{{\\sigma },0}F_{{\\sigma }1}^{\\rm lw},{\\cal T}^{-1*}_{{\\sigma }^{\\prime },0}F_{{\\sigma }^{\\prime } 0}\\right]\\nonumber \\\\[5pt]\\hspace{28.45274pt}+\\sum _{\\sigma ^{\\prime }}\\frac{\\lambda _{\\sigma }}{\\lambda _{{\\sigma }^{\\prime }}}{\\cal T}_{{\\sigma },0}^* C_{{\\sigma }{\\sigma }^{\\prime }}\\left[{\\cal T}^{-1*}_{{\\sigma },0}F_{{\\sigma }0},{\\cal T}^{-1*}_{{\\sigma }^{\\prime },0}F_{{\\sigma }^{\\prime } 1}^{\\rm lw}\\right],$ where $\\Upsilon _{\\sigma }$ is defined in (REF ).", "Equation (REF ) is equivalent to $(u\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}- \\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B\\partial _u)\\Bigg [F_{{\\sigma }1}^{\\rm lw}\\nonumber \\\\[5pt]\\hspace{28.45274pt}+\\Bigg \\lbrace \\frac{Z_{\\sigma }\\lambda _{\\sigma }}{T_{\\sigma }}\\varphi _1^{\\rm lw}+\\frac{Iu}{B}\\left(\\frac{Z_{\\sigma }}{T_{\\sigma }}\\partial _\\psi \\varphi _0+\\Upsilon _{\\sigma }\\right)\\Bigg \\rbrace F_{{\\sigma }0}\\Bigg ]\\nonumber \\\\[5pt]\\hspace{28.45274pt}=\\sum _{\\sigma ^{\\prime }}{\\cal T}_{{\\sigma },0}^* C_{{\\sigma }{\\sigma }^{\\prime }}\\left[{\\cal T}^{-1*}_{{\\sigma },0}F_{{\\sigma }1}^{\\rm lw},{\\cal T}^{-1*}_{{\\sigma }^{\\prime },0}F_{{\\sigma }^{\\prime } 0}\\right]\\nonumber \\\\[5pt]\\hspace{28.45274pt}+\\sum _{\\sigma ^{\\prime }}\\frac{\\lambda _{\\sigma }}{\\lambda _{{\\sigma }^{\\prime }}}{\\cal T}_{{\\sigma },0}^* C_{{\\sigma }{\\sigma }^{\\prime }}\\left[{\\cal T}^{-1*}_{{\\sigma },0}F_{{\\sigma }0},{\\cal T}^{-1*}_{{\\sigma }^{\\prime },0}F_{{\\sigma }^{\\prime } 1}^{\\rm lw}\\right].$ The definition of the new function $G_{{\\sigma }1}^{\\rm lw}$ given in (REF ) seems appropriate.", "Employing that the collision operator vanishes when acting on Maxwellians with the same temperature, and using property (REF ), the dependence on $\\varphi _0$ and $\\varphi _1^{\\rm lw}$ is removed and equation (REF ) is obtained." ], [ "Computation of the turbulent piece of the collision operator", "We have to calculate $[{\\cal T}_{{\\sigma },1}^*C_{{\\sigma }{\\sigma }^{\\prime }}^{\\rm sw}]^{\\rm lw}$ appearing in (REF ) and this appendix is devoted to that end.", "Then, $\\left[ {\\cal T}_{{\\sigma },1}^*C_{{\\sigma }{\\sigma }^{\\prime }}^{\\rm sw}\\right]^{\\rm lw}=\\nonumber \\\\[5pt]\\hspace{14.22636pt}\\Bigg [\\frac{Z_{\\sigma }\\lambda _{\\sigma }}{B}\\Bigg (-\\frac{1}{B}\\left(\\hat{\\mathbf {b}}\\times \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\tilde{\\Phi }_{\\sigma }^{\\rm sw}\\right)\\cdot \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}-\\tilde{\\phi }_{{\\sigma }1}^{\\rm sw}\\partial _\\mu +\\partial _\\mu \\tilde{\\Phi }_{{\\sigma }1}^{\\rm sw}\\partial _\\theta \\Bigg )\\nonumber \\\\[5pt]\\hspace{14.22636pt}\\Bigg \\lbrace {\\cal T}_{NP, {\\sigma }}^* C_{\\sigma \\sigma ^\\prime }\\left[{\\mathbb {T}_{{\\sigma },0}}F_{\\sigma 1}^{\\rm sw}-\\frac{Z_{\\sigma }\\lambda _{\\sigma }}{T_{\\sigma }}{\\mathbb {T}_{{\\sigma },0}}\\tilde{\\phi }_{{\\sigma }1}^{\\rm sw}{\\cal T}_{{\\sigma },0}^{-1*} F_{{\\sigma }0}, {\\cal T}_{{\\sigma }^{\\prime },0}^{-1*}F_{{\\sigma }^{\\prime } 0}\\right]\\nonumber \\\\[5pt]\\hspace{14.22636pt}+ \\ \\frac{\\lambda _{\\sigma }}{\\lambda _{{\\sigma }^{\\prime }}}{\\cal T}_{NP, {\\sigma }}^*C_{\\sigma \\sigma ^\\prime } \\Bigg [ {\\cal T}_{{\\sigma },0}^{-1 *}F_{{\\sigma }0} ,{\\mathbb {T}_{{\\sigma }^{\\prime },0}}F_{\\sigma ^{\\prime } 1}^{\\rm sw}\\nonumber \\\\[5pt]\\hspace{14.22636pt}-\\frac{Z_{{\\sigma }^{\\prime }}\\lambda _{{\\sigma }^{\\prime }}}{T_{{\\sigma }^{\\prime }}}{\\mathbb {T}_{{\\sigma }^{\\prime },0}}\\tilde{\\phi }_{{\\sigma }^{\\prime }1}^{\\rm sw}{\\cal T}_{{\\sigma }^{\\prime },0}^{-1 *}F_{{\\sigma }^{\\prime } 0}\\Bigg ]\\Bigg \\rbrace \\Bigg ]^{\\rm lw}.$ But the first term on the right side of (REF ) does not contribute in the long-wavelength limit because, for any $g = O(1)$ , $\\left[\\left(\\hat{\\mathbf {b}}\\times \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\tilde{\\Phi }_{\\sigma }^{\\rm sw}\\right)\\cdot \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }} g^{\\rm sw}\\right]^{\\rm lw}=\\nonumber \\\\[5pt]\\hspace{28.45274pt}-\\hat{\\mathbf {b}}\\cdot \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\times \\left[g^{\\rm sw}\\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\tilde{\\Phi }_{\\sigma }^{\\rm sw}\\right]^{\\rm lw}=O(\\epsilon _s).$ Therefore, $\\left[ {\\cal T}_{{\\sigma },1}^*C_{{\\sigma }{\\sigma }^{\\prime }}^{\\rm sw}\\right]^{\\rm lw}=\\Bigg [\\frac{Z_{\\sigma }\\lambda _{\\sigma }}{B}\\Bigg (-\\tilde{\\phi }_{{\\sigma }1}^{\\rm sw}\\partial _\\mu +\\partial _\\mu \\tilde{\\Phi }_{{\\sigma }1}^{\\rm sw}\\partial _\\theta \\Bigg )\\nonumber \\\\[5pt]\\hspace{14.22636pt}\\Bigg \\lbrace {\\cal T}_{NP, {\\sigma }}^* C_{\\sigma \\sigma ^\\prime }\\left[{\\mathbb {T}_{{\\sigma },0}}F_{\\sigma 1}^{\\rm sw}-\\frac{Z_{\\sigma }\\lambda _{\\sigma }}{T_{\\sigma }}{\\mathbb {T}_{{\\sigma },0}}\\tilde{\\phi }_{{\\sigma }1}^{\\rm sw}{\\cal T}_{{\\sigma },0}^{-1*} F_{{\\sigma }0}, {\\cal T}_{{\\sigma }^{\\prime },0}^{-1*}F_{{\\sigma }^{\\prime } 0}\\right]\\nonumber \\\\[5pt]\\hspace{14.22636pt} + \\ \\frac{\\lambda _{\\sigma }}{\\lambda _{{\\sigma }^{\\prime }}}{\\cal T}_{NP, {\\sigma }}^*C_{\\sigma \\sigma ^\\prime } \\Bigg [ {\\cal T}_{{\\sigma },0}^{-1 *}F_{{\\sigma }0} ,{\\mathbb {T}_{{\\sigma }^{\\prime },0}}F_{\\sigma ^{\\prime } 1}^{\\rm sw}\\nonumber \\\\[5pt]\\hspace{14.22636pt}-\\frac{Z_{{\\sigma }^{\\prime }}\\lambda _{{\\sigma }^{\\prime }}}{T_{{\\sigma }^{\\prime }}}{\\mathbb {T}_{{\\sigma }^{\\prime },0}}\\tilde{\\phi }_{{\\sigma }^{\\prime }1}^{\\rm sw}{\\cal T}_{{\\sigma }^{\\prime },0}^{-1 *}F_{{\\sigma }^{\\prime } 0}\\Bigg ]\\Bigg \\rbrace \\Bigg ]^{\\rm lw}.$ As for its gyroaverage, $\\Big \\langle \\Big [ &{\\cal T}_{{\\sigma },1}^*C_{{\\sigma }{\\sigma }^{\\prime }}^{\\rm sw}\\Big ]^{\\rm lw}\\Big \\rangle =-\\partial _\\mu \\Bigg \\langle \\Bigg [\\frac{Z_{\\sigma }\\lambda _{\\sigma }}{B}\\tilde{\\phi }_{{\\sigma }1}^{\\rm sw}\\nonumber \\\\[5pt]&\\Bigg \\lbrace {\\cal T}_{NP, {\\sigma }}^* C_{\\sigma \\sigma ^\\prime }\\left[{\\mathbb {T}_{{\\sigma },0}}F_{\\sigma 1}^{\\rm sw}-\\frac{Z_{\\sigma }\\lambda _{\\sigma }}{T_{\\sigma }}{\\mathbb {T}_{{\\sigma },0}}\\tilde{\\phi }_{{\\sigma }1}^{\\rm sw}{\\cal T}_{{\\sigma },0}^{-1*} F_{{\\sigma }0}, {\\cal T}_{{\\sigma }^{\\prime },0}^{-1*}F_{{\\sigma }^{\\prime } 0}\\right]\\nonumber \\\\[5pt]& + \\ \\frac{\\lambda _{\\sigma }}{\\lambda _{{\\sigma }^{\\prime }}}{\\cal T}_{NP, {\\sigma }}^*C_{\\sigma \\sigma ^\\prime } \\Bigg [ {\\cal T}_{{\\sigma },0}^{-1 *}F_{{\\sigma }0} ,{\\mathbb {T}_{{\\sigma }^{\\prime },0}}F_{\\sigma ^{\\prime } 1}^{\\rm sw}\\nonumber \\\\[5pt]&-\\frac{Z_{{\\sigma }^{\\prime }}\\lambda _{{\\sigma }^{\\prime }}}{T_{{\\sigma }^{\\prime }}}{\\mathbb {T}_{{\\sigma }^{\\prime },0}}\\tilde{\\phi }_{{\\sigma }^{\\prime }1}^{\\rm sw}{\\cal T}_{{\\sigma }^{\\prime },0}^{-1 *}F_{{\\sigma }^{\\prime } 0}\\Bigg ]\\Bigg \\rbrace \\Bigg ]^{\\rm lw}\\Bigg \\rangle .$ In order to get the last expression we have integrated by parts in $\\theta $ and $\\mu $ ." ], [ "Computation of the last term of (", "with $[{\\cal T}_{\\sigma ,1}^{-1 *} F_{{\\sigma }1}^{\\rm lw}]^{\\rm lw}= -{\\cal T}_{{\\sigma },0}^{-1 *}\\Bigg \\lbrace \\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}\\nonumber \\\\[5pt]\\hspace{14.22636pt}+\\Bigg (u\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\cdot \\mbox{$\\rho $}+\\frac{B}{4}[\\mbox{$\\rho $}(\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}})+(\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}})\\mbox{$\\rho $}]:\\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\nonumber \\\\[5pt]\\hspace{14.22636pt}-\\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}\\Bigg )\\partial _u+\\Bigg (-\\frac{\\mu }{B}\\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}B\\nonumber \\\\[5pt]\\hspace{14.22636pt}-\\frac{u}{4}[\\mbox{$\\rho $}(\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}})+(\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}})\\mbox{$\\rho $}]:\\nabla _\\mathbf {R}\\hat{\\mathbf {b}}+\\frac{u\\mu }{B}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}\\nonumber \\\\[5pt]\\hspace{14.22636pt}- \\frac{u^2}{B}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\cdot \\mbox{$\\rho $}- \\frac{Z_{\\sigma }}{B}\\mbox{$\\rho $}\\cdot \\nabla \\varphi _0\\Bigg )\\partial _\\mu \\Bigg \\rbrace F_{{\\sigma }1}^{\\rm lw},$ $[{\\cal T}_{\\sigma ,1}^{-1 *} F_{\\sigma 1}^{\\rm sw}]^{\\rm lw}=- {\\cal T}_{{\\sigma },0}^{-1 *}\\Bigg [\\Bigg ( \\frac{Z_{\\sigma }\\lambda _{\\sigma }}{B^2}(\\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\widetilde{\\Phi }_{\\sigma 1}^{\\rm sw}\\times \\hat{\\mathbf {b}}) \\cdot \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\nonumber \\\\[5pt]\\hspace{14.22636pt}-\\frac{Z_{\\sigma }\\lambda _{\\sigma }\\widetilde{\\phi }_{{\\sigma }1}^{\\rm sw}}{B}\\frac{\\partial }{\\partial \\mu } \\Bigg ) F_{{\\sigma }1}^{\\rm sw}\\Bigg ]^{\\rm lw}.$ Here to obtain (REF ) and (REF ) we have used the results in .", "The term $[{\\cal T}_{\\sigma ,2}^{-1 *}F_{{\\sigma }0}]^{\\rm lw}$ is calculated in .", "Now, let us write the gyroaverage of (REF ): $\\Big \\langle {\\cal T}_{{\\sigma }, 0}^{*}C_{{\\sigma }{\\sigma }^{\\prime }}^{(2){\\rm lw}}\\Big \\rangle ={\\cal T}_{{\\sigma }, 0}^{*} C_{\\sigma \\sigma ^\\prime }\\Big [{\\cal T}_{{\\sigma }, 0}^{-1*}\\left\\langle F_{\\sigma 2}^{\\rm lw}\\right\\rangle +{\\cal T}_{{\\sigma }, 0}^{-1*}\\left\\langle {\\cal T}_{{\\sigma }, 0}^{*} [{\\cal T}_{\\sigma ,1}^{-1 *} F_{\\sigma 1}^{\\rm lw}]^{\\rm lw}\\right\\rangle \\nonumber \\\\\\hspace{14.22636pt}+{\\cal T}_{{\\sigma }, 0}^{-1*}\\left\\langle {\\cal T}_{{\\sigma }, 0}^{*}[{\\cal T}_{\\sigma ,2}^{-1 *} F_{{\\sigma }0}]^{\\rm lw}\\right\\rangle ,{\\cal T}^{-1*}_{{\\sigma }^{\\prime },0}F_{{\\sigma }^{\\prime } 0} \\Big ]+\\left(\\frac{\\lambda _{\\sigma }}{\\lambda _{{\\sigma }^{\\prime }}}\\right)^2{\\cal T}_{{\\sigma }, 0}^{*}C_{\\sigma \\sigma ^\\prime }\\Big [{\\cal T}^{-1*}_{{\\sigma },0}F_{{\\sigma }0},\\nonumber \\\\\\hspace{14.22636pt}{\\cal T}_{{\\sigma }^\\prime , 0}^{-1*}\\left\\langle F_{\\sigma ^{\\prime } 2}^{\\rm lw}\\right\\rangle +{\\cal T}_{{\\sigma }^\\prime , 0}^{-1*}\\left\\langle {\\cal T}_{{\\sigma }^\\prime , 0}^{*}[{\\cal T}_{\\sigma ^{\\prime },1}^{-1 *} F_{\\sigma ^{\\prime } 1}^{\\rm lw}]^{\\rm lw}\\right\\rangle +{\\cal T}_{{\\sigma }^\\prime , 0}^{-1*}\\left\\langle {\\cal T}_{{\\sigma }^\\prime , 0}^{*}[{\\cal T}_{\\sigma ^{\\prime },2}^{-1 *} F_{{\\sigma }^{\\prime } 0}]^{\\rm lw}\\right\\rangle \\Big ] \\nonumber \\\\\\hspace{14.22636pt}+\\frac{\\lambda _{\\sigma }}{\\lambda _{{\\sigma }^{\\prime }}}\\left\\langle {\\cal T}_{{\\sigma }, 0}^{*}C_{\\sigma \\sigma ^\\prime }\\left[{\\cal T}^{-1*}_{{\\sigma },0} F_{\\sigma 1}^{\\rm lw}+ [{\\cal T}_{\\sigma ,1}^{-1 *} F_{\\sigma 0}]^{\\rm lw},{\\cal T}^{-1*}_{{\\sigma }^{\\prime },0}F_{\\sigma ^\\prime 1}^{\\rm lw}+[{\\cal T}_{\\sigma ^{\\prime },1}^{-1 *} F_{\\sigma ^{\\prime } 0}]^{\\rm lw}\\right]\\right\\rangle \\nonumber \\\\\\hspace{14.22636pt}+\\frac{\\lambda _{\\sigma }}{\\lambda _{{\\sigma }^{\\prime }}}\\Bigg \\langle {\\cal T}_{{\\sigma }, 0}^{*} \\Bigg [C_{\\sigma \\sigma ^\\prime } \\Bigg [{\\mathbb {T}_{{\\sigma },0}}F_{{\\sigma }1}^{\\rm sw}-\\frac{Z_{\\sigma }\\lambda _{\\sigma }}{T_{\\sigma }}{\\mathbb {T}_{{\\sigma },0}}\\widetilde{\\phi }_{{\\sigma }1}^{\\rm sw}{\\cal T}_{{\\sigma },0}^{-1 *}F_{{\\sigma }0},{\\mathbb {T}_{{\\sigma }^{\\prime },0}}F_{{\\sigma }^{\\prime } 1}^{\\rm sw}\\nonumber \\\\\\hspace{14.22636pt}-\\frac{Z_{{\\sigma }^{\\prime }}\\lambda _{{\\sigma }^{\\prime }}}{T_{{\\sigma }^{\\prime }}}{\\mathbb {T}_{{\\sigma }^{\\prime },0}}\\widetilde{\\phi }_{{\\sigma }^{\\prime } 1}^{\\rm sw}{\\cal T}_{{\\sigma }^{\\prime },0}^{-1 *}F_{{\\sigma }^{\\prime } 0}\\Bigg ] \\Bigg ]^{\\rm lw}\\Bigg \\rangle ,$ where we have used that $\\left\\langle {\\cal T}_{{\\sigma }, 0}^{*}[{\\cal T}_{\\sigma ,1}^{-1 *} F_{\\sigma 1}^{\\rm sw}]^{\\rm lw}\\right\\rangle = 0$ .", "Here, $\\left\\langle {\\cal T}_{{\\sigma }, 0}^{*}[{\\cal T}_{\\sigma ,1}^{-1 *} F_{{\\sigma }1}^{\\rm lw}]^{\\rm lw}\\right\\rangle =\\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}\\left(\\partial _u-\\frac{u}{B}\\partial _\\mu \\right)F_{{\\sigma }1}^{\\rm lw},$ and $\\left\\langle {\\cal T}_{{\\sigma }, 0}^{*} [{\\cal T}_{\\sigma ,2}^{-1 *} F_{{\\sigma }0}]^{\\rm lw}\\right\\rangle =\\nonumber \\\\\\hspace{28.45274pt} \\frac{\\mu }{2B} (\\leftrightarrow \\over {\\mathbf {I}}- \\hat{\\mathbf {b}}\\hat{\\mathbf {b}}) : \\Bigg [\\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\ln n_{\\sigma }+ \\left( \\frac{u^2/2 + \\mu B}{T_{\\sigma }} -\\frac{3}{2} \\right) \\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\ln T_{\\sigma }\\Bigg ] F_{{\\sigma }0}\\nonumber \\\\\\hspace{28.45274pt}- \\frac{\\mu }{B} \\frac{Z_{\\sigma }}{T_{\\sigma }^2}\\nabla _\\mathbf {R}\\varphi _0 \\cdot \\nabla _\\mathbf {R}T_{\\sigma }F_{{\\sigma }0} -\\frac{\\mu }{2B} \\frac{u^2/2 + \\mu B}{T_{\\sigma }^3} |\\nabla _\\mathbf {R}T_{\\sigma }|^2F_{{\\sigma }0} \\nonumber \\\\\\hspace{28.45274pt}+ \\frac{\\mu }{2B} \\Bigg |\\frac{\\nabla _\\mathbf {R}n_{\\sigma }}{n_{\\sigma }} + \\frac{Z_{\\sigma }\\nabla _\\mathbf {R}\\varphi _0}{T_{\\sigma }} + \\left( \\frac{u^2/2 + \\mu B}{T_{\\sigma }} - \\frac{3}{2}\\right) \\frac{\\nabla _\\mathbf {R}T_{\\sigma }}{T_{\\sigma }} \\Bigg |^2 F_{{\\sigma }0}\\nonumber \\\\\\hspace{28.45274pt}- \\frac{\\mu }{2B^2} \\nabla _{\\mathbf {R}_\\bot } B \\cdot \\left(\\frac{\\nabla _\\mathbf {R}n_{\\sigma }}{n_{\\sigma }} + \\frac{Z_{\\sigma }\\nabla _\\mathbf {R}\\varphi _0}{T_{\\sigma }} + \\left( \\frac{u^2/2 + \\mu B}{T_{\\sigma }} - \\frac{3}{2}\\right) \\frac{\\nabla _\\mathbf {R}T_{\\sigma }}{T_{\\sigma }} \\right) F_{{\\sigma }0}\\nonumber \\\\\\hspace{28.45274pt}+ \\frac{Z_{\\sigma }^2\\lambda _{\\sigma }^2}{2T_{\\sigma }^2}\\left[ \\left\\langle (\\widetilde{\\phi }_{{\\sigma }1}^{\\rm sw})^2 \\right\\rangle \\right]^{\\rm lw}F_{{\\sigma }0} + \\frac{1}{T_{\\sigma }} \\Bigg [ - \\frac{Z_{\\sigma }^2}{2B^2}|\\nabla _\\mathbf {R}\\varphi _0|^2 \\nonumber \\\\\\hspace{28.45274pt}- \\frac{Z_{\\sigma }^2\\lambda _{\\sigma }^2}{2B}\\partial _\\mu \\left[ \\left\\langle (\\widetilde{\\phi }_{{\\sigma }1}^{\\rm sw})^2 \\right\\rangle \\right]^{\\rm lw}- \\frac{3 Z_{\\sigma }\\mu }{2 B^2} \\nabla _{\\mathbf {R}_\\bot } B \\cdot \\nabla _\\mathbf {R}\\varphi _0\\nonumber \\\\\\hspace{28.45274pt}- \\frac{Z_{\\sigma }u^2}{B^2} \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\varphi _0 + \\Psi _{B, {\\sigma }}\\nonumber \\\\\\hspace{28.45274pt}+\\frac{Z_{\\sigma }\\mu }{B} (\\leftrightarrow \\over {\\mathbf {I}}- \\hat{\\mathbf {b}}\\hat{\\mathbf {b}}): \\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\varphi _0 \\Bigg ] F_{{\\sigma }0}.$ This last result has been obtained by gyroaveraging (REF )." ], [ "Second-order inverse transformation of a Maxwellian", "The calculation of $C_{{\\sigma }{\\sigma }^{\\prime }}^{(2){\\rm lw}}$ in requires $[\\mathcal {T}^{-1\\ast }_{{\\sigma }, 2} F_{{\\sigma }0}]^{\\rm lw}$ .", "We start by using that $F_{{\\sigma }0}$ is a Maxwellian that depends on $\\mathbf {R}$ and $u^2/2 + \\mu B(\\mathbf {R})$ , giving $\\left[ \\mathcal {T}^{-1\\ast }_{{\\sigma }, 2} F_{{\\sigma }0} \\right]^{\\rm lw}=\\nonumber \\\\\\hspace{14.22636pt}\\frac{1}{2B^2} (\\mathbf {v}\\times \\hat{\\mathbf {b}}) (\\mathbf {v}\\times \\hat{\\mathbf {b}}) : \\Bigg [\\nabla _\\mathbf {r}\\nabla _\\mathbf {r}\\ln n_{\\sigma }+ \\left( \\frac{v^2}{2T_{\\sigma }} -\\frac{3}{2} \\right) \\nabla _\\mathbf {r}\\nabla _\\mathbf {r}\\ln T_{\\sigma }\\nonumber \\\\\\hspace{14.22636pt}-\\frac{v^2}{2T_{\\sigma }^3} \\nabla _\\mathbf {r}T_{\\sigma }\\nabla _\\mathbf {r}T_{\\sigma }+\\Bigg ( \\frac{\\nabla _\\mathbf {r}n_{\\sigma }}{n_{\\sigma }} + \\left( \\frac{v^2}{2T_{\\sigma }}- \\frac{3}{2} \\right) \\frac{\\nabla _\\mathbf {r}T_{\\sigma }}{T_{\\sigma }} \\Bigg ) \\Bigg ( \\frac{\\nabla _\\mathbf {r}n_{\\sigma }}{n_{\\sigma }}\\nonumber \\\\\\hspace{14.22636pt}+ \\left( \\frac{v^2}{2T_{\\sigma }} -\\frac{3}{2} \\right) \\frac{\\nabla _\\mathbf {r}T_{\\sigma }}{T_{\\sigma }} \\Bigg ) \\Bigg ]{\\cal T}_{{\\sigma },0}^{-1*}F_{{\\sigma }0}\\nonumber \\\\\\hspace{14.22636pt}+ {\\mathbf {R}}_{02}^{\\rm lw}\\cdot \\left( \\frac{\\nabla _\\mathbf {r}n_{\\sigma }}{n_{\\sigma }} + \\left( \\frac{v^2}{2T_{\\sigma }} - \\frac{3}{2} \\right) \\frac{\\nabla _\\mathbf {r}T_{\\sigma }}{T_{\\sigma }} \\right) {\\cal T}_{{\\sigma },0}^{-1*}F_{{\\sigma }0}\\nonumber \\\\\\hspace{14.22636pt}-\\frac{1}{B} H_{01}^{\\rm lw}(\\mathbf {v}\\times \\hat{\\mathbf {b}}) \\cdot \\Bigg (\\frac{\\nabla _\\mathbf {r}n_{\\sigma }}{n_{\\sigma }} + \\left( \\frac{v^2}{2T_{\\sigma }} -\\frac{5}{2} \\right) \\frac{\\nabla _\\mathbf {r}T_{\\sigma }}{T_{\\sigma }} \\Bigg )\\frac{{\\cal T}_{{\\sigma },0}^{-1*} F_{{\\sigma }0}}{T_{\\sigma }}\\nonumber \\\\\\hspace{14.22636pt}+ \\frac{1}{2} \\left[ H_{01}^2\\right]^{\\rm lw}\\frac{{\\cal T}_{{\\sigma },0}^{-1*}F_{{\\sigma }0}}{T_{\\sigma }^2} - H_{02}^{\\rm lw}\\frac{{\\cal T}_{{\\sigma },0}^{-1*}F_{{\\sigma }0}}{T_{\\sigma }},$ where the functions $\\mathbf {R}_{02}$ , $H_{01}$ and $H_{02}$ are given by $\\mathbf {R}= \\mathbf {r}+ \\frac{\\epsilon _{\\sigma }}{B} \\mathbf {v}\\times \\hat{\\mathbf {b}}+\\epsilon _{\\sigma }^2 \\mathbf {R}_{02} + O(\\epsilon _{\\sigma }^3)$ and $\\frac{u^2}{2} + \\mu B(\\mathbf {R}) = \\frac{v^2}{2} + \\epsilon _{\\sigma }H_{01}+ \\epsilon _{\\sigma }^2 H_{02}+ O(\\epsilon _{\\sigma }^3).$ In what follows we calculate $\\mathbf {R}_{02}$ , $H_{01}$ and $H_{02}$ .", "To compute $\\mathbf {R}_{02}$ we use that $\\mathbf {r}= \\mathbf {R}_g + \\epsilon _{\\sigma }\\mbox{$\\rho $}(\\mathbf {R}_g, \\mu _g, \\theta _g).$ Employing the results in (REF ) it is easy to see that $\\mathbf {r}= \\mathbf {R}_g - \\frac{\\epsilon _{\\sigma }}{B} \\mathbf {v}\\times \\hat{\\mathbf {b}}+\\epsilon _{\\sigma }^2 \\mathcal {T}_{{\\sigma }, 0}^{-1\\ast } \\Bigg [ - \\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}\\mbox{$\\rho $}\\nonumber \\\\[5pt]\\hspace{28.45274pt}+ \\left(\\frac{\\mu }{B}\\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}B + u(\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}})\\mbox{$\\rho $}:\\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\right) \\partial _\\mu \\mbox{$\\rho $}\\nonumber \\\\[5pt]\\hspace{28.45274pt}- \\left(\\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {e}}_2 \\cdot \\hat{\\mathbf {e}}_1 -\\frac{u}{2\\mu }\\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\cdot \\mbox{$\\rho $}\\right)\\partial _\\theta \\mbox{$\\rho $}\\Bigg ] + O(\\epsilon _{\\sigma }^3).$ Using $\\nabla _\\mathbf {R}\\mbox{$\\rho $}= - (2B)^{-1} \\nabla _\\mathbf {R}B \\mbox{$\\rho $}-(\\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\cdot \\mbox{$\\rho $}) \\hat{\\mathbf {b}}+ (\\nabla _\\mathbf {R}\\hat{\\mathbf {e}}_2 \\cdot \\hat{\\mathbf {e}}_1) \\mbox{$\\rho $}\\times \\hat{\\mathbf {b}}$ , $\\partial _\\mu \\mbox{$\\rho $}= (2\\mu )^{-1}\\mbox{$\\rho $}$ and $\\partial _\\theta \\mbox{$\\rho $}= - \\mbox{$\\rho $}\\times \\hat{\\mathbf {b}}$ , we obtain $\\mathbf {r}= \\mathbf {R}_g - \\frac{\\epsilon _{\\sigma }}{B} \\mathbf {v}\\times \\hat{\\mathbf {b}}+\\frac{\\epsilon _{\\sigma }^2}{B^2} \\Bigg [ \\frac{1}{B} (\\mathbf {v}\\times \\hat{\\mathbf {b}})(\\mathbf {v}\\times \\hat{\\mathbf {b}}) \\cdot \\nabla _\\mathbf {r}B \\nonumber \\\\\\hspace{28.45274pt}+ (\\mathbf {v}\\times \\hat{\\mathbf {b}}) \\cdot \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}\\cdot (\\mathbf {v}\\times \\hat{\\mathbf {b}}) \\hat{\\mathbf {b}}+ v_{||} (\\mathbf {v}\\times \\hat{\\mathbf {b}}) \\cdot \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}\\times \\hat{\\mathbf {b}}\\Bigg ]\\nonumber \\\\\\hspace{28.45274pt}+O(\\epsilon _{\\sigma }^3).$ Finally, since $\\mathbf {R}_g = \\mathbf {R}+ \\epsilon _{\\sigma }^2 \\mathbf {R}_2 +O(\\epsilon _{\\sigma }^3)$ with $\\mathbf {R}_2$ given in (REF ), we obtain $\\mathbf {R}_{02} = \\frac{1}{B} \\Bigg [ \\left( v_{||} \\hat{\\mathbf {b}}+\\frac{1}{4} \\mathbf {v}_\\bot \\right) \\mathbf {v}\\times \\hat{\\mathbf {b}}\\nonumber \\\\\\hspace{28.45274pt}+ \\mathbf {v}\\times \\hat{\\mathbf {b}}\\left( v_{||} \\hat{\\mathbf {b}}+ \\frac{1}{4} \\mathbf {v}_\\bot \\right) \\Bigg ] {\\scriptstyle {{_{\\displaystyle \\cdot }}\\atop \\times }}$ r( bB )         + v||B2 vrb+ v||B2 bbrbv         + b8B2 [ vv- (vb) (vb) ]: rb         +Z B2 bT,0(R/)         + v22B3 bbrB - v24B3 r B, where ${\\mathbf {a}}{\\mathbf {b}} $$$ M= a(bM)$.", "The long-wavelength component is\\begin{eqnarray}\\mathbf {R}_{02}^{\\rm lw}= \\frac{1}{B} \\Bigg [ \\left( v_{||} \\hat{\\mathbf {b}}+\\frac{1}{4} \\mathbf {v}_\\bot \\right) \\mathbf {v}\\times \\hat{\\mathbf {b}}\\nonumber \\\\\\hspace{28.45274pt}+ \\mathbf {v}\\times \\hat{\\mathbf {b}}\\left( v_{||} \\hat{\\mathbf {b}}+ \\frac{1}{4} \\mathbf {v}_\\bot \\right) \\Bigg ] {\\scriptstyle {{_{\\displaystyle \\cdot }}\\atop \\times }}\\end{eqnarray} \\nabla _\\mathbf {r}\\left( \\frac{\\hat{\\mathbf {b}}}{B} \\right)\\nonumber \\\\\\hspace{28.45274pt}+\\frac{v_{||}}{B^2} \\mathbf {v}_\\bot \\cdot \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}+ \\frac{v_{||}}{B^2}\\hat{\\mathbf {b}}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}\\cdot \\mathbf {v}_\\bot \\nonumber \\\\\\hspace{28.45274pt}+\\frac{\\hat{\\mathbf {b}}}{8B^2} [ \\mathbf {v}_\\bot \\mathbf {v}_\\bot - (\\mathbf {v}_\\bot \\times \\hat{\\mathbf {b}})(\\mathbf {v}_\\bot \\times \\hat{\\mathbf {b}}) ]: \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}\\nonumber \\\\\\hspace{28.45274pt}+\\frac{v_\\bot ^2}{2B^3} \\hat{\\mathbf {b}}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}B -\\frac{v_\\bot ^2}{4B^3} \\nabla _{\\mathbf {r}_\\bot } B.$ To obtain $H_{01}$ and $H_{02}^{\\rm lw}$ , we use that the expressions of the Hamiltonian in the two different sets of variables are related (with some abuse of notation) by $\\frac{v^2}{2} + Z_{\\sigma }\\lambda _{\\sigma }\\epsilon _{\\sigma }\\varphi (\\mathbf {r},\\mathbf {v},t)=\\nonumber \\\\[5pt]\\hspace{28.45274pt}\\frac{u^2}{2} + \\mu B(\\mathbf {R})+ Z_{\\sigma }\\lambda _{\\sigma }\\epsilon _{\\sigma }\\langle \\phi _{\\sigma }\\rangle (\\mathbf {R},\\mu ,t)\\nonumber \\\\[5pt]\\hspace{28.45274pt}+ Z_{\\sigma }^2\\lambda _{\\sigma }^2\\epsilon _{\\sigma }^2 \\Psi _{\\phi ,{\\sigma }} + Z_{\\sigma }\\lambda _{\\sigma }\\epsilon _{\\sigma }^2\\Psi _{\\phi B,{\\sigma }} + \\epsilon _{\\sigma }^2 \\Psi _{B,{\\sigma }}\\nonumber \\\\[5pt]\\hspace{28.45274pt}-\\frac{Z_{\\sigma }\\lambda _{\\sigma }\\epsilon _{\\sigma }^2}{B}\\partial _t \\widetilde{\\Phi }_{\\sigma }+ O(\\epsilon _{\\sigma }^3).$ Let us give a more detailed explanation of the last equation.", "As shown in reference [16], and to the order of interest, the Hamiltonian in gyrokinetic coordinates, $\\overline{H}_{\\sigma }$ , is the Hamiltonian in cartesian coordinates, $H^{\\mathbf {X}}_{\\sigma }$ , after a change of coordinates and the addition of the partial derivative with respect to time of a gauge function.", "This function is $-S_{NP,{\\sigma }}-\\epsilon _{\\sigma }^2 S_{P,{\\sigma }}^{(2)}$ , where $S_{NP,{\\sigma }}$ (which does not depend on time) and $S_{P,{\\sigma }}^{(2)}$ are given in equations (81) and (108) of reference [16], respectively.", "As a result, $\\overline{H}_{\\sigma }={\\cal T}_{\\sigma }^* H_{\\sigma }^{\\mathbf {X}} -\\epsilon _{\\sigma }^2 \\partial _t S_{P,{\\sigma }}^{(2)}+ O(\\epsilon _{\\sigma }^3),$ This is the origin of the last term in (REF ).", "The function $\\langle \\phi _{\\sigma }\\rangle (\\mathbf {R}, \\mu , t)$ is $\\langle \\phi _{\\sigma }\\rangle (\\mathbf {R}, \\mu , t)= \\langle \\phi _{\\sigma }\\rangle (\\mathbf {R}_g, \\mu _g, t)\\nonumber \\\\[5pt]\\hspace{14.22636pt}- \\epsilon _{\\sigma }\\left( \\mathbf {R}_2 \\cdot \\nabla _{\\mathbf {R}_{g\\perp }/\\epsilon _{\\sigma }} \\langle \\phi _{\\sigma }\\rangle +\\mu _1 \\partial _{\\mu _g} \\langle \\phi _{\\sigma }\\rangle \\right)\\nonumber \\\\[5pt]\\hspace{14.22636pt}+ O(\\epsilon _{\\sigma }^2).$ Here it is worth distinguishing between long-wavelength and short-wavelength pieces.", "For the long-wavelength potential, $\\langle \\phi _{\\sigma }^{\\rm lw}\\rangle (\\mathbf {R}_g, \\mu _g, t) =\\frac{1}{\\epsilon _{\\sigma }\\lambda _{\\sigma }} \\varphi _0(\\mathbf {R}_g, t) +\\varphi _1^{\\rm lw}(\\mathbf {R}_g, t)\\nonumber \\\\[5pt]\\hspace{14.22636pt}+ \\frac{\\epsilon _{\\sigma }\\mu _g}{2 \\lambda _{\\sigma }B (\\mathbf {R}_g)} \\left(\\leftrightarrow \\over {\\mathbf {I}}- \\hat{\\mathbf {b}}(\\mathbf {R}_g) \\hat{\\mathbf {b}}(\\mathbf {R}_g)\\right):\\nabla _{\\mathbf {R}_g} \\nabla _{\\mathbf {R}_g}\\varphi _0(\\mathbf {R}_g, t) + O(\\epsilon _{\\sigma }^2).$ This has to be written in $(\\mathbf {r}, \\mathbf {v})$ variables.", "Employing the results in (REF ) gives $\\langle \\phi _{\\sigma }^{\\rm lw}\\rangle (\\mathbf {R}_g, \\mu _g, t) =\\frac{1}{\\epsilon _{\\sigma }\\lambda _{\\sigma }} \\varphi _0 +\\varphi _1^{\\rm lw}+ \\frac{1}{\\lambda _{\\sigma }B} (\\mathbf {v}\\times \\hat{\\mathbf {b}}) \\cdot \\nabla _\\mathbf {r}\\varphi _0\\nonumber \\\\[5pt]\\hspace{14.22636pt}+ \\frac{\\epsilon _{\\sigma }}{B} (\\mathbf {v}\\times \\hat{\\mathbf {b}}) \\cdot \\nabla _\\mathbf {r}\\varphi _1^{\\rm lw}\\nonumber \\\\[5pt]\\hspace{14.22636pt}-\\frac{1}{\\lambda _{\\sigma }} \\mathcal {T}_{{\\sigma }, 0}^{-1\\ast }\\Bigg [ - \\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}\\mbox{$\\rho $}+ \\left(\\frac{\\mu }{B}\\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}B + u(\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}})\\mbox{$\\rho $}:\\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\right) \\partial _\\mu \\mbox{$\\rho $}\\nonumber \\\\[5pt]\\hspace{14.22636pt}- \\left(\\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {e}}_2 \\cdot \\hat{\\mathbf {e}}_1 -\\frac{u}{2\\mu }\\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\cdot \\mbox{$\\rho $}\\right)\\partial _\\theta \\mbox{$\\rho $}\\Bigg ] \\cdot \\nabla _\\mathbf {r}\\varphi _0\\nonumber \\\\[5pt]\\hspace{14.22636pt}+\\frac{\\epsilon _{\\sigma }}{2\\lambda _{\\sigma }B^2} \\left[ (\\mathbf {v}\\times \\hat{\\mathbf {b}})(\\mathbf {v}\\times \\hat{\\mathbf {b}}) + \\frac{v_\\bot ^2}{2} (\\leftrightarrow \\over {\\mathbf {I}}- \\hat{\\mathbf {b}}\\hat{\\mathbf {b}}) \\right]: \\nabla _\\mathbf {r}\\nabla _\\mathbf {r}\\varphi _0 +O(\\epsilon _{\\sigma }^2),$ where on the right-hand side everything is evaluated at $\\mathbf {r}$ .", "With this result, we find that to lowest order $\\frac{u^2}{2} + \\mu B(\\mathbf {R}) &= \\frac{v^2}{2} -\\frac{Z_{\\sigma }\\epsilon _{\\sigma }}{B}(\\mathbf {v}\\times \\hat{\\mathbf {b}}(\\mathbf {r})) \\cdot \\nabla _\\mathbf {r}\\varphi _0(\\mathbf {r},t) \\nonumber \\\\[5pt]&+ Z_{\\sigma }\\lambda _{\\sigma }\\epsilon _{\\sigma }{\\mathbb {T}_{{\\sigma },0}}\\widetilde{\\phi }_{{\\sigma }1}^{\\rm sw}(\\mathbf {r},\\mathbf {v},t) + O(\\epsilon _{\\sigma }^2),$ giving $H_{01}^{\\rm lw}(\\mathbf {r},\\mathbf {v},t) = - \\frac{Z_{\\sigma }}{B}(\\mathbf {v}\\times \\hat{\\mathbf {b}}(\\mathbf {r})) \\cdot \\nabla _\\mathbf {r}\\varphi _0(\\mathbf {r},t)$ and $\\left[ H_{01}^2 \\right]^{\\rm lw}(\\mathbf {r},\\mathbf {v},t)&= \\frac{Z_{\\sigma }^2}{B^2}\\left[ (\\mathbf {v}\\times \\hat{\\mathbf {b}}(\\mathbf {r})) \\cdot \\nabla _\\mathbf {r}\\varphi _0(\\mathbf {r},t) \\right]^2\\nonumber \\\\[5pt]& +Z_{\\sigma }^2\\lambda _{\\sigma }^2\\mathcal {T}^{-1\\ast }_{{\\sigma },0}\\left[(\\widetilde{\\phi }_{{\\sigma }1}^{\\rm sw})^2 \\right]^{\\rm lw}(\\mathbf {r},\\mathbf {v},t).$ Going to higher order, we find $H_{02}^{\\rm lw}(\\mathbf {r},\\mathbf {v},t)= - \\frac{Z_{\\sigma }\\lambda _{\\sigma }}{B} (\\mathbf {v}\\times \\hat{\\mathbf {b}}) \\cdot \\nabla _\\mathbf {r}\\varphi _1^{\\rm lw}-Z_{\\sigma }\\mathbf {R}_{02}^{\\rm lw}\\cdot \\nabla _\\mathbf {r}\\varphi _0\\nonumber \\\\\\hspace{14.22636pt}- Z_{\\sigma }^2\\lambda _{\\sigma }^2 \\mathcal {T}^{-1\\ast }_{{\\sigma },0}\\Psi _{\\phi ,{\\sigma }}^{\\rm lw}- Z_{\\sigma }\\lambda _{\\sigma }\\mathcal {T}^{-1\\ast }_{{\\sigma },0}\\Psi _{\\phi B,{\\sigma }}^{\\rm lw}\\nonumber \\\\\\hspace{14.22636pt}- \\mathcal {T}^{-1\\ast }_{{\\sigma },0}\\Psi _{B,{\\sigma }} -\\frac{Z_{\\sigma }}{2B^2} \\left[ (\\mathbf {v}\\times \\hat{\\mathbf {b}}) (\\mathbf {v}\\times \\hat{\\mathbf {b}}) + \\frac{v_\\bot ^2}{2} (\\leftrightarrow \\over {\\mathbf {I}}- \\hat{\\mathbf {b}}\\hat{\\mathbf {b}}) \\right]: \\nabla _\\mathbf {r}\\nabla _\\mathbf {r}\\varphi _0\\nonumber \\\\\\hspace{14.22636pt}+\\frac{Z_{\\sigma }^2\\lambda _{\\sigma }^2}{B}\\mathcal {T}^{-1\\ast }_{{\\sigma },0} \\left[\\left((\\nabla _{\\mathbf {R}_\\bot /\\epsilon _{\\sigma }} {\\widetilde{\\Phi }_{\\sigma }^{\\rm sw}}) \\times \\hat{\\mathbf {b}}\\right) \\cdot \\nabla _{\\mathbf {R}_\\bot /\\epsilon _{\\sigma }}\\langle {\\phi _{\\sigma }^{\\rm sw}} \\rangle \\right]^{\\rm lw}\\nonumber \\\\\\hspace{14.22636pt}-\\frac{Z_{\\sigma }^2\\lambda _{\\sigma }^2}{B}\\mathcal {T}^{-1\\ast }_{{\\sigma },0} \\left[{\\widetilde{\\phi }_{{\\sigma }1}^{\\rm sw}}\\partial _\\mu \\langle {\\phi _{{\\sigma }1}^{\\rm sw}} \\rangle \\right]^{\\rm lw}.$ Note that $[ (\\nabla _{(\\mathbf {R}_\\bot /\\epsilon _{\\sigma })}\\widetilde{\\Phi }_{\\sigma }^{\\rm sw}\\times \\hat{\\mathbf {b}}) \\cdot \\nabla _{\\mathbf {R}_\\bot /\\epsilon _{\\sigma }}\\langle \\phi _{\\sigma }^{\\rm sw}\\rangle ]^{\\rm lw}=O(\\epsilon _{\\sigma })$ and can be neglected, giving $H_{02}^{\\rm lw}(\\mathbf {r},\\mathbf {v},t)= - \\frac{Z_{\\sigma }\\lambda _{\\sigma }}{B} (\\mathbf {v}\\times \\hat{\\mathbf {b}}) \\cdot \\nabla _\\mathbf {r}\\varphi _1^{\\rm lw}-{Z_{\\sigma }} \\mathbf {R}_{02}^{\\rm lw}\\cdot \\nabla _\\mathbf {r}\\varphi _0\\nonumber \\\\[5pt]\\hspace{14.22636pt}- Z_{\\sigma }^2\\lambda _{\\sigma }^2 \\mathcal {T}^{-1\\ast }_{{\\sigma },0}\\Psi _{\\phi ,{\\sigma }}^{{\\rm lw}} - Z_{\\sigma }\\lambda _{\\sigma }\\mathcal {T}^{-1\\ast }_{{\\sigma },0}\\Psi _{\\phi B,{\\sigma }}^{{\\rm lw}}- \\mathcal {T}^{-1\\ast }_{{\\sigma },0}\\Psi _{B,{\\sigma }}\\nonumber \\\\[5pt]\\hspace{14.22636pt}-{\\frac{Z_{\\sigma }}{2B^2}} \\left[ (\\mathbf {v}\\times \\hat{\\mathbf {b}}) (\\mathbf {v}\\times \\hat{\\mathbf {b}}) + \\frac{v_\\bot ^2}{2} (\\leftrightarrow \\over {\\mathbf {I}}- \\hat{\\mathbf {b}}\\hat{\\mathbf {b}}) \\right]: \\nabla _\\mathbf {r}\\nabla _\\mathbf {r}\\varphi _0\\nonumber \\\\[5pt]\\hspace{14.22636pt}-\\frac{Z_{\\sigma }^2\\lambda _{\\sigma }^2}{B}\\mathcal {T}^{-1\\ast }_{{\\sigma },0}\\left[{\\widetilde{\\phi }_{{\\sigma }1}^{\\rm sw}}\\partial _\\mu \\langle {\\phi _{{\\sigma }1}^{\\rm sw}}\\rangle \\right]^{\\rm lw}.$ Combining all these results we obtain $ \\left[ \\mathcal {T}^{-1\\ast }_{{\\sigma }, 2} F_{{\\sigma }0} \\right]^{\\rm lw}= \\frac{1}{2B^2} (\\mathbf {v}\\times \\hat{\\mathbf {b}}) (\\mathbf {v}\\times \\hat{\\mathbf {b}}) : \\Bigg [\\nabla _\\mathbf {r}\\nabla _\\mathbf {r}\\ln n_{\\sigma }+{\\frac{Z_{\\sigma }}{T_{\\sigma }}} \\nabla _\\mathbf {r}\\nabla _\\mathbf {r}\\varphi _0\\nonumber \\\\\\hspace{14.22636pt}-{\\frac{Z_{\\sigma }}{T_{\\sigma }^2}} ( \\nabla _\\mathbf {r}\\varphi _0 \\nabla _\\mathbf {r}T_{\\sigma }+ \\nabla _\\mathbf {r}T_{\\sigma }\\nabla _\\mathbf {r}\\varphi _0 ) + \\left( \\frac{v^2}{2T_{\\sigma }} - \\frac{3}{2} \\right)\\nabla _\\mathbf {r}\\nabla _\\mathbf {r}\\ln T_{\\sigma }\\nonumber \\\\\\hspace{14.22636pt} - \\frac{v^2}{2T_{\\sigma }^3}\\nabla _\\mathbf {r}T_{\\sigma }\\nabla _\\mathbf {r}T_{\\sigma }\\Bigg ] {\\cal T}_{{\\sigma },0}^{-1*}F_{{\\sigma }0}+ \\frac{1}{2B^2} \\Bigg [ (\\mathbf {v}\\times \\hat{\\mathbf {b}}) \\cdot \\Bigg ( \\frac{\\nabla _\\mathbf {r}n_{\\sigma }}{n_{\\sigma }}\\nonumber \\\\\\hspace{14.22636pt} +{\\frac{Z_{\\sigma }\\nabla _\\mathbf {r}\\varphi _0}{T_{\\sigma }}}+ \\left(\\frac{v^2}{2T_{\\sigma }} - \\frac{3}{2} \\right) \\frac{\\nabla _\\mathbf {r}T_{\\sigma }}{T_{\\sigma }} \\Bigg ) \\Bigg ]^2 {\\cal T}_{{\\sigma },0}^{-1*}F_{{\\sigma }0} \\nonumber \\\\\\hspace{14.22636pt}+\\mathbf {R}_{02}^{\\rm lw}\\cdot \\left( \\frac{\\nabla _\\mathbf {r}n_{\\sigma }}{n_{\\sigma }} +{\\frac{Z_{\\sigma }\\nabla _\\mathbf {r}\\varphi _0}{T_{\\sigma }}}+ \\left(\\frac{v^2}{2T_{\\sigma }} - \\frac{3}{2} \\right) \\frac{\\nabla _\\mathbf {r}T_{\\sigma }}{T_{\\sigma }} \\right) {\\cal T}_{{\\sigma },0}^{-1*}F_{{\\sigma }0}\\nonumber \\\\\\hspace{14.22636pt}+\\frac{Z_{\\sigma }^2\\lambda _{\\sigma }^2}{2T_{\\sigma }^2}\\mathcal {T}^{-1\\ast }_{{\\sigma },0}\\left[(\\widetilde{\\phi }_{{\\sigma }1}^{\\rm sw})^2 \\right]^{\\rm lw}F_{{\\sigma }0}+ \\frac{1}{T_{\\sigma }} \\Bigg [ \\frac{Z_{\\sigma }\\lambda _{\\sigma }}{B} (\\mathbf {v}\\times \\hat{\\mathbf {b}}) \\cdot \\nabla _\\mathbf {r}\\varphi _1^{\\rm lw}\\nonumber \\\\\\hspace{14.22636pt}+ Z_{\\sigma }^2\\lambda _{\\sigma }^2\\mathcal {T}^{-1\\ast }_{{\\sigma },0} \\Psi _{\\phi ,{\\sigma }}^{\\rm lw}+Z_{\\sigma }\\lambda _{\\sigma }\\mathcal {T}^{-1\\ast }_{{\\sigma },0} \\Psi _{\\phi B,{\\sigma }}^{{\\rm lw}}\\nonumber \\\\\\hspace{14.22636pt}+ \\mathcal {T}^{-1\\ast }_{{\\sigma },0} \\Psi _{B,{\\sigma }} +{\\frac{Z_{\\sigma }v_\\bot ^2}{4B^2} }(\\leftrightarrow \\over {\\mathbf {I}}- \\hat{\\mathbf {b}}\\hat{\\mathbf {b}}): \\nabla _\\mathbf {r}\\nabla _\\mathbf {r}\\varphi _0\\nonumber \\\\\\hspace{14.22636pt}+ \\frac{Z_{\\sigma }^2\\lambda _{\\sigma }^2}{B}\\mathcal {T}^{-1\\ast }_{{\\sigma },0}\\left[{\\widetilde{\\phi }_{{\\sigma }1}^{\\rm sw}}\\partial _\\mu \\langle {\\phi _{{\\sigma }1}^{\\rm sw}}\\rangle \\right]^{\\rm lw}\\Bigg ] {\\cal T}_{{\\sigma },0}^{-1*}F_{{\\sigma }0}.$" ], [ "Calculations for the Fokker-Planck equation to\n$O(\\epsilon _{\\sigma }^2)$", "Start with (REF ).", "Employing the definition of $G_{\\sigma 1}$ in (REF ), $\\partial _\\zeta F_{{\\sigma }1}^{\\rm lw}\\equiv 0$ , and using the identities $(\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}\\varphi _0) \\cdot \\nabla _\\mathbf {R}G_{{\\sigma }1}^{\\rm lw}= \\partial _\\psi \\varphi _0 I \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}G_{{\\sigma }1}^{\\rm lw},$ $(\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}B) \\cdot \\nabla _\\mathbf {R}G_{{\\sigma }1}^{\\rm lw}=\\partial _\\psi B I \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}G_{{\\sigma }1}^{\\rm lw}\\nonumber \\\\[5pt]\\hspace{28.45274pt}- \\partial _\\psi G_{{\\sigma }1}^{\\rm lw}I\\, \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B,$ $[\\hat{\\mathbf {b}}\\times (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}) ] \\cdot \\nabla _\\mathbf {R}G_{{\\sigma }1}^{\\rm lw}=\\nonumber \\\\[5pt]\\hspace{28.45274pt}(\\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}) \\cdot \\nabla _\\mathbf {R}G_{{\\sigma }1}^{\\rm lw}- (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}) \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}G_{{\\sigma }1}^{\\rm lw}=\\nonumber \\\\[5pt]\\hspace{28.45274pt}\\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\left(\\frac{I}{B}\\partial _\\psi G_{{\\sigma }1}^{\\rm lw}\\right) \\nonumber \\\\[5pt]\\hspace{28.45274pt} - \\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta \\partial _\\psi \\left( \\frac{1}{\\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta } I \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}G_{{\\sigma }1}^{\\rm lw}\\right)\\nonumber \\\\[5pt]\\hspace{28.45274pt} -(\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}) \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}G_{{\\sigma }1}^{\\rm lw},$ $\\hat{\\mathbf {b}}\\times (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}) \\cdot \\left( \\mu \\nabla _\\mathbf {R}B + Z_{\\sigma }\\nabla _{\\mathbf {R}_\\perp } \\varphi _0\\right) =\\nonumber \\\\[5pt]\\hspace{28.45274pt}\\nabla _\\mathbf {R}\\cdot \\left( \\mu \\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}B +Z_{\\sigma }\\hat{\\mathbf {b}}\\times \\nabla _{\\mathbf {R}_\\perp } \\varphi _0\\right) \\nonumber \\\\\\hspace{28.45274pt}- \\mu (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}})\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B\\nonumber \\\\\\hspace{28.45274pt}= \\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\left[ \\frac{I}{B}\\left( \\mu \\partial _\\psi B +Z_{\\sigma }\\partial _\\psi \\varphi _0 \\right)\\right] \\nonumber \\\\\\hspace{28.45274pt}- \\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta \\partial _\\psi \\left( \\frac{I}{\\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta } \\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B \\right)\\nonumber \\\\\\hspace{28.45274pt}- \\mu (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}) \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B,$ $Z_{\\sigma }\\lambda _{\\sigma }\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\varphi _1^{\\rm lw}\\partial _u\\left(\\frac{Z_{\\sigma }\\lambda _{\\sigma }\\varphi _1^{\\rm lw}}{T_{\\sigma }}F_{{\\sigma }0}\\right)=\\nonumber \\\\\\hspace{28.45274pt}-\\left(u\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}-\\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B\\partial _u\\right)\\left[\\frac{1}{2}\\left(\\frac{Z_{\\sigma }\\lambda _{\\sigma }\\varphi _1^{\\rm lw}}{T_{\\sigma }}\\right)^2F_{{\\sigma }0}\\right],$ $\\frac{1}{B}\\Bigg [ - Z_{\\sigma }\\nabla _\\mathbf {R}\\varphi _0 \\times \\hat{\\mathbf {b}}+\\mu \\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}B\\nonumber \\\\\\hspace{28.45274pt}+u^2 \\hat{\\mathbf {b}}\\times (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}})\\Bigg ] \\cdot \\nabla _\\mathbf {R}\\left(\\frac{Z_{\\sigma }\\lambda _{\\sigma }\\varphi _1^{\\rm lw}}{T_{\\sigma }}F_{ \\sigma 0}\\right)\\nonumber \\\\\\hspace{28.45274pt}-Z_{\\sigma }\\lambda _{\\sigma }\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\varphi _1^{\\rm lw}\\partial _u\\Bigg [\\frac{Iu}{B}\\Bigg (\\frac{Z_{\\sigma }}{ T_\\sigma }\\partial _\\psi \\varphi _0 + \\Upsilon _{\\sigma }\\Bigg )F_{{\\sigma }0}\\Bigg ]\\nonumber \\\\\\hspace{28.45274pt}-\\frac{u}{B}\\left[\\hat{\\mathbf {b}}\\times \\left(\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\right)\\right]\\cdot \\big (\\mu \\nabla _\\mathbf {R}B\\nonumber \\\\\\hspace{28.45274pt}+Z_{\\sigma }\\nabla _\\mathbf {R}\\varphi _0\\big )\\partial _u\\left(\\frac{Z_{\\sigma }\\lambda _{\\sigma }\\varphi _1^{\\rm lw}}{T_{\\sigma }}F_{ \\sigma 0}\\right) =\\nonumber \\\\\\hspace{28.45274pt}-\\frac{Z_{\\sigma }\\lambda _{\\sigma }I}{B}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\varphi _1^{\\rm lw}\\Upsilon _{\\sigma }F_{{\\sigma }0}\\nonumber \\\\\\hspace{28.45274pt}+\\frac{Z_{\\sigma }\\lambda _{\\sigma }}{BT_{\\sigma }}F_{{\\sigma }0}\\big [\\mu \\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}B+u^2 \\hat{\\mathbf {b}}\\times (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}})\\big ]\\cdot \\nabla _\\mathbf {R}\\varphi _1^{\\rm lw}\\nonumber \\\\\\hspace{28.45274pt}+\\left(u\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}-\\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B\\partial _u\\right)\\Bigg [\\frac{Z_{\\sigma }\\lambda _{\\sigma }I u }{T_{\\sigma }B}\\varphi _1^{\\rm lw}F_{ \\sigma 0}\\Bigg (\\frac{Z_{\\sigma }}{ T_\\sigma }\\partial _\\psi \\varphi _0\\nonumber \\\\\\hspace{28.45274pt}+ \\Upsilon _{\\sigma }-\\frac{\\partial _\\psi T_{\\sigma }}{T_{\\sigma }}\\Bigg )\\Bigg ],$ $\\Bigg [ - \\frac{Z_{\\sigma }}{B}\\nabla _{\\mathbf {R}} \\varphi _0 \\times \\hat{\\mathbf {b}}+\\frac{\\mu }{B} \\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}B\\nonumber \\\\\\hspace{28.45274pt}+\\frac{u^2}{B} \\hat{\\mathbf {b}}\\times (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}})\\Bigg ]\\cdot \\nabla _\\mathbf {R}\\left( \\frac{Iu}{B} \\right)\\nonumber \\\\\\hspace{28.45274pt}- \\frac{u}{B} [\\hat{\\mathbf {b}}\\times (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}})] \\cdot \\left( \\mu \\nabla _\\mathbf {R}B+ Z_{\\sigma }\\nabla _{\\mathbf {R}} \\varphi _0 \\right)\\partial _u \\left( \\frac{Iu}{B}\\right) \\nonumber \\\\\\hspace{28.45274pt}=- \\frac{u}{B} (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}) \\left[ u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\left(\\frac{Iu}{B} \\right) - \\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B\\partial _u \\left( \\frac{Iu}{B}\\right) \\right],$ and $\\frac{1}{B}\\Bigg [ - Z_{\\sigma }\\nabla _\\mathbf {R}\\varphi _0 \\times \\hat{\\mathbf {b}}+\\mu \\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}B\\nonumber \\\\\\hspace{28.45274pt}+u^2 \\hat{\\mathbf {b}}\\times (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}})\\Bigg ] \\cdot \\nabla _\\mathbf {R}\\left(\\Bigg (\\frac{Z_{\\sigma }}{ T_\\sigma }\\partial _\\psi \\varphi _0 + \\Upsilon _{\\sigma }\\Bigg )F_{ \\sigma 0}\\right)\\nonumber \\\\\\hspace{28.45274pt}-\\frac{u}{B}\\left[\\hat{\\mathbf {b}}\\times \\left(\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\right)\\right]\\cdot \\big (\\mu \\nabla _\\mathbf {R}B\\nonumber \\\\\\hspace{28.45274pt}+Z_{\\sigma }\\nabla _\\mathbf {R}\\varphi _0\\big )\\partial _u\\left(\\Bigg (\\frac{Z_{\\sigma }}{ T_\\sigma }\\partial _\\psi \\varphi _0 + \\Upsilon _{\\sigma }\\Bigg )F_{ \\sigma 0}\\right)=\\nonumber \\\\\\hspace{28.45274pt}\\left[ u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\left(\\frac{Iu}{B} \\right) - \\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B\\partial _u \\left( \\frac{Iu}{B}\\right) \\right]\\times \\nonumber \\\\\\hspace{28.45274pt}\\Bigg [\\partial _\\psi \\left(\\frac{Z_{\\sigma }}{T_{\\sigma }}\\partial _\\psi \\varphi _0\\right)-\\frac{Z_{\\sigma }}{T_{\\sigma }^2}\\partial _\\psi \\varphi _0\\partial _\\psi T_{\\sigma }+\\partial _\\psi \\Upsilon _{\\sigma }\\nonumber \\\\\\hspace{28.45274pt}-\\frac{\\mu }{T_{\\sigma }^2}\\partial _\\psi B\\partial _\\psi T_{\\sigma }+\\Bigg (\\frac{Z_{\\sigma }}{ T_\\sigma }\\partial _\\psi \\varphi _0 + \\Upsilon _{\\sigma }\\Bigg )^2\\Bigg ]F_{{\\sigma }0},$ we obtain $\\frac{1}{B}\\Bigg [ - Z_{\\sigma }\\nabla _\\mathbf {R}\\varphi _0 \\times \\hat{\\mathbf {b}}+\\mu \\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}B\\nonumber \\\\\\hspace{28.45274pt}+u^2 \\hat{\\mathbf {b}}\\times (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}})\\Bigg ] \\cdot \\nabla _\\mathbf {R}F_{ \\sigma 1}^{\\rm lw}\\nonumber \\\\\\hspace{28.45274pt}- \\Bigg \\lbrace Z_{\\sigma }\\lambda _{\\sigma }\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\varphi _1^{\\rm lw}+\\frac{u}{B} [\\hat{\\mathbf {b}}\\times (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}})]\\nonumber \\\\\\hspace{28.45274pt}\\cdot \\left( \\mu \\nabla _\\mathbf {R}B + Z_{\\sigma }\\nabla _\\mathbf {R}\\varphi _0 \\right) \\Bigg \\rbrace \\partial _uF_{\\sigma 1}^{\\rm lw}\\nonumber \\\\\\hspace{28.45274pt}= \\frac{1}{B}\\left( Z_{\\sigma }\\partial _\\psi \\varphi _0 + \\mu \\partial _\\psi B\\right) I \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}G_{\\sigma 1}^{\\rm lw}\\nonumber \\\\\\hspace{28.45274pt}-\\frac{I\\mu }{B} \\partial _\\psi G_{\\sigma 1}^{\\rm lw}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B \\nonumber \\\\\\hspace{28.45274pt} - \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\Theta \\partial _\\psi \\left( \\frac{I u^2}{\\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta } \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}G_{\\sigma 1}^{\\rm lw}\\right)\\nonumber \\\\\\hspace{28.45274pt}+ u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\left(\\frac{I u}{B} \\partial _\\psi G_{\\sigma 1}^{\\rm lw}\\right)\\nonumber \\\\\\hspace{28.45274pt}- Z_{\\sigma }\\lambda _{\\sigma }\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\varphi _1^{\\rm lw}\\partial _u G_{\\sigma 1}^{\\rm lw}\\nonumber \\\\\\hspace{28.45274pt}- u\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\left[ \\frac{I}{B} \\left( \\mu \\partial _\\psi B + Z_{\\sigma }\\partial _\\psi \\varphi _0 \\right) \\right]\\partial _u G_{\\sigma 1}^{\\rm lw}\\nonumber \\\\\\hspace{28.45274pt}+ \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\Theta \\partial _\\psi \\left(\\frac{Iu\\mu }{\\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta }\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B \\right) \\partial _uG_{\\sigma 1}^{\\rm lw}\\nonumber \\\\\\hspace{28.45274pt}- \\frac{u}{B}(\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}) \\left( u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}- \\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B \\partial _u \\right)\\Bigg \\lbrace G_{\\sigma 1}^{\\rm lw}\\nonumber \\\\\\hspace{28.45274pt}- \\frac{Iu}{B}F_{\\sigma 0} \\left(\\frac{Z_{\\sigma }}{ T_\\sigma }\\partial _\\psi \\varphi _0 + \\Upsilon _{\\sigma }\\right)\\Bigg \\rbrace \\nonumber \\\\\\hspace{28.45274pt}+\\frac{Z_{\\sigma }\\lambda _{\\sigma }I}{B}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\varphi _1^{\\rm lw}F_{\\sigma 0}\\Upsilon _{\\sigma }\\nonumber \\\\\\hspace{28.45274pt}-\\frac{Z_{\\sigma }\\lambda _{\\sigma }}{T_{\\sigma }B} F_{{\\sigma }0}\\left[\\mu \\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}B +u^2 \\hat{\\mathbf {b}}\\times (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}})\\right]\\cdot \\nabla _\\mathbf {R}\\varphi _1^{\\rm lw}\\nonumber \\\\\\hspace{28.45274pt}- \\left( u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}- \\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B\\partial _u \\right) \\Bigg \\lbrace \\frac{1}{2} \\left(\\frac{Z_{\\sigma }\\lambda _{\\sigma }\\varphi _1^{\\rm lw}}{T_{\\sigma }} \\right)^2 F_{\\sigma 0}\\nonumber \\\\\\hspace{28.45274pt}+\\frac{Z_{\\sigma }\\lambda _{\\sigma }Iu}{T_{\\sigma }B}\\varphi _1^{\\rm lw}F_{{\\sigma }0}\\Bigg [ \\frac{Z_{\\sigma }}{T_\\sigma }\\partial _\\psi \\varphi _0+\\Upsilon _{\\sigma }-\\frac{1}{T_\\sigma } \\partial _\\psi T_\\sigma \\Bigg ]\\nonumber \\\\\\hspace{28.45274pt}+\\frac{1}{2} \\left( \\frac{Iu}{B} \\right)^2F_{{\\sigma }0}\\Bigg [\\partial _\\psi \\Upsilon _{\\sigma }-\\frac{\\mu \\partial _\\psi B}{T_{\\sigma }}\\partial _\\psi \\ln T_{\\sigma }\\nonumber \\\\\\hspace{28.45274pt}+ \\partial _\\psi \\left(\\frac{Z_{\\sigma }}{T_{\\sigma }} \\partial _\\psi \\varphi _0 \\right)-\\frac{Z_{\\sigma }}{T_\\sigma ^2}\\partial _\\psi \\varphi _0\\partial _\\psi T_\\sigma \\nonumber \\\\\\hspace{28.45274pt}+\\Bigg (\\frac{Z_{\\sigma }}{T_{\\sigma }} \\partial _\\psi \\varphi _0 +\\Upsilon _{\\sigma }\\Bigg )^2\\Bigg ]\\Bigg \\rbrace .$ To simplify this expression we use $\\frac{1}{B} \\left( Z_{\\sigma }\\partial _\\psi \\varphi _0 + \\mu \\partial _\\psi B \\right) I \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}G_{\\sigma 1}^{\\rm lw}=\\nonumber \\\\[5pt]\\hspace{28.45274pt}\\partial _u \\left[ \\frac{I}{B}\\left(Z_{\\sigma }\\partial _\\psi \\varphi _0 + \\mu \\partial _\\psi B\\right) u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}G_{\\sigma 1}^{\\rm lw}\\right] \\nonumber \\\\[5pt]\\hspace{28.45274pt}- \\frac{Iu}{B}\\left( Z_{\\sigma }\\partial _\\psi \\varphi _0 + \\mu \\partial _\\psi B \\right)\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\left(\\partial _u G_{\\sigma 1}^{\\rm lw}\\right),$ $u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\left( \\frac{I u}{B}\\partial _\\psi G_{\\sigma 1}^{\\rm lw}\\right) -\\frac{I}{B} \\partial _\\psi G_{\\sigma 1}^{\\rm lw}\\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B =\\nonumber \\\\[5pt]\\hspace{28.45274pt}\\left( u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}-\\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B \\partial _u \\right)\\left( \\frac{I u}{B} \\partial _\\psi G_{\\sigma 1}^{\\rm lw}\\right)\\nonumber \\\\\\hspace{28.45274pt}+ \\frac{Iu}{B} \\partial _{\\psi }\\partial _\\mu G_{\\sigma 1}^{\\rm lw}\\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B,$ $- Z_{\\sigma }\\lambda _{\\sigma }\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\varphi _1^{\\rm lw}\\partial _u G_{\\sigma 1}^{\\rm lw}=\\nonumber \\\\[5pt]\\hspace{28.45274pt}- \\left( u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}- \\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B \\partial _u \\right) \\left( \\frac{Z_{\\sigma }\\lambda _{\\sigma }\\varphi _1^{\\rm lw}}{u}\\partial _u G_{\\sigma 1}^{\\rm lw}\\right) \\nonumber \\\\\\hspace{28.45274pt} +\\partial _u\\left[\\frac{Z_{\\sigma }\\lambda _{\\sigma }\\varphi _1^{\\rm lw}}{u}\\left( u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}G_{\\sigma 1}^{\\rm lw}- \\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B \\partial _u G_{\\sigma 1}^{\\rm lw}\\right) \\right],$ $- u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\left[ \\frac{I}{B}\\left( \\mu \\partial _\\psi B+ Z_{\\sigma }\\partial _\\psi \\varphi _0 \\right)\\right] \\partial _u G_{\\sigma 1}^{\\rm lw}=\\nonumber \\\\\\hspace{28.45274pt}-\\left( u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}- \\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B\\partial _u \\right) \\left[\\frac{I}{B} \\left( \\mu \\partial _\\psi B + Z_{\\sigma }\\partial _\\psi \\varphi _0 \\right) \\partial _uG_{\\sigma 1}^{\\rm lw}\\right] \\nonumber \\\\\\hspace{28.45274pt}-\\partial _u \\left[ \\frac{I}{B} \\left( \\mu \\partial _\\psi B+Z_{\\sigma }\\partial _\\psi \\varphi _0 \\right) \\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B \\partial _u G_{\\sigma 1}^{\\rm lw}\\right] \\nonumber \\\\\\hspace{28.45274pt}+ \\frac{I u}{B} \\left( \\mu \\partial _\\psi B+ Z_{\\sigma }\\partial _\\psi \\varphi _0 \\right) \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\left( \\partial _u G_{\\sigma 1}^{\\rm lw}\\right),$ $\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\Theta \\partial _\\psi \\left( \\frac{Iu\\mu }{\\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta }\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B \\right)\\partial _u G_{\\sigma 1}^{\\rm lw}=\\nonumber \\\\\\hspace{28.45274pt}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\Theta \\partial _\\psi \\left( \\frac{Iu\\mu }{\\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta }\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B \\partial _u G_{\\sigma 1}^{\\rm lw}\\right)\\nonumber \\\\\\hspace{28.45274pt}- \\frac{Iu\\mu }{B} \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B\\partial _\\psi \\partial _u G_{\\sigma 1}^{\\rm lw},$ and $\\frac{Z_{\\sigma }\\lambda _{\\sigma }I}{B}\\hat{\\mathbf {b}}& \\cdot \\nabla _\\mathbf {R}\\varphi _1^{\\rm lw}F_{{\\sigma }0}\\Upsilon _{\\sigma }\\nonumber \\\\& -\\frac{Z_{\\sigma }\\lambda _{\\sigma }}{T_\\sigma B} F_{{\\sigma }0}\\left[\\mu \\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}B +u^2 \\hat{\\mathbf {b}}\\times (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}})\\right] \\cdot \\nabla _\\mathbf {R}\\varphi _1^{\\rm lw}\\nonumber \\\\&= \\frac{Z_{\\sigma }\\lambda _{\\sigma }}{B} (\\nabla _\\mathbf {R}\\varphi _1^{\\rm lw}\\times \\hat{\\mathbf {b}}) \\cdot \\nabla _\\mathbf {R}F_{\\sigma 0}\\nonumber \\\\&+ \\frac{Z_{\\sigma }\\lambda _{\\sigma }u}{B} [\\hat{\\mathbf {b}}\\times (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}})] \\cdot \\nabla _\\mathbf {R}\\varphi _1^{\\rm lw}\\partial _uF_{\\sigma 0}.$ Employing these results in (REF ) gives $\\frac{1}{B}\\Bigg [ - Z_{\\sigma }\\nabla _\\mathbf {R}\\varphi _0 \\times \\hat{\\mathbf {b}}+\\mu \\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}B\\nonumber \\\\\\hspace{14.22636pt}+u^2 \\hat{\\mathbf {b}}\\times (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}})\\Bigg ] \\cdot \\nabla _\\mathbf {R}F_{ \\sigma 1}^{\\rm lw}\\nonumber \\\\\\hspace{14.22636pt}- \\Bigg \\lbrace Z_{\\sigma }\\lambda _{\\sigma }\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\varphi _1^{\\rm lw}+\\frac{u}{B} [\\hat{\\mathbf {b}}\\times (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}})]\\nonumber \\\\\\hspace{14.22636pt}\\cdot \\left( \\mu \\nabla _\\mathbf {R}B + Z_{\\sigma }\\nabla _\\mathbf {R}\\varphi _0 \\right) \\Bigg \\rbrace \\partial _uF_{\\sigma 1}^{\\rm lw}\\nonumber \\\\\\hspace{14.22636pt}= -\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\Theta \\partial _\\psi \\Bigg [\\frac{Iu}{\\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta }\\left( u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}- \\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B\\partial _u \\right)G_{{\\sigma }1}^{\\rm lw}\\Bigg ]\\nonumber \\\\\\hspace{14.22636pt}+\\partial _u\\Bigg \\lbrace \\Bigg [\\frac{I}{B}(Z_{\\sigma }\\partial _\\psi \\varphi _0 + \\mu \\partial _\\psi B)+\\frac{Z_{\\sigma }\\lambda _{\\sigma }}{u}\\varphi _1^{\\rm lw}\\Bigg ]\\nonumber \\\\\\hspace{14.22636pt}\\times \\left( u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}- \\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B\\partial _u \\right)G_{{\\sigma }1}^{\\rm lw}\\Bigg \\rbrace \\nonumber \\\\\\hspace{14.22636pt}-\\frac{u}{B}\\left(\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}\\right)\\left( u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}- \\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B\\partial _u \\right)\\Bigg [G_{{\\sigma }1}^{\\rm lw}\\nonumber \\\\\\hspace{14.22636pt}-\\frac{Iu}{B}\\left(\\frac{Z_{\\sigma }}{T_{\\sigma }}\\partial _\\psi \\varphi _0 + \\Upsilon _{\\sigma }\\right)F_{{\\sigma }0}\\Bigg ]\\nonumber \\\\\\hspace{14.22636pt}+\\left( u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}- \\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B\\partial _u \\right)\\Bigg \\lbrace \\frac{Iu}{B}\\partial _\\psi G_{{\\sigma }1}^{\\rm lw}-\\frac{Z_{\\sigma }\\lambda _{\\sigma }}{u}\\varphi _1^{\\rm lw}\\partial _uG_{{\\sigma }1}^{\\rm lw}\\nonumber \\\\\\hspace{14.22636pt}-\\frac{I}{B}(Z_{\\sigma }\\partial _\\psi \\varphi _0 + \\mu \\partial _\\psi B)\\partial _uG_{{\\sigma }1}^{\\rm lw}-\\frac{1}{2}\\left(\\frac{Z_{\\sigma }\\lambda _{\\sigma }\\varphi _1^{\\rm lw}}{T_{\\sigma }}\\right)^2F_{{\\sigma }0}\\nonumber \\\\\\hspace{14.22636pt}-\\frac{Z_{\\sigma }\\lambda _{\\sigma }Iu}{T_{\\sigma }B}\\varphi _1^{\\rm lw}F_{{\\sigma }0}\\Bigg (\\frac{Z_{\\sigma }}{T_{\\sigma }}\\partial _\\psi \\varphi _0 +\\Upsilon _{\\sigma }-\\partial _\\psi \\ln T_{\\sigma }\\Bigg )\\nonumber \\\\\\hspace{14.22636pt}-\\frac{1}{2}\\left(\\frac{Iu}{B}\\right)^2\\Bigg [\\partial _\\psi \\Upsilon _{\\sigma }-\\frac{\\mu \\partial _\\psi B}{T_{\\sigma }}\\partial _\\psi \\ln T_{\\sigma }+\\partial _\\psi \\left(\\frac{Z_{\\sigma }}{T_{\\sigma }}\\partial _\\psi \\varphi _0\\right)\\nonumber \\\\\\hspace{14.22636pt}-\\frac{Z_{\\sigma }}{T_{\\sigma }^2}\\partial _\\psi \\varphi _0\\partial _\\psi T_{\\sigma }+\\Bigg (\\frac{Z_{\\sigma }}{T_{\\sigma }}\\partial _\\psi \\varphi _0 +\\Upsilon _{\\sigma }\\Bigg )^2\\Bigg ]\\Bigg \\rbrace \\nonumber \\\\\\hspace{14.22636pt}+\\frac{Z_{\\sigma }\\lambda _{\\sigma }}{B}\\left(\\nabla _\\mathbf {R}\\varphi _1^{\\rm lw}\\times \\hat{\\mathbf {b}}\\right)\\cdot \\nabla _\\mathbf {R}F_{{\\sigma }0}\\nonumber \\\\\\hspace{14.22636pt}+\\frac{Z_{\\sigma }\\lambda _{\\sigma }u}{B}\\left[\\hat{\\mathbf {b}}\\times \\left(\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\right)\\right]\\cdot \\nabla _\\mathbf {R}\\varphi _1^{\\rm lw}\\partial _u F_{{\\sigma }0}.$ We also manipulate the terms containing $\\mathbf {K}$ (defined by equation (REF )) in (REF ).", "First, note that $- \\frac{u \\mu }{B}&(\\nabla _\\mathbf {R}\\times \\mathbf {K})_\\bot \\cdot \\nabla _\\mathbf {R}F_{\\sigma 0}\\nonumber \\\\&+\\frac{\\mu }{B} (\\nabla _\\mathbf {R}\\times \\mathbf {K})_\\bot \\cdot \\left( \\mu \\nabla _\\mathbf {R}B+ Z_{\\sigma }\\nabla _\\mathbf {R}\\varphi _0 \\right) \\partial _uF_{\\sigma 0}=\\nonumber \\\\&-\\frac{u\\mu }{B}\\left(\\frac{Z_{\\sigma }}{T_\\sigma } \\partial _\\psi \\varphi _0+\\Upsilon _{\\sigma }\\right)(\\nabla _\\mathbf {R}\\times \\mathbf {K})\\cdot \\nabla _\\mathbf {R}\\psi F_{{\\sigma }0}.$ Using one finds $(\\nabla _\\mathbf {R}\\times \\mathbf {K}) \\cdot \\nabla _\\mathbf {R}\\psi = \\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Bigg ( \\frac{I}{2B} \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}\\nonumber \\\\\\hspace{28.45274pt}+ \\frac{R}{|\\nabla _\\mathbf {R}\\psi |^2} \\hat{\\mbox{$\\zeta $}}\\cdot \\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\psi \\cdot (\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}\\psi ) \\Bigg ),$ so $- \\frac{u \\mu }{B}(\\nabla _\\mathbf {R}\\times \\mathbf {K})_\\bot \\cdot \\nabla _\\mathbf {R}F_{\\sigma 0}\\nonumber \\\\\\hspace{28.45274pt}+\\frac{\\mu }{B} (\\nabla _\\mathbf {R}\\times \\mathbf {K})_\\bot \\cdot \\left( \\mu \\nabla _\\mathbf {R}B+ Z_{\\sigma }\\nabla _\\mathbf {R}\\varphi _0 \\right) \\partial _uF_{\\sigma 0}=\\nonumber \\\\\\hspace{28.45274pt} - \\left( u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}- \\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B \\partial _u \\right)\\Bigg \\lbrace {\\mu F_{\\sigma 0}}\\nonumber \\\\\\hspace{28.45274pt}\\times \\Bigg (\\frac{Z_{\\sigma }}{T_{\\sigma }}\\partial _\\psi \\varphi _0+\\Upsilon _{\\sigma }\\Bigg )\\Bigg ( \\frac{I}{2B} \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}\\nonumber \\\\\\hspace{28.45274pt}+ \\frac{R}{|\\nabla _\\mathbf {R}\\psi |^2} \\hat{\\mbox{$\\zeta $}}\\cdot \\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\psi \\cdot (\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}\\psi ) \\Bigg )\\Bigg \\rbrace .$ Hence, employing (REF ) and (REF ), and reorganizing, the second-order Fokker-Planck equation, (REF ), becomes $ -B\\partial _\\theta F_{{\\sigma }3}^{\\rm lw}+ \\frac{\\lambda _{\\sigma }^2}{\\tau _{\\sigma }}\\partial _{\\epsilon _s^2 t} F_{{\\sigma }0} +\\Bigg ( u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}- \\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B \\partial _u \\Bigg )\\Bigg \\lbrace F_{\\sigma 2}^{\\rm lw}\\nonumber \\\\\\hspace{14.22636pt}+\\left[\\frac{Z_{\\sigma }\\lambda _{\\sigma }^2}{T_\\sigma }\\varphi _2^{\\rm lw}+ \\frac{Z_{\\sigma }}{T_\\sigma }\\frac{\\mu }{2B}(\\leftrightarrow \\over {\\mathbf {I}}- \\hat{\\mathbf {b}}\\hat{\\mathbf {b}}) : \\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\varphi _0\\right] F_{{\\sigma }0}\\nonumber \\\\\\hspace{14.22636pt}+\\frac{1}{T_{\\sigma }}\\left(\\Psi _{B,{\\sigma }} + Z_{\\sigma }\\lambda _{\\sigma }\\Psi _{\\phi B,{\\sigma }}^{\\rm lw}+ Z_{\\sigma }^2\\lambda _{\\sigma }^2\\Psi _{\\phi ,{\\sigma }}^{\\rm lw}\\right)F_{{\\sigma }0}\\nonumber \\\\\\hspace{14.22636pt}+\\frac{Iu}{B}\\partial _\\psi G_{\\sigma 1}^{\\rm lw}-\\frac{Z_{\\sigma }\\lambda _{\\sigma }\\varphi _1^{\\rm lw}}{u} \\partial _uG_{\\sigma 1}^{\\rm lw}\\nonumber \\\\\\hspace{14.22636pt} -\\frac{I}{B} \\left( \\mu \\partial _\\psi B+ Z_{\\sigma }\\partial _\\psi \\varphi _0 \\right)\\partial _u G_{\\sigma 1}^{\\rm lw}- \\frac{1}{2}\\left(\\frac{Z_{\\sigma }\\lambda _{\\sigma }\\varphi _1^{\\rm lw}}{T_\\sigma }\\right)^2F_{\\sigma 0}\\nonumber \\\\\\hspace{14.22636pt}- \\frac{Z_{\\sigma }\\lambda _{\\sigma }I u}{T_{\\sigma }B}\\varphi _1^{\\rm lw}F_{{\\sigma }0} \\Bigg [\\frac{Z_{\\sigma }}{T_\\sigma }\\partial _\\psi \\varphi _0 +\\Upsilon _{\\sigma }-\\frac{1}{T_\\sigma } \\partial _\\psi T_\\sigma \\Bigg ] \\nonumber \\\\\\hspace{14.22636pt} - \\frac{1}{2}\\left( \\frac{Iu}{B} \\right)^2 F_{{\\sigma }0} \\Bigg [\\partial _\\psi \\Upsilon _{\\sigma }-\\frac{\\mu \\partial _\\psi B}{T_{\\sigma }}\\partial _\\psi \\ln T_{\\sigma }\\nonumber \\\\\\hspace{14.22636pt}+\\partial _\\psi \\left(\\frac{Z_{\\sigma }}{T_\\sigma }\\partial _\\psi \\varphi _0\\right) - \\frac{Z_{\\sigma }}{ T_\\sigma ^2}\\partial _\\psi \\varphi _0\\partial _\\psi T_\\sigma \\nonumber \\\\ \\hspace{14.22636pt} + \\Bigg (\\frac{Z_{\\sigma }}{T_\\sigma }\\partial _\\psi \\varphi _0 + \\Upsilon _{\\sigma }\\Bigg )^2 \\Bigg ]- {\\mu F_{{\\sigma }0}}\\Bigg ( \\frac{I}{2B} \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}\\nonumber \\\\\\hspace{14.22636pt}+ \\frac{R}{|\\nabla _\\mathbf {R}\\psi |^2} \\hat{\\mbox{$\\zeta $}}\\cdot \\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\psi \\cdot (\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}\\psi ) \\Bigg )\\left(\\frac{Z_{\\sigma }}{T_\\sigma }\\partial _\\psi \\varphi _0 + \\Upsilon _{\\sigma }\\right)\\nonumber \\\\\\hspace{14.22636pt} + \\left[ \\frac{Z_{\\sigma }\\lambda _{\\sigma }}{u \\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta } F_{\\sigma 1}^{\\rm sw}(\\hat{\\mathbf {b}}\\times \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle ) \\cdot \\nabla _\\mathbf {R}\\Theta \\right]^{\\rm lw}\\Bigg \\rbrace \\nonumber \\\\\\hspace{14.22636pt}-\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\Theta \\partial _\\psi \\left[\\frac{Iu}{\\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta }\\left(u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}- \\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B \\partial _u\\right)G_{{\\sigma }1}^{\\rm lw}\\right] \\nonumber \\\\\\hspace{14.22636pt} + \\partial _u \\left[\\frac{I}{B} \\left( \\mu \\partial _\\psi B+ Z_{\\sigma }\\partial _\\psi \\varphi _0 \\right)\\left(u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}- \\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B \\partial _u\\right)G_{{\\sigma }1}^{\\rm lw}\\right] \\nonumber \\\\\\hspace{14.22636pt} +\\partial _u\\left[\\frac{Z_{\\sigma }\\lambda _{\\sigma }\\varphi _1^{\\rm lw}}{u}\\left(u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}- \\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B \\partial _u\\right)G_{{\\sigma }1}^{\\rm lw}\\right] \\nonumber \\\\\\hspace{14.22636pt} - \\frac{u}{B}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}\\left(u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}- \\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B \\partial _u\\right)G_{{\\sigma }1}^{\\rm lw}\\nonumber \\\\\\hspace{14.22636pt}+\\frac{Z_{\\sigma }\\lambda _{\\sigma }}{B}\\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta \\partial _\\psi \\left[ \\frac{F_{{\\sigma }1}^{\\rm sw}}{\\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta }\\left(\\hat{\\mathbf {b}}\\times \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\right)\\cdot \\nabla _\\mathbf {R}\\psi \\right]^{\\rm lw}\\nonumber \\\\\\hspace{14.22636pt} - Z_{\\sigma }\\lambda _{\\sigma }\\partial _u\\Bigg [\\Bigg (\\frac{\\mu }{ u B}(\\hat{\\mathbf {b}}\\times \\partial _\\Theta B\\nabla _\\mathbf {R}\\Theta )\\cdot \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}\\rangle ^{\\rm sw}\\nonumber \\\\\\hspace{14.22636pt}+ \\frac{u}{B}\\left[\\hat{\\mathbf {b}}\\times (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}})\\right]\\cdot \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}\\rangle ^{\\rm sw}\\nonumber \\\\\\hspace{14.22636pt}+\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\langle \\phi _{{\\sigma }1}\\rangle ^{\\rm sw}\\Bigg )F_{{\\sigma }1}^{\\rm sw}\\Bigg ]^{\\rm lw}\\nonumber \\\\\\hspace{14.22636pt}=\\sum _{{\\sigma }^{\\prime }}{\\cal T}_{{\\sigma },0}^{*}C_{{\\sigma }{\\sigma }^{\\prime }}^{(2){\\rm lw}}+\\sum _{{\\sigma }^{\\prime }}\\left[{\\cal T}_{{\\sigma },1}^*C_{{\\sigma }{\\sigma }^{\\prime }}^{(1)}\\right]^{\\rm lw}.$ Finally, defining $G_{{\\sigma }2}^{\\rm lw}& = \\langle F_{\\sigma 2}^{\\rm lw}\\rangle \\nonumber \\\\&+\\left[\\frac{Z_{\\sigma }\\lambda _{\\sigma }^2}{T_\\sigma }\\varphi _2^{\\rm lw}+ \\frac{Z_{\\sigma }}{T_\\sigma }\\frac{\\mu }{2B}(\\leftrightarrow \\over {\\mathbf {I}}- \\hat{\\mathbf {b}}\\hat{\\mathbf {b}}) : \\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\varphi _0\\right] F_{{\\sigma }0}\\nonumber \\\\&+\\frac{1}{T_{\\sigma }}\\left(\\Psi _B + Z_{\\sigma }\\lambda _{\\sigma }\\Psi _{\\phi B, {\\sigma }}^{\\rm lw}+ Z_{\\sigma }^2\\lambda _{\\sigma }^2\\Psi _{\\phi ,{\\sigma }}^{\\rm lw}\\right)F_{{\\sigma }0}\\nonumber \\\\&+\\frac{Iu}{B}\\partial _\\psi G_{\\sigma 1}^{\\rm lw}-\\frac{Z_{\\sigma }\\lambda _{\\sigma }\\varphi _1^{\\rm lw}}{u} \\partial _uG_{\\sigma 1}^{\\rm lw}\\nonumber \\\\& -\\frac{I}{B} \\left( \\mu \\partial _\\psi B + Z_{\\sigma }\\partial _\\psi \\varphi _0 \\right)\\partial _u G_{\\sigma 1}^{\\rm lw}- \\frac{1}{2}\\left(\\frac{Z_{\\sigma }\\lambda _{\\sigma }\\varphi _1^{\\rm lw}}{T_\\sigma }\\right)^2F_{\\sigma 0}\\nonumber \\\\&- \\frac{Z_{\\sigma }\\lambda _{\\sigma }I u}{T_{\\sigma }B}\\varphi _1^{\\rm lw}F_{{\\sigma }0} \\Bigg (\\frac{Z_{\\sigma }}{T_\\sigma }\\partial _\\psi \\varphi _0 +\\Upsilon _{\\sigma }- \\frac{1}{T_\\sigma } \\partial _\\psi T_\\sigma \\Bigg ) \\nonumber \\\\& - \\frac{1}{2}\\left( \\frac{Iu}{B} \\right)^2 F_{{\\sigma }0} \\Bigg [\\partial _\\psi \\Upsilon _{\\sigma }-\\frac{\\mu \\partial _\\psi B}{T_{\\sigma }}\\partial _\\psi \\ln T_{\\sigma }\\nonumber \\\\&+\\partial _\\psi \\left(\\frac{Z_{\\sigma }}{T_\\sigma }\\partial _\\psi \\varphi _0\\right) - \\frac{Z_{\\sigma }}{T_\\sigma ^2}\\partial _\\psi \\varphi _0\\partial _\\psi T_\\sigma + \\Bigg (\\frac{Z_{\\sigma }}{T_\\sigma }\\partial _\\psi \\varphi _0 +\\Upsilon _{\\sigma }\\Bigg )^2 \\Bigg ]\\nonumber \\\\&- {\\mu F_{{\\sigma }0}}\\Bigg ( \\frac{I}{2B} \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}\\nonumber \\\\&+ \\frac{R}{|\\nabla _\\mathbf {R}\\psi |^2} \\hat{\\mbox{$\\zeta $}}\\cdot \\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\psi \\cdot (\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}\\psi ) \\Bigg )\\Bigg (\\frac{Z_{\\sigma }}{T_\\sigma }\\partial _\\psi \\varphi _0 +\\Upsilon _{\\sigma }\\Bigg )\\nonumber \\\\& + \\left[ \\frac{Z_{\\sigma }\\lambda _{\\sigma }}{u \\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta } F_{\\sigma 1}^{\\rm sw}(\\hat{\\mathbf {b}}\\times \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle ) \\cdot \\nabla _\\mathbf {R}\\Theta \\right]^{\\rm lw}\\nonumber \\\\& -\\left\\langle \\frac{1}{ u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\Theta } \\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}\\Theta \\sum _{\\sigma ^\\prime }{\\cal T}_{{\\sigma },0}^*C_{\\sigma \\sigma ^\\prime }^{(1){\\rm lw}} \\right\\rangle ;$ using the results in and $B^{-1}\\nabla _\\mathbf {R}\\cdot (B\\mbox{$\\rho $})+\\partial _u\\hat{u}_1+\\partial _\\mu \\hat{\\mu }_1+\\partial _\\theta \\hat{\\theta }_1=\\frac{u}{B}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}$ to write $\\left\\langle \\left[{\\cal T}_{\\sigma ,1}^\\ast C_{\\sigma \\sigma ^\\prime }^{(1)}\\right]^{\\rm lw}\\right\\rangle = \\nonumber \\\\\\hspace{14.22636pt}\\partial _u\\left(\\left\\langle u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\cdot \\mbox{$\\rho $}\\,{\\cal T}_{{\\sigma },0}^*C_{\\sigma \\sigma ^\\prime }^{(1){\\rm lw}}\\right\\rangle -\\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}\\left\\langle {\\cal T}_{{\\sigma },0}^*C_{\\sigma \\sigma ^\\prime }^{(1){\\rm lw}}\\right\\rangle \\right)\\nonumber \\\\\\hspace{14.22636pt}+ \\partial _\\mu \\Bigg \\lbrace \\frac{u \\mu }{B} \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}\\left\\langle {\\cal T}_{{\\sigma },0}^*C_{\\sigma \\sigma ^\\prime }^{(1){\\rm lw}}\\right\\rangle \\nonumber \\\\\\hspace{14.22636pt}-\\left\\langle \\left(\\frac{Z_{\\sigma }}{B} \\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}\\varphi _0 + \\frac{\\mu }{B} \\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}B +\\frac{u^2}{B} \\hat{\\mathbf {b}}\\cdot \\nabla \\hat{\\mathbf {b}}\\cdot \\mbox{$\\rho $}\\right){\\cal T}_{{\\sigma },0}^*C_{\\sigma \\sigma ^\\prime }^{(1){\\rm lw}}\\right\\rangle \\Bigg \\rbrace \\nonumber \\\\\\hspace{14.22636pt}- \\frac{u}{B}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}\\left\\langle {\\cal T}_{{\\sigma },0}^*C_{\\sigma \\sigma ^\\prime }^{(1){\\rm lw}}\\right\\rangle + \\frac{1}{B} \\nabla _\\mathbf {R}\\cdot \\left\\langle B\\mbox{$\\rho $}\\,{\\cal T}_{{\\sigma },0}^*C_{\\sigma \\sigma ^\\prime }^{(1){\\rm lw}}\\right\\rangle \\nonumber \\\\\\hspace{14.22636pt} +\\left[{\\cal T}_{{\\sigma },1}^\\ast C_{\\sigma \\sigma ^\\prime }^{(1){\\rm sw}}\\right]^{\\rm lw};$ and employing (REF ) and (REF ), we can write the gyroaveraged, long-wavelength second-order Fokker-Planck equation as in (REF ).", "However, for some purposes, mainly in connection with the long-wavelength gyrokinetic quasineutrality equation, it is useful to recast equation (REF ) in a different fashion.", "After some straightforward algebra one gets $G_{{\\sigma }2}^{\\rm lw}& = \\langle F_{\\sigma 2}^{\\rm lw}\\rangle +\\frac{Z_{\\sigma }\\lambda _{\\sigma }^2}{T_\\sigma }\\varphi _2^{\\rm lw}F_{{\\sigma }0}+\\frac{Iu}{B}\\partial _\\psi G_{\\sigma 1}^{\\rm lw}-\\frac{Z_{\\sigma }\\lambda _{\\sigma }\\varphi _1^{\\rm lw}}{u} \\partial _uG_{\\sigma 1}^{\\rm lw}\\nonumber \\\\& -\\frac{I}{B} \\left( \\mu \\partial _\\psi B + Z_{\\sigma }\\partial _\\psi \\varphi _0 \\right)\\partial _u G_{\\sigma 1}^{\\rm lw}- \\frac{1}{2}\\left(\\frac{Z_{\\sigma }\\lambda _{\\sigma }\\varphi _1^{\\rm lw}}{T_\\sigma }\\right)^2F_{\\sigma 0}\\nonumber \\\\&- \\frac{Z_{\\sigma }\\lambda _{\\sigma }I u}{T_{\\sigma }B}\\varphi _1^{\\rm lw}F_{{\\sigma }0} \\Bigg [\\frac{Z_{\\sigma }}{T_\\sigma }\\partial _\\psi \\varphi _0+\\Upsilon _{\\sigma }- \\frac{1}{T_\\sigma } \\partial _\\psi T_\\sigma \\Bigg ]\\nonumber \\\\&-\\frac{1}{2B^2}\\left(\\left({Iu} \\right)^2+\\mu B|\\nabla _\\mathbf {R}\\psi |^2\\right)\\Bigg [- \\frac{2Z_{\\sigma }}{T_\\sigma ^2}\\partial _\\psi \\varphi _0\\partial _\\psi T_\\sigma \\nonumber \\\\&- \\frac{u^2/2 + \\mu B}{T_\\sigma }\\left( \\partial _\\psi \\ln T_\\sigma \\right)^2+ \\Bigg (\\frac{Z_{\\sigma }}{T_\\sigma }\\partial _\\psi \\varphi _0 +\\Upsilon _{\\sigma }\\Bigg )^2\\Bigg ]F_{{\\sigma }0}\\nonumber \\\\&- \\frac{1}{2}\\left( \\frac{Iu}{B} \\right)^2 F_{{\\sigma }0} \\Bigg [\\partial ^2_\\psi \\ln n_\\sigma +\\frac{Z_{\\sigma }}{T_\\sigma }\\partial _\\psi ^2 \\varphi _0\\nonumber \\\\&+ \\left(\\frac{u^2/2 + \\mu B}{T_\\sigma } - \\frac{3}{2} \\right)\\partial ^2_\\psi \\ln T_\\sigma \\Bigg ] \\nonumber \\\\&+\\mu \\Bigg (\\frac{1}{2 B^2}\\nabla _\\mathbf {R}B\\cdot \\nabla _\\mathbf {R}\\psi -\\frac{I}{2B} \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}\\nonumber \\\\&- \\frac{R}{|\\nabla _\\mathbf {R}\\psi |^2} \\hat{\\mbox{$\\zeta $}}\\cdot \\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\psi \\cdot (\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}\\psi )\\Bigg )\\left(\\frac{Z_{\\sigma }}{T_\\sigma }\\partial _\\psi \\varphi _0+\\Upsilon _{\\sigma }\\right)F_{{\\sigma }0}\\nonumber \\\\& + \\left[ \\frac{Z_{\\sigma }\\lambda _{\\sigma }}{u \\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta } F_{\\sigma 1}^{\\rm sw}(\\hat{\\mathbf {b}}\\times \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle ) \\cdot \\nabla _\\mathbf {R}\\Theta \\right]^{\\rm lw}\\nonumber \\\\& -\\left\\langle \\frac{1}{ u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\Theta } \\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}\\Theta \\sum _{\\sigma ^\\prime } {\\cal T}_{{\\sigma }, 0}^{*} C_{\\sigma \\sigma ^\\prime }^{(1){\\rm lw}} \\right\\rangle \\nonumber \\\\&-\\frac{\\mu }{2B} (\\leftrightarrow \\over {\\mathbf {I}}- \\hat{\\mathbf {b}}\\hat{\\mathbf {b}}) : \\Bigg [\\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\ln n_{\\sigma }\\nonumber \\\\&+ \\left( \\frac{u^2/2+\\mu B}{T_{\\sigma }} -\\frac{3}{2} \\right) \\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\ln T_{\\sigma }\\nonumber \\\\&+\\frac{Z_{\\sigma }}{T_{\\sigma }}\\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\varphi _0\\Bigg ] F_{{\\sigma }0}- \\frac{Z_{\\sigma }^2\\lambda _{\\sigma }^2}{2T_{\\sigma }^2}\\left[ \\left\\langle (\\widetilde{\\phi }_{{\\sigma }1}^{\\rm sw})^2 \\right\\rangle \\right]^{\\rm lw}F_{{\\sigma }0}\\nonumber \\\\&+\\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}\\left(\\frac{u}{B}\\partial _\\mu -\\partial _u\\right)F_{{\\sigma }1}^{\\rm lw}\\nonumber \\\\&+\\left\\langle {\\cal T}_{{\\sigma },0}^{*}\\left[{\\cal T}_{{\\sigma },1}^{-1*}F_{{\\sigma }1}^{\\rm lw}\\right]^{\\rm lw}\\right\\rangle +\\left\\langle {\\cal T}_{{\\sigma },0}^{*}\\left[{\\cal T}_{{\\sigma }, 2}^{-1*}F_{{\\sigma }0}\\right]^{\\rm lw}\\right\\rangle ,$ where $\\left\\langle {\\cal T}_{{\\sigma }, 0}^{*} \\left[{\\cal T}_{{\\sigma },1}^{-1*}F_{{\\sigma }1}^{\\rm lw}\\right]^{\\rm lw}\\right\\rangle $ is given in (REF ) and $\\left\\langle {\\cal T}_{{\\sigma }, 0}^{*}\\left[{\\cal T}_{{\\sigma }, 2}^{-1*}F_{{\\sigma }0}\\right]^{\\rm lw}\\right\\rangle $ is given in (REF ).", "A less obvious calculation transforms the previous equation into $G_{{\\sigma }2}^{\\rm lw}& = \\langle F_{\\sigma 2} \\rangle ^{\\rm lw}+\\frac{Z_{\\sigma }\\lambda _{\\sigma }^2}{T_\\sigma }\\varphi _2^{\\rm lw}F_{{\\sigma }0}+\\frac{Iu}{B}\\partial _\\psi G_{\\sigma 1}^{\\rm lw}-\\frac{Z_{\\sigma }\\lambda _{\\sigma }\\varphi _1^{\\rm lw}}{u} \\partial _uG_{\\sigma 1}^{\\rm lw}\\nonumber \\\\& -\\frac{I}{B} \\left( \\mu \\partial _\\psi B+ Z_{\\sigma }\\partial _\\psi \\varphi _0 \\right)\\partial _u G_{\\sigma 1}^{\\rm lw}- \\frac{1}{2}\\left(\\frac{Z_{\\sigma }\\lambda _{\\sigma }\\varphi _1^{\\rm lw}}{T_\\sigma }\\right)^2F_{\\sigma 0}\\nonumber \\\\&- \\frac{Z_{\\sigma }\\lambda _{\\sigma }I u}{T_{\\sigma }B}\\varphi _1^{\\rm lw}F_{{\\sigma }0} \\Bigg (\\frac{Z_{\\sigma }}{T_\\sigma }\\partial _\\psi \\varphi _0+\\Upsilon _{\\sigma }-\\frac{1}{T_\\sigma } \\partial _\\psi T_\\sigma \\Bigg )\\nonumber \\\\&-\\frac{1}{2B^2}\\left(\\left({Iu} \\right)^2+\\mu B|\\nabla _\\mathbf {R}\\psi |^2\\right)\\Bigg [- \\frac{2Z_{\\sigma }}{T_\\sigma ^2}\\partial _\\psi \\varphi _0\\partial _\\psi T_\\sigma \\nonumber \\\\&+ \\Bigg (\\frac{Z_{\\sigma }}{T_\\sigma }\\partial _\\psi \\varphi _0 +\\Upsilon _{\\sigma }\\Bigg )^2+\\partial _\\psi ^2\\ln n_{\\sigma }\\nonumber \\\\&+\\left(\\frac{u^2/2 +\\mu B}{T_{\\sigma }}-\\frac{3}{2}\\right)\\partial _\\psi ^2\\ln T_{\\sigma }\\nonumber \\\\&-\\frac{u^2/2 +\\mu B}{T_{\\sigma }}(\\partial _\\psi \\ln T_{\\sigma })^2+\\frac{Z_{\\sigma }}{T_\\sigma }\\partial _\\psi ^2 \\varphi _0\\Bigg ]F_{{\\sigma }0}\\nonumber \\\\& + \\left[ \\frac{Z_{\\sigma }\\lambda _{\\sigma }}{u \\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta } F_{\\sigma 1}^{\\rm sw}(\\hat{\\mathbf {b}}\\times \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle ) \\cdot \\nabla _\\mathbf {R}\\Theta \\right]^{\\rm lw}\\nonumber \\\\& -\\left\\langle \\frac{1}{ u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\Theta } \\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}\\Theta \\sum _{\\sigma ^\\prime }{\\cal T}_{{\\sigma },0}^*C_{\\sigma \\sigma ^\\prime }^{(1){\\rm lw}} \\right\\rangle \\nonumber \\\\&- \\frac{Z_{\\sigma }^2\\lambda _{\\sigma }^2}{2T_{\\sigma }^2}\\left[\\left\\langle (\\widetilde{\\phi }_{{\\sigma }1}^{\\rm sw})^2 \\right\\rangle \\right]^{\\rm lw}F_{{\\sigma }0}+\\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}\\left(\\frac{u}{B}\\partial _\\mu -\\partial _u\\right)G_{{\\sigma }1}^{\\rm lw}\\nonumber \\\\&+\\left\\langle {\\cal T}_{{\\sigma },0}^{*}\\left[{\\cal T}_{{\\sigma },1}^{-1*}F_{{\\sigma }1}^{\\rm lw}\\right]^{\\rm lw}\\right\\rangle +\\left\\langle {\\cal T}_{{\\sigma },0}^{*}\\left[{\\cal T}_{{\\sigma },2}^{-1*}F_{{\\sigma }0}\\right]^{\\rm lw}\\right\\rangle .$ To obtain (REF ) from (REF ) we used equation (REF ) and $ \\frac{1}{B} \\nabla _\\mathbf {R}B \\cdot \\nabla _\\mathbf {R}\\psi + I \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}\\nonumber \\\\\\hspace{28.45274pt}- \\frac{2RB}{|\\nabla _\\mathbf {R}\\psi |^2} \\hat{\\mbox{$\\zeta $}}\\cdot \\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\psi \\cdot (\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}\\psi )\\nonumber \\\\\\hspace{28.45274pt}- \\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\psi : (\\leftrightarrow \\over {\\mathbf {I}}- \\hat{\\mathbf {b}}\\hat{\\mathbf {b}}) = 0.$ Let us prove this.", "First, we have that $\\nabla _\\mathbf {R}B \\cdot \\nabla _\\mathbf {R}\\psi = \\frac{I}{R^2 B}\\nabla _\\mathbf {R}I \\cdot \\nabla _\\mathbf {R}\\psi \\nonumber \\\\[5pt]\\hspace{28.45274pt}+ \\frac{1}{R^2 B} \\nabla _\\mathbf {R}\\psi \\cdot \\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\psi \\cdot \\nabla _\\mathbf {R}\\psi -\\frac{B}{R} \\nabla _\\mathbf {R}R \\cdot \\nabla _\\mathbf {R}\\psi ,$ where we have employed that $B^2 = (I^2 + |\\nabla _\\mathbf {R}\\psi |^2)/R^2$ .", "Noting that $\\partial _\\zeta (\\nabla _\\mathbf {R}\\psi ) =(\\nabla _\\mathbf {R}\\psi \\cdot \\nabla _\\mathbf {R}R) \\hat{\\mbox{$\\zeta $}}$ we derive the following identities $\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}= \\frac{1}{B^2} \\mathbf {B}\\cdot (\\nabla _\\mathbf {R}\\times \\mathbf {B}) \\nonumber \\\\\\hspace{28.45274pt}= \\frac{1}{B^2} \\mathbf {B}\\cdot \\left[ \\nabla _\\mathbf {R}I \\times \\nabla _\\mathbf {R}\\zeta + \\nabla _\\mathbf {R}\\cdot \\left(\\nabla _\\mathbf {R}\\psi \\nabla _\\mathbf {R}\\zeta \\right) - \\nabla _\\mathbf {R}\\cdot \\left(\\nabla _\\mathbf {R}\\zeta \\nabla _\\mathbf {R}\\psi \\right) \\right] \\nonumber \\\\\\hspace{28.45274pt}=\\frac{1}{B^2} \\mathbf {B}\\cdot \\left[ \\nabla _\\mathbf {R}I \\times \\nabla _\\mathbf {R}\\zeta + \\left( \\nabla _\\mathbf {R}^2 \\psi - \\frac{2}{R} \\nabla _\\mathbf {R}\\psi \\cdot \\nabla _\\mathbf {R}R \\right) \\nabla _\\mathbf {R}\\zeta \\right]\\nonumber \\\\\\hspace{28.45274pt}= - \\frac{1}{R^2 B^2} \\nabla _\\mathbf {R}I \\cdot \\nabla _\\mathbf {R}\\psi + \\frac{I}{R^2 B^2} \\nabla _\\mathbf {R}^2 \\psi - \\frac{2I}{R^3 B^2}\\nabla _\\mathbf {R}\\psi \\cdot \\nabla _\\mathbf {R}R,$ $\\hat{\\mbox{$\\zeta $}}\\cdot \\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\psi \\cdot (\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}\\psi ) = - \\frac{|\\nabla _\\mathbf {R}\\psi |^2}{R B} \\hat{\\mbox{$\\zeta $}}\\cdot \\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\psi \\cdot \\hat{\\mbox{$\\zeta $}}\\nonumber \\\\\\hspace{28.45274pt}+ \\frac{I}{B} \\hat{\\mbox{$\\zeta $}}\\cdot \\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\psi \\cdot (\\nabla _\\mathbf {R}\\zeta \\times \\nabla _\\mathbf {R}\\psi ) \\nonumber \\\\\\hspace{28.45274pt}= - \\frac{|\\nabla _\\mathbf {R}\\psi |^2}{R^2 B}\\nabla _\\mathbf {R}R \\cdot \\nabla _\\mathbf {R}\\psi ,$ $\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\psi \\cdot \\hat{\\mathbf {b}}=\\frac{I^2}{R^2 B^2} \\hat{\\mbox{$\\zeta $}}\\cdot \\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\psi \\cdot \\hat{\\mbox{$\\zeta $}}\\nonumber \\\\\\hspace{28.45274pt}+\\frac{2I}{RB^2} \\hat{\\mbox{$\\zeta $}}\\cdot \\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\psi \\cdot (\\nabla _\\mathbf {R}\\zeta \\times \\nabla _\\mathbf {R}\\psi ) \\nonumber \\\\\\hspace{28.45274pt}+\\frac{1}{B^2} (\\nabla _\\mathbf {R}\\zeta \\times \\nabla _\\mathbf {R}\\psi ) \\cdot \\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\psi \\cdot (\\nabla _\\mathbf {R}\\zeta \\times \\nabla _\\mathbf {R}\\psi ) \\nonumber \\\\\\hspace{28.45274pt}= \\frac{I^2}{R^3 B^2} \\nabla _\\mathbf {R}R \\cdot \\nabla _\\mathbf {R}\\psi \\nonumber \\\\\\hspace{28.45274pt}+ \\frac{1}{B^2} (\\nabla _\\mathbf {R}\\zeta \\times \\nabla _\\mathbf {R}\\psi ) \\cdot \\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\psi \\cdot (\\nabla _\\mathbf {R}\\zeta \\times \\nabla _\\mathbf {R}\\psi ),$ and $(\\nabla _\\mathbf {R}\\zeta \\times \\nabla _\\mathbf {R}\\psi ) \\cdot \\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\psi \\cdot (\\nabla _\\mathbf {R}\\zeta \\times \\nabla _\\mathbf {R}\\psi )\\nonumber \\\\\\hspace{28.45274pt}= - (\\nabla _\\mathbf {R}\\zeta \\times \\nabla _\\mathbf {R}\\psi ) \\cdot \\nabla _\\mathbf {R}(\\nabla _\\mathbf {R}\\zeta \\times \\nabla _\\mathbf {R}\\psi ) \\cdot \\nabla _\\mathbf {R}\\psi \\nonumber \\\\\\hspace{28.45274pt}= - \\nabla _\\mathbf {R}\\psi \\cdot \\nabla _\\mathbf {R}(\\nabla _\\mathbf {R}\\zeta \\times \\nabla _\\mathbf {R}\\psi ) \\cdot (\\nabla _\\mathbf {R}\\zeta \\times \\nabla _\\mathbf {R}\\psi ) \\nonumber \\\\\\hspace{28.45274pt}+ \\nabla _\\mathbf {R}\\psi \\cdot \\lbrace (\\nabla _\\mathbf {R}\\zeta \\times \\nabla _\\mathbf {R}\\psi ) \\times [ \\nabla _\\mathbf {R}\\times (\\nabla _\\mathbf {R}\\zeta \\times \\nabla _\\mathbf {R}\\psi ) ] \\rbrace \\nonumber \\\\\\hspace{28.45274pt}=- \\nabla _\\mathbf {R}\\psi \\cdot \\nabla _\\mathbf {R}\\left( \\frac{|\\nabla _\\mathbf {R}\\psi |^2}{2 R^2} \\right)\\nonumber \\\\\\hspace{28.45274pt}+ \\frac{|\\nabla _\\mathbf {R}\\psi |^2}{R^2} \\left(\\nabla _\\mathbf {R}^2 \\psi - \\frac{2}{R} \\nabla _\\mathbf {R}R \\cdot \\nabla _\\mathbf {R}\\psi \\right) \\nonumber \\\\\\hspace{28.45274pt}= - \\frac{1}{R^2} \\nabla _\\mathbf {R}\\psi \\cdot \\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\psi \\cdot \\nabla _\\mathbf {R}\\psi \\nonumber \\\\\\hspace{28.45274pt}+\\frac{|\\nabla _\\mathbf {R}\\psi |^2}{R^2} \\nabla _\\mathbf {R}^2 \\psi -\\frac{|\\nabla _\\mathbf {R}\\psi |^2}{R^3} \\nabla _\\mathbf {R}R \\cdot \\nabla _\\mathbf {R}\\psi .$ Using relations (REF ), (REF ), (REF ), (REF ), and (REF ), it is trivial to check that (REF ) is satisfied." ], [ "Proof of (", "First, $(\\nabla _\\mathbf {R}\\times \\mathbf {K}) \\cdot \\nabla _\\mathbf {R}\\psi =\\nonumber \\\\\\hspace{28.45274pt}\\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta \\partial _\\Theta \\left[\\frac{1}{\\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta } \\mathbf {K}\\cdot (\\nabla _\\mathbf {R}\\psi \\times \\nabla _\\mathbf {R}\\Theta ) \\right]\\nonumber \\\\\\hspace{28.45274pt}+\\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta \\partial _\\zeta \\left[\\frac{1}{\\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta } \\mathbf {K}\\cdot (\\nabla _\\mathbf {R}\\psi \\times \\nabla _\\mathbf {R}\\zeta ) \\right].$ Employing that $\\partial _\\zeta \\mathbf {R}= (\\nabla _\\mathbf {R}\\psi \\times \\nabla _\\mathbf {R}\\Theta )/(\\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta ) = R\\hat{\\mbox{$\\zeta $}}$ and $\\partial _\\Theta \\mathbf {R}= - (\\nabla _\\mathbf {R}\\psi \\times \\nabla _\\mathbf {R}\\zeta )/(\\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta )$ , we find $(\\nabla _\\mathbf {R}\\times \\mathbf {K}) \\cdot \\nabla _\\mathbf {R}\\psi =\\nonumber \\\\\\hspace{28.45274pt}\\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta \\partial _\\Theta \\left(\\partial _\\zeta \\mathbf {R}\\cdot \\mathbf {K}\\right) - \\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta \\partial _\\zeta \\left(\\partial _\\Theta \\mathbf {R}\\cdot \\mathbf {K}\\right)\\nonumber \\\\\\hspace{28.45274pt}= \\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\left( \\frac{I}{2B} \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}\\right) - \\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta \\partial _\\Theta \\left( \\partial _\\zeta \\hat{\\mathbf {e}}_2 \\cdot \\hat{\\mathbf {e}}_1 \\right) \\nonumber \\\\\\hspace{28.45274pt}+ \\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta \\partial _\\zeta \\left(\\partial _\\Theta \\hat{\\mathbf {e}}_2 \\cdot \\hat{\\mathbf {e}}_1 \\right) =\\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\left( \\frac{I}{2B} \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}\\right) \\nonumber \\\\\\hspace{28.45274pt}- \\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Theta \\left(\\partial _\\zeta \\hat{\\mathbf {e}}_2 \\cdot \\partial _\\Theta \\hat{\\mathbf {e}}_1 - \\partial _\\Theta \\hat{\\mathbf {e}}_2 \\cdot \\partial _\\zeta \\hat{\\mathbf {e}}_1 \\right).$ Now, with the help of the relations $\\nabla _\\mathbf {R}\\hat{\\mathbf {e}}_1 = \\nabla _\\mathbf {R}\\hat{\\mathbf {e}}_1\\cdot \\hat{\\mathbf {b}}\\hat{\\mathbf {b}}+ \\nabla _\\mathbf {R}\\hat{\\mathbf {e}}_1 \\cdot \\hat{\\mathbf {e}}_2 \\hat{\\mathbf {e}}_2$ and $\\nabla _\\mathbf {R}\\hat{\\mathbf {e}}_2 = \\nabla _\\mathbf {R}\\hat{\\mathbf {e}}_2 \\cdot \\hat{\\mathbf {b}}\\hat{\\mathbf {b}}+ \\nabla _\\mathbf {R}\\hat{\\mathbf {e}}_2 \\cdot \\hat{\\mathbf {e}}_1 \\hat{\\mathbf {e}}_1$ , one gets $\\partial _\\zeta \\hat{\\mathbf {e}}_2 \\cdot \\partial _\\Theta \\hat{\\mathbf {e}}_1 - \\partial _\\Theta \\hat{\\mathbf {e}}_2 \\cdot \\partial _\\zeta \\hat{\\mathbf {e}}_1=\\nonumber \\\\\\hspace{28.45274pt}\\left(\\partial _\\zeta \\hat{\\mathbf {e}}_2 \\cdot \\hat{\\mathbf {b}}\\right) \\left(\\partial _\\Theta \\hat{\\mathbf {e}}_1 \\cdot \\hat{\\mathbf {b}}\\right) -\\left( \\partial _\\Theta \\hat{\\mathbf {e}}_2 \\cdot \\hat{\\mathbf {b}}\\right) \\left( \\partial _\\zeta \\hat{\\mathbf {e}}_1 \\cdot \\hat{\\mathbf {b}}\\right) \\nonumber \\\\\\hspace{28.45274pt}= \\left(\\partial _\\zeta \\hat{\\mathbf {b}}\\cdot \\hat{\\mathbf {e}}_2 \\right) \\left(\\partial _\\Theta \\hat{\\mathbf {b}}\\cdot \\hat{\\mathbf {e}}_1 \\right) - \\left( \\partial _\\Theta \\hat{\\mathbf {b}}\\cdot \\hat{\\mathbf {e}}_2 \\right) \\left(\\partial _\\zeta \\hat{\\mathbf {b}}\\cdot \\hat{\\mathbf {e}}_1 \\right)\\nonumber \\\\\\hspace{28.45274pt}= \\left( \\partial _\\Theta \\hat{\\mathbf {b}}\\times \\partial _\\zeta \\hat{\\mathbf {b}}\\right) \\cdot \\hat{\\mathbf {b}}.$ Since this quantity is independent of the choice of $\\hat{\\mathbf {e}}_1$ and $\\hat{\\mathbf {e}}_2$ , we can use $\\hat{\\mathbf {e}}_1 = \\nabla _\\mathbf {R}\\psi /|\\nabla _\\mathbf {R}\\psi |$ and $\\hat{\\mathbf {e}}_2 = (\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}\\psi )/|\\nabla _\\mathbf {R}\\psi |$ without loss of generality, giving $\\partial _\\zeta \\hat{\\mathbf {e}}_2 \\cdot \\partial _\\Theta \\hat{\\mathbf {e}}_1 - \\partial _\\Theta \\hat{\\mathbf {e}}_2 \\cdot \\partial _\\zeta \\hat{\\mathbf {e}}_1=\\partial _\\zeta \\left( \\frac{\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}\\psi }{|\\nabla _\\mathbf {R}\\psi |} \\right) \\cdot \\partial _\\Theta \\left( \\frac{\\nabla _\\mathbf {R}\\psi }{|\\nabla _\\mathbf {R}\\psi |} \\right)\\nonumber \\\\\\hspace{28.45274pt}- \\partial _\\Theta \\left(\\frac{\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}\\psi }{|\\nabla _\\mathbf {R}\\psi |} \\right) \\cdot \\partial _\\zeta \\left( \\frac{\\nabla _\\mathbf {R}\\psi }{|\\nabla _\\mathbf {R}\\psi |} \\right)\\nonumber \\\\\\hspace{28.45274pt}= - \\partial _\\Theta \\left(\\frac{1}{|\\nabla _\\mathbf {R}\\psi |^2} \\frac{\\partial \\nabla _\\mathbf {R}\\psi }{\\partial \\zeta } \\cdot (\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}\\psi ) \\right).$ Thus, $(\\nabla _\\mathbf {R}\\times \\mathbf {K}) \\cdot \\nabla _\\mathbf {R}\\psi = \\mathbf {B}\\cdot \\nabla _\\mathbf {R}\\Bigg ( \\frac{I}{2B} \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\times \\hat{\\mathbf {b}}\\nonumber \\\\\\hspace{28.45274pt}+ \\frac{R}{|\\nabla _\\mathbf {R}\\psi |^2} \\hat{\\mbox{$\\zeta $}}\\cdot \\nabla _\\mathbf {R}\\nabla _\\mathbf {R}\\psi \\cdot (\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}\\psi ) \\Bigg ).$" ], [ "Some computations related to the long-wavelength\nquasineutrality equation", "Firstly, let us show that (REF ) can be rewritten as in (REF ).", "Employ the relation (recall () and ()) $\\partial _\\mu \\mu _{{\\sigma },1}+\\partial _\\theta \\theta _{{\\sigma },1} =\\frac{1}{B}\\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}B,$ the identity $\\langle \\mbox{$\\rho $}\\mbox{$\\rho $}\\rangle = \\frac{\\mu }{B}(\\leftrightarrow \\over {\\mathbf {I}}-\\hat{\\mathbf {b}}\\hat{\\mathbf {b}}),$ and the long-wavelength limit of (REF ) and (), $\\mathbf {R}_{{\\sigma },2}^{\\rm lw}&=& - \\frac{2u }{B} \\hat{\\mathbf {b}}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\cdot (\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}}) - \\frac{1}{8} \\hat{\\mathbf {b}}\\left[ \\mbox{$\\rho $}\\mbox{$\\rho $}-(\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}}) (\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}}) \\right]: \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\nonumber \\\\&-&\\frac{u }{B} \\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}\\cdot \\mbox{$\\rho $}- \\frac{1}{2B} \\mbox{$\\rho $}\\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}B +O(\\epsilon _{\\sigma }),$ $\\mu _{{\\sigma },1}^{\\rm lw}&=& - \\frac{u^2}{B} \\hat{\\mathbf {b}}\\cdot \\nabla _{\\mathbf {R}} \\hat{\\mathbf {b}}\\cdot \\mbox{$\\rho $}+ \\frac{u}{4} \\left[ \\mbox{$\\rho $}(\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}}) +(\\mbox{$\\rho $}\\times \\hat{\\mathbf {b}}) \\mbox{$\\rho $}\\right]: \\nabla _{\\mathbf {R}} \\hat{\\mathbf {b}}\\\\[5pt]&-&\\frac{Z_{\\sigma }}{B}\\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {R}\\varphi _0 +O(\\epsilon _{\\sigma }),$ to recast (REF ) into $\\sum _{\\sigma }\\frac{Z_{\\sigma }}{\\lambda _{\\sigma }^2}\\int (BF_{{\\sigma }2}^{\\rm lw}-\\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\times \\mathbf {K}\\, F_{{\\sigma }0}+u\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\times \\hat{\\mathbf {b}}\\, F_{{\\sigma }1}^{\\rm lw})\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\nonumber \\\\[5pt]\\hspace{28.45274pt}+2\\pi \\sum _{\\sigma }\\frac{Z_{\\sigma }}{\\lambda _{\\sigma }^2}\\Bigg [\\nabla _\\mathbf {r}\\cdot \\left(\\frac{3}{2B} \\nabla _{\\mathbf {r}_\\bot } B\\int \\mu F_{{\\sigma }0}\\mbox{d}u\\mbox{d}\\mu \\right)\\nonumber \\\\[5pt]\\hspace{28.45274pt}+\\frac{1}{2}\\nabla _\\mathbf {r}\\nabla _\\mathbf {r}:\\left((\\leftrightarrow \\over {\\mathbf {I}}-\\hat{\\mathbf {b}}\\hat{\\mathbf {b}})\\int \\mu F_{{\\sigma }0}\\mbox{d}u\\mbox{d}\\mu \\right)\\nonumber \\\\[5pt]\\hspace{28.45274pt}-\\nabla _\\mathbf {r}\\cdot \\left(\\frac{1}{B}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}\\int u^2\\mu \\partial _\\mu F_{{\\sigma }0}\\mbox{d}u\\mbox{d}\\mu \\right)\\nonumber \\\\[5pt]\\hspace{28.45274pt}-\\nabla _\\mathbf {r}\\cdot \\left(\\frac{Z_{\\sigma }}{B}(\\leftrightarrow \\over {\\mathbf {I}}-\\hat{\\mathbf {b}}\\hat{\\mathbf {b}})\\cdot \\nabla _\\mathbf {r}\\varphi _0\\int \\mu \\partial _\\mu F_{{\\sigma }0}\\mbox{d}u\\mbox{d}\\mu \\right)\\Bigg ] =0.$ We can get more explicit expressions by noting that the integrals containing $F_{{\\sigma }0}$ can be worked out analytically.", "Namely, if $F_{{\\sigma }0}=\\frac{n_{{\\sigma }}}{(2\\pi T_{{\\sigma }})^{3/2}}\\exp \\left(-\\frac{\\mu B + u^2/2}{T_{{\\sigma }}}\\right),$ then $\\partial _{\\mu }F_{{\\sigma }0}=-\\frac{B}{T_{{\\sigma }}}F_{{\\sigma }0},$ and $\\int \\mu F_{{\\sigma }0}\\mbox{d}u\\mbox{d}\\mu =\\frac{n_{{\\sigma }}T_{{\\sigma }}}{2\\pi B^2},\\quad \\int u^2\\mu F_{{\\sigma }0}\\mbox{d}u\\mbox{d}\\mu =\\frac{n_{{\\sigma }}T_{{\\sigma }}^2}{2\\pi B^2}.$ With these results, equation (REF ) finally becomes (REF ).", "Now, we proceed to recast (REF ) into (REF ) by using the function $G_{{\\sigma }2}^{\\rm lw}$ defined in (REF ).", "A simple rewriting of (REF ) in terms of $G_{{\\sigma }2}^{\\rm lw}$ gives $\\sum _{\\sigma }\\frac{Z_{\\sigma }}{\\lambda _{\\sigma }^2}\\int B\\Bigg \\lbrace G_{{\\sigma }2}^{\\rm lw}-\\frac{Z_{\\sigma }\\lambda _{\\sigma }^2}{T_\\sigma }\\varphi _2^{\\rm lw}F_{{\\sigma }0}-\\frac{Iu}{B}\\partial _\\psi G_{\\sigma 1}^{\\rm lw}\\nonumber \\\\[5pt]\\hspace{28.45274pt}+\\frac{Z_{\\sigma }\\lambda _{\\sigma }\\varphi _1^{\\rm lw}}{u} \\partial _uG_{\\sigma 1}^{\\rm lw}+\\frac{1}{2}\\left(\\frac{Z_{\\sigma }\\lambda _{\\sigma }\\varphi _1^{\\rm lw}}{T_\\sigma }\\right)^2F_{\\sigma 0}\\nonumber \\\\[5pt]\\hspace{28.45274pt}+\\frac{1}{2B^2}\\left(\\left({Iu} \\right)^2+\\mu B|\\nabla _\\mathbf {r}\\psi |^2\\right)\\Bigg [- \\frac{2Z_{\\sigma }}{T_\\sigma ^2}\\partial _\\psi \\varphi _0\\partial _\\psi T_\\sigma \\nonumber \\\\[5pt]\\hspace{28.45274pt}+ \\Bigg (\\frac{Z_{\\sigma }}{T_\\sigma }\\partial _\\psi \\varphi _0 +\\Upsilon _{\\sigma }\\Bigg )^2+\\partial _\\psi ^2\\ln n_{\\sigma }\\nonumber \\\\[5pt]\\hspace{28.45274pt}+\\left(\\frac{u^2/2+\\mu B}{T_{\\sigma }}-\\frac{3}{2}\\right)\\partial _\\psi ^2\\ln T_{\\sigma }\\nonumber \\\\[5pt]\\hspace{28.45274pt}-\\frac{u^2/2+\\mu B}{T_{\\sigma }}(\\partial _\\psi \\ln T_{\\sigma })^2+\\frac{Z_{\\sigma }}{T_\\sigma }\\partial _\\psi ^2 \\varphi _0\\Bigg ]F_{{\\sigma }0}\\nonumber \\\\[5pt]\\hspace{28.45274pt}- \\left[ \\frac{Z_{\\sigma }}{u \\mathbf {B}\\cdot \\nabla _\\mathbf {r}\\Theta } F_{\\sigma 1}^{\\rm sw}(\\hat{\\mathbf {b}}\\times \\nabla _{\\mathbf {r}_\\perp /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle ) \\cdot \\nabla _\\mathbf {r}\\Theta \\right]^{\\rm lw}\\nonumber \\\\[5pt]\\hspace{28.45274pt}+\\left\\langle \\frac{1}{ u \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\Theta } \\mbox{$\\rho $}\\cdot \\nabla _\\mathbf {r}\\Theta \\sum _{\\sigma ^\\prime }{\\cal T}_{{\\sigma },0}^*C_{\\sigma \\sigma ^\\prime }^{(1){\\rm lw}} \\right\\rangle \\nonumber \\\\[5pt]\\hspace{28.45274pt}+ \\frac{Z_{\\sigma }^2\\lambda _{\\sigma }^2}{2T_{\\sigma }^2}\\left[ \\left\\langle (\\widetilde{\\phi }_{{\\sigma }1}^{\\rm sw})^2 \\right\\rangle \\right]^{\\rm lw}F_{{\\sigma }0}\\Bigg \\rbrace \\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\nonumber \\\\[5pt]\\hspace{28.45274pt}+\\sum _{\\sigma }\\frac{Z_{\\sigma }}{\\lambda _{\\sigma }^2}\\int u\\hat{\\mathbf {b}}\\cdot (\\nabla _\\mathbf {r}\\times \\hat{\\mathbf {b}})G_{{\\sigma }1}^{\\rm lw}\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta =0.$ Here we have used (REF ) to write $\\int B {\\cal T}_{{\\sigma },0}^*\\left\\langle \\left[{\\cal T}_{{\\sigma },1}^{-1 *}F_{{\\sigma }1}^{\\rm lw}\\right]^{\\rm lw}\\right\\rangle \\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\nonumber \\\\[5pt]\\hspace{28.45274pt}=\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\times \\hat{\\mathbf {b}}\\int u F_{{\\sigma }1}^{\\rm lw}\\,\\,\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta ,$ and we have employed the result in : $ \\int B\\left\\langle {\\cal T}_{{\\sigma },0}^*\\left[\\mathcal {T}^{-1\\ast }_{{\\sigma }, 2} F_{{\\sigma }0} \\right]^{\\rm lw}\\right\\rangle \\,\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta =\\nonumber \\\\[5pt]\\hspace{28.45274pt} \\nabla _\\mathbf {r}\\nabla _\\mathbf {r}: \\left[ \\frac{n_{\\sigma }T_{\\sigma }}{2B^2} (\\leftrightarrow \\over {\\mathbf {I}}- \\hat{\\mathbf {b}}\\hat{\\mathbf {b}}) \\right] + \\nabla _\\mathbf {r}\\cdot \\left( \\frac{Z_{\\sigma }n_{\\sigma }}{B^2} \\nabla _{\\mathbf {r}_\\perp } \\varphi _0\\right)\\nonumber \\\\[5pt]\\hspace{28.45274pt} + \\nabla _\\mathbf {r}\\cdot \\left( \\frac{3n_{\\sigma }T_{\\sigma }}{2B^3} \\nabla _{\\mathbf {r}_\\perp } B \\right) + \\nabla _\\mathbf {r}\\cdot \\left( \\frac{n_{\\sigma }T_{\\sigma }}{B^2} \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}\\right)\\nonumber \\\\[5pt]\\hspace{28.45274pt} - \\frac{n_{\\sigma }T_{\\sigma }}{B^2} \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\times \\mathbf {K}.$ Equation (REF ) can be simplified even more employing $\\int B \\left((Iu)^2+\\mu B|\\nabla _\\mathbf {r}\\psi |^2\\right) F_{{\\sigma }0}\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\nonumber \\\\[5pt]= (RB)^2 n_{\\sigma }T_{\\sigma }\\nonumber \\\\[5pt]\\int B \\left((Iu)^2+\\mu B|\\nabla _\\mathbf {r}\\psi |^2\\right)\\frac{u^2/2 + \\mu B}{T_{\\sigma }}F_{{\\sigma }0}\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\nonumber \\\\[5pt]= \\frac{5}{2}(RB)^2 n_{\\sigma }T_{\\sigma }\\nonumber \\\\[5pt]\\int B \\left((Iu)^2+\\mu B|\\nabla _\\mathbf {r}\\psi |^2\\right)\\left(\\frac{u^2/2 + \\mu B}{T_{\\sigma }}-\\frac{3}{2}\\right)F_{{\\sigma }0}\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\nonumber \\\\[5pt]=(RB)^2 n_{\\sigma }T_{\\sigma }\\nonumber \\\\[5pt]\\int B \\left((Iu)^2+\\mu B|\\nabla _\\mathbf {r}\\psi |^2\\right)\\left(\\frac{u^2/2 + \\mu B}{T_{\\sigma }}-\\frac{3}{2}\\right)^2F_{{\\sigma }0}\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\nonumber \\\\[5pt]=\\frac{7}{2}(RB)^2 n_{\\sigma }T_{\\sigma },$ finally giving (REF )." ], [ "Integral of the second-order piece of the\ntransformation of the Maxwellian", "In this Appendix we calculate the integral in velocity space (REF ).", "The integrand $\\left\\langle {\\cal T}_{{\\sigma }, 0}^{*} \\left[ \\mathcal {T}^{-1\\ast }_{{\\sigma }, 2} F_{{\\sigma }0} \\right]^{\\rm lw}\\right\\rangle $ is given in (REF ).", "Using $\\frac{1}{T_{\\sigma }} \\int B\\Psi _{B,{\\sigma }} F_{{\\sigma }0}\\, \\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta =\\nonumber \\\\\\hspace{14.22636pt}- \\frac{3 n_{\\sigma }T_{\\sigma }}{2B^3}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}B + \\frac{n_{\\sigma }T_{\\sigma }}{2B^3}(\\leftrightarrow \\over {\\mathbf {I}}- \\hat{\\mathbf {b}}\\hat{\\mathbf {b}}) : \\nabla _\\mathbf {r}\\nabla _\\mathbf {r}\\mathbf {B}\\cdot \\hat{\\mathbf {b}}\\nonumber \\\\\\hspace{14.22636pt}-\\frac{3 n_{\\sigma }T_{\\sigma }}{2B^4} |\\nabla _{\\mathbf {r}_\\bot } B|^2 + \\frac{n_{\\sigma }T_{\\sigma }}{2B^2} \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}: \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}\\nonumber \\\\\\hspace{14.22636pt}- \\frac{n_{\\sigma }T_{\\sigma }}{2B^2} (\\nabla _\\mathbf {r}\\cdot \\hat{\\mathbf {b}})^2$ and $\\int B\\Bigg ( \\frac{Z_{\\sigma }^2\\lambda _{\\sigma }^2}{2T_{\\sigma }^2} \\left[ \\left\\langle (\\widetilde{\\phi }_{{\\sigma }1}^{\\rm sw})^2 \\right\\rangle \\right]^{\\rm lw}\\nonumber \\\\[5pt]\\hspace{14.22636pt}-\\frac{Z_{\\sigma }^2\\lambda _{\\sigma }^2}{2B T_{\\sigma }}\\partial _\\mu \\left[ \\left\\langle (\\widetilde{\\phi }_{{\\sigma }1}^{\\rm sw})^2 \\right\\rangle \\right]^{\\rm lw}\\Bigg ) F_{{\\sigma }0}\\, \\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta =\\nonumber \\\\[5pt]\\hspace{14.22636pt}- \\frac{Z_{\\sigma }^2\\lambda _{\\sigma }^2}{2T_{\\sigma }} \\int \\partial _{\\mu } \\left(\\left[ \\left\\langle (\\widetilde{\\phi }_{{\\sigma }1}^{\\rm sw})^2 \\right\\rangle \\right]^{\\rm lw}F_{{\\sigma }0} \\right)\\, \\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta = 0,$ we obtain $\\int B \\left\\langle {\\cal T}_{{\\sigma }, 0}^{*}\\left[ \\mathcal {T}^{-1\\ast }_{{\\sigma }, 2} F_{{\\sigma }0} \\right]^{\\rm lw}\\right\\rangle \\, \\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta \\nonumber \\\\\\hspace{28.45274pt}= \\frac{1}{2B^2} (\\leftrightarrow \\over {\\mathbf {I}}- \\hat{\\mathbf {b}}\\hat{\\mathbf {b}}) : \\nabla _\\mathbf {r}\\nabla _\\mathbf {r}( n_{\\sigma }T_{\\sigma }) + \\nabla _\\mathbf {r}\\cdot \\left(\\frac{Z_{\\sigma }n_{\\sigma }}{B^2} \\nabla _{\\mathbf {r}_\\bot } \\varphi _0 \\right)\\nonumber \\\\\\hspace{28.45274pt}-\\frac{1}{2B^3} \\nabla _{\\mathbf {r}_\\bot } B \\cdot \\nabla _{\\mathbf {r}_\\bot }( n_{\\sigma }T_{\\sigma }) -\\frac{3 n_{\\sigma }T_{\\sigma }}{2B^3} \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}B\\nonumber \\\\\\hspace{28.45274pt}+ \\frac{n_{\\sigma }T_{\\sigma }}{2B^3} (\\leftrightarrow \\over {\\mathbf {I}}- \\hat{\\mathbf {b}}\\hat{\\mathbf {b}}) : \\nabla _\\mathbf {r}\\nabla _\\mathbf {r}\\mathbf {B}\\cdot \\hat{\\mathbf {b}}- \\frac{3 n_{\\sigma }T_{\\sigma }}{2B^4}|\\nabla _{\\mathbf {r}_\\bot } B|^2\\nonumber \\\\\\hspace{28.45274pt}+ \\frac{n_{\\sigma }T_{\\sigma }}{2B^2} \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}:\\nabla _\\mathbf {r}\\hat{\\mathbf {b}}- \\frac{n_{\\sigma }T_{\\sigma }}{2B^2} (\\nabla _\\mathbf {r}\\cdot \\hat{\\mathbf {b}})^2.$ Using $(\\leftrightarrow \\over {\\mathbf {I}}- \\hat{\\mathbf {b}}\\hat{\\mathbf {b}}) : \\nabla _\\mathbf {r}\\nabla _\\mathbf {r}\\mathbf {B}\\cdot \\hat{\\mathbf {b}}=\\nonumber \\\\\\hspace{28.45274pt}(\\leftrightarrow \\over {\\mathbf {I}}-\\hat{\\mathbf {b}}\\hat{\\mathbf {b}}): \\nabla _\\mathbf {r}\\nabla _\\mathbf {r}B - B \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}: (\\nabla _\\mathbf {r}\\hat{\\mathbf {b}})^\\mathrm {T} + B | \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}|^2$ and $\\frac{1}{2B^2} (\\leftrightarrow \\over {\\mathbf {I}}- \\hat{\\mathbf {b}}\\hat{\\mathbf {b}}) : \\nabla _\\mathbf {r}\\nabla _\\mathbf {r}(n_{\\sigma }T_{\\sigma }) =\\nonumber \\\\\\hspace{14.22636pt}\\nabla _\\mathbf {r}\\nabla _\\mathbf {r}: \\left[ \\frac{n_{\\sigma }T_{\\sigma }}{2B^2}(\\leftrightarrow \\over {\\mathbf {I}}- \\hat{\\mathbf {b}}\\hat{\\mathbf {b}}) \\right] + \\frac{2}{B^3} \\nabla _{\\mathbf {r}_\\bot } B \\cdot \\nabla _{\\mathbf {r}_\\bot } (n_{\\sigma }T_{\\sigma })\\nonumber \\\\\\hspace{14.22636pt}+ \\frac{1}{B^2} \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}(n_{\\sigma }T_{\\sigma }) + \\frac{n_{\\sigma }T_{\\sigma }}{B^3} (\\leftrightarrow \\over {\\mathbf {I}}- \\hat{\\mathbf {b}}\\hat{\\mathbf {b}}): \\nabla _\\mathbf {r}\\nabla _\\mathbf {r}B\\nonumber \\\\\\hspace{14.22636pt}+ \\frac{5 n_{\\sigma }T_{\\sigma }}{2 B^2} (\\nabla _\\mathbf {r}\\cdot \\hat{\\mathbf {b}})^2 - \\frac{2 n_{\\sigma }T_{\\sigma }}{B^3}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}B\\nonumber \\\\\\hspace{14.22636pt}+ \\frac{n_{\\sigma }T_{\\sigma }}{B^2}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}( \\nabla _\\mathbf {r}\\cdot \\hat{\\mathbf {b}})+ \\frac{n_{\\sigma }T_{\\sigma }}{2B^2} \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}: \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}\\nonumber \\\\\\hspace{14.22636pt}- \\frac{3 n_{\\sigma }T_{\\sigma }}{B^4} |\\nabla _{\\mathbf {r}_\\bot } B|^2,$ equation (REF ) can be rewritten as $\\int B \\left\\langle {\\cal T}_{{\\sigma }, 0}^{*}\\left[ \\mathcal {T}^{-1\\ast }_{{\\sigma }, 2} F_{{\\sigma }0} \\right]^{\\rm lw}\\right\\rangle \\, \\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta =\\nonumber \\\\\\hspace{14.22636pt}\\nabla _\\mathbf {r}\\nabla _\\mathbf {r}: \\left[ \\frac{n_{\\sigma }T_{\\sigma }}{2B^2} (\\leftrightarrow \\over {\\mathbf {I}}- \\hat{\\mathbf {b}}\\hat{\\mathbf {b}}) \\right] + \\nabla _\\mathbf {r}\\cdot \\left(\\frac{Z_{\\sigma }n_{\\sigma }}{B^2} \\nabla _{\\mathbf {r}_\\bot } \\varphi _0 \\right)\\nonumber \\\\\\hspace{14.22636pt}+ \\frac{3}{2B^3} \\nabla _{\\mathbf {r}_\\bot } B \\cdot \\nabla _{\\mathbf {r}_\\bot }(n_{\\sigma }T_{\\sigma }) + \\frac{1}{B^2} \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}( n_{\\sigma }T_{\\sigma })\\nonumber \\\\\\hspace{14.22636pt}- \\frac{7 n_{\\sigma }T_{\\sigma }}{2B^3} \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}B+ \\frac{3n_{\\sigma }T_{\\sigma }}{2B^3} (\\leftrightarrow \\over {\\mathbf {I}}- \\hat{\\mathbf {b}}\\hat{\\mathbf {b}}) : \\nabla _\\mathbf {r}\\nabla _\\mathbf {r}B\\nonumber \\\\\\hspace{14.22636pt}- \\frac{n_{\\sigma }T_{\\sigma }}{2B^2} \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}: (\\nabla _\\mathbf {r}\\hat{\\mathbf {b}})^\\mathrm {T} +\\frac{n_{\\sigma }T_{\\sigma }}{2B^2} | \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}|^2\\nonumber \\\\\\hspace{14.22636pt}- \\frac{9 n_{\\sigma }T_{\\sigma }}{2B^4} |\\nabla _{\\mathbf {r}_\\bot } B|^2 +\\frac{n_{\\sigma }T_{\\sigma }}{B^2} \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}: \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}+\\frac{2n_{\\sigma }T_{\\sigma }}{B^2} (\\nabla _\\mathbf {r}\\cdot \\hat{\\mathbf {b}})^2\\nonumber \\\\\\hspace{14.22636pt}+\\frac{n_{\\sigma }T_{\\sigma }}{B^2} \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}( \\nabla _\\mathbf {r}\\cdot \\hat{\\mathbf {b}}).$ With further manipulations, we find $\\int B \\left\\langle {\\cal T}_{{\\sigma }, 0}^{*}\\left[ \\mathcal {T}^{-1\\ast }_{{\\sigma }, 2} F_{{\\sigma }0} \\right]^{\\rm lw}\\right\\rangle \\, \\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta =\\nonumber \\\\\\hspace{14.22636pt}\\nabla _\\mathbf {r}\\nabla _\\mathbf {r}: \\left[ \\frac{n_{\\sigma }T_{\\sigma }}{2B^2} (\\leftrightarrow \\over {\\mathbf {I}}- \\hat{\\mathbf {b}}\\hat{\\mathbf {b}}) \\right] + \\nabla _\\mathbf {r}\\cdot \\left(\\frac{Z_{\\sigma }n_{\\sigma }}{B^2} \\nabla _{\\mathbf {r}_\\bot } \\varphi _0 \\right)\\nonumber \\\\\\hspace{14.22636pt}+ \\nabla _\\mathbf {r}\\cdot \\left( \\frac{3n_{\\sigma }T_{\\sigma }}{2B^3}\\nabla _{\\mathbf {r}_\\bot } B \\right) + \\nabla _\\mathbf {r}\\cdot \\left( \\frac{n_{\\sigma }T_{\\sigma }}{B^2} \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}\\right)\\nonumber \\\\\\hspace{14.22636pt}-\\frac{n_{\\sigma }T_{\\sigma }}{2B^2} \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}: (\\nabla _\\mathbf {r}\\hat{\\mathbf {b}})^\\mathrm {T}+ \\frac{n_{\\sigma }T_{\\sigma }}{2B^2} | \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}|^2\\nonumber \\\\\\hspace{14.22636pt}+\\frac{n_{\\sigma }T_{\\sigma }}{2B^2} (\\nabla _\\mathbf {r}\\cdot \\hat{\\mathbf {b}})^2.$ Finally, we show that we can combine the last three terms of the previous equation to give a more recognizable term.", "Employing $\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\times \\mathbf {K}= \\frac{1}{2} (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\times \\hat{\\mathbf {b}})^2 - (\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {r}\\hat{\\mathbf {e}}_1) : (\\nabla _\\mathbf {r}\\hat{\\mathbf {e}}_2)^\\mathrm {T},$ $\\nabla _\\mathbf {r}\\hat{\\mathbf {e}}_1 \\cdot (\\nabla _\\mathbf {r}\\hat{\\mathbf {e}}_2)^\\mathrm {T} = (\\nabla _\\mathbf {r}\\hat{\\mathbf {e}}_1 \\cdot \\hat{\\mathbf {b}}) (\\nabla _\\mathbf {r}\\hat{\\mathbf {e}}_2 \\cdot \\hat{\\mathbf {b}}) =\\nonumber \\\\(\\nabla _\\mathbf {r}\\hat{\\mathbf {b}}\\cdot \\hat{\\mathbf {e}}_1) (\\nabla _\\mathbf {r}\\hat{\\mathbf {b}}\\cdot \\hat{\\mathbf {e}}_2) = \\frac{1}{2} \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}\\cdot (\\nabla _\\mathbf {r}\\hat{\\mathbf {b}}\\times \\hat{\\mathbf {b}})^\\mathrm {T},$ and $\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}\\times \\hat{\\mathbf {b}}= (\\nabla _\\mathbf {r}\\hat{\\mathbf {b}})^\\mathrm {T}- (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}) \\hat{\\mathbf {b}}- (\\nabla _\\mathbf {r}\\cdot \\hat{\\mathbf {b}}) (\\leftrightarrow \\over {\\mathbf {I}}-\\hat{\\mathbf {b}}\\hat{\\mathbf {b}}),$ one finds the identity $\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\times \\mathbf {K}= \\frac{1}{2} (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\times \\hat{\\mathbf {b}})^2 + \\frac{1}{2} \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}: \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}-\\frac{1}{2} (\\nabla _\\mathbf {r}\\cdot \\hat{\\mathbf {b}})^2.$ Since $\\nabla _\\mathbf {r}\\hat{\\mathbf {b}}: (\\nabla _\\mathbf {r}\\hat{\\mathbf {b}})^\\mathrm {T} - \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}:\\nabla _\\mathbf {r}\\hat{\\mathbf {b}}= |\\nabla _\\mathbf {r}\\times \\hat{\\mathbf {b}}|^2$ and $\\nabla _\\mathbf {r}\\times \\hat{\\mathbf {b}}=\\hat{\\mathbf {b}}\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\times \\hat{\\mathbf {b}}+ \\hat{\\mathbf {b}}\\times (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\hat{\\mathbf {b}})$ , we obtain $\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\times \\mathbf {K}= \\frac{1}{2} \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}: (\\nabla _\\mathbf {r}\\hat{\\mathbf {b}})^\\mathrm {T} - \\frac{1}{2} |\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {r}\\hat{\\mathbf {b}}|^2 - \\frac{1}{2} (\\nabla _\\mathbf {r}\\cdot \\hat{\\mathbf {b}})^2,$ giving equation (REF ).", "We point out that $\\nabla _\\mathbf {r}\\times \\mathbf {K}$ was computed for the first time by Littlejohn in reference [42]." ], [ "Proof of (", "In this Appendix we prove (REF ).", "To do so, we take the short-wavelength quasineutrality equation to first order, given in (REF ), apply the operator $\\partial _t + B^{-1} (\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {r}\\varphi _0(\\mathbf {r}, t)) \\cdot \\nabla _{\\mathbf {r}_\\bot /\\epsilon _s}$ , and multiply it by $\\varphi _1^{\\rm sw}(\\mathbf {r}, t)$ to find $\\sum _{\\sigma }\\frac{Z_{\\sigma }}{\\lambda _{\\sigma }}\\varphi _1^{\\rm sw}(\\mathbf {r}, t) \\left( \\partial _t + \\frac{Z_{\\sigma }\\tau _{\\sigma }}{B} (\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {r}\\varphi _0(\\mathbf {r}, t) ) \\cdot \\nabla _{\\mathbf {r}_\\bot /\\epsilon _{\\sigma }} \\right) \\int B\\Bigg [ \\nonumber \\\\[5pt]\\hspace{14.22636pt} -Z_{\\sigma }\\lambda _{\\sigma }\\widetilde{\\phi }_{{\\sigma }1}^{\\rm sw}\\left(\\mathbf {r}-\\epsilon _{\\sigma }\\mbox{$\\rho $}(\\mathbf {r},\\mu ,\\theta ),\\mu ,\\theta ,t\\right)\\frac{F_{{\\sigma }0}(\\mathbf {r},u,\\mu ,t)}{T_{{\\sigma }}(\\mathbf {r},t)}\\nonumber \\\\[5pt]\\hspace{14.22636pt}+ F_{{\\sigma }1}^{\\rm sw}\\left(\\mathbf {r}-\\epsilon _{\\sigma }\\mbox{$\\rho $}(\\mathbf {r},\\mu ,\\theta ),u,\\mu ,t\\right)\\Bigg ]\\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta = 0.$ Since $ \\varphi _1^{\\rm sw}(\\mathbf {r},t) =\\phi _{{\\sigma }1}^{\\rm sw}(\\mathbf {r}-\\epsilon _{\\sigma }\\mbox{$\\rho $}(\\mathbf {r},\\mu ,\\theta ),\\mu ,\\theta ,t)+ O(\\epsilon _{\\sigma }),$ $\\nabla _\\mathbf {r}\\varphi _0 (\\mathbf {r}, t) = \\nabla _\\mathbf {R}\\varphi _0(\\mathbf {R}, t) + O(\\epsilon _{\\sigma })$ and $\\nabla _{\\mathbf {r}_\\bot /\\epsilon _\\sigma } = \\nabla _{\\mathbf {R}_\\bot /\\epsilon _{\\sigma }} + O(\\epsilon _{\\sigma }),$ we find that (REF ) becomes $\\sum _{\\sigma }\\frac{Z_{\\sigma }}{\\lambda _{\\sigma }} \\int B\\Bigg [\\phi _{{\\sigma }1}^{\\rm sw}\\Bigg ( \\partial _t + \\frac{Z_{\\sigma }\\tau _{\\sigma }}{B} (\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}\\varphi _0 ) \\cdot \\nabla _{\\mathbf {R}_\\bot /\\epsilon _{\\sigma }} \\Bigg ) \\nonumber \\\\[5pt]\\hspace{14.22636pt}\\Bigg (-\\frac{Z_{\\sigma }\\lambda _{\\sigma }\\widetilde{\\phi }_{{\\sigma }1}^{\\rm sw}}{T_{{\\sigma }}} F_{{\\sigma }0}+ F_{{\\sigma }1}^{\\rm sw}\\Bigg )\\Bigg ]^{\\rm lw}\\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta = O(\\epsilon _s).$ In this expression, the functions $\\phi _{{\\sigma }1}^{\\rm sw}$ and $F_{{\\sigma }1}^{\\rm sw}$ are evaluated at $\\mathbf {R}= \\mathbf {r}- \\epsilon _{\\sigma }\\mbox{$\\rho $}(\\mathbf {r}, \\mu , \\theta )$ , but after the coarse-grain average we can Taylor expand and, to lowest order, they can be evaluated at $\\mathbf {R}= \\mathbf {r}$ .", "Thus, we find $-\\sum _{\\sigma }Z_{\\sigma }^2 \\int \\frac{B F_{{\\sigma }0} }{2 T_{{\\sigma }}}\\Bigg ( \\partial _t \\left[ (\\widetilde{\\phi }_{{\\sigma }1}^{\\rm sw})^2\\right]^{\\rm lw}\\nonumber \\\\[5pt] \\hspace{14.22636pt}+ \\frac{Z_{\\sigma }\\tau _{\\sigma }}{B} (\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}\\varphi _0 ) \\cdot \\nabla _{\\mathbf {R}_\\bot /\\epsilon _{\\sigma }}\\left[ (\\widetilde{\\phi }_{{\\sigma }1}^{\\rm sw})^2\\right]^{\\rm lw}\\Bigg )\\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta \\nonumber \\\\[5pt] \\hspace{14.22636pt}+ \\sum _{\\sigma }\\frac{Z_{\\sigma }}{\\lambda _{\\sigma }} \\int B \\Bigg [\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\Bigg ( \\partial _t F_{{\\sigma }1}^{\\rm sw}\\nonumber \\\\[5pt] \\hspace{14.22636pt}+ \\frac{Z_{\\sigma }\\tau _{\\sigma }}{B} (\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}\\varphi _0 ) \\cdot \\nabla _{\\mathbf {R}_\\bot /\\epsilon _{\\sigma }}F_{{\\sigma }1}^{\\rm sw}\\Bigg )\\Bigg ]^{\\rm lw}\\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta = O(\\epsilon _s).$ Employing that the time derivative of a long-wavelength contribution is small by $\\epsilon _s^2$ and that $\\nabla _{\\mathbf {R}_\\bot /\\epsilon _{\\sigma }}g^{\\rm lw}\\sim \\epsilon _{\\sigma }$ , we obtain that $\\sum _{\\sigma }\\frac{Z_{\\sigma }}{\\lambda _{\\sigma }}\\int B \\Bigg [\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\Bigg ( \\partial _tF_{{\\sigma }1}^{\\rm sw}\\nonumber \\\\[5pt] \\hspace{14.22636pt}+ \\frac{Z_{\\sigma }\\tau _{\\sigma }}{B} (\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}\\varphi _0 ) \\cdot \\nabla _{\\mathbf {R}_\\bot /\\epsilon _{\\sigma }}F_{{\\sigma }1}^{\\rm sw}\\Bigg )\\Bigg ]^{\\rm lw}\\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta = O(\\epsilon _s).$ Now, we use (REF ) in (REF ), getting $- \\sum _{\\sigma }\\int B \\Bigg [\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\Bigg ( u\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}F_{{\\sigma }1}^{\\rm sw}\\nonumber \\\\[5pt] \\hspace{14.22636pt}+\\frac{u^2}{B} (\\hat{\\mathbf {b}}\\times (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}))\\cdot \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}F_{{\\sigma }1}^{\\rm sw}\\nonumber \\\\[5pt] \\hspace{14.22636pt} +\\frac{\\mu }{B}(\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}B)\\cdot \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}F_{{\\sigma }1}^{\\rm sw}\\Bigg ) \\Bigg ]^{\\rm lw}\\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta \\nonumber \\\\[5pt]\\hspace{14.22636pt}+ \\sum _{{\\sigma },{\\sigma }^{\\prime }} \\int B\\Bigg [\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\Bigg ( {\\cal T}_{NP, {\\sigma }}^* C_{\\sigma \\sigma ^\\prime }\\Bigg [{\\mathbb {T}_{{\\sigma },0}}F_{\\sigma 1}^{\\rm sw}\\nonumber \\\\[5pt] \\hspace{14.22636pt}-\\frac{Z_{\\sigma }\\lambda _{\\sigma }}{T_{\\sigma }}{\\mathbb {T}_{{\\sigma },0}}\\tilde{\\phi }_{{\\sigma }1}^{\\rm sw}{\\cal T}_{{\\sigma },0}^{-1*} F_{{\\sigma }0}, {\\cal T}_{{\\sigma }^{\\prime },0}^{-1*}F_{{\\sigma }^{\\prime } 0}\\Bigg ]\\nonumber \\\\[5pt]\\hspace{14.22636pt}+ \\ \\frac{\\lambda _{\\sigma }}{\\lambda _{{\\sigma }^{\\prime }}}{\\cal T}_{NP, {\\sigma }}^*C_{\\sigma \\sigma ^\\prime } \\Bigg [ {\\cal T}_{{\\sigma },0}^{-1 *}F_{{\\sigma }0} ,{\\mathbb {T}_{{\\sigma }^{\\prime },0}}F_{\\sigma ^{\\prime } 1}^{\\rm sw}\\nonumber \\\\[5pt] \\hspace{14.22636pt}-\\frac{Z_{{\\sigma }^{\\prime }}\\lambda _{{\\sigma }^{\\prime }}}{T_{{\\sigma }^{\\prime }}}{\\mathbb {T}_{{\\sigma }^{\\prime },0}}\\tilde{\\phi }_{{\\sigma }^{\\prime }1}^{\\rm sw}{\\cal T}_{{\\sigma }^{\\prime },0}^{-1 *}F_{{\\sigma }^{\\prime } 0}\\Bigg ] \\Bigg ) \\Bigg ]^{\\rm lw}\\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta =O(\\epsilon _s).$ Here, we have used the fact that $\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle $ does not depend on $u$ , and the relations $\\left[ \\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\nabla _{\\mathbf {R}_\\bot /\\epsilon _{\\sigma }} \\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\right]^{\\rm lw}= \\frac{1}{2} \\nabla _{\\mathbf {R}_\\bot /\\epsilon _{\\sigma }} \\left[ \\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle ^2 \\right]^{\\rm lw}= O(\\epsilon _{\\sigma })$ and $\\left[ \\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle (\\hat{\\mathbf {b}}\\times \\nabla _{\\mathbf {R}_\\bot /\\epsilon _{\\sigma }} \\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle ) \\cdot \\nabla _{\\mathbf {R}_\\bot /\\epsilon _{\\sigma }} F_{{\\sigma }1}^{\\rm sw}\\right]^{\\rm lw}= \\nonumber \\\\\\hspace{28.45274pt}- \\frac{1}{2} \\hat{\\mathbf {b}}\\cdot \\nabla _{\\mathbf {R}_\\bot /\\epsilon _{\\sigma }} \\times \\left[ F_{{\\sigma }1}^{\\rm sw}\\nabla _{\\mathbf {R}_\\bot /\\epsilon _{\\sigma }} \\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle ^2 \\right]^{\\rm lw}=O( \\epsilon _{\\sigma }).$ Finally, to relate (REF ) to (REF ), we employ that, up to terms of order $\\epsilon _{\\sigma }$ , $ \\int B \\left[ \\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle {\\cal T}_{NP, {\\sigma }} C_{{\\sigma }{\\sigma }^\\prime }^{\\rm sw}\\right]^{\\rm lw}\\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta =\\nonumber \\\\[5pt]\\hspace{42.67912pt}- \\int B \\left[\\tilde{\\phi }_{{\\sigma }1}^{\\rm sw}{\\cal T}_{NP, {\\sigma }} C_{{\\sigma }{\\sigma }^\\prime }^{\\rm sw}\\right]^{\\rm lw}\\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta ,$ where $C_{{\\sigma }{\\sigma }^\\prime }^{\\rm sw}(\\mathbf {r}, \\mathbf {v}, t)$ is the collision operator applied on a function with wavelengths on the order of the sound gyroradius.", "To prove this, we begin with the particle conservation property of the collision operator, that gives $\\left[ \\varphi _1^{\\rm sw}(\\mathbf {r}, t) \\int C_{{\\sigma }{\\sigma }^\\prime }^{\\rm sw}(\\mathbf {r}, \\mathbf {v}, t) \\mbox{d}^3 v \\right]^{\\rm lw}= 0.$ Using (REF ) this equation becomes $\\int B \\Bigg [ \\phi _{{\\sigma }1}^{\\rm sw}(\\mathbf {r}- \\epsilon _{\\sigma }\\mbox{$\\rho $}(\\mathbf {r}, \\mu , \\theta ), \\mu , \\theta , t)\\times \\nonumber \\\\[5pt]\\hspace{14.22636pt}C_{{\\sigma }{\\sigma }^\\prime }^{\\rm sw}(\\mathbf {r}, u \\hat{\\mathbf {b}}+ \\mbox{$\\rho $}(\\mathbf {r}, \\mu , \\theta ) \\times \\mathbf {B}, t)\\Bigg ]^{\\rm lw}\\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta = O(\\epsilon _s).$ Since we are only considering the long wavelength component, we can Taylor expand around $\\mathbf {r}$ , leaving $\\int B \\Bigg [ \\phi _{{\\sigma }1}^{\\rm sw}(\\mathbf {r}, \\mu , \\theta , t)\\times \\nonumber \\\\[5pt]\\hspace{14.22636pt}C_{{\\sigma }{\\sigma }^\\prime }^{\\rm sw}(\\mathbf {r}+ \\epsilon _{\\sigma }\\mbox{$\\rho $}(\\mathbf {r}, \\mu , \\theta ), u \\hat{\\mathbf {b}}+ \\mbox{$\\rho $}(\\mathbf {r}, \\mu , \\theta ) \\times \\mathbf {B}, t) \\Bigg ]^{\\rm lw}\\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta \\nonumber \\\\[5pt]\\hspace{14.22636pt}=O(\\epsilon _s),$ which is equivalent to (REF ).", "Substituting (REF ) into (REF ), flux surface averaging and integrating by parts finally yields $\\sum _{\\sigma }\\Bigg \\langle \\int B \\Bigg [F_{{\\sigma }1}^{\\rm sw}\\Bigg ( u\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\nonumber \\\\[5pt] \\hspace{14.22636pt}+\\frac{u^2}{B} (\\hat{\\mathbf {b}}\\times (\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}\\hat{\\mathbf {b}}))\\cdot \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\nonumber \\\\[5pt] \\hspace{14.22636pt} +\\frac{\\mu }{B}(\\hat{\\mathbf {b}}\\times \\nabla _\\mathbf {R}B)\\cdot \\nabla _{\\mathbf {R}_\\perp /\\epsilon _{\\sigma }}\\langle \\phi _{{\\sigma }1}^{\\rm sw}\\rangle \\Bigg ) \\Bigg ]^{\\rm lw}\\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta \\Bigg \\rangle _\\psi \\nonumber \\\\[5pt]\\hspace{14.22636pt}- \\sum _{{\\sigma },{\\sigma }^{\\prime }} \\Bigg \\langle \\int B\\Bigg [\\tilde{\\phi }_{{\\sigma }1}^{\\rm sw}\\Bigg ( {\\cal T}_{NP, {\\sigma }}^* C_{\\sigma \\sigma ^\\prime }\\Bigg [{\\mathbb {T}_{{\\sigma },0}}F_{\\sigma 1}^{\\rm sw}\\nonumber \\\\[5pt] \\hspace{14.22636pt}-\\frac{Z_{\\sigma }\\lambda _{\\sigma }}{T_{\\sigma }}{\\mathbb {T}_{{\\sigma },0}}\\tilde{\\phi }_{{\\sigma }1}^{\\rm sw}{\\cal T}_{{\\sigma },0}^{-1*} F_{{\\sigma }0}, {\\cal T}_{{\\sigma }^{\\prime },0}^{-1*}F_{{\\sigma }^{\\prime } 0}\\Bigg ]\\nonumber \\\\[5pt]\\hspace{14.22636pt}+ \\ \\frac{\\lambda _{\\sigma }}{\\lambda _{{\\sigma }^{\\prime }}}{\\cal T}_{NP, {\\sigma }}^*C_{\\sigma \\sigma ^\\prime } \\Bigg [ {\\cal T}_{{\\sigma },0}^{-1 *}F_{{\\sigma }0} ,{\\mathbb {T}_{{\\sigma }^{\\prime },0}}F_{\\sigma ^{\\prime } 1}^{\\rm sw}\\nonumber \\\\[5pt] \\hspace{14.22636pt}-\\frac{Z_{{\\sigma }^{\\prime }}\\lambda _{{\\sigma }^{\\prime }}}{T_{{\\sigma }^{\\prime }}}{\\mathbb {T}_{{\\sigma }^{\\prime },0}}\\tilde{\\phi }_{{\\sigma }^{\\prime }1}^{\\rm sw}{\\cal T}_{{\\sigma }^{\\prime },0}^{-1 *}F_{{\\sigma }^{\\prime } 0}\\Bigg ] \\Bigg ) \\Bigg ]^{\\rm lw}\\mbox{d}u \\mbox{d}\\mu \\mbox{d}\\theta \\Bigg \\rangle _\\psi .$ From this expression and (REF ) we obtain (REF ) by integrating by parts in $\\mu $ ." ], [ "Solvability conditions of the gyrokinetic\nFokker-Planck equation of any order", "We want to prove that the solvability conditions in subsection REF are the only ones to second order.", "We do this by showing that, to general order, only the flux-surface averaged zeroth and second moments of the Fokker-Planck equation can give solvability conditions.", "To order $\\epsilon _s^j$ the gyrokinetic Fokker-Planck equation for species ${\\sigma }$ can be written as (we drop the superindex lw in this appendix) $\\tau _{\\sigma }\\lambda _{\\sigma }^{-j}&\\left(u\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}-\\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B\\partial _u\\right) G_{{\\sigma }j}\\nonumber \\\\[5pt]&-\\tau _{\\sigma }\\sum _{{\\sigma }^{\\prime }}\\Bigg ({\\cal T}_{{\\sigma },0}^*C_{{\\sigma }{\\sigma }^{\\prime }}\\Bigg [\\lambda _{\\sigma }^{-j}{\\cal T}_{{\\sigma },0}^{-1*}G_{{\\sigma }j},{\\cal T}_{{\\sigma }^{\\prime },0}^{-1*}F_{{\\sigma }^{\\prime } 0}\\Bigg ]\\nonumber \\\\[5pt]&+{\\cal T}_{{\\sigma },0}^*C_{{\\sigma }{\\sigma }^{\\prime }}\\Bigg [{\\cal T}_{{\\sigma },0}^{-1*}F_{{\\sigma }0},\\lambda _{{\\sigma }^{\\prime }}^{-j}{\\cal T}_{{\\sigma }^{\\prime },0}^{-1*}G_{{\\sigma }^{\\prime } j}\\Bigg ]\\Bigg )=\\tau _{\\sigma }\\lambda _{\\sigma }^{-j} R_{{\\sigma }j},$ where $R_{{\\sigma }j}$ collects terms that do not contain $\\langle F_{{\\sigma }j}\\rangle $ for any ${\\sigma }$ .", "To any order, $\\langle G_{{\\sigma }j}\\rangle = G_{{\\sigma }j}$ differs from $\\langle F_{{\\sigma }j}\\rangle $ , at most, in terms that have been determined by lowest order equations.", "We recall that the gyrophase-dependent piece of the distribution function to order $O(\\epsilon _s^j)$ , $F_{{\\sigma }j} - \\langle F_{{\\sigma }j}\\rangle $ , has been determined by the Fokker-Planck equation of order $O(\\epsilon _s^{j-1})$ .", "Also, we point out that we have introduced the factor $\\tau _{\\sigma }\\lambda _{\\sigma }^{-j}$ in (REF ) because it is convenient for the proof that follows.", "We must study the solvability conditions for the set of equations (REF ) when ${\\sigma }$ runs from 1 to $N$ , with $N$ the number of different species.", "To this end, it is appropriate to work in the vector space ${\\cal F}^N :={\\cal F}({\\cal P}_1)\\times \\dots \\times {\\cal F}({\\cal P}_{\\sigma })\\times \\dots \\times {\\cal F}({\\cal P}_N) ,$ which is the cartesian product of the sets of functions on the phase spaces of the different species.", "Define $G_j = [ G_{1 j},\\dots , G_{{\\sigma }j},\\dots , G_{N j}]$ and $S_j = [\\tau _1 \\lambda _1^{-j}R_1,\\dots ,\\tau _{\\sigma }\\lambda _{\\sigma }^{-j} R_{\\sigma },\\dots , \\tau _N\\lambda _N^{-j} R_N] \\in {\\cal F}^N$ .", "On ${\\cal F}^N$ , the set of equations (REF ) can be rewritten as ${\\cal L}_j G_j = S_j,$ where $({\\cal L}_j G_j)_{\\sigma }&:=\\tau _{\\sigma }\\lambda _{\\sigma }^{-j}\\left(u\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}-\\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B\\partial _u\\right) G_{{\\sigma }j}\\nonumber \\\\[5pt]&-\\tau _{\\sigma }\\sum _{{\\sigma }^{\\prime }}\\Bigg ({\\cal T}_{{\\sigma },0}^*C_{{\\sigma }{\\sigma }^{\\prime }}\\Bigg [\\lambda _{\\sigma }^{-j}{\\cal T}_{{\\sigma },0}^{-1*}G_{{\\sigma }j},{\\cal T}_{{\\sigma }^{\\prime },0}^{-1*}F_{{\\sigma }^{\\prime } 0}\\Bigg ]\\nonumber \\\\[5pt]&+{\\cal T}_{{\\sigma },0}^*C_{{\\sigma }{\\sigma }^{\\prime }}\\Bigg [{\\cal T}_{{\\sigma },0}^{-1*}F_{{\\sigma }0},\\lambda _{{\\sigma }^{\\prime }}^{-j}{\\cal T}_{{\\sigma }^{\\prime },0}^{-1*}G_{{\\sigma }^{\\prime } j}\\Bigg ]\\Bigg ).$ The solvability conditions are defined by functions $K\\in {\\cal F}^N$ satisfying $\\sum _{\\sigma }\\int B K_{\\sigma }({\\cal L}_j G_j)_{\\sigma }\\mbox{d}^3 R\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta = 0$ for every $G_j\\in {\\cal F}^N$ .", "Then, equation (REF ) implies that $\\sum _{\\sigma }\\tau _{\\sigma }\\lambda _{\\sigma }^{-j}\\int B K_{\\sigma }R_{{\\sigma }j}\\mbox{d}^3 R\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta = 0.$ Let us denote by $K =[K_1,\\dots ,K_{\\sigma },\\dots , K_N]$ and $G =[G_1,\\dots ,G_{\\sigma },\\dots , G_N]$ two arbitrary elements of ${\\cal F}^N$ .", "Then, a natural scalar product is defined by $(K \\vert G)=\\sum _{\\sigma }\\int B(\\mathbf {R})K_{\\sigma }(\\mathbf {R},u,\\mu ,\\theta )G_{\\sigma }(\\mathbf {R},u,\\mu ,\\theta )\\mbox{d}^3 R\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta .$ The question about solvability conditions can be expressed in terms of the scalar product.", "Our aim is to find those $K\\in {\\cal F}^N$ such that $(K\\vert {\\cal L}_j G_j) = 0$ for every $G_j\\in {\\cal F}^N$ .", "Since the scalar product is non-degenerate, the condition for $K$ is equivalent to $({\\cal L}_j^\\dagger K \\vert G_j) = 0$ , where ${\\cal L}_j^\\dagger $ is the adjoint of ${\\cal L}_j$ .", "Therefore, the solvability conditions derived to $j$ th order are given by the equations $(K\\vert S_j)= 0, \\quad K\\in \\mbox{Ker}({\\cal L}_j^\\dagger ).$ Of course, it might happen that some of these equations be trivial identities that do not add new conditions on lower-order quantities.", "The important point is that every non-trivial solvability condition is found by calculating all of the equations (REF ).", "We turn to compute ${\\cal L}_j^\\dagger $ .", "It is obvious that the piece of ${\\cal L}_j$ associated to parallel streaming, ${\\cal L}_{j||}$ , defined by $({\\cal L}_{j ||} G_j)_{\\sigma }:=\\tau _{\\sigma }\\lambda _{\\sigma }^{-j}\\left(u\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}-\\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B\\partial _u\\right) G_{{\\sigma }j},$ is antisymmetric.", "That is, $(K\\vert {\\cal L}_{j ||} G_j) = - ({\\cal L}_{j ||} K\\vert G_j),$ for any $K\\in {\\cal F}^N$ .", "In other words, ${\\cal L}_{j ||}^\\dagger =-{\\cal L}_{j ||}$ .", "It will also be useful to note that $({\\cal L}_{j ||} G_j)_{\\sigma }= F_{{\\sigma }0}({\\cal L}_{j ||} {\\hat{G}}_j)_{\\sigma },$ with ${\\hat{G}}_{{\\sigma }j} := G_{{\\sigma }j}/F_{{\\sigma }0}$ , and $F_{{\\sigma }0}(\\mathbf {R},u,\\mu )=\\frac{n_{{\\sigma }}(\\psi )}{(2\\pi T_{{\\sigma }}(\\psi ))^{3/2}}\\exp \\left(-\\frac{\\mu B(\\psi ,\\Theta ) + u^2/2}{T_{{\\sigma }}(\\psi )}\\right).$ In order to find the adjoint of the piece of ${\\cal L}_j$ corresponding to collisions, ${\\cal L}_{j \\, coll} = {\\cal L}_j -{\\cal L}_{j ||},$ we need to prove a preliminary property.", "Define ${\\hat{C}}_{{\\sigma }{\\sigma }^{\\prime }}[{\\hat{G}}_{{\\sigma }j},{\\hat{G}}_{{\\sigma }^{\\prime } j}]&=C_{{\\sigma }{\\sigma }^{\\prime }}\\Bigg [\\lambda _{\\sigma }^{-j}{\\cal T}_{{\\sigma },0}^{-1*}G_{{\\sigma }j},{\\cal T}_{{\\sigma }^{\\prime },0}^{-1*}F_{{\\sigma }^{\\prime } 0}\\Bigg ]\\nonumber \\\\[5pt]&+C_{{\\sigma }{\\sigma }^{\\prime }}\\Bigg [{\\cal T}_{{\\sigma },0}^{-1*}F_{{\\sigma }0},\\lambda _{{\\sigma }^{\\prime }}^{-j}{\\cal T}_{{\\sigma }^{\\prime },0}^{-1*}G_{{\\sigma }^{\\prime } j}\\Bigg ].$ From definition (REF ) one obtains ${\\hat{C}}_{{\\sigma }{\\sigma }^{\\prime }}[{\\hat{G}}_{{\\sigma }j},{\\hat{G}}_{{\\sigma }^{\\prime } j}]=\\nonumber \\\\[5pt]\\hspace{14.22636pt}\\gamma _{{\\sigma }{\\sigma }^{\\prime }}\\nabla _{\\mathbf {v}}\\cdot \\int \\leftrightarrow \\over {\\mathbf {W}}(\\tau _{\\sigma }\\mathbf {v}- \\tau _{{\\sigma }^{\\prime }}\\mathbf {v}^{\\prime })\\cdot \\nonumber \\\\[5pt]\\hspace{14.22636pt}\\Bigg (\\frac{\\tau _{\\sigma }}{\\lambda _{\\sigma }^{j}}\\nabla _\\mathbf {v}\\hat{g}_{\\sigma }-\\frac{\\tau _{{\\sigma }^{\\prime }}}{\\lambda _{{\\sigma }^{\\prime }}^{j}}\\nabla _{\\mathbf {v}^{\\prime }} \\hat{g}_{{\\sigma }^{\\prime }}\\Big )f_{{\\sigma }0}(\\mathbf {v})f_{{\\sigma }^{\\prime } 0}(\\mathbf {v}^{\\prime })\\mbox{d}^3v^{\\prime }.$ Here, to ease the notation, we understand $f_{{\\sigma }0} \\equiv {\\cal T}_{{\\sigma },0}^{-1*}F_{{\\sigma }0}$ and $\\hat{g}_{\\sigma }\\equiv {\\cal T}_{{\\sigma },0}^{-1*}{\\hat{G}}_{{\\sigma }j}$ .", "To get (REF ) we have used $\\nabla _\\mathbf {v}f_{{\\sigma }0} = -\\frac{1}{T_{\\sigma }}\\mathbf {v}f_{{\\sigma }0},$ $T_{\\sigma }= T_{{\\sigma }^{\\prime }}, \\mbox{ for every pair ${\\sigma }$, ${\\sigma }^{\\prime }$,}$ and $\\leftrightarrow \\over {\\mathbf {W}}(\\tau _{\\sigma }\\mathbf {v}- \\tau _{{\\sigma }^{\\prime }}\\mathbf {v}^{\\prime })\\cdot (\\tau _{\\sigma }\\mathbf {v}- \\tau _{{\\sigma }^{\\prime }}\\mathbf {v}^{\\prime }) \\equiv 0,\\mbox{ for every pair ${\\sigma }$, ${\\sigma }^{\\prime }$.", "}$ The operator ${\\hat{C}}_{{\\sigma }{\\sigma }^{\\prime }}[{\\hat{G}}_{{\\sigma }j},{\\hat{G}}_{{\\sigma }^{\\prime }j}]$ does not have nice symmetry properties with respect to the scalar product, but its symmetrization in ${\\sigma }$ and ${\\sigma }^{\\prime }$ does.", "A simple integration by parts yields the following symmetric expression for any pair of functions $k_{\\sigma }(\\mathbf {v})$ and $k_{{\\sigma }^{\\prime }}(\\mathbf {v})$ : $\\tau _{\\sigma }\\int k_{\\sigma }(\\mathbf {v}){\\cal T}_{{\\sigma },0}^{*}{\\hat{C}}_{{\\sigma }{\\sigma }^{\\prime }}[{\\hat{G}}_{{\\sigma }j},{\\hat{G}}_{{\\sigma }^{\\prime } j}]\\mbox{d}^3 v\\nonumber \\\\[5pt]\\hspace{14.22636pt}+\\tau _{{\\sigma }^{\\prime }} \\int k_{{\\sigma }^{\\prime }}(\\mathbf {v}){\\cal T}_{{\\sigma }^{\\prime },0}^{*}{\\hat{C}}_{{\\sigma }^{\\prime }{\\sigma }}[{\\hat{G}}_{{\\sigma }^{\\prime } j},{\\hat{G}}_{{\\sigma }j}]\\mbox{d}^3 v=\\nonumber \\\\[5pt]\\hspace{14.22636pt}-\\gamma _{{\\sigma }{\\sigma }^{\\prime }}\\int \\left(\\tau _{\\sigma }\\nabla _\\mathbf {v}k_{\\sigma }- \\tau _{{\\sigma }^{\\prime }}\\nabla _{\\mathbf {v}^{\\prime }} k_{{\\sigma }^{\\prime }} \\right)\\cdot \\leftrightarrow \\over {\\mathbf {W}}(\\tau _{\\sigma }\\mathbf {v}- \\tau _{{\\sigma }^{\\prime }}\\mathbf {v}^{\\prime })\\cdot \\nonumber \\\\[5pt]\\hspace{14.22636pt}\\left( \\tau _{\\sigma }\\nabla _\\mathbf {v}\\left(\\lambda _{\\sigma }^{-j} \\hat{g}_{\\sigma }\\right) -\\tau _{{\\sigma }^{\\prime }}\\nabla _{\\mathbf {v}^{\\prime }}\\left(\\lambda _{{\\sigma }^{\\prime }}^{-j} \\hat{g}_{{\\sigma }^{\\prime }}\\right)\\right)f_{{\\sigma }0}(\\mathbf {v})f_{{\\sigma }^{\\prime } 0}(\\mathbf {v}^{\\prime })\\mbox{d}^3v\\mbox{d}^3v^{\\prime }.$ Hence, denoting $K_{\\sigma }\\equiv {\\cal T}_{{\\sigma },0}^{*}k_{\\sigma }$ , we can write $\\tau _{\\sigma }\\int k_{\\sigma }(\\mathbf {v}){\\cal T}_{{\\sigma },0}^{*}{\\hat{C}}_{{\\sigma }{\\sigma }^{\\prime }}[{\\hat{G}}_{{\\sigma }j},{\\hat{G}}_{{\\sigma }^{\\prime } j}]\\mbox{d}^3 v\\nonumber \\\\[5pt]\\hspace{14.22636pt}+\\tau _{{\\sigma }^{\\prime }}\\int k_{{\\sigma }^{\\prime }}(\\mathbf {v}){\\cal T}_{{\\sigma }^{\\prime },0}^{*}{\\hat{C}}_{{\\sigma }^{\\prime }{\\sigma }}[{\\hat{G}}_{{\\sigma }^{\\prime } j},{\\hat{G}}_{{\\sigma }j}]\\mbox{d}^3 v=\\nonumber \\\\[5pt]\\hspace{14.22636pt}\\frac{\\tau _{\\sigma }}{\\lambda _{\\sigma }^{j}}\\int {\\cal T}_{{\\sigma },0}^{*}{\\hat{C}}_{{\\sigma }{\\sigma }^{\\prime }}\\left[\\lambda _{\\sigma }^{j} K_{\\sigma },\\lambda _{{\\sigma }^{\\prime }}^{j} K_{{\\sigma }^{\\prime }}\\right]\\hat{g}_{\\sigma }(\\mathbf {v})\\mbox{d}^3 v\\nonumber \\\\[5pt]\\hspace{14.22636pt}+\\frac{\\tau _{{\\sigma }^{\\prime }}}{\\lambda _{{\\sigma }^{\\prime }}^{j}} \\int {\\cal T}_{{\\sigma }^{\\prime },0}^{*}{\\hat{C}}_{{\\sigma }^{\\prime }{\\sigma }}\\left[\\lambda _{{\\sigma }^{\\prime }}^{j} K_{{\\sigma }^{\\prime }},\\lambda _{{\\sigma }}^{j} K_{{\\sigma }}\\right]\\hat{g}_{{\\sigma }^{\\prime }}(\\mathbf {v})\\mbox{d}^3 v.$ Thus, for every $K\\in {\\cal F}^N$ , $(K\\vert {\\cal L}_j G)=\\nonumber \\\\[5pt]\\hspace{14.22636pt}-\\sum _{\\sigma }\\int \\Bigg [\\frac{\\tau _{\\sigma }B}{\\lambda _{\\sigma }^j F_{{\\sigma }0}}\\left(u\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}-\\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B\\partial _u\\right)F_{{\\sigma }0}K_{\\sigma }\\Bigg ]G_{{\\sigma }j}\\mbox{d}^3 R\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\nonumber \\\\[5pt]\\hspace{14.22636pt}-\\sum _{{\\sigma },{\\sigma }^{\\prime }}\\int \\frac{\\tau _{\\sigma }B}{\\lambda _{{\\sigma }}^{j}F_{{\\sigma }0}}{\\cal T}_{{\\sigma },0}^{*}{\\hat{C}}_{{\\sigma }{\\sigma }^{\\prime }}\\left[\\lambda _{{\\sigma }}^{j}K_{{\\sigma }},\\lambda _{{\\sigma }^{\\prime }}^{j}K_{{\\sigma }^{\\prime }}\\right]G_{{\\sigma }j}\\mbox{d}^3 R\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta .$ This means that for any $K\\in {\\cal F}^N$ , the action of the adjoint of ${\\cal L}_j$ is given by $({\\cal L}_j^\\dagger K)_{\\sigma }=-\\Bigg [\\frac{\\tau _{\\sigma }}{\\lambda _{\\sigma }^j F_{{\\sigma }0}}\\left(u\\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}-\\mu \\hat{\\mathbf {b}}\\cdot \\nabla _\\mathbf {R}B\\partial _u\\right)F_{{\\sigma }0}K_{\\sigma }\\Bigg ]\\nonumber \\\\[5pt]\\hspace{14.22636pt}-\\sum _{{\\sigma }^{\\prime }}\\frac{\\tau _{\\sigma }}{\\lambda _{{\\sigma }}^{j}F_{{\\sigma }0}}{\\cal T}_{{\\sigma },0}^{*}{\\hat{C}}_{{\\sigma }{\\sigma }^{\\prime }}\\left[\\lambda _{{\\sigma }}^{j}K_{{\\sigma }},\\lambda _{{\\sigma }^{\\prime }}^{j}K_{{\\sigma }^{\\prime }}\\right].$ An entropy argument similar to the one employed in subsection REF can be used here to obtain the solutions of ${\\cal L}_j^\\dagger K = 0$ .", "Multiplying the equation ${\\cal L}_j^\\dagger K = 0$ by $\\lambda _{\\sigma }^j B K_{\\sigma }F_{{\\sigma }0}$ , integrating over $u,\\mu $ , and $\\theta $ , flux-surface averaging, and summing over all of the species gives $-\\left\\langle \\sum _{{\\sigma },{\\sigma }^{\\prime }}\\tau _{{\\sigma }}\\int B K_{\\sigma }{\\cal T}_{{\\sigma },0}^{*}{\\hat{C}}_{{\\sigma }{\\sigma }^{\\prime }}\\left[\\lambda _{{\\sigma }}^{j}K_{{\\sigma }},\\lambda _{{\\sigma }^{\\prime }}^{j}K_{{\\sigma }^{\\prime }}\\right]\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\right\\rangle _\\psi = 0,$ that can be recasted into $\\Bigg \\langle \\sum _{{\\sigma },{\\sigma }^{\\prime }}\\frac{\\gamma _{{\\sigma }{\\sigma }^{\\prime }}}{2}\\int \\left(\\tau _{\\sigma }\\nabla _\\mathbf {v}k_{\\sigma }-\\tau _{{\\sigma }^{\\prime }}\\nabla _{\\mathbf {v}^{\\prime }} k_{{\\sigma }^{\\prime }}\\right)\\cdot \\leftrightarrow \\over {\\mathbf {W}}(\\tau _{\\sigma }\\mathbf {v}- \\tau _{{\\sigma }^{\\prime }}\\mathbf {v}^{\\prime })\\cdot \\nonumber \\\\[5pt]\\hspace{28.45274pt}\\left(\\tau _{\\sigma }\\nabla _\\mathbf {v}k_{\\sigma }-\\tau _{{\\sigma }^{\\prime }}\\nabla _{\\mathbf {v}^{\\prime }} k_{{\\sigma }^{\\prime }}\\right)f_{{\\sigma }0}f_{{\\sigma }^{\\prime } 0}\\mbox{d}^3v\\mbox{d}^3v^{\\prime }\\Bigg \\rangle _\\psi = 0.$ This equation has the following types of solutions: $k_{\\sigma }=q_{\\sigma }(\\mathbf {r})$ , $k_{\\sigma }= \\tau _{\\sigma }^{-1} \\mathbf {V}(\\mathbf {r})\\cdot \\mathbf {v}$ , and $k_{\\sigma }= Q(\\mathbf {r}) \\mathbf {v}^2/2$ .", "Again, in analogy with the calculation of subsection REF , it is easy to show that ${\\cal L}^\\dagger _j K = 0$ implies that $\\mathbf {V}(\\mathbf {R})\\equiv 0$ and that $\\lbrace q_{\\sigma }$ , ${\\sigma }= 1,\\dots ,N\\rbrace $ and $Q$ are flux functions, but otherwise arbitrary.", "In other words, every $K\\in \\mbox{Ker}({\\cal L}_j^\\dagger )$ can be written as a linear combination of elements of the form $[q_1(\\psi ),0,\\dots ,0],\\, \\dots ,\\, [0,\\dots , q_{\\sigma }(\\psi ),\\dots ,0],\\,\\dots , \\, [0,\\dots , q_N(\\psi )],\\nonumber \\\\[5pt]Q(\\psi )\\left(u^2/2 + \\mu B\\right)[1,\\, \\dots ,1,\\, \\dots , \\, 1],$ where the functions $\\lbrace q_{\\sigma }$ , ${\\sigma }= 1,\\dots ,N\\rbrace $ and $Q$ are arbitrary.", "Then, the solvability conditions are given by (REF ).", "Equivalently, due to the arbitrariness of the functions $\\lbrace q_{\\sigma }$ , ${\\sigma }= 1,\\dots ,N\\rbrace $ and $Q$ , the solvability conditions can be expressed as $\\sum _{\\sigma }\\left\\langle \\tau _{\\sigma }\\lambda _{\\sigma }^{-j}\\int B K_{\\sigma }R_{{\\sigma }j}\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\right\\rangle _\\psi = 0, \\quad K\\in \\mbox{Ker}({\\cal L}_j^\\dagger ).$ More concretely, all the solvability conditions of the Fokker-Planck equation to order $O(\\epsilon _s^j)$ are obtained by working out $\\left\\langle \\tau _{\\sigma }\\lambda _{\\sigma }^{-j} \\int B R_{{\\sigma }j}\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\right\\rangle _\\psi = 0, \\mbox{ for each ${\\sigma }$, and}\\nonumber \\\\[5pt]\\left\\langle \\sum _{\\sigma }\\tau _{\\sigma }\\lambda _{\\sigma }^{-j}\\int B \\left(u^2/2 + \\mu B\\right)R_{{\\sigma }j}\\mbox{d}u\\mbox{d}\\mu \\mbox{d}\\theta \\right\\rangle _\\psi = 0.$ The proof in this appendix guarantees that transport equations for particle and total energy density are the only solvability conditions for the long-wavelength second-order Fokker-Planck equation, (REF ).", "Finally, the reader can immediately check that when (REF ) is applied to the first-order equations, (REF ), no condition is obtained." ] ]

[ [ "One-sided FKPP travelling waves in the context of homogeneous\n fragmentation processes" ], [ "Abstract In this paper we introduce the one-sided FKPP equation in the context of homogeneous fragmentation processes.", "The main result of the present paper is concerned with the existence and uniqueness of one-sided FKPP travelling waves in this setting.", "Moreover, we prove some analytic properties of such travelling waves.", "Our techniques make use of fragmentation processes with killing, an associated product martingale as well as various properties of L\\'evy processes." ], [ "Introduction", "This paper deals with an integro-differential equation that is defined in terms of the dislocation measure of a fragmentation process.", "Given its probabilistic interpretation we consider this equation as an analogue of the one-sided Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP) travelling wave equation in the context of fragmentation processes.", "In particular, we are concerned with the existence and uniqueness of solutions to this equation in the setting of conservative or dissipative homogeneous fragmentation processes and we derive certain analytic properties of the solutions when they exist.", "The FKPP travelling wave equation in the context of fragmentations has a similar probabilistic interpretation as the classical FKPP travelling wave equation whose probabilistic interpretation is related to branching Brownian motions, see Section .", "In this respect we also refer to [1], where the two-sided FKPP travelling wave equation for conservative homogeneous fragmentations is studied.", "It turns out that solutions in the setting of fragmentation processes have similar properties as their classical counterparts.", "However, the techniques of proving existence and uniqueness results are very different between the two cases, not least because the equations differ significantly.", "Indeed, whereas the classical FKPP travelling wave equation is a differential equation of second order, the FKPP travelling wave equation in our setting is an integro-differential equation of first order.", "This difference results from the non-diffusive behaviour of fragmentation processes and the more complicated jump structure of fragmentations in comparison with branching Brownian motions.", "In the context of homogeneous fragmentation processes we prove the existence and uniqueness of one-sided travelling waves within a certain range of wave speeds.", "More precisely, the problem we are concerned with in this paper can be roughly described as follows.", "Consider the integro-differential equation $cf^{\\prime }(x)+\\int _{\\mathcal {P}}\\left(\\prod _{n}f(x+\\ln (|\\pi _n|))-f(x)\\right)\\mu (\\text{d}\\pi )=0$ for certain $c\\in {\\mathbb {R}}^+:=(0,\\infty )$ and all $x\\in {\\mathbb {R}}^+_0:=[0,\\infty )$ , where the product is taken over all $n\\in {\\mathbb {N}}$ with $|\\pi _n|\\in {\\mathbb {R}}^+$ .", "Here the space $\\mathcal {P}$ is the space of partitions $(\\pi _n)_{n\\in {\\mathbb {N}}}$ of ${\\mathbb {N}}$ and $\\mu $ is the so-called dislocation measure on $\\mathcal {P}$ .", "This notation is introduced in more detail in the next section.", "We are interested in solutions $f:{\\mathbb {R}}\\rightarrow [0,1]$ of the above equation that satisfy $f|_{{\\mathbb {R}}^+_0}\\in C^1({\\mathbb {R}}^+_0,[0,1])\\qquad \\text{and}\\qquad f|_{(-\\infty ,0)}\\equiv 1$ as well as the boundary condition $\\lim _{x\\rightarrow \\infty }f(x)=0.$ Roughly speaking, the main result of this paper states that there is some constant $c_0>0$ such that there exists a unique solution of the above boundary value problem for every $c>c_0$ and there does not exist such a solution for any $c\\le c_0$ .", "Our approach is based on using fragmentation processes with killing at an exponential barrier.", "These processes have been studied in [19] and we briefly describe the corresponding concepts below.", "The outline of this paper is as follows.", "In the next section we give a brief introduction to homogeneous fragmentation processes as well as appropriately killed versions of such fragmentations.", "Subsequently, in Section  we introduce the one-sided FKPP equation in our setting and we state our main results.", "Afterwards, in the fourth section we provide some motivation for the problems considered in this paper by explaining some related results that are known in the literature on fragmentation processes and branching Brownian motions, respectively.", "In Section  we show how the existence and uniqueness of one-sided FKPP travelling waves for fragmentation processes can be obtained if the dislocation measure is finite.", "This provides some motivation for the existence and uniqueness result in the setting of general fragmentation processes, which this paper is mainly concerned with.", "The subsequent three sections are devoted to the proofs of our main results.", "Throughout the present paper we adopt the notation ${\\mathbb {R}}_\\infty :=[-\\infty ,\\infty )$ as well as the conventions $\\ln (0):=-\\infty $ and $\\inf (\\emptyset ):=\\infty $ .", "The notation $C^n$ , $n\\in {\\mathbb {N}}_0:={\\mathbb {N}}\\cup \\lbrace 0\\rbrace $ , refers to the set of $n$ -times continuously differentiable functions.", "The integral of a real-valued function $f$ with respect to the Lebesgue measure on a set $[s,t]\\subseteq {\\mathbb {R}}$ is denoted by $\\int _{[s,t]}f(u)\\text{d}u$ and $\\int _s^tf(u)\\text{d}u$ denotes the Riemann integral.", "The operators $\\wedge $ and $\\vee $ refer to the minimum and maximum, respectively.", "Furthermore, we shall use the abbreviation DCT for the dominated convergence theorem.", "All the random objects are assumed to be defined on a complete probability space $(\\Omega ,F,\\mathbb {P})$ ." ], [ "Homogeneous fragmentation processes with killing", "In this section we provide a brief introduction to partition-valued fragmentation processes and we present the main tools that we need in the subsequent sections.", "In addition, we introduce a specific killing mechanism for these processes.", "The advantage of partition-valued fragmentation processes compared to so-called mass fragmentations is their explicit genealogical structure of blocks.", "This structure is crucial for the killing mechanism that we introduce below.", "Regarding the state space of partition-valued fragmentation processes let $\\mathcal {P}$ be the space of partitions $\\pi =(\\pi _n)_{n\\in {\\mathbb {N}}}$ of ${\\mathbb {N}}$ , where the blocks of $\\pi $ are ordered by their least element such that $\\inf (\\pi _i)<\\inf (\\pi _j)$ if $i<j$ .", "For every $\\pi \\in \\mathcal {P}$ let $(|\\pi |^\\downarrow _n)_{n\\in {\\mathbb {N}}}$ be the decreasing reordering of the sequence given by $|\\pi _n|:=\\limsup _{k\\rightarrow \\infty }\\frac{\\sharp (\\pi _n\\cap \\lbrace 1,\\ldots ,k\\rbrace )}{k}$ for every $n\\in {\\mathbb {N}}$ , where $\\sharp $ denotes the counting measure on ${\\mathbb {N}}$ .", "Throughout this paper we consider a homogeneous $\\mathcal {P}$ -valued fragmentation process $\\Pi := (\\Pi (t))_{t\\in {\\mathbb {R}}^+_0}$ , where $\\Pi (t) = (\\Pi _n(t))_{n\\in {\\mathbb {N}}}$ , and we denote by $F:=(F_t)_{t\\in {\\mathbb {R}}^+_0}$ the completion of the filtration generated by $\\Pi $ .", "Homogeneous $\\mathcal {P}$ -valued fragmentations are exchangeable Markov processes that were introduced in [4], see also [6].", "Bertoin showed in [4] that the distribution of $\\Pi $ is determined by some constant $d\\in {\\mathbb {R}}^+_0$ (the rate of erosion which describes the drift of $\\Pi $ ) and a $\\sigma $ -finite measure $\\nu $ (the so-called dislocation measure that indirectly describes the jumps of $\\Pi $ ) on the infinite simplex $\\mathcal {S}:=\\left\\lbrace {\\bf s}:=(s_n)_{n\\in {\\mathbb {N}}}:s_1\\ge s_2\\ldots \\ge 0,\\,\\sum _{n\\in {\\mathbb {N}}}s_n\\le 1\\right\\rbrace ,$ such that $\\nu (\\lbrace (1,0,\\ldots )\\rbrace )=0$ and $\\int _{\\mathcal {S}}(1-s_1)\\nu (d{\\bf s})<\\infty .$ The process $\\Pi $ is said to be conservative if $\\nu (\\sum _{n\\in {\\mathbb {N}}}s_n<1)=0$ , i.e.", "if there is no loss of mass by sudden dislocations, and dissipative otherwise.", "In this paper we allow for both of these cases.", "Throughout this paper we assume that $d=0$ as well as $\\nu ({\\bf s}\\in \\mathcal {S}:s_2=0)=0$ .", "In view of the forthcoming assumption (REF ) this enables us to resort to the results of [19], where the same assumptions are made.", "Let us mention that the assumption $d=0$ does not result in any loss of generality, see Remark REF .", "Consider the exchangeable partition measure $\\mu $ on $\\mathcal {P}$ given by $\\mu (d\\pi ) = \\int _{\\mathcal {S}}\\varrho _{\\bf s}(d\\pi )\\nu (d{\\bf s}),$ where $\\varrho _{\\bf s}$ is the law of Kingman's paint-box based on ${\\bf s}\\in \\mathcal {S}$ .", "Similarly to $\\nu $ the measure $\\mu $ describes the jumps of $\\Pi $ , although more directly, and is also referred to as dislocation measure.", "In [4] Bertoin showed that the homogeneous fragmentation process $\\Pi $ is characterised by a Poisson point process.", "More precisely, there exists a $\\mathcal {P}\\times \\mathbb {N}$ -valued Poisson point process $(\\pi (t),\\kappa (t))_{t\\in {\\mathbb {R}}^+_0}$ with characteristic measure $\\mu \\otimes \\sharp $ such that $\\Pi $ changes state only at the times $t\\in {\\mathbb {R}}^+_0$ for which an atom $(\\pi (t),\\kappa (t))$ occurs in $(\\mathcal {P}\\setminus ({\\mathbb {N}},\\emptyset ,\\ldots ))\\times \\mathbb {N}$ .", "At such a time $t\\in {\\mathbb {R}}^+_0$ the sequence $\\Pi (t)$ is obtained from $\\Pi (t-)$ by replacing its $\\kappa (t)$ -th term, $\\Pi _{\\kappa (t)}(t-)\\subseteq {\\mathbb {N}}$ , with the restricted partition $\\pi (t)|_{\\Pi _{\\kappa (t)}(t-)}$ and reordering the terms such that the resulting partition of ${\\mathbb {N}}$ is an element of $\\mathcal {P}$ .", "We denote the possible random jump times of $\\Pi $ , i.e.", "the times at which the abovementioned Poisson point process has an atom in $(\\mathcal {P}\\setminus ({\\mathbb {N}},\\emptyset ,\\ldots ))\\times \\mathbb {N}$ , by $(t_i)_{i\\in \\mathcal {I}}$ , where the index set $\\mathcal {I}\\subseteq {\\mathbb {R}}^+_0$ is countable.", "Moreover, by exchangeability, the limit $|\\Pi _n(t)|:=\\lim _{k\\rightarrow \\infty }\\frac{\\sharp (\\Pi _n(t)\\cap \\lbrace 1,\\ldots ,k\\rbrace )}{k},$ referred to as asymptotic frequency, exists $\\mathbb {P}$ -a.s. simultaneously for all $t\\in {\\mathbb {R}}^+_0$ and $n\\in {\\mathbb {N}}$ .", "Let us point out that the concept of asymptotic frequencies provides us with a notion of size for the blocks of a $\\mathcal {P}$ -valued fragmentation process.", "In Theorem 3 of [4] Bertoin showed that the process $(-\\ln (|\\Pi _1(t)|))_{t\\in {\\mathbb {R}}^+_0}$ is a killed subordinator, a fact we shall make use of below.", "For the time being, let $x\\in {\\mathbb {R}}_\\infty $ .", "In this paper we are concerned with a specific procedure of killing blocks of $\\Pi $ , see Figure REF , that was introduced in [19].", "More precisely, for $c>0$ a block $\\Pi _n(t)$ is killed, with cemetery state $\\emptyset $ , at the moment of its creation $t\\in {\\mathbb {R}}^+_0$ if $|\\Pi _n(t)|<e^{-(x+ct)}$ .", "We denote the resulting fragmentation process with killing by $\\Pi ^x:=(\\Pi ^x(t))_{t\\in {\\mathbb {R}}^+_0}$ and the cemetery state of $\\Pi ^x$ is $(\\emptyset ,\\ldots )$ .", "Note that possibly $\\Pi ^x(t)\\notin \\mathcal {P}$ as $\\bigcup _{n\\in {\\mathbb {N}}}\\Pi ^x_n(t)\\subsetneq {\\mathbb {N}}$ is possible due to the killing of blocks.", "We denote by $\\zeta ^x$ , $x\\in {\\mathbb {R}}_\\infty $ , the random extinction time of $\\Pi ^x$ , i.e.", "$\\zeta ^x$ is the supremum of all the killing times of individual blocks.", "The question whether $\\zeta ^x$ is finite or infinite was considered in Theorem 2 of [19], see also Proposion REF below.", "Furthermore, define a function $\\varphi :{\\mathbb {R}}\\rightarrow [0,1]$ by $\\varphi (x):=\\mathbb {P}(\\zeta ^x<\\infty )$ for all $x\\in {\\mathbb {R}}_\\infty $ .", "The function $\\varphi $ will be of utmost interest in the present paper.", "Let us point out that $\\varphi $ depends on the drift $c>0$ of the killing line.", "However, in order to keep the notation as simple as possible, we omit this dependence in the notation as the constant $c$ does not vary within results or proofs.", "Note that if $x<0$ , then $\\zeta ^x=0$ and thus $\\varphi (x)=1$ .", "Let us remark that we could choose a non-zero rate $d$ of erosion by changing the slope $c$ of the killing line: Remark 1 The results of this paper remain valid if we omit the assumption $d=0$ and replace the slope $c>0$ by $c_d:=c+d$ .", "Consequently, the assumption $d=0$ is merely made for the sake of simplicity, but does not restrict the generality of our results.", "$\\lozenge $ Set $\\underline{p}:=\\inf \\left\\lbrace p\\in {\\mathbb {R}}:\\int _{\\mathcal {S}}\\left|1-\\sum _{n\\in {\\mathbb {N}}}s^{1+p}_n\\right|\\nu (\\text{d}{\\bf s})<\\infty \\right\\rbrace \\in [-1,0]$ and for any $p>\\underline{p}$ define $\\Phi (p):=\\int _{\\mathcal {S}}\\left(1-\\sum _{n\\in {\\mathbb {N}}}s^{1+p}_n\\right)\\nu (\\text{d}{\\bf s})$ as well as $\\Phi (\\underline{p}):=\\lim _{p\\downarrow \\underline{p}}\\Phi (p).$ Moreover, for each $p\\in [\\underline{p},\\infty )$ set $c_p:=\\frac{\\Phi (p)}{1+p}.$ Throughout this paper we assume that there exists some $p\\in (\\underline{p},\\infty )$ such that $(1+p)\\Phi ^{\\prime }(p)>\\Phi (p),$ where $\\Phi ^{\\prime }$ denotes the derivative of $\\Phi $ .", "Let us point out that a sufficient condition for (REF ) is the existence of some $p^*\\in [\\underline{p},\\infty )$ such that $\\Phi (p^*)=0$ .", "In particular, (REF ) holds if $\\Pi $ is conservative.", "In view of (REF ) the same line of argument as in Lemma 1 of [5] yields the existence of a unique solution of the equation $(1+p)\\Phi ^{\\prime }(p)=\\Phi (p)$ on $(\\underline{p},\\infty )$ .", "We denote this unique solution of (REF ) by $\\bar{p}$ .", "The definition in (REF ) then entails that $c_{\\bar{p}}=\\Phi ^{\\prime }(\\bar{p})$ .", "According to [19] the fragmentation process with killing survives with positive probability if the drift of the killing line is greater than $c_{\\bar{p}}$ and becomes extinct almost surely otherwise.", "Proposition 2 (Theorem 2 of [19]) If $c>c_{\\bar{p}}$ , then $\\varphi (x)\\in (0,1)$ for all $x\\in {\\mathbb {R}}^+_0$ .", "If, on the other hand, $c\\le c_{\\bar{p}}$ , then $\\varphi \\equiv 1$ .", "For any $t\\in {\\mathbb {R}}^+_0$ we denote by $B_n(t)$ the block of $\\Pi (t)$ that contains the element $n\\in {\\mathbb {N}}$ .", "According to Theorem 3 (ii) in [4] it follows by means of the exchangeability of $\\Pi $ that under $\\mathbb {P}$ the process $\\xi _n:=(-\\ln (|B_n(t)|))_{t\\in {\\mathbb {R}}^+_0}\\,,$ cf.", "Figure REF , is a killed subordinator with Laplace exponent $\\Phi $ , cemetery state $\\infty $ and killing rate $\\int _{\\mathcal {S}}\\left(1-\\sum _{k\\in {\\mathbb {N}}}s_k\\right)\\nu (\\text{d}{\\bf s}).$ Hence, the process $X_n:=(X_n(t))_{t\\in {\\mathbb {R}}^+_0}$ , defined by $X_n(t):=ct-\\xi _n(t)$ for all $t\\in {\\mathbb {R}}^+_0$ , is a spectrally negative Lévy process of bounded variation.", "Let $\\mathcal {I}_n\\subset \\mathcal {I}$ be such that the jump times of $X_n$ are given by $(t_i)_{i\\in \\mathcal {I}_n}$ .", "Note that $(t_i)_{i\\in \\mathcal {I}_n}$ are precisely the times when the subordinator $\\xi _n$ jumps.", "For $n\\in {\\mathbb {N}}$ and $x\\in {\\mathbb {R}}^+_0$ we shall make use of the shifted and killed process $X^x_n:=(X^x_n(t))_{t\\in {\\mathbb {R}}^+_0}$ , see Figure REF , given by $X^x_n(t):=(X_n(t)+x)1_{\\lbrace \\tau ^-_{n,x}>t\\rbrace }-\\infty \\cdot 1_{\\lbrace \\tau ^-_{n,x}\\le t\\rbrace }=\\left(x+ct+\\ln (|B_n(t)|)\\right)1_{\\lbrace \\tau ^-_{n,x}>t\\rbrace }-\\infty \\cdot 1_{\\lbrace \\tau ^-_{n,x}\\le t\\rbrace }$ for each $t\\in {\\mathbb {R}}^+_0$ , where $\\tau ^-_{n,x}:=\\inf \\lbrace t\\in {\\mathbb {R}}^+_0:X_n(t)<-x\\rbrace \\qquad \\text{as well as}\\qquad \\infty \\cdot 0:=0.$ Figure: Illustration of X n x X^x_n.For every $t\\in {\\mathbb {R}}^+_0$ set $\\mathcal {N}^x_t:=\\left\\lbrace n\\in {\\mathbb {N}}:\\left[t<\\tau ^-_{n,x}\\right]\\wedge \\left[\\exists \\,k\\in {\\mathbb {N}}:n=\\min \\Pi ^x_k(t)\\right]\\right\\rbrace .$ That is to say, $\\mathcal {N}^x_t$ consists of all the indices of blocks $B_n(t)$ that are not yet killed by time $t$ .", "Let us remark that the first condition “$t<\\tau ^-_{n,x}$ ” ensures that the block containing $n\\in {\\mathbb {N}}$ is still alive at time $t$ and the second condition “$\\exists \\,k\\in {\\mathbb {N}}:n=\\min \\left(\\Pi ^x_k(t)\\right)$ ” is used to avoid considering the same block multiple times.", "More precisely, for a block $B_n(t)$ that is alive at time $t\\in {\\mathbb {R}}^+_0$ only its least element is an element of $\\mathcal {N}^x_t$ .", "Without this condition all elements of $B_n(t)$ would be in $\\mathcal {N}^x_t$ .", "Note that in [19] the notation $\\mathcal {N}^x_t$ refers to a different ordering of the same set of blocks and the set of indices that we denote here by $\\mathcal {N}^x_t$ is the quotient space $\\widetilde{\\mathcal {N}}_t^x/\\sim $ in [19].", "Throughout this paper we shall repeatedly need an estimate regarding the number of fragments alive at a given time $t\\in {\\mathbb {R}}^+_0$ .", "To this end, set $N^x_t:=\\text{card}(\\mathcal {N}^x_t)$ and observe that $N^x_t<\\infty $ $\\mathbb {P}$ -almost surely.", "Indeed, since $\\sum _{n\\in {\\mathbb {N}}}|\\Pi _n(t)|\\le 1$ we infer that $|\\Pi _n(t)|\\ge e^{-(x+ct)}$ for at most $e^{x+ct}$ -many $n\\in {\\mathbb {N}}$ .", "Hence, $N^x_t\\le e^{x+ct}.$" ], [ "The one-sided FKPP equation for fragmentations", "In this section we establish the set-up for our considerations by defining the FKPP travelling wave equation in the context of fragmentation processes.", "Moreover, this section is devoted to presenting our main results.", "The main problem addressed in this paper is to find a range of wave speeds for which we can prove the existence of a unique travelling wave solution to the one-sided FKPP equation for homogeneous fragmentations as defined below.", "In order to tackle this problem we shall derive a connection between solutions of the abovementioned FKPP equation and a product martingale that was introduced in [19].", "Furthermore, we aim at studying some analytic properties of such travelling waves." ], [ "Set-up", "For any function $f$ on some subset of ${\\mathbb {R}}_\\infty $ set $\\mathcal {C}_f:=\\left\\lbrace x\\in {\\mathbb {R}}^+:f^{\\prime }(x)\\text{ exists}\\right\\rbrace ,$ where $f^{\\prime }$ denotes the derivative of $f$ .", "Since we do not know a priori whether the functions $f$ we are interested in are differentiable, we need to define the integro-differential equations below for arguments in $\\mathcal {C}_f$ .", "Regarding solutions of such an equation we shall be particularly concerned with the function $\\varphi $ , given by (REF ), and our main results show in particular that $\\mathcal {C}_\\varphi ={\\mathbb {R}}^+$ .", "For functions $u:{\\mathbb {R}}^+_0\\times {\\mathbb {R}}_\\infty \\rightarrow [0,1]$ , with $u(t,\\cdot )|_{[-\\infty ,ct)}\\equiv 1$ for each $t\\in {\\mathbb {R}}^+_0$ and $u(0,\\cdot )|_{{\\mathbb {R}}^+_0}=g$ for some continuous function $g:{\\mathbb {R}}^+_0\\rightarrow [0,1]$ , consider the initial value problem $\\frac{\\partial u}{\\partial t}(t,x)=\\int _{\\mathcal {P}}\\left(\\prod _{n\\in {\\mathbb {N}}}u(t,x+\\ln (|\\pi _n|))-u(t,x)\\right)\\mu (\\text{d}\\pi )$ for all $x\\in {\\mathbb {R}}^+_0$ and $t\\in \\mathcal {C}_{u(\\cdot ,x)}$ .", "We call this initial value problem one-sided FKPP equation for fragmentation processes.", "Here we are interested in the so-called FKPP travelling wave solutions of (REF ) with wave speed $c\\in {\\mathbb {R}}^+_0$ , that is in solutions of (REF ) which are of the form $u(t,x)=f(x-ct)$ for all $t,x\\in {\\mathbb {R}}^+_0$ with $x-ct\\ge 0$ .", "Definition 3 A one-sided FKPP travelling wave for fragmentation processes is a monotone function $f:{\\mathbb {R}}_\\infty \\rightarrow [0,1]$ , with $f|_{[-\\infty ,0)}\\equiv 1$ , that satisfies the following one-sided FKPP travelling wave equation $cf^{\\prime }(x)+\\int _{\\mathcal {P}}\\left(\\prod _{n\\in {\\mathbb {N}}}f(x+\\ln (|\\pi _n|))-f(x)\\right)\\mu (\\text{d}\\pi )=0$ for all $x\\in \\mathcal {C}_f$ with the boundary condition $\\lim _{x\\rightarrow \\infty }f(x)=0.$ Let us now introduce an operator whose definition is inspired by the integro-differential equation (REF ).", "To this end, let $\\mathcal {D}_L$ be the set of all functions $f:{\\mathbb {R}}_\\infty \\rightarrow [0,1]$ , with $f|_{[-\\infty ,0)}\\equiv 1$ , for which the mapping $\\pi \\mapsto \\prod _{n\\in {\\mathbb {N}}}f(x+\\ln (|\\pi _n|)-f(x)$ on $\\mathcal {P}$ is $\\mu $ -integrable.", "Then we define an integral operator $L$ with domain $\\mathcal {D}_L$ by $Lf(x):=\\int _{\\mathcal {P}}\\left(\\prod _{n\\in {\\mathbb {N}}}f(x+\\ln (|\\pi _n|)-f(x)\\right)\\mu (\\text{d}\\pi )$ for each $f\\in \\mathcal {D}_L$ and all $x\\in {\\mathbb {R}}^+_0$ .", "Recall that the upper Dini derivative (from above) of a function $f:{\\mathbb {R}}_\\infty \\rightarrow [0,1]$ is defined by $f^{\\prime }_+(x):=\\limsup _{h\\downarrow 0}\\frac{f(x+h)-f(x)}{h}$ for all $x\\in {\\mathbb {R}}$ .", "On this note, observe that the Dini derivative is well defined, but may take the value $\\infty $ or $-\\infty $ .", "The following class of monotone functions plays a crucial role in the analysis of the one-sided FKPP travelling wave equation.", "Definition 4 We denote by $\\mathcal {T}$ the set consisting of all nonincreasing functions $f:{\\mathbb {R}}_\\infty \\rightarrow [0,1]$ , with $f|_{[-\\infty ,0)}\\equiv 1$ and such that $f|_{{\\mathbb {R}}^+}$ is continuous, that satisfy (REF ) as well as $\\sup _{x\\in [s,t]}|f^{\\prime }_+(x)|<\\infty $ for all $s,t\\in {\\mathbb {R}}^+$ ." ], [ "Main results", "Here we present the main results of this paper.", "For this purpose, let us consider the following process which several of our proofs make use of: For any function $f:{\\mathbb {R}}_\\infty \\rightarrow [0,1]$ and $x\\in {\\mathbb {R}}^+_0$ let $Z^{x,f}:=(Z^{x,f}_t)_{t\\in {\\mathbb {R}}^+_0}$ be given by $Z^{x,f}_t:=\\prod _{n\\in \\mathcal {N}^x_t}f\\left(X^x_n(t)\\right)$ for all $t\\in {\\mathbb {R}}^+_0$ .", "This process was considered in Section 5 of [19] and in some proofs we shall resort to Theorem 10 in [19], which is concerned with the martingale property of $Z^{x,f}$ .", "In this spirit, the first main result of the present paper reads as follows: Theorem 1 Let $c>c_{\\bar{p}}$ .", "In addition, let $f\\in \\mathcal {T}$ and assume that $Z^{x,f}$ is a martingale.", "Then $f$ solves (REF ).", "The above theorem will be proven in Section .", "The following result, whose proof will be provided in Section , deals with some analytic properties of one-sided FKPP travelling waves.", "Theorem 2 Every one-sided FKPP travelling wave $f\\in \\mathcal {T}$ is right-continuous at 0 and the function $f|_{{\\mathbb {R}}^+}$ is strictly monotonically decreasing and continuously differentiable.", "In particular, it follows from Theorem REF that $-cf^{\\prime }=Lf$ on ${\\mathbb {R}}^+$ for every one-sided FKPP travelling wave $f\\in \\mathcal {T}$ , where $c>0$ denotes the wave speed.", "The main goal of this paper is to establish the existence of a unique travelling wave in $\\mathcal {T}$ to (REF ) with wave speed $c$ for $c>c_{\\bar{p}}$ as well as the nonexistence of such a travelling wave with wave speed $c\\le c_{\\bar{p}}$ .", "More specifically, the following result states that the extinction probability of the fragmentation process with killing solves equation (REF ) with boundary condition (REF ) for $c>c_{\\bar{p}}$ and, moreover, is the only such function.", "Recall the function $\\varphi $ defined in (REF ).", "Theorem 3 If $c>c_{\\bar{p}}$ , then there exists a unique one-sided FKPP travelling wave in $\\mathcal {T}$ with wave speed $c$ , given by $\\varphi $ .", "On the other hand, if $c\\le c_{\\bar{p}}$ , then there is no one-sided FKPP travelling wave in $\\mathcal {T}$ with wave speed $c$ .", "We shall prove Theorem REF in Section .", "In view of the forthcoming Remark REF this theorem shows that one-sided FKPP travelling waves exist precisely for those positive wave speeds for which there do not exist two-sided travelling waves.", "Moreover, in view of [19], Theorem REF shows that travelling wave solutions exist exactly for those wave speeds that are larger than the asymptotic decay of the largest fragment in the fragmentation process with killing on the event of survival.", "Let us further point out that if $\\Pi $ is conservative, i.e.", "if loss of mass by sudden dislocations is excluded, then in view of (3) in [18] equation (REF ) can be written as $cf^{\\prime }(x)+\\int _{\\mathcal {S}}\\left(\\prod _{n\\in {\\mathbb {N}}}f(x+\\ln (s_n))-f(x)\\right)\\nu (\\text{d}{\\bf s})=0,$ which is the analogue of the two-sided FKPP equation considered in [1] in the conservative setting, cf.", "(REF )." ], [ "Motivation – The classical FKPP equation", "In order to present the framework in which Theorem REF should be seen let us now briefly mention some known results that are related to our work.", "To this end we denote by $C^{1,2}({\\mathbb {R}}^+_0\\times A,[0,1])$ , $A\\subseteq {\\mathbb {R}}$ , the space of all functions $f:{\\mathbb {R}}^+_0\\times A\\rightarrow [0,1]$ such that $f(x,\\cdot )\\in C^2(A,[0,1])$ and $f(\\cdot ,y)\\in C^1({\\mathbb {R}}^+_0,[0,1])$ for all $x\\in {\\mathbb {R}}^+_0$ and $y\\in A$ .", "The classical FKPP equation in the form that is of most interest for us, cf.", "[24], is the following parabolic partial differential equation: $\\frac{\\partial u}{\\partial t}=\\frac{1}{2}\\frac{\\partial ^2u}{\\partial x^2}+\\beta (u^2-u)$ with $u\\in C^{1,2}({\\mathbb {R}}^+_0\\times {\\mathbb {R}},[0,1])$ .", "This equation, which first arose in the context of a genetics model for the spread of an advantageous gene through a population, was originally introduced by Fisher [13], [14] as well as by Kolmogorov, Petrovskii and Piscounov [20].", "Since then it has attracted much attention by analysts and probabilists alike.", "In fact, several authors showed that this equation is closely related to dyadic branching Brownian motions, e.g.", "[24] (see also [25]), thus establishing a link of this analytic problem to probability theory.", "In this probabilistic interpretation the term “$\\frac{1}{2}\\frac{\\partial ^2u}{\\partial x^2}$ ” corresponds to the motion of the underlying Brownian motion, the “$\\beta $ ” is the rate at which the particles split and the term “$u^2-u$ ” results from the binary branching, where two particles replace one particle at each branching time.", "A solution $u$ of equation (REF ) can be interpreted in different ways.", "The classical work concerning this partial differential equation, such as [13], [14] and [20], describes the wave of advance of advantageous genes.", "More precisely, there are two types of individuals (or genes) in a population and $u(t,x)$ measures the frequency or concentration of the advantageous type at the time-space point $(t,x)$ .", "In the setting regarding the abovementioned probabilistic interpretation, that links (REF ) with a dyadic branching Brownian motion, let $u(t,x)$ be the probability that at time $t$ the largest particle of the branching Brownian motion has a value less than $x$ .", "Then $u$ satisfies equation (REF ), see (7) in [24].", "That is to say, in [13], [14] and [20] the FKPP equation (REF ) describes the bulk of a population, in [24] it describes the most advanced particle of a branching Brownian motion.", "The classical FKPP travelling waves are solutions of (REF ) of the form $u(t,x)=f(x-c t)$ for some $f\\in C^2({\\mathbb {R}},[0,1])$ and some constant $c\\in {\\mathbb {R}}$ .", "This leads to the so-called FKPP travelling wave equation with wave speed $c\\in {\\mathbb {R}}$ , $\\frac{1}{2}f^{\\prime \\prime }+cf^{\\prime }+\\beta (f^2-f) &= 0\\\\[0.5ex]\\lim _{x\\rightarrow -\\infty }f(x) &= 0\\\\[0.5ex]\\lim _{x\\rightarrow \\infty }f(x) &= 1,$ where $\\beta >0$ .", "This travelling wave boundary value problem was studied by various authors, using both analytic as well as probabilistic techniques, and it is known that is has a unique (up to additive translation) solution $f\\in C^2({\\mathbb {R}},[0,1])$ if $|c|\\ge \\sqrt{2\\beta }$ .", "In the opposite case that $0\\le |c|<\\sqrt{2\\beta }$ there is no travelling wave solution.", "Regarding probabilistic approaches to the classical FKPP travelling wave equation we also refer for instance to the work of Bramson [7], [8], Chauvin and Rouault [10], [11], Uchiyama [28], [29] as well as [17], [22] and [26].", "Interesting with regard to our work is that the above boundary value problem was extended to continuous-time branching random walks, cf.", "[21], and to conservative homogeneous fragmentation processes, see [1].", "In the context of such fragmentation processes the corresponding partial integro-differential equation, referred to as FKPP equation, is given by $\\frac{\\partial u}{\\partial t}(t,x)=\\int _{\\mathcal {S}}\\left(\\prod _{n\\in {\\mathbb {N}}}u(t,x+\\ln (s_n))-u(t,x)\\right)\\nu (\\text{d}{\\bf s})$ for certain $u:{\\mathbb {R}}^+_0\\times {\\mathbb {R}}\\rightarrow [0,1]$ .", "Note that (REF ) looks quite different compared to the classical FKPP equation (REF ).", "This difference results from the fact that fragmentation processes have no spatial motion except at jump times and from the more complicated jump structure of fragmentations in comparison with dyadic branching Brownian motions.", "However, despite the difference of the above equation compared to the classical FKPP equation, the name FKPP equation for (REF ) stems from the similarity in terms of the probabilistic interpretation these two equations have.", "Of particular interest to us are the FKPP travelling waves to (REF ) with wave speed $c\\in {\\mathbb {R}}$ , i.e.", "solutions of (REF ) which are of the form $u(t,x)=f(x-ct)$ for all $t\\in {\\mathbb {R}}^+_0$ and $x\\in {\\mathbb {R}}$ .", "These travelling wave solutions are functions $f\\in C^1({\\mathbb {R}},[0,1])$ that satisfy the following FKPP travelling wave equation $cf^{\\prime }(x)+\\int _{\\mathcal {S}}\\left(\\prod _{n\\in {\\mathbb {N}}}f(x+\\ln (s_n)-f(x)\\right)\\nu (\\text{d}{\\bf s})=0$ for all $x\\in {\\mathbb {R}}$ with boundary conditions $\\lim _{x\\rightarrow -\\infty }f(x)=0\\qquad \\text{and}\\qquad \\lim _{x\\rightarrow \\infty }f(x)=1.$ Remark 5 For every $p\\in (\\underline{p},\\bar{p}]$ let $\\mathcal {T}_2(p)$ denote the space of monotonically increasing functions $f\\in C^1({\\mathbb {R}},[0,1])$ satisfying the boundary conditions $\\lim _{x\\rightarrow -\\infty }f(x)=0$ as well as $\\lim _{x\\rightarrow \\infty }f(x)=1$ and such that the mapping $x\\mapsto e^{(1+p)x}(1-f(x))$ is monotonically increasing.", "In Theorem 1 of [1] Berestycki et.", "al.", "showed that for $p\\in (\\underline{p},\\bar{p}]$ there exists a unique (up to additive translation) FKPP travelling wave solution in $\\mathcal {T}_2(p)$ with wave speed $c_p$ , cf.", "(REF ).", "According to Lemma 1 of [5] the mapping $p\\mapsto {\\Phi (p)}{(1+p)}=c_p$ is monotonically increasing on $(\\underline{p},\\bar{p}]$ and thus it follows in view of Theorem 3 (ii) in [1] that $c_{\\bar{p}}$ is the maximal wave speed for two-sided travelling waves.", "$\\lozenge $ In this paper we are interested in the one-sided counterpart of the abovementioned FKPP equation.", "In the classical setting the one-sided FKPP equation is the following partial differential equation $\\frac{\\partial u}{\\partial t}=\\frac{1}{2}\\frac{\\partial ^2u}{\\partial x^2}+\\beta (u^2-u)$ on ${\\mathbb {R}}^+\\times {\\mathbb {R}}^+$ with $u\\in C^{1,2}({\\mathbb {R}}^+_0\\times {\\mathbb {R}}^+_0,[0,1])$ .", "Observe that this equation is the analogue of (REF ) for functions defined on ${\\mathbb {R}}^+_0\\times {\\mathbb {R}}^+_0$ .", "The corresponding one-sided FKPP travelling wave equation with wave speed $c\\in {\\mathbb {R}}$ is given by the differential equation $\\frac{1}{2}f^{\\prime \\prime }+cf^{\\prime }+\\beta (f^2-f) = 0$ on ${\\mathbb {R}}^+$ for $f\\in C^2({\\mathbb {R}}^+_0,[0,1])$ satisfying the boundary conditions $\\lim _{x\\rightarrow 0}f(x) = 1\\qquad \\text{as well as}\\qquad \\lim _{x\\rightarrow \\infty }f(x) = 0.$ By considering killed branching Brownian motion with drift, killed upon hitting the origin, Harris et.", "al.", "proved in [16] that solutions of the one-sided FKPP travelling wave boundary value problem (REF ) and (REF ) exist and are unique (up to additive translation) for all $c\\in (-\\sqrt{2\\beta },\\infty )$ and there is no such travelling wave solution for $c\\in (-\\infty ,-\\sqrt{2\\beta }]$ .", "Notice that the one-sided travelling wave solutions for negative $c$ are precisely those wave speeds for which there does not exist a two-sided travelling wave.", "With regard to the one-sided FKPP travelling wave equation in the classical setting we refer also to [30], concerning existence of a solution, as well as [27] for existence and uniqueness of a solution of (REF ) and (REF ) obtained by means of analytic techniques." ], [ "The finite activity case", "In this section we prove existence of a one-sided FKPP travelling wave in the situation of a finite dislocation measure $\\nu $ , sometimes referred to as the finite activity case.", "In this respect note that a homogeneous fragmentation process with finite $\\nu $ may still have infinitely many jumps in any finite time interval after the first jump, because infinitely many blocks may be present at any such time and each block fragments with the same rate.", "However, in this setting of a finite dislocation measure every block $B_n$ , $n\\in {\\mathbb {N}}$ , has only finitely many jumps up to any $t\\in {\\mathbb {R}}^+_0$ .", "In particular, this implies that the fragmentation process with killing and with finite $\\nu $ has finite activity in bounded time intervals, since at any time there are only finitely many blocks alive.", "Therefore, in this finite activity situation it is possible to consider the time $\\tau _k:\\Omega \\rightarrow {\\mathbb {R}}^+\\cup \\lbrace \\infty \\rbrace $ , $k\\in {\\mathbb {N}}$ , of the $k$ -th jump of the killed process $\\Pi ^x$ , a fact we shall make use of below.", "An approach to solve the classical one-sided FKPP equation with boundary condition $u(0,x)=g(x)$ for some suitable function $g:{\\mathbb {R}}^+_0\\rightarrow [0,1]$ is to show that the function $u:{\\mathbb {R}}^+_0\\times {\\mathbb {R}}^+_0\\rightarrow [0,1]$ , given by $u(t,x)=\\mathbb {E}\\left(\\prod _{n\\in {\\mathbb {N}}}g(x+Y_n(t))\\right)$ for all $t,x\\in {\\mathbb {R}}^+_0$ , is a solution of the considered boundary value problem, where the $Y_n(t)$ are the positions of the particles at time $t$ in a dyadic branching Brownian motion.", "In this section we show that for fragmentations with a finite dislocation measure $\\nu $ a similar approach as above works for the initial value problem (REF ).", "More precisely, we prove that for certain functions $g:{\\mathbb {R}}_\\infty \\rightarrow [0,1]$ the function $u:{\\mathbb {R}}^+_0\\times {\\mathbb {R}}_\\infty \\rightarrow [0,1]$ , defined by $\\forall \\,x\\in [ct,\\infty ):\\,u(t,x)=\\mathbb {E}\\left(\\prod _{n\\in \\mathcal {N}^{x-ct}_t}g(x+\\ln (|B_n(t)|))\\right)\\qquad \\text{and}\\qquad u(t,\\cdot )|_{[-\\infty ,ct)}\\equiv 1$ for all $t\\in {\\mathbb {R}}^+_0$ , solves equation (REF ) with boundary condition $u(0,\\cdot )=g$ .", "Proposition 6 Assume that $\\nu (\\mathcal {S})<\\infty $ and let $c>0$ .", "Then every function $u:{\\mathbb {R}}^+_0\\times {\\mathbb {R}}_\\infty \\rightarrow [0,1]$ defined by (REF ), for some function $g:{\\mathbb {R}}_\\infty \\rightarrow [0,1]$ with $g|_{{\\mathbb {R}}^+_0}\\in C^0({\\mathbb {R}}^+_0)$ and $g|_{[-\\infty ,0)}\\equiv 1$ , satisfies the boundary condition $u(0,\\cdot )=g$ and solves equation (REF ) for any $x\\in {\\mathbb {R}}^+_0$ .", "In particular, any such function $u$ of the form $u(t,x)=f(x-ct)$ for some $f:{\\mathbb {R}}_\\infty \\rightarrow [0,1]$ , with $f|_{[-\\infty ,0)}\\equiv 1$ , and all $t\\in {\\mathbb {R}}^+_0$ and $x\\in {\\mathbb {R}}_\\infty $ solves (REF ).", "The following corollary of Proposition REF provides a short proof that the extinction probability $\\varphi $ of the fragmentation process with killing solves equation (REF ) in the special case of a finite dislocation measure.", "Corollary 7 Assume that $\\nu (\\mathcal {S})<\\infty $ and let $c>c_{\\bar{p}}$ .", "Then $\\varphi $ is an FKPP travelling wave with wave speed $c$ .", "Proof Let us first show that $\\varphi $ solves (REF ).", "For this purpose, observe that the fragmentation property, in conjunction with the tower property of conditional expectations, yields that $\\varphi (x-ct) &= \\mathbb {E}\\left(\\prod _{n\\in \\mathcal {N}^{x-ct}_t}\\left.\\mathbb {P}\\left(\\zeta ^{x-ct+ct+y}<\\infty \\right)\\right|_{y=\\ln (|B_n(t)|)}\\right)\\\\[0.5ex]&= \\mathbb {E}\\left(\\prod _{n\\in \\mathcal {N}^{x-ct}_t}\\varphi (x+\\ln (|B_n(t)|))\\right),$ and thus $u:{\\mathbb {R}}^+_0\\times {\\mathbb {R}}_\\infty \\rightarrow [0,1]$ , given by $u(t,x):=\\varphi (x-ct)$ , satisfies (REF ) with $g=\\varphi $ .", "Hence, according to Proposition REF the function $\\varphi $ solves (REF ).", "Since $c>c_{\\bar{p}}$ , it follows from Theorem 10 in [19] that $\\varphi $ also satisfies the boundary condition (REF ), which completes the proof.", "$\\square $ The major part of this paper, cf.", "Theorem REF , is concerned with the proof that the conclusion of Corollary REF holds true also in the general case of an infinite dislocation measure.", "Proof of Proposition  REF   The proof is based on a decomposition according to the first and second jump times of a killed fragmentation.", "Treating these parts separately we obtain the desired expression for the right derivative of $u(\\cdot ,x)$ , $x\\in {\\mathbb {R}}^+_0$ .", "Let $g:{\\mathbb {R}}_\\infty \\rightarrow [0,1]$ be some function that satisfies $g|_{{\\mathbb {R}}^+_0}\\in C^0({\\mathbb {R}}^+_0)$ and $g|_{[-\\infty ,0)}\\equiv 1$ .", "Further, consider the function $u:{\\mathbb {R}}^+_0\\times {\\mathbb {R}}_\\infty \\rightarrow [0,1]$ defined by (REF ) and fix some $x\\in {\\mathbb {R}}^+_0$ as well as $t\\in \\mathcal {C}_{u(\\cdot ,x)}$ .", "In the light of the càdlàg paths of $\\Pi $ and the DCT, note first that $u$ satisfies the boundary condition (REF ), since $|B_1(0)|=1$ and $|B_n(0)|=0$ , i.e.", "$g(x+\\ln (|B_n(0)|))=1$ , for all $n\\in {\\mathbb {N}}\\setminus \\lbrace 1\\rbrace $ .", "In order to prove that $u$ solves (REF ) Lebesgue-a.e., let $(t_i)_{i\\in \\mathcal {I}^x}$ be the jump times of $\\Pi ^x$ and in view of the finiteness of the dislocation measure and $N^x_t\\le e^{x+ct}$ , cf.", "(REF ), we assume without loss of generality that $\\mathcal {I}^x={\\mathbb {N}}$ and that $0<t_i< t_j$ for any $i,j\\in {\\mathbb {N}}$ with $i<j$ .", "Since $t_1$ is exponentially distributed with parameter $\\mu (\\mathcal {P})$ , we have $\\lim _{h\\downarrow 0}\\frac{\\mathbb {P}\\left(t_1\\le h\\right)}{h}= \\lim _{h\\downarrow 0}\\frac{1-e^{-h\\mu (\\mathcal {P})}}{h}= \\mu (\\mathcal {P})$ and deduce by resorting to the strong fragmentation property of $\\Pi $ that $\\begin{aligned}\\lim _{h\\downarrow 0}\\frac{\\mathbb {P}(t_2\\le h)}{h}&\\le \\lim _{h\\downarrow 0}\\frac{\\mathbb {P}(t_1\\le h)}{h}\\lim _{h\\downarrow 0}\\mathbb {E}\\left(\\left.\\mathbb {P}\\left(\\mathfrak {e}_{\\mu (\\mathcal {P})e^{x+c t}}\\le h\\right)\\right|_{t=t_1}\\right)\\\\[0.5ex]&= \\mu (\\mathcal {P})\\mathbb {E}\\left(\\lim _{h\\downarrow 0}\\left(1-e^{-h\\mu (\\mathcal {P})e^{x+ct_1}}\\right)\\right)\\\\[0.5ex]&=0,\\end{aligned}$ where $\\mathfrak {e}_{\\mu (\\mathcal {P})e^{x+c t_1}}$ denotes a random variable that is exponentially distributed with parameter $\\mu (\\mathcal {P})e^{x+c t_1}$ .", "Consequently, $\\lim _{h\\downarrow 0}\\frac{\\mathbb {P}(t_1\\le h<t_2)}{h}= \\lim _{h\\downarrow 0}\\frac{\\mathbb {P}(t_1\\le h)}{h}-\\lim _{h\\downarrow 0}\\frac{\\mathbb {P}(t_2\\le h)}{h}= \\mu (\\mathcal {P}).$ By means of the strong Markov property, the fact that the distrubution of $\\pi (t_1)$ is given by ${\\mu (\\cdot )}{\\mu (\\mathcal {P})}$ and the independence between $\\pi (t_1)$ and the random vector $\\left(\\begin{array}{c}t_1\\\\t_2\\end{array}\\right)$ , see Proposition 2 in Section 0.5 of [2], we have $& \\mathbb {E}\\left(\\prod _{n\\in \\mathcal {N}^{x-c(t+h)}_{t+h}}g(x+\\ln (|B_n(t+h)|))1_{\\left\\lbrace t_1\\le h<t_2\\right\\rbrace }\\right)\\\\[0.75ex]&= \\mathbb {E}\\Bigg (\\prod _{n\\in {\\mathbb {N}}}\\Bigg (1_{\\lbrace x-c(t+h)+ct_1<-\\ln (|\\pi _n(t_1)|)\\rbrace }+1_{\\lbrace x-c(t+h)+ct_1\\ge -\\ln (|\\pi _n(t_1)|)\\rbrace }\\\\[0.5ex]&\\qquad \\cdot \\mathbb {E}\\Bigg (\\prod _{k\\in \\mathcal {N}^{x-c(t+h)+ct_1+\\ln (|\\pi _n(t_1)|)}_t}g\\left(x+\\ln (|\\pi _n(t_1)|)+\\ln \\left(\\left|B^{(n)}_k(t)\\right|\\right)\\right)1_{\\left\\lbrace t_1\\le h<t_2\\right\\rbrace }\\Bigg |F_{t_1}\\Bigg )\\Bigg )\\Bigg )\\\\[0.75ex]&= \\mathbb {E}\\Bigg (\\prod _{n\\in {\\mathbb {N}}}\\Bigg (1_{\\lbrace x-c(t+h-t_1)<-\\ln (|\\pi _n(t_1)|)\\rbrace }+1_{\\lbrace x-c(t+h-t_1)\\ge -\\ln (|\\pi _n(t_1)|)\\rbrace }\\\\[0.5ex]&\\qquad \\cdot \\mathbb {E}\\Bigg (1_{\\left\\lbrace t_1\\le h<t_2\\right\\rbrace }\\mathbb {E}\\Bigg (\\prod _{k\\in \\mathcal {N}^{x-c(t+h-t_1)+\\ln (|\\pi _n(t_1)|)}_t}g\\left(x+\\ln (|\\pi _n(t_1)|)+\\ln \\left(\\left|B^{(n)}_k(t)\\right|\\right)\\right)\\Bigg |F_h\\Bigg )\\Bigg |F_{t_1}\\Bigg )\\Bigg )\\Bigg )\\\\[0.75ex]&= \\mathbb {E}\\Bigg (1_{\\left\\lbrace t_1\\le h<t_2\\right\\rbrace }\\prod _{n\\in {\\mathbb {N}}}\\Bigg (1_{\\lbrace x-c(t+h-t_1)<-\\ln (|\\pi _n(t_1)|)\\rbrace }+1_{\\lbrace x-c(t+h-t_1)\\ge -\\ln (|\\pi _n(t_1)|)\\rbrace }\\\\[0.5ex]&\\qquad \\cdot \\mathbb {E}\\Bigg (\\prod _{k\\in \\mathcal {N}^{x+\\ln (u)-ct}_t}g\\left(x+\\ln \\left(u\\right)+\\ln (|B_k(t)|)\\right)\\Bigg )\\Bigg |_{u=|\\pi _n(t_1)|}\\Bigg )\\Bigg )\\\\[0.75ex]&= \\mathbb {P}(t_1\\le h<t_2)\\mathbb {E}\\left(\\prod _{n\\in {\\mathbb {N}}}u_h(t,x+\\ln (|\\pi _n(t_1)|))\\right)\\\\[0.75ex]&= \\mathbb {P}(t_1\\le h<t_2)\\int _\\mathcal {P}\\prod _{n\\in {\\mathbb {N}}}u_h(t,x+\\ln (|\\pi _n|))\\frac{\\mu (\\text{d}\\pi )}{\\mu (\\mathcal {P})}\\, ,$ where $u_h(t,\\cdot )|_{[c(t+h-t_1),\\infty )}:= u|_{[c(t+h-t_1),\\infty )}\\qquad \\text{as well as}\\qquad u_h(t,\\cdot )|_{[-\\infty ,c(t+h-t_1))}:\\equiv 1.$ Therefore, (REF ) yields that $& \\lim _{h\\downarrow 0}\\mathbb {E}\\left(\\frac{1}{h}\\prod _{n\\in \\mathcal {N}^{x-c(t+h)}_{t+h}}g(x+\\ln (|B_n(t+h)|))1_{\\left\\lbrace t_1\\le h<t_2\\right\\rbrace }\\right) \\\\[0.75ex]&= \\int _\\mathcal {P}\\prod _{n\\in {\\mathbb {N}}}\\lim _{h\\downarrow 0}u_h(t,x+\\ln (|\\pi _n|))\\mu (\\text{d}\\pi )\\frac{1}{\\mu (\\mathcal {P})}\\lim _{h\\downarrow 0}\\frac{\\mathbb {P}(t_1\\le h<t_2)}{h}\\\\[0.75ex]&= \\int _\\mathcal {P}\\prod _{n\\in {\\mathbb {N}}}u(t,x+\\ln (|\\pi _n|))\\mu (\\text{d}\\pi ).$ Moreover, $&\\mathbb {E}\\left(\\prod _{n\\in \\mathcal {N}^{x-ct}_t}g(x+\\ln (|B_n(t)|))1_{\\left\\lbrace t^{(n)}_1\\le h<t^{(n)}_2\\right\\rbrace }\\right)\\\\[0.5ex]&= \\mathbb {E}\\left(\\prod _{n\\in \\mathcal {N}^{x-ct}_t}g(x+\\ln (|B_n(t)|))\\mathbb {P}\\left(\\left.t^{(n)}_1\\le h<t^{(n)}_2\\right|F_t\\right)\\right)\\\\[0.5ex]&=u(t,x)\\mathbb {P}(t_1\\le h<t_2)\\\\[0.5ex]&=\\int _\\mathcal {P}u(t,x)\\mu (\\text{d}\\pi )\\frac{1}{\\mu (\\mathcal {P})}\\mathbb {P}(t_1\\le h<t_2)$ holds for all $h>0$ , where conditionally on $F_t$ the $t^{(n)}_1$ and $t^{(n)}_2$ are independent copies of $t_1 $ and $t_2$ , respectively.", "Hence, $\\lim _{h\\downarrow 0}\\mathbb {E}\\left(\\frac{1}{h}\\prod _{n\\in \\mathcal {N}^{x-ct}_t}g(x+\\ln (|B_n(t)|))1_{\\left\\lbrace t^{(n)}_1\\le h<t^{(n)}_2\\right\\rbrace }\\right) &= \\int _\\mathcal {P}u(t,x)\\mu (\\text{d}\\pi )\\frac{1}{\\mu (\\mathcal {P})}\\lim _{h\\downarrow 0}\\frac{\\mathbb {P}(t_1\\le h<t_2)}{h}\\\\[0.5ex]&= \\int _\\mathcal {P}u(t,x)\\mu (\\text{d}\\pi ).$ Since $|B_1(h)|=1$ and $\\mathcal {N}^{x-c(t+h)}_h=\\lbrace 1\\rbrace $ on $\\lbrace t_1> h\\rbrace $ , we deduce with $t^{(n)}_1$ being defined as above that $\\begin{aligned}& \\mathbb {E}\\left(\\prod _{n\\in \\mathcal {N}^{x-c(t+h)}_{t+h}}g(x+\\ln (|B_n(t+h)|))1_{\\left\\lbrace t_1> h\\right\\rbrace }\\right)\\\\[0.75ex]&= \\mathbb {E}\\left(\\left.\\mathbb {E}\\left.\\left(\\prod _{n\\in \\tilde{\\mathcal {N}}^{x-c(t+h)+ch}_t}g(x+\\ln (\\gamma )+\\ln (|B^{(n)}(t)|))\\right|F_h\\right)\\right|_{\\gamma =|B_1(h)|}1_{\\left\\lbrace t_1> h\\right\\rbrace }\\right)\\\\[0.75ex]&= \\mathbb {E}\\left(\\mathbb {E}\\left(\\prod _{n\\in \\mathcal {N}^{x-ct}_t}g(x+\\ln (|B_n(t)|))\\right)1_{\\left\\lbrace t_1> h\\right\\rbrace }\\right)\\\\[0.75ex]&= \\mathbb {E}\\left(\\prod _{n\\in \\mathcal {N}^{x-ct}_t}g(x+\\ln (|B_n(t)|))\\right)\\mathbb {P}\\left(t_1> h\\right)\\\\[0.75ex]&= \\mathbb {E}\\left(\\prod _{n\\in \\mathcal {N}^{x-ct}_t}g(x+\\ln (|B_n(t)|))\\mathbb {P}\\left(t_1> h\\right)\\right)\\\\[0.75ex]&= \\mathbb {E}\\left(\\prod _{n\\in \\mathcal {N}^{x-ct}_t}g(x+\\ln (|B_n(t)|))\\mathbb {P}\\left(\\left.t^{(n)}_1> h\\right|F_t\\right)\\right)\\\\[0.75ex]&=\\mathbb {E}\\left(\\prod _{n\\in \\mathcal {N}^{x-ct}_t}g(x+\\ln (|B_n(t)|))1_{\\left\\lbrace t^{(n)}_1> h\\right\\rbrace }\\right)\\end{aligned}$ holds for each $h>0$ , where conditionally on $F_h$ the $\\tilde{\\mathcal {N}}^{(\\cdot )}_t$ and $B^{(n)}$ are independent copies of $\\mathcal {N}^{(\\cdot )}_t$ and $B_n$ , respectively.", "Furthermore, note that (REF ) results in $\\lim _{h\\downarrow 0}\\frac{\\mathbb {P}\\left(t^{(n)}_2\\le h\\right)}{h}=\\lim _{h\\downarrow 0}\\frac{\\mathbb {E}\\left(\\mathbb {P}\\left(\\left.t^{(n)}_2\\le h\\right|F_t\\right)\\right)}{h}=\\lim _{h\\downarrow 0}\\frac{\\mathbb {P}\\left(t_2\\le h\\right)}{h}=0,$ where $t^{(n)}_2$ is defined as above.", "Bearing in mind that $|g|\\le 1$ it follows from the DCT in conjunction with (REF ) and (REF ), respectively, that $\\lim _{h\\downarrow 0}\\mathbb {E}\\left(\\prod _{n\\in \\mathcal {N}^{x-c(t+h)}_{t+h}}g(x+\\ln (|B_n(t+h)|))1_{\\left\\lbrace t_2\\le h\\right\\rbrace }\\right)&=0\\\\[0.5ex]&=\\lim _{h\\downarrow 0}\\mathbb {E}\\left(\\prod _{n\\in \\mathcal {N}^{x-ct}_t}g(x+\\ln (|B_n(t)|))1_{\\left\\lbrace t^{(n)}_2\\le h\\right\\rbrace }\\right).$ Since $u(t+h,x) &= \\mathbb {E}\\left(\\prod _{n\\in \\mathcal {N}^{x-c(t+h)}_{t+h}}g(x+\\ln (|B_n(t+h)|))1_{\\left\\lbrace t_1>h\\right\\rbrace }\\right)\\\\[0.75ex]&\\qquad +\\mathbb {E}\\left(\\prod _{n\\in \\mathcal {N}^{x-c(t+h)}_{t+h}}g(x+\\ln (|B_n(t+h)|))1_{\\left\\lbrace t_2\\le h\\right\\rbrace }\\right)\\\\[0.75ex]&\\qquad +\\mathbb {E}\\left(\\prod _{n\\in \\mathcal {N}^{x-c(t+h)}_{t+h}}g(x+\\ln (|B_n(t+h)|))1_{\\left\\lbrace t_1\\le h<t_2\\right\\rbrace }\\right)$ and $u(t,x) &= \\mathbb {E}\\left(\\prod _{n\\in \\mathcal {N}^{x-ct}_t}g(x+\\ln (|B_n(t)|))1_{\\left\\lbrace t^{(n)}_1> h\\right\\rbrace }\\right)+ \\mathbb {E}\\left(\\prod _{n\\in \\mathcal {N}^{x-ct}_t}g(x+\\ln (|B_n(t)|))1_{\\left\\lbrace t^{(n)}_2\\le h\\right\\rbrace }\\right)\\\\[0.5ex]&\\qquad +\\mathbb {E}\\left(\\prod _{n\\in \\mathcal {N}^{x-ct}_t}g(x+\\ln (|B_n(t)|))1_{\\left\\lbrace t^{(n)}_1\\le h<t^{(n)}_2\\right\\rbrace }\\right)$ hold for every $h>0$ , it thus follows from (REF ), (REF ) and (REF ) that $\\lim _{h\\downarrow 0}\\frac{u(t+h,x)-u(t,x)}{h}= \\int _\\mathcal {P}\\left(\\prod _{n\\in {\\mathbb {N}}}u(t,x+\\ln (|\\pi _n|))-u(t,x)\\right)\\mu (\\text{d}\\pi ),$ which completes the proof, since $t\\in \\mathcal {C}_{u(\\cdot ,x)}$ .", "$\\square $" ], [ "Sufficiency criterion for the existence of travelling waves", "The goal of this section is to provide the proof of Theorem REF .", "A first approach to try proving Theorem REF might be to pursue a line of argument along the lines of the proof of Theorem 1 in [1].", "But that proof relies on $f$ being continuously differentiable and in our situation we cannot use any differentiability assumption.", "In fact, even if we knew that $f$ is differentiable with a bounded derivative $f^{\\prime }$ , we would at least need that the set of discontinuities of $f^{\\prime }$ is a Lebesgue null set.", "However, in general the set of such discontinuities may have positive Lebesgue measure, cf.", "Example 3.5 in [31].", "Let us start with the following auxiliary result.", "Lemma 8 Let $f\\in \\mathcal {T}$ as well as $a,b\\in {\\mathbb {R}}^+$ .", "Then we have $f(a)-f(b)\\le (b-a)\\sup _{x\\in (a,b)}|f^{\\prime }_+(x)|.$ Proof Define a function $\\phi :[a,b]\\rightarrow {\\mathbb {R}}$ by $\\phi (x):=f(x)-\\frac{f(b)-f(a)}{b-a}(x-a)$ for all $x\\in [a,b]$ .", "Let us first show that there exists some $x_0\\in (a,b)$ such that $\\limsup _{h\\downarrow 0}\\frac{\\phi (x_0+h)-\\phi (x_0)}{h}\\le 0.$ To this end, assume $\\phi ^{\\prime }_+(x):=\\limsup _{h\\downarrow 0}\\frac{\\phi (x+h)-\\phi (x)}{h}>0$ for each $x\\in (a,b)$ .", "Then for every $x\\in (a,b)$ there exists some $\\epsilon _x>0$ such that for every $\\epsilon \\in (0,\\epsilon _x]$ we have $\\frac{\\phi (x+h)-\\phi (x)}{h}>0$ for some $h\\in (0,\\epsilon )$ .", "We now show that this implies that $\\phi $ is nondecreasing on $(a,b)$ .", "For this purpose, consider $c,d\\in [a,b]$ and assume $\\max _{x\\in [c,d]}\\phi (x)\\ne \\phi (d),$ where the existence of this maximum follows from the continuity of $\\phi $ , which in turn follows from $f\\in \\mathcal {T}$ being continuous.", "Then there exists some $x_0\\in [c,d)$ such that $\\max _{x\\in [c,d]}\\phi (x)=\\phi (x_0).$ However, this implies that $\\phi (x_0)\\ge \\phi (x)$ for all $x\\in (x_0,(x_0+\\epsilon _{x_0})\\wedge d)$ , which contradicts (REF ).", "Hence, (REF ) cannot be true and consequently we infer that $\\max _{x\\in [c,d]}\\phi (x)=\\phi (d)$ for all $c,d\\in [a,b]$ under assumption (REF ).", "Note that $\\phi $ not being nondecreasing on $[a,b]$ would entail that there exist $c,d\\in [a,b]$ , with $c<d$ , such that $\\phi (c)>\\phi (d)$ , which contradicts (REF ).", "Therefore, we conclude that $\\phi $ is nondecreasing and nonconstant on $[a,b]$ if (REF ) holds.", "This, however, contradicts the fact that $\\phi (a)=f(a)=\\phi (b).$ We thus deduce that (REF ) cannot hold and hence there exists some $x_0\\in (a,b)$ such that (REF ) holds.", "With $x_0\\in (a,b)$ given by (REF ) we obtain $0\\ge \\phi ^{\\prime }_+(x_0)=f^{\\prime }_+(x_0)-\\frac{f(b)-f(a)}{b-a},$ which results in $0\\le \\sup _{x\\in (a,b)}|f^{\\prime }_+(x)|-\\frac{f(a)-f(b)}{b-a}$ and thus $f(a)-f(b)\\le (b-a)\\sup _{x\\in (a,b)}|f^{\\prime }_+(x)|.$ $\\square $ We proceed by establishing two auxiliary results, which in spirit are analogues of respective results in [1].", "Afterwards we provide a lemma giving conditions under which only the block containing 1 is alive in the fragmentation process with killing.", "Finally, having all these auxiliary results at hand, we finish this section with the proof of Theorem REF .", "Observe first that a straightforward argument by induction yields that $\\left|\\prod _{n\\in {\\mathbb {N}}}a_n-\\prod _{n\\in {\\mathbb {N}}}b_n\\right|\\le \\sum _{n\\in {\\mathbb {N}}}|a_n-b_n|$ holds for all sequences $(a_n)_{n\\in {\\mathbb {N}}},(b_n)_{n\\in {\\mathbb {N}}}\\in [0,1]^{\\mathbb {N}}$ .", "The following lemma, whose proof is based on (REF ), shows in particular that $f\\in \\mathcal {D}_L$ and, moreover, that $Lf$ is bounded on compact sets for any $f\\in \\mathcal {T}$ .", "Lemma 9 Let $f\\in \\mathcal {T}$ and let $a,b\\in {\\mathbb {R}}^+$ .", "Then $\\int _{\\mathcal {P}}\\sup _{x\\in [a,b]}\\left|\\prod _{n\\in {\\mathbb {N}}}f(x+\\ln (|\\pi _n|))-f(x)\\right|\\mu (\\text{d}\\pi )<\\infty .$ Proof By means of (REF ) we have $& \\int _{\\mathcal {P}}\\sup _{x\\in [a,b]}\\left|\\prod _{n\\in {\\mathbb {N}}}f(x+\\ln (|\\pi _n|))-f(x)\\right|\\mu (\\text{d}\\pi )\\\\[0.5ex]&\\le \\int _{\\mathcal {P}}\\sup _{x\\in [a,b]}\\left|f(x+\\ln (|\\pi |^\\downarrow _1))-f(x)\\right|\\mu (\\text{d}\\pi )+\\int _{\\mathcal {P}}\\sum _{n\\in {\\mathbb {N}}\\setminus \\lbrace 1\\rbrace }\\sup _{x\\in [a,b]}\\left|f(x+\\ln (|\\pi |^\\downarrow _n))-1\\right|\\mu (\\text{d}\\pi ).$ Since $\\frac{\\text{d}}{\\text{d}x}[\\ln (x)+2(1-x)]=\\frac{1}{x}-2$ and $\\ln (1)+2(1-1)=0$ , we deduce that $-\\ln (x)\\le 2(1-x)$ holds for all $x\\in [{1}{2},1]$ .", "Therefore, for every $\\epsilon \\in (0,{1}{2}]$ we have $-\\ln (|\\pi |^\\downarrow _1)\\le 2(1-|\\pi |^\\downarrow _1)$ for all $\\pi \\in \\mathcal {P}$ with $1-|\\pi |^\\downarrow _1\\le \\epsilon $ .", "Moreover, by means of Lemma REF we have for any $x\\in {\\mathbb {R}}^+$ and $\\pi \\in \\mathcal {P}$ with $|\\pi |^\\downarrow _1>e^{-x}$ the estimate $\\left|f\\left(x+\\ln (|\\pi |^\\downarrow _1)\\right)-f(x)\\right|\\le -\\ln (|\\pi |^\\downarrow _1)\\sup _{y\\in \\left(x+\\ln (|\\pi |^\\downarrow _1),\\,x\\right)}|f^{\\prime }_+(y)|.$ Furthermore, for every $\\gamma \\in (0,a)$ define $A_{a,\\gamma }:=\\left\\lbrace \\pi \\in \\mathcal {P}:a+\\ln (|\\pi |^\\downarrow _1)\\in [0,\\gamma )\\right\\rbrace =\\left\\lbrace \\pi \\in \\mathcal {P}:|\\pi |^\\downarrow _1\\in [e^{-a},e^{\\gamma -a})\\right\\rbrace $ and observe that in view of $\\gamma -a<0$ and (REF ) we have $\\mu \\left(A_{a,\\gamma }\\right)\\le \\mu \\left(\\lbrace \\pi \\in \\mathcal {P}:|\\pi |^\\downarrow _1<e^{\\gamma -a}\\rbrace \\right)<\\infty .$ Hence, resorting to (REF ), (REF ) and (REF ) as well as (REF ) we conclude in the light of (3) in [18] and $f(x)\\in [0,1]$ for every $x>0$ that $&\\int _{\\mathcal {P}}\\sup _{x\\in [a,b]}\\left|f(x+\\ln (|\\pi |^\\downarrow _1))-f(x)\\right|\\mu (\\text{d}\\pi )\\\\[0.5ex]&\\le \\int _{\\lbrace \\pi \\in \\mathcal {P}:1-|\\pi |^\\downarrow _1>\\epsilon \\rbrace \\cup A_{a,\\gamma }}\\sup _{x\\in [a,b]}\\left|f(x+\\ln (|\\pi |^\\downarrow _1))-f(x)\\right|\\mu (\\text{d}\\pi )\\\\[0.5ex]&\\qquad +\\int _{\\lbrace \\pi \\in \\mathcal {P}:1-|\\pi |^\\downarrow _1\\le \\epsilon \\rbrace \\setminus A_{a,\\gamma }}\\sup _{x\\in [a,b]}\\left|f(x+\\ln (|\\pi |^\\downarrow _1))-f(x)\\right|\\mu (\\text{d}\\pi )\\\\[0.5ex]&\\le \\mu \\left(\\lbrace \\pi \\in \\mathcal {P}:1-|\\pi |^\\downarrow _1>\\epsilon \\rbrace \\cup A_{a,\\gamma }\\right)+\\int _{\\lbrace \\pi \\in \\mathcal {P}:1-|\\pi |^\\downarrow _1\\le \\epsilon \\rbrace \\setminus A_{a,\\gamma }}-\\ln (|\\pi |^\\downarrow _1)\\sup _{y\\in (a+\\ln (|\\pi |^\\downarrow _1),b)}|f_+^{\\prime }(y)|\\,\\mu (\\text{d}\\pi )\\\\[0.5ex]&\\le \\mu \\left(\\lbrace \\pi \\in \\mathcal {P}:|\\pi |^\\downarrow _1<1-\\epsilon \\rbrace \\right)+\\mu \\left(A_{a,\\gamma }\\right)+2\\sup _{y\\in [\\gamma ,b)}|f_+^{\\prime }(y)|\\int _{\\mathcal {P}}(1-|\\pi |^\\downarrow _1)\\,\\mu (\\text{d}\\pi )\\\\[0.5ex]&< \\infty $ for any $\\epsilon \\in (0,{1}{2}]$ , which shows that the first term on the right-hand side of (REF ) is finite.", "In order to deal with the second term on the right-hand side of (REF ), note that the monotonicity of $f$ together with $f|_{[-\\infty ,0)}\\equiv 1$ and $f|_{[0,\\infty )}\\in [0,1]$ yields that $\\int _{\\mathcal {P}}\\sum _{n\\in {\\mathbb {N}}\\setminus \\lbrace 1\\rbrace }\\sup _{x\\in [a,b]}|1-f(x+\\ln (|\\pi |^\\downarrow _n))|\\mu (\\text{d}\\pi )&\\le \\int _{\\mathcal {P}}\\sum _{n\\in {\\mathbb {N}}\\setminus \\lbrace 1\\rbrace }|1-f(b+\\ln (|\\pi |^\\downarrow _n))|\\mu (\\text{d}\\pi )\\\\[0.5ex]&\\le \\int _{\\mathcal {P}}\\sum _{n\\in {\\mathbb {N}}\\setminus \\lbrace 1\\rbrace }e^{(b+\\ln (|\\pi |^\\downarrow _n))}\\mu (\\text{d}\\pi )\\\\[0.5ex]&= e^b\\int _{\\mathcal {P}}\\sum _{n\\in {\\mathbb {N}}\\setminus \\lbrace 1\\rbrace }|\\pi |^\\downarrow _n\\mu (\\text{d}\\pi )\\\\[0.5ex]&< \\infty $ for all $x>0$ .", "Observe that the finiteness holds, since $\\int _{\\mathcal {P}}\\sum _{n\\in {\\mathbb {N}}\\setminus \\lbrace 1\\rbrace }|\\pi |^\\downarrow _n\\mu (\\text{d}\\pi ) =\\int _{\\mathcal {P}}\\left(\\left(1-|\\pi |^\\downarrow _1\\right)+\\left(\\sum _{n\\in {\\mathbb {N}}}|\\pi |^\\downarrow _n-1\\right)\\right)\\mu (\\text{d}\\pi )\\le \\int _{\\mathcal {P}}(1-|\\pi |^\\downarrow _1)\\mu (\\text{d}\\pi )< \\infty .$ Consequently, also the second term on the right-hand side of (REF ) is finite.", "$\\square $ As already mentioned, the previous lemma implies that $Lf$ exists for each $f\\in \\mathcal {T}$ .", "The next lemma goes a step further for that it shows that $Lf$ is continuous for every $f\\in \\mathcal {T}$ .", "Lemma 10 Let $f\\in \\mathcal {T}$ .", "Then the function $Lf$ is continuous on ${\\mathbb {R}}^+$ .", "Proof Fix some $x\\in {\\mathbb {R}}^+$ and let $(x_k)_{k\\in {\\mathbb {N}}}$ be a sequence in ${\\mathbb {R}}^+$ with $x_k\\rightarrow x$ as $k\\rightarrow \\infty $ .", "In addition, fix some $\\epsilon \\in (0,x)$ and let $k_\\epsilon \\in {\\mathbb {N}}$ be such that $|x-x_k|\\le \\epsilon $ for all $k\\ge k_\\epsilon $ .", "Observe that $& \\int _{\\mathcal {P}}\\sup _{k\\ge k_\\epsilon }\\left|\\prod _{n\\in {\\mathbb {N}}}f(x+\\ln (|\\pi _n|))-f(x)-\\prod _{n\\in {\\mathbb {N}}}f(x_k+\\ln (|\\pi _n|))+f(x_k)\\right|\\mu (\\text{d}\\pi )\\\\[0.5ex]&\\le \\int _{\\mathcal {P}}\\left|\\prod _{n\\in {\\mathbb {N}}}f(x+\\ln (|\\pi _n|))-f(x)\\right|\\mu (\\text{d}\\pi )+\\int _{\\mathcal {P}}\\sup _{y\\in [x-\\epsilon ,\\,x+\\epsilon ]}\\left|\\prod _{n\\in {\\mathbb {N}}}f(y+\\ln (|\\pi _n|))-f(y)\\right|\\mu (\\text{d}\\pi ).$ According to Lemma REF both of the integrals on the right-hand side of (REF ) are finite.", "Hence, we can apply the DCT and deduce that $&\\lim _{k\\rightarrow \\infty }|Lf(x)-Lf(x_k)|\\\\[0.5ex]&= \\int _{\\mathcal {P}}\\lim _{k\\rightarrow \\infty }\\left|\\prod _{n\\in {\\mathbb {N}}}f(x+\\ln (|\\pi _n|))-f(x)-\\prod _{n\\in {\\mathbb {N}}}f(x_k+\\ln (|\\pi _n|))+f(x_k)\\right|\\mu (\\text{d}\\pi )\\\\[0.5ex]&= \\int _{\\mathcal {P}}\\left|\\prod _{n\\in {\\mathbb {N}}}f(x+\\ln (|\\pi _n|))-\\prod _{n\\in {\\mathbb {N}}}\\lim _{k\\rightarrow \\infty }f(x_k+\\ln (|\\pi _n|))-f(x)+\\lim _{k\\rightarrow \\infty }f(x_k)\\right|\\mu (\\text{d}\\pi )\\\\[0.5ex]&= 0,$ where the final equality follows from $f\\in \\mathcal {T}$ being continuous on ${\\mathbb {R}}^+$ .", "Notice that we can interchange the limit and the product in the penultimate equality, since only finitely many factors of the product differ from 1.", "Hence, we have proven the continuity of $Lf$ at $x$ and since $x\\in {\\mathbb {R}}^+$ was chosen arbitrarily, this completes the proof.", "$\\square $ Recall the process $Z^{x,f}$ that we defined in (REF ) and set $\\Delta Z^{x,f}_t:=Z^{x,f}_t-Z^{x,f}_{t-}$ for every $t>0$ .", "We are now in a position to prove Theorem REF .", "Proof of Theorem  REF Throughout the proof let $x\\in \\mathcal {C}_f$ and let $(a_n)_{n\\in {\\mathbb {N}}}$ be a sequence in $(0,1)$ with $a_n\\downarrow 0$ as $n\\rightarrow \\infty $ .", "Moreover, Consider the following stopping time $\\delta :=\\inf \\left\\lbrace t>0:x+ct+\\ln \\left(|\\Pi (t)|^\\downarrow _2\\right)>0\\right\\rbrace \\wedge \\tau ^-_{1,0}\\wedge 1.$ Recall from Lemma REF that $f\\in \\mathcal {D}_L$ .", "The idea of the proof is to consider an appropriate decomposition of the limit of $\\mathbb {E}(Z^{x,f}_{\\delta \\wedge a_n}-Z^{x,f}_{0})a_n^{-1}$ as $n\\rightarrow \\infty $ , which by the martingale property of $Z^{x,f}$ equals 0.", "In this spirit the proof deals with the jumps and drift that contribute to the difference $Z^{x,f}_{\\delta \\wedge a_n}-Z^{x,f}_{0}$ separately and eventually combines these considerations in order to prove the assertion.", "Let us first deal with the jumps of $\\Pi ^x$ that contribute to the difference $Z^{x,f}_{\\delta \\wedge a_n}-Z^{x,f}_{0}$ .", "To this end, we start by pointing out that $\\delta >0$ $\\mathbb {P}$ -almost surely.", "Indeed, if $\\delta =\\tau ^-_{1,0}\\wedge 1$ , then the $\\mathbb {P}$ -a.s. positivity of $\\delta $ follows, since for $X_n$ the point 0 is irregular for $(-\\infty ,0)$ .", "In order to deal with the case $\\delta <\\tau ^-_{1,0}\\wedge 1$ , note that $x+c\\tau +\\ln \\left(|\\Pi (\\tau )|^\\downarrow _1\\right)\\ge x+c\\tau +\\ln \\left(|B_1(\\tau )|\\right)=X^x_1(\\tau )\\ge x\\Longrightarrow |\\Pi (\\tau )|^\\downarrow _1\\ge e^{-c\\tau }$ and $x+c\\tau +\\ln \\left(|\\Pi (\\tau )|^\\downarrow _2\\right)\\ge 0\\Longrightarrow |\\Pi (\\tau )|^\\downarrow _2\\ge e^{-(x+c\\tau )}$ hold on the event $\\lbrace \\delta <\\tau ^-_{1,0}\\wedge 1\\rbrace $ for any random time $\\tau \\le \\delta $ .", "Therefore, on this event we have $e^{-c\\tau }\\le |\\Pi (\\tau )|^\\downarrow _1\\le 1-e^{-(x+c\\tau )}$ for every random time $\\tau \\le \\delta $ , which implies that $\\delta \\ge \\tau \\ge \\frac{1}{c}\\ln \\left(1+e^{-x}\\right)>0$ on $\\lbrace \\delta <\\tau ^-_{1,0}\\wedge 1\\rbrace $ .", "The compensation formula for Poisson point processes yields that $& \\frac{1}{a_n}\\mathbb {E}\\left(\\sum _{i\\in \\mathcal {I}}1_{(0,\\delta \\wedge a_n]}(t_i)\\Delta Z^{x,f}_{t_i}\\right)\\\\[0.5ex]&= \\frac{1}{a_n}\\mathbb {E}\\left(\\sum _{i\\in \\mathcal {I}}1_{(0,\\delta \\wedge a_n]}(t_i)\\prod _{l\\in {\\mathbb {N}}}f(X^x_1(t_i-)+\\ln (|\\pi _l(t_i)|))-f(X^x_1(t_i-))\\right)\\\\[0.5ex]&= \\frac{1}{a_n}\\mathbb {E}\\left(\\int _{(0,\\delta \\wedge a_n]}\\int _\\mathcal {P}\\prod _{l\\in {\\mathbb {N}}}f(X^x_1(t-)+\\ln (|\\pi _l|))-f(X^x_1(t-))\\mu (\\text{d}\\pi )\\text{d}t\\right)\\\\[0.5ex]&= \\mathbb {E}\\left(\\frac{1}{a_n}\\int _{(0,\\delta \\wedge a_n]}Lf\\left(X^x_1(t-)\\right)\\text{d}t\\right).$ Since, $\\min _{y\\in [x,\\,x+ca_n]}Lf(y)\\mathbb {E}\\left(\\frac{\\delta \\wedge a_n}{a_n}\\right)\\le \\mathbb {E}\\left(\\frac{1}{a_n}\\int _{(0,\\delta \\wedge a_n]}Lf\\left(X^x_1(t-)\\right)\\text{d}t\\right)\\le \\max _{y\\in [x,\\,x+ca_n]}Lf(y)\\mathbb {E}\\left(\\frac{\\delta \\wedge a_n}{a_n}\\right),$ we thus infer by means of Lemma REF and the DCT that $\\lim _{n\\rightarrow \\infty }\\frac{\\mathbb {E}\\left(\\sum _{i\\in \\mathcal {I}}1_{(0,\\delta \\wedge a_n]}(t_i)\\Delta Z^{x,f}_{t_i}\\right)}{a_n}=Lf(x).$ Let us now deal with the remaining contribution to the difference $Z^{x,f}_{a_n}-Z^{x,f}_{0}$ .", "For this purpose, consider the process $(\\hat{Z}^{x,f}_t)_{t\\in {\\mathbb {R}}^+_0}$ , given by $\\hat{Z}^{x,f}_t:=Z^{x,f}_t-\\sum _{i\\in \\mathcal {I}:t_i\\le t}\\Delta Z^{x,f}_{t_i}.$ Since, according to Lemma REF and (REF ), $\\mathbb {E}\\left(\\sup _{n\\in {\\mathbb {N}}}\\left|\\frac{f(x+c(\\delta \\wedge a_n))-f(x)}{c(\\delta \\wedge a_n)}\\cdot \\frac{\\delta \\wedge a_n}{a_n}\\right|\\right)\\le \\mathbb {E}\\left(\\sup _{y\\in (x,x+c(\\delta \\wedge a_n))}|f^{\\prime }_+(y)|\\right)\\le \\sup _{y\\in (x,x+c)}|f^{\\prime }_+(y)|<\\infty ,$ we deduce by applying the DCT that $\\begin{aligned}\\lim _{n\\rightarrow \\infty }\\frac{\\mathbb {E}\\left(\\hat{Z}^{x,f}_{\\delta \\wedge a_n}-\\hat{Z}^{x,f}_0\\right)}{a_n}&= \\lim _{n\\rightarrow \\infty }\\mathbb {E}\\left(\\frac{f(x+c(\\delta \\wedge a_n))-f(x)}{a_n}\\right)\\\\[0.5ex]&= c\\,\\lim _{n\\rightarrow \\infty }\\mathbb {E}\\left(\\frac{f(x+c(\\delta \\wedge a_n))-f(x)}{c(\\delta \\wedge a_n)}\\cdot \\frac{\\delta \\wedge a_n}{a_n}\\right)\\\\[0.5ex]&= c\\,\\mathbb {E}\\left(\\lim _{n\\rightarrow \\infty }\\frac{f(x+c(\\delta \\wedge a_n))-f(x)}{c(\\delta \\wedge a_n)}\\cdot \\lim _{n\\rightarrow \\infty }\\frac{\\delta \\wedge a_n}{a_n}\\right)\\\\[0.5ex]&=cf^{\\prime }(x).\\end{aligned}$ Combining (REF ) with (REF ) yields that $0 &=\\lim _{n\\rightarrow \\infty }\\frac{\\mathbb {E}\\left(Z^{x,f}_{\\delta \\wedge a_n}-Z^{x,f}_{0}\\right)}{a_n}\\\\[0.5ex]&= \\lim _{n\\rightarrow \\infty }\\frac{\\mathbb {E}\\left(\\sum _{i\\in \\mathcal {I}}1_{\\lbrace t_i\\in (0,\\delta \\wedge a_n]\\rbrace }\\Delta Z^{x,f}_{t_i}\\right)}{a_n}+\\lim _{n\\rightarrow \\infty }\\frac{\\mathbb {E}\\left(\\hat{Z}^{x,f}_{\\delta \\wedge a_n}-\\hat{Z}^{x,f}_0\\right)}{a_n}\\\\[0.5ex]&= Lf(x)+cf^{\\prime }(x)$ holds for all $x\\in \\mathcal {C}_f$ , where the first equality results from the martingale property of $Z^{x,f}$ in conjunction with the optional sampling theorem.", "Consequently, $f$ solves (REF ), which completes the proof.", "$\\square $" ], [ "Analytic properties of one-sided FKPP travelling waves", "In this section we provide the proof of Theorem REF .", "For this purpose we shall resort to the following version of the fundamental theorem of calculus for Dini derivatives, taken from [15].", "Proposition 11 (Theorem 11 in [15]) Let $a,b\\in {\\mathbb {R}}$ with $a<b$ .", "If f is a continuous function that has a finite Dini derivative $f^{\\prime }_+(y)$ for every $y\\in [a,b]$ , then $f(b)-f(a) = \\int _{[a,b]} f^{\\prime }_+(y)\\text{ d}y,$ provided that $f^{\\prime }_+$ is Lebesgue integrable over $[a, b]$ .", "This version of the fundamental theorem of calculus for Dini derivatives will be used in the proof of Proposition REF that we are now going to present.", "Furthermore, we shall resort to Proposition REF also in the proof of Theorem REF , where we show that travelling waves are continuously differentiable.", "Let us point out that $f$ having finite Dini derivatives is essential in Proposition REF .", "Indeed, for singular functions $f$ , such as the Cantor function, the equality in (REF ) does not hold true, since in that case $f^{\\prime }_+=0$ Lebesgue-a.e.", "but $f$ is not a constant function.", "Let us proceed with the following proposition that shows uniqueness of one-sided FKPP travelling waves in $\\mathcal {T}$ with wave speed $c>c_{\\bar{p}}$ .", "Our method of proof for this result makes use of Proposition REF .", "Proposition 12 Any one-sided FKPP travelling wave $f\\in \\mathcal {T}$ with wave speed $c>c_{\\bar{p}}$ satisfies $f=\\varphi .$ Proof Let $f\\in \\mathcal {T}$ be a function that solves (REF ) and fix some $x>0$ .", "Notice first that the map $t\\mapsto \\hat{Z}^{x,f}_t:=Z^{x,f}_t-\\sum _{i\\in \\mathcal {I}:t_i\\le t}\\Delta Z^{x,f}_{t_i}$ is continuous.", "In addition, recall from (REF ) the definition $\\delta :=\\inf \\left\\lbrace t>0:x+ct+\\ln \\left(|\\pi (t)|^\\downarrow _2\\right)>0\\right\\rbrace \\wedge \\tau ^-_{1,0}\\wedge 1$ and observe by means of (REF ) that the Dini derivative $\\hat{Z}^{x,f}_+$ , given by $\\hat{Z}^{x,f}_+(s):=\\limsup _{h\\downarrow 0}\\frac{\\hat{Z}^{x,f}_{s+h}-\\hat{Z}^{x,f}_s}{h} =c f^{\\prime }_+(X^x_1(s))$ for all $s\\in [0,\\delta ]$ , is a finite Lebesgue measurable function, since $f^{\\prime }_+$ and $s\\mapsto X^x_1(s)$ are Lebesgue measurable.", "Moreover, in view of (REF ) we also infer that $\\int _{[0,\\delta ]}\\left|\\hat{Z}^{x,f}_+(s)\\right|\\text{d}s\\le c\\int _{[0,\\delta )}\\left|f^{\\prime }_+(X^x_1(s))\\right|\\text{d}s\\le c \\int _{[0,1)}\\sup _{y\\in [x,\\,x+c]}\\left|f^{\\prime }_+(y)\\right|\\text{d}s= c\\sup _{y\\in [x,\\,x+c]}\\left|f^{\\prime }_+(x)\\right|<\\infty $ holds $\\mathbb {P}$ -almost surely.", "Hence, $\\hat{Z}^{x,f}_+(s)$ is Lebesgue integrable over $[0,\\delta ]$ .", "According to Proposition REF we thus obtain that $Z^{x,f}_{\\delta \\wedge a_n}-Z^{x,f}_0=\\hat{Z}^{x,f}_{\\delta \\wedge a_n}-\\hat{Z}^{x,f}_0+\\sum _{i\\in \\mathcal {I}:t_i\\le \\delta \\wedge a_n}\\Delta Z^{x,f}_{t_i}=\\int _{[0,\\delta \\wedge a_n]} \\hat{Z}^{x,f}_+(s)\\text{ d}s+\\sum _{i\\in \\mathcal {I}:t_i\\le \\delta \\wedge a_n}\\Delta Z^{x,f}_{t_i}$ for all $n\\in {\\mathbb {N}}$ , where $(a_n)_{n\\in {\\mathbb {N}}}$ is a sequence in $(0,1)$ with $a_n\\downarrow 0$ as $n\\rightarrow \\infty $ .", "With $\\eta $ being the Poisson random measure on ${\\mathbb {R}}^+_0\\otimes \\mathcal {P}$ that determines $X_1$ , we deduce from (REF ), in conjunction with the compensation formula for Poisson point processes and Fubini's theorem that $&\\mathbb {E}\\left(Z^{x,f}_{\\delta \\wedge a_n}\\right)-\\mathbb {E}\\left(Z^{x,f}_0\\right)\\\\[0.5ex]&= \\mathbb {E}\\left(\\int \\limits _{[0,\\delta \\wedge a_n]} \\hat{Z}^{x,f}_+(s)\\text{ d}s\\right)\\\\&\\qquad +\\mathbb {E}\\left(\\int \\limits _{[0,1]\\times \\mathcal {P}}1_{[0,\\delta \\wedge a_n]}(s)\\left(\\prod _{k\\in {\\mathbb {N}}}f(X^x_1(s-)+\\ln (|\\pi _k|))-f(X^x_1(s-))\\right)\\eta (\\text{d}s,\\text{d}\\pi )\\right)\\\\[0.5ex]&= \\mathbb {E}\\left(\\int \\limits _{[0,\\delta \\wedge a_n]} \\hat{Z}^{x,f}_+(s)\\text{ d}s\\right)+\\mathbb {E}\\left(\\int \\limits _{[0,\\delta \\wedge a_n]}\\int \\limits _{\\mathcal {P}}\\left(\\prod _{k\\in {\\mathbb {N}}}f(X^x_1(s-)+\\ln (|\\pi _k|))-f(X^x_1(s-))\\right)\\mu (\\text{d}\\pi )\\text{ d}s\\right)\\\\[0.5ex]&= \\mathbb {E}\\left(\\int \\limits _{[0,\\delta \\wedge a_n]} \\hat{Z}^{x,f}_+(s)\\text{ d}s\\right)+\\mathbb {E}\\left(\\int \\limits _{[0,\\delta \\wedge a_n]}Lf(X^x_1(s-)\\text{ d}s\\right)$ for all $n\\in {\\mathbb {N}}$ .", "Observe that $f$ being monotone and $X^x_1$ having only at most countably many jumps $\\mathbb {P}$ -a.s. implies that $X^x_1(s)\\in C_f$ for Lebesgue-a.a. $s\\in (0,1)$ $\\mathbb {P}$ -almost surely.", "By means of (REF ) and (REF ) as well as the fact that any $u\\in (0,1)$ is $\\mathbb {P}$ -a.s. not a jump time of $\\Pi $ this results in $\\mathbb {E}\\left(Z^{x,f}_{\\delta \\wedge a_n}\\right)-\\mathbb {E}\\left(Z^{x,f}_0\\right)= \\mathbb {E}\\left(\\int _{[0,\\delta \\wedge a_n]}\\left(c f^{\\prime }_++Lf\\right)(X^x_i(s))\\text{d}s\\right)= 0,$ i.e.", "$\\mathbb {E}\\left(Z^{x,f}_{\\delta \\wedge a_n}\\right)=\\mathbb {E}\\left(Z^{x,f}_0\\right)=f(x).$ By means of the strong fragmentation property of $\\Pi $ we thus conclude that $\\mathbb {E}\\left(\\left.Z^{x,f}_{\\tau +\\delta \\wedge a_n}\\right|F_\\tau \\right)=\\prod _{n\\in \\mathcal {N}^x_\\tau }\\left.\\mathbb {E}\\left(Z^{y,f}_{\\delta \\wedge a_n}\\right)\\right|_{y=X^x_n(\\tau )}=\\prod _{n\\in \\mathcal {N}^x_\\tau }f(X^x_n(\\tau ))=Z^{x,f}_\\tau $ holds $\\mathbb {P}$ -a.s. for every finite stopping time $\\tau $ .", "Therefore, $\\begin{aligned}\\mathbb {E}\\left(\\left.Z^{x,f}_{t+k(\\delta \\wedge a_n)}\\right|F_t\\right)&= \\mathbb {E}\\left(\\left.Z^{x,f}_{t+\\delta }+\\sum _{n=2}^k\\mathbb {E}\\left(\\left.Z^{x,f}_{t+n\\delta }-Z^{x,f}_{t+(n-1)\\delta }\\right|F_{t+(n-1)\\delta }\\right)\\right|F_t\\right)\\\\&= \\mathbb {E}\\left(\\left.Z^{x,f}_{t+\\delta }\\right|F_t\\right)\\\\&= Z^{x,f}_t\\end{aligned}$ $\\mathbb {P}$ -a.s. for all $t\\in {\\mathbb {R}}^+_0$ and every measurable $k:\\Omega \\rightarrow {\\mathbb {N}}$ .", "For any $s,t\\in {\\mathbb {R}}^+_0$ set $k_s:=\\left\\lfloor \\frac{s}{\\delta \\wedge a_n}\\right\\rfloor \\qquad \\text{as well as}\\qquad r_s:=\\frac{s}{\\delta \\wedge a_n}-k_s\\in (0,1)$ and observe that $Z^{x,f}$ is $\\mathbb {P}$ -a.s. left-continuous at $t+s$ , since $t+s$ is $\\mathbb {P}$ -a.s. not a jump time of $\\Pi $ .", "In conjunction with the DCT for conditional expectations and (REF ) this implies that $\\mathbb {E}\\left(\\left.Z^{x,f}_{t+s}\\right|F_t\\right)=\\lim _{n\\rightarrow \\infty }\\mathbb {E}\\left(\\left.Z^{x,f}_{t+s-r_s(\\delta \\wedge a_n)}\\right|F_t\\right)=\\lim _{n\\rightarrow \\infty }\\mathbb {E}\\left(\\left.Z^{x,f}_{t+k_s(\\delta \\wedge a_n)}\\right|F_t\\right)=Z^{x,f}_t$ for all $s,t\\in {\\mathbb {R}}^+_0$ .", "Hence, $Z^{x,f}$ is a martingale and thus we deduce from Theorem 10 in [19] that $f=\\varphi $ .", "$\\square $ Proposition REF shows that in order to derive analytic properties of one-sided FKPP travelling waves in $\\mathcal {T}$ we only need to consider the function $\\varphi $ .", "Bearing this in mind we proceed to prove Theorem REF .", "In order to obtain strict monotonicity of $\\varphi $ we shall use the following result.", "Lemma 13 Let $c>c_{\\bar{p}}$ .", "For any $0\\le x<y<\\infty $ there exists some $\\alpha _{x,y}>0$ such that $\\varphi (x)-\\varphi (x+h)\\ge \\alpha _{x,y}\\left(\\varphi (y)-\\varphi (y+h)\\right)$ for all $h>0$ .", "Proof In the first part of this proof we show that for every deterministic time $t>0$ the probability that $X_1$ reaches level $x>0$ before time $t$ is positive.", "In the second part we use this fact in order to obtain a lower bound of the probability that for some $n\\in {\\mathbb {N}}$ the process $X^x_n$ hits a given level $y>x$ before some deterministic time $s>0$ .", "Subsequently, we combine this lower bound with the estimate (REF ) of the number of blocks that are alive at a given time and with the positivity of the probability of extinction.", "Part I For every $x\\in {\\mathbb {R}}^+_0$ set $\\quad \\tau ^+_{1,x}:=\\inf \\lbrace t\\in {\\mathbb {R}}^+_0:X_1(t)>x\\rbrace .$ According to Corollary 3.14 in [23] we have that $(\\tau ^+_{1,x})_{x\\in {\\mathbb {R}}^+_0}$ is a subordinator with either killing at an independent exponential “time” $\\mathfrak {e}$ or with no killing in which case we set $\\mathfrak {e}:=\\infty $ .", "Moreover, by means of Proposition 1.7 in [3] we thus infer that $\\mathbb {P}\\left(\\tau ^+_{1,x}< t\\right)=\\mathbb {P}\\left(\\lbrace \\tilde{\\tau }^+_{1,x}< t\\rbrace \\cap \\lbrace x<\\mathfrak {e}\\rbrace \\right)=\\mathbb {P}\\left(\\tilde{\\tau }^+_{1,x}< t\\right)\\mathbb {P}\\left(x< \\mathfrak {e}\\right)>0$ holds for all $t>0$ and $x\\in {\\mathbb {R}}^+_0$ , where $(\\tilde{\\tau }^+_{1,x})_{x\\in {\\mathbb {R}}^+_0}$ is some non-killed subordinator, independent of $\\mathfrak {e}$ , satisfying $\\tilde{\\tau }^+_{1,x}1_{\\lbrace x< \\mathfrak {e}\\rbrace }=\\tau ^+_{1,x}1_{\\lbrace x< \\mathfrak {e}\\rbrace }.$ For the time being, fix some $x\\in {\\mathbb {R}}^+_0$ .", "Let us now show that $\\forall \\,t>0:\\,\\mathbb {P}(\\tau ^+_{1,x}<\\tau ^-_{1,0}\\wedge t)>0.$ To this end, assume we have $\\exists \\,t_0>0:\\,\\mathbb {P}(\\tau ^+_{1,x}<\\tau ^-_{1,0}\\wedge t_0)=0.$ Our goal is to show that this results in a contradiction.", "For this purpose, set $\\tau ^2_0:=\\tilde{\\tau }^2_0:=0$ and for every $n\\in {\\mathbb {N}}$ define $\\tilde{\\tau }^1_n := \\inf \\lbrace t\\ge \\tilde{\\tau }^2_{n-1}:X_1(t)<0\\rbrace \\qquad \\text{as well as}\\qquad \\tilde{\\tau }^2_n := \\inf \\lbrace t\\ge \\tilde{\\tau }^1_n:X_1(t)=0\\rbrace .$ In addition, set $n^*:=\\sup \\left\\lbrace n\\in {\\mathbb {N}}:\\tilde{\\tau }^2_n<\\infty \\right\\rbrace $ as well as $\\tau ^1_n := \\inf \\lbrace t\\ge \\tau ^2_{n-1}:X_1(t)<0\\rbrace \\qquad \\text{and}\\qquad \\tau ^2_n := \\inf \\lbrace t\\ge \\tau ^1_n:X_1(t)=0\\rbrace \\wedge \\tilde{\\tau }^2_{n^*},$ where $\\tilde{\\tau }^2_\\infty :=\\infty $ .", "Since for $X_1$ the point 0 is irregular for $(-\\infty ,0)$ , there exists some $\\varepsilon >0$ such that $\\mathbb {P}(\\tau ^-_{1,0}\\ge \\varepsilon )>0$ and consequently we obtain by means of the strong Markov property of $\\Pi $ that $\\sum _{n\\in {\\mathbb {N}}}\\mathbb {P}\\left(\\tau ^1_n-\\tau ^2_{n-1}\\ge \\varepsilon \\left|F_{\\tau ^2_{n-1}}\\right.\\right)=\\sum _{n\\in {\\mathbb {N}}}\\mathbb {P}\\left(\\tau ^-_{1,0}\\ge \\varepsilon \\right)=\\infty $ $\\mathbb {P}$ -almost surely.", "Since $\\lbrace \\tau ^1_n-\\tau ^2_{n-1}\\ge \\varepsilon \\rbrace $ is $F_{\\tau ^2_{n}}$ -measurable, we can apply an extended Borel-Cantelli lemma (see e.g.", "[12] or [9]) to deduce that $\\left\\lbrace \\lbrace \\tau ^1_n-\\tau ^2_{n-1}\\ge \\varepsilon \\rbrace \\text{ holds for infinitely many $n\\in {\\mathbb {N}}$}\\right\\rbrace =\\left\\lbrace \\sum _{n\\in {\\mathbb {N}}}\\mathbb {P}\\left(\\tau ^1_n-\\tau ^2_{n-1}\\ge \\varepsilon \\left|F_{\\tau ^2_{n-1}}\\right.\\right)=\\infty \\right\\rbrace .$ Thus (REF ) implies that $\\tau ^2_n\\rightarrow \\infty $ $\\mathbb {P}$ -a.s. on the event $\\lbrace n^*=\\infty \\rbrace $ as $n\\rightarrow \\infty $ .", "With $t_0$ given by (REF ) another application of the strong Markov property therefore yields that $\\mathbb {P}\\left(\\tau ^+_{1,x}< t_0\\right) \\le \\mathbb {E}\\left(\\sum _{n\\in {\\mathbb {N}}}\\mathbb {P}\\left(\\tau ^3_{n,x}<\\tau ^1_n\\wedge t_0\\left|F_{\\tau ^2_{n-1}}\\right.\\right)\\right)= \\sum _{n\\in {\\mathbb {N}}}\\mathbb {P}\\left(\\tau ^+_{1,x}<(\\tau ^-_{1,0}\\wedge t_0)\\right)=0,$ where $\\tau ^3_{n,x} := \\inf \\lbrace t\\ge \\tau ^2_{n-1}:X_1(t)>x\\rbrace $ for all $n\\in {\\mathbb {N}}$ .", "Since (REF ) contradicts (REF ), we conclude that (REF ) does indeed hold true.", "Part II Let $0\\le x<y<\\infty $ and for any $t\\in {\\mathbb {R}}^+_0$ set $R^x_1(t):=\\sup _{n\\in {\\mathbb {N}}}X^x_n(t)$ .", "In addition, we define $\\tau ^+_y(x):=\\inf \\left\\lbrace t\\in {\\mathbb {R}}^+_0:R^x_1(t)\\ge y\\right\\rbrace .", "$ Note that $R^x_1(\\tau ^+_y(x))=y$ if $\\tau ^+_y(x)<\\infty $ , since $R^x_1$ does not jump upwards and thus creeps over the level $y$ .", "Furthermore, let $s>0$ and set $\\gamma :=e^{x+cs}-1$ as well as $\\alpha _{x,y}:=\\mathbb {P}\\left(\\tau ^+_y(x)<\\zeta ^x\\wedge s\\right)\\mathbb {P}(\\zeta ^{y}<\\infty )^\\gamma .$ Observe that (REF ) and Proposition REF imply that $\\alpha _{x,y}>0$ , since $\\mathbb {P}\\left(\\tau ^+_y(x)<\\zeta ^x\\wedge s\\right)\\ge \\mathbb {P}\\left(\\tau ^+_{1,y-x}<\\tau ^-_{1,0}\\wedge s\\right).$ By means of the strong fragmentation property of $\\Pi $ we deduce that $\\varphi (x)-\\varphi (x+h) &= \\mathbb {P}(\\zeta ^x<\\infty )-\\mathbb {P}(\\zeta ^{x+h}<\\infty )\\\\[0.5ex]&\\overset{(*)}{\\ge }\\mathbb {P}\\left(\\tau ^+_y(x)<\\zeta ^x\\wedge s\\right)\\mathbb {P}(\\zeta ^y<\\infty )^\\gamma \\left(\\mathbb {P}(\\zeta ^y<\\infty )-\\mathbb {P}(\\zeta ^{y+h}<\\infty )\\right)\\\\[0.5ex]&= \\alpha _{x,y}\\left(\\varphi (y)-\\varphi (y+h)\\right)$ holds true for any $h>0$ , where the exponent $\\gamma $ in $(*)$ results from the estimate $N^x_{\\tau ^+_y(x)}\\le e^{x+c\\tau ^+_y(x)}< e^{x+cs}=\\gamma +1$ $\\mathbb {P}$ -a.s. on $\\lbrace \\tau ^+_y(x)<s\\rbrace $ .", "Notice that in $(*)$ we have used that the value of $X^x_n$ , $n\\in {\\mathbb {N}}$ , at time $\\tau ^+_y(x)$ is less than or equal to $y$ as well as the monotonicity of the probability of extinction.", "$\\square $ Observe that $\\varphi $ is clearly a monotone function.", "However, even though monotonicity is trivial, it is not obvious whether $\\varphi $ is strictly monotone.", "The following lemma answers the question regarding strict monotonicity of $\\varphi $ affirmatively.", "Lemma 14 Let $c>c_{\\bar{p}}$ .", "Then $\\varphi $ is strictly monotonically decreasing on ${\\mathbb {R}}^+_0$ .", "Proof Let $x\\in {\\mathbb {R}}^+_0$ and set $\\gamma _x:=\\ln \\left(|\\pi (\\zeta ^x)|^\\downarrow _1\\cdot \\left|\\Pi ^x_{\\kappa (\\zeta ^x)}(\\zeta ^x-)\\right|\\right).$ According to Proposition REF we have $\\mathbb {P}(\\zeta ^x<\\infty )>0$ and hence $& \\mathbb {P}\\left(\\lbrace \\zeta ^x<\\infty \\rbrace \\cap \\bigcup _{n\\in {\\mathbb {N}}}\\lbrace x+c\\zeta ^x+\\gamma _x\\in (-n,0)\\rbrace \\right)\\\\[0.5ex]&= \\mathbb {P}(\\lbrace \\zeta ^x<\\infty \\rbrace \\cap \\lbrace x+c\\zeta ^x+\\gamma _x\\in (-\\infty ,0)\\rbrace )\\\\[0.5ex]&= \\mathbb {P}(\\zeta ^x<\\infty )\\\\[0.5ex]&> 0.$ Therefore, there exists some $z>0$ such that $\\mathbb {P}\\left(\\lbrace \\zeta ^x<\\infty \\rbrace \\cap \\lbrace x+c\\zeta ^x+\\gamma _x\\in (-z,0)\\rbrace \\right)>0$ and thus the strong fragmentation property, in conjunction with Proposition REF , yields that $\\mathbb {P}(\\lbrace \\zeta ^x<\\infty \\rbrace \\cap \\lbrace \\zeta ^{x+z}=\\infty \\rbrace )\\ge \\mathbb {P}(\\lbrace \\zeta ^x<\\infty \\rbrace \\cap \\lbrace x+c\\zeta ^x+\\gamma _x\\in (-z,0)\\rbrace )\\mathbb {P}(\\zeta ^0=\\infty )> 0.$ Consequently, there exists some $z>0$ such that $\\mathbb {P}(\\zeta ^x<\\infty ) &= \\mathbb {P}(\\lbrace \\zeta ^x<\\infty \\rbrace \\cap \\lbrace \\zeta ^{x+z}=\\infty \\rbrace )+\\mathbb {P}(\\lbrace \\zeta ^x<\\infty \\rbrace \\cap \\lbrace \\zeta ^{x+z}<\\infty \\rbrace )\\\\[0.5ex]&> \\mathbb {P}(\\zeta ^{x+z}<\\infty ),$ where the final estimate follows from $\\lbrace \\zeta ^{x+z}<\\infty \\rbrace \\subseteq \\lbrace \\zeta ^x<\\infty \\rbrace $ .", "Observe that (REF ) implies that for every $h>0$ there exists some $y\\ge x$ such that $\\varphi (y)>\\varphi (y+h).$ According to Lemma REF , for all $h>0$ and $y\\ge x$ satisfying (REF ) there exists some $\\alpha _{x,y}>0$ such that $\\varphi (x)-\\varphi (x+h)\\ge \\alpha _{x,y}\\left(\\varphi (y)-\\varphi (y+h)\\right)>0.$ Since $x\\in {\\mathbb {R}}^+_0$ was chosen arbitrarily, this proves the assertion that $\\varphi $ is strictly monotonically decreasing on ${\\mathbb {R}}^+_0$ .", "$\\square $ In the proof of Theorem REF we shall make use of the theory of scale functions for spectrally negative Lévy processes.", "For this purpose, let $W$ be the scale function of the spectrally negative Lévy process $X_1$ .", "That is to say, $W$ is the unique continuous and strictly monotonically increasing function $W:{\\mathbb {R}}^+_0\\rightarrow {\\mathbb {R}}^+_0$ , whose Laplace transform satisfies $\\int _{(0,\\infty )}e^{-\\beta x}W(x)\\text{ d}x=\\frac{1}{\\psi (\\beta )}$ for all $\\beta >\\Psi (0)$ , where $\\psi $ denotes the Laplace exponent of $X_1$ and $\\Psi (0):=\\sup \\lbrace \\lambda >0:\\psi (\\lambda )=0\\rbrace $ .", "Let us now tackle the proof of Theorem REF .", "Proof of Theorem  REF This proof is divided into two parts.", "In the first part we show that $\\varphi \\in \\mathcal {T}$ and that $\\varphi $ is right-continuous at 0.", "Subsequently, in the second part we use the continuity of $\\varphi |_{{\\mathbb {R}}^+}$ in order to prove that $\\varphi |_{{\\mathbb {R}}^+}$ is continuously differentiable, if $\\varphi $ solves (REF ).", "According to Proposition REF and Lemma REF the proof is then complete.", "Part I Recall the definition of $\\mathcal {T}$ in Definition REF and note that Theorem 10 in [19] yields that $\\varphi $ satisfies (REF ).", "Hence, since $\\varphi $ is nonincreasing, in order to prove $\\varphi \\in \\mathcal {T}$ it remains to show that $\\varphi |_{{\\mathbb {R}}^+}$ is continuous and that (REF ) holds.", "To this end, let $\\mathfrak {n}$ denote the excursion measure of excursions $({\\bf e}_s)_{s\\in {\\mathbb {R}}^+_0}$ indexed by the local time at the running maximum of the Lévy process $X_1$ .", "Furthermore, for any such excursion ${\\bf e}$ let $\\bar{\\bf e}$ denote the height of this excursion.", "According to Lemma 8.2 in [23] the scale function $W$ has a right-derivative on ${\\mathbb {R}}^+$ given by $W^{\\prime }_+(x)=W(x)\\mathfrak {n}(\\bar{\\bf e}> x)$ for any $x\\in {\\mathbb {R}}^+$ .", "Note that $\\mathfrak {n}$ being $\\sigma $ -finite (cf.", "Theorem 6.15 in [23]) implies that $\\sup _{y\\ge x}\\mathfrak {n}(\\bar{\\bf e}> y)=\\mathfrak {n}(\\bar{\\bf e}> x)<\\infty .$ for every $x\\in {\\mathbb {R}}^+_0$ .", "Moreover, we have $\\mathbb {P}(\\xi ^x=\\infty )\\ge \\mathbb {P}_x\\left(\\tau ^+_{1,y}<\\tau ^-_{1,0}\\right)\\mathbb {P}(\\xi ^y=\\infty )$ for all $x,y\\in {\\mathbb {R}}^+_0$ with $x<y$ , where under $\\mathbb {P}_x$ the process $X_1$ is shifted to start in $x$ .", "Therefore, we obtain $\\varphi (x)-\\varphi (x+h)\\le (1-\\varphi (x))\\left(\\frac{1}{\\mathbb {P}_x\\left(\\tau ^+_{1,x+h}<\\tau ^-_{1,0}\\right)}-1\\right)\\le \\frac{1}{\\mathbb {P}_x\\left(\\tau ^+_{1,x+h}<\\tau ^-_{1,0}\\right)}-1$ for all $h\\in {\\mathbb {R}}^+$ and $x\\in {\\mathbb {R}}^+_0$ .", "By means of (8.8) in Theorem 8.1 of [23] we have $\\frac{1}{\\mathbb {P}_x\\left(\\tau ^+_{1,x+h}<\\tau ^-_{1,0}\\right)}-1=\\frac{W(x+h)-W(x)}{W(x)}$ for any $x\\in {\\mathbb {R}}^+_0$ .", "Consequently, $\\sup _{y\\ge x}\\left|\\varphi ^{\\prime }_+(y)\\right|\\le \\sup _{y\\ge x}\\limsup _{h\\downarrow 0}\\left(\\frac{1}{W(y)}\\frac{W(y+h)-W(y)}{h}\\right)=\\sup _{y\\ge x}\\mathfrak {n}(\\bar{\\bf e}> y)<\\infty $ holds for every $x\\in {\\mathbb {R}}^+$ .", "Moreover, in view of (REF ), (REF ) and the continuity of $W|_{{\\mathbb {R}}^+_0}$ we deduce that $\\varphi |_{{\\mathbb {R}}^+}$ is continuous.", "Therefore, we conclude that $\\varphi \\in \\mathcal {T}$ .", "Since $X_1$ has bounded variation, we infer by means of Lemma 8.6 in [23] that $W(0)>0$ and thus the above line of argument also yields that $\\varphi $ is right-continuous at 0.", "Part II Assume that $\\varphi $ satisfies (REF ) on $\\mathcal {C}_\\varphi $ .", "In view of Part I it follows from Lemma REF that $L\\varphi $ is continuous.", "Moreover, we deduce from (REF ) and the monotonicity of $\\varphi $ that $\\varphi ^{\\prime }_+=-c^{-1}L\\varphi $ Lebesgue-almost everywhere on ${\\mathbb {R}}^+$ .", "Since the upper Dini derivative $\\varphi ^{\\prime }_+$ is bounded on any interval $[a,b]\\subseteq {\\mathbb {R}}^+$ , it thus follows from Proposition REF and Lebesgue's integrability criterion for Riemann integrals that $\\varphi (b)-\\varphi (a)=\\int _a^b\\varphi ^{\\prime }_+(x)\\text{ d}x=-\\frac{1}{c}\\int _a^bL\\varphi (x)\\text{ d}x=F(b)-F(a)$ for Lebesgue-almost all $a,b\\in {\\mathbb {R}}^+$ , where $F\\in C^1({\\mathbb {R}}^+,{\\mathbb {R}})$ is an antiderivative of $-c^{-1}L\\varphi $ on ${\\mathbb {R}}^+$ .", "Hence, we have $\\varphi =F+\\text{const.", "}$ Lebesgue-almost everywhere on ${\\mathbb {R}}^+$ .", "Since $\\varphi |_{{\\mathbb {R}}^+_0}$ and $F$ are continuous, this implies that $\\varphi |_{{\\mathbb {R}}^+}=F+\\text{const.", "}$ and consequently $\\varphi |_{{\\mathbb {R}}^+}\\in C^1({\\mathbb {R}}^+,[0,1])$ .", "In the light of Proposition REF and Lemma REF this proves the assertion.", "$\\square $" ], [ "Existence and uniqueness of one-sided travelling waves", "This section is devoted to the proof of Theorem REF .", "Our method of proof makes use of the results that we developed in the previous two sections.", "Proof of Theorem  REF The first part of the proof shows the nonexistence of one-sided FKPP travelling waves in $\\mathcal {T}$ for wave speeds $c\\le c_{\\bar{p}}$ and the second part proves the existence of such travelling waves for wave speeds above the critical value $c_{\\bar{p}}$ .", "The uniqueness was shown in Proposition REF .", "Part I Fix some $c\\le c_{\\bar{p}}$ as well as $x\\in {\\mathbb {R}}^+_0$ and let $f\\in \\mathcal {T}$ .", "Further, assume that $f$ satisfies (REF ).", "Then the proof of Proposition REF shows that $(Z^{x,f}_t)_{t\\in {\\mathbb {R}}^+_0}$ is a uniformly integrable martingale and hence the $\\mathbb {P}$ –a.s.", "martingale limit $Z^{x,f}_\\infty :=\\lim _{t\\rightarrow \\infty }Z^{x,f}_t$ satisfies $\\mathbb {E}\\left(Z^{x,f}_\\infty \\right)=\\mathbb {E}\\left(Z^{x,f}_0\\right)=f(x).$ Since $c\\le c_{\\bar{p}}$ , we have according to Proposition REF that $\\mathbb {P}(\\zeta ^x<\\infty )=1$ , that is to say $\\mathcal {N}^x_t\\rightarrow \\emptyset $ $\\mathbb {P}$ -a.s. as $t\\rightarrow \\infty $ .", "Because the empty product equals 1, we thus infer that $Z^{x,f}_\\infty =\\lim _{t\\rightarrow \\infty }\\prod _{n\\in \\mathcal {N}^x_t}f(X^x_n(t))=1$ $\\mathbb {P}$ -almost surely.", "In view of (REF ) this implies that $f\\equiv 1$ , which is a contradiction to $f\\in \\mathcal {T}$ , since every $f\\in \\mathcal {T}$ satisfies (REF ).", "Consequently, there does not exist a function $f\\in \\mathcal {T}$ that satisfies (REF ).", "Part II Now let $c>c_{\\bar{p}}$ and $x\\in {\\mathbb {R}}^+_0$ .", "In the light of Proposition REF it only remains to show that $\\varphi $ is indeed a one-sided FKPP travelling wave with wave speed $c$ .", "Since $\\varphi \\in \\mathcal {T}$ satisfies the boundary condition (REF ), we only have to deal with (REF ).", "In order to prove that $\\varphi $ solves (REF ) we aim at applying Theorem REF .", "To this end, observe that the fragmentation property of $\\Pi $ yields that $\\varphi (x)=\\mathbb {E}(\\mathbb {P}(\\zeta ^x<\\infty |F_t))=\\mathbb {E}\\left(\\prod _{n\\in \\mathcal {N}^x_t}\\varphi (X^x_n(t))\\right)=\\mathbb {E}\\left(Z^{x,\\varphi }_t\\right)$ for every $t\\in {\\mathbb {R}}^+_0$ .", "By means of another application of the fragmentation property we therefore deduce that $\\mathbb {E}\\left(\\left.Z^{x,\\varphi }_{t+s}\\right|F_t\\right)=\\prod _{n\\in \\mathcal {N}^x_t}\\left.\\mathbb {E}\\left(Z^{y,\\varphi }_s\\right)\\right|_{y=X^x_n(t)}=\\prod _{n\\in \\mathcal {N}^x_t}\\varphi (X^x_n(t))=Z^{x,\\varphi }_t$ holds $\\mathbb {P}$ -a.s. for all $s,t\\in {\\mathbb {R}}^+_0$ .", "Hence, $Z^{x,\\varphi }$ is a $\\mathbb {P}$ -martingale.", "In the proof of Theorem REF we have shown that $\\varphi \\in \\mathcal {T}$ and consequently we infer from Theorem REF that $\\varphi $ solves the integro-differential equation (REF ).", "$\\square $" ] ]

[ [ "Elastic pp Scattering at LHC Energies" ], [ "Abstract We consider the first LHC data for elastic pp scattering in the framework of Regge theory with multiple Pomeron exchanges.", "The simplest eikonal approach allows one to describe differential elastic cross sections at LHC, as well as pp and $\\bar{p}p$ scattering at lower collider energies, on a reasonable level." ], [ "Introduction", "In Regge theory the Pomeron exchange dominates the high energy soft hadron interaction.", "The Pomeron has vacuum quantum numbers, so the difference in $pp$ and $\\bar{p}p$ should disappear.", "At LHC energies the contributions of all other exchanges to the elastic scattering amplitude becomes negligible, and then one can directly extract the Pomeron parameters from the experimental data.", "In the present paper we consider the first LHC data (TOTEM Collaboration [1]) for $pp$ small angle elastic scattering and we compare them with the simplest approaches of Regge theory and with the results ed for other lower collider energies.", "The experimental elastic cross section is well described by a pure exponential form in the interval of momentum transfer $\\vert t \\vert = 0 - 0.3$ GeV$^2$ .", "In this interval the cross section falls down more than 400 times.", "The experimental ratio of $\\sigma _{el}/\\sigma _{tot}$ is equal to $\\sim 0.25$ and an intersting point to be analysed is whether in the framework of a conventional Regge theory we have a chance to describe such a large elastic cross section without introducing, either a second Pomeron pole with a large intercept $\\alpha _P(0)=1.362$ , as in [2], a rather non-trivial spatial $b_t$ -distribution of the matter in the proton with a deep minimum at $b_t=0$ , like it was done in [3] (see the form of $\\gamma (b)$ in Eq.", "(9) of [3])Note that the total cross section obtained in [3] for $\\sqrt{s}=7$  TeV is $\\sigma _{tot}=90.9$  mb, much smaller than that measured by TOTEM., or more complicated approaches, such as the three-channel eikonal model [4] or the model [5] in which uses the general parton distributions." ], [ "Elastic Scattering Amplitude at LHC energies", "Let us consider elastic $pp$ ($\\bar{p}p$ ) scattering at very high energies in the framework of Regge-Gribov theory [6], where only Pomeron exchanges should be accounted for.", "It is suitable to use the following normalization of the elastic scattering amplitude $A(s,t)$ : $\\sigma ^{tot} = 8 \\pi \\cdot Im A(s,t=0)\\;, \\;\\;\\frac{d\\sigma }{dt} = 4\\pi \\cdot \\vert A(s,t) \\vert ^2 \\;.$ The simplest contribution to the elastic scattering amplitude is the one-Pomeron, $P$ , exchange, that can be written as: $A^{(1)}(s,t) = \\gamma (t) \\cdot \\left(\\frac{s}{s_0}\\right)^{\\alpha _P(t) - 1}\\cdot \\eta (\\Theta ) \\;,$ where $\\gamma (t) = g_1(t)\\cdot g_2(t)$ , $g_1(t)$ and $g_2(t)$ are the couplings of a Pomeron to the beam and target hadrons, $\\alpha _P(t) = \\alpha _P(0) + \\alpha ^{\\prime }_P\\cdot t$ is the Pomeron trajectory, $\\alpha _P(0)$ (intercept) and $\\alpha ^{\\prime }_P$ (slope) are some numbers, and $\\eta (\\Theta )$ is the signature factor which determines the complex structure of the scattering amplitude ($\\Theta $ equal to +1 and to $-1$ for Reggeon with positive and negative signature, respectively).", "Specifically for Pomeron exchange ($\\Theta = +1$ ): $\\eta (\\Theta ) = \\frac{1 + \\Theta \\cdot exp [-i \\pi \\alpha _P(t)]}{\\sin {[\\pi \\alpha _P(t)]}} = i - \\tan ^{-1}\\left(\\frac{\\pi \\alpha _P(t)}{2}\\right) \\;.$ In the case of a Pomeron trajectory with $\\alpha _P(0) > 1$ , the correct asymptotic behavior $\\sigma _{tot} \\sim \\ln ^2s$ [7], [8] compatible with the Froissart bound [9], can only be obtained by taking into account the multipomeron cuts.", "Indeed, for the Pomeron trajectory $\\alpha _P(t) = 1 + \\Delta + \\alpha ^{\\prime }_P\\cdot t\\;,\\,\\, \\Delta > 0 \\;,$ the one-Pomeron contribution to $\\sigma ^{tot}_{hN}$ rises with energy as $s^{\\Delta }$ .", "To comply with the $s$ -channel unitarity and, in particular, with the Froissart bound, this contribution should be screened by the multipomeron discontinuities shown in Fig. 1.", "Figure: Regge-pole theory diagrams: (a) single, (b) double, and (c) triplePomeron exchange in elastic pppp scattering.A simple quasi-eikonal treatment [10] allows one to present the total elastic scattering amplitude $A(s,t)$ as a series $A(s,t) = A^{(1)}(s,t) + A^{(2)}(s,t) + A^{(3)}(s,t) + ... \\;,$ where each $A^{(n)}(s,t)$ contribution corresponds to the exchange of $n$ Pomerons.", "The value of $A^{(1)}(s,t)$ is given by Eq.", "(2), and $A^{(2)}(s,t) = \\frac{1}{2!}", "\\int \\frac{d^2\\vec{q}_1}{\\pi }\\cdot A^{(1)}(s,\\vec{q}_1) \\cdot i \\cdot A^{(1)}(s,\\vec{q}-\\vec{q}_1) \\,$ $A^{(3)}(s,t) = \\frac{1}{3!}", "\\int \\frac{d^2\\vec{q}_1}{\\pi }\\cdot \\frac{d^2\\vec{q}_2}{\\pi }\\cdot A^{(1)}(s,\\vec{q}_1) \\cdot i \\cdot A^{(1)}(s,\\vec{q}_2) \\cdot i \\cdot A^{(1)}(s,\\vec{q}-\\vec{q}_1-\\vec{q}_2) \\;,$ where all $q_i$ are two-dimensional vectors in the perpendicular plane to the beam axis, $t = -{\\vec{q}}^2$ .", "The results of the integrations in Eqs.", "(6), (7), etc., depend on the assumption about the form of the function $\\gamma (t)$ , with $t=-q^2$ .", "These integrations can be analytically performed in the simplest case of Gaussian functions: $\\gamma (q^2) = \\gamma _0\\cdot e^{-R^2 \\cdot q^2} \\;.$ In this case the total elastic scattering amplitude of Eq.", "(5) is equal to $A(s,t) = \\eta _P\\cdot \\gamma _0\\cdot e^{\\Delta \\xi }\\cdot \\sum _{n=1}^{\\infty }\\frac{1}{n\\cdot n!}", "\\left(i\\cdot C\\cdot \\frac{\\eta _P\\cdot (q^2/n^2)\\cdot \\gamma _0}{\\lambda }\\cdot e^{\\Delta \\cdot \\xi } \\right)^{n-1} \\cdot exp\\left[-\\frac{\\lambda }{n} q^2\\right] \\;,$ where $C$ is the quasi-eikonal enhancement coefficient (see [10]), $\\lambda = R^2 + \\alpha ^{\\prime }_P\\cdot \\xi $ , $\\xi = \\ln {s/s_0}$ , $s_0$ = 1 GeV$^2$ .", "At asymptotically high energies, $s \\rightarrow \\infty $ , the amplitude of Eq.", "(9) leads to the Froissart behaviour of the total cross section, $\\sigma ^{tot}(s) \\sim \\ln ^2{s}$ .", "On the other hand, it is well-known that the form of the function $\\gamma (q^2)$ in Eq.", "(8) is in contradiction with the experimental data on the shape of the differential elastic cross section, so we have also used the parametrization of $\\gamma (q^2)$ as a sum of two gaussians: $\\gamma (q^2) = \\gamma _0\\cdot (a \\cdot e^{-R^2_1 \\cdot q^2} +(1-a) \\cdot e^{-R^2_2 \\cdot q^2}) \\;,$ that leads to a better agreement with the data." ], [ "Comparison with the experimental data", "The results of the calculation of $d\\sigma /dt$ at $\\sqrt{s}$ = 7 TeV, obtained with the one-exponential parametrization of $\\gamma (q^2)$ in Eq.", "(8), are presented in Fig. 2.", "The values of $\\gamma _0$ were fixed by the value of $\\sigma ^{tot}$ at the same energy measured by TOTEM Collaboration [1].", "The two theoretical curves correspond to the values $C = 1.5$ (quasi-eikonal approach) and $C = 1$ (eikonal approach), and both are in total disagreement with the experimental data (several experimental points presented in Fig.", "2 are taken from [12]).", "Figure: The differential cross section of elastic pppp scattering at s\\sqrt{s} = 7TeV calculated in both eikonal (A) and quasi-eikonal (B) approaches, with theone-exponential parametrization of γ(q 2 )\\gamma (q^2) in Eq. (8).", "The experimentalpoints are taken from .The main reason of the disagreements of the two theoretical curves in Fig.", "2 with the experimental data comes from the rather large rescattering contributions (exchanges of several Pomerons) in Eq. (9).", "These contributions transform the bare Gaussian $t$ -dependence of $d\\sigma /dt$ given by Eq.", "(8) into functions faster decreasing with $q^2$ , whereas the experimental LHC data [1], [12] practically show a Gaussian $t$ -dependence.", "The simplest way to avoid this problem is to use a two-exponential form for the function $\\gamma (q^2)$ as the one given by Eq. (10).", "All the integrals in Eqs.", "(6), (7), etc., can be analytically calculated, giving an expression for $A^{(n)}$ : $A^{(n)}(s,q^2) & = & \\frac{i^{(n-1)}}{n!", "}\\cdot \\left[\\eta _P\\cdot (q^2/n^2)\\cdot \\gamma _0 e^{\\Delta \\cdot \\xi } \\right]^n\\cdot \\left[\\frac{a^n}{n\\cdot \\lambda _1^{(n-1)}} + \\frac{(a-1)^n}{n\\cdot \\lambda _2^{(n-1)}} \\right.", "+\\\\ \\nonumber & + & \\sum _{k=1}^{n-1} C^k_n\\cdot \\frac{a^{(n-k)}\\cdot (1-a)^k}{\\lambda _1^{(n-k-1)}\\cdot \\lambda _2^{(k-1)}\\cdot [k\\cdot \\lambda _1 +(n-k)\\cdot \\lambda _2]} \\cdot \\\\ \\nonumber & \\cdot & \\left.", "exp\\left(-\\frac{q^2}{(n-k)\\cdot \\beta _1 + k\\cdot \\beta _2}\\right) \\right] \\;, \\\\ \\nonumber C^k_n & = & \\frac{n!", "}{k!\\cdot (n-k)!}", "\\;, \\;\\; \\lambda _i = R^2_i +\\alpha ^{\\prime }_P\\cdot \\xi \\;, \\;\\; \\beta _i = 1/\\lambda _i \\;.$ Figure: The differential cross section of elastic pppp scattering at s\\sqrt{s} = 7 TeVcalculated in the eikonal approach, C = 1 (solid curve), and the contribution todifferential cross section of elastic pppp scattering at s\\sqrt{s} = 7 TeV ofonly the real part of the amplitude (dashed curve) by using the two-exponentialparametrization of the function γ(q 2 )\\gamma (q^2) given in Eq. (10).", "The resultsobtained for the differential cross section in the two quasi-eikonalapproaches with C = 1.5 and C = 0.8 are also shown.", "The experimental pointshave been taken from ..", "The results of the calculation of $d\\sigma /dt$ at $\\sqrt{s}$ = 7 TeV obtained with the parametrization of the function $\\gamma (q^2)$ given in Eq.", "(10) are presented in Fig. 3.", "The quasi-eikonal case in which C = 1.5 leads again to a too fast decrease and it gives a too small slope at low $q^2$ .", "Instead, the eikonal approach, C = 1, leads to a reasonable description of the data.", "The agreement of our calculations with the experimental data [1] at small $q^2$ comes from the facts that both the calculated and the experimental $q^2$ -dependences are close to Gaussians and that the calculated value of $\\sigma ^{tot}$ is in agreement with the experimental result [1] (see below).", "One important point to be stressed is that in the diffraction minimum, or in the beginning of the “shoulder”, the cross section $d\\sigma /dt$ is practically determined by only the real part of the amplitude (see solid and dashed curves in Fig. 3).", "The solid curve in Fig.", "3 was calculated with the following values of the parameters: $\\Delta & = & 0.115 , \\;\\; \\alpha ^{\\prime }_P = 0.23 \\; {\\rm GeV}^{-2}, \\;\\;\\gamma = 1.9 \\; {\\rm GeV}^{-2}, \\\\ \\nonumber a & = & 0.48, \\;\\; R^2_1 = 8.5 \\; {\\rm GeV}^{-2}, \\;\\; R^2_2 = 0.9 \\;{\\rm GeV}^{-2} \\;.$ The quality of the description is even better in the quasi-eikonal case with C = 0.8.", "However, value of C smaller than 1 seem to be in contradiction with the Reggeon unitarity condition [13].", "The differential cross section of elastic $\\bar{p}p$ scattering at $\\sqrt{s}$ = 62 GeV, $\\sqrt{s}$ = 546 GeV, and $\\sqrt{s}$ = 1.8 TeV calculated with the values of the parameters given in Eq.", "(12) are presented in Fig. 4.", "At the energy $\\sqrt{s}$ = 62 GeV the theoretical curves are slightly below the experimental points, probably due to the contribution of the $f$ -Reggeon exchange, that has not been accounted for in our calculations.", "Figure: The differential cross section of elastic p ¯p\\bar{p}p scattering ats\\sqrt{s} = 62 GeV, , , s\\sqrt{s} = 546 GeV , , ,and s\\sqrt{s} = 1.8 TeV , calculated in the eikonal (C = 1)approach with the two-exponential parametrization of the function γ(q 2 )\\gamma (q^2)given in Eq.", "(10).The calculated values of total cross sections $\\sigma ^{tot}$ , of $d\\sigma /dt(t=0)$ , and of the slope of the elastic scattering cone parameter $B_{el}$ ($d\\sigma /dt \\sim exp(-B_{el}\\cdot q^2)$ ) are presented in Table 1, together with the experimental data.", "It is necessary to note that the slope parameter was calculated in the interval $q^2 = 0-0.1$ GeV$^2$ .", "Table: NO_CAPTIONTable 1.", "The comparison of the calculated values of total cross sections $\\sigma ^{tot}$ , of $d\\sigma /dt(t=0)$ , and of the slope parameter $B$ with the corresponding experimental data [1], [21], [22], [23].", "The general energy dependence of the differential elastic $pp$ ($\\bar{p}p$ ) cross sections is shown in Fig. 5.", "At the energy $\\sqrt{s} = 62$  GeV some contribution of $f$ -Reggeon should be present.", "Figure: The differential cross section of elastic pppp scattering at s\\sqrt{s} =7 TeV (solid curve), s\\sqrt{s} = 546 GeV (dottedd curve), and s\\sqrt{s} =62 GeV (dashed curve) calculated in the eikonal approach (C = 1) withthe two-exponential parametrization of the funtion γ(q 2 )\\gamma (q^2) given inEq.", "(10).However, in the complete Reggeon diagram technique [6] not only Regge-poles and cuts, but also more complicated diagrams, e.g.", "the so-called enhanced diagrams, should be taken into account.", "In the numerical calculation of such diagrams some new uncertainties appear, since the vertices of the coupling of multireggeon systems are unknown.", "The common feature of such calculations results in the additional increase of the Pomeron intercept $\\alpha _P(0) = 1+\\Delta $ ." ], [ "Conclusion", "We obtain a general description of elastic $pp$ scattering that seems to be successful, as one can see from Figs.", "3 and 4, and from Table 1.", "To do so we only use the three parameters shown in Eq.", "(12), namely $\\gamma $ , which determines the normalization of total $pp$ cross section, $\\Delta $ , which determines the increase of the total $pp$ cross section with energy, and $\\alpha ^{\\prime }_P$ which determines the increase of the diffractive slope cone parameter.", "These parameters are practically not correlated.", "Another three parameters, $a$ , $R_1^2$ , and $R_2^2$ are related to the geometrical shape of the proton and they should be determined from the experiment in the same way as we determine the geometrical shape of atomic nuclei.", "The exact values of the position of the diffractive dip and of the elastic cross section in the dip-region strongly depend of the particular form, like those in Eqs.", "(8) and (10), choosen to parmetrize the $q^2$ -dependence of the Pomeron-nucleon coupling.", "With our oversimplified parametrization we did not succeed in describing the dip-region.", "On the other hand, it is sure this can be done by using a more complicated vertex $\\gamma (q^2)$ with a larger number of parameters, as the parametrization used in [3], and that, strangely enough, shows a minimum at $b_t=0$ , or the one in Eq.", "(11c) of reference [2], which needs a not well justified additional term in order to describe the dip.", "Acknowledgements We are grateful to M.G.", "Ryskin for useful discussions and comments.", "This paper was supported by Ministerio de Educación y Ciencia of Spain under the Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042) and by project FPA 2005–01963, by Xunta de Galicia (Spain) and by University of Santiago de Compostela (Spain), and, in part, by grant RFBR 11-02-00120-a." ] ]

[ [ "Physical properties of Lyman-alpha emitters at $z\\sim 0.3$ from\n UV-to-FIR measurements" ], [ "Abstract The analysis of the physical properties of low-redshift Ly$\\alpha$ emitters (LAEs) can provide clues in the study of their high-redshift analogues.", "At $z \\sim 0.3$, LAEs are bright enough to be detected over almost the entire electromagnetic spectrum and it is possible to carry out a more precise and complete study than at higher redshifts.", "In this study, we examine the UV and IR emission, dust attenuation, SFR and morphology of a sample of 23 GALEX-discovered star-forming (SF) LAEs at $z \\sim 0.3$ with direct UV (GALEX), optical (ACS) and FIR (PACS and MIPS) data.", "Using the same UV and IR limiting luminosities, we find that LAEs at $z\\sim 0.3$ tend to be less dusty, have slightly higher total SFRs, have bluer UV continuum slopes, and are much smaller than other galaxies that do not exhibit Ly$\\alpha$ emission in their spectrum (non-LAEs).", "These results suggest that at $z \\sim 0.3$ Ly$\\alpha$ photons tend to escape from small galaxies with low dust attenuation.", "Regarding their morphology, LAEs belong to Irr/merger classes, unlike non-LAEs.", "Size and morphology represent the most noticeable difference between LAEs and non-LAEs at $z \\sim 0.3$.", "Furthermore, the comparison of our results with those obtained at higher redshifts indicates that either the Ly$\\alpha$ technique picks up different kind of galaxies at different redshifts or that the physical properties of LAEs are evolving with redshift." ], [ "Introduction", "In recent years, a substantial number of Ly$\\alpha $ -emitting galaxies (LAEs) have been discovered over a wide range of redshifts, from the local Universe up to $z \\sim 7$ , and even beyond.", "At $z \\gtrsim 2.0$ , the Ly$\\alpha $ emission is located in the optical or near-IR and LAEs are mostly found using the narrow-band technique, where a combination of narrow- and broad-band filters are employed to sample the Ly$\\alpha $ emission and constrain its nearby continuum, respectively [32], [5], [60], [18], [28], [30], [32], [74], [48], [61], [49], [75].", "At $z\\lesssim 2$ , the Ly$\\alpha $ line is in the UV and, so far, LAEs can only be found via Galaxy Evolution Explorer [47] grism spectroscopy or other space-borne UV observatories.", "In this case, the selection technique is based on searching for Ly$\\alpha $ emission in the UV spectrum of objects with a measured UV continuum [19], [16], [17].", "The physical properties of star-forming (SF) LAEs have been studied by classically fitting [6] (hereafter BC03) templates to their observed spectral energy distribution (SED) [57], [41], [25], [52], [40], [65], [54], [53], [31].", "With this method, the stellar mass and age of galaxies can be reasonably constrained with good sampling of the rest-frame UV to mid-IR SEDs, but other physical properties, such as dust attenuation, star formation rate (SFR), and metallicity tend to suffer from large uncertainties: metallicities can be obtained only by using rest-frame optical emission lines, and information about the dust emission in the FIR is essential for obtaining accurate values of dust attenuation and SFR.", "At $z \\sim 0.3$ , [16], [23] and [17] agree in their optical spectroscopic studies that LAEs are metal-poor galaxies with low dust attenuation.", "[17] also report that LAEs with higher EW(H$\\alpha $ ) have bluer colors, lower metallicities and less extinctions, consistent with their being at a primeval evolutionary stage.", "Employing a SED- fitting procedure, [17] and [23] report that LAEs are mainly young galaxies with median values of 100 and 300 Myr, respectively.", "With the launch of the Herschel Space Telescope [64] and the data taken with the Photodetector Array Camera and Spectrometer [66], we are in possession of deep FIR data that allow us to constrain the FIR SED of LAEs.", "Unfortunately, few FIR counterparts for LAEs at $z \\gtrsim 2.0$ have been reported so far; therefore, a deep analysis has not yet been possible.", "In fact, in [59] we only found four SF LAEs at $2.0 \\lesssim z \\lesssim 3.5$ with FIR counterparts out of a sample of 140, only two of them having a Ly$\\alpha $ rest-frame equivalent width (Ly$\\alpha $ EW$_{rest-frame}$ ) above 20Å, the typical minimum value for LAEs found via narrow-band imaging.", "Despite this low number, the detection of some LAEs in the FIR reveals that some of them are red and dusty objects, thus dust and Ly$\\alpha $ emission are not mutually exclusive [13].", "At $z \\sim 0.3$ , the typical FIR-observed fluxes of LAEs make them probably detectable under the limiting fluxes of PACS and MIPS–24$\\mu $ m observations.", "In [58], we report PACS-FIR detections and study dust attenuation in a sample of twelve spectroscopically GALEX-selected LAEs at $z \\sim 0.3$ and $\\sim 1.0$ .", "In this paper, we expand the work started in [58] about the physical properties of LAEs at z$\\sim $ 0.3.", "Specifically, we obtain UV and total IR luminosities, dust attenuation, SFR, size and examine the morphology of 23 IR-detected LAEs.", "We also compare the results with those obtained for a sample of galaxies which, under the same limiting UV and IR luminosities, do not show Ly$\\alpha $ emission in their rest-frame UV spectra.", "The structure of this paper is as follows.", "In Section we present the LAE data sample, reporting their PACS and MIPS–24$\\mu $ m counterparts in Section .", "The comparison sample is presented in Section .", "Section gives the results of our study and in Section we show the main conclusions of this work.", "Throughout this paper we assume a flat universe with $(\\Omega _m, \\Omega _\\Lambda , h_0)=(0.3, 0.7, 0.7)$ , and all magnitudes are listed in the AB system [56]." ], [ "LAE sample", "In this study, we use a sample of GALEX-discovered LAEs at $z \\sim 0.3$ built in [16] by looking for an emission line in the FUV spectra of objects with a measured UV continuum.", "Since the FUV spectra become very noisy at the edges of the spectral range, only those objects with Ly$\\alpha $ emission within 1452.5–1750 Å were selected.", "In the redshift space, this wavelength range implies that LAEs are distributed within $z=0.195$ –0.44, with a median value of $z \\sim 0.3$ .", "In [16], LAEs were classified as SF or AGN via rest-frame UV emission-line diagnostics (shape and width of the Ly$\\alpha $ line and the presence/absence of AGN ionization lines) and their X-ray measurements.", "In this study, we consider only those LAEs with an SF nature, ruling out the AGN.", "Among the fields studied in [16] we focus on COSMOS, ECDF-South, ELAIS-S1, and Lockman.", "These fields were selected by their wealth of FIR data from PACS-Herschel or MIPS-Spitzer (see Section for more details).", "With the aim of comparing with narrow-band selected high-redshift LAEs, we only include in the sample those Ly$\\alpha $ -emitting galaxies whose Ly$\\alpha $ EW$_{\\rm rest-frame}$ are above 20 Å, which is the typical threshold in narrow-band searches.", "In this sense, we define LAEs as those Ly$\\alpha $ -emitting galaxies whose Ly$\\alpha $ EW$_{\\rm rest-frame}$ are above 20 Å, using this definition throughout the study.", "All the previous considerations render a sample of 30 SF LAEs.", "This sample is nearly complete up to $m_{\\rm NUV} \\sim 21.5$ mag, which represents the UV limiting magnitude of the sample.", "This implies a UV limiting magnitude of $\\log (L_{\\rm UV}/L_\\odot ) = 9.9$ at $z \\sim 0.3$ , which is similar to that in the samples of high-redshift LAEs found via the narrow-band technique.", "All our LAEs have Ly$\\alpha $ luminosities below 10$^{43}$ erg s$^{-1}$ , the typical median value for LAEs at $z \\gtrsim 2.0$ .", "Indeed, the lower values of the Ly$\\alpha $ luminosities for LAEs at $z \\sim 0.3$ are an indication of an evolution in the physical properties of LAEs between that redshift and $z\\gtrsim 2.0$ [17]." ], [ "PACS-FIR and MIPS–24$\\mu $ m counterparts of LAEs", "The PACS-FIR observations used in this study were taken within the framework of PACS Evolutionary Probe project (PEP, PI D. Lutz).", "PEP is the Herschel Guaranteed Time Key-Project to obtain the best profit from studying FIR galaxy evolution with Herschel instrumentation [45].", "Among the fields where LAEs catalogued in [16] are located, COSMOS, GOODS-South and ECDF-South have been already observed in PACS–100$\\mu $ m and PACS–160$\\mu $ m bandsThe Lockman field has also been observed with PACS, but the only SF LAE at z$\\sim $ 0.3 catalogued in [16] in that field has no detection in any of its bands..", "In this study, we use PACS fluxes extracted by using MIPS–24$\\mu $ m priors, with limiting (100$\\mu $ m, 160$\\mu $ m) fluxes in COSMOS, GOODS-South and ECDF-South (5.0, 11.0) mJy, (1.1, 2.0) mJy and (4.2, 8.2) mJy, respectively.", "The PACS-FIR counterparts of the LAEs at $z \\sim 0.3$ in COSMOS and GOODS-South have already been reported in [58], by looking for a PACS detection within 2$^{\\prime \\prime }$ around the location of each LAE, which is the typical astrometric uncertainty in the position of the FIR sources.", "In this study, we add the PACS counterparts in ECDF-South to the previous set, following the same matching criterion.", "We find one extra LAE detected in both PACS–100$\\mu $ m and PACS–160$\\mu $ m, assembling, in total, a sample of six PACS-detected SF LAEs.", "All these FIR counterparts represent a direct measurement of the FIR dust emission in LAEs at $z \\sim 0.3$ .", "As an example, we show in Figure REF the FIR SED of the three LAEs with counterparts both in PACS–100$\\mu $ m and PACS–160$\\mu $ m. Both observational FIR data points and best [14] and [67] fitted templates are represented.", "MIPS–24$\\mu $ m measurements allow us to constrain the FIR SED of PACS-undetected but MIPS–24$\\mu $ m detected LAEs.", "In [58] we report MIPS–24$\\mu $ m counterparts for LAEs in the COSMOS field by using data taken from the S–COSMOS survey [71].", "We found that 90% of LAEs at $z \\sim 0.3$ were detected in that band.", "Here, we extend the MIPS–24$\\mu $ m counterparts with data coming from the Spitzer Wide-area InfraRed Extragalactic survey [44] in the ECDF-South, Lockman and ELAIS-South fields, using a matching radius of 2$^{\\prime \\prime }$ as well.", "We find thirteen extra MIPS-detections.", "Summarizing, among the 30 LAEs with MIPS/PACS coverage, 23 are detected in MIPS/PACS.", "Nine of them belong to the sample in [58].", "Therefore, we find that more than 75% of LAEs at $z \\sim 0.3$ are detected in the mid–IR/FIR.", "Figure REF shows the detection fraction in MIPS–24$\\mu $ m and PACS bands of a sample of UV-selected galaxies at $z \\sim 0.3$ (see Section ).", "It can clearly be seen that UV-brighter galaxies are more likely to be detected in the IR.", "This tendency is similar to that found for MIPS–24$\\mu $ m detections in high-redshfift galaxies [69].", "There is a decrease in the detection fraction at the highest UV luminosities but in those cases the number of galaxies in each luminosity bin is low and the error bars are too large to draw any conclusion.", "As commented above, the limiting UV luminosity associated with the limiting magnitude of the LAEs in the sample studied is about 9.9 at $z \\sim 0.3$ .", "From Figure REF it can be deduced that UV luminosities above that value correspond to a high detection fraction in the IR, which is compatible with the large percentage of IR-detected LAEs found in this study.", "Figure: MIPS–24μ\\mu m (blue dots) and PACS (red squares) non-detection fractions as a function of UV luminosity for a sample of UV-selected galaxies at z∼0.3z\\sim 0.3.", "Error bars are obtained by assuming Poisson statistics.According to the limiting fluxes of the MIPS–24$\\mu $ m observations, MIPS-detected LAEs at $z \\sim 0.3$ have $\\log (L_{\\rm IR}/L_\\odot ) \\gtrsim 10.0$ in COSMOS and $\\gtrsim 10.40$ in the SWIRE fields.", "Regarding PACS observations, the limiting luminosities are $\\log (L_{\\rm IR}/L_{\\odot }) \\sim 10.5$ , $\\sim 9.76$ and $\\sim 10.4$ in COSMOS, GOODS-South and ECDF-South, respectively.", "In this sense, we adopt a limiting luminosity of $\\log (L_{\\rm IR}/L_{\\odot }) \\sim 10.4$ for the whole sample.", "Note that, although PACS observations in GOODS-South and MIPS observations in COSMOS are deeper than that threshold, there is only one LAE with $L_{\\rm IR}$ below $10^{10.4} L_\\odot $ and it is due to its lower redshift, $z \\sim 0.2$ .", "The adopted IR limiting luminosity represents an IR star formation rate of SFR$_{\\rm IR} \\sim 4.5 M_\\odot \\textrm {yr}^{-1}$ , according to the [38] calibration." ], [ "The control sample", "We also aim at comparing the derived physical properties of our IR-detected SF LAEs with those of other IR-detected SF galaxies in the same redshift range and with the same UV and IR limiting luminosities but which do not exhibit Ly$\\alpha $ emission in their UV spectrum.", "To do that, we define a control sample focusing on the COSMOS field and using data from GALEX [33], [76] and PACS observations plus the photometric redshifts, $z_{\\rm phot}$ , of the COSMOS photometric catalog [11].", "At $z \\sim 0.3$ , $z_{\\rm phot}$ are quite reliable and can be used instead of spectroscopic surveys, which contain much fewer objects.", "In this way, we select all the sources with $z_{\\rm phot}$ between 0.2 and 0.4 with measurements both in GALEX and PACS and whose UV luminosities are higher than the UV limiting luminosity of LAEs at $z \\sim 0.3$ .", "In order to avoid possible contamination from AGN, we rule out from the sample those samples which are detected in X-rays with CHANDRA.", "From now on, we refer to those galaxies as non-LAEs, and they will be the main source of comparison to study the differentiating characteristic of LAEs.", "The sample contains 135 galaxies.", "This sample also contains Ly$\\alpha $ -emitting galaxies whose Ly$\\alpha $ EW$_{\\rm rest-frame}$ are below 20 Å; therefore, although they exhibit a Ly$\\alpha $ line, it is not strong enough to be selected in narrow-band searches.", "Furthermore, we also retain those galaxies with z$_{\\rm phot}$ between 0.2 and 0.4, which are detected in GALEX and PACS but have UV luminosities fainter than the UV limiting luminosity of LAEs.", "We call these UV-faint SF galaxies.", "These sources will not be directly compared with LAEs because of their different selection criterion in the UV, but will be used to place LAEs within a more general scenario of SF galaxies at the same redshift.", "Note that, owing to GALEX spectroscopic limitations, we do not have UV spectra for these sources and we therefore do no know whether they exhibit Ly$\\alpha $ emission.", "It should be noted that, for the same reasons as pointed out in Section , there is no bias in the IR selection between LAEs and the galaxies in the control sample, all of them being limited to the same IR luminosity, $\\log (L_{\\rm IR}/L_{\\odot }) \\sim 10.4$ .", "To summarize, we have a control sample formed by two kinds of galaxies: i) non-LAEs, which are galaxies that do not show Ly$\\alpha $ emission with Ly$\\alpha $ EW$_{\\rm rest-frame}$ above 20 Å, are in the same redshift range of our LAEs, and have the same UV and IR limiting luminosities; ii) UV-faint SF galaxies: formed by galaxies which are at the same redshift as our LAEs, have the same IR limiting luminosity, but are fainter in the UV than our LAEs and non-LAEs.", "Only the non-LAEs will be used directly to compare the properties of LAEs with those of other galaxies without Ly$\\alpha $ emission.", "The UV-faint galaxies will be used only to place LAEs and non-LAEs within a more general framework of SF galaxies at $z \\sim 0.3$ ." ], [ "Results", "The UV and mid-IR/FIR measurements, which represent the emitted light from young populations and the re-emission of the light absorbed by dust in the UV, respectively, can be used to analyze the physical properties of our galaxies.", "The UV/IR combination allows an accurate determination of dust attenuation and total SFR, SFR$_{\\rm total}$ .", "Furthermore, FIR detections themselves enable us to examine the IR nature of galaxies." ], [ "UV and IR luminosities.", "$L_{\\rm UV}$ , expressed as $\\nu L_{\\nu }$ , is obtained for each galaxy from its observed magnitude in the NUV band and using the assumed cosmology.", "We choose NUV band because at $z \\sim 0.3$ , it covers the continuum near the Ly$\\alpha $ line and is not contaminated by the Ly$\\alpha $ emission.", "On the IR side, $L_{\\rm IR}$ is defined as the integrated IR luminosity between 8 and 1000 $\\mu $ m in the rest-frame and is obtained by using different calibrations depending on whether an object is PACS-detected or PACS-undetected but MIPS–24$\\mu $ m-detected.", "In the first case, $L_{\\rm IR}$ is calculated using calibrations between PACS bands and total IR luminosities (Eqns REF and REF ): $\\log {L_{\\rm IR}} = 0.99 \\log {L_{100\\,\\mu {\\rm m}}} + (0.44 \\pm 0.25)$ $\\log {L_{\\rm IR}} = 0.96 \\log {L_{160\\,\\mu {\\rm m}}} + (0.77 \\pm 0.21)$ where all the luminosities are in solar units and $L_{100\\,\\mu \\textrm {m}}$ and $L_{160\\,\\mu \\textrm {m}}$ are defined as $\\nu L_{\\nu }$ .", "The calibration employed for each galaxy is that associated with the reddest PACS band where it is detected.", "This would represent the nearest measurement to the dust emission peak, ensuring a better determination of $L_{\\rm IR}$ .", "To derive these calibrations, we focus on the COSMOS field and select all the PACS-detected objects both at 100 $\\mu $ m and 160 $\\mu $ m, which are spectroscopically catalogued to be at $z \\lesssim 0.5$ in the zCOSMOS survey [43].", "This condition enables us to carry out accurate FIR SED fits with [14] (hereafter CE01) templates since at that $z \\lesssim 0.5$ the dust emission peak is well sampled with those FIR bands.", "The fits are performed with the Zurich Extragalactic Bayesian Redshift Analyzer [22] in the maximum likelihood mode and the $L_{\\rm IR}$ for each galaxy is obtained by integrating its best fitted template between 8 and 1000 $\\mu $ m in the rest-frame.", "The calibrations were built by comparing the $L_{\\rm IR}$ of each source with that in the PACS bands and the errors are twice the standard deviation in the fittings.", "In this process, we ruled out AGN via X-ray emission diagnosis.", "For the PACS-undetected but MIPS–24$\\mu $ m-detected objects, we convert MIPS–24$\\mu $ m fluxes into $L_{\\rm IR}$ by fitting the fluxes to CE01 templates.", "These templates are built in such a way that for a given flux and redshift a unique solution for $L_{\\rm IR}$ exists.", "The determination of $L_{\\rm IR}$ from single FIR band extrapolations has also been employed in other studies [21], [55], [20].", "[21] analyze the applicability of this procedure, finding that MIPS–24$\\mu $ m extrapolations to $L_{\\rm IR}$ are valid for galaxies up to $z \\sim 1.5$ and which fall below the ULIRG limit.", "Both conditions are met in our case, both in LAEs and in the galaxies belonging to the control sample.", "In Figure REF we present the $L_{\\rm UV}$ and $L_{\\rm IR}$ for LAEs at $z \\sim 0.3$ and those for the control sample, both non-LAEs and UV-faint galaxies.", "On the UV side, it can be seen that the histogram for LAEs includes larger values than that for non-LAEs, indicating that LAEs are UV-brighter than non-LAEs.", "Median $\\log {(L_{\\rm UV})}$ for LAEs and non-LAEs are 10.1 and 9.9, respectively.", "On the IR side, $L_{\\rm IR}$ values for LAEs and non-LAEs are in the same range, mainly spanning from $\\log {(L_{\\rm IR}/L_{\\odot })} = 10.4$ to 11.2.", "Even for UV-faint galaxies, the histogram of $L_{\\rm IR}$ is similar to that for LAEs and non-LAEs, despite their difference in the UV selection.", "According to their $L_{\\rm IR}$ , galaxies can be classified into three types: normal SF galaxies: $L_{\\rm IR} <10^{11}\\,L_{\\odot }$ , luminous infrared galaxies (LIRGs): $10^{11}< L_{\\rm IR}$ [$L_{\\odot }$ ] $<$ 10$^{12}$ and ultra-luminous infrared galaxies (ULIRGs): $L_{\\rm IR}> 10^{12}\\,L_{\\odot }$ .", "As can be seen in Figure REF , most of our IR-detected LAEs are in the normal SF regime.", "Due to the correlation between MIPS–24$\\mu $ m luminosity and $L_{\\rm IR}$ , IR-undetected LAEs with FIR coverage have $L_{\\rm IR}$ less than the IR limiting luminosity ($\\sim $ 10$^{10.4}\\,L_{\\odot }$ ), being normal SF galaxies too.", "In this way, considering the 30 LAEs with IR coverage (both IR-detected and IR-undetected), we find that at $z\\sim 0.3$ more than 80% of LAEs are SF galaxies.", "Only six LAEs have $10^{11} < L_{IR}[L_{\\odot }] < 10^{11.5}$ , and none of them have $L_{\\rm IR}> 10^{11.5}$ nor fall in the ULIRG class.", "Note that, considering both IR-detected and undetected LAEs, we work with a nearly complete sample of LAEs up to $m_{\\rm NUV}\\sim 21.5$ mag [16], and therefore, the non-existence of ULIRGs in our sample of LAEs is an unbiased result under that limit.", "If there were LAEs with an ULIRG nature at $z\\sim 0.3$ , they would have to be fainter in the UV than those analyzed in the present work.", "No ULIRG is found in the control sample (non-LAEs and UV-faint galaxies) either, most galaxies being in the normal SF regime as well.", "Indeed, most recent studies of the FIR properties of galaxies with Herschel do not find many ULIRGs at $z\\sim 0.3$ , but they begin appearing from $z\\sim 1.0$ [20], [21], [7], [45].", "At $z \\gtrsim 2$ , [13] found Ly$\\alpha $ emission in the optical spectra of a sample of 850 $\\mu $ m detected SF sub-mm galaxies with a ULIRG nature.", "[51], [53] also suggest that some LAEs at $z \\sim 2$ have a ULIRG nature.", "In this way, the discovery of Ly$\\alpha $ emission in ULIRGs at $z \\gtrsim 2$ and the fact that most of our LAEs at have $L_{\\rm IR} \\lesssim 11.2 L_\\odot $ suggests that the IR nature of the galaxies found via their Ly$\\alpha $ emission is changing from $z \\sim 2$ to $z \\sim 0.3$ : there is a population of red and dusty LAEs with a ULIRG nature at $z \\gtrsim 2$ , which is not seen at $z \\sim 0.3$ .", "This trend is similar to that found in previous studies for general galaxies detected in MIPS–24$\\mu $ m and to the trend of cosmic star formation [35], [36], [37], [63], [42].", "[17] find a dramatic evolution in the maximum of the Ly$\\alpha $ luminosity function between $z \\sim 0.3$ and $z \\gtrsim 1.0$ .", "This result, together with the evolution in the IR emission of LAEs suggested above, indicates that either the properties of galaxies selected via their Ly$\\alpha $ emission are evolving over cosmic time or the Ly$\\alpha $ selection technique is not tracing the same kind of objects at different redshifts." ], [ "Dust attenuation", "The ratio between $L_{\\rm IR}$ and $L_{\\rm UV}$ is a good tracer of dust attenuation in galaxies.", "Here, we adopt the calibration between the IR/UV ratio and dust attenuation found in [8] (Equation REF ) to obtain dust attenuation for our IR-detected LAEs at $z \\sim 0.3$ and for the galaxies in the control sample, both non-LAEs and UV-faint galaxies: $A_{\\rm NUV} = -0.0495x^3 + 0.4718x^2 + 0.8998x + 0.169$ where $A_{\\rm NUV}$ is the dust attenuation in the NUV band and $x=\\log {\\left(L_{\\rm IR}/L_{\\rm UV}\\right)}$ .", "The conversion from $A_{\\rm NUV}$ to the dust attenuation at 1200Å, $A_{1200\\,Å}$ , is made by using the [10] reddening law.", "As in [9], we do not apply $K$ -correction to the UV luminosities given that the $L_{\\nu }$ spectrum of galaxies is quite flat in the UV range.", "In this way, we assume that the NUV fluxes observed at $z\\sim 0.3$ is the same as those observed at $z\\sim 0$ in the NUV band.", "In Figure REF we represent the dust attenuation of our LAEs versus their $L_{\\rm IR}$ along with the dust attenuation distribution, which has a median value of $A_{1200Å}\\sim 1.5$ mag.", "We also plot in that figure the dust attenuation for non-LAEs and for the UV-faint galaxies.", "It can be seen that, for each value of $L_{\\rm IR}$ , LAEs are slightly less dusty than non-LAEs.", "In fact, the median value of the dust attenuation distribution for non-LAEs is $A_{1200Å}\\sim 2.0$ mag.", "Furthermore, for both LAEs and non-LAEs there is a trend towards brighter galaxies in the IR being dustier.", "[17] find, comparing the UV spectral index to the H$\\alpha $ /H$\\beta $ flux ratio, that the dust attenuation in most LAEs at $z\\sim 0.3$ can be also described by [12] or [26] laws, i.e., the stellar extinction is better represented as a uniform screen rather than a patchy distribution.", "The inclusion of those laws here would change the final numbers in the sense that the dust attenuation would be lower than that obtained with the [10] law, although the relation between the dust attenuation between different populations would remain the same.", "LAEs with a fainter UV continuum than the limiting magnitude are not included in the sample and they could be dustier than those presented in this study.", "Therefore, there could be some LAEs at $z\\sim 0.3$ which are dustier and have larger rest-frame Ly$\\alpha $ EW equivalent widths than those presented here.", "The finding of a Ly$\\alpha $ emission in the UV spectra of a dusty object at $z\\sim 0.3$ could indicate the presence of a clumpy ISM [50].", "The possible existence of this and other kinds of geometries has been reported to be able to recover the observed Ly$\\alpha $ /H$\\alpha $ and H$\\alpha $ /H$\\beta $ ratios and the observed SEDs of LAEs at different redshifts [23], [72], [31], [25]." ], [ "Dust attenuation and UV continuum slope", "Dust attenuates the rest-frame UV light of a galaxy in a wavelength-dependent way, this attenuation being larger for shorter UV wavelengths.", "Therefore, dust produces an increase in the rest-frame UV continuum slope, $\\beta $ , from negative to less negative or even positive values.", "On the other hand, part of the absorbed light in the UV is in turn re-emitted in the FIR regime.", "In this way, dust attenuation and $\\beta $ are expected to be related.", "Indeed, many studies employ different relations between dust attenuation and $\\beta $ to obtain the dust attenuation in galaxies that are not detected in the FIR.", "In this section we compare dust attenuation and $\\beta $ for the IR-detected LAEs, non-LAEs, and UV-faint galaxies.", "Shown in Figure REF is the IRX-$\\beta $ diagram for the three kinds of galaxies.", "$\\beta $ values are obtained by employing the [39] recipe, as in [7].", "In this way, we assume that the rest-frame UV continuum can be described by a power law in the form $f_{\\lambda }\\sim \\lambda ^\\beta $ and obtain $\\beta $ from GALEX photometry in the FUV and NUV channels.", "It can be clearly seen that LAEs have UV continuum slopes much bluer than non-LAEs, which is compatible with their being among the least dusty galaxies at their redshift (see Section REF ).", "The finding of larger (redder) values of $\\beta $ for the UV-faint galaxies than for LAEs and non-LAEs is due to their faintness in the UV.", "Shown in Figure REF are also the curves of [39] and [2] corresponding to normal SF and starburst (SB) galaxies, respectively.", "It can be seen that although many LAEs are distributed around the curve corresponding to the normal SF galaxies, some of them occupy a different location in the IRX-$\\beta $ diagram, quite near to the SB relation.", "Some studies use a relation between dust attenuation and $\\beta $ to obtain the dust attenuation from the values of the UV continuum slope for objects which are not detected in the FIR.", "The reason for this procedure is that, whereas the UV continuum slope is relatively easy to obtain from broad-band photometry over a wide range of redshifts, the detection rate in the FIR of certain kinds of galaxies is very low, mainly at the highest redshifts.", "This is typically the case for LAEs [59].", "But it should be noted that the reliability of this technique is based on that the relation assumed between dust attenuation and $\\beta $ applies for the population of galaxies under study.", "What we find here is that this is not the case for LAEs at $z \\sim 0.3$ , which are distributed between the SB and the SF relations so that a unique relation cannot be applied to the whole population to recover the dust attenuation from $\\beta $ .", "Furthermore, it has been reported that the location of galaxies in the IRX-$\\beta $ diagram also depends on the $L_{\\rm IR}$ and age.", "For example, at $z \\sim 2$ , galaxies with ages below 100 Myr tend to be less reddened at a given UV continuum slope than the values predicted by the SB relation [70], [69], [68].", "This tendency might have a significant influence on LAEs, since most of them have been reported to be young galaxies in a wide range of redshifts.", "On the other hand, non-LAEs are mostly distributed around the normal SF curve, although in a zone associated with higher $\\beta $ values than those for LAEs.", "There is a very low percentage of non-LAEs near the SB curve.", "From another point of view, for each value of dust attenuation, LAEs tend to have bluer UV continuum slopes than non-LAEs.", "This could indicate a different star formation mode between LAEs and non-LAEs." ], [ "Dust attenuation and redshift", "Two open questions in the study of LAEs are their dust attenuation and its possible evolution with redshift.", "In the previous sections we provided additional clues to answer the former question for LAEs at $z\\sim 0.3$ by using UV and mid-IR/FIR measurements.", "Now, in an attempt to answer the latter question, we compare our values with those at different redshifts reported in previous studies [23], [17], [31], [53], [28], [41], [28], [57], [52], [25], [24], [65].", "Figure REF shows the representation of the results reported in those works, along with the curve of the redshift evolution of a general population of galaxies found in [34].", "We also include the line associated with a dust attenuation of $A_{1200Å}$ = 1.0 mag at all redshifts in order to clarify the differences in dust attenuation between LAEs at different redshifts.", "It can be seen that most LAEs at $z\\gtrsim 2.5$ have dust attenuation below 1.0 mag in 1200 Å.", "Actually, most results indicate that the SEDs of LAEs at those redshifts are compatible with dust attenuation below 0.5 mag at 1200 Å[41], [28], [27].", "At $z\\sim 0.3$ , we find a dust attenuation distribution centered in $A_{1200\\,Å}$ = 1.5 mag.", "This is compatible with the results reported by [23] and [17] even though the results were obtained with rest-frame optical spectroscopy and SED fitting and consequently different methodologies.", "At $z \\sim 2.0$ [31] and [53] report high dust attenuation values, higher than those reported at $z \\sim 0.3$ and $z \\gtrsim 2.5$ .", "Although these median values are high, the dust attenuation distributions found in both studies are quite wide, which makes it difficult to establish a clear tendency between $z \\sim 2.0$ and other redshifts.", "However, the high median values themselves are an indication that there should exist a population of dusty LAEs at $z \\sim 2.0$ that are not seen at other redshifts.", "Furthermore, the redshift distribution of the red and dusty objects selected via their sub-mm emission is centred around $z \\sim 2.3$ [13].", "Some of those galaxies are SF ULIRGs exhibiting Ly$\\alpha $ emission in the spectra, which reinforces the idea that, at those redshifts, the Ly$\\alpha $ selection technique can also segregate dusty galaxies, in contrast with the classical idea of LAEs as galaxies with low dust attenuation.", "[1] find the emergence of a small population of red galaxies at $z < 3$ (in terms of their UV continuum slope) in their study of integral-field espectroscopically selected LAEs at $1.9 < z < 3.8$ .", "Moreover, in [59] we report the detection in PACS–160$\\mu $ m of a sample of LAEs at $2.0 \\lesssim z \\lesssim 3.5$ and derive their dust attenuation by employing the IR/UV ratio [8].", "As result, we obtain values of $A_{1200\\,Å} \\gtrsim 4.5$ mag, higher than the median values reported in [31] and [53] but compatible with the width of the distribution in [53].", "Note that the values of dust attenuation reported in [59] represent the upper limit on LAEs at that redshift, since we only work with FIR-detected sources.", "At $z \\sim 3.1$ , most LAEs are found to be nearly dust-free objects [28], [27], [41], representing a great difference from $z\\sim 3.1$ to $z \\sim 2.1$ –2.3.", "However, it should be considered that at $z\\gtrsim 2.5$ , the dust attenuation in most studies has been obtained by using a stacking analysis of the sample.", "[53] compared the dust attenuation obtained for LAEs at $z \\sim 2.3$ when performing a stacking analysis with those found when considering individual objects.", "They found that dust attenuation is considerably reduced when stacking the total sample and a subsample of old LAEs, indicating that dust attenuation is very sensitive to stacking.", "Despite the uncertainties of stacking, the evolution of dust attenuation between $z\\gtrsim 3$ and $z\\sim 2$ could be related to the evolution in other properties of LAEs.", "[15] report that there is a significant evolution in the LAE luminosity function between $z \\sim 3.1$ and $z \\sim 2.1$ .", "[3] claim for an evolution in the size of LAEs from $z\\sim 3.1$ to $z\\sim 2.1$ , raising the median half-light radius from 1.0 kpc at $z\\sim 3.1$ to 1.4 kpc at $z\\sim 2.1$ .", "[54] also found evolution in the physical properties of LAEs between $z\\sim 3$ and $\\sim $ 2, in the sense that at $z\\sim 2$ LAEs have redder SEDs, which indicates that they are more evolved (dustier, older and more massive) than those at $z\\sim 3$ .", "At $z\\gtrsim 4$ , the samples studied contain very few objects and the results are not as statistically significant as at lower redshifts; therefore, although most studies have reported that the SED of LAEs is compatible with some levels of dust, more work is needed to confirm that behavior.", "Therefore, the general trend is that LAEs at $z\\gtrsim 2.5$ tend to be slightly less dusty than their low-redshift analogues.", "Furthermore, it can be seen that LAEs at $z\\gtrsim 2.0$ tend to follow the dust evolution of galaxies with redshift, while at $z\\sim 0.3$ , they deviate from that behavior.", "Apart from indicating a possible evolution of dust attenuation with redshift for LAEs, Figure REF also shows the lack of knowledge that we still have about the physical properties of those galaxies.", "While the best method of determining dust attenuation is the combination of direct UV and FIR measurement used in this study, different procedures have been used to derive that parameter.", "However, it has not been checked yet whether they provide consistent results at a given redshift.", "Furthermore, at the highest redshifts, either the number of objects studied is low, or stacking analysis had to be done due to the non-detection of LAEs in many photometric bands." ], [ "Star Formation Rates", "The combination of UV and IR data also provides the most accurate determination of SFR in galaxies.", "Adopting the [38] calibrations, the SFR associated with the observed UV and IR luminosities are given by the expressions: $\\textrm {SFR}_{\\rm UV,uncorrected}[M_{\\odot }\\textrm {yr}^{-1}] = 1.4\\cdot 10^{-28}L_{\\nu ,\\rm observed}$ $\\textrm {SFR}_{\\rm IR}[M_{\\odot }\\textrm {yr}^{-1}] = 4.5\\cdot 10^{-44}L_{IR}$ where $L_{\\nu ,\\rm observed}$ is in units of erg/s/Hz and $L_{\\rm IR}$ is defined in the same way as in Section REF .", "In order to obtain the SFR$_{\\rm total}$ for our galaxies we assume that all the light absorbed in the UV is in turn re-radiated in the FIR.", "In this scenario, SFR$_{\\rm total}$ can be calculated as the sum of a dust-uncorrected component, SFR$_{\\rm UV,uncorrected}$ and the correction term shown in Eqn.", "REF .", "Thus, it can be written: $\\textrm {SFR}_{\\rm total} = \\textrm {SFR}_{\\rm UV, uncorrected} + \\textrm {SFR}_{\\rm IR}$ The distributions of the SFR$_{\\rm total}$ for IR-detected LAEs and the galaxies in the control sample are shown in Figure REF .", "It can be seen that IR-detected LAEs have SFR$_{\\rm total}$ , ranging mainly from 10 to 40 $M_{\\odot }$ yr$^{-1}$ , with a median of 18 $M_{\\odot }$ yr$^{-1}$ , and peaking around 13 $M_{\\odot }$ yr$^{-1}$ .", "The median value of SFR$_{\\rm total}$ for non-LAEs is 15 $M_{\\odot }$ yr$^{-1}$ , with a distribution peaking around 8 $M_{\\odot }$ yr$^{-1}$ , which tends to be shifted to lower values of SFR$_{\\rm total}$ than that for LAEs.", "The median value found in this study is larger than that reported in [17] by using extinction-corrected H$\\alpha $ luminosities, SFR$_{\\rm H\\alpha -corrected}\\sim 6 M_{\\odot }$ yr$^{-1}$ .", "Figure REF shows the ratio between SFR$_{\\rm IR}$ and SFR$_{\\rm total}$ against $L_{\\rm UV}$ and $L_{\\rm IR}$ for LAEs and the general population of galaxies at $z\\sim 0.3$ .", "Considering the galaxies in the control sample, it can be seen that the ratio has a strong dependence on NUV luminosity: the IR contribution is lower with increasing $L_{\\rm UV}$ .", "In Section we found that UV-bright galaxies at $z\\sim 0.3$ are more like those detected in the FIR than UV-faint ones.", "Now, we have found that, for IR-detected galaxies, UV-bright ones have a lower IR contribution to SFR$_{\\rm total}$ (are more transparent) than UV-faint ones.", "There is a limit in the NUV up to which SFR$_{\\rm IR}$ is a good estimator of SFR$_{\\rm total}$ .", "Considering ratios of 80% and 90%, SFR$_{\\rm IR}$ is a good indicator of SFR$_{\\rm total}$ for galaxies at $z\\sim 0.3$ with $L_{\\rm NUV}\\lesssim 9\\,L_{\\odot }$ and $\\lesssim 9.5L_{\\odot }$ , respectively.", "IR-detected LAEs at $z\\sim 0.3$ are all above the mentioned thresholds; therefore, both the UV and IR light contribute significantly to SFR$_{\\rm total}$ .", "Despite this, the FIR contribution to SFR$_{\\rm total}$ for most LAEs is greater than 60%, indicating that rest-frame UV-based methods would underestimate SFR$_{\\rm total}$ by a factor greater than two.", "This is an indication of the great importance of FIR measurements when calculating the SFR: even in galaxies with low dust attenuation, such as LAEs, FIR emission makes a significant contribution; therefore, FIR data must be used to obtain accurate results.", "Figure: The ratio between SFR IR _{\\rm IR} and SFR total _{\\rm total} against UV luminosity for our LAEs (red dots), non-LAEs (blue triangles), and other UV-faint galaxies (yellow squares).", "Dashed and dashed-dotted horizontal lines limit the zones where the SFR IR _{\\rm IR} represent 90% and 80% of SFR total _{\\rm total}, respectively.The distribution of SFR IR _{\\rm IR}/SFR total _{\\rm total} is also represented for the three kinds of galaxies, with the same color code.", "Histograms have been normalized to the maximum to clarify the representation." ], [ "Morphology", "In the previous sections, we have analyzed the physical properties of LAEs with direct IR and UV measurements.", "We now examine whether there is a difference between LAEs and non-LAEs regarding their morphology in the optical side of their SED.", "In order to carry out a precise morphological analysis, high spatial resolution images (i.e., HST/ACS) are required.", "In the case of non-LAEs, given that they are all located in the COSMOS field, there is ACS information for almost all of them, 127 (about 95%) to be precise.", "However, the IR-detected LAEs are distributed in different fields, some of which do not have available HST images.", "This limits the morphological study to eight IR-detected LAEs.", "We take $12^{\\prime \\prime }\\times 12^{\\prime \\prime }$ $I$ -band ACS cut-outs of our sources (LAEs and control sample) and follow both an analytical and a visual procedure.", "The analytical one is aimed at obtaining the physical sizes of the galaxies, whereas the visual one has the objective of classifying the galaxies within the different types of the Hubble sequence.", "Note that we use the images of the galaxies in the same passband and, since they are all in the same redshift range, we are analyzing the same zone of their SED, with no need for morphological $K$ -correction.", "In order to obtain the sizes of our galaxies we fit Sersic profiles [73] to their light distributions by using GALFIT [62].", "Figure REF shows the light-half radii of LAEs and the galaxies in the control sample.", "It can clearly be seen that there is a remarkable difference: LAEs tend to be significantly smaller than non-LAEs.", "The median values of $R_{\\rm eff}$ for LAEs, non-LAEs, and UV-faint galaxies are 1.5, 4.1, and 3.4 kpc, respectively.", "Note that non-LAEs are bigger than UV-faint galaxies owing to the difference in their UV brightness.", "The difference in size between LAEs and other UV-selected galaxies is also present at higher redshifts.", "[46] report that LAEs tend to be smaller than LBGs at the same redshift from $z\\sim 2$ up to $z\\sim 4$ .", "From $z\\gtrsim 5$ , LAEs and LBGs show similar sizes and have similar properties.", "Figure: Distribution of the effective radius for our LAEs (red-shaded), non-LAEs (blue,) and other UV-faint galaxies (yellow).", "Histograms have been normalized to the maximum in order to clarify the representation.At $z\\gtrsim 2$ , narrow-band selected LAEs have been reported to be compact [4], [3], [46].", "[4], and [3] report median values for $R_{\\rm eff}$ of 1.0 and 1.4 kpc for their LAEs at $z\\sim 3.1$ and $\\sim $ 2.1, respectively.", "The median value for LAEs that we obtain in this study is not significantly different from the values reported at higher redshifts; hence, a clear evolution in the median value of the physical size of LAEs is not seen from $z\\sim 2$ down to $z\\sim 0.3$ .", "Figures REF and REF show the ACS cut-outs of LAEs and non-LAEs, respectively, with available ACS information.", "It can be seen that the vast majority of non-LAEs have clear disk-like morphologies, most of them belonging to the Sb, Sc or Sd classes.", "However, LAEs tend to depart from that tendency.", "Although two of them seem to be disk-like galaxies, the other six LAEs have compact or interacting/merging morphologies.", "As was pointed out before, we only have eight LAEs with available ACS information, so the results are not statistically significant.", "However, despite the scarcity, the morphology of LAEs seem to be more heterogeneous than that of non-LAEs and this is an indication of a possible morphological differentiation.", "[17] also studied the sizes and morphology of their LAEs and UV-selected galaxies at $z \\sim 0.3$ , examining the dependence of size upon the H$\\alpha $ equivalent width (EW(H$\\alpha $ )) as well.", "They report that LAEs with higher EW(H$\\alpha $ ) are generally unresolved at the 1” resolution of the CFHT MegaPrime U-band images, while the galaxies with lower EW(H$\\alpha $ ) are generally extended.", "Furthermore, at the same EW(H$\\alpha $ ), they found that LAEs are significantly smaller than galaxies without Ly$\\alpha $ emission.", "This agrees with the result we find here for our IR-detected LAEs, despite the results were obtained with optical observations in different bands.", "[16] analyzed the morphology of a sample of LAEs and other galaxies without Ly$\\alpha $ emission by using i'-band ground-based data from the CFHT MegaPipe database.", "In agreement with our result, they found that the LAE sample contains a much larger fraction of mergers and compact galaxies then the NUV-continuum selected sample.", "These significant morphological differences, along with the results found in previous sections, indicate that Ly$\\alpha $ photons tend to escape preferentially from irregular or interacting galaxies of small size, low dust attenuation, and high SFRs." ], [ "Conclusions", "In this study, we have obtained fundamental physical properties (UV and IR emission, dust attenuation, SFR, and morphology) of LAEs at $z\\sim 0.3$ , defined as those Ly$\\alpha $ -emitting galaxies with Ly$\\alpha $ EW$_{\\rm rest-frame}$ above 20 Å, the typical threshold in the narrow-band searches at higher redshifts.", "Furthermore, we have compared LAEs with non-LAEs, defined as galaxies at the same redshift as LAEs which, with the same UV and IR limiting luminosities, show no Ly$\\alpha $ emission with Ly$\\alpha $ EW$_{\\rm rest-frame}\\gtrsim $ 20 Å in their spectrum.", "The main conclusions of our work are: We find that a large percentage ($\\sim $ 75%) of our LAEs have MIPS/PACS IR counterparts under a limiting luminosity of $\\log (L_{\\rm IR}/L_{\\odot })\\sim 10.4$ .", "These mid-IR/FIR detections are a direct measurement of their dust emission.", "We find that 80% of the LAEs studied at $z\\sim 0.3$ , both IR-detected and undetected, are normal SF galaxies, $L_{\\rm IR}<10^{11}\\,L_{\\odot }$ .", "We find only six LAEs with 10$^{11}\\lesssim L_{\\rm IR}[L_{\\odot }]\\lesssim 10^{11.5}$ and none with $L_{\\rm IR}>10^{11.6}L_\\odot $ or with a ULIRG nature.", "The finding of a noticeable number of ULIRGs at $z\\sim 2.5$ exhibiting Ly$\\alpha $ emission suggests that the IR nature of objects selected by means of their Ly$\\alpha $ emission changes with redshift.", "For each value of $L_{\\rm IR}$ , LAEs are among the least dusty galaxies at $z\\sim 0.3$ .", "The distribution of the dust attenuation in 1200 Å  of LAEs and non-LAEs are centered around 1.5 and 2.0, respectively.", "In this study we have obtained dust attenuation by combining UV and FIR measurements without the uncertainties of rest-frame UV/optically based methods, which do not take into account the dust emission in the FIR.", "The dust attenuation of objects selected via their Ly$\\alpha $ emission is evolving with redshift, from dust-free objects at $z\\sim 3.0$ to LAEs with high and low/moderate dust attenuation at $z\\sim 2.3$ and $\\sim $ 0.3, respectively.", "However, it should be noted that the procedures followed in the different studies are not the same and that at $z\\gtrsim 3$ very few LAEs have been individually studied without the uncertainties of stacking.", "The suggested evolution of dust attenuation of LAEs along with the finding that the UV and IR nature of LAEs is changing with redshift, indicates that the physical properties of galaxies selected at different redshifts from their Ly$\\alpha $ emission are not the same: either they are evolving or the technique is picking up galaxies of a different nature.", "LAEs have a much bluer UV continuum slope than non-LAEs, which is compatible with their being among the least dusty objects at their redshift.", "Furthermore, while most non-LAEs follow the trend of normal SF galaxies, some LAEs seem to be starbursts galaxies, indicating a possible difference in their mode of star formation.", "The distribution of LAEs between the SF and SB relations indicates that a unique relation between UV continuum slope and dust attenuation does not apply for the whole population.", "Therefore, the determination of dust attenuation from $\\beta $ fails for LAEs at $z \\sim 0.3$ .", "The SFR$_{\\rm total}$ for LAEs tends to be larger than for non-LAEs.", "We find a noticeable contribution of the IR emission to the SFR$_{\\rm total}$ in LAEs and non-LAEs, about 60%.", "Therefore, although LAEs have a bright UV continuum and are among the least dusty galaxies at $z\\sim 0.3$ , it is essential to take into consideration their dust emission in the FIR for obtaining accurate values of their SFR$_{\\rm total}$ .", "The size of LAEs tend to be smaller than those of non-LAEs, with median values of 1.5 and 4.1 kpc, respectively.", "Despite the low number of LAEs with available ACS information, a visual inspection of their morphologies reveals that they tend to be compact, disk-like, or merging/interacting galaxies (i.e., there is a heterogeneity in morphology) in opposition to non-LAEs, which are mainly disk-like galaxies.", "These are the most noticeable differences between LAEs and non-LAEs.", "We thank the anonymous referee for the comments provided, which have helped us to improve the manuscript.", "This work was supported by the Spanish Plan Nacional de Astrononomía y Astrofísica under grant AYA2008-06311-C02-01.", "Some of the data presented in this paper were obtained from the Multimission Archive at the Space Telescope Science Institute (MAST).", "STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.", "Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX09AF08G and by other grants and contracts.", "Based on observations made with the European Southern Observatory telescopes obtained from the ESO/ST-ECF Science Archive Facility.", "Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.", "The Herschel spacecraft was designed, built, tested, and launched under a contract to ESA managed by the Herschel/Planck Project team by an industrial consortium under the overall responsibility of the prime contractor Thales Alenia Space (Cannes), and including Astrium (Friedrichshafen) responsible for the payload module and for system testing at spacecraft level, Thales Alenia Space (Turin) responsible for the service module, and Astrium (Toulouse) responsible for the telescope, with in excess of a hundred subcontractors.", "PACS has been developed by a consortium of institutes led by MPE (Germany) and including UVIE (Austria); KUL, CSL, IMEC (Belgium); CEA, OAMP (France); MPIA (Germany); IFSI, OAP/AOT, OAA/CAISMI, LENS, SISSA (Italy); IAC (Spain).", "This development has been supported by the funding agencies BMVIT (Austria), ESA- PRODEX (Belgium), CEA/CNES (France), DLR (Germany), ASI (Italy), and CICYT/MICINN (Spain).", "Figure: ACS I-band 12”x12” cut-outs of the 8 IR-detected LAEs with available ACS observations.", "LAEs are the galaxies in the center of each image.", "At z∼\\sim 0.3, the scale is 4.45 kpc/”, which means that the diameter of each cut-out represents 53.4 kpc.Figure: ACS I-band 12 '' ×12 '' 12^{\\prime \\prime }\\times 12^{\\prime \\prime } cut-outs of the non-LAEs with available ACS information, which are the sources in the center of each image.", "At z∼0.3z\\sim 0.3, the scale is 4.45 kpc/ '' ^{\\prime \\prime }, which means that the diameter of each cut-out represents 53.4 kpc." ] ]

[ [ "Properties of Bulgeless Disk Galaxies II. Star Formation as a Function\n of Circular Velocity" ], [ "Abstract We study the relation between the surface density of gas and star formation rate in twenty moderately-inclined, bulgeless disk galaxies (Sd-Sdm Hubble types) using CO(1-0) data from the IRAM 30m telescope, HI emission line data from the VLA/EVLA, H-alpha data from the MDM Observatory, and PAH emission data derived from Spitzer IRAC observations.", "We specifically investigate the efficiency of star formation as a function of circular velocity (v_circ).", "Previous work found that the vertical dust structure and disk stability of edge-on, bulgeless disk galaxies transition from diffuse dust lanes with large scale heights and gravitationally-stable disks at v_circ < 120 km/s (M_star <~ 10^10 M_sun) to narrow dust lanes with small scale heights and gravitationally-unstable disks at v_circ > 120 km/s.", "We find no transition in star formation efficiency (Sigma_SFR/Sigma_HI+H2) at v_circ = 120 km/s, or at any other circular velocity probed by our sample (v_circ = 46 - 190 km/s).", "Contrary to previous work, we find no transition in disk stability at any circular velocity in our sample.", "Assuming our sample has the same dust structure transition as the edge-on sample, our results demonstrate that scale height differences in the cold interstellar medium of bulgeless disk galaxies do not significantly affect the molecular fraction or star formation efficiency.", "This may indicate that star formation is primarily affected by physical processes that act on smaller scales than the dust scale height, which lends support to local star formation models." ], [ "Introduction", "High resolution studies of the Milky Way and nearby galaxies show that star formation mainly occurs in giant molecular clouds (GMCs), with typical GMC masses between about $10^{3}$ and $10^{7} \\, M_{\\odot }$ and radii between about ten and several hundred parsecs [26].", "It is challenging to study molecular gas on these scales in even the nearest galaxies [10] and studies of more distant galaxies can only currently investigate the average properties of gas and stars on larger scales.", "An advantage of more distant galaxies is that they exhibit a much wider range of physical properties, which enables investigation into the influence of environment (e.g., mid-plane pressure, metallicity, and scale height) and processes that act on large scales (e.g., shear and large-scale gravitational instabilities) on star formation.", "Extragalactic studies of star formation have found that the star formation rate (SFR) surface density ($\\Sigma _{\\rm SFR}$ ) and gas surface density ($\\Sigma _{\\rm gas}$ ) are correlated in the form of the Kennicutt-Schmidt law: $\\Sigma _{\\rm SFR} \\propto \\Sigma _{\\rm gas}^{N}$ [71], [37].", "This star formation law has been studied by averaging over scales as small as about $100 \\, {\\rm pc}$ and as large as the entire optical disk.", "Recent high-resolution studies have found that the SFR surface density is more strongly correlated with the molecular gas surface density ($\\Sigma _{\\rm H_{2}}$ ) than with the atomic gas surface density ($\\Sigma _{\\rm HI}$ ), with $\\Sigma _{\\rm SFR} \\propto \\Sigma _{\\rm H_{2}}^{N}$ and $N$ between 0.8 and 1.5 [87], [38], [5], [6], [72], [51], [64].", "This result confirms, from an extragalactic perspective, that stars form from molecular gas and has led to an expansion in the scope of many star formation studies to investigate how environment and large-scale processes affect the molecular fraction in the interstellar medium (ISM).", "[49] addressed environmental effects on the molecular fraction and star formation efficiency (SFE) with high-quality, $750\\, {\\rm pc}$ -resolution observations of $\\Sigma _{\\rm SFR}$ , $\\Sigma _{\\rm HI}$ , and $\\Sigma _{\\rm H_{2}}$ over the optical disk of 23 nearby galaxies.", "They compared these observations to many star formation models and thresholds and concluded that no model fit the data sufficiently well to be declared a clear favorite.", "This result led them to suggest that physics that acts on scales smaller than their resolution is most important for setting the molecular fraction and SFE.", "While no model was an ideal fit to the data presented in [49], the best fit was arguably with a model in which the ratio of molecular to atomic surface density is related to the mid-plane pressure ($P_{\\rm h}$ ): $R_{\\rm mol} \\equiv \\Sigma _{\\rm H_{2}}/\\Sigma _{\\rm HI} \\propto P_{\\rm h}^{\\alpha }$ , and the molecular SFE (${\\rm SFE[H_{2}}] \\equiv \\Sigma _{\\rm SFR}/\\Sigma _{\\rm H_{2}}$ ) is constant, such that ${\\rm SFE} \\equiv \\Sigma _{\\rm SFR}/\\Sigma _{\\rm HI+H_{2}} = {\\rm SFE({\\rm H_{2}})} \\, \\frac{R_{\\rm mol}}{R_{\\rm mol}+1}$ .", "[49] noted that a model where the molecular SFE is constant requires the population of GMCs in any region to be sampled from the same distribution of properties (e.g., size and mass), independent of environment.", "Furthermore, once GMCs form, the general environment cannot have a strong influence on their properties.", "Finally, one must compare the model to observations with a number of GMCs per resolution element so as to average over evolutionary effects.", "[24] predicted that $R_{\\rm mol}$ should depend on the mid-plane pressure and the interstellar radiation field ($j$ ): $R_{\\rm mol} \\propto P_{\\rm h}^{2.2} \\,j^{-1}$ .", "Assuming $\\Sigma _{\\rm SFR} \\propto \\Sigma _{\\rm H_{2}}$ and $j \\propto \\Sigma _{\\rm SFR}$ , the model predicts $R_{\\rm mol} \\propto P_{\\rm h}^{\\alpha }$ , with $\\alpha = 1.2$ .", "[87] studied seven molecular-dominated spiral galaxies and found $\\alpha =0.8$ , while [8] found $\\alpha =0.92$ in their study of fourteen galaxies with a large range of $R_{\\rm mol}$ .", "Values between $\\alpha =0.5$ and 1.2 encompass most of the [49] data on the SFE versus $P_{\\rm h}$ plane.", "[59] recently presented a star formation model that produces an approximately linear relationship between $R_{\\rm mol}$ and $P_{\\rm h}$ .", "The authors divided the ISM into a diffuse component and a gravitationally bound component.", "The fraction of gas in each component is set by the requirement that gas pressure in the diffuse component is balanced by the gravity of stars, dark matter, and gas (both diffuse and bound), while heating (mainly by ultraviolet (UV) photons from O and B stars formed in the bound component) balances cooling.", "The model assumes that the SFE within the gravitationally bound component is constant.", "[11] added a metallicity-dependent heating term to the [59] model, which resulted in agreement between the large H1 surface densities observed in the Small Magellanic Cloud and the surface density of diffuse gas calculated with the adjusted [59] model.", "Another leading star formation model is that of [46], where the molecular fraction is determined by processes that act on scales no larger than $\\sim 100 \\, {\\rm pc}$ , which is the size of atomic-molecular complexes in the model.", "Specifically, the molecular fraction is set by the balance between the formation of molecular hydrogen on the surfaces of dust grains and the destruction of molecular hydrogen by UV photons.", "Both dust shielding and ${\\rm H_{2}}$ self-shielding contribute to the survival of molecular hydrogen in the interior of the complexes.", "Stars form only from molecular gas and the model produces a constant molecular SFE because properties of molecular clouds, like $\\Sigma _{\\rm H_{2}}$ , are independent of general ISM conditions, at least while gas surface densities in the general ISM are less than GMC densities ($\\sim 85 \\,M_{\\odot } \\, {\\rm pc}^{-2}$ ).", "In this paper, we study star formation in a sample of 20 bulgeless disk galaxies.", "Bulgeless galaxies are interesting from a number of perspectives.", "Their existence in relatively large numbers [35], [42] provides an important constraint on hierarchical galaxy formation models, in which galaxies generally have rich merger histories that lead to bulge growth.", "The agreement between models and observations is becoming better as feedback, high gas fractions, and satellite mergers on radial orbits are included in the models (Robertson et al.", "2006, Hopkins et al.", "2008, Brook et al.", "2011; but see also Scannapieco et al.", "2011).", "Under the assumption that bulges do form when significant merger events occur, bulgeless galaxies are a suitable sample to study secular evolution, where internal processes like star formation lead to changes in the galaxies, such as bulge growth [43] A number of works have studied the components of star formation in late-type disk galaxies.", "[9] found that their sample of 47 late-type spirals are similar to earlier-type spirals in that they fall on the approximately linear correlation between far-infrared (FIR) luminosity, which traces star formation, and the molecular hydrogen mass within the central few kpc.", "Furthermore, [53] found that low-surface brightness, late-type disks lie on this same relation.", "[16] studied the dust and cold ISM structure in a sample of 49 edge-on bulgeless disk galaxies.", "They inferred that there is a sharp transition in dust lane structure with circular velocity ($v_{\\rm circ}$ ) based on measurements of $R-K$ color versus height above the midplane.", "Galaxies with $v_{\\rm circ} <120 \\, {\\rm km \\, s^{-1}}$ (we also refer to these as low-$v_{\\rm circ}$ galaxies) appear to have no dust lanes while galaxies with $v_{\\rm circ} > 120 \\, {\\rm km \\, s^{-1}}$ (high-$v_{\\rm circ}$ galaxies) have well-defined dust lanes [1].", "The authors concluded that the transition is likely due to a transition in dust scale height rather than due to a sharp transition in the quantity of dust present: low-$v_{\\rm circ}$ galaxies have diffuse dust lanes with large scale heights while high-$v_{\\rm circ}$ galaxies have dust lanes with smaller scale heights.", "They came to this conclusion because the dust structure transition occurs over a relatively narrow range in circular velocity ($\\sim 10 \\, {\\rm km \\,s^{-1}}$ ), and therefore over a relatively narrow range in gas and total mass, where the dust-to-gas ratio (DGR) does not vary substantially.", "[16] also found that disk stability, parametrized by a generalized Toomre Q parameter including both gas and stars [63], is correlated with the dust structure, with low-$v_{\\rm circ}$ galaxies generally stable and high-$v_{\\rm circ}$ galaxies generally unstable.", "Furthermore, they concluded that a sharp change in the contribution of turbulence to the stability parameter is the likely cause of the stability and dust scale height transitions.", "The authors suggested that high-$v_{\\rm circ}$ galaxies have turbulence dominated by supernovae explosions and gravitational instabilities while low-$v_{\\rm circ}$ galaxies have turbulence dominated by only supernovae.", "Independent of the source of the turbulence, the turbulent velocities must be lower in the high-$v_{\\rm circ}$ galaxies to explain the stability results.", "An alternative interpretation for the dust structure transition is that it is due to differences in stellar surface density and dust opacity, as suggested by [31].", "[16] noted that the cold, star-forming gas should have a similar distribution to the dust, with larger scale heights in low-$v_{\\rm circ}$ galaxies compared to high-$v_{\\rm circ}$ galaxies.", "They hypothesized that a transition in SFE might accompany the transition in dust scale height and stability if the volume density of gas is the relevant quantity for setting the SFR.", "A low-$v_{\\rm circ}$ galaxy with a larger scale height but the same $\\Sigma _{\\rm gas}$ relative to a high-$v_{\\rm circ}$ galaxy will have a lower gas volume density ($\\rho _{\\rm gas}$ ).", "The low-$v_{\\rm circ}$ galaxy likely also has a lower gas pressure because pressure is proportional to the gas volume density ($P \\propto \\rho _{\\rm gas} \\, \\sigma _{\\rm gas}^2$ , where $\\sigma _{\\rm gas}$ is the gas velocity dispersion).", "In the context of the star formation model where the molecular fraction is set by the mid-plane pressure, we then expect lower $R_{\\rm mol}$ , $\\Sigma _{\\rm SFR}$ , and SFE in the low-$v_{\\rm circ}$ galaxy.", "In this paper, we address whether there is a SFE transition at $v_{\\rm circ} = 120 \\, {\\rm km \\, s^{-1}}$ in our sample of bulgeless disk galaxies (Section REF ).", "We also investigate whether scale height differences affect the SFE and discuss implications for the scale of physical processes that are important for setting the molecular fraction and the SFE.", "We examine our results in light of recent star formation models such as the the mid-plane pressure model and the model of [46] (Sections REF and REF ).", "To carry out this study, we trace molecular gas with CO(1–0) data from the Institut de Radioastronomie Millimétrique (IRAM) $30 \\, {\\rm m}$ telescope and atomic gas with H1 $21 \\, {\\rm cm}$ data from the Very Large Array (VLA; Watson et al.", "2011, hereafter Paper I).", "We trace the SFR with H$\\alpha $ data from the 2.4 m Hiltner Telescope of the MDM Observatory combined with polycyclic aromatic hydrocarbon (PAH) emission data derived from Spitzer Space Telescope Infrared Array Camera (IRAC) observations.", "We also estimate the stellar mass from the Spitzer IRAC data.", "These observations, and measurements derived from the data, are described in Section .", "We describe the quantities that we derive from these measurements in Section .", "Our results are in Section  and we discuss these results in Section ." ], [ "Sample Summary, Observations, Data Reductions, and Measurements", "Our sample is composed of 20 Sd-Sdm galaxies within $32 \\, {\\rm Mpc}$ , with circular velocities between 46 and $190 \\, {\\rm km \\, s^{-1}}$ .", "These properties are well matched to the [16] sample.", "However, in contrast to the [16] sample of edge-on galaxies, we selected our galaxies to be moderately inclined, with inclinations between 16 and 56, such that we can accurately measure the SFR and gas surface densities and place the galaxies on the star formation law.", "Section 2 in Paper I and Table  provide a description of the sample selection.", "Many of the measurements described in this section were carried out to derive surface densities – of gas, SFR, and stars.", "These surface densities must be measured over the same area.", "The IRAM CO(1–0) data are the limiting factor, as they are single-beam, ${\\rm FWHM} =21$ measurements centered on each galaxy.", "Therefore, we measured the emission within a 21$$ -diameter circular aperture centered on the IRAM pointing center, the coordinates of which are listed in Table , for the following datasets: the H1 data from the VLA, the H$\\alpha $ data from the MDM Observatory, and the PAH and $4.5 \\, \\mu {\\rm m}$ data from Spitzer IRAC." ], [ "IRAM 30 m CO(1–0)", "Thirteen of our objects were observed in the CO(1–0) and CO(2–1) lines at 115 and 230 GHz in May 2007 with the IRAM $30 \\, {\\rm m}$ telescope on Pico Veleta.", "Dual polarization receivers were used at both frequencies with the 512 $\\times $ 1 MHz filterbanks on the CO(1–0) line and the 256 $\\times $ 4 MHz filterbanks on the CO(2–1) line.", "The observations were carried out in wobbler switching mode with a wobbler throw of 200 in the azimuthal direction.", "At the beginning of the observations the frequency tuning was checked by observing a bright galaxy at a similar redshift.", "Observations of the same calibration source on different days allowed us to check the relative calibration, which was excellent (better than 10%) for CO(1–0).", "The calibration in CO(2–1) was equally good, except for one day when the calibration observation was different by $\\sim 35\\%$ .", "Pointing was monitored on nearby quasars, Mars, or Jupiter every 60 – 90 minutes.", "During the observation period, the weather conditions were generally good, with pointing better than 3.", "The typical system temperature was 300-500 K at 115 GHz and 500–1000 K at 230 GHz on the $T_{\\rm A}^*$ scale.", "At 115 GHz (230 GHz), the IRAM forward efficiency, $F_{\\rm eff}$ , was 0.95 (0.91), the beam efficiency, $B_{\\rm eff}$ , was 0.75 (0.54), and the half-power beam size is 21$$ (11$$ ).", "All CO spectra and line intensities are presented on the main beam temperature scale ($T_{\\rm mb}$ ) which is defined as $T_{\\rm mb} = (F_{\\rm eff}/B_{\\rm eff})\\times T_{\\rm A}^*$ .", "For the data reduction, we selected the observations with good quality (taken during satisfactory weather conditions and showing a flat baseline), averaged the spectra from the individual scans of the source, and subtracted a constant continuum for the CO(1–0) spectra and a linear continuum for the CO(2–1) spectra.", "Figure REF shows the CO spectra for the objects observed in May 2007.", "The black spectrum shows the CO(1–0) data and the red spectrum shows the CO(2-1) data.", "We did not use the CO(2-1) data in this paper, but show it for completeness and to corroborate some of the weaker CO(1–0) detections.", "The solid black horizontal line is the velocity range over which we integrated to derive the CO(1–0) line intensity.", "The dashed horizontal line is centered on the systemic velocity of the galaxy, derived from velocity field modeling of our H1 emission line data.", "The width of the dashed line is the width of the H1 emission line at 20% of the peak flux density ($W_{20}$ ; see Table  for these values).", "The integrated CO(1–0) line intensities are presented in Table .", "We did not detect ESO 544-G03, ESO 418-G008, ESO 501-G023, or UGC 6446.", "For these galaxies, we quoted the CO line intensities as upper limits, computed with $I_{\\rm CO} < 3 \\, \\sigma _{\\rm rms} \\,(W_{20} \\, \\Delta v)^{1/2}$ , where $\\sigma _{\\rm rms}$ is the noise in the spectrum in K and $\\Delta v$ is the CO(1–0) spectrum channel width, which is $10 \\, {\\rm km \\, s^{-1}}$ in the Hanning-smoothed spectra of Figure REF .", "CO(1–0) and CO(2–1) spectra and line intensities for six of our objects – PGC 3853, NGC 2805, NGC 3906, NGC 4519, NGC 5964, and NGC 6509 – were presented in [9], also based on IRAM $30\\, {\\rm m}$ observations and reduced in the same manner.", "We did not obtain CO data for ESO 555-G027.", "Figure: CO(1–0) (black) and CO(2–1) (red) spectra for the objectsobserved in May 2007.", "See for the remainingspectra.", "The black horizontal line designates the velocity rangeover which we integrated to derive the CO(1–0) line intensity.ESO 544-G03, ESO 418-G008, ESO 501-G023, and UGC 6446 do not havethis line because they were undetected.", "The dashed horizontal lineis centered on the systemic velocity derived from velocity fieldmodeling of our H1 data, and has a width of W 20 W_{20}, whichwe derived from the integrated H1 line profile.The H1 $21 \\, {\\rm cm}$ data for our sample were obtained from the VLA/Expanded VLA, operated by the National Radio Astronomy ObservatoryThe National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc., for projects AZ0133 (carried out in 2001 August), AL0575 (carried out in 2002 June and November), AM0873 (carried out in 2006 October and November), and AM0942 (carried out in 2008 May and 2009 July and August).", "The galaxies were observed in the C or CnB configurations, which provide a nominal angular resolution of 13.", "The channel width is generally $5.2 \\, {\\rm km \\, s^{-1}}$ .", "The observations and reductions are described further in Paper I.", "We measured the integrated H1 line intensity from velocity-integrated intensity maps (with units of ${\\rm Jy \\, beam^{-1} \\, km \\, s^{-1}}$ ) created from naturally-weighted data cubes.", "The beam major axis FWHM (${B_{\\rm maj}}$ ) and minor axis FWHM (${B_{\\rm min}}$ ) for each data cube are listed in Table 3 of Paper I.", "Figure: (Continued)Ten of our objects have $B_{\\rm maj}$ and $B_{\\rm min} < 21 $ .", "To adjust the H1 beam to match the CO(1–0) beam, we used the AIPS task CONVL to convolve the integrated intensity map so the output Gaussian beam has ${B_{\\rm maj}} = {B_{\\rm min}} = 21 $ .", "To approximately match the H1 beam to the CO(1–0) beam for the remaining ten objects with $B_{\\rm maj} > 21 $ (on average, $B_{\\rm maj} = 29$ ) and $B_{\\rm min} < 21 $ , we either used the original integrated intensity map or convolved the map to have $B_{\\rm min} = 21 $ and the beam major axis approximately equal to the original $B_{\\rm maj}$ .", "Using the resulting maps, we measured the average H1 line intensity in ${\\rm Jy \\, beam^{-1} \\, km \\, s^{-1}}$ within the 21-diameter circular aperture ($\\langle {I_{\\rm HI}} \\rangle $ ) with the AIPS task IMEAN.", "These values are listed in Table .", "The integrated H1 line intensity in ${\\rm K \\, km \\, s^{-1}}$ is given by: $\\langle {I_{\\rm HI}} \\rangle \\, [{\\rm K \\, km \\, s^{-1}}] =\\frac{6.07 \\times 10^{5} \\ \\langle I_{\\rm HI} \\rangle \\, [{\\rm Jy \\,beam^{-1} \\, km \\, s^{-1}}]}{B_{\\rm maj} \\, B_{\\rm min}},$ where ${B_{\\rm maj}}$ and ${B_{\\rm min}}$ are in arcseconds and now refer to the beam of the map on which we made the measurements.", "These values are listed in Table .", "For the objects with ${\\rm B_{maj}} > 21 $ , we assumed that the average line intensity within a 21 beam is equal to the measured line intensity from our image with a larger beam.", "We estimated the uncertainty introduced by this assumption by convolving the integrated intensity maps of three objects with ${\\rm B_{maj}} <21 $ such that the convolved beams match those of the ten objects with ${\\rm B_{maj}} > 21 $ .", "We measured the average line intensity within the 21-diameter circular aperture and compared this to the true value measured from the map with a 21 beam.", "We found that the line intensities in our test cases differ from the true values by up to 11% (in the case of the NGC 4519 beam, which has a $B_{\\rm maj} = 51.91$ ) and by 4% on average.", "The test measurements both over and under estimate the true value depending on the emission distribution.", "Therefore, we included this in our uncertainty estimate, but make no correction.", "The main contributors to the final uncertainty in our H1 line intensities are flux calibration (5%), aliasing (up to 11% and described in Paper I), and using an image with a beam larger than 21 to estimate the line intensity within 21 (on average 4%).", "Not all objects are subject to the latter two uncertainties.", "Nonetheless, we conservatively assigned the quadrature sum of these uncertainties (13%) as the generic uncertainty associated with our H1 line intensities within the 21-diameter aperture.", "NGC 6509 shows H1 in absorption on the east side of the galaxy because it is in the foreground of the radio source 4C +06.63.", "The average line intensity within 21 is unaffected because the eastern edge of the aperture and the western edge of the radio lobe are separated by about 20." ], [ "Epicyclic Frequency", "We calculated a representative epicyclic frequency ($\\kappa $ ) for the 21-diameter circular aperture to use in the stability analysis of Section REF .", "We used $\\kappa =\\sqrt{2(1+\\beta )} \\, (v/r)$ , where $\\beta = d{\\rm log}(v)/d{\\rm log}(r)$ .", "We determined $\\beta $ and the rotation velocity, $v$ , at a radius, $r$ , of $5.25$ (half the radius of the 21-diameter aperture) from the H1 rotation curves presented in Paper I, where we fit a tilted ring model to the data to derive $v$ at radii every $(B_{\\rm maj} B_{\\rm min})^{0.5}$ , beginning at $(B_{\\rm maj} B_{\\rm min})^{0.5}/2$ .", "We simply fit a line between the origin and the first point in the rotation curve, the average radius of which is 8.3$$ .", "We used the slope of the line as an estimate of $dv/dr$ and evaluated $v$ at $5.25$ .", "There may be inaccuracies introduced to $\\kappa $ evaluated in this manner because the origin and first point in the rotation curve are within a single beam.", "Therefore, we also calculated the epicyclic frequency at $10.5$ using the same method as above, but by fitting the line between the first and second points from the rotation curve, where the average second radius is $25.0$ .", "The epicyclic frequencies calculated at $10.5$ are smaller than the values at $5.25$ by 30% on average.", "We used the epicyclic frequencies evaluated at $5.25$ in our stability analysis (and list these in Table ), but include 30% uncertainties on the values.", "Beam smearing, where many velocity components are within the spatial beam, is likely in effect in this region.", "Beam smearing leads to underestimated velocities and gradients [78] and therefore $\\kappa $ may also be underestimated.", "To account for this, we use $\\kappa $ evaluated at $5.25$ , as these values are larger." ], [ "MDM H$\\alpha $", "H$\\alpha $ and continuum images of the galaxies were obtained at the MDM 2.4m Hiltner telescope over the course of four observing runs in January 2007, November 2007, May/June 2007, and January 2008.", "Each galaxy was observed for between 30 min and 2.5 hours through a pair of matched, custom-made $15 \\, {\\rm nm}$ wide narrowband filters centered at $663 \\, {\\rm nm}$ and $693 \\, {\\rm nm}$ , hereafter the 663bp15 and 693bp15 filters, respectively.", "The H$\\alpha $ emission line falls within the 663bp15 bandpass for all of the galaxies in this sample.", "Observations were obtained with the Direct CCD camera “Echelle,” which has 2048x2048 pixels.", "The CCD was binned over 2x2 pixels to produce a plate scale of $0.55\"$ /pixel, which was well-matched to the $1^{\\prime \\prime }-1.5^{\\prime \\prime }$ image quality measured on most nights.", "The field of view was $9.4^{\\prime }$ .", "Conditions were photometric for most of the observations and a series of spectrophotometric standards were observed for flux calibration, as were a series of twilight flats.", "The exposure times and observation dates are listed in Table .", "Overscan subtraction, flat fielding, cosmic ray rejection, and bad pixel removal were performed with IRAFIRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under cooperative agreement with the National Science Foundation.. All of the 663bp15 and 693bp15 images of each galaxy were registered to a common coordinate system with SCAMP [2] and a combined image for each filter was constructed with SWARP [4].", "Observations of spectrophotometric standard stars were used to determine the absolute flux calibration for the 663bp15 filter (including an airmass correction) and the relative throughput of the two filters.", "We calculated the expected ratio of stellar flux in these filters for late-type galaxies by using the SYNPHOT package in IRAF to convolve the filter transmission functions with a series of Bruzual & Charlot 1995 [13] stellar population synthesis models with both continuous and exponentially declining ($\\tau = 1$ Gyr) star formation histories.", "The range of flux ratios from these models and the relative throughput of the two filters were used to scale the 693bp15 image and subtract it from the 663bp15 image to create an H$\\alpha $ image for each galaxy.", "We measured the H$\\alpha $ flux for each galaxy within the 21-diameter circular aperture and within a circle of diameter $D_{25}$ , the B-band major isophotal diameter at $25 \\, {\\rm mag \\,arcsec^{-2}}$ , with aperture photometry.", "All H$\\alpha $ flux measurements were multiplied by a factor of 0.75 to account for emission from [N2] $\\lambda \\lambda 6548,6584$ in the 663np15 bandpass [36].", "The H$\\alpha $ flux measurements were also corrected for Galactic extinction using the extinction law of [57] assuming $R_{V} = 3.1$ and reddening values from [70] and tabulated on NEDThe NASA/IPAC Extragalactic Database (NED) is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration..", "The final values within the 21 aperture are listed in Table .", "A number of our galaxies show substantial variation in the H$\\alpha $ flux within the 21 aperture if we vary the center of the aperture by the $\\sim 2 - 3$ pointing accuracy of the IRAM $30\\, {\\rm m}$ .", "Including this uncertainty, as well as uncertainty in the [NII] correction and absolute calibration uncertainties, we estimate the uncertainty in the H$\\alpha $ flux within the 21-diameter aperture to be about 30%.", "Figure REF compares our H$\\alpha $ fluxes measured within $D_{25}$ relative to published values from [41], [32], [56], and [25], where we corrected the published values for [N2] emission and Galactic extinction if necessary.", "Our values are on average 20% smaller than the published values.", "This systematic offset is comparable to our calibration uncertainty and may be due to differences in the filter profiles, the angular extent of the apertures, and the treatment of bright stars with potentially large subtraction residuals.", "We note that this offset corresponds to an uncertainty in the star formation rate surface density of less than $0.1 \\, {\\rm dex}$ , which is smaller than the $0.2 \\, {\\rm dex}$ uncertainty that we assign due to the variable contribution to dust heating from non-star forming populations.", "Figure: Comparison between our Hα\\alpha fluxes measured within acircle of diameter D 25 D_{25} and published Hα\\alpha fluxes.", "Bothvalues are corrected for [N2] emission and Galacticextinction.", "The dotted line shows equality.", "The largest outlier isNGC 0337." ], [ "Fourteen of the 20 galaxies were observed as part of our Spitzer Cycle 5 Program 50102; the remaining six (NGC 337, UGC 1862, ESO 418-G008, NGC 2805, NGC 4519, and NGC 4561) were observed for various other programs.", "Observations for Program 50102 consisted of five, dithered observations with a frame time of 30s in all four IRAC channels ($3.6 \\, \\mu {\\rm m}$ , $4.5 \\, \\mu {\\rm m}$ , $5.8 \\, \\mu {\\rm m}$ , and $8 \\, \\mu {\\rm m}$ ).", "Data for the other six archival datasets were comparable to our observations, with the exception that the data for UGC 1862 only include the galaxy in channels 2 and 4.", "In the analysis described below, we use the channel 1, 2 and 4 data.", "The Basic Calibrated Data (BCD) for all 20 galaxies were processed with Sean Carey's artifact mitigation softwarehttp://spider.ipac.caltech.edu/staff/carey/irac_artifacts/, which corrects for a variety of effects such as muxbleed, column pulldown/pullup, electronic banding, and first frame effect.", "We then created mosaics for each channel with the Spitzer Science Center's MOPEX (MOasicker and Point source EXtractor) package.", "This package corrects individual images for background variations and optical distortions, and then projects them onto an output mosaic image for each channel.", "These mosaic images were used for five measurements for each galaxy: the inclination ($i$ ), position angle of the major axis (PA), the PAH flux density, the $4.5 \\, \\mu {\\rm m}$ flux density, and the exponential disk scale length." ], [ "Inclination and Position Angle", "We estimated the inclination and PA for each galaxy with the IRAF ellipse task, which fits elliptical isophotes with the iterative method described by [34].", "The $3.6 \\, \\mu {\\rm m}$ data ($4.5 \\, \\mu {\\rm m}$ for UGC 1862) were fit because this channel has the greatest sensitivity to the old stellar population, yet is relatively insensitive to dust.", "We found that the isophote fits, particularly in the outer, low-surface brightness regions, were relatively sensitive to the presence of bright stars.", "We therefore created masks of these stars with the SExtractor package [3] and included this mask as an input to the ellipse task.", "These masks also excluded regions with relatively poor coverage in the IRAC mosaic.", "We averaged the PA and ellipticity values for the largest isophotes to derive our final PAs and inclinations.", "These values were used as inputs for the rotation curve analysis described in Paper I and were reported in Table 5 of that work." ], [ "PAH Flux Density", "We calculated the PAH flux density within the 21-diameter circular aperture using the $8 \\, \\mu {\\rm m}$ images, which are dominated by the 7.7 and $8.6 \\, \\mu {\\rm m}$ PAH features, and the stellar emission-dominated $3.6 \\, \\mu {\\rm m}$ images.", "We measured the $8 \\, \\mu {\\rm m}$ and $3.6 \\, \\mu {\\rm m}$ flux densities using the IRAF task phot.", "In both measurements, we subtracted from each pixel the median sky background, which we measured within a large-radius annulus centered on the galaxy.", "We applied the band-specific extended source aperture correction to the $8 \\, \\mu {\\rm m}$ and $3.6 \\, \\mu {\\rm m}$ flux densities (described in the IRAC Instrument Handbook; for the 21-diameter circular aperture, the correction is 0.985 and 0.896 for the $3.6 \\, \\mu {\\rm m}$ and $8 \\, \\mu {\\rm m}$ bands, respectively).", "Finally, the PAH flux density is the $8 \\, \\mu {\\rm m}$ flux density, less the stellar emission contribution, which we estimated by scaling the $3.6 \\, \\mu {\\rm m}$ flux density by 0.255 [39].", "For UGC 1862, we used the $4.5 \\, \\mu {\\rm m}$ flux density, scaled by a factor of 0.389, to remove the stellar contribution to the $8 \\, \\mu {\\rm m}$ band.", "This scale factor is derived from values quoted in [27], except we assumed that all the $4.5 \\, \\mu {\\rm m}$ -band emission is stellar.", "The uncertainties on the calibrated $3.6 \\, \\mu {\\rm m}$ and $8 \\, \\mu {\\rm m}$ flux densities are about 10% [29].", "We quote PAH flux density uncertainties that are simple error propagated values.", "Our PAH flux densities are presented in Table ." ], [ "$4.5 \\, \\mu {\\rm m}$ Flux Density", "We calculated the total $4.5 \\, \\mu {\\rm m}$ flux density and the flux density within the 21-diameter circular aperture such that we can derive the total stellar mass and the stellar mass surface density in Section REF .", "We first masked out bright foreground stars with the IRAF APPHOT and DAOPHOT packages, in particular the phot and substar tasks.", "We used large $4.5 \\, \\mu {\\rm m}$ to $8.0 \\, \\mu {\\rm m}$ flux ratios to identify foreground stars.", "For a few bright stars that were not adequately subtracted, we manually replaced the affected pixels with values from a region at a similar radius from the center of the galaxy.", "We used the IRAF task phot to measure the flux density within the 21-diameter circular aperture.", "For the total flux density, we used the IRAF task ellipse to measure the flux density within an ellipse where the semi-major axis (SMA) is $D_{25}/2$ and the PA and ellipticity ($e = 1-{\\rm cos} \\, i$ ) were derived from a combination of rotation curve analyses on the VLA H1 data and the ellipse fits of the IRAC $3.6 \\, \\mu {\\rm m}$ data, described in Section REF (PA and $i$ are given in Table ).", "In both measurements, we subtracted the median sky background from each pixel, measured within a large-radius annulus centered on the galaxy.", "To confirm that the $D_{25}$ aperture is appropriate for the total flux density measurement, we identified the SMA where the surface brightness profile flattens (specifically, where the surface brightness decreases by, on average, less than 2% over a number of apertures).", "The flux density enclosed within this ellipse was generally within 7% of the flux density within the $D_{25}$ ellipse, except in two cases: UGC 6446 and UGC 6930, where the flux density within the $D_{25}$ ellipse was larger by a factor of 1.9 and 1.5, respectively.", "We henceforth use the flux density within the $D_{25}$ ellipse as the total flux density.", "We applied the extended source aperture correction to the 21 and $D_{25}$ flux density measurements, with a value of 1.013 for the 21 aperture and an average value of 0.944 for the $D_{25}$ measurements.", "We did not apply a color correction.", "The 21 and total $4.5 \\, \\mu {\\rm m}$ flux densities are listed in Table ." ], [ "Exponential Disk Scale Length", "We estimated the exponential scale length of each galaxy using the IRAF ellipse task on the $4.5 \\, \\mu {\\rm m}$ data, allowing the center, PA, and ellipticity to vary as a function of semi-major axis.", "We fit an exponential to the mean isophotal intensity profile to derive the central surface brightness and scale length.", "We excluded PSF and/or bar components from the profile fit based on visual examination of the images and a provisional GALFIT [61] analysis.", "We created a two-dimensional image representing the ellipse profile with bmodel, subtracted the model from the original image, and found that the standard deviation in the region of the galaxy within the residual image is typically less than about ten times the standard deviation in a galaxy-free region of the original image.", "Given the small-scale structures present in most of the galaxies, we accepted these values and fits.", "The scale lengths are listed in Table  and are between 0.69 and $3.4 \\,{\\rm kpc}$ .", "We assigned an error of 20% to the scale lengths, based on comparing the scale lengths computed as described above to scale lengths computed from ellipse runs where we held the PA and ellipticity fixed as a function of semi-major axis at the values from Table .", "We confirmed that scale lengths derived from the $3.6 \\, \\mu {\\rm m}$ data are generally consistent with the values derived from the $4.5 \\, \\mu {\\rm m}$ data within the uncertainty (they differ by 9% on average)." ], [ "Derived Quantities", "For all the surface density calculations below, we used the 21-diameter aperture, to match the beam of our CO(1–0) data from IRAM.", "Table  lists the physical size of 21 (0.7 - $3.2 \\, {\\rm kpc}$ ) and the parameters derived in the following sections.", "The surface densities are all within the deprojected area of the aperture, which we calculated with the inclinations from Table ." ], [ "Atomic, Molecular, and Total Hydrogen Surface Density", "The H1 surface density is given by: $\\Sigma _{\\rm HI} \\, [M_{\\odot } \\, {\\rm pc^{-2}}] = 0.015 \\, \\langle I_{\\rm HI} \\rangle \\, {\\rm cos}(i),$ where $\\langle I_{\\rm HI} \\rangle $ is the average integrated H1 line intensity within the 21-diameter circular aperture in ${\\rm K \\, km \\, s^{-1}}$ from Section REF .", "We did not include a correction for He.", "The $\\Sigma _{\\rm HI}$ uncertainty is dominated by the contribution from the line intensity uncertainty and the typical uncertainty in ${\\rm log} \\, (\\Sigma _{\\rm HI})$ is $0.06 \\, {\\rm dex}$ .", "The ${\\rm H_{2}}$ surface density is given by: $\\Sigma _{\\rm H_{2}} \\, [M_{\\odot } \\, {\\rm pc^{-2}}] = \\\\3.2 \\, \\frac{X_{\\rm CO}}{2.0 \\times 10^{20} \\, {\\rm cm^{-2} \\, (K \\,km \\, s^{-1})^{-1}}} \\, I_{\\rm CO} \\, {\\rm cos}(i),$ where $X_{\\rm CO}$ is the CO-to-${\\rm H}_{2}$ conversion factor and $I_{\\rm CO}$ is the CO line intensity in ${\\rm K \\, km \\, s^{-1}}$ from Section REF .", "We used a constant $X_{\\rm CO}$ of $2.8\\times 10^{20} \\, {\\rm cm^{-2} \\, (K \\, km \\, s^{-1})}^{-1}$ .", "We used this value rather than the Milky Way value of $2.0 \\times 10^{20} \\,{\\rm cm^{-2} \\, (K \\, km \\, s^{-1})}^{-1}$ so we can plot the [37] total hydrogen star formation law relative to our data.", "Again, we did not include a correction for He.", "The typical uncertainty in ${\\rm log} \\, (\\Sigma _{\\rm H_{2}})$ is $0.07 \\, {\\rm dex}$ , due mainly to the CO line intensity uncertainty.", "We did not include uncertainty due to $X_{\\rm CO}$ .", "[50] studied five local group galaxies and concluded that $X_{\\rm CO}$ is relatively constant at the solar value for $12+{\\rm log} \\, (O/H)\\gtrsim 8.4$ and increases with decreasing oxygen abundance below $12+{\\rm log} \\, (O/H) \\sim 8.2 - 8.4$ .", "We have only one galaxy where we estimated that the oxygen abundance is below $12+{\\rm log} \\, (O/H)= 8.4$ (Section REF ), so we do not expect much $X_{\\rm CO}$ variation in our sample.", "We also use the total hydrogen surface density, $\\Sigma _{\\rm HI+H_{2}}= \\Sigma _{\\rm HI} + \\Sigma _{\\rm H_{2}}$ .", "The typical uncertainty in ${\\rm log} \\, (\\Sigma _{\\rm HI+H_{2}})$ is $0.05 \\, {\\rm dex}$ ." ], [ "Star Formation Rate Surface Density", "We used H$\\alpha $ emission to trace the unobscured SFR and PAH emission to trace the obscured SFR.", "The H$\\alpha $ emission is due to recombination in H2 regions, which are ionized by O and early B stars.", "PAHs are small dust grains (or large molecules) that are primarily excited by single UV photons [74], [48].", "Because ionizing radiation is not required to heat PAHs [27], [60], PAH emission traces lower-mass, longer-lived stars than those ultimately responsible for the H$\\alpha $ emission.", "PAH emission has been used to trace the total SFR [91], but there is variation in the $8 \\, \\mu {\\rm m}$ luminosity at a given SFR due to environment, especially metallicity [14].", "[39] used 75 galaxies to calibrate SFR estimates based on H$\\alpha $ and PAH emission by comparing the combined H$\\alpha $ and PAH luminosity to the H$\\alpha $ luminosity corrected for extinction using the Balmer decrement.", "The authors found that SFRs calculated with H$\\alpha $ and PAH emission agreed with their reference SFRs with as little scatter as SFRs calculated with H$\\alpha $ and $24 \\, \\mu {\\rm m}$ emission.", "We used the SFR calibration of [39] to calculate the SFR surface density within the 21-diameter circular aperture: $\\Sigma _{\\rm SFR} [M_{\\odot } \\, {\\rm yr^{-1} \\, kpc^{-2}}]= \\\\7.30 \\times 10^{10}(F_{\\rm H\\alpha } + 1.1 \\times 10^{-28} \\, \\nu \\,F_{\\rm PAH}) \\, {\\rm cos}(i),$ where $F_{H\\alpha }$ and $F_{\\rm PAH}$ are the H$\\alpha $ flux in ${\\rm erg \\, s^{-1} \\, cm^{-2}}$ and the PAH flux density in mJy within the 21-diameter circular aperture, $\\nu $ is the central frequency of the $8\\, \\mu {\\rm m}$ IRAC band in Hz, and the constant includes the aperture area.", "This assumes the initial mass function (IMF) from [14], which is similar to that presented in [45].", "The [39] SFR was calibrated for galaxy-averaged data, but should be appropriate for our data because we generally probe regions that are a couple of square kpc in area and should therefore contain a number of star forming regions.", "Even using both H$\\alpha $ and PAHs to trace the SFR, galaxies with low metallicity could have underestimated SFRs because the fraction of dust mass in PAHs decreases at low metallicity, particularly below $Z_{\\odot }/4$ [22], [77].", "In the following section we estimate the oxygen abundance of each galaxy from the stellar mass and find that only one galaxy (ESO 501-G023) has $12+{\\rm log} \\, (O/H) < 8.3$ .", "Therefore, we do not expect significant SFR underestimates in our sample.", "[39] discussed that the dominant source of uncertainty in their SFRs is due to varying contributions to dust heating from older stellar populations ($\\gtrsim 100 \\, {\\rm Myr}$ ).", "Based on their suggestions, and including the fact that our sample has a limited range in morphology and therefore star formation history, we assigned a general uncertainty of $0.2 \\, {\\rm dex}$ to our SFR surface densities.", "This dominates over the contribution due to H$\\alpha $ flux and PAH flux density measurement uncertainties.", "In upcoming sections, we use the SFE defined as $\\Sigma _{\\rm SFR}/\\Sigma _{\\rm HI+H_{2}}$ in ${\\rm yr}^{-1}$ .", "The typical uncertainty in the SFE is $0.2 \\, {\\rm dex}$ ." ], [ "Stellar Mass Surface Density, Total Stellar Mass, and\nOxygen Abundance", "We derived the stellar mass surface density within the 21-diameter circular aperture and the total stellar mass from the $4.5 \\, \\mu {\\rm m}$ flux densities.", "We chose to use the $4.5 \\,\\mu {\\rm m}$ data over the $3.6 \\, \\mu {\\rm m}$ data because there are no PAH emission features in the $4.5 \\, \\mu {\\rm m}$ band.", "To estimate the stellar mass, we used a relationship between K-band mass-to-light ratio and color from [1], derived from stellar population synthesis modeling: ${\\rm log} \\, (\\Upsilon _{\\ast }^{\\rm K}) = 1.43(J-K_{\\rm s})-1.53,$ where $\\Upsilon _{\\ast }^{\\rm K}$ is the mass-to-light ratio in the K band in $M_{\\odot }/L_{\\rm K,\\odot }$ , $L_{\\rm K,\\odot }$ is the solar luminosity in the K band, and the $J$ and $K_{\\rm s}$ magnitudes are from the Two Micron All Sky Survey [33] and are listed in Table .", "The original relationship uses Johnson $K$ magnitudes, but use of $K_{\\rm s}$ magnitudes does not introduce significant error to the mass.", "This relation is a linear combination of the $\\Upsilon _{\\ast }^{\\rm K}$ - $(V-J)$ and $\\Upsilon _{\\ast }^{\\rm K}$ - $(V-K)$ relations presented in Table 1 of [1] and we subtracted $0.15 \\, {\\rm dex}$ to convert from the scaled Salpeter IMF that the authors use to a [45] IMF.", "For three galaxies that do not have 2MASS data, we used: ${\\rm log} \\, \\Upsilon _{\\ast }^{\\rm K} = 0.21(B-K_{\\rm s})-1.11 =0.21(B-[4.5]) -1.23,$ where we used B-band magnitudes from the Third Reference Catalogue of Bright Galaxies [20].", "This equation is similar to that used in [47], but we used the [1] Table 1 model results, we have converted to a [45] IMF, and we assumed $(K_{\\rm s}-[4.5]) = 0.58$ .", "This color is the average value from the eleven galaxies in our sample where the 2MASS aperture radius (“r_m_ext”) and our aperture SMA ($D_{25}/2$ ) differ by less than 40% of our SMA.", "Six galaxies have spuriously large/red $(K_{\\rm s}-[4.5])$ colors because the 2MASS aperture radius is much smaller than our aperture radius [17].", "We then calculated the stellar mass within the 21-diameter circular aperture and the total stellar mass with: ${\\rm log} \\, (M_{\\ast }/M_{\\odot }) = {\\rm log} \\, \\Upsilon _{\\ast }^{\\rm K} + {\\rm log} \\, (L_{4.5}/L_{4.5,\\odot }) - 0.4(K_{\\rm s}-[4.5]),$ as in [47].", "To calculate the luminosity of the galaxy, $L_{4.5}$ , we used either the 4.5$\\, \\mu {\\rm m}$ flux density within the 21-diameter circular aperture or the total 4.5$\\, \\mu {\\rm m}$ flux density from Table , the distances presented in Table , the absolute 4.5$ \\, \\mu {\\rm m}$ magnitude of the Sun, $M_{4.5, \\odot } = 3.3$ [47], [58], and the zero-magnitude flux density of the 4.5$\\, \\mu {\\rm m}$ band, $F_{4.5}^0 = 179.7 \\, {\\rm Jy}$ [65].", "This equation uses the fact that $(K-[4.5])\\sim 0$ for the Sun.", "We again assumed $(K_{\\rm s}-[4.5]) = 0.58$ , for all galaxies in this case.", "For the stellar mass surface density ($\\Sigma _{\\ast }$ ), we then divided by the deprojected area of the 21-diameter circular aperture.", "The average offset between masses computed with Equations REF and REF is 0.2$\\, {\\rm dex}$ , with masses computed with Equation REF generally smaller.", "This offset is smaller than our estimate of the uncertainty in the mass, described further below.", "The masses computed with Equation REF above are in good agreement with masses computed using the [58] relationship between K-band and 4.5$\\,\\mu {\\rm m}$ band mass-to-light ratios, derived from stellar population synthesis modeling.", "[1] cite uncertainties of $0.1-0.2\\, {\\rm dex}$ in their mass-to-light ratios, which includes model and dust uncertainties and allows for small star formation bursts.", "The uncertainty introduced to the stellar mass by uncertainties in $L_{4.5}$ is of similar order, mainly due to distance uncertainties.", "We assigned a general uncertainty of 0.3$\\, {\\rm dex}$ to the stellar masses, which is the quadrature sum of the above uncertainties.", "This is in agreement with the [15] estimate of typical stellar mass uncertainties from stellar population synthesis modeling.", "The uncertainty in the stellar mass surface density is dominated by the contribution from the mass-to-light ratio.", "We therefore assigned a typical uncertainty of $0.2 \\, {\\rm dex}$ .", "We estimated a representative oxygen abundance for each galaxy to help assess the accuracy and precision of our molecular hydrogen and SFR surface densities (Sections REF and REF ) and also so we can compare our data to the [46] model (Section REF ).", "We estimated the oxygen abundance with the total stellar masses calculated from the 4.5$\\, \\mu {\\rm m}$ flux density and the mass-metallicity relation of [81] (their Equation 3), which assumes a [45] IMF, consistent with the rest of our analysis.", "We derive an average uncertainty in $12+{\\rm log}(O/H)$ of $0.11 \\, {\\rm dex}$ ." ], [ "Stability Parameters and Mid-Plane Pressure", "In Section REF , we study trends between SFE and stability, parametrized by a generalized Toomre Q parameter [80], [63].", "We calculated the stability parameters within the 21-diameter circular aperture.", "The stability parameter for the gas and stellar components of the disk are $Q_{\\rm gas} = \\frac{\\kappa \\sigma _{\\rm gas}}{\\pi {\\rm G} \\Sigma _{\\rm gas}}, {\\rm and}$ $Q_{\\rm stars} = \\frac{\\kappa \\sigma _{\\ast , \\rm r}}{\\pi {\\rm G}\\Sigma _{\\ast }},$ respectively, where $\\kappa $ is the epicyclic frequency and $\\sigma _{\\rm gas}$ and $\\sigma _{\\ast , \\rm r}$ are the gas and radial stellar velocity dispersions.", "[63] calculated the stability parameter that includes both gas and stars in a thin rotating disk: $\\frac{1}{Q_{\\rm gas+stars}} = \\frac{2}{Q_{\\rm stars}} \\, \\frac{q}{1+q^{2}} +\\frac{2}{Q_{\\rm gas}} \\, R \\, \\frac{q}{1+q^{2}R^{2}},$ where $q = k \\sigma _{\\ast , \\rm r}/ \\kappa $ , $R = \\sigma _{\\rm gas}/\\sigma _{\\ast , \\rm r}$ , and $k$ is a free parameter that represents the wavenumber of the perturbation.", "In all cases, the instability condition is when $Q < 1$ (except, strictly speaking, $Q_{\\rm stars} < 1.07$ ).", "[67] included disk thickness in the stability estimation by incorporating a factor related to the ratio of the vertical to radial velocity dispersion, but we settled on the [63] parameter for ease of comparison with other works.", "We generally made similar assumptions for the components of Q as [49].", "We assumed $\\sigma _{\\rm gas} = 11 \\, {\\rm km \\,s^{-1}}$ , which is appropriate for the warm neutral medium, $\\sigma _{\\ast , \\rm r} = 1.67 \\sigma _{\\ast , \\rm z}$ , and $\\sigma _{\\ast ,\\rm z} = (2 \\pi \\, G \\, l_{\\ast } \\, \\Sigma _{\\ast }/7.3)^{0.5}$ , where $\\sigma _{\\ast , \\rm z}$ is the vertical stellar velocity dispersion and $l_{\\ast }$ is the stellar scale length [79], [76], [84], [85].", "This $\\sigma _{\\ast }$ equation assumes that the disk is isothermal in the z direction, $h_{\\ast }$ is constant as a function of radius, and $l_{\\ast }/h_{\\ast } = 7.3$ [44], where $h_{\\ast }$ is the stellar scale height.", "Note that the latter two assumptions are not in conflict with a sample like [16] because those authors found only a transition in dust scale height with circular velocity, not in stellar scale height.", "We used the scale lengths and stellar mass surface densities from Sections REF and REF to derive $\\sigma _{\\ast , {\\rm r}}$ values between 9 and $78 \\, {\\rm km \\,s^{-1}}$ .", "For $\\Sigma _{\\rm gas}$ , we multiplied the total-hydrogen surface density ($\\Sigma _{\\rm HI+H_{2}}$ ) by a factor of 1.36 to include helium.", "For $\\Sigma _{\\ast }$ , we used the value derived in Section REF .", "For the epicyclic frequency, we used the value from Section REF , evaluated at $5.25$ .", "For the wavenumber of the perturbation, $k = 2 \\pi /\\lambda $ , where $\\lambda $ is the wavelength of the perturbation, we used the common method of varying $\\lambda $ to find the minimum $Q_{\\rm gas+stars}$ ($Q_{\\rm gas+stars, min}$ ) for the region [89], [90].", "This typically results in smaller, less stable $Q_{\\rm gas+stars}$ compared to $Q_{\\rm gas}$ and $Q_{\\rm stars}$ .", "We found $\\lambda = 0.5 - 3.8 \\, {\\rm kpc}$ at $Q_{\\rm gas+stars, min}$ .", "We propagated the uncertainties in $\\kappa $ , $l_{\\ast }$ , $\\Sigma _{\\rm gas}$ , and $\\Sigma _{\\ast }$ to derive a typical uncertainty in $Q_{\\rm gas}$ of 30%, in $Q_{\\rm stars}$ of 60%, and in $Q_{\\rm gas+stars}$ of 40%.", "Figure: SFR surface density versus atomic hydrogen surface density.Red squares represent galaxies with v circ <120 km s -1 v_{\\rm circ} < 120 \\, {\\rm km\\, s^{-1}}.", "Blue triangles represent galaxies with v circ >120 km s -1 v_{\\rm circ} >120 \\, {\\rm km \\, s^{-1}}.", "The circular velocities are fromH1 rotation curve fits (Paper I).", "A representative errorbar is shown in the lower right corner (seeSections  and for details).", "Thevertical dashed line is at 9M ⊙ pc -2 9 \\, M_{\\odot } \\, {\\rm pc}^{-2}, thetypical maximum density for atomic hydrogen.", "For comparison, thesmall black points represent measurements from fromseven spiral galaxies sampled at 750 pc 750 \\, {\\rm pc} resolution.We also calculated the mid-plane pressure ($P_{\\rm h}$ ) with the following equation from [23]: $P_{\\rm h} \\approx \\frac{\\pi }{2} \\, G \\, \\Sigma _{\\rm gas} (\\Sigma _{\\rm gas} + \\frac{\\sigma _{\\rm gas}}{\\sigma _{\\ast , \\rm z}} \\,\\Sigma _{\\ast })$ We propagated the uncertainties in $l_{\\ast }$ , $\\Sigma _{\\rm gas}$ , and $\\Sigma _{\\ast }$ to derive a typical uncertainty in $P_{\\rm h}$ of 40%." ], [ "Bulgeless Disk Galaxies on the Kennicutt-Schmidt Law", "In this section, we determine whether our galaxy sample follows the various versions of the Kennicutt-Schmidt law, i.e.", "the relation between the surface density of gas (atomic, molecular, or the sum of both) and SFR (Sections REF - REF ).", "In Section REF , we use these results to show that there is no transition in SFE at a circular velocity of $120 \\, {\\rm km \\, s^{-1}}$ , or at any other circular velocity probed by our sample ($v_{\\rm circ} = 46 - 190 \\, {\\rm km \\,s^{-1}}$ ).", "The circular velocities were derived from H1 rotation curve fits in Paper I and we include them in Table  for convenience.", "Figure REF shows the relationship between the star formation rate surface density and the atomic hydrogen surface density within the 21-diameter circular aperture, which corresponds to physical diameters of 0.7 - $3.2 \\, {\\rm kpc}$ .", "Red squares denote galaxies with $v_{\\rm circ} < 120 \\, {\\rm km \\, s^{-1}}$ and blue triangles denote galaxies with $v_{\\rm circ} > 120 \\, {\\rm km \\,s^{-1}}$ .", "The vertical dashed line is at $9 \\, M_{\\odot } \\, {\\rm pc}^{-2}$ and represents the typical maximum H1 surface density observed in most nearby galaxies [5].", "For comparison, we have shown as small dots the $750 \\, {\\rm pc}$ -diameter regions from the seven spiral galaxies studied in [5] and [49].", "Consistent with our assumptions, [5] used the same Kroupa-type IMF and do not include He in their gas surface densities.", "An important difference between our datasets is that [5] derived their SFR surface densities from a combination of FUV (1350 - 1750$ \\, {\\rm Å}$ ) and $24 \\, \\mu {\\rm m}$ data while we use H$\\alpha $ and PAH emission.", "The Spearman rank correlation coefficient for our data is 0.5.", "For comparison, the coefficient for the [5] data is 0.4.", "Our data are more correlated than the [5] data, primarily because our sample includes several galaxies with large H1 surface densities.", "Figure: SFR surface density versus molecular hydrogen surfacedensity.", "Symbols are as in Figure .", "The solidline shows the fit to the molecular hydrogen starformation law: Σ SFR ∝Σ H 2 \\Sigma _{\\rm SFR} \\propto \\Sigma _{\\rm H_{2}}.", "Wediscuss the offset of our data relative to this fit inSection ." ], [ "Molecular Hydrogen Kennicutt-Schmidt Law", "Figure REF shows the relationship between the star formation rate surface density and the molecular hydrogen surface density.", "The symbols are as in Figure REF .", "We convert the [5] $\\Sigma _{\\rm H_{2}}$ values to $X_{\\rm CO} = 2.8 \\times 10^{20} \\, {\\rm cm^{-2} \\, (K \\, km \\, s^{-1})}^{-1}$ to match our assumptions.", "The solid line shows the [5] fit, also converted to the above $X_{\\rm CO}$ .", "The Spearman rank correlation coefficient for our data is 0.7, including upper limits.", "For comparison, the coefficient for the [5] data is 0.8.", "Our sample of bulgeless disk galaxies, both the low- and high-$v_{\\rm circ}$ objects, appears to lie offset to higher star formation rate surface densities relative to the [5] fit.", "In what follows, we investigate the significance of and possible reasons for this offset.", "Under our assumptions (He not included in the gas surface densities, $X_{\\rm CO} = 2.8 \\times 10^{20} {\\rm cm^{-2} \\, [K \\, km \\,s^{-1}]^{-1}}$ , and the Kroupa-type IMF), the [5] fit is ${\\rm log} \\, (\\Sigma _{\\rm SFR}$ $[M_{\\odot } \\, {\\rm yr^{-1} \\,kpc^{-2}}]) = a + b\\, {\\rm log} \\, (\\Sigma _{\\rm H_{2}} \\, [M_{\\odot }\\, {\\rm pc^{-2}}])$ with $a=-3.3 \\pm 0.2$ and $b=1.0 \\pm 0.2$ .", "We fit for the intercept of our data assuming the Bigiel slope of $b=1.0$ and including our upper limits in the fit and find $a=-3.0$ with an rms of $0.3 \\, {\\rm dex}$ .", "We determine the significance of the offset between our sample of bulgeless disk galaxies and the [5] fit to the molecular hydrogen star formation law by randomly selecting nineteen measurements from [5], allowing multiple selections of the same point [62].", "We assume a slope of 1.0, calculate the intercept, and repeat this process $10^{6}$ times.", "We find that the probability of measuring a intercept greater than or equal to $-3.0$ is $4 \\times 10^{-6}$ .", "We can exclude two possible reasons for the offset.", "First, the offset is not likely due to the measurements being central values because we confirmed that the centers of the [5] galaxies are not offset from the general trend (this can also be seen in Figure 10 of Bigiel et al.", "2008).", "Second, our assumption of a single CO-to-${\\rm H}_{2}$ conversion factor probably does not lead to underestimated molecular surface densities because our sample includes only one galaxy for which $X_{\\rm CO}$ may be underestimated because of a low oxygen abundance, as discussed in Section REF .", "Furthermore, we do not find that galaxies with lower stellar mass, and by implication lower-$(O/H)$ , are more offset from the [5] fit.", "Note, however, that [73] did observe that lower-oxygen abundance (down to $12+{\\rm log}(O/H) =8.25$ ) galaxies are offset to higher molecular SFE in their study of 33 galaxies with directly-measured oxygen abundances.", "They discussed that more observations are needed to determine if the offset is due to $X_{\\rm CO}$ variation or if it represents true SFE variation.", "The quoted [5] intercept uncertainty ($0.2 \\, {\\rm dex}$ ) takes into account variations in star formation tracers, uncertainty introduced by estimating the CO(1–0) line intensity from CO(2-1) data, and scatter in the data.", "The first of these is particularly relevant in comparing our datasets because we trace star formation with H$\\alpha $ and PAH data and [5] trace star formation with FUV and $24 \\, \\mu {\\rm m}$ data.", "Our intercept is larger than the [5] intercept by only $1.2\\sigma $ , if we take their error bar as $\\sigma $ .", "In summary, our data are significantly offset from the [5] fit to the molecular hydrogen star formation law in terms of statistical uncertainties, but are nearly consistent if systematic errors are included.", "[38] discussed that offsets are expected between star formation laws derived from observations at different spatial resolution if the power-law index of the star formation law is not equal to one ($\\Sigma _{\\rm SFR} \\propto \\Sigma _{\\rm gas}^{N}$ with $N\\ne 1$ ).", "In transitioning from high to lower resolution, the SFR and gas surface density will be decreased by approximately the same factor.", "If $N=1$ , the lower resolution data will still lie on the same relation as the higher-resolution data, but at lower gas and SFR surface densities.", "In contrast, if $N > 1$ ($N < 1$ ), the lower resolution observations will be positively (negatively) offset from the higher resolution observations.", "This effect could contribute to our observed offset if $N>1$ because our data probe up to $3.2 \\, {\\rm kpc}$ scales and the [5] data has $750 \\, {\\rm pc}$ resolution.", "We cannot provide a firm explanation for the offset between our data and the [5] fit, but possible reasons for the offset include the use of different star formation tracers and resolutions, $X_{\\rm CO}$ variation, and true SFE differences." ], [ "Total Hydrogen Kennicutt-Schmidt Law", "Figure REF shows the relationship between the star formation rate surface density and the total hydrogen surface density.", "The vertical dashed line again shows the typical maximum H1 density that is observed in most nearby galaxies.", "The Spearman rank correlation coefficient for our data and also the [5] data is 0.8.", "This correlation is stronger than the correlation between SFR surface density and either atomic or molecular hydrogen surface density.", "In their study of star formation in the atomic-dominated regime, [73] found that the rank correlation coefficient between SFR surface density and total hydrogen surface density is similar to or somewhat larger than the value for the correlation with the molecular hydrogen surface density.", "However, they noted that data can be correlated with a strong rank statistic even if the parameters are related by different functions in the atomic- and molecular-dominated regimes.", "As in the Schruba et al.", "study, our stronger total hydrogen correlation is not likely related to fundamental physics.", "The solid black line shows the [46] model with $0.4 \\,Z_{\\odot }$ metallicity.", "The model assumes a clumping factor, $c$ , that is the inverse of the filling factor of $\\sim 100 \\, {\\rm pc}$ -sized atomic-molecular complexes in the beam.", "This value is not constrained well by data, so we assume the same value as [46]: $c=5$ .", "The magenta dot-dashed line shows the [37] fit to a sample of normal and starburst galaxies, where the measurements are averages over the entire optical disk.", "The blue dotted line shows the [38] fit to regions in M51 that were studied at $520 \\, {\\rm pc}$ resolution.", "[38] attribute the offset between the blue and magenta lines to dilution effects when the power-law index of the star formation law is not 1.0 (as discussed in Section REF ).", "Figure: SFR surface density versus total hydrogen surface density.Symbols are as in Figure .", "The magentadot-dashed line shows the fit to a sample of 61normal spirals and 36 starburst galaxies, where the measurements areaverages over the optical disk.", "The dotted line shows the fit to M51 data at 520 pc 520 \\, {\\rm pc} resolution.The solid black line shows the model at 0.4Z ⊙ 0.4 \\,Z_{\\odot } metallicity." ], [ "No Transition in SFE with Circular Velocity", "In Section , we discussed that there could be a SFE transition in bulgeless disk galaxies at $v_{\\rm circ} = 120 \\, {\\rm km \\, s^{-1}}$ , depending on the star formation model assumed.", "In this section we show that there is no offset between the low- and high-$v_{\\rm circ}$ galaxies on the star formation law.", "In Section , we interpret this result to discuss the relationship between SFE and the scale height of the cold ISM.", "In Figures REF , REF , and REF , there is a slight trend for low-${v_{\\rm circ}}$ galaxies to lie at lower surface densities compared to high-$v_{\\rm circ}$ galaxies, with average low- versus high-$v_{\\rm circ}$ surface densities as follows: ${\\rm log} \\langle \\Sigma _{\\rm HI} \\rangle = 0.9$ versus 1.0, ${\\rm log} \\langle \\Sigma _{\\rm H_{2}}\\rangle < 0.6$ versus $= 0.9$ , ${\\rm log} \\langle \\Sigma _{\\rm HI+H_{2}}\\rangle = 1.1$ versus 1.2, and ${\\rm log} \\langle \\Sigma _{\\rm SFR}\\rangle = -2.4$ versus -2.2.", "However, the differences are not significant; the Kolmogorov-Smirnov (K-S) test probabilities that the low- versus high-$v_{\\rm circ}$ galaxies are drawn from the same population in $\\Sigma _{\\rm H_{2}}$ , $\\Sigma _{\\rm HI+H_{2}}$ , and $\\Sigma _{\\rm SFR}$ are 0.04, 0.2, and 0.2, respectively.", "Figure REF shows the distribution of SFE in the low-$v_{\\rm circ}$ (red solid) and high-$v_{\\rm circ}$ (blue dashed) galaxies.", "If the slope of the star formation law is not 1.0, a sample that follows the relation would have SFE that varies as a function of $\\Sigma _{\\rm HI+H_{2}}$ , which will tend to broaden the distributions.", "However, our low- and high-$v_{\\rm circ}$ objects have similar $\\Sigma _{\\rm HI+H_{2}}$ distributions, so this should not lead to spurious offsets between the samples.", "Furthermore, the $\\Sigma _{\\rm HI+H_{2}}$ range is not so great that it would hide any differences between the samples.", "We find no significant difference between the SFE distributions, with a K-S test probability of 0.8 that the low- and high-$v_{\\rm circ}$ galaxies are drawn from the same population.", "Figure: SFE histograms of the low-v circ v_{\\rm circ} (red solid) andhigh-v circ v_{\\rm circ} (blue dashed) samples.", "There is no significantdifference between the distributions.We conclude that our sample of bulgeless disk galaxies does not show a strong transition in SFE at $v_{\\rm circ} = 120 \\, {\\rm km \\,s^{-1}}$ , where [16] concluded that there is a strong transition in dust scale height.", "In Figure REF we show that there is no circular velocity at which there is a transition in SFE.", "Furthermore, there is no strong trend of SFE with circular velocity within the range we probe.", "Finally, we find no difference in the ratio of the molecular to atomic surface density between the low- and high-$v_{\\rm circ}$ galaxies (the K-S test probability that the samples are drawn from the same $R_{\\rm mol}$ population is 0.4), nor do we find a transition in $R_{\\rm mol}$ at any circular velocity (Figure REF ).", "Figure: SFE versus circular velocity.", "There is no circular velocityat which a transition in SFE occurs." ], [ "SFE Trends with Stability and Mid-plane Pressure", "In this section, we address whether our sample shows a transition or trends in stability or mid-plane pressure, using the $Q_{\\rm gas}$ , $Q_{\\rm stars}$ , $Q_{\\rm gas+stars, min}$ , and $P_{\\rm h}$ values calculated in Section REF .", "Figure REF shows the SFE versus $Q_{\\rm gas}$ (left), $Q_{\\rm stars}$ (middle), and $Q_{\\rm gas+stars, min}$ (right).", "We find no unstable regions.", "The SFE generally decreases with larger, more stable Q values, although the correlation is not strong.", "For SFE versus $Q_{\\rm gas}$ , $Q_{\\rm stars}$ , and $Q_{\\rm gas+stars, min}$ , we find Spearman rank correlation coefficients of -0.2, -0.3, and -0.2, respectively.", "The sign of this correlation is as expected, but one might instead have expected a sharp decrease in SFE when Q rises above 1.", "These results are qualitatively similar to those in [49].", "Figure: Ratio of the molecular to atomic gas surface density (R mol R_{\\rm mol}) versus circular velocity.", "There is no circular velocity atwhich a transition in R mol R_{\\rm mol} occurs.", "[16] concluded that galaxies with ${\\rm v_{circ}} < 120\\, {\\rm km \\, s^{-1}}$ are generally stable while galaxies with ${\\rm v_{circ}} > 120 \\, {\\rm km \\, s^{-1}}$ are generally unstable, especially in the central ($r<l_{\\ast }$ ) regions.", "Contrary to these results, we see no evidence for a transition in stability at ${\\rm v_{circ}} = 120 \\, {\\rm km \\, s^{-1}}$ ; the K-S test probability that the low- and high-$v_{\\rm circ}$ galaxies are drawn from the same population of $Q_{\\rm gas}$ , $Q_{\\rm stars}$ , and $Q_{\\rm gas+stars,min}$ is 0.5, 0.2, and 0.8, respectively.", "There are two principal differences between our assumptions for the stability inputs and those in [16].", "First, we use a constant gas velocity dispersion and they used the quadrature sum of the velocity dispersion of atomic and molecular gas (10 and $5 \\, {\\rm km \\, s^{-1}}$ , respectively), weighted by the relative mass surface densities of the components.", "We use these assumptions and still find no difference in stability between the low- and high-$v_{\\rm circ}$ galaxies.", "Second, [16] estimated the molecular hydrogen surface densities for their sample from a scaling with circular velocity ($\\Sigma _{\\rm H_{2}} = (v_{\\rm circ}/47.1 \\, {\\rm km\\, s^{-1}})^{2.49}$ ).", "This scaling relation was derived from a similar sample of galaxies with $\\Sigma _{\\rm H_{2}}$ values from [68] and $v_{\\rm circ}$ estimated from single-dish H1 data.", "Molecular hydrogen surface densities calculated using the scaling with $v_{\\rm circ}$ are too large by a factor of six for our sample of high-$v_{\\rm circ}$ galaxies and too large by a factor of two for our sample of low-$v_{\\rm circ}$ galaxies.", "This discrepancy leads to smaller, less stable $Q_{\\rm gas}$ and $Q_{\\rm gas+stars}$ in the high-$v_{\\rm circ}$ galaxies relative to the low-$v_{\\rm circ}$ galaxies.", "However, even under these assumptions there is no significant difference between the low- and high-$v_{\\rm circ}$ galaxies in their $Q$ values.", "Note that the difference in our molecular hydrogen surface densities compared to the values derived from the scaling with $v_{\\rm circ}$ may also be related to resolution differences.", "Our $\\Sigma _{\\rm H_{2}}$ measurements are within the central 21 while the [68] data have 45 resolution and a similar distance distribution.", "We test whether a stability transition occurs at any circular velocity in Figure REF , where we plot $Q_{\\rm gas+stars, min}$ versus $v_{\\rm circ}$ .", "There is no circular velocity below which the regions are stable and above which the regions are unstable.", "If any trend is present, it is that higher circular velocity objects are more stable.", "Figure: SFE (Σ SFR /Σ HI +H 2 \\Sigma _{\\rm SFR} / \\Sigma _{\\rm HI+H_{2}}) versusstability parameters Q gas Q_{\\rm gas} (left), Q stars Q_{\\rm stars} (middle),and Q gas + stars , min Q_{\\rm gas+stars, \\, min} (right).", "The symbols are as inFigure .", "Note that the condition forinstability is Q≲1Q \\lesssim 1, so none of our galaxies are strictlyunstable.", "We describe our derivation of the errorbar inSections  and .We plot the SFE versus the mid-plane pressure in Figure REF .", "As seen in [49], we find an increase in SFE with increasing mid-plane pressure, but we do not probe high enough $P_{\\rm h}$ values or have enough data points to sample the constant-SFE region of the diagram that is clear in [49].", "We find no difference between the low- and high-$v_{\\rm circ}$ galaxies in their $P_{\\rm h}$ distributions, with a K-S test probability of 0.7 that they are drawn from the same population.", "Even focusing only on galaxies with the same $\\Sigma _{\\rm HI+H_{2}}$ (within the uncertainty in the parameter), the median pressure is the same in low- and high-$v_{\\rm circ}$ galaxies.", "Figure: Stability parameter, including the gas and stellarcontribution, versus circular velocity.", "There is no circularvelocity at which a transition in stability occurs.", "We describeour derivation of the vertical errorbar inSection ." ], [ "Dependence of Star Formation on Metallicity", "In this section, we use the oxygen abundance estimated from the stellar mass to study how star formation depends on metallicity and compare these results to recent theoretical work by [46].", "At a given total gas surface density, the [46] model predicts lower SFE at lower metallicity because ${\\rm H_{2}}$ survival requires a higher column density as ${\\rm H_{2}}$ self-shielding becomes more important than shielding by dust.", "This metallicity dependence distinguishes the model from other leading models, such as the model where mid-plane pressure determines where the ISM is molecular.", "Figure: SFE versus mid-plane pressure.", "Symbols are as inFigure .Figure: Left: SFR surface density versus total hydrogen surfacedensity, as in Figure , but showing a largerrange to demonstrate the model trends.", "The data are divided byoxygen abundance, where orange pentagons represent galaxies with12+ log (O/H)<8.712+{\\rm log} \\, (O/H) < 8.7 and purple circles represent galaxieswith 12+ log (O/H)>8.712+{\\rm log} \\, (O/H) > 8.7.", "The lines show the model with a metallicity of 0.1Z ⊙ 0.1 \\, Z_{\\odot }(orange solid) and 1.4Z ⊙ 1.4 \\, Z_{\\odot } (purple dashed), whichcorrespond to the lowest and highest metallicities in our sample.Right: Histograms of the offset of the data from the model with a metallicity of 0.4Z ⊙ 0.4 \\, Z_{\\odot }.The orange line represents the distribution of galaxies with12+ log (O/H)<8.712+{\\rm log} \\, (O/H) < 8.7 and the purple line represents thedistribution of galaxies with 12+ log (O/H)>8.712+{\\rm log} \\, (O/H) > 8.7.", "Thereis no significant difference between the distributions.The left panel of Figure REF shows the total hydrogen star formation law, as plotted in Figure REF , except the galaxies are divided into low-$(O/H)$ and high-$(O/H)$ , with the division at $12+{\\rm log} (O/H)= 8.7$ .", "The orange solid and purple dashed lines show the model of [46] at $0.1 \\, Z_{\\odot }$ and $1.4 \\, Z_{\\odot }$ , respectively.", "These metallicities correspond to the extrema of our dataset, assuming $Z/Z_{\\odot } = (O/H)/(O/H)_{\\odot }$ and $12+{\\rm log}(O/H)_{\\odot } = 8.86$ [19].", "We see no evidence that the low- and high-$(O/H)$ galaxies cluster towards the low- and high-$Z$ models, respectively.", "The right panel of Figure REF shows the offset of our data from the $0.4 \\, Z_{\\odot }$ [46] model, which provides the best fit to our dataset as a whole (although note that we would expect the best fit to be the $0.7 \\, Z_{\\odot }$ model, as that corresponds to the average $(O/H)$ of our full sample).", "The orange solid and purple dashed lines show the distributions of our low-$(O/H)$ and high-$(O/H)$ galaxies, as divided in the left panel.", "If our data clearly followed the [46] model, high-$(O/H)$ galaxies would be positively offset from the $0.4 \\,Z_{\\odot }$ model and low-$(O/H)$ galaxies would be negatively offset.", "We find no significant difference between the offset of the low- versus high-$(O/H)$ galaxies (the K-S test probability that the samples are drawn from the same offset population is 0.14).", "A second order effect is that the offset from the $0.4 \\, Z_{\\odot }$ model is expected to decrease with increasing $\\Sigma _{\\rm HI+H_{2}}$ .", "This may be related to the narrower offset distribution for the high-$(O/H)$ galaxies, which tend to have higher $\\Sigma _{\\rm HI+H_{2}}$ .", "Figure: SFE versus oxygen abundance, scaled to the solar value.", "Thesolid line demonstrates the SFE trend with metallicity for the model at log (Σ HI +H 2 )=0.8{\\rm log}(\\Sigma _{\\rm HI+H_{2}}) =0.8.Figure REF shows that there is a correlation between SFE and oxygen abundance in our data.", "For comparison, the line represents the [46] model, where the SFE depends on metallicity, $\\Sigma _{\\rm gas}$ , and the filling factor.", "We have plotted the [46] model with ${\\rm log} \\, (\\Sigma _{\\rm HI+H_{2}}) =0.8$ (no He).", "This is a low $\\Sigma _{\\rm HI+H_{2}}$ value compared to our data.", "Higher $\\Sigma _{\\rm HI+H_{2}}$ models reach constant SFE at lower metallicity.", "The mismatch between the average $\\Sigma _{\\rm HI+H_{2}}$ in the data and the best-fit $\\Sigma _{\\rm HI+H_{2}}$ used in the model is likely due to our assumptions for $X_{\\rm CO}$ and the IMF in the data, our filling factor assumption in the model, and to uncertainties in the metallicity scale.", "For the latter, various strong-line metallicity methods return $12+{\\rm log}(O/H)$ values as disparate as $0.5 \\, {\\rm dex}$ [40], and it is unknown which calibration best aligns with the solar value.", "Therefore, the model line in Figure REF only represents the expected trend of lower SFE at lower metallicity, at a given $\\Sigma _{\\rm gas}$ .", "At this level, the data do show the expected trend, with a Spearman correlation coefficient of 0.6.", "However, upon closer inspection, this agreement is mainly due to the fact that lower-$(O/H)$ galaxies tend to have lower $\\Sigma _{\\rm HI+H_{2}}$ (this is most obvious in Figure REF , although also note that the K-S test probability is 0.05 that the $\\Sigma _{\\rm HI+H_{2}}$ distributions of the low- and high-$(O/H)$ samples are drawn from the same population, which is not strong evidence for a difference).", "Because of their lower $\\Sigma _{\\rm HI+H_{2}}$ values, the low-$(O/H)$ galaxies are closer to the atomic-dominated regime of the star formation law where SFEs are lower.", "This is of course consistent with the [46] model, but is not a discriminating test of the model because many other properties correlate with gas surface density [49].", "A sample with a range of directly-measured metallicities within a small range of gas surface density would provide a more discriminating test.", "In summary, we see no clear evidence to support the [46] model, but our data are also not inconsistent with it." ], [ "Discussion", "In Section REF , we found that there is no transition in molecular fraction or SFE at any circular velocity probed by our sample.", "In Section REF , we found that all our galaxies are formally stable.", "While we did find a general trend of decreasing SFE with larger, more stable Q values, we found no sharp transition in stability at any circular velocity.", "Finally in Section REF , we found that SFE decreases at lower oxygen abundance, but the trend is not a particularly constraining test of the [46] model.", "In this section, we first address our assumption that there is a transition in dust scale height at $v_{\\rm circ} = 120 \\, {\\rm km \\, s^{-1}}$ , rather than a transition in dust content.", "We then interpret our results to discuss the relationship between SFE and the scale height of the cold ISM.", "Finally, we comment on the scale of physical processes that affect star formation and discuss our results in the context of leading star formation models." ], [ "A Transition in Dust Scale Height versus Dust Content", "In this section, we investigate the argument that the dust structure transition observed by [16] is due to a transition in dust scale height rather than due to a transition in the amount of dust present.", "[16] came to this conclusion because the dust structure transition occurs over a narrow range in circular velocity, where a large change in the DGR, and therefore dust content, is unexpected.", "All but one galaxy in our sample are detected at Infrared Astronomical Satellite 60 and 100$\\, \\mu {\\rm m}$ , which indicates that there is at least some dust in the low-$v_{\\rm circ}$ galaxies.", "We estimated a DGR proxy as the ratio of the total infrared luminosity ($L_{\\rm TIR}$ ), calculated with IRAS 25, 60, and 100$\\,\\mu {\\rm m}$ data [54] and the [18] relation, to the combined atomic and molecular hydrogen mass.", "We find no transition in this DGR proxy at $v_{\\rm circ} = 120 \\, {\\rm km \\,s^{-1}}$ , nor do we find any correlation between the DGR proxy and circular velocity.", "$L_{\\rm TIR}$ is not necessarily proportional to the dust mass because the dust temperature may not be the same for all the galaxies; nevertheless, there is clearly a significant amount of dust in the low-$v_{\\rm circ}$ galaxies.", "Two recent studies have found convincing evidence that low-$v_{\\rm circ}$ spirals have large dust scale heights.", "[75] carried out a resolved stellar population study of six edge-on, late-type spirals with $v_{\\rm circ} = 67 - 131 \\, {\\rm km \\, s^{-1}}$ and found that the scale height of the young ($\\lesssim 10^{8} \\, {\\rm yr}$ ) stellar population, which presumably formed from the cold ISM, is larger than in the Milky Way.", "[52] modeled the spectral energy distribution (SED) of three edge-on, low surface brightness galaxies with $v_{\\rm circ} = 88 - 105 \\, {\\rm km \\,s^{-1}}$ and found that a significant amount of dust must be present to account for the FIR (70 and 160$ \\, \\mu {\\rm m}$ ) emission, but the dust must have a large scale height such that it does not significantly obscure the optical emission.", "The authors concluded that the galaxies have dust scale heights greater than or equal to the stellar scale heights.", "This is in contrast to modeling studies of high surface brightness galaxies, which concluded that the dust scale height is about half the stellar scale height [88].", "[31] provided an alternative explanation for the dust structure transition observed by [16].", "The authors studied a large sample of irregular galaxies and found that the average B-band surface brightness is smaller than in higher-mass spiral galaxies.", "Based on this result, they suggested that lower stellar surface density is the cause of the larger scale heights in low-$v_{\\rm circ}$ galaxies.", "They discussed that disk stability could correlate with the dust structure but not be the cause of the transition because the gas scale height [7] shares many of the same parameters with the stability parameter.", "The authors discussed that dust opacity may also contribute to the observed difference in dust structure because of two effects.", "First, lower-$v_{\\rm circ}$ objects should have lower metallicity, and therefore lower dust content.", "Second, the scale length of a lower-mass galaxy is smaller and therefore a low-$v_{\\rm circ}$ , edge-on galaxy will have less depth from the edge to the center over which to accumulate dust column density compared to a high-$v_{\\rm circ}$ galaxy.", "Note that no scale height transition is needed in this interpretation.", "However, dust opacity is not likely the sole cause of the dust structure transition because of the reasons listed above.", "[31] suggested that the observed dust structure transition is likely due to a combination of stellar surface density and dust opacity.", "The authors agreed that a scale height transition does occur, so if this interpretation is correct, we can still constrain the effect of scale height differences on SFE.", "In Sections REF and REF , we assume that the dust scale heights in low-$v_{\\rm circ}$ galaxies are a factor of two larger than in high-$v_{\\rm circ}$ galaxies.", "[16] estimated this factor by examining the dust morphology in Hubble Space Telescope images of a couple edge-on, late-type galaxies.", "This factor is consistent with the SED modeling results of [52] and [88] if the stellar scale height distributions of low- and high-$v_{\\rm circ}$ galaxies have significant overlap, which [16] found to be true (although there are low-$v_{\\rm circ}$ galaxies where the stellar scale height is about half the value in high-$v_{\\rm circ}$ galaxies, in which case the dust scale heights may be comparable)." ], [ "The Relationship between SFE and Scale Height", "In Section REF , we found no transition in SFE at $v_{\\rm circ} = 120 \\, {\\rm km \\, s^{-1}}$ or at any circular velocity probed by our sample.", "In this section, we interpret this result to discuss the relationship between SFE and the scale height of the cold ISM.", "For this discussion, we make a number of assumptions.", "First, we assume that our sample of bulgeless disk galaxies is similar to that of [16] in that galaxies with $v_{\\rm circ} > 120 \\,{\\rm km \\, s^{-1}}$ have narrow dust lanes while galaxies with $v_{\\rm circ} < 120 \\, {\\rm km \\, s^{-1}}$ have no obvious dust lanes.", "This is a reasonable assumption because it was our main consideration in choosing the sample, but because our galaxies are moderately inclined rather than edge-on, we are unable to directly measure the vertical dust structure.", "Second, as in [16], we assume that the dust scale heights in the low-$v_{\\rm circ}$ galaxies are larger than in the high-$v_{\\rm circ}$ galaxies (Section REF addresses this assumption).", "Finally, we assume that the molecular gas and dust scale heights are comparable.", "[16] suggested that there may be a SFE transition associated with the dust scale height transition at $v_{\\rm circ} =120 \\, {\\rm km \\, s^{-1}}$ .", "The authors gave an example that assumes the true star formation law is a correlation between the volume density of gas ($\\rho _{\\rm gas}$ ) and the SFR volume density ($\\rho _{\\rm SFR}$ ): $\\rho _{\\rm SFR} \\propto \\rho _{\\rm gas}^{N}$ .", "The larger scale heights of low-$v_{\\rm circ}$ galaxies lead to lower gas volume densities.", "Depending on the index, $N$ , a low-$v_{\\rm circ}$ galaxy with the same gas surface density as a high-$v_{\\rm circ}$ galaxy can have a lower SFR surface density and therefore a lower SFE.", "To consider this point, we first assume the same $\\Sigma _{\\rm gas}$ for a low- and high-$v_{\\rm circ}$ galaxy and the following relationships between the surface and volume density of gas and SFR: $\\Sigma _{\\rm gas} \\propto \\rho _{\\rm gas} \\, h$ , and $\\Sigma _{\\rm SFR}\\propto \\rho _{\\rm SFR} \\, h$ , where $h$ is the scale height of the star forming gas and newly formed stars and the proportionality constant is the same for both relationships.", "We set $\\beta $ equal to the ratio of dust scale heights in low-$v_{\\rm circ}$ (lv) versus high-$v_{\\rm circ}$ (hv) galaxies: $\\beta = h_{\\rm lv}/h_{\\rm hv}$ .", "[16] very approximately estimated this ratio to be about two.", "Under these assumptions, $\\Sigma _{\\rm SFR, hv} =\\beta ^{N-1} \\, \\Sigma _{\\rm SFR, lv}$ , where $\\Sigma _{\\rm SFR, hv}$ and $\\Sigma _{\\rm SFR, lv}$ are the star formation rate surface densities of the low- and high-$v_{\\rm circ}$ galaxy with the same $\\Sigma _{\\rm gas}$ .", "[16] discussed the case where $\\beta =2$ and $N=1.5$ , where we expect the high-$v_{\\rm circ}$ galaxy to have a $\\Sigma _{\\rm SFR}$ that is a factor of 1.4 larger than the low-$v_{\\rm circ}$ galaxy.", "Note that if $N=1$ there is no expected difference in star formation rate surface density.", "In the star formation law plots of Figures REF , REF , and REF , we are sensitive to offsets in intercept between the low- and high-$v_{\\rm circ}$ samples that are greater than the uncertainty in the intercept, which is $\\sim 0.3 - 0.4 \\, {\\rm dex}$ , assuming an uncertainty in $\\Sigma _{\\rm SFR}$ of $0.2 \\, {\\rm dex}$ and the number of galaxies in our low- and high-$v_{\\rm circ}$ samples.", "In the case discussed by [16], we expect the low-$v_{\\rm circ}$ galaxies to be offset to lower $\\Sigma _{\\rm SFR}$ by $0.15 \\, {\\rm dex}$ .", "Therefore, we cannot exclude offsets at the level expected by [16].", "The star formation law assumed above is not likely correct given that recent studies have found a strong molecular star formation law and no atomic gas star formation law.", "However, we can also determine the expected $\\Sigma _{\\rm SFR}$ offset if the molecular fraction is set by the mid-plane pressure and the molecular SFE is constant (see also Section ).", "We assume the same $\\Sigma _{\\rm HI+H_{2}}$ for a low- and high-$v_{\\rm circ}$ galaxy, $R_{\\rm mol} \\propto P_{\\rm h}$ , and $P_{\\rm h} \\propto \\rho _{\\rm HI+H_{2}} \\sigma _{\\rm gas}^{2}$ , where $\\rho _{\\rm HI+H_{2}}$ is the total hydrogen volume density and the velocity dispersion of the gas is constant.", "In this scenario, $\\rho _{\\rm HI+H_{2}}$ , $P_{\\rm h}$ , and $R_{\\rm mol}$ are lower by a factor of $\\beta $ in the low-$v_{\\rm circ}$ galaxy compared to the high-$v_{\\rm circ}$ galaxy.", "The expected offset in $\\Sigma _{\\rm SFR}$ depends on $R_{\\rm mol}$ , which varies from 0.1 to 2.7 in our sample.", "If $\\beta = 2$ and $R_{\\rm mol}$ of the high-$v_{\\rm circ}$ galaxy is 0.1 (2.7), we expect $\\Sigma _{\\rm SFR}$ to be lower by $0.3 \\, {\\rm dex}$ ($0.1 \\, {\\rm dex}$ ) in the low-$v_{\\rm circ}$ galaxy compared to the high-$v_{\\rm circ}$ galaxy.", "Given our uncertainties, this level of offset would also be difficult to detect.", "However, we can reject our assumption that $P_{\\rm h}$ is lower by a factor of two in low-$v_{\\rm circ}$ galaxies.", "This is evident from Section REF , where we found no significant difference in the mid-plane pressure distributions of the low- and high-$v_{\\rm circ}$ galaxies, even when considering only objects with the same $\\Sigma _{\\rm HI+H_{2}}$ .", "Furthermore, we found no difference in the $R_{\\rm mol}$ distributions of the low- versus high-$v_{\\rm circ}$ galaxies (Section REF ).", "The two scenarios explored above predict offsets in intercept between the low- and high-$v_{\\rm circ}$ samples that are less than the uncertainty.", "Therefore, we cannot clearly exclude these options.", "Nonetheless, our data show no evidence for a strong transition in SFE at any circular velocity.", "A simple interpretation that is consistent with our data is that low-$v_{\\rm circ}$ galaxies have a lower number of molecular clouds per unit volume compared to high-$v_{\\rm circ}$ galaxies at the same $\\Sigma _{\\rm HI+H_{2}}$ , but lower only by the ratio of the cold ISM scale heights in low- versus high-$v_{\\rm circ}$ galaxies.", "This results in the same total number of molecular clouds within the beam for a low- and high-$v_{\\rm circ}$ galaxy at the same $\\Sigma _{\\rm HI+H_{2}}$ and thus gives the same $\\Sigma _{\\rm H_{2}}$ and $\\Sigma _{\\rm SFR}$ (assuming the molecular clouds have the same density and that we average over evolutionary effects).", "Note that the above applies in the molecular-dominated regime.", "We have few data points below $\\Sigma _{\\rm HI+H_{2}} \\sim 9 \\, M_{\\odot } \\, {\\rm pc}^{-2}$ , but expect that $\\Sigma _{\\rm SFR}$ can vary substantially for galaxies with the same $\\Sigma _{\\rm HI+H_{2}}$ , depending on the physical processes that affect the molecular fraction (e.g., those processes discussed in Krumholz et al.", "2009 and Ostriker et al.", "2010).", "In conclusion, we interpret our result that there is no transition in SFE at any circular velocity as evidence that scale height differences at the level of about a factor of two do not significantly affect the molecular fraction or SFE in bulgeless disk galaxies.", "However, offsets in SFE below our uncertainty level are still possible." ], [ "Comparison to Star Formation Models", "[16] very approximately estimated that the dust scale heights of low-$v_{\\rm circ}$ galaxies are about a factor of two larger than high-$v_{\\rm circ}$ galaxies.", "Assuming our sample has a similar range in scale height, our results indicate that these scale height differences, which lead to gas volume density differences also at the level of a factor of about two, do not lead to obvious differences in the SFE.", "Our results favor star formation models where small-scale physical processes are more important than processes that act on larger scales, of order the dust and cold gas scale height (10s to $100 \\, {\\rm pc}$ ).", "Based on their comparison to many star formation models without an obvious favorite, [49] discussed that physics below their resolution of $750 \\, {\\rm pc}$ is likely most important for determining the SFE.", "We contribute with a further constraint that the SFE is likely affected primarily by processes that act on scales smaller than the cold gas and dust scale height.", "We cannot exclude all star formation models that include large-scale physics because there are processes that may affect star formation but are neither affected by nor affect the scale height.", "For example, our sample has no power to constrain the effects of large-scale radial processes, like shear [30], on star formation.", "In addition, star formation may be affected by environmental properties that depend on the gas volume density but also depend on other variables that counteract a variable volume density.", "For example, the pressure in the ISM is related to the gas volume density and velocity dispersion: $P \\propto \\rho _{\\rm gas} \\,\\sigma _{\\rm gas}^2$ .", "While we expect the volume density to be lower in low-$v_{\\rm circ}$ galaxies, the velocity dispersion may be larger.", "With the right combination of $\\rho _{\\rm gas}$ and $\\sigma _{\\rm gas}$ , there could be no difference in pressure between low- and high-$v_{\\rm circ}$ galaxies.", "While the latter argument illustrates our limitations, we note that there is currently no strong observational reason to assume different gas velocity dispersions between the low- and high-$v_{\\rm circ}$ galaxies.", "[79] found some variation in the central H1 velocity dispersion in eleven galaxies ranging from early-type spirals to irregulars, but more observations are needed.", "Would leading star formation models have predicted a difference in SFE in galaxies with scale heights that differ by about a factor of two?", "In the [46] model, $\\Sigma _{\\rm SFR}$ is a function of metallicity, $\\Sigma _{\\rm gas}$ , and the beam filling factor of $\\sim 100 \\, {\\rm pc}$ -sized atomic-molecular complexes.", "There is no direct dependence on the scale height, so we would not expect a transition in SFE unless there is a transition in the metallicity or filling factor with scale height.", "We find no transition in oxygen abundance at any circular velocity in our data, but there should be a correlation between these two properties given the mass-metallicity relation [81], which we also do not see.", "The filling factor of star-forming complexes is not well constrained, although there is no a priori reason to suppose that it would be different in low- versus high-$v_{\\rm circ}$ objects.", "[46] do not predict a difference in SFE in galaxies with different scale heights and the fact that we did not find a transition in SFE at $v_{\\rm circ} = 120 \\, {\\rm km \\, s^{-1}}$ is not a strong constraint on this model.", "In the model where the mid-plane pressure sets the molecular fraction and the molecular SFE is constant, the pressure is proportional to the gas volume density, which we expect to vary between the low- and high-$v_{\\rm circ}$ galaxies.", "If the gas velocity dispersion is fixed, we would expect the low-$v_{\\rm circ}$ galaxies to have lower molecular to atomic surface density ratios and lower SFEs relative to the high-$v_{\\rm circ}$ galaxies .", "However, we found no difference between the mid-plane pressure, molecular to atomic surface density ratio, or SFE distributions of the low- and high-$v_{\\rm circ}$ galaxies.", "If our assumptions are correct, our result is inconsistent with this model.", "However, the offset in SFE expected for galaxies with cold gas scale heights that differ by about a factor of two may be less than our SFE uncertainties.", "Furthermore, the [59] model relates the molecular fraction (or in their terms, the fraction of gas in gravitationally bound complexes) to the pressure of the diffuse component of the ISM.", "We have a constraint only on the scale height and volume density of the cold component of the ISM; therefore the [59] model may not predict molecular fraction and SFE differences in our sample.", "In general, our results are somewhat more consistent with local models of star formation, like the [46] model, but we do not find conclusive evidence for or against either the [46] or [59] model.", "One final matter to address is whether central measurements are sufficient to determine if there is a transition in molecular fraction, SFE, and/or stability at the dust structure transition of $v_{\\rm circ} = 120 \\, {\\rm km \\, s^{-1}}$ .", "One might question the use of central measurements because dust structure, SFE, and stability may be affected by the higher gas and stellar densities and shorter dynamical times characteristic of these regions.", "The only approach that will fully address this concern is to obtain off-center measurements of the above properties.", "Our single-beam CO data currently limit us from carrying out this analysis.", "Meanwhile, there is some evidence that our central pointings are sufficient to address these questions.", "First, our measurements trace a significant fraction of the disk: the 21 aperture probes physical scales of $0.7 -3.2 \\, {\\rm kpc}$ , which is similar to the disk scale lengths of our sample ($0.7 - 3.4 \\, {\\rm kpc}$ ).", "Second, the central regions of bulgeless galaxies are more morphologically and kinematically similar to the outskirts than in a galaxy with a bulge.", "Finally, we expect a galaxy to be less stable against gravitational collapse in the center compared to the outer disk because the gas and stellar surface densities are larger.", "However, we find that both the low- and high-$v_{\\rm circ}$ galaxies are stable.", "This suggests that there would also not be a stability transition at $v_{\\rm circ} = 120 \\,{\\rm km \\, s^{-1}}$ in off-center measurements because both the low- and high-$v_{\\rm circ}$ galaxies would be more stable." ], [ "Summary", "We have presented a study of star formation in twenty moderately-inclined, bulgeless disk galaxies.", "We found no transition in star formation efficiency ($\\Sigma _{\\rm SFR}/\\Sigma _{\\rm HI+H_{2}}$ ) or disk stability at $v_{\\rm circ} = 120 \\, {\\rm km \\,s^{-1}}$ .", "This circular velocity was previously found to be associated with a transition in the vertical dust structure of edge-on, bulgeless disk galaxies that is most likely due to a transition in the scale height of the cold ISM.", "We also found no transition in star formation efficiency or disk stability at any circular velocity probed by our sample.", "Our results demonstrate that the scale height of the cold ISM does not play a major role in setting the molecular fraction or the star formation efficiency.", "We also found decreasing star formation efficiency with lower oxygen abundance, which we estimated from the stellar mass.", "This result is consistent with the recent [46] model, but a sample with a large range of metallicities within a small range of gas surface density would provide a more constraining test of the model.", "In general, our results are most consistent with local models of star formation that include physical processes that act on smaller scales than the dust and cold gas scale height (10s to $100 \\, {\\rm pc}$ ).", "We thank Frank Bigiel for providing us with his data.", "We are also grateful to Todd A. Thompson for helpful comments and discussion, Richard W. Pogge for supplying the narrowband filters used in the H$\\alpha $ observations, Roberto J. Assef, David W. Atlee, and Katharine J. Schlesinger for obtaining some of the H$\\alpha $ observations, and the referee for comments that improved this work.", "L.C.W.", "gratefully acknowledges support from an NSF Graduate Research Fellowship and an Ohio State University Distinguished University Fellowship.", "PM is grateful for support from the NSF via award AST-0705170.", "U.L.", "acknowledges financial support from the research projects AYA2007-67625-C02-02 and AYA2011-24728 from the Spanish Ministerio de Ciencia y Educación and from the Junta de Andalucía.", "This work is based in part on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA.", "Support for this work was provided by NASA through an award issued by JPL/Caltech.", "This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.", "lcccccccccccc 0pt General Galaxy Properties Source RA DEC D $D_{25}$ $B$ $M_{B}$ Type $V_{\\rm sys}$ $v_{\\rm circ}$ PA i $W_{20}$ (hh:mm:ss.s) (dd:mm:ss) (Mpc) (arcsec) (mag) (mag) (${\\rm km\\, s^{-1}}$ ) (${\\rm km\\, s^{-1}}$ ) ($$ ) ($$ ) (${\\rm km \\, s^{-1}}$ ) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) NGC 0337 00:59:50.0 -07:34:41 20.7 [T88] 173 11.44 -20.14 7.0 $1646 \\pm 2 $ $145 ^{+5 }_{-4 } $ $118 \\pm 5 $ $44 \\pm 2 $ 261 PGC 3853 01:05:04.8 -06:12:46 11.4 [T08] 250 11.98 -18.30 7.0 $1094.7 \\pm 0.4 $ $128.1 ^{+1.6 }_{-2 } $ $105.3 \\pm 0.2 $ $41.4 \\pm 1.1 $ 192 PGC 6667 01:49:10.3 -10:03:45 24.6 [T88] 173 12.92 -19.03 6.7 $1989.2 \\pm 0.6 $ $155 ^{+4 }_{-4 } $ $122.9 \\pm 1.8 $ $34.0 \\pm 1.1 $ 198 ESO 544-G030 02:14:57.2 -20:12:40 13.9 [T08] 123 13.25 -17.47 7.7 $1608.4 \\pm 1.0$ $100.9 ^{+1.4 }_{-2 } $ $107.6 \\pm 1.1 $ $48.5 \\pm 1.2 $ 146 UGC 1862 02:24:24.8 -02:09:41 22.3 [T08] 99.6 13.47 -18.27 7.0 $1382.9 \\pm 0.4 $ $55 ^{+6 }_{-5 } $ $21.7 \\pm 1.7 $ $43 \\pm 4 $ 125 ESO 418-G008 03:31:30.8 -30:12:46 23.6 [T08] 70.5 13.65 -18.21 8.0 $1195.4 \\pm 0.3 $ $74.1 ^{+0.6 }_{-0.6 } $ $317.9 \\pm 1.1 $ $55.6 \\pm 1.4 $ 140 ESO 555-G027 06:03:36.6 -20:39:17 24.3 [T88] 138 13.18 -18.75 7.0 $1978.7 \\pm 0.4 $ $190 ^{+40 }_{-30 } $ $221.5 \\pm 0.3 $ $21 \\pm 4 $ 162 NGC 2805 09:20:20.4 +64:06:12 28.0 [T88] 379 11.17 -21.07 7.0 $1732.6 \\pm 0.6 $ $81 ^{+7 }_{-8 } $ $300 \\pm 3 $ $38 \\pm 4 $ 120 ESO 501-G023 10:35:23.6 -24:45:21 7.01 [T08] 208 12.86 -16.37 8.0 $1046.8 \\pm 0.7 $ $46 ^{+18 }_{-8 } $ $224 \\pm 2 $ $37 \\pm 12 $ 83 UGC 6446 11:26:40.6 +53:44:58 18.0 [T08] 213 13.30 -17.98 7.0 $645.5 \\pm 0.6 $ $79.7 ^{+1.3 }_{-0.8 } $ $189.4 \\pm 0.5 $ $52.5 \\pm 1.9 $ 150 NGC 3794 11:40:54.8 +56:12:10 19.2 [T08] 134 13.23 -18.19 6.5 $1384.9 \\pm 0.7 $ $103.3 ^{+1.1 }_{-1.0 } $ $123.1 \\pm 1.1 $ $54.8 \\pm 1.3 $ 182 NGC 3906 11:49:40.2 +48:25:30 18.3 [...] 112 13.50 -17.81 7.0 $959.44 \\pm 0.7 $ $65 ^{+30 }_{-16 } $ $180 \\pm 20 $ $16 \\pm 5 $ 49 UGC 6930 11:57:17.2 +49:17:08 17.0 [T88] 262 12.38 -18.77 7.0 $776.7 \\pm 0.7 $ $121 ^{+20 }_{-15 } $ $39.5 \\pm 0.5 $ $25 \\pm 4 $ 140 NGC 4519 12:33:30.5 +08:39:16 19.6 [T08] 190 12.15 -19.31 7.0 $1218.1 \\pm 1.0 $ $112 ^{+8 }_{-7 } $ $355 \\pm 2 $ $42 \\pm 3 $ 218 NGC 4561 12:36:08.6 +19:19:26 12.3 [T88] 90.8 12.82 -17.63 8.0 $1402.2 \\pm 0.9 $ $57 ^{+6 }_{-5 } $ $227 \\pm 8 $ $34 \\pm 4 $ 171 NGC 4713 12:49:58.1 +05:18:39 14.9 [T08] 162 11.85 -19.02 7.0 $654.5 \\pm 0.5 $ $110.9 ^{+2 }_{-1.8 } $ $274.0 \\pm 0.6 $ $45.2 \\pm 1.2 $ 176 NGC 4942 13:04:19.2 -07:39:00 28.5 [T88] 112 13.27 -19.00 7.0 $1741 \\pm 2 $ $124 ^{+6 }_{-5 } $ $137.3 \\pm 0.8 $ $37 \\pm 2 $ 177 NGC 5964 15:37:36.3 +05:58:28 24.7 [T88] 250 12.28 -19.68 7.0 $1447.1 \\pm 1.2 $ $168 ^{+18 }_{-14 } $ $136.7 \\pm 1.2 $ $32 \\pm 3 $ 208 NGC 6509 17:59:24.9 +06:17:12 28.2 [T88] 95.1 12.12 -20.13 7.0 $1811.0 \\pm 0.4 $ $153 ^{+12 }_{-9 } $ $280.8 \\pm 1.1 $ $41 \\pm 4 $ 266 IC 1291 18:33:51.5 +49:16:45 31.5 [T88] 109 13.28 -19.21 8.0 $1951.0 \\pm 1.1 $ $189 ^{+17 }_{-14 } $ $131 \\pm 2 $ $28 \\pm 3 $ 209 Column 1: Object name; Column 2 and 3: Right ascension and declination (J2000.0) from [20]; Column 4: Distance and distance reference.", "Distances are derived using the Tully-Fisher relation, except for NGC 3906.", "T08: [83], T88: [82], and the NGC 3906 distance is from the [20] heliocentric velocity, corrected for Virgo infall using [55] and using $H_{0} = 71 \\, {\\rm km \\, s^{-1} \\,Mpc^{-1}}$ .", "Column 5: Major isophotal diameter at $25 \\, {\\rm mag\\, arcsec^{-2}}$ in the B band, from [20].", "Column 6: Apparent blue magnitude, corrected for Galactic and internal extinction and redshift.", "Values are from [20], except for NGC 4942 and PGC 6667, which are from [21] and are only corrected for Galactic extinction.", "Column 7: Absolute blue magnitude, calculated from the apparent magnitude in Column 6 and the distance in Column 4.", "Column 8: Morphological type from [20].", "Column 9: Systemic velocity, corrected to the heliocentric reference frame.", "Column 10: Circular velocity.", "Columns 9 and 10 were derived from the VLA H1 rotation curve analysis in Paper I.", "Column 11: Position angle of the major axis (degrees N to E to receding side).", "Column 12: Inclination.", "Columns 11 and 12 were derived from a combination of the rotation curve analyses on the H1 data and ellipse fits of the IRAC $3.6 \\,\\mu {\\rm m}$ data.", "Columns 9-12 were originally presented in Table 6 of Paper I.", "Column 13: Width of the HI line at 20% of the peak flux density, corrected for the spectral resolution but not turbulent broadening.", "The uncertainty in $W_{20}$ is $5\\, {\\rm km\\, s^{-1}}$ for all objects except UGC 6446 and NGC 3906, where the uncertainty is $3 \\, {\\rm km \\, s^{-1}}$ and $10\\, {\\rm km \\,s^{-1}}$ , respectively.", "Column 13 was originally presented in Table 4 of Paper I. lcccccccccccc 0pt Measured and Literature Properties Source RA DEC $F_{\\rm H\\alpha }$ $F_{\\rm PAH}$ $\\langle I_{\\rm HI} \\rangle $ $B_{\\rm maj}$ $B_{\\rm min}$ $I_{\\rm CO}$ $F_{4.5}^{21}$ $F_{4.5}^{D25}$ $J$ $K_{\\rm s}$ (hh:mm:ss.s) (dd:mm:ss) ($10^{-14} \\, {\\rm erg \\, s^{-1} \\, cm^{-2}}$ ) (mJy) (${\\rm Jy \\, beam^{-1} \\, km \\, s^{-1}}$ ) (arcsec) (arcsec) (${\\rm K \\, km\\, s^{-1}}$ ) (mJy) (mJy) (mag) (mag) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) NGC 0337 00:59:49.9 -07:34:44 $ 10.2 \\pm 0.05 $ $ 73 \\pm 8 $ $1.28 \\pm 0.17 $ 25.77 15.06 $ 4.7 \\pm 0.3 $ $ 11.7 \\pm 1.2 $ $64 \\pm 6 $ $ 9.876 \\pm 0.025 $ $ 9.059 \\pm 0.045$ PGC 3853 01:05:04.9 -06:12:45 $ 0.98 \\pm 0.04 $ $ 6.9 \\pm 0.8 $ $0.36 \\pm 0.05 $ 25.33 15.47 $ 1.39 \\pm 0.18 $ $ 3.4 \\pm 0.3 $ $44 \\pm 4 $ $ 10.031 \\pm 0.037 $ $ 9.280 \\pm 0.080$ PGC 6667 01:49:10.3 -10:03:40 $ 2.37 \\pm 0.02 $ $ 8.5 \\pm 0.9 $ $0.90 \\pm 0.12 $ 26.00 21.00 $ 0.71 \\pm 0.13 $ $ 2.4 \\pm 0.2 $ $15.2 \\pm 1.5 $ $ 11.678 \\pm 0.037 $ $ 10.951 \\pm 0.080$ ESO 544-G030 02:14:56.8 -20:12:44 $ 0.85 \\pm 0.02 $ $ 5.3 \\pm 0.6 $ $0.59 \\pm 0.08 $ 27.78 12.01 $ <0.4 $ $ 2.5 \\pm 0.3 $ $12.8 \\pm 1.3 $ $ ... $ $ ... $ UGC 1862 02:24:24.8 -02:09:44 $ 1.97 \\pm 0.02 $ $ 4.2 \\pm 0.5 $ $0.24 \\pm 0.03 $ 23.97 15.32 $ 0.60 \\pm 0.10 $ $ 2.2 \\pm 0.2 $ $12.4 \\pm 1.2 $ $ 11.927 \\pm 0.023 $ $ 11.177 \\pm 0.054$ ESO 418-G008 03:31:30.7 -30:12:48 $ 1.84 \\pm 0.03 $ $ 6.7 \\pm 0.8 $ $0.92 \\pm 0.12 $ 32.00 21.00 $ <0.9 $ $ 2.7 \\pm 0.3 $ $6.7 \\pm 0.7 $ $ 12.752 \\pm 0.049 $ $ 12.169 \\pm 0.126$ ESO 555-G027 06:03:36.8 -20:39:15 $ 1.42 \\pm 0.05 $ $ 9.3 \\pm 1.0 $ $0.44 \\pm 0.06 $ 29.00 21.00 $ ... $ $ 2.4 \\pm 0.2 $ $16.1 \\pm 1.6 $ $ 11.984 \\pm 0.045 $ $ 11.271 \\pm 0.103$ NGC 2805 09:20:20.3 +64:06:11 $ 2.53 \\pm 0.017 $ $ 15.5 \\pm 1.7 $ $0.34 \\pm 0.04 $ 21.00 21.00 $ 2.34 \\pm 0.10 $ $ 4.2 \\pm 0.4 $ $41 \\pm 4 $ $ 10.827 \\pm 0.026 $ $ 10.117 \\pm 0.046$ ESO 501-G023 10:35:23.3 -24:45:15 $ 0.76 \\pm 0.03 $ $ 1.04 \\pm 0.14 $ $0.134 \\pm 0.017 $ 30.30 12.51 $ <0.6 $ $ 0.90 \\pm 0.09 $ $9.9 \\pm 1.0 $ $ ... $ $ ... $ UGC 6446 11:26:40.4 +53:44:48 $ 1.19 \\pm 0.02 $ $ 0.44 \\pm 0.09 $ $0.43 \\pm 0.06 $ 21.00 21.00 $ <0.4 $ $ 0.97 \\pm 0.10 $ $9.5 \\pm 1.0 $ $ ... $ $ ... $ NGC 3794 11:40:54.3 +56:12:07 $ 2.37 \\pm 0.016 $ $ 7.3 \\pm 0.9 $ $0.49 \\pm 0.06 $ 21.00 21.00 $ 0.72 \\pm 0.15 $ $ 3.5 \\pm 0.4 $ $13.0 \\pm 1.3 $ $ 11.871 \\pm 0.039 $ $ 11.005 \\pm 0.072$ NGC 3906 11:49:39.9 +48:25:32 $ 2.25 \\pm 0.02 $ $ 7.7 \\pm 0.9 $ $0.25 \\pm 0.03 $ 21.00 21.00 $ 0.37 \\pm 0.12 $ $ 2.8 \\pm 0.3 $ $14.0 \\pm 1.4 $ $ 11.788 \\pm 0.045 $ $ 11.007 \\pm 0.067$ UGC 6930 11:57:17.4 +49:16:58 $ 0.76 \\pm 0.03 $ $ 6.8 \\pm 0.8 $ $0.25 \\pm 0.03 $ 21.00 21.00 $ 1.31 \\pm 0.19 $ $ 2.6 \\pm 0.3 $ $29 \\pm 3 $ $ 11.752 \\pm 0.037 $ $ 11.153 \\pm 0.068$ NGC 4519 12:33:30.3 +08:39:18 $ 3.0 \\pm 0.03 $ $ 44 \\pm 5 $ $0.99 \\pm 0.13 $ 51.91 18.77 $ 2.9 \\pm 0.3 $ $ 6.9 \\pm 0.7 $ $36 \\pm 4 $ $ 10.499 \\pm 0.037 $ $ 9.548 \\pm 0.059$ NGC 4561 12:36:08.2 +19:19:22 $ 4.55 \\pm 0.03 $ $ 7.9 \\pm 1.0 $ $1.02 \\pm 0.13 $ 21.00 21.00 $ 0.68 \\pm 0.12 $ $ 4.1 \\pm 0.4 $ $14.8 \\pm 1.5 $ $ 11.480 \\pm 0.046 $ $ 10.617 \\pm 0.074$ NGC 4713 12:49:58.0 +05:18:41 $ 5.35 \\pm 0.03 $ $ 35 \\pm 4 $ $0.69 \\pm 0.09 $ 21.00 21.00 $ 4.5 \\pm 0.2 $ $ 7.1 \\pm 0.7 $ $44 \\pm 4 $ $ 10.367 \\pm 0.026 $ $ 9.737 \\pm 0.053$ NGC 4942 13:04:19.1 -07:38:58 $ 3.55 \\pm 0.04 $ $ 10.1 \\pm 1.1 $ $0.49 \\pm 0.06 $ 22.00 21.00 $ 1.4 \\pm 0.2 $ $ 3.4 \\pm 0.3 $ $12.9 \\pm 1.3 $ $ 11.617 \\pm 0.058 $ $ 10.428 \\pm 0.092$ NGC 5964 15:37:36.2 +05:58:27 $ 2.11 \\pm 0.05 $ $ 9.0 \\pm 1.0 $ $0.30 \\pm 0.04 $ 21.00 21.00 $ 0.89 \\pm 0.17 $ $ 3.4 \\pm 0.3 $ $40 \\pm 4 $ $ 12.384 \\pm 0.058 $ $ 11.794 \\pm 0.113$ NGC 6509 17:59:25.2 +06:17:11 $ 4.62 \\pm 0.03 $ $ 41 \\pm 5 $ $0.50 \\pm 0.07 $ 21.00 21.00 $ 6.0 \\pm 0.2 $ $ 10.6 \\pm 1.1 $ $42 \\pm 4 $ $ 10.284 \\pm 0.024 $ $ 9.663 \\pm 0.043$ IC 1291 18:33:52.5 +49:16:42 $ 6.02 \\pm 0.02 $ $ 9.3 \\pm 1.0 $ $0.70 \\pm 0.09 $ 21.00 21.00 $ 0.37 \\pm 0.08 $ $ 2.4 \\pm 0.2 $ $10.2 \\pm 1.0 $ $ 13.125 \\pm 0.052 $ $ 12.689 \\pm 0.152$ Column 1: Object name.", "Column 2 and 3: RA and DEC (J2000.0) of the pointing center of the IRAM 30m CO(1-0) observations.", "All measurements within a 21$$ -diameter circular aperture are centered on these coordinates.", "Column 4: H$\\alpha $ flux within a 21$$ -diameter circular aperture, corrected for Galactic extinction and N2 emission.", "We quote only measurement uncertainties here.", "Column 5: PAH flux density within a 21$$ -diameter circular aperture.", "Column 6: Average integrated H1 line intensity within a 21$$ -diameter circular aperture, measured from image convolved to have beam major and minor axes given in Column 7 and 8.", "Column 7 and 8: Beam major and minor axes of image used for HI line intensity measurement.", "Column 9: Integrated CO(1-0) line intensity.", "Non-detections are quoted as $3 \\sigma $ upper limits.", "Column 10: 4.5$\\mu {\\rm m}$ flux density within a 21$$ -diameter circular aperture.", "Column 11: Total 4.5$\\mu {\\rm m}$ flux density.", "Column 12: Total $J$ -band magnitude from 2MASS, corrected for Galactic extinction.", "Column 13: Total $K_{\\rm s}$ -band magnitude from 2MASS, corrected for Galactic extinction.", "lcccc 5 0pt H$\\alpha $ Observations Galaxy $t_{663nb15}$ (min) Run(s) $t_{693nb15}$ (min) Run(s) (1) (2) (3) (4) (5) NGC0337 2400 3 2400 3 PGC3853 1800 4 1800 1 PGC6667 4500 1,3 4500 1,3 ESO544-G030 5400 1,3 5400 1,3 UGC1862 4500 1,3 5400 1,3 ESO418-G008 2700 1 2700 1 ESO555-G027 2700 1 2700 1 NGC2805 2400 3 2400 3 ESO501-G023 3600 1 3600 1 UGC6446 3900 2 4200 2 NGC3794 6000 1,3 6000 1,3 NGC3906 3200 1,2,3 3000 1,2 UGC6930 3600 2 3600 2 NGC4519 3600 2 3600 2 NGC4561 3600 1 3600 1 NGC4713 2400 1,2 2400 1,2 NGC4942 4500 1,2 3600 2 NGC5964 3600 2 3600 2 NGC6509 4800 2 3600 2 IC1291 3600 2 3600 2 Column 1: Object name.", "Column 2: Total exposure time in the 663nb15 filter.", "Column 3: Observing run code.", "Column 4: Total exposure time in the 693nb15 filter.", "Column 5: Observing run code.", "Observing run codes are: (1) Jan 2007; (2) May/June 2007; (3) Nov 2007; (4) Jan 2008. lccccccccccccc 0pt Derived Properties Source $R_{21}$ ${\\rm log} \\, \\Sigma _{\\rm HI}$ ${\\rm log} \\, \\Sigma _{\\rm H2}$ ${\\rm log} \\, \\Sigma _{\\rm SFR}$ ${\\rm log} \\, \\Sigma _{\\ast }$ ${\\rm log} \\, M_{\\ast }$ $12 + {\\rm log}(O/H)$ $l_{\\ast }$ $\\kappa $ $Q_{\\rm gas}$ $Q_{\\ast }$ $Q_{\\rm gas+stars}$ ${\\rm log} \\, P_{\\rm h}$ (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) NGC 0337 2.11 1.33 1.18 -1.67 2.49 9.9 8.93 1.7 0.17 2.7 2.9 2.0 5.32 PGC 3853 1.16 0.78 0.67 -2.68 1.88 9.1 8.68 1.6 0.22 12 8 5 4.42 PGC 6667 2.51 1.09 0.42 -2.44 1.73 9.3 8.75 2.4 0.12 4.8 6 3.2 4.50 ESO 544-G030 1.42 1.02 $<0.09$ -2.83 1.51 8.6 8.46 1.1 0.12 6.0 5 2.9 4.39 UGC 1862 2.27 0.62 0.29 -2.70 1.67 9.2 8.70 1.5 0.07 7 3.0 2.3 4.04 ESO 418-G008 2.40 0.84 $<0.34$ -2.72 1.40 8.7 8.50 1.1 0.07 4.5 3 2.0 4.21 ESO 555-G027 2.47 0.78 ... -2.44 1.77 9.3 8.74 1.9 0.12 ... 5 ... ... NGC 2805 2.85 0.74 0.92 -2.28 1.93 9.8 8.91 2.3 0.041 1.8 1.6 1.1 4.52 ESO 501-G023 0.71 0.40 $<0.32$ -3.16 1.01 7.8 8.00 0.72 0.12 16 8 5 3.77 UGC 6446 1.83 0.73 $<-0.008$ -3.21 1.01 8.6 8.47 1.1 0.07 6.5 5 2.9 3.88 NGC 3794 1.96 0.75 0.27 -2.64 1.93 9.2 8.72 1.5 0.15 12 5 3.9 4.26 NGC 3906 1.86 0.69 0.20 -2.42 1.94 9.1 8.67 0.93 0.09 8 2.1 1.8 4.27 UGC 6930 1.73 0.66 0.72 -2.63 1.62 9.1 8.66 2.1 0.18 11 10 6 4.24 NGC 4519 2.00 0.82 0.99 -1.93 2.46 9.8 8.90 2.0 0.22 8 4 3.7 4.83 NGC 4561 1.25 1.23 0.40 -2.32 2.15 8.9 8.58 0.69 0.14 4.2 2.3 1.7 4.98 NGC 4713 1.52 0.99 1.15 -1.98 1.99 9.2 8.70 1.4 0.25 6.3 7 4.0 4.94 NGC 4942 2.90 0.88 0.69 -2.35 2.52 10.0 8.96 1.8 0.12 5.8 2.1 1.9 4.72 NGC 5964 2.52 0.71 0.53 -2.44 1.70 9.5 8.82 3.4 0.055 3.9 3 2.2 4.11 NGC 6509 2.87 0.89 1.31 -1.92 2.18 9.7 8.88 1.7 0.22 4.7 6 3.3 5.06 IC 1291 3.21 1.10 0.17 -2.19 1.35 8.9 8.60 2.1 0.18 8 13 5 4.37 Column 1: Object name.", "Column 2: Physical size of 21 (kpc).", "All surface density measurements are within a 21-diameter aperture.", "Column 3: HI mass surface density ($M_{\\odot } \\, {\\rm pc^{-2}}$ ).", "The typical uncertainty is $0.06 \\,{\\rm dex}$ .", "Column 4: ${\\rm H_{2}}$ mass surface density ($M_{\\odot }\\, {\\rm pc^{-2}}$ ), computed with $X_{\\rm CO} = 2.8 \\times 10^{20} \\,{\\rm cm^{-2} \\, (K \\, km \\, s^{-1})^{-1}}$ .", "The typical uncertainty is $0.07 \\, {\\rm dex}$ .", "Column 5: SFR surface density, computed assuming the Kroupa-type IMF from [14] ($M_{\\odot } \\, {\\rm yr^{-1} \\, kpc^{-2}}$ ).", "We assume a typical uncertainty of $0.2 \\,{\\rm dex}$ .", "Column 6: Stellar mass surface density ($M_{\\odot } \\,{\\rm pc^{-2}}$ ).", "We assign a a typical uncertainty of $0.2 \\, {\\rm dex}$ .", "Column 7: Total stellar mass ($M_{\\odot }$ ).", "We assign typical uncertainty of $0.3 \\, {\\rm dex}$ .", "Column 8: Oxygen abundance derived from the total stellar mass and the mass-metallicity relation of [81].", "We derive a typical statistical uncertainty of $0.11 \\, {\\rm dex}$ , but note that strong-line metallicity methods return $12+{\\rm log}(O/H)$ values as disparate as $0.5 \\, {\\rm dex}$ .", "Column 9: Stellar scale length (kpc).", "We estimate the uncertainty to be 20%.", "Column 10: Epicyclic frequency (km/s/pc), evaluated at $5.25$ .", "We estimate the uncertainty to be 30%.", "Column 11: Gas stability parameter.", "The typical uncertainty is 30%.", "Column 12: Stellar stability parameter.", "The typical uncertainty is 60%.", "Column 13: Combined gas and stellar stability parameter.", "The typical uncertainty is 40%.", "Column 14: Mid-plane pressure (${\\rm K \\,cm^{-3}}$ ).", "The typical uncertainty is $0.17 \\, {\\rm dex}$ ." ] ]

[ [ "The Exponential Mechanism for Social Welfare: Private, Truthful, and\n Nearly Optimal" ], [ "Abstract In this paper we show that for any mechanism design problem with the objective of maximizing social welfare, the exponential mechanism can be implemented as a truthful mechanism while still preserving differential privacy.", "Our instantiation of the exponential mechanism can be interpreted as a generalization of the VCG mechanism in the sense that the VCG mechanism is the extreme case when the privacy parameter goes to infinity.", "To our knowledge, this is the first general tool for designing mechanisms that are both truthful and differentially private." ], [ "Introduction", "In mechanism design a central entity seeks to allocate resources among a set of selfish agents in order to optimize a specific objective function such as revenue or social welfare.", "Each agent has a private valuation for the resources being allocated, which is commonly referred to as her type.", "A major challenge in designing mechanisms for problems of resource allocation among selfish agents is getting them to reveal their true types.", "While in principle mechanisms can be designed to optimize some objective function even when agents are not truthful, the analysis of such mechanisms is complicated and the vast majority of mechanisms are designed to incentivize agents to be truthful.", "One reason that an agent might not want to be truthful is that lying gives her a better payoff.", "Research in algorithmic mechanism design has mostly focused on this possibility and has successfully designed computationally-efficient incentive-compatible mechanisms for many problems , i.e., mechanisms where each agent achieves optimal payoff by bidding truthfully (see [24] for a survey of results).", "However, a second reason that an agent might not bid truthfully is that the privacy of her type might itself be of value to her.", "Bidding truthfully could well result in an outcome that reveals the private type of an agent.", "Consider for example, a matching market in which $n$ oil companies are bidding for $n$ oil fields.", "A company may have done extensive research in figuring out its valuations for each field.", "It may regard this information as giving it competitive advantage and seek to protect its privacy.", "If it participates in a traditional incentive-compatible mechanism, say, the VCG mechanism, it has two choices – 1) bid truthfully, get the optimum payoff but potentially reveal private information or 2) introduce random noise into its bid to (almost) preserve privacy, but settle for a suboptimal payoff.", "In this and more generally in multi-agent settings where each agent's type is multidimensional, we aim to answer the following question: Can we design mechanisms that simultaneously achieve near optimal social welfare, are incentive compatible, and protect the privacy of each agent?", "The notion of privacy we will consider is differential privacy, which is a paradigm for private data analysis developed in the past decade, aiming to reveal information about the population as a whole, while protecting the privacy of each individual (E.g., see surveys [13], [14] and the reference therein)." ], [ "Our Results and Techniques", "Our main contribution is a novel instantiation of the exponential mechanism for any mechanism design problem with payments, that aims to maximize social welfare.", "We show that our version of the exponential mechanism is incentive compatible and individually rationalHere, we consider individual rationality in expectation.", "Achieving individual rationality in the ex-post sense is impossible for any non-trivial private mechanism since the probability of a non-zero price would have to jump by an infinitely large factor as an agent changes from zero valuation to non-zero valuation., while preserving differential privacy.", "In fact, we show that the exponential mechanism can be interpreted as a natural generalization of the VCG mechanism in the sense that the VCG mechanism is the special case when the privacy parameter goes to infinity.", "Alternatively, our mechanism can be viewed as an affine maximum-in-distributed-range mechanism with Shannon entropy providing the offsets.", "We will formally define affine maximum-in-distributed-range mechanisms in sec:prelim and more details on this observation are deferred to sec:gibbs.", "Readers are referred to [8], [10], [11], [9] for recent applications of maximum-in-distributed-range mechanisms in algorithmic mechanism design.", "Our proof is by connecting the exponential mechanism to the Gibbs measure and free energy in statistical mechanics.", "We exploit this connection to provide a simple proof of the incentive compatibility of the mechanism.", "We believe this intriguing connection is of independent interest and may lead to new ways of understanding the exponential mechanism and differential privacy.", "While we do not have an efficient way of computing the allocation and prices of the exponential mechanism in general (this is also not known for VCG), we do show that in special cases such as multi-item auctions and procurement auctions for spanning tree, we can efficiently implement the exponential mechanism either exactly or approximately.", "Further, we show that the trade-off between privacy and social welfare in the exponential mechanism is asymptotically optimal in these two cases, even if we compare to mechanisms that need not be truthful.", "We also include another application of the exponential mechanism for the combinatorial public project problem where the social welfare is close to optimal for an arbitrarily small constant $\\epsilon $ .", "Interestingly, our implementation of the exponential mechanism for multi-item auctions has further implications in the recent work on blackbox reductions in Bayesian mechanism design [17], [3].", "Combining our exponential mechanism for the matching market with the blackbox reduction procedure in [17], [3], we can get a blackbox reduction that converts any algorithm into BIC, differentially private mechanisms.", "We will leave further discussions to the relevant section.", "McSherry and Talwar [23] first proposed using differentially private mechanisms to design auctions by pointing out that differential privacy implies approximate incentive compatibility as well as resilience to collusion.", "In particular, they study the problem of revenue maximization in digital auctions and attribute auctions.", "They propose the exponential mechanism as a solution for these problems.", "McSherry and Talwar also suggest using the exponential mechanism to solve mechanism design problems with different objectives, such as social welfare.The main difference between our instantiation of the exponential mechanism and that by McSherry and Talwar is that we use properly chosen payments to incentivize agents to report truthfully.", "Their instantiation of the exponential mechanism is differentially private, but only approximately truthful.", "Nissim et al.", "[25] show how to convert differentially private mechanisms into exactly truthful mechanism in some settings.", "However, the mechanism loses its privacy property after such conversion.", "Xiao [29] seeks to design mechanisms that are both differentially private and perfectly truthful and proposes a method to convert any truthful mechanism into a differentially private and truthful one when the type space is small.", "Unfortunately, it does not seem possible to extend the results in [25], [29] to more general mechanism design problems, while our result applies to any mechanism design problem (with payments).", "Xiao [29] also proposed to explicitly model the agents' concern for privacy in the utilities by assuming agent $i$ has a disutility that depends on the amount of information $\\epsilon _i$ leaked by the mechanism.", "Chen et al.", "[7] and Nissim et al.", "[25] explored this direction and introduced truthful mechanisms for some specific problems.", "Exact evaluation of an agent's dis-utility usually requires knowledge of the types of all agents and hence this kind of mechanism can only be private if agents do not need to exactly compute their own dis-utility.", "The above works circumvent this issue by designing strictly truthful and sufficiently private mechanisms such that any agent's gain in privacy by lying is outweighed by the loss in the usual notion of utility, regardless of the exact value of dis-utility for privacy.", "Finally, Ghosh and Roth [16] study the problem of selling privacy in auctions, which can be viewed as an orthogonal approach to combining mechanism design and differential privacy." ], [ "Preliminaries", "A mechanism design problem is defined by a set of $n$ agents and a range $R$ of feasible outcomes.", "Throughout this paper we will assume the range $R$ to be discrete, but all our results can be easily extended to continuous ranges with appropriate integrability.", "Each agent $i$ has a private valuation $v_i : R \\mapsto [0, 1]$ .", "A central entity chooses one of the outcomes based on the agents' (reported) valuations.", "We will let $\\mathbf {0}$ denote the all-zero valuation and let $v_{-i}$ denote the valuations of every agent except $i$ .", "For the sake of presentation, we will assume that the agents' valuations can be any functions mapping the range of feasible outcomes to the interval $[0, 1]$ .", "It is worth noting that since our mechanisms are incentive compatible in this setting, they are also automatically incentive compatible for more restricted valuations (e.g., submodular valuations for a combinatorial public project problem).", "A mechanism $M$ consists of an allocation rule $x(\\cdot )$ and a payment rule $p(\\cdot )$ .", "The mechanism first lets the agents submit their valuations.", "However, an agent may strategically submit a false valuation if that is beneficial to her.", "We will let $b_1, \\dots , b_n: R\\mapsto [0, 1]$ denote the reported valuations (bids) from the agents and let $b$ denote the vector of these valuations.", "After the agents submit their bids, the allocation rule $x(\\cdot )$ chooses a feasible outcome $r = x(b) \\in R$ and the payment rule $p(\\cdot )$ chooses a vector of payments $p(b) \\in \\mathbb {R}^n$ .", "We will let $p_i(b)$ denote the payment for agent $i$ .", "Note that both $x(\\cdot )$ and $p(\\cdot )$ may be randomized.", "We will consider the standard setting of quasi-linear utility: given the allocation rule, the payment rule, and the reported valuations $b$ , for each $i \\in [n]$ , the utility of agent $i$ is $u_i(v_i, x(b), p_i(b)) = v_i(x(b)) - p_i(b) \\hspace{5.0pt}.$ We will assume the agents are risk-neutral and aim to maximize their expected utilities.", "The goal is to design polynomial time mechanisms $M$ that satisfy various objectives.", "In this paper, we will focus on the problem of maximizing the expected social welfare, which is defined to be the sum of the agents' valuations: $\\mathop {\\bf E}[\\sum _{i=1}^n v_i(x(b))]$ .", "Besides the expected social welfare, we take into consideration the strategic play of utility-maximizing agents and their concern about the mechanism leaking non-trivial information about their private data.", "Thus, we will restrict our attention to mechanisms that satisfy several game-theoretic requirements and have a privacy guarantee that we will define in the rest of this section." ], [ "Game-Theoretical Solution Concepts", "A mechanism is incentive compatible (IC) if truth-telling is a dominant strategy, i.e., by reporting the true values an agent always maximizes her expected utility regardless of what other agents do - $v_i\\in \\arg \\max _{b_i} \\mathop {\\bf E}[v_i(x(b_i, b_{-i})) - p_i(b_i, b_{-i})]$ .", "We will also consider an approximate notion of truthfulness.", "A mechanism is $\\gamma $ -incentive compatible ($\\gamma $ -IC) if no agent can get more than $\\gamma $ extra utility by lying.", "Further, a mechanism is individually rational (IR) if the expected utility of each agent is always non-negative, assuming this agent reports truthfully: $\\mathop {\\bf E}[v_i(x(v_i, b_{-i})) - p_i(v_i, b_{-i})] \\ge 0$ .", "We seek to design mechanisms that are incentive compatible and individually rational." ], [ "Affine Maximum-In-Distributed-Range", "An allocation rule $x(\\cdot )$ is an affine maximum-in-distributed-range allocation if there is a set $S$ of distributions over feasible outcomes, parameters $a_1, \\dots , a_n \\in \\mathbb {R}^+$ , and an offset function $c : S \\mapsto \\mathbb {R}$ , such that the $x(v_1, \\dots , v_n)$ always chooses the distribution $\\nu \\in S$ that maximizes $\\mathop {\\bf E}_{r \\sim \\nu } \\left[ \\sum ^n_{i = 1} a_i v_i(r) \\right] + c(\\nu ) \\hspace{5.0pt}.$ In this paper, we are particularly interested in the case when $a_i = 1$ , $\\forall i \\in [n]$ , and $c$ is the Shannon entropy of the distribution scaled by an appropriate parameter.", "The affine maximum-in-distributed-range mechanisms can be interpreted as slight generalizations of the well-studied maximum-in-distributed-range mechanisms.", "If $a_i = 1$ for every $i \\in [n]$ and $c(\\cdot ) = 0$ , then such allocation rules are referred to as maximum-in-distributed-range (MIDR) allocations.", "There are well-known techniques for charging proper prices to make MIDR allocations and their affine generalizations incentive compatible.", "The resulting mechanisms are called MIDR mechanisms.", "MIDR mechanisms are important tools for designing computationally efficient mechanisms that are incentive compatible and approximate social welfare well (e.g., see [8], [10], [9], [11])." ], [ "Differential Privacy", "Differential privacy is a notion of privacy that has been studied the most in the theoretical computer science community over the past decade.", "It requires the distribution of outcomes to be nearly identical when the agent profiles are nearly identical.", "Formally, Definition 1 A mechanism is $\\epsilon $ -differentially private if for any two valuation profiles $v= (v_1, \\dots , v_n)$ and $v^{\\prime } = (v_1^{\\prime },\\dots , v_n^{\\prime })$ such that only one agent has different valuations in the two profiles, and for any set of outcomes $S \\subseteq R$ , we have $\\mathop {\\bf Pr}[x(v) \\in S] \\le \\exp (\\epsilon ) \\cdot \\mathop {\\bf Pr}[x(v^{\\prime })\\in S] \\hspace{5.0pt}.$ This definition of privacy has many appealing theoretical properties.", "Readers are referred to [13], [14] for excellent surveys on the subject.", "We will also consider a standard variant that defines a more relaxed notion of privacy.", "Definition 2 A mechanism is $(\\epsilon , \\delta )$ -differentially private if for any two valuation profile $v= (v_1, \\dots , v_n)$ and $v^{\\prime } =(v_1^{\\prime }, \\dots , v_n^{\\prime })$ such that only one agent has different valuations in the two profiles, and for any set of outcomes $S\\subseteq R$ , $\\mathop {\\bf Pr}[x(v) \\in S] \\le \\exp (\\epsilon ) \\cdot \\mathop {\\bf Pr}[x(v^{\\prime }) \\in S] + \\delta \\hspace{5.0pt}.$ Typically, we will consider very small values of $\\delta $ , say, $\\delta =\\exp (-n)$ ." ], [ "Differentially Private Payment", "In the above definitions, we only consider the privacy of the allocation rule.", "We note that in practice, the payments need to be differentially private as well.", "We can handle privacy issues in the payments by the standard technique of adding Laplace noise.", "In particular, if the payments are implemented via secure channels (e.g., the same channels that the agents use to submit their bids) such that the each agent's payment is accessible only by the agent herself and the central entity, then adding independent Laplace noise with standard deviation $O(\\epsilon ^{-1})$ is sufficient to guarantee $\\epsilon $ -differentially private payments.", "Since the techniques used to handle payments are quite standard, we will defer the extended discussion of this subject to the appendix." ], [ "The Exponential Mechanism", "One powerful tool in the differential privacy literature is the exponential mechanism of McSherry and Talwar [23].", "The exponential mechanism is a general technique for constructing differentially private algorithms over an arbitrary range $R$ of outcomes and any objective function $Q(D, r)$ (often referred to as the quality function in the differential privacy literature) that maps a pair consisting of a data set $D$ and a feasible outcome $r \\in R$ to a real-valued score.", "In our setting, $D$ is a (reported) valuation profile and the quality function $Q(v, r) = \\sum _{i=1}^n v_i(r)$ is the social welfare.", "Given a range $R$ , a data set $D$ , a quality function $Q$ , and a privacy parameter $\\epsilon $ , the exponential mechanism $\\textsc {Exp}(R, D, Q, \\epsilon )$ chooses an outcome $r$ from the range $R$ with probability $\\mathop {\\bf Pr}\\left[\\textsc {Exp}(R, D, Q, \\epsilon ) = r\\right] \\propto \\exp \\left(\\frac{\\epsilon }{2\\Delta } Q(D, r)\\right) \\hspace{5.0pt},$ where $\\Delta $ is the Lipschitz constant of the quality function $Q$ , that is, for any two adjacent data set $D_1$ and $D_2$ , and for any outcome $r$ , the score $Q(D_1, r)$ and $Q(D_2, r)$ differs by at most $\\Delta $ .", "In out setting, the Lipschitz constant of the social welfare function is 1.", "We sometimes use $\\textsc {Exp}(D, \\epsilon )$ for short when the range $R$ and the quality function $Q$ is clear from the context.", "We will use the following theorem about the exponential mechanism.", "Theorem 1 (E.g., [23], [28]) The exponential mechanism is $\\epsilon $ -differentially private and ensures that $\\mathop {\\bf Pr}\\left[ Q(D, \\textsc {Exp}(D, \\epsilon )) < \\max _{r \\in R} Q(D, r) - \\frac{\\ln \\left|R \\right|}{\\epsilon } - \\frac{t}{\\epsilon } \\right] \\le \\exp (-t) \\hspace{5.0pt}.$" ], [ "The Exponential Mechanism is Incentive Compatible", "In this section, we will show that if we choose the social welfare to be the quality function, then the exponential mechanism can be implemented in an incentive compatible and individually rational manner.", "Formally, for any range $R$ and any privacy parameter $\\epsilon > 0$ , the exponential mechanism $\\textsc {Exp}^R_\\epsilon $ with its pricing scheme is presented in fig:expo.", "Our main theorem is the following: Figure: Exp ϵ R \\textsc {Exp}^R_\\epsilon : the incentive-compatible exponentialmechanism.Theorem 2 The exponential mechanism with our pricing scheme is IC and IR.", "Our proof of thm:main relies on the connection between the exponential mechanism and a well known probability measure in probability and statistical mechanics called the Gibbs measure.", "Once we have established this connection, the proof of thm:main becomes very simple." ], [ "The Exponential Mechanism and the Gibbs Measure", "The Gibbs measure, also known as the Boltzmann distribution in chemistry and physics, is formally defined as follows: Definition 3 (Gibbs measure) Suppose we have a system consisting of particles of a gas.", "If the particles have $k$ states $1, \\dots , k$ , possessing energy $E_1, \\dots , E_k$ respectively, then the probability that a random particle in the system has state $i$ follows the Gibbs measure: $\\mathop {\\bf Pr}[\\text{state} = i] \\propto \\exp \\left( - \\frac{1}{k_B T} E_i \\right) \\hspace{5.0pt},$ where $T$ is the temperature, and $k_B$ is the Boltzmann constant.", "Note that the Gibbs measure asserts that nature prefers states with lower energy level.", "Indeed, if $T \\rightarrow 0$ , then almost surely we will see a particle with lowest-energy state.", "On the other hand, if $T \\rightarrow +\\infty $ , then all states are equally likely to appear.", "Thus the temperature $T$ is a measure of uncertainty in the system: the lower the temperature, the less uncertainty in the system, and vice versa." ], [ "Gibbs Measure vs. Exponential Mechanism", "It is not difficult to see the analogy between the Gibbs measure and the exponential mechanism.", "Firstly, the quality $Q(r)$ of an outcome $r \\in R$ (in our instantiation, $Q(r)$ is the social welfare $\\sum _i v_i(r)$ ) is an analog of the energy (more precisely, the negative of the energy) of a state $i$ .", "In the exponential mechanism the goal is to maximize the expected quality of the outcome, while in physics nature tries to minimize the expected energy.", "Second, the privacy parameter $\\epsilon $ is an analogue of the inverse temperature $T^{-1}$ , both measuring the level of uncertainty in the system.", "The more privacy we want in the mechanism, the more uncertainty we need to impose in the distribution of outcomesWe note that the privacy guarantee $\\epsilon $ is not necessarily a monotone function of the entropy of the outcome distribution.", "So the statement above is only for the purpose of establishing a high-level connection between the Gibbs measure and the exponential mechanism..", "Finally, the Lipschitz constant $\\Delta $ and Boltzmann constant $k_B$ are both scaling factors that come from the environment.", "tab:gibbs summarize this connection between the Gibbs measure and the exponential mechanism.", "Table: A high-level comparison between the Gibbs measure and the exponential mechanism" ], [ "Gibbs Measure Minimizes Free Energy", "It is well-known that the Gibbs measure maximizes entropy given the expected energy.", "In fact, a slightly stronger claim (e.g., see [22]) states that the Gibbs measure minimizes free energy.", "To be precise, suppose $T$ is the temperature, $\\nu $ is a distribution over the states, and $S(\\nu )$ is the Shannon entropy of $\\nu $ .", "Then the free energy of the system is $F(\\nu , T) = \\mathop {\\bf E}_{i \\sim \\nu }[E_i] - k_B T \\cdot S(\\nu ) \\hspace{5.0pt}.$ The following result is well known in the statistical physics literature.", "Theorem 3 (E.g., see [22]) $F(\\nu , T)$ is minimized when $\\nu $ is the Gibbs measure.", "For self-containedness, we include the proof of thm:freeenergy as follows.", "Note that the free energy can be written as $F(\\nu , T) & = & \\mathop {\\bf E}_{i \\sim \\nu }[E_i] - k_B T \\cdot S(\\nu ) \\\\& = & \\sum _i \\mathop {\\bf Pr}_\\nu [i] E_i + k_B T \\sum _i \\mathop {\\bf Pr}_\\nu [i] \\ln \\mathop {\\bf Pr}_\\nu [i] \\hspace{5.0pt}.", "$ Further, the first term of the right hand side can be rewritten as $\\sum _i \\mathop {\\bf Pr}_\\nu [i] E_i & = & k_B T \\sum _i \\mathop {\\bf Pr}_\\nu [i] \\frac{1}{k_B T} E_i \\\\& = & - k_B T \\sum _i \\mathop {\\bf Pr}_\\nu [i] \\ln \\left(\\exp \\left(-\\frac{1}{k_B T} E_i \\right)\\right) \\\\& = & - k_B T \\sum _i \\mathop {\\bf Pr}_\\nu [i] \\ln \\left( \\frac{\\exp \\left( - \\frac{1}{k_B T} E_i \\right)}{\\sum _j \\exp \\left( - \\frac{1}{k_B T} E_j \\right)} \\right) - k_B T \\ln \\left( \\sum _j \\exp \\left( - \\frac{1}{k_B T} E_j \\right) \\right) \\\\& = & - k_B T \\sum _i \\mathop {\\bf Pr}_\\nu [i] \\ln \\left( \\mathop {\\bf Pr}_{\\textit {\\rm Gibbs}}[i] \\right) - k_B T \\ln \\left( \\sum _j \\exp \\left( - \\frac{1}{k_B T} E_j \\right) \\right) \\hspace{5.0pt}.", "$ By eq:2 and eq:3, the free energy equals $F(\\nu , T) = k_B T \\cdot D_{KL}(\\nu \\, || \\, \\textit {\\rm Gibbs}) - k_B T \\ln \\left( \\sum _j \\exp \\left( - \\frac{1}{k_B T} E_j \\right) \\right) \\hspace{5.0pt}.$ Note that the second term is independent of $\\nu $ .", "By basic properties of the KL-divergence, the above is minimized when $\\nu $ is the Gibbs measure." ], [ "Proof of thm:main", "By the connection between Gibbs measure and exponential mechanism and thm:freeenergy, we have the following analogous lemma for our instantiation of the exponential mechanism.", "Lemma 4 The free social welfare, $\\mathop {\\bf E}_{r \\sim \\nu }\\left[\\sum _i v_i(r)\\right] + \\frac{2}{\\epsilon } \\cdot S(\\nu ) ~,$ is maximized when $\\nu = \\textsc {Exp}^R_\\epsilon (v_1, \\dots , v_n)$ ." ], [ "Incentive Compatibility", "Let us consider a particular agent $i$ , and fix the bids $b_{-i}$ of the other agents.", "Suppose agent $i$ has value $v_i$ and bids $b_i$ .", "For notational convenience, we let $b(r) = \\sum _{k=1}^n b_k(r)$ and let $h_i(b_{-i}) = \\frac{2}{\\epsilon } \\ln \\left( \\sum _{r \\in R}\\exp \\left( \\frac{\\epsilon }{2} \\sum _{k \\ne i} v_k(r) \\right) \\right) \\hspace{5.0pt}.$ Using the price $p_i$ charged to agent $i$ as in fig:expo, her utility when she bids $b_i$ is $\\mathop {\\bf E}_{r \\sim \\textsc {Exp}^R_\\epsilon (b_i, b_{-i})}[v_i(r) + \\sum _{k \\ne i} b_k(r)] + \\frac{2}{\\epsilon } \\cdot S(\\textsc {Exp}^R_\\epsilon (b_i, b_{-i})) - h_i(b_{-i}) \\hspace{5.0pt},$ which equals the free social welfare plus a term that does not depend on agent $i$ 's bid.", "By lem:freeenergy, the free social welfare is maximized when we use the outcome distribution by the exponential mechanism with respect to agent $i$ 's true value.", "Therefore, truthful bidding is a utility-maximizing strategy for agent $i$ ." ], [ "Individual Rationality", "We first note that for any agent $i$ , it is not difficult to verify that $p_i = 0$ when $v_i = \\mathbf {0}$ regardless of bidding valuations of other agents.", "Therefore, by bidding $\\mathbf {0}$ agent $i$ could always guarantee non-negative expected utility.", "Since we have shown that the exponential mechanism is truthful-in-expectation, we get that the utility of agent $i$ when she truthfully reports her valuation is always non-negative.", "Remark 1 We notice that lem:freeenergy implies that the allocation rule of the exponential mechanism is affine maximum-in-distributed-range.", "As a result, there are standard techniques to charge prices so that the mechanisms is IC and IR as presented above.", "Remark 2 Alternatively, one can prove thm:main via the procedure developed by Rochet [27]: first prove the cyclic monotonicity of the exponential allocation rule, which is known to be the necessary and sufficient condition for being the allocation rule of a truthful mechanism; then derive the pricing scheme that rationalizes the exponential allocation rule via Rochet's characterization.", "We will omit further details of this proof in this extended abstract." ], [ "Generalization", "In the original definition by McSherry and Talwar [23], the exponential mechanism is defined with respect to a prior distribution $\\mu (\\cdot )$ over the feasible range $R$ .", "More precisely, the exponential mechanism given $\\mu $ , $\\textsc {Exp}_{\\mu }(R, D, Q, \\epsilon )$ , chooses an outcome $r$ from the range $R$ with probability $\\mathop {\\bf Pr}\\left[ \\textsc {Exp}_{\\mu }(R, D, Q, \\epsilon ) = r \\right] \\propto \\mu (r) \\exp \\left(\\frac{\\epsilon }{2\\Delta } Q(D, r) \\right) \\hspace{5.0pt}.$ When $\\mu $ is chosen to be the uniform distribution over the feasible range, we recover the definition in sec:prelim.", "Using a different $\\mu $ can improve computational efficiency as well as the trade-off between privacy and the objective for some problems (e.g., [5]).", "In every use of the (generalized) exponential mechanism, to our knowledge, $\\mu $ is taken to be the uniform distribution over a sub-range that forms a geometric covering of the feasible range.", "But in general, this need not be the optimal choice.", "We observe that our result can be extended to the above generalized exponential mechanism as well.", "More precisely, we can show that the generalized exponential mechanism is affine maximum-in-distributed-range as well.", "Theorem 5 For any range $R$ , any quality function $Q$ , any privacy parameter $\\epsilon $ , any prior distribution $\\mu $ , and any database $D$ , the generalized exponential mechanism satisfies $\\textsc {Exp}_{\\mu }(R, D, Q, \\epsilon ) = \\arg \\max _{\\nu } \\mathop {\\bf E}_{r \\sim \\nu }[Q(D, r)] - \\frac{2}{\\epsilon } D_{KL}(\\nu || \\mu ) \\hspace{5.0pt}.$ Corollary 6 For any mechanism design problem for social welfare and any prior distribution $\\mu $ over the feasible range, the generalized exponential mechanism (w.r.t.", "$\\mu $ ) is IC and IR with appropriate payment rule.", "The proof of thm:general and deriving the pricing scheme in cor:general is very similar to the corresponding parts in sec:expo and hence omitted." ], [ "Applications", "Our result in thm:main applies to a large family of problems.", "In fact, it can be used to derive truthful and differentially private mechanisms for any problem in mechanism design (with payments) that aims for social welfare maximization.", "In this section, we will consider three examples – the combinatorial public project problem (CPPP), the multi-item auction, and the procurement auction for a spanning tree.", "The exponential mechanism for the combinatorial public project problem is incentive compatible, $\\epsilon $ -differentially private, and achieves nearly optimal social welfare for any constant $\\epsilon > 0$ .", "However, we cannot implement the exponential mechanism in polynomial time for CPPP in general because implementing VCG for CPPP is known to be $\\textit {\\bf NP}$ -hard and the exponential mechanism is a generalization of VCG.", "For the other two applications, we manage to implement the exponential mechanism in polynomial time, where the implementation for multi-item auction is only approximate so that it is only approximately truthful and approximately differentially private, and the implementation for procurement auction for spanning trees is exact.", "The social welfare for these two cases, however, is nearly optimal only when the privacy parameter $\\epsilon $ is super-constantly large.", "Nonetheless, we show that the trade-offs between privacy and social welfare of the exponential mechanism in these two applications are asymptotically optimal." ], [ "Combinatorial Public Project Problem", "The first interesting application of our result is a truthful and differentially private mechanism for the Combinatorial Public Project Problem (CPPP) originally proposed by Papadimitriou et al. [26].", "In CPPP, there are $n$ agents and $m$ public projects.", "Each agent $i$ has a private valuation function $v_i$ that specifies agent $i$ 's value (between 0 and 1) for every subset of public projects.", "The objective is to find a subset $S$ of public projects to build, of size at most $k$ (a parameter), that maximizes the social welfare, namely, $\\sum _iv_i(S)$ .", "This problem has received a lot of attention in the algorithmic game theory literature because strong lower bounds can be shown for the approximation ratio of this problem by any truthful mechanism when the valuations are submodular (e.g., see [26], [12]).", "Further, the CPPP is of practical interest as well.", "The following is a typical CPPP scenario in the real world.", "Suppose some central entity (e.g., the government) wants to build several new hospitals where there are $m$ potential locations to choose from.", "Due to resource constraints, the government can only build $k$ hospitals.", "Each citizen has a private value for each subset of locations that may depend on the distance to the closest hospital and the citizen's health status.", "Note that the agents may be concerned about their privacy if they choose to participate in the mechanism because their valuations typically contain sensitive information.", "For example, the citizens who have high values for having a hospital close by in the above scenario are more likely to have health problems.", "Therefore, it would be interesting to design mechanisms for the CPPP that are not only truthful but also differentially private.", "The size of the range of outcomes is ${m \\atopwithdelims ()k} = O(m^k)$ .", "So by thm:expo and thm:main, we have the following.", "Theorem 7 For any $\\epsilon > 0$ , the exponential mechanism $\\textsc {Exp}^{\\textit {\\rm CPPP}}_{\\epsilon }$ for CPPP is IC, $\\epsilon $ -differentially private, and ensures $\\mathop {\\bf Pr}\\left[\\sum _{i=1}^n v_i \\left(\\textsc {Exp}^{\\textit {\\rm CPPP}}_\\epsilon \\right) < \\mathsf {opt}- \\frac{k \\ln m}{\\epsilon } -\\frac{t}{\\epsilon } \\right] \\le \\exp (-t) \\hspace{5.0pt}.$ It is known that the exponential mechanism achieves the optimal trade-off between privacy and social welfare for CPPP (e.g., [28]).", "Further, note that the optimal social welfare could be as large as $n$ .", "Moreover, the number of projects $k \\le m$ is typically much smaller than the number of agents $n$ .", "Therefore, the exponential mechanism achieves social welfare that is close to optimal.", "However, it is worth noting that we only requires $k$ and $m$ to be mildly smaller than $n$ (e.g., $O(n^{1-c})$ for any small constant $c > 0$ ), in which cases the size of the type space, which is exponential in $k$ and $m$ , is still quite large so that the approach in [29] does not apply.", "In some scenarios such as the one above where the government wants to build a few new hospitals, $k$ is sufficiently small so that it is acceptable to have running time polynomial in the size of the range of outcomes.", "In such cases, it is easy to see that the exponential mechanism for CPPP can be implemented in time polynomial in $n$ and ${m \\atopwithdelims ()k}$ ." ], [ "Multi-Item Auction", "Next we consider a multi-item auction.", "Here, the auctioneer has $n$ heterogeneous items (one copy of each item) that she wishes to allocate to $n$ different agentsThe case when the number of items is not the same as the number of agents can be reduced to this case by adding dummy items or dummy agents.", "So our setting is w.l.o.g..", "Agent $i$ has a private valuation $v_i = (v_{i1}, \\dots , v_{ik})$ , where $v_{ij}$ is her value for item $j$ .", "We will assume the agents are unit-demand, that is, each agent wants at most one item.", "It is easy to see that each feasible allocation of the multi-item auction is a matching between agents and items.", "We will let the $R_M$ denote the range of multi-item auction, that is, the set $\\Pi _n$ of all permutations on $[n]$ .", "The multi-item auction and related problems are very well-studied in the algorithmic game theory literature (e.g., [7], [4]).", "They capture the motivating scenario of allocating oil fields and many other problems that arise from allocating public resources.", "The VCG mechanism can be implemented in polynomial time to maximize social welfare in this problem since max-matching can be solved in polynomial time.", "The new twist in our setting is to design mechanisms that are both truthful and differentially private and have good social welfare guarantee." ], [ "Approximate Implementation of the Exponential Mechanism", "Unfortunately, exactly sampling matchings according to the distribution specified in the exponential mechanism seems hard due to its connection to the problem of computing the permanent of non-negative matrices (e.g., see [18]), which is $\\#P$ -complete.", "Instead, we will sample from the desired distribution approximately.", "Moreover, we show that there is an efficient approximate implementation of the payment scheme.", "As a result of the non-exact implementation, we only get $\\gamma $ -IC instead of perfect IC, $(\\epsilon , \\delta )$ -differential privacy instead of $\\epsilon $ -differential privacy, and lose an additional $n \\gamma $ additive factor in social welfare.", "Here, $\\gamma $ will be inverse polynomially small.", "The discussion of this approximate implementation of the exponential mechanism is deferred to the full version.", "Note that the size of the range of feasible outcomes of multi-item auction is $n!$ .", "By thm:expo, we have the following: Theorem 8 For any $\\delta \\in (0, 1)$ , $\\epsilon > 0$ , $\\gamma > 0$ , there is a polynomial time (in $n$ , $\\epsilon ^{-1}$ , $\\gamma ^{-1}$ , and $\\log (\\delta ^{-1})$ ) approximate implementation of the exponential mechanism, $\\widehat{\\textsc {Exp}}^{R_M}_\\epsilon $ that is $\\gamma $ -IC, $(\\epsilon ,\\delta )$ -differentially private, and ensures that $\\mathop {\\bf Pr}\\left[\\sum _{i=1}^n v_i \\left( \\widehat{\\textsc {Exp}}^{R_M}_\\epsilon \\right) < \\mathsf {opt}- \\gamma n - \\frac{\\ln (n!", ")}{\\epsilon } - \\frac{t}{\\epsilon } \\right] \\le \\exp (-t) \\hspace{5.0pt}.$ Note that here we are achieving $\\gamma $ -IC and $(\\epsilon , \\delta )$ -differentially privacy while in the instantiation of the exponential mechanism by McSherry and Talwar [23] is $\\epsilon $ -IC and $\\epsilon $ -differentially private.", "Our result in thm:matching is better in most applications since typically $\\epsilon $ is large, usually a constant or occasionally a super-constant, while $\\gamma $ is small, usually requires to be $1 / \\textrm {poly}$ for $\\gamma $ -IC to be an appealing solution concept.", "The trade-off between privacy and social welfare in thm:matching can be interpreted as the follows: if we want to achieve social welfare that is worse than optimal by at most an $O(n)$ additive term, then we need to choose $\\epsilon = \\Omega (\\log n)$ .", "The next theorem shows that this is tight.", "The proof is deferred to the full version.", "Theorem 9 Suppose $M$ is an $\\epsilon $ -differentially private mechanism for the multi-item auction problem and the expected welfare achieve by $M$ is at least $\\mathsf {opt}- \\frac{n}{10}$ .", "Then $\\epsilon = \\Omega (\\log n)$ .", "Note that in this theorem, we do not restrict $M$ to be incentive compatible.", "In other word, this lower bound holds for arbitrary differentially private mechanisms.", "So there is no extra cost for imposing the truthfulness constraint." ], [ "Implication in BIC Blackbox Reduction", "Recently, Hartline et al.", "[17] and Bei and Huang [3] introduce blackbox reductions that convert any algorithm into nearly Bayesian incentive-compatible mechanisms with only a marginal loss in the social welfare.", "Both approach essentially create a virtual interface for each agent which has the structure of a matching market and then run VCG in the virtual matching markets.", "By running the exponential mechanism instead of the VCG mechanism, we can obtain a blackbox reduction that converts any algorithm into a nearly Bayesian incentive-compatible and differentially private mechanism.", "We will defer more details to the full version of this paper." ], [ "Procurement Auction for Spanning Trees", "Another interesting application is the procurement auction for a spanning tree (e.g., see [6]).", "Procurement auctions (also known as reverse auctions) are a type of auction where the roles of buyers and sellers are reversed.", "In other word, the central entity seeks to buy, instead of sell, items or services from the agents.", "In particular in the procurement auction for spanning trees, consider $n = {k \\atopwithdelims ()2}$ selfish agents own edges in a publicly known network of $k$ nodes.", "We shall imagine the nodes to be cities and the edges as potential highways connecting cities.", "Each agent $i$ has a non-negative cost $c_i$ for building a highway along the corresponding edge.", "The central entity (e.g., the government) wants to purchase a spanning tree from the network so that she can build highways to connect the cities.", "The goal is to design incentive compatible and differentially private mechanisms that provide good social welfare (minimizing total cost).", "Although this is a reverse auction in which agents have costs instead of values and the payments are from the central entity to the agents, by interpreting the costs as the negative of the valuations (i.e.", "$v_i = -c_i$ if the edge is purchased and $v_i = 0$ otherwise), we can show that the exponential mechanism with the same payment scheme is incentive compatible for procurement auctions via almost identical proofs.", "We will omit the details in this extended abstract.", "Next, we will discuss how to efficiently implement the exponential mechanism." ], [ "Sampling Spanning Trees", "There has been a large body of literature on sampling spanning tree (e.g., see [21] and the reference therein).", "Recently, Asadpour et al.", "[1] have developed a polynomial time algorithm for sampling entropy-maximizing distributions, which is exactly the kind of distribution used by the exponential mechanism.", "Therefore, the allocation rule of the exponential mechanism can be implemented in polynomial time for the spanning tree auction." ], [ "Implicit Payment Scheme by Babaioff, Kleinberg, and Slivkins {{cite:fa80efaaa5c02770e1e5fea59ef8f2c0734a0813}}", "Although we can efficiently generate samples from the desired distribution, it is not clear how to compute the exact payment explicitly.", "Fortunately, Babaioff et al.", "[2], [20] provide a general method of computing an unbiased estimator for the payment given any rationalizable allocation ruleAlthough the result in [2] only applies to single-parameter problems, Kleinberg [20] pointed out the same approach can be extended to multi-parameter problems if the type space is convex..", "Hence, we can use the implicit payment method in [2], [20] to generate the payments in polynomial time.", "Note that the size of the range of feasible outcomes of spanning tree auction is the number of different spanning tree in a complete graph with $k$ vertices, which equals $k^{k-2}$ .", "By thm:expo we have the following: Theorem 10 For any $\\epsilon > 0$ , the exponential mechanism $\\textsc {Exp}^{\\textrm {tree}}_\\epsilon $ runs in polynomial time (in $k$ and $\\epsilon ^{-1}$ ), is IC, $\\epsilon $ -differentially private, and ensures that $\\mathop {\\bf Pr}\\left[\\sum _{i=1}^n c_i \\left( \\widehat{\\textsc {Exp}}^{\\text{tree}}_\\epsilon \\right) > \\mathsf {opt}+ \\frac{(k-2) \\log k}{\\epsilon } + \\frac{t}{\\epsilon } \\right] \\\\\\le \\exp (-t) \\hspace{5.0pt}.$ This trade-off between privacy and social welfare in thm:tree essentially means that we need $\\epsilon =\\Omega (\\log k)$ in order to get $\\mathsf {opt}+ O(k)$ guarantee on expected total cost.", "The next theorem shows that this tradeoff is also tight.", "The proof is deferred to the full version due to space constraint.", "Theorem 11 Suppose $M$ is an $\\epsilon $ -differentially private mechanism for the procurement auction for spanning tree and the expected total cost by $M$ is at most $\\mathsf {opt}+ \\frac{k}{24}$ .", "Then $\\epsilon = \\Omega (\\log k)$ .", "Similar to the case in the multi-item auction, the above lower bound does not restrict $M$ to be incentive compatible.", "So the exponential mechanism is optimal even if we compare it to non-truthful ones." ], [ "Acknowledgement", "The authors would like to thank Aaron Roth for many useful comments and helpful discussions.", "In this section, we will discuss what is the amount of noise one needs to add to the payments in order to achieve $\\epsilon $ -differential privacy.", "We will consider two different models depending on how the payments are implemented: the public payment model and the private payment model.", "In the public payment model, the payments of the agents will become public information at the end of the auction, that is, the adversary who tries to learn the private valuations of the agents can see all the payments.", "Therefore, a payment scheme is $\\epsilon $ -differentially private in the public payment model if and only if for any $i \\in [n]$ , any value profiles $v= (v_1, \\dots , v_n)$ and $v^{\\prime } = (v_1, \\dots ,v^{\\prime }_i, \\dots , v_n)$ that differ only in the valuation of agent $i$ , and any possible payment profile $p$ , the probability $& \\mathop {\\bf Pr}[p_1(v), \\dots , p_n(v) = p] \\\\\\le ~ & \\exp (\\epsilon ) \\mathop {\\bf Pr}[p_1(v^{\\prime }), \\dots , p_n(v^{\\prime }) = p] \\hspace{5.0pt}.$ In the private payment model, we will assume the payments are implemented via secure channels such that the payment of each agent is only known to the corresponding agent and a few trusted parties, e.g., the central entity who runs the mechanism and/or the bank.", "Here, there are two cases based on what information the adversary can learn from the payments.", "If the adversary is not one of the agents, then by our assumption, he cannot see any of the payments and therefore cannot learn any information from the payments.", "If the adversary is one of the agents, then the only information of the payments that he will have access to is his own payment.", "Therefore, a payment scheme is $\\epsilon $ -differentially private in the public payment model if and only if for any $i \\ne j \\in [n]$ , any value profiles $v= (v_1, \\dots , v_n)$ and $v^{\\prime } = (v_1, \\dots , v^{\\prime }_i, \\dots ,v_n)$ that differ only in the valuation of agent $i$ , and any possible payment $p$ of agent $j$ , the probability $\\mathop {\\bf Pr}[p_j(v) = p] \\le \\exp (\\epsilon ) \\mathop {\\bf Pr}[p_j(v^{\\prime }) = p] \\hspace{5.0pt}.$ We will measure the amount of noise in the payments using $L_2$ norm, that is, we aim to minimize the total variance of the agents' payments in the worst-case: $\\max _{v} \\sum _{i=1}^n \\mathop {\\bf Var}[p_i(v)]$ .", "Next, we will proceed to analyze the amount of noise needed in each of the two models.", "We will start with an upper bound on the sensitivity of each agent's payment as a function of the bids.", "Lemma 12 For any $i, j \\in [n]$ , and any value profiles $v= (v_1, \\dots ,v_n)$ and $v^{\\prime } = (v_1, \\dots , v^{\\prime }_i, \\dots , v_n)$ that only differ in the valuation of agent $i$ , we have $|p_j(v) - p_j(v^{\\prime })| \\le 1$ .", "Note that by thm:main, the exponential mechanism is individual rational.", "It is also easy to see that it has no positive transfer for that otherwise the zero-value agent could gain by lying.", "So by our assumption that the agents' valuations are always between 0 and 1, we have $0 \\le p_j(v), p_j(v^{\\prime }) \\le 1$ .", "So lem:paymentsensitivity follows trivially.", "In the public payment model, the mechanism has to reveal a vector of $n$ real numbers (the payments) at the end of the auction, where each entry has sensitivity 1 by lem:paymentsensitivity.", "Therefore, we can use the standard treatment for answering numerical queries, namely, adding independent Laplace noise $\\textsc {Lap}(\\frac{n}{\\epsilon })$ to each entry, where $\\textsc {Lap}(b)$ is the Laplace distribution with p.d.f.", "$f_{\\textsc {Lap}(b)}(x) = \\frac{1}{2b} \\exp \\left(-\\frac{|x|}{b}\\right)$ .", "More precisely, we can show the following theorem.", "Theorem 13 In the public payment model, the following payment scheme is $\\epsilon $ -differentially private and has total variance $O(n^{3/2}\\epsilon ^{-1})$ , while maintaining the IC and IR in expectation: let $p_1, \\dots ,p_n$ be the payments specified in the exponential mechanism (fig:expo); let $x_1, \\dots , x_n$ be i.i.d.", "variables following the Laplace distribution $\\textsc {Lap}(\\frac{n}{\\epsilon })$ ; use payment scheme $(p_1 + x_1, \\dots , p_n + x_n)$ .", "The proof follows by standard analysis of the Laplace mechanism (e.g., see [15]).", "So we will omit the details in this extended abstract.", "It is worth mentioning that since the problem of designing payment scheme in the public payment model is a special case of answering $n$ non-linear numerical queries, it may be possible to reduce the amount of noise by using more specialized scheme on a problem-by-problem basis.", "However, we feel this is less insightful than the other results we have in this paper, so we will focus on general mechanisms and payment schemes that work for all mechanism design problems.", "Now let us turn to the private payment model.", "By our previous discussion, the mechanism only need to release at most one real number to each potential adversary in this model.", "So one may expect much less noise is needed in this model.", "Indeed, we could again use the standard treatment of adding Laplace noise, but this time it suffices to add independent Laplace noise $\\textsc {Lap}(\\frac{1}{\\epsilon })$ to each entry.", "Theorem 14 In the private payment model, the following payment scheme is $\\epsilon $ -differentially private and has total variance $O(\\sqrt{n} \\epsilon ^{-1})$ , while maintaining the IC and IR: in expectation: let $p_1, \\dots ,p_n$ be the payments specified in the exponential mechanism (fig:expo); let $x_1, \\dots , x_n$ be i.i.d.", "variables following the Laplace distribution $\\textsc {Lap}(\\frac{1}{\\epsilon })$ ; use payment scheme $(p_1 + x_1, \\dots , p_n + x_n)$ ." ], [ "Approximate Implementation for Multi-Item Auction", "In this section, we will explain how to approximately implement the exponential mechanism in the multi-item auction setting.", "The main technical tool in this section is the seminal work of Jerrem, Sinclair, and Vigoda [19] on approximating the permanent of non-negative matrices, which can be phrased as follows: Lemma 15 (FPRAS for permanent of non-negative matrices [19]) For any $\\gamma > 0$ and any $\\delta \\in (0, 1)$ , there is an algorithm that computes the permanent of an arbitrary $n \\times n$ matrix $A = \\lbrace a_{ij}\\rbrace _{i, j \\in [n]}$ up to a multiplicative factor of $\\exp (\\gamma )$ with probability at least $1 - \\delta $ .", "The running time is polynomial in $n$ , $\\gamma ^{-1}$ , $\\log (\\delta ^{-1})$ , and $\\log (\\max _{i,j\\in [n]} a_{ij} / \\min _{i,j\\in [n]}a_{ij})$ .", "To see the connection between the permanent of non-negative matrices and implementation of the exponential mechanism in the multi-item auction setting, we point out that the normalization factor in the outcome distribution of the exponential mechanism is the permanent of a non-negative matrix: $\\sum _{r \\in R_M} \\exp \\left( \\frac{\\epsilon }{2} \\sum _{i=1}^n v_i(r)\\right) = \\sum _{\\pi \\in \\Pi _n} \\prod _{i=1}^n \\exp \\left(\\frac{\\epsilon }{2} v_{i\\pi [i]} \\right) = \\textrm {perm}\\left(\\left\\lbrace \\exp \\left(\\frac{\\epsilon }{2} v_{ij} \\right) \\right\\rbrace _{i, j \\in [n]}\\right)\\hspace{5.0pt}.$ We will let $A(v)$ denote the matrix $\\lbrace \\exp (\\frac{\\epsilon }{2} v_{ij})\\rbrace _{i, j \\in [n]}$ .", "Moreover, we let $A_{-i,-j}(v)$ denote the $(n-1)\\times (n-1)$ matrix obtained by removing the $i^{th}$ row and the $j^{th}$ column of $A(v)$ ." ], [ "Approximate Sampler", "Now we are ready to introduce the approximate sampler for the multi-item auction.", "Lemma 16 For any $\\delta \\in (0, 1)$ and $\\gamma > 0$ , there is a sampling algorithm whose running time is polynomial in $n$ , $\\epsilon ^{-1}$ $\\gamma ^{-1}$ , and $\\log \\delta ^{-1}$ , such that with probability at least $1 - \\delta $ , it chooses an outcome $r$ with probability $\\mathop {\\bf Pr}[r] \\in [\\exp (-\\gamma ), \\exp (\\gamma )] \\mathop {\\bf Pr}[\\textsc {Exp}^{R_M}_\\epsilon = r] \\hspace{5.0pt}.$ We will recursively decide which item we will allocate to agent $i$ for $i = 1, 2, \\dots , n$ by repeatedly computing an accurate estimation of the marginal distribution.", "Concretely, the algorithm is given as follows: Use the FPRAS in lem:fpras to compute $\\textrm {perm}(A_{-1,-j}(v))$ up to a multiplicative factor of $\\exp (\\frac{\\gamma }{2n})$ with success probability at least $1 - \\frac{\\delta }{n^2}$ .", "Let $x_j$ denote the approximate value.", "Sample an item $j$ with probability $\\mathop {\\bf Pr}[j] \\propto x_j$ .", "Allocate item $j$ to agent 1 and recurse on the remaining $n-1$ agents and $n-1$ items.", "First we note that for each allocation $\\pi \\in \\Pi _n$ , the probability that $\\pi $ is chosen as the outcome can be decomposed into $n$ stages by Bayes' rule: $\\mathop {\\bf Pr}[\\textsc {Exp}^{R_M}_\\epsilon (v) = \\pi ] \\, = \\, &\\mathop {\\bf Pr}\\big [\\text{agent $1$ gets $\\pi [1]$}\\,\\big ] \\cdot \\mathop {\\bf Pr}\\big [\\text{agent $2$ gets $\\pi [2]$} \\,|\\, \\pi [1]\\,\\big ] \\\\& \\cdots \\mathop {\\bf Pr}\\big [\\text{agent $n$ gets $\\pi [n]$} \\,|\\, \\pi [1],\\dots , \\pi [n-1]\\,\\big ] \\hspace{5.0pt}.$ In the first recursion of our algorithm, we use the distribution $\\mathop {\\bf Pr}[\\text{agent $1$ gets item $j$}] \\propto x_j \\approx \\textrm {perm}(A_{-1, -j}(v)) \\hspace{5.0pt}.$ Further, in the exponential mechanism $\\mathop {\\bf Pr}[\\text{agent $1$ gets item $j$ in $\\textsc {Exp}^{R_M}_\\epsilon $}]& \\propto & \\sum _{\\pi : \\pi [1] = j} \\exp \\left(\\frac{\\epsilon }{2} \\sum _{k=1}^n v_{k\\pi [k]} \\right) \\\\& = & \\exp \\left(\\frac{\\epsilon }{2} v_{1j}\\right)\\textrm {perm}(A_{-1, -j}(v)) \\hspace{5.0pt}.$ Since $x_j$ approximate $\\textrm {perm}(A_{-1,-j}(v))$ up to an $\\exp (\\frac{\\gamma }{2n})$ factor, we know the probability that item $j$ is allocated to agent 1 in our algorithm approximate the correct marginal up to an $\\exp (\\frac{\\gamma }{n})$ multiplicative factor.", "Similar claim holds for the rest of the $n-1$ stages as well.", "So the probability that we samples a permutation $\\pi \\in R_M$ differs from the correct distribution by at most a $\\exp (\\frac{\\gamma }{n})^n =\\exp (\\gamma )$ factor.", "Moreover, by union bound the failure probability is at most $\\delta $ ." ], [ "Approximate Payments", "Next, we will turn to approximate implementation of the payment scheme.", "First, recall that the payment for agent $i$ is $p_i & = & \\mathop {\\bf E}_{r \\sim \\textsc {Exp}^{R_M}_\\epsilon (v)} [v_i(r)] -\\frac{2}{\\epsilon } \\ln \\left(\\sum _{r \\in R_M} \\exp \\left(\\frac{\\epsilon }{2} \\sum _{k=1}^n v_k(r) \\right) \\right) +\\frac{2}{\\epsilon } \\ln \\left( \\sum _{r \\in R_M} \\exp \\left(\\frac{\\epsilon }{2} \\sum _{k \\ne i} v_k(r) \\right) \\right) \\\\& = & \\mathop {\\bf E}_{r \\sim \\textsc {Exp}^{R_M}_\\epsilon (v)} [v_i(r)] -\\frac{2}{\\epsilon } \\ln \\left(\\textrm {perm}\\left(A(v_i,v_{-i})\\right) \\right) + \\frac{2}{\\epsilon } \\ln \\left(\\textrm {perm}\\left(A(\\mathbf {0}, v_{-i})\\right) \\right)\\hspace{5.0pt}.$ The next lemma states that we can efficiently compute an estimator for the payment $p_i$ with inverse polynomially small bias.", "Lemma 17 For any $\\delta \\in (0, 1)$ and $\\gamma \\in (0, 1)$ , we can compute in polynomial time (in $n$ , $\\epsilon ^{-1}$ , and $\\gamma ^{-1}$ ) a random estimator $\\hat{p}_i$ for $p_i$ such that the bias is small: $\\left|\\mathop {\\bf E}[\\hat{p}_i] - p_i \\right| \\le \\gamma $ .", "By lem:fpras, we can efficiently estimate $\\textrm {perm}(A(v_i, v_{-i}))$ and $\\textrm {perm}(A(\\mathbf {0},v_{-i}))$ up to an multiplicative factor of $\\exp (\\frac{\\gamma }{6})$ with success probability at least $1 - \\frac{\\gamma }{6}$ .", "Hence, we can compute $\\ln (\\textrm {perm}(A(v_i, v_{-i}))$ and $\\ln (\\textrm {perm}(A(\\mathbf {0}, v_{-i})))$ up to additive bias of $\\frac{\\gamma }{6}$ with probability $1 - \\frac{\\gamma }{6}$ .", "Note that the total bias introduced if the FPRAS fails is at most 1 and that could happens with probability at most $\\frac{\\gamma }{6}$ .", "So the total bias from estimating $\\ln (\\textrm {perm}(A(v_i, v_{-i})))$ and $\\ln (\\textrm {perm}(A(\\mathbf {0}, v_{-i})))$ is at most $\\frac{\\gamma }{2}$ .", "It remains to compute an estimator for $\\mathop {\\bf E}_{r \\sim \\textsc {Exp}^{R_M}_\\epsilon (v)}[v_i(r)]$ with bias less than $\\frac{\\gamma }{2}$ .", "In order to do so, we will use the algorithm in lem:approxsampler to sample an outcome $r^*$ from a distribution whose probability mass function differs from that of $\\textsc {Exp}^{R_M}_\\epsilon (v)$ by at most a $\\exp (\\frac{\\gamma }{6})$ factor point-wise, with success probability at least $1 -\\frac{\\gamma }{6}$ .", "Then we will use $v_i(r^*)$ as our estimator.", "Note that conditioned on the sampler runs correctly, we have $\\left|\\mathop {\\bf E}[v_i(r^*)] - \\mathop {\\bf E}_{r \\sim \\textsc {Exp}^{R_M}_\\epsilon (v)} [v_i(r)] \\right|\\le \\left(\\exp \\left(\\frac{\\gamma }{6}\\right) - 1\\right) \\mathop {\\bf E}_{r \\sim \\textsc {Exp}^{R_M}_\\epsilon (v)}[v_i(r)] \\le \\left(\\exp (\\frac{\\gamma }{6}) - 1\\right) \\le \\frac{\\gamma }{3} \\hspace{5.0pt}.$ Moreover, the maximum bias conditioned on the failure of the sampler is at most 1, which happens with probability at most $\\frac{\\gamma }{6}$ .", "So the total bias from the estimator for $\\mathop {\\bf E}_{r \\sim \\textsc {Exp}^{R_M}_\\epsilon (v)}[v_i(r)]$ is at most $\\frac{\\gamma }{2}$ ." ], [ "Lower Bound for Multi-Item Auction", "[Proof of thm:lbmultiitem] Let us first define some notations.", "For any $j^* \\in [n]$ , we will let $e^{j^*}$ denote the valuation profile such that $e^{j^*}_j = 1$ if $j = j^*$ and $e^{j^*}_j = 0$ if $j \\ne j^*$ .", "That is, an agent with valuation $e^{j^*}$ is single-minded who only value getting item $j^*$ (with value 1) and has no interest in getting any other item.", "We will say $j^*$ is the critical item for this agent.", "Suppose $M$ is an $\\epsilon $ -differentially private mechanism such that $M$ always obtain at least $\\mathsf {opt}- \\frac{n}{10}$ expected social welfare.", "Let us consider the following randomly chosen instance: each agent's valuation is chosen from $e^1, \\dots , e^n$ independently and uniformly at random.", "Let us consider the social welfare we get by running mechanism $M$ on this randomly constructed instance.", "We first note that $\\mathop {\\bf E}_{v}[\\mathsf {opt}(v)] = (1 - e^{-1}) n$ for that each item has probability $1 - e^{-1}$ of being the critical item of at least one of the agents.", "By our assumption, the expected welfare obtained by $M$ shall be at least $(1 - e^{-1})n - \\frac{n}{10} >\\frac{n}{2}$ .", "Therefore, we have $\\sum _{i=1}^n \\sum _{j=1}^n \\mathop {\\bf Pr}[\\text{$M$ allocate $j$ to agent$i$} \\,|\\, \\text{$j$ is critical for $i$}] \\mathop {\\bf Pr}[\\text{$j$ iscritical for $i$}] \\ge \\frac{n}{2} \\hspace{5.0pt}.$ Note that $\\mathop {\\bf Pr}[\\text{$j$ is critical for $i$}] = \\frac{1}{n}$ for all $i, j \\in [n]$ , we get that the average probability that a critical item-agent pair is allocated is at least half: $ \\frac{1}{n^2} \\sum _{i=1}^n \\sum _{j=1}^n \\mathop {\\bf Pr}[\\text{$M$ allocate$j$ to agent $i$} \\,|\\, \\text{$j$ is critical for $i$}] \\ge \\frac{1}{2} \\hspace{5.0pt}.$ Similarly, we have $\\sum _{i=1}^n \\sum _{j=1}^n \\mathop {\\bf Pr}[\\text{$M$ allocate $j$to agent $i$} \\,|\\, \\text{$j$ is not critical for $i$}]\\mathop {\\bf Pr}[\\text{$j$ is not critical for $i$}] \\le \\frac{n}{2} \\hspace{5.0pt}.$ Note that $\\mathop {\\bf Pr}[\\text{$j$ is not critical for $i$}] = \\frac{n-1}{n}$ for all $i, j \\in [n]$ , we get that the average probability that the average probability that a non-critical item-agent pair is chosen in the allocation is very small: $ \\frac{1}{n^2} \\sum _{i=1}^n \\sum _{j=1}^n \\mathop {\\bf Pr}[\\text{$M$ allocate$j$ to agent $i$} \\,|\\, \\text{$j$ is not critical for $i$}] \\le \\frac{1}{2(n-1)} \\hspace{5.0pt}.$ By eq:lbmatching1 and eq:lbmatching2, we have $\\frac{ \\sum _{i=1}^n \\sum _{j=1}^n \\mathop {\\bf Pr}[\\text{$M$ allocate $j$ toagent $i$} \\,|\\, \\text{$j$ is critical for $i$}]}{\\sum _{i=1}^n \\sum _{j=1}^n \\mathop {\\bf Pr}[\\text{$M$ allocate $j$ to agent $i$}\\,|\\, \\text{$j$ is not critical for $i$}]} \\ge n-1\\hspace{5.0pt}.$ In particular, we know there exists a $(i, j)$ pair such that $\\frac{ \\mathop {\\bf Pr}[\\text{$M$ allocate $j$ toagent $i$} \\,|\\, \\text{$j$ is critical for $i$}]}{ \\mathop {\\bf Pr}[\\text{$M$ allocate $j$ to agent $i$}\\,|\\, \\text{$j$ is not critical for $i$}]} \\ge n-1\\hspace{5.0pt}.$ Since $M$ is $\\epsilon $ -differentially private, we get that $\\exp (\\epsilon ) \\ge n-1$ , and thus $\\epsilon = \\Omega (\\log n)$ ." ], [ "Lower Bound for Procurement Auction for Spanning Trees", "[Proof of thm:lbspanningtree] Suppose $M$ is an $\\epsilon $ -differentially private mechanism whose expected total cost is at most $\\mathsf {opt}+ \\frac{k}{24}$ .", "We will consider the following randomly generated instance.", "Each agent $i$ 's cost value $c_i$ is independently chosen as $c_i = \\left\\lbrace \\begin{aligned}& 1 & & \\text{ , w.p. }", "1 - \\frac{1}{2k} \\\\& 0 & & \\text{ , w.p. }", "\\frac{1}{2k}\\end{aligned}\\right.", "$ If an agent has cost 0, we say this agent and the corresponding edge are critical.", "Let us first analyze the expected value of $\\mathsf {opt}$ for such randomly generated instances.", "Intuitively, we want to pick as many critical edges as possible.", "In particular, when there are no cycles consists of only critical edges, the minimum spanning tree shall pick all critical edges, which comprise a forest in the graph, and then pick some more edges to complete the spanning tree.", "Lemma 18 With probability at least $\\frac{1}{2}$ , there are no cycle consists of only critical edges.", "[Proof of lem:st1] For each cycle of length $t$ , the probability that all edges on this cycle are critical is $(2k)^{-t}$ .", "Note that the number of cycles of length $t$ is at most ${k \\atopwithdelims ()t} (t-1)!", "\\le k^t$ .", "Here ${k\\atopwithdelims ()t}$ is the number of subsets of $t$ vertices and $(t-1)!$ is the number of different Hamiltonian cycles among $t$ vertices.", "Hence, by union bound, the probability that there is any cycle consists of only critical edges is at most $\\sum _{t=2}^k (2k)^{-t} \\cdot k^t = \\sum _{t=2}^k 2^{-t} <\\frac{1}{2}$ .", "Moreover, by Chernoff-Höeffding bound, we have that the number of critical edges is at least $\\frac{k}{3}$ with probability at least $\\frac{3}{4}$ .", "Therefore, by union bound, with probability at least $\\frac{1}{4}$ , we have that there are at least $\\frac{k}{3}$ critical edges and there are no cycle consists of only critical edges.", "So in this case, we have $\\mathsf {opt}\\le k - \\frac{k}{3} =\\frac{2k}{3}$ .", "Therefore, the expectation of the optimal total cost is at most $\\mathop {\\bf E}[\\mathsf {opt}] \\le \\frac{3}{4} k + \\frac{1}{4} \\frac{2k}{3} =\\frac{11k}{12}$ .", "By our assumption on $M$ , we get that the expected total cost of the outcome chosen by $M$ is at most $\\frac{11k}{12} + \\frac{k}{24} =\\frac{23k}{24}$ .", "In other words, the expected number of critical edges chosen by $M$ is at least $\\frac{k}{24}$ .", "That is, $\\sum _{i=1}^n \\mathop {\\bf Pr}[\\text{edge $i$ is chosen} \\,|\\, \\text{edge $i$ iscritical}] \\mathop {\\bf Pr}[\\text{edge $i$ is critical}] \\ge \\frac{k}{24}\\hspace{5.0pt}.$ Note that $\\mathop {\\bf Pr}[\\text{edge $i$ is critical}] = \\frac{1}{2k}$ for all $i \\in [n]$ and $n = {k \\atopwithdelims ()2} = \\frac{k(k-1)}{2}$ , we get that on average a critical edge is chosen with at least constant probability $\\frac{1}{n} \\sum _{i=1}^n \\mathop {\\bf Pr}[\\text{edge $i$ is chosen} \\,|\\,\\text{edge $i$ is critical}] \\ge \\frac{1}{6} \\hspace{5.0pt}.$ On the other hand, it is easy to see $\\sum _{i=1}^n \\mathop {\\bf Pr}[\\text{edge $i$ is chosen} \\,|\\, \\text{edge $i$ isnot critical}] \\mathop {\\bf Pr}[\\text{edge $i$ is not critical}] \\le k\\hspace{5.0pt}.$ By $\\mathop {\\bf Pr}[\\text{edge $i$ is not critical}] = 1 - \\frac{1}{2k}$ and $n= {k \\atopwithdelims ()2}$ , we get that on average a non-critical edge is chosen with very small probability $\\frac{1}{n} \\sum _{i=1}^n \\mathop {\\bf Pr}[\\text{edge $i$ is chosen} \\,|\\,\\text{edge $i$ is not critical}] \\le \\frac{2k^2}{(2k-1)n} =\\frac{4k}{(k-1)(2k-1)} \\le \\frac{8}{2k-1} \\hspace{5.0pt}.$ Therefore, we have $\\frac{\\sum _{i=1}^n \\mathop {\\bf Pr}[\\text{edge $i$ is chosen} \\,|\\,\\text{edge $i$ critical}]}{\\sum _{i=1}^n \\mathop {\\bf Pr}[\\text{edge $i$ ischosen} \\,|\\, \\text{edge $i$ is not critical}]} \\ge \\frac{2k-1}{48}\\hspace{5.0pt}.$ In particular, there exists an agent $i$ , such that $\\frac{\\mathop {\\bf Pr}[\\text{edge $i$ is chosen} \\,|\\, \\text{edge $i$critical}]}{\\mathop {\\bf Pr}[\\text{edge $i$ is chosen} \\,|\\, \\text{edge $i$ isnot critical}]} \\ge \\frac{2k-1}{48} \\hspace{5.0pt}.$ However, the above amount is upper bounded by $\\exp (\\epsilon )$ since $M$ is $\\epsilon $ -differentially private.", "So we conclude that $\\epsilon = \\Omega (k)$ ." ] ]

[ [ "Tail-Constraining Stochastic Linear-Quadratic Control: Large Deviation\n and Statistical Physics Approach" ], [ "Abstract Standard definition of the stochastic Risk-Sensitive Linear-Quadratic (RS-LQ) control depends on the risk parameter, which is normally left to be set exogenously.", "We reconsider the classical approach and suggest two alternatives resolving the spurious freedom naturally.", "One approach consists in seeking for the minimum of the tail of the Probability Distribution Function (PDF) of the cost functional at some large fixed value.", "Another option suggests to minimize the expectation value of the cost functional under constraint on the value of the PDF tail.", "Under assumption of the resulting control stability, both problems are reduced to static optimizations over stationary control matrix.", "The solutions are illustrated on the examples of scalar and 1d chain (string) systems.", "Large Deviation self-similar asymptotic of the cost functional PDF is analyzed." ], [ "Introduction", "Stochastic differential equations are used both in control [1], [2], [3], [4], [5], [6], [7] and statistical physics [8], [9], [10], [11], [12] to state the problems.", "The two fields also use similar mathematical methods to analyze these equations.", "However, and in spite of the commonalities, there were relatively few overlaps between the disciplines in the past, even though the communications between two communities improved in the recent years.", "Some new areas in control, for example stochastic path integral control [13], [14], [15], [16], have emerged influenced by analogies, intuition and advances in statisical/theoretical physics.", "Vice versa, many practical experimental problems in physics, chemistry and biology dealing with relatively small systems (polymers, membranes, etc), which are driven and experience significant thermal fluctuations, can now be analyzed and manipulated/controled with accuracy and quality unheard of in the past, see for example [17], [18].", "Besides, approaches from both control theory and statistical physics started to be applied to large natural and engineered networks, like chemical, bio-chemical and queuing networks [19], [20], [21], [22].", "Dynamics over these networks is described by stochastic differential equations, the networks have enough of control knobs, and they function under significant fluctuations which need to be controlled to prevent rare but potentially devastating failures.", "Related setting of stochastic optimization, i.e.", "optimization posed under uncertainty, has also came recently in the spot light of statistical physics inspired algorithms and approaches [23].", "Convergence of these are related ideas motivated the manuscript, where we discuss analysis and control of rare events in the simplest possible, but practically rather widespread universal and general, linear setting.", "We realize that the general topic of linear control is well studied and many (if not all) possible questions, e.g.", "related to proper way of accounting for risk (rare events), were discussed in the field in the past.", "In spite of that, we still hope that this manuscript may also be useful not only to physicists, who may wish to explore new and largely unusual (in physics) formulations, but also to control theorists.", "Consider first order (in time derivatives) stochastic linear dynamics of a vector $x=(x_i|i=1,\\cdots ,N)$ over time interval $t^{\\prime }\\in [t;T]$ $\\frac{d}{dt^{\\prime }}x=Ax+Bu+\\xi (t^{\\prime }),$ where $A$ and $B$ are constant matrices; $u(t^{\\prime })$ is the control vector applied at the moment of time $t^{\\prime }$ ; and $\\lbrace \\xi \\rbrace =(\\xi (t^{\\prime })|t^{\\prime }\\in [t;T])$ is the zero mean, short-correlated noise with covariance $V$ $\\langle \\xi _i(t^{\\prime })\\rangle =0,\\quad \\langle \\xi _i(t^{\\prime })\\xi _j(t^{\\prime \\prime })\\rangle =\\delta (t^{\\prime }-t^{\\prime \\prime })V_{ij},\\quad i,j=1,\\cdots ,N$ where one utilizes \"statistical physics\" notations for the expectation value (average) over noise, $\\langle \\cdots \\rangle $ .", "Here in Eq.", "(REF ) and below the averages are over multiple possible realizations of the noise, each generating a new trajectory of the system, $\\lbrace x\\rbrace =(x(t^{\\prime })|t^{\\prime }\\in [t;T])$ , under given control $\\lbrace u\\rbrace =(u(t^{\\prime })|t^{\\prime }\\in [t;T])$ .", "The Eq.", "(REF ) is causal, thus assuming retarded (Stratonovich) regularization of the noise on the right-hand-side of the discreet version of Eq.", "(REF ).", "The physical meaning of the vectors and matrices in Eq.", "(REF ) is as follows.", "$A$ is the matrix explaining stretching, shearing and rotation of the system trajectory in the $N$ -dimensional space if the control and external noise would not be applied.", "Matrix $B$ describes possible limitations on the degrees of freedom in the system one can control.", "To simplify notations we consider signal, control and noise vectors having the same dimension, $N$ , where thus $B$ is quadratic.", "The setting of Eqs.", "(REF ,REF ) is classic one in the control theory.", "It describes the so-called Linear-Quadratic (LQ) stochastic control problem, which was introduced in [2], [3], [5], [6] and became foundational for the control theory as a field, see e.g.", "[24], [25] and references therein.", "In the classical formulation one seeks to solve the following optimization, $t\\in [0;T]$ : $&& \\min _{\\lbrace u\\rbrace } \\Biggl \\langle J(t;T;\\lbrace u\\rbrace , \\lbrace x\\rbrace )\\Biggr \\rangle ,\\\\&& J(t;T;\\lbrace u\\rbrace , \\lbrace x\\rbrace )\\equiv \\frac{1}{2}x^*(T)F x(T)+\\frac{1}{2}\\int _t^T dt^{\\prime } \\left( x^*(t^{\\prime })Q x(t^{\\prime })+u^*(t^{\\prime })R u(t^{\\prime })\\right),\\nonumber $ where $Q,R$ and $F$ are pre-defined stationary (time independent) symmetric positive matrices and one uses the super-script asterisk, $*$ , to mark transposition.", "$J(t;T;\\lbrace u\\rbrace , \\lbrace x\\rbrace )$ (later on, and when it is not confusing, we will use the shortcut notation $J$ ) is a scalar quadratic cost functional of the state vector $\\lbrace x\\rbrace =(x(t^{\\prime })|t^{\\prime }\\in [t;T])$ , and the control vector, $\\lbrace u\\rbrace =(u(t^{\\prime })|t^{\\prime }\\in [t;T])$ evaluated for all intermediate times $t^{\\prime }$ from the $[0;T]$ interval.", "Here in Eqs.", "(REF ) (and everywhere below in the manuscript) the average over noise $\\lbrace \\xi \\rbrace $ includes conditioning to Eq.", "(REF ), i.e.", "$\\lbrace x\\rbrace $ is dependent on realization of the noise, $\\lbrace \\xi \\rbrace $ , and on the control, $\\lbrace u\\rbrace $ , according to Eq.", "(REF ).", "It is assumed that the stochastic LQ control is evaluated off-line, i.e.", "the optimal solution $u_*(t;x(t))$ of Eq.", "(REF ) is computed and saved prior to executing actual experiment for any initial condition $x(t)$ at any $t$ .", "Then in the course of the actual experiment (execution of the dynamics) $x(t)$ is measured at any time $t$ and respective $u_*(t;x(t))$ is applied.", "(When observation of $x(t)$ is partial and noisy one needs to generalize the stochastic LQ control, for example considering the stochastic Linear Quadratic Gaussian (LQG) control, see e.g.", "[25] for details.)", "We also assume (and the details will be clarified below) that the optimal control succeeds, i.e.", "the systems stabilizes and $J$ does not grow with $T$ faster than linearly.", "An unfortunate caveat of the LQ setting (REF ) is in the lack of fluctuations control: even though the LQ solution is optimal in terms of minimizing the expectation value of the cost functional it may generate very significant fluctuations when it comes to analysis of the ${\\cal J}\\gg \\langle J\\rangle $ tail of the Probability Distribution Function, ${\\cal P}({\\cal J})$ , of the cumulative cost ${\\cal J}(T;x(0))\\equiv J(0;T;\\lbrace u_*\\rbrace , \\lbrace x\\rbrace )$ .", "Stochastic Risk Sensitive LQ (RS-LQ) scheme [26], [27], [28] was introduced to improve control of the abnormal fluctuations of $J$ .", "RS-LQ constitutes the following generalization of the LQ scheme (REF ) $&& \\max _{\\lbrace u\\rbrace } \\langle \\exp \\left(-\\theta J\\right)\\rangle ,$ where $\\theta $ is a pre-defined parameter.", "Intuitively one relates the case of positive $\\theta $ to a risk-avert optimum.", "It is assumed within the standard RS-LQ scheme that $\\theta $ is fine tuned by some additional considerations.", "Note that, as shown in [29], the RS-LQ control is also ultimately related to the so-called $H_\\infty $ -norm robust control.", "(See also [30] for further discussion of the relation.)", "In this paper we analyze two natural modifications of the stochastic RS-LQ control.", "The two schemes can both be interpreted in terms of the RS-LQ approach supplemented by an additional optimization over $\\theta $ .", "Our first, Tail-Optimum (TO), scheme consists in the following modification of the LQ (REF ) and RS-LQ (REF ) ones $\\min _{\\lbrace u\\rbrace } {\\cal P}(J=j\\cdot (T-t)|\\lbrace u\\rbrace ).$ In words, the TO-LQ control minimizes (at any time $t$ and given the current observation $x(t)$ ) the probability of the current value of the cost functional $J(t;T;\\lbrace u\\rbrace , \\lbrace x\\rbrace )$ evaluated at a predefined value, $j\\cdot (T-t)$ , where thus $j$ is the only external parameter left in the formulation.", "Another strategy, which we call Chance-Constrained LQ (CC-LQ), in reference to similar formulations in optimization theory [31], [32], [33], consists in minimizing the mean of the cost functional under condition that the tail probability evaluated at $j\\cdot (T-t)$ does not exceed the prescribed threshold value $\\varepsilon (t;T)$ $&\\min \\limits _{\\lbrace u\\rbrace } &\\langle J\\rangle \\\\&& \\mbox{s.t. }", "{\\cal P}(J=j\\cdot (T-t)|\\lbrace u\\rbrace )\\le \\varepsilon (t;T).\\nonumber $ Main objectives, and consequently results of this study, are $\\bullet $ To extend the asymptotic, $T\\rightarrow \\infty $ , approach, developed in the past for LQ (REF ) and RS-LQ (REF ) optimal controls to the new TO-LQ (REF ) and CC-LQ (REF ) optimal settings.", "At $T\\rightarrow \\infty $ the optimal control takes the following universal linear in $x$ form $u_*(t;x)=-Kx, $ where $K$ is $t$ -independent but model dependent matrix.", "The condition of the system stability, intuitively translating into the expectation that ${\\cal J}$ grows not faster than linearly with $T$ , naturally requires that all the eigenvalues of the stability matrix, $\\mu =BK-A$ , have positive real part.", "The linearity of the optimal control (REF ) in $x$ is a direct consequence of the linearity of the initial dynamical system.", "Time-indepedence and initial condition-independence of the optimal control (REF ) are asymptotic: they are achieved at $T\\gg \\tau _*$ , where $\\tau _*$ can be estimated as the inverse of the absolute value of the $(BK-A)$ 's eignevalue with the smallest real part.", "System of algebraic equations defining $K$ implicitly for the TO-LQ and CC-LQ cases are presented and then juxtaposed against the previously analyzed cases of the LS and RS-LQ controls.", "(See Eqs.", "(REF ,REF ,REF ).)", "Finding optimal control is reduced to optimization over time-independent, $K$ .", "The resulting dependencies are homogeneous in time, with $t$ and $T$ always enter in the $T-t$ combination.", "(This also simplifies the analysis allowing to set $t=0$ .)", "$\\bullet $ To analyze statistics of the optimal cost functional, ${\\cal J}$ , in the Large Deviation (LD) regime, i.e.", "at large but finite $T$ .", "We show that in the stable regime the PDF of ${\\cal J}$ attains the following universal LD form $\\log {\\cal P}({\\cal J})\\sim -T {\\cal S}({\\cal J}/T),$ where the LD function, ${\\cal S}(j)$ , is a convex function of its argument found implicitly (in a closed algebraic form, which may or may not yield an efficient algorithm) for the four cases (of LS, RS-LQ, TO-LQ and CC-LQ controls) considered.", "The LD function shows a universal, ${\\cal S}(j)\\rightarrow a j$ , tail at large (i.e.", "larger than typical) $j$ , where the value of positive $a$ depends on the model.", "This suggests, in particular, that it is natural to choose in the CC optimization (REF ), $\\varepsilon (t;T)=\\exp (-c(T-t))$ , for the threshold, with $c$ been a constant.", "To derive compact algebraic expressions for the LD function we, first, analyze the generating function of ${\\cal J}$ evaluated at linear $u$ parameterized by $K$ as in Eq.", "(REF ), $&& {\\cal Z}(\\theta ;K)\\equiv \\langle \\exp \\left(-\\theta {\\cal J}\\right)\\rangle _*,$ then express the optimization/control objective as a convolution of the integral or differential operator/kernel in $\\theta $ (the choice will depend on the model) and ${\\cal Z}(\\theta ;K)$ , and finally evaluate optimization over $K$ in the asymptotic LD approximation.", "Here in Eq.", "(REF ) the low asterisk mark $*$ in the expectation/average (over noise and constrained to Eq.", "(REF )) indicates that the control vector is taken in the form of the linear ansatz, $u\\rightarrow - K x$ , where $K$ is left yet undetermined.", "The remainder of the manuscript is organized as follows.", "We start discussing the deterministic case (of zero noise) in Section .", "This regime is of interest for two reasons.", "First, in the asymptotic of zero noise the four, generally different, control schemes become equivalent.", "Besides, and as well known from the classical papers [2], [24], [25], optimal control in the bare LQ case (correspondent to minimization of the cost function average) is not sensitive (and thus independent of) the level of the noise.", "Section is devoted to analysis of the generating function (REF ), the average value of the cost function and the tail of the cost function distribution restricted to yet unspecified value of $K$ .", "Optimization over $K$ , resulting in the known RS-LQ optimal relations and also derivation of the new optimal relations for $K$ in the TO-LQ and CC-LQ cases, is discussed in Section .", "We describe and compare asymptotic Large Deviation forms of the cost function PDF, ${\\cal P}({\\cal J})$ , in the optimal regimes.", "In this and preceding Section we also discuss many times the illustrative \"scalar\" example, where $x$ and $u$ are scalars.", "An infinite system example, of a \"string\" formed from a linear 1d chain, is discussed in Section .", "We conclude and discuss related future challenges in Section ." ], [ "Deterministic Case and LQ-optimal control", "We start this Section from a disclaimer: all results reported here are classical, described in [2], [24], [25], [26], [27], [28], [29] and latter papers and books, see e.g.", "[34], [35], [36].", "We present it here only for making the whole story of the manuscript self-explanatory and coherent.", "When the noise is ignored, Eq.", "(REF ) should be considered as a deterministic constraint, reducing any of the optimal control schemes (REF ,REF ,REF ,REF ) to a simple variation of the cost functional () over $u$ .", "Using the standard variational technique with a time dependent Lagrangian multiplier for the constraint, and then excluding the multiplier one derives the equation $&& \\frac{d}{dt^{\\prime }}u^*+u^*RB^{-1}ABR^{-1}=x^*QBR^{-1},$ which should be supplied by the boundary condition (also following from the variation), $u^*(T)+x^*(T)FBR^{-1}=0$ .", "(Let us remind that we choose the notations where the dimensionality of $u$ coincides with the dimensionality of $x$ .", "We also assume that inverses of all the matrices involved in the formulation are well defined.", "This assumption is not critical and is made here only to simplify the notations.", "In the general case when some of the matrices, in particular $R$ , are not full rank, one can generalize the formulas properly, using a proper notion of the pseudo-inverse.)", "Substituting, $u=-R^{-1}B^*\\Pi x$ , in Eq.", "(REF ) one arrives at the following equation for $\\Pi $ $\\frac{d}{dt^{\\prime }}\\Pi +\\Pi A+A^*\\Pi -\\Pi B R^{-1}B^*\\Pi +Q=0.$ with the boundary condition $\\Pi (T)=F$ .", "Eq.", "(REF ), solved backwards in time, results in $\\Pi (t)$ and then, $u_*(t;x)=-R^{-1}B^*\\Pi (t)x=-Kx$ .", "To gain a qualitative understanding of the backwards in time dynamics of $\\Pi $ , let us briefly discuss the simplest possible case with all the matrixes entering Eq.", "(REF ) replaced by scalars, then yielding the following analytic solution for the optimal $K$ $K=\\frac{A-\\sqrt{A^2+\\frac{QB^2}{R}}\\left(\\tanh \\left(\\sqrt{A^2+\\frac{QB^2}{R}}(t-T_0)\\right)\\right)^{\\pm 1}}{B},$ where $T_0$ and $\\pm 1$ are chosen to satisfy the boundary condition, $K(T)=BF/R$ .", "When $T\\gg \\tau =1/\\sqrt{A^2+QB^2/R}$ , the backwards in time dynamics saturates (after a short $\\sim \\tau $ transient) to a $F$ -independent constant, resulting from replacing $\\tanh $ in Eq.", "(REF ) by $-1$ .", "Therefore, in the stationary regime, $T\\rightarrow \\infty $ the optimal control is with the constant in time, frozen $K$ .", "One also finds that the optimal control in the one dimensional deterministic case is always stable, $\\mu =KB-A>0$ .", "Returning to the general (finite vector) case one concludes that when $T$ is sufficiently large the optimal control is of the form described by Eq.", "(REF ), i.e.", "it is linear in $x$ and asymptotically time independent, with $K=R^{-1}B^*\\Pi _0$ where $\\Pi _0$ solves Eq.", "(REF ) with the first term replaced by zero.", "It is well known in the control theory that (under some standard common sense assumptions on $B$ and $R$ matrices) stable solution of the system of the algebraic Riccati equations is unique and moreover it can be found efficiently.", "(See e.g.", "Chapter 12 of Sec[37] and references therein.)", "Let us now discuss the bare LQ control, now in the presence of the noise.", "Since Eq.", "(REF ) is linear, one can naturally split the full solution into a sum, $x=x_1+x_2$ , where $x_1$ satisfies Eq.", "(REF ) without noise and it is equivalent to the noise-less solution, just discussed in this Section.", "Then, the second term satisfies, $dx_2/dt^{\\prime }=A x_2+\\xi $ , with $x_2(t)=0$ .", "However, since the noise is zero mean, $\\langle \\xi \\rangle =0$ , $x_2$ is zero mean too, i.e.", "$\\langle x_2\\rangle =0$ .", "Next, let us analyze the split of term in $\\langle J\\rangle $ , which is the optimization objective of the LQ scheme.", "Since, $x_1$ and $x_2$ are independent (by construction) and because $x_2$ is zero mean, $\\langle J\\rangle $ , splits into two terms, $\\langle J_1\\rangle +\\langle J_2\\rangle $ , each dependent on $x_1$ and $x_2$ vectors only.", "$\\langle J_1\\rangle $ is simply equivalent to $J$ analyzed above in the deterministic case, while $\\langle J_2\\rangle $ is $u$ -independent, thus not contributing the optimization at all.", "To summarize, the LQ optimal control is not sensitive to the noise and it is thus equivalent to the deterministic (noiseless) case described above." ], [ "Generating Function", "Consider the Generating Function (GF), ${\\cal Z}(\\theta ;K)$ , defined by Eq.", "(REF ).", "${\\cal Z}(\\theta ;K)$ is of an obvious relevance to the RS-LQ scheme, but it is also useful for analysis of other schemes as well, because of the following (Laplace transform) relation to the PDF of ${\\cal J}$ : ${\\cal Z}(\\theta ;K)=\\int \\limits _0^\\infty d{\\cal J}\\ \\exp (-\\theta {\\cal J}) {\\cal P}_*({\\cal J}),$ where (as before) the asterisk in the sub-script indicates that the PDF was evaluated at $u=Kx$ , with $K$ being yet undefined constant matrix.", "The inverse of Eq.", "(REF ) is ${\\cal P}_*({\\cal J})=\\int \\limits _{c-i\\infty }^{c+i\\infty } \\frac{d\\theta }{2\\pi i}\\exp (\\theta {\\cal J}) {\\cal Z}(\\theta ;K),$ where it is assumed that the integration contour, considered in the complex plain of $\\theta $ , goes on the right from all the singularities (poles and cuts) of ${\\cal Z}(\\theta ;K)$ .", "In the path integral representation GF gets the following form $&& {\\cal Z}(\\theta ;K)\\sim \\int {\\cal D}x\\ {\\cal D}p\\ \\exp \\Biggl (\\int _0^T dt \\Biggl ( -\\frac{\\theta }{2}x^*\\tilde{Q} x+p^* (\\partial _t x+\\mu x) +\\frac{1}{2}p^*Vp \\Biggr )\\Biggr ),\\\\&& \\tilde{Q}=Q+K^*RK,\\quad \\mu =BK-A,$ where $p$ is an auxiliary vector variable (momentum).", "Here and everywhere below we assume that, even if the dynamics was not stable before application of the control, control stabilizes it.", "Formally, this means that $\\mu $ , defined by Eq.", "(), has no eigenvalues with negative real values.", "The \"boundary\" ($F$ -dependent term) in Eq.", "(REF ) was ignored, assuming that (like in the one-dimensional LQ case discussed above) it may only influence how the optimum is approached (backwards in time) but remains inessential for describing asymptotic behavior of the optimal control.", "This path integral is (most conveniently) evaluated by changing to the Fourier (frequency) domain, expressing pair correlation function as the frequency integral, and then relating it to the derivative of the log-GF over $\\theta $ , $&& \\langle x_i x_j\\rangle =\\int \\limits _{-\\infty }^{+\\infty }\\frac{d\\omega }{2\\pi }\\left(\\omega ^2 V^{-1}+\\mu ^*V^{-1}\\mu +\\theta \\tilde{Q}\\right)^{-1}_{ij},\\\\&& \\frac{\\partial \\log {\\cal Z}(\\theta ;K)}{\\partial \\theta }=-\\frac{T}{2}\\langle x^*\\tilde{Q}x\\rangle .$ Here in Eqs.", "(REF ,) the averaging is over the path integral measure described by Eq.", "(REF ).", "Further, evaluating the integral over $\\theta $ , fixing normalization, ${\\cal Z}(0;K)=1$ , and using the standard formula of matrix calculus, $d/d\\theta \\log \\det (\\theta \\cdot 1+D)=\\mbox{tr}((\\theta \\cdot 1+D)^{-1})$ , where 1 stands for the unit matrix, one arrives at the following expression $\\log ({\\cal Z}(\\theta ;K))=-\\frac{T}{2}\\int \\limits _{-\\infty }^{+\\infty }\\frac{d\\omega }{2\\pi }\\log \\frac{\\det \\left(\\omega ^2 V^{-1}+\\mu ^*V^{-1}\\mu +\\theta \\tilde{Q}\\right)}{\\det \\left(\\omega ^2 V^{-1}+\\mu ^*V^{-1}\\mu \\right)},$ which is asymptotically exact at $T\\rightarrow \\infty $ .", "Moreover, one can show that for any (spatially) finite system next order corrections to the rhs of Eq.", "(REF ) are $O(1)$ .", "Note that this representation (REF ) of the log-GF, as an integral over frequency of a log-det, is similar to the relation discussed in Section 3 of [29] in the context of linking the RS-LQG control to the maximum entropy formulation of the $H_\\infty $ control.", "The log-det has also appeared in [38] where statistics of currents were analyzed in general non-equilibrium (off-detailed-balance) linear system.", "To gain intuition let us first analyze Eq.", "(REF ) in the simple scalar case where the integral on the rhs can be evaluated analytically $\\log ({\\cal Z}(\\theta ;K)) =\\frac{T}{2}\\left(\\mu -\\sqrt{\\mu ^2+\\theta \\tilde{Q} V}\\right).$ Substituting this expression into Eq.", "(REF ) and estimating the integral over $\\theta $ in a saddle-point approximation (justified when $T$ is large) one arrives at the LD expression (REF ) where ${\\cal S}_*(j)=\\frac{V(Q+RK^2)}{16 j}+\\frac{(BK-A)^2j}{V(Q+RK^2)}-\\frac{BK-A}{2}.$ The LD function is obviously convex and it is defined only for positive $j$ .", "(The asterisk marks, as before, that the average and the probability are computed conditioned to yet unspecified $K$ .)", "${\\cal S}_*(j)$ achieves its minimum at, $\\langle j \\rangle _* =-T^{-1}\\left.", "\\partial \\log {\\cal Z}/\\partial \\theta \\right|_{\\theta =0}=V\\tilde{Q}/(4\\mu )$ , and shows linear asymptotic, ${\\cal S}_*(j)\\approx j \\mu ^2/(V\\tilde{Q})$ , at $j\\gg \\langle j\\rangle $ .", "Note, that the aforementioned asymptotic is associated with the cut-singularity in the complex $\\theta $ plane of the GF expression (REF ).", "Indeed, substituting Eq.", "(REF ) into Eq.", "(REF ) and shifting the integration contour to the left, thus forcing it to surround anti-clockwise the $]-\\infty ;\\theta _*=-\\mu ^2/(V\\tilde{Q})]$ cut, and then estimating the integral by a small part of the contour surrounding vicinity of the cut tip at $\\theta _*$ , we arrive at the aforementioned $j\\gg \\langle j\\rangle $ asymptotic, ${\\cal S}(j)\\approx -j\\theta _*$ .", "Returning back to analysis of the general formulas (REF ,REF ), one observes that even though to reconstructing $S_*(j)$ in its full integrity explicitly as a function of $K$ does not look feasible, we can still, motivated by the scalar case analysis, make some useful general statements about both the average, $\\langle j\\rangle _*$ , and the $j\\gg \\langle j\\rangle _*$ asymptotic of ${\\cal S}_*(j)$ .", "We will start from the latter problem.", "For analysis of the tail the key object of interest is the $\\det $ in Eq.", "(REF ) considered at zero frequency, $\\omega =0$ .", "Specifically, one aims to find the zero of the determinant with the largest real value: $\\theta _*=\\max \\limits _\\theta \\mbox{Re}\\left(\\theta \\right)_{\\det \\left(\\mu ^*V^{-1}\\mu +\\theta \\tilde{Q}\\right)=0}.$ Indeed, any zero (there might be many of these in the general matrix case) marks the tip of the respective cut singularity of ${\\cal Z}(\\theta ;K)$ in the complex $\\theta $ -plane.", "Then, the tail, $j\\gg \\langle j\\rangle _*$ , asymptotic of the LD function becomes, ${\\cal S}_*(j)=-j\\theta _*$ .", "Note, that this linear in $j$ estimation is valid only in the case of a finite system, when the set (spectrum) of zeros (defined by the condition in Eq.", "(REF ) is discrete.", "In the case of an infinite system, when the spectrum of zeros becomes quasi-continuous, one needs to account for the multiple zeros, as illustrated in the \"string\" example of Section .", "To evaluate $\\langle j\\rangle _*$ (as a function of $K$ ) in the general case one first analyzes it in the time representation.", "Substituting the $u=-Kx$ ansatz with constant $K$ in Eq.", "(REF ), expressing $x(t)$ formally as an integral over time (for a given realization of the noise), substituting the result into Eq.", "(), averaging over noise, and then taking the $T\\rightarrow \\infty $ limit one arrives at $\\langle j\\rangle _*=\\frac{1}{2}\\int \\limits _0^\\infty \\mbox{tr}\\left(Ve^{-\\mu ^*t}\\tilde{Q}e^{-\\mu t}\\right) dt=\\frac{1}{2}\\left.\\mbox{tr}\\left(V\\Pi \\right)\\right|_{\\mu ^*\\Pi +\\Pi \\mu =\\tilde{Q}},$ where the latter expression is implicit (as the condition is a matrix one, thus not resolvable explicitly in general) function of $K$ .", "It is straightforward but tedious to check (introducing matrix Lagrangian multiplier for the condition in Eq.", "(REF ) and making variation over $K$ and $\\Pi $ ) that optimization of Eq.", "(REF ) over $K$ results in the algebraic Riccatti equation equivalent to Eq.", "(REF ) with the first term ignored.", "Note that the fact that the optimal control derived from the optimization of the average cost function in the stochastic case coincides with the result of the deterministic optimization (ignoring stochasticity) is the fact very well known in the control theory.", "This fact is akin to a similar statistical physics statement naturally emerging in analysis of the linear stochastic systems driven by white-Gaussian noise and characterized by flux/current (see e.g.", "[38]): the average flux corresponds to the minimum of a functional quadratic in the flux.", "The optimal value of the functional in the deterministic case saturates to a constant at $T\\rightarrow \\infty $ , while in the stochastic case the average optimal cost grows with time linearly.", "Asymptotic convergence of the two seemingly different schemes to the same optimal control is thus an indication of the asymptotic self-consistency of the linear ansatz (REF ).", "Differentiating Eq.", "(REF ) over $\\theta $ and then setting $\\theta $ to zero, one derives an alternative (to Eq.", "(REF )) representation for the average rate of the cost function conditioned to $K$ $\\langle j\\rangle _*=\\int \\limits _{-\\infty }^{+\\infty }\\frac{d\\omega }{4\\pi }\\mbox{tr}\\left(\\left(\\omega ^2V^{-1}+\\mu ^*V^{-1}\\mu \\right)^{-1}\\tilde{Q}\\right).$ Note that comparison of Eqs.", "(REF ,REF ,REF ) also allows to derive expression for the derivative of the log-GF as a time integral, and then have it presented in an implicit algebraic form $-T^{-1}\\partial _\\theta \\log {\\cal Z}(\\theta ;K)&=&\\int \\limits _{-\\infty }^{+\\infty }\\frac{d\\omega }{4\\pi }\\mbox{tr}\\left(\\left(\\omega ^2V^{-1}+\\mu ^*V^{-1}\\mu +\\theta \\tilde{Q}\\right)^{-1}\\tilde{Q}\\right)\\nonumber \\\\&=&\\frac{1}{2}\\mbox{tr}\\left(\\tilde{V}\\Pi \\right)_{\\mu ^*\\Pi +\\Pi \\mu =\\tilde{Q}}\\\\&=&\\frac{1}{2}\\int \\limits _0^\\infty \\mbox{tr}\\left(\\tilde{V}e^{-\\mu ^*t}\\tilde{Q}e^{-\\mu t}\\right) dt,$ where $\\tilde{V}=V(1+\\theta V(\\mu ^*)^{-1}\\tilde{Q}\\mu ^{-1})^{-1}$ ." ], [ "Optimal Asymptotic Controls", "In this Section we formulate the RS-LQ, TO-LQ and CC-LQ asymptotic schemes in the general vector/matrix form as an optimization over $K$ .", "(Note that the asymptotic LQ scheme was already stated as a minimum of Eq.", "( REF ), or equivalently of Eq.", "(REF ) in the preceding Section.)", "Then we illustrate these formulations on the scalar example.", "$K_{\\mbox{RS}}$ , which is asymptotically optimal for the RS-LQ control considered at $\\theta >0$ , is found by maximizing ${\\cal Z}(\\theta ;K)$ .", "Using Eq.", "(REF ) one derives $\\min \\limits _K \\left.\\int \\limits _{-\\infty }^{+\\infty }d\\omega \\log \\frac{\\det \\left(\\omega ^2 V^{-1}+(BK-A)^*V^{-1}(BK-A) +\\theta (Q+K^*RK)\\right)}{\\det \\left(\\omega ^2 V^{-1}+(BK-A)^*V^{-1}(BK-A)\\right)}\\right|_{\\mbox{Re}(\\lambda (BK-A))>0},$ where $\\mbox{Re}(\\lambda (BK-A))>0$ denotes the stability condition ensuring that the real values of all the eigen-values of $BK-A$ are positive.", "Note that constancy of the stationary RS-LQ optimal control was proven in [26], therefore making our approach self-consistent.", "An alternative, but obviously equivalent, formulation of the RS-LQ optimal control consists in minimizing $-T^{-1}\\partial _\\theta \\log {\\cal Z}(\\theta ;K)$ .", "Going along this path and utilizing Eq.", "() one arrives at $\\min _{K,\\Pi }\\frac{1}{2}\\mbox{tr}\\left(V(1+\\theta V(\\mu ^*)^{-1}\\tilde{Q}\\mu ^{-1})^{-1}\\Pi \\right)_{\\mu ^*\\Pi +\\Pi \\mu =Q+K^*RK},$ generalizing the LQ formulation stated in the preceding Section as the minimization of Eq.", "(REF ).", "Solving Eq.", "(REF ) is reduced to analysis of the respective generalization of the Riccati equations which can than be turned into a linear eigen-value problem described within the so-called Hamiltonian approach to the RS-LQ problem discussed in [28].", "From Eq.", "(REF ), and assuming time-independence of the control, one can state the general asymptotic TO-LQ optimum utilizing Eqs.", "(REF ,REF ) as an optimization of a double integral over frequency and $\\theta $ .", "However, in practice one is interested to discuss the TO-LQ optimization only at sufficiently large values of the cost, $j T$ .", "Using analysis of the preceding Section one derives the desired double asymptotic (valid at large $T$ and large $j$ ) and simpler to state expression describing $K_{\\mbox{TO}}$ $\\min \\limits _K \\max \\limits _\\theta \\mbox{Re}\\left(\\theta \\right)_{\\begin{array}{c}\\mbox{Re}(\\lambda (BK-A))>0\\\\\\det \\left(\\mu ^*V^{-1}\\mu +\\theta \\tilde{Q}\\right)=0\\end{array}},$ where $\\max $ is over complex $\\theta $ and the optimal LD value of the PDF tail is exponential, $\\log {\\cal P}_{\\mbox{TO}}({\\cal J})\\approx -\\mbox{Re}(\\theta _{\\mbox{TO}}){\\cal J},$ with $\\theta _{\\mbox{TO}}$ solving Eq.", "(REF ).", "Note that the $\\det =0$ condition in Eq.", "(REF ) is reminiscent of the $\\mu $ -measure which is the key element of the robust control approach, see [35], [37] and references therein.", "In the same double asymptotic (large $T$ and large $j$ ) regime the optimal CC-LQ control (REF ) is given by $\\left.\\min \\limits _K \\mbox{tr}\\left(V\\Pi \\right)\\right|_{\\begin{array}{c}\\mbox{Re}(\\lambda (BK-A))>0\\\\\\mu ^*\\Pi +\\Pi \\mu =\\tilde{Q},\\\\\\max \\limits _\\theta (\\mbox{Re}(\\theta ))_{\\det (\\mu ^*V^{-1}\\mu +\\theta \\tilde{Q})=0}\\ge \\frac{\\log (1/\\varepsilon )}{jT}.\\end{array}}$ Note that unlike Eqs.", "(REF ,REF ), Eq.", "(REF ) does not have valid solutions for any value of the $\\log (1/\\varepsilon )/{\\cal J}$ ratio.", "In fact, it is clear from Eq.", "(REF ) that to have a nonempty solution of Eq.", "(REF ) one needs to require that $\\mbox{Re}(\\theta _{\\mbox{TO}})jT\\le \\log (1/\\varepsilon )$ .", "Once the optimum solution is found, one estimates the LD asymptotic of the cost function PDF by an expression similar to the one given by Eq.", "(REF ), with $\\mbox{TO}$ subscript replaced by the $\\mbox{CC}$ one." ], [ "Scalar case", "In the remainder of this Section we illustrate all of the aforementioned formulas on the scalar example.", "In this simple case integral on the rhs of Eq.", "( REF ) is equal to $2\\pi \\left(\\sqrt{(BK-A)^2+V\\theta (Q+RK^2)}-\\left(BK-A\\right)\\right),$ resulting in the following optimal value $K_\\theta =\\frac{A/B+\\sqrt{{A^2}/{B^2}+{Q}/{R}+QV\\theta /B^2}}{1+VR\\theta /B^2}.$ The large deviation tail of the PDF of $j$ at a given $K$ can be extracted from Eq.", "(REF ): $j\\gg \\langle j\\rangle :\\quad {\\cal S}_*(j)\\approx \\frac{(BK-A)^2}{V(Q+RK^2)} j +o(j).$ Optimizing the PDF over $K$ we find two different cases depending on the sign of $A$ .", "At $A>0$ coefficient in front of the linear in $j$ term on the rhs of Eq.", "(REF ) grows monotonically with $K$ from the $(A/B,+\\infty )$ interval.", "To find the optimal value of $K$ in this case one has to take into the $O(j)$ term thus deriving : $A>0:\\quad K_{\\mbox{TO}}=\\sqrt{4Aj}{RV},\\quad \\log {\\cal P}_{\\mbox{TO}}\\approx -\\frac{B^2jT}{RV}.$ In the other case of $A=-|A|<0$ the linear coefficient in Eq.", "(REF ) reaches its maximum at $K=BQ/(R|A|)$ , thus resulting in $A<0:\\quad \\log {\\cal P}_{\\mbox{TO}}\\approx -\\frac{jT}{RVQ}\\left(RA^2+B^2Q\\right).$ Finally, the CC-optimal formula (REF ) has no solution if $B^2j/(RV)<c$ in the $A>0$ case and if $j\\left(RA^2+B^2Q\\right)/(RVQ)<c$ in the $A<0$ case.", "(Here we assume, as above, that $\\epsilon (0;T)=\\exp (-c T)$ .)", "When $\\epsilon $ is chosen sufficiently small (i.e.", "$c$ is sufficiently large), the feasibility domain in Eq.", "(REF ) is not empty and one distinguishes two regimes depending on how $K_\\varepsilon $ , defined by $K_\\varepsilon =\\frac{1}{1-\\kappa }\\left(\\frac{A}{B}+\\sqrt{\\kappa \\left(\\frac{A^2}{B^2}+(1-\\kappa )\\frac{Q}{R}\\right)}\\right),\\quad \\kappa =\\frac{cVR}{B^2j},$ compares with $K_0$ , which is the bare LQ optimal value correspondent to $K_\\theta $ from Eq.", "(REF ) evaluated at $\\theta =0$ .", "One derives $K_{\\mbox{CC}}=\\max \\left(K_\\varepsilon , K_0\\right),$ where of the two regimes one is achieved within the interior of the optimization domain (tail constraint is not restrictive) while the other one corresponds to the tail imposed by the boundary of the domain.", "It is worth noting that (REF ) is valid for both signs of $A$ ." ], [ "Example of a String", "In this Section we discuss an explicitly solvable example of an infinite system where the set of zeros (of the determinant in the condition of Eq.", "(REF )) forms a quasi-continuous spectrum.", "Consider a string, defined as an over-damped system of multiple bids on a line connected to each other by elastic springs of strength $D$ , stretched by the linear force of the strength $A$ and subject to Langevien driving: $&& \\partial _t x_j=Ax_j +D(x_{j+1}+x_{j-1}-2x_j)+B u_j +\\xi _j, \\\\&& J=\\frac{1}{2}\\int _0^T dt\\ \\sum _j (Qx_j^2+R u_j^2),$ where $j=1,\\cdots ,N$ , $x_j$ marks position of the $j$ -th bid of the string, and the zero-mean white-Gaussian noise is distributed as in Eq.", "(REF ) with $V_{ij} = V\\delta _{ij}$ .", "$u_j$ in Eq.", "(REF ) stands for control.", "We are looking for a time-independent linear in $x$ control, assuming that the control acts uniformly on all bids of the string, i.e.", "$u_j= - K x_j$ .", "Let us also assume that the string is periodic with the period $N$ .", "Then, solution of Eq.", "(REF ) allows expansion in the series over spatial harmonics $x_j=\\sum _{j=1}^N \\exp (i q(j/N)) x_q,$ with the wave vector, $q$ , from the interval, $-\\pi <q<\\pi $ , and resulting in the following separated equations for the individual harmonics $\\partial _t x_q=Ax_q -2D(1-\\cos q) x_q- B K x_q +\\xi _q.$ Repeating the steps leading to (REF ) one arrives at $\\log {\\cal Z}=\\frac{T}{2}\\sum _q\\left(BK-A +2D(1-\\cos q)-\\sqrt{(BK-A+2D(1-\\cos q))^2+V(Q+RK^2)\\theta }\\right).$ We choose to analyze only the most interesting regime, $D\\gg BK- A$ , when a nontrivial collective behavior emerges.", "Then, in the long wave-length, $1-\\cos q \\rightarrow q^2/2$ , and continuous, $\\sum _q\\rightarrow (N/2\\pi ) \\int dq$ , limits one derives $&& \\log {\\cal Z}=-\\frac{TN}{3\\pi }\\frac{(BK-A)^{3/2}}{D^{1/2}}(1+s)^{1/4} i\\left((1+\\sqrt{1+s}){\\cal K}\\left(\\frac{1}{2}-\\frac{1}{2\\sqrt{1+s}}\\right)-2 {\\cal E}\\left(\\frac{1}{2}-\\frac{1}{2\\sqrt{1+s}}\\right) \\right), \\\\ &&s=\\frac{V(Q+RK^2)\\theta }{(BK-A)^2},$ where one utilizes the standard ${\\cal K},{\\cal E}$ notations for the elliptic functions.", "Expression on the rhs of Eq.", "(REF ) shows a singularity at $s=-1$ , coinciding with the singularity (in the complex $\\theta $ plane) observed in the scalar case at $\\theta _*$ .", "Substituting Eq.", "(REF ) into Eq.", "(REF ) and evaluating the integral over $\\theta $ in the saddle-point approximation one arrives at ${\\cal S}_*(j)\\approx \\frac{\\sqrt{2}\\ N(BK-A)^{3/2}}{3\\pi D^{1/2}}+\\theta _* j.$ Juxtaposing the string expression Eq.", "(REF ) to the scalar one Eq.", "(REF ) one notes different behaviors with respect to $BK-A$ .", "Optimizing Eq.", "(REF ) over $K$ at a given large value $J=jT$ , one obtains the same tail expression, second formula in (REF ), however with another optimal control $K^{5/2}_{\\mbox{str}}=\\frac{2^{3/2}\\pi A D^{1/2}}{RVTN B^{1/2}} J,$ replacing the first formula in Eq.", "(REF ).", "Note that in the string case the optimal $K$ scales as $J^{2/5}$ which should be contrasted to the $J^{1/2}$ scaling in the scalar case from Eq.", "(REF )." ], [ "Conclusions and Path Forward", "This manuscript contributes the subject in control theory - designing control scheme with some guarantees not only on the average of the cost functions but also on fluctuations, specifically extreme fluctuations related to the tail of the cost function PDF.", "We consider linear, first order in time derivative, stochastic system of the Langevien type subject to minimization of a quadratic cost function and also with (chance) constraints imposed on the tail of the cost function PDF.", "In the stationary regime of large time, when control is sufficient to make the system stable, we reduce the stochastic dynamic problem of the \"field theory\" type to static optimization analysis with objectives and constraints stated in a matrix form.", "This type of reduction is unusual in the system lacking the fine-tuned Fluctuation Dissipation relation between relaxational and stochastic terms.", "On the other hand, the progress made is linked to linearity of the underlying stochastic systems which allowed, as in some problems of passive scalar turbulence [39], [40], [41] and driven linear-elastic systems [42], [38], to formally express solution for the system trajectory as an explicit function of the noise realization.", "Besides that, main technical ingredients, which allowed us to derive the results, consisted in making plausible assumption about the structure of the control (linear in the state variable and frozen in time), and then performing asymptotic evaluations of the cost functions statistics conditioned to the value of the cost matrix.", "Techniques of path integral, spectral analysis and large deviation estimations were used.", "We tested results on the simple scalar case and illustrated utility of the method on an exemplary high-dimensional system (1d chain of particles connected in a string).", "We plan to continue exploring the interface between control theory and statistical physics addressing the following challenges.", "Computational feasibility of the main formulas of the paper, stating RS-, TO- and CC- controls in Eqs.", "(REF ,REF ,REF ) as static optimization problems, need to be analyzed for large systems and networks.", "After all main efforts in the applied control theory go into designing efficient algorithms for discovering optimal, or close to optimal, control, and we do plan to contribute this important task.", "Therefore, further analysis is required to answer the important practical question: if the static formulations of the newly introduced TO-QG and CC-QG controls allow computationally favorable exact or approximate expressions in terms of convex optimizations?", "We also plan to study weakly non-linear stochastic systems through a singular perturbation stochastic diagrammatic technique of the Martin-Siggia-Rose type [43].", "Besides, some of the methods we used in the manuscript, especially related to large deviation analysis, are not restricted to linear systems.", "Our preliminary tests show that effects of the non-linearity on the PDF tail are seriously enhanced in comparison with how the same nonlinearity influences the average case control.", "It will be interesting to study TO- and CC- versions of the path-integral nonlinear control problems discussed in [13], [14], [15], [16].", "These problems, in their standard min-cost formulations, allow reduction (under some Fluctuation-Dissipation-Theorem like relations between the form of control, covariance matrix of the noise and the cost function) from the generally non-linear Hamilton-Jacobi-Bellman equations for the optimal cost function to a linear equation of a Schrödinger type.", "The effects of partial observability and noise in the observations can be easily incorporated in both TO- and CC- schemes discussed in the paper.", "In fact this type of generalization is standard and widespread in the control theory, where for example the LQG (Linear-Quadratic-Gaussian) control generalizes the LQ control.", "In terms of relevance to an application, this work was motivated by recent interest and discussions related to developing new optimization and control paradigms for power networks, so-called smart grids.", "In this application, strong fluctuations associated with loads and renewable generation, electro-mechanical control of generation, desire to make the energy production cheaper while also (and most importantly) maintaining probabilistic security limitations of the chance-constrained type - all of the above make the theoretical model discussed in this paper an ideal framework to consider.", "In particular, we plan to extend the approaches of [44], [45] and modify and apply the theory developed in this manuscript to design a multi-objective Chance Constrained Optimum Power Flow including better control of generation, loads and storage resources in power grids.", "We also anticipate that some of the models and results discussed in the paper are of interest for problems in statistical micro- and bio- fluidics, focusing on adjusting characteristics of individual molecules (polymers, membranes, etc) and also aimed at modifying properties of the medium (non-Newtonian flows) macroscopically.", "Time independent and linear nature of the control schemes discussed in the paper make them especially attractive for these applications.", "Natural constrains, e.g.", "associated with the force-field (optical or mechanical) as well as with some other physical limitations, could be incorporated into control as single- or multi-objective cost functions.", "We are thankful to D. Bienstock, L. Gurvits, H.J.", "Kappen, K. Turitsyn and participants of the \"Optimization and Control Theory for Smart Grids\" project at LANL for motivating discussions and remarks.", "Research at LANL was carried out under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No.", "DE C52-06NA25396." ] ]

[ [ "Introduction to the nonequilibrium functional renormalization group" ], [ "Abstract In these lectures we introduce the functional renormalization group out of equilibrium.", "While in thermal equilibrium typically a Euclidean formulation is adequate, nonequilibrium properties require real-time descriptions.", "For quantum systems specified by a given density matrix at initial time, a generating functional for real-time correlation functions can be written down using the Schwinger-Keldysh closed time path.", "This can be used to construct a nonequilibrium functional renormalization group along similar lines as for Euclidean field theories in thermal equilibrium.", "Important differences include the absence of a fluctuation-dissipation relation for general out-of-equilibrium situations.", "The nonequilibrium renormalization group takes on a particularly simple form at a fixed point, where the corresponding scale-invariant system becomes independent of the details of the initial density matrix.", "We discuss some basic examples, for which we derive a hierarchy of fixed point solutions with increasing complexity from vacuum and thermal equilibrium to nonequilibrium.", "The latter solutions are then associated to the phenomenon of turbulence in quantum field theory." ], [ "Introduction", "Thermal equilibrium properties of many-body or field theories are known to be efficiently classified in terms of renormalization group fixed points.", "A particularly powerful concept is the notion of infrared fixed points which are characterized by universality.", "These correspond to critical phenomena in thermal equilibrium, where the presence of a characteristic large correlation length leads to independence of long-distance properties from details of the underlying microscopic theory.", "In contrast, a classification of properties of theories far from thermal equilibrium in terms of renormalization group fixed points is much less developed.", "$\\nonumber $ The notion of universality or criticality far from equilibrium is to a large extent unexplored, in particular, in relativistic quantum field theories.", "Here, the strong interest is mainly driven by theoretical and experimental advances in our understanding of early-universe cosmology as well as relativistic collision experiments of heavy nuclei in the laboratory.", "In the latter contexts, a particular class of nonthermal fixed points has attracted much interest in recent years.", "It is associated to the phenomenon of turbulence in quantum field theory, where a universal power-law behavior describes the transport of conserved quantities.", "Traditionally, turbulence is associated mostly with the dynamics of vortices in fluids [1] but also nonlinear waves can show turbulent behavior [2].", "In particular, it is well-known that interacting quantum field theories can lead to nonlinear dynamics and wave turbulence, even for very weakly coupled theories.", "Among the best studied theoretical examples in relativistic quantum field theory are scalar inflaton models for the dynamics of the early universe [3].", "In a large class of models, the strongly accelerated expansion of the universe after the Big Bang is followed by turbulent behavior of the inflaton field before thermal equilibrium is achieved.", "The connection to the well-established phenomenon of weak wave turbulence in the presence of small nonlinearities has been studied in great detail [4].", "Here, weak wave turbulence is associated to an energy cascade from small to high wave numbers.", "Only recently it has been realized that the direct energy transport towards higher wavenumbers is part of a dual cascade, in which also an inverse particle flux towards the infrared at small wave numbers occurs [5].", "One of the striking consequences is Bose condensation far from equilibrium [6].", "Similar scaling phenomena may also occur for gauge field dynamics in the context of heavy-ion collisions at sufficiently high energies [7], [8], [9], [10], or also for the nonrelativistic dynamics of ultracold atoms [11], [12], [13].", "The emergence of same macroscopic scaling phenomena from very different underlying microscopic physics is a formidable manifestation of universality far from equilibrium.", "Understanding the dominant collective phenomena in quantum field theories far from equilibrium represents a major challenge.", "Important phenomena, such as the infrared particle cascade and subsequent Bose condensation mentioned above, are genuinely nonperturbative and require suitable approximation techniques.", "Here a nonequilibrium functional renormalization group approach [14], [15], [16], [17], [18], [19], [20], [21], or related real-time functional integral techniques based on $n$ -particle irreducible (nPI) effective actions [22], can serve as a very useful means to gain analytic understanding.", "Other implementations of the renormalization group idea in this context include the so-called numerical renormalization group approach [23], the time-dependent density matrix renormalization group [24], the real-time renormalization group in Liouville space [25], or flow equations describing infinitesimal unitary transformations [26].", "These approaches can be complemented by numerical simulations in (classical-statistical) nonequilibrium lattice theories [27], [28], [29], [30] or using kinetic descriptions [31] in their respective range of applicability.", "In these lectures we discuss basic properties of the nonequilibrium renormalization group for the scale dependent generating functional of 1PI correlation functions in relativistic quantum field theory, following closely Ref. [18].", "Concentrating on the explicit example of a $N$ -component scalar field theory allows us to focus on the relevant differences to Euclidean treatments in vacuum or thermal equilibrium, without introducing too much formalism and abstract notation.", "With the help of standard references from the functional renormalization group in Euclidean spacetime [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], one can apply these considerations in a similar way to include fermions or gauge fields.", "We begin in section by emphasizing some important differences between thermal equilibrium and nonequilibrium.", "This will help us to understand why there exists a hierarchy of fixed point solutions with increasing complexity from vacuum and thermal equilibrium to nonequilibrium.", "In section we introduce the notion of scaling behavior far from equilibrium in the context of weak wave turbulence for the description of stationary transport of conserved quantities.", "The nonequilibrium functional renormalization group is introduced in section , where we derive the relevant flow equations.", "In section we solve the flow equations in an approximation based on a resummed large-$N$ expansion to next-to-leading order.", "The results are used in section to determine scaling exponents for nonthermal fixed points.", "We summarize and give an outlook in section .", "All information about a quantum theory is encoded in its correlation functions for a given density matrix $\\varrho $ .", "In thermal equilibrium, for the case of a canonical ensemble with inverse temperature $\\beta \\equiv 1/T$ and Hamilton operator $H$ , the density matrix is given by $\\varrho ^{\\rm (eq)} \\sim e^{-\\beta H}$ and it is normalized such that $\\operatorname{Tr}\\varrho ^{\\rm (eq)} = 1$ .", "A thermal real-time correlation function for a Heisenberg field operator $\\Phi (x)$ is given by the time-ordered trace rCl G(eq)(x-y) = Tr{(eq)  T0 (x) (y) } = (x) (y)eq  (x0-y0) + (y) (x)eq  (y0-x0)  , for a two-point function, and involves $n$ fields for a $n$ -point function.", "Here, $\\textrm {T}_0$ is the time-ordering operator, $x=(x^0,\\mathit {x})$ denotes the time $x^0$ and space $\\mathit {x}$ variablesWe use a metric with signature $(+,-,-,-)$ .", "and, to be specific, we consider real scalar fields.", "Nonequilibrium typically requires the specification of a density matrix $\\varrho (t_0)$ at some initial time $t_0$ where $\\varrho (t_0) \\ne \\varrho ^{\\rm (eq)}$ .", "The task of nonequilibrium quantum field theory is then to determine the real-time evolution of correlation functions such as the two-point function $ G(x,y) = \\operatorname{Tr}\\left\\langle {\\varrho (t_0) \\,\\textrm {T}_0 \\Phi (x) \\Phi (y)}\\right\\rangle \\equiv \\left\\langle { \\textrm {T}_0 \\Phi (x) \\Phi (y)}\\right\\rangle ~, $ for times $x^0, y^0 > t_0$ .", "In general, translational invariance does not hold out of equilibrium and thus, two-point functions will depend on both $x$ and $y$ separately, i.e.", "$ G(x,y) = F(x,y) - \\frac{i}{2} \\rho (x,y) \\operatorname{sgn}(x^{0} - y^{0}) ~.", "$ In the second line of (REF ) we introduced the statistical two-point function $F(x,y)$ and the spectral function $\\rho (x,y)$ .", "These are given by the expectation values of the anti-commutator of the scalar fieldIn this section, we will assume for simplicity that we are in the symmetric phase where the field expectation value $\\phi (x) \\equiv \\left\\langle {\\Phi (x)}\\right\\rangle $ vanishes.", "Otherwise the connected statistical function is given by $F(x,y) = \\frac{1}{2} \\left\\langle {\\lbrace \\Phi (x), \\Phi (y) \\rbrace }\\right\\rangle - \\phi (x) \\phi (y)$ .", "$ F(x,y) = \\frac{1}{2} \\left\\langle {\\lbrace \\Phi (x), \\Phi (y) \\rbrace }\\right\\rangle ~, $ and the commutator $ \\rho (x,y) = i \\left\\langle {[ \\Phi (x) , \\Phi (y)]}\\right\\rangle ~, $ respectively.", "The decomposition (REF ) follows from (REF ) with the elementary properties of the Heaviside step function, $\\theta (x^{0} - y^{0}) + \\theta (y^{0} - x^{0}) = 1$ and $\\operatorname{sgn}(x^{0} - y^{0}) = \\theta (x^{0} - y^{0}) - \\theta (y^{0} - x^{0})$ .", "The spectral function is also related to the retarded and advanced Green's function by $G^R(x,y) = \\theta (x^0-y^0) \\rho (x,y) = G^A(y,x)$ , respectively.", "Loosely speaking, the spectral function $\\rho (x,y)$ determines which states are available while the statistical function $F(x,y)$ contains the information about how often a state is occupied [22]." ], [ "Absence of a fluctuation-dissipation relation", "In contrast to the general nonequilibrium case, it is an important simplification of thermal or vacuum theories that the statistical (REF ) and spectral function (REF ) are related by the so-called fluctuation-dissipation relation.", "We discuss this relation for the two-point correlation function with a canonical density matrix $\\varrho ^{\\rm (eq)} \\sim e^{-\\beta H}$ .", "For times $t > 0$ the thermal two-point function (REF ) reads rCl Tr{ e-H (t,x) (0,y) } = Tr{ e-H eH (0,y) e-H=  (-i ,y) (t,x) }  .", "In the second line we have used the invariance of the trace under cyclic changes and inserted $e^{-\\beta H}e^{\\beta H} = 1$ .", "From the real-time Heisenberg evolution $\\Phi (t,\\mathit {x}) = e^{i H t} \\Phi (0,\\mathit {x}) e^{- i H t}$ , we may define in complete analogy $\\Phi (-i \\beta ,\\mathit {x}) \\equiv e^{\\beta H} \\Phi (0,\\mathit {x}) e^{- \\beta H}$ as an extension to `imaginary times' [43].", "We can state the above relation directly in terms of the statistical (REF ) and spectral function (REF ) using $\\langle \\Phi (t,\\mathit {x}) \\Phi (0,\\mathit {y}) \\rangle =F(t,0;\\mathit {x},\\mathit {y}) - \\frac{i}{2} \\rho (t,0;\\mathit {x},\\mathit {y})$ , and correspondingly for the r.h.s.", "of (REF ).", "Since equilibrium is spacetime translation invariant, (REF ) then reads rCl ( F(eq)(x-y) - i2(eq)(x-y) )|y0 = 0 = ( F(eq)(x-y) + i2(eq)(x-y) )|y0 = -i   .", "Translation invariance also makes it convenient to consider the Fourier transform with real four-momentum $p=(p^0,\\mathit {p})$ , that is $ F^{\\rm (eq)}(x-y) = \\int \\!\\!\\frac{d^{d+1} p}{(2\\pi )^{d+1}} e^{-i p (x-y)} F^{\\rm (eq)}(p) ~, $ and equivalently for $\\rho ^{\\rm (eq)}(x-y)$ .", "Taking for a moment for granted that these integrals can be properly regularized and defined for the considered quantum field theory, we obtain from (REF ) rCl F(eq)(p) - i2 (eq) (p) = ep0 ( F(eq)(p) + i2 (eq) (p) )  .", "This can be written in the form of a fluctuation-dissipation relation as $ F^{\\rm (eq)}(p) = -i \\left( n_{\\rm BE}(p^{0}) + \\frac{1}{2} \\right) \\rho ^{\\rm (eq)} (p) ~, $ where $ n_{\\rm BE}(p^{0}) = \\frac{1}{e^{\\beta p^{0}} - 1} ~, $ is the Bose-Einstein distribution function.", "Since this distribution depends on frequency $p^0$ and not on spatial momenta $\\mathit {p}$ , it relates the anti-commutator expectation value of fields ($F$ ) and the respective commutator ($\\rho $ ) in a nontrivial way.", "Moreover, at zero temperature the distribution function $n_{BE}$ is zero and the anti-commutator and commutator are directly proportional to each other.", "In contrast, for a nonequilibrium density matrix no such constraints exist in general.", "As a consequence, $F$ and $\\rho $ are linearly independent out of equilibrium.", "In particular, the absence of a fluctuation-dissipation relation will allow us to observe new types of scaling solutions, beyond those known from thermal equilibrium or the vacuum." ], [ "Infrared fixed points", "Renormalization group fixed points correspond to scaling solutions for correlation functions.", "Before computing them from first principles in the main part of these lectures, we illustrate here heuristically possible scaling behaviors in and out of equilibrium.", "For that purpose, we consider translationally invariant spectral and statistical two-point functions, $F(x-y)$ and $\\rho (x-y)$ , assuming also spatial isotropy to limit the number of possible scaling exponents.", "The Fourier transforms, $F(p^0,\\mathit {p})$ and $\\rho (p^0,\\mathit {p})$ , then depend on real frequency and momenta, where any scaling ansatz has to take into account the different possible scalings of spatial momenta vs. frequencies.", "This difference is described by the `dynamical critical exponent' $z$ .", "Furthermore, we denote the overall scaling of the statistical two-point function by the `occupation number exponent' $\\kappa $ and of the spectral function by the `anomalous dimension' $\\eta $ .", "The scaling behavior may then be described as rCl F(p0,p) = s2 + F(sz p0 , s p)  , (p0 , p) = s2- (sz p0, s p)  , for any real scaling parameter $s > 0$ .", "Before we consider such a scaling behavior for the nonequilibrium case, it is very instructive to first insert (REF ) and (REF ) into the fluctuation-dissipation relation (REF ) in order to see how this constraint relates the different exponents.", "We restrict the discussion to frequencies (and momenta) much smaller than the temperature $T$ .", "This would be the relevant range for scaling behavior, for instance, near second-order phase transitions in thermal equilibrium.", "In the infrared, where $p^{0} \\ll T$ , we see that the distribution function (REF ) assumes the scaling form $ n_{\\rm BE} (p^{0}) \\sim \\frac{T}{p^{0}}~.", "$ In particular, occupation numbers become large such that the `proportionality factor' $n_{\\rm BE}(p^0) + 1/2$ appearing in (REF ) can always be replaced by (REF ) for sufficiently small $p^0$ .", "Using (REF ) and (REF ) we can then directly read off from (REF ) the values of $\\kappa $ for the vacuum and thermal cases: rCl Vacuum  (T = 0) :    = -  , Thermal  (T 0) :    = - + z  .", "As was mentioned in the introduction for the example of wave turbulence, also far from equilibrium there exist important scaling solutions.", "These solutions can be spacetime translation invariant, however, they are in general not constrained by a fluctuation-dissipation relation.", "In that case, we may always write down a relation of the form $ F(p) = -i \\left( n(p) + \\frac{1}{2} \\right) \\rho (p) ~, $ with some generalized `distribution function' $n(p)$ defined from the ratio of $F(p)$ and $\\rho (p)$ .", "However, in contrast to (REF ) for the case of thermal equilibrium, here $n(p)$ will in general depend both on frequency $p^{0}$ and spatial momentum $\\mathit {p}$ .", "Then from (REF ) and (REF ) itself no relation between the exponents follows, but only that the distribution function $n(p)$ scales as $ n(p^{0},\\mathit {p}) = s^{\\kappa +\\eta }\\, n(s^{z} p^{0} , s \\mathit {p}) ~, $ for $n(p) \\gg 1/2$ .", "It will require an actual calculation to determine $\\kappa $ at a nonthermal fixed point, and we will show from the nonequilibrium renormalization group in a large-$N$ approximation to next-to-leading order that possible scaling solutions are [5], [18] rCl Nonthermal :    = - + z + d   , = - + 2z + d   .", "Here, $d$ is the spatial dimension, and we will see that they describe the phenomenon of strong turbulence associated to particle (REF ) and energy (REF ) cascades, respectively [5], [44], [13], [6].", "Because the dimensionality of space enters, the nonthermal values for the exponent $\\kappa $ and the corresponding fluctuations can be very large depending on $d$ , which has been confirmed also in lattice field theory simulations [5], [44], [45], [13].", "Comparing the nonthermal results with (REF ) and (REF ), one observes a hierarchy of possible fixed point solutions with increasing complexity from vacuum, and thermal equilibrium, to nonequilibrium." ], [ "Boltzmann transport", "Scaling behavior far from equilibrium is typically discussed with the help of kinetic theory or a Boltzmann equation, which can describe transport properties of dilute, weakly interacting many-body systems.", "To make contact with the literature, we discuss the basic concepts of stationary transport in that way before we start from the nonequilibrium renormalization group in quantum field theory in section .", "A Boltzmann equation describes the dynamics in terms of a single-particle distribution function $n(t,{\\mathit {p}})$ , which depends on time $t$ and spatial momentum ${\\mathit {p}}$ for spatially homogeneous systems.", "The rate of change in the distribution of particles equals the difference between the rates at which particles in a phase space region are generated or lost due to collisions.", "For bosons there is an enhancement of the rate if the final state is already occupied.", "The collision terms may be expressed in terms of scattering cross sections and distribution functions.", "A Boltzmann equation describing $2 \\leftrightarrow 2$ scatterings involving four particles reads rCl t np(t) = 1,2,3 dp 1 2 3   [ ( np + 1 ) ( n1 + 1 ) n2 n3 - np n1 ( n2 + 1 ) ( n3 + 1 ) ] C22(t,p) , where $\\int _{\\mathit {1},\\mathit {2},\\mathit {3}}\\!d\\Gamma _{p 1 \\leftrightarrow 2 3}$ denotes the relativistically invariant measure to be specified below.", "We have written $n_{\\mathit {p}} = n(t,\\mathit {p})$ and $n_{\\mathit {i}} = n(t,\\mathit {k}_{i})$ as a compact notation for the distribution functions.", "Equation (REF ) can be understood as arising from the lowest-order perturbative contribution of the respective quantum field theory [22].", "Since the functional renormalization group treatment starting with section will include also the perturbative behavior, we only give here some relations to facilitate comparisons with the literature.", "We consider the example of a relativistic, real scalar field theory with mass $m$ and quartic self-interaction $\\lambda $ , whose Lagrangian density is $ \\mathcal {L} = \\frac{1}{2} ( \\partial _{\\mu } \\Phi )^{2} - \\frac{1}{2} m^{2} \\Phi ^{2} - \\frac{\\lambda }{4!}", "\\Phi ^{4} ~.", "$ We introduce center and relative (Wigner) coordinates $ X = \\frac{x + y}{2} ~, \\quad r = x - y ~, $ and Fourier transform both the statistical (REF ) and spectral function (REF ) with respect to the relative coordinates rCl F(X,p) = dd+1 r  ei p r F(X+r/2, X-r/2)  , (X,p) = dd+1 r  ei p r (X+r/2, X-r/2)  .", "The Fourier transform of the statistical function is real and that of the spectral function is purely imaginary due to their anti-commutator and commutator definitions, respectively.", "For spatially homogeneous systems, the correlation functions only depend on time $t \\equiv X^0$ and four-momentum $p$ .", "We can define a time-dependent `distribution function' $n(t,p)$ , depending on four-momentum $p$ , by writing $ F(t,p) = - i \\left( n(t,p) + \\frac{1}{2} \\right) \\rho (t,p)~, $ which reduces to (REF ) for time translation invariant systems.", "If the spectral function is taken to be of the translation invariant lowest-order (free-field) form $ \\rho ^{(0)}(p) = 2 \\pi i \\operatorname{sgn}\\big (p^{0}\\big ) \\, \\delta \\big ( (p^{0})^{2} - \\omega _{\\mathit {p}}^{2} \\big ) ~, $ with single-particle energy $\\omega _{\\mathit {p}}$ , then $ n_{\\mathit {p}}( t) \\equiv - i \\int _{0}^{\\infty } \\frac{dp^{0}}{2\\pi }\\, 2 p^{0} \\rho ^{(0)}(p)\\, n(t,p) ~, $ corresponds to the distribution function employed in the Boltzmann equation (REF ).", "The definition (REF ) ensures that the single-particle distribution function $n_{\\mathit {p}}(t)$ is evaluated for on-shell four-momentum with only positive energy, i.e.", "$p^0 = \\omega _{\\mathit {p}}$ .", "The relativistically invariant measure appearing in (REF ) is then, for the theory (REF ), given by rCl 1,2,3dp 1 2 3 = 26 12 p  (i = 13 dd ki(2 )d 12 i ) (2 )d+1 (d+1)(p + k1 - k2 - k3)  , where to lowest order $\\omega _{\\mathit {p}} = \\sqrt{\\mathit {p}^2 + m^2}$ and we write $\\omega _{\\mathit {i}} = \\omega (\\mathit {k}_{i})$ .", "More precisely, in a gradient expansion to lowest order in the number of derivatives with respect to the center coordinate $X^0$ and in powers of the relative coordinate $r^0$ , the spectral function for spatially homogeneous systems obeys [22] $p^0 \\frac{\\partial }{\\partial X^0} \\rho (X^0,p) = 0 ~ ,$ in agreement with the constant free-field form (REF ).", "In contrast, the statistical function $F(t,p)$ to this order can be time-dependent and its evolution is given by rCl 0d p02  2 p0 t F(t,p) = -i 0d p02  2 p0 p  t n(t,p) = C(t,p)  , which defines the collision term $C(t,\\mathit {p})$ describing the effects of interactions to lowest order in the gradient expansion.", "The restriction of the collision term to $2 \\leftrightarrow 2$ scatterings, which are the leading perturbative contributions $\\sim \\lambda ^2$ , gives the Boltzmann equation (REF ).", "We emphasize that the Boltzmann equation has a limited range of applicability.", "In particular, it is restricted to the perturbative regime.", "Even for weak coupling, $\\lambda \\ll 1$ , nonperturbative corrections can play a crucial role once the momentum modes become highly occupied.", "More precisely, equation (REF ) can only be applied for parametrically small occupancies, $n_{\\mathit {p}} \\ll 1/\\lambda $ .", "Nonperturbatively large occupation numbers $n_{\\mathit {p}} \\sim 1/\\lambda $ lead to important corrections, since the `thin-gas' approximation underlying the Boltzmann equation no longer holds and multiparticle scatterings have to be taken into account.", "The calculation of these nonperturbative corrections will be a central topic for the nonequilibrium functional renormalization group below.", "However, for the moment we will consider what one expects from perturbation theory.", "More precisely, we will analyze (REF ) in the `classical' regime of occupation numbers $1/\\lambda \\gg n_{\\mathit {p}} \\gg 1$, where the collision term can be approximated by rCl C22(cl)(t,p) = 1,2,3 dp 1 2 3   [ ( np + n1 ) n2 n3 - np n1 ( n2 + n3 ) ] .", "For $n_{\\mathit {p}} \\lesssim 1$ quantum or `dissipative' processes will play an important role, which will obstruct scaling at sufficiently high momenta.", "The situation for the real scalar field theory is schematically summarized in Fig.", "REF , which sketches the occupation number as a function of momentum on a double-logarithmic scale such that straight lines correspond to power-law behavior.", "Different slopes can occur in different momentum regions.", "The fact that each scaling exponent is associated to an approximately conserved quantity, such as energy or particle number, will be explained next for the perturbative regime using the Boltzmann equation.", "Figure: Illustration of the dual cascade for scalar quantum field theory.", "Different scaling exponents can occur in different momentum regimes, which are associated to stationary transport of conserved quantities." ], [ "Weak wave turbulence", "We may write the Boltzmann equation (REF ) formally as a continuity equation $ \\partial _{t} \\varepsilon + \\nabla _{\\mathit {p}} \\cdot \\mathit {j}_{\\mathit {p}} = 0 ~, $ for the energy density $\\varepsilon (\\mathit {p},t) = \\omega _{\\mathit {p}} n(\\mathit {p},t)$ in momentum space, where the divergence of the energy flux $\\mathit {j}_{\\mathit {p}}$ is given for the collision term (REF ) by $ \\nabla _{\\mathit {p}} \\cdot \\mathit {j}_{\\mathit {p}} = - \\omega _{\\mathit {p}} \\,C_{2 \\leftrightarrow 2}^{\\rm (cl)} (t,\\mathit {p})~.", "$ Since total energy is conserved for the relativistic quantum field theory, we may ask for its effect on the dynamics.", "We consider the case of an isotropic system where the only nonvanishing part of the flux is given by its radial component.", "Integrating the continuity equation (REF ) over the volume of the sphere $B(k)$ of radius $k$ in momentum space, we obtain $ \\int _{B(k)} d^{d}p \\, \\partial _{t} \\varepsilon = -(2\\pi )^{d} A(k) ~, $ which states that the change of net energy contained in $B(k)$ is given by the flux $A(k)$ through the boundary $\\partial B(k)$ : $ -(2\\pi )^{d} A(k) = \\frac{2 \\pi ^{d/2}}{\\Gamma \\left(d/2\\right)} \\int _{0}^{k} d|\\mathit {p}| \\, |\\mathit {p}|^{d-1} \\omega _{\\mathit {p}} \\,C_{2 \\leftrightarrow 2}^{\\rm (cl)} (t,\\mathit {p}) ~.", "$ Thermal equilibrium is characterized by the vanishing of the collision term and a zero net flux $A(k)$ .", "Here, however, we are interested in possible stationary solutions of (REF ) where the distribution $n_{\\mathit {p}}$ characterizes some steady state that is far from equilibrium, known as weak wave turbulence [46].", "For such a steady state to exist, the distribution function $n_{\\mathit {p}}$ needs to satisfy the stationarity condition $ C_{2\\leftrightarrow 2}^{\\rm (cl)} [n] = 0~.", "$ However, in contrast to thermal equilibrium, it has a nonvanishing flux.", "Such a `flux state' describes the stationary transport of conserved quantities [46].", "Here, we show that these stationary states correspond to the situation where $A(k)$ becomes scale-independent.", "To investigate the behavior of the energy flux under scaling transformations, we make for the single-particle energy a scaling ansatz $ \\omega (s \\mathit {p}) = s \\omega (\\mathit {p}) ~, $ using the linear dispersion for the relativistic scalar field theory (REF ) at sufficiently high momenta $\\mathit {p}^2 \\gg m^2$.", "Similarly, the occupation number distribution is taken to obey the scaling form $ n(s \\mathit {p}) = s^{-\\kappa } n(\\mathit {p}) ~, $ with the occupation number scaling exponent $\\kappa $ introduced already in section .", "Together with the scaling property of the measure, rCl 1,2,3 dp 12 3 (s p , s k1 , s k2 , s k3 ) = s4 1,2,3 dp 12 3 (p , k1 , k2, k3)  , with $\\mu _{4} = (3d - 4) - (d+1) = 2 d - 5$ for the scalar field theory case (REF ), we see that the collision integral satisfies $ C_{2\\leftrightarrow 2}^{\\rm (cl)} (s \\mathit {p}) = s^{-3 \\kappa + \\mu _{4}}\\, C_{2\\leftrightarrow 2}^{\\rm (cl)} (\\mathit {p}) ~.", "$ This result can easily be generalized for $m$ -particle scattering processes $ C_{m}^{\\rm (cl)} (s \\mathit {p}) = s^{-\\kappa (m-1) + \\mu _{m}} C_{m}^{\\rm (cl)} (\\mathit {p}) ~, $ where $C_{m}^{\\rm (cl)}$ denotes the collision term for the case of $m$ -particle scattering in the classical regime, and $\\mu _{m} = (m-2) d - m - 1$ .", "Using, accordingly, $\\omega (\\mathit {p}) = |\\mathit {p}|\\, \\omega (1)$ and $n(\\mathit {p}) = |\\mathit {p}|^{-\\kappa }\\, n(1)$ etc.", "for the isotropic system, the flux (REF ) can be seen to give $ A(k) ~ \\sim ~ \\frac{\\omega (1)\\, C_{m}^{\\rm (cl)} (1)}{d+1 - \\kappa (m-1) + \\mu _{m}} \\, k^{d + 1 - \\kappa (m-1) + \\mu _{m}} ~.", "$ For $A(k)$ to become independent of the scale $k$ up to logarithmic corrections, the $k$ -exponent $d + 1 - \\kappa (m-1) + \\mu _{m}$ must vanish.", "For the scaling exponent $\\kappa $ this gives $ \\kappa = \\frac{\\mu _{m} + d + 1}{m-1} = d - \\frac{m}{m-1}~.", "$ Since the denominator of (REF ) also vanishes in this case, the limit $ \\lim _{\\kappa \\rightarrow d - m/(m-1)}\\, \\frac{C_{m}^{\\rm (cl)}}{d+ 1 - \\kappa (m-1) + \\mu _{m}} = const.", "~ $ has to exist.", "Therefore, the collision integral must have a zero of first degree in $d+ 1 - \\kappa (m-1) + \\mu _{m}$ [46], [18].", "Here, we will assume that the stationary state exists and refer to the literature for further details.", "Specifically, for the scalar theory with quartic self-interaction in $d = 3$ spatial dimensions, we have the scaling exponent $ d = 3 ~: \\quad \\kappa = \\frac{5}{3} ~, $ for the transport of energy over some range of momentum scales.", "This is the so-called energy cascade, which is well-known from the perturbative theory of weak wave turbulence [46], [4].", "We also note that from the relativistic scaling ansatz one observes that the scaling exponent associated to momentum conservation is the same as for the energy cascade.", "If the dynamics is approximated by the Boltzmann equation (REF ), i.e.", "if only the perturbatively leading $2\\leftrightarrow 2$ scattering processes are taken into account, then particle number is conserved.", "Of course, there are total particle number changing processes in the considered relativistic quantum field theory.", "However, these processes appear at higher order in the coupling, such that for $\\lambda \\ll 1$ inelastic scattering rates are much smaller than the elastic ones [47].", "Because of this separation of scales, there can be important consequences of approximate particle number conservation for the phenomenon of weak wave turbulence.", "Since the perturbative particle flux is simply given by the momentum integral of the collision integral $C_{2\\leftrightarrow 2}^{\\rm (cl)}$ , going through the corresponding steps as above one may easily verify that the scaling exponent $ \\kappa = d - \\frac{m+1}{m-1} ~, $ describes the stationary transport of particles.", "For the special case of $m = 4$ , and $d = 3$ , this particle cascade is characterized by the exponent $ d = 3 ~: \\quad \\kappa = \\frac{4}{3}~.", "$ So far, we have only considered effects of $2\\rightarrow 2$ particle scattering.", "For stationary turbulence in quantum field theories, however, the dynamics can lead to a nonvanishing field expectation value or Bose condensation far from equilibrium [6].", "This gives rise to an effective three-vertex ($m=3$ ) such that the energy cascade in three dimensions is characterized by the scaling exponent [4] $ d = 3 ~: \\quad \\kappa = \\frac{3}{2} ~.", "$ In these lectures, we will consider only the case of a vanishing field expectation value for simplicity.", "From the above discussion we have observed that there are two types of perturbative power-law distributions corresponding to stationary transport of energy and particle number, respectively.", "These may be realized in different regions of momentum space.", "However, the simple analysis cannot determine in which regions of momentum space the different scaling solutions are realized, or whether the cascades describe transport from small to large wavenumbers or vice versa.", "A thorough discussion leads to the picture of a dual cascade in scalar quantum field theory as illustrated in Fig.", "REF  [5], [6], [12], [45], [13].", "Its quantitative description at low momenta requires, however, to go beyond perturbation theory.", "It has been found that the approximate conservation of an effective particle number can be applied to the nonperturbative regime of scalar field theories at low momenta, which leads to different values of turbulent scaling exponents than the perturbative analysis suggests [5], [44].", "Thus, we are in need of reliable nonperturbative techniques to access all characteristic momentum regions of the nonequilibrium dynamics.", "In the following, we will consider the functional renormalization group as an ideal tool to access a wide range of scales in a unified framework." ], [ "Functional renormalization group", "We want to obtain information about scaling solutions for correlation functions far from equilibrium.", "In the previous section, starting from perturbative kinetic theory or the Boltzmann equation, we illustrated how such a scaling behavior may arise in a weakly coupled scalar theory.", "However, in the infrared, where occupation numbers become large and the system is strongly correlated, the perturbative description is insufficient.", "Thus, we need a general framework to calculate correlation functions from first principles.", "Here, we write down a generating functional for nonequilibrium correlation functions [48], [22], which serves as a starting point to define the nonequilibrium functional renormalization group on a closed time-path following the presentation of Ref.", "[18]." ], [ "Generating functional", "All information about a nonequilibrium quantum field theory can be efficiently described in terms of the nonequilibrium generating functional for correlation functions [48], [22].", "For given density matrix $\\varrho _0 \\equiv \\varrho (t_{0})$ at some initial time $t_{0}$ , correlation functions can be obtained from the generating functional rCl Z[J, R; 0] = Tr0    TC i   { x,C (x) J(x) .", "+ .", "12 x,y,C (x) R(x,y) (y) }  , for a scalar field theory as described by (REF ) in the presence of sources $J$ and $R$ .", "The introduction of the bilinear source term $\\sim R$ will be convenient for the derivation of the functional renormalization group equation, which is explained below.", "Here the time integration is taken over a closed time path $\\mathcal {C}$ , i.e.", "$\\int _{x,\\mathcal {C}} \\equiv \\int _{\\mathcal {C}} dx^{0}\\int d^{d} x$ , displayed in Fig.", "REF [49], [50].", "The closed time-path appears because we want to compute correlation functions, which are given as the trace over the density matrix with time-ordered products of Heisenberg field operators, as exemplified in section .", "Representing the trace as a path integral will require a time path where the initial and final times are identified, which is discussed in more detail below.", "Above, $\\,\\textrm {T}_{\\mathcal {C}}$ denotes time-ordering along the contour $\\,\\mathcal {C}$ .", "As seen from Fig.", "REF , this contour consists of an upper $\\mathcal {C}^{+}$ and a lower branch $\\mathcal {C}^{-}$ where the time-ordering on the lower branch is reversed.", "To extract correlation functions efficiently, the field $\\Phi (x)$ may be written in terms of $\\Phi ^{\\pm }(x^{0},\\mathit {x})$ where the $\\pm $ -index denotes on which part of the contour $\\mathcal {C}^{\\pm }$ the time argument is located.", "E.g.", "the contour integration for the source term in (REF ) takes the form rCl x,C (x) J(x) x,C+ +(x) J+(x) + x,C- -(x) J-(x) = t0 dx0 ddx ( +(x) J+(x) - -(x) J-(x) )  , where the minus sign comes from the reversed time-ordering along $\\mathcal {C}^{-}$ .", "Setting the sources $J$ and $R$ to zero in (REF ) we obtain the partition sum $ Z[J,R; \\varrho _{0}] \\big |_{J,R = 0} = \\operatorname{Tr}\\varrho _{0} = 1 ~, $ from the normalization of the density matrix.", "Figure: Closed time path 𝒞\\mathcal {C}.For instance, taking the second functional derivative of the generating functional (REF ) with respect to the classical source $J^{+}$ defined on $\\mathcal {C}^{+}$ and setting all sources $J$ and $R$ to zero, we obtain $ \\left.", "\\frac{\\delta ^{2} Z[J , R ; \\varrho _{0}]}{i\\delta J^{+}(x) \\, i \\delta J^{+}(y)} \\right|_{J,R = 0} = \\left\\langle \\textrm {T}_0 \\Phi (x) \\Phi (y) \\right\\rangle \\equiv G^{++}(x,y)~.", "$ Here we have used that $\\textrm {T}_{\\mathcal {C}}$ is identical to standard time ordering $\\textrm {T}_0$ in this case, since both $x^{0}$ and $y^{0}$ lie on the upper part of the contour, i.e.", "on $\\mathcal {C}^+$ .", "We also introduced the notation $G^{++}(x,y)$ in order to distinguish this correlator from the other possible second functional derivatives with respect to the sources $J^{+}$ ,$J^{-}$ and setting $J,R = 0$ afterwards.", "These can be written as: rCl G++(x,y) = (x) (y)   (x0 - y0) .", ".", "+ (y) (x)   (y0 - x0)  , G– (x,y) = (x) (y)   (y0 - x0) .", ".", "+ (y) (x)   (x0 - y0)  , G+- (x,y) = (y) (x)  , G-+ (x,y) = (x) (y)  .", "We emphasize that not all of the above two-point functions are independent.", "In particular, using the property $\\theta (x^{0}-y^{0}) + \\theta (y^{0}-x^{0}) = 1$ of the Heaviside step function one obtains the algebraic identity: $ G^{++} (x,y) + G^{--}(x,y) = G^{+-}(x,y) + G^{-+}(x,y)~.", "$ This identity will be of use later on." ], [ "Functional integral", "To simplify the evaluation of correlation functions we write the generating functional (REF ) in terms of a functional integral representation.", "For an intuitive presentation we follow standard techniques (see e.g.", "[51]): We evaluate the trace using eigenstates of the Heisenberg field operators $\\Phi ^{\\pm }$ at initial time $t_0$ , $ \\Phi ^{\\pm }(t_{0},\\mathit {x}) \\, |\\,{\\varphi ^{\\pm }}\\rangle = \\varphi _{0}^{\\pm }(\\mathit {x}) \\,|\\,{\\varphi ^{\\pm }}\\rangle ~, $ such that (REF ) may be written as rCl Z[J,R; 0] = [d0+]   + | 0   TC  i { x, C (x) J(x) .", "+ .", "12 x,y, C (x) R(x,y) (y) } | +  .", "Here the integration measure is given by $ \\int [d\\varphi _{0}^{\\pm }] \\, \\equiv \\int \\prod _{\\mathit {x}} d\\varphi _{0}^{\\pm }(\\mathit {x}) ~.", "$ With the insertion $ \\int [d\\varphi _{0}^{-}] \\:|\\,{\\varphi ^{-}}\\rangle \\langle {\\varphi ^{-}}\\,| = {1}~, $ we may bring (REF ) to a form rCl Z[J,R; 0] = [d0+] [d0-] + |   0   | - ( -,t0  |  +,t0 )J,R  .", "Here the transition amplitude in the presence of the sources is given by rCl ( -,t0  |  +,t0 )J,R - |   TC  i { x, C (x) J(x) .", "+ .", "12 x,y, C (x) R(x,y) (y) }   | +  .", "This matrix element can be written as a functional integral over the fields $\\varphi ^{\\pm }$ rCl ( -,t0  |  +,t0 )J,R = [d+]' [d-]' i { S[] + x,C (x) J(x) .", "+ .", "12 x,y,C (x) R(x,y) (y) } , which is essentially the same procedure as employed to obtain standard path integral expressions for vacuum of equilibrium matrix elements [43].", "Here $S[\\varphi ]$ is the classical action for the scalar theory, and the measure is given by $ \\int [d\\varphi ^{\\pm }]^{\\prime } \\, \\equiv \\int \\!\\prod _{\\mathit {x}; \\, x^{0} \\,>\\, t_{0}}\\!", "d\\varphi ^{\\pm } (x^{0},\\mathit {x}) ~.", "$ Here, the functional integration goes over the field configurations $\\varphi ^{\\pm }(x^{0},\\mathit {x})$ that satisfy the boundary condition $\\varphi ^{\\pm }(x^{0} = t_{0},\\mathit {x}) = \\varphi _{0}^{\\pm }(\\mathit {x})$ .", "The above expression (REF ) together with (REF ) displays two important ingredients entering nonequilibrium quantum field theory: the quantum fluctuations described by the functional integral with action $S$ that determines the transition matrix element, and the statistical fluctuations encoded in the averaging procedure over the initial conditions as specified by the initial density matrix $\\varrho _{0}$ ." ], [ "$N$ -component scalar field theory", "So far, we have not specified any internal field degrees of freedom.", "In the following, we consider a $N$ -component vector field $\\Phi _{a}$ with $a = 1, \\ldots , N$ for an $O(N)$ -symmetric theory.", "The number of field components will later serve as an expansion parameter that allows for a controlled approximation of renormalization group equations.", "The classical action for the $O(N)$ -model with a quartic self-interaction is given by rCl S[] = 12 x,y,C a (x) i D-1a b (x,y) b (y) - 4 N!", "x,C a (x) a (x) b (x) b (x)  , where the time integration runs over the contour and $i D_{a b}^{-1}$ denotes the free inverse propagator satisfying $ \\left( - \\Box _{x} - m^{2} \\right) i D_{a b} (x,y) = \\delta _{a b} \\delta _{\\mathcal {C}}^{(d+1)}(x-y) ~.", "$ Using again the $\\pm $ -index to denote on which part of the contour $\\mathcal {C}^{\\pm }$ the time argument is located, the free part of the action takes the form rCl S0[+,-] = 12 x,y ( a+ (x) i D-1a b (x,y) b+ (y) .", ".", "- a- (x) i D-1a b (x,y) b-(y) )  .", "The interaction term is written as rCl Sint [+,-] = - 4 N!", "x ( a+(x) a+(x) b+(x) b+(x) .", ".", "- a-(x) a-(x) b-(x) b-(x) ) .", "The time integration is implicitly defined along the positive branch of the contour, i.e.", "$\\int _{x} \\, \\equiv \\int _{t_{0}}^{\\infty } dx^{0} \\int d^{d} x$ and the minus sign in front of the $\\varphi ^{-}$ -components of the action comes from the reversed time ordering along the lower branch $\\mathcal {C}^{-}$ ." ], [ "Quantum vs. classical dynamics", "In order to discuss the different origins of quantum and of classical-statistical fluctuations, it is convenient to introduceIn order to prevent a proliferation of symbols, the notation does not distinguish the new definition of $\\varphi _a$ from its earlier use since no confusion for the following can occur.", "$ \\varphi _{a} = \\frac{1}{2} \\left( \\varphi _{a}^{+} + \\varphi ^{-}_{a} \\right) ~, \\quad {\\tilde{\\varphi }}_{a} = \\varphi _{a}^{+} - \\varphi _{a}^{-} ~.", "$ For the rest of the lectures we will use this basis for our calculations.", "The action (REF ) takes the following form $ S[\\varphi ,{\\tilde{\\varphi }}] = S_{0} [\\varphi , {\\tilde{\\varphi }}] + S_{\\rm int} [\\varphi ,{\\tilde{\\varphi }}] ~, $ where the free part is given by $ S_{0} [\\varphi ,{\\tilde{\\varphi }}] = \\int _{x,y} {\\tilde{\\varphi }}_{a} (x) i D_{a b}^{-1} (x,y) \\varphi _{b} (y) ~, $ and the interaction part reads rCl Sint[,] = - 6 N x a (x) a (x) b (x) b (x) - 24 N x a (x) a (x) b (x) b (x)  , Finally, the linear source term can be written in this basis takes the form rCl x ( +a(x) J+a (x) - -a (x) J-a (x) ) = x ( a(x) Ja (x) + a (x) Ja (x) ) , whereas the bilinear source term is written as rCl x,y ( +a (x)  ,  -a (x) ) R++a b (x,y) - R+-a b (x,y) - R-+a b (x,y) R–a b (x,y) +b (y) b- (y) = x,y ( a (x)  ,  a (x) ) RFa b (x,y) RAa b (x,y) RRa b (x,y) RFa b (x,y) b (y) b (y)  .", "We have added the indices $\\mathrm {R},\\mathrm {A},F,{\\tilde{F}}$ to the bilinear sources that suggest a relation to the retarded, advanced, and statistical components in the new basis.", "This connection will be made more explicit in the following subsections.", "The two types of vertices appearing in the interaction part (REF ) are illustrated in Fig.", "REF .", "To understand their role for the dynamics, it is important to note that one can also write down a functional integral for the corresponding nonequilibrium classical-statistical field theory.", "The standard derivation of the latter employs that the classical field equation of motion can be obtained from $S[\\varphi ,{\\tilde{\\varphi }}]$ by functional differentiation with respect to $\\tilde{\\varphi }$ , rCl .", "S[,]a(x) | = 0 = -( x + m2 ) a (x) - 3 N a (x) b(x) b(x) = 0 , where the derivative had to be evaluated for $\\tilde{\\varphi }=0$ to eliminate terms originating from the part of the interaction term (REF ) that is cubic in $\\tilde{\\varphi }$ .", "This classical dynamics can then be implemented as a functional integral using the representation of the $\\delta $ -functional rCl [ S[,]a(x) | = 0 ] = [d ] ( i x S[,]a(x) | = 0a(x) )  , with the help of the `auxiliary' field $\\tilde{\\varphi }$ and further steps involving the integration with respect to $\\varphi $ [52], [53], [54].", "We emphasize that in the exponent on the r.h.s.", "of (REF ) only terms linear in $\\tilde{\\varphi }$ appear.", "In particular, the interaction term $\\sim \\tilde{\\varphi }^3$ , appearing with (REF ) in the functional integral of the quantum theory, does not occur for the classical theory.", "In summary, the generating functionals for correlation functions are very similar in the quantum and the classical statistical theory.", "A crucial difference is that the quantum theory is characterized by an additional vertex.", "In a diagrammatic language, if loop corrections involving only the classical vertex dominate over those involving the quantum vertex or a combination of both, then the quantum dynamics can be approximately described by the classical-statistical field theory.", "This has been analyzed in great detail in recent years for nonequilibrium phenomena [55], [56], [5], and we will come back to this point in the context of turbulence below.", "Figure: Classical (left) and quantum vertex (right) in the scalar field theory." ], [ "Connected one- and two-point functions", "The generating functional for connected correlation functions is given by $ W = - i \\ln Z ~.", "$ We may calculate field expectation values by functional differentiation with respect to the sources $J$ and $\\tilde{J}$ , which we denote as $ \\frac{\\delta W}{\\delta J_{a} (x)} = {\\tilde{\\phi }}_{a} (x) ~,\\quad \\frac{\\delta W}{\\delta \\tilde{J}_{a} (x)} = \\phi _{a} (x) ~, $ for fixed $J$ and $R$ .", "The connected two-point correlation functions are given by the second functional derivatives rCl 2 WJa (x) Jb (y) = GRa b (x,y)  , 2 WJa (x) Jb (y) = GAa b (x,y)  , 2 WJa (x) Jb (y) = i Fa b (x,y)  , 2 WJa (x) Jb (y) = i Fa b (x,y) , where $G^{\\mathrm {R},\\mathrm {A}}$ are the retarded/advanced propagators, $F$ is the statistical propagator, and ${\\tilde{F}}$ is the `anomalous' propagator.", "We note that the retarded and advanced propagators satisfy the symmetry property $ G_{a b}^{\\mathrm {A}} (x,y) = G^{\\mathrm {R}}_{b a} (y,x) ~, $ and for the statistical propagators, we have $ F_{a b} (x,y) = F_{b a} (y,x) ~, \\quad {\\tilde{F}}_{a b} (x,y) = {\\tilde{F}}_{b a} (y,x) ~.", "$ These properties follow directly from the definition of the propagators in terms of the second functional derivatives with respect to $J$ and ${\\tilde{J}}$ .", "The spectral function $\\rho $ is given by the difference of the retarded and advanced propagators $ \\rho _{a b} (x,y) = G^{\\mathrm {R}}_{a b} (x,y) - G_{a b}^{\\mathrm {A}} (x,y) ~, $ and $G^{\\mathrm {R}}_{a b} (x,y) = \\rho _{a b} (x,y) \\theta (x^{0}-y^{0})$ .", "It is important to note that the anomalous propagator ${\\tilde{F}}$ vanishes in the limit where the external sources are set to zero.", "This is a consequence of the algebraic identity (REF ), since ${\\tilde{F}}= G^{++} + G^{--} - G^{-+} - G^{+-}$ , as one may readily check by changing basis.", "More generally, in the absence of sources, we have [51] rCl WJa (x) |J,J , RR,A,F,F = 0 = a (x) = 0  , 2 WJa (x)  Jb (y) |J,J ,RR,A,F,F = 0 = i Fa b (x,y) = 0  , and, correspondingly, arbitrary functional derivatives of the generating functional with respect to $J$ vanish in the absence of sources." ], [ "Functional renormalization group", "A most convenient derivation of the functional renormalization group equation starts from the two-particle irreducible (2PI) effective action [57].", "The latter is obtained as a Legendre transform of the generating functional (REF ) with respect to the linear and bilinear source terms $J,R$ .", "One obtains a functional of the field expectation values $\\phi ,{\\tilde{\\phi }}$ , and the propagators $G^{\\mathrm {R}},G^{\\mathrm {A}},F,{\\tilde{F}}$ : rCl 2PI[,,GR,GA,F,F] = W - x { a (x) Ja(x) + a (x) Ja (x) } - 12 x,y { RAa b (x,y) [ a (x) b (y) - i GRa b (x,y) ]          + RRa b (x,y) [ a (x) b (y) - i GAa b (x,y) ]          + RFa b (x,y)   [ a (x) b (y) + Fa b (x,y) ]          + RFa b (x,y) [ a (x) b (y) + Fa b (x,y) ] }  , where the sources depend on the field expectation values and the propagators, i.e.", "$J = J\\left[\\phi ,{\\tilde{\\phi }}, G^{\\mathrm {R}},G^{\\mathrm {A}},F,{\\tilde{F}}\\right]$ etc.", "We may partially undo the Legendre transform for the bilinear sources $R^{\\mathrm {R},\\mathrm {A},F,{\\tilde{F}}}$ to obtain an expression for the one-particle irreducible (1PI) effective action in the presence of these sources, i.e.", "rCl [,,RR,A,F,F] = 2PI - i2 Tr{ GR RR + GA RA .", ".", "+ i F RF + i F RF }  .", "Here the propagators depend on the fields and bilinear sources, i.e.", "$G^{\\mathrm {R}}=G^{\\mathrm {R}}\\left[\\phi , {\\tilde{\\phi }},R^{\\mathrm {R},\\mathrm {A},F,\\tilde{F}}\\right]$ etc.", "The above equation is our starting point for the construction of the functional renormalization group.", "We take the bilinear sources to depend on some characteristic momentum scale $k \\ge 0$ , which renders the effective action scale-dependent, $ \\Gamma _{k} [\\phi , {\\tilde{\\phi }}] \\equiv \\Gamma \\left[ \\phi , {\\tilde{\\phi }} , R^{\\mathrm {R},\\mathrm {A},F,\\tilde{F}}_{k} \\right].", "$ Taking now the derivative with respect to the scale $k$ , denoting $ \\dot{\\Gamma }_{k} [\\phi , {\\tilde{\\phi }}] \\equiv k \\frac{\\partial \\Gamma _{k} [\\phi , {\\tilde{\\phi }}]}{\\partial k} ~, $ we get from (REF ) the renormalization group equation for the scale-dependent effective action: $ \\hspace{-2.0pt}{\\dot{\\Gamma }}_{k} = - \\frac{i}{2} \\operatorname{Tr}\\left\\lbrace G_{k}^{\\mathrm {R}} {\\dot{R}}_{k}^{\\mathrm {R}} + G_{k}^{\\mathrm {A}} {\\dot{R}}_{k}^{\\mathrm {A}} + i F_{k} {\\dot{R}}_{k}^{\\tilde{F}}+ i \\tilde{F}_{k} {\\dot{R}}_{k}^F \\right\\rbrace ~, $ Here we have used the fact that $\\Gamma _{2\\mathrm {PI}}$ in (REF ) is independent of the bilinear sources $R_{k}^{\\mathrm {R},\\mathrm {A},F,\\tilde{F}}$ .", "The flow equation (REF ) corresponds to the Wetterich equation for the effective average action [58], however, now evaluated on the closed time path." ], [ "Cutoff functions", "With suitable cutoff functions, the effective average action $\\Gamma _k$ may be viewed as a coarse grained effective action, which includes all quantum-statistical fluctuations with characteristic momenta above the scale $k$ .", "If the infrared cutoff scale $k$ is sent to zero, one recovers the standard 1PI effective action with all fluctuations included, i.e.", "$ \\Gamma _{k=0} [\\phi ,{\\tilde{\\phi }}] = \\Gamma \\left[\\phi , {\\tilde{\\phi }},R_{k=0}^{\\mathrm {R},\\mathrm {A},F,{\\tilde{F}}}=0\\right] ~.", "$ Therefore, the sources $R_{k}^{\\mathrm {R},\\mathrm {A},F,{\\tilde{F}}}$ must vanish in the infrared limit.", "With this property, the renormalization group equation may be used to describe the flow starting from some high momentum scale $\\Lambda $ , where the microscopic physics is characterized by some classical action $S$ , i.e.", "$ \\lim _{k \\rightarrow \\Lambda } \\Gamma _{k} [\\phi ,{\\tilde{\\phi }}] \\simeq S [\\phi , {\\tilde{\\phi }}] ~.", "$ The renormalization group flow then interpolates between the classical action and the full quantum effective action that appears when all fluctuations have been taken into account.", "We have seen that the renormalization group on the closed time path requires the specification of the various cutoff functions $R_k^{\\mathrm {R},\\mathrm {A},F,{\\tilde{F}}}$ , which seems to allow for a wider class of possible choices than in the corresponding Euclidean field theories.", "In order to discuss what constraints can be obtained from the requirement (REF ), it is useful to start from the – now $k$ -dependent – generating functional (REF ): rCl Zk [J,J] = [d] [d] i { S [, ] + x { a (x) Ja (x) + a (x) Ja (x) } + 12 x,y { a (x) RAk,a b (x,y) b (y)          + a (x) RRk, a b (x,y) b (y)          + a (x) RFk, a b (x,y) b (y)          + a (x) RFk, a b (x,y) b (y) } } .", "Here we have hidden the averaging over the initial density matrix $\\varrho _{0}$ in the notation, since we will not be concerned with the special choice of the initial conditions for the present purposes.", "Later on we will comment on the role of the initial conditions.", "The affective average action (REF ) can then be written as rCl k [,] = - i Zk - x { a (x) Ja (x) + a (x) Ja (x) } - 12 x,y { a (x) RAk,a b (x,y) b (y)          + a (x) RRk, a b (x,y) b (y)          + a (x) RFk, a b (x,y) b (y)          + a (x) RFk, a b (x,y) b (y) } .", "In the following, we employ the notation $ \\Gamma _{k,a}^{\\phi } (x) \\equiv \\frac{\\delta \\Gamma _{k}}{\\delta \\phi _{a} (x)} ~, \\quad \\Gamma ^{{\\tilde{\\phi }}}_{k, a} (x) \\equiv \\frac{\\delta \\Gamma _{k}}{\\delta {\\tilde{\\phi }}_{a} (x)} ~.", "$ By functional differentiation of (REF ) with respect to the fields $\\phi _{a}$ and $\\tilde{\\phi }_{a}$ , we obtain the equations of motion for the fields $\\phi $ and $\\tilde{\\phi }$ : rCl k,a (x) = - Ja (x) - y { RFk, a b (x,y) b (y) .", "+ .", "12 RAk, a b (x,y) b (y) + 12 b (y) RRk, b a (y, x) }, k, a (x) = - Ja (x) - y { RFk, a b (x,y) b (y) .", "+ .", "12 RRk, a b (x,y) b (y) + 12 b (y) RAk, b a (y,x) } .", "With these we may eliminate the sources $\\tilde{J}_{a} (x)$ and $J_{a} (x)$ from (REF ), expressing them in terms of the first functional derivatives of $\\Gamma _{k}$ .", "We then shift the fields in the generating functional (REF ) by $ \\varphi \\rightarrow \\phi + \\varphi ~, \\quad {\\tilde{\\varphi }} \\rightarrow {\\tilde{\\phi }} + {\\tilde{\\varphi }}~, $ observing that the measure is invariant under these transformations.", "That way, from (REF ), we finally arrive at the following functional integro-differential equation for the effective action: rCl k[,] = S[,] - i [d] [d] i   { S[+ , + ]      - S [, ] + x { a (x) k,a (x) + a (x) k,a (x) }      - 12 x,y { a (x) RAk,a b (x,y) b (y) .", "+ a (x) RRk, a b (x,y) b (y)             + a (x) RFk, a b (x,y) b (y) .", "+ a (x) RFk, a b (x,y) b (y) } } .", "With the following representation for the $\\delta $ -functional $ \\delta [\\varphi ] = \\int [d{\\tilde{\\varphi }}] \\exp \\left\\lbrace i \\int _{x} \\varphi _{a} (x) {\\tilde{\\varphi }}_{a} (x) \\right\\rbrace ~, $ and equivalently for $\\delta [{\\tilde{\\varphi }}]$ , one observes that the property (REF ) can be efficiently achieved with a class of cutoff functions chosen as rCl RR,Ak, a b (x,y) = Rk (-x) (x-y) a b  , RF,Fk, a b(x,y) = 0 , with the property $ \\lim _{k \\rightarrow \\Lambda } R_{k} \\rightarrow \\infty ~.", "$ This ensures that the bilinear source terms in (REF ) act as $\\delta $ -constraints in the limit $k \\rightarrow \\Lambda $ , thus suppressing fluctuations in the fields $\\varphi $ and $\\tilde{\\varphi }$ .", "More generally, possible nonzero $R_{k}^{F,\\tilde{F}}$ should not be chosen to grow as fast as $R^{\\mathrm {R},\\mathrm {A}}_{k}$ for $k \\rightarrow \\Lambda $ in order to comply with (REF ) [18].", "The choice of vanishing $R_{k}^{F,{\\tilde{F}}}$ greatly simplifies the structure of the flow equations.", "In this case the exact flow equation for the effective average action (REF ) becomes $ {\\dot{\\Gamma }}_{k} [\\phi ,{\\tilde{\\phi }}] = - \\frac{i}{2} \\operatorname{Tr}\\left\\lbrace G^{\\mathrm {R}}_{k} {\\dot{R}}^{\\mathrm {R}}_{k}+ G^{\\mathrm {A}}_{k} {\\dot{R}}^{\\mathrm {A}}_{k} \\right\\rbrace ~.", "$ Furthermore, from the symmetry properties (REF ) of the propagators $G_{k}^{\\mathrm {R},\\mathrm {A}}$ , and the cutoff functions $R_{k}^{\\mathrm {R},\\mathrm {A}}$ , one observes that $\\operatorname{Tr}\\left\\lbrace G^{\\mathrm {R}}_{k} [\\phi ,{\\tilde{\\phi }}] {\\dot{R}}^{\\mathrm {R}}_{k} \\right\\rbrace = \\operatorname{Tr}\\left\\lbrace G^{\\mathrm {A}}_{k} [\\phi ,{\\tilde{\\phi }}] {\\dot{R}}^{\\mathrm {A}}_{k} \\right\\rbrace $ holds.", "However, it proves to be convenient to keep both the retarded and advanced functions in (REF ) as it simplifies the diagrammatic rules that will be introduced below." ], [ "Propagators", "Above we have derived the exact flow equation for the effective average action $\\Gamma _{k}$ that depends on the retarded and advanced propagators $G_{k}^{\\mathrm {R}}$ and $G_{k}^{\\mathrm {A}}$ .", "It remains to relate the two-point functions to functional derivatives of $\\Gamma _{k}$ .", "Here, we want to illustrate how to obtain these relations starting from simple identities for the sources $J$ and ${\\tilde{J}}$ .", "As an example, we may consider the following identity rCl (d+1)(x-y) a b = Ja (x)Jb (y) = z { Ja (x)c (z) c (z)Jb (y) + Ja (x)c (z)c (z)Jb (y) }  , where we have used that $J = J[\\phi ,{\\tilde{\\phi }}]$ is a functional of the field expectation values.", "Using the equation of motion $ \\frac{\\delta \\Gamma _{k}}{\\delta {\\tilde{\\phi }}_{a} (x)} = - J_{a} (x) - \\int _{y} R^{\\mathrm {R}}_{k, a b} (x,y) \\phi _{b} (y) ~, $ we may write the functional derivatives $\\delta J_{a} (x)/\\delta \\phi _{c} (z)$ and $\\delta J_{a} (x)/\\delta {\\tilde{\\phi }}_{c} (z)$ in terms of second functional derivatives of the effective average action.", "Here we have exploited the symmetry property of the cutoff functions $R_{k}^{\\mathrm {R},\\mathrm {A}}$ and the vanishing of $R_{k}^{F,{\\tilde{F}}}$ to write (REF ) in a somewhat simpler form.", "Furthermore, the functional derivatives of the field expectation values $ \\frac{\\delta \\phi _{c} (z)}{\\delta J_{b} (y)} = \\frac{\\delta ^{2} W_{k}}{\\delta \\tilde{J}_{c} (z)\\,\\delta J_{b} (y)} ~, \\quad \\frac{\\delta {\\tilde{\\phi }}_{c} (z)}{\\delta J_{b} (y)} = \\frac{\\delta ^{2} W_{k}}{\\delta J_{c} (z) \\,\\delta J_{b} (y)} ~, $ can be expressed in terms of the $k$ -dependent generating functional $W_{k}$ .", "That way, we may rewrite (REF ) as rCl (d+1)(x-y) a b = z { ( - 2 ka (x)  c (z) - RRk, a c (x,z) )   2 WkJc (z)  Jb (y) .", ".", "- 2 ka (x) c (z) 2 WkJc (z)   Jb (y) } .", "With (REF ) this can be written in a compact matrix form, $ \\left( \\Gamma _{k}^{{\\tilde{\\phi }}\\phi } + R^{\\mathrm {R}}_{k} \\right) G^{\\mathrm {R}}_{k} + \\Gamma ^{{\\tilde{\\phi }}{\\tilde{\\phi }}}_{k} i \\tilde{F}_{k} = - {1} ~, $ where we have used the notation $ \\Gamma ^{{\\tilde{\\phi }} \\phi }_{k, a b} (x,y) \\equiv \\frac{\\delta ^{2} \\Gamma _{k} [\\phi ,{\\tilde{\\phi }}]}{\\delta {\\tilde{\\phi }}_{a} (x) \\, \\delta \\phi _{b} (y)} ~, $ for the functional derivatives of the effective average action.", "Starting from the remaining three identities for the functional derivatives of the sources, that is $ \\frac{\\delta \\tilde{J}_{a} (x)}{\\delta \\tilde{J}_{b} (y)} = \\delta ^{(d+1)} (x-y) \\delta _{a b} ~, $ and $ \\frac{\\delta J_{a} (x)}{\\delta \\tilde{J}_{b} (y)} = 0~, \\quad \\frac{\\delta \\tilde{J}_{a} (x)}{\\delta J_{b} (y) } = 0~, $ we obtain the set of equations rCl ( k + RAk ) GAk + k i Fk = - 1  , ( k + RRk ) i Fk + k GAk = 0  , ( k + RAk ) i Fk + k GRk = 0  .", "Together with our result from above, this linear system can be solved for the propagators $G^{\\mathrm {R},\\mathrm {A}}_{k}, F_{k}$ , and ${\\tilde{F}}_{k}$ : rCl GkR = - [ ( k + RkR ) - k (k + RkA)-1 k ]-1 , GkA = - [ ( k + RkA ) - k (k + RkR)-1 k ]-1 , i Fk = - [ k - ( k + RkA ) ( k )-1 (k + RkR)]-1 , i Fk = - [ k - ( k + RkR ) ( k )-1 (k + RkA)]-1 , where the propagators depend on the field expectation values $\\phi _{a}$ and ${\\tilde{\\phi }}_{a}$ , i.e.", "$G_{k}^{\\mathrm {R},\\mathrm {A}} = G_{k}^{\\mathrm {R},\\mathrm {A}} [\\phi ,{\\tilde{\\phi }}]$ , etc.", "We emphasize that the above is valid for the choice $R^{F,{\\tilde{F}}}=0$ , and for nonvanishing $R^{F,{\\tilde{F}}}$ the propagators take a different form [18])." ], [ "Diagrammatics", "From the flow equation for the effective average action (REF ), we can construct the flow equations for arbitrary $n$ -point functions by functional differentiation.", "As an example, we consider rCl k,a b (x,y) = - i2 Tr{ 2 GRk[,]a (x)  b (y) RRk .", ".", "+ 2 GAk[,]a (x)   b (y)RAk } , involving functional derivatives of the retarded and advanced propagators.", "For the retarded propagator we have, e.g.", "rCl 2 GRk,a b [,]c  d = GRk,a e k, e f c d GRk,f b - GRk, a c k, e f c d (k + RAk)-1f g k, g h GRk, h b + GRk, a e k, e f ( k + RAk )-1f g k, g h c d ( k + RAk )-1k, h i k, i j GRk, j b - GRk, a e k, e f (k + RAk )-1f g k, g h c d GRk, h b + plus terms involving three-vertices  .", "This seems rather complicated at first sight.", "However, after taking derivatives, we can set the sources $\\tilde{J} = J = 0$ to zero, which corresponds to evaluating all expressions at the field configuration $(\\phi =\\phi _0(k),\\tilde{\\phi }=0)$ that extremize the effective average action.", "Here $\\phi _{0}(k=0)$ can be nonzero in the case of spontaneous symmetry breaking.", "According to the equation of motions, this configuration fulfills with (REF ) for $\\tilde{J} = J = 0$ (and $R_k^{F,{\\tilde{F}}}=0$ ): $ \\left.", "\\Gamma _{k}^{\\phi }\\right|_{\\phi =\\phi _0(k),\\tilde{\\phi }=0} (x) = 0 ~.", "$ More generally, all the functional derivatives of the effective average action with respect to $\\phi $ vanish at the extremum [51], i.e.", "$ \\left.", "\\frac{\\delta ^{n} \\Gamma _{k} [\\phi ,{\\tilde{\\phi }}]}{\\delta \\phi (x_{1}) \\, \\delta \\phi (x_{2}) \\cdots \\,\\delta \\phi (x_{n})} \\right|_{\\phi _0(k) ,{\\tilde{\\phi }} = 0} = 0 ~.", "$ In particular, we have $\\Gamma ^{\\phi \\phi }_{k} [\\phi =\\phi _0(k) ,{\\tilde{\\phi }} = 0] = 0$ so that the (REF ) simplify considerably.", "The anomalous statistical propagator ${\\tilde{F}}_{k}[\\phi =\\phi _0(k) ,{\\tilde{\\phi }} = 0]$ is zero and the nonvanishing propagators are given by rCl GRk,a b = - ( k + RRk )-1a b = (344,-193)[lb]$a$ (371,-193)[lb]$b$ (383,-197)[lb], 0.5 Black 1.0 (342,-195)(356,-195) (356,-195)(376,-195)2 GAk, a b = - ( k + RAk )-1a b = 0.5 Black (344,-193)[lb]$a$ (371,-193)[lb]$b$ (383,-197)[lb], 1.0 (342,-195)(356,-195)2 (358,-195)(376,-195) i Fk, a b =  ( GRk k GAk )a b   = 0.5 Black (344,-193)[lb]$a$ (371,-193)[lb]$b$ (383,-197)[lb], 1.0 (342,-195)(376,-195) where we have also introduced their diagrammatic representation.", "In the following, for simplicity we will work in the symmetric phase where the macroscopic field expectation value vanishes, i.e.", "$\\phi _{0}(k) = 0$ .", "As a consequence, also all three-vertices vanish for the considered theory with interaction (REF ) and using our above example we get the comparably compact expression: rCl .", "2 GRk,a b [,]c (x) d (y) |, = 0 = GRk, a e k, e f c d GRk, f b + i Fk, a g k, g h c d GRk, h b  .", "We will use a diagrammatic representations, where the retarded and advanced cutoff functions $R_{k}^{\\mathrm {R},\\mathrm {A}}$ are denoted by the insertion of a cross, i.e.", "rCl 0.5 Black (83,-198)[lb]$\\displaystyle {\\dot{R}}_{k,ab}^{\\mathrm {R}} \\,=\\, $ (125,-188)[lb]$a$ (170,-188)[lb]$b$ (179,-195)[lb]  , 1.0 (145,-193)(151,-187)(145,-187)(151,-193) (151,-190)(124,-190)2 (151,-190)(175,-190) 0.5 Black (191,-198)[lb]$\\displaystyle {\\dot{R}}_{k,ab}^{\\mathrm {A}} \\,=\\, $ (233,-188)[lb]$a$ (278,-188)[lb]$b$ (290,-195)[lb]  , 1.0 (256,-193)(262,-187)(256,-187)(262,-193) (259,-190)(232,-190) (259,-190)(286,-190)2 and the proper vertices, indicated by the full dot, are given by $ \\begin{picture}(20,29) (280,-158){0.5}{Black}(196,-153)[lb]{\\displaystyle \\Gamma _{k,abcd}^{{\\tilde{\\phi }}\\phi \\phi \\phi } \\,=\\, }(392,-150)[lb]{}(244,-135)[lb]{c}(273,-135)[lb]{d}(243,-161)[lb]{a}(273,-161)[lb]{b}(280,-152)[lb]{~,}(300,-153)[lb]{\\displaystyle \\Gamma _{k,abcd}^{{\\tilde{\\phi }}{\\tilde{\\phi }}\\phi \\phi } \\,=\\, }(350,-135)[lb]{c}(378,-135)[lb]{d}(349,-161)[lb]{a}(378,-161)[lb]{b}{1.0}(260,-145){3.4}(250,-135)(270,-155)(260,-145)(270,-135)(260,-145)(250,-155){2}{1.0}(365,-145){3.4}(355,-135)(365,-145)(365,-146)(375,-135)(365,-145)(355,-155){2}(375,-155)(365,-145){2}\\end{picture} $ and equivalently for the remaining four-vertices.", "For the exact flow equation for the effective average action the one-loop form is written diagrammatically as $ \\begin{picture}(73,26) (24,-167){0.5}{Black}(-6,-167)[lb]{\\displaystyle {\\dot{\\Gamma }}_k \\,=\\, -\\frac{i}{2} \\,\\bigg \\lbrace }(75,-160)[lb]{\\displaystyle +}(118,-167)[lb]{\\displaystyle \\bigg \\rbrace ~.", "}{1.0}(53,-148)(59,-142)(53,-142)(59,-148)(56,-155)(10,90,270)(56,-155)(10,-90,90){2}{1.0}(98,-148)(104,-142)(98,-142)(104,-148)(101,-155)(10,90,270){2}(101,-155)(10,-90,90)\\end{picture} $ Similarly, using a compact notation for the derivatives the flow equation for the two-point function takes the form rCl 0.5 Black (127,-39)[lb]$+$ (26,-48)[lb]$\\displaystyle {\\dot{\\Gamma }}_{k,ab}^{(2)}\\,\\, =\\,\\, -\\frac{i}{2} \\, \\bigg \\lbrace $ (95,-55)[lb]$a$ (111,-55)[lb]$b$ (145,-55)[lb]$a$ (161,-55)[lb]$b$ 1.0 (105,-34)(10,100,180) (105,-34)(10,180,-80)2 (105,-34)(10,0,100)2 (105,-34)(10,-80,0) 0.5 (105,-44)3.4 1.0 (103,-21)(108,-27)(103,-27)(108,-21) (95,-45)(115,-45) (155,-34)(10,90,270) (155,-34)(10,0,90)2 (155,-34)(10,-90,0) 0.5 (155,-44)3.4 1.0 (152,-21)(158,-27)(152,-27)(158,-21) (145,-45)(165,-45) 0.5 Black (79,-39)[lb]$+$ (127,-39)[lb]$+$ (178,-48)[lb]$\\displaystyle \\bigg \\rbrace ~,$ (95,-55)[lb]$a$ (111,-55)[lb]$b$ (145,-55)[lb]$a$ (161,-55)[lb]$b$ 1.0 (105,-34)(10,90,180)2 (105,-34)(10,180,270) (105,-34)(10,0,90) (105,-34)(10,-90,0)2 0.5 (105,-44)3.4 1.0 (102,-21)(108,-27)(102,-27)(108,-21) (95,-45)(115,-45) (155,-34)(10,180,270) (155,-34)(10,90,180)2 (155,-34)(10,-90,90) 0.5 (155,-44)3.4 1.0 (152,-21)(158,-27)(152,-27)(158,-21) (165,-45)(145,-45) and for the four-point function we have rCl 0.5 Black (50,-50)[lb]$\\displaystyle {\\dot{\\Gamma }}_{k,abcd}^{(4)} \\, = $ 0.5 Black (124,-42)[lb]$\\displaystyle +$ (174,-42)[lb]$\\displaystyle +$ (226,-42)[lb]$\\displaystyle +$ (60,-50)[lb]$ -\\frac{i}{8} \\, \\bigg \\lbrace $ (83,-60)[lb]$a$ (96,-60)[lb]$b$ (105,-60)[lb]$c$ (115,-60)[lb]$d$ (133,-60)[lb]$a$ (145,-60)[lb]$b$ (155,-60)[lb]$c$ (165,-60)[lb]$d$ (183,-60)[lb]$a$ (195,-60)[lb]$b$ (205,-60)[lb]$c$ (215,-60)[lb]$d$ (233,-60)[lb]$a$ (245,-60)[lb]$b$ (255,-60)[lb]$c$ (265,-60)[lb]$d$ 1.0 (102,-34)(10,90,140) (102,-34)(10,140,-80)2 (102,-34)(10,20,90)2 (102,-34)(10,-80,20) 0.5 (94,-39)3.4 (110,-39)3.4 1.0 (99,-21)(105,-27)(99,-27)(105,-21) (94,-39)(87,-51) (94,-39)(97,-51) (110,-39)(107,-51) (110,-39)(117,-51) (152,-34)(10,90,140) (152,-34)(10,20,90)2 (152,-34)(10,140,210)2 (152,-34)(10,-30,20) (152,-34)(10,210,260) (152,-34)(10,260,-30)2 0.5 (144,-39)3.4 (160,-39)3.4 1.0 (149,-21)(155,-27)(149,-27)(155,-21) (144,-39)(137,-51) (144,-39)(147,-51) (160,-39)(157,-51) (160,-39)(167,-51) (202,-34)(10,90,140) (202,-34)(10,20,90)2 (202,-34)(10,140,210)2 (202,-34)(10,210,20) 0.5 (194,-39)3.4 (210,-39)3.4 1.0 (199,-21)(205,-27)(199,-27)(205,-21) (194,-39)(187,-51) (194,-39)(197,-51) (210,-39)(207,-51) (210,-39)(217,-51) 1.0 (252,-34)(10,90,210) (252,-34)(10,210,280)2 (252,-34)(10,20,90)2 (252,-34)(10,-80,20) 0.5 (244,-39)3.4 (260,-39)3.4 1.0 (249,-21)(255,-27)(249,-27)(255,-21) (244,-39)(237,-51) (244,-39)(247,-51) (260,-39)(257,-51) (260,-39)(267,-51) 0.5 Black (83,-42)[lb]$\\displaystyle +$ (133,-42)[lb]$\\displaystyle +$ (183,-42)[lb]$\\displaystyle +$ (234,-42)[lb]$\\displaystyle +$ (91,-60)[lb]$a$ (103,-60)[lb]$b$ (113,-60)[lb]$c$ (123,-60)[lb]$d$ (141,-60)[lb]$a$ (153,-60)[lb]$b$ (163,-60)[lb]$c$ (173,-60)[lb]$d$ (191,-60)[lb]$a$ (203,-60)[lb]$b$ (213,-60)[lb]$c$ (223,-60)[lb]$d$ (241,-60)[lb]$a$ (253,-60)[lb]$b$ (263,-60)[lb]$c$ (273,-60)[lb]$d$ 1.0 (110,-34)(10,90,260) (110,-34)(10,260,330)2 (110,-34)(10,20,90)2 (110,-34)(10,-30,20) 0.5 (102,-39)3.4 (118,-39)3.4 1.0 (107,-21)(113,-27)(107,-27)(113,-21) (102,-39)(95,-51) (102,-39)(105,-51) (118,-39)(115,-51) (118,-39)(125,-51) (160,-34)(10,90,140) (160,-34)(10,20,90)2 (160,-34)(10,90,20) 0.5 (152,-39)3.4 (168,-39)3.4 1.0 (157,-21)(163,-27)(157,-27)(163,-21) (152,-39)(145,-51) (152,-39)(155,-51) (168,-39)(165,-51) (168,-39)(175,-51) (210,-34)(10,40,90) (210,-34)(10,-30,40)2 (210,-34)(10,90,160)2 (210,-34)(10,160,210) (210,-34)(10,210,280)2 (210,-34)(10,-80,-30) 0.5 (202,-39)3.4 (218,-39)3.4 1.0 (207,-21)(213,-27)(207,-27)(213,-21) (202,-39)(195,-51) (202,-39)(205,-51) (218,-39)(215,-51) (218,-39)(225,-51) 1.0 (260,-34)(10,40,90) (260,-34)(10,260,40)2 (260,-34)(10,160,260) (260,-34)(10,90,160)2 0.5 (252,-39)3.4 (268,-39)3.4 1.0 (257,-21)(263,-27)(257,-27)(263,-21) (252,-39)(245,-51) (252,-39)(255,-51) (268,-39)(265,-51) (268,-39)(275,-51) 0.5 Black (133,-42)[lb]$\\displaystyle +$ (183,-42)[lb]$\\displaystyle +$ (234,-42)[lb]$\\displaystyle +$ (83,-42)[lb]$\\displaystyle +$ (91,-60)[lb]$a$ (103,-60)[lb]$b$ (113,-60)[lb]$c$ (123,-60)[lb]$d$ (141,-60)[lb]$a$ (153,-60)[lb]$b$ (163,-60)[lb]$c$ (173,-60)[lb]$d$ (191,-60)[lb]$a$ (203,-60)[lb]$b$ (213,-60)[lb]$c$ (223,-60)[lb]$d$ (241,-60)[lb]$a$ (253,-60)[lb]$b$ (263,-60)[lb]$c$ (273,-60)[lb]$d$ 1.0 (110,-34)(10,40,90) (110,-34)(10,-30,40)2 (110,-34)(10,160,-30) (110,-34)(10,90,160)2 0.5 (102,-39)3.4 (118,-39)3.4 1.0 (107,-21)(113,-27)(107,-27)(113,-21) (102,-39)(95,-51) (102,-39)(105,-51) (118,-39)(115,-51) (118,-39)(125,-51) 1.0 (160,-34)(10,-80,90) (160,-34)(10,90,160)2 (160,-34)(10,160,210) (160,-34)(10,210,280)2 0.5 (152,-39)3.4 (168,-39)3.4 1.0 (157,-21)(163,-27)(157,-27)(163,-21) (152,-39)(145,-51) (152,-39)(155,-51) (168,-39)(165,-51) (168,-39)(175,-51) (210,-34)(10,-30,90) (210,-34)(10,260,-30)2 (210,-34)(10,160,260) (210,-34)(10,90,160)2 0.5 (202,-39)3.4 (218,-39)3.4 1.0 (207,-21)(213,-27)(207,-27)(213,-21) (202,-39)(195,-51) (202,-39)(205,-51) (218,-39)(215,-51) (218,-39)(225,-51) (260,-34)(10,160,90) (260,-34)(10,90,160)2 (252,-39)3.4 (268,-39)3.4 1.0 (257,-21)(263,-27)(257,-27)(263,-21) (252,-39)(245,-51) (252,-39)(255,-51) (268,-39)(265,-51) (268,-39)(275,-51) 0.5 Black (80,-42)[lb]$\\displaystyle + \\quad P(a,b,c,d) \\,\\, \\bigg \\rbrace $ 0.5 Black (130,-44)[lb]$\\displaystyle +$ (180,-44)[lb]$\\displaystyle +$ (230,-45)[lb]$\\displaystyle +$ (280,-50)[lb]$\\bigg \\rbrace ~.$ (73,-50)[lb]$\\displaystyle - \\frac{i}{2}\\, \\bigg \\lbrace $ (93,-63)[lb]$a$ (102,-63)[lb]$b$ (113,-63)[lb]$c$ (122,-63)[lb]$d$ (143,-63)[lb]$a$ (152,-63)[lb]$b$ (163,-63)[lb]$c$ (172,-63)[lb]$d$ (193,-63)[lb]$a$ (202,-63)[lb]$b$ (213,-63)[lb]$c$ (222,-63)[lb]$d$ (243,-63)[lb]$a$ (252,-63)[lb]$b$ (263,-63)[lb]$c$ (272,-63)[lb]$d$ 1.0 (110,-34)(10,90,180) (110,-34)(10,180,270)2 (110,-34)(10,0,90)2 (110,-34)(10,-90,0) 0.5 (110,-44)3.4 1.0 (107,-21)(113,-27)(107,-27)(113,-21) (97,-55)(110,-44) (105,-55)(110,-44) (115,-55)(110,-44) (123,-55)(110,-44) (160,-34)(10,90,270) (160,-34)(10,0,90)2 (160,-34)(10,-90,0) 0.5 (160,-44)3.4 1.0 (157,-21)(163,-27)(157,-27)(163,-21) (147,-55)(160,-44) (155,-55)(160,-44) (165,-55)(160,-44) (173,-55)(160,-44) (210,-34)(10,90,180)2 (210,-34)(10,180,270) (210,-34)(10,0,90) (210,-34)(10,-90,0)2 0.5 (210,-44)3.4 1.0 (207,-21)(213,-27)(207,-27)(213,-21) (197,-55)(210,-44) (205,-55)(210,-44) (215,-55)(210,-44) (223,-55)(210,-44) (260,-34)(10,180,90) (260,-34)(10,90,180)2 0.5 (260,-44)3.4 1.0 (257,-21)(263,-27)(257,-27)(263,-21) (247,-55)(260,-44) (255,-55)(260,-44) (265,-55)(260,-44) (273,-55)(260,-44) Here, $P(a,b,c,d)$ denotes all possible permutations of the indices on legs of the respective diagrams.", "The diagrammatic representation of the flow equations for $n$ -point functions follows the standard construction rules: draw all combinations of propagators $G^{\\mathrm {R},\\mathrm {A}}_{k}$ and $F_{k}$ , and vertices $\\Gamma ^{(n)}_{k}$ and attach the appropriate symmetry factors, taking into account that certain diagrams may either vanish or are identical.", "We emphasize, however, again that in the diagrammatic representation the propagators are evaluated at an extremum of the effective average action.", "We may simplify the construction rules of the flow equations even further by introducing the derivative operator $\\tilde{\\partial }_{k}$ , $ \\tilde{\\partial }_{k} \\Gamma _{k} [\\phi ,{\\tilde{\\phi }}] \\equiv \\operatorname{Tr}\\left[ {\\dot{R}}^{\\mathrm {R}}_{k} \\frac{\\delta }{\\delta R^{\\mathrm {R}}_{k}} + {\\dot{R}}^{\\mathrm {A}}_{k} \\frac{\\delta }{\\delta R^{\\mathrm {A}}_{k}} \\right] \\Gamma _{k}~.", "$ With this we may reduce expressions like $ \\left( \\Gamma ^{{\\tilde{\\phi }}\\phi }_{k} + R^{\\mathrm {R}}_{k} \\right)^{-1} {\\dot{R}}^{\\mathrm {R}}_{k} \\left( \\Gamma ^{{\\tilde{\\phi }}\\phi }_{k} + R^{\\mathrm {R}}_{k} \\right)^{-1}= - \\tilde{\\partial }_{k} \\left( \\Gamma ^{{\\tilde{\\phi }}\\phi } + R^{\\mathrm {R}}_{k} \\right)^{-1} ~, $ to a simple form.", "For instance, the flow equation for the two-point function can be written as $ \\begin{picture}(249,28) (15,-54){0.5}{Black}(120,-42)[lb]{\\displaystyle +}(170,-42)[lb]{\\displaystyle +}(221,-47)[lb]{\\bigg \\rbrace ~.", "}(16,-47)[lb]{\\displaystyle {\\dot{\\Gamma }}_{k,ab}^{(2)}\\, =\\, -\\, \\frac{i}{2} \\, {\\tilde{\\partial }}_k \\,\\, \\bigg \\lbrace }(90,-56)[lb]{a}(107,-56)[lb]{b}(140,-56)[lb]{a}(157,-56)[lb]{b}(190,-56)[lb]{a}(207,-56)[lb]{b}{1.0}(100,-34)(10,90,-90){2}(100,-34)(10,-90,90){0.5}(100,-44){3.4}{1.0}(90,-45)(110,-45)(150,-34)(10,90,270)(150,-34)(10,-90,90){2}{0.5}(150,-44){3.4}{1.0}(140,-45)(160,-45){1.0}(200,-34)(10,90,-90)(200,-34)(10,-90,90){0.5}(200,-44){3.4}{1.0}(190,-45)(210,-45)\\end{picture} $ Equivalently, for the four-point function $\\Gamma _{k}^{(4)}$ we have rCl 0.5 Black (27,-45)[lb]$\\displaystyle {\\dot{\\Gamma }}_{k,abcd}^{(4)} \\, = \\, $ 0.5 Black (125,-37)[lb]$\\displaystyle +$ (187,-37)[lb]$\\displaystyle +$ (30,-45)[lb]$ -\\frac{i}{16} \\, \\tilde{\\partial }_k\\, \\bigg \\lbrace $ (72,-28)[lb]$a$ (72,-47)[lb]$b$ (117,-28)[lb]$c$ (117,-47)[lb]$d$ (135,-28)[lb]$a$ (135,-47)[lb]$b$ (180,-28)[lb]$c$ (180,-47)[lb]$d$ (196,-28)[lb]$a$ (196,-47)[lb]$b$ (241,-28)[lb]$c$ (241,-47)[lb]$d$ 1.0 (97,-34)(10,-90,90) (97,-34)(10,90,270)2 0.5 (87,-34)3.4 (107,-34)3.4 1.0 (87,-34)(79,-25) (87,-34)(79,-43) (107,-34)(115,-25) (107,-34)(115,-43) (159,-34)(10,-90,0)2 (159,-34)(10,180,-90) (159,-34)(10,90,180)2 (159,-34)(10,0,90) 0.5 (149,-34)3.4 (169,-34)3.4 1.0 (149,-34)(141,-25) (149,-34)(141,-43) (169,-34)(177,-25) (169,-34)(177,-43) (220,-34)(10,180,90) (220,-34)(10,90,180)2 0.5 (210,-34)3.4 (230,-34)3.4 1.0 (210,-34)(202,-25) (210,-34)(202,-43) (230,-34)(238,-25) (230,-34)(238,-43) 0.5 Black (125,-37)[lb]$\\displaystyle +$ (187,-37)[lb]$\\displaystyle +$ (58,-37)[lb]$\\displaystyle +$ (72,-28)[lb]$a$ (72,-47)[lb]$b$ (117,-28)[lb]$c$ (117,-47)[lb]$d$ (135,-28)[lb]$a$ (135,-47)[lb]$b$ (180,-28)[lb]$c$ (180,-47)[lb]$d$ (196,-28)[lb]$a$ (196,-47)[lb]$b$ (241,-28)[lb]$c$ (241,-47)[lb]$d$ 1.0 (97,-34)(10,0,90)2 (97,-34)(10,90,180) (97,-34)(10,180,270)2 (97,-34)(10,270,360) 0.5 (87,-34)3.4 (107,-34)3.4 1.0 (87,-34)(79,-25) (87,-34)(79,-43) (107,-34)(115,-25) (107,-34)(115,-43) (159,-34)(10,-90,90)2 (159,-34)(10,90,270) 0.5 (149,-34)3.4 (169,-34)3.4 1.0 (149,-34)(141,-25) (149,-34)(141,-43) (169,-34)(177,-25) (169,-34)(177,-43) (220,-34)(10,90,360) (220,-34)(10,0,90)2 0.5 (210,-34)3.4 (230,-34)3.4 1.0 (210,-34)(202,-25) (210,-34)(202,-43) (230,-34)(238,-25) (230,-34)(238,-43) 0.5 Black (125,-37)[lb]$\\displaystyle +$ (187,-37)[lb]$\\displaystyle +$ (58,-37)[lb]$\\displaystyle +$ (72,-28)[lb]$a$ (72,-47)[lb]$b$ (117,-28)[lb]$c$ (117,-47)[lb]$d$ (135,-28)[lb]$a$ (135,-47)[lb]$b$ (180,-28)[lb]$c$ (180,-47)[lb]$d$ (196,-28)[lb]$a$ (196,-47)[lb]$b$ (241,-28)[lb]$c$ (241,-47)[lb]$d$ 1.0 (97,-34)(10,-90,180) (97,-34)(10,180,270)2 0.5 (87,-34)3.4 (107,-34)3.4 1.0 (87,-34)(79,-25) (87,-34)(79,-43) (107,-34)(115,-25) (107,-34)(115,-43) (159,-34)(10,0,270) (159,-34)(10,270,360)2 0.5 (149,-34)3.4 (169,-34)3.4 1.0 (149,-34)(141,-25) (149,-34)(141,-43) (169,-34)(177,-25) (169,-34)(177,-43) (220,-34)(10,0,360) 0.5 (210,-34)3.4 (230,-34)3.4 1.0 (210,-34)(202,-25) (210,-34)(202,-43) (230,-34)(238,-25) (230,-34)(238,-43) 0.5 Black (58,-44)[lb]$\\displaystyle + \\quad P(a,b,c,d) \\, \\bigg \\rbrace $ 0.5 Black (115,-44)[lb]$\\displaystyle +$ (175,-44)[lb]$\\displaystyle +$ (232,-50)[lb]$\\bigg \\rbrace ~.$ (36,-50)[lb]$\\displaystyle - \\frac{i}{2}\\,\\, \\tilde{\\partial }_k\\,\\, \\bigg \\lbrace $ (73,-63)[lb]$a$ (83,-63)[lb]$b$ (93,-63)[lb]$c$ (102,-63)[lb]$d$ (133,-63)[lb]$a$ (143,-63)[lb]$b$ (153,-63)[lb]$c$ (162,-63)[lb]$d$ (193,-63)[lb]$a$ (203,-63)[lb]$b$ (213,-63)[lb]$c$ (222,-63)[lb]$d$ 1.0 (90,-34)(10,-90,90) (90,-34)(10,90,270)2 0.5 (90,-44)3.4 1.0 (77,-55)(90,-44) (85,-55)(90,-44) (95,-55)(90,-44) (103,-55)(90,-44) (150,-34)(10,90,270) (150,-34)(10,-90,90)2 0.5 (150,-44)3.4 1.0 (137,-55)(150,-44) (145,-55)(150,-44) (155,-55)(150,-44) (163,-55)(150,-44) (210,-34)(10,90,270) (210,-34)(10,-90,90) 0.5 (210,-44)3.4 1.0 (197,-55)(210,-44) (205,-55)(210,-44) (215,-55)(210,-44) (223,-55)(210,-44)" ], [ "Solving truncated flow equations", "By calculating the functional derivatives of the effective average action, we extracted above the flow equations for $n$ -point functions.", "We will use these flow equations in the following to study the behavior in the vicinity of possible nonthermal fixed points.", "Fixed points correspond to translationally invariant scaling solutions for $n$ -point functions both in space and time.", "Therefore, we are interested in the limit $t_{0} \\rightarrow - \\infty $ for which the dependence on the details about the initial conditions encoded in $\\varrho (t_0)$ are lost.", "As discussed in section , conserved quantities will play an important role and we will not impose a fluctuation-dissipation relation to be able to describe nonthermal fixed points." ], [ "Stationarity condition", "The presence of nonthermal scaling solutions was discussed in section based on a stationarity condition for the nonequilibrium time evolution equations.", "An equivalent (scale-dependent) stationarity condition can be obtained from the functional renormalization group, where it appears as a nontrivial identity relating the various second functional derivatives of $\\Gamma _k[\\phi ,{\\tilde{\\phi }}]$ .", "For spacetime translation invariant systems, it is very convenient to consider the correlation functions in Fourier space.", "We start by writing down the following identity in momentum space, rCl i k, a c (p) { GRk, c a (p) - GAk, c a (p) } = - i GRk, a c (p) k, c d GAk, d e (p) { ( GRk )-1e a (p) - ( GAk )-1e a (p) }  .", "In order to interpret further the different combinations of terms appearing in this equation, we write the two-point functions $\\Gamma ^{(2)}_{k}$ in the form rCl k k k k = 0 - ( GAk )-1 - RAk - ( GRk )-1 - RRk ( GRk )-1 i Fk ( GAk )-1 0 i D-1 i D-1 0 - 0 Ak Rk i Fk  , where $i D^{-1}$ is the free inverse propagator.", "The second line defines the self-energies $\\Sigma ^{\\mathrm {R},\\mathrm {A},F}_{k}$ , which in turn are given by rCl Fk, a b i k, a b  , k, a b Rk, a b - Ak, a b = k, a b - k, ab  , where we have used that $R^{\\mathrm {R}}_{k} = R^{\\mathrm {A}}_{k}$ for the considered class of cutoff functions (REF ).", "With these identifications, the identity (REF ) can be written in terms of the self-energies as $ \\Sigma ^F_{k, a b} (p)\\, \\rho _{k, b a } (p) - F_{k, a b} (p)\\, \\Sigma ^{\\rho }_{k, b a} (p) = 0~.", "$ This equation is well-known in nonequilibrium physics.", "In the language of Boltzmann dynamics employed in section , it essentiallyUsing the Wigner coordinates employed in section , the nonequilibrium time evolution of the statistical function is given by [59] $ 2p^\\mu \\partial _{X^\\mu } F_{ab}(X,p) = i \\left( \\Sigma ^F_{ac} \\rho _{cb} - F_{ac} \\Sigma ^\\rho _{cb} \\right)(X,p).", "$ The condition (REF ) is related to this by taking the trace at a stationary point, where the correlation functions become independent of $X$ .", "states that `gain terms' equal `loss terms' for which stationarity is achieved [22].", "This aspect will be analyzed in detail in section .", "Of course, the condition (REF ) is trivially fulfilled if the fluctuation-dissipation relation holds, i.e.", "in thermal equilibrium where rCl Fk(eq) (p) = -i ( nBE(p0) + 12 ) k(eq) (p)  , F  (eq)k (p) = -i ( nBE(p0) + 12 )  (eq)k (p)  , for the propagators and self-energies [22].", "These relations will not be assumed in the following.", "Using the representation of the self-energies in terms of two-point functions we may immediately write down the flow equations for the statistical component $\\Sigma ^{F}_{k}$ , which is given by rCl 0.5 Black (142,-42)[lb]$\\displaystyle +$ (192,-42)[lb]$\\displaystyle +$ (238,-47)[lb]$\\, \\bigg \\rbrace ~,$ (40,-47)[lb]$\\displaystyle {\\dot{\\Sigma }}^{F}_{k, a b} \\,\\,=\\,\\, \\frac{1}{2} \\, {\\tilde{\\partial }}_k \\,\\, \\bigg \\lbrace \\,$ (110,-56)[lb]$a$ (127,-56)[lb]$b$ (160,-56)[lb]$a$ (177,-56)[lb]$b$ (210,-56)[lb]$a$ (227,-56)[lb]$b$ 1.0 (120,-34)(10,90,-90)2 (120,-34)(10,-90,90) 0.5 (120,-44)3.4 1.0 (110,-45)(130,-45)2 (170,-34)(10,90,270) (170,-34)(10,-90,90)2 0.5 (170,-44)3.4 1.0 (160,-45)(180,-45)2 1.0 (220,-34)(10,90,-90) (220,-34)(10,-90,90) 0.5 (220,-44)3.4 1.0 (210,-45)(230,-45)2 reading in momentum space rCl Fk, a b (p) = 12 k q   { k, a b c d (p,-p,q)   GRk, d c (q)        + k,a b c d (p,-p,q)   GAk, d c (q)        + k, a b c d (p,-p,q)   i Fk, d c (q) }  .", "The spectral self-energy satisfies the flow equation rCl 0.5 Black (8,-47)[lb]$\\displaystyle {\\dot{\\Sigma }}^{\\rho }_{k, a b} \\,\\,=\\,\\, {\\dot{\\Gamma }}^{\\phi {\\tilde{\\phi }}}_{k,ab} \\, -\\, {\\dot{\\Gamma }}^{{\\tilde{\\phi }}\\phi }_{k,ab}$ 0.5 Black (137,-42)[lb]$\\displaystyle +$ (187,-42)[lb]$\\displaystyle +$ (40,-48)[lb]$\\,\\, =\\,\\, - \\frac{i}{2} \\,\\, {\\tilde{\\partial }}_k \\,\\, \\bigg \\lbrace $ (105,-56)[lb]$a$ (122,-56)[lb]$b$ (155,-56)[lb]$a$ (172,-56)[lb]$b$ (205,-56)[lb]$a$ (222,-56)[lb]$b$ 1.0 (115,-34)(10,90,-90)2 (115,-34)(10,-90,90) 0.5 (115,-44)3.4 1.0 (105,-45)(115,-45) (115,-45)(125,-45)2 (165,-34)(10,90,270) (165,-34)(10,-90,90)2 0.5 (165,-44)3.4 1.0 (155,-45)(165,-45) (165,-45)(175,-45)2 1.0 (215,-34)(10,90,-90) (215,-34)(10,-90,90) 0.5 (215,-44)3.4 1.0 (205,-45)(215,-45) (215,-45)(225,-45)2 0.5 Black (90,-42)[lb]$\\displaystyle -$ (137,-42)[lb]$\\displaystyle -$ (187,-42)[lb]$\\displaystyle -$ (232,-48)[lb]$\\,\\, \\bigg \\rbrace ~.$ (105,-56)[lb]$a$ (122,-56)[lb]$b$ (155,-56)[lb]$a$ (172,-56)[lb]$b$ (205,-56)[lb]$a$ (222,-56)[lb]$b$ 1.0 (115,-34)(10,90,-90)2 (115,-34)(10,-90,90) 0.5 (115,-44)3.4 1.0 (115,-45)(125,-45) (105,-45)(115,-45)2 (165,-34)(10,90,270) (165,-34)(10,-90,90)2 0.5 (165,-44)3.4 1.0 (165,-45)(175,-45) (155,-45)(165,-45)2 1.0 (215,-34)(10,90,-90) (215,-34)(10,-90,90) 0.5 (215,-44)3.4 1.0 (215,-45)(225,-45) (205,-45)(215,-45)2" ], [ "Resummed $1/N$ -expansion", "In general, flow equations for $n$ -point functions are expressed in terms of $(n+2)$ -point functions, and to find approximate solutions one has to truncate the infinite hierarchy of coupled flow equations.", "Here, we use an ansatz for the four-point functions, which is equivalent to a resummed large-$N$ expansion of the 2PI effective action to next-to-leading order (NLO) [60].", "Using $O(N)$ symmetry, we can always write the two-point functions as, for instance, $ \\Gamma ^{{\\tilde{\\phi }}\\phi }_{k, a b} (x,y) = \\Gamma ^{{\\tilde{\\phi }}\\phi }_{k} (x,y) \\delta _{a b} ~.", "$ The four-point functions can be decomposed into the contributions from the different channels, e.g.", "for $\\Gamma _{k}^{{\\tilde{\\phi }}\\phi \\phi \\phi }$ , we have [18] rCl k, a b c d (x,y,z,w) = , k (x,z) (d+1)(x-y) (d+1)(z-w) a b c d + , k (z,y) (d+1)(z-x) (d+1)(y-w) a c b d + , k (y,z) (d+1)(y-z) (d+1)(x-w) b c a d  .", "In the resummed large-$N$ expansion we write these individual contributions diagrammatically as rCl 0.5 Black (52,-232)[lb]$\\Gamma _k^{\\tilde{\\phi }\\phi , \\phi \\phi }(x,y)\\,=\\,$ (117,-228)[lb]$x$ (158,-229)[lb]$y$ (166,-232)[lb]  , 1 (140,-224)(3,10)(0)0 (129,-224)(121,-214)2 (129,-224)(121,-234) (151,-224)(159,-214) (151,-224)(159,-234) 0.5 Black (52,-232)[lb]$\\Gamma _k^{\\tilde{\\phi }\\phi , \\phi \\phi }(x,y)\\,=\\,$ (117,-228)[lb]$x$ (158,-229)[lb]$y$ (166,-232)[lb]  , 1 (140,-224)(3,10)(0)0 (129,-224)(121,-214)2 (129,-224)(121,-234)2 (151,-224)(159,-214)2 (151,-224)(159,-234) where the full blobs denote the resummed bubble chain with full propagators $G^{\\mathrm {R},\\mathrm {A}}, F,{\\tilde{F}}$ inserted on the internal lines connected by the bare vertices given by $ \\Gamma ^{{\\tilde{\\phi }}\\phi \\phi \\phi }_{\\Lambda } = - \\frac{\\lambda }{3N} ~, \\quad \\Gamma ^{{\\tilde{\\phi }}{\\tilde{\\phi }}{\\tilde{\\phi }}\\phi }_{\\Lambda } = - \\frac{\\lambda }{12 N}~.", "$ For instance, rCl 0.5 Black (64,-227)[lb]$\\displaystyle = $ (99,-229)[lb]$\\displaystyle - \\,\\, iN $ 1 (45,-224)(3,6)(0)0 (38,-224)(30,-216) (38,-224)(30,-232) (52,-224)(60,-216)2 (52,-224)(60,-232) 1 (85,-224)(77,-216) (85,-224)(77,-232) (85,-224)(93,-216)2 (85,-224)(93,-232) 1 (132,-224)(124,-216) (132,-224)(124,-232) (142,-224)(10,180,90) (142,-224)(10,90,180)2 (152,-224)(160,-216)2 (152,-224)(160,-232) 0.5 Black (166,-229)[lb]$\\displaystyle + \\,\\, (-iN)^2 $ (271,-228)[lb]$\\displaystyle + \\,\\, ... $ 1 (215,-224)(207,-214) (215,-224)(207,-234) (225,-224)(10,180,90) (225,-224)(10,90,180)2 (246,-224)(10,180,90) (246,-224)(10,90,180)2 (256,-224)(264,-216)2 (256,-224)(264,-232) 0.5 Black (64,-227)[lb]$\\displaystyle = $ (99,-229)[lb]$\\displaystyle - \\,\\, iN $ (205,-229)[lb]$\\displaystyle .$ 1 (85,-224)(77,-216) (85,-224)(77,-232) (85,-224)(93,-216)2 (85,-224)(93,-232) 1 (145,-224)(3,6)(0)0 (139,-224)(131,-216) (139,-224)(131,-232) (163,-224)(10,180,90) (163,-224)(10,90,180)2 (181,-224)(3,6)(0)0 (188,-224)(196,-234) (188,-224)(196,-214)2 Each vertex contributes a factor $1/N$ and each closed loop gives a factor of $N = \\delta _{a b} \\delta _{b a}$ .", "Thus, all shown diagrams contribute at the same order, and there are no other diagrams that contribute at this order in the expansion.", "Notably, all $n$ -point functions $\\Gamma _{k}^{(n)}$ with $n > 4$ are of higher order in the large-$N$ expansion and the infinite hierarchy of flow equations is closed at the level of the four-point functions [17], [18].", "Diagrammatically this can be expressed in terms of the flow equation $ \\begin{picture}(73,26) (90,-239){0.5}{Black}(82,-227)[lb]{\\displaystyle = }(98,-229)[lb]{\\displaystyle - \\,\\, i N \\,\\, {\\tilde{\\partial }}_{k} \\,\\,}(205,-228)[lb]{\\displaystyle ,}{1}(55,-214){0.8}(55,-224)(3,6)(0){0}(48,-224)(40,-216)(48,-224)(40,-232)(62,-224)(70,-216){2}(62,-224)(70,-232){1}(151,-224)(3,6)(0){0}(144,-224)(136,-216)(144,-224)(136,-232)(168,-224)(10,180,90)(168,-224)(10,90,180){2}(186,-224)(3,6)(0){0}(193,-224)(201,-234)(193,-224)(201,-214){2}\\end{picture} $ which has only the four-vertex $\\Gamma ^{(4)}_{k}$ appearing on the r.h.s.", "We also note that the scale derivative represents a total derivative in this approximation [17], [18], [61], which can directly be understood from our derivation of the exact flow equation with the help of the 2PI effective action in section .", "Similarly, one obtains for the other four-vertices rCl 0.5 Black (64,-227)[lb]$\\displaystyle = $ (99,-229)[lb]$\\displaystyle - \\,\\, iN $ 1 (45,-224)(3,6)(0)0 (38,-224)(30,-216)2 (38,-224)(30,-232)2 (52,-224)(60,-216)2 (52,-224)(60,-232) 1 (85,-224)(77,-216)2 (85,-224)(77,-232)2 (85,-224)(93,-216)2 (85,-224)(93,-232) 1 (132,-224)(124,-216)2 (132,-224)(124,-232)2 (142,-224)(10,180,90) (142,-224)(10,90,180)2 (152,-224)(160,-216)2 (152,-224)(160,-232) 0.5 Black (166,-229)[lb]$\\displaystyle + \\,\\, (-iN)^2 $ (271,-228)[lb]$\\displaystyle + \\,\\, ... $ 1 (217,-224)(209,-214)2 (217,-224)(209,-234)2 (227,-224)(10,180,90) (227,-224)(10,90,180)2 (248,-224)(10,180,90) (248,-224)(10,90,180)2 (258,-224)(266,-216)2 (258,-224)(266,-232) 0.5 Black (64,-227)[lb]$\\displaystyle = $ (99,-229)[lb]$\\displaystyle - \\,\\, iN $ (205,-228)[lb]$\\displaystyle ,$ 1 (85,-224)(77,-216)2 (85,-224)(77,-232)2 (85,-224)(93,-216)2 (85,-224)(93,-232) 1 (145,-224)(3,6)(0)0 (139,-224)(131,-216)2 (139,-224)(131,-232)2 (163,-224)(10,180,90) (163,-224)(10,90,180)2 (181,-224)(3,6)(0)0 (188,-224)(196,-234) (188,-224)(196,-214)2 and rCl Black 1 (38,-224)(3,6)(0)0 (31,-224)(23,-216)2 (31,-224)(23,-232) (45,-224)(53,-216)2 (45,-224)(53,-232) 0.5 Black (57,-236)[lb]$\\displaystyle = \\, -\\frac{i}{2}N \\,\\, \\bigg \\lbrace $ (153,-229)[lb]$\\displaystyle + $ (210,-229)[lb]$\\displaystyle + $ (264,-236)[lb]$\\displaystyle \\,\\, \\bigg \\rbrace \\, $ 1 (115,-224)(105,-216)2 (115,-224)(105,-232) (125,-224)(10,-90,90) (125,-224)(10,90,-90)2 (135,-224)(145,-216)2 (135,-224)(145,-232) 1 (173,-224)(165,-216)2 (173,-224)(165,-232) (184,-224)(10,90,-90) (184,-224)(10,-90,90)2 (195,-224)(203,-216)2 (195,-224)(203,-232) 1 (230,-224)(222,-216)2 (230,-224)(222,-232) (241,-224)(10,90,-90) (241,-224)(10,-90,90) (252,-224)(260,-216)2 (252,-224)(260,-232) 0.5 Black (63,-236)[lb]$\\displaystyle + \\, \\frac{1}{2}(-iN)^2 \\, \\bigg \\lbrace \\,$ (185,-229)[lb]$\\displaystyle + $ 1 (129,-224)(121,-216)2 (129,-224)(121,-232) (140,-224)(10,90,0) (140,-224)(10,0,90)2 (161,-224)(10,-90,90) (161,-224)(10,90,-90)2 (172,-224)(180,-216)2 (172,-224)(180,-232) 1 (203,-224)(195,-216)2 (203,-224)(195,-232) (214,-224)(10,-90,90) (214,-224)(10,90,-90)2 (235,-224)(10,180,90) (235,-224)(10,90,180)2 (246,-224)(254,-216)2 (246,-224)(254,-232) 0.5 Black (110,-229)[lb]$\\displaystyle + $ (185,-229)[lb]$\\displaystyle + $ 1 (129,-224)(121,-216)2 (129,-224)(121,-232) (140,-224)(10,90,0) (140,-224)(10,0,90)2 (161,-224)(10,90,-90) (161,-224)(10,-90,90)2 (172,-224)(180,-216)2 (172,-224)(180,-232) 1 (203,-224)(195,-216)2 (203,-224)(195,-232) (214,-224)(10,90,-90) (214,-224)(10,-90,90)2 (235,-224)(10,180,90) (235,-224)(10,90,180)2 (246,-224)(254,-216)2 (246,-224)(254,-232) 0.5 Black (110,-229)[lb]$\\displaystyle + $ (185,-229)[lb]$\\displaystyle + $ (257,-236)[lb]$\\displaystyle \\, \\bigg \\rbrace + ... $ 1 (129,-224)(121,-216)2 (129,-224)(121,-232) (140,-224)(10,90,0) (140,-224)(10,0,90)2 (161,-224)(10,180,90) (161,-224)(10,90,180) (172,-224)(180,-216)2 (172,-224)(180,-232) 1 (203,-224)(195,-216)2 (203,-224)(195,-232) (214,-224)(10,-90,90) (214,-224)(10,90,-90) (235,-224)(10,180,90) (235,-224)(10,90,180)2 (246,-224)(254,-216)2 (246,-224)(254,-232) 0.5 Black (57,-236)[lb]$\\displaystyle = \\, -\\frac{i}{2}N \\, \\bigg \\lbrace \\,$ (175,-229)[lb]$\\displaystyle + $ 1 (118,-224)(3,6)(0)0 (111,-224)(103,-216)2 (111,-224)(103,-232) (135,-224)(10,-90,90) (135,-224)(10,90,-90)2 (153,-224)(3,6)(0)0 (159,-224)(167,-216)2 (159,-224)(167,-232) 1 (203,-224)(3,6)(0)0 (196,-224)(188,-216)2 (196,-224)(188,-232) (220,-224)(10,90,-90) (220,-224)(10,-90,90)2 (237,-224)(3,6)(0)0 (244,-224)(252,-216)2 (244,-224)(252,-232) 0.5 Black (90,-229)[lb]$\\displaystyle +$ (172,-236)[lb]$\\displaystyle \\, \\bigg \\rbrace \\,~.", "$ 1 (118,-224)(3,6)(0)0 (111,-224)(103,-216)2 (111,-224)(103,-232) (135,-224)(10,90,-90) (135,-224)(10,-90,90) (153,-224)(3,6)(0)0 (159,-224)(167,-216)2 (159,-224)(167,-232) The loop contributions appearing in the chain of bubbles diagrams are given by the expressions $ \\Pi ^{\\mathrm {R}, \\mathrm {A}}_{k} (p) = \\frac{\\lambda }{3} \\int _{q} F_{k} (p-q) G^{\\mathrm {R}, \\mathrm {A}}_{k} (q) ~, $ and $ \\Pi ^{F}_{k} (p) = \\frac{\\lambda }{6} \\int _{q} \\Big [ F_{k} (p-q) F_{k} (q) - \\frac{1}{4} \\rho _{k} (p-q) \\rho _{k} (q) \\Big ] ~.", "$ In terms of these one-loop expressions the four-point functions read rCl 0.5 Black (65,-234)[lb]$\\displaystyle = \\,\\, \\Gamma ^{{\\tilde{\\phi }}\\phi \\phi \\phi }_{\\Lambda } \\left\\lbrace \\, 1 - \\Pi ^{\\mathrm {A}}_{k} +\\left( \\Pi ^{\\mathrm {A}}_{k} \\right)^{2} + \\ldots \\, \\right\\rbrace $ 1 (45,-224)(3,6)(0)0 (38,-224)(30,-216) (38,-224)(30,-232) (52,-224)(60,-216)2 (52,-224)(60,-232) 0.5 Black (65,-234)[lb]$\\displaystyle = \\,\\, - \\frac{\\lambda }{3 N} + \\frac{\\lambda _{\\text{eff},k}}{3 N} \\Pi ^{\\mathrm {A}}_{k} ~,$ 0.5 Black (65,-234)[lb]$\\displaystyle = \\,\\, i \\frac{\\lambda }{3 N} \\left\\lbrace \\, 1 - \\Pi ^{\\mathrm {R}}_{k} - \\Pi ^{\\mathrm {A}}_{k} +\\ldots \\, \\right\\rbrace $ 1 (45,-224)(3,6)(0)0 (38,-224)(30,-216)2 (38,-224)(30,-232) (52,-224)(60,-216)2 (52,-224)(60,-232) 0.5 Black (65,-234)[lb]$\\displaystyle = \\,\\, i \\frac{\\lambda _{\\text{eff},k}}{3 N} \\Pi ^{F}_{k} ~,$ 0.5 Black (65,-234)[lb]$\\displaystyle = \\,\\, \\Gamma ^{{\\tilde{\\phi }}{\\tilde{\\phi }}{\\tilde{\\phi }}\\phi }_{\\Lambda } \\left\\lbrace \\, 1 - \\Pi ^{\\mathrm {A}}_{k} +\\left( \\Pi ^{\\mathrm {A}}_{k} \\right)^{2} + \\ldots \\, \\right\\rbrace $ 1 (45,-224)(3,6)(0)0 (38,-224)(30,-216)2 (38,-224)(30,-232)2 (52,-224)(60,-216)2 (52,-224)(60,-232) 0.5 Black (65,-234)[lb]$\\displaystyle = \\,\\, - \\frac{\\lambda }{12 N} + \\frac{\\lambda _{\\text{eff},k}}{12 N} \\Pi ^{\\mathrm {A}}_{k} ~,$ where we have defined the momentum-dependent effective coupling $ \\lambda _{\\text{eff},k} (p) = \\frac{\\lambda }{\\big [ 1 + \\Pi ^{\\mathrm {R}}_{k} (p) \\big ] \\big [ 1 + \\Pi ^{\\mathrm {A}}_{k} (p) \\big ]} ~, $ to express the set of resummed diagrams.", "In order to observe the equivalence with the previous expressions above, we note that, e.g., for the set of diagrams contributing to the vertices $\\Gamma ^{\\phi \\phi \\phi {\\tilde{\\phi }}}_{k}$ and $\\Gamma ^{{\\tilde{\\phi }}{\\tilde{\\phi }}{\\tilde{\\phi }}\\phi }_{k}$ , we have rCl 11+Rk (p) 11+Ak (p) = 1 - Rk (p) - Ak (p) + ( Rk (p) )2 + ( Ak (p) )2 + ... , where mixed products of the type $\\Pi ^{\\mathrm {R}}_{k} (p) \\Pi ^{\\mathrm {A}}_{k} (p)$ vanish in the expansion, since there are no vertices $\\Gamma ^{{\\tilde{\\phi }}{\\tilde{\\phi }}\\phi \\phi }$ that would allow for that particular combination of retarded and advanced one-loop diagrams $\\Pi ^{\\mathrm {R},\\mathrm {A}}$ .", "Plugging the vertices into the flow equation for the self-energies $\\Sigma ^{\\rho ,F}_{k}$ and integrating the total scale derivative gives the final result for the statistical component rCl 0.5 Black (26,-234)[lb]$\\displaystyle \\Sigma _k^{\\rho \\,} \\,\\,=\\,\\, -i \\,\\, \\bigg \\lbrace $ (134,-227)[lb]$\\displaystyle +$ 1 (102,-224)(3,10)(0)0 (80,-224)(90,-224) (114,-224)(124,-224)2 (102,-224)(12,0,90) (102,-224)(12,90,180)2 1 (172,-224)(3,10)(0)0 (150,-224)(160,-224) (194,-224)(184,-224)2 (172,-224)(12,0,180) 0.5 Black 1 (65,-227)[lb]$\\displaystyle -$ (134,-227)[lb]$\\displaystyle -$ (201,-232)[lb]$\\displaystyle \\bigg \\rbrace ~, $ (102,-224)(3,10)(0)0 (80,-224)(90,-224)2 (114,-224)(124,-224) (102,-224)(12,0,90)2 (102,-224)(12,90,180) 1 (172,-224)(3,10)(0)0 (150,-224)(160,-224)2 (194,-224)(184,-224) (172,-224)(12,0,180) and the spectral component rCl 0.5 Black (25,-230)[lb]$\\displaystyle \\Sigma _k^{F} \\,\\,=\\,\\, $ (113,-228)[lb]$\\displaystyle +$ 1 (80,-224)(3,10)(0)0 (58,-224)(68,-224)2 (92,-224)(102,-224)2 (80,-224)(12,0,90) (80,-224)(12,90,180)2 1 (155,-224)(3,10)(0)0 (133,-224)(143,-224)2 (177,-224)(167,-224)2 (155,-224)(12,0,90)2 (155,-224)(12,90,180) 0.5 Black (195,-228)[lb]$\\displaystyle +$ (260,-228)[lb]$\\displaystyle .$ 1 (230,-224)(3,10)(0)0 (208,-224)(218,-224)2 (252,-224)(242,-224)2 (230,-224)(12,0,180) Writing these expression explicitly in momentum space, we have rCl Fk (p) = - 13 N q eff,k (p-q) { Fk (p-q) Fk (q) .", ".", "- 14 k (p-q) k (q) }  , k (p) = - 13 N q eff,k (p-q) { Fk (p-q) k (q) .", ".", "+ k (p-q) Fk (p) }  , which has the structure of a two-loop self-energy, however, with a momentum-dependent coupling $\\lambda _{\\text{eff},k}$ .", "To summarize, we have found that in the large-$N$ expansion to NLO the infinite hierarchy of flow equations is closed on the level of four-point diagrams.", "These can be solved directly by integrating the total scale derivative $\\tilde{\\partial }_{k}$ .", "Thereby we obtain the full expressions for the self-energies and vertices to this order.", "In the following section we will investigate the scaling behavior of the self-energies that enter the stationarity condition (REF ).", "That way we can extract the scaling exponent $\\kappa $ characterizing different types of fixed points." ], [ "Nonperturbative stationary transport", "From the integrated self-energies (REF ) and (REF ) we can directly classify the scaling solutions in the limit $k \\rightarrow 0$ , where the regulator is sent to zero.", "In order to better compare the nonperturbative aspects of these results to the perturbative discussion of section , we write without loss of generality for $k=0$ : $ F(p) = -i \\left( n(p) + \\frac{1}{2} \\right) \\rho (p) ~, $ and the presentation follows to a large extent Ref. [44].", "Equivalently to (REF ), the function $n(p)$ depends on the four-momentum $p = (p^{0},\\mathit {p})$ , in contrast to the case of thermal equilibrium (see section ).", "It satisfies the symmetry property $ n(-p) = - \\left( n(p) + 1 \\right)~, $ which follows from $F(-p) = F(p)$ , and $\\rho (-p) = - \\rho (p)$ .", "We then write the stationarity condition (REF ) in terms of $n(p)$ and the spectral function $\\rho (p)$ using the identity rCl (p) F(p) - F (p) (p) = i ( F (p) - i2 (p) ) ( F(p) + i2 (p)) - i ( F (p) + i2 (p) ) ( F(p) - i2 (p))  , where by (REF ) we have rCl F(p) + i2 (p) = -i n(p) (p)  , F(p) - i2 (p) = -i ( n(p) + 1 ) (p)  .", "From the self-energies (REF ) and (REF ) we construct the linear combinations rCl F (p) i2 (p) = - 18 N 1,2 eff (p-k1) ( F(p-k1-k2) i2 (p-k1-k2)) ( F1 i2 1 ) ( F2 i2 2 )  , that enter the stationarity equation (REF ).", "In terms of $n(p)$ this reads rCl F (p) - i2 (p) = - i 18 N 1,2 eff (p-k1) ( n(p-k1-k2) + 1 ) (p-k1-k2) ( n1 + 1 ) 1 ( n2 + 1 ) 2  , F (p) + i2 (p) = - i 18 N 1,2 eff (p-k1) n(p-k1-k2) (p-k1-k2) n1 1   n2 2  .", "Finally, putting everything together we obtain the following form for the stationarity condition: rCl - i ( F - F ) (p) = - 18 N 1,2,3 (2 )d+1 (d+1) (p-k1-k2-k2) eff (p-k1) { ( n1 + 1 ) ( n2 +1) ( n3 + 1) np               - n1 n2 n3 ( np + 1 ) }   1 2 3 p  .", "We may bring this expression into a form which can be directly compared to kinetic or Boltzmann descriptions by mapping onto positive frequencies $p^{0}$ .", "In particular, we consider the `collision integral' $ -i \\int ^{\\infty }_{0} \\frac{dp^{0}}{2 \\pi } \\left( \\Sigma ^{\\rho } F - \\Sigma ^{F} \\rho \\right) (p^{0},\\mathit {p}) \\equiv C^{\\textrm {NLO}} (\\mathit {p}) ~, $ which will allow us to relate this discussion to the presentation in section .", "Here, the upper index $\\textrm {NLO}$ indicates that this is accurate to next-to-leading order in the large-$N$ expansion, including processes to all orders in the coupling constant, in contrast to the perturbative discussion in section .", "After performing the positive frequency integral, we get rCl CNLO(p) = d22   {( np + 1 ) ( n1 + 1 ) n2 n3 - np n1 ( n2 + 1 ) ( n3 + 1 ) } + d13(a)   { ( np + 1 ) ( n1 + 1 ) ( n2 + 1 ) n3 - np n1 n2 ( n3 + 1 ) } + d13(b)   { ( np + 1 ) n1 n2 n3 - np ( n1 + 1 ) ( n2 + 1 ) ( n3 + 1 ) } + d04   { ( np + 1 ) ( n1 + 1 ) ( n2 + 1 ) ( n3 + 1 ) - np n1 n2 n3 }  , with the $2^{3} = 8$ contributions from the different orthants in frequency space and rCl d22 = 18 N 1,2,3(>) (2 )d+1 (d+1) ( p + k1 - k2 - k3) [ eff (p+k1) + eff (p-k2)+ eff (p-k3) ] 1 2 3 p  , rCl d13(a) = 18 N 1,2,3(>) (2 )d+1 (d+1) ( p + k1 + k2 - k3 ) [ eff ( p + k1 ) + eff ( p + k2 ) + eff ( p - k3 ) ] 1 2 3 p  , rCl d13(b) = 18 N 1,2,3(>) (2 )d+1 (d+1) ( p - k1 - k2 - k3 ) eff ( p - k1 )   1 2 3 p  , rCl d04 = 18 N 1,2,3(>) (2 )d+1 (d+1) ( p + k1 + k2 + k3 ) eff ( p + k1 )   1 2 3 p  .", "Here, the $(>)$ sign on the integrals indicates, that the integrals run over positive frequencies, i.e.", "$ \\int ^{(>)}_{1,2,3} \\equiv \\int _{0}^{\\infty } \\left( \\frac{dp^{0}}{2 \\pi } \\prod _{i = 1,2,3} \\frac{dk_{i}^{0}}{2 \\pi } \\right) \\int \\left(\\prod _{i = 1,2,3} \\frac{d^{d}k_{i}}{(2 \\pi )^{d}}\\right) ~.", "$ From the Boltzmann description of the weakly interacting theory we clearly recognize from the first contribution above $2\\leftrightarrow 2$ scattering processes, however, with an effective coupling $\\lambda _{\\text{eff}}(p)$ .", "This momentum-dependent coupling is a consequence of the summation of an infinite number of processes, which will be crucial in order to be able to discuss the nonperturbative regime of strong turbulence at low momenta as is explained in the following.", "All other processes are `off-shell' and turn out not to play an important role in this context [5], [18], [12]." ], [ "Scaling solutions", "We want to study the nonequilibrium steady states and their scaling properties.", "Along the lines of our discussion in section , we can extract these properties from a scaling analysis.", "For a discussion in terms of a direct determination of the principal zeros of the collision integral (REF ), we refer to Refs.", "[5], [18], [12].Ref.", "[18] tacitly assumes the absence of particle number changing processes for the derivation of the particle cascade.", "For a proper discussion of this aspect see Ref. [12].", "From the quantities entering the collision integral (REF ) it is clear that we need the scaling properties of $n(p)$ , the effective coupling $\\lambda _{\\text{eff}} (p)$ , and the measure.", "For that purpose, we consider first the scaling behavior of $n(p)$ .", "Extending the discussion of section , we also take into account a possible dynamic critical exponent $z$ different from one and a nontrivial anomalous dimension $\\eta $ following section .", "With the scaling ansatz given by (REF ) and (REF ) for the statistical and spectral function, respectively, we have $ n(p^{0} , \\mathit {p}) = s^{\\kappa + \\eta } n(s^{z} p^{0} , s \\mathit {p}) ~, $ again assuming that $n(p) \\gg 1/2$ .", "For the scaling analysis of the momentum-dependent effective coupling $\\lambda _{\\text{eff}}(p)$ , we use $ \\left( \\Pi ^{\\mathrm {A}} (p) \\right)^{\\ast } = \\Pi ^{\\mathrm {A}} (-p) = \\Pi ^{\\mathrm {R}} (p) ~, $ to write (REF ) as rCl eff (p) = | 1 + R (p) |2  .", "To extract its scaling, we need the scaling behavior of the one-loop diagram $ \\Pi ^{\\mathrm {R}} (p) = \\frac{\\lambda }{3} \\int _{q} F(p-q) G^{\\mathrm {R}} (q) ~, $ which is obtained from the statistical propagator $F$ and the representation of retarded propagator in terms of the spectral function $G^{\\mathrm {R}}(x,y) = \\rho (x,y)\\theta (x^{0}-y^{0})$ , i.e.", "$ G^{\\mathrm {R}} (p^{0},\\mathit {p}) = s^{2-\\eta } G^{\\mathrm {R}} (s^{z} p^{0} , s \\mathit {p}) ~.", "$ Thus, we have rCl R (p) = 3 q F(p-q) GR (q) = 3 q s2+ F(sz (p0 - q0) , s ( p - q ) ) s2 - GR (sz q0 , s q)  , for the one-loop diagram $\\Pi ^{\\mathrm {R}}$ .", "By an appropriate rescaling of the measure of this one-loop diagram, i.e., taking $q^{0} \\rightarrow s^{-z} q^{0}$ and $\\mathit {q} \\rightarrow s^{-1} \\mathit {q}$ , it can be brought to the form rCl R (p) = sR (sz p0 , s p )  , with the scaling exponent $ \\Delta = 4 + \\kappa - \\eta - z - d \\,.", "$ For positive $\\Delta > 0$ one observes that $\\Pi ^{\\mathrm {R}}(p) \\gg 1$ for sufficiently low momenta, such that fluctuations become important in the infrared.", "Inserting this result into the expression (REF ) for the effective coupling, we finally get $ \\Delta > 0 : \\quad \\lambda _{\\text{eff}} (p^{0}, \\mathit {p} ) = s^{-2 \\Delta } \\lambda _{\\text{eff}} (s^{z} p^{0} , s \\mathit {p}) \\, .", "$ In contrast, for the case $\\Delta \\le 0$ the scaling of the effective coupling is trivial.", "This case will be relevant at sufficiently high momenta, where $\\Pi ^{\\mathrm {R}}(p)$ is small.", "Then, from (REF ) it follows that the effective coupling is essentially equivalent to the perturbative coupling, $ \\Delta \\le 0 : \\quad \\lambda _{\\text{eff}} (p) \\simeq \\lambda ~.", "$ Table: NO_CAPTIONIt remains to determine the scaling behavior of the measure $\\int d\\Gamma $ .", "We consider, for instance, the $2\\leftrightarrow 2$ processes: rCl d22 = 18 N 0 (dp02 i = 1,2,3 dki02 ) (i = 1,2,3 ddki(2 )d ) (2 )d+1 (d+1) (p + k1 - k2 - k3) [ eff (p + k1) + eff (p - k2) + eff (p - k3) ] 1 2 3 p  .", "From the scaling analysis, we obtain rCl d22(sz p0 , s p,sz k10 , s k1,sz k20 , s k2,sz k30 , s k3) = s-4 z - 3 d + z + d - 2 + 4 (2- ) d22 (p,k1,k2,k3)  , where the momentum dependencies in the measure are indicated explicitly.", "With a positive exponent for the one-loop diagram (REF ), $\\Delta > 0$ , the scaling of the measure can be written as $\\sim s^{-2 \\kappa - 2 \\eta - z}$ .", "In contrast, for $\\Delta \\le 0$ , where the coupling is given by $\\lambda _{\\text{eff}} \\simeq \\lambda $ , one finds the scaling $\\sim s^{-3 z - 2 d + 8 - 4 \\eta }$ .", "The same scaling properties are also obtained for the remaining measures for the `off-shell' $1\\leftrightarrow 3$ and $0\\leftrightarrow 4$ processes, which may be neglected for the following discussion.", "Putting everything together, in the infrared where strong fluctuations dominate the dynamics, we obtain $ \\Delta > 0 : \\quad C^{\\textrm {NLO}} (\\mathit {p}) = s^{\\kappa + \\eta - z} C^{\\textrm {NLO}}(s \\mathit {p}) ~.", "$ In contrast, in the case where the scaling of the coupling is trivial, i.e.", "$\\lambda _{\\text{eff}}(p) \\simeq \\lambda $ , relevant at high momenta we have the scaling behavior $ \\Delta \\le 0 : \\quad C^{\\textrm {NLO}} (\\mathit {p}) = s^{3 \\kappa - \\eta - 3 z - 2 d + 8} C^{\\textrm {NLO}} (s \\mathit {p}) ~.", "$ First, we consider the implications of the scaling behavior in the high momentum region where $\\Delta \\le 0$ such that $\\lambda _{\\text{eff}}(p) \\simeq \\lambda $ .", "In this case we recover the perturbative discussion of section and the momentum integral $ \\int _{0}^{k} d|\\mathit {p}| |\\mathit {p}|^{d-1} C^{\\textrm {NLO}} (\\mathit {p}) ~, $ describes the particle flux through a sphere of radius $k$ .", "Performing the scaling transformation, we obtain $ \\Delta \\le 0 : \\quad \\kappa = d + z + \\frac{\\eta - 8}{3}~, $ for the scaling exponent $\\kappa $ .", "Setting the anomalous dimension $\\eta = 0$ and the dynamical critical exponent $z = 1$ one may immediately verify that this reproduces the same weak wave turbulence exponent $\\kappa = \\frac{4}{3}$ for the particle cascade in $d = 3$ spatial dimensions as in section .", "In contrast, in the infrared scaling region with positive values of $\\Delta $ , we obtain the exponent $ \\Delta > 0: \\quad \\kappa = d + z - \\eta ~, $ for the particle flux.", "This scaling behavior in the nonlinear, low-momentum regime is also known as strong turbulence, which is displayed schematically in Fig.", "REF in section  .", "Of course, we can also extract the scaling behavior associated with the energy cascade.", "The scaling solutions for the relativistic scalar $N$ -component theory, where $z = 1$ and $\\eta = 0$ are summarized in the table.", "The predicted values for the exponents have been tested using classical-statistical simulations on the lattice in various dimensions [5], [44], [6].", "Similar studies have also been performed for nonrelativistic scalar theories [12], [45], [13], [62], or in the context of non-abelian gauge theories [7], [8], [10]." ], [ "Summary and outlook", "In these lectures we have discussed different aspects of the nonequilibrium functional renormalization group.", "As an example, we considered the physics of wave turbulence in relativistic scalar $N$ -component field theory.", "We have seen that the standard treatment based on Boltzmann equations can only describe the perturbative regime of the associated stationary transport of conserved charges.", "In contrast, the functional renormalization group can efficiently describe the perturbative regime as well as the nonperturbative physics at low momenta.", "We have approximately solved the renormalization group equations using a (2PI) resummed large-$N$ expansion to NLO.", "This approximation is particularly suitable, since it provides an accurate description of the physics at not too small $N$ , while it is still analytically treatable.", "In particular, the scale derivatives are total derivatives in this approximation, which can be trivially integrated.", "This allowed us to efficiently discuss the close relations and differences of the renormalization group approach with kinetic theory and 2PI effective action techniques in nonequilibrium physics.", "The renormalization group provides powerful nonperturbative approximation schemes beyond a resummed large-$N$ expansion, such as derivative expansions or expansions in powers of fields with momentum-dependent vertices which we have not considered in these lectures.", "While this has been explored to a large extent mainly in Euclidean spacetime for physics in thermal equilibrium [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], much less is known in the context of far-from-equilibrium problems in quantum field theory.", "Here the functional renormalization group may give important insights also for the nonequilibrium dynamics of, in particular, non-abelian gauge theories, where suitable large-$N$ techniques are difficult to implement." ], [ "Acknowledgments", "We thank T. Gasenzer, G. Hoffmeister, A. Rothkopf, C. Scheppach, J. Schmidt, D. Sexty and J. Stockemer for collaborations on related work." ] ]

[ [ "Multiband effects on beta-FeSe single crystals" ], [ "Abstract We present the upper critical fields Hc2(T) and Hall effect in beta-FeSe single crystals.", "The Hc2(T) increases as the temperature is lowered for field applied parallel and perpendicular to (101), the natural growth facet of the crystal.", "The Hc2(T) for both field directions and the anisotropy at low temperature increase under pressure.", "Hole carriers are dominant at high magnetic fields.", "However, the contribution of electron-type carriers is significant at low fields and low temperature.", "Our results show that multiband effects dominate Hc2(T) and electronic transport in the normal state." ], [ "Introduction", "The discovery of iron-based superconductors has generated a great deal of interests because of rather high transition temperature $T_{c}$ and high upper critical fields $\\mu _{0}H_{c2}$ .", "Crystal structures of iron-based superconductors can be mainly categorized into several types: FePn-1111 type (REOFePn, RE = rare earth; Pn = P or As), FePn-122 type (AFe$_{2}$ As$_{2}$ , A = alkaline or alkaline-earth metals), FePn111 type (AFeAs), FeCh-11 type (FeCh, Ch = S, Se, Te), FeCh-122 type (A$_{x}$ Fe$_{2-y}$ Ch$_{2}$ ), and other structures with more complex oxide layers.", "[1]$^{-}$[6] Despite structural similarity, i.e., shared FePn or FeCh tetrahedron layers, iron-based superconductors exhibit diverse physical properties.", "These include possible differences in pairing symmetry,[7]$^{-}$[10] relation to competing or coexisting order states (spin density wave vs. superconductivity),[11]$^{-}$[13] and diverse normal state properties.", "[1]$^{,}$[6] FeCh-11 type materials are of special interest because their crystal structure has no blocking layers in between FeCh layers, yet they have similar calculated Fermi surface topology when compared to other iron-based superconductors.", "[14] Furthermore, they also exhibit some exotic features, such as significant pressure effect,[15]$^{,}$[16] and excess Fe with local moment according to theoretical calculation.", "[17] The $\\mu _{0}H_{c2}$ gives some important information on fundamental superconducting properties: coherence length, anisotropy, details of underlying electronic structures and dimensionality of superconductivity as well as insights into the pair-breaking mechanism.", "Previous studies on FeTe$_{1-x}$ Se$_{x}$ and FeTe$_{1-x}$ S$_{x}$ single crystals indicate that the spin-paramagnetic effect is the main pair-breaking mechanism.", "[18]$^{-}$[20] However, for FePn-1111 and FePn-122 type superconductors the two-band effect with high diffusivity ratio between different bands dominates $\\mu _{0}H_{c2}(T)$ .", "[21]$^{-}$[23] On the other hand, magnetic penetration depth study of $\\beta $ -FeSe polycrystal indicates that $\\beta $ -FeSe is a two-band superconductor.", "[24] Therefore, it is of interest to investigate multiband and spin paramagnetic effects on the $\\mu _{0}H_{c2}$ of $\\beta $ -FeSe.", "An extremely complex binary alloy phase diagram and associated difficulties in single crystal preparation impeded the growth of pure $\\beta $ -FeSe single crystals.", "[25] Hence, systematic studies of anisotropy in $\\mu _{0}H_{c2}(T)$ and pair breaking mechanism in high magnetic field are still lacking.", "In this work, we report on the upper critical fields of pure $\\beta $ -FeSe single crystals in dc high magnetic fields up to 35 T at ambient and high pressures.", "The results shows that two-band features dominate the pair breaking with additional influence of spin paramagnetic effect." ], [ "Experiment", "Details of crystal synthesis and characterization are explained elsewhere.", "[25] The $\\mu _{0}H_{c2}$ is determined by measuring the magnetic field dependence of radio frequency (rf) contactless penetration depth in a static magnet up to 35 T at the National High Magnetic Field Laboratory (NHMFL) in Tallahassee, Florida.", "The rf technique is very sensitive to small changes in the rf penetration depth (about 1-5 nm) in the mixed state and thus is an accurate method for determining the $\\mu _{0}H_{c2}$ of superconductors.", "[26] At certain magnetic field, the probe detects the transition to the normal state by tracking the shift in resonant frequency, which is proportional to the change in penetration depth as $\\Delta \\lambda $ $$ $\\Delta F$ .", "Small single crystals were chosen and the sample was placed in a circular detection coil.", "More details can be found in Refs.", "29 and 30.", "For measurement under pressure, the sample was placed in a 15 turn coil within the gasket hole of a turnbuckle diamond anvil cell (DAC) made of beryllium copper and containing diamonds with 1.2 mm culets.", "[27] The pressure was calibrated at $\\sim $ 4 K by comparing the fluorescence of a small chip of ruby within the DAC with an ambient ruby at the same temperature.", "[28] The small dimensions of the DAC allow for angular rotation with respect to the applied magnetic field so H$\\parallel $ (101) and H$\\perp $ (101) orientations can be explored in situ.", "Using four-probe configuration of Hall measurement, the Hall resistivity was extracted from the difference of transverse resistance measured at the positive and negative fields, i.e., $\\rho _{xy}(H)=[\\rho (+H)-\\rho (-H)]/2$ , which can effectively eliminate the longitudinal resistivity component due to voltage probe misalignment." ], [ "Results and Discussion", "As shown in the main panel of Fig.", "1(a) and (b), the rf shift ($\\Delta F$ ) at 10 K (above $T_{c}$ ) shows a smooth and almost linear magnetic-field dependence without any steep changes for both field directions.", "In the normal state the rf shift is sensitive to the magnetoresistance of the sample and detection coil.", "[31] However, when the temperature is below $T_{c}$ , there is a sudden increase of $\\Delta F(H)$ which deviates from the background signal.", "This corresponds to entry to the mixed state.", "Moreover, with decreasing temperature, the inflexion points of the $\\Delta F(H)$ curves shift to higher field for both field directions, consistent with the higher $\\mu _{0}H_{c2}(T)$ at lower temperature.", "The temperature dependence of $\\mu _{0}H_{c2}(T)$ for H$\\parallel $ (101) and H$\\perp $ (101) is determined from the intersections of $\\Delta F(H)$ curves between the extrapolated slopes of the rf signals below inflexion points and the normal-state backgrounds ($T$ = 10 K) (insets in Fig.", "1(a) and (b)).", "[31] The difference between this and other criterion, e.g.", "the intersection of extrapolated slopes below and above inflexion points in each $\\Delta F(H)$ curve, is taken as the error bar of $\\mu _{0}H_{c2}(T)$ .", "In order to compare the upper critical fields determined from different measurement methods, the $\\mu _{0}H_{c2}(T)$ obtained from $\\Delta F(T,H)$ curves and $\\rho (T,H)$ data with different criteria are plotted together (Fig.", "2(a)).", "[32] In the low field region, the temperature dependence of $\\mu _{0}H_{c2}(T)$ determined from the rf shift is almost linear with slight upturn near $T_{c}$ ($H$ = 0 T).", "This is close to the $\\mu _{0}H_{c2,zero}(T)$ determined from 10% $\\rho _{n}(T,H)$ .", "It is consistent with the results reported in the literature.", "[33] Assuming $\\mu _{0}H_{c2}(T$ = 0.35 K$)$ $\\approx $ $\\mu _{0}H_{c2}(0)$ , the zero temperature limit of upper critical fields are 17.4(2) and 19.7(4) T for H$\\parallel $ (101) and H$\\perp $ (101), respectively.", "On the other hand, according to the Werthamer-Helfand-Hohenberg (WHH) theory, orbital pair breaking field $\\mu _{0}H_{c2}(0)$ = -0.693$T_{c}$ ($d\\mu _{0}H_{c2}/dT|_{T_{c}}$ ),[34] and using the initial slopes $d\\mu _{0}H_{c2}/dT|_{T_{c}}$ at low fields obtained from $\\rho (T,H)$ data (-2.54(4) T/K for H$\\parallel $ (101) and -2.55(4) T/K for H$\\perp $ (101)) with $T_{c}$ = 8.7 K,[32] we obtain the $\\mu _{0}H_{c2}(0)$ are 15.3(2) and 15.4(2) T for H$\\parallel $ (101) and H$\\perp $ (101), respectively.", "This is smaller than experimental results.", "The deviation from WHH model is clearly seen in Fig.", "2(b), where the $\\mu _{0}H_{c2}(T)$ becomes gradually larger than expected values from theory.", "The enhancement of the $\\mu _{0}H_{c2}$ in the low temperature and high field region implies that multiband effect are not negligible.", "On the other hand, assuming the electron-phonon coupling parameter $\\lambda _{e-ph}$ = 0.5 (typical value for weak-coupling BCS superconductors),[35] the Pauli limiting field $\\mu _{0}H_{p}(0)$ = 1.86$T_{c}(1+\\lambda _{e-ph})^{1/2}$ is 19.8 T.[36] This is nearly the same as the $\\mu _{0}H_{c2,H\\perp (101)}$ ($T=$ 0.35 K) and larger than values for H$\\parallel $ (101) or the orbital pair breaking fields.", "It suggests that the spin-paramagnetic effect might also have some influence on the upper critical fields.", "This is rather different from other FeCh-11 superconductors where the Pauli limiting fields are much smaller than orbital pair-breaking fields and therefore the spin-paramagnetic effect governs $\\mu _{0}H_{c2}(T)$ .", "[19]$^{,}$[20] Figure: (a) Temperature dependence of μ 0 H c2 (T)\\protect \\mu _{0}H_{c2}(T) of β\\protect \\beta -FeSe single crystal for H∥\\parallel (101) (closed symbols)and H⊥\\perp (101) (open symbols) obtained from ρ(T)\\protect \\rho (T) and ΔF\\Delta F curves.", "(b) Fits of μ 0 H c2 (T)\\protect \\mu _{0}H_{c2}(T) for H∥\\parallel (101) using eq.", "(1) for different pairing scenarios: (1) WHH; (2) ϖ\\protect \\varpi >> 0, λ 11 \\protect \\lambda _{11} = 0.241, λ 22 \\protect \\lambda _{22} =0.195, λ 12 \\protect \\lambda _{12} = λ 21 \\protect \\lambda _{21} = 0.01, η\\protect \\eta = D 2 _{2}/D 1 _{1} = 0.40; (3) ϖ\\protect \\varpi >> 0, λ 11 \\protect \\lambda _{11} = λ 22 \\protect \\lambda _{22} = 0.5, λ 12 \\protect \\lambda _{12} = λ 21 \\protect \\lambda _{21} = 0.25, η\\protect \\eta = D 2 _{2}/D 1 _{1} = 0.44; (4)ϖ\\protect \\varpi >> 0, λ 11 \\protect \\lambda _{11} = 0.8 λ 22 \\protect \\lambda _{22} = 0.34, λ 12 \\protect \\lambda _{12} = λ 21 \\protect \\lambda _{21} = 0.18, η\\protect \\eta = D 2 _{2}/D 1 _{1} = 0.32; and (5) ϖ\\protect \\varpi << 0, λ 11 \\protect \\lambda _{11} = λ 22 \\protect \\lambda _{22} = 0.49, λ 12 \\protect \\lambda _{12} = λ 21 \\protect \\lambda _{21} = 0.5, η\\protect \\eta = D 2 _{2}/D 1 _{1} =0.35;According to the two-band BCS model in the dirty limit with orbital pair breaking and negligible interband scattering,[37] $\\mu _{0}H_{c2}$ is given by $a_{0}[\\text{ln}t+U(h)][\\text{ln}t+U(\\eta h)]+a_{2}[\\text{ln}t+U(\\eta h)] \\\\+a_{1}[\\text{ln}t+U(h)]=0$ where $t=T/T_{c}$ , $U(x)=\\psi (1/2+x)-\\psi (x)$ , $\\psi (x)$ the digamma function, $\\eta =D_{2}/D_{1}$ , $D_{1}$ and $D_{2}$ are intraband diffusivities of the bands 1 and 2, $h=H_{c2}D_{1}/(2\\phi _{0}T)$ , $\\phi _{0}$ the magnetic flux quantum.", "$a_{0}$ , $a_{1}$ , and $a_{2}$ are constants described with intraband- and interband coupling strength, $a_{0}=2\\varpi /\\lambda _{0}$ , $a_{1}=1+\\lambda _{-}/\\lambda _{0}$ , and $a_{1}=1-\\lambda _{-}/\\lambda _{0}$ , where $\\varpi =\\lambda _{11}\\lambda _{22}-\\lambda _{12}\\lambda _{21}$ , $\\lambda _{0}=(\\lambda _{-}^{2}+4\\lambda _{12}\\lambda _{21})^{1/2}$ , and $\\lambda _{-}=\\lambda _{11}-\\lambda _{22}$ .", "Terms $\\lambda _{11}$ and $\\lambda _{22}$ are the intraband couplings in the bands 1 and 2 and $\\lambda _{12}$ and $\\lambda _{21}$ describe the interband couplings between bands 1 and 2.", "It should be noted that if $\\eta $ = 1, eq.", "(1) will reduce to the simplified WHH equation for single-band dirty superconductors.", "[34] By using the coupling constants determined from an $\\mu SR$ experiment with very small interband coupling,[24] the combined $\\mu _{0}H_{c2,H\\parallel (101)}(T)$ data from both rf and resistivity measurements can be very well explained (Fig.", "2(a) fit lines).", "The ratio of band diffusivities is $\\eta $ = 0.40, which is similar to the value of FeAs-122 but much larger than that of other two-band iron-based superconductors, such as FeAs-1111.", "[21]$^{-}$[23] With current coupling constants, it leads to the similar shape of $\\mu _{0}H_{c2,H\\parallel (101)}(T)$ when compared to the FeAs-122,[23] but significantly different from FeAs-1111 where there is an obvious upturn at low temperature.", "[21]$^{,}$[22] We have also performed fits for different values of coupling constants:[21]$^{-}$[23] (1) dominant intraband coupling, $\\varpi >0$ and (2) dominant interband coupling, $\\varpi <0$ .", "The different sets of fitting parameters result in almost identical result, fitting the experimental data well (Fig.", "2(b)).", "The derived $\\eta $ is in the range of 0.32-0.44, suggesting that the fitting results are insensitive to the choice of coupling constants.", "Thus, either interband and intraband coupling strength are comparable or their difference is below the resolution of our experiment.", "Figure: (a) Field dependence of ρ xy (H)\\protect \\rho _{xy}(H) at varioustemperatures.", "Solid lines are the fitting results using eq.", "(2).", "for T<T< 60K and single-band model for T=T= 60 K. In order to exhibit data clearly, theρ xy (H)\\protect \\rho _{xy}(H) at different temperatures are shifted along verticalaxis with certain values.", "(b) Temperature dependence of carrier density n(=n h -n e )n(=n_{h}-n_{e}) of β\\protect \\beta -FeSe crystal.In order to further investigate multiband characteristics in $\\beta $ -FeSe, we studied the Hall effect of $\\beta $ -FeSe (Fig.", "3).", "According to the band calculations, at least four bands originated from Fe 3d orbitals cross the Fermi level.", "[14] Two bands are hole type and the other two are electron type.", "[14] We use a simplified two-carrier model including one electron type with electron density $n_{e}$ and mobility $\\mu _{e}$ and one hole type with hole density $n_{h}$ and mobility $\\mu _{h}$ .", "According to the classical expression for the Hall coefficient including both electron and hole type carriers,[38] $\\rho _{xy}/\\mu _{0}H= \\\\R_{H}=\\frac{1}{e}\\frac{(\\mu _{h}^{2}n_{h}-\\mu _{e}^{2}n_{e})+(\\mu _{h}\\mu _{e})^{2}(\\mu _{0}H)^{2}(n_{h}-n_{e})}{(\\mu _{e}n_{h}+\\mu _{h}n_{e})^{2}+(\\mu _{h}\\mu _{e})^{2}(\\mu _{0}H)^{2}(n_{h}-n_{e})^{2}}$ Once there are two carrier types present, the field dependence of $\\rho _{xy}(H)$ will become nonlinear.", "Moreover, eq.", "(2) gives $R_{H}=e^{-1}\\cdot (\\mu _{h}^{2}n_{h}-\\mu _{e}^{2}n_{e})/(\\mu _{e}n_{h}+\\mu _{h}n_{e})^{2}$ when $\\mu _{0}H\\rightarrow $ 0, and $R_{H}=e^{-1}\\cdot 1/(n_{h}-n_{e})$ when $\\mu _{0}H\\rightarrow $ $\\infty $ .", "As shown in inset (a) of Fig.", "3, $\\rho _{xy}(H)$ is positive and almost linear in $\\mu _{0}H$ at T = 60 K, indicating the hole type carrier is dominant.", "However, $\\rho _{xy}(H)$ exhibits obvious nonlinear behavior below 50 K and even changes sign in low fields at 15 K (inset (b) in Fig.", "3).", "This is a signature of coexistence of electron and hole type carriers.", "The $\\rho _{xy}(H)$ can be described very well using a linear relation for $T$ $=$ 60 K and eq.", "(2) for $T$ $\\leqslant $ 50 K as shown with the solid fit lines in the inset (a) and (b) of Fig.", "3.", "The obtained carrier density $n(=n_{h}-n_{e})$ changes from 1.93$\\times $ 10$^{21}$ cm$^{-3}$ (15 K) to 4.7$\\times $ 10$^{21}$ cm$^{-3}$ (60 K) gradually.", "The change of sign of $\\rho _{xy}(H)$ in the low field region at 15 K indicates $(\\mu _{h}^{2}n_{h}-\\mu _{e}^{2}n_{e})$ $<$ 0.", "Because $n_{h}-n_{e}>0$ at higher field, it indicates that the $\\mu _{e}>\\mu _{h}$ at low temperature, consistent with the band structure calculation results.", "[14] Moreover, the negative Seebeck coefficients in $\\beta $ -FeSe below $\\sim $ 250 K also confirm that the electron band is dominant at low temperature.", "[41] Figure: (a) Temperature dependence of μ 0 H c2 (T)\\protect \\mu _{0}H_{c2}(T) for H∥\\parallel (101) (closed symbols) and H⊥\\perp (101) (open symbols) atambient pressure and PP = 0.51 GPa obtained from ΔF\\Delta F curves.", "Inset(a) field dependence of ΔF\\Delta F at 0 and 0.51 GPa for H∥\\parallel (101)at T = 0.35 K. Inset (b) The temperature dependence of the anisotropy of μ 0 H c2 (P,T)\\protect \\mu _{0}H_{c2}(P,T) at PP = 0 and 0.51 Gpa.Since there is remarkable pressure effect on $T_{c}$ for $\\beta $ -FeSe,[15]$^{,}$[16] it is instructive to study the pressure dependence of upper critical fields.", "As shown in the inset of Fig.", "4, under pressure ($P$ = 0.51 GPa), the inflexion point of $\\Delta F(H)$ curve shifts to higher field when compared to the ambient pressure curve, suggesting that the $\\mu _{0}H_{c2}(T)$ is enhanced with pressure.", "It is consistent with the significant positive pressure effect of $T_{c}$ for $\\beta $ -FeSe.", "[15]$^{,}$[16] The temperature dependence of $\\mu _{0}H_{c2}(T)$ for H$\\parallel $ (101) and H$\\perp $ (101) shows that the upper critical fields for both field directions are enhanced in the whole measured temperature region under pressure.", "The $\\mu _{0}H_{c2}(T=$ 0.45 K$)$ for H$\\perp $ (101) is about 24.6 T, close to the estimated value at 1.48 GPa using linear extrapolation.", "[15] It suggests that $\\mu _{0}H_{c2}(0)$ at 0.51 GPa should be larger than linear-extrapolated value.", "This could originate from the difference in sample purity between our single crystals and polycrystals or intrinsic multiband effect.", "As shown in the inset (b) of Fig.", "4, at ambient pressure, the anisotropy of $\\mu _{0}H_{c2}(T)$ , $\\gamma $ ($P$ = 0 GPa$,T$ ) = $\\mu _{0}H_{c2,H\\perp (101)}(P$ = 0 GPa$,T)/\\mu _{0}H_{c2,H\\parallel (101)}$ ($P$ = 0 GPa$,T$ ), is smaller than in other iron based superconductors, especially at high temperature.", "Moreover, the temperature dependence of $\\gamma $ ($P,T$ ) increases at high temperature and decreases when $T\\ll T_{c}$ , which is different from other iron based superconductors in which the $\\gamma $ ($T$ ) usually decreases with temperature.", "The increase of $\\gamma $ ($P,T$ ) with temperature has also been observed in two-band superconductor MgB$_{2}$ .", "This may be due to the higher contribution of the band with lower band anisotropy at low temperature.", "The decrease of $\\gamma $ ($P,T$ ) with temperature when temperature is far from Tc may be related to the possible spin-paramagnetic effect.", "On the other hand, under pressure, the $\\gamma $ ($P$ = 0.51 GPa$,T$ ) increases when compared to the value at ambient pressure.", "This could originate from the pressure-induced Fermi surface changes that increase the anisotropy of Fermi velocity (diffusivity) of dominant band.", "It should be noted that in order to study the anisotropy and pressure evolution of $\\mu _{0}H_{c2}(T)$ more clearly, the pressure dependence of $\\mu _{0}H_{c2}(T)$ along crystallographic axes should be measured in the future." ], [ "Conclusion", "In summary, we studied the upper critical field of $\\beta $ -FeSe crystals.", "The results indicate that the two band effects dominate the $\\mu _{0}H_{c2}(T)$ , with possible influence of spin-paramagnetic effect.", "A nonlinear field dependence of $\\rho _{xy}(H)$ at low temperature also confirms the existence of multiple bands in electronic transport.", "The dominant carriers are hole-type in high field but electron type carriers become important in low field due to either increased carrier density or enhanced mobility.", "The $\\mu _{0}H_{c2}(T)$ is enhanced for both field directions and the anisotropy of $\\mu _{0}H_{c2}(0)$ is also increased under pressure." ], [ "Acknowledgements", "Work at Brookhaven is supported by the U.S. DOE under Contract No.", "DE-AC02-98CH10886 (R. Hu and H. Ryu) and in part by the Center for Emergent Superconductivity, an Energy Frontier Research Center funded by the U.S. DOE, Office for Basic Energy Science (H. Lei and C. Petrovic).", "Work at the National High Magnetic Field Laboratory is supported by the DOE NNSA DEFG52-10NA29659 (S. W.T.", "and D. G.), by the NSF Cooperative Agreement No.", "DMR-0654118 and by the state of Florida.", "$^{\\ast }$ Present address: Center for Nanophysics & Advanced Materials and Department of Physics, University of Maryland, College Park MD 20742-4111, USA." ] ]

[ [ "Brane Geometry and Dimer Models" ], [ "Abstract The field content and interactions of almost all known gauge theories in AdS_5/CFT_4 can be expressed in terms of dimer models or bipartite graphs drawn on a torus.", "Associated with the fundamental cell is a complex structure parameter tau_R.", "Based on the brane realization of these theories, we can specify a special Lagrangian (SLag) torus fibration that is the natural candidate to be identified as the torus on which the dimer lives.", "Using the metrics known in the literature, we compute the complex structure tau_G of this torus.", "For the theories on C^3 and the conifold and for orbifolds thereof tau_R = tau_G.", "However, for more complicated examples, we show that the two complex structures cannot be equal and yet, remarkably, differ only by a few percent.", "We leave the explanation for this extraordinary proximity as an open challenge." ], [ "Introduction", "The story of D3-branes probing conical Calabi–Yau threefolds (CY$_3$ ) giving rise to examples of AdS$_5$ /CFT$_4$ duality is a theme of central importance to modern physics during the last two decades.", "The supersymmetric gauge theory lives on the worldvolume of the D3-brane while the gravitational description is obtained by considering the background sourced by $N$ D3-branes placed at the tip of the CY$_3$ .", "The Calabi–Yau manifolds under consideration are cones over a five-dimensional Sasaki–Einstein base ${\\cal B}$ , and the tip of the cone is what the D3-branes probe.", "The near-brane region of the geometry is AdS$_5\\times {\\cal B}$ .", "This implies that the gauge theory on the worldvolume of the branes is a four-dimensional superconformal field theory generically with $\\mathcal {N}=1$ supersymmetry.", "To date, almost all known explicit pairs of AdS/CFT belong to a particular subclass of non-compact Calabi–Yau manifolds, the so-called toric manifolds, of which infinite families have been constructed.", "The toric description facilitates the algebraic geometry, the differential geometry, as well as the physics: the geometry is encoded entirely into the combinatorics of certain lattice polytopes and classes of explicit metrics have been constructed; so too can the worldvolume physics be succinctly described in terms of a two-dimensional linear sigma model.", "We will thus focus on toric Calabi–Yau threefolds.", "Even though various techniques for constructing the dual worldvolume field theory given a toric diagram have been developed using the tool of D-brane partial resolution of singularities since the early days [1], [2], [3], it was not until [4], [5], [6] that it was realized that the most powerful way of understanding AdS$_5$ /CFT$_4$ for toric Calabi–Yau threefolds, is through dimer models, or, equivalently, brane tilings.", "Using dimer models, the gauge theory of interest can be neatly encoded in a bipartite graph drawn on a torus.", "This graph expresses the complete information about both the field content and the interactions of the theory: the gauge groups are represented by polygonal faces in the graph, the fields by edges, and the superpotential terms by vertices, which are colored either black and white.", "Furthermore, it is possible to encode dynamical data such as the scaling dimensions of the fields at the conformal fixed point in the infrared in terms of the angles between the edges on the bipartite graph, which, in turn, fixes a particular shape for the fundamental cell on the torus.", "Quite surprisingly, this shape for the torus — encoded by the complex structure of the unit cell — is, upon the obvious action of $SL(2,\\,\\mathbb {Z})$ , an invariant: all the toric (Seiberg dual) phases of the theory have the same complex structure [7].", "This triggers the suspicion that the complex structure of the bipartite graph might be read off directly from the geometry of the Calabi–Yau.", "In order to test this hypothesis, one should identify in the CY$_3$ geometry the dimer itself.", "This is an open question, in the end related to the underlying reason for the coding of scaling dimensions in terms of angles.", "In this letter, we take a first step toward realizing this goal.", "Indeed, this small but crucial step will constitute the first investigation of dimer models and tilings from the differential geometry of the bulk Calabi–Yau.", "In short, inspired by the proposal [6] of identifying the torus in which the dimer resides as part of the $\\mathbb {T}^3$ in the SYZ prescription of mirror symmetry [8], we will attempt to find this torus explicitly.", "Indeed, mirror symmetry [8] can be thought of as $T$ -duality once we exhibit the CY$_3$ as a supersymmetric torus fibration.", "This suggests that we should regard the CY$_3$ as a special Lagrangian (SLag) torus fibration, identifying the $U(1)^2$ with the relevant $\\mathbb {T}^2$ on which the dimer lives.", "This suggests a way to metrically identify the torus, in particular allowing us to compute its complex structure parameter.", "On the other hand, as discussed above, field theory arguments suggest this complex structure will take a certain value for each ${\\cal N}=1$ SCFT, namely the one determined by the R-charges in a so-called isoradial embedding of the dimer.", "It is thus natural to guess that the complex structure of the $\\mathbb {T}^2$ identified in the geometry, which we denote as $\\tau _G$ , will match the complex structure computed in field theory, which we denote as $\\tau _R$ .", "This comparison will be the heart of our present investigation.", "Indeed, we find the match $\\tau _G = \\tau _R$ to be true for the spaces ${\\mathbb {C}}^3$ and the conifold, as well as for symmetric orbifolds of these spaces.", "Quite interestingly, this naïvely expected equality of complex structures is not quite realized in general SCFTs as the geometrical torus is slightly different from the field theoretic torus.", "This “not quite” is in fact remarkably fascinating.", "We find that the Klein $j$ -invariants of the two tori are tantalizingly close numerically.", "Given the extremely complicated nature of the $j$ -invariant, this numerical proximity is highly non-trivial.", "The two $\\tau $ -parameters are bound not to agree due to the fact that the field theory $\\tau _R$ , can be argued to be, generically, a transcendental number using the four exponentials conjecture while the geometrical $\\tau _G$ , computed from the explicit metrics known in the literature, can be shown to be an algebraic number.", "This leaves a very interesting open question as for the origin of this small mismatch.", "In the remainder of the paper we will further explain the above ideas, in particular showing explicitly the SLags in the geometry, which is per se an interesting mathematical problem, and demonstrate how they fail by a tiny amount to reproduce the field theory result.", "This paper is organized as follows.", "We begin with Section , in which we briefly review dimer models, their description in terms of branes and SLags.", "In Section , we then construct the SLag for ${\\mathbb {C}}^3$ corresponding to the ${\\cal N}=4$ super-Yang–Mills theory and find that $\\tau _R = \\tau _G$ .", "We next do the same for the Klebanov–Witten theory on the conifold in Section , and find agreement once again.", "Section  briefly comments on the concordance of $\\tau _R$ and $\\tau _G$ for orbifolds of ${\\mathbb {C}}^3$ and the conifold.", "Section  examines the more general case of $L^{a,b,c}$ and $Y^{p,q}$ spaces.", "We again verify that for the spaces $Y^{p,0}$ and $Y^{p,p}$ , the agreement between $\\tau _R$ and $\\tau _G$ continues to hold.", "This must be the case as these are the orbifolds that we discussed in the previous section.", "However, for $L^{a,b,a}$ spaces there is a small mismatch between the two $\\tau $ parameters.", "Section  speculates on the origin of the mismatch and presents avenues for further investigation." ], [ "Toric CFTs, five-branes, and SLags", "As described in Section , the theories we are interested in are encoded as dimer models.", "In a nutshell, dimer models are bipartite graphs drawn on a torus consisting on black and while nodes linked in a certain way.", "Each face of the graph represents a gauge group $SU(N)$ of the corresponding gauge theory.", "Links join black and white nodes and separate two faces; these correspond to bifundamental fields charged under the gauge groups associated to the faces that they separate, with the definition of fundamental and antifundamental defined by the orientation of the genus one Riemann surface.", "We number the links to label the fields.", "The dimer models reflect the toric character of the theory expressed in a superpotential that can be written as $W = W_+ - W_-$ , where each field appears exactly once in each of $W_+$ and $W_-$ .", "In the dimer, vertices correspond to the superpotential terms.", "Going around the black nodes clockwise, the fields label a monomial whose trace appears in $W_+$ .", "Going around the white nodes counterclockwise, the fields label a monomial whose trace appears in $W_-$ .", "In this way, we capture the field content of the ${\\cal N}=1$ SCFT and the terms in the superpotential.", "The canonical example of ${\\cal N}=4$ super-yang-Mills theory is, in fact, representable as a dimer model.", "In Figure REF , we illustrate the above rules diagrammatically with this important case.", "The “clover quiver” with the three adjoints $\\phi ^{1,2,3}_1$ is shown at the far left (the subscript 1 is to emphasize that all these three fields are charged under the group corresponding to face “1” in the dimer model, shown in the middle.", "To the right, we include the planar toric diagram for $\\mathbb {C}^3$ for completeness.", "Below the diagrams we show the famous trivalent superpotential.", "To this theory and many more we shall shortly return.", "Figure: The quiver, dimer and toric diagram for 𝒩=4{\\cal N}=4 super-Yang-Mills theory in 4-dimensions, corresponding to the toric Calabi–Yau threefold ℂ 3 \\mathbb {C}^3." ], [ "Physical origin of the dimer", "The existence of an underlying torus may at first seem mysterious and was initially forced upon us by the curious fact that for all toric quiver gauge theories we have $N_W - N_E + N_G = 0 ~.$ Here $N_W$ is the number of monomials in the superpotential, $N_E$ is the total number of fields, and $N_G$ is the total number of gauge group factors.", "In the dimer, these are, respectively, the number of nodes, edges, and faces in the fundamental region.", "Of course, we recognize this as the Euler relation for a genus one Riemann surface, whence the torus.", "The physical motivation of the dimer construction, whereby explaining why AdS$_5$ /CFT$_4$ should obey this topological condition was expounded in [6].", "The answer turns out to be mirror symmetry.", "We know that upon mirror symmetry, the original IIB setup of D3-branes at the tip of the Calabi–Yau cone ${\\cal M}$ gets mapped to a system of intersecting D6-branes on the mirror Calabi–Yau ${\\cal W}$ .", "The mirror ${\\cal W}$ is given by the following complete intersection in $\\mathbb {C}[x,y,u,v,z]$ : $u\\,v=z ~, \\qquad P(x,\\,y)=z ~, \\qquad z\\in \\mathbb {C} ~,$ where $P(x,\\,y)$ is the Newton polynomial of the toric diagram of the ${\\cal M}$ .", "This polynomial is constructed starting from the toric diagram of the CY$_3$ , which we recall is a collection of lattice two-vectors $\\lbrace (p_i, q_i) \\rbrace \\in \\mathbb {C}^2$ .", "Then the Newton polynomial is given by $P(x,y) = \\sum _{\\lbrace {\\rm points\\ in\\ toric\\ diagram}\\rbrace }\\,a_i\\,x^{p_i}\\,y^{q_i} ~,$ where the $a_i$ are complex numbers parameterizing the complex moduli of ${\\cal W}$ , and hence the Kähler moduli of the original ${\\cal M}$ .", "The mirror ${\\cal W}$ is in fact a double fibration over $\\mathbb {C}$ .", "The equation $P(x,\\,y)=z$ defines, for each point $z$ , a certain Riemann surface $\\Sigma _z$ .", "The other fibration contains an $S^1$ corresponding to $\\lbrace u,\\,v\\rbrace \\rightarrow \\lbrace e^{i\\,\\theta }\\,u,\\,e^{-i\\,\\theta }\\,v\\rbrace $ ; obviously this $U(1)$ collapses at $z=0$ .", "The surface $\\Sigma _z$ develops singularities at some critical points $z_* = z^{cr}_i$ where $\\partial _x P= \\partial _y P = 0$ .", "At these points $z_i^{cr}$ , a one-cycle of $\\Sigma _z$ pinches off.", "Hence, over the segment on the $z$ -plane joining $z=0$ and $z_i^{cr}$ there is a $U(1)^2$ , which is pinching off at the ends in a certain way.", "This is topologically an $S^3$ where the D6-branes are wrapped.", "There will be one $S^3$ for each critical point of $\\Sigma _z$ , which meet at the point $z=0$ .", "This is illustrated in part (a) of Figure REF .", "The dimer itself is then the intersection of these $S^3$ cycles at the origin of the $z$ -plane, as some finite graph $\\Gamma $ ; this is shown in part (b) of the figure.", "Figure: (a) The mirror of the Calabi–Yau threefold as a double fibration over ℂ\\mathbb {C}.", "(b) The S 3 S^3 cycles meet at the origin in the zz-plane on a finite graph, which is the dimer model.From the expression of the Newton polynomial $P(x,y)$ , it is clear that each monomial on $P$ specifies a $U(1)$ as $(x,\\,y)\\rightarrow (e^{-i\\,q_i\\,\\theta }\\,x,\\,e^{i\\,p_i\\,\\theta }\\,y)$ , which can be seen as follows.", "The critical points $(x_cr,y_cr)$ satisfy $dP(x,y)|_{(x_{cr},y_{cr})} = 0 \\quad \\Longrightarrow \\quad \\sum _i a_i\\,p_i\\,y_{cr}\\,x_{cr}^{p_i}\\,y_{cr}^{q_i}=0 ~, \\qquad \\sum _i \\, a_i\\,q_i\\,x_{cr}\\,x_{cr}^{p_i}\\,y_{cr}^{q_i}=0 ~.$ Thus, for each monomial we can find a critical point by setting $-q_i\\,x_{cr}+p_i\\,y_{cr}=0$ , hence finding on the $(x,\\,y)$ plane a complex line with a slope given by $(p_i,\\,q_i)$ .", "Since $x$ and $y$ are complex variables, there is a $\\mathbb {T}^2$ associated to their phases in the natural way $(x,\\,y)=(r_x\\,e^{i\\,\\theta _x},\\,r_y\\,e^{i\\,\\theta _y})\\rightarrow (\\theta _x,\\,\\theta _y)$ .", "Each monomial therefore defines a one-cycle winding on $\\mathbb {T}^2$ as specified by $(p_i,\\,q_i)$ , and this one-cycle serves as base for a cylinder which develops around each critical point of $P$ .", "As discussed above, the critical points in turn correspond to D6-branes.", "This thus serves as a natural way to identify their winding on $\\Sigma _0$ .", "In fact, recalling that the so-called $(p,\\,q)$ -web is the graph dual to the toric diagram, as it was realized as far back as [9], $\\Sigma _0$ is nothing but the thickened $(p,\\,q)$ -web associated to the CY$_3$ .", "In the language of [6], the $(p,q)$ -web is the spine of the amœba projection of $\\Sigma _0$ .", "Once we have identified the D6-branes on $\\Sigma _0$ , it is natural to consider their projection to the $\\mathbb {T}^2$ defined by $(\\theta _x,\\,\\theta _y)$ .", "This defines the so-called alga map, and it indeed shows the dimer in an explicit way [6].", "It was later understood [10] that the construction in [6] can be related to a brane tiling of a $\\mathbb {T}^2$ .", "This can be heuristically understood starting with the original IIB configuration of $N$ D3-branes probing a toric CY$_3$ and performing two $T$ -dualities along two coordinates of the toric fiber.", "As the toric fibers shrink somewhere on the base, these $T$ -dualities produce a certain arrangement of NS5-branes winding around the torus.", "In turn, the D3-branes will map into D5-branes wrapping the torus.", "However, as is well-known, NS5-branes and D5-branes must join into $(p,\\,q)$ five-branes running at certain angles in the web plane so as to preserve supersymmetry.", "Thus, the system of NS5-branes and D5-branes become a tiling of the $\\mathbb {T}^2$ .", "It turns out that the $(p,\\,q)$ five-branes separating diverse regions of the tiling do form the $(p,\\,q)$ -web of the CY$_3$ .", "Indeed, these two pictures above are related by $T$ -duality: by $T$ -dualizing the collapsing $S^1$ in the $\\lbrace u,\\,v\\rbrace $ plane in the construction of [6], the D6-branes get mapped to D5-branes.", "But, as the $S^1$ is collapsing, $T$ -duality produces a configuration of NS5-branes following the $(p,\\,q)$ -web, thus recovering the picture in [10].", "For excellent reviews with further details we refer the reader to [11], [12].", "As described in [12], the gauge theory can be read off from the brane system directly." ], [ "R-charges and $\\tau _R$", "Our discussions above identify the dimer topologically as living on a $\\mathbb {T}^2$ part of the $\\mathbb {T}^3$ fibration in mirror symmetry.", "We can further fix this torus, and this distinguishes the so-called isoradial embedding of the dimer.", "Now, the R-charges of the fields in the theory are determined through the standard $a$ -maximization procedure [13].", "First of all, we demand that the sum of the R-charges of fields that appear in a monomial in the superpotential is two; this is just so that the Lagrangian is well-defined when written as a superspace integral: $\\sum _{\\mbox{around each vertex}} R_i = 2 ~.$ Secondly, associated to each face of the dimer is a gauge group, whose $\\beta $ function is $\\beta _G = \\frac{3N}{2(1-\\frac{g^2 N}{8\\pi ^2})} \\left( 2 - \\sum _{\\mbox{around each face}} (1 - R_i) \\right) ~,$ and which vanishes for conformality.", "The $R_i$ in the expression is an R-charge of a field in fundamental or antifundamental representations of $G$ .As usual, adjoints are regarded as both a fundamental and an antifundamental field and counted twice.", "Subject to these constraints, we maximize the central charge $a = \\frac{3}{32} \\left( 3 \\sum _i (R_i - 1)^3 - \\sum _i (R_i - 1) \\right) ~,$ whereby fixing the values of all the R-charges of all the fields in the gauge theory.", "With these R-charges, we can fix the dimer (and hence the torus).", "First, we draw lines drawn from the center of each face to a vertex to be of unit length.", "This is called an isoradial embedding because now each face is inscribed by a unit circle.", "There is a moduli space of such embeddings.", "Now, the R-charge of a field, which is an edge in the dimer, is geometrically interpreted as the angle at the origin of each unit circle subtended by the equilateral triangle defined by the edge: $\\theta _i = \\pi R_i$ .", "In this way, condition (REF ) is just that as we circumnavigate any of the vertices the angle sum is $2\\pi $ , and thus the dimer is truly planar.", "Similarly, condition (REF ) dictates that lines joining all centers of faces prescribe rhombi and we have a rhombus tiling of the plane.", "Furthermore, the length of an edge is easily seen as $2\\cos \\frac{\\pi }{2} R_i$ .", "With $R_i$ being the R-charge of the associated field as determined by $a$ -maximization, we select a particular dimer from the moduli space of the isoradial dimers [14].", "The complex structure parameter of the dimer drawn in this way is $\\tau _R$ .", "We will hence forth use this particular complex parameter from the field theory.$\\endcsname $A bipartite graph on a torus $\\mathbb {T}^2$ can be encoded in terms of a Belyi pair, consisting of an elliptic curve $\\Sigma _1$ together with a holomorphic map $\\beta : \\Sigma _1 \\rightarrow \\mathbb {P}^1$ branched over three points on the $\\mathbb {P}^1$  [15].", "The elliptic curve that is the source for this map has its own complex structure parameter $\\tau _B$ .", "For $\\mathbb {C}^3$ and the conifold and orbifolds thereof, $\\tau _R = \\tau _B$  [15].", "This is not, however, true generally [16]." ], [ "Special Lagrangian fibrations and $\\tau _G$", "The object of our investigation is to explicitly identify the $\\mathbb {T}^2$ on which the dimer lives.", "As briefly reviewed above, this $\\mathbb {T}^2$ is topologically identified with the complex structure of the Newton polynomial in the mirror, or, equivalently, with the $\\mathbb {T}^2$ where the brane tiling lives.", "In order to make the identification more precise, let us focus on the mirror symmetry transformation of [6].", "As described in [8], mirror symmetry on toric Calabi–Yau threefolds can be understood as fiberwise $T$ -duality on the original CY$_3$ .$\\endcsname $To the best of our knowledge, this is however still a heuristic picture in that its precise mathematical characterization remains to be fully understood.", "Besides, our Calabi–Yau is non-compact, which adds a further subtlety.", "Following this inspiration, in our non-compact setup, we are instructed to regard the CY$_3$ as a special Lagrangian (SLag) fibration.", "Recall that the SLag cycle $L$ in a CY$_3$ is a middle-dimensional manifold (i.e., a three-cycle) satisfying ${\\rm Im}\\,({\\rm P}_L[\\Omega ]\\,)=0 ~, \\qquad {\\rm P}_L[\\omega ]=0 ~,$ where $\\Omega $ and $\\omega $ are, respectively, the holomorphic three-form and the Kähler form of the CY$_3$ and ${\\rm P}_L$ denotes the pull-back onto the SLag cycle.", "It is then natural to regard the CY$_3$ itself as a fibration $f:\\,{\\mathrm {C}Y}_3\\,\\rightarrow \\mathbb {R}^3$ over $\\mathbb {R}^3$ such that each fiber is a SLag cycle.", "Our geometries, being toric, admit the action of an $U(1)^3$ .", "However, only a $U(1)^2$ subgroup will leave invariant $\\Omega $ and $\\omega $ .", "It is then natural to concentrate on SLag fibrations invariant under this $U(1)^2$ .", "Then, we can piece together the pictures of [6] and [10]: $T$ -dualizing this $U(1)^2$ will lead to the brane tiling of [10], while a further $T$ -duality will be analogous to mirror symmetry and take us to the picture in [6].", "This is in fact summarized in Figure 82 of [12].", "This further suggests that the $U(1)^2$ defining the SLag also defines the $\\mathbb {T}^2$ on which the dimer lives, the object of our primary interest.", "Due to the $U(1)^2$ invariance, we can be more precise in defining our SLag fibration, which is defined in terms of the moment maps $\\mu _i$ of the two $U(1)$ actions as $f=(f_0,\\,\\mu _1,\\,\\mu _2)\\in \\mathbb {R}^3$ .", "The $f_0$ is a “generalized moment map” for $\\Omega $ , appropriately chosen so that it ensures that the fibration is SLag; we will define this more precisely for explicit examples later.", "The $U(1)^2$ defining the SLag will generically collapse in a certain way on $\\mathbb {R}^3$ .", "The generic fiber over a collapsing locus is topologically $\\mathbb {R}^+\\times \\mathbb {T}^2$ .", "Seen on $\\mathbb {R}^3$ , these form a set of lines supported on a $\\mathbb {R}^2$ (on the plane $x_1=0$ in $\\mathbb {R}^3$ ) that in fact coincides with the $(p,\\,q)$ -web on the Calabi–Yau.", "The lines forming the web meet at a single point at the origin of $\\mathbb {R}^2$ , where the fiber is in fact metrically a torus over $\\mathbb {T}^2$ .", "At this point, the collapsing torus coincides with the $U(1)^2$ , which leaves the SLag invariant.", "Hence, it is the point we will be most interested in understanding.", "This $\\mathbb {T}^2$ is naturally identified with the elliptic curve that supports the dimer.", "By pulling back the CY$_3$ metric to this $\\mathbb {T}^2$ , we will be able to compute its associated complex structure, which we will call $\\tau _G$ , and which we will use to compare with the many field theory $\\tau _R$ computed in [7].", "In summary, given the toric CY$_3$ , we follow the following algorithm: find the metric where possible; explicitly identify the SLag from the coordinates; find the $U(1)^2$ -invariant part of the SLag from the moment maps; this should be the torus on which the dimer lives; pull back the metric to this torus and compute its complex parameter $\\tau _G$ compare with $\\tau _R$ from the isoradial dimer." ], [ "Example: $\\mathbb {C}^3$", "Let us begin with the simplest example, $\\mathbb {C}^3$ .", "The dimer model is that of ${\\cal N}=4$ super-Yang-Mills, as described in Figure REF .", "In a physics related context, a detailed discussion can be found in [17].", "We now follow the prescription of [18].", "Letting the complex coordinates of $\\mathbb {C}^3$ be $z_{1,2,3}$ , the holomorphic three-form and the Kähler form are simply $\\Omega = dz_1\\wedge dz_2\\wedge dz_3 ~, \\qquad \\omega =\\frac{i}{2}\\,\\sum _{i=1}^3\\,dz_i\\wedge d\\bar{z}_i ~.$ The $U(1)^3$ toric fiber is generated by the rotations $z_i\\rightarrow e^{i\\,\\theta _i}\\,z_i$ .", "However, it is clear that only a two-dimensional subspace, namely a $U(1)^2$ , will leave invariant both $\\Omega $ and $\\omega $ .", "We can choose $\\nonumber U(1)_1\\,:\\,&& (z_1,\\,z_2,\\,z_3)\\,\\rightarrow \\, (e^{i\\,\\theta _1}\\,z_1,\\,z_2,\\,e^{-i\\,\\theta _1}\\,z_3)\\\\U(1)_2\\,:\\,&& (z_1,\\,z_2,\\,z_3)\\,\\rightarrow \\, (z_1,\\,e^{i\\,\\theta _2}\\,z_2,\\,e^{-i\\,\\theta _2}\\,z_3)$ as the $U(1)^2$ action.", "Note that the fixed points of the $U(1)^2$ are $z_1=z_3=0$ , where $U(1)_1$ collapses; $z_2=z_3=0$ , where $U(1)_2$ collapses, and $z_1=z_2=0$ , where $U(1)_1-U(1)_2$ collapses.", "The moment maps associated to $U(1)_{1,2}$ are, respectively, $\\mu _1=|z_1|^2-|z_3|^2 ~, \\qquad \\mu _2=|z_2|^2-|z_3|^2 ~.$ Then, the SLag fibration reads $f=({\\rm Im}\\,(z_1\\,z_2\\,z_3),\\,|z_1|^2-|z_3|^2,\\,|z_2|^2-|z_3|^2) ~,$ where the first entry has been chosen so that every fiber is SLag.", "Indeed, $f$ maps the non-compact CY$_3$ , here just $\\mathbb {C}^3$ , to $\\mathbb {R}^3$ and the fibers are $\\mathbb {R}\\times U(1)^2$ .", "Let us now look at the fixed loci, as mentioned above.", "These are the following: $\\mathcal {S}_1$ : $U(1)_1$ collapses and the fixed loci are $z_1=z_3=0$ .", "Thus, $f=(0,\\,0,\\,x) ~, \\qquad x\\in \\mathbb {R}^+ ~;$ $\\mathcal {S}_2$ : $U(1)_2$ collapses and the fixed loci are $z_2=z_3=0$ .", "Thus, $f=(0,\\,x,\\,0) ~, \\qquad x\\in \\mathbb {R}^+ ~;$ $\\mathcal {S}_3$ : $U(1)_1-U(1)_2$ collapse and the fixed loci are $z_1=z_2=0$ .", "Thus, $f=(0,\\,-x,\\,-x) ~, \\qquad x\\in \\mathbb {R}^+ ~.$ Defining the set $PQ=\\mathcal {S}_1\\,\\cup \\, \\mathcal {S}_2\\,\\cup \\, \\mathcal {S}_3$ , we can readily construct the $(p,\\,q)$ -web for $\\mathbb {C}^3$ .", "This is just given by the three directional vectors in the above, viz., $(0,0,1)$ , $(0,1,0)$ and $(0,-1,-1)$ .", "Indeed, this is just the planar graph dual to the toric diagram given in Figure REF , illustrating our discussion above that the collapsing cycles should give the spine of the amœba.", "Topologically, every SLag fiber is a cone over a $\\mathbb {T}^2$ .", "However, there is a special fiber at the center $z_1=z_2=z_3=0$ of the $(p,\\,q)$ -web which is metrically a cone over a $\\mathbb {T}^2$ .", "This follows because at the origin of the web $f=0$ , a scaling symmetry $f\\rightarrow \\lambda \\,f$ appears.", "As this is the point over which all legs of the web meet, this is the fiber whose $\\mathbb {T}^2$ is the subject of our investigation.", "As explained above, this is where the dimer should reside.", "Let us denote this special fiber as $L_0$ .", "It is fairly easy to see explicitly.", "Writing $z_i=r_i\\,e^{i\\psi _i}$ , the origin of the web is at $\\psi _1+\\psi _2+\\psi _3=0,\\,\\pi ~; \\qquad r_1=r_2=r_3=r ~.$ In order to manifestly exhibit the conelike structure, we now note that the $\\mathbb {C}^3$ metric can be written as $ds^2=\\sum \\limits _{i=1}^3 ( dr_i^2+r_i^2\\,d\\psi _i^2 ) ~.$ Introducing $\\rho =\\sqrt{3}\\,r$ and using (REF ), the pull-back of the metric to $L_0$ is $ds^2=d\\rho ^2+\\frac{1}{3}\\,\\rho ^2\\,\\Big [ d\\psi _1^2+d\\psi _2^2+(d\\psi _1+d\\psi _2)^2\\Big ] ~,$ where the cone structure of $\\mathbb {C}^3$ is now apparent: $\\rho $ is now the lateral side of the cone, and the base is the $\\mathbb {T}^2$ whose metric is $ds_{\\mathbb {T}^2}^2=\\frac{1}{3}\\,\\Big [ d\\psi _1^2+d\\psi _2^2+(d\\psi _1+d\\psi _2)^2\\Big ] ~.$ Our proposal, as discussed in the previous section, is that this $\\mathbb {T}^2$ is where the dimer lives.", "What is its complex structure?", "We can easily determine it and refer the reader to Appendix  on how to find the $\\tau $ -parameter in general.", "Our metric in (REF ) corresponds to the case in the appendix where $A=B=C=\\frac{2}{3}$ and we readily obtain the complex structure $\\tau _G=\\frac{1}{2}\\,\\Big (1+i\\,\\sqrt{3}\\Big )=e^{i\\frac{\\pi }{3}} ~.$ Very nicely, this exactly matches $\\tau _R$ from the isoradial embedding (cf.", "[15])." ], [ "An alternative point of view", "In order to better understand the physics as well as the mathematics, let us do the same computation for $\\mathbb {C}^3$ in a slightly different language, which we will later use to compute more complicated examples.", "The space $\\mathbb {C}^3$ can be thought of as a cone over $S^5$ , which in turn is a $U(1)$ bundle over $\\mathbb {P}^2$ .", "In order to see this, let us introduce $z_0=r\\,\\cos \\phi _1\\,e^{i\\chi } ~, \\qquad z_1=r\\,\\sin \\phi _1\\,\\cos \\frac{\\phi _2}{2}\\,e^{i\\frac{2\\,\\chi +\\psi +\\phi _3}{2}} ~, \\qquad z_2=r\\,\\sin \\phi _1\\,\\sin \\frac{\\phi _2}{2}\\,e^{i\\frac{2\\,\\chi +\\psi -\\phi _3}{2}} ~,$ so that $\\sum \\limits _{i=0}^2 |z_i|^2=r^2$ .", "The homogeneous coordinates on $\\mathbb {P}^2$ are given by $\\hat{z}_1=\\frac{z_1}{z_0}$ and $\\hat{z}_2=\\frac{z_2}{z_0}$ .", "The range of the coordinates here is $\\phi _1\\,\\in \\,[0,\\,\\frac{\\pi }{2}]$ , $\\phi _2\\,\\in \\,[0,\\,\\pi ]$ , and $\\phi _3\\,\\in \\,[0,\\,2\\,\\pi ]$ , while $\\psi \\,\\in \\,[0,\\,4\\,\\pi ]$ and $\\chi \\,\\in \\,[0,\\,2\\,\\pi ]$ .", "By substituting the angular expressions (REF ) into the standard expressions for the $\\mathbb {C}^3$ metric, Kähler form, and holomorphic form it is straightforward to construct the corresponding expressions in angular coordinates.", "Specifically, we have that $ds^2=(d\\chi -A)^2+ds_{\\mathbb {P}^2}^2 ~,$ where $A=-\\frac{1}{2}\\,\\sin ^2\\phi _1\\,\\Big (d\\psi +\\cos \\phi _2\\,d\\phi _3\\Big )$ and $ds_{\\mathbb {P}^2}^2=d\\phi _1^2+\\frac{1}{4}\\,\\sin ^2\\phi _1\\,\\Big [\\,\\cos ^2\\phi _1\\,\\Big (d\\psi +\\cos \\phi _2\\,d\\phi _3\\Big )^2+d\\phi _2^2+\\sin ^2\\phi _2\\,d\\phi _3^2\\,\\Big ] ~.$ Naïvely, the relevant $U(1)^2$ is given in terms of $\\lbrace \\partial _{\\phi _3},\\,\\partial _{\\psi }\\rbrace $ .", "In fact, the center of the SLag is now given by $\\phi _2=\\frac{\\pi }{2}$ , $\\phi _1=\\frac{1}{2}\\,\\arccos \\frac{1}{3}$ and $\\psi =-3\\,\\chi $ .", "Taking the pull-back of the metric we find $ds^2_c=\\frac{1}{6}\\,\\Big (d\\phi _3^2+3\\,d\\chi ^2\\Big ) ~,$ which clearly does not yield the correct $\\tau $ .", "This is due to somewhat subtle global issues that we have not taken into account.", "Let us first start by noticing that for fixed $\\phi _1$ , in the $\\mathbb {P}^2$ base we find, using (REF ), locally an $S^3$ .", "The $\\mathbb {T}^2$ is written in terms of the $\\varphi _i$ angles defined by $\\psi =2\\,\\varphi _1+\\varphi _2 ~, \\qquad \\phi _3=\\varphi _2 ~,$ so that $\\partial _{\\varphi _1}=2\\,\\partial _{\\psi } ~, \\qquad \\partial _{\\varphi _2}=\\partial _{\\psi }+\\partial _{\\phi _3} ~.$ See, e.g., Section 4 of [19].", "Writing the $z_i$ as $z_i=r_i\\,e^{i\\,\\psi _i}$ , the toric $\\mathbb {T}^3$ is nothing but the three $\\psi _i$ coordinates.", "In terms of the $\\varphi _i$ , the transformation reads $\\psi _1=\\chi ~, \\qquad \\psi _2=\\chi +\\varphi _1+\\varphi _2 ~, \\qquad \\psi _3=\\chi +\\varphi _1 ~,$ or, in matrix form $\\left(\\begin{array}{c} \\psi _1 \\\\ \\psi _2 \\\\ \\psi _3 \\end{array}\\right) = M\\,\\left(\\begin{array}{c} \\chi \\\\ \\varphi _1 \\\\ \\varphi _2 \\end{array}\\right) ~, \\qquad M = \\left( \\begin{array}{ccc} 1 & 0 & 0 \\\\ 1 & 1 & 1 \\\\ 1 & 1 & 0\\end{array}\\right) ~.$ Note that ${\\rm det}M=-1$ , so this is an $SL(3,\\,\\mathbb {Z})$ transformation.The overall sign is just due to orientation.", "Upon simply sending $\\chi $ to $-\\chi $ we would recover the unit determinant.", "Thus, we see that the set of coordinates $(\\chi ,\\,\\varphi _1,\\,\\varphi _2)$ are a good global basis for the toric $\\mathbb {T}^3$ as well as the $\\mathbb {T}^2$ .", "In terms of these coordinates the metric at the center of the SLag is $ds_c^2=\\frac{2}{9}\\,\\Big (d\\varphi _1^2+d\\varphi _2^2+d\\varphi _1\\,d\\varphi _2\\Big ) ~,$ which leads to the expected $\\tau _G = \\tau _R = \\exp ( \\frac{\\pi i }{3})$ ." ], [ "Example: the conifold", "Encouraged by the success of the matching for the simplest case of $\\mathbb {C}^3$ , let us now look at a more involved example, namely the conifold theory.", "For reference, the quiver, dimer, and toric diagram are given in Figure REF .", "Figure: The quiver, dimer and toric diagram for the conifold theory.The quartic superpotential is given underneath.The metric for the conifold is the earliest known example for a (non-compact) Calabi–Yau manifold [20]: $ds^2=dr^2+\\frac{r^2}{9}\\,g_5^2+\\frac{r^2}{6}\\,\\Big (\\sum _{i=1,\\,2}\\,e_{\\theta _i}^2+e_{\\phi _i}^2\\Big ) ~,$ where we have defined $g_5=d\\psi _R-\\cos \\theta _1\\,d\\phi _1-\\cos \\theta _2\\,d\\phi _2 ~, \\qquad e_{\\theta _i}=d\\theta _i ~, \\qquad e_{\\phi _i}=\\sin \\theta _i\\,d\\phi _i ~.$ Here $\\phi _i\\,\\in \\,[0,\\,2\\,\\pi ]$ , $\\theta _i\\,\\in \\,[0,\\,\\pi ]$ , and $\\psi _R\\,\\in \\,[0,\\,4\\,\\pi ]$ .", "By forming the following combinations $e_1=e^{i\\psi _R}\\,\\Big (dr+i\\,\\frac{r}{3}\\,g_5\\Big ) ~, \\qquad e_2=\\frac{r}{\\sqrt{6}}\\,\\Big (e_{\\theta _1}+i\\,e_{\\phi _1}\\Big ) ~, \\qquad e_3=\\frac{r}{\\sqrt{6}}\\,\\Big (e_{\\theta _2}+i\\,e_{\\phi _2}\\Big ) ~,$ we see that the metric is just $ds^2=\\sum \\limits _{i=1}^3\\,|e_i|^2$ , so that the Kähler and top holomorphic forms take the standard $\\mathbb {C}^3$ form.", "In particular, the Kähler form is $\\omega =\\frac{i}{2}\\,\\sum \\,e_i\\wedge \\bar{e}_i=\\frac{r}{6}\\,\\Big ( 2\\,dr\\wedge g_5+r\\,e_{\\theta _1}\\wedge e_{\\phi _1}+e_{\\theta _2}\\wedge e_{\\phi _2}\\Big ) ~.$ Therefore, as we have done above, we can find the $U(1)^2$ invariant part of the $\\mathbb {T}^3$ with the action of the $U(1)^2$ generated by $\\lbrace \\partial _{\\phi _1},\\,\\partial _{\\phi _2}\\rbrace $ .", "It is not difficult to construct their moment maps, which read $\\mu _{\\phi _1}=-\\frac{r^2}{6}\\,\\cos \\theta _1 ~, \\qquad \\mu _{\\phi _2} =- \\frac{r^2}{6}\\,\\cos \\theta _1 ~.$ The SLag sits at $\\mu _{\\phi _i}=x_i$ , where $x_i$ are real constants.", "However, since we are interested on the center of the $(p,\\,q)$ -web, we can set $x_i=0$ , that is, $\\theta _i=\\frac{\\pi }{2}$ .", "Furthermore, we have $\\Omega \\sim e^{i\\,\\psi _R}$ when substituting (REF ) into $e_1\\wedge e_2\\wedge e_3$ .", "This just shows that $\\psi _R$ is identified with the field theory R-symmetry.", "The SLag condition demands that $\\psi _R=0$ .", "In summary then, we find, upon taking $\\psi _R=0$ , $dr=0$ , and $\\theta _1=\\theta _2=\\frac{\\pi }{2}$ both moment maps vanish while ${\\rm Im}\\, \\Omega ={\\rm P}_L[\\omega ]=0$ .", "Thus, this corresponds to the $U(1)^2$ -invariant SLag at the center of the web where the dimer should live.", "The pull-back of the metric there is $ds_{\\mathbb {T}^2}^2=\\frac{r^2}{6}\\,\\Big (d\\phi _1^2+d\\phi _2^2\\Big ) ~,$ which is just a square torus, so we find $\\tau _G=i$ .", "Note that in this case there are no further global issues, as $\\lbrace \\partial _{\\phi _i}\\rbrace $ do indeed cover a torus.Strictly speaking, the globally well-defined Killing vectors are $\\partial _{\\phi _i}+\\partial _{\\psi _R}$ , but for our purposes we can just consider $\\partial _{\\phi _i}$ since $\\psi _R=0$ .", "Once again, the geometric complex structure $\\tau _G$ so obtained is the same as the complex structure $\\tau _R$ of the isoradial dimer with edge lengths fixed by the R-charges [15]." ], [ "Orbifolds", "An immediate consequence for the above examples is that we can directly compute the $\\tau _G$ for all their orbifolds.", "As the local form for the metric will be unaffected by the orbifolding procedure, we can just borrow the study of the SLag submanifolds from the unorbifolded geometries.", "However, the global analysis will be different, as orbifolding will change the periodicity of the angles.", "The simplest example is the non-chiral orbifold of the conifold, which corresponds to taking $\\phi _1\\,\\in \\,[0,\\,\\pi ]$ .", "We can borrow the above result (REF ) provided we redefine $\\phi _2=\\frac{\\tilde{\\phi }_2}{2}$ so that $\\tilde{\\phi }_2\\,\\in \\,[0,\\,2\\,\\pi ]$ so that we can use the formulæ in the appendix.", "The metric is then $ds_{\\mathbb {T}^2}^2=\\frac{r^2}{6}\\,\\Big (d\\phi _1^2+\\frac{1}{4}\\,d\\tilde{\\phi }_2^2\\Big ) ~,$ which leads to $\\tau _G=2\\,i$ .", "This is precisely the expected result along the lines of [15], [16].", "Moreover, from here, it is immediate that the orbifold pattern for $\\tau _G$ will follow the dimer pattern for $\\tau _R$ , and so if the unorbifolded space agrees, all higher orbifolds will also show the $\\tau _R=\\tau _G$ agreement." ], [ "$Y^{p,q}$ and {{formula:91b26663-3f44-4751-813f-b330bac0c978}} manifolds", "The examples above indicate that global issues should be relevant when tackling more involved manifolds, such as the $Y^{p,q}$ spaces constructed in [19] and the $L^{a,b,c}$ spaces constructed in [21], [19].", "Instead of considering local expressions for the metrics where in the above we encountered subtleties involving the well-definedness and periodicity of the angular coordinates, we will use the more powerful approach based on the symplectic structure of these manifolds that was developed in [22]." ], [ "Symplectic coordinates", "The key observation is that, upon introducing suitable coordinates $\\lbrace y^i,\\,\\phi _i\\rbrace $ , the metric can be encoded in terms of a symplectic potential $G=G(y)$ such that $ds^2=G_{ij}\\,dy^i\\,dy^j+G^{ij}\\,d\\phi _i\\,d\\phi _j ~,$ where $G_{ij}=\\frac{\\partial ^2\\,G}{\\partial y^i\\,\\partial y^j} ~.$ Furthermore, the Kähler form reads $\\omega = dy^i\\wedge d\\phi _i ~,$ while the top holomorphic form is $\\Omega = e^{i\\,\\phi _1}\\cdots ~.$ We must set $\\phi _1=0$ so that ${\\rm Im}\\,\\Omega =0$ , while the $U(1)^2$ will be generated by $\\partial _{\\phi _2}$ and $\\partial _{\\phi _3}$ .", "These are globally well defined, and the associated moment maps are $\\mu _2=y_2$ and $\\mu _3=y_3$ , respectively.", "The center of the web will sit at $y_2=y_3=0$ .", "Thus, in all, the $U(1)^2$ -invariant sub-torus of the SLag at the origin will be given by the metric: $ds_{\\mathbb {T}^2}=G^{IJ}(y_2=y_3=0)\\,d\\phi _I\\,d\\phi _J ~, \\qquad I,\\,J=2,\\,3 ~.$ We shall then need to compute the complex structure $\\tau _G$ of this torus.", "We are left with the rather non-trivial task of constructing the symplectic potentials $G$ .", "Luckily this problem has been solved in [22] and [23], where it is shown that $G$ can be entirely constructed from the toric data and must be of the form $G=G^{{\\rm can}}+G^{{\\rm b}}+g ~,$ where $G^{{\\rm can}}$ is related to the canonical part of the metric as in [24], $G^{{\\rm b}}$ is associated to Reeb vector moduli described in [22], and $g$ is a remainder function that is a homogeneous and degree one rational function in the $y$ coordinates.", "In particular, $g$ satisfies a Heun equation found in [23].", "Note that for the geometric analog of $a$ -maximization, viz.", "volume minimization, the $g$ drops out and plays no rôle.", "As we shall see, however, in the case at hand, it has crucial significance." ], [ "Warmup: $Y^{p,q}$", "Let us begin with the simpler case of the $Y^{p,q}$ spaces.", "The dimer model and hence the quiver and superpotential are nicely given in [5] and the isoradial $\\tau _R$ were listed for a few cases in [7].", "As one can see, the theories are already very complicated, and $\\tau _R$ typically lives in highly non-trivial field extensions of $\\mathbb {Q}$ .", "Let us now use the notation of [23] and use the toric diagram defined by the following outward normal lattice vectors — one can check that these vectors are actually co-planar, as required for a Calabi–Yau:$\\endcsname $We emphasize that we have given the normals rather than the actually vectors of the toric diagram, i.e., we are describing the $(p,\\,q)$ -webs.", "$\\lbrace v_1 = (1,-1,-p) ~,v_2 = (1,0,0) ~,v_3 = (1,-1,0) ~,v_4 = (1,-2,-p+q)\\rbrace ~.$ Moreover, one defines the quantities $\\ell = \\frac{q}{3 q^2 - 2 p^2 + p \\sqrt{4 p^2 - 3 q^2}} ~,B = \\lbrace 3,-3,-\\frac{3}{2} (p-q+\\frac{\\ell }{3})\\rbrace ~,v_5 = B - v_1 - v_3 ~,v_6 = - v_2 - v_4 ~,$ where $B$ is the Reeb vector after $Z$ -minimization.As a caveat lector, we remark that in [22] the coordinates $\\lbrace (1,0,0), (1-,p-q-1,p-q),(1,p,p), (1,1,0)\\rbrace $ are used; we can see that the two are related by an $SL(3;\\mathbb {Z})$ transformation.", "The ones in (REF ) can be taken to these by, for example, $A = {\\scriptsize \\left( \\begin{array}{ccc} 1 & 0 & 0 \\\\-1 & -1 & 1 \\\\0 & q-p & p-q-1 \\end{array} \\right)}$ .", "Using these data, we can readily determine the symplectic potential to be $G(y_1,y_2,y_3) =\\frac{1}{2} \\sum \\limits _{i=1}^6 A \\cdot v_i \\cdot \\left( y_1, y_2, y_3 \\right)^T\\log \\left| A \\cdot v_i \\cdot \\left( y_1, y_2, y_3 \\right)^T \\right| ~,$ where we have used the matrix $A$ to rotate to [22].", "We stress that this is the form for the full symplectic potential as found in [23], including in particular the remainder function $g$ .", "The latter is associated to the $v_5$ and $v_6$ vectors, and so dropping them would only lead to the $G^{{\\rm can}}+G^{{\\rm b}}$ piece.", "From this, as explained above, we can find the metric of the required torus to be the $I=J=2,3$ submatrix of $G^{IJ} =\\left.", "\\left( \\frac{\\partial G}{\\partial y_i \\partial y_j} \\right)^{-1}\\right|_{y_2 = y_3 = 0} ~.$ From this expression and Appendix , we can determine the $\\tau _G$ .", "Let us tabulate some of the results below: $\\nonumber \\tau _G(Y^{1,0}) = i ~, \\tau _G(Y^{2,0}) = 2 i ~, \\quad \\tau _G(Y^{3,0}) = \\exp (\\frac{2\\pi i}{3}) ~, \\tau _G(Y^{4,0}) = 2 i ~, \\quad \\tau _G(Y^{2,2}) = \\frac{2 i}{\\sqrt{3}} ~.$ Comparing with the known results for $\\tau _R$ , we find exact agreement.", "This should not surprise us given our success in the previous sections.", "The spaces $Y^{p,0}$ are simply $\\mathbb {Z}_p$ orbifolds of the conifold, and $Y^{1,0}$ is just the conifold.", "The space $Y^{p,p}$ are cones over lens spaces and are simply the quotient $\\mathbb {C}^3 / \\mathbb {Z}_{2p}$ .", "As orbifolds of the geometries that we have explicitly studied above, it is guaranteed that $\\tau _G = \\tau _R$ .", "At this point, it is interesting to recall that for the geometric counterpart of $a$ -maximization the remainder function $g$ does not play any role and simply drops.", "On the contrary, in our case we generically do explicitly need the full form of the symplectic potential — i.e., including $g$ .", "More explicitly, while for the conifold and its orbifolds, $g$ plays no role, for the $Y^{p,p}$ spaces we find the correct result only upon considering the full $G$ including $g$ .", "What about something non-symmetric like $Y^{3,1}$ ?", "We find that $\\begin{array}{l}\\tau _G(Y^{3,1}) = -1+\\frac{1}{3} i \\sqrt{2+\\sqrt{\\frac{11}{3}}} ~, \\\\\\tau _R(Y^{3,1}) = -\\frac{1}{2} i \\left(1-2 \\cos \\left(\\sqrt{\\frac{11}{3}} \\pi \\right)+2 \\cos \\left(2\\sqrt{\\frac{11}{3}} \\pi \\right)+2 \\cos \\left(\\sqrt{33} \\pi \\right)\\right) \\csc \\left(\\sqrt{33} \\pi \\right) ~.\\end{array}$ These are glaringly different expressions.", "Can they be related to each other an $SL(2; \\mathbb {Z})$ transformation which would mean that the tori are really the same?", "In order to do so, we compute the Klein $j$ -invariant for both cases.$\\endcsname $Strictly speaking, we compute Klein's absolute invariant, without the conventional 1728 prefactor so that $j(i) = 1$ .", "We obtain that, numerically, $j(\\tau _G(Y^{3,1})) \\simeq 8.3796$ and $j(\\tau _R(Y^{3,1})) \\simeq 8.4126$ .", "These are rather close real numbers, and given the complicated nature of the $j$ -invariant, and the agreement in the other cases, this can not be a mere coincidence.", "Let us for now bear this discrepancy in mind and accumulate more data." ], [ "$L^{a,b,c}$ , an extraordinary puzzle", "The next infinite family of affine Calabi–Yau manifolds well-known to the AdS/CFT community is the $L^{a,b,c}$ toric spaces [25], [26].", "We again follow the nomenclature of [23].", "The symplectic potential can be computed starting with the outward pointing normal primitive vectors for $L^{a,b,c}$ : $v_1=(1,\\,1,\\,0) ~, \\quad v_2=(1,\\,a\\,k,\\,b) ~, \\quad v_3=(1,\\,-\\,a\\,l,\\,c) ~, \\quad v_4=(1,\\,0,\\,0) ~,$ where $k,\\,l$ are integers such that $k\\,c+b\\,l=1$ .", "The Reeb vector $B$ is found by the minimization technique described in [22]; this is a vector $B = (3, b_2, b_3)$ which minimizes the functional $Z(b_2,b_3) = \\frac{1}{24} \\sum \\limits _{i=1}^4\\frac{\\det (\\lbrace w_{i-1}, w_i, w_{i+1}\\rbrace )}{\\det (\\lbrace B,w_{i-1},w_i\\rbrace ) \\det (\\lbrace B,w_i,w_{i+1}\\rbrace )} ~,$ where $w_{1,2,3,4} = v_{1,2,3,4}$ and cyclically, $w_0 = v_4$ , $w_5=v_1$ , and where the notation $\\det (\\lbrace a,\\,b,\\,c\\rbrace )$ means the determinant of the matrix constructed by arranging $a,\\,b,\\,c$ as its rows.", "The extremization will give some rather complicated quartics in $b_2$ and $b_3$ , which we solve.", "Finally, introducing $v_5=B-v_1-v_3 ~, \\qquad v_6=B-v_2-v_4 ~,$ for the extremized $B$ values, we obtain the full symplectic potential (REF ), including the summand $g$ , in terms of coordinates $y=(y_1,\\,y_2,\\,y_3)$ , $G=\\frac{1}{2}\\,\\langle B,\\,y\\rangle \\,\\log \\langle B,\\,y\\rangle +\\frac{1}{2}\\,\\sum _{m=1}^3\\,\\langle v_{2\\,m-1},\\,y\\rangle \\,\\log |\\bar{x}-\\bar{x}_m|+\\frac{1}{2}\\,\\sum _{m=1}^3\\,\\langle v_{2\\,m},\\,y\\rangle \\,\\log |\\bar{y}-\\bar{y}_m| ~,$ where $\\langle \\cdot ,\\,\\cdot \\rangle $ stands for the usual Cartesian product and $&&\\bar{x}_1=-\\frac{\\det (\\lbrace v_1,\\,v_5,\\,v_6\\rbrace )}{\\det (\\lbrace v_3,\\,v_5,\\,v_6\\rbrace )} ~, \\qquad \\bar{x}_2=1 ~, \\qquad \\bar{x}_3=-\\frac{\\det (\\lbrace v_1,\\,v_5,\\,v_6\\rbrace )}{\\det (\\lbrace v_1,\\,v_3,\\,v_6\\rbrace )} ~, \\\\&&\\bar{y}_1=1 ~, \\qquad \\bar{y}_2=-1 ~, \\qquad \\bar{y}_3=\\frac{\\beta +\\alpha }{\\beta -\\alpha } ~,$ where $\\alpha =1+\\frac{\\det (\\lbrace v_2,\\,v_3,\\,v_4\\rbrace )}{\\det (\\lbrace v_3,\\,v_4,\\,v_6\\rbrace )} ~, \\qquad \\beta =1+\\frac{\\det (\\lbrace v_2,\\,v_3,\\,v_4\\rbrace )}{\\det (\\lbrace v_2,\\,v_3,\\,v_6\\rbrace )} ~.$ Finally, $x$ and $y$ are implicitly defined by the equations $\\langle v_2,\\,y\\rangle =\\frac{\\langle B,\\,y\\rangle }{2\\,\\alpha }\\,(\\alpha -\\bar{x})\\,(1-\\bar{y}) ~, \\qquad \\langle v_4,\\,y\\rangle =\\frac{\\langle B,\\,y\\rangle }{2\\,\\beta }\\,(\\beta -\\bar{x})\\,(1+\\bar{y}) ~.$ Now, we have that $L^{a,a,a}=Y^{a,0}$ , the orbifolds of the conifold (the simplest case $L^{1,1,1}=Y^{1,0}$ is just the conifold, where we recovered above the expected result $\\tau _G=i$ ).", "Indeed, for the higher $a$ cases, we also have $\\tau _G = \\tau _R$ .", "For these cases, as described above, the remainder function $g$ does not play a rôle, and in fact with just $G^{{\\rm can}}+G^{{\\rm b}}$ , we can reproduce this result.", "However, moving to the next simplest case $Y^{p,p}$ , the orbifolds of $\\mathbb {C}^3$ , we in fact have need the function $g$ to match $\\tau _R$ and $\\tau _G$ .", "Indeed, once we know the SLag fibration structure for a certain CY$_3$ , its non-chiral orbifolds immediately follow.", "By construction, these correspond to changing the period of the $\\mathbb {T}^2$ coordinates.", "Hence, we find that the complex structure of the relevant $\\mathbb {T}^2$ follows the same pattern as in [16].", "As remarked above, knowing that $\\tau _G = \\tau _R$ for $\\mathbb {C}^3$ and the conifold, it is not a surprise that the same applies to their symmetric orbifolds.", "Let us now move to a less symmetric and new example, namely that of $L^{1,2,1}$ , otherwise known as the suspended pinched point (SPP).", "This corresponds to D3-branes on the generalized conifold with the defining equation $xy=uv^2$ .", "Using the procedure above, we find that the geometric $\\tau _G$ of SPP is $\\tau _G=-\\frac{1}{2}+\\frac{i}{2}\\,\\sqrt{3\\,(2+\\sqrt{3})} \\quad \\Longrightarrow \\quad j(\\tau _G)\\approx -20.8416 ~.$ Luckily the analytic expression for the complex structure of the isoradial dimer has been computed from field theory [15].", "It turns out that the $L^{a,b,a}$ subfamily has fields with R-charges [27] $R(u_1) = R(y) = \\frac{1}{3} \\frac{b-2a+w}{b-a} ~, \\qquad R(u_2) = R(z) = \\frac{1}{2} R(v_1) = \\frac{1}{3} \\frac{2b-a-w}{b-a} ~,$ where $w = \\sqrt{a^2+b^2-ab}$ .", "Defining $\\chi _i = \\exp (i\\frac{\\pi }{2}R(u_i))$ , we compute [15] $\\tau _R = \\frac{(\\chi _1 + \\chi _1^{-1})(1 + \\chi _2^{-2})}{b(\\chi _1 + \\chi _1^{-1})\\chi _2^{-2} - a(\\chi _2 + \\chi _2^{-1})\\chi _1\\chi _2 - (b-a)(\\chi _2^2 + \\chi _2^{-2})\\chi _1} ~.$ For $L^{1,2,1}$ , we find $\\tau _R(L^{1,2,1}) = \\frac{1}{2} i \\left(i+2 \\tan \\left(\\frac{\\pi }{2 \\sqrt{3}}\\right)+\\cot \\left(\\frac{\\pi }{2 \\sqrt{3}}\\right)\\right) ~,$ and hence $j(\\tau _R)\\approx -20.3559$ .", "Thus, once more we find the remarkable proximity between $j(\\tau _G)$ and $j(\\tau _R)$ .", "We can at this point ask ourselves whether all this is due to numerical error.", "After all, Klein's invariant $j$ -function is a complicated non-linear function of its argument.", "To settle this matter, we recall the discussion in Section 7 of [15].", "Making use of the four exponentials conjecture, it was argued that $\\tau _R$ for SPP arising from the dimer construction is a transcendental number.", "For reference, we leave a detailed account of this proof for the current case to Appendix .", "On the other hand, it is clear from our discussion above that whatever result we get from the geometrical construction, $\\tau _G$ is going to be an algebraic number simply because $G$ only contains algebraic numbers and all subsequent manipulation to find $\\tau _G$ involve only algebraic (in fact, quadratic) functions.", "Now the $j$ -invariant of two complex numbers are the same if and only if the two numbers are related by an $SL(2;\\mathbb {Z})$ Möbius transformation $\\tau \\mapsto \\frac{a \\tau + b}{c \\tau + d}$ with $ad-bc=1$ and $a,b,c,d \\in \\mathbb {Z}$ .", "Therefore, clearly, an algebraic number can not have the same $j$ -invariant as a transcendental one.", "Thus, assuming the four exponentials conjecture is true — no counterexample to this conjecture is known to exist and it is widely accepted in the mathematical community — $j(\\tau _G)$ and $j(\\tau _R)$ cannot match in this case, and we exclude numerical errors.", "Nevertheless, it is truly remarkable that the geometry gives a torus so close to the field theory one.", "Is there a fundamental explanation for this discrepancy?", "In order to quantify the difference between $\\tau _G$ and $\\tau _R$ , we focus on the $L^{1,b,1}$ family.", "We can prove analytically here that $\\mbox{Re}(\\tau _G) = \\frac{b-1}{2} ~,$ while $\\mbox{Im}(\\tau _G)$ lives in some high even-degree extension of $\\mathbb {Q}$ .", "Nevertheless, (REF ) guarantees that $j$ will at least be real.$\\endcsname $To see this, we need merely look at the Laurent expansion of the $j$ -function in terms of $q = e^{2\\pi i\\tau }$ .", "We see that $q$ and therefore $j(\\tau )$ is real whenever the real part of $\\tau $ is an integer or a half-integer.", "By a further modular transformation, we can always put $\\mbox{Re}(\\tau _G)$ to 0 or $\\frac{1}{2}$ .", "In turn, from the field theory side, $\\tau _R$ will be very complicated transcendental numbers.", "While the real part of $\\tau $ is of course not an $SL(2,{\\mathbb {Z}})$ invariant and therefore not in itself a physically meaningful quantity, it may be convenient to set the real parts of $\\tau _R$ and $\\tau _G$ equal to each other via a modular transformation in order to facilitate a direct comparison of the two.", "Quite surprisingly, the $SL(2,\\,\\mathbb {Z})$ transformation with $\\lbrace a=2,\\,b=1,\\,c=-1,\\,d=0\\rbrace $ (plus repeated action with the $T$ generator) brings the real part to the form $\\mbox{Re}(\\tau _R) = \\frac{b-1}{2} ~.$ Hence, we are ensured that also the field theory $j$ will also be real.", "Thus, we can, in the precise sense described above, associate the discrepancy between $\\tau _R$ and $\\tau _G$ to a small mismatch in their imaginary parts.", "We can compare the $j$ -invariant of $\\tau _G$ and $\\tau _R$ for $L^{1,b,1}$ for various values of $b$ .", "In Figure REF , we plot the quantity of the absolute value of the ratio of differences and find a beautiful fit to $\\left| \\frac{j(\\tau _G) - j(\\tau _R)}{j(\\tau _G) + j(\\tau _R)} \\right|\\simeq -0.02+0.05\\,\\arctan \\,0.4\\,b ~.$ Figure: For the spaces L 1,b,1 L^{1,b,1}, a measure of the difference between geometric torus, with complex structure τ G \\tau _G, and the QFT torus, with complex structure τ R \\tau _R.One interesting consequence is that the difference between $\\tau _G$ and $\\tau _R$ , at least for the generalized conifold family under consideration, asymptotically saturates." ], [ "Conclusions and prospects", "Dimer models have played a prominent role in the understanding of $\\mathcal {N}=1$ SCFTs dual to D3-branes probing toric CY$_3$ singularities.", "Their physical appearance was understood in [6].", "In particular, the torus where the dimer lives was identified.", "Our purpose in this letter is to take this identification further and metrically identify the torus.", "Once the metric is known, it is natural to compute the complex structure, which might be expected to match the complex structure obtained from field theory considerations [7].", "Indeed, this is the very first step towards a full metric identification of the dimer from a geometrical perspective, hoping for a deeper understanding of some of its still somewhat mysterious properties, such as the nature of the isoradial embedding.", "In [15] the combinatorial properties of dimers were first explored.", "Since the dimer can be encoded in a very economical way in three permutations — each capturing, respectively black vertices, white vertices, and faces — through the Belyi theorem, it was argued that all the information can in fact be encoded in a Belyi pair consisting on a “worldsheet” torus together with a map from this “worldsheet” torus into a “target” $\\mathbb {P}^1$ ramified only over $\\lbrace 0,\\,1,\\,\\infty \\rbrace $ such that the ramification data reproduces in a specific sense the combinatorial structure.", "(For further details see [15].)", "The beauty of the Belyi construction is that the existence of such map ensures that the worldsheet torus can be defined over $\\overline{\\mathbb {Q}}$ .", "Furthermore, this worldsheet torus is rigid, and so we naturally obtain, associated to each SCFT through ths torus serving as worldsheet for its Belyi map, another complex structure parameter $\\tau _B$ .", "While in [15] it was speculated that $\\tau _B$ might equal $\\tau _R$ , it was further shown in [16] not to be true.", "Thus, in view of our findings in this note, we have a triple of complex structures assigned to each SCFT, namely $\\lbrace \\tau _R,\\,\\tau _G,\\,\\tau _B\\rbrace $ .", "We have seen that, in a precise sense, $\\tau _R\\sim \\tau _G$ .", "It remains to fully clarify the nature of this agreement/disagreement as well the hypothetical connection to $\\tau _B$ .", "In our case, a very simple reasoning led us to propose a certain $\\mathbb {T}^2$ arising from the SLag fibration structure of the CY$_3$ as that where the dimer lives.", "Indeed, for the simplest cases we found the torus expected from the field theory arguments.", "Furthermore, we find that this torus behaves in the appropriate way under orbifolding thus reproducing the field theory patters for its complex structure.", "However, moving to generic geometries we found this proposed torus to be almost but not exactly the expected one.", "Indeed, focusing on the complex structure, we found that an “$SL(2,\\,\\mathbb {Z})$ frame” exists where both $\\tau _R$ and $\\tau _G$ have the same real part, while the imaginary part differs by a very small amount.", "This is reflected in a more $SL(2,\\,\\mathbb {Z})$ invariant way in that the Klein $j$ -invariants come to be surpassingly close to each other.", "One must ask whether this discrepancy just signals that we have simply identified the torus incorrectly.", "However, given the highly non-linear nature of the Klein invariant, or the tiny discrepancy in only the imaginary part of the complex structure, it would seem a cosmic coincidence to repeatedly have this almost matching purely by chance.", "We note that the constructions in [6] and [10] lie, respectively, three and two $T$ -dualities away from the original IIB setup of D3-branes at the tip of the Calabi–Yau cone.", "This raises the question as to which frame the $\\mathbb {T}^2$ on which the dimer is drawn actually lives.", "Naïvely, two $T$ -dualities would leave $\\tau $ invariant up to an $SL(2,\\,\\mathbb {Z})$ transformation, thus suggesting that considering the original CY$_3$ should be enough.", "In any case, this is the Calabi–Yau that is technically accessible in an explicit way.", "Moreover, one would be naturally inclined to consider the original CY$_3$ as it is only in this frame that we have an AdS$_5$ space.", "Recall that the complex structure of the $\\mathbb {T}^2$ is fixed in field theory by the R-charges of the fields at the SCFT point, which on the other hand matches volumes in the IIB geometry.", "However, we stress that one possible reason for the disagreement might simply be that we are looking to the wrong “duality frame.” At any rate, it is very surprising how close our “wrong torus” comes.", "It is only slightly different in the imaginary part of the complex structure.", "This demands an explanation.", "Uncovering the nature of the reason for the small disagreement would be extremely interesting.", "It seems one fundamental problem we face is that it is hard to find a “microscopic quantification” of the disagreement.", "In other words, should we consider the Klein invariant as we have done in the paper?", "Or should we rather consider some other modular invariant?", "In fact, motivated by the fact that the $\\tau _R$ comes in terms of transcendental numbers of the form $e^{i\\,\\frac{\\pi }{2\\,\\sqrt{3}}}$ , one natural place to look for such corrections might be in terms of instanton contributions.", "These would arise upon resolving the singularity at the tip of the cone to introduce Kähler moduli.", "However, it is unclear what exactly those instantons would contribute and what their exact nature is.", "We leave this very interesting problem open." ], [ "Acknowledgements", "We wish to thank James Sparks for being generous with his time and offering important insights into the geometric analysis.", "We are grateful to Amihay Hanany, Jurgis Pasukonis, and Sanjaye Ramgoolam for collaboration on related topics.", "YHH would like to thank the Science and Technology Facilities Council, UK, for an Advanced Fellowship and grant ST/J00037X/1, the Chinese Ministry of Education, for a Chang-Jiang Chair Professorship at NanKai University, as well as City University, London and Merton College, Oxford, for their enduring support.", "YHH and VJ jointly acknowledge NSF grant CCF-1048082.", "The work of VJ is based upon research supported by the South African Research Chairs Initiative of the Department of Science and Technology and National Research Foundation.", "VJ as well thanks Queen Mary, University of London and STFC grant ST/G000565/1 for supporting his work during the early stages of this project.", "DRG is supported by the Aly Kaufman fellowship.", "He also acknowledges partial support from the Israel Science Foundation through grant 392/09 and from the Spanish Ministry of Science through the research grant FPA2009-07122 and Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042)." ], [ "Complex structures of tilted tori", "Let us consider a generic $\\mathbb {T}^2$ .", "The most general form for its metric is $ds^2=A\\,d\\phi _1^2+B\\,d\\phi _2^2+C\\,d\\phi _1\\,d\\phi _2 ~.$ Massaging the expression converts this into $ds^2=\\frac{4\\,A\\,B-C^2}{4\\,B}\\,d\\phi _1^2+B\\,\\Big ( d\\phi _2+\\frac{C}{2\\,B}\\,d\\phi _1 \\Big )^2 ~.$ It is now convenient to introduce $\\psi _1=\\frac{\\sqrt{4\\,A\\,B-C^2}}{2\\,\\sqrt{B}}\\,\\phi _1 ~, \\qquad \\psi _2=\\sqrt{B}\\,\\phi _2$ whose identifications are $\\psi _i\\sim \\psi _i+\\Delta _i$ with $\\Delta _1=\\frac{\\sqrt{4\\,A\\,B-C^2}}{2\\,\\sqrt{B}}\\,2\\,\\pi ~, \\qquad \\Delta _2=\\sqrt{B}\\,2\\,\\pi ~.$ In these coordinates, $ds^2=d\\psi _1^2+\\Big (d\\psi _2+\\gamma \\,\\,d\\psi _1\\Big )^2 ~, \\qquad \\gamma =\\frac{C}{\\sqrt{4\\,A\\,B-C^2}} ~.$ This corresponds to a torus generated by $\\vec{\\ell }_1=\\Delta _1\\,(1,\\,-\\gamma ) ~, \\qquad \\vec{\\ell }_2= \\Delta _2\\,(0,\\,1) ~.$ Hence, $\\tau =\\frac{\\Delta _1\\,\\gamma }{\\Delta _2}\\,(1+i\\,\\gamma ^{-1}) ~.$ In terms of $A,\\,B,\\,C$ we find $\\tau =\\frac{C}{2\\,B}\\,\\Big (1+i\\,\\frac{\\sqrt{4\\,A\\,B-C^2}}{C}\\Big ) ~.$" ], [ "The four exponentials conjecture", "The four exponentials conjecture is one of the key consequences of Schanuel's conjecture and would constitute one of the most important results in number theory.", "It states that given two pairs of complex numbers $(x_1, x_2)$ and $(y_1, y_2)$ such that each pair is linearly independent over $\\mathbb {Q}$ , then at least one of the numbers in the list $\\lbrace e^{x_1 y_1} ~, e^{x_1 y_2} ~, e^{x_2 y_1} ~, e^{x_2 y_2} \\rbrace ~,$ is transcendental.", "Now, consider our case of $L^{1,2,1}$ .", "We have that $\\tau _R = \\frac{1}{2} i \\left(i+2 \\tan \\left(\\frac{\\pi }{2 \\sqrt{3}}\\right)+\\cot \\left(\\frac{\\pi }{2 \\sqrt{3}}\\right)\\right) = \\frac{1-3x^2}{x^4-1} ~,$ where $x = \\exp ( i \\frac{\\pi }{2 \\sqrt{3}})$ .", "Noting that this is an algebraic (in fact, rational) function in the single complex number $x$ , it suffices to show that $x$ is transcendental to imply that $\\tau _R$ is also.", "Let $(x_1, x_2) = (1, \\frac{1}{2 \\sqrt{3}})$ and $(y_1,y_2) = (\\pi i , \\frac{\\pi i}{2 \\sqrt{3}})$ , we form the list of three number ($x$ appears twice): $e^{\\pi i} = -1 ~,e^{i \\frac{\\pi }{2 \\sqrt{3}}} = x ~,e^{\\frac{\\pi i}{12}} = \\frac{\\sqrt{3}+1}{2 \\sqrt{2}}+i \\left(\\frac{\\sqrt{3}-1}{2 \\sqrt{2}}\\right) ~.$ Now, the conjecture states that at least one of these must be transcendental and seeing the first and last to be clearly algebraic, $x$ must thus be the transcendental one.", "Rewriting (REF ), we have: $\\tau _R \\, x^4 + 3x^2 - ( \\tau _R + 1) = 0 ~.$ If $\\tau _R$ were algebraic, then $x$ must be algebraic since $\\overline{\\mathbb {Q}}$ is algebraically closed.", "Because $x$ is not algebraic assuming the four exponentials conjecture, the contrapositive applies and $\\tau _R$ must be transcendental." ] ]

[ [ "Bounded Counter Languages" ], [ "Abstract We show that deterministic finite automata equipped with $k$ two-way heads are equivalent to deterministic machines with a single two-way input head and $k-1$ linearly bounded counters if the accepted language is strictly bounded, i.e., a subset of $a_1^*a_2^*... a_m^*$ for a fixed sequence of symbols $a_1, a_2,..., a_m$.", "Then we investigate linear speed-up for counter machines.", "Lower and upper time bounds for concrete recognition problems are shown, implying that in general linear speed-up does not hold for counter machines.", "For bounded languages we develop a technique for speeding up computations by any constant factor at the expense of adding a fixed number of counters." ], [ "Introduction", "The computational model investigated in the present work is the two-way counter machine as defined in [4].", "Recently, the power of this model has been compared to quantum automata and probabilistic automata [17], [14].", "We will show that bounded counters and heads are equally powerful for finite deterministic devices, provided the languages under consideration are strictly bounded.", "By equally powerful we mean that up to a single head each two-way input head of a deterministic finite machine can be simulated by a counter bounded by the input length and vice versa.", "The condition that the input is bounded cannot be removed in general, since it is known that deterministic finite two-way two-head automata are more powerful than deterministic two-way one counter machines if the input is not bounded [2].", "The special case of equivalence between deterministic one counter machines and two-head automata over a single letter alphabet has been shown with the help of a two-dimensional automata model as an intermediate step in [10], see also [11].", "Adding resources to a computational model should intuitively increase its power.", "This is true in the case of time and space hierarchies for Turing machines, see the text book [15].", "The language from [2] cited above is easily acceptable with two counters and thus shows that two counters are more powerful than one.", "A further growing number of unbounded counters does however not increase the power of these machines due to the classical result of Minsky [8], showing that machines with two counters are universal.", "Thus the formally defined hierarchy of language classes accepted by machines with a growing number of counters collapses to the second level.", "A tight hierarchy is obtained if the counters are linearly bounded [9] or if the machines are working in real-time [4].", "In the latter case the machines are allowed to make one step per input symbol and there is obviously no difference in accepting power between one-way and two-way access to the input.", "If we restrict the input to be read one-way (sometimes called on-line [5]), a hierarchy in exponential time can be shown [12].", "The starting point of our investigation of time hierarchies is Theorem 1.3 of [4], where the authors show that the language of marked binary palindromes has time complexity $\\Theta (n^2/\\log n)$ on two-way counter machines.", "By techniques from descriptional complexity [7] for the lower bounds we are able to separate classes of machines with different numbers of counters.", "A main motivation for this work is of course a fundamental interest in the way the capabilities of a computational device influence its power.", "There are however other more technical consequences.", "Special cases of strictly bounded languages are languages over a single letter alphabet.", "Our simulation of heads by counters thus eliminates the need for hierarchy results separating $k+1$ bounded counters from $k$ bounded counters for deterministic devices with the help of single letter alphabet languages; the hierarchy for deterministic devices stated in Theorem 3 of [9] follows from the corresponding result for multi-head automata in Theorem 1 of [9].", "In comparison with multi-head automata, counter machines appear to be affected by slight technical changes of the definition.", "There are, e.g., several natural ways to define counter machines with counters bounded by the input length [13].", "One could simply require that the counters never overflow, with the drawback that this property is undecidable in general.", "Alternatively the machine could block in the case of overflow, acceptance then being based on the current state, a specific error condition could be signaled to the machine, with the counter being void, or the counter could simply remain unchanged.", "The latter model is clearly equivalent to the known concept of a simple multi-head automaton.", "All these variants, which seem to be slightly different in power, can easily be simulated with the help of heads and therefore coincide, at least for strictly bounded input.", "Regarding time bounded computations, we present an algorithm for the recognition of marked palindromes working with only two counters, while the upper bound outlined in the proof of [4] requires at least three counters (one for storing $\\log _2 m$ and two for encoding portions of the input).", "We show that counter machines lack general linear speed-up.", "Other models of computation with this property include Turing machines with tree storages [6] and Turing machines with a fixed alphabet and a fixed number of tapes [1].", "By adapting the witness language, we disprove a claim that these machines satisfy speed-up for polynomial time bounds [5].", "Finally we present a class of languages where linear speed-up can be achieved by adding a fixed number of counters." ], [ "Definitions", "A language is bounded if it is a subset of $w_1^*w_2^*\\cdots w_m^*$ for a fixed sequence of words $w_1, w_2\\ldots , w_m$ (which are not necessarily distinct).", "We call $w_1^*w_2^*\\cdots w_m^*$ the bound of the language.", "A language is strictly bounded if it is a subset of $a_1^*a_2^*\\cdots a_m^*$ for distinct symbols $a_1, a_2, \\ldots , a_m$ .", "A maximal sequence of symbols $a_i$ in the input will be called a block.", "Formal definitions of variants of counter machines can be found in [4].", "We only point out a few essential features of the models.", "The main models of computation investigated here are the $k$ -head automaton, the bounded counter machine with $k$ counters, and the register machine with $k$ bounded registers, cf.", "[9].", "The two former types of automata have read-only input tapes bordered by end-markers.", "Therefore on an input of length $n$ there are $n+2$ different positions that can be read.", "The $k$ -head automaton is equipped with $k$ two-way heads that may move independently on the input tape and transmit the symbols read to the finite control.", "Note that informally two heads of the basic model cannot see each other, i.e., the machine has no way of finding out that they happen to be reading the same input square.", "Heads with the capability “see” each others are called sensing.", "The bounded counter machine is equipped with a single two-way head and $k$ counters that can count up to the input length.", "The operations it can perform on the counters are increment, decrement, and zero-test.", "The machines start their operation with all heads next to the left end-marker (on the first input symbol if the input is not empty) and all counters set to zero.", "Acceptance is by final state and can occur with the input head at any position.", "The register machine receives an input number in its first register, all other registers are initially zero.", "We compare register machines with the other types of machines by identifying nonnegative integers and strings over a single letter alphabet of a corresponding length.", "All machines have deterministic and nondeterministic variants and accept by entering a final state.", "The set of marked palindromes over a binary alphabet is $L = \\lbrace x\\$ x^T \\mid x\\in \\lbrace 0, 1\\rbrace ^*\\rbrace ,$ where $x^T$ denotes the reversal of $x$ .", "With the restriction that at least approximately half of the input is filled by zeros, we obtain $L^{\\prime } = \\lbrace x0^{|x|}\\$ 0^{|x|}x^T \\mid x\\in \\lbrace 0, 1\\rbrace ^*\\rbrace ,$ and a further restriction to only a logarithmic information content leads to the family $L_m = \\lbrace x0^{2^{|x|/m}-|x|}\\$ 0^{2^{|x|/m}-|x|}x^T \\mid x\\in \\lbrace 0, 1\\rbrace ^*\\rbrace $ for $m\\ge 1$ ." ], [ "Equivalence of Heads and Counters for Deterministic Machines", "In order to simplify the presentation we assume below without loss of generality that a multi-head automaton moves exactly one head in every step.", "Suppose a multi-head automaton operates on a word from a strictly bounded language.", "We call a step in which a head passes from one block of identical symbols to a neighboring block or an end-marker an event.", "The head moved in this step is said to cause the event.", "Let the input of a deterministic multi-head automaton be strictly bounded.", "It is possible to determine whether a head will cause the next event (under the assumption that no other head does) by inspecting an input segment of fixed length around this head independently of the input size.", "If it can cause the next event it will also be determined at which boundary of the scanned block it will happen.", "Let the automaton have $r$ internal states.", "If the head moves to a square at least $r$ positions away from its initial position within the same block before the next event, then some state must have been repeated (since all other heads keep reading the same symbols) and the automaton, receiving the same information from its heads while no event occurs, will continue to work in a cycle until a head causes an event.", "Therefore it suffices to simulate the machine on a segment of $2r-1$ symbols under and around the head under consideration (or less symbols if the segment exceeds the boundaries of the input tape).", "We assume that no other head causes the next event, therefore at most $2r^2-r$ different partial configurations consisting of state, symbols read by the heads and the position on the segment are possible before one of the following happens: The head leaves the current block causing an event.", "The head leaves the segment around the initial position of the head.", "The automaton gets into a loop repeating partial configurations within the segment.", "In the the two former cases the boundary at which the next event is possibly caused by the head under cosideration is clear.", "In the latter case the head cannot cause the next event.", "Deterministic multi-head automata with $k$ two-way heads and deterministic bounded counter machines with $k-1$ counters are equivalent over strictly bounded languages.", "Any bounded counter machine with $k-1$ counters can easily be simulated by a $k$ -head automaton, independently of the structure of the input string.", "The multi-head automaton simulates the input head of the counter machine with one of its heads and encodes the values stored by the counters as the distances of the remaining head positions from the left end-marker.", "For the converse direction we will describe the simulation of a deterministic multi-head automaton $M$ by a deterministic bounded counter machine $C$ and call $C$ 's single input head its pointer, thus avoiding some ambiguities.", "The counters and the pointer of $C$ are assigned to the heads of $M$ , this assignment varies during the simulation.", "The counters will store distances between head positions and boundaries of blocks of the input, where distances to left or right boundaries may occur in the course of the simulation.", "The finite amount of information consisting of the assignment and the type of distance for each counter is stored in $C$ 's finite control.", "Depending on the type of distance stored for a head, movements of $M$ 's heads are translated into the corresponding increment and decrement operations.", "If a distance to a left boundary is stored, a left movement causes a decrement and a right movement an increment operation on the counter.", "For distances to a right boundary the operations are reversed.", "We divide the computation of the multi-head automaton $M$ into intervals.", "Each interval starts with a configuration in which at least one head is next to a boundary (i.e., on one of the two positions left or right of the boundary between blocks), one of these heads being represented by $C$ 's pointer.", "Notice that the initial configuration satisfies this requirement.", "Each interval except the last one ends when the next event occurs.", "After this event the machine is again in a configuration suitable for a new interval.", "Counter machine $C$ always updates the symbols read by the heads of $M$ (initially the first input symbol or the right end-marker) and keeps this information in its finite control.", "A counter assigned to a head encodes the number of symbols within the block that are to the left resp.", "right of the head position.", "The counter machine also maintains the information which of these two numbers is stored.", "We start our description of the algorithm that $C$ executes in a configuration with the property that at least one head of $M$ is next to a boundary.", "One of these heads is represented by the single pointer of $C$ .", "First $C$ moves its pointer to every block that is read by some head of $M$ .", "It can determine these blocks uniquely from the symbols stored in the finite control.", "It moves its pointer next to the boundary that is indicated by the type of distance stored on the counter assigned to the head under consideration.", "While the counter is not zero it decrements the counter and moves the pointer towards the head position.", "Then it determines whether this head could cause the next event, provided that no other head does, by applying Lemma .", "For this purpose the pointer reads the surrounding segment of length $2r-1$ without losing the head position.", "Then the test is carried out in $C$ 's finite control.", "Suppose the head could cause the next event at some boundary.", "Then $C$ updates the contents of the counter to reflect the distance of $M$ 's head from that boundary.", "If the block has the form $a_i^{x_i}$ and the head is on position $n\\ge 1$ in this block, then the counter machine moves its pointer towards the boundary where the event possibly occurs and measures the distance with the help of the counter.", "Thus $C$ updates the contents of the counter with either $n-1$ or $x_i-n$ , respectively, depending on whether the event can occur at the left or right boundary.", "If no event can be caused by the head currently considered, one of the distances is stored, say to the left boundary.", "These operations are carried out for every head.", "Finally $C$ moves its pointer back to the initial position, which is possible since it is next to a boundary.", "Then it starts to simulate $M$ step by step, translating head movements into counter operations according to the distance represented by the counter contents.", "If $M$ gets into an accepting state, $C$ accepts.", "If the pointer leaves its block the next interval starts.", "If a counter is about to be decremented from zero this operation is not carried out.", "Instead the current pointer position is recorded in this counter and the roles of pointer and counter are interchanged.", "The symbols read by the heads that pointer and counter are now assigned to, as well as the internal state of $M$ are updated.", "This information can clearly be kept in $C$ 's finite control.", "A new interval starts.", "The initial configuration of $C$ has all counters set to zero with the pointer and all simulated heads reading the first input symbol.", "The assignment of heads to counters is arbitrary, all counters store the distance to the left boundary.", "The equivalence of heads and counters implies that the intermediate concept of simple heads — two-way heads that cannot distinguish different input symbols — also coincides in power with counters over strictly bounded languages.", "It is open whether the analogous equivalence holds over arbitrary input or, as conjectured in [10], [11], simple heads are more powerful than counters.The $n$ -bounded counters of [10], [16] can count from 0 up to $n$ and can be tested for these values.", "They are easily seen to be equivalent to simple heads.", "The class of deterministic two-way machines equipped with $k$ such counters is denoted $C(k)$ in [10], [16], while machines with $k$ unbounded counters are denoted by $D(k)$ .", "Finding a candidate for the separation even of one simple head from a counter seems to be difficult.", "In recognizing the language $L_4 = \\lbrace ww \\mid w\\in \\lbrace 0, 1\\rbrace ^*\\rbrace $ the full power of a simple head was used in [16], but this is not necessary.", "A counter machine can first check that the input contains an even number of symbols.", "Starting with the first symbol it then stores the distance of the current symbol to the left end-marker on the counter.", "Sweeping its head over the entire input it increments the counter in every second step and thus computes the offset of the corresponding symbol, moves its head to this position and compares the symbols.", "Then it reverses the computation to return to the initial position and moves its head to the next symbol.", "If all corresponding symbols are equal it accepts.", "Finally we compare the power of register machines and multi-head automata over a single letter alphabet.", "For the simulation we will identify lengths of input strings and input numbers.", "In Lemma 5 of [9] a simulation of $k$ -head automata over a single letter alphabet by $k+1$ register machines is given.", "The simulation is rather specialized, since it applies only to subsets of words that have a length which is a power of two.", "We will generalize this simulation to arbitrary languages over a single letter alphabet.", "Every deterministic (nondeterministic) $k$ -head automaton over a single letter alphabet can be simulated by a deterministic (nondeterministic) $k+1$ register machine.", "The heads are even allowed to see each others.", "First we normalize multi-head automata that can detect heads scanning the same square such that the heads appear in a fixed left-to-right sequence on the tape (if some heads are on the same square we allow any sequence, which includes this fixed one).", "This is easily achieved because the automata can internally switch the roles of two heads which are about to be transposed.", "The register machine simulating a $k$ -head automaton with the help of $k+1$ registers stores in its registers the distances between neighboring heads or the end-marker, where the distance is the number of steps to the right a head would have to carry out in order to reach the next head or end-marker.", "Register 1 represents the distance of the last head to the right end-marker.", "Whenever a head moves the register machine updates the two related distances.", "A small technical problem is the distance to the left end-marker, which formally should be $-1$ in the initial configuration.", "The register machine stores the information whether the left-most heads scan the left end-marker in its finite control.", "In this way all distances can be bounded by the input length." ], [ "Time-Bounds for Counter Machines", "The purpose of this section is to establish lower and upper time-bounds on counter machines for concrete recognition problems.", "The recognition of the language $L^{\\prime }$ of marked palindromes with desert requires $\\frac{n^2-16n\\log _2n-dn}{8(\\log _2n+2\\log _2s+1)}$ steps for input strings of length $n$ on 1-counter machines and $\\frac{n^2-16n\\log _2n-dn}{8(2k\\log _2n+\\log _2s)}$ steps on $k$ -counter machines in the worst case and for $n$ sufficiently large, where $s$ and $d$ are constants depending on the specific counter machine.", "Let $M$ be a 1-counter machine with $s\\ge 2$ states accepting $L^{\\prime }$ and let $x$ be an incompressible string with $|x| = m \\ge 1$ .", "Consider the accepting computation of $M$ on $x0^{|x|}\\$ 0^{|x|}x^T$ and choose position $i$ adjacent to or within the central portion $0^{|x|}\\$ 0^{|x|}$ with a crossing sequence $c$ having $\\ell $ entries of minimum length.", "Notice that for 1-counter machines the counter is bounded from above by $s(n+2)< 2sn$ , since $n\\ge 2m+1$ .", "String $x$ can be reconstructed from the following information: A description of $M$ ($O(1)$ bits).", "A self-delimiting encoding of the length of $x$ ($2\\log _2n$ bits).", "Position $i$ of $c$ ($\\log _2n$ bits).", "Length $\\ell $ of $c$ ($\\log _2(4s^2n) $ bits).", "Crossing sequence $c$ recording the counter contents and the state $M$ enters when crossing position $i$ ($\\ell (\\log _2n+2\\log _2s+1)$ bits).", "A formalized description of the reconstruction procedure outlined below ($O(1)$ bits).", "For reconstructing $x$ from the above data, a simulator sets up a section of length $|x|$ followed by all symbols (0 or $\\$ $ ) up to position $i$ .", "Then the simulator cycles through all strings $y$ of length $|x|$ and simulates $M$ step by step.", "Whenever position $i$ is reached, it is checked that the current entry of the crossing sequence matches state and counter contents.", "If not, the current $y$ is discarded and the next string is set up.", "If it matches, the simulation continues from state and counter contents of the next entry of the crossing sequence.", "If $M$ accepts, the encoded $x$ has been found and the simulation terminates.", "Since $x$ is incompressible, for some constant $d$ compensating the $O(1)$ contributions we must have: $ |x| = (n-1)/4 \\le \\ell (\\log _2n+2\\log _2s+1) + 4\\log _2n + d/4 - 1/4$ and thus $ \\ell \\ge \\frac{n-16\\log _2n-d}{4(\\log _2n+2\\log _2s+1)}.$ There are $(n-1)/2+2\\ge n/2$ positions of crossing sequences with length at least $\\ell $ , thus we get $T(n) \\ge \\frac{n^2-16n\\log _2n-dn}{8(\\log _2n+2\\log _2s+1)}.$ For machines with $k\\ge 2$ counters we bound the counter contents by the coarse upper bound $n^2$ (since the asymptotical bound grows more slowly, this bound suffices).", "This increases the bound on the length of the encoding of crossing sequences to $\\ell (2k\\log _2n+\\log _2s)$ bits.", "The time bound becomes: $T(n) \\ge \\frac{n^2-16n\\log _2n-dn}{8(2k\\log _2n+\\log _2s)}.$ Next we present an upper bound for the full language $L$ and give an algorithm that uses only two counters in comparison to at least three in [4].", "We conjecture that this cannot be reduced to one counter for subquadratic algorithms.", "Language $L$ of marked palindromes can be accepted in $O(n^2/\\log n)$ steps by a two-counter machine.", "We describe informally the work of a machine $M$ accepting $L$ on an input of length $n\\ge 1$ .", "The idea is to encode segments of length $\\log _2 n$ and iteratively compare segment by segment.", "First $M$ scans the input and counts the symbols before the $\\$ $ (if no $\\$ $ is found, $M$ rejects).", "After the $\\$ $ the counter is decremented and the input is rejected, if zero is not reached on the right end-marker or a second $\\$ $ is encountered.", "The first scan takes $n$ steps if $M$ starts on the leftmost input-symbol as defined in [4].", "First $M$ puts 1 on counter 1, repeatedly reads a symbol, doubles the counter contents (exchanging roles for each bit read), and adds 1 if the symbol read was 1.", "Notice that after such a doubling one of the counter contents is zero.", "Then $M$ makes excursions to the left and to the right counting up on the empty counter and down on the counter holding the encoding until the latter counter becomes empty or $\\$ $ resp.", "an end-marker is reached.", "The net effect is an (attempted) subtraction of $n/2$ from the encoding.", "If the counter becomes zero, the initial encoding was less than $n/2$ .", "After each of the excursions, the other counter is used for returning to the initial position and the encoding is restored.", "If the encoding exceeds $n/2$ , the process stops and the segment is compared to the corresponding portion to the right of $\\$ $ .", "Using the empty counter, $M$ moves to the corresponding portion and in a symmetrical way as for the encoding decodes the segment.", "Since the encoding has a 1 as its most significant bit, no excursions are necessary.", "In order to return to the last position in the segment, $M$ repeats the encoding process.", "Then it continues with the next segment.", "The iterations stop if the marker $\\$ $ is reached.", "For the time analysis we omit constant and linear contributions to the number of steps, these are accounted for by an appropriately chosen constant factor of the leading term.", "The initial scan is clearly linear.", "By the doubling procedure the amortized cost of encoding and decoding is linear per segment.", "Notice that the excusions are aborted if the counter holding the encoding is empty and thus also the excursions have linear complexity per segment.", "Since the number of segments is $O(n/\\log n)$ , the bound follows.", "For $k$ -counter machines with fixed $k\\ge 2$ accepting $L$ in $O(n^2/\\log n)$ steps there is no linear speed-up.", "The lower bounds for a subset of $L$ in Theorem  carry over to the full language.", "These absolute lower bounds show that the algorithm from Theorem  cannot be accelerated by an arbitrary factor.", "By adding counters and encoding larger segments of the input a speed-up for the recognition of $L$ is possible.", "We now adapt the witness language to the bounds considered in [5].", "There speed-up results for deterministic and nondeterminstic two-way machines with $r$ counters and time bounds of the form $pn^k$ with $p > 1$ and $k\\ge 1$ are stated.", "No formal proofs are given, but the preceding section contains a reference to [4].", "We note here that the speed-up results in [4] are based on Theorem 1.2, which appears before the definition of two-way machines and therefore applies to one-way models only.", "In the special case of linear time bounds we will disprove the claimed speed-up.", "For at least quadratic bounds the technique does not apply, since the type of languages considered can be accepted in quadratic time comparing bit by bit and constant speed-up by forming blocks of constant size can clearly be achieved.", "Whether a general linear speed-up is possible is open, since there seems to be no efficient way to compress the contents of the input tape.", "The recognition of the language $L_m$ requires $(m/13)n$ steps on 4-counter machines in the worst case and for $n$ sufficiently large.", "The proof is adapted from the one given for Theorem  and we only describe the differences.", "We assume $|x| > m\\log _2|x|$ in the following.", "Since the part $x$ is now only a logarithmic portion of the input, $2\\log _2\\log _2n$ bits suffice for encoding the length.", "The time bound is the linear function $pn$ , therefore the crossing sequence can be described in $\\ell (4(\\log _2n+\\log _2p)+\\log _2s)$ bits.", "For an incompressible $x$ we obtain: $m\\log _2n - 2m \\le |x| & = & m\\log _2((n-1)/2) \\\\& \\le &\\ell (4(\\log _2n+\\log _2p)+\\log _2s) + 2\\log _2n + d \\\\& = & \\ell (6\\log _2n) + d^{\\prime }$ with constants $d, d^{\\prime }$ and $ \\ell \\ge m/6 - o(1).$ Since the desert is at least $n/2$ symbols long for $|x| > m\\log _2|x|$ we get the time bound $T(n) \\ge (m/12)n - o(n).$ We now have to show that there is a linear recognition algorithm for $L_m$ .", "The language $L_m$ can be accepted in $(2m+3)n + o(n) $ steps by a 4-counter machine.", "In a first left-to-right scan, an acceptor $M_m$ for $L_m$ determines the length of the part of the input before the $\\$ $ , compares it to the part after $\\$ $ and at the same time counts that length again in another counter.", "By repeatedly dividing by two, it computes $|x|/m$ in at most $n$ steps and checks in a right-to left scan, whether the middle portion of the input contains only 0's.", "While doing so, $M_m$ preserves $|x|/m$ and computes $2^{|x|/m}-|x|$ for checking the left half of the input.", "Then $M_m$ starts to encode blocks of $|x|/2m$ bits on two counters while using the other two counters for checking the length of the block.", "Since $|x| \\le m\\log _2n$ , the encoding can be done in $O(\\sqrt{n})$ steps.", "Then $M_m$ moves its input head to the other half of the input, decodes the stored information, and contiues with the next iteration.", "The number of $2m$ iterations of $n + o(n)$ steps each can be counted in the finite control.", "As an example take $L_{39}$ , which can be accepted in $82n$ steps by Theorem  for sufficiently large $n$ , but a linear speed-up to $2n$ is impossible by Theorem ." ], [ "Speed-Up for Counter Machines on Bounded Input", "In Section  we have shown that linear speed-up for certain recognition problems on counter machines can only be achieved by adding counters.", "If the input is compressible, the situation is different and a fixed number of counters depending on the structure of the language suffices for speeding up by any constant factor.", "We will apply the classical result of Fine and Wilf on periodicity of sequences: [[3]] Let $(f_n)_{n\\ge 0}$ and $(g_n)_{n\\ge 0}$ be two periodic sequences of period $h$ and $k$ , respectively.", "If $f_n = g_n$ for $h + k - \\mbox{gcd}(h, k)$ consecutive integers, then $f_n = g_n$ for all $n$ .", "For every $k$ -counter machine accepting a bounded language with $m$ blocks and operating in time $t(n)$ there is an equivalent counter machine with $k+m$ counters operating in time $n + c t(n)$ for any constant $c > 0$ .", "The strategy is to encode the input of a counter machine $M$ with $k$ counters on $m$ counters of a simulator $M^{\\prime }$ in a first stage and simulate the two-way machine $M$ with speed-up using the method suitable for one-way machines from the proof of Theorem 5.3 in [4].", "Let the accepted language be a subset of $w_1^*w_2^*\\cdots w_m^*$ .", "Note that in general the exponents $k_1, \\ldots , k_m$ of $w_1, \\ldots , w_m$ for a given input are not easily recognizable, since the borders between the factors might not be evident.", "We will show that an encoding of the input is nevertheless possible.", "The encoding of the input will work in stages.", "At the end of stage $i$ the encoding covers at least the first $i$ blocks.", "Let $\\mu = \\max \\lbrace |w_j| \\mid 1 \\le j \\le m\\rbrace $ .", "At the start of stage $i$ simulator $M^{\\prime }$ reads $2\\mu $ additional input symbols if possible (if the end of the input is reached, the suffix is recorded in the finite control).", "Machine $M^{\\prime }$ records this string $y$ in the finite control.", "Then for each conjugate $vu$ of $w_i = uv$ the input segment $y$ is compared to a prefix of length $2\\mu $ of $(vu)^{\\omega }$ .", "If no match is found, the segment $y$ and its position is stored in the finite control and the next stage starts.", "If a match to some $vu$ is found, $M^{\\prime }$ assigns a counter to this $vu$ , stores the number of copies of $vu$ within $y$ (including a trailing prefix of $vu$ if necessary) are recorded in the finite control, and then $M^{\\prime }$ continues to count the number of copies of $vu$ found in the input until the end of the input is reached or the next $|vu|$ symbols do not match $vu$ .", "These $|vu|$ symbols are stored in the finite control.", "The process ends when the input is exhausted.", "Then $M^{\\prime }$ has stored an encoding of the input, which can be recovered by concatenating the segments stored in the finite control and the copies of the conjugates of the $w_i$ .", "We now argue that at the end of stage $i$ the encoding has reached the end of block $i$ .", "Initially the claim holds vacuously.", "Suppose by induction that the claim holds for stage $i-1$ .", "The next $2\\mu $ symbols are beyond block $i-1$ and if they do not match a power of a conjugate, then the string is not embedded into a block.", "Thus it extends over the end of block $i$ and the claim holds.", "Otherwise a match between some $vu$ and $y$ is found.", "String $y$ is a factor of some $w_j^\\omega $ and by the Fine and Wilf Result (Theorem ) both are powers of the same $z$ since $2\\mu \\ge h + k - \\mbox{gcd}(h, k)$ with $h = |vu|$ and $k = |w_j|$ .", "Therefore $M^{\\prime }$ is able to encode all of the copies of $w_j$ on the counter.", "Notice that $vu$ is not necessarily a conjugate of $w_j$ .", "Thus the claim also holds in this case.", "Since each stage requires at most one counter, the $m$ additional counters suffice.", "After encoding the input, the two-way input-head of $M$ is simulated by $M^{\\prime }$ with the help of $m$ counters and its input-head, which is used as an initially empty counter measuring the distance to the right end-marker.", "Whenever the simulated head enters a block $i$ , the simulator starts to decrement the corresponding counter and increment a counter available (since it has just been zero).", "The input head is simulated on $w_i$ , where the position of the input head position modulo $|w_i|$ is kept in the finite control of $M^{\\prime }$ .", "Also the assignment of counters to blocks is dynamic and stored in the finite control.", "Now $M^{\\prime }$ is replaced by $M^{\\prime \\prime }$ with compressed counter contents and operating with speed-up according to the proof of Theorem 5.3 from [4]." ] ]

[ [ "New Wrinkles on an Old Model: Correlation Between Liquid Drop Parameters\n and Curvature Term" ], [ "Abstract The relationship between the volume and surface energy coefficients in the liquid drop A^{-1/3} expansion of nuclear masses is discussed.", "The volume and surface coefficients in the liquid drop expansion share the same physical origin and their physical connection is used to extend the expansion with a curvature term.", "A possible generalization of the Wigner term is also suggested.", "This connection between coefficients is used to fit the experimental nuclear masses.", "The excellent fit obtained with a smaller number of parameters validates the assumed physical connection." ], [ "New Wrinkles on an Old Model: Correlation Between Liquid Drop Parameters and Curvature Term L. G. Moretto P. T. Lake L. Phair Lawrence Berkeley Laboratory, One Cyclotron Road, Berkeley, CA 94720 J.", "B. Elliott Lawrence Livermore Laboratory, 7000 East Avenue, Livermore, CA 94550 The relationship between the volume and surface energy coefficients in the liquid drop $A^{-1/3}$ expansion of nuclear masses is discussed.", "The volume and surface coefficients share the same physical origin and their physical connection is used to extend the expansion with a curvature term.", "A possible generalization of the Wigner term is also suggested.", "This connection between coefficients is used to fit the experimental nuclear masses.", "The excellent fit obtained with a smaller number of parameters validates the assumed physical connections and the usefulness of the curvature term.", "Introduction.", "Nuclear masses and their dependence on atomic and mass number gave essential information about the nature of nuclear forces.", "They also led to the formulation of the liquid drop model, arguably the most precise and easily interpretable description of the masses themselves [1].", "Empirical trends and scientific intuition led to the formulation of the liquid drop model.", "In its traditional form, the liquid drop model approximates the binding energy of a given nucleus of mass number $A$ and charge $Z$ as [1]: $E_B(A,Z)=-a_v A +a_s A^{2/3} + a_c \\frac{Z(Z-1)}{A^{1/3}} \\nonumber \\\\+ a_a \\frac{(A-2Z)^2}{A} \\pm \\frac{\\delta }{\\sqrt{A}}.$ The five terms in this equation are associated with five independent aspects of nuclei expected to affect the binding energy.", "These aspects are the nuclear volume, surface, Coulomb repulsion, proton-neutron asymmetry, and pairing.", "A fit of this equation to nuclear masses gives the coefficients and reproduces the experimental values to within 1% or $\\sim $ 10 MeV for heavy nuclei.", "This is an outstanding result that attests to the profound physical content of the overall equation and to the interpretation of its individual terms.", "The residual 1% discrepancy is due to shell structure.", "The shell corrections, evaluated according to the Strutinsky procedure [2] and grafted onto the liquid drop model, permit an accurate evaluation of nuclear masses and fission barriers to within 1-2 MeV [3], [4], [5], [6].", "This hybrid approach remains to this day the yet unmatched paragon for more sophisticated models such as Hartree-Fock-Bogoliubov [7], [8].", "Motivated by this early success of the liquid drop model, many additional terms have been suggested, each with its own physical interpretation.", "An example of this is found in Myers and Swiatecki [3] who suggested: $E_B(A,Z)= -a_v &\\left(1-k\\frac{I^2}{A^2}\\right) A +a_s \\left(1-k\\frac{I^2}{A^2}\\right) A^{2/3} \\nonumber \\\\&+ a_c \\frac{Z(Z-1)}{A^{1/3}} + W\\frac{|I|}{A} - C_4\\frac{Z^2}{A}\\pm \\frac{\\delta }{\\sqrt{A}},$ where $I{=}A-2Z$ .", "The main difference between Eq.", "(REF ) and Eq.", "(REF ) is the extension of the neutron-proton asymmetry to the surface energy term.", "Also, a term linear in $|I|$ was introduced.", "The further physical insight found in writing the asymmetry energy as in Eq.", "(REF ) is the connection it implies between volume and surface energies.", "The authors argued that the change of the volume energy due to the neutron-proton asymmetry $I$ should be reflected in the surface energy of the system as well, though stating that this was done without empirical evidence [3].", "The natural implication is that the surface and volume energies are related through their common origin.", "A term linear in $|I|$ was originally suggested by Wigner in considering the exchange force of nucleons [9].", "An empirical observation of such a dependency in the masses was reported by Myers and Swiatecki, hence its addition in the above equation [3].", "The last alteration was the addition of a term proportional to $Z^2/A$ , which is to account for the difference in the Coulomb energy due to the diffuse nuclear surface.", "Again, this trend was not observed in the nuclear data, but was added because it would be anticipated.", "We will not discuss the role of this term in the nuclear binding since it requires an additional fitting parameter to describe what is ambiguous in the data.", "In this paper we extend and generalize the insights discussed above.", "In particular, we argue that the relation between volume energy and the surface energy is strong enough that their coefficients should not be taken as independent variables.", "Furthermore, we discuss the need of a third term arising from the same physics, proportional to $A^{1/3}$ , in order to create a consistent physical picture of the nuclear binding energy.", "Finally, a linear term in $|I|$ is naturally introduced when treating the asymmetry term as the expectation value of the isospin, $T^2$ .", "A revised liquid drop model is fit to the experimental binding energies of the nuclides to test these considerations.", "The importance of the revised terms is assessed by comparing the fit of the original and revised models.", "The Liquid Drop Formula: A Truncated Series Expansion.", "The first term of Eq.", "(REF ) is aptly called the volume term.", "Its proportionality to $A$ indicates saturating forces leading to constant density and binding energy per nucleon.", "The obvious similarity to molecular fluids led naturally to the introduction of the second term, the surface term.", "Its proportionality to $A^{2/3}$ speaks to the lack of saturation on the nuclear surface, whose area, through the constant density of the fluid, should indeed be proportional to $A^{2/3}$ .", "Progressing along the same line, it was widely appreciated that the surface term is a finite size correction and that additional terms in the expansion might be needed, such as a curvature term.", "Generally, we can think of a generalized liquid drop formula as a rapidly converging series expansion in powers of $A^{-1/3}$ , known as the leptodermous expansion [10]: $E_B = -a_v A + a_s A^{2/3} + a_r A^{1/3} + ...$ It is left to be determined how many terms in the expansion are necessary to describe the physics of the nuclear system.", "The incorporation of a curvature term, with its coefficient $a_r$ , proportional to $A^{1/3}$ is almost demanded by the truly small size of nuclei ($A \\le 300$ ) compared to the size of the drops typically considered in molecular fluids, such as aerosols, where $A \\ge 10^6$ .", "Higher order terms also may be of importance due to the small size of nuclei, but would be intractable without an understanding of the lower order curvature term.", "The role of the curvature term in nuclear systems was considered only recently and has yielded ambiguous results [3], [4], [5], [6].", "The increased number of parameters and the ability of the traditional liquid drop formula without curvature to fit the data made the problem of identifying the magnitude of this term rather difficult.", "We believe that it is possible to shed additional light on this subject by considering the physical origin of the various terms.", "Volume and surface terms both arise from the same physical property of nuclear forces: saturation, and the lack thereof.", "Thus, surface and volume terms should be related to one another, being themselves different effects of the same cause.", "Furthermore, the experimental surface and volume coefficients turn out to be approximately equal.", "Is this an accident or could they possibly be equal?", "To answer this question, consider a system of small sticky cubes used to build larger, composite cubes.", "These cubes interact only when in direct contact.", "The system is characterized by some bond strength, $\\epsilon $ , when two faces are touching.", "The energy of a cube of $A$ constituents is equal to a volume energy minus a surface energy, just as in the nuclear case.", "Counting the number of bonds in a cube of size $A$ reveals: $E_B^{(cube)}(A)=-3A\\epsilon + 3A^{2/3}\\epsilon .$ Thus, in this model the volume and surface energy coefficients are exactly equal with $a_v{=}a_s{=}3\\epsilon $ .", "Even though this is a simplified model in comparison to a nucleus, it exemplifies the fact that the volume and surface terms are strongly connected.", "This insight motivates setting $a_v{=}a_s$ without any loss of information.", "Table: Fits of the nuclear masses with Eq.", "() using different mass ranges and setting a r =0a_r{=}0.", "All the parameters in units of MeV.", "The value in the parentheses is the uncertainty in the last digit.One difference between this simple model system and a nucleus is the diffuseness of the nuclear surface.", "What effect does a diffuse surface have on the binding energy of a drop?", "Since the volume energy is a property of the bulk system it would remain unchanged.", "The fact that the system naturally becomes diffuse means that it gains a larger binding energy in doing so.", "The surface energy would then be lowered in comparison to the sharp surface system.", "This implies that the surface energy coefficient should be equal to or smaller than the volume energy coefficient, contrary to what is observed in traditional liquid drop fits to the nuclear masses.", "As the system is made smaller, more terms in the leptodermous expansion may be needed to properly predict binding energies.", "If one were to fit the expansion with an insufficient number of terms, what ailments would be observed?", "The terms included in the equation would have to change from their nominal values to accommodate the lack of higher order terms.", "Also, the deviation from the nominal value would be worse for smaller masses, where the higher order terms are more important.", "As an example, consider nuclear binding energies in various mass ranges.", "Each mass range can be fit with Eq.", "(REF ), using the fitting procedure that is described in the following section.", "Table REF shows the results of such an exercise.", "Most terms do not vary systematically as the mass range is changed, their variation being of the order of 1%.", "The exceptions are the surface energy and the pairing energy.", "The pairing energy is of unrelated physics and is not discussed here.", "The surface energy coefficient decreases as the mass range is incremented.", "This trend indicates that the $A^{2/3}$ term is not sufficient in describing the lack of saturation in the system.", "As the masses used in the fit increase, the surface term tends to the value of the volume coefficient.", "Hence, both the need of a curvature term and setting $a_v{=}a_s$ are motivated.", "Figure: Schematic representation of the surface energy.", "The system on the left represents a flat surface of an infinite liquid and the system on the right is a finite liquid drop.", "The surface area of a constituent of radius r n r_n exposed on the surface of a liquid drop of radius RR is more than that of a surface particle at the flat surface, as emphasized by the bold curve.Now that we see the need for a curvature term, one may ask: What is its origin?", "To answer, let us consider a simple liquid with spherical molecules of radius $r_n$ .", "As shown on the left side of Fig.", "REF , on a flat liquid surface the molecules should protrude half way on average, losing half of their binding energy.", "If the liquid surface is curved like that of a sphere, as in the right side of Fig.", "REF with a drop radius of $R$ , the molecules protrude more, losing additional binding.", "Thus, the curvature and surface terms arise from the same physical effect and their coefficients should be related.", "In order to obtain a quantitative estimate of this effect, consider a model that is geometric in nature.", "The surface energy is considered to be proportional to the protruding surface area of a constituent residing on the surface times the number present on the surface.", "As a function of nuclear radius $R$ , the resulting exposed surface area $S$ of a constituent on the surface is: $S=2\\pi r_n^2\\left(1+\\frac{r_n}{2R}\\right),$ with the limiting case of a planar system, $S{=}2\\pi r_n^2$ .", "The number of particles on the nuclear surface is proportional to $A^{2/3}$ .", "The overall surface energy is then: $E_s = a_s A^{2/3} \\left(1+\\frac{r_n}{2R}\\right).$ Since nuclei exhibit a saturation density, the nuclear radius is approximated as $R{=}r_0A^{1/3}$ , with $r_0$ being a constant.", "Inserting this relation into Eq.", "(REF ) yields: $E_s = a_s A^{2/3} + a_s \\frac{r_n}{2r_0}A^{1/3}.$ Here we identify the usual surface term proportional to $A^{2/3}$ .", "Furthermore, we notice a curvature term proportional to $A^{1/3}$ with a coefficient that is dependent on the surface energy coefficient and the ratio of the “molecule” radius to $r_0$ , which is directly related to the saturation density.", "The above equation is reminiscent of the Tolman correction to the surface tension [11].", "This term can be interpreted as the Tolman correction for the nuclear system in its ground state.", "Naturally, deviations from sphericity of the molecules would involve a (temperature dependent) reorientation on the surface.", "This would alter the simple relationship between volume, surface and curvature energies.", "We will limit the discussion to the case of an isotropic force for the model presented here.", "We may check the model further by putting experimental values into Eq.", "(REF ).", "Taking $r_0 \\simeq 1.2$ fm [5] and the radius of a free nucleon to be $r_n \\simeq 0.9$ fm [12], yields: $a_r \\simeq a_s \\frac{0.9}{2.4} \\simeq \\frac{3}{8}a_s.$ These geometric arguments thus give a first order approximation as to the sign and magnitude of the curvature term.", "Other aspects might influence the actual value of the curvature term in the nuclear system, but it would be notable if a proper fit to nuclear masses were to produce a value close to the above estimate.", "To further appreciate the significance of the relation between volume, surface and curvature energies, consider the following.", "What information is gained in knowing the leptodermous expansion for an arbitrary liquid in its ground state?", "First, consider the volume energy.", "The volume energy gives no information of the internal structure of the system.", "It is just the scale which sets the size of the rest of the terms in the leptodermous expansion.", "Now a measurement of the system's surface energy is made.", "The particle density of the system can be deduced by comparing the surface and volume energies.", "This is done by anticipating that the two coefficients will be the same in terms of $A$ and $A^{2/3}$ , respectively.", "Avogadro's number could thus be inferred.", "Finally, the curvature energy is determined and from it the size of a single particle in the liquid can be estimated.", "This is shown in Eq.", "(REF ).", "Here we see how the hierarchy of terms in the leptodermous expansion can be related to the internal structure of a fluid.", "Even though this exercise is pedagogical in nature, it demonstrates the physical significance of each term.", "It could have allowed Democritus to prove his atomic theory, had he been inclined to do so.", "Table: Fits from the four different mass equations as described in the text.", "All parameters are in units of MeV.", "The value in the parentheses is the uncertainty in the last digit.Nuclear Mass Fit Results.", "We use a set of 2076 masses, corrected for microscopic effects according to Möller et al.", "[13].", "These microscopic corrections account for the shell effects along with the effects associated with nuclear deformation.", "The masses considered in the fits correspond to nuclear masses from reference [14] with $N>7$ , $Z>7$ , and with experimental uncertainties less than 150 keV.", "The lower limit of neutron and proton numbers is chosen to ensure that the included nuclei are large enough to be considered as liquid drops.", "The restriction on the experimental uncertainties is not only due to the error of the mass, but also to the reliability of the shell correction for masses far away from stability.", "The binding energy, $E_B$ , of each nucleus is defined as: $E_B(A,Z)=Z m_p + (A-Z) m_n - M(A,Z) + \\Delta _{shell}(A,Z),$ with $m_p$ and $m_n$ being the mass of a proton and neutron, respectively, $M$ is the experimental mass of the nucleus, and $\\Delta _{shell}$ is the shell correction.", "The liquid drop formula is fit to this binding energy with each nucleus given an equal weight.", "The mean square deviation of the fit is used to evaluate its goodness: $\\chi ^2=\\frac{\\sum (E_i^{(ex)}-E_i^{(th)})^2}{N}.$ We use the following liquid drop formula: $E_B =& (-a_v A + a_s A^{2/3} + a_r A^{1/3})\\left(1-k\\left(\\frac{|I|(|I|+2)}{A^2}\\right)\\right) \\nonumber \\\\&+ a_c \\frac{Z(Z-1)}{A^{1/3}} \\pm \\frac{\\delta }{\\sqrt{A}},$ where we insert the mass asymmetry dependence $I{=}A-2Z$ both in the volume and surface terms according to Myers and Swiatecki [3].", "If the mass asymmetry term is interpreted as an “isospin” dependence, the term linear with $I^2$ should be treated as $T^2$ , with $T{=}|I|/2$ .", "This “isospin” presents itself as the square $T^2$ , which we rewrite (with a possibly unjustified quantal sensitivity) as $\\langle T^2 \\rangle {=} T(T+1) {=} |I|(|I|+2)/4$ .", "This introduces a linear term in $|I|$ without the addition of a new parameter, as opposed to a freely varying Wigner term [9].", "The following fits are performed: $a_v$ and $a_s$ vary independently without a curvature term.", "Same as above, but forcing $a_v{=}a_s$ .", "$a_v$ and $a_s$ vary independently with a curvature term.", "Same as above, but forcing $a_v{=}a_s$ .", "$a_v$ and $a_s$ vary independently without a curvature term.", "Same as above, but forcing $a_v{=}a_s$ .", "$a_v$ and $a_s$ vary independently with a curvature term.", "Same as above, but forcing $a_v{=}a_s$ .", "The Coulomb, mass asymmetry and pairing coefficients are left as free parameters in all of the above fits.", "The results are shown in Table REF and are discussed below.", "Fig.", "REF shows plots of the residual masses of the fits, the exact binding energy with shell corrections included minus the binding energy predicted from the fitted formula.", "Figure: The residual mass from the corresponding fits.", "The label in the top left corner of each plot corresponds to the fits listed in the text.", "The connected lines represent chains of isotopes.Comparing fits A and B shows that setting $a_v{=}a_s$ without the curvature term does not ameliorate the situation.", "Quite to the contrary, the $\\chi ^2$ value is 8 times larger and the plot of the residual masses shows clear deviations.", "Left without constraint, the surface term incorporates the curvature effects and becomes larger.", "Comparing fits A and C, we observe that the introduction of the curvature term as a free parameter improves the resulting fits as expected.", "But is the value of $a_r$ physically meaningful and how does it compare to the expectations of the geometric model?", "Rearranging Eq.", "(REF ) gives the radius of the nucleon as $r_n=2.4\\frac{a_r}{a_s},$ in units of fm.", "Using this equation, the nucleon radius is found to be 0.60(5) fm, smaller than the experimental value of 0.84 fm [12].", "This size of deviation is not unexpected from the crude approximations used, and it remains impressive that both the sign and relative magnitude are predicted.", "Furthermore, the surface energy coefficient moves within error of the volume energy coefficient.", "The other parameters change within 2% between the two fits, showing consistent results.", "By forcing $a_v{=}a_s$ with the presence of the curvature correction, as in fit D, the $\\chi ^2$ changes by a fraction of a percent.", "Also, the parameters not associated with the saturating nuclear force are left unchanged.", "Thus, no physics is lost with setting $a_v{=}a_s$ .", "Without taking into account the curvature term, the volume and surface parameters will tend to be irreconcilably different to be considered equal.", "This explains the reason why the two terms have previously been treated as independent values.", "The addition of the curvature term corrects this discrepancy, and it is found that the surface and volume energies are close to being equal, giving no visible difference in the fitting of the experimental data.", "Another fit was performed using $\\langle I^2 \\rangle {=} |I|(|I|+x)$ , with the added fit parameter $x$ .", "This addition is equivalent to introducing an adjustable Wigner term linear in isospin.", "Table REF shows the fit with and without letting $x$ vary.", "None of the other fit parameters change substantially.", "As for $x$ itself, it is found to be 1.51(3), which slightly lowers the $\\chi ^2$ of the fit.", "When written in the form presented by Myers and Swiatecki[3], this corresponds to a congruence energy of 41.3(8) MeV, which agrees with the value 42 MeV which they report.", "With most of the parameters changing less than 1%, the same physics is still captured by setting $\\langle I^2 \\rangle {=} |I|(|I|+2)$ .", "Table: Fits of the nuclear masses to the liquid drop model using different isospin dependencies.", "The first sets 〈I 2 〉=|I|(|I|+2)\\langle I^2 \\rangle {=} |I|(|I|+2), where as the second represents a fit to 〈I 2 〉=|I|(|I|+x)\\langle I^2 \\rangle {=} |I|(|I|+x).", "All parameters are in units of MeV.", "The value in the parentheses is the uncertainty in the last digit.Implications of the curvature term.", "The existence of a curvature energy, especially important in light nuclei, may imply effects hitherto undiscovered.", "We give here two examples.", "The curvature of the surface in the nuclear deformation landscape, and in particular at the fission saddle point, exhibits large variations going from positive to negative.", "Therefore, the prediction of fission saddle point configurations and masses will be affected by the presence of a curvature term, which will acquire a tensorial form.", "The fragment distribution predicted by the Fisher model [15] is dependent on the surface energy of the clusters.", "The theory uses a term proportional to $A^\\sigma $ for this purpose.", "Since the fragment yields are weighted heavily towards lighter fragments away from the critical temperature, the introduction of a curvature term would seem imperative.", "Thus, the curvature term could alter predictions of the critical temperature in an unknown way.", "Conclusion.", "Previous efforts have addressed the need of a curvature term in the liquid drop expansion of nuclear masses, but no consistent interpretation was made.", "Some works state that it is unnecessary, and that it is enough to stop the expansion at the level of a surface term [3].", "Other studies give conflicting results, and even the sign of the curvature correction remains ambiguous [4], [5], [6].", "Some of these references do give results that agree with the ones here, but do not offer a physical picture.", "We demonstrate that the surface energy coefficient in the traditional liquid drop formula changes when different mass ranges are considered.", "The decreasing trend in the surface energy coefficient with increasing mass number is consistent with the presence of a curvature term.", "We present a consistent description of the curvature term's nature, determine its sign and demonstrate its presence in the nuclear masses.", "Simple physical arguments predict that the volume and surface energy coefficients should be equal.", "Without the introduction of the curvature term, the volume and surface energy coefficients appear to differ from each other.", "With the addition of the curvature term, the two coefficients agree within error.", "The nature of the “Wigner” term linear with isospin is also considered.", "A slight change in the definition of the squared isospin, possibly quantum mechanical in nature, captures its relative magnitude without introducing an additional parameter.", "What is gained through these considerations is a streamlined physical picture of the liquid drop model.", "Consider the difference of the original liquid drop model in Eq.", "(REF ) to the final equation presented here in Eq.", "(REF ).", "Even though the latter appears more complicated, there are the same number of free fit parameters as the former.", "Instead of adding more and more terms to produce more and more exact representations of the nuclear masses, we have added a geometric physical picture and kept the same number of variables to obtain a more accurate result.", "The lessons learned with this equation are more telling than letting all the parameters free.", "Acknowledgments.", "This work was performed by by Lawrence Berkeley National Laboratory and was supported by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of Nuclear Physics, of the U.S. Department of Energy under Contract DE-AC02-05CH11231.", "This work also performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344." ] ]

[ [ "Critical Points of Correlated Percolation in a Gravitational Link-adding\n Network Model" ], [ "Abstract Motivated by the importance of geometric information in real systems, a new model for long-range correlated percolation in link-adding networks is proposed with the connecting probability decaying with a power-law of the distance on the two-dimensional(2D) plane.", "By overlapping it with Achlioptas process, it serves as a gravity model which can be tuned to facilitate or inhibit the network percolation in a generic view, cover a broad range of thresholds.", "Moreover, it yields a set of new scaling relations.", "In the present work, we develop an approach to determine critical points for them by simulating the temporal evolutions of type-I, type-II and type-III links(chosen from both inter-cluster links, an intra-cluster link compared with an inter-cluster one, and both intra-cluster ones, respectively) and corresponding average lengths.", "Numerical results have revealed objective competition between fractions, average lengths of three types of links, verified the balance happened at critical points.", "The variation of decay exponents $a$ or transmission radius $R$ always shifts the temporal pace of the evolution, while the steady average lengths and the fractions of links always keep unchanged just as the values in Achlioptas process.", "Strategy with maximum gravity can keep steady average length, while that with minimum one can surpass it.", "Without the confinement of transmission range, $\\bar{l} \\to \\infty$ in thermodynamic limit, while $\\bar{l}$ does not when with it.", "However, both mechanisms support critical points.", "In two-dimensional free space, the relevance of correlated percolation in link-adding process is verified by validation of new scaling relations with various exponent $a$, which violates the scaling law of Weinrib's." ], [ "Introduction", "Correlated percolation[1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14] is a useful theoretic model in statistical physics.", "It provides us with fundamental understanding of spread processes of message, disease and matter in nature and society.", "Linking probability between any two nodes in it takes the form of $p(r)\\sim r^{-a}$ , where $r$ is $d$ -dimensional distance between the nodes, and $a$ is a positive real number, namely, distance-decay exponent of links.", "Weinrib and Halperin[15] analytically studied whether the correlations change the percolation behavior or not.", "Weinrib[16] pointed out, for $a<d$ , the correlations are relevant if $a\\nu -2<0$ , where $\\nu $ is the percolation-length exponent for uncorrelated percolation; while for $a>d$ the correlations are relevant if $d\\nu -2<0$ .", "It is a generalization of the Harris criterion[17] appears earlier.", "Recently, network models referring to correlated percolation have gradually appeared[11].", "Achlioptas process(AP)[18] for link-adding networks, which is an attractive topic at present[19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], could be viewed as a new kind of correlated percolation if we put all nodes uniformly on a two-dimensional(2D) plane.", "Starting from a set of isolated nodes, two candidate links are put to nodes randomly at every time step, but only the link with smaller product $m_i m_j$ is retained, where $m_i$ (or $m_j$ ) is the mass (the number of nodes) of the cluster that node i(or j) belongs to, which is called Product Rule(PR).", "A link chosen with PR from both inter-cluster candidates is called a type-I link.", "When two candidate-links are of different types, i.e., one is an inter-cluster link, the other is a intra-cluster one, always the later is retained, and it is called a type-II link.", "While for both intra-cluster ones, the retained link is arbitrarily chosen no matter they are in the same or different clusters, and it is called a type-III link.", "Generally speaking, in the way of AP, network percolation is inhibited, which postpones the appearance of the threshold $T_c$ at which a giant component G starts to grow, and results in a sharp growth of G called an explosive percolation.", "In our point of view, if we put AP on a 2D plane, it gives rise to a new mechanism of long-range correlation for the nodes based on co-evolutionary growing masses of components they connected.", "The selective rule for topological links relies on mass-product instead of 2D geometric length of them, which prevents the property exhibited in the previous correlated percolation.", "And correlation feature in AP-type of percolation has not been revealed up till now.", "A recent model of ours[39] based on the observation of phenomena in different real systems[40], [41], , [43], [44], [45] describes another kind of correlated percolation in growing networks, which could be viewed as a overlapping of traditional correlated percolation with AP in a 2D space.", "The link-occupation function in the model takes the form $p(r)\\sim m_im_j/r_{ij}^a$ which looks like Newton's gravity rule.", "It resumes the classical Erdos-Renyi(ER) random graph model when exponent $a \\rightarrow \\infty $ , and it gives another extreme of AP when $a \\rightarrow 0$ .", "Different properties of such kind of new correlated percolation are expected, since ER random graph grows without any bias, AP takes a strong bias to inhibit network percolation independent of geometric distance, while the new model with gravity-like rule have some distance-related relax on such bias, which produces a new type of correlated percolation.", "In this paper, we report the simulation results on objective competition between type-I, II and III links in both gravity model[43], [45] and AP model, and we point out a new mechanism to support critical points, which bears the scaling relations revealed in our recent work.", "Different saturation effect is manifested, which distinguishes it from the traditional correlated percolations." ], [ "Model", "Suppose $N$ isolated nodes are uniformly scattered on a 2D plane.", "For convenience of calculating distance, the plane is discretized with a triangular lattice, each minimal edge with the length of two units for the convenience of algorithm.", "Each vertex of the triangles is occupied by a node so that we exclude all possible biases except link-adding rules.", "For any two pairs of nodes $i$ and $j$ possibly with the same product $m_i m_j$ , a type-I link connects the the pair with longer distance if both the links' ends hit the nodes belonging to different clusters; while a type-II link connects the nodes inside the same cluster if the other one is an inter-cluster link; a type-III one connects arbitrarily chosen pair of nodes if both candidate links are intra-cluster ones.", "Parallel to PR, we pick randomly two pairs [$(i,j)$ and $(k,l)$ ] of nodes in the plane at every time step.", "For the pair $(i,j)$ (and for $(k,l)$ likewise), we calculate the generalized gravity defined by $g_{ij} \\equiv m_i m_j/r^a_{ij}$ , where $r_{ij}$ is the geometric distance between $i$ and $j$ , and $a$ is an adjustable decay exponent.", "Once we have $g_{ij}$ and $g_{kl}$ , we have two choices in selecting which pair to connect.", "For the case of the maximum gravity strategy (we call it $G_{\\rm max}$ ) we connect the pair with the larger value of the gravity, e.g., the link $(i,j)$ is made if $g_{ij} > g_{kl}$ and the link $(k,l)$ otherwise.", "We also use the minimum gravity strategy ($G_{\\rm min}$ ) in which we favor the smaller gravity pair to make connection.", "The two strategies, $G_{\\rm max}$ and $G_{\\rm min}$ , lead the link-adding networks to evolve along the opposite paths of percolation processes.", "Generally speaking, $G_{\\rm max}$ facilitates the percolation process, whereas $G_{\\rm min}$ inhibits it.", "All such generalized gravity values are calculated inside the circular transmission range with the radius $R$ centered at one of nodes $i$ and $j$ as the speaking node [46] in a mobile ad hoc network[41], , [45].", "For the different limits of parameters $R$ and $d$ , we have three cases in the model.", "Case I: With the transmission range $R\\rightarrow \\infty $ , we have a generalized gravitation rule which is an extension  of widely used gravitation model [43] $(a=1)$ with the tunable decaying exponent $a$ .", "Case II: With the exponent $a=0$ , we assume that node pairs can be linked with PR topologically inside the transmission range with a limited radius $R$ .", "Case III: With both limited values of radius $R$ and exponent $a$ , we have the gravity rule inside the transmission range.", "It can describe the communication or traffics with constrained power or resources.", "For case I and case III in the model, three scaling relations have been found with large scale simulations.", "When strategy $G_{max}$ is adopted in 2D free space(case I), we have $C\\sim a^{-\\alpha } f( t a^{\\epsilon })$ where $t=(T-T_0)/T_0$ is dimensionless time-step with $T_0=0.78$ , $a$ is the decay exponent of connection probability.", "$\\alpha =0.01$ , $\\epsilon =0.20$ , and $f(x)$ is a universal function.", "When strategy $G_{min}$ is adopted inside the transmission range with radius $R$ (case III), we have $C\\sim (a/a_0)^{-\\theta } h[t(a/a_0)^{\\phi }]$ for certain parameter ranges of $a$ and $R$ , where $t=(T-T_0)/T_0$ , $T_{0}=1.0$ , $\\theta =0.005$ , $\\phi =-0.50$ , $a_0=0.5$ , and $h(x)$ is a universal function.", "In addition, when strategy $G_{max}$ for case III is adopted inside transmission range defined by $R$ , we have another scaling relation $C\\sim R^{-\\delta } H( t \\rho ^{\\eta })$ for $R>3$ , where $\\rho =(R-R_0)/R_0$ , $R_{0}=2$ , $\\eta =-0.10$ , $\\delta =-0.005$ , $T_0=1.0$ and $H(x)$ is a universal function.", "To understand three scaling relations above, we should look into the mechanism of the evolution processes underlying $C(T)$ .", "To see what happens in such critical points $T_{0}$ , and what are particular of them in certain link-adding processes, we count the temporal link fractions $F(T)$ , and calculate the average lengths $\\bar{l}$ of links which is defined as the summation of all lengths of links for a certain type over its number in a window $\\Delta T=20$ time-steps.", "By observation of the time-dependent behaviors of fractions of type-I, II and III links, new properties were found out for our gravity-like model together with AP producing explosive percolations." ], [ "3. Simulation results", "All simulations are carried out on the triangular lattice of the size $N = L\\times L$ with $L=32, 64, 128$ and 256, respectively.", "We simulate either of strategy $G_{\\rm max}$ or $G_{\\rm min}$ for either case I or III.", "The total number of links equating to that of time-steps is divided by $N$ , which is defined as $T$ .", "The mass of the largest component divided by $N$ makes up the observable $C$ , the node fraction of the largest component.", "All results presented in this work are obtained from 5000 different realizations of network configurations with $L=128$ if not specially indicated.", "Figure: (color online)Evolution of fractions oftype-I, type-II and type-III links in Achlioptas process on 2Dplane.Figure: (color online)Evolution of averagelengths of type-I, type-II and type-III links in Achlioptas processon 2D plane.Inspired by Cho and Kahng's [33] work and a referee of ref.", "[39], we have gone further by calculating fractions of three types of links and arithmetic average lengths of their links.", "Our attention was pointed at AP first.", "In Fig.1 we illustrated the evolution of fractions of 3 types of links.", "Just at the threshold the fraction of type-I$(F_{I})$ links has a sharp drop-down, meanwhile that of type-II $(F_{II})$ shoots up, crossing $F_I$ at $T_c=0.888$ .", "A little after it, $F_I$ crosses with growing fraction of type-III links ($F_{III}$ ) at the level $F_I=F_{III}=0.25$ , while $F_{II}$ gets its summit($F_{II}=0.5$ ) at the same point, which has not been concerned by previous works.", "However, it is this property that pervades all cases in the present correlated percolation.", "In Fig.2, the average lengths of type-II($\\bar{l}_{II}$ ) merges that of type-III($\\bar{l}_{III}$ ) at $T_c$ after an abrupt growth, $\\bar{l}_{II}$ starts to grow earlier than $\\bar{l}_{III}$ .", "The level of $\\bar{l}$ for both of them keep invariant for $T>T_{c}$ , while $\\bar{l}_{I}$ starts to decrease from $T_c$ .", "We see from both the figures that in explosive percolation the system undergoes a sharp transition from a type-I link dominant phase into a type-II and III dominant phase at $T_c$[33].", "Besides, average lengths undergo a parallel transition at the same point.", "Actually, 3 levels of $\\bar{l}$ go to infinity in dynamic limit from finite size scaling transformation(not shown).", "Figure: (color online)Evolution offractions of type-I, type-II and type-III links with G max G_{max} andprobability decay exponent a=0.2,0.5,1.0,2.0,3.0and5.0a=0.2,0.5, 1.0, 2.0, 3.0 and 5.0 incase I (R→∞R\\rightarrow \\infty ).Figure: (color online) Evolution ofaverage lengths l ¯\\bar{l} of type-I, type-II and type-III links withG max G_{max} and probability decay exponent a=0.2,0.5,1.0,2.0,3.0and5.0a=0.2,0.5, 1.0, 2.0, 3.0and 5.0 in case I.Figure: (color online) Node fractionC(T)C(T) of the largest component with strategy G max G_{\\rm max} and thesame parameters in Fig.3.Now we turn to the fraction of 3 types of links in case I of the present gravity model[39].", "With strategy $G_{max}$ in free 2D space, we have scaling relation (1) for distance-decay exponent $a \\in [0.2, 2.0]$ .", "Fig.3 shows the evolution of fractions corresponding to it.", "Curves for $F_I$ cross those of $F_{II}$ with $a=0.5, 1.0$ and $2.0$ around $T=0.78$ quite clearly, even those with $a=0.2$ barely cross near it.", "But fractions for $a=3.0$ and $5.0$ shift the cross point rightward obviously.", "On the other hand, the curves for $F_{I}$ cross $F_{III}$ at $T=1.0$ for almost all values $a$ except $a=0.2$ .", "To determine which one would be the candidate of another critical point, we cast ourselves on the assistance from the observation of average lengths of links.", "In Fig.4 $\\bar{l}_{II}(T)$ merges $\\bar{l}_{II}(T)$ for $a=3.0$ and $5.0$ at $T=1.0$ , separating themselves from others.", "However, almost all other ones collect at $T=0.78$ , which gives hint to us for $T_0$ .", "(Here a better resolution is needed in further calculation).", "Correspondingly, in Fig.5 for scaling relation (1) with exponents $\\epsilon =0.2$ , $\\alpha =0.01$ , and $T_{0}=0.78$ , $C(T)$ for $\\alpha \\in [0.5, 2.0]$ collapse into the universal function very well, with that for $a=0.2$ barely collapsing onto it.", "But those $C(T)$ for $a=3.0$ and $5.0$ do not behave well in collapse.", "The separation from others at the turning middle part indicates the deviation of their $T_0$ from $0.78 $ with which others share.", "In the description of the average lengths for case I with $G_{max}$ , simulated results $\\bar{l}(T)$ in Fig.4 with all values $a$ demonstrate the same steady level ($\\bar{l}\\simeq 131.50$ for $L=128$ ).", "Variation of parameter $a$ only shifts starting points of up-growing $F_{II}$ and $F_{III}$ as $a$ increases.", "The saturation effect of large decay exponents $(a=3.0, 5.0)$ appears clearly and is shown by dash lines, which demonstrates the inheritance from traditional correlated percolation[16].", "In this case with $G_{max}$ , special level of fractions at cross point of $F_{I}(T)$ and $F_{III}(T)$ keeps 0.25 just as in AP without any distance-decay included, so does $F_{II}$ at hiking its summit.", "Figure: (color online)(a)Susceptibilityχ(T)\\chi (T) of the system with G max G_{max} in Case I. L=32,64,128L=32, 64, 128and 256.", "(b)Percolation thresholds T c T_c for a=0.2,0.5,0.8,1.2,2.0,3.0a=0.2, 0.5, 0.8, 1.2,2.0, 3.0 and 5.05.0 with G max G_{max} in Case I.Figure: (color online)Size-dependentaverage lengths l ¯\\bar{l} of type-I, type-II and type-III links withstrategy G max G_{\\rm max} in case I. Parameters are the same as inFig.6As in a usual way, we determine the critical point $T_c$ of percolation by observation of tips of susceptibility $\\chi (T)$ (Fig.6).", "Comparing $T_c$ in Fig.6 with $\\bar{l}(T)$ in Fig.4, we find that these $T_c$ approximately hit the horizontal coordinates of middle point of growing fraction of type-II links, which means that $T_c$ is the transition point from the inter-cluster-link dominant phase to the intra-cluster-link dominant phase.", "Besides $T_c$ , we have another (sub)-critical point $T_0$ which is in certain range independent of decay exponent $a$ in gravity model, and $T_0$ indicates the balance between the fractions of type-I and type-II links, yielding a new scaling behavior of $C(T,a)$ in formula (1) not revealed by previous works.", "Moreover, the steady level $\\bar{l} = 131.5$ is always $a-$ independent, type-independent which takes the inherited value of that in AP.", "Actually, 131.5 is the value for L=128 only.", "We have size effect since a free boundary condition instead of a periodic one is adopted.", "The finite size effect is shown in Fig.7 which gives that $\\bar{l}\\sim L $ , i.e., $\\bar{l} \\sim N^{1/2}$ where $N$ is the number of nodes on the 2D plane.", "Hopefully it goes towards infinity in thermodynamic limit.", "However, the finite size effect of $C(T,N)$ (Fig.8 for an example) is not strong enough for us to identify the scaling exponents $\\nu , \\beta $ as usual.", "Therefore, we can check the validation of scaling laws presented by Weinrib [16] for correlated percolation in the present model only by rescaling susceptibility $\\chi (a,T)$ .", "Fig.", "9 illustrates the results of it for examples $a=0.5$ and $2.0$ , respectively.", "We have scaling relation Figure: (color online)Node fractionC(T)C(T) of the largest component with strategy G max G_{\\rm max} in CaseI.", "(a)a=0.5a=0.5; (b)a=2.0a=2.0.", "N=32,64,128N=32, 64, 128 and 256.Figure: (color online) Scaling ofsusceptibility χ(T)\\chi (T) in case I for (a) a=0.5a=0.5; (b) a=2.0a=2.0.N=32,64,128N=32, 64, 128 and 256.$\\chi \\sim N^{\\gamma /\\nu }G((T-T_c)N^{1/\\nu })$ where $1/\\nu =0.3$ , $\\gamma /\\nu =0.75$ , and $G(x)$ is a universal function.", "With these values of scaling exponents, the scaling law $\\nu _{long}=2/a$ is not applicable to the present model, where $\\nu _{long}$ is correlation-length exponent in the long-range case.", "With power-law form for the correlation function g(r), Weinrib had derived the extended Harris criterion: the long-range nature of the correlations is relevant if $a\\nu -2 < 0$ , which means the correlations change the percolation critical behavior.", "It has been violated since now they all behave differently from traditional short range percolation in a 2D triangular lattice ($\\nu =4/3$ ) and the correlations are relevant no matter $a \\nu -2$ is less(Fig.9(a)) or larger(Fig.9b) than zero.", "This is because in strategy $G_{max}$ we have overlapped the power-law correlation function g(r) with AP which is another kind of autocorrelation process with positive feed back effect of mass-growing.", "Figure: (color online)Evolution offractions of type-I, type-II and type-III links with G min G_{min} andprobability decay exponent a=0.2,1.2,2.0a=0.2,1.2,2.0 and 5.05.0 in case I.L=256L=256.Figure: (color online)Evolution ofaverage lengths of type-I, type-II and type-III links with G min G_{min}and probability decay exponent a=0.2,1.2,2.0a=0.2,1.2,2.0 and 5.05.0 in case I.L=256L=256.For the strategy $G_{min}$ which prefers smaller gravity, it tends to retain a longer link under the comparison of the same product of masses $m_{i}m_{j} $ .", "That is to say, long range links have predominance.", "Evolution of $F_{I}(T)$ and $F_{II}(T)$ links do not cross at any common point.", "In Fig.10, the cross points for $F_{I}$ and $F_{II}$ shift leftward from $T_c$ of AP as decay exponent $a$ increases, which means that $G_{min}$ as a correlated percolation mechanism weaken the explosive effect caused by AP.", "But the starting position of fraction-II still provides hints of thresholds $T_c$ .", "However, it is hard to locate a common cross point for $F_{I}$ and $F_{III}$ in a range of $a$ .", "Therefore, we have no scaling relation for them.", "The steady average lengths of links (Fig.11) have the same level of $G_{max}$ and AP cases, and size-effect (not shown) also tells the divergence of $\\bar{l}$ , but all of them do not tell any possible hint for critical points.", "Generally speaking, strategy $G_{min}$ facilitates longer links for certain geometric distribution of clusters or nodes.", "In the evolution, strategy $G_{min}$ emphasis the assignments for different types of links, encourage longer and intra-cluster links.", "Humps above the steady $F_{II}$ and $F_{III}$ implies out-of-pace growing of link lengths of $F_{I}$ .", "That is, $F_{I}$ surpasses the growing speed of the giant component.", "Here, it is the geometric distance-dependent strategy that makes $G_{min}$ alleviate effect of AP.", "With smaller $a$ (e.g., a=0.2 in Fig.11)the strategy has the opportunity to exhaust long links before $T_c$ ; while with middle values of $a$ (e.g., $a=1.2$ and $2.0$ ) it may take longer time to exhaust them.", "However, with too large $a$ (e.g., $a=5.0$ ), $G_{min}$ fall off quicker than the natural dimension, we can only see the AP-type short-range effect of saturation.", "In this limit, i.e., $a\\le 3.0$ , percolations are no longer relevant, which causes saturation of curves $C(T)$ in Fig.1b [39].", "However, we should not expect the short-range percolation exponent $\\nu =4/3$ of correlation length here for 2D triangular lattice, since AP has been included in the strategy $G_{min}$ .", "Figure: (color online)Evolution offractions of type-I, type-II and type-III links with strategyG min G_{min} and probability decay exponent a=0.2,0.5,1.0,1.5,2.0,3.0,5.0a=0.2, 0.5, 1.0, 1.5, 2.0,3.0, 5.0 and 10.010.0 in case III.", "R=4R=4.Figure: (color online)Evolution ofaverage lengths of type-I, type-II and type-III links with strategyG min G_{min} and probability decay exponent a=0.2,0.5,1.0,1.5,2.0,3.0,5.0a=0.2, 0.5, 1.0, 1.5, 2.0,3.0, 5.0 and 10.010.0 in case III.", "R=4R=4.Figure: (color online)Scaling of nodefractions C(T)C(T) with strategy G min G_{min}, a=1.0,2.0a=1.0, 2.0 and 3.03.0inCase III.", "R=4R=4.Figure: (color online)Pseudo-scaling ofnode fractions C(T)C(T) with strategy G min G_{min}, a=0.2,0.5,1.0,1.5,2.0,3.0,5.0a=0.2, 0.5, 1.0,1.5, 2.0, 3.0, 5.0 and 10.010.0 in Case III.", "R=4R=4.Figure: (color online)Evolution offractions of type-I, type-II and type-III links with strategyG min G_{min} and probability decay exponent a=0.2,0.5,1.0,1.5,2.0,3.0a=0.2, 0.5, 1.0, 1.5, 2.0,3.0 and 5.05.0 in case III.", "R=8R=8.The distinct feature in case III for both $G_{min}$ and $G_{max}$ is that the candidate links are selected not only by comparing gravities, but also constrained inside a transmission range $R$ ($r=2R$ in geometric distance), which ruins the effect comes from the divergence of average link lengths.", "It is well known that all possible singularities at critical points come from the singularity of correlation length.", "However, here no length could goes to infinity in any way, which ruins possible common cross point relies on the balance between $F_{I}$ and $F_{II}$ as in the 2D free space (case I of the present model)and induces the possibility to yield novel scaling relation other than any previous ones.", "For possible critical point, we seek help from the evolution of link fractions of 3 types.", "Fig.12 shows the behaviors of $F_{I}$ , $F_{II}$ and $F_{III}$ with $G_{min}$ for all simulated distance-decay exponents $a$ and $R=4$ .", "Correspondingly, Fig.13 shows the behaviors of $\\bar{l}_{I}$ , $\\bar{l}_{II}$ and $\\bar{l}_{III}$ the the same set of parameters.", "Critical point $T_{0}=1.0$ ($T_{0}=0.99$ to be precise) distinguishes itself from others by intuitive observation.", "The cross point for $a=5.0$ and $a=10.0$ go rightward from which others share, which means they could not share the same $T_{0}$ hence the same scaling relation with other decay exponents.", "Besides, in rescaling process for $C(T,a)$ , curves for $a=0.2$ and $a=0.5$ failed in collapse, because smaller transmission range $R$ inhibits the effect of slower(long-range) decay for connection probability.", "The rescaled function $C(t)$ for $R=4$ is shown in Fig.14.", "It seems that we could go further with $\\phi =-1.5$ to include more exponents $a$ in the scaling as shown in Fig.15.", "However, it is meaningless in physics due to above mentioned reasons.", "Actually, scaling behaviors are R-dependent, but exponents $\\phi $ and $\\theta $ need not to vary.", "The variation of $R$ only shifts $T_{0}$ as illustrated in Fig.16 for $R=8$ .", "Therefore, we keep $\\phi =-0.5$ , $\\theta =0.005$ for all values of $R$ with $G_{min}$ , but take $T_{0}=0.99$ for $R=4$ , $T_{0}=0.92$ for $R=8$ , and so on.", "The humps above the steady level of $\\bar{l}_{II}$ and $\\bar{l}_{III}$ (all independent of exponents $a$ ) come from similar mechanism as in free 2D space but now at much lower level constrained by transmission radius $R$ , and they are independent of size $L$ of the system.", "Figure: (color online)Evolution of fractions of type-I, type-II andtype-III links with strategy G max G_{max} and probability decayexponent a=2.0a=2.0 for R=4,6R=4,6 and 8 in case III.", "L=256L=256.", "500realizations of network configurations.Figure: (color online)Evolution offractions of type-I, type-II and type-III links with strategyG max G_{max} and probability decay exponent a=5.0a=5.0 for R=3,4,6R=3, 4, 6 and8 in case III.Figure: (color online)Evolution ofaverage lengths of type-I, type-II and type-III links with strategyG max G_{max} and probability decay exponent a=2.0a=2.0 for R=4,6R=4,6 and 8in case III.", "L=256L=256.", "500 realizations of network configurations.Figure: (color online)Evolution ofaverage lengths of type-I, type-II and type-III links with strategyG max G_{max} and probability decay exponent a=5.0a=5.0 for R=3,4,6R=3, 4, 6 and8 in case III.Figure: (color online)Scaling of nodefractions C(T)C(T) with strategy G max G_{max}, a=2.0a=2.0 for R=4,6R=4, 6 and8 in Case III.L=256L=256.", "500 realizations of networkconfigurations.Figure: (color online)Scaling of nodefractions C(T)C(T) with strategy G max G_{max}, a=5.0a=5.0 for R=3,4,6R=3, 4, 6and 8 in Case III.Scaling relation (3) for case III with strategy $G_{max}$ inside transmission range with radius $R$ is checked for various decay exponents $a$ and for different sizes (L=32, 64, 128 and 256).", "Its validation is independent of size $L$ simulated.", "In Fig.17 and Fig.18, the evolution of $F_{I}$ , $F_{II}$ and $F_{III}$ for both $a=2.0$ and $a=5.0$ behave much similarly.", "$F_{I}$ and $F_{II}$ cross at the level a little bit lower than $0.45$ , while $F_{I}$ crosses $F_{III}$ at the level $0.25$ , which keeps the same as in all previous cases.", "The changes of exponent $a$ and $R$ only shift fractions along horizontal direction of figures, i.e., to change starting points and growing/dropping speed instead of levels of them.", "However, $T_0=1.0$ keeps as their common fixed point for $F_{I}$ and $F_{III}$ to cross.", "In Fig.19 and Fig.20, $F_{I}$ and $F_{III}$ for both $a=2.0$ and $a=5.0$ arrive at the same level hitting $T_{0}=1.0$ , which distinguishes this point from totally 3 cross points, and makes up a candidate of critical point $T_0$ for scaling relations.", "The scaling exponents $\\eta =-0.10$ and $\\delta =-0.005$ have been checked for lower values of parameter $a$ ($0.5\\le a \\le 3.0$ , Fig.21).", "However, for $a=5.0$ , we have to choose a new set of exponents: $\\eta =-0.25$ and $\\delta =-0.01$ (Fig.22).", "It is not strange that steady levels of $\\bar{l}$ keep unchanged for certain $R$ , independent of $a$ or $L$ , just as that with $G_{min}$ in case III.", "$\\bar{l}$ inside a circle defined by $R$ can not go to infinity under any circumstance, but still support a critical point, which distinguishes correlated percolation in case III from case I and traditional models.", "It deserves further investigation." ], [ "Discussion and Conclusions", "In this paper, we have proposed a new network model of correlated percolation in which geometric distance-dependent power-law decay connection probability overlaps Achlioptas process to form a gravity model.", "It can be tuned to facilitate or inhibit percolation with strategy $G_{max}$ or $G_{min}$ , cover a wide range of thresholds $T_c$ , yield a set of new scaling relations.", "And it provides a scheme for better description of practical processes in complex systems.", "We have developed a new approach to find out candidate critical points with physical meanings other than that of traditional ones.", "There are objective competition and balance between type-I and type-II, type-I and type-III links, meanwhile, competition of average lengths between type-II and type-III links.", "Along this line threshold $T_c$ is found to overlap the balance point between factions $F_I$ and $F_{II}$ in the explosive percolation of Achlioptas process, and the steady average lengths of three types of links are all divergent to infinity in thermodynamic limit.", "The percolation is indeed a transition from type-I link dominant phase to type-II and type-III dominant phase.", "By observing evolutions of fractions of type-I, type-II and type-III links, a candidate critical point can be chosen combined with the message on evolutions of average lengths of them.", "With strategy $G_{max}$ in 2D triangular lattice, fraction $F_{I}$ get balance with $F_{II}$ , makes up a critical point $T_0$ which supports scaling relation (1) in case I of the model.", "With strategy $G_{min}$ inside certain transmission range with radius $R$ , a duet balance exists for $F_{I}$ and $F_{III}$ meanwhile $\\bar{l}_{II}$ and $\\bar{l}_{III}$ , makes up another critical point $T_0$ which supports scaling relation (2) in case III.", "With strategy $G_{min}$ and certain range $0.5 \\le a \\le 3.0$ of decay exponent $a$ , again a duet balance exists for $F_{I}$ and $F_{III}$ meanwhile $\\bar{l}_{II}$ and $\\bar{l}_{III}$ , makes up another critical point $T_0$ which supports scaling relation (3) for a mini-scale of $R$ in case III.", "This approach serves an assistant tool in seeking critical points of order parameter $C(T)$ which is usually not easy to determine in an intuitive way.", "In numerical calculations, besides percolation threshold $T_c$ , two fixed points, $T_0=0.78$ and $T_0=1.0$ emerge as distinct points not only for special temporal crux but also for unchanged levels of $F_I$ , $F_{II}$ and $F_{III}$ inherited from AP, which is expected to be further proved in analytical ways.", "However, they have different physical meanings.", "The former corresponds to a divergent average length of links, while the later corresponds to confined average lengths by transmission range $R$ , which distinguishes itself from traditional critical points in percolations.", "Correlated percolations are relevant since long-range correlation drastically changes the critical properties.", "The validation ranges of decay exponents $a$ with various strategies in different cases define the relevance of correlation.", "They have demonstrated novel scaling relations different from traditional 2D short-range percolation in triangular lattice.", "The intervention of distance-dependent power-law decay ingredients alleviates the explosive effect of percolation transition by horizontal adjustment of evolutions along temporal axis, separates $T_0$ from $T_c$ , while the overlapped AP included in the present gravity model always conquers the vertical levels of three fractions and average lengths, which are found neither in traditional correlated percolations of continuities in 2D space nor in complex networks.", "Moreover, the node fraction $C(T)$ of the largest component, fractions $F(T)$ , and average lengths $\\bar{l}(T)$ of three types of links all show saturation phenomena as pointed out by Weinrib but with different values of exponent of $a$ since now AP overlaps in the present gravity model.", "And scaling law of Weinrib is no longer obeyed according to the evidence of numerical results of average-length exponents.", "We are indebt to anonymous referees for stimulating comments.", "Zhu thanks H. Park, P. Holm, X.-S. Chen and Z.-M. Gu for useful discussion.", "We acknowledge financial support from National Natural Science Foundation of China (NNSFC) under Grants No.", "11175086, 10775071 and 10635040.", "Kim was supported by the National Research Foundation of Korea (NRF) funded by the Korea government (MEST) under Grant No.", "2011-0015731." ] ]

[ [ "Late time evolution of the gravitational wave damping in the early\n Universe" ], [ "Abstract An analytical solution for time evolution of the gravitational wave damping in the early Universe due to freely streaming neutrinos is found in the late time regime.", "The solution is represented by a convergent series of spherical Bessel functions of even order and was possible with the help of a new compact formula for the convolution of spherical Bessel functions of integer order." ], [ "Introduction", "Thorough analysis of cosmic microwave background (CMB) radiation provides a unique test of a standard inflationary cosmological model [1], [2], [3], [4], [5], [6].", "While scalar fluctuations of CMB serve as an invaluable source for exploring density of matter and radiation and large-scale structure of the universe, [7], [8], [9], [10], [11], [12], observations of tensor fluctuations of CMB open a window for searching after a signature of gravitational waves [13], [14], [15], [16].", "The CMB observations done by Wilkinson Microwave Anisotropy Probe (WMAP) [17] generally support theoretical predictions based on the standard inflationary cosmological model.", "The detailed analysis of the experimental data gives more and more accurate values [18], [19], [20] for the most valuable cosmological parameters such as baryon density, total matter density, Hubble constant, and age of the Universe.", "Independently of WMAP measurements, there is a long quest for a direct observation of cosmological gravitational waves [21].", "Specially designed for this task Laser Interferometer Gravitational Wave Observatory (LIGO) puts a major effort in this experimental challenge [22].", "A direct observation of cosmological gravitational waves would serve as a decisive test for validity of the Einstein general theory of relativity in the same way as the Michelson–-Morley experiment served as a major proof of the Einstein special theory of relativity.", "As in the experimental case, cosmological tensor fluctuations pose a challenge also from a theoretical side [23].", "Following S. Weinberg [23] we argue that “The particles of both the cold dark matter and baryonic plasma move too slowly to contribute any anisotropic inertia.", "In tensor modes there are no perturbations to densities or streaming velocities, so there are no perturbations to either the cold dark matter or baryonic plasma that need to be followed here.\"", "Therefore the only contributions to the anisotropic inertia tensor are due to photons and neutrinos.", "Further simplification comes from the following argument by S. Weinberg [23], [24], [25]: “The anisotropic inertia tensor is the sum of the contributions from photons and neutrinos, but photons have a short mean free time before the era of recombination, and make only a small contribution to the total energy density afterwards, so their contribution to the anisotropic inertia is small.", "This leaves neutrinos (including antineutrinos), which have been traveling essentially without collisions since the temperature dropped below about $10^{10}$ K, and which make up a good fraction of the energy density of the universe until cold dark matter becomes important, at a temperature about $10^{4}$ K. The tensor part of the anisotropic inertia tensor is given by l ij(u)=-4(u)0uK(u-U)hij'(U)dU, so the gravitational wave equation l d2dt2hij+3aaddthij-a2hij=16Gij, now becomes an integro-differential equation: l d2dt2hij+3aaddthij-a2hij= -64G(u)0uK(u-U)hij'(U)dU.\"", "The impact of neutrino source on gravitational wave damping has been thoroughly considered [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33].", "In this paper we report an analytical solution for the damping of gravitational waves in the early Universe due to freely streaming neutrinos, eq.", "(), in the late time regime, $u\\gg {Q}\\gg {1}$ .", "The solution is represented by an infinite series of spherical Bessel functions of even order.", "First we shall explain each term in the introduced equations (-)." ], [ "Notations", "We are interested in the time evolution of $h_{ij}(\\textbf {x},t)$ that is the tensor perturbation to the metric $g_{\\mu \\nu }$ : lll g00=-1, gi0=0, gij(x,t)=a2(t)[ij+hij(x,t)], where $a(t)$ is the time–dependent Robertson–-Walker scale factor.", "The kernel $K(u)$ in eq.", "() is represented by the sum of three spherical Bessel functions $j_{n}(u)$ , lll K(u)=115j0(u)+221j2(u)+135j4(u).", "The anisotropic stress tensor $\\pi _{ij}(u)$ is obtained from the solution of the Boltzmann equation for freely streaming neutrinos [24], [25] which defines the stress–-energy tensor for the Einstein field equations.", "The functions $\\overline{\\rho }_{\\nu }(u)$ and $\\overline{\\rho }_{\\gamma }(u)$ give the unperturbed equilibrium neutrino and photon energy density, correspondingly, which define the ratio: lll f(0)=+= 3(78)(411)4/3 1+3(78)(411)4/3 0.40523.", "The variable $u$ is the product of the wave number $k$ and the conformal time, lll u(t)=k0tdt'a(t').", "The boundary condition to eq.", "() is lll h'ij(0)=0, and it is assumed that we can parameterize the tensor $h_{ij}(u)$ as lll hij(u)=hij(0)(u).", "Introducing the dimensionless quantity lll ya(t)aeq, aeq=a(t=teq), where $t_{eq}$ is the time of matter-radiation equality, the general eq.", "() can be written as [24] lll (1+y)d2(y)dy2+( 2(1+y)y+12 )d(y)dy+Q2(y)= -24 f(0)y20y K(2Q(1+y-1+y'))d(y')dy'dy', with the boundary conditions lll (0)=1, .d(y)dy|y=0=0.", "Eq.", "() can be further simplified by the change of variable, lll y=u(u+4Q)4Q2, into lll d2(u)du2+(4(u+2Q)u(u+4Q))d(u)du+(u)= -24 f(0)(4Qu(u+4Q))20uK(u-U)d(U)dUdU.", "Here $Q$ is defined by the ratio of the wave number to its value at the time of matter-radiation equality, lll Q=2kkeq, keq=aeqHeq, H(t)=a(t)a(t)." ], [ "Time evolution of the gravitational wave damping", "In the late time regime, $u\\gg {Q}\\gg {1}$ , the general eq.", "() simplifies into lll d2(u)du2+4ud(u)du+(u)= Cu40uK(u-U)d(U)dUdU, where $\\hat{C}\\equiv {-24 f_{\\nu }(0)(4Q)^{2}}$ .", "Below we present the analytical solution for eq.", "() together with boundary conditions ().", "Clearly we need to find a specific function that would “absorb\" all derivatives of $u$ and all powers of $u$ on the left hand side of eq.", "().", "For the differential operator that appears in the left hand side of eq.", "(), lll L=u4(d2du2+4uddu+1), these conditions can be satisfied with the function lll fn(u)=n (n+3)2n-1[jn-2(u)+jn(u)]+(n-2) (n+1)2n+3[jn(u)+jn+2(u)].", "Applying the differential operator () to the function () we obtain a single spherical Bessel function lll L[fn(u)]=n(n-2)(n+1)(n+3)(2 n+1)jn(u) which is exactly what we are looking for.", "Therefore we can look for the solution of eq.", "() in terms of the expansion: lll (u)=n=0cn(n (n+3)2n-1[jn-2(u)+jn(u)]+(n-2) (n+1)2n+3[jn(u)+jn+2(u)]).", "The left hand side of eq.", "() transforms into lll n=0n(n-2)(n+1)(n+3)(2 n+1)cnjn(u).", "The regular at the origin solution for the homogeneous part of eq.", "(), lll ”0(u)+4u'0(u)+0(u)=0, is the sum of the two spherical Bessel functions, lll 0(u)=j0(u)+j2(u).", "The homogenous part $\\chi _{0}(u)$ of the general solution $\\chi (u)$ can be already seen as a linear combination of the first two terms in the expansion ($\\ref {ex1}$ ) for $n=0$ and $n=2$ .", "Figure: Graphical representation for theupper but one triangular structure of the matrix B 2k,2l B_{2k,2l}, eq.", "().The matrix indices ll and kk define a position of the matrix element in the xyxy-plane, whilealong zz-axe we plot its value.The right hand side of eq.", "() is represented by the convolution of the kernel () with the first derivative of the unknown function $\\chi (u)$ which we are looking for in terms of a series ().", "Clearly one needs a mathematical tool that relates a convolution of spherical Bessel functions to a series of those.", "In Appendix A we prove a useful formula for the convolution of spherical Bessel functions, that is not presented in the mathematical literature: lll Jn,m(u)0udUjm(u-U)jn(U)= =4(-i)n+m2i l=0(2l+1)iljl(u) ( L=|l-m|L-n-10L=l+m l,0,m,0|L,02L+L2-n (1+n) +L=|l-n|m-L-10L=l+n l,0,n,0|L,02L+L2-m (1+m) ), where $\\langle {l,0,m,0|L,0\\rangle }$ are the Clebsch–Gordan coefficients.", "The right hand side of eq.", "() is lll C0uK(u-U)d(U)dUdU= C l=0 levenjl(u)k=0 kevenl+2Bk,lck.", "Here the matrix $B_{k,l}$ is generated by the convolution of the first derivative of the function $\\chi (u)$ , lll '(u)=ncn( jn-3(u)(-2+n) n (3+n)(-3+2 n) (-1+2 n) +jn-1(u)n (-3-4 n+n2)(-3+2 n) (3+2 n)+.", ".", "jn+1(u)(-1-n) (2+6 n+n2)(-1+2 n) (5+2 n)+ jn+3(u)(-1-n) (-2+n) (3+n)(3+2 n) (5+2 n) ), with the kernel () by means of eq.", "(), and for integer $k\\in [0,10]$ and $l\\in [0,10]$ is lll B2k,2l ( ccccccccccc 0 115 110 13350 174900 -13780 119404 -291981980 1193050 -3717381650 4142031990 0 -13 -12 -1370 -17980 1756 -519404 29396396 -138610 373476330 -418406398 0 415 755 -567117325 -1052344100 -73313860 -1453640332 -2977297220 1735855850 -172791095043950 658878322334020 0 0 1855 7933465 -30138820 -1350741580 -207723716 -1247220220 3531351350 -114756156100 -80773627684060 0 0 0 88225 68323209475 -661117955 -13433234 -365263730060030 -4375681491891400 522924710220410200 -2453033530687160 0 0 0 0 2657 16613933 -81703204930 -45208398918910 -39881257400 -677217295631654600 174227819231512200920 0 0 0 0 0 1223 74315143451 -244553567567 -302161504504 -62891453334486152 -958782956355236888 0 0 0 0 0 0 238405 7010171142505 -23558935077800 -57738138353800 -701153331744440 0 0 0 0 0 0 0 304465 1153916275 -119899240975 -14348771831410 0 0 0 0 0 0 0 0 1825 3999749725 -14059732645370 0 0 0 0 0 0 0 0 0 92117 602003669123 ) We should notice an unpleasant feature of the matrix of coefficients (): the first row up to a factor $k=-{5}$ is identical to the second row.", "This is a direct response to the symmetry of the introduced function $f_{n}(u)$ , eq.", "().", "The functions $f_{n}(u)$ for $n=0$ and $n=2$ are exactly the same up to the factor $k=-{5}$ , lll f0(u)= -23(j0(u)+j2(u)), f2(u)= 103(j0(u)+j2(u)).", "Therefore the rank of the matrix () is ${\\tt Rank}[B]=N-1$ .", "This is a real obstacle because it leads to inconsistency with the boundary conditions.", "Indeed, the boundary conditions () are met if we set lll -23c0+103c2=1.", "On the other hand, the linear dependence of the matrix is equivalent to lll -23c0+103c2=0.", "In order to avoid this unpleasant feature we can start summation in the series () from $n=2$ instead of $n=0$ which is equivalent to setting lll c00, and thus the boundary conditions () are met if we set lll c2=310.", "Absence of the $j_{2}(u)$ term in the left hand side of eq.", "() leads to a restriction on the first coefficients in eq.", "(): lll -13 c2+415 c4=0.", "Finally we get the system of linear equations lll n(n-2)(n+1)(n+3)(2 n+1)cn=Ck=0kevenn+2Bk,nck that returns solution (for even $n$ and $k$ ) lll cn+2=n(n-2)(n+1)(n+3)(2 n+1)cn-Ck=0kevennBk,nckCBn+2,n.", "In the limit $Q\\gg {1}$ and owing to $\\hat{C}\\equiv {-24 f_{\\nu }(0)(4Q)^{2}}$ we have $Q$ -independent solution: lll cn+2=-k=0kevennBk,nckBn+2,n, and therefore for $n\\in {[0,9]}$ we have $c_{2n}=\\left\\lbrace 0,\\frac{3}{10},\\frac{3}{8},\\frac{5}{16},\\frac{35}{128},\\frac{63}{256},\\frac{231}{1024},\\frac{429}{2048},\\frac{6435}{32768},\\frac{12155}{65536}\\right\\rbrace $ which completes our series solution eq.", "().", "Figure: Late time evolution of the gravitational wave damping χ(u)\\chi (u) in the early Universe" ], [ "Conclusion", "We have analyzed the problem of gravitational wave damping in the early Universe due to freely streaming neutrinos in the late time regime $u\\gg {Q}\\gg {1}$ .", "As in the opposite limit $u\\ll {Q}$ [26], the solution is represented by a convergent series of spherical Bessel functions of even order and is independent of the $Q$ -value.", "Thus we conclude that the problem gravitational wave damping in the early Universe due to freely streaming neutrinos is completely solved in an analytical way in both early and late time limits." ], [ "Acknowledgements", "I am thankful to W.W. Repko and V.G.", "Zelevinsky for an encouragement and many helpful suggestions.", "I am indebted to N.O.", "Birge, J.T.", "Linnemann, C. Schmidt, M. Shapiro, and A.L.", "Volberg for valuable remarks.", "Here we derive the convolution integral () of spherical Bessel functions for integer orders $n,m$ .", "Theoretical efforts for such a convolution integral were put forward in [26] but here we report a new compact formula which has a clear exchange symmetry ($n{\\longleftrightarrow }m$ ) and can be readily applied for a practical calculations.", "Starting with the integral lll Jn,m(u)0udUjm(u-U)jn(U) we prove ().", "First, we represent a spherical Bessel function as a Fourier transformation of the Legendre polynomial $P_{n}(z)$ , lll jn(u)=(-i)n2-11ds(ius)Pn(s).", "Substitution of (REF ) into (REF ) leads to lll Jn,m(u)=(-i)n+m4-11ds-11dt(iut)Pm(s)Pn(t) 0udU(iUs-iUt)= = (-i)n+m4i -11ds-11dtPm(s)Pn(t)(ius)-(iut)(s-t).", "Now we employ the Legendre function of the second kind $Q_{n}(z)$ defined as lll Qn(z)=12-11dz'Pn(z')z-z'.", "Performing the integrations over $t$ and $s$ we obtain lll (-i)n+m4i -11ds-11dtPm(s)Pn(t)(ius)-(iut)(s-t)= = (-i)n+m2i-11dt(iut) [Pn(t)Qm(t)+Pm(t)Qn(t)].", "Further we reincarnate spherical Bessel functions by decomposing plane waves in terms of Legendre polynomials: lll (iut)=l=0(2l+1)iljl(u)Pl(t), which leads to lll (-i)n+m2i-11dt(iut) [Pn(t)Qm(t)+Pm(t)Qn(t)]= = (-i)n+m2i l=0(2l+1)(-i)-ljl(u) -11dtPl(t) [Pn(t)Qm(t)+Pm(t)Qn(t)].", "The angular momentum coupling simplifies the product of Legendre polynomials, lll Pl(x)Pm(x)=L=|l-m|L=l+ml,0,m,0|L,02PL(x), in terms of the Clebsch–-Gordan coefficients $\\langle {l,0,m,0|L,0\\rangle }$ .", "Introducing lll Wm-1(z)=2k=0[m-12](m-2k-1)(2k+1)(m-k)Pm-2k-1(z), and using the analog of eq.", "(REF ), lll Pl(z)Qm(z)=L=|l-m|L=l+ml,0,m,0|L,02 ( QL(z)+WL-1(z)) -Pl(z)Wm-1(z), we come to llll -11dtPl(t) [Pn(t)Qm(t)+Pm(t)Qn(t)]= = -11dt [Pn(t)Pl(t)Qm(t)I+Pl(t)Pm(t)IIQn(t)]= = -11dt [ Pn(t)(L=|l-m|L=l+ml,0,m,0|L,02 ( QL(t)+WL-1(t))-Pl(t)Wm-1(t)).", "+ .L=|l-m|L=l+ml,0,m,0|L,02 PL(t)Qn(t) ], where in the first term we have decomposed the product $P_{l}(t)Q_{m}(t)$ , whereas in the second term we decomposed the product $P_{l}(t)P_{m}(t)$ .", "Using the parity identity, lll -11dt Pn(t)QL(t)=- -11dtPL(t)Qn(t), we obtain lll -11dtPl(t) [Pn(t)Qm(t)+Pm(t)Qn(t)]= = -11dt [ Pn(t)(L=|l-m|L=l+ml,0,m,0|L,02 WL-1(t)-Pl(t)Wm-1(t))].", "Therefore lll -11dtL=|l-m|L=l+ml,0,m,0|L,02Pn(t)WL-1(t)= = -11dtL=|l-m|L=l+ml,0,m,0|L,02 k=0[(L-1)/2]2L-4k-1(2k+1)(L-k)PL-2k-1(t)Pn(t)= 4L=|l-m|L-n-10L=l+m l,0,m,0|L,02L+L2-n (1+n).", "On the other hand, lll -11dtPl(t)Wm-1(t)Pn(t)= = -11dtk=0[(m-1)/2]2m-4k-1(2k+1)(m-k)Pm-2k-1(t)Pl(t)Pn(t)= = -11dtk=0[(m-1)/2]2m-4k-1(2k+1)(m-k)Pm-2k-1(t) L=|l-n|L=l+nl,0,n,0|L,02PL(t)= -4L=|l-n|m-L-10L=l+n l,0,n,0|L,02L+L2-m (1+m).", "Thus, the identity () is demonstrated, lll -11dt L=|l-m|L=l+ml,0,m,0|L,02 Pn(t)WL-1(t)- -11dtPn(t)Pl(t)Wm-1(t)= = 4L=|l-m|L-n-10L=l+m l,0,m,0|L,02L+L2-n (1+n) +4L=|l-n|m-L-10L=l+n l,0,n,0|L,02L+L2-m (1+m)." ] ]

[ [ "Masers in star forming regions" ], [ "Abstract Maser emission plays an important role as a tool in star formation studies.", "It is widely used for deriving kinematics, as well as the physical conditions of different structures, hidden in the dense environment very close to the young stars, for example associated with the onset of jets and outflows.", "We will summarize the recent observational and theoretical progress on this topic since the last maser symposium: the IAU Symposium 242 in Alice Springs." ], [ "Introduction", "Cosmic masers are known as a unique tool in star-formation studies and are one of the first observed signpost of high-mass star formation, particularly the hydroxyl (OH), water (H$_2$ O) and methanol (CH$_3$ OH) masers, that are common and intense.", "This is demonstrated by the many results presented in this volume.", "At the last IAU Symposium 242 in Alice Springs, [21] summarized the relation between masers and star-formation in the following way \"maser observations are at the vanguard of star formation research: yesterdays observations can be explained by complementary data and theory today, and todays observations lay the groundwork for the breakthroughs that will be achieved in the context of tomorrow.\"", "In this review we will summarize some of the achievements and discoveries in the area of star-formation masers presented in the literature since the Australian Symposium.", "It is important to evaluate what \"yesterday's tomorrow\" has unveiled in the area of star formation and identify the possible \"today's tomorrow\" breakthroughs." ], [ "Population studies", "It was relatively well established from earlier surveys of masers in our Galaxy that massive star-forming regions can be associated with OH, H$_2$ O and Class II CH$_3$ OH masers (e.g., [12], [67]).", "However, there is still a great need for verifying what the relation is between specific stages and classes of star formation and different masers.", "For this, complete and ever more sensitive surveys with better astrometric precision are most valuable.", "For example, since our previous meeting [35] discovered that high-mass star formation (HMSF) is present in both the far and near 3 kpc arms through 49 detections of 6.7 GHz methanol masers.", "The Red MSX Sources (RMS) based survey by [73] investigated the statistical correlation of water masers with early-stage of massive star-formation.", "They found similar detection rates for UC H II regions and MYSOs, suggesting that the conditions needed for maser activity are equally likely in these two stages of star formation.", "Nowadays, more instruments have become available, especially to focus systematically on different maser transitions like the higher excited methanol masers from both Class I (collisional excitation) and II (radiative excitation) methanol, as well as silicon monoxide (SiO), ammonia (NH$_3$ ) and formaldehyde (H$_2$ CO) masers.", "Details are presented by e.g.", "Kurtz, Kalenskii, Voronkov, Sjouwerman, Wootten, Booth, Kim, Pestalozzi, Brogan and their collaborators in these proceedings.", "In addition, yesterday's key-questions, which were deemed essential in order to use masers for studying the physics of star formation, are still not fully answered: Is there an evolutionary sequence based on maser occurrence?", "Are Class I and Class II methanol masers associated?", "Where, when and how exactly do masers arise?", "What physical conditions are needed to produce the maser(s)?", "These questions require systematic studies of a large number of sources, possibly at high angular resolution, as well as observations of specific sources using multi-wavelength observations, in order to converge and refine our hypotheses.", "For example, single-dish studies suggested at first that the methanol Class I and II are coincident, but later interferometric images showed they are not co-spatial on arcsecond scales, even though they may be driven by the same YSO [17]." ], [ "Methanol masers, the most widespread masers in HMSFRs", "Methanol masers have been widely studied, particularly after the discovery of the widespread, bright 6.7 GHz transition [42].", "Many transitions have been found to maser from both the A and E types and they have been classified into Class I and Class II, which are collisionally and radiatively excited, respectively [14].", "In general, the Class I (e.g., 36 and 44 GHz lines) are likely associated with outflows (lying further from the central objects) while the Class II (e.g., 6.7 and 12.2 GHz transitions) often coincide with hot molecular cores, UC H II regions, OH masers and near-IR sources.", "The most common Class II masers are 6.7 and 12.2 GHz lines that according to the models (e.g., [15]) should co-propagate quite often, as confirmed by observations (e.g., [8]).", "Both masers are strongly associated with HMSFRs and enable us to probe the dense environments where stars are being born.", "A large sample of 113 sources with known 6.7 GHz masers and 1.2-mm dust clumps was searched at 12.2 GHz by [7].", "These authors found out that when the 6.7 GHz emission is more luminous, the the evolutionary stage of the central object tends to be more advanced, while also the 12.2 GHz is often associated with more evolved regions.", "Based on this an evolutionary sequence for masers associated with massive star formation regions was proposed, which is consistent with conclusions of e.g.", "the survey of Class I by [54].", "However, some discussion continues to refine this relation, e.g.", "[24] and [13].", "One of the key research areas over the past few years has been on relatively rare methanol masers.", "[81] found two new detections of 9.9 GHz Class I masers.", "To date we know of 5 masers at 9.9 GHz (additionally from [62], Voronkov et al.", "2006, 2011).", "The detection rate is likely so low because of the strong dependence of the maser brightness on the physical conditions ([63]).", "This maser is believed to trace shocks caused by different phenomena (e.g., expanding H II regions, outflows).", "A particularly interesting case is G331.13$-$ 00.24, which shows periodic variability at 6.7 GHz with a period of 500 days ([30]).", "There is an obvious urgency to verify whether the variations of both lines correlate, pointing to a common origin of the seed radiation and providing an estimate of the physical conditions for that.", "For more details see Voronkov et al.", "(these proceedings).", "There are now also first arcsecond-resolution images of the 36 GHz methanol masers in HMSFRs thanks to the upgrades of both ATCA and EVLA, e.g.", "in M8E ([58]), Sgr A ([61]), G309.38$-$ 0.13 ([80]) and DR21 [23]).", "In the latter case it was found that surprisingly the Class I 36 GHz and 229 GHz masers appear in close proximity (also in velocity) with the Class II 6.7 GHz maser, while the 44 GHz Class I masers is absent.", "According to the model by such cases require an intermixed environment of dust and gas at a lowish temperature of $\\approx $ 60 K. One may wonder whether we have come closer to answering the question \"when do Class I masers appear?\".", "[13] searched 192 EGOs (the candidates associated with ongoing outflows) for 95 GHz methanol masers, resulting in a 55 per cent detection rate.", "These detections are likely associated with the redder GLIMPSE point-source colors.", "There are two possible explanations, either the Class I objects are associated with lower stellar masses or they are associated with more than one evolutionary phase during high-mass star formation, apparently contradicting the most straightforward schemes [7].", "[40] compared the physical conditions by observing several molecular tracers in both weak and bright mid-IR emitting massive dense cores.", "The methanol Class I maser at 84.5 GHz was found to be strongly anti-correlated with the 12 $\\mu $ m source brightness, leading to an interpretation that these represent more embedded mid-IR sources with a spherically symmetric distribution of the envelope material.", "[20] searched for rare and weak methanol masers at 37.7, 38.3, 38.5 GHz Class II methanol masers towards 70 HMSFRs.", "They detected 13 at 37.7 GHz and 3 at 38.3/5 GHz and found that the 37.7 GHz masers are associated with the most luminous 6.7 and 12.2 GHz masers, likely representing a short (of 1000–4000 years) period in an advanced stage of the evolution.", "Therefore, the 37.7 GHz methanol masers may be called the horsemen of the apocalypse for the Class II methanol maser phase." ], [ "The morphology of 6.7 GHz masers", "More sensitive VLBI surveys have led to the discovery of more complex 6.7 GHz maser structures, including several that show a ring-like morphology (Bartkiewicz et al.", "2005, 2009).", "Kinematics of the maser spots revealed that outflow/infall dominates over the possible Keplerian rotation in a disc.", "A similar morphology with a similar kinematic signature was found in the well-known HMSFR Cep A, where, due to additional constraints on the orientation, the radial motions are more likely resulting from infall ([70], [64]).", "Moreover, it seems that the magnetic field plays a role in shaping this morphology ([77]).", "Such ring-like characteristics were also seen in water masers associated with slightly more advanced stages, where masers were likely tracing an accretion disc or its remnant [47].", "[71] analysed some of these ring-like maser sources using thermal emission at arcsec scale and found that mostly the distribution of the methanol gas peaks at the maser position with the larger scale gas showing a modest outflow velocity.", "They argued that the methanol gas has a single origin in these sources, possibly associated with an accretion shock.", "ALMA resolution is necessary for probing the regions of interest at size scales of 1000 AU.", "For all of these studies it is important to remember that the VLBI techniques resolves out some of the emission.", "[51] noted that more complex morphologies and often larger structures become apparent when using shorter baseline interferometers (EVLA, MERLIN) compared to VLBI, analyzing a study of 72 sources from the Arecibo Methanol Masers Galactic Survey.", "Thus, the 29% detection of the methanol rings of [4] may be biased by observational effects.", "Similarly, [17] also noted that shorter baselines observations resulted in more complex and extended emission for two targets from the EVN sample of [4].", "On the other hand, comparing [51] and [4] results, we note that in three out five cases the emission is very similar on EVN and MERLIN images, but this does not include any of the ring sources." ], [ "Gas kinematics through proper motion studies of masers", "There has also been significant progress on maser proper motion studies at mas scale.", "These result from multi-epoch observations that often have two simultaneous objectives: proper motions of the maser features in order to derive the kinematics of the gas in the direct environment of the YSOs and accurate direct distances by means of detecting the parallax.", "The parallax measurements are summarized and presented in this volume by e.g., Reid, Honma et al., Nagayama et al., Choi et al.", "(these proceedings) Here we focus on the dynamics studies.", "In a series of papers, Moscadelli et al.", "(e.g., 2007, 2011a) demonstrated the power of multi-epoch VLBI for tracing the 3D kinematics close to an YSO towards nine well-studied HMSFRs.", "Combining observations of 22 GHz water and 6.7 GHz methanol masers within a time-span of a few years they detected various motions such as outflow, rotation, infall, all happening in the direct environments of these YSOs.", "They pointed out that these velocity gradients on mas scale still reflect large-scale (100-1000 AU) motions ([46]).", "In some cases such studies can constrain the YSO position and mass.", "For example, [28] directly measured these different phenomena going on within 400 AU from the high-mass protostar AFGL 5142; the gas infall is traced by the 3D velocities of the methanol masers, while a slow, massive, collimated, bipolar outflow is detectable through the water masers.", "Very detailed dynamics were registered by who used multi-epoch data of water masers towards Cep A HW2, noticing morphological changes at scales of 70 AU in a time-span of 5 years.", "They also argued that the R5 expanding bubble structure has been dissipating in the circumstellar medium and that a slow, wide-angle outflow at the scale of 1000 AU co-exists with the well-known high-velocity jets.", "In addition to these methanol and water maser observations, there are unique data from SiO masers, although they are quite rare around YSOs.", "[41] observed the Orion Source I at both 43.1 and 42.8 GHz transitions, resulting in a most detailed view of the inner 20-100 AU of a MYSO.", "The SiO masers lie in an X-shaped structure, with clearly separated blue- and redshifted emission, while bridges of intermediate-velocity emission connects both sides.", "They proposed that these masers are related to a wide-angle bipolar wind emanating from a rotating, edge-on disc.", "This is providing direct evidence of the formation of a MYSO via disc-mediated accretion.", "Other examples and more explanations can be found in contributions by e.g.", "Sanna et al., Goddi et al., Sawada-Satoh et al., Sugiyama et al.", "(these proceedings)." ], [ "Physical conditions for maser emission", "In order to answer the key question where, when and how exactly do masers arise?, we must probe the physical conditions in which they form.", "Such studies concern multiple maser transitions and studying the masing regions at a wide range of wavelengths.", "Both surveys of a large number of sources, as well as detailed individual source observations are needed to complete the scenario of maser formation.", "A very good example is the result obtained by [16] that Class I and II methanol masers coincide with so-called extended green objects (EGOs) which are indicators of outflows and are a promising starting point for identifying MYSOs (Cyganowski, these proceedings).", "[6] investigated the OH/H$_2$ O/CH$_3$ OH relation for a large sample of HMSFRs and noticed a closer similarity of the velocities of OH and methanol masers than of either of these species compared to the water maser peak velocity.", "In spite of the different pumping schemes of water and methanol masers, they both show a similar, 80% detection rate association with OH sources.", "It also has been found by comparing high-luminosity masers with low-luminosity ones that the high brightness ones are related to lower NH3(1,1) excitation temperatures, smaller densities, but three times larger column densities.", "Moreover, the high-luminosity sources are associated with 10 times more massive molecular cores, larger outflows and their internal motions are more pronounced [85].", "Interestingly, [50] showed that the continuum of the counterparts of 6.7 GHz methanol masers is consistent with rapidly accreting massive YSOs ($>$ 0.001 M$_\\odot $  yr$^{-1}$ ) by constraining their SEDs.", "Only a minority of the sample, 30%, coincided with H II regions that are usually ultra- or hyper compact.", "The latter was also confirmed by [56] and [59].", "Indeed the majority of 6.7 GHz masers seems to appear before the H II stage of MYSOs, as was suggested by earlier studies, e.g., [83].", "Alternatively, we may still not have been able to reach the proper sensitivity for such conclusions.", "Studies of specific sources in multiple maser transitions and their counterparts in other tracers are of special value.", "In the well known ON 1 source OH transitions at 1.612, 1.665, 1.667, 1.720, 6.031 and 6.035 GHz lie in a similar region as 6.7 GHz methanol masers.", "[33] concluded that they possibly trace a shock front in the form of a torus/ring around the YSO.", "That interpretation is also supported by polarization angles and velocity gradients.", "In the HMSFR NGC 7538 IRS 1 new masers at 12.2 GHz were found ([44]) and in addition, 23.1 GHz Class II methanol masers were accurately registered ([27]).", "They appear closely associated with 4.8 GHz H$_2$ CO masers, indicating that the conditions must be similar for both of these relatively rare masers.", "It is possible that they are excited by the free-free emission from an H II region.", "However, surprisingly, they are not accompanied by any 6.7 or 12.2 GHz methanol masers.", "Although we are collecting more and more information, the origin of maser structures in high-mass star formation is still not clear.", "A long-term question what do linearly distributed methanol masers trace, an edge-on disc or an outflow?", "is still open.", "[18] showed that orientations of SiO outflows were not consistent with the methanol masers delineating a disk orientation.", "Moreover, for the methanol rings the proposed morphology could generally not be confirmed by infrared high resolution imaging (De Buizer et al., these proceedings).", "[5], using NH$_3$ as a tracer towards methanol Class II masers found that if Keplerian accretion disks exist, they should be confined to regions smaller than 1000 AU.", "Therefore, ALMA-resolution observations are really needed in order to reach the relevant scales in the direct environment of MYSOs." ], [ "Masers as a signpost of star formation", "Masers are readily usable as a diagnostic in complex SFRs, for example as indicators that star formation has begun.", "[55] investigated the NGC 3576 region and verified the evolutionary status of the various molecular components.", "Water masers were found towards the NH$_3$ emission peaks, lying in the arms of the filament.", "In the HMSFR G19.61$-$ 0.23 water masers trace the outflow/jet associated with the most massive core, SMA 1, also traced by H$^{13}$ CO$^{+}$ emission ([26]).", "The massive cold dense core G333.125–0.562 showed water and methanol masers, as well as SiO thermal emission, but remained undetected at wavelengths shorter than 70 $\\mu $ m [39].", "Moreover, 44 GHz methanol masers coincide with presumably masing 23 GHz NH$_3$ emission in the EGO G35.03$+$ 0.35 [10].", "The latter project is based on simultaneous observations of continuum emission and a comprehensive set of lines, something that has become possible with the Expanded Very Large Array (EVLA) and should contribute significantly to our understanding of star formation (Brogan et al., this volume)." ], [ "Variability of masers", "The 6.7 and 12.2 GHz methanol masers have been monitored and unexpectedly periodic variations were discovered from some masers in HMSFRs (e.g.", "[29]).", "Such studies are possible with single-dish observations and often require long-term commitments.", "Monitoring can provide important clues about which phenomena are responsible for the variability, but also about the more general physical conditions in the masing regions or the background radiation field.", "Recently, [32] summarized nine years of monitoring of G12.89$+$ 0.49 at the 6.7 and 12.2 GHz transitions and suggested that the stability of the period is best explained by assuming an underlying binary system.", "In G9.62$+$ 0.20E three methanol lines at 6.7, 12.2 and 107 GHz showed flaring [74] and a colliding-wind binary (CWB) scenario is found to explain periodicity through variations in the seed photon flux and/or the pumping radiation field [75].", "Follow-up studies were required in order to provide more details about the source; VLBI imaging revealed the maser distribution [31] and multi-epoch observations enabled the direct estimation of its distance of 5.2$\\pm $ 0.6 kpc via trigonometric parallax ([57]).", "Recently, [68] discovered a similar case of variability in G22.357$+$ 0.066 that can also be explained by changes in the background free-free emission.", "A period of 179 days was derived from single dish monitoring.", "The time delays seen between maser features can be combined with the VLBI imaging to construct the 3D structure of the maser region.", "Another example is G33.641$-$ 0.228 where the 6.7 GHz methanol bursts originate from a region of 70 AU [25].", "The authors interpret this as coming from an impulsive energy release like a stellar flare.", "By monitoring many different objects we may also find more newly appearing masers as was the case with 6.7 GHz emission in IRAS 22506$+$ 5944 [84].", "Variability was also detected for other maser transitions.", "In G353.273$+$ 0.641 intermittent 22 GHz maser flare activity appeared to be accompanied by structural changes, likely indicating that the excitation is linked to an episodic radio jet [48].", "[38] presented a catalog of 22 GHz H$_2$ O spectra monitored over 30 years towards G34.3$+$ 0.15 (aka W44C).", "They detected a long-term variability with an average period of 14 years and two series of flares that are likely associated with some cyclic activity of the protostar in the UC H II.", "1720 MHz OH masers towards W75N also showed flaring, possibly related to the very dense molecular material that is excited and slowly accelerated by the outflow [22].", "A surprising event was registered in IRAS18566$+$ 0408 by [1]: the 4.8 GHz H$_2$ CO maser showed flaring and a period correlated with the 6.7 GHz methanol outbursts.", "Both regions are separated spatially, so both phenomena likely indicate variations in the infrared radiation field, maybe related to periodic accretion events." ], [ "Masers in low and intermediate-mass protostars", "Because low and intermediate-mass stars are more common and evolve more slowly, they should in principle be easier to study, at closer distances, maybe less embedded and in less confusing environments.", "It should therefore be possible to study the star-formation process in more detail.", "However, masers are not so common in these kinds of objects and appear possibly in different stages.", "[2] found that only 9 and 6% of a sample of 180 intermediate-mass YSOs showed 22 GHz water and 44 GHz methanol maser emission, respectively.", "Water is likely related to the inner parts of outflows and can be highly variable, while methanol is possibly associated with the interfaces of the outflows with the ambient dense gas.", "The detection rates of both masers rapidly decreases as the central (proto)stars evolve and the excitations of the two masers appear closely related.", "The most embedded (Class 0-like) intermediate-mass YSOs known to date are all associated with water masers (e.g., ; de Oliveira Alves, these proceedings).", "However, an intriguing object, IRAS 00117$+$ 6412 was found with water masers in MM2, where no outflow seems present [49].", "In order to understand this case, more observations with the best sensitivity and resolution are needed.", "The 44 and 36 GHz Class I methanol masers rarely appear in lower-mass YSOs.", "[36] found only four 44 GHz and one 36 GHz maser, while no emission was detected towards the remaining 39 outflows.", "They also noticed the masers have lower luminosity compared to those in HMSFRs.", "Imaging of L1157 indicates that the 44 GHz maser may form in thin layers of turbulent post-shock gas or in collapsing clumps ([37]; Kalenskii et al.", "these proceedings)." ], [ "HMSFRs in the Galactic Centre and beyond the Galaxy", "Recent searches towards Sgr A revealed that the 36 GHz Class I methanol masers correlate with NH$_3$ (3,3) density peaks and outline regions of cloud-cloud collisions, maybe just before the onset of local massive star formation [60].", "The 44 GHz masers correlate with the 36 GHz locations, but less with the OH masers at 1720 MHz [52], which are associated with the interaction between the supernova remnant Sgr A East and the interstellar medium [53].", "One particularly interesting group of 44 GHz masers was found that do not overlap with any 36 GHz emission.", "These may signal the presence of a hotter and denser environment than the material swept up from the shock, maybe related to advanced star formation [52].", "Kilomasers are masers beyond our Galaxy, with luminosities comparable to the brightest galactic maser, which either amplify a background AGN or originate from star-forming regions.", "These Galactic analog H$_2$ O masers have become a great tool in studies of young super-star cluster formation with high angular resolution.", "They are detected in a few nearby galaxies only (e.g., [11]).", "[9] found that water masers in the Antennae Galaxies are associated with star-formation, as they show kinematic and spatial agreement with massive and dense CO molecular clouds.", "The various early stages of star formation in the components of the Antennae Galaxies was confirmed by [72].", "The Large and Small Magellanic Clouds were also searched for maser emission.", "[34] detected four 6.7 GHz methanol (of which one is new) and two 6.035 GHz OH (of which one is new) masers in the LMC.", "Both transitions indicate much more modest maser populations compared to the Milky Way, likely originating from lower oxygen and carbon abundances.", "Moreover, [19] found the first 12.2 GHz methanol maser towards the LMC.", "They also detected 22 GHz water and 6.7 GHz methanol masers that are associated with more luminous and redder YSOs.", "The first 6.7 GHz methanol spectrum towards Andromeda Galaxy (M31) was presented by [61].", "More on kilo- and also mega-masers is presented by e.g.", "Tarchi (these proceedings)." ], [ "Magnetic field", "Masers are a particularly unique tool for studying the magnetic field studies in HMSFRs (Vlemmings, these proceedings).", "This is an important subject in star formation, as the magnetic field could be a dominant force in the process by supporting the molecular cloud against gravitational collapse, regulating the accretion and shaping the outflows as has been argued for Cepheus A ([76]).", "We have already mentioned the work by [33] who found a shock front in the form of a torus/ring around the YSO in ON 1.", "That scenario is also supported by the measured polarization angles of the masers.", "A magnetic field strength of a few mG was detected through the Zeeman splitting of OH and methanol masers.", "Linearly and circularly polarized emission of 22 GHz water masers was used for measuring the orientation and strength of magnetic fields in W75N.", "The magnetic fields around the young massive protostar VLA2 are found to be well ordered around an expanding gas shell [66].", "In NGC 7538–IRS 1 the water masers did not show significant Zeeman splitting while the 6.7 GHz methanol masers indicated a possible range of magnetic field strength of 50 mG $< | B_{\\parallel } | <$ 500 mG, depending on the value of Zeeman-splitting coefficient.", "These masers likely are related to the outflow and the interface between infall and the large-scale torus, respectively [65]." ], [ "Summary", "It is clear that many new results and discoveries have been obtained in our attempts to understand star formation and the associated masers.", "Are we closer to answer the key-questions?", "What is the current state of masers in SFRs?", "We note, that: The time sequence for masers seems to take shape and is confirmed by (most) new methanol transition measurements, however, some issues still exists, Convergence can be seen on the issue where water, methanol (both Classes), OH and SiO arise, but testing the hypotheses with high-resolution observations and at other wavelengths is critical, One may hope that in synergy with ALMA the role of masers to study small scale dynamics will be strengthened, Monitoring programs are starting to give important clues about co-evolving binaries, There should be more focus given to low-mass stars, More work on models to get accurate physical conditions is needed.", "We started the review with the popular statement that masers are an unique tool in star-formation studies and in the end we are more convinced that they really are.", "In order to progress the use of masers and solving the detailed questions we just need better instruments and of course the patience to work on data to obtain more and more interesting and even surprising results.", "We all can wait (and work) with curiosity for the further discoveries that will be presented in the next maser symposium.", "AB acknowledges support by the Polish Ministry of Science and Higher Education through grant N N203 386937." ] ]

[ [ "Partial LLL Reduction" ], [ "Abstract The Lenstra-Lenstra-Lovasz (LLL) reduction has wide applications in digital communications.", "It can greatly improve the speed of the sphere decoding (SD) algorithms for solving an integer least squares (ILS) problem and the performance of the Babai integer point, a suboptimal solution to the ILS problem.", "Recently Ling and Howgrave-Graham proposed the so-called effective LLL (ELLL) reduction.", "It has less computational complexity than LLL, while it has the same effect on the performance of the Babai integer point as LLL.", "In this paper we propose a partial LLL (PLLL) reduction.", "PLLL avoids the numerical stability problem with ELLL, which may result in very poor performance of the Babai integer point.", "Furthermore, numerical simulations indicated that it is faster than ELLL.", "We also show that in theory PLLL and ELLL have the same effect on the search speed of a typical SD algorithm as LLL." ], [ "Introduction", "In a multiple-input and multiple-output (MIMO) system, often we have the following linear model: ${y}=H{x}+v,$ where ${y}\\in \\mathbb {R}^n$ is the channel output vector, $v\\in \\mathbb {R}^n$ is the noise vector following a normal distribution $\\mathcal {N}(0,\\sigma ^2 I)$ , $H\\in \\mathbb {R}^{n\\times m}$ is the channel matrix, and ${x}\\in \\mathbb {Z}^m$ is the unknown integer data vector.", "In some applications where complex ILS problems may need to be solved instead, we can first transform the complex ILS problems to equivalent real ILS problems.", "For simplicity, like [1], in this paper we assume $m=n$ and $H$ is nonsingular.", "To estimate ${x}$ , one solves an integer least squares problem $\\min _{{x}\\in {Z}^n}\\Vert {y}-H{x}\\Vert _2^2,$ which gives the maximum-likelihood estimate of ${x}$ .", "It has been proved that the ILS problem is NP-hard [2].", "For applications which have high real-time requirement, an approximate solution of (REF ) is usually computed instead.", "A often used approximation method is the nearest plane algorithm proposed by Babai [3] and the produced approximate integer solution is referred to as the Babai integer point.", "In communications, a method for finding this approximate solution is referred to as a successive interference cancellation decoder.", "A typical method to solve (REF ) is a sphere decoding (SD) algorithm, such as the Schnorr-Euchner algorithm (see [4] and [5]) or its variants (see, e.g., [6] and [7]).", "A SD algorithm has two phases.", "First the reduction phase transforms (REF ) to an equivalent problem.", "Then the search phase enumerates integer points in a hyper-ellipsoid to find the optimal solution.", "The reduction phase makes the search phase easier and more efficient.", "The Lenstra-Lenstra-Lovasz (LLL) reduction [8] is the mostly used reduction in practice.", "An LLL reduced basis matrix has to satisfy two conditions.", "One is the size-reduction condition and the other is the Lovasz condition (see Section for more details).", "Recently Ling and Howgrave-Graham [1] argued geometrically that the size-reduction condition does not change the performance of the Babai integer point.", "Then they proposed the so-called effective LLL reduction (to be referred to as ELLL) which mostly avoids size reduction.", "They proved that their ELLL algorithm has less time complexity than the original LLL algorithm given in [8].", "However, as implicitly pointed out in [1], the ELLL algorithm has a numerical stability problem.", "Our simulations, presented in Section , will indicate that ELLL may give a very bad estimate of ${x}$ than the LLL reduction due to its numerical stability problem.", "In this paper, we first show algebraically that the size-reduction condition of the LLL reduction has no effect on a typical SD search process.", "Thus it has no effect on the performance of the Babai integer point, the first integer point found in the search process.", "Then we propose a partial LLL reduction algorithm, to be referred to as PLLL, which avoids the numerical stability problem with ELLL.", "Numerical simulations indicate that it is faster than ELLL and is as numerically stable as LLL." ], [ "LLL Reduction", "In matrix language, the LLL reduction can be described as a QRZ factorization [9]: $Q^T HZ= R,$ where $Q\\in \\mathbb {R}^{n\\times n}$ is orthogonal, $Z\\in \\mathbb {Z}^{n\\times n}$ is a unimodular matrix (i.e., $\\det (Z)=\\pm 1$ ), and $R\\in \\mathbb {R}^{n\\times n}$ is upper triangular and satisfies the following two conditions: $\\begin{split}& | r_{i,j}| \\le | r_{i,i}|/2, \\ \\ 1\\le i<j \\le n \\\\& \\delta \\, r_{i-1,i-1}^2 \\le r^2_{i-1,i}+r^2_{i,i}, \\ \\ 1<i \\le n,\\end{split}$ where the parameter $\\delta \\in (1/4, 1]$ .", "The first condition in (REF ) is the size-reduction condition and the second condition in (REF ) is the Lovasz condition.", "Define ${\\bar{{y}}}=Q^T{y}$ and $z=Z^{-1}{x}$ .", "Then it is easy to see that the ILS problem (REF ) is reduced to $\\min _{z\\in \\mathbb {Z}^n}\\Vert {\\bar{{y}}}-Rz\\Vert ^2_2.$ If $\\hat{z}$ is the solution of the reduced ILS problem (REF ), then $\\hat{{x}}=Z\\hat{z}$ is the ILS solution of the original problem (REF ).", "The LLL algorithm first applies the Gram-Schmidt orthogonalization (GSO) to $H$ , finding the QR factors $Q$ and $R$ (more precisely speaking, to avoid square root computation, the original LLL algorithm gives a column scaled $Q$ and a row scaled $R$ which has unit diagonal entries).", "Two types of basic unimodular matrices are then implicitly used to update $R$ so that it satisfies (REF ): integer Gauss transformations (IGT) matrices and permutation matrices, see below.", "To meet the first condition in (REF ), we can apply an IGT: $Z_{ij}=I-\\zeta e_ie_j^T.$ where $e_i$ is the $i$ -th column of $I_n$ .", "It is easy to verify that $Z_{ij}$ is unimodular.", "Applying $Z_{ij}\\ (i < j)$ to $R$ from the right gives ${\\bar{R}}= RZ_{ij} = R-\\zeta Re_ie_j^T,$ Thus ${\\bar{R}}$ is the same as $R$ , except that ${\\bar{r}}_{kj} = r_{kj} - \\zeta r_{ki},\\ k = 1, \\ldots , i$ .", "By setting $\\zeta = \\lfloor r_{ij}/r_{ii} \\rceil $ , the nearest integer to $r_{ij}/r_{ii}$ , we ensure $|{\\bar{r}}_{ij}|\\le |{\\bar{r}}_{ii}|/2$ .", "To meet the second condition in (REF ), we permutations columns.", "Suppose that we interchange columns $i-1$ and $i$ of $R$ .", "Then the upper triangular structure of $R$ is no longer maintained.", "But we can bring $R$ back to an upper triangular matrix by using the GSO technique (see [8]): ${\\bar{R}}=G_{i-1,i}RP_{i-1,i},$ where $G_{i-1,i}$ is an orthogonal matrix and $P_{i-1,i}$ is a permutation matrix.", "Thus, ${\\bar{r}}_{i-1,i-1}^2&= r^2_{i-1,i}+r^2_{i,i},\\\\{\\bar{r}}^2_{i-1,i}+{\\bar{r}}^2_{i,i}&= r_{i-1,i-1}^2.", "\\nonumber $ If $\\delta \\, r_{i-1,i-1}^2 > r^2_{i-1,i}+r^2_{i,i}$ , then the above operation guarantees $\\delta \\, {\\bar{r}}_{i-1,i-1}^2 < {\\bar{r}}^2_{i-1,i}+{\\bar{r}}^2_{i,i}$ .", "The LLL reduction process is described in Algorithm .", "LLL reduction [1] apply GSO to obtain $H=QR$ ; set $Z=I_n$ , $k=2$ ; $k\\le n$ apply IGT $Z_{k-1,k}$ to reduce $r_{k-1,k}$ : $R=RZ_{k-1,k}$ ; update $Z$ : $Z=ZZ_{k-1,k}$ ; $\\delta \\, r_{k-1,k-1}^2> \\left(r^2_{k-1,k}+r^2_{k,k}\\right)$ permute and triangularize $R$ : $R\\!=\\!G_{k-1,k}RP_{k-1,k}$ ; update $Z$ : $Z=ZP_{k-1,k}$ ; $k=k-1$ , when $k>2$ ; $i=k-2,\\dots ,1$ apply IGT $Z_{ik}$ to reduce $r_{ik}$ : $R=RZ_{ik}$ ; update $Z$ : $Z=ZZ_{i,k}$ ; $k=k+1$ ;" ], [ "SD Search Process and Babai Integer Point", "For later use we briefly introduce the often used SD search process (see, e.g., [7]), which is a depth-first search (DFS) through a tree.", "The idea of SD is to search for the optimal solution of (REF ) in a hyper-ellipsoid defined as follow: $\\Vert {\\bar{{y}}}-Rz\\Vert _2^2< \\beta .$ Define $\\begin{split}& c_n = {\\bar{y}}_n/r_{nn},\\\\& c_k =\\big ({\\bar{y}}_k-\\sum _{j=k+1}^n r_{kj}z_j \\big )/r_{kk}, \\ \\ k=n-1,\\ldots ,1.\\end{split}$ Then it is easy to show that (REF ) is equivalent to $\\mbox{level } k: \\qquad r_{kk}^2(z_k-c_k)^2<\\beta -\\sum _{j=k+1}^n r_{jj}^2(z_j-c_j)^2,$ where $k=n, n-1,\\dots ,1$ .", "Suppose $z_n, z_{n-1}, \\ldots , z_{k+1}$ have been fixed, we try to determine $z_k$ at level $k$ by using (REF ).", "We first compute $c_k$ and then take $z_k=\\lfloor c_k \\rceil $ .", "If (REF ) holds, we move to level $k-1$ to try to fix $z_{k-1}$ .", "If at level $k-1$ , we cannot find any integer for $z_{k-1}$ such that (REF ) (with $k$ replaced by $k-1$ ) holds, we move back to level $k$ and take $z_k$ to be the next nearest integer to $c_k$ .", "If (REF ) holds for the chosen value of $z_k$ , we again move to level $k-1$ ; otherwise we move back to level $k+1$ , and so on.", "Thus after $z_n,\\dots ,z_{k+1}$ are fixed, we try all possible values of $z_k$ in the following order until (REF ) dose not hold anymore and we move back to level $k+1$ : $\\begin{split}& \\lfloor c_k \\rceil ,\\lfloor c_k \\rceil -1,\\lfloor c_k \\rceil +1,\\lfloor c_k \\rceil -2,\\dots , ~\\text{if } c_k\\le \\lfloor c_k \\rceil , \\\\& \\lfloor c_k \\rceil ,\\lfloor c_k \\rceil +1,\\lfloor c_k \\rceil -1,\\lfloor c_k \\rceil +2,\\dots , ~\\text{if }c_k>\\lfloor c_k \\rceil .\\end{split}$ When we reach level 1, we compute $c_1$ and take $z_1=\\lfloor c_1 \\rceil $ .", "If (REF ) (with $k=1$ ) holds, an integer point, say $\\hat{z}$ , is found.", "We update $\\beta $ by setting $\\beta =\\Vert {y}-R\\hat{z}\\Vert _2^2$ and try to update $\\hat{z}$ to find a better integer point in the new hyper-ellipsoid.", "Finally when we cannot find any new value for $z_n$ at level $n$ such that the corresponding inequality holds, the search process stops and the latest found integer point is the optimal solution we seek.", "At the beginning of the search process, we set $\\beta =\\infty $ .", "The first integer point $z$ found in the search process is referred to as the Babai integer point." ], [ "Effects of size reduction on search", "Ling and Howgrave-Graham [1] has argued geometrically that the performance of the Babai integer point is not affected by size reduction (see the first condition in (REF )).", "This result can be extended.", "In fact we will prove algebraically that the search process is not affected by size reduction.", "We stated in Section that the size-reduction condition in (REF ) is met by using IGTs.", "It will be sufficient if we can show that one IGT will not affect the search process.", "Suppose that two upper triangular matrices $R\\in \\mathbb {R}^{n\\times n}$ and ${\\bar{R}}\\in \\mathbb {R}^{n\\times n}$ have the relation: ${\\bar{R}}=RZ_{st}, \\ \\ Z_{st}=I-\\zeta e_s e_t^T, \\ \\ s<t.$ Thus, ${\\bar{r}}_{kt}&= r_{kt}-\\zeta r_{ks}, \\qquad & \\text{if } & k \\le s, \\\\{\\bar{r}}_{kj}&= r_{kj}, & \\text{if } & k >s \\text{ or } j \\ne t. $ Let ${\\bar{z}}= Z_{st}^{-1}z$ .", "Then the ILS problem (REF ) is equivalent to $\\min _{{\\bar{z}}\\in \\mathbb {Z}^n} \\Vert {\\bar{{y}}}-{\\bar{R}}{\\bar{z}}\\Vert _2^2.$ For this ILS problem, the inequality the search process needs to check at level $k$ is $\\mbox{level } k: \\ \\ {\\bar{r}}_{kk}^2({\\bar{z}}_k-{\\bar{c}}_k)^2<\\beta -\\sum _{j=k+1}^n {\\bar{r}}_{jj}^2({\\bar{z}}_j-{\\bar{c}}_j)^2,$ Now we look at the search process for the two equivalent ILS problems.", "Suppose ${\\bar{z}}_n, {\\bar{z}}_{n-1}, \\ldots , {\\bar{z}}_{k+1}$ and $z_n, z_{n-1}, \\ldots , z_{k+1}$ have been fixed.", "We consider the search process at level $k$ under three different cases.", "Case 1: $k > s$ .", "Note that ${\\bar{R}}_{k:n,k:n}=R_{k:n,k:n}$ .", "It is easy to see that we must have ${\\bar{c}}_i=c_i$ and ${\\bar{z}}_i=z_i$ for $i=n, n-1, \\ldots , k+1$ .", "Thus at level $k$ , ${\\bar{c}}_k=c_k$ and the search process takes an identical value for ${\\bar{z}}_k$ and $z_k$ .", "For the chosen value, the two inequalities (REF ) and (REF ) are identical.", "So both hold or fail at the same time.", "Case 2: $k=s$ .", "According to Case 1, we have ${\\bar{z}}_{i}=z_i$ for $i=n,n-1,\\ldots ,s+1$ .", "Thus ${\\bar{c}}_k&= \\frac{{\\bar{y}}_k-\\sum _{j=k+1}^n {\\bar{r}}_{kj}{\\bar{z}}_j}{{\\bar{r}}_{kk}}\\\\&= \\frac{{\\bar{y}}_k-\\sum _{j=k+1,j\\ne t}^n r_{kj}z_j-(r_{kt}-\\zeta r_{kk})z_t}{r_{kk}}\\\\&= c_k+\\zeta z_t,$ where $\\zeta $ and $z_t$ are integers.", "Note that $z_k$ and ${\\bar{z}}_k$ take on values according to (REF ).", "Thus values of $z_k$ and ${\\bar{z}}_k$ taken by the search process at level $k$ must satisfy ${\\bar{z}}_k=z_k+\\zeta z_t$ .", "In other words, there exists one-to-one mapping between the values of $z_k$ and ${\\bar{z}}_k$ .", "For the chosen values of ${\\bar{z}}_k$ and $z_k$ , ${\\bar{z}}_k-{\\bar{c}}_k=z_k-c_k$ .", "Thus, again the two inequalities (REF ) and (REF ) are identical.", "Therefore both inequalities hold or fail at the same time.", "Case 3: $k <s$ .", "According to Case 1 and Case 2, ${\\bar{z}}_i=z_i$ for $i=n,n-1,\\ldots ,s+1$ and ${\\bar{z}}_s=z_s+\\zeta z_t$ .", "Then for $k=s-1$ , ${\\bar{c}}_{k}&= \\frac{{\\bar{y}}_{k}-\\sum _{j=k+1}^n {\\bar{r}}_{kj}{\\bar{z}}_j}{{\\bar{r}}_{kk}} \\\\&= \\frac{{\\bar{y}}_k-\\sum _{j=k+2,j\\ne t}^n r_{kj}z_j-r_{ks}{\\bar{z}}_s-{\\bar{r}}_{kt}z_t}{r_{kk}}\\\\&= \\frac{{\\bar{y}}_k-\\sum _{j=k+1}^n r_{kj}z_j-\\zeta r_{ks}z_t+\\zeta r_{ks}z_t}{r_{kk}}\\\\&= c_{k}.$ Thus the search process takes an identical value for ${\\bar{z}}_k$ and $z_k$ when $k=s-1$ .", "By induction we can similarly show this is true for a general $k<s$ .", "Thus, again the two inequalities (REF ) and (REF ) are identical.", "Therefore they hold or fail at the same time.", "In the above we have proved that the search process is identical for both ILS problems (REF ) and (REF ) (actually the two search trees have an identical structure).", "Thus the speed of the search process is not affected by the size-reduction condition in (REF ).", "For any two integer points ${\\bar{z}}^\\ast $ and $z^\\ast $ found in the search process at the same time for the two ILS problems, we have seen that ${\\bar{z}}_i^\\ast =z_i^\\ast $ for $i=n, \\ldots , s+1, s-1, \\ldots , 1$ and ${\\bar{z}}_s^\\ast =z_s^\\ast +\\zeta z_t^\\ast $ , i.e., ${\\bar{z}}^\\ast =Z_{st}^{-1}z^\\ast $ .", "Then $\\Vert {\\bar{{y}}}-{\\bar{R}}{\\bar{z}}^\\ast \\Vert _2^2 = \\Vert {\\bar{{y}}}-Rz^\\ast \\Vert _2^2.$ Thus, the performance of the Babai point is not affected by the size-reduction condition in (REF ) either, as what [1] has proved from a geometric perspective.", "However, the IGTs which reduce the super-diagonal entries of $R$ are not useless when they are followed by permutations.", "Suppose $|r_{i-1,i}|>\\frac{|r_{i-1,i-1}|}{2}$ .", "If we apply $Z_{i-1,i}$ to reduce $r_{i-1,i}$ , permute columns $i-1$ and $i$ of $R$ and triangularize it, we have from (REF ) and (REF ) that ${\\bar{r}}_{i-1,i-1}^2&= \\left(r_{i-1,i}-\\left\\lfloor \\frac{r_{i-1,i}}{r_{i-1,i-1}}\\right\\rceil r_{i-1,i-1}\\right)^2+r^2_{ii}\\\\& < r_{i-1,i}^2+r^2_{ii}.$ From (REF ) we observe that the IGT can make $|r_{i-1,i-1}|$ smaller after permutation and triangularization.", "Correspondingly $|r_{i,i}|$ becomes larger, as it is easy to prove that $|r_{i-1,i-1}r_{i,i}|$ remains unchanged after the above operations.", "The ELLL algorithm given in [1] is essentially identical to Algorithm after lines –, which reduce other off-diagonal entries of $R$ , are removed." ], [ "Numerical stability issue", "We have shown that in the LLL reduction, an IGT is useful only if it reduces a super-diagonal entry.", "Theoretically, all other IGTs will have no effect on the search process.", "But simply removing those IGTs can causes serious numerical stability problem even $H$ is not ill conditioned.", "The main cause of the stability problem is that during the reduction process, some entries of $R$ may grow significantly.", "For the following $n\\times n$ upper triangular matrix $H=\\begin{bmatrix}1 &2 &4 & & & \\\\&1 &2 &0 & & \\\\& &1 &2 &4 & \\\\& & &1 &2 & \\ddots \\\\& & & &1 & \\ddots \\\\& & & & & \\ddots \\\\\\end{bmatrix},$ when $n=100$ , the condition number $\\kappa _2(H) \\approx 34$ .", "The LLL reduction will reduce $H$ to an identity matrix $I$ .", "However, if we apply the ELLL reduction, the maximum absolute value in $R$ will be $2^{n-1}$ .", "When $n$ is big enough, an integer overflow will occur.", "In the ELLL algorithm, the super-diagonal entries are always reduced.", "But if a permutation does not occur immediately after the size reduction, then this size reduction is useless in theory and furthermore it may help the growth of the other off-diagonal entries in the same column.", "Therefore, for efficiency and numerical stability, we propose a new strategy of applying IGTs in Algorithm .", "First we compute $\\zeta =\\lfloor r_{k-1,k}/r_{k-1,k-1} \\rceil $ .", "Then we test if the following inequality $\\delta \\, r^2_{k-1,k-1} > \\left(r_{k-1,k}-\\zeta r_{k-1,k-1}\\right)^2 + r^2_{kk}$ holds.", "If it does not, then the permutation of columns $k-1$ and $k$ will not occur, no IGT will be applied, and the algorithm moves to column $k+1$ .", "Otherwise, if $\\zeta \\ne 0$ , the algorithm reduces $r_{k-1,k}$ and if $|\\zeta |\\ge 2$ , the algorithm also reduces all $r_{i,k}$ for $i=k-2, k-3, \\dots ,1$ for stability consideration.", "When $|\\zeta |=1$ , we did not notice any stability problem if we do not reduce the above size of $r_{i,k}$ for $i=k-2, k-3, \\dots ,1$ ." ], [ "Householder QR with minimum column pivoting", "In the original LLL reduction and the ELLL reduction, GSO is used to compute the QR factorization of $H$ and to update $R$ in the later steps.", "The cost of computing the QR factorization by GSO is $2n^3$ flops, larger than $4n^3/3$ flops required by the QR factorization by Householder reflections (note that we do not need to form the $Q$ factor explicitly in the reduction process); see, e.g., [10].", "Thus we propose to compute the QR factorization by Householder reflections instead of GSO.", "Roughly speaking, the reduction would like to have small diagonal entries at the beginning and large diagonal entries at the end.", "In our new reduction algorithm, the IGTs are applied only when a permutation will occur.", "The less occurrences of permutations, the faster the new reduction algorithm runs.", "To reduce the occurrences of permutations in the reduction process, we propose to compute the QR factorization with minimum-column-pivoting: $Q^T HP= R$ where $P\\in \\mathbb {Z}^{n \\times n}$ is a permutation matrix.", "In the $k$ -th step of the QR factorization, we find the column in $H_{k:n,k:n}$ , say column $j$ , which has the minimum 2-norm.", "Then we interchange columns $k$ and $j$ of $H$ .", "After this we do what the $k$ -th step of a regular Householder QR factorization does.", "Algorithm REF describes the process of the factorization.", "QR with minimum-column-pivoting [1] set $R=H,P=I_n$ ; compute $l_k=\\Vert r_k\\Vert _2^2$ , $k=1\\dots ,n$ ; $k=1, 2, \\ldots , n$ find $j$ such that $l_j$ is the minimum among $l_k,\\dots ,l_n$ ; exchange columns $k$ and $j$ of $R$ , $l$ and $P$ ; apply a Householder reflection $Q_k$ to eliminate $r_{k+1,k}, r_{k+2,k}, \\dots ,r_{n,k}$ ; update $l_j$ by setting $l_j=l_j-r_{k,j}^2$ , $j=k+1,\\dots ,m$ ; Note that the cost of computation of $l_j$ in the algorithm is negligible compared with the other cost.", "As Givens rotations have better numerical stability than GSO, in line of Algorithm , we propose to use a Givens rotation to do triangularization." ], [ "PLLL reduction algorithm", "Now we combine the strategies we proposed in the previous subsections and give a description of the reduction process in Algorithm REF , to be referred to as a partial LLL (PLLL) reduction algorithm.", "PLLL reduction [1] compute the Householder QR factorization with minimum pivoting: $Q^THP=R$ ; set $Z=P$ , $k=2$ ; $k\\le n$ $\\zeta =\\lfloor r_{k-1,k}/r_{k-1,k-1} \\rceil $ , $\\alpha =(r_{k-1,k}-\\zeta r_{k-1,k-1})^2$ ; $\\delta \\, r_{k-1,k-1}^2> (\\alpha +r^2_{k,k})$ $\\zeta \\ne 0 $ apply the IGT $Z_{k-1,k}$ to reduce $r_{k-1,k}$ ; $|\\zeta |\\ge 2$ $i=k-1,\\dots ,1$ apply the IGT $Z_{i,k}$ to reduce $r_{i,k}$ ; update $Z$ : $Z=ZZ_{i,k}$ ; permute and triangularize: $R=G_{k-1,k}RP_{k-1,k}$ ; update $Z$ : $Z=ZP_{k-1,k}$ ; $k=k-1$ , when $k>2$ ; $k=k+1$ ;" ], [ "Numerical Experiments", "In this section we give numerical test results to compare efficiency and stability of LLL, ELLL and PLLL.", "Our simulations were performed in MATLAB 7.8 on a PC running Linux.", "The parameter $\\delta $ in the reduction was set to be $3/4$ in the experiments.", "Two types of matrices were tested.", "Type 1.", "The elements of $H$ were drawn from an i.i.d.", "zero-mean, unit variance Gaussian distribution.", "Type 2.", "$H=U^T$ , where $U$ and $ are the Q-factors of the QR factorizations of random matricesand $ is a diagonal matrix, whose first half diagonal entries follow an i.i.d.", "uniform distribution over 10 to 100, and whose second half diagonal entries follow an i.i.d.", "uniform distribution over 0.1 to 1.", "So the condition number of $H$ is bounded up by 1000.", "For matrices of Type 1, we gave 200 runs for each dimension $n$ .", "Figure REF gives the average flops of the three reduction algorithms, and Figure REF gives the average relative backward error $\\Vert H-Q_cR_cZ_c^{-1}\\Vert _2/\\Vert H\\Vert _2$ , where $Q_c$ , $R_c$ and $Z_c^{-1}$ are the computed factors of the QRZ factorization produced by the reduction.", "From Figure 1 we see that PLLL is faster than both LLL and ELLL.", "From Figure 2 we observe that the relative backward error for both LLL and PLLL behaves like $O(nu)$ , where $u \\approx 10^{-16}$ is the unit round off.", "Thus the two algorithms are numerically stable for these matrices.", "But ELLL is not numerically stable sometimes.", "Figure: Matrices of Type 1 - FlopsFigure: Matrices of Type 1 - Backward ErrorsFor matrices of Type 2, Figure REF displays the average flops of the three reduction algorithms over 200 runs for each dimension $n$ .", "Again we see that PLLL is faster than both LLL and ELLL.", "To see how the reduction affects the performance of the Babai integer point, for Type 2 of matrices, we constructed the linear model ${y}=H{x}+v$ , where ${x}$ is an integer vector randomly generated and $v\\sim \\mathcal {N}(0,0.2^2 I)$ .", "Figure REF shows the average bit error rate (BER) over 200 runs for each dimension $n$ .", "Form the results we observe that the computed Babai points obtained by using LLL and PLLL performed perfectly, but the computed Babai points obtained by using ELLL performed badly when the dimension $n$ is larger than 15.", "Our simulations showed that the computed ILS solutions obtained by using the three reduction algorithms behaved similarly.", "All these indicate that ELLL can give a very poor estimate of ${x}$ due to its numerical stability problem.", "Figure: Matrices of Type 2 - FlopsFigure: Matrices of Type 2 - BER" ] ]

[ [ "Analytical Quantum Dynamics in Infinite Phase Space" ], [ "Abstract We develop a dynamical theory, based on a system of ordinary differential equations describing the motion of particles which reproduces the results of quantum mechanics.", "The system generalizes the Hamilton equations of classical mechanics to the quantum domain, and turns into them in the classical limit $\\hbar\\rightarrow 0$.", "The particles' motions are completely determined by the initial conditions.", "In this theory, the wave function $\\psi$ of quantum mechanics is equal to the exponent of an action function, obtained by integrating some Lagrangian function along particle trajectories, described by equations of motion.", "Consequently, the equation for the logarithm of a wave function is related to the equations of motion in the same way as the Hamilton-Jacobi equation is related to the Hamilton equations in classical mechanics.", "We demonstrate that the probability density of particles, moving according to these equations, should be given by a standard quantum-mechanical relation, $\\rho=|\\psi|^2$.", "The theory of quantum measurements is presented, and the mechanism of nonlocal correlations between results of distant measurements with entangled particles is revealed.", "In the last section, we extend the theory to particles with nonzero spin." ], [ "Introduction", "According to standard quantum mechanicsIn this article we will use, in order of their appearance, the following abbreviations: QM — quantum mechanics, ODE — ordinary differential equation, PDE — partial differential equation, QHJE — quantum Hamilton-Jacobi equation, $\\cal P$ AQD — analytical quantum dynamics in infinite phase space, OSFI — one-step Feynman integral, CD — Cartan distribution, HC — Hamiltonian conditions, DBBT — de Broglie - Bohm theory, FDS — full description space, RDS — reduced description space.", "(QM), the state of every physical system is described by a wave function, whose time evolution is determined by the Schrödinger equation.", "In this paper we will consider only closed systems, for which the description by a wave function is sufficient.", "We know how to set up experiments with a known initial wave function, and then using the Schrödinger equation we can calculate it at any later moment.", "However, contrary to, say, an electric field in an electromagnetic wave, or a field of pressure in a sound wave, the wave function is not an object of observation and measurement.", "Consequently, besides Schrödinger's equation, the theory additionally includes a set of rules, specifying the results of experiments with quantum systems in terms of their wave functions.", "These rules were developed in late 1920-s and collectively named the (statistical) interpretation of QM.", "Thus, the theory has two parts: Schrödinger's equation and interpretation.", "Such structure of the theory may be viewed in various ways.", "The standard attitude consists of the faith that the described construction constitutes the desired complete and fundamental law of nature.", "However, there are a number of objections that may be raised against this point of view: – It seems natural to expect from a fundamental theory that it reflects all observable elements of physical reality and gives the law of evolution for them.", "Thus the very fact that QM is formulated in terms of wave functions, which cannot be directly observed, and requires an interpretation that establishes a connection between wave functions and results of experiments creates doubts in its fundamental character.", "– This interpretation is a separate and independent part of the theory's foundation, whereas it seems desirable for a fundamental theory to allow the derivation of all its experimental consequences by pure math from the dynamical laws of evolution alone.", "– By necessity the interpretation, which describes the response of an approximately classical apparatus to its interaction with a quantum system, is expressed in classical terms.", "However the behavior of any apparatus, which is just a physical object built up of atoms, should be derivable from QM.", "Consequently, QM contains an unacceptable for a fundamental theory logical vicious circle: in the words of a classic textbook [1]: “... quantum mechanics occupies a very unusual place among physical theories: it contains classical mechanics as a limiting case, yet at the same time it requires this limiting case for its own formulation.\"", "– The interpretation happens to be probabilistic, thus employing a series of similar experiments, possibly performed in different places and at different times, to establish the meaning of a wave function in the experiment at hand.", "The wave function in this particular experiment, however, certainly appears relevant.", "It therefore seems desirable for the theory to define the meaning of a wave function in every individual experiment, without reference to its repetitions (especially when such repetitions are clearly impossible, such as when discussing the wave function of the universe) which QM fails to do.", "The fact that a wave function must have a nonstatistical interpretation in “internal\" terms also clearly follows from the utility of a concept of a wave function of quarks confined inside hadrons.", "– According to this interpretation, during a measurement the wave function abandons the unitary law of evolution, which it normally follows, and suffers a collapse.", "However, the conditions under which this change of a character of evolution happens, are not specified in QM, and attempts to formulate such conditions have not been convincingly successful.", "– The collapse occurs randomly into different possible states, but QM does not explain the reason for this randomness.", "Consequently, the values of corresponding probabilities (which one would expect to see among the results of a fundamental theory) in QM are not derived, but postulated, or, in other words, are taken from experiment.", "– According to QM, unless a system is in an eigenstate of a measured quantity, the result of its measurement does not exist before the measurement is done, but is rather created during the measurement.", "In some cases this is completely obvious — see a discussion of spin measurement for a particle with spin 3/2 in [2].", "This means that a measurement is not a fundamental unanalyzable primitive, but a nontrivial physical process for which QM fails to give an adequate description.", "– Moreover, this process of measurement produces nonlocally correlated results for measurements performed with space-like separated entangled particles, but QM does not describe any mechanism which causes these correlations.", "Thus it appears that the standard combination of Schrödinger's equation and statistical interpretation is too complicated, artificial and, in the words of John Bell “unprofessionally vague and ambiguous\" [3], while the nature obviously prefers simple, natural, and clear fundamental laws.", "Consequently, we suggest in this paper to regard the situation in the following alternative way: The statistical interpretation does not, of course, follow from the Schrödinger equation (simply because the latter only describes the behavior of a wave function), but is a generalization of results of observations and experiments.", "Our trust in statistical interpretation is based on its agreement with experiment, and only on this agreement.", "Therefore, the interpretation is a phenomenological part of quantum theory, and so the whole existing theory is semi-phenomenological.", "Then to this semi-phenomenological theory the above objections are inapplicable, while at the same time there remains a possibility that the nature is ruled by the other, “simple, natural, and clear\" fundamental theory, from which statistical interpretation (and maybe Schrödinger's equation as well) follows.", "The conclusion about the phenomenological (or, as it is often called, “pragmatic\" [4]) nature of existing QM may also be drawn from the works devoted to its foundations.", "We read, for example, in Bohr [5]: “Strictly speaking, the mathematical formalism of quantum mechanics and electrodynamics merely offers rules of calculation for the deduction of expectations about observations obtained under well-defined experimental conditions specified by classical physical concepts\", or, in a frequently quoted more recent paper [6]: “...quantum theory does not describe physical reality.", "What it does is provide an algorithm for computing probabilities for the macroscopic events\", “...the time dependence of the wavefunction does not represent the evolution of a physical system.", "It only gives the evolution of our probabilities for the outcomes of potential experiments on that system.\"", "Thus according to these works, QM describes the results of our observations of electrons, atoms, etc.", "Which theory, then, describes these particles, which we observe, themselves?", "Of course, neither these quotations, nor the arguments presented above, can prove that a better theory is needed.", "They can, however, motivate a search for such a theory.", "Indeed, it is hard to help feeling that peculiar features of QM are the consequences of a fact that it misses some important part of a complete theory, a part which is substituted by a phenomenological description of the way it works.", "In this searched-for complete theory, the wave function must have a definite meaning in every individual experiment, and the theory must explain the nature of randomness and derive the standard quantum-mechanical expression for probability.", "The expression for probability will thus become just a property of a wave function, rather than a basis for its interpretation.", "This theory should also explain the properties of quantum measurements, describe the mechanism which creates nonlocal correlations, and fill all other gaps listed above; in particular, it should contain an image of every observable element of reality and predict its behavior directly from the theory's dynamical laws, without the need for any special interpretation.", "Compared to QM, such a theory would be much less vulnerable to suspicions of being a mere semi-phenomenology, and it is a goal of this paper to present a theory which appears to satisfy these demands.", "Before discussing this theory, we recall the Hamilton-Jacobi equation $\\frac{\\partial p}{\\partial t} + \\frac{1}{2m} \\, (\\nabla p)^2 + U = 0 $ for an action function $p({\\bf x },t)$ in classical mechanics.", "This function does not describe any individual trajectory and motion of a particle along it; rather, it describes a family of such trajectories, of which none can be singled out given an action function alone.", "Individual trajectories and particles' motions along them are described in classical mechanics by Hamilton ordinary differential equations (ODEs).", "Given these trajectories, the action function, which solves the Hamilton-Jacobi equation, may be obtained by integrating a Lagrangian function along them.", "On the other hand, the trajectories of a family, described by an action function $p({\\bf x },t)$ , may be reconstructed from this function using the equality ${\\bf p }({\\bf x },t) = \\nabla p({\\bf x },t)$ , which says that the momentum ${\\bf p }$ at point $({\\bf x },t)$ of any trajectory is equal to the gradient of an action function at this point.", "Thus the Hamilton and Hamilton-Jacobi equations represent two parts of the same theory — classical mechanics — the former describing particle motion along classical trajectories, and the latter, properties of the families or ensembles of these trajectories.", "Now we formulate the basic idea of the present approach.", "Its initial step is purely mathematical.", "Namely, it is shown in a second section of the paper, that similar to the case of first order partial differential equations (PDEs) with one unknown function, such as just discussed Hamilton-Jacobi equation, for a large class of PDEs of second and higher orders the solution of equation may be represented as an action function, i.e.", "the value of the function $p({\\bf x },t)$ that solves the equation may in every point be obtained as an integral from some “Lagrangian\" function along the curve that leads to this point and is completely and uniquely determined by some system of ODEs.", "As is well known [7], [8], [9], [10], for PDEs of higher than first order the system of ODEs with such properties neither exists in the usual phase space with coordinates $t,x^i, p_i$ , where $p_i = \\partial p/\\partial x^i$ are first derivatives of an unknown function, nor even in the same space extended by adding to its coordinates the derivatives of an unknown function up to any finite order.", "Such a system, however, exists in an infinite phase space, the coordinates of which include all possible partial derivatives of an unknown function, and a corresponding mathematical theory is developed in section 2.", "Although the very possibility of solving higher-order PDEs in this way was known for quite some time [11], the specific form of solution presented in section 2 seems to be new.", "In spite of the presence of an infinite number of variables and equations in the theory, it happens to be quite transparent and manageable; in fact, the theory is remarkably similar to the Hamiltonian formalism in classical mechanics and reproduces all its essential features.", "The theory of first order PDEs also can be formulated in an infinite phase space and turns out to be a special case of our theory, but in this case ODEs for $x^i$ and $p_i$ , i.e.", "for coordinates in the usual phase space, decouple from other equations and can be considered independently.", "Thus, we obtain a general Hamiltonian formalism that covers a large class of PDEs of first as well as higher orders on equal grounds.", "Returning now to physics consider, along with a wave function $\\psi $ , a function $p = (\\hbar /i)\\ln \\psi $ .", "Clearly, this function contains the same information as $\\psi $ , and may be used instead of it in all discussions.", "For a spinless particle of mass $m$ in external potential $U$ we have from the Schrödinger equation the following PDE for $p({\\bf x },t)$ : $\\frac{\\partial p}{\\partial t} + \\frac{1}{2m} \\, (\\nabla p)^2 + U + \\frac{\\hbar }{2im} \\Delta p = 0 \\, .$ Except for the last term, proportional to $\\hbar $ , this is the Hamilton-Jacobi equation for an action function in classical mechanics.", "On the other hand, this equation happens to belong to the class of PDEs considered in section 2, which have a solution in the form of an action function.", "In this situation the following main idea of the present approach emerges with an absolute inevitability: consider Eq.", "(REF ) as an equation for the action function in a new, quantum, theory, the wave function — as an exponent of the new action function (multiplied by $i/\\hbar $ ), the curves along which the Lagrangian function should be integrated to produce the action function — as particle's trajectories in the new theory, and ODEs that determine these curves — as new equations of motion, which correct Hamilton's equations.", "In exact analogy with classical mechanics, the resulting theory will have two sides: ODE side, represented by the equations of motion of particles along their trajectories, and PDE one, represented by Eq.", "(REF ) for the action function that describes, along with the wave function, ensembles of trajectories.", "In the following, Eq.", "(REF ) will be called “quantum Hamilton-Jacobi equation\" (QHJE).", "The motion of particles takes place in an infinite phase space, $\\cal P$ , defined in section 2, and the theory will be called analytical quantum dynamics in infinite phase space, or $\\cal P$ AQD.", "The general structure of $\\cal P$ AQD and classical mechanics is presented in Fig.", "REF .", "Figure: The structure of classical mechanics and 𝒫\\cal PAQD.", "The relations betweenequations of motion and equations for action functions are the same for both theories, and in thelimit ℏ→0\\hbar \\rightarrow 0 quantum equations turn into classical ones in both columns.", "In standardQM the lower left rectangle is absent and substituted by the statistical interpretation.In view of described above new mathematical possibility of dealing with Eq.", "(REF ), it seems difficult to dispute that this approach certainly appears quite natural; one could say that by the very form of Eq.", "(REF ) the nature pushes us in this direction.", "Had the possibility of solving second order PDEs in the way described above been known in 1926-1927, it is hard to doubt that this work would be done right then!", "Further, being based on equations of motion and only on them, the theory should be considered simple; for the same reason it promises to be clear and unambiguous.", "Thus, it seems worth the efforts to investigate the possibilities which may open in this direction; in doing so we will also finish the job left unfinished eighty five years ago due to such historical accident as an absence of a proper mathematical formalism at that time.", "Last but not least, we note that the theory is completely fixed by Eq.", "(REF ) for its action function (that is — fixed by the Schrödinger equation) and doesn't contain any additional freedom to improve its agreement with the second part of standard QM, the statistical interpretation.", "Therefore the fact that, as we will soon see, such agreement is nevertheless achieved (or, in other words, that statistical interpretation is deduced from the present theory) should be considered as a weighty argument in the theory's favor.", "The equations of motion of the theory are explicitly written down in the beginning of section 3, and the rest of the paper is devoted to demonstrating that QM may be understood as a theory of particles moving according to these equations, the difference between classical and quantum mechanics being the result of the different form and number of equations in these theories.", "Equations of motion are followed in section 3 by their general discussion.", "As in classical mechanics, these equations are self-sufficient: given initial conditions, they define the particle's motion unambiguously, without any need for using a wave function or the Schrödinger equation.", "In the classical limit $\\hbar \\rightarrow 0$ , the equations of motion turn directly into the Hamilton equations of classical mechanics.", "The equations and particle trajectories live in an infinite phase space.", "We show that projections of these trajectories to physical space coincide with “Bohmian trajectories,\" introduced by de Broglie and Bohm [12], [13] on very different grounds, and discuss the relation between the present theory and that of de Broglie and Bohm.", "The class of PDEs covered by the theory in section 2 is very large, and so the question arises: what singles Eq.", "(REF ) out of this class?", "It is shown in section 4, that equations of motion may be obtained from a “one-step Feynman integral\" (OSFI), combined with an appropriate variational principle.", "OSFI provides, therefore, an alternative starting point of the theory, bypassing the geometric theory of section 2.", "Moreover, OSFI is a functional of a Lagrangian function, which is necessarily classical, i.e., depends on the position and velocity of a particle only.", "Consequently, OSFI may be considered as a general source of quantum theories, obtained by “quantization\" of corresponding classical theories, represented by Lagrangian functions.", "As further discussed in section 4, the theories, obtained in this way, will automatically exhibit familiar features of QM: superposition principle, path-integral representation, and wave-particle duality.", "We note that in a mathematical derivation of the latter feature, an infinite number of variables and equations in our theory, which initially appears to be a theory's disadvantage, plays a crucial role.", "Another consequence of OSFI is that a corresponding PDE may be obtained from a variational principle.", "This is shown in the beginning of section 5.1.", "By Noether's theorem, the symmetries of such a PDE lead to conservation laws.", "We then use a fundamental invariance of all PDEs, considered in section 2, with respect to a shift of the unknown function by a constant to derive a continuity equation.", "In section 5.1 this is done for a standard Hamiltonian of the Schrödinger equation, and in section 5.2 — in a general case, without using an explicit form of a Hamiltonian.", "In section 5.3, we use the current conservation to prove that a form $|\\psi |^2 dV$ is an integral invariant of our equations of motion, which replaces the canonical integral invariant (Liouville measure) $d^3\\!p\\, dV$ of the Hamilton equations in classical mechanics.", "Using the invariance of the form $|\\psi |^2 dV$ , section 6 demonstrates that a probability density in configuration space should be equal to $|\\psi |^2$ .", "We give two proofs, the second one using the maximization of a specially introduced functional of probability density, analogous to the Gibbs entropy.", "We compare the situation in QM to the one in classical statistics.", "A brief review of equilibrium and nonequilibrium classical statistics is presented in the Appendix in a form convenient for such comparison.", "It is shown there that the repetition of steps which led to the expression of Gibbs entropy in QM leads in classical statistics (where for invariant measure one uses the form $d^3\\!p\\, dV$ , rather than $|\\psi |^2 dV$ ) to its standard classical expression.", "The probability density $|\\psi |^2$ that maximizes Gibbs entropy in QM has, therefore, a status identical to that of a microcanonical distribution, which maximizes Gibbs entropy in classical statistics.", "Note that the difference between these distributions results from the difference between corresponding invariant measures, which, in turn, follows from the difference in equations of motion.", "Regarding the claims [12], [14] that the $|\\psi |^2$ distribution in QM may arise in a way similar to relaxation to statistical equilibrium in classical statistics, the Appendix also shows that this relaxation is related to the growth of Boltzmann, rather than Gibbs, entropy, and is caused by the properties of macroscopic systems which cannot have any analogs in a one-particle theory.", "In section 7, the one-particle theory of the previous sections is generalized to multiparticle systems.", "We also discuss how the standard physical picture of quantum particles in a potential created by classical macroscopic objects emerges from our theory.", "Section 8 considers the theory of quantum measurements.", "Von Neumann's measurements with discrete and continuous spectra are considered in sections 8.1 and 8.2.", "The theory discussed there is a $\\cal P$ AQD-adaptation of the theory developed by Bohm [12], [13].", "In section 8.3, the measurement of a particle's position by a photographic plate or in a bubble chamber, which is not a von Neumann's measurement, is considered, and its properties are discussed.", "In the end of this section we analyze the double-slit experiment discussed by Feynman [15] and compare its results with $\\cal P$ AQD predictions.", "Section 9 considers the mechanism of nonlocal correlations between the results of measurements, performed with space-like separated, but entangled, particles.", "We argue that the relativistic version of $\\cal P$ AQD, although nonlocal, will be Lorentz invariant.", "Section 10 considers particles with spin.", "We show that their theory, which adequately generalizes the theory of spinless particles, may be developed based on extended configuration space, which includes, besides the particle's space position, also its “internal\" SU(2) coordinates.", "Finally in Conclusion, we give a brief review of our theory, compare it with standard QM, and finish with several general remarks." ], [ "Basic definitions and notation", "In this section we discuss the question of when the solution of a PDE system may be obtained, as in the case of the Hamilton-Jacobi equation, via solving some related system of ODEs.", "Consider evolutionary PDEs of the form $\\frac{\\partial p^r}{\\partial t} + H^r = 0\\,, \\,\\,\\,\\, r = 1, \\ldots , m \\, , $ where the $p^r$ are $m$ unknown functions (“dependent variables\") of $n$ space variables $q^1, \\ldots , q^n$ combined into a vector ${\\bf q }$ , and time $t$ (“independent variables\"), and $H^r$ are functions of $t$ , ${\\bf q }$ , and partial derivatives of unknown functions with respect to space variables up to some finite order.", "Denote these derivatives by corresponding multi-indices, as in $p^r_{ij} = \\partial ^2 p^r/ \\partial q^i \\partial q^j$ , and include in the set of all possible multi-indices an empty one, denoted as $\\mbox{ø}$ , which will correspond to the function $p^r$ itself.", "Use $i,j,k$ for space indices, running from 1 to $n$ , use $r$ and $s$ for function indices, running from 1 to $m$ , and use Greek letters for multi-indices.", "The order of indices in a multi-index is arbitrary, and two multi-indices which differ only by permutation are considered to be the same.", "Correspondingly, only one such multi-index will be assumed to be included in a summation over all possible multi-indices.", "If $\\sigma = i_1 i_2 \\ldots i_k$ , let $\\sigma i$ or $i\\sigma $ be the “extended\" multi-index $i_1 i_2 \\ldots i_k i$ , and if $\\mu =j_1\\ldots j_l$ , let $\\sigma \\mu $ or $\\mu \\sigma $ be the multi-index $i_1\\ldots i_kj_1\\ldots j_l$ .", "Let $\\sigma _i$ , $i = 1,\\ldots ,n$ , denote the number of times index $i$ is found in the multi-index $\\sigma $ , so that $\\sigma $ may be represented as $\\sigma _1$ ones, followed by $\\sigma _2$ twos, etc.", "It is useful to think of the multi-index $\\sigma $ as an $n$ -dimensional vector with nonnegative integer components $\\sigma _i$ .", "Summation over all possible multi-indices $\\sigma $ then reduces to summation over all $\\sigma _i$ : $\\sum _\\sigma = \\sum _{\\sigma _1,\\ldots ,\\sigma _n=0}^\\infty \\,$ .", "Let $|\\sigma |$ denote the total number of indices in multi-index $\\sigma $ , so $|\\sigma | = \\sum _{i=1}^n \\sigma _i$ .", "Let $\\partial _0 = \\partial _t = \\partial / \\partial t$ .", "For every multi-index $\\sigma $ , let $\\sigma !=\\prod _{i=1}^n\\sigma _i!$ and $\\partial _\\sigma =\\prod _{i=1}^n (\\partial / \\partial q^i)^{\\sigma _i}$ .", "For any $n$ -dimensional vector ${\\bf x }$ , let ${\\bf x }^\\sigma =\\prod _{i=1}^n (x^i)^{\\sigma _i}$ .", "Let $\\lbrace p\\rbrace $ denote the set of all unknown functions and all their derivatives.", "By analogy with classical mechanics, functions $p^r$ will be called action functions, or just actions, and their derivatives, momentums.", "Denote the set of all momentums, i.e., all $p^r_\\sigma $ with $\\sigma \\ne \\mbox{ø}$ , by ${\\bf p }$ , so $H^r = H^r(t,{\\bf q },{\\bf p })$ .", "Denote the space of independent variables ${\\bf q }$ and $t$ (“base space\") by $M$ , and the space of vectors ${\\bf q }$ alone (“configuration space\") by $Q$ .", "Let $J^\\infty $ (“infinite jet space\") be the space with coordinates $t$ , ${\\bf q }$ , $\\lbrace p\\rbrace $ , i.e., all independent as well as dependent variables and all their space derivatives.", "Call the similar space $\\cal P$ with coordinates $t$ , ${\\bf q }$ , ${\\bf p }$ “infinite phase space.\"", "We assume that $H^r$ depends analytically on its arguments, and consider only analytic or real-analytic solutions of Eq.", "(REF ).", "A mathematically rigorous treatment of geometry of analytic jets in $J^\\infty $ may be found in [16].", "Denote by $J^k$ a jet space of $k$ -times continuously differentiable functions, which includes derivatives only up to $k$ -th order (“k-jets space\") [7], [8], [9].", "We will also use the notation $J^k_m$ when it is necessary to indicate the number $m$ of unknown functions explicitly, and when $m=1$ we will drop the function number superscript in equations.", "We will consider every solution of Eq.", "(REF ), with all its space partial derivatives, as creating a graph in $J^\\infty $ .", "Denote such a graph by $\\Gamma $ and note it is an $n+1$  – dimensional surface in $J^\\infty $ .", "We want to find out when such graphs can be usefully considered as formed by a congruence of curves described by a system of ODEs." ], [ "Curves in the graph of a PDE solution", "It is easy to write an equation for an arbitrary curve which lies in the graph.", "Let ${\\bf q }(t)$ be the projection of the curve to the base space $M$ .", "Consider the operator of total differentiation $D_i = \\frac{\\partial }{\\partial q^i} + \\sum _{r = 1}^m \\sum _\\sigma p^r_{\\sigma i}\\frac{\\partial }{\\partial p^r_\\sigma }\\, , $ where the second summation runs over all possible multi-indices $\\sigma $ .", "At every point of $J^\\infty $ , the operator $D_i$ raises the partial derivative $\\partial /\\partial q^i$ to a graph of an analytic function of ${\\bf q }$ , which passes through the point (see [7]).", "In other words, if $\\Theta $ is such a graph, $\\lbrace p_\\Theta ({\\bf q })\\rbrace $ is a set of values of unknown functions and their derivatives at a point of $\\Theta $ with a base coordinate ${\\bf q }$ , and $F(t,{\\bf q },\\lbrace p\\rbrace )$ is some function in $J^\\infty $ , then $\\frac{\\partial }{\\partial q^i}\\, F\\big (t,{\\bf q },\\lbrace p_\\Theta ({\\bf q })\\rbrace \\big ) = D_i F\\!\\left.\\big (t,{\\bf q },\\lbrace p\\rbrace \\big )\\right|_{\\lbrace p\\rbrace = \\lbrace p_\\Theta ({\\bf q })\\rbrace }\\, .", "$ By consecutive differentiation of Eq.", "(REF ) with respect to space variables we now obtain equations (“prolongations\" of (REF )) which describe the behavior of space derivatives $p^r_\\sigma $ of the solution $\\frac{\\partial p^r_\\sigma }{\\partial t} + H^r_\\sigma = 0 \\, , $ where $H^r_\\sigma = D_\\sigma H^r$ and $D_\\sigma $ is repeated total differentiation, i.e., $D_\\sigma = \\prod _{i=1}^n (D_i)^{\\sigma _i}$ .", "Eq.", "(REF ) gives the value of the partial derivative of $p^r_\\sigma $ with respect to $t$ , while its partial derivative with respect to $q^i$ is, by definition, $p^r_{\\sigma i}$ .", "Consequently, the time dependence of the $\\Gamma $ -image of the point ${\\bf q }(t)$ on the base is described by the system of ODEsSummation over repeated indices and multi-indices is assumed here and below unless stated otherwise.", "The summation will not be assumed when the expression with repeated indices or multi-indices stands on one side of an equality if the other side of the equality also has the same (multi-)indices used only once.", "$\\begin{array}{ccl}\\displaystyle \\dot{p}^r_\\sigma & = & \\displaystyle \\frac{\\partial p^r_\\sigma }{\\partial q^i}\\, \\dot{q}^i + \\frac{\\partial p^r_\\sigma }{\\partial t} \\\\[0.4cm]& = & \\displaystyle p^r_{\\sigma i}\\, \\dot{q}^i - H^r_\\sigma \\, , \\end{array}$ where by dot we denote the total derivative of a corresponding value with respect to $t$ along the curve ${\\bf q }(t)$ (see Fig.", "REF ).", "Figure: A graph Γ\\Gamma of solution of PDE () in the infinite jet space.", "When apoint moves in the configuration space, the set of its coordinates 𝐪{\\bf q } and values {p}\\lbrace p\\rbrace ofsolution and solutions's derivatives at 𝐪{\\bf q } moves in Γ\\Gamma according to the system ofODEs ().Given the PDE (REF ), we constructed the ODE system (REF ), which defines the evolution of values of unknown functions and their derivatives at a point ${\\bf q }(t)$ moving (in an arbitrary way) through the base.", "It is easy to show that there also exists an inverse correspondence: if there is a point ${\\bf q }(t)$ moving through the base, and a sequence of functions of time $p^s_\\sigma (t)$ , which satisfies the system of ODEs (REF ), then functions $p^s({\\bf r},t)$ , which are the sums of Taylor series in a point ${\\bf q }(t)$ with coefficients $p^s_\\sigma (t)$ , will satisfy PDE (REF ).", "Indeed, these Taylor series are $p^s({{\\bf r }},t)\\, = \\sum _\\sigma \\frac{1}{\\sigma !}", "\\, p^s_\\sigma (t)\\big ({\\bf r }- {\\bf q }(t)\\big )^\\sigma .", "$ Taking the time derivative and using Eq.", "(REF ) for $\\dot{p}^s_\\sigma $ , one obtains, after some cancellations, $\\frac{\\partial }{\\partial t}\\, p^s({\\bf r },t)\\, = \\,- \\sum _\\sigma \\frac{1}{\\sigma !}", "\\, D_\\sigma H^s \\!\\!\\left.", "\\big (t,{\\bf q },\\lbrace p\\rbrace \\big )\\right|_{\\lbrace p\\rbrace = \\lbrace p^s_\\sigma (t)\\rbrace }\\big ({\\bf r }- {\\bf q }(t) \\big )^\\sigma , $ which by Eq.", "(REF ) and analyticity of $H^s$ is equal to $-H^s\\big (t,{\\bf r },\\lbrace p_\\Gamma ({\\bf r })\\rbrace \\big )$ , where $\\Gamma $ is the graph of functions (REF ) in $J^\\infty $ , as required.", "We see that there exists a simple and general relation between the evolution of analytic functions, described by PDE (REF ), on one hand, and the evolution of coordinates ${\\bf q }(t)$ and momentums $p^r_\\sigma (t)$ , described by ODE system (REF ), on the other.", "This relation is just a straightforward consequence of the structure of Eqs.", "(REF ) and (REF ), and is satisfied for an arbitrary curve ${\\bf q }(t)$ .", "However, if one adds to system (REF ) an additional equation, expressing $\\dot{q}^i$ through other variables, then this extended system of ODEs will be closed with respect to the evolution of all variables involved, and may be viewed as representing equations of motion that are generated by “Hamiltonians\" $H^r$ in the same way as Hamilton equations are generated by a Hamiltonian in classical mechanics.", "Consequently, to obtain potential equations of motion, some plausible condition capable of providing such an equation is needed." ], [ "Generalized Hamiltonian fields and equations of motion", "To formulate the condition described above, we will use concepts and notation of the geometric theory of PDEsA reader unfamiliar with this theory may skip directly to Eq.", "(REF ), which is a desired expression for velocity.", "An elementary justification of this expression is presented in a footnote immediately after it.", "[7], [8], [9], [10], [17], [18].", "The set $\\left\\lbrace D_i, \\, \\partial /\\partial p^r_\\sigma , \\, \\partial /\\partial t \\right\\rbrace $ is a basis for the tangent bundle $T(J^\\infty )$ .", "Define the sequence of 1-forms $\\widetilde{\\omega }^r_\\sigma = \\mathrm {d}p^r_\\sigma - p^r_{\\sigma i}\\mathrm {d}q^i .", "$ The set $\\left\\lbrace \\mathrm {d}q^i, \\, \\widetilde{\\omega }^r_\\sigma , \\, \\mathrm {d}t \\right\\rbrace $ is a basis for the cotangent bundle $T^*(J^\\infty )$ .", "The duality relations $\\begin{array}{lcllcllcl}D_i \\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\mathrm {d}q^j & = & \\delta _{ij} \\, ,\\quad & D_i \\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\widetilde{\\omega }^r_\\sigma & = & 0 \\, ,\\quad & D_i \\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\mathrm {d}t & = & 0 \\, , \\\\[0.3cm]\\displaystyle \\frac{\\partial }{\\partial p^r_\\sigma } \\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\mathrm {d}q^i & = & 0 \\, ,\\quad & \\displaystyle \\frac{\\partial }{\\partial p^r_\\sigma } \\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\widetilde{\\omega }^s_\\nu & = & \\delta _{rs} \\delta _{\\sigma \\nu } \\, ,\\quad & \\displaystyle \\frac{\\partial }{\\partial p^r_\\sigma } \\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\mathrm {d}t & = & 0 \\, , \\\\[0.5cm]\\displaystyle \\frac{\\partial }{\\partial t} \\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\mathrm {d}q^i & = & 0 \\, ,\\quad & \\displaystyle \\frac{\\partial }{\\partial t} \\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\widetilde{\\omega }^r_\\sigma & = & 0\\, ,\\quad & \\displaystyle \\frac{\\partial }{\\partial t} \\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\mathrm {d}t & = & 1 \\, ,\\end{array}$ then follow at every point of $J^\\infty $ .", "In the following, let $T^r_\\sigma $ be the distribution generated by the fields $D_i$ and $\\partial /\\partial p^r_{\\sigma i}$ , $i = 1,\\ldots ,n$ .", "The curves (REF ) are now recognized as integral curves of a vector field $\\begin{array}{ccl}X & = & \\displaystyle \\frac{\\partial }{\\partial t} + \\dot{q}^i \\frac{\\partial }{\\partial q^i} + \\dot{p}^r_\\sigma \\frac{\\partial }{\\partial p^r_\\sigma } \\\\[0.4cm]& = & \\displaystyle \\frac{\\partial }{\\partial t} + \\dot{q}^i D_i - H^r_\\sigma \\frac{\\partial }{\\partial p^r_\\sigma } \\, , \\end{array}$ which cancels the Pfaff system of differential forms (“Cartan forms\") $\\omega ^r_\\sigma = \\mathrm {d}p^r_\\sigma - p^r_{\\sigma i}\\, \\mathrm {d}q^i + H^r_\\sigma \\, \\mathrm {d}t \\, , $ i.e., satisfies equations $X \\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\omega ^r_\\sigma = 0, \\quad \\mbox{for all} \\;\\, r, \\sigma \\, .", "$ The system $\\lbrace \\omega ^r_\\sigma \\rbrace $ defines in $J^\\infty $ a distribution (“Cartan distribution\" or CD), and the graphs $\\Gamma $ of solutions of (REF ) are the integral manifolds of this distribution, which is a necessary and sufficient condition for every surface in $J^\\infty $ to satisfy two requirements: First, to truly represent a graph of some function, i.e., to ensure agreement between coordinates $p^r_\\sigma $ in $J^\\infty $ and the values of corresponding partial derivatives of a function $p^r({\\bf q }, t)$ represented by this graph.", "And second, to guarantee that this function satisfies Eq.", "(REF ) [7], [8], [9].", "In the case of first order PDEs with $m=1$ , considered in a space of one-jets $J^1_1$ , the trajectories, described by Hamilton equations, are the characteristic curves of a corresponding exterior differential system.", "These curves are uniquely defined in $J^1_1$ , and so one and only one of them passes through every point on the graph $\\Gamma $ of the PDE solution.", "It is therefore natural to expect that the desired trajectories in $J^\\infty $ are the characteristic curves of the system $\\lbrace \\omega ^r_\\sigma \\rbrace $ .", "This condition, however, happens to not be sufficient for the selection of $\\dot{{\\bf q }}$ .", "Indeed, by direct calculation it is easy to check that the differential forms (REF ) satisfy the equation $\\mathrm {d}\\omega ^r_\\sigma = \\mathrm {d}q^i \\wedge \\omega ^r_{\\sigma i}- \\frac{\\partial H^r_\\sigma }{\\partial p^s_\\nu }\\,\\, \\mathrm {d}t \\wedge \\omega ^s_\\nu \\,.", "$ From Eqs.", "(REF ) and (REF ) we immediately obtain that $X \\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\mathrm {d}\\omega ^r_\\sigma $ is a linear combination of forms (REF ).", "Therefore, for every $\\dot{{\\bf q }}$ the vector field $X$ not only belongs to CD, but is also its characteristic field, and so in sharp contrast with the case of first order PDEs, there is a continuum of characteristic curves of CD passing through every point of $\\Gamma $ in $J^\\infty $ .", "It is instructive to consider the source of this difference.", "By analyticity, the graph of one and only one analytic function can pass in $J^\\infty $ through every point, which defines all spatial partial derivatives.", "Therefore, $J^\\infty $ is split into a foliation, each leaf of which is a graph $\\Gamma $ of the analytic solution of Eq.", "(REF ), so that every point of $J^\\infty $ belongs to one and only one leaf.", "These leaves are the integral manifolds of CD, so CD in the infinite jet space of analytic functions is completely integrable.We note that this complete integrability cannot be considered a consequence of the Frobenius theorem and Eq.", "(REF ), for the Frobenius theorem requires finite dimensionality of the space, and in infinite-dimensional space is no longer true [8].", "In contrast to $J^\\infty $ , in $J^1_1$ CD is not completely integrable.", "Consequently, the dimension of the graphs $\\Gamma $ there is less than the dimension of CD, and so through every point of $J^1_1$ different integral manifolds of CD (i.e.", "graphs of solutions $\\Gamma $ ) pass, namely, with different values $p^{\\scriptscriptstyle \\Gamma }_{ik}$ of the second derivatives on $\\Gamma $ .", "On the other hand, graphs $\\Gamma $ are formed by characteristic curves, so at every point these curves should belong to every graph that passes through this point.", "It turns out that this condition alone is sufficient to uniquely specify the characteristic field $X$ at this point, including the value of $\\dot{{\\bf q }}$ .", "Indeed, the corresponding PDE has the form $\\frac{\\partial p}{\\partial t} + H = 0\\,.", "$ On a graph $\\Gamma $ of a given solution of (REF ) in $J^1_1$ , the operator of the total derivative is $D^{\\scriptscriptstyle \\Gamma }_i = \\frac{\\partial }{\\partial q^i} + p_i \\, \\frac{\\partial }{\\partial p} +p^{\\scriptscriptstyle \\Gamma }_{ik} \\, \\frac{\\partial }{\\partial p_k} \\, .", "$ Similar to (REF ), the curves on $\\Gamma $ should satisfy a system of ODEs: $\\dot{p} & = & \\displaystyle p_i \\, \\dot{q}^i - H , \\\\[0.1cm]\\dot{p}_i & = & \\displaystyle p^{\\scriptscriptstyle \\Gamma }_{ik}\\,\\dot{q}^k -D^{\\scriptscriptstyle \\Gamma }_i H \\nonumber \\\\[0.1cm]& = & \\displaystyle p^{\\scriptscriptstyle \\Gamma }_{ik} \\left(\\dot{q}^k - \\frac{\\partial H}{\\partial p_k} \\right) - \\left( \\frac{\\partial }{\\partial q^i} +p_i \\frac{\\partial }{\\partial p} \\right) H .", "$ Now, as characteristic curves should belong to every such graph, the dependence on $p^{\\scriptscriptstyle \\Gamma }_{ik}$ for them must disappear.", "Recalling also that $H$ does not depend on $p$ , we then obtain from Eq.", "() the standard Hamilton equations for $\\dot{q}^i$ and $\\dot{p}_i$ : $\\dot{q}^i = \\frac{\\partial H}{\\partial p_i} \\, , \\quad \\; \\dot{p}_i = -\\frac{\\partial H}{\\partial q^i} \\, .", "$ Thus, the fact that in $J^1_1$ characteristics of CD have a Hamiltonian form is based on the specifics of $J^1_1$ , namely, on the lack of complete integrability of CD there, which explains why this property cannot be generalized to $J^\\infty $ .", "Fortunately, however, the requirement of being characteristics is not the only one which distinguishes Hamiltonian curves from all other curves that lie on the graphs of solutions of Eq.", "(REF ) in $J^1_1$ .", "Let $\\omega = \\mathrm {d}p - p_i \\mathrm {d}q^i + H \\mathrm {d}t$ be the (only) Cartan form in $J^1_1$ .", "The Hamiltonian curves in $J^1_1$ may then be defined as integral curves of a vector field $X$ , which cancels the 2-form $\\mathrm {d}\\omega $ , i.e., satisfies the condition $X \\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\mathrm {d}\\omega = 0$ [19].", "This condition may of course be rewritten as $\\xi \\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}(X \\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\mathrm {d}\\omega ) = 0$ , for all $\\xi \\in T\\big (J^1_1\\big )$ .", "Now, in $J^\\infty $ , CD is defined by a sequence of forms $\\lbrace \\omega ^r_\\sigma \\rbrace $ , and a fruitful generalization of the above condition in $J^1_1$ is to require $\\xi \\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}(X \\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\mathrm {d}\\omega ^r_\\sigma ) = 0, \\quad \\mbox{for all} \\;\\, r,\\sigma , \\xi \\in T^r_\\sigma ,$ in $J^\\infty $ , which is a condition satisfied by the usual Hamiltonian curves of first-order PDEs with $m=1$ after their prolongation from $J^1_1$ to $J^\\infty _1$ .", "In the following, we will call fields $X$ that satisfy conditions (REF ) generalized Hamiltonian or, for short, simply Hamiltonian as they are called in $J^1_1$ .", "To find their form we observe that the internal product of $X$ , given by the first line of Eq.", "(REF ), with $\\mathrm {d}\\omega ^r_\\sigma $ , is $X \\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\mathrm {d}\\omega ^r_\\sigma = \\sum _i \\left( \\dot{q}^i - \\frac{\\partial H^r_\\sigma }{\\partial p^r_{\\sigma i}} \\right) \\omega ^r_{\\sigma i}\\,- \\, \\sum _i \\left( \\dot{p}^r_{\\sigma i}- l^r_{\\sigma i}\\right) \\mathrm {d}q^i \\, - \\,{\\sum _{s,\\, \\nu }}^{\\prime } \\frac{\\partial H^r_\\sigma }{\\partial p^s_\\nu } \\, \\omega ^s_\\nu \\, + \\, \\sum _{s,\\, \\nu }\\left( \\dot{p}^s_\\nu - l^s_\\nu \\right) \\frac{\\partial H^r_\\sigma }{\\partial p^s_\\nu } \\, \\mathrm {d}t \\, , $ where all summations are explicit, $\\sum ^{\\prime }_{s,\\, \\nu }$ omits terms with $(s,\\nu ) =(r, \\sigma i)$ for all $i$ , and where we introduced functions $l^r_\\sigma = l^r_\\sigma (t,{\\bf q },{\\bf p },\\dot{{\\bf q }})$ by the relations $l^r_\\sigma = p^r_{\\sigma i}\\dot{q}^i - H^r_\\sigma \\, .", "$ Now if the field $\\xi $ belongs to $T^r_\\sigma $ , i.e.", "is a linear combination of $D_i$ and $\\partial / \\partial p^r_{\\sigma i}$ , then only the first two terms in (REF ) contribute to $\\xi \\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}(X \\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\mathrm {d}\\omega ^r_\\sigma )$ , and the condition (REF ) gives $\\dot{q}^i & = & \\displaystyle \\frac{\\partial H^r_\\sigma }{\\partial p^r_{\\sigma i}} \\, , \\\\[0.2cm]\\dot{p}^r_{\\sigma i}& = & l^r_{\\sigma i}, $ where in Eq.", "(REF ) summation over $r$ and $\\sigma $ is not assumed.", "Equations () (with arbitrary $\\sigma $ ) just reproduce Eqs.", "(REF ) (with $\\sigma \\ne \\mbox{ø}$ ), while Eq.", "(REF ) gives the desired expression for velocity.Note that for first order PDEs with one unknown function this expression does not depend on $\\sigma $ and gives the same value of $\\dot{q}^i$ as in Eq.", "(REF ).", "Consequently, one can bypass the geometric consideration above by simply postulating (REF ) in a general case.", "For all PDEs such that the right hand side of (REF ) does not depend on $r$ and $\\sigma $ , the resulting theory will be a generalization of the usual Hamiltonian formalism for first order PDEs with one unknown function.", "For this expression to make sense, we must additionally require that the derivative there be independent of $r$ and $\\sigma $ .", "We will prove the following: $\\partial H^r_\\sigma /\\partial p^r_{\\sigma i}$ does not depend on $\\sigma $ if $H^r$ satisfies the condition $D_k \\, \\frac{\\partial H^r}{\\partial p^r_{\\nu k}} = 0, \\quad \\mbox{for all} \\;\\, k, \\nu \\ne \\mbox{ø} $ (no summation over $r$ and $k$ here!", "), as, for example, if $H^r$ is linear in second and higher order derivatives of $p^r$ with constant coefficients.", "First we show that if any function $F(t,{\\bf q },\\lbrace p\\rbrace )$ depends on derivatives $p^r_\\sigma $ only up to some finite order, and satisfies the condition $D_k F = 0$ , then it is a function of $t$ and $q^j$ , $j\\ne k$ , only.", "Indeed, let $r, \\nu $ be such that $\\partial F / \\partial p^r_\\sigma = 0$ for all $\\sigma $ such that $|\\sigma | > |\\nu |$ .", "Then applying the obvious identity $\\frac{\\partial }{\\partial p^r_\\nu } = \\frac{\\partial }{\\partial p^r_{\\nu k}} D_k - D_k \\frac{\\partial }{\\partial p^r_{\\nu k}} \\, ,$ where summation over $k$ is not assumed, to the function $F$ , we obtain immediately that $\\partial F / \\partial p^r_\\nu = 0$ also.", "Consequently, $F$ cannot depend on any $p^r_\\nu $ at all, and is a function of $t$ and ${\\bf q }$ only, so that the equalities $0 = D_k F = \\partial F /\\partial q^k$ prove our statement.", "Now for a function $H^r$ which satisfies condition (REF ), this means that for any nonempty multi-index $\\nu $ and any $k$ , $\\partial H^r / \\partial p^r_{\\nu k}$ may only be a function of $q^j$ , $j\\ne k$ , and $t$ .", "Consequently, $H^r$ has the form $H^r = H^r_1 + H^r_2$ , where $H^r_1 = H^r_1\\big (t,{\\bf q },p^r_i,\\lbrace p^s_\\sigma , s \\ne r\\rbrace \\big )$ , and $H^r_2=\\sum _{|\\nu |>1} a_\\nu p^r_\\nu $ , where $a_\\nu $ depends on $t$ and $q^j$ , $j \\notin \\nu $ , only.", "It can then be easily seen that $H^r_{2\\sigma }$ does not contain $p^r_{\\sigma i}$ at all, while for any $\\sigma \\ne \\mbox{ø}$ the only term with $p^r_{\\sigma i}$ in $H^r_{1\\sigma }$ is an additive term equal to $p^r_{\\sigma i}\\,\\partial H^r_1 / \\partial p^r_i$ , and so all $\\partial H^r_\\sigma /\\partial p^r_{\\sigma i}$ are equal to $\\partial H^r_1 / \\partial p^r_i$ .", "We conclude that functions $H^r$ produce a Hamiltonian field if they satisfy the following “Hamiltonian conditions\" (HC): First, for all $r$ , the $H^r$ satisfy Eq.", "(REF ) (HC1).", "Second, given $i$ , $\\partial H^r/ \\partial p^r_i$ is independent of $r$ (HC2).", "Equation (REF ) then gives for $\\dot{q}^i$ the values that are the same for all $r$ and $\\sigma $ , and, along with Eq.", "(REF ) for $\\dot{p}^r_\\sigma $ , constitutes the desired system of equations of motion.", "We will now discuss the structure of the resulting theory in more detail.", "First note that the right hand sides of Eqs.", "(REF ) and (REF ) depend only on $p^r_\\sigma $ with $\\sigma \\ne \\mbox{ø}$ , and not on $p^r$ .", "Consequently, this system of equations splits into two subsystems.", "The first subsystem consists of Eq.", "(REF ) and Eq.", "(REF ) with $\\sigma \\ne \\mbox{ø}$ , which determine time histories (trajectories) ${\\bf q }(t)$ , ${\\bf p }(t)$ .", "The second subsystem, consisting of equations (REF ) with $\\sigma =\\mbox{ø}$ , shows how actions $p^r$ vary along these trajectories.", "The equations of the first subsystem, which is equivalent to the system (REF ) and (), constitute the desired set of equations of motion.", "After the equations of motion are solved, equations of the second subsystem allow us to obtain expressions for the actions $p^r$ along corresponding trajectories by quadrature.", "As will be discussed later, if one makes equations of motion a starting point of a theory, then these expressions may be considered as defining actions for given trajectories.", "We will now show that Eq.", "() may be presented in a “Hamiltonian\" form, similar to the second Eq.", "(REF ).", "Indeed, using Eq.", "(REF ) and making all summations explicit, we have $\\begin{array}{ccl}l^r_{\\sigma i}& = & \\displaystyle \\sum _j p^r_{\\sigma i j} \\dot{q}^j - D_i H^r_\\sigma \\\\[0.2cm]& = & \\displaystyle \\sum _j p^r_{\\sigma i j} \\dot{q}^j - \\left(\\sum _j p^r_{\\sigma j i}\\frac{\\partial }{\\partial p^r_{\\sigma j}} + \\frac{\\partial }{\\partial q^i} + {\\sum _{s,\\, \\nu }}^{\\prime }p^s_{\\nu i} \\frac{\\partial }{\\partial p^s_\\nu } \\right) H^r_\\sigma \\\\[0.4cm]& = & \\displaystyle - \\left(\\frac{\\partial }{\\partial q^i} + {\\sum _{s,\\, \\nu }}^{\\prime }p^s_{\\nu i} \\frac{\\partial }{\\partial p^s_\\nu } \\right) H^r_\\sigma \\, ,\\end{array}$ where $\\sum ^{\\prime }_{s,\\, \\nu }$ omits terms with $(s,\\nu ) = (r, \\sigma j)$ for all $j$ .", "Now consider $H^r_\\sigma $ as a function of $p^r_{\\sigma j}$ , $j = 1,\\ldots ,n$ , and all its other arguments: $H^r_\\sigma =H^r_\\sigma (t,{\\bf q },{\\bf p }^{\\prime },p^r_{\\sigma j})$ , where ${\\bf p }^{\\prime }$ is the set of all $p^s_\\nu $ such that $\\nu \\ne \\mbox{ø}$ and for any $j$ , $(s,\\nu ) \\ne (r, \\sigma j)$ .", "Let $\\Theta $ be the graph of a $t$ -independent analytic function of ${\\bf q }$ passing through a point with $\\cal P$ -coordinates $(t, {\\bf q }, {\\bf p })$ in $J^\\infty $ .", "Let ${\\bf p }^{\\prime }_\\Theta ({\\bf q })$ be the set of values of ${\\bf p }^{\\prime }$ at a point of $\\Theta $ with a base coordinate ${\\bf q }$ .", "Then using (REF ), Eq.", "() may be written as $\\dot{p}^r_{\\sigma i}= - \\frac{\\partial }{\\partial q^i} H^r_\\sigma \\big (t,{\\bf q }, {\\bf p }^{\\prime }_\\Theta ({\\bf q }), p^r_{\\sigma j}\\big ) \\, .", "$ Although the right hand side of this equation is nothing but the last line of Eq.", "(REF ) rewritten less explicitly, we prefer this form because of its obvious analogy with the standard Hamilton equation for momentum.", "Making the reference to graph $\\Theta $ implicit, the system of equations of motion (REF ) and () can now be presented in the form $\\dot{q}^i = \\displaystyle \\frac{\\partial }{\\partial p^r_{\\sigma i}}\\, H^r_\\sigma \\big (t,{\\bf q }, {\\bf p }^{\\prime }({\\bf q }), p^r_{\\sigma j}\\big )\\,, \\quad \\; \\dot{p}^r_{\\sigma i}=- \\frac{\\partial }{\\partial q^i}\\, H^r_\\sigma \\big (t,{\\bf q }, {\\bf p }^{\\prime }({\\bf q }), p^r_{\\sigma j}\\big ) \\, .", "$ We will informally refer to variables $q^i$ , $p^r_\\sigma $ , and $p^r_{\\sigma i}$ as forming an $r$ -$\\sigma $ sector of the theory.", "Evidently, in every $r$ -$\\sigma $ sector the theory looks like a standard theory of characteristics for the Hamilton-Jacobi equation with the Hamiltonian $H^r_\\sigma $ in $J^1_1$ — the same conclusion that may be drawn by comparison of Eqs.", "(REF ) and (REF ), or (REF ) and (REF ).", "We will next explore other aspects of this similarity." ], [ "Variational principles", "If one introduces “action forms\" $\\rho = p_i\\, \\mathrm {d}q^i - H\\, \\mathrm {d}t$ in $J^1_1$ and $\\rho ^r_\\sigma = p^r_{\\sigma i}\\, \\mathrm {d}q^i - H^r_\\sigma \\,\\mathrm {d}t$ in $J^\\infty $ , then the corresponding Cartan forms may be written as $\\omega = \\mathrm {d}p - \\rho $ and $\\omega ^r_\\sigma = \\mathrm {d}p^r_\\sigma - \\rho ^r_\\sigma $ .", "For an arbitrary curve $C$ in $J^1_1$ , the difference of $p$ at its ends is $\\Delta p = \\int _C \\mathrm {d}p$ .", "Now if curve $C$ lies on the graph $\\Gamma $ of the solution, then the tangent vector $X$ at an arbitrary point of $C$ cancels the 1-form $\\omega $ , and so $X\\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\mathrm {d}p = X\\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\rho $ .", "For such curves, therefore, the difference of $p$ at their ends is given by the invariant Hilbert integral $\\Delta p = \\int _C \\rho $ [20], [21].", "Similarly, in $J^\\infty $ , the difference of $p^r_\\sigma $ at the ends of any curve $C$ that lies on the graph of solution is $\\Delta p^r_\\sigma = \\int _C \\rho ^r_\\sigma \\, .", "$ In $J^1_1$ , the condition that the integral $\\int _C \\rho $ remains stationary with respect to any variations of curve $C$ that do not change the space and time coordinates of its ends may be used to select the curves (trajectories) which satisfy the Hamilton equations of motion [19].", "Similarly, in $J^\\infty $ , the trajectories that satisfy Eqs.", "(REF ) may be selected by the condition that for every $r$ and $\\sigma $ , the integral $\\int _C \\rho ^r_\\sigma $ is stationary with respect to variations of $C$ that do not change the base coordinates of its ends and are generated by any vector field which belongs to $T^r_\\sigma $ .", "Indeed, let $C$ go from point $A$ to point $B$ in $J^\\infty $ , and let an infinitesimal $\\varepsilon $ -variation transform $C$ into the curve $C^{\\prime }$ , which goes from $A^{\\prime }$ to $B^{\\prime }$ .", "By the condition stated above, the base coordinates of $A^{\\prime }$ and $B^{\\prime }$ are the same as of $A$ and $B$ , and so $\\int _A^{A^{\\prime }} \\rho ^r_\\sigma = \\int _{B^{\\prime }}^B \\rho ^r_\\sigma = 0$ , where the integrals are taken along straight segments connecting $A$ with $A^{\\prime }$ and $B$ with $B^{\\prime }$ .", "Consequently, the variation of the integral is equal to the integral over the closed loop $AA^{\\prime }B^{\\prime }BA$ , which in turn, using Stokes' theorem, may be represented as an integral over an area $D$ inside the loop: $\\delta \\int _C \\rho ^r_\\sigma \\,\\,=\\,\\, \\int _{C^{\\prime }} \\rho ^r_\\sigma - \\int _C \\rho ^r_\\sigma \\,\\,=\\,\\, \\oint _{AA^{\\prime }B^{\\prime }BA} \\rho ^r_\\sigma \\,\\,=\\,\\, \\int _D \\mathrm {d}\\rho ^r_\\sigma \\, .$ If $C$ is an integral curve of a vector field $X$ , parameterized by a base coordinate $t$ , and the variation is generated by a vector field $V$ , then the above integral is equal to $\\varepsilon \\int _{t_A}^{t_B} V \\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}(X \\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\mathrm {d}\\rho ^r_\\sigma ) \\mathrm {d}t$ .", "Since $\\mathrm {d}\\rho ^r_\\sigma =- \\mathrm {d}\\omega ^r_\\sigma $ , the integrand here has the same form as the left hand side of Eq.", "(REF ).", "Consequently, repeating the derivation that follows (REF ) we conclude that this integral vanishes for arbitrary field $V \\in T^r_\\sigma $ if and only if Eqs.", "(REF ) and () are satisfied (or, equivalently, Eq.", "(REF ) is satisfied).", "Clearly, this variational principle is completely equivalent to our original condition (REF ), and may be considered to be its restatement.", "As in $J^1_1$ , our trajectories may be also obtained from a different variational principle, a Lagrangian one.", "For that, we introduce the Legendre transformations of Hamiltonians $H^r_\\sigma $ , i.e.", "the functions $\\begin{array}{rcl}L^r_\\sigma & = & \\displaystyle l^r_\\sigma \\left(t,{\\bf q },{\\bf p },\\frac{\\partial H^r_\\sigma }{\\partial p^r_{\\sigma i}}\\right) \\\\[0.4cm]& = & \\displaystyle p^r_{\\sigma i}\\, \\frac{\\partial H^r_\\sigma }{\\partial p^r_{\\sigma i}} - H^r_\\sigma \\, .", "\\end{array}$ For Hamiltonians satisfying condition (REF ), when $\\sigma \\ne \\mbox{ø}$ , this transformation is of a trivial nature: As was discussed above, in these cases $H^r_\\sigma $ contains $p^r_{\\sigma i}$ only in an additive term which is linear in it, so in Eq.", "(REF ) this term gets canceled, and $L^r_\\sigma $ is simply equal to the sum of the remaining terms with a minus sign.", "When $\\sigma =\\mbox{ø}$ , however, $H^r$ does not have to be linear in $p^r_i$ .", "We assume that there are some values of $r$ such that $H^r$ is a nonlinear function of $p^r_i$ , and will consider only those values of $r$ below.", "For those values of $r$ , velocities $v^i = \\partial H^r /\\partial p^r_i$ are nontrivial functions of momentums $p^r_i$ .", "These relations between velocities $v^i$ and momentums $p^r_i$ are supposed to be resolved with respect to $p^r_i$ , expressing them through ${\\bf v }$ as $p^r_i = \\varphi ^r_i(t,{\\bf q },{\\bf p }^{\\prime },{\\bf v })$ with some functions $\\varphi ^r_i$ , where, here and in what follows, ${\\bf p }^{\\prime }$ is a set of all $p^s_\\nu $ such that $\\nu \\ne \\mbox{ø}$ and $(s,\\nu )\\ne (r,i)$ .", "The obtained expression for $p^r_i$ should then be substituted into Eq.", "(REF ) with $\\sigma =\\mbox{ø}$ , resulting in the definition $L^r(t,{\\bf q },{\\bf p }^{\\prime },{\\bf v }) = v^i \\varphi ^r_i(t,{\\bf q },{\\bf p }^{\\prime },{\\bf v }) - H^r\\big (t,{\\bf q },{\\bf p }^{\\prime },\\varphi ^r_i(t,{\\bf q },{\\bf p }^{\\prime },{\\bf v })\\big )\\, .$ The above-defined Lagrangians $L^r$ can now be used to derive the principle of stationary action in its Lagrangian form from the invariant Hilbert integral.", "The derivation closely follows the one for the first order PDE [21]; nevertheless, it is presented here for completeness and because of the complicating presence of (absent in the first order case) higher momentums.", "Let $A$ and $B$ be two points on the graph $\\Gamma $ of the solution that both belong to the same “true\" (i.e., obtained by solution of Eqs.", "(REF ) and (REF )) trajectory $C_{AB}$ .", "Let ${\\bf q }(t)$ be an arbitrary curve through the base, connecting the base projections of $A$ and $B$ .", "The difference between the values of $p^r$ in $B$ and $A$ may then be expressed, as in Eq.", "(REF ), by the integral $\\Delta p^r_{AB} = \\displaystyle \\int _{C_{\\bf q }} \\big [p^r_i \\dot{q}^i - H^r(t,{\\bf q },{\\bf p }^{\\prime },p^r_i)\\big ] \\mathrm {d}t \\, , $ where $C_{\\bf q }$ is the image of the curve ${\\bf q }(t)$ on the graph $\\Gamma $ .", "In this integral, $p^r_i$ and $H^r$ may be expressed [22] through $L^r$ as $p^r_i = \\frac{\\partial }{\\partial v^i}\\, L^r(t,{\\bf q },{\\bf p }^{\\prime },{\\bf v }) \\, , \\quad \\,H^r(t,{\\bf q },{\\bf p }^{\\prime },p^r_i) = p^r_i v^i - L^r(t,{\\bf q },{\\bf p }^{\\prime },{\\bf v }) \\, ,$ where the value of $v^i$ in these formulas should be set to a known value $\\partial H^r(t,{\\bf q },{\\bf p }^{\\prime },p^r_i) / \\partial p^r_i$ , i.e., to the velocity of a true trajectory, passing through the corresponding point ${\\bf q }$ .", "Substituting these expressions into Eq.", "(REF ), obtain $\\Delta p^r_{AB} = \\displaystyle \\int _{C_{\\bf q }} \\left[ L^r(t,{\\bf q },{\\bf p }^{\\prime },{\\bf v }) + (\\dot{q}^i - v^i)\\frac{\\partial }{\\partial v^i}\\, L^r(t,{\\bf q },{\\bf p }^{\\prime },{\\bf v }) \\right] \\mathrm {d}t \\, .$ Now the integrand here contains the first two terms of a Taylor series expansion of $L^r(t,{\\bf q },{\\bf p }^{\\prime },\\dot{{\\bf q }})$ in powers of $\\dot{q}^i - v^i$ .", "When $\\dot{q}^i$ is close to $v^i$ , i.e., when the curve $C_{\\bf q }$ is close to the true trajectory $C_{AB}$ , we have $\\displaystyle \\int _{C_{\\bf q }} L^r(t,{\\bf q },{\\bf p }^{\\prime },\\dot{{\\bf q }}) \\mathrm {d}t - \\Delta p^r_{AB} = \\displaystyle \\frac{1}{2}\\int _{C_{\\bf q }}(\\dot{q}^i - v^i)(\\dot{q}^j - v^j) \\frac{\\partial ^2}{\\partial v^i \\partial v^j}\\,L^r(t,{\\bf q },{\\bf p }^{\\prime },{\\bf v }) \\mathrm {d}t + O \\left( (\\dot{q}^i - v^i)^3 \\right) ,$ which is of second order with respect to $\\dot{{\\bf q }}-{\\bf v }$ and, therefore, with respect to the deviation of trajectory $C_{\\bf q }$ from the true one $C_{AB}$ .", "Consequently, the “action integral\" $\\int _{C_{\\bf q }} L^r(t,{\\bf q },{\\bf p }^{\\prime },\\dot{{\\bf q }}) \\mathrm {d}t$ is stationary with respect to variations of $C_{\\bf q }$ around $C_{AB}$ , which is the way the principle of stationary action is formulated in $J^\\infty $ .", "The stationarity of the action integral is, therefore, an alternative condition, which may be used to select true trajectories, which satisfy Eqs.", "(REF ) and (REF ), from arbitrary curves on $\\Gamma $ , which satisfy Eq.", "(REF ) only.", "Eq.", "(REF ) with $\\sigma =\\mbox{ø}$ then becomes the Euler-Lagrange equation in the usual way, and the difference between the values of the action function $p^r$ in two points may be expressed, as for first order PDE, as an integral from the Lagrangian along the true trajectory that connects these points: $\\Delta p^r_{AB} = \\displaystyle \\int _{C_{AB}} L^r(t,{\\bf q },{\\bf p }^{\\prime },{\\bf v })\\, \\mathrm {d}t \\, .", "$ It is clear from Eqs.", "(REF ) and (REF ) that a similar expression may be written for $\\Delta p^r_\\sigma $ with arbitrary $r$ and $\\sigma $ , the only difference being that when $H^r_\\sigma $ is linear in $p^r_{\\sigma i}$ , the corresponding Lagrangian $L^r_\\sigma $ does not depend on ${\\bf v }$ and so is a function of $t$ , ${\\bf q }$ , and $\\lbrace p^s_\\nu \\!", ": \\nu \\ne \\mbox{ø}, (s,\\nu ) \\ne (r,\\sigma i)\\rbrace $ only.", "Therefore, we always have $\\Delta p^r_{\\sigma _{AB}} = \\displaystyle \\int _{C_{AB}} L^r_\\sigma \\, \\mathrm {d}t \\, .", "$" ], [ "From ODEs to PDE: infinite phase space formulation", "So far, we started with the PDE (REF ) and developed the system of ODEs (REF ).", "Now we take system (REF ) as the starting point and will construct a corresponding PDE from it.", "The system lives in an infinite phase space $\\cal P$ , which has a geometry almost identical to that of $J^\\infty $ : The operator of total differentiation $D_i$ (REF ), basis forms $\\widetilde{\\omega }^r_\\sigma $ (REF ), duality relations (REF ), and Cartan forms $\\omega ^r_\\sigma $ (REF ) in $\\cal P$ will be the same as in $J^\\infty $ , with the only difference that in $\\cal P$ all multi-indices in these formulas and in all summations should be nonempty.", "Thus forms $\\omega ^r_\\sigma $ and $\\widetilde{\\omega }^r_\\sigma $ exist only for $\\sigma \\ne \\mbox{ø}$ , and integral manifolds of CD, defined by forms $\\omega ^r_\\sigma $ , are graphs of derivatives of analytic solutions of Eq.", "(REF ).", "Also, vector field $X$ , generated by Eq.", "(REF ), is tangent to these graphs, has the same form (REF ) with $q^i = \\partial H^r / \\partial p^r_i$ (no contribution with $\\sigma = \\mbox{ø}$ there!", "), and satisfies Eq.", "(REF ).", "Equation (REF ) with $\\sigma =\\mbox{ø}$ then may be considered as defining the actions $S^r_{AB}$ , corresponding to arbitrary curve $C_{AB}$ , and for $r$ such that $H^r$ is not linear in $p^r_i$ , Eq.", "(REF ) with $\\sigma = \\mbox{ø}$ will ensure that the Euler-Lagrange equations for these actions are satisfied, and so the principle of stationary action for them holds.", "The action functions $S^r({\\bf q },t)$ with given initial condition $S^r_0({\\bf q })$ at $t=t_0$ are defined as in classical mechanics [19]: Namely, let ${S^r_0}_\\sigma = \\partial _\\sigma S^r_0$ , then for every space vector ${\\bf q }_0$ consider a trajectory in $\\cal P$ that is a solution of Eq.", "(REF ) with initial conditions ${\\bf q }(t_0) = {\\bf q }_0\\, , \\quad \\; p^r_\\sigma (t_0) = {S^r_0}_\\sigma ({\\bf q }_0) \\, , $ and define functions $S^r({\\bf q },t)$ by $S^r({\\bf q },t) = \\displaystyle S^r_0({\\bf q }_0) + \\int _{C_{AB}} L^r\\, \\mathrm {d}t \\, , $ where $C_{AB}$ is the trajectory that ends at time $t$ in a point $B\\in \\cal P$ with a space coordinate ${\\bf q }$ and ${\\bf q }_0$ is a space coordinate of a starting point $A\\in \\cal P$ of this trajectory.", "We will assume that trajectories don't intersect, and so this definition is unambiguous.", "While equations of motion (REF ) and their solutions describe individual trajectories, action functions $S^r({\\bf q },t)$ describe a family of trajectories selected by Eq.", "(REF ).", "Consequently on trajectories that form the family, dependence of the initial momentums ${p^r_\\sigma }_0=p^r_\\sigma (t_0)$ of the trajectory on its initial space coordinate ${\\bf q }_0$ is given by ${p^r_\\sigma }_0 ({\\bf q }_0) = {S^r_0}_\\sigma ({\\bf q }_0) \\, .", "$ For the just-defined functions $S^r({\\bf q },t)$ , the following generalizations of classical results hold: First, for any $\\sigma \\ne \\mbox{ø}$ the functions $S^r_\\sigma = \\partial _\\sigma S^r$ satisfy $S^r_\\sigma ({\\bf q },t) = p^r_\\sigma ({\\bf q },t) \\, , $ where $p^r_\\sigma ({\\bf q },t)$ is a $p^r_\\sigma $ -coordinate of point $B$ .", "Second, for any $\\sigma $ , including $\\sigma =\\mbox{ø}$ , the $S^r_\\sigma $ satisfy the Hamilton-Jacobi-type equation (REF ) $\\frac{\\partial }{\\partial t}\\,S^r_\\sigma ({\\bf q },t) + H^r_\\sigma (t,{\\bf q },{\\bf S }) = 0 \\, , $ where ${\\bf S }$ is a set of all partial derivatives $S^r_\\sigma $ , $\\sigma \\ne \\mbox{ø}$ , of functions $S^r$ .", "As in classical mechanics, Eq.", "(REF ) means that the values $p^r_\\sigma $ , which originally were independent variables evolving according to Eq.", "(REF ), become also partial derivatives of action functions $S^r$ .", "As was the case with the principle of stationary action, the proof is similar to the standard one [19], but we present it because there are some additional complications.", "Let $C_{AB}$ and $C_{A^{\\prime }B^{\\prime }}$ be two close trajectories, with the base coordinates of $A$ and $A^{\\prime }$ being $({\\bf q }_0,t_0)$ and $({\\bf q }^{\\prime }_0,t_0)$ and of $B$ and $B^{\\prime }$ being $({\\bf q },t)$ and $({\\bf q }^{\\prime },t^{\\prime })$ .", "These trajectories are integral curves of the vector field $X$ .", "Let action forms $\\rho ^r_\\sigma $ for any $\\sigma $ , including $\\sigma =\\mbox{ø}$ , be defined as in $J^\\infty $ .", "Now connect point $A$ with $A^{\\prime }$ and point $B$ with $B^{\\prime }$ by straight segments, and consider an integral of $\\rho ^r_\\sigma $ along the closed loop $AA^{\\prime }B^{\\prime }BA$ .", "By Stokes' theorem we have $\\oint _{AA^{\\prime }B^{\\prime }BA} \\rho ^r_\\sigma \\,\\,=\\,\\, \\int _D \\mathrm {d}\\rho ^r_\\sigma \\, , $ where $D$ is the region inside the loop.", "The difference $V(t)$ between points of $C_{AB}$ and $C_{A^{\\prime }B^{\\prime }}$ with the same $t$ is given by the vector $\\overrightarrow{AA^{\\prime }}$ , dragged (with parameter $t-t_0$ ) by a flow of the vector field $X$ .", "Now, the vector $\\overrightarrow{AA^{\\prime }}$ is the vector ${\\bf q }^{\\prime }_0 - {\\bf q }_0$ raised to the graph of the analytic function $S^r_0({\\bf q })$ in $\\cal P$ .", "Consequently, if ${\\bf q }^{\\prime }_0 - {\\bf q }_0 = \\varepsilon ^i \\partial / \\partial q^i$ , then $\\overrightarrow{AA^{\\prime }} =\\varepsilon ^i D_i$ , so $\\overrightarrow{AA^{\\prime }}$ is a linear combination of $D_i$ .", "It is easy to calculate that the Lie derivative of $D_i$ in the direction of $X$ is $[X,D_i] = - \\,(D_i \\dot{q}^j) D_j \\, ,$ i.e., also a linear combination of $D_j$ , and therefore, so is the difference $V(t)$ .", "Now, the integral on the right hand side of Eq.", "(REF ) is equal to $\\int _{t_A}^{t_B} V(t) \\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}(X \\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\mathrm {d}\\rho ^r_\\sigma )\\, \\mathrm {d}t$ , but by Eq.", "(REF ) $X \\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\mathrm {d}\\rho ^r_\\sigma $ is a linear combination of Cartan forms that are canceled by any $D_i$ and, therefore, by $V(t)$ .", "We have, eventually, that for all $\\sigma $ this integral vanishes, and with it the integral of $\\rho ^r_\\sigma $ along $AA^{\\prime }B^{\\prime }BA$ , and so $\\int _B^{B^{\\prime }} \\rho ^r_\\sigma = \\left(\\int _B^A + \\int _A^{A^{\\prime }} + \\int _{A^{\\prime }}^{B^{\\prime }}\\right) \\rho ^r_\\sigma \\, .$ But for $\\sigma \\ne \\mbox{ø}$ , as in Eq.", "(REF ), $\\int _B^A \\rho ^r_\\sigma = p^r_\\sigma (A) - p^r_\\sigma (B)$ , and similarly for $\\int _{A^{\\prime }}^{B^{\\prime }} \\rho ^r_\\sigma $ .", "On $AA^{\\prime }$ , $t=t_0$ and so $p^r_\\sigma = {S^r_0}_\\sigma $ , $\\overrightarrow{AA^{\\prime }}\\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\mathrm {d}t = 0$ , and for all $\\sigma $ $\\int _A^{A^{\\prime }}\\rho ^r_\\sigma = \\int _A^{A^{\\prime }} p^r_{\\sigma i}\\mathrm {d}q^i = \\int _A^{A^{\\prime }} {S^r_0}_{\\sigma i} \\mathrm {d}q^i= {S^r_0}_\\sigma (A^{\\prime }) - {S^r_0}_\\sigma (A) \\, , $ which for $\\sigma \\ne \\mbox{ø}$ is equal to $p^r_\\sigma (A^{\\prime }) - p^r_\\sigma (A)$ .", "Consequently, for $\\sigma \\ne \\mbox{ø}$ Eq.", "(REF ) becomes $\\int _B^{B^{\\prime }} p^r_{\\sigma i}\\mathrm {d}q^i - H^r_\\sigma \\mathrm {d}t = p^r_\\sigma (B^{\\prime }) - p^r_\\sigma (B) \\, ,$ which in the limit $\\varepsilon ^i \\rightarrow 0$ , $\\Delta t \\rightarrow 0$ gives $\\frac{\\partial }{\\partial q^i}\\, p^r_\\sigma ({\\bf q },t) = p^r_{\\sigma i}({\\bf q },t)\\, , \\quad \\; \\frac{\\partial }{\\partial t}\\, p^r_\\sigma (B) = - H^r_\\sigma (B) \\, .$ Now if for all $|\\sigma | = N$ , Eqs.", "(REF ) are true, then for such $\\sigma $ and all $i$ , $S^r_{\\sigma i}({\\bf q },t) = \\frac{\\partial }{\\partial q^i} \\, S^r_\\sigma ({\\bf q },t) = \\frac{\\partial }{\\partial q^i} \\, p^r_\\sigma ({\\bf q },t) = p^r_{\\sigma i}({\\bf q },t)\\, ,$ and so (REF ) is also true for $|\\sigma | = N+1$ .", "Then for $\\sigma =\\mbox{ø}$ , since trajectories $C_{AB}$ and $C_{A^{\\prime }B^{\\prime }}$ satisfy Eq.", "(REF ), we have $\\int _B^A\\rho ^r = \\int _B^A L^r \\mathrm {d}t = S^r(A) - S^r(B) \\, ,$ and similarly for $\\int _{A^{\\prime }}^{B^{\\prime }} \\rho ^r$ , while $\\int _A^{A^{\\prime }}\\rho ^r$ is given by Eq.", "(REF ) with $\\sigma =\\mbox{ø}$ .", "Eq.", "(REF ) now gives $\\int _B^{B^{\\prime }} p^r_i \\mathrm {d}q^i - H^r \\mathrm {d}t = S^r(B^{\\prime }) - S^r(B),$ or in the limit $\\varepsilon ^i \\rightarrow 0$ , $\\Delta t \\rightarrow 0$ , $\\frac{\\partial }{\\partial q^i} \\, S^r({\\bf q },t) = p^r_i({\\bf q },t)\\, , \\quad \\; \\frac{\\partial }{\\partial t}\\, S^r(B) = - H^r(B) \\, ,$ which completes the proof of Eq.", "(REF ), and then the second relations in Eqs.", "(REF ) and (REF ) prove Eq.", "(REF ).", "We see again in Eq.", "(REF ) that solution of the PDE (REF ) may be obtained from solutions of the ODEs (REF ).", "Conversely, any sequence of functions $S^r_\\sigma ({\\bf q },t)$ which satisfy $\\frac{\\partial S^r_\\sigma }{\\partial q^i} = S^r_{\\sigma i}, \\quad \\mbox{for all} \\;\\, \\sigma ,i, $ and which also satisfy Eq.", "(REF ) with initial conditions corresponding to a family of trajectories with given initial distribution of momentums (REF ), may be used for integration of the equations of motion for trajectories of this family: Equation (REF ), read from right to left, gives for all $t$ the distribution on the family's trajectories of momentums which satisfy Eq.", "(REF ).", "Indeed, we have for these momentums $\\partial p^r_{\\sigma i}/\\partial t = \\partial _i\\partial _t S^r_\\sigma $ .", "Using Eq.", "(REF ), it is then easy to show that if a point ${\\bf q }(t)$ moves with velocity $\\dot{q}^k$ given by the first equation in (REF ), then the time derivative $\\dot{p}^r_{\\sigma i}({\\bf q }(t),t) = \\frac{\\partial p^r_{\\sigma i}}{\\partial t} + \\frac{\\partial p^r_{\\sigma i}}{\\partial q^k} \\, \\dot{q}^k$ of a function $p^r_{\\sigma i}$ at a point ${\\bf q }(t)$ is given by the second equation in (REF ).", "For Hamiltonians of first order, this is the basis of a Jacobi method of integration of equations of motion, and in the following we will call it the “generalized Jacobi method\" for arbitrary Hamiltonians.", "Now, when this method is used, it is obviously desirable to make it applicable to as large a class of trajectory families as possible.", "From this point of view, the formulation we used above is unnecessarily restrictive and may be generalized.", "Indeed, initial conditions that define the family's trajectories are given by Eq.", "(REF ).", "In this equation, ${S^r_0}_\\sigma $ are derivatives of the functions $S^r_0$ .", "However, we saw that the sequence $S^r_\\sigma $ that is used in the generalized Jacobi method does not contain $S^r$ and includes only functions $S^r_\\sigma $ with $\\sigma \\ne \\mbox{ø}$ .", "Therefore, all these functions are derivatives of $S^r_i$ , $i=1,\\ldots ,n$ , while the functions $S^r_i$ themselves and, consequently, their initial values $S^r_{0i}$ , do not have to be derivatives of any other functions.", "On the other hand, we have from Eq.", "(REF ) that $\\frac{\\partial {S^r_0}_i}{\\partial q^j} = {S^r_0}_{ij} = \\frac{\\partial {S^r_0}_j}{\\partial q^i} \\, .$ This means, that 1-forms ${S^r_0}_i \\mathrm {d}q^i$ should be closed, $\\mathrm {d}({S^r_0}_i \\mathrm {d}q^i) = 0$ , which will allow us to define the functions $S^r_0$ by $S^r_0({\\bf q }) = \\int _{{\\bf q }_0}^{\\bf q }{S^r_0}_i \\mathrm {d}q^i + S^r_0({\\bf q }_0) \\, , $ and so ${S^r_0}_i$ will be their derivatives.", "The integration in (REF ) runs along arbitrary curves in configuration space $Q$ that connect points ${\\bf q }_0$ and ${\\bf q }$ , and the value $S^r_0({\\bf q }_0)$ , as well as the vector ${\\bf q }_0$ itself, are also arbitrary.", "Thus from the very beginning, the functions $S^r_0$ are defined up to an arbitrary additive constant; moreover, they will be usual, single-valued functions only if configuration space $Q$ is simply connected.", "If the fundamental group of $Q$ is nontrivial, then in general Eq.", "(REF ) defines functions $S^r_0$ as multi-valued, or single-valued on the universal covering space of $Q$ .", "The branches of $S^r_0$ may differ only by a constant, and so they all have the same derivatives ${S^r_0}_\\sigma $ , $\\sigma \\ne \\mbox{ø}$ .", "Therefore, these derivatives will be single-valued as they should be because the family has one, and only one, trajectory starting at every point of configuration space at $t=t_0$ , and the functions ${S^r_0}_\\sigma $ , $\\sigma \\ne \\mbox{ø}$ , define initial momentums of these trajectories.", "Consequently, in the currently considered statement of the problem, which starts with equations of motion in the infinite phase space $\\cal P$ , the functions $S^r$ and their initial values $S^r_0$ in the generalized Jacobi method are defined up to an additive constant, and in cases where the configuration space $Q$ is not simply connected, may be multi-valued.", "Note that these conclusions are purely topological, not dynamical — they do not depend on the form of the Hamiltonians $H^r_\\sigma $ or on their order.", "The simplest example is a family of trajectories on a circle that all have the same initial velocity $v$ .", "The function $S_0$ is then equal to $vr\\varphi +\\mbox{const}$ , where $r$ is the radius of the circle, and $\\varphi $ is the angular coordinate on it.", "When $v\\ne 0$ , this function is multi-valued on the circle, but single-valued on the universal covering space ${\\rm R}^1$ ." ], [ "The case of complex-valued solutions", "We now allow complex-valued solutions of the PDE (REF ).", "We only consider the case of one complex function $p({\\bf q },t)$ , the generalization to the situation when there are several of them being obvious.", "Let $p^{1,2}$ be this function's real and imaginary parts, so that $p = p^1 + ip^2$ , and similarly $H = H^1 + iH^2$ .", "The conjugated values are $\\bar{p} = p^1 - ip^2$ and $\\bar{H} = H^1 - iH^2$ .", "By setting $m = 2$ , the theory of the previous subsections may be applied directly to the functions $p^1$ and $p^2$ , treated as independent real functions with Hamiltonians $H^1$ and $H^2$ .", "However, it is often more convenient to express the same results via complex functions $p$ and $\\bar{p}$ , because in this representation they behave as if they were independent and also because the equations for $\\bar{p}$ are simply the conjugated equations for $p$ .", "As is usually done, introduce vector fields $\\frac{\\partial }{\\partial p} = \\frac{1}{2}\\left(\\frac{\\partial }{\\partial p^1} - i \\frac{\\partial }{\\partial p^2}\\right) \\, , \\quad \\,\\frac{\\partial }{\\partial \\bar{p}} = \\frac{1}{2}\\left(\\frac{\\partial }{\\partial p^1} + i \\frac{\\partial }{\\partial p^2}\\right) $ and 1-forms $\\mathrm {d}p = \\mathrm {d}p^1 + i \\mathrm {d}p^2 \\, , \\quad \\, \\mathrm {d}\\bar{p} = \\mathrm {d}p^1 - i \\mathrm {d}p^2 \\, ,$ which satisfy duality relations $\\begin{array}{lcllcl}\\displaystyle \\frac{\\partial }{\\partial p} \\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\mathrm {d}p & = & 1 \\, ,\\quad & \\displaystyle \\frac{\\partial }{\\partial p} \\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\mathrm {d}\\bar{p} & = & 0 \\, ,\\\\[0.3cm]\\displaystyle \\frac{\\partial }{\\partial \\bar{p}} \\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\mathrm {d}p & = & 0 \\, ,\\quad & \\displaystyle \\frac{\\partial }{\\partial \\bar{p}} \\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\mathrm {d}\\bar{p} & = & 1 \\, .\\end{array}$ We have $\\sum _{r = 1}^2 p^r_{\\sigma i}\\frac{\\partial }{\\partial p^r_\\sigma } = p_{\\sigma i} \\frac{\\partial }{\\partial p_\\sigma } + \\bar{p}_{\\sigma i}\\frac{\\partial }{\\partial \\bar{p}_\\sigma }$ and so the operator of total differentiation may be written as $D_i = \\frac{\\partial }{\\partial q^i} + p_{\\sigma i} \\frac{\\partial }{\\partial p_\\sigma } + \\bar{p}_{\\sigma i}\\frac{\\partial }{\\partial \\bar{p}_\\sigma } \\, .", "$ Similarly, vector field $X$ , Eq.", "(REF ), may be written as $X = \\displaystyle \\frac{\\partial }{\\partial t} + \\dot{q}^i D_i - H_\\sigma \\frac{\\partial }{\\partial p_\\sigma } -\\bar{H}_\\sigma \\frac{\\partial }{\\partial \\bar{p}_\\sigma } \\, .$ Since $H$ is an analytic function of $p_j\\,$ , it satisfies the Cauchy-Riemann equations $\\frac{\\partial H^1}{\\partial p^1_j} = \\frac{\\partial H^2}{\\partial p^2_j}\\, ,\\quad \\, \\frac{\\partial H^1}{\\partial p^2_j} = -\\,\\frac{\\partial H^2}{\\partial p^1_j} \\, .$ The first of these equations means that if $p^{1,2}$ are considered as independent real functions with Hamiltonians $H^1$ and $H^2$ , then HC2 are automatically satisfied with the corresponding velocity $\\dot{q}^j \\, = \\,\\frac{\\partial H^1}{\\partial p^1_j} \\, = \\, \\frac{\\partial H^2}{\\partial p^2_j} \\, .$ Expressing here $\\partial / \\partial p^{1,2}$ through $\\partial / \\partial p$ and $\\partial / \\partial \\bar{p}$ , and similarly $H^{1,2}$ through $H$ and $\\bar{H}$ , and taking into account that, being an analytic function, $H$ depends on $p^{1,2}$ only through the combination $p=p^1+ip^2$ , and $\\bar{H}$ depends on $p^{1,2}$ only through $\\bar{p} = p^1-ip^2$ , obtain $\\dot{q}^j \\,= \\, \\frac{\\partial H^1}{\\partial p^1_j} \\,= \\, \\left(\\frac{\\partial }{\\partial p_j} + \\frac{\\partial }{\\partial \\bar{p}_j}\\right)\\frac{H + \\bar{H}}{2} \\,= \\, \\frac{1}{2}\\left(\\frac{\\partial H}{\\partial p_j} + \\frac{\\partial \\bar{H}}{\\partial \\bar{p}_j}\\right) \\, .$ It is easy to see that $\\partial H^2 / \\partial p^2_j$ gives the same expression for the velocity.", "For application to a theory of particles with spin, we also need to consider the case of a complex analytic function $p(w,t)$ of complex coordinate $w = w^1 + i w^2$ with a Hamiltonian $H(p_w)$ , where $p_w = \\partial p/\\partial w$ .", "If $w$ and $p$ were real, the $w$ 's velocity would be equal to $\\dot{w} = \\frac{\\partial H}{\\partial p_w} \\, .", "$ It is easy to see that due to analyticity of all the functions involved and the corresponding Cauchy-Riemann equations, the same expression for $\\dot{w}$ remains true in a complex case.", "Indeed, $p_w$ is given by the standard expressions $p_w = p^1_{w^1} + i p^2_{w^1} = p^2_{w^2} - i p^1_{w^2} \\, .$ From that, and using the Cauchy-Riemann equations, we have for the real and imaginary parts of (REF ) $\\Re \\left(\\frac{\\partial H}{\\partial p_w}\\right) = \\frac{\\partial H^1}{\\partial p^1_{w^1}} = \\frac{\\partial H^2}{\\partial p^2_{w^1}} \\, ,\\quad \\,\\Im \\left(\\frac{\\partial H}{\\partial p_w}\\right) = \\frac{\\partial H^2}{\\partial p^2_{w^2}} = \\frac{\\partial H^1}{\\partial p^1_{w^2}} \\, .$ On the other hand, considering $p^{1,2}$ as independent real functions of real variables $w^{1,2}$ with Hamiltonians $H^{1,2}$ , we have $\\dot{w}^1 = \\frac{\\partial H^1}{\\partial p^1_{w^1}} = \\frac{\\partial H^2}{\\partial p^2_{w^1}} \\, ,\\quad \\,\\dot{w}^2 = \\frac{\\partial H^1}{\\partial p^1_{w^2}} = \\frac{\\partial H^2}{\\partial p^2_{w^2}} \\, ,$ where again we used the Cauchy-Riemann equations, and the first (resp.", "second) representation of $p_w$ in (REF ) for calculation of $\\dot{w}^1$ (resp.", "$\\dot{w}^2$ ).", "Thus, as it was for velocity $\\dot{q}^j$ , the HC2 for $\\dot{w}^{1,2}$ are automatically satisfied due to the Cauchy-Riemann equations, and the $\\dot{w}^{1,2}$ are equal to $\\Re (\\partial H/\\partial p_w)$ and $\\Im (\\partial H/\\partial p_w)$ in Eq.", "(REF ), which proves (REF )." ], [ "Discussion", "The following general picture emerges from the above development.", "As for the standard case of first order equations, the PDE (REF ) with $H^r$ satisfying HC allows the introduction of a corresponding system of ODEs (REF ).", "The values whose dynamical evolution is governed by this system are coordinates of a point, moving in a base, and partial derivatives at this point of unknown functions.", "On the other hand, as was just discussed, Eq.", "(REF ) may be considered on its own, as Hamilton equations are in classical mechanics, with the action functions being introduced later.", "In general, the system (REF ) is an infinite hierarchical system of coupled equations.", "We will not attempt its solution in this work; what will be important for us here is that this system has solutions whenever Eq.", "(REF ) does, and as for any system of first order ODEs, this solution is unique.", "The system then defines in $J^\\infty $ and $\\cal P$ some trajectories, which lie in graphs $\\Gamma $ of solutions of (REF ) and are the characteristic curves of a corresponding exterior differential system.", "Like any characteristic curves [18], these trajectories express solutions of Eq.", "(REF ) with given initial conditions and their derivatives as integrals of $\\rho ^r_\\sigma $ or $L^r_\\sigma \\mathrm {d}t$ along them.", "The direct proof of this statement, which doesn't use the theory of characteristics, is presented in section 2.5.", "The theory splits the whole jet space $J^\\infty $ into $r$ -$\\sigma $ sectors with coupled dynamics, described by Hamiltonians $H^r_\\sigma $ .", "The sectors share common coordinates $q^i$ , but there are no conflicts, because, thanks to HC, the dynamic they all define for these coordinates is the same.", "The structure of the theory in every $r$ -$\\sigma $ sector is similar to the one in $J^1_1$ , but the value $p^r_{\\sigma i}$ has a dual meaning: while on one hand, in an $r$ -$\\sigma i$ sector it plays the role of an action, obtainable from the above integrals, on the other hand, in sector $r$ -$\\sigma $ it is an $i$ -th component of momentum, evolving according to the corresponding “Hamilton equation\" with Hamiltonian $H^r_\\sigma $ .", "In our equations, this duality may be seen especially clearly in the comparison of Eq.", "(), which describes the evolution of $p^r_{\\sigma i}$ as an action in an $r$ -$\\sigma i$ sector, with the second equation of (REF ), where it evolves as the $i$ -th component of momentum in sector $r$ -$\\sigma $ .", "As the second equation of (REF ) is just a different form of (), sectors $r$ -$\\sigma $ and $r$ -$\\sigma i$ obviously agree on the dynamics of a variable $p^r_{\\sigma i}$ which they share.", "A variable $p^r$ belongs to only one sector $r$ -$\\mbox{ø}$ , and so for its time derivative we have only the “action form\" representation, given by Eq.", "(REF ) with $\\sigma =\\mbox{ø}$ .", "First order evolutionary PDEs with $m=1$ satisfy HC automatically, and so the whole theory is completely applicable to them.", "They are different, however, from higher-order equations in the following important aspect: For any $N \\ge 1$ , the resulting equations of motion for ${\\bf q }$ and $p_\\sigma $ , $|\\sigma |\\le N$ , form a closed subsystem, and the corresponding geometric theory may be formulated in a space of $N$ -jets $J^N_1$ .", "Indeed, if the order of the PDE (REF ) is equal to $k$ , then $H_\\sigma $ contains the derivatives of orders up to $|\\sigma |+k$ .", "Since the Cartan forms $\\omega _\\sigma $ are expressed through $p_{\\sigma i}$ , which is of order $|\\sigma |+1$ , and $H_\\sigma $ , for first order equations, i.e., $k=1$ , and any integer $N\\ge 1$ the system of exterior equations $\\lbrace \\omega _\\sigma = 0\\, , |\\sigma | < N\\rbrace $ is closed with respect to the set of derivatives of $p$ it includes.", "For this system, the space of $N$ -jets $J^N_1$ is sufficient, and the space $J^\\infty _1$ is not necessary.", "On the contrary, for, say, second order equations, $H_\\sigma $ in the exterior equation $\\omega _\\sigma = 0$ contains the variables $p_\\nu $ with $|\\nu | = |\\sigma | + 2$ .", "In order to ensure that these variables do indeed describe corresponding derivatives of the solution, which is represented by a graph in a jet space, we need to require that this graph also solves an exterior equation $\\omega _{\\sigma ^{\\prime }} = 0$ with $|\\sigma ^{\\prime }| = |\\sigma | + 1$ .", "But then $H_{\\sigma ^{\\prime }}$ in $\\omega _{\\sigma ^{\\prime }}$ will contain variables $p_{\\nu ^{\\prime }}$ with $|\\nu ^{\\prime }| = |\\sigma ^{\\prime }|+2 = |\\sigma |+3$ , and so the process will never stop, and the use of the infinite jet space $J^\\infty _1$ becomes inevitable.Another reason to use an infinite jet space is Bäcklund's theorem [9], [10], from which it follows that CD, defined by 1-forms (REF ) with Hamiltonians $H^r$ of higher-than-first order, cannot have characteristic fields in any finite jet space $J^k$ , $k<\\infty $ .", "Similar considerations show that, while for a higher order PDE the expression for $\\dot{p}_\\sigma $ with $\\sigma \\ne \\mbox{ø}$ contains $p_\\nu $ with $|\\nu | > |\\sigma |$ , for a first order PDE it doesn't, and so for it a system of equations $\\dot{p}_\\sigma = l_\\sigma $ with $|\\sigma | \\le N$ is closed for any $N\\ge 1$ .", "The system corresponding to $N=1$ is the simplest possible, but it still describes the evolution of the most important variables: ${\\bf q }, \\, p$ , and $p_i$ .", "A theory of this system in $J^1_1$ is a usual Hamiltonian theory of first order evolutionary PDE, which is a part of their “full\", i.e.", "including all derivatives, theory, while the latter is a special case of our theory of satisfying HC evolutionary PDE of arbitrary order and with arbitrary $m$ ." ], [ "Hamiltonian flow of quantum Hamilton-Jacobi equation", "Now we apply the technique developed above to non-relativistic quantum theory.", "We start with the one-particle case and consider the multi-particle situation later.", "By expressing the wave function as $\\psi ({\\bf x },t) = \\displaystyle \\exp \\left(\\frac{i}{\\hbar }\\,\\,p({\\bf x },t) \\right) , \\quad \\quad p({\\bf x },t) = S({\\bf x },t) +\\frac{\\hbar }{i}\\,\\,R({\\bf x },t) \\, , $ where $p$ is complex and $S$ and $R$ are real functions of position and time, the one-particle Schrödinger equation $i \\hbar \\, \\frac{\\partial \\psi }{\\partial t} = -\\frac{\\hbar ^2}{2 m}\\, \\Delta \\psi + U({\\bf x },t) \\psi $ may be equivalently presented as an evolutionary PDE for $p$ as $\\frac{\\partial p}{\\partial t} + H = 0 \\, , $ or as a system of evolutionary PDE for $R$ and $S$ as $\\frac{\\partial S}{\\partial t} + H^S = 0\\, , \\quad \\quad \\frac{\\partial R}{\\partial t} + H^R = 0 \\, ,$ The Hamiltonian functions $H$ , $H^S$ , and $H^R$ in the above equations are $H & = & \\frac{1}{2 m}\\, p_j^2 + U + \\frac{\\hbar }{2 i m}\\, p_{jj} \\, , \\\\[0.2cm]H^S & = & \\frac{1}{2 m}\\, S_j^2 + U - \\frac{\\hbar ^2}{2 m}\\, \\left(R_j^2 +R_{jj}\\right) , \\\\[0.2cm]H^R & = & \\frac{1}{m} \\left(S_j\\, R_j + \\frac{1}{2}\\, S_{jj}\\right), $ where the indices denote corresponding partial derivatives and we extend the summation rule to expressions like $p_j^2 = p_j p_j$ .", "Along with (REF ), Eq.", "(REF ) and system (REF ) will be also called “quantum Hamilton-Jacobi equation(s)\" (QHJE).", "Obviously, QHJE is equivalent to the Schrödinger equation (REF ), and the action function $p$ carries the same information as the wave function $\\psi $ .", "In the following, for convenience, we will often discuss only one of these functions/equations, with the understanding that our conclusions may be applied, with proper modifications, to the other.", "Also, since there is only one wave function, we will always say “action function,\" even when there are several (two) of them.", "We can now see immediately that Eqs.", "(REF ) and (REF ) satisfy HC1.", "This is obvious in Cartesian coordinates used in Eqs.", "(REF )-(), and is instructive to verify in the cylindrical and spherical coordinate systems.", "As they should (see section 2.6), Eqs.", "(REF ) and (REF ) also satisfy HC2 with corresponding velocity $v^j = \\frac{\\partial H^S}{\\partial S_j} = \\frac{\\partial H^R}{\\partial R_j} = \\frac{1}{m} \\, S_j\\, , $ which, in agreement with Eq.", "(REF ), may also be expressed as $v^j = \\frac{1}{2m}\\,(p_j + \\bar{p}_j)\\, .", "$ Consequently, the theory of the previous section may be used.", "It means that in a space of analytic jets, corresponding to PDE (REF ) (or to a system of PDEs (REF )), there exist trajectories, described by the system of ODEs (REF ), such that the solutions of the PDE and their derivatives may be obtained from the initial conditions by integrating the corresponding Lagrangians along these trajectories (see Eq.", "(REF )).", "The very existence of such ODEs and trajectories is just a mathematical fact, proven in the previous section.", "However, it raises an inevitable physical question: do the particles indeed move along these trajectories?", "Or, more practically: can peculiar features of quantum mechanics be understood, and its predictions reproduced, by assuming so?", "There are more questions.", "As we discussed, if the solution of the PDE is known, then the values of $p^r_\\sigma $ and ${\\bf v }$ may be obtained from it by the generalized Jacobi method.", "Therefore, the ODE and PDE formulations should be considered as two faces of the same theory, exactly like Hamilton equations and Hamilton-Jacobi equation in classical mechanics.", "But in classical mechanics, the roles of these equations are very different: while Hamilton (or Newton) equations provide the description of individual trajectories, the Hamilton-Jacobi equation describes an evolution of the (action) function which does not correspond to any particular trajectory, but is associated with a family of them.", "As was discussed at length in section 2.5, similar roles are played by the equations of motion (REF ) and the PDE (REF ) in the mathematical theory of higher order equations.", "Now the other question is whether the situation in quantum mechanics is the same, so that the action or wave functions describe families or ensembles of trajectories, while the description of individual events/trajectories is provided by the system of ordinary differential equations of motion (REF ).", "In the rest of this work we defend a positive answer to these questions.", "As we already mentioned in the Introduction, we call this approach an analytical quantum dynamics in infinite phase space ($\\cal P$ AQD).", "We now start with its general description.", "As is clear from the previous section, the state of a particle at some moment $t$ in $\\cal P$ AQD is defined by a triple $({\\bf x },{\\bf S },{\\bf R })$ or, equivalently, $({\\bf x },{\\bf p },\\bar{{\\bf p }})$ .", "Here ${\\bf x }$ is the position of a particle at time $t$ , and ${\\bf S }$ , ${\\bf R }$ , ${\\bf p }$ and $\\bar{{\\bf p }}$ are sets of all the derivatives of the corresponding action functions in ${\\bf x }$ at this time, so that ${\\bf S }$ is a set of all $S_\\sigma ({\\bf x },t)$ with $\\sigma \\ne \\mbox{ø}$ and similarly for ${\\bf R }$ , ${\\bf p }$ and $\\bar{{\\bf p }}$ .", "However, in the framework of $\\cal P$ AQD, they are just a set of independent fundamental variables, identified by their multi-indices, which describe the state of a particle at time $t$ exactly like components of momentum in classical mechanics.", "The evolution of a state is described by the equations of motion (REF ) (the second of these equations is easier to use in the form ()), where the functions $p^r$ in that equation are now $p^1=S$ and $p^2=R$ or $p^1=p$ and $p^2=\\bar{p}$ .", "For future reference, we present here expressions for Hamiltonians and some equations of motion.", "The first Eq.", "(REF ), i.e.", "the equation for $\\dot{{\\bf x }}$ , takes the form of Eq.", "(REF ) for the $(S,R)$ formulation and Eq.", "(REF ) for the $(p,\\bar{p})$ formulation.", "The operator of total differentiation for the $(S,R)$ formulation is $D_i = \\frac{\\partial }{\\partial x^i} + S_{\\sigma i} \\frac{\\partial }{\\partial S_\\sigma } + R_{\\sigma i} \\frac{\\partial }{\\partial R_\\sigma }\\,.$ Expressions for $H^S$ and $H^R$ are given in Eqs.", "() and () above.", "We also have $H^S_i & = & \\frac{1}{m}\\, S_j S_{ji} + U_i - \\frac{\\hbar ^2}{m} \\left(R_j R_{ji}+ \\frac{1}{2}\\, R_{jji}\\right) , \\\\[0.2cm]H^R_i & = & \\frac{1}{m} \\left(S_j R_{ji} + S_{ji} R_j + \\frac{1}{2}\\, S_{jji}\\right) .$ The time derivatives of the actions $S$ and $R$ and their first momentums are $\\dot{S} & = & \\frac{1}{2 m}\\, S_j^2 - U + \\frac{\\hbar ^2}{2 m} \\left(R_j^2 +R_{jj}\\right) , \\\\[0.2cm]\\dot{S}_i & = & -\\, U_i + \\frac{\\hbar ^2}{m} \\left(R_j R_{ji} + \\frac{1}{2}\\,R_{jji}\\right) , \\\\[0.2cm]\\dot{R} & = & - \\frac{1}{2 m} \\, S_{jj}\\, , \\\\[0.2cm]\\dot{R}_i & = & - \\frac{1}{m} \\left(S_{ji} R_j + \\frac{1}{2}\\, S_{jji}\\right) .$ For the $(p,\\bar{p})$ formulation, we only need equations for $\\dot{p}_\\sigma $ and $H_\\sigma $ , since the equations for $\\dot{\\bar{p}}_\\sigma $ and $\\bar{H}_\\sigma $ are obtained from them by conjugation in an obvious way.", "The Hamiltonian $H$ is given by Eq.", "(REF ), and the operator of total differentiation by Eq.", "(REF ).", "Then for $H_i$ and $H_{ik}$ we have $H_i & = & \\frac{1}{m}\\, p_j p_{ji} + U_i + \\frac{\\hbar }{2 i m}\\, p_{jji} \\, , \\\\[0.2cm]H_{ik} & = & \\frac{1}{m}\\, (p_j p_{jik} + p_{jk} p_{ji}) + U_{ik} + \\frac{\\hbar }{2 i m}\\,p_{jjik} \\, ,$ while the time derivatives of the action and first momentums are $\\dot{p} & = & \\frac{1}{2 m} \\, p_j \\bar{p}_j - U - \\frac{\\hbar }{2 i m}\\, p_{jj}\\, , \\\\[0.2cm]\\dot{p}_i & = & \\frac{1}{2 m} \\,(\\bar{p}_j - p_j) p_{ji} - U_i - \\frac{\\hbar }{2 i m}\\,p_{jji}\\, ,\\\\[0.2cm]\\dot{p}_{ik} & = & \\frac{1}{2 m} \\,(\\bar{p}_j - p_j) p_{jik} - \\frac{1}{m} \\, p_{ji} p_{jk}- U_{ik} - \\frac{\\hbar }{2 i m}\\, p_{jjik}\\, .$ It is not difficult to derive a general expression for $H_\\sigma $ .", "We say that a multi-index $\\nu $ is a subindex of the multi-index $\\sigma $ , and write $\\nu \\subset \\sigma $ , if there exists a multi-index $\\mu $ such that $\\sigma = \\nu \\mu $ .", "This multi-index $\\mu $ will then be denoted as $\\sigma \\setminus \\nu $ .", "Every multi-index is its own subindex, and the empty multi-index is a subindex of every multi-index.", "We also say that the multi-index $\\nu \\subset \\sigma $ is chosen from the multi-index $\\sigma $ if $\\nu $ is obtained from $\\sigma $ in the following way: write $\\sigma $ as a sequence of indices $i_1,\\ldots ,i_{|\\sigma |}$ , then with this sequence fixed select $|\\nu |$ members of the sequence to form $\\nu $ , and the others form $\\sigma \\setminus \\nu $ .", "Denote the summation over all such choices from a fixed sequence by $\\sum _{\\nu \\prec \\sigma }$ .", "With this definition, it is easy to prove by induction that $H_\\sigma = \\frac{1}{2m} \\,\\sum _{\\nu \\prec \\sigma } p_{j\\nu } p_{j\\sigma \\setminus \\nu }+ U_\\sigma + \\frac{\\hbar }{2 i m}\\, p_{jj\\sigma } \\, , $ where the factor $1/2$ accounts for the fact that in the sum over $\\nu $ every term appears twice.", "If not all indices in $\\sigma $ are different, then the same subindex $\\nu \\subset \\sigma $ may be chosen from $\\sigma $ in different ways.", "Consequently, there will be different choices that give the same (i.e., with the same $\\nu $ ) contribution to the sum in (REF ).", "For example, this will always happen when configuration space is one-dimensional, and the reader is encouraged to write formulas for $H_\\sigma $ with $|\\sigma | = 1,2,3,\\ldots $ in this case.", "It may be useful to present the summation in (REF ) in a form that contains only different contributions.", "Since the number of ways by which $\\nu _i$ indices $i$ may be chosen from $\\sigma _i$ of them in a multi-index $\\sigma $ is equal to $C_{\\sigma _i}^{\\nu _i} = \\sigma _i!/\\nu _i!\\,(\\sigma _i - \\nu _i)!$ , the total number of ways by which a multi-index $\\nu \\subset \\sigma $ may be chosen from $\\sigma $ is $C_\\sigma ^\\nu =\\prod _{i=1}^n C_{\\sigma _i}^{\\nu _i}$ .", "Therefore, Eq.", "(REF ) may be rewritten as $H_\\sigma = \\frac{1}{2m} \\,\\sum _{\\nu \\subset \\sigma } C_\\sigma ^\\nu p_{j\\nu }p_{j\\sigma \\setminus \\nu } + U_\\sigma + \\frac{\\hbar }{2 i m}\\, p_{jj\\sigma } \\, , $ where the summation now is over all different subindices $\\nu $ of $\\sigma $ .", "Correspondingly, the equations of motion for the $p_\\sigma $ , $\\sigma \\ne \\mbox{ø}$ , become $\\dot{p}_\\sigma = \\frac{1}{2 m} \\,(\\bar{p}_j - p_j) p_{j\\sigma } - \\frac{1}{2m} \\,{\\sum _{\\nu \\subset \\sigma }}^{\\prime } C_\\sigma ^\\nu p_{j\\nu } p_{j\\sigma \\setminus \\nu }- U_\\sigma - \\frac{\\hbar }{2 i m}\\, p_{jj\\sigma }\\, .", "$ where the summation $\\sum _{\\nu \\subset \\sigma }^{\\prime }$ excludes terms with $\\nu =\\mbox{ø}$ and $\\nu =\\sigma $ .", "The emerging theory is in many respects similar to classical mechanics, but there are also important differences.", "As in classical mechanics, the particles in $\\cal P$ AQD move along well-defined trajectories, with definite values of position, velocity, and all momentums at every moment of time.", "The states of the particle belong to an infinite phase space $\\cal P$ , and the equations of motion (REF ) describe the evolution of these states in terms of Hamiltonian flow in $\\cal P$ .", "For the $(p,\\bar{p})$ formulation, these equations take the form of (REF ) and (REF ), and the corresponding Hamiltonian flow is generated by the vector field $\\begin{array}{ccl}X & = & \\displaystyle \\frac{\\partial }{\\partial t} + v^i \\frac{\\partial }{\\partial x^i} + \\dot{p}_\\sigma \\frac{\\partial }{\\partial p_\\sigma }+ \\dot{\\bar{p}}_\\sigma \\frac{\\partial }{\\partial \\bar{p}_\\sigma } \\\\[0.4cm]& = & \\displaystyle \\frac{\\partial }{\\partial t} + v^i D_i - H_\\sigma \\frac{\\partial }{\\partial p_\\sigma }- \\bar{H}_\\sigma \\frac{\\partial }{\\partial \\bar{p}_\\sigma } \\, , \\end{array}$ where $v^i$ , $\\dot{p}_\\sigma $ , $D_i$ , and $H_\\sigma $ are given by Eqs.", "(REF ), (REF ), (REF ), and (REF ) respectively, $\\dot{\\bar{p}}_\\sigma $ and $\\bar{H}_\\sigma $ are obtained by conjugation, and summation over $\\sigma $ does not include $\\sigma =\\mbox{ø}$ .", "As in classical mechanics, initial value of the state uniquely determines its future evolution.", "The action function and QHJE are not needed for solution of the equations of motion.", "The action function is brought into use as an additional mathematical structure either by introducing a Taylor series (REF ) or via Eqs.", "(REF ) and (REF ) of the previous section.", "For the $(p,\\bar{p})$ formulation, Eqs.", "(REF ) with initial condition $p({\\bf x }_0,t_0)=p_0({\\bf x }_0)$ take the form $p({\\bf x },t) = \\displaystyle p_0({\\bf x }_0) + \\int _{C_{AB}} L\\, \\mathrm {d}t \\, , $ and conjugated equation for $\\bar{p}({\\bf x },t)$ , and similarly for the $(S,R)$ formulation, they take the form $\\begin{array}{ccl}S({\\bf x },t) & = & \\displaystyle S_0({\\bf x }_0) + \\int _{C_{AB}} L^S\\, \\mathrm {d}t \\, , \\\\[0.4cm]R({\\bf x },t) & = & \\displaystyle R_0({\\bf x }_0) + \\int _{C_{AB}} L^R\\, \\mathrm {d}t \\, , \\end{array}$ where $L$ , $L^S$ , and $L^R$ are given by the right hand sides of Eqs.", "(REF ), (REF ), and () respectively, and $C_{AB}$ is the particle's trajectory, connecting points $A=({\\bf x }_0,t_0)$ and $B=({\\bf x },t)$ .", "As was discussed in section 2.5, in the spaces with a nontrivial fundamental group, the action function may be multi-valued, and it is always defined up to an additive constant.", "Consequently, only the derivatives of the action function are relevant, and so this function may be represented by a graph in $\\cal P$ .", "It then describes a family of trajectories, determined by the given momentums at each position at some initial time.", "The same is true in classical mechanics; the important difference, however, is that while in $\\cal P$ AQD Eqs.", "(REF ) fix all momentums/derivatives, the corresponding classical equations [19] fix only the first of them.", "As a result, in classical mechanics the action function cannot be considered as characterizing the individual state/trajectory of a particle: a given trajectory may belong to any of a continuum of different families of trajectories, with different action functions.", "Contrary to that, in $\\cal P$ AQD, if the state of a particle belongs to some family, described by an action function, then by Eq.", "(REF ) it determines all derivatives of this function at a point where the particle is.", "As an action function is analytic, it is equal, up to a constant, to the sum of a corresponding Taylor series.", "Consequently, in $\\cal P$ AQD the state of a particle determines the action/wave function of a family, which includes it, and is, therefore, described or characterized by this function.", "This description, however, is not complete: since an analytic function can be expanded in a Taylor series at any point of space, there are different (i.e., with different ${\\bf q }$ ) members of a family that all have the same action/wave function.", "Thus a complete description of particle's state may be given either by a point in $\\cal P$ or, equivalently, by a point in a base and an action function, defined up to an additive constant (or wave function, defined up to a constant factor).", "The action function introduced in this way satisfies QHJE, which expresses its time derivative through this function itself, regardless of which particular trajectory is responsible for its appearance, and so the action function obtains its own dynamics.", "Nevertheless, it is clear that in the framework of $\\cal P$ AQD, on the fundamental level of equations of motion, the action function is a useful, but purely mathematical entity: for determination of particle's trajectory, its use is neither necessary nor sufficient.", "However, the action function, or rather the wave function, gains physical significance when a family of trajectories described by it gains physical significance.", "This will be the case when one considers the preparation of an experiment.", "Namely, as we will see later, using macroscopic control tools one can usually fix the wave function, but not the trajectory (i.e., not the specific $\\cal P$ AQD state) of a particle.", "This means, that with every macroscopically identical repetition of an experiment, the wave function of a prepared particle will be reproduced, but with a different specific trajectory.", "These trajectories belong to the just-described family, and make up an ensemble that the wave function is associated with.", "Thus the wave function reflects the preparation procedure and describes the properties of an emergent ensemble, but not individual events (trajectories) in it, in agreement with Einstein's views (see corresponding discussion in [23]).", "The statistical distribution of trajectories in this ensemble will be discussed later.", "When a particle moves in an infinite phase space, its position in configuration space moves with the velocity given by Eq.", "(REF ).", "This is the same velocity that is attributed to the particle in the de Broglie - Bohm theory (DBBT) [12], [13], where the wave function and particle's position are considered as fundamental elements of physical reality.", "It is then postulated that the wave function evolves according to Schrödinger's equation (REF ) and guides the motion of a particle according to Eq.", "(REF ).", "Alternatively, it is assumed that relation (REF ) is satisfied at some initial moment of time, and then the particle moves according to Newton's law, but under the influence of an additional “quantum potential\", which is created by the wave function and is given by the part of $H^S$ , Eq.", "(), proportional to $\\hbar ^2$ .", "With an additional assumption about initial statistical distribution of particles, DBBT is known to reproduce experimental predictions of QM.", "From the $\\cal P$ AQD point of view, the relation between $\\cal P$ AQD, DBBT, and standard QM is as follows: While $\\cal P$ AQD develops both the ODE part of the theory, describing the particle's motion, and the PDE part, which describes the evolution of the action function, the standard QM restricts itself to the PDE part, thus being an analog of the Hamilton-Jacobi part of classical mechanics without its Newton/Hamilton ODE part.", "Consequently, to compensate for this missing part of the theory, QM employs the statistical interpretation, which postulates the missing part's results.", "The progress achieved by DBBT is based on the observation that the need for the interpretation disappears if one postulates just described dynamical law of particle's motion, for all experimental predictions of QM can be deduced from this law mathematically.", "However, in the absence of a full geometric picture and the theory of equations of motion, developed in section 2, this modification of the theory required a promotion of the wave function to the rank of a real physical field that guides the particle or acts on it (but is not acted upon) with a quantum potential.", "As a result, DBBT drew a picture of the world so alien to the generally accepted ideas about a possible structure of physical theory, that the majority of the physical community found it too hard to accept, in spite of the theory's success with some difficult issues of QM, such as the measurement problem.", "As was discussed above, far from declaring the wave function a real physical field, $\\cal P$ AQD may deal without it at all.", "However, using the wave function may be convenient from the practical point of view.", "Thus for $\\cal P$ AQD, DBBT just implements the generalized Jacobi method: rather than solve the ordinary differential equations of motion, one can instead solve the Schrödinger equation or QHJE, and then get the particle's velocity from Eq.", "(REF ), where momentum $S_j$ is obtained from the real part of the action function by a simple differentiation.", "The same procedure works in classical mechanics, and so for $\\cal P$ AQD the DBBT program sounds exactly like a suggestion to consider classical mechanics as a theory of particles and real physical “action field\" $S$ that evolves according to the Hamilton-Jacobi equation and guides particles, forcing them to move with the velocity ${\\bf v }= \\nabla S/m$ .", "Besides different physical picture, $\\cal P$ AQD also differs from DBBT by an extra requirement of analyticity, which will become increasingly important in what follows.", "Nevertheless, the particles in $\\cal P$ AQD move along the same “Bohmian trajectories\" with velocity (REF ) as in DBBT, which will allow us to use, with proper modifications, some of its important results.", "The classical limit of $\\cal P$ AQD is best seen in the $(S,R)$ formulation.", "The Hamiltonian $H^S$ and the equations of motion for the action function $S$ and its derivatives contain terms proportional to $\\hbar ^2$ .", "When these terms are small compared to other, “classical\" ones, they may be neglected.", "The equations for $S$ and $S_\\sigma $ then decouple from the equations for $R$ and $R_\\sigma $ , and directly turn into the system of equations for the theory with a Hamiltonian, given by the first two terms of $H^S$ , Eq. ().", "This is a first-order Hamiltonian of classical mechanics, and the theory is classical mechanics, prolonged from the classical space of 1-jets $J^1_1$ to the corresponding infinite jet space $J^\\infty _1$ of the “full\" theory, which describes all derivatives of the action function.", "As was discussed in section 2.7, in the infinite system of equations of this theory the standard equations of classical mechanics form a closed “classical\" subsystem, which provides full information about the evolution of the action function $S$ , its first derivatives, i.e., components of classical momentum, and, most importantly, the particle's position ${\\bf q }$ .", "If the equations of the classical subsystem are solved, the higher derivatives $S_\\sigma $ can be obtained from the solution either by quadrature (REF ) or simply by direct differentiation of the action function $S$ .", "Thus in a classical limit (or in a formal limit $\\hbar \\rightarrow 0$ ) $\\cal P$ AQD dramatically simplifies, both conceptually and in terms of its complexity, and reduces to this subsystem, i.e., to classical mechanics." ], [ "The form of Hamiltonian, superposition principle, pathintegration, and wave-particle\nduality", "The form of Hamiltonian, superposition principle, path integration, and wave-particle duality The evolution of the wave function $\\psi ({\\bf x },t)$ over an infinitesimal time interval $\\varepsilon $ may be represented by a one-step Feynman integral as $\\psi ({\\bf x },t+\\varepsilon ) = \\int \\exp \\left[ \\frac{i}{\\hbar }\\,\\varepsilon L\\left( \\frac{{\\bf x }-{\\bf y }}{\\varepsilon },{\\bf y }, t \\right)\\right] \\psi ({\\bf y },t) \\prod _{i=1}^n \\frac{dy^i}{A}\\,+\\,{\\cal O}(\\varepsilon ^2)\\,,$ where $L({\\bf v },{\\bf y },t)=m {\\bf v }^2/2 - U({\\bf y },t)$ is a classical Lagrangian, and $A=\\sqrt{2\\pi i\\hbar \\varepsilon /m}$ is a normalization constant [24].", "Let ${\\bf q }(t)$ be some curve in the base space, and $\\lbrace p(t)\\rbrace $ be the set of values of the action function $p$ and its derivatives at time $t$ at the point ${\\bf q }(t)$ : $\\lbrace p(t)\\rbrace = \\big \\lbrace \\partial _\\sigma (\\hbar /i) \\ln \\psi \\big ({\\bf q }(t),t\\big ) \\big \\rbrace $ .", "At any time, we have then the wave function $\\psi ({\\bf x },t) = \\exp \\left[ \\frac{i}{\\hbar }\\,\\sum _\\sigma \\frac{1}{\\sigma !", "}\\, p_\\sigma (t)\\big ({\\bf x }- {\\bf q }(t)\\big )^\\sigma \\right] .", "$ At time $t=0$ , let the curve pass through a point ${\\bf q }=0$ with velocity ${\\bf v }$ , so that ${\\bf q }(0)=0$ and ${\\bf q }(\\varepsilon )={\\bf v }\\varepsilon $ .", "Using Eq.", "(REF ) and letting ${\\bf z }={\\bf x }-{\\bf v }\\varepsilon $ , $p_\\sigma =p_\\sigma (0)$ , and $p^{\\prime }_\\sigma = p_\\sigma (\\varepsilon )$ , we have from Eq.", "(REF ): $\\sum _\\sigma \\frac{1}{\\sigma !", "}\\, p^{\\prime }_\\sigma \\, {\\bf z }^\\sigma = \\frac{\\hbar }{i}\\,\\ln \\int \\exp \\left[ \\frac{i}{\\hbar }\\,\\varepsilon L\\left( \\frac{{\\bf z }-{\\bf y }}{\\varepsilon }+{\\bf v },{\\bf y }, t \\right) + \\frac{i}{\\hbar }\\,\\sum _\\sigma \\frac{1}{\\sigma !", "}\\, p_\\sigma {\\bf y }^\\sigma \\right] \\prod _{i=1}^n \\frac{dy_i}{A} \\,\\,+\\,\\, {\\cal O}(\\varepsilon ^2)\\, .", "$ Schrödinger's equation is a consequence of Eq.", "(REF ), therefore, Eq.", "(REF ) with the Hamiltonian function (REF ), and then Eq.", "(REF ) follow from it as well.", "It is, however, instructive to obtain that the evolution of momentums $p_\\sigma $ along the curve ${\\bf q }(t)$ corresponds to Eq.", "(REF ), i.e., that $p^{\\prime }_\\sigma = p_\\sigma + \\varepsilon ( p_{\\sigma i} v^i - H_\\sigma ) + {\\cal O}(\\varepsilon ^2)$ with $H_\\sigma $ given by Eq.", "(REF ), directly from Eq.", "(REF ).", "For that, we need to find the coefficients of the expansion of the integral in (REF ) in powers of $z^i$ .", "It is convenient to introduce one more variable ${\\bf u }= {\\bf y }- {\\bf z }$ and, using the expression for the Lagrangian, rewrite this integral as $\\frac{\\hbar }{i}\\,\\ln \\int \\exp \\left[ \\frac{im}{2\\hbar \\varepsilon } \\, u^2 - \\frac{im}{\\hbar } \\, {\\bf u }{\\bf v }+ \\frac{im}{2\\hbar } \\, v^2 \\varepsilon - \\frac{i\\varepsilon }{\\hbar } \\, \\sum _\\sigma \\frac{1}{\\sigma !}", "\\, U_\\sigma ({\\bf z }+{\\bf u })^\\sigma + \\frac{i}{\\hbar }\\,\\sum _\\sigma \\frac{1}{\\sigma !}", "\\,p_\\sigma ({\\bf z }+{\\bf u })^\\sigma \\right] \\prod _{i=1}^n \\frac{du^i}{A} \\, .", "$ The integral here is of the kind that may be evaluated using standard rules of the diagram technique [25].", "The logarithm in front of the integral means that we should include only connected diagrams.", "The first term in the exponent defines a contraction $\\langle u^j u^k \\rangle =-(\\varepsilon \\hbar /im)\\,\\delta _{jk}$ .", "Since we are only interested in the zero-order and first-order contributions of $\\varepsilon $ , and the contraction is proportional to $\\varepsilon $ , we have to consider only diagrams with one contraction or with no contractions at all.", "Then the expression in Eq.", "(REF ) will become the sum of the following contributions: The contraction of the term $-(im/\\hbar ){\\bf u }{\\bf v }$ in the exponent with itself gives $-m v^2 \\varepsilon /2$ , where $1/2$ is a symmetry factor, and cancels the contribution of the third term in the exponent.", "As the potential term in the exponent already has a coefficient $\\varepsilon $ in front of it, we can write there ${\\bf z }^\\sigma $ instead of $({\\bf z }+{\\bf u })^\\sigma $ , and then the contribution of this term to (REF ) will be equal to $- \\varepsilon \\sum _\\sigma (1/\\sigma !)", "\\, U_\\sigma {\\bf z }^\\sigma $ .", "In the last term in the exponent, it is sufficient to expand $({\\bf z }+{\\bf u })^\\sigma $ up to the second power of $u^i$ .", "This term then becomes equal to $(i/\\hbar )\\sum _\\sigma ({\\bf z }^\\sigma /\\sigma !", ")\\,(p_\\sigma + p_{\\sigma i}u^i + p_{\\sigma ij}u^iu^j/2)$ .", "Now the contribution to (REF ) of the term $p_\\sigma $ here is equal to $\\sum _\\sigma ({\\bf z }^\\sigma /\\sigma !", ")p_\\sigma $ , the contribution of the contraction of $p_{\\sigma i}u^i$ with $-(im/\\hbar ){\\bf u }{\\bf v }$ is equal to $\\varepsilon \\sum _\\sigma ({\\bf z }^\\sigma /\\sigma !", ")p_{\\sigma i}v^i$ , the contribution of the contraction of $p_{\\sigma i}u^i$ with itself is equal to $-(\\varepsilon /2m)\\sum _{\\nu \\mu }({\\bf z }^\\nu {\\bf z }^\\mu /\\nu !\\mu !", ")p_{\\nu i}p_{\\mu i}$ , where $1/2$ is a symmetry factor, and finally the contribution of the contraction of $u^i$ with $u^j$ in $p_{\\sigma ij}u^iu^j$ is equal to $-(\\varepsilon \\hbar /2im)\\sum _\\sigma ({\\bf z }^\\sigma /\\sigma !", ")p_{\\sigma jj}$ .", "Now, collecting all terms and comparing the coefficients for equal powers of $z^i$ in both parts of Eq.", "(REF ), we obtain Eq.", "(REF ) with $H_\\sigma $ given by Eq.", "(REF ).", "We have thus demonstrated that the whole system of equations (REF ) with Hamiltonian (REF ) may be compactly represented by one equation (REF ).", "On the other hand, starting from the ODEs (REF ) with Hamiltonian (REF ), and reversing the above arguments, we can derive Eq.", "(REF ) in the framework of $\\cal P$ AQD.", "Equation (REF ), therefore, is equivalent to the system (REF ) with Hamiltonian (REF ), and, being augmented with the variational principles of section 2.4 for determination of the velocity ${\\bf v }$ , may be taken as an alternative starting point of the theory.", "Equation (REF ) then will fit in the general scheme of $\\cal P$ AQD as a separate postulate, restricting the possible form of the Hamiltonians in (REF ).", "For more general situations than the just-considered motion of a particle in a flat space under the influence of a potential force, the form of the Lagrangian function in (REF ) will be different, for example, in magnetic field it will include a linear in velocity term $(e/c){\\bf A }\\cdot ({\\bf x }-{\\bf y })/\\varepsilon $ , where ${\\bf A }$ is a vector potential evaluated at the “midpoint\" $({\\bf x }+{\\bf y })/2$ [26].", "The Hamilton operator in Schrödinger's equation and the Hamiltonian function $H$ in Eq.", "(REF ) will then be determined by this Lagrangian function in the same way as for the standard case above, and to be able to develop $\\cal P$ AQD we have to require that the function $H$ satisfies HC1, Eq.", "(REF ).", "In the development based on Eq.", "(REF ), this additional condition appears completely arbitrary and artificial.", "We will see in the next two sections, however, that it is also necessary for a derivation of the probabilistic interpretation of the wave function.", "As was just mentioned, the quantum Hamiltonian function in Eq.", "(REF ), which describes the dynamics of a particle in $\\cal P$ AQD, is determined by the Lagrangian function in (REF ), or by the corresponding classical Hamiltonian, obtained from it in the limit $\\varepsilon \\rightarrow 0$ .", "This last Hamiltonian will necessarily be of first order, i.e., it will depend only on position and the usual momentums $p_i$ , and not on any $p_\\sigma $ with $|\\sigma |>1$ .", "Thus the approach that starts from Eq.", "(REF ) automatically reduces the variety of possible quantum Hamiltonians in (REF ), which were previously restricted by the Hamiltonian conditions only, to those which are obtainable in the described way from some classical Hamiltonian of the first order.", "As was discussed above, this classical Hamiltonian will then describe the classical limit of the corresponding quantum theory, which in turn will become its quantization.", "However, this quantization does not have to be unique.", "Indeed, while the quantum Hamiltonian in (REF ) is determined by a Lagrangian function in (REF ), which is written for finite $\\varepsilon $ , the corresponding classical Hamiltonian is obtained from this function in the limit $\\varepsilon \\rightarrow 0$ .", "Consequently, there might be cases when different Lagrangian functions in (REF ) define different quantum Hamiltonians, but the same classical Hamiltonian in the $\\varepsilon \\rightarrow 0$ limit.", "For example, the quantum Hamiltonian in magnetic field would be different, if the vector potential in the term $(e/c){\\bf A }\\cdot ({\\bf x }-{\\bf y })/\\varepsilon $ in Lagrangian was evaluated at other point than $({\\bf x }+{\\bf y })/2$ ; this ambiguity reflects operator ordering ambiguity in canonical quantization [26].", "In such cases, we will regard these different Lagrangians as defining physically different quantum theories that nevertheless share a common classical limit.", "In other words, the more fundamental quantum theory must uniquely define its classical approximation, but not the other way around.", "In such situations, if competing theories are supposed to describe nature, then not more than one of them can do it right, and it should be chosen based on its phenomenological success.", "The next observation regarding the approach based on Eq.", "(REF ) as a foundation of the theory is that it automatically introduces the wave function, which in $\\cal P$ AQD is defined by Eq.", "(REF ), as an object with a linear law of evolution (REF ).", "The superposition principle then follows immediately.", "As was discussed above, the particle's state in $\\cal P$ AQD may be described by its position and a wave function, and it is the wave function part of this description that is the subject of the superposition principle: if at some initial time $t=t_0$ , the wave function $\\psi $ , Eq.", "(REF ), is equal to a linear combination of other functions $\\psi _k$ , $k=1,\\ldots ,n_k$ , of the form (REF ) with the same $q(t_0)$ as $\\psi $ , then it continues to be that combination as time evolves.", "The position $q(t)$ of the particle then evolves according to the equations of motion with the wave function $\\psi $ .", "There is no such concept as superposition of a particle's position, and the time evolution of this position in a state with the wave function $\\psi $ is not related in any simple way to evolutions in states with wave functions $\\psi _k$ .", "Equation (REF ) is equivalent to Eq.", "(REF ), from which Schrödinger's equation (REF ) immediately follows.", "An even more important property of the approach that selects Hamiltonians in (REF ) using Eq.", "(REF ) is that while the integral in (REF ) is mathematically well defined, and does not suffer from any difficulties that are usually associated with the path integration, the iteration of Eq.", "(REF ) leads to a Feynman path-integral representation of wave and action functions.", "Thus the value of the action function at some point of the base space may be obtained in two seemingly very different ways: either as action integrals (REF ) and (REF ) along a particle's well-defined trajectory, coming to this point, or as a logarithm of a sum over paths.", "Obviously, this remarkable duality is a consequence of the fact that the quantum Hamiltonian (REF ), which determines the particle's dynamics in $\\cal P$ AQD, was obtained from the Lagrangian function of the path integral via Eq.", "(REF ).", "Still, it is not immediately clear how the action integrals (REF ) and (REF ), which operate only with the values defined directly on a particle's trajectory, and not anywhere else, can reproduce the sum over paths, which is obviously affected by the whole neighborhood of the trajectory.", "The answer is that the trajectory itself, and therefore the action integrals, are determined by an infinite system of (ordinary differential) equations, which depend on all derivatives $p_\\sigma $ of the action and $U_\\sigma $ of the potential.", "But the action function is analytic (this is one of the postulates of $\\cal P$ AQD), and we also assume that the potential function is analytic (and believe it always is analytic in nature).", "Consequently the trajectory, using these derivatives, obtains the full knowledge of the action and potential functions everywhere, and with it the ability to reproduce results obtained by path integration.", "In other words, the action integrals (REF ) and (REF ) utilize the information about analytic action and potential functions which is contained in their derivatives at the points of the actual particle's trajectory, while summation over paths uses the values of these functions on the whole space directly.", "But path integration provides a purely wave description of a particle's behavior, which naturally explains such characteristically wave phenomena as interference and diffraction.", "Therefore, it is because of a special form of the Hamiltonian, obtained from Eq.", "(REF ), and analyticity of the action/wave function, that the particle, which moves along a single trajectory, exhibits at the same time the characteristics of a wave, thus possessing the property of wave-particle duality.", "For example, in agreement with conclusions of [27], in a two-slit experiment the motion of a particle, passing through one slit, may depend crucially on whether the other slit is open or closed, even when the difference between the classical forces acting on the particle in these two cases is negligible.", "This is of course a purely quantum effect, completely impossible in classical theory, where a particle's trajectory is determined by a finite system of equations that depend only on the first derivative of the potential.", "Therefore, the wave-particle duality in the quantum domain receives a simple and natural mathematical explanation in $\\cal P$ AQD." ], [ "Variational principle, continuity equation, and invariantmeasure", "Variational principle, continuity equation, and invariant measure Equations (REF ), () for $R({\\bf x },t)$ may be rewritten in the form of a continuity equation $\\frac{\\partial j^0}{\\partial t} + \\mbox{div} {\\bf j }= 0, $ where $j^0 = |\\psi |^2 = e^{2R}$ , ${\\bf j }= j^0 {\\bf v }$ , and ${\\bf v }= \\nabla S/m$ is the particle velocity.", "The invariance of the measure, associated with conserved current $(j^0,{\\bf j }) = |\\psi |^2(1,{\\bf v })$ , is used in the next section to demonstrate that the particle's probability density is equal to $|\\psi |^2$ .", "Therefore, the conservation of this current is a very important element of the theory, and in this section we present several different proofs of it, which will allow us to better elucidate its origin.", "We will also derive an important expression for a corresponding invariant measure." ], [ "Variational principle and current conservation", "Equation (REF ) implies that Schrödinger's equation may be obtained from a stationary action principle.", "Indeed, consider the value $\\mathbf {H}(\\psi ,\\psi ^*) = \\frac{d}{d\\varepsilon } \\int \\psi ^*({\\bf x }) \\exp \\left[ \\frac{i}{\\hbar }\\,\\varepsilon L\\left( \\frac{{\\bf x }-{\\bf y }}{\\varepsilon },{\\bf y }, t \\right)\\right] \\psi ({\\bf y }) \\prod _i \\left.\\frac{dx^idy^i}{A} \\,\\, \\right|_{\\varepsilon =0} .", "$ Since the integration with a factor $\\exp (i\\varepsilon L/\\hbar )$ propagates $\\psi ({\\bf y })$ from time $t$ to time $t+\\varepsilon $ , and $\\psi ^*({\\bf x })$ from $t$ to $t-\\varepsilon $ , we obtain, obviously, $\\dot{\\psi }({\\bf x }) = \\frac{\\delta }{\\delta \\psi ^*({\\bf x })}\\, \\mathbf {H}(\\psi ,\\psi ^*) \\, , \\quad \\;\\dot{\\psi ^*}({\\bf x }) = -\\frac{\\delta }{\\delta \\psi ({\\bf x })}\\, \\mathbf {H}(\\psi ,\\psi ^*) \\, ,$ so that $\\mathbf {H}(\\psi ,\\psi ^*)$ is a Hamiltonian function with respect to the canonical field coordinates $\\psi ({\\bf x })$ and their conjugate momentums $\\psi ^*({\\bf x })$ (or coordinates $\\psi ^*({\\bf x })$ and momentums $-\\psi ({\\bf x })$ , which differs just by a canonical transformation).", "Being Hamilton equations, Eqs.", "(REF ), which are equivalent to Schrödinger's equation and its conjugate, follow, after standard discretization, from a stationary action principle in a Hamiltonian form $\\delta \\!\\int \\!", "\\mathbf {L}(\\psi ,\\psi ^*)\\,dt = 0$ , where $\\mathbf {L}(\\psi ,\\psi ^*) = \\frac{\\hbar }{i}\\,\\left[\\int \\psi ^*({\\bf x })\\dot{\\psi }({\\bf x }) \\prod _i dx^i -\\mathbf {H}(\\psi ,\\psi ^*)\\right]$ (see [19] and section 2.4) and the factor $\\hbar /i$ is introduced for convenience.", "To calculate $\\mathbf {H}(\\psi ,\\psi ^*)$ , integrate over $\\prod _i dy^i$ in Eq.", "(REF ) to get $\\mathbf {H}(\\psi ,\\psi ^*) = -\\frac{i}{\\hbar }\\,\\int \\psi ^*({\\bf x }) \\widehat{H} \\psi ({\\bf x }) \\prod _i dx^i \\, ,$ where $\\widehat{H}$ is a Hamilton operator, and so $\\mathbf {L}(\\psi ,\\psi ^*) = \\int \\!", "{\\cal L}\\prod _idx^i$ , where the Lagrangian density ${\\cal L}$ is ${\\cal L}= \\psi ^*({\\bf x }) \\left(\\frac{\\hbar }{i}\\,\\, \\partial _t + \\widehat{H} \\right) \\psi ({\\bf x })\\, .", "$ For the standard Hamiltonian of Schrödinger's equation (REF ) we have then ${\\cal L}= \\psi ^*({\\bf x }) \\left(\\frac{\\hbar }{i}\\,\\, \\partial _t - \\frac{\\hbar ^2}{2m}\\,\\partial _j^2 + U \\right)\\psi ({\\bf x }) \\, .$ As is well known [7], [28], both equations of motion and conservation laws, considered below, remain invariant with respect to adding a total divergence to the Lagrangian density.", "By adding appropriate terms to the above expression, we then obtain a familiar variational principle $\\delta \\int {\\cal L}\\prod _i dx^i \\, dt = 0$ with symmetric Lagrangian density ${\\cal L}= \\frac{\\hbar }{2i} \\left(\\psi ^*\\partial _t\\psi - \\psi \\partial _t\\psi ^*\\right) +\\frac{\\hbar ^2}{2m}\\, \\partial _i \\psi ^* \\partial _i \\psi + U \\psi ^* \\psi \\, , $ and it is indeed easy to verify directly that the corresponding Euler-Lagrange equation $\\frac{\\partial {\\cal L}}{\\partial \\psi ^*} - \\sum _{i=0}^n \\partial _i \\frac{\\partial {\\cal L}}{\\partial \\,(\\partial _i \\psi ^*)} = 0 \\, ,$ is equivalent to the Schrödinger equation (REF ).", "By Noether's theorem [7], [10], [28], if a transformation $\\psi \\rightarrow \\psi + \\alpha \\Delta $ , $\\psi ^* \\rightarrow \\psi ^* + \\alpha \\Delta ^*$ with infinitesimal real parameter $\\alpha $ changes the Lagrangian density ${\\cal L}$ just by adding a total divergence to it, ${\\cal L}\\rightarrow {\\cal L}+ \\alpha \\sum _{i=0}^n \\partial _i \\Lambda ^i \\, , $ then the solutions of the equations of motion (i.e., in our case Schrödinger's equation (REF )) satisfy a local conservation law $\\partial _0 j^0 + \\partial _k j^k = 0 $ with a current $j^i$ which is, for a first order Lagrangian ${\\cal L}$ , equal to $j^i = \\frac{\\partial {\\cal L}}{\\partial \\,(\\partial _i \\psi )} \\,\\Delta + \\frac{\\partial {\\cal L}}{\\partial \\,(\\partial _i \\psi ^*)} \\,\\Delta ^* -\\Lambda ^i \\, , \\quad i = 0,\\ldots ,n. $ As was discussed at the end of section 2.5 and in section 3, by their very construction the action function is defined up to an additive constant and the wave function up to a constant factor.", "Therefore, we should expect that the corresponding transformation does not change the equations of motion, and so the original and transformed Lagrangian densities differ by a total divergence only.", "Since the wave function is complex, we should consider two different transformations.", "Under the scale transformation $\\psi \\rightarrow e^\\alpha \\psi $ , $\\psi ^* \\rightarrow e^\\alpha \\psi ^*$ , when the wave function satisfies the Schrödinger equation, the Lagrangian density ${\\cal L}$ indeed changes as in (REF ) with $(\\Lambda ^0,{\\bf \\Lambda }) = \\left(0,\\, \\frac{\\hbar }{2m}\\,(\\psi ^* \\nabla \\psi + \\psi \\nabla \\psi ^*)\\right),$ but the sum of the first two terms in Eq.", "(REF ) in this case is equal to $\\Lambda ^i$ , and so the total current $j^i$ vanishes and the scale invariance does not lead to any conservation law.", "The phase transformation $\\psi \\rightarrow e^{i\\alpha /\\hbar }\\psi $ , $\\psi ^*\\rightarrow e^{-i\\alpha /\\hbar }\\psi ^*$ is more useful.", "Lagrangian density ${\\cal L}$ is invariant with respect to it, i.e., satisfies Eq.", "(REF ) with $\\Lambda ^i=0$ .", "Consequently, the corresponding current $(j^0, {\\bf j }) = \\left(\\psi ^*\\psi ,\\, \\frac{\\hbar }{2im}\\,(\\psi ^* \\nabla \\psi - \\psi \\nabla \\psi ^*)\\right) $ is conserved, i.e., satisfies Eq.", "(REF ), which in this case coincides with a continuity equation (REF ).", "Therefore, the current $(j^0,{\\bf j })$ is equal to $|\\psi |^2(1,{\\bf v })$ , where the velocity ${\\bf v }$ is given by Eqs.", "(REF ) or (REF ).", "To obtain this form of the conserved current, we used an explicit form of a Hamiltonian here.", "However, in section 5.2 we will show that this result has a much more general character, namely, for a wide range of possible Lagrangian functions $L$ in Eq.", "(REF ), the current, which is conserved due to the phase invariance, is equal to $|\\psi |^2(1,{\\bf v })$ with ${\\bf v }$ given by Eq.", "(REF ) (which for the standard Hamiltonian coincides with (REF ))." ], [ "Current conservation for Hamiltonian operators of general form", "It is desirable to derive the conservation of the current $(j^0,{\\bf j }) = |\\psi |^2(1,{\\bf v })$ under more general assumptions than above where we used an explicit form of a standard Hamiltonian.", "Here we will show that this conservation follows from the phase invariance of the Lagrangian density ${\\cal L}$ , Eq.", "(REF ), for an arbitrary quadratic in velocity Lagrangian function $L$ in Eq.", "(REF ), provided it satisfies some simple conditions.", "First substitute into (REF ) the representation (REF ) to get ${\\cal L}= |\\psi ({\\bf x })|^2 \\left(p_t + H\\right) , $ where the Hamiltonian $H$ , which corresponds to the Lagrangian function $L$ in (REF ), is a function of the space derivatives of the action function $p$ .", "We have from (REF ) and (REF ) $H = - \\left.\\frac{dW}{d\\varepsilon }\\,\\right|_{\\varepsilon = 0} , \\quad \\quad W = \\frac{\\hbar }{i}\\,\\ln \\int \\exp \\left[\\frac{i}{\\hbar }\\,\\varepsilon L + \\frac{i}{\\hbar }\\,p({\\bf x }+ {\\bf u }) \\right] \\prod _i \\frac{du^i}{A} \\, , $ where we introduced ${\\bf u }= {\\bf y }- {\\bf x }$ .", "As in section 4, expand the exponent in (REF ) in powers of $u^i$ and consider $W$ as a generating function for connected diagrams.", "We will assume that similar to the case of a standard Hamiltonian, the corresponding contraction $\\langle u^j u^k \\rangle $ is purely imaginary and proportional to $\\varepsilon $ .", "To account for a possible presence of a magnetic field, we allow the product $\\varepsilon L$ to have a vector potential term $\\,-(e/c) A_k u^k$ inside it [26], but assume that there are no other $\\varepsilon $ -independent and linear in $u^k$ terms there.", "The Lagrangian density (REF ) will then be a function of the wave functions $\\psi $ and $\\psi ^*$ and derivatives of $\\psi $ up to a second order.", "Using the corresponding formulas for second-order Lagrangian functions [7], [10], [28], the current, which conserves due to invariance of the Lagrangian density ${\\cal L}$ , Eq.", "(REF ), with respect to the phase transformation $\\psi \\rightarrow e^{i\\alpha /\\hbar }\\psi $ , $\\psi ^*\\rightarrow e^{-i\\alpha /\\hbar }\\psi ^*$ , will then be equal to $(j^0,{\\bf j })$ , where $\\begin{array}{ccl}j^0 & = & \\displaystyle \\frac{i}{\\hbar }\\,\\psi \\frac{\\partial {\\cal L}}{\\partial \\psi _t} \\, , \\\\[0.4cm]j^k & = & \\displaystyle \\frac{i}{\\hbar }\\,\\psi \\left(\\frac{\\partial {\\cal L}}{\\partial \\psi _k} - D_l \\frac{\\partial {\\cal L}}{\\partial \\psi _{kl}}\\right) +\\frac{i}{\\hbar }\\,\\psi _l \\frac{\\partial {\\cal L}}{\\partial \\psi _{kl}} \\, .", "\\end{array}$ We now want to rewrite these expressions in terms of derivatives of $H$ over $p_k$ and $p_{kl}$ .", "We have from Eq.", "(REF ) the following relations between partial derivatives of $p$ and $\\psi $ : $\\begin{array}{ccl}p_k & = & \\displaystyle \\frac{\\hbar }{i}\\,\\frac{\\psi _k}{\\psi } \\, , \\\\[0.3cm]p_{kl} & = & \\displaystyle \\frac{\\hbar }{i}\\,\\left(\\frac{\\psi _{kl}}{\\psi } - \\frac{\\psi _k \\psi _l}{\\psi ^2}\\right) .\\end{array}$ Let $p_{kl}$ and $p_{lk}$ enter the expression for ${\\cal L}$ symmetrically.", "We have then from (REF ) $\\begin{array}{ccl}\\displaystyle \\frac{\\partial {\\cal L}}{\\partial \\psi _k} & = & \\displaystyle \\frac{\\hbar }{i}\\,\\,\\frac{1}{\\psi }\\left(\\frac{\\partial {\\cal L}}{\\partial p_k} - 2 \\frac{i}{\\hbar }\\,p_l\\frac{\\partial {\\cal L}}{\\partial p_{kl}}\\right) , \\\\[0.4cm]\\displaystyle \\frac{\\partial {\\cal L}}{\\partial \\psi _{kl}} & = & \\displaystyle \\frac{\\hbar }{i}\\,\\,\\frac{1}{\\psi }\\, \\frac{\\partial {\\cal L}}{\\partial p_{kl}} \\, ,\\end{array}$ where the factor of 2 in the first equation compensates for the dropped contribution of $\\partial {\\cal L}/\\partial p_{lk}$ .", "Substituting these expressions into Eq (REF ) and using Eq.", "(REF ) and condition (REF ), we obtain for the current $\\begin{array}{ccl}j^0 & = & |\\psi |^2 , \\\\[0.1cm]j^k & = & \\displaystyle |\\psi |^2 \\left[\\frac{\\partial H}{\\partial p_k} + \\frac{i}{\\hbar }\\,\\left(\\bar{p}_l - p_l\\right)\\frac{\\partial H}{\\partial p_{kl}} \\right] , \\end{array}$ so that $j^0$ has the right form, and we need to evaluate the derivatives of $H$ in an expression for $j^k$ .", "For every function $B$ of coordinates $u^i$ , we denote by $\\langle B\\rangle $ the corresponding sum of connected diagrams produced by the generating function $W$ : $\\langle B\\rangle = \\frac{\\displaystyle \\int \\exp \\left[\\frac{i}{\\hbar }\\,\\varepsilon L + \\frac{i}{\\hbar }\\,\\left(p + p_k u^k + \\frac{1}{2}\\,p_{kl} u^k u^l + \\cdots \\right) \\right] B \\prod _i \\frac{du^i}{A}}{\\displaystyle \\int \\exp \\left[\\frac{i}{\\hbar }\\,\\varepsilon L + \\frac{i}{\\hbar }\\,\\left(p + p_k u^k + \\frac{1}{2}\\, p_{kl} u^k u^l + \\cdots \\right) \\right] \\prod _i \\frac{du^i}{A}}\\, ,$ where the derivatives $p_\\sigma $ are taken at the point ${\\bf x }$ where the Lagrangian density (REF ) is evaluated.", "We have then from Eq.", "(REF ) $\\frac{\\partial H}{\\partial p_k} = - \\left.\\frac{d}{d\\varepsilon }\\, \\langle u^k \\rangle \\,\\right|_{\\varepsilon = 0} =- \\left.\\frac{d}{d\\varepsilon }\\, \\frac{i}{\\hbar }\\,\\left(p_l - \\frac{e}{c}\\, A_l\\right) \\langle u^k u^l \\rangle \\,\\right|_{\\varepsilon = 0} \\, ,$ where we used the fact that the only nonzero contribution to $d \\langle u^k \\rangle /d \\varepsilon $ for $\\varepsilon = 0$ comes from contraction of $u^k$ with $\\varepsilon $ -independent terms in the exponent.", "On the other hand, we have $2\\, \\frac{\\partial H}{\\partial p_{kl}} = -\\left.\\frac{d}{d\\varepsilon }\\,\\langle u^k u^l \\rangle \\,\\right|_{\\varepsilon = 0}\\,,$ and so $\\begin{array}{ccc}\\displaystyle \\frac{\\partial H}{\\partial p_k} & = & \\displaystyle 2\\, \\frac{i}{\\hbar }\\,\\left(p_l - \\frac{e}{c}\\,A_l\\right)\\frac{\\partial H}{\\partial p_{kl}} \\, ,\\\\[0.4cm]\\displaystyle \\frac{\\partial \\bar{H}}{\\partial \\bar{p}_k} & = & \\displaystyle 2\\, \\frac{i}{\\hbar }\\,\\left(\\bar{p}_l - \\frac{e}{c}\\,A_l\\right)\\frac{\\partial H}{\\partial p_{kl}}\\, ,\\end{array}$ where to obtain the second equation we conjugated the first one and used that $\\partial H/\\partial p_{kl}$ is purely imaginary.", "Now using these equalities in Eq.", "(REF ) for $j^k$ , we obtain $j^k = \\displaystyle \\frac{1}{2}\\,|\\psi |^2 \\left(\\frac{\\partial H}{\\partial p_k} + \\frac{\\partial \\bar{H}}{\\partial \\bar{p}_k}\\right)= \\displaystyle |\\psi |^2 v^k , $ where ${\\bf v }$ is the particle's velocity (REF ), as was required." ], [ "Invariant measure", "For an arbitrary current $(i^0,{\\bf i })$ in the base space $M$ that satisfies a continuity equation $\\partial _t i^0 + \\partial _k i^k = 0$ , a form $\\nu = i^0\\Omega $ , $\\Omega = \\mathrm {d}x^1 \\wedge \\cdots \\wedge \\mathrm {d}x^n$ , integrated over any subspace of $M$ corresponding to a fixed time $t$ , is invariant with respect to a vector field $Y = \\partial _t + u^k \\partial _k$ with ${\\bf u }= {\\bf i }/i^0$ , so that this form defines on $M$ a measure that is invariant with respect to the flow of $Y$ .", "Indeed, let $D$ be an arbitrary cell in configuration space, every point of which moves with velocity ${\\bf u }$ .", "If $ds$ is an element of the boundary of $D$ , orthogonal to a unit vector ${\\bf n }$ pointing outside, then by the continuity equation over a time interval $dt$ the integral $\\int _D \\nu $ will reduce by ${\\bf i }{\\bf n }\\, dsdt$ due to the current ${\\bf i }$ through $ds$ .", "On the other hand, since the element $ds$ moves with velocity ${\\bf u }$ , during time $dt$ a volume ${\\bf u }{\\bf n }\\,dsdt$ will be added to $D$ , and with it a value $i^0{\\bf u }{\\bf n }\\,dsdt$ added to $\\int _D \\nu $ .", "Therefore, this integral will not change, and so $\\nu $ is invariant with respect to $Y$ .", "The same result may also be obtained by direct calculation.", "Indeed, using equalities $Y(\\nu ) =\\mathrm {d}(Y\\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\nu ) + Y\\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\mathrm {d}\\nu $ and $\\mathrm {d}\\Omega = 0$ , we obtain $\\begin{array}{ccl}Y(\\nu ) & = & \\mathrm {d}(i^0Y\\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\Omega ) + Y\\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\mathrm {d}(i^0\\Omega ) \\\\[0.2cm]& = & \\mathrm {d}i^0\\wedge (Y\\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\Omega ) + i^0\\mathrm {d}(Y\\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\Omega ) + Y\\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\mathrm {d}i^0\\wedge \\Omega \\, .\\end{array}$ In the last term, substitute $Y\\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\mathrm {d}i^0\\wedge \\Omega = Y(i^0)\\,\\Omega - \\mathrm {d}i^0\\wedge (Y\\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\Omega )$ to get $Y(\\nu ) = i^0\\mathrm {d}(Y\\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\Omega ) + Y(i^0)\\,\\Omega \\, .", "$ In the second term of this equation, we have $Y(i^0) = \\partial _t i^0 + u^k\\partial _k i^0$ , and in the first term $Y\\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\Omega = \\sum _k (-1)^{k-1}u^k(\\mathrm {d}x^1\\wedge \\cdots \\wedge \\mathrm {d}x^n)^{\\prime },$ where $(\\,)^{\\prime }$ means that the factor $\\mathrm {d}x^k$ in the product is dropped.", "From this, we have $\\mathrm {d}(Y\\hspace{1.0pt} \\mbox{\\vrule depth-0.1pt height0.53pt width 6.7pt\\vrule depth-0.1pt height5.5pt} \\hspace{3.5pt}\\Omega ) = \\Omega \\,\\partial _k u^k + \\mathrm {d}t \\sum _k(-1)^{k-1}\\partial _t u^k (\\mathrm {d}x^1\\wedge \\cdots \\wedge \\mathrm {d}x^n)^{\\prime }.$ We will integrate $Y(\\nu )$ over surfaces with fixed $t$ in the base space $M$ , so we are interested in a pullback $\\pi _t^*Y(\\nu )$ , where $\\pi _t\\!", ":\\,Q\\rightarrow M$ is the natural embedding that maps configuration space $Q$ into such surfaces.", "Then obviously the term with $\\mathrm {d}t$ does not contribute to such integrals, $\\pi _t^*\\mathrm {d}t= 0$ , and collecting remaining terms we have $\\pi _t^*Y(\\nu ) = \\Omega \\,(\\partial _ti^0 + u^k\\partial _k i^0+i^0 \\partial _k u^k)$ .", "But the last two terms sum to $\\partial _k(u^k i^0) =\\partial _k i^k$ , and so from the continuity equation we obtain $\\pi _t^*Y(\\nu ) = 0$ .", "The vector flow $X$ in $\\cal P$ AQD is defined by the first line of Eq.", "(REF ).", "Its first two terms have the form of the just-considered vector field $Y$ with respect to a current (REF ), and so conserve the form $\\omega = j^0\\Omega $ , while the last two terms, where $\\sigma \\ne \\mbox{ø}$ , when acting on this form give zero.", "However, the flow $X$ is defined in the infinite phase space $\\cal P$ , rather than in the base space $M$ .", "Consequently, to formulate the invariance condition, we use the map $\\pi _t\\!", ":\\,Q\\rightarrow \\cal {P}$ , which projects configuration space $Q$ into a part of the graph of a solution of Schrödinger's equation with given $t$ .", "With so-defined $\\pi _t$ , we have then the desired identity $\\pi _t^*X(\\omega ) = 0$ , which expresses the invariance of the form $\\omega $ and corresponding measure with respect to the Hamiltonian flow (REF )." ], [ "Probability density", "In the previous sections we studied the mathematical structure of $\\cal P$ AQD.", "Here we start considering its physical implications, i.e., experimental consequences of the assumption that particles move along the trajectories that we discussed.", "It is then natural to think, and we will confirm it later, that in $\\cal P$ AQD the experimentally measured particle's position should be equal to its position in $\\cal P$ .", "In this case, what can $\\cal P$ AQD say about the distribution of this measurement's results?", "We believe that in every repetition of an experiment, in which the particle is described by a wave function $\\psi $ , its position coordinate in $\\cal P$ assumes a random value, determined by a specific history of this particular repetition.", "In the mathematical limit of an infinite number of such repetitions, the results form an ensemble that determines the particle's probability density $\\rho $ : the probability of finding the particle in any volume element is equal to the relative number of ensemble members with the particle inside that element.", "It seems natural to assume, in agreement with experiment, that this probability density is determined by the wave function only, and not by the way in which the ensemble with this wave function was created.", "Then once created at some time $t$ , the ensemble remains representative for all future time, for one of the ways to create an ensemble at any time $t^{\\prime } > t$ is to create it at time $t$ and let it evolve till time $t^{\\prime }$ .", "But during such evolution, every volume element $dV$ , propagating with $\\cal P$ AQD Hamiltonian flow, continues to contain the same ensemble members, and so the probability $\\rho dV$ to find the particle in this element remains constant, which means that the change in the probability density $\\rho $ in the element is inversely proportional to $dV$ .", "On the other hand, as was demonstrated in the previous section, the product $|\\psi |^2 dV$ in this element also remains constant.", "Consequently, along any given trajectory, the probability density $\\rho $ should be proportional to $|\\psi |^2$ .", "The coefficient of proportionality can, by this reasoning, depend on trajectory.", "However, we note that once selected, this coefficient should remain fixed in the presence of any external fields that may be applied to the particle in the future.", "Since such fields can shuffle trajectories in an arbitrary way, but the coefficient should remain a continuous function of trajectory, it is clear that for all trajectories it must be the same.", "Moreover, even if for some reason configuration space $Q$ splits into two subspaces $Q_1$ and $Q_2$ such that trajectories never cross from one of them to the other, according to the way the wave function is brought into $\\cal P$ AQD, that will only mean that rather than being defined up to one constant factor in $Q$ , the wave function is now defined up to two independent constant factors in $Q_1$ and $Q_2$ .", "Obviously, these factors can be chosen in such a way as to make the coefficients of proportionality between $\\rho $ and $|\\psi |^2$ in $Q_1$ and $Q_2$ equal.", "We conclude that the probability density $\\rho $ should be proportional to $|\\psi |^2$ with a constant coefficient, or equal to it if $\\int |\\psi |^2 dV =1$ , i.e., if the wave function is normalized.", "Thus in $\\cal P$ AQD this classic relation between the wave function and probability density becomes just a property of the wave function, rather than its main physical meaning.", "It is worth recalling that this property is a consequence of such fundamental elements of the theory as the possibility of obtaining the Schrödinger equation from a variational principle (which in turn follows from the quantization via one-step Feynman integral — see the corresponding discussion at the beginning of section 5) and the definition of an action function, which leaves the freedom to add an arbitrary constant to it and which, therefore, for the theory following from a variational principle, results in a corresponding current conservation.", "Also, in the process of proving that the space part of the current has the desired form (REF ) for a general Hamiltonian function $H$ obtained from the one-step Feynman integral, we had to require that this function $H$ satisfies HC1 (REF ).", "Thus for any theory based on the one-step Feynman integral, the condition (REF ) is needed for the probabilistic interpretation of the wave function as well as for the very possibility to develop $\\cal P$ AQD in the first place.", "The same expression for the probability density may be also obtained in a different manner, if we count the number of possible ways by which a given ensemble can be created.", "This means the following: We break configuration space $Q$ into small cells, so that in every cell the probability density $\\rho $ can be considered constant.", "The ensemble of $N$ points, representing the particle in a state with a wave function $\\psi $ , is described by the numbers $N_i$ of points in every cell, $\\sum _i N_i = N$ .", "The series of $N$ experiments, which form an ensemble, is then characterized by the sequence $i_1,\\ldots ,i_N$ of cells the particle was found in in each experiment, so that such ensemble can be created in ${\\cal N}= N!/\\prod _i N_i!$ different ways.", "These numbers $N_i$ and consequently ${\\cal N}$ , depend on the specific way the space $Q$ is split into cells, and so some reasonable prescription for a way the splitting is done should be made.", "We will demand, as we did above, that once created, the ensemble should remain representative for all future time.", "The splitting of configuration space $Q$ into cells, therefore, should be such that once it is done and fixed, the number ${\\cal N}$ for every ensemble, i.e., the number of ways this ensemble can be created, remains constant with time.", "But the image of every cell $i$ , corresponding to the particle's flow over arbitrary time $t$ , has the same invariant measure $|\\psi |^2 dV$ and contains the same $N_i$ ensemble members as the cell itself.", "It is then clear that the demand will be satisfied, if in the limit of vanishing cell volumes, they will all have the same invariant measure, which we denote as $\\Delta \\Gamma $ .", "For our ensemble, described by the distribution of points in configuration space, we need now a characteristic of the number of its realizations that remains finite as $N\\rightarrow \\infty $ and $\\Delta \\Gamma \\rightarrow 0$ (in this order).", "The number of cells in a small area of configuration space $Q$ of a particle is proportional to $1/\\Delta \\Gamma $ , and in a small area of configuration space $Q^N$ of the ensemble — to $1/(\\Delta \\Gamma )^N$ .", "As ${\\cal N}$ is proportional to this number we need to factor it out, i.e., to consider $(\\Delta \\Gamma )^N{\\cal N}$ .", "The function of this last number that stays finite as $N\\rightarrow \\infty $ is the $N$ th root of it, and so we come to considering $\\Delta \\Gamma {\\cal N}^{1/N}$ .", "The value $\\begin{array}{ccl}S_G(\\rho ) & = & \\displaystyle \\lim _{\\Delta \\Gamma \\rightarrow 0} \\lim _{N\\rightarrow \\infty } \\ln \\left(\\Delta \\Gamma \\,{\\cal N}^{1/N}\\right) \\\\[0.3cm]& = & \\displaystyle \\lim _{\\Delta \\Gamma \\rightarrow 0} \\left(\\ln \\Delta \\Gamma + \\lim _{N\\rightarrow \\infty }\\frac{1}{N}\\,\\ln {\\cal N}\\right) \\end{array}$ will be called the Gibbs entropy of the ensemble, representing probability density $\\rho $ in configuration space $Q$ .", "We have then, using Stirling's formula, $\\lim _{N\\rightarrow \\infty }\\frac{1}{N}\\ln {\\cal N}= \\lim _{N\\rightarrow \\infty }\\frac{1}{N}\\ln \\frac{N!", "}{\\prod _i N_i!}", "= -\\lim _{N\\rightarrow \\infty } \\sum _i \\frac{N_i}{N}\\ln \\frac{N_i}{N}\\, .$ Further, $\\lim _{N\\rightarrow \\infty } N_i/N = \\rho _i \\Delta V_i = (\\rho _i/|\\psi _i|^2) \\Delta \\Gamma $ and $\\sum _i \\rho _i \\Delta V_i = \\lim _{N\\rightarrow \\infty } \\sum _i N_i/N = 1$ , therefore $\\begin{array}{ccl}S_G(\\rho ) & = & \\displaystyle \\lim _{\\Delta \\Gamma \\rightarrow 0} \\left[\\ln \\Delta \\Gamma - \\sum _i \\Delta V_i \\rho _i \\ln \\left(\\frac{\\rho _i}{|\\psi _i|^2}\\Delta \\Gamma \\right)\\right] \\\\[0.4cm]& = & \\displaystyle \\lim _{\\Delta V_i\\rightarrow 0} \\sum _i \\Delta V_i \\rho _i \\ln \\frac{|\\psi _i|^2}{\\rho _i} \\\\[0.5cm]& = & \\displaystyle \\int \\rho \\ln \\frac{|\\psi |^2}{\\rho }\\, dV \\, .", "\\end{array}$ In the absence of circumstances that make some cells preferable compared to others, every sequence $i_1,\\ldots ,i_N$ should be assigned equal probability.", "In the limit $N\\rightarrow \\infty $ , the emerging ensemble (“Gibbs ensemble\") should then maximize ${\\cal N}$ or $S_G(\\rho )$ .", "Indeed, in the limit $N\\rightarrow \\infty $ , the relative frequency of emergence of two ensembles with $S_G^{(2)} <S_G^{(1)}$ is $\\lim _{N\\rightarrow \\infty } \\frac{{\\cal N}_2}{{\\cal N}_1} = \\lim _{N\\rightarrow \\infty } \\exp \\left[N \\left(S_G^{(2)} - S_G^{(1)} \\right) \\right] = 0 \\, .", "$ Now, since $\\ln x \\le x - 1$ , we have from Eq.", "(REF ), when the wave function is normalized, $S_G(\\rho ) \\le \\int \\rho \\left(\\frac{|\\psi |^2}{\\rho } - 1\\right) dV = 0 \\, ,$ $S_G(\\rho )$ achieving its maximum possible value of 0 for $\\rho = |\\psi |^2$ , which will, therefore, be the observed probability density.", "As was discussed in section 3, the particles in $\\cal P$ AQD move with the same velocity (REF ) as in the theory of de Broglie - Bohm.", "It was shown by Bohm [12] that this law of motion preserves the standard quantum form of the probability density: if $\\rho $ is equal to $|\\psi |^2$ at some initial time $t_0$ , then it will stay equal to it for all $t>t_0$ .", "It was hypothesized [12], [14] that an arbitrary initial distribution would converge to the stable density $|\\psi |^2$ for $t$ of the order of some “relaxation time,\" in the same way as macroscopic systems converge to thermal equilibrium.", "The derivation, presented above, shows that this hypothesis is unnecessary.", "Nevertheless, especially because we are using the concept of entropy, and looking for a distribution which maximizes it, it is instructive to compare the situation in $\\cal P$ AQD with that in classical statistics.", "We present a brief sketch of statistical distribution and entropy growth in classical statistics, based mostly on the works [29], [30], in a form convenient for such comparison in the Appendix.", "From the discussion there, the following conclusions may be drawn: – The convergence to thermal equilibrium in classical statistics is related to the growth of Boltzmann entropy $S_B$ , rather than Gibbs entropy $S_G$ , which is maximized in $\\cal P$ AQD.", "– The growth of Boltzmann entropy is related to such properties of macroscopic systems as possibility of their crude, but adequate, description; as typicality (i.e., practical equality of observable magnitudes of additive physical values to their averages over microcanonical ensembles); and as possibility of replacement of one ensemble by the other in the process of these systems' time evolution (see details in the Appendix).", "These properties exist only in macroscopic systems that consist of enormous number of particles, and don't have any analogs in one-particle dynamics, classical or quantum.", "– The nature of the quantum distribution $|\\psi |^2$ is identical to that of the microcanonical distribution in classical statistics.", "Both distributions maximize the corresponding Gibbs entropies, and emerge not because of the large number of particles in a system, but because of the infinite number of systems, be they one- or multi-particle, in the Gibbs ensemble.", "According to Eq.", "(REF ), the ensemble with less than maximum Gibbs entropy has zero probability to arise.", "Consequently, all observed distributions automatically have the maximum possible values of their Gibbs entropy, in contrast to macroscopic systems' Boltzmann entropy, which grows due to the physical process of thermalization." ], [ "Multiparticle systems and quantum particles in a macroscopic classical environment", "We now extend our approach to multiparticle systems.", "For a system of $n_p$ particles in $n$ -dimensional space, we do it by directly combining $n_p$ one-particle $n$ -dimensional configuration spaces into $n_p n$ -dimensional configuration space $Q$ of a system.", "The theory of previous sections will be generalized in a straightforward way to look like a one-particle theory with corresponding Hamiltonian in an $n_p n$ -dimensional space $Q$ .", "In particular, the wave function of a system is related to the corresponding action function as in Eq.", "(REF ): $\\begin{array}{ccl}\\psi ({\\bf x }_1,\\ldots ,{\\bf x }_{n_p},t) & = & \\displaystyle \\exp \\left(\\frac{i}{\\hbar }\\,\\,p({\\bf x }_1,\\ldots ,{\\bf x }_{n_p},t)\\right),\\\\[0.4cm]p({\\bf x }_1,\\ldots ,{\\bf x }_{n_p},t) & = & \\displaystyle S({\\bf x }_1,\\ldots ,{\\bf x }_{n_p},t)+ \\frac{\\hbar }{i}\\,\\,R({\\bf x }_1,\\ldots ,{\\bf x }_{n_p},t) \\, , \\end{array}$ and momentums in the infinite phase space are partial derivatives of the action function with respect to the components of ${\\bf x }_1,\\ldots ,{\\bf x }_{n_p}$ .", "The Hamiltonian function (the multiparticle analog of the one-particle Hamiltonian (REF )) $H = \\sum _{k=1}^{n_p} \\frac{p^2_{j_k}}{2 m_k} + U + \\frac{\\hbar }{i}\\,\\sum _{k=1}^{n_p}\\frac{p_{j_k j_k}}{2 m_k} \\, , $ where summation over repeating indices $j_k$ is from 1 to $n$ , is obtained from the multiparticle Schrödinger equation in the same way as in section 3, and defines the evolution of momentums and particle velocities by the equations of motion (REF ), so that in particular the velocity of the $k$ -th particle is ${\\bf v }_k = \\nabla _k S({\\bf x }_1,\\ldots ,{\\bf x }_{n_p},t)/m_k$ .", "For macroscopic systems, the part of the action function related to their directly observable macroscopic degrees of freedom is much larger than Planck's constant $\\hbar $ .", "As was discussed at the end of section 3, in the corresponding equations of motion the terms with $\\hbar $ may be dropped, and then these equations reduce to those of classical mechanics, so that these degrees of freedom will exhibit a classical behavior.", "The wave function describing these classical degrees of freedom is given by Eq.", "(REF ) with the action function $S$ that solves the multiparticle analog of the classical part of Eqs.", "(REF ), (), i.e., the classical Hamilton-Jacobi equation $\\frac{\\partial S}{\\partial t} + H_c\\left({\\bf q },\\, \\frac{\\partial S}{\\partial {\\bf q }}\\right) = 0\\,, $ where the macroscopic degrees of freedom are combined into the vector ${\\bf q }$ , $H_c({\\bf q },{\\bf p })$ is a corresponding classical Hamiltonian, and where by derivative with respect to a vector we understand a vector made from derivatives over the corresponding components.", "As was discussed in section 5, the wave function amplitude $A = e^R$ always satisfies a continuity equation, which in this case has the form $\\frac{\\partial A^2}{\\partial t} + \\sum _i \\frac{\\partial }{\\partial q_i} \\left[ A^2 \\left.", "\\frac{\\partial H_c({\\bf q },\\, {\\bf p })}{\\partial p_i} \\,\\right|_{{\\bf p }= \\partial S/\\partial {\\bf q }} \\right] = 0 \\,, $ The standard quantum-mechanical derivation of equations (REF ), (REF ) for the action function and amplitude in the quasiclassical case may be found, for example, in [31].", "We now want to consider a combined system, consisting of macroscopic objects interacting with quantum particles.", "The same consideration applies to the interaction of macroscopic objects with their own internal (like electrons' or phonons') microscopic degrees of freedom, which should be described quantum-mechanically.", "In fact, it will be sufficient for our analysis to consider an extremely simplified situation where a macroscopic object is represented by one particle with a large (macroscopic) mass $M$ in the limit $M \\rightarrow \\infty $ interacting with a quantum particle with a fixed (microscopic) mass $m$ .", "Let the Hamiltonian of this system be $H = \\frac{p_x^2}{2M} + \\frac{p_y^2}{2m} + U(x) + V(x,y) \\, ,$ where $x$ and $y$ are the particle coordinates (their dimensionality will be irrelevant for us, so we may consider them one-dimensional), $U(x)$ is the potential energy of the heavy particle that scales proportionally to $M$ as $M \\rightarrow \\infty $ , and $V(x,y)$ is the potential of the particle interaction and of the light particle alone and is independent of $M$ .", "The wave function $\\Psi (x,y,t)$ of the system satisfies the Schrödinger equation $\\frac{\\hbar }{i}\\,\\frac{\\partial \\Psi }{\\partial t} \\,-\\, \\frac{\\hbar ^2}{2M}\\, \\frac{\\partial ^2 \\Psi }{\\partial x^2}\\,-\\, \\frac{\\hbar ^2}{2m}\\, \\frac{\\partial ^2 \\Psi }{\\partial y^2} \\,+\\, \\big [U(x) + V(x,y)\\big ] \\Psi \\,=\\, 0 \\, .", "$ We take the point in $(x,y)$ -configuration space where the system is at initial time $t=0$ , as a coordinate system's origin.", "Then the initial action function $p(x,y)$ is a power series in $x$ and $y$ , and it may be presented as a sum $p(x,y) = p_M(x) + p_m(x,y)$ , where $p_M$ collects all the terms of the series with the powers of $x$ alone, and $p_m$ the remaining terms, which contain nonzero powers of $y$ .", "The wave function $\\Psi (x,y,t)$ may always be represented as a product $A(x,t)e^{(i/\\hbar )S(x,t)}\\phi (x,y,t)$ with real functions $A(x,t)$ and $S(x,t)$ .", "We have then from Eq.", "(REF ) $\\begin{array}{l}\\displaystyle \\phi (x,y,t) \\left(\\frac{\\hbar }{i}\\,\\, \\frac{\\partial }{\\partial t} \\,-\\, \\frac{\\hbar ^2}{2M}\\, \\frac{\\partial ^2}{\\partial x^2} \\,+\\, U\\right) A e^{\\frac{i}{\\hbar }\\,S} \\\\[0.4cm]\\displaystyle \\quad + \\, A e^{\\frac{i}{\\hbar }\\,S} \\left[ \\left(\\frac{\\hbar }{i}\\,\\, \\frac{\\partial }{\\partial t} \\,-\\, \\frac{\\hbar ^2}{2m}\\,\\frac{\\partial ^2}{\\partial y^2} \\,+\\, V\\right)\\phi \\,+\\, \\left(\\frac{\\hbar }{i}\\,\\frac{1}{M} \\,\\frac{\\partial S}{\\partial x} \\,-\\, \\frac{\\hbar ^2}{M}\\, \\frac{1}{A}\\, \\frac{\\partial A}{\\partial x} \\right) \\frac{\\partial \\phi }{\\partial x} \\,-\\,\\frac{\\hbar ^2}{2M}\\, \\frac{\\partial ^2 \\phi }{\\partial x^2} \\right] \\,=\\, 0\\,.", "\\end{array}$ Let now the function $A(x,t)e^{(i/\\hbar )S(x,t)}$ cancel the first term in (REF ), i.e., it satisfies the equation $\\displaystyle \\left(\\frac{\\hbar }{i}\\,\\, \\frac{\\partial }{\\partial t} \\,-\\, \\frac{\\hbar ^2}{2M}\\, \\frac{\\partial ^2}{\\partial x^2} \\,+\\, U\\right) A e^{\\frac{i}{\\hbar }\\,S} \\,=\\, 0 \\, , $ with initial condition $A(x,0)e^{(i/\\hbar )S(x,0)} = e^{(i/\\hbar )p_M(x)}$ .", "The function $\\phi (x,y,t)$ must then cancel the second term in (REF ), i.e., satisfy an equation $\\displaystyle \\left(\\frac{\\hbar }{i}\\,\\, \\frac{\\partial }{\\partial t} \\,-\\, \\frac{\\hbar ^2}{2m}\\,\\frac{\\partial ^2}{\\partial y^2} \\,+\\, V\\right)\\phi \\,+\\, \\left(\\frac{\\hbar }{i}\\,\\frac{1}{M} \\,\\frac{\\partial S}{\\partial x} \\,-\\, \\frac{\\hbar ^2}{M}\\, \\frac{1}{A}\\, \\frac{\\partial A}{\\partial x} \\right) \\frac{\\partial \\phi }{\\partial x} \\,-\\,\\frac{\\hbar ^2}{2M}\\, \\frac{\\partial ^2 \\phi }{\\partial x^2} \\,=\\, 0\\,.", "$ To investigate the $M \\rightarrow \\infty $ limit, expand the functions $S$ , $A$ , and $\\phi $ in powers of $1/M$ as $S = S^{(c)} + \\sum _{k=0}^\\infty \\frac{S^{(k)}}{M^k}\\,, \\quad \\quad A = \\sum _{k=0}^\\infty \\frac{A^{(k)}}{M^k}\\,, \\quad \\quad \\phi = \\sum _{k=0}^\\infty \\frac{\\phi ^{(k)}}{M^k}\\,,$ where $S^{(c)}(x,t)$ is proportional to $M$ while coefficients $S^{(k)}(x,t)$ , $A^{(k)}(x,t)$ , and $\\phi ^{(k)}(x,y,t)$ are $M$ -independent, and neglect all contributions with positive powers of $1/M$ .", "For $S^{(c)}(x,t)$ , we have then the classical Hamilton-Jacobi equation $\\frac{\\partial S^{(c)}}{\\partial t} + \\frac{1}{2M} \\left(\\frac{\\partial S^{(c)}}{\\partial x}\\right)^2 + U = 0\\,, $ and for $A^{(0)}(x,t)$ , a continuity equation $\\frac{\\partial {A^{(0)}}^2}{\\partial t}+\\frac{1}{M}\\,\\frac{\\partial }{\\partial x}\\left({A^{(0)}}^2\\,\\frac{\\partial S^{(c)}}{\\partial x}\\right)=0\\,.$ The solution of Eq.", "(REF ) is given by integrals of the Lagrangian function along classical trajectories in the potential $U(x)$ , and so $S^{(c)}(x,t)$ will scale proportionally to $M$ as $M \\rightarrow \\infty $ , as expected.", "The velocity of the heavy particle will converge for $M\\rightarrow \\infty $ to an $M$ -independent limit $v(x,t)=(1/M)\\,\\partial S^{(c)}(x,t)/\\partial x$ , and since the action $S^{(c)}(x,t)$ satisfies the Hamilton-Jacobi equation, this particle will exhibit a classical motion in the potential $U(x)$ .", "Let now $x(t)$ be the trajectory of the heavy particle.", "Since it represents a macroscopic object, this trajectory is directly observable and, as such, known.", "The behavior of the light particle is described by momentums $p_\\sigma $ with multi-indices $\\sigma $ that include $y$ at least once.", "These momentums are derivatives of $(\\hbar /i) \\ln \\Psi (x,y,t)$ taken at $x=x(t)$ .", "That means that the light particle is described by the wave function $\\phi (x,y,t)$ at a point $x(t)$ , i.e., in the $M \\rightarrow \\infty $ limit, by the function $\\psi (y,t) = \\phi ^{(0)}\\big (x(t),y,t\\big )$ .", "From (REF ), the function $\\phi ^{(0)}$ satisfies the equation $\\displaystyle \\left(\\frac{\\hbar }{i}\\,\\, \\frac{\\partial }{\\partial t} \\,-\\, \\frac{\\hbar ^2}{2m}\\, \\frac{\\partial ^2}{\\partial y^2}\\,+\\, V\\right)\\phi ^{(0)} \\,+\\, \\frac{\\hbar }{i}\\,\\, v(x,t)\\, \\frac{\\partial \\phi ^{(0)}}{\\partial x} \\,=\\, 0\\, .", "$ Combining the last term in (REF ) with the first one, and letting $W(y,t) =V\\big (x(t),y,t\\big )$ , we then obtain the equation for $\\psi (y,t)$ : $\\frac{\\hbar }{i}\\,\\, \\frac{\\partial \\psi }{\\partial t} \\,-\\, \\frac{\\hbar ^2}{2m}\\, \\frac{\\partial ^2 \\psi }{\\partial y^2}\\,+\\, W(y,t)\\, \\psi \\,=\\, 0 \\, ,$ which is the Schrödinger equation for a light particle in the potential $W(y,t)$ created by a heavy particle moving along the classical trajectory $x(t)$ .", "Thus in $\\cal P$ AQD, the experimentally observed separation of reality into a macroscopic world that behaves classically and a microscopic one that exhibits quantum behavior in a classical macroscopic environment is not postulated as in standard quantum mechanics, but obtained as a direct consequence of its equations of motion." ], [ "The theory of quantum measurements", "Besides different equations of motion, the difference in the measurement procedure is probably the most important difference between classical and quantum theory.", "For every physical quantity, quantum mechanics specifies a corresponding linear hermitian operator $O$ .", "In $\\cal P$ AQD, $O\\psi ({\\bf x })/\\psi ({\\bf x })$ may be interpreted as a numerical value, which this quantity has if a particle with wave function $\\psi $ happens to be at a point ${\\bf x }$ .", "If $\\psi $ is an eigenstate of $O$ , then this value is the same for all ${\\bf x }$ (i.e., for all possible trajectories of the particle) and is a corresponding real eigenvalue of $O$ .", "If, on the other hand, $\\psi $ is not an eigenstate, then this value will be different for different ${\\bf x }$ , and for a given point ${\\bf x }$ will in general be an arbitrary complex number that would have been the result of a measurement of $O$ , if this measurement had its classical meaning.", "In quantum theory, however, the situation is more complicated.", "Indeed, in contrast to classical theory, which deals with macroscopic objects, quantum theory describes microscopic ones, whose properties are usually not directly observable.", "In order to find the value of any physical quantity that such objects possess, one has to produce the interaction of this quantity with another one that is  observable, and to infer the value of the quantity of interest from the reaction of that observable quantity.", "The observable quantity is a characteristic of the “apparatus\", and may have a macroscopic character, like the position of a pointer, or a microscopic one, as in a Stern-Gerlach experiment, where the measured quantity is a particle's spin and the observable (or rather, in this case, detectable) quantity is this particle's position, and the role of an apparatus is played by the particle itself.", "Thus the measurement procedure in the quantum domain is highly indirect, which causes its peculiar properties.", "To analyze them, we will apply our theory to the combination of a particle and an apparatus.", "We will identify several different kinds of quantum measurements, and consider them in turn." ], [ "von Neumann's measurements with discrete spectrum", "The measurements of the first kind were originally investigated by von Neumann [32], and so we will call them von Neumann's measurements.", "In this subsection we will consider the case where the spectrum of a measured observable $O$ is discrete.", "According to von Neumann, if the apparatus performs a measurement of this observable, and the particle's state is its eigenstate $\\psi _i$ (which is assumed to be normalized, $\\int |\\psi _i|^2 dx = 1$ ), corresponding to an eigenvalue $O_i$ (so that, in $\\cal P$ AQD, the quantity $O$ has the value of $O_i$ for arbitrary position ${\\bf x }$ of the particle) then the reading of the apparatus should have the corresponding $i$ -th value, clearly distinguishable from others.", "In more detail this means the following: Before the measurement, at initial time $t=0$ , the apparatus is set into the state $\\varphi (y)$ , where $y$ is the apparatus coordinate, which is assumed to be directly observable.", "We also assume that $\\varphi (y)$ is centered at $y=0$ and has width $\\Delta y$ .", "Since before the measurement a particle and an apparatus are independent, if a particle is in a state $\\psi _i$ , then an initial wave function of the combined particle-apparatus system is $\\varphi (y)\\psi _i({\\bf x })$ .", "If $\\Psi ^{(i)}(y,{\\bf x },t)$ is the result of an evolution of this state during the measurement, then it is required that for $t$ larger than the duration of measurement $\\Delta t$ , $\\Psi ^{(i)}(y,{\\bf x },t)$ should be centered around some $y_i(t)$ and have such a width that the overlap of $\\Psi ^{(i)}(y,{\\bf x },t)$ and $\\Psi ^{(j)}(y,{\\bf x },t)$ in $y$ -space could be neglected for all $j \\ne i$ (in $\\cal P$ AQD, $\\Psi ^{(i)}$ and $\\Psi ^{(j)}$ are analytic functions, and so they always overlap, but we can require each of them to be negligibly small in the area where the other one is centered).", "In other words, over the measurement time $\\Delta t$ different packets $\\Psi ^{(i)}$ should diverge in $y$ -space far enough to make their overlap negligible.", "By observing the value of $y$ after the measurement, we can then infer the value of $O$ before it.", "In particular, if the measurement time $\\Delta t$ is so short, and particle-apparatus interaction Hamiltonian $H_{int}$ is so strong, that during the measurement all other terms in the total Hamiltonian of the combined particle-apparatus system may be neglected compare to $H_{int}$ , and if $H_{int}$ is proportional to $O$ , then the wave function of the system $\\Psi ^{(i)}(y,{\\bf x },\\Delta t)$ immediately after the measurement will have the form $\\varphi ^{(i)}(y) \\psi _i({\\bf x })$ , i.e., the particle after the measurement will remain in an eigenstate $\\psi _i$ of $O$ .", "But this is not necessary.", "Explicit models of such a measurement are considered in [32] and, in great detail, in [33].", "If such an apparatus is built, then an interesting situation occurs when, before the measurement, a particle in not  in an eigenstate of $O$ , i.e., if its wave function is $\\psi ({\\bf x }) = \\sum _ic_i \\psi _i({\\bf x })$ with more than one nonzero coefficient $c_i$ .", "We assume that this wave function is normalized, so that $\\sum _i |c_i|^2 = 1$ .", "By the linearity of Schrödinger's equation, in this case the initial wave function of the combined system $\\varphi (y)\\psi ({\\bf x })$ evolves during the measurement into $\\sum _i c_i \\Psi ^{(i)}(y,{\\bf x },t)$ , in direct contradiction with experiment, from which we know that in fact the combined system will end up in one of the states $\\Psi ^{(i)}(y,{\\bf x },t)$ .", "To save the theory, von Neumann postulated, besides the unitary evolution described by the Schrödinger equation, the second law of evolution, which acts only during the measurements: a random, unpredictable, and unanalyzable collapse of the linear combination $\\sum _i c_i\\Psi ^{(i)}(y,{\\bf x },t)$ into one of $\\Psi ^{(i)}(y,{\\bf x },t)$ with experimentally observed probability  $|c_i|^2$ .", "Nobody, however, was able to formulate convincingly when the unitary evolution should be replaced by the collapse (or, in other words, what exactly allows us to qualify an experiment as being a measurement).", "Similar issues arise in other orthodox approaches to the interpretation of quantum theory.", "This is the essence of the quantum measurement problem, which found a simple and natural resolution in the framework of DBBT [12], [13].", "We now reproduce Bohm's solution of the problem using the language of $\\cal P$ AQD.", "In $\\cal P$ AQD, the state of the combined system is characterized by its position in configuration space and all its momentums, all of which evolve according to the corresponding equations of motion.", "As a consequence of this evolution, the action function, which is just a corresponding Taylor series, evolves according to the quantum Hamilton-Jacobi equation, while the wave function evolves according to the Schrödinger equation as was described above.", "During this process, the combined evolution of the system's position and wave function is such that the system normally stays in the areas of configuration space where the wave function is not small.", "Consequently, when the packets $\\Psi ^{(i)}(y,{\\bf x },t)$ start to diverge, the apparatus position $y(t)$ will end up in the area where one of them, say the $k$ -th, is not small, i.e., near $y_k(t)$ .", "Now, the momentums are derivatives of the system's action function, i.e., $(\\hbar /i) \\ln \\sum _i c_i \\Psi ^{(i)}(y,{\\bf x },t)$ , at the point $\\big (y(t),{\\bf x }(t)\\big )$ (where ${\\bf x }(t)$ is the particle's position) in configuration space, and the further the packets move away from each other the closer are these derivatives to the ones of $(\\hbar /i) \\ln \\Psi ^{(k)}(y,{\\bf x },t)$ .", "The measurement ends when the overlap of the packets becomes negligible, and with it the difference between the exact momentums and the derivatives of $(\\hbar /i) \\ln \\Psi ^{(k)}(y,{\\bf x },t)$ becomes negligible also.", "Consequently, although the wave function is still equal to $\\sum _i c_i \\Psi ^{(i)}(y,{\\bf x },t)$ , the motion of the “physical\" variables, i.e., the system's position in configuration space and momentums, will be the same as if the wave function was equal to $\\Psi ^{(k)}(y,{\\bf x },t)$ , in agreement with experiment.", "This explains the apparent wave function collapse.", "The probability of observing the $k$ -th result of the measurement is calculated according to the general rules of section 6 as an integral from $\\big |\\sum _i c_i \\Psi ^{(i)}(y,{\\bf x },t)\\big |^2$ over the area where $\\Psi ^{(k)}(y,{\\bf x },t)$ is not small, i.e., around $y_k(t)$ in $y$ -space and all ${\\bf x }$ -space, and since all $\\Psi ^{(i)}(y,{\\bf x },t)$ are normalized and don't overlap, this integral, again in agreement with experiment, is equal to $|c_k|^2$ .", "The following features of von Neumann's measurement procedure deserve special mention: – Although measurement statistics are determined by the wave function of the particle alone, the result of every individual measurement (unless the particle was in the eigenstate of $O$ before it) is determined by the full $\\cal P$ AQD states (i.e., positions and all momentums, or positions and wave functions) of both particle and apparatus.", "– If the paticle was not in the eigenstate of $O$ , then the measurement's result $O_k$ is completely unrelated to the value $O\\psi ({\\bf x })/\\psi ({\\bf x })$ (where ${\\bf x }$ is the particle's position) of observable $O$ before the measurement.", "This and the previous note mean that unless the particle was in a corresponding eigenstate, $O$ 's observed value is not really measured, but rather created by the particle and apparatus jointly in the process of a measurement.", "What is  measured (by the corresponding relative frequencies of a series of measurements) is a set of values of the squared amplitudes $|c_i|^2$ .", "– Unless the interaction Hamiltonian is proportional to $O$ and satisfies other requirements discussed above, after the measurement the particle doesn't have to be in a state with a definite $O$ value, let alone the state with $O$ equal to the measured eigenvalue $O_k$ .", "– To successfully perform a measurement, the apparatus doesn't have to be macroscopic.", "The only necessary condition is that the packets $\\Psi ^{(i)}(y,{\\bf x },t)$ with different $i$ do not overlap after some time $\\Delta t$ (the duration of the measurement).", "In a Stern-Gerlach experiment, where the apparatus is the particle itself, the measurement ends and the wave function collapses not when the particle is detected after passing the magnet and we learn the spin measurement's result, but earlier, when the wave packets corresponding to the different spins cease to overlap.", "See, however, the next note.", "– Although we are discussing the wave function collapse, the “empty\" packets $\\Psi ^{(i)}(y,{\\bf x },t)$ with $i\\ne k$ do not disappear, but just move away from the “active\" packet $\\Psi ^{(k)}(y,{\\bf x },t)$ , so that their contribution to momentums and, therefore, their influence on the dynamics of the system vanishes.", "If, in their future evolution, all or some of the packets $\\Psi ^{(i)}$ have again overlapped with $\\Psi ^{(k)}$ , then the measurement would be “undone\", the wave function would “uncollapse\", and the value of $O$ would again become undetermined (in a sense that instead of being equal to $\\Psi ^{(k)}$ , the wave function would become equal to the linear combination of $\\Psi ^{(k)}$ and overlapping packets $\\Psi ^{(i)}$ ).", "This overlap, however, should happen in an ${\\bf x }$ -$y$ space of dimensionality $\\mathrm {dim}\\,{\\bf x }+ \\mathrm {dim}\\,y$ .", "For the purpose of this argument, $y$ should include all coordinates of the apparatus and its environment that are connected by a chain of nonnegligible interactions.", "Consequently, while $\\mathrm {dim}\\,y$ is small (as before the particle is detected in a Stern-Gerlach experiment) such reversion of the measurement can, in principle, be accomplished.", "However, as soon as $\\mathrm {dim}\\,y$ becomes macroscopically large (as when the particle is detected or observed by any macroscopic, conscious or not, observer) the reversion becomes practically impossible, and its possibility may be neglected.", "– The set of possible final states $\\big \\lbrace \\Psi ^{(i)}(y,{\\bf x },t)\\big \\rbrace $ of the system is predetermined by the measurement apparatus and does not depend on the initial wave function of the particle.", "Consequently, after the wave function collapses into one of these states, all information about the particle's initial state, and all influence of this state on the future history of the system is lost.", "On a positive side, that means that von Neumann's measurements are convenient for experiment preparation.", "Indeed, after the observable $O$ is measured and found equal to $O_k$ , say, we know that the system is prepared in the state $\\Psi ^{(k)}(y,{\\bf x },t)$ , regardless of the initial wave function of a particle." ], [ "von Neumann's measurements with continuous spectrum", "An analysis, similar to that just presented, is also possible when the spectrum of a measured observable $O$ is continuous.", "First consider the case when $O$ is not a particle's position.", "Here it will be easier to use an explicit consideration, based on a particle-apparatus interaction Hamiltonian $H_{int}$ proportional to $O$ .", "Following [12], [13], [32], [33], choose it in the form $H_{int} = g(t) O p_y$ , where $p_y = -i\\hbar \\partial /\\partial y$ is the momentum conjugate to the apparatus position $y$ , and the factor $g(t)$ represents the switching of the interaction on and off.", "Assume it has an impulsive character, so that $g(t) = g_0$ for $0<t<\\tau $ and $g(t) = 0$ for $t<0$ and $t>\\tau $ , where $\\tau $ is the duration of the measurement.", "Consider the limit of very small $\\tau $ and large $g_0$ .", "The influence of the particle's and apparatus' own Hamiltonians on the evolution of the wave function during the measurement may then be neglected compared to the influence of $H_{int}$ , so that between $t=0$ and $t=\\tau $ the Schrödinger equation may be approximated by $i \\hbar \\,\\frac{\\partial \\Psi }{\\partial t}\\, = \\, H_{int} \\Psi \\,=\\, - i \\hbar g_0 O \\,\\frac{\\partial \\Psi }{\\partial y} \\, .$ Let $\\psi _a(x)$ be eigenfunctions of $O$ , $\\,O\\psi _a(x) = a\\psi _a(x)$ , normalized so that $\\int \\psi ^*_{a^{\\prime }}(x) \\psi _{a^{\\prime \\prime }}(x) dx = \\delta (a^{\\prime } - a^{\\prime \\prime }) ,$ and $c(a)$ be the coefficients of an expansion of the initial particle's wave function $\\psi (x,t=0)$ in an integral over them: $\\psi (x,0) = \\int c(a) \\psi _a(x)\\, da \\, .$ As before, assume the initial wave function $\\varphi (y)$ of the apparatus to be centered at $y=0$ , have width $\\Delta y$ , and be normalized, $\\int |\\varphi (y)|^2 dy = 1$ .", "The total wave function of a system $\\Psi (x,y,t)$ may be expanded as an integral over $\\psi _a(x)$ as $\\Psi (x,y,t) = \\int c_a(y,t) \\psi _a(x) \\, da \\, .$ It is easy to see that the expansion coefficients $c_a(y,t)$ must satisfy the equation $\\frac{\\partial c_a(y,t)}{\\partial t} = - g_0 a \\, \\frac{\\partial c_a(y,t)}{\\partial y}$ with the initial condition $c_a(y,0) = c(a) \\varphi (y)$ .", "Then the solution for $c_a$ is $c_a(y,t) = c_a(y - g_0 a t, 0) = c(a) \\varphi (y - g_0 a t) ,$ so that at the moment $t=\\tau $ at the end of the measurement, the system's wave function will be $\\Psi (x,y,\\tau ) \\, = \\, \\int c(a) \\psi _a(x) \\varphi (y - g_0 \\tau a) \\, da \\, .$ It is convenient to introduce a new apparatus coordinate $\\widetilde{y}= y / (g_0 \\tau )$ and new function $\\widetilde{\\varphi }(z) = \\varphi (g_0 \\tau z)$ , which becomes negligible when $|z| > \\sigma $ , where the half-width $\\sigma = \\Delta y / (2 g_0 \\tau )$ .", "The system's final wave function can then be written as $\\Psi (x,y,\\tau ) = \\int c(a) \\psi _a(x) \\widetilde{\\varphi }(\\widetilde{y}- a) \\, da \\, .", "$ The directly observable coordinate $\\widetilde{y}$ plays now the role of a pointer for the measurement of $O$ .", "Indeed, if the initial wave function of a particle $\\psi (x,0)$ is an eigenstate of $O$ , say $\\psi _{a_0}(x)$ , then $c(a) = \\delta (a - a_0)$ and $\\Psi (x,y,\\tau ) = \\psi _{a_0}(x) \\widetilde{\\varphi }(\\widetilde{y}-a_0)$ .", "Since $\\widetilde{\\varphi }(\\widetilde{y}- a_0)$ vanishes for $|\\widetilde{y}- a_0| > \\sigma $ , the value of $\\widetilde{y}$ at $t = \\tau $ will be between $a_0 - \\sigma $ and $a_0 + \\sigma $ , so that $\\widetilde{y}$ points to the correct value of $O$ with precision $\\sigma $ .", "It is assumed, that parameters $\\Delta y$ , $g_0$ , and $\\tau $ may be chosen at will, and so $\\sigma $ can be made arbitrary small.", "Consequently, although never exact, the measurement of $O$ can be made arbitrarily precise.", "In a general situation, when $\\psi (x,0)$ is not an eigenstate of $O$ , the scaled position $\\widetilde{y}$ evolves during the measurement according to the equations of motion, and ends at $t = \\tau $ at some $\\widetilde{y}_0$ .", "Similar to the discrete spectrum case, because of the properties of the function $\\widetilde{\\varphi }$ , the evolution of the system's coordinates and momentums will then be the same as if the wave function at $t = \\tau $ instead of being $\\Psi (x,y,\\tau )$ , Eq.", "(REF ), was equal to $\\Psi ^{(\\widetilde{y}_0)}(x,y,\\tau ) = \\int _{\\widetilde{y}_0 - \\sigma }^{\\widetilde{y}_0 + \\sigma } c(a) \\psi _a(x)\\widetilde{\\varphi }(\\widetilde{y}- a)\\,da\\, .", "$ Since $c(a)$ and $\\psi _a(x)$ are smooth (analytic) functions of $a$ , for sufficiently small $\\sigma $ the function $\\Psi ^{(\\widetilde{y}_0)}(x,y,\\tau )$ may be approximated with arbitrary precision as $\\Psi ^{(\\widetilde{y}_0)}(x,y,\\tau ) = c(\\widetilde{y}_0) \\psi _{\\widetilde{y}_0}(x) f(\\widetilde{y}) \\, , \\quad \\; f(\\widetilde{y}) =\\int _{\\widetilde{y}_0 - \\sigma }^{\\widetilde{y}_0 + \\sigma } \\widetilde{\\varphi }(\\widetilde{y}- a)\\,da\\, ,$ so that the function $\\Psi ^{(\\widetilde{y}_0)}(x,y,\\tau )$ , to which the system appears to collapse, is an eigenstate of $O$ with the eigenvalue $\\widetilde{y}_0$ .", "The probability $p(a_0, da_0)$ to find the value of $O$ (i.e., the value of $\\widetilde{y}_0$ ) between $a_0$ and $a_0 + da_0$ is equal to $\\rho (a_0) da_0$ , where $\\rho (a_0)$ is the corresponding probability density.", "By the general rules of section 6 we have for it $\\rho (a_0) = g_0 \\tau \\int \\big |\\Psi (x, g_0 \\tau a_0, \\tau )\\big |^2 dx = g_0 \\tau \\int |c(a)|^2 |\\widetilde{\\varphi }(a_0 - a)|^2 da \\, ,$ where we used the normalization condition for the functions $\\psi _a(x)$ and the factor $g_0 \\tau $ appears because $dy = g_0 \\tau \\,d\\widetilde{y}$ .", "For sufficiently small $\\sigma $ , $|c(a)|^2$ in the integrand may be again approximated by $|c(a_0)|^2$ with negligible error, and using normalization of the function $\\widetilde{\\varphi }$ , we obtain, in agreement with von Neumann's postulate and experiment, the standard result $\\rho (a_0) = |c(a_0)|^2$ .", "To summarize, von Neumann's measurement procedure of an observable with continuous spectrum that is not a particle's position is similar to the one with discrete spectrum, and has the same, listed above, properties.", "In particular, unless the particle was initially in the eigenstate of $O$ , the measurement's result is unrelated to the value $O\\psi ({\\bf x })/\\psi ({\\bf x })$ of $O$ before the measurement, and with arbitrary precision the set of possible final states of the particle-apparatus system is predetermined and does not depend on the initial state of the particle.", "For the measurement procedure, considered above, the corresponding set of possible particle final states is just the set $\\lbrace \\psi _a(x)\\rbrace $ of eigenstates of $O$ .", "We now consider von Neumann's measurement of a particle's position, and show that, in contrast to other physical quantities, this measurement results in the true $\\cal P$ AQD particle position.", "Indeed, we have, obviously, for the eigenfunctions $\\psi _a(x)$ of an operator $O = x$ and coefficients $c(a)$ of expansion of the particle's wave function $\\psi (x,0)$ in this case, $\\psi _a(x) =\\delta (x-a)$ and $c(a) = \\psi (a,0)$ .", "Although $\\delta (x-a)$ is not an analytic function of $x$ , and so cannot be considered a legitimate $\\cal P$ AQD wave function, we still can use it in intermediate mathematical transformations.", "The integral over $a$ in Eq.", "(REF ) can then be immediately calculated to give $\\Psi (x,y,\\tau ) = \\psi (x,0) \\widetilde{\\varphi }(\\widetilde{y}- x)$ .", "Let now the particle's position before the measurement be $x_0$ .", "Assume that the function $\\varphi (y)$ is real.", "Then the motion of the particle during the measurement may be neglected.", "The final value $\\widetilde{y}_0$ of $\\widetilde{y}$ will now be such that $\\Psi (x_0,g_0\\tau \\widetilde{y}_0,\\tau )$ does not vanish.", "Due to the properties of the function $\\widetilde{\\varphi }$ , that means that $\\widetilde{y}_0$ should be in a $\\sigma $ -vicinity of $x_0$ , $|\\widetilde{y}_0 -x_0| \\le \\sigma $ , as was asserted.", "The collapsed wave function $\\Psi ^{(\\widetilde{y}_0)}(x,y,\\tau )$ , Eq.", "(REF ), is easily calculated to be equal to $\\psi (x,0) \\widetilde{\\varphi }(\\widetilde{y}- x)$ for $|x - \\widetilde{y}_0|\\le \\sigma $ and equal to zero for $|x - \\widetilde{y}_0| > \\sigma $ .", "It is, therefore, contained in a $\\sigma $ -vicinity of $\\widetilde{y}_0$ , and for $\\sigma \\rightarrow 0$ , as in other cases of von Neumann's measurements, loses the memory of the particle's initial state." ], [ "Position measurements of the second kind and the double-slit experiment", "In practice, von Neumann's measurement is never used for a particle's position.", "The real position measurement is carried out by such devices as a bubble chamber or photographic plate.", "As we will see, wave function collapse and some other important features of this measurement are significantly different from those of von Neumann's, which justifies calling it the measurement of the second kind.", "The measurement of a particle's position by a photographic plate or in a bubble chamber may be described as follows.", "The physical state is filled with microscopic detectors (molecules of photo-emulsion for photographic plate, or of overheated liquid for bubble chamber), which change their state (chemical changes in emulsion, ionization in a liquid) if the measured particle passes in close vicinity to them.", "Due to the special physics of detectors, this changed microstate evolves then in such a way as to produce directly observable macroscopic changes (dark spot on developed plate, bubble in a chamber).", "The detectors with such changed state mark the position of a particle.", "In our analysis of this procedure, we again use the fact that the wave function in $\\cal P$ AQD has its own dynamics, the same as in standard quantum mechanics, and so its evolution may be analyzed without reference to a particular particle's trajectory that is responsible for this wave function's existence, and which may be included in the analysis later.", "Then the following crude model may be suggested for the description of a position measurement.", "Consider first just one detector, fixed at a point with position $\\tilde{x}$ .", "Let the detector itself be characterized by the parameter $y$ , which in the initial state is close to zero, so that the initial wave function of the detector is, for example, $\\exp (-y^2/4)$ (we will not worry about wave function normalization here).", "Let the particle's wave function be $\\psi (x)$ , so that an initial wave function of the particle-detector system is $\\psi (x)\\exp (-y^2/4)$ .", "Let the physics of the detector and its interaction with the particle be such that within a short measurement time $\\Delta t$ , $y$ moves from the vicinity of zero to the vicinity of some $Y\\gg 1$ , so that the detector's wave function becomes, for example, $\\exp \\big [-(y-Y)^2/4\\big ]$ , if during this time the particle's distance from the detector was less than some characteristic distance $\\sigma $ .", "The evolution of a particle's wave function during the measurement time due to its own Hamiltonian, i.e., without interaction with the detector, will be of no importance for us.", "We can, therefore, consider the measurement to be instantaneous, i.e., $\\Delta t$ to be so small that the change of the particle's wave function during the measurement due to its own dynamics is negligible.", "The wave function of the particle-detector system immediately after the measurement can then be written in a general form as $\\widetilde{\\Psi } = \\psi (x) \\left\\lbrace a(x - \\tilde{x},y) \\exp \\left[-\\frac{1}{4}\\,(y-Y)^2\\right]+ b(x - \\tilde{x},y) \\exp \\left(-\\frac{1}{4}\\, y^2\\right) \\right\\rbrace \\, .$ The functions $a$ and $b$ reflect the physics of the particle-detector interaction.", "All we know about them is that $a(x-\\tilde{x},y)$ vanishes and $b(x-\\tilde{x},y)$ converges to 1 when the distance from $x$ to $\\tilde{x}$ becomes larger than $\\sigma $ , and that $b(x-\\tilde{x},y)$ vanishes when this distance is smaller than $\\sigma $ .", "The dependence on $y$ is included in $a$ and $b$ for generality, and is supposed to leave the general character of $y$ -dependence of the corresponding terms intact, i.e., the probability density is concentrated near $y=Y$ in the first term, and near $y=0$ in the second.", "Now if there are many detectors like that, then the initial wave function will be $\\psi (x) \\prod _i\\exp (-y_i^2/4)$ , and the wave function after the measurement will be $\\Psi = \\psi (x) \\prod _i \\left\\lbrace a(x - x_i,y_i) \\exp \\left[-\\frac{1}{4}\\,(y_i-Y)^2\\right]+ b(x - x_i,y_i) \\exp \\left(-\\frac{1}{4}\\, y_i^2\\right) \\right\\rbrace \\, , $ where $y_i$ is the $y$ -coordinate of the $i$ -th detector, and $x_i$ is its position.", "To avoid unnecessary complications, we will make the simplifying assumption that $\\sigma $ -vicinities of different detectors do not overlap and, at the same time, do not leave any places in the $x$ -space uncovered.", "The after-measurement wave function may then be rewritten as $\\Psi = \\psi (x) \\sum _j a(x - x_j, y_j) \\exp \\left[-\\frac{1}{4}\\,(y_j-Y)^2\\right] \\prod _{i \\ne j}\\exp \\left(-\\frac{1}{4}\\, y_i^2\\right).", "$ Let now the particle's position $x_p$ at this moment happen to be in a $\\sigma $ -vicinity of the $k$ -th detector, $x_p\\approx x_k$ .", "Then obviously in the sum over $j$ in (REF ), all terms except the $k$ -th, will give a negligible contribution to momentums (i.e., derivatives of $(\\hbar /i)\\ln \\Psi $ at that $x_p$ and proper $y_i$ 's) and so the future motion of the particle and detectors will proceed as if the wave function was equal to this $k$ -th term, i.e., underwent a collapse $\\Psi \\longrightarrow \\psi (x)\\, a(x - x_k, y_k) \\exp \\left[-\\frac{1}{4}\\,(y_k-Y)^2\\right]\\prod _{i \\ne k} \\exp \\left(-\\frac{1}{4}\\, y_i^2\\right).", "$ In this state, the $y$ -coordinate of the $k$ -th detector will then be found near $Y$ , and all others will be near zero, and the particle's probability density, although far from being a delta-function centered at $x_p$ , will be concentrated in $x_p$ 's and the $k$ -th detector's $\\sigma $ -vicinity, where $\\sigma $ may be considered as a measurement precision.", "With this precision, therefore, a position measurement of the second kind, like its von Neumann's counterpart, measures the true $\\cal P$ AQD position of a particle.", "We note, however, that although for both kinds of measurements the final selection of a member of a linear superposition, to which the wave function would collapse, is made by some variable which may have one, and only one value, for von Neumann's measurement this variable is the apparatus position $y$ , while for a position measurement of the second kind, it is the measured particle's position $x_p$ .", "Also, for a position measurement of the second kind, the precision $\\sigma $ is fixed by the physics of detectors and so cannot be made arbitrary small.", "Consequently, while for an arbitrarily precise von Neumann measurement, the final wave function becomes equal to one possible function from the predetermined set of them, the final wave function after a position measurement of the second kind does depend on the initial wave function of the particle.", "Indeed, its $x$ -dependence is essentially given by the product $\\psi (x)\\, a(x - x_k, Y)$ , i.e., is equal to the initial function $\\psi (x)$ modulated by a factor $a(x - x_k, Y)$ .", "If this factor is smooth enough, and if the characteristic wavelengths in $\\psi (x)$ are much smaller than $\\sigma $ , then the packet $\\psi (x)\\, a(x - x_k, Y)$ will keep propagating without spreading much along a trajectory that is close to the one the particle would have by itself, i.e., if its position was not measured.", "In a bubble chamber, this packet will then trigger other detectors, thus producing a track which approximates the particle's unperturbed trajectory.", "The same consideration may be also applied to the double-slit experiment discussed by Feynman [15].", "In this case, variable $x$ in Eqs.", "(REF )-(REF ) will denote the coordinate on the screen, and $x_i$ , $i=1,2$ — the position of the $i$ -th slit.", "Without detectors, the wave function of the particle immediately behind the screen would be equal to $\\psi (x) \\sum _{j=1,2}a(x-x_j)$ , where $\\psi (x)$ is the wave function in front of the screen, and the “shadow function\" $a(x-x_j)$ is nonzero only for $x$ inside the $j$ -th slit.", "Propagating away from the screen, the waves from the two slits would overlap and create an interference pattern.", "On the other hand, in the presence of detectors the wave function will be given by Eq.", "(REF ) with indices $i,j$ there taking the values of 1 and 2.", "The condition that detectors work well and allow to determine through which slit the particle have passed means then exactly that the packets from the two slits remain well separated with respect to coordinates $y_1$ and $y_2$ , and so the interference between them is impossible.", "As was explained above, if the particle have passed the slit $k$ and was detected there, then its future motion and the motion of detectors will be the same as in the state with the wave function (REF ), i.e., as if the other slit was closed.", "In agreement with [15], the observation of an interference is, therefore, incompatible with the detection of the path chosen by the particle.", "These two operations are just mutually exclusive: the interference happens when the packets overlap, while the detection of the path requires them to be well separated.", "Note that this conclusion remains perfectly valid even when the detectors are microscopic, like the one-bit detectors discussed in [34]." ], [ "Nonlocality, analyticity, and covariance", "Although in $\\cal P$ AQD, as in classical mechanics, particles move along well defined trajectories, the equations of motion in these theories are fundamentally different.", "The only momentums that contribute to the equations of classical mechanics are the first order momentums $p_{j_k}$ , where $k$ runs from 1 to the number of particles $n_p$ , and for every $k$ , $j_k$ runs from 1 to the dimension of physical space $n$ .", "Every momentum $p_{j_k}$ , therefore, is “bound\" to a corresponding particle $k$ , and changes only due to the presence of forces, described by a potential function $U$ .", "In nonrelativistic mechanics, the forces corresponding to this potential normally vanish with distance, while in relativistic cases the potential propagates with finite speed, which is not larger than the speed of light $c$ .", "Classical mechanics is, therefore, local: to predict the behavior of a particle separated by a large distance from others during some time $\\Delta t$ , one doesn't have to know what happens further than the distance of $c\\Delta t$ from it.", "This locality, we see, is a consequence of the fact that in classical mechanics particles influence each other only through the action of the potential, which has the described properties.", "In $\\cal P$ AQD the situation is different.", "To avoid tedious manipulations with a multiparticle Hamiltonian (REF ), we may simply make all masses $m_k$ equal to each other and denote them as $m$ .", "The Hamiltonian (REF ) will then look exactly like the one-particle Hamiltonian (REF ), but in an $n_p n$ , rather than in an $n$ -dimensional space.", "Correspondingly, Eq.", "(REF ) for the evolution of momentums will hold, with summation over repeating indices $j$ there running from 1 to $n_p n$ .", "Now if particles are entangled, i.e., the system's action function is not equal to the sum of separate particles' actions (or system's wave function to the product of separate particles' wave functions) then there exist nonzero momentums $p_\\mu $ with “mixed\" multi-indices $\\mu $ , which include indices from different particles.", "Eq.", "(REF ) will then interconnect the time evolutions of all possible momentums $p_\\mu $ , and with them of particles' velocities.", "Since the momentum-dependent part of (REF ) does not depend on the particle positions, and all momentums are taken at the same time, they obtain the status of global variables: each momentum affects the time evolution of all others at the same moment of time, independently of the particle positions and the distances between them.", "Thus in this new (i.e., nonclassical, “nonpotential\") way, the particles in $\\cal P$ AQD influence each other on the whole hypersurface $t=\\mbox{const}$ instantaneously, and over arbitrary distance.", "Clearly, the reason for this nonlocality is that an analytic function is a fundamentally nonlocal object — the set of its derivatives in any point of space determines its behavior arbitrarily far from this point.", "The nonlocal kind of behavior described above, is, according to Bell's theorem, necessary for any theory that dynamically derives experimentally observed nonlocal correlations between entangled particles, rather than just predicts them, as does standard quantum mechanics [35], [36].", "$\\cal P$ AQD is built as an “ODE side\" of quantum mechanics, which always agrees, of course, with its “PDE side\", i.e., the Schrödinger equation and the conventional theory based on it.", "As such, $\\cal P$ AQD must be nonlocal: if it were local, so would the standard quantum mechanics.", "Note also, that $\\cal P$ AQD does not conflict with our intuition: indeed, our intuition is classical, but the classical limit of $\\cal P$ AQD is just the usual, completely local classical mechanics!", "$\\cal P$ AQD thus has the desired feature of being a fundamentally nonlocal theory with a local classical limit.", "The described nonlocal behavior was first discovered in the framework of DBBT and discussed extensively there [13].", "It was soon realized that DBBT's nonlocality is in perfect accord with the requirements of Bell's theorem and is, in this respect, welcomed [35].", "There remained, however, a difficult question about the theory's relativistic invariance.", "The influence, propagating with infinite speed, seems to be in an obvious conflict with the requirements of special relativity theory.", "This concern is addressed in [13], where it is proved that such influence cannot be used for transmission of superluminal signals.", "Still, there is the other concern: propagation of influence with infinite speed requires a selection of preferred reference frame, in which this propagation happens along surfaces $t = \\mbox{const}$ , in contradiction with the spirit of the theory of relativity, which demands that physical laws must be the same in every inertial frame of reference.", "This is generally considered to be a serious problem for DBBT [37].", "We will now show, that, thanks to the additional requirement of analyticity, $\\cal P$ AQD may be formulated in an arbitrary analytic foliation of space-time, and will have the same form in each of them.", "Our consideration will be nonrelativistic.", "It will be argued at the end of this section, however, that its relativistic version, although still nonlocal, will be not only Lorentz invariant, but can be also made generally covariant.", "Indeed, consider an arbitrary analytic foliation of space-time, generated by a single-valued analytic function $f({\\bf x },\\tau )$ , i.e., a partition of space-time into 3-dimensional hypersurfaces (leaves of foliation) $t = f({\\bf x },\\tau ) \\, , $ where $\\tau $ parameterizes hypersurfaces (we can, for example, conveniently require $\\tau =f(0,\\tau )$ ) monotonically, so that $\\partial f({\\bf x },\\tau )/\\partial \\tau > 0$ for all ${\\bf x }$ and $\\tau $ , and such that the whole space-time is covered (so that every point $({\\bf x }_0,t_0)$ belongs to some hypersurface, i.e., $t_0 = f({\\bf x }_0,\\tau _0)$ with some $\\tau _0$ ).", "In relativistic theory we require the surfaces $\\tau = \\mbox{const}$ to be space-like.", "We will call this foliation $f$ -foliation.", "The standard partition of space-time into surfaces $t = \\mbox{const}$ (“standard foliation\") corresponds to a function $f({\\bf x },\\tau ) = \\tau $ .", "We now want to introduce wave functions, defined on surfaces $\\tau = \\mbox{const}$ , rather than $t = \\mbox{const}$ .", "For a one-particle case, the wave function $\\psi ({\\bf x },t)$ was introduced as a solution of a Schrödinger equation, analytic with respect to ${\\bf x }$ for every $t$ .", "It is then also analytic with respect to both ${\\bf x }$ and $t$ , and so the function $\\psi ^{(f)}({\\bf x },\\tau ) = \\psi \\big ({\\bf x },f({\\bf x },\\tau )\\big )$ is analytic with respect to ${\\bf x }$ and $\\tau $ .", "To define a wave function on surfaces $\\tau =\\mbox{const}$ in a multi-particle case, we will borrow from relativistic theory the multi-time formalism [38], where each particle has its own individual time, and the multi-time wave function of $n_p$ particles $\\psi ({\\bf x }_1,t_1,\\ldots ,{\\bf x }_{n_p},t_{n_p})$ depends on positions and times of all of them.", "Detailed analysis of the physical meaning of this wave function and of corresponding analytical quantum dynamics will be a subject of relativistic consideration.", "In nonrelativistic theory, where the interaction between particles is mediated by an instantaneous potential function, the multi-time formalism can be defined only for particles that do not interact with each other (but can interact with an external potential).", "It will be sufficient for our purpose, however, to consider such noninteracting particles, because here we are only interested in nonlocal correlations, caused by entanglement, and not in correlations due to an interparticle interaction.", "The multi-time wave function then satisfies the system of equations $\\left(i\\hbar \\frac{\\partial }{\\partial t_k}\\, -\\, \\hat{H}_k\\right)\\psi ({\\bf x }_1,t_1,\\ldots ,{\\bf x }_{n_p},t_{n_p}) = 0 \\,, \\,\\,\\,\\, k = 1,\\ldots ,n_p \\, , $ where $\\hat{H}_k = -\\frac{\\hbar ^2}{2m_k} \\, \\Delta _k + U_k({\\bf x }_k,t_k)$ is the Hamilton operator of the $k$ -th particle in the external potential $U_k({\\bf x },t)$ , $\\Delta _k$ being a Laplace operator, acting on the coordinates of the $k$ -th particle ${\\bf x }_k$ .", "The wave function $\\psi ({\\bf x }_1,t_1,\\ldots ,{\\bf x }_{n_p},t_{n_p})$ may be obtained by path integration over all the paths such that for every $k = 1,\\ldots ,n_p$ , the paths for the $k$ -th particle terminate in a point $({\\bf x }_k,t_k)$ .", "The action functions $S({\\bf x }_1,t_1,\\ldots ,{\\bf x }_{n_p},t_{n_p})$ and $R({\\bf x }_1,t_1,\\ldots ,{\\bf x }_{n_p},t_{n_p})$ , defined from $\\psi ({\\bf x }_1,t_1,\\ldots ,{\\bf x }_{n_p},t_{n_p})$ as in Eq.", "(REF ), satisfy a system of quantum Hamilton-Jacobi equations $\\frac{\\partial S}{\\partial t_k} + H^S_k = 0\\, , \\quad \\quad \\frac{\\partial R}{\\partial t_k} + H^R_k = 0 \\, ,\\quad \\,\\,k = 1,\\ldots ,n_p \\, , $ similar to Eq.", "(REF ), with $H^S_k$ and $H^R_k$ given by Eqs.", "() and (), where the index $j$ in those equations means the derivative with respect to the $j$ -th coordinate of the $k$ -th particle, and the potential $U$ is understood as $U_k({\\bf x }_k,t_k)$ .", "As will become clear soon, it is appropriate to postulate, in a straightforward generalization of Eq.", "(REF ) and corresponding one-time theory, that if in a state described by a wave function $\\psi ({\\bf x }_1,t_1,\\ldots ,{\\bf x }_{n_p},t_{n_p})$ , the particles have space-time positions $({\\bf x }_1,t_1),\\ldots ,({\\bf x }_{n_p},t_{n_p})$ , then their velocities are given by ${\\bf v }_k = \\frac{\\partial H^S_k}{\\partial S_{{\\bf x }_k}} = \\frac{\\partial H^R_k}{\\partial R_{{\\bf x }_k}} = \\frac{1}{m_k} \\,S_{{\\bf x }_k}({\\bf x }_1,t_1,\\ldots ,{\\bf x }_{n_p},t_{n_p})\\, , $ where $S_{{\\bf x }_k}({\\bf x }_1,t_1,\\ldots ,{\\bf x }_{n_p},t_{n_p}) = \\partial S({\\bf x }_1,t_1,\\ldots ,{\\bf x }_{n_p},t_{n_p})/\\partial {\\bf x }_k$ , and similarly for $R_{{\\bf x }_k}$ .", "With so-defined multi-time wave function, the wave function on a hypersurface $\\tau =\\mbox{const}$ of any $f$ -foliation is obtained by placing each particle on this hypersurface, $\\psi ^{(f)}({\\bf x }_1,\\ldots ,{\\bf x }_{n_p},\\tau ) = \\psi \\big ({\\bf x }_1,f({\\bf x }_1,\\tau ),\\ldots ,{\\bf x }_{n_p},f({\\bf x }_{n_p},\\tau )\\big ) \\, ,$ and it is an analytic function of all ${\\bf x }_k$ and $\\tau $ .", "The whole theory developed above for a standard foliation may then be reproduced for an arbitrary analytic $f$ -foliation.", "The $\\tau $ -evolution of the wave function $\\psi ^{(f)}({\\bf x }_1, \\ldots ,{\\bf x }_{n_p},\\tau )$ is governed by the equation $i\\hbar \\frac{\\partial \\psi ^{(f)}}{\\partial \\tau } = \\hat{H}^{(f)} \\psi ^{(f)} .$ The transformation of space coordinates does not affect our nonrelativistic analysis, and so we will use the same coordinates in all foliations.", "Then $\\hat{H}^{(f)} = \\sum _{k=1}^{n_p} \\hat{H}^{(f)}_k \\, , \\quad \\; \\hat{H}^{(f)}_k = f_\\tau ({\\bf x }_k,\\tau ) \\hat{H}_k \\, ,$ where $f_\\tau $ denotes the derivative of $f$ with respect to its second argument, i.e., $f_\\tau ({\\bf x }_k,\\tau ) = \\partial f({\\bf x }_k,\\tau )/\\partial \\tau $ .", "In what follows, we also need the derivative $f_{{\\bf x }}$ of $f$ with respect to its first argument, $f_{{\\bf x }}({\\bf x }_k,\\tau ) = \\partial f({\\bf x }_k,\\tau )/\\partial {\\bf x }_k$ .", "The action functions $p^{(f)}({\\bf x }_1,\\ldots ,{\\bf x }_{n_p},\\tau )$ , $S^{(f)}({\\bf x }_1,\\ldots ,{\\bf x }_{n_p},\\tau )$ , and $R^{(f)}({\\bf x }_1,\\ldots ,{\\bf x }_{n_p},\\tau )$ on $f$ -foliation are again obtained from $\\psi ^{(f)}$ as in Eq.", "(REF ), and then the quantum Hamilton-Jacobi equation, the momentums $p^{(f)}_\\sigma $ , $S^{(f)}_\\sigma $ , and $R^{(f)}_\\sigma $ and the equations of motion for them are introduced in the same way as for a standard foliation.", "In particular, the quantum Hamilton-Jacobi equation for $S^{(f)}$ is $\\frac{\\partial S^{(f)}}{\\partial \\tau } + H^{(f)^S} = 0 \\, ,$ where $H^{(f)^S} = \\sum _{k=1}^{n_p} H^{(f)^S}_k \\, , \\quad \\; H^{(f)^S}_k = f_\\tau ({\\bf x }_k,\\tau ) H^S_k \\, , $ and particle velocities in the $f$ -foliation, i.e., with respect to a new “time\" $\\tau $ , are given by the usual relation ${\\bf v }^{(f)}_k = \\partial H^{(f)^S} / \\partial S^{(f)}_{{\\bf x }_k}$ , where $S^{(f)}_{{\\bf x }_k} = \\partial S^{(f)}/\\partial {\\bf x }_k$ , or, using (REF ), ${\\bf v }^{(f)}_k = f_\\tau ({\\bf x }_k,\\tau )\\, \\frac{\\partial H^S_k}{\\partial S^{(f)}_{{\\bf x }_k}} \\, .", "$ Note, that the derivative over ${\\bf x }_k$ in $S^{(f)}_{{\\bf x }_k}$ is taken along the leaf of the $f$ -foliation, i.e., for $\\tau = \\mbox{const}$ , contrary to the derivative in $S_{{\\bf x }_k}$ , which is taken for $t = \\mathrm {const}$ .", "We can now show that velocities ${\\bf v }^{(f)}_k$ , Eq.", "(REF ), and ${\\bf v }_k$ , Eq.", "(REF ), correspond to the same motion of the $k$ -th particle.", "We note first that if this particle moves from point ${\\bf x }_k$ on leaf $\\tau $ to point ${\\bf x }_k + d{\\bf x }_k$ on leaf $\\tau +d\\tau $ , so that its $\\tau $ -velocity is ${\\bf v }^{(f)}_k = d{\\bf x }_k/d\\tau $ , then by Eq.", "(REF ) we have for a corresponding time interval $\\begin{array}{ccl}dt_k & = & f_\\tau ({\\bf x }_k,\\tau )\\, d\\tau + f_{{\\bf x }}({\\bf x }_k,\\tau )\\, d{\\bf x }_k \\\\[0.2cm]& = & \\big ( f_\\tau ({\\bf x }_k,\\tau ) + f_{{\\bf x }}({\\bf x }_k,\\tau )\\, {\\bf v }^{(f)}_k \\big )\\, d\\tau \\, .\\end{array}$ Consequently, $t$ -velocity ${\\bf v }_k = d{\\bf x }_k/dt_k$ should be equal to ${\\bf v }^{(f)}_k/\\big (f_\\tau + f_{{\\bf x }}\\, {\\bf v }^{(f)}_k\\big )$ , or ${\\bf v }^{(f)}_k = {\\bf v }_k\\, \\big (f_\\tau + f_{{\\bf x }}\\, {\\bf v }^{(f)}_k\\big ) \\, .", "$ To demonstrate that this relation between ${\\bf v }^{(f)}_k$ and ${\\bf v }_k$ does indeed take place, we need to express $S_{{\\bf x }_k}$ in Eq.", "(REF ) through $S^{(f)}_{{\\bf x }_k}$ in Eq.", "(REF ).", "For the space-time of the $k$ -th particle, consider the surface $\\tau =\\mbox{const}$ , or $t_k =f({\\bf x }_k,\\tau )$ .", "We have for the derivatives of the action function $S({\\bf x }_1,t_1,\\ldots ,{\\bf x }_{n_p},t_{n_p})$ along this surface $\\begin{array}{ccl}\\displaystyle \\left.\\frac{\\partial S}{\\partial {\\bf x }_k}\\, \\right|_{\\tau =\\mathrm {const}} & = & \\displaystyle \\left.\\frac{\\partial S}{\\partial {\\bf x }_k}\\,\\right|_{t_k = \\mathrm {const}}+\\,\\,\\left.\\frac{\\partial S}{\\partial t_k}\\, \\right|_{{\\bf x }_k=\\mathrm {const}} \\cdot \\,\\,\\left.\\frac{\\partial t_k}{\\partial {\\bf x }_k}\\, \\right|_{\\tau = \\mathrm {const}} \\\\[0.6cm]& = & S_{{\\bf x }_k} - H^S_k f_{\\bf x }({\\bf x }_k,\\tau ) \\, .\\end{array}$ But $\\partial S/\\partial {\\bf x }_k|_{\\tau =\\mathrm {const}} = S^{(f)}_{{\\bf x }_k}$ , and so $S_{{\\bf x }_k} =S^{(f)}_{{\\bf x }_k} + f_{{\\bf x }}H^S_k$ , and therefore $\\frac{\\partial H^S_k}{\\partial S^{(f)}_{{\\bf x }_k}} = \\frac{\\partial H^S_k}{\\partial S_{{\\bf x }_k}} \\left(1 + f_{{\\bf x }}\\frac{\\partial H^S_k}{\\partial S^{(f)}_{{\\bf x }_k}} \\right) .$ Now multiplying this equation by $f_\\tau ({\\bf x }_k,\\tau )$ and using Eqs.", "(REF ), (REF ), we immediately obtain the desired relation (REF ).", "Thus, for any function $f$ , the description provided by a foliation-based wave function $\\psi ^{(f)}$ agrees with the one provided by a multi-time wave function $\\psi $ .", "We have, therefore, the following situation.", "In every foliation, the state of a system is described by particle positions and by all momentums.", "Being the derivatives of the logarithm of a wave function along the leaves of the foliation, momentums depend on the foliation chosen, and so in any given system's state, there are different sets of momentums, corresponding to different possible foliations.", "As for a standard foliation, by the equations of motion momentums, corresponding to any foliation, are global variables — they are bound to the leaves of their foliation, rather than to the points on these leaves, and influence each other (and, consequently, the particle velocities) over the whole leaves of this foliation instantaneously.", "$\\cal P$ AQD, therefore, does not require a preferred frame of reference: in every foliation, the theory, expressed through the foliation's momentums and Hamiltonian function, looks the same.", "At the same time, considerations, based on different foliations, agree with each other in terms of actual motion of particles, because they all predict the same motion as the consideration based on a common object, the multi-time wave function $\\psi ({\\bf x }_1,t_1,\\ldots ,{\\bf x }_{n_p},t_{n_p})$ , as was just discussed.", "On the other hand, this multi-time wave function in all $R^{4n_p}$ may be uniquely obtained, for any $f$ and $\\tau _0$ , from a function $\\psi ^{(f)}_{\\tau _0}({\\bf x }_1,\\ldots ,{\\bf x }_{n_p}) = \\psi ^{(f)}({\\bf x }_1,\\ldots ,{\\bf x }_{n_p},\\tau _0)$ by path integration or by solving equations (REF ), and so every function $\\psi ^{(f)}_{\\tau _0}$ contains the same information as the multi-time wave function $\\psi $ in all $R^{4n_p}$ .", "In relativistic theory, the leaves of foliations corresponding to different Lorentz frames are flat, and the angles between different foliations' leaves correspond to relative velocities of respective frames.", "In addition to a time transformation, a Lorentz transformation of the space coordinates inside the leaves should be done.", "It seems then reasonable to expect that in relativistic theory the set of momentums, corresponding to each Lorentz frame, will behave as described above, i.e., momentums will affect each other instantaneously in this frame, and in each frame the theory will be the same.", "Also, if there are two space-like separated entangled measurements, then neither of them can be considered as causing the result of the other.", "Indeed, in different foliations their time order will be different, and their results are unambiguously determined by the state (i.e., positions and all momentums, or positions and wave function) on any leave of an arbitrary foliation.", "Foliations with nonflat leaves (i.e., leaves that in different space-time points are angled differently with respect to the leaves of “inertial\" foliations) will appear, when the theory is formulated in reference frames with local accelerations.", "Additional terms corresponding to inertial forces, proportional to particle masses, will then appear in Hamiltonian functions, so that every foliation will come with its own field of these forces.", "The equivalence of different foliations, i.e., the general covariance of the theory, can then be restored in a standard way by introducing a gauge field, which would adsorb the potential of inertial forces, in what seems to be a natural route leading to a gauge theory of gravitation [39]." ], [ "Analytical quantum dynamics of particles with spin", "   Analytical quantum dynamics of particles with spin In this section we show how to describe in $\\cal P$ AQD particles with spin.", "Since the wave function of a particle with spin $s$ is a $2s+1$ -component spinor,We will denote particle's spin by the small letter $s$ , to distinguish it from the real part $S$ of the action function $p$ .", "there seem to be two possible ways to include spin in the theory.", "The first one is to somehow define corresponding $2s+1$ complex or $2(2s+1)$ real action functions, satisfying evolutionary equations of (REF ) type with Hamiltonians that depend only on derivatives of the action and satisfy Hamiltonian conditions.", "This, however, does not seem to be possible.", "Indeed, for the Schrödinger equation we passed from the wave function to its logarithm, the action function, in order to obtain an evolutionary (namely, quantum Hamilton-Jacobi) equation with Hamiltonian that depends only on derivatives of the action function, rather than on this function itself.", "For a multi-component wave function, this simple trick will work only in a trivial case when every component satisfies its own equation, independent of others.", "Moreover, in case of several, say $n_p$ , particles, one would have to find not $2s+1$ complex action functions, which satisfy equations of the required form, but $(2s+1)^{n_p}$ of them!", "Clearly, this approach doesn't appear promising.", "The second possible approach is to transform a system of $2s+1$ equations for spinor components into an equivalent equation of the desired form for one new wave function.", "This can be done by using spin coherent states, and this is the approach that we will employ here.", "Spin coherent states are defined with the help of a spin rotating operator that rotates the spin state through an angle $\\alpha $ about direction ${\\bf n }$ .", "The explicit form of this operator is $\\exp \\left(-i\\alpha {\\bf n }{\\bf s }\\right)$ , where ${\\bf s }$ is a spin operator in units of $\\hbar $ .", "The rotation ${\\cal R}(\\chi ,\\theta ,\\varphi )$ , corresponding to Euler angles $\\chi $ , $\\theta $ , $\\varphi $ , is obtained as a rotation through the angle $\\chi $ about axis $Oz$ , followed by rotation through angle $\\theta $ about axis $Oy$ , followed by another rotation through angle $\\varphi $ about $Oz$ , and is described by the product of the three corresponding operators: ${\\cal R}(\\chi ,\\theta ,\\varphi ) = e^{-i\\varphi s_z} e^{-i\\theta s_y}e^{-i\\chi s_z}$ .", "Let $|s,m\\rangle $ be a standard eigenstate of the spin operators: ${\\bf s }^2 |s,m\\rangle = s(s+1) |s,m\\rangle $ , $s_z |s,m\\rangle = m |s,m\\rangle $ .", "Then the spin coherent state is defined as the maximally polarized state $|s,s\\rangle $ rotated by the operator ${\\cal R}(\\chi ,\\theta ,\\varphi )$ : $|\\chi ,\\theta ,\\varphi \\rangle = e^{-i\\varphi s_z} e^{-i\\theta s_y} e^{-i\\chi s_z} |s,s\\rangle $ .", "The explicit representation of the coherent state is $|\\chi ,\\theta ,\\varphi \\rangle = \\sqrt{(2s)!}", "\\sum _{m=-s}^s \\frac{u^{s+m} v^{s-m}}{\\sqrt{(s+m)!(s-m)!", "}}\\, |s,m\\rangle \\, , $ where complex parameters $u$ and $v$ are defined as $u = \\cos \\frac{\\theta }{2}\\,\\, e^{-i(\\varphi +\\chi )/2} \\, , \\quad \\, v = \\sin \\frac{\\theta }{2}\\,\\,e^{i(\\varphi -\\chi )/2}.", "$ Clearly, if $u = u_1 + iu_2$ and $v = v_1 + iv_2$ , where $u_{1,2}$ and $v_{1,2}$ are real, then $|u|^2 + |v|^2 = u_1^2 + u_2^2 + v_1^2 + v_2^2 = 1$ , so that parameters $u$ and $v$ (or $u_1$ , $u_2$ , $v_1$ , $v_2$ ) live on the three-dimensional unit sphere $S^3$ in the four-dimensional real space $R^4$ .", "Denote the set $(\\chi ,\\theta ,\\varphi )$ , or corresponding sets $(u,v)$ or $(u_1,u_2,v_1,v_2)$ , as $\\Omega $ , and the coherent state (REF ) as $|\\Omega \\rangle $ .", "The expression (REF ) for it may be easily derived, for example, by using the Schwinger bosons representation of spin operators [40].", "It is well known [40], [41], [42] that the system of spin coherent states is not orthogonal, overcomplete, and allows a resolution of unity in a Hilbert space of states with spin $s$ : $\\frac{2s+1}{\\pi ^2} \\int d\\Omega |\\Omega \\rangle \\langle \\Omega | = \\sum _m |s,m\\rangle \\langle s,m| \\, , $ where $d \\Omega = \\frac{1}{8} \\, \\sin \\theta \\, d \\chi d \\theta d \\varphi $ is the area element on $S^3$ .", "If $w_1,w_2,w_3,w_4$ are Cartesian coordinates in $R^4$ that are related to angular coordinates $\\chi ,\\theta ,\\varphi $ and radial distance $r$ by $w_{1,2} = r u_{1,2} \\, , \\quad \\, w_{3,4} = r v_{1,2} \\, , $ then the integration measure in $R^4$ is related to $d\\Omega $ by $d w_1 d w_2 d w_3 d w_4 = r^3 d r d\\Omega \\, ,$ which follows from the expression (most easily obtained by direct calculation with Mathematica) for the jacobian of the transformation (REF ) $\\det \\frac{\\partial (w_1,w_2,w_3,w_4)}{\\partial (\\chi ,\\theta ,\\varphi ,r)} =\\frac{1}{8} \\, r^3 \\sin \\theta \\, .$ The action of the spin operators $s_\\pm = s_x \\pm i s_y$ and $s_z$ on spin coherent states is described by equations [40] $\\begin{array}{lcl}s_+ |\\Omega \\rangle & = & \\displaystyle v \\partial _u |\\Omega \\rangle , \\\\[0.15cm]s_- |\\Omega \\rangle & = & \\displaystyle u \\partial _v |\\Omega \\rangle , \\\\[0.1cm]s_z |\\Omega \\rangle & = & \\displaystyle \\frac{1}{2}\\, (u \\partial _u - v \\partial _v) |\\Omega \\rangle .\\end{array}$ Using spin coherent states, the one-component wave function, corresponding to a spin state  $|\\psi \\rangle $ with spin $s$ , is defined as a scalar product $\\psi (\\Omega ) = \\langle \\Omega |\\psi \\rangle $ .", "It is clear from the resolution of unity (REF ) that using its wave function, the state $|\\psi \\rangle $ may be expanded over spin coherent states as $|\\psi \\rangle = \\frac{2s+1}{\\pi ^2} \\int d\\Omega \\, \\psi (\\Omega ) |\\Omega \\rangle \\, ,$ so that all information about the state is contained in its wave function and vice versa.", "The argument $\\Omega $ in the wave function describes a rotation with respect to Cartesian coordinates in three-dimensional space, and so when the coordinate system itself is rotated the wave function transforms accordingly.", "The rotations are elements of the three-dimensional rotation group SO(3), where the spin wave function is defined.", "As is well known, SO(3) is not simply connected: its fundamental group is cyclic group of order 2.", "As was discussed in section 2.5 and section 3, this means that the spin wave function defined in SO(3) may be double-valued, as it indeed is when spin $s$ is half-integer.", "The universal cover of SO(3) is the group SU(2) that covers SO(3) two-to-one, and so the spin wave function is single-valued in SU(2).", "Elements of SO(3) and SU(2) are parameterized by points on the sphere $S^3$ considered above [41], and the area element $d\\Omega $ , Eq.", "(REF ), is the Haar measure of these groups, so the spin wave function may be considered as defined on $S^3$ .", "Now if $|\\psi \\rangle = \\sum _{m=-s}^s \\psi _m({\\bf x },t)|s,m\\rangle $ is a state of a particle with spin $s$ , then the corresponding wave function is $\\psi ({\\bf x },\\Omega ,t) = \\sqrt{(2s)!}", "\\sum _{m=-s}^s \\frac{\\bar{u}^{s+m} \\, \\bar{v}^{s-m}}{\\sqrt{(s+m)!(s-m)!}}", "\\, \\psi _m({\\bf x },t) \\, .", "$ Thus a wave function of a particle with spin $s$ is defined in configuration space $Q = R^3 \\times S^3$ and is an analytic function of $\\bar{u}$ and $\\bar{v}$ , selected from arbitrary analytic functions of these variables by the condition of being a homogeneous function of power $2s$ , i.e., by condition $(\\bar{u}\\partial _{\\bar{u}}+ \\bar{v}\\partial _{\\bar{v}}) \\psi ({\\bf x },\\Omega ,t) = 2 s \\psi ({\\bf x },\\Omega ,t) \\, .", "$ In the space of analytic functions of $\\bar{u}$ and $\\bar{v}$ , an operator $(1/2)(\\bar{u}\\partial _{\\bar{u}}+\\bar{v}\\partial _{\\bar{v}})$ plays, therefore, the role of a total spin operator.", "From Eq.", "(REF ) we obtain the action of spin operators on a wave function: $\\begin{array}{lcl}\\langle \\Omega |s_+|\\psi \\rangle & = & \\bar{u}\\partial _{\\bar{v}}\\psi (x,\\Omega ,t) , \\\\[0.2cm]\\langle \\Omega |s_-|\\psi \\rangle & = & \\bar{v}\\partial _{\\bar{u}}\\psi (x,\\Omega ,t) , \\\\[0.1cm]\\langle \\Omega |s_z|\\psi \\rangle & = & \\displaystyle \\frac{1}{2}\\,(\\bar{u}\\partial _{\\bar{u}}-\\bar{v}\\partial _{\\bar{v}}) \\psi (x,\\Omega ,t) ,\\end{array}$ where we used that operators $s_+$ and $s_-$ are hermitian conjugates of each other.", "Note, that spin operators make a complete set of non-trivial first order differential operators that leave a wave function in the form (REF ).", "The only remaining operator, $\\bar{u}\\partial _{\\bar{u}}+ \\bar{v}\\partial _{\\bar{v}}$ , gives $2s$ by Eq.", "(REF ).", "The behavior of a particle of charge $e$ in an electric field with scalar potential $A_0$ and magnetic field ${\\bf B }$ with vector potential ${\\bf A }$ is described by $i\\hbar \\frac{\\partial \\psi }{\\partial t} = \\left[\\frac{1}{2m} \\left(\\frac{\\hbar }{i}\\,\\nabla - \\frac{e}{c}\\, {\\bf A }\\right)^2 + eA_0- \\gamma {\\bf B }{\\bf s }\\right] \\psi \\, , $ where $c$ is the speed of light.", "For a particle with spin $1/2$ , Dirac theory gives for a constant $\\gamma $ the value of $e\\hbar /mc$ .", "If $\\psi $ is the just-defined wave function in a spin coherent state representation, then the term ${\\bf B }{\\bf s }$ expands as $\\begin{array}{ccl}{\\bf B }{\\bf s }& = & \\displaystyle \\frac{1}{2}\\, \\left(B_+ s_- + B_- s_+\\right) + B_z s_z \\\\[0.3cm]& = & \\displaystyle \\frac{1}{2}\\, \\big [B_+ \\bar{v}\\partial _{\\bar{u}}+ B_- \\bar{u}\\partial _{\\bar{v}}+ B_z (\\bar{u}\\partial _{\\bar{u}}-\\bar{v}\\partial _{\\bar{v}})\\big ] \\, , \\end{array}$ where $B_\\pm = B_x \\pm i B_y$ .", "Using matrix notations and standard Pauli matrices, we have then $\\begin{array}{ccl}{\\bf B }{\\bf s }& = & \\displaystyle \\frac{1}{2}\\, (\\bar{u}, \\bar{v})\\left(\\begin{array}{cc}B_z & B_- \\\\B_+ & -B_z\\end{array} \\right)\\left(\\begin{array}{c}\\partial _{\\bar{u}}\\\\\\partial _{\\bar{v}}\\end{array} \\right)\\\\[0.35cm]& = & \\displaystyle \\frac{1}{2}\\, (\\bar{u}, \\bar{v}) \\, {\\bf B }\\sigma \\left(\\begin{array}{c}\\partial _{\\bar{u}}\\\\\\partial _{\\bar{v}}\\end{array} \\right).", "\\end{array}$ As in the spinless case, expressing the wave function as $\\psi ({\\bf x },\\Omega ,t) = \\displaystyle \\exp \\left(\\frac{i}{\\hbar }\\,\\,p({\\bf x },\\Omega ,t) \\right) , \\quad \\quad p({\\bf x },\\Omega ,t)= S({\\bf x },\\Omega ,t) + \\frac{\\hbar }{i}\\,\\,R({\\bf x },\\Omega ,t) $ introduce the action function $p({\\bf x },\\Omega ,t)$ and its real and imaginary parts $S({\\bf x },\\Omega ,t)$ and $-\\hbar R({\\bf x },\\Omega ,t)$ .", "It is convenient to use a gauge $\\mbox{div} {\\bf A }= 0$ .", "Momentums that correspond to spin variables, such as $\\bar{u}$ , $u_1$ , or $\\chi $ (i.e., partial derivatives of the action with respect to these variables) will be denoted by the corresponding indices.", "The derivatives with respect to complex variables are understood as in Eq.", "(REF ).", "Then substituting (REF ) into (REF ), obtain for a particle with spin a quantum Hamilton-Jacobi equation (REF ) with Hamiltonian function $H = \\frac{1}{2m} \\left(p_j - \\frac{e}{c} \\, A_j\\right)^2 + eA_0 + \\frac{\\hbar }{2im} \\,p_{jj} - \\frac{i\\gamma }{2\\hbar } \\, (\\bar{u}, \\bar{v}) \\, {\\bf B }\\sigma \\left(\\begin{array}{c}p_{\\bar{u}} \\\\p_{\\bar{v}}\\end{array} \\right).", "$ The Hamiltonian (REF ) is of the first order with respect to the spin variables.", "Consequently, HC1 is satisfied for it automatically, while HC2 is satisfied due to analyticity, as was discussed in section 2.6.", "The whole theory of section 2, therefore, is applicable, but this time in configuration space $Q = R^3 \\times S^3$ , so that at any time the particle has its space position in $R^3$ , “internal\" SU(2) position on $S^3$ , and all corresponding momentums.", "To guarantee that the particle's spin is equal to $s$ , $S^3$ positions and momentums should satisfy $\\frac{i}{\\hbar }\\,(\\bar{u}p_{\\bar{u}} + \\bar{v}p_{\\bar{v}}) = 2s \\, , $ which follows from Eq.", "(REF ).", "Since spin operators change only the projections of spin, and not its value, for any Hamiltonian that, as in Eq.", "(REF ), depends only on spin operators, it is sufficient if this condition is satisfied at the initial moment of time.", "According to the general theory of section 2, particle velocity in physical space $R^3$ is given by Eq.", "(REF ), i.e., $v^j = \\frac{1}{2m}\\, (p_j + \\bar{p}_j) - \\frac{e}{mc}\\, A_j \\, , $ while SU(2) variables evolve (see section 2.6) according to $(\\dot{\\bar{u}}, \\dot{\\bar{v}}) = \\left(\\frac{\\partial H}{\\partial p_{\\bar{u}}}\\, , \\frac{\\partial H}{\\partial p_{\\bar{v}}}\\right) =- \\frac{i\\gamma }{2\\hbar } \\, (\\bar{u}, \\bar{v}) \\, {\\bf B }\\sigma $ or, after hermitian conjugation, $\\left(\\begin{array}{c}\\dot{u} \\\\\\dot{v}\\end{array}\\right) = \\frac{i\\gamma }{2\\hbar } \\, {\\bf B }\\sigma \\left(\\begin{array}{c}u \\\\v\\end{array}\\right) .", "$ This is an equation of spinor rotation with angular velocity $\\omega = - (\\gamma /\\hbar ){\\bf B }$ .", "The time evolution of spinor $(u,v)^T$ (where $T$ indicates transposition), composed of SU(2) coordinates of a particle, is, therefore, very simple: at any moment it rotates with this angular velocity, ${\\bf B }$ being the magnetic field at the current particle's position.", "Using Eqs.", "(REF ), (REF ), it is easy to demonstrate that the value $|u|^2 + |v|^2$ is conserved along the spinor's trajectory, and so remains equal to one, if it was equal to it initially.", "Equations of motion for momentums couple all kinds of them: “space\" momentums, with multi-indices composed of $x$ , $y$ , and $z$ , “spin\" momentums with multi-indices composed of $\\bar{u}$ and $\\bar{v}$ , and “mixed\" momentums, with multi-indices composed of both kinds of variables.", "These equations decouple if the magnetic field ${\\bf B }$ is spatially uniform, and the initial wave function factorizes in the form $\\psi ({\\bf x },\\Omega ,0) = \\psi ^{(x)}({\\bf x },0) \\psi ^{(s)}(\\Omega ,0)$ or $p({\\bf x },\\Omega ,0) = p^{(x)}({\\bf x },0) + p^{(s)}(\\Omega ,0)$ , where the indices $x$ and $s$ mark the space and spin parts.", "The mixed momentums then remain equal to zero and the wave function remains factorized at all times.", "The space part of the wave/action function satisfies the equations for a spinless particle, and so the particle moves in the physical space as if it didn't have any spin.", "Using Eq.", "(REF ), the equation $\\frac{\\partial p^{(s)}}{\\partial t} - \\frac{i\\gamma }{2\\hbar } \\, (\\bar{u}, \\bar{v}) \\, {\\bf B }\\sigma \\left(\\begin{array}{c}p^{(s)}_{\\bar{u}} \\\\p^{(s)}_{\\bar{v}}\\end{array} \\right) = 0$ for a spin part of an action function may be written in the form $\\dot{p}^{(s)} = 0$ , where $\\dot{p}^{(s)} = \\partial p^{(s)}/\\partial t + \\dot{\\bar{u}} \\partial _{\\bar{u}}p^{(s)} + \\dot{\\bar{v}} \\partial _{\\bar{v}}p^{(s)}$ .", "As for every homogeneous PDE of the first order [43], the solution $p^{(s)}$ , therefore, remains constant along the equation's characteristic curve, i.e., along the trajectory $\\big (u(t),v(t)\\big )$ in $S^3$ described by Eq.", "(REF ).", "Along with $p^{(s)}$ , the wave function $\\psi ^{(s)}$ also remains constant, i.e., $\\langle u(t),v(t)|\\psi ^{(s)}(t)\\rangle =\\mbox{const}$ .", "Consequently, like a spinor $(u,v)^T$ , the spin part $\\psi ^{(s)}$ of the wave function rotates with angular velocity $\\omega = - (\\gamma /\\hbar ){\\bf B }$ , exhibiting the well-known spin precession in a spatially uniform magnetic field.", "Like a spinless Schrödinger equation, Eq.", "(REF ) may be obtained from a variational principle $\\delta \\int {\\cal L}\\, d\\Omega ^{(u,v)} \\prod _{j=1}^3 dx^j dt = 0 $ with Lagrangian density ${\\cal L}= \\frac{\\hbar }{2i} \\left(\\psi ^*\\partial _t\\psi - \\psi \\partial _t\\psi ^*\\right) +\\frac{\\hbar ^2}{2m}\\, \\left(\\partial _j\\psi ^* + \\frac{ie}{\\hbar c} A_j \\psi ^*\\right) \\left(\\partial _j \\psi - \\frac{ie}{\\hbar c} A_j \\psi \\right) + e A_0 \\psi ^* \\psi - \\gamma \\psi ^*{\\bf B }{\\bf s }\\psi \\, .", "$ The spin part of the integration measure in Eq.", "(REF ) is $d\\Omega ^{(u,v)} = \\mathrm {d}u_1 \\wedge \\mathrm {d}u_2 \\wedge \\mathrm {d}v_1 \\wedge \\mathrm {d}v_2 = - (1/4) \\mathrm {d}u \\wedge \\mathrm {d}\\bar{u}\\wedge \\mathrm {d}v \\wedge \\mathrm {d}\\bar{v}$ , and integration over $u_1$ , $u_2$ , $v_1$ , $v_2$ runs over the whole space $R^4$ , so that this variational principle defines Eq.", "(REF ) in the whole space $R^4$ , and not only on the unit sphere $S^3$ .", "The phase invariance of the Lagrangian density ${\\cal L}$ leads, by Noether's theorem, to the corresponding conservation law, which now has the form $\\partial _0 j^0 + \\mbox{div} {\\bf j }+ \\partial _{\\bar{u}}j^{\\bar{u}} + \\partial _{\\bar{v}}j^{\\bar{v}} = 0 \\, , $ where the components of the current are given by Eq.", "(REF ), but with $i=0, \\ldots ,3,\\bar{u},\\bar{v}$ this time.", "Substituting there $\\Delta = (i/\\hbar )\\psi $ , $\\Delta ^* =-(i/\\hbar )\\psi ^*$ , $\\Lambda ^i=0$ , and using Eq.", "(REF ) for ${\\bf B }{\\bf s }$ , one gets for the current $\\begin{array}{ccl}\\left(j^0,{\\bf j },j^{\\bar{u}},j^{\\bar{v}}\\right) &=&\\displaystyle \\left(|\\psi |^2, \\,\\frac{\\hbar }{2im}\\, (\\psi ^*\\nabla \\psi - \\psi \\nabla \\psi ^*) - \\frac{e}{mc} \\, {\\bf A }|\\psi |^2, \\,- \\frac{i\\gamma }{2\\hbar } \\,|\\psi |^2 (\\bar{u}, \\bar{v}) \\, {\\bf B }\\sigma \\right) \\\\[0.4cm]& = & |\\psi |^2 (1,{\\bf v },\\dot{\\bar{u}},\\dot{\\bar{v}}) \\, .\\end{array}$ Note, that since this current doesn't have a radial component in $R^4$ , the conservation law (REF ) is satisfied on every sphere with the center in the origin there, including a unit sphere $S^3$ , where we need it.", "The results of the previous sections can now be immediately generalized to the case of particles with spin.", "Most importantly, $|\\psi |^2$ becomes the probability density in configuration space $R^3 \\times S^3$ with respect to a measure $d\\Omega \\prod _{j=1}^3dx_j$ , where $d\\Omega $ is a measure (REF ) on $S^3$ , and the measurement of spin-related physical quantities is described by the same theory of section 8 as for space-related quantities.", "Although, as was discussed in section 2.6, the above derivation in complex coordinates $\\bar{u}, \\bar{v}$ is equivalent to the one that uses coordinates $u_1, u_2, v_1, v_2$ , it may be instructive to present a direct derivation in these real coordinates.", "For that, it is convenient to present the last term in Eq.", "(REF ) in the form $- \\gamma {\\bf B }{\\bf s }\\psi = \\frac{\\hbar }{i}\\,\\big (U^*\\partial _{\\bar{u}}+ V^*\\partial _{\\bar{v}}\\big )\\psi \\, ,$ where we introduced $U^* = U_1 - iU_2$ and $V^* = V_1 - iV_2$ for which, by Eq.", "(REF ), we have $\\begin{array}{ccrcr}U_1 & = & \\displaystyle -\\frac{\\gamma }{2\\hbar }\\,\\Re \\big [i(B_+\\bar{v}+ B_z\\bar{u})\\big ] & = & \\displaystyle -\\frac{\\gamma }{2\\hbar }\\,(B_x v_2 - B_y v_1 + B_z u_2) \\, , \\\\[0.4cm]U_2 & = & \\displaystyle \\frac{\\gamma }{2\\hbar }\\,\\Im \\big [i(B_+\\bar{v}+ B_z\\bar{u})\\big ] & = & \\displaystyle \\frac{\\gamma }{2\\hbar }\\,(B_x v_1 + B_y v_2 + B_z u_1) \\, , \\\\[0.4cm]V_1 & = & \\displaystyle -\\frac{\\gamma }{2\\hbar }\\,\\Re \\big [i(B_-\\bar{u}- B_z\\bar{v})\\big ] & = & \\displaystyle -\\frac{\\gamma }{2\\hbar }\\,(B_x u_2 + B_y u_1 - B_z v_2) \\, , \\\\[0.4cm]V_2 & = & \\displaystyle \\frac{\\gamma }{2\\hbar }\\,\\Im \\big [i(B_-\\bar{u}- B_z\\bar{v})\\big ] & = & \\displaystyle \\frac{\\gamma }{2\\hbar }\\,(B_x u_1 - B_y u_2 - B_z v_1) \\, .\\end{array}$ Since $\\psi $ is an analytic function of $\\bar{u}$ and $\\bar{v}$ , we have, using the Cauchy-Riemann equations, $\\partial _{\\bar{u}}\\psi = \\partial _{u_1}\\psi = i\\partial _{u_2}\\psi $ , and so $U^*\\partial _{\\bar{u}}\\psi = U_1\\partial _{u_1}\\psi - iU_2\\, i\\partial _{u_2}\\psi = U_1 \\partial _{u_1}\\psi + U_2 \\partial _{u_2}\\psi $ and also $V^*\\partial _{\\bar{v}}\\psi = V_1 \\partial _{v_1}\\psi + V_2 \\partial _{v_2}\\psi $ .", "Equation (REF ) can now be written in the form $i\\hbar \\frac{\\partial \\psi }{\\partial t} = \\left[\\frac{1}{2m} \\left(\\frac{\\hbar }{i}\\,\\nabla - \\frac{e}{c}\\, {\\bf A }\\right)^2 + eA_0+\\frac{\\hbar }{i}\\,\\big (U_1 \\partial _{u_1} + U_2 \\partial _{u_2} + V_1 \\partial _{v_1} + V_2 \\partial _{v_2}\\big )\\right] \\psi \\, , $ and after substituting (REF ), we obtain for the action function a quantum Hamilton-Jacobi equation (REF ) with Hamiltonian function $H = \\frac{1}{2m} \\left(p_j - \\frac{e}{c} \\, A_j\\right)^2 + eA_0 + \\frac{\\hbar }{2im} \\,p_{jj} + U_1 p_{u_1} + U_2 p_{u_2} + V_1 p_{v_1} + V_2 p_{v_2} \\, ,$ so that the spin coordinates have velocities $\\dot{u}_{1,2} = U_{1,2}$ , $\\dot{v}_{1,2} =V_{1,2}$ , which agrees with Eq.", "(REF ).", "Equation (REF ) may be obtained from the variational principle (REF ) with the same Lagrangian density ${\\cal L}$ as in Eq.", "(REF ), but with the spin term $-\\gamma \\psi ^*{\\bf B }{\\bf s }\\psi $ there presented as $(\\hbar /i)\\psi ^*(U_1\\partial _{u_1} + U_2 \\partial _{u_2} + V_1 \\partial _{v_1} + V_2 \\partial _{v_2})\\psi $ .", "The phase invariance of ${\\cal L}$ leads then to the conservation law $\\partial _0 j^0 + \\mbox{div} {\\bf j }+ \\partial _{u_1} j^{u_1} + \\partial _{u_2} j^{u_2} +\\partial _{v_1} j^{v_1} + \\partial _{v_2} j^{v_2} = 0$ with current $\\begin{array}{ccl}\\left(j^0,{\\bf j },j^{u_1},j^{u_2},j^{v_1},j^{v_2}\\right) & = & \\displaystyle \\left(|\\psi |^2,\\,\\frac{\\hbar }{2im}\\, (\\psi ^*\\nabla \\psi - \\psi \\nabla \\psi ^*) - \\frac{e}{mc} \\, {\\bf A }|\\psi |^2,\\, |\\psi |^2 \\big (U_1,U_2,V_1,V_2\\big ) \\right) \\\\[0.4cm]& = & |\\psi |^2 (1,{\\bf v },\\dot{u}_1,\\dot{u}_2,\\dot{v}_1,\\dot{v}_2)\\end{array}$ and, therefore, to the probabilistic interpretation of $|\\psi |^2$ and the measurement theory of section 8.", "Finally, we present the theory in “natural\" coordinates $\\chi , \\theta , \\varphi $ on $S^3$ .", "For that, note that $u_1, u_2, v_1, v_2$ in Eq.", "(REF ) are just Cartesian coordinates in $R^4$ , running from $-\\infty $ to $\\infty $ .", "To avoid confusion, rename them as $w_1,w_2,w_3,w_4$ and make the transformation (REF ), where now $u_{1,2}$ and $v_{1,2}$ are real and imaginary parts of complex coordinates $u$ and $v$ , Eq.", "(REF ), on $S^3$ .", "Let $J =\\partial (w_1,w_2,w_3,w_4)/\\partial (\\chi ,\\theta ,\\varphi ,r)$ be the jacobian matrix of this transformation.", "The spin term in the Schrödinger equation (REF ) may then be presented, using its form in (REF ), as $- \\gamma {\\bf B }{\\bf s }\\psi = \\frac{\\hbar }{i}\\,\\big (U_\\chi \\partial _\\chi + U_\\theta \\partial _\\theta +U_{\\varphi }\\partial _{\\varphi } + U_r\\partial _r\\big )\\psi \\, ,$ where $\\big (U_\\chi ,U_\\theta ,U_{\\varphi },U_r\\big ) = \\big (U_1,U_2,V_1,V_2\\big ) \\left(J^{-1}\\right)^T\\big |_{r=1} \\, .$ Since we only need this transformation on $S^3$ , i.e., for $r=1$ , $U_{1,2}$ and $V_{1,2}$ here are given by Eq.", "(REF ) where $u_{1,2}$ and $v_{1,2}$ are real and imaginary parts of $u$ and $v$ , Eq.", "(REF ).", "Direct calculation using Mathematica then gives $\\begin{array}{ccl}U_\\chi & = & \\displaystyle -\\frac{\\gamma }{\\hbar }\\,\\frac{1}{\\sin \\theta }\\,(B_x \\cos \\varphi +B_y \\sin \\varphi ) ,\\\\[0.4cm]U_\\theta & = & \\displaystyle \\frac{\\gamma }{\\hbar }\\, (B_x \\sin \\varphi - B_y \\cos \\varphi ) ,\\\\[0.4cm]U_{\\varphi } & = & \\displaystyle \\frac{\\gamma }{\\hbar } \\left[\\frac{\\cos \\theta }{\\sin \\theta }\\, (B_x\\cos \\varphi + B_y \\sin \\varphi ) - B_z\\right] ,\\end{array}$ and, as expected, $U_r=0$ .", "The Schrödinger equation now has the form $i\\hbar \\frac{\\partial \\psi }{\\partial t} = \\left[\\frac{1}{2m} \\left(\\frac{\\hbar }{i}\\,\\nabla - \\frac{e}{c}\\, {\\bf A }\\right)^2 + eA_0+ \\frac{\\hbar }{i}\\,\\big (U_\\chi \\partial _\\chi + U_\\theta \\partial _\\theta + U_{\\varphi } \\partial _{\\varphi }\\big )\\right] \\psi \\, , $ and the Hamiltonian function in a quantum Hamilton-Jacobi equation (REF ) will become $H = \\frac{1}{2m} \\left(p_j - \\frac{e}{c} \\, A_j\\right)^2 + eA_0 + \\frac{\\hbar }{2im} \\,p_{jj} + U_\\chi p_\\chi + U_\\theta p_\\theta + U_{\\varphi } p_{\\varphi } \\, , $ so that $\\dot{\\chi } = U_\\chi $ , $\\dot{\\theta } = U_\\theta $ , $\\dot{\\varphi } = U_{\\varphi }$ .", "Equation (REF ) may be obtained from a variational principle $\\delta \\int {\\cal L}\\,d\\Omega \\prod _{j=1}^3dx^jdt = 0$ , where the measure $d\\Omega $ is given by Eq.", "(REF ) and the Lagrangian density ${\\cal L}$ by Eq.", "(REF ) with spin term $-\\gamma \\psi ^*{\\bf B }{\\bf s }\\psi $ there presented as $(\\hbar /i)\\psi ^*(U_\\chi \\partial _\\chi + U_\\theta \\partial _\\theta + U_{\\varphi }\\partial _{\\varphi })\\psi $ .", "The measure $d\\Omega $ is not homogeneous — it is equal to the product of differentials of independent variables times the function $f=\\sin \\theta $ .", "In such cases, to ensure the possibility of all necessary integrations by parts, the derivation of the equations of motion and conservation laws from the variational principle differs by using instead of the usual derivatives $\\partial _k$ the operator $\\partial _k^{(f)} = (1/f)\\partial _kf$ , which acts on any function $p$ as $\\partial _k^{(f)}p = (1/f)\\partial _k(fp)$ [44].", "In our case, when $f$ is a function of only one variable $\\theta $ , all derivatives except $\\partial _\\theta $ remain unchanged.", "The correct form of a conservation law that follows from the phase invariance of ${\\cal L}$ in a space $R^3\\times S^3$ with an inhomogeneous integration measure $d\\Omega \\prod _{j=1}^3dx^j$ , and has the usual meaning and consequences there, is $\\partial _0 j^0 + \\mbox{div} {\\bf j }+ \\partial _\\chi j^\\chi + \\frac{1}{\\sin \\theta }\\,\\partial _\\theta \\big (\\sin \\theta j^\\theta \\big ) + \\partial _{\\varphi } j^{\\varphi } = 0$ where $\\begin{array}{ccl}\\left(j^0,{\\bf j },j^\\chi ,j^\\theta ,j^{\\varphi }\\right) & = & \\displaystyle \\left(|\\psi |^2,\\,\\frac{\\hbar }{2im}\\, (\\psi ^*\\nabla \\psi - \\psi \\nabla \\psi ^*) - \\frac{e}{mc}\\, {\\bf A }|\\psi |^2,\\,|\\psi |^2 \\big (U_\\chi ,U_\\theta ,U_{\\varphi }\\big ) \\right) \\\\[0.4cm]& = & |\\psi |^2 (1,{\\bf v },\\dot{\\chi },\\dot{\\theta },\\dot{\\varphi }) .\\end{array}$ The conservation law (REF ) may also be obtained directly by substituting there expressions (REF ) for components of the current, noticing that $|\\psi |^2 = e^{2R}$ , and using quantum Hamilton-Jacobi equation with Hamiltonian function (REF ).", "We thus demonstrated that the analytical quantum dynamics of particles with spin can be developed by an extension of particle configuration space from $R^3$ to $R^3\\times S^3$ .", "The de Broglie - Bohm - style theory of spin in $R^3\\times S^3$ , in which particles are considered as a point limit of extended rigid objects, is developed in chapter 10 of Holland's book [13].", "Our spin theory uses an infinite phase space over $R^3\\times S^3$ , and all the theory of the previous sections is applicable to it.", "In particular, particles move in $R^3\\times S^3$ along trajectories that are well defined by the equations of motion, $|\\psi |^2$ is the probability density in $R^3 \\times S^3$ , and the measurement of a spin component in a Stern-Gerlach experiment is a typical example of von Neumann's measurement procedure with discrete spectrum." ], [ "Conclusion", "   Conclusion Let us summarize the main points of $\\cal P$ AQD.", "It is straightforward to verify that for a sum $p^s$ of a Taylor series (REF ) to satisfy PDE (REF ), it is necessary and sufficient if the Taylor coefficients, or momentums, $p^r_\\sigma $ satisfy the ODEs (REF ), where $\\dot{{\\bf q }}$ is the velocity of an expansion point moving in the configuration space.", "Consequently, we have a simple and universal connection between evolutionary PDE (REF ) and the dynamical ODEs (REF ), where the velocity $\\dot{{\\bf q }}$ is still arbitrary.", "If, further, the Hamiltonian function $H^r$ in Eq.", "(REF ) satisfies the Hamiltonian conditions of section 2.3, then there exists a special velocity, given by Eq.", "(REF ), which leads to Eq.", "(REF ), the variational principles of section 2.4, and the hierarchical Hamiltonian structure of the whole theory.", "Thus, there is a general ODE/PDE Hamiltonian formalism that may be filled with different physical contents, depending on the form of a Hamiltonian function.", "In classical mechanics, the Hamiltonian function is of the first order.", "As was explained in section 2.7, the ODE part of the theory in this case simplifies into an ordinary Hamiltonian mechanics in the usual phase space.", "Quantum theory utilizes the second available option, with a Hamiltonian of a higher order and ODEs residing in an infinite phase space.", "More specifically, it appears that in a nonrelativistic domain for spinless particles, nature builds quantum theories by the recipe of section 4, so that any theory of this kind is defined by Eq.", "(REF ) with some Lagrangian function, quadratic in velocity and such that the corresponding Hamiltonian function satisfies Hamiltonian condition (REF ).", "The theory will then automatically exhibit the superposition principle, path-integral representation, wave-particle duality (which is shown to be possible only in the infinite phase space), and the classical limit, described by Hamilton-Jacobi PDE and Hamilton ODEs with a Hamiltonian, corresponding to the Lagrangian function in Eq.", "(REF ).", "Also, the resulting Schrödinger equation will be automatically obtainable from a variational principle, so its symmetries will lead to corresponding conservation laws.", "Since Hamiltonian functions in our theory depend only on derivatives of unknown functions, they are automatically invariant with respect to shifts of these functions by arbitrary constants.", "This symmetry leads to a current conservation, a current being defined with the correct velocity (REF ), and to invariance of a measure $|\\psi |^2dV$ with respect to equations of motion.", "This invariance leads then to the probability density $\\rho =|\\psi |^2$ in the same way as invariance of the Liouville measure leads to the microcanonical distribution in classical statistics, the difference in probability densities resulting from different forms of equations of motion.", "The probabilistic interpretation of the wave function is, therefore, deduced in $\\cal P$ AQD rather than being postulated.", "The multiparticle generalization of the theory leads to the standard picture of quantum particles in a classical macroscopic environment, and being applied to specially constructed apparatuses, to the quantum theory of measurement.", "The measurements of classical quantities that may be used as parts of particle-apparatus interaction Hamiltonians, appear to have the desired features in this theory.", "On the other hand, the measurement and observation of nonlocal momentums, such as $p_{x_iy_j}$ , where $i\\ne j$ are particle indices, is impossible, because Hamiltonians built by the rules of section 4 cannot contain such terms.", "The presence of such momentums makes the whole multiparticle theory nonlocal, and explains the mechanism of nonlocal correlations.", "On the other hand, their nonobservability prevents using them for the transmission of superluminal signals.", "As was discussed in section 9, in spite of being nonlocal, the relativistic version of the theory seems to be presentable in a Lorentz invariant and even generally covariant way.", "Finally, the theory of particles with nonzero spin resides in configuration space that includes, along with particle's space coordinates, its internal SU(2) degrees of freedom.", "Table: The comparison of 𝒫\\cal PAQD and QM.In Table 1, different aspects of $\\cal P$ AQD and QM are compared in a self-explanatory form.", "The comparison clearly demonstrates that in spite of experimental agreement, the two theories are evidently different and draw different pictures of the physical reality.", "Several additional remarks may be useful.", "First, as was discussed in sections 2.5 and 3, the actual solution of the equations of motion may be obtained by the generalized Jacobi method from a known wave function.", "In the spinless case, the particles will then move along Bohmian trajectories.", "The equations of motion, however, determine the motion completely and unambiguously themselves, and so a technique of their direct solution, without any use of a wave function, should be possible to develop if desirable.", "Second, the theory is formulated in terms of momentums, and as was just discussed not all of them are observable.", "We note, however, that all coordinates and momentums that are observable (for example in such devices as bubble chambers) are reflected in the theory, and nonobservable momentums are nonobservable not because they are postulated to be such, but because this is a property of an observation/measurement procedure that follows from the basic equations of the theory.", "The situation here should be compared with the one in QM, which is formulated entirely in terms of nonobservable wave functions, and brings in the observable quantities (coordinates and momentums) only through the interpretational part of the theory.", "Third, the theory of von Neumann's measurements, presented in section 8, exhibits all the properties attributed to the measurement procedure in standard QM, in particular such measurements must satisfy the uncertainty relations.", "However, contrary to QM, $\\cal P$ AQD gives detailed description of the behavior of both observed system and observing apparatus before, during, and after their interaction.", "In this situation, it is not unthinkable to speculate that new “measurement-like\" procedures may be found, which will provide more information than is permitted by the uncertainty principle, or will generate experimental situations (especially when mesoscopic objects are involved) in which QM (but not $\\cal P$ AQD) fails to give unambiguous predictions.", "The investigation of such possibilities lies, however, outside of the scope of the present work.", "To summarize, the mathematical theory developed in section 2 allows to give a simple description of quantum phenomena as resulting from generalized Hamiltonian motion of particles.", "The present theory does not suffer from the shortcomings discussed in the Introduction.", "It appears especially important, that although the specific form of the theory is completely determined by the Schrödinger equation, it nevertheless allows to simultaneously deduce the statistical interpretation, which in existing quantum theory is described by the separate and independent axioms.", "We conclude, that it seems not unreasonable to believe that $\\cal P$ AQD may indeed provide the true and fundamental description of nature." ], [ "Acknowledgements", "tocsectionAcknowledgements I am grateful to Joseph Krasil'shchik for critical reading of section 2 of this work and valuable comments." ], [ "Statistical distribution and entropy growth in classical statistics", "           Statistical distribution and entropy growth in classical statistics To facilitate the comparison in section 6 of statistical distribution in $\\cal P$ AQD and in the classical theory, here we review the basics of classical equilibrium and nonequilibrium statistics in a convenient for this comparison form.", "A classical system is represented by a point in a phase space $P$ of dimension $2n_pn_s$ , where $n_p$ is the number of particles and $n_s$ is the dimension of the physical space.", "Denote a general point of $P$ by $\\gamma $ .", "Invariant measure in $P$ is a Liouville measure $\\prod _{i=1}^{n_p}d^{n_s}r_id^{n_s}p_i$ , and we denote it by $d\\Gamma $ .", "Besides the description in a phase space $P$ , or the space of microstates that we will also call “full description space\" (FDS), classical statistics uses a much cruder description of systems by sets of kinetic or thermodynamic parameters forming “reduced description space\" (RDS) or the space of macrostates of dimension $n_R \\ll 2n_pn_s$ .", "The coordinates $R_i$ of this space are the values of some functions $r_i(\\gamma )$ , $i=1,\\ldots ,n_R$ of the microstate, and the macrostate is considered fully specified by the known values of all the $R_i$ , i.e., by the set $R = \\lbrace R_i, i = 1,\\ldots ,n_R\\rbrace $ .", "In other words, phase space $P$ is broken into subspaces $P_R$ that correspond to small cells in RDS: $P_R = \\lbrace \\gamma \\in P: \\, R_i \\le r_i(\\gamma ) \\le R_i+\\delta _i, i=1,\\ldots ,n_R\\rbrace $ , and a crude description of a system at any time $t$ is given by the corresponding $R(t)$ , i.e., by specifying a subspace that the point $\\gamma $ resides in at this moment.", "Different points of $P_R$ represent then different microstates, compatible with the same macrostate $R$ , so that every time the system is prepared in this macrostate, its microstate will be represented by some random point $\\gamma \\in P_R$ .", "As it was with a quantum particle, in the mathematical limit of an infinite number of such preparations, these points will form an ensemble $A_R$ with probability density $\\rho _R$ that maximizes the corresponding Gibbs entropy $S_G^{(R)}(\\rho )$ .", "Repeating the steps that led from (REF ) to (REF ), but using the Liouville measure this time, it is easy to show that $S_G^{(R)}(\\rho ) = - \\int _{P_R}\\!\\!", "\\rho \\ln \\rho \\,d\\Gamma \\, .", "$ We call the density $\\rho _R$ that maximizes this $S_G^{(R)}(\\rho )$ a microcanonical density, and corresponding ensemble $A_R$ a microcanonical ensemble.", "Let us show that the microcanonical density is constant: $\\rho _R = \\mbox{const} = \\frac{1}{\\Gamma (P_R)} \\, , $ where $\\Gamma (P_R) = \\int _{P_R} d\\Gamma $ is a phase volume of $P_R$ .", "Indeed, from $\\rho _R =\\mbox{const}$ and $\\ln x \\le x-1$ , we have for any other normalized probability density $\\rho $ in $P_R$ : $\\begin{array}{ccl}\\displaystyle - \\int _{P_R}\\!\\!", "\\rho _R \\ln \\rho _R \\,d\\Gamma & = & \\displaystyle - \\int _{P_R}\\!\\!", "\\rho \\ln \\rho \\,d\\Gamma - \\int _{P_R}\\!\\!", "\\rho \\ln \\frac{\\rho _R}{\\rho }\\, d\\Gamma \\\\[0.5cm]& \\ge & \\displaystyle - \\int _{P_R}\\!\\!", "\\rho \\ln \\rho \\,d\\Gamma + \\int _{P_R}\\!\\!", "\\rho \\left(1 -\\frac{\\rho _R}{\\rho }\\right) d\\Gamma \\\\[0.5cm]& = & \\displaystyle - \\int _{P_R}\\!\\!", "\\rho \\ln \\rho \\,d\\Gamma \\, .\\end{array}$ Due to the constancy of $\\rho _R$ , the averaging over $A_R$ , i.e., over a set of systems randomly created in a macrostate $R$ , reduces to the averaging over $P_R$ with the measure $d\\Gamma $ .", "Consequently, the need for introducing ensembles $A_R$ disappears; instead, we will use as ensembles the corresponding subsets $P_R$ of FDS.", "The maximum possible value of the Gibbs entropy on $P_R$ will be called the Boltzmann entropy of $R$ and denoted as $S_B(R)$ .", "We have, obviously, $S_B(R) = \\max _{\\lbrace \\rho \\rbrace } S_G^{(R)}(\\rho ) = S_G^{(R)}(\\rho _R) = -\\ln \\rho _R= \\ln \\Gamma (P_R) \\, .", "$ While Gibbs entropy is a function of the probability density $\\rho $ in FDS, Boltzmann entropy is a function of a set $R$ that belongs to a RDS and describes a macroscopic state of a system.", "The corresponding functions $r_i$ may be, for example, the particle, energy, and momentum densities in small cells, covering the volume of a system or, for another example, the densities of particles in the cells that cover a six-dimensional one-particle phase space (i.e., essentially, the values of the Boltzmann distribution function in different points of this space).", "The fact that the Boltzmann entropy $S_B$ depends on the macroscopic state $R$ of a system, rather than on its microscopic state $\\gamma $ , is a manifestation of its “anthropomorphic\" nature, as was discussed by Jaynes [29]: by its very thermodynamic definition, the difference of a system's entropy between different states depends on which parameters were held fixed and which were allowed to change during the transition from one state to the other.", "The relations (REF ) and (REF ) are based only on the $N\\rightarrow \\infty $ limit (see Eq.", "(REF )).", "They are, therefore, universally applicable to any system and any functions $r_i$ , even if they don't have a macroscopic character.", "In macroscopic systems, however, functions $r_i$ may be selected in a way that allows them to possess additional important properties [30].", "Namely, functions $r_i$ give a crude description of a system; as was just discussed, usually they represent the properties of particles in small cells in coordinate or phase spaces.", "Then in macroscopic systems, with their enormous quantity of particles, the size of these cells may be chosen in such a way that on one hand they are sufficiently small, so that the parts of the system inside them appear homogeneous and further division does not lead to more refined description, while on the other hand they are large enough to still contain a macroscopic number of particles.", "The behavior of all physical values that are additive with respect to contributions of separate particles or small groups of particles, will then be regulated by a central limit theorem: in a typical microstate $\\gamma \\in P_R$ , these values, as well as their time derivatives, will be close to the corresponding averages over $P_R$ with negligible dispersion.", "The averaging over $P_R$ serves, therefore, as a method of calculation of typical, i.e., observable, magnitudes of thermodynamic values, with the microcanonical ensemble often being replaced by a canonical one for calculational convenience.", "The existence of such intermediate scale of description refinement, and of corresponding typical behavior, practically identical to average one, is the first important property of macroscopic systems.", "Another property is related to the character of their time evolution.", "Namely, let at time $t_0$ the system be described by parameter set $R(t_0)$ , so that its microstate belongs to an ensemble (i.e., subspace of FDS) $P_{R(t_0)}$ .", "We let all states of this ensemble evolve until the time $t_1>t_0$ and denote the resulting ensemble $P_{R(t_0)}(t_1)$ .", "As was just discussed, typical values of the functions $r_i$ in this ensemble will be close, with negligible dispersion, to corresponding averages over it, which we will denote by $R_i(t_1)$ .", "The typical, i.e., observable, macrostate $R(t_1)$ for $t_1>t_0$ is obtained, therefore, by direct averaging of the equations of motion over $P_{R(t_0)}$ .", "We refer to the corresponding averaged equations as generalized kinetic equations.", "Let us now compare the ensemble $P_{R(t_0)}(t_1)$ with a microcanonical ensemble $P_{R(t_1)}$ that corresponds to the values of the parameters $R$ , observed at time $t_1$ .", "Neglecting extremely rare nontypical microstates, we can say that the values of $R$ in all states $\\gamma \\in P_{R(t_0)}(t_1)$ are equal to $R(t_1)$ , so that $P_{R(t_0)}(t_1)$ is a subset of $P_{R(t_1)}$ , $P_{R(t_0)}(t_1)\\subset P_{R(t_1)}$ .", "We need to elaborate in what sense the microstates of $P_{R(t_0)}(t_1)$ that do not belong to $P_{R(t_1)}$ are “extremely rare.\"", "Note that phase volumes that are essential for our conclusions are such that their logarithms are extensive, i.e., scale proportionally to the number of particles $n_p$ in a system.", "Consequently, we are only interested in the logarithms of these volumes per particle in the limit as $n_p\\rightarrow \\infty $ .", "Then the inclusion $P_{R(t_0)}(t_1)\\subset P_{R(t_1)}$ should be understood as $\\lim _{n_p\\rightarrow \\infty } \\frac{1}{n_p} \\Big ( \\ln \\Gamma \\left(P_{R(t_0)}(t_1)\\right) -\\ln \\Gamma \\left(P_{R(t_0)}(t_1){\\textstyle \\bigcap } P_{R(t_1)}\\right) \\Big ) = 0 \\, ,$ so that the part of $P_{R(t_0)}(t_1)$ that falls outside of $P_{R(t_1)}$ is inessential in this sense of logarithm per particle.", "This consideration also explains why we should not care about the exact values of the $\\delta _i$ in the definition of $P_R$ — they are inessential in the same sense.", "We are especially interested in the situation when at time $t_0$ the system was in a nonequilibrium state.", "The states of the ensemble $P_{R(t_0)}(t_1)$ will not be typical for $P_{R(t_1)}$ in this case, for they will have nontypical for $P_{R(t_1)}$ correlations.", "These correlations will manifest themselves under time inversion: the states of $P_{R(t_0)}(t_1)$ will return back to the macrostate $R(t_0)$ , which is further from equilibrium than $R(t_1)$ , while the typical states of $P_{R(t_1)}$ will approach equilibrium (modulus tiny thermodynamic fluctuations) with deviation of time from $t_1$ in both directions.", "Also, if we wait for the whole Poincaré cycle to pass, then we will observe another manifestation: the states of $P_{R(t_0)}(t_1)$ will deviate off equilibrium as far as to $R(t_0)$ , while the typical states of $P_{R(t_1)}$ will deviate less — up to $R(t_1)$ .", "It happens, however, and this is the second important property of macroscopic systems, that the correlations, which are different in $P_{R(t_0)}(t_1)$ and $P_{R(t_1)}$ , are $n$ -particle correlations with macroscopically large $n$ (“large-$n$ correlations\"), while “small-$n$ correlations\" (i.e., one-particle densities and correlations between a small number of particles) in $P_{R(t_0)}(t_1)$ and $P_{R(t_1)}$ are practically the same.", "At the end of the Poincaré cycle nontypical large-$n$ correlations will conspire to coherently affect small-$n$ ones and drive the system into an abnormally (for $P_{R(t_1)}$ ) nonequilibrium state $R(t_0)$ , but before that the influence of large-$n$ correlations on small-$n$ ones may be neglected.", "But it is only these small-$n$ correlations, and not large-$n$ ones, that contribute to observable and measurable physical values of interest.", "Consequently, in the normal physical experiment, when time goes only forward, but not as far forward as for the length of the Poincaré cycle, nontypical for $P_{R(t_1)}$ large-$n$ correlations in $P_{R(t_0)}(t_1)$ do not manifest themselves, and we can regard the states of $P_{R(t_0)}(t_1)$ as typical states of $P_{R(t_1)}$ .", "In other words, the origin of the current microstate $\\gamma (t_1)$ of the system, reflected in the ensemble $P_{R(t_0)}(t_1)$ , becomes irrelevant, the only important question being in which subspace $P_R$ of $P$ point $\\gamma $ resides now.", "This means that for every time $t>t_0$ , ensemble $P_{R(t_0)}(t)$ , representing the system, can (and must, for that makes the calculations much simpler) be replaced by a microcanonical ensemble $P_{R(t)}$ , where $R(t)$ is obtained from initial $R(t_0)$ by solving the generalized kinetic equations.", "Now from the inclusion $P_{R(t_0)}(t)\\subset P_{R(t)}$ and the invariance of the Liouville measure, we have for $t>t_0$ $\\Gamma \\big (P_{R(t_0)}\\big ) = \\Gamma \\big (P_{R(t_0)}(t)\\big ) \\le \\Gamma \\big (P_{R(t)}\\big )$ and so $S_B\\big (R(t)\\big ) \\ge S_B\\big (R(t_0)\\big )$ , i.e., Boltzmann entropy never decreases and achieves its maximum in equilibrium, when the system's macrostate $R$ ceases to change." ] ]

[ [ "Quantized Charge Pumping through a Carbon Nanotube Double Quantum Dot" ], [ "Abstract We demonstrate single-electron pumping in a gate-defined carbon nanotube double quantum dot.", "By periodic modulation of the potentials of the two quantum dots we move the system around charge triple points and transport exactly one electron or hole per cycle.", "We investigate the pumping as a function of the modulation frequency and amplitude and observe good current quantization up to frequencies of 18 MHz where rectification effects cause the mechanism to break down." ], [ "Quantized Charge Pumping through a Carbon Nanotube Double Quantum Dot S.J.", "Chorley J. Frake C.G.", "Smith G.A.C.", "Jones M.R.", "[email protected] Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom We demonstrate single-electron pumping in a gate-defined carbon nanotube double quantum dot.", "By periodic modulation of the potentials of the two quantum dots we move the system around charge triple points and transport exactly one electron or hole per cycle.", "We investigate the pumping as a function of the modulation frequency and amplitude and observe good current quantization up to frequencies of 18 MHz where rectification effects cause the mechanism to break down.", "73.63.Fg, 73.63.Kv, 73.23.Hk Fast and accurate control of the charge state of quantum dots is important in research areas varying from quantum information processing to quantum metrology.", "Quantum computation schemes based on charge or spin qubits defined in quantum dots [1], for example, require manipulation of the charge state well within the coherence times of the qubits which, in practise, implies nanosecond control.", "In quantum metrology, efforts towards the development of a current standard require the transfer of single electrons at nanosecond timescales with a precision of 0.1 part per million or better [2].", "The ability of fast control of the charge states of carbon nanotube quantum dots is of particular interest in these respects.", "In quantum information processing, carbon nanotube quantum dots are attractive as electron spin coherence times are expected to be long and because spin-orbit interaction [3], [4], [5] allows for electrical or even optical [6] control of the spin states.", "Carbon nanotube quantum dots are also promising as single electron pumps in quantum metrology for use in a current standard.", "The ultimate precision of single-electron pumps is widely regarded to be dependent on the strength of electron-electron interactions, suppressing errors due to co-tunneling events [7].", "These interactions are exceptionally strong in carbon nanotubes.", "In addition, when combined with electron-hole recombination, the ability to transfer single electrons in semiconducting nanotubes at well defined intervals has potential in quantum optics as an electrically-driven on-demand single-photon source in the infrared frequency range [8], [9].", "Here we demonstrate single-electron pumping in a gate-defined carbon nanotube double quantum dot by periodic modulation of the potentials of the two quantum dots around charge triple points in the stability diagram.", "We investigate the pumping as a function of the modulation frequency and amplitude and show quantized charge pumping up to frequencies of 18 MHz, corresponding to a period of $\\sim 55$ nanoseconds.", "The device we consider is a single-walled carbon nanotube grown by chemical vapour deposition on degenerately doped Si terminated by 300 nm SiO$_2$ , see Fig. 1(a).", "To define a double quantum dot, the nanotube is contacted by evaporated source and drain electrodes of Ti/Au which form the outer tunnel barriers of the quantum dots.", "A capacitively coupled top gate, separated from the nanotube by $\\sim 3$ nm of AlO$_x$ is used to control the tunnel coupling between the dots.", "The outer two plunger gates, set back from the nanotube, control the electron number on each dot.", "Figure: (color online) (a) Scanning electronmicrograph of a typical carbon nanotube (CNT) device with simplifiedelectrical setup.", "(b) Charge stability diagram of the carbonnanotube double quantum dot for V sd =0.5V_{sd} = 0.5 mV measured at T=40T=40 mK.The ordered pairs (n,m)(n,m) represent the excess electrons on the two dots.", "The circle shows a typicalpumping trajectory around a triple point.The device was bonded onto microwave printed circuit board and mounted on the tail of a $40\\,$ mK dilution refrigerator.", "The two plunger gates are connected to semirigid coaxial cables via bias tees at the mixing chamber, allowing dc and rf signals to be applied.", "The source is connected to a resonant circuit with dc connection to allow simultaneous transport and rf reflectometry measurements [10].", "High frequency synthesized waveforms were applied to the gates using a Tektronix AWG5014 arbitrary waveform generator and the current through the nanotube device is measured by a Keithley 6514 electrometer.", "Figure: (color online) Pumped current stabilitydiagram as a function of gate oscillation amplitude.", "For theleftmost measurement, the amplitude of the oscillation voltages onV L V_L and V R V_R are 25 and 87.5 mV, respectively .The applied potentials are then increased in steps as indicated,keeping the ratio fixed.", "The applied bias V sd =0V_{sd} = 0.", "The linetraces show the current for the paths through the electron and holepumping cycles, as indicated in the top panels by the dashed lines.The current is quantized for both polarities.At low temperatures, quantum dots form between each contact and the central barrier.", "For appropriate settings of the Si back gate voltage $V_{BG}=1$ V and top gate voltage $V_T = -0.05$ V, the charge stability diagram displays the characteristic honeycomb pattern of a double quantum dot [11], [12], see Fig. 1(b).", "The electron occupation number of the quantum dots is indicated by the ordered pairs $(n,m)$ .", "A finite conductance is observed at the triple points where three different charge states are degenerate.", "Weak co-tunneling is observed for some charge transitions of the left quantum dot but mostly suppressed otherwise.", "The charging energies of the two quantum dots can be obtained from the charge stability diagrams and yield $U_L \\approx 6$ and $U_R \\approx 5$ meV for the left and right quantum dot, respectively.", "The interdot charging energy $U^{\\prime } \\approx 1.3$ meV.", "Figure: (color online) (a) Pumping trajectories around the triplepoints in the double dot stability diagram.", "The color indicates the polarity of the current which is oppositefor the two triple points.", "(b) Pumping trajectories that cross the triple points.", "The color indicates the polarity of the current due to rectification.Following a technique first demonstrated for metallic single-electron tunneling devices [13], the occupation number $(n,m)$ of the quantum dots is varied periodically by applying a small sinusoidal voltage to each plunger gate with a 90$^\\circ $ phase shift between them such that the gates trace out approximately circular paths on the stability diagram [14].", "The centers of the circular paths are then varied by addition of a dc signal offset while the source-drain bias $V_{sd}$ is kept at zero.", "Figure 2 shows the resulting current around a representative triple point pair of our nanotube double quantum dot for a modulation frequency $f=6$ MHz and a sequence of increasing amplitude.", "The line traces show the current along the dashed lines in the top panels for each measurement.", "For the lowest pumping amplitude, current peaks of opposite polarity are observed at the two triple points and the current is zero elsewhere.", "As the amplitude of the signal is increased, clear current plateaus of about 0.96 pA are observed.", "The plateaus are flat within our measurement accuracy of $\\Delta I \\sim 50$ fA.", "These observations can be understood considering the pumping sequence illustrated in Fig. 3(a).", "When a path on the stability diagram encircles a triple point, exactly one electron or hole moves between the electrodes per cycle and the current is expected to be quantized as $I=ef$ , where $e$ is the electron charge [13], [15].", "For the leftmost trajectory in Fig.", "3(a), for example, electrons are moved in the sequence $(n,m)\\rightarrow (n+1,m)\\rightarrow (n,m+1)\\rightarrow (n,m)$ .", "When the same path is encircling the other triple point of a pair, the polarity of the current is reversed.", "For the modulation frequency $f=6$ MHz used in Fig.", "2, this corresponds to the observed currents of 0.96 pA.", "The area in the stability diagram in which a quantized current is observed increases with pumping amplitude up to a maximum set by $U^{\\prime }$ .", "As expected, the magnitude of the current observed on the plateaus in Fig.", "2 does not increase with pumping amplitude and depends on the modulation frequency only.", "With increasing pumping amplitudes (rightmost plots), there are trajectories that include both triple points such that no charge is transferred between the dots but only between the dots and leads, i.e., following a sequence $(n,m)\\rightarrow (n+1,m) \\rightarrow (n+1,m+1) \\rightarrow (n,m+1) \\rightarrow (n,m)$ .", "This results in an area of zero current in between the plateaus.", "In the rightmost plots of Fig.", "2, another effect becomes visible where non-quantized currents are observed above and below the quantized regions.", "These positions in the stability diagram correspond to those trajectories that cross the triple points where the double dot has a finite conductance and a non-quantized current flows when $V_{sd} \\ne 0$ .", "The effect can therefore be understood as rectification where modulation of the source and drain potentials by the plunger gates - to which they are capacitively coupled - induces a current.", "The polarity of the rectified currents depends on the direction of the trajectories in the stability diagram near the triple points [16], as indicated in Fig. 3(b).", "Rectification effects become stronger for larger amplitudes and frequencies preventing us from observing a clean quantized current above $\\sim 18$ MHz.", "These observations are further illustrated by Fig.", "4 which shows the current at the triple points for increasing frequencies.", "The current follows the expected relation $I=ef$ , indicated by the dashed lines, upto 18 MHz where it starts to deviate [17] as rectification effects become dominant.", "We note that 18 MHz is a much smaller frequency than the intrinsic dynamic limit of the device which is given by its $RC$ time constant [7], [18] and which we estimate to be of order 1-10 GHz [19].", "This suggests that current quantization could be observed for much larger modulation frequencies if rectification can be sufficiently suppressed.", "Indeed, in previous work in which potential modulation of nanotube quantum dots was achieved by surface-acoustic-waves (SAW) - avoiding direct coupling between the rf source and nanotube contacts - quantized currents in the GHz range were observed [20], [21].", "We believe rectification can be reduced in the present double dot devices by a different choice of substrate, e.g., undoped Si or quartz, and stronger coupling between the quantum dots and plunger gates.", "Figure: (color online) Pumped current as a functionof frequency for the electron (blue) and hole cycle (red) for clockwise trajectoriesin the stability diagram.", "Theamplitude of the gate modulation is as used in the middle panel ofFig. 2.", "The dashed lines correspond to I=±efI=\\pm ef.In conclusion, we demonstrate charge pumping in a carbon nanotube double quantum by periodic modulation of the dot potentials.", "We investigate the pumping as a function of the modulation amplitude and frequency and observe quantized currents up to frequencies of $\\sim 18$ MHz.", "Above this frequency, rectification prevents us from observing the quantization cleanly.", "We believe that for an optimized device geometry accurately quantized currents in the GHz range are experimentally feasible.", "We thank David Cobden and Jiang Wei for the carbon nanotube growth and Karl Petersson, Victoria Russell, and Mamta Thangaraj for technical assistance.", "This work was supported by EPSRC, the Newton Trust and the Royal Society (M.R.B.", ")." ] ]

[ [ "Evidence for 9 planets in the HD 10180 system" ], [ "Abstract We re-analyse the HARPS radial velocities of HD 10180 and calculate the probabilities of models with differing numbers of periodic signals in the data.", "We test the significance of the seven signals, corresponding to seven exoplanets orbiting the star, in the Bayesian framework and perform comparisons of models with up to nine periodicities.", "We use posterior samplings and Bayesian model probabilities in our analyses together with suitable prior probability densities and prior model probabilities to extract all the significant signals from the data and to receive reliable uncertainties for the orbital parameters of the six, possibly seven, known exoplanets in the system.", "According to our results, there is evidence for up to nine planets orbiting HD 10180, which would make this this star a record holder in having more planets in its orbits than there are in the Solar system.", "We revise the uncertainties of the previously reported six planets in the system, verify the existence of the seventh signal, and announce the detection of two additional statistically significant signals in the data.", "If of planetary origin, these two additional signals would correspond to planets with minimum masses of 5.1$^{+3.1}_{-3.2}$ and 1.9$^{+1.6}_{-1.8}$ M$_{\\oplus}$ on orbits with 67.55$^{+0.68}_{-0.88}$ and 9.655$^{+0.022}_{-0.072}$ days periods (denoted using the 99% credibility intervals), respectively." ], [ "Introduction", "Over the recent years, radial velocity surveys or nearby stars have provided detections of several exoplanet systems with multiple low-mass planets, even few Earth-masses, in their orbits [19], [21], [22].", "These systems include a four planet system around the M-dwarf GJ 581 [21], which has been proposed to possibly have a habitable planet in its orbit [30], [32], a system of likely as many as seven planets orbiting HD 10180 [19], and several systems with 3-4 low mass planets, e.g.", "HD 20792 with minimum planetary masses of 2.7, 2.4, and 4.8 M$_{\\oplus }$ [24] and HD 69830 with three Neptune mass planets in its orbit [18].", "Currently, one of the most accurate spectrographs used in these surveys, is the High Accuracy Radial Velocity Planet Searcher (HARPS) mounted on the ESO 3.6m telescope at La Silla, Chile [20].", "In this article, we re-analyse the HARPS radial velocities of HD 10180 published in [19].", "These measurements were reported to contain 6 strong signatures of low-mass exoplanets in orbits ranging from 5 days to roughly 2000 days and a possible seventh signal at 1.18 days.", "These planets include five 12 to 25 $M_{\\oplus }$ planets classified in the category of Neptune-like planets, a more massive outer planet with a minimum mass of 65 M$_{\\oplus }$ , and a likely super-Earth with a minimum mass of 1.35 $M_{\\oplus }$ orbiting the star in close proximity [19].", "While the confidence in the existence of the six more massive companions in this system is rather high, it is less so for the innermost super-Earth [4].", "Yet, even if the radial velocity signal corresponding to this low-mass companion was an artefact caused by noise and data sampling or periodic phenomena of the stellar surface, the HD 10180 system would be second only to the Solar system with respect to the number of planets in its orbits, together with the transiting Kepler-11 6-planet system [15].", "In this article, we re-analyse the radial velocity data of HD 10180 using posterior samplings and model probabilities.", "We perform these analyses to verify the results of [19] with another data analysis method, to calculate accurate uncertainty estimates for the planetary parameters, and to see if this data set contains additional statistically significant periodic signals that could be interpreted as being of planetary origin." ], [ "Observations of HD 10180 planetary system", "The G1 V star HD 10180 is a relatively nearby and bright target for radial velocity surveys with a Hipparcos parallax of 25.39$\\pm $ 0.62 mas and $V=7.33$ [19].", "It is a very inactive ($\\log R_{HK} = -5.00$ ) Solar-type star with similar mass and metallicity ($m_{\\star } = 1.06 \\pm 0.05$ , [Fe/H]$=0.08\\pm 0.01$ ) and does not appear to show any well-defined activity cycles based on the HARPS observations [19].", "When announcing the discovery of the planetary system around HD 10180, [19] estimated the excess variations in the HARPS radial velocities, usually referred to as stellar jitter, to be very low, approximately 1.0 ms$^{-1}$ .", "These properties make this star a suitable target for radial velocity surveys and enable the detection of very low-mass planets in its orbit.", "[19] announced in 2010 that HD 10180 is host to six Neptune-mass planets in its orbit with orbital periods of 5.76, 16.36, 49.7, 123, 601, and 2200 days, respectively.", "In addition, they reported a 1.18 days power in the periodogram of the HARPS radial velocities that, if caused by a planet orbiting the star, would correspond to a minimum mass of only 35% more than that of the Earth.", "These claims were based on 190 HARPS measurements of the variations in the stellar radial velocity between November 2003 and June 2009.", "The HARPS radial velocities have a baseline of more than 2400 days, which enabled [19] to constrain the orbital parameters of the outer companion in the system with almost similar orbital period.", "In addition, these HARPS velocities have an estimated average instrument uncertainty of 0.57 ms$^{-1}$ and a relatively good phase coverage with only seven gaps of more than 100 days, corresponding to the annual visibility cycle of the star in Chile." ], [ "Statistical analyses", "We analyse the HD 10180 radial velocities using a simple model that contains $k$ Keplerian signals assumed to be caused by non-interacting planets orbiting the star.", "We also assume that any post-Newtonian effects are negligible in the timescale of the observations.", "Our statistical models are then those described in e.g.", "[26] and [27], where each radial velocity measurement was assumed to be caused by the Keplerian signals, some unknown reference velocity about the data mean, and two Gaussian random variables with zero means representing the instrument noise with a known variance as reported for the HD 10180 data by [19] together with the radial velocities, and an additional random variable with unknown variance that we treat as a free parameter of our model.", "This additional random variable describes the unknown excess noise in the data caused by the instrument and the telescope, atmospheric effects, and the stellar surface phenomena.", "Clearly, the assumption that measurement noise has a Gaussian distribution might be limiting in case it was actually more centrally concentrated, had longer tails, was skewed, or was dependent on time and other possible variables, such as stellar activity levels.", "However, with “only“ 190 radial velocities it is unlikely that we could spot non-Gaussian features in the data reliably.", "For this reason, and because as far as we know the Gaussian one is the only noise model used when analysing radial velocity data, we restrict our statistical models to Gaussian ones." ], [ "Posterior samplings", "We analyse the radial velocities of HD 10180 using the adaptive Metropolis posterior sampling algorithm of [11] because it appears to be efficient in drawing a statistically representative sample from the parameter posterior density in practice [27], [29].", "This algorithm is essentialy an adaptive version of the famous Metropolis-Hastings algorithm [23], [12], that adapts the proposal density to the information gathered up to the $i$ th member of the chain when proposing the $i+1$ th member.", "It uses a Gaussian multivariate proposal density for the parameter vector $\\theta $ and updates its covariance matrix $C_{i+1}$ using $C_{i+1} = \\frac{i-1}{i}C_{i} + \\frac{s}{i} \\bigg [ i \\bar{\\theta }_{i-1}\\bar{\\theta }_{i-1}^{T} - (i+1) \\bar{\\theta }_{i}\\bar{\\theta }_{i}^{T} + \\theta _{i}\\theta _{i}^{T} + \\epsilon I \\bigg ] ,$ where $\\bar{\\theta }$ is the mean of the parameter vector, $( \\cdot )^{T}$ is used to denote the transpose, $I$ is identity matrix of suitable dimension, $\\epsilon $ is some very small number that enables the correct ergodicity properties of the resulting chain [11], and parameter $s$ is a scaling parameter that can be chosen as 2.4$^{2} K^{-1}$ , where $K$ is the dimension of $\\theta $ , to optimise the mixing properties of the chain [6].", "We calculate the marginal integrals needed in model selection using the samples from posterior probability densities with the one block Metropolis-Hastings (OBMH) method of [2], also discussed in [3].", "However, since the adaptive Metropolis algorithm is not exactly a Markovian process, only asymptotically so [11], the method of [2] does not necessarily yield reliable results.", "Therefore, after a suitable burn-in period used to find the global maximum of the posterior density, during which the proposal density also converges to a multivariate Gaussian that approximates the posterior, we fix the covariance matrix to its present value, and continue the sampling with the Metropolis-Hastings algorithm, that enables the applicability of the OBMH method." ], [ "Prior probability densities", "The prior probability densities of Keplerian models describing radial velocity data have received little attention in the literature.", "[5] proposed choosing the Jeffreys' prior for the period ($P$ ) of the planetary orbit, the radial velocity amplitude ($K$ ), and the amplitude of “jitter“, i.e.", "the excess noise in the measurements ($\\sigma _{J}$ ).", "This choice was justified because they make the logarithms of these parameters evenly distributed [5], [8].", "We use this functional form of prior densities for the orbital period and choose the cutoff periods such that they correspond to the 1 day period, below which we do not expect to find any signals in this work, and a period of 10$T_{obs}$ , where $T_{obs}$ is the time span of the observations.", "We choose this upper limit because it enables the detection of linear trends in the data corresponding to long-period companions whose orbital period cannot be constrained, but is not much greater than necessary in practice [28], which could slow down the posterior samplings by increasing the hypervolume of the parameter space with reasonably high likelihood values.", "Anyhow, if the period of the outermost companion cannot be constrained from above, it would violate our detection criterion of the previous subsection.", "Unlike in [5], we do not use the Jeffreys' prior for the radial velocity amplitudes nor the excess noise parameter.", "Instead, because the HARPS data of HD 10180 deviate about their mean less than 20 ms$^{-1}$ , we use uniform priors for these parameters as $\\pi (K_{i}) = \\pi (\\sigma _{J}) = U(0, a_{RV})$ , for all $i$ , and use a similar uniform prior for the reference velocity ($\\gamma $ ) about the mean as $\\pi (\\gamma ) = U(-a_{RV}, a_{RV})$ , where we choose the parameter of these priors as $a_{RV} = 20$ ms$^{-1}$ .", "While the radial velocity amplitudes could in principle have values greater than 20 ms$^{-1}$ while the data would still not deviate more than that about the mean, we do not consider that possibility a feasible one.", "Following [5], we choose uniform priors for the two angular parameters in the Keplerian model, the longitude of pericentre ($\\omega $ ) and the mean anomaly ($M_{0}$ ).", "However, we do not set the prior of orbital eccentricity ($e$ ) equal to a uniform one between 0 and 1.", "Instead, we expect high eccentricities to be less likely in this case because there are at least six, likely as many as seven, known planets ofbiting HD 10180.", "Therefore, high eccentricities would result in instability and therefore we do not consider their prior probabilities to be equal to the low eccentricity orbits.", "Our choice is then a Gaussian prior for the eccentricity, defined as $\\pi (e_{i}) \\propto \\mathcal {N}(0, \\sigma _{e}^{2})$ (with the corresponding scaling in the unit interval), where the parameter of this prior model is set as $\\sigma _{e} = 0.3$ .", "This choice penalises the high eccentricity orbits in practice but still enables them if the data insists so.", "In practice, with respect to the weight this prior puts on zero eccentricity, it gives the eccentricities of 0.2, 0.4, and 0.6 relative weights of 0.80, 0.41, and 0.14, indicating that this prior can only have a relatively minor effect on the posterior densities.", "Finally, we required that the planetary systems corresponding to out Keplerian solution to the data did not have orbital crossings between any of the companions.", "We used this condition as additional prior information by estimating that the likelihood of having any two planets in orbits that cross oneanother is zero.", "We could have used a more restrictive criteria, such as the requirement that the planets do not enter each others Hill spheres at any given time, but decided to keep the situation simpler because we wanted to see whether the orbital periods of the proposed companions get constrained by data instead of stability criteria, as described in the next subsection.", "This choice of restricting the solutions in such a way that the corresponding planetary system does not suffer from destabilising orbital crossings also helps reducing the computational requirements by making the posterior samplings simpler.", "Having found $k$ Keplerian signals in the data, we simply search for additional signals between them by limiting the period space of the additional signals between these $k$ periods.", "We set the initial periods of the $k$ planets close to the solution of the $k$ -Keplerian model and perform $k+1$ samplings where each begins with the period of the $k+1$ th signal in different ”gaps“ around the previously found $k$ signals, i.e.", "corresponding to planet inside the orbit of the innermost one, between the two innermost ones, and so forth.", "If a significant $k+1$ th signal is not found in one of these ”gaps“, the corresponding solution can simply be neglected.", "However, if there are signals in two or more gaps, it is straightforward to determine the most significant one because they can be treated as different models containing the same exact number of parameters.", "We then choose the most significant periodicity as the $k+1$ th one and continue testing whether there are additional signals in the data.", "In this way, the problem of being able to rearrange the signals in any order, that would cause the posterior density to be actually highly multimodal [4], actually disappears because in a given solution the orbital crossings are forbidden and the ordering of the companions remains fixed." ], [ "Detection threshold", "While the Bayesian model probabilities can be used reliably in assessing the relative posterior probabilities of models with differing numbers of Keplerian signals [5], [10], [17], [26], [27], we introduce additional criteria to make sure that the signals we detect can be interpreted as being of planetary origin and not arising from unmodelled features in the measurement noise or as spurious signals caused by measurement sampling.", "Our basic criterion is that the posterior probability of a model with $k+1$ Keplerian signals has to exceed 150 times that of a model with only $k$ Keplerian signals to claim there are $k+1$ planets orbiting the target star.", "We choose this threshold probability based on the considerations of [14].", "We require that the signals we detect in the measurements have radial velocity amplitudes, $K_{i}$ for all $i$ , statistically distinguishable from zero.", "In practice this means, that not only the maximum a posteriori estimate is clearly greater than zero, but that the corresponding Bayesian $\\delta $ credibility sets, as defined in e.g.", "[26], do not allow the amplitude to be negligible with $\\delta = 0.99$ , i.e.", "with a probability of 99%.", "The second criterion is that the periods of all signals ($P_{i}$ ) are well defined by the posterior samples in the sense, that they can be constrained from above and below and do not get constrained purely by the condition that orbital crossings corresponding to the planetary orbits are not allowed.", "To further increase the confidence of our solutions, we do not set the prior probabilities of the different models equal in our analyses.", "We suspect a priori, that detecting $k+1$ planets would be less likely than detecting $k$ planets in any given system.", "In other words, we estimate that any set of radial velocity data would be more likely to contain $k$ Keplerian signals than $k+1$ .", "Therefore, we set the a priori probabilities of models with $k$ and $k+1$ Keplerian signals such that $P(k) = 2P(k+1)$ for all $k$ , i.e.", "we penalise the model with one additional planet by a factor of two.", "Because of this subjective choice, if the model with $k+1$ Keplerian signals receives a posterior probability that exceeds our detection threshold of being 150 times greater than that of the corresponding model with $k$ Keplerian signals, we are likely underestimating the confidence level of the model with $k+1$ Keplerian signals relative to a uniform prior.", "Physically, this choice of prior probabilities for different models corresponds to the fact that the more planets there are orbiting a star, the less stable orbits there will be left.", "Therefore, we estimate that if $k$ planets are being detected, there is naturally ”less room“ for an additional $k+1$ th companion.", "However, this might be true statistically, not in an individual case, which leaves room for discussion.", "Yet, this and the benefit that we underestimate the significance of any detected signals encourages us to use this prior." ], [ "Frequentist and Bayesian detection thresholds", "In addition to the Bayesian analyses described in the previous subsections, we analyse the residuals of each model using the Lomb-Scargle periodograms [16], [25].", "As in [19], we plot the 10%, 1%, and 0.1% false alarm probabilities (FAP's) to the periodograms to see the significance of the powers they contain.", "However, because [19] calculated the FAP's in a frequentist manner by performing random permutations to the residuals while keeping the exact epochs of the data fixed and by seeing how often this random permutation produces the observed powers, we first put this periodogram approach to its philosophical context.", "Generating $N$ random permutations of the residuals for each model aims at simulating a situation, where it would be possible to receive $N$ independent sets of measurements from the system of interest and seeing how often the measurement noise generates the signals corresponding to the highest powers in the periodogram.", "While they would not have been independent even in the case this process tries to simulate, because the exact epochs are fixed making the measurements actually dependent through the dimension of time (the measurement is actually a vector of two numbers, radial velocity and time), this approach suffers from another more significant flaw.", "The uncertainties of the signals removed from the data cannot be taken into account, which means that the method assumes the removed signals were known correctly.", "Obviously this is not the case even with the strongest signals, and even less so for the weaker ones, making the process prone to biases.", "Therefore, while likely producing reliable results when the signals are clear and their periods get accurate constraints, this method cannot be expected to provide reliable results in the case of extremely weak signals with large (and unknown) uncertainties.", "As noted by [19], when testing the significance of the 600 days signal, they could not take into account the uncertainties of the parameters of each Keplerian signal, that of the reference velocity, nor the uncertainty in estimating the excess noise in the data correctly.", "The above ”frequentist“ way of performing the analyses and intepreting the consequent results is different from the Bayesian one.", "Because we only received one set of data, we have to base all our results on that and not some hypothetical data that would have corresponded to repetition of the original measurements.", "The Bayesian philosophy is to inference all the information from the data by combining it with our prior beliefs on what might be producing it.", "For instance, as described above when discussing our choice of priors, we can expect tightly packed multiplanet systems to be more likely to contain planets on close-circular orbits than on very eccentric ones.", "Also, with the powerful posterior sampling algorithms available, it is possible to take the uncertainties in every parameter into account simultaneously, which enables the detection of weak signals in the data [7], [8], [9], [26] and prevents the detection of false positives, as happened in the case of Gliese 581 [31], [10], [27].", "Yet, despite the above problems in the traditional periodogram analyses, we take advantage of the power spectra of the residuals in our posterior samplings.", "The highest peaks in the periodogram can be used very efficiently together with Bayesian methods by using the corresponding periodicities as initial states of the Markov chains in the adaptive Metropolis algorithm.", "Because of this choice, the initial parameter vector of the Markov chain starts very close to the likely maximum a posteriori (MAP) solution, which makes its convergence to the posterior density reasonably rapid and helps reducing computational requirements.", "When drawing a sample from the parameter posterior density and using it to calculate the corresponding model probabilities, it became crucial that this sample was a statistically representative one.", "While posterior samplings generally provide a global solutions, it is always possible that the chain converges to a local maximum and stays in its vicinity within a sample of finite size.", "To make sure that we indeed received the global solution, we calculated several Markov chains starting from the vicinity of the apparent MAP solution and compared them to see that they indeed corresponded to the same posterior probability density.", "In practice, sampling the parameter spaces was computationally demanding because the probability that the parameter vectors drawn from the Gaussian proposal density are close to the posterior maximum decreases rapidly when the number of parameters with non-Gaussian probability densities increases.", "Therefore, while models with 0-6 planets were reasonably easy to sample and we received acceptance rates of 0.1-0.3, these rates decreased when increasing the number of signals in the model further.", "As a result, for models with 8-9 Keplerian signals, the acceptance rates decreased below 0.1 and forced us to increase the chain lenghts by two orders of magnitude from a typical $1.0 \\times 10^{6}$ to as high as $1.0 \\times 10^{8}$ .", "In the following subsections, the results are based on several samplings that yielded the same posterior densities, and also consistent model probabilities." ], [ "The number of significant periodicities", "The posterior probabilities of the different models provide information on the number of Keplerian signals ($k$ ) in the data.", "[19] were confident that there are six planets orbiting HD 10180 based on their periodogram-based analyses of model residuals and the corresponding random permutations of them when calculating the significance levels of the periodogram powers.", "They also concluded that the six planets in the system with increasing periods of 5.75962$\\pm $ 0.00028, 16.3567$\\pm $ 0.0043, 49.747$\\pm $ 0.024, 122.72$\\pm $ 0.20, 602$\\pm $ 11, and 2248$^{+102}_{-106}$ days comprise a stable system given that the masses of the planets are within a factor of $\\sim $ 3 from the minimum masses of 13.70$\\pm $ 0.63, 11.94$\\pm $ 0.75, 25.4$\\pm $ 1.4, 23.6$\\pm $ 1.7, 21.4$\\pm $ 3.0, and 65.3$\\pm $ 4.6 M$_{\\oplus }$ , respectively.", "Because [19] pointed out that there are actually two peaks in the periodogram corresponding to the seventh signal, i.e.", "1.18 and 6.51 days periodicities that are the one day aliases of each other, but noted that the 6.51 signal, if corresponding to a planet, would cause the system to be unstable in short timescales, we adopted the 1.18 periodicity as the seventh signal in the data.", "The relative probabilities of the models with $k = 0, ..., 9$ are shown in Table REF together with the period ($P_{s}$ ) of the next Keplerian signal added to the model.", "Table: The relative posterior probabilities of models with k=0,...,9k=0, ..., 9 Keplerian signals (ℳ k \\mathcal {M}_{k}) given radial velocities of HD 10180 (or data, dd) together with the periods (P s P_{s}) of the signals added to the model when increasing the number of signals in the model by one.", "Also shown are the logarithmic Bayesian evidences (P(d|ℳ k )P(d | \\mathcal {M}_{k})) and their uncertainties as standard deviations and the root mean square (RMS) values of the residuals for each model.According to the model probabilities in Table REF , the 8- and 9-Keplerian models are the most probable descriptions of the processes producing the data out of those considered.", "Improving the statistical model by adding the seventh signal, with a period of 1.18 days, increases the model probability by a factor of more than $10^{6}$ , which makes the credibility of this seventh signal high.", "Adding two more signals corresponding to 67.6 and 9.66 days periodicities increases the model probabilities even further.", "As a result, the 9-Keplerian model receives the greatest posterior probability of slightly more than 150 times more than the next best model, the 8-Keplerian model.", "This enables us to conclude, that there is strong evidence in favour of the 67.6 and 9.66 days periodicities not being produced by random processes, i.e.", "measurement noise." ], [ "Periodograms of residuals", "We subtracted the models with 6 to 9 periodic signals from the data and calculated the Lomb-Scargle periodograms [16], [25] of these residuals (Fig.", "REF ) together with the standard analytica FAP's.", "As already seen in [19], the two strong powers corresponding to 1.18 days periodicity and its 1 day alias at 6.51 days are strong in the residuals of the 6-Keplerian model (top panel in Fig.", "REF ) and exceed the 1% FAP.", "These peaks get also removed from the residuals of the 7-Keplerian model (2nd panel).", "However, it can be seen that 9.66 and 67.6 days periods have strong powers in these residuals, yet, neither of them exceeds even the 10% FAP level.", "Modelling these periodicities as Keplerian signals and plotting the periodograms of the corresponding residuals of the 9-Keplerian model (bottom panel) shows that there are no strong powers left in the residuals.", "While this does not indicate that these two periodicities are significant, it shows that they are clearly present in the periodograms of the residuals and support the findings in the previous subsection based on the model probabilities.", "Figure: The Lomb-Scargle periodograms of the HD 10180 radial velocities for the residuals of the models with 6 (top) to 9 (bottom) periodic signals.", "The dotted, dashed, and dot-dashed lines indicate the 10%, 1%, and 0.1% FAPs, respectively.We note that there appear to be two almost equally strong peaks in the 8-Keplerian model residuals (Fig.", "REF , panel 3).", "However, these powers corresponding to periods of 9.66 and 1.11 days are one day aliases of oneanother.", "This aliasing is clear as the 1.11 days power is absent in the periodogram of the 9-Keplerian model (Fig.", "REF , bottom panel)." ], [ "Planetary interpretation: orbital parameters", "Because of the samplings of parameter posterior densities of each statistical model, we were able to calculate the estimated shapes of the parameter distributions for each model and use these to estimate the features in the corresponding densities.", "We describe these densities using three numbers, the MAP estimates and the corresponding 99% Bayesian credibility sets (BCS's), as defined in e.g.", "[26].", "The simple MAP point estimates and the corresponding 99% BCS's do not represent these dentities very well because some of the parameters are highly skewed and have tails on one or both sides.", "However, we list the parameters interpreted as being of planetary origin in this subsection.", "When calculating the semimajor axes and minimum masses of the planets, we took the uncertainty in the stellar mass into account by treating it as an independent random variable.", "We assumed that this random variable had a Gaussian density with mean equal to the estimate given by [19] of 1.06 M$_{\\odot }$ and a standard deviation of 5% of this estimate.", "As a consequence, the densities of these parameters are broader than they would be if using a fixed value for the stellar mass, which indicates greater uncertainty in their values." ], [ "The 6 planet solution", "The parameter estimates of our 6-Keplerian model are listed in Table REF[19] use letter b for the 1.18 days periodicity not present in the 6-Keplerian model.", "Therefore, we use letters c - h in Table 2 to have the letters denote the same signals.", "Yet, the solution of [4] denotes the 5.76 days signal with letter b..", "This solution is consistent with the solution reported by [19] but the uncertainties are slightly greater likely because we took into account the uncertainty in the jitter parameter $\\sigma _{j}$ and because [19] used more conservative uncertainty estimates from the covariance matrix of the parameters that does not take the nonlinear correlations between the parameters into account.", "The greatest difference is therefore in the uncertainty of the orbital period of the outermost companion, whose probability density has a long tail towards longer periods and periods as high as 2670 days cannot be ruled out with 99% confidence (the supremum of the 99% BCS).", "Table: The 6-planet solution of HD 10180 radial velocities.", "MAP estimates of the parameters and their 99% BCS's." ], [ "The 9 planet solution", "Assuming that all the periodic signals in the data are indeed caused by planetary companions orbiting the star, the parameters of out 9-Keplerian solution are listed in Table REF and the phase folded orbits of the 9 Keplerian signals are plotted in Fig.", "REF .", "Table: The 9-planet solution of HD 10180 radial velocities.", "MAP estimates of the parameters and their 99% BCS's.Figure: The phase-folded Keplerian signals of the 9-planet solution with the other 8 signals removed.Because the simple estimates in Table REF can be very misleading in practice, especially in the case of nonlinear correlations between the parameters, we also plotted the projected distributions of some of the parameters in the Appendix.", "The distributions of $P_{i}$ , $K_{i}$ , and $e_{i}$ , for each planet $i = 1, ..., 9$ (with indice 1 (9) referring to the shortest (longest) period), are shown in Fig.", "REF and show that the periods of all Keplerian signals are indeed well constrained, the radial velocity amplitudes differ significantly from zero, and the orbital eccentricities peak at or close to zero indicating likely circular orbits.", "We also plotted the parameter describing the magnitude of the excess noise, or jitter, in Fig REF to demonstrate that the MAP estimate of this parameter of 1.15 ms$^{-1}$ , with a BCS of [0.92, 1.42] ms$^{-1}$ , is consistent with the estimate of [19] of 1.0 ms$^{-1}$ , while this is not the case for the 6-Keplerian model for which the $\\sigma _{j}$ receives a MAP estimate of 1.40 ms$^{-1}$ with a BCS of [1.18, 1.70] ms$^{-1}$ (Table REF ).", "The periods of all companions get constrained well but that of the 9.66 days signal remains bimodal with two peaks at 9.66 and 9.59 days (Fig.", "REF ).", "The 9.59 days peak was found to have lower probability based on our posterior samplings because the Markov chain had good mixing properties in the sense that it visited both maxima several times during all the samplings, the posterior density had always a global maximum at 9.66 days." ], [ "Dynamically allowed orbits", "We perform tests of dynamical stability within the context of Lagrange stability to see whether the two additional signals in the HD 10180 radial velocities could correspond to low-mass planets orbiting the star.", "We use the analytical approximated Lagrange stability criterion of [1] to test the stability of each subsequent pairs of planets in the system.", "While this analytical criterion is only a rough approximation and only applicable for two planet systems, it can nevertheless provide useful information on the likely stability or instability of the system.", "According to [1], the orbits of two planets (denoted using subindices 1 and 2, respectively) satisfy approximately the Lagrange stability criterion if $\\alpha ^{-3} \\bigg ( \\mu _{1} - \\frac{\\mu _{2}}{\\delta ^{2}} \\bigg ) \\big ( \\mu _{1} \\gamma _{1} + \\mu _{2} \\gamma _{2} \\delta \\big )^{2} > 1 + \\mu _{1}\\mu _{2} \\bigg ( \\frac{3}{\\alpha } \\bigg )^{4/3} ,$ where $\\mu _{i} = m_{i} M^{-1}$ , $\\alpha = \\mu _{1} + \\mu _{2}$ , $\\gamma _{i} = \\sqrt{1 - e^{2}_{i}}$ , $\\delta = \\sqrt{a_{2}/a_{1}}$ , $M = m_{\\star } + m_{1} + m_{2}$ , $e_{i}$ is the eccentricity, $a_{i}$ is the semimajor axis, $m_{i}$ is the planetary mass, and $m_{\\star }$ is stellar mass.", "Using the above relation, we calculate the threshold curves for the $i$ th planet with both, the next planet inside its orbit ($i-1$ th) and the next planet outside its orbit ($i+1$ th).", "We use the MAP parameter estimates for the $i$ th planet and calculate the allowed eccentricities of the $i-1$ th and $i+1$ th planets as a function of their semimajor axes by using the MAP estimates for their masses.", "As [19] found their 6 planet solution stable, we use it as a test case when calculating the Lagrange stability threshold curves.", "We use the 6 planet solution in Table REF , and plot the threshold curves together with the orbital parameters in Fig.", "REF .", "In this Fig., the shaded areas indicate the likely unstable parameter space and the red circles indicate the positions of the modelled planets in the system.", "Figure: The approximated Lagrange stability thresholds between each two planets and the MAP orbital parameters of the 6 planet solution (Table ).It can be seen in Fig.", "REF , that the $i$ th planet has orbital parameters that keeps it inside the Lagrange stability region of the neighbouring planetary companions for all $i = 1, ..., 6$ .", "This result then agrees with the numerical integrations of [19] and, while only a rough approximation of the reality, encourages us to use the criterion in Eq.", "(REF ) for our 7-, 8-, and 9-companion solutions as well.", "Fig.", "REF also suggests, that there might be stable regions between the orbits of these 6 planets for additional low-mass companions.", "When interpreting all the signals in our 9-planet solution as being of planetary origin, the stability thresholds show some interesting features (Fig.", "REF ).", "The periodicities at 9.66 and 67.6 days would correspond to planets that satisfy the condition in Eq.", "(REF ) if the orbital eccentricities of all the companions were close to or below the MAP estimate, which, according to the probability densities in Fig.", "REF , appears to be likely based on the data alone.", "This means that the planetary origin of these periodicities cannot be ruled out by this analysis.", "Figure: The approximated Lagrange stability thresholds between each two planets and the MAP orbital parameters of the 9 planet solution (Table ).In reality, the stability constraints are necessarily more limiting than those described by the simple Eq.", "REF because of the gravitational interactions between all the planets, not only the nearby ones.", "Also, it does not take the stabilising or destabilising effects of mean motion resonances.", "However, the numerical integrations of the orbits of the seven planets performed by [19], actually do not rule the 0.09 and 0.33 AU orbits (Table REF ) out as unstable but show that there are regions of at least ”reasonable stability“ in the vicinity of these orbits given that they are close-circular and that the planetary masses in these orbits are small.", "When interpreted as being of planetary origin, the periodic signals at roughly 9.66 and 67.6 days satisfy these requirements.", "We note, that the selected prior density for the orbital eccentricities, namely $\\pi (e_{i}) \\propto \\mathcal {N}(0, \\sigma _{e}^{2})$ for all $i$ , in fact helps slightly in removing a priori unstable solutions from the parameter posterior density.", "However, this effect is not very significant in this case, because the prior requirement that the signals in the data do not correspond to planets with crossing orbits constrains the eccentricities much more strongly and the parameter posteriors we would receive with uniform eccentricity prior would therefore not differ significantly from those reported in Table REF and Figs.", "REF and REF ." ], [ "Avoiding unconstrained solutions", "To further emphasise our confidence in the 9 periodic signals in the data, we tried finding additional signals in the gaps between the 9-Keplerians, especially, between the 123 and 600 days orbits.", "This part of the period-space is interesting because any habitable planet in the system would have its orbital period in this space and because the stability thresholds allow the existence of low-mass planets in close-circular orbits in this region (see Fig.", "REF ).", "As could already be suspected based on the periodograms of residuals in Fig.", "REF , we could not find any signals between the periods of 123 and 600 days.", "The sampling of the parameter space of this 10-Keplerian model was much more difficult than that of the models with fewer signals because the orbital period of this hypothetical 10th signal was only constrained by the fact that a priori we did not allow orbital crossings (Fig.", "REF ).", "As a result, the probability density of the orbital period did not have a clear maximum but several small maxima, whose relative significance is not known because we cannot be sure whether the Markov chain converged to the posterior in this case (Fig.", "REF ).", "Therefore, the distribution of the orbital period in Fig.", "REF is only a rough estimate of what the density might look like.", "Figure: The posterior densities of the minimum mass and orbital period of the planet that could exist in the habitable zone of HD 10180 without having been detected using the current data.", "The solid curve is a Gaussian density with the same mean (μ\\mu ) and variance (σ 2 \\sigma ^{2}) as the parameter distribution has.Because different samplings yielded similar but not equal densities for the orbital period, we could not be sure whether the chain had indeed converged to the posterior or not.", "For this reason, we did not consider the corresponding posterior probability of this model trustworthy and do not show it in Table REF .", "The posterior probabilities we received were roughly 1-5% of that of the 9-Keplerian model.", "However, these samplings still provide some interesting information in the sense that we can put an upper limit to the planetary masses that could exist between the 123 and 600 days periods and still not be detected confidently by the current observations.", "According to the samplings of the parameter space, the probability of there being Keplerian signals between 123 and 600 days periods with radial velocity amplitudes in excess of 1.1 ms$^{-1}$ is less than 1%.", "Our MAP solution for this amplitude is 0.12 ms$^{-1}$ with a BCS of [0.00, 1.10] ms$^{-1}$ .", "This means that we can rule out the existence of planets more massive than approximately 12.1 M$_{\\oplus }$ in this period space because such companions could have been detected by the current data.", "Therefore, even the fact that a signal was not detected can help constraining the properties of the system, as seen in the probability density of the minimum mass of this hypothetical planet in Fig.", "REF – clearly this signal is indistinguishable from one with negligible amplitude, as the density is peaking close to zero.", "This result means that an Earth mass planet could exist in the habitable zone of HD 10180 and most likely, if there is a low-mass companion orbiting the star between orbital periods of 123 and 600 days, it has a minimum mass of less than 12.1 M$_{\\oplus }$ .", "However, we cannot say much about the possible orbit of this hypothetical companion, because all the orbits between 123 and 600 days without orbital crossings are almost equally probable (Fig.", "REF ).", "All we can say, is that orbital crossings limit the allowed periodicities and yield a 99% BCS of [128, 534] days for this orbital period, though, further dynamical constraints would narrow this interval even further, as shown in Fig.", "REF and Fig.", "12 of [19]." ], [ "Detectability of Keplerian signals", "To further emphasise the significance of the 9 signals we detect in the HD 10180 radial velocities, we generated an artificial data set to see if known signals could be extracted from it confidently given the definition of our criteria for detection threshold.", "This data set was generated by using the same 190 epochs as in the HD 10180 data of [19].", "We generated the radial velocities corresponding to these epochs by using a superposition of the 9 signals with parameters roughly as in Table REF .", "Further, we added three noise components, Gaussian noise with zero mean as described by the uncertainty estimates of each original radial velocity of [19], Gaussian noise with zero mean and $\\sigma = 1.1$ ms$^{-1}$ and uniform noise as a random number drawn from the interval [-0.2, 0.2] ms$^{-1}$ .", "The lattter two produce together the observed excess noise in the data of roughly 1.15 ms$^{-1}$ when modelled as pure Gaussian noise.", "We used this different noise model when generating the data to not commit an inverse crime, i.e.", "to not generate the data using the same model used to analyse it which would correspond to studying the properties of the model only [13].", "Analysing this artificial data yielded results confirming the trustworthiness of our methods.", "Using the detection criteria defined above, we could extract all 9 signals from the artificial data with well constrained amplitudes and periods that were consistent with the added signals in the sense that the 99% BCS's of the parameters contained the values of the added signals.", "The MAP estimates did differ from the added signals but not significantly so given the uncertainties as described by distributions corresponding to the parameter posterior density.", "This simply represents the statistical nature of the solutions based on posterior samplings.", "As an example, we show the periods and amplitudes of the three weakest signals with the lowes radial velocity amplitudes as probability distributions (Fig.", "REF ).", "This Fig.", "indicates that if the radial velocity noise is indeed dominated by Gaussian noise, the low-amplitude signals we report can be detected confidently.", "This conclusion was also supported by the corresponding model probabilities we received for the artificial data set.", "These probabilities indicated that the 9-Keplerian model had the greatest posterior probability exceedind the threshold of 150 times greater than the probability of the 8-Keplerian model.", "Figure: The distributions of the periods and amplitudes received for the three weakest signals in the artificially generated data.", "The added signals had P i =P_{i} = 1.18, 9.66, and 67.55 days, and K i =K_{i} = 0.78, 0.53, and 0.75 ms -1 ^{-1} for i=1,3,6i = 1, 3, 6, respectively." ], [ "Comparison with earlier results", "The analyses of [19] of the same radial velocities yielded differing results, i.e.", "the number of significant powers in periodogram was found to be 7 instead of the 9 significant periodic signals reported here.", "While this difference is likely due to the fact that the power spectrums are calculated by fixing the parameters of the previous signals to some point estimates when searching for additional peaks, the analyses of [4] suffer from similar sources of bias.", "The Bayesian approach of [4] was basically a search of $k+1$ th signal in the residuals of the model with $k$ Keplerian signals.", "They derived the probability densities of the residuals by assuming they were uncorrelated, which is unlikely to be the case, especially, if there are signals left in the residuals.", "Therefore, any significance test, in this case the comparison of Bayesian evidences of a null hypothesis and an alternative one with a model containing one more signal, is similarly biased by the fact that these correlations are not fully accounted for.", "While this source of bias might be relatively small, in the approach of [4], the effective number of parameters is artificially decreased by the very fact that residuals are being analysed.", "This decrease, in turn, might make any comparisons of Bayesian evidences biased.", "Modifying the uncertainties corresponding to the posterior density of the model residuals does not necessarily account for this decrease in the dimension when the weakest signals among the $k$ detected ones are at or below the residual uncertainty and their contribution to the total uncertainty of the residuals becomes negligible in the first place.", "In fact, we could replicate the results of [4] using the OBMH estimates of marginal integrals.", "Using their simple method, we received a result that the 6-Keplerian model residuals could not be found to contain one more signal.", "While the Bayesian evidence for this additional signal did not exceed that of having the 6-Keplerian model residuals consist of purely Gaussian noise, the amplitude of this additional signal in the residuals was also found indistinguishable from zero, in accordance with our criteria of not detecting a signal.", "Therefore, the results we present here do not actually conflict with those of [4].", "As an example, the log-Bayesian evidences of 6 and 7-Keplerian models (Table REF ) were -358.36 and -343.73, respectively.", "Analysing the residuals of the model with $k=6$ using 0- and 1-Keplerian models should yield the similar numbers (up to differences in the priors), if the method of [4] were trustworthy.", "Instead, we received values of roughly -338 for both models.", "This means, that the null-hypothesis is exaggerated because of the fact that the corresponding model contains only two free parameters, the jitter magnitude and the reference velocity.", "The Occamian principle cannot therefore penalise this model as much as it should, and the results are biased in favour of the null-hypothesis, which effectively prevents the detection of low-amplitude signals.", "We also tested the method of [4] in analysing the artificial test data described in the previous subsection.", "The results were almost similar: the log-Bayesian evidences given the residuals of a model with $k$ Keplerian signals were found to favour the 6-Keplerian interpretation, while the artificial data was known to contain 9 signals.", "This further emphasises the fact, that the method of [4], while capable of detecting the strongest signals in the data, cannot be considered trustworthy if it fails to make a positive detection of a low-amplitude signal.", "Yet, it is likely trustworthy if it provides a positive detection." ], [ "Conclusions and Discussion", "We have re-analysed the 190 HARPS radial velocities of HD 10180 published in [19] and report our findings in this article.", "First, we have revised the orbital parameters of the proposed 6 planetary companions to this star and calculated realistic uncertainty estimates based on samplings of the parameter space.", "We also verified the significance of the 1.18 days signal reported by [19] and interpreted as arising from a planetary companion with this orbital period and a minimum mass of as low as 1.3 M$_{\\oplus }$ .", "In addition to these seven signals, we report two additional periodic signals that are, according to our model probabilities in Table REF , statistically significant and unlikely to be caused by noise or data sampling or poor phase-coverage of the observations.", "Their amplitudes are well constrained and differ statistically from zero, which would not be the case unless they corresponded to actual periodicities in the data.", "We can also constrain their periods from above and below reasonably accurately.", "A related analysis of the same radial velocities was recently carried out by [4] but they received results differing in the number of significant periodicities.", "They claimed that only 6 signals can be detected reliably in the data, as opposed to 9 detected in the current work.", "However, they first analysed the data using a model with $k$ Keplerian signals and analysed the remaining residuals to see if they contained one more, $k+1$ th signal.", "While, as in the analyses of [19], this approach does not fully account for the uncertainties of the first $k$ signals when they have low amplitudes and their contribution to the residual uncertainty is negligible, it also actually assumes that the $k$ -Keplerian model is a correct one and then tests if it is not so, which is a clear contradiction and, while useful in case of strong signals, as demonstrated by [4], likely prevents the detection of weak signals in the data.", "This is underlined by the fact that [4] assume the residual vector to have an uncorrelated multivariate Gaussian distribution – this clearly cannot be the case if there are signals left in the residuals.", "Our analyses are not prone to similar weaknesses.", "Because planetary companions orbiting the star would produce the kind of periodicities we observe in the radial velocities, the interpretation of the two new signals as two new low-mass planets seems reasonable.", "As noted by [19], the star is a very quiet one without clear activity-induced periodicities, which makes it unlikely that one or some of the periodic signals in the data were caused by stellar phenomena.", "Also, the periodicities we report, namely 9.66 and 67.6 days, do not coincide with any periodicities arising from the movement of the bodies in the Solar system.", "Therefore, we consider the interpretation of these two new signals of being of planetary origin to be the most credible explanation.", "If this was the case, these two signals would correspond to planets on close-circular orbits with minimum masses of 1.9$^{+1.6}_{-1.8}$ and 5.1$^{+3.1}_{-3.2}$ M$_{\\oplus }$ , respectively, enabling the classification of them as super-Earths.", "Apart from the significance of the signals we observe, there is another rather strong argument in favour of the interpretation that all nine signals in the data are actually of planetary origin.", "Assuming that they were not, which based on stability reasons is the case with the 6.51 days signal that is quite certainly an alias of the 1.18 days periodicity likely caused by a planet [19], we would expect the weakest signals to be at random periods independent of the six strong periodicities in the data and the seventh 1.18 periodicity.", "Instead, this is not the case but the two additional signals reported in the current work appear at periods that fall in between the existing ones and, if interpreted as being of planetary origin, likely have orbits that enable long-term stability of the system (Fig.", "REF ) if their orbital eccentricities are close to or below the estimates in Table REF .", "As stated by [19], there are ”empty“ places in the HD 10180 system that allow dynamical stability of low-mass planets in the orbits corresponding to these empty places in the orbital parameter space, especially in the $a-e$ space.", "The two periodic signals we observe in the data correspond exactly to those empty places if interpreted as being of planetary origin.", "Additional measurements are needed to verify the significance of the two new periodic signals in the radial velocities of HD 10180 and to set tighter constraints to the orbital parameters of the planets in the system.", "Also, the possibility that all nine signals in the data correspond to planets should be tested by full-scale numerical integrations of their orbits.", "If all the configurations allowed by our solution in Table REF were found to correspond to unstable systems in any timescales of less than the estimated stellar age of 4.3$\\pm $ 0.5 Gyr [19], it would be a strong argument against the planetary interpretation of one or both of the signals we report in this article or the third low-amplitude 1.18 days signal.", "However, the results we present here and those in [19] suggest that this planetary interpretation of all the signals cannot be ruled out by dynamical analyses of the system.", "If the significance of these signals increases when additional high-precision radial velocities become available, and their interpretation as being of planetary origin is confirmed, the planetary system aroung HD 10180 will be the first one to top the Solar system in terms of number of planets in its orbits.", "Further, according to the rough dynamical considerations of the current work and the more extensive numerical integrations of [19], there are stable orbits for a low-mass companion in or around the habitable zone of the star.", "If such a companion exists, its minimum mass is unlikely to exceed 12.1 M$_{\\oplus }$ according to our posterior samplings of the corresponding parameter space.", "M. Tuomi is supported by RoPACS (Rocky Planets Around Cools Stars), a Marie Curie Initial Training Network funded by the European Commission's Seventh Framework Programme.", "The author would like to thank the two anonymous referees for constructive comments and suggestions that resulted in significant improvements in the article." ] ]

[ [ "Left-right symmetry at LHC and precise 1-loop low energy data" ], [ "Abstract Despite many tests, even the Minimal Manifest Left-Right Symmetric Model (MLRSM) has never been ultimately confirmed or falsified.", "LHC gives a new possibility to test directly the most conservative version of left-right symmetric models at so far not reachable energy scales.", "If we take into account precise limits on the model which come from low energy processes, like the muon decay, possible LHC signals are strongly limited through the correlations of parameters among heavy neutrinos, heavy gauge bosons and heavy Higgs particles.", "To illustrate the situation in the context of LHC, we consider the \"golden\" process $pp \\to e^+ N$.", "For instance, in a case of degenerate heavy neutrinos and heavy Higgs masses at 15 TeV (in agreement with FCNC bounds) we get $\\sigma(pp \\to e^+ N)>10$ fb at $\\sqrt{s}=14$ TeV which is consistent with muon decay data for a very limited $W_2$ masses in the range (3008 GeV, 3040 GeV).", "Without restrictions coming from the muon data, $W_2$ masses would be in the range (1.0 TeV, 3.5 TeV).", "Influence of heavy Higgs particles themselves on the considered LHC process is negligible (the same is true for the light, SM neutral Higgs scalar analog).", "In the paper decay modes of the right-handed heavy gauge bosons and heavy neutrinos are also discussed.", "Both scenarios with typical see-saw light-heavy neutrino mixings and the mixings which are independent of heavy neutrino masses are considered.", "In the second case heavy neutrino decays to the heavy charged gauge bosons not necessarily dominate over decay modes which include only light, SM-like particles." ], [ "Introduction", "In general there are two ways in which non-standard models can be tested.", "In the first approach, Standard Model (discovered) processes or observables can be calculated very accurately by taking into account radiative corrections of the non-standard model.", "In the second approach we can look into completely new effects (new processes) which are not present in the Standard Model (SM) but exist in its extensions.", "Their detections would be a clear signal for the non-standard physics.", "Here radiative corrections beyond leading order are, at least at first approximation, not necessary.", "At the LHC era it is interesting to think closer how these two approaches could be joined and how we can profit from this situation.", "It is not a common strategy, especially as Grand Unified Theories (GUT) are concerned.", "Here we calculate 1-loop radiative corrections at low energies consistently in the framework of the non-standard model (not only in its SM subset, this issue of consistency has been explored intensively in [1], [2], [3], see also [4], [5], [6]).", "In the next step we are looking into some specific non-standard process at LHC, taking into account obtained earlier precise low energy predictions for parameters of the model.", "We consider left-right symmetric model based on the $SU(2)_L \\otimes SU(2)_R \\otimes U(1)_{B-L}$ gauge group [7], [8] in its most restricted form, so-called Minimal Left-Right Symmetric Model ($MLRSM$ ).", "We choose to explore the most popular version of the model with a Higgs representation with a bidoublet $\\Phi $ and two (left and right) triplets $\\Delta _{L,R}$ [9].", "We also assume that the vacuum expectation value of the left-handed triplet $\\Delta _{L}$ vanishes, $\\langle \\Delta _{L}\\rangle =0$ and the CP symmetry can be violated by complex phases in the quark and lepton mixing matrices.", "Left and right gauge couplings are chosen to be equal, $g_L=g_R$ .", "For reasons discussed in [1] and more extensively in [10], we discuss see-saw diagonal light-heavy neutrino mixings.", "It means that $W_1$ couples mainly to light neutrinos, while $W_2$ couples to the heavy ones.", "$Z_1$ and $Z_2$ turn out to couple to both of them [11], [12].", "$W_L-W_R$ mixing is neglected hereAs an interesting detail, the most stringent data comes from astrophysics through the supernova explosion analysis [13], [14], $\\xi <3\\cdot 10^{-5}$ , typically $\\xi <0.05$ [15]..", "Taking such a restricted model, easier its parametrization and less extra parameters are involved in phenomenological studies.", "However, it does not mean that it is easier to confirm or falsify it, in fact, despite of many interesting studies and constraints, the model has not been ruled out so far (though many interesting questions and problems calling for consistency of the model have been arose [1], [2], [3].", "PDG [15] gives $M_{W_2}>1$ TeV for standard couplings decaying to $e \\nu $ , recently the CMS collaboration established the generic bound [16] $M_{W_2}>1.4$ TeV.", "Moreover, CMS published exclusion limits for LR model [17], they excluded large region in parameter space ($M_N$ , $M_{W_2}$ ) which extends up to $M_{W_2}=1.7$ TeV.", "Similarly, ATLAS collaboration gives exclusion limits on both $M_N$ and $M_{W_2}$ .", "They obtained that $M_{W_2}> 1.8$ TeV for difference in mass of $M_N$ and $M_{W_2}$ larger than $0.3$ TeV [18] (for 34 $pb^{-1}$ ).", "The very last ATLAS analysis [19] based on the integrated luminosity of 2.1 $fb^{-1}$ pushed it even further, for some neutrino mass ranges it reaches already 2.3 TeV.", "These exclusion searches assume generally that $M_{W_2}> M_N$ , however in LR model the situation can be different i.e.", "$M_{W_2} < M_N$ .", "Let us note that $K_L-K_S$ data gives for the minimal LR model a strong theoretical limit, which is (at least) at the level of 2.5 TeV [20], [21].", "In further studies we take then the rough $K_L-K_S$ limit for $W_2$ mass (to which the LHC analysis approaches quickly, and rather sooner than later will overcome it) $M_{W_2}>2.5\\; {\\rm TeV}.$ For heavy neutrino limit $M_N>780$ GeV [18], but it must be kept in mind that bounds on $M_N$ and $M_{W_2}$ are not independent from each other.", "Let us mention that simultaneous fit to low energy charge and neutral currents give $M_{W_2}>715$ GeV [15], [23].", "Neutrinoless double beta decay allows for heavy neutrinos with relatively light masses, if Eq.", "(REF ) holds, for more detailed studies, see e.g.", "[21], [22].", "Detailed studies which take into account potential signals with $\\sqrt{s}=14$ TeV at LHC conclude that heavy gauge bosons and neutrinos can be found with up to 4 and 1 TeV, respectively, for typical LR scenarios [24], [25], [26].", "Anyway, such a relatively low (TeV) scale of the heavy sector is theoretically possible, even if GUT gauge unification is demanded, for a discussion, see e.g.", "[27] and [28].", "As far as one loop corrections are concerned, there are not many papers devoted to the LR model.", "Apart from [1], [2], [3], [12] in which one of the authors of this paper has been involved (MLRSM model), there are other papers: [29] (limits on $W_2$ mass coming from the $K_L-K_S$ process (finite box diagrams, renormalization not required), [30] (LEP physics), [31] (process $b\\rightarrow s \\gamma $ ).", "Some interesting results are included also in papers [32] where the problem of decoupling of heavy scalar particles in low energy processes has been discussed.", "On the other hand, the LHC collider gives us a new opportunity to investigate LR models and to look for possible direct signals.", "Lately a few interesting papers analysed possible signals connected with the LR model [24], [26], [21], [33], [34], [35].", "As we are looking for non-standard signals, we restrict here calculations at high energies to the first approximation (tree level).", "In the next section we will discuss low energy limits on right sector of MLRSM which come from precise calculation of the muon decay.", "In section some representative LR signals at LHC will be discussed, taking into account severe limits coming from the muon decay analysis.", "We end up with conclusions.", "We have decided to skip most of the details connected with definition of fields, interactions and parameters in the MLRSM.", "All these details can be found in [12] and [2] (especially the Appendix there)." ], [ "One-loop low energy constraints on the right sector in MLRSM ", "Four-fermion interactions describe low energy processes in the limit $\\frac{q^2}{M^2} << 1$ , where $q$ is the transfer of four momentum and $M$ is the mass of the gauge boson involved in the interactions.", "This is an effective approximation of the fundamental gauge theory.", "This construction allows to replace the complete interaction by the point interaction with the effective coupling constant (which depends on the model).", "Independently, the model can be postulated with universal constant coupling (e.g.", "Fermi model with universal constant $G_F$ ).", "Next, taking into account the perturbation, corrections to so defined constants can be calculated at higher levels.", "Both effective and universal procedures describe the same process, so the corrections calculated in this way must be the same.", "This fact can be used to constrain parameters of the tested model.", "In the SM, all radiative corrections are embedded in the $\\Delta r$ term [15] $\\frac{G_F}{\\sqrt{2}}=\\frac{e^2}{8(1-M_W^2/M_Z^2)M_W^2}(1+\\Delta r).$ With the present values of the coupling constants and masses [15] $&& G_F = 1.166364(5)\\cdot 10^{-5}\\;{\\rm GeV}^{-2}, \\;\\;\\;\\;1/\\alpha = 137.0359976 \\pm 0.00000050, \\nonumber \\\\&& M_{W} = 80.399 \\pm 0.023 \\; {\\rm GeV},\\;\\;\\;\\;M_{Z} = 91.1876 \\pm 0.0021 \\; {\\rm GeV},$ experimental fits to the $\\Delta r$ parameter in SM give [15]The error has decreased about 3 times during last decade or so, mostly due to improvements in $W$ boson mass measurement.", "$\\Delta r \\equiv \\Delta r_0 \\pm \\Delta r_{\\sigma }= 0.0362 \\pm 0.0006.$ Matching for the muon decay and the structure of $ \\Delta r$ in the MLRSM model at the 1-loop level has been discussed in [2], see also [36] for more details on the matching in the context of SM.", "Figure: NO_CAPTION$\\Delta r$ as function of ${\\rm v}_R$ for different masses of heavy Higgs particles 5 TeV, 10 TeV and 15 TeV, see Eqs.", "(REF ,REF ).", "Solid (dashed) lines are for neutrino heavy masses with $h_M=0.1$ $(h_M=1)$ , see Eq.", "(REF ).", "Bold horizontal lines show the 3$\\sigma $ C.L.", "constraint on $\\Delta r$ , see Eq.", "(REF ).", "Excluded region comes from the bound on $W_2$ , Eq.", "(REF ).", "In Fig.", "$\\Delta r$ as function of ${\\rm v}_R$ for different masses of heavy Higgs particles and heavy neutrinos is shown.", "While plotting we have considered the variations of $\\Delta r$ with respect to ${\\rm v}_R$ , as the heavy gauge boson masses are directly proportional to this parameter, $M_{W_2} \\simeq 0.47 \\;{\\rm v}_R,\\;\\;\\; M_{Z_2}\\simeq 0.78 \\; {\\rm v}_R,$ see Fig.5 in [2].", "Mass of the lightest neutral Higgs scalar is assumed to be $M_{H_0^0}=120$ GeV ($\\Delta r$ is not sensitive to this mass, see Fig.6 in [2]).", "Masses of remaining heavy Higgs particles $H \\equiv \\lbrace H_1^0,H_2^0,H_3^0,A_1^0,A_2^0,H_1^+,H_2^+,\\delta _L^{++}\\equiv H_1^{++},\\delta _R^{++}\\equiv H_2^{++} \\rbrace $ $M_{H} & \\equiv & M_{H_1^0}=M_{H_3^0}=M_{A_1^0}=M_{A_2^0}=M_{H_1^+}=M_{H_2^+}= M_{H_1}^{++}= M_{H_2^0}=M_{H_2}^{++} $ are assumed to be equal, $m_H ={\\rm v}_R.", "$ Heavy neutrino masses $M_N = \\sqrt{2} h_M {\\rm v}_R,$ are taken in the range $h_M \\in [0.1,1]$ .", "$h_M$ is the Yukawa coupling connected with the right-handed Higgs triplet.", "$h_M<0.1$ are not forbidden, however attention should be paid to the limits coming from direct experimental searches (LEP $Z_1$ decays, ATLAS, CMS), especially for a region of small ${\\rm v}_R$ which we explore.", "On the other hand, $h_M>1$ reaches non-perturbative region.", "We can see, as expected in the framework of GUT models to which MLRSM belongs, that for given $m_H$ and $M_N$ there is a very narrow space for ${\\rm v}_R$ which are consistent with muon data (fine-tuning).", "Table: NO_CAPTION Set A.", "Values of ${\\rm v}_R$ for which various Higgs masses give $\\Delta r$ in agreement with Eq.", "(REF ).", "The ranges of $v_R$ are achieved by varying heavy neutrino masses in the domain $h_M \\in [0.1, 1]$ , see Eq.", "(REF ).", "For ${({\\rm v}_R)}_{min}$ depicted by an asterisk in the last column, and corresponding $(M_N)_{min}$ for which $h_M<0.1$ , see the main text.", "Set B.", "${\\rm v}_R$ is fixed in addition, leaving as the only free MLRSM parameter the neutrino mass $M_N$ , see Fig..", "Values obtained for $m_H \\le 5$ TeV do not fulfil direct LHC experimental search limits, the same is true for $m_H=10$ TeV if the limit Eq.", "(REF ) is applied.", "Figure: NO_CAPTIONScheme for limited parameters in Table .", "${({\\rm v}_R)}_B$ is a fixed value of ${\\rm v}_R$ for which set B is defined with maximal value of degenerate heavy neutrino mass (in the perturbative region, $h_M=1$ ).", "Figure: NO_CAPTION$\\Delta r$ as function of ${\\rm v}_R$ for three different masses of heavy Higgs particles, 5 TeV, 10 TeV and 15 TeV.", "Cases (a) and (b) are different by the heavy neutrino mass spectrum.", "In the case (a) a mass of $N_4$ is fixed, $M_{N_4}=800$ GeV and masses of $N_5,N_6$ neutrinos vary with ${\\rm v}_R$ , Eq.", "(REF ).", "In the case (b) all $N_4,N_5,N_6$ neutrinos have masses which obey Eq.", "(REF ).", "For the solid lines with $h_M=0.1$ the neutrino cases (a) and (b) give the same predictions.", "Bold horizontal lines show the 3$\\sigma $ C.L.", "constraint on $\\Delta r$ , see Eq.", "(REF ).", "Table  describes the situation more precisely.", "Set A shows ranges of ${\\rm v}_R$ which fit at 3$\\sigma $ C.L.", "to Eq.", "(REF ) for varying heavy neutrino masses in the range $h_M \\in [0.1, 1]$ , see Eq.", "(REF ).", "The upper limit of ${\\rm v}_R$ corresponds to neutrino masses with $h_M=1$ and $\\Delta r_{max}=\\Delta r_0+3 \\Delta r_{\\sigma }$ , the lower limit of ${\\rm v}_R$ corresponds to $h_M=0.1$ and $\\Delta r_{min}=\\Delta r_0-3 \\Delta r_{\\sigma }$ , see Fig.. We can see that the heavy degenerate neutrinos can be relatively light having masses below 1 TeV.", "A minimal heavy neutrino mass for $({\\rm v}_R)_{min}$ depicted with asterisk in the last column could be even smaller (if $h_M<0.1$ ).", "For instance, ${\\rm v}_R=6398$ GeV ($M_{W_2} \\simeq 3$ TeV) and muon data in the range $\\Delta r_0 \\pm 3 \\Delta r_{\\sigma }$ restricts allowed heavy neutrino masses to the region $100 \\le M_N \\le 2210$ [in GeV] (it means that $(h_M)_{min}\\simeq 0.01$ ).", "Set B describes a range of $M_N$ which fits at 3$\\sigma $ C.L.", "to Eq.", "(REF ) where in addition also ${\\rm v}_R$ is fixed.", "Here a fixed point is chosen to be a value of ${\\rm v}_R$ which for given $m_H$ and a neutrino mass with $h_M=1$ gives $\\Delta r_{min}=\\Delta r_0 -3 \\Delta r_{\\sigma }$ (crossing with lower of horizontal lines in Fig.).", "Then we are looking for $h_M<1$ which still covers $3 \\sigma $ C.L.", "region constraint by Eq.", "(REF ) and we get the range of neutrino masses written in the Table , see Fig.. For Set B possible values of $M_N$ are of course even more restricted than for Set A.", "Results in Table  are compatible with Eq.", "(REF ) for the last column, $m_H=15$ TeV.", "If we take into account FCNC, neutral heavy Higgs mass should be larger than 10-15 TeV, going down to a few TeV only in some special cases (for references and update discussion, see [37]).", "So, from now on, let us focus on the last column, $m_H=15$ TeV.", "If we start with some other value of ${\\rm v}_R$ instead $({\\rm v}_R)_{B}$ , e.g.", "${\\rm v}_R=6500$ GeV ($M_{W_2} \\simeq 3055$ GeV) and muon data in the range $\\Delta r_0 \\pm 3 \\Delta r_{\\sigma }$ restricts allowed heavy neutrino masses to the region $2654 \\le M_N \\le 3232$ GeV.", "To discuss a case with non-degenerate neutrinos, in Fig.", "we let one of the heavy neutrinos to be much lighter, $M_{N_4}=800$ GeV (for $N_5,N_6$ we keep masses through the relation Eq.", "(REF )).", "We call it the case (a).", "For the case (b) we vary all three heavy masses with ${\\rm v}_R$ , in accordance with Eq.", "(REF ) (degeneracy, the same $h_M$ ).", "In the case $h_M=0.1$ there is only one line, as two cases (a) and (b) give the same predictions.", "We can see that lines change slightly with chosen neutrino mass spectrum, but not dramatically, values of allowed ${\\rm v}_R$ are relatively stable and well constrained.", "In summary, heavy ($m_H > 10$ TeV) Higgs masses are allowed and follow roughly ${ \\rm v}_R$ scale (allowed ${\\rm v}_R$ increases with increasing $m_H$ ).", "However, the most important for the LHC phenomenology is the fact that still light (at the level of hundreds of GeV) heavy neutrinos are allowed in the framework of MLRSM.", "Let us discuss it more carefully." ], [ "Decay widths and branching ratios of the heavy LR spectrum", "Figure: NO_CAPTIONFigure: NO_CAPTIONDecay branching fractions and total widths for $W_2$ decays.", "Symbol $q \\bar{q}$ on this and next plots stands for a sum of all quark flavours, $q \\bar{q}\\equiv \\sum \\limits _{i, i^{\\prime }=u,d,s,b,c,t} q_i \\bar{q_{i^{\\prime }}}$ .", "Similarly, $lN \\equiv \\sum \\limits _{i=4}^6 l_{i-3}N_i$ , $l \\nu \\equiv \\sum \\limits _{i=1}^3 l_{i}\\nu _i$ .", "Experimental limits on $W_L-W_R$ mixing angle $\\xi $ are very severe and, similarly as in the muon decay case, we neglect it here.", "Second, as already mentioned in Introduction, we assume MLRSM with diagonal light-heavy neutrino mixings of the \"see-saw\" type $|U_{\\nu _i j}| \\simeq \\frac{|\\langle M_D \\rangle |}{M_{N_j}} \\delta _{i,j-3},\\;\\;\\; i=1,2,3,\\;\\;\\;j=4,5,6$ where $\\langle M_D \\rangle $ is an order of magnitude of the Dirac neutrino mass matrix and $\\nu _i$ stands for 3 light neutrinos.", "These two are conservative assumptions, on the other hand they are very natural and we can see what signals we can get at LHC for such harsh model conditions.", "For instance, analyzed in [39] signals which stem from the gauge boson triple vertices including heavy gauge bosons are absent completely in our scenario.", "In Fig.REF we can see that heavy gauge boson decay is dominated by quark channelsIn Fig.REF and the next we do not depict explicitly exclusion regions (e.g.", "Eq.", "(REF )), as the limits for the heavy particle spectrum change quickly with increasing LHC luminosity, see e.g.", "[18] vs. [19].. Second of importance is $W_2$ decay to heavy neutrinos, that is why these two channels make the \"golden\" process considered in the next section large.", "As the mixing in Eq.", "(REF ) becomes smaller, the $l \\nu $ decay mode falls, e.g.", "for $M_D=0.1$ GeV we obtain $Br(W_2 \\rightarrow l \\nu )\\simeq 10^{-11}$ .", "These are a kind of textbook results, see e.g.", "[40] and references therein.", "However, there are scenarios in which branching ratios can be different and heavy particles can decay dominantly to the light particles, so not through the right-handed currents.", "This is a case of non see-saw models where mixing angles are independent of heavy neutrino masses, see e.g.", "[10].", "Let us assume then that light-heavy neutrino mixing defined in Eq.", "(REF ) is independent of the heavy neutrino mass, experimental limits on elements of this mixing read (this limit has improved substantially over the last decade) [43] $\\sum _{j=4,5,6} U_{\\nu _1, {j-3}} U_{ \\nu _1, j-3}^{\\ast } = U_{\\nu _1, 4} U_{\\nu _1, 4}^{\\ast } \\le 0.003 \\equiv \\kappa ^2_{max}.$ In this case, the $l \\nu $ branching ratio in Fig.REF will enhanceIn a case where more than one heavy neutrino state exists (which is true in MLRSM), the maximal light-heavy neutrino mixing defined in Eq.", "(REF ) is constrained further among others by neutrinoless double beta decay measurements to be less than $\\kappa ^2_{max}/2$ [41].", "We take then this parameter in our considerations for non-decoupling light-heavy neutrino mixings., BR($W_2 \\rightarrow l \\nu = 5\\cdot 10^{-4}$ ).", "Still, it is not large.", "$q \\bar{q}$ and $l N$ modes dominate.", "Figure: NO_CAPTIONFigure: NO_CAPTIONDecay branching fractions and total widths for $Z_2$ decays.", "Figure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONDecay branching fractions and total widths for heavy neutrino decays with see-saw type of mixing, Eq.", "(REF ), first row ($\\langle M_D \\rangle $ = 0.1 GeV (on left) and 1 GeV (on right)).", "In the second row, on left, branching ratios with maximal type of mixings are calculated, Eq.", "(REF ).", "On right the total widths are given.", "$M_{W_2}$ is fixed at 2.5 TeV.", "In Fig.REF decays of the $Z_2$ boson are shown.", "Also here results are practically independent of light-heavy mixing scenarios, Eqs.", "(REF ,REF ).", "$Z_2$ heavy boson decays are also dominated by quark channels.", "Here the situation is more complicated and up to a per mil level, a few channels contribute.", "Interestingly, also $Z_2$ decay to a pair of light neutrinos or bosons as well as to the $Z_1 H_0^0$ pair are substantial.", "However, the situation changes with respect to light-heavy mixing scenarios for the case of heavy neutrino decays.", "Decays of the first of heavy neutrinos $N_4$ in Fig.REF are dominated by the $e^{\\pm } W_1^{\\mp }$ (we neglect here the mixing between different generations) mode till the threshold where $W_2$ production is open.", "Mass of $W_2$ is fixed at 2.5 TeV.", "Still $e^{\\pm } W_1^{\\mp }$ option is large, even if $W_2$ mass would be smaller ($1.5\\; {\\rm TeV} \\le M_{W_2}\\le 2.5$ TeV).", "Changing the mixing Eq.", "(REF ) affects mainly $\\nu _e Z_2$ mode (which is negligible).", "If we took maximal possible mixing, $\\kappa ^2_{max}/2=0.0015$ , then the branching ratios for heavy neutrino decays change qualitatively (left figure in the second row in Fig.REF ).", "We can see that the $N_4 \\rightarrow e W_1$ and $N_4 \\rightarrow \\nu Z_1$ decays dominate over decay channels to the heavy states $W_2,Z_2$ in the kinematically allowed regions.", "The reason is that although decay amplitudes for light boson modes are proportional to the small light-heavy neutrino mixing, the helicity summed amplitudes for gauge boson modes are suppressed in addition by the masses of gauge bosons, which is a stronger effect in a case of heavy gauge bosons.", "The difference between both scenarios of neutrino mixings is clearly visible on the last plot in Fig.REF where total decay widths are given.", "In order to show influence of the Higgs sector we deliberately distorted heavy Higgs mass spectrum to include some lighter Higgs masses such that Higgs particles show up in the neutrino decay.", "However, open in this way Higgs decay modes contribute well below per mille level in total and are negligible." ], [ "LR signals at LHC, a sample", "The so-called \"golden\" process where the left-right symmetry signal is not suppressed due to small light-heavy neutrino mixings is depicted in Fig.REF (here heavy neutrino couples directly to $W_2$ which decays hadronically, Fig.REF ).", "Thus the final state consists of the same sign di-leptons and jets which also carries a clear signature of lepton number violation.", "Even if we consider the leptonic decay modes of $W_2$ , we can have 3-leptons and $missing~~ energy$ as our signal events.", "The presence of one missing energy source allows to reconstruct $W_2$ fully, and then the reconstruction of the right-handed neutrino, $N_{i}$ , helps to reduce the combinatorial backgrounds for this process.", "In [21] it has been discussed that the dominant background for this process is coming from $t \\bar{t}$ events and is negligible beyond the TeV scale.", "In the other case where $W_2$ decays hadronically with the largest branching fraction, the invariant mass of the hardest jets plus one(two) lepton(s) also allows to reconstruct in a clean way the heavy neutrino $N$ and $W_2$ masses.", "Figure: NO_CAPTIONA tree level basic diagram for the $u\\bar{d} \\rightarrow e^{+}e^{+} W_2^-$ process.", "The process is not suppressed if $W_2^+$ decays to right-handed quarks forming jets (it is suppressed if it decays to standard, left-handed leptons).", "Figure: NO_CAPTIONCross-sections for processes $pp\\rightarrow e^{\\pm } N_4 \\rightarrow e ^{\\pm } e^{\\pm } W_2^{\\mp }$ for sets of parameters in Set B, $m_H=3,4,5$ TeV, respectively.", "As discussed in the last Section, muon decay data restricts very much possible values of ${\\rm v}_R$ (and through the relation Eq.", "(REF ) masses of heavy gauge bosons) for chosen spectrum of Higgs and neutrino masses.", "Let us then assume a scenario for LHC potential discoveries with $M_{W_2} \\simeq 2.5$ TeV (then ${\\rm v}_R \\simeq 5$ TeV).", "If we choose the most uniform scenario defined by Set B in Table  (with the same masses for all Higgs particles and also for all heavy neutrinos), then muon decay data sets the heavy neutrino masses of the order 7 TeV (and masses of Higgs particles of the order of 10 TeV).", "We have computed the cross-sectionFor numerical results we use CalcHEP [44] and Madgraph5 [45] with our own implementation of the MLRSM model in Feynrules [46].", "We made a couple of cross checks for correctness of implementations for neutrino and gauge boson mixings.", "Results for $e^-e^+ \\rightarrow \\nu N$ [11], $e^-e^- \\rightarrow W^-_1 W^-_1$ [38], $e^- \\gamma \\rightarrow N W_{1}^-$ [41] and $pp \\rightarrow l W_2$ [26] have been recovered, among others.", "for the process $pp\\rightarrow e^{\\pm } N_4 \\rightarrow e ^{\\pm } e^{\\pm } W_2^{\\mp }$ for the sets of parameters given in Set B (Table 1, $v_R=1661$ GeV is for $m_H=3$ TeV).", "Signatures for heavy neutrinos and charged gauge bosons in hadron colliders have been discussed already some decades ago, for a first paper on these kind of signals, see [42].", "Results are shown in Fig.REF .", "As can be seen in this case the cross-section is very small for LHC operating at 7 TeV, results for higher $m_H$ will give even smaller values.", "Going to 14 TeV of course improve the situation but still this scenario is very unlikely to be discovered.", "Figure: NO_CAPTIONCross-section for process $pp\\longrightarrow e^{+} N_4$ as a function of heavy neutrino mass $M_N$ for different sets of $W_2$ masses, $\\sqrt{s}=14$ TeV.", "Excluded region of $M_N$ depends on $M_{W_2}$ , see the plots in [18], [19].", "We have just fixed it safely at $M_N=780$ GeV.", "Luckily, other scenarios are possible where one of heavy right-handed neutrinos has smaller mass, e.g.", "800 GeV, but other two are very heavy having masses $\\sim $ 5 TeV.", "There is also an option with 3 degenerate heavy neutrinos but with smaller ${\\rm v}_R$ , e.g.", "${\\rm v}_R=6398$ GeV ($M_{W_2}\\simeq 3$ TeV), see the last column in Table .", "These scenarios are still compatible with muon decay data (though relatively light heavy gauge boson is required).", "It gives much bigger cross-section, see Fig.REF , with anticipated luminosity this is a detectable process.", "From the above plot it is clear that as the mass of the heavy neutrino and the scale ${\\rm v}_R$ increase, the production cross-section falls rapidly and then the further decays of the $N$ followed by the decay of $W_2$ suppress the effective cross-section for this \"golden\" process.", "Figure: NO_CAPTIONCross-section for process $pp\\longrightarrow e^{+} N_4$ for different sets of heavy neutrino masses $M_N$ , $\\sqrt{s}=14$ TeV.", "The results are for degenerate heavy neutrino and Higgs particle masses.", "$m_H$ masses are fixed at 10 TeV and $m_H=15$ TeV.", "The whole shaded bands correspond to parameters labeled as Set A in Table  and Fig..", "In addition, for each ${\\rm v}_R$ between $({\\rm v}_R)_{min}$ and $({\\rm v}_R)_B$ in Fig.", ", heavy neutrino mass spectrum which is in agreement with muon decay data is obtained (we call it Set B).", "In this way a possible cross-section for allowed $M_{W_2}-M_N$ masses is constrained dramatically.", "These regions are denoted by almost vertical and thin black stripes within the wider shaded regions.", "However, in Fig.REF we show more carefully how precise low energy data from Table  restricts a space of possible cross-section for this process.", "Let us assume that Higgs masses are degenerate, at the level of 10 TeV and 15 TeV (the first case is almost excluded, see Eqs.", "(REF ),(REF ) and Table 1).", "Then vertical bands restrict regions of possible cross-sections for given $M_N$ masses.", "If we assume in addition that heavy neutrino masses are also degenerate, then the black, thin strips inside these bands give for each ${\\rm v}_R$ very narrow intervals of possible heavy neutrino masses, consequently, region of possible cross-sections is very limited.", "With an assumed luminosity of tens of inverse femtobarns at $\\sqrt{s}=14$ TeV, $\\sigma (pp \\rightarrow e^+ N)\\simeq 10$ fb would give hundreds of events, which we take as a safety discovery limit for this process.", "Relevant experimental conditions do not spoil signals, for a discussion on kinematical cuts and a background for this process, see e.g.", "[26].", "In this case, without muon data, possible ${\\rm v}_R$ values give $W_2$ mass in the range (1 TeV, 3.5 TeV) for heavy neutrino masses up to 1 TeV.", "Muon data shrinks the region very much, $1970\\; {\\rm GeV} \\le M_{W_2} \\le 2050\\; {\\rm GeV}$ (for $m_H=10$ TeV) and $3008\\; {\\rm GeV} \\le M_{W_2} \\le 3040\\; {\\rm GeV}$ (for $m_H=15$ TeV).", "From Fig.REF it should be clear that increasing heavy Higgs masses would shift the $v_R$ scale to higher level, decreasing further cross sections for the considered process.", "In summary, for the left-right LHC phenomenology, Higgs mass spectrum is optimal in vicinity of 15 TeV region.", "For a case of degenerate heavy neutrinos, heavy Higgs particles with masses at about 10 TeV and below are practically excluded by muon data.", "On the other hand, MLRSM scenarios with Higgs particles masses at about 20 TeV (and above) are allowed by muon data, however, low energy muon restrictions constraint heavy gauge boson and neutrino masses in such a way that $\\sigma (pp \\rightarrow e^+ N)<10$ fb." ], [ "Conclusions", "It is very important to take into account low energy data in phenomenological analysis of non-standard models at LHC.", "This is a quite common action in supersymmetric models, e.g.", "precise $(g-2)_\\mu $ analysis is very important for pinning down parameter space for supersymmetry collider searches [47], [48].", "These kind of analysis are less popular in GUT models (it is justified if a decoupling of heavy states occurs).", "We should acknowledge the last work [33] where a connection between neutrinoless doubly beta decay and LHC for LR models is undertaken (this is however by its nature purely \"tree level\" calculation and connection).", "Here we show the interplay between fermion-boson heavy spectrum of the MLRSM model in the muon decay.", "As it is typical for GUT models, it is also true for MLRSM that \"extensions of the SM in most cases end up in a fine tuning problem, because decoupling of new heavy states, in theories where masses are generated by spontaneous symmetry breaking, is more the exception than the rule\" (quotation from [49]).", "As shown in Section , fixing heavy gauge boson masses and Higgs particle masses, the region of possible heavy neutrino mass spectrum is restricted by the muon decay.", "However, there is still a way to get at least one relatively light heavy neutrino, which can be explored at LHC.", "This is possible as heavy particles effects are effectively \"weighted\" at the 1-loop level (for virtual particles the effects are summed up which means that effects of 3 heavy degenerate neutrinos can be equivalent to the effects of one relatively light and two heavier heavy neutrino masses).", "In this case, there is still a way that left-right symmetry is broken at low enough energy scale such that LR models can be discovered directly at LHC (see Section ).", "Let us note, that the situation gets more interesting if LHC finds heavy particles which appear in the spectrum of the LR model.", "Then analysis could be reversed – the obtained physical parameters can be helpful to further pin down remaining parameters for a part of the spectrum which can not be directly constrained at LHC, through the low energy precise analysis like the muon decay.", "For instance, knowledge of both the mass of the lightest of heavy neutrinos and of the scale ${\\rm v}_R$ ($W_2$ boson mass reconstruction) will restrict masses of heavier neutrinos in $\\Delta r$ .", "This will be a great hint for searches of remaining particles since we would be able to predict where to look for them.", "We think that this kind of low-high energy analysis is important and should be further explored, for instance including 1-loop level calculations in MLRSM for lepton flavour violating processes.", "In general, when making numerical predictions for any model beyond the SM, as many as possible of low energy observables and precision LEP observables should be taken into account." ], [ "Acknowledgements", "We would like to thank Henryk Czyż, Fred Jegerlehner, Miha Nemev$š$ ek and Marek Zrałek for useful discussions and comments.", "Work supported by the Research Executive Agency (REA) of the European Union under the Grant Agreement number PITN-GA-2010-264564 (LHCPhenoNet) and by the Polish Ministry of Science under grant No.", "N N202 064936." ] ]

[ [ "Simple variational approaches to quantum wells" ], [ "Abstract We discuss two simple variational approaches to quantum wells.", "The trial harmonic functions analyzed in an earlier paper give reasonable results for all well depths and are particularly suitable for deep wells.", "On the other hand, the exponential functions proposed here are preferable for shallow wells.", "We compare the shallow-well expansions for both kind of functions and show that they do not exhibit the cubic term appearing in the exact series.", "It is also shown that the deep-well expansion for the harmonic functions agree with the first terms of perturbation theory." ], [ "Introduction", "In a recent paper we discussed the application of the variational method to a Gaussian well.", "[1] We showed that harmonic variational functions yield reasonably accurate results for all well depths despite the fact that the shallow-well expansion for the ground-state variational energy does not agree with the exact expansion beyond the leading term.", "Students at introductory courses on quantum mechanics are commonly encouraged to solve the Schrödinger equation for a particle in a one-dimensional square box with finite walls.", "They thus learn that the exact solution decays exponentially as $|x|\\rightarrow \\infty $ .", "After such an experience they would find it unreasonable the choice of harmonic variational functions that do not exhibit the behavior at infinity expected for the exact solutions to the Schrödinger equation for the Gaussian well.", "For this reason we think that it is interesting to show an alternative variational calculation based on trial functions with the expected exponential behavior at infinity.", "In Sec.", "we develop the variational method for a general single-well potential-energy function.", "In Sec.", "we discuss the application of the harmonic and exponential trial functions to the Gaussian well, compare the approximate and exact energies for the ground and first-excited state as well as their shallow-well expansions.", "In Sec.", "we compare the variational and exact deep-well expansions that we did not consider in our previous paper.", "[1] Finally, in Sec.", "we discuss the main results and draw conclusions." ], [ "Simple variational method", "We are interested in the solutions to the Schrödinger equation $\\hat{H}\\psi _{n}=\\epsilon _{n}\\psi _{n},\\,n=0,1,\\ldots $ for the dimensionless Hamiltonian operator $\\hat{H}=\\hat{T}+v(x)=-\\frac{1}{2}\\frac{d^{2}}{dx^{2}}+v(x)$ In our previous paper we showed how to convert the Schrödinger equation into a dimensionless eigenvalue equation and we do not repeat it here.", "[1] For concreteness we choose the dimensionless potential-energy function to be of the form $v(x)=-v_{0}f(x)$ , where $v_{0}>0$ and the function $f(x)$ exhibits a maximum at $x=0$ and tends to zero when $|x|\\rightarrow \\infty $ .", "Since we restrict ourselves to single wells we also assume that $f^{\\prime }(x)>0$ if $x<0$ and $f^{\\prime }(x)<0$ if $x>0$ although we do not make use of this condition explicitly.", "We will discuss a particular example later on.", "The reader will find some references about quantum-mechanical wells in our earlier paper.", "[1] It is our purpose to obtain simple solutions for some states of the quantum well by means of the variational method.", "To this end we choose a variational function $\\varphi (a,x)$ that depends on a variational parameter $a$ .", "If we assume that $\\epsilon _{0}<\\epsilon _{1}<\\epsilon _{2}<\\cdots $ then the variational principle gives us an upper bound to the smallest eigenvalue $W(a)=\\frac{\\left\\langle \\varphi \\right| \\hat{H}\\left| \\varphi \\right\\rangle }{\\left\\langle \\varphi \\right| \\left.", "\\varphi \\right\\rangle }\\ge \\epsilon _{0} $ and we realize that the optimum value of $a$ is given by the minimum of $W(a) $ .", "If we write $W(a)=F(a)-v_{0}G(a)$ where $F(a)=\\left\\langle \\varphi \\right| \\hat{T}\\left| \\varphi \\right\\rangle /\\left\\langle \\varphi \\right|\\left.", "\\varphi \\right\\rangle $ and $G(a)=\\left\\langle \\varphi \\right|f(x)\\left| \\varphi \\right\\rangle /\\left\\langle \\varphi \\right| \\left.\\varphi \\right\\rangle $ then $W^{\\prime }(a)=0$ becomes $F^{\\prime }(a)-v_{0}G^{\\prime }(a)=0 $ Commonly, solving this equation for $a$ may not be possible or may lead to cumbersome expressions.", "In that case we can resort to a simple parametric expression for the approximate energy in the form $v_{0} &=&\\frac{F^{\\prime }(a)}{G^{\\prime }(a)} \\nonumber \\\\W(a) &=&\\frac{F(a)G^{\\prime }(a)+F^{\\prime }(a)G(a)}{G^{\\prime }(a)}$ If $f(-x)=f(x)$ then the states have definite parity $\\psi _{n}(-x)=(-1)^{n}\\psi _{n}(x)$ and the variational principle applies to $\\epsilon _{0}$ or $\\epsilon _{1}$ if the variational function is even or odd, respectively.", "Therefore, the variational equations (REF ) apply to both the ground and first excited states." ], [ "The Gaussian well", "In order to illustrate the performance of the variational method on quantum-mechanical wells we choose a particularly simple example that allows us to calculate the functions $F(a)$ and $G(a)$ analytically.", "As in our earlier paper we select the Gaussian well given by $f(x)=e^{-x^{2}} $ and begin present discussion with the calculation based on Harmonic-oscillator-like variational functions.", "[1] The simplest variational function for the ground state is $\\varphi _{0h}(a,x)=e^{-ax^{2}} $ and we easily obtain $v_{0} &=&\\frac{\\sqrt{2}\\sqrt{a}\\left( 2a+1\\right) ^{3/2}}{2} \\nonumber \\\\W_{0h}(a) &=&-\\frac{a\\left( 4a+1\\right) }{2} $ We showed that $W_{0h}\\rightarrow 0$ as $v_{0}\\rightarrow 0$ according to[1] $W_{0h}=-v_{0}^{2}+4v_{0}^{4}-48v_{0}^{6}+832v_{0}^{8}-17408v_{0}^{10}+\\ldots \\,.", "$ while the exact expansion is[1] $\\epsilon _{0}=-\\frac{\\pi v_{0}^{2}}{2}+\\sqrt{2}\\,\\pi v_{0}^{3}-\\frac{\\pi \\left( 2\\pi +3\\sqrt{3}+3\\right) v_{0}^{4}}{3}+\\frac{\\sqrt{2}\\,\\pi \\left(2\\pi +3\\sqrt{3}\\right) v_{0}^{5}}{3}-\\ldots \\,.", "$ Note that the cubic term is missing in the approximate expansion (REF ).", "We have also shown that $\\varphi _{1h}(a,x)=xe^{-ax^{2}} $ is a suitable variational function for the first excited state, and in this case we have $v_{0} &=&\\frac{\\sqrt{2}\\sqrt{2a+1}\\left( 4a^{2}+4a+1\\right) }{4\\sqrt{a}}\\nonumber \\\\W_{1h}(a) &=&-\\frac{a\\left( 8a^{2}+2a-1\\right) }{2\\left(2a+1\\right) } $ It is well known that $\\epsilon _{1}(v_{0})\\rightarrow 0^-$ as $v_{0}$ approaches a critical well strength $v_{0,1}\\approx 1.342$ from above[1], [2].", "We can estimate the critical strength by means of equations (REF ) in a quite simple way.", "We first obtain a positive root of $W_{1h}(a_{c})=0$ and then $v_{0}(a_{c})$ ; the result is $a_{c}=1/4$ and $v_{0}(1/4)=9\\sqrt{3}/8\\approx 1.95$ .", "Since the variational energy is an upper bound to the exact one for all $v_{0}$ then it is not surprising that $v_{0}(a_{c})>v_{0,1}$ .", "The error for the critical strength obtained with this variational function is rather large: $45\\%$ .", "We may try to improve those results by means of more convenient trial functions.", "Since $v(x\\rightarrow \\pm \\infty )=0$ we know that the eigenfunctions behave as $\\psi \\sim e^{-\\alpha |x|}$ for sufficiently large $|x|$ , where $\\alpha =\\sqrt{-2\\epsilon }$ .", "In order to avoid the function $|x|$ in our calculations we work on the half positive line $x>0$ and take into account the boundary conditions at origin: $\\psi (0)\\ne 0$ , $\\psi ^{\\prime }(0)=0$ for the even states and $\\psi (0)=0$ , $\\psi ^{\\prime }(0)\\ne 0$ for the odd ones.", "Thus, the simplest trial exponential function for the ground state appears to be $\\varphi _{0e}(a,x)=(1+ax)e^{-ax},\\,x>0 $ The calculation of the integrals is straightforward and we obtain $v_{0} &=&\\frac{a}{\\sqrt{\\pi }e^{a^{2}}\\left( 4a^{6}+4a^{4}-5a^{2}+2\\right)[1-\\mathrm {erf}{\\left( a\\right) ]}-2a\\left( 2a^{4}+a^{2}-2\\right) }\\nonumber \\\\W_{0e}(a) &=&\\frac{a^{2}\\left\\lbrace \\sqrt{\\pi }e^{a^{2}}\\left(4a^{6}+a^{2}-2\\right) \\left[ \\mathrm {erf}{\\left( a\\right) }-1\\right]+2a\\left( 2a^{4}-a^{2}+2\\right) \\right\\rbrace }{10\\left\\lbrace \\sqrt{\\pi }e^{a^{2}}\\left( 4a^{6}+4a^{4}-5a^{2}+2\\right) \\left[ \\mathrm {erf}{\\left(a\\right) }-1\\right] +2a\\left( 2a^{4}+a^{2}-2\\right) \\right\\rbrace }$ where $\\mathrm {erf}(z)$ is the error function.", "In order to obtain the $v_{0}$ -series for $W_{0e}$ we first expand $v_{0}(a)$ into an $a$ -series: $v_{0}=\\frac{a}{2\\sqrt{\\pi }}+\\frac{3a^{3}}{4\\sqrt{\\pi }}-\\frac{4a^{4}}{3\\pi }+\\frac{9a^{5}}{8\\sqrt{\\pi }}+\\ldots $ that we invert to obtain the $v_{0}$ -series for $a$ $a=2\\sqrt{\\pi }v_{0}-12\\pi ^{3/2}v_{0}^{3}+\\frac{128\\pi ^{3/2}}{3}v_{0}^{4}+144\\pi ^{5/2}v_{0}^{5}+\\ldots $ Then, we expand $W_{0e}$ in a Taylor series about $a=0$ $W_{0e}=-\\frac{a^{2}}{10}-\\frac{a^{4}}{5}+\\frac{2a^{5}}{5\\sqrt{\\pi }}+\\ldots $ and substitute the series (REF ) to obtain the shallow-well expansion $W_{0e}=-\\frac{2\\pi v_{0}^{2}}{5}+\\frac{8\\pi ^{2}v_{0}^{4}}{5}-\\frac{64\\pi ^{2}v_{0}^{5}}{15}+\\ldots $ The calculation is straightforward but extremely tedious if carried out by hand.", "For this reason it is a good exercise for showing the students the advantage and power of computer algebra.", "Such software even offer a command for obtaining the inverted series (REF ) in one step.", "For sufficiently small $v_{0}$ this expansion is slightly better than the one derived in our earlier paper, namely Eq.", "(REF ), as follows from comparing the leading terms of the variational and exact $v_{0}$ -series: $W_{0h}\\approx -v_{0}^{2}$ , $W_{0e}\\approx -1.26v_{0}^{2}$ and $\\epsilon _{0}\\approx -1.57v_{0}^{2} $ .", "However, the cubic term that appears in the exact expansion (REF ) is also missing in the variational treatment based on the exponential function (REF ).", "From those results we conclude that $\\epsilon _{0}<W_{0e}<W_{0h}$ for sufficiently small $v_{0}$ .", "In other words, the trial function with the correct asymptotic behavior at infinity yields a more accurate variational energy for sufficiently shallow wells.", "Fig REF shows $W_{0h}$ , $W_{0e}$ and $\\epsilon _{0}$ (obtained by numerical integration[2]) for some values of $v_{0}$ .", "The variational curves are almost indistinguishable in the scale of the Figure.", "The numerical results show that they cross at $v_{0}=v_{c}$ , where $2.4022<v_{c}<2.4023$ , so that $W_{0e}<W_{0h}$ if $v_{0}<v_{c}$ and $W_{0e}>W_{0h}$ if $v_{0}>v_{c}$ .", "In other words: $W_{0e}$ is more accurate for shallow wells as argued above and $W_{0h}$ is more accurate for deep ones.", "The former inequality is consistent with the previous comparison of the leading terms of the $v_{0}$ -expansions.", "In the next section we discuss the deep-well limit.", "For the first excited state we propose the simple trial function $\\varphi _{1e}(a,x)=xe^{-ax},\\,x>0 $ and obtain $v_{0} &=&\\frac{1}{a\\left[ \\sqrt{\\pi }e^{a^{2}}\\left( 4a^{4}+12a^{2}+3\\right)\\left[ 1-\\mathrm {erf}\\left( a\\right) \\right] -2a\\left( 2a^{2}+5\\right)\\right] } \\nonumber \\\\W_{1e}(a) &=&\\frac{a^{2}\\left\\lbrace \\sqrt{\\pi }e^{a^{2}}\\left(4a^{4}+8a^{2}+1\\right) \\left[ \\mathrm {erf}\\left( a\\right) -1\\right]+2a\\left( 2a^{2}+3\\right) \\right\\rbrace }{2\\left\\lbrace \\sqrt{\\pi }e^{a^{2}}\\left(4a^{4}+12a^{2}+3\\right) \\left[ \\mathrm {erf}\\left( a\\right) -1\\right]+2a\\left( 2a^{2}+5\\right) \\right\\rbrace } $ The variational energy vanishes at $a_{c}\\approx 0.550$ that leads to the approximate critical parameter $v(a_{c})\\approx 1.56$ with an error of $17\\%$ , quite smaller than that for the harmonic function (REF ).", "Once again we appreciate that the exponential function leads to more accurate results close to threshold (shallow well).", "Fig.", "REF shows $W_{1h}$ , $W_{1e}$ and $\\epsilon _{1}$ (obtained by numerical integration[2]) for a range of $v_{0}$ values.", "In this case the variational curves are not so close each other and the crossing takes place at $3.5154<v_{c}<3.51541$ .", "We realize that the exponential function is preferable for the description of shallow wells as suggested by the errors in the variational critical well strengths.", "On the other hand, the harmonic function leads to more accurate results for deep wells" ], [ "The deep-well case", "The deep-well expansion for a general well can be easily derived by means of perturbation theory.", "[3] It follows from those results that the first terms of the expansion for the Gaussian well are $\\epsilon _{n}=-v_{0}+\\left( n+\\frac{1}{2}\\right) \\sqrt{2v_{0}}-\\frac{3}{16}\\left( 1+2n+2n^{2}\\right) +\\mathit {O}\\left( v_{0}^{-1/2}\\right)$ where $n=0,1,\\ldots $ is the harmonic-oscillator quantum number.", "In order to obtain the deep-well expansion for $W_{0h}$ first note that the expression for $v_{0}$ in equation (REF ) shows that $a\\rightarrow \\infty $ as $v_{0}\\rightarrow \\infty $ and that the leading term is $a\\approx \\frac{\\sqrt{2v_{0}}}{2}$ .", "Therefore, we substitute $a=\\frac{\\sqrt{2v_{0}}}{2}+a_{1}+\\frac{a_{2}}{\\sqrt{v_{0}}}$ into the expression for $v_{0}$ and set the unknown coefficients $a_{1}$ and $a_{2}$ in order to remove the leading powers of $v_{0}$ .", "In this way we obtain $a=\\frac{\\sqrt{2v_{0}}}{2}-\\frac{3}{8}+\\frac{3\\sqrt{2}}{128\\sqrt{v_{0}}}+\\mathit {O}\\left( v_{0}^{-3/2}\\right)$ Finally, we substitute this equation into the expression for $W_{0h}(a)$ , expand and keep the leading terms; the result is $W_{0h}=-v_{0}+\\frac{\\sqrt{2v_{0}}}{2}-\\frac{3}{16}+\\mathit {O}\\left(v_{0}^{-1/2}\\right) $ Note that this expression agrees with (REF ) for $n=0$ .", "Proceeding exactly in the same way with equation (REF ) we obtain $a=\\frac{\\sqrt{2v_{0}}}{2}-\\frac{5}{8}-\\frac{5\\sqrt{2}}{128\\sqrt{v_{0}}}+\\mathit {O}\\left( v_{0}^{-3/2}\\right)$ and $W_{1h}=-v_{0}+\\frac{3\\sqrt{2v_{0}}}{2}-\\frac{15}{16}+\\mathit {O}\\left(v_{0}^{-1/2}\\right) $ that agrees with (REF ) when $n=1$ .", "It is clear that the harmonic variational functions are suitable for the description of deep wells which is in agreement with the results in figures REF and REF ." ], [ "Conclusions", "We have calculated the first two energy levels of the Gaussian well by means of two types of variational functions.", "The harmonic trial functions lead to simpler expressions and their results are reasonable for all values of the well depth.", "The approximate energies exhibit a slightly accurate expansion for shallow wells and a more satisfactory one for deep wells.", "In the latter case they provide the first three dominant terms of perturbation theory.", "On the other hand, the expressions for the energy obtained from exponential functions are rather complicated and their results are less accurate for deep wells.", "However, they are useful if one is interested in the behavior of the system near threshold (shallow wells).", "Present results suggest that the appropriate behavior of the wavefunction at infinity does not always guarantee the greatest accuracy.", "In the case of a deep well the wavefunction is strongly localized and compressed about $x=0$ and the description of the neighborhood of the origin is more important than what happens at larger $|x|$ .", "A most intriguing result is that both variational approaches fail to yield the cubic term in the shallow-well expansion.", "We have carried out a similar calculation for the particle in the square box with finite walls and found that the expansion of the variational energy given by the trial function (REF ) does not exhibit the cubic term that appears in the exact series.", "In this case one can obtain the exact $v_{0}$ -series for $\\epsilon _{0}$ from the transcendental equation that determines the energy levels of that model.", "We do not have a satisfactory explanation for this failure of the variational method.", "Figure: (Color online) Variational energies W 0e W_{0e} (solid line, blue) andW 0h W_{0h} (dashed line, red) and numerical ones (circles)Figure: (Color online) Variational energies W 1e W_{1e} (solid line, blue) andW 1h W_{1h} (dashed line, red) and numerical ones (circles)" ] ]

[ [ "Symmetrization of plurisubharmonic and convex functions" ], [ "Abstract We show that Schwarz symmetrization does not increase the Monge-Ampere energy for $S^1$-invariant plurisubharmonic functions in the ball.", "As a result we derive a sharp Moser-Trudinger inequality for such functions.", "We also show that similar results do not hold for general balanced domains except for complex ellipsoids and discuss related questions for convex functions." ], [ "Introduction.", "If $\\phi $ is a real valued function defined in a domain $\\Omega $ in ${\\mathbb {R}}^n$ , its Schwarz symmetrization, see [4], is a radial function, $\\hat{\\phi }(x)=f(|x|)$ , with $f$ increasing, that is equidistributed with $\\phi $ .", "The latter requirement means that for any real $t$ , the measure of the corresponding sublevel sets of $\\phi $ and $\\hat{\\phi }$ are equal, i e $|\\lbrace \\phi <t\\rbrace |=|\\lbrace \\hat{\\phi }<t\\rbrace |=:\\sigma (t).$ Notice that since $\\hat{\\phi }$ is radial, its natural domain of definition is a ball, $B$ .", "Moreover, as $t$ goes to infinity, $\\sigma (t)$ tends to the volume of $\\Omega $ and also to the volume of $B$ .", "Thus the volume of $B$ equals the volume of $\\Omega $ .", "Since $\\phi $ and $\\hat{\\phi }$ are equidistributed, any integrals of the form $\\int _\\Omega F(\\phi ) dx$ and $\\int _B F(\\hat{\\phi }) dx,$ where $F$ is a measurable function of a real variable, are equal.", "One fundamental property of symmetrization is that many other quantities measuring the 'size' of a function, decrease under symmetrization.", "The prime examples of this are energy integrals $\\int _\\Omega |\\nabla \\phi |^p,$ for $p\\ge 1$ , see [4].", "By the Polya-Szegö theorem $\\int _B |\\nabla \\hat{\\phi }|^p\\le \\int _\\Omega |\\nabla \\phi |^p ,$ if $\\phi $ vanishes on the boundary of $\\Omega $ .", "This means that e g the study of Sobolev type inequalities $(\\int _\\Omega |\\phi |^q)^{1/q}\\le A(\\int _\\Omega |\\nabla \\phi |^p)^{1/p}$ is immediately reduced to the radial case, which is a one-variable problem.", "Before we go on we remark that the inequality (1.1) is strongly related to the isoperimetric inequality.", "Indeed, if we take $p=1$ and $\\phi $ to be the characteristic function of $\\Omega $ , then as noted above, the corresponding ball has the same volume as $\\Omega $ .", "On the other hand, The $L^1$ -norm of $\\nabla \\phi $ (taken in the sense of distributions), is the area of the boundary of $\\Omega $ .", "It follows that the area of the boundary of $\\Omega $ is not smaller than the area of the sphere bounding the same volume, which is the isoperimetric inequality.", "The isoperimetric inequality is also the main ingredient in the proof of (1.1).", "In this paper we will investigate analogs of (1.1) for another type of energy functional which is of interest in connection with convex and plurisubharmonic functions.", "In the case of convex functions, the functional is ${\\mathcal {E}}(\\phi ):=\\int (-\\phi ) MA(\\phi ),$ where $MA(\\phi )$ is the Monge-Ampere measure of $\\phi $ , defined as $MA(\\phi ):=\\det (\\phi _{j k}) dx$ when $\\phi $ is twice differentiable.", "We will only consider this functional when $\\phi $ vanishes on the boundary.", "In the one-dimensional case we can then integrate by parts, so that ${\\mathcal {E}}(\\phi )=\\int |d\\phi |^2$ is the classical energy.", "In the general case we can also integrate by parts, and then find that ${\\mathcal {E}}$ is still an $L^2$ -norm of $d\\phi $ , but the norm of the differential is measured by the Hessian of $\\phi $ .", "This is why this functional makes sense primarily for convex functions.", "We also denote by $\\phi $ the corresponding functional for plurisubharmonic functions.", "$\\Omega $ is then a domain in ${\\mathbb {C}}^n$ and we let ${\\mathcal {E}}(\\phi ):=\\frac{1}{n+1}\\int _\\Omega (-\\phi ) (dd^c\\phi )^n,$ called the pluricomplex or Monge-Ampere energy.", "(Notice that our normalization here is slightly different from the real case.", "It also differs from the definition used in [1] by a sign; here we have chosen signs so that the energy is nonnegative.)", "It is defined for plurisubharmonic functions, vanishing on the boundary, satisfying some extra condition so that the complex Monge-Ampere measure is well defined.", "Just like in the real case, the pluricomplex energy equals the classical (logarithmic) energy when the complex dimension is one.", "We start with the case of plurisubharmonic functions.", "The first problem is that the Schwarz symmetrization of a plurisubharmonic function is not necessarily plurisubharmonic, so the Aubin-Yau energy is not naturally defined.", "Indeed, already when the complex dimension is one and we take $\\phi (z)=\\log |(z-a)/(1-\\bar{a} z)|$ to be the Green kernel, $\\hat{\\phi }$ is subharmonic only if $a=0$ , so that $\\phi $ is already radial.", "(We thank Joaquim Ortega and Pascal Thomas for providing us with this simple example.)", "Our first observation is that if we consider only functions (and domains) that are $S^1$ -invariant, i e invariant under the map $z\\mapsto e^{i\\theta }z$ , then the symmetrization, $\\hat{\\phi }$ , of a plurisubharmonic function, $\\phi $ , is again plurisubharmonic.", "Thus it is meaningful to consider its energy and the main result we prove is that ${\\mathcal {E}}(\\hat{\\phi })\\le {\\mathcal {E}}(\\phi )$ when $\\Omega $ is a ball.", "The condition of $S^1$ -symmetry of course makes this result trivial when $n=1$ , but notice that it is a rather weak restriction in high dimensions as it only means invariance under a one dimensional group.", "In section 4, we study the corresponding problems for convex functions.", "In that case convexity is preserved under Schwarz symmetrization (this must be well known but we include a proof in section 4), so we need no extra condition (like $S^1$ -invariance).", "We then show that for convex functions in the ball, vanishing on the boundary, symmetrization decreases the Monge-Ampere energy, just like in the complex case, and following a similar argument.", "It is natural to ask if these symmetrization results also hold for other domains than the ball.", "In the classical case of the Polya-Szegö theorem one symmetrizes the domain and the function at the same time and it is the symmetrization of the domain that is most clearly linked to the isoperimetric inequality.", "It turns out that the counterpart to this for Monge-Ampere energy does not hold.", "Indeed, in section 2 we prove the somewhat surprising fact that our symmetrization result in the complex case holds if and only if the domain $\\Omega $ is an ellipsoid, i e the image of the Euclidean ball under a complex linear transformation.", "The proof is based on the interpretation of $S^1$ -invariant domains as unit disk bundles of line bundles over projective space, and the proof uses the Bando-Mabuchi uniqueness theorem for Kähler-Einstein metrics on $\\mathbb {P}^n$ .", "In the real case the situation is a little bit more complicated.", "It was first shown by Tso, [8], that the symmetrization inequality fails in general: There is a convex domain and a convex function vanishing on the boundary of that domain, whose Schwarz symmetrization has larger energy.", "In section 4 we first give a general form of Tso's example and relate it to Santalò's inequality.", "We show that if the symmetrization inequality holds for a certain domain $\\Omega $ , then $\\Omega $ must be a maximizer for the Mahler volume, i e for the product of the volume of $\\Omega $ with the volume of its polar body, $\\Omega ^{\\circ }$ .", "By (the converse to) Santalò's inequality this means that $\\Omega $ is an ellipsoid.", "Thus we arrive at the same conclusion as in the complex case, but this time for a completely different reason, that seems to have no counterpart in the complex setting.", "We then argue that if we redefine the Monge-Ampere energy in the real setting by dividing by the Mahler volume we get an energy functional that behaves more like in the complex setting, and for which the phenomenon discovered by Tso disappears.", "That the symmetrization inequality holds for this renormalized energy is thus a weaker statement.", "Nevertheless we show, by an argument similar to the one used in the complex case, that even the weaker inequality holds only for ellipsoids.", "The origin of this paper is our previous article [1], where we studied Moser-Trudinger inequalities of the form $\\log \\int _\\Omega e^{-\\phi }\\le A {\\mathcal {E}}(\\phi ) +B,$ for plurisubharmonic functions in $\\Omega $ that vanish on the boundary.", "It follows immediately from (1.2) that, when $\\Omega $ is a ball and $\\phi $ is $S^1$ -invariant, the proof of inequalities of this type can be reduced to the case of radial functions.", "In [1] we proved a Moser-Trudinger inequality using geodesics in the space of plurisubharmonic functions.", "Here we will use instead symmetrization, but we point out that the proof of our main result (1.2) also uses geodesics.", "By the classical results of Moser, [5], we then obtain in section 3 a sharpening of the Moser-Trudinger inequality from [1].", "Symmetrization was the main tool used by Moser to study the real variable Moser-Trudinger inequality, and it is interesting to note that symmetrization applied to (1.3) leads to the same one variable inequality as in Moser's case.", "As a result we deduce that if $\\phi $ is $S^1$ -invariant and has finite energy that we can normalize to be equal to one , then $\\int _B e^{n(-\\phi )^{(n+1)/n}} < \\infty .$ We do not know if this estimate holds without the assumption of $S^1$ -symmetry." ], [ "Symmetrization of plurisubharmonic functions", "The proofs in this section are based on a result from [2] that we first recall.", "We consider a pseudoconvex domain $\\mathcal {D}$ in ${\\mathbb {C}}^{n+1}$ and its $n$ -dimensional slices $D_t=\\lbrace z\\in {\\mathbb {C}}^n; (t,z)\\in \\mathcal {D}\\rbrace $ where $t$ ranges over (a domain in ) ${\\mathbb {C}}$ .", "We say that a domain $D$ in ${\\mathbb {C}}^n$ is $S^1$ -invariant if $D$ is invariant under the map $z\\mapsto e^{i\\theta } z=( e^{i\\theta }z_1, ...e^{i\\theta }z_n)$ for all $\\theta $ in ${\\mathbb {R}}$ .", "A function (defined in a $S^1$ -invariant domain) is $S^1$ -invariant if $f(e^{i\\theta } z)=f(z)$ for all real $\\theta $ .", "Theorem 2.1 Assume that $\\mathcal {D}$ is a pseudoconvex domain in ${\\mathbb {C}}^{n+1}$ such that all its slices $D_t$ are connected and $S^1$ -invariant.", "Assume also that the origin belongs to $D_t$ when $t$ lies in a domain $U$ in ${\\mathbb {C}}$ .", "Then $\\log |D_t|$ is a superharmonic function of $t$ in $U$ .", "Theorem 2.1 is a consequence of the main result in [2] which says that if $B_t(z,z)$ is the Bergman kernel on the diagonal for the domain $D_t$ , then $\\log B_t(z,z)$ is plurisubharmonic in $\\mathcal {D}$ .", "The hypotheses on $D_t$ in the theorem imply that $B_t(0,0)=|D_t|^{-1}$ which gives the theorem.", "Theorem 2.1 can be seen as a complex variant of the (multiplicative form of) the Brunn-Minkowski inequality, which says that if $\\mathcal {D}$ is instead convex in ${\\mathbb {R}}^{n+1}$ , and the $n$ -dimensional slices are defined in the same way then $\\log |D_t|$ is a concave function of $t$ , without any extra assumptions on the slices.", "The Brunn-Minkowski inequality replaces the use of Theorem 2.1 when we later consider energies of convex functions.", "Then we will use the stronger fact that even $\\sigma ^{1/n}$ is concave in the real setting.", "In the proofs below we will use the following lemma on symmetrizations of subharmonic functions.", "Lemma 2.2 Let $u$ be a smooth subharmonic function defined in an open set $U$ in ${\\mathbb {R}}^N$ , and assume that $u$ vanishes on the boundary of $U$ .", "Let $\\sigma (t):=\\lbrace x; u(x)<t\\rbrace |$ for $t<0$ .", "Then $\\sigma $ is strictly increasing on the interval $(\\min u, 0)$ and the Schwarz symmetrization of $u$ , $\\hat{u}$ , equals $ g(|x|)$ where $g(r)=\\sigma ^{-1}(c_N r^N)$ where $c_N$ is the volume of the unit ball in ${\\mathbb {R}}^N$ , when $c_N r^N>|U_{\\min (u)}|.$ When $c_N r^N\\le |U_{\\min (u)}|,$ $g(r)=\\min (u)$ Denote by $U_t$ the domain where $u<t$ .", "Assume that $|U_t|=|U_{t+\\epsilon }|$ for some $\\epsilon >0$ .", "By Sard's lemma, some $s$ between $t$ and $t+\\epsilon $ is a regular value of $u$ , so the boundary of $U_s$ is smooth.", "By the Hopf lemma, the gradient of $u$ does not vanish on the boundary of $U_s$ unless $u$ is constant in $U_s$ .", "In the latter case $s\\le \\min u$ .", "If this is not the case, i e if $s>\\min u$ , the coarea formula gives $\\sigma ^{\\prime }(s)=\\int _{\\partial U_s} dS/|\\nabla u|>0$ which contradicts that $\\sigma $ is constant on $(t, t+\\epsilon )$ .", "This proves the first part of the lemma.", "The second part follows since $\\sigma ((g(r))=|\\lbrace g(|x|)<g(r)\\rbrace |= c_N r^N.$ We can now easily prove the next basic result.", "We say that a domain $\\Omega $ in ${\\mathbb {C}}^n$ is balanced if for any $\\lambda $ in ${\\mathbb {C}}$ with $|\\lambda |\\le 1$ and $z$ in $\\Omega $ , $\\lambda z$ also lies in $\\Omega $ .", "Theorem 2.3 Let $\\Omega $ be a balanced domain in ${\\mathbb {C}}^n$ .", "Let $\\phi $ be an $S^1$ -invariant plurisubharmonic function in $\\Omega $ .", "Then $\\hat{\\phi }$ , the Schwarz symmetrization of $\\phi $ is plurisubharmonic.", "We may of course assume that $\\phi $ is smooth so that the previous lemma applies.", "By definition, $\\hat{\\phi }$ can be written $\\hat{\\phi }(z)=f(\\log |z|),$ so what we need to prove is that $f$ is convex.", "Since $\\phi $ and $\\hat{\\phi }$ are equidistributed, for any real $t$ , $\\sigma (t):=|\\lbrace z\\in \\Omega ; \\phi (z)<t\\rbrace |=|\\lbrace z\\in \\Omega ; \\hat{\\phi }(z)<t\\rbrace =|\\lbrace z; |z|<\\exp (f^{-1}(t))\\rbrace |.$ Hence $f^{-1}(t)=n^{-1}\\log \\sigma (t) +b_n.$ Since $\\sigma $ is increasing, $f^{-1} $ is also increasing.", "Therefore $f$ is convex precisely when $f^{-1}$ is concave, i e when $\\log \\sigma $ is concave.", "Consider the domain in ${\\mathbb {C}}^{n+1}$ $\\mathcal {D}=\\lbrace (\\tau ,z); z\\in \\Omega \\quad \\text{and} \\quad \\phi (z)-{\\rm Re\\, }\\tau <0\\rbrace .$ Then, if $t={\\rm Re\\, }\\tau $ , $\\sigma (t)=|D_\\tau |$ .", "Note that $\\mathcal {D}$ is pseudoconvex since $\\phi -{\\rm Re\\, }\\tau $ is plurisubharmonic and we claim that $\\mathcal {D}$ also satisfies all the other conditions of Theorem 2.1 .", "Let $z$ lie in $D_\\tau $ for some $\\tau $ .", "The function $\\gamma (\\lambda ):=\\phi (\\lambda z)$ is then subharmonic in the unit disk, and moreover it is radial, i e $\\gamma (\\lambda )=g(|\\lambda |),$ where $g$ is increasing.", "Therefore the whole disk $\\lbrace \\lambda z\\rbrace $ is contained in $D_\\tau $ .", "In particular the origin lies in any $D_\\tau $ , and the origin can be connected with $z$ by a curve, so ${\\mathcal {D}}_{\\tau }$ is connected.", "Thus Theorem 2.1 can be applied and we conclude that $\\log \\sigma ({\\rm Re\\, }\\tau )=\\log |D_\\tau |$ is a superharmonic function of $\\tau $ .", "Since this function only depends on ${\\rm Re\\, }\\tau $ it is actually concave, and the proof is complete.", "The next theorem is the main result of this paper, and here we need to assume that $\\Omega $ is a ball.", "See the remarks below for a discussion of the problem with considering more general domains.", "Theorem 2.4 Let $\\phi $ be plurisubharmonic in the unit ball, and assume that $\\phi $ extends continuously to the closed ball with zero boundary values.", "Assume also that $\\phi $ is $S^1$ -invariant, and let $\\hat{\\phi }$ be the Schwarz symmetrization of $\\phi $ .", "Then ${\\mathcal {E}}(\\hat{\\phi })\\le {\\mathcal {E}}(\\phi ).$ In the proof of Theorem 2.1 we used the geometrically obvious fact that the inverse of an increasing concave function is convex.", "We will need a generalization of this that we state as a lemma.", "Lemma 2.5 Let $a(s,t)$ be a concave function of two real variables.", "Assume $a$ is strictly increasing with respect to $t$ , and let $t=k(s,x)$ be the inverse of $a$ with respect to the second variable for $s$ fixed, so that $a(s,k(s,x))=x$ .", "Then $k$ is convex as a function of both variables $s$ and $x$ .", "Assume not.", "After choosing a new origin, there is then a point $p=(s_{0},x_{0})$ such that $k(0,0)>(k(p)+k(-p))/2.$ Since $a$ is strictly increasing with respect to $t$ $0=a(0,k(0,0))>a((s_{0}-s_{0})/2,(k(p)+k(-p))/2)\\ge $ $\\ge (a(s_{0},k(p))+a(-s_{0},k(-p)))/2=(x_{0}-x_{0})/2=0.$ This is a contradiction .", "In the sequel we shall use well known facts about geodesics and subgeodesics in the space of plurisubharmonic functions on the ball, see [1] for proofs.", "These are curves $\\phi _t(z)=\\phi (t,z)$ where $t$ is real parameter, here varying between 0 and 1.", "By definition, $\\phi _t$ is a subgeodesic if $\\phi ({\\rm Re\\, }\\tau ,z)$ is plurisubharmonic as a function of $(\\tau ,z)$ , and it is a geodesic if moreover this plurisubharmonic function solves the homogenuous complex Monge-Ampere equation $(dd^c\\phi )^{n+1}=0.$ We also assume throughout that $\\phi _t$ vanishes for $|z|=1$ .", "It is not hard to see that if $\\phi _0$ and $\\phi _1$ are two continuous plurisubharmonic functions in the ball, vanishing on the boundary, then they can be connected with a bounded geodesic, see [1].", "Here we shall first assume that $\\phi _0$ and $\\phi _1$ are smooth and can be connected by a geodesic of class $C^1$ , and then get the inequality for general $\\phi $ by approximation.", "We will use the following three facts, for which we refer to [1].", "First, ${\\mathcal {E}}(\\phi _t)$ is an affine function of $t$ along any bounded geodesic.", "Second, ${\\mathcal {E}}(\\phi _t)$ is concave along a bounded subgeodesic.", "On the other hand, if $\\phi _0$ and $\\phi _1$ are plurisubharmonic and vanish on the boundary and we let $\\phi _t=t\\phi _1 +(1-t)\\phi _0$ for $t$ between 0 and 1, then ${\\mathcal {E}}(\\phi _t)$ is convex.", "Finally, if $\\phi _t$ is of class $C^1$ , then ${\\mathcal {E}}(\\phi _t)$ is differentiable with derivative $\\frac{d}{dt}{\\mathcal {E}}(\\phi _t)=\\int _B-\\dot{\\phi _t}(dd^c_z\\phi _t)^n.$ In the proof of Theorem 2.3 we fix a plurisubharmonic $\\phi $ in the ball, that we assume smooth.", "We put $\\phi =\\phi _1$ and connect it with $\\phi _0$ , chosen to satisfy an equation $(dd^c\\phi _0)^n= F(\\phi _0),$ where $F$ is some smooth function of a real variable.", "Actually, it is not hard to check that any smooth increasing radial function satisfies such an equation.", "We also first assume that $\\phi =\\phi _1$ and $\\phi _0$ can be connected with a $C^1$ geodesic $\\phi _t$ .", "We then take the Schwarz symmetrization of each $\\phi _t$ and obtain another curve $\\hat{\\phi }_t$ .", "The next proposition shows that the new curve is a subgeodesic.", "Proposition 2.6 Let $\\phi _{t}$ be a subgeodesic of $S^{1}$ -invariant plurisubharmonic functions.", "Then $\\hat{\\phi }_{t}$ is also a subgeodesic.", "Let $\\phi _{s}$ be a subgeodesic which we may assume to be smooth.", "Let $A(s,t)=|\\lbrace z,\\phi _{s}(z)<t\\rbrace |.$ It follows again from Theorem 2.1 that $a:=\\log A$ is a concave function of $s$ and $t$ together.", "As in the proof of Theorem 2.3 all we need to prove is that the inverse of $a$ with respect to $t$ (for $s$ fixed), $k(s,x)$ ) is convex with respect to $s$ and $t$ jointly.", "But this is precisely the content of Lemma 2.4.", "We now first sketch the principle of the argument and fill in some details and change the set up a little bit afterwords.", "Consider the energy functionals along the two curves $\\phi _t$ and $\\hat{\\phi }_t$ , ${\\mathcal {E}}(\\phi _t)=:g(t)$ and ${\\mathcal {E}}(\\hat{\\phi }_t)=h(t)$ .", "Since $\\phi _0$ is already radial, $g(0)=h(0)$ , and we want to prove that $g(1)\\ge h(1)$ .", "We know that $g$ is affine and that $h$ is concave, so this follows if we can prove that $g^{\\prime }(0)=h^{\\prime }(0)$ .", "But $g^{\\prime }(0)=\\int -\\dot{\\phi }_0 (dd^c\\phi _0)^n,$ since the geodesic is $C^1$ .", "We also claim that we can arrange things so that $h^{\\prime }(0)=\\int -\\frac{d\\hat{\\phi }_t }{dt}|_{t=0}\\,(dd^c\\phi _0)^n.$ By the choice of $\\phi _0$ , $g^{\\prime }(0)=\\int -\\dot{\\phi }_0 F(\\phi _0)=\\frac{d}{dt}|_{t=0}\\int -G(\\phi _t),$ if $G^{\\prime }=F$ .", "Similarily $h^{\\prime }(0)=\\frac{d}{dt}|_{t=0}\\int -G(\\hat{\\phi }_t).$ But, since $\\phi _t$ and $\\hat{\\phi }_t$ are equidistributed $\\int -G(\\phi _t)=\\int -G(\\hat{\\phi }_t)$ for all $t$ .", "Hence $g^{\\prime }(0)=h^{\\prime }(0)$ and the proof is complete.", "It remains to see why we can assume that the geodesic $\\phi _t$ is $C^1$ and to motivate the claim about the derivative of $h$ .", "First, since we have assumed that $\\phi _0$ and $\\phi _1$ are smooth up to the boundary, we can by a max construction assume that they are booth equal to $A \\log ((1+|z|^2)/2)$ for some large $A>0$ , when $|z|>(1-\\epsilon )$ .", "Then $\\phi _0$ and $\\phi _1$ can be extended to psh functions in all of ${\\mathbb {C}}^n$ , equal to $A \\log ((1+|z|^2)/2)$ outside of the unit ball.", "In fact, we can even consider them as metrics on a line bundle $\\mathcal {O}(A)$ over $\\mathbb {P}^n$ .", "It the follows from Chen's theorem, [3], that they can be connected by a $C^1$ geodesic in the space of metrics on $\\mathcal {O}(A)$ .", "It is easy to see that this geodesic must in fact be equal to $A \\log ((1+|z|^2)/2)$ for $|z|> 1-\\epsilon $ for some positive $\\epsilon $ .", "In particular, it vanishes on the boundary of the ball, and $\\dot{\\phi }_t$ is identically zero near the boundary.", "To handle the claim about the derivative of $h$ we change the setup a little bit.", "$\\hat{\\phi }_0=\\phi _0$ is smooth and we can approximate $\\hat{\\phi }_1$ from above by a smooth radial plurisubharmonic function.", "Now connect these two smooth functions by a geodesic, $\\psi _t$ , which can be take to be $C^{(1,1)}$ by the above argument.", "(As a matter of fact it will even be smooth, since geodesics between radial functions come from geodesics between smooth convex functions, which are smooth).", "Let ${\\mathcal {E}}(\\psi _t)=:k(t).$ Since $\\psi _t\\ge \\hat{\\phi }_t$ , $-\\dot{\\psi }_0\\le -\\dot{\\hat{\\phi }}_0$ .", "We then apply the above argument to $k$ instead of $h$ and find that $k(1)\\le g(1)$ .", "Taking limits as $\\psi _1$ tends to $\\hat{\\phi }_1$ we conclude the proof." ], [ "Other domains.", "Let us consider a smoothly bounded balanced domain $\\Omega $ in ${\\mathbb {C}}^n$ which we can write as $\\Omega =\\lbrace z; u_\\Omega (z)<0\\rbrace $ where $u_\\Omega $ is logarithmically homogenuous , i e $u_\\Omega (\\lambda z)=\\log |\\lambda | +u_\\Omega (z)$ .", "Indeed, $u_\\Omega $ is the logarithm of the Minkowski functional for $\\Omega $ .", "We first claim that if $\\Omega $ is pseudoconvex then $u_\\Omega $ is plurisubharmonic.", "Lemma 2.7 Let $u$ be a smooth function such that $D:=\\lbrace (w,z); u(z)-{\\rm Re\\, }w<0\\rbrace $ is pseudoconvex.", "Then $u$ is plurisubharmonic.", "At a point $z$ where $du=0$ the Levi form of the boundary of $D$ is precisely $dd^c u$ so if $D$ is pseudoconvex $dd^c u\\ge 0$ at such points.", "The general case is reduced to this by subtracting a linear form ${\\rm Re\\, }a\\cdot z$ from $u$ and considering the biholomorphic transformation $(w,z)\\mapsto (w+a\\cdot z, z)$ .", "Since the set $\\lbrace u_\\Omega (z)-{\\rm Re\\, }w<0\\rbrace =\\lbrace u_\\Omega (z e^{-w})<0\\rbrace $ is pseudoconvex it follows from the lemma that $u_\\Omega $ is plurisubharmonic.", "Let us now consider $S^1$ -invariant functions in $\\Omega $ of the form $\\phi (z)=f(u_\\Omega )$ where $f$ is convex.", "If we normalize so that the volume of $\\Omega $ equals the volume of the unit ball it is clear that the Schwarz symmetrization of $\\phi $ , $\\hat{\\phi }$ is $f(\\log |z|)$ .", "Proposition 2.8 If $\\phi =f(u_\\Omega )$ with $f$ convex, the Monge-Ampere energy of $\\phi $ equals $\\int _\\Omega (-\\phi )(dd^c\\phi )^n=2^{-n}\\int _{-\\infty }^0 (f^{\\prime })^{n+1}(t)dt.$ In particular, the energy of $\\phi $ is equal to the energy of $\\hat{\\phi }$ , the Schwarz symmetrization of $\\phi $ .", "In the proof we use the next lemma.", "Lemma 2.9 If $\\phi =f(u_\\Omega )$ with $f$ convex $\\int _{u_\\Omega <s} (dd^c\\phi )^{n}= 2^{-n}f^{\\prime }(s)^n.$ $\\int _{u_\\Omega <s} (dd^c\\phi )^{n}=\\int _{u_\\Omega =s}d^c\\phi \\wedge (dd^c\\phi )^{n-1}=f^{\\prime }(s)^n\\int _{u_\\Omega =s}d^cu_\\Omega \\wedge (dd^c u_\\Omega )^{n-1}.$ But $\\int _{u_\\Omega =s}d^cu_\\Omega \\wedge (dd^c u_\\Omega )^{n-1}=\\int _{u_\\Omega <s}(dd^c u_\\Omega )^n.$ Since $u_\\Omega $ is log homogenous it satsifies the homogenous Monge-Ampere equation outside of the origin, so $(dd^c u_\\Omega )^n$ is a Dirac mass at the origin.", "But $u_\\Omega -\\log |z|$ is bounded near the origin so this point mass must be same as $(dd^c\\log |z|)^n= 2^{-n}.$ Proof of Proposition 2.8 First assume that $f(s)$ is constant for $s$ sufficiently large negative.", "Let $\\sigma (s):= \\int _{u_\\Omega <s} (dd^c\\phi )^n.$ Then ${\\mathcal {E}}(\\phi )=\\int ^0 -f(s)d\\sigma (s)=\\int ^0 \\sigma (s)f^{\\prime }(s)ds,$ so the formula for the energy folows from the previous lemma.", "The general case, when $f$ is not constant near $-\\infty $ follows from approximation.", "The last statement, that ${\\mathcal {E}}(\\phi )={\\mathcal {E}}(\\hat{\\phi })$ then follows if $|\\Omega |$ equals the volume of the unit ball, since then $\\hat{\\phi }=f(\\log |z|)$ .", "But then the same thing must hold in general since the energy is invariant under scalings.", "$\\Box $ Let us now define the '$\\Omega $ -symmetrization', $S_\\Omega (\\phi )$ , of a plurisubharmonic function $\\phi $ in $\\Omega $ , vanishing on the boundary, $\\phi $ , as the unique function of the form $f(u_\\Omega )$ that is equidistributed with $\\phi $ .", "Notice that if the $\\Omega $ -symmetrization of $\\phi $ equals $f(u_\\Omega )$ , then the Schwarz symmetrization is given by $\\hat{\\phi }= f(\\log (|z|/R)$ , if $R$ is chosen so that the volume of $\\Omega $ equals the volume of the ball of radius $R$ .", "The last part of proposition 2.8 then says that ${\\mathcal {E}}_\\Omega (S_\\Omega (\\phi ))={\\mathcal {E}}_B(\\hat{\\phi }),$ where we have put subscripts on ${\\mathcal {E}}$ to emphasize over which domain we compute the energy, and $B$ denotes a ball of the same volume as $\\Omega $ .", "Notice also that it follows from Theorem 2.3 that $S_\\Omega (\\phi )$ is plurisubharmonic if $\\phi $ is plurisubharmonic.", "Indeed, Theorem 2.3 says that $\\hat{\\phi }=f(\\log |z|)$ is plurisubharmonic, i e that $f$ is convex and increasing, from which it follows that $f(u_\\Omega )$ is plurisubharmonic.", "We therefore see that to prove that the Schwarz symmetrization of a function $\\phi $ on $\\Omega $ has smaller Monge-Ampere energy than $\\phi $ is equivalent to proving that the $\\Omega $ -symmetrization of $\\phi $ has smaller energy than $\\phi $ .", "One might try to prove this by following the same method as in the proof of Theorem 2.4.", "The point where the proof breaks down is that we need to choose a reference function on $\\Omega $ that satisfies an equation of the form $(dd^c\\phi _0)^n=F(\\phi _0)$ where $\\phi _0$ is of the form $\\phi _0=f(u_\\Omega )$ .", "This is easy if $\\Omega $ is the ball so that $u_\\Omega =\\log |z|$ since $(dd^c\\phi _0)^n$ then is invariant under the unitary group if $\\phi _0$ is, and hence must be radial.", "Nothing of the sort holds for other domains.", "Since, outside the origin, $(dd^c\\phi _0)^n=f^{\\prime \\prime }(u_\\Omega )f^{\\prime }(u_\\Omega )^{n-1}du_\\Omega \\wedge d^cu_\\Omega \\wedge (dd^c u_\\Omega )^{n-1},$ what we want is that the determinant of the Leviform $du_\\Omega \\wedge d^cu_\\Omega \\wedge (dd^c u_\\Omega )^{n-1}$ be constant on all level surfaces of $u_\\Omega $ .", "This is clearly true if $\\Omega $ is a ball, and therefore also true if $\\Omega $ is the image of a ball under a complex linear transformation.", "We shall next see that these are the only cases when this holds.", "Proposition 2.10 Let $\\Omega $ be a balanced domain in $\\mathbb {C}^n$ and let $u_\\Omega $ be the uniquely determined logarithmically homogenous (plurisubharmonic) function that vanishes on the boundary of $\\Omega $ .", "Assume $u_\\Omega $ satisfies the condition that (2.3) be constant on some, and therefore every , level surface of $u_\\Omega $ .", "Then $\\Omega $ is an ellipsoid $\\Omega =\\lbrace z; \\sum a_{j k} z_j\\bar{z}_k <1\\rbrace $ for some positively definite matrix $A=(a_{j k})$ .", "We will use the relation between logarithmically homogenous functions on $\\mathbb {C}^n$ and metrics on the tautological line bundle $\\mathcal {O}(-1)$ on $\\mathbb {P}^{n-1}$ .", "Recall that $\\mathbb {P}^{n-1}$ is the quotient of $\\mathbb {C}^n\\setminus \\lbrace 0\\rbrace $ under the equivalence relation $z\\sim \\lambda z$ if $\\lambda $ is a nonzero complex number.", "Let $p(z)=[z]$ be the projection map from $\\mathbb {C}^n\\setminus \\lbrace 0\\rbrace $ to $\\mathbb {P}^{n-1}$ ; $[z]$ being the representation of a point in homogenous coordinates.", "Then $\\mathbb {C}^n\\setminus \\lbrace 0\\rbrace $ can be interpreted as the total space of $\\mathcal {O}(-1)$ , minus its zero section.", "A logarithically homogenous function like $u_\\Omega $ can then be written $u_\\Omega =\\log |z|_h$ for some metric $h$ on $\\mathcal {O}(-1)$ .", "In an affine chart $[z]=[(1,\\zeta )]$ on $\\mathbb {P}^{n-1}$ with associated trivialization of $\\mathcal {O}(-1)$ ; $z=(\\lambda , \\lambda \\zeta )$ , $|z|^2_h=|\\lambda |^2 e^{\\psi (\\zeta )}$ where $-\\psi $ is a local representative for the metric $h$ on $\\mathcal {O}(-1)$ .", "Hence $u_\\Omega =\\log |\\lambda | +(1/2)\\psi (\\zeta ).$ Let us now look at the form (2.3).", "If it is constant on level surfaces it must be equal to $ F(u) dz\\wedge d\\bar{z}$ for some function $F(u)$ .", "By log-homogenuity, the form is moreover homogenous of degree $-2n$ , so we must have $F(u)= Ce^{-2n u}.$ Changing coordinates to $(\\lambda , \\lambda \\zeta )$ we get $du_\\Omega \\wedge d^cu_\\Omega \\wedge (dd^c u_\\Omega )^{n-1}= C e^{-2n u} |\\lambda |^{2n-2} d\\lambda \\wedge d\\bar{\\lambda }\\wedge d\\zeta \\wedge d\\bar{\\zeta }.$ On the other hand, by (2.4), $du_\\Omega \\wedge d^cu_\\Omega \\wedge (dd^c u_\\Omega )^{n-1}=C^{\\prime } |\\lambda |^{-2} d\\lambda \\wedge d\\bar{\\lambda }\\wedge (dd^c\\psi )_{n-1}.$ Hence $(dd^c\\psi )_{n-1}= C^{\\prime \\prime } e^{-n\\psi } d\\zeta \\wedge d\\bar{\\zeta }.$ This means precisely that the metric $n\\psi $ on the anticanonical line bundle $\\mathcal {O}(n)$ on $\\mathbb {P}^{n-1}$ solves the Kähler-Einstein equation .", "But all such metrics can be written (on the total space) $\\log |\\chi |_h^2= n\\log |A z|^2$ where $z$ are the standard coordinates on $\\mathbb {C}^n$ and $A$ is a positively definite matrix.", "(This follows e g from the Bando-Mabuchi uniqueness theorem, which says that any Kähler-Einstein metric can be obtained from the standard metric $\\log |z|^2$ via a holomorphic automorphism.)", "Hence $u_\\Omega =\\log |Az|,$ so $\\Omega $ is an ellipsoid.", "We shall now finally show that the relevance of the form (2.3) is not just an artefact of the proof.", "Indeed, we shall show that if the symmetrization inequality ${\\mathcal {E}}_B(\\hat{\\phi })\\le {\\mathcal {E}}_\\Omega (\\phi )$ holds for all $S^1$ -invariant plurisubharmonic functions $\\phi $ in $\\Omega $ that vanish on the boundary, then $\\Omega $ must satisfy the hypothesis of proposition 2.10, and therefore be an ellipsoid.", "Let $\\psi _0=f_0(\\log |z|)$ be a function in the ball that solves a Kahler-Einstein type equation $(dd^c\\psi )^n= ce^{-\\psi } i^{n^2}dz\\wedge d\\bar{z}.$ Then $\\psi _0$ is a critical point for a functional of the type ${\\mathcal {F}}_B(\\phi ):=\\log \\int _B e^{-\\phi }-c^{\\prime }{\\mathcal {E}}_B(\\phi ) ,$ see [1] for more on this.", "From this it follows that ${\\mathcal {F}}_B(\\psi ^{\\prime })\\le {\\mathcal {F}}_B(\\psi _0)$ for all $S^1$ -invariant plurisubharmonic functions in the ball that vanish on the boundary.", "This is explained in [1], so we just indicate the argument here.", "The point is that the functional ${\\mathcal {F}}_B$ is concave along geodesics in the space of $S^1$ -invariant plurisubharmonic functions that vanish on the boundary.", "This follows from two facts.", "First, the energy term is affine along geodesics, and second, the function $\\log \\int _B e^{-\\phi _t}$ is concave under (sub)geodesics.", "The latter fact follows again from the main result in [2] on plurisubharmonic variation of Bergman kernels, since $( \\int _B e^{-\\phi _t})^{-1}$ is the Bergman kernel at the origin for the weight $\\phi _t$ if $\\phi _t$ is $S^1$ -invariant.", "Given the concavity of ${\\mathcal {F}}_B$ it then follows that a critical point is a maximum, i e (2.5) holds.", "Let us now consider the analogous functional defined on functions on $\\Omega $ , ${\\mathcal {F}}_\\Omega (\\phi ):=\\log \\int _\\Omega e^{-\\phi }-c^{\\prime }{\\mathcal {E}}_\\Omega (\\phi ) .$ Assume, to get a contradiction that ${\\mathcal {E}}_\\Omega (S_\\Omega (\\phi )\\le {\\mathcal {E}}_\\Omega (\\phi )$ .", "Then ${\\mathcal {F}}_\\Omega $ increases under $\\Omega $ -symmetrization.", "Moreover, by proposition 2.8, ${\\mathcal {F}}_\\Omega (S_\\Omega (\\phi ))={\\mathcal {F}}_B(\\hat{\\phi })$ so ${\\mathcal {F}}_\\Omega (\\phi )\\le {\\mathcal {F}}_B(\\hat{\\phi })\\le {\\mathcal {F}}_B(\\psi _0),$ where $\\psi _0$ is the Kähler-Einstein potential discussed above.", "Hence the maximum of the left hand side over all $\\phi $ is attained for $\\phi =f_0(u_\\Omega ).$ But then it is easy to see that $\\phi $ solves the same Kahler-Einstein equation as $\\psi _0$ .", "Indeed, at least if $\\Omega $ is strictly pseudoconvex, $\\phi $ is strictly plurisubharmonic ouside the origin.", "Therefore small perturbations of $\\phi $ are still plurisubharmonic and the variational equation for ${\\mathcal {F}}_\\Omega $ is just the Kähler-Einstein equation.", "In particular, $\\phi $ solves an equation of type (2.2) in $\\Omega $ , which we have seen is possible only if $\\Omega $ is an ellipsoid.", "We summarize the discussion in the next theorem.", "Theorem 2.11 Let $\\Omega $ be a strictly pseudoconvex balanced domain for which the symmetrization inequality ${\\mathcal {E}}_B(\\hat{\\phi })\\le {\\mathcal {E}}_\\Omega (\\phi )$ holds for all $S^1$ -invariant plurisubharmonic $\\phi $ that vanish on the boundary.", "Then $\\Omega $ is an ellipsoid." ], [ "A sharp Moser-Trudinger inequality for $S^1$ -invariant functions.", "Our results in the previous section, together with Moser's inequality imply rather easily the next estimate.", "Theorem 3.1 Let $\\phi $ be a smooth $S^1$ -invariant plurisubharmonic function in the unit ball that vanishes on the boundary.", "Let ${\\mathcal {E}}:={\\mathcal {E}}(\\phi ).$ Then $\\int _{\\mathcal {B}}e^{n{\\mathcal {E}}^{-{1/n}} (-\\phi )^{n+1)/n}}\\le C$ where $C$ is an absolute constant.", "In the proof we may by our main result on symmetrization assume that $\\phi (z)=f(\\log |z|)$ is a radial function.", "The main result of Moser, [5], is that if $w$ is an increasing function on $(-\\infty , 0)$ that vanishes when $t$ goes to zero and satisfies $\\int _{-\\infty }^0 (-w^{\\prime })^{n+1} dt\\le 1$ then $\\int _{-\\infty }^0 e^{(-w)^{(n+1)/n}} e^{t} dt \\le C,$ where $C$ is an absolute constant.", "Applying this to $w_\\kappa (s):=\\kappa ^{n/(n+1)}w(s/\\kappa )$ we obtain that $\\int _{-\\infty }^0 e^{\\kappa (-w)^{(n+1)/n}} e^{\\kappa t} dt \\le C/\\kappa ,$ under the same hypothesis.", "Next we have the following lemma.", "Lemma 3.2 Let $f$ be an increasing convex function on $(-\\infty , 0]$ with $f(0)=0$ , and let $\\phi (z)=f(\\log |z|)$ .", "Let $F$ be a nonnegative measurable function of one real variable.", "Then (a) $\\int _{\\mathcal {B}}F\\circ \\phi =a_n\\int _{-\\infty }^0 F\\circ f e^{2n t} dt$ (with $a_n$ being the area of the unit sphere in ${\\mathbb {C}}^n$ ).", "and (b) ${\\mathcal {E}}= 2^{-n}\\int _{-\\infty }^0 (f^{\\prime })^{n+1} dt.$ The first formula follows from $\\int _BF(\\phi )=\\int ^0 F\\circ f d\\sigma (t)$ where $\\sigma =|\\lbrace z; |z|\\le e^t\\rbrace = \\pi ^n/n!", "e^{2nt}$ .", "The second formula is a special case of Proposition 2.8.", "Applying the scaled version of Moser's result with $-w= f E^{-1/(n+1)}2^{-n/(n+1)}$ and $\\kappa =2n$ the theorem follows.", "To relate this to Moser-Trudinger inequalities of the form studied in [1] we start with the elementary inequality for positive numbers $x$ and $\\xi $ $x\\xi \\le \\frac{1}{n+1} x^{n+1} +\\frac{n}{n+1}\\xi ^{(n+1)/n}$ (valid since $(n+1)$ and $(n+1)/n$ are dual exponents).", "This implies $\\xi \\le \\frac{1}{n+1} x^{n+1} +\\frac{n}{n+1}\\xi ^{(n+1)/n}/x^{(n+1)/n}.$ Choose $x$ so that $x^{n+1}=E/(n+1)^n$ and take $\\xi =(-\\phi )$ .", "Then $-\\phi \\le \\frac{1}{(n+1)^{n+1}}E +n E^{-1/n}(-\\phi )^{(n+1)/n}.$ Therefore Theorem 3.1 implies the sharp Moser-Trudinger inequality for $S^1$ -invariant functions from [1] $\\log \\int e^{-\\phi }\\le \\frac{1}{(n+1)^{n+1}}{\\mathcal {E}}(\\phi ) + B,$ with $B=\\log C$ , $C$ the universal constant in Moser's estimate." ], [ "Symmetrization of convex functions", "First we note the following analog of Theorem 2.2.", "Theorem 4.1 Let $\\phi $ be a convex function defined in a convex domain $\\Omega $ in ${\\mathbb {R}}^n$ and let $\\hat{\\phi }$ be its Schwarz symmetrization.", "Then $\\hat{\\phi }$ is also convex.", "This fact should be well known but we include a proof in order to emphazise the similarity with Theorem 2.2.", "By definition $\\hat{\\phi }(x)=g(|x|)$ for some increasing function $g$ and we need to prove that $g$ is convex (notice the change in convention as compared with the complex case where we wrote $\\hat{\\phi }(z)=f(\\log |z|)$ ).", "As before $\\sigma (t):=|\\lbrace x\\in \\Omega ; \\phi (x)<t\\rbrace |= a_n (g^{-1}(t))^n,$ so it suffices to prove that $\\sigma ^{1/n}$ is concave.", "But if we put $\\mathcal {D}:=\\lbrace (t,x); x\\in \\Omega \\, \\text{and}\\, \\phi (x)-t<0\\rbrace ,$ $\\sigma (t)$ is the volume of the slices $D_t$ .", "By the Brunn-Minkowski theorem (see section 2) it follows that $\\sigma ^{1/n}$ is concave, and we are done.", "We next state the real variable analog of Theorem 2.3.", "Theorem 4.2 Let $\\phi $ be a convex function in the ball, continuous on the closed ball and vanishing on the boundary.", "Let $\\hat{\\phi }$ be its Schwarz symmetrization.", "Then ${\\mathcal {E}}(\\hat{\\phi })\\le {\\mathcal {E}}(\\phi ).$ This is proved in a way completely parallell to the complex case, so we shall not give the details.", "We define geodesics and subgeodesics in the space of convex functions as before.", "Then the real energy is concave along subgeodesics and affine along geodesics as before and the analog of the formula for the first order derivative also holds.", "We can therefore repeat the proof practically verbatim." ], [ " Other domains", "We have already seen in section 2 that in the complex case, the energy does not in general decrease under Schwarz symmetrization if we consider functions defined on domains different than the ball.", "In the real setting, the first counterexample to the same effect was given by Tso, [8].", "We shall first discuss Tso's counterexample, and start by giving the example in a more general form.", "In the next theorem appears the Mahler volume of a convex set $\\Omega $ containing the origin.", "It is defined as $M(\\Omega ):=|\\Omega ||\\Omega ^\\circ |,$ where $\\Omega ^\\circ $ is the polar body of $\\Omega $ .", "In the sequel we will write ${\\mathcal {E}}_\\Omega $ for the energy of functions defined in $\\Omega $ .", "Theorem 4.3 Let $\\Omega $ be a bounded convex domain in ${\\mathbb {R}}^n$ containing the origin, and let $\\mu _\\Omega $ be the Minkowski functional of $\\Omega $ .", "Let $u$ be a convex function in $\\Omega $ of the form $u(x)=f(\\mu _\\Omega (x))$ , and let $\\hat{u}$ be its Schwarz symmetrization.", "Then $M(\\Omega )^{-1}{\\mathcal {E}}_\\Omega (u)=M(B_\\Omega )^{-1}{\\mathcal {E}}_B(\\hat{u})$ (where $B_\\Omega $ is the ball of the same volume as $\\Omega $ ).", "Notice that we could as well have divided by just $|\\Omega ^\\circ |$ instead of the Mahler volume, since the volumes of $\\Omega $ and $B_\\Omega $ are automatically equal, but the Mahler volume seems to simplify a little below.", "From the theorem we see that if ${\\mathcal {E}}_B(\\hat{u})\\le {\\mathcal {E}}_\\Omega (u)$ it follows that we have an inequality for the Mahler volumes $M(B)\\le M(\\Omega ).$ This inequality fails in a very strong way.", "Indeed, if we assume that $\\Omega $ is also symmetric so that $-\\Omega =\\Omega $ , then Santalò's inequality, [7], says that the opposite is true $M(B)\\ge M(\\Omega ).$ (Tsos's counterexample is the case of Theorem 4.3 when $\\Omega $ is a simplex.)", "It therefore seems that in the real case it is natural to normalize the energy by dividing by the Mahler volume of the domain.", "Notice that this is a difference as compared to the complex setting, where Theorem 4.3 holds without normalization.", "The reason for this is that in the case of ${\\mathbb {R}}^n$ , the Minkowski functional of a convex domain $\\Omega $ satisfies the equation $MA(\\mu _\\Omega )= |\\Omega ^0|.$ On the other hand in ${\\mathbb {C}}^n$ we have if $\\Omega $ is a balanced domain that $(dd^c\\log \\mu _\\Omega )^n=(dd^c\\log |z|)^n= 2^{-n}\\delta _0,$ where $\\delta _0$ is a point mass at the origin, and thus is independent of the domain.", "The question then becomes if $M(\\Omega )^{-1}{\\mathcal {E}}_\\Omega (u)\\ge M(B)^{-1}{\\mathcal {E}}_B(\\hat{u})$ for any convex function $u$ on $\\Omega $ that vanishes on the boundary.", "Just like in the complex case we define for a convex function $u$ defined on some convex domain $L$ , its $\\Omega $ -symmetrization $S_\\Omega (u)$ as the unique function, equidistributed with $u$ which can be written $S_\\Omega (u)=f(\\mu _\\Omega ).$ Note that the $\\Omega $ -symmetrization of $u$ is the same as the $\\Omega ^{\\prime }$ -symmetrization if $\\Omega $ and $\\Omega ^{\\prime }$ are homothetic.", "Moreover, since $|\\lbrace S_\\Omega (u)<0\\rbrace |=|\\lbrace u<0\\rbrace |$ $S_\\Omega (u)$ vanishes on the boundary of a multiple $s\\Omega $ of $\\Omega $ , with $s$ chosen so that $s\\Omega $ has the same volume as $L$ , if $u$ vanishes on the boundary of $L$ .", "Notice that if $\\Omega $ is a ball, centered at the origin, $S_\\Omega $ is just the Schwarz symmetrization.", "In terms of $\\Omega $ -symmetrizations, Theorem 4.3 says that the normalized energy of all $\\Omega $ -symmetrizations coincide: $M(\\Omega )^{-1}{\\mathcal {E}}_\\Omega (S_\\Omega (u))=M(\\Omega ^{\\prime })^{-1}{\\mathcal {E}}_{\\Omega ^{\\prime }}(S_{\\Omega ^{\\prime }}(u)),$ if $\\Omega $ and $\\Omega ^{\\prime }$ are two convex domains.", "The desired inequality (4.1) thus means that ${\\mathcal {E}}_\\Omega (S_\\Omega (u))\\le {\\mathcal {E}}_\\Omega (u)$ for convex functions $u$ on $\\Omega $ that vanish on the boundary.", "This would be the analog of Theorem 4.2 for general convex domains, and it is precisely the same question that we discussed in the complex case.", "Just like in the complex case we shall now see that this holds only for ellipsoids.", "Most of the argument is completely parallell to the complex case and will be omitted.", "Only the last part, involving Kähler-Einstein metrics has to be changed and we shall now describe how this is done.", "As in the complex case we see that if the symmetrization inequality holds, then $\\mu _\\Omega $ , the Minkowski functional of $\\Omega $ satsifies a condition of the form: There is a convex function of $\\mu _\\Omega $ such that $u=f(\\mu _\\Omega )$ satisfies an equation $MA(u)= F(u)$ for some function $F$ .", "To see the meaning of this more explicitly we resort to the complex formalism.", "Define $u$ and $\\mu _\\Omega $ on $\\mathbb {C}^n$ by putting $u(z)=u(x)$ etc, i e by letting all functions involved be independent of the imaginary part of $z$ .", "Then $MA(u)d\\lambda (z)= C(dd^c u)^n$ where $d\\lambda $ is the standard volume form on $\\mathbb {C}^n$ .", "Since $MA(\\mu _\\Omega )=0$ outside the origin, it follows if $u=f(\\mu _\\Omega )$ that $c^{\\prime }(dd^c u)^n=(f^{\\prime }(\\mu ))^{n-1}f^{\\prime \\prime }(\\mu )d\\mu \\wedge d^c \\mu \\wedge (dd^c \\mu )^{n-1}.$ Hence we see that $d\\mu \\wedge d^c \\mu \\wedge (dd^c \\mu )^{n-1}=G(\\mu ) d\\lambda $ for $x\\ne 0$ , where we write $\\mu $ instead of $\\mu _\\Omega $ since $\\Omega $ is now fixed.", "Since $\\mu $ is homogenous of degree 1, $d\\mu $ is homogenous of degree zero, and $dd^c\\mu $ is homogenous of degree -1.", "Therefore the left hand side is homogenous of degree $(n-1)$ so we can take $G(\\mu )=\\mu ^{1-n}$ .", "It also follows from this equation that any function $u=f(\\mu )$ , with $f$ convex and strictly increasing must satsify an equation $MA(u)= F(u)$ for some function $F$ .", "Take $u=\\mu ^2$ .", "Then $F(u)$ must be homogenous of degree zero, so $F(u)$ is a constant.", "All in all, $u=\\mu ^2$ is outside of the origin a smooth convex function that satisfies $MA(u)= C.$ Moreover, the second derivatives of $u$ stay bounded near the origin, so $u$ solves the same Monge-Ampere equation on all of $\\mathbb {R}^n$ in a generalized sense.", "We can then apply a celebrated theorem by Jörgens, Calabi and Pogorelov, (see [6]) to conclude that $u$ is a quadratic form.", "We have thus proved the next theorem.", "Theorem 4.4 Let $\\Omega $ be a convex domain containing the origin.", "Assume that for any convex function in $\\Omega $ , $v$ that vanishes on the boundary the symmetrization inequality $M(\\Omega )^{-1}{\\mathcal {E}}_\\Omega (v)\\ge M(B)^{-1}{\\mathcal {E}}_B(\\hat{v})$ holds.", "Then $v$ is an ellipsoid.", "Remark: We saw above that the condition on our domain is that $\\mu =\\mu _\\Omega $ satisfies an equation $d\\mu \\wedge d^c \\mu \\wedge (dd^c \\mu )^{n-1}= C \\mu ^{1-n} d\\lambda .$ One can show that this is equivalent to the condition that $\\Omega $ is a stationary point for the Mahler functional $M(\\Omega )=|\\Omega ||\\Omega ^\\circ |.$ Thus it follows from the Jörgens- Calabi-Pogorelov theorem that any such stationary point is an ellipsoid.", "Notice that the two results are not equivalent though: in the case of the Mahler functional we know beforehand that our function $u=\\mu ^2$ grows quadratically at infinity, whereas the Jörgens- Calabi-Pogorelov theorem applies to any convex solution.", "At any rate, the analogy between the Kähler-Einstein condition in the complex case and the Mahler volume in the real case seems quite interesting.", "$\\Box $ We conclude with the proof of Theorem 4.3, which is proved more or less as in the complex case.", "Notice that the appearance of the factor $|\\Omega ^\\circ |$ in the lemma is the main difference as compared to Proposition 2.8.", "Lemma 4.5 Let $\\Omega $ be a smoothly bounded convex domain containing the origin, with Minkowski functional $\\mu _\\Omega $ .", "Let $u$ be a smooth convex function in $\\Omega $ of the form $u(x)=f(\\mu _\\Omega (x))$ , vanishing on the boundary so that $f(1)=0$ .", "Then $\\sigma (s):=\\int _{\\mu _\\Omega <s} MA(u)= f^{\\prime }(s)^n |\\Omega ^\\circ |.$ We may assume that $u$ is strictly convex.", "Then the map $x\\mapsto \\nabla u(x)$ is a diffeomorphism from $\\lbrace \\mu _\\Omega \\le s\\rbrace $ to a domain $U_s$ in ${\\mathbb {R}}^n$ , and by the change of variables formula $\\sigma (s)= |U_s|.$ But $U_s$ only depends on the gradient map restricted to the boundary of the set $\\Omega _s$ where $\\mu _\\Omega <s$ , i e on the value of $f^{\\prime }(s)$ .", "We may therefore take $f(s)=as$ and even, by homogenuity, take $a=1$ .", "Then the boundary of $\\Omega _s$ is mapped to the boundary of $\\Omega ^\\circ $ , so the volume is $|\\Omega _\\circ |$ .", "Lemma 4.6 Under the same hypotheses as in the previous lemma, ${\\mathcal {E}}(u)=\\int ^1 f^{\\prime }(s)^{n+1}ds |\\Omega ^\\circ |.$ We have ${\\mathcal {E}}(u)=-\\int ^1 f(s)d\\sigma (s)=\\int ^1 f^{\\prime }(s)\\sigma (s)ds,$ so this follows from the previous lemma.", "Lemma 4.7 Let $\\Omega $ and $u$ be as in the previous lemmas, and let $B$ be a ball centered at the origin of the same volume as $\\Omega $ .", "Then $S_B(u)=f(\\mu _B).$ By definition, $S_B(u)=g(\\mu _B)$ and $|\\lbrace g(\\mu _B)<t\\rbrace |=|\\lbrace f(\\mu _\\Omega \\rbrace )<t\\rbrace |.$ The left hand side here is $|B| (g^{-1}(t))^n$ and the right hand side is $|\\Omega | (f^{-1}(t))^n.$ Since $|B|=|\\Omega |$ , $g^{-1}=f^{-1}$ , so we are done.", "Combining Lemma 4.6 and Lemma 4.5 we see that ${\\mathcal {E}}(u)/|\\Omega ^\\circ |={\\mathcal {E}}(S_B)/|B^\\circ |.$ This proves Theorem 4.3." ] ]

[ [ "Unirationality and existence of infinitely transitive models" ], [ "Abstract We study unirational algebraic varieties and the fields of rational functions on them.", "We show that after adding a finite number of variables some of these fields admit an infinitely transitive model.", "The latter is an algebraic variety with the given field of rational functions and an infinitely transitive regular action of a group of algebraic automorphisms generated by unipotent algebraic subgroups.", "We expect that this property holds for all unirational varieties and in fact is a peculiar one for this class of algebraic varieties among those varieties which are rationally connected." ], [ "Introduction", "This article aims to relate unirationality of a given algebraic variety with the property of being a homogeneous space with respect to unipotent algebraic group action.", "More precisely, let $X$ be an algebraic variety defined over a field $\\mathbf {k}$ , and $\\text{Aut}(X)$ be the group of regular automorphisms of $X$ .", "Let also $\\mathop {\\rm SAut}(X) \\subseteq \\text{Aut}(X)$ be the subgroup generated by algebraic groups isomorphic to the additive group $\\mathbb {G}_a$ .", "Definition 1.1 (cf.", "[1]) We call variety $X$ infinitely transitive if for any $k\\in \\mathbb {N}$ and any two collections of points $\\lbrace P_1,\\ldots ,P_k\\rbrace $ and $\\lbrace Q_1,\\ldots ,Q_k\\rbrace $ on $X$ there exists an element $g\\in \\mathop {\\rm SAut}(X)$ such that $g(P_i)=Q_i$ for all $i$ .", "Similarly, we call $X$ stably infinitely transitive if $X \\times \\mathbf {k}^m$ is infinitely transitive for some $m$ .", "Recall that in Birational Geometry adding a number $m$ of algebraically independent variables to the function field $\\mathbf {k}(X)$ is referred to as stabilization.", "Geometrically this precisely corresponds to taking the product $X \\times \\mathbf {k}^m$ with the affine space.", "Note also that if $X$ is infinitely transitive, then it is unirational, i.e., $\\mathbf {k}(X) \\subseteq \\mathbf {k}(y_1,....y_m)$ for some $\\mathbf {k}$ -transcendental elements $y_i$ (see [1]).", "This suggests to regard (stable) infinite transitivity as a birational property of $X$ (in particular, we will usually assume the test variety $X$ to be smooth and projective): Definition 1.2 We call variety $X$ stably b-infinitely transitive if the field $\\mathbf {k}(X)(y_1,....y_m)$ admits an infinitely transitive model (not necessarily smooth or projective) for some $m$ and $\\mathbf {k}(X)$ -transcendental elements $y_i$ .", "If $m=0$ , we call $X$ b-infinitely transitive.", "Example 1.3 The affine space $X:=\\mathbf {k}^{\\dim X}$ is stably infinitely transitive (and infinitely transitive when $\\dim X \\ge 2$ ), see [9].", "More generally, any rational variety is stably b-infinitely transitive, and it is b-infinitely transitive if the dimension $\\ge 2$ .", "Example REF suggests that being stably b-infinitely transitive does not give anything interesting for rational varieties.", "In the present article, we put forward the following: Conjecture 1.4 Any unirational variety $X$ is stably b-infinitely transitive.", "Thus, Conjecture REF together with the above mentioned result from [1] provides a (potential) characterization of unirational varieties among all those which are rationally connected.", "Note also that the class of rationally connected varieties contains all stably b-infinitely transitive varieties.", "We think that not every rationally connected variety is stably birationally infinitely transitive.", "In particular we expect that generic Fano hypersurfaces from the family considered by Kollar in [11] are not stably birationally infinitely transitive.", "These are generic smooth hypersurfaces of degree $d$ in $\\mathbb {P}^{n+1}$ , $d>\\displaystyle \\frac{2}{3}(n+3)$ .", "Our expectations are based on the Kollar's fundamental observation (see [11]) which yields strong restrictions on any surjective map of a uniruled variety of the same dimension on such a hypersurface.", "Remark 1.5 Originally, the study of infinitely transitive varieties was initiated in the paper [9].", "We also remark one application of these varieties to the Lüroth problem in [1], where a non-rational infinitely transitive variety was constructed.", "See [4] for the properties of locally nilpotent derivations (LNDs for short), [16] for the Makar-Limanov invariant, and [2], [5], [6], [10], [13], and [15] for other results, properties and applications of infinitely transitive (and related) varieties.", "We are going to verify Conjecture REF for some particular cases of $X$ (see Theorems REF , REF and Propositions REF , REF and REF below).", "At this stage, one should note that it is not possible to lose the stabilization assumption in Conjecture REF : Example 1.6 Any three-dimensional algebraic variety $X$ with an infinitely transitive model is rational.", "Indeed, let us take a one-dimensional algebraic subgroup $G\\subset \\mathop {\\rm SAut}(X)$ acting on $X$ with a free orbit.", "Then $X$ is birationally isomorphic to $G \\times Y$ (see Remark REF below), where $Y$ is a rational surface (since $X$ is unirational).", "On the other hand, if $X :=X_3 \\subset \\mathbb {P}^4$ is a smooth cubic hypersurface, then it is unirational but not rational (see [3]).", "However, Conjecture REF is true as stated for $X_3$ , because $X_3$ is stably b-infinitely transitive (see Proposition REF below).", "In this context, it would be also interesting to settle down the case of the quartic hypersurface $X_4$ in $\\mathbb {P}^4$ (or, more generally, in $\\mathbb {P}^n$ for arbitrary $n$ ), which relates our subject to the old problem of (non-)unirationality of (generic) $X_4$ (cf.", "Remark REF below).", "Notations 1.7 Throughout the paper we keep up with the following: $\\mathbf {k}$ is an algebraically closed field of characteristic zero and $\\mathbf {k}^\\times $ is the multiplicative group of $\\mathbf {k}$ ; $X_1\\approx X_2$ denotes birational equivalence between two algebraic varieties $X_1$ and $X_2$ ; we abbreviate infinite transitivity (transitive, transitively, etc.)", "to inf. trans.", "Acknowledgment.", "The second author would like to thank Courant Institute for hospitality.", "The second author has also benefited from discussions with I. Cheltsov, Yu.", "Prokhorov, V. Shokurov, and K. Shramov.", "The third author would like to thank I. Arzhantsev for fruitful discussions.", "The authors are grateful to the referee for valuable comments.", "The authors thank the organizers of the summer school and the conference in Yekaterinburg (2011), where the work on the article originated.", "The result was presented at the Simons Symposium “Geometry Over Non-Closed Fields” in February 2012.", "The first author wants to thank Simons Foundation for support and participants of the Symposium for useful discussions.", "In particular, the question of J.-L. Colliot-Thélène (see below) was raised during the Symposium." ], [ "The set-up", "The goal of the present section is to prove the following: Theorem 2.1 Let $K := \\mathbf {k}(X)$ for some (smooth projective) algebraic variety $X$ of dimension $n$ over $\\mathbf {k}$ .", "We assume there are $n$ presentations (we call them cancellations (of $K$ or $X$ )) $K = K^{\\prime }(x_i)$ for some $K^{\\prime }$ -transcendental elements $x_i$ , algebraically independent over $\\mathbf {k}$ .", "Then there exists an inf.", "trans.", "model of $K(y_1,\\ldots ,y_n)$ for some $K$ -transcendental elements $y_i$ .", "Let us put Theorem REF into a geometric perspective.", "Namely, the presentation $K = K^{\\prime }(x_i)$ reads as there exists a model of $K$ , say $X_i^n$ , with a surjective regular map $\\pi _i : X_i^n \\rightarrow Y_i^{n-1}$ and general fiber $\\simeq \\mathbb {P}^1$ such that $\\pi _i$ admits a section over an open subset in $Y_i^{n-1}$ .", "Moreover, by resolving indeterminacies, we may assume $X_i^n := X$ fixed for all $i$ .", "Then, since $K$ admits $n$ cancellations, $n$ vectors, each tangent to a fiber of some $\\pi _i$ , span the tangent space to $X$ at the general point.", "Indeed, we have a map to $\\mathbb {P}^n$ $X \\dasharrow \\mathbb {P}^n, \\quad x\\mapsto (1:x_1(x):\\ldots :x_n(x)).$ It is dominant since elements $x_1,\\ldots ,x_n$ are algebraically independent over $\\mathbf {k}$ , and the tangent map is surjective at the general point.", "So we obtain the geometric counterpart of Theorem REF : Theorem 2.2 Let $X$ be a smooth projective variety of dimension $n$ .", "Assume that there exist $n$ morphisms $\\pi _i :X \\rightarrow Y_i$ satisfying the following: (1) $Y_i$ is a (normal) projective variety such that $\\pi _i$ admits a section over an open subset in $Y_i$ ; for the general point $\\zeta \\in X$ and the fiber $F_i=\\mathbb {P}^1_i := \\pi _i^{-1}(\\pi _i(\\zeta )) \\simeq \\mathbb {P}^1$ , vector fields $T_{F_1,\\,\\zeta }, \\ldots , T_{F_n,\\,\\zeta }$ span the tangent space $T_{X,\\,\\zeta }$ .", "Then $X$ is stably b-inf.", "trans.", "Note that existence of a section over an open subset on $Y_i$ means (almost by definition) birational triviality of the fibration $\\pi _i$ .", "In Sections REF and REF we illustrate our arguments by considering the cases when $\\dim X = 1$ and 2, respectively.", "In higher dimensions we additionally need the following: (3) for some ample line bundles $H_i$ on $Y_i$ and their pullbacks $\\pi _i^*H_i$ to $X$ , the $n\\times n$ -matrix (REF ) $(\\pi _i^*H_i\\cdot \\mathbb {P}^1_i)$ is of maximal rank (in particular, the classes of $\\pi _1^*H_1,\\ldots ,\\pi _n^*H_n$ in $\\mathop {\\rm Pic}(X)$ are linearly independent).", "In particular, this means that the fibers $\\mathbb {P}^1_1$ , $\\mathbb {P}^1_2, \\ldots $ , $\\mathbb {P}^1_n$ are linearly independent in $H_2(X)$ .", "In Sections REF , REF and REF we prove Theorem REF , assuming that the condition REF is satisfied.", "Furthermore, adding new variables (i.e., forming the product of $X$ and an affine space) and passing to a (good) birational model, we may assume that REF holds, see Sections REF and  REF .", "One-dimensional case Variety ${\\mathcal {O}}(m)^\\times _{\\mathbb {P}^1}$ (or, equivalently, ${\\mathcal {O}}(-m)^\\times _{\\mathbb {P}^1}$ ) is just an affine cone minus the origin over a rational normal curve of degree $m$ .", "Thus ${\\mathcal {O}}(m)^\\times _{\\mathbb {P}^1}$ is a quasiaffine toric variety, so it is infinitely transitive by [2].", "Indeed, we can use only those automorphisms which preserve the origin, i.e., for $m$ -transitivity on $\\mathbf {k}^2\\setminus \\lbrace 0\\rbrace $ we use $(m+1)$ -transitivity on $\\mathbf {k}^2$ .", "Two-dimensional case Let us study now the next simplest case when $X=\\mathbb {P}^1 \\times \\mathbb {P}^1$ .", "Choose $H_2:={\\mathcal {O}}(1)$ on the first factor $\\mathbb {P}^1$ and, similarly, $H_1:={\\mathcal {O}}(1)$ on the second factor $\\mathbb {P}^1$ .", "Now take the pullbacks $\\pi _1^* H_1$ and $\\pi _2^*H_2$ to $X$ and throw away their zero sections.", "We obtain a toric bundle over $X$ isomorphic to $(\\mathbf {k}^2\\setminus \\lbrace 0\\rbrace )\\times (\\mathbf {k}^2\\setminus \\lbrace 0\\rbrace )$ .", "The latter is inf.", "trans.", "since $\\mathbf {k}^2\\setminus \\lbrace 0\\rbrace $ is (cf.", "the one-dimensional case above).", "More generally, if one starts with $H_2={\\mathcal {O}}(m_2)$ and $H_1={\\mathcal {O}}(m_1)$ for some $m_i\\geqslant 1$ , then the resulting variety will be $((\\mathbf {k}^2 \\setminus \\lbrace 0\\rbrace )/(\\mathbb {Z}/m_1 \\mathbb {Z}))\\times ((\\mathbf {k}^2 \\setminus \\lbrace 0\\rbrace )/(\\mathbb {Z}/m_2 \\mathbb {Z}))$ .", "It is again inf-transitive being the product of two inf-transitive varieties.", "Indeed, for $m_i>1$ the corresponding variety is just the smooth locus on the corresponding toric variety, and its inf-transitivity is shown in [2].", "Remark 2.3 The product of two (quasiaffine or affine) inf-transitive varieties is inf-transitive.", "Indeed, we call variety $X$ flexible if the tangent space at every smooth point on $X$ is generated by the tangent vectors to the orbits of one-parameter unipotent subgroups in $\\operatorname{Aut}(X)$ .", "It was shown in [1] that for affine $X$ being flexible is equivalent to inf-transitivity.", "But clearly the product of two flexible varieties is again flexible.", "Construction of an inf.", "transitive model in the simplest case Recall the setting.", "In the notation of Theorem REF , we choose very ample line bundles $H_i$ on each $Y_i$ , $i=1,\\ldots , n$ , take their pullbacks $\\pi _i^*H_i$ to $X$ , put $m_{ij}:=(\\pi _i^*H_i)|_{\\mathbb {P}^1_j}$ , and form the intersection matrix $M_n=M_n(X)=(m_{ij})_{1\\leqslant i,j \\leqslant n}, \\quad m_{ij}=(\\pi _i^*H_i)|_{\\mathbb {P}^1_j} \\,.$ Clearly, for all $i$ we have $(\\pi _i^*H_i)|_{\\mathbb {P}^1_i}=0$ ; however, for $i \\ne j$ , $(\\pi _i^*H_i)|_{\\mathbb {P}^1_j}>0$ , being equal the restriction of $H_i$ to an image of a generic $\\mathbb {P}^1_j$ via $p_i$ (i.e.", "the restriction of a bundle on the variety $Y_i$ ).", "The matrix $M_n$ defines a linear map from a subgroup of the Pickard group $\\mathop {\\rm Pic}X$ to $\\mathbb {Z}^n$ .", "In this section we suppose that the classes of $\\mathbb {P}^1_1, \\ldots $ , $\\mathbb {P}^1_n$ in $H_2(X)$ are linearly independent, and also that $\\det M_n\\ne 0$ .", "Our goal is to construct a quasiaffine variety $\\mathfrak {T}_X$ , $\\mathfrak {T}_X\\approx X \\times k^N$ for some $N$ , equipped with a collection of projections to quasiaffine varieties $\\bar{Y}_i$ with generic fibers being equal to $(\\mathbf {k}^2 \\setminus \\lbrace 0\\rbrace )/(\\mathbb {Z}/m \\mathbb {Z})$ , and such that an open subset of $\\mathfrak {T}_X$ is inf-transitive, cf.", "Section REF .", "The existence of a good open subset will be shown in Section REF .", "To start with, let us set $\\bar{Y}_i := \\mbox{the affine cone }{\\mathcal {O}}_{Y_i}(H_i)^{\\times }\\mbox{ minus the origin}$ over $Y_i$ embedded via $H_i$ , $1\\leqslant i \\leqslant n$ .", "It is a quasiaffine variety.", "Technical step – adding one more coordinate We already embedded $Y_i$ into affine varieties, now we also need to embed $X$ .", "For this purpose, we take a very ample line bundle $H_0$ on $X$ , replace $X$ with $X^{\\prime }=X\\times \\mathbb {P}^1$ , and $Y_i$ with $Y^{\\prime }_i=Y_i\\times \\mathbb {P}^1$ .", "Let also $Y_0=X$ , clearly we have $X^{\\prime }\\rightarrow Y_0=X$ , which makes the situation absolutely symmetric with respect to indices $0,1,\\ldots ,n$ .", "We modify the set of $H_i$ s in the following way: for every $i>0$ , we construct $H_i^{\\prime }$ on $Y_i^{\\prime }$ being the sum of the trivial lift of $H_i$ from $Y_i$ and ${\\mathcal {O}}(1)$ on the new $\\mathbb {P}^1$ (in fact here we can take any ${\\mathcal {O}}(n_i)$ ).", "Now the intersection matrix $M_{n+1}(X^{\\prime })$ takes the form $M_{n+1}(X^{\\prime })=\\left(\\begin{array}{*{20}r}0 & k_1 & \\ldots & k_n \\\\1& & & \\\\ \\vdots & & M_n & \\\\1 & & & \\\\\\end{array}\\right) .$ Here $k_i := H_0 \\cdot \\mathbb {P}^1_i$ .", "We further denote $X^{\\prime }$ just by $X$ and $n+1=\\dim X^{\\prime }$ just by $n$ , keeping in mind that one of our projections is just a trivial projection.", "We also assume that one column of our matrix contains only 1s (and one 0 on the diagonal).", "The construction of $\\mathfrak {T}_X$ We construct a vector bundle $H_1\\times _X \\pi _2^*H_2\\times _X \\pi _3^*H_3\\times _X \\ldots \\times _X \\pi _n^*H_n$ and furthermore a toric bundle $\\mathfrak {T}_X=(H_1)^\\times \\times _X (\\pi _2^*H_2)^\\times \\times _X \\ldots \\times _X (\\pi _n^*H_n)^\\times .$ We denote by $\\delta $ the canonical projection $\\mathfrak {T}_X\\rightarrow X$ .", "Line operations in the intersection matrix (REF ) correspond to base changes in this toric bundle (Neron-Severi torus).", "For our convenience, we fix below the following set of line bundles $L_1, \\ldots , L_n \\in \\langle H_1, \\pi _2^*H_2, \\ldots , \\pi _n^*H_n \\rangle $ : (i) each of them should be primitive in the lattice $\\mathbb {Z}(H_1, \\pi _2^*H_2, \\ldots , \\pi _n^*H_n)$ , (ii) all in total, they should be linearly independent in the lattice $\\mathbb {Z}(H_1, \\pi _2^*H_2, \\ldots , \\pi _n^*H_n).$ They can be chosen in the following way.", "There is a map ${\\begin{matrix}\\mathbb {Z}(H_1,\\ldots ,H_n) &\\xrightarrow{}& \\mathbb {Z}^n &\\xrightarrow{}&\\mathbb {Z}.\\end{matrix}}$ Its kernel has dimension $n-1$ , and $H_i$ itself belongs to the kernel.", "So in fact there is a map $\\mathbb {Z}(H_1, \\ldots , \\check{H}_i,\\ldots ,H_n)\\rightarrow \\mathbb {Z}$ , and $L_i$ is any covector defining this map.", "All in total, they can be chosen linearly independent.", "Construction of local 2-dimensional coordinates Recall that we denote by $\\bar{Y}_i$ the total space of $H_i^\\times \\rightarrow Y_i$ .", "Lemma 2.4 For each $i$ , there is a fibration $\\varphi _i: \\mathfrak {T}_X\\rightarrow \\bar{Y}_i$ such that its general fiber equals $((\\mathbf {k}^2 \\setminus \\lbrace 0\\rbrace )/(\\mathbb {Z}/m_i \\mathbb {Z}))\\times T^{n-2}_i$ , where $T^{n-2}_i\\simeq (\\mathbf {k}^\\times )^{n-2}$ .", "Choose a basis $\\langle H_i, L_i, H^{\\prime }_1,\\ldots ,H^{\\prime }_{n-2} \\rangle $ and a linear map $\\mathbb {Z}^n \\rightarrow \\mathbb {Z}^2$ which is just taking the two first coordinates in the new basis.", "Its kernel will correspond precisely to a $(n-2)$ -dimensional torus, the bundle $H_i$ will provide us with the affine cone $\\bar{Y}_i$ over $Y_i$ , and the bundle $L_i$ restricted to $\\mathbb {P}^1_i$ will form a quasiaffine fiber of form $(\\mathbf {k}^2 \\setminus \\lbrace 0\\rbrace )/(\\mathbb {Z}/m \\mathbb {Z})$ over a general point of $\\bar{Y}_i$ .", "We have a commutative diagram.", "${\\begin{matrix}\\mathfrak {T}_X&\\xrightarrow{}& L_i^{\\times } \\times H_i^{\\times }\\\\\\downarrow && \\mathbox{mphantom}{\\scriptstyle {(\\mathbf {k}^2 \\setminus \\lbrace 0\\rbrace )/(\\mathbb {Z}/n_i \\mathbb {Z})}\\times \\ldots }\\downarrow {\\scriptstyle {(\\mathbf {k}^2 \\setminus \\lbrace 0\\rbrace )/(\\mathbb {Z}/n_i \\mathbb {Z})}\\times \\ldots }&&\\\\X &\\xrightarrow{}& Y_i\\end{matrix}}$ $\\text{where }\\;{\\begin{matrix}\\mathfrak {T}_X&\\xrightarrow{}& \\bar{Y}_i &\\xrightarrow{}&Y_i\\end{matrix}}$ This realization will be intensively used below.", "Note that the fibration is trivial over any open subset $U$ in $Y_i$ such that all the fibers of $\\pi _i$ are $\\mathbb {P}^1$ s over $U$ and the restriction of all $H$ s are generic on these fibers, and respectively over $\\bar{Y}_i$ .", "So if one fixes a finite number of points $P_1,\\ldots ,P_s$ in $\\bar{Y}_i$ , we can choose an open subset $U^{\\prime }$ in $\\bar{Y}_i$ containing the fibers passing through all these points (since it is quasiaffine).", "Lemma 2.5 At the general point $x$ on $\\mathfrak {T}_X$ , local coordinates on $((\\mathbf {k}^2 \\setminus \\lbrace 0\\rbrace )/(\\mathbb {Z}/m_i \\mathbb {Z})$ -fibers from Lemma REF , $i=1,\\ldots , n$ , form a system of local coordinates on $\\mathfrak {T}_X$ at $x$ .", "In the notation of Lemma REF , tangent space to each fiber of $\\varphi _i$ is spanned by a pair of the tangent vectors to $(\\mathbf {k}^2 \\setminus \\lbrace 0\\rbrace )/(\\mathbb {Z}/m_i \\mathbb {Z})$ and by tangent vectors to $T^{n-2}_i$ .", "By the condition REF of Theorem REF and by non-degeneracy of matrix $M_n$ , the tangent vectors to $(\\mathbf {k}^2 \\setminus \\lbrace 0\\rbrace )/(\\mathbb {Z}/m_i \\mathbb {Z})$ are linearly independent, which proves the assertion.", "Quasiaffineness of $\\mathfrak {T}_X$ Here we exploit the projections (REF ) and the technique from the proof of Lemma REF .", "Lemma 2.6 The variety $\\mathfrak {T}_X$ is quasiaffine.", "The bundle $H_1$ gives an embedding of $X$ to a projective space $\\mathbb {P}^{N_1}$ , and every $H_i$ , $i=2,\\ldots ,n$ , embeds $Y_i$ to a $\\mathbb {P}^{N_i}$ .", "The variety $\\mathfrak {T}_X$ is now $\\lbrace (x,l_2,\\ldots ,l_n)\\rbrace $ such that $x\\in X$ , $l_i\\in \\mathop {\\rm cone}(\\pi _i(x))$ in ${\\mathbb {A}}^{N_1+N_2+\\ldots +N_n}$ .", "Note that $\\mathfrak {T}_X \\rightarrow X$ is a principal toric bundle which has a section (the diagonal), and all the fibers are isomorphic to $(\\mathbf {k}^\\times )^{n}$ (see formula (REF )).", "In particular, we have $\\mathfrak {T}_X \\approx X\\times \\mathbf {k}^{n}$ .", "Idea of further proof Proposition 2.7 The variety $\\mathfrak {T}_X$ is stably b-inf.", "trans.", "Its proof will be given in Section REF .", "We use the ideas from [9], [2] and [1] to move an $m$ -tuple of general (in the sence of Section REF ) points to another such $m$ -tuple.", "Stratification on $X$ Let $q\\in X$ be an arbitrary point.", "We denote by $X(q)$ the locus of all points on $X$ connected to $q$ by a sequence of smooth fibers $\\mathbb {P}^1_i$ of the projections $\\pi _i$ , $1\\leqslant i \\leqslant n$ .", "Lemma 2.8 Let $Z$ be an irreducible subvariety of $X$ .", "Consider all smooth fibers $\\mathbb {P}^1_i$ passing through the points of $Z$ and the union $Z^{\\prime }$ of all such fibers.", "Then either $\\dim Z^{\\prime } > \\dim Z$ or all smooth fibers $\\mathbb {P}^1_i$ which contain points in $Z$ are actually contained in the closure $\\bar{Z}$ .", "If the curve $\\mathbb {P}^1_i$ intersects $Z$ but is not contained in $Z$ then the curves in the same family intersect an open subvariety in $Z$ since the subvariety $\\tilde{X}_i$ consisting of curves $\\mathbb {P}^1_i$ is an open subvariety of $X$ .", "Hence in the latter case $\\dim Z^{\\prime } > Z$ .", "Otherwise all the smooth fibers $\\mathbb {P}^1_i$ which contain points in $Z$ are actually contained in the closure of $Z$ .", "Note that the same holds even if a line $\\mathbb {P}^1_i$ intersects the closure $\\bar{Z}$ but is not contained in $\\bar{Z}$ .", "Corollary 2.9 Every point in $X(q)$ is connected to $q$ by a chain of $\\mathbb {P}^1_i$ of length at most $n^2$ .", "Indeed, let $X_p(q)$ be a subvariety obtained after adding the points connected by the chains of curves of length at most $p$ .", "It is a union of algebraic subvarieties of $X$ of dimension $\\leqslant p$ .", "Then by adding the curves from all $n$ families of $\\mathbb {P}^1_i$ we either increase the dimension of every component of maximal dimension, or one of them $X_p^0(q)$ is invariant, i.e.", "all smooth fibers $\\mathbb {P}^1_i$ which contain points in $X_p^0(q)$ are actually contained in the closure of $X_p^0(q)$ .", "Note that in the latter case since $q\\in X_p^0(q)$ , all other components are contained in $X_p^0(q)$ , and hence $X_p^0(q)= X(q)$ .", "Thus after adding lines from different families we obtain either $X(q)$ or a variety $X_{p+n}(q)$ with maximal component of greater dimension.", "Thus we will need at most $n^2$ lines to get $X(q)$ .", "Remark 2.10 If we started with a generic point $q\\in X$ , then it follows from Lemma REF that $\\dim X(q)=n$ .", "Indeed, the condition REF of Theorem REF implies that the tangent vectors to the smooth $\\mathbb {P}^1_i$ -fibers in $q$ generate the full tangent space in $q$ , and if $X(q)$ was of lower dimension then the tangent space would also be of lower dimension.", "Remark 2.11 The bound in Corollary REF is not effective.", "By a more thorough examination one can show that the sequence $X_0(q) \\subseteq X_1(q) \\subseteq \\ldots $ stabilizes earlier than at the $n^2$ th step.", "Corollary 2.12 We can apply the same in reverse.", "Consider $x\\in X(q)$ .", "Then all points in $X$ which are connected to $x\\in X(q)$ can be connected by a chain of length at most $n^2 +n$ .", "Indeed, for any such point $x^{\\prime }$ we have $X_{n^2}(x^{\\prime })$ of dimension $n$ and hence contains an open subvariety in $X$ .", "It may take at most $n$ $\\mathbb {P}^1_i$ s to connect to $x$ .", "Thus the variety obtained from general points in $X$ in $n^2$ steps coincides with the subvariety of all points in $X$ connected to general point by a chain of smooth lines.", "Construction of a big open subset in $\\mathfrak {T}_X$ Now we pass from stratification on $X$ to stratification on $\\mathfrak {T}_X$ .", "We stratify $\\mathfrak {T}_X$ in the following way: we take toric preimages for every strata in $X$ .", "For our needs we take the toric preimage of $X(q)$ for a general point $q\\in X$ .", "Note that for every fiber $\\mathbb {P}^1_i$ of $\\pi _i$ its preimage is $(((\\mathbf {k}^2 \\setminus \\lbrace 0\\rbrace )/(\\mathbb {Z}/m \\mathbb {Z}))\\times T^{n-2}$ , and for every chain $P_1 - P_2 -\\ldots - P_k$ connecting two points in $X$ there is a chain $\\bar{P}_1 - \\bar{P}_2 -\\ldots - \\bar{P}_k$ in $\\mathfrak {T}_X$ such that every two adjacent points belong to the same $((\\mathbf {k}^2 \\setminus \\lbrace 0\\rbrace )/(\\mathbb {Z}/m \\mathbb {Z}))$ for one of the projections, Proof of Proposition  REF Let a variety $\\mathfrak {T}_X$ be as in (REF ).", "For each $i$ we have a fibration with the quasiaffine base and fiber being $((\\mathbf {k}^2 \\setminus \\lbrace 0\\rbrace )/(\\mathbb {Z}/m_i \\mathbb {Z})\\times T^{n-2}$ , see (REF ).", "Definition 2.13 For the points $C_1,\\ldots ,C_r$ in the base of projection (REF ), let $\\mathop {\\rm Stab}_{C_1,\\ldots ,C_r}$ be the subgroup in $\\mathop {\\rm SAut}(\\mathfrak {T}_X)$ preserving all the fibers of the projection and fixing pointwise the fibers above $C_1,\\ldots ,C_r$ .", "Now we need some technique concerning locally nilpotent derivations (LNDs).", "To lift automorphisms, we need to extend an LND on $(\\mathbf {k}^2 \\setminus \\lbrace 0\\rbrace )/(\\mathbb {Z}/m \\mathbb {Z})$ to a LND on $\\mathfrak {T}_X$ (i.e.", "to a locally nilpotent derivation of the algebra $\\mathbf {k}[\\mathfrak {T}_X]$ ).", "More precisely, suppose that we chose a fiber of form $(\\mathbf {k}^2 \\setminus \\lbrace 0\\rbrace )/(\\mathbb {Z}/m \\mathbb {Z})$ of the projection (REF ) and some other fibers that we want to fix.", "We can project all these subvarieties to $\\bar{Y}_i$ and then take a regular function on $\\bar{Y}_i$ which equals 1 at the projection of the first fiber and 0 in the projections of other fibers.", "If we multiply the LND by this function (obviously belonging to the kernel of the derivation) and trivially extend it to the toric factor, we will obtain a rational derivation well-defined on an open subset $U$ of $\\mathfrak {T}_X$ corresponding to the smooth locus of the corresponding $\\pi _i$ .", "Now we can take a regular function on $\\mathfrak {T}_X$ (lifted from a regular function on $\\bar{Y}_i$ ) such that its zero locus contains the singular locus of the projection, multiply the derivation by some power of this function and obtain a regular LND on $\\mathfrak {T}_X$ .", "For a given $m$ -tuple of points $P_1,P_2,\\ldots ,P_m$ , we need the following lemma: Lemma 2.14 In the notation as above (REF ), let $C_0$ be a point on a base such that the fiber over this point is general, and $P_1,\\ldots ,P_s$ be some points from this fiber with different projections to $\\bar{Y}_i$ .", "Let also $C_1,\\ldots ,C_r$ be some other points of the base.", "Then the subgroup $Stab_{C_1,\\ldots ,C_r}$ acts infinitely transitively on the fiber over $C_0$ , i.e.", "can map $P_1,\\ldots ,P_s$ in any other subset in the same fiber.", "By [2], the fiber is infinitely transitive.", "For every one-parameter unipotent subgroup of automorphisms on this fiber, we can lift it to $\\mathfrak {T}_X$ , fixing pointwise a given finite collection of fibers, see above.", "Now it remains to prove infinite transitivity for $\\mathfrak {T}_X$ .", "There are two ways to show it.", "Way 1 Lemma 2.15 For $m+1$ points $P_1, P^{\\prime }_1, P_2,P_3,\\ldots ,P_m$ projecting to the chosen above open subset in $X$ , there exists an automorphism mapping $P_1$ to $P^{\\prime }_1$ and preserving all the other points.", "There always exists a small automorphism which moves the initial set to a set where for all $i$ all the $(\\mathbf {k}^2 \\setminus \\lbrace 0\\rbrace )/(\\mathbb {Z}/m_i \\mathbb {Z})$ -coordinates of the given points are different.", "Let us connect the projections of $P_1$ and $P^{\\prime }_1$ by a chain of smooth $\\mathbb {P}^1_i$ -curves in $X$ .", "We denote by $Q_1,\\ldots , Q_s$ the intersection points of these curves, $Q_i\\in X$ , $Q_1=\\delta (P_1)$ , $Q_s=\\delta (P^{\\prime }_1)$ .", "For $i=2,\\ldots ,(s-1)$ we take some lifts $R_i\\in \\mathfrak {T}_X$ of these points in such a way that all their $(\\mathbf {k}^2 \\setminus \\lbrace 0\\rbrace )/(\\mathbb {Z}/m_i \\mathbb {Z})$ -coordinates do not coincide with the corresponding coordinates of the previous points.", "Let $R_1:=P_1$ and $R_s=P^{\\prime }_1$ .", "For every $i$ , $1\\leqslant i\\leqslant (s-1)$ , we want to map $R_i$ to $R_{i+1}$ by an automorphism of $\\mathfrak {T}_X$ preserving all the other points in the given set.", "We may assume that $R_i$ and $R_{i+1}$ belong to one two-dimensional fiber of form $(\\mathbf {k}^2 \\setminus \\lbrace 0\\rbrace )/(\\mathbb {Z}/m \\mathbb {Z})$ of one of the projections to $\\bar{Y}_j \\times T_j^{n-2}$ (the toric fibration is generated by $L_i$ s, and we can densify the sequence of $R_i$ s if needed to change only one $L_i$ -direction at every step to fulfill this condition).", "Every such two-dimensional fiber is inf-transitive.", "Now we need to lift the corresponding automorphism to $\\mathfrak {T}_X$ .", "We need two following observations.", "First, if we are lifting a curve with respect to the projection $\\pi _i$ , then the resulting automorphism is well defined over the singular fibers and is trivial there.", "Second, all the two-dimensional fibers belonging to the same fiber of $\\varphi _i$ move together, and if several $P_j$ belong to the same fiber as the $R_i$ which we are moving, then we use that their projections to the two-dimensional fiber are different and also different from the projection of $R_{i+1}$ , and we use inf-transitivity (not only 1-transitivity) of the corresponding fiber.", "Now for every $i$ we lift the corresponding automorphism of the 2-dimensional fiber to an automorphism of $\\mathfrak {T}_X$ from the corresponding subgroup $\\mathop {\\rm Stab}$ fixing the points from the other fibers of $\\delta $ , and multiply all these automorphisms.", "It does not change $P_2,\\ldots ,P_m$ and maps $P_1$ to $P^{\\prime }_1$ .", "This ends the proof.", "Now infinite transitivity follows easily: to map $P_1,P_2,\\ldots ,P_m$ to $Q_1,Q_2,\\ldots $ , $Q_m$ , we map $P_1$ to $Q_1$ fixing $P_2,\\ldots ,P_m,Q_2,\\ldots ,Q_m$ , etc.", ".", "Way 2 The other way to finish the proof is as follows.", "It is enough to show 1-transitivity while fixing some other points of a given finite set.", "Let us consider automorphisms of bounded degree fixing $P_2,\\ldots ,P_m$ and the orbits of $P_1$ and $P^{\\prime }_1$ under this group.", "Clearly, by flexibility every orbit is an open subset in $\\mathfrak {T}_X$ , and every two dominant subsets should have a nonempty intersection.", "So there is a common point, which means that $P_1$ can be mapped to $P^{\\prime }_1$ by a subgroup in $\\mathop {\\rm SAut}(\\mathfrak {T}_X)$ fixing $P_2,\\ldots ,P_m$ .", "Remark 2.16 Conversely, in view of Theorem REF , given a b-inf.", "trans.", "variety $X$ there exist $\\dim X$ cancellations of $X$ .", "Indeed, for general point $\\zeta \\in X$ we can find $\\dim X$ tangent vectors spanning $T_{X, \\zeta }$ , such that each vector generates a copy of $\\mathbb {G}_a =: G_i \\subseteq \\mathop {\\rm SAut}(X)$ , $1 \\le i \\le n$ .", "Let $\\mathfrak {G}\\subseteq \\mathop {\\rm SAut}(X)$ be the subgroup generated by the groups $G_2, \\ldots , G_n$ .", "Then we have $X \\approx G_1 \\times \\mathfrak {G} \\cdot \\zeta $ .", "Increasing the rank of the corresponding subgroup in $H_2$ We want to treat here the case when $\\mathop {\\rm rk}\\langle \\mathbb {P}^1_1, \\ldots , \\mathbb {P}^1_n \\rangle $ is $t$ , $t<n$ , as of a subgroup in $H_2(X)$ .", "Remark 2.17 Here we precise the ancient construction of $\\mathfrak {T}_X$ .", "Indeed, if the rank is not maximal, then the toric bundle contains a trivial part, and we need to get rid of it.", "One way is to change it with the trivial vector bundle part.", "However here we give another construction which uses stabilization.", "Lemma 2.18 There is a stabilization $X^{\\prime }$ of $X$ such that $\\dim X^{\\prime } - t(X^{\\prime })< (n-t)$ .", "We assume that the cycles $\\mathbb {P}^1_i$ are dependent and in particular that an integer multiple of $\\mathbb {P}^1_n$ is contained in the envelope of $\\mathbb {P}^1_i$ , $i < n$ , on $X$ .", "There is a natural projection $p_{n,n+1}: X\\times \\mathbb {P}^1\\rightarrow Y_n$ with a generic fiber $\\mathbb {P}^1\\times \\mathbb {P}^1$ .", "Let us take $\\mathbb {P}^1\\times \\mathbb {P}^1$ and blow it up at 3 points.", "Thus we will have $\\mathbb {P}^2$ with 5 blown up points.", "For every 4 points there is a pencil of conics passing through four points.", "Indeed, if we fix two smooth conics, there is a pencil of conics passing through the intersection.", "So on $\\mathbb {P}^1\\times \\mathbb {P}^1$ blown up at three points we can choose two different 4-tuples of points on $\\mathbb {P}^2$ and define two projections $\\bar{\\pi }_i : Bl_{Q_1,Q_2,Q_3}(\\mathbb {P}^1\\times \\mathbb {P}^1)\\rightarrow \\mathbb {P}^1$ .", "Now we can extend them to $\\mathbb {P}^1\\times \\mathbb {P}^1\\times B$ by blowing up three constant sections and similarly extend projections.", "The projections $\\bar{\\pi }_1$ , $\\bar{\\pi }_2$ provide cancellations with new $\\bar{\\mathbb {P}}^1_i$ , $i=1,2$ , independent with generic $\\mathbb {P}^1_j$ , $j\\ne 1,2$ on the blown up $X$ .", "We denote the resulting variety by $\\tilde{X}$ , it is a smooth model of $X\\times \\mathbb {P}^1$ .", "Here the rank $t^1 = t+2$ .", "Increasing the rank of the matrix $M$ For a variety $X$ with a given set of cancellations and corresponding bundles, we constructed (REF ) a matrix $M_n$ of restrictions.", "To prove birational stable infinite transitivity, we need the rank of this matrix to be full.", "The aim of this section is to prove the following lemma.", "Lemma 2.19 Let the rank of the subgroup generated by $\\mathbb {P}^1_i$ in $H_2(X,Z)$ be $n$ , and let matrix $M_n=M(X)=(m_{i,j})$ be as in (REF ) and its rank be $s < n= \\dim X$ .", "Then there exists a birational model $\\tilde{X}$ for $X\\times \\mathbb {P}^1$ with $n+1$ projections corresponding to cancellations and a family $H^1_i$ , $i=1,2,\\ldots ,(n+1)$ , such that $s_1 \\ge n+2$ for the new matrix $M(\\tilde{X})$ .", "By Lemma REF , we may assume that all the classes $[\\mathbb {P}^1_1]$ , $[\\mathbb {P}^1_2], \\ldots $ , $[\\mathbb {P}^1_n]$ are independent in $H_2(X)$ .", "If $s < n= \\dim X$ , then due to Hodge duality there is a divisor with nonzero positive pairings with all the fibers $\\mathbb {P}^1_i$ , i.e.", "there is an ample divisor $H_{n+1}$ on $X$ such that it defines an element in the lattice $\\mathbb {Z}^n$ which is not contained in $M (\\mathbb {Z}(H_1,H_2,\\ldots ,H_n))$ (here we identify $H_i$ with the elements of the standard basis in $\\mathbb {Z}^n$ ).", "Let us define in this case $\\pi _i^1 : X\\times \\mathbb {P}^1\\rightarrow (Y_i\\times \\mathbb {P}^1)$ ; take $H_i^1= H_i + {\\mathcal {O}}_{\\mathbb {P}^1}(n_i)$ for some positive numbers $n_i$ ; $\\pi _{n+1} : X\\times \\mathbb {P}^1\\rightarrow X$ the trivial projection; and $H_{n+1}$ chosen above.", "Then if the restriction of $H_{n+1}$ on $\\mathbb {P}^1_i$ is ${\\mathcal {O}}(t_i)$ , the new matrix $M(\\tilde{X})$ is as follows: $M(\\tilde{X})=\\left(\\begin{array}{*{20}r} &&& n_1 \\\\ &M(X)&& \\vdots \\\\ &&& n_n\\\\ t_1 &\\ldots &t_n&0\\end{array}\\right).$ Note that all the diagonal elements $m_{i,i}= 0$ .", "The matrix $M(\\tilde{X})$ in this case has rank $s+2$ for some choice of $n_i$ .", "Indeed, the last row of $M(\\tilde{X})$ is independent with other rows by the assumption on $H_{n+1}$ .", "Now we can add $n_i$ in such a way that the rank of $M(\\tilde{X})$ will be $(\\mathop {\\rm rk}M+1)+1$ (if $\\mathop {\\rm rk}M < n$ ).", "Hence $\\mathop {\\rm rk}M_1= s+2$ in this case.", "Corollary 2.20 In finite number of steps (not more than $2n$ ), using Lemmas REF and REF , we obtain a model $\\tilde{X}$ of $X\\times \\mathbb {P}^r$ with $\\mathop {\\rm rk}M (\\tilde{X})= \\dim (X\\times \\mathbb {P}^r)=\\dim (\\tilde{X})$ .", "Examples Here we collect several examples and properties of (stably) b-inf.", "trans.", "varieties.", "Quotients Let us start with the projective space $\\mathbb {P}^n$ , $n \\ge 2$ , and a finite group $G\\subset PGL_{n+1}(\\mathbf {k})$ .", "Notice that the quotient $\\mathbb {P}^n/G$ is stably b-inf.", "trans.", "Indeed, let us replace $G$ by its finite central extension $\\tilde{G}$ acting linearly on $V :=\\mathbf {k}^{n+1}$ , so that $V/\\tilde{G} \\approx \\mathbb {P}^n/G \\times \\mathbb {P}^1$ .", "Further, form the product $V \\times V$ with the diagonal $\\tilde{G}$ -action, and take the quotient $V^{\\prime } := (V \\times V)/\\tilde{G}$ .", "Then, projecting on the first factor we get $V^{\\prime }\\approx V \\times V/\\tilde{G}$ , and similarly for the second factor.", "This implies that $V^{\\prime }$ admits $2n+2$ cancellations (cf.", "Theorem REF ).", "Hence $V^{\\prime }$ is stably b-inf.", "trans.", "by Theorem REF .", "The argument just used can be summarized as follows: Lemma 3.1 Let $X \\rightarrow S$ be a $\\mathbb {P}^m$ -fibration for some $m\\in \\mathbb {N}$ .", "Then the product $X \\times _S X \\approx X \\times \\mathbf {k}^m$ admits $2m$ algebraically independent cancellations over $S$ .", "Note that $X \\times _S X$ has two projections (left and right) onto $X$ , both having a section (the diagonal $\\Delta _X \\subset X\\times _S X$ ), hence the corresponding $\\mathbb {P}^m$ -fibrations are birational (over $S$ ) to $X \\times \\mathbf {k}^m$ .", "This gives $2m$ algebraically independent cancellations over $S$ .", "Corollary 3.2 Assume that $X$ carries a collection of distinct birational structures of $\\mathbb {P}^{m_i}$ -bundles, $\\pi _i: X\\rightarrow S_i$ , with the condition that the tangent spaces of generic fibers of $\\pi _i$ span the tangent space of $X$ at the generic point.", "Then $X$ is stably b-inf.", "trans.", "Indeed, after multiplying by the maximum of $m_i$ we may assume that all $\\mathbb {P}^{m_i}$ -bundles provide with at least $2m_i$ different cancellations (see Lemma REF ).", "We can now apply Theorem REF .", "Remark 3.3 It seems plausible that given an inf.", "trans.", "variety $X$ and a finite group $G\\subset \\text{Aut}(X)$ , variety $X/G$ is stably b-inf.", "trans.", "(though the proof of this fact requires a finer understanding of the group $\\mathop {\\rm SAut}(X)$ ).", "At this stage, note also that if $G$ is cyclic, then there exists a $G$ -fixed point on $X$ .", "Indeed, since $X$ is unirational (cf.", "Section ), it has trivial algebraic fundamental group $\\pi ^{\\text{alg}}_1(X)$ (see [12]).", "Then, if the $G$ -action is free on $X$ , we get $G \\subset \\pi ^{\\text{alg}}_1(X/G) = \\lbrace 1\\rbrace $ for $X/G$ smooth unirational, a contradiction.", "This fixed-point-non-freeness property of $X$ relates $X$ to homogeneous spaces, and it would be interesting to investigate whether this is indeed the fact, i.e., in particular, does $X$ , after stabilization and passing to birational model, admit a uniformization which is a genuine (finite dimensional) algebraic group?This question was suggested by J.-L. Colliot-Thélène in connection with Conjecture REF .", "However, there are reasons to doubt the positive answer, since, for example, it would imply that $X$ is (stably) birationally isomorphic to $G/H$ , where both $G, H$ are (finite dimensional) reductive algebraic groups.", "Even more, up to stable birational equivalence we may assume that $X = G^{\\prime }/H^{\\prime }$ , where $H^{\\prime }$ is a finite group and $G^{\\prime }$ is the product of a general linear group, Spin groups and exceptional Lie groups.", "The latter implies, among other things, that there are only countably many stable birational equivalence classes of unirational varieties, but we could not develop a rigorous argument to bring this to contradiction.", "Cubic hypersurfaces Let $X_3 \\subset \\mathbb {P}^{n + 1}$ , $n \\ge 2$ , be a smooth cubic.", "Then Proposition 3.4 $X_3$ is stably b-inf.", "trans.", "Smooth cubic $X_3$ contains a two-dimensional family of lines which span $\\mathbb {P}^4$ .", "Let $L \\subset X_3$ be a line and $\\pi : X_3 \\dashrightarrow \\mathbb {P}^{n- 1}$ the linear projection from $L$ .", "Let us resolve the indeterminacies of $\\pi $ by blowing up $X_3$ at $L$ .", "We arrive at a smooth variety $X_L$ together with a morphism $\\pi _L : X_L \\rightarrow \\mathbb {P}^{n - 1}$ whose general fiber is $\\mathbb {P}^1$ ($\\simeq $ a conic in $\\mathbb {P}^2$ ).", "Varying $L \\subset X_3$ , we then apply Lemma REF and Corollary REF to get that $X_3$ is stably b-inf.", "trans.", "Quartic hypersurfaces Let $X_4\\subset \\mathbb {P}^n$ , $n \\ge 4$ , be a quartic hypersurface with a line $L\\subset X_4$ of double singularities.", "Then Proposition 3.5 $X_4$ is stably b-inf.", "trans.", "Consider the cone $\\mathfrak {X}_4\\subset \\mathbb {P}^{n+1}$ over $X_4$ .", "Then $\\mathfrak {X}_4$ contains a plane $\\Pi $ of double singularities.", "Pick up a (generic) line $L^{\\prime }\\subset \\Pi $ and consider the linear projection $\\mathfrak {X}_4\\dashrightarrow \\mathbb {P}^{n-1}$ from $L^{\\prime }$ .", "This induces a conic bundle structure on $\\mathfrak {X}_4$ , similarly as in the proof of Proposition REF , and varying $L^{\\prime }$ in $\\Pi $ as above we obtain that $\\mathfrak {X}_4$ is stably b-inf.", "trans.", "Then, since $\\mathfrak {X}_4 \\approx X_4 \\times \\mathbf {k}$ , Proposition REF follows.", "Complete intersections Let $X_{2\\cdot 2\\cdot 2} \\subset \\mathbb {P}^6$ be the smooth complete intersection of three quadrics.", "Then Proposition 3.6 $X_{2\\cdot 2\\cdot 2}$ is stably b-inf.", "trans.", "The threefold $X_{2\\cdot 2\\cdot 2}$ contains at least a one-dimensional family of lines.", "Let $L \\subset X_{2\\cdot 2\\cdot 2}$ be a line and $X_L \\rightarrow X_{2\\cdot 2\\cdot 2}$ the blowup of $L$ .", "Then the threefold $X_L$ carries the structure of a conic bundle (see [7]).", "Now, varying $L$ and applying the same arguments as in the proof of Proposition REF , we obtain that $X_{2\\cdot 2\\cdot 2}$ is stably b-inf.", "trans.", "Remark 3.7 Fix $n, r \\in \\mathbb {N}$ , $n \\gg r$ , and a sequence of integers $0<d_1\\le \\ldots \\le d_m$ , $m \\ge 2$ .", "Let us assume that $(n-r)(r+1)\\ge \\displaystyle \\sum _{i=1}^m{d_i+r\\atopwithdelims ()r}$ .", "Consider the complete intersection $X:=H_1\\cap \\ldots \\cap H_m$ of hypersurfaces $H_i\\subset \\mathbb {P}^n$ of degree $d_i$ .", "Then it follows from the arguments in [14] that $X$ contains a positive dimensional family of linear subspaces $\\simeq \\mathbb {P}^r$ .", "Moreover, $X$ is unirational, provided $X$ is generic.", "It would be interesting to adopt the arguments from the proofs of Propositions REF , REF and REF to this more general setting in order to prove that $X$ is stably b-inf.", "trans.", "Remark 3.8 Propositions REF , REF and REF (cf.", "Remark REF ) provide an alternative method of proving unirationality of complete intersections (see [7] for recollection of classical arguments).", "Note also that (generic) $X_{2\\cdot 2\\cdot 2}$ is non-rational (see for example [17]), and (non-)rationality of the most of other complete intersections considered above is not known.", "At the same time, verifying stable b-inf.", "trans.", "property of other (non-rational) Fano manifolds (cf.", "[7]) is out of reach for our techniques at the moment." ], [ "Examples", "Here we collect several examples and properties of (stably) b-inf.", "trans.", "varieties." ], [ "Quotients", "Let us start with the projective space $\\mathbb {P}^n$ , $n \\ge 2$ , and a finite group $G\\subset PGL_{n+1}(\\mathbf {k})$ .", "Notice that the quotient $\\mathbb {P}^n/G$ is stably b-inf.", "trans.", "Indeed, let us replace $G$ by its finite central extension $\\tilde{G}$ acting linearly on $V :=\\mathbf {k}^{n+1}$ , so that $V/\\tilde{G} \\approx \\mathbb {P}^n/G \\times \\mathbb {P}^1$ .", "Further, form the product $V \\times V$ with the diagonal $\\tilde{G}$ -action, and take the quotient $V^{\\prime } := (V \\times V)/\\tilde{G}$ .", "Then, projecting on the first factor we get $V^{\\prime }\\approx V \\times V/\\tilde{G}$ , and similarly for the second factor.", "This implies that $V^{\\prime }$ admits $2n+2$ cancellations (cf.", "Theorem REF ).", "Hence $V^{\\prime }$ is stably b-inf.", "trans.", "by Theorem REF .", "The argument just used can be summarized as follows: Lemma 3.1 Let $X \\rightarrow S$ be a $\\mathbb {P}^m$ -fibration for some $m\\in \\mathbb {N}$ .", "Then the product $X \\times _S X \\approx X \\times \\mathbf {k}^m$ admits $2m$ algebraically independent cancellations over $S$ .", "Note that $X \\times _S X$ has two projections (left and right) onto $X$ , both having a section (the diagonal $\\Delta _X \\subset X\\times _S X$ ), hence the corresponding $\\mathbb {P}^m$ -fibrations are birational (over $S$ ) to $X \\times \\mathbf {k}^m$ .", "This gives $2m$ algebraically independent cancellations over $S$ .", "Corollary 3.2 Assume that $X$ carries a collection of distinct birational structures of $\\mathbb {P}^{m_i}$ -bundles, $\\pi _i: X\\rightarrow S_i$ , with the condition that the tangent spaces of generic fibers of $\\pi _i$ span the tangent space of $X$ at the generic point.", "Then $X$ is stably b-inf.", "trans.", "Indeed, after multiplying by the maximum of $m_i$ we may assume that all $\\mathbb {P}^{m_i}$ -bundles provide with at least $2m_i$ different cancellations (see Lemma REF ).", "We can now apply Theorem REF .", "Remark 3.3 It seems plausible that given an inf.", "trans.", "variety $X$ and a finite group $G\\subset \\text{Aut}(X)$ , variety $X/G$ is stably b-inf.", "trans.", "(though the proof of this fact requires a finer understanding of the group $\\mathop {\\rm SAut}(X)$ ).", "At this stage, note also that if $G$ is cyclic, then there exists a $G$ -fixed point on $X$ .", "Indeed, since $X$ is unirational (cf.", "Section ), it has trivial algebraic fundamental group $\\pi ^{\\text{alg}}_1(X)$ (see [12]).", "Then, if the $G$ -action is free on $X$ , we get $G \\subset \\pi ^{\\text{alg}}_1(X/G) = \\lbrace 1\\rbrace $ for $X/G$ smooth unirational, a contradiction.", "This fixed-point-non-freeness property of $X$ relates $X$ to homogeneous spaces, and it would be interesting to investigate whether this is indeed the fact, i.e., in particular, does $X$ , after stabilization and passing to birational model, admit a uniformization which is a genuine (finite dimensional) algebraic group?This question was suggested by J.-L. Colliot-Thélène in connection with Conjecture REF .", "However, there are reasons to doubt the positive answer, since, for example, it would imply that $X$ is (stably) birationally isomorphic to $G/H$ , where both $G, H$ are (finite dimensional) reductive algebraic groups.", "Even more, up to stable birational equivalence we may assume that $X = G^{\\prime }/H^{\\prime }$ , where $H^{\\prime }$ is a finite group and $G^{\\prime }$ is the product of a general linear group, Spin groups and exceptional Lie groups.", "The latter implies, among other things, that there are only countably many stable birational equivalence classes of unirational varieties, but we could not develop a rigorous argument to bring this to contradiction." ], [ "Cubic hypersurfaces", "Let $X_3 \\subset \\mathbb {P}^{n + 1}$ , $n \\ge 2$ , be a smooth cubic.", "Then Proposition 3.4 $X_3$ is stably b-inf.", "trans.", "Smooth cubic $X_3$ contains a two-dimensional family of lines which span $\\mathbb {P}^4$ .", "Let $L \\subset X_3$ be a line and $\\pi : X_3 \\dashrightarrow \\mathbb {P}^{n- 1}$ the linear projection from $L$ .", "Let us resolve the indeterminacies of $\\pi $ by blowing up $X_3$ at $L$ .", "We arrive at a smooth variety $X_L$ together with a morphism $\\pi _L : X_L \\rightarrow \\mathbb {P}^{n - 1}$ whose general fiber is $\\mathbb {P}^1$ ($\\simeq $ a conic in $\\mathbb {P}^2$ ).", "Varying $L \\subset X_3$ , we then apply Lemma REF and Corollary REF to get that $X_3$ is stably b-inf.", "trans." ], [ "Quartic hypersurfaces", "Let $X_4\\subset \\mathbb {P}^n$ , $n \\ge 4$ , be a quartic hypersurface with a line $L\\subset X_4$ of double singularities.", "Then Proposition 3.5 $X_4$ is stably b-inf.", "trans.", "Consider the cone $\\mathfrak {X}_4\\subset \\mathbb {P}^{n+1}$ over $X_4$ .", "Then $\\mathfrak {X}_4$ contains a plane $\\Pi $ of double singularities.", "Pick up a (generic) line $L^{\\prime }\\subset \\Pi $ and consider the linear projection $\\mathfrak {X}_4\\dashrightarrow \\mathbb {P}^{n-1}$ from $L^{\\prime }$ .", "This induces a conic bundle structure on $\\mathfrak {X}_4$ , similarly as in the proof of Proposition REF , and varying $L^{\\prime }$ in $\\Pi $ as above we obtain that $\\mathfrak {X}_4$ is stably b-inf.", "trans.", "Then, since $\\mathfrak {X}_4 \\approx X_4 \\times \\mathbf {k}$ , Proposition REF follows." ], [ "Complete intersections", "Let $X_{2\\cdot 2\\cdot 2} \\subset \\mathbb {P}^6$ be the smooth complete intersection of three quadrics.", "Then Proposition 3.6 $X_{2\\cdot 2\\cdot 2}$ is stably b-inf.", "trans.", "The threefold $X_{2\\cdot 2\\cdot 2}$ contains at least a one-dimensional family of lines.", "Let $L \\subset X_{2\\cdot 2\\cdot 2}$ be a line and $X_L \\rightarrow X_{2\\cdot 2\\cdot 2}$ the blowup of $L$ .", "Then the threefold $X_L$ carries the structure of a conic bundle (see [7]).", "Now, varying $L$ and applying the same arguments as in the proof of Proposition REF , we obtain that $X_{2\\cdot 2\\cdot 2}$ is stably b-inf.", "trans.", "Remark 3.7 Fix $n, r \\in \\mathbb {N}$ , $n \\gg r$ , and a sequence of integers $0<d_1\\le \\ldots \\le d_m$ , $m \\ge 2$ .", "Let us assume that $(n-r)(r+1)\\ge \\displaystyle \\sum _{i=1}^m{d_i+r\\atopwithdelims ()r}$ .", "Consider the complete intersection $X:=H_1\\cap \\ldots \\cap H_m$ of hypersurfaces $H_i\\subset \\mathbb {P}^n$ of degree $d_i$ .", "Then it follows from the arguments in [14] that $X$ contains a positive dimensional family of linear subspaces $\\simeq \\mathbb {P}^r$ .", "Moreover, $X$ is unirational, provided $X$ is generic.", "It would be interesting to adopt the arguments from the proofs of Propositions REF , REF and REF to this more general setting in order to prove that $X$ is stably b-inf.", "trans.", "Remark 3.8 Propositions REF , REF and REF (cf.", "Remark REF ) provide an alternative method of proving unirationality of complete intersections (see [7] for recollection of classical arguments).", "Note also that (generic) $X_{2\\cdot 2\\cdot 2}$ is non-rational (see for example [17]), and (non-)rationality of the most of other complete intersections considered above is not known.", "At the same time, verifying stable b-inf.", "trans.", "property of other (non-rational) Fano manifolds (cf.", "[7]) is out of reach for our techniques at the moment." ] ]

[ [ "Universal properties and the first law of black hole inner mechanics" ], [ "Abstract We show by explicit computations that the product of all the horizon areas is independent of the mass, regardless of the topology of the horizons.", "The universal character of this relation holds for all known five dimensional asymptotically flat black rings, and for black strings.", "This gives further evidence for the crucial role that the thermodynamic properties at each horizon play in understanding the entropy at the microscopic level.", "To this end we propose a \"first law\" for the inner Cauchy horizons of black holes.", "The validity of this formula, which seems to be universal, was explicitly checked in all cases." ], [ "39" ] ]

[ [ "Majorization in spaces with a curved geometry" ], [ "Abstract The Hardy-Littlewood-P?olya majorization theorem is extended to the framework of some spaces with a curved geometry (such as the global NPC spaces and the Wasserstein spaces).", "We also discuss the connection between our concept of majorization and the subject of Schur convexity." ], [ " The Hardy-Littlewood-Pólya majorization theorem is extended to the framework of some spaces with a curved geometry (such as the global NPC spaces and the Wasserstein spaces).", "We also discuss the connection between our concept of majorization and the subject of Schur convexity.", "In 1929, G. H. Hardy, J. E. Littlewood and G. Pólya [9], [10] have proved an important characterization of convex functions in terms of a partial ordering of vectors $x=(x_{1},...,x_{n})$ in $\\mathbb {R}^{n}$ .", "In order to state it we need a preparation.", "We denote by $x^{\\downarrow }$ the vector with the same entries as $x$ but rearranged in decreasing order, $x_{1}^{\\downarrow }\\ge \\cdots \\ge x_{n}^{\\downarrow }.$ Then $x$ is weakly majorized by $y$ (abbreviated, $x\\prec _{\\ast }y)$ if $\\sum _{i\\,=\\,1}^{k}\\,x_{i}^{\\downarrow }\\le \\sum _{i\\,=\\,1}^{k}\\,y_{i}^{\\downarrow }\\quad \\text{for }k=1,...,n \\qquad \\mathrm {(1)}$ and $x$ is majorized by $y$ (abbreviated, $x\\prec y)$ if in addition $\\sum _{i\\,=\\,1}^{n}\\,x_{i}^{\\downarrow }=\\sum _{i\\,=\\,1}^{n}\\,y_{i}^{\\downarrow }\\,.", "\\qquad \\mathrm {(2)}$ Intuitively, $x\\prec y$ means that the components in $x$ are less spread out than the components in $y$ .", "As is shown in Theorem 1 below, the concept of majorization admits an order-free characterization based on the notion of doubly stochastic matrix.", "Recall that a matrix $A\\in \\,$ M$_{n}(\\mathbb {R})$ is doubly stochastic if it has nonnegative entries and each row and each column sums to unity.", "Theorem 1 (Hardy, Littlewood and Pólya [9], Theorem 8).", "Let $x$ and $y$ be two vectors in $\\mathbb {R}^{n}$ , whose entries belong to an interval $I.$ Then the following statements are equivalent: $i)$ $x\\prec y;$ $ii)$ There is a doubly stochastic matrix $A=(a_{ij})_{1\\le i,j\\le n}$ such that $x=Ay;$ $iii)$ The inequality $\\sum _{i=1}^{n}f(x_{i})\\le \\sum _{i=1}^{n}f(y_{i})$ , holds for every continuous convex function $f:I\\rightarrow \\mathbb {R}$ .", "The proof of this result is also available in the recent monographs [15] and [18].", "Remark 1 M. Tomić [25] and H. Weyl [26] have noticed the following characterization of weak majorization: $x\\prec _{\\ast }y$ if and only if $\\sum _{i=1}^{n}f(x_{i})\\le \\sum _{i=1}^{n}f(y_{i})$ for every continuous nondecreasing convex function $f$ defined on an interval containing the components of $x$ and $y.$ The reader will find the details in [15], Proposition B2, p. 157.", "Nowadays there are known many important applications of majorization to matrix theory, numerical analysis, probability, combinatorics, quantum mechanics etc.", "See [3], [15], [18], [21], and [22].", "They were made possible by the constant growth of the theory, able to uncover the most diverse situations.", "In what follows we will be interested in a simple but basic extension of the concept of majorization as was mentioned above: the weighted majorization.", "Indeed, the entire subject of majorization can be switched from vectors to Borel probability measures by identifying a vector $x=(x_{1},...,x_{n})$ in $\\mathbb {R}^{n}$ with the discrete measure $\\frac{1}{n}\\sum _{i=1}^{n}\\delta _{x_{i}}$ acting on $\\mathbb {R}$ .", "By definition, $\\frac{1}{n}\\sum _{i=1}^{n}\\delta _{x_{i}}\\prec \\frac{1}{n}\\sum _{i=1}^{n}\\delta _{y_{i}}$ if the conditions (1) and (2) above are fulfilled, and Theorem 1 can be equally seen as a characterization of this instance of majorization.", "Choquet's theory made available a very general framework of majorization by allowing the comparison of Borel probability measures whose supports are contained in a compact convex subset of a locally convex separated space.", "The highlights of this theory are presented in [22] and refer to a concept of majorization based on condition $iii)$ in Theorem 1 above.", "Of interest to us is the particular case of discrete probability measures on the Euclidean space $\\mathbb {R}^{N},$ that admits an alternative approach via condition $ii)$ in the same Theorem 1.", "Indeed, in this case one can introduce a relation of the form $\\sum _{i=1}^{m}\\lambda _{i}\\delta _{x_{i}}\\prec \\sum _{j=1}^{n}\\mu _{j}\\delta _{y_{j}} \\qquad \\mathrm {(3)}$ by asking the existence of a $m\\times n$ -dimensional matrix $A=(a_{ij})_{i,j}$ such that $a_{ij}\\ge 0,\\text{ for all }i,j\\\\\\sum _{j=1}^{n}a_{ij}=1,\\text{\\quad }i=1,...,m\\\\\\mu _{j}=\\sum _{i=1}^{m}a_{ij}\\lambda _{i}\\text{,\\quad }j=1,...,n \\qquad \\mathrm {(4,5,6)}$ and $x_{i}=\\sum _{j=1}^{n}a_{ij}y_{j}\\text{,\\quad }i=1,...,n \\qquad \\mathrm {(7)}$ The matrices verifying the conditions (4)&(5) are called stochastic on rows.", "When $m=n$ and all weights $\\lambda _{i}$ and $\\mu _{j}$ are equal, the condition (6) assures the stochasticity on columns, so in that case we deal with doubly stochastic matrices.", "The fact that (3) implies $\\sum _{i=1}^{m}\\lambda _{i}f(x_{i})\\prec \\sum _{j=1}^{n}\\mu _{j}f(y_{j}),$ for every continuous convex function $f$ defined on a convex set containing all points $x_{i}$ and $y_{i},$ is covered by a general result due to S. Sherman [23].", "See also the paper of J. Borcea [5] for a nice proof and important applications.", "It is worth noticing that the extended definition of majorization given by (3) is related, via equality (7), to an optimization problem as follows: $x_{i}=\\arg \\min _{z\\in \\mathbb {R}^{N}}\\frac{1}{2}\\sum _{j=1}^{n}a_{ij}\\left\\Vert z-y_{j}\\right\\Vert ^{2},\\text{\\quad for }i=1,...,m.$ The aim of the present paper is to discuss the analogue of the relation of majorization (3) within certain classes of spaces with curved geometry.", "We will start with the spaces with global nonpositive curvature (abbreviated, global NPC spaces).", "The subject of majorization in these spaces was touched in [17] via a different concept of majorization.", "Central to us here is the generalization of Theorem 1.", "Definition 1 A global NPC space is a complete metric space $M=(M,d)$ for which the following inequality holds true: for each pair of points $x_{0},x_{1}\\in M$ there exists a point $y\\in M$ such that for all points $z\\in M,$ $d^{2}(z,y)\\le \\frac{1}{2}d^{2}(z,x_{0})+\\frac{1}{2}d^{2}(z,x_{1})-\\frac{1}{4}d^{2}(x_{0},x_{1}).", "\\qquad \\mathrm {(8)}$ These spaces are also known as the Cat 0 spaces.", "See [6].", "In a global NPC space, each pair of points $x_{0},x_{1}\\in M$ can be connected by a geodesic (that is, by a rectifiable curve $\\gamma :[0,1]\\rightarrow M$ such that the length of $\\gamma |_{[s,t]}$ is $d(\\gamma (s),\\gamma (t))$ for all $0\\le s\\le t\\le 1)$ .", "Moreover, this geodesic is unique.", "In a global NPC space, the geodesics play the role of segments.", "The point $y$ that appears in Definition 1 is the midpoint of $x_{0}$ and $x_{1}$ and has the property $d(x_{0},y)=d(y,x_{1})=\\frac{1}{2}d(x_{0},x_{1}).$ Every Hilbert space is a global NPC space.", "Its geodesics are the line segments.", "The upper half-plane H$=\\left\\lbrace z\\in \\mathbb {C}:\\operatorname{Im}z>0\\right\\rbrace $ , endowed with the Poincaré metric, $ds^{2}=\\frac{dx^{2}+dy^{2}}{y^{2}},$ constitutes another example of a global NPC space.", "In this case the geodesics are the semicircles in H perpendicular to the real axis and the straight vertical lines ending on the real axis.", "A Riemannian manifold $(M,g)$ is a global NPC space if and only if it is complete, simply connected and of nonpositive sectional curvature.", "Besides manifolds, other important examples of global NPC spaces are the Bruhat-Tits buildings (in particular, the trees).", "See [6].", "More information on global NPC spaces is available in [2], [12], and [24].", "See also our papers [17] and [20].", "Definition 2 A set $C\\subset M$ is called convex if $\\gamma ([0,1])\\subset C$ for each geodesic $\\gamma :[0,1]\\rightarrow M$ joining $\\gamma (0),\\gamma (1)\\in C$ .", "A function $\\varphi :C\\rightarrow \\mathbb {R}$ is called convex if $C$ is a convex set and for each geodesic $\\gamma :[0,1]\\rightarrow C$ the composition $\\varphi \\circ \\gamma $ is a convex function in the usual sense, that is, $\\varphi (\\gamma (t))\\le (1-t)\\varphi (\\gamma (0))+t\\varphi (\\gamma (1))$ for all $t\\in [0,1].$ The function $\\varphi $ is called concave if $-\\varphi $ is convex.", "The distance function on a global NPC space $M=(M,d)$ verifies not only the inequality (REF ), but also the following stronger version of it, $d^{2}(z,x_{t})\\le (1-t)d^{2}(z,x_{0})+td^{2}(z,x_{1})-t(1-t)d^{2}(x_{0},x_{1});$ here $z\\ $ is any point in $C$ and $x_{t}$ is any point on the geodesic $\\gamma $ joining $x_{0},x_{1}\\in C$ .", "In terms of Definition REF , this shows that all the functions $d^{2}(\\cdot ,z)$ are uniformly convex.", "In particular, they are convex and the balls are convex sets.", "In a global NPC space $M=(M,d)$ the distance function $d$ is convex on $M\\times M$ and also convex are the functions $d(\\cdot ,z).$ See [24], Corollary 2.5, for details.", "Recall that the direct product of metric spaces $M_{i}=(M_{i},d_{i})$ ($i=1,...,n)$ is the metric space $M=(M,d_{M})$ defined by $M={\\displaystyle \\prod \\nolimits _{i=1}^{n}}M_{i}$ and $d_{M}(x,y)=\\left( \\sum _{i=1}^{n}d_{i}(x_{i},y_{i})^{2}\\right) ^{1/2}.$ It is a global NPC space if all factors are global NPC spaces.", "When $x_{1},...,x_{m},y_{1},...,y_{n}$ are points of a global NPC space $M,$ and $\\lambda _{1},...,\\lambda _{m}\\in [0,1]$ are weights that sum to 1, we will define the relation of majorization $\\sum _{i=1}^{m}\\lambda _{i}\\delta _{x_{i}}\\prec \\sum _{j=1}^{n}\\mu _{j}\\delta _{y_{j}} \\qquad \\mathrm {(9)}$ by asking the existence of an $m\\times n$ -dimensional matrix $A=(a_{ij})_{i,j}$ that is stochastic on rows and verifies the following two conditions: $\\mu _{j}=\\sum _{i=1}^{m}a_{ij}\\lambda _{i},\\quad j=1,...,n \\qquad \\mathrm {(10)}$ and $x_{i}=\\arg \\min _{z\\in M}\\frac{1}{2}\\sum _{j=1}^{n}a_{ij}d^{2}(z,y_{j}),\\text{\\quad }i=1,...,m. \\qquad \\mathrm {(11)}$ The existence and uniqueness of the problems of optimization (11) is assured by the fact that the objective functions are uniformly convex and positive.", "See [12], Section 3.1, or [24], Proposition 1.7, p. 3.", "Notice that the above definition agrees with the usual one in the Euclidean case.", "It is also related to the definition of the barycenter of a Borel probability measure $\\mu $ defined on a global NPC space $M$ .", "Precisely, if $\\mu \\in \\mathcal {P}_{2}(M)$ (the set of those probability measures under which all functions $d^{2}(\\cdot ,z)$ are integrable), then its barycenter is defined by the formula $\\operatornamewithlimits{bar}(\\mu )=\\arg \\min _{z\\in M}\\frac{1}{2}\\int _{M}d^{2}(z,x)d\\mu (x).$ This definition, due to E. Cartan [7], was inspired by Gauss' Least Squares Method.", "A larger approach of the notion of barycenter is offered by the recent paper of Sturm [24].", "The particular case of discrete probability measures $\\lambda =\\sum _{i=1}^{n}\\lambda _{i}\\delta _{x_{i}}$ is of special interest because the barycenter of $\\lambda $ can be seen as a good analogue for the convex combination (or weighted mean) $\\lambda _{1}x_{1}+\\cdots +\\lambda _{n}x_{n}.$ Indeed, $\\operatornamewithlimits{bar}(\\lambda )=\\arg \\min _{z\\in M}\\frac{1}{2}\\sum _{i=1}^{n}\\lambda _{i}d^{2}(z,x_{i}),$ and the way $\\operatornamewithlimits{bar}(\\lambda )$ provides a mean with nice features was recently clarified by Lawson and Lim [14].", "As an immediate consequence one obtains the relation $\\delta _{\\operatornamewithlimits{bar}(\\lambda )}\\prec \\lambda .$ A word of caution when denoting $\\operatornamewithlimits{bar}(\\lambda )$ as $\\lambda _{1}x_{1}+\\cdots +\\lambda _{n}x_{n}$ .", "Probably a notation like $\\lambda _{1}x_{1}\\boxplus \\cdots \\boxplus \\lambda _{n}x_{n}$ suits better because $\\operatornamewithlimits{bar}(\\lambda )$ can be far from the usual the arithmetic mean.", "Consider for example the case where $M$ is the space $\\operatornamewithlimits{Sym}^{++}(n,\\mathbb {R)}$ (of all positively definite matrices with real coefficients), endowed with the trace metric, $d_{\\operatornamewithlimits{trace}}(A,B)=\\left( \\sum _{k=1}^{n}\\log ^{2}\\lambda _{k}\\right) ^{1/2},$ where $\\lambda _{1},\\dots ,\\lambda _{n}$ are the eigenvalues of $AB^{-1}$ .", "In this case $\\frac{1}{2}A\\boxplus \\frac{1}{2}B=A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2},$ that is, it coincides with the geometric mean of $A$ and $B.$ See [4], Section 6.3, or [13], for details.", "Since the convex combinations within a global NPC space lack in general the property of associativity, $\\sum _{i=1}^{n+1}\\lambda _{i}x_{i}=(1-\\lambda _{n+1})\\left( \\sum _{i=1}^{n}\\frac{\\lambda _{i}}{1-\\lambda _{n+1}}x_{i}\\right) +\\lambda _{n+1}x_{n+1},$ the proof of Jensen's inequality is not trivial even in the discrete case.", "This explains why this inequality was first stated in this context only in 2001 by J. Jost [11].", "We recall it here in the formulation of Eells and Fuglede [8], Proposition 12.3, p. 242: Theorem 2 (Jensen's Inequality).", "For any lower semicontinuous convex function $f:M\\rightarrow \\mathbb {R}$ and any Borel probability measure $\\mu \\in \\mathcal {P}_{2}(M)$ we have the inequality $f(\\operatornamewithlimits{bar}(\\mu ))\\le \\int _{M}f(x)d\\mu (x),$ provided the right hand side is well-defined.", "The proof of Eells and Fuglede is based on the following remark concerning barycenters: If a probability measure $\\mu $ is supported by a convex closed set $K$ , then its barycenter $\\operatornamewithlimits{bar}(q)$ lies in $K$ .", "A probabilistic approach of Theorem 2 is due to Sturm [24].", "An immediate consequence of Theorem 2 is the following couple of inequalities that work for any points $z,x_{1},...,x_{n},\\,y_{1},...,y_{n}$ in a global NPC space: $d^{2}\\left( \\frac{1}{n}x_{1}\\boxplus \\cdots \\boxplus \\frac{1}{n}x_{n},z\\right)\\le \\frac{d^{2}(x_{1},z)+\\cdots +d^{2}(x_{n},z)}{n}$ and $d\\left( \\frac{1}{n}x_{1}\\boxplus \\cdots \\boxplus \\frac{1}{n}x_{n},\\frac{1}{n}y_{1}\\boxplus \\cdots \\boxplus \\frac{1}{n}y_{n}\\right) \\le \\frac{d(x_{1},y_{1})+\\cdots +d(x_{n},y_{n})}{n}.$ The next theorem offers a partial extension of Hardy-Littlewood-Pólya Theorem to the context of global NPC spaces.", "Theorem 3 If $\\sum _{i=1}^{m}\\lambda _{i}\\delta _{x_{i}}\\prec \\sum _{j=1}^{n}\\mu _{j}\\delta _{y_{j}},$ in the global NPC space $M,$ then, for every continuous convex function $f$ defined on a convex subset $U\\subset M$ containing all points $x_{i}$ and $y_{j}$ we have $\\sum _{i=1}^{m}\\lambda _{i}f(x_{i})\\le \\sum _{j=1}^{n}\\mu _{j}f(y_{j}).$ By our hypothesis, there is an $m\\times n$ -dimensional matrix $A=(a_{ij})_{i,j}$ that is stochastic on rows and verifies the conditions (10) and (11).", "The last condition, shows that each point $x_{i}$ is the barycenter of the probability measure $\\sum _{j=1}^{n}a_{ij}\\delta _{y_{j}}$ , so by Jensen's inequality we infer that $f(x_{i})\\le \\sum _{j=1}^{n}a_{ij}f(y_{j}).$ Multiplying each side by $\\lambda _{i}$ and then summing up over $i$ from 1 to $m,$ we conclude that $\\sum _{i=1}^{m}\\lambda _{i}f(x_{i}) & \\le \\sum _{i=1}^{m}\\left( \\lambda _{i}\\sum _{j=1}^{n}a_{ij}f(y_{j})\\right) \\\\& =\\sum _{j=1}^{n}\\left( \\sum _{i=1}^{m}a_{ij}\\lambda _{i}\\right) f(y_{j})\\\\& =\\sum _{j=1}^{n}\\mu _{j}f(y_{j}).$ By our hypothesis, there is an $m\\times n$ -dimensional matrix $A=(a_{ij})_{i,j}$ that is stochastic on rows and verifies the conditions (10) and (11).", "The last condition, shows that each point $x_{i}$ is the barycenter of the probability measure $\\sum _{j=1}^{n}a_{ij}\\delta _{y_{j}}$ , so by Jensen's inequality we infer that $f(x_{i})\\le \\sum _{j=1}^{n}a_{ij}f(y_{j}).$ Multiplying each side by $\\lambda _{i}$ and then summing up over $i$ from 1 to $m,$ we conclude that $\\sum _{i=1}^{m}\\lambda _{i}f(x_{i}) & \\le \\sum _{i=1}^{m}\\left( \\lambda _{i}\\sum _{j=1}^{n}a_{ij}f(y_{j})\\right) \\\\& =\\sum _{j=1}^{n}\\left( \\sum _{i=1}^{m}a_{ij}\\lambda _{i}\\right) f(y_{j})\\\\& =\\sum _{j=1}^{n}\\mu _{j}f(y_{j}).$ In a global NPC space the distance function from a convex set is a convex function.", "See [24], Corollary 2.5.", "Combining this fact with Theorem REF we infer the following result.", "Corollary 1 If $\\sum _{i=1}^{m}\\lambda _{i}\\delta _{x_{i}}\\prec \\sum _{j=1}^{n}\\mu _{j}\\delta _{y_{j}},$ and all coefficients $\\lambda _{i}$ are positive, then $\\lbrace x_{1},...,x_{m}\\rbrace $ is contained in the convex hull of $\\lbrace y_{1},...,y_{n}\\rbrace .$ In particular, the points $x_{i}$ spread out less than the points $y_{j}.$ Another application of Theorem REF yields a new set of inequalities verified by the functions $d(\\cdot ,$ $z)$ in a global NPC space $M$ .", "These functions are convex and the same is true for the functions $f(d(\\cdot ,$ $z))$ whenever $f$ is a continuous nondecreasing convex function defined on $\\mathbb {R}_{+}.$ According to Theorem REF , if $\\frac{1}{n}\\sum _{i=1}^{n}\\delta _{x_{i}}\\prec \\frac{1}{n}\\sum _{i=1}^{n}\\delta _{y_{i}}$ in $M,$ then $\\sum _{i=1}^{n}f(d(x_{i},z))\\le \\sum _{i=1}^{n}f(d(y_{i},z)).$ Taking into account Remark 1 we arrive at the following result: Corollary 2 If $\\frac{1}{n}\\sum _{i=1}^{n}\\delta _{x_{i}}\\prec \\frac{1}{n}\\sum _{i=1}^{n}\\delta _{y_{i}}$ in $M=(M,d),$ then for all $z\\in M,$ $\\left( d(x_{1},z),...,d(x_{n},z)\\right) \\prec _{\\ast }\\left( d(y_{1},z),...,d(y_{n},z)\\right)$ According to a result due to Ando (see [15], Theorem B.3a, p. 158), the converse of Corollary REF works when $M=\\mathbb {R}$ .", "The entropy function, $H(t)=-t\\log t,$ is concave and decreasing for $t\\in [1/e,\\infty ),$ so by Corollary REF we infer that ${\\displaystyle \\prod \\nolimits _{i=1}^{n}}d(x_{i},z)^{d(x_{i},z)}\\ge {\\displaystyle \\prod \\nolimits _{i=1}^{n}}d(y_{i},z)^{d(y_{i},z)},$ whenever $\\frac{1}{n}\\sum _{i=1}^{n}\\delta _{x_{i}}\\prec \\frac{1}{n}\\sum _{j=1}^{n}\\delta _{y_{j}}$ and all the points $x_{i}$ and $y_{i}$ are at a distance $\\ge 1/e$ from $z.$ Many other inequalities involving distances in a global NPC space can be derived from Corollary REF and the following result due to Fan and Mirsky: if $x,y\\in \\mathbb {R}_{+}^{n},$ then $x\\prec _{\\ast }y$ if and only if $\\Phi (x)\\le \\Phi (y)$ for all functions $\\Phi :\\mathbb {R}^{n}\\rightarrow \\mathbb {R}$ such that: $\\Phi (x)>0$ when $x\\ne 0;$ $\\Phi (\\alpha x)=\\left|\\alpha \\right|\\Phi (x)$ for all real $\\alpha ;$ $\\Phi (x+y)\\le \\Phi (x)+\\Phi (y);$ $\\Phi (x_{1},...,x_{n})=\\Phi (\\varepsilon _{1}x_{\\pi (1)},...,\\varepsilon _{n}x_{\\pi (n)})$ whenever each $\\varepsilon _{i}$ belongs to $\\lbrace -1,1\\rbrace $ and $\\pi $ is any permutation of $\\left\\lbrace 1,...,n\\right\\rbrace .$ $\\Phi (x)>0$ when $x\\ne 0;$ $\\Phi (\\alpha x)=\\left|\\alpha \\right|\\Phi (x)$ for all real $\\alpha ;$ $\\Phi (x+y)\\le \\Phi (x)+\\Phi (y);$ $\\Phi (x_{1},...,x_{n})=\\Phi (\\varepsilon _{1}x_{\\pi (1)},...,\\varepsilon _{n}x_{\\pi (n)})$ whenever each $\\varepsilon _{i}$ belongs to $\\lbrace -1,1\\rbrace $ and $\\pi $ is any permutation of $\\left\\lbrace 1,...,n\\right\\rbrace .$ For details, see [15], Proposition B6, p. 160.", "It is worth noticing the connection between our definition of majorization and the subject of Schur convexity (as presented in [15]): Theorem 4 Suppose that $\\frac{1}{n}\\sum _{i=1}^{n}\\delta _{x_{i}}\\prec \\frac{1}{n}\\sum _{i=1}^{n}\\delta _{y_{i}}$ in the global NPC space $M=(M,d),$ and $f:M^{n}\\rightarrow \\mathbb {R}$ is a continuous convex function invariant under the permutation of coordinates.", "Then $f(x_{1},...,x_{n})\\le f(y_{1},...,y_{n}).$ For the sake of simplicity we will restrict here to the case where $n=3.$ According to the definition of majorization, if $\\frac{1}{3}\\sum _{i=1}^{3}\\delta _{x_{i}}\\prec \\frac{1}{3}\\sum _{i=1}^{3}\\delta _{y_{i}}$ , then there exists a doubly stochastic matrix $A=(a_{ij})_{i,j=1}^{3}$ such that $x_{i}=\\operatornamewithlimits{bar}(\\sum _{j=1}^{n}a_{ij}\\delta _{y_{j}})\\quad \\text{for}i=1,...,n.$ As $A$ can be uniquely represented under the form $A=\\left(\\begin{array}[c]{ccc}\\lambda _{1}+\\lambda _{2} & \\lambda _{3}+\\lambda _{5} & \\lambda _{4}+\\lambda _{6}\\\\\\lambda _{3}+\\lambda _{4} & \\lambda _{1}+\\lambda _{6} & \\lambda _{2}+\\lambda _{5}\\\\\\lambda _{5}+\\lambda _{6} & \\lambda _{2}+\\lambda _{4} & \\lambda _{1}+\\lambda _{3}\\end{array}\\right) ,$ where all $\\lambda _{k}$ are nonnegative and $\\sum _{k=1}^{6}\\lambda _{k}=1$ (a simple matter of linear algebra) we can represent the elements $x_{j}$ as $x_{1} & =\\operatornamewithlimits{bar}((\\lambda _{1}+\\lambda _{2})\\delta _{y_{1}}+(\\lambda _{3}+\\lambda _{4})\\delta _{y_{2}}+(\\lambda _{5}+\\lambda _{6})\\delta _{y_{3}}),\\\\x_{2} & =\\operatornamewithlimits{bar}((\\lambda _{3}+\\lambda _{5})\\delta _{y_{1}}+(\\lambda _{1}+\\lambda _{6})\\delta _{y_{2}}+(\\lambda _{2}+\\lambda _{4})\\delta _{y_{3}}),\\\\x_{3} & =\\operatornamewithlimits{bar}\\left( (\\lambda _{4}+\\lambda _{6})\\delta _{y_{1}}+(\\lambda _{2}+\\lambda _{5})\\delta _{y_{2}}+(\\lambda _{1}+\\lambda _{3})\\delta _{y_{3}}\\right) .$ It is easy to see that $(x_{1},x_{2},x_{3})$ is the barycenter of $\\mu & =\\lambda _{1}\\delta _{(y_{1},y_{2},y_{3})}+\\lambda _{2}\\delta _{(y_{1},y_{3},y_{2})}+\\lambda _{3}\\delta _{(y_{2},y_{1},y_{3})}\\\\& +\\lambda _{4}\\delta _{(y_{2},y_{3},y_{1})}+\\lambda _{5}\\delta _{(y_{3},y_{1},y_{2})}+\\lambda _{6}\\delta _{(y_{3},y_{2},y_{1})},$ so by Jensen's inequality and the symmetry of $f$ we get $f(x_{1},...,x_{n})\\le \\lambda _{1}f(y_{1},y_{2},y_{3})+\\lambda _{2}f(y_{1},y_{3},y_{2})+\\lambda _{3}f(y_{2},y_{1},y_{3})\\\\+\\lambda _{4}f(y_{2},y_{3},y_{1})+\\lambda _{5}f(y_{3},y_{1},y_{2})+\\lambda _{6}f(y_{3},y_{2},y_{1})\\\\=(\\lambda _{1}+\\cdots +\\lambda _{6})f(y_{1},y_{2},y_{3})=f(y_{1},y_{2},y_{3}).$ For the sake of simplicity we will restrict here to the case where $n=3.$ According to the definition of majorization, if $\\frac{1}{3}\\sum _{i=1}^{3}\\delta _{x_{i}}\\prec \\frac{1}{3}\\sum _{i=1}^{3}\\delta _{y_{i}}$ , then there exists a doubly stochastic matrix $A=(a_{ij})_{i,j=1}^{3}$ such that $x_{i}=\\operatornamewithlimits{bar}(\\sum _{j=1}^{n}a_{ij}\\delta _{y_{j}})\\quad \\text{for}i=1,...,n.$ As $A$ can be uniquely represented under the form $A=\\left(\\begin{array}[c]{ccc}\\lambda _{1}+\\lambda _{2} & \\lambda _{3}+\\lambda _{5} & \\lambda _{4}+\\lambda _{6}\\\\\\lambda _{3}+\\lambda _{4} & \\lambda _{1}+\\lambda _{6} & \\lambda _{2}+\\lambda _{5}\\\\\\lambda _{5}+\\lambda _{6} & \\lambda _{2}+\\lambda _{4} & \\lambda _{1}+\\lambda _{3}\\end{array}\\right) ,$ where all $\\lambda _{k}$ are nonnegative and $\\sum _{k=1}^{6}\\lambda _{k}=1$ (a simple matter of linear algebra) we can represent the elements $x_{j}$ as $x_{1} & =\\operatornamewithlimits{bar}((\\lambda _{1}+\\lambda _{2})\\delta _{y_{1}}+(\\lambda _{3}+\\lambda _{4})\\delta _{y_{2}}+(\\lambda _{5}+\\lambda _{6})\\delta _{y_{3}}),\\\\x_{2} & =\\operatornamewithlimits{bar}((\\lambda _{3}+\\lambda _{5})\\delta _{y_{1}}+(\\lambda _{1}+\\lambda _{6})\\delta _{y_{2}}+(\\lambda _{2}+\\lambda _{4})\\delta _{y_{3}}),\\\\x_{3} & =\\operatornamewithlimits{bar}\\left( (\\lambda _{4}+\\lambda _{6})\\delta _{y_{1}}+(\\lambda _{2}+\\lambda _{5})\\delta _{y_{2}}+(\\lambda _{1}+\\lambda _{3})\\delta _{y_{3}}\\right) .$ It is easy to see that $(x_{1},x_{2},x_{3})$ is the barycenter of $\\mu & =\\lambda _{1}\\delta _{(y_{1},y_{2},y_{3})}+\\lambda _{2}\\delta _{(y_{1},y_{3},y_{2})}+\\lambda _{3}\\delta _{(y_{2},y_{1},y_{3})}\\\\& +\\lambda _{4}\\delta _{(y_{2},y_{3},y_{1})}+\\lambda _{5}\\delta _{(y_{3},y_{1},y_{2})}+\\lambda _{6}\\delta _{(y_{3},y_{2},y_{1})},$ so by Jensen's inequality and the symmetry of $f$ we get $f(x_{1},...,x_{n})\\le \\lambda _{1}f(y_{1},y_{2},y_{3})+\\lambda _{2}f(y_{1},y_{3},y_{2})+\\lambda _{3}f(y_{2},y_{1},y_{3})\\\\+\\lambda _{4}f(y_{2},y_{3},y_{1})+\\lambda _{5}f(y_{3},y_{1},y_{2})+\\lambda _{6}f(y_{3},y_{2},y_{1})\\\\=(\\lambda _{1}+\\cdots +\\lambda _{6})f(y_{1},y_{2},y_{3})=f(y_{1},y_{2},y_{3}).$ The following consequence of Theorem REF relates the majorization of measures to the dispersion of their supports.", "Corollary 3 If $\\frac{1}{n}\\sum _{i=1}^{n}\\delta _{x_{i}}\\prec \\frac{1}{n}\\sum _{i=1}^{n}\\delta _{y_{i}}$ in the global NPC space $M=(M,d)$ , then $\\sum _{1\\le i<j\\le n}d^{\\alpha }(x_{i},x_{j})\\le \\sum _{1\\le i<j\\le n}d^{\\alpha }(y_{i},y_{j})$ for every $\\alpha \\ge 1$ .", "Alert readers have probably already noticed that essential for the theory of majorization presented above is the occurrence of the following two facts: the existence of a unique minimizer for the functionals of the form $J(x)=\\frac{1}{2}\\sum _{i=1}^{m}\\lambda _{i}d^{2}(x,x_{i})$ (thought of as the barycenter $\\operatornamewithlimits{bar}$ ($\\lambda )$ of the discrete probability measure $\\lambda =\\sum _{i=1}^{m}\\lambda _{i}\\delta _{x_{i}});$ the Jensen type inequality, $f(\\operatornamewithlimits{bar}(\\lambda ))\\le \\int fd\\lambda =\\sum _{i=1}^{m}\\lambda _{i}f(x_{i}),$ for $f$ in our class of generalized convex functions.", "the existence of a unique minimizer for the functionals of the form $J(x)=\\frac{1}{2}\\sum _{i=1}^{m}\\lambda _{i}d^{2}(x,x_{i})$ (thought of as the barycenter $\\operatornamewithlimits{bar}$ ($\\lambda )$ of the discrete probability measure $\\lambda =\\sum _{i=1}^{m}\\lambda _{i}\\delta _{x_{i}});$ the Jensen type inequality, $f(\\operatornamewithlimits{bar}(\\lambda ))\\le \\int fd\\lambda =\\sum _{i=1}^{m}\\lambda _{i}f(x_{i}),$ for $f$ in our class of generalized convex functions.", "The recent paper of Agueh and Carlier [1] shows that such a framework is available also in the case of certain Borel probability measures, equipped with the Wasserstein metric.", "More precisely they consider the space $\\mathcal {P}_{2}(\\mathbb {R}^{N})$ (of all Borel probability measures on $\\mathbb {R}^{N}$ having finite second moments) endowed with the Wasserstein metric, $\\mathcal {W}_{2}(\\mu ,\\nu )=\\inf \\left( \\int _{\\mathbb {R}^{N}\\times \\mathbb {R}^{N}}\\left\\Vert x-y\\right\\Vert ^{2}d\\gamma (x,y)\\right) ^{1/2},$ where the infimum is taken over all Borel probability measures $\\gamma $ on $\\mathbb {R}^{N}\\times \\mathbb {R}^{N}$ with marginals $\\mu $ and $\\nu $ .", "The barycenter $\\operatornamewithlimits{bar}(\\sum _{i=1}^{m}\\lambda _{i}\\delta _{\\nu _{i}}),$ of a discrete probability measure $\\sum _{i=1}^{m}\\lambda _{i}\\delta _{\\nu _{i}},$ is defined as the minimizer of the functional $J(\\nu )=\\frac{1}{2}\\sum _{i=1}^{m}\\lambda _{i}\\mathcal {W}_{2}^{2}(\\nu _{i},\\nu ).$ This minimizer is unique when at least one of the measures $\\nu _{i}$ vanishes on every Borel set of Hausdorff dimension $N-1$ .", "See [1], Proposition 2.2 and Proposition 3.5.", "The natural class of convex function on the Wasserstein space is that of functions convex along barycenters.", "According to [1], Definition 7.1, a function $\\mathcal {F}:$ $\\mathcal {P}_{2}(\\mathbb {R}^{N})\\rightarrow \\mathbb {R}$ is said to be convex along barycenters if for any discrete probability measure $\\sum _{i=1}^{m}\\lambda _{i}\\delta _{\\nu _{i}}$ on $\\mathcal {P}_{2}(\\mathbb {R}^{N})$ we have $\\mathcal {F}(\\operatornamewithlimits{bar}(\\sum _{i=1}^{m}\\lambda _{i}\\delta _{\\nu _{i}}))\\le \\sum _{i=1}^{m}\\lambda _{i}\\mathcal {F}(\\nu _{i}).$ This notion of convexity coincides with the notion of displacement convexity introduced by McCann [16] if $N=1,$ and is stronger than this in the general case.", "However, the main examples of displacement convex functions (such as the the internal energy, the potential energy and the interaction energy) are also examples of functions convex along barycenters.", "See [1], Proposition 7.7.", "Theorem 5 The concept of majorization and all results noticed in the case of global NPC spaces (in particular, Theorem 3 and Theorem 4) remain valid in the context discrete probability measures on $\\mathcal {P}_{2}(\\mathbb {R}^{N})$ having unique barycenters and the functions $\\mathcal {F}:$ $\\mathcal {P}_{2}(\\mathbb {R}^{N})\\rightarrow \\mathbb {R}$ convex along barycenters.", "We end our paper with an open problem that arises in connection to Rado's geometric characterization of majorization in $\\mathbb {R}^{n}:$ $(x_{1},...,x_{n})\\prec (y_{1},...,y_{n})$ in $\\mathbb {R}^{n}$ if and only if $(x_{1},...,x_{n})$ lies in the convex hull of the $n!$ permutations of $(y_{1},...,y_{n})$ .", "See [15], Corollary B.3, p. 34.", "A relation of majorization of this kind can be introduced in the power space $M^{n}$ (of any global NPC space $M=(M,d)$ as well as of $\\mathcal {P}_{2}(\\mathbb {R}^{N}))$ by putting $(x_{1},...,x_{n})\\prec (y_{1},...,y_{n})\\text{ }$ if $\\frac{1}{n}\\sum _{i=1}^{n}\\delta _{x_{i}}\\prec \\frac{1}{n}\\sum _{i=1}^{n}\\delta _{y_{i}}.$ The proof of Theorem REF yields immediately the necessity part of Rado's characterization: if $(x_{1},...,x_{n})\\prec (y_{1},...,y_{n})$ in $M^{n},$ then $(x_{1},...,x_{n})$ lies in the convex hull of the $n!$ permutations of $(y_{1},...,y_{n})$ .", "Do the converse work?", "We know that the answer is positive if $M$ is a Hilbert space but the general case remains open.", "Acknowledgement.", "This paper is supported by a grant of the Romanian National Authority for Scientific Research, CNCS – UEFISCDI, project number PN-II-ID-PCE-2011-3-0257." ] ]

[ [ "Comparing simultaneous measurements by two high-resolution imaging\n spectropolarimeters: the `Goettingen' FPI@VTT and CRISP@SST" ], [ "Abstract In July 2009, the leading spot of the active region NOAA11024 was observed simultaneously and independently with the 'Goettingen' FPI at VTT and CRISP at SST, i.e., at two different sites, telescopes, instruments and using different spectral lines.", "The data processing and data analysis have been carried out independently with different techniques.", "Maps of physical parameters retrieved from 2D spectro-polarimetric data observed with 'Goettingen' FPI and CRISP show an impressive agreement.", "In addition, the 'Goettingen' FPI maps also exhibit a notable resemblance with simultaneous TIP (spectrographic) observations.", "The consistency in the results demonstrates the excellent capabilities of these observing facilities.", "Besides, it confirms the solar origin of the detected signals, and the reliability of FPI-based spectro-polarimeters." ], [ "Introduction", "The improvement of solar observational capabilities has revealed a wealth of unresolved structures present in the solar atmosphere whose physical driver(s) is not yet established.", "For the study of these small-scale solar structures, there are various observational requirements to be fulfilled: an appropriate field of view, an angular resolution below 0.1 arcsec, a spectral resolution of at least 300 000 and a temporal resolution of 10 sec or less.", "In view of these requirements, [8] compares the performance of FPI- and grating-based spectrometers.", "The conclusion is that FPI systems are better suited when a combination of high spatial, spectral and temporal resolution is pursued.", "However, the reliability of the spectro(-polarimetric) capabilities of the FPI-based instruments is still under debate, mainly due to the complexity of the optical requirements, of the processing of the large data sets acquired during the observations and other facts like the degradation of image quality intrinsic to any FPI imaging system.", "Yet, 2D spectro-polarimetry is a powerful technique which offers the possibility of recovering spectral line profiles in each pixel across the field of view, within a short time interval.", "This kind of measurements allow one to derive the spatial distribution of physical parameters such as temperature, velocity and magnetic field, together with the temporal evolution [3].", "In addition, 2D filtergrams can be restored from $seeing$ effects, leading to the highest spatial resolution.", "For this contribution, two independent observing teams have compared the data sets recorded simultaneously with two different FPI imaging systems in order to check the validity of their measurements, and in consequence, the performance of this kind of post-focus instruments." ], [ "Observations and Data Analysis", "On 4 July 2009, two independent observing campaigns took place at the German Vacuum Tower Telescope (VTT, Observatorio del Teide, Tenerife) and the Swedish 1-m Tower Telescope (SST, Observatorio de El Roque de los Muchachos, La Palma), respectively.", "The common target was the leading spot of the active region NOAA 11024 ($\\vartheta $  $\\sim $  28$^{\\circ }$ ).", "The observations were recorded with the `Göttingen' FPI [12], [1] at the VTT and with CRISP [16], [17] at the SST.", "The specifications of the observations presented here are collected in Table REF .", "Table: Specifications of the observations at the VTT and SST telescopes." ], [ "Observations with the `Göttingen' FPI at the VTT", "At the time of the observations, the `Göttingen' FPICurrently updated to GFPI [11] and attached to the 1.5 m GREGOR telescope.", "(G-FPI, from now on) was operational at the VTT.", "The full Stokes vector was retrieved from four polarimetric measurements of the incoming sunlight taken with a modulator system, based on ferro-electric liquid crystals (FLCs), of estimated efficiencies $\\epsilon _Q$ =0.41, $\\epsilon _U$ =0.49, and $\\epsilon _V$ =0.58.", "The data were recorded in speckle mode, i.e., (seven) frames of short exposure (20 ms) per wavelength and polarimetric state.", "This procedure allows one to restore the images by applying speckle reconstruction techniques [6] and speckle interferometry [7], [2].", "Further detailed information on the data and the data analysis procedure can be found in [14]." ], [ "CRISP at the SST", "CRISP, operational at the SST, is likewise an imaging spectropolarimeter based on two FPI etalons, yet it is a telecentric mounting, i.e., the etalons are placed close to the focal plane.", "The differences between a telecentric and a collimated mounting and their optical requirements are discussed in [9] and [16].", "Together with the present data, quasi-simultaneous observations in the Fe i 630.15 nm line and other wavelengths (filters) not discussed here, were recorded.", "The data were restored applying Multi-Object Multi-Frame Blind Deconvolution [18].", "CRISP measures the polarimetric states of the incoming light with a modulator system based on two nematic liquid crystals (NLCs).", "The physical parameters were retrieved applying the SIR code [15] to an area of 26$^{\\prime \\prime }$  $\\times $  27$^{\\prime \\prime }$ centred in the spot.", "Figure: Intensity maps of NOAA 11024 leading spot observed at 09:50 UT.", "Left: ∼\\sim  50\"×\\times  50\" intensity map recorded with the G-band channel at the VTT.", "Right: continuum (630.25 nm) intensity map (∼\\sim  55\"×\\times  55\") recorded with CRISP.", "For a better comparison, the CRISP image has been rotated by 74 ∘ ^{\\circ } counterclockwise in order to match the VTT image orientation." ], [ "Comparing the Retrieved Physical Parameters", "Let us emphasise that the observations, as the data analysis, were carried out by two independent groups with independent scientific goals, hence, the diversity and differences in the observations and data analysis methods.", "The aim of this contribution is to compare qualitatively the results from the two independent approaches." ], [ "Intensity Maps", "Simultaneously to the G-FPI scans, images in a G band channel (430 nm, 1 nm bandwidth) were recorded.", "Figure REF shows intensity maps from the G band (VTT) and narrow-band continuum at $\\sim $  630.2 nm (SST).", "The quality of the images is comparable.", "This is expected since the diffraction limit of the VTT at 430 nm and the SST at 630 nm are similar.", "The higher contrast shown by the G band image (11%) with respect to the continuum image (6%) is inherent to the formation in the deep photosphere of the molecular band.", "Nevertheless, the filigrees and other bright fine structures characteristic of G band images can also be identified in the continuum data.", "A deeper comparison of the images after proper co-alignment shows a one-to-one correlation, even at the smallest structures.", "This guarantees the solar origin of the observed structures and discards them to be artifacts introduced by, e.g., the image reconstruction techniques." ], [ "Doppler Maps", "Figure REF shows velocity maps from the G-FPI (left) and CRISP (right) data.", "The G-FPI map has been obtained by measuring Doppler shifts of the centre-of-gravity (COG) of the Stokes $I$ profiles (at 617.34 nm) with respect to the average profile from the ambient granulation.", "The result is the line-of-sight component of the velocity field.", "In the case of the CRISP data, the ${v}_{LOS}$ has been obtained as an output from the inversions (at 630.25 nm).", "Figure: Velocity maps of NOAA 11024 leading spot (09:50 UT).", "Left: `Göttingen FPI' Doppler map measured with the COG method from Stokes I profiles.", "Right: CRISP velocity map retrieved from inversions.", "The maps are clipped to the velocity values given in the color bars.", "For a better comparison, the CRISP map has been rotated by 74 ∘ ^{\\circ } counterclockwise in order to match the VTT map orientation.Despite the difference in the spatial resolution, due to the different throughput of the instruments and the telescope apertures, and the different measurement approaches, the resemblance is evident, i.e., the correlation of the redshifts (clear areas) and blueshifts (dark areas) is consistent.", "Distinct features like, e.g., the redshifts observed in the G-FPI map at ($x$ , $y$ ) = (10$^{\\prime \\prime }$ , 12$^{\\prime \\prime }$ ) and (9$^{\\prime \\prime }$ , 14$^{\\prime \\prime }$ ) and co-spatial with the upper light bridge of the spot, are clearly seen in the CRISP data.", "This confirms the reliability of the observations as well as the different data-analysis approaches." ], [ "Line-of-sight Component of the Magnetic Field", "The LOS component of the magnetic field has been measured from the COG method [13] applied to the $I+V$ and $I-V$ profiles for the G-FPI data.", "For comparison, we have calculated LOS magnetograms from the CRISP magnetic field strength and inclination maps inferred from the SIR inversions.", "Note that these maps correspond to a different moment in the sunspot formation than the previous maps.", "They also differ by about 2 min between them.", "However, once more, the similitude of the magnetograms, clipped to the same values, leaves no doubt about the solar nature of the observed signal.", "Figure: LOS magnetograms of NOAA 11024 leading spot.", "Left: `Göttingen FPI' LOS magnetogram measured with the COG method (10:11 UT).", "Right: CRISP LOS magnetogram inferred from inversions (10:13 UT).", "The maps are clipped to the values given in the color bars.", "For a better comparison, the CRISP map has been rotated by 74 ∘ ^{\\circ } counterclockwise in order to match the VTT map orientation." ], [ "Comparison of 2D and 1D Spectro-polarimetric VTT Data", "During the observations with the G-FPI, additional data with the Tenerife Infrared Polarimeter [5], attached to the Echelle spectrograph, were recorded in the IR wavelength range around the Fe i line at 1089.6 nm ($g_\\textrm {\\scriptsize eff}$ =1.5).", "A large (spatial) scan with TIP II covered the whole sunspot while simultaneously scanning in wavelength with the G-FPI.", "The correspondence of the simultaneous and co-spatial maps of the physical parameters with the 2D and the 1D spectro-polarimeters is excellent [14].", "This is a crucial evidence of the ability of the G-FPI to measure spectro-polarimetric signals." ], [ "Conclusions", "We have compared qualitatively some of the results obtained from simultaneous but independent observations which differ in many aspects (observing sites, telescopes, instruments, image reconstruction techniques and data analysis).", "The resemblance of the intensity, as well as the spectroscopic and polarimetric signals, despite the differences in spatial resolution intrinsic to the optical systems, is remarkable.", "The similitude of the different maps, even at the smallest scales, assures the reliability of the findings and validates the performance of the imaging spectro-polarimeters based on FPI etalons.", "Spectropolarimeters based on FPI imaging systems are often used today, e.g., GFPI@GREGOR, TESOS@VTT [19], IBIS@DST [4], CRISP@SST and IMaX@Sunrise [10], and are planned for future large solar telescopes, e.g., VTF@ATST and the FPI systems for EST.", "With this contribution, we demonstrate the reliability and high-quality performance of such systems as spectro-polarimeters when the combination of highest spatial, spectral and temporal resolution is pursued.", "The VTT is operated by the Kiepenheuer-Institut für Sonnenphysik at the Spanish Observatorio del Teide.", "The SST is operated by the Institute for Solar Physics of the Royal Swedish Academy of Sciences in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias.", "NBG acknowledges financial support by the DFG grant Schm 1168/9-1." ] ]

[ [ "The strength of Ramsey Theorem for coloring relatively large sets" ], [ "Abstract We characterize the computational content and the proof-theoretic strength of a Ramsey-type theorem for bi-colorings of so-called {\\em exactly large} sets.", "An {\\it exactly large} set is a set $X\\subset\\Nat$ such that $\\card(X)=\\min(X)+1$.", "The theorem we analyze is as follows.", "For every infinite subset $M$ of $\\Nat$, for every coloring $C$ of the exactly large subsets of $M$ in two colors, there exists and infinite subset $L$ of $M$ such that $C$ is constant on all exactly large subsets of $L$.", "This theorem is essentially due to Pudl\\`ak and R\\\"odl and independently to Farmaki.", "We prove that --- over Computable Mathematics --- this theorem is equivalent to closure under the $\\omega$ Turing jump (i.e., under arithmetical truth).", "Natural combinatorial theorems at this level of complexity are rare.", "Our results give a complete characterization of the theorem from the point of view of Computable Mathematics and of the Proof Theory of Arithmetic.", "This nicely extends the current knowledge about the strength of Ramsey Theorem.", "We also show that analogous results hold for a related principle based on the Regressive Ramsey Theorem.", "In addition we give a further characterization in terms of truth predicates over Peano Arithmetic.", "We conjecture that analogous results hold for larger ordinals." ], [ "Introduction", "A finite set $X\\subseteq \\mathbf {N}$ is large if $\\textrm {card}(X) > \\min (X)$ .", "A finite set $X\\subseteq \\mathbf {N}$ is exactly large if $\\textrm {card}(X) = \\min (X)+1$ .", "The concept of large set was introduced by Paris and Harrington [10] and is the key ingredient of the famous Paris-Harrington principle, also known as the Large Ramsey Theorem.", "The latter is the first example of a natural theorem of finite combinatorics that is unprovable in Peano Arithmetic.", "We are interested in the following extension of the Infinite Ramsey Theorem to bicolorings of exactly large sets.", "Theorem 1 (Pudlàk-Rödl [25] and Farmaki [8], [9]) For every infinite subset $M$ of $\\mathbf {N}$ , for every coloring $C$ of the exactly large subsets of $\\mathbf {N}$ in two colors, there exists an infinite set $L\\subseteq M$ such that every exactly large subset of $L$ gets the same color by $C$ .", "We refer to the statement of the above Theorem as $\\textsc {RT}(!\\omega )$ (the `!'", "is mnemonic for `exactly', while the reason for the use of `$\\omega $ ' is that large sets are also known as `$\\omega $ -large sets').", "By an instance of $\\textsc {RT}(!\\omega )$ we indicate a pair $(M,C)$ of the appropriate type.", "Theorem REF — with slightly different formulations — has been essentially proved by Pudlàk and Rödl [25] and independently by Farmaki [8], [9].", "Pudlàk and Rödl's version is stated in terms of `uniform families'.", "Farmaki's version is in terms of Schreier families.", "Schreier families, originally defined in [29], play an important role in the theory of Banach spaces.", "The notion has been generalized to countable ordinals in [2], [1], [33].", "In fact, both [25] and [8] prove a generalization of the above theorem to any countable ordinal (see infra for more details).", "As observed in [9], Schreier families turn out to essentially coincide with the concept of exactly large set.", "The classical Schreier family is defined as follows $ \\lbrace s = \\lbrace n_1,\\dots ,n_k\\rbrace \\subseteq \\mathbf {N}\\;:\\; n_1 < \\dots < n_k \\mbox{ and } n_1 \\ge k\\rbrace ,$ while the `thin Schreier family' $\\mathcal {A}_\\omega $ is defined by imposing $n_1=k$ (see, e.g., [9]).", "Thus, the Schreier family $\\mathcal {A}_\\omega $ is just an inessential variant of the family of exactly large subsets of $\\mathbf {N}$ .", "In the present paper we investigate the computational and proof-theoretical content of $\\textsc {RT}(!\\omega )$ .", "That is, we characterize the complexity of homogeneous sets witnessing the truth of computable instances of $\\textsc {RT}(!\\omega )$ and we characterize the theorem in terms of formal systems of arithmetic (in the spirit of Reverse Mathematics [31]).", "In particular, we show that there are computable colorings of the exactly large subsets of $\\mathbf {N}$ in two colors all of whose homogeneous sets compute the Turing degree $0^{(\\omega )}$ .", "The degree $0^{(\\omega )}$ is well-known to be the degree of arithmetical truth, i.e., of the first-order theory of the structure $(\\mathbf {N},+,\\times )$ (see, e.g., [27]).", "We show also a reversal of these results by proving that a solution to an instance of $\\textsc {RT}(!\\omega )$ can always be found within the $\\omega $ th Turing jump of the instance.", "Our proofs are such that we obtain as corollaries of the just described computability results the following proof-theoretical results.", "First, we show that — over Computable Mathematics — $\\textsc {RT}(!\\omega )$ implies closure under the $\\omega $ -jump (or, equivalenty, under arithmetic truth): in terms of Reverse Mathematics, we prove that $\\textsc {RT}(!\\omega )$ implies — over $\\textsc {RCA}_0$ — the axiom stating the existence of the $\\omega $ th Turing jump of $X$ for every set $X$ .", "As a reversal we obtain that $\\textsc {RT}(!\\omega )$ is provable in Computable Mathematics ($\\textsc {RCA}_0$ ) augmented by closure under the $\\omega $ Turing jump.", "The system obtained from $\\textsc {RCA}_0$ by adding the axiom stating the closure under the $\\omega $ Turing jump is denoted in the literature as $\\textsc {ACA}_0^+$ .", "By analogy with $\\textsc {RT}(!\\omega )$ we formulate and prove a version of Kanamori-McAloon's Regressive Ramsey Theorem [12] for regressive colorings of exactly large sets and study its effective content.", "We prove analogous results as for $\\textsc {RT}(!\\omega )$ .", "In addition, we present a natural characterization of $\\textsc {RT}(!\\omega )$ in terms of truth predicates over Peano Arithmetic.", "We believe that our results are interesting from the point of view of Computable Mathematics and of the Proof Theory of Arithmetic.", "By Computable Mathematics we here mean the task of measuring the computational complexity of solutions of computable instances of combinatorial problems.", "We give a complete characterization of the strength of $\\textsc {RT}(!\\omega )$ in terms of Computability Theory.", "Our results also yield a characterization of $\\textsc {RT}(!\\omega )$ in terms of proof-theoretic strength as measured by equivalence to subsystems of second order arithmetic, in the spirit of Reverse Mathematics.", "Ramsey's Theorem has been intensively studied from both the viewpoint of Computable Mathematics and of the Proof Theory of Arithmetic, and our characterizations nicely extend the known relations between Ramsey Theorem for coloring finite hypergraphs and the finite Turing jump.", "On the other hand, natural combinatorial theorems at the level of first-order arithmetical truth are not common.", "Our results show that going from colorings of sets of a fixed finite cardinality to colorings of large sets correspondingly boosts the complexity of a coloring principle from hardness with respect to fixed levels of the arithmetical hierarchy to hardness with respect to the whole hierarchy.", "Thus, moving from finite dimensions to exactly large sets acts as a uniform transfer principle corresponding to the move from the finite Turing jumps to the $\\omega $ Turing jump.", "It might be the case that a similar effect can be obtained in other computationally more tame contexts.", "We note that some natural isomorphism problems for computationally tame structures (e.g., the isomorphism problem for automatic graphs and for automatic linear orders) have been recently characterized as being at least as hard as $0^{(\\omega )}$ (see [19]).", "Our results might have interesting connections with this line of research to the extent that graph isomorphism can be related to homogeneity." ], [ "$\\textsc {RT}(!\\omega )$ and Ramsey Theorem", "We first give a combinatorial proof of $\\textsc {RT}(!\\omega )$ featuring an infinite iteration of the finite Ramsey Theorem.", "This proof will be used as a model for our upper bound proof in Section .", "We then recall what is known about the effective content of Ramsey Theorem and establish the easy fact that $\\textsc {RT}(!\\omega )$ implies Ramsey Theorem for all finite exponents.", "We denote by $[X]^{!\\omega }$ the set of exactly large subsets of $X$ .", "For the rest we follow standard partition-calculus notation from combinatorics.", "[Proof of Theorem REF ] Let $M$ be an infinite subset of $\\mathbf {N}$ , let $C:[\\mathbf {N}\\setminus \\lbrace 1,\\dots ,a\\rbrace ]^{!\\omega }\\rightarrow 2$ .", "We build an infinite homogeneous subset $L\\subseteq M$ for $C$ in stages.", "We keep in mind the fact that the family of all exactly large subsets of $M$ can be decomposed based on the minimum element of the set, in the sense that $S\\in [\\mathbf {N}]^{!\\omega }$ if and only if $S=\\lbrace s_1,s_2,\\dots ,s_m\\rbrace $ and $\\lbrace s_2,\\dots ,s_m\\rbrace \\in [\\mathbf {N}- \\lbrace 1,\\dots ,s_1\\rbrace ]^{s_1}$ .", "Let ${C_a}\\colon {[\\mathbf {N}]^{a}}\\rightarrow {2}$ be defined as $C_a(x_1,\\dots ,x_a)=C(a,x_1,\\dots ,x_a)$ .", "We define a sequence ${\\lbrace (a_i,X_i)\\rbrace }_{i\\in \\mathbf {N}}$ such that $a_0= \\min (M)$ , $X_{i+1}\\subseteq X_i\\subseteq M$ , $X_i$ is an infinite and $C_{a_i}$ –homogeneous and $a_i<\\min (X_i)$ , $a_{i+1}=\\min X_i$ .", "At the $i$ -th step of the construction we use Ramsey Theorem for coloring $a_i$ –tuples from the infinite set $X_{i-1}$ (where $X_{-1}=M$ ).", "We finally apply Ramsey Theorem for coloring singletons in two colors (i.e., the Infinite Pigeonhole Principle) to the sequence ${\\lbrace a_i\\rbrace }_{i\\in \\mathbf {N}}$ to get an infinite $C$ –homogeneous set.", "Note that the above proof ostensibly uses induction on $\\Sigma ^1_1$ -formulas.", "We will show below how to transform the above proof into a proof using only induction on arithmetical formulas with second order parameters.", "We now recall what is known about the computational content of Ramsey Theorem and establish a first, easy comparison with $\\textsc {RT}(!\\omega )$ .", "For $n\\in \\mathbf {N}$ , we denote by $\\textsc {RT}^n$ the standard Ramsey Theorem for colorings of $n$ -tuples in two colors, i.e., the assertion that every coloring $C$ of $[\\mathbf {N}]^n$ in two colors admits an infinite homogeneous set.", "With a notable exception, the status of Ramsey's Theorem with respect to computational content is well-known, as summarized in the following theorems.", "Theorem 2 (Jockusch, [11]) $\\,$ For each $n\\ge 2$ there exists a computable coloring $C:[\\mathbf {N}]^n\\rightarrow 2$ admitting no infinite homogeneous set in $\\Sigma _n^0$ .", "For each $n$ , for each computable coloring $C:[\\mathbf {N}]^n\\rightarrow 2$ , there exists an infinite $C$ -homogeneous set in $\\Pi _n^0$ .", "For each $n\\ge 2$ there exists a computable coloring $C:[\\mathbf {N}]^n\\rightarrow 2$ all of whose homogeneous sets compute $0^{(n-2)}$ .", "Points (1), (2), (3) of the above Theorem are Theorem 5.1, Theorem 5.5 and Theorem 5.7 in [11], respectively.", "Essentially drawing on the above results, Simpson proved the following Theorem (Theorem III.7.6 in [31]).", "Theorem 3 (Simpson, [31]) The following are equivalent over $\\textsc {RCA}_0$ .", "$\\textsc {RT}^3$ , $\\textsc {RT}^n$ for any $n\\in \\mathbf {N}$ , $n\\ge 3$ , $\\forall X\\exists Y (Y = X^{\\prime })$ .", "In (3) above, the expression $\\forall X\\exists Y (Y = X^{\\prime })$ is a formalization of the assertion that the Turing jump of $X$ exists (and is $Y$ ).", "Details on how this formalization is carried out in $\\textsc {RCA}_0$ will be presented when needed.", "It is also known that the three statements of the previous Theorem are equivalent to the system $\\textsc {ACA}_0$ (i.e., the system obtained by adding to $\\textsc {RCA}_0$ all instances of the comprehension axiom for arithmetical formulas).", "One of the major open problems in the Proof Theory of Arithmetic is whether Ramsey's Theorem for colorings of pairs implies the totality of the Ackermann function over $\\textsc {RCA}_0$ (see [30], [4]).", "The strength of the full Ramsey Theorem (with syntactic universal quantification over all exponents) has been established by McAloon [20].", "Theorem 4 (McAloon, [20]) The following are equivalent over $\\textsc {RCA}_0$ .", "$\\forall n \\textsc {RT}^n$ , $\\forall n \\forall X \\exists Y(Y=X^{(n)})$ .", "In (2) above the expression $\\forall n \\forall X \\exists Y(Y=X^{(n)})$ denotes a formalization of the assertion that the $n$ -th Turing jump of $X$ exists for all $n$ .", "Details on how this formalization is carried out in $\\textsc {RCA}_0$ will be presented when needed.", "Our main result — Theorem REF below — is that an analogous relation holds between $\\textsc {RT}(!\\omega )$ and closure under the $\\omega $ -jump.", "Theorem REF establishes the equivalence of $\\forall n \\textsc {RT}^n$ with the system $\\textsc {ACA}_0^{\\prime }$ consisting of $\\textsc {RCA}_0$ augmented by an axiom stating that for every $n$ and for every set $X$ the $n$ -th jump of $X$ exists for all sets $X$ .", "As a corollary of our computability-theoretic analysis we will obtain that $\\textsc {RT}(!\\omega )$ is equivalent to the system $\\textsc {ACA}_0^+$ consisting of $\\textsc {RCA}_0$ augmented by an axiom stating that for every set $X$ the $\\omega $ -jump of $X$ exists.", "The following easy Proposition relates $\\textsc {RT}(!\\omega )$ to the standard Ramsey Theorem.", "Proposition 1 $\\textsc {RT}(!\\omega )$ implies $\\forall n \\textsc {RT}^n$ over $\\textsc {RCA}_0$ .", "Let $n\\ge 1$ and $C:[\\mathbf {N}]^n\\rightarrow 2$ be given.", "We construct $C^{\\prime }:[\\mathbf {N}]^{!\\omega }\\rightarrow 2$ from $C$ as follows.", "Let $s=\\lbrace s_0,\\dots ,s_m\\rbrace $ be an exactly large set (then $m=s_0$ ).", "We set $C^{\\prime }(s)={\\left\\lbrace \\begin{array}{ll}C(s_0,\\dots ,s_{n-1}) & \\mbox{ if } s_0\\ge n,\\\\0 & \\mbox{ otherwise.}\\end{array}\\right.", "}$ Let $H$ be an infinite $C^{\\prime }$ -homogeneous set as given by $\\textsc {RT}(!\\omega )$ .", "Let $i\\in \\lbrace 0,1\\rbrace $ be the color of $[H]^{!\\omega }$ .", "Let $H^{\\prime } = H \\cap [n,\\infty )$ .", "Let $s\\in [H^{\\prime }]^n$ .", "Thus $\\min (s)\\ge n$ .", "Let $s^{\\prime }$ be any exactly large set extending $s$ in $H^{\\prime }$ .", "Then $C(s)=C^{\\prime }(s^{\\prime })=i$ .", "Thus $H^{\\prime }$ is $C$ -homogeneous of color $i$ .", "We will see below that $\\textsc {RT}(!\\omega )$ is in fact strictly stronger than $\\forall n \\textsc {RT}(n)$ ." ], [ "$\\textsc {RT}(!\\omega )$ and Second Order Arithmetic with {{formula:bbf5e5ab-79cc-4bd4-a456-12614562f36e}} -jumps", "We prove the following Theorem, characterizing the strength of $\\textsc {RT}(!\\omega )$ over Computable Mathematics.", "Theorem 5 The following are equivalent over $\\textsc {RCA}_0$ .", "$\\textsc {RT}(!\\omega )$ , $\\forall X \\exists Y(Y=X^{(\\omega )})$ .", "In (2) above, the expression $\\forall X\\exists Y (Y = X^{(\\omega )})$ is a formalization of the assertion that the $\\omega $ th Turing jump of $X$ exists.", "Details on how this formalization is carried out in $\\textsc {RCA}_0$ will be presented when needed.", "The implication from $1.$ to $2.$ follows from Theorem REF below.", "The implication from $2.", "$ to $1.", "$ follows from Theorem REF below.", "The system consisting of $\\textsc {RCA}_0$ plus the axiom $\\forall X \\exists Y (Y = X^{(\\omega )})$ is known as $\\textsc {ACA}_0^+$ .", "From the viewpoint of Computable Mathematics, the implication from $1.", "$ to $2.", "$ is essentially based on a purely computability-theoretic result showing that $\\textsc {RT}(!\\omega )$ has computable instances all of whose solutions compute $0^{(\\omega )}$ (see Theorem REF and Proposition REF below)." ], [ "Lower Bounds", "Our first result is that $\\textsc {RT}(!\\omega )$ admits a computable instance that does not admit arithmetical solutions.", "This is obtained by a Shoenfield's Limit Lemma construction based on the colorings from Jockusch's original proof of Theorem REF point (1).", "Our second main result is that $\\textsc {RT}(!\\omega )$ admits a computable instance all of whose solutions compute $0^{(\\omega )}$ .", "Recall that there exists sets that are incomparable with all $0^{(i)}$ with $i\\ge 1$ (see, e.g., [27]).", "We actually prove that $\\textsc {RT}(!\\omega )$ implies $\\forall X\\exists Y(Y=X^{(\\omega )})$ over $\\textsc {RCA}_0$ .", "Note that for the hardness result we do not use Jockusch's proof of Theorem REF point (3) (i.e., essentially, Lemma 5.9 in [11]).", "Instead we provide an explicit construction of a family of suitable colorings.", "The construction mimics some model-theoretic constructions of indicators for classes of $\\Sigma ^0_n$ formulas.", "For a very nice and short introduction into this method we refer to [18].", "In addition, we show how to adapt the proof of Proposition 4.4 in the recent [6] to get a computable instance of $\\textsc {RT}(!\\omega )$ all of whose solutions compute all levels of the arithmetical hierarchy.", "We fix the following computability-theoretic notation.", "Let $\\varphi $ be a fixed acceptable numbering [27] for a class of all recursive functions By definition, the acceptable programming systems for a class are those which contain a universal simulator and into which all other universal programming systems for the class can be compiled.", "Acceptable systems are characterized as universal systems with an algorithmic substitutivity principle called S-m-n and satisfy self-reference principles such as Recursion Theorems [27].", "We write $\\lbrace e\\rbrace ^X(x)=y$ to indicate that the $\\varphi $ -program with index $e$ and oracle $X$ outputs $y$ on input $x$ .", "We write $\\lbrace e\\rbrace ^X(x){\\downarrow }$ if there exists a $y$ such that $\\lbrace e\\rbrace ^X(x)=y$ .", "Following notation from [32] (Definition III 1.7), we write $\\lbrace e\\rbrace ^X_s(x) =y$ if $x,y,e<s$ and $s>0$ and a $\\varphi $ -program with an index $e$ and oracle $X$ outputs $y$ on input $x$ within less than $s$ steps of computation and the computation only uses numbers smaller than $s$ .", "We say that such an $s$ bounds the use of the computation.", "We occasionally write $\\varphi _{e,s}^X(x) =y$ for $\\lbrace e\\rbrace ^X_s(x) =y$ .", "For the sake of our proof-theoretic results to follow we assume to have fixed a formalization of the assertion $\\lbrace e\\rbrace ^X_s(x) =y$ .", "We write $\\lbrace e\\rbrace ^X_s(x){\\downarrow }$ (or $\\varphi _{e,s}(x){\\downarrow }$ ) if $\\exists y(\\lbrace e\\rbrace ^X_s(x) =y)$ .", "$W_{e,s}^X$ denotes the domain of $\\lbrace e\\rbrace ^X_s$ .", "A set $X$ is Turing-reducible to a set $Y$ (denoted $X\\le _T Y$ ) if and only if there exist $i,j$ such that $(\\forall x)(x\\in Y \\leftrightarrow \\exists s (\\lbrace i\\rbrace ^X_s(x){\\downarrow }))$ and $(\\forall x)(x\\notin Y \\leftrightarrow \\exists s (\\lbrace j\\rbrace ^X_s(x){\\downarrow }))$ .", "Once a suitable formalization of the assertion $\\lbrace e\\rbrace ^X_s(x) =y$ is fixed, the above definition of $X\\le _T Y$ can be formalized in Computable Mathematics ($\\textsc {RCA}_0$ ).", "We choose not to distinguish notationally between the real concept and its formalization, and we define the two at once.", "We take care of defining the relevant computability-theoretic notions (e.g., the Turing jumps) in such a way as to make it clear how they formalize in subsystems of second order arithmetic.", "We first show how to define a computable coloring of exactly large sets such that all all homogeneous sets avoid all levels of the Arithmetical Hierarchy.", "Our first step towards this goal is the following relativized version of a result of Jockusch's [11].", "Lemma 1 There exists a $X$ -computable coloring ${e^X}\\colon {[\\mathbf {N}]^2}\\rightarrow {{\\lbrace 0,1\\rbrace }}$ such that whenever $X$ is a $\\Sigma _i^0$ –complete set then $e^X$ has no homogeneous set in $\\Sigma _{i+2}^0$ .", "A straightforward relativization of Theorem 3.1. of [11].", "In our construction below we make use of Shoenfield's Limit Lemma [28].", "This result is usually stated as follows (see, e.g., [32] for a standard textbook treatment).", "If $B$ is computably enumerable in $A$ and $f\\le _T B$ then there exists a binary $A$ -computable function $h(x,s)$ such that $f(x)=\\lim _{s} h(x,s)$ , for every $x$ .", "In our application below we will have $B=A^{\\prime }$ .", "On the other hand, we will need more uniformity, as we now indicate.", "Let $g^X(i,e,s,x)$ be defined as follows.", "$g^X(i,e,s,x)={\\left\\lbrace \\begin{array}{ll}\\lbrace e\\rbrace _s^{W_{i,s}^X}(x) & \\mbox{ if } \\lbrace e\\rbrace _s^{W_{i,s}^X}(x){\\downarrow },\\\\0 & \\mbox{ otherwise.}\\end{array}\\right.", "}$ For each fixed $X$ , $g^X$ is $X$ -computable.", "Let $B$ be computably enumerable in $A$ and let $f$ be computable in $B$ .", "Let $i$ and $e$ be such that $B = W_i^A$ and $f=\\lbrace e\\rbrace ^B$ .", "Then $ f(x) = \\lim _s g^A(e,i,x,s).$ In fact, in our application, we will have $B=K_{i+1}$ and $A=K_i$ , where ${\\lbrace K_i\\rbrace }_{i\\in \\mathbf {N}}$ is a fixed sequence of sets such that $K_0=\\emptyset $ and, for each $i\\ge 1$ , $K_i$ is a $\\Sigma ^0_i$ –complete set.", "For the sake of uniformity of our construction below, we take $K_{i+1}$ to be a halting problem for machines with oracle $K_{i}$ , for $i\\ge 0$ .", "So, e.g., $K_1$ is just the halting problem for standard Turing machines.", "We fix an index $h$ such that for every $i\\ge 0$ , $K_{i+1}= W_h^{K_i}$ .", "In our application of Shoenfield's Limit Lemma to $B=K_{i+1}$ and $A=K_i$ , we can thus get rid of the argument $i$ in $g^X$ by freezing it to $h$ throughout.", "Theorem 6 There exists a computable sequence of functions $e_n^X\\colon [\\mathbf {N}]^{n+2} \\rightarrow {\\lbrace 0,1\\rbrace }$ such that for any $n\\ge 0$ , for every $i\\in \\mathbf {N}$ , $e^{K_i}_n$ is $K_i$ -computable and computes a coloring with no homogeneous set in $\\Sigma _{i+n+2}^0$ .", "We present a recursive procedure for constructing the sequence.", "For $n=0$ we take the function from Lemma REF .", "Let us assume that we have defined a sequence with the desired properties up through $e^X_n$ .", "We show how to compute the machine ${e^X_{n+1}}\\colon {[N]^{n+3}}\\rightarrow {{\\lbrace 0,1\\rbrace }}$ .", "To ensure the desired properties of $e_{n+1}^X$ it is enough that for each $i\\ge 0$ if $X=K_{i}$ , any homogeneous set for $e_{n+1}^{K_{i}}$ is a homogeneous set for $e_{n}^{K_{i+1}}$ .", "Moreover, $e^X_{n+1}$ should be obtained effectively from an index for $e_{n}^X$ .", "We use the same idea as in Proposition 2.1 of Jockusch' paper [11].", "We take $g^X(e, x_1,\\dots ,x_{n+2},s)$ such that $\\lim _{s\\rightarrow \\infty }g^{K_i}(e_n,x_1,\\dots ,x_{n+2},s)=e^{K_{i+1}}_n(x_1,\\dots ,x_{n+2}).$ As observed above, such $g^X$ is a fixed function.", "Now, we define $e^X_{n+1}$ as follows.", "$e^{X}_{n+1}(x_1,\\dots ,x_{n+2},s) := g^{X}(e_n,x_1,\\dots ,x_{n+2},s).$ Now, if $Y$ is an infinite homogeneous set for $e^{K_i}_{n+1}$ colored 0, then it is easy to see that any tuple $(x_1,\\dots ,x_{n+2})\\in [Y]^{n+2}$ has to be colored 0 by $e^{K_{i+1}}_n$ (and similarly for $Y$ colored 1).", "This concludes the proof.", "Theorem 7 There exists a computable coloring ${C}\\colon {[\\mathbf {N}]^{!\\omega }}\\rightarrow {2}$ such that any infinite homogeneous set for $C$ is not $\\Sigma ^0_i$ , for any $i\\in \\mathbf {N}$ .", "Let $S={\\lbrace s_1,\\dots , s_{\\textrm {card}(S)}\\rbrace }$ be an exactly large set.", "Then $\\textrm {card}(S)=s_1+1$ .", "We define $C(S)=e^{K_0}_{s_1-1}(s_1,\\dots ,s_{\\textrm {card}(S)}).$ Then any infinite homogeneous set $Y$ for $C$ has to be also homogeneous for $e^{K_0}_{a-1}$ , for each $a\\in Y$ .", "By Theorem REF such a set is not in $\\Sigma ^0_{a+1}$ .", "Since $Y$ is infinite, $Y$ is not $\\Sigma ^0_i$ , for any $i\\ge 0$ .", "We next show that for each set $A$ the principle $\\textsc {RT}(!\\omega )$ has computable in $A$ instances all of whose solutions compute $A^{(\\omega )}$ .", "It follows as a corollary that $\\textsc {RT}(!\\omega )$ proves over $\\textsc {RCA}_0$ that for every set $X$ the $\\omega $ -jump of $X$ exists.", "We give two proofs of this result.", "The construction in the first one mimics some indicator constructions for $\\Sigma ^0_n$ classes of formulas.", "The second proof is obtained by adapting a recent proof by Dzhafarov and Hirst [6] in combination with an old result by Enderton and Putnam [7].", "Theorem 8 For each set $A$ there exists a computable in $A$ coloring $C_\\omega :[\\mathbf {N}]^{!\\omega }\\rightarrow 2$ such that all infinite homogeneous sets for $C_\\omega $ compute $A^{(\\omega )}$ .", "We fix the following definitions of Turing jumps for the sake of the present proof.", "For a set $X$ we denote by $X^{\\prime }$ the set of indices of Turing machines which stop on input 0 with $X$ as an oracle: $X^{\\prime }={\\lbrace e: {\\lbrace e\\rbrace }^X(0){{\\downarrow }}\\rbrace }.$ We denote the $n$ -th jump of $X$ by $X^{(n)}$ .", "For formalization issues, saying that `$X^{(n)}$ exists' is conveniently read as saying that there exists a set $X \\subseteq \\lbrace 0,\\dots ,n\\rbrace \\times \\mathbf {N}$ such that for each $i<n$ , $\\lbrace a\\,:\\, (i+1,a)\\in X\\rbrace $ is the jump of $\\lbrace b\\,:\\, (i,b)\\in X\\rbrace $ .", "The $\\omega $ jump, $X^{(\\omega )}$ , of a set $X$ is the set $X^\\omega ={\\lbrace (i,j) : j \\in X^{(i)}\\rbrace }.$ For formalization issues, saying that `$X^{(\\omega )}$ exists' is conveniently read as saying that a set $Y$ exists such that, for all $n\\in \\mathbf {N}$ , the $n$ -th projection of $Y$ is equal to $X^{(n)}$ .", "Let $A$ be an arbitrary set.", "We define a family of computable in $A$ colorings $C_n:[\\mathbf {N}]^{n+1} \\rightarrow \\lbrace 0,1\\rbrace $ , for $n\\in \\mathbf {N}$ and $n\\ge 2$ , and Turing machines $M_n(x,y)$ such that for any $n\\ge 2$ , the following three points hold.", "All infinite homogeneous sets for $C_n$ have color 1.", "If $X$ is an infinite homogeneous set for $C_n$ then for any $a_1 <\\dots < a_{n+1}\\in X$ it holds that if $a$ is a code for a sequence $(a_1,\\dots , a_{n+1})$ then $M_n(x, a)$ decides $A^{(n-1)}$ for machines with indices less than or equal to $a_1$ .", "Machines $M_n$ are total.", "If their inputs are not from an infinite homogeneous set for $C_n$ then we have no guarantee on the correctness of their output.", "The second condition is a kind of uniformity condition.", "It states that no matter how we choose a sequence $a=(a_1,\\dots , a_{n-1})$ from an infinite $C_n$ –homogeneous set we can decide $A^{(n-1)}$ below $a_1$ with one, recursively constructed machine $M_n$ which is given $a$ as an oracle.", "We fix a pairing function $ \\frac{x(x+1)}{2}+y$ which is a bijection between $\\mathbf {N}^2$ and $\\mathbf {N}$ and denote it by ${\\langle x,y \\rangle }$ .", "We define $C_2$ as $C_2(k, y,z) = \\left\\lbrace \\begin{array}{ll}1 & \\text{if }\\forall e \\le k ({\\lbrace e\\rbrace }_y^{A}(0){\\downarrow }\\Leftrightarrow {\\lbrace e\\rbrace }_z^{A}(0){\\downarrow })\\\\0 & \\text{otherwise.", "}\\end{array}\\right.$ Now, if $X$ is an infinite $C_2$ –homogeneous set then it has to be colored 1.", "If $k \\in X$ then let us take a bound $b\\in X$ such that for each Turing machine $e\\le k$ ${\\lbrace e\\rbrace }^{A}(0){\\downarrow }\\Leftrightarrow {\\lbrace e\\rbrace }^{A}_b(0){\\downarrow }.$ Such a bound exists since $X$ is infinite and there are only finitely many Turing machines below $k$ .", "It follows that any $y\\in X$ greater than $b$ has the above property too.", "Therefore, the color of any tuple $\\lbrace k,y, y^{\\prime }\\rbrace \\in [X]^3$ , where $y,y^{\\prime }\\ge b$ has to be 1.", "It follows that the whole $X$ has to be colored 1.", "Let us also observe that it is easy to construct a machine $M_2(e,(k,b,b^{\\prime }) )$ that searches for a computation of $e$ below $b$ , provided that $e\\le k$ .", "Such a machine decides $A^{\\prime }$ up to $k$ if it is given $k$ and $b >k $ which belongs to some infinite $C_2$ –homogeneous set.", "Now, let us assume that we have constructed $C_n$ and $M_n$ for some $n\\ge 2$ .", "We obtain $C_{n+1}$ and $M_{n+1}$ as follows.", "We set $C_{n+1}(a_1,\\dots , a_{n+2})=$ ${\\left\\lbrace \\begin{array}{ll}1 & \\mbox{if }\\lbrace a_1,\\dots , a_{n+2}\\rbrace \\mbox{ is } C_n\\mbox{--homogeneous and}\\\\& \\forall e\\le a_1 ({\\lbrace e\\rbrace }^{Y}_{a_2}(0){\\downarrow }\\Leftrightarrow {\\lbrace e\\rbrace }^{Y}_{a_3}(0){\\downarrow }),\\text{ where}\\\\& Y={\\lbrace i \\le a_2 : M_n(i,(a_2,\\dots , a_{n+2})) \\text{ accepts,}\\rbrace }\\\\0 & \\text{otherwise}.\\end{array}\\right.", "}$ Ideally, we would like to replace the condition in the second line of the above definition by $\\forall e\\le a_1 ({\\lbrace e\\rbrace }^{A^{(n-1)}}_{a_2}(0){\\downarrow }\\Leftrightarrow {\\lbrace e\\rbrace }^{A^{(n-1)}}_{a_3}(0){\\downarrow }).$ However, such a condition would lead to a coloring which may be non-recursive in $A$ .", "Thus, instead of checking ${\\lbrace e\\rbrace }^{A^{(n-1)}}_{z}(0){\\downarrow }$ we use approximations of these sets computed by machines $M_{n}$ .", "Now, let an infinite set $X$ be $C_{n+1}$ –homogeneous and assume, towards a contradiction, that it is colored 0.", "Let us take an infinite $Z\\subseteq X$ such that $Z$ is colored 1 by $C_n$ .", "For a given $a_1\\in Z$ let $a_2$ be so large that $M_n$ can correctly decide all oracles queries for machines below $a_1$ on input 0.", "Let us take $a_3,\\dots , a_{n+2}\\in Z$ such that $\\forall e \\le a_1 ({\\lbrace e\\rbrace }^Y_{a_2}(0){\\downarrow }\\Leftrightarrow {\\lbrace e\\rbrace }^Y_{a_3}(0){\\downarrow }),$ where $Y={\\lbrace i \\le a_2 : M_n(i,(a_2,\\dots , a_{n+2})) \\text{ accepts}\\rbrace }$ .", "Again, such $a_2,\\dots , a_{n+1}$ exists since there are only finitely many machines below $a_1$ and $M_n(i, (a_1,\\dots ,a_{n+2}))$ correctly decides $A^{(n-1)}$ below $a_2$ .", "Thus, we have equivalence $\\forall e \\le a_1 ( {\\lbrace e\\rbrace }^Y_{a_2}(0){\\downarrow }\\Leftrightarrow {\\lbrace e\\rbrace }^{A^{(n-1)}}(0){\\downarrow }).$ Now, it is easy to see that the color of $C_{n+1}(a_1,\\dots , a_{n+2})=1$ and, consequently, the whole $X$ is colored 1.", "Now, let us describe a Turing machine $M_{n+1}(e, (a_1,\\dots , a_{n+2}))$ which decides $A^{(n)}$ below $a_1$ if $(a_1,\\dots , a_{n+2})$ is a sequence from an infinite $C_{n+1}$ –homogeneous set.", "We use the fact that for each $a_1 <a_2$ from an infinite $C_{n+1}$ –homogeneous set and for all $e<a_1$ we have ${\\lbrace e\\rbrace }^{A^{(n-1)}}_{a_1}(0){\\downarrow }\\Leftrightarrow {\\lbrace e\\rbrace }^{A^{(n-1)}}_{a_2}(0){\\downarrow }$ and consequently, by infinity of the given $C_{n+1}$ –homogeneous set, ${\\lbrace e\\rbrace }^{A^{(n-1)}}_{a_1}(0){\\downarrow }\\Leftrightarrow {\\lbrace e\\rbrace }^{A^{(n-1)}}(0){\\downarrow }.$ In the first part of the computation $M_{n+1}(e, (a_1,\\dots , a_{n+2}))$ computes the set $Y={\\lbrace i \\le a_2: \\text{$M_n(i, (a_2,\\dots , a_{n+1}))$ accepts}\\rbrace }.$ Then, it checks whether ${\\lbrace e\\rbrace }^Y_{a_2}{\\downarrow }$ and if this holds, $M_{n+1}$ accepts.", "Now, we may turn our attention to colorings of $!\\omega $ –large sets.", "We construct a computable coloring $C_\\omega $ and a Turing machine $M_\\omega (e,a)$ such that All infinite homogeneous sets for $C_\\omega $ are colored 1.", "If $X$ is an infinite homogeneous set for $C_\\omega $ then for any for any $a_1 <\\dots < a_{k}\\in X$ it holds that if ${\\lbrace a_1,\\dots , a_k\\rbrace }$ is an exactly $\\omega $ –large set and $a$ is a code for the sequence $(a_1,\\dots , a_{k})$ then $M_\\omega (x, a)$ decides $A^{(\\omega )}$ for pairs $(i,j)$ such that $i, j\\le a_1$ .", "Machine $M_\\omega $ stops on all inputs.", "If the inputs are not from an infinite homogeneous set for $C_\\omega $ then we have no guarantee on the correctness of the output.", "We define $C_\\omega $ as follows.", "$C_{\\omega }(a_1,\\dots , a_k) = C_{a_1}(a_1,\\dots , a_k).$ For a sequence $a=(a_1,\\dots , a_k)$ , we define $M_{\\omega }(e, a) = M_{a_1}(e, a).$ Since any infinite $C_\\omega $ –homogeneous set $X$ is also $C_n$ –homogeneous for any $n\\in \\mathbf {N}$ one can easily show that $C_\\omega $ and $M_\\omega $ have the required properties.", "Finally, we define a machine $M(x)$ which decides $A^{(\\omega )}$ with any infinite $C_\\omega $ –homogeneous set $X$ given as an oracle.", "Let us fix a recursive sequence of recursive functions $f_{i,j}$ , for $i\\le j$ , such that $f_{i,j}$ is a many–one reduction from $A^{(i)}$ to $A^{(j)}$ .", "The machine $M$ on input $(i,j)$ searches for an element $a_1 \\in X$ such that $i,j <a_1$ .", "Then, it searches for the next $a_1$ elements of $X$ , be they $a_2, \\dots , a_k$ .", "After constructing such a sequence $M$ simulates $M_{a_1}( f_{i,a_1}(j), (a_1,\\dots , a_k))$ and outputs the result of this simulation.", "Let us observe that if we want $M$ to be provably total in some theory $T$ we need $T$ to prove that for each infinite set $X$ and for each $y$ there exists an $\\omega $ –large subset of $X$ with $y$ as a minimum.", "But this is obviously true in the case of $\\omega $ –large sets.", "It is interesting to observe that a proof from the recent [6] can be easily adapted to show that $\\textsc {RT}(!\\omega )$ has a computable instance all of whose solutions compute $0^{(i)}$ for all $i\\in \\mathbf {N}$ .", "This gives, in combination with a property of least upper bounds of sequences of degrees as we will see, an alternative proof of our Reverse Mathematics corollary of Theorem REF .", "We now give the necessary details, which illustrate a strict analogy between model-theoretic-like constructions as in the proof of Theorem REF and computability-theoretic constructions.", "The proof of the following proposition is modeled after the proof of Proposition 4.4 in [6].", "Although the latter proof is for a different principle (the so-called Polarized Ramsey Theorem), the gist of it is to show directly that $\\forall n\\textsc {RT}^n$ implies $\\forall n \\forall X \\exists Y (Y=X^{(n)})$ without the need of formalizing the proof of Theorem REF point (3).", "This turns out to be surprisingly well-suited for our purposes.", "We denote by $2\\mathbf {N}$ the set of even natural numbers.", "Note that the proof can be carried out in $\\textsc {ACA}_0^{\\prime }$ , which is available under the assumption of $\\textsc {RT}(!\\omega )$ by virtue of Proposition REF .", "Proposition 2 For every set $X$ there exists a computable coloring $C^X:[\\mathbf {N}]^{!\\omega }\\rightarrow 2$ such that if $H\\subseteq 2\\mathbf {N}$ is an infinite homogeneous set for $C$ then $H$ computes $X^{(n-1)}$ for every $2n\\in H$ .", "For the sake of the present argument we define/formalize the assertion $Y=X^{\\prime }$ stating that $Y$ is the Turing jump of $X$ as follows.", "$\\forall x \\forall e (\\langle x,e\\rangle \\in Y \\leftrightarrow \\exists s (\\lbrace e\\rbrace ^X_s(x){\\downarrow })$ The definition of the $n$ th jump is then as in the proof of Theorem REF .", "Following [6] we define the following approximations of the finite jumps (where [6] use $\\Phi $ we use $W$ , $\\Phi $ being traditionally reserved for Blum Complexity Measures).", "For any set $X$ and integer $s$ define $ X^{\\prime }_s = \\lbrace \\langle m,e\\rangle \\;:\\; (\\exists t < s) m \\in W_{e,t}^X\\rbrace .$ For integers $u_1,\\dots ,u_n$ and $s$ define $X^{(0)}=X$ , and $ X^{(n+1)}_{u_n,\\dots ,u_1,s} = (X^{(n)}_{u_n,\\dots ,u_1})^{\\prime }_s.$ $C^X$ is defined as follows.", "Let $A = \\lbrace a_0,\\dots ,a_p\\rbrace $ be exactly large, i.e., $a_0 = p$ .", "If $a_0 = 2n$ for some $n$ let $C^X(A)=1$ if there exist $1\\le i \\le n$ and $\\exists (e,m) < a_{n-i}$ such that $ \\lnot ((m,e) \\in X^{(i)}_{a_n,\\dots ,a_{n-i+1}} \\leftrightarrow (m,e) \\in X^{(i)}_{a_{2n},\\dots ,a_{2n-i+1}})$ and $C^X(A)=0$ otherwise.", "If $a_0 = 2n+1$ then $C^X(A)=0$ (the value is irrelevant in this case).", "Let $H$ be an infinite homogeneous set for $C^X$ as given by $\\textsc {RT}(!\\omega )$ applied to $C^X$ and $M=[2,\\infty )\\cap 2\\mathbf {N}$ .", "We first claim that the color of $C^X$ on $[H]^{!\\omega }$ is 0.", "Suppose otherwise.", "Let $A\\in [H]^{!\\omega }$ such that $C^X(A)= 1$ .", "Then there exists $i\\le n$ such that $\\exists (e,m) < a_{n-i}$ such that $ \\lnot ((m,e) \\in X^i_{a_n,\\dots ,a_{n-i+1}} \\Leftrightarrow (m,e) \\in X^i_{a_{2n},\\dots ,a_{2n-i+1}})$ where $n$ is such that $A=\\lbrace 2n,a_1,\\dots ,a_{2n}\\rbrace $ .", "Now consider the coloring obtained by coloring $B=\\lbrace b_1,\\dots ,b_{2n}\\rbrace \\in [H\\cap (2n,\\infty )]^{2n}$ with the least $i\\le n$ such that $\\exists (e,m) < b_{n-i}$ such that $ \\lnot ((m,e) \\in X^{(i)}_{b_n,\\dots ,b_{n-i+1}} \\Leftrightarrow (m,e) \\in X^{(i)}_{b_{2n},\\dots ,b_{2n-i+1}}).$ By Ramsey Theorem $\\textsc {RT}^{2n}_n$ , this coloring admits an infinite homogeneous set $H^{\\prime }\\subseteq H\\cap (2n,\\infty )$ .", "Then we argue exactly as in [6] to obtain a contradiction.", "Now we claim that for every $h\\in H$ , $X^{(n-1)}$ is computable in $H$ , where $h = 2n$ .", "In fact we show that $X^{(n-1)}$ is definable by recursive comprehension from $H$ .", "We define a finite sequence $(X_0,\\dots ,X_{n-1})$ as follows.", "$X_0=X$ .", "For each $i\\in [1,n)$ , $(m,e)\\in X_i$ if and only if $(m,e) \\in X^{(i)}_{a_n,a_{n-1},\\dots ,a_{n-i+1}}$ where $(2n,a_1,\\dots ,a_n,a_{n+1},\\dots ,a_{2n})$ is the lexicographically least exactly large set in $H$ such that $(m,e) < a_{n-i}$ .", "We claim that for each $i < n-1$ , $X_{i+1}=X_i^{\\prime }$ .", "First we show that $X_{i+1}\\subseteq X_i^{\\prime }$ .", "Suppose $(m,e)\\in X_{i+1}$ .", "By definition of $X_{i+1}$ , $(m,e) \\in X^{(i+1)}_{a_n,a_{n-1},\\dots ,a_{n-i}}$ where $(2n,a_1,\\dots ,a_n,a_{n+1},\\dots ,a_{2n})$ is the lexicographically least exactly large set in $H$ such that $(m,e) < a_{n-i-1}$ .", "Thus $(m,e)\\in (X^{(i)}_{a_n,\\dots ,a_{n-i+1}})^{\\prime }_{a_{n-i}}$ , and so $(\\exists t < a_{n-i})(m\\in W_{e,t}^{X^{(i)}_{a_n,\\dots ,a_{n-i+1}}})$ .", "Since $a_{n-i}$ bounds the use of the computation, and by homogeneity of $H$ , it follows that $X^{(i)}_{a_n,\\dots ,a_{n-i+1}}$ and $X_i$ agree below $a_{n-i}$ .", "Therefore $(\\exists t < a_{n-i})(m\\in W_{e,t}^{X_i})$ .", "Next we show that $X_i^{\\prime }\\subseteq X_{i+1}$ .", "Suppose $(m,e)\\in X_i^{\\prime }$ .", "Then there exists $t$ such that $m\\in W_{e,t}^{X_i}$ .", "Let $(2n,a_1,\\dots ,a_n,a_{n+1},\\dots ,a_{2n})$ be the lexicographically least exactly large set in $H$ such that $(m,e) < a_{n-i-1}$ .", "Choose $b_{n-i} \\in H$ such that $b_{n-i} > \\max \\lbrace t,a_{n-i-1}\\rbrace $ .", "Choose an increasing tuple $(b_{n-i+1},\\dots ,b_n)$ in $H$ with $b_{n-i} < b_{n-i+1}$ .", "By the homogeneity of $H$ and the definition of $X_i$ , the sets $X_i$ and $X^{(i)}_{b_n,\\dots ,b_{n-i+1}}$ agree on elements below $b_{n-i}$ .", "Thus $(\\exists w < b_{n-i})(m\\in W_{e,t}^{b_n,\\dots ,b_{n-i+1}})$ , i.e., $(m,e)\\in (X^{(i)}_{b_n,\\dots ,b_{n-i+1}})^{\\prime }_{b_{n-i}}$ , and the latter set is equal to $X^{(i+1)}_{b_n,\\dots ,b_{n-i}}$ .", "By homogeneity of $H$ we then have that $(m,e) \\in X^{(i+1)}_{a_n,\\dots ,a_{n-i}}$ , hence $(m,e)\\in X_{i+1}$ .", "It is well-known that $\\lbrace 0^{(i)} \\,:\\, i\\in \\mathbf {N}\\rbrace $ has no least upper bound.", "Yet we can obtain from the previous proposition a result about $0^{(\\omega )}$ by the following result by Enderton and Putnam [7].", "The original proof needs the existence of the double jump, which is well-within our current base theory by virtue of Proposition REF .", "Lemma 2 (Enderton-Putnam, [7]) Let $I$ be an infinite set.", "Let $X$ be a set.", "Let $Y$ be a set such that for every $i\\in I$ , $X^{(i)} \\le _T Y$ .", "Then, $X^{(\\omega )}$ is many-one reducible to $Y^{(2)}$ .", "We can now derive our main proof-theoretical result of the present section.", "Theorem 9 $\\textsc {RT}(!\\omega )$ implies $\\forall X \\exists Y (Y=X^{(\\omega )})$ over $\\textsc {RCA}_0$ .", "The result can be obtained by formalization of the proof of Theorem REF .", "Alternatively, we can argue as follows.", "The proof of Proposition REF is so devised as to formalize in $\\textsc {ACA}_0^{\\prime }$ which is well within our hypotheses $\\textsc {RCA}_0+\\textsc {RT}(!\\omega )$ (see Proposition REF .", "Let $X$ be a computable set and $C^X$ be as in Proposition REF .", "Then, by that proposition, every homogeneous set for $H$ computes $0^{(i)}$ for all $i\\in \\mathbf {N}$ .", "Let $H$ be such an infinite homogeneous set for $C^X$ .", "Such an $H$ exists by $\\textsc {RT}(!\\omega )$ applied to the instance $(2\\mathbf {N},C^X)$ .", "Then by Lemma REF , $H^{(2)}$ computes $X^{(\\omega )}$ .", "So it remains to show that $\\textsc {RT}(!\\omega )$ implies that $H^{(2)}$ exists.", "But this is obvious since $\\textsc {RT}(!\\omega )$ implies $\\forall n \\textsc {RT}^n$ , by Proposition REF , and $\\forall n \\textsc {RT}^n$ implies $\\forall X \\forall n \\exists Y (Y = X^{(n)})$ , by Theorem REF ." ], [ "Upper Bounds", "We show a reversal of Theorem REF .", "Theorem 10 $\\forall X \\exists Y (Y = X^{(\\omega )})$ implies $\\textsc {RT}(!\\omega )$ over $\\textsc {RCA}_0$ .", "The idea of the proof is the following.", "We take the proof of $\\textsc {RT}(!\\omega )$ in Theorem REF as a starting point.", "We replace the sets $X_i$ by Turing machines with oracles from $C^{(a)}$ , for $a$ an element of a model of $\\textsc {RCA}_0$ .", "These Turing machines are constructed in a uniform way.", "These machines are designed so as to compute the sets $X_i$ and thus turn the induction in the proof of Theorem REF into a first-order induction.", "Moreover, since they will need as oracles the sets $C^{(a)}$ the whole construction will be recursive in $C^{(\\omega )}$ .", "The Lemma below presents the basic construction which replaces the use of sets $X_i$ by constructing Turing machines with oracles.", "We do not tailor for optimality of the oracles used, rather for clarity of the construction and we only take care that all oracles used are below $C^{(\\omega )}$ .", "We begin by recalling the definition of the Erdős-Rado tree associated to a coloring.", "Definition 1 (Erdős-Rado tree) Let $a\\ge 1$ .", "Let $C:[\\mathbf {N}]^{a+1}\\rightarrow 2$ .", "The Erdős-Rado tree $T$ of $C$ is the set of finite sequences $t$ of natural numbers defined as follows.", "If $t$ is of length $\\ell > n$ , $t(n)$ is the least $j$ such that the following two conditions hold.", "For all $m<n$ , $t(m)<j$ , and For all $m_1< \\dots < m_a < m \\le n$ , $C(t(m_1),\\dots , t(m_a),j) = C(t(m_1),\\dots ,t(m_a),t(m))$ .", "It is easy to see that $T$ is a finitely branching tree and computable in $C$ .", "We denote by $A\\oplus B$ the join of $A$ and $B$ .", "Lemma 3 Let $a\\ge 1$ .", "Let ${C}\\colon {[U]^a}\\rightarrow {2}$ .", "One can find effectively a machine $f_a$ with oracle $(C\\oplus U)^{(2a)}$ such that $f_a$ computes a $C$ –homogeneous set.", "For $a=1$ , the machine $f_1$ needs to ask the $\\Pi ^0_2(C\\oplus U)$ oracle whether $\\forall n\\exists k\\ge n (C(k)=0\\wedge U(k))$ .", "If the answer is yes, then $f_1$ computes the set $C(x)=0\\wedge U(x)$ , otherwise it computes the set $C(x)=1\\wedge U(x)$ .", "Now, let us consider the induction step for $a+1$ .", "Machine $f_{a+1}$ first constructs the Erdős–Rado tree $T_a$ for the function ${C}\\colon {[U]^{a+1}}\\rightarrow {2}$ .", "The tree $T_a$ is computable in $C\\oplus U$ .", "Then, we can obtain an index for a machine $e_p$ which computes the leftmost infinite path $P$ of $T_a$ using a $\\Pi ^0_2(C\\oplus U)$ –complete oracle.", "Indeed, a sequence $\\langle b_0,\\dots , b_k\\rangle \\in P$ if and only if $\\forall n\\ge k \\exists \\langle b_{k+1},\\dots ,b_n\\rangle \\textrm {\\ such that\\ } \\langle b_0,\\dots ,b_n\\rangle \\in T_a$ and $\\exists n \\ge k$ such that $\\forall \\langle b^{\\prime }_0,\\dots ,b^{\\prime }_k\\rangle \\le _{\\textrm {lex}}\\langle b_0,\\dots ,b_k\\rangle $ , $\\forall \\langle b^{\\prime }_{k+1},\\dots , b^{\\prime }_{n}\\rangle $ , the following holds $\\langle b^{\\prime }_0,\\dots , b^{\\prime }_n\\rangle \\notin T_a.$ The crucial property of elements from $P$ is that the color of any $(a+1)$ –tuple from $P$ does not depend on the last element of the tuple.", "Thus, if we restrict the domain of the coloring $C$ to $P$ , we can treat the coloring $C$ as a coloring of $a$ –tuples.", "Let us call this restricted coloring $C^{\\prime }$ .", "Then, we construct a machine $f_a$ (which may be obtained by inductive hypothesis) and use it with oracle $(C^{\\prime }\\oplus P)^{(2a)}$ .", "Any infinite $C^{\\prime }$ –homogeneous subset of $P$ computed by $f_a$ is also $C$ –homogeneous.", "Moreover, since $P$ is recursive in $\\Pi ^0_2(C\\oplus U)$ , the complexity of the oracle is $(C\\oplus U)^{(2(a+1))}$ as required.", "This completes the recursive construction and the proof of the Lemma.", "[Proof of Theorem REF ] Once the machine $f_a$ are constructed as in Lemma REF we can replace oracles they use by one oracle $C^{(\\omega )}$ .", "At each step of the construction machines query only a finite fragments of $C^{(\\omega )}$ but to make a construction uniform we can replace calls to different oracles by calls to $C^{(\\omega )}$ .", "Now, we can replace the $\\Sigma ^1_1$ –induction in the proof of Theorem REF by first-order induction.", "As in the proof of Theorem REF , for a coloring $C\\colon [{\\bf N}]^{!\\omega }\\rightarrow 2$ we define $C_a(x_1,\\dots , x_a)= C(a,x_1,\\dots , x_a)$ , for $a< x_1<\\dots < x_a$ .", "If $f_a$ is a function that computes $C_a$ homogeneous set, we can refer to this set as the range of $f_a$ , ${\\rm rg}(f_a)$ .", "We formulate the first order induction in the following form: for each $n$ there exists a sequence ${\\lbrace (a_i, f_{a_i}) : i\\le n\\rbrace }$ such that for each $i<n$ , $a_0=2$ , ${\\rm rg}(f_{a_{i+1}}) \\subseteq {\\rm rg}(f_{a_i}) \\subseteq {\\bf N}$ , ${\\rm rg}(f_{a_i})$ is infinite and $C_{a_i}$ –homogenous, $a_{i+1} = \\min ({\\rm rg}(f_{a_i}) \\cap {\\lbrace x \\in {\\bf N}\\colon x>a_i\\rbrace })$ .", "The reader may want to compare these conditions with the conditions used in the proof of Theorem REF (cfr.", "second column of page 2).", "Instead of sets $X_i$ we use indexes of machines $f_{a_i}$ computing $C_{a_i}$ –homogeneous sets.", "Then, using arithmetical comprehension (which is available since $\\forall X \\exists Y (Y = X^{(\\omega )})$ implies $\\textsc {ACA}_0$ and more) we may carry out the induction and prove that there exists infinite sequence ${\\lbrace (a_i,f_{a_i})\\rbrace }_{i\\in \\mathbf {N}}$ with the above properties.", "By construction, the set the set $\\lbrace a_i: i\\in \\mathbf {N}\\rbrace $ is $C$ –homogeneous.", "Let us observe than we could not carry out the above proof from the assumption $\\forall n \\textsc {RT}^n$ even though we could perform each step of the induction.", "The problem is that we would not have just one oracle $C^{(\\omega )}$ in the whole construction but we could be forced to use stronger and stronger oracles at each step.", "So, the construction could not be expressed as a single arithmetical formula." ], [ "The Regressive Ramsey Theorem for coloring exactly large sets", "In this section we formulate and analyze an analogue of $\\textsc {RT}(!\\omega )$ based on Kanamori-McAloon's principle (also known as the Regressive Ramsey Theorem) [12].", "This principle is well-studied (see, e.g., [22], [14], [15], [3]) and is one of the most natural examples of a combinatorial statement independent of Peano Arithmetic.", "The idea for studying the analogue principle for colorings of exactly large sets came from the analysis of the proof of Proposition 4.4 in [6].", "The natural way of glueing together the colorings used in that proof gives rise to a regressive function on exactly large sets.", "To state the Regressive Ramsey Theorem we need a bit of terminology.", "A coloring $C$ is called regressive if for every $S\\subseteq \\mathbf {N}$ of the appropriate type, $C(S)<\\min (S)$ whenever $\\min (S)>0$ .", "We denote by $\\mathsf {KM}^d$ the following statement: For every regressive coloring $C:[\\mathbf {N}]^d\\rightarrow \\mathbf {N}$ there exists an infinite $H\\subseteq \\mathbf {N}$ such that the color of elements of $[H]^d$ only depends on their minimum, i.e., if $s,s^{\\prime }\\in [H]^d$ are such that $\\min (s)=\\min (s^{\\prime })$ then $C(s)=C(s^{\\prime })$ .", "A set such as $H$ is called min-homogeneous for $C$ .", "We formulate a natural infinite version of the Regressive Ramsey Theorem for coloring large sets as follows.", "For every regressive coloring $C:[\\mathbf {N}]^{!\\omega }\\rightarrow \\mathbf {N}$ , for every infinite $M\\subseteq \\mathbf {N}$ , there exists an infinite $H\\subseteq M$ that is min-homogeneous for $C$ .", "We denote this statement by $\\mathsf {KM}(!\\omega )$ .", "A combinatorial proof of $\\mathsf {KM}(!\\omega )$ can be given along exactly the same lines as the proof of $\\textsc {RT}(!\\omega )$ in Theorem REF above.", "In fact, as we now prove, $\\mathsf {KM}(!\\omega )$ is equivalent to $\\textsc {RT}(!\\omega )$ over $\\textsc {RCA}_0$ .", "Proposition 3 Over $\\textsc {RCA}_0$ , $\\mathsf {KM}(!\\omega )$ and $\\textsc {RT}(!\\omega )$ are equivalent.", "We first prove that $\\mathsf {KM}(!\\omega )$ implies $\\textsc {RT}(!\\omega )$ .", "This is almost trivial.", "Let $C:[\\mathbf {N}]^{!\\omega }\\rightarrow 2$ be given.", "Then $C$ is regressive on $[\\mathbf {N}\\setminus \\lbrace 0,1\\rbrace ]^{!\\omega }$ .", "Let $H$ be an infinite min-homogeneous set for $C$ .", "Define $C^{\\prime }:[H]\\rightarrow 2$ as follows.", "$C^{\\prime }(h)=i$ if all exactly large sets in $H$ with minimum $h$ have color $i$ .", "By the Infinite Pigeonhole Principle, let $H^{\\prime }\\subseteq H$ be an infinite $C^{\\prime }$ -homogeneous set.", "Then $H^{\\prime }$ is $C$ -homogeneous.", "Now, we prove that $\\textsc {RT}(!\\omega )$ implies $\\mathsf {KM}(!\\omega )$ .", "Let $C:[\\mathbf {N}]^{!\\omega }\\rightarrow \\mathbf {N}$ be a regressive coloring.", "We define $C^{\\prime }:[\\mathbf {N}]^{!\\omega }\\rightarrow \\lbrace 0,1\\rbrace $ in such a way that if $X$ is an infinite $C^{\\prime }$ –homogenous set then $Y={\\lbrace x-1 \\colon x\\in X\\rbrace }$ is $\\min $ –homogenous for $C$ .", "For a tuple $A=(a_0, \\dots , a_k)\\in [{\\bf N}]^{k+1}$ , where $a_0\\ge 1$ and $k=a_0$ , we define $C^{\\prime }(A)$ as 1 if all tuples $(a_0 -1, c_1, \\dots , c_{k-1}) \\in [{\\lbrace a_i - 1 \\colon 0\\le i \\le k\\rbrace }]^{a_0)}$ get the same color under $C$ .", "Otherwise, we define $C^{\\prime }(A)$ as 0.", "(We define $C^{\\prime }((0))$ arbitrarily.)", "It is easy to prove by $\\textsc {RCA}_0$ induction that for each infinite $C^{\\prime }$ –homogenous set $X$ has the stated above property.", "It is instructive to observe how the proof of Proposition REF goes through almost unchanged.", "The details diverge from the proof of Proposition 4.4. in [6] in a different point.", "Proposition 4 For every set $X$ there exists a computable regressive coloring $C^X:[\\mathbf {N}]^{!\\omega }\\rightarrow 2$ such that if $H\\subseteq 2\\mathbf {N}$ is an infinite min-homogeneous set for $C$ then $H$ computes $X^{(n-1)}$ for every $2n\\in H$ .", "Let $X$ be a set.", "Define $C^X:[\\mathbf {N}]^{!\\omega }\\rightarrow \\mathbf {N}$ as follows.", "Let $A = \\lbrace a_0,\\dots ,a_p\\rbrace $ be exactly large, i.e., $a_0 = p$ .", "If $a_0 = 2n$ for some $n$ then $A=\\lbrace 2n,a_1,\\dots ,a_n,a_{n+1},\\dots ,a_{2n}\\rbrace $ .", "Let $C^X(A)$ be the least $i\\in [1,n]$ such that there exists $(m,e) < a_{n-i}$ such that $ \\lnot ((m,e) \\in X^i_{a_n,\\dots ,a_{n-i+1}} \\Leftrightarrow (m,e) \\in X^i_{a_{2n},\\dots ,a_{2n-i+1}})$ if such an $i$ exists, and $C^X(A)=0$ otherwise.", "If $a_0 = 2n+1$ then $C^X(A)=0$ (the value is irrelevant in this case).", "Note that $C^X$ is a regressive coloring.", "Let $H$ be an infinite min-homogeneous set for $C^X$ as given by $\\mathsf {KM}(!\\omega )$ applied to $C^X$ and $M=[2,\\infty )\\cap 2\\mathbf {N}$ .", "We first claim that the color of $C^X$ restricted to $[H]^{!\\omega }$ is 0 and $H$ is indeed homogeneous.", "Suppose otherwise by way of contradiction.", "Let $i>0$ be such that for some $n$ , $A=\\lbrace 2n,a_1,\\dots ,a_n,a_{n+1},\\dots ,a_{2n}\\rbrace \\in [H]^{!\\omega }$ and $C^X(A)=i$ .", "Note that $i \\le n$ and that the color is $i$ for every exactly large set in $H$ with minimum $2n$ .", "Let $H=\\lbrace 2n_j\\rbrace _{j\\in J}$ for some $J$ .", "Let $n=n_{j}$ .", "Let $h=n_{j+n-i}$ .", "We claim that there exists $(m_0,e_0) < 2h$ such that for all $B\\in [H\\cap (2n,\\infty )]^{2n}$ $ \\lnot ((m_0,e_0)\\in X^i_{b_n,\\dots ,b_{n-i+1}} \\Leftrightarrow (m_0,e_0)\\in X^i_{b_{2n},\\dots ,b_{2n-i+1}}) $ where $B=\\lbrace b_1,\\dots ,b_n,b_{n+1},\\dots ,b_{2n}\\rbrace $ .", "We get the existence of $(m_0,e_0)$ by coloring $[H\\cap (2n,\\infty )]^{2n}$ according to the least $(m,e) < 2h$ witnessing the color is $i$ (i.e., by an application of a finite Ramsey Theorem of suitable dimension).", "Fix such a $B$ .", "By minimality of $i$ it must be the case that $X^i_{b_n,\\dots ,b_{n-i+2}}$ agrees with $X^i_{b_{2n},\\dots ,b_{2n-i+2}}$ on values below $b_{n-i+1}$ .", "Therefore $(m_0,e_0)\\in X^i_{b_n,\\dots ,b_{n-i+1}} \\rightarrow (m_0,e_0)\\in X^i_{b_{2n},\\dots ,b_{2n-i+1}},$ since $b_{n-i+1} < b_{2n-i+1}$ .", "Then by choice of $(m_0,e_0)$ the converse implication must fail.", "Therefore $(m_0,e_0)\\in X^{i}_{b_{2n},\\dots ,b_{2n-i+1}}$ holds unconditionally.", "Thus, $(\\exists t < b_{2n-i+1})(m_0\\in W_{e_0,t}^{X^{i-1}_{b_{2n},\\dots ,b_{2n-i+2}}}).$ Choose $(b_1^*,\\dots ,b_n^*,b_{n+1}^*,\\dots ,b_{2n}^*)$ in $[H\\cap (2n,\\infty )]^{2n}$ with $b_{n-i+1}^* > b_{2n-i+1}$ and $b_{2n-i+2}^* \\ge b_{2n-i+2}$ .", "By the same argument as above applied to the sequence $(b_1,\\dots ,b_{2n-i+1},b_{2n-i+2}^*,\\dots ,b_{2n}^*)$ we have that $(\\exists t < b_{2n-i+1})(m_0\\in W_{e_0,t}^{X^{i-1}_{b_{2n}^*,\\dots ,b_{2n-i+2}^*}}).$ But by minimality of $i$ we have that $X^{i-1}_{b_n^*,\\dots ,b_{n-i+2}^*}$ and $X^{i-1}_{b_{2n}^*,\\dots ,b_{2n-i+2}^*}$ must agree on all elements below $b_{n-i+1}^*$ and therefore also on all elements below $b_{2n-i+1}$ .", "But $b_{2n-i+1}$ bounds the use of the computation showing $m_0\\in W_{e_0}^{X^{i-1}}$ and since $b_{2n-i+1} < b_{n-i+1}^*$ we have that $ (\\exists t < b_{n+i-1}^*)(m_0\\in W_{e_0,t}^{X^{i-1}_{b_{n}^*,\\dots ,b_{n-i+2}^*}}),$ and on the other hand, since $b_{2n-i+2}^* \\ge b_{2n-i+2}$ , we have that $ (\\exists t < b_{2n+i-1}^*)(m_0\\in W_{e_0,t}^{X^{i-1}_{b_{2n}^*,\\dots ,b_{2n-i+2}^*}}).$ But these two facts contradict the choice of $(m_0,e_0)$ .", "We then claim that for every $h\\in H$ , $X^n$ is computable in $H$ , where $h = 2n$ .", "Since $H$ is homogeneous of color 0, the argument goes through unchanged as in the proof of Proposition REF .", "We next observe without proof that an analogue of Proposition REF holds for $\\mathsf {KM}(!\\omega )$ .", "The proof is similar to that of Theorem REF .", "Theorem 11 $\\forall X \\exists Y (Y = X^{(\\omega )})$ implies $\\mathsf {KM}(!\\omega )$ over $\\textsc {RCA}_0$ ." ], [ "Peano Arithmetic with $\\omega $ inductive truth predicates", "In this section we compare the strength of $\\textsc {RT}(!\\omega )$ with Peano Arithmetic augmented by a hierarchy of truth predicates.", "We establish a close correspondence between these theories.", "Let $\\alpha $ be an ordinal and let $\\textsc {PA}({\\lbrace \\mathsf {Tr} {}_\\beta : \\beta <\\alpha \\rbrace })$ be Peano arithmetic extended by axioms which express, for each $\\beta <\\alpha $ , that $\\mathsf {Tr} {}_\\beta (x)$ is a truth predicate for the language with predicates $\\mathsf {Tr} {}_\\gamma $ , for $\\gamma <\\beta $ and with full induction in the extended language.", "The axioms for being a truth predicate for a language $L_\\beta $ are the usual Tarski condition for compositional definitions of the truth values for connectives and quantifiers.", "They may be presented as follows.", "Let $L_n$ be a language with truth predicates $\\mathsf {Tr} {}_{0}, \\dots , \\mathsf {Tr} {}_{n-1}$ .", "Then, for each $n\\in {\\bf N}$ we put in $\\textsc {PA}({\\lbrace \\mathsf {Tr} {}_\\beta : \\beta <\\omega \\rbrace })$ $\\forall (t=t^{\\prime })\\in L_n (\\mathsf {Tr} {}_n(t=t^{\\prime }) \\equiv {\\rm val}(t) = {\\rm val}(t^{\\prime }))$ , $\\forall (t\\le t^{\\prime })\\in L_n (\\mathsf {Tr} {}_n(t\\le t^{\\prime }) \\equiv {\\rm val}(t) \\le {\\rm val}(t^{\\prime }))$ , for all $i<n$ we have $\\forall x (\\mathsf {Tr} {}_n(\\mathsf {Tr} {}_i(\\bar{x})) \\equiv \\mathsf {Tr} {}_i(x))$ , $\\forall \\varphi \\in L_n (\\mathsf {Tr} {}_n(\\lnot \\varphi ) \\equiv \\lnot \\mathsf {Tr} {}_n(\\varphi ))$ , $\\forall \\varphi ,\\psi \\in L_n (\\mathsf {Tr} {}_n(\\varphi \\Rightarrow \\psi ) \\equiv (\\mathsf {Tr} {}_n(\\varphi )\\Rightarrow \\mathsf {Tr} {}_n(\\psi )))$ , $\\forall \\varphi (\\mathsf {Tr} {}_n(\\exists x \\varphi (x)) \\equiv \\exists x\\,\\mathsf {Tr} {}_n(\\varphi (\\bar{x})))$ , where $\\bar{x}$ is the $x$ -th numeral which is a name for an element $x$ in the model, $t$ and $t^{\\prime }$ are closed terms and ${\\rm val}$ is an arithmetical function which computes a value of a closed term.", "The laws for other propositional connectives and for the universal quantifier may be easily proved from the above axioms.", "For more on theories with truth predicates, also called satisfaction classes we refer to [16] and [17].", "Theorem 12 The following theories are equivalent over the language of Peano arithmetic: $\\textsc {RCA}_0 + \\textsc {RT}(!\\omega )$ , $\\textsc {PA}(\\lbrace \\mathsf {Tr} {}_i: i\\in \\mathbf {N}\\rbrace )$ .", "For the direction from $1.$ to $2.$ , we use the fact that the truth for arithmetical formulas with second order parameters say $P_0, \\dots , P_n$ is many–one reducible to the the $\\omega $ -jump of $P_1\\oplus \\cdots \\oplus P_n = {\\lbrace (i,j)\\in {\\bf N}^2 \\colon j\\in P_i\\rbrace }$ .", "Now, if $d$ is a proof in $\\textsc {PA}(\\lbrace \\mathsf {Tr} {}_i: i\\in \\mathbf {N}\\rbrace )$ then it uses only finitely many truth predicates, say $T_1,\\dots , T_n$ .", "We can define them using $0^{(\\omega )},\\dots , 0^{(\\omega n)}$ and carry out the proof in $\\textsc {RCA}_0 + \\textsc {RT}(!\\omega )$ proving the axioms for truth theory of these $\\mathsf {Tr} {}_1,\\dots , \\mathsf {Tr} {}_n$ .", "For the other direction, if $M\\models \\textsc {PA}(\\lbrace \\mathsf {Tr} {}_i: i\\in \\mathbf {N}\\rbrace )$ then we can extend $M$ to a model of $\\textsc {RCA}_0 + \\textsc {RT}(!\\omega )$ without changing its first-order part.", "We simply construct a sequence of models $M_i$ , for $i\\in \\mathbf {N}$ as follows.", "As $M_0$ we take just $M$ and for $M_{i+1}$ we take all sets which are $\\Delta ^0_1$ -definable from the language with truth predicates $\\mathsf {Tr} {}_0,\\dots , \\mathsf {Tr} {}_i$ .", "The sum of all $M_i$ is obviously closed on $\\omega $ jumps since it is closed on taking arithmetical truth for each sequence of second order parameters $P_0,\\dots , P_n$ .", "It follows that such obtained model satisfies $\\textsc {RCA}_0 + \\textsc {RT}(!\\omega )$ and since the first-order part of both models is the same we get conservativity in the language of $\\textsc {PA}$ .", "In [17] the authors characterize the arithmetical strength of Peano arithmetic with one predicate axiomatized as a truth predicate and with induction for the full language.", "Let $\\alpha $ be an ordinal.", "We define $\\omega _0(\\alpha )=\\alpha $ and let $\\omega _{k+1}(\\alpha )=\\omega ^{\\omega _k(\\alpha )}$ .", "Now, for an ordinal $\\alpha $ let $\\varepsilon _\\alpha $ be the $\\alpha $ th ordinal $\\beta $ with the property $\\omega ^\\beta =\\beta $ .", "Thus, the first such ordinal, $\\varepsilon _0$ , is the limit of $\\omega _k(0)$ and $\\varepsilon _{\\alpha +1}$ is the limit of $\\omega _k(\\varepsilon _\\alpha +1)$ , where $k\\in \\mathbf {N}$ .", "For limit $\\lambda $ , one may prove that $\\varepsilon _{\\lambda }$ is the limit of $\\varepsilon _{\\lambda _k}$ , where $\\lambda _k$ is a sequence of ordinals converging to $\\lambda $ .", "Of course, in order to define such ordinals in arithmetic one needs to define also a coding system which would represent such ordinals as natural numbers.", "After representing the ordering up to $\\alpha $ in arithmetic, one can define the principle of transfinite induction up to $\\alpha $ , $\\mathbf {TI}(\\alpha )$ .", "In [17] the following theorem is proved.", "Theorem 13 ([17]) The arithmetical consequences of $\\textsc {PA}(\\mathsf {Tr} {}_0)$ are exactly the consequences of the theory $\\textsc {PA}+ \\lbrace \\mathbf {TI}(\\varepsilon _{\\omega _k(0)})\\colon k\\in \\mathbf {N}\\rbrace $ .", "Our results allows us to characterize the arithmetical strength of Peano arithmetic with $\\omega $ many truth predicates.", "Let us define a sequence $\\alpha _0=\\varepsilon _0$ and $\\alpha _{k+1}=\\varepsilon _{\\alpha _k}$ , for $k\\in \\mathbf {N}$ .", "The limit of this sequence is usually denoted by $\\varphi _2(0)$ in the Veblen notation system for ordinals.", "The proof theoretic ordinal of the theory $\\textsc {ACA}^+_0$ is $\\varphi _2(0)$ (see [26] for a proof).", "The arithmetical equivalence of this theory with $\\textsc {PA}(\\lbrace \\mathsf {Tr} {}_i\\colon i\\in \\mathbf {N}\\rbrace )$ allows us to characterize the latter theory by transfinite induction.", "Theorem 14 The arithmetical consequences of $\\textsc {PA}(\\lbrace \\mathsf {Tr} {}_i\\colon i\\in \\mathbf {N}\\rbrace )$ are exactly the consequences of the theory $\\textsc {PA}+ \\lbrace \\mathbf {TI}(\\alpha )\\colon \\alpha < \\varphi _2(0) \\rbrace $ ." ], [ "Conclusion and Future Research", "We have characterized the effective and the proof-theoretical content of a natural combinatorial Ramsey-type theorem due to Pudlàk and Rödl [25] and, independently, to Farmaki [9].", "We have proved that the theorem has computable instances all of whose solutions compute $0^{(\\omega )}$ , the Turing degree of arithmetic truth.", "Moreover, we have shown that the theorem exactly captures closure under $\\omega $ -jump over Computable Mathematics.", "The theorem is interestingly related to Banach space theory because of its equivalent formulation in terms of Schreier families.", "We now indicate two natural directions for future work on the subject.", "First, we conjecture that our results generalize to the transfinite generalizations above $\\omega $ of the notions of large set, Schreier family, and Turing jump.", "The notions of $\\alpha $ -large set, $\\alpha $ -Schreier family, and $\\alpha $ -Turing jump are all well-defined and studied for every countable ordinal (see, respectively, [13], [9], and [27] for definitions).", "As mentioned in the introduction, $\\textsc {RT}(!\\omega )$ generalizes nicely to colorings of $\\alpha $ -Schreier families, or, equivalently, of exactly $\\alpha $ -large sets.", "We conjecture that a modification of our arguments will show that, for each fixed $\\alpha $ , the principle $\\textsc {RT}(!\\alpha )$ generalizing $\\textsc {RT}(!\\omega )$ to colorings of $!\\alpha $ -large sets is equivalent — over Computable Mathematics — to the closure under the $\\alpha $ -th Turing jump.", "Thus, the full theorem $\\forall \\alpha \\textsc {RT}(!\\alpha )$ would be equivalent to the system $\\textsc {ATR}_0$ (Arithmetical Transfinite Recursion, see [31]).", "Provability in $\\textsc {ATR}_0$ can be easily proved by inspection of the proof by Pudlàk and Rödl [25] (using Nash-Williams Theorem) or else by using the $\\Sigma ^0_1$ -Ramsey Theorem.", "A second direction for future work is the following.", "Since $\\textsc {RT}(!\\omega )$ is at least as strong as Ramsey's Theorem it is obviously possible to obtain finite independence results for Peano Arithmetic by imposing a suitable largeness condition (see [5] for a concrete example).", "A corollary of our results is that $\\textsc {RT}(!\\omega )$ implies over $\\textsc {RCA}_0$ the well-ordering of the proof-theoretic ordinal of the system $\\textsc {ACA}_0^+$ , i.e., $\\varphi _2(0)$ in Veblen notation.", "Using (as of now standard) techniques of miniaturization it is then possible to extract from $\\textsc {RT}(!\\omega )$ finite first-order independence results in the spirit of the Paris-Harrington principle [10] but for the much stronger principle $\\textsc {ACA}_0^+$ .", "The hope that finite independence results for systems stronger than Peano Arithmetic could be extracted from $(\\forall \\alpha )\\textsc {RT}(!\\alpha )$ is expressed in [9].", "Our results for $\\textsc {RT}(!\\omega )$ confirm this expectation already for $\\alpha =\\omega $ .", "Details will be reported elsewhere." ] ]

[ [ "Efficient quantum communication under collective noise" ], [ "Abstract We introduce a new quantum communication protocol for the transmission of quantum information under collective noise.", "Our protocol utilizes a decoherence-free subspace in such a way that an optimal asymptotic transmission rate is achieved, while at the same time encoding and decoding operations can be efficiently implemented.", "The encoding and decoding circuit requires a number of elementary gates that scale linearly with the number of transmitted qudits, m. The logical depth of our encoding and decoding operations is constant and depends only on the channel in question.", "For channels described by an arbitrary discrete group G, i.e.", "with a discrete number, |G|, of possible noise operators, perfect transmission at a rate m/(m+r) is achieved with an overhead that scales at most as $\\mathcal{O}(d^r)$ where the number of auxiliary qudits, r, depends solely on the group in question.", "Moreover, this overhead is independent of the number of transmitted qudits, m. For certain groups, e.g.", "cyclic groups, we find that the overhead scales only linearly with the number of group elements |G|." ], [ "Introduction", "The transmission of quantum information between several communication partners is a central element of many applications of quantum information theory, including quantum cryptography and quantum key distribution [1], quantum networks [2], [3], and distributed quantum computation [4].", "In a realistic set-up quantum communication is subject to noise and imperfections leading to imperfect communication channels.", "Several methods are known to deal with such a situation, including teleportation-based communication utilizing entanglement purification [5], or encoding of quantum information using quantum error-correcting codes [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16].", "In cases where only restricted types of errors occur, i.e.", "when parties lack a shared phase or spatial reference frame (see [17] and references therein), or when quantum communication is subject to collective noise, error-avoiding schemes are also available [18], [19], [20], [21], [22], [23], [24], [25].", "In such schemes information is stored and transmitted in a decoherence-free subspace (DFS).", "A similar problem occurs when considering the storage of quantum information where the source of collective noise is, for example, a globally fluctuating magnetic field.", "In this paper we propose a novel quantum communication and storage scheme capable of dealing with collective noise associated to an arbitrary finite group $G$ .", "The scheme is efficient with respect to the required encoding and decoding operations, and is capable of achieving an optimal transmission or storage rate in the asymptotic limit.", "Our scheme is mainly based on the usage of a DFS, however elements from standard error-correcting codes are also utilized.", "Whereas error-avoiding schemes encode quantum information directly into a DFS, error-correcting schemes usually involve measurements of ancilla systems whose outcome allows one to determine the kind of error that occurred and deduce the required correction operations.", "We use ancilla systems for the storage or transmission of $m$ qudits in such a way that we construct a joint state of system plus ancilla that lies within a DFS.", "Similar to standard error correction schemes we measure the ancilla system and use the measurement outcome to determine the required correction operation.", "However, in our scheme the measurement does not reveal any information about the channel or the kind of error that occurred.", "In addition, the correction operation is local, i.e.", "consists of single-system operations only, making our protocol suitable in multiparty communication scenarios with multiple receivers.", "Our protocol is an extension of the measure and re-align protocol proposed in [26] for communicating quantum information between parties lacking a shared frame of reference.", "Such a protocol was shown to be optimal in [27].", "Contrary to the protocol in [26], where Bob measures an auxiliary system to learn the relative transformation between his and Alice's frame of reference, our protocol hides Alice's direction from Bob while still allowing perfect communication of quantum information.", "We show that for channels described by an arbitrary discrete group $G$ , i.e.", "with a discrete number, $\\vert G\\vert $ , of possible noise operators, the number of ancillary systems, $r$ , is independent of the number of qudits, $m$ , to be transmitted.", "Our protocol achieves a transmission rate of $m/(m+r)$ , i.e.", "$m+r$ physical qudits need to be send through a noisy channel to faithfully transmit $m$ logical qudits, that is asymptotically optimal approaching unity in the large $m$ -limit.", "Furthermore, we provide an explicit encoding and decoding circuit that is efficient.", "In particular, the required number of single and two-qubit gates (henceforth referred to as elementary gates) for any finite group $G$ scales as $\\mathcal {O}(m,|G|\\log (|G|), d^r)$ .", "The additional overhead of $\\mathcal {O}(d^r)$ gates is required in order to prepare the joint state of system plus ancilla in a DFS, where $r$ is some (finite) integer that depends solely on the channel in question Note that $r$ itself depends on the number of group elements and the specific group in question, which may lead to a worse than linear scaling in $|G|$ ..", "In addition, the logical depth of the circuit, the amount of time required by the circuit to generate the desired state, is $\\mathcal {O}(|G|\\log (|G|),d^r)$ , independent of $m$ .", "For the case where $G$ is a cyclic group this overhead can be shown to scale only linearly ($\\mathcal {O}(|G|)$ ) with the number of group elements.", "Since the number of elementary gates required to implement our protocol scales linearly with the number, $m$ , of logical qudits our protocol is more efficient than the best error-avoiding communication scheme [28].", "The latter scales as $N\\mathrm {poly}(\\log (N),d)$ , where $N$ is the total number of $d$ -dimensional physical systems.", "Recently, an alternative implementation for encoding and decoding in a DFS was proposed that also scales linearly in the number of logical qubits being transmitted, but achieves an asymptotic rate of $1/2$  [29] Private communication with Mikio Nakahara.. Due to the asymptotically optimal rate of transmission and linear scaling of our protocol a practical implementation seems feasible.", "This paper is organized as follows.", "In Sec.", "we review the concept of DFS and some basic results within group representation theory.", "We also introduce our notation in this section and review previous encoding schemes utilizing a DFS [28].", "In Sec.", "we present a novel method to encode information in a DFS for collective noise described by an arbitrary finite group, $G$ , where we make use of a finite number, $r$ , of auxiliary systems.", "In Sec.", "we explicitly construct encoding and decoding circuit implementations for our protocol and discuss the required resources, i.e.", "the number of elementary gates, and the logical depth of these circuits.", "A direct comparison with other methods [28], [29] shows that our scheme is more efficient.", "We also consider the transmission rate of our protocol and show its asymptotic optimality.", "We summarize and conclude in Sec.", "." ], [ "Decoherence-Free Subspace", "In this section we review some basic results of group representation theory, in particular the concept of a DFS.", "Moreover, we outline the basic principles underpinning the best known quantum communication protocols which utilize DFS [28].", "The problem we are considering is the transmission of quantum information through a quantum channel with collective noise.", "One party, the sender, prepares $N$ , $d$ -dimensional quantum systems in some state, $\\rho \\in {\\mathcal {B}}({\\mathcal {H}}_d^{\\otimes N})$ , and sends them through a noisy quantum channel to one or more parties, the receivers.", "The noise of the quantum channel is described by a set of $M$ operators, $\\lbrace U_{g_i},\\, i\\in (0,\\ldots ,M-1)\\rbrace $ , and a probability distribution, $\\lbrace p_{g_i}\\rbrace $ with $p_{g_i}>0, \\forall i\\in (0,\\ldots ,M-1)$ where $\\sum _{i=0}^{M-1} p_{g_i}=1$ .", "The noise of the channel is assumed to be collective.", "That is, the same noise operator acts on each of the transmitted $d$ -dimensional systems.", "After transmission through the channel the quantum state of the $N$ systems is given by $\\nonumber {\\mathcal {E}}[\\rho ]&=&\\sum _{i=0}^{M-1} p_{g_i} \\left(U^{(1)}_{g_i}\\otimes U^{(2)}_{g_i}\\otimes \\cdots \\otimes U^{(N)}_{g_i}\\right)[\\rho ]\\\\&&\\times \\left(U^{(1)}_{g_i}\\otimes U^{(2)}_{g_i}\\otimes \\cdots \\otimes U^{(N)}_{g_i}\\right)^{\\dagger },$ where $U^{(k)}_{g_i}$ denotes the operator $U_{g_i}$ acting on the $k^{\\mathrm {th}}$ quantum system.", "Hence, the receiver(s) obtain corrupt quantum data.", "Our task is to construct a communication protocol that allows for efficient, error-free communication of quantum information through quantum channels subject to collective noise.", "The fact that the noise of the channel is collective allows for the construction of error-avoiding protocols [18], [19], [20], [21], [22], [23], [24], [25], i.e.", "protocols that can protect quantum information from being corrupted, as we now review.", "We can assume, without loss of generality, that the set of operations, $\\lbrace U_{g_i},\\, i\\in (0,\\ldots ,M-1)\\rbrace $ , forms a unitary representation of a symmetry group, $G$ , acting on the $d$ -dimensional Hilbert space, ${\\mathcal {H}}_d$  If the set $\\lbrace U_{g_i},\\,i\\in (0,\\ldots , M-1)\\rbrace $ does not form a group we can always supplement it with additional operations whose probability of occurrence is zero.. We focus only on unitary representations as it is known that every representation of a finite group is equivalent to a unitary representation [30].", "Two representations, $U$ and $T$ , of a group $G$ are equivalent if there exists an invertible matrix, $V$ , such that for all $g_i\\in G$ , $VU_{g_i}V^\\dagger =T_{g_i}$ .", "Due to Schur's lemmata [30], the collective representation, $U_{g_i}^{\\otimes N}\\equiv U^{(1)}_{g_i}\\otimes \\cdots \\otimes U^{(N)}_{g_i},\\, g_i\\in G$ , can be decomposed into irreducible representations (irreps) of $G$ , $U^{(\\lambda )}$ , as follows $U_{g_i}^{\\otimes N}=\\bigoplus _\\lambda \\alpha ^{(\\lambda )}\\, U_{g_i}^{(\\lambda )}, \\, \\forall g_i\\in G.$ Here, $\\lambda $ labels the inequivalent irreps of $G$ , and $\\alpha ^{(\\lambda )}$ denotes the multiplicity of irrep $U^{(\\lambda )}$ .", "Let us denote by $\\lbrace \\left| \\lambda ,m,\\beta \\right>\\rbrace $ the orthonormal basis in which all matrices $U_{g_i}^{\\otimes N},\\, i\\in (0,\\ldots ,M-1)$ , are block diagonal.", "As before, $\\lambda $ labels the inequivalent irreps of $G$ , and $m$ labels an orthonormal basis of the space, ${\\mathcal {M}}^{(\\lambda )}$ , on which $U_{g_i}^{(\\lambda )}$ acts upon.", "Note that the dimension of ${\\mathcal {M}}^{(\\lambda )}$ coincides with the dimension of the irrep $U^{(\\lambda )}$ .", "Moreover, $\\beta $ labels an orthonormal basis of ${\\mathcal {N}}^{(\\lambda )}\\equiv \\hbox{$I$\\hspace{-6.99997pt}$C$}^{\\alpha ^{(\\lambda )}}$ , the space of dimension $\\alpha ^{(\\lambda )}$ , on which $\\lbrace U_{g_i}^{\\otimes N},\\, g_i\\in G\\rbrace $ acts trivially for any $g_i\\in G$ .", "The total Hilbert space, ${\\mathcal {H}}_d^{\\otimes N}$ , can be conveniently written with respect to the block diagonal basis, $\\lbrace \\left| \\lambda ,m,\\beta \\right>\\rbrace $ , as ${\\mathcal {H}}_d^{\\otimes N}=\\bigoplus _\\lambda {\\mathcal {H}}^{(\\lambda )}=\\bigoplus _\\lambda {\\mathcal {M}}^{(\\lambda )}\\otimes {\\mathcal {N}}^{(\\lambda )}.$ Writing an arbitrary state, $\\left| \\psi \\right>\\in {\\mathcal {H}}_d^{\\otimes N}$ , in the basis $\\lbrace \\left| \\lambda ,m,\\beta \\right>\\rbrace $ , i.e.", "$\\left| \\psi \\right>=\\sum _{\\lambda ,m,\\beta } v_{\\lambda ,m,\\beta } \\left| \\lambda ,m,\\beta \\right>,$ where $v_{\\lambda ,m,\\beta }\\in \\hbox{$I$\\hspace{-6.99997pt}$C$}$ satisfy $\\sum _{\\lambda ,m,\\beta }\\vert v_{\\lambda ,m,\\beta }\\vert ^2=1$ , and acting on this state with the operator $U_{g_i}^{\\otimes N}$ yields $U_{g_i}^{\\otimes N}\\left| \\psi \\right>=\\sum _{\\lambda ,m,m^{\\prime },\\beta } v_{\\lambda ,m,\\beta }\\, u^{(\\lambda )}_{m^{\\prime },m}(g_i)\\left| \\lambda ,m^{\\prime },\\beta \\right>,$ where $u^{(\\lambda )}_{m^{\\prime },m}(g_i)\\equiv \\left< \\lambda , m^{\\prime }\\right|U_{g_i}^{\\otimes N}\\left| \\lambda , m\\right>$ .", "Therefore, the action of the collective noise operations, $\\lbrace U_{g_i}^{\\otimes N},\\, g_i\\in G\\rbrace $ , affects the index associated to the carrier spaces, ${\\mathcal {M}}^{(\\lambda )}$ , but not the index associated to the multiplicity spaces, ${\\mathcal {N}}^{(\\lambda )}$ .", "Using the results reviewed above it can be shown that, if the probability distribution, $\\lbrace p_{g_i}\\rbrace $ , is the uniform prior Eq.", "(REF ) reduces to ${\\mathcal {E}}[\\rho ]=\\sum _\\lambda \\left({\\mathcal {D}}_{{\\mathcal {M}}^{(\\lambda )}}\\otimes {\\mathcal {I}}_{{\\mathcal {N}}^{(\\lambda )}}\\right)\\circ {\\mathcal {P}}^{(\\lambda )}[\\rho ],$ where ${\\mathcal {D}}$ is the completely depolarizing map, ${\\mathcal {D}}(A)=\\frac{\\mbox{tr}(A)}{\\mathrm {dim}({\\mathcal {H}})} \\mbox{$1 \\hspace{-2.84526pt} {\\bf l}$},\\forall A\\in {\\mathcal {B}}({\\mathcal {H}})$ , ${\\mathcal {I}}$ is the identity map, and ${\\mathcal {P}}^{(\\lambda )}(A)=\\Pi _\\lambda A\\Pi _\\lambda $ , where $\\Pi _\\lambda $ is the projector onto the space ${\\mathcal {H}}^{(\\lambda )}$  If $\\lbrace p_{g_i}\\rbrace $ is not the uniform prior then the map ${\\mathcal {D}}$ is a partially decohering map..", "Such a situation is encountered, for instance, if the collective noise is due to the complete lack of a shared frame of reference associated with the symmetry group $G$  [17].", "The discussion above shows that in the presence of collective noise the total Hilbert space, ${\\mathcal {H}}_d^{\\otimes N}$ , can be decomposed into sectors, ${\\mathcal {H}}^{(\\lambda )}$ , that allow for the possibility of noiseless encoding and decoding of information.", "Using Eq.", "(REF ) the sectors ${\\mathcal {H}}^{(\\lambda )}$ are the Hilbert spaces arising from the composition of two virtual quantum systems with corresponding state spaces ${\\mathcal {M}}^{(\\lambda )},\\,{\\mathcal {N}}^{(\\lambda )}$  [31].", "As the collective noise of the channel acts only on the subsystem associated with ${\\mathcal {M}}^{(\\lambda )}$ this subsystem is a decoherence-full subsystem.", "On the other hand, the subsystem associated with ${\\mathcal {N}}^{(\\lambda )}$ is a decoherence-free, or noiseless, subsystem.", "The sector ${\\mathcal {H}}^{(\\lambda )}$ is a decoherence-free subspace [18] if for any state $\\left| \\psi ^{(\\lambda )}\\right>\\in {\\mathcal {H}}^{(\\lambda )}$ , $U^{(\\lambda )}_{g_i}\\left| \\psi ^{(\\lambda )}\\right>=u^{(\\lambda )}(g_i)\\left| \\psi ^{(\\lambda )}\\right>, \\quad \\forall g_i\\in G,$ where $u^{(\\lambda )}(g_i)\\in \\hbox{$I$\\hspace{-6.99997pt}$C$}$ with $\\vert u^{(\\lambda )}(g_i)\\vert =1$ .", "It follows that ${\\mathcal {H}}^{(\\lambda )}$ is a decoherence-free subspace if and only if the dimension of the decoherence-full subsystem, ${\\mathcal {M}}^{(\\lambda )}$ , is trivial.", "In this case one makes use of the entire subspace, ${\\mathcal {H}}^{(\\lambda )}$ , to store and transmit quantum information.", "If the decoherence-full subsystems are of non-trivial dimension then according to Eq.", "(REF ) no state $\\left| \\psi ^{(\\lambda )}\\right>\\in {\\mathcal {H}}^{(\\lambda )}$ is invariant under collective noise.", "However, the states $\\left| \\beta \\right>\\in {\\mathcal {N}}^{(\\lambda )}$ , associated with the decoherence-free subsystem, are unaffected by the noise of the channel.", "In this case one makes use of a decoherence-free subsystem to store and transmit quantum information.", "Henceforth, we abbreviate both decoherence-free subspaces and subsystems as DFS." ], [ "Example: $\\mathrm {SU}(2)$", "We illustrate the use of a DFS for the most general type of collective noise on $N$ , two-dimensional quantum systems that is associated with $\\mathrm {SU}(2)$ (see appendix  for more details).", "The state $\\left| j_1,m_1\\right>\\otimes \\cdots \\otimes \\left| j_N,m_N\\right>$ can be written, using the familiar Clebsh-Gordan decomposition, as a linear superposition of the orthonormal basis states $\\lbrace \\left| J,M,\\beta \\right>\\rbrace $ , where $J$ labels the total angular momentum, $M$ labels the projection of total angular momentum onto the $z$ -axis, and $\\beta $ labels the various ways $N$ spins can add to a particular total angular momentum $J$ .", "As we explain in appendix , the smallest non-trivial DFS occurs for the case of three qubits.", "Using the logical basis $\\left| 0_L\\right>\\equiv c_1\\left| J=\\frac{1}{2},M=\\frac{1}{2},\\beta =0\\right>+c_2\\left| J=\\frac{1}{2},M=\\frac{-1}{2},\\beta =0\\right>,\\\\\\left| 1_L\\right>\\equiv d_1\\left| J=\\frac{1}{2},M=\\frac{1}{2},\\beta =1\\right>+d_2\\left| J=\\frac{1}{2},M=\\frac{-1}{2},\\beta =1\\right>,$ where $|c_1|^2+|c_2|^2=1$ and $|d_1|^2+|d_2|^2=1$ , one logical qubit can be transmitted noiselessly through the channel.", "It follows that the rate of transmission of quantum information is $1/3$ , i.e.", "three physical qubits are required in order to transmit one logical qubit.", "In the general case where $N$ , $d$ -dimensional systems are used several DFS exist.", "However, as the action of collective noise of the channel destroys coherences between different irrep sectors, ${\\mathcal {H}}^{(J)}$ , of the total Hilbert space only a single DFS can be used to transmit quantum data.", "Hence, to achieve the maximum possible transmission rate the sender and receiver must utilize the DFS with the largest dimension.", "In the limit where $N\\rightarrow \\infty $ it is known that the rate of transmission of quantum data using the largest available DFS is $1-\\mathcal {O}\\left(\\log _2(N)\\right)/N$  [24].", "To encode the logical qubit one needs to be able to perform the transformation that maps the tensor product basis, $\\left| j_1,m_1\\right>\\otimes \\cdots \\otimes \\left| j_N,m_N\\right>$ , to the block diagonal basis, $\\left| J,M,\\beta \\right>$ .", "This basis transformation is accomplished by the Schur transform.", "The latter is the matrix whose columns are $\\left| J,M,\\beta \\right>$ .", "It was also shown that the Schur transform, for encoding and decoding information in a DFS, can be efficiently implemented, up to an arbitrary error $\\epsilon $ , using a number of elementary gates that grows as $N\\cdot \\mathrm {poly}(\\log (N),d,\\log (\\epsilon ^{-1}))$ , where $N$ is the total number of systems, and $d$ is their dimension [28].", "This is achieved by first performing the Clebsch-Gordan transformation described above on the first three systems, whose output is a linear superposition between the states of a $J=1/2$ system and a $J=3/2$ system.", "A second application of the Clebsch-Gordan transformation, between two systems whose joint state is in a linear superposition of a $J=1/2$ system and a $J=3/2$ system, and the forth qubit results in a linear superposition between states of systems with $J=0,1,2$ .", "Continuing this way, the final Clebsch-Gordan transformation couples the $N^{\\mathrm {th}}$ qubit with two systems whose joint state is in a linear superposition of $0\\le J\\le \\left(\\frac{N-1}{2}\\right)$ .", "This results in a cascade of $N$ applications of the Clebsch-Gordan transformation mapping the uncoupled basis, $\\left| j_1,m_1\\right>\\otimes \\left| j_2,m_2\\right>\\otimes \\ldots \\otimes \\left| j_N,m_N\\right>$ , to the total angular momentum basis, $\\left| J,M,\\beta \\right>$ .", "The resulting quantum circuit implementing the Schur transform requires a total of $N\\cdot \\mathrm {poly}(\\log (N),d,\\log (\\epsilon ^{-1}))$ elementary gates in order to be implemented (see [32] for more details).", "Before we introduce our protocol for transmitting quantum data through channels subject to collective noise, we will need a few useful tools from representation theory (see [30] for a good exposition on the subject), which we review here briefly.", "Two elements, $g_i,g_k\\in G$ , are said to be conjugate if there exists an element, $g_l\\in G$ , such that $g_l\\cdot g_i\\cdot g_l^{-1}=g_k$ .", "One can define an equivalence relation on the group $G$ : each element within the set $[g_i]\\equiv \\lbrace g_k\\in G\\;\\mathrm {for\\, which}\\;\\exists \\, g_l\\in G\\;\\mathrm {such\\,that}\\;g_k=g_l\\cdot g_i\\cdot g_l^{-1}\\rbrace $ is equivalent to $g_i$ .", "The set $[g_i]$ is called the conjugacy class of $g_i$ .", "Furthermore, every element of $G$ belongs to one and only one conjugacy class.", "Therefore, the conjugacy classes, $\\lbrace [g_1],\\ldots ,[g_s]\\rbrace $ , form a partition of $G$ .", "Here and in the following $s$ denotes the number of conjugacy classes.", "An important result from representation theory is that the number of inequivalent irreps of any finite or compact Lie group, $G$ , is equal to the number of conjugacy classes of the group.", "Another important concept we will use frequently is that of the character of a representation.", "As we are concerned with channels associated with finite groups it suffices to consider only unitary representations, since every representation of a finite group is equivalent to a unitary representation [30].", "The latter are homomorphisms between $G$ and $\\mathbb {U}(n)$ , the group of $n\\times n$ unitary matrices.", "It follows that the representation of $g_l\\cdot g_i\\cdot g_l^{-1}$ is of the form $U_{g_l}U_{g_i}U_{g_l}^{-1}$ .", "The character of $U_{g_i}$ is defined as $\\chi _{g_i}\\equiv \\mathrm {tr}(U_{g_i})$ .", "Since $\\mathrm {tr}(U_{g_l}U_{g_i}U_{g_l}^{-1})=\\mathrm {tr}(U_{g_i})$ , all elements in the same conjugacy class, $[g_i]$ , have the same character, which we denote by $\\chi _{[g_i]}$ .", "As there are $s$ conjugacy classes, $\\lbrace [g_1],\\ldots ,[g_s]\\rbrace $ , the compound character, $\\chi $ , of a representation, $U$ , is an $s$ -dimensional vector whose entries, $\\chi _i$ , are $\\chi _{[g_i]}$ .", "The compound character of an irrep, $U^{(\\lambda )}$ , is denoted by $\\chi ^{(\\lambda )}$ and is again an $s$ -dimensional vector whose entries, $\\chi ^{(\\lambda )}_i$ , are $\\chi ^{(\\lambda )}_{[g_i]}$ .", "The character table for a finite group, $G$ , is a convenient way to display the compound characters of all irreducible representations, $U^{(\\lambda )}$ of $G$ , where each row, indexed by the irrep label $\\lambda $ , in the character table contains the characters $\\chi ^{(\\lambda )}_{[g_i]}$ for $i\\in (1,\\ldots , s)$ .", "It is known that the characters of the irreps of a finite group satisfy the following orthogonality relation $\\frac{1}{|G|}\\sum _{i=1}^s \\vert [g_i]\\vert \\chi ^{(\\lambda )}_i\\chi ^{(\\lambda ^\\prime )*}_i=\\delta _{\\lambda ,\\lambda ^\\prime },$ where $\\vert [g_i]\\vert $ denotes the number of elements belonging to the conjugacy class $[g_i]$ , and $\\chi ^{(\\lambda ^\\prime )*}_i$ denotes the complex conjugate of $\\chi ^{(\\lambda ^\\prime )}_i$ .", "Knowing $\\chi $ , and all $\\chi ^{(\\lambda )}$ for a representation, $U$ , of a finite group $G$ is sufficient to decompose $U$ into a direct sum of irreps as we now explain.", "Let $U$ be a representation of a group $G$ that has $s$ conjugacy classes.", "Then by the linearity of the trace the compound character, $\\chi $ , of $U$ is given by $\\chi =\\sum _{\\lambda =1}^{s} \\alpha ^{(\\lambda )}\\chi ^{(\\lambda )},$ where $\\alpha ^{(\\lambda )}$ are positive integers corresponding to the multiplicities of the irreps $U^{(\\lambda )}$ .", "Knowing the compound characters, $\\chi ^{(\\lambda )}$ , for all the irreps of a group $G$ and $|[g_i]|$ allows to determine all $\\alpha ^{(\\lambda )}$ 's in Eq.", "(REF ) using Eq.", "(REF ).", "One particular representation of importance in this work is the regular representation, ${\\mathcal {R}}$ , of a finite group $G$ .", "Let ${\\mathcal {H}}$ be a $\\vert G\\vert $ -dimensional Hilbert space and associate to every element, $g_i\\in G$ , one computational basis vector of ${\\mathcal {H}}$ , which we denote by $\\left| g_i\\right>$ .", "In this basis, ${\\mathcal {R}}_{g_k}$ , for any $ g_k\\in G$ , is a $\\vert G\\vert \\times \\vert G\\vert $ permutation matrix which maps the set $\\lbrace \\left| g_i\\right>\\rbrace _{g_i\\in G}$ into itself.", "More precisely, we have ${\\mathcal {R}}_{g_k}\\left| g_i\\right>=\\left| g_k\\cdot g_i\\right>\\in \\lbrace \\left| g_l\\right>\\rbrace _{g_l\\in G},\\forall g_k, g_i\\in G,$ where $g_k\\cdot g_i$ is the group product between the elements $g_k,\\,g_i\\in G$ .", "As the regular representation acts by permuting the computational basis vectors amongst themselves it follows that $\\mathrm {tr}({\\mathcal {R}}(g_i))=\\left\\lbrace \\begin{array}{l l}|G| & \\quad \\mbox{if $g_i=e$}\\\\0 & \\quad \\mbox{otherwise.", "}\\\\ \\end{array} \\right.$ Thus, the character of the regular representation is a vector whose first entry is $|G|$ and the rest are zero.", "It can be easily shown, using Eq (REF ) and the fact that $|[e]|=1$ , that the regular representation contains every irrep, $U^{(\\lambda )}$ , a number of times equal to the dimension, $d_\\lambda $ , of $U^{(\\lambda )}$ .", "That is ${\\mathcal {R}}_{g_i}=\\bigoplus _\\lambda d_\\lambda U^{(\\lambda )}_{g_i}.$ Computing the characters on both sides of Eq.", "(REF ) gives a useful relation between the order of the group and the dimensions of the irreps: $\\nonumber \\chi ({\\mathcal {R}}_e)=|G|&=\\sum _\\lambda d_\\lambda \\chi ^{(\\lambda )}_e\\\\&=\\sum _\\lambda d_\\lambda ^2.$" ], [ "The protocol", "In this section we describe our communication protocol where we consider collective noise described by an arbitrary discrete group, $G$ , with a number of elements equal to the order, $\\vert G\\vert $ , of the group (Sec.", "REF ).", "We show how to encode and decode $m$ logical qudits using $m+r$ physical qudits prepared in a DFS.", "We use $r$ auxiliary qudits to construct a set of perfectly distinguishable states in such a way that the collective noise maps these states into each other.", "In order to encode $m$ logical qudits of quantum information an entangled state is prepared, between the $r$ auxiliary qudits and the $m$ logical qudits, that remains invariant under collective noise.", "The decoding procedure consists of a unitary correction on the $m$ logical qudits which is conditioned on the outcome of a projective measurement on the first $r$ auxiliary qudits.", "In Sec.", "REF , we show how our protocol is implemented for a few examples of collective noise channels." ], [ "Encoding and decoding of quantum data.", "We begin by considering a quantum channel whose noise is described by a unitary representation, $U$ , of a finite group, $G$ , acting on the Hilbert space, ${\\mathcal {H}}_d$ .", "We say that the representation, $U$ , of a finite group, $G$ , is isomorphic to $G$ if there is a one-to-one correspondence between the elements, $g_i\\in G$ , and the matrix representation of these elements, $U_{g_i}$ .", "For an $r$ -qudit state $\\left| \\psi \\right>\\in {\\mathcal {H}}_d^{\\otimes r}$ and a representation $U$ , we define the set of states $S^{(U)}_{(r,\\left| \\psi \\right>)}\\equiv \\lbrace \\left| \\psi (g_i)\\right>=U_{g_i}^{\\otimes r}\\left| \\psi \\right>,\\,g_i\\in G\\rbrace $ .", "We will be interested in those sets $S^{(U)}_{(r,\\left| \\psi \\right>)}$ for which the following two conditions are fulfilled.", "For any pair $g_i,\\,g_k\\in G$ , it holds that: the states $\\left| \\psi (g_i)\\right>,\\,\\left| \\psi (g_k)\\right>\\in S^{(U)}_{(r,\\left| \\psi \\right>)}$ are mutually orthogonal, i.e.", "$\\left\\langle \\psi (g_i)|\\psi (g_k)\\right\\rangle =\\delta _{ik}$ , $U_{g_k}^{\\otimes r}\\left| \\psi (g_i)\\right>=\\left| \\psi (g_k\\cdot g_i)\\right>\\in S^{(U)}_{(r,\\left| \\psi \\right>)}$ , where $g_k\\cdot g_i$ denotes the group product between $g_i,g_k\\in G$ .", "In particular, the set $S^{(U)}_{(r,\\left| \\psi \\right>)}$ is closed under the action of $U^{\\otimes r}$ .", "We refer to a set, $S^{(U)}_{(r,\\left| \\psi \\right>)}$ , fulfilling both conditions as the set of token states.", "We now show how the existence of a set of token states allows the construction of a protocol for error-free transmission of quantum information through a noisy channel.", "Let the noise of the channel be described by the set of operators $\\lbrace U_{g_i},\\,g_i\\in G\\rbrace $ .", "Let us assume that there exists an integer, $r$ , and a state, $\\left| \\psi \\right>\\in {\\mathcal {H}}_d^{\\otimes r}$ , such that $S^{(U)}_{(r,\\left| \\psi \\right>)}$ fulfills condition $(1)$ and $(2)$ .", "The sender, Alice, wishes to transmit a state, $\\left| \\phi \\right>\\in {\\mathcal {H}}_d^{\\otimes m}$ , to the receiver, Bob.", "To that end Alice prepares $m+r$ qudits in the state $\\left| \\chi _{\\phi }\\right>=\\frac{1}{\\sqrt{|G|}}\\sum _{g_i\\in G} \\left| \\psi (g_i)\\right>\\otimes (U_{g_i}^{\\otimes m}\\left| \\phi \\right>),$ with $\\left| \\psi (g_i)\\right>\\in S^{(U)}_{(r,\\left| \\psi \\right>)}$ .", "Sending the $r+m$ qudits prepared in the state of Eq.", "(REF ) through the channel yields $\\nonumber \\left| \\chi ^{\\prime }_{\\phi }\\right>&=&U_{g_k}^{\\otimes (r+m)}\\left| \\chi _{\\phi }\\right> \\\\ \\nonumber &=&\\frac{1}{\\sqrt{|G|}}\\sum _{g_i\\in G} U_{g_k}^{\\otimes r}\\left| \\psi (g_i)\\right>\\otimes (U_{g_k}U_{g_i})^{\\otimes m}\\left| \\phi \\right>),\\\\$ for some $g_k\\in G$ .", "Since $U^{\\otimes r}_{g_k}\\left| \\psi (g_i)\\right>=\\left| \\psi (g_k\\cdot g_i)\\right>$ (condition (2)), and $U_{g_k}U_{g_i}=U_{g_k\\cdot g_i}=U_{g_l}$ , for some $l\\in \\lbrace 0,\\ldots , |G|-1\\rbrace $ , Eq.", "(REF ) yields $\\nonumber U_{g_k}^{\\otimes (r+m)}\\left| \\chi _{\\phi }\\right>&=\\frac{1}{\\sqrt{|G|}}\\sum _{g_l\\in G} \\left| \\psi (g_l)\\right>\\otimes (U_{g_l}^{\\otimes m}\\left| \\phi \\right>)\\\\&=\\left| \\chi _\\phi \\right>.$ Thus, Bob receives the same state $\\left| \\chi _{\\phi }\\right>$ .", "Therefore, the state of Eq.", "(REF ) lies in a DFS as it is invariant under the action $U_{g_k}^{\\otimes (m+r)}$ for all $g_k\\in G$ .", "As we will show below, the number of auxiliary systems, $r$ , required for the construction of the token states is independent of the number of logical qudits, $m$ .", "To decode the quantum data Bob performs the measurement described by $\\lbrace A_i=|\\psi (g_i)\\rangle \\langle \\psi (g_i)|\\, \\mathrm {for}\\; i\\in (0,\\ldots , \\vert G\\vert -1), \\,A_\\perp =I-\\sum _{i=0}^{\\vert G\\vert -1} A_i\\rbrace $ on the first $r$ qudits and obtains outcome $i\\in (0,\\ldots ,\\vert G\\vert -1)$ .", "Note that outcome $A_\\perp $ has zero probability of occurring.", "Conditioned on the outcome, $i$ , the unitary $U_{g_i^{-1}}^{\\otimes m}$ is performed on the remaining $m$ qudits to retrieve $\\left| \\phi \\right>\\in {\\mathcal {H}}_d^{\\otimes m}$ .", "Unlike error-correcting codes, no information about the error is obtained, only about which unitary transformation is needed in order to retrieve the message.", "We also note that our protocol is naturally suited to a communication scenario involving multiple receivers.", "Suppose Alice wishes to distribute the state $\\left| \\phi \\right>\\in {\\mathcal {H}}_d^{\\otimes m}$ to multiple parties through communication channels whose noise is described by $\\lbrace U_{g_i},\\, g_i\\in G\\rbrace $ , where the same operation, $U_{g_i}$ , is applied on all transmitted qudits.", "Alice prepares the state in Eq.", "(REF ), and sends the first $r$ qudits to a single receiver while distributing the remaining $m$ qudits amongst multiple receivers.", "The party holding the first $r$ qudits performs the collective measurement described above, and communicates the outcome to the remaining parties who can each apply the appropriate correction locally on their individual systems.", "Alternatively, Alice and Bob can perform a measure and re-align protocol [26].", "Alice prepares the state $\\left| \\chi _\\phi \\right>=\\left| \\psi (g_i)\\right>\\otimes U_{g_i}^{\\otimes m}\\left| \\phi \\right>$ , for some $g_i\\in G$ , and sends this state through the channel to Bob.", "Bob's description of the state sent to him by Alice is given by the ensemble of states $\\lbrace p_k,\\, \\left| \\psi (g_k)\\right>\\otimes U_{g_k}^{\\otimes m}\\left| \\phi \\right>\\rbrace $ , where $g_k\\in G$ .", "By performing the same measurement described above on the first $r$ qudits, Bob obtains the outcome $i$ and performs the correction, $U_{g_i^{-1}}$ , on the remaining $m$ qudits to retrieve the state $\\left| \\phi \\right>$ with certainty.", "Since two parties sharing a collective noise channel, associated with a finite group, $G$ , is equivalent to two parties sharing a perfect quantum channel but lacking a shared frame of reference associated with the finite group $G$ , the protocol described in this paragraph is equivalent to a reference frame alignment protocol [33], [34], [35].", "Indeed, it is known that for reference frame alignment protocols associated with finite groups there exist states, prepared by Alice, and measurements, performed by Bob, that allow the two parties to perfectly align their respective frames of reference [36].", "However, the sender and receiver might wish for their respective reference frames to remain hidden, as a party's reference frame might serve as a way of identifying themselves to some third party.", "Furthermore, there are instances where it is advantageous to maintain the noise of a quantum channel, as noisy channels can improve the performance of certain quantum information primitives such as bit commitment [37], [38].", "For these cases, it is beneficial for Alice and Bob to utilize a communication protocol that does not allow either party to learn about the other party's reference frame (or equivalently about the action of the channel).", "In this paper our goal is to construct an error-avoiding protocol whose implementation, in terms of elementary gates, is more efficient than the best currently known protocols utilizing a DFS.", "Therefore, we will focus mainly on the DFS protocol outlined above (Eq.", "(REF )).", "In determining the number of gates required to implement the DFS protocol (Sec. )", "we will also determine an upper bound on the number of gates required to perform the measure and re-align protocol.", "We now show how the set of token states, $S^{(U)}_{(r,\\left| \\psi \\right>)}$ , can be constructed using the following theorem, which we prove in Appendix .", "Theorem 1 Let $U$ be a representation of some finite group, $G$ , on a Hilbert space, ${\\mathcal {H}}_d$ , that is isomorphic to $G$ .", "Then there exists an integer, $r$ , and a state $\\left| \\psi \\right>\\in {\\mathcal {H}}_d^{\\otimes r}$ , such that $S^{(U)}_{(r,\\left| \\psi \\right>)}$ satisfies both conditions (1) and (2), if and only if $U^{\\otimes r}$ contains the regular representation, ${\\mathcal {R}}$ , of $G$ .", "Using Theorem REF , the state $\\left| \\psi \\right>\\in {\\mathcal {H}}_d^{\\otimes r}$ can then be chosen to be $\\left| \\psi \\right>=\\sum _\\lambda \\sqrt{\\frac{d_\\lambda }{\\vert G\\vert }}\\sum _{n=1}^{d_\\lambda }\\left| \\xi ^{(\\lambda )}_n\\right>\\otimes \\left| \\zeta ^{(\\lambda )}_n\\right>,$ where the sum is taken over all irreps.", "The set $\\left\\lbrace \\left| \\xi ^{(\\lambda )}_n\\right>\\right\\rbrace $ $\\left(\\left\\lbrace \\left| \\zeta ^{(\\lambda )}_n\\right>\\right\\rbrace \\right)$ denotes an orthonormal basis of ${\\mathcal {M}}^{(\\lambda )}$ $\\left({\\mathcal {N}}^{(\\lambda )}\\right)$ , and $d_\\lambda =\\mathrm {dim}({\\mathcal {M}}^{(\\lambda )})$ .", "We note that the state of Eq.", "(REF ) was shown to optimize the maximum likelihood of a correct guess in a reference frame alignment protocol [39], [40].", "Recall that ${\\mathcal {M}}^{(\\lambda )}$ denotes the space on which the irrep, $U^{(\\lambda )}$ , of $G$ acts non-trivially, whereas ${\\mathcal {N}}^{(\\lambda )}$ denotes the space on which $U^{\\otimes r}$ acts trivially.", "Note that, since $U^{\\otimes r}$ contains the regular representation, the dimension of the multiplicity space, ${\\mathcal {N}}^{(\\lambda )}$ , is at least $d_\\lambda $ .", "We will show in Appendix  that it is always possible to choose $r$ sufficiently large (and independent of $m$ ) such that there exists a state $\\left| \\psi \\right>\\in {\\mathcal {H}}_d^{\\otimes r}$ so that $S^{(U)}_{(r,\\left| \\psi \\right>)}$ fulfills condition (1) and (2)." ], [ "Examples ", "In this subsection we explicitly construct the set of token states, $S^{(U)}_{(r,\\left| \\psi \\right>)}$ , and the state $\\left| \\chi _\\phi \\right>$ of Eq.", "(REF ), for the case of the Pauli channel, and for the cyclic groups $\\mathbb {Z}_3$ and $\\mathbb {Z}_N$ ." ], [ "The Pauli channel", "Suppose that Alice and Bob share a Pauli channel, i.e.", "a channel whose noise is described by the set of operators $\\lbrace e,x,y,z\\rbrace \\equiv \\lbrace I,\\sigma _x,i\\sigma _y, \\sigma _z\\rbrace $ .", "Using the protocol above, we need to construct a set of four token states, one for each element of the Pauli group, that are orthonormal and closed under the action of the Pauli operators.", "Thus, we require that $r\\ge 2$ .", "Infact, $r=2$ suffices as we now prove.", "The set of tensor products of Pauli operators, $\\lbrace I^{\\otimes 2}, \\sigma _x^{\\otimes 2},(i\\sigma _y)^{\\otimes 2},\\sigma _z^{\\otimes 2}\\rbrace $ , forms a four-dimensional representation of the Klein group $K_4$ —the smallest, non-cyclic, finite abelian group.", "The character table for the inequivalent irreps, $U^{(\\lambda )}$ , of $K_4$ is shown in Table REF .", "Table: The character table for K 4 K_4As all the irreps of the Klein group are one-dimensional [30], computing the compound character, $\\chi $ , of $\\lbrace U_{g_i}^{\\otimes 2},\\, g_i\\in K_4\\rbrace $ , one finds that $\\chi =(4,0,0,0)$ .", "As the latter is the character of the regular representation, ${\\mathcal {R}}=U^{(0)}\\oplus U^{(1)}\\oplus U^{(2)}\\oplus U^{(3)}$ , of $K_4$ given by $\\nonumber {\\mathcal {R}}_{g_0}=\\left(\\begin{matrix} 1&0&0&0\\\\0&1&0&0\\\\0&0&1&0\\\\0&0&0&1\\end{matrix}\\right),\\, {\\mathcal {R}}_{g_1}=\\left(\\begin{matrix} 1&0&0&0\\\\0&1&0&0\\\\0&0&-1&0\\\\0&0&0&-1\\end{matrix}\\right),\\\\{\\mathcal {R}}_{g_2}=\\left(\\begin{matrix} 1&0&0&0\\\\0&-1&0&0\\\\0&0&1&0\\\\0&0&0&-1\\end{matrix}\\right),\\,{\\mathcal {R}}_{g_3}=\\left(\\begin{matrix} 1&0&0&0\\\\0&-1&0&0\\\\0&0&-1&0\\\\0&0&0&1\\end{matrix}\\right),$ it follows that the collective action of the Pauli operators on two qubits is equivalent to the regular representation of $K_4$ .", "We now construct the fiducial state, $\\left| \\psi \\right>\\in {\\mathcal {H}}_2^{\\otimes 2}$ , of Eq.", "(REF ).", "Consider the action of $U_{g_i}^{\\otimes 2}$ on the state $\\left| \\Phi ^+\\right>=(\\left| 00\\right>+\\left| 11\\right>)/\\sqrt{2}$ , where $\\left| ab\\right>\\equiv \\left| a\\right>\\otimes \\left| b\\right>$ .", "For any $g_i\\in K_4$ it holds that $U_{g_i}^{\\otimes 2}\\left| \\Phi ^+\\right>=\\left| \\Phi ^+\\right>$ .", "Hence, the state $\\left| \\Phi ^+\\right>$ belongs to the space ${\\mathcal {H}}^{(0)}$ on which the trivial representation of $K_4$ acts.", "Similarly, it can be shown that $\\lbrace \\left| \\Psi ^+\\right>=(\\left| 01\\right>+\\left| 10\\right>)/\\sqrt{2}, \\, \\left| \\Psi ^-\\right>=(\\left| 01\\right>-\\left| 10\\right>)/\\sqrt{2},\\, \\left| \\Phi ^-\\right>=(\\left| 00\\right>-\\left| 11\\right>)/\\sqrt{2}\\rbrace $ belong in the spaces ${\\mathcal {H}}^{(1)}, {\\mathcal {H}}^{(2)}, {\\mathcal {H}}^{(3)}$ respectively.", "Thus, using Eq.", "(REF ), $\\left| \\psi \\right>\\in {\\mathcal {H}}_2^{\\otimes 2}$ is given by $\\left| \\psi \\right>=\\frac{\\left| \\Phi ^+\\right>+\\left| \\Psi ^+\\right>+\\left| \\Psi ^-\\right>+\\left| \\Phi ^-\\right>}{2}=\\left| 0+\\right>,$ where $\\left| \\pm \\right>=(\\left| 0\\right>\\pm \\left| 1\\right>)/\\sqrt{2}$ .", "It follows that the set of token states is given by $S^{(U)}_{(2,\\left| 0+\\right>)}\\equiv \\lbrace \\left| 0+\\right>, \\left| 1+\\right>,-\\left| 1-\\right>, \\left| 0-\\right>\\rbrace $ , corresponding to $\\lbrace \\left| \\psi (g_0)\\right>,\\left| \\psi (g_1)\\right>, \\left| \\psi (g_2)\\right>,\\left| \\psi (g_3)\\right>\\rbrace $ respectively The negative sign in front of the state $\\left| 1-\\right>$ is needed to ensure that the state $\\left| \\chi _\\phi \\right>$ remains in a DFS.. Notice that the set of token states is perfectly distinguishable and closed under the action of $\\lbrace U_{g_i}^{\\otimes 2},\\, g_i\\in K_4\\rbrace $ .", "Alice wants to send a quantum state, $\\left| \\phi \\right>\\in {\\mathcal {H}}_2^{\\otimes m}$ , to Bob.", "Following the protocol in Sec.", "REF , Alice encodes her $m$ -partite quantum state by preparing $\\nonumber \\left| \\chi _{\\phi }\\right>&=\\frac{1}{2}\\sum _{g_i\\in K_4}\\left| \\psi (g_i)\\right>\\otimes U_{g_i}^{\\otimes m}\\left| \\phi \\right>\\\\ \\nonumber &=\\frac{1}{2}\\left(\\left| 0+\\right>\\otimes \\left| \\phi \\right>+\\left| 1+\\right>\\otimes \\sigma _x^{\\otimes m}\\left| \\phi \\right>\\right.\\\\&\\left.-\\left| 1-\\right>\\otimes (i\\sigma _y)^{\\otimes m}\\left| \\phi \\right>+\\left| 0-\\right>\\otimes \\sigma _z^{\\otimes m}\\left| \\phi \\right>\\right).$ The state in Eq.", "(REF ) is invariant under the action of $\\sigma _x^{\\otimes (2+m)}$ as $\\nonumber \\sigma _x^{\\otimes (2+m)}\\left| \\chi _\\phi \\right>&=\\frac{1}{2}\\left(\\sigma _x^{\\otimes 2}\\left| 0+\\right>\\otimes (\\sigma _x\\cdot I)^{\\otimes m}\\left| \\phi \\right>\\right.\\\\ \\nonumber &\\left.+\\sigma _x^{\\otimes 2}\\left| 1+\\right>\\otimes (\\sigma _x\\cdot \\sigma _x)^{\\otimes m}\\left| \\phi \\right>\\right.\\\\ \\nonumber &\\left.-\\sigma _x^{\\otimes 2}\\left| 1-\\right>\\otimes (\\sigma _x\\cdot i\\sigma _y)^{\\otimes m}\\left| \\phi \\right>\\right.\\\\&\\left.+\\sigma _x^{\\otimes 2}\\left| 0-\\right>\\otimes (\\sigma _x\\cdot \\sigma _z)^{\\otimes m}\\left| \\phi \\right>\\right)$ which gives $\\nonumber \\sigma _x^{\\otimes (2+m)}\\left| \\chi _\\phi \\right>&=\\frac{1}{2}\\left(\\left| 1+\\right>\\otimes \\sigma _x^{\\otimes m}\\left| \\phi \\right>+\\left| 0+\\right>\\otimes \\left| \\phi \\right>\\right.\\\\ \\nonumber &\\left.+\\left| 0-\\right>\\otimes \\sigma _z^{\\otimes m}\\left| \\phi \\right>-\\left| 1-\\right>\\otimes (i\\sigma _y)^{\\otimes m}\\left| \\phi \\right>\\right)\\\\&=\\left| \\chi _\\phi \\right>.$ A similar calculation shows that the state $\\left| \\chi _\\phi \\right>$ is invariant under $U_{g_i}^{\\otimes 2+m}$ for all $g_i\\in K_4$ .", "Thus, if Alice sends $2+m$ qubits, prepared in the state $\\left| \\chi _\\phi \\right>$ , through the channel, Bob will receive $2+m$ qubits in the state $\\left| \\chi _\\phi \\right>$ .", "Note that this is true for any probability distribution $\\lbrace p_{g_i},\\, g_i\\in (e,x,y,z)\\rbrace $ .", "To decode the quantum data Bob simply performs the measurement $\\lbrace A_i=|\\psi (g_i)\\rangle \\langle \\psi (g_i)|,\\, i\\in (0,1,2,3)\\rbrace $ on the first two qubits.", "Upon obtaining outcome $i$ , Bob simply applies $U_{g_i^{-1}}^{\\otimes m}$ on the remaining $m$ qubits and retrieves the state $\\left| \\phi \\right>\\in {\\mathcal {H}}_2^{\\otimes m}$ .", "Notice that the measurement corresponds to single qubit measurements in the $z$ and $x$ basis on auxiliary qubit 1 and 2 respectively, and that the required correction operations are local." ], [ "Channel associated with the discrete cyclic group $\\mathbb {Z}_3$", "For our second example we consider the case where the noise of the channel is given by the two-dimensional representation of $\\mathbb {Z}_3$ , the cyclic group of three elements, $U_{g_i}=\\sum _{n=0}^{1}\\omega ^{ng_i}|n\\rangle \\langle n|\\quad g_i\\in (0,1,2),$ where $\\omega =e^{i2\\pi /3}$ .", "The character table for $\\mathbb {Z}_3$ is given in Table.", "REF .", "Table: The character table for ℤ 3 \\mathbb {Z}_3As $\\mathbb {Z}_3$ is abelian, all its irreps are one dimensional and are given by $U_{g_i}^{(\\lambda )}=\\omega ^{\\lambda g_i}$ .", "It follows that the regular representation, ${\\mathcal {R}}$ , of $\\mathbb {Z}_3$ is given by ${\\mathcal {R}}_{g_i}=U^{(0)}_{g_i}\\oplus U^{(1)}_{g_i}\\oplus U^{(2)}_{g_i}=\\left(\\begin{matrix}1&0&0\\\\0&\\omega ^{g_i}&0\\\\0&0&\\omega ^{2g_i}\\end{matrix}\\right).$ Using Eq.", "(REF ), $\\lbrace U_{g_i}^{\\otimes 2},\\, g_i\\in \\mathbb {Z}_3\\rbrace $ is given by $U_{g_i}^{\\otimes 2}=\\sum _{n_1,n_2=0}^1\\omega ^{(n_1+n_2)g_i}|n_1,n_2\\rangle \\langle n_1,n_2|\\quad g_i\\in (0,1,2),$ where $n_1$ and $n_2$ are added modulo three.", "Let us denote by $\\lbrace \\left| \\lambda ,\\beta \\right>\\rbrace _{\\beta =1}^{\\alpha ^{(\\lambda )}}$ the set of orthogonal states $\\lbrace \\left| n_1,n_2\\right>\\rbrace _{(n_1+n_2){\\mathrm {mod}\\,3}=\\lambda }$ .", "Here, $\\alpha ^{(\\lambda )}=\\binom{2}{\\lambda }$ denotes the number of states corresponding to the same $\\lambda $ .", "Then Eq.", "(REF ) can be written as $\\nonumber U_{g_i}^{\\otimes 2}&=\\sum _{\\lambda =0}^2\\sum _{\\beta =1}^{\\alpha ^{(\\lambda )}}\\omega ^{\\lambda g_i}|\\lambda ,\\beta \\rangle \\langle \\lambda ,\\beta |\\\\&=\\bigoplus _{\\lambda =0}^2 U_{g_i}^{(\\lambda )}\\otimes I_{\\alpha ^{(\\lambda )}}, \\,\\forall g_i\\in (0,1,2),$ where $I_{\\alpha ^{(\\lambda )}}$ is the ${\\alpha ^{(\\lambda )}}$ -dimensional identity.", "It follows that $\\left| 0\\right>\\equiv \\left| 00\\right>\\in {\\mathcal {H}}^{(0)}$ , the subspace upon which $U^{(0)}$ acts, $\\left| 2\\right>\\equiv \\left| 11\\right>\\in {\\mathcal {H}}^{(2)}$ , the subspace upon which $U^{(2)}$ acts, and $\\lbrace \\left| 1,1\\right>\\equiv \\left| 01\\right>,\\left| 1,2\\right>\\equiv \\left| 10\\right>\\rbrace \\in {\\mathcal {H}}^{(1)}$ , the subspace upon which $U^{(1)}$ acts.", "Since every inequivalent irrep of $\\mathbb {Z}_3$ is present in $\\lbrace U_{g_i}^{\\otimes 2},\\, g_i\\in \\mathbb {Z}_3\\rbrace $ the latter contains the regular representation as a sub-representation.", "We now construct the set of token states for such a channel.", "The subspace ${\\mathcal {H}}^{(1)}$ , upon which irrep $U^{(1)}$ acts, is a two-dimensional DFS.", "However, one state from this subspace is sufficient to construct our token states since for all abelian groups $d_\\lambda =\\mathrm {dim}({\\mathcal {M}}^{(\\lambda )})=1$ in Eq.", "(REF ).", "Hence, we can choose $\\left| 1,1\\right>=\\left| 01\\right>$ as our standard state from the ${\\mathcal {H}}^{(1)}$ subspace.", "Thus, the state $\\left| \\psi \\right>\\in {\\mathcal {H}}_2^{\\otimes 2}$ of Eq.", "(REF ) reads $\\left| \\psi \\right>=\\frac{1}{\\sqrt{3}}\\left(\\left| 00\\right>+\\left| 01\\right>+\\left| 11\\right>\\right)$ or, in the $\\lbrace \\left| \\lambda ,\\beta \\right>\\rbrace $ basis, $\\left| \\psi \\right>=\\frac{1}{\\sqrt{3}}\\left(\\left| 0\\right>+\\left| 1,1\\right>+\\left| 2\\right>\\right).$ In what follows we explicitly use Eq.", "(REF ) but of course the same reasoning would hold if one uses Eq.", "(REF ) instead.", "The set of token states is given by $S^{(U)}_{(2,\\left| \\psi \\right>)}\\equiv \\lbrace \\left| \\psi (g_i)\\right>=\\frac{1}{\\sqrt{3}}\\left(\\left| 00\\right>+\\omega ^{g_i}\\left| 01\\right>+\\omega ^{2g_i}\\left| 11\\right>\\right), \\, g_i\\in (0,1,2)\\rbrace $ .", "To communicate an arbitrary $m$ -partite state, $\\left| \\phi \\right>\\in {\\mathcal {H}}_2^{\\otimes m}$ , Alice prepares the state $\\nonumber \\left| \\chi _{\\phi }\\right>&=\\frac{1}{\\sqrt{3}}\\sum _{g_i\\in \\mathbb {Z}_3}\\left| \\psi (g_i)\\right>\\otimes U_{g_i}^{\\otimes m}\\left| \\phi \\right>\\\\ \\nonumber &=\\frac{1}{3}\\left((\\left| 00\\right>+\\left| 01\\right>+\\left| 11\\right>)\\otimes U_{g_0}^{\\otimes m}\\left| \\phi \\right>\\right.\\\\ \\nonumber &\\left.+(\\left| 00\\right>+\\omega \\left| 01\\right>+\\omega ^2\\left| 11\\right>)\\otimes U_{g_1}^{\\otimes m}\\left| \\phi \\right>\\right.\\\\&\\left.+(\\left| 00\\right>+\\omega ^2\\left| 01\\right>+\\omega \\left| 11\\right>)\\otimes U_{g_2}^{\\otimes m}\\left| \\phi \\right>\\right).$ Suppose that the channel performs $U_{g_2}$ on all the qubits.", "Then $\\nonumber U_{g_2}^{\\otimes (2+m)}&\\left| \\chi _{\\phi }\\right>=\\frac{1}{3}\\left(U_{g_2}^{\\otimes 2}(\\left| 00\\right>+\\left| 01\\right>+\\left| 11\\right>)\\right.\\\\ \\nonumber &\\left.\\otimes (U_{g_2}U_{g_0})^{\\otimes m}\\left| \\phi \\right>+U_{g_2}^{\\otimes 2}(\\left| 00\\right>+\\omega \\left| 01\\right>+\\omega ^2\\left| 11\\right>)\\right.\\\\ \\nonumber &\\left.\\otimes (U_{g_2}U_{g_1})^{\\otimes m}\\left| \\phi \\right>+U_{g_2}^{\\otimes 2}(\\left| 00\\right>+\\omega ^2\\left| 01\\right>+\\omega \\left| 11\\right>)\\right.\\\\&\\left.\\otimes (U_{g_2}U_{g_2})^{\\otimes m}\\left| \\phi \\right>\\right).$ As representations are homomorphisms, $U_{g_k}U_{g_i}=U_{g_{k+i}}$ , Eq.", "(REF ) gives $\\nonumber U_{g_2}^{\\otimes (2+m)}\\left| \\chi _{\\phi }\\right>&=\\frac{1}{3}\\left((\\left| 00\\right>+\\omega ^2\\left| 01\\right>+\\omega \\left| 11\\right>)\\otimes U_{g_2}^{\\otimes m}\\left| \\phi \\right>\\right.\\\\ \\nonumber &\\left.+(\\left| 00\\right>+\\left| 01\\right>+\\left| 11\\right>)\\otimes U_{g_0}^{\\otimes m}\\left| \\phi \\right>\\right.\\\\ \\nonumber &\\left.+(\\left| 00\\right>+\\omega \\left| 01\\right>+\\omega ^2\\left| 11\\right>)\\otimes U_{g_1}^{\\otimes m}\\left| \\phi \\right>\\right)\\\\&=\\left| \\chi _\\phi \\right>.$ A similar calculation shows that the state $\\left| \\chi _\\phi \\right>$ is invariant for all $g_i\\in \\mathbb {Z}_3$ .", "Thus, if Alice sends $2+m$ qubits, prepared in the state $\\left| \\chi _\\phi \\right>$ , through the channel Bob will receive $2+m$ qubits in the state $\\left| \\chi _{\\phi }\\right>$ .", "Note that this is true for any probability distribution $\\lbrace p_{g_i},\\, g_i\\in \\mathbb {Z}_3\\rbrace $ .", "To decode the state $\\left| \\phi \\right>\\in {\\mathcal {H}}_2^{\\otimes m}$ , Bob performs the measurement $\\lbrace A_i=|\\psi (g_i)\\rangle \\langle \\psi (g_i)|,\\, \\mathrm {for}\\; i\\in (0,1,2),\\, A_\\perp =I-\\sum _i A_i \\rbrace $ on the first two qubits.", "Note that in this example $A_\\perp =|10\\rangle \\langle 10|$ so that the set of measurements is complete on ${\\mathcal {H}}_2^{\\otimes 2}$ , and that the probability of obtaining this measurement outcome is zero.", "Upon obtaining outcome $i$ , Bob implements $U_{g_i^{-1}}^{\\otimes m}$ on the remaining $m$ qubits and retrieves $\\left| \\phi \\right>\\in {\\mathcal {H}}_2^{\\otimes m}$ ." ], [ "Channel associated with the discrete cyclic group $\\mathbb {Z}_N$", "The above example can be easily generalized to the case where the channel's noise is associated to the group $\\mathbb {Z}_N$ , the cyclic group of $N$ elements, with its action on a $d$ -dimensional Hilbert space, ${\\mathcal {H}}_d$ , given by $U_{g_i}=\\sum _{n=0}^{d-1}\\omega ^{ng_i}|n\\rangle \\langle n|,\\quad g_i\\in (0,\\ldots , N-1),$ where $\\omega =e^{i2\\pi /N}$ .", "Notice that here $N$ denotes the number of elements of the group.", "The character table for $\\mathbb {Z}_N$ is given in Table.", "REF .", "Table: The character table for ℤ N \\mathbb {Z}_NAs $\\mathbb {Z}_N$ is abelian all its inequivalent irreps are one-dimensional and are given by $U_{g_i}^{(\\lambda )}=\\omega ^{\\lambda g_i}$ .", "If $N\\le d$ , then the representation, $\\lbrace U_{g_i},\\, g_i\\in \\mathbb {Z}_N\\rbrace $ , in Eq.", "(REF ) contains the regular representation.", "If $N>d$ , then we need to tensor the representation of Eq.", "(REF ) with itself a number of times, $r$ , such that every inequivalent irrep appears at least once.", "In Appendix , we show that $r$ is finite and depends only on the representation, $U$ , of $G$ .", "In Appendix  we show how to explicitly compute $r$ using some examples and here we will derive it for $\\mathbb {Z}_N$ .", "Let $N>d$ and consider the representation $\\lbrace U_{g_i}^{\\otimes r},\\, g_i\\in \\mathbb {Z}_N\\rbrace $ .", "Using Eq.", "(REF ), the latter is given by $U_{g_i}^{\\otimes r}=\\sum _{n_1\\ldots n_r=0}^{d-1} \\omega ^{(n_1+\\ldots +n_r)g_i}|n_1\\ldots n_r\\rangle \\langle n_1\\ldots n_r|$ for all $g_i\\in \\mathbb {Z}_N$ , where $n_1,\\ldots , n_r\\in (0,\\ldots , d-1)$ and $n_1+\\ldots +n_r$ are added modulo $N$ .", "Similarly to the $\\mathbb {Z}_3$ example above we use the notation $\\lbrace \\left| \\lambda ,\\beta \\right>\\rbrace _{\\beta =0}^{\\alpha ^{(\\lambda )}}\\equiv \\lbrace \\left| n_1,\\ldots ,n_r\\right>\\rbrace _{(n_1+\\ldots +n_r){\\mathrm {mod}\\,N}=\\lambda }$ , where $\\alpha ^{(\\lambda )}$ denotes the number of states corresponding to the same $\\lambda $ .", "Using this notation, the operators $U_{g_i}^{\\otimes r}$ may be re-written as $\\nonumber U_{g_i}^{\\otimes r}&=\\sum _{\\lambda =0}^{r(d-1)}\\sum _{\\beta =1}^{\\alpha ^{(\\lambda )}}\\, \\omega ^{\\lambda g_i}|\\lambda ,\\beta \\rangle \\langle \\lambda ,\\beta |\\\\&=\\bigoplus _{\\lambda =0}^{r(d-1)} U_{g_i}^{(\\lambda )}\\otimes I_{\\alpha ^{(\\lambda )}},\\,\\forall g_i\\in \\mathbb {Z}_N.$ Since $\\mathbb {Z}_N$ has $N$ inequivalent irreps, it follows from Eq.", "(REF ) that $\\lbrace U_{g_i}^{\\otimes r},\\,g_i\\in \\mathbb {Z}_N\\rbrace $ contains the regular representation whenever $r(d-1) \\ge N-1$ .", "That is, the space ${\\mathcal {H}}_{{\\mathcal {R}}}\\equiv {\\mathcal {H}}_2^{\\otimes r^{\\prime }}$ , with $r^{\\prime }=\\log _2 N$ , on which the regular representation acts is embedded in the higher dimensional space ${\\mathcal {H}}_d^{\\otimes r}$ , with $r=\\lceil \\frac{N-1}{d-1}\\rceil $ .", "Notice that this corresponds to an exponential increase of required resources: $r=\\lceil \\frac{N-1}{d-1}\\rceil $ qudits are required to ensure that the set $S^{(U)}_{(r,\\left| \\psi \\right>)}$ containing $r^{\\prime }=\\log _2 N$ states satisfies properties (1) and (2).", "Despite this exponential overhead we find an efficient implementation for the group $\\mathbb {Z}_N$ that scales only linearly with the number of group elements $N$ .", "We now construct the set of token states for such a channel.", "Notice that the states $\\lbrace \\left| \\lambda ,\\beta \\right>\\rbrace _{\\beta =1}^{\\alpha ^{(\\lambda )}}$ , for a given $\\lambda $ , form an $\\alpha ^{(\\lambda )}$ -dimensional DFS.", "However, one state from this subspace is sufficient to construct our token states since for all abelian groups $d_\\lambda =\\mathrm {dim}({\\mathcal {M}}^{(\\lambda )})=1$ in Eq.", "(REF ).", "Without loss of generality, we choose $\\left| \\lambda ,1\\right>$ for each irrep $\\lambda $ .", "In the computational basis the state $\\left| \\lambda ,1\\right>$ corresponds to the $r$ -qudit state where the first $\\lambda $ qudits are in state $\\left| 1\\right>$ , and the remaining $r-\\lambda $ qudits are in the state $\\left| 0\\right>$ .", "Hence, the fiducial state, $\\left| \\psi \\right>\\in {\\mathcal {H}}_d^{\\otimes r}$ , of Eq.", "(REF ) is given by $\\left| \\psi \\right>=\\sqrt{\\frac{1}{N}}(\\left| 0\\ldots 00\\right>+\\left| 0\\ldots 01\\right>+\\left| 0\\ldots 11\\right>+\\ldots + \\left| 1\\ldots 11\\right>),$ or in the $\\lbrace \\left| \\lambda ,\\beta \\right>\\rbrace $ basis $\\left| \\psi \\right>=\\sqrt{\\frac{1}{N}}\\sum _{\\lambda =0}^{N-1} \\left| \\lambda ,1\\right>.$ The set of token states, $S^{(U)}_{(r,\\left| \\psi \\right>)}$ is therefore given by $\\left\\lbrace \\left| \\psi (g_i)\\right>=\\sqrt{\\frac{1}{N}}\\sum _{\\lambda =0}^{N-1}\\omega ^{\\lambda \\cdot g_i}\\left| \\lambda ,1\\right>,\\, g_i\\in \\mathbb {Z}_N\\right\\rbrace ,$ where we have chosen the more compact form of Eq.", "(REF ) for the state $\\left| \\psi \\right>\\in {\\mathcal {H}}_d^{\\otimes r}$ .", "To communicate an arbitrary $m$ -partite state, $\\left| \\phi \\right>\\in {\\mathcal {H}}_d^{\\otimes m}$ , Alice prepares the state $\\left| \\chi _{\\phi }\\right>=\\frac{1}{N}\\sum _{g_i\\in \\mathbb {Z}_N} \\sum _{\\lambda =0}^{N-1}\\omega ^{\\lambda \\cdot g_i}\\left| \\lambda ,1\\right>\\otimes U_{g_i}^{\\otimes m}\\left| \\phi \\right>.$ Similarly to the previous example, it can be shown that $\\left| \\chi _{\\phi }\\right>$ is invariant under $U_{g_i}^{\\otimes (r+m)}$ for any $g_i\\in \\mathbb {Z}_N$ .", "Thus, if Alice sends $r+m$ qudits, prepared in the state $\\left| \\chi _\\phi \\right>$ , through the channel, Bob will receive the $r+m$ qudits in the state $\\left| \\chi _{\\phi }\\right>$ independently of the probability distribution $\\lbrace p_{g_i},\\, g_i\\in \\mathbb {Z}_N\\rbrace $ .", "As before, the decoding of the quantum data is achieved by performing the measurement $\\lbrace A_i=|\\psi (g_i)\\rangle \\langle \\psi (g_i)|,\\, \\mathrm {for}\\; i\\in (0,\\ldots , N-1),\\,A_\\perp =I-\\sum _iA_i\\rbrace $ .", "Conditioned on the outcome, $i$ , of this measurement the correction $U_{g_i^{-1}}^{\\otimes m}$ to the remaining qudits is applied Like before, the outcome $A_\\perp $ has zero probability of occurrence..", "In the next section we explicitly calculate the number of elementary gates required to encode and decode quantum data using the protocol described above." ], [ " Implementation of our protocol", "In this section we analyze the required resources for encoding and decoding quantum data transmitted through collective noise channels described by an arbitrary discrete group, $G$ .", "As mentioned before, whereas Alice and Bob can communicate using the measure and re-align protocol of [26], such a protocol is undesirbale since it requires Bob to learn the action of the channel.", "As the noise of the channel can be used to offer security [41], or to improve the performance of certain quantum information primitives [37], [38], it is advantageous to utilize an error-avoiding protocol that reveals no information about the noise of the channel.", "In the following we show how to implement the protocol of Sec. .", "In particular, we discuss the required number of elementary gates of the encoding and decoding circuit, as well as the logical depth of the circuit.", "A direct comparison to previously introduced DFS schemes [28] reveals that our method is more efficient, and also achieves the optimal transmission rate in the asymptotic limit.", "We also provide an upper bound on the number of elementary gates needed to implement the measure and re-align protocol." ], [ "Encoding circuit", "For ease of exposition we shall assume throughout that all the physical systems used in the protocol are qubits.", "We determine the number of elementary gates for the case of qudits at the end of this section.", "Recall that our protocol encodes quantum information contained in an $m$ -qubit state, $\\left| \\phi \\right>\\in {\\mathcal {H}}_2^{\\otimes m}$ , by preparing the state of Eq.", "(REF ), where $\\left| \\psi \\right>\\in {\\mathcal {H}}_2^{\\otimes r}$ for finite $r$ , is given by Eq.", "(REF ).", "Notice that there are $r^{\\prime } = \\log _2 |G|$ orthogonal token states, which are, however, encoded into $r \\ge r^{\\prime }$ qubits to ensure the proper behavior under $\\lbrace U_{g_i}^{\\otimes r},\\,g_i\\in G\\rbrace $ .", "First we associate to each group element $g_i\\in G$ a computational basis vector, $\\left| g_i\\right>\\equiv \\left| i_{r^{\\prime }},\\ldots i_{1}\\right>\\in {\\mathcal {H}}_2^{\\otimes r^{\\prime }},$ where $i=\\sum _{k=1}^{r^{\\prime }} 2^{k-1} i_k$ .", "Then we define the unitary operation $T$ in such a way that $T\\left| 0\\right>^{\\otimes r-r^{\\prime }}\\left| g_i\\right>=\\left| \\psi (g_i)\\right>,$ i.e.", "a computational basis state $\\left| g_i\\right>$ of $r^{\\prime }$ qubits, embedded into ${\\mathcal {H}}_2^{\\otimes r}$ , is transformed to a token state $\\left| \\psi (g_i)\\right>$ of $r$ qubits.", "We can now re-write Eq.", "(REF ) as $\\left| \\chi _\\phi \\right>= \\big (T \\otimes \\mbox{$1 \\hspace{-2.84526pt} {\\bf l}$}\\big ) \\left| 0\\right>^{\\otimes r-r^{\\prime }}\\frac{1}{\\vert G\\vert }\\sum _{g_i\\in G}\\left| g_i\\right>\\otimes U_{g_i}^{\\otimes m}\\left| \\phi \\right>.$ The encoding of quantum information takes place in two steps.", "One first prepares the $r^{\\prime }+m$ qubit state $\\frac{1}{\\vert G\\vert }\\sum _{g_i\\in G}\\left| g_i\\right>\\otimes U_{g_i}^{\\otimes m}\\left| \\phi \\right>$ , followed by the $r$ qubit operation $T$ .", "The second step can be implemented using at most ${\\cal O}(2^r)$ elementary gates.", "The latter is an upper bound on the number of gates required to implement the measure and re-align strategy of [41].", "In the following we concentrate on the first step, in particular the efficient implementation of the unitary operation, $W$ , acting on $r^{\\prime }+m$ qubits, defined as $W\\equiv \\sum _{g_i\\in G}\\left| g_i\\right>\\left< g_i\\right| \\otimes U_{g_i}^{\\otimes m}.$ To present a circuit implementation of the gate in Eq.", "(REF ) we define the unitary operators $W_{g_i}&=&(\\mbox{$1 \\hspace{-2.84526pt} {\\bf l}$}-\\left| g_i\\right>\\left< g_i\\right|)\\otimes \\mbox{$1 \\hspace{-2.84526pt} {\\bf l}$}+\\left| g_i\\right>\\left< g_i\\right|\\otimes U_{g_i},\\\\W_{g_i}^m&=&(\\mbox{$1 \\hspace{-2.84526pt} {\\bf l}$}-\\left| g_i\\right>\\left< g_i\\right|)\\otimes \\mbox{$1 \\hspace{-2.84526pt} {\\bf l}$}+\\left| g_i\\right>\\left< g_i\\right|\\otimes U_{g_i}^{\\otimes m}.$ The gate $W_{g_i}$ implements a unitary operation $U_{g_i}$ only if the control register is in state $\\left| g_i\\right>$ .", "In case $\\lbrace \\left| g_i\\right>\\rbrace $ forms a complete orthonormal basis on ${\\mathcal {H}}_2^{\\otimes r^{\\prime }}$ (i.e.", "$\\vert G\\vert =2^{r^{\\prime }}$ ), we have $W\\equiv \\prod _{g_i\\in G} W_{g_i}^m,$ and therefore $W\\left(\\left| +\\right>^{\\otimes r^{\\prime }}\\otimes \\left| \\phi \\right>\\right)=\\frac{1}{\\vert G\\vert }\\sum _{g_i\\in G} \\left| g_i\\right>\\otimes U_{g_i}^{\\otimes m}\\left| \\phi \\right>,$ where $\\left| +\\right>^{\\otimes r^{\\prime }}\\equiv \\frac{1}{\\vert G\\vert }\\sum _{g_i\\in G}\\left| g_i\\right>$  Note that if $\\lbrace \\left| g_i\\right>\\rbrace $ does not form a complete orthonormal basis, that is $\\vert G\\vert <2^{r^{\\prime }}$ , we have to apply $W$ to a state which is the superposition of $\\vert G\\vert $ computational basis states (which does not coincide with $\\left| +\\right>^{r^{\\prime }}$ ).", "Such an input state can be easily generated in the following way.", "Let $\\tilde{r}<r^{\\prime }$ be such that $2^{\\tilde{r}-1}<|G|<2^{\\tilde{r}}$ .", "Then, $\\left| \\Psi \\right>=\\otimes _{i=2}^{\\tilde{r}} U_{1i}\\left| +\\right>^{\\otimes r^{\\prime }}$ , where $U_{1i}=\\left| 0\\right>\\left< 0\\right|\\otimes \\mbox{$1 \\hspace{-2.84526pt} {\\bf l}$}+\\left| 1\\right>\\left< 1\\right|\\otimes U_i$ .", "Choosing $U_i$ , which acts on qubit $i$ , either as $U_i=\\mbox{$1 \\hspace{-2.84526pt} {\\bf l}$}$ , or $U_i\\left| +\\right>=\\left| 0\\right>$ allows one to generate any desired superposition of the form $|0\\rangle |+\\ldots ++\\rangle + |1\\rangle |+0 \\ldots 0+\\rangle $ with $\\tilde{r}$ gates.", "Note that not all terms in this sum have the same weight when we write the state in the computational basis.", "This can however easily be corrected by preparing the first qubit in a state $\\cos \\alpha |0\\rangle + \\sin \\alpha |1\\rangle $ rather than $|+\\rangle $ and choosing $\\alpha $ appropriately.. Notice that $W$ corresponds to the sequence of controlled-unitary gates, $W_{g_i}^m$ , for all possible values of $g_i\\in G$ .", "We will now outline a circuit implementing the gate $W$ of Eq.", "(REF ).", "First, note that $\\left| g_i\\right>$ is a binary representation of the value $i\\le 2^{r^{\\prime }}$ corresponding to the group element $g_i\\in G$ .", "The gate $W_{g_i}^m$ , for some fixed $g_i\\in G$ , can be implemented by applying local unitaries $\\sigma _x^{(i_{r^{\\prime }})}\\otimes \\ldots \\otimes \\sigma _x^{(i_{1})}$ to the first $r^{\\prime }$ qubits such that $\\sigma _x^{(i_{r^{\\prime }})}\\otimes \\ldots \\otimes \\sigma _x^{(i_{1})}\\left| i_{r^{\\prime }}\\ldots i_{1}\\right>=\\left| 1\\right>^{\\otimes r^{\\prime }}$ and then applying the gate $V_{g_i}^m \\equiv (\\mbox{$1 \\hspace{-2.84526pt} {\\bf l}$}-\\left| 1\\right>\\left< 1\\right|^{\\otimes r^{\\prime }})\\otimes \\mbox{$1 \\hspace{-2.84526pt} {\\bf l}$}+ \\left| 1\\right>\\left< 1\\right|^{\\otimes r^{\\prime }}\\otimes U_{g_i}^{\\otimes m}.$ The latter can in turn be implemented by applying the gate $V_{g_i}=(\\mbox{$1 \\hspace{-2.84526pt} {\\bf l}$}-\\left| 1\\right>\\left< 1\\right|^{\\otimes r^{\\prime }})\\otimes \\mbox{$1 \\hspace{-2.84526pt} {\\bf l}$}+ \\left| 1\\right>\\left< 1\\right|^{\\otimes r^{\\prime }}\\otimes U_{g_i}$ $m$ times, where the control qubits remain the same but the target qubit is always a new one.", "The implementation of the gates $V_{g_i}^m$ , $W_{g_i}^m$ , and $W$ is shown in Figs.", "(REF , REF , REF ) respectively." ], [ "Resources", "We will now count how many elementary gates are required in order to implement $V_{g_i}$ .", "In [42] it has been shown that a control gate of the form $\\Lambda _{r^{\\prime }} (U)=(\\mbox{$1 \\hspace{-2.84526pt} {\\bf l}$}-\\left| 1\\right>\\left< 1\\right|^{\\otimes r^{\\prime }})\\otimes \\mbox{$1 \\hspace{-2.84526pt} {\\bf l}$}+ \\left| 1\\right>\\left< 1\\right|^{\\otimes r^{\\prime }}\\otimes U$ , where $r^{\\prime }$ denotes the number of control qubits, can be implemented using $40(r^{\\prime }-2)+1$ elementary gates The factor $4(r^{\\prime }-2)$ is the number of Toffoli gates required to implement a $r^{\\prime }$ -control-X gate, and the factor 10 comes from applying these gates twice (before and after), and using 5 CNOTs per Toffoli gate.", "The $+1$ is the controlled-U operation (see Lemma 7.11 and Lemma 7.2 in [42].", "In order to apply a controlled-$U_{g_i}^{\\otimes m}$ gate we simply apply the $m$ controlled-$U_{g_i}$ gates with different target qubits in between the two $r^{\\prime }$ -controlled-$\\sigma _x$ gates (see Figs.", "(REF , REF )).", "Figure: The quantum circuit implementation of the encoding operation V g i m V_{g_i}^m.", "We note that the circuit implementation of the r ' r^{\\prime }-controlled Toffoli gates in this circuit can be found in Lemma 7.2 of .Figure: The quantum circuit for W g i m W_{g_i}^m for any state g i =i r ' ...i 1 \\left| g_i\\right>=\\left| i_{r^{\\prime }}\\ldots i_{1}\\right>, a binary representation of the group element g i ∈Gg_i\\in G. The gate (σ x ) i m ⊕1 (\\sigma _x)_{i_m\\oplus 1} flips the m th m^{\\mathrm {th}} qubit of the input state, if the m th m^{\\mathrm {th}} digit, i m i_m, in the binary representation of g i ∈Gg_i\\in G is zero.", "After implementing the gate V g i m V_{g_i}^m the bit string is restored to its original value.Thus, $f(r^{\\prime })\\equiv 40(r^{\\prime }-2)+m$ elementary gates are required to implement $V_{g_i}^m$ .", "Therefore, the number of gates required to implement $W_{g_i}^m$ is $M \\equiv 40(r^{\\prime }-2)+m+r^{\\prime }$ (another $r^{\\prime }$ operations for local basis change in the control register Notice that there would actually be $2r^{\\prime }$ operations for the basis change, $r^{\\prime }$ before and $r^{\\prime }$ after the application of $V_{11 \\ldots 1}^m$ .", "However, the second set of $r^{\\prime }$ operations only returns the control register to its initial value, which is not necessary for a single gate $W_{g_j}^m$ and can in fact be combined with the first set of $r^{\\prime }$ operation of the subsequent gate $W_{g_l}^m$ .", ").", "Since a total of $|G|$ different gates $W_{g_i}^m$ need to be applied to implement $W$ (one for each group element $g_i\\in G$ - see Eq.", "(REF ) and Fig.", "REF ), we find that the total number of elementary gates required to implement $W$ is given by $\\vert G\\vert M=\\vert G\\vert (41r^{\\prime }-80+m),$ where $|G|=2^{r^{\\prime }}$ .", "That is, the resources required to encode the quantum data are linear in the number of qubits, $m$ , to be transmitted and scale as $\\vert G\\vert \\log (\\vert G\\vert )$ .", "After performing the number of gates in Eq.", "(REF ) an additional ${\\cal O}(2^r)$ gates are required to implement the unitary $T$ that maps the computational basis states $\\lbrace \\left| g_i\\right>\\rbrace $ to the set of token states $\\lbrace \\left| \\psi ({g_i})\\right>\\rbrace $  This is an upper bound on the number of gates required to implement $T$ ..", "Figure: The circuit implementation of the encoding circuit W=∑ g i ∈G W g i m W=\\sum _{g_i\\in G} W_{g_i}^m, where g i ∈Gg_i\\in G is written in binary notation.Finally, we note that if the dimension of the representation, $U$ , of $G$ is $d$ , i.e.", "if the operators $\\lbrace U_{g_i},\\, g_i\\in G\\rbrace $ act on $d$ -dimensional systems, then the number of elementary gates required to implement $W$ only increases by a factor which is independent of $m$ and $\\vert G\\vert $ .", "In this case the unitary transformation, $T$ , requires at most ${\\cal O}(d^r)$ elementary gates in order to be implemented.", "In the following, we consider some special groups and show that the required resources can be significantly reduced." ], [ "Abelian groups", "We now discuss a method to implement the gate $W$ , given in Eq.", "(REF ), for the case of quantum channels whose collective noise is associated with a finite abelian group.", "Denoting by $g_1,\\ldots , g_k$ the generators of the group, then for any $g\\in G$ there exists a string, $(l_1(g),\\ldots , l_k(g))$ , with $l_j(g)\\in N$ , such that $g=g_1^{l_1(g)}\\cdots g_k^{l_k(g)}$ .", "Since we are dealing with finite groups we have that for any $j$ there exists a $L_j$ such that $l_j(g)\\le L_j$ for any $g\\in G$ .", "Writing, $\\left| g\\right>=\\left| l_1(g)\\ldots l_k(g)\\right>$ , where $l_j(g)$ is represented in binary notation, the gate $W$ in Eq.", "(REF ) becomes $W= \\sum _{g\\in G}\\left| g\\right>\\left< g\\right| \\otimes U_{g}^{\\otimes m}=\\prod _{i=1}^k U_i,$ with $U_i=\\sum _{l_i=0}^{L_i}\\left| l_i\\right>\\left< l_i\\right| \\otimes (U_{g_i}^{l_i})^{\\otimes m}$ .", "Note that each gate $U_i$ is acting on a $L_i$ -dimensional control system (i.e.", "$\\log {L_i}$ control qubits) and $m$ target qubits (the first control system controls how often $g_1$ is applied, the second how often $g_2$ etc.).", "Since $U_i$ is acting on $(\\log L_i+m)$ qubits and requires at most $L_i$ control gates, the number of elementary gates required to implement $U_i$ is at most $L_i f(\\log {L_i})=L_i[40(\\log {L_i}-2)+m]$ (see Eq.", "(REF )).", "Thus, the total number of required elementary gates is $\\sum _{i=1}^k L_i f(\\log {L_i}) \\le k\\,\\mbox{max}_i\\lbrace L_i(f(\\log {L_i})\\rbrace $ , which might be substantially smaller than $|G|(41r^{\\prime }-80+m)$ gates required in the general case.", "Notice that the unitary basis change, $T$ , could also be implemented more efficiently (i.e.", "with less than ${\\cal O}(2^r)$ gates) in certain cases." ], [ "Cyclic groups", "Let us now consider the particular situation where the collective noise of the channel is associated with a general cyclic group, i.e.", "a group with only one generating element $h\\equiv g_1$ .", "Then, the group elements are given by $h^j$ where $1 \\le j \\le L$ with $L=2^{r^{\\prime }}$ .", "As before, we associate to each group element, $h^j$ , the number $j$ , which we write in binary notation as $j= j_{r^{\\prime }}\\ldots j_{1}$ , with $j=\\sum _{i=1}^{r^{\\prime }} 2^{i-1} j_i$ and $U_{h^j}=U^j=\\prod _{i=1}^{r^{\\prime }} U^{2^{i-1} j_i}$ .", "The last equation is the key to an efficient implementation of the operation $W$ (Eq.", "(REF )).", "Rather than implementing a product of $2^{r^{\\prime }}$ controlled unitary operations $W_{g_i}^m$ (Eq.", "(REF )), it suffices to perform $r^{\\prime }$ controlled unitary operations that use the $i^{\\rm th}$ qubit of the first register as the control, and perform the operation $(U^{2^i j_i})^{\\otimes m}$ on the message qubits if the bit value is one.", "That is, the first control qubit controls whether $\\mbox{$1 \\hspace{-2.84526pt} {\\bf l}$}$ or $U^{\\otimes m}$ is applied, the second whether $\\mbox{$1 \\hspace{-2.84526pt} {\\bf l}$}$ or $(U^2)^{\\otimes m}$ , the third whether $\\mbox{$1 \\hspace{-2.84526pt} {\\bf l}$}$ or $(U^4)^{\\otimes m}$ etc.", "In total, this leads to the implementation of the operation $(U^j)^{\\otimes m}$ if the control state is given by $\\left| j\\right> = \\left| j_{r^{\\prime }}\\ldots j_{2}j_1\\right>$ , corresponding exactly to the operation $W$ .", "As each of these gates consists of $m$ two-qubit gates, we have a total of $m \\log {L}=m r^{\\prime }$ gates.", "This leads to a significant reduction of the required resources for cyclic groups, i.e.", "for $\\mathbb {Z}_N$ , with $N = 2^{r^{\\prime }} \\in \\mathbb {N}$ , we only require $m \\log N$ gates.", "Notice that for cyclic groups the implementation of the unitary basis change to the token basis, i.e.", "the unitary operation, $T$ , in Eq.", "(REF ), can also be done much more efficiently than the upper bound of $\\mathcal {O}(d^r)$ operations.", "We consider $d=2$ , i.e. qubits.", "First, we notice that for the group $\\mathbb {Z}_N$ , the required number of qubits to store the token states is given by $r =|G|-1=N-1$ (see Eq.", "(REF )), while only $r^{\\prime }= \\log _2 N$ qubits are required to label the group elements.", "The implementation of $T$ then consists of a Schur transformation that maps the computational basis states to the Schur basis states (see Eq.", "(REF )), followed by the Fourier transformation (see Eq.", "(REF )).", "Notice that the order of the operations can be exchanged and, furthermore, the Fourier transformation just acts on standard basis states of $r^{\\prime } < r$ qubits.", "Both operations can be implemented efficiently; the Fourier transformation using ${\\cal O}(r^{\\prime } \\log r^{\\prime })$ gates, and the Schur transformation using ${\\cal O}(r\\mathrm {poly}(\\log r))$ resources (following the results of [28]).", "For the group $\\mathbb {Z}_N$ the change from the computational basis to the $\\left| \\lambda ,1\\right>$ basis (see Eq.", "(REF )) can in fact be implemented using only $r+r^{\\prime }$ elementary gates as we now show.", "As mentioned above, since all irreps of $\\mathbb {Z}_N$ are one-dimensional, we simply need to construct one state, $\\left| \\lambda ,\\beta \\right>$ , for each $\\lambda $ which is a computational basis state containing $\\lambda $ ones.", "In order to do so, we take $r^{\\prime }$ qubits (the first register) containing the computational basis states $\\left| j_{r^{\\prime }}, j_{r^{\\prime }-1},\\ldots , j_{1}\\right>$ , and an additional $r=2^{r^{\\prime }}-1$ qubits (the second register) that we partition into $r^{\\prime }$ groups $A_{m}$ .", "Each group, $A_m$ , in the second register corresponds to the $m^{th}$ qubit of the first register and contains $2^{m-1}$ qubits (corresponding to its value in binary representation).", "To construct the required states we proceed in two steps.", "Firstly, we perform $m$ CNOT operations with the $m^{\\rm th}$ qubit in the first register as the control and the $2^{m-1}$ qubits in the group $A_m$ of the second register as targets.", "The $m$ CNOT operations cause all qubits within the group, $A_m$ , in the second register to flip if the $m^{\\rm th}$ qubit in the first register is in the state $\\left| j_m\\right> = \\left| 1\\right>$ , and does nothing to the qubits in group $A_m$ if $\\left| j_m\\right> = \\left| 0\\right>$ .", "Secondly, we apply $r^{\\prime }$ CNOT operations with one of the qubits in $A_m$ of the second register as the control qubit, and the $m^{\\rm th}$ qubit of the first register as the target.", "This ensures that the first register is in the state $\\left| 0\\right>^{\\otimes r^{\\prime }}$ , while the state of the second register contains a total number of ones corresponding to the value of the bit-string $j_{r^{\\prime }} j_{r^{\\prime }-1} \\ldots j_{1}$ .", "For example, the elements of $\\mathbb {Z}_N$ can be represented in binary notation using $r^{\\prime }=\\log _2 N$ bits.", "Without loss of generality we assume that $N$ is an exact power of two If $N$ is not an exact power of two then $r^{\\prime }=\\lceil \\log _2 N\\rceil $ ..", "Thus, the first register consists of the $r^{\\prime }$ qubit computational basis states of the form $\\left| j_{r^{\\prime }},j_{r^{\\prime }-1},\\ldots ,j_2,j_1\\right>$ , where $j_i\\in (0,1)$ for $i\\in (1,\\ldots , r^{\\prime })$ .", "The second register consists of $r=2^{r^{\\prime }}-1=N-1$ qubits, initially in the state $\\left| 0\\right>$ , so that the initial state of both registers is given by $\\left(\\left| j_{r^{\\prime }}j_{r^{\\prime }-1}\\ldots j_2j_1\\right>\\right)\\otimes \\left(\\left| 0\\right>^{\\otimes \\frac{N}{2}}\\left| 0\\right>^{\\otimes \\frac{N}{4}}\\ldots \\left| 0\\right>^{\\otimes 2}\\left| 0\\right>\\right),$ where we have partitioned the $r$ qubits in the second register into $r^{\\prime }$ groups, $A_m$ , each containing $2^{m-1}$ qubits.", "After applying the first $m$ CNOT gates, with the second register as target, the state of the two registers is $\\left(\\left| j_{r^{\\prime }}j_{r^{\\prime }-1}\\ldots j_2j_1\\right>\\right)\\otimes \\left(\\left| j_{r^{\\prime }}\\right>^{\\otimes \\frac{N}{2}}\\left| j_{r^{\\prime }-1}\\right>^{\\otimes \\frac{N}{4}}\\ldots \\left| j_2\\right>^{\\otimes 2}\\left| j_1\\right>\\right),$ and after an additional $r^{\\prime }$ CNOT gates, with the first register as target, the final state of the two registers is $\\left(\\left| 0\\right>^{\\otimes r^{\\prime }}\\right)\\otimes \\left(\\left| j_{r^{\\prime }}\\right>^{\\otimes \\frac{N}{2}}\\left| j_{r^{\\prime }-1}\\right>^{\\otimes \\frac{N}{4}}\\ldots \\left| j_2\\right>^{\\otimes 2}\\left| j_1\\right>\\right).$ From our discussion above, it follows that the total number of CNOT operations required to implement the basis change is given by $r+r^{\\prime }$ .", "In addition, the Fourier transformation needs to be applied before this basis change, which involves ${\\cal O}(r^{\\prime } \\log r^{\\prime })$ gates.", "Thus, the overhead for implementing the operation $T$ (Eq.", "(REF )) for the case of finite cyclic groups is $r+r^{\\prime }+{\\cal O}(r^{\\prime } \\log r^{\\prime })=N-1+\\log _2 N+{\\cal O}(\\log _2 N\\log (\\log _2 N)) = {\\cal O}(N)$ , i.e.", "only linear with the number of group elements $N$ , despite the exponential increase of $r$ compared to $r^{\\prime }$ required for the token states corresponding to the group $\\mathbb {Z}_N$ .", "Together with the efficient implementation of the operation $W$ discussed above we find that for cyclic groups the encoding requires ${\\cal O}(m \\log N,N)$ elementary gates." ], [ "Logical depth", "It should be noted that the logical depth of our protocol is independent of the number of transmitted qubits $m$ .", "This is in contrast to the DFS-based communication scheme put forward in [28].", "It is easy to see that the logical depth of the circuit to implement $W$ is given by $\\vert G\\vert (41r^{\\prime }-80+1)$ , since all the control gates occurring in $V_{g_i}^m$ , acting on $m$ qubits originally prepared in $\\left| \\phi \\right>$ , can be implemented in parallel.", "Also, the unitary basis change, $T$ in Eq.", "(REF ), only requires at most ${\\cal O}(2^r)$ gates, where $r$ depends only on the representation, $U$ , of $G$ , leading to a logical depth that is independent of $m$ .", "The same is true for the more efficient implementations for abelian and cyclic groups discussed above.", "This allows for a very efficient implementation of our communication scheme." ], [ "Decoding", "In order to decode the information Bob simply measures in the basis $\\lbrace \\left| \\psi (g_i)\\right>\\rbrace $ and applies, depending on the outcome, one of the operations $U_{g_i^{-1}}^{\\otimes m}$ in order to retrieve the state $\\left| \\phi \\right>\\in {\\mathcal {H}}_2^{\\otimes m}$ .", "In practice, this can be done by implementing the inverse of the unitary operation $T$ , appearing in Eq.", "(REF ), that maps the product basis to the token state basis, i.e.", "$T^\\dagger $ , followed by $r^{\\prime }$ single qubit measurements in the computational basis.", "Notice that $T^\\dagger $ can also be implemented with at most ${\\cal O}(2^r)$ basic gates, independent of $m$ , or more efficiently for certain groups as shown above.", "The final correction operation, $U_{g_i^{-1}}^{\\otimes m}$ , is comprised of single-qubit operations that can be performed independently and in parallel.", "This makes possible a multi-receiver scenario as described in Sec. .", "Notice however, that in general the auxiliary systems need to be transmitted to a single party who also performs the measurement, and then communicates the classical measurement outcome to the different receivers." ], [ "Asymptotic transmission rate", "We now compare the asymptotic rate of transmission of quantum information of our protocol to that of a DFS code.", "First, let us compute the rate of transmission for the latter.", "As not all ${\\mathcal {N}}^{(\\lambda )}$ can be simultaneously utilized, the maximum number of logical qubits, $m$ , that can be transmitted using a number of physical qubits, $n$ , is $m=\\log _2(\\mathrm {max}_\\lambda \\,\\mathrm {dim}({\\mathcal {N}}^{(\\lambda )}))$ .", "Hence, the rate of transmission for a DFS code is given by $R_{DFS}=\\frac{\\log _2(\\mathrm {max}_\\lambda \\mathrm {dim}({\\mathcal {N}}^{(\\lambda )})}{n},$ and it is known that $\\lim _{n\\rightarrow \\infty }R_{DFS}\\rightarrow 1-\\frac{O(\\log n)}{n}$  [24].", "We now calculate the asymptotic rate of transmission, $R$ , using the protocol described in Sec. .", "As for any finite group $G$ , $m$ logical qudits can be perfectly transmitted (i.e.", "with unit fidelity of transmission) using $r+m$ physical qudits, the rate of transmission is given by $R=\\frac{m}{r+m}.$ However, as $G$ is a finite group the number of qudits, $r$ , required to construct our token states is finite and depends only on the representation, $U$ , of $G$ (see Appendix ).", "Hence, in the limit $m\\rightarrow \\infty $ , Eq.", "(REF ) tends to unity and our protocol achieves the optimal transmission rate.", "In conclusion, we have introduced a new protocol for transmitting quantum information through channels with collective noise.", "We have shown how to transmit $m$ logical qudits using $m+r$ physical qudits at a rate that is optimal in the asymptotic limit.", "The protocol makes use of ideas both from DFS and error correction.", "On the one hand, a specific state of system plus ancilla qubits is used that lies in a DFS of the joint system.", "On the other hand, the ancilla qubits are measured to determine the required correction operation similar to error correction.", "However, in our protocol no information about the channel and hence the actual error is revealed.", "In the case of channels associated with a finite group $G$ , the $m$ logical qudits can be transmitted with perfect fidelity, and can be efficiently encoded and decoded.", "We find that the number of elementary gates required for the encoding and decoding circuit scales as $\\mathcal {O}(m,|G|\\log |G|, d^r)$ , where $r$ is an integer that depends solely on the channel in question, and local measurements.", "For the case of finite cyclic groups, $\\mathbb {Z}_N$ , we discover that the encoding and decoding operations can be efficiently implemented with at most $m\\log N+ \\mathcal {O}(N)$ operations, where $N$ is the order of the cyclic group.", "Moreover, the logical depth of the encoding and decoding circuit for finite groups is independent of the number of logical qudits, $m$ .", "As the required number of elementary gates scales only linearly in the number of logical qudits, our protocol is more efficient than the best currently known DFS protocols [28], [29].", "Based on our findings, a practical implementation of our protocol seems feasible.", "Whereas the implementation of our protocol for finite abelian groups is very efficient, it is not obvious if an efficient implementation of our protocol is feasible for the case of non-abelian groups.", "This is due to the fact that, in general, a $\\mathcal {O}(d^r)$ overhead is required to implement the unitary operator in Eq.", "(REF ), which performs a basis change from the computational basis to the token-state basis.", "Even though we have explicitly shown that for any finite group it is always possible to find such a token state basis, the required overhead depends on the group in question.", "For a group with $\\vert G\\vert $ elements $r^{\\prime }=\\log _2\\vert G\\vert $ qubits are required to label the elements, however $r \\ge r^{\\prime }$ qubits are needed to construct a token basis with desired properties.", "While for the Pauli group we find that $r=r^{\\prime }$ , in the case of finite cyclic groups, we discover that $r= (\\vert G\\vert -1)/(d-1)$ , i.e.", "an exponential overhead.", "However, despite this exponential increase for cyclic groups, we have shown that the unitary operator in Eq.", "(REF ) can be efficiently implemented using $\\mathcal {O}(r)$ operations, i.e.", "with an overhead that scales only linear in the number of group elements.", "Whether such an exponential overhead of $r$ also occurs for other groups, and whether this can also be compensated by a more efficient implementation of the operation $T$ , is presently unknown.", "The implementation of our protocol to the case of collective noise channels associated with continuous groups remains an interesting open problem." ], [ "Acknowledgements", "We would like to thank Giulio Chiribella for his helpful comments and for pointing out an alternative proof of Lemma REF .", "Michael Skotiniotis would like to thank the Institute for Theoretical Physics at the University of Innsbruck for their gracious hospitality while this work was being conducted.", "The research was funded by the Austrian Science Fund (FWF): Y535-N16, SFB F40-FoQus, P20748-N16, P24273-N16 and the European Union (NAMEQUAM), NSERC Canada, EU-Canada exchange, and USARO." ], [ "Representation theory of $\\mathrm {SU}(2)$", "In this appendix we show how DFS arise in the presence of the most general type of collective noise that is associated with the group $\\mathrm {SU}(2)$ on $N$ , two-dimensional quantum systems In particular the most general collective noise is given by $\\lbrace p_g, \\,g\\in \\mathrm {SU}(2)\\rbrace $ with $0\\le p_g<1$ satisfying $\\int \\, p_g\\mathrm {d}g=1$ ..", "Since any $U\\in \\mathrm {SU}(2)$ can be written as $e^{-i\\frac{\\theta }{2} \\vec{n}\\cdot \\vec{\\sigma }}$ , where $ \\vec{n}$ denotes a three-dimensional vector and $\\vec{\\sigma }$ denotes the Pauli vector, $U^{\\otimes N}=e^{-i\\theta \\vec{n}\\cdot \\vec{J}}$ , with $\\vec{J}=\\frac{1}{2} \\sum _i \\vec{\\sigma }_i$ denoting the total angular momentum operator.", "Hence, $U^{\\otimes N}$ commutes with $J^2=\\vec{J}\\cdot \\vec{J}$ for any $U\\in \\mathrm {SU}(2)$ .", "Thus, any $U^{\\otimes N}$ is block-diagonal in the eigenbasis of $J^2$ .", "Denote by $\\lbrace \\left| J,M,\\beta \\right>\\equiv \\left| J,M\\right>\\otimes \\left| \\beta \\right>\\rbrace $ an orthonormal basis for the $2^N$ -dimensional Hilbert space, ${\\mathcal {H}}_2^{\\otimes N}$ , where $\\lbrace \\left| J,M\\right>\\rbrace _{M=-J}^J$ is the joint eigenbasis of $J^2$ and the $z$ -component of the total angular momentum operator, $J_z$ , and $\\beta \\in (1,\\ldots ,\\alpha ^{(J)})$ is a degeneracy (multiplicity) index, with $\\alpha ^{(J)}$ the number of orthonormal states of total angular momentum $J$ and $J_z=M$ .", "In this basis $U^{\\otimes N}$ may be conveniently written in block diagonal form as $U^{\\otimes N}=\\oplus _{J}\\,U^{(J)}\\otimes I_{\\alpha ^{(J)}}$ , where $J$ is the total angular momentum, $U^{(J)}$ are the irreps of $\\mathrm {SU}(2)$ , and $I_{\\alpha ^{(J)}}$ denotes the $\\alpha ^{(J)}$ -dimensional identity operator.", "Consequently, we can decompose the total Hilbert space, ${\\mathcal {H}}_2^{\\otimes N}$ , into orthogonal subspaces, ${\\cal H}^{(J)}=\\mbox{span}\\, \\lbrace \\left| J,M,\\beta \\right>\\rbrace $ , where $-J\\le M\\le J$ and $1\\le \\beta \\le \\alpha ^{(J)}$ , as ${\\cal H}=\\oplus _J H^{(J)}=\\oplus _J{\\mathcal {M}}^{(J)}\\otimes {\\mathcal {N}}^{(J)}$ .", "Here, ${\\mathcal {M}}^{(J)}$ is the space upon which the irrep $U^{(J)}$ of $\\mathrm {SU}(2)$ acts, and ${\\mathcal {N}}^{(J)}$ is the multiplicity space upon which the trivial representation, $I_{\\alpha ^{(J)}}$ , of $\\mathrm {SU}(2)$ acts.", "Moreover, as the dimension of the irrep $U^{(J)}$ coincides with $\\mathrm {dim}({\\mathcal {M}}^{(J)})$ given by $2J+1$ , all irreps, $U^{(J)}$ , are inequivalent.", "It follows that the only irrep of trivial dimension is $U^{(J=0)}$ which, using the rules for addition of angular momenta, occurs in the decomposition of $U^{\\otimes N}$ only if $N$ is even.", "Hence, for $N$ even, the sector ${\\mathcal {H}}^{(J=0)}$ is a decoherence-free subspace.", "For $J>0$ , the irreps, $U^{(J)}$ are of non-trivial dimension, and the sectors ${\\mathcal {H}}^{(J)}$ are no longer decoherence-free.", "For $J>0$ a decoherence-free subsystem exists if $\\alpha ^{(J)}>1$ .", "The smallest number of spin-1/2 systems that admits a non-trivial noiseless encoding for a general $U\\in \\mathrm {SU}(2)$ occurs for the case of three qubits.", "The representation, $U^{\\left(\\frac{1}{2}\\right)}_{(\\theta ,\\phi ,\\psi )}$ , where $(\\theta ,\\phi ,\\psi )$ are the Euler angles, acting on a two-dimensional Hilbert space is given by $U^{\\left(\\frac{1}{2}\\right)}_{(\\theta ,\\phi ,\\psi )}=\\left(\\begin{array}{cc}e^{-i\\frac{1}{2}(\\theta +\\psi )}\\cos (\\phi /2) & -e^{-i\\frac{1}{2}(\\theta -\\psi )}\\sin (\\phi /2)\\\\e^{i\\frac{1}{2}(\\theta -\\psi )}\\sin (\\phi /2) & e^{i\\frac{1}{2}(\\theta +\\psi )}\\cos (\\phi /2)\\end{array}\\right).$ As mentioned above, the representation $U^{\\left(\\frac{1}{2}\\right)\\otimes 3}_{(\\theta ,\\phi ,\\psi )}$ can be decomposed into orthogonal sectors, labeled by the total angular momentum $J$ , as $U^{\\left(\\frac{1}{2}\\right)\\otimes 3}_{(\\theta ,\\phi ,\\psi )}=U^{\\left(\\frac{3}{2}\\right)}_{(\\theta ,\\phi ,\\psi )}\\bigoplus U^{\\left(\\frac{1}{2}\\right)}_{(\\theta ,\\phi ,\\psi )}\\otimes I_2$ , where $I_2$ is the two-dimensional identity operator and the representation $U^{\\left(\\frac{3}{2}\\right)}_{(\\theta ,\\phi ,\\psi )}$ is given by $\\left< \\frac{3}{2},m^{\\prime }\\right|U^{\\left(\\frac{3}{2}\\right)}_{(\\theta ,\\phi ,\\psi )}\\left| \\frac{3}{2},m\\right>=e^{-i m^{\\prime }\\theta }\\,d^{\\left(\\frac{3}{2}\\right)}_{m^{\\prime },m}(\\phi )\\,e^{-i m\\psi }.$ The matrix $d^{\\left(\\frac{3}{2}\\right)}(\\phi )$ in Eq.", "(REF ) is the Wigner small-$d$ matrix given by $\\nonumber d^{\\left(\\frac{3}{2}\\right)}_{3/2,3/2}(\\phi )&=\\frac{1+\\cos \\phi }{2}\\cos \\frac{\\phi }{2} \\\\ \\nonumber d^{\\left(\\frac{3}{2}\\right)}_{3/2,1/2}(\\phi )&=-\\sqrt{3}\\frac{1+\\cos \\phi }{2}\\sin \\frac{\\phi }{2}\\\\ \\nonumber d^{\\left(\\frac{3}{2}\\right)}_{3/2,-1/2}(\\phi )&=\\sqrt{3}\\frac{1-\\cos \\phi }{2}\\cos \\frac{\\phi }{2}\\\\ \\nonumber d^{\\left(\\frac{3}{2}\\right)}_{3/2,-3/2}(\\phi )&=-\\frac{1-\\cos \\phi }{2}\\sin \\frac{\\phi }{2}\\\\ \\nonumber d^{\\left(\\frac{3}{2}\\right)}_{1/2,1/2}(\\phi )&=\\frac{3\\cos \\phi -1}{2}\\cos \\frac{\\phi }{2}\\\\d^{\\left(\\frac{3}{2}\\right)}_{1/2,-1/2}(\\phi )&=-\\frac{3\\cos \\phi +1}{2}\\sin \\frac{\\phi }{2},$ where the matrix elements, $d^{\\left(\\frac{3}{2}\\right)}_{m^{\\prime },m}(\\phi )$ , satisfy the relation $d^{\\left(\\frac{3}{2}\\right)}_{m,m^{\\prime }}(\\phi )=(-1)^{m-m^{\\prime }}d^{\\left(\\frac{3}{2}\\right)}_{m^{\\prime },m}(\\phi )=d^{\\left(\\frac{3}{2}\\right)}_{-m^{\\prime },m}(\\phi )$ .", "Consequently, the total Hilbert space, ${\\mathcal {H}}_2^{\\otimes 3}$ , decomposes into orthogonal sectors, labelled by the total angular momentum quantum number $J$ , with orthonormal basis vectors $\\nonumber \\left| J=\\frac{3}{2},M=\\frac{3}{2}\\right>&=&\\left| 000\\right>\\\\ \\nonumber \\left| J=\\frac{3}{2},M=\\frac{1}{2}\\right>&=&\\frac{1}{\\sqrt{3}}\\left(\\left| 001\\right>+\\left| 010\\right>+\\left| 100\\right>\\right)\\\\ \\nonumber \\left| J=\\frac{3}{2},M=-\\frac{1}{2}\\right>&=&\\frac{1}{\\sqrt{3}}\\left(\\left| 110\\right>+\\left| 101\\right>+\\left| 011\\right>\\right)\\\\ \\nonumber \\left| J=\\frac{3}{2},M=-\\frac{3}{2}\\right>&=&\\left| 111\\right>\\\\ \\nonumber \\left| J=\\frac{1}{2},M=\\frac{1}{2},\\beta =0\\right>&=&\\frac{1}{\\sqrt{2}}\\left(\\left| 100\\right>-\\left| 010\\right>\\right)\\\\ \\nonumber \\left| J=\\frac{1}{2},M=-\\frac{1}{2},\\beta =0\\right>&=&\\frac{1}{\\sqrt{2}}\\left(\\left| 011\\right>-\\left| 101\\right>\\right)\\\\ \\nonumber \\left| J=\\frac{1}{2},M=\\frac{1}{2},\\beta =1\\right>&=&\\sqrt{\\frac{2}{3}}\\left| 001\\right>-\\frac{\\left| 010\\right>+\\left| 100\\right>}{\\sqrt{6}}\\\\ \\nonumber \\left| J=\\frac{1}{2},M=-\\frac{1}{2},\\beta =1\\right>&=&\\sqrt{\\frac{2}{3}}\\left| 110\\right>-\\frac{\\left| 101\\right>+\\left| 011\\right>}{\\sqrt{6}},\\\\$ where the degeneracy index, $\\beta $ , keeps track of whether $J=1/2$ arose due to the coupling of the first two qubits in a spin-1 or spin-0 state of total angular momentum.", "Thus, the sector ${\\mathcal {H}}^{\\left(J=\\frac{1}{2}\\right)}$ contains a two-dimensional DFS.", "Defining the logical basis $\\left| 0_L\\right>\\equiv c_1\\left| J=\\frac{1}{2},M=\\frac{1}{2},\\beta =0\\right>+c_2\\left| J=\\frac{1}{2},M=\\frac{-1}{2},\\beta =0\\right>,\\\\\\left| 1_L\\right>\\equiv d_1\\left| J=\\frac{1}{2},M=\\frac{1}{2},\\beta =1\\right>+d_2\\left| J=\\frac{1}{2},M=\\frac{-1}{2},\\beta =1\\right>,$ where $|c_1|^2+|c_2|^2=1$ and $|d_1|^2+|d_2|^2=1$ , one can transmit one logical qubit noiselessly through the channel." ], [ " Proof of Theorem ", "In this appendix we provide a detailed proof of Theorem REF in Sec.", "regarding the construction of a set of token states, $S^{(U)}_{(r,\\left| \\psi \\right>)}$ .", "First, we show that for a representation, $U$ , of a group, $G$ , that is isomorphic to $G$ , there exists an integer, $r$ , such that $U^{\\otimes r}$ contains the regular representation, ${\\mathcal {R}}$ , of $G$ as a sub-representation.", "Next we show that there exists an $r$ and a state, $\\left| \\psi \\right>$ , such that the set of states $S^{(U)}_{(r,\\left| \\psi \\right>)}$ satisfies conditions (1) and (2) in Sec.", "if and only if $U^{\\otimes r}$ contains the regular representation, ${\\mathcal {R}}$ , of $G$ as a sub-representation.", "Finally, we demonstrate that the state $\\left| \\psi \\right>\\in {\\mathcal {H}}_d^{\\otimes r}$ in the definition of $S^{(U)}_{(r,\\left| \\psi \\right>)}$ can be chosen to be of the form given by Eq.", "(REF ).", "Lemma 1 Let $U$ be a representation of $G$ that is isomorphic to $G$ .", "Then there exists a finite integer $r$ , such that $U^{\\otimes r}$ contains the regular representation as a sub-representation.", "To prove Lemma REF we will make use of the following theorem, whose proof can be found in [43].", "An alternative proof, as well as several bounds on the integer, $r$ , can also be found in [44] We became aware of the fact that Lemma REF had already been proved by [44] after this work was completed.. Theorem 2 Let $U$ be a representation of $G$ that is isomorphic.", "Then, there exists an integer $n$ , such that $U^{\\otimes n}$ contains every irrep of $G$ at least once.", "(Lemma REF ).", "Write $U$ as the sum of inequivalent irreps, $U^{(\\lambda )}$ , $U=\\sum _\\lambda \\alpha ^{(\\lambda )} U^{(\\lambda )},$ where $\\alpha ^{(\\lambda )}$ is the multiplicity of irrep $U^{(\\lambda )}$ .", "The character of the representation $U$ , on the conjugacy class $[g_i]$ , $\\chi _{[g_i]}$ , is given by $\\chi _{[g_i]}=\\sum _\\lambda \\alpha ^{(\\lambda )} \\chi ^{(\\lambda )}_{[g_i]}.$ Since $G$ is a finite group, and $U$ is isomorphic to $G$ , it follows from Theorem REF that there exists an integer, $n$ , such that $U^{\\otimes n}$ contains every irrep of $G$ at least once.", "Defining $\\Gamma =\\bigoplus _{\\lambda =1}^s U^{(\\lambda )}$ , where $s$ denotes as before the number of inequivalent irreps, and using Theorem REF it follows that $U^{\\otimes n}=\\Gamma \\bigoplus V,$ where $V$ is a representation of $G$ .", "Now consider the decomposition of $U^{\\otimes nm}$ where $m$ is some integer.", "This may be written as $U^{\\otimes nm}=\\left(\\Gamma \\bigoplus V\\right)^{\\otimes m}.$ If two matrices $A$ and $B$ are block diagonal, then $A\\otimes B$ is also block diagonal, and if $A$ and $B$ are representations of $G$ , then so is $A\\otimes B$ .", "Moreover, $A\\otimes B$ is reducible, so that each block of $A\\otimes B$ can be reduced further into sub-blocks.", "Consider only the block $\\Gamma ^{\\otimes m}$ from Eq.", "(REF ).", "This block can be written as $\\nonumber \\Gamma ^{\\otimes m}&=\\left(\\bigoplus _{\\lambda =1}^sU^{(\\lambda )}\\right)\\otimes \\Gamma ^{\\otimes m-1}\\\\&=\\bigoplus _{\\lambda =1}^sU^{(\\lambda )}\\otimes \\left(\\bigoplus _{\\lambda ^{\\prime }=1}^s U^{(\\lambda ^{\\prime })}\\right)^{\\otimes m-1}.$ Each block, labeled by $\\lambda $ , in Eq.", "(REF ) consists of sub-blocks given by $U^{(\\lambda )}\\bigotimes _{i=1}^{m-1}U^{(\\nu _i)}$ , where $\\nu _i$ can take any value from the set of irrep labels $\\lbrace 1,\\ldots ,s\\rbrace $ .", "One such sub-block is the one where all $\\nu _i=\\lambda $ , that is $U^{(\\lambda )\\otimes m}$ .", "If $U^{(\\lambda )}$ is isomorphic to $G$ , then by Theorem REF there exists an integer $m$ such that $U^{(\\lambda )\\otimes m}=\\Gamma \\bigoplus V^{\\prime }$ , where $V^{\\prime }$ is a representation of $G$  For irreps that are not isomorphic to $G$ one can produce a similar argument.", "Let $U^{(\\lambda )}$ , be one such irrep.", "Then, $U^{(\\lambda )}\\otimes \\Gamma =\\bigoplus _\\lambda a^{(\\lambda )}U^{(\\lambda )}$ , and $U^{(\\lambda )}\\otimes \\Gamma ^{\\otimes m-1}= \\left(\\bigoplus _\\lambda a^{(\\lambda )}U^{(\\lambda )}\\right)\\otimes \\Gamma ^{\\otimes m-2}$ .", "Using a similar argument as above it follows that for suitable integer $m$ the block $U^{(\\lambda )}\\otimes \\Gamma ^{\\otimes m}$ will contain $\\Gamma $ as a sub-representation..", "Hence $\\Gamma ^{\\otimes m}= \\kappa \\Gamma \\bigoplus T,$ where $\\kappa >1$ is an integer, and $T$ is a representation of $G$ .", "This process can be carried as far as we like increasing the multiplicity of every irrep as much as we like.", "Note that this is not the most efficient way to increase the multiplicities of every irrep, as we focused only on the sub-block, $U^{(\\lambda )\\otimes m}$ , in Eq.", "(REF ) and neglected all other irrep products.", "As the dimensions of all irreps are finite, there exists an $r_\\lambda $ for each $\\lambda $ such that the irrep $U^{(\\lambda )}$ occurs at least $\\mathrm {dim}(U^{(\\lambda )})$ times in $U^{\\otimes r_\\lambda }$ .", "Choosing $r=\\mathrm {max}\\lbrace r_\\lambda \\rbrace $ ensures that $U^{\\otimes r}$ contains the regular representation, ${\\mathcal {R}}$ , as a sub-representation.", "In Appendix  we illustrate how one can compute $r$ for some examples.", "We are now ready to prove that there exists an $r$ and a state $\\left| \\psi \\right>\\in {\\mathcal {H}}^{\\otimes r}$ such that the set $S^{(U)}_{(r,\\left| \\psi \\right>)}$ satisfies conditions (1) and (2) in Sec.", ", if and only if the representation $U^{\\otimes r}$ contains the regular representation, ${\\mathcal {R}}$ of $G$ , as a sub-representation (Theorem 1).", "(Theorem REF ).", "To prove the backward implication assume that $U^{\\otimes r}={\\mathcal {R}}\\bigoplus _\\lambda \\alpha _\\lambda U^{(\\lambda )}$ .", "The total Hilbert space, ${\\mathcal {H}}^{\\otimes r}$ , decomposes as ${\\mathcal {H}}^{\\otimes r}={\\mathcal {H}}_{{\\mathcal {R}}}\\bigoplus _\\lambda {\\mathcal {H}}^{(\\lambda )}$ , where ${\\mathcal {H}}_{{\\mathcal {R}}}$ is the Hilbert space on which the regular representation is acting.", "Denote by $\\lbrace \\left| \\psi (g_i)\\right>;\\, i=0,\\ldots ,\\vert G\\vert -1\\rbrace $ the first $\\vert G\\vert $ standard basis vectors in ${\\mathcal {H}}^{\\otimes r}$ , i.e.", "the computational basis of ${\\mathcal {H}}_{{\\mathcal {R}}}$ embedded in the $2^r$ -dimensional Hilbert space ${\\mathcal {H}}^{\\otimes r}$ .", "From the definition of the regular representation it follows that $U_{g_k}^{\\otimes r}\\left| \\psi (g_i)\\right>=\\left| \\psi (g_k\\cdot g_i)\\right>=\\left| \\psi (g_l)\\right>,$ where $g_k\\cdot g_i$ is the group product, and $l\\in (0,\\ldots ,\\vert G\\vert -1)$ is such that $g_k\\cdot g_i=g_l$ .", "Thus, the set of states $\\lbrace \\left| \\psi (g_i)\\right>;\\, i=0,\\ldots , \\vert G\\vert -1\\rbrace $ satisfies properties (1) and (2).", "To prove the forward implication assume that the set of states, $S^{(U)}_{(r,\\left| \\psi \\right>)}$ , where $\\left| \\psi \\right>\\in {\\mathcal {H}}^{\\otimes r}$ , satisfies properties (1) and (2).", "Define $\\nonumber P_G&\\equiv \\frac{1}{|G|}\\sum _{g_i\\in G}U_{g_i}^{\\otimes r}|\\psi \\rangle \\langle \\psi |U_{g_i}^{\\otimes r\\,\\dagger }\\\\&=\\frac{1}{\\vert G\\vert }\\sum _{g_i\\in G}|\\psi (g_i)\\rangle \\langle \\psi (g_i)|.$ Then by property (2) $\\nonumber U_{g_k}^{\\otimes r}P_GU_{g_k}^{\\otimes r\\,\\dagger }&=\\frac{1}{|G|}\\sum _{g_i\\in G}U_{g_k}^{\\otimes r}|\\psi (g_i)\\rangle \\langle \\psi (g_i)|U_{g_k}^{\\otimes r\\,\\dagger }\\\\&=\\frac{1}{|G|}\\sum _{g_i\\in G}|\\psi (g_k\\cdot g_i)\\rangle \\langle \\psi (g_k\\cdot g_i)|.$ Denoting $g_k\\cdot g_i=g_l\\in G$ , Eq.", "(REF ) can be written as $U_{g_k}^{\\otimes r}P_GU_{g_k}^{\\otimes r\\,\\dagger }=\\frac{1}{|G|}\\sum _{g_l\\in G}|\\psi (g_l)\\rangle \\langle \\psi (g_l)|=P_G.$ Since $p_g$ in Eq.", "(REF ) is the Haar measure, it follows form Schur's first lemma that $P_G$ is a multiple of the $G$ -dimensional identity.", "Using Eq.", "(REF ), Eq.", "(REF ) may also be written as $P_G=\\sum _\\lambda \\left({\\mathcal {D}}_{{\\mathcal {M}}^{(\\lambda )}}\\otimes {\\mathcal {I}}_{{\\mathcal {N}}^{(\\lambda )}}\\right)\\circ {\\mathcal {P}}^{(\\lambda )}[|\\psi \\rangle \\langle \\psi |],$ where $\\lambda $ labels the irreps present in the decomposition of $U^{\\otimes r}$ , ${\\mathcal {D}}$ is the completely depolarizing map, ${\\mathcal {D}}(A)=\\frac{\\mbox{tr}(A)}{\\mathrm {dim}({\\mathcal {H}})}\\mbox{$1 \\hspace{-2.84526pt} {\\bf l}$}, \\,\\forall A\\in {\\mathcal {B}}({\\mathcal {H}})$ , ${\\mathcal {I}}$ is the identity map, and ${\\mathcal {P}}^{(\\lambda )}(A)=\\Pi _\\lambda A\\Pi _\\lambda $ , where $\\Pi _\\lambda $ is the projector onto the space ${\\mathcal {H}}^{(\\lambda )}$ .", "Write $\\left| \\psi \\right>=\\sum _\\lambda c_\\lambda \\left| \\psi ^{(\\lambda )}\\right>$ , where $c_\\lambda \\in \\hbox{$I$\\hspace{-6.99997pt}$C$}$ satisfy $\\sum _\\lambda \\vert c_\\lambda \\vert ^2=1$ , and $\\left| \\psi ^{(\\lambda )}\\right>=\\Pi ^{(\\lambda )}\\left| \\psi \\right>$ .", "Using the Schmidt decomposition, and defining $\\left\\lbrace \\left| \\xi ^{(\\lambda )}_n\\right>\\right\\rbrace $ and $\\left\\lbrace \\left| \\zeta ^{(\\lambda )}_n\\right>\\right\\rbrace $ as orthonormal basis for ${\\mathcal {M}}^{(\\lambda )}$ and ${\\mathcal {N}}^{(\\lambda )}$ respectively, we may write $\\left| \\psi ^{\\lambda }\\right>=\\sum _{n=1}^{\\tilde{d}_\\lambda }\\mu ^{(\\lambda )}_n\\left| \\xi ^{(\\lambda )}_n\\right>\\left| \\zeta ^{(\\lambda )}_n\\right>,$ where $\\tilde{d}_\\lambda \\le \\mathrm {min}\\lbrace d_\\lambda =\\mathrm {dim}({\\mathcal {M}}^{(\\lambda )}),\\,\\mathrm {dim}({\\mathcal {N}}^{(\\lambda )})\\rbrace $ and $0\\ne \\mu ^{(\\lambda )}_n\\in \\mathbb {R} $ are the Schmidt coefficients.", "Substituting Eq.", "(REF ) into Eq.", "(REF ) gives $\\nonumber P_G&=\\sum _\\lambda |c_\\lambda |^2\\sum _{n,n^{\\prime }=1}^{\\tilde{d}_\\lambda }\\mu ^{(\\lambda )}_n\\mu ^{(\\lambda )*}_{n^{\\prime }}\\left({\\mathcal {D}}_{{\\mathcal {M}}^{(\\lambda )}}\\left[\\left| \\xi ^{(\\lambda )}_n\\right>\\left< \\xi ^{(\\lambda )}_{n^{\\prime }}\\right|\\right]\\right.\\\\ \\nonumber &\\left.\\otimes \\, {\\mathcal {I}}\\left[\\left| \\zeta ^{(\\lambda )}_n\\right>\\left< \\zeta ^{(\\lambda )}_{n^{\\prime }}\\right|\\right]\\right)\\\\&=\\sum _\\lambda |c_\\lambda |^2\\sum _{n=1}^{\\tilde{d}_\\lambda }|\\mu ^{(\\lambda )}_n|^2\\frac{\\mbox{$1 \\hspace{-2.84526pt} {\\bf l}$}_{d_\\lambda }}{d_\\lambda }\\otimes |\\zeta ^{(\\lambda )}_n\\rangle \\langle \\zeta ^{(\\lambda )}_n|.$ Computing the rank on both sides of Eq.", "(REF ), and using the notation $\\rho _{{\\mathcal {N}}^{(\\lambda )}}=\\mathrm {tr}_{{\\mathcal {M}}^{(\\lambda )}}[|\\psi ^{(\\lambda )}\\rangle \\langle \\psi ^{(\\lambda )}|]$ , one obtains $|G|=\\sum _\\lambda d_\\lambda \\, \\mathrm {rk}\\left(\\rho _{{\\mathcal {N}}^{(\\lambda )}}\\right).$ As $|G|=\\sum _{\\lambda } d_\\lambda ^2$ (see Eq(REF )), and $\\mathrm {rk}\\left(\\rho _{{\\mathcal {N}}^{(\\lambda )}}\\right)=\\tilde{d}_\\lambda \\le d_\\lambda $ , $\\forall \\lambda $ , it follows that $\\mathrm {rk}\\left(\\rho _{{\\mathcal {N}}^{(\\lambda )}}\\right)=\\tilde{d}_\\lambda =d_\\lambda $ .", "Hence $\\tilde{d}_\\lambda = \\mathrm {min}\\lbrace d_\\lambda =\\mathrm {dim}({\\mathcal {M}}^{(\\lambda )}),\\,\\mathrm {dim}({\\mathcal {N}}^{(\\lambda )})\\rbrace $ and therefore the multiplicity of each irrep, ${\\mathrm {dim}}({\\mathcal {N}}^{(\\lambda )})$ , occurs a number of times greater than or equal to its dimension, $d_\\lambda $ .", "Thus, $U^{\\otimes r}$ contains the regular representation, ${\\mathcal {R}}$ , of $G$ as a sub-representation.", "This completes the proof.", "In order to construct the set of token states, $S^{(U)}_{(r,\\left| \\psi \\right>)}$ , for an isomorphic representation, $U$ , and an $r$ chosen such that $U^{\\otimes r}$ contains the regular representation, one can choose the state $\\left| \\psi \\right>\\in {\\mathcal {H}}_d^{\\otimes r}$ to be of the form in Eq.", "(REF ) as we now show.", "Since $\\mathrm {tr}(P_G)=1$ we have $P_G=\\frac{1}{|G|} \\mbox{$1 \\hspace{-2.84526pt} {\\bf l}$}$ , where $\\mbox{$1 \\hspace{-2.84526pt} {\\bf l}$}$ is the $\\vert G\\vert $ -dimensional identity operator.", "Writing $\\left| \\psi \\right>=\\sum _\\lambda c_\\lambda \\left| \\psi ^{(\\lambda )}\\right>$ , with $\\left| \\psi ^{(\\lambda )}\\right>$ given by the Schmidt decomposition, Eq.", "(REF ), and using Eq.", "(REF ) we have $\\nonumber P_G=\\frac{1}{|G|}\\sum _\\lambda I_{{\\mathcal {M}}^{(\\lambda )}}\\otimes I_{{\\mathcal {N}}^{(\\lambda )}}&=\\sum _\\lambda |c_\\lambda |^2\\left({\\mathcal {D}}_{{\\mathcal {M}}^{(\\lambda )}}\\otimes {\\mathcal {I}}_{{\\mathcal {N}}^{(\\lambda )}}\\right)\\\\&\\left[|\\psi ^{(\\lambda )}\\rangle \\langle \\psi ^{(\\lambda )}|\\right].$ As both ${\\mathcal {D}}$ and ${\\mathcal {I}}$ are trace-preserving quantum operations, looking at a single sector, $\\lambda $ , and computing the trace on both ${\\mathcal {M}}^{(\\lambda )}$ and ${\\mathcal {N}}^{(\\lambda )}$ one obtains $\\frac{1}{|G|}d_\\lambda ^2=|c_\\lambda |^2.$ Inserting this value for $|c_\\lambda |^2$ in the expression given in Eq.", "(REF ) for $P_G$ leads to $P_G=\\frac{1}{|G|}\\sum _\\lambda I_{{\\mathcal {M}}^{(\\lambda )}}\\otimes \\sum _{n=1}^{d_\\lambda } |\\mu _n^{(\\lambda )}|^2 |\\zeta ^{(\\lambda )_n}\\rangle \\langle \\zeta ^{(\\lambda )_n}|.$ In addition, $\\nonumber P_G^2&=\\sum _{\\lambda ,\\lambda ^{\\prime }}\\vert c_\\lambda \\vert ^2\\vert c_{\\lambda ^{\\prime }}\\vert ^2\\sum _{n,m}\\vert \\mu ^{(\\lambda )}_n\\vert ^2\\vert \\mu ^{(\\lambda ^{\\prime })}_m\\vert ^2\\left\\langle \\xi ^{(\\lambda )}_n|\\xi ^{(\\lambda ^{\\prime })}_m\\right\\rangle \\\\ \\nonumber &\\times \\left\\langle \\zeta ^{(\\lambda )}_n|\\zeta ^{\\lambda ^{\\prime })}_m\\right\\rangle \\frac{\\left| \\xi ^{(\\lambda )}_n\\right>\\left< \\xi ^{(\\lambda ^{\\prime })}_m\\right|}{d_\\lambda d_\\lambda ^{\\prime }}\\otimes \\left| \\zeta ^{(\\lambda )}_n\\right>\\left< \\zeta ^{\\lambda ^{\\prime })}_m\\right|\\\\&=\\sum _{\\lambda }\\vert c_\\lambda \\vert ^4\\sum _n\\vert \\mu ^{(\\lambda )}_n\\vert ^4\\frac{|\\xi ^{(\\lambda )}_n\\rangle \\langle \\xi ^{(\\lambda )}_n|}{d_\\lambda ^2}\\otimes |\\zeta ^{\\lambda )}_n\\rangle \\langle \\zeta ^{\\lambda )}_n|,$ where we have used the fact that $\\left\\langle \\xi ^{(\\lambda )}_n|\\xi ^{(\\lambda ^{\\prime })}_m\\right\\rangle =\\delta _{\\lambda ,\\lambda ^{\\prime }}\\delta _{n,m}$ .", "As $P_G^2=\\frac{1}{|G|}P_G$ , equating the terms of Eq.", "(REF ) and Eq.", "(REF ) yields $\\nonumber &\\frac{\\vert c_\\lambda \\vert ^4\\vert \\mu ^{(\\lambda )}_n\\vert ^4}{d_\\lambda ^2}=\\frac{\\vert c_\\lambda \\vert ^2\\vert \\mu ^{(\\lambda )}_n\\vert ^2}{|G|d_\\lambda }\\\\&\\frac{\\vert c_\\lambda \\vert ^2\\vert \\mu ^{(\\lambda )}_n\\vert ^2}{d_\\lambda }\\left(\\frac{\\vert c_\\lambda \\vert ^2\\vert \\mu ^{(\\lambda )}_n\\vert ^2}{d_\\lambda }-\\frac{1}{|G|}\\right)=0.$ Hence, using Eq.", "(REF ), $\\mu ^{(\\lambda )}_n=d_\\lambda ^{-1/2}$ $\\forall \\lambda ,n$ and thus, the state $\\left| \\psi \\right>\\in {\\mathcal {H}}_d^{\\otimes r}$ can be chosen as $\\left| \\psi \\right>=\\sum _\\lambda \\sqrt{\\frac{d_\\lambda }{|G|}}\\sum _n^{d_\\lambda }\\left| \\xi ^{(\\lambda )}_n\\right>\\left| \\zeta ^{(\\lambda )}_n\\right>.$" ], [ " Calculating the number of auxiliary systems $r$", "In this appendix we show how to calculate the number of auxiliary systems, $r$ , required in our protocol such that $U^{\\otimes r}$ , where $U$ is a isomorphic representation of a group $G$ , contains the regular representation, ${\\mathcal {R}}$ , of $G$ .", "In particular, we calculate $r$ in the case where the collective noise of the channel is associated with the cyclic group $\\mathbb {Z}_3$ , and the symmetric group on three elements, $S_3$ .", "As a first step we review how the multiplicity of the irreps can be computed (see Sec.", "II).", "Let $U$ be a isomorphic representation of a finite group, $G$ .", "Then by Lemma REF , there exists an integer $n$ , such that $U^{\\otimes n}$ contains every irrep of $G$ at least once.", "That is $U^{\\otimes n}=\\sum _\\lambda \\gamma ^{(\\lambda )}_n U^{(\\lambda )},$ where $\\gamma ^{(\\lambda )}_n\\ge 1$ is the number of times the irrep $U^{(\\lambda )}$ appears.", "Since $\\mathrm {tr}(U_{g_i}^{\\otimes n})=(\\mathrm {tr}(U_{g_i}))^n$ , it follows that $(\\chi _{[g_i]})^n=\\sum _\\lambda \\gamma ^{(\\lambda )}_n \\chi ^{\\lambda }_{[g_i]}.$ As explained in Sec.", "II the multiplicities, $\\gamma ^{(\\lambda )}_n$ , can be easily computed using Eq.", "(REF ).", "We obtain $\\sum _{i=1}^s|[g_i]|\\chi ^{(\\lambda ^\\prime )*}_{[g_i]}(\\chi _{[g_i]})^n=|G|\\gamma ^{(\\lambda ^\\prime )}_n.$ Thus, given the compound character of the representation $U$ , as well as the compound characters of all the irreducible representations of $G$ and $|[g_i]|$ , one can compute the multiplicities $\\gamma ^{(\\lambda )}_n$ for any integer $n$ .", "To ensure that the regular representation is contained in $U^{\\otimes n}$ , $\\gamma ^{(\\lambda )}_n\\ge d_\\lambda $ must hold.", "We are going to show next how large $n$ must be chosen to ensure the validity of the condition above for some examples." ], [ "Example 1: $\\mathbb {Z}_3$", "For our first example we consider a channel whose collective noise is given by the two dimensional representation of $\\mathbb {Z}_3$ of Eq.", "(REF ).", "Note that this representation is isomorphic to $\\mathbb {Z}_3$ .", "Even though we have already shown in Sec.", "III that if $U$ is a $d$ –dimensional representation of $\\mathbb {Z}_N$ it is sufficient to chose $r=\\lceil \\frac{N-1}{d-1}\\rceil $ , we explicitly compute $r$ here for the case $d=2,N=3$ .", "The character table for $\\mathbb {Z}_3$ is given in Table REF , and the regular representation, ${\\mathcal {R}}$ , of $\\mathbb {Z}_3$ is given in Eq.", "(REF ).", "The compound character of the representation given in Eq.", "(REF ) can be easily computed to be $\\chi =\\left(\\begin{array}{c} 2\\\\-\\omega ^2\\\\-\\omega \\end{array}\\right),$ where $\\omega =e^{\\frac{i2\\pi }{3}}$ .", "Using Eq.", "(REF ), Table REF , and the fact that $|[g_i]|=1,\\, \\forall i\\in (0,1,2)$ , Eq.", "(REF ) reads, for $n=1$ , $\\nonumber \\gamma ^{(0)}_1&=\\frac{1}{3}\\left(2-\\omega ^2-\\omega \\right)=1\\\\ \\nonumber \\gamma ^{(1)}_1&=\\frac{1}{3}\\left(2+\\omega ^2(-\\omega ^2)+\\omega (-\\omega )\\right)=1\\\\\\gamma ^{(2)}_1&=\\frac{1}{3}\\left(2+\\omega (-\\omega ^2)+\\omega ^2(-\\omega )\\right)=0.$ Thus, the decomposition of $U$ , given by Eq.", "(REF ), into irreps is $U=U^{(0)}\\oplus U^{(1)}$ .", "Using Eq.", "(REF ) with $n=2$ we obtain $\\nonumber \\gamma ^{(0)}_2&=\\frac{1}{3}\\left(4+(-\\omega ^2)^2+(-\\omega )^2\\right)=1\\\\ \\nonumber \\gamma ^{(1)}_2&=\\frac{1}{3}\\left(4+\\omega ^2(-\\omega ^2)^2+\\omega (-\\omega )^2\\right)=2\\\\\\gamma ^{(2)}_2&=\\frac{1}{3}\\left(4+\\omega (-\\omega ^2)^2+\\omega ^2(-\\omega )^2\\right)=1.$ Thus $U^{\\otimes 2}=U^{(0)}\\oplus 2U^{(1)}\\oplus U^{(2)}$ .", "As all the irreps are one-dimensional, and they all appear at least once, $U^{\\otimes 2}$ contains the regular representation." ], [ "Example 2: $S_3$", "As a second example we consider a channel whose collective noise is described by the group $S_3$ , the symmetric group on three elements.", "Using cycle notation, the elements of $S_3$ are $\\left\\lbrace (1)(2)(3), (123), (132), (12)(3), (13)(2), (23)(1)\\right\\rbrace ,$ where $(ab)(c)$ denotes the permutation where symbol $c$ remains fixed and symbols $a, b$ are interchanged ($a\\rightarrow b\\rightarrow a,\\, c$ ), and $(abc)$ denotes the cyclic permutation where $a$ moves to the place of $b$ , $b$ moves to the place of $c$ , and $c$ moves to the place of $a$ ($a\\rightarrow b\\rightarrow c\\rightarrow a$ ).", "It is known that there is a one-to-one correspondence between the conjugacy classes of the symmetric group and the number of different cycle structures of the group [30].", "Hence, $S_3$ has three conjugacy classes; $[(1)(2)(3)]$ which contains only one element, $[(123)]$ which contains two elements, and $[(12)(3)]$ containing three elements.", "Furthermore, we note that $S_3$ can be generated by two elements such as $(123)$ and $(12)(3)$ .", "As $S_3$ has three conjugacy classes, there exist three inquivalent irreps of $S_3$ , two one-dimensional irreps, given by $U^{(0)}_{g_i}=1$ for all $g_i\\in S_3$ and $U^{(1)}_{g_i}=\\left\\lbrace \\begin{array}{l l}1 & \\quad \\mbox{if $g_i=(123)$}\\\\-1 & \\quad \\mbox{if $g_i=(12)(3)$}\\\\ \\end{array} \\right.,$ and a two-dimensional irrep, $U^{(2)}$ , given by $U^{(2)}_{(123)}=\\left(\\begin{matrix}-\\frac{1}{2}&-\\frac{\\sqrt{3}}{2}\\\\\\frac{\\sqrt{3}}{2}&-\\frac{1}{2}\\end{matrix}\\right),\\,U^{(2)}_{(12)(3)}=\\left(\\begin{matrix}1&0\\\\0&-1\\end{matrix}\\right),$ where the remaining elements of $S_3$ are generated by $U_{(12)(3)}^aU_{(123)}^b$ , with $a\\in (0,1)$ , and $b\\in (0,1,2)$ .", "The character table for $S_3$ is given in Table REF .", "Table: The character table for 𝕊 3 \\mathbb {S}_3The regular representation, ${\\mathcal {R}}$ , of $S_3$ is ${\\mathcal {R}}_{(123)}&=\\left(\\begin{matrix}1&0&0&0&0&0\\\\0&1&0&0&0&0\\\\0&0&-\\frac{1}{2}&-\\frac{\\sqrt{3}}{2}&0&0\\\\0&0&\\frac{\\sqrt{3}}{2}&-\\frac{1}{2}&0&0\\\\0&0&0&0&-\\frac{1}{2}&-\\frac{\\sqrt{3}}{2}\\\\0&0&0&0&\\frac{\\sqrt{3}}{2}&-\\frac{1}{2}\\end{matrix}\\right),\\\\{\\mathcal {R}}_{(12)(3)}&=\\left(\\begin{matrix}1&0&0&0&0&0\\\\0&-1&0&0&0&0\\\\0&0&1&0&0&0\\\\0&0&0&-1&0&0\\\\0&0&0&0&1&0\\\\0&0&0&0&0&-1\\end{matrix}\\right),$ where the remaining elements are generated by ${\\mathcal {R}}_{(12)(3)}^a{\\mathcal {R}}_{(123)}^b$ with $a\\in (0,1)$ , and $b\\in (0,1,2)$ .", "Suppose that the action of our channel is given by Eq.", "(REF ), i.e.", "$U=U^{(2)}$ , with compound character given by $\\chi =\\left(\\begin{array}{c}2\\\\-1\\\\0\\end{array}\\right).$ Obviously we have $\\gamma ^{(0)}_1=\\gamma ^{(1)}_1=0$ and $\\gamma ^{(2)}_1=1$ .", "Using Eq.", "(REF ) for $n=2$ we obtain $\\nonumber \\gamma ^{(0)}_2&=\\frac{1}{6}\\left(1\\cdot 1\\cdot 2^2+2\\cdot 1\\cdot (-1)^2+0\\right)=1\\\\ \\nonumber \\gamma ^{(1)}_2&=\\frac{1}{6}\\left(1\\cdot 1\\cdot 2^2+2\\cdot 1\\cdot (-1)^2+0\\right)=1\\\\\\gamma ^{(2)}_2&=\\frac{1}{3}\\left(1\\cdot 2\\cdot 2^2+2\\cdot (-1)\\cdot (-1)^2+0\\right)=1.$ Hence, $U^{\\otimes 2}=U^{(0)}\\oplus U^{(1)}\\oplus U^{(2)}$ .", "This is not the regular representation as $U^{(2)}$ is two-dimensional but appears only once.", "Evaluating Eq.", "(REF ) a third time for $n=3$ yields $\\nonumber \\gamma ^{(0)}_3&=\\frac{1}{6}\\left(1\\cdot 1\\cdot 2^3+2\\cdot 1\\cdot (-1)^3+0\\right)=1\\\\ \\nonumber \\gamma ^{(1)}_3&=\\frac{1}{6}\\left(1\\cdot 1\\cdot 2^3+2\\cdot 1\\cdot (-1)^3+0\\right)=1\\\\\\gamma ^{(2)}_3&=\\frac{1}{3}\\left(1\\cdot 2\\cdot 2^3+2\\cdot (-1)\\cdot (-1)^3+0\\right)=3.$ Hence $U^{\\otimes 3}=U^{(0)}\\oplus U{(1)}\\oplus 3U^{(2)}$ .", "As each irrep appears a number of times equal to, or greater than its dimension, $U^{\\otimes 3}$ contains the regular representation, ${\\mathcal {R}}$ , of $S_3$ ." ] ]

[ [ "Cycles of the magnetic activity of the Sun and solar-type stars and\n simulation of their fluxes" ], [ "Abstract The application of the Wavelet analysis and Fourier analysis to the dataset of variations of radiation fluxes of solar-like stars and the Sun is examined.", "In case of the Sun the wavelet-analysis helped us to see a set of values of periods of cycles besides \"11-year\" cycle: the long-duration cycles of 22-year, 40-50 year and 100-120 year and short-duration cycles of 2-3,5 years and 1,3-year.", "We present a method of the chromospheric flux simulation using the 13 late-type stars, which have well-determined cyclic flux variations similar to the 11-year solar activity cycle.", "Our flux prediction is based on the chromospheric calcium emission time series measurements from the Mount Wilson Observatory and comparable solar dataset.", "We show that solar three - component modeling well explains the stellar chromospheric observations." ], [ "Cycles of the magnetic activity of the Sun and solar-type stars and simulation of their fluxes E.A.", "Bruevich $^{a}$ , I.K.", "Rozgacheva $^{b}$ Sternberg Astronomical Institute, Moscow, Russia Moscow State Pedagogical University, Russia E-mail: $^a$ [email protected], $^b$ [email protected] Abstract.", "The application of the Wavelet analysis and Fourier analysis to the dataset of variations of radiation fluxes of solar-like stars and the Sun is examined.", "In case of the Sun the wavelet-analysis helped us to see a set of values of periods of cycles besides \"11-year\" cycle: the long-duration cycles of 22-year, 40-50 year and 100-120 year and short-duration cycles of 2-3,5 years and 1,3-year.", "We present a method of the chromospheric flux simulation using the 13 late-type stars, which have well-determined cyclic flux variations similar to the 11-year solar activity cycle.", "Our flux prediction is based on the chromospheric calcium emission time series measurements from the Mount Wilson Observatory and comparable solar dataset.", "We show that solar three - component modeling well explains the stellar chromospheric observations.", "KEY WORDS: the Sun, solar-type stars, active areas, cyclic activity, chromospheric emission.", "1 .", "Introduction.", "The solar-type magnetic activity among the stars Magnetic activity of the Sun is called the complex of composite electromagnetic and hydrodynamic processes in the solar atmosphere.", "They create a local active area: plages and spots in the photosphere, calcium flocculi in the chromosphere and prominences in the corona of the Sun.", "The analysis of active regions is necessary to study the magnetic field of the Sun and the physics of magnetic activity.", "This task is of fundamental importance for astrophysics of the Sun and the stars.", "Its applied meaning is connected with the influence of solar flares on the Earth's magnetic field and the need to solar activity variations and flares forecasts.", "It is difficult to predict the the evolution of each active region in details in present time.", "However, it has long been established that the total change of the active areas is cyclical.", "The cyclical nature of solar activity allows you to predict the state of the global activity of the Sun.", "The duration of the \"eleven-year cycle of solar activity ranged from 7 to 17 years according to 160 years of direct solar observations.", "For the first time the quasi-biennial variations of solar activity have been described in the work of the (Vitinsky et al.", "1986).", "In the recent studies on the subject of quasi-biennial variations of solar radiation (Ivanov-Kholodnyj & Chertoprud 2008; Bruevich & Ivanov-Kholodnyj 2011) the importance of this problem study were emphasized.", "It turned out that quasi-biennial solar cycles are closely associated with various quasi-biennial processes on the Earth, in particular with quasi-biennial variations of the velocity of the Earth rotation and speed of the stratospheric wind.", "The main methods of quasi-biennial solar cycles and the results of their study were more fully described in the monograph (Rivin 1989).", "Application of modern mathematical methods (modification of the method of main components, the \"Singular Spectrum Analysis\") for the treatment of long time series of 3032 the monthly averaged values of Wolf numbers gave an opportunity to draw some conclusions about the duration of the cycle and the form of individual quasi-biennial oscillations depending on the duration and power of eleven-year cycle (Khramova et al.", "2002).", "It was discovered the activity with the cyclicity period of 1,3 years during the last eight of the 11-year cycle.", "It is best manifested in the phase of maximum of the 11-year cycle and in the early phase of its decline.", "The axis of the dipole large-scale field at this time is located in the plane of the solar equator (Livshits & Obrydko 2006).", "These facts show on the independent existence of large-scale magnetic field and its influence on the processes of local magnetic activity.", "A comparison of magnetic activity of solar-type stars of different age allows us to check the basic representations of the internal structure and evolution of convective shells of these stars.", "Among the hundreds of thousands of stars in the vicinity of the Sun only few thousands fall under the definition of solar-type stars.", "The relative paucity of such stars is a consequence of their low luminosity.", "This prevents their detection and observation at a considerable distance from the Sun.", "Photometric observations of stars with active atmospheres are regularly held in the optical range since the mid of XX century.", "These observations include the measurement of their radiation in different ranges of the electromagnetic spectrum during the long intervals of time.", "In the present work we study the low-amplitude cyclical variability of solar and star's fluxes, as well as the duration of cycles and their dependence on the physical parameters of the Sun and solar-type stars.", "1.1 \"HK-project\" of Mount Wilson observatory One of the first and still the most outstanding program of observations of solar-type stars is the observation programme \"HK-project\" of Mount Wilson observatory.", "Implementation of this project has led to the discovery of \"11-cycles\" activity in solar-type stars.", "This observation program lasts for more than 40 years.", "First O. Wilson began this program in 1965.", "He attached great importance to the long-standing systematic observations of cycles in the stars (Baliunas et al.", "1995; Lockwood et al.", "2007).", "For this project the stars were carefully chosen according to those physical parameters, which were most close to the Sun: cold, single stars-dwarfs, belonging to the Main sequence.", "Close binary systems are excluded.", "All the stars of \"HK-project\" almost evenly are distributed on the celestial sphere.", "In the framework of the \"HK-project\" information about the chromospheric fluxes variations of the Sun was obtained from ground-based observations of the global index of solar activity (the 10,7 cm radio flux) - F10,7, and these observations were subsequently adapted to the same values of star's fluxes in the chromospheric lines of $H$ (396.8nm) and $K$ (393.4nm) $Ca II$ .", "Thus, in our work we use for stars and the Sun S-index (or $S_{CaII}$ ) - the relationship of radiation fluxes in the centers of emission lines $H$ and $K$ (396,8 nm and 393,4 nm) to radiation fluxes in the near-continuum (400,1 and 390,1 nm).", "This index is a sensitive indicator of the chromospheric activity of the Sun and the stars.", "The first results concerning the observations of 91 stars, were published in the (Wilson 1978).", "Since 1974 the head of the project became A. Vaughan.", "He constructed a \"HK-spectrometer\" of the next generation with the use of more modern technologies and continued observation of several hundred stars.", "At the same time observations of the same stars with help of the 60-inch telescope took place to determine their periods of rotation.", "The results of the joint observations of the radiation fluxes and periods of rotation gave the opportunity for the first time in stellar astrophysics to detect the rotational modulation of the observed fluxes (Noyes et al.", "1984).", "This meant that on the surface of the star there are inhomogeneities those were living and evolving in several periods of rotation of the stars around its axis.", "In addition, the evolution of the periods of rotation of the stars in time clearly pointed to the fact of existence of the star differential rotations similar to the Sun differential rotations.", "Radiation fluxes in the lines of $H$ and $K$ $Ca II$ from the Sun are formed on the upper levels of the solar of chromosphere and they are good indicators of active regions - areas with high magnetic activity.", "Because of the remoteness of the stars from us we can't distinguish the different active regions on their disks.", "However, the study of the radiation fluxes in the lines of $H$ and $K$ $Ca II$ ., normalized on a nearby continuum, gives us the indirect information about the numbers and sizes of the active regions in the atmospheres of the stars.", "To study the cycles of magnetic activity of the atmospheres of the stars we use new dataset of simultaneous observations of variations of the photospheric and chromospheric radiation fluxes of the Sun and of 33 stars of \"HK-project\".", "These data were obtained during the last 20 years in Lowell observatory (photometric observations) and during 40 years of observation in chromospheric lines of 111 stars in Smithsonian observatory of Stanford University (Baliunas et al.", "1995; Radick et al.", "1998; Lockwood et al.", "2007).", "The authors of \"HK-project\" with the help of frequency analysis of the 40-year observations have discovered (Baliunas et al.", "1995; Lockwood et al.", "2007) the periods of 11-year cyclic activity vary little in size for the same star.", "So they have determined the resistant variations of chromospheric activity.", "The durations of cycles vary from 7 to 20 years for different stars.", "The stars with cycles represent about 30% of the total number of \"HK-project\" stars.", "The determination of quasi-biennial activity of the cyclical nature of the stars by these authors did not take place.", "The heterogeneities on the disk of the star, which are responsible for the atmospheric activity outside the flares are spots and plages in the photosphere, flocculi in the chromosphere, prominences and coronal mass ejections in the corona (by analogy with the thoroughly studied formations in the Sun).", "For the duration of the existence these active regions vary greatly in time - they can be observed from several hours to several months.", "The contribution from the active regions should be taken into account in the analysis of observational \"HK-project\" data, both as in the lines of H and K, as in photospheric fluxes from the entire disk in broadband filters of photometric system of Lowell observatory, close to the standard UBV - system.", "1.2 Cycles of magnetic activity of the Sun Variations of different indices of solar activity characterize the variation of the radiation fluxes of the solar atmosphere at different altitudes.", "In the works (Kane 2002; Li & Sofia 2001) was found a strong correlation of the index of EUV (short-wave ultraviolet radiation, satellite observations of AE-E (1977-1980 years), Pioneer Venus (1979-1992 years) and SEM/SOHO (1996-2001 years) with the fluxes in the hydrogen lines , radio fluxes F10,7 and radiation fluxes in the lines of $MgII$ (Beer 2000).", "Also, all the indices of solar activity reasonably well correlated with Wolf numbers.", "In Fig.", "1 the annual average variations of Wolf numbers from 1700 to 2005 years are shown.", "The values of W in the period from 1850 to 2005 years were received as the result of a direct observations of the solar activity with ground-based observatories, and from 1750 to 1850 these values were obtained as a result of indirect estimations.", "Figure: The time series of annual average of Wolf numbers from 1700to 2005 years.", "Used data of the National Geophysical Data CenterSolar and Terrestrial Physics.The result of wavelet - analysis (Daubechies wavelet) of series of observations of average annual Wolf numbers (Fig.", "2) in the form of many of isolines is shown in Fig.", "2.", "For each isolines the value of the wavelet-coefficients are of the same.", "The isolines specify the maximum values of wavelet-coefficients, which corresponds to the most likely value of the period of the cycle.", "There are three well-defined cycles of activity: - the main cycle of activity is approximately equal to a 10 - 11 years; - 40-50- year cyclicity; - 100 to 120-year-old (ancient) cyclicity.", "Figure: Wavelet-analysis (Daubechies wavelet) of time series ofannual averages of Wolf numbers.", "The ordinate axis is the durationof the cycles (Cyclicity, years), the abscissa axis is the time(years)To identify a more short cycles we use the more accurate measurements of Wolf numbers according to observations from 1950 to 2011 years.", "In Fig.", "3 presents monthly averages of Wolf numbers.", "These values exclude almost rotational modulation of the Sun's observations, because the rotation period of the Sun on the \"active latitudes\" is about 27 days, close to the interval on which performed averaging.", "Figure: The time series of monthly mean values of Wolf numbers from1950 to 2011 years, the cycles 19 - 23.", "Used data of the NationalGeophysical Data Center Solar and TerrestrialPhysics.Fourier-analysis of time series of monthly average Wolf numbers can not detect quasi-biennial cycles because of the considerable variation in the duration of quasi-biennial cycles during the \"eleven-year\" cycle.", "Wavelet-analysis gives us that opportunity.", "Wavelet-analysis of a number of observations average monthly Wolf numbers with the help of the wavelet Morley (Fig.", "4) shows that there is the main cycle of variations is about a 10 - 11 years.", "We clearly see in Fig.", "4 that there are also short cycles: - 5,5- year-old cycles, - quasi-biennial cycles The values of the duration and amplitude of these short cycles are changing in different intervals of observations.", "In Fig.", "4 shows that the duration of the quasi-biennial cycle is reduced from 3,5 years at the beginning of each \"eleven-year cycle\" of up to 2 years at the end of it.", "Figure: Wavelet-analysis (wavelet Morley) of the time series ofmonthly average Wolf numbers.", "The ordinate axis is the duration ofthe cycles (Cyclicity, years), the abscissa axis is the time(years)In Fig.5 and Fig.6 we show the results of our wavelet-analysis for the numbers of the Wolf W and fluxes of radio emission F10,7 - the most common indices of solar activity.", "On the axis of ordinates postponed the duration of the quasi-biennial cycle, the x - axis the time of observation in years.", "Figure: Wavelet-analysis of observations of Wolf numbers, data set1950 - 2011 years.", "Used data of the National Geophysical Data CenterSolar and Terrestrial Physics.The ordinate axis is the duration ofthe cycles (Cyclicity, years), the abscissa axis is the time(years)Figure: Wavelet-analysis of observations of fluxes of radioemission F 10,7 F_{10,7}, data set 1950 - 2011 years.", "Used data of theNational Geophysical Data Center Solar and Terrestrial Physics.Theordinate axis is the duration of the cycles (Cyclicity, years), theabscissa axis is the time (years)The results of the wavelet-analysis, presented in Fig.", "5 and Fig.", "6 show that, along with the quasi-biennial cyclicity of solar radiation we can also see and cycles with a period of about 1,3, found earlier in (Livshits & Obrydko 2006) 2.", "The cyclic activity of solar-type stars on the \"11-year\" and quasi-biennial time scales In the last we can see a large number of works, where low-amplitude variations of the radiation of the stars with active atmospheres are investigated with the help of modern approaches.", "In the work (Kolah & Olach 2009) were analyzed variations of the radiation of EI Eri - one of the bright solar-type stars.", "Data of observations were studied with help of Fourier analysis and wavelet-analysis.", "Variations of the fluxes of radiation of this star are quite stable.", "Study with help a Fourier analysis or wavelet-analysis gives the similar results: the cycle of variations of the fluxes of radiation is an average of 2,7 years.", "Table: Results of our calculations of T 11 T_{11} and T 2 T_{2} valuesand \"HK-project\" T 11 HK T^{HK}_{11} calculations for 29 stars and theSunTable 1 and Table 2 present the results of our calculation of the periods of variations of chromospheric radiation of the stars (Bruevich & Kononovich 2011).", "The method of fast Fourier transform was used to determine periods of the cyclicity of stars.", "Table: Table 1 - continued.", "Results of our calculations ofT 11 T_{11} and T 2 T_{2} values and \"HK-project\" T 11 HK T^{HK}_{11}calculations for 23 starsWe used data of the observation of (Lockwood et al.", "2007) for frequency analysis of time series of S-index of the Sun and 33 of the stars.", "In the work (Bruevich and Kononovich 2011) the values of periods of resistant \"11-year\" cycles ($T_{11}$ - column 7 in Table 1 and Table 2) and periods of quasi-biennial cycles ($T_2$ - column 8 in Table 1 and Table 2) for the stars from column 2 of Table 1 and Table 2 were determined.", "A dash in columns 7 and 8 means that the cyclicity was not detected.", "Table 1 and Table 2 contain the values of the \"11-year\" cycles $T^{HK}_{11}$ (column 6), which were determined by authors of the \"HK-project\" during the conduct of primary spectral analysis of the data (Baliunas et al.", "1995).", "Table 1 and Table 2 also show the spectral classes of stars (column 3), their periods of rotation $P_{rot}$ (column 4) and their effective temperatures $T_{eff}$ (column 5).", "The data analysis of Table 1 and Table 2 shows that among the stars of the earlier spectral types, and accordingly, faster rotating, the quasi-biennial cycles are found more confident.", "Note that these cycles we found among the stars, as in the case of existing of \"11-year\" cycles, so without them.", "3.", "The cyclic variations of fluxes of radiation from different layers of the solar atmosphere 3.1 Indices of solar activity The first index, describing the solar activity, was the index of sun spots.", "At the present time it is called the index of \"Wolf numbers \".", "This index of solar activity has the most long-term history of the direct observations.", "In addition, the rows of Wolf numbers were restored since 1750 according to indirect data, see.", "Fig.", "1.", "This index characterizes the state of the photosphere of the Sun, so as spots are such heterogeneities of that characterize the active region at the level of photosphere.", "Much more objective index of activity of photosphere is the index Total Solar Irradiance (TSI).", "In addition to the fluxes of photosphere's radiation (attributable to the visible range of the spectrum) which give the main contribution into the TSI.", "The fluxes of all the available spectral intervals from x-ray to infrared are summarized in this index.", "The most reliable are the data of TSI observations from 1978, with help of the equipment installed on satellites HF, ACRIM-I,II and VIRGO.", "Creation of the combined database of TSI observations Judith Lean currently coordinates, author of a three-component model of EUV (extreme ultraviolet) radiation of the Sun (Lean et al.", "1986).", "For monitoring of variations of chromosphere's radiation of the Sun on the satellites of the NOAA series from 1978 year to the present time the measurements of the $Mg II$ (280 nm, core-to wing ratio) index are made.", "This activity index is very similar to the S-index of solar-type stars, observed in the \"HK-project\", with the difference that in the case of the solar index $Mg II$ (280 nm) observations are carried out in the range of radiation, out of reach for the ground-based observatories.", "Observations of solar radiation from the top of chromosphere and the bottom of the corona the - index $F_{10,7}$ (radio flux at 10,7 cm wavelength) are very important for solar activity study.", "This activity index has also rather lengthy series of regular direct observations with ground-based observatories (in Ottawa and other) and correlates well with all the rest of the solar indices.", "Archive data of index $F_{10,7}$ are detailed and accessible.", "In addition, he is sensitive to changes in solar weather, the variation of this index in the cycle of activity make up about 200%, from (75 to 240)$\\cdot 10^{-22}$ Watt/m2.", "Therefore, $F_{10,7}$ is used for the prognosis and monitoring of the solar activity more often than the other indices.", "For observations of coronal activity of the Sun the index of x-ray radiation of 0,1 - 0,8 nm (background of the radiation without flares) is used.", "This activity index is very variable, even within a small interval of time, because after even small flares the background radiation remains elevated for some time.", "In a cycle of activity this index varies by two-three orders of magnitude from $10^{-9}$ to $(1-5)$$\\cdot 10^{-7}$ Watt/m2.", "Also there are another activity indexes associated with the flare activity of the Sun, they are also well correlated with the main solar indices.", "All activity indexes, characterizing fluxes of radiation from different layers of the solar atmosphere are connected among themselves and are defined ultimately from the major parameter of the activity - activity of the magnetic field.", "3.2 Empirical basis for the forecasts of the amplitude of cyclic activity of stars For the Sun one of the most important tasks is to predict the amplitude of cyclic activity, affecting a number of earth processes.", "Contribution to the low-amplitude variations of solar radiation occurs from the active processes, simultaneously in the three layers of the atmosphere - photosphere, chromosphere and the corona.", "It is known that different indices of solar activity are rather closely interrelated.", "This fact allows us to predict the variation of the solar radiation from different layers, according to the observations of the change in the index of activity from only one layer.", "In the present work we use the ideology of the works (Lean et al.", "1983; Lean 2001) for the prediction of \"11-year\" activity of the stars.", "We present the full amplitude of the radiation heat flow in the chromosphere's lines of stars in the form of the composition of three main components of fluxes of radiation, two of which are the constant and inconstant background with the grid of supergranulation, the third is the contribution from the active areas, which smoothly change during the cycle.", "For the prediction of the amplitudes of variations of the fluxes of radiation from the stars we chose the values of the fluxes in the lines $H$ and $K$ $CaII$ of the \"HK-project\" stars with the well-identified \"11-year-old\" cycles (from 8 to 13 years).", "It turned out that the cyclical nature of these fluxes is so similar to the solar cycle, that for the forecast of amplitudes of fluxes of radiation from these stars are quite applicable methods used in the practice of solar forecasts (Borovik et al.", "1997).", "For the successful prediction of variations of the fluxes of radiation from the stars, we should take into account the following characteristics of their \"11-year\" activity cycles: (a) The chromospheric emission lines $H$ and $K$ , visible on the background of wide and deep profiles of absorption lines contain information about the temperature of the atmosphere.", "It is established, that the profiles of the lines differ markedly depending on the photospheric substrate (is it the photosphere's background, or the spot and the plage).", "The profiles of lines, averaged over the disc of the Sun, depend on the phase of the solar cycle; (b) According to observations, sun spots are often collected in groups, surrounded by photospheric plages and flocculi fields in chromosphere.", "The area of spot is in several times less than the total area of the chromospheric flocculi and coronal condensations associated with this spot.", "The total area of solar spots in the epoch of solar activity maximum occupies an area of 0,5 % of the hemisphere.", "The total area of the spots in solar-type stars can reach 15 - 20 %; (c) The surface brightness of the active area increases with its increasing in size and with increase in the number of spots in it.", "On average, according to measurements at Skylab, (Schriver et al.", "1985; Van Driel-Gesztelyi 2006) the surface brightness of active regions is in 3 - 5 times higher than the brightness of the chromospheric net, and a relative increase in luminance (contrast) depends on the wave length; (d) The variation of shortwave radiation are affected the existence of \"bright spots\" - areas of the small bipolar region, in which there are usually 2 - 3 magnetic loops, about 2500 km in diameter and 12,000 km in length (Sheely & Golub 1979).", "Comparison of x-ray photographs of the Sun, with simultaneous observations in the lines of calcium K and $H_{\\alpha }$ showed that \"the bright x-ray points\" localized primarily on the borders of the cells of the chromospheric net and in 83% of cases coincide with prominent elements of the net (Egamberdiev 1983).", "This testifies to the connection between \"bright x-ray points\" and \" bright points\" of the net; (e) Basically, the sources of x-ray radiation of the Sun (in contrast to the ultraviolet radiation) are situated in the coronal loops, which are concentrated in active regions.", "According to observations of the satellite Skylab in the ultraviolet and x-ray spectral bands all the coronal loops can be grouped into three main groups, (Orral 1981): the first group are the small arches, coming mainly across the photospheric section of the magnetic polarity, their temperature is about $2\\cdot 10^6 K$ , the concentration of electrons $n_e\\approx 3,5\\cdot 10^{9}cm^{-3}$ (in the surrounding space $n_e\\approx 2,5\\cdot 10^{9}cm^{-3}$ ); the second group are the loops which connect the magnetic poles out of the line of section polarities, their temperature of $10^5 <T<10^6$ ; the third group are the cold loops (chromospheric ribbons), which are the bases of the first group, their temperature $T <5\\cdot 10^4K$ (these loops are visible in $L_{\\alpha }$ and other chromospheric emission lines).", "According to the results of observations of the satellite AE-E Nimbus 7 the three-component model for the prediction of short-wave radiation of the Sun was proposed (Lean et al.", "1986).", "In this model it is assumed that the flux of radiation is defined by three components: (I) is the constant component - the (BASAL) component, the sources of which are evenly distributed on the solar disk, and do not change in a cycle of activity; (II) is evenly changing component - the radiation of the \"active\" chromospheric net (which are also assumed to be uniformly distributed over the disk, but its occurrence is considered to be caused by the collapse of the Active Regions (AR) and, consequently, its intensity should be proportional to their total areas; (III) is the rapidly changing component - the intense radiation from the plage areas, coinciding with the active regions.", "In accordance with that the flux of radiation is calculated by the formula (Lean et.", "al 1986): $ I = I_{\\lambda Q} \\Big \\lbrace 1 + f_N \\Big ( C_{\\lambda N}- 1 \\Big ) \\Big \\rbrace + 2 \\pi F_ {\\lambda Q}(1) \\Sigma {A_i \\mu _i R_{\\lambda }(\\mu _i)}\\Big ( C_{p \\lambda } W_i -1 \\Big ) \\qquad \\mathrm {(1)}$ where $I$ is the full flux of chromospheric emission, $I_{\\lambda Q}$ is the contribution of the constant component (BASAL), $C_{p \\lambda }$ is the values of AR contrasts and they are similar to contrasts from (Cook et all., 1980), $C_{N \\lambda }$ is the value of \"active network\" contrast: they are equal to $0.5 \\cdot C_{p \\lambda } $ for continuum and $1/3 \\cdot C_{p \\lambda }$ for lines, $f_{N}$ is part of disk (without AR) that is occupied by the \"active network\".", "The second member in the right part of (1) describes emission from all AR on the disk; $A_i$ are values of their squares, $\\mu _i$ describes the AR position: $\\mu _i = {cos {\\phi _i} cos {\\theta _i}}$ (where $\\phi _i$ and $\\theta _i$ are the coordinates of AR number $i$ ).", "$R_\\lambda (\\mu _i)$ describes the relative change of the surface brightness $F_{\\lambda Q}(\\mu _i)$ with moving from center to edge of disk.", "The relative adding AR contribution to full flux from the different AR is determined by the factor $W_i$ that is linearly changed from the value $0.76$ to $1.6$ depending of the brightness ball of flocculus (according to ball flocculae changes from 1 to 5).", "So the \"active network\" part in all the surface without AR is determined by the AR decay, the next relationship between $f_N$ in time moment $t$ and average values $A_i$ in earlier time is right: $ f_N(t)= 13.3 \\cdot 10^{-5} \\cdot < \\Sigma A_i(t-27)> \\qquad \\mathrm {(2)}$ where the time-averaging is taken for 7 previous rotation periods, $A_i$ is measured in one million parts of the disk.", "Note that: first, the 27-day variation of the radiation caused by the movement and evolution of active regions; secondly, the 11-year cyclic changes in the significant contribution to make \"active network\".", "4.", "An empirical model of cyclic variations of the values of fluxes of star's radiation in the $CaII$ lines of $H$ and $K$ From the analysis of long-term variations of the solar streams in the lines of ionized calcium $H$ and $K$ , it follows that the total fluxes of star's and Sun's radiation consist of three main components: From the analysis of long-term variations of the solar streams in the lines of ionized calcium $H$ and $K$ , it follows that the total fluxes of star's and Sun's radiation consist of three main components: (I) is the constant component, the so-called \"BASAL\" in solar physics (below for the stars BASAL = $P_{min}$ ); (II) is slowly changing in a cycle of activity background, including the constant component $P_{min}$ , (for chromospheric calcium emission of the stars - $P_{CaII}(t)$ ); (III) - active regions on the disk (let's call this part of the flux of chromospheric radiation of the stars - $S_{CaII}(t)$ ).", "The total flux is equal to $ S_{CaII}(t) = P_{CaII}(t) +S_{AR}(t) \\qquad \\mathrm {(3)}$ Between the values of the fluxes of chromospheric radiation for the Sun and the stars $P_{CaII}(t)$ and $S_{CaII}(t)$ , obviously, there is a close connection that will be used in the future.", "According to the data of observations (Lean et al.", "1983; Lean 2005), the maximum variation of the background of fluxes of solar radiation in the radio band is about 20%.", "This agrees well with the model of radiation of the Sun in the line of hydrogen (Borovik et al.", "1997), where the maximum amplitude of the background radiation is also the order of 20%.", "We introduce the factor of similarity k to forecast of cyclical variations of chromospheric radiation of the stars.", "It equals to the ratio of maximum amplitudes of the background radiation to the maximum amplitude of the flux of star's radiation in a cycle of activity: $k= (P_{CaII}^{max} - P_{min})/(S_{CaII}^{max} -P_{min}) \\qquad \\mathrm {(4)}$ Figure: Variations of fluxes of radiation of the Sun inchromospheric lines, data set from (Radick et al.", "1998)Figure: Variation of fluxes of the radiation of the HD 81809 starin chromospheric lines, data set from (Radick et al.", "1998)In accordance with this, each value $S_{CaII}(t)$ set in compliance the value $P_{CaII}(t)= k \\cdot S_{CaII}(t)$ .", "Here the value $S_{CaII}(t)$ was obtained from observation data and was presented for the 13 stars with well-determined cycles of chromospheric activity (star's group \"EXCELLENT\") and the Sun.", "In Fig.", "7 and Fig.", "8 we present the smoothed curves of variations of the fluxes for the Sun and for one of the studied here stars - the star HD 81809, belonging to the group of \"EXCELLENT\" class.", "From the observational data of chromospheric radiation (Radick et al.", "1998) and knowing the values of the periods of cycles $T=T_{cyc}$ (see Table 2) for all 13 of the stars of the group of \"EXCELLENT\" (see for example the data for the HD 81809 star in Figure 8) we can define individual values of the flux of the radiation $S_{CaII}(t/T)$ (with fairly good accuracy, determined by the quality of observations).", "Further, with a coefficient of similarity $k$ we can estimate these values $P_{CaII}(t/T)$ .", "Then for each star we compute the coefficients $a$ and $b$ of the regression equation (which will be further used for the forecasts of the required moment of time in the future): $ S_{CaII}(t)= a \\cdot P_{CaII}(t) + b \\qquad \\mathrm {(5)}$ Then when predicting the variations of the chromospheric fluxes for the 13 stars (Table 3), we apply the same approach as that used in the work (Bocharova & Nusinov 1983) for the prediction of the solar background radiation in the \"11-year\" cycle.", "Modifying the formula of (Bocharova & Nusinov 1983), we obtain the following expression for the analytical approximation of the values for the background radiation of the stars (Bruevich 1999): $ P_{CaII}(t)=P_{\\min }\\cdot \\Big (1+\\sin ^4 \\cdot { {\\pi \\cdot t}\\over {T}} \\Big ) \\cdot e^{-{{\\pi \\cdot t}\\over {T}} } \\qquad \\mathrm {(6)}$ where is the minimum value of the background flow (BASAL), corresponding to the lowest value of the flux of radiation of the stars during the cycles of chromospheric activity and is determined from observations (on Fig.", "9 and Fig.", "10 the level of BASAL is shown with help of the solid line), $T$ - the period of the cycle chromospheric activity.", "Table: Observed parameters of 13 \"EXCELLENT\" class stars andregression coefficients aa and bb calculated from (5)For the stars $T=T_{cyc}$ can vary from 6 to 23 years, according to data(Baliunas et al.", "1995).", "However, for stars of the group \"EXCELLENT\" this value varies in a narrow framework and is closed to the solar period in 11 years.", "The variable $t$ is the current time from the beginning of the cycle, expressed as a fraction of cycle (for example,$t = 0,1\\cdot T, 0,2\\cdot T... 0,9\\cdot T, T$ ).", "Simulated flux $ S_{CaII}(t)$ for solar-type stars we can calculate from the regression equation (5), using values $a$ and $b$ , as well as having calculated in advance the value $ P_{CaII}(t)$ with help of the formula (6).", "In the Table 3 we show the results of our calculation of the coefficients $a$ and $b$ for 13 stars, having a well-determined long-term cyclical variability of their fluxes $ S_{CaII}$ .", "Also we presented the values of relative maximum variation of fluxes in a cycle of activity, as a total fluxes $ {\\Delta S_{CaII}^{max}}/ {P_{min}}$ , as well as of that part of the fluxes, which is responsible only for the contribution to the radiation from active regions on the disk $ {\\Delta S_{AR}^{max}}/ {P_{min}}$ .", "In Table 3 we present also the relative full flux variation in activity cycle maximum: ($ {\\Delta S_{CaII}^{max}}/ {P_{min}}$ ) and relative AR adding flux in activity cycle maximum: ($\\Delta S_{AR}^{max} / P_{min}$ ).", "The value $P_{mim}$ - that is equal to BASAL emission for different we can determine from (Baliunas et al/ 1995) data, see Fig.", "9 and Fig.", "10.", "Figure: Variations of fluxes of radiation of the Sun inchromospheric lines according to observations the Observatory MountWilson (Lockwood et al.", "1997) - crosses and calculated according tothe formulas (5) and (6) the model curves for S CaII S_{CaII} andP CaII P_{CaII}Figure: Variations of fluxes of radiation of the HD 81809 star inchromospheric lines according to observations the Observatory MountWilson (Lockwood et al.", "1997) - crosses and calculated according tothe formulas (5) and (6) the model curves for S CaII S_{CaII} andP CaII P_{CaII}In Table 3 and at Fig.", "9 and Fig.", "10 we use the following notations: $P_{\\min }$ - the constant component of the radiation of the stars - (BASAL), $ P_{CaII}$ - the background radiation, changing in a cycle of activity, $ S_{CaII}$ - the complete flux of radiation , ${\\Delta S_{CaII}^{\\max }}$ - the maximum amplitude of the variations of radiation in a cycle of activity ${\\Delta S_{AR}^{\\max }}$ - the contribution to the total flux of the active regions.", "Thus, the procedure of the forecast of model curves of cyclic variations of fluxes of chromospheric radiation of stars in the 11-year time scale consists of the following: 1.", "For the prediction of the radiation intensity of use data from Table 2 (take the value $P_{\\min }$ for the selected stars from column 5).", "In Fig.", "9 and Fig.", "10 - the level of BASAL=$P_{\\min }$ .", "2.", "From Fig.", "10 (star HD 81809) and from (Lockwood et al.", "2007) for the stars of the group \"EXCELLENT\" we define the moment of the beginning of the chromospheric cycle and $t/T$ for the selected us time of the forecasting flux.", "3.", "Then using equation (6) calculate the value $ P_{CaII}(t)$ for the moment $t$ .", "4.", "Using equation (5) we calculate the forecasting flux $S_{CaII}(t)$ for the moment $t$ .", "5.", "Conclusions The large amount of data of regular observation of solar radiation in different spectral ranges does preferred for us the conduct of wavelet-analysis to study the cycles of magnetic activity of the Sun.", "The wavelet-analysis helped us to see a set of values of periods of cycles besides 11-year cycle: - long duration cycles at 22-year, 40-50 year and 100-120 year time scales; - short duration cycles at quasi-biennial and 1,3-year time scales.", "Unfortunately in case of stars we have time series of observations in one - two spectral ranges which are less informative and short in time.", "This often forcing us to be limited to Fourier analysis in the study of cyclic recurrence of radiation from stars.", "When we analyze results of our predictions (in Table 3 we presented the observed values that we're discussed in this issue and our estimations as $ {\\Delta S_{CaII}^{max}}/ {P_{min}}$ and $\\Delta S_{AR}^{max} / P_{min}$ ) some conclusions can be made: - we can see from the last column of Table 3 that, among the stars of the group \"EXCELLENT\" of spectral class K (with the best quality of cyclicity) the active regions may provide the excess above the background flux (consisting of constant for each star BASAL-radiation and of radiation from changing the brightness during the cycle of chromospheric network) in the maximum of the cycle of up to 10-20%; - the most bright flocculi, which are in 2 times brighter than the unperturbed of chromosphere (cell centers of chromospheric network) should occupy the area of 5 - 10% of the surface of the star; - in case of the young stars of spectral classes K and M without cycles this value (area that bright flocculi should occupy) is much larger (up to 50%), and for the stars of spectral type G with the well determined cycles of activity, much less (1-3 %).", "For the Sun this value is only 0,1 %.", "References 1.", "Baliunas, S.L., Donahue, R.A., et al.", "(1995) Astrophys.", "J., 438, 269.", "2.", "Beer J., (2000), Space Sci.", "Rev., 94, 53.", "3.", "Borovik, V.N., Livshitz, M.A., Medar, V.G.", "(1997) Astronomy Reports, 41, N6, 836.", "4.", "Bruevich, E.A., Kononovich E.V.", "(2011) Moscow University Physics Bull., 66, N1, 72; ArXiv e-prints, (arXiv:1102.3976v1) 5.", "Bruevich, E.A., Ivanov-Kholodnyj G.S.", "(2011) ArXiv e-prints, (arXiv:1108.5432v1).", "6.", "Bruevich, E.A.", "(1999) Mosc.", "Univ.", "Phys.", "Bull., Ser.", "3, No6, 48.", "7.", "Cook, J.W., Brueckner, G.E., Van Hoosier, M.E., (1980) J.Geophys.Res., A85, N5, 2257.", "8.", "Van Driel-Gesztelyi, L. (2006) in: V. Bothmer, A.A.", "Hady(eds.", "), Solar Activity and its Magnetic Origin, IAUS 233, 205.", "9.", "Egamberdiev, S. A.", "1983, Soviet Astronomy Letters, 9, 385.", "10.", "Ivanov-Kholodnyj, G.S.", "& Chertoprud, V.E., (2008) Sol.-Zemn.", "Fiz., 2, 291.", "11.", "Khramova, M.N., Kononovich, E.V.", "& Krasotkin, S.A., (2002) Astron.", "Vestn., 36, 548.", "12.", "Kollath, Z., Olah, K. (2009) Astron.", "and Astrophys, 501, 695.", "13.", "Lean J.L., Scumanich A., (1983) J. Geophys.", "Res., A88, N7, 5751.", "14.", "Lishits, I.M., Obridko, V.N., (2006) Astronomy Reports, 83, N 11, 1031.", "15.", "Lockwood, G.W., Skif, B.A., Radick R.R., Baliunas, S.L., Donahue, R.A. and Soon W., (2007) Astrophysical Journal Suppl., 171, 260.", "16.", "Noyes, R. W.; Hartmann, L. W.; Baliunas, S. L.; Duncan, D. K.; Vaughan, A. 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[ [ "Knot Universes in Bianchi Type I Cosmology" ], [ "Abstract We investigate the trefoil and figure-eight knot universes from Bianchi-type I cosmology.", "In particular, we construct several concrete models describing the knot universes related to the cyclic universe and examine those cosmological features and properties in detail.", "Finally some examples of unknotted closed curves solutions (spiky and Mobius strip universes) are presented." ], [ "Introduction", "Inflation is one of the most important phenomena in modern cosmology and has been confirmed by recent observations on cosmic microwave background (CMB) radiation [1].", "Furthermore, it is suggested by the cosmological and astronomical observations of Type Ia Supernovae [2], CMB radiation [1], large scale structure (LSS) [3], baryon acoustic oscillations (BAO) [4], and weak lensing [5] that the expansion of the current universe is accelerating.", "In order to explain the late time cosmic acceleration, we need to introduce so-called dark energy in the framework of general relativity or modify the gravitational theory, which can be regarded as a kind of geometrical dark energy (for reviews on dark energy, see, e.g., [6]-[11], and for reviews on modified gravity, see, e.g., [12]-[17]).", "It is considered that there happened a Big Bang singularity in the early universe.", "In addition, at the dark energy dominated stage, the finite-time future singularities will occur [18]-[23].", "There also exists the possibility that a Big Crunch singularity will happen.", "To avoid such cosmological singularities, there are various proposals such as the cyclic universe [24]-[26] (in other approach of the cyclic universe, see [27]), the ekpyrotic scenario [28], and the bouncing universe [29].", "On the other hand, as a related theory to the cyclic universe, the trefoil and figure-eight knot universes have been explored in Ref. [30].", "In the homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) and the homogeneous and anisotropic Bianchi-type I cosmologies, the geometrical description of these knot theories corresponds to oscillating solutions of the gravitational field equations.", "Note that the terms \"the trefoil knot universe\" and \"the figure-eight knot universe\" were introduced for the first time in Ref. [30].", "Moreover, the Weierstrass $\\wp (t)$ , $\\zeta (t)$ and $\\sigma (t)$ functions and the Jacobian elliptic functions have been applied to solve several issues on astrophysics and cosmology [31].", "In particular, very recently, by combining the reconstruction method in Refs.", "[12], [22], [33], [34] with the Weierstrass and Jacobian elliptic functions, the equation of state (EoS) for the cyclic universes [35] and periodic generalizations of Chaplygin gas type models [36]-[38] for dark energy [39] have been examined.", "This procedure can be considered to a novel approach to cosmological models in order to investigate the properties of dark energy.", "In this paper, we explore the cosmological features and properties of the trefoil and figure-eight knot universes from Bianchi-type I cosmology in detail.", "In particular, we construct several concrete models describing the trefoil and figure-eight knot universes based on Bianchi-type I spacetime.", "In our previous work [30], the models of the knot universes from the homogeneous and isotropic FLRW spacetime were studied.", "By using the equivalent procedure, as continuous investigations, in this work we explicitly demonstrate that the knot universes can be constructed by Bianchi-type I spacetime.", "In other words, our purpose is to establish the formalism which can describe the knot universes.", "It is significant to emphasize that according to the recent cosmological data analysis [1], it is implied that the universe is homogeneous and isotropic.", "In fact, however, recently the feature of anisotropy of cosmological phenomena such as anisotropic inflation [46] has also been studied in the literature.", "In such a cosmological sense, it can be regarded as reasonable to consider the anisotropic universe including Bianchi-type I spacetime.", "The units of the gravitational constant $8 \\pi G = c =1$ with $G$ and $c$ being the gravitational constant and the seed of light are used.", "The organization of the paper is as follows.", "In Sec.", "II, we explain the model and derive the basic equations.", "In Sec.", "III, we investigate the trefoil knot universe.", "Next, we study the figure-eight knot universe in Sec.", "IV.", "In Sec.", "V we present some unknotted closed curve solutions of the model.", "Finally, we give conclusions in Sec.", "VI." ], [ "The model", "In this section we briefly review some basic facts about the Einstein's field equation.", "We start from the standard gravitational action (chosen units are $c=8\\pi G=1$ ) $S=\\frac{1}{4}\\int d^{4}x\\sqrt{-g}(R-2\\Lambda +L_m),$ where $R$ is the Ricci scalar, $\\Lambda $ is the cosmological constant and $L_m$ is the matter Lagrangian.", "For a general metric $g_{\\mu \\nu }$ , the line element is $ds^2=g_{\\mu \\nu }dx^\\mu dx^\\nu , \\ \\ (\\mu ,\\nu =0,1,2,3).$ The corresponding Einstein field equations are given by $R_{\\mu \\nu }+\\Big (\\Lambda -\\frac{1}{2}R\\Big )g_{\\mu \\nu }=- \\kappa ^2T_{\\mu \\nu },$ where $R_{\\mu \\nu }$ is the Ricci tensor.", "This equation forms the mathematical basis of the theory of general relativity.", "In (REF ), $T_{\\mu \\nu }$ is the energy-momentum tensor of the matter field defined as $T_{\\mu \\nu }=\\frac{2}{\\sqrt{-g}}\\frac{\\delta L_m}{\\delta g^{\\mu \\nu }},$ and satisfies the conservation equation $\\nabla _\\mu T^{\\mu \\nu }=0,$ where $\\nabla _\\mu $ is the covariant derivative which is the relevant operator to smooth a tensor on a differentiable manifold.", "Eq.", "(REF ) yields the conservations of energy and momentums, corresponding to the independent variables involved.", "The general Einstein equation (REF ) is a set of non-linear partial differential equations.", "We consider the Bianchi - I metric $ds^2=-d\\tau ^2+A^2dx_1^2+B^2dx_2^2+C^2dx_3^2,$ where we assume that $\\tau =t/t_0, x_i=x_i^{\\prime }/x_{i0}, A, B, C$ are dimensionless (usually we put $t_0=x_{i0}=1$ ).", "Here the metric potentials $A,B$ and $C$ are functions of $\\tau =t$ alone.", "This insures that the model is spatially homogeneous.", "The statistical volume for the anisotropic Bianchi type-I model can be written as $V=ABC.$ The Ricci scalar is $R=g^{ij}R_{ij}=2\\left(\\frac{\\ddot{A}}{A}+\\frac{\\ddot{B}}{B}+\\frac{\\ddot{C}}{C}+\\frac{\\dot{A}\\dot{B}}{AB}+\\frac{\\dot{A}\\dot{C}}{AC}+\\frac{\\dot{B}\\dot{C}}{BC}\\right),$ where $\\dot{A}=dA/d\\tau $ and so on.", "The non-vanishing components of Einstein tensor $G_{ij}=R_{ij}-0.5g_{ij}R$ are $G_{00}&=&\\frac{\\dot{A}\\dot{B}}{AB}+\\frac{\\dot{A}\\dot{C}}{AC}+\\frac{\\dot{B}\\dot{C}}{BC},\\\\G_{AA}&=&-A^2\\left(\\frac{\\ddot{B}}{B}+\\frac{\\ddot{C}}{C}+\\frac{\\dot{B}\\dot{C}}{BC}\\right),\\\\G_{BB}&=&-B^2\\left(\\frac{\\ddot{A}}{A}+\\frac{\\ddot{C}}{C}+\\frac{\\dot{A}\\dot{C}}{AC}\\right),\\\\G_{CC}&=&-C^2\\left(\\frac{\\ddot{B}}{B}+\\frac{\\ddot{A}}{A}+\\frac{\\dot{B}\\dot{A}}{BA}\\right).$ We define $a=(ABC)^{\\frac{1}{3}}$ as the average scale factor so that the average Hubble parameter may be defined as $H=\\frac{\\dot{a}}{a}=\\frac{1}{3}\\left(\\frac{\\dot{A}}{A}+\\frac{\\dot{B}}{B}+\\frac{\\dot{C}}{C}\\right).$ We write this average Hubble parameter $H$ sometimes as $H=\\frac{1}{3}\\left(H_{1}+H_{2}+H_{3}\\right),$ where $H_{1}=\\frac{\\dot{A}}{A},\\quad H_{2}=\\frac{\\dot{B}}{B},\\quad H_{3}=\\frac{\\dot{C}}{C}$ are the directional Hubble parameters in the directions of $x_1, x_2$ and $x_3$ respectively.", "Hence we get the important relations $A=A_0e^{\\int H_{1}dt},\\quad B=B_0e^{\\int H_{2}dt},\\quad C=C_0e^{\\int H_{3}dt},$ where $A_0, B_0, C_0$ are integration constants.", "The other important cosmological quantity is the deceleration parameter $q$ , which for our model reads as $q=-\\frac{a\\ddot{a}}{\\dot{a}^{2}}.$ Next, we assume that the energy-momentum tensor of fluid has the form $T_{ij}=diag[T_{00},T_{11},T_{22},T_{33}]=diag[\\rho ,-p_{1},-p_{2},-p_{3}].$ Here $p_{i}$ are the pressures along the $x_i$ axes recpectively, $\\rho $ is the proper density of energy.", "Then the Einstein equations (with gravitational units, $8\\pi G=1$ and $c=1$ ) read as $R_{ij}-\\frac{1}{2}Rg_{ij}=-T_{ij},$ where we assumed $\\Lambda =0$ .", "For the metric (REF ) these equations take the form $\\frac{\\dot{A}\\dot{B}}{AB}+\\frac{\\dot{B}\\dot{C}}{BC}+\\frac{\\dot{C}\\dot{A}}{CA}-\\rho &=&0,\\\\\\frac{\\ddot{B}}{B}+\\frac{\\ddot{C}}{C}+\\frac{\\dot{B}\\dot{C}}{BC}+p_1&=&0,\\\\\\frac{\\ddot{C}}{C}+\\frac{\\ddot{A}}{A}+\\frac{\\dot{C}\\dot{A}}{CA}+p_2&=&0,\\\\\\frac{\\ddot{A}}{A}+\\frac{\\ddot{B}}{B}+\\frac{\\dot{A}\\dot{B}}{AB}+p_3&=&0.$ In terms of the Hubble parameters this system takes the form $H_1H_2+H_2H_3+H_1H_3-\\rho &=&0,\\\\\\dot{H}_2+\\dot{H}_3+H^2_2+H^2_3+H_2H_3+p_1&=&0,\\\\\\dot{H}_3+\\dot{H}_1+H^2_3+H^2_1+H_3H_1+p_2&=&0,\\\\\\dot{H}_1+\\dot{H}_2+H^2_1+H^2_2+H_1H_2+p_3&=&0.$ Also we can introduce the three EoS parameters as $\\omega _1=\\frac{p_1}{\\rho },\\quad \\omega _2=\\frac{p_2}{\\rho },\\quad \\omega _3=\\frac{p_3}{\\rho }$ and the deceleration parameters $q_1=-\\frac{\\ddot{A}A}{\\dot{A}^2},\\quad q_2=-\\frac{\\ddot{B}B}{\\dot{B}^2},\\quad q_3=-\\frac{\\ddot{C}C}{\\dot{C}^2}.$ Finally we want present the equation $2\\dot{H}+6H^2=\\rho -p,$ where $p=\\frac{p_1+p_2+p_3}{3}$ is the average pressure.", "Hence we can calculate the average papameter of the EoS as $\\omega =\\frac{p}{\\rho }=\\frac{\\omega _1+\\omega _2+\\omega _3}{3}.$ Let us also we present the expression of $R$ in terms of $H_i$ .", "From (REF ) and (REF ) follows $R=2\\left(\\dot{H}_{1}+\\dot{H}_{2}+\\dot{H}_{3}+H_1^2+H_2^2+H_3^2+H_1H_2+H_1H_3+H_2H_3)\\right).$ Now we want present the knot and unknotted universe solutions of the system (REF )–() or its equivalent (REF )–().", "Consider some examples." ], [ "The trefoil knot universe", "Our aim in this section is to construct simplest examples of the knot universes, namely, the trefoil knot universes.", "Consider examples." ], [ "Example 1.", "Let us we assume that our universe is filled by the fluid with the following parametric EoS $p_1&=&-\\frac{D_1}{E_1},\\\\p_2&=&-\\frac{D_2}{E_2},\\\\p_3&=&-\\frac{D_3}{E_3},\\\\\\rho &=&\\frac{D_0}{E_0},$ where $D_1&=&(-12\\sin ^2(3\\tau )+36\\cos (3\\tau )+18\\cos ^2(3\\tau ))\\cos (2\\tau )-49\\sin (2\\tau )\\\\& &(26/49+\\cos (3\\tau ))\\sin (3\\tau ),\\\\E_1&=&\\sin (3\\tau )(2+\\cos (3\\tau ))\\sin (2\\tau ),\\\\D_2&=&-18\\sin (2\\tau )\\cos ^2(3\\tau )+(-49\\sin (3\\tau )\\cos (2\\tau )-36\\sin (2\\tau ))\\cos (3\\tau )-\\\\& &26\\sin (3\\tau )\\cos (2\\tau )+12\\sin ^2(3\\tau )\\sin (2\\tau ),\\\\E_2&=&\\sin (3\\tau )(2+\\cos (3\\tau ))\\cos (2\\tau ),\\\\D_3&=&-30\\sin (3\\tau )(2+\\cos (3\\tau ))\\cos ^2(2\\tau )-38\\sin (2\\tau )(\\cos ^2(3\\tau )-(27/38)\\sin ^2(3\\tau )+\\\\& &(58/19)\\cos (3\\tau )+40/19)\\cos (2\\tau )+30\\sin (3\\tau )\\sin ^2(2\\tau )(2+\\cos (3\\tau )),\\\\E_3&=&(2+\\cos (3\\tau ))^2\\cos (2\\tau )\\sin (2\\tau ),\\\\D_0&=&(6\\cos ^2(2\\tau )-6\\sin ^2(2\\tau ))\\cos ^3(3\\tau )+(24\\cos ^2(2\\tau )-22\\sin (3\\tau )\\sin (2\\tau )\\cos (2\\tau )-\\\\& &24\\sin ^2(2\\tau ))\\cos ^2(3\\tau )+((-6\\sin ^2(3\\tau )+24)\\cos ^2(2\\tau )-52\\sin (3\\tau )\\sin (2\\tau )\\\\& &\\cos (2\\tau )+(6\\sin ^2(3\\tau )-24)\\sin ^2(2\\tau ))\\cos (3\\tau )-(12(\\cos (2\\tau )-(3/4)\\sin (3\\tau )\\\\& &\\sin (2\\tau )))(\\sin (3\\tau )\\cos (2\\tau )+(4/3)\\sin (2\\tau ))\\sin (3\\tau ),\\\\E_0&=&(2+\\cos (3\\tau ))^2\\cos (2\\tau )\\sin (2\\tau )\\sin (3\\tau ).$ Substituting these expressions for the pressures and density of energy into the system (REF )–(), we obtain the following its solution $A&=&A_0+[2+\\cos (3\\tau )]\\cos (2\\tau ), \\\\B&=&B_0+[2+\\cos (3\\tau )]\\sin (2\\tau ), \\\\C&=&C_0+\\sin (3\\tau ),$ where $A_0, B_0, C_0$ are some real constants.", "We see that this solution describes the trefoil knot.", "In fact the solution (REF )–() is the parametric equation of the trefoil knot.", "In Fig.", "REF we plot the trefoil knot for Eqs.", "(REF )–(), where we assume $A_0=B_0=C_0=0$ and the initial conditions are $A(0)=3, B(0)=C(0)=0$ .", "The Hubble parameters for the solution (REF )–() with (REF ) read as Figure: The trefoil knot for Eqs.", "()–(), where A 0 =B 0 =C 0 =0A_0=B_0=C_0=0.$H_1&=&-2\\tan (2\\tau )-\\frac{2\\sin (3\\tau )}{2+\\cos (3\\tau )}, \\\\H_2&=&-2\\cot (2\\tau )-\\frac{2\\sin (3\\tau )}{2+\\cos (3\\tau )}, \\\\H_3&=&3\\cot (3\\tau ).$ Figure: The evolution of the Hubble parameters for Eqs.", "()–().In Fig.REF we plot the evolution of $H_i$ for the solution (REF )–() with (REF ).", "It is interesting to study the evolution of the volume of the trefoil knot universe.", "For our case it is given by $V=[2+\\cos (3\\tau )]^2\\cos (2\\tau )\\sin (2\\tau )\\sin (3\\tau ).$ Figure: The evolution of the volume of the trefoil knot universe with respect to the cosmic time τ\\tau for Eq.", "().In Fig.", "REF we plot the evolution of the volume of the trefoil knot universe with respect to the cosmic time $\\tau $ for Eq.", "(REF ) with (REF ).", "To get $V\\ge 0$ , we must consider $A_0, B_0, C_0>0$ , if exactly e.g.", "as $A_0>3, B_0>3, C_0>1$ .", "But below for simplicity we take the case (3.16).", "The other interesting quantity is the scalar curvature.", "For the trefoil knot solution (3.13)-(3.15), it has the form $ R&=&(6(12\\sin ^2(3\\tau )\\sin ^2(2\\tau )-28\\sin (3\\tau )\\cos (2\\tau )\\sin (2\\tau )+3\\sin ^3(3\\tau )\\cos (2\\tau )\\sin (2\\tau )-\\\\& &-12\\sin ^2(3\\tau )\\cos ^2(2\\tau )-8\\cos (3\\tau )\\sin ^2(2\\tau )-8\\cos ^2(3\\tau )\\sin ^2(2\\tau )-2\\cos ^3(3\\tau )\\sin ^2(2\\tau )+\\\\& &+8\\cos (3\\tau )\\cos ^2(2\\tau )+8\\cos ^2(3\\tau )\\cos ^2(2\\tau )+2\\cos ^3(3\\tau )\\cos ^2(2\\tau )-\\\\& &-52\\sin (3\\tau )\\cos (2\\tau )\\cos (3\\tau )\\sin (2\\tau )-19\\cos (2\\tau )\\sin (2\\tau )\\sin (3\\tau )\\cos ^2(3\\tau )+\\\\& &+6\\sin ^2(2\\tau )\\sin ^2(3\\tau )\\cos (3\\tau )-\\\\& &-6\\cos ^2(2\\tau )\\sin ^2(3\\tau )\\cos (3\\tau )))/(\\sin (2\\tau )\\cos (2\\tau )(2+\\cos (3\\tau ))^2\\sin (3\\tau )).$ Figure: The evolution of the RR with respect of the cosmic time τ\\tau for Eq.", "()In Fig.REF we plot the evolution of the $R$ with respect of the cosmic time $\\tau $ .", "So we have shown that the universe can live in the trefoil knot orbit according to the solution (3.13)-(3.15).", "It is interesting to note that this trefoil knot solution admits infinite number accelerated and decelerated expansion phases of the universe.", "To show this, as an example let us consider the solution for $C$ from (3.13)-(3.15) that is $C=C_0+\\sin (3\\tau )$ .", "In this case we have $\\ddot{C}=-9\\sin (3\\tau )$ so that $\\ddot{C}>0$ (accelerating phase) as $\\tau \\in (\\frac{\\pi }{3}+\\frac{2n\\pi }{3},\\frac{2\\pi }{3}+\\frac{2n\\pi }{3})$ and $\\ddot{C}<0$ (decelerating phase) as $\\tau \\in (\\frac{2n\\pi }{3}, \\frac{\\pi }{3}+\\frac{2n\\pi }{3})$ with the transion points $\\dot{C}=3\\cos (3\\tau _i)=0$ as $\\tau _i=(0.5 \\pi +n\\pi )/3$ , where $n$ is integer that is $n=0, \\pm 1, \\pm 2, \\pm 3, ...$ ." ], [ "Example 2.", "Now we consider the following parametric EoS $p_1&=&-\\frac{D_1}{E_1},\\\\p_2&=&-\\frac{D_2}{E_2},\\\\p_3&=&-\\frac{D_3}{E_3},\\\\\\rho &=&\\frac{D_0}{E_0},$ where $D_1&=&-\\sin ^2(2\\tau )\\cos ^2(3\\tau )+(-2\\cos (2\\tau )-4\\sin ^2(2\\tau )-3-\\sin (3\\tau )\\sin (2\\tau ))\\cos (3\\tau )+\\\\&&+\\sin (3\\tau )\\sin (2\\tau )-4\\cos (2\\tau )-\\sin ^2(3\\tau )-4\\sin ^2(2\\tau ),\\\\E_1&=&1,\\\\D_2&=&-(2+\\cos ^2(3\\tau ))\\cos ^2(2\\tau )-\\sin (3\\tau )(-1+\\cos (3\\tau ))\\cos (2\\tau )+\\\\&&+(-3+2\\sin (2\\tau ))\\cos (3\\tau )+4\\sin (2\\tau )-\\sin ^2(3\\tau ),\\\\E_2&=&1,\\\\D_3&=&-(2+\\cos ^2(3\\tau ))\\cos ^2(2\\tau )+(-\\sin (2\\tau )\\cos ^2(3\\tau )+(-4\\sin (2\\tau )-2)\\cos (3\\tau )+3\\sin (3\\tau )-\\\\&&-4-4\\sin (2\\tau ))\\cos (2\\tau )+2\\sin (2\\tau )\\cos (3\\tau )+(4+3\\sin (3\\tau ))\\sin (2\\tau )-\\sin ^2(3\\tau ),\\\\E_3&=&1,\\\\D_0&=&(2+\\cos (3\\tau ))(((2+\\cos (3\\tau ))\\sin (2\\tau )+\\sin (3\\tau ))\\cos (2\\tau )+\\sin (3\\tau )\\sin (2\\tau )),\\\\E_0&=&1.$ Substituting these expressions for the pressures and density of energy into the system (REF )–(), we obtain the following its solution $H_1&=&[2+\\cos (3\\tau )]\\cos (2\\tau )=2\\cos (2\\tau )+0.5[\\cos (5\\tau )+\\cos (\\tau )], \\\\H_2&=&[2+\\cos (3\\tau )]\\sin (2\\tau )=2\\sin (2\\tau )+0.5[\\sin (5\\tau )-\\sin (\\tau )], \\\\H_3&=&\\sin (3\\tau ).$ We see that this solution again describes the trefoil knot but for the \"coordinates\" $H_i$ .", "Note that the scale factors we can recovered from (REF ).", "We get $A&=&A_0e^{\\sin (2\\tau )+0.1\\sin (5\\tau )+0.5\\sin (\\tau )}, \\\\B&=&B_0e^{-[\\cos (2\\tau )+0.1\\cos (5\\tau )-0.5\\cos (\\tau )]}, \\\\C&=&C_0e^{-\\frac{1}{3}\\cos (3\\tau )},$ where $A_0, B_0, C_0$ are some real constants.", "In Fig.REF we plot the evolution of $A,B,C$ accordingly to (REF )–() and for the initial conditions $A(0)=1, B(0)=e^{-0.6}, C(0)=e^{-1/3}$ , where we assume that $A_0=B_0=C_0=1$ .", "Figure: The evolution of A,B,CA,B,C accordingly to ()–(), [t∈0..2πt \\in 0..2\\pi ].For this example, the volume of the universe is given by $V=V_0e^{\\lbrace \\sin (2\\tau )+0.1\\sin (5\\tau )+0.5\\sin (\\tau )-[\\cos (2\\tau )+0.1\\cos (5\\tau )-0.5\\cos (\\tau )]-\\frac{1}{3}\\cos (3\\tau )\\rbrace }.$ The evolution of the volume for (REF ) is presented in Fig.", "REF for $A_0=B_0=C_0=V_0=1$ and for the intial condition $V(0)=e^{-14/15}$ .", "Figure: The evolution of the volume for () with A 0 =B 0 =C 0 =V 0 =1A_0=B_0=C_0=V_0=1.The scalar curvature has the form $R&=&(2\\cos ^2(2\\tau )+2\\sin ^2(2\\tau )+2\\cos (2\\tau )\\sin (2\\tau ))\\cos ^2(3\\tau )+(8\\cos ^2(2\\tau )+(2\\sin (3\\tau )+4+\\\\& &+8\\sin (2\\tau ))\\cos (2\\tau )+6+8\\sin ^2(2\\tau )+(-4+2\\sin (3\\tau ))\\sin (2\\tau ))\\cos (3\\tau )+8\\cos ^2(2\\tau )+\\\\& &+(-2\\sin (3\\tau )+8+8\\sin (2\\tau ))\\cos (2\\tau )+8\\sin ^2(2\\tau )+\\\\& &+(-8-2\\sin (3\\tau ))\\sin (2\\tau )+2\\sin ^2(3\\tau ).$ Figure: The evolution of the RR with respect of the cosmic time τ\\tau for Eq.", "()In Fig.REF we plot the evolution of the $R$ with respect of the cosmic time $\\tau $ .", "Finally we conclude that the Einstein equations for the Bianchi I type metric admit the trefoil knot solution of the form (3.34)-(3.36) or (3.37)-(3.39).", "These solutions describe the accelerated and decelerated phases of the expansion of the universe." ], [ "Example 3.", "Now we present a new kind of the trefoil knot universes.", "Let the system (REF )–() has the solution $A&=&A_0+[2+\\mbox{cn}(3\\tau )]\\mbox{cn}(2\\tau ), \\\\B&=&B_0+[2+\\mbox{cn}(3\\tau )]\\mbox{sn} (2\\tau ), \\\\C&=&C_0+\\mbox{sn}(3\\tau ),$ where $\\mbox{cn}(t)\\equiv \\mbox{cn}(t,k)$ and $\\mbox{sn}(t)\\equiv \\mbox{sn}(t,k)$ are the Jacobian elliptic functions which are doubly periodic functions, $k$ is the elliptic modulus.", "Fig.", "REF shows the knotted closed curve corresponding to the solution (REF )–() with (REF ).", "Substituting the formulas (REF )–() into the system (REF )–() we get the corresponding expressions for $\\rho $ and $p_i$ that gives us the parametric EoS.", "This parametric EoS reads as Figure: The knotted closed curve corresponding to the solution ()–() with (), t∈[0,4πt \\in [0,4\\pi ], k=1/3k=1/3.$p_1&=&-\\frac{D_1}{E_1},\\\\p_2&=&-\\frac{D_2}{E_2},\\\\p_3&=&-\\frac{D_3}{E_3},\\\\\\rho &=&\\frac{D_0}{E_0},$ where $D_1&=&9k^2cn(3\\tau , k)sn^3(3\\tau , k)sn(2\\tau , k)-12\\frac{\\partial }{\\partial \\tau }am(3\\tau , k)sn^2(3\\tau , k)cn(2\\tau , k)\\frac{\\partial }{\\partial \\tau }am(2\\tau , k)-\\\\&&-4sn(2\\tau , k)((9/4)cn^3(3\\tau , k)k^2+(9/2)cn^2(3\\tau , k)k^2+\\\\&&+((45/4)\\frac{\\partial }{\\partial \\tau }am^2(3\\tau , k)+cn^2(2\\tau , k)k^2+\\frac{\\partial }{\\partial \\tau }am^2(2\\tau , k))cn(3\\tau , k)+2\\frac{\\partial }{\\partial \\tau }am^2(2\\tau , k)+\\\\&&+(9/2)\\frac{\\partial }{\\partial \\tau }am^2(3\\tau , k)+2cn^2(2\\tau , k)k^2)sn(3\\tau , k)+18cn(3\\tau , k)\\frac{\\partial }{\\partial \\tau }am(3\\tau , k)\\\\&&(2+cn(3\\tau , k))cn(2\\tau , k)\\frac{\\partial }{\\partial \\tau }am(2\\tau , k),\\\\E_1&=&(2+cn(3\\tau , k))sn(2\\tau , k)sn(3\\tau , k),\\\\D_2&=&9k^2cn(3\\tau , k)sn^3(3\\tau , k)cn(2\\tau , k)+12\\frac{\\partial }{\\partial \\tau }am(3\\tau , k)sn^2(3\\tau , k)\\frac{\\partial }{\\partial \\tau }am(2\\tau , k)sn(2\\tau , k)-\\\\&&-9cn(2\\tau , k)cn^3(3\\tau , k)k^2+2cn^2(3\\tau , k)k^2+((4/9)\\frac{\\partial }{\\partial \\tau }am^2(2\\tau , k)+5\\frac{\\partial }{\\partial \\tau }am^2(3\\tau , k)-\\\\&&-(4/9)k^2sn^2(2\\tau , k))cn(3\\tau , k)+(8/9)\\frac{\\partial }{\\partial \\tau }am^2(2\\tau , k)+2\\frac{\\partial }{\\partial \\tau }am^2(3\\tau , k)-\\\\&&-(8/9)k^2sn^2(2\\tau , k))sn(3\\tau , k)-\\\\&&-18cn(3\\tau , k)\\frac{\\partial }{\\partial \\tau }am(3\\tau , k)(2+cn(3\\tau , k))\\frac{\\partial }{\\partial \\tau }am(2\\tau , k)sn(2\\tau , k),\\\\E_2&=&sn(3\\tau , k)(2+cn(3\\tau , k))cn(2\\tau , k),\\\\D_3&=&-4k^2sn(2\\tau , k)(2+cn(3\\tau , k))^2cn^3(2\\tau , k)-30\\frac{\\partial }{\\partial \\tau }am(2\\tau , k)sn(3\\tau , k)\\frac{\\partial }{\\partial \\tau }am(3\\tau , k)\\\\&&(2+cn(3\\tau , k))cn^2(2\\tau , k)+4sn(2\\tau , k)(k^2(2+cn^2(3\\tau , k))sn^2(2\\tau , k)+\\\\&&+(-(9/2)\\frac{\\partial }{\\partial \\tau }am^2(3\\tau , k)+(9/2)k^2sn^2(3\\tau , k)-5\\frac{\\partial }{\\partial \\tau }am^2(2\\tau , k))cn^2(3\\tau , k)+\\\\&&+(-20\\frac{\\partial }{\\partial \\tau }am^2(2\\tau , k)+9k^2sn^2(3\\tau , k)-9\\frac{\\partial }{\\partial \\tau }am^2(3\\tau , k))cn(3\\tau , k)+\\\\&&+(27/4)sn^2(3\\tau , k)\\frac{\\partial }{\\partial \\tau }am(3\\tau , k)^2-20\\frac{\\partial }{\\partial \\tau }am^2(2\\tau , k))cn(2\\tau , k)+\\\\&&+30\\frac{\\partial }{\\partial \\tau }am(3\\tau , k)sn(3\\tau , k)\\frac{\\partial }{\\partial \\tau }am(2\\tau , k)sn^2(2\\tau , k)(2+cn(3\\tau , k)),\\\\E_3&=&(2+cn(3\\tau , k))^2cn(2\\tau , k)sn(2\\tau , k),\\\\D_0&=&-4sn(3\\tau , k)sn(2\\tau , k)cn(2\\tau , k)(2+cn(3\\tau , k))^2\\frac{\\partial }{\\partial \\tau }am^2(2\\tau , k)+6\\frac{\\partial }{\\partial \\tau }am(\\tau , k)(2+cn(3\\tau , k))\\\\&&(cn^2(3\\tau , k)+2cn(3\\tau , k)-sn^2(3\\tau , k))(cn(2\\tau , k)-sn(2\\tau , k))(cn(2\\tau , k)+sn(2\\tau , k))\\\\&&\\frac{\\partial }{\\partial \\tau }am(2\\tau , k)-18sn(3\\tau , k)sn(2\\tau , k)(-(1/2)sn^2(3\\tau , k)+cn^2(3\\tau , k)+\\\\&&+2cn(3\\tau , k))cn(2\\tau , k)\\frac{\\partial }{\\partial \\tau }am^2(3\\tau , k),\\\\E_0&=&(2+cn^2(3\\tau , k))cn(2\\tau , k)sn(2\\tau , k)sn(3\\tau , k).$ The volume of the universe for the solution (REF )–() with (REF ) looks like $V=[2+\\mbox{cn}(3\\tau )]^2\\mbox{cn}(2\\tau )\\mbox{sn}(2\\tau )\\mbox{sn}(3\\tau ).$ The evolution of the volume for (REF ) is presented in Fig.", "REF Figure: The evolution of the volume of the trefoil knot universe with respect to the cosmic time τ\\tau for Eq.", "()The scalar curvature has the form $ R&=&(-8\\mbox{sn}(2\\tau , k)\\mbox{sn}(3\\tau , k)k^2(2+\\mbox{cn}(3\\tau , k))^2\\mbox{cn}^3(2\\tau , k)+\\\\& &+12\\mbox{dn}(2\\tau , k)\\mbox{dn}(3\\tau , k)(2+\\mbox{cn}(3\\tau , k))(\\mbox{cn}^2(3\\tau , k)+\\\\& &+2\\mbox{cn}(3\\tau , k)-3\\mbox{sn}^2(3\\tau , k))\\mbox{cn}^2(2\\tau , k)-18\\mbox{sn}(3\\tau , k)(-(4/9)k^2(2+\\mbox{cn}(3\\tau , k))^2\\mbox{sn}^2(2\\tau , k)+\\\\& &+(-4k^2\\mbox{cn}(3\\tau , k)-2\\mbox{cn}^2(3\\tau , k)k^2-\\mbox{dn}^2(3\\tau , k))\\mbox{sn}^2(3\\tau , k)+\\\\& &+(2+\\mbox{cn}(3\\tau , k))(\\mbox{cn}^3(3\\tau , k)k^2+2\\mbox{cn}^2(3\\tau , k)k^2+\\\\& &+(5\\mbox{dn}^2(3\\tau , k)+(4/3)\\mbox{dn}^2(2\\tau , k))\\mbox{cn}(3\\tau , k)+\\\\& &+(8/3)\\mbox{dn}^2(2\\tau , k)+2\\mbox{dn}^2(3\\tau , k)))\\mbox{sn}(2\\tau , k)\\mbox{cn}(2\\tau , k)-\\\\& &-12\\mbox{dn}(2\\tau , k)\\mbox{sn}^2(2\\tau , k)\\mbox{dn}(3\\tau , k)(2+\\mbox{cn}(3\\tau , k))(\\mbox{cn}^2(3\\tau , k)+\\\\& &+2\\mbox{cn}(3\\tau , k)-3\\mbox{sn}^2(3\\tau , k)))/(\\mbox{cn}(2\\tau , k)\\mbox{sn}(2\\tau , k)(2+\\mbox{cn}(3\\tau , k))^2\\mbox{sn}(3\\tau , k)).$ Figure: The evolution of the RR with respect of the cosmic time τ\\tau for Eq.", "()In Fig.REF we plot the evolution of the $R$ with respect of the cosmic time $\\tau $ ." ], [ "Example 4.", "Our fourth example is given by $H_1&=&[2+\\mbox{cn}(3\\tau )]\\mbox{cn}(2\\tau ), \\\\H_2&=&[2+\\mbox{cn}(3\\tau )]\\mbox{sn} (2\\tau ), \\\\H_3&=&\\mbox{sn}(3\\tau )$ which again the knotted closed curve in Fig.", "REF but for the \"coordinates\" $H_i$ .", "Note that the corresponding parametric EoS looks like $p_1&=&-\\frac{D_1}{E_1},\\\\p_2&=&-\\frac{D_2}{E_2},\\\\p_3&=&-\\frac{D_3}{E_3},\\\\\\rho &=&\\frac{D_0}{E_0},$ where $D_1&=&-(2+cn(3\\tau , k))^2sn^2(2\\tau , k)-sn(3\\tau , k)(-3\\frac{\\partial }{\\partial \\tau }am(3\\tau , k)+2+cn(3\\tau , k))sn(2\\tau , k)+\\\\&&+(-2cn(2\\tau , k)\\frac{\\partial }{\\partial \\tau }am(2\\tau , k)-3\\frac{\\partial }{\\partial \\tau }am(3\\tau , k))cn(3\\tau , k)-4cn(2\\tau , k)\\frac{\\partial }{\\partial \\tau }am(2\\tau , k)-\\\\&&-sn^2(3\\tau , k),\\\\E_1&=&1,\\\\D_2&=&-(2+cn(3\\tau , k))^2cn^2(2\\tau , k)-sn(3\\tau , k)(-3\\frac{\\partial }{\\partial \\tau }am(3\\tau , k)+2+cn(3\\tau , k))cn(2\\tau , k)+\\\\&&+(-3\\frac{\\partial }{\\partial \\tau }am(3\\tau , k)+2\\frac{\\partial }{\\partial \\tau }am(2\\tau , k)sn(2\\tau , k))cn(3\\tau , k)+4\\frac{\\partial }{\\partial \\tau }am(2\\tau , k)sn(2\\tau , k)-\\\\&&-sn^2(3\\tau , k),\\\\E_2&=&1,\\\\D_3&=&-(2+cn(3\\tau , k))^2cn^2(2\\tau , k)+(-sn(2\\tau , k)cn^2(3\\tau , k)+(-4sn(2\\tau , k)-2\\frac{\\partial }{\\partial \\tau }am(2\\tau , k))\\\\&&cn(3\\tau , k)+3sn(3\\tau , k)\\frac{\\partial }{\\partial \\tau }am(3\\tau , k)-4\\frac{\\partial }{\\partial \\tau }am(2\\tau , k)-4sn(2\\tau , k))cn(2\\tau , k)+\\\\&&+2\\frac{\\partial }{\\partial \\tau }am(2\\tau , k)sn(2\\tau , k)cn(3\\tau , k)+(4\\frac{\\partial }{\\partial \\tau }am(2\\tau , k)+3sn(3\\tau , k)\\frac{\\partial }{\\partial \\tau }am(3\\tau , k))sn(2\\tau , k)-\\\\&&-sn^2(3\\tau , k),\\\\E_3&=&1,\\\\D_0&=&(((2+cn(3\\tau , k))sn(2\\tau , k)+sn(3\\tau , k))cn(2\\tau , k)+sn(2\\tau , k)sn(3\\tau , k))\\\\&&(2+cn(3\\tau , k)),\\\\E_0&=&1.$ Figure: The evolution of p i ,ρp_i, \\rho for ()–(), t∈[0,2πt \\in [0,2\\pi ], k=1/3k=1/3, ρ\\rho (red), p 1 p_1(blue), p 2 p_2(green), p 3 p_3(black).In Fig.", "REF we plot the evolution of $p_i, \\rho $ for (REF )–().", "The scalar curvature has the form $R&=&2(2+\\mbox{cn}(3\\tau , k))^2\\mbox{cn}^2(2\\tau , k)+(2\\mbox{sn}(2\\tau , k)\\mbox{cn}^2(3\\tau , k)+(8\\mbox{sn}(2\\tau , k)+4\\mbox{dn}(2\\tau , k)+\\\\& &+2\\mbox{sn}(3\\tau , k))\\mbox{cn}(3\\tau , k)+8\\mbox{sn}(2\\tau , k)+(4-6\\mbox{dn}(3\\tau , k))\\mbox{sn}(3\\tau , k)+\\\\& &+8\\mbox{dn}(2\\tau , k))\\mbox{cn}(2\\tau , k)+2\\mbox{sn}^2(2\\tau , k)\\mbox{cn}^2(3\\tau , k)+(8\\mbox{sn}^2(2\\tau , k)+\\\\& &+(-4\\mbox{dn}(2\\tau , k)+2\\mbox{sn}(3\\tau , k))\\mbox{sn}(2\\tau , k)+6\\mbox{dn}(3\\tau , k))\\mbox{cn}(3\\tau , k)+\\\\& &+8\\mbox{sn}^2(2\\tau , k)+((4-6\\mbox{dn}(3\\tau , k))\\mbox{sn}(3\\tau , k)-8\\mbox{dn}(2\\tau , k))\\mbox{sn}(2\\tau , k)+2\\mbox{sn}(3\\tau , k)^2$ Figure: The evolution of the RR with respect of the cosmic time τ\\tau for Eq.", "()In Fig.REF we plot the evolution of the $R$ with respect of the cosmic time $\\tau $ ." ], [ "The figure-eight knot universe", "Our aim in this section is to demonstrate some examples the figure-eight knot universes for the Bianchi type I metric (REF ).", "We give some particular figure-eight knot universe models." ], [ "Example 1.", "Again, let us we assume that our universe is filled by the fluid with the following parametric EoS $\\rho &=&\\frac{D_8}{E_8},\\\\p_1&=&-\\frac{D_9}{E_9},\\\\p_2&=&-\\frac{D_{10}}{E_{10}},\\\\p_3&=&-\\frac{D_{11}}{E_{11}},$ where $D_8&=&(-2\\sin (2\\tau )\\cos (3\\tau )-(3(2+\\cos (2\\tau )))\\sin (3\\tau ))(-2\\sin (2\\tau )\\sin (3\\tau )+\\\\& &(3(2+\\cos (2\\tau )))\\cos (3\\tau ))\\sin (4\\tau )+12\\cos (3\\tau )((2+\\cos (2\\tau ))\\cos (3\\tau )-\\\\& &(2/3)\\sin (2\\tau )\\sin (3\\tau ))(2+\\cos (2\\tau ))\\cos (4\\tau )-(12(2+\\cos (2\\tau )))\\cos (4\\tau )*\\\\& &((2+\\cos (2\\tau ))\\sin (3\\tau )+(2/3)\\sin (2\\tau )\\cos (3\\tau ))\\sin (3\\tau ),\\\\E_8&=&(2+\\cos (2\\tau ))^2\\cos (3\\tau )\\sin (3\\tau )\\sin (4\\tau ),\\\\D_9&=&((72+36\\cos (2\\tau ))\\cos (4\\tau )-12\\sin (4\\tau )\\sin (2\\tau ))\\cos (3\\tau )-(29((24/29)*\\\\& &\\cos (4\\tau )\\sin (2\\tau )+(cos(2\\tau )+50/29)\\sin (4\\tau )))\\sin (3\\tau ),\\\\E_9&=&\\sin (4\\tau )(2+\\cos (2\\tau ))\\sin (3\\tau ),\\\\D_{10}&=&(-24\\cos (4\\tau )\\sin (2\\tau )+\\sin (4\\tau )(-29\\cos (2\\tau )-50))\\cos (3\\tau )-\\\\& &(36((2+\\cos (2\\tau ))\\cos (4\\tau )-(1/3)\\sin (4\\tau )\\sin (2\\tau )))\\sin (3\\tau ),\\\\E_{10}&=&\\sin (4\\tau )(2+\\cos (2\\tau ))\\cos (3\\tau ),\\\\D_{11}&=&-(30(2+\\cos (2\\tau )))\\sin (2\\tau )\\cos (3\\tau )^2+\\sin (3\\tau )(12\\sin (2\\tau )^2-196\\cos (2\\tau )-\\\\& &180-53\\cos (2\\tau )^2)\\cos (3\\tau )+30\\sin (2\\tau )\\sin (3\\tau )^2(2+\\cos (2\\tau )),\\\\E_{11}&=&(2+\\cos (2\\tau ))^2\\cos (3\\tau )\\sin (3\\tau ).$ Substituting these expressions for the pressuries and the density of energy into the system (REF )–(), we obtain the following its solution [30] $A&=&A_0+[2+\\cos (2\\tau )]\\cos (3\\tau ), \\\\B&=&B_0+[2+\\cos (2\\tau )]\\sin (3\\tau ), \\\\C&=&C_0+\\sin (4\\tau ).$ This solution is nothing but the parametric equation of the figure-eight knot as we can see from Fig.", "REF , where we assume that $A_0=B_0=C_0=0$ and the initial conditions have the form $A(0)=3, B(0)=0, C(0)=0$ .", "And for that reason in [30] we called such models as the figure-eight knot universes.", "Figure: The figure-eight knot for ()–() with ().Note that the \"coordinates\" $A,B,C$ with (3.16) satisfy the equation $4(h-2)^4-4(h-2)^2+z^2=0,$ where $h=2+\\cos (2\\tau )$ .", "Let us calculate the volume of the universe.", "For our case it is given by $V=[2+\\cos (2\\tau )]^2\\cos (3\\tau )\\sin (3\\tau )\\sin (4\\tau ),$ where we used (REF ).", "Figure: The evolution of the volume for the solution ()–() with (), t∈[0,2πt \\in [0,2\\pi ].In Fig.9 we present the evolution of the volume for the solution (REF )–() with (REF ).", "The scalar curvature has the form $R&=&((24(\\cos (4\\tau )\\cos (2\\tau )-(3/2)\\sin (4\\tau )\\sin (2\\tau )+2\\cos (4\\tau )))(2+\\cos (2\\tau ))\\cos ^2(3\\tau )-\\\\& &-102\\sin (3\\tau )(\\sin (4\\tau )\\cos ^2(2\\tau )+((188/51)\\sin (4\\tau )+(16/51)\\cos (4\\tau )\\sin (2\\tau ))\\cos (2\\tau )+\\\\& &+(32/51)\\cos (4\\tau )\\sin (2\\tau )+(172/51)\\sin (4\\tau )-\\\\& & -(4/51)\\sin ^2(2\\tau )\\sin (4\\tau ))\\cos (3\\tau )-24\\sin ^2(3\\tau )(\\cos (4\\tau )\\cos (2\\tau )-\\\\& &-(3/2)\\sin (4\\tau )\\sin (2\\tau )+2\\cos (4\\tau ))(2+\\cos (2\\tau )))/(\\sin (3\\tau )\\cos (3\\tau )*\\\\& &*(2+\\cos (2\\tau ))^2\\sin (4\\tau )).$ Figure: The evolution of the RR with respect of the cosmic time τ\\tau for Eq.", "()In Fig.REF we plot the evolution of the $R$ with respect of the cosmic time $\\tau $ .", "So we found the figure-eight knot solution of the Einstein equations which again describe the accelerated and decelerated expansion phases of the universe." ], [ "Example 2", "Now we consider the system (REF )–().", "Let its solution is given by $H_1&=&[2+\\cos (2\\tau )]\\cos (3\\tau )=2\\cos (3\\tau )+\\cos (5\\tau )+\\cos (\\tau ), \\\\H_2&=&[2+\\cos (2\\tau )]\\sin (3\\tau )=2\\sin (3\\tau )+\\sin (\\tau )+\\sin (5\\tau ), \\\\H_3&=&\\sin (4\\tau ).$ Then the coorresponding scale factors read as $A&=&A_0e^{\\frac{2}{3}\\sin (3\\tau )+0.2\\sin (5\\tau )+\\sin (\\tau )}, \\\\B&=&B_0e^{-[\\frac{2}{3}\\cos (3\\tau )+0.2cos(5\\tau )+\\cos (\\tau )]}, \\\\C&=&C_0e^{-0.25\\cos (4\\tau )}.$ For this solution the parametric EoS looks like $\\rho &=&\\frac{D_0}{E_0},\\\\p_1&=&-\\frac{D_1}{E_1},\\\\p_2&=&-\\frac{D_2}{E_2},\\\\p_3&=&-\\frac{D_3}{E_3},$ where $D_0&=&(((2+\\cos (2\\tau ))\\sin (3\\tau )+\\sin (4\\tau ))\\cos (3\\tau )+\\sin (3\\tau )\\sin (4\\tau ))(2+\\cos (2\\tau )),\\\\E_0&=&1,\\\\D_1&=&-(2+\\cos (2\\tau ))^2\\sin ^2(3\\tau )+(2\\sin (2\\tau )-2\\sin (4\\tau )-\\sin (4\\tau )\\cos (2\\tau ))\\sin (3\\tau )-\\\\&&-6\\cos (3\\tau )-3\\cos (3\\tau )\\cos (2\\tau )-4\\cos (4\\tau )-\\sin ^2(4\\tau ),\\\\E_1&=&1,\\\\D_2&=&-(2+\\cos (2\\tau ))^2\\cos ^2(3\\tau )+(2\\sin (2\\tau )-2\\sin (4\\tau )-\\sin (4\\tau )\\cos (2\\tau ))\\cos (3\\tau )-\\\\&&-4\\cos (4\\tau )+6\\sin (3\\tau )+3\\sin (3\\tau )\\cos (2\\tau )-\\sin ^2(4\\tau ),\\\\E_2&=&1,\\\\D_3&=&-3\\sin (\\tau )-64\\sin (\\tau )\\cos ^9(\\tau )+36\\sin (\\tau )\\cos ^5(\\tau )+40\\sin (\\tau )\\cos ^4(\\tau )+4\\sin (\\tau )\\cos ^3(\\tau )-\\\\&&-6\\sin (\\tau )\\cos ^2(\\tau )-3\\sin (\\tau )\\cos (\\tau )-25\\cos ^2(\\tau )+5\\cos (\\tau )-40\\cos ^5(\\tau )-64\\cos ^10(\\tau )+\\\\&&+96\\cos ^8(\\tau )-84\\cos ^6(\\tau )+68\\cos ^4(\\tau )+26\\cos ^3(\\tau ),\\\\E_3&=&1.$ Figure: The plot of the EoS ()–(), t∈[0,2πt \\in [0, 2\\pi ], ρ\\rho (red), p 1 p_1(blue), p 2 p_2(green), p 3 p_3(black).In Fig.", "REF we plot the EoS (REF )–().", "For this example, the evolution of the volume of the universe is given by $V=V_0e^{\\frac{2}{3}\\sin (3\\tau )+0.2\\sin (5\\tau )+\\sin (\\tau )-\\frac{2}{3}\\cos (3\\tau )-0.2cos(5\\tau )-\\cos (\\tau )-0.25\\cos (4\\tau )}.$ The evolution of the volume is presented in Fig.REF for $A_0=B_0=C_0=V_0=1$ and for the intial condition $V(0)=e^{-127/60}$ .", "Figure: The evolution of the volume for the expression () with V 0 =1V_0=1, t∈[0,2πt \\in [0,2\\pi ].The scalar curvature has the form $R&=&2(2+\\cos (2\\tau ))^2\\cos ^2(3\\tau )+(2(2+\\cos (2\\tau ))^2\\sin (3\\tau )+(6+2\\sin (4\\tau ))\\cos (2\\tau )+\\\\& &+12-4\\sin (2\\tau )+4\\sin (4\\tau ))\\cos (3\\tau )+2(2+\\cos (2\\tau ))^2\\sin ^2(3\\tau )+\\\\& &+((-6+2\\sin (4\\tau ))\\cos (2\\tau )+4\\sin (4\\tau )-4\\sin (2\\tau )-12)*\\\\& &*\\sin (3\\tau )+2\\sin ^2(4\\tau )+8\\cos (4\\tau ).$ Figure: The evolution of the RR with respect of the cosmic time τ\\tau for Eq.", "()In Fig.REF we plot the evolution of the $R$ with respect of the cosmic time $\\tau $ .", "Again we have shown that the Einstein equations admit the figure-eight knot solution and it again describe the accelerated and decelerated expansion phases of the universe." ], [ "Example 3.", "Now we present the figure-eight knot universe induced by the Jacobian elliptic functions.", "Let the system (REF )–() has the solution $A&=&A_0+[2+\\mbox{cn}(2\\tau )]\\mbox{cn}(3\\tau ), \\\\B&=&B_0+[2+\\mbox{cn}(2\\tau )]\\mbox{sn} (3\\tau ), \\\\C&=&C_0+\\mbox{sn}(4\\tau ).$ Figure: The knotted closed curve corresponding to the solution ()–() with (), t∈[0,4πt \\in [0,4\\pi ], k=1/3k=1/3.Note that $\\mbox{cn}(t)$ and $\\mbox{sn}(t)$ are the doubly periodic Jacobian elliptic functions.", "Fig.REF shows the knotted closed curve corresponding to the solution (REF )–() with (REF ).", "Substituting the formulas (REF )–() into the system (REF )–() we get the corresponding expressions for $\\rho $ and $p_i$ that gives us the parametric EoS.", "The evolution of the volume of the universe for (REF ) reads as $V=[2+\\mbox{cn}(2\\tau )]^2\\mbox{cn}(3\\tau )\\mbox{sn}(3\\tau )\\mbox{sn}(4\\tau ).$ The scalar curvature has the form $R&=&(-18\\mbox{sn}(3\\tau , k)\\mbox{sn}(4\\tau , k)k^2(2+\\mbox{cn}(2\\tau , k))^2\\mbox{cn}^3(3\\tau , k)+(24(-(3/2)\\mbox{sn}(2\\tau , k)\\mbox{dn}(2\\tau , k)\\mbox{sn}(4\\tau , k)+\\\\& &+\\mbox{cn}(4\\tau , k)\\mbox{dn}(4\\tau , k)(2+\\mbox{cn}(2\\tau , k))))(2+\\mbox{cn}(2\\tau , k))\\mbox{dn}(3\\tau , k)\\mbox{cn}^2(3\\tau , k)-\\\\& &-(32(-(9/16)\\mbox{sn}(4\\tau , k)k^2(2+\\mbox{cn}(2\\tau , k))^2\\mbox{sn}^2(3\\tau , k)+((\\mbox{cn}^2(4\\tau , k)k^2+(27/16)\\mbox{dn}^2(3\\tau , k)+\\\\& &+\\mbox{dn}^2(4\\tau , k)-(1/2)k^2\\mbox{sn}^2(2\\tau , k)+(1/2)\\mbox{dn}^2(2\\tau , k))\\mbox{cn}^2(2\\tau , k)+(-k^2\\mbox{sn}^2(2\\tau , k)+\\\\& &+(27/4)\\mbox{dn}^2(3\\tau , k)+\\mbox{dn}^2(2\\tau , k)+4\\mbox{dn}^2(4\\tau , k)+4\\mbox{cn}^2(4\\tau , k)k^2)\\mbox{cn}(2\\tau , k)+4\\mbox{dn}^2(4\\tau , k)+\\\\& &+(27/4)\\mbox{dn}^2(3\\tau , k)-(1/4)\\mbox{dn}^2(2\\tau , k)\\mbox{sn}^2(2\\tau , k)+4\\mbox{cn}^2(4\\tau , k)k^2)\\mbox{sn}(4\\tau , k)+\\mbox{cn}(4\\tau , k)\\mbox{dn}(4\\tau , k)\\\\& &\\mbox{dn}(2\\tau , k)\\mbox{sn}(2\\tau , k)(2+\\mbox{cn}(2\\tau , k))))\\mbox{sn}(3\\tau , k)\\mbox{cn}(3\\tau , k)-(24(-(3/2)\\mbox{sn}(2\\tau , k)\\mbox{dn}(2\\tau , k)\\\\& &\\mbox{sn}(4\\tau , k)+\\mbox{cn}(4\\tau , k)\\mbox{dn}(4\\tau , k)(2+\\mbox{cn}(2\\tau , k))))(2+\\mbox{cn}(2\\tau , k))\\mbox{sn}^2(3\\tau , k)\\mbox{dn}(3\\tau , k))/(\\mbox{cn}(3\\tau , k)\\\\& &\\mbox{sn}(3\\tau , k)(2+\\mbox{cn}(2\\tau , k))^2\\mbox{sn}(4\\tau , k)).$ Figure: The evolution of the RR with respect of the cosmic time τ\\tau for Eq.", "()In Fig.REF we plot the evolution of the $R$ with respect of the cosmic time $\\tau $ ." ], [ "Example 4.", "We now consider the following solution of the system (REF )–(): $H_1&=&[2+\\mbox{cn}(2\\tau )]\\mbox{cn}(3\\tau ), \\\\H_2&=&[2+\\mbox{cn}(2\\tau )]\\mbox{sn} (3\\tau ), \\\\H_3&=&\\mbox{sn}(4\\tau )$ which again the trefoil knot universe as shown in Fig.REF but for the \"coordinates\" $H_i$ .", "The corresponding parametric EoS reads as $\\rho &=&\\frac{D_0}{E_0},\\\\p_1&=&-\\frac{D_1}{E_1},\\\\p_2&=&-\\frac{D_2}{E_2},\\\\p_3&=&-\\frac{D_3}{E_3},$ where $D_0&=&(((2+cn(2\\tau , k))sn(3\\tau , k)+sn(4\\tau , k))cn(3\\tau , k)+sn(3\\tau , k)sn(4\\tau , k))\\\\&&(2+cn(2\\tau , k)),\\\\E_0&=&1,\\\\D_1&=&2\\frac{\\partial }{\\partial \\tau }am(2\\tau , k)sn(2\\tau , k)sn(3\\tau , k)-(3(2+cn(2\\tau , k)))cn(3\\tau , k)\\frac{\\partial }{\\partial \\tau }am(3\\tau , k)-\\\\&&-4cn(4\\tau , k)\\frac{\\partial }{\\partial \\tau }am(4\\tau , k)-(2+cn(2\\tau , k))^2sn(3\\tau , k)^2-sn(4\\tau , k)^2-\\\\&&-(2+cn(2\\tau , k))sn(3\\tau , k)sn(4\\tau , k),\\\\E_1&=&1,\\\\D_2&=&-4cn(4\\tau , k)\\frac{\\partial }{\\partial \\tau }am(4\\tau , k)+2\\frac{\\partial }{\\partial \\tau }am(2\\tau , k)sn(2\\tau , k)cn(3\\tau , k)+(3(2+cn(2\\tau , k)))\\\\&&\\frac{\\partial }{\\partial \\tau }am(3\\tau , k)sn(3\\tau , k)-sn^2(4\\tau , k)-(2+cn(2\\tau , k))^2cn^2(3\\tau , k)-\\\\&&-(2+cn(2\\tau , k))cn(3\\tau , k)sn(4\\tau , k),\\\\E_2&=&1,\\\\D_3&=&-(2+cn(2\\tau , k))^2cn^2(3\\tau , k)+(-sn(3\\tau , k)cn^2(2\\tau , k)+(-4sn(3\\tau , k)-\\\\&&-3\\frac{\\partial }{\\partial \\tau }am(3\\tau , k))cn(2\\tau , k)+2\\frac{\\partial }{\\partial \\tau }am(2\\tau , k)sn(2\\tau , k)-6\\frac{\\partial }{\\partial \\tau }am(3\\tau , k)-\\\\&&-4sn(3\\tau , k))cn(3\\tau , k)+3\\frac{\\partial }{\\partial \\tau }am(3\\tau , k)sn(3\\tau , k)cn(2\\tau , k)+\\\\&&+(6\\frac{\\partial }{\\partial \\tau }am(3\\tau , k)+2\\frac{\\partial }{\\partial \\tau }am(2\\tau , k)sn(2\\tau , k))sn(3\\tau , k)-sn^2(4\\tau , k),\\\\E_3&=&1.$ Figure: The plot of the EoS ()–(), t∈[0,2πt \\in [0,2\\pi ], k=1/3 k=1/3, ρ\\rho (red), p 1 p_1(blue), p 2 p_2(green), p 3 p_3(black).Its plot we give in Fig.REF .", "The scalar curvature has the form $R&=&2(2+\\mbox{cn}(2\\tau , k))^2\\mbox{cn}(3\\tau , k)^2+(2(2+\\mbox{cn}(2\\tau , k))^2\\mbox{sn}(3\\tau , k)+(6\\mbox{dn}(3\\tau , k)+2\\mbox{sn}(4\\tau , k))\\mbox{cn}(2\\tau , k)+\\\\& &+12\\mbox{dn}(3\\tau , k)-4\\mbox{dn}(2\\tau , k)\\mbox{sn}(2\\tau , k)+4\\mbox{sn}(4\\tau , k))\\mbox{cn}(3\\tau , k)+2(2+\\mbox{cn}(2\\tau , k))^2\\mbox{sn}(3\\tau , k)^2+\\\\& &+((-6\\mbox{dn}(3\\tau , k)+2\\mbox{sn}(4\\tau , k))\\mbox{cn}(2\\tau , k)+4\\mbox{sn}(4\\tau , k)-4\\mbox{dn}(2\\tau , k)\\mbox{sn}(2\\tau , k)-12\\mbox{dn}(3\\tau , k))*\\\\& &*\\mbox{sn}(3\\tau , k)+2\\mbox{sn}(4\\tau , k)^2+8\\mbox{cn}(4\\tau , k)\\mbox{dn}(4\\tau , k).$ Figure: The evolution of the RR with respect of the cosmic time τ\\tau for Eq.", "()In Fig.REF we plot the evolution of the $R$ with respect of the cosmic time $\\tau $ ." ], [ "Other unknotted models of the universe", "In this section we would like to present some unknotted but closed curve solutions of the Einstein equation for the Bianchi I type metric.", "As an examples we consider the spiky and Mobious strip universe solutions." ], [ "Spiky universe solutions", "Our aim in this subsection is to present some unknotted closed curve solutions namely the spiky universe solutions." ], [ "Example 1", "Let our universe is filled by the fluid with the following parametric EoS $\\rho &=&\\frac{D_8}{E_8},\\\\p_1&=&-\\frac{D_9}{E_9},\\\\p_2&=&-\\frac{D_{10}}{E_{10}},\\\\p_3&=&-\\frac{D_{11}}{E_{11}},$ where $D_8&=&-[\\alpha \\sin ((n-1)\\tau )(n-1)+\\alpha (n-1)\\sin (\\tau )][\\alpha \\cos ((n-1)\\tau )(n-1)-\\\\& &\\alpha (n-1)\\cos (\\tau )]\\sin (\\tau )+[\\alpha \\cos ((n-1)\\tau )(n-1)-\\alpha (n-1)\\cos (\\tau )]\\cos (\\tau )\\times \\\\& &[\\alpha \\cos ((n-1)\\tau )+\\alpha (n-1)\\cos (\\tau )]-[\\alpha \\sin ((n-1)\\tau )(n-1)+\\alpha (n-1)\\sin (\\tau )]\\times \\\\& &[\\alpha \\sin ((n-1)\\tau )-\\alpha (n-1)\\sin (\\tau )]\\cos (\\tau ),\\\\E_8&=&[\\alpha \\cos ((n-1)\\tau )+\\alpha (n-1)\\cos (\\tau )][\\alpha \\sin ((n-1)\\tau )-\\alpha (n-1)\\sin (\\tau )]\\sin (\\tau ),\\\\D_9&=&[-\\alpha \\sin ((n-1)\\tau )(n-1)^2+\\alpha (n-1)\\sin (\\tau )]\\sin (\\tau )-[\\alpha \\sin ((n-1)\\tau )-\\alpha (n-1)\\sin (\\tau )]\\sin (\\tau )+\\\\& &[\\alpha \\cos ((n-1)\\tau )(n-1)-\\alpha (n-1)\\cos (\\tau )]\\cos (\\tau ),\\\\E_9&=&[\\alpha \\sin ((n-1)\\tau )-\\alpha (n-1)\\sin (\\tau )]\\sin (\\tau ),\\\\D_{10}&=&[\\alpha \\cos ((n-1)\\tau )+\\alpha (n-1)\\cos (\\tau )]\\sin (\\tau )+\\sin (\\tau )[\\alpha \\cos ((n-1)\\tau )(n-1)^2+\\alpha (n-1)\\cos (\\tau )]+\\\\& &[\\alpha \\sin ((n-1)\\tau )(n-1)+\\alpha (n-1)\\sin (\\tau )]\\cos (\\tau ),\\\\E_{10}&=&-[\\alpha \\cos ((n-1)\\tau )+\\alpha (n-1)\\cos (\\tau )]\\sin (\\tau ),\\\\D_{11}&=&[\\alpha \\sin ((n-1)\\tau )-\\alpha (n-1)\\sin (\\tau )][-\\alpha \\cos ((n-1)\\tau )(n-1)^2-\\alpha (n-1)\\cos (\\tau )]+\\\\& &[\\alpha \\cos ((n-1)\\tau )+\\alpha (n-1)\\cos (\\tau )][-\\alpha \\sin ((n-1)\\tau )(n-1)^2+\\alpha (n-1)\\sin (\\tau )]-\\\\& &[\\alpha \\sin ((n-1)\\tau )(n-1)+\\alpha (n-1)\\sin (\\tau )][\\alpha \\cos ((n-1)\\tau )(n-1)-\\alpha (n-1)\\cos (\\tau )],\\\\E_{11}&=&[\\alpha \\cos ((n-1)\\tau )+\\alpha (n-1)\\cos (\\tau )][\\alpha \\sin ((n-1)\\tau )-\\alpha (n-1)\\sin (\\tau )].$ Substituting these expressions for the pressuries and the density of energy into the system (REF )–(), we obtain the following its solution $A&=&\\alpha \\cos [(n-1)\\tau ]+\\alpha (n-1)\\cos [\\tau ], \\\\B&=&\\alpha \\sin [(n-1)\\tau ]-\\alpha (n-1)\\sin [\\tau ], \\\\C&=&\\sin (\\tau ).$ It is the spiky like solution so that such solutions we call the spike universe.", "Its plot presented in Fig.REF for the initial conditions $A(0)=\\alpha n=10, B(0)=0, C(0)=0$ .", "Figure: The spiky universe for ()–(), n=10,α=1n=10, \\alpha =1.Let us calculate the volume of this universe.", "It is given by $V=\\alpha ^2[\\cos [(n-1)\\tau ]+(n-1)\\cos [\\tau ]][\\sin [(n-1)\\tau ]-(n-1)\\sin [\\tau ]]\\sin (\\tau ).$ In Fig.REF shown the evolution of the volume for (REF ), $n=10, \\alpha =1$ .", "Figure: The evolution of the volume for (5.16), n=10n=10, α=1\\alpha =1.The scalar curvature has the form $R&=&(-2\\cos (\\tau )(n-1)\\cos ^2((n-1)\\tau )+((6(4/3-2n+n^2))\\sin (\\tau )\\sin ((n-1)\\tau )-\\\\& &-(2((n-2)\\cos (\\tau )^2+\\sin ^2(\\tau )(n^2-3n+4)))(n-1))\\cos ((n-1)\\tau )+2\\cos (\\tau )(\\sin ^2((n-1)\\tau )+\\\\& &+\\sin (\\tau )(n^2-4n+6)\\sin ((n-1)\\tau )+(\\cos (\\tau )^2-5\\sin ^2(\\tau ))(n-1))(n-1))/((\\cos ((n-1)\\tau )+\\\\& &+\\cos (\\tau )(n-1))(-\\sin ((n-1)\\tau )+(n-1)\\sin (\\tau ))\\sin (\\tau )).$ Figure: The evolution of the RR with respect of the cosmic time τ\\tau for Eq.", "()In Fig.REF we plot the evolution of the $R$ with respect of the cosmic time $\\tau $ .", "In this example, we have shown that the Einstein equations admit the spike-like solution.", "We can show this solution describes the accelerated and decelerated expansion phases of the universe." ], [ "Example 2", "The system (REF )–() admits the following solution $H_1&=&\\alpha \\cos [(n-1)\\tau ]+\\alpha (n-1)\\cos [\\tau ], \\\\H_2&=&\\alpha \\sin [(n-1)\\tau ]-\\alpha (n-1)\\sin [\\tau ], \\\\H_3&=&\\sin (\\tau ).$ The corresponding EoS takes the form $\\rho &=&\\frac{D_{12}}{E_{12}},\\\\p_1&=&-\\frac{D_{13}}{E_{13}},\\\\p_2&=&-\\frac{D_{14}}{E_{14}},\\\\p_3&=&-\\frac{D_{15}}{E_{15}},$ where $D_{12}&=&[\\alpha \\cos ((n-1)\\tau )+\\alpha (n-1)\\cos (\\tau )][\\alpha \\sin ((n-1)\\tau )+[1-\\alpha (n-1)]\\sin (\\tau )]+\\\\& &[\\alpha \\sin ((n-1)\\tau )-\\alpha (n-1)\\sin (\\tau )]\\sin (\\tau ),\\\\E_{12}&=&1,\\\\D_{13}&=&\\alpha (n-1)[\\cos ((n-1)\\tau )-\\cos (\\tau )]+\\cos (\\tau )+[\\alpha \\sin ((n-1)\\tau )-\\alpha (n-1)\\sin (\\tau )]^2+\\\\& &[\\alpha \\sin ((n-1)\\tau )+[1-\\alpha (n-1)]\\sin (\\tau )]\\sin (\\tau ),\\\\E_{13}&=&1,\\\\D_{14}&=&-\\alpha \\sin ((n-1)\\tau )(n-1)-\\alpha (n-1)\\sin (\\tau )+\\cos (\\tau )+[\\alpha \\cos ((n-1)\\tau )+\\alpha (n-1)\\cos (\\tau )]^2+\\\\& &\\sin (\\tau )^2+[\\alpha \\cos ((n-1)\\tau )+\\alpha *(n-1)\\cos (\\tau )]\\sin (\\tau ),\\\\E_{14}&=&1,\\\\D_{15}&=&\\alpha (n-1)[\\cos ((n-1)\\tau )-\\cos (\\tau )-\\sin ((n-1)\\tau )-\\sin (\\tau )]+\\\\& &[\\alpha \\sin ((n-1)\\tau )-\\alpha (n-1)\\sin (\\tau )]^2+[\\alpha \\cos ((n-1)\\tau )+\\alpha (n-1)\\cos (\\tau )]^2+\\\\& &[\\alpha \\cos ((n-1)\\tau )+\\alpha (n-1)\\cos (\\tau )][\\alpha \\sin ((n-1)\\tau )-\\alpha (n-1)\\sin (\\tau )],\\\\E_{15}&=&1.$ The scalar curvature has the form $R&=&2\\alpha ^2\\cos ((n-1)\\tau )^2+4\\alpha ((1/2)\\alpha \\sin ((n-1)\\tau )+(1/2+(-(1/2)n+1/2)\\alpha )\\sin (\\tau )+\\\\& &+(n-1)(\\alpha \\cos (\\tau )+1/2))\\cos ((n-1)\\tau )+2\\alpha ^2\\sin ((n-1)\\tau )^2+2\\alpha ((1+(2-2n)\\alpha )\\sin (\\tau )+\\\\& &+(\\alpha \\cos (\\tau )-1)(n-1))\\sin ((n-1)\\tau )+(2+2\\alpha ^2(n-1)^2+(2-2n)\\alpha )\\sin (\\tau )^2-(2(n-1))\\alpha *\\\\& &*(1+(-1+\\alpha (n-1))\\cos (\\tau ))\\sin (\\tau )+(2(\\alpha ^2(n-1)^2\\cos (\\tau )+1+\\alpha (-n+1)))\\cos (\\tau ).$ Figure: The evolution of the RR with respect of the cosmic time τ\\tau for Eq.", "()In Fig.REF we plot the evolution of the $R$ with respect of the cosmic time $\\tau $ ." ], [ "Example 3", "Our next solution for the system (REF )–() is given by $H_1&=&\\alpha \\cos [(n-1)\\tau ]-\\alpha (n-1)\\cos [\\tau ], \\\\H_2&=&\\alpha \\sin [(n-1)\\tau ]-\\alpha (n-1)\\sin [\\tau ], \\\\H_3&=&\\sin (\\tau ).$ In Fig.27 we plot this spiky type solution.", "Figure: The evolution of the spiky type solution (5.34)-(5.36) with n=10,α=1n=10, \\alpha =1.The corresponding EoS takes the form $\\rho &=&\\frac{D_{16}}{E_{16}},\\\\p_1&=&-\\frac{D_{17}}{E_{17}},\\\\p_2&=&-\\frac{D_{18}}{E_{18}},\\\\p_3&=&-\\frac{D_{19}}{E_{19}},$ where $D_{16}&=&[\\alpha \\cos ((n-1)\\tau )-\\alpha (n-1)\\cos (\\tau )][\\alpha \\sin ((n-1)\\tau )+[1-\\alpha (n-1)]\\sin (\\tau )]+\\\\& &[\\alpha \\sin ((n-1)\\tau )-\\alpha (n-1)\\sin (\\tau )]\\sin (\\tau ),\\\\E_{16}&=&1,\\\\D_{17}&=&\\alpha (n-1)[\\cos ((n-1)\\tau )-\\cos (\\tau )]+\\cos (\\tau )+[\\alpha \\sin ((n-1)\\tau )-\\alpha (n-1)\\sin (\\tau )]^2+\\\\& &[\\alpha \\sin ((n-1)\\tau )+[1-\\alpha (n-1)]\\sin (\\tau )]\\sin (\\tau ),\\\\E_{17}&=&1,\\\\D_{18}&=&-\\alpha \\sin ((n-1)\\tau )(n-1)+\\alpha (n-1)\\sin (\\tau )+\\cos (\\tau )+[\\alpha \\cos ((n-1)\\tau )-\\alpha (n-1)\\cos (\\tau )]^2+\\\\& &\\sin (\\tau )^2+[\\alpha \\cos ((n-1)\\tau )-\\alpha *(n-1)\\cos (\\tau )]\\sin (\\tau ),\\\\E_{18}&=&1,\\\\D_{19}&=&\\alpha (n-1)[\\cos ((n-1)\\tau )-\\cos (\\tau )-\\sin ((n-1)\\tau )+\\sin (\\tau )]+\\\\& &[\\alpha \\sin ((n-1)\\tau )-\\alpha (n-1)\\sin (\\tau )]^2+[\\alpha \\cos ((n-1)\\tau )-\\alpha (n-1)\\cos (\\tau )]^2+\\\\& &[\\alpha \\cos ((n-1)\\tau )-\\alpha (n-1)\\cos (\\tau )][\\alpha \\sin ((n-1)\\tau )-\\alpha (n-1)\\sin (\\tau )],\\\\E_{19}&=&1.$ The scalar curvature has the form $R&=&2\\alpha ^2\\cos ((n-1)\\tau )^2-(4(-(1/2)\\alpha \\sin ((n-1)\\tau )+(-1/2+((1/2)n-1/2)\\alpha )\\sin (\\tau )+\\\\& &+(n-1)(-1/2+\\alpha \\cos (\\tau ))))\\alpha \\cos ((n-1)\\tau )+2\\alpha ^2\\sin ((n-1)\\tau )^2-\\\\& &-(2((-1+(-2+2n)\\alpha )\\sin (\\tau )+(\\alpha \\cos (\\tau )+1)(n-1)))\\alpha \\sin ((n-1)\\tau )+\\\\& &+(2+2\\alpha ^2(n-1)^2+(-2n+2)\\alpha )\\sin (\\tau )^2+(2(n-1))(1+(-1+\\alpha (n-1))*\\\\& &*\\cos (\\tau ))\\alpha \\sin (\\tau )+2\\cos (\\tau )(\\alpha ^2(n-1)^2\\cos (\\tau )+1+(1-n)\\alpha ).$ Figure: The evolution of the RR with respect of the cosmic time τ\\tau for Eq.", "()In Fig.REF we plot the evolution of the $R$ with respect of the cosmic time $\\tau $ ." ], [ "M$\\ddot{o}$ bius strip universe solutions", "If we consider the model with the \"cosmological constant\", then the systems (REF )–() and (REF )–() take the form, respectively $\\frac{\\dot{A}\\dot{B}}{AB}+\\frac{\\dot{B}\\dot{C}}{BC}+\\frac{\\dot{C}\\dot{A}}{CA}-\\rho -\\Lambda &=&0,\\\\\\frac{\\ddot{B}}{B}+\\frac{\\ddot{C}}{C}+\\frac{\\dot{B}\\dot{C}}{BC}+p_1-\\Lambda &=&0,\\\\\\frac{\\ddot{C}}{C}+\\frac{\\ddot{A}}{A}+\\frac{\\dot{C}\\dot{A}}{CA}+p_2-\\Lambda &=&0,\\\\\\frac{\\ddot{A}}{A}+\\frac{\\ddot{B}}{B}+\\frac{\\dot{A}\\dot{B}}{AB}+p_3-\\Lambda &=&0$ and $H_1H_2+H_2H_3+H_1H_3-\\rho -\\Lambda &=&0,\\\\\\dot{H}_2+\\dot{H}_3+H^2_2+H^2_3+H_2H_3+p_1-\\Lambda &=&0,\\\\\\dot{H}_3+\\dot{H}_1+H^2_3+H^2_1+H_3H_1+p_2-\\Lambda &=&0,\\\\\\dot{H}_1+\\dot{H}_2+H^2_1+H^2_2+H_1H_2+p_3-\\Lambda &=&0.$ Now we want to present some solutions of these systems.", "Consider examples." ], [ "Example 1", "One of the simplest solutions of (REF )–() is given by $A&=&A_0+\\left(1+\\frac{1}{2}\\Lambda \\cos \\frac{\\tau }{2}\\right)\\cos \\tau , \\\\B&=&B_0+\\left(1+\\frac{1}{2}\\Lambda \\cos \\frac{\\tau }{2}\\right)\\sin \\tau ,\\\\C&=&C_0+\\frac{1}{2}\\Lambda \\sin \\frac{\\tau }{2}.$ It is the parametric equation of the M$\\ddot{o}$ bius strip and, hence, such model we call the M$\\ddot{o}$ bius strip universe.", "Its plot was presented in Fig.REF .", "Figure: The plot of the Mo ¨\\ddot{o}bius strip universe for ()–() with () and τ=0→2π\\tau =0\\rightarrow 2\\pi and Λ=[-1.1]\\Lambda =[-1.", "1]The evolution of the volume of the M$\\ddot{o}$ bius strip universe for (REF )–() with (REF ) reads as $V=0.5\\Lambda \\left(1+\\frac{1}{2}\\Lambda \\cos \\frac{\\tau }{2}\\right)^2\\cos \\tau \\sin \\tau \\sin \\frac{\\tau }{2}.$ The evolution of the volume with(REF ) and $\\alpha =\\Lambda =1$ presented in Fig.REF .", "Figure: The evolution of the volume of the Mo ¨\\ddot{o}bius strip universe for ()–() with () and α=Λ=1\\alpha =\\Lambda =1.The corresponding EoS takes the form $\\rho &=&\\frac{D_{20}}{E_{20}},\\\\p_1&=&-\\frac{D_{21}}{E_{21}},\\\\p_2&=&-\\frac{D_{22}}{E_{22}},\\\\p_3&=&-\\frac{D_{23}}{E_{23}},$ where $D_{20}&=&[\\frac{1}{4}\\Lambda \\sin (\\frac{\\tau }{2})\\cos (\\tau )+(1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2}))\\sin (\\tau )][\\frac{1}{4}\\Lambda \\sin (\\frac{\\tau }{2})\\sin (\\tau )-(1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2})\\cos (\\tau )]\\times \\\\& &[C_0+\\frac{1}{2}\\Lambda \\sin (\\frac{\\tau }{2})]+\\frac{\\Lambda }{4}[-\\frac{1}{4}\\Lambda \\sin (\\frac{\\tau }{2})\\sin (\\tau )+(1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2}))\\cos (\\tau )]\\cos (\\frac{\\tau }{2})\\times \\\\& &[A_0+(1+\\frac{1}{2})\\Lambda \\cos (\\frac{\\tau }{2})\\cos (\\tau )]+\\frac{\\Lambda }{4}[-\\frac{1}{4}\\Lambda \\sin (\\frac{\\tau }{2})\\cos (\\tau )-(1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2}))\\sin (\\tau )]\\cos (\\frac{\\tau }{2})\\times \\\\& &[B_0+(1+\\frac{1}{2})\\Lambda \\cos (\\frac{\\tau }{2}))\\sin (\\tau )]-\\Lambda [A_0+(1+\\frac{1}{2})\\Lambda \\cos (\\frac{\\tau }{2}))\\cos (\\tau )]\\times \\\\& &[B_0+(1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2}))\\sin (\\tau )][C_0+\\frac{1}{2}\\Lambda \\sin (\\frac{\\tau }{2})],\\\\E_{20}&=&[A_0+(1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2}))\\cos (\\tau )][B_0+(1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2}))\\sin (\\tau )][C_0+\\frac{1}{2}\\Lambda \\sin (\\frac{\\tau }{2})],\\\\D_{21}&=&[C_0+\\frac{1}{2}\\Lambda \\sin (\\frac{\\tau }{2})][-\\frac{1}{8}\\sin (\\tau )\\Lambda \\cos (\\frac{\\tau }{2})-\\frac{1}{2}\\Lambda \\sin (\\frac{\\tau }{2})\\cos (\\tau )-(1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2}))\\sin (\\tau )]-\\\\& &\\frac{\\Lambda }{8}[B_0+(1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2}))\\sin (\\tau )]\\sin (\\frac{\\tau }{2})+\\frac{\\Lambda }{4}[-\\frac{1}{4}\\Lambda \\sin (\\frac{\\tau }{2})\\sin (\\tau )+(1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2}))\\times \\\\& &\\cos (\\tau )]\\cos (\\frac{\\tau }{2})-\\Lambda [B_0+(1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2}))\\sin (\\tau )][C_0+\\frac{1}{2}\\Lambda \\sin (\\frac{\\tau }{2})],\\\\E_{21}&=&[B_0+(1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2}))\\sin (\\tau )][C_0+\\frac{1}{2}\\Lambda \\sin (\\frac{\\tau }{2})],\\\\D_{22}&=&-\\frac{\\Lambda }{8}[A_0+(1+\\frac{1}{2}\\cos (\\frac{\\tau }{2}))\\cos (\\tau )]\\Lambda \\sin (\\frac{\\tau }{2})+[C_0+\\frac{1}{2}\\Lambda \\sin (\\frac{\\tau }{2})][-\\frac{\\Lambda }{8}\\cos (\\tau )\\cos (\\frac{\\tau }{2})+\\\\& &\\frac{1}{2}\\Lambda \\sin (\\frac{\\tau }{2})\\sin (\\tau )-(1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2}))\\cos (\\tau )]+\\frac{1}{4}[-\\frac{1}{4}\\Lambda \\sin (\\frac{\\tau }{2})\\cos (\\tau )-\\\\& &(1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2}))\\sin (\\tau )]\\Lambda \\cos (\\frac{\\tau }{2})-\\Lambda [A_0+(1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2}))\\cos (\\tau )][C_0+\\frac{1}{2}\\Lambda \\sin (\\frac{\\tau }{2})],\\\\E_{22}&=&[A_0+(1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2}))\\cos (\\tau )][C_0+\\frac{1}{2}\\Lambda \\sin (\\frac{\\tau }{2})],\\\\D_{23}&=&[B_0+(1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2}))\\sin (\\tau )][-\\frac{\\Lambda }{8}\\cos (\\tau )\\cos (\\frac{\\tau }{2})+\\frac{1}{2}\\Lambda \\sin (\\frac{\\tau }{2})\\sin (\\tau )-(1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2}))\\times \\\\& &\\cos (\\tau )]+[A_0+(1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2}))\\cos (\\tau )][-\\frac{\\Lambda }{8}\\sin (\\tau )\\cos (\\frac{\\tau }{2})-\\frac{1}{2}\\Lambda \\sin (\\frac{\\tau }{2})\\cos (\\tau )-\\\\& &(1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2}))\\sin (\\tau )]-[\\frac{1}{4}\\Lambda \\sin (\\frac{\\tau }{2})\\cos (\\tau )+(1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2}))\\sin (\\tau )]\\times \\\\& &[-\\frac{1}{4}\\Lambda \\sin (\\frac{\\tau }{2})\\sin (\\tau )+(1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2}))\\cos (\\tau )]-\\\\& &\\Lambda [A_0+(1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2}))\\cos (\\tau )][B_0+(1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2}))\\sin (\\tau )],\\\\E_{23}&=&[B_0+(1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2}))\\sin (\\tau )][A_0+(1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2}))\\cos (\\tau )].$ The scalar curvature has the form $R&=&((-2\\sin (\\tau )^2\\Lambda ^2+2\\cos (\\tau )^2\\Lambda ^2)\\cos ((1/2)\\tau )^3+\\\\& &+(-8\\sin (\\tau )^2\\Lambda -17\\cos (\\tau )\\sin (\\tau )\\sin ((1/2)\\tau )\\Lambda ^2+\\\\& &+8\\cos (\\tau )^2\\Lambda )\\cos ((1/2)\\tau )^2+((6\\sin (\\tau )^2\\Lambda ^2-6\\cos (\\tau )^2\\Lambda ^2)\\sin ((1/2)\\tau )^2-\\\\& &-60\\cos (\\tau )\\sin (\\tau )\\sin ((1/2)\\tau )\\Lambda +8\\cos (\\tau )^2-8\\sin (\\tau )^2)\\cos ((1/2)\\tau )+\\\\& &+(\\sin ((1/2)\\tau )^2\\Lambda ^2\\cos (\\tau )\\sin (\\tau )+(12\\sin (\\tau )^2\\Lambda -12\\cos (\\tau )^2\\Lambda )\\sin ((1/2)\\tau )-\\\\& &-52\\cos (\\tau )\\sin (\\tau ))\\sin ((1/2)\\tau ))/(\\sin (\\tau )\\cos (\\tau )(2+\\Lambda \\cos ((1/2)\\tau ))^2\\sin ((1/2)\\tau )).$ Figure: The evolution of the RR with respect of the cosmic time τ\\tau for Eq.", "()In Fig.REF we plot the evolution of the $R$ with respect of the cosmic time $\\tau $ .", "In this subsubsection, we have shown that the Einstein equations have the M$\\ddot{o}$ bius strip universe solution.", "Again we can show that this solution describes the accelerated and decelerated expansion phases of the universe." ], [ "Example 2", "For the system (REF )–() the Mobious solution reads as $H_1&=&\\left(1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2})\\right)\\cos (\\tau ), \\\\H_2&=&\\left(1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2})\\right)\\sin (\\tau ),\\\\H_3&=&\\frac{1}{2}\\Lambda \\sin (\\frac{\\tau }{2}).$ The corresponding EoS takes the form $\\rho &=&\\frac{D_{24}}{E_{24}},\\\\p_1&=&-\\frac{D_{25}}{E_{25}},\\\\p_2&=&-\\frac{D_{26}}{E_{26}},\\\\p_3&=&-\\frac{D_{27}}{E_{27}},$ where $D_{24}&=&[1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2})]^2\\cos (\\tau )\\sin (\\tau )+\\frac{\\Lambda }{2}[1+\\frac{\\Lambda }{2}\\cos (\\frac{\\tau }{2})][\\sin (\\tau )+\\cos (\\tau )]\\sin (\\frac{\\tau }{2})-\\Lambda ,\\\\E_{24}&=&1,\\\\D_{25}&=&-\\frac{1}{4}\\Lambda \\sin (\\frac{\\tau }{2})\\sin (\\tau )+[1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2})][\\cos (\\tau )+ \\frac{\\Lambda }{2}\\sin (\\tau )\\sin (\\frac{\\tau }{2})] +\\frac{1}{4}\\Lambda \\cos (\\frac{\\tau }{2})+\\\\& &[1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2})]^2\\sin ^2(\\tau )+\\frac{\\Lambda ^2}{4}\\sin ^2(\\frac{\\tau }{2})-\\Lambda ,\\\\E_{25}&=&1,\\\\D_{26}&=&-\\frac{1}{4}\\Lambda \\sin (\\frac{\\tau }{2})\\cos (\\tau )-[1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2})][\\sin (\\tau )+\\frac{\\Lambda }{2}\\cos (\\tau )\\sin (\\frac{\\tau }{2})]+\\frac{1}{4}\\Lambda \\cos (\\frac{\\tau }{2})+\\\\& &[1+\\frac{1}{2}\\Lambda \\cos (\\frac{\\tau }{2})]^2\\cos ^2(\\tau )+\\frac{\\Lambda ^2}{4}\\sin ^2(\\frac{\\tau }{2})-\\Lambda ,\\\\E_{26}&=&1,\\\\D_{27}&=&[1+\\frac{\\Lambda }{2}\\cos (\\frac{\\tau }{2})-\\frac{\\Lambda }{4}\\sin (\\frac{\\tau }{2})][\\cos (\\tau )+\\sin (\\tau )]+\\\\& &[1+\\frac{\\Lambda }{2}\\cos (\\frac{\\tau }{2})]^2[1+\\cos (\\tau )\\sin (\\tau )]-\\Lambda ,\\\\E_{27}&=&1.$ The scalar curvature has the form $R&=&(1/2)\\Lambda ^2(\\cos ^2(\\tau )+\\sin ^2(\\tau )+\\cos (\\tau )\\sin (\\tau ))\\cos ^2((1/2)\\tau )+\\\\& &+(1/2((\\cos (\\tau )+\\sin (\\tau ))\\Lambda \\sin ((1/2)\\tau )+4\\cos ^2(\\tau )+(2+4\\sin (\\tau ))\\cos (\\tau )+\\\\& &+1-2\\sin (\\tau )+4\\sin ^2(\\tau )))\\Lambda \\cos ((1/2)\\tau )+(1/2)\\Lambda ^2\\sin ^2((1/2)\\tau )+\\\\& &+(1/2(\\cos (\\tau )+\\sin (\\tau )))\\Lambda \\sin ((1/2)\\tau )+\\\\& &+2\\cos ^2(\\tau )+(1/2(4+4\\sin (\\tau )))\\cos (\\tau )-2\\sin (\\tau )+2\\sin ^2(\\tau )$ Figure: The evolution of the RR with respect of the cosmic time τ\\tau for Eq.", "()In Fig.REF we plot the evolution of the $R$ with respect of the cosmic time $\\tau $ ." ], [ "Example 1", "Now we want present some solutions in terms of the Jacobian elliptic functions.", "In fact, the system (REF )–() has the following particular solution $A&=&A_0+\\left(1+\\frac{1}{2}\\Lambda \\mbox{cn} \\frac{\\tau }{2}\\right)\\mbox{cn} \\tau , \\\\B&=&B_0+\\left(1+\\frac{1}{2}\\Lambda \\mbox{cn}\\frac{\\tau }{2}\\right)\\mbox{sn}\\tau ,\\\\C&=&C_0+\\frac{1}{2}\\Lambda \\mbox{sn} \\frac{\\tau }{2}.$ The corresponding EoS takes the form $\\rho &=&\\frac{D_{28}}{E_{28}},\\\\p_1&=&-\\frac{D_{29}}{E_{29}},\\\\p_2&=&-\\frac{D_{30}}{E_{30}},\\\\p_3&=&-\\frac{D_{31}}{E_{31}},$ where $D_{28}&=&[\\frac{1}{4}\\Lambda \\mbox{dn}\\frac{\\tau }{2}\\ \\mbox{sn}\\frac{\\tau }{2}\\ \\mbox{cn}\\tau +(1+\\frac{1}{2}\\Lambda \\mbox{cn}\\frac{\\tau }{2})\\ \\mbox{dn}\\tau \\ \\mbox{sn}\\tau ] [\\frac{1}{4}\\Lambda \\mbox{dn}\\frac{\\tau }{2}\\ \\mbox{sn}\\frac{\\tau }{2}\\ \\mbox{sn}\\tau -(1+\\frac{1}{2}\\Lambda \\mbox{cn}\\frac{\\tau }{2})\\ \\mbox{cn}\\tau \\ \\mbox{dn}\\tau ]\\times \\\\& & [C_0+\\frac{1}{2}\\Lambda \\mbox{sn}\\frac{\\tau }{2}]+\\frac{\\Lambda }{4} [-\\frac{1}{4}\\Lambda \\mbox{dn}\\frac{\\tau }{2}\\ \\mbox{sn}\\frac{\\tau }{2}\\ \\mbox{sn}\\tau +(1+\\frac{1}{2}\\Lambda \\mbox{cn}\\frac{\\tau }{2})\\ \\mbox{cn}\\tau \\ \\mbox{dn}\\tau ]\\mbox{cn}\\frac{\\tau }{2}\\ \\mbox{dn}\\frac{\\tau }{2}\\times \\\\& &[A_0+(1+\\frac{1}{2}\\Lambda \\mbox{cn}\\frac{\\tau }{2})\\mbox{cn}\\tau ]-\\frac{1}{4} [\\frac{1}{4}\\Lambda \\mbox{dn}\\frac{\\tau }{2}\\ \\mbox{sn}\\frac{\\tau }{2}\\ \\mbox{cn}\\tau +(1+\\frac{1}{2}\\Lambda \\mbox{cn}\\frac{\\tau }{2})\\ \\mbox{dn}\\tau \\ \\mbox{sn}\\tau ]\\Lambda \\mbox{cn}\\frac{\\tau }{2}\\ \\mbox{dn}\\frac{\\tau }{2}\\times \\\\& & [B_0+(1+\\frac{1}{2}\\Lambda \\mbox{cn}\\frac{\\tau }{2})\\ \\mbox{sn}\\tau ]-\\Lambda [A_0+(1+\\frac{1}{2}\\Lambda \\mbox{cn}\\frac{\\tau }{2})\\ \\mbox{cn}\\tau ] [B_0+(1+\\frac{1}{2}\\Lambda \\mbox{cn}\\frac{\\tau }{2})\\ \\mbox{sn}\\tau ]\\times \\\\& & [C_0+\\frac{1}{2}\\Lambda \\mbox{sn}\\frac{\\tau }{2}],\\\\E_{28}&=&[A_0+(1+\\frac{1}{2}\\Lambda \\mbox{cn}\\frac{\\tau }{2})\\ \\mbox{cn}\\tau ] [B_0+(1+\\frac{1}{2}\\Lambda \\mbox{cn}\\frac{\\tau }{2})\\ \\mbox{sn}\\tau ] [C_0+\\frac{1}{2}\\Lambda \\mbox{sn}\\frac{\\tau }{2}],\\\\D_{29}&=&[C_0+\\frac{1}{2}\\Lambda \\mbox{sn}\\frac{\\tau }{2}] [\\frac{1}{8}\\Lambda \\mbox{cn}\\frac{\\tau }{2}\\ \\mbox{sn}^2\\frac{\\tau }{2}\\ \\mbox{sn}\\tau -\\frac{1}{8}\\Lambda \\mbox{dn}^2\\frac{\\tau }{2}\\ \\mbox{cn}\\frac{\\tau }{2}\\ \\mbox{sn}\\tau -\\frac{1}{2}\\Lambda \\mbox{dn}\\frac{\\tau }{2}\\ \\mbox{sn}\\frac{\\tau }{2}\\ \\mbox{cn}\\tau \\ \\mbox{dn}\\tau -\\\\& & (1+\\frac{1}{2}\\Lambda \\mbox{cn}\\frac{\\tau }{2})\\ \\mbox{dn}^2\\tau \\ \\mbox{sn}\\tau -[1+\\frac{1}{2}\\Lambda \\mbox{cn}\\frac{\\tau }{2}]\\ \\mbox{cn}^2\\tau \\ \\mbox{sn}\\tau ]+[B_0+(1+\\frac{1}{2}\\Lambda \\mbox{cn}\\frac{\\tau }{2})\\ \\mbox{sn}\\tau ]\\times \\\\& & [-\\frac{1}{8}\\Lambda \\mbox{dn}^2\\frac{\\tau }{2}\\ \\mbox{sn}\\frac{\\tau }{2}-\\frac{1}{8}\\Lambda \\mbox{cn}^2\\frac{\\tau }{2} \\mbox{sn}\\frac{\\tau }{2}]-\\frac{\\Lambda }{4} [\\frac{1}{4}\\Lambda \\mbox{dn}\\frac{\\tau }{2}\\ \\mbox{sn}\\frac{\\tau }{2}\\ \\mbox{sn}\\tau -(1+\\frac{1}{2}\\Lambda \\mbox{cn}\\frac{\\tau }{2})\\ \\mbox{cn}\\tau \\ \\mbox{dn}\\tau ]\\times \\\\& & \\mbox{cn}\\frac{\\tau }{2}\\ \\mbox{dn}\\frac{\\tau }{2}-\\Lambda [B_0+(1+\\frac{1}{2}\\Lambda \\mbox{cn}\\frac{\\tau }{2})\\ \\mbox{sn}\\tau ] [C_0+\\frac{1}{2}\\Lambda \\mbox{sn}\\frac{\\tau }{2}],\\\\E_{29}&=&[B_0+(1+\\frac{1}{2}\\Lambda \\mbox{cn}\\frac{\\tau }{2})\\ \\mbox{sn}\\tau ] [C_0+\\frac{1}{2}\\Lambda \\mbox{sn}\\frac{\\tau }{2}],\\\\D_{30}&=&-[A_0+(1+\\frac{1}{2}\\Lambda \\ \\mbox{cn}\\frac{\\tau }{2})\\ \\mbox{cn}\\tau ] [\\frac{1}{8}\\Lambda \\ \\mbox{dn}^2\\frac{\\tau }{2}\\ \\mbox{sn}\\frac{\\tau }{2}+\\frac{1}{8}\\Lambda \\ \\mbox{cn}^2\\frac{\\tau }{2}\\ \\mbox{sn}\\frac{\\tau }{2}]+[C_0+\\frac{1}{2}\\Lambda \\ \\mbox{sn}\\frac{\\tau }{2}]\\times \\\\& &[\\frac{1}{8}\\Lambda \\ \\mbox{cn}\\frac{\\tau }{2}\\ \\mbox{sn}^2\\frac{\\tau }{2}\\ \\mbox{cn}\\tau -\\frac{1}{8}\\Lambda \\ \\mbox{dn}^2\\frac{\\tau }{2}\\ \\mbox{cn}\\frac{\\tau }{2}\\ \\mbox{cn}\\tau +\\frac{1}{2}\\Lambda \\ \\mbox{dn}\\frac{\\tau }{2}\\ \\mbox{sn}\\frac{\\tau }{2}\\ \\mbox{dn}\\tau \\ \\mbox{sn}\\tau +\\\\& &(1+\\frac{1}{2}\\Lambda \\ \\mbox{cn}\\frac{\\tau }{2})(\\mbox{cn}\\tau \\ \\mbox{sn}^2\\tau - \\mbox{dn}^2\\tau \\ \\mbox{cn}\\tau )]-\\frac{\\Lambda }{4} [\\frac{1}{4}\\Lambda \\ \\mbox{dn}\\frac{\\tau }{2}\\ \\mbox{sn}\\frac{\\tau }{2}\\ \\mbox{cn}\\tau +(1+\\frac{1}{2}\\Lambda \\ \\mbox{cn}\\frac{\\tau }{2})\\times \\\\& & \\mbox{dn}\\tau \\ \\mbox{sn}\\tau ]\\ \\mbox{cn}\\frac{\\tau }{2}\\ \\mbox{dn}\\frac{\\tau }{2}-\\Lambda [A_0+(1+\\frac{1}{2}\\Lambda \\ \\mbox{cn}\\frac{\\tau }{2})\\ \\mbox{cn}\\tau ] [C_0+\\frac{1}{2}\\Lambda \\ \\mbox{sn}\\frac{\\tau }{2}],\\\\E_{30}&=&[A_0+(1+\\frac{1}{2}\\Lambda \\ \\mbox{cn}\\frac{\\tau }{2})\\ \\mbox{cn}\\tau ] [C_0+\\frac{1}{2}\\Lambda \\ \\mbox{sn}\\frac{\\tau }{2}],\\\\D_{31}&=&[B_0+(1+\\frac{1}{2}\\Lambda \\ \\mbox{cn}\\frac{\\tau }{2})\\ \\mbox{sn}\\tau ] [\\frac{1}{8}\\Lambda \\ \\mbox{cn}\\frac{\\tau }{2}\\ \\mbox{sn}^2\\frac{\\tau }{2}\\ \\mbox{cn}\\tau -\\frac{1}{8}\\Lambda \\ \\mbox{dn}^2\\frac{\\tau }{2}\\ \\mbox{cn}\\frac{\\tau }{2}\\ \\mbox{cn}\\tau +\\\\& &\\frac{1}{2}\\Lambda \\ \\mbox{dn}\\frac{\\tau }{2}\\ \\mbox{sn}\\frac{\\tau }{2}\\ \\mbox{dn}\\tau \\ \\mbox{sn}\\tau +(1+\\frac{1}{2}\\Lambda \\ \\mbox{cn}\\frac{\\tau }{2}) (\\mbox{sn}^2\\tau -\\ \\mbox{dn}^2\\tau )\\ \\mbox{cn}\\tau ]+[A_0+(1+\\frac{1}{2}\\Lambda \\ \\mbox{cn}\\frac{\\tau }{2})\\ \\mbox{cn}\\tau ]\\times \\\\& &[\\frac{1}{8}\\Lambda \\ \\mbox{cn}\\frac{\\tau }{2}\\ \\mbox{sn}\\tau (\\mbox{sn}^2\\frac{\\tau }{2}-\\mbox{dn}^2\\tau )-\\frac{1}{2}\\Lambda \\ \\mbox{dn}\\frac{\\tau }{2}\\ \\mbox{sn}\\frac{\\tau }{2}\\ \\mbox{cn}\\tau \\ \\mbox{dn}\\tau -(1+\\frac{1}{2}\\Lambda \\ \\mbox{cn}\\frac{\\tau }{2})\\ \\mbox{dn}^2\\tau \\ \\mbox{sn}\\tau -\\\\& &(1+\\frac{1}{2}\\Lambda \\ \\mbox{cn}\\frac{\\tau }{2})\\ \\mbox{cn}^2\\tau \\ \\mbox{sn}\\tau ]+[\\frac{1}{4}\\Lambda \\ \\mbox{dn}\\frac{\\tau }{2}\\ \\mbox{sn}\\frac{\\tau }{2}\\ \\mbox{cn}\\tau +(1+\\frac{1}{2}\\Lambda \\ \\mbox{cn}\\frac{\\tau }{2})\\ \\mbox{dn}\\tau \\ \\mbox{sn}\\tau ]\\times \\\\& &[\\frac{1}{4}\\Lambda \\ \\mbox{dn}\\frac{\\tau }{2}\\ \\mbox{sn}\\frac{\\tau }{2}\\ \\mbox{sn}\\tau -(1+\\frac{1}{2}\\Lambda \\ \\mbox{cn}\\frac{\\tau }{2})\\ \\mbox{cn}\\tau \\ \\mbox{dn}\\tau ]-\\Lambda [A_0+(1+\\frac{1}{2}\\Lambda \\ \\mbox{cn}\\frac{\\tau }{2})\\ \\mbox{cn}\\tau ]\\times \\\\& & [B_0+(1+\\frac{1}{2}\\Lambda \\ \\mbox{cn}\\frac{\\tau }{2})\\ \\mbox{sn}\\tau ],\\\\E_{31}&=&[B_0+(1+\\frac{1}{2}\\Lambda \\ \\mbox{cn}\\frac{\\tau }{2})\\ \\mbox{sn}\\tau ] [A_0+(1+\\frac{1}{2}\\Lambda \\ \\mbox{cn}\\frac{\\tau }{2})\\ \\mbox{cn}\\tau ].$ The evolution of the volume of the universe for (REF ) reads as ($A_0=B_0=C_0=0$ ) $V=\\frac{1}{2}\\Lambda \\left(1+\\frac{1}{2}\\Lambda \\mbox{cn} \\frac{\\tau }{2}\\right)^2\\mbox{cn} \\tau \\ \\mbox{sn}\\tau \\ \\mbox{sn} \\frac{\\tau }{2}.$ The evolution of the volume with (REF ) and $\\Lambda =1$ presented in Fig.REF .", "Figure: The evolution of the volume of the trefoil knot universe with respect to the cosmic time τ\\tau for Eq.", "()The scalar curvature has the form $R&=&(2\\Lambda (2k^2\\mbox{cn}((1/2)\\tau , k)+\\Lambda k^2\\mbox{cn}^2((1/2)\\tau , k)+\\\\& &+(1/2)\\Lambda \\mbox{dn}^2((1/2)\\tau , k))\\mbox{cn}(\\tau , k)\\mbox{sn}(\\tau , k)\\mbox{sn}((1/2)\\tau , k)^3-\\\\& &-6\\Lambda \\mbox{dn}(\\tau , k)\\mbox{dn}((1/2)\\tau , k)(\\mbox{cn}(\\tau , k)-\\mbox{sn}(\\tau , k))(\\mbox{cn}(\\tau , k)+\\\\& &+\\mbox{sn}(\\tau , k))(2+\\Lambda \\mbox{cn}((1/2)\\tau , k))\\mbox{sn}^2((1/2)\\tau , k)-\\\\& &-(4(2+\\Lambda \\mbox{cn}((1/2)\\tau , k)))((1/4)\\mbox{cn}^3((1/2)\\tau , k)k^2\\Lambda +\\\\& &+(1/2)\\mbox{cn}^2((1/2)\\tau , k)k^2+\\Lambda (3\\mbox{dn}^2(\\tau , k)+\\mbox{cn}^2(\\tau , k)k^2+(5/4)\\mbox{dn}^2((1/2)\\tau , k)-\\\\& &-\\mbox{sn}^2(\\tau , k)k^2)\\mbox{cn}((1/2)\\tau , k)+6\\mbox{dn}^2(\\tau , k)+(1/2)\\mbox{dn}^2((1/2)\\tau , k)-\\\\& &-2\\mbox{sn}^2(\\tau , k)k^2+2\\mbox{cn}^2(\\tau , k)k^2)\\mbox{cn}(\\tau , k)\\mbox{sn}(\\tau , k)\\mbox{sn}((1/2)\\tau , k)+\\\\& &+2\\mbox{cn}((1/2)\\tau , k)\\mbox{dn}((1/2)\\tau , k)\\mbox{dn}(\\tau , k)(\\mbox{cn}(\\tau , k)-\\mbox{sn}(\\tau , k))(\\mbox{cn}(\\tau , k)+\\\\& &+\\mbox{sn}(\\tau , k))(2+\\Lambda \\mbox{cn}((1/2)\\tau , k))^2)/(\\mbox{cn}(\\tau , k)\\mbox{sn}(\\tau , k)*\\\\& &*(2+\\Lambda \\mbox{cn}^2((1/2)\\tau , k))\\mbox{sn}((1/2)\\tau , k)).$ Figure: The evolution of the RR with respect of the cosmic time τ\\tau for Eq.", "()In Fig.REF we plot the evolution of the $R$ with respect of the cosmic time $\\tau $ ." ], [ "Example 2", "Similarly, we can show that the system (REF )–() has the following solution $H_1&=&\\left(1+\\frac{1}{2}\\Lambda \\mbox{cn} \\frac{\\tau }{2}\\right)\\mbox{cn} \\tau , \\\\H_2&=&\\left(1+\\frac{1}{2}\\Lambda \\mbox{cn}\\frac{\\tau }{2}\\right)\\mbox{sn}\\tau ,\\\\H_3&=&\\frac{1}{2}\\Lambda \\mbox{sn} \\frac{\\tau }{2}.$ The corresponding EoS takes the form $\\rho &=&\\frac{D_{32}}{E_{32}},\\\\p_1&=&-\\frac{D_{33}}{E_{33}},\\\\p_2&=&-\\frac{D_{34}}{E_{34}},\\\\p_3&=&-\\frac{D_{35}}{E_{35}},$ where $D_{32}&=&[1+\\frac{1}{2}\\ \\Lambda \\ \\mbox{cn}\\frac{\\tau }{2}]^2 \\ \\mbox{cn}\\tau \\ \\mbox{sn}\\tau +\\frac{\\Lambda }{2}\\ [1+\\frac{1}{2} \\Lambda \\ \\mbox{cn}\\frac{\\tau }{2}][\\mbox{sn}\\tau +\\mbox{cn}\\tau ] \\ \\mbox{sn}\\frac{\\tau }{2}-\\Lambda ,\\\\E_{32}&=&1,\\\\D_{33}&=&\\frac{1}{4}\\Lambda \\mbox{dn}\\frac{\\tau }{2}\\ \\mbox{sn}\\frac{\\tau }{2}[1-\\mbox{sn}\\tau ]+[1+\\frac{1}{2} \\Lambda \\ \\mbox{cn}\\frac{\\tau }{2}] [\\mbox{cn}\\tau \\ \\mbox{dn}\\tau +\\frac{\\Lambda }{2}\\ \\mbox{sn}\\tau \\ \\mbox{sn}\\frac{\\tau }{2}]+[1+\\frac{1}{2} \\Lambda \\mbox{cn}\\frac{\\tau }{2}]^2\\ \\mbox{sn}^2\\tau +\\\\& &\\frac{1}{4}\\Lambda ^2\\mbox{sn}^2\\frac{\\tau }{2}-\\Lambda ,\\\\E_{33}&=&1,\\\\D_{34}&=&-\\frac{1}{4} \\Lambda \\ \\mbox{dn}\\frac{\\tau }{2} \\ \\mbox{sn}\\frac{\\tau }{2}\\ \\mbox{cn}\\tau -(1+\\frac{1}{2} \\Lambda \\ \\mbox{cn}\\frac{\\tau }{2})[\\mbox{dn}\\tau \\ \\mbox{sn}\\tau +\\frac{\\Lambda }{2}\\ \\mbox{cn}\\tau \\ \\mbox{sn}\\frac{\\tau }{2}]+\\frac{1}{4} \\Lambda \\ \\mbox{cn}\\frac{\\tau }{2} \\ \\mbox{dn}\\frac{\\tau }{2}+\\\\& &[1+\\frac{1}{2} \\Lambda \\ \\mbox{cn}\\frac{\\tau }{2}]^2 \\ \\mbox{cn}^2\\tau +\\frac{1}{4}\\ \\Lambda ^2\\ \\mbox{sn}^2\\frac{\\tau }{2}-\\Lambda ,\\\\E_{34}&=&1,\\\\D_{35}&=&-\\frac{1}{4}\\Lambda \\ \\mbox{dn}\\frac{\\tau }{2}\\ \\mbox{sn}\\frac{\\tau }{2}[\\mbox{cn}\\tau +\\mbox{sn}\\tau ]+[1+\\frac{1}{2} \\Lambda \\ \\mbox{cn}\\frac{\\tau }{2}][\\mbox{cn}\\tau -\\mbox{sn}\\tau ] \\mbox{dn}\\tau +\\\\& &[1+\\frac{1}{2}\\ \\Lambda \\ \\mbox{cn}\\frac{\\tau }{2}]^2[\\mbox{sn}^2\\tau + \\mbox{cn}^2\\tau + \\mbox{cn}\\tau \\ \\mbox{sn}\\tau ]-\\Lambda ,\\\\E_{35}&=&1.$ The scalar curvature has the form $R&=&(1/2)\\Lambda ^2(\\mbox{cn}^2(\\tau , k)+\\mbox{sn}^2(\\tau , k)+\\mbox{cn}(\\tau , k)\\mbox{sn}(\\tau , k))\\mbox{cn}^2((1/2)\\tau , k)+\\\\& &+(1/2(\\Lambda (\\mbox{sn}(\\tau , k)+\\mbox{cn}(\\tau , k))\\mbox{sn}((1/2)\\tau , k)+\\mbox{dn}((1/2)\\tau , k)+4\\mbox{cn}^2(\\tau , k)+\\\\& &+(4\\mbox{sn}(\\tau , k)+2\\mbox{dn}(\\tau , k))\\mbox{cn}(\\tau , k)+4\\mbox{sn}^2(\\tau , k)-\\\\& &-2\\mbox{dn}(\\tau , k)\\mbox{sn}(\\tau , k)))\\Lambda \\mbox{cn}((1/2)\\tau , k)+(1/2)\\Lambda ^2\\mbox{sn}^2((1/2)\\tau , k)-\\\\& &-(1/2)\\Lambda (\\mbox{dn}((1/2)\\tau , k)-2)(\\mbox{sn}(\\tau , k)+\\\\& &+\\mbox{cn}(\\tau , k))\\mbox{sn}((1/2)\\tau , k)+2\\mbox{cn}^2(\\tau , k)+(1/2(4\\mbox{dn}(\\tau , k)+\\\\& &+4\\mbox{sn}(\\tau , k)))\\mbox{cn}(\\tau , k)-2\\mbox{sn}(\\tau , k)(-\\mbox{sn}(\\tau , k)+\\mbox{dn}(\\tau , k)).$ Figure: The evolution of the RR with respect of the cosmic time τ\\tau for Eq.", "()In Fig.REF we plot the evolution of the $R$ with respect of the cosmic time $\\tau $ ." ], [ "Integrable models", "The system (2.21)-(2.24) contents 4 equations for 7 unknown functions.", "This means we can add 3 new additional equations.", "It gives us in particular to construct integrable Bianchi models.", "In this subsection we present two examples of such integrable models." ], [ "Euler top equation", "Let us we assume that $A=A_1, B=A_2, C=A_3$ obey the Euler top equation.", "The simple Euler top equation reads as $\\dot{A}_{1}&=&A_2A_3, \\\\\\dot{A}_{2}&=&A_3A_1, \\\\\\dot{A}_{3}&=&A_1A_2.$ The system of equations (2.21)-(2.24) and (5.124)-(5.126) we call the Bianchi I-Euler model which is integrable." ], [ "Heisenberg ferromagnet equation", "Our second example is the Heisenberg ferromagnet equation (HFE).", "Here we assume that the variables $A=S_1, B=S_2, C=S_3$ satisfy the equation $iS_{t}+\\frac{1}{\\omega }[S, W]=0,\\quad iW_{x}+\\omega [S, W]=0,$ where $S=S_i\\sigma _i$ , $W=W_i\\sigma _i$ and $\\sigma _i$ are Pauli's matrices.", "It the principal chiral type equation with the $U=-i\\lambda S,\\quad V=-\\frac{i\\lambda }{\\omega (\\lambda +\\omega )}W.$ .", "Note this principal chiral type equation is the particular case of the following (2+1)-dimensional M-XCIX equation [47] (see also [48]) $iS_{t}+0.5([S, S_{y}]+uS)_{x}+\\frac{1}{\\omega }[S, W]&=&0,\\\\u_x-0.5S\\cdot [S_x,S_y]&=&0,\\\\iW_{x}+\\omega [S, W]&=&0 $ or (that equivalent) $iS_{t}+0.5[S, S_{xy}]+uS_{x}+\\frac{1}{\\omega }[S, W]&=&0,\\\\u_x-0.5S\\cdot [S_x,S_y]&=&0,\\\\iW_{x}+\\omega [S, W]&=&0.", "$ This M-XCIX equation is integrable by the following Lax representation [47] $\\Phi _{x}&=&U\\Phi ,\\\\\\Phi _{t}&=&2\\lambda \\Phi _y+V\\Phi ,$ with ($S^2=I$ ) $U&=&-i\\lambda S, \\\\V&=&\\lambda V_{1}+\\frac{i}{\\lambda +\\omega }W-\\frac{i}{\\omega }W$ and $S&=&\\begin{pmatrix} S_3&S^{-}\\\\S^{+}& -S_3\\end{pmatrix},\\\\V_1&=&0.25([S, S_{y}]+uS),\\\\W&=&\\begin{pmatrix} W_3&W^{-}\\\\W^{+}& -W_3\\end{pmatrix}.$" ], [ "Conclusion", "In the present paper, we have constructed several concrete models describing the trefoil and figure-eight knot universes from Bianchi-type I cosmology and examined the cosmological features and properties in detail.", "To realize the cyclic universes, it is necessary to a non-canonical scalar field with non well-defined vacuum in the context of the quantum field theory or extend gravity, e.g., with adding higher order derivative terms and $f(R)$ gravity [25].", "Indeed, however, these modified gravity theories have to satisfy the tests on the solar system scale as well as cosmological constraints so that those can be alternative gravitational theories to general relativity.", "The significant cosmological consequence of our approach is that we have shown the possibility to obtain the knot universes related to the cyclic universes from Bianchi-type I spacetime within general relativity.", "Furthermore, recently it has been pointed out that the asymmetry of the EoS for the universe can lead to cosmological hysteresis [26].", "On the other hand, Bianchi-type I spacetime describes the spatially anisotropic cosmology and hence the EoS for the universe has the asymmetry in the oscillating process through the expanding and contracting behaviors.", "Accordingly, it is considered that in the constructed models of the knot universes cosmological hysteresis could occur.", "The observation of this phenomenon in our models is one of our future works on the knot universes.", "Finally, it should be remarked that by summarizing the results of our previous [30] and this works, the knot universes describing the cyclic universes can be realized from the homogeneous and isotropic FLRW spacetime as well as the homogeneous and anisotropic Bianchi-type I cosmology.", "In these series of works, the formulations of model construction method of the knot universes have been established.", "Thus, it can be expected that the presented formalism is useful to realize the universes with other features from both the isotropic and anisotropic spacetimes.", "Finally we would like to note that all solutions presented above describe the accelerated and decelerated expansion phases of the universe." ] ]

[ [ "The large N limit of M2-branes on Lens spaces" ], [ "Abstract We study the matrix model for N M2-branes wrapping a Lens space L(p,1) = S^3/Z_p.", "This arises from localization of the partition function of the ABJM theory, and has some novel features compared with the case of a three-sphere, including a sum over flat connections and a potential that depends non-trivially on p. We study the matrix model both numerically and analytically in the large N limit, finding that a certain family of p flat connections give an equal dominant contribution.", "At large N we find the same eigenvalue distribution for all p, and show that the free energy is simply 1/p times the free energy on a three-sphere, in agreement with gravity dual expectations." ], [ "Introduction and summary", "One of the most exciting features of supersymmetric gauge theories is that one can compute certain protected quantities exactly.", "The most fundamental of these quantities is the partition function.", "In recent years localization techniques have been developed that allow the computation of partition functions for supersymmetric gauge theories in different dimensions, and on different backgrounds, starting with the work of [1], [2].", "These results have also led to new tests of conjectured dualities between theories.", "Three-dimensional $\\mathcal {N}=2$ gauge theories form a particularly fertile ground in which to develop these ideas.", "The simplest compact manifold on which one can define $d=3$ supersymmetric field theories is the round three-sphere, originally studied in [2].", "This generalizes to other three-manifolds $M_3$ , including $S^1\\times S^2$ and certain one-parameter families of squashed three-spheres [3], [4].", "In this paper we study $\\mathcal {N}=2$ theories on the Lens spaces $L(p,1)$ , which are free quotients of $S^3$ by $\\mathbb {Z}_p$ .", "In the first part of the paper we derive a formula for the full localized partition function of a three-dimensional $\\mathcal {N}=2$ $U(N)$ Chern-Simons-matter theory on such a Lens space.", "Several of the ingredients have already appeared in previous papers, including [5], [6], [7].", "The partition function reduces to a matrix model integral, where the potential function depends non-trivially on $p$ .", "In the second part of the paper we consider the partition function in the large $N$ limit, keeping the Chern-Simons levels fixed.", "The motivation for this is that, for appropriate matter content, one expects to be able to reproduce these results from a dual M-theory gravity computation.", "In particular, we focus on the low energy effective theory on $N$ M2-branes, described by the ABJM theory [8].", "A new feature that arises when $M_3$ has non-trivial fundamental group is that one must sum over different topological sectors in the partition function.", "In the present case, different sectors are labelled by a diagonal $N\\times N$ matrix with entries in $\\mathbb {Z}_p$ .", "In the large $N$ limit, we show that one can in fact focus on the contribution from matrices proportional to the identity.", "This drastically simplifies the analysis.", "At large $N$ we may use a saddle point approximation to the matrix model, following [9].", "The leading contribution to the free energy arises from a specific eigenvalue distribution.", "In order to gain some intuition we study this distribution numerically, for a number of values of $p$ .", "These numerical results lead to a simple ansatz for the eigenvalue distribution at large $N$ .", "We then use this ansatz to obtain analytic results for the free energy, as well as for the eigenvalue distribution and corresponding density.", "We find that the eigenvalue behaviour is in fact independent of $p$ , with a free energy that is simply $1/p$ times the free energy on a three-sphere.", "This is in agreement with gravity dual expectations, where $L(p,1)$ arises as the conformal boundary of AdS$_4/\\mathbb {Z}_p$ .", "The organization of this paper is as follows.", "In section we derive the full localized partition function for a three-dimensional $\\mathcal {N}=2$ Chern-Simons-matter theory on the Lens space $L(p,1)$ .", "Section contains the numerical results, which are compared to corresponding analytic results in section .", "We mention some open problems in the outlook section .", "Finally, some technical results are relegated to the appendices." ], [ "The localized partition function on $L(p,1)$", "In this section we derive a formula (REF ) for the full localized partition function of a three-dimensional $\\mathcal {N}=2$ Chern-Simons-matter theory on the Lens space $L(p,1)=S^3/\\mathbb {Z}_p$ .", "We then specialize to the ABJM theory [8] on $N$ M2-branes of interest." ], [ "The Lens space $L(p,1)$", "The Lens space $L(p,1)=S^3/\\mathbb {Z}_p$ is a certain freely-acting quotient of the round $S^3$ by a group of order $p$ .", "Regarding $S^3$ as a unit sphere in Euclidean 2, with complex coordinates $z_1,z_2$ , the $\\mathbb {Z}_p$ action is generated by $2 \\ \\ni \\ (z_1,z_2) \\ \\mapsto \\ ( \\omega _p\\, z_1, \\omega _p^{-1} z_2)~,$ where $\\omega _p=\\mathrm {e}^{2\\pi \\mathrm {i}/p}$ is a primitive $p$ -th root of unity.", "Notice this is simply a $\\mathbb {Z}_p$ quotient along the $S^1$ fibre of the Hopf fibration: $S^1\\hookrightarrow S^3\\rightarrow S^2$ .", "Quotienting $S^3$ by the free action (REF ) leads to a smooth three-manifold $L(p,1)$ with $\\pi _1(L(p,1))\\cong \\mathbb {Z}_p$ .", "There are then $p$ topologically inequivalent complex line bundles $L$ over $L(p,1)$ , labelled by their first Chern class $c_1(L)\\in H^2(L(p,1);\\mathbb {Z})\\cong \\mathbb {Z}_p$ .", "Each $L$ admits a flat $U(1)$ connection, which plays an important role in studying gauge theory on $L(p,1)$ .", "For example, rather than complex-valued functions on $L(p,1)$ , it will be important to consider more generally sections of $L$ .", "Concretely, one can construct such sections as certain projections of functions on the covering space $S^3$ .", "For example, we may expand complex-valued functions on $S^3$ in terms of hyperspherical harmonics $Y_{\\ell ,m,n}(\\theta ,\\phi ,\\psi ) &=& y_{\\ell ,m,n}(\\theta )\\mathrm {e}^{\\mathrm {i}m\\phi } \\mathrm {e}^{\\mathrm {i}n\\psi }~,$ where $(\\theta ,\\phi ,\\psi )$ are standard Euler angles on $S^3$ .", "Here $\\ell \\in \\mathbb {Z}_{\\ge 0}$ while $m,n\\in \\lbrace -\\frac{\\ell }{2},-\\frac{\\ell }{2}+1,\\ldots ,\\frac{\\ell }{2}\\rbrace $ , with $\\ell $ labelling the $(\\frac{\\ell }{2},\\frac{\\ell }{2})$ spin representation of $SU(2)_L\\times SU(2)_R$ acting on $S^3$ .", "For example, $Y_{\\ell ,m,n}$ has eigenvalue $-\\ell (\\ell +2)$ under the Laplacian.", "If $z_i=r_i\\mathrm {e}^{\\mathrm {i}\\varphi _i}$ denote polar coordinates on each copy of $ in $ 2$,then $ =1+2$, $ =1-2$, andin terms of Euler anglesthe generator (\\ref {Zpaction}) thus acts as\\begin{eqnarray}\\psi & \\mapsto & \\psi + \\frac{4\\pi }{p}~.\\end{eqnarray}The complex-valued functions on $ L(p,1)$ are preciselythe $ Zp$-invariant functions on $ S3$, and thus (\\ref {Euleraction})and (\\ref {hyperspherical}) imply that functions on$ L(p,1)$ are spanned by the modes (\\ref {hyperspherical}) satisfying\\begin{eqnarray}2n &\\equiv & 0 \\ \\mbox{mod}\\ p~.\\end{eqnarray}More generally, sections of $ L$ are spanned by the modes (\\ref {hyperspherical})satisfying\\begin{eqnarray}2n &\\equiv & c_1(L) \\ \\mbox{mod}\\ p~,\\end{eqnarray}where we are using the isomorphism $ H2(L(p,1);Z)Zp$.This follows since the holonomy of the flat connection $ A$ on $ L$ aroundthe generator $$ of $ 1(L(p,1))$ is\\begin{eqnarray}\\exp \\left[\\mathrm {i}\\int _\\gamma A\\right] &=& \\mathrm {e}^{2\\pi \\mathrm {i}c_1(L)/p}~.\\end{eqnarray}Here $$ is represented by a circle fibre in $ S1L(p,1)S2$.Sections of $ L$ must then also pick up this phase around $$.$ Another issue, important for considering supersymmetric field theories, concerns the Killing spinors.", "The 4 Killing spinors on $S^3$ transform in the $(\\mathbf {2},\\mathbf {1})\\oplus (\\mathbf {1},\\mathbf {2})$ representation of $SU(2)_L\\times SU(2)_R\\cong \\mathrm {Spin}(4)$ .", "The spinor used for localization in [2] is in the $(\\mathbf {2},\\mathbf {1})$ representation.", "In this language the $\\mathbb {Z}_p$ action in () is contained in $U(1)_R\\subset SU(2)_L\\times SU(2)_R$ , and hence the Killing spinor used for localization on $S^3$ projects down to a Killing spinor on $L(p,1)$ .", "We pause here to make some comments on more general Lens spaces $L(p,q)$ , with $q>1$ .", "These are defined as the free quotient of $S^3\\subset 2$ by the action $2 \\ \\ni \\ (z_1,z_2) \\ \\mapsto \\ ( \\omega _p^q\\, z_1, \\omega _p^{-1} z_2)~,$ with $q$ relatively prime to $p$ .", "The $\\mathbb {Z}_p$ action () now becomes $\\phi & \\mapsto & \\phi + \\frac{2\\pi (q-1)}{p}~,\\nonumber \\\\\\psi & \\mapsto & \\psi + \\frac{2\\pi (q+1)}{p}~,$ on the Euler angles.", "In particular, there is therefore no invariant spinor, unless $q=1$ or $q=p-1$ .", "This makes the treatment for this case more involved, and we therefore leave it for future work." ], [ "Localization of the path integral", "The localization of the path integral on $L(p,1)$ is very similar to the original computation for $S^3$ in [2].", "Indeed, locally the spaces are identical, so one just needs to keep track of how global differences affect formulae.", "For example, for a $U(N)$ gauge theory the path integral still localizes onto flat connections $A$ , but on $L(p,1)$ there are non-trivial flat connections that one must then sum over.", "A flat $U(N)$ connection on a manifold $M$ is determined by its holonomies, which define a homomorphism $\\varrho :\\pi _1(M)\\rightarrow U(N)$ .", "Gauge transformations act by conjugation, so that flat $U(N)$ connections are in 1-1 correspondence with $\\mathrm {Hom}(\\pi _1(M)\\rightarrow U(N))/\\mathrm {conjugation}~.$ Since $\\pi _1(L(p,1))\\cong \\mathbb {Z}_p$ , specifying a flat connection is equivalent to specifying the holonomy around the generator $\\gamma $ of $\\pi _1(L(p,1))\\cong \\mathbb {Z}_p$ $\\mathrm {hol}_\\gamma \\left(A\\right) &=& \\mathrm {diag}\\left(\\omega _p^{m_1},\\ldots ,\\omega _p^{m_N}\\right)~,$ where $0\\le m_i <p$ , and $i=1,\\ldots ,N$ runs over the generators of the Cartan $U(1)^N$ subgroup of $U(N)$ .", "Here we order $m_1\\le m_2\\le \\ldots \\le m_N$ (conjugation permutes the entries).", "The localized path integral will then give a sum over topological sectors $\\mathbf {m}=\\mathrm {diag}(m_1,\\ldots ,m_N)$ , for each $U(N)$ gauge group.", "Apart from this, as for $S^3$ all fields localize to zero except for the D-term and scalar $\\sigma $ in the $\\mathcal {N}=2$ vector multiplet, which are related via $D &=& -\\sigma ~.$ The scalar $\\sigma $ must be covariant constant.", "Writing the flat gauge field defined by $\\mathbf {$ } as $A_\\mathbf {=}-\\mathrm {i}g_\\mathbf {^}{-1}\\mathrm {d}g_\\mathbf {$ }, this implies that $\\sigma =g_\\mathbf {^}{-1}\\sigma _0g_\\mathbf {$ } where $\\sigma _0$ is a constant $N\\times N$ Hermitian matrix satisfying $[\\sigma _0,\\mathbf {]}&=& 0~.$ For a Chern-Simons gauge theory, this saddle point solution gives a standard classical contribution to the saddle point approximation of the path integral $\\exp \\left[-S_{\\mathrm {classical}}(\\sigma _0)\\right] &=& \\exp \\left[\\frac{\\mathrm {i}\\pi k}{p}\\mathrm {Tr} (\\sigma _0^2)\\right]~,$ coming from the supersymmetric completion of the Chern-Simons interaction, evaluated on (REF ).", "Here $k\\in \\mathbb {Z}$ is the Chern-Simons level.", "The $p$ -dependence in (REF ) simply arises because $\\mathrm {Vol}(L(p,1))=\\mathrm {Vol}(S^3)/p$ .", "The path integral then reduces to a matrix integral over $\\sigma _0$ , as well as the discrete sum over $\\mathbf {m}$ labelling flat $U(N)$ gauge fields.", "One must also include the Chern-Simons action for the flat gauge field: $\\exp \\left[-S_{CS}(A)\\right] & = & \\exp \\left[-\\frac{\\mathrm {i}k}{4\\pi }\\int _{L(p,1)} \\mathrm {Tr}\\left(A\\wedge \\mathrm {d}A+\\frac{2}{3}A^3\\right)\\right]\\nonumber \\\\&=& \\exp \\left[-\\frac{\\mathrm {i}\\pi k }{p} \\mathrm {Tr}\\, (\\mathbf {m}^2)\\right]~.$ One computes (REF ) in a standard way: choose a four-manifold $M_4$ with boundary $\\partial M_4 = L(p,1)$ , and an extension of the bundle and (flat) connection $A$ on $L(p,1)$ to corresponding data over $M_4$ .", "The Chern-Simons action is then in fact defined as $-(\\mathrm {i}k/4\\pi )\\int _{M_4}\\mathrm {Tr}\\, (F\\wedge F)$ , which can be shown to be independent of choices, modulo $2\\pi \\mathrm {i}$ .", "For example, in the present case one can take $M_4=$ total space of $\\mathcal {O}(p)\\rightarrow \\mathbb {CP}^1$ , and note that the restriction map $\\mathbb {Z}\\cong H^2(M_4;\\mathbb {Z}) \\rightarrow H^2(L(p,1);\\mathbb {Z})\\cong \\mathbb {Z}_p$ is simply reduction mod $p$ .", "Having summarized the localization, we next turn to the effect on the one-loop contributions around the saddle points specified by $(\\sigma _0,\\mathbf {m})$ .", "Due to the remarks in section REF , the spectra of operators that contribute to the one-loop determinants reduce to an appropriate projection of the full spectra on $S^3$ ." ], [ "Matter multiplet", "We consider here the contribution of a chiral matter field $\\Phi $ , in the representation $\\mathcal {R}$ of the gauge group, to the one-loop determinant around the classical background labelled by $(\\sigma _0,\\mathbf {m})$ .", "We denote the R-charge of $\\Phi $ as $\\Delta =\\Delta (\\Phi )$ – the canonical value is $\\Delta =\\frac{1}{2}$ – and the weights of $\\mathcal {R}$ by $\\rho $ .", "The bosonic contribution to the one-loop determinant is then [2], [10], [11] $\\mathrm {det}_{\\ell /2,m}(D_{\\mathrm {boson}}) &=& \\ell (\\ell +2)- 4m(1-\\Delta )+\\Delta ^2+\\rho (\\sigma _0)^2~,$ where $\\ell ,m$ label the same quantum numbers as in section REF , so that $\\ell \\ge 0$ and $m\\in \\lbrace -\\frac{\\ell }{2},-\\frac{\\ell }{2}+1,\\ldots ,\\frac{\\ell }{2}\\rbrace $ .", "In particular, the $\\ell (\\ell +2)$ term simply comes from the eigenvalue under (minus) the scalar Laplacian.", "On $S^3$ there are $\\ell +1$ such modes, labelled by the quantum number $n\\in \\lbrace -\\frac{\\ell }{2},-\\frac{\\ell }{2}+1,\\ldots ,\\frac{\\ell }{2}\\rbrace $ , while on $L(p,1)$ we should keep only those modes satisfying $2n &\\equiv & \\rho (\\mathbf {)} \\ \\mbox{mod}\\ p~,$ as follows from ().", "The fermionic contribution to the one-loop determinant is also given by (REF ), but now with $m\\in \\lbrace -\\frac{\\ell }{2},-\\frac{\\ell }{2}+1,\\ldots ,\\frac{\\ell }{2}-1\\rbrace $ , and with an additional contribution of $(-1)^\\ell \\left(\\ell +\\Delta +\\mathrm {i}\\rho (\\sigma _0)\\right)\\left(\\ell +2-\\Delta +\\mathrm {i}\\rho (\\sigma _0)\\right)$ .", "Again, on $S^3$ there is a degeneracy of $\\ell +1$ , labelled by $n$ , while on $L(p,1)$ we should keep only those modes satisfying (REF ).", "Since the one-loop determinant is a ratio of fermionic and bosonic determinants, we thus see that for fixed $\\ell , m$ and $\\mathbf {m}$ , for every choice of $n$ satisfying (REF ) the contributions from fermionic and bosonic determinants will cancel, except for the “missing” fermionic mode with $m=\\tfrac{\\ell }{2}$ – this remains uncancelled in the bosonic determinant.", "We thus conclude that, for fixed $\\ell $ , we have $\\frac{\\mathrm {det}_{\\ell /2}(D_{\\mathrm {fermion}})}{\\mathrm {det}_{\\ell /2}(D_{\\mathrm {boson}})} &=&(-1)^\\ell \\frac{ \\left(\\ell +\\Delta +\\mathrm {i}\\rho (\\sigma _0)\\right)\\left(\\ell +2-\\Delta +\\mathrm {i}\\rho (\\sigma _0)\\right)}{ \\ell (\\ell +2)- 2\\ell (1-\\Delta )+\\Delta ^2+\\rho (\\sigma _0)^2}\\nonumber \\\\&=& (-1)^\\ell \\frac{\\ell + 2-\\Delta +\\mathrm {i}\\rho (\\sigma _0)}{\\ell +\\Delta -\\mathrm {i}\\rho (\\sigma _0)}~,$ where the degeneracy is the number of half-integers $n\\in \\lbrace -\\frac{\\ell }{2},-\\frac{\\ell }{2}+1,\\ldots ,\\frac{\\ell }{2}\\rbrace $ satisfying $2n &\\equiv & \\rho (\\mathbf {m}) \\ \\mbox{mod} \\ p~.$ This degeneracy was denoted by $N_\\rho (\\ell )$ in [7].", "Thus in total $Z^{\\mathrm {matter}}_{\\mathrm {1-loop}}(\\sigma _0,\\mathbf {)} &=& \\prod _{\\rho \\in \\mathcal {R}}\\prod _{\\ell \\ge 0}\\left(\\frac{\\ell +2-\\Delta +\\mathrm {i}\\rho (\\sigma _0)}{\\ell +\\Delta -\\mathrm {i}\\rho (\\sigma _0)}\\right)^{N_\\rho (\\ell )}~.$" ], [ "Vector multiplet", "The analysis for the one-loop contribution of the vector multiplet is very similar.", "Here it is more convenient to follow the analysis in [3], where rather than working out the full spectrum, most of which then cancels in the ratio of determinants, instead one isolates the uncancelled modes from the outset.", "These are precisely the eigenmodes which are not paired with a superpartner.", "We shall refer to these as the uncancelled modes.", "The uncancelled gaugino modes on $S^3$ have eigenvalues $\\mu &=& n_1+n_2+\\mathrm {i}\\alpha (\\sigma _0)$ under the relevant Dirac operator, where $n_i$ denote the charges under $\\partial _{\\varphi _i}$ , where recall that $\\varphi _1$ , $\\varphi _2$ are azimuthal angles on each copy of $ in $ 2S3$.The normalizable modes are $ {n1,n20}{(n1,n2)=(0,0)}$.The corresponding charges under $$, $$are then $ 12(n1-n2)$, $ 12(n1+n2)$,respectively (see just before equation (\\ref {Euleraction})), so that the projection condition becomes\\begin{eqnarray}n_1 &\\equiv & n_2 + \\alpha (\\mathbf {)} \\ \\mathrm {mod} \\ p~.\\end{eqnarray}The uncancelled transverse vector modes also have eigenvalues of the form(\\ref {mus}), except now $ n1,n2-1$.", "Thus\\begin{eqnarray}Z^{\\mathrm {vector}}_{\\mathrm {1-loop}}(\\sigma _0,\\mathbf {)} \\ =\\ \\prod _\\alpha \\left[ \\frac{{\\displaystyle \\prod }_{\\mbox{\\tiny $\\begin{array}{c}n_1,n_2\\ge 0 \\\\ (n_1,n_2)\\ne (0,0) \\\\ n_1\\equiv n_2 + \\alpha (\\mathbf {)} \\ \\mbox{mod}\\ p\\end{array}$}} [n_1+n_2+\\mathrm {i}\\alpha (\\sigma _0)]}{{\\displaystyle \\prod }_{\\mbox{\\tiny $\\begin{array}{c}n_1,n_2\\ge 1 \\\\n_1\\equiv n_2 - \\alpha (\\mathbf {)} \\ \\mbox{mod}\\ p\\end{array}$}}[-n_1-n_2+\\mathrm {i}\\alpha (\\sigma _0)]}\\right]~.\\end{eqnarray}We then rewrite this as a product over only the positive roots $ >0$, while at the same time multiplying by thesame expression with $ -$.", "In doing this,one sees that all the terms in the numerators and denominators cancel, \\emph {except}for the numerator contributions of $ {n1=0,n21}$, $ {n11,n2=0}$, which are left uncancelled.", "We are thus left with\\begin{eqnarray}Z^{\\mathrm {vector}}_{\\mathrm {1-loop}}(\\sigma _0,\\mathbf {)} & =& \\prod _{\\alpha >0} \\Bigg [\\prod _{r\\ge 1}\\left[-\\alpha (\\mathbf {)}+pr+\\mathrm {i}\\alpha (\\sigma _0)\\right]\\frac{\\prod _{r\\ge 0}\\left[\\alpha (\\mathbf {)}+pr-\\mathrm {i}\\alpha (\\sigma _0)\\right]}{(-\\mathrm {i}\\alpha (\\sigma _0))^{\\delta _{\\alpha (\\mathbf {)},0}}}\\times \\nonumber \\\\&&\\frac{\\prod _{r\\ge 0}\\left[\\alpha (\\mathbf {)}+pr+\\mathrm {i}\\alpha (\\sigma _0)\\right]}{(\\mathrm {i}\\alpha (\\sigma _0))^{\\delta _{\\alpha (\\mathbf {)},0}}}\\prod _{r\\ge 1}\\left[-\\alpha (\\mathbf {)}+pr-\\mathrm {i}\\alpha (\\sigma _0)\\right]\\Bigg ]~.\\end{eqnarray}Notice that here, in a slight abuse of notation, we have assumed that $ 0()<p$.The last equation may then be rewritten\\begin{eqnarray}Z^{\\mathrm {vector}}_{\\mathrm {1-loop}}(\\sigma _0,\\mathbf {)} & =& \\prod _{\\alpha >0}\\left[\\prod _{r=1}^\\infty (pr)^4\\right](\\alpha (\\mathbf {)}-\\mathrm {i}\\alpha (\\sigma _0))\\prod _{r=1}^\\infty \\left[1+\\frac{\\left(\\alpha (\\sigma _0)+\\mathrm {i}\\alpha (\\mathbf {)}\\right)^2}{r^2p^2}\\right]\\times \\nonumber \\\\&&(\\alpha (\\mathbf {)}+\\mathrm {i}\\alpha (\\sigma _0)) \\prod _{r=1}^\\infty \\left[1+\\frac{\\left(\\alpha (\\sigma _0)-\\mathrm {i}\\alpha (\\mathbf {)}\\right)^2}{r^2p^2}\\right]\\cdot \\frac{1}{(\\alpha (\\sigma _0)^2)^{\\delta _{\\alpha (\\mathbf {)},0}}}\\nonumber \\end{eqnarray}Zeta function regularizing $ r=1(pr)4 zeta= (2)2/p2$ and using the infinite product formulafor $$, we obtain\\cite {Gang:2009wy, Benini:2011nc}\\begin{eqnarray}Z^{\\mathrm {vector}}_{\\mathrm {1-loop}}(\\sigma _0,\\mathbf {)} \\ =\\ \\prod _{\\alpha >0} \\frac{2\\sinh \\left[\\frac{\\pi }{p}\\left(\\alpha (\\sigma _0)+\\mathrm {i}\\alpha (\\mathbf {)}\\right)\\right]2\\sinh \\left[\\frac{\\pi }{p}\\left(\\alpha (\\sigma _0)-\\mathrm {i}\\alpha (\\mathbf {)}\\right)\\right]}{\\left(\\alpha (\\sigma _0)^2\\right)^{\\delta _{\\alpha (\\mathbf {)}},0}}~.\\end{eqnarray}$" ], [ "Partition function", "Putting everything together from the previous sections, we arrive at the final formula for the localized partition function of an $\\mathcal {N}=2$ Chern-Simons-matter theory on the Lens space $L(p,1)$ $ Z \\ =\\ \\sum _{\\mathbf {}\\int \\mathrm {d}\\sigma _0 \\exp \\left[\\frac{\\mathrm {i}\\pi k}{p}\\left(\\mathrm {Tr} (\\sigma _0^2)-\\mathrm {Tr}\\, (\\mathbf {^}2)\\right)\\right]Z^{\\mathrm {vector}}_{\\mathrm {1-loop}}(\\sigma _0,\\mathbf {)} \\, Z^{\\mathrm {matter}}_{\\mathrm {1-loop}}(\\sigma _0,\\mathbf {)}~,}where the one-loop vector and matter contributions are given by (\\ref {vector}), (\\ref {matter}), respectively.$ Recall that for a $U(N)$ gauge group, $\\sigma _0$ is a constant $N\\times N$ Hermitian matrix that commutes with $\\mathbf {$ }.", "We may thus diagonalize $\\sigma _0 &=& \\left(\\frac{\\lambda _1}{2\\pi },\\ldots ,\\frac{\\lambda _N}{2\\pi }\\right)~,$ where $\\lambda _i/2\\pi $ , $i=1,\\ldots ,N$ , are the real eigenvalues of $\\sigma _0$ , and we order $\\lambda _1\\le \\cdots \\le \\lambda _N$ .", "A choice of positive roots for $G$ is then $\\alpha _{ij}(\\sigma _0) &=& \\frac{\\lambda _i-\\lambda _j}{2\\pi }~,$ with $i<j$ .", "Notice that the Vandermonde determinant then contributes a factor to the integrand of (REF ) when rewriting $\\int \\mathrm {d}\\sigma _0 &=& \\int \\prod _{i=1}^N\\frac{\\mathrm {d}\\lambda _i}{2\\pi } \\prod _{i<j \\, |\\, m_i=m_j}\\left(\\frac{\\lambda _i-\\lambda _j}{2\\pi }\\right)^2~,$ which precisely cancels the denominator in ().", "We now specialise to the particular gauge theory of interest, namely the ABJM theory on $N$ M2-branes [8].", "This is a $U(N)\\times U(N)$ Chern-Simons gauge theory with Chern-Simons levels $(k,-k)$ for the two gauge group factors, two chiral matter fields in the bifundamental representation $(\\mathbf {N},\\overline{\\mathbf {N}})$ , and two chiral matter fields in the conjugate $(\\overline{\\mathbf {N}},\\mathbf {N})$ representation.", "More precisely, this is the low energy worldvolume theory on $N$ M2-branes transverse to $4/\\mathbb {Z}_k$ , where the $\\mathbb {Z}_k$ acts with weights $(1,1,-1,-1)$ on the four complex coordinates.", "The R-charges/scaling dimensions of the 4 chiral fields all take the canonical value of $\\Delta =\\frac{1}{2}$ .", "We may thus introduce eigenvalues $\\lambda _i$ , $\\tilde{\\lambda }_i$ , $i=1,\\ldots ,N$ , for the two gauge group factors, and correspondingly matrices $\\mathbf {$ }, $\\mathbf {\\tilde{m}}$ specifying the flat connections for each copy of $U(N)$ .", "Note that the weights for the bifundamental representation $(\\mathbf {N},\\overline{\\mathbf {N}})$ are $\\rho _{ij}(\\sigma _0) &=& \\frac{\\lambda _i-\\tilde{\\lambda }_j}{2\\pi }~,$ with minus this for the conjugate representation.", "Thus the partition function for the ABJM theory on $L(p,1)$ is $Z & =& \\sum _{\\mathbf {,}\\tilde{\\mathbf {}}\\frac{1}{N!^2}\\int \\prod _{i=1}^N \\frac{\\mathrm {d}\\lambda _i}{2\\pi }\\frac{\\mathrm {d}\\tilde{\\lambda }_i}{2\\pi }\\exp \\left[\\frac{\\mathrm {i}k}{4\\pi p}\\left(\\sum _{i=1}^N \\left( \\lambda _i^2-\\tilde{\\lambda }_i^2\\right)-(2\\pi )^2\\sum _{i=1}^N\\left(m_i^2-\\tilde{m_i}^2\\right)\\right)\\right]\\times \\nonumber \\\\&&\\prod _{i<j}2 \\sinh \\left[\\frac{\\lambda _i-\\lambda _j+2\\pi \\mathrm {i}(m_i-m_j)}{2p}\\right]2\\sinh \\left[\\frac{\\lambda _i-\\lambda _j-2\\pi \\mathrm {i}(m_i-m_j)}{2p}\\right]\\times \\nonumber \\\\&& \\prod _{i<j}2 \\sinh \\left[\\frac{\\tilde{\\lambda }_i-\\tilde{\\lambda }_j+2\\pi \\mathrm {i}(\\tilde{m}_i-\\tilde{m}_j)}{2p}\\right]2\\sinh \\left[\\frac{\\tilde{\\lambda }_i-\\tilde{\\lambda }_j-2\\pi \\mathrm {i}(\\tilde{m}_i-\\tilde{m}_j)}{2p}\\right]\\times \\nonumber \\\\&& \\prod _{i,j}\\left[P_{p}^{m_i-\\tilde{m}_j}\\bigg (\\frac{\\lambda _i-\\tilde{\\lambda }_j}{2\\pi }\\bigg )\\right]^2~,}where we have defined\\begin{eqnarray}P_{p}^\\kappa (z) &:= & \\prod _{\\ell =0}^\\infty \\left(\\frac{\\ell +\\frac{3}{2}+\\mathrm {i}z}{\\ell +\\frac{1}{2}-\\mathrm {i}z}\\right)^{N_\\kappa (\\ell )}\\left(\\frac{\\ell +\\frac{3}{2}-\\mathrm {i}z}{\\ell +\\frac{1}{2}+\\mathrm {i}z}\\right)^{N_{p-\\kappa }(\\ell )}~.\\end{eqnarray}The latter is precisely the contribution from \\emph {one} (\\mathbf {N},\\overline{\\mathbf {N}})chiral field, and \\emph {one} (\\overline{\\mathbf {N}},\\mathbf {N}) field, andthese correspond to the respective factors in the product (\\ref {P}).As before, the notation N_\\kappa (\\ell ) means the number ofhalf-integers n\\in \\lbrace -\\frac{\\ell }{2},-\\frac{\\ell }{2}+1,\\ldots ,\\frac{\\ell }{2}\\rbrace such that 2n\\equiv \\kappa mod p.The square at the very end of (\\ref {ABJM}) thenaccounts for the fact there are two of each type ofbifundamental field.$" ], [ "Matter potentials", "The matter potentials $V_{p}^\\kappa (z) &:= & \\log P_{p}^\\kappa (z)~,$ where $P_{p}^\\kappa (z)$ is defined by (), play an important role in the dynamics of the matrix model (REF ).", "A general discussion of these potentials, which in general involve polygamma functions, may be found in appendix .", "In particular, the products in () are divergent and must be regularized, and we do this using zeta function regularization.", "The resulting functions simplify somewhat in particular cases.", "In this subsection we give a few examples, for low values of $p$ .", "Recall that for fixed $p\\ge 1$ , we have $0\\le \\kappa <p$ .", "The (regularized) matter potentials for small $p$ then simplify to $V_{1}^{0}(z) & = & -\\log \\left[2\\cosh (\\pi z)\\right]~,\\nonumber \\\\V_{2}^0(z) &=& -\\frac{1}{2} \\log [2 \\cosh (z\\pi )] + 2z\\cot ^{-1} \\mathrm {e}^{\\pi z} +\\frac{\\mathrm {i}}{\\pi } \\left[\\mathrm {Li}_2(-\\mathrm {i}\\mathrm {e}^{-z\\pi }) - \\mathrm {Li}_2(\\mathrm {i}\\mathrm {e}^{-z\\pi }) \\right]~,\\nonumber \\\\V_{2}^1(z) &=& -\\frac{1}{2} \\log (2 \\cosh (z\\pi ) ) -2z\\cot ^{-1} \\mathrm {e}^{\\pi z} -\\frac{\\mathrm {i}}{\\pi } \\left[ \\mathrm {Li}_2(-\\mathrm {i}\\mathrm {e}^{-z\\pi }) - \\mathrm {Li}_2(\\mathrm {i}\\mathrm {e}^{-z\\pi }) \\right]~,\\nonumber \\\\V_{3}^0(z) &=& \\log \\frac{2 \\cosh ^2(\\pi z/3)}{\\cosh \\pi z}~,\\nonumber \\\\V_{3}^1(z) &=&V_{3,1}^2(z) \\ = \\ -\\log [2 \\cosh (\\pi z/3)]~.$ The reader is referred to the Appendices and for more detail." ], [ "The large $N$ limit: numerical results", "Our aim in the remainder of the paper is to compute the M-theory limit of the ABJM partition function (REF ), which means fixed Chern-Simons level $k$ and $N\\rightarrow \\infty $ .", "The partition function (REF ) is an extremely complicated object, and to gain some intuition we will begin in this section by solving the matrix model numerically for large values of $N$ .", "We will do this by extending the saddle-point methods of [9] to the present case.", "The behaviour is simple enough to suggest an ansatz for the eigenvalue distribution, precisely as for the ABJM model on $S^3$ studied in [9], which in section we then analytically show reproduces the numerics.", "Moreover, this analytic result agrees with the expected large $N$ gravity dual result for the free energy of $F_{\\mathrm {M}-\\mathrm {theory}} = \\frac{\\pi \\sqrt{2k}}{3p}N^{3/2}$ ." ], [ "General discussion", "The matrix model partition function (REF ) of $N$ M2-branes on a Lens space $L(p,1)$ has the following form $Z & = & \\sum _{\\mathbf {,}{\\tilde{\\mathbf {}}} Z_{\\mathbf {,}\\mathbf {\\tilde{m}}} \\ = \\ \\sum _{\\mathbf {,}\\mathbf {\\tilde{m}}} \\int \\left( \\prod _{i=1}^N \\mathrm {d}\\lambda _i \\mathrm {d}\\tilde{\\lambda }_i \\right) \\mathrm {e}^{-F_{\\mathbf {,}\\mathbf {\\tilde{m}}}(\\lambda _i,\\tilde{\\lambda }_i) }~,}where recall that \\mathbf {=}\\mathrm {diag}(m_1,\\ldots ,m_N) has entries m_1\\le m_2\\le \\ldots \\le m_N with 0\\le m_i<p, and specifiesthe flat connection for the first U(N) gauge group, while tilded quantities refer to the second U(N) gauge group.The basic idea is that when the number N of eigenvalues \\lambda _i is large, each contribution Z_{\\mathbf {,}\\mathbf {\\tilde{m}}} can be well approximated in the saddle-point limit by Z=\\mathrm {e}^{-F}, where the free energy F is an extremum of F_{{\\bf m},{\\bf \\tilde{m}}}(\\lambda _i,\\tilde{\\lambda }_i) with respect to \\lambda _i and \\tilde{\\lambda }_i.$ Given $F(\\lambda _i,\\tilde{\\lambda }_i)$ (in what follows, we suppress the lower indices ${\\bf m,\\tilde{m}}$ ) the saddle-point equations are $\\frac{\\partial F}{\\partial \\lambda _i} \\ = \\ 0~,~~~~~\\frac{\\partial F}{\\partial \\tilde{\\lambda }_i} \\ = \\ 0~.$ The extremum of the free energy is then given by $F(\\lambda ^{0}_i,\\tilde{\\lambda }^{0}_i)$ , where $\\lambda ^{0}_i,\\tilde{\\lambda }^{0}_i$ are the solutions of the saddle-point equations.", "This then gives the leading contribution to $Z_{\\bf m,\\tilde{m}}$ at large $N$ and fixed $k$ .", "As for $p=1$ , it will turn out that the saddle-point solution has complex eigenvalues, which means we deform the real integrals over $\\lambda _i,$ $\\tilde{\\lambda }_i$ in (REF ) into the complex plane.", "Even though highly non-trivial, the equations (REF ) can be solved numerically.", "It is convenient to view these equations as describing the equilibrium configuration of $2N$ particles, whose two-dimensional coordinates are given by the complex numbers $\\lambda _i$ and $\\tilde{\\lambda }_i$ .", "This equilibrium configuration can be found by introducing a “time dependence”, so that $\\lambda _i,~\\tilde{\\lambda }_i \\rightarrow \\lambda _i(t),~\\tilde{\\lambda }_i(t)$ , and writing down equations of motion for $\\lambda _i(t)$ and $\\tilde{\\lambda }_i(t)$ such that their solutions approach the equilibrium configuration for late times:In order for the eigenvalues to go to the correct attractor point as $t \\rightarrow \\infty $ , a priori one might need to multiply the left hand sides of (REF ) by a complex number.", "For the case at hand in fact this is unnecessary.", "$\\frac{\\mathrm {d}\\lambda _i}{\\mathrm {d}t}\\ = \\ -\\frac{\\partial F}{\\partial \\lambda _i}~,~~~~~\\frac{\\mathrm {d}\\tilde{\\lambda }_i}{\\mathrm {d}t}\\ = \\ - \\frac{\\partial F}{\\partial \\tilde{\\lambda }_i}~.$ In the following we will solve these equations numerically.", "From this we will extract generic behaviour that will lead to a corresponding analytic computation in section ." ], [ "Flat connection dependence", "A new ingredient in the partition function on Lens spaces, with respect to the case on $S^3$ , is the sum over different flat connections labelled by $\\mathbf {$ }, $\\mathbf {\\tilde{m}}$ .", "We are interested in the large $N$ limit, and in this limit we expect $Z\\ =\\ \\sum _{{\\bf m},{\\bf \\tilde{m}}} \\mathrm {e}^{-F_{{\\bf m},{\\bf \\tilde{m}}}} \\ \\longrightarrow \\ \\mathrm {e}^{-F_{\\mathrm {M-theory}}} \\ = \\ Z_{\\mathrm {M-theory}} ~.$ In the supergravity approximation to M-theory, we are computing the log of the partition function in the large $N$ limit, and we are interested in the leading term only.", "Hence, even though we have the sum of many terms on the left hand side of (REF ), we expect only certain terms to contribute.", "More precisely, we may focus on the contribution of $(\\mathbf {,}\\mathbf {\\tilde{m}})=(\\mathbf {_}0,\\mathbf {\\tilde{m}}_0)$ with least $F_{{\\bf m}_0,{\\bf \\tilde{m}}_0}$ , in the large $N$ limit.", "Note that contributions of $\\mathrm {e}^{-F_{{\\bf m},{\\bf \\tilde{m}}}}$ for other choices of $(\\mathbf {,}\\mathbf {\\tilde{m}})$ do not need to be suppressed with respect to that for $(\\mathbf {_}0,\\mathbf {\\tilde{m}}_0)$ : if they give a similar contribution, this will simply lead to a logarithmic, and hence subleading, correction at large $N$ , since we are taking the logarithm to obtain the free energy.", "We have performed a numerical evaluation of $F_{{\\bf m},{\\bf \\tilde{m}}}$ , for several values of $N,p$ and all choices of ${\\bf m}, {\\bf \\tilde{m}}$ .", "In all cases the leading contribution comes from ${\\bf m} ={\\bf \\tilde{m}} = \\mathrm {diag}(c,c,c,...,c) = c \\cdot 1_{N\\times N}$ , where $c$ is an integer with $0\\le c<p$ .", "Hence the first lesson we draw from the numerical analysis is that we may focus on the specific case ${\\bf m} ={\\bf \\tilde{m}} = \\mathrm {diag}(c,c,c,...,c)$ if we are only interested in the large $N$ limit.", "This simplifies the problem, and its treatment, enormously.", "Furthermore, note that with these choices of flat connection the eigenvalues will respect certain symmetries, discussed below, but that this would not be true for any other choice of flat connection.", "So from now on we focus on this case.Other choices of $\\mathbf {$ }, $\\mathbf {\\tilde{m}}$ , presumably important for computing subleading corrections, are briefly discussed in Appendix B." ], [ "Numerical plots", "Figure REF shows the distribution of eigenvalues for the case $p=2$ and $N=100$ .", "From this distribution we can draw several conclusions.", "First, we see that the eigenvalue distribution is invariant under $\\lambda _i \\rightarrow - \\lambda _i$ , $\\tilde{\\lambda }_i \\rightarrow - \\tilde{\\lambda }_i$ .", "Furthermore, for the equilibrium configuration $\\lambda _i$ and $\\tilde{\\lambda }_i$ are complex conjugates of each other.", "To be more precise, we find that $\\lambda _{i} = - \\lambda _{N-i+1}$ (with the same for $\\tilde{\\lambda }_i$ ) and $\\bar{\\lambda }_i = {\\tilde{\\lambda }}_i$ .", "As for the $p=1$ case, these are symmetries of the equations of motion, so we expect these symmetries for the equilibrium distributions as well.", "As already mentioned, however, these symmetries will not be present for more general choices of ${\\bf m},{\\bf \\tilde{m}} $ .", "Figure: Distribution of eigenvalues for p=2p=2, N=100N=100.Other features are that for large values of $N$ the density of eigenvalues is relatively uniform, the real part of the eigenvalues grows with $N$ , while the imaginary part stays bounded.", "The numerics are consistent with the real part growing as $N^{1/2}$ , while the imaginary part of the eigenvalues stays bounded between $-\\pi /2$ and $\\pi /2$ .", "As we increase the Chern-Simons level $k$ the slope also increases.", "The analytic treatment in section (after assumptions justified by the numerics) predicts a slope proportional to $\\sqrt{k}$ – in fact precisely the same slope as for $p=1$ .", "This is also consistent with the numerical results – see Figure REF .", "Figure: Distribution of eigenvalues for p=1,2,3p=1,2,3 and N=100N=100." ], [ "The large $N$ limit: analytic results", "As in the previous section, the idea is to compute the partition function (REF ) in a saddle-point approximation, focusing on the contribution from $\\mathbf {=}\\mathbf {\\tilde{m}}=c\\cdot 1_{N\\times N}$ , which from the numerics we see determines the free energy in the large $N$ limit.", "As the number of eigenvalues $N$ for each gauge group tends to infinity, one has a continuum limit in which one can replace the sums over eigenvalues in the potential by integrals.", "In particular, one can then separate the interactions between eigenvalues into “long range forces,” for which the interaction between eigenvalues is non-local, plus a local interaction.", "A key point is that these long range forces automatically cancel.", "An appropriate ansatz for $\\lambda _i$ will then lead to a simple local action for the eigenvalues, which may be solved in the saddle-point approximation exactly in the large $N$ limit.", "This analytic result may then be checked against the numerical results, and we find excellent agreement.", "We will also comment on the relation to the gravity dual." ], [ "Long range forces", "Let us focus first on the long range forces, which come from the leading terms in an asymptotic expansion of the $\\sinh $ and matter potential $V_{p}^{\\kappa =0}(z)$ in (REF ).Recall here that since $\\mathbf {=}\\mathbf {\\tilde{m}}$ is proportional to the identity matrix, we have $\\kappa =0$ in all cases.", "In the former case we define $\\left[\\log 2\\sinh z\\right]^{\\mathrm {asymp}} &:= & z\\, \\mathrm {sign}\\left(\\mathrm {Re}\\, z\\right)~.$ The point here is that for $\\mathrm {Re}\\, z>0$ we have the series $\\log \\left[2\\sinh z\\right] &=& {z} - \\sum _{\\ell =1}^\\infty \\frac{1}{\\ell }\\mathrm {e}^{-2\\ell z}~.$ while for $\\mathrm {Re}\\, z<0$ we have $\\log \\left[2\\sinh z\\right] &=& \\mathrm {i}\\pi -{z} -\\sum _{\\ell =1}^\\infty \\frac{1}{\\ell }\\mathrm {e}^{2\\ell z}~.$ We shall see momentarily that the constant $\\mathrm {i}\\pi $ term in (REF ) does not contribute to the long range force computation, which is why we omit this constant in the definition (REF ).", "The sums of exponential terms in (REF ), (REF ) will be of relevance momentarily.", "The matter potentials $V_{p}^0(z)$ depend in a complicated way on $p$ .", "The relevant asymptotic expansions are discussed in appendix .", "In particular, this leads to $\\left[V_{p}^0(z)\\right]^\\mathrm {asymp}&:= & -\\frac{\\pi z}{p}\\, \\mathrm {sign}\\left(\\mathrm {Re}\\, z\\right)~.$ We then have the following general form of the expansions for $V_{p}^0(z)$ $V_{p}^0(z) \\, = \\, \\left[V_{p}^0(z)\\right]^\\mathrm {asymp}+ \\pi z\\, \\mathrm {sign}(\\mathrm {Re}\\, z)\\sum _{\\ell \\in \\frac{1}{p}\\mathbb {N}} c_\\ell \\, \\mathrm {e}^{-\\ell \\pi z\\, \\mathrm {sign}(\\mathrm {Re}\\, z)} + \\sum _{\\ell \\in \\frac{1}{p}\\mathbb {N}} d_\\ell \\, \\mathrm {e}^{-\\ell \\pi z\\, \\mathrm {sign}(\\mathrm {Re}\\, z)} ~,$ for appropriate constants $c_\\ell $ , $d_\\ell $ depending on $p$ .", "In the large $N$ limit we then take a continuous limit, in which sums over $i=1,\\ldots ,N$ become Riemann integrals $\\frac{1}{N}\\sum _{i=1}^N & \\longrightarrow & \\int _{x_{\\mathrm {min}}}^{x_{\\mathrm {max}}} \\rho (x)\\, \\mathrm {d}x.$ The numerical results of section then suggest we make the following ansatz for the eigenvalues $\\lambda (x) &=& N^\\alpha x + \\mathrm {i}y(x)~, \\qquad \\tilde{\\lambda }(x) \\ = \\ N^\\alpha x - \\mathrm {i}y(x)~,$ where $\\alpha >0$ , and these formulae are understood to be correct to order $N^{-\\epsilon }$ , for some $\\epsilon >0$ .", "Notice we have deformed the real eigenvalues of the Hermitian matrix $\\sigma $ into the complex plane in (REF ), anticipating a complex saddle point, and that the function $\\rho (x)$ describes the density of the eigenvalues.", "Also recall that $\\bar{\\lambda }_i\\leftrightarrow \\tilde{\\lambda }_i$ is a symmetry of the system – in (REF ) we have imposed that the solution is invariant under this symmetry, which is again supported by the numerical results.", "The long range forces are then, by definition, determined by the leading asymptotic terms in the potential.", "Substituting (REF ), (REF ) into the logarithim of the partition function (REF ) and taking the continuum limit (REF ), we obtain $-F_{\\mathrm {asymp}} &=& N^2 \\int _{x_{\\mathrm {min}}}^{x_{\\mathrm {max}}} \\rho (x)\\, \\mathrm {d}x \\int _{x_{\\mathrm {min}}}^{x_{\\mathrm {max}}} \\rho (x^{\\prime })\\, \\mathrm {d}x^{\\prime }\\, \\mathrm {sign}(x-x^{\\prime })\\Bigg [\\frac{1}{2p}(\\lambda (x)-\\lambda (x^{\\prime })) \\nonumber \\\\&& + \\frac{1}{2p}(\\tilde{\\lambda }(x)-\\tilde{\\lambda }(x)^{\\prime }) - 2\\times \\frac{\\lambda (x)-\\tilde{\\lambda }(x^{\\prime })}{2p}\\Bigg ]~.$ Here one sees from (REF ) that one substitutes $z=(\\lambda (x)-\\tilde{\\lambda }(x^{\\prime }))/2\\pi $ into (REF ).", "The factor of 2 in the last term of (REF ) accounts for the two copies of chiral fields in the $(\\mathbf {N},\\overline{\\mathbf {N}})$ and $(\\overline{\\mathbf {N}},\\mathbf {N})$ representations.", "Also notice that the original sum in the vector multiplet contribution to (REF ) is over $i<j$ , which means $x-x^{\\prime }<0$ in the continuum limit.", "In writing (REF ) we have simply extended this to a sum over $i>j$ by replacing $\\lambda _i-\\lambda _j$ by $\\lambda _j-\\lambda _i$ .", "It is then straightforward to see from the ansatz (REF ) that all the terms in (REF ) cancel: the real parts simply cancel inside the square bracket, while the imaginary parts contribute zero on using the anti-symmetry under $x\\leftrightarrow x^{\\prime }$ implied by the $\\mathrm {sign}(x-x^{\\prime })$ term.", "Thus $F_\\mathrm {asymp}=0$ , and the long range forces indeed cancel for $L(p,1)$ ." ], [ "Local action", "It follows that only the exponential sums in (REF ), (REF ), (REF ) contribute to the partition function in the large $N$ limit.", "Since the latter depend on $\\mathrm {sign}\\left(\\mathrm {Re}\\, z\\right)$ , which is equal to $\\mathrm {sign}(x-x^{\\prime })$ , we first split the double integrals as $N^2 \\int _{x_{\\mathrm {min}}}^{x_{\\mathrm {max}}} \\rho (x)\\, \\mathrm {d}x \\int _{x_{\\mathrm {min}}}^{x_{\\mathrm {max}}} \\rho (x^{\\prime })\\, \\mathrm {d}x^{\\prime }&\\longrightarrow & N^2 \\int _{x_{\\mathrm {min}}}^{x_{\\mathrm {max}}} \\rho (x)\\, \\mathrm {d}x \\int _{x_{\\mathrm {min}}}^{x} \\rho (x^{\\prime })\\, \\mathrm {d}x^{\\prime }\\nonumber \\\\&& + N^2 \\int _{x_{\\mathrm {min}}}^{x_{\\mathrm {max}}} \\rho (x)\\, \\mathrm {d}x \\int _{x}^{x_{\\mathrm {max}}} \\rho (x^{\\prime })\\, \\mathrm {d}x^{\\prime }~,$ so that $x-x^{\\prime }>0$ for the first term on the right hand side, while $x-x^{\\prime }<0$ for the second term.", "We will then apply the general formula $\\int _{x_\\mathrm {min}}^x\\mathrm {d}x^{\\prime } \\, \\mathrm {e}^{-\\beta N^\\alpha (x-x^{\\prime })}\\, f(x,x^{\\prime }) &=& \\frac{1}{\\beta N^\\alpha }\\left[\\mathrm {e}^{-\\beta N^\\alpha (x-x^{\\prime })}\\, f(x,x^{\\prime })\\right]_{x_{\\mathrm {min}}}^x\\nonumber \\\\&& - \\frac{1}{\\beta N^\\alpha }\\int _{x_{\\mathrm {min}}}^x \\mathrm {d}x^{\\prime }\\, \\mathrm {e}^{-\\beta N^\\alpha (x-x^{\\prime })}\\, \\frac{\\mathrm {d}}{\\mathrm {d}x^{\\prime }} f(x,x^{\\prime })~,$ which follows trivially from an integration by parts.", "The first term on the right hand side is simply $\\frac{1}{\\beta N^{\\alpha }}f(x,x)$ , plus a term which is exponentially suppressed in the large $N$ limit.", "The formula (REF ), with a similar formula applying for $x-x^{\\prime }<0$ , amount to the representation $\\delta (x) &=& \\lim _{c\\rightarrow \\infty } \\frac{c}{2}\\, \\mathrm {e}^{-c|x|}~,$ thus reducing the integral over $x$ , $x^{\\prime }$ to an integral over $x$ , in the large $N$ limit.", "Applying this to the sums over exponentials in (REF ), (REF ), (REF ) is a straightforward task.", "For the vector multiplet and matter multiplet contributions, we obtain to leading order $-F_{\\mathrm {vector}}&=& -\\frac{4p\\pi ^2}{6}N^{2-\\alpha }\\int _{x_{\\mathrm {min}}}^{x_{\\mathrm {max}}} \\rho (x)^2\\, \\mathrm {d}x +o(N^{2-\\alpha })~,\\\\-F_{\\mathrm {matter}} &=& 8N^{2-\\alpha }\\int _{x_{\\mathrm {min}}}^{x_{\\mathrm {max}}} \\rho (x)^2\\, \\mathrm {d}x \\Bigg \\lbrace \\sum _{\\ell \\in \\frac{1}{p}\\mathbb {N}}\\frac{c_\\ell }{\\ell }y(x)\\sin \\left[\\ell y(x)\\right]+ \\frac{c_\\ell }{\\ell ^2}\\cos \\left[\\ell y(x)\\right] \\nonumber \\\\&&+ \\frac{d_\\ell }{\\ell }\\cos \\left[\\ell y(x)\\right]\\Bigg \\rbrace +o(N^{2-\\alpha })~.$ The term in curly brackets is denoted $J_{p}[y(x)]$ in Appendix , and may be evaluated by Fourier summation to give $-F_{\\mathrm {matter}}&=& 8N^{2-\\alpha }\\int _{x_{\\mathrm {min}}}^{x_{\\mathrm {max}}} \\rho (x)^2\\, \\mathrm {d}x\\left[\\frac{\\pi ^2}{24}\\left(2p-\\frac{3}{p}\\right) + \\frac{y(x)^2}{2p}\\right] +o(N^{2-\\alpha })~.$ Combining with (REF ), we thus obtain the leading order result $F_{\\mathrm {one-loop}} \\ = \\ F_{\\mathrm {vector}} + F_{\\mathrm {matter}} \\, = \\, \\frac{N^{2-\\alpha }}{p}\\int _{x_{\\mathrm {min}}}^{x_{\\mathrm {max}}} \\rho (x)^2\\, \\left[\\pi ^2 - 4y(x)^2\\right]\\, \\mathrm {d}x+o(N^{2-\\alpha })~.$ It remains the add the contribution of the classical terms in (REF ).", "This is a trivial modification of the $p=1$ computation, the only difference being the factor of $1/p=\\mathrm {Vol}(L(p,1))/\\mathrm {Vol}(S^3)$ : $F_{\\mathrm {classical}} &=& \\frac{kN^{1+\\alpha }}{p\\pi }\\int _{x_{\\mathrm {min}}}^{x_{\\mathrm {max}}} xy(x)\\rho (x)\\, \\mathrm {d}x + o(N^{1+\\alpha })~.$ The total free energy action is then to leading order $F &=& \\frac{kN^{1+\\alpha }}{p\\pi }\\int _{x_{\\mathrm {min}}}^{x_{\\mathrm {max}}} xy(x)\\rho (x)\\, \\mathrm {d}x + \\frac{N^{2-\\alpha }}{p}\\int _{x_{\\mathrm {min}}}^{x_{\\mathrm {max}}} \\rho (x)^2\\, \\left[\\pi ^2-4y(x)^2\\right] \\, \\mathrm {d}x~.$ As for the case of the round sphere with $p=1$ , non-trivial saddle points will require both terms to be of the same order, so that $\\alpha =\\frac{1}{2}$ and hence $\\lambda (x)=N^{1/2}x+\\mathrm {i}y(x)$ .", "Remarkably, we see that the action $F$ in (REF ) is simply $1/p$ times the action for $p=1$ in reference [9].", "In particular, the saddle point equations derived from (REF ) are identical to those in reference [9], which allows us to simply write down that the density $\\rho (x)$ is constant $\\rho (x) &=& \\frac{k}{2\\sqrt{2}\\pi }~,$ and the imaginary part of the eigenvalues $y(x)$ is linear $y(x) &=& \\frac{\\sqrt{k}}{2\\sqrt{2}}\\, x~,$ with $-x_{\\mathrm {min}}=x_{\\mathrm {min}}= \\pi \\sqrt{2/k}$ , so that $y(x)\\in \\left[-\\frac{\\pi }{2},\\frac{\\pi }{2}\\right]$ .", "Of course, this is perfectly consistent with the numerical results in Figures REF and REF .", "The dependence on $p$ only enters in the free energy $F$ evaluated on this saddle point solution, which is $F &=& N^{3/2}\\frac{\\pi \\sqrt{2k}}{3p} +o(N^{3/2})~.$ Again, this is consistent with the numerics.", "The formula (REF ) is expected from the supergravity dual solution AdS$_4/\\mathbb {Z}_p\\times S^7/\\mathbb {Z}_k$ , since the quotient by $\\mathbb {Z}_p$ simply divides the overall supergravity action by $p$ .", "The only slight subtlety here is that AdS$_4/\\mathbb {Z}_p$ has a $\\mathbb {Z}_p$ orbifold singularity at the “centre”.", "In principle there might exist degrees of freedom at this singularity which then contribute to the leading order large $N$ free energy, but the field theory result we have obtained implies this is not the case." ], [ "Outlook", "In this paper we considered the large $N$ limit of the partition function of $N$ M2-branes on the Lens space $L(p,1)$ .", "Some open problems include: The partition function (REF ) is valid for all $N, k, p$ , and it would be interesting to study this more generally, for example at finite $N$ , or in the 't Hooft limit in which $N/k$ is held fixed.", "One might also consider squashed Lens spaces, for which there are supergravity dual solutions [12].", "Another interesting open question is whether these theories have a description in terms of a Fermi gas, as for the ABJM theory on $S^3$ [13].", "This may be a useful method for computing subleading corrections.", "The generalization of these results to more general Lens space $L(p,q)$ will be addressed in a forthcoming publication." ], [ "Acknowledgments", "L. F. A and J. F. S. would like to thank the Isaac Newton Institute for hospitality during the completion of this work.", "J. F. S. is supported by a Royal Society Research Fellowship." ], [ "Computation of potentials", "In this appendix we present analytical expressions for the infinite products that enter into the partition function for the Lens spaces $L(p,1)$ .", "These infinite products have the following form: $P_{p}^\\kappa (z) \\ = \\ \\prod _{\\ell =0}^\\infty \\left(\\frac{\\ell +\\tfrac{3}{2}+\\mathrm {i}z}{\\ell +\\tfrac{1}{2}-\\mathrm {i}z} \\right)^{N_\\kappa (\\ell )} \\left(\\frac{\\ell +\\tfrac{3}{2}-\\mathrm {i}z}{\\ell +\\tfrac{1}{2}+\\mathrm {i}z} \\right)^{N_{p-\\kappa }(\\ell )}~,$ where $\\kappa =0,1,...,p-1$ and $N_\\kappa (\\ell )$ denotes the number of integers $m=\\lbrace -\\ell ,-\\ell +2,...,\\ell -2,\\ell \\rbrace $ such that $m\\equiv \\kappa $ mod $p$ .", "For computing the free energy the log of this product is relevant.", "Hence we introduce the potentials $V_{p}^\\kappa $ : $V_{p}^\\kappa \\ :=\\ \\log P_{p}^\\kappa ~.$" ], [ "$p=1$", "Let us explain in detail how to obtain the potential for $p=1$ .", "This result is already known [2], but it is instructive to recover it.", "In this case we have $\\kappa =0$ and the potential reduces to $V_1(z) \\ = \\ \\sum _{\\ell =0}^\\infty (\\ell +1) \\log \\left(\\frac{\\ell +\\tfrac{3}{2}+\\mathrm {i}z}{\\ell +\\tfrac{1}{2}-\\mathrm {i}z}\\cdot \\frac{\\ell +\\tfrac{3}{2}-\\mathrm {i}z}{\\ell +\\tfrac{1}{2}+\\mathrm {i}z} \\right)~.$ This sum is divergent, hence in order to compute it we need to regularize it.", "A standard procedure is to take the derivative of the potential and perform the sum.", "We obtain $V^{\\prime }_1(z) \\ = \\ -\\pi \\tanh \\, (\\pi z)~.$ Of course, in taking the derivative we are dropping an additive constant (which could be infinite).", "Integrating back we find $V_1(z) \\ =\\ -\\log \\cosh \\, (\\pi z) +c ~.$ The integration constant can be fixed by zeta function regularization.", "This is explained in detail for instance in appendix A of [14].", "The constant $c$ is defined as the value of $V_1(z)$ at $z=0$ .", "We obtain $\\frac{1}{2}c \\ = \\ \\sum _{\\ell =0}^\\infty (\\ell +1) \\log \\left(\\frac{\\ell +\\tfrac{3}{2} }{\\ell +\\tfrac{1}{2}} \\right)~.$ We will compute this divergent sum by using $\\zeta -$ function regularization.", "Let us define $\\zeta _Z(s) &=& \\sum _{\\ell =0}^\\infty \\left( \\frac{\\ell +1}{(\\ell +\\tfrac{3}{2})^s}- \\frac{\\ell +1}{(\\ell +\\tfrac{1}{2})^s} \\right)~.$ Hence the quantity we wish to compute is just $-\\zeta ^{\\prime }_Z(0)$ .", "These sums are by definition zeta functions, and their generalization, Hurwitz zeta functions $\\zeta _a(s) \\ = \\ \\sum _{\\ell =0}^\\infty \\frac{1}{(\\ell +a)^s}~.$ Sums with factors of $\\ell $ in the numerator are easily obtained, since a factor of $\\ell +a$ in the numerator can be absorbed by a shift $s \\rightarrow s-1$ .", "For the particular case at hand we obtain $\\zeta _Z(s) &=& -(2^s - 1) \\zeta (s)~.$ Hence $-\\zeta ^{\\prime }_Z(0)= -\\frac{\\log 2}{2}$ .", "This implies $c=-\\log 2$ , and the following result for $V_1$ : $V_1(z) \\ = \\ -\\log (2 \\cosh \\, (\\pi z))~,$ which coincides with the known result." ], [ "General $p$ and {{formula:c38c5f06-d9bb-4ea4-8993-c03bbce49d85}}", "Let us start by introducing some more notation: $f(\\ell ) \\ := \\ \\log \\left(\\frac{\\ell +\\tfrac{3}{2}+\\mathrm {i}z}{\\ell +\\tfrac{1}{2}-\\mathrm {i}z} \\right)~.$ The potentials will be given by the sum of two terms $V_{p}^\\kappa (z) \\ = \\ U_{p}^\\kappa (z) +U_{p}^{p-\\kappa }(-z) ~.$ By working out some explicit examples, one can convince oneself that the general form of each contribution $U$ is as follows $U_{p}^\\kappa (z) = \\sum _{\\ell =0}^\\infty \\left[ s_0 f(p \\ell ) +s_1 f(p \\ell +1) +...+s_{p-1} f(p \\ell +(p-1))\\right]~,$ where $s_0,...,s_{\\ell -1}$ depend on $p, \\kappa $ and $\\ell $ .", "The important fact is that they are always of the form $s_i = a_i+ b_i \\ell $ .", "$V_{p}^\\kappa (z)$ can be computed in two steps.", "First we use the intermediate result $\\sum _{\\ell =0}^\\infty (a + b \\ell ) \\log (p \\ell +c +\\mathrm {i}z) &=& \\frac{1}{p} \\Big [ (b c- a p+\\mathrm {i}zb) \\log \\Gamma \\left(\\frac{c+\\mathrm {i}z}{p} \\right) \\nonumber \\\\&& - b p~\\psi ^{(-2)}\\left(\\frac{c+\\mathrm {i}z}{p} \\right) \\Big ]~,$ where $\\psi ^{(-2)}\\ $ is the polygamma function, and we have dropped a term that can later be fixed (for the final result) by using zeta function regularization.", "All our expressions are the sum of such building blocks.", "In order to assemble the correct building blocks, we just have to compute $s_i=a_i+b_i \\ell $ for the fixed value of $p, \\kappa $ that we are interested in.", "Finally, once we have computed the $s_i$ , we can compute the correct integration constants by using zeta function regularization, as shown above.", "It is straightforward to write a Mathematica code that computes the final potential $V_{p}^\\kappa (z)$ for any choice of $p, \\kappa $ .The code is available upon request.", "The point is that since $s_i$ is at most linear in $\\ell $ , we can compute $s_i$ by looking at the terms with $0 \\le \\ell \\le 2p-1$ ." ], [ "Some explicit examples", "Below we present some explicit results that are used in the numerics.", "Even though the general answer depends on polygamma functions, for some cases the final expression can be simplified: $V_{1}^{0}(z) & = & -\\log \\left[2\\cosh (\\pi z)\\right]~,\\nonumber \\\\V_{2}^0(z) &=& -\\frac{1}{2} \\log [2 \\cosh (z\\pi )] + 2z\\cot ^{-1} \\mathrm {e}^{\\pi z} +\\frac{\\mathrm {i}}{\\pi } \\left[\\mathrm {Li}_2(-\\mathrm {i}\\mathrm {e}^{-z\\pi }) - \\mathrm {Li}_2(\\mathrm {i}\\mathrm {e}^{-z\\pi }) \\right]~,\\nonumber \\\\V_{2}^1(z) &=& -\\frac{1}{2} \\log (2 \\cosh (z\\pi ) ) -2z\\cot ^{-1} \\mathrm {e}^{\\pi z} -\\frac{\\mathrm {i}}{\\pi } \\left[ \\mathrm {Li}_2(-\\mathrm {i}\\mathrm {e}^{-z\\pi }) - \\mathrm {Li}_2(\\mathrm {i}\\mathrm {e}^{-z\\pi }) \\right]~,\\nonumber \\\\V_{3}^0(z) &=& \\log \\frac{2 \\cosh ^2(\\pi z/3)}{\\cosh \\pi z}~,\\nonumber \\\\V_{3}^1(z) &=&V_{3,1}^2(z) \\ = \\ -\\log [2 \\cosh (\\pi z/3)]~.$ A general formula is given in appendix C for the case $\\kappa =0$ .", "Some comments are in order.", "We see that the sum of potentials over all $\\kappa $ for fixed $p$ satisfies a completeness condition $\\sum _{\\kappa =0}^{p-1} V_{p}^\\kappa \\ = \\ V_1 ~.$ This is of course expected, since fixing $\\kappa $ projects over certain terms in the sum giving $V_1$ .", "Another comment is that from the structure of the sums we expect $V_{p}^0(z)$ to be an even function and $V_{p}^\\kappa (z)=V_{p}^{p-\\kappa }(-z)$ ." ], [ "A wave of eigenvalues", "In the body of the paper we have shown that in the large $N$ limit we can focus on the case ${\\bf m} = {\\bf \\tilde{m}} = c \\cdot 1_{N \\times N} $ .", "Furthermore, we have analyzed numerically the distribution of eigenvalues for this case.", "One can use the numerics to analyze the eigenvalue distribution for other choices of ${\\bf m}$ and ${\\bf \\tilde{m}} $ .", "These will presumably be important if one wants to compute subleading corrections to our result.", "An interesting distribution of eigenvalues is obtained if ${\\bf m}={\\bf \\tilde{m}} \\ne c \\cdot 1_{N \\times N}$ – see Figure REF .", "Figure: Distribution of eigenvalues for N=100,p=2N=100,~p=2 and n=3n=3, 10, 20, 30, 50, 70, 80, 90, 97 from left to right and top to bottom.We have shown the eigenvalue distribution for $p=2$ and $N=100$ , for the cases ${\\bf m} = {\\bf \\tilde{m}} = (0, \\ldots ,0, 1,\\ldots , 1)$ , with $n$ zeros and $N-n=100-n$ ones.", "From left to right, top to bottom, we show the eigenvalue distribution for $n=3$ , 10, 20, 30, 50, 70, 80, 90 and 97.", "The distribution of eigenvalues is reminiscent of a wave moving from left to right, with the location of the kink at the boundary between the group of zeros and the group of ones.", "Equivalently, we see that for $p=2$ and ${\\bf m} = {\\bf \\tilde{m}}$ the eigenvalues distribute in two segments.", "The length of the segments is equal to the quantity of zeros and ones, respectively.", "The numerics seem to suggest that when the number of zeros and ones is “macroscopic” (i.e.", "of the same order as $N$ ) there is a finite “jump” between the segments.", "This feature is also present for other cases.", "For instance, Figure REF shows the eigenvalue distribution for $p=3$ , $N=100$ and ${\\bf m} = {\\bf \\tilde{m}} = (0,\\ldots ,0, 1,\\ldots ,1, 2,\\ldots ,2)$ , with 30 zeros, 30 ones and 40 twos.", "It would be interesting to understand whether the finite “jump” between segments is really there for large $N$ or an artifact of $N$ being not large enough.", "Figure: Distribution of eigenvalues for N=100,p=3N=100,~p=3 and 𝐦=𝐦 ˜=(0,...,0,1,...,1,2,...,2){\\bf m} = {\\bf \\tilde{m}} = (0,\\ldots ,0, 1,\\ldots , 1, 2,\\ldots , 2).", "We see that the eigenvalues distribute in three segments." ], [ "Asymptotic expansions", "In this appendix we present the asymptotic expansions for the potentials found above.", "We will present a detailed analysis for $\\kappa =0$ , since, as discussed above, this is enough to compute the free energy in the large $N$ limit.", "Proceeding as explained in Appendix A we find the following expressions for $p$ odd and even respectively: $\\partial _zV_{p=\\mathrm {odd}} & = & \\frac{\\mathrm {i}}{p} \\sum _{n=0}^{p-1} (-1)^n \\left[ \\psi \\left(\\frac{1+2n+2 \\mathrm {i}z}{2p}\\right) - \\psi \\left(\\frac{1+2n-2 \\mathrm {i}z}{2p}\\right) \\right]~, \\nonumber \\\\\\partial _zV_{p=\\mathrm {even}} &=& \\frac{1}{p^2} \\sum _{n=0}^{p-1} (-1)^n \\Bigg [ (2z+\\mathrm {i}(2n+1-p)) \\psi \\left(\\frac{1+2n+2 \\mathrm {i}z}{2p}\\right) \\nonumber \\\\&& + (2z-\\mathrm {i}(2n+1-p)) \\psi \\left(\\frac{1+2n-2 \\mathrm {i}z}{2p}\\right) \\Bigg ]~.$ For $p$ odd these expressions can be given in terms of trigonometric functions, but the present form is more uniform.", "Now we would like to compute the asymptotic expansions of such expressions, when the real part of $z$ is very large.", "We have the following for $|z| \\rightarrow \\infty $ and $\\arg (z)$ very close to $\\pi $ : $\\psi (z) \\ = \\ \\log (z) -\\frac{1}{2z} - \\sum _{k=1}^\\infty \\frac{B_{2k}}{2k z^{2k}} + \\frac{1}{2} \\mathrm {i}\\pi (\\mathrm {i}\\cot (\\pi z)-1) ~,$ where $B_{2k}$ are the Bernoulli numbers.", "The coefficient in front of $ \\mathrm {i}\\pi (\\mathrm {i}\\cot (\\pi z)-1) $ is actually one if $|\\arg (z)|>\\pi $ , and is zero otherwise.", "For the case of real functions, as we are considering, these two average to $1/2$ .", "It is now easy to compute the asymptotic expansion, substituting the expansion for $\\psi $ into the expression for $\\partial _zV_{p}$ .", "For each case we obtain:" ], [ "$p$ odd", "This gives the following expansion for $ V_{p=\\mathrm {odd}}$ : $V_{p=\\mathrm {odd}} \\ = \\ -2 \\frac{\\pi }{p} z+ \\sum _{\\ell =1}^\\infty \\frac{\\mathrm {e}^{-2 \\frac{\\pi }{p} z\\ell }}{\\ell \\cos (\\frac{\\pi \\ell }{p})}~.$ Quite remarkably, zeta function regularization implies a value for $V(z=0)$ such that the asymptotic expansion doesn't have a constant term.", "Given an expansion of the form $V \\ = \\ z\\pi \\left(c_0 +\\sum _{\\ell } c_\\ell \\, \\mathrm {e}^{-\\pi z\\ell } \\right) +\\sum _\\ell d_{\\ell }\\, \\mathrm {e}^{-\\pi z\\ell }~,$ we have seen in the body of the paper that the contribution relevant at large $N$ is $J\\ = \\ \\sum _{\\ell } \\left(\\frac{c_\\ell }{ \\ell } y \\sin (\\ell y)+\\frac{c_\\ell }{ \\ell ^2} \\cos (\\ell y)+\\frac{d_\\ell }{\\ell } \\cos (\\ell y) \\right)~.$ For the present case we have $J_{p=\\mathrm {odd}} \\ = \\ \\frac{p}{2} \\sum _{\\ell =1}^\\infty \\frac{\\cos (\\frac{2\\ell }{p}y)}{\\ell ^2 \\cos (\\ell \\frac{\\pi }{p})} \\ = \\ \\frac{2p^2-3}{24p}\\pi ^2+\\frac{y^2}{2p}~,$ where we have assumed that $y$ lies in the range $[-\\pi /2,\\pi /2]$ .", "This is justified by the numerics." ], [ "$p$ even", "For $p$ even it is convenient to focus on the contribution for a fixed value of $n$ in the general expression for $\\partial _zV_{p,1}$ .", "We obtain $\\partial _zV_{p}|_n &= & 2(-1)^n\\frac{1+2n-p}{p^2} \\pi -\\frac{\\pi z}{p} \\sum _{\\ell =1}^\\infty \\frac{4 (-1)^{n} \\sin (\\frac{\\pi (2n+1)}{p} \\ell ) }{p} \\mathrm {e}^{-2 \\frac{\\pi }{p} z\\ell } \\nonumber \\\\&&+ \\frac{2(-1)^n (1+2n-p) \\pi }{p^2} \\sum _{\\ell =1}^\\infty \\cos \\left(\\frac{(2n+1)\\pi }{p}\\ell \\right) \\mathrm {e}^{-2 \\frac{\\pi }{p} z\\ell }~.$ The contribution from this term to $J$ , which we denote $J|_n$ , can be computed as above.", "We obtain $J|_n \\ = \\ (-1)^n \\left(-\\frac{(1+2n-p)(1+4n^2-4n(p-1)+2p(p-1))}{24p^2} \\pi ^2 +\\frac{p-1-2n}{2p^2} y^2 \\right)~.$ Summing over $n$ from zero to $p-1$ and using that $p$ is even we obtain $J_{p=\\mathrm {even}} \\ =\\ \\frac{2p^2-3}{24p}\\pi ^2+\\frac{y^2}{2p}~.$ This has exactly the same form as for $p$ odd!", "We thus arrive at the following result, valid for all values of $p$ : $J_{p} \\ = \\ \\frac{2p^2-3}{24p}\\pi ^2+\\frac{y^2}{2p}~.$" ] ]

[ [ "Generalized Measures of Fault Tolerance in (n,k)-star Graphs" ], [ "Abstract This paper considers a kind of generalized measure $\\kappa_s^{(h)}$ of fault tolerance in the $(n,k)$-star graph $S_{n,k}$ and determines $\\kappa_s^{(h)}(S_{n,k})=n+h(k-2)-1$ for $2 \\leqslant k \\leqslant n-1$ and $0\\leqslant h \\leqslant n-k$, which implies that at least $n+h(k-2)-1$ vertices of $S_{n,k}$ have to remove to get a disconnected graph that contains no vertices of degree less than $h$.", "This result contains some known results such as Yang et al.", "[Information Processing Letters, 110 (2010), 1007-1011]." ], [ "Introduction", "It is well known that interconnection networks play an important role in parallel computing/communication systems.", "An interconnection network can be modeled by a graph $G=(V, E)$ , where $V$ is the set of processors and $E$ is the set of communication links in the network.", "The connectivity $\\kappa (G)$ of a graph $G$ is an important measurement for fault-tolerance of the network, and the larger $\\kappa (G)$ is, the more reliable the network is.", "A subset of vertices $S$ of a connected graph $G$ is called a vertex-cut if $G-S$ is disconnected.", "The connectivity $\\kappa (G)$ of $G$ is defined as the minimum cardinality over all vertex-cuts of $G$ .", "Because $\\kappa $ has many shortcomings, one proposes the concept of the $h$ -super connectivity of $G$ , which can measure fault tolerance of an interconnection network more accurately than the classical connectivity $\\kappa $ .", "A subset of vertices $S$ of a connected graph $G$ is called an $h$ -super vertex-cut, or $h$ -cut for short, if $G-S$ is disconnected and has the minimum degree at least $h$ .", "The $h$ -super connectivity of $G$ , denoted by $\\kappa ^{(h)}_s(G)$ , is defined as the minimum cardinality over all $h$ -cuts of $G$ .", "It is clear that, if $\\kappa _s^{(h)}(G)$ exists, then $\\kappa (G)=\\kappa _s^{(0)}(G)\\leqslant \\kappa _s^{(1)}(G)\\leqslant \\kappa _s^{(2)}(G)\\leqslant \\cdots \\leqslant \\kappa _s^{(h-1)}(G)\\leqslant \\kappa _s^{(h)}(G).$ For any graph $G$ and integer $h$ , determining $\\kappa _s^{(h)}(G)$ is quite difficult.", "In fact, the existence of $\\kappa _s^{(h)}(G)$ is an open problem so far when $h\\geqslant 1$ .", "Only a little knowledge of results have been known on $\\kappa _s^{(h)}$ for particular classes of graphs and small $h$ 's.", "This paper is concerned about $\\kappa _s^{(h)}$ for the $(n,k)$ -star graph $S_{n,k}$ .", "For $k=n-1$ , $S_{n,n-1}$ is isomorphic to a star graph $S_n$ , Cheng and Lipman [3], Hu and Yang [5], Nie et al.", "[6] and Rouskov et al.", "[7], independently, determined $\\kappa _s^{(1)}(S_n)=2n-4$ for $n\\geqslant 3$ .", "Very recently, Yang et al.", "[9] have showed that if $2\\leqslant k\\leqslant n-2$ then $\\kappa _s^{(1)}(S_{n,k})=n+k-3$ for $n\\geqslant 3$ and $\\kappa _s^{(2)}(S_{n,k})=n+2k-5$ for $n\\geqslant 4$ .", "We, in this paper, will generalize these results by proving that $\\kappa _s^{(h)}(S_{n,k})=n+h(k-2)-1$ for $2\\leqslant k\\leqslant n-1$ and $0\\leqslant h\\leqslant n-k$ .", "The proof of this result is in Section 3.", "In Section 2, we recall the structure of $S_{n,k}$ and some lemmas used in our proofs." ], [ "Definitions and lemmas", "For given integer $n$ and $k$ with $1\\leqslant k\\leqslant n-1$ , let $I_n=\\lbrace 1,2,\\ldots ,n\\rbrace $ and $P(n,k)=\\lbrace p_{1}p_{2}\\ldots p_{k}:\\ p_{i}\\in I_n, p_{i}\\ne p_{j}, 1\\leqslant i\\ne j\\leqslant k\\rbrace $ , the set of $k$ -permutations on $I_n$ .", "Clearly, $|P(n,k)|=n\\,!/(n-k)\\,!$ .", "Definition 2.1 The $(n,k)$ -star graph $S_{n,k}$ is a graph with vertex-set $P(n,k)$ .", "The adjacency is defined as follows: a vertex $p=p_{1}p_{2}\\ldots p_{i}\\ldots p_{k}$ is adjacent to a vertex (a) $p_{i}p_{2}\\cdots p_{i-1}p_{1}p_{i+1}\\cdots p_{k}$ , where $2\\leqslant i\\leqslant k$ (swap $p_{1}$ with $p_{i}$ ).", "(b) $\\alpha p_{2}p_{3}\\cdots p_{k}$ , where $\\alpha \\in I_n\\setminus \\lbrace p_{i}:\\ 1\\leqslant i\\leqslant k\\rbrace $ (replace $p_{1}$ by $\\alpha $ ).", "The vertices of type $(a)$ are referred to as swap-neighbors of $p$ and the edges between them are referred to as swap-edge or $i$ -edges.", "The vertices of type $(b)$ are referred to as unswap-neighbors of $p$ and the edges between them are referred to as unswap-edges.", "Clearly, every vertex in $S_{n,k}$ has $k-1$ swap-neighbors and $n-k$ unswap-neighbors.", "Usually, if $x=p_1p_2\\dots p_k$ is a vertex in $S_{n,k}$ , we call $p_i$ the $i$ -th bit for each $i\\in I_k$ .", "The $(n,k)$ -star graph $S_{n,k}$ is proposed by Chiang and Chen [4] who showed that $S_{n,k}$ is $(n-1)$ -regular $(n-1)$ -connected.", "Lemma 2.2 For any $\\alpha =p_1p_2\\cdots p_{k-1}\\in P(n,k-1)$ $(k \\geqslant 2)$ , let $V_\\alpha =\\lbrace p\\alpha :\\ p\\in I_n\\setminus \\lbrace p_i:\\ i\\in I_{k-1}\\rbrace \\rbrace $ .", "Then the subgraph of $S_{n,k}$ induced by $V_\\alpha $ is a complete graph of order $n-k+1$ , denoted by $K^\\alpha _{n-k+1}$ .", "Proof.", "For any two vertices $p\\alpha $ and $q\\alpha $ in $V_\\alpha $ with $p\\ne q$ , by the condition $(b)$ of Definition REF , $p\\alpha $ and $q\\alpha $ are linked in $S_{n,k}$ by an unswap-edge.", "Thus, the subgraph of $S_{n,k}$ induced by $V_\\alpha $ is a complete graph $K_{n-k+1}$ .", "By Lemma REF , the vertex-set $P(n,k)$ of $S_{n,k}$ can be decomposed into $|P(n,k-1)|$ subsets, each of which induces a complete graph $K_{n-k+1}$ .", "It is clear that, for any two distinct elements $x$ and $y$ in $P(n,k)$ , if they are in different complete subgraphs $K^\\alpha _{n-k+1}$ and $K^\\beta _{n-k+1}$ $(\\alpha \\ne \\beta )$ , then there is at most one edge between $x$ and $y$ in $S_{n,k}$ , which is a swap-edge if and only if $\\alpha $ and $\\beta $ differ in only one bit.", "Thus, we have the following conclusion.", "Lemma 2.3 The vertex-set of $S_{n,k}$ can be partitioned into $|P(n,k-1)|$ subsets, each of which induces a complete graph of order $n-k+1$ .", "Furthermore, there is at most one swap edge between any two complete graphs.", "Let $S^{t:i}_{n-1,k-1}$ denote a subgraph of $S_{n,k}$ induced by all vertices with the $t$ -th bit $i$ for $2\\leqslant t\\leqslant k$ .", "The following lemma is a slight modification of the result of Chiang and Chen [4].", "Lemma 2.4 For a fixed integer $t$ with $2\\leqslant t\\leqslant k$ , $S_{n,k}$ can be decomposed into $n$ subgraphs $S^{t:i}_{n-1,k-1}$ , which is isomorphic to $S_{n-1,k-1}$ , for each $i\\in I_n$ .", "Moreover, there are $\\frac{(n - 2)!", "}{(n - k)!", "}$ independent swap-edges between $S^{t:i}_{n-1,k-1}$ and $S^{t:j}_{n-1,k-1}$ for any $i,j\\in I_n$ with $i\\ne j$ .", "Lemma 2.5 (Chen et al.", "[2]) In an $S_{n,k}$ , a cycle has a length at least 6 if it contains a swap-edge." ], [ "Main results", "In this section, we present our main results, that is, we determine the $h$ -super connectivity of the $(n,k)$ -star graph $S_{n,k}$ .", "Since $S_{n,1}\\cong K_n$ , we only consider the case of $k\\geqslant 2$ in the following discussion.", "Lemma 3.1 $\\kappa _s^{(h)}(S_{n,k})\\leqslant n+h(k-2)-1$ for $2 \\leqslant k \\le n-1$ and $0\\leqslant h \\leqslant n-k$ .", "Proof.", "By our hypothesis of $h \\leqslant n-k$ , for any $\\alpha \\in P(n,k-1)$ , we can choose a subset $X\\subseteq V(K^\\alpha _{n-k+1})$ such that $|X|=h+1$ .", "Then the subgraph of $K^\\alpha _{n-k+1}$ induced by $X$ is a complete graph $K_{h+1}$ .", "Let $S$ be the neighbor-set of $X$ in $S_{n,k}-X$ .", "Clearly, $V(K^\\alpha _{n-k+1}-X)\\subseteq S$ , that is, $X$ has exactly $n-k+1-|X|$ unswap-neighbors in $V(K^\\alpha _{n-k+1}-X)\\cap S$ .", "Since $S_{n,k}$ is $(n-1)$ -regular, every vertex of $X$ has exactly $(k-1)$ swap-neighbors are not in $K^\\alpha _{n-k+1}$ .", "Moreover, any two swap-neighbors of $X$ are different from each other by Lemma REF .", "It follows that $|S|=n-k+1-|X|+|X|(k-1)=n+h(k-2)-1.$ We now need to show that $S$ is an $h$ -cut of $S_{n,k}$ .", "Clearly, $S$ is a vertex-cut of $S_{n,k}$ since $S_{n,k}$ is not a complete graph for $k\\geqslant 2$ .", "We only need to show that every vertex of $S_{n,k}-(X\\cup S)$ has degree at least $h$ .", "Let $u$ be a vertex in $S_{n,k}-(X\\cup S)$ .", "If $u$ has a neighbor $v$ in $S\\cap V(K^\\alpha _{n-k+1})$ , then $u$ is a swap-neighbor of $v$ since all the unswap-neighbors of $v$ are in $V(K^\\alpha _{n-k+1})$ .", "If $u$ has a neighbor $v$ in $S\\setminus V(K^\\alpha _{n-k+1})$ , then $v$ has a swap-neighbor in $V(K^\\alpha _{n-k+1})$ .", "Moreover, if $u$ has two neighbor $v,v^{\\prime }$ in $S$ , then three vertices $u, v$ and $v^{\\prime }$ are concluded in a cycle of length at most 5 and containing at least one swap-edge, which contradicts with Lemma REF .", "Thus, $u$ has at most one neighbor in $S$ .", "In other words, $u$ has at least $n-2$ neighbor in $S_{n,k}-S$ .", "Since $n-2\\geqslant n-k\\geqslant h$ for $k\\geqslant 2$ , $u$ has degree at least $h$ in $S_{n,k}-S$ .", "By the arbitrariness of $u\\in S_{n,k}-(X\\cup S)$ , $S$ is an $h$ -cut of $S_{n,k}$ , and so $\\kappa _s^{(h)}(S_{n,k})\\leqslant |S|=n+h(k-2)-1$ as required.", "The lemma follows.", "Corollary 3.2 $\\kappa _s^{(h)}(S_{n,2})=n-1$ for $0 \\leqslant h\\leqslant n-2$ .", "Proof.", "On the one hand, $\\kappa _s^{(h)} (S_{n,2})\\leqslant n-1$ by Lemma REF when $k=2$ .", "On the other hand, $\\kappa _s^{(h)}(S_{n,2})\\ge \\kappa (S_{n,2})=n-1$ .", "To state and prove our main results, we need some notations.", "Let $S$ be an $h$ -cut of $S_{n,k}$ and $X$ be the vertex-set of a connected component of $S_{n,k}-S$ .", "For a fixed $t\\in I_k\\setminus \\lbrace 1\\rbrace $ and any $i\\in I_n$ , let $\\begin{array}{l}Y=V(S_{n,k}-S-X),\\\\X_i=X\\cap V(S^{t:i}_{n-1,k-1}),\\\\Y_i=Y\\cap V(S^{t:i}_{n-1,k-1})\\ {\\rm and}\\\\S_i=S\\cap V(S^{t:i}_{n-1,k-1}),\\end{array}$ and let $\\begin{array}{l}J=\\lbrace i\\in I_n:\\ X_i\\ne \\emptyset \\rbrace ,\\\\J^{\\prime }=\\lbrace i\\in J:\\ Y_i\\ne \\emptyset \\rbrace \\ \\ {\\rm and} \\\\T=\\lbrace i\\in I_n:\\ Y_i\\ne \\emptyset \\rbrace .\\end{array}$ Lemma 3.3 Let $S$ be a minimum $h$ -cut of $S_{n,k}$ and $X$ be the vertex-set of a connected component of $S_{n,k}-S$ .", "If $3\\leqslant k\\leqslant n-1$ and $1\\leqslant h\\leqslant n-k$ then, for any $t\\in I_k\\setminus \\lbrace 1\\rbrace $ , (a) $S_i$ is an $(h-1)$ -cut of $S^{t:i}_{n-1,k-1}$ for any $i\\in J^{\\prime }$ , (b) $\\kappa _s^{(h)}(S_{n,k})\\geqslant |J^{\\prime }|\\ \\kappa _s^{(h-1)}(S_{n-1,k-1})$ , (c) $J\\cup T =I_n$ .", "Proof.", "(a) By the definition of $J^{\\prime }$ , $S_i$ is a vertex-cut of $S^{t:i}_{n-1,k-1}$ for any $i\\in J^{\\prime }$ .", "For any vertex $x$ in $S^{t:i}_{n-1,k-1}-S_i$ , since $x$ has degree at least $h$ in $S_{n,k}-S$ and has exactly one neighbor outsider $S^{t:i}_{n-1,k-1}$ , $x$ has degree at least $h-1$ in $S^{t:i}_{n,k}-S_i$ .", "This fact shows that $S_i$ is an $(h-1)$ -cut of $S^{t:i}_{n-1,k-1}$ for any $i\\in J^{\\prime }$ .", "(b) By the assertion (a), we have $|S_i|\\geqslant \\kappa _s^{(h-1)}(S_{n-1,k-1})$ , and so $\\kappa _s^{(h)}(S_{n,k})=|S|\\geqslant \\sum _{i\\in J^{\\prime }} |S_i|\\geqslant |J^{\\prime }|\\kappa _s^{(h-1)}(S_{n-1,k-1}).$ (c) If $J\\cup T \\ne I_n$ , that is, $I_n\\setminus (J\\cup T)\\ne \\emptyset $ , then there exists an $i_0\\in I_n$ such that $V(S^{t:i_0}_{n-1,k-1})=S_{i_0}$ .", "Thus, we have $\\begin{array}{rl}\\kappa _s^{(h)}(S_{n,k}) &=|S|\\geqslant |S_{i_0}| = \\frac{(n-1)!}{(n-k)!", "}\\\\&\\geqslant (n-1)(n-2)\\\\&=n+(n-1)(n-3)-1\\\\&>n+(n-3)(n-3)-1\\\\&\\geqslant n+h(k-2)-1,\\end{array}$ which contradicts to Lemma REF .", "Thus, $J\\cup T =I_n$ .", "The Lemma follows.", "Theorem 3.4 $\\kappa _s^{(h)}(S_{n,k})=n+h(k-2)-1$ for $2\\leqslant k\\leqslant n-1$ and $0\\leqslant h\\leqslant n-k$ .", "Proof.", "By Lemma REF , we only need to prove that, for $2\\leqslant k\\leqslant n-1$ and $0\\leqslant h\\leqslant n-k$ , $\\kappa _s^{(h)}(S_{n,k})\\geqslant n+h(k-2)-1.$ We proceed by induction on $k\\geqslant 2$ and $h\\geqslant 0$ .", "The inequality (REF ) is true for $k=2$ and any $h$ with $0\\leqslant h\\leqslant n-2$ by Corollary REF .", "The inequality (REF ) is also true for $h=0$ and any $k$ with $2\\leqslant k\\leqslant n-1$ since $\\kappa _s^{(0)}(S_{n,k})=\\kappa (S_{n,k})=n-1$ .", "Assume the induction hypothesis for $k-1$ with $k\\geqslant 3$ and for $h-1$ with $h\\geqslant 1$ , that is, $\\kappa _s^{(h-1)}(S_{n-1,k-1})\\geqslant n+(h-1)(k-3)-2.$ Let $S$ be a minimum $h$ -cut of $S_{n,k}$ and $X$ be the vertex-set of a minimum connected component of $S_{n,k}-S$ .", "Use notations defined in (REF ) and (REF ).", "Choose $t\\in I_k\\setminus \\lbrace 1\\rbrace $ such that $|J|$ is as large as possible.", "For each $i\\in I_n$ , we write $S^i_{n-1,k-1}$ for $S^{t:i}_{n-1,k-1}$ for short.", "We consider three cases depending on $|J^{\\prime }|=0$ , $|J^{\\prime }|=1$ or $|J^{\\prime }|\\geqslant 2$ .", "Case 1.", "$|J^{\\prime }|=0$ , In this case, $X_i\\ne \\emptyset $ and $Y_i=\\emptyset $ for each $i\\in J$ , that is, $J\\cap T=\\emptyset $ .", "By Lemma REF (c), $|J|\\geqslant 2$ or $|T|\\geqslant 2$ since $n\\geqslant 4$ .", "Clearly, $J\\ne \\emptyset $ and $T\\ne \\emptyset $ .", "Without loss of generality, assume $|J|\\geqslant 2$ , $\\lbrace i_{1},i_2\\rbrace \\subseteq J$ and $i_3\\in T$ .", "By Lemma REF , there are $\\frac{(n - 2)!", "}{(n - k)!", "}$ independent swap-edges between $S^{i_1}_{n-1,k-1}$ (resp.", "$S^{i_2}_{n-1,k-1}$ ) and $S^{i_3}_{n-1,k-1}$ , each edge of which has at least one end-vertex in $S$ .", "Since $J\\cap T=\\emptyset $ and $S_{i_1}\\cap S_{i_2}=\\emptyset $ , we have that $\\begin{array}{c}|S|\\geqslant 2\\ \\frac{(n-2)!}{(n-k)!", "}.\\end{array}$ Noting that, for $k=3$ , $\\begin{array}{c}2\\ \\frac{(n-2)!}{(n-k)!", "}\\geqslant 2(n-2)\\geqslant n+(n-3)-1\\geqslant n+h(k-2)-1,\\end{array}$ and, for $k\\geqslant 4$ , $\\begin{array}{c}2\\ \\frac{(n-2)!}{(n-k)!", "}\\geqslant 2(n-2)(n-3)\\geqslant n+(n-3)(n-3)-1\\geqslant n+h(k-2)-1,\\end{array}$ we have that $\\begin{array}{c}2\\ \\frac{(n-2)!}{(n-k)!", "}\\geqslant n+h(k-2)-1\\ \\ {\\rm for}\\ k\\geqslant 3.\\end{array}$ It follows from (REF ) and (REF ) that $\\begin{array}{rl}\\kappa _s^{(h)}(S_{n,k})& =|S|\\geqslant 2\\ \\frac{(n-2)!}{(n-k)!", "}\\geqslant n+h(k-2)-1.\\end{array}$ Case 2.", "$|J^{\\prime }|=1$ , Without loss of generality, assume $J^{\\prime }=\\lbrace 1\\rbrace $ .", "By Lemma REF (a), $S_1$ is an $(h-1)$ -cut of $S^1_{n-1,k-1}$ .", "Let $S^{\\prime }=S\\setminus S_1$ .", "If $|S^{\\prime }|\\geqslant n-2$ then, by (REF ), we have that $\\begin{array}{rl}\\kappa _s^{(h)}(S_{n,k})& =|S| = |S_1|+|S^{\\prime }| \\geqslant \\kappa _s^{(h-1)}(S_{n-1,k-1})+(n-2) \\\\&\\geqslant (n+(h-1)(k-3)-2)+(n-2)\\\\&\\geqslant (n+(h-1)(k-3)-2)+(h+k-2)\\\\&= n+h(k-2)-1\\end{array}$ We now assume $|S^{\\prime }|\\leqslant n-3$ .", "We claim $|J|=1$ .", "Suppose to the contrary $|J|\\geqslant 2$ .", "If $|T|= 1$ then, by Lemma 3.3 (c), we have $|J|=n$ .", "Then $|S_i|\\geqslant 1$ for $i\\in J$ , otherwise there exists $i\\in J\\setminus J^{\\prime }$ such that $X_i=V(S^i_{n-1,k-1})$ , then $|X| >|X_i|=|V(S^i_{n-1,k-1})|>|Y|$ , which contradicts to the minimality of $X$ .", "Therefore, $|S^{\\prime }|\\geqslant n-1$ , a contradiction.", "If $|T|\\geqslant 2$ , assume that $i_1\\in J\\setminus J^{\\prime }$ and $i_2\\in T\\setminus J^{\\prime }$ , then $X_{i_1}\\ne \\emptyset ,Y_{i_1}=\\emptyset ,X_{i_2}=\\emptyset , Y_{i_2}\\ne \\emptyset $ .", "By Lemma REF , there are $\\frac{(n - 2)!", "}{(n - k)!", "}$ independent swap-edges between $S^{i_1}_{n-1,k-1}$ and $S^{i_2}_{n-1,k-1}$ , each edge of which must have one end-vertex in $S^{\\prime }$ .", "Thus, we have $\\begin{array}{c}|S^{\\prime }|\\geqslant \\frac{(n- 2)!", "}{(n - k)!", "}\\geqslant n-2\\ \\ {\\rm for}\\ k\\geqslant 3,\\end{array}$ a contradiction.", "Thus, $|J|=1$ .", "We have $J=\\lbrace 1\\rbrace $ since $\\lbrace 1\\rbrace =J^{\\prime }\\subseteq J$ .", "Then $X_1=X$ and $|X_1|\\geqslant h+1$ .", "By the choice of $t$ , the $i$ -th ($i\\ne 1$ ) bits of all vertices in $X_1$ are same, and so $X_1$ a complete graph.", "Thus, as computed in (REF ), we have that $\\begin{array}{rl}\\kappa _s^{(h)}(S_{n,k})& =|S|=n+(|X_1|-1)(k-2)-1\\geqslant n+h(k-2)-1.\\end{array}$ Case 3.", "$|J^{\\prime }|\\geqslant 2$ .", "By Lemma 3.3 (b) and (REF ), we have that $\\begin{array}{rl}\\kappa _s^{(h)}(S_{n,k})=|S| &\\geqslant |J^{\\prime }|\\kappa _s^{(h-1)}(S_{n-1,k-1})\\\\&\\geqslant 2(n+(h-1)(k-3)-2)\\\\&\\geqslant n+(h+k)+2(h-1)(k-3)-4\\\\&=n+h(k-2)+(h-1)(k-3)-1\\\\&\\geqslant n+h(k-2)-1.\\end{array}$ By the induction principle, the theorem follows.", "Corollary 3.5 (Yang et al.", "[9]) If $2\\leqslant k\\leqslant n-2$ then $\\kappa _s^{(1)}(S_{n,k})=n+k-3$ for $n\\geqslant 3$ and $\\kappa _s^{(2)}(S_{n,k})=n+2k-5$ for $n\\geqslant 4$ .", "As we have known, when $k=n-1$ , $S_{n,n-1}$ is isomorphic to the star graph $S_n$ .", "Akers and Krishnamurthy [1] determined $\\kappa (S_n)=n-1$ for $n\\geqslant 2$ ; Cheng and Lipman [3], Hu and Yang [5], Nie et al.", "[6] and Rouskov et al.", "[7], independently, determined $\\kappa _s^{(1)}(S_n)=2n-4$ for $n\\geqslant 3$ .", "All these results can be obtained from our result by setting $k=n-1$ and $h=0,1$ , respectively.", "Corollary 3.6 $\\kappa (S_n)=n-1$ for $n\\geqslant 2$ and $\\kappa _s^{(1)}(S_n)=2n-4$ for $n\\geqslant 3$ .", "Remark 3.7 Wan and Zhang [8] determined $\\kappa _s^{(2)}(S_n)=6(n-3)$ for $n\\geqslant 4$ .", "Thus, our result is invalid for $\\kappa _s^{(h)}(S_n)$ when $h\\geqslant 2$ .", "Thus, determining $\\kappa _s^{(h)}(S_n)$ for $h\\geqslant 2$ needs other technique." ] ]

[ [ "Improvements in the computation of ideal class groups of imaginary\n quadratic number fields" ], [ "Abstract We investigate improvements to the algorithm for the computation of ideal class groups described by Jacobson in the imaginary quadratic case.", "These improvements rely on the large prime strategy and a new method for performing the linear algebra phase.", "We achieve a significant speed-up and are able to compute ideal class groups with discriminants of 110 decimal digits in less than a week." ], [ "Introduction", "Given a fundamental discriminant $\\Delta $ , it is known that the corresponding ideal class group $\\operatorname{Cl}(\\Delta )$ of the order $\\mathcal {O}_{\\Delta }$ of discriminant $\\Delta $ in $\\mathbb {K}= \\mathbb {Q}(\\sqrt{\\Delta })$ is a finite abelian group that can be decomposed as $\\operatorname{Cl}(\\Delta ) \\simeq \\bigoplus _i \\mathbb {Z}/d_i\\mathbb {Z},$ where the divisibility condition $d_{i}|d_{i+1}$ holds.", "In this paper we investigate improvements in the computation of the group structure of $\\operatorname{Cl}(\\Delta )$ : that is, determining the $d_i$ , which is of both cryptographic and number theoretic interest.", "Indeed some cryptographic protocols relying on the difficulty of solving the discrete logarithm problem (DLP) in imaginary quadratic orders have been proposed [3], [9], and solving instances of the DLP is closely related to finding the group structure of $\\operatorname{Cl}(\\Delta )$ .", "In 1968 Shanks [19] proposed an algorithm relying on the baby-step giant-step method in order to compute the structure of the ideal class group of an imaginary quadratic number field in time $O\\left( |\\Delta |^{1/4 + \\epsilon } \\right)$ , or $O\\left( |\\Delta |^{1/5 + \\epsilon } \\right)$ under the extended Riemann hypothesis [13].", "This allows us to compute class groups of discriminants having up to 20 or 25 decimal digits.", "Then a subexponential strategy was described in 1989 by Hafner and McCurley [8].", "The expected running time of this method is $e^{ \\left( \\sqrt{2}+o(1)\\right) \\sqrt{\\log |\\Delta |\\log \\log |\\Delta |} }.$ Buchman and Düllmann [2] computed class groups with discriminants of around 50 decimal digits using an implementation of this algorithm.", "An improvement of this method was published by Jacobson in 1999 [10].", "He achieved a significant speed-up by using sieving strategies to generate the matrix of relations.", "He was able to compute the structure of class groups of discriminants having up to 90 decimal digits.", "More recently Sutherland [22] used generic methods in order to compute class groups with discriminants having 100 decimal digits.", "Unlike the previous algorithms, this one relies heavily on the particular structure of $\\operatorname{Cl}(\\Delta )$ thus obtaining variable performances depending on the values of $\\Delta $ .", "Our approach is based on that of Jacobson, using new techniques to accelerate both the sieving phase and the linear algebra phase; we have obtained the group structure of class groups of 110 decimal digit discriminants." ], [ "The ideal class group", "In this section we give essential results concerning the ideal class group and the subexponential strategies for computing its structure.", "For a more detailed description of the theory of ideal class groups we refer to [5] and [17].", "In the following, $\\Delta $ is a non-square integer congruent to 0 or 1 modulo 4, and the quadratic order of discriminant $\\Delta $ is defined as the $\\mathbb {Z}$ -module $\\mathcal {O}_{\\Delta } = \\mathbb {Z}+ \\frac{\\Delta + \\sqrt{\\Delta }}{2}\\mathbb {Z}.$ We also denote by $\\mathbb {K}$ the field $\\mathbb {Q}(\\sqrt{\\Delta })$ ." ], [ "Description", "Elements of $\\operatorname{Cl}(\\Delta )$ are obtained from fractional ideals of $\\mathcal {O}_{\\Delta }$ , which are $\\mathbb {Z}$ -modules of $\\mathbb {K}$ of the form: $\\mathfrak {a} = q \\left( a\\mathbb {Z}+ \\frac{b + \\sqrt{\\Delta }}{2}\\mathbb {Z}\\right), $ where $a$ and $b$ are integers with $b\\equiv \\Delta \\ \\text{mod}\\ 2$ and $q$ is a rational number.", "The prime ideals are the fractional ideals for which there exists a prime number $p$ such that: $\\mathfrak {p}= p\\mathbb {Z}+ \\frac{b_p+\\sqrt{\\Delta }}{2}\\mathbb {Z}\\ \\ \\text{or}\\ \\ \\mathfrak {p}= p\\mathbb {Z}\\ (p \\text{ inert in } \\mathbb {K}).$ Definition 2.1 (Ideal Class group) Let $\\mathcal {I}_{\\Delta }$ be the set of invertible fractional ideals of $\\mathcal {O}_{\\Delta }$ , and $\\mathcal {P}_{\\Delta }=\\left\\lbrace (\\alpha )\\in \\mathcal {I}_{\\Delta },\\alpha \\in \\mathbb {K}\\right\\rbrace $ the subset of principal ideals.", "We define the ideal class group of $\\Delta $ as : $\\operatorname{Cl}(\\Delta ) := \\mathcal {I}_{\\Delta }/\\mathcal {P}_{\\Delta },$ where the group law is the one derived from the multiplication of $\\mathbb {Z}$ -modules.", "For every $\\mathfrak {a}\\in \\mathcal {I}_{\\Delta }$ , there exist uniquely determined prime ideals $\\mathfrak {p}_1,\\hdots ,\\mathfrak {p}_n$ and exponents $e_1,\\hdots , e_n$ in $\\mathbb {Z}$ such that $\\mathfrak {a} = \\mathfrak {p}_1^{e_1}\\hdots \\mathfrak {p}_n^{e_n}.$ Unlike $\\mathcal {I}_{\\Delta }$ , the ideal class group $\\operatorname{Cl}(\\Delta )$ is a finite group.", "Its order is called the class number and usually denoted by $h(\\Delta )$ .", "It grows like $|\\Delta |^{1/2+\\epsilon }$ , as shown in [20]." ], [ "Computing the group structure", "The algorithm for computing the group structure of $\\operatorname{Cl}(\\Delta )$ is divided into two major phases: relation collection and linear algebra.", "In the first phase, we begin by precomputing a factor base $\\mathcal {B} = \\left\\lbrace \\mathfrak {p}_1,\\hdots ,\\mathfrak {p}_n\\right\\rbrace $ of non-inert prime ideals satisfying $\\mathcal {N}\\left( \\mathfrak {p}_i\\right) \\le B$ , where $B$ is a smoothness bound.", "Then we look for relations of the form $\\left( \\alpha \\right) = \\mathfrak {p}_1^{e_1}\\hdots \\mathfrak {p}_n^{e_n}, $ where $\\alpha \\in \\mathbb {K}$ .", "Every $n$ -tuple $[e_1,\\hdots ,e_n]$ collected becomes a row of what we will refer to as the relation matrix $A\\in \\mathbb {Z}^{m\\times n}$ .", "We have from [1] the following important result: Theorem 2.2 Let $\\Lambda $ be the lattice spanned by the set of the possible relations.", "Assuming GRH, if $B\\ge 6\\log ^2\\Delta $ , then we have $\\operatorname{Cl}(\\Delta ) \\simeq \\mathbb {Z}^n/\\Lambda .$ After the relation collection phase we can test if $A$ has full rank and if its rows generate $\\Lambda $ using methods described in §REF .", "If it is not the case then we have to compute more relations.", "From now on we assume that $A$ has full rank and that its rows generate $\\Lambda $ .", "The linear algebra phase consists of computing the Smith Normal Form (SNF) of $A$ .", "Any matrix $A$ in $\\mathbb {Z}^{n\\times n}$ with non zero determinant can be written as $ A = V^{-1}\\left( \\begin{array}{cccc}d_1 & 0 & \\hdots & 0 \\\\0 & d_2 & \\ddots & \\vdots \\\\\\vdots & \\ddots & \\ddots & 0 \\\\0 & \\hdots & 0 & d_n \\end{array} \\right) U^{-1}$ , where $d_{i+1} | d_i$ for all $1 \\le i < n$ and $U$ and $V$ are unimodular matrices in $\\mathbb {Z}^{n\\times n}$ .", "The matrix $\\text{diag}(d_1,\\hdots ,d_n)$ is called the SNF of $A$ .", "If $m=n$ and $\\text{diag}(d_1,\\hdots ,d_n) = \\text{SNF}(A)$ then $\\operatorname{Cl}(\\Delta ) \\simeq \\bigoplus _{i=1}^{n} \\mathbb {Z}/d_i\\mathbb {Z}.$ This reduces the problem of computing the group structure of $\\operatorname{Cl}(\\Delta )$ to computing the SNF of a relation matrix $A$ in $\\mathbb {Z}^{n\\times n}$ .", "For an arbitrary $A$ in $\\mathbb {Z}^{m\\times n}$ we start by computing the Hermite Normal Form (HNF) of $A$ .", "A matrix $H$ is said to be in HNF if it has the shape $ H = \\left(\\begin{BMAT}(@)[2pt,3cm,3cm]{c}{c.c}\\begin{BMAT}(e){cccc}{cccc}h_{1,1}& 0 & \\hdots & 0 \\\\\\vdots & h_{2,2}& \\ddots & \\vdots \\\\\\vdots & \\vdots & \\ddots & 0 \\\\* & * & \\hdots & h_{n,n}\\end{BMAT} \\\\\\begin{BMAT}[2pt,3cm,1cm]{c}{c}(0)\\end{BMAT}\\end{BMAT}\\right) $ , where $0\\le h_{ij} < h_{ii}$ for all $j<i$ and $h_{ij}=0$ for all $j>i$ .", "For each matrix $A$ in $\\mathbb {Z}^{m\\times n}$ there exists a matrix $H$ in HNF and a unimodular matrix $W$ in $\\mathbb {Z}^{m\\times m}$ such that $H = WA.$ The upper block of $H$ is a $n\\times n$ relation matrix whose SNF provides us the group structure of $\\operatorname{Cl}(\\Delta )$ .", "There is an index $l$ such that $h_{i,i} = 1$ for every $i\\ge l$ .", "The upper left $l\\times l$ submatrix of $H$ is called the essential part of $H$ .", "In order to compute the group structure of $\\operatorname{Cl}(\\Delta )$ it suffices to compute the SNF of the essential part of $H$ , which happens to have small dimension in our context." ], [ "The use of sieving for computing the relation matrix", "The use of sieving to create the relation matrix was first described by Jacobson [10].", "Here we follow the approach of [11] Chap.13, which relies on the following lemma: Lemma 2.3 If $\\mathfrak {a} = \\left( a\\mathbb {Z}+ \\frac{b + \\sqrt{\\Delta }}{2}\\mathbb {Z}\\right)$ with $a>0$ , then for all $x,y$ in $\\mathbb {Z}$ there exists $\\mathfrak {b}\\in \\mathcal {I}_{\\Delta }$ such that $\\mathfrak {a}\\mathfrak {b}\\in \\mathcal {P}$ and $\\mathcal {N}(\\mathfrak {b}) = ax^2 + bxy + \\frac{b^2 - \\Delta }{4a}y^2.$ The strategy for finding relations is the following: We start with $\\mathfrak {a}=\\prod _i \\mathfrak {p}_i^{e_i} =: \\left( a\\mathbb {Z}+ \\frac{b + \\sqrt{\\Delta }}{2}\\mathbb {Z}\\right),$ whose norm is $B$ -smooth.", "Then we choose a sieve radius $R$ satisfying $R\\approx \\sqrt{|\\Delta |/2}/\\mathcal {N}(\\mathfrak {a})$ and we look for values of $x\\in [-R,R]$ such that $\\varphi (x,1)$ is $B$ -smooth where $\\varphi (x,y) = ax^2 + bxy + \\frac{b^2 - \\Delta }{4a}y^2,$ which allows us to find $\\mathfrak {b} = \\prod _i\\mathfrak {p}_i^{f_i}$ satisfying $\\mathfrak {a}\\mathfrak {b}=(\\gamma )$ for some $\\gamma $ in $\\mathbb {K}$ .", "The $\\mathfrak {p}_i$ and $f_i$ are deduced from the decomposition $\\varphi (x,1)=\\prod _ip_i^{v_i}$ .", "For more details we refer to [11], Chap 13.", "This method yields the relation $(\\gamma ) = \\prod _i \\mathfrak {p}_i^{e_i+f_i}.$ Now given a binary quadratic form $\\varphi (x,y)=ax^2+bxy+cy^2$ of discriminant $\\Delta $ , we are interested in finding values of $x\\in [-R,R]$ such that $\\varphi (x,1)$ is $B$ -smooth.", "This can be done trivially by testing all the possible values of $x$ , but there is a well-known method for pre-selecting some values of $x$ in $[-R,R]$ that are going to be tested, namely the quadratic sieve (introduced by Pomerance [18]).", "It consists in initializing to 0 an array $S$ of length $2R+1$ and precomputing the roots $r_i^{\\prime }$ and $r_i^{\\prime \\prime }$ , or the double root $r_i^{\\prime }$ , of $\\varphi (x,1)\\mod {p}_i$ for each $p_i\\le B$ such that $\\left( \\frac{\\Delta }{p_i} \\right) \\ne -1$ .", "Then for each $x$ in $[-R,R]$ of the form $x=r_i+kp_i$ for some $k$ , we add $\\lfloor \\log p_i\\rfloor $ to $S[x]$ .", "At the end of this procedure, if $\\varphi (x,1)$ is $B$ -smooth, then $S[x]\\approx \\log \\varphi (x,1)$ .", "As $\\varphi (x,1)\\approx \\sqrt{\\Delta /2}R$ , we set a bound $F = \\log \\left( \\sqrt{\\frac{\\Delta }{2}}R\\right) -T\\log (p_n),$ where $T$ is a number representing the tolerance to rounding errors due to integer approximations.", "We then perform a trial division test on every $\\varphi (x,1)$ such that $S[x]\\ge F$ .", "In this section we describe the improvements that allowed us to achieve a significant speed-up with respect to the existing algorithm and the computation of class group structures of large discriminants.", "Our contribution is to take advantage of the large prime variants, of an algorithm due to Vollmer [23] for the SNF which had not been implemented in the past, and of special Gaussian elimination techniques." ], [ "Large prime variants", "The large prime variants were developed in the context of integer factorization to speed up the relation collection phase in both the quadratic sieve and the number field sieve.", "Jacobson considered analogous variants for class group computation [10], but the speed-up of the relation collection phase was achieved at the price of such a slow-down of the linear algebra that it did not significantly improve the overall time.", "The main idea is the following: We define the “small primes\" to be the prime ideals in the factor base and the small prime bound as the corresponding bound $B_1=B$ .", "Then we define a large prime bound $B_2$ .", "During the relation collection phase we choose not to restrict ourselves to relations only involving primes $\\mathfrak {p}$ in $\\mathcal {B}$ but we also keep relations of the form $(\\alpha )=\\mathfrak {p}_1\\hdots \\mathfrak {p}_n \\mathfrak {p}\\ \\ \\text{and}\\ \\ (\\alpha )=\\mathfrak {p}_1\\hdots \\mathfrak {p}_n \\mathfrak {p}\\mathfrak {p}^{\\prime }$ for $\\mathfrak {p}_i$ in $\\mathcal {B}$ , and for $\\mathfrak {p},\\mathfrak {p}^{\\prime }$ of norm less than $B_2$ .", "We will respectively refer to them as 1-partial relations and 2-partial relations.", "Keeping partial relations only involving one large prime is the single large prime variant, whereas keeping two of them is the double large prime variant which was first described by Lenstra and Manasse [12].", "In this paper we do not consider the case of more large primes, but it is a possibility that has been studied in the context of factorization [14].", "Partial relations may be identified as follows.", "Let $m$ be the residue of $\\varphi (x,1)$ after the division by all primes $p\\le B_1$ , and assume that $B_2 < B_1^2$ .", "If $m=1$ then we have a full relation.", "If $m\\le B_2$ then we have a 1-partial relation.", "We can see here that detecting 1-partial relations is almost for free.", "If we also intend to collect 2-partial relations then we have to consider the following possibilities: $m > B_2^2$ ; $m$ is prime and $m > B_2$ ; $m \\le B_2$ ; $m$ is composite and $B_1^2 < m \\le B_2^2$ .", "In Cases 1 and 2 we discard the relation.", "In Case 3 we have a 1-partial relation, and in Case 4 we have $m=pp^{\\prime }$ where $p = \\mathcal {N}(\\mathfrak {p})$ and $p^{\\prime } = \\mathcal {N}(\\mathfrak {p}^{\\prime })$ .", "After testing if we are in Cases 1, 2, or 3 we have to factorize the residue.", "We have done that using Milan's implementation of the SQUFOF algorithm [15] based on the theoretical work of [7].", "Even though we might have to factor the residue, collecting a partial relation is much faster than collecting a full relation because the probability that $\\mathcal {N}(\\mathfrak {b})$ is $B_2$ -smooth is much greater than the probability that it is $B_1$ -smooth.", "This improvement in the speed of the relation collection phase comes at a price: The number of columns in the relation matrix is much greater, thus preventing us from running the linear algebra phase directly on the resulting relation matrix and forcing us to find many more relations since we have to produce a full rank matrix.", "We will see in §REF how to reduce the dimensions of the relation matrix using Gaussian elimination techniques and in § how to optimize the parameters to make the creation of the relation matrix faster, even though there are many more relations to be found." ], [ "Gaussian elimination techniques", "Traditionally rows were recombined to give full relations as follows: In the case of 1-partial relations, any pair of relations involving the same large prime $\\mathfrak {p}$ were recombined into a full relation.", "In the case of 2-partial relations, Lenstra [12] described the construction of a graph whose vertices were the relations and whose edges linked vertices having one large prime in common.", "Finding independent cycles in this graph allows us to find recombinations of partial relations into full relations.", "In this paper we rather follow the approach of Cavallar [4], developed for the number field sieve, which uses Gaussian elimination on columns without distinguishing those corresponding to the large primes from the others.", "One of the main differences between our relation matrices and the matrices produced in the number field sieve is that our entries are in $\\mathbb {Z}$ rather than $\\mathbb {F}_2$ , thus obliging us to monitor the evolution of the size of the coefficients.", "Indeed, eliminating columns at the price of an explosion of the size of the coefficients can be counter-productive in preparation for the HNF algorithm.", "In what follows we will use a few standard definitions that we briefly recall here.", "First, subtracting two rows is called merging.", "This is because rows are stored as lists of the non-zero entries sorted with respect to the corresponding columns and subtracting them corresponds to merging the two sorted lists.", "If two rows $r_1$ and $r_2$ share the same prime $\\mathfrak {p}$ with coefficients $c_1$ and $c_2$ respectively then multipling $r_1$ by $c_2$ and $r_2$ by $c_1$ and merging is called pivoting.", "Finally, finding a sequence of pivots leading to the elimination of a column of Hamming weight $k$ is a $k$ -way merge.", "We aim to reduce the dimension of the relation matrix by performing $k$ -way merges on the columns of weight $k=1,\\hdots ,w$ in increasing order for a certain bound $w$ .", "Unfortunately, the density of the rows and the size of the coefficients increase during the course of the algorithm, thus obliging us to use optimized pivoting strategies.", "In what follows we describe an algorithm performing $k$ -way merges to minimize the growth of both the density and the size of the coefficients.", "First we have to define a cost function defined over the set of the rows encapsulating the difficulty induced for the HNF algorithm.", "In factorization, we want to find a vector in the kernel of the relation matrix which is defined over $\\mathbb {F}_2$ ; the only property of the row that really matters is its Hamming weight.", "In our context, we need to minimize the Hamming weight of the row, but we also have to take into account the size of the coefficients.", "Different cost functions lead to different elimination strategies.", "Our cost function was determined empirically: We took the number of non-zero entries, counting $c$ times those whose absolute value was above a bound $Q$ , where $c$ is a positive number.", "If $r = [e_1,\\hdots ,e_n]$ corresponds to $(\\alpha )=\\prod _i\\mathfrak {p}_i^{e_i}$ then $C(r) = \\sum _{1\\le |e_i|\\le Q}1 + c\\sum _{|e_j| > Q}1.$ Indeed as we will see, matrices with small entries are better suited for the HNF algorithm described in §REF .", "Let us assume now that we are to perform a $k$ -way merge on a given column.", "We construct a complete graph $\\mathcal {G}$ of size $k$ as follows: The vertices are the rows $r_i$ .", "Every edge linking $r_i$ and $r_j$ is labeled by $C(r_{ij})$ , where $r_{ij}$ is obtained by pivoting $r_i$ and $r_j$ .", "Finding the best sequence of pivots with respect to the cost function $c$ we chose is equivalent to finding the minimum spanning tree $\\mathcal {T}$ of $\\mathcal {G}$ , and then recombining every row $r$ with its parent starting with the leaves of $\\mathcal {T}$ .", "Unfortunately, some coefficients might grow during the course of column eliminations despite the use of this strategy.", "Once a big coefficient is created in a given row $r$ , it is likely to spread to other rows once $r$ is involved in another column elimination.", "We must therefore discard such rows as quickly as possible.", "In our implementation we chose to do it regularly: Once we have performed all the $k$ -way merges for $k\\le 10\\cdot i$ and $i=1,\\hdots ,w/10$ we discard a fixed number $K$ of the rows containing the largest coefficients.", "We show in Table REF the effect of the use of a cost function taking into account the size of the coefficients and the regular discard of the worst rows for $\\Delta = -4(10^{70}+1)$ with $c = 100$ , $Q = 8$ and $K = 10$ .", "We kept track of the evolution of the dimensions of the matrix, the average Hamming weight of the rows, and the maximum and minimum size of the coefficients.", "In the first case we use the traditional cost function that only takes into account the Hamming weight of the rows and we keep deleting the worst rows regularly; this corresponds to taking $c=1$ and $K=10$ .", "In the second case, we use the cost function described above but without row elimination by setting $c=100$ and $K=0$ .", "In the third case, we combine the two ($c=100$ and $K=10$ ).", "We clearly see that the coefficients are properly monitored only in the latter case.", "Indeed using a cost function that does not take into account the size of the coefficients and just discarding the worst rows regularly seems more efficient in terms of reduction of the matrix dimension, but the row corresponding to $i=12$ (that is to say after all the 120-way merges) clearly shows that we run the risk of an explosion of the coefficients.", "Figure: Comparative table of elimination strategies" ], [ "Vollmer's algorithm for computing the HNF", "In [10] it has been observed that the algorithm used to compute the HNF of the relation matrix relied heavily on the sparsity of the matrix.", "While recombinations of the kind described in [12] or the techniques of §REF reduce the dimensions of the matrix, they also dramatically increase the density of the matrix, thus slowing down the computation of the HNF.", "We had to find an HNF algorithm whose features were adapted to our situation.", "Vollmer described in [23] an algorithm of polynomial complexity depending on the capacity to solve diophantine linear systems, but not on the density of the matrix.", "It was not implemented at the time because there was no efficient diophantine linear system solver available.", "We implemented Vollmer's algorithm using the IML [21] library provided by Storjohann.", "Here we give a brief description of the algorithm (for more details we refer to [23]).", "We assume we have created an $m\\times n$ relation matrix $A$ of full rank.", "For each $i\\le n$ , we define two matrices $ A_i = \\left(\\begin{BMAT}(e)[2pt,3cm,3cm]{ccc}{ccc}a_{1,1} & \\hdots & a_{m,1}\\\\\\vdots & & \\vdots \\\\a_{1,i} & \\hdots & a_{m,i}\\end{BMAT}\\right) \\ \\ \\text{and}\\ \\ e_i = \\left(\\begin{BMAT}(e)[2pt,0pt,3cm]{c}{cccc}0 \\\\ \\vdots \\\\ 0 \\\\ 1\\end{BMAT}\\right).", "$ For each $i$ , we define $h_i$ to be the minimal denominator of a rational solution of the system $A_ix = e_i;$ this is computed using the function MinCertifiedSol of IML, which is an implementation of (Special)MinimalSolution from [16], and used in [23] for the complexity analysis.", "In [23] it is shown that $h(\\Delta ) = \\prod _i h_i.$ Fortunately, analytic formulae allow us to compute a bound $h_*$ such that $h_*\\le h(\\Delta ) < 2h_*,$ so we do not have to compute $h_i$ for every $i\\in [1,n]$ .", "In addition, the matrices produced for the computation of the group structure of $\\operatorname{Cl}(\\Delta )$ have small essential part, which keeps the number of diophantine systems to solve small (about the same size as the number of columns of the essential part) as shown in [23].", "[H] Computation of the class number $\\Delta $ , relation matrix $A$ of full rank and $h_*$ $h (\\Delta ) $ $h\\leftarrow 1$ $i\\leftarrow n$ $h < h_*$ Compute the minimal denominator $h_i$ of a solution of $A_i\\cdot x=e_i$ $h\\leftarrow h\\cdot h_i$ $i\\leftarrow i-1$ $h$ We can compute the essential part of the HNF of $A$ with a little extra effort involving only modular reductions of coefficients; we refer to [23] for more details.", "This part of the algorithm is highly dependent on the performance of the diophantine solver we use, which in turn is mostly influenced by the number of columns of the matrix and the size of the coefficients.", "The bechmarks available [21] show that the algorithm runs much faster on matrices with 3-bit coefficients, which is why we took coefficient size into account in the cost function for the Gaussian elimination." ], [ "Optimization of the parameters", "In this section we proceed to optimize the parameters involved in the relation collection phase.", "Each parameter has an effect on the overall time taken to compute the group structure of $\\operatorname{Cl}(\\Delta )$ .", "Recall (REF ) giving the bound $F$ ; when we collect partial relations it should be adapted in the following way: $F = \\log \\left( \\sqrt{\\frac{\\Delta }{2}} R \\right) -T\\log B_2,$ where $B_2$ is the large prime bound." ], [ "Optimization of $T$", "The parameter $T$ represents the tolerance to rounding errors in the traditional sieving algorithms.", "Its value is empirically determined, and usually lies in the interval $[1,2]$ .", "In the large prime variant it also encapsulates the number of large primes we want to allow.", "Indeed, if there were no rounding errors one would expect this value to be 1 for one large prime and 2 for two large primes.", "In practice, we can exhibit an optimum value which differs slightly from what we would expect.", "In figure REF we show the overall running time of the algorithm when the parameter $T$ varies between 1.5 and 3.5 for the discriminant $\\Delta = -4(10^{75}+1)$ .", "The size of the factor base taken is 3250, the ratio $B_2/B_1$ equals 120, and we allow two large primes.", "Figure: Optimum value of TTOne of the main issues for determining the optimal value of $T$ is that it tends to shift when one modifies the value of $B_1$ , the rest being unchanged.", "Indeed, if for example $B_2/B_1 = 120$ then $F = \\log \\left( \\sqrt{\\frac{\\Delta }{2}} R \\right) -T\\log 120B_1,$ so when we increase $B_1$ we have to lower $T$ to compensate.", "Figure REF illustrates this phenomenon on the example $\\Delta = -4(10^{75}+1)$ , with two large primes.", "Figure: Effect of |ℬ||\\mathcal {B}| on the optimal value of TTIn Figure REF we study the evolution of the optimal value of $T$ for the single and double large prime variants on discriminants of the form $-4(10^n+1)$ where $n$ ranges between 60 and 80.", "It appears that, as we expected, the optimal value for the double large prime variant is greater than the one corresponding to the single large prime variant.", "This value is between 2 and 2.3 for one large prime and around 2.7 when we allow two large primes.", "Figure: Optimal value of TT when nn varies" ], [ " The size of the factor base", "The optimal size of the factor base reflects the trade-off between the time spent on the relation collection phase and on the linear algebra phase.", "This optimum is usually not the size that minimizes the time spent on the relation collection phase.", "To illustrate this, Figure REF shows the time taken by the algorithm for $\\Delta = -4(10^{75}+1)$ with $B_2/B_1 = 120$ and the corresponding optimal $T$ .", "Figure: Optimal value of |ℬ||\\mathcal {B}|The optimal size of the factor base increases with the size of the discriminant.", "Figure REF shows the optimal size of the factor base for discriminants of the form $-4(10^n+1)$ as $n$ ranges between 60 and 80 for both one large prime and two large primes.", "We notice that the single large prime variant requires smaller factor bases than without large primes, and bigger factor bases than the double large prime variant.", "Figure: Optimal value of |ℬ||\\mathcal {B}| when nn varies" ], [ " The ratio $B_2/B_1$ ", "Theoretically $B_2$ should not exceed $B_1^2$ .", "In practice, when the ratio $B_2/B_1$ is too high we lose time taking into account partial relations involving primes that are so large that they are very unlikely to occur twice and to lead to a recombination.", "This phenomenon is known in the context of factorization, and 120 is a common choice of value of $B_2/B_1$ (see [6]).", "We ran experiments using 12, 120 and 1200 as values for the ratio $B_2/B_1$ .", "Figure REF shows the results for $\\Delta = -4(10^{75}+1)$ with two large primes.", "We give the optimum timings for each value of the size of the factor base, and compare those values for the three different ratios.", "It appears that 120 is indeed the best choice, but the performance of the algorithm is not highly dependent on this parameter.", "Figure: Comparative table of B 2 /B 1 B_2/B_1" ], [ "Comparative timings", "In Figure REF we give comparative timings in seconds between no large primes and the large prime variants for discriminants of the form $-4(10^n+1)$ , for $n$ between 60 and 80.", "We used 2.4GHz Opterons with 16GB of memory, and the NTL library with GMP.", "It appears that we achieved a significant speed-up by using the large prime strategy.", "Direct comparison with previous methods based on sieving is hard since the timings available in [10] were obtained on 296 MHz UtraSPARC-II processors; therefore we just quote that the computation of the group structure corresponding to $\\Delta = -4(10^{80}+1)$ took 5.37 days (463968 CPU time) at the time.", "We also notice that the double large prime variant does not provide an impressive improvement on the overall time for the sizes of discriminant involved.", "The performance is comparable for discriminants of 60 decimal digits and starts showing an improvement when we reach 75 digit discrimants.", "Figure: Comparative table of the performances (CPU time)" ], [ "Large discriminants", "In the imaginary case, the largest class groups that had been computed using relation collection methods had 90 digits; some 100 decimal digit discriminant class group structures could be computed using the techniques of [22].", "With the techniques described in this paper, we achieved the computation of a class group with a 110 decimal digit discriminant.", "We used 100 Core2 Duo 2.4GHz Pentium IV processors with 2 GB of memory each for the sieving, and one 2.66 GHz Opteron with 64 GB of memory for the linear algebra, which is the real bottleneck of this algorithm.", "Indeed, the sieving phase can be trivially parallelized for as many processors as we have and does not require much memory, whereas the linear algebra can only be parallelized into the number of factors of $h$ that we get from Vollmer's algorithm (around 10 in our examples) and requires a lot of memory.", "Indeed the limit in terms of matrix dimensions for the diophantine solver on a 64GB memory computer seems to be around 10000 columns.", "For comparison, in the case of the 110 decimal digit discriminant we had to handle an 8000-column matrix (after the Gaussian reduction).", "Figure: Decomposition of Cl(Δ)\\operatorname{Cl}(\\Delta ) for Δ=-4(10 n +1)\\Delta = -4(10^n+1)" ], [ "Acknowledgements", "The author thanks Andreas Enge for his support on this project, the fruitful discussions we had and a careful reading of this article.", "We thank Nicolas Thériault and all the organizing comitee of the conference CHILE 2009 where the original results of this paper were first presented.", "We also thank Jérôme Milan for his support on issues regarding implementation, especially with the TIFA library.", "Received June 2009; revised December 2009.", "E-mail address: [email protected]" ] ]

[ [ "Equal coefficients and tolerance in coloured Tverberg partitions" ], [ "Abstract The coloured Tverberg theorem was conjectured by B\\'ar\\'any, Lov\\'{a}sz and F\\\"uredi and asks whether for any d+1 sets (considered as colour classes) of k points each in R^d there is a partition of them into k colourful sets whose convex hulls intersect.", "This is known when d=1,2 or k+1 is prime.", "In this paper we show that (k-1)d+1 colour classes are necessary and sufficient if the coefficients in the convex combination in the colourful sets are required to be the same in each class.", "We also examine what happens if we want the convex hulls of the colourful sets to intersect even if we remove any r of the colour classes.", "Namely, if we have (r+1)(k-1)d+1 colour classes of k point each, there is a partition of them into k colourful sets such that they intersect using the same coefficients regardless of which r colour classes are removed.", "We also investigate the relation of the case k=2 and the Gale transform, obtaining a variation of the coloured Radon theorem." ], [ "Introduction", "Tverberg's theorem is a very well known result in discrete geometry about partitions of points and intersection of convex hulls.", "It says the following, Theorem (Tverberg's theorem [16]) Given a set of $(k-1)(d+1)+1$ points in $\\mathbb {R}^d$ , there is a partition of them into $k$ sets $A_1, A_2, \\ldots , A_k$ such that their convex hulls intersect.", "This is a generalisation of Radon's theorem from 1921 [13], which treats the case $k=2$ .", "Moreover, the number $(k-1)(d+1)+1$ cannot be improved.", "A colourful generalisation of this theorem was conjectured by Bárány, Füredi and Lovász in 1989 [2].", "However, the conjecture below in its full strength was first proposed by Bárány and Larman [3].", "The colourful version states the following, Conjecture (Coloured Tverberg's theorem) Let $F_1, F_2, \\ldots , F_{d+1}$ be sets (which we consider as colour classes) of $k$ points each in $\\mathbb {R}^d$ .", "Then there is a partition of their union into $k$ pairwise disjoint colourful sets $A_1, A_2, \\ldots , A_k$ such that their convex hulls intersect.", "With a colourful set we mean a set that has exactly one element of each colour class.", "Note that this is not a colourful version of Tverberg's theorem in the same way as the colourful versions of Helly's theorem or Carathéodory's theorem are [1], since if all the colour classes are equal this does not yield the original theorem.", "By a colourful $k$ -partition we refer to a family of $k$ pairwise disjoint colourful sets (even if every colour class has more than $k$ elements).", "Historically this conjecture asked if there was a number $t=t(k,d)$ such that if $F_1, F_2, \\ldots , F_{d+1}$ are sets of $t$ points each in $\\mathbb {R}^d$ , this type of $k$ -partition always exists.", "The existence of $t$ was first shown by Bárány, Füredi and Lovász for $k=3$ , $d=2$ [2].", "The general case was settled by Živaljević and Vrećica, who showed that if $k$ was prime, $2k-1$ points were enough [19] (giving a bound of $4k-3$ points for all $k$ ).", "Stated this way, the coloured Tverberg theorem is still open for most cases.", "If $k+1$ is prime, this was solved by Blagojević, Matschke and Ziegler [4] with topological methods (equivariant obstruction theory).", "At the core of their argument is the computation of degrees for a simplicial pseudomanifold; this was made explicit by Vrećica and Živaljević [17].", "By now there is also a purely geometrical proof of this fact by Matoušek, Tancer and Wagner [11], which still follows the same scheme.", "Blagojević et al.", "also gave an alternative proof using different, advanced topological tools (equivariant cohomology, index theory) in [5].", "The optimal solution for the case $k+1$ prime gives the bound $t(k,d) \\le 2k-2$ for all $k$ , as noted in [4].", "The case $d=1,2$ was solved by Bárány and Larman [3] for any value of $k$ .", "For more references on this problem and historical notes we recomend [11].", "The case $k=2$ (also known as the coloured Radon theorem) was originally solved by László Lovász by constructing a function from $\\mathbb {S}^d$ to $\\mathbb {R}^d$ that depended on the pairs of points and then applying the Borsuk-Ulam theorem.", "His proof is contained in [3].", "Here we show a different proof of this fact using only basic linear algebra, which is a simplification of the methods that will be used later on.", "In section we present a third proof, using different methods (but still non-topological).", "Theorem (Coloured Radon) Given $d+1$ pairs of points $F_1, F_2, \\ldots , F_{d+1}$ in $\\mathbb {R}^d$ , we can partition them into two disjoint colourful sets whose convex hulls have non-empty intersection.", "Writing $F_i = \\lbrace x_i, y_i\\rbrace $ in an arbitrary way, the points $x_i - y_i$ are linearly dependent.", "Then there is a linear combination $ \\sum _{i=1}^{d+1} \\alpha _i (x_i - y_i) = 0$ where not all the coefficients are 0.", "After proper relabelling of the points, we may suppose that all the coefficients are non-negative and, by a scalar multiplication, their sum is 1.", "Thus, $\\sum _{i=1}^{d+1} \\alpha _i x_i = \\sum _{i=1}^{d+1} \\alpha _i y_i,$ as we wanted.", "This proof has the advantage that it gives an algorithm to find the partition and the point of intersection since it only involves finding a linear dependence.", "Note that this proof not only gives the partition we wanted, but also shows that to find the point of intersection we may ask that the corresponding points have the same coefficients.", "This also happens in Lovász's poof, since the images of antipodal points in his construction have this property.", "Thus it seems natural to ask whether this could also be possible in the coloured Tverberg Theorem.", "To state this in a more precise way, let $F_1, F_2, \\ldots , F_n$ be sets of at least $k$ points each in $\\mathbb {R}^d$ and $A_1, A_2, \\ldots , A_k$ a colourful $k$ -partition of them.", "We can denote the elements of each $A_i$ by $A_i = \\lbrace x^i_j : x^i_j \\in F_j \\rbrace $ .", "We say that the convex hulls of the $A_i$ intersect with equal coefficients if there are coefficients $\\alpha _1, \\alpha _2, \\ldots , \\alpha _n$ of a convex combination such that $\\sum _{j} \\alpha _j x^i_j $ is the same point for all $i$ .", "In this paper we show that this extension of the colourful Tverberg theorem is not possible with $d+1$ colour classes, regardless of the number of points each class has.", "The following theorem gives the optimal number of colour classes and the optimal number of points per class.", "Namely, Theorem 1 Let $F_1, F_2, \\ldots , F_{n}$ be sets of $t$ points each in $\\mathbb {R}^d$ .", "If $n=(k-1)d+1$ and $t = k$ , we can find a colourful $k$ -partition $A_1, A_2, \\ldots , A_k$ of them such that their convex hulls intersect with equal coefficients.", "Moreover, if $n < (k-1)d+1$ this theorem is false regardless of the value of $t$ .", "In section we show that for $n=(k-1)d+1$ and $t=k$ , there are at least $(k-1)!^{(k-1)d}$ such partitions.", "It is conjectured that for the classical Tverberg theorem there are always at least $(k-1)!^d$ good partitions.", "This is known as Sierksma's conjecture or the Dutch cheese conjecture.", "Bounds for the number of Tverberg partitions have been obtained when $k$ is a prime power ([18], [8]).", "The only non-trivial case that has been solved is $d=2, k=3$ by Hell [9].", "We find it surprising that in this aspect the classical Tverberg theorem seems more resistant.", "Recently many theorems with tolerance have appeared in this kind of settings.", "We say that a property $P$ of points in $\\mathbb {R}^d$ is true in $F$ with tolerance $r$ if $P(F\\backslash C)$ is true for all $C \\subset F$ of size $r$ .", "For example, $P$ can be “captures the origin”.", "There are now versions of the classical Helly, Carathéodory and Tverberg theorems with a tolerance condition [12], [15].", "We show that the previous theorem also has a version with tolerance, but in this case we do not know that the number of colour classes is optimal.", "Theorem 2 Let $d \\ge 2$ , $n= (r+1)(k-1)d+1$ and $F_1, F_2, \\ldots , F_n$ be sets of $k$ points each in $\\mathbb {R}^d$ .", "Then we can find a colourful $k$ -partition $A_1, A_2, \\ldots , A_k$ of them such that for any set $C$ of $r$ colour classes, the convex hulls of $A_1 \\backslash C$ , $A_2 \\backslash C$ , $\\ldots $ , $A_k \\backslash C$ intersect with equal coefficients.", "The proofs of Theorems REF and REF are given in section .", "If $k=2$ , theorem REF gives a nice version of the coloured Radon theorem with tolerance.", "Corollary 3 (Coloured Radon theorem with tolerance) Let $d \\ge 2$ , and $F_1$ , $F_2$ , $\\ldots $ , $F_{(r+1)d+1}$ be pairs of points in $\\mathbb {R}^d$ .", "Then we can split them into two disjoint colourful sets $A_1, A_2$ such that, if we remove any $r$ colour classes, the convex hulls of what is left in $A_1$ and $A_2$ intersect.", "It would be interesting to know if the number of parts in the corollary above is enough for a coloured Tverberg theorem with tolerance (without equal coefficients).", "Namely, Conjecture 4 (Coloured Tverberg theorem with tolerance) Let $d \\ge 2$ , and $F_1, F_2, \\ldots , F_{(r+1)d+1}$ be sets of $k$ points each in $\\mathbb {R}^d$ .", "Then we can split them into $k$ disjoint colourful sets $A_1, A_2, \\ldots , A_k$ such that, if we remove any $r$ colour classes, the convex hulls of what is left in $A_1, A_2, \\ldots , A_k$ intersect.", "In section we investigate the relation between the Gale transform and the coloured Radon theorem.", "With this we are able to obtain the following variation.", "Theorem 5 Given a set of $k+d+2$ points in $\\mathbb {R}^d$ of $k$ possible colours, we can find two disjoint sets $A$ and $B$ such that they have the same number of points of each colour and their convex hulls intersect.", "Note that the theorem above does not imply coloured Radon.", "This is due to the fact that the theorem does not take into consideration how the colours are distributed.", "However, the coloured Radon theorem can be proved with the same method if this extra information is considered." ], [ "Preliminaries", "The proof of theorem REF will be in the same spirit of Sarkaria's proof of Tverberg's theorem [14] but without lifting the points to $\\mathbb {R}^{d+1}$ .", "We are able to use the additional structure on the type of partitions we want in order to reduce the number of dimensions we need.", "Let $F_1, F_2, \\ldots , F_n$ be $n$ sets of $t$ points each in $\\mathbb {R}^d$ and $k \\le t$ a positive integer.", "For convenience we consider each $F_j$ as an ordered set and denote its elements by $F_j = \\lbrace z(j)_1, z(j)_2, \\ldots , z(j)_t\\rbrace $ .", "This order given to each $F_j$ is not important at the moment, but it will be when counting the number of good partitions.", "Denote by $\\Sigma _{k,t}$ the set of injective functions from $[k]=\\lbrace 1,2,\\ldots , k\\rbrace $ to $[t]$ .", "We can consider the colourful $k$ -partitions $(A_1, A_2, \\ldots , A_k)$ as vectors $(\\sigma _1, \\sigma _2, \\ldots , \\sigma _n)$ in $(\\Sigma _{k,t})^n$ by defining $\\sigma _j (i) = m \\ \\mbox{if and only if } x^i_j = z(j)_m$ or equivalently $A_i = \\lbrace x^i_j : x^i_j = z(j)_{\\sigma _j (i)}\\rbrace $ We call $(\\sigma _1, \\sigma _2, \\ldots , \\sigma _n)$ the function representation of $(A_1, A_2, \\ldots , A_k)$ .", "If $t=k$ we call this the permutation representation of $(A_1, A_2, \\ldots , A_k)$ .", "To see this in a simpler way, write the elements of each $F_j$ $k$ times in $k$ rows.", "Then choose from the first row the elements of $A_1$ , in the second row the elements of $A_2$ and so on.", "Then the function representation of $(A_1, A_2, \\ldots , A_k)$ becomes apparent, as in the figure below.", "The reverse operation can also be performed.", "unit=0.45cm (0,15) (-3,6) [fillstyle=solid, fillcolor=gray](2,4.5)(4,5.4) [fillstyle=solid, fillcolor=gray](8,3.5)(9.9,4.5) [fillstyle=solid, fillcolor=gray](4,2.5)(6,3.5) [fillstyle=solid, fillcolor=gray](0.1,0.5)(2,1.5) [fillstyle=solid, fillcolor=gray](15,4.5)(17,5.4) [fillstyle=solid, fillcolor=gray](13,3.5)(15,4.5) [fillstyle=solid, fillcolor=gray](19,2.5)(20.9,3.5) [fillstyle=solid, fillcolor=gray](11.1,0.5)(13,1.5) [fillstyle=solid, fillcolor=gray](23.1,4.5)(25,5.4) [fillstyle=solid, fillcolor=gray](27,3.5)(29,4.5) [fillstyle=solid, fillcolor=gray](31,2.5)(32.9,3.5) [fillstyle=solid, fillcolor=gray](25,0.5)(27,1.5) (1,5)$z(1)_1$ (1,4)$z(1)_1$ (1,3)$z(1)_1$ (1,1)$z(1)_1$ (3,5)$z(1)_2$ (3,4)$z(1)_2$ (3,3)$z(1)_2$ (3,1)$z(1)_2$ (5,5)$z(1)_3$ (5,4)$z(1)_3$ (5,3)$z(1)_3$ (6,2.2)$\\vdots $ (5,1)$z(1)_3$ (7,5)$\\ldots $ (7,4)$\\ldots $ (7,3)$\\ldots $ (7,1)$\\ldots $ (9,5)$z(1)_t$ (9,4)$z(1)_t$ (9,3)$z(1)_t$ (9,1)$z(1)_t$ (0,0.4)(10,5.5) [arrows=->](5,0)(5,-2) (5,-3)$\\sigma _1 (1) = 2, \\sigma _1 (2) = t, \\sigma _1(3)=3$ (5,-4)$ \\ldots , \\sigma _1(k) = 1$ (5,6)$F_1$ (12,5)$z(1)_1$ (12,4)$z(2)_1$ (12,3)$z(2)_1$ (12,1)$z(2)_1$ (14,5)$z(2)_2$ (14,4)$z(2)_2$ (14,3)$z(2)_2$ (14,1)$z(2)_2$ (16,5)$z(2)_3$ (16,4)$z(2)_3$ (16,3)$z(2)_3$ (16,1)$z(2)_3$ (18,5)$\\ldots $ (18,4)$\\ldots $ (18,3)$\\ldots $ (18,1)$\\ldots $ (17,2.2)$\\vdots $ (20,5)$z(2)_t$ (20,4)$z(2)_t$ (20,3)$z(2)_t$ (20,1)$z(2)_t$ (11,0.4)(21,5.5) [arrows=->](16,0)(16,-2) (16,-3)$\\sigma _2 (1) = 3, \\sigma _2 (2) = 2, \\sigma _2(3)=t$ (16,-4)$ \\ldots , \\sigma _2(k) = 1$ (22,3)$\\cdots $ (16,6)$F_2$ (24,5)$z(n)_1$ (24,4)$z(n)_1$ (24,3)$z(n)_1$ (24,1)$z(n)_1$ (26,5)$z(n)_2$ (26,4)$z(n)_2$ (26,3)$z(n)_2$ (26,1)$z(n)_2$ (28,5)$z(n)_3$ (28,4)$z(n)_3$ (28,3)$z(n)_3$ (28,1)$z(n)_3$ (30,5)$\\ldots $ (30,4)$\\ldots $ (30,3)$\\ldots $ (30,1)$\\ldots $ (29,2.2)$\\vdots $ (32,5)$z(n)_t$ (32,4)$z(n)_t$ (32,3)$z(n)_t$ (32,1)$z(n)_t$ (23,0.4)(33,5.5) [arrows=->](28,0)(28,-2) (28,-3)$\\sigma _n (1) = 1, \\sigma _n (2) = 3, \\sigma _n(3)=t$ (28,-4)$ \\ldots , \\sigma _n(k) = 2$ (28,6)$F_n$ (34,5)$A_1$ (34,4)$A_2$ (34,3)$A_3$ (34,1)$A_k$ Let $u_1, u_2, \\ldots , u_k$ be the vertices of a regular simplex in $\\mathbb {R}^{k-1}$ centred at the origin.", "We use these sets to represent the distribution in the $k$ -partition.", "Given $\\sigma \\in \\Sigma _{k,t}$ we define $F_j (\\sigma ) \\in \\mathbb {R}^{(k-1)d}$ as $F_j (\\sigma ) = \\sum _{i=1}^{k} u_i \\otimes z(j)_{\\sigma (i)}$ where $\\otimes $ represents the tensor product.", "Lemma 6 Let $F_1, F_2, \\ldots , F_n$ be sets of $t$ points each in $\\mathbb {R}^d$ and $(A_1, A_2, \\ldots , A_k)$ be a colourful $k$ -partition of them.", "Then for coefficients $\\alpha _1, \\alpha _2, \\ldots , \\alpha _n$ we have that $\\sum _{j=1}^n \\alpha _j x^i_j$ is the same point for all $i$ if and only if $\\sum _{j=1}^n \\alpha _j F_j(\\sigma _j) = 0$ , where $(\\sigma _1, \\sigma _2, \\ldots , \\sigma _n)$ is the function representation of $(A_1, A_2, \\ldots , A_k)$ .", "It suffices to prove this for $d=1$ since we can repeat the same argument for each coordinate in $\\mathbb {R}^d$ .", "In this case $u_i \\otimes z(j)_m$ is simply $z(j)_m u_i$ .", "Note that, for scalars $\\beta _1, \\beta _2, \\ldots , \\beta _k$ , we have that $\\sum _{i=1}^k \\beta _i u_i = 0$ if and only if $\\beta _1 = \\beta _2 = \\ldots = \\beta _k$ .", "Thus $\\sum _{j=1}^n \\alpha _j F_j(\\sigma _j) = 0$ if and only if $\\sum _{j=1}^n \\alpha _j z(j)_{\\sigma _j (i)}$ is the same for all $i$ , as we wanted." ], [ "Proofs of theorems ", "[Proof of theorem REF ] We first show that for $n = (k-1)d+1$ and $t=k$ there are such coefficients.", "For this it suffices to note that each of the sets $F_j (\\Sigma _{k,k})$ captures the origin.", "We have $n$ sets that capture the origin in $\\mathbb {R}^{n-1}$ , so by the Bárány colourful version of Carathéodory's theorem [1], there are permutations $\\sigma _1, \\sigma _2, \\ldots , \\sigma _n$ such that the set $\\lbrace F_1 (\\sigma _1), F_2 (\\sigma _2), \\ldots , F_n (\\sigma _n)\\rbrace $ captures the origin.", "Using lemma REF we are done.", "Now suppose that $n \\le (k-1)d$ .", "If $\\sigma _1$ is fixed, note that by varying $F_1$ , then $F_1(\\sigma _1)$ can be any point of $\\mathbb {R}^{(k-1)d}$ .", "Suppose we are given $F_2, F_3, \\ldots , F_n$ and we want to find all the choices for $F_1$ that satisfy the theorem.", "Given any coloured $k$ -partition $(A_1, A_2, \\ldots , A_k)$ , by applying the same permutation to each colour class, we can assume that in its function representation $\\sigma _1$ is the identity.", "If we can find injective functions $\\sigma _2, \\sigma _3, \\ldots , \\sigma _n$ such that the set $\\lbrace F_1 (\\sigma _1), F_2 (\\sigma _2), \\ldots , F_n (\\sigma _n)\\rbrace $ captures the origin then $F_1 (\\sigma _1)$ must be in the $(n-1)$ -flat generated by ${0, F_2(\\sigma _2),\\ldots , F_n(\\sigma _n)}$ .", "Note that there is only a finite number of possible choices for $\\sigma _2, \\sigma _3, \\ldots , \\sigma _n$ .", "Since a finite number of flats of codimension at least 1 cannot cover $\\mathbb {R}^{(k-1)d}$ , we are done.", "To show the versions with tolerance with this method we would need some version of the colourful Carathéodory theorem with tolerance.", "Conveniently this was done in Lemma 1 in the proof of Tverberg's theorem with tolerance [15].", "Here we re-write this lemma to fit the current notation.", "Lemma 7 Let $p \\ge 1$ and $r \\ge 0$ be integers, $n=(r+1)p+1$ , $F_1, F_2, \\ldots , F_n \\subset \\mathbb {R}^p$ subsets of $\\mathbb {R}^p$ that capture the origin, $F = \\bigcup _{j=1}^n F_j$ and $G$ a group with $|G| \\le p$ .", "Suppose there is an action of $G$ in each of $F_1, F_2, \\ldots , F_n$ such that the following holds.", "For all $x \\in F_j$ , $Gx$ captures the origin, for all $j$ .", "For all $A \\subset F$ , if $A$ captures the origin then so does $gA$ , for all $g \\in G$ .", "Then we can find elements $x_1 \\in F_1, x_2 \\in F_2, \\ldots , x_n \\in F_n$ such that $\\lbrace x_1, x_2, \\ldots , x_n\\rbrace $ captures the origin with tolerance $r$ .", "[Proof of theorem REF ] Note that there is an action of $\\Sigma _{k,k}$ (as the symmetric group) in $F_j(\\Sigma _{k,k})$ given by $\\sigma F_j (\\tau ) = F_j (\\sigma \\tau )$ .", "In particular, this induces an action of $\\mathbb {Z}_k$ given by $m F_j(\\tau ) = F_j (\\beta ^m \\tau )$ where $\\beta $ is a cycle of length $k$ .", "Consider $p=(k-1)d$ and $F_1 (\\Sigma _{k,k})$ , $F_2 (\\Sigma _{k,k})$ , $\\ldots $ , $F_n(\\Sigma _{k,k})$ with their action of $\\mathbb {Z}_k$ (all using the same $\\beta $ ).", "If $d \\ge 2$ , these sets with their actions of $\\mathbb {Z}_k$ satisfy the conditions of lemma REF .", "Using this lemma and lemma REF we obtain the result." ], [ "Remarks on the proof", "In the general case of the coloured Tverberg theorem, it seems tempting to extend Lovász's argument for the case $k >2$ .", "Namely, we consider the $(d+1)$ -fold join $U = \\underbrace{G \\ast G \\ast \\cdots \\ast G}_{d+1}$ of a discrete set with $k$ elements and try to prove that for any linear function $f: U \\longrightarrow \\mathbb {R}^d$ there are points $x_1, x_2, \\ldots , x_k$ in disjoint faces of $U$ such that $f(x_1)=f(x_2)=\\ldots = f(x_k)$ .", "Then it seems natural to use the combinatorial properties of $U$ (or a further refinement of this space) to treat this as an equivariant topology problem with the natural action of $G$ in $U$ , as it is done in cases like the topological Tverberg theorem [6].", "However, a simple approach of this kind is bound to give points $x_1, x_2, \\ldots , x_k$ which have the same coefficients when written as combination of the $d+1$ copies of $G$ .", "Since we showed that for this case at least $(k-1)d+1$ colour classes are necessary, other topological methods are needed for the general case.", "In the proof of theorem REF , the final argument was that each $F_j (\\Sigma _{k,k})$ captured the origin.", "However, a smaller set would suffice, namely $F_j (\\lbrace \\beta , \\beta ^2, \\ldots , \\beta ^k\\rbrace )$ captures the origin for any $\\beta $ a cycle of length $k$ .", "For simplicity we can take $\\beta $ the permutation that shifts every element to the right, except the last one.", "Fix the order of $F_1$ and assign a cyclic order (that is, an order up to iterated applications of $\\beta $ ) to each $F_j$ with $j>1$ .", "Note that the partitions we obtain for each way to assign cyclic orders to these families of points are all different.", "Thus, there are at least $(k-1)!^{(k-1)d}$ of these partitions.", "This is similar to the conjectured $(k-1)!^d$ different partitions for the typical Tverberg theorem in the sense that both are roughly $(k-1)!^{{m}/{k}}$ where $m$ is the number of points that are given in the theorem.", "The last step towards showing that $(k-1)d+1$ colour classes are necessary for Theorem REF relies on the fact that a finite number of flats of codimension at least 1 cannot cover the whole space.", "However, more can be said.", "Let $n\\le (k-1)d$ and $F_1, F_2, \\ldots , F_n$ be sets of $t$ random points each in $\\mathbb {R}^d$ where each point is chosen according to a (possibly different) distribution where hyperplanes have measure 0.", "Then, with probability 1, there is no colourful $k$ -partition $A_1, A_2, \\ldots , A_k$ such that the convex hulls of the $A_i$ intersect with equal coefficients." ], [ "Coloured Radon and the Gale transform", "Here we present a third proof of the coloured Radon, again without topological tools.", "This is done via the Gale transform.", "The Gale transform of a set of $n$ points $a_1, a_2, \\ldots , a_n$ in $\\mathbb {R}^d$ that are not all contained in a hyperplane is a set of $n$ points $b_1, b_2, \\ldots , b_n$ in $\\mathbb {R}^{n-d-1}$ such that the following two conditions hold $\\sum _i b_i = 0$ and for every two disjoint subsets $X, Y \\subset [n]$ , the convex hull of the sets $\\lbrace a_i : i \\in X\\rbrace $ and $\\lbrace a_i : i \\in Y \\rbrace $ intersect if and only if there is a hyperplane $H$ through the origin in $\\mathbb {R}^{n-d-1}$ that leaves $\\lbrace b_i : i \\in X\\rbrace $ in one (closed) side, $\\lbrace b_i : i \\in Y\\rbrace $ in the other (closed) side and goes through every other $b_i$ .", "See, for example, [7] for a complete exposition.", "We are now ready to prove the coloured Radon theorem.", "Let $F_1, F_2, \\ldots , F_{d+1}$ be the sets of pairs of points in $\\mathbb {R}^d$ and $F = \\cup _i F_i$ .", "Let $J$ be the Gale transform of $F$ .", "Then $J$ consists of $d+1$ pairs of points $J_1, J_2, \\ldots , J_{d+1}$ in $\\mathbb {R}^{d+1}$ .", "We want to find a hyperplane $H$ through the origin that splits every pair in $J$ simultaneously.", "Let $m_i$ be the midpoints of the $J_i$ .", "Note that since the sum of all the points in $J$ is 0, then the sum of all $m_i$ is 0.", "With this the hyperplane $H$ through $m_1, m_2, \\ldots , m_{d+1}$ goes through the origin, and splits each pair.", "With this idea in mind we can prove theorem REF .", "We use the ham-sandwich theorem (see [10]) in addition to the Gale transform.", "[Proof of theorem REF ] Consider a set $X$ of $k+d+2$ points in $\\mathbb {R}^d$ each painted with one of $k$ possible colors.", "Its Gale transform is a set $Y$ in $\\mathbb {R}^{k+1}$ .", "If we consider the origin of $\\mathbb {R}^{k+1}$ painted with a new color, we have $k+1$ coloured finite sets.", "Using the ham-sandwich theorem, there is a hyperplane that splits them all evenly.", "This hyperplane must go through the origin, and thus gives the two subset of $X$ we we looking for.", "If $k=1$ this gives the following corollary Corollary 8 Given $d+3$ points in $\\mathbb {R}^d$ , there are two disjoint subset $A$ and $B$ of the same size such that their convex hulls intersect.", "It would be interesting to find an analogous statement for Tverberg partitions.", "Namely, finding the smallest $n=n(d,k)$ such that for every set of $n$ points in $\\mathbb {R}^d$ we can find $k$ pairwise disjoint subsets of the same size whose convex hulls intersect.", "Clearly $(k-1)(d+1) < n \\le k(d+1)$ ." ], [ "Acknowledgements", "We would like to thank Imre Bárány for the helpful discussions and observations on this subject and his help on simplifying the proof of the coloured Radon theorem in the introduction." ] ]

[ [ "Multi-plectoneme phase of double-stranded DNA under torsion" ], [ "Abstract We use the worm-like chain model to study supercoiling of DNA under tension and torque.", "The model reproduces experimental data for a much broader range of forces, salt concentrations and contour lengths than previous approaches.", "Our theory shows, for the first time, how the behavior of the system is controlled by a multi-plectoneme phase in a wide range of parameters.", "This phase does not only affect turn-extension curves but also leads to a non-constant torque in the plectonemic phase.", "Shortcomings from previous models and inconsistencies between experimental data are resolved in our theory without the need of adjustable parameters." ], [ "colorlinks=true Multi-plectoneme phase of double-stranded DNA under torsion Marc Emanuel Instituut Lorentz voor de theoretische natuurkunde, Universiteit Leiden,P.O.", "Box 9506, NL-2300 RA Leiden, The Netherlands Institute of Complex Systems II, Forschungszentrum Jülich, Jülich 52425, Germany Giovanni Lanzani Helmut Schiessel Instituut Lorentz voor de theoretische natuurkunde, Universiteit Leiden,P.O.", "Box 9506, NL-2300 RA Leiden, The Netherlands 64.70.km,87.10.Ca,87.15.ad We use the worm-like chain model to study supercoiling of DNA under tension and torque.", "The model reproduces experimental data for a much broader range of forces, salt concentrations and contour lengths than previous approaches.", "Our theory shows, for the first time, how the behavior of the system is controlled by a multi-plectoneme phase in a wide range of parameters.", "This phase does not only affect turn-extension curves but also leads to a non-constant torque in the plectonemic phase.", "Shortcomings from previous models and inconsistencies between experimental data are resolved in our theory without the need of adjustable parameters.", "The DNA contained in every cell of all higher organisms is hundred times longer than the cell diameter: to fit inside it has to fold.", "This is a challenging problem since DNA is a semi-flexible polymer, making it hard to confine it in small spaces.", "On the other hand, local unfolding of DNA has to be efficient, as it plays a key role in the transcription and replication of the genome.", "Unfolding is achieved by stretching and twisting the molecule: unraveling how DNA reacts to them is crucial to understand cellular activities.", "The relevant mechanical properties of DNA have been studied with single molecule techniques, where individual molecules are stretched and/or twisted under physiological conditions.", "The stretching and bending elasticity, in the absence of twisting, has been investigated through measurements of the force-extension relation of DNA [1] and theories based on the worm-like chain (WLC) model have successfully explained the experimental results [2] [3].", "The WLC model [4] is a coarse-grained approximation where the particular sequence of basepairs (bp) is hidden by treating the DNA as a homogeneous semiflexible polymer.", "Figure: Geometry of the plectoneme.To understand stretched DNA under torsional stress [5], [6], [7] several models based on the WLC framework were proposed.", "However, they were either purely mechanical [8], involved non-linear elasticity [9], phenomenological [10], aimed only at a certain region of the experimental data [11] or had to invoke a non-canonically reduction of the DNA charge density [12].", "The outcome of the experiments still remains poorly understood.", "In this Letter we present a theory, based on the WLC model, without any of the aforementioned shortcomings.", "The results describe experimental data accurately (see Fig.", "REF for an example).", "Up to now it has been assumed that under high twist DNA reduces its torque through the formation of a single plectoneme, see Fig.", "REF .", "We show here for the first time how thermal fluctuations lead to a multiple plectoneme phase instead.", "We demonstrate its impact on the turn-extension slope, see Fig.", "REF , and on the torque, see Fig.", "REF .", "Figure: The turn-extension plots of a 600600\\,nm DNA chain at 320320\\,mM ionicstrength for various tensions between 0.250.25\\, and 44\\,pN.", "Comparisonbetween theory and experiments.", "Experimental data from , .Figure: The results of the theory with and without the possibility to formmore than one plectoneme are presented alongside the experimental results(33\\,pN, 2020\\,mM, Experimental data from , .In the experiments the DNA is anchored at one end to a surface and at the other end to a magnetic [7] or optical [6] tweezer.", "This allows to control the tension and the linking number ($n$ ), the number of turns inserted, at the same time.", "Increasing $n$ at constant tension yields turn-extension plots like the ones shown in Fig.", "REF .", "Initially most of $n$ goes into twist ($T\\!w$ ) of the molecule while the end-to-end distance remains approximately constant.", "Then a transition — dependent on the salt concentration, the DNA length and the applied force and often accompanied by a jump [6] — signals the formation of a plectoneme.", "From this point onward all the additional $n$ is stored inside the growing plectoneme as writhe ($W\\!r$ ), a quantity related to the path of the polymer.", "Writhe, twist and linking number are connected through White's relation [14], , $n=T\\!w+W\\!r$ .", "We assume that the legs and the end loop form a homoclinic solution, characterized by the parameter $t \\in [0,1]$ , as described in [16].", "$t=0$ corresponds to a straight rod, and $t=1$ to the homoclinic loop.", "Inside the homoclinic solutions, at the point of (non-zero) minimum distance of symmetric points, we insert a plectoneme.", "Such a minimum exists in the range $0.804\\gtrsim t>1 $ ; half this distance sets the plectoneme radius $R(t)$ .", "The homoclinic solution stores some fixed amount of writhe, $W\\!r_l= 2 (\\arcsin t) /\\pi $ .", "Moreover its bending and potential energy sum up to $E_l= 8 F \\lambda t$ .", "Here $ F$ is the tension, $P_b=A/k_B T$ the DNA bending persistence length and $\\lambda = \\sqrt{A/F}$ .", "On the other hand, the plectoneme has a bending energy density $e_b = A \\cos ^4\\alpha /2 R^2(t)$ and a contribution to the potential energy density of $f $ where $\\alpha $ denotes the angle of the plectoneme, Fig.", "REF .", "When properly accounting for the presence of the end loop, the plectoneme writhe density is exactly given by $wr_p(t)= \\sin \\alpha \\cos \\alpha / 2 \\pi R(t)$ .", "The electrostatic interaction in the plectoneme has a free energy density $f_{\\mathrm {el,0}}$ described by Ubbink and Odijk [17].", "The effective charge density in this contribution is calculated on the basis of a charge density of 2 charges/$0.34\\,$ nm with the method described in [18].", "The zero-temperature energy density of a DNA chain with contour $ L_c $ containing $m$ plectonemes of total relative length $l_p \\equiv L_p / L_c$ is given by $e_0 (m,l_p) &= l_p e_b + m \\frac{E_l}{L_c}+l_p(F+f_{\\mathrm {el,0}})\\\\&+2 \\pi ^2 P_C k_B T(\\frac{n}{L_c}-m\\frac{W\\!r_l}{L_c}- l_pwr_p)^2$ where the last term is the twist contribution to the energy, and $P_C$ is the torsional persistence length.", "However, to properly account for the experimental situation the zero-temperature analysis is not sufficient.", "Thermal fluctuations lead to three different contributions: the first affects the DNA legs, the second acts within plectonemes and the last is related to the number of plectonemes and their position and length distribution.", "Outside the plectonemes, thermal fluctuations modify the shape of the DNA; for a given torque $ \\tau $ they shorten the DNA end-to-end distance by a factor $\\rho (F,\\tau )$  [19] and give origin to thermal writhe that lowers the twist energy density, which can be expressed as a renormalized torsional persistence length $ P_C^{\\mathrm {eff}}(\\lambda )$  [19].", "Inside the plectonemes, following Ref.", "[17] we consider fluctuations in two directions.", "We denote by $\\sigma _r$ ($ \\sigma _p $ ) the standard deviation in the radial (pitch) direction.", "The fluctuations in the pitch direction are set by the geometry: $\\sigma _p = \\pi R\\sin \\alpha $ , see Fig.", "REF , while in the other direction they are set by the electrostatic repulsion.", "Their contribution changes the electrostatic energy, up to first order, to $f_{\\mathrm {el}} =f_{\\mathrm {el,0}} \\times \\exp (4 \\kappa ^2\\sigma _r^2)$  [17], where $\\kappa ^{-1}$ is the Debije length.", "The presence of twist couples the two directions of the fluctuations, an effect not studied before Marc Emanuel, in preparation..", "This result in a new effective deflection length: $\\bar{\\lambda }&=2\\frac{\\lambda _r^3\\lambda _p+\\lambda _r^2\\lambda _p^2+\\lambda _r\\lambda _p^3}{(\\lambda _r+\\lambda _p)(\\lambda _r^2+\\lambda _p^2)}$ where $\\lambda _{r,p}= (P_b \\langle \\sigma _{r,p}^2 \\rangle )^{1/3} $ are the deflection lengths of confinement as defined in Ref. [21].", "The resulting twist energy density, renormalized with $ P_C^{\\mathrm {eff}}(\\bar{\\lambda }) $ , depends on the confinement.", "Since twist diffusion happens on a very short time scale, the twist energy density in the legs and in the plectonemes should be the same.", "This non-trivially couples the linking number density between legs and plectonemes.", "As a consequence of thermal fluctuations, the plectonemes' path is shortened by $\\rho _{\\mathrm {str}}$ , its bending energy density ($ e_b $ ) and bare writhe density ($ wr_b $ ) are modified; the confinement of the polymer contributes an additional $f_{\\mathrm {conf}}$ to the free energy [22].", "Together with the change from $ f_{\\mathrm {el},0} $ to $f_{\\mathrm {el}} $ , these modify $ e_0(m, l_p) $ to $ f(m, l_p) $ .", "In the infinite chain limit, as long as the number of plectonemes stays small, the plectoneme parameters $ R,\\,\\alpha $ and $ \\sigma _r$ become independent of $ l_p $ since end loop contributions drop out.", "The final contribution of the fluctuations comes from two combinatorial factors in the partition function.", "They arise from the number of ways the total length of the plectonemes can be distributed between them, and the number of ways the plectonemes can be placed along the DNA.", "Since the quantities involved are continuous we need to impose a $\\xi $ -cutoff in our calculations which we assume in the following to be given by the DNA helical repeat, $\\xi =3.4\\,$ nm.", "Assuming hard-core interactions between the plectonemes, this results in the partition function $Z &= Z_0 + \\sum _{m=1} \\int \\mathrm {d}L_p G(m,L_p) e^{-\\beta f(m,l_p) L_c}\\\\G &= \\frac{(\\rho (L_c-L_p)-mL_{\\mathrm {loop}})^m}{\\xi ^m m!", "}\\frac{L_p^{m-1}}{\\xi ^{m-1}(m-1)!", "}.$ Here $Z_0$ is the partition function when $m=0$ and $L_{\\mathrm {loop}}$ is the length of a single end loop.", "The first factor of $G$ is the number of ways one can arrange $m$ plectonemes along the DNA; the second factor is the number of ways the length $L_p$ can be distributed between $m$ plectonemes.", "When $ \\partial _n m L_{\\mathrm {loop}} \\ll \\partial _n L_p $ the system is in the single-plectoneme state.", "On the other hand, for $ \\partial _n mL_{\\mathrm {loop}} \\approx \\partial _n L_p $ the physics of increasing $n$ cannot be described by the notion of plectoneme length growth alone, but a full multi-plectoneme approach is needed.", "To characterize these two states we introduce the multi-plectoneme parameter $ \\eta $ as the difference between the writhe efficiencies of loops and plectonemes $\\eta \\equiv \\frac{E_l}{W\\!r_l} - \\frac{f_p}{wr_p}$ where $ f_p $ is the free energy density difference between plectoneme and legs.", "$ \\eta $ is important because it enters exponentially the multi-plectoneme parameter $ \\zeta $ $\\zeta \\equiv e^{- W\\!r_l \\eta } \\left( \\frac{W\\!r_l/L_l}{wr_p} \\right)^2.$ One can show that $ \\zeta \\ll 1 $ corresponds to a single-plectoneme state, whereas $ \\zeta \\approx 1 $ signals the multi-plectoneme phase.", "In the inset of Fig REF , $ \\zeta $ is displayed as a function of salt concentration for different tensions.", "Figure: Phase diagram of the average number of plectonemes as a function of tension and salt concentration for a 7.2 μ\\mu m long chain.", "Note the shift ofthe maximum from low tension at high salt to high tension at low salt.", "The inset shows the multi-plectoneme parameter versus salt concentration for 11\\,pN (blue), 22\\,pN(green) and 33\\,pN (red).Theories without the possibility of multiple plectonemes, typically predict, for low salt concentrations, a too steep slope of the linear part of the turn-extension curves.", "As can be seen in Fig.", "REF the multi-plectoneme phase accurately describes the experiments, even for very low salt concentrations ($ 20\\, $ mM, data from [12], ).", "Figure: The supercoiling density-torque plots of a 56005600\\,nm DNA chain at 100100\\,mM ionicstrength for 0.910.91\\,pN, 1.81.8\\,pN and 33\\,pN tension.", "Comparisonbetween theory and torques inferred from extension measurements.", "Data taken from Figure: The supercoiling density-torque plots of a 725725\\,nm DNA chain at 100100\\,mM ionicstrength for 11\\,pN, 22\\,pN and 33\\,pN tension.", "Comparisonbetween theory and torques measured using a specially crafted cylinder.", "Data taken from Besides the slope, the multi-plectoneme phase influences the torque after the transition.", "In fact, after the transition into the single-plectoneme phase, $n $ is transferred at a fixed rate into $ l_p $ .", "This results in a small bump in the torque at the transition, caused by the use of a number of turns clamp, followed by a constant plectoneme torque.", "On the other hand, in the multi-plectoneme phase the entropic contribution to the free energy ceases to be linear in $ n $ .", "This explains the difference between torques measured in optical tweezer experiments [6] and calculated using Maxwell relations in a magnetic tweezer setup [7].", "The latter method is based on the assumption of a constant torque after the transition.", "However, for the multi-plectoneme phase our theory predicts a non-constant torque.", "In Fig.", "REF we show what our model predicts for the data presented in [7].", "To facilitate comparison with the original paper, not the linking number, but the supercoiling density is used, which is defined as the ratio of the linking number density to the linking density of the two strands of free DNA.", "As can be seen in Fig.", "REF , the assumption of constant torque underestimates the torque difference between the high and low tension curves.", "The multi-plectoneme phase prediction, however, correctly reproduces the torque directly measured in [6], as is shown in Fig.", "REF .", "A final consequence of the multi-plectoneme phase is the change in the dynamics of plectonemes in a chain torsionally loaded.", "Twist mediated plectoneme length redistribution over the plectonemes makes a fast diffusion of plectonemes possible also in the crowded environment of the plasmoid in bacteria or a dense chromatin fiber in eukaryotes.", "The implications for cellular processes from transcription to segregation are significant.", "To conclude: the results of the theory show how the multi-plectoneme phase is crucial to understand the static and dynamic behavior of DNA under tension and torque.", "The torque in fact is much higher than what was previously computed [7], a fact that could be crucial in the understanding of the life processes in which DNA is involved.", "The authors thank Ralf Seidel for providing us with experimental data and Theo Odijk and Ralf Seidel for fruitful discussions.", "We acknowledge discussions with Cees Dekker and Marijn van Loenhout." ] ]

[ [ "A Generalized Apery Series" ], [ "Abstract The inverse central binomial series {equation}S_k(z)=\\sum_{n=1}^{\\infty}\\frac{n^k z^n}{\\binom{2n}{n}}{equation} popularized by Ap\\'ery and Lehmer is evaluated for positive integers $k$ along with the asymptotic behavior for large $k$.", "It is found that value $z=2$, as commented on by D. H. Lehmer provides a unique relation to $\\pi$." ], [ "Introduction", "Since the appearance of $S_{-3}(1)$ in Apéry's famous proof [1] in 1979 that $\\zeta (3)$ is irrational, an extensive literature has been devoted to the series $S_k(z)=\\sum _{n=1}^{\\infty }\\frac{n^k z^n}{\\binom{2n}{n}}$ For example, in 1985 Lehmer [2] presented a number of special cases which could be obtained from the Taylor series for $f(x)=x^{-1/2}(1-x)^{-1/2}\\sin ^{-1}x$ using only elementary calculus.", "In passing, he noted that when $k$ is a positive integer, $S_k(2)$ had the form $a_k-b_k\\pi $ and that the rational number $a_k/b_k$ “is a close approximation to $\\pi $ .", "This remark was recently taken up by Dyson et al.", "[3], who proved that $|a_k/b_k-\\pi |=O(Q^{-k})$ as $k\\rightarrow \\infty $ where $Q=\\sqrt{1+(2\\pi /\\ln 2)^2}$ .", "Lehmer also showed that for positive integer $k$ $S_k(z)=\\frac{2^{k+z^{5/2}}z^{1/2}}{(4-z)^{k+{3/2}}}( A_k(z/4)\\sin ^{-1}(\\sqrt{z/4})+\\sqrt{z(4-z)}B_k(z/4))$ where $A_k$ and $B_k$ are recursively defined polynomials.", "It was apparently not until 2005 that (REF ) was evaluated explicitly, for $z=1$ , by J. Borwein and P. Girgensohn [4] who showed $S_k(1)=\\frac{1}{2}(-1)^{k+1}\\sum _{j=1}^{k+1} (-1)^j j!S(j+1,j) 3^{-j}\\binom{2j}{j}\\left(\\sum _{i=1}^{j-1}\\frac{3^i}{(2i+1) \\binom{2i}{i}}+ \\frac{2}{3 \\sqrt{3}} \\pi \\right).$ where the Stirling numbers of the second kind are defined by $S(k,j)=\\frac{(-1)^j}{j!", "}\\sum _{m=0}^j(-1)^m m^k\\binom{j}{m}.$ The aim of the present note is to extend (REF ) to complex $z$ and thus to continue (REF ) analytically beyond its circle of convergence $|z|=4$ ." ], [ "Calculation", "We begin with the observation that $\\left(m\\binom{2m}{m}\\right)^{-1} =B(m,m+1)$ , where B denotes Euler's beta integral.", "Hence, $S_k(z)=\\int _0^1\\frac{dt}{t}\\sum _{m=1}^{\\infty }m^{k+1}(zt(1-t))^m.$ Next, equation (21) of Girgensohn and Borwein [4], $\\sum _{m=1}^{\\infty }m^pX^m=\\sum _{n=1}^p\\sum _{m=1}^n(-1)^{m+n}\\binom{n}{m} m^pX^n(1-X)^{-n-1},$ gives $S_k(z)=\\sum _{n=1}^{k+1}\\sum _{m=1}^n(-1)^{m+n}\\binom{n}{m}m^{k+1}\\int _0^1\\frac{dt}{t}\\frac{(zt(1-t))^n}{(1-zt(1-t))^{n+1}}.$ In the appendix it is shown that $\\int _0^1\\frac{dt}{t}\\frac{(zt(1-t))^n}{(1-zt(1-t))^{n+1}}=\\frac{\\sqrt{\\pi }\\Gamma (n)}{\\Gamma (n+1/2)}X^n\\;_2F_1(-1/2,n;n+1/2;-X)$ where $X=z/(4-z)$ , so $S_k(z)=\\sum _{n=1}^{k+1}n!B(n,1/2)S(k+1,n)X^n\\;_2F_1(-1/2,n;n+1/2;-X).$ By induction, starting with the tabulated value for $n=1$ and using Gauss' contiguity relations we find (some details are given in the appendix) $_2F_1(-1/2,n;n+1/2;-X) =\\nonumber $ $\\left( \\frac{1}{2} \\right)_n \\left(\\frac{1}{n!", "}+\\frac{1}{\\sqrt{\\pi }\\Gamma (n)}\\sum _{k=0}^{n-1}\\frac{(-1)^k\\Gamma (k+1/2)}{(k+1)!", "}\\binom{n-1}{k}\\left(\\frac{X+1}{X}\\right)^{k+1}\\times \\right.", "\\nonumber \\\\\\left.\\left[\\sqrt{X}\\sin ^{-1}\\sqrt{\\frac{X}{X+1}}-\\frac{1}{2}\\sum _{l=1}^k\\frac{(l-1)!", "}{(1/2)_l}\\left(\\frac{X}{X+1} \\right)^l \\right] \\right).$ (We have used the ascending factorial notation $(a)_n=\\Gamma (a+n)/\\Gamma (a)$ ).", "Therefore we have the principal result $S_k(z)=\\sum _{n=1}^{k+1}n!\\left(\\frac{z}{4-z}\\right)^nS(k+1,n)\\times \\nonumber $ $\\left(\\frac{1}{n}+\\sum _{p=0}^{n-1}(-1)^p\\frac{(1/2)_p}{(p+1)!", "}\\binom{n-1}{p}\\left(\\frac{4}{z}\\right)^{p+1} \\left(\\sqrt{\\frac{z}{4-z}}\\sin ^{-1}\\frac{\\sqrt{z}}{2}-\\frac{1}{2}\\sum _{l=1}^p\\frac{\\Gamma (l)}{(1/2)_l}\\left(\\frac{z}{4}\\right)^l\\right) \\right)$ Equation (REF ) is rather condensed; in unpacking it, sums with upper limit less than the lower limit are to be interpreted as 0.", "It is clear from (REF ) that for rational $z$ $\\sum _{m=1}^{\\infty }\\frac{n^kz^n}{\\binom{2n}{n}}=R_1(z,k)+R_2(z,k)\\sqrt{\\frac{z}{4-z}}\\sin ^{-1}\\frac{\\sqrt{z}}{2},$ where $R_j$ is a rational number.", "One sees from (REF ) that $ S_k(z)$ is analytic on the two-sheeted Riemann surface formed by two planes cut and rejoined along the real half-line $x>4$ .", "The numbers in (REF ) have the explicit expressions $R_1(z,k)= $ $\\sum _{n=1}^{k+1}n!S(k+1,n)\\left(\\frac{z}{4-z}\\right)^n\\left(\\frac{1}{n}-\\frac{1}{2}\\sum _{p=1}^{n-1}\\sum _{l=1}^p\\frac{(-1)^p(1/2)_p}{(p+1)!", "(1/2)_l}\\binom{n-1}{p}\\Gamma (l)\\left(\\frac{4}{z}\\right)^{p-l+1}\\right),\\nonumber $ $R_2(z,k)=\\sum _{n=1}^{k+1}n!S(k+1,n)\\sum _{p=0}^{n-1}\\frac{(-1)^p}{(p+1)!", "}\\binom{n-1}{p}\\left(\\frac{4}{z}\\right)^{p+1}.$" ], [ "Asymptotics", "It is convenient to work in terms of the exponential generating function $G(z,t):=\\sum _{k=0}^{\\infty }S_k(z)\\frac{t^k}{k!", "}=S_0(ze^t)=\\frac{z}{4-ze^t}+\\frac{4\\sqrt{z}e^{t/2}}{(4-ze^t)^{3/2}}\\sin ^{-1}\\frac{\\sqrt{z}e^{t/2}}{2}$ To find the generating functions $\\rho _j(z,t):=\\sum R_j(z,k)t^k/k!$ , it would be simplest to start with a series $D_k(z)=R_1(z,k)-R_2(z,k)\\sqrt{\\frac{z}{4-z}}\\sin ^{-1}\\sqrt{z}/2$ , work out its generating function $D(z,t)$ and by taking the sum and difference identify $\\rho _1$ and $\\rho _2$ .", "However, this series has not been found and there is nothing to guarantee its existence in tractable form.", "Therefore, the $\\rho _j$ were evaluated directly from (REF ) and (REF ).", "The details are omitted as the results $\\rho _1(z,t) & = & \\frac{ze^t}{4-ze^t}+ \\nonumber \\\\& & \\frac{8}{\\pi }\\sqrt{\\frac{ze^t}{(4-ze^t)^{3/2}}}\\left(\\sin ^{-1}\\frac{\\sqrt{z}e^{t/2}}{2}\\cos ^{-1}\\frac{\\sqrt{z}}{2}-\\cos ^{-1}\\frac{\\sqrt{z}e^{t/2}}{2}\\sin ^{-1}\\frac{\\sqrt{z}}{2}\\right),$ $\\rho _2(z,t)=4\\sqrt{\\frac{(4-z)e^t}{(4-ze^t)^3}}$ are easily verified.", "In the case $z=2$ , (REF ) and (REF ) are identical to Dyson's formulas [3], [5] obtained empirically.", "In view of the prominent role that the ratio $R_1(z,t)/R_2(z,t)$ plays in Dyson et al.", "[3] for $z=2$ it is interesting to examine it for general $z$ .", "From (REF ) we have $R_2(z,k)=\\frac{2k!\\sqrt{4-z}}{\\pi i}\\oint \\frac{ds}{s^{k+1}}\\frac{e^{s/2}}{(4-ze^s)^{3/2}}.$ The non-zero singularity closest to $s=0$ is $s_0=\\ln (4/z)$ and it dominates the asymptotic behavior.", "Ignoring the other singularities, distorting the contour to a small circle about $s_0$ and translating back to the origin by $t=s-s_0$ , we have $R_2(z,k)\\sim -\\frac{k!\\sqrt{4-z}}{zs_0^{k+1}}\\oint \\frac{dt}{2\\pi i}\\frac{e^{t/2}}{(1-e^t))^{3/2}}.$ The exact value of the integral in (REF ) is $-(2/\\pi )\\sqrt{e/(e-1)}$ , and so $R_2(z,k)\\sim \\frac{k!", "}{(\\ln (4/z))^{k+1}}\\frac{2}{\\pi }\\sqrt{\\frac{e(4-z)}{z(e-1)}}.$ In the same way we obtain $R_1(z,k)\\sim \\frac{k!", "}{(\\ln (4/z))^{k+1}}\\left(\\sqrt{2}+\\frac{2}{\\pi }\\left(\\sqrt{\\frac{}{}}{e}{e-1}-\\sqrt{2}\\right)\\cos ^{-1}\\frac{\\sqrt{z}}{2}-\\frac{2^{3/2}}{\\pi }\\sin ^{-1}\\frac{\\sqrt{z}}{2}\\right).$" ], [ "Discussion", "From (REF ) and (REF ) we find $\\lim _{k\\rightarrow \\infty }\\left(\\frac{R_1(z,k)}{R_2(z,k)}-\\sqrt{\\frac{z}{4-z}}\\sin ^{-1}\\frac{\\sqrt{z}}{2}\\right)=\\sqrt{\\frac{z}{4-z}}\\left(\\cos ^{-1}\\frac{\\sqrt{z}}{2}-\\sin ^{-1}\\frac{\\sqrt{z}}{2}\\right).$ It thus appears that Lehmer's choice, $z=2$ , is the unique permissible case for which the limit (REF ) vanishes.", "(Also the Lehmer limit, as defined by Dyson et al.", "[3], relates to $\\pi /4$ here rather than $\\pi $ ).", "Finally, for negative integer indices, since $2 S_{-k}(z)=\\;_{k+1}F_k(1,\\dots ,1;\\tfrac{3}{2},2,\\dots ,2;\\tfrac{1}{4}z),$ the fact that $S_{-k}(z)$ can be obtained from $S_0(z)$ by successive integrations with respect to $z$ and the explicit evaluations by Lehmer [2], Borwein and Girgensohn [4] and others [6], [7], [8], [9], [10] it should be possible to obtain explicit values for sundry generalized hypergeometric functions." ], [ " Appendix: Derivation of Equations (", "Let us consider, for any integrable function $F$ , $I=\\int _0^1\\frac{dt}{t}F(t(1-t))\\nonumber $ Let $u=t(1-t)$ , so $u(0)=u(1)=0;\\;\\; u(1/2)=1/4$ .", "Then there are two expressions for $t$ : $t_+=\\frac{1}{2}(1+\\sqrt{1-4t})\\quad \\mbox{ for }\\frac{1}{2}\\le t\\le 1, \\, \\text{ with }\\frac{dt_+}{t_+}=\\left(1-\\frac{1}{\\sqrt{1-4u}}\\right)\\frac{du}{u}\\nonumber $ and $t_-=\\frac{1}{2}(1-\\sqrt{1-4t})\\quad \\mbox{ for }0\\le t\\le \\frac{1}{2},\\, \\text{ with }\\frac{dt_+}{t_+}=\\left(1+\\frac{1}{\\sqrt{1-4u}}\\right)\\frac{du}{u}.\\nonumber $ Consequently, $I & = & \\int _0^{1/2}\\frac{dt_-}{t_-}F(u)+\\int _{1/2}^1\\frac{dt_+}{t_+}F(u)=2\\int _0^{1/4}\\frac{du}{u\\sqrt{1-4u}}F(u) \\nonumber \\\\& = &2\\int _0^1\\frac{dx}{x\\sqrt{1-x}}F \\left(\\frac{1}{4}x \\right)=2 \\int _0^1\\frac{dt}{(1-t)\\sqrt{t}}F\\left(\\frac{1-t}{4}\\right)\\nonumber $ and, with $t=x^2$ , $I=4\\int _0^1\\frac{dx}{1-x^2}F\\left(\\frac{1-x^2}{4}\\right).\\nonumber $ Therefore, $L=\\int _0^1\\frac{dt}{t}\\frac{(zt(1-t))^{\\alpha }}{(1-zt(1-t))^{\\beta }}=2\\left(\\frac{z}{4}\\right)^{\\alpha -\\beta }\\int _0^1dx\\frac{(1-x^2)^{\\alpha -1}}{(a^2+x^2)^{\\beta }},\\nonumber $ where $a^2=1/X=(4-z)/z$ .", "From standard references $\\int _0^1 dx \\cos (xy)(1-x^2)^{\\alpha -1}=\\sqrt{\\frac{\\pi y}{8}}\\left(\\frac{2}{y}\\right)^{\\alpha }\\Gamma (\\alpha )J_{\\alpha -1/2}(y),\\nonumber $ $\\int _0^{\\infty }dx \\cos (xy)(a^2+x^2)^{-\\beta }=\\frac{\\sqrt{\\pi }}{\\Gamma (\\beta )}\\left(\\frac{y}{2a}\\right)^{\\beta -1/2}K_{\\beta -1/2}(ay)\\nonumber $ so, by the Parseval relation for the Fourier transform $\\int _0^1dx\\frac{(1-x^2)^{\\alpha -1}}{(a^2+x^2)^{\\beta }}=\\frac{2^{\\alpha -\\beta }}{a^{\\beta -1/2}}\\frac{\\Gamma (\\alpha )}{\\Gamma (\\beta )}\\int _0^{\\infty }dy y^{\\beta -\\alpha }J_{\\alpha -1/2}(y)K_{\\beta -1/2}(ay).\\nonumber $ This is a tabulated Hankel Transform and yields $L=\\sqrt{\\pi }\\left(\\frac{4}{z}\\right)^{\\beta -\\alpha }X^{\\beta }\\frac{\\Gamma (\\alpha )}{\\Gamma (\\alpha +1)}\\;_2F_1 \\left(\\frac{1}{2},\\beta ;\\alpha +\\frac{1}{2};-X \\right).\\nonumber $ Consequently $\\int _0^1\\frac{dx}{x}\\frac{(zt(1-t))^n}{(1-zt(1-t))^{n+1}}=\\sqrt{\\pi }\\left(\\frac{4}{z}\\right)X^{n+1}\\;_2F_1 \\left(\\frac{1}{2},n+1;n+\\frac{1}{2};-X \\right)\\nonumber $ However, since $\\;_2F_1(a,b;c;z)=(1-z)^{c-a-b}\\;_2F_1(c-a,c-b;c;z)$ , $\\;_2F_1 \\left(\\tfrac{1}{2},n+1;n+\\tfrac{1}{2};-X \\right)=(1+X)^{-1}\\;_2F_1 \\left(-\\tfrac{1}{2};n;n+\\tfrac{1}{2};-X \\right)\\nonumber $ Next, we note that [11] $\\;_2F_1(-1/2,1;3/2;z)=\\frac{1}{2} \\left(1+(1-z)\\frac{\\tanh ^{-1}\\sqrt{z}}{\\sqrt{z}} \\right).\\nonumber $ With $z\\rightarrow -z$ , noting that $-i \\tanh ^{-1} iw=\\sin ^{-1}\\sqrt{\\frac{w}{1+w}}$ one has $\\;_2F_1(-1/2,1;3/2;-z)=\\frac{1}{2}(1+(1+z)\\frac{\\sin ^{-1}\\sqrt{\\frac{z}{1+z}}}{\\sqrt{z}}).$ We next apply Gauss' differentiation formula $\\frac{d}{dz}((1+z)^k\\;_2F_1(-1/2,k;k+1/2;-z))=\\nonumber $ $\\frac{2k(k+1)}{2k+1}(1+z)\\;_2F_1(-1/2,k+1;k+3/2;-z).$ Iteration of (REF ) starting with (REF ), after a great deal of tedious algebra, aided by Mathematica, results in (REF )." ], [ "Acknowledgements", "The author is grateful to Profs.", "N. E. Frankel for suggesting this problem and F. J. Dyson for enlightening correspondence.", "2000 Mathematics Subject Classification: Primary 11B65; Secondary 33B05.", "Keywords: Binomial coefficient, infinite series, generating function.", "(Concerned with sequences A008277, A145557 Received ; revised version received Return to Journal of Integer Sequences home pagehttp://www.cs.uwaterloo.ca/journals/JIS/." ] ]

[ [ "Hitting sbottom in natural SUSY" ], [ "Abstract We compare the experimental prospects of direct stop and sbottom pair production searches at the LHC.", "Such searches for stops are of great interest as they directly probe for states that are motivated by the SUSY solution to the hierarchy problem of the Higgs mass parameter - leading to a \"Natural\" SUSY spectrum.", "Noting that sbottom searches are less experimentally challenging and scale up in reach directly with the improvement on b-tagging algorithms, we discuss the interplay of small TeV scale custodial symmetry violation with sbottom direct pair production searches as a path to obtaining strong sub-TeV constraints on stops in a natural SUSY scenario.", "We argue that if a weak scale natural SUSY spectrum does not exist within the reach of LHC, then hopes for such a spectrum for large regions of parameter space should sbottom out.", "Conversely, the same arguments make clear that a discovery of such a spectrum is likely to proceed in a sbottom up manner." ], [ "Introduction.", "Supersymmetry (SUSY)See Refs.", "[1], [2] for reviews.", "has been the leading paradigm of physics Beyond the Standard Model (BSM) for the last three decades, and searching for SUSY is the leading BSM priority of the experimental particle physics community.", "Unfortunately, despite extensive and continued searches at generations of colliders and underground detectors, no direct evidence of weak scale SUSY particles have been found to date.", "LEP, and now the Large Hadron Collider (LHC) has pushed the direct production bounds on SUSY particles away from the natural expectation for their mass scale, proximate to the weak scale, with the bounds on SUSY colored particles now in the TeV range in large regions of parameter space.", "The interpretation of experimental results in missing transverse energy ($E_T \\!", "\\!", "\\!", "\\!", "\\!", "\\!", "\\!", "/$   ) searches in terms of the scale of SUSY particles is model dependent, but it is difficult (although clearly not impossibleSee Refs.", "[3], [4], [5], [6], [7], [8] for some discussion of SUSY scenarios consistent with the most recent experimental bounds.", "In order to reduce the experimental profile of a SUSY spectrum, to be compatible with the non observation of superparticles to date, a minimal effective SUSY spectrum which is composed only of the third generation scalar superpartners, gauginos and Higgsinos is generally entertained.", "See also Refs.", "[9], [10], [11].)", "to design models which compellingly explain the basic experimental result; namely, that SUSY partners have not been found, to date, residing at mass scales where they naturally should be.", "The question of whether a SUSY theory is natural, in that it avoids excessive fine tuning, is largely tied to the mass scale of third generation sfermion partners to the SM fermions with the largest couplings to the electroweak symmetry breaking (EWSB) sector.", "The couplings of the third generation sfermions transmit the soft SUSY breaking mass scale $\\rm M_{SUSY}$ to this sector, which leads to large perturbative corrections to Lagrangian parameters linked to the EWSB scale $v \\sim 246 \\, {\\rm GeV}$ without fine tuning.For a recent discussion on fine-tuning in traditional GUT-based SUSY models in light of the recent LHC results, see Ref. [12].", "The question of naturalness when considering natural SUSY sfermion spectra has largely focused on the bounds on the mass scale of stops to date, and in particular direct production bounds on stops.See however Ref.", "[6] for a rather comprehensive summary of the current experimental limits on stop and sbottoms.", "However, missing energy based signal isolation is not as powerful for such searches, as the stop typically decays into final states too similar to SM top production backgrounds which also has a missing energy component due to the decay $t \\rightarrow b \\, \\bar{\\nu } \\, \\ell $ .", "Overcoming SM backgrounds is a major challenge in stop searches, which also do not scale well with the invariant mass of the stop due to the complex decay topology, as we will discuss.See Ref.", "[13] for an attempt to tame the large SM background associated with the generically difficult stop signal.", "Indeed, whereas broad bounds on first and second generation squarks exist [14], [15], and on sbottoms up to $390 \\, {\\rm GeV}$ at $95 \\%$ CL for neutralino masses below $60 \\, {\\rm GeV}$ [16], generic searches of this form for directly produced stops decaying into $t \\, \\bar{t}$ +$E_T \\!", "\\!", "\\!", "\\!", "\\!", "\\!", "\\!", "/$    are not sensitive to the cross sections expected for stops to date.See Refs.", "[17], [18], [19], [20] for searches related to direct stop production.", "This is due to a combination of signal efficiency and $t \\, \\bar{t}$ background contamination constraints.", "However, one can relate sbottom searches to the stop sector, as the splitting of the left-handed components of stops and sbottoms are bounded by precision measurements of the $W$ boson's properties at LEP.", "Hence, as the sbottom searches become more sensitive, the bound on the stop sector becomes stronger through an interplay of direct sbottom limits and Electroweak Precision Data (EWPD).", "A relationship of this form between sbottom and stop exclusion regions also follows from the fact that soft SUSY masses are invariant under $\\rm SU_L(2)$ .", "In this way, sbottom direct search limits can strongly drive stop limits in a manner that can provide more experimental reach than direct stop searches alone, particularly in the regime of stop masses most of interest in natural SUSY.", "As a result, there is a tension between naturalness, which limits the stop scale from above, and sbottom searches, which restrict the stop sector from below, which might become acute in the 2012 data set.", "The paper is outlined as follows.", "In Section we discuss the strong theoretical link between the sbottom and stop masses due to theoretical consistency and EWPD, and also discuss how naturalness concerns influence the expected stop masses in combination with Higgs mass constraint.", "In Section we will discuss prospects for exclusion limits on direct pair production of sbottoms and stops.", "In Section we demonstrate the interplay of indirect constraints on stop masses and show how this leads to our conclusions." ], [ "Minimal Consistency Constraints.", "In this section we consider three sources of minimal consistency constraints on the stop sector before discussing the experimental prospects of stop and sbottom searches in the following section.", "The first constraint is associated with the splitting between stops and sbottoms and the contribution to the $\\Delta \\rho $ parameter, which restricts the amount of custodial symmetry – $\\rm SU_C(2)$ – violation.", "The second restriction comes from the paradigm of natural SUSY with minimal fine tuning, namely that the stop sector should be close to the electroweak symmetry breaking scale.", "Finally, one can draw further conclusions on the stop sector by looking at the Higgs mass in the minimal SUSY model (MSSM).", "This last requirement is more model dependent, as the Higgs may receive contributions to lift its mass in non minimal SUSY models - such as from an SM singlet as in the next to minimal SUSY model (NMSSM) [21], [22]." ], [ "$\\rm SU_C(2)$ Violation", "In a natural SUSY model, the squarks are characterized by the following mass matrix $\\mathcal {L}_{m_{\\tilde{f}}} = - \\frac{1}{2} \\left(\\tilde{f}^\\dagger _L , \\tilde{f}^\\dagger _R \\right) \\, {\\bf \\mathcal {Z}} \\left(\\begin{array}{c}\\tilde{f}_L \\\\\\tilde{f}_R \\\\\\end{array} \\right),$ where ${\\bf \\mathcal {Z}} = \\left(\\begin{array}{cc}\\cos ^2 \\theta _{\\tilde{f}} \\, m_{\\tilde{f}_1}^2 + \\sin ^2 \\theta _{\\tilde{f}} \\, m_{\\tilde{f}_2}^2 & \\quad \\, \\sin \\theta _{\\tilde{f}} \\cos \\theta _{\\tilde{f}} \\left(m_{\\tilde{f}_1}^2 - m_{\\tilde{f}_2}^2 \\right)\\\\\\sin \\theta _{\\tilde{f}} \\cos \\theta _{\\tilde{f}} \\left(m_{\\tilde{f}_1}^2 - m_{\\tilde{f}_2}^2 \\right) & \\quad \\sin ^2 \\theta _{\\tilde{f}} \\, m_{\\tilde{f}_1}^2 + \\cos ^2 \\theta _{\\tilde{f}} \\, m_{\\tilde{f}_2}^2 \\\\\\end{array} \\right).$ The cosine ($c_{\\tilde{f}}$ ) and sine ($s_{\\tilde{f}}$ ) of the squark mixing angles $\\theta _{\\tilde{f}}$ and the physical masses $m_{\\tilde{f}_1},m_{\\tilde{f}_2}$ are derivable from the initial soft SUSY breaking chiral squark masses in the Lagrangian corresponding to the fields $\\tilde{f}_{L/R}$ .", "We also define a mass difference for later convenience $\\delta m^2 = m_{\\tilde{t}_2}^2 - m_{\\tilde{t}_1}^2>0$ .", "The off-diagonal entry in the squark mass matrix is proportional to the corresponding fermion mass, as such we will neglect sbottom mixing, implicitly restricting ourselves to a moderate value of $\\tan \\beta $ regime in this paper.", "Explicitly, our convention is that $m_{\\tilde{f}_1}$ corresponds to the left handed sfermion in the limit of no mixing, and this is the limit we adopt for the sbottom states.", "This choice is conservative, as we will relate the experimental bound on the lightest sbottom to the stop sector.", "If the lightest sbottom experimentally bounded was purely right-handed, the bounds that we will discuss would actually be stronger, as the left handed sbottom would then be heavier, enforcing stronger (although unquantified) bounds on the stop sector.", "This is true so long as the spectrum is such that left handed sbottoms are not experimentally inaccessible compared to right handed sbottoms.", "We will quantify this condition on the sbottom branching ratio in what follows.", "In the MSSM, a relationship is enforced between sbottom and stop direct search bounds because soft SUSY masses are $\\rm SU_L(2)$ invariantThe mass relation is still true of the non-minimal SUSY models if there is no additional EWSB other than the MSSM Higgs sector.. At leading order, this results in the well known relation $m_{\\tilde{b}_1}^2 \\approx \\cos ^2 \\theta _{\\tilde{t}} \\, m_{\\tilde{t}_1}^2 + \\sin ^2 \\theta _{\\tilde{t}} \\, m_{\\tilde{t}_2}^2 - m_t^2 - m_W^2 \\, \\cos (2 \\, \\beta ).$ Here we have considered small sbottom mixing, $\\cos ^2 \\theta _{\\tilde{b}} \\sim 1$ .", "We will neglect perturbative corrections to this relationship.", "This relation does not tie the sbottom limit to the lightest stop ($\\tilde{t}_1$ ) necessarily, but only to the left-handed composition – $\\tilde{t}_L$ .", "In the worst case scenario, where the lightest stop would be purely right-handed $m_{\\tilde{t}_2}^2 >m_{\\tilde{b}, min}^2 +m_t^2$ , and there is no direct prediction on the lightest stop from this relation.", "In the next section we will show how naturalness in minimal models– a key motivation of natural SUSY – changes this picture, and can lead to stronger conclusions as it selects for a nearly degenerate stop spectrum.", "Besides this relation, there is a simple interplay in a minimal sfermion spectrum between EWPD and direct sbottom and stop production searches.", "Limits from EWPD quantifies the bounds on non SM interactions that modify the vacuum polarizations of the $W^\\pm , Z$ bosons, characterized by the $\\rm STU$ parameters [23], [24], [25].", "Fits to these parameters can be re-interpreted if the Higgs hints at $\\sim 125 \\, {\\rm GeV}$ are confirmed.", "With a Higgs mass fixed to this prior value, EWPD then gives a direct constraint (or direct measure) on $\\rm SU_C(2)$ breaking physics in a natural SUSY sfermion spectrum.", "For EWPD fits, we use the results of the ${\\it Gfitter}$ [26] $S = 0.02 \\pm 0.11, \\quad \\quad T = 0.05 \\pm 0.12, \\quad \\quad U =0.07 \\pm 0.12 \\;.$ We include a correction to $\\rm STU$ of the form $(\\Delta S, \\Delta T, \\Delta U)_{mh = 125} = (0.004,-0.003,-0.0001)$ due to shifting the best fit value of the Higgs mass in these fit results from 120 GeV to $125 \\, {\\rm GeV}$ using the one-loop Higgs boson contribution to $\\rm STU$ .", "The relevant quantity for constraining non SM $\\rm SU_C(2)$ violation is $(\\Delta \\rho _0)^{\\pm }_L &=& (\\rho _0)_{mh = 125} - 1, \\nonumber \\\\&=& \\hat{\\alpha }(m_z) \\, (T + (\\Delta T)_h), \\\\&=&(3.67 \\pm 8.82) \\times 10^{-4}.$ Here we have taken the PDG value $\\hat{\\alpha }(m_z) = 127.916 \\pm 0.015$ .", "The dominant one loop contribution to this quantity in natural SUSY spectra arises from the one loop scalar top and bottom contribution.", "Explicitly it is given by the following expression [28], [29], [30], [27] where we neglect terms proportional to small sbottom mixing angles $\\Delta \\rho _0^{SUSY} &\\approx \\frac{3\\, G_F \\, \\cos ^2 \\theta _{\\tilde{t}}}{8 \\, \\sqrt{2} \\, \\pi ^2} \\, \\left\\lbrace - \\sin ^2 \\theta _{\\tilde{t}} \\, F_0[m^2_{\\tilde{t}_1}, m^2_{\\tilde{t}_2}]+ F_0[m^2_{\\tilde{t}_1}, m^2_{\\tilde{b}_1}] +\\tan ^2 \\theta _{\\tilde{t}} \\, F_0[m^2_{\\tilde{t}_2}, m^2_{\\tilde{b}_1}] \\right\\rbrace .$ The function $F_0$ is defined as $F_0[x, y] = x + y - \\frac{2 \\, x \\, y}{x - y} \\, \\log \\frac{x}{y}.$ It is instructive to consider the constraint $\\Delta \\rho _0^{SUSY} \\lesssim (\\Delta \\rho _0)^+_L$ .", "We show this constraint in Fig.", "(1) when a lower bound on $m_{\\tilde{b}_1}$ is fixed to various values.", "Figure: The allowed (m t ˜ 1 ,δm)(m_{\\tilde{t}_1}, \\delta m) consistent with the Δρ\\Delta \\rho constraint and an imposed lower bound on m b ˜ 1 m_{\\tilde{b}_1}, for various stop mixing angles.", "The colour coding of the plotsis systematic in this section.The top left figure, in red, corresponds to the allowed regions considering these constraints for(tanβ,sinθ t )=(10,0.2)(\\tan \\beta , \\sin \\theta _t) = (10,0.2) and various m b ˜ 1 m_{\\tilde{b}_1} lower bounds, the top right figure shows a set of blue regions with (10,0.3)(10,0.3), the bottom left figure shows a set of green regions where (10,0.5)(10,0.5) and the bottom right plot shows allowed regions in brown where (10,1/2)(10,1/\\sqrt{2}) - this last case corresponding to maximal mixing.", "These plots are not sensitive to tanβ\\tan \\beta .Due to EWPD constraints and/or simply insisting on $\\rm SU_L(2)$ preserving soft masses, raising the direct exclusion limit of $m_{\\tilde{b}_1}$ indirectly yields an exclusion constraint in the space of $(m_{\\tilde{t}_1},m_{\\tilde{t}_2}, \\theta _{\\tilde{t}})$ .", "The constraints on this space are likely to be driven by sbottom searches for large regions of parameter space, as we will discuss.", "This is fortunate for efficiently raising stop mass limits and addressing the question of when a natural SUSY paradigm for particular parameters in the stop sector is experimentally ruled out.", "In this manner, the parameter space for such natural SUSY sfermion spectra can sbottom out experimentally.", "Conversely, these same arguments make clear that a sbottom up discovery of the third generation sfermion spectra is a scenario experimentally favoured in large regions of parameter space." ], [ "Natural SUSY and the stop/sbottom splitting", "A natural SUSY spectrum must also confront theoretical consistency in the form of fine tuning considerations.", "Although it is difficult to define a uniquely compelling fine tuning measure, or argue what degree of fine tuning is clearly unacceptable, a popular fine tuning measure is based on the required cancelation of the tree level and loop contributions to the $Z$ boson mass.", "An ominous level of fine tuning is widely considered to be $\\sim 1\\%$ .", "We use a fine tuning measure inspired by [31], [32].", "The $Z$ mass can receive contributions from many SUSY breaking sources, but unavoidably, the stop sector contributes to the $Z$ via the stop contributions to the up-type Higgs, $\\delta m_{H_u}$ .", "Therefore, when using this measure, we will consider the theory to be at least fine-tuned by the splitting between stops and $m_Z$ .", "This fine tuning measure is given by $\\Delta _Z > \\Delta _Z^t = \\left| \\frac{\\delta _t \\, m_Z^2}{m_Z^2} \\right|.$ Restricting ourselves to the moderate value of $\\tan \\beta $ and further assuming a degree of degeneracy in the mass spectrum of heavier squarks so that the impact of the stop loops remains largest, one finds [33], [34], [35], [36] at one loop $\\delta _t \\, m_Z^2 = \\frac{3}{16 \\, \\pi ^2} \\, \\left(y_t^2 (m_{\\tilde{t}_1}^2 + m_{\\tilde{t}_2}^2 - 2 m_t^2) + \\frac{(m_{\\tilde{t}_1}^2 - m_{\\tilde{t}_2}^2)^2}{4 \\, v^2 \\, \\sin ^2 \\beta } 4 \\, c_{\\tilde{t}}^2 \\, s_{\\tilde{t}}^2 \\right) \\, \\log \\left(\\frac{2 \\, \\Lambda ^2}{m_{\\tilde{t}_1}^2 + m_{\\tilde{t}_2}^2} \\right).", "$ Here $\\Lambda $ is the scale associated with new states required to cut off the logarithmic divergence resulting form the splitting of the stop and top masses and is associated with the messenger scale.", "For numerical purposes we conservatively consider the cut off scale to be taken to be a factor of 100 above the expected approximate geometric mean of the stop masses $ \\sim 1 \\, {\\rm TeV}$ .", "Note that the Eqn.", "(REF ) neglects $1/\\tan ^2 \\beta $ corrections, so that our analysis is essentially restricted to a range, $2 \\lesssim \\tan \\beta \\lesssim 20$ , when we impose this constraint.", "The upper limit follows from the assumed dominance of stop loops in Eqn.", "(REF ) and can be relaxed.", "In a natural SUSY spectrum, one also expects light Higgsinos as their mass is driven by the $\\mu $ parameter, and relatively light gluinos with a mass scale $\\lesssim 1 \\, {\\rm TeV}$ .", "This later expectation follows from the one loop correction that gluinos generate for stop masses, which contributes to Eqn.", "(REF ) at two loops.", "The contributions from these particles to this fine tuning measure are sub dominant and neglected.", "This is a conservative choice.", "Figure: Fine-tuning measure (in %\\%) for different bounds on sbottom particles.", "Left: general case.", "Right: maximal mixing case.", "Here tanβ=10\\tan \\beta =10 and Λ=100\\Lambda =100 TeV.In Fig.", "(REF ) we show the maximum value of $\\Delta _Z^t$ as a function of the sbottom limit, using the constraints from $\\Delta \\rho $ and relation Eq.", "(REF ).", "We varied the stop parameters in a completely general way, (i.e.", "we do not impose the MSSM Higgs mass constraint here) and also show how the finetuning scales with the sbottom bounds faster as we approach maximal mixing.", "If the sbottom limit is increased to 500 GeV, one already knows that the tuning of the theory is at least at the level of 5% according to this measure, whereas pushing the limits of sbottoms to $2 \\, {\\rm TeV}$ translates in an increase of the tuning to the 0.5% region.", "Our approach is not to invoke any particular UV model dictating a full SUSY spectrum.", "And we note that our parameter choices, such as the numerical values of $\\Lambda $ , are conservative.", "However, the argument we advance can be made more precise at the cost of more assumptions in the UV structure of the theory.", "For example, in minimal gauge mediation [37], soft masses can be generated at a low messenger scale around 100 TeV so that soft scalar masses and the gaugino masses are determined by $m^2_i=2N\\sum _a C_a (\\frac{\\alpha _a}{4\\pi })^2\\frac{F^2}{\\Lambda ^2}$ and $M_a=N \\frac{\\alpha _a}{4\\pi }\\frac{F}{\\Lambda }$ , with $F<\\Lambda ^2$ , respectively, where $N$ is twice the Dynkin index of the messenger fields, $C_a$ is the quadratic Casimir invariant of group $G_a$ , and $F$ is the SUSY breaking F-term.", "For perturbative unification of gauge couplings, $N\\le 5$ for $\\Lambda =100$ TeV.", "There is no one-loop trilinear messenger contribution to the A-terms but they are generated by the renormalization group evolution proportional to the gaugino masses.", "We note that in natural SUSY, one must invoke the SUSY breaking for the first two generation sfermions beyond minimal gauge mediation.", "A large splitting between stop/sbottom masses is possible for a large Bino gaugino mass.", "In this case, a small $\\mu $ term, which can be a consequence of natural SUSY, leads to a Higgsino-like MSSM lightest SUSY particle (LSP).", "On the other hand, if the stop/sbottom mass splitting is small, a Bino gaugino can be the LSP.", "We assume that the MSSM LSP is long-lived such that it decays outside the detector.", "This is the case when the gravitino mass is of sub keV [37].", "Then, as will be discussed later, we can apply the bounds from direct production of sbottoms with the analysis of missing transverse energy plus b-jets." ], [ "MSSM Higgs and the stop/sbottom ratio", "As noted by many authors a large Higgs mass consistent with current experimental hints is challenging to accommodate in a minimal SUSY scenario.", "Working in the decoupling limit and lifting the Higgs mass through one loop stop corrections, one has the relationship [33], [34], [35], [2] $m_h^2 &= m_Z^2 \\cos ^2 (2 \\, \\beta ) + \\frac{3}{4 \\, \\pi ^2} \\, \\sin ^2 \\, \\beta \\, y_t^2 \\, \\left[m_t^2 \\, \\log \\left(\\frac{m_{\\tilde{t}_1} \\, m_{\\tilde{t}_2}}{m_t^2} \\right) + c_{\\tilde{t}}^2 \\, s_{\\tilde{t}}^2 (m_{\\tilde{t}_2}^2 - m_{\\tilde{t}_1}^2) \\, \\log \\left(m_{\\tilde{t}_2}^2/m_{\\tilde{t}_1}^2\\right) \\right.", "\\\\&\\hspace{170.71652pt} \\left.", "{} \\hspace{-14.22636pt}+ \\frac{c_{\\tilde{t}}^4 \\, s_{\\tilde{t}}^4}{m_t^2} \\left((m_{\\tilde{t}_2}^2 - m_{\\tilde{t}_1}^2)^2 - \\frac{1}{2} (m_{\\tilde{t}_2}^4 - m_{\\tilde{t}_1}^4) \\, \\log \\left(m_{\\tilde{t}_2}^2/m_{\\tilde{t}_1}^2\\right)\\right) \\right].", "\\nonumber $ It is instructive to consider the interplay of imposing that the Higgs mass is lifted by stops and the fine tuning constraint.", "This illustrates the experimentally derived tension built into a natural SUSY sfermion spectrum.", "We plot this relation in Fig.", "(REF ) where we treat the stop masses and the stop mixing angle as free parameters.", "Figure: The figures illustrate the required stop mass and mass differences to obtain a 125 GeV 125 \\, {\\rm GeV} Higgs for various parameter values (left), the restriction of less than 1%1\\% fine tuning on the Z mass according to the defined measure with Λ=100\\Lambda = 100 TeV (middle) and in the right figure we show the interplay of the fine tuning, Higgs mass and sinθ t \\sin \\theta _t dependence.As in Fig.", "(1), the red solid line corresponds to(tanβ,sinθ t )=(10,0.2)(\\tan \\beta , \\sin \\theta _t) = (10,0.2),the blue dashed line is (10,0.3)(10,0.3), the green dot-dashed line is (10,0.5)(10,0.5) and the brown dotted line is (10,1/2)(10,1/\\sqrt{2}),the parameter space below the corresponding line in the middle plot has the defined fine tuning measure ≲1%\\lesssim 1 \\%.", "In the rightmost figure the black solid line is m h =125 GeV m_h = 125 \\, {\\rm GeV} using Eqn.", "() with μ=400 GeV ,m t ˜ 1 =400 GeV \\mu = 400\\,{\\rm GeV}, m_{\\tilde{t}_1} = 400 \\, {\\rm GeV}, below the black dashed line is the region of parameter space where the Colour and Charge preserving vacuum condition is satisfied, and the shaded regions are the 1%1\\% (lighter shaded region) an 5%5\\% fine tuning regions (darker shaded region).As $\\tan \\beta $ becomes smaller, the fine-tuning measure at given stop masses scales down mildly due to a smaller top Yukawa coupling.", "However, due to the Higgs mass constraint, the smaller $\\tan \\beta $ , the larger stop masses we need for the Higgs mass, in turn leading to a more fine-tuned situation.", "As $\\Lambda $ is taken to be larger, the fine tuning measure logarithmically scales to require less mass splitting.", "The parameter space most consistent with these constraints is the scenario where $\\tan \\beta $ takes a moderate value greater than about 5 and stops masses are nearly degenerate.", "These general considerations support the point that sbottom exclusions strongly drive stop exclusions though EWPD constraints and $\\rm SU_L(2)$ preserving SUSY soft masses quite generally; a split stop spectrum with $m_{\\tilde{t}_1} \\gg m_{\\tilde{t}_2}$ is disfavoured.", "Also note that considering the condition of a colour and charge preserving vacuum further constraints the parameter space consistent with a $\\sim 125 \\, {\\rm GeV}$ Higgs mass.", "Insisting on absolute stability when considering this constraint in the minimal MSSM, and attempting to minimize fine tuning to the level of $5 \\, \\%$ (as shown in Fig.", "(REF , right)), supports a maximal mixing scenario with small mass splitting amongst the stop states.We have checked the effects of two loop corrections in the case of degenerate spectra and such corrections do not significantly effect the argument for the sub TeV masses we are interested in excluding or discovering.", "One could avoid the constraints associated with raising the Higgs mass through a non-minimal scenario such as the NMSSM [38], [39], [40], [41], [42], [43].", "So long as the mechanism invoked to raise the Higgs mass does not violate the effective symmetries that are known to be present at the weak scale, any number of mechanisms can be invoked to raise the Higgs mass.", "However, even if one remains agnostic about the actual mechanism by which the Higgs mass is raised, one still has fine tunings issues that can be minimized by a somewhat degenerate stop spectra with large mixing, and the argument we will advance is still supported by these considerations.", "Searches for third generation squarks are a very active area in the SUSY groups at ATLAS and CMS.", "Several searches with 2011 data have been done, and we briefly review them in this section, with the aim of illustrating the relative ease of sbottom searches for sub TeV squark masses.", "Sbottom searches use a combination of jets, leptons and missing transverse energy plus b-jets.", "The searches are either gluino-assisted [44], [45] or based on direct production [16], [45].", "Generally speaking, searches for squarks depend strongly on the gluino mass.", "The limits from gluino assisted production are model dependent, and lose validity if the gluino becomes so heavy that pair production becomes negligible.", "For first and second generation squarks, and at moderate values of the gluino mass, t-channel exchange of a gluino is the dominant production mechanism.", "Hence, setting limits on a simplified model without gluinos, effectively decoupling this particle from the production, leads to weaker bounds than the light squark-gluino searches, see for example Ref.", "[46], [47].", "With the full 2011 dataset, the bound on the first two generation squarks, independently of the gluino mass is above 1 TeV However, one should keep in mind that the simplified model analysis for multijets and $E_T \\!", "\\!", "\\!", "\\!", "\\!", "\\!", "\\!", "/$   is summing over six degenerate squarks to impose a mass bound.. With third generation squarks, the t-channel gluino diagram is absent, or very suppressed by PDFs, but searches do still depend on the gluino, if the gluino pair production is significant, and gluinos decay to stops and sbottoms [44].", "Gluino masses are also limited by naturalness, due to their naturalness constraints on Eqn.", "(10) at two loops.", "The gluino is still expected to be in the mass range $m_{\\tilde{g}} \\sim 1 \\, {\\rm TeV}$ in a natural SUSY scenario.", "The overall production cross sections for sbottoms and stops are numerically very similar for typical natural SUSY third generation squarks.", "Although the hadro-production of these states are differentiated by the existence of the $b\\bar{b} \\rightarrow \\tilde{b} \\, \\tilde{b}$ process with a t-channel gluino exchange, for gluinos $\\gtrsim 1 \\, {\\rm TeV}$ and stop and sbottom masses $\\lesssim 1 \\, {\\rm TeV}$ , that are of interest in natural SUSY spectra, the impact of this extra process is $< 1\\, \\%$ of the overall rate [48].", "Direct production searches are the key ingredient to set a bound on stops and sbottoms which have such similar cross sections for coincident mass scales.", "Robust bounds from direct production of sbottoms can be obtained using triggers on final states with 2 b-jets and $E_T \\!", "\\!", "\\!", "\\!", "\\!", "\\!", "\\!", "/$  .", "A search of this form is interpreted in the context of a simplified model with two parameters (lightest sbottom and LSP masses), assuming ${\\rm BR}(\\tilde{b}_1\\rightarrow b$ LSP)=1.", "With these assumptions, at $2.05 fb^{-1}$ one can exclude sbottom masses up to 390 GeV with an LSP below 60 GeV.", "The search is most sensitive for $\\Delta m=m_{\\tilde{b}}-m_{LSP}> 130 $ GeV.", "See Ref.", "[16] for details.", "We show typical Feynman diagrams appropriate for searches of this form (when decays to a neutralino ($\\tilde{\\chi }^0$ ) are kinematically accessible) for the production and decay of stops and sbottoms in Fig.", "(REF ).", "Figure: Direct production and decay of b ˜b ˜\\tilde{b} \\, \\tilde{b} and t ˜t ˜\\tilde{t} \\, \\tilde{t}.The limit on the lightest sbottom depends on the $\\tilde{b}_L$ and $\\tilde{b}_R$ admixture in the mass eigenstate and on the nature of the LSP.", "If $\\tilde{b}_1$ has a large component of $\\tilde{b}_L$ , the sbottom would also decay to charginos ($\\tilde{\\chi }^\\pm $ ), hence reducing the BR to the LSP– provided there is a large mass gap between $\\tilde{\\chi }^\\pm $ and the LSP.", "In Fig.", "(REF ), we quantify how the limit on the sbottom mass depends on the BR to the LSP, assuming a bound on the sbottom mass of 400 GeV with LHC at 7 TeV.", "We kept the mass of the $\\tilde{\\chi }^0$ fixed to 60 GeV, and the separation $\\Delta m>$ 130 GeV.", "The current search is then sensitive to sbottom masses in the 200-400 GeV range, or ${\\rm BR}(\\tilde{b}_1\\rightarrow b$ LSP)$>$ 0.13.", "Figure: Expected limits on sbottom masses as a function of the χ ˜ ± \\tilde{\\chi }^\\pm branching ratio.Using this approach direct sbottom searches are rather inclusive and range over many SUSY scenarios, however direct stop searches using this approach are not sensitive to date to the expected stop production cross section [17], [18], [19], [20].", "The reason is essentially that the $t \\bar{t}$ +jets backgrounds are challenging, even with the additional handle on $E_T \\!", "\\!", "\\!", "\\!", "\\!", "\\!", "\\!", "/$   and the tagging efficiencies and signal isolate is not sensitive enough.", "The searches that have been performed that quote stop mass bounds are limited to date, with two kinds of searches available.", "One assumes gluino assisted production, hence leading to no bound on the stop mass unless a gluino mass measurement is achieved [44].", "The other stop search focuses on a very specific scenario within GMSB, with a stop decaying into a b-jet and $\\tilde{\\chi }^\\pm $ or, if kinematically allowed, into a top and $\\tilde{\\chi }^0$ .", "The $\\tilde{\\chi }^0$ is the NLSP, and decays to a Z and gravitino.", "The search is then based on a final state with two jets (where at least one is tagged as a b-jet), leptonic Z and $E_T \\!", "\\!", "\\!", "\\!", "\\!", "\\!", "\\!", "/$  .", "With these assumptions, one can impose bounds on the $(\\tilde{t}, \\tilde{\\chi }^0)$ mass parameter space.", "See Ref.", "[49] for details." ], [ "Reach on stop/sbottom masses with 2012 data", "Direct stop searches are also limited by combinatorics and decay topology.", "Multi-top final states are busy signatures, and new sources of missing energy just add to the complexity of the event.", "Moreover, at high stop mass, the top becomes boosted, and its decay products tend to merge.", "In W leptonic decays, isolation criteria would fail to keep the event, and in the W hadronic channel, the jets would tend to merge, tampering with reconstruction and also with b-tagging procedures.", "To illustrate this, in Fig.", "(REF ) we plot the minimum separation in $R$ space between two jets from the top decay, when the top comes from a stop of masses 300 to 800 GeV.", "Figure: Minimum separation in RR space between two jets of the top decay coming from a stop.The plots have been generated with Madgraph5 [50] for LHC at 8 TeV, with parton level cuts of 20 GeV for all the jets.", "The jets are parton level objects, where a smearing in energy and momentum has been applied, but no hadronization.", "The $\\Delta R$ we plot is then the separation of the partons: it is an optimistic view of the issue of merging, as hadronization and parton showering, plus clustering, would worsen the plot.", "Note that at $m_{\\tilde{t}}\\simeq $ 500 GeV, the merging becomes sizable.", "The issue of merging is addressed with boosted top techniques.", "Unfortunately, these techniques have a limited range of efficiencies, as a function of top $p_T$ .", "The efficiency reaches 45% for a range of $p_T>$ 600 GeV.", "Stops of mass 500 GeV would lead to 3% of the tops in that range, whereas for stops of mass 1 TeV, 30% of the tops would satisfy this bound on $p_T$  [51].", "Since with 2012 data due to PDF effects and limited statistics at large invariant mases, one could expect reaching production cross section limits on third generation for masses in the few hundreds of GeVs, an efficiency of few percent seems rather discouraging.", "On the other hand, sbottom searches based on b-jets+$E_T \\!", "\\!", "\\!", "\\!", "\\!", "\\!", "\\!", "/$    are more easy to scale to higher values of sbottom masses in the interesting range for natural SUSY theories.", "b-tagging efficiency is relatively stable with transverse momentum ($p_T$ ), ranging from 60-80% for $p_T$ in the range below 670 GeV [52].", "Commissioning for b-tagging for the 8 TeV run is not yet public, but we will assume the efficiencies would remain in that range.", "In Fig.", "(REF ) we plot the $p_T$ distribution of the b-jet coming from the sbottom decays when the neutralino mass is fixed to 100 GeV.", "In the range of $p_T$ shown here, we expect that the known b-tagging algorithms would be applicable.", "Figure: p T p_T distribution of the b-jet coming from the sbottom decays.", "The neutralino mass is fixed to 100 GeV.One can estimate the reach on sbottom masses in the search described in Ref.", "[16] with the 2011 full dataset, assuming there are no improvements in the efficiency so that the limit on the total cross section just scales with $\\sqrt{ {\\cal L}}$ .", "The result of this exercise leads to an increase from 390 GeV to 420 GeV.", "The running of 2012 at 8 TeV will further increase the sbottom bound, as that pair production cross section from 7 to 8 TeV increase by a factor 2(3) for a 500 GeV(1 TeV) sbottom.", "Assuming $20 \\, fb^{-1}$ of data per experiment, and estimating that the background fraction would not increase significantly from the 7 TeV run, one can forcast a reach of $\\lesssim 800 \\, {\\rm GeV}$ .", "An increase in efficiency of the analysis would probably be required to access sbottom masses above TeV with the 2012 run.", "To summarize the experimental situation we show in Table REF the efficiencies for basic cuts in stop and sbottom searches, for a range of masses up to 1 TeV.", "Table: Estimated efficiencies ϵ i \\epsilon _i for basic cuts in searches for sbottoms and stops, for LHC at 8 TeV.In the first column, we use the information in Ref.", "[52] on b-tagging.", "In this note from CMS, different b-tagging algorithms are compared in the 7 TeV run (no similar study is available for the 8 TeV run).", "Those algorithms achieve a 60-80% tagging efficiency (in $t\\bar{t}$ samples) with a mistag rate in the range of few percent.", "Only the region of b-jet $p_T< $ 670 GeV contains enough statistics to study the algorithms.", "One could worry that, for high sbottoms masses, the b-jet could be very boosted, beyond what is described in the CMS note.", "We take a conservative approach and ask for a cut on b-jet $p_T<$ 670 GeV, the reach of the CMS study.", "Even at 1 TeV, the cut only reduces $\\sim $ 20% of the signal.", "Assuming the commissioning of b-tagging at 8 TeV is as efficient as in the 7 TeV run, the issue of tagging b-jets coming from the decay of sbottoms up to TeV would have a stable efficiency, in the range 50-80% per b-tag.", "In the second column of Table.", "REF , we discuss the stop searches without boosted techniques.", "There is no public experimental study in this regime, and we cannot estimate the efficiency of the cuts required to reduce the backgrounds to acceptable levels.", "We quote a cut on lepton isolation which will be basic in all leptonic studies: top backgrounds are a major issue in these searches, and would require b-tagging and possibly leptonic decays, which require some isolation cut.", "Usual isolation cuts are $\\Delta R>0.7$ , a criteria which is harder to satisfy as the stop mass increases Note that a $p_T$ dependent cut could improve the situation [53], but to our knowledge there is no experimental study on the efficiency of varying the $\\Delta R$ cut based on the $p_T$ of the objects..", "Indeed, whereas for masses below 300 GeV, the cut is more than 50% efficient, for masses above 700 GeV, only 25% of the events would pass isolation requirements.", "Those numbers correspond to one of the tops decay products.", "Asking for both tops products passing the isolation cuts would correspond to an efficiency $\\epsilon _{\\Delta R}^2$ .", "Finally, in the last column we discuss stop searches using boosted techniques.", "Those are very promising for large stop masses [51].", "There are many proposals of top-tagging techniques, and we are going to focus on the Johns Hopkins algorithm [54] Note that for low top $p_T$ , the HEP algorithm [55] may be more efficient than the Johns Hopkins [54]..", "They reach an efficiency in the 40-50% level for tops with $p_T>$ 600 GeV.", "Below this value, the efficiency degrades very fast.", "We then quote what is the efficiency for one of the tops from the stop decays to pass this cut, for several values of stop masses.", "Again, asking for both tops in that range would lead to an efficiency of $\\epsilon _{p_T^t>600}^2$ .", "Note that for the range of masses 300-700 GeV, the efficiency is very low ($10^{-5}-10^{-3}$ for two top-tags), as compared with the b-tagging efficiencies from sbottom searches ($\\gtrsim 0.3$ for two b-tags).", "Top-tagging techniques could improve dramatically this year, but they should do so by orders of magnitude to reach the sensitivity of sbottoms searches at the same mass point.", "These considerations show that in the interesting mass range for stops in natural SUSY, $m_{\\tilde{t}_1}, m_{\\tilde{t}_2} \\lesssim 1 \\, {\\rm TeV}$ , the related bounds on sbottoms discussed are likely to be the most sensitive experimental probe for large regions of the $(m_{\\tilde{t}_1},m_{\\tilde{t}_2}, \\theta _{\\tilde{t}})$ space, and hopes for natural SUSY can hit sbottom in the 2012 run." ], [ "Natural MSSM Higgs and the stop/sbottom mass limits.", "In this section, we explore in more detail the interplay of the three sources of constraints on stops discussed in Sec.", ": $\\rm SU_C(2)$ violation, naturalness and a MSSM Higgs.", "$\\rm SU_C(2)$ violation bounds from $\\Delta \\rho $ already relates the stop and sbottom sectors, but accommodating a natural MSSM Higgs at $m_h \\sim 125 \\, {\\rm GeV}$ in the theory adds an even stronger correlation between the two sectors.", "We illustrate this point in Fig.", "(REF ), where we plot the bound on the lightest stop derived from a bound on the lightest sbottom.", "The blue line corresponds to imposing $\\rm SU_C(2)$ constraints, whereas the red line corresponds to adding the constraints of accommodating a natural Higgs in the MSSM.", "We chose the maximal stop mixing case, and a value of finetuning of 1%.", "The end-point of the red line corresponds to the situation where no solutions with less than 1% fine-tuning are obtained.", "Figure: Left:The sbottom bound versus the stop bound using restrictions from custodial violations (blue line) and a natural MSSM Higgs, with a finetuning at the level of 1%.", "Maximal mixing case.Right:The sbottom bound versus stop bound imposing constaints from violations of custodial symmetry for different stop mixing angles.One may wonder how those constraints vary with the stop mixing angle, as we know the custodial constraints are weakened when the lightest stop is purely right-handed.", "In Fig.", "(REF ), we show the effect of this variation, when constraints from violations of custodial symmetry are applied.", "If the lightest stop is purely right-handed, there is no correlation between a bound on sbottoms and the lightest stop.", "This bound would only be correlated with the heaviest stop.", "But if the lightest stop has any admixture of left-handed stop, improvements on the sbottom bounds lead to a push of the lightest stop mass.", "If we also imposed a constraint on naturalness, or the MSSM Higgs, even the case of the light right-handed stop becomes correlated with sbottom searches as we have discussed, as a nearly degenerate spectra is selected for.", "Note that mixing angles are not renormalization group invariant.", "Invoking particular mixing angles to disassociate the stop and sbottom sectors requires further tuning of parameters." ], [ "Conclusions", "The testing ground for natural SUSY is widely considered to be direct production in the stop sector, but direct access to this sector is extremely experimentally challenging.", "In this paper, we have exploited the minimal consistency constraints associated with experimentally motivated limits of custodial symmetry violation to link the stops– and the issue of fine-tuning– to the comparatively cleaner and more promising searches for sbottoms.", "For example, a sbottom bound of 500 GeV translates into a degree of fine-tuning in the theory of at least 5%, whereas setting a sbottom bound on the 1.5 TeV range, pushes the fine-tuning below the 1% level.", "Even if EWPD is ignored, a strong relationship between the mass scale of sbottoms and stops follows from only assuming that soft SUSY masses are $\\rm SU_L(2)$ invariant.", "These links between direct sbottom searches and the stop parameter space of interest in natural SUSY scenarios are made even stronger when an MSSM Higgs in the 125 GeV region is itself associated with the stop spectrum.", "Although unknown mixing angles mean that a mapping of the excluded sbottom space is related to a range of stop masses, the relationship between the sectors is strong enough that sbottom searches can be reasonable expected to largely drive the exclusion of the stop parameter space in the 2012 run.", "Hopes for a natural SUSY may thus sbottom out experimentally.", "Conversely, if weak scale natural SUSY reveals itself in the 2012 run, a sbottom up discovery of natural SUSY is favoured by these same arguments.", "“I'll speak in a monstrous little voice.\"", "(s?", ")bottom – A Midsummers Night's Dream Act I, Scene ii" ], [ "Acknowledgments", "MT thanks James Wells for a helpful conversation concerning radical naturalists.", "We thank G. Salam for helpful comments on top and bottom tagging." ] ]

[ [ "Moving Multi-Channel Systems in a Finite Volume with Application to\n Proton-Proton Fusion" ], [ "Abstract The spectrum of a system with multiple channels composed of two hadrons with nonzero total momentum is determined in a finite cubic volume with periodic boundary conditions using effective field theory methods.", "The results presented are accurate up to exponentially suppressed corrections in the volume due to the finite range of hadronic interactions.", "The formalism allows one to determine the phase shifts and mixing parameters of pipi-KK isosinglet coupled channels directly from Lattice Quantum Chromodynamics.", "We show that the extension to more than two channels is straightforward and present the result for three channels.", "From the energy quantization condition, the volume dependence of electroweak matrix elements of two-hadron processes is extracted.", "In the non-relativistic case, we pay close attention to processes that mix the 1S0-3S1 two-nucleon states, e.g.", "proton-proton fusion (pp -> d+ e^+ + nu_e), and show how to determine the transition amplitude of such processes directly from lattice QCD." ], [ "Introduction", "Scattering processes in hadronic physics provide useful information about the properties of particles and their interactions.", "Some of these processes are well investigated in experiments with reliable precision.", "However, there are interesting two-body hadronic processes whose experimental determinations continue to pose challenges.", "They mainly include two-body hadronic scatterings near or above the kinematic threshold with the possibility of the occurrence of resonances.", "Here we discuss two pertinent cases in Quantum Chromodynamics (QCD), the first of which is the scalar sector, whose nature is still puzzling (see for example Ref.", "[1] and references therein).", "While some phenomenological models suggest the scalar resonances to be tetraquark states (as first proposed by Jaffe [2]), others propose these to be weakly bound mesonic molecular states.", "The most famous of which are the flavorless $a_{0}(980)$ and $f_{0}(980)$ , that are considered to be candidates for a $K\\bar{K}$ molecular states [3], [4], [5].", "In order to shine a light on the nature of these resonances, it would be necessary to perform model-independent multi-channel calculations including the $\\lbrace \\pi \\pi ,\\pi \\pi \\pi \\pi , K\\bar{K}, \\eta \\eta \\rbrace $ scattering states directly from the underlying theory of QCD.", "In the baryonic sector, observations of the strong attractive nature of the isosinglet $\\bar{K}N$ scattering channel led to the postulation of kaon condensation in dense nuclear matter [6].", "However, extracting $\\bar{K}N$ scattering parameters is a rather challenging task, due to the presence of the $\\Sigma \\pi $ scattering channel and the $\\Lambda \\left(1405\\right)$ resonance below the $\\bar{K}N$ threshold, see for example Ref.", "[7].", "Previous chiral perturbation theory calculations have found inconsistency between experimental determination of the $\\bar{K}N$ scattering length from scattering data and kaonic hydrogen level shifts [8], [9], [10], [11], [12], but as with any low-energy effective field theory (EFT) calculation, there are unaccountable systematic errors associated with the large number of unknown low-energy coefficients (LECs) needed to perform accurate calculations of multi-channel processes.", "In addition to these strongly coupled scattering processes, there are weak processes involving multi-hadron states that require further investigation.", "For instance, Lattice QCD (LQCD) calculations have recently shown further evidence of $\\Xi ^{-}\\Xi ^{-}$ and $\\Lambda \\Lambda $ ($H$ -dibaryon) shallow bound states [13], [14], [15].", "This will certainly reignite experimental searches for evidences of these states.", "Among the possible weak decays of the $H$ -dibaryon include $H\\rightarrow (n\\Lambda ,n\\Sigma ^{0},p\\Sigma ^{-},nn)$ [16].", "In hyper-nuclear physics, there has been much interest in a definitive determination of the contribution of non-mesonic weak decays ($\\Lambda N\\rightarrow NN$ , $\\Lambda NN\\rightarrow NNN$ ) to the overall decay of hyper-nuclei.", "In particular, as discussed in Ref.", "[17] (and references therein), there was a long standing puzzle regarding the theoretical underestimation of the ratio of the decay widths $\\Gamma (\\Lambda n\\rightarrow nn)/\\Gamma (\\Lambda p\\rightarrow np)$ as compared to the experimental value.", "Certainly a great deal of progress has been made by meson-exchange models in order to close this gap, however a model-independent calculation directly from QCD would give further insight into the mechanism of these decays.", "These two cases illustrate processes where it is necessary to evaluate weak matrix elements between multi-hadronic states.", "Currently, LQCD provides the most reliable method for performing calculations of low-energy QCD observables.", "LQCD calculations are necessarily performed in a Eucledian and finite spacetime volume.", "Although the former forbids one to calculate the physical scattering amplitudes from their Euclidean counterparts away from the kinematic threshold due to the Maiani-Testa theorem [18], the latter is proven to be a useful tool in extracting the physical scattering quantities from lattice calculations.", "In his prominent work, Lüscher showed how one can obtain the infinite volume scattering phase shifts by calculating energy levels of interacting two-body systems in the finite volume [19], [20].", "The Lüscher method which was later generalized to the moving frames in Refs.", "[21], [22], [23], is only applicable to scattering processes below the inelastic threshold.", "Therefore it cannot be used near the inelastic threshold where new channels open up, and a generalized formalism has to be developed to deal with the coupled (multi)-channel processes.", "A direct calculation of the near threshold scattering quantities using LQCD can lead to the identification of resonances in QCD such as those discussed above, and provide reliable predictions for their masses and their decay widths.", "One such generalization was developed by Liu $et\\; al.$ in the context of quantum mechanical two-body scattering [24], [25].", "There, the authors have been able to deduce the relation between the infinite volume coupled-channel S-matrix elements and the energy shifts of the interacting particles in the finite volume by solving the coupled Schrodinger equation both in infinite volume and on a torus.", "The idea is that as long as the exponential volume corrections are sufficiently small, the polarization effects, as well as other field theory effects, are negligible.", "Therefore after replacing the non-relativistic (NR) dispersion relations with their relativistic counterparts, the quantum mechanical result of Liu $et\\; al.$ [24], [25] is speculated to be applicable to the massive field theory.", "In another approach, Lage $et\\; al.$ considered a two-channel Lippman-Schwinger equation in a NR effective field theory.", "They presented the mechanism for obtaining the $\\bar{K}N$ scattering length, and studying the nature of the $\\Lambda \\left(1405\\right)$ resonance from LQCD [26].", "Later on, Bernard $et\\; al.$ generalized this method to the relativistic EFT which would be applicable for coupled $KK-\\pi \\pi $ channels [1].", "Unitarized chiral perturbation theory provides another tool to study a variety of resonances in the coupled-channel scatterings.", "This method uses the Bethe-Salpeter equation for a coupled-channel system to dynamically generate the resonances in both light-meson sector and meson-baryon sector in the infinite volume, see for example Refs.", "[4], [27], [28], [29].", "When applied in the finite volume, the volume-dependent discrete energy spectrum can be produced, and by fitting the parameters of the chiral potential to the measured energy spectrum on the lattice, the resonances can be located by solving the scattering equations in the infinite volume.", "This method has been recently used to study the resonances $f_0(600)$ , $f_0(980)$ and $a_0(980)$ in Refs.", "[30], [31], $\\Lambda (1405)$ in Ref.", "[32], $a_1(1260)$ in Ref.", "[33], $\\Lambda _c(2595)$ in Ref.", "[34], and $D^*_{s 0}(2317)$ in Ref.", "[35] in the finite volume.", "One should note that in contrast with the single-channel scattering system, coupled-channel scattering requires determination of three independent scattering parameters which would require in the very least three measurements of the energy levels in the finite volume.", "As proposed in Refs.", "[1], [30], one can impose twisted boundary conditions in the lattice calculation to be able to increase the number of measurements by varying the twist angle and further constrain the scattering parameters.", "Another tool to circumvent this problem is the use of asymmetric lattices as is investigated in Refs.", "[1], [30], [32].", "Alternatively, one can perform calculations with different boost momenta [32], [33].", "The goal of this paper, as discussed in the following paragraphs, is two-fold.", "First, in section we present a model-independent fully relativistic framework for determining the finite volume (FV) coupled-channel spectrum in a moving frame.", "Secondly, in section we show how to extract the matrix element of the current operator between two-hadron states directly from from a FV calculation of the matrix elements, for both relativistic and NR processes.", "These two are small stepping-stones towards one of the overarching goals of hadronic physics, which is to determine properties of multi-hadron systems directly from the underlying theory of QCD.", "This paper is structured as follows.", "In the first section we present the result for a scalar field theory model, which illustrates all the features of the problem at hand and allows us to derive the quantization condition (QC) for energy eigenvalues using a diagrammatic expansion.", "Although, in deriving the QC for this toy model we make a series of approximations, it is shown that this result is in perfect agreement with the exact QC which is obtained from the generalization of the work by Kim $et\\; al.$ [22], [36].", "In section REF , we derive the general form of the FV quantization condition for $N$ arbitrarily strongly coupled two-body states.", "This result has been independently derived and confirmed in a parallel work by Hansen and Sharpe [37], [38].", "In this paper, most of the emphasis will be placed on the N=2 case, but the result for the N=3 case will be also explicitly shown.", "After developing the FV coupled-channel formalism, we extend our work to be able to determine electroweak matrix elements in the two-hadron sector.", "The formalism for extracting the physical transition amplitude for $K\\rightarrow \\pi \\pi $ from the FV matrix elements of the weak Hamiltonian in the finite volume, has been developed by Lellouch and Lüscher [39].", "This former quantity has been shown to be proportional to the latter at the leading order in the weak coupling, and the proportionality factor (the LL factor), is shown to be related to the derivative of the $\\pi \\pi $ scattering phase shift.", "The generalization of the LL factor to the moving frame is given in Refs.", "[22], [36], [23].", "Ultimately, it would be desirable to calculate electroweak matrix elements of states containing an arbitrary number of hadrons, but for the time being we restrict ourselves to the two-body sector.", "Furthermore, we pay close attention to processes involving two nucleons.", "As is discussed in Ref.", "[40], the weak disintegration of deuteron in processes such as $\\nu d\\rightarrow \\nu pn$ and $\\nu _{e}d\\rightarrow nne^{+}$ is of great importance in some neutrino experiments.", "These processes entail weak mixing between the $^{1}S_{0}- {^{3}}S_{1}$ two-nucleon channels, and for energies below the pion-production threshold, the dynamics of this system can be described using a pionless EFT (EFT$\\left(\\lnot \\hspace{-2.27626pt}{\\pi }\\right)$ ) [41], [42], [43].", "The result presented gives a relation between the FV weak matrix element and the infinite volume LECs theat parametrize the one-body and two-body weak axial-vector currents.", "In deriving the result it has been assumed that the FV two-nucleon ground states predominantly described by NR S-wave phase shifts, and we have also used degenerate perturbation theory in the derivation.", "Both of these are reasonable approximations that will require further investigation to correctly access the size of the corrections." ], [ "Meson-Meson Coupled Channels ", "The goal of this section is to present the quantization condition for N coupled-channel system in a moving frame.", "We present two independent ways to obtain the quantization condition desired.", "In section REF , we present a toy model that illustrates the features of the problem at hand and allows us to determine the scattering matrix using a diagrammatic expansion.", "In order to derive the quantization condition using this approach, it is convenient to consider the case where the parameter responsible for mixing two channels is small.", "However, as shown in section REF , the result presented is in fact exact for an arbitrary mixing parameter.", "In section REF , we show how to obtain the quantization condition for N arbitrarily strongly coupled two-body channels in a moving frame by solving the Bethe-Salpeter equation.", "This case is the most relevant if we want to consider systems coupled via QCD interactions, e.g.", "the $I=0$ system $\\pi \\pi \\rightarrow (K\\bar{K},\\eta \\eta )\\rightarrow \\pi \\pi $ .It is important to mention that we are not making any claims about the relevance of the four-pion channel in the light-scalar sector of QCD.", "At this point, it is not evident how to incorporate such states into the calculation, therefore if one would choose to use the formalism presented here to study the light-scalar sector of QCD, there will be an overall systematic error that must be accounted for through other means." ], [ "Toy problem: two weakly coupled two-meson channels in the moving frame ", "In this section, we consider a two-meson coupled system with total angular momentum equal to zero.", "The two channels will be labeled $I$ and $II$ .", "In general the four mesons can have different masses and quantum numbers, but for the time being we restrict ourselves to the case where two mesons in each channel are identical.", "Using the “barred\" parameterization [44], the time-reversal invariant S-matrix describing this system can be written as $S_2=\\begin{pmatrix}e^{i2\\delta _I}\\cos {2\\overline{\\epsilon }}&ie^{i(\\delta _I+\\delta _{II})}\\sin {2\\overline{\\epsilon }}\\\\ie^{i(\\delta _I+\\delta _{II})}\\sin {2\\overline{\\epsilon }}&e^{i2\\delta _{II}}\\cos {2\\overline{\\epsilon }} \\\\\\end{pmatrix},$ where $\\delta _{I}$ and $\\delta _{II}$ are the phase shifts corresponding to the scattering in channels $I$ and $II$ respectively, and $\\bar{\\epsilon }$ is a parameter which characterizes the mixing between the channels.", "The subscript 2 on $S$ denotes the number of coupled channels.", "At energies below the four-meson threshold, the dynamics of such system can be described by a simple scalar effective field theory (EFT) $\\mathcal {L}=\\sum _{i={I,II}} \\phi _{i}^\\dag (\\partial ^2-m_i^2)\\phi _{i}-\\left(\\frac{\\phi _I\\phi _I}{2} \\hspace{8.5359pt}\\frac{\\phi _{II}\\phi _{II}}{2}\\right)^\\dag \\begin{pmatrix}c_I&{g}/{2}\\\\{g}/{2}&c_{II} \\\\\\end{pmatrix}\\begin{pmatrix}{\\phi _{I}\\phi _{I}}/{2}\\\\{\\phi _{II}\\phi _{II}}/{2} \\\\\\end{pmatrix}+\\cdots ,$ where $\\phi _i$ is the meson annihilation operator for the $i^{th}$ channel, and $c_i$ and g are the LECs of the theory.", "Ellipsis denotes higher derivative four-meson terms which will be neglected in this section.", "Our goal is to determine the coupled-channel spectrum in a finite cubic volume with the periodic boundary conditions (PBCs) that falls within the p-regime of LQCD [45], [46], defined by $\\frac{m_\\pi L}{2\\pi }\\gg 1$ where $m_\\pi $ is the pion mass, and $L$ is the spatial extent of the volume.", "In this regime, FV corrections to the single-particle dressed propagator are exponentially suppressed [19], as are FV contributions from t- and u-channel scattering diagrams [19], [20], [39].", "The leading order (LO) volume effects in the two-body sector arise from the presence of poles in the s-channel scattering diagrams.", "These lead to power-law volume corrections to the two-particle spectrum [19], [20].", "Therefore in the following discussion we will restrict ourselves to the contribution of such diagrams to the scattering matrix $\\mathcal {M}$ , whose $\\mathcal {M}_{ij}$ matrix element corresponds to the scattering amplitude from the $i^{th}$ channel to the $j^{th}$ channel.", "Certainly, the t (u)-channel diagrams contribute to the renormalization of the theory and therefore to the definition of the LECs, but for energies below the four-particle threshold, their FV effects are exponentially suppressed.", "It is convenient to redefine the LECs to absorb the contributions from all diagrams with exponentially suppressed FV effects, such as those arising from the t- and u-channel diagrams.", "When considering momentum-independent interactions, the infinite volume loops contributing to the s-channel diagrams are $G^{\\infty }_{i}\\equiv \\frac{i}{2}\\int \\frac{d^4k}{(2\\pi )^4}\\frac{1}{\\left[(k-P)^2-m_i^2+i\\epsilon \\right][k^2-m_i^2+i\\epsilon ]},$ where $P=(E,\\textbf {P})$ is the total four-momentum of the system.", "Figure: Shown are the diagrammatic representations of the full scattering amplitudes.", "Solid (dashed) lines represent single particle propagators of the II (IIII) channel.", "The first two diagrams require the LECs to reproduce the scattering amplitude for the II channel.", "The grey circle (diamond) is the full scattering amplitude of the II (IIII) channel.", "The black dot denotes the effective coupling for the II channel, which includes an infinite series of intermediate IIII bubble diagrams.", "There are two other diagrams for the IIII channel that require the LECs to recover ℳ II,II \\mathcal {M}_{II,II} and have not been shown.", "The third diagram ensures that the mixing term gg reproduces the off-diagonal scattering matrix elements.It is straightforward to write down the equations that $\\lbrace c_I,c_{II},g\\rbrace $ must simultaneously satisfy in order to reproduce the $\\mathcal {M}$ -matrix elements of the theory, similar to the approach of Ref.", "[47] in examining the role of $\\Delta \\Delta $ intermediate states in the ${^1}\\hspace{-1.70709pt}{S_0}$ (nucleon-nucleon) NN scattering.", "These equations, which are diagrammatically shown in Fig.", "(REF ), can be written as follows $c^{*\\infty }_{i} &=&c_{i}+g^2\\sum _{j\\ne i}\\frac{G^{\\infty }_{j}}{1-c^{*\\infty }_{j}G^{\\infty }_{j}}=-\\frac{\\mathcal {M}_{i,i}}{1-\\mathcal {M}_{i,i}G^{\\infty }_{i}},\\\\g=&-&|\\mathcal {M}_{I,II}|e^{i(\\delta _I+\\delta _{II})} {({1-c^{*\\infty }_{I}G^{\\infty }_{I}})({1-c^{*\\infty }_{II}G^{\\infty }_{II}})},$ where $\\mathcal {M}_{i,i}$ is the full relativistic S-wave scattering amplitude for the $i^{th}$ channel, and $\\mathcal {M}_{I,II}$ is the amplitude describing the mixing between the two channels.", "$c^{*\\infty }_{i(j)}$ is an effective coupling for the $i (j)$ channel, which includes an infinite series of intermediate $j (i)$ bubble diagrams as shown in Fig.", "(REF ).", "Solving these coupled equations leads to a renormalized theory and determines the scale-dependence of the LECs.", "Once this is done, one can study physics in a finite volume.", "In particular, we are interested in the energy eigenvalues of the meson-meson system placed in a finite volume with the PBCs.", "The spectrum can be determined by requiring the real part of the inverse of the FV scattering amplitude to vanish.One should note that using the notion of FV scattering amplitude is merely for the mathematical convenience.", "As there is no asymptotic state by which one could define the scattering amplitude in a finite volume, one should in principle look at the pole locations of the two-body correlation function.", "However, one can easily show that both correlation function and the so-called FV scattering amplitude have the same pole structure, so we use the latter for the sake of simpler representation.", "In a periodic volume, the integrals over the spatial momenta appearing in Feynman diagrams are replaced by a sum over discretized three-momenta.", "In finite volume the integral in Eq.", "(REF ) is replaced by $G^{V}_{i}\\equiv \\frac{i}{2L^3}\\sum _{\\mathbf {k}}\\int \\frac{dk^0}{2\\pi }\\frac{1}{\\left[(k-P)^2-m_i^2+i\\epsilon \\right][k^2-m_i^2+i\\epsilon ]},$ where the spatial momenta are quantized due to the PBCs $\\mathbf {k}=2\\pi \\mathbf {n}/L$ for $\\textbf {n}\\in Z^3$ , while the temporal extent of the Minkowski space remains infinite.", "This sum suffers from the same UV divergence of Eq.", "(REF ), and the difference of the two, $\\delta {G^{V}_{i}}\\equiv {G^{V}_{i}}-{G^{\\infty }_{i}}$ , is finite.", "It is simplest to consider the case where the mixing coupling, $g$ is small, and keep our expressions to leading order in $g$ .", "Using this and the definitions of the LECs in Eq.", "(REF ), the FV scattering amplitude of the channel $I$ can be written as $(\\mathcal {M}_{I,I})_V\\approx \\frac{i\\mathcal {M}_{I,I}}{1+\\mathcal {M}_{I,I}\\delta G^{V}_{I}+\\delta G^{V}_{II}\\frac{|\\mathcal {M}_{I,II}|^2 e^{i2(\\delta _I+\\delta _{II})}}{\\mathcal {M}_{I,I}(1+\\mathcal {M}_{II,II}\\delta G^{V}_{II})}}.$ Finally, we obtain the quantization condition for the coupled-channel problem at LO in the mixing parameter $\\mathcal {R}e\\left\\lbrace |\\mathcal {M}_{I,II}|^2 e^{i2(\\delta _I+\\delta _{II})}-\\left(\\mathcal {M}_{I,I}+\\frac{1}{\\delta G^{V}_{I}}\\right)\\left(\\mathcal {M}_{II,II}+\\frac{1}{\\delta G^{V}_{II}}\\right)\\right\\rbrace =0,$ which is equivalent to the result of Ref.", "[37], [38].Note that the FV loop function for channel $i$ , $F_{i}$ , defined in Eq.", "(24) of Ref.", "[37], [38] is equal to $\\left(-i\\right)$ times the FV loop function $\\delta G_{i}^{V}$ as defined in Eq.", "(REF ) below.", "At this point we have refrained from using explicit expressions for the FV integrals and scattering matrix elements; these details will be presented in the following section.", "It is important to note that despite the simplicity of this toy model, it illustrates all the features of the problem under the investigation, and as will be shown in the next section, the result for the weakly coupled channels presented above is the same to the strongly coupled case when only the S-wave contribution to the two-body scattering in the cubic lattice is taken into account." ], [ "N strongly coupled two-body channels in a moving frame ", "In the previous section, the following assumptions have been made.", "First of all, the higher order four-meson terms in the derivative expansion of the effective Lagrangian have been neglected.", "Secondly, the scattering is restricted to two coupled channels composed of identical particles, and only the S-wave scattering is considered.", "Most importantly, the mixing term was assumed to be small in deriving the quantization condition.", "As mentioned earlier, the latter condition is the most relevant when considering systems coupled via QCD interactions.", "In this section we will simultaneously remove all of this assumptions.", "Most of the details associated with generalizing to a moving frame have been developed by Kim $et\\; al.$ [22], [36], which will be briefly reviewed here for completeness (see also Refs.", "[21], [23] for alternative derivations).", "A system with total energy and momentum, $E$ and $\\mathbf {P}$ , in the laboratory frame has a CM energy $E^*=\\sqrt{E^2-\\mathbf {P}^2}$ .", "For the $i^{th}$ channel with two mesons each having masses $m_{i,1}$ and $m_{i,2}$ , the CM relative momentum is $q^{*2}_i=\\frac{1}{4}\\left(E^{*2}-2(m_{i,1}^2+m_{i,2}^2)+\\frac{(m_{i,1}^2-m_{i,2}^2)^2}{E^{*2}}\\right),$ which simplifies to $\\frac{E^{*2}}{4}-m_{i}^2$ when $m_{i,1}=m_{i,2}=m_{i}$ .", "Because the $S$ -matrix for the $l^{th}$ partial wave is a N-dimensional matrix, the scattering amplitude is also necessarily a N-dimensional matrix.", "In order to have a fully relativistic result that holds for all possible energies below four-particle threshold, the scattering amplitude must include all possible diagrams, i.e.", "contributions from s-, t- and u-channels as well as self-energy corrections.", "Fig.", "REF depicts the FV analogue of the scattering amplitude, $\\mathcal {M}^V$ , for the special case of $N=2$ channels.", "This amplitude is written in a self-consistent way in terms of the Bethe-Salpeter kernel, $\\mathcal {K}$ , which is the sum of all s-channel two-particle irreducible diagrams, Fig.", "REF .", "For energies below the four-particle threshold, the intermediate particles in the kernel and the self-energy diagrams, Fig.", "REF , cannot go on-shell, and therefore these are exponentially close to their infinite-volume counterparts.", "In fact, only in the s-channel diagrams can all intermediate particles be simultaneously put on-shell.", "Figure: a) The fully-dressed FV two-particle propagator, ℳ V \\mathcal {M}^V can be written in a self-consistent way in terms of the Bethe-Salpeter Kernel, 𝒦\\mathcal {K} and the FV s-channel bubble 𝒢 V \\mathcal {G}^V.", "Note that the only difference between the diagrammatic expansion of the scattering amplitude in the coupled-channel case and the single-channel case is that now the amplitude, kernels and two-particle propagators should be promoted to matrices in the basis of the channels that are kinematically allowed.", "b) Shown is the 𝒦 I,I \\mathcal {K}_{I,I}-component of the kernel, which sums all s-channel two-particle irreducible diagrams for channel II.", "c) The fully dressed one-particle propagator is the sum of all one-particle irreducible diagrams and is denoted by a black dot on the propagator lines.Having upgraded the kernel and the two-particle propagators to matrices in the space of open channels, it is straightforward to obtain a non-perturbative quantization condition for the energy levels of the system.", "It is important to note that the channels only mix by off-diagonal terms in the kernel, which implies that in the absence of interactions a two-pion state continues to propagate as a two-pion state.", "With this in mind, in the presence of momentum-dependent vertices Eq.", "(REF ) is replaced by $\\left[iG^{V}(\\mathbf {p}_a,\\mathbf {p}_b)\\right]_{ab}&\\equiv &\\frac{n_i}{L^3}\\sum _{\\mathbf {k}}\\int \\frac{dk^0}{2\\pi }\\frac{[\\mathcal {K}(\\mathbf {p}_a,\\mathbf {k})]_{ai}~[\\mathcal {K}(\\mathbf {k},\\mathbf {p}_b)]_{ib}}{[(k-P)^2-m_{i,1}^2+i\\epsilon ][k^2-m_{i,2}^2+i\\epsilon ]},$ where the subscripts $a, i, b$ denote the initial, intermediate and final states, respectively, and $n_i$ is ${1}/{2}$ if the particles in the $i^{th}$ loop are identical and 1 otherwise.", "The sum over all intermediate states, and therefore index $i$ is assumed.", "Since the FV corrections arise from the pole structure of the intermediate two-particle propagator, one would expect that the difference between this loop and the its infinite volume counterpart should depend on the on-shell momentum.", "The on-shell condition fixes the magnitude of the momentum running through the kernels but not its direction.", "Therefore it is convenient to decompose the product of the kernels into spherical harmonics.", "These depend not only on the directionality of the intermediate momentum but also on those of the incoming and outgoing momenta.", "Also, one may represent the N two-body propagators as a diagonal matrix $\\mathcal {G}=diag(\\mathcal {G}_{1},\\mathcal {G}_{2},\\cdots , \\mathcal {G}_{N})$ as depicted in Fig.", "REF .", "These are infinite-dimensional matrices with with matrix elements [22], [36] $(\\delta G^{V}_i)_{l,m;l^{\\prime },m^{\\prime }}\\equiv (G^{V}_i-G^{\\infty }_i)_{l,m;l^{\\prime },m^{\\prime }}=-i\\left(\\mathcal {K} \\delta \\mathcal {G}^{V}_i \\mathcal {K}\\right)_{l,m;l^{\\prime },m^{\\prime }},$ where $(\\delta \\mathcal {G}^{V}_{i})_{l_1,m_1;l_2,m_2}&=&i \\frac{q^*_in_i}{8\\pi E^*}\\left(\\delta _{l_1,l_2}\\delta _{m_1,m_2}+i\\frac{4\\pi }{q_i^*}\\sum _{l,m}\\frac{\\sqrt{4\\pi }}{q_i^{*l}}c^{\\textbf {P}}_{lm}(q_i^{*2})\\int d\\Omega ^*Y^*_{l_1m_1}Y^*_{lm}Y_{l_2m_2} \\right),\\nonumber \\\\$ and the function $c^{\\textbf {P}}_{lm}$ is defined asNote that our definition of the $c_{lm}^{\\textbf {P}}$ function differs that of Ref.", "[22] by an overall sign.", "$c^{\\textbf {P}}_{lm}(x)=\\frac{1}{\\gamma }\\left[\\frac{1}{ L^3}\\sum _{\\textbf {k}}-\\mathcal {P}\\int \\frac{d^3\\mathbf {k}}{(2\\pi )^3}\\right]\\frac{\\sqrt{4\\pi }Y_{lm}(\\hat{k}^*)~k^{*l}}{{k}^{*2}-x} \\ .$ $\\mathcal {P}$ in this relation denotes the principal value of the integral, and $\\mathbf {k}^*={\\gamma }^{-1}(\\mathbf {k}_{||}-\\alpha \\mathbf {P})+\\mathbf {k}_{\\perp }$ , where $\\mathbf {k}_{||}$ ($\\mathbf {k}_{\\perp }$ ) denotes the component of momentum vector $\\mathbf {k}$ that is parallel (perpendicular) to the boost vector $\\mathbf {P}$ , $\\alpha =\\frac{1}{2}\\left[1+\\frac{m_1^2-m_2^2}{E^{*2}}\\right]$ , $E^*$ is the CM energy of the system, $E^*=\\sqrt{q^*+m_1^2}+\\sqrt{q^*+m_2^2}$ , and the relativistic $\\gamma $ factor is defined by $\\gamma =E/E^*$  [48], [49], [50].", "This reduces to the NR value of $\\alpha =\\frac{m_1}{m_1+m_2}$ as is presented in Ref. [51].", "Note that this result is equivalent to the NR limit of the result obtained in Refs.", "[21], [22], [23] for the boosted systems of particles with identical masses.The kinematic function $c^{\\textbf {P}}_{lm}(q_i^{*2})$ can also be written in terms of the three-dimensional Zeta function, $\\mathcal {Z}^d_{lm}$ , $\\nonumber c^{\\textbf {P}}_{lm}(q^{*2})=\\frac{\\sqrt{4\\pi }}{\\gamma L^3}\\left(\\frac{2\\pi }{L}\\right)^{l-2}\\mathcal {Z}^d_{lm}[1;(q^*L/2\\pi )^2],\\hspace{28.45274pt}\\mathcal {Z}^d_{lm}[s;x^2]=\\sum _{\\mathbf {r} \\in P_d}\\frac{Y_{l,m}(\\mathbf {r})}{(r^2-x^2)^s},$ where the sum is performed over $P_d=\\left\\lbrace \\mathbf {r}\\in \\textbf {R}^3\\hspace{2.84544pt} | \\hspace{2.84544pt}\\mathbf {r}={\\gamma }^{-1}(\\mathbf {m}_{||}-\\alpha \\mathbf {d})+\\mathbf {m}_{\\perp } \\text{,}m\\in \\textbf {Z}\\right\\rbrace $ , $\\mathbf {d}$ is the normalized boost vector $\\mathbf {d}=\\mathbf {P}L/2\\pi $ , and $\\alpha $ is defined above [48], [49], [50].", "The generalization of the quantization conditions for N channels that are coupled via an arbitrarily strong interaction is now straightforward given such upgrading of the kernel, $\\mathcal {K}$ , to not just be a matrix over angular momentum but also over N channels as discussed before.", "The kernel is assured to reproduce the infinite volume scattering matrix ($\\mathcal {M}$ ) by solving the following matrix equation $i\\mathcal {M}&=&-i\\mathcal {K}-i\\mathcal {K}\\mathcal {G^{\\infty }} \\mathcal {K}-i\\mathcal {K}\\mathcal {G^{\\infty }} \\mathcal {K}\\mathcal {G^{\\infty }}\\mathcal {K}+\\cdots =-i\\mathcal {K}\\frac{1}{1-\\mathcal {G^{\\infty }}\\mathcal {K}}\\Rightarrow \\mathcal {K}=-\\mathcal {M}\\frac{1}{1-\\mathcal {G^{\\infty }}\\mathcal {M}}.$ With this definition of the kernel, one can proceed to evaluate poles of the N-channels FV scattering matrix by replacing the infinite volume loops $\\mathcal {G}^\\infty $ with their FV $\\mathcal {G}^V$ counterparts, $-i\\mathcal {M}^{V}&=&-i\\mathcal {K}-i\\mathcal {K}\\mathcal {G}^{V} \\mathcal {K}-i\\mathcal {G}^{V}\\mathcal {K}\\mathcal {G}^{V}\\mathcal {K}+\\cdots =-i\\mathcal {K}\\frac{1}{1-\\mathcal {G}^{V}\\mathcal {K}}\\\\& = & -i\\frac{1}{1-\\mathcal {M}\\mathcal {G}^{\\infty }}\\mathcal {M}\\frac{1}{1+\\delta \\mathcal {G}^{V}\\mathcal {M}}({1-\\mathcal {M}\\mathcal {G}^{\\infty }}).$ Finally arriving at the quantization condition $\\mathcal {R}e\\left\\lbrace \\det (\\mathcal {M}^{-1}+\\delta \\mathcal {G}^{V})\\right\\rbrace =\\mathcal {R}e\\left\\lbrace {\\rm {det}}_{\\rm {oc}}\\left[\\rm {det}_{\\rm {pw}}\\left[\\mathcal {M}^{-1}+\\delta \\mathcal {G}^{V}\\right]\\right]\\right\\rbrace =0 ,$ where the determinant $\\rm {det}_{\\rm {oc}}$ is over the N open channels and the determinant $\\rm {det}_{\\rm {pw}}$ is over the partial waves, and both $\\mathcal {M}$ and $\\delta \\mathcal {G}^V$ functions are evaluated on the on-shell value of the momenta.", "We have taken the real part of the determinant in Eq.", "(REF ), but as it will be shown shortly, this determinant condition gives rise to only one single real condition for both single channel and two coupled-channel cases with $l_{max}=0$ , so we omit the notion of the real part in the QC from now on.", "For a general proof of the reality of quantization condition with any number of coupled channels see Refs.", "[37], [38].", "For N=1, one reproduces the result first obtained by Rummukainen and Gottlieb [21] and later confirmed by Kim $et\\; al.$ [22] and Christ et al.", "[23] for the case of single-channel moving frame two-particle systems as follows.", "First note that it is convenient to evaluate the determinant using the spherical harmonic basis of $\\delta \\mathcal {G}^{V}$ , Eq.", "(REF ), and the on-shell scattering amplitude $\\mathcal {M}_i$ [22] $(\\mathcal {M}_{i})_{l_1,m_1;l_2,m_2}&=&\\delta _{l_1,l_2}\\delta _{m_1,m_2}\\frac{8\\pi E^*}{n_iq^*_i}\\frac{e^{2i\\delta ^{(l)}_i(q^*_i)}-1}{2i}.$ If the two equal-mass meson interpolating operator is in the $A_1^+$ irreducible representation of the cubic group, the energy eigenstates of the system have overlap with the $l=0,4,6,\\ldots $ angular momentum states at zero total momentum, making the truncation at $l_{max}=0$ a rather reasonable approximation in the low-energy limit.", "When $\\textbf {P}\\ne 0$ , the symmetry group is reduced, and at low energies the $l=0$ will mix with the $l=2$ partial wave as well as with higher partial waves [21].", "For two mesons with different masses, the symmetry group is even further reduced in the boosted frame, making the mixing to occur between $l=0$ and $l=1$ states as well as with higher angular momentum states [49].", "An easy way to see the latter is to note that in contrast with the case of degenerate masses, the kinematic function $c_{lm}^{\\textbf {P}}$ as defined in Eq.", "(REF ) is non-vanishing for odd $l$ when the masses are different.", "As a result even and odd angular momenta can mix in the quantization condition.", "This however does not indicate that the spectrum of the system is not invariant under parity.", "As long as all interactions between the particles are parity conserving, the spectrum of the system and its parity transformed counterpart are the same.", "One should note that the determinant condition, Eq.", "(REF ), guarantees this invariance: any mechanism, for example, which takes an S-wave scattering state to an intermediate P-wave two-body state, would take it back to the final S-wave scattering state, and the system ends up in the same parity state.Note that under parity $\\mathcal {Z}^d_{lm}\\rightarrow \\left(-1\\right)^{l}\\mathcal {Z}^d_{lm}$ .", "Note also that under the interchange of particles $\\mathcal {Z}^d_{lm}\\rightarrow \\left(-1\\right)^{l}\\mathcal {Z}^d_{lm}$ , so that for degenerate masses the $c_{lm}^{P}$ functions vanish for odd $l$ .", "This is expected since the parity transformation in the CM frame is equivalent to the interchange of particles.", "However, as is explained above for the case of parity transformation, despite the fact that $\\delta \\mathcal {G}^{V}$ is not symmetric with respect to the particle masses, the quantization condition is invariant under the interchange of the particles.", "Nevertheless, let us assume that the contributions from higher partial waves to the scatterings are negligible, so that one can truncate the determinant over the angular momentum at $l_{max}=0$ .", "Then the familiar quantization condition for the S-wave scattering, $q_i^*\\cot ({\\delta _i^0})=4\\pi c_{00}^{P}(q^{*2}_i),$ is recovered.", "It is convenient to introduce a pseudo-phase defined by ${q^*_i}\\cot ({\\phi ^P_i})\\equiv -4\\pi { c_{00}^P(q^{*2}_i)}$ to rewrite the quantization condition as $\\cot ({\\delta _i})=-\\cot ({\\phi ^P_i})\\Rightarrow \\delta _i+\\phi ^P_i=m\\pi ,$ where $m$ is an integer.", "In this form, the quantization condition is manifestly real.", "For the N=2 case, the expression for the scattering amplitude in Eq.", "(REF ) is modified, as it now depends on the mixing angle $\\bar{\\epsilon }$ , and the scattering matrix is no longer diagonal, while still symmetric.", "By labeling the off-diagonal terms as $\\mathcal {M}_{I,II}$ , and using the definition of the S-matrix for the coupled-channel system, Eq.", "(REF ), the scattering matrix elements can be written as $(\\mathcal {M}_{i,i})_{l_1,m_1;l_2,m_2}&=&\\delta _{l_1,l_2}\\delta _{m_1,m_2}\\frac{8\\pi E^*}{n_iq^*_i}\\frac{\\cos (2\\bar{\\epsilon })e^{2i\\delta ^{(l_1)}_i(q^*_i)}-1}{2i},\\\\(\\mathcal {M}_{I,II})_{l_1,m_1;l_2,m_2}&=&\\delta _{l_1,l_2}\\delta _{m_1,m_2}\\frac{8\\pi E^*}{\\sqrt{n_In_{II}q^*_Iq^*_{II}}}\\sin (2\\bar{\\epsilon })\\frac{e^{i(\\delta ^{(l_1)}_I(q^*_I)+\\delta ^{(l_1)}_{II}(q^*_{II}))}}{2},$ where the usual relativistic normalization of the states is used in evaluating the S-matrix elements.", "From Eq.", "(REF ) one obtains $\\det \\begin{pmatrix}1+\\delta \\mathcal {G}^{V}_I\\mathcal {M}_{I,I}&\\delta \\mathcal {G}^{V}_I\\mathcal {M}_{I,II}\\\\\\delta \\mathcal {G}^{V}_{II}\\mathcal {M} _{I,II}&1+\\delta \\mathcal {G}^{V}_{II}\\mathcal {M}_{II,II}\\\\\\end{pmatrix}=0,$ where the determinant is not only over the number of channels but also over angular momentum which is left implicit.", "In deriving this result we have made no assumption about the relative size of the scattering matrix elements, but when $l_{max}=0$ , we recover the LO result in Eq.", "(REF ).", "For $l_{max}=0$ one can use the pseudo-phase definition in Eq.", "(REF ) to rewrite the quantization condition in a manifestly real form, $\\cos {2\\bar{\\epsilon }}\\cos {\\left(\\phi ^P_1+\\delta _1-\\phi ^P_2-\\delta _2\\right)}=\\cos {\\left(\\phi ^P_1+\\delta _1+\\phi ^P_2+\\delta _2\\right)},$ which is equivalent to the result given in Refs.", "[24], [25] in the CM frame.The agreement between Eq.", "(REF ) and Eq.", "(37) of Ref.", "[24] can be achieved by noting that the pseudo-phase $\\phi ^P_{i}$ as defined in Eq.", "(REF ) is equivalent to the negative $\\Delta _{i}$ as defined in Eq.", "(36) of Ref.", "[24].", "On the other hand, the mixing parameter $\\overline{\\epsilon }$ as defined in Eq.", "(REF ) is related to the mixing parameter $\\eta _{0}$ defined in Eq.", "(14) of Ref.", "[24] through $\\eta _{0}=\\cos 2\\bar{\\epsilon }$ .", "It is easy to see that in the $\\bar{\\epsilon }\\rightarrow 0$ limit, one recovers the decoupled quantization conditions for both channels $I$ and $II$ , Eq.", "(REF ).", "The extension to a larger number of coupled channels is straightforward.", "As an example, we consider the N=3 case.", "Unitarity as well as time-reversal invariance allow us to parametrize the S-matrix using three phases shifts $\\lbrace \\delta _I, \\delta _{II}, \\delta _{II}\\rbrace $ and three mixing angles $\\lbrace \\bar{\\epsilon }_1, \\bar{\\epsilon }_2, \\bar{\\epsilon }_3\\rbrace $ $S_3=\\begin{pmatrix}e^{i2\\delta _I}c_1&ie^{i(\\delta _I+\\delta _{II})}s_1c_3&ie^{i(\\delta _I+\\delta _{III})}s_1s_3\\\\ie^{i(\\delta _I+\\delta _{II})}s_1c_2&e^{i2\\delta _{II}}\\left(c_1c_2c_3-s_2s_3\\right)&ie^{i(\\delta _I+\\delta _{III})}\\left(c_1c_2s_3+s_2c_3\\right)\\\\ie^{i(\\delta _I+\\delta _{III})}s_1s_2&ie^{i(\\delta _{II}+\\delta _{III})}\\left(c_1s_2c_3+c_2s_3\\right)&ie^{i2\\delta _{III}}\\left(c_1s_2s_3-c_2c_3\\right)\\\\\\end{pmatrix},$ where $c_i=\\cos (2\\bar{\\epsilon }_i)$ , $s_i=\\sin (2\\bar{\\epsilon }_i)$ .", "Note that in the limit ${\\epsilon }_2={\\epsilon }_3=0$ the third channel decouples, and one obtains a block diagonal matrix composed of $S_2$ corresponding to the $I-II$ coupled channel, as well as a single element corresponding to the scattering in the uncoupled channel $III$ .", "The spectrum of three-coupled channel is obtained from $\\det \\begin{pmatrix}1+\\delta \\mathcal {G}^{V}_I\\mathcal {M}_{I,I}&\\delta \\mathcal {G}^{V}_I\\mathcal {M}_{I,II}&\\delta \\mathcal {G}^{V}_I\\mathcal {M}_{I,III}\\\\\\delta \\mathcal {G}^{V}_{II}\\mathcal {M}_{II,I}&1+\\delta \\mathcal {G}^{V}_{II}\\mathcal {M}_{II,II}&\\delta \\mathcal {G}^{V}_{II}\\mathcal {M}_{II,III}\\\\\\delta \\mathcal {G}^{V}_{III}\\mathcal {M}_{III,I}&\\delta \\mathcal {G}^{V}_{III}\\mathcal {M}_{III,II}&1+\\delta \\mathcal {G}^{V}_{III}\\mathcal {M}_{III,III}\\end{pmatrix}=0,$ where the scattering matrix elements can be determined from Eq.", "(REF ) using the relationship between the scattering amplitudes and the S-matrix elements, $(\\mathcal {M}_{i,j})_{l_1,m_1;l_2,m_2}=\\delta _{l_1,l_2}\\delta _{m_1,m_2}\\frac{8\\pi E^*}{\\sqrt{n_in_{j}q^*_iq^*_{j}}}\\frac{(S^{(l_1)}_3)_{i,j}-\\delta _{i,j}}{2i}.$" ], [ "Two-Body electroweak Matrix Elements in a finite volume ", "As discussed in the introduction, electroweak processes in the two-hadron sector of QCD encompass a variety of interesting processes, so it is desirable to calculate the electroweak matrix elements directly from LQCD.", "One of the very first attempts to develop a formalism for such processes from a FV Euclidean calculation is due to Lellouch and Lüscher.", "In their seminal work [39], they restricted the analysis to $K\\rightarrow \\pi \\pi $ decay in the kaon's rest frame, and showed that the absolute value of the transition matrix element in an Euclidian FV is proportional to the physical transition matrix element.", "This proportionality factor is known as the LL-factor.", "This formalism was then generalized to moving frames in Refs.", "[22], [23].", "Here we present the generalization of Lellouch and Lüscher formalism to processes where the initial and final states are composed of two-hadron S-wave states.", "In the relativistic case, the coupled-channel result, Eq.", "(REF ), is used to derived the $2\\rightarrow 2$ LL-factor for boosted systems, while the contributions from one-body currents to the processes are neglected.", "In section REF we discuss a particular case, namely $pp\\rightarrow d+e^++\\nu _e$ , where the one-body current is in fact the dominant contribution to the weak transition amplitude.", "Using EFT($\\lnot \\hspace{-2.27626pt}{\\pi }$ ), the relationship between the FV matrix element and the pertinent LECs of the weak interaction is found.", "It is shown that in the presence of a one-body current, the FV and infinite volume weak matrix elements are no longer proportional to one another.", "Nevertheless, the infinite volume result can still be determined from the FV calculation of matrix elements upon determining the LECs of the EFT describing the process as become evident shortly." ], [ "Relativistic 2-Body LL-Factor ", "In order to derive the relativistic two-body LL-factor, one first notes that in the absence of the weak interaction, the two states decouple, and as a result the $S$ -matrix becomes diagonal.", "As is pointed out by Lellouch and Lüscher, there is a simple trick to obtain the desired relation between infinite volume and FV matrix elements by assuming the initial and final states to be nearly degenerate with energy $E^*_0$ (each satisfying Eq.", "(REF )) when there is no weak interaction.", "Once the perturbative weak interaction is turned on, the degeneracy is lifted, and the energy eigenvalues are $E^*_{\\pm }=E^*_0\\pm V|\\mathcal {M}^V_{I,II}|\\equiv E^*_0\\pm \\Delta E^*,$ where $\\mathcal {M}^V_{I,II}$ is the FV matrix element of the weak Hamiltonian density.", "Consequently, the CM momenta and the scattering phase shifts acquire perturbative corrections of the form $\\Delta q^*_i=\\frac{1}{4q_i^*}\\left(E^*_0-\\frac{(m_{i,1}^2-m_{i,2}^2)^2}{ E_0^{*3}}\\right)V|\\mathcal {M}^V_{I,II}|\\equiv \\Delta \\tilde{q}^*_i \\hspace{2.84544pt}V|\\mathcal {M}^V_{I,II}|,$ and $\\Delta \\delta _i({q}^*_i )=\\delta ^{\\prime }_i({q}^*_i ) \\Delta \\tilde{q}^*_i \\hspace{2.84544pt}V|\\mathcal {M}^V_{I,II}|,$ where $\\delta ^{\\prime }_i({q}^*_i )$ denotes the derivative of the phase shift with respect to the momentum evaluated at the free CM momentum, and $V=L^3$ .", "The perturbed energy necessarily satisfies the quantization condition, Eq.", "(REF ).", "The generalized LL-factor for $2\\rightarrow 2$ scattering is then obtained by Taylor expanding Eq.", "(REF ) to leading order in the weak matrix element about the free energy solution, $|\\mathcal {M}^\\infty _{I,II}|^2&=&V^2 \\left\\lbrace \\Delta \\tilde{q}^*_I \\Delta \\tilde{q}^*_{II}\\left(\\frac{8 \\pi E^*_0}{n_Iq^*_I}\\right)\\left(\\frac{8\\pi E^*_{0}}{n_{II}q^*_{II}}\\right)\\left({\\phi ^{P}_{I}}^{\\prime }(q^*_{I})+\\delta _{I}^{\\prime }(q^*_{I})\\right)\\left({\\phi ^{P}_{II}}^{\\prime }(q^*_{II})+\\delta _{II}^{\\prime }(q^*_{II})\\right)\\right\\rbrace |\\mathcal {M}^V_{I,II}|^2,\\nonumber \\\\$ where ${\\phi ^{P}_{i}}^{\\prime }({q}^*_i )$ denotes the derivative of the pseudo-phase with respect to the momentum evaluated at the free CM momentum.", "Note that we arrived at the generalization of the LL factor for two-body matrix elements using the degeneracy of states argument.", "Lin $et\\; al.$ [52] showed that the LL-factor for $K\\rightarrow \\pi \\pi $ can also be derived using the density of states in the large volume limit, and this argument was then generalized by Kim $et\\; al.$ [22] to boosted systems.", "Here it will be shown that the result in Eq.", "(REF ) is also consistent with the derivation based on the Kim $et\\; al.$ 's work.", "Let $\\sigma _i\\left(\\mathbf {x},t\\right)$ be the two-particle annihilation operator for the $i^{th}$ channel.", "Then the two particle correlation function in FV can be written as $C_{\\mathbf {P},i}^{V}\\left(t\\right)&\\equiv &\\int d^3x\\hspace{2.84544pt}e^{i\\mathbf {P}\\cdot \\mathbf {x}}\\left\\langle 0\\right|\\sigma _i\\left(\\mathbf {x},t\\right)\\sigma ^\\dag _i\\left(\\mathbf {0},0\\right)\\left|0\\right\\rangle _V= V\\sum _m e^{-E_mt}\\left|\\left\\langle 0\\right|\\sigma \\left(\\mathbf {0},0\\right)\\left|i;\\mathbf {P},m\\right\\rangle _V \\right|^{2}\\nonumber \\\\&\\stackrel{L\\rightarrow \\infty }{\\longrightarrow }&V \\int dE\\rho _{V,i}(E)e^{-Et}\\left|\\left\\langle 0\\right|\\sigma \\left(\\mathbf {0},0\\right)\\left|i;\\mathbf {P},E\\right\\rangle _V \\right|^{2},$ where a complete set of states is being inserted in the first equality.", "In the second equality, we have introduced the density of states for the $i^{th}$ channel, $\\rho _{V,i}(E)$ , which is defined as $\\rho _{V,i}(E)=dm_i/dE$ .", "Using Eqs.", "(REF ), (REF ) the density of states can be written as $\\rho _{V,i}(E^*)=\\left({\\phi ^{P}_{i}}^{\\prime }(q^*_{i})+\\delta _{i}^{\\prime }(q^*_{i})\\right)\\Delta \\tilde{q}^*_{i}/\\pi $ .", "In the infinite volume the two-particle correlation function is [52] $C_{\\mathbf {P},i}^{\\infty }\\left(t\\right)=\\frac{n_i}{8\\pi ^{2}}\\int dE\\frac{q^{*}_i}{E^{*}}e^{-Et}\\left|\\left\\langle 0\\right|\\sigma \\left(\\mathbf {0},0\\right)\\left|i;\\mathbf {P},E\\right\\rangle _\\infty \\right|^{2},$ where the factor of $n_i$ has been introduced to account for the double counting of the phase space when the particles are identical.", "It is straightforward to show that this relation still holds when the two particles have different masses.", "From Eqs.", "(REF ), (REF ) the relationship between the states of infinite and asymptotically large (yet finite) volume can be deduced, $\\left|i;\\mathbf {P},E\\right\\rangle _\\infty \\Leftrightarrow 2\\pi \\sqrt{\\frac{2V\\rho _{V,i}E_0^*}{n_iq_i^*}}\\left|i;\\mathbf {P},E\\right\\rangle _V.$ This relation therefore recovers the LL-factor as given in Eq.", "(REF ).", "It also demonstrates that the LL-factor accounts for different normalizations of the states in the finite volume and infinite volume in the presence of interactions.", "It is important to understand that this derivation strongly relies on the assumptions that there are no one-body currents that mix the initial and final states.", "In reality most systems are sensitive to one-body currents.", "In order to find a relation between the FV and infinite volume electroweak matrix elements it is necessary to understand the contributions from the one-body currents.", "In the following section we consider one pertinent problem and show that in fact one-body currents introduce additional FV contributions." ], [ "Proton-proton fusion in EFT($\\lnot \\hspace{-2.27626pt}{\\pi }$ ) ", "In this section we discuss the weak interaction in the two-nucleon sector.", "This sector has been previously studied by Detmold and Savage [40] in the finite volume.", "They considered a novel idea of studying electroweak matrix elements using a background field.", "Since evaluating matrix elements of electroweak currents between NN states, e.g.", "$\\left\\langle d\\right|\\left.A^{\\mu }\\right.\\left|np\\right\\rangle $ , is naively one or two orders of magnitude more difficult than performing NN four-point functions, they present a procedure for extracting the relevant LECs of the pionless EFT, EFT($\\lnot \\hspace{-2.27626pt}{\\pi }$ ) [43], by calculating four-point functions of nucleons in a finite volume in the presence of a background electroweak field.", "This would be a project worth pursuing with great benefits, namely a five-point function is replaced by a four-point function, thereby dramatically reducing the number of propagator contractions.", "For isovector quantities, this procedure comes at a small cost, since for perturbatively small background fields, the QCD generated gauge links get modified by a multiplicative factor that couples the valence quarks to the external field, $U^{QCD}_{\\mu }(x)\\rightarrow U^{QCD}_{\\mu }(x)U_{\\mu }^{ext}(x)$ .", "On the other hand, for isoscalar quantities this approach would require the generation of gauge configurations in the presence of the background field.", "For both isovector and isoscalar quantities, one would need to perform calculations at a range of background field strengths in order to precisely discern the contribution of the coupling between the background field and the baryonic currents to the NN spectrum.", "Additionally, the nature of this background field will differ depending on the physics one is interested in.", "Alternatively, one can always evaluate matrix elements of electroweak currents with gauge configurations that solely depend on the QCD action, which is the case considered here.", "With the improvement in the computational resources available for LQCD calculations, the studies of nucleonic matrix elements will become feasible shortly, and therefore their connection to the physical matrix elements should be properly addressed.", "The goal is to explore FV corrections of weak matrix elements in the two-nucleon sector.", "In particular, we will consider the proton-proton fusion process, $(pp\\rightarrow d+e^++\\nu _e)$ , which mixes the ${^1}\\hspace{-1.70709pt}{S_0}-{^3}\\hspace{-1.70709pt}{S_1}$ channels.", "In order to do this calculation the mechanism of EFT($\\lnot \\hspace{-2.27626pt}{\\pi }$ ) [41], [42], [43] will be used.", "The methodology is similar to the one used for the toy problem considered in the first section, except the fields are now non-relativistic and carry isospin and spin indices.", "The presence of a weak interaction, leads to a contribution to the Lagrangian that couples the axial-vector current $A^{\\mu =3}=\\frac{1}{2}\\left(\\bar{u}\\gamma ^3\\gamma ^5u-\\bar{d}\\gamma ^3\\gamma ^5d\\right)$ to an external weak current.", "In terms of the low-energy degrees of freedom, the axial current will receive one-body and two-body contributions.", "At energies well below the pion-production threshold, the EFT($\\lnot \\hspace{-2.27626pt}{\\pi }$ ) Lagrangian density including weak interactions can be written as [41], [42], [43], [53], [54], [55] $\\mathcal {L}&=&N^\\dag \\left(i\\partial _0+\\frac{\\nabla ^2}{2M}-\\frac{W_3g_A}{2}\\sigma ^3\\tau ^3\\right)N-C^{{\\left({^1}\\hspace{-1.70709pt}{S_0}\\right)}}_0\\left(N^TP_1^aN\\right)^\\dag \\left(N^TP_1^aN\\right)\\nonumber \\\\&&-C^{{\\left({^3}\\hspace{-1.70709pt}{S_1}\\right)}}_0\\left(N^TP_3^jN\\right)^\\dag \\left(N^TP_3^jN\\right)-{L_{1,A}}W_3\\left[\\left(N^TP_1^3N\\right)^\\dag \\left(N^TP_3^3N\\right)+h.c.\\right]+\\cdots ,$ where $N$ is the nucleon annihilation operator with bare mass $M$ , $\\lbrace C^{{\\left({^1}\\hspace{-1.70709pt}{S_0}\\right)}}_0, C^{{\\left({^3}\\hspace{-1.70709pt}{S_1}\\right)}}_0, g_A,{L_{1,A}}\\rbrace $ are the LECs of the theory, $g_A = 1.26$ is the nucleon axial coupling constant, $W_3$ is the external weak current, and $\\lbrace P_1^a, P_3^j\\rbrace $ are the standard $\\lbrace {^1}\\hspace{-1.70709pt}{S_0},{^3}\\hspace{-1.70709pt}{S_1}\\rbrace $ -projection operators, $P_1^a=\\frac{1}{\\sqrt{8}}\\tau _2\\tau ^a\\sigma _2,\\hspace{28.45274pt}P_3^j=\\frac{1}{\\sqrt{8}}\\tau _2\\sigma _2\\sigma ^j,$ where $\\tau $ ($\\sigma $ ) are the Pauli matrices which act in isospin (spin) space.", "In Eq.", "(REF ) the ellipsis denotes an infinite tower of higher order operators.", "The $\\mathcal {O}(p^{2n})$ -operator for the $\\lbrace {^1}\\hspace{-1.70709pt}{S_0}, {^3}\\hspace{-1.70709pt}{S_1}\\rbrace $ state will have corresponding LECs $\\lbrace C^{{\\left({^1}\\hspace{-1.70709pt}{S_0}\\right)}}_{2n}, C^{{\\left({^3}\\hspace{-1.70709pt}{S_1}\\right)}}_{2n}\\rbrace $ , which are included in this calculation.", "In this section we only consider NN systems in the S-wave channel.", "As discussed in section REF this introduces a systematic error in FV systems that is negligible near the kinematic threshold.", "At leading order, a weak transition between the isosinglet and isotriplet two-nucleon channels is described by an insertion of the single body current (which is proportional to $g_A$ ) and the bubble chain of the $C_0^{({^3}\\hspace{-1.70709pt}{S_1})}$ and $C_0^{({^1}\\hspace{-1.70709pt}{S_0})}$ contact interactions on the corresponding nucleonics legs as discussed in Ref. [56].", "At next to leading order (NLO), the hadronic matrix element of $pp\\rightarrow d+e^++\\nu _e$ receives contributions from one insertion of the the $C_2p^2$ operator along with one insertion of the single-body operator proportional to $g_A$ [57].", "At the same order, a single insertion of the two-body current that is proportional to ${L_{1,A}}$ also contributes to the transition amplitude [55].", "In both of these contributions the dressing of the NN states with the corresponding bubble chain of the LO contact interactions must be assumed.", "As is discussed in Ref.", "[55], the two-body contribution is estimated to give rise to a few-percent correction to the hadronic matrix element, and its corresponding LEC, $L_{1A}$ , is known to contribute to the elastic and inelastic neutrino-deuteron scattering cross sections as well [53], [55].", "Of course, the electromagnetism plays a crucial role in the initial state interactions in the pp-fusion process, but as is shown in Refs.", "[56], [57], [55], the ladder QED diagrams can be summed up to all orders nonperturbatively.", "Since LQCD calculations of the matrix elements of the axial-vector current involving two-nucleons would allow for a determination of ${L_{1,A}}$ , one will achieve tighter theoretical constraints on the cross section of these processes.", "Furthermore, having obtained the one-body and two-body LECs of the weak sector will allow for the determination of the few-body weak observables.", "In the absence of weak interactions, the on-shell scattering amplitude for both channels can be determined exactly in terms of their corresponding LECs by performing a geometric series over all the bubble diagrams [43] $\\mathcal {M}^0=-\\frac{\\sum _{n=0}^{\\infty } C_{2n}q^{*2n}}{1-G^\\infty _0\\sum _{n=0}^{\\infty } C_{2n}q^{*2n}},$ where the on-shell relative momentum in the CM frame is related to the total NR energy and momentum of the two-nucleon system via, $q^*=\\sqrt{ME-\\frac{1}{4}{\\mathbf {P}^2}}$ , and $G^\\infty _0$ denotes the loop integral $G^{\\infty }_0&=&\\left(\\frac{\\mu }{2}\\right)^{4-D}\\int \\frac{d^3\\mathbf {k}}{(2\\pi )^3}\\frac{1}{E-\\frac{\\mathbf {k}^2}{2M}-\\frac{(\\mathbf {P}-\\mathbf {k})^2}{2M}+i\\epsilon }$ which is linearly divergent.", "In order to preserve Galilean invariance and maintain a sensible power-counting scheme for NR theories with an unnaturally large scattering length, the power-divergence subtraction (PDS) scheme is used to regularize the integral [41], [42], [58].", "Using PDS, the integral above becomes $G^{\\infty }_0&=&-\\frac{M}{4\\pi }\\left(\\mu +i\\sqrt{ME-{\\mathbf {P}}^2/4}\\right)=-\\frac{M}{4\\pi }\\left(\\mu +iq^*\\right),$ where $\\mu $ is the renormalization scale.", "When the volume is finite, the integral above is replaced by its FV counterpart, $G^{V}_{0}$ .", "It is straightforward to find the relation between the FV correction $\\delta G^{V}_{0}=G^{V}_{0}-G^{\\infty }_{0}$ and the NR version of the kinematic function $c_{00}^{P}$ defined in Eq.", "(REF ), $c^{{P}}_{00}(q^{*2})=\\left[\\frac{1}{L^3}\\sum _{\\textbf {k}}-\\mathcal {P}\\int \\frac{d^3\\mathbf {k}}{(2\\pi )^3}\\right]\\frac{1}{({\\mathbf {k}-\\frac{\\mathbf {P}}{2}})^{2}-q^{*2}} \\ ,$ where we have used the fact that for degenerate particles $\\alpha $ that is defined after Eq.", "(REF ) is $\\frac{1}{2}$ , and in the NR limit the relativistic factor $\\gamma $ is equal to one.", "One can arrive at the desired relation by adding and subtracting the infinite volume two-particle propagator, Eq.", "(REF ), to $G^{V}_{0}$ .", "One of them can be evaluated using PDS, Eq.", "(REF ), and the other one can be written in terms of a regularized principle value integral, leading to $G^V_0(E,{P})&=&-\\frac{M}{4\\pi }\\mu -c_{00}^{P}(q^{*2}),$ therefore arriving at $\\delta G^V_0(E,{P})&=&=\\frac{M}{4\\pi }\\left(q^*\\cot \\phi ^{P}+i q^*\\right),$ where we have used the pseudo-phase definition, Eq.", "(REF ).", "The goal is to find a relation between the FV matrix elements of the axial-vector current and the LECs that parametrize the weak interaction, namely $\\lbrace g_A,{L_{1,A}}\\rbrace $ , following a procedure analogous to section REF .", "The first step is to find the QC satisfied by the energy eigenvalues of the two-nucleon system in presence of an external weak field as was also considered in Ref.", "[40].The main distinction between the result that will be obtained here and that of Ref.", "[40] is that we will consider the case where the two-nucleon system has arbitrary momentum below inelastic thresholds, while Ref.", "[40] only considered the two-nucleon system at rest.", "After obtaining the QC for this theory, the trick by Lellouch and Lüscher [39] can be utilized to obtain an expression for the FV weak matrix element.", "The main difference between the problem considered here and the problem discussed in the previous section is that the dominant contribution to the weak processes in the NN sector comes from the one-body current, namely the term proportional to the axial charge in Lagrangian, Eq.", "(REF ).", "In fact this contribution modifies the nucleon propagator and therefore the on-shell condition.", "To avoid complications associated with the modification of external legs appearing in the FV analogue of the scattering amplitude, $\\mathcal {M}^{V}$ , we obtain the QC for this system by looking at the pole structure of the NN correlation function in the presence of the weak field.", "As before a $2\\times 2$ kernel $\\mathcal {K}$ can be formed that incorporates the tree-level $2\\rightarrow 2$ transitions, $i\\mathcal {K}&=&\\begin{pmatrix}-i\\sum C^{\\left({^3}\\hspace{-1.70709pt}{S_1}\\right)}_{2n}q^{*2n} &-i{L_{1,A}}\\\\-i{L_{1,A}}&-i\\sum C^{\\left({^1}\\hspace{-1.70709pt}{S_0}\\right)}_{2n}q^{*2n}\\\\\\end{pmatrix}.$ The FV function $\\mathcal {G}^V$ can be still expressed as a $2\\times 2$ matrix in the basis of channels, except it will attain off-diagonal elements due to the presence of the single-body operator in contrast with the the scalar sector studied before, $\\delta \\mathcal {G}^V=\\left(\\begin{array}{ccc}G_{+}^{V} & & G_{-}^{V}\\\\\\\\G_{-}^{V} & & G_{+}^{V}\\end{array}\\right),$ where FV functions $G^V_+$ and $G^V_-$ are defined as $G^{V}_{\\pm }=\\frac{M}{2L^3}\\sum _{\\mathbf {k}}\\left[\\frac{1}{E-\\frac{\\mathbf {k}^2}{2M}-\\frac{(\\mathbf {P}-\\mathbf {k})^2}{2M}-W_3g_A}\\pm \\frac{1}{E-\\frac{\\mathbf {k}^2}{2M}-\\frac{(\\mathbf {P}-\\mathbf {k})^2}{2M}+W_3g_A}\\right].$ Since we only aim to present the result up to NLO in the EFT expansion according to the power counting discussed above, it suffices to keep only the LO terms in $g_A$ when expanding these FV functions in powers of the weak coupling.", "Explicitly, $G^V_+=G^V_0(E,{P})+\\mathcal {O}(W_3^2g_A^2)$ where $G^V_0$ is defined in Eq.", "(REF ), and $G^V_-=W_3g_A~G^V_1(E,{P})+\\mathcal {O}(W_3^3g_A^3)$ with $G^{V}_1=\\frac{M}{L^3}\\sum _{\\mathbf {k}}\\frac{1}{((\\mathbf {k}-\\frac{\\mathbf {p}}{2})^2-q^{*2})^2}.$ In order to form the NN correlation function, let us also introduce a diagonal matrix $\\mathcal {A}_{NN}$ , whose each diagonal element denotes the overlap between the two-nucleon interpolating operators in either isosinglet or isotriplet channels and the vacuum.", "With theses ingredients, the NN correlation function in the presence of the external weak field can be easily evaluated, as is diagrammatically presented in Fig.", "(REF ).", "It is important to note that in evaluating the FV loops, one should pay close attention to the pole structure of $G^V_{\\pm }$ , Eq.", "(REF ).", "In other words, the on-shell condition for the free two-nucleon system is modified in the presence of the single-body weak current, namely, $q^{*2}\\rightarrow q^{*2}\\pm MW_3g_A$ .", "Figure: Shown is the NN correlation function in the isosinglet (isotriplet) channel in the presence of an external weak field.", "𝒜 NN \\mathcal {A}_{NN} denotes the overlap between the NN interpolating operators and the vacuum.", "The two-dimensional kernel is denoted by 𝒦\\mathcal {K}.", "The diagonal terms of the kernel correspond to the strong part of the interactions, while off-diagonal terms depict contributions that arise from the weak interaction, namely L 1,A L_{1,A}.", "Unlike the scalar sector considered before, the FV function, δ𝒢 V \\delta \\mathcal {G}^V, has diagonal and off-diagonal contributions due to the presence of the single-body current.Then it is straightforward to show that after keeping only terms up to $\\mathcal {O}(C_2q^{*2}W_3g_A,W_3L_{1,A})$ , the QC obtained from the pole structure of the NN correlation function reads $\\left[q^*\\cot \\delta ^{({^1}\\hspace{-1.70709pt}{S_0})}+q^*\\cot \\phi ^P\\right]\\left[q^*\\cot \\delta ^{({^3}\\hspace{-1.70709pt}{S_1})}+q^*\\cot \\phi ^P\\right]=\\left[\\frac{4\\pi }{M}W_3{\\widetilde{L}_{1,A}}+\\frac{4\\pi }{M}W_3g_AG_1^V\\right]^2,$ where ${\\widetilde{L}_{1,A}}$ that is defined as ${\\widetilde{L}_{1,A}}&=&\\frac{1}{C^{\\left({^3}\\hspace{-1.70709pt}{S_1}\\right)}_0C^{\\left({^1}\\hspace{-1.70709pt}{S_0}\\right)}_0}\\left[{{L}_{1,A}}-\\frac{g_AM}{2}\\left(C^{\\left({^3}\\hspace{-1.70709pt}{S_1}\\right)}_2+C^{\\left({^1}\\hspace{-1.70709pt}{S_0}\\right)}_2\\right)\\right],$ is a renormalization scale independent quantity [53], [54], [55], [40] $\\mu \\frac{d}{d\\mu }{\\widetilde{L}_{1,A}}=0.$ Before proceeding let us compare this result with the one presented in Ref. [40].", "As discussed, the authors of Ref.", "[40] have obtained the same quantization condition for two-nucleon systems in the presence of an external weak field using a dibaryon formalism.", "The advantage of this formalism is that the diagrammatic representation of the processes of interest are greatly simplified using an auxiliary field with quantum numbers of two nucleons.", "In fact, the full di-nucleon propagator sums up all $2\\rightarrow 2$ interactions nonperturbatively.", "In the QC presented in Ref.", "[40], the contributions of the axial charge current to all orders have been kept, but as the higher order operators that contribute to the weak transition have not been included in their calculation, their result is only valid up to $\\mathcal {O}(g_AC_2q^{*2})$  [53], [54], [55].", "In the dibaryon formalism, the two-body weak current is parametrized by $l_{1,A}$ which is related to ${\\widetilde{L}_{1,A}}$ in this work viaNote that Eq.", "(31) of Ref.", "[40] defines $l_{1,A}$ as $\\frac{8\\pi }{M}{\\widetilde{L}_{1,A}}$ , but we suspect this discrepancy is only due to a typo in their result.", "$l_{1,A}=\\frac{16\\pi }{M}{\\widetilde{L}_{1,A}}.$ Using this relation between the LECs of both theories, and keeping in mind the order up to which the resuslt of both calculations are valid, one will find agreement between the result presented in Eq.", "(REF ) and that of Ref.", "[40] after setting the momentum of the CM to zero.", "Having obtained the QC for this system, Eq.", "(REF ), it is straightforward to obtain the relationship between the FV matrix elements of the Hamiltonian density and the LECs.", "In the absence of weak interactions, the two NN states are degenerate with energy $E_0^*$ and on-shell momentum $q^*_0$ , satisfying the free quantization condition $\\cot (\\phi ^P)=-\\cot (\\delta )$ .", "As the weak interaction is turned on, the degeneracy is lifted, leading to a shift in energy equal to $\\Delta E^*=V|\\mathcal {M}^V_{{^1}\\hspace{-1.70709pt}{S_0}-{^3}\\hspace{-1.70709pt}{S_1}}|$ , where $|\\mathcal {M}^V_{{^1}\\hspace{-1.70709pt}{S_0}-{^3}\\hspace{-1.70709pt}{S_1}}|$ is the FV matrix element of the Hamiltonian density between the $^1S_0$ and $^3S_1$ states.", "Note that this matrix element is proportional to $W_3$ .", "Therefore it is convenient to define the purely hadronic matrix element $|\\mathcal {M}^V_{W}|=|\\mathcal {M}^V_{{^1}\\hspace{-1.70709pt}{S_0}-{^3}\\hspace{-1.70709pt}{S_1}}|/W_3$ which is in fact what would be calculated via LQCD.", "Expanding the Eq.", "(REF ) about the free energy, and keeping LO terms in the weak interaction, one obtains $\\left(\\frac{MV}{2}\\right)^2\\csc ^2{\\delta ^{({^1}\\hspace{-1.70709pt}{S_0})}}\\csc ^2{\\delta ^{({^3}\\hspace{-1.70709pt}{S_1})}}\\left(\\phi ^{\\prime }+\\delta ^{({^1}\\hspace{-1.70709pt}{S_0})^{\\prime }} \\right)\\left(\\phi ^{\\prime }+\\delta ^{({^3}\\hspace{-1.70709pt}{S_1})^{\\prime }}\\right)|\\mathcal {M}^V_{W}|^2=\\left(\\frac{4\\pi }{M}{\\widetilde{L}_{1,A}}+\\frac{4\\pi }{M}g_AG_1^V\\right)^2.$ This result shows that in order to determine weak matrix elements in the NN sector, not only it is necessary to determine the derivatives of the phase shifts in the ${^1}\\hspace{-1.70709pt}{S_0}$ and ${^3}\\hspace{-1.70709pt}{S_1}$ channels with respect to the on-shell momenta, but also it is necessary to determine the nucleon axial coupling constant.", "There is no clear crosscheck for this result, since it is not clear how to implement the density of states approach for this problem.", "The presence of the one-body operator makes the mixing between the two states non-trivial, therefore one would expect a more complicated relationship between the FV and infinite volume states than the one predicted via the density of states approach.", "Although it would be desirable to obtain a generalization of Lellouch and Lüscher's result for $2\\rightarrow 2$ systems, this example demonstrates that in the two-body sector, one-body currents lead to large FV corrections.", "In principle, the FV matrix elements depend on the nature of the problem that is considered, and each weak hadronic process must be separately studied." ], [ "Summary and Conclusion", "In this paper, we have presented and derived the FV quantization condition for a system of multi-coupled channels each composed of two hadrons in a moving frame.", "In the second section, the quantization condition at LO in mixing phase, Eq.", "(REF ) is derived for the S-wave scattering, where a rather simple EFT toy model for scalar particles is used.", "Using the techniques developed by Kim $et\\; al.$ [22], the quantization condition for the boosted systems with arbitrary mixing angles, Eq.", "(REF ) has been obtained.", "It then became evident that the toy model used is in perfect agreement with the nonperturbative result when the angular momenta is truncated at the S-wave.", "From the generalized results, we have also derived the quantization condition for N=3 coupled channels composed of two hadrons, Eq.", "(REF ).", "The advantage of this result is that it allows the lattice practitioners to perform coupled-channel calculations at multiple boosted momenta in a periodic volume, thereby increasing the number of measurements in order to best constrain the S-matrix elements.", "When N=2, the S-matrix can be parametrized by three real parameters (two scattering phase shifts and one mixing angle), therefore one needs to perform at least three measurements at each CM momentum, which can be done by using combinations of different boost momenta and different volumes.", "The N=3 case would require six measurements for each CM momentum to constrain the three phase shifts and three mixing angles.", "We have also derived the relationship between FV matrix elements and infinite volume matrix elements in the two-body sector, Eq.", "(REF ).", "This was first calculated for $K\\rightarrow \\pi \\pi $ in the rest frame by Lellouch and Lüscher [39], and later extended to the boosted system in Refs.", "[22], [23].", "Here we have shown two ways to obtain the extension of LL-factor for $2\\rightarrow 2$ relativistic processes.", "The first entails expanding the coupled quantization, Eq.", "(REF ), about the free energy of the system when the two channels are decoupled.", "This method assumes the two states to be degenerate in the absence of the weak interaction, as was first developed by Lellouch and Lüscher [39].", "Kim $et\\; al.$ [22] generalized the method of [52] and derived the relationship between FV and infinite volume two-particle states in the moving frame, Eq.", "(REF ), which has been shown to agree with our result of the two-body relativistic LL-factor, Eq.", "(REF ).", "This derivation strongly depends on the fact that the two channels only mix in the presence of two-body currents.", "In reality most systems are in fact sensitive to one-body currents as well.", "We have used EFT($\\lnot \\hspace{-2.27626pt}{\\pi }$ ) [43], [41], [42], [59], [60], [60] to determine the extension of the LL-factor for NR baryonic systems.", "In particular, we consider processes that mix the ${^1}\\hspace{-1.70709pt}{S_0}-{^3}\\hspace{-1.70709pt}{S_1}$ NN channels.", "This is pertinent for performing calculations of proton-proton fusion, among other interesting processes, directly from LQCD, [40].", "The channels in this system are mixed not only by a two-body operator but also by a one-body operator.", "As it is shown, FV effects arising from the insertion of a one-body operator are sizable and therefore must be included.", "Unlike any previous case, the FV and infinite volume weak matrix elements are not simply proportional to each other.", "The result demonstrates that in fact the FV matrix element is proportional to a linear combination of the LO and NLO LECs that parametrize the weak interactions in the NN sector, which can in turn be used to constrain the proton-proton fusion transition amplitude at the few-percent level [56], [57], [54]." ], [ "Acknowledgment", "We would like to thank Martin J.", "Savage and William Detmold for fruitful discussions, and for the feedback on the first manuscript of this paper.", "We also thank Stephen R. Sharpe and Maxwell T. Hansen for useful conversations, and for communicating their results with us prior to publication.", "RB and ZD were supported in part by the DOE grant DE-FG02-97ER41014." ] ]

[ [ "Calculation of the phase of hidden rotating antiferromagnetic order" ], [ "Abstract The phase of the rotating order parameter in rotating antiferromagnetism is calculated using a combination of mean-field theory and Heisenberg equation.", "This phase shows a linear time dependence, which allows us to interpret rotating antiferromagnetism as a synchronized Larmor-like precession of all the spins in the system or as an unusual ${\\bf q}=(\\pi,\\pi)$ spin-wave around a zero local magnetization.", "We discuss implications for the pseudogap state of high-$T_C$ superconducting materials.", "Rotating antiferromagnetism has been proposed to model the pseudogap state in these materials." ], [ "Introduction", "According to several researchers, the puzzling pseudogap (PG) phenomenon in high-$T_C$ superconductors (HTSC) is caused by some sort of hidden order.", "This is supported by the observation of a depression in the density of states at the Fermi level, with no order parameter responsible for this depression observed yet [1], [2], [3], [4].", "Rotating antiferromagnetism (RAF) has been recently proposed as a possible candidate for this hidden order, and several physical quantities have already been calculated within the RAF theory (RAFT) with good agreement with available experimental data [5], [6], [7].", "RAF is one of several other proposals for the PG (see Ref.", "[4] for a discussion).", "Contrary to theories of circulating currents [8], [9], [10], RAF is based on the concept of an order parameter that has a finite magnitude below a critical temperature but a time-dependent phase [11].", "Note that all the physical quantities that have so far been calculated within RAFT do not depend on the phase of the order parameter in RAF [5], [6], [7], [12], [13], [14], [15].", "The lack of the time dependence profile for this phase limited however the full understanding of the nature of RAF.", "The purpose of this work is to calculate this phase as a function of time using a combination of RAFT and the Heisenberg equation.", "We show that it varies linearly with time.", "As a consequence of this time dependence, RAF can be interpreted as a $(\\pi ,\\pi )$ unusual spin wave around a zero local magnetization or as a synchronized Larmor-like precession of all the spins in the system.", "Because the phase of this order parameter is time dependent, it was not possible to calculate it in RAFT alone, which is a mean-field approach.", "This paper is organized as follows.", "First in Sec.", "REF we rederive RAFT using the spin ladder operators, which are necessary for the phase calculation.", "In Sec.", "REF , we review RAFT.", "Then in Sec.", "REF we use the Heisenberg equation to get the time dependence for the spin ladder operators, which yields the time dependence of the phase of the rotating order parameter.", "In Sec.", "REF , the interpretation of RAF as an unusual $(\\pi ,\\pi )$ spin wave is explained.", "Finally, conclusions are drawn in Sec.", "." ], [ "Approach", "As we are only interested in understanding the nature of the PG phase of HTSCs in this work, we restrict ourselves to the non superconducting phase.", "Consider the $t$ -$t^{\\prime }$ Hubbard model in two dimensions: $H&=&-t\\sum _{\\langle i,j\\rangle \\sigma }c^{\\dag }_{i,\\sigma }c_{j,\\sigma }-t^{\\prime }\\sum _{\\langle \\langle i,j\\rangle \\rangle \\sigma }c^{\\dag }_{i,\\sigma }c_{j,\\sigma }+ {\\rm h.c.}\\cr &&-\\mu \\sum _{i,\\sigma }n_{i,\\sigma } + U\\sum _in_{i\\uparrow }n_{i\\downarrow },$ where $\\langle i,j\\rangle $ and $\\langle \\langle i,j\\rangle \\rangle $ designate summation over nearest and second-nearest neighboring sites, respectively.", "$t$ and $t^{\\prime }$ are electron hopping energies to nearest and second-nearest neighbors, respectively.", "Because the phase of RAF is related to the spin ladder operators, it is useful to rewrite Hamiltonian (REF ) using these operators." ], [ "Rewriting the Hamiltonian using the spin ladder operators", "Using the spin ladder operator written in second quantization $S_i^+=c^{\\dag }_{i,\\uparrow }c_{i,\\downarrow }$ , the onsite Coulomb repulsion term $Un_{i\\uparrow }n_{i\\downarrow }$ can on one hand be cast in the form $Un_{i\\uparrow }n_{i\\downarrow }=Un_{i\\uparrow } - US_{i}^+S_i^-$ and on the other hand as $Un_{i\\uparrow }n_{i\\downarrow }=Un_{i\\downarrow } - US_{i}^-S_i^+$ .", "Summing and dividing by 2 yields the symmetrized expression $Un_{i\\uparrow }n_{i\\downarrow }=\\frac{U}{2}(n_{i\\uparrow } + n_{i\\downarrow }) -\\frac{U}{2}(S_{i}^+S_i^- + S_{i}^-S_i^+)$ .", "The latter can be proved by calculating the action of each side of the equality on the possible states $\\lbrace |0\\rangle ,|\\uparrow \\rangle ,|\\downarrow \\rangle ,|\\uparrow \\downarrow \\rangle \\rbrace $ , and noting that $S_{i}^+S_i^-|\\uparrow \\downarrow \\rangle =S_{i}^-S_i^+|\\uparrow \\downarrow \\rangle =0$ due to Pauli exclusion principle, and $S_i^+|0\\rangle \\equiv c_{i\\uparrow }^\\dag c_{i\\downarrow }|0\\rangle =0$ .", "For our many-body system, sites are neither full nor empty, but are on average occupied by a density smaller than 1 away from half filling.", "Therefore, the terms $S_{i}^+S_i^-$ and $S_{i}^-S_i^+$ , which are responsible for onsite spin-flip excitations, will contribute by lowering energy for the sites that are partially occupied by the same density of spin up and down electrons.", "One can decouple this term in mean-field theory using $\\langle S_i^-\\rangle \\equiv \\langle c^{\\dag }_{i,\\downarrow }c_{i,\\uparrow }\\rangle $ , which leads to a collective behavior for the spin-flips, and the results obtained in this way are the same as in RAFT [5], [6], [7], [12], [13], [14].", "In this state, a spin flip process at site $i$ is simultaneously accompanied by another one at another site $j$ ; the occurrence of the spin flips is synchronized.", "Thermal motion has obviously an effect on this order as it does on conventional orders; i.e., above a critical temperature (identified with the PG temperature) the spin-flip processes become uncorrelated, leading to the disappearance of the long-range non conventional order.", "The spin-flip processes, which are purely quantum, continue to exist even above this critical temperature, but in an incoherent disordered manner.", "The occurrence in RAFT of a second-order phase transition at the PG temperature is consistent with experimental data supporting its existence [4]." ], [ "Review of RAFT", "We rederive RAFT, which deals with the static part (magnitude) of the order parameter $\\langle S_i^\\pm \\rangle $ , using the spin ladder operators then for the dynamic (phase) part we will use the Heisenberg equation to find its time dependence.", "To the best of our knowledge the combination of mean-field theory and the Heisenberg equation of quantum mechanics constitutes a novel approach for the PG in HTSCs.", "The parameter $Q_i={\\langle c_{i,\\uparrow }c^{\\dag }_{i,\\downarrow }\\rangle }=-\\langle S_i^-\\rangle \\equiv |Q|e^{i\\phi _i}$ is defined in order to carry on a mean-field decoupling of the $t$ -$t^{\\prime }$ Hubbard model.", "Consider the ansatz where $\\phi _i-\\phi _j=\\pi $ , with $i$ and $j$ labeling any two adjacent lattice sites.", "Except for this difference of $\\pi $ between the phases of the order parameter on two adjacent sites, the phases $\\phi _i\\equiv \\phi $ are site independent and assume any value in $[0,2\\pi ]$ .", "The normal state Hamiltonian in RAFT [5], [6], [7] is $H\\approx \\sum _{{\\bf k}\\in RBZ}\\Psi ^{\\dag }_{\\bf k}{\\cal H}\\Psi _{\\bf k}+NUQ^2-NUn^2,$ where $N$ is the number of sites, and $n=\\langle n_{i,\\sigma }\\rangle $ is the expectation value of the number operator.", "Because of antiferromagnetic correlations the lattice consists of two sublattices $A$ and $B$ , even though there is no long-range static antiferromagnetic order.", "The summation runs over the reduced Brillouin zone (RBZ).", "The Nambu spinor is $\\Psi ^{\\dag }_{\\bf k}=(c^{A\\dag }_{{\\bf k}\\uparrow }\\ c^{B\\dag }_{{\\bf k}\\uparrow }\\ c^{A\\dag }_{{\\bf k}\\downarrow }\\ c^{B\\dag }_{{\\bf k}\\downarrow })$ , and the Hamiltonian matrix is ${\\cal H}=\\left(\\begin{array}{ccccc}&-\\mu ^{\\prime } &\\epsilon &Qe^{i\\phi }&0 \\\\&\\epsilon & -\\mu ^{\\prime }&0 & -Qe^{i\\phi } \\\\&Qe^{-i\\phi }&0 &-\\mu ^{\\prime } &\\epsilon \\\\&0 & -Qe^{-i\\phi }&\\epsilon &-\\mu ^{\\prime }\\\\\\end{array}\\right),$ yielding the energy spectra $E_{\\pm }({\\bf k})=-\\mu ^{\\prime }({\\bf k})\\pm E_q({\\bf k}),$ where $\\mu ^{\\prime }({\\bf k})=\\mu -Un +4t^{\\prime }\\cos k_x \\cos k_y$ , $E_q({\\bf k})=\\sqrt{\\epsilon ^2({\\bf k})+(UQ)^2}$ , and $\\epsilon ({\\bf k})=-2t(\\cos k_x+\\cos k_y)$ .", "Because the energy spectra $E_\\pm ({\\bf k})$ do not depend on the phase $\\phi $ one should be able to transform $\\cal H$ to a matrix that does not depend on the phase.", "This can indeed be done using the spin-dependent gauge transformation $c_{i,\\uparrow }\\rightarrow e^{i\\phi /2}c_{i,\\uparrow }$ and $c_{i,\\downarrow }\\rightarrow e^{-i\\phi /2}c_{i,\\downarrow }$ .", "This transformation is equivalent to performing a rotation by angle $-\\phi $ about the $z$ axis for the $x$ and $y$ components of the spin operator.", "Indeed, upon using this gauge transformation, the spin ladder operators transform according to $S^+_{i}\\rightarrow e^{-i\\phi }S^+_{i}$ and $S^-_{i}\\rightarrow e^{i\\phi }S^-_{i}$ , which yields: $\\left(\\begin{array}{cc}&S_i^x\\\\&S_i^y\\end{array}\\right)\\rightarrow \\left(\\begin{array}{cc}\\cos \\phi &\\sin \\phi \\\\-\\sin \\phi &\\cos \\phi \\end{array}\\right)\\left(\\begin{array}{cc}&S_i^x\\\\&S_i^y\\end{array}\\right).$ The thermal averages of $S_i^x$ and $S_i^y$ are given by $\\frac{\\langle S_i^x\\rangle }{\\hbar } &=& Q\\cos \\phi ,\\ \\frac{\\langle S_i^y\\rangle }{\\hbar } = -Q\\sin \\phi ,\\ \\ i \\in A, \\ {\\rm or}\\cr \\frac{\\langle S_i^x\\rangle }{\\hbar } &=& -Q\\cos \\phi ,\\ \\frac{\\langle S_i^y\\rangle }{\\hbar } = Q\\sin \\phi ,\\ \\ i \\in B.$ Note that $\\langle S_i^z\\rangle =0$ for $i$ in both sublattices.", "Because the phase $\\phi $ assumes any value between 0 and $2\\pi $ (see below), rotational symmetry will not look broken for times greater than the period of rotation.", "However if the typical time scale of a probe is much smaller than this period symmetry may appear broken." ], [ "Calculation of the time dependence of phase $\\phi $", "The magnitude $Q$ , which was calculated using the minimization of the mean-field free energy [5], [6], [7], behaves as in a second-order phase transition in agreement with experimental evidence in [4].", "Next we calculate the phase using the Heisenberg equation $\\frac{dS_j^+}{d\\tau }=\\frac{1}{i\\hbar }[S_j^+,H]$ .", "We consider the limit where electron hopping is neglected in comparison to $\\frac{U}{2}(S_j^+S_j^- + S_j^-S_j^+)$ .", "The limit considered here is $U\\sim 3t$ -$5t$ ; this is an intermediate coupling limit where $U>t$ but smaller than the bandwidth $\\sim 8t$ when $t^{\\prime }\\ll t$ .", "It is justified to use this approximation because spin dynamics is faster than charge dynamics; i.e., an onsite spin flip needs a time $\\tau \\sim \\hbar /U$ to be realized, while a charge hopping between adjacent sites takes a longer time $\\tau \\sim \\hbar /t$ , ($U>t$ ).", "In the Heisenberg equation the undecoupled interaction is used instead of RAFT's Hamiltonian (REF ) in order to treat as best as possible quantum fluctuations.", "To carry on the calculation, we keep in mind that any site $j$ is on average only partially occupied, and that $|\\langle S_j^\\pm \\rangle |<\\hbar /2$ .", "For this reason, terms like $S_j^+S_j^+S_j^-$ and $S_j^+S_j^-S_j^+$ should be kept until the end (these terms normally give zero when acting on a spin up state, but a nonzero contribution is expected when applied to a partially occupied state where thermal averages are meaningful and suitable).", "In the commutator of the Heisenberg equation $[S_j^+,H]\\approx -\\frac{U}{2}[S_j^+,(S_j^+S_j^- + S_j^-S_j^+)]$ , we need to calculate $[S_j^+,(S_j^+S_j^- + S_j^-S_j^+)]=[S_j^+,S_j^+S_j^-]+[S_j^+,S_j^-S_j^+]=2\\hbar (S_j^+S_j^z + S_j^zS_j^+)$ .", "Using the fundamental commutation relation $[S_j^z,S_j^+]=\\hbar S_j^+$ , one gets $S_j^+S_j^z + S_j^zS_j^+=\\hbar S_j^+ + 2 S_j^+S_j^z$ , which leads to $\\frac{dS_j^+}{d\\tau }= iS_j^+\\bigg (\\frac{U}{\\hbar }+ \\frac{2U}{\\hbar ^2}S_j^z\\bigg ),\\ \\ \\tau \\ {\\rm is\\ time}.$ Again we stress that this equation is obtained in the intermediate coupling limit ($U$ smaller than the bandwidth but higher than hopping energies), where spin dynamics is not governed by the Heisenberg exchange coupling $\\sim t^2/U$ suitable for the strong coupling limit.", "Eq.", "(REF ) gives zero when acting on state $|\\uparrow \\rangle $ or $|\\downarrow \\rangle $ .", "However, for a collective state where any site is only partially occupied, one has to take the thermal average of Eq.", "(REF ).", "One then replaces $S_i^z$ by its RAFT's thermal average, which is zero.", "Integrating Eq.", "(REF ) gives for the thermal average $\\langle S_j^+(\\tau )\\rangle \\approx \\langle S_j^+(0)\\rangle e^{iU\\tau /\\hbar },$ which yields $\\phi =U\\tau /\\hbar $ modulo $2\\pi $ when $\\langle S_j^+(0) \\rangle $ is identified with $|\\langle S_j^+(\\tau )\\rangle |$ , ($-|\\langle S_j^+(\\tau )\\rangle |$ ), for sublattice $A$ , ($B$ ), and $e^{i\\phi }$ with $e^{iU\\tau /\\hbar }$ .", "The angular frequency is thus $\\omega _{sf}=U/\\hbar $ , and period $T_{sf}=2\\pi \\hbar /U$ is the time required to perform a spin-flip process, or the time needed for the rotating order parameter $\\langle S_i^{x(y)} \\rangle $ to complete a $2\\pi $ revolution in a classical point of view.", "The magnetic configuration (REF ) takes on the following form $\\langle S_i^x\\rangle /\\hbar = Q\\cos (\\omega _{sf} \\tau )$ , $\\langle S_i^y\\rangle /\\hbar = -Q\\sin (\\omega _{sf} \\tau )$ for $i$ in sublattice $A$ or $\\langle S_i^x\\rangle /\\hbar = -Q\\cos (\\omega _{sf} \\tau )$ , $\\langle S_i^y\\rangle /\\hbar = Q\\sin (\\omega _{sf} \\tau )$ for $i$ in sublattice $B$ , and $\\langle S_i^z\\rangle = 0$ for $i$ in sublattice $A$ or $B$ ." ], [ "Interpretation of rotating antiferromagnetism", "To illustrate well rotating magnetic order, consider first the much simpler example of the time evolution of a single spin in a magnetic field $B$ along the $z$ -axis, with the initial state given for a spin pointing in the positive $x$ -direction by $| S_x,+\\rangle = {\\frac{1}{\\sqrt{2}}}(|+\\rangle + |-\\rangle )$ .", "The time-dependent expectation values of the spin components are $\\langle S^x\\rangle =\\frac{\\hbar }{2}\\cos (\\omega t)$ , $\\langle S^y\\rangle =\\frac{\\hbar }{2}\\sin (\\omega t)$ , and $\\langle S^z\\rangle =0$ , with $\\omega =\\frac{|e|B}{m_ec}$ .", "$e$ and $m_e$ are the charge and mass of the electron, respectively, and $c$ is the speed of light.", "Classically speaking, the spin is confined to rotate about the $z$ -axis in the $xy$ plane with Larmor angular frequency $\\omega $ .", "A rotating ferromagnetic state can be realized by placing $N$ such states with the same frequency on a lattice made of $N$ sites.", "For a rotating antiferromagetic state, opposite initial states ($\\pm | S_x,+\\rangle $ : spins point in opposite directions on the $x$ -axis) are required on each two adjacent sites of the lattice.", "To relate RAF to spin flip processes, we note that $\\langle S^\\pm \\rangle =\\langle S^x\\rangle \\pm i\\langle S^y\\rangle =\\frac{\\hbar }{2}e^{\\pm i\\omega t}$ in this example.", "Note that in this example is model independent, which may indicate that all model parameters will do is changing multiplying physical factors, not the physics itself.", "In a given model, a coupling is necessary for providing the building bloc for RAF, which is the precession of a spin (with no local magnetization) for each lattice site.", "The RAF state constructed in this way shows a hidden order that can be realized even at finite temperature without violating the Mermin-Wagner theorem [16].", "The example above allows us to interpret RAF as a state where spins precess collectively in a synchronized manner in the spins' $xy$ plane around an effective staggered magnetic field $B=m_ecU/\\hbar |e|$ caused by onsite Coulomb repulsion.", "For our many-body system, $\\hbar /2$ in $\\langle S^\\pm \\rangle =\\frac{\\hbar }{2}e^{\\pm i\\omega t}$ is replaced by the magnitude of the RAF order parameter $Q$ , which can assume values smaller than $1/2$ due to thermal averaging.", "This state is strongly doping dependent.", "When doping increases, $Q$ rapidly decreases then vanishes at a doping identified as the quantum critical point underneath the superconducting dome [5], [6], [7], [12], [13], [14].", "In comparison to ordinary spin waves in an antiferromagnet, RAF's state could be viewed as a ${\\bf q}=(\\pi ,\\pi )$ spin wave in an antiferromagnet with zero magnetization.", "Note however that for our system (where $\\langle S_i^z\\rangle =0$ ), spin-wave theory is not applicable because the spin-wave theory is built around a stable nonzero $\\langle S_i^z\\rangle $ state." ], [ "conclusion", "The rotating antiferromagnetism theory and Heisenberg equation are combined in order to calculate the phase of the rotating order parameter.", "This phase behaves linearly in time.", "This allows us to interpret rotating antiferromagnetism in terms of a Larmor-like spin precession about an effective magnetic field, which is proportional to onsite Coulomb repulsion.", "Another way to see rotating antiferromagnetism is as an unusual spin-wave at ${\\bf q}=(\\pi ,\\pi )$ around a zero magnetization.", "This work was necessary for unveiling the nature of rotating antiferromagnetism, which has been proposed for explaining the pseudogap behavior in high-$T_C$ materials.", "Rotating antiferromagnetic order is an example of hidden order, which is a serious candidate for the PG state in HTSCs.", "This is supported by the good success of the rotating antiferromagnetism theory in the calculation of thermodynamics [5], [6], [7], optical conductivity [12], [15], Raman [13], and angle-resolved photoemission spectroscopy properties [14].", "Author wishes to thank A.-M. S. Tremblay for helpful comments on the manuscript." ] ]

[ [ "Gravitating tensor monopole in a Lorentz-violating field theory" ], [ "Abstract We present a solution of the coupled Einstein and rank-two antisymmetric tensor field equations where Lorentz symmetry is spontaneously broken, and we discuss its observational signatures.", "Especially, the deflection angles have important qualitative differences between tensor and scalar monopoles.", "If a monopole were to be detected, it would be discriminated whether or not to correspond to a tensor one.", "This phenomenon might open up new direction in the search of Lorentz violation with future astrophysical observations." ], [ "INTRODUCTION", "Lorentz symmetry is cornerstone in the foundation of modern physics.", "The experimental tests of Lorentz violation are also interested for a decade (see Ref.", "[1] and references therein).", "The possibility of Lorentz-violating field theory were intensively studied in the various contexts, including Riemann-Cartan geometry [2], Riemann-Finsler geometry [3], string theory [4], and noncommutative geometry [5].", "Especially, a tensor field theory with dynamical Lorentz symmetry violating such that the manifold of equivalent vacua after the violation is not shrinkable to a point may contain monopole solutions [6], [7].", "There exist monopole solutions in the minimal model coupled to gravity [6], [7] for antisymmetric 2-tensor field, in which the far-field approximation and Bogomol'nyi-Prasad-Sommerfield (BPS) limit are used.", "It is worth noting that above-mentioned solution is not exact one for the metric around a tensor monopole since it is not the solution of the equation of the tensor field.", "One must still find a solution of the coupled equations of motion valid throughout space.", "On the other hand, if the symmetry that is broken is a global symmetry of scalar fields, the gravitational effect of monopole configuration [8] is equivalent to that of a deficit solid angle in the metric, plus that of a negative mass at the origin [9], [10].", "The properties of scalar monopoles have been investigated in the various space-time [11].", "Monopoles could be produced by the phase transition in the early Universe and their existence has important implications in cosmology.", "It is possible that the monopole still exist as relic object in the Universe today, since isolated topological defect is stable.", "If a tensor monopole were to be detected, it would offer precious enlightenment on fundamental symmetries in physics.", "For the scalar case, the internal symmetry is spontaneously broken, and Lorentz symmetry is exact.", "On the contrary, Lorentz symmetry will be broken by the vacuum solution in the tensor case.", "However, the signature of tensor monopole is effectively the same as a scalar one in the Seifert's approximation.", "It is of course not possible to use tensor monopole set-ups to assess the existence of Lorentz violation in this approximation.", "In this paper, we propose a novel approach that might relax the above problem and open new possibilities for the detection of Lorentz violation in future astrophysical observations.", "We show the realistic solution for the coupled system, and discuss its observational signatures.", "Using standard techniques, we have calculated the light ray propagating in these backgrounds.", "Note that the deflection angles are dependent upon the \"apparent impact parameter\".", "From this point of view, the signature of an antisymmetric tensor monopole can be distinguished from two species of monopoles in the future tests.", "Moreover, the tensor monopole would provide inestimable insight into the role played by Lorentz symmetry in physics." ], [ " The field equations", "We consider the 1+3 dimensional action $S=\\int { d^{4}x\\sqrt{-g}(\\frac{R}{16\\pi G}-\\mathcal {L}_m)},$ where the gravity part of the action is the usual Einstein-Hilbert action with the gravitational coupling constant $G$ and curvature scalar $R$ .", "$\\mathcal {L}_m$ is the Lagrangian of an antisymmetric 2-tensor field which takes on a background expected value [6], $\\mathcal {L}_m=-\\frac{1}{6}F^{\\mu \\nu \\rho }F_{\\mu \\nu \\rho }-\\frac{\\lambda }{2}(B^{\\mu \\nu }B_{\\mu \\nu }-b^2)^2,$ where $B_{\\mu \\nu }$ is an antisymmetric tensor field and $F_{\\mu \\nu \\rho }=3\\partial _{[\\mu }B_{\\nu \\rho ]}$ is its associated field strength.", "Sometime, $B_{\\mu \\nu }$ is known as the Kalb-Ramond field [12], [13].", "For the metric, the spherically symmetric ansätz in Schwarzschild-like coordinates reads: $ds^2=-E(r)dt^2+F(r)dr^2+r^2(d\\theta ^2+\\sin ^2\\theta d\\varphi ^2),$ while for the Kalb-Ramond field, we also choose the spherically symmetric ansätz [6]: $B_{tr}=-B_{rt}=0, B_{\\theta \\varphi }=-B_{\\varphi \\theta }=g(r)r^2\\sin ^2\\theta .$ Using Eqs.", "(1)-(4), the equation of motion for the Kalb-Ramond field can be reduced to $\\frac{1}{2}(\\frac{E^{\\prime }}{E}-\\frac{F^{\\prime }}{F})(g^{\\prime }+\\frac{2}{r}g)+\\frac{\\partial }{\\partial r}(g^{\\prime }+\\frac{2}{r}g)-2\\lambda Fg(2g^2-b^2)=0$ where primes denote differentiation with respect to $r$ .", "As a general feature, a solution with $g(0)=0$ and $g(r)\\rightarrow b/\\sqrt{2}$ as $r\\rightarrow \\infty $ corresponds to a monopole configuration since the vacuum manifold contains a non-contractible two-sphere (i.e., $\\pi _2(\\mathcal {M}_{vac})=\\mathbb {Z}$ ).", "Let us mention, by the way, that the equation of motion is analogous between the tensor monopole and $O(3)$ scalar monopole.", "In the both cases, no exact expression is known, although series method [9] or numerical calculation [10] can be used to approximate it for the $O(3)$ monopole.", "The energy-momentum tensor of a tensor monopole configuration is $T_{t}^{t}&=&-\\frac{1}{F}(g^{\\prime }+\\frac{2}{r}g)^2-\\frac{\\lambda }{2}(2g^2-b^2)^2,\\\\T_{r}^{r}&=&\\frac{1}{F}(g^{\\prime }+\\frac{2}{r}g)^2-\\frac{\\lambda }{2}(2g^2-b^2)^2,\\\\T_{\\theta }^{\\theta }&=&T_{\\varphi }^{\\varphi }=\\frac{1}{F}(g^{\\prime }+\\frac{2}{r}g)^2-\\frac{\\lambda }{2}(2g^2-b^2)^2+4\\lambda g^2(2g^2-b^2).\\nonumber \\\\$ Varying the action (1) with respect to the metric fields gives the Einstein equations $-\\frac{1}{F}(\\frac{1}{r^2}-\\frac{F^{\\prime }}{Fr})+\\frac{1}{r^2}&=&\\frac{\\epsilon }{2b^2}[\\frac{1}{F}(g^{\\prime }+2\\frac{g}{r})^2+\\frac{\\lambda }{2}(2g^2-b^2)^2]\\nonumber \\\\\\\\-\\frac{1}{F}(\\frac{1}{r^2}+\\frac{E^{\\prime }}{Er})+\\frac{1}{r^2}&=&\\frac{\\epsilon }{2b^2}[-\\frac{1}{F}(g^{\\prime }+2\\frac{g}{r})^2+\\frac{\\lambda }{2}(2g^2-b^2)^2]\\nonumber \\\\$ where the dimensionless quantity $\\epsilon \\equiv 16\\pi G b^2$ .", "In order to solve the system of equations (5),(9) and (10) uniquely, we have to introduce 6 boundary conditions, which we choose to be $&g&(0)=0,F(0)=1,E(0)=e_0,\\nonumber \\\\&g&(r)|_{r\\rightarrow \\infty }=\\frac{b}{\\sqrt{2}},E(r)r^{-2\\epsilon }|_{r\\rightarrow \\infty }=(2\\lambda b^2)^{\\epsilon },\\nonumber \\\\&F&(r)|_{r\\rightarrow \\infty }=1+\\epsilon .$" ], [ "THIN-WALL APPROXIMATION", "We start our discussion with a simplified model for the monopole configuration, just to show the main features of the exact solution in a simple manner.", "Let us modeling the monopole configuration in the thin-wall limit $g= \\left\\lbrace \\begin{array}{ll}0&\\textrm {if r<\\delta }\\\\\\frac{b}{\\sqrt{2}}&\\textrm {if r>\\delta }\\end{array}\\right.$ where $\\delta $ is the core radius.", "Einstein equations inside the core are solved by a de Sitter metric $ds^2=-(1-\\frac{\\lambda \\epsilon b^2r^2}{12})dt^2+\\frac{dr^2}{1-\\frac{\\lambda \\epsilon b^2r^2}{12}}+r^2d\\Omega ^2.$ The exterior solution is given by $ds^2=&-&(\\sqrt{2\\lambda }br)^{2\\epsilon }(1-\\frac{2GM}{(\\sqrt{2\\lambda }b)^\\epsilon r^{1+\\epsilon }})dt^2 \\nonumber \\\\&+&\\frac{1+\\epsilon }{1-\\frac{2GM}{(\\sqrt{2\\lambda }b)^\\epsilon r^{1+\\epsilon }}}dr^2+r^2d\\Omega ^2,$ where $M$ is an arbitrary constant of integration.", "Both $\\delta $ and $M$ are determined by Eqs.", "(9) and (10) at the boundary between the interior and exterior region, which correspond to the continuity of the metric.", "The result is $\\delta &=&\\frac{1}{\\sqrt{2\\lambda }b}(\\frac{1}{1+\\epsilon })^\\frac{1}{2\\epsilon }\\\\M&=&-\\frac{8\\pi b}{\\sqrt{2\\lambda }}[1-\\frac{(1+\\epsilon )^{1-\\frac{1}{\\epsilon }}}{24}](\\frac{1}{1+\\epsilon })^{\\frac{1+\\epsilon }{2\\epsilon }}$ We argue that it is possible to match an interior de Sitter solution to an exterior tensor monopole solution, but only for $M<0$ .", "This property is consistent with the negative mass of scalar monopole [9].", "Furthermore, we have $\\frac{1}{\\sqrt{2\\lambda e}b}\\le \\delta \\le \\frac{1}{2\\sqrt{\\lambda }b}$ and $-\\frac{23\\pi b}{\\sqrt{72\\lambda }}\\le M\\le -(8e^{-1/2}-\\frac{1}{3}e^{-3/2})\\frac{\\pi b}{\\sqrt{2\\lambda }}$ for $0\\le \\epsilon \\le 1$ , where $e=2.71828\\cdots $ is base of natural logarithm.", "It is worth noting that the solution of BPS limit [6] is not exact solutions for the metric around a tensor monopole, since it is not the solution of Eq.(5).", "In other words, Seifert's result [6] only describes the scene of far-field.", "Eq.", "(12) is an approximative solution of the Kalb-Ramond field in the thin-wall limit.", "Therefore, the simplified model shares some features of the realistic solution for the coupled Einstein-Kalb-Ramond system of equations (5), (9) and (10), as we shall rigorously confirm in the next section." ], [ "THE SOLUTION FOR THE COUPLED SYSTEM", "The asymptotic form of the functions $E(r)$ , $F(r)$ and $g(r)$ can be systematically constructed in both regions, near the origin and for $r\\rightarrow \\infty $ .", "Expanding the functions around the origin gives: $E(r)&=&e_0(1+(\\frac{3\\epsilon }{b^2}g_1^2-\\frac{b^2\\epsilon \\lambda }{12})r^2 \\nonumber \\\\&+&[\\frac{27\\epsilon ^2}{10b^4}g_1^4+(\\frac{7\\epsilon \\lambda }{10}+\\frac{3\\epsilon ^2\\lambda }{40})g_1^2]r^4+O(r^6))\\\\F(r)&=&1+(\\frac{3\\epsilon }{2b^2}g_1^2+\\frac{b^2\\epsilon \\lambda }{12})r^2 \\nonumber \\\\&+&[(\\frac{13\\epsilon ^2\\lambda }{40}-\\frac{4\\epsilon \\lambda }{5})g_1^2-\\frac{9\\epsilon ^2}{20b^4}g_1^4+\\frac{b^4\\epsilon ^2\\lambda ^2}{144}]r^4+O(r^6)\\nonumber \\\\ \\\\g(r)&=&g_1(r+[(\\frac{\\epsilon }{20}-\\frac{1}{5})\\lambda b^2-\\frac{9\\epsilon }{20b^2}g_1^2]r^3 \\nonumber \\\\ &+&[(\\frac{1}{5}-\\frac{13\\epsilon }{60}+\\frac{\\epsilon ^2}{24})\\frac{\\lambda ^2b^4}{14}+(1-\\frac{3\\epsilon }{10}-\\frac{3\\epsilon ^2}{20})\\frac{\\lambda g_1^2}{7}\\nonumber \\\\&+&\\frac{27\\epsilon ^2g_1^4}{160b^4}]r^5+O(r^7))$ where $g_1$ and $e_0\\equiv E(0)$ are free parameters to be determined numerically.", "The asymptotic behavior ($r\\gg (\\sqrt{2\\lambda }b)^{-1}$ ) is given by $E(r)&=&(\\sqrt{2\\lambda }br)^{2\\epsilon }(1+\\frac{(1-\\epsilon )\\epsilon }{4\\lambda b^2(1+\\epsilon )}\\frac{1}{r^2}\\nonumber \\\\&-&\\frac{\\epsilon (1-\\epsilon )^3}{8\\lambda ^2 b^4(3-\\epsilon )(1+\\epsilon )^2}\\frac{1}{r^4}+O(\\frac{1}{r^6})),\\\\F(r)&=&(1+\\epsilon )(1-\\frac{\\epsilon (1-\\epsilon )}{4\\lambda b^2(1+\\epsilon )}\\frac{1}{r^2}\\nonumber \\\\&+&\\frac{\\epsilon (\\epsilon -1)(\\epsilon ^3-4\\epsilon ^2-5\\epsilon +16)}{16\\lambda ^2b^4(3-\\epsilon )(1+\\epsilon )^2}\\frac{1}{r^4}+O(\\frac{1}{r^6})),\\\\g(r)&=&\\frac{b}{\\sqrt{2}}-\\frac{1-\\epsilon }{2\\sqrt{2}\\lambda b(1+\\epsilon )}\\frac{1}{r^2}\\nonumber \\\\&-&\\frac{(1-\\epsilon )(3-\\epsilon ^2)}{8\\sqrt{2}\\lambda ^2b^3(1+\\epsilon )^2}\\frac{1}{r^4}+O(\\frac{1}{r^6}).$ It is obviously that $F(r)$ will converge to $(1+\\epsilon )$ , but $E(r)$ grows without bound as $r\\rightarrow \\infty $ and $E(r)\\propto r^{2\\epsilon }$ .", "From Eqs.", "(6)-(8), we have $\\rho +p_r+2p_\\theta =\\frac{4b^2}{r^2}+\\frac{(1-\\epsilon )^3}{(1+\\epsilon )^2\\lambda }\\frac{1}{r^4}+O(\\frac{1}{r^6})$ which is proportional to the $tt$ component of the trace-reversed energy-momentum tensor and couples to the $tt$ component of the metric in the linearized approximation [6].", "On the contrary, $\\rho +p_r+2p_\\theta $ falls off as $r^{-4}$ for the $O(3)$ scalar monopole.", "Therefore, their gravitational fields have essential differences.", "If the mass scale $b$ is well below the Planck scale, the far-field shall become sufficiently flat so that the solution of tensor monopole can be embedded in one describing the suitable large-scale structure.", "The limit of flat space is recovered for $\\epsilon =0$ , $e_0=1$ and $E(r)=F(r)=1$ in Eqs.", "(17)-(22), and $g_1$ is determined numerically.", "We do that through a fourth-order Runge-Kutta method for the quantity $\\tilde{g}(\\tilde{r})$ , where $\\tilde{g}\\equiv \\frac{g}{b}$ and $\\tilde{r}=\\sqrt{2\\lambda }br$ is a dimensionless parameter.", "We impose the initial conditions at the origin $\\tilde{g}(0)=0$ and $\\dot{\\tilde{g}}(0)=\\frac{g_1}{\\sqrt{2\\lambda }b^2}$ , where overdot denotes differentiation with respect to $\\tilde{r}$ .", "$g_1$ is adjusted so that $\\tilde{g}\\rightarrow \\frac{1}{\\sqrt{2}}$ for large $\\tilde{r}$ using shooting routine.", "We display $\\tilde{g}(\\tilde{r})$ in Fig.1 for the case of flat spacetime.", "Next, we present the numerical solutions of the full system of field equations coupled to gravity.", "These solutions are the gravitating generalization of the flat spacetime one.", "To evaluate the solutions of full system by numerical method, the boundary conditions (11) can be reduced to $&\\tilde{g}&(0)=0,F(0)=1,E(0)=e_0,\\nonumber \\\\&\\tilde{g}&(\\tilde{R})=\\frac{1}{\\sqrt{2}},E(\\tilde{R})=\\tilde{R}^{2\\epsilon },F(\\tilde{R})=1+\\epsilon ,$ up to $O(\\tilde{R}^{-2})$ order for large $\\tilde{R}$ , where $\\tilde{R}=\\sqrt{2\\lambda }bR$ .", "To use shooting method, we impose the initial conditions at the origin $\\tilde{g}(0)=0$ , $\\tilde{E}(0)=0$ , $\\tilde{F}(0)=0$ and $\\dot{\\tilde{F}}(0)=1$ .", "The values of $\\dot{\\tilde{g}}(0)$ and $\\dot{\\tilde{E}}(0)$ are adjusted so that $\\tilde{g}(\\tilde{R})=\\frac{1}{\\sqrt{2}}$ and $\\tilde{E}(\\tilde{R})=\\tilde{R}^{1-2\\epsilon }$ , where $\\tilde{E}\\equiv \\frac{\\tilde{r}}{E}$ and $\\tilde{F}\\equiv \\frac{\\tilde{r}}{F}$ .", "In Fig.1, we display $\\tilde{g}(\\tilde{r})$ for $\\epsilon =0$ and $10^{-2}$ .", "The profile of $\\tilde{g}(\\tilde{r})$ is insensitive to $\\epsilon $ for $0\\le \\epsilon \\le 1$ not only asymptotically, but also close to the origin.", "In Fig.2, the components of metric $E(\\tilde{r})$ and $F(\\tilde{r})$ are plotted vs the dimensionless coordinate $\\tilde{r}=\\sqrt{2\\lambda }br$ for different $\\epsilon $ .", "Moreover, both $E(\\tilde{r})$ and $F(\\tilde{r})$ increase with $\\epsilon $ increasing.", "Figure: The function of g ˜(r ˜)\\tilde{g}(\\tilde{r}) corresponding to the monopole configuration, is plotted vs the dimensionless coordinate r ˜=2λbr\\tilde{r}=\\sqrt{2\\lambda }br for ϵ=0\\epsilon =0 (dash line) and 10 -2 10^{-2} (solid line).", "The shape of the curve is quite insensitive to the value ϵ\\epsilon in the range of 0≤ϵ≤10\\le \\epsilon \\le 1.Figure: E(r ˜)E(\\tilde{r}) (solid line) and F(r ˜)F(\\tilde{r}) (dash line) are plotted vs the dimensionless coordinate r ˜=2λbr\\tilde{r}=\\sqrt{2\\lambda }br for different values of ϵ\\epsilon .", "It is clearly seen from this figure that both E(r ˜)E(\\tilde{r}) and F(r ˜)F(\\tilde{r}) increase with ϵ\\epsilon increasing." ], [ "OBSERVATIONAL SIGNATURES", "We now study the motion of test photons around a tensor monopole.", "Eqs.", "(20)-(22) are good approximation unless we were interested in the test photons moving right into the monopole core $\\delta \\sim (\\sqrt{2\\lambda }b)^{-1}$ .", "In the case of BPS limit, Seifert have pointed out that the gravitational redshift experienced by a photon in the background of tensor monopole is within no more than $\\epsilon ^2$ order if the mass scale $b$ is well below the Planck scale [7].", "It is still kept that redshift effect will be negligible for the realistic solution of coupled system.", "However, the effect for the deflection of light by the gravitational field is more interesting.", "A null geodesics equation in the plane $\\theta =\\pi /2$ reads $-E(r)\\dot{t}^2+F(r)\\dot{r}^2+r^2\\dot{\\varphi }^2=0$ where dot denotes the derivative with respect to some affine parameter on the worldline.", "Since the metric is spherically symmetric and static, there are two Killing vector field $t^\\mu $ and $\\varphi ^\\mu $ leading to two constants of the motion: $\\mathcal {E}=E^2\\dot{t}$ and $\\mathcal {J}=r^2\\dot{\\varphi }$ .", "From Eq.", "(25), we have $\\frac{d\\varphi }{dr}=\\pm \\frac{1}{r^2}\\frac{1}{\\sqrt{\\beta ^{-2}\\frac{1}{EF}-\\frac{1}{Fr^2}}}$ where $\\beta =\\mathcal {J}/\\mathcal {E}$ .", "If $\\mathcal {E}^2<(2+\\frac{2}{3}\\epsilon -\\frac{7}{9}\\epsilon ^2+\\frac{14}{9}\\epsilon ^3)\\lambda b^2\\mathcal {J}^2$ , we have $r_m>(\\sqrt{2\\lambda }b)^{-1}$ , where $r_m$ is the value of $r$ for which the denominator of Eq.", "(26) vanishes.", "In other words, $r_m$ is the largest root of the equation $\\beta ^2E(r_m)=r_m^2$ and is larger than the core radius of the monopole.", "The orbit of the light ray will have a \"turning point\" at $r=r_m$ .", "In this case, we have approximate expression of the total angular deflection up to $\\tilde{r}_m^{-4}$ ($\\tilde{r}_m=\\sqrt{2\\lambda }br_m$ ) order $\\Delta \\varphi &=&\\frac{\\sqrt{1+\\epsilon }}{1-\\epsilon }(\\pi -2Arcsin[\\sqrt{\\frac{\\epsilon (1-\\epsilon )}{2(1+\\epsilon )}}\\frac{1}{\\tilde{r}_m}\\nonumber \\\\&-&(\\frac{\\sqrt{\\epsilon }(1-\\epsilon )^{5/2}}{2\\sqrt{2}(3-\\epsilon )(1+\\epsilon )^{3/2}}+\\frac{\\epsilon ^{3/2}(1-\\epsilon )^{3/2}}{4\\sqrt{2}(1+\\epsilon )^{3/2}})\\frac{1}{\\tilde{r}_m^3}])\\nonumber \\\\&+&\\frac{1}{2}\\frac{\\epsilon (1-\\epsilon )}{\\sqrt{1+\\epsilon }(5-3\\epsilon )}\\frac{1}{\\tilde{r}_m^2}-\\frac{1}{2}\\frac{\\epsilon (1-\\epsilon )}{(1+\\epsilon )^{3/2}}\\nonumber \\\\ &\\times &[\\frac{3-\\epsilon ^2}{5-\\epsilon }+\\frac{11-7\\epsilon -\\epsilon ^2+\\epsilon ^3}{2(3-\\epsilon )(7-3\\epsilon )}+\\frac{3\\epsilon (1-\\epsilon )}{4(5-3\\epsilon )}]\\frac{1}{\\tilde{r}_m^4}.\\nonumber \\\\$ Defining $\\delta \\varphi \\equiv \\Delta \\varphi -\\pi $ to be the angle between the \"unperturbed\" and \"perturbed\" directions of propagation up to $\\epsilon $ order $\\delta \\varphi \\approx \\frac{3}{2}\\pi \\epsilon -\\sqrt{2\\epsilon }\\tilde{r}_m^{-1}+\\frac{\\epsilon }{10}\\tilde{r}_m^{-2}+\\frac{\\sqrt{2\\epsilon }}{6}\\tilde{r}_m^{-3}-\\frac{181\\epsilon }{420}\\tilde{r}_m^{-4}$ Obviously, $\\delta \\varphi \\approx \\frac{3}{2}\\pi \\epsilon $ in the case of $\\tilde{r}_m\\gg \\epsilon ^{-1}$ , i.e, we repeat Seifert's approximation [7].", "By using same techniques, we obtain the angular deflection $\\delta \\varphi _s$ for the case of $O(3)$ scalar monopole [14] $\\delta \\varphi _s\\approx \\frac{\\pi }{4}\\epsilon _s-\\sqrt{3\\epsilon _s/2}\\tilde{r}_m^{-1/2}+\\frac{25\\epsilon _s}{96}\\tilde{r}_m^{-2}+\\sqrt{2\\epsilon _s/3}\\tilde{r}_m^{-5/2}-\\frac{\\epsilon _s}{8}\\tilde{r}_m^{-4}$ where $\\epsilon _s=16\\pi G\\eta ^2$ and $\\eta $ is mass scale in the $O(3)$ scalar monopole.", "For contrasting the two species of monopoles, we take $\\eta =\\sqrt{3/2}b$ , Eq.", "(29) can be rewritten as $\\delta \\varphi _s\\approx \\frac{3}{2}\\pi \\epsilon -3\\sqrt{\\epsilon }\\tilde{r}_m^{-1/2}+\\frac{25\\epsilon }{16}\\tilde{r}_m^{-2}+2\\sqrt{\\epsilon }\\tilde{r}_m^{-5/2}-\\frac{3\\epsilon }{4}\\tilde{r}_m^{-4}$ Therefore, the deflection angles have important qualitative differences between the tensor and scalar monopoles.", "It furnishes a possibility that two species are discriminated by the observation of light rays in these backgrounds.", "In Fig.3, we plotted the $\\delta \\varphi $ and $\\delta \\varphi _s$ vs the parameter $r_m$ for a typical grand unification scale $b\\sim 10^{16}GeV$ .", "By the numerical calculation, we show that $\\delta \\varphi $ is quite insensitive and $\\delta \\varphi _s$ is sensitive in the same interval of $r_m$ .", "In Fig.4, we plotted the $\\delta \\varphi $ and $\\delta \\varphi _s$ vs the apparent impact parameter $\\beta $ for $\\epsilon =10^{-2}$ .", "Both tensor and scalar monopoles have tiny core radius $\\delta \\sim \\delta _s\\sim (\\sqrt{2\\lambda }b)^{-1}$ , therefore Eqs.", "(28) and (30) are very accurate expressions when $r_m$ is far larger than the core radius.", "Set $\\theta _{max}$ is the maximum of $|\\delta \\varphi _s-\\delta \\varphi |$ , we have $\\theta _{max}\\sim \\sqrt{\\epsilon \\delta /r_m}$ .", "If the source of light, the monopole, and the observer are aligned exactly, all the rays that pass at the appropriate parameter $r_m$ around the monopole, at any azimuth, reach the position of the observer.", "Under this special circumstance, the observer sees an infinite number of images, which form a ring around the monopole.", "Assuming the source is much farther from the monopole, its rays are then nearly parallel to the line of alignment, and the deflection angle required for the ray to reach the observer is $r_m/D$ , where $D$ is the distance from the monopole to the observer.", "Thus, the angular radius of Einstein ring is $r_m/D\\approx \\frac{3}{2}\\pi \\epsilon $ unless a monopole is nearing the solar system, which leads to $\\theta _{max}\\ll 10^{-9}$ radians.", "For $\\epsilon =10^{-2}$ and $D\\approx 10^4$ light-years, we have $r_m/D\\approx 0.05$ radians and $\\theta _{max}\\approx 10^{-25}$ radians.", "By means of observation of Einstein ring, a monopole is able to find but it is powerless to determine whether or not to correspond to a tensor one.", "Once that the Einstein rings are discovered, we have to go a step further by the deflection of light near the monopole.", "For a light ray just grazing the monopole, the effect is quite evident.", "If $r_m\\sim 10^{18}\\epsilon \\delta $ , we have $\\theta _{max}\\sim 10^{-9}$ radians.", "This angle is at the limit of resolution of telescope at present, so it can be observed when $r_m<10^{18}\\epsilon \\delta $ .", "On the other hand, the star near the Sun are visible only during a total eclipse of the Sun, and even then brightness of the solar corona restricts observations to $r_m>2R_\\odot $ .", "In the case of monopole, there do not exist these difficulties since the monopole is a cold and dark object.", "In conclusion, we have found that the deflection angles have important qualitative differences between tensor and scalar monopoles.", "This phenomenon might open up new direction in the search of Lorentz violation.", "If a monopole were detected, it would be discriminated whether or not corresponding to tensor one.", "Furthermore, tensor monopole would provide insight into the roles played by Lorentz symmetry in physics.", "Figure: The functions δϕ\\delta \\varphi and δϕ s \\delta \\varphi _s are plotted for ϵ=16πGb 2 =10 -6 \\epsilon =16\\pi Gb^2=10^{-6}, where the unit of r m r_m is 1 2λbϵ\\frac{1}{\\sqrt{2\\lambda }b\\epsilon } and unit of deflection angle is ϵ\\epsilon .", "The shape of the δϕ\\delta \\varphi curve is quite insensitive to the value of r m r_m in the interval 1 8λbϵ≤r m ≤25 8λbϵ\\frac{1}{\\sqrt{8\\lambda }b\\epsilon }\\le r_m\\le \\frac{25}{\\sqrt{8\\lambda }b\\epsilon }.", "On the contrary, the shape of δϕ s \\delta \\varphi _s is sensitive in this interval.Figure: The variations of δϕ\\delta \\varphi and δϕ s \\delta \\varphi _s with β\\beta are plotted for ϵ=16πGb 2 =0.01\\epsilon =16\\pi Gb^2=0.01, where the unit of β\\beta is 1 2λb\\frac{1}{\\sqrt{2\\lambda }b}.The shape of the δϕ s \\delta \\varphi _s curve is more sensitive to β\\beta than that of δϕ\\delta \\varphi , although both curves tend to 3 2πϵ\\frac{3}{2}\\pi \\epsilon when β\\beta is large enough.This work is supported by National Education Foundation of China under grant No.", "200931271104 and Innovation Program of Shanghai Municipal Education Commission (12YZ089)." ] ]

[ [ "Practical learning method for multi-scale entangled states" ], [ "Abstract We describe a method for reconstructing multi-scale entangled states from a small number of efficiently-implementable measurements and fast post-processing.", "The method only requires single particle measurements and the total number of measurements is polynomial in the number of particles.", "Data post-processing for state reconstruction uses standard tools, namely matrix diagonalisation and conjugate gradient method, and scales polynomially with the number of particles.", "Our method prevents the build-up of errors from both numerical and experimental imperfections." ], [ "Introduction", "Quantum state tomography [1] is a method to learn a quantum state from measurements performed on many identically prepared systems.", "This task is crucial not only to assess the degree of control exhibited during the preparation and transformation of quantum states, but also in comparing theoretical predictions to real-life systems.", "For instance, numerical methods are used to compute the ground states or thermal states of model quantum systems.", "Quantum state tomography could be used to check that the experimental state corresponds to the predicted one, thus providing an essential link between theory and experiments.", "For example, one could in principle use tomography to settle the question [2] of which states correctly describe the quantum Hall fluid at various filling parameters.", "In practice however, the state of $n$ particles is described by a number of parameters that scales exponentially with $n$ .", "Therefore, tomography requires an exponential number of identically prepared systems on which to perform exponentially many measurements needed to span a basis of observables that completely characterizes the state.", "Furthermore, solving the inference problem to determine the quantum state that is compatible with all these measurement outcomes requires an exponential amount of classical post-processing.", "These factors limit tomography to at most a few tens of particles.", "While this is unavoidable for a generic state, many states encountered in nature have special properties that could be exploited to simplify the task of tomography.", "In fact, the overwhelming majority of tomographic experiments performed to date [3], [4], [5], [6], [7], [8], [9] were used to learn state described with only a few parameters.", "Such variational states—family of states specified with only a few parameters—are widespread in many-body physics because they are tailored for numerical calculations and can predict many phenomena observed in nature (Kondo effect, superconductivity, fractional statistics, etc).", "One example, familiar to the quantum information community, is matrix product states (MPS) [10], [11], [12], [13] that are at the heart of the density matrix renormalisation group (DMRG) numerical method, suitable for the description of one-dimensional quantum systems with finite correlation length [14].", "Recently, we and others have demonstrated [15] that tomography can be performed efficiently on MPS, i.e., such states can be learned from a small number of simple measurements and efficient classical post-processing.", "Here, we take this result one step further and demonstrate that it is possible to efficiently learn the states associated to the multi-scale renormalization ansatz (MERA) introduced by Vidal [16], for which efficient numerical algorithms to minimize the energy of local Hamiltonians exist [17].", "As opposed to MPS, these MERA states are not restricted to one dimension and can describe systems with long-range correlation.", "This last distinction is important because one of the most interesting phenomena in physics, quantum phase transitions, leads to a diverging correlation length and are therefore not suitably described by MPS.", "In contrast, MERA have been successfully used to study numerous many-body models, such as the critical Ising model in 1D [17] and 2D [18], and can also accurately describe systems with topological order [19], [20], [21], [22], [23], [24].", "In this work, we present two related methods to learn the one-dimensional MERA description of a state using tomographic data obtained from local measurements performed on several copies of the states.", "Our learning methods for MERA are based on the identification of the unitary gates in the quantum circuit that outputs the MERA state.", "In that regard, this Article is a continuation of our work on MPS and is reminiscent of early methods to numerically optimise MERA tensors [25].", "However, going from MPS to MERA is non-trivial because MERA exhibits a spatial arrangement of gates that is more elaborate.", "Since MERA is a powerful numerical tool, our learning method bridges the gap between numerical simulations and experiments by allowing the direct comparison of numerical predictions to experimental states.", "The first method we present requires unitary control of the system and the ability to perform tomography on blocks of a few particles, which can be realized using the correlations between single-particle measurements.", "Crucially, the size of those blocks does not depend on the total size of the system, making it a scalable method.", "In an experiment, one cannot know beforehand if the state in the lab is a MERA.", "However, our method contains a built-in certification procedure from which one can assess the proper functioning of the method as the experiments are performed and conclusively determine if the state is well described by the MERA.", "The second method builds on the first one, but completely circumvents the need for unitary control.", "Thus, this MERA learning method can be implemented with existing technologies.", "The drawback of this simplified method is that it does not come with a built-in certification procedure.", "Certification in this case can be realized using the Monte Carlo scheme [26], which requires the same experimental toolbox.", "The rest of the paper is organised as follows.", "We first present the proposed method for MERA learning in section .", "Subsection REF explains how to identify the disentanglers.", "We start by deriving a necessary condition for the existence of suitable disentangler and then turn this criterion into a heuristic objective function that we minimize numerically over unitary space.", "In subsection REF , we carefully analyze the buildup of errors in our procedure and show that errors only accumulate linearly with the size of the system.", "In subsection REF , we present numerical benchmarks of our tomography method.", "In subsection, REF , we present the simplified method by demonstrating that it is not necessary to apply the disentanglers to the experimental state since we can simulate the effect of those disentanglers numerically, albeit at the cost of more repeated measurements and a slightly worse error scaling (analyzed in appendix ).", "In section , we discuss potential issues for our numerical scheme and suggest modifications to prevent them.", "Finally, we present in appendix a tool to contract two different MERA states, which allows for the efficient comparison of a MERA whose parameters have been identified experimentally using our method to a predicted theoretical MERA state.", "MERA states can be described as the output of a quantum circuit [16] whose structure is represented on Fig.", "REF (as seen with inputs on the top and output at the bottom).", "For simplicity, we will focus on a one dimensional binary MERA circuit for qubits, but our method generalizes to all 1D MERA states, i.e., particles could have more internal states thus accounting for a larger MERA refinement parameter $\\chi $ and isometries could renormalize several particles to one effective particle.", "The circuit contains three classes of unitaries.", "Disentanglers (represented as $\\square $ ) are two-qubit unitary gates; isometries (represented as $\\triangle $ ) are also two-qubit gates but with with one input qubit always in the $|0\\rangle $ state; the top tensor (represented as $\\bigcirc $ ) is a special case of isometry that takes as input two qubits in the $|00\\rangle $ state.", "Each renormalisation layer is made of a row of disentanglers and a row of isometries.", "Disentanglers remove the short-scale entanglement between adjacent blocks of two qubits while isometries renormalise each pair of qubits to a single qubit.", "Each renormalisation layer performs theses operations on a different lengthscale.", "The quantum circuit thus mirrors the renormalisation procedure that underlies the MERA.", "Learning a MERA amounts to identifying the various gates in that circuit, which turns the experimental state into the all $|0\\rangle $ state.", "The intuitive idea behind our scheme is to proceed by varying the isometries and disentanglers until the “ancillary” qubits reach the state $|0\\rangle $ for each row of isometries.", "We will exploit this feature to numerically determine each disentangler.", "Consider the $n$ qubits at the lowest layer of the MERA.", "Figure: The optimal disentangler U ˜\\tilde{U}can be computed from the tomographic estimation of the density matrixρ 123 \\rho _{123} on the first three qubits.", "Once applied, the resultingstate ρ ˜ 12 [U ˜]\\tilde{\\rho }_{12}[\\tilde{U}] is very close to a rank 2 matrix.Thus, there exist a unitary VV such that ρ ˜ 12 [U ˜]\\tilde{\\rho }_{12}[\\tilde{U}]can be rotated such that the first qubit is almost in the state |0〉|0\\rangle .Let $\\rho _{123}$ be the reduced density matrix on the three first qubits (see Fig.", "REF ).", "If the state is exactly a MERA, there exists a unitary $U_{23}$ acting on qubits 2 and 3 (see left of Fig.", "REF ) such that applying this unitary and tracing out the 3rd qubit yields a density matrix $\\rho _{12}\\left[U_{23}\\right]=\\mbox{Tr}_{3}\\left[\\left(\\mathbb {I}_{1}\\otimes U_{23}\\right)\\rho _{123}\\left(\\mathbb {I}_{1}\\otimes U_{23}^{\\dagger }\\right)\\right]$ of rank at most 2.", "Indeed, if the rank was strictly greater than 2, it would be impossible for the isometry $V$ (see left of Fig.", "REF ) to map the density matrix $\\rho _{12}\\left[U_{23}\\right]$ to a state with one of the qubit in the state $|0\\rangle $ because the dimension of the space $|0\\rangle \\otimes \\mathbb {C}^{2}$ would be strictly smaller than the dimension of the support of the density matrix.", "Hence, we have the necessary criterion $\\exists \\tilde{U}_{23}\\quad \\rho _{12}\\left[\\tilde{U}_{23}\\right]\\mbox{ has rank less or equal than 2$.$}$ To find a unitary that fulfills this criterion, it is necessary to know the state $\\rho _{123}$ , and this can be achieved by brute-force tomography on these three qubits.", "Once the original state on the three qubits is known, one has to perform a search over the space of unitaries to find a suitable disentangler.", "To do this, we will define an objective function to minimise numerically.", "Once this optimal unitary operator $\\tilde{U}$ has been found numerically, it is necessary to consider how it modifies the quantum state before learning the other elements of the circuit.", "One obvious way to do so is to apply the unitary transformation to the experimental state and continue the procedure on the transformed state.", "This amounts to undoing the circuit, and should in the end map the experimental state to the all $|0\\rangle $ state.", "For simplicity, we will first present our scheme assuming that the state is transformed at every step this way.", "Of course, such unitary control increases the complexity of the scheme and could be out of the reach of current technologies.", "However, in section REF , we will explain how this unitary transformation can be circumvented at the cost of a slight increase in the number of measurements.", "After the optimal disentangler $\\tilde{U}$ has been applied to the state, we need to identify the unitary $V$ that rotates the density matrix on the first two qubits such that the first qubit is brought to the $|0\\rangle $ state, c.f.", "Fig REF left.", "This does not require any additional tomographic estimate since we already know the descriptions of the state on the three first qubits and the disentangler.", "We can thus compute the state on the first two qubits $\\rho _{12}[\\tilde{U}]$ and diagonalise it to obtain the eigenvectors corresponding to its two non-zero eigenvalues.", "The unitary $V$ is chosen to map those two eigenvectors to the space $|0\\rangle \\otimes \\mathbb {C}^{2}$ , i.e., $V$ rotates the qubits such that the support of the density matrix is mapped to a space where the first qubit is in the $|0\\rangle $ state.", "All other disentanglers of this layer can be found by recursing the above procedure.", "Once a disentangler has been identified, it is physically applied to the system and brute-force tomogrography is performed on the next block of three qubits.", "Notice that for the last block, a single unitary is responsible for minimising the rank of two density matrices, $\\rho _{n-3,n-2}$ and $\\rho _{n-1,n}$ .", "One possible way to handle this is to get a tomographic estimate of the state on the last four qubits and to try to minimise the rank of both reduced matrices.", "Another way, for which we have opted in our numerical simulations, is to perform multiple sweeps over the layer.", "For instance, the disentanglers will first be identified from left to right and then the next sweep will be performed from right to left, using the disentanglers found in the first sweep as initial guesses in the space of unitaries (see Fig.", "REF ).", "Figure: Identification of the disentanglersusing two successive sweeps of the chain.", "Dotted regions cover particleson which brute-force tomography is performed.", "The first sweep (reddotted regions) finds unitaries starting from the left end of thechain.", "Those unitaries will be used as initial guesses for the secondsweep (blue striped regions) that starts from the right end of thechain.The number of sweeps can be increased for better accuracy but each additional sweep requires to extract the tomographic estimates again.", "Multiple sweeps would also allow to apply our method to MERA states with periodic boundary conditions in 1D and could be useful for 2D-MERA states.", "While this would be an interesting continuation of our work, we focus on 1D-MERA for the rest of the article." ], [ "Heuristic objective functions", "One of the steps in our protocol consists in identifying the unitary $\\tilde{U}$ that minimizes the rank of $\\rho _{12}[U]$ , c.f.", "eq.", "(REF ).", "There are many distinct ways this can be done and in this section, we present a practical heuristic to accomplish this task.", "Minimising the rank of the density matrix $\\rho _{12}\\left[U\\right]$ is not a suitable numerical task because, even if the experimental state is an exact MERA, the inferred density matrix will typically have full rank due to machine precision and the imperfect tomographic estimation of $\\rho _{123}$ .", "Thus, we turn the problem of finding $\\tilde{U}_{23}$ into an optimization problem by considering the eigendecomposition of $\\rho _{12}\\left[U\\right]=\\sum _{k}\\lambda _{k}|\\psi _{k}\\rangle \\langle \\psi _{k}|$ where the eigenvalues are sorted in decreasing order $\\lambda _{1}\\ge \\lambda _{2}\\ge \\lambda _{3}\\ge \\lambda _{4}$ .", "If $\\rho _{12}[U]$ has most of its support on a two-dimensional space, it will have two small eigenvalues that are typically non-zero due to imperfections.", "We thus consider the objective function $f\\left(U,\\rho _{123}\\right)=\\sum _{k>2}\\lambda _{k}$ and we perform a minimisation over the space of unitaries to determine the optimal unitary $\\tilde{U}$ .", "This objective function has a well-defined operational meaning – it is the probability of measuring the disentangled qubit in the $|1\\rangle $ state after the isometry $V$ has been applied.", "We will see in section REF that this property can be used to certify the distance between the experimental and the reconstructed states.", "Another way to think about this objective function is to consider the characteristic polynomial $P[X]$ of $\\rho _{12}\\left[U\\right]$ which is of the form $P[X]=X^{4}-X^{3}+aX^{2}-bX+c$ where the coefficients $a$ , $b$ and $c$ are positive since they correspond to the sum of product of the positive eigenvalues of the density matrix.", "In particular, coefficient $b$ is the sum of all products of three eigenvalues, i.e., $b=\\lambda _{1}\\lambda _{2}\\lambda _{3}+\\lambda _{1}\\lambda _{2}\\lambda _{4}+\\lambda _{1}\\lambda _{3}\\lambda _{4}+\\lambda _{2}\\lambda _{3}\\lambda _{4}$ .", "In order for the rank of the density matrix to be 2, it is sufficient for all 4 products of three eigenvalues to vanish, i.e, $\\rho _{12}\\left[U\\right]\\mbox{ of rank less than 2} & \\Longleftrightarrow & b=0\\mbox{.", "}$ Thus, another suitable objective function is the positive coefficient $b$ , which is a polynomial in the entries of $\\rho _{12}\\left[U\\right]$ .", "Indeed, using Bocher formula, coefficient $b$ can be expressed as $6b=1-3\\mbox{Tr}A^{2}+2\\mbox{Tr}A^{3}$ where $A=\\rho _{12}\\left[U\\right]$ .", "Thus, $b$ is a well-behaved function with respect to the density matrix.", "Note also that $b$ can be expressed without diagonalising the density matrix $\\rho _{12}[U]$ .", "We will focus on minimising (REF ) in all subsequent numerical discussion and results." ], [ "Numerical minimisation over unitary space", "Minimisations of (REF ) is performed using a conjugate gradient method.", "We first have to account for the fact that the unitary manifold is not a vector space.", "To get around this problem, we go to the Hermitian space by writing any unitary $U$ as the result of a Hamiltonian evolution, i.e., there exists a Hermitian matrix $H$ such that $U=e^{iH}$ .", "It is then possible to use the standard conjugate gradient method.", "Let us sketch the algorithm in more details.", "First, we select a unitary $U_{0}$ either at random or from an initial guess (provided for instance by a previous sweep).", "We will search the unitary space by generating a sequence of unitaries $\\left\\lbrace U_{k}\\right\\rbrace $ .", "At the $k$th step of the minimisation, the algorithm is the following.", "We center the unitary space at point $U_{k-1}$ by defining $\\rho _{k}=(\\mathbb {I}\\otimes U_{k-1})\\rho _{k-1}(\\mathbb {I}\\otimes U_{k-1})^{\\dagger }$ .", "We compute the gradient $G^{(k)}$ by parametrizing the Hamiltonian $H$ on 2 qubits by its decomposition on the Pauli group $H=\\sum _{\\mu \\nu }h_{\\mu \\nu }\\sigma _{\\mu }\\otimes \\sigma _{\\nu }$ where $\\sigma _{\\mu }\\in \\lbrace \\mathbb {I},\\sigma _{x},\\sigma _{y},\\sigma _{z}\\rbrace $ is a Pauli matrix.", "We successively evaluate the component of the gradient $G^{(k)}$ in the direction $(\\mu ,\\,\\nu )$ by looking at the effect of the test unitary $U_{\\mu ,\\nu }=\\mathbb {I}+i\\epsilon \\sigma _{\\mu }\\otimes \\sigma _{\\nu }$ on the objective function, i.e., $G_{\\mu ,\\,\\nu }^{(k)}=\\frac{f(U_{\\mu ,\\,\\nu },\\,\\rho _{k})-f(\\mathbb {I},\\,\\rho _{k})}{\\epsilon }$ where $\\epsilon $ is a small number.", "Instead of following the gradient, which would generally undo some of the minimization performed in the previous steps, we use a conjugate gradient method where the new direction of search $\\tilde{G}^{(k)}$ is optimized by taking into account the direction used in the previous step $\\tilde{G}^{(k-1)}$ through the Polak-Ribiï¿œre formula.", "More precisely, $\\tilde{G}^{(k)}=G^{(k)}+\\beta \\tilde{G}^{(k-1)}$ in which the real parameter $\\beta $ is defined as $\\beta =\\max \\left(0,\\,\\frac{G^{(k)}\\cdot \\left(\\tilde{G}^{(k-1)}-G^{(k)}\\right)}{\\tilde{G}^{(k-1)}\\cdot \\tilde{G}^{(k-1)}}\\right)$ .", "We perform a line search along the direction $\\tilde{G}^{(k)}$ by considering the family of unitaries $\\exp \\left(-it\\sum _{\\mu ,\\nu }\\tilde{G}_{\\mu ,\\nu }\\sigma _{\\mu }\\otimes \\sigma _{\\nu }\\right)$ and optimizing the parameter $t$ to find $t_{opt}$ .", "We then define $U_{k}=\\exp \\left(-it_{opt}\\sum _{\\mu ,\\nu }\\tilde{G}_{\\mu ,\\nu }\\sigma _{\\mu }\\otimes \\sigma _{\\nu }\\right)U_{k-1}$ which ends the $k$th iteration.", "We iterate until the objective function is close enough to zero or that improvement has stopped.", "This method is heuristic since the objective functions present no characteristic that would ensure the convergence of the conjugate gradient method.", "In particular, our search over unitary space depends on the starting point, i.e., the unitary chosen in the first iteration.", "Indeed, some starting points will lead the heuristic to a local minima where it will get stuck.", "In order to avoid this phenomenon, we can repeat the overall search by picking at random (according to the unitary Haar measure) different initial points which lead to potentially different minima and keep the smallest of those minima, which we expect to be the global minimum.", "In any case, this is a minimization problem over a spapce of constant dimension, so the method used to solve it does not affect the scaling with the number of particles $n$ .", "Ultimately, we can always use a finite mesh over the unitary space and use brute-force search.", "Nevertheless, we found numerically that this heuristic works well.", "It is also possible that a choice of unitary that is optimal locally, in the sense that it minimizes (REF ), is not optimal globally as it might lead to a state for which it is impossible to find a disentangler obeying (REF ) elsewhere in the circuit.", "This is a phenomenon that is more likely to occur when the minimum is degenerate, i.e., there exists several distinct (modulo gauge) exact disentanglers for the state.", "However, we have performed numerical experiments on randomly generated MERA states as well as physically motivated states and found that the conjugate gradient performs well (see section REF )." ], [ "Error analysis", "In practice, due to numerical and experimental imperfections, the disentangled qubits will not be exactly in the $|0\\rangle $ state, but merely close to it.", "This situation arises from the conjunction of three causes : i) the experimental state of the system is not exactly a MERA, but merely close to one, ii) the tomographic estimate of the density matrices on blocks of three qubits are slightly inaccurate due to noisy measurements and experimental finite precision, iii) the numerical minimization did not find the exact minimum." ], [ "Isolating each elementary steps to prevent error buildup", "Our error analysis will show that the buildup of errors is linear in the number of disentanglers of the MERA circuit which is itself linearly proportional to the number of particles in the experimental state.", "Essentially, the distance between the reconstructed state and the experimental state is the sum of the error made at each elementary step when estimating a disentangler and an isometry.", "Fortunately, the error made at each elementary step does not depend on the errors made at previous steps.", "The key to isolate each step from the others is to measure the qubit that should have been disentangled in the computational basis.", "With high probability the qubit will be found in the $|0\\rangle $ state.", "While the probability of measuring the $|0\\rangle $ outcome depends on previous errors, the post-selected state is now free from previous errors.", "The interest of this postselection is two-fold.", "First, it forbids errors in previous steps to contaminate the state and amplify the error made at the current step, thus limiting the error propagation.", "Second, by accumulating statistics on this measurement, we can estimate the probability of the all-0 outcome and use it to bound the distance of the reconstructed state to the actual state in the lab.", "Therefore, our procedure comes with a built-in certification process.", "Let us now describe the error analysis in more details." ], [ "Error at each elementary step", "Recall the notation of Fig.", "REF .", "Due to numerical and experimental imperfections, the state on qubits 1, 2 and 3 after applying the disentangler $\\tilde{U}_{1}$ and the isometry $V_{1}$ is not exactly in the $|0\\rangle \\otimes \\mathbb {C}^{2(n-1)}$ subspace but contains a small component orthogonal to that space.", "Thus, it has the form $V_{1}\\tilde{U}_{1}|\\psi \\rangle =\\frac{|0\\rangle |\\eta _{1}\\rangle +|e_{1}\\rangle }{\\sqrt{1+\\langle e_{1}|e_{1}\\rangle }}$ where $|\\eta _{1}\\rangle $ is the normalised pure state on qubits 2 to $n$ if qubit 1 had been completely disentangled from the chain and $|e_{1}\\rangle $ is some subnormalized vector supported on the subspace $|1\\rangle \\otimes \\mathbb {C}^{2(n-1)}$ .", "The isometry $V_{1}$ is chosen to minimize the norm of $|e_{1}\\rangle $ , i.e., to minimize $\\epsilon _{1}\\equiv \\langle e_{1}|e_{1}\\rangle $ .", "Further along the layer, the state after applying $k$ disentanglers and $k$ isometries will be of the form $V_{k}\\tilde{U}_{k}\\dots V_{1}\\tilde{U}_{1}|\\psi \\rangle =\\frac{|0\\rangle ^{\\otimes k}|\\eta _{k}\\rangle +|e_{k}^{cm}\\rangle }{\\sqrt{1+\\epsilon _{k}^{cm}}}$ where the first term $|0\\rangle ^{\\otimes k}|\\eta _{k}\\rangle $ is the normalised state had the $k$ qubits in position 1, 3$\\dots 2k-3$ been completely disentangled from the chain and $|e_{k}^{cm}\\rangle $ is the accumulated error vector orthogonal to the space where those $k$ qubits are in the $|0\\rangle ^{\\otimes k}$ state.", "In order to find the optimal disentangler and isometry, we measure the last disentangled qubit in the computational basis and post-select on the $|0\\rangle $ outcome, which occurs with probability $(1+\\epsilon _{k}^{cm})^{-1}$ .", "We then perform brute force tomography and identify numerically the disentangler and the isometry that minimimizes the norm of the error vector $|e_{k+1}\\rangle $ such that $V_{k+1}\\tilde{U}_{k+1}|\\eta _{k}\\rangle =\\frac{|0\\rangle |\\eta _{k+1}\\rangle +|e_{k+1}\\rangle }{\\sqrt{1+\\varepsilon _{k+1}}}\\mbox{.", "}$ Applying this disentangler and isometry to the whole state of the chain, one gets $V_{k+1}\\tilde{U}_{k+1}\\dots V_{1}\\tilde{U}_{1}|\\psi \\rangle =\\frac{|0\\rangle ^{\\otimes k+1}|\\eta _{k+1}\\rangle +|e_{k+1}^{cm}\\rangle }{\\sqrt{1+\\epsilon _{k+1}^{cm}}}$ where the accumulated error vector at step $k+1$ is $|e_{k+1}^{cm}\\rangle =|e_{k+1}\\rangle +\\sqrt{1+\\epsilon _{k+1}}V_{k+1}\\tilde{U}_{k+1}|e_{k}^{cm}\\rangle $ and the square if its norm $\\epsilon _{k+1}^{cm}=\\langle e_{k+1}^{cm}|e_{k+1}^{cm}\\rangle $ obeys the recurrence relation $1+\\epsilon _{k+1}^{cm}=\\left(1+\\epsilon _{k+1}\\right)\\left(1+\\epsilon _{k}^{cm}\\right)$ since the elementary error vector $|e_{k+1}\\rangle $ , for which all previous ancillary particles have been disentangled, is orthogonal to the vector $V_{k+1}\\tilde{U}_{k+1}|e_{k}^{cm}\\rangle $ .", "Thus, $1+\\epsilon _{k+1}^{cm}=\\prod _{i=1}^{k+1}\\left(1+\\epsilon _{i}\\right)\\mbox{.", "}$" ], [ "Global error", "After the choice of $m$ disentanglers and $m$ isometries, the reconstructed state is $|\\phi \\rangle =V_{m}^{\\dagger }\\tilde{U}_{m}^{\\dagger }\\dots V_{1}^{\\dagger }\\tilde{U}_{1}^{\\dagger }|0\\rangle ^{\\otimes m+1}|\\eta _{m}\\rangle $ .", "Its difference to the actual experimental state $|\\psi \\rangle $ can be stated in terms of the (in) fidelity as $1-\\left|\\langle \\phi |\\psi \\rangle \\right|^{2} & = & 1-1/\\left(1+\\epsilon _{m}^{cm}\\right)\\nonumber \\\\& = & 1-1/\\prod _{i=1}^{m}1+\\epsilon _{i}\\mbox{.", "}$ Practically, one is interested in guaranteeing that the reconstructed state is close to the experimental up to global error $E$ , i.e., to guarantee that $1-\\left|\\langle \\phi |\\psi \\rangle \\right|^{2}\\le E$ .", "Suppose that all error vectors are bounded, i.e.", "that for all step $i$ , we have $\\epsilon _{i}\\le \\varepsilon $ .", "Inverting (REF ), it suffices that $\\varepsilon \\le (1-E)^{-1/m}-1\\simeq E/m$ in the limit where the tolerable global error $E$ is small.", "Thus, we see that errors accumulate linearly and that a precision inversely proportionnal to the number of disentanglers is sufficient to ensure a constant global error.", "Furthermore, statistics on the post-selection performed at each step allows to estimate each $\\epsilon _{k}^{cm}$ and in particular $\\varepsilon _{m}^{cm}$ that gives direct access to the distance betweem the reconstructed and experimental states.", "Finally, from this measurement data, one can estimate the error $\\epsilon _{i}$ performed at each step in order to identify steps that have gone wrong.", "This information can be used to turn the scheme into an adaptative one.", "Suppose the error is particularly large for a given step.", "This might be due to an important amount of entanglement concentrated in one region of space, e.g.", "near a defect, which can be accounted for by increasing the MERA refinement parameter $\\chi $ locally, i.e.", "by using disentanglers acting on a larger number of qubits.", "In principle, $\\chi $ could be increased until the error is below some threshold." ], [ "Benchmarking results", "We have performed numerical simulations to benchmark the performances of the MERA learning method.", "We have generated random MERA states —by picking each unitary gate in the circuit from the unitary group Haar measure—, simulated the experiment on those states, and use our algorithm to infer the initial MERA state.", "We did not introduce noise in measurements to simulate experimental errors since the error analysis indicates how those errors would build up.", "As mentioned before, there is no guarantee that our minimization procedure converges to the true minimum, resulting in small imperfections in the reconstructed state.", "Figure: (top) Infidelity to the “experimentalstate”, i.e, 1-〈ψ tomo |ψ〉 2 1-\\left|\\langle \\psi _{tomo}|\\psi \\rangle \\right|^{2}where |ψ〉|\\psi \\rangle is a random MERA on nn qubits and |ψ tomo 〉|\\psi _{tomo}\\rangle is the state reconstructed from the MERA learning method using threesweeps.", "(bottom) Processing time (on a standard laptop) to performMERA learning using three sweeps.", "Both figures exhibit 10 runs foreach number of qubits n∈{8,12,16,20,24}n\\in \\lbrace 8,\\,12,\\,16,\\,20,\\,24\\rbrace .", "In both figures,each ×\\times represents results for one random MERA.", "The full linesrepresent median for each number of qubits.", "The dashed line on thetop figure is the linear approximation to the median.", "Notice thatthe numerical minimisation can fail to converge as illustrated bythe atypical data points.", "For instance, for one of the 20-qubit MERA,the processing time was 338.3 seconds and the infidelity to the truestate is large, 1-〈ψ tomo |ψ〉 2 =3.916×10 -3 1-\\left|\\langle \\psi _{tomo}|\\psi \\rangle \\right|^{2}=3.916\\times 10^{-3}.Figure (REF , top) shows the distance between the reconstructed state and the actual state.", "As indicated by the dashed line, these results are in good agreement with a linear scaling of the error where the source of errors is due to finite machine precision and approximate minimisation of the objective function.", "The inference algorithm's complexity is dominated by the conjugate gradient descents, and therefore scales linearly with the number of disentanglers or the number of particles in the system.", "Figure (REF , bottom) shows the actual runtime of the inference algorithm for different randomly chosen MERA states and of various sizes.", "Once again, we see a good agreement with a linear dependence with the system size.", "Systems of up to 24 qubits can easily be handled in a few minutes of computation and requires 1197 different measurement settings for each sweep of the 24 qubit system.", "This is to be contrasted with the 656,100 experiments needed to reconstruct the state of 8 qubits in [3] and the post-processing of the data that took approximately a week [27].", "Additional sweeps allow the method to converge as showed on Fig.", "REF .", "Figure: Infidelity to a 20 qubit state using a reconstructedmethod with a variable number of sweeps.", "Each line corresponds toa different random MERA.We also tested how our method behaved on a physical model, namely the 1D Ising model with transverse field at the critical point.", "The results obtained where coherent with what is expected from the approximation of the true ground state with a MERA with refinement parameter $\\chi =2$ ." ], [ "Possible improvements", "Note the presence of isolated points on the graphs of Fig.", "REF that achieve a lower fidelity and required a longer running time.", "These cases appear because the heuristic fails to find a global minimum.", "It appears that in some cases, a unitary transformation $U_{23}$ meeting criterion (REF ) is not sufficient to guarantee that it will be possible to find subsequent disentanglers obeying (REF ).", "Put another way, locally minimising the objective function might not lead to a global optimum.", "Indeed, consider the following example.", "Let $|\\psi \\rangle $ be a MERA state whose first qubit is disentangled from the rest of the chain, i.e.", "$|\\psi \\rangle =|0\\rangle |\\phi \\rangle $ .", "The rank of the density matrix on the first two qubits is at most 2 and that remains true after any unitary is applied on qubits 2 and 3.", "Thus, any choice of disentangler minimises the objective function (REF ), including the identity, i.e., applying no disentangler at all.", "However, suppose the state $|\\phi \\rangle $ on qubits 2 to $n$ is highly entangled and that removing part of this entanglement between qubits 2 and 3 was crucial to be able to reconstruct its MERA description.", "In this case, applying the identity on qubits 2 and 3, even if locally optimal, was not globally optimal.", "Hence, minimizing the objective function (REF ) seems to be necessary but not sufficient to successively identify all disentanglers.", "Although our numerical simulations suggest that this situation is rather atypical, it is possible to overcome this problem by imposing additional constraint on the disentangler.", "For instance, one can demand that the second qubit be in a state as pure as possible, effectively minimizing the entanglement between the last qubit of one block and the first qubit of the next block.", "This corresponds to the following modified objective function $f\\left(\\tilde{\\rho }_{12}\\left[U\\right]\\right)=\\sum _{k>2}\\lambda _{k}+\\epsilon \\lambda _{2}$ i.e., we add a small perturbation that will only take action when the two smallest eigenvalues of $\\tilde{\\rho }_{12}[U_{23}]$ are very small and will further constrain the search.", "This slight modification solved the problematic situation we considered, and there exist many other heuristics to improve the method." ], [ "Trading unitary control for repeated\nmeasurements", "For pedagogical reasons, we presented our learning method in a way that required disentanglers and isometries to be physically applied to the experimental state in order to unravel the circuit.", "In this section, we will show how to circumvent unitary control at the price of slightly more elaborate numerical processing and consuming more copies of the state.", "The main idea is to numerically simulate how measurements performed on the original, unaltered experimental system would be transformed if the unraveling circuit had been applied." ], [ "Simulating measurements on renormalized state", "Recall that a MERA is an ansatz that corresponds to a renormalization procedure.", "Each renormalisation step maps a state to another one on fewer particles and schematically corresponds to a layer of the MERA circuit.", "Applying the first layer and removing the ancillary particles that have been (approximately) disentangled maps the experimental state $\\rho _{0}$ on $n$ particles to a state $\\rho _{1}$ on fewer particles (see Fig REF ).", "Figure: MERA as a renormalisation procedurethat creates a sequence of states ρ τ τ \\left\\lbrace \\rho _{\\tau }\\right\\rbrace _{\\tau }.Recursively, this procedure constructs a sequence of states $\\left\\lbrace \\rho _{\\tau }\\right\\rbrace _{\\tau }$ .", "To get from $\\rho _{\\tau -1}$ to $\\rho _{\\tau }$ , one can either perform this mapping physically by experimentally applying the gates corresponding to the MERA layer, or one can compute the function mapping $\\rho _{\\tau -1}$ to $\\rho _{\\tau }$ from the description of the gates.", "As in [17], define a ascending superoperator $\\mathcal {A}$ that maps an operator $O_{\\tau -1}$ acting on layer $\\tau -1$ to the an operator $O{}_{\\tau }$ acting on the next layer $\\tau $ $O_{\\tau }=\\mathcal {A}_{\\tau }(O_{\\tau -1})$ such that $\\mbox{Tr}[\\rho _{\\tau }\\mathcal {A}_{\\tau }(O_{\\tau -1})]=\\mbox{Tr}[\\rho _{\\tau -1}O_{\\tau -1}]\\mbox{.", "}$ This recursively carries over to the experimental state $\\rho _{0}$ $\\mbox{Tr}[\\rho _{\\tau }\\mathcal {A}_{\\tau }\\circ \\dots \\circ \\mathcal {A}_{1}(O_{0})]=\\mbox{Tr}[\\rho _{0}O_{0}]\\mbox{.", "}$ Thus, in order to extract information from a density matrix $\\rho _{\\tau }$ , one can measure the expectation value of several observables $O_{0}^{i}$ on the density matrix $\\rho _{0}$ .", "Measuring those observables will effectively amount to measuring the observables $O_{\\tau }^{i}\\equiv \\mathcal {A}_{\\tau }\\circ \\dots \\circ \\mathcal {A}_{1}(O_{0}^{i})$ on the density matrix $\\rho _{\\tau }$ .", "The ascending superoperator can be computed from the knowledge of the disentanglers and isometries.", "Its exact form depends on the physical support of the observable.", "For instance, for ternary MERA, we can restrict to ascending superoperator that only depends on the isometries of the MERA [22] (see Fig.", "REF ).", "Figure: Ascending superoperator and renormalizedobservables for a ternary MERA.", "a) Ternary MERA with one site observableO 0 O_{0} that is transformed into a renormalized observable 𝒜(O 0 )\\mathcal {A}(O_{0})on the renormalized state.", "b) Tensor contraction corresponding tothe ascending superoperator 𝒜\\mathcal {A}.This is a simple example where an experimental observable on one particle is mapped to observable on one particle.", "More generally, observables on many sites will be ascended to observables on fewer sites.", "Any choice of observables is valid as long as the renormalized observables $\\left\\lbrace O_{\\tau }^{i}\\right\\rbrace _{i}$ span the support of the reduced density matrix $\\rho _{\\tau }$ ." ], [ "Overhead in the number of measurements", "This procedure leads to a overhead in the total number of measurements because renormalized observables are less efficient at extracting information.", "Suppose (for clarity) that we measure Pauli observables $\\left\\lbrace O_{0}^{i}\\right\\rbrace _{i}$ on the experimental states.", "These observables are orthonormal for the Hilbert-Schmidt inner product and thus maximize information extraction.", "However, the renormalized observables $O_{1}^{i}\\equiv \\mathcal {A}_{1}(O_{0}^{i})$ need not be orthonormal.", "Consider their Gram matrix $G_{ij}=\\mbox{Tr}\\left[O_{1}^{i}\\left(O_{1}^{j}\\right)^{\\dagger }\\right]$ which can be diagonalised by a unitary matrix $Z$ .", "Its normalised eigenvectors $R_{1}^{i}=\\frac{1}{\\sqrt{\\lambda _{i}}}\\sum _{j}Z_{ij}O_{1}^{j}$ are orthornormal observables but cannot be directly measured because they do not correspond to simple observables on the experimental state, but instead to linear combination of them.", "Thus, to reconstruct the density matrix $\\rho _{1}=\\sum _{i}r_{1}^{i}R_{1}^{i}$ , the expectation values $r_{1}^{i}=\\mbox{Tr}\\rho _{1}R_{1}^{i}$ have to be computed by taking a linear combination of the expectation values $o_{0}^{j}\\equiv \\mbox{Tr}\\rho _{0}O_{0}^{j}$ on the experimental state $r_{1}^{i}=\\frac{1}{\\sqrt{\\lambda _{i}}}\\sum _{j}Z_{ij}\\mbox{Tr}\\rho _{1}O_{1}^{j}=\\frac{1}{\\sqrt{\\lambda _{i}}}\\sum _{j}Z_{ij}o_{0}^{j}\\mbox{.", "}$ Due to limited number of repeated measurements, estimation of each $o_{0}^{i}$ will present a variance $\\mathbb {V}(o_{0}^{i})$ .", "Suppose that measurements are repeated enough times to ensure that all variances are below a precision threshold, i.e., $\\mathbb {V}(o_{0}^{i})\\le \\epsilon $ .", "Since $r_{1}^{i}$ is a linear combination of those measurements, it will have a variance $\\mathbb {V}(r_{1}^{i})=\\frac{1}{\\lambda _{i}}\\sum _{j}|Z_{ij}|^{2}\\mathbb {V}(o_{0}^{j})\\le \\epsilon /\\lambda _{i}$ .", "Therefore, in order to ensure a precision $\\epsilon $ on the estimate of $r_{1}^{i}$ , this imprecision needs to be compensated by multiplying the number of repeated measurements by the conditioning factor$\\lambda _{i}^{-1}$ .", "When scaling operators on $\\tau $ layers, the conditioning factors for each layer will multiply (in the worst case) but we expect the conditioning for each layer to be a constant independant of system size.", "Thus, the total number of measurements will remain polynomial in the number of particles since there is only a logarithmic number of renormalisation layers.", "We can make this argument rigorous for critical systems that exhibit scale-invariance, precisely the physical systems for which MERA was introduced.", "Due to scale-invariance, the ascending operator $\\mathcal {A}_{\\tau }$ will not depend on the index of the layer and we refer to it as the scaling superoperator $\\mathcal {S}$ [22].", "Its diagonalization yields eigenvectors $\\phi _{\\alpha }$ called scaling operators associated to eigenvalues $\\mu _{\\alpha }$ .", "In [22], it was shown that those eigenvalues are related to the scaling dimensions $\\Delta _{\\alpha }$ of the underlying conformal field theory (CFT) by $\\Delta _{\\alpha }=\\log _{3}\\mu _{\\alpha }$ where the basis of the log depends on the MERA type (here we consider a ternary MERA for clarity).", "Scaling operators $\\phi _{\\alpha }$ can be used as observables to extract information about states in higher level of the MERA.", "Indeed, one can simulate a measurement of $\\mathcal {S}^{\\tau }(\\phi _{\\alpha })$ on $\\rho _{\\tau }$ by measuring the observable $\\phi _{\\alpha }$ on $\\rho _{0}$ .", "We can analyze the increase in the number of measurements by distinguishing two sources of imprecision.", "First, to reconstruct $\\rho _{\\tau }$ one has to use normalised operator $\\phi _{\\alpha }^{[\\tau ]}=3^{\\tau \\Delta _{\\alpha }}\\mathcal {S}^{\\tau }(\\phi _{\\alpha })$ whose increased statistical fluctuations have to be compensated by performing additional measurements.", "Second, diagonalising the Gram matrix of the $\\phi _{\\alpha }^{[\\tau ]}$ will introduce another conditioning factor.", "However, this Gram matrix is independant of the layer since $G_{\\alpha \\beta }^{[\\tau ]}=\\mbox{Tr}\\left[\\phi _{\\alpha }^{[\\tau ]}\\phi _{\\beta }^{[\\tau ]}\\right]=\\mbox{Tr}\\left[\\phi _{\\alpha }\\phi _{\\beta }\\right]$ .", "Thus, the conditioning factor for layer $\\tau $ will be the product of a factor exponential in the number of layers and a constant factor coming from the orthonormalisation.", "This modified scheme circumvents the need of unitary control, but looses some of the features of the original scheme.", "First, because the system is not physically disentangled, we cannot certify directly the fidelity of the reconstruction.", "Second, as explained in appendix , the errors build up quadratically." ], [ "Discussion", "In this work, we have presented a tomography method that allows to efficiently learn the MERA description of a state by patching together tomography experiments on a few particles and using fast numerical processing.", "The method is heuristic but works very well in numerical simulations.", "A complete analytical understanding of how to find an optimal disentangler at each step would be desirable, but may well be intractable.", "With regards to experimental use, the method should be tought of as a proof of principle and is flexible enough to be adapted to accomodate many experimental constraints.", "One issue of fundamental interest raised by our work is the relationship between the numerical tractability of a variational class of states and the ability to learn efficiently the variational parameters.", "In order to be interesting, variational class of states must not only be described by a small number of parameters, but also allow for the efficient numerical computation of quantities of interest, such as the energy of the system, correlation functions, or more generally expectation values of local observables.", "On its own, an efficient representation is of limited computational usefulness.", "For instance, the Gibbs state or thermal state of a local Hamiltonian is described by a few parameters — a temperature and a local Hamiltonian — but does not allow to extract physical quantities of interest efficiently.", "Another example is the variational class of projected entangled pair states or PEPS [28], the generalization of MPS to system in more than one dimension.", "While PEPS have been instrumental in better understanding of quantum many-body systems, they are in general intractable numerically [29].", "Is there a relation between numerical tractability and efficient tomography?", "The method presented in [15] to learn a MPS from local measurements made explicit use of the energy minimization algorithm for MPS; namely DMRG [30].", "This example suggests that numerical tractability could imply that learning the variational parameters is possible.", "In that regard, MERA are intriguing states because they live at the frontier of tractability.", "Indeed, in more than 1 dimension, MERA states are a subclass of PEPS [31] with a bond dimension independant of system size [32].", "While the computation of expectation values of local observables is believed to be intractable for PEPS, it is efficient for MERA.", "In one dimension, MERA can be seen as MPS if one allows the bond dimension to grow polynomially with the size of the system (while MPS are usually required to have a constant bond dimension).", "Thus, while MPS manipulations typically have a computational cost linear in the number of particles, 1D-MERA manipulations have a computational cost which is superlinear (but yet polynomial).", "Beyond MPS and MERA, one could consider states obtained from a quantum circuit where the positions of the gates are known and try to identify those gates.", "An interesting question is then to characterize what topology of circuits makes it possible to learn gates efficiently.", "This could lead to formal methods for the testing and verification of quantum hardware.", "OLC acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) through a Vanier scholarship.", "OLC wishes to thank Andy Ferris for stimulating discussions and sharing numerical data on scale-invariant MERAs.", "DP acknowledges financial support by the Lockheed Martin Corporation." ], [ "Error analysis without\npost-selection", "The modified scheme that circumvents the need of unitary control modifies the error propagation.", "Indeed, the scaling of the overall error increases since the error at each step will depend on previous errors.", "Essentially, to find the optimal disentangler, the algorithm will not receive the perfect state $|\\eta _{k}\\rangle $ (see eq.", "(REF )) but the state $\\frac{1}{1+E_{k}^{cm}}|\\eta _{k}\\rangle \\langle \\eta _{k}|+\\frac{E_{k}^{cm}}{1+E_{k}^{cm}}|E_{k}^{cm}\\rangle \\langle E_{k}^{cm}|$ where $|E_{k}^{cm}\\rangle $ is a subnormalised error vector resulting from the accumulation of all previous errors whose square norm is $E_{k}^{cm}\\equiv \\langle E_{k}^{cm}|E_{k}^{cm}\\rangle $ .", "Thus, the numerical minimisation returns a unitary that is not the perfect disentangler.", "In the degenerate case —when there are many unitaries reaching roughly the same minimum value of the objective function—, this might change the disentangling unitary drastically.", "Indeed, we note that the existence of degenerate local minima affects both of our tomography methods, the one with and the one without unitary control of the system.", "In such degenerate cases, exploring many local minima by selecting random initial guesses could get around the problem.", "However, it is likely that these instances are intrinsically hard and that our algorithm does not converge to the right answer in those cases, c.f.", "the atypical data points on Fig.", "REF .", "In the non-degenerate case, we can argue that the accumulation of errors would be quadratic in the number of particles.", "We proceed in three steps.", "First, we analyze how the modification of the input state will affect the disentangling unitary returned by the algorithm.", "Second, we evaluate how this imperfect disentangler impacts the error propagation.", "Third, we bound the error to show the quadratic scaling.", "Our algorithm returns the unitary $\\tilde{U}=e^{i\\tilde{H}}$ that minimizes the objective function (REF ) for a given state $\\rho $ .", "If we don't post-select on the ancillary particles being disentangled, this minimization is not performed on the the perfect state $\\rho _{0}=|\\eta _{k}\\rangle \\langle \\eta _{k}|$ but rather on the state $(1-\\varepsilon )\\rho _{0}+\\varepsilon \\sigma $ where $\\varepsilon =\\frac{E_{k}^{cm}}{1+E_{k}^{cm}}$ and $\\sigma =|E_{k}^{cm}\\rangle \\langle E_{k}^{cm}|$ .", "We want to know how much $\\tilde{U}=\\arg \\min _{U}f(U,\\rho )$ changes when $\\rho $ changes.", "Using the chain rule, we formally write $\\frac{\\partial \\tilde{U}}{\\partial \\rho }=\\frac{\\partial \\tilde{U}}{\\partial f}\\frac{\\partial f}{\\partial \\rho }$ .", "The first term quantifies how much $\\tilde{U}$ changes when the objective function changes for a given $\\rho $ .", "In the non-degenerate case, we expect this term to be bounded in norm by a Lipschitz constant $\\eta $ .", "The second term evaluates how the objective function changes when going from $\\rho _{0}$ to $(1-\\varepsilon )\\rho _{0}+\\varepsilon \\sigma $ .", "Recalling that the objective function is a sum of eigenvalues and using non-degenerate perturbation theory, this term is going to be proportional to $\\varepsilon $ .", "Thus, instead of $\\tilde{U}=e^{i\\tilde{H}}$ , the minimization algorithm returns $e^{i(\\tilde{H}+\\varepsilon \\eta A)}\\approx W\\tilde{U}$ where $W=e^{i\\varepsilon \\eta A}$ .", "As a consequence, eq.", "(REF ) is modified to read $W_{k+1}\\frac{|0\\rangle ^{\\otimes k+1}|\\eta _{k+1}\\rangle +|e_{k+1}^{cm}\\rangle }{\\sqrt{1+\\epsilon _{k+1}^{cm}}}=e^{i\\theta }\\frac{|0\\rangle ^{\\otimes k+1}|\\eta _{k+1}\\rangle +|E_{k+1}^{cm}\\rangle }{\\sqrt{1+E_{k+1}^{cm}}}\\mbox{.", "}$ Taking into account the anomalous unitary $W$ , we get $1+E_{k+1}^{cm}=1+\\epsilon _{k+1}^{cm}/\\beta ^{2}=(1+\\epsilon _{k+1})(1+E_{k}^{cm})/\\beta ^{2}$ where $\\beta ^{2}=\\left|\\langle \\eta _{k+1}|\\langle 0|^{\\otimes k+1}W|0\\rangle ^{\\otimes k+1}|\\eta _{k+1}\\rangle \\right|^{2}=\\left|\\langle W\\rangle \\right|^{2}$ .", "Using $W=e^{i\\varepsilon \\eta A}$ , calculations show that $\\beta ^{2}=1-\\varepsilon ^{2}\\eta ^{2}\\left(\\langle A^{2}\\rangle -\\langle A\\rangle ^{2}\\right)=1-\\varepsilon ^{2}\\eta ^{2}\\Delta ^{2}$ where the variance $\\Delta ^{2}$ of $A$ with respect to state $|0\\rangle ^{\\otimes k+1}|\\eta _{k+1}\\rangle $ appears.", "Recalling that $\\varepsilon =\\frac{E_{k}^{cm}}{1+E_{k}^{cm}}$ , one gets $\\beta ^{2}=\\frac{\\left(1+E_{k}^{cm}\\right)^{2}-\\left(E_{k}^{cm}\\right)^{2}\\eta ^{2}\\Delta ^{2}}{\\left(1+E_{k}^{cm}\\right)^{2}}\\ge \\frac{1}{\\left(1+E_{k}^{cm}\\right)^{2}}$ for any $E_{k}^{cm}$ if $\\eta ^{2}\\Delta ^{2}\\le 1$ or for small $E_{k}^{cm}$ otherwise.", "Thus, the error magnitude $E_{k+1}^{cm}$ obeys the recurrence relation $1+E_{k+1}^{cm} & \\le & \\left(1+E_{k}^{cm}\\right)^{3}(1+\\epsilon _{k+1})\\\\& \\le & \\left(1+\\epsilon _{1}\\right)^{3k}\\dots \\left(1+\\epsilon _{k+1}\\right)\\mbox{.", "}$ Assuming that the error at each step is bounded $\\epsilon _{k}\\le \\epsilon $ , the total error scales as $E_{m}^{cm}\\le \\frac{3}{2}m^{2}\\epsilon \\mbox{.", "}$" ], [ "Comparing a reconstructed MERA to a predicted\nMERA", "In this section, we describe a polynomial algorithm to contract two MERA states, thus allowing to compute their fidelity.", "This algorithm is of practical interest for comparing a MERA whose parameters have been identified experimentally using our method to a predicted MERA state –found by numerical optimisation for instance.", "Notice that contracting two different MERA states also allows to compute expectation values of tensor product of local observables $\\bigotimes _{i}A_{i}$ since it suffices to contract the original state $|\\psi \\rangle $ and the modified state $|\\phi \\rangle =\\bigotimes _{i}A_{i}|\\psi \\rangle $ , which is also a MERA state.", "Defining a method to contract two MERA states is equivalent to giving a prescription on how to sequentially contract the tensor network resulting from joining two MERA states.", "Recall that contracting two tensors $(M)_{i_{a}j_{b}}$ and $(N)_{k_{b}\\ell _{c}}$ to obtain $T_{i_{a}\\ell _{c}}=\\sum _{j_{b}}M_{i_{a}j_{b}}N_{j_{b}\\ell _{c}}$ has a computational cost of $a\\times b\\times c$ where $a$ is the number of values that the index $i_{a}$ can take $b$ and $c$ are defined in the same way with respect to $j_{b}$ and $\\ell _{c}$ .", "In a tensor network, every tensor is usually represented with a number of bonds that each represent an index that has the same maximal number of possible values.", "For a MERA, this maximal bond dimension is usually denoted by $\\chi $ .", "The main idea to contract efficiently two MERA states is essentially to turn them into two MPS before contracting them.", "We look at the MERA circuit as having $n/2$ columns of gates vertically and $\\log _{\\chi }n-1$ renormalisation layers horizontally.", "The sequence of contraction is to sequentially contract every tensor in the leftmost column to create a tensor with a large number of bonds that will then contract with every tensor in the next column.", "The maximal number of bonds that this leftmost tensor will have throughout the contraction of the network is given by the maximal number of bonds that are opened when taking a vertical cut in the tensor network.", "For a single MERA, cutting through each of the $\\log _{\\chi }n-1$ layer opens up two bonds, one for the righmost incoming edge of the isometry and one for the outgoing edge of the isometry.", "Thus, for the contraction of two MERAs, the maximum number of bonds for a vertical cut is bounded by $\\max \\#=2\\times 2\\times \\log _{\\chi }n=4\\log _{\\chi }n$ , which is verified numerically (see top of Fig.", "REF ).", "Figure: (top) Maximum number of bonds during the contractionprocedure as a function of the logarithm of the number of qubits nn.Numerical results (solid blue line) are consistent with the expectedbound of 4log 2 n4\\log _{2}n. (bottom) Contraction cost CC as a functionof the number nn of qubits on a logscale.", "Numerical results (solidblue line) are consistent with the O(n 5 )O(n^{5}) bound (dot dashed greenline) but linear approximation (dashed red line) indicate that thecost scales like a smaller power of nn, namely C≃n 4.3 C\\simeq n^{4.3}.Since at every contraction step, the leftmost tensor with a large number of bonds contract with another tensor that has at most two bonds in addition to the ones being contracted, the maximum cost of one contraction is $\\chi ^{\\max \\#}\\chi ^{2}=\\chi ^{2}n^{4}$ .", "Finally, there are $O(n)$ disentanglers and isometries to contract so the total cost of contracting the network is bounded by $O(n^{5})$ .", "Actual numerical simulations show that this bound is probably not tight (see bottom of Fig.", "REF )." ] ]

[ [ "Multidegrees of Tame automorphisms with one prime number" ], [ "Abstract Let $3\\leq d_1\\leq d_2\\leq d_3$ be integers.", "We show the following results: (1) If $d_2$ is a prime number and $\\frac{d_1}{\\gcd(d_1,d_3)}\\neq2$, then $(d_1,d_2,d_3)$ is a multidegree of a tame automorphism if and only if $d_1=d_2$ or $d_3\\in d_1\\mathbb{N}+d_2\\mathbb{N}$; (2) If $d_3$ is a prime number and $\\gcd(d_1,d_2)=1$, then $(d_1,d_2,d_3)$ is a multidegree of a tame automorphism if and only if $d_3\\in d_1\\mathbb{N}+d_2\\mathbb{N}$.", "We also relate this investigation with a conjecture of Drensky and Yu, which concerns with the lower bound of the degree of the Poisson bracket of two polynomials, and we give a counter-example to this conjecture." ], [ "Introduction", "Throughout this paper, let $F=(F_1,\\dots ,F_n): k^n\\rightarrow k^n$ be a polynomial map, where $k$ is a field of characteristic 0.", "Denote by $\\mathrm {Aut}~k^n$ the group of all polynomial automorphisms of $k^n$ .", "Denote by $\\mathrm {mdeg}F:=(\\deg F_1,\\ldots ,\\deg F_n)$ the multidegree of $F$ and by $\\mathrm {mdeg}$ the mapping from the set of all polynomial maps into the set $\\mathbb {N}^n$ , where $\\mathbb {N}$ denotes the set of all nonnegative integers.", "A polynomial automorphism $F=(F_1,\\ldots ,F_n)$ of $k^n$ is called elementary if $F=(x_1,\\ldots ,x_{i-1}, \\alpha x_i+f(x_1,\\ldots ,x_{i-1},x_{i+1},\\ldots ,x_n), x_{i+1},\\ldots ,x_n)$ for $\\alpha \\in k^*$ .", "Denote by $\\mathrm {Tame}~k^n$ the subgroup of $\\mathrm {Aut}~k^n$ that is generated by all elementary automorphisms.", "The element in $\\mathrm {Tame}~k^n$ is called tame automorphism.", "The classical Jung-van der Kulk theorem [4], [15] shows that every polynomial automorphism of $k^2$ is tame.", "For many years people believe that $\\mathrm {Aut}~k^n$ is equal to $\\mathrm {Tame}~k^n$ .", "However, in 2004, Shestakov and Umirbaev [12], [13] proved the famous Nagata conjecture, that is, the Nagata automorphism on $k^3$ is not tame.", "The multidegree plays an important role in the description of polynomial automorphisms.", "For example, the Jacobian conjecture is equivalent to the assert that if $(F_1, F_2)$ is a polynomial map satisfying the Jacobian condition, then $\\mathrm {mdeg}F=(\\deg F_1, \\deg F_2)$ is principal, that is, $\\deg F_1\\mid \\deg F_2$ or $\\deg F_2\\mid \\deg F_1$ [1].", "But it is difficult to describe the multidegrees of polynomial maps in higher dimensions, even in dimension three.", "Recently, Karaś present a series of papers concerning with multidegrees of tame automorphisms in dimension three, see [5], [7], [8], [9].", "In [5], Karaś proposed the following conjecture.", "Conjecture 1.1 [5] Let $3\\le p_1\\le d_2\\le d_3$ be integers with $p_1$ a prime number.", "Then $(p_1,d_2,d_3)\\in \\mathrm {mdeg}(\\mathrm {Tame}~k^3)$ if and only if $p_1\\mid d_2$ or $d_3\\in p_1\\mathbb {N}+d_2\\mathbb {N}$ .", "In [6], Karaś showed that if $\\frac{d_3}{d_2}\\ne \\frac{3}{2}$ or $\\frac{d_3}{d_2}=\\frac{3}{2}$ and $d_2>2p_1-4$ , then Conjecture REF is valid.", "In [14], Sun and Chen also proved that if one of the following conditions is satisfied: (i) $\\frac{d_2}{\\gcd (d_2,d_3)}\\ne 2$ ; (ii) $\\frac{d_3}{\\gcd (d_2,d_3)}\\ne 3$ ; (iii) $d_2>2p_1-5$ , then Conjecture REF is true.", "In this paper, we consider a variation of the conjecture of Karaś.", "Let $3\\le d_1\\le d_2\\le d_3$ be integers.", "We show the following results: (1) If $d_2$ is a prime number and $\\frac{d_1}{\\gcd (d_1,d_3)}\\ne 2$ , then $(d_1,d_2,d_3)\\in \\mathrm {mdeg}(\\mathrm {Tame}~k^3)$ if and only if $d_1=d_2$ or $d_3\\in d_1\\mathbb {N}+d_2\\mathbb {N}$ ; (2) If $d_3$ is a prime number and $\\gcd (d_1,d_2)=1$ , then $(d_1,d_2,d_3)\\in \\mathrm {mdeg}(\\mathrm {Tame}~k^3)$ if and only if $d_3\\in d_1\\mathbb {N}+d_2\\mathbb {N}$ .", "We also relate this investigation to a conjecture of Drensky and Yu, which concerns with the lower bound of the degree of the Poisson bracket of two polynomials, and we give a counter-example of Drensky and Yu's conjecture." ], [ "Preliminaries", "Recall that a pair $f, g \\in \\mathbb {C}[x_1,\\ldots ,x_n]$ is called $*$ -reduced in [12], [13] if $f, g$ are algebraically independent; $\\bar{f}, \\bar{g}$ are algebraically dependent, where $\\bar{f}$ denotes the highest homogeneous component of $f$ ; $\\bar{f}\\notin \\langle \\bar{g} \\rangle $ and $\\bar{g}\\notin \\langle \\bar{f} \\rangle $ .", "The following inequality plays an important role in the proof of the Nagata conjecture in [12], [13] and is also essential in our proofs.", "Theorem 2.1 ([12]).", "Let $f, g \\in k[x_1,\\ldots ,x_n]$ be a $*$ -reduced pair, and $G(x,y)\\in k[x,y]$ with $\\deg _y G(x,y)=pq+r,\\ 0\\le r<p$ , where $p=\\frac{\\deg f}{\\gcd (\\deg f, \\deg g)}$ .", "Then $\\deg G(f,g)\\ge q(p\\deg g-\\deg f-\\deg g+\\deg [f,g])+r \\deg g.$ Note that $[f,g]$ means the Poisson bracket of $f$ and $g$ defined by $[f,g]=\\sum _{1\\le i<j\\le n}(\\frac{\\partial f}{\\partial x_i}\\frac{\\partial g}{\\partial x_j}-\\frac{\\partial f}{\\partial x_j}\\frac{\\partial g}{\\partial x_i})[x_i,x_j].$ By definition, $\\deg [x_i,x_j]=2$ for $i\\ne j$ and $\\deg 0=-\\infty $ , $\\deg [f,g]=\\max _{1\\le i<j\\le n}\\deg \\lbrace (\\frac{\\partial f}{\\partial x_i}\\frac{\\partial g}{\\partial x_j}-\\frac{\\partial f}{\\partial x_j}\\frac{\\partial g}{\\partial x_i})[x_i,x_j]\\rbrace .$ It is shown that $[f,g]=0$ if and only if $f,g$ are algebraically dependent.", "If $f, g$ are algebraically independent, then $\\deg [f,g]=2+\\max _{1\\le i<j\\le n}\\deg (\\frac{\\partial f}{\\partial x_i}\\frac{\\partial g}{\\partial x_j}-\\frac{\\partial f}{\\partial x_j}\\frac{\\partial g}{\\partial x_i})\\ge 2.$ Remark 2.2 It is easy to shown (see [8] for example) that Theorem REF is true even if $f,g$ just satisfy: (1) $f, g$ are algebraically independent; (2) $\\bar{f}\\notin \\langle \\bar{g} \\rangle $ and $\\bar{g}\\notin \\langle \\bar{f} \\rangle $ .", "Theorem 2.3 ([13]).", "Let $F=(F_1,F_2,F_3)$ be a tame automorphism of $k^3$ .", "If $\\deg F_1+\\deg F_2+\\deg F_3>3$ , then $F$ admits either an elementary reduction or a reduction of types I-IV (see [13]).", "Remark 2.4 It is shown by Kuroda that there is no tame automorphism on $k[x,y,z]$ admitting reductions of type IV, see [11].", "Recall that we say a polynomial automorphism $F=(F_1,F_2,F_3)$ admits an elementary reduction if there exists a polynomial $g\\in k[x,y]$ and a permutation $\\sigma $ of the set $\\lbrace 1,2,3\\rbrace $ such that $\\deg (F_{\\sigma (1)}-g(F_{\\sigma (2)},F_{\\sigma (3)}))<\\deg F_{\\sigma (1)}$ .", "In this paper, we consider when $(d_1,d_2,d_3)$ is a multidegree of a tame automorphism on $k^3$ .", "Note that if $(F_1,F_2,F_3)$ with multidegree $(d_1,d_2,d_3)$ is a tame automorphism, then after a permutation $\\sigma $ , $(F_{\\sigma (1)},F_{\\sigma (2)},F_{\\sigma (3)})$ is also a tame automorphism.", "It is also shown that if $d_1<3$ , then $(d_1,d_2,d_3)\\in \\mathrm {mdeg}(\\mathrm {Tame}~k^3)$ , see [8].", "Thus, without loss of generality, we can assume that $3\\le d_1\\le d_2\\le d_3$ ." ], [ "multidegree $(d_1,p_2,d_3)$ with {{formula:dd826fba-24b6-4b6d-9e3a-9c882aab04b7}} a prime number", "In this section, let $3\\le d_1\\le p_2\\le d_3$ be integers with $p_2$ a prime number.", "We start with some lemmas.", "Lemma 3.1 [2] If $a$ , $b$ are positive integers that $\\gcd (a,b)=1$ , then $l\\in a\\mathbb {N}+b\\mathbb {N}$ for all integers $l\\ge (a-1)(b-1)$ .", "Lemma 3.2 If $(d_1,p_2,d_3)\\in \\mathrm {mdeg}(\\mathrm {Tame}~k^3)$ , then there exists a tame automorphism with multidegree $(d_1,p_2,d_3)$ which admits an elementary reduction.", "Let $F$ be a tame automorphism with $\\mathrm {mdeg}~F=(d_1,p_2,d_3)$ .", "By Theorem REF and Remark REF , $F$ admits an elementary reduction or a reduction of types I-III.", "If $F$ admits a reduction of type III, then after a permutation, by [13] there exists $n\\in \\mathbb {N}$ such that $&(3.1)\\quad n<d_1\\le \\frac{3}{2}n,\\ p_2=2n,\\ d_3=3n;\\quad \\text{or}\\\\&(3.2)\\quad d_1=\\frac{3}{2}n,\\ p_2=2n,\\ \\frac{5n}{2}<d_3\\le 3n.$ Since $p_2$ is a prime number greater than 3, (3.1) and (3.2) can not be satisfied.", "Thus, $F$ admits no reduction of type III.", "By the definitions of reductions of types I and II, or see [6], if $F$ admits a reduction of type I or II, then there exists a tame automorphism admitting an elementary reduction with the same multidegree.", "We are now in a position to show our main result in this section.", "Theorem 3.3 Let $3\\le d_1\\le p_2\\le d_3$ be integers with $p_2$ a prime number.", "If $\\frac{d_1}{\\gcd (d_1,d_3)}\\ne 2$ , then $(d_1,p_2,d_3)\\in \\mathrm {mdeg}(\\mathrm {Tame}~k^3)$ if and only if $d_1=p_2$ or $d_3\\in d_1\\mathbb {N}+p_2\\mathbb {N}$ .", "(1) If $d_1=p_2$ or $d_3\\in d_1\\mathbb {N}+p_2\\mathbb {N}$ , then by [8], $(d_1,p_2,d_3)\\in \\mathrm {mdeg}(\\mathrm {Tame}~k^3)$ .", "(2) Now suppose that $d_1<p_2$ and $d_3\\notin d_1\\mathbb {N}+p_2\\mathbb {N}$ .", "Moreover, by Lemma REF , $d_3<(d_1-1)(p_2-1)$ .", "If there exists a tame automorphism with multidegree $(d_1,p_2,d_3)$ , then by Lemma REF , there exists a tame automorphism $F=(F_1,F_2,F_3)$ with $\\mathrm {mdeg}~F=(d_1,p_2,d_3)$ admitting an elementary reduction.", "Now the proof proceeds into three cases.", "Case 1: If $F$ admits an elementary reduction of the form $(F_1,F_2,F_3-g(F_1,F_2))$ such that $\\deg (F_3-g(F_1,F_2))<\\deg F_3$ , then $\\deg F_3=\\deg g(F_1,F_2)$ .", "Since $F$ is a polynomial automorphism, it follows that $F_i, F_j$ $(i,j=1,2,3)$ are algebraically independent, and hence $\\deg [F_i,F_j]\\ge 2$ .", "Moreover, $\\bar{F_i}\\notin \\langle \\bar{F_j} \\rangle $ since otherwise we have $\\deg F_i \\mid \\deg F_j$ , which contradicts to the fact that $d_1\\nmid p_2$ and $d_3\\notin d_1\\mathbb {N}+p_2\\mathbb {N}$ .", "Note that $p=\\frac{\\deg F_1}{\\gcd (\\deg F_1,\\deg F_2)}=d_1$ .", "Set $\\deg _y g(x,y)=d_1q+r,\\ 0\\le r<d_1$ .", "By Theorem REF and Remark REF , $d_3&=\\deg F_3=\\deg g(F_1,F_2)\\\\&\\ge q(d_1p_2-d_1-p_2+\\deg [F_1,F_2])+rp_2\\\\&\\ge q(d_1p_2-d_1-p_2+2)+rp_2.$ Since $d_3<(d_1-1)(p_2-1)$ , we have $q=0$ .", "Note that $0\\le r<d_1$ , we can suppose that $g(x,y)=g_0(x)+g_1(x)y+\\cdots +g_{d_1-1}(x)y^{d_1-1}$ .", "It follows from $\\gcd (d_1,p_2)=1$ that the sets $d_1\\mathbb {N},~ d_1\\mathbb {N}+p_2,~\\dots ,~d_1\\mathbb {N}+(d_1-1)p_2$ are disjoint.", "Thus, $d_3&=\\deg g(F_1,F_2)=\\deg (g_0(F_1)+g_1(F_1)F_2+\\cdots +g_{d_1-1}(F_1)F_2^{d_1-1})\\\\&=\\max _{0\\le i\\le d_1-1}\\lbrace \\deg F_1\\deg g_i+i\\deg F_2\\rbrace =\\max _{0\\le i\\le d_1-1}\\lbrace d_1\\deg g_i+ip_2\\rbrace ,$ which contradicts $d_3\\notin d_1\\mathbb {N}+p_2\\mathbb {N}$ .", "Case 2: If $F$ admits an elementary reduction of the form $(F_1-g(F_2,F_3),F_2,F_3)$ , then $\\deg F_1=\\deg g(F_2,F_3)$ .", "$p=\\frac{\\deg F_2}{\\gcd (\\deg F_2,\\deg F_3)}=p_2$ .", "Set $\\deg _y g(x,y)=p_2q+r,\\ 0\\le r<p_2$ .", "Then $d_1&=\\deg F_1=\\deg g(F_2,F_3)\\\\&\\ge q(p_2d_3-p_2-d_3+\\deg [F_2,F_3])+rd_3\\\\&\\ge q(3d_3-p_2-d_3+2)+rd_3\\\\&\\ge q((d_3-p_2)+d_3+2)+rd_3.$ Since $d_1<(d_3-p_2)+d_3+2$ and $d_1<d_3$ , it follows that $q=r=0$ .", "Suppose that $g(F_2,F_3)=g_1(F_2)$ .", "Then $d_1=\\deg F_1=\\deg g_1(F_2)\\in p_2\\mathbb {N}$ , contrary to $d_1<p_2$ .", "Case 3: If $F$ admits an elementary reduction of the form $(F_1,F_2-g(F_1,F_3),F_3)$ , then $\\deg F_2=\\deg g(F_1,F_3)$ .", "It follows from $d_3\\notin d_1\\mathbb {N}+p_2\\mathbb {N}$ that $\\gcd (d_1,d_3)\\ne d_1$ , whence $p=\\frac{d_1}{\\gcd (d_1,d_3)}\\ge 2$ .", "Moreover, since $\\frac{d_1}{\\gcd (d_1,d_3)}\\ne 2$ , $p\\ge 3$ .", "Let $\\deg _yg(x,y)=pq+r,\\ 0\\le r<p$ .", "Then $p_2&=\\deg F_2=\\deg g(F_1,F_3)\\\\&\\ge q(pd_3-d_1-d_3+\\deg [F_1,F_3])+rd_3\\\\&\\ge q(3d_3-d_1-d_3+2)+rd_3\\\\&=q((d_3-d_1)+d_3+2)+rd_3.$ Thus, $q=r=0$ .", "Suppose that $g(F_1,F_3)=g_1(F_1)$ .", "Then $p_2=\\deg F_2=\\deg g_1(F_1)\\in d_1\\mathbb {N}$ , a contradiction.", "Therefore, $F$ can not admit any elementary reduction, the contradiction implies that there exists no tame automorphism with multidegree $(d_1,p_2,d_3)$ if $d_1<p_2$ and $d_3\\notin d_1\\mathbb {N}+p_2\\mathbb {N}$ .", "We claim that the condition $\\frac{d_1}{\\gcd (d_1,d_3)}\\ne 2$ in Theorem REF can not be removed.", "Indeed, Kuroda construct some tame automorphisms, after a permutation, with multidegree $(2m,2pm+p+1,(2p+1)m)$ admitting reductions of type I in [10].", "Particularly, let $p=2$ , $m=5$ or 11.", "Then Example 3.4 There exist tame automorphisms with multidegree $(10,23,25)$ and $(22,47,55)$ .", "Moreover, using the method in [10], we can get a tame automorphism $F=(f_1,f_2,f_3)$ with $\\mathrm {mdeg}~F=(10,23,25)$ admitting reductions of type I, where $\\left\\lbrace \\begin{array}{ll}f_1=x+y^2-g^2, \\\\f_2=\\frac{256}{25}f_1^5+g+h^2, \\\\f_3=f_2+h,\\end{array}\\right.$ $g=z+3x^2y+3xy^3+y^5$ and $h=y-6(x+y^2)^2g+8(x+y^2)g^3-\\frac{16}{5}g^5$ .", "In the proof of Theorem REF , we observe that if a more precise lower bound of $\\deg [F_1,F_3]$ is given, then we can give a better description of $\\mathrm {mdeg}(\\mathrm {Tame}~k^3)$ .", "This is closely related to a conjecture of Drensky and Jie-Tai Yu.", "Conjecture 3.5 [3] Let $f$ and $g$ be algebraically independent polynomials in $k[x_1,\\ldots ,x_n]$ such that the homogeneous components of maximal degree of $f$ and $g$ are algebraically dependent, $f$ and $g$ generate their integral closures $C(f)$ and $C(g)$ in $k[x_1,\\ldots ,x_n]$ , respectively, and neither $\\deg f | \\deg g$ nor $\\deg g | \\deg f$ .", "Then $\\deg [f,g]>\\min \\lbrace \\deg (f), \\deg (g)\\rbrace .$ Although some counter-examples of Conjecture REF are given in [3], it is still of great interest to find a meaningful lower bound of $\\deg [f,g]$ , and such a bound will give a nice description of $\\mathrm {Tame}~k^n$ and $\\mathrm {Aut}~k^n$ .", "We observe that, from Example REF , we can construct some counter-examples of Conjecture REF .", "Example 3.6 $F=(f_1,f_2,f_3)=(x+y^2-g^2,\\frac{256}{25}f_1^5+g+h^2,h)$ is a tame automorphism admitting an elementary reduction, where $g=z+3x^2y+3xy^3+y^5$ and $h=y-6(x+y^2)^2g+8(x+y^2)g^3-\\frac{16}{5}g^5$ .", "Moreover, $\\mathrm {mdeg}~F=(10,23,25)$ , $\\deg [f_1,f_3]=8<\\min \\lbrace \\deg f_1,\\deg f_3\\rbrace $ .", "Thus, $(f_1,f_3)$ is a counter-example of Conjecture REF .", "It follows from Example REF that $F^{\\prime }=(x+y^2-g^2,\\frac{256}{25}f_1^5+g+h^2,f_2+h)$ is a tame automorphism with $\\mathrm {mdeg}~F^{\\prime }=(10,23,25)$ admitting an reduction of type I.", "By [13], after composing an affine automorphism $(x,y,z-y)$ , $F=(f_1,f_2,f_3)=(x+y^2-g^2, \\frac{256}{25}f_1^5+g+h^2, h)$ is a tame automorphism with $\\mathrm {mdeg}~F=(10,23,25)$ admitting an elementary reduction.", "Moreover, $\\deg [f_1,f_3]= (-30x^2y^4-54x^3y^2-18x^4-6y^3z-12xyz+1)[x,y]\\\\-(6y^4+12xy^2+6x^2)[x,z]+(-10y^5-18xy^3-6x^2y+2z)[y,z].$ Thus, $\\deg [f_1,f_3]=8<\\min \\lbrace \\deg f_1,\\deg f_3\\rbrace $ , whence the homogeneous components of maximal degree of $f_1$ and $f_3$ are algebraically dependent.", "Furthermore, since $(f_1,f_2,f_3)$ is a polynomial automorphism, it follows that $f_1$ and $f_3$ are algebraically independent irreducible polynomials.", "Thus, $(f_1,f_3)$ is a counter-example of Conjecture REF ." ], [ "multidegree $(d_1,d_2,p_3)$ with {{formula:e343040c-6e77-4336-b7ee-a9d071494ac9}} a prime number", "In this section, let $3\\le d_1\\le d_2\\le p_3$ be integers with $\\gcd (d_1,d_2)=1$ and $p_3$ a prime number.", "Lemma 4.1 If $(d_1,d_2,p_3)\\in \\mathrm {mdeg}(\\mathrm {Tame}~k^3)$ with $\\gcd (d_1,d_2)=1$ and $p_3$ a prime number, then there exists a tame automorphism with multidegree $(d_1,d_2,p_3)$ which admits an elementary reduction.", "Let $F$ be a tame automorphism with $\\mathrm {mdeg}~F=(d_1,d_2,p_3)$ .", "By Theorem REF and Remark REF , $F$ admits an elementary reduction or a reduction of types I-III.", "If $F$ admits a reduction of type III, then after a permutation, by [13] there exists $n\\in \\mathbb {N}$ such that $&(4.1)\\quad n<d_1\\le \\frac{3}{2}n,\\ d_2=2n,\\ p_3=3n;\\quad \\text{or}\\\\&(4.2)\\quad d_1=\\frac{3}{2}n,\\ d_2=2n,\\ \\frac{5n}{2}<p_3\\le 3n.$ Since $p_3$ is a prime number greater that 3, (4.1) can not be satisfied.", "If $(d_1,d_2,p_3)$ satisfies (4.2), it follows from $\\gcd (d_1,d_2)=1$ that $n=2$ , and hence $5<p_3\\le 6$ , contrary to the fact that $p_3$ is a prime number.", "Thus, $F$ admits no reduction of type III.", "By [6], if $F$ admits a reduction of type I or II, then there exists a tame automorphism with the same multidegree that admits an elementary reduction.", "We can now formulate our main result in this section.", "Theorem 4.2 Let $3\\le d_1\\le d_2\\le p_3$ be integers with $\\gcd (d_1,d_2)=1$ and $p_3$ a prime number.", "Then $(d_1,d_2,p_3)\\in \\mathrm {mdeg}(\\mathrm {Tame}~k^3)$ if and only if $p_3\\in d_1\\mathbb {N}+d_2\\mathbb {N}$ .", "(1) If $p_3\\in d_1\\mathbb {N}+d_2\\mathbb {N}$ , by [8], $(d_1,d_2,p_3)\\in \\mathrm {mdeg}(\\mathrm {Tame}~k^3)$ .", "(2) Now suppose that $p_3\\notin d_1\\mathbb {N}+d_2\\mathbb {N}$ , whence $p_3<(d_1-1)(d_2-1)$ by Lemma REF .", "If there exists a tame automorphism with multidegree $(d_1,d_2,p_3)$ , then by Lemma REF , there exists a tame automorphism $F=(F_1,F_2,F_3)$ with $\\mathrm {mdeg}~F=(d_1,d_2,p_3)$ admitting an elementary reduction.", "Now the proof will be divided into three cases.", "Case 1: If $F$ admits an elementary reduction of the form $(F_1-g(F_2,F_3),F_2,F_3)$ , then $\\deg F_1=\\deg g(F_2,F_3)$ .", "$p=\\frac{\\deg F_2}{\\gcd (\\deg F_2,\\deg F_3)}=d_2$ .", "Set $\\deg _y g(x,y)=d_2q+r,\\ 0\\le r<d_2$ .", "Then $d_1&=\\deg F_1=\\deg g(F_2,F_3)\\\\&\\ge q(d_2p_3-d_2-p_3+\\deg [F_2,F_3])+rp_3\\\\&\\ge q(3p_3-d_2-p_3+2)+rp_3\\\\&\\ge q((p_3-d_2)+p_3+2)+rp_3.$ Thus, $q=r=0$ .", "Suppose that $g(F_2,F_3)=g_1(F_2)$ .", "Then $d_1=\\deg F_1=\\deg g_1(F_2)\\in d_2\\mathbb {N}$ , contrary to $d_1<d_2$ .", "Case 2: If $F$ admits an elementary reduction of the form $(F_1,F_2-g(F_1,F_3),F_3)$ , then $\\deg F_2=\\deg g(F_1,F_3)$ .", "$p=\\frac{\\deg F_1}{\\gcd (\\deg F_1,\\deg F_3)}=d_1$ .", "Set $\\deg _yg(x,y)=d_1q+r,\\ 0\\le r<d_1$ .", "Then $d_2&=\\deg F_2=\\deg g(F_1,F_3)\\\\&\\ge q(d_1p_3-d_1-p_3+\\deg [F_1,F_3])+rp_3\\\\&\\ge q(3p_3-d_1-p_3+2)+rp_3\\\\&=q((p_3-d_1)+p_3+2)+rp_3.$ Thus, $q=r=0$ .", "Suppose that $g(F_1,F_3)=g_1(F_1)$ .", "Then $d_2=\\deg F_2=\\deg g_1(F_1)\\in d_1\\mathbb {N}$ , a contradiction.", "Case 3: If $F$ admits an elementary reduction of the form $(F_1,F_2,F_3-g(F_1,F_2))$ such that $\\deg (F_3-g(F_1,F_2))<\\deg F_3$ , then $\\deg F_3=\\deg g(F_1,F_2)$ .", "It follows from $\\gcd (d_1,d_2)=1$ that $p=\\frac{\\deg F_1}{\\gcd (\\deg F_1,\\deg F_2)}=d_1$ .", "Set $\\deg _y g(x,y)=d_1q+r,\\ 0\\le r<d_1$ .", "Then $p_3&=\\deg F_3=\\deg g(F_1,F_2)\\\\&\\ge q(d_1d_2-d_1-d_2+\\deg [F_1,F_2])+rd_2\\\\&\\ge q(d_1d_2-d_1-d_2+2)+rd_2.$ Since $p_3<(d_1-1)(d_2-1)$ , we have $q=0$ .", "Note that $0\\le r<d_1$ , we can suppose that $g(x,y)=g_0(x)+g_1(x)y+\\cdots +g_{d_1-1}(x)y^{d_1-1}$ .", "It follows from $\\gcd (d_1,d_2)=1$ that the sets $d_1\\mathbb {N},~ d_1\\mathbb {N}+d_2,~\\dots ,~d_1\\mathbb {N}+(d_1-1)d_2$ are disjoint.", "Thus, $p_3&=\\deg g(F_1,F_2)=\\deg (g_0(F_1)+g_1(F_1)F_2+\\cdots +g_{d_1-1}(F_1)F_2^{d_1-1})\\\\&=\\max _{0\\le i\\le d_1-1}\\lbrace \\deg F_1\\deg g_i+i\\deg F_2\\rbrace =\\max _{0\\le i\\le d_1-1}\\lbrace d_1\\deg g_i+id_2\\rbrace ,$ which contradicts $p_3\\notin d_1\\mathbb {N}+d_2\\mathbb {N}$ .", "Thus, $F$ admits no elementary reduction, the contradiction implies that there exists no tame automorphism with multidegree $(d_1,d_2,p_3)$ if $p_3\\notin d_1\\mathbb {N}+d_2\\mathbb {N}$ ." ] ]

[ [ "Mott Transition and Spin Structures of Spin-1 Bosons in Two-Dimensional\n Optical Lattice at Unit Filling" ], [ "Abstract We study the ground state properties of spin-1 bosons in a two-dimensional optical lattice, by applying a variational Monte Carlo method to the S=1 Bose-Hubbard model on a square lattice at unit filling.", "A doublon-holon binding factor introduced in the trial state provides a noticeable improvement in the variational energy over the conventional Gutzwiller wave function and allows us to deal effectively with the inter-site correlations of particle densities and spins.", "We systematically show how spin-dependent interactions modify the superfluid-Mott insulator transitions in the S=1 Bose-Hubbard model due to the interplay between the density and spin fluctuations of bosons.", "Furthermore, regarding the magnetic phases in the Mott region, the calculated spin structure factor elucidates the emergence of nematic and ferromagnetic spin orders for antiferromagnetic ($U_2>0$) and ferromagnetic ($U_2<0$) couplings, respectively." ], [ "Mott Transition and Spin Structures of Spin-1 Bosons in Two-Dimensional Optical Lattice at Unit Filling Yuta TogaE-mail address: [email protected], Hiroki Tsuchiura, Makoto Yamashita$^{1,2}$ , Kensuke Inaba$^{1,2}$ , and Hisatoshi Yokoyama$^{3}$ Department of Applied Physics, Tohoku University, Sendai 980-8579, Japan $^{1}$ NTT Basic Research Laboratories, NTT Corporation, Atsugi, Kanagawa 243-0198, Japan $^{2}$ Japan Science and Technology Agency, CREST, Chiyoda, Tokyo 102-0075, Japan $^{3}$ Department of Physics, Tohoku University, Sendai 980-8578, Japan We study the ground state properties of spin-1 bosons in a two-dimensional optical lattice, by applying a variational Monte Carlo method to the $S=1$ Bose-Hubbard model on a square lattice at unit filling.", "A doublon-holon binding factor introduced in the trial state provides a noticeable improvement in the variational energy over the conventional Gutzwiller wave function and allows us to deal effectively with the inter-site correlations of particle densities and spins.", "We systematically show how spin-dependent interactions modify the superfluid-Mott insulator transitions in the $S=1$ Bose-Hubbard model due to the interplay between the density and spin fluctuations of bosons.", "Furthermore, regarding the magnetic phases in the Mott region, the calculated spin structure factor elucidates the emergence of nematic and ferromagnetic spin orders for antiferromagnetic ($U_2>0$ ) and ferromagnetic ($U_2<0$ ) couplings, respectively.", "Mott transition, superfluid, insulating state, nematic phase, ferromagnetism, $S=1$ Bose-Hubbard model, doublon-holon binding, variational Monte Carlo method Recent progress in ultracold atom experiments has offered unprecedented opportunities for exploring fundamental quantum phenomena in strongly correlated many-body systems that have been largely ignored in conventional condensed-matter physics.", "A prominent example of such phenomena is the quantum phase transition from a superfluid (SF) to a Mott insulator (MI), demonstrated using cold bosons with frozen spin degrees of freedom trapped in optical lattices[1].", "Furthermore, quantum gas microscope techniques [2], [3], [4], [5] have opened the door to the detection and manipulation of single bosons at a single site level in an optical lattice, just like scanning tunneling microscopy in solid state physics.", "Quite recently, Endres et al.", "[6] used this technique to track the SF-MI transition in more detail, and found that correlated pairs consisting of a doubly populated site (doublon, D) and an unpopulated site (holon, H), which represent the excitations in an MI, fundamentally determine the properties of the SF-MI transition, which is consistent with recent numerical studies [7], [8].", "Now theoretical interest is naturally moving towards the quantum phase transitions in bosons with unfrozen spin degrees of freedom[9], [10] trapped in optical lattices.", "The doublon, which is one fragment of the elementary excitation in an MI, will have internal spin structures unlike spinless (spin-frozen) systems.", "Thus, we can strongly expect the interplay between spin correlations and the SF-MI transition to play key roles in the ground state of the systems.", "The simplest of such systems will be $S=1$ bosons on an optical lattice, whose properties are well captured by the $S=1$ Bose-Hubbard model (BHM)[11] ${\\cal H} &=& -t\\sum _{\\langle i,j\\rangle }\\sum _{\\alpha }\\left( \\hat{a}_{i,\\alpha }^{\\dagger }\\hat{a}_{j,\\alpha }+ \\hat{a}_{j,\\alpha }^{\\dagger }\\hat{a}_{i,\\alpha } \\right)- \\mu \\sum _{i}\\hat{n}_{i}\\nonumber \\\\& &+ \\frac{U_{0}}{2}\\sum _{i}\\hat{n}_{i}(\\hat{n}_{i}-1)+ \\frac{U_{2}}{2}\\sum _{i}\\left( \\hat{S}_{i}^{2}- 2\\hat{n}_{i} \\right) ,$ where $a_{j,\\alpha }$ is an annihilation operator of a boson of spin $\\alpha $ at the site $j$ , $\\hat{n}_{j} = \\sum _\\alpha \\hat{a}_{j,\\alpha }^{\\dagger }\\hat{a}_{j,\\alpha }$ and $t, U_0 > 0$ .", "Here, $\\alpha =-1,0,1$ , and $\\langle i,j \\rangle $ denotes a nearest-neighbor-site pair; the definition of $t$ is a half of that reported in some studies.", "The spin-dependent (last) term in eq.", "(REF ) induces spin mixing and enriches the physics of this model, compared with spinless models.", "In cold-atom systems, the values of $U_0$ and $U_2$ depend on the $s$ -wave scattering wavelength characteristic of the atom species; $U_2>0$ ($<0$ ) for Na (Rb) atoms.", "In contrast to more familiar Fermi-Hubbard models with $S=1/2$ where an antiferromagnetic superexchange interaction prevails in the strongly correlated regime, the $S=1$ BHM exhibits complicated effective inter-site spin interactions that lead to exotic magnetic phases[12], [13], owing to the absence of the Pauli exclusion principle.", "Thus far, the phase diagram of this model has been studied based on mean-field type theories[14], [15] including a Gutzwiller approximation (GA)[16], [17], and on density matrix renormalization group[18], [19] and quantum Monte Carlo (QMC) methods for the one-dimensional system[20], [21].", "Alternatively, at the cost of the density fluctuation, an effective spin Hamiltonian obtained by a strong-coupling expansion [12], [15], [22] was studied to explore the magnetic structures in the MI phase using QMC calculations in two and three dimensions[23].", "In this letter, we provide a consistent description of the ground state properties and the phase transition from SF to MI of the $S = 1$ BHM on a square lattice, focusing on correlated pair excitations based on a variational Monte Carlo (VMC) approach, which is beyond conventional mean-field and GA techniques.", "We consider the simplest case of an SF-MI transition so that we restrict the particle density $\\rho = N/N_{s}$ ($N$ : particle number, $N_{s}$ : the total number of sites) to $\\rho =1$ (unit filling) and put $\\mu = 0$ .", "We consider the behavior for other odd commensurate densities ($\\rho =3,5,\\cdots $ ) to be essentially the same.", "We will report even-$\\rho $ cases separately.", "To implement VMC calculations, we employ an occupation number representation at each site, $|n_1, n_0, n_{-1}\\rangle $ .", "As a variational wave function, we use a Jastrow-type, $|\\Psi _{\\rm DH}\\rangle ={\\mathcal {P}}_{\\rm DH} {\\mathcal {P}}_{\\rm G} |\\Phi \\rangle $ .", "Here $|\\Phi \\rangle $ is the one-body part: $|\\Phi \\rangle =\\frac{1}{\\sqrt{N_{\\rm s}!", "}}\\left(\\hat{a}_{{\\bf 0},1}^\\dag +\\hat{a}_{{\\bf 0},0}^\\dag +\\hat{a}_{{\\bf 0},-1}^\\dag \\right)^{N_{\\rm s}}|0\\rangle ,$ where, $\\hat{a}_{{\\bf 0},1}^\\dag $ indicates the ${k}={\\bf 0}$ component of the Fourier transformation of $\\hat{a}_{j,1}^\\dag $ .", "Since the total $S_{z}$ is well conserved in cold atom systems, we work in the subspace of $\\sum _{j}S_{j}^{z} = 0$ .", "As an onsite correlation factor, the Gutzwiller projection is extended so that it depends on the spin configurations in the site, [17] ${\\mathcal {P}}_{\\rm G}=\\prod _j\\gamma (n_{j,1}, n_{j,0}, n_{j,-1})|n_1, n_0, n_{-1}\\rangle _j\\ _j\\langle n_1,n_0,n_{-1}|,$ where coefficients $\\gamma $ are variational parameters controlling $n_{j,\\alpha }$ , the occupation particle number of spin $\\alpha $ on the site $j$ .", "The dependence on spin configuration is necessary for $U_2\\ne 0$ .", "Since we have confirmed that the probability $P(n)$ for $n\\ge 4$ becomes negligible for the value of interest, $U_{0}/t\\ $ >$$$\\sim $ 10$,we impose a restriction, $ (nj,1, nj,0, nj,-1)=0$,on the total occupation number ($ n$) at each site for$ njnj,1+nj,0+nj,-14$.The D-H correlation factor $ PDH(,)$ used here is the same as that introduced in refs.", "{manuela} and {yokoB},where $$ ($ '$) is a variational parameter ($ 01$) that controls the strength of theD-H binding between nearest-neighbor (lattice-diagonal) sites.For $ =1$, isolated doublons and holons are prohibited.We assume $$ and $ '$ are independent of the onsite spin configuration for simplicity.It should be noted here that in effect a multiply populated site means a doublon for $ U0/t $>$$\\sim $ 20$, because$ P(n)$ for $ n3$ almost vanishes.$ In the VMC calculations, we first optimized the variational parameters using a quasi-Newton method, and calculated the quantities for sets of model parameters ($U_0/t$ , $U_2/t$ ) with several million configurations for several system sizes of $N_s=L\\times L$ sites with $L=8$ -24.", "Figure: (Color online) The total energy per site of |Ψ G 〉|\\Psi _{\\rm G}\\rangle and |Ψ DH 〉|\\Psi _{\\rm DH}\\rangle is compared as a function of U 0 /tU_0/t for three system sizes.The VMC data of |Ψ G 〉|\\Psi _{\\rm G}\\rangle for finite LL's will converge to the GA result,the exact analytic result of |Ψ G 〉|\\Psi _{\\rm G}\\rangle for L=∞L=\\infty .The ratio U 2 /U 0 U_2/U_0 is fixed at 0.1, and the result is similar to thatof U 2 =0U_2=0.We start by discussing the features of a Mott transition in $|\\Psi _{\\rm DH}\\rangle $ by comparing those obtained with a Gutzwiller wave function $|\\Psi _{\\rm G}\\rangle =\\mathcal {P}_{\\rm G}|\\Phi \\rangle $ when the ratio $U_2/U_0$ is fixed at $0.1$ .", "Figure REF shows how the total energy $E/t$ is improved by employing $|\\Psi _{\\rm DH}\\rangle $ in the region of intermediate correlation strength.", "The value of $|\\Psi _{\\rm G}\\rangle $ arrives at zero at the Brinkman-Rice transition point[24], $U_0^{\\rm BR}/t\\sim 24.3$ .", "On the other hand, $E/t$ of $|\\Psi _{\\rm DH}\\rangle $ is considerably less than that of GA, and remains finite even for large $U_0/t$ values.", "Figure: (Color online) (a) Condensate fraction or momentum distribution functionat k=(0,0){k} = (0,0) and (b) density fluctuation, as a function ofU 0 /tU_0/t for |Ψ G 〉|\\Psi _{\\rm G}\\rangle and of |Ψ DH 〉|\\Psi _{\\rm DH}\\rangle for U 2 /U 0 =0.1U_2/U_0=0.1.The inset in (a) shows an enlarged view near the Mott transition point.In Fig.", "REF (a), the condensate fraction or ${k}=(0,0)$ element of the momentum distribution function, $n({k})=\\sum _\\alpha \\langle \\hat{a}_{{k},\\alpha }^\\dag \\hat{a}_{{k},\\alpha }\\rangle ,$ is plotted as a function of $U_0/t$ .", "This quantity is finite in the SF phase, but should vanish as $1/N_{\\rm s}$ in the MI phase.", "The value of $|\\Psi _{\\rm DH}\\rangle $ exhibits a sudden drop (for $L\\ge 16$ ) at $U_0=U_{\\rm 0c}\\sim 22t$ and vanishes as $1/N_{\\rm s}$ for $U_0>U_{\\rm 0c}$ in the inset shown in Fig.", "REF (a).", "$n({k}={0})/N$ is regarded as an order parameter of the Mott transition here, and so this anomaly indicates a first-order SF-MI transition.", "In fact, we confirmed that the small-$|{q}|$ behavior of the density correlation function $N({q})$ (not shown) changes suddenly from linear to quadratic in momentum $|{q}|$ at $U_0=U_{\\rm 0c}$ .", "These features are substantially identical to those in the spinless case on the same D-H binding mechanism[8].", "Next, let us look at the density fluctuation, $\\sigma _0^2=\\langle \\hat{n}^2\\rangle -\\langle \\hat{n}\\rangle ^2$ , which is shown in Fig.", "REF (b).", "As is known, the $\\sigma _0^2$ of GA completely vanishes at $U_0=U_0^{\\rm BR}$ , which corroborates the view that each site is occupied by exactly one particle in the MI phase.", "On the other hand, $\\sigma _0^2$ of $|\\Psi _{\\rm DH}\\rangle $ exhibits a small step at $U_0=U_{\\rm 0c}$ and remains finite for $U_0>U_{\\rm 0c}$ .", "This finite density fluctuation is reflected in the small but finite values of $P(0)\\sim P(2)$ .", "In the following, we show that the doublon plays a crucial role for the spin structure.", "Figure: (Color online) (a) U 2 /U 0 U_2/U_0 dependence of optimized D-H binding parameter η\\eta as a function of U 0 /tU_0/t.", "(b),(c) show the number fraction of atoms with each spin-component and (d),(e) show the probabilities of doublons with each spin state around U 0c /tU_{0c}/t for U 2 /U 0 =±0.1U_2/U_0=\\pm 0.1.Now, we turn to the spin-dependent features caused by the $U_{2}$ -term in ${\\cal H}$ .", "In Fig.", "REF (a), the optimized values of the D-H binding parameter $\\eta $ are plotted to recognize the effects of the $U_{2}$ -term on the SF-MI transition.", "Recall that $\\eta $ controls the strength of the D-H binding between nearest-neighbor sites, and $\\eta = 1$ means that each doublon is tightly bound to an adjacent holon.", "We find that the SF-MI transition point $U_{0c}/t$ shifts to noticeably larger values for $U_{2}/U_{0} = 0.3$ and $-0.1$ , whereas the shift for $U_{2}/U_{0} = 0.1$ is very small.", "First, let us consider the difference in the $U_{0c}$ shift between for $U_{2}/U_{0} = \\pm 0.1$ .", "Since the system is in the vicinity of the MI phase ($U_{0}\\gg t$ ), we may restrict the Fock space to $n_{j} = 0, 1, 2$ at each site; actually we confirmed that $n_j\\ge 3$ is negligible for $U_{0}/t\\ $ >$$$\\sim $ 20$.To minimize the ground state energy for $ U20$, not only the numberof doubly populated ($ n=2$) sites but also their spin structures have tobe optimized.To this end, it is convenient to employ the eigenstates of$ Sj2$, $ |Sj,Sjz$, where$ Sj$ and $ Sjz$ indicate the magnitude and $ z$-componentof the total spin at the site-$ j$, respectively; $ Sj$ is even (odd)when an even (odd) number of atoms occupy the site-$ j$.Using $ |Sj,Sjz$, our basis set$ |nj,1,nj,0,nj,-1$ for $ n=2$ is represented as\\begin{eqnarray}|2,0,0\\rangle &=&|2,2\\rangle \\rangle , ~~~|1,1,0\\rangle = |2,1\\rangle \\rangle , \\nonumber \\\\|0,2,0\\rangle &=& \\frac{1}{\\sqrt{3}}\\left( \\sqrt{2}|2,0\\rangle \\rangle + |0,0\\rangle \\rangle \\right), \\nonumber \\\\|1,0,1\\rangle &=& \\frac{1}{\\sqrt{3}}\\left( |2,0\\rangle \\rangle - \\sqrt{2}|0,0\\rangle \\rangle \\right).\\end{eqnarray}According to eq.", "(\\ref {S0S2}), the on-site energies of $ n=2$ sitesare calculated as$ U0+U2$ for $ |2,0,0$ and $ |1,1,0$,$ U0$ for $ |0,2,0$, and$ U0-U2$ for $ |1,0,1$.We also define the probability $ p = N/Ns$($ = -1, 0, 1$), where $ N$ denotes the total number ofparticles of spin $$; in the present case, $ p$ obeysthe conditions $ 0 p 1$ and $ p = 1$.Since we assume $ j Sjz = 0$, the relation $ p1 = p-1$ holds,resulting in $ 2p1 + p0 = 1$.Using these formulae, the classical statistical weights for the $ n=2$configurations can be calculated as$ (p1)2$ for $ |2,0,0$, $ 2(p1)2$ for $ |1,0,1$,$ (p0)2$ for $ |0,2,0$,and $ 2p0p1$ for $ |1,1,0$.For $ U2 t$, the expectation value of the $ U2$-term per siteis similarly obtained as\\begin{equation}E_{2}(p_{1}) \\propto -U_{2} p_{1}\\left( p_{1} - 1/2 \\right) .\\end{equation}This energy has the minimum value $ U2/2$ at $ p1 = 1/4$ for $ U2 < 0$,and $ 0$ at $ p1 = 0$ and $ 1/2$ for $ U2 > 0$.Actually, the VMC results in Figs.", "\\ref {dh-D}(b) and \\ref {dh-D}(c) show$ p1 = N1/Ns 1/4$ for $ U2/U0 = -0.1$,and $ p1 0.5$ for $ U2/U0 = 0.1$.This causes imbalanced spin population for $ n=2$ sites, as shownin Figs.~\\ref {dh-D}(d) and \\ref {dh-D}(e).Thus, we find through eq.~(\\ref {E2}) that the on-site energies of$ n=2$ sites are renormalized to $ U0 + U2/2$ for $ U2 < 0$,while they are not renormalized for $ U2 > 0$.Since the onsite energies of $ n=2$ sites primarily govern the SF-MItransition, this is an appropriate explanation of the difference in the$ U0c/t$ shifts between for $ U2/U0 = 0.1$.Finally, we point out that the degeneracy in $ U2$-energy between$ p1 = 0$ and $ 0.5$ for $ U2 > 0$ found in eq.", "(\\ref {E2}) isowing to the present spin-1 rotational symmetry, namely,$ |Sx = 0= ( |Sz = 1+ |Sz = -1)/2$.Therefore, we can generate a ground state with $ p1 0$ using VMCif we choose a certain initial condition with $ p0 p1$.$ Next, we discuss the difference in the $U_{0c}/t$ shifts between for $U_{2}/U_{0} = 0.1$ and $0.3$ .", "In the above discussion, we adopted a $classical$ statistical weighting [$E_2$ in eq.", "()], which ignores the effect of spin fluctuation caused by the $U_{2}$ -term, because the spin fluctuation (or singlet formation) is suppressed by the particle hopping for small $|U_2|/U_0$ 's.", "This is not the case for large $|U_2|/U_0$ 's.", "When the $U_2$ -term becomes predominant over the hopping term, the singlet state $|0,0\\rangle \\rangle =1/\\sqrt{3}\\left(|0,2,0\\rangle -\\sqrt{2}|1,0,1\\rangle \\right)$ becomes significant in $n=2$ sites, in order to reduce further the $U_{2}$ -energy through the spin-exchange processes in the $\\hat{{S}}^{2}$ -term.", "In this case, assuming $U_2\\gg t$ , we ignore the hopping term and directly diagonalize the interaction part in ${\\cal H}$ using $|S_{j},S_{j}^{z}\\rangle \\rangle $ .", "As a result, we find that the on-site energy of $n=2$ sites is renormalized toward $U_{0} - 2U_{2}$ , which vanishes for $U_{2}/U_{0} = 0.5$ .", "Thus, the value of $U_{0c}/t$ is bound to grow rapidly as $U_{2}/U_{0}$ approaches 0.5, which explains the pronounced shift for $U_{2}/U_{0} = 0.3$ in Fig.", "REF (a).", "Now, we study the magnetic structures and the spin correlations in the ground state around the SF-MI transition.", "For $U_2/U_0>0$ , the imbalance of the spin populations found in Fig.", "REF (d) suggests that the ground state of the system exhibits the spin-nematic property.", "To detect this with regard to the present case, it is useful to study not only spin-correlation functions but also a spin-nematic parameter $Q_{\\alpha }$ defined as $Q_{\\alpha } = \\frac{1}{N_{\\rm s}}\\sum _{j}\\left\\langle \\left(\\hat{S}_{j}^{\\alpha }\\right)^{2}- \\frac{1}{3}\\hat{{S}}_{j}^{2} \\right\\rangle ~.$ The maximum value of $Q_{\\alpha }$ for a single atom with $S=1$ is 1/3, which is realized in, e.g., $|1,0,0\\rangle $ and $|0,0,1\\rangle $ , and also in the eigenstates of $\\hat{S}^{x}$ with $S^{x} = 0$ , which can be expressed as a coherent superposition of $|1,0,0\\rangle $ and $|0,0,1\\rangle $ .", "Of course, isotropic spin gives $Q_{\\alpha } = 0$ .", "Figure REF shows $Q_{z}$ for two positive values of $U_2/U_0$ .", "Reflecting the imbalanced spin population shown in Fig.", "REF (d), large values of $Q_{z}$ are observed in the whole range of $U_{0}/t > 0$ , that is, both in the SF and MI phases.", "In the MI phase, the single site state is not a coherent superposition of $|1,0,0\\rangle $ and $|0,0,1\\rangle $ due to the restriction of $\\sum _{j}S_{j}^{z} = 0$ .", "Thus, the spins in the MI phase exhibit a rod-like nematic structure with $Q_{z}\\sim 1/3$ and $Q_{x} = Q_{y} \\sim -1/6$ .", "Here, as mentioned in the previous paragraph, the population of doublon $|0,2,0\\rangle $ and thus $N_{0}$ increases as $U_{2}/U_{0}$ increases, which leads to lower $Q_z$ values.", "This is made manifest in that $Q_{z}$ for $U_{2}/U_{0} = 0.3$ is smaller than that for $U_{2}/U_{0} = 0.1$ .", "The minima of $Q_{z}$ at $U_0/t\\sim 10$ have the same cause, i.e., in the lower $U_0$ region, $Q_z$ decreases with increase in $U_0/t$ due to an accompanying increase of $U_2$ , but, for $U_0/t\\ $ >$$$\\sim $ 10$,$ Qz$ enhances as $ U0/t$ increases as a result of decrease of doublon density.In the same context, $ Qz$ is slightly reduced from the full-moment value$ 1/3$ in the MI phase, where the D-H factor in $ |DH$induces a particle density fluctuation, in contrast to GA.Incidentally, the spin directions remain arbitrary and indefinitefor $ U2=0$ or $ U0=0$, indicating that the point of $ U2/U0=0$ is singular.In addition, we show the spin structure factor,$ S(q)=1Nsj, SjSeiqRj, $for $ U2/U0=0.1$ in Fig.~\\ref {sq}(a).The constant $ S(q)$ indicates that there isno spin correlation, and the spins take random directions.Here, the decrease in $ S(q=0)$ is reflected in the restriction of $ jSjz = 0$.Thus, we find a nematic order is realized for $ U2/U0>0$.$ Figure: (Color online) Spin nematic parameter Q z Q_z for two positive valuesof U 2 /U 0 U_2/U_0.For comparison, the GA result is also plotted, where,in fact, Q z Q_{z} cannot be determined for U 0 >U 0 BR U_0>U_0^{\\rm BR}.Figure: (Color online) Example of spin structure factor for (a) antiferromagneticcoupling (U 2 >0)U_{2}>0) and (b) ferromagnetic coupling (U 2 <0U_{2}<0).The phase for U 0 /t=10U_0/t=10 (30) is SF (MI).The inset in (b) shows enlarged views of the two phases.For $U_2/U_0<0$ , $Q_{z}$ becomes negative and its absolute value decreases monotonically as $U_0/t$ increases for $U_0<U_{0{\\rm c}}$ , and is almost constant $\\sim -1/6$ in the MI phase (not shown), indicating that the spins are polarized in the $x$ -$y$ plane.", "In Fig.", "REF (b), $S({q})$ for $U_2/U_0=-0.1$ is plotted in the two phases.", "By considering that the value of $S({q}={0})$ is extremely large with almost full moment and diverges proportionally to $N_{\\rm s}$ , a ferromagnetic (FM) long-range order is realized in the $x$ -$y$ plane.", "The magnitude of the FM moment is almost constant in both the SF and MI phases (not shown but expected from Fig.", "REF (b)).", "The results reported above (for $U_{2} \\gtrless 0$ ) are consistent with those at the weak- (SF, $t \\gg U_0 \\gg |U_2|$ )[17] and strong- (MI, $t \\rightarrow 0$ )[23] interaction limits.", "In summary, the $S=1$ Bose-Hubbard model [eq.", "(REF )] on a square lattice at unit filling is studied, using a variational Monte Carlo method.", "A doublon-holon binding factor $\\mathcal {P}_{\\rm DH}$ , which is the essence of Mott transitions, not only improves the variational energy considerably upon the GA, but enables us to study the details of the spin structure directly even in the MI phase without resorting to an effective spin Hamiltonian.", "For $U_2>0$ ($<0$ ), a spin nematic (ferromagnetic) phase is stabilized from SF to MI phases.", "The present results broadly support the phase diagrams[14], [15], [16] and spin structure[17], [23] as regards $S=1$ BHM proposed in previous studies.", "Some of the numerical computations were carried out at the Yukawa Institute Computer Facility and at the Cyberscience Center, Tohoku University.", "This work is supported by a Grant-in-Aid for Scientific Research (C) and also by the Next Generation Supercomputing Project, Nanoscience Program, from MEXT of Japan." ] ]

[ [ "On the Value of Multiple Read/Write Streams for Data Compression" ], [ "Abstract We study whether, when restricted to using polylogarithmic memory and polylogarithmic passes, we can achieve qualitatively better data compression with multiple read/write streams than we can with only one.", "We first show how we can achieve universal compression using only one pass over one stream.", "We then show that one stream is not sufficient for us to achieve good grammar-based compression.", "Finally, we show that two streams are necessary and sufficient for us to achieve entropy-only bounds." ], [ "Introduction", "Massive datasets seem to expand to fill the space available and, in situations where they no longer fit in memory and must be stored on disk, we may need new models and algorithms.", "Grohe and Schweikardt [21] introduced read/write streams to model situations in which we want to process data using mainly sequential accesses to one or more disks.", "As the name suggests, this model is like the streaming model (see, e.g., [28]) but, as is reasonable with datasets stored on disk, it allows us to make multiple passes over the data, change them and even use multiple streams (i.e., disks).", "As Grohe and Schweikardt pointed out, sequential disk accesses are much faster than random accesses — potentially bypassing the von Neumann bottleneck — and using several disks in parallel can greatly reduce the amount of memory and the number of accesses needed.", "For example, when sorting, we need the product of the memory and accesses to be at least linear when we use one disk [27], [20] but only polylogarithmic when we use two [9], [21].", "Similar bounds have been proven for a number of other problems, such as checking set disjointness or equality; we refer readers to Schweikardt's survey [34] of upper and lower bounds with one or more read/write streams, Heinrich and Schweikardt's paper [23] relating read/write streams to classic complexity theory, and Beame and Huynh's paper [4] on the value of multiple read/write streams for approximating frequency moments.", "Since sorting is an important operation in some of the most powerful data compression algorithms, and compression is an important operation for reducing massive datasets to a more manageable size, we wondered whether extra streams could also help us achieve better compression.", "In this paper we consider the problem of compressing a string $s$ of $n$ characters over an alphabet of size $\\sigma $ when we are restricted to using $\\log ^{\\mathcal {O} (1)} n$ bits of memory and $\\log ^{\\mathcal {O} (1)} n$ passes over the data.", "Throughout, we write $\\log $ to mean $\\log _2$ unless otherwise stated.", "In Section , we show how we can achieve universal compression using only one pass over one stream.", "Our approach is to break the string into blocks and compress each block separately, similar to what is done in practice to compress large files.", "Although this may not usually significantly worsen the compression itself, it may stop us from then building a fast compressed index (see [29] for a survey) unless we somehow combine the indexes for the blocks, or clustering by compression [11] (since concatenating files should not help us compress them better if we then break them into pieces again).", "In Section  we use a vaguely automata-theoretic argument to show one stream is not sufficient for us to achieve good grammar-based compression.", "Of course, by `good' we mean here something stronger than universal compression: we want to build a context-free grammar that generates $s$ and only $s$ and whose size is nearly minimum.", "In a paper with Gawrychowski [17] we showed that with constant memory and logarithmic passes over a constant number of streams, we can build a grammar whose size is at most quadratic in the minimum.", "Finally, in Section  we show that two streams are necessary and sufficient for us to achieve entropy-only bounds.", "Along the way, we show we need two streams to find strings' minimum periods or compute the Burrows-Wheeler Transform.", "As far as we know, this is the first paper on compression with read/write streams, and among the first papers on compression in any streaming model; we hope the techniques we have used will prove to be of independent interest." ], [ "Universal compression", "An algorithm is called universal with respect to a class of sources if, when a string is drawn from any of those sources, the algorithm's redundancy per character approaches 0 with probability 1 as the length of the string grows.", "The class most often considered, and which we consider in this section, is that of stationary, ergodic Markov sources (see, e.g., [12]).", "Since the $k$ th-order empirical entropy $H_k (s)$ of $s$ is the minimum self-information per character of $s$ with respect to a $k$ th-order Markov source (see [33]), an algorithm is universal if it stores any string $s$ in $n H_k (s) + o (n)$ bits for any fixed $\\sigma $ and $k$ .", "The $k$ th-order empirical entropy of $s$ is also our expected uncertainty about a randomly chosen character of $s$ when given the $k$ preceding characters.", "Specifically, $H_k (s) = \\left\\lbrace \\begin{array}{ll}(1 / n) \\sum _a \\mathsf {occ} (a, s) \\log \\frac{n}{\\mathsf {occ}(a, s)} &\\mbox{if $k = 0$,}\\\\[1ex](1 / n) \\sum _{|w| = k} |w_s| H_0 (w_s) &\\mbox{otherwise,}\\end{array} \\right.$ where $\\mathsf {occ} (a, s)$ is the number of times character $a$ occurs in $s$ , and $w_s$ is the concatenation of those characters immediately following occurrences of $k$ -tuple $w$ in $s$ .", "In a previous paper [19] we showed how to modify the well-known LZ77 compression algorithm [35] to use sublinear memory while still storing $s$ in $n H_k (s) + \\mathcal {O} \\hspace{-2.125pt} \\left( {n \\log \\log n / \\log n} \\right)$ bits for any fixed $\\sigma $ and $k$ .", "Our algorithm uses nearly linear memory and so does not fit into the model we consider in this paper, but we mention it here because it fits into some other streaming models (see, e.g., [28]) and, as far as we know, was the first compression algorithm to do so.", "In the same paper we proved several lower bounds using ideas that eventually led to our lower bounds in Sections  and  of this paper.", "[Gagie and Manzini, 2007] We can achieve universal compression using one pass over one stream and $\\mathcal {O} \\hspace{-2.125pt} \\left( {n / \\log ^2 n} \\right)$ bits of memory.", "To achieve universal compression with only polylogarithmic memory, we use a algorithm due to Gupta, Grossi and Vitter [22].", "Although they designed it for the RAM model, we can easily turn it into a streaming algorithm by processing $s$ in small blocks and compressing each block separately.", "[Gupta, Grossi and Vitter, 2008] In the RAM model, we can store any string $s$ in $n H_k (s) + \\mathcal {O} \\hspace{-2.125pt} \\left( {\\sigma ^k \\log n} \\right)$ bits, for all $k$ simultaneously, using $\\mathcal {O} \\hspace{-2.125pt} \\left( {n} \\right)$ time.", "We can achieve universal compression using one pass over one stream and $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log ^{1 + \\epsilon } n} \\right)$ bits of memory.", "We process $s$ in blocks of $\\log ^\\epsilon n$ characters, as follows: we read each block into memory, apply Theorem  to it, output the result, empty the memory, and move on to the next block.", "(If $n$ is not given in advance, we increase the block size as we read more characters.)", "Since Gupta, Grossi and Vitter's algorithm uses $\\mathcal {O} \\hspace{-2.125pt} \\left( {n} \\right)$ time in the RAM model, it uses $\\mathcal {O} \\hspace{-2.125pt} \\left( {n \\log n} \\right)$ bits of memory and we use $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log ^{1 + \\epsilon } n} \\right)$ bits of memory.", "If the blocks are $s_1, \\ldots , s_b$ , then we store all of them in a total of $\\sum _{i = 1}^b \\left( |s_i| H_k (s_i) + \\mathcal {O} \\hspace{-2.125pt} \\left( {\\sigma ^k \\log \\log n} \\right) \\right)\\le n H_k (s) + \\mathcal {O} \\hspace{-2.125pt} \\left( {\\sigma ^k n \\log \\log n / \\log ^\\epsilon n} \\right)$ bits for all $k$ simultaneously.", "Therefore, for any fixed $\\sigma $ and $k$ , we store $s$ in $n H_k (s) + o (n)$ bits.", "A bound of $n H_k (s) + \\mathcal {O} \\hspace{-2.125pt} \\left( {\\sigma ^k n \\log \\log n / \\log ^\\epsilon n} \\right)$ bits is not very meaningful when $k$ is not fixed and grows as fast as $\\log \\log n$ , because the second term is $\\omega (n)$ .", "Notice, however, that Gupta et al.", "'s bound of $n H_k (s) + \\mathcal {O} \\hspace{-2.125pt} \\left( {\\sigma ^k \\log n} \\right)$ bits is also not very meaningful when $k \\ge \\log n$ , for the same reason.", "As we will see in Section , it is possible for $s$ to be fairly incompressible but still to have $H_k (s) = 0$ for $k \\ge \\log n$ .", "It follows that, although we can prove bounds that hold for all $k$ simultaneously, those bounds cannot guarantee good compression in terms of $H_k (s)$ when $k \\ge \\log n$ .", "By using larger blocks — and, thus, more memory — we can reduce the $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\sigma ^k n \\log \\log n / \\log ^\\epsilon n} \\right)$ redundancy term in our analysis, allowing $k$ to grow faster than $\\log \\log n$ while still having a meaningful bound.", "Specifically, if we process $s$ in blocks of $c$ characters, then we use $\\mathcal {O} \\hspace{-2.125pt} \\left( {c \\log n} \\right)$ bits of memory and achieve a redundancy term of $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\sigma ^k n \\log c\\,/\\,c} \\right)$ , allowing $k$ to grow nearly as fast as $\\log _\\sigma c$ while still having a meaningful bound.", "We will show later, in Theorem , that this tradeoff is nearly optimal: if we use $m$ bits of memory and $p$ passes over one stream and our redundancy term is $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\sigma ^k r} \\right)$ , then $m p r = \\Omega (n / f (n))$ for any function $f$ that increases without bound.", "It is not clear to us, however, whether we can modify Corollary  to take advantage of multiple passes.", "Open Problem 1 With multiple passes over one stream, can we achieve better bounds on the memory and redundancy than we can with one pass?" ], [ "Grammar-based compression", "Charikar et al.", "[8] and Rytter [32] independently showed how to build a nearly minimal context-free grammar APPROX that generates $s$ and only $s$ .", "Specifically, their algorithms yield grammars that are an $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log n} \\right)$ factor larger than the smallest such grammar OPT, which has size $\\Omega (\\log n)$ bits.", "[Charikar et al., 2005; Rytter, 2003] In the RAM model, we can approximate the smallest grammar with $|\\mathsf {APPROX}| = \\mathcal {O} \\hspace{-2.125pt} \\left( {|\\mathsf {OPT}|^2} \\right)$ using $\\mathcal {O} \\hspace{-2.125pt} \\left( {n} \\right)$ time.", "In this section we prove that, if we use only one stream, then in general our approximation must be superpolynomially larger than the smallest grammar.", "Our idea is to show that periodic strings whose periods are asymptotically slightly larger than the product of the memory and passes, can be encoded as small grammars but, in general, cannot be compressed well by algorithms that use only one stream.", "Our argument is based on the following two lemmas.", "If $s$ has period $\\ell $ , then the size of the smallest grammar for that string is $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\ell \\log \\sigma + \\log n \\log \\log n} \\right)$ bits.", "Let $t$ be the repeated substring and $t^{\\prime }$ be the proper prefix of $t$ such that $s = t^{\\lfloor n / \\ell \\rfloor } t^{\\prime }$ .", "We can encode a unary string $X^{\\lfloor n / \\ell \\rfloor }$ as a grammar $G_1$ with $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log n} \\right)$ productions of total size $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log n \\log \\log n} \\right)$ bits.", "We can also encode $t$ and $t^{\\prime }$ as grammars $G_2$ and $G_3$ with $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\ell } \\right)$ productions of total size $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\ell \\log \\sigma } \\right)$ bits.", "Suppose $S_1$ , $S_2$ and $S_3$ are the start symbols of $G_1$ , $G_2$ and $G_3$ , respectively.", "By combining those grammars and adding the productions $S_0 \\rightarrow S_1 S_3$ and $X \\rightarrow S_2$ , we obtain a grammar with $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\ell + \\log n} \\right)$ productions of total size $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\ell \\log \\sigma + \\log n \\log \\log n} \\right)$ bits that maps $S_0$ to $s$ .", "Consider a lossless compression algorithm that uses only one stream, and a machine performing that algorithm.", "We can compute any substring from its length; for each pass, the machine's memory configurations when it reaches and leaves the part of the stream that initially holds that substring; all the output the machine produces while over that part.", "Let $t$ be the substring and assume, for the sake of a contradiction, that there exists another substring $t^{\\prime }$ with the same length that takes the machine between the same configurations while producing the same output.", "Then we can substitute $t^{\\prime }$ for $t$ in $s$ without changing the machine's complete output, contrary to our specification that the compression be lossless.", "Lemma  implies that, for any substring, the size of the output the machine produces while over the part of the stream that initially holds that substring, plus twice the product of the memory and passes (i.e., the number of bits needed to store the memory configurations), must be at least that substring's complexity.", "Therefore, if a substring is not compressible by more than a constant factor (as is the case for most strings) and asymptotically larger than the product of the memory and passes, then the size of the output for that substring must be at least proportional to the substring's length.", "In other words, the algorithm cannot take full advantage of similarities between substrings to achieve better compression.", "In particular, if $s$ is periodic with a period that is asymptotically slightly larger than the product of the memory and passes, and $s$ 's repeated substring is not compressible by more than a constant factor, then the algorithm's complete output must be $\\Omega (n)$ bits.", "By Lemma , however, the size of the smallest grammar that generates $s$ and only $s$ is bounded in terms of the period.", "With one stream, we cannot approximate the smallest grammar with $|\\mathsf {APPROX}| \\le |\\mathsf {OPT}|^{\\mathcal {O} (1)}$ .", "Suppose an algorithm uses only one stream, $m$ bits of memory and $p$ passes to compress $s$ , with $m p = \\log ^{\\mathcal {O} (1)} n$ , and consider a machine performing that algorithm.", "Furthermore, suppose $s$ is binary and periodic with period $m p \\log n$ and its repeated substring $t$ is not compressible by more than a constant factor.", "Lemma  implies that the machine's output while over a part of the stream that initially holds a copy of $t$ , must be $\\Omega (m p \\log n - m p) = \\Omega (m p \\log n)$ .", "Therefore, the machine's complete output must be $\\Omega (n)$ bits.", "By Lemma , however, the size of the smallest grammar that generates $s$ and only $s$ is $\\mathcal {O} (m p \\log n + \\log n \\log \\log n) \\subset \\log ^{\\mathcal {O} (1)} n$ bits.", "Since $n = \\log ^{\\omega (1)} n$ , the algorithm's complete output is superpolynomially larger than the smallest grammar.", "As an aside, we note that a symmetric argument shows that, with only one stream, in general we cannot decode a string encoded as a small grammar.", "To see why, instead of considering a part of the stream that initially holds a copy of the repeated substring $t$ , consider a part that is initially blank and eventually holds a copy of $t$ .", "(Since $s$ is periodic and thus very compressible, its encoding takes up only a fraction of the space it eventually occupies when decompressed; without loss of generality, we can assume the rest is blank.)", "An argument similar to the proof of Lemma  shows we can compute $t$ from the machine's memory configurations when it reaches and leaves that part, so the product of the memory and passes must again be greater than or equal to $t$ 's complexity.", "With one stream, we cannot decompress strings encoded as small grammars.", "Theorem  also has the following corollary, which may be of independent interest.", "With one stream, we cannot find strings' minimum periods.", "Consider the proof of Theorem .", "Notice that, if we could find $s$ 's minimum period, then we could store $s$ in $\\log ^{\\mathcal {O} (1)} n$ bits by writing $n$ and one copy of its repeated substring $t$ .", "It follows that we cannot find strings' minimum periods.", "Corollary  may at first seem to contradict work by Ergün, Muthukrishnan and Sahinalp [13], who gave streaming algorithms for determining approximate periodicity.", "Whereas we are concerned with strings which are truly periodic, however, they were concerned with strings in which the copies of the repeated substring can differ to some extent.", "To see why this is an important difference, consider the simple case of checking whether $s$ has period $n / 2$ (i.e., whether or not it is a square).", "Suppose we know the two halves of $s$ are either identical or differ in exactly one position, and we want to determine whether $s$ truly has period $n / 2$ ; then we must compare each corresponding pair of characters and, by a crossing-sequences argument (see, e.g., [27] for details of a similar argument), this takes $\\Omega (n / m)$ passes.", "Now suppose we care only whether the two halves of $s$ match only in nearly all positions; then we need compare only a few randomly chosen pairs to decide correctly with high probability.", "With one stream, we cannot even check strings' minimum periods.", "In the conference version of this paper we left as an open problem proving whether or not multiple streams are useful for grammar-based compression.", "As we noted in the introduction, in a subsequent paper with Gawrychowski [17] we showed that with constant memory and logarithmic passes over a constant number of streams, we can approximate the smallest grammar with $|\\mathsf {APPROX}| = \\mathcal {O} \\hspace{-2.125pt} \\left( {|\\mathsf {OPT}|^2} \\right)$ , answering our question affirmatively." ], [ "Entropy-only bounds", "Kosaraju and Manzini [25] pointed out that proving an algorithm universal does not necessarily tell us much about how it behaves on low-entropy strings.", "In other words, showing that an algorithm encodes $s$ in $n H_k (s) + o (n)$ bits is not very informative when $n H_k (s) = o (n)$ .", "For example, although the well-known LZ78 compression algorithm [36] is universal, $|\\mathsf {LZ78} (1^n)| = \\Omega (\\sqrt{n})$ while $n H_0 (1^n) = 0$ .", "To analyze how algorithms perform on low-entropy strings, we would like to get rid of the $o (n)$ term and prove bounds that depend only on $n H_k (s)$ .", "Unfortunately, this is impossible since, as the example above shows, even $n H_0 (s)$ can be 0 for arbitrarily long strings.", "It is not hard to show that only unary strings have $H_0 (s) = 0$ .", "For $k \\ge 1$ , recall that $H_k (s) = (1 / n) \\sum _{|w| = k} |w_s| H_0 (w_s)$ .", "Therefore, $H_k (s) = 0$ if and only if each distinct $k$ -tuple $w$ in $s$ is always followed by the same distinct character.", "This is because, if a $w$ is always followed by the same distinct character, then $w_s$ is unary, $H_0 (w_s) = 0$ and $w$ contributes nothing to the sum in the formula.", "Manzini [26] defined the $k$ th-order modified empirical entropy $H_k^* (s)$ such that each context $w$ contributes at least $\\lfloor \\log |w_s| \\rfloor + 1$ to the sum.", "Because modified empirical entropy is more complicated than empirical entropy — e.g., it allows for variable-length contexts — we refer readers to Manzini's paper for the full definition.", "In our proofs in this paper, we use only the fact that $n H_k (s) \\le n H_k^* (s) \\le n H_k (s) + \\mathcal {O} \\hspace{-2.125pt} \\left( {\\sigma ^k \\log n} \\right)\\,.$ Manzini showed that, for some algorithms and all $k$ simultaneously, it is possible to bound the encoding's length in terms of only $n H_k^* (s)$ and a constant $g_k$ that depends only on $\\sigma $ and $k$ ; he called such bounds `entropy-only'.", "In particular, he showed that an algorithm based on the Burrows-Wheeler Transform (BWT) [7] stores any string $s$ in at most $(5 + \\epsilon ) n H_k^* (s) + \\log n + g_k$ bits for all $k$ simultaneously (since $n H_k^* (s) \\ge \\log (n - k)$ , we could remove the $\\log n$ term by adding 1 to the coefficient $5 + \\epsilon $ ).", "[Manzini, 2001] Using the BWT, move-to-front coding, run-length coding and arithmetic coding, we can achieve an entropy-only bound.", "The BWT sorts the characters in a string into the lexicographical order of the suffixes that immediately follow them.", "When using the BWT for compression, it is customary to append a special character $ that is lexicographically less than any in the alphabet.", "For a more thorough description of the BWT, we again refer readers to Manzini's paper.", "In this section we first show how we can compute and invert the BWT with two streams and, thus, achieve entropy-only bounds.", "We then show that we cannot achieve entropy-only bounds with only one stream.", "In other words, two streams are necessary and sufficient for us to achieve entropy-only bounds.", "One of the most common ways to compute the BWT is by building a suffix array.", "In his PhD thesis, Ruhl introduced the StreamSort model [31], [2], which is similar to the read/write streams model with one stream, except that it has an extra primitive that sorts the stream in one pass.", "Among other things, he showed how to build a suffix array efficiently in this model.", "[Ruhl, 2003] In the StreamSort model, we can build a suffix array using $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log n} \\right)$ bits of memory and $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log n} \\right)$ passes.", "With two streams, we can compute the BWT using $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log n} \\right)$ bits of memory and $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log ^2 n} \\right)$ passes.", "We can compute the BWT in the StreamSort model by appending $ to $s$ , building a suffix array, and replacing each value $i$ in the array by the $(i - 1)$ st character in $s$ (replacing either 0 or 1 by $, depending on where we start counting).", "This takes $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log n} \\right)$ bits of memory and $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log n} \\right)$ passes.", "Since we can sort with two streams using $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log n} \\right)$ bits memory and $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log n} \\right)$ passes (see, e.g., [34]), it follows that we can compute the BWT using $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log n} \\right)$ bits of memory and $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log ^2 n} \\right)$ passes.", "We note as an aside that, once we have the suffix array for a periodic string, we can easily find its minimum period.", "To see why, suppose $s$ has minimum period $\\ell $ , and consider the suffix $u$ of $s$ that starts in position $\\ell + 1$ .", "The longest common prefix of $s$ and $u$ has length $n - \\ell $ , which is maximum; if another suffix $v$ shared a longer common prefix with $s$ , then $s$ would have period $n - |v| < \\ell $ .", "It follows that, if the first position in the suffix array contains $i$ , then the $(\\ell + 1)$ st position contains $i - 1$ (assuming $s$ terminates with $, so $u$ is lexicographically less than $s$ ).", "With two streams we can easily find the position $\\ell + 1$ that contains $i - 1$ and then check that $s$ is indeed periodic with period $\\ell $ .", "With two streams, we can compute a string's minimum period using $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log n} \\right)$ bits and $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log ^2 n} \\right)$ passes.", "Now suppose we are given a permutation $\\pi $ on $n + 1$ elements as a list $\\pi (1), \\ldots , \\pi (n + 1)$ , and asked to rank it, i.e., to compute the list $\\pi ^0 (1), \\ldots , \\pi ^n (1)$ .", "This problem is a special case of list ranking (see, e.g., [3]) and has a surprisingly long history.", "For example, Knuth [24] described an algorithm, which he attributed to Hardy, for ranking a permutation with two tapes.", "More recently, Bird and Mu [5] showed how to invert the BWT by ranking a permutation.", "Therefore, reinterpreting Hardy's result in terms of the read/write streams model gives us the following bounds.", "[Hardy, c. 1967] With two streams, we can rank a permutation using $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log n} \\right)$ bits of memory and $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log ^2 n} \\right)$ passes.", "With two streams, we can invert the BWT using $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log n} \\right)$ bits of memory and $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log ^2 n} \\right)$ passes.", "The BWT has the property that, if a character is the $i$ th in $\\mathsf {BWT} (s)$ , then its successor in $s$ is the lexicographically $i$ th in $\\mathsf {BWT} (s)$ (breaking ties by order of appearance).", "Therefore, we can invert the BWT by replacing each character by its lexicographic rank, ranking the resulting permutation, replacing each value $i$ by the $i$ th character of $\\mathsf {BWT} (s)$ , and rotating the string until $ is at the end.", "This takes $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log n} \\right)$ memory and $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log ^2 n} \\right)$ passes.", "Since we can compute and invert move-to-front, run-length and arithmetic coding using $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log n} \\right)$ bits of memory and $\\mathcal {O} \\hspace{-2.125pt} \\left( {1} \\right)$ passes over one stream, by combining Theorem  and Corollaries  and  we obtain the following theorem.", "With two streams, we can achieve an entropy-only bound using $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log n} \\right)$ bits of memory and $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log ^2 n} \\right)$ passes.", "It follows from Theorem  and a result by Hernich and Schweikardt [23] that we can achieve an entropy-only bound using $\\mathcal {O} \\hspace{-2.125pt} \\left( {1} \\right)$ bits of memory, $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log ^3 n} \\right)$ passes and four streams.", "It follows from their theorem below that, with more streams, we can even reduce the number of passes to $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log n} \\right)$ .", "[Hernich and Schweikardt, 2008] If we can solve a problem with logarithmic work space, then we can solve it using $\\mathcal {O} \\hspace{-2.125pt} \\left( {1} \\right)$ bits of memory and $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log n} \\right)$ passes over $\\mathcal {O} \\hspace{-2.125pt} \\left( {1} \\right)$ streams.", "With $\\mathcal {O} \\hspace{-2.125pt} \\left( {1} \\right)$ streams, we can achieve an entropy-only bound using $\\mathcal {O} \\hspace{-2.125pt} \\left( {1} \\right)$ bits of memory and $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log n} \\right)$ passes.", "To compute the $i$ th character of $\\mathsf {BWT} (s)$ , we find the $i$ th lexicographically largest suffix.", "To find this suffix, we loop though all the suffixes and, for each, count how many other suffixes are lexicographically less.", "Comparing two suffixes character by character takes $\\mathcal {O} \\hspace{-2.125pt} \\left( {n^2} \\right)$ time, so we use a total of $\\mathcal {O} \\hspace{-2.125pt} \\left( {n^4} \\right)$ time; it does not matter now how much time we use, however, just that we need only a constant number of $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log n} \\right)$ -bit counters.", "Since we can compute the BWT with logarithmic work space, it follows from Theorem  that we can compute it — and thereby achieve an entropy-only bound — with $\\mathcal {O} \\hspace{-2.125pt} \\left( {1} \\right)$ bits of memory and $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log n} \\right)$ passes over $\\mathcal {O} \\hspace{-2.125pt} \\left( {1} \\right)$ streams.", "Although we have not been able to prove an $\\Omega (\\log n)$ lower bound on the number of passes needed to achieve an entropy-only bound with $\\mathcal {O} \\hspace{-2.125pt} \\left( {1} \\right)$ streams, we have been able to prove such a bound for computing the BWT.", "Our idea is to reduce sorting to the BWT, since Grohe and Schweikardt [21] showed we cannot sort $n$ numbers with $o (\\log n)$ passes over $\\mathcal {O} \\hspace{-2.125pt} \\left( {1} \\right)$ streams.", "It is trivial, of course, to reduce sorting to the BWT if the alphabet is large enough — e.g., linear in $n$ — but our reduction is to the more reasonable problem of computing the BWT of a ternary string.", "With $\\mathcal {O} \\hspace{-2.125pt} \\left( {1} \\right)$ streams, we cannot compute the BWT using $o (\\log n)$ passes.", "Suppose we are given a sequence of $n$ numbers $x_1, \\ldots , x_n$ , each of $2 \\log n$ bits.", "Grohe and Schweikardt showed we cannot generally sort such a sequence using $o (\\log n)$ passes over $\\mathcal {O} \\hspace{-2.125pt} \\left( {1} \\right)$ tapes.", "We now use $o (\\log n)$ passes to turn $x_1, \\ldots , x_n$ into a ternary string $s$ such that, by calculating $\\mathsf {BWT} (s)$ , we sort $x_1, \\ldots , x_n$ .", "It follows from this reduction that we cannot compute the BWT using $o (\\log n)$ passes, either.", "With one pass, $O (\\log n)$ bits of memory and two tapes, for $1 \\le i \\le n$ and $1 \\le j \\le 2 \\log n$ , we replace the $j$ th bit $x_i [j]$ of $x_i$ by $x_i [j]\\ 2\\ x_i\\ i\\ j$ , writing 2 as a single character, $x_i$ in $2 \\log n$ bits, $i$ in $\\log n$ bits and $j$ in $\\log \\log n + 1$ bits; the resulting string $s$ is of length $2 n \\log n (3 \\log n + \\log \\log n + 2)$ .", "The only characters followed by 2s in $s$ are the bits at the beginning of replacement phrases, so the last $2 n \\log n$ characters of $\\mathsf {BWT} (s)$ are the bits of $x_1, \\ldots , x_n$ ; moreover, since the lexicographic order of equal-length binary strings is the same as their numeric order, the $x_i [j]$ bits will be arranged by the $x_i$ values, with ties broken by the $i$ values (so if $x_i = x_{i^{\\prime }}$ with $i < i^{\\prime }$ , then every $x_i [j]$ comes before every $x_{i^{\\prime }} [j^{\\prime }]$ ) and further ties broken by the $j$ values; therefore, the last $2 n \\log n$ bits of the transformed string are $x_1, \\ldots , x_n$ in sorted order.", "To show we need at least two streams to achieve entropy-only bounds, we use De Bruijn cycles in a proof similar to the one for Theorem .", "A $\\sigma $ -ary De Bruijn cycle of order $k$ is a cyclic sequence in which every possible $k$ -tuple appears exactly once.", "For example, Figure REF shows binary De Bruijn cycles of orders 3 and 4.", "Our argument this time is based on Lemma  and the results below about De Bruijn cycles.", "We note as a historical aside that Theorem  was first proven for the binary case in 1894 by Flye Sainte-Marie [15], but his result was later forgotten; De Bruijn [6] gave a similar proof for that case in 1946, then in 1951 he and Van Aardenne-Ehrenfest [1] proved the general version we state here.", "Figure: Examples of binary De Bruijn cycles of orders 3 and 4.If $s \\in d^*$ for some binary $\\sigma $ -ary De Bruijn cycle $d$ of order $k$ , then $n H_k^* (s) = \\mathcal {O} \\hspace{-2.125pt} \\left( {\\sigma ^k \\log n} \\right)$ .", "By definition, each distinct $k$ -tuple is always followed by the same distinct character; therefore, $n H_k (s) = 0$ and $n H_k^* (s) = \\mathcal {O} \\hspace{-2.125pt} \\left( {\\sigma ^k \\log n} \\right)$ .", "[Van Aardenne-Ehrenfest and De Bruijn, 1951] There are $\\left( \\sigma !^{\\sigma ^{k - 1}} / \\sigma ^k \\right)$ $\\sigma $ -ary De Bruijn cycles of order $k$ .", "We cannot store most $k$ th-order De Bruijn cycles in $o (\\sigma ^k \\log \\sigma )$ bits.", "By Stirling's Formula, $\\log \\left( \\sigma !^{\\sigma ^{k - 1}} / \\sigma ^k \\right) = \\Theta (\\sigma ^k \\log \\sigma )$ .", "Since there are $\\sigma ^k$ possible $k$ -tuples, $k$ th-order De Bruijn cycles have length $\\sigma ^k$ , so Corollary  means that we cannot compress most De Bruijn cycles by more than a constant factor.", "Therefore, we can prove a lower bound similar to Theorem  by supposing that $s$ 's repeated substring is a De Bruijn cycle, then using Lemma  instead of Lemma .", "With one stream, we cannot achieve an entropy-only bound.", "As in the proof of Theorem , suppose an algorithm uses only one stream, $m$ bits of memory and $p$ passes to compress $s$ , with $m p = \\log ^{\\mathcal {O} (1)} n$ , and consider a machine performing that algorithm.", "This time, however, suppose $s$ is binary and periodic with period $m p\\,f (n)$ , where $f (n) = \\mathcal {O} \\hspace{-2.125pt} \\left( {\\log n} \\right)$ is a function that increases without bound; furthermore, suppose $s$ 's repeated substring $t$ is a $k$ th-order De Bruijn cycle, $k = \\log (m p\\,f (n))$ , that is not compressible by more than a constant factor.", "Lemma  implies that the machine's output while over a part of the stream that initially holds a copy of $t$ , must be $\\Omega (m p\\,f (n) - m p) = \\Omega (m p\\,f (n))$ .", "Therefore, the machine's complete output must be $\\Omega (n)$ bits.", "By Lemma , however, $n H_k^* (s) = \\mathcal {O} \\hspace{-2.125pt} \\left( {2^k \\log n} \\right) = \\mathcal {O} \\hspace{-2.125pt} \\left( {m p\\,f (n) \\log n} \\right) \\subset \\log ^{\\mathcal {O} (1)} n$ .", "Recall that in Section  we asserted the following claim, which we are now ready to prove.", "If we use $m$ bits of memory and $p$ passes over one stream and achieve universal compression with an $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\sigma ^k r} \\right)$ redundancy term, for all $k$ simultaneously, then $m p r = \\Omega (n / f (n))$ for any function $f$ that increases without bound.", "Consider the proof of Theorem : $n H_k (s) = 0$ but we must output $\\Omega (n)$ bits, so $r = \\Omega (n / \\sigma ^k) = \\Omega (n / (m p\\,f (n)))$ .", "Notice Theorem  also implies a lower bound for computing the BWT: if we could compute the BWT with one stream then, since we can compute move-to-front, run-length and arithmetic coding using $\\mathcal {O} \\hspace{-2.125pt} \\left( {\\log n} \\right)$ bits of memory and $\\mathcal {O} \\hspace{-2.125pt} \\left( {1} \\right)$ passes over one stream, we could thus achieve an entropy-only bound with one stream, contradicting Theorem .", "With one stream, we cannot compute the BWT.", "In the conference version of this paper [16] we closed with a brief discussion of three entropy-only bounds that we proved with Manzini [18].", "Our first bound was an improved analysis of the BWT followed by move-to-front, run-length and arithmetic coding (which lowered the coefficient from $5 + \\epsilon $ to $4.4 + \\epsilon $ ), but our other bounds (one of which had a coefficient of $2.69 + \\epsilon $ ) were analyses of the BWT followed by algorithms which we were not sure could be implemented with $\\mathcal {O} \\hspace{-2.125pt} \\left( {1} \\right)$ streams.", "We now realize that, since both of these other algorithms can be computed with logarithmic work space, it follows from Theorem  that they can indeed be computed with $\\mathcal {O} \\hspace{-2.125pt} \\left( {1} \\right)$ streams.", "After having proven that we cannot compute the BWT with one stream, we promptly start working with Ferragina and Manzini on a practical algorithm [14] that does exactly that.", "However, that algorithm does not fit into the streaming models we have considered in this paper; in particular, the product of the internal memory and passes there is $\\mathcal {O} \\hspace{-2.125pt} \\left( {n \\log n} \\right)$ bits, but we use only $n$ bits of workspace on the disk.", "The existence of a practical algorithm for computing the BWT in external memory raises the question of whether we can query BWT-based compressed indexes quickly in external memory.", "Chien et al.", "[10] proved lower bounds for indexed pattern matching in the external-memory model, but that model allows does not distinguish between sequential and random access to blocks.", "The read/write-streams model is also inappropriate for analyzing the complexity of this task, since we can trivially use only one pass over one stream if we leave the text uncompressed and scan it all with a classic sequential pattern-matching algorithm.", "Orlandi and Venturini [30] recently showed how we can store a sample of the BWT that lets us estimate what parts of the full BWT we need to read in order to answer a query.", "If we modify their data structure slightly, we can make it recursive; i.e., with a smaller sample we can estimate what parts of the sample we need to read in order to estimate what parts of the full BWT we need to read.", "Suppose we store on disk a set of samples whose sizes increase exponentially, finishing with the BWT itself.", "We use each sample in turn to estimate what parts of the next sample we need to read, then read them into internal memory using only one pass over the next sample.", "This increases the size of the whole index only slightly and lets us answer queries by reading few blocks and in the order they appear on disk.", "We are currently working to optimize and implement this idea." ], [ "Acknowledgments", "Many thanks to Ferdinando Cicalese, Paolo Ferragina, Paweł Gawrychowski, Roberto Grossi, Ankur Gupta, Andre Hernich, Giovanni Manzini, Jens Stoye and Rossano Venturini, for helpful discussions.", "This research was done while the author was at the University of Eastern Piedmont in Alessandria, Italy, supported by the Italy-Israel FIRB Project “Pattern Discovery Algorithms in Discrete Structures, with Applications to Bioinformatics”, and at Bielefeld University, Germany, supported by the Sofja Kovalevskaja Award from the Alexander von Humboldt Foundation and the German Federal Ministry of Education and Research." ] ]

[ [ "Absence of broken inversion symmetry phase of electrons in bilayer\n graphene under charge density fluctuations" ], [ "Abstract On a lattice model, we study the possibility of existence of gapped broken inversion symmetry phase (GBISP) of electrons with long-range Coulomb interaction in bilayer graphene using both self-consistent Hartree-Fock approximation (SCHFA) and the renormalized-ring-diagram approximation (RRDA).", "RRDA takes into account the charge-density fluctuations beyond the mean field.", "While GBISP at low temperature and low carrier concentration is predicted by SCHFA, we show here the state can be destroyed by the charge-density fluctuations.", "We also present a numerical algorithm for calculating the self-energy of electrons with the singular long-range Coulomb interaction on the lattice model." ], [ "introduction", "Because of its tunable band gap, which can be changed through an external gate voltage, bilayer graphene is a promising material with a great potential for application to new electronic devices.", "[1], [2], [3], [4] In the low-carrier-doping regime of bilayer graphene, electrons are strongly coupled via Coulomb interactions.", "The phase of bilayer graphene in this low carrier concentration, and at low temperature, is still not completely understood.", "Several candidates have been suggested for the ground state, such as a ferroelectric-layer asymmetric state,[5], [6], [7], [8], [9] a layer-polarized antiferromagnetic state,[10], [11] a quantum anomalous Hall state,[8], [12], [13] a quantum spin Hall state,[8], [13] a quantum valley Hall state,[14] a charge density wave state,[15] and the possibility of gapless states, such as the nematic state.", "[16], [17] The experimental observations on the ground state of bilayer graphene, all performed on high quality suspended samples, are also controversial.", "Some experimental results showed that the system is gapped at the neutrality point,[18], [19], [20], [21] whereas one experiment found a gapless state.", "[22] So far, most of the theoretical studies are based on the self-consistent Hartree-Fock approximation (SCHFA),[5], [9], [14] many-body perturbation theory,[7] and the renormalization group approach.", "[17], [16], [11] All the above approaches have been applied to the simplified two-[5], [9], [14], [7], [17], [11] and four-band[16] continuum models.", "It is well known that the SCHFA usually overestimates the order parameter characterizing a broken symmetry phase and the transition temperature because it neglects the fluctuations of the effective one-body interaction field and of other one-body observables such as the charge density.", "Since the understanding of the electronic state of bilayer graphene at low carrier doping and low temperature is a fundamental issue for graphene physics, it is necessary to investigate the state with a more sophisticated approach that takes into account the effect of charge density fluctuations on top of the mean-field ground state.", "In this work, we study the existence of a gapped broken inversion symmetry phase (GBISP) using both the SCHFA and the renormalized ring-diagram approximation (RRDA).", "[23] The RRDA takes into account the charge density fluctuation (CDF) effect beyond the mean field and satisfies the microscopic conservation laws.", "[24] For an electron system with long-range Coulomb interactions, CDF is the predominant contribution to the self-energy of electrons.", "It has been shown[25] that the RRDA results for the ground-state energy of two- and three-dimensional interacting electron gases are more accurate than the random-phase approximation (RPA) results when compared with Monte Carlo simulations.", "In the RRDA, the Green's function and self-energy are self-consistently determined by coupled integral equations.", "The self-consistent calculation of the self-energy in momentum space involves carrying out many convolutions, which are numerically expensive.", "In order to reduce the computational time required by our approach, we convert the convolutions in momentum space to multiplications in real space.", "Since the continuum model is the low-energy limiting case of the lattice model, the momentum of the electrons is confined within two valleys around the Dirac points.", "[26], [27] Because of the finite momentum cutoff for each valley, the conversion of the convolution from momentum space to real space is no longer valid for the two- and four-band continuum models.", "Instead of modeling bilayer graphene with an effective continuum model, we therefore sketch it as a bilayer of a hexagonal lattice model.", "The lattice model does not require a momentum-space cutoff, and is therefore immune to the aforementioned problems of the continuum models.", "The key problem in calculating the self-energy is to manage to deal with the long-range Coulomb interaction between electrons accurately.", "For the two-dimensional system under consideration, this interaction is inversely proportional to the momentum transfer $q$ in the long-wavelength limit.", "In a continuum model, one can transform the $1/q$ singularity to the logarithmic one after performing the azimuthal integration[28] and then get rid of the logarithmic singularity by special treatment.", "In a lattice model, however, we cannot perform the azimuthal integration analytically and must face the $1/q$ singularity.", "Since dealing with the long-range Coulomb interaction is inevitable in many-body problems, we now present a numerical algorithm to tackle the interaction divergence issues systematically." ], [ "Lattice model", "The lattice structure of bilayer graphene is shown in Fig.", "1.", "The unit cell in each layer is represented by a diamond.", "The unit cell of the bilayer system contains four atoms denoted as a$_1$ , b$_1$ , a$_2$ and b$_2$ .", "The lattice constant of monolayer graphene is defined as the distance between two nearest corner atoms in the diamond and is given by $a \\approx 2.4$ Å .", "The interlayer distance is $z_0 = 3.34$ Å  $\\approx 1.4 a$ .", "The energy of electron hopping between the nearest-neighbor (NN) carbon atoms in each layer is $t \\approx 2.82$ eV,[29] while the interlayer NN hopping is $t_1 \\approx 0.39$ eV.", "[30] The Hamiltonian describing the electrons is given by $H=-\\sum _{ij\\sigma }t_{ij}c^{\\dagger }_{i\\sigma }c_{j\\sigma }+\\frac{1}{2}\\sum _{ij}\\delta n_iv_{ij}\\delta n_j $ where $c^{\\dagger }_{i\\sigma }$ creates an electron at site $i$ with spin $\\sigma $ , $\\delta n_j=n_j-n$ with $n_j$ as the electron density operator at site $j$ and $n$ the average occupation number of electrons per site (which is also the charge number of the neutralizing background), and $v_{ij}$ is the Coulomb interaction between electrons at sites $i$ and $j$ .", "The model is restricted to NN hopping within the same layer and between the adjacent sites on top and bottom layers as shown in Fig.", "1.", "As described by Eq.", "(REF ), we consider here only the charge-charge interactions.", "Since the long-range antiferromagnetic order is prohibited[31] in two-dimensional space, we neglect the antiferromagnetic coupling due to the on-site repulsion in the present work.", "We now consider the behavior of Coulomb interaction $v_{ij}$ between two electrons at sites $i$ and $j$ .", "At long distance, $v_{ij}$ is given by $v_{ij} = e^2/\\epsilon r_{ij}$ with $\\epsilon $ the dielectric constant in the high-frequency limit of the system and $r_{ij}$ the distance.", "However, at short distances, because of the spread of the $\\pi $ -orbital wave function of the conduction electrons, $v_{ij}$ is weakened from the behavior $1/r_{ij}$ .", "Taking the effect of the wave function spread into account, we model the interaction as $v_{ij}=\\frac{e^2}{\\epsilon r_{ij}}[1-\\exp (-r_{ij}/r_0)] $ with $r_0 = a$ .", "Clearly, $v_{ij}$ behaves as $e^2/\\epsilon r_{ij}$ at large $r_{ij}$ , while it is suppressed from the `bare' Coulomb interaction ($e^2/\\epsilon r_{ij}$ ) at small $r_{ij}$ .", "In particular, at $r_{ij} = 0$ , it is given by a finite value $e^2/\\epsilon r_0$ .", "For the present electron system with long-range Coulomb interactions, the final result under consideration should not be sensitively dependent upon the details of the short-range behavior of the interaction.", "This can be understood from the behavior of its Fourier component in momentum space.", "The Fourier component is singular at the long-wavelength limit and the singular part is independent of the short-range behavior.", "At low carrier concentration, the electrons state is mainly determined by the singular part of the interaction.", "We use the dimensionless constant $g \\equiv e^2/\\epsilon at$ to denote the strength of Coulomb coupling.", "The range $0.4 \\le g < 1.8$ covers the cases of various experimental setups, from suspended bilayer graphene (BLG) to BLG placed on substrates[32] as SiO$_2$ and ice.", "Figure: Left: Structure of Bernal stacking bilayer graphene.", "Right: Top view of the bilayer graphene.", "The parameters tt and t 1 t_1 are the electron hopping energies between one atom and its nearest neighbor belonging to the same layer, and to the neighboring layer above or below, respectively.", "The unit cell of each layer is represented by the green-sided diamond.The system defined by Eq.", "(REF ) satisfies the particle-hole symmetry.", "To see this, we denote the doped electron concentration per carbon atom as $\\delta $ and have $n = 1+\\delta $ .", "Under the transformation $\\delta \\rightarrow -\\delta $ and $c_{j,\\sigma } \\rightarrow +(-) c^{\\dagger }_{j,\\sigma }$ and $c^{\\dagger }_{j,\\sigma } \\rightarrow +(-) c_{j,\\sigma }$ for electrons at a$_j$ (b$_j$ ) sites, $H$ is unchanged.", "Furthermore, $K = H -\\mu (\\hat{N} - N_0)$ ($\\hat{N}$ being the total electron number operator and $N_0$ being the total number of lattice sites, so that the operator $N - N_0$ refers to the total number of doped electrons) is also unchanged under the above electron-hole transformation, provided $\\mu \\rightarrow -\\mu $ .", "Thus, the chemical potential $\\mu $ must be an odd function of $\\delta $ .", "The Green's function $G$ of the electron system is defined as $G(i,j,\\tau -\\tau ^{\\prime }) = -\\langle T_{\\tau }C_{i\\sigma }(\\tau )C^{\\dagger }_{j\\sigma }(\\tau ^{\\prime })\\rangle $ where $C^{\\dagger }_{j\\sigma } = (c^{\\dagger }_{a_1j\\sigma },c^{\\dagger }_{b_1j\\sigma },c^{\\dagger }_{a_2j\\sigma },c^{\\dagger }_{b_2j\\sigma })$ with $c^{\\dagger }_{a_{l}(b_{l})j\\sigma }$ creating an electron of spin $\\sigma $ at site a$_{l}$ (b$_{l}$ ) of the $l$ th (= 1,2, respectively, for top and bottom) layer of the $j$ th unit cell.", "In momentum-frequency space, $G$ (a 4$\\times $ 4 matrix) can be expressed in terms of the self-energy $\\Sigma (k,i\\omega _{\\ell })$ as $G(k,i\\omega _{\\ell }) = [i\\omega _{\\ell } +\\mu -h_k-\\Sigma (k,i\\omega _{\\ell })]^{-1} $ with $h_k = \\begin{pmatrix}0& \\epsilon _k&0&0\\\\\\epsilon _k^{\\ast }&0&-t_1&0\\\\0&-t_1&0&\\epsilon _k\\\\0&0&\\epsilon _k^{\\ast }&0\\\\\\end{pmatrix}$ where $\\omega _{\\ell } = (2\\ell +1)\\pi T$ is the fermionic Matsubara frequency with $\\ell $ as integer number and $T$ the temperature, and $\\epsilon _k = -t[1+\\exp (-ik_x)+\\exp (-ik_y)]$ .", "Here $\\mu $ is the chemical potential and is determined by $n = \\frac{2T}{N_0}\\sum _{k\\ell }{\\rm Tr}G(k,i\\omega _{\\ell })\\exp (i\\omega _{\\ell }\\eta ), $ where the factor 2 stems from the spin degeneracy and $\\eta $ is an infinitesimally small positive constant.", "To proceed, we need to provide an approximation for $\\Sigma (k,i\\omega _{\\ell })$ .", "In the following sections, we investigate the possibility of the existence of the GBISP using the SCHFA and the RRDA for the self-energy, respectively." ], [ "Studying the existence of the GBISP using the SCHFA", "Let us first consider the physical meaning of the GBISP.", "As can be seen in Fig.", "1, supposing the origin is at the middle point of a b$_1$ a$_2$ bond, when changing each atom at site $r_j$ to $-r_j$ , the whole lattice is unchanged.", "This transformation is equivalent to interchanging the top and bottom layers and then rotating the lattice by an angle $\\pi $ around the b$_1$ a$_2$ bond.", "In the non symmetry-broken state, the electron system is unchanged with respect to such a transformation.", "However, when the strong Coulomb interactions drive the system to a GBISP, the two layers cease to be equivalent by inversion symmetry, and the electrons experience different fields on the two layers.", "Specifically, there may exist net electronic charge accumulation at each atom.", "We denote the deviations of the electronic charge density from the average value $n$ at each of the four sites of the unit cell as ($\\delta _1,\\delta _2,-\\delta _2,-\\delta _1$ ).", "Figure: Self-energy of the SCHFA.", "Left: Hartree term.", "Right: Fock exchange term.", "The solid line with an arrow denotes the Green's function.", "The wavy line is the Coulomb interaction.Under the SCHFA or the mean-field approximation, the self-energy is diagrammatically given by Fig.", "2.", "The Hartree term is diagonal, $\\Sigma ^H_{\\mu \\nu } = \\Delta _{\\mu }\\delta _{\\mu \\nu }$ , with $\\Delta _{\\mu } &=& \\delta _1u_{\\mu 1}+\\delta _2u_{\\mu 2} \\\\u_{\\mu 1} &=& \\lim _{q\\rightarrow 0}[v_{\\mu 1}(q)-v_{\\mu 4}(q)]\\\\u_{\\mu 2} &=& \\lim _{q\\rightarrow 0}[v_{\\mu 2}(q)-v_{\\mu 3}(q)]$ where $v_{\\mu \\nu }(q)$ (with the subscripts $\\mu \\nu $ being the same as those used in the definition of the Green's function, denoting the four sublattices a$_1$ , b$_1$ , a$_2$ and b$_2$ ) is the Fourier component of the Coulomb interaction.", "In the long-wavelength limit, $v_{\\mu \\nu }(q)$ behaves like $v_{\\mu \\nu }(q) \\rightarrow \\frac{2\\pi e^2}{S_0\\epsilon Q}\\exp (-z_{\\mu \\nu }Q) + \\tilde{v}_{\\mu \\nu }, ~~~~q \\rightarrow 0 $ where $S_0 = \\sqrt{3}a^2/2$ is the area of the two-dimensional unit cell of monolayer graphene, and $Q$ is the magnitude of the vector $\\vec{Q} = \\hat{M} \\vec{q}$ with[33] $\\hat{M} = \\begin{pmatrix}1& 0\\\\-\\frac{1}{\\sqrt{3}}&\\frac{2}{\\sqrt{3}}\\\\\\end{pmatrix}$ and where the components of $\\vec{q}$ are along the nonorthogonal axes of the diamond-shaped Brillouin zone.", "The value of $z_{\\mu \\nu } = 0$ or $z_0$ (the distance of the two layers) depends on $\\mu \\nu $ denoting the same layer or two different layers.", "The last term in Eq.", "(REF ), $\\tilde{v}_{\\mu \\nu }$ , is the regular part of the Coulomb potential for $q \\rightarrow 0$ .", "The $q$ dependence in Eq.", "(REF ) is different from the conventional form because the coordinate axes of the reciprocal lattice where $\\vec{q}$ is defined are nonorthogonal.", "The wave vector $\\vec{Q}$ is defined in an orthogonal basis.", "[33] The relations $\\Delta _1 = -\\Delta _4$ and $\\Delta _2=-\\Delta _3$ can be easily checked.", "The Fock exchange term is given by $\\Sigma ^F_{\\mu \\nu }(k) = -\\frac{1}{M}\\sum _qv_{\\mu \\nu }(q)\\tilde{n}_{\\mu \\nu }(k-q) $ where $M = N_0/4$ is the total number of unit cells in one layer, and $\\tilde{n}_{\\mu \\nu }(k)$ is given as $\\tilde{n}_{\\mu \\nu }(k) = T\\sum _{\\ell }G_{\\mu \\nu }(k,i\\omega _{\\ell })\\exp (i\\omega _{\\ell }\\eta )-\\delta _{\\mu \\nu }/2, $ which corresponds to the quasiparticle distribution function, the term $-\\delta _{\\mu \\nu }/2$ stemming from the non-normal order of the electronic interaction operator.", "Under the mean-field approximation, the self-energy $\\Sigma _{\\mu \\nu }(k) = \\Sigma ^H_{\\mu \\nu }+\\Sigma ^F_{\\mu \\nu }(k)$ is independent of the frequency.", "By diagonalizing the effective Hamiltonian $h_k + \\Sigma (k)$ , one can explicitly carry out the frequency summation in Eq.", "(REF ).", "The parameters $\\delta _1$ and $\\delta _2$ are determined by $\\delta _1 &=& \\frac{1}{M}\\sum _k[\\tilde{n}_{11}(k)-\\tilde{n}_{44}(k)], \\\\\\delta _2 &=& \\frac{1}{M}\\sum _k[\\tilde{n}_{22}(k)-\\tilde{n}_{33}(k)].", "$ So far, all the components of self-energy and parameters are self-consistently determined by Eqs.", "(REF )-().", "The magnitude of $\\delta _1$ is larger than that of $\\delta _2$ .", "To see it, consider temporarily the isolated b$_1$ and a$_2$ atoms without Coulomb interaction.", "Since they are bonded by $t_1$ , their atomic degenerate states are split in two bonding-antibonding states with eigenvalue $\\pm t_1$ .", "Therefore, the states of the b$_1$ and a$_2$ sublattices contribute mostly to the eigenstates corresponding to the noninteracting energy bands of overall energy separation $\\pm t_1$ from the zero energy.", "At low temperature, the lower band is occupied while the upper band is empty.", "On the other hand, the valence and conduction bands close to zero energy have eigenvectors which are composed predominantly of the linear combination of atomic states of the a$_1$ and b$_2$ sublattices.", "The atoms of these two latter sublattices are the first to be affected by the Coulomb interaction, and they are subject to the most charge accumulation in the case of the GBISP.", "The two parameters $\\delta _1$ and $\\delta _2$ are not independent, but are correlated through the Green's functions as described by Eqs.", "(REF ) and (REF )-().", "We can chose $\\delta _1$ as the independent order parameter of the GBISP.", "Figure: Equation for the particle-hole propagator D(k)D(k).", "The triangle denotes D(k)D(k).", "The effective interaction between particles and holes is obtained by disconnecting a Green's function line in the self-energy given in Fig.", "2.To determine the GBISP phase boundary, that is the relation between the critical temperature $T_0$ and the carrier doping concentration $\\delta $ , we expand the self-energy and the Green's function to first order in the order parameter $\\delta _1$ .", "Let us define the matrix $D(k) = \\frac{\\partial }{\\partial \\delta _1}[\\Sigma (k)-S^{\\dagger }\\Sigma ^{\\ast }(k)S]/2 $ with $S = \\begin{pmatrix}0& 0&0&1\\\\0&0&1&0\\\\0&1&0&0\\\\1&0&0&0\\\\\\end{pmatrix}.$ Notice that $\\Sigma ^{\\ast }(k) = \\Sigma ^{t}(k)$ (the transpose of $\\Sigma $ ) since $\\Sigma ^{\\dagger }(k) = \\Sigma (k)$ .", "By this symmetry relation and by the definition in Eq.", "(REF ), $D(k)$ has the following structure: $D = \\begin{pmatrix}D_{11}& D_{12}&D_{13}&0\\\\D^{\\ast }_{12}&D_{22}&0&-D_{13}\\\\D^{\\ast }_{13}&0&-D_{22}&-D_{12}\\\\0&-D^{\\ast }_{13}&-D^{\\ast }_{12}&-D_{11}\\\\\\end{pmatrix}.$ Therefore, only four elements $D_{11}$ , $D_{12}$ , $D_{13}$ and $D_{22}$ need to be determined.", "Under the mean-field approximation, we have the following equation for $D(k)$ : $D_{\\mu \\nu }(k) = d_{\\mu }\\delta _{\\mu \\nu }-\\frac{1}{M}\\sum _{k^{\\prime }\\lambda \\lambda ^{\\prime }}v_{\\mu \\nu }(k-k^{\\prime })f^{\\lambda \\lambda ^{\\prime }}_{\\mu \\nu }(k^{\\prime })D_{\\lambda \\lambda ^{\\prime }}(k^{\\prime }) \\nonumber \\\\ $ with $d_{\\mu } &=& u_{\\mu 1}+u_{\\mu 2}\\frac{\\partial \\delta _2}{\\partial \\delta _1}, \\\\f^{\\lambda \\lambda ^{\\prime }}_{\\mu \\nu }(k) &=& T\\sum _{\\ell }G_{\\mu \\lambda }(k,i\\omega _{\\ell })G_{\\lambda ^{\\prime }\\mu }(k,i\\omega _{\\ell }), $ where $G(k,i\\omega _{\\ell })$ 's are the normal-state Green's functions in which $\\delta _1 = \\delta _2 = 0$ .", "Again, the frequency summation in Eq.", "() can be performed analytically.", "For the normal state, the Green's functions satisfy the relation $G_{\\mu \\nu } = G_{\\bar{\\nu }\\bar{\\mu }}$ with $\\bar{\\mu }= 5-\\mu $ .", "We therefore have $f^{\\lambda \\lambda ^{\\prime }}_{\\mu \\nu } = f^{\\bar{\\lambda }^{\\prime }\\bar{\\lambda }}_{\\bar{\\nu }\\bar{\\mu }}$ .", "By noting these relations, $\\partial \\delta _2/\\partial \\delta _1$ can be expressed as $\\frac{\\partial \\delta _2}{\\partial \\delta _1} = \\frac{2}{M}\\sum _{k\\lambda \\lambda ^{\\prime }}f^{\\lambda \\lambda ^{\\prime }}_{22}(k)D_{\\lambda \\lambda ^{\\prime }}(k).$ Taking the partial derivative of Eq.", "(REF ) with respect to $\\delta _1$ , we obtain the condition for the phase transition, $\\frac{2}{M}\\sum _{k\\lambda \\lambda ^{\\prime }}f^{\\lambda \\lambda ^{\\prime }}_{11}(k)D_{\\lambda \\lambda ^{\\prime }}(k) = 1.", "$ Note that $d_{\\mu }$ can be formally expressed as $d_{\\mu } &=& \\frac{2}{M}\\sum _{k\\lambda \\lambda ^{\\prime }}[u_{\\mu 1}f^{\\lambda \\lambda ^{\\prime }}_{11}(k)+ u_{\\mu 2}f^{\\lambda \\lambda ^{\\prime }}_{22}(k)]D_{\\lambda \\lambda ^{\\prime }}(k) \\nonumber \\\\&=& \\frac{2}{M}\\sum _{k\\kappa \\lambda \\lambda ^{\\prime }}v_{\\mu \\kappa }(0)f^{\\lambda \\lambda ^{\\prime }}_{\\kappa \\kappa }(k)D_{\\lambda \\lambda ^{\\prime }}(k), \\nonumber $ where in the second line, the definition of $u_{\\mu 1(2)}$ , $f^{\\lambda \\lambda ^{\\prime }}_{\\mu \\nu } = f^{\\bar{\\lambda }^{\\prime }\\bar{\\lambda }}_{\\bar{\\nu }\\bar{\\mu }}$ and $D_{\\bar{\\lambda }\\bar{\\lambda }^{\\prime }}(k) = -D_{\\lambda ^{\\prime }\\lambda }(k)$ has been used.", "(The factor 2 originates from the spin degeneracy.)", "Putting this result into Eq.", "(REF ), one obtains the coupled linear equations for $D$ 's.", "The equations are diagrammatically shown in Fig.", "3.", "The function $D(k)$ is actually the particle-hole propagator.", "The effective interaction between particles and holes is the result of disconnecting a Green's function in the self-energy as given in Fig.", "2, by following the procedure explained in Fig.", "3.", "Clearly, these equations are equivalent to solving the problem of a particle-hole propagator with a unity eigenvalue.", "Instead of solving the eigenvalue equations as given in Fig.", "3, $D(k)$ 's can be determined more easily from Eqs.", "(REF ), (REF ) and (REF ) by self-consistent iteration.", "For a given carrier doping concentration $\\delta $ , the transition temperature $T_0$ can be found by gradually lowering temperature $T$ from a value higher than the critical one, and solving the equations for $D$ 's in the normal state at each step of the process.", "When the left-hand side of Eq.", "(REF ) becomes equal to 1, the critical temperature $T_0$ is reached.", "Figure: Phase diagram of bilayer graphene in the SCHFA for coupling constant g=1g = 1.", "The symbols are the numerical solution for transition points.", "The dashed line is an extrapolation of the finite-temperature results to low temperature.To numerically solve Eqs.", "(REF ), (REF ) and (REF )-(REF ), we need to carefully treat the convolution of the Coulomb interaction $v_{\\mu \\nu }(q)$ and the function $\\tilde{n}_{\\mu \\nu }(k-q)$ as appearing in Eq.", "(REF ) [and the similar one appearing in Eq.", "(REF )] because $v_{\\mu \\nu }(q)$ is singular at $q = 0$ .", "In Appendix A, we present an algorithm to deal with this problem.", "In Fig.", "4, we show the phase diagram of the electron system in the $\\delta -T$ plane for coupling constant $g = 1$ .", "At low temperature and low carrier doping, the system is in the GBISP.", "The transition temperature as a function of $\\delta $ is uniquely defined only at low $\\delta < 0.24\\times 10^{-4}$ .", "However, in the region $0.24\\times 10^{-4} < \\delta < 0.3\\times 10^{-4}$ , each $\\delta $ corresponds to two transition temperatures.", "In the latter case, the phase boundary was determined by adjusting $\\delta $ for every given $T$ .", "The numerical results for the order parameters $\\delta _1$ and $\\delta _2$ as functions of $T$ at $\\delta = 0$ for coupling constants $g = 0.5$ , 1 and 1.8 are shown in Fig.", "5.", "The SCHFA results are denoted as HF.", "We notice that $|\\delta _1| > |\\delta _2|$ , but $\\delta _2$ is not negligibly small, which is different from what has been assumed in the two-band model.", "[5] From Fig.", "5 one can understand that the charge configuration at the four sites in the unit cell is $(-|\\delta _1|,|\\delta _2|,-|\\delta _2|,|\\delta _1|)$ [another solution is $(|\\delta _1|,-|\\delta _2|,|\\delta _2|,-|\\delta _1|)$ ].", "The signs of $\\delta _1$ and $\\delta _2$ are the opposite of each other because with such a charge distribution, the Coulomb interaction between sites a and b in the same plane is attractive and stabilizes the GBISP.", "It is also seen from Fig.", "5 that the transition temperature is higher for a system with stronger coupling.", "Figure: Order parameters δ 1 \\delta _1 and δ 2 \\delta _2 as functions of temperature TT at δ=0\\delta = 0 for coupling constants g=0.5g = 0.5, 1 and 1.8.", "The symbols refer to numerical results.", "Circles and squares denoted as HF are the SCHFA results for δ 1 \\delta _1 and δ 2 \\delta _2, respectively.", "The RRDA results denoted by triangles (δ 1 \\delta _1) and inverse triangles (δ 2 \\delta _2) are vanishingly small.Our lattice model is different from both the two- and four-band continuum effective models.", "[5], [9], [26], [27] The two- and four-band continuum models are established under the consideration that the energy scale of quasiparticle spectral resonances involved in the problem is small with respect to a characteristic energy taken from bilayer graphene noninteracting band structure.", "For the four-band continuum model, the energy should be much less than the bandwidth of the $\\pi $ orbitals of graphene.", "The two-band model for BLG is accurate only in the case in which the quasiparticle energy is much smaller than the gap $t_1$ .", "In the presence of long-range Coulomb interaction $v(q)$ , the energy transfer at small $q$ is very large and the assumption for the validity of the two- and four-band continuum models becomes problematic.", "In this sense, the lattice model appears to be more reasonable.", "Another important difference between the lattice model and the two- and four-band continuum models relates to the valley physics in the Brillouin zone.", "In the two- and four-band continuum models, the two valleys are independent of each other and the valley index is treated as an overall degeneracy index.", "On the contrary, within the lattice model two states belonging to different valleys can be connected by nonzero intervalley Hamiltonian matrix elements." ], [ "Suppression of the GBISP in the RRDA", "The order parameters $\\delta _1$ and $\\delta _2$ so obtained by the SCHFA are overestimated because charge fluctuations have been ignored.", "We here reexamine the possibility of the existence of the GBISP using the RRDA.", "Under the RRDA, besides the Hartree-Fock terms given in Fig.", "2, the additional part of the self-energy, denoted by $\\Sigma ^c(k,i\\omega _{\\ell })$ , is shown in Fig.", "6.", "Each bubble in Fig.", "6 is composed of two Green's functions $G$ , representing the charge polarizability.", "In terms of $G$ , the elements of $\\Sigma ^c(k,i\\omega _{\\ell })$ are expressed as $\\Sigma ^c_{\\mu \\nu }(k,i\\omega _{\\ell }) &=& -\\frac{T}{M}\\sum _{q,m} G_{\\mu \\nu }(k-q,i\\omega _{\\ell }-i\\nu _m)W^c_{\\mu \\nu }(q,i\\nu _m)\\nonumber $ where $\\nu _m$ is the bosonic Matsubara frequency, and $W^c_{\\mu \\nu }(q,i\\nu _m)$ is an effective interaction.", "The matrix form of $W^c$ is given by $W^c(q,i\\nu _m) = [1-v(q)\\chi (q,i\\nu _m)]^{-1}v(q)-v(q) $ with $\\chi _{\\mu \\nu }(q,i\\nu _m)=\\frac{2T}{M}\\sum _{k,\\ell } G_{\\mu \\nu }(k,i\\omega _{\\ell })G_{\\nu \\mu }(k-q,i\\omega _{\\ell }-i\\nu _m) \\nonumber $ and $v(q)$ is the Fourier component ($4\\times 4$ matrix) of the Coulomb interaction.", "The total self-energy is given by $\\Sigma _{\\mu \\nu }(k,i\\omega _{\\ell })=\\Delta _{\\mu }\\delta {\\mu \\nu }+ \\Sigma ^F_{\\mu \\nu }(k)+ \\Sigma ^c_{\\mu \\nu }(k,i\\omega _{\\ell }).$ The Green's function $G$ is self-consistently determined and satisfies the microscopic conservation laws.", "[24] Figure: Additional part of the self-energy besides the Hartree-Fock terms.Note that $\\Sigma ^c$ is a convolution of $G$ and $W^c$ , and $\\chi $ is a convolution of two $G$ 's in momentum and frequency space.", "The easy way to calculate them is by Fourier transform.", "At low temperature, the summations index over the Matsubara frequencies should run up to a large frequency cutoff.", "To reduce the requirement of computer memory storage and accelerating the numerical computation, the special algorithm of Ref.", "YanE can be used.", "The interaction $W^c(q,i\\nu _m)$ vanishes for $m \\rightarrow \\infty $ .", "For finite $\\nu _m$ , we need to carefully deal with the singularity at $q = 0$ .", "The Fourier transform $W^c(q,i\\nu _m)$ to $W^c(r,i\\nu _m)$ is discussed in Appendix B.", "At the ground state for $T = 0$ , the Matsubara frequencies $\\omega _{\\ell }$ and $\\nu _m$ are treated as the continuous variables $\\omega $ and $\\nu $ , respectively, and the summations over them are replaced by integrals, $T\\sum _{\\omega _{\\ell }}&\\rightarrow & \\int _{-\\infty }^{\\infty }\\frac{d\\omega }{2\\pi } \\nonumber \\\\T\\sum _{\\nu _m}&\\rightarrow & \\int _{-\\infty }^{\\infty }\\frac{d\\nu }{2\\pi }.", "\\nonumber $ We computed the Green's function within the RRDA.", "The results for the order parameters $\\delta _1$ and $\\delta _2$ for $g = 0.5$ , 1 and 1.8 at $\\delta = 0$ are shown in Fig.", "5 and compared with the SCHFA.", "The doping concentration chosen corresponds to $\\delta = 0$ , for which the SCHFA transition temperature reaches its maximum.", "Though $\\delta = 0$ is the most favorable case for the GBISP predicted by the SCHFA, the two order parameters are substantially suppressed by CDF in the RRDA; the magnitude of the two parameters is three orders smaller than that of the SCHFA.", "For inspecting the GBISP ordering at low temperature, the RRDA calculation for $g = 1.8$ is performed down to $T = 0$ .", "From the numerical results, we conclude that there is no the GBISP in systems with $g \\le 1.8$ under the RRDA.", "We also computed the Green's function within RPA, in which the polarizability $\\chi (q,i\\nu _m)$ in $W^c(q,i\\nu _m)$ [see Eq.", "(REF )] is replaced by the polarizability for noninteracting electrons.", "At $\\delta = 0$ , similar to the RRDA, the parameters $\\delta _1$ and $\\delta _2$ so obtained are vanishingly small.", "In the RRDA, a replacement of the bare Coulomb interaction in the Hartree term by the screened one is prohibited because where the ring diagrams are equivalent to a self-energy insertion to the Green's function to be renormalized.", "When performing the RPA calculation, we also need to keep the bare Coulomb interaction in the Hartree term.", "The reason for the suppression of the GBISP under the RRDA is that the exchange interaction is significantly weakened by the screening due to electronic charge-density fluctuations while the Hartree term opposing the charge transfer between the two layers[5] is not changed.", "At low temperature, in a wide range of Matsubara frequencies, the exchange interaction is short-ranged and weakened and does not favor the GBISP transition.", "We point out that the suppression of the GBISP here is not due to prohibition by the Mermin-Wagner theorem.", "[31] The theorem applies to a system with continuous symmetry; if the symmetry were broken, there would be a logarithmically diverging number of long-wavelength collective fluctuations accompanying the excitations on top of the broken symmetry ground state of the two-dimensional system.", "In the present case, the inversion is a discrete symmetry operation, and there is no diverging long-wavelength collective fluctuation arising from the breaking of such a symmetry." ], [ "Summary", "In summary, we have studied the physics of interacting electrons in bilayer graphene using the lattice model.", "The possibility of the existence of a GBISP at low temperature and low-carrier-doping concentration is reinvestigated with both the SCHFA and the RRDA.", "The latter approach takes into account the charge density fluctuations beyond the SCHFA or the mean-field approximation.", "Under the RRDA, the exchange interaction is weakened substantially, and the existence of a GBISP becomes unsustainable.", "We have also presented the numerical method for dealing with convolution of a singular Coulomb interaction and the Green's function on the lattice model.", "This numerical method should be usable for solving problems in many-particle systems.", "This work was supported by the National Basic Research 973 Program of China under Grants No.", "2011CB932700 and No.", "2012CB932300, NSFC under Grant No.", "10834011, and the Robert A. Welch Foundation under Grant No.", "E-1146.", "Appendix A: Calculation of the momentum-space convolution of Coulomb interaction with a smooth function for a lattice model For solving problems of two-dimensional electron system in the presence of long-range Coulomb interaction, we sometimes need to deal with the convolution $C(k)=\\frac{1}{M}\\sum _{q} V(q)f(k-q) $ where the $q$ summation runs over the first Brillouin zone, $V(q)$ is the Coulomb interaction, and $f(k)$ is a smooth function of $k$ .", "On a lattice, an analytical expression for $V(q)$ is not available but its long-wavelength behavior is known.", "For the honeycomb lattice, it is given by Eq.", "(REF ).", "$V(q)$ can be divided into long-range and short-range parts.", "For the honeycomb lattice under consideration, define $v^l(q)= \\sum _{n} \\frac{c}{|\\vec{Q}_n+\\vec{Q}|}\\exp (-a_0|\\vec{Q}_n+\\vec{Q}|) $ where $c=2\\pi e^2/S_0\\epsilon $ is the same factor that appeared in Eq.", "(REF ), $\\vec{Q}_n$ is the reciprocal lattice vector, $\\vec{Q} = \\hat{M} \\vec{q}$ is as given in the text, and $a_0$ is an auxiliary parameter.", "By taking $a_0 = 2a$ , the summation in Eq.", "(REF ) converges quickly and only a few terms need to be summed up.", "Clearly, $v^l(q)$ represents a long-range interaction.", "With $v^l(q)$ , $V(q)$ can be written as $V(q) = v^l(q) +v^s(q)$ where $v^s(q)$ is so defined by the equation and is the short-range part of $V(q)$ .", "Note that both $v^l(q)$ and $v^s(q)$ are periodic functions of $q$ .", "Equation (REF ) now is given by $C(k)=\\frac{1}{M}\\sum _{q} v^s(q)f(k-q)+\\frac{1}{M}\\sum _{q} v^l(q)f(k-q).", "$ The first integral in Eq.", "(REF ) can be safely performed by Fourier transform.", "In the second integral, the singularity appears at $q = 0$ .", "To find out an auxiliary function for this integral, we pay attention to the expanding form of $f(k-q)$ $f(k-q) \\rightarrow f(k) -q_x f_x(k)-q_yf_y(k) $ where $f_{x(y)}(k)= df(k)/dk_{x(y)}$ .", "Define two auxiliary functions $v_x(q)$ and $v_y(q)$ by $v_{x(y)}(q)= \\sum _{n} \\frac{c[q_{x(y)}+(\\hat{M}^{-1} \\vec{Q}_n)_{x(y)}]}{|\\vec{Q}_n+\\vec{Q}|}\\exp (-a_0|\\vec{Q}_n+\\vec{Q}|), \\nonumber $ where $v_{x(y)}(q)$ is periodic and odd under $\\vec{q} \\rightarrow -\\vec{q}$ .", "The second integral in Eq.", "(REF ) can be written as $\\frac{1}{M}\\sum _{q} v^l(q)f(k-q)& =& \\frac{1}{M}\\sum _{k^{\\prime }} \\lbrace v^l(k-k^{\\prime })[f(k^{\\prime })-f(k)]\\nonumber \\\\&&+v_x(k-k^{\\prime })f_x(k)\\nonumber \\\\&&+v_y(k-k^{\\prime })f_y(k)\\rbrace \\nonumber \\\\&&+ f(k)v^l(r)|_{r=0}.", "$ Now, there is no singularity in the integrand in Eq.", "(REF ).", "The leading term of $v^l(k-k^{\\prime })[f(k^{\\prime })-f(k)]$ as $k^{\\prime } \\rightarrow k$ is proportional to the derivative of $f$ multiplied with a sign factor since $v^l(k-k^{\\prime }) \\propto 1/|\\hat{M}(\\vec{k}^{\\prime }-\\vec{k})|$ .", "This leading term varies discontinuously at $k^{\\prime } = k$ .", "The discontinuity is canceled out by the remaining term $v_x(k-k^{\\prime })f_x(k)+v_y(k-k^{\\prime })f_y(k)$ .", "As a result, the integrand is a smooth function.", "The integral can then be carried out numerically.", "The last term stems from the introduction of the auxiliary functions to the integrand.", "The value $v^l(r)|_{r=0}$ is given by $v^l(r)|_{r=0} = \\frac{1}{M}\\sum _{q} v^l(q), $ which can be calculated explicitly.", "Replace $q$ -summation by $\\frac{1}{M}\\sum _{q} \\rightarrow \\frac{S_0}{V}\\sum _{Q} = S_0\\int _{\\rm BZ}\\frac{d\\vec{Q}}{(2\\pi )^2}$ where $S_0 = \\sqrt{3}a^2/2$ is the area of the unit cell of the honeycomb lattice as appearing in the text, and BZ means the integral is performed over the first Brillouin zone.", "The combination of the integration over BZ and the $Q_n$ -summation in the definition of $v^l(q)$ equals the integration of the function $c\\exp (-a_0Q)/Q$ over the total space of $Q$ , $\\ v^l(r)|_{r=0} &=& S_0\\int \\frac{d\\vec{Q}}{(2\\pi )^2}\\frac{c}{Q}\\exp (-a_0Q) \\nonumber \\\\&=& e^2/a_0.$ The function $f(k)$ is assumed to be smooth here.", "However, for calculating the Fock exchange self-energy, $f(k)$ corresponds to the distribution function and varies dramatically at the Fermi surface at low temperature.", "In this case, extremely dense grids in a momentum regime covering the Fermi surface should be used to denote the variation of $f(k)$ .", "The term $v_x(k-k^{\\prime })f_x(k)+v_y(k-k^{\\prime })f_y(k)$ was introduced in the right-hand side of Eq.", "(REF ) in order to smooth the integrand.", "Because $v_x(q)$ and $v_y(q)$ are periodic and odd functions of $q$ , the contribution from the integral of $v_x(k-k^{\\prime })f_x(k)+v_y(k-k^{\\prime })f_y(k)$ is zero.", "At $T = 0$ , there is a discontinuity in $f(k)$ at the Fermi surface and its derivatives $f_x(k)$ and $f_y(k)$ are $\\delta $ functions.", "Therefore, the use of this term at $T = 0$ is unworthy.", "At $T = 0$ , this term should be removed, keeping the discontinuity in the integrand.", "The cost is to use dense grids near the Fermi surface to ensure the accuracy of the result.", "Appendix B: Fourier transform of the screening potential $W^c(q,i\\nu _m)$ To take the Fourier transform of $W^c(q,i\\nu _m)$ given by Eq.", "(REF ) from momentum space to real space, we first pay attention to its singularity at $q = 0$ .", "For small $\\nu _m$ , because $\\chi (q,i\\nu _m)$ is finite, the singularity exists only in the second term $v(q)$ on the right-hand side of Eq.", "(REF ).", "Its real space form is known as that given by Eq.", "(REF ) for its elements.", "However, at large $\\nu _m$ , because $\\chi (q,i\\nu _m)$ is vanishingly small, there is also a singularity in the first term on right-hand side of Eq.", "(REF ) and both terms cancel with each other.", "We need a systematic numerical scheme for the Fourier transform at any $\\nu _m$ .", "Note that in the limit $q \\rightarrow 0$ , $v(q) \\rightarrow v_0(q)\\hat{A}$ with $v_0 (q) = c/Q$ (again $c=2\\pi e^2/S_0\\epsilon $ and $Q = |\\hat{M} \\vec{q}|$ ) and $\\hat{A} = \\begin{pmatrix}1& 1&1&1\\\\1&1&1&1\\\\1&1&1&1\\\\1&1&1&1\\\\\\end{pmatrix}.", "\\nonumber $ In the same limit, we have $W^c(q,i\\nu _m) &\\rightarrow & -\\frac{\\alpha _mc}{Q(Q+\\alpha _m)}\\hat{A} \\nonumber \\\\&=& W_m(Q)\\hat{A},$ with $\\alpha _m = -c\\sum _{\\mu \\nu }\\chi _{\\mu \\nu }(0,i\\nu _m)$ and $W_m(Q)$ so defined by Eq.", "(REF ).", "By observing this asymptotic form, we take the auxiliary function for the Fourier transform as $W_a(q) = \\sum _nW_m(|\\vec{Q} + \\vec{Q}_n|)\\exp (-a_0|\\vec{Q} + \\vec{Q}_n|) $ where $a_0$ again is a parameter for fast convergence of the summation over the reciprocal lattice vectors $\\vec{Q}_n$ .", "The Fourier transform of $W^c(q,i\\nu _m)$ is separated into two parts, $[W^c(q,i\\nu _m)-W_a(q)\\hat{A}]$ and $W_a(q)\\hat{A}$ .", "There is no singularity in the first one and it can be safely transformed by numerical computation.", "For the second one, $W_a(q)$ is transformed as $W_a(r) &=& a^2\\int _{BZ}\\frac{d\\vec{q}}{(2\\pi )^2}W_a(q)\\exp (i\\vec{q} \\cdot \\vec{r}) \\nonumber \\\\&=& S_0\\int _{BZ}\\frac{d\\vec{Q}}{(2\\pi )^2}W_a(q)\\exp (i\\vec{Q} \\cdot \\vec{R}) \\nonumber \\\\&=& S_0\\int \\frac{d\\vec{Q}}{(2\\pi )^2}W_m(Q)\\exp (i\\vec{Q} \\cdot \\vec{R}-a_0Q) \\nonumber \\\\&=& -\\frac{S_0\\alpha _mc}{2\\pi }\\int ^{\\infty }_0dQ\\frac{\\exp (-a_0Q)}{Q+\\alpha _m}J_0(QR) $ where the first line is the definition with $\\vec{q}$ and $\\vec{r}$ given in the quadrilateral coordinate system, the second line converts $\\vec{q}$ to $\\vec{Q} =\\hat{M}\\vec{q}$ and $\\vec{R} = (\\hat{M}^{t})^{-1}\\vec{r}$ (with $\\hat{M}^{t}$ the transpose of $\\hat{M}$ ) in the orthogonal systems with $d\\vec{q} = d\\vec{Q}/|\\hat{M}| = \\sqrt{3}d\\vec{Q}/2$ , the third line comes from the definition of $W_a(q)$ given by Eq.", "(REF ), the last line is obtained after the azimuthal integration, and $J_0(QR)$ is the Bessel function.", "Now the singularity in the integrand exists only when $\\alpha _m =0$ , but $\\alpha _m$ also appears in the front factor and the integral vanishes.", "However, for large $R$ , $J_0(QR)$ oscillates rapidly with $Q$ .", "By observing the large-$QR$ behavior of $J_0(QR)$ , we choose the auxiliary function[28] $J_A(z) &=& \\sqrt{\\frac{1}{\\pi z +1}}\\lbrace [1+\\frac{\\pi ^2z}{8(\\pi z +1)^2}]\\sin (z)\\nonumber \\\\&&+[1-\\frac{\\pi ^2z}{8(\\pi z +1)^2}]\\cos (z)\\rbrace $ and separate $J_0(QR)$ to $J_0(QR)-J_A(QR)$ and $J_A(QR)$ .", "By replacing $J_0(QR)$ with $J_0(QR)-J_A(QR)$ in Eq.", "(REF ), the integral can be accurately carried out by simple numerical method.", "The remaining integral with $J_0(QR)$ replaced by $J_A(QR)$ can be performed using Filon's method." ] ]

[ [ "Superburst Models for Neutron Stars with Hydrogen and Helium-Rich\n Atmospheres" ], [ "Abstract Superbursts are rare day-long Type I X-ray bursts due to carbon flashes on accreting neutron stars in low-mass X-ray binaries.", "They heat the neutron star envelope such that the burning of accreted hydrogen and helium becomes stable, and the common shorter X-ray bursts are quenched.", "Short bursts reappear only after the envelope cools down.", "We study multi-zone one-dimensional models of the neutron star envelope, in which we follow carbon burning during the superburst, and we include hydrogen and helium burning in the atmosphere above.", "We investigate both the case of a solar composition and a helium-rich atmosphere.", "This allows us to study for the first time a wide variety of thermonuclear burning behavior as well as the transitions between the different regimes in a self-consistent manner.", "For solar composition, burst quenching ends much sooner than previously expected.", "This is because of the complex interplay between the 3-alpha, hot CNO, and CNO breakout reactions.", "Stable burning of hydrogen and helium transitions via marginally stable burning (mHz QPOs) to less energetic bursts with short recurrence times.", "We find a short-lived bursting mode where weaker and stronger bursts alternate.", "Eventually the bursting behavior changes back to that of the pre-superburst bursts.", "Because of the scarcity of observations, this transition has not been directly detected after a superburst.", "Using the MINBAR burst catalog we identify the shortest upper limit on the quenching time for 4U 1636-536, and derive further constraints on the time scale on which bursts return." ], [ "Introduction", "Superbursts are day-long flares observed from neutron stars in low-mass X-ray binaries (LMXBs; [8], [53]), that are attributed to the unstable thermonuclear burning of a carbon-rich layer ([11]; see [7] for a discussion on alternative types of fuel).", "The thermonuclear runaway starts at a column depth of $y=10^{11}\\,\\mathrm {g\\, cm^{-2}}$ to $y=10^{12}\\,\\mathrm {g\\, cm^{-2}}$ , close to the outer crust ([13]).", "Because of the high temperature dependence of carbon burning (e.g., [3]), the ignition is sensitive to crustal heating, and hence superburst observations place constraints on the combined crustal heating and neutrino cooling of the crust and core.", "For example, the superburst from the classical transient X-ray source 4U 1608–522 requires the crust to have heated up faster than predicted by current models ([36]).", "Because of the long typical recurrence time of approximately a year, superbursts are rare: so far 22 (candidates) have been detected from 13 sources (e.g., [35]; see [39]; [5]; [25]; [52]; [2] for recent discoveries; see also Table REF ).", "The carbon fuel is expected to be produced by thermonuclear burning of accreted hydrogen and/or helium higher up in the atmosphere (e.g., [53]).", "All known superbursting sources exhibit unstable hydrogen/helium burning observed as short (up to $\\sim 100\\,\\mathrm {s}$ duration) Type I X-ray bursts, as well as stable burning, evidenced by high values of the $\\alpha $ parameter, the ratio of the integrated persistent emission between two bursts and the burst fluence ([29]).", "This burning takes place at a column depth of $y\\simeq 10^{8}\\,\\mathrm {g\\, cm^{-2}}$ .", "Whereas stable hydrogen/helium burning through the hot CNO cycle and the $3\\alpha $ -process produce carbon, it is destroyed in bursts by CNO-cycle breakout reactions, followed by a series of helium captures catalyzed by protons — the $\\alpha $ p-process — and proton captures with subsequent $\\beta $ decays — the rp-process — creating heavier elements ([50], [51]).", "Current models produce burst ashes with a carbon mass fraction of typically $5\\%$ ([60]), whereas a slightly higher mass fraction is found for models that accrete material with a higher metallicity at a lower rate.", "Superburst observations, however, indicate a carbon content that is closer to $20\\%$ ([13]).", "[45] recently suggested that chemical separation by freeze out close to the outer crust may increase the carbon fraction.", "If the superburst ignites in a sufficiently thick layer, carbon burning initially proceeds as a detonation ([59]).", "This produces a shock that travels to the surface, pushing the envelope outward.", "The hydrogen/helium-rich atmosphere is heated by the shock ([58]) and the subsequent fall-back of the outer layers ([34]).", "This produces a bright precursor burst.", "Furthermore, the ignition conditions for a hydrogen/helium flash are reached.", "X-ray emission from this flash adds to the precursor burst, or it may account for the entire precursor emission, in case there is no strong shock ([34]).", "Precursor bursts have been identified in all cases when the start of the superburst was observed and the data were of sufficient quality (e.g., [38]).", "After the precursor burst, the envelope is sufficiently hot for all subsequently accreted hydrogen and helium to burn stably at the same rate at which it is accreted: the bursts are quenched ([8], [11], [41]).", "After the envelope has cooled down, unstable burning resumes, and Type I bursts return.", "Quenching times of one to several weeks are predicted ([12]).", "Because of non-contiguous observing schedules and the presence of frequent data gaps due to Earth occultations and South Atlantic Anomaly passages, the end of the quenching period and the reappearance of bursts have most likely not been observed.", "Only upper limits of more than one month have been derived for the quenching time (e.g., [38]).", "The return of hydrogen/helium bursts after a superburst provides a unique opportunity to study the transition of stable to unstable burning in the neutron star atmosphere.", "This transition is observed in a number of LMXBs, and is associated with changes in the mass accretion rate (e.g., [9]).", "A higher rate implies a hotter envelope as well as a faster accumulation of accreted material, which leads to steady-state burning of hydrogen and helium.", "Current burst models predict that this transition occurs at a 10 times higher accretion rate than inferred from observations (e.g., [23]), although increased crustal heating (e.g., [20]) and rotationally induced turbulent mixing ([47]) may reduce the discrepancy ([37]).", "Alternatively, it may be the result of a higher local accretion rate.", "The transition of the thermonuclear burning behavior in the atmosphere after a superburst is solely due to the temperature change, whereas the mass accretion rate remains constant.", "Note that in practice the mass accretion rate can vary on the time scale of a superburst, which would provide an effect additional to the change in the burning behavior.", "Because the cooling after a superburst sets the thermal profile of the envelope, this poses strict constraints on the environment in which the first bursts ignite.", "Detecting the transition of stable to unstable burning after a superburst will allow us to determine the column depth and temperature at which the transition occurs.", "These are important ingredients for improving the current models.", "Unfortunately, the observations of bursts after a superburst are scarce, and at the moment of writing the actual transition has not been observed.", "In this paper we provide one-dimensional multi-zone models of a neutron star envelope that undergoes a superburst.", "We create models with, respectively, a carbon-rich, a helium-rich , and a solar-composition atmosphere.", "We self-consistently simulate both the superburst and the burning processes in the atmosphere.", "The burst quenching is followed, as well as the transition to unstable burning and the return of bursts.", "We focus on the nuclear reactions that are responsible for the different phenomena in the light curves.", "Finally, we combine a large set of observations to place constraints on burst quenching." ], [ "Numerical method and observations", "The one-dimensional models of the neutron star envelope presented in this paper are similar to the superburst models by [34].", "We refer to that study for a detailed description of the code employed and the setup of the models.", "Here we describe the main properties of our simulations, as well as the differences with respect to the models by [34].", "Most notable are the use of a different prescription for electron conductivity and a different implementation of accretion, which may cause small differences in the ignition conditions of superbursts." ], [ "Stellar Evolution Code", "We create and evolve one-dimensional models of the neutron star envelope using the implicit hydrodynamics stellar evolution code KEPLER ([57]).", "The version of KEPLER that we use is similar to the version used in recent studies (e.g., [61], [60], [22]).", "We employ an adaptive one-dimensional Lagrangian grid in the radial direction, under the assumption of spherical symmetry.", "To follow the chemical evolution we use a large adaptive nuclear network ([48]) including the hot-CNO cycle ([56]), the $3\\alpha $ -, rp-, and $\\alpha $ p-processes, carbon fusion, and photodisintegration.", "We take into account neutrino energy loss ([30]), as well as radiative opacity and electron conductivity ([24]).", "Mixing of the chemical composition by convection, semiconvection, and thermohaline circulation is implemented as a diffusive process using mixing-length theory (e.g., [6]).", "Mass accretion is implemented by increasing the pressure of the outermost zone in the model at each time step (see also [62], [55]).", "Periodically a zone is added on top.", "Because of the details of the implementation, this causes a small dip in the light curve.", "We carefully check that this does not influence any of the conclusions that we draw about the bursting behavior.", "The light curves in this paper are corrected by omitting the brief time intervals when the dips occur." ], [ "Model Setup", "We model the neutron star envelope above a radius of $10\\,\\mathrm {km}$ and an enclosed mass of $1.4\\,\\mathrm {M_{\\odot }}$ , up to the surface.", "The bottom of the surface zone is at a column depth of $y=10^{6}\\,\\mathrm {g\\, cm^{-2}}$ .", "This is well below the typical column depth for hydrogen/helium bursts of $y\\simeq 10^{8}\\,\\mathrm {g\\, cm^{-2}}$ .", "The inner part of the model is formed by a $2\\times 10^{27}\\,\\mathrm {g}$ iron substrate.", "It serves as a buffer into which heat generated by the superburst can flow, and diffuse outward on longer time scales.", "This ensures a correct late-time light curve.", "Crustal heating is taken into account by a constant luminosity at the inner boundary equivalent to $Q_{\\mathrm {b}}=0.2\\,\\mathrm {MeV\\, nucleon^{-1}}$ .", "Because of neutrino cooling in the substrate, the effective amount of crustal heating at the top of the substrate, close to the superburst ignition depth, is $Q_{\\mathrm {b}}=0.1\\,\\mathrm {MeV\\, nucleon^{-1}}$ .", "This is in the range of typically assumed values for superbursters at accretion rates above $10\\,\\%$ of the Eddington limit (e.g., [21], [13]).", "On top of the substrate we accrete a mixture of $80\\,\\%$ $^{56}\\mathrm {Fe}$ and $20\\,\\%$ $\\mathrm {^{12}C}$ .", "The latter is the typical mass fraction of carbon that [13] found from fits to observed light curves of hydrogen accreting superbursters, and $^{56}\\mathrm {Fe}$ is the most abundant isotope in the ashes of hydrogen-rich bursts (e.g., [60]).", "The ashes of helium-rich bursts consist mostly of somewhat lighter isotopes: $^{28}\\mathrm {Si}$ , $^{32}\\mathrm {S}$ , $^{36}\\mathrm {Ar}$ , $^{40}\\mathrm {Ca}$ (e.g., [32]).", "For our models, however, we use $^{56}\\mathrm {Fe}$ , in order to have a consistent opacity of all envelope models, giving comparable ignition conditions for the superburst.", "Accretion of a hydrogen or helium-rich atmosphere is done only briefly before the superburst, because the computational expense of simulating the many short bursts that occur before the superburst is prohibitive.", "From a model where only a carbon-rich mixture was accreted we know already the moment of superburst ignition.", "$12.7\\,\\mathrm {hr}$ before this time, we replace the accretion composition by a solar ($71\\,\\%$ by mass $^{1}$ H, $27\\,\\%$ $^{4}$ He, $2\\,\\%$ $^{14}$ N), or a helium-rich mixture ($98\\,\\%$ $^{4}$ He, $2\\,\\%$ $^{14}$ N).", "This allows for several hydrogen and/or helium bursts to take place before the superburst, ensuring equilibrium is reached for the effects of chemical inertia ([60]).", "The change of accretion composition does not affect the ignition of the superburst, because it depends on the pressure at the bottom of the carbon layer, which continues to increase at the some rate.", "Moreover, at that time the initial phase of the carbon runaway has already started.", "An accretion rate of $\\dot{M}=5.25\\times 10^{-9}\\,\\mathrm {M_{\\odot }}\\,\\mathrm {yr^{-1}}$ is used.", "For an atmosphere of solar composition on a neutron star of $1.4\\,\\mathrm {M_{\\odot }}$ this corresponds to $30\\,\\%$ of the Eddington-limited rate $\\dot{M}_{\\mathrm {Edd}}=1.75\\times 10^{-8}\\,\\mathrm {M_{\\odot }}\\,\\mathrm {yr^{-1}}$ .", "The presented results are not corrected for the redshift due to the neutron star's gravity (see also [34]).", "Our Newtonian model has the same surface gravity as when general relativity is taken into account for a star with the same mass and a $11.2\\,\\mathrm {km}$ radius, which has a gravitational redshift of $z=0.26$ (e.g., [60]).", "Based on the atmosphere composition, we refer to the simulations as model “C”, “He”, and “H” (Table REF )." ], [ "MINBAR Catalog of Observations", "To derive observational constraints on the bursting behavior of superbursters we employ version 0.51 of the Multi-INstrument Burst ARchive (MINBAR; [33]; see http://users.monash.edu.au/$\\sim $ dgallow/minbar for more details).", "This catalog contains the results of the analysis of 4,192 observed Type I X-ray bursts from 72 sources as well as 27,340 pointings on 84 sources.", "The observations have been performed with the Wide-Field Cameras (WFCs) on board the Beppo Satellite per Astronomia X (BeppoSAX; [9]) and the Proportional Counter Array (PCA) on board the Rossi X-Ray Timing Explorer (RXTE; [19]).", "Both instruments are sensitive in a similar energy range above $2\\,\\mathrm {keV}$ .", "Because of its larger collecting area, the PCA is more sensitive to faint bursts (e.g., [33]).", "MINBAR comprises the largest collection of X-ray bursts available, but we refer to the literature if more constraining observations have been reported." ], [ "Hydrogen/Helium Atmosphere Models", "Initially the accretion of carbon-rich material is simulated to self-consistently build up a neutron star envelope close to the ignition of a superburst.", "Approximately half a day before the thermonuclear runaway of carbon burning sets in, the accretion composition is changed.", "We create three different models: one accreting solar composition (model H), one with helium-rich material (model He), and for comparison one where we retain the carbon-rich atmosphere (model C).", "From that moment we follow both the carbon burning in the ocean and the hydrogen/helium burning in the atmosphere.", "Several normal X-ray bursts occur before the superburst (Fig.", "REF , REF ).", "During the superburst decay, hydrogen and helium burn stably: bursts are quenched.", "Once the envelope has cooled sufficiently, unstable burning resumes.", "At the transition, oscillatory burning takes place (marginally stable burning).", "The first bursts after the superburst are less energetic and have shorter recurrence times than the bursts before the superburst.", "When the envelope cools down further, the recurrence times become longer and the bursts become as energetic as they were before the superburst.", "For the models H and He we calculate over 900 bursts per model." ], [ "Superburst", "The superburst occurs $1.28\\,\\mathrm {years}$ after the start of the simulations, after accreting a column of $y=1.1\\times 10^{12}\\,\\mathrm {g\\, cm^{-2}}$ .", "Its ignition is not in-phase with the occurrence of hydrogen/helium bursts (Fig.", "REF , REF ): flashes in the envelope do not trigger the runaway carbon burning.", "The superburst occurs slightly earlier in the models H and He compared to C: 1.1 minutes for helium accretion and 5.2 minutes for solar composition.", "This may be due to heating from hydrogen and helium burning, but the differences in recurrence time, and therefore ignition depth, are too small to cause substantial differences in, for example, the superburst fluence or other observables.", "The amount of time before the superburst during which we simulate the hydrogen/helium bursts may not have been long enough to bring the model into thermal equilibrium, and the deeper layers could still be increasing somewhat in temperature because of heating by the bursts.", "Because most of the energy produced by the bursts is radiated away from the surface, however, this introduces only a minor deviation in the temperature at the carbon ignition depth, which can be modeled by a slightly higher effective $Q_{\\mathrm {b}}$ .", "The superbursts start with a detonation, which drives a shock toward the surface, and produces a brief shock breakout peak in the light curve (Fig.", "REF ), followed on a dynamical time-scale by a precursor burst.", "The superburst peak is reached $6.4$  minutes after the superburst onset.", "Stable hydrogen (helium) burning produces a peak luminosity, $L_{\\mathrm {peak}}$ , that is $5\\,\\%$ ($0.4\\,\\%$ ) higher for model H (He) compared to C (Table REF ).", "The decays follow the same two-component power-law profile, with again hydrogen burning raising the luminosity by a few percent.", "The light curves for the models H and C deviate from the cooling curve of C when the luminosity becomes comparable to that from stable hydrogen or helium burning.", "The hydrogen or helium burning contribution to the total luminosity exceeds $10\\,\\%$ after roughly $\\sim 2\\times 10^{4}\\,\\mathrm {s}$ for model H, and after $\\sim 1\\times 10^{5}\\,\\mathrm {s}$ for He.", "For all models a small amount of carbon burns at the bottom of the newly accreted material.", "The burning time scale, however, is much longer than the accretion time, so this is not steady-state burning, but leads up to the next superburst.", "Table: Properties of Superbursts andSubsequent Bursts for Different Atmosphere Compositions" ], [ "Shock Breakout", "After the superburst detonation, a shock travels outward from the bottom of the carbon-rich layer.", "Once it reaches the surface, a shock breakout peak is produced in the light curve (Fig.", "REF ).", "We determine its maximum luminosity, $L_{\\mathrm {shock\\, br}}$ , and its fluence, $E_{\\mathrm {shock\\, br}}$ , as measured within $5\\times 10^{-6}\\,\\mathrm {s}$ after the onset (Table REF ).", "The differences in $L_{\\mathrm {shock\\, br}}$ trace variations in the opacity of the outer atmosphere.", "The shock accelerates the outer zones, where the density is lowest, and the shock over-pressure is highest, to a substantial fraction of the speed of light within a very short time interval.", "This behavior is likely not well resolved by our model, and at times introduces large variability in, for example, the luminosity in the outer zones during the shock breakout and the start of the subsequent precursor burst.", "During this part of the superburst we take the mean luminosity of the 10 outermost zones to reduce the effect on the light curve, although some of the introduced variability remains visible (Fig.", "REF )." ], [ "Precursor", "Most of the energy of the shock is used to expand the outer layers.", "They fall back on a dynamical time scale of $10^{-5}\\,\\mathrm {s}$ , and dissipate that energy into heat, which powers a precursor burst that reaches the Eddington limit.", "At the start of the precursor burst, the fallback causes the material to settle while undergoing a damped oscillation.", "This creates corresponding variability of super-Eddington luminosity as high as $3.7\\times 10^{40}\\,\\mathrm {erg}\\mathrm {\\, s^{-1}}$ (Fig.", "REF , REF ).", "The precursor light curves are dependent on the atmospheric composition (Fig.", "REF ).", "The heating of the atmosphere instigates the burning of hydrogen and/or helium upon fallback (Fig.", "REF , REF ), leading to a thermonuclear runaway.", "This adds to the precursor fluence $E_{\\mathrm {precursor}}$ (Table REF ).", "In model H $26\\,\\%$ of the $7.4\\times 10^{7}\\,\\mathrm {g}$ column of solar composition is burned, and in model He $67\\,\\%$ of the $9.0\\times 10^{7}\\,\\mathrm {g}$ helium column.", "Because of the difference in energy yield of the hydrogen and helium burning nuclear reactions, the total energy released and, hence, the fluence is very close for these particular compositions.", "The luminosity reaches a plateau caused by photospheric radius expansion (PRE, Fig.", "REF ).", "We compute the duration of the PRE phase as the time from the precursor onset to the time when the luminosity drops below $90\\%$ of the plateau value (Table REF ).", "After the PRE phase, the luminosity drops quickly, reaching a minimum at $t_{\\mathrm {minimum}}$ (Table REF ), and subsequently climbing to the superburst peak at approximately $400\\,\\mathrm {s}$ .", "The precursor light curve contains some irregularities.", "In the figures we have filtered out strong variations occurring from one time step to the next, that are due to the outer few grid points, and not representative of the overall behavior of the model." ], [ "Bursts in a Helium-rich Atmosphere", "First we discuss the effect of the superburst on the thermonuclear processes in model He.", "After the superburst precursor, helium burning in the atmosphere proceeds in a stable manner.", "Bursts only reappear once the envelope has cooled down sufficiently after $11.3\\,\\mathrm {days}$ (Fig.", "REF ).", "Before bursts resume, the burning is marginally stable when the temperature in the atmosphere drops to $3\\times 10^{8}\\,\\mathrm {K}$ , leading to oscillations in the light curve with a period close to $20\\,\\mathrm {minutes}$ (Fig.", "REF ).", "The oscillations have a small amplitude of at most approximately $10\\,\\%$ of the average luminosity, and numerical noise dampens the oscillations during certain periods.", "Figure: Peak luminosity L peak L_{\\mathrm {peak}} and recurrencetime t recur t_{\\mathrm {recur}} of bursts after the quenching period formodel He.", "The dotted lines indicate the mean values for bursts beforethe superburst and the luminosity of the accretion process.Once the bursts start, the burst peak flux increases, and the third burst already has a peak luminosity comparable to the bursts before the superburst (Fig.", "REF , REF ).", "During $2.3$  days we find both bright bursts with a peak luminosity of $L_{\\mathrm {peak}}\\simeq 4\\times 10^{38}\\,\\mathrm {erg\\, s^{-1}}$ (the Eddington luminosity for a hydrogen-deficient atmosphere) and weaker bursts with $L_{\\mathrm {peak}}\\simeq 3\\times 10^{37}\\,\\mathrm {erg\\, s^{-1}}$ .", "The bright as well as the weak bursts have a relatively slow rise of the light curve.", "During this time $3\\alpha $ is the predominant nuclear process, producing $^{12}$ C. In the bright bursts $3\\alpha $ raises the temperature sufficiently for $\\alpha $ -captures to take over, causing a faster rise of the luminosity, and producing mostly $^{28}$ Si.", "The bright bursts heat the envelope sufficiently for additional stable helium burning to take place, followed by a weak burst.", "Stable helium burning reduces the helium content of the atmosphere, leading to a burst with a lower $L_{\\mathrm {peak}}$ .", "The weak burst does not heat the envelope enough for additional stable burning, such that the next burst is again bright, and ignites after a shorter recurrence time.", "While the envelope continues to cool from the superburst, the number of bright bursts in between weak bursts increases, until after $2.3$  days the weak bursts disappear.", "Figure: Comparison of light curves of helium burstsaligned on the peak.", "The smallest burst occurred 34 minutes afterbursts resumed, and the one with the bump before the peak occurredafter 2.3 days.", "The burst with the longest decay occurred beforethe superburst.", "Superburst emission has been subtracted for all bursts.Comparing the profiles of individual bursts after the quenching period, the weak bursts have a relatively long rise of $\\sim 5\\,\\mathrm {s}$ (Fig.", "REF ).", "The early bright bursts share this slow rise to a similar luminosity as the peak of the weak bursts, but then transition in a fast sub-second rise to the Eddington luminosity.", "Later bright bursts show an initial “bump” that decreases in duration over time.", "The bursts from before the superburst show a slow rise component with a duration of $0.4\\,\\mathrm {s}$ .", "The decay of the bursts becomes longer as the recurrence time increases, reflecting the longer thermal time scales of increasing ignition depths.", "The flux from the accretion process at a rate of $\\dot{M}=5.25\\times 10^{-9}\\,\\mathrm {M_{\\odot }}\\,\\mathrm {yr^{-1}}$ for a neutron star of $1.4\\,\\mathrm {M_{\\odot }}$ with a $10\\,\\mathrm {km}$ radius is $4.35\\times 10^{37}\\,\\mathrm {erg\\, s^{-1}}$ (assuming isotropic emission and a $100\\,\\%$ efficient accretion process).", "The initial oscillations and weak bursts have a lower $L_{\\mathrm {peak}}$ , but the brighter bursts outshine the accretion flux by approximately a factor 10.", "The superburst burns most carbon out to $y\\simeq 7\\times 10^{8}\\,\\mathrm {g\\, cm^{-2}}$ .", "Between this depth and the ignition depth of the helium bursts at $y\\simeq 1\\times 10^{8}\\,\\mathrm {g\\, cm^{-2}}$ , a carbon mass fraction of $0.1$ survives the runaway burning, but burns on a longer time scale.", "Carbon burns through $^{12}\\mathrm {C}(^{12}\\mathrm {C},\\alpha )^{20}\\mathrm {Ne}$ and subsequent $\\alpha $ -capture reactions, producing predominantly $^{28}$ Si.", "When we stop the simulation, the carbon mass fraction of this material varies from $2\\times 10^{-3}$ down to $2\\times 10^{-4}$ .", "Directly after the superburst, during the burst quenching period, helium burns stably to carbon by the $3\\alpha $ -process.", "Captures of $\\alpha $ on $^{12}$ C and $^{14}$ N from the accretion composition produce a limited fraction of $^{16}$ O, $^{18}$ F, and heavier isotopes up to magnesium (Fig.", "REF ).", "Over time, as the atmosphere cools down, the $\\alpha $ -capture rates reduce, and only lighter elements up to neon are produced.", "The main product of nuclear burning during the quenching period, however, is $^{12}$ C, with a mass fraction of $95\\,\\%$ .", "After the quenching period, this material is compressed to higher densities, and $^{12}\\mathrm {C}(^{12}\\mathrm {C},\\alpha )^{20}\\mathrm {Ne}$ reduces the carbon mass fraction.", "Subsequent $\\alpha $ -capture reactions produce mainly $^{28}$ Si, $^{32}$ S, and $^{36}$ Ar.", "The next superburst ignition occurs close to the bottom of this layer, and depends on the remaining carbon fraction.", "When we end the simulation, the $^{12}$ C fraction is $23\\,\\%$ , and still dropping.", "Figure: Composition of the envelope at the end of theburst quenching period, down to the bottom of the layer of ashes fromstable helium burning.", "Only the most abundant isotopes are shown.During the quenching period, most burning takes place close to $y\\simeq 10^{8}\\,\\mathrm {g\\, cm^{-2}}$ , but not all helium is burned.", "Some survives down to a depth of $y\\simeq 10^{10}\\,\\mathrm {g\\, cm^{-2}}$ .", "The low mass fraction ($Y\\lesssim 10^{-3}$ ) makes $3\\alpha $ inefficient, but series of $\\alpha $ -captures ($^{12}\\mathrm {C}(\\alpha ,\\gamma ){}^{16}\\mathrm {O}(\\alpha ,\\gamma ){}^{20}\\mathrm {Ne}(\\alpha ,\\gamma )^{24}\\mathrm {Mg}$ ) still occur.", "After stable burning at $y\\simeq 10^{8}\\,\\mathrm {g\\, cm^{-2}}$ ends, the $\\alpha $ -captures in the deeper layers continue until helium is fully depleted.", "During this time, even though the helium mass fraction is low, because of the large total mass down to $y\\simeq 10^{10}\\,\\mathrm {g\\, cm^{-2}}$ , the captures contribute a substantial fraction of the total generated energy at any given time.", "This effectively slows down the cooling for up to $8.1\\,\\mathrm {days}$ , keeping the recurrence time of the bursts constant.", "Once helium is depleted at these depths, the cooling continues, and the burst recurrence time increases over time.", "$46.3$  days after burst resumption the recurrence time is 88 minutes, whereas the pre-superburst recurrence time was 100 minutes.", "We stop the simulation here, but extrapolating the trend, the original recurrence time will be recovered approximately 115 days after burst resumption.", "The model produces bursts with a spread of $\\sim 50\\,\\%$ in $L_{\\mathrm {peak}}$ (Fig.", "REF ) with a quasi-periodic behavior.", "To a lesser extend these variations can also be seen in $t_{\\mathrm {recur}}$ .", "This is caused by the dependence of the ignition conditions of a burst on the previous bursts, based on, e.g., the fraction of helium that was burned, the heat deposition, and compositional inertia." ], [ "Bursts in a Solar-composition Atmosphere", "Burst quenching is also present in model H (Fig.", "REF , REF ).", "Directly following the superburst, hydrogen and helium burning is stable.", "Helium burns by the $3\\alpha $ -process, providing CNO elements which facilitate hydrogen burning by the hot-CNO cycle.", "Break-out from this cycle by $^{15}\\mathrm {O}(\\alpha ,\\gamma )^{19}\\mathrm {Ne}$ allows for a rapid series of proton captures and $\\beta $ -decays (rp-process) to produce mostly $^{66}\\mathrm {Ga}$ and $^{66}\\mathrm {Ge}$ , whereas a large number of isotopes with mass numbers in the range of 59–72 contribute mass fractions of several percents (Fig.", "REF ).", "Similarly to model He, after the superburst a layer with a $^{12}$ C mass fraction of $10\\,\\%$ remains between $y\\simeq 1\\times 10^{8}\\,\\mathrm {g\\, cm^{-2}}$ and $y\\simeq 7\\times 10^{8}\\,\\mathrm {g\\, cm^{-2}}$ .", "It burns on a longer time scale down to a mass fraction of $2\\times 10^{-4}$ .", "Hydrogen and helium burning during the burst quenching period produce carbon at a mass fraction of $5\\,\\%$ .", "Nuclear burning on longer time scales reduces the carbon mass fraction in this layer to $10^{-3}$ .", "Burst quenching is much shorter than in the model with a helium-rich atmosphere: bursts resume after $1.1\\,\\mathrm {days}$ .", "As the superburst cools, the temperature drops below $4.3\\times 10^{8}\\,\\mathrm {K}$ , and the breakout reaction $^{15}\\mathrm {O}(\\alpha ,\\gamma )^{19}\\mathrm {Ne}$ becomes less efficient than the $3\\alpha $ -process.", "Whereas at higher temperatures the break-out reactions quickly removed CNO elements, now the CNO mass fraction is growing.", "This causes an increase in the helium production through the CNO cycle, and results in the increase of the energy generation rate of both the CNO cycle and the $3\\alpha $ -process.", "The involved reactions raise the atmosphere temperature until once more the $^{15}\\mathrm {O}(\\alpha ,\\gamma )^{19}\\mathrm {Ne}$ break-out can efficiently remove most of the CNO elements, providing seed nuclei for the rp-process, which captures most of the hydrogen.", "With hydrogen and the CNO isotopes gone, the CNO cycle and rp-process switch off, reducing the helium production, and thereby the $3\\alpha $ rate, allowing the atmosphere to cool down.", "Note that while hydrogen is depleted, helium is not.", "As fresh hydrogen and helium are accreted and mixed in from layers closer to the surface, the cycle repeats itself.", "This produces a series of oscillations in the light curve, that announces the end of the steady-state burning of the burst quenching phase (Fig.", "REF ).", "The time scale for the oscillations is $5.0\\,\\mathrm {min}$ .", "The oscillations increase in amplitude over time.", "After $2.1\\,\\mathrm {hr}$ the $^{15}\\mathrm {O}(\\alpha ,\\gamma )^{19}\\mathrm {Ne}$ breakout and subsequent rp-process increase the temperature during the oscillations sufficiently for the $^{14}\\mathrm {O}(\\alpha ,p)^{17}\\mathrm {F}(p,\\gamma )^{18}\\mathrm {Ne}$ breakout reactions initiate the $\\alpha $ p-process, which causes a faster rise of the luminosity, producing small bursts instead of oscillations.", "For $\\sim 1.5\\,\\mathrm {hr}$ oscillations and small bursts alternate, until an equilibrium is reached and only small bursts occur (Fig.", "REF ).", "The bursts following an oscillation have a shorter recurrence time.", "Initially the bursts have a slow rise, similar to the oscillations, and a faster decay.", "Over time, the rise shortens, and the decay lengthens (Fig.", "REF ).", "Figure: Comparison of light curves of hydrogen/heliumbursts aligned on the peak.", "From small peak to high, the bursts occurred1.5 hr, 4.6 hr, 2.7 days, and 13.1 days after bursting resumed,respectively.", "The brightest burst occurred before the superburst.Superburst luminosity has been subtracted for all bursts.When the oscillations have disappeared, and only bursts remain, the recurrence time is constant at $8.2\\,\\mathrm {minutes}$ until approximately $1.2\\,\\mathrm {days}$ after burst resumption.", "As in model He, this is due to the burning of residual helium at greater depths, which delays the cooling of the atmosphere.", "Although the recurrence time is constant, the peak luminosity $L_{\\mathrm {peak}}$ increases with time.", "Afterward both the recurrence time and $L_{\\mathrm {peak}}$ increase with time (Fig.", "REF ).", "After 13 days $t_{\\mathrm {recur}}=39\\,\\mathrm {minutes}$ is reached.", "The burst peak luminosity is then at $77\\,\\%$ of the pre-superburst level.", "The burst light curve approaches the profiles of the bursts before the superburst.", "At this time we end the simulation.", "Extrapolating, the recurrence time of $52\\,\\mathrm {minutes}$ of the bursts before the superburst will be reached approximately 35 days after burst resumption.", "Figure: Peak luminosity L peak L_{\\mathrm {peak}} and recurrencetime t recur t_{\\mathrm {recur}} of bursts after the quenching period formodel H. The dotted lines indicate the mean values for bursts beforethe superburst and the luminosity of the accretion process.For $1\\times 10^{5}\\lesssim t\\lesssim 4\\times 10^{5}\\,\\mathrm {s}$ there are some `oscillatory' variations in both $t_{\\mathrm {recur}}$ and $L_{\\mathrm {peak}}$ , which are likely of similar origin as the late-time variations in the burst properties in the helium-rich atmosphere (Fig.", "REF )." ], [ "Observational Limits on Burst Quenching", "The first normal burst observed from a source after a superburst provides an upper limit to the burst quenching time.", "For all known (candidate) superbursts, we identify the first detected burst either from MINBAR, or from the literature (Table REF ).", "The superbursts from 4U 1735–444 and 4U 1820–303 occurred when (or close to when) the source exhibited a persistent flux where no or very few bursts have been detected for the respective sources.", "Most probably hydrogen and helium were undergoing steady-state burning already before the superburst started.", "For this reason we exclude these superbursts when we investigate constraints on burst quenching.", "Furthermore, we do not consider the recent superbursts from EXO 1745–248, SAX J1747.0–2853, and SAX J1828.5–1037, which are still being analyzed at the moment of writing, nor those from GX 17+2, which has an atypically high mass accretion rate ([26]).", "Table: Observational Limits on BurstQuenching for All Known (Candidate) SuperburstsTwo superbursters are so-called ultracompact X-ray binaries (UCXBs; e.g., [27]), where the accreted material is thought to be hydrogen deficient, but may contain helium: 4U 0614+091 ([40]) and 4U 1820–303 ([53], [25]; see also [10]).", "From 4U 0614+091 two bursts were observed in the period after the superburst: after $18.6$ and $34.9$ days, respectively ([40]).", "Both bursts reach a bolometric peak flux that is consistent within $1\\sigma $ .", "The first burst is more symmetric with a rise of $5\\,\\mathrm {s}$ and a decay of $2.0\\,\\mathrm {s}$ , whereas the second one has a faster rise of $1\\,\\mathrm {s}$ and a slower decay of $13.0\\,\\mathrm {s}$ .", "It should be noted, however, that the rise and decay time scales were derived from different energy bands, with the first burst being observed at higher energies, causing part of the shorter decay time.", "The other superbursting sources are thought to accrete hydrogen-rich material.", "As the model with the hydrogen-rich envelope predicts normal bursting behavior to return in $35\\,\\mathrm {days}$ , we identify the bursts from MINBAR observed in that period, as well as the net exposure time, $t_{\\mathrm {exp}}$ , of all observations, including those without bursts (Fig.", "REF , Table REF ).", "Figure: Cumulative net exposure time during 35 daysafter a superburst for hydrogen-accreting sources.", "We exclude sourceswith less than 3 hr 3\\,\\mathrm {hr} of exposure time, and those for whichhydrogen/helium burning is stable, as well as GX 17+2 (see also Table ).For each superburst we indicate the source name and the time of thesuperburst (MJD).", "Circles indicate the detection of normal bursts.The earliest observed normal burst occurred $15.0$  days after the superburst from 4U 1636–536 on MJD $51324.21$ .", "After most superbursts the total exposure time is short, and observations are infrequent.", "Therefore, we combine the data for these sources in $1\\,\\mathrm {day}$ time bins to be able to distinguish changes in the burst behavior on similar time scales as exhibited by our models.", "To place constraints on the burst recurrence time, $t_{\\mathrm {recur}}$ , we use the combined $t_{\\mathrm {exp}}$ and number of bursts during each day, taking into account the presence of data gaps by Earth occultations and by passages through the South Atlantic Anomaly.", "Data gaps reduce the fraction of the observation time $t_{\\mathrm {obs}}$ when bursts can be observed: $t_{\\mathrm {exp}}=\\eta t_{\\mathrm {obs}}$ , with $\\eta $ the observation efficiency.", "For $\\eta <1$ bursts can be missed, and when $N$ bursts occur, $n\\le N$ are detected.", "Using a binomial distribution, the probability $P$ of detecting $n$ bursts out of $N$ is $P(n;N,\\eta )=\\binom{N}{n}\\eta ^{n}(1-\\eta )^{N-n}.$ Since $n$ and $\\eta $ are known for each observation, we use $P$ to identify those values of $N$ for which the probability of detecting $n$ is largest.", "After normalization, we determine the 90% confidence region for $N$ , which we use to constrain $t_{\\mathrm {recur}}=t_{\\mathrm {obs}}/N$ .", "As additional constraint, if no bursts are observed, we require that $t_{\\mathrm {recur}}$ is longer than the uninterrupted part of a pointing.", "For a $96\\,\\mathrm {minute}$ satellite orbit we estimate this as $(96\\,\\mathrm {minutes})\\times \\eta $ .", "Furthermore, in our superburst selection only for the superburst on $51324.21$ from 4U 1636–536 more than one burst was observed in the following month.", "The closest pair was separated by $17.8\\,\\mathrm {hr}$ .", "We use this as upper limit for $t_{\\mathrm {recur}}$ at 24 days after the superburst onset.", "The combined constraints are presented in Fig.", "REF .", "Figure: Observational limits on the burst recurrencetime t recur t_{\\mathrm {recur}} from combined observations after the superburstsfrom Fig.", "in 1 day 1\\,\\mathrm {day} time bins.", "Theregion between the gray areas is the 90% confidence region for t recur t_{\\mathrm {recur}}.The solid line is the net exposure time t exp t_{\\mathrm {exp}} per day,and the hatched region indicates the range of minimum values of t recur t_{\\mathrm {recur}}observed for bursts at similar persistent flux as at the time of thesuperbursts.To determine the expected $t_{\\mathrm {recur}}$ before the superbursts, we search MINBAR for the shortest time interval between two subsequent bursts that occurred at a similar level of persistent flux as the superburst.", "For six superbursts we find time intervals between $1.4\\,\\mathrm {hr}$ and $3.6\\,\\mathrm {hr}$ , whereas for others we find time intervals of $8.0\\,\\mathrm {hr}$ and longer.", "The longer $t_{\\mathrm {recur}}$ are most likely because of a short total exposure time on a given source at the level of persistent flux of interest, and not representative of the actual $t_{\\mathrm {recur}}$ .", "Indeed, the shorter time intervals are found for most of the superbursts in Fig.", "REF , which have the longest cumulative exposure times.", "Therefore, we take $1.4$ –$3.6\\,\\mathrm {hr}$ to be the range of expected values for $t_{\\mathrm {recur}}$ .", "Comparing the expected pre-superburst $t_{\\mathrm {recur}}$ to the limits on $t_{\\mathrm {recur}}$ (Fig.", "REF ), we see that $t_{\\mathrm {recur}}\\le 5.5\\,\\mathrm {hr}$ is strongly disfavored during the first 4 days, with $t_{\\mathrm {recur}}$ likely exceeding $12.3\\,\\mathrm {hr}$ during the first day.", "Therefore, these superbursts caused a substantial increase in the burst recurrence time, or possibly quenched bursts altogether.", "Furthermore, for most $1\\,\\mathrm {day}$ time bins the lower limit for $t_{\\mathrm {recur}}$ is at least $0.5$ –$1.0\\,\\mathrm {hr}$ , which disfavors bursts with shorter $t_{\\mathrm {recur}}$ .", "The strongest constraints are due to a burst from 4U 1636–536 on day 26: $0.5\\,\\mathrm {hr}\\le t_{\\mathrm {recur}}\\le 1.5\\,\\mathrm {hr}$ .", "This suggests $t_{\\mathrm {recur}}$ is back at pre-superburst values after 26 days.", "Only 5 days earlier, the lower limit was $t_{\\mathrm {recur}}\\ge 4.2\\,\\mathrm {hr}$ , and on day 34 $t_{\\mathrm {recur}}\\ge 3.2\\,\\mathrm {hr}$ : both at the upper part of the pre-superburst range.", "Therefore, the allowed values for $t_{\\mathrm {recur}}$ within the pre-superburst range vary somewhat with time, which may be due to differences in the persistent flux from one superburst to the next as well as variations during the month after a given superburst.", "The majority of the bursts indicated in Fig.", "REF followed the superburst on MJD 51324.21 from 4U 1636–536.", "All those bursts have similar properties, and exhibit PRE, indicating that the Eddington limit was reached.", "The persistent flux was approximately $6\\,\\%$ of the burst peak flux.", "The burst properties are typical for bursts from this source at that level of persistent flux.", "Burst properties not changing between days 15 and 26 suggest that normal bursting behavior was resumed at most two weeks after the superburst." ], [ "Discussion", "We present one-dimensional multi-zone simulations of the neutron star envelope, where we study the effect of a superburst on a helium-rich and on a solar-composition atmosphere, where matter is accreted at a rate of $\\dot{M}=5.25\\times 10^{-9}\\,\\mathrm {M_{\\odot }}\\mathrm {yr^{-1}}$ .", "The simulations are similar to those presented by [34] where a carbon-rich layer is build up by accretion until a superburst ignites.", "In the present work we replace the accretion composition shortly before the superburst to build an atmosphere of either a helium-rich or a solar composition.", "The results are summarized in Table REF .", "We find that the hydrogen/helium-rich atmosphere changes the shock breakout and precursor peak height and duration, whereas the superburst itself does not differ substantially from models with a carbon-rich atmosphere.", "After the superburst all burning in the atmosphere is stable, and bursts are quenched until the envelope has cooled down from the superburst.", "At that point burning first becomes marginally stable, followed by weak bursts with short recurrence times.", "Over time, recurrence times lengthen, and the burst properties return to those from before the superburst." ], [ "Superburst", "[34] found for a model with the same accretion rate and a slightly higher $Q_{\\mathrm {b}}=0.13\\,\\mathrm {MeV\\, nucleon^{-1}}$ a recurrence time of $1.70$  years, $33\\,\\%$ longer than the recurrence time found in the present study.", "With $Q_{\\mathrm {b}}=0.1\\,\\mathrm {MeV\\, nucleon^{-1}}$ one would expect our cooler model to produce a somewhat longer recurrence time.", "For the models in the current paper, we use a different implementation of accretion and, most importantly, of the compressional heating from the accreted material.", "A further difference is the prescription used for electron conductivity.", "The different phases displayed by the superburst are the same as described by [34], including the shock breakout, the precursor, and the two-component power-law decay.", "A difference lies in the height of the shock breakout peak and the oscillations at the start of the precursor, where super-Eddington luminosities are reached.", "The peak luminosity has a strong dependence on the resolution at the surface of the model ([34]).", "The bottom of the outer zone of our models is at a column depth of $y\\simeq 10^{6}\\,\\mathrm {g\\, cm^{-2}}$ , whereas the photosphere of a neutron star is typically located at $y\\simeq 1\\,\\mathrm {g\\, cm^{-2}}$ , so it is likely that our simulations have not resolved this fully.", "A previous study also finds the shock breakout to be super-Eddington, but likewise does not fully resolve the photosphere ([58]).", "This underlines the importance of the treatment of the outer layers during these hydrodynamic events.", "Whereas our non-relativistic simulations employ diffusive radiation transport, a relativistic hydrodynamic model that includes full radiation transport will be much better suited to simulate these processes in detail.", "Such a model will also be able to determine whether any material is lost from the neutron star during the super-Eddington luminosity phases." ], [ "Precursor", "After the initial oscillations, the precursor luminosity settles at the Eddington limit, $L_{\\mathrm {Edd}}$ , producing a plateau in the light curve.", "The height of the plateau is different for the three atmosphere compositions because of the dependence of $L_{\\mathrm {Edd}}$ on the opacity, $\\kappa $ : $L_{\\mathrm {Edd}}\\propto \\kappa ^{-1}$ .", "The opacity in the neutron star photosphere is typically assumed to be dominated by Thomson scattering, which depends exclusively on the hydrogen mass-fraction.", "Both models He and C are devoid of hydrogen, but they reach the Eddington limit at slightly different luminosities.", "The opacity in the outer zones of the models continue to increase toward the surface.", "It is likely that the true photospheric values are not yet reached, which may explain the variations in $L_{\\mathrm {Edd}}$ .", "The precursor in model C is predominantly powered by the fallback of shock-heated material.", "The fluence of this precursor is about half that of the precursors in the models He and H, indicating that even in those cases fallback has a substantial contribution to the precursor, with hydrogen and helium burning accounting for the rest of the fluence.", "The superburst ignited at a relatively early phase in the cycle of hydrogen/helium bursts.", "If it had ignited at a later phase, the respective helium-rich and solar composition columns would have been several times larger, allowing for somewhat longer precursor bursts.", "Note that the brightness of the precursors would not change, as it is set by the Eddington limit for the atmosphere composition.", "Precursor bursts that reach the Eddington limit with durations of several seconds have been observed in the few instances when the start of the superburst was observed and the data were of sufficient quality (e.g., [54], [53]).", "Previous models of a superburst below a helium atmosphere skipped the fallback, losing an important energy source for the precursor ([58]).", "The shock breakout left a flat temperature profile, delaying the burst from helium burning by a thermal time scale of several seconds.", "The precursor of model He arrives on a dynamical time scale of $\\sim 10^{-5}\\,\\mathrm {s}$ , and is powered by both the fallback and the subsequent thermonuclear burning of helium.", "Furthermore, [58] argue that the thermonuclear runaway of helium is triggered either by the shock if the helium layer is sufficiently thick, or by the carbon deflagration reaching the helium layer.", "In our simulations the carbon deflagration does not reach that far out, and the shock does not trigger the helium burst immediately.", "It is rather the fallback that heats the atmosphere sufficiently to burn helium, even if the helium layer is thin (Fig.", "REF )." ], [ "Burst Quenching", "The superburst heats the atmosphere such that all subsequently accreted hydrogen and helium burn stably, and no bursts are produced.", "Burst quenching continues until the envelope cools down sufficiently for burning to become unstable again.", "Using the analytic estimate from [12], one predicts for the conditions of our model a quenching time of $5.5\\,\\mathrm {days}$ for model H and $2.2\\,\\mathrm {days}$ for model He.", "Here it is assumed that $3\\alpha $ is solely responsible for the stability of thermonuclear burning.", "Model H, however, yields the shortest quenching time of $1.1\\,\\mathrm {days}$ , whereas in model He bursts are quenched for $11.3\\,\\mathrm {days}$ .", "The latter is over five times longer than the analytic estimate, which may indicate that the approximations employed in the estimate are too strong.", "In contrast, we find a five times shorter quenching time for model H. In this case $3\\alpha $ is not the sole process responsible for the resumption of bursts, but the CNO breakout reaction $^{15}\\mathrm {O}(\\alpha ,\\gamma )^{19}\\mathrm {Ne}$ plays a key role.", "The importance of the $^{15}\\mathrm {O}(\\alpha ,\\gamma )^{19}\\mathrm {Ne}$ reaction for regulating the CNO abundance in the context of X-ray bursts was stressed before by [16], and indicated as an important factor in the stability of the burning processes ([18], [17]).", "Our study shows its importance in determining the duration of the quenching period.", "Mixing due to rotation, or a rotationally induced magnetic field, can also influence the stability of thermonuclear burning ([47], [37]).", "Rotation was not taken into account in this study, but could lead to a somewhat longer quenching period.", "For the ignition column depth of our simulations, $y=1.1\\times 10^{12}\\,\\mathrm {g\\, cm^{-2}}$ , [34] showed that the fraction of the superburst energy that is lost in neutrinos is small, whereas for ignition at larger depths neutrino losses are substantial, and the photon fluence reaches a maximum value.", "This maximum fluence implies that there is a maximum burst quenching time, which is longer than the values found in our simulations." ], [ "Marginally Stable Burning", "When burst quenching ends, burning becomes marginally stable, producing oscillations in the light curve.", "This has been observed from a small number of sources as mHz quasi-periodic oscillations (mHz QPOs; [49], [1], [42]), and is associated with a burning mode at the transition of stable and unstable burning ([23]).", "Using the one-zone analytic approximation from Equation (11) in [23], and substituting the values appropriate for model He ($E_{\\star }=594\\,\\mathrm {keV\\, nucleon^{-1}}$ , $T_{8}=0.54$ , $y_{8}=1.0$ , and $\\dot{m}=0.3\\,\\dot{m}_{\\mathrm {Edd}}$ ), one obtains the oscillation period $P_{\\mathrm {osc}}=25\\,\\mathrm {minutes}$ , which is close to the $20\\,\\mathrm {minute}$ period in our simulations.", "Marginally stable burning in model H has a period of $5\\,\\mathrm {minutes}$ , which is close to $P_{\\mathrm {osc}}=6\\,\\mathrm {minutes}$ predicted by the analytic approximation with $T_{8}=4.2$ , $y_{8}=0.60$ , and $\\dot{m}=0.3\\,\\dot{m}_{\\mathrm {Edd}}$ .", "The analytic estimate from [23] considers only accretion, radiative cooling, and a single thermonuclear process, the rate of which has a similar temperature dependence as the cooling rate.", "In the case of model He, where $3\\alpha $ is the single dominant thermonuclear process, this approximation seems valid.", "It is interesting that it also provides a good estimate of $P_{\\mathrm {osc}}$ in model H, where the nuclear reactions proceed through a complex interplay between the $3\\alpha $ , hot-CNO, and CNO breakout processes." ], [ "Burst Resumption", "As the envelope continues to cool, the oscillatory burning transitions into bursts.", "At first their fluence is relatively low and the recurrences times are short.", "Over time (almost four months for model He; one month for H) the burst properties regain their pre-superburst values.", "The light curves of the later bursts, as well as of the pre-superburst bursts, are qualitatively similar to those reported from recent multi-zone simulations (e.g., [32], [60], [17], [31]), which provide very good agreement with observed bursts ([22], [28], [14]).", "The transition from stable burning, to bursts, via marginally stable burning (mHz QPOs), and weak bursts, has not been observed directly after a superburst.", "In this respect the bursting behavior of IGR J17480-2446 in the globular cluster Terzan 5 is interesting ([46], [44], [4], [42]).", "During an outburst in late 2010, its accretion rate varied by more than a factor five, and it displayed a continuous transition from bright bursts to weak bursts, to mHz QPOs, and back.", "The changes in the burst properties are quite similar to those in model H. The weak bursts have a longer rise and recurrence times as short as a few minutes.", "The bursts become brighter with longer recurrence times and longer decay profiles.", "The decrease of the accretion rate causes the atmosphere to cool, similar to the cooling after the superburst in the models, but it also changes the burst fuel accumulation time.", "This is more complex than the situation in the simulations, and we cannot compare the time scale on which the changes in burst behavior take place, as it is set by the variations in the mass accretion rate, instead of the cooling time scale of the envelope." ], [ "Carbon Production and Destruction", "One of the biggest challenges in superburst theory is the creation of the correct amount of carbon.", "Fits to superburst light curves determine the carbon mass fraction to be close to $20\\,\\%$ ([12], [13]), which is the amount we adopted for our models.", "Models of helium bursts, or mixed hydrogen/helium bursts, however, produce less than half of that (e.g., [60]).", "It is speculated that both bursts and stable burning of hydrogen and helium are required to produce enough carbon for superbursts.", "In fact, most superbursting sources exhibit this combination of burning behavior ([29]).", "Our models exhibit both burning regimes, and the next superburst will ignite at the bottom of the layer that is accreted just after the simulated superburst.", "Because of the relatively high temperature in the envelope after the superburst, however, most of the produced carbon burns once it is compressed to higher densities over time.", "Simulations over a longer time are required to determine how much carbon can survive when the envelope cools down further, and the carbon fusion reaction rate is reduced.", "Model He might produce a carbon-rich layer where the next superburst can ignite, whereas in model H most carbon is destroyed.", "[45] recently suggested that a larger carbon fraction can be obtained at the superburst ignition depth by separating out the carbon from heavier elements by freezing at the crust–ocean interface." ], [ "Observations after Superbursts", "The detection of a burst is a certain upper limit to the quenching time.", "Table REF presents the current observational upper limits on burst quenching for all known superbursts, based on the first detection of a burst following each superburst.", "We identify an upper limit of $15.0\\,\\mathrm {days}$ for 4U 1636–536 using the MINBAR burst catalog.", "This is the shortest reported value apart from the $2.2\\,\\mathrm {day}$ limit for GX 17+2, which accretes at a much higher rate than all other superbursters.", "4U 1636–536 accretes hydrogen-rich material at a rate of approximately $0.1\\,\\dot{M}_{\\mathrm {Edd}}$ .", "The $15.0\\,\\mathrm {day}$ upper limit is well above the $1.1\\,\\mathrm {day}$ quenching time we find for model H, which has a three time higher mass accretion rate.", "The shortest upper limit for UCXBs, which may accrete helium-rich material, is $18.6\\,\\mathrm {days}$ for 4U 0614+091 ([40]).", "Model He has a lower quenching time of $t_{\\mathrm {quench}}=11.3\\,\\mathrm {days}$ , but has a 30 times larger mass accretion rate.", "[12] estimate the dependence of $t_{\\mathrm {quench}}$ on the accretion rate to be $t_{\\mathrm {quench}}\\propto \\dot{M}^{-3/4}$ .", "Using this to correct for the differences in accretion rate between the models and the observations, model H's $t_{\\mathrm {quench}}=6.6\\,\\mathrm {days}$ is still below the observational upper limit, but model He's $t_{\\mathrm {quench}}=28.9\\,\\mathrm {days}$ lies above the upper limit.", "Considering, however, that the analytic approximation does not reproduce the results from our multi-zone models well, the accretion rate dependence here employed requires further scrutiny before conclusions can be drawn.", "The two bursts following the superburst from 4U 0614+091 have the same peak flux and suggest a trend towards longer bursts at later times, just as predicted by model He.", "For the hydrogen-accreting sources we combine the observations following superbursts, that are reported in MINBAR.", "There is a strong indication for burst quenching during the first 4 days after superbursts.", "A burst observed after $26\\,\\mathrm {days}$ firmly constrains the $t_{\\mathrm {recur}}$ at a pre-superburst value.", "Furthermore, the lack in variation of the properties of bursts observed from 4U 1636–536 suggests that pre-superburst bursting behavior had returned already after $15\\,\\mathrm {days}$ , which is shorter than the $35\\,\\mathrm {days}$ predicted by model H. [13] derive from fits to the light curve of a different superburst from the same source an ignition column depth that is lower by a factor $0.45$ compared to the ignition depth in our models.", "This implies a shorter cooling time scale after the superburst ($t_{\\mathrm {cool}}\\propto y_{\\mathrm {ign}}^{3/4}$ ; [12]), and hence a faster return to normal bursts of $19.3\\,\\mathrm {days}$ , which is closer to what we infer from the observations.", "For all but a few days, we derive $t_{\\mathrm {recur}}\\ge 1\\,\\mathrm {hr}$ .", "This leaves little room for the short recurrence times predicted by our models and observed from IGR J17480-2446.", "The majority of the observations during the first week after superbursts were, however, performed with the BeppoSAX WFCs, which did not have enough sensitivity to detect oscillatory behavior or weak bursts (see, e.g., [33] for a comparison of the detection of weak bursts with the RXTE PCA and BeppoSAX WFCs).", "Alternatively, the bursting behavior after a superburst may resemble that close to the transition to stable burning observed at higher $\\dot{M}$ , when the burst rate decreases (e.g., [9]).", "Both models H and He predict a period when weak and brighter bursts alternate.", "This behavior has not been observed, which is not surprising considering the short time that it is expected to take place.", "For model H this burning mode lasts a few hours, whereas for model He it continues for $2.3$  days.", "It is, therefore, easiest to observe this behavior from a UCXB.", "Because observations after a superburst are often few and far apart (Fig.", "REF ), we have to combine the data for superbursts which differ by up to a factor 6 in $y_{\\mathrm {ign}}$ ([13]), and have corresponding different quenching and cooling times.", "A campaign of frequent and longer observations during the month following a superburst would, therefore, be very important in improving our current inability to accurately predict changes in bursting behavior.", "Now that RXTE has ceased its operations, there is no observatory available with a sufficiently large effective area and time resolution to study the mHz QPOs and weak bursts, as well as the brief superburst precursors.", "A future mission such as LOFT, that greatly improves on collecting area, will be able to make important steps forward in our understanding of thermonuclear processes on neutron stars ([15])." ], [ "Conclusions", "We create one-dimensional multi-zone models of a neutron star envelope undergoing a superburst (carbon flash), in the presence of an atmosphere of either pure helium or of solar composition.", "The latter is the first model of its kind.", "After the superburst we continue the simulations to study burst quenching as well as the return of normal Type I X-ray bursts, simulating over 900 hydrogen or helium flashes per model.", "The heating of the atmosphere by fallback of shocked material generates a precursor burst that reaches the Eddington limit.", "Any available hydrogen or helium ignites, which extends the duration of the precursor.", "After the superburst, the atmosphere is sufficiently hot for bursts to be quenched, and all hydrogen and helium to burn in a stable manner.", "In a pure helium layer mostly carbon is produced, and in a layer of solar composition stable rp-process burning creates mostly germanium and gallium.", "As the envelope cools down from the superburst, bursts reappear in the light curve.", "In the helium atmosphere this happens after $11.3\\,\\mathrm {days}$ , when the $3\\alpha $ -process becomes unstable.", "We find that in a solar composition quenching ends much sooner, after only $1.1\\,\\mathrm {days}$ .", "In this case it is the interplay between $3\\alpha $ , the hot CNO cycle, and the $^{15}\\mathrm {O}(\\alpha ,\\gamma )^{19}\\mathrm {Ne}$ breakout reaction that leads to unstable burning.", "In both models, at the transition from stable burning to unstable burning, oscillations are produced in the light curve due to marginally stable burning ([23]), followed for a brief time by alternating weak and brighter bursts, which have longer and shorter recurrence times, respectively.", "The latter bursting mode has not been observed yet.", "Eventually the weak bursts disappear.", "In the helium atmosphere the bright bursts are immediately as bright as those before the superburst, whereas in the solar atmosphere burst peak luminosities grow as the envelope cools further.", "For a few days left-over helium above the burst ignition column depth burns, pausing the cooling.", "Afterward the burst durations and recurrence times increase back to the pre-superburst values over the course of one month for solar composition, and four months for helium composition.", "The transition from burst quenching to bursts after a superburst has not been directly observed.", "Using the MINBAR catalog we identify the shortest reported upper limit to the quenching time of $15.0\\,\\mathrm {days}$ for 4U 1636–536 (with the exception of GX 17+2), and we derive further constraints on the time scales for quenching and the return of bursts.", "The short recurrence times found by the simulations are disfavored, but not excluded.", "The transition between the different burning regimes that we describe exhibits, however, strong similarities with bursts observed from the transient burster IGR J17480–2446 in Terzan 5 (e.g., [42]).", "The authors thank K. Chen for helpful discussions.", "This paper uses preliminary analysis results from the Multi-INstrument Burst ARchive (MINBAR), which is supported under the Australian Academy of Science's Scientific Visits to Europe program, and the Australian Research Council's Discovery Projects and Future Fellowship funding schemes.", "The authors thank the International Space Science Institute in Bern for hosting an International Team on Type I X-ray bursts.", "L.K.", "is supported by the Joint Institute for Nuclear Astrophysics (JINA, grant PHY08-22648), a National Science Foundation Physics Frontier Center.", "A.H. acknowledges support from the DOE Program for Scientific Discovery through Advanced Computing (SciDAC, DE-FC02-09ER41618) and by the US Department of Energy under grant DE-FG02-87ER40328." ] ]

[ [ "Gas and Metal Distributions within Simulated Disk Galaxies" ], [ "Abstract We highlight two research strands related to our ongoing chemodynamical Galactic Archaeology efforts: (i) the spatio-temporal infall rate of gas onto the disk, drawing analogies with the infall behaviour imposed by classical galactic chemical evolution models of inside-out disk growth; (ii) the radial age gradient predicted by spectrophometric models of disk galaxies.", "In relation to (i), at low-redshift, we find that half of the infall onto the disk is gas associated with the corona, while half can be associated with cooler gas streams; we also find that gas enters the disk preferentially orthogonal to the system, rather than in-plane.", "In relation to (ii), we recover age gradient troughs/inflections consistent with those observed in nature, without recourse to radial migrations." ], [ "Introduction", "The infall of gas onto galaxies is a fundamental constituent of any cosmologically-motivated models of galaxy evolution, whether they be classic galactic chemical evolution models (e.g.", "Lineweaver et al.", "2004; Renda et al.", "2004) or hydrodynamical simulations (e.g.", "Kawata & Gibson 2003; Brook et al.", "2004).", "The shape of the metallicity distribution function of a given ensemble of stars can be a powerful tool to constrain the (otherwise, little known) interplay between infalling and outflowing material (e.g.", "Fenner & Gibson 2003; Pilkington et al.", "2012b).", "In the local Universe, we often associate (rightly or wrongly) this infalling fuel for future star formation with the high-velocity clouds which permeate our halo (e.g.", "Gibson et al.", "2001; Pisano et al.", "2004).", "Classic chemical evolution models, constrained by both the metallicity distribution function and gradients (abundance and surface density) in the disk, suggest “inside-out” growth of the disk is required (e.g.", "Chiappini et al.", "2001; Fenner & Gibson 2003; Mollá & Díaz 2005; Pilkington et al.", "2012a,b).", "By “inside-out”, we mean a scenario in which the timescale for gas infall onto the disk increases as a function of galactocentric radius; whether this timescale is linear (e.g.", "Chiappini et al.", "2001) or a higher-order parametrisation (e.g.", "Mollá & Díaz 2005) is less important than the fact that (a) the overall infall rate is (roughly) exponential, and (b) the timescale increases with radius.", "Fig REF shows one such parametrisation from the chemical evolution model of Renda et al.", "(2004).", "Figure: Gas infall rate onto the disk as a function of time (xx-axis)and radius (ranging from inner to outer disk, going from the upperto the lower curves).", "The infall timescale here (from the modelsof Renda et al.", "2004) grows linearly with galactocentric radius.As alluded to above, energetic outflows (driven by a combination of thermal/kinetic energy from supernovae and massive star winds/radiation) also play a critical role in regulating star formation, infall of both fresh and re-cycled disk material, and setting the chemistry of the resulting system.", "The heating and chemical profiles of the halo are an ideal place to examine the veracity of one's feedback scheme through a comparison of the radial profiles of various neutral and ionised species (e.g.", "Hi, Ovi, Mgii) with those observed in nature.", "We have made significant strides in this area (Stinson et al.", "2012), and future works in this series will examine these radial distributions using a statistical sample of cosmolgical disk simulations in a variety of environments, from field to groups.", "The spatio-temporal infall pattern of gas onto the disk and the predicted age gradients within spectrophotometric disk models are touched upon in the following sub-sections.", "These are each, very much, works in progress, rather than finished products, so we ask the reader to bear that in mind!" ], [ "How Does Gas Get Into Galaxies?", "Making use of Ramses-CH (Few et al.", "2012), a new self-consistent implementation of chemical evolution within the Ramses cosmological adaptive mesh refinement code (Teyssier 2002), we re-simulate the disk described by Sánchez-Blázquez et al.", "(2009) and analyse the temporal and spatial infall rates of hot/coronal and cooler/stream gas onto the disk.", "Our task is a (seemingly) straightforward one: confirm/refute the aforementioned fundamental tenet of chemical evolution, that the gas infall onto simulated disks (in a cosmological context) proceeds in an inside-out fashion.", "In the upper-left panel of Fig 2, we show in black (magenta) the inflowing (outflowing) gas flux through parallel (0.5 kpc thick) slabs situated $\\pm $ 5 kpc from the mid-plane (extending to a galactocentric radius of 25 kpc),The choice of $\\pm $ 5 kpc heights is a compromise between being as close to the disk as possible, without being `swamped' by the galactic fountain/re-circulation signal (Gibson et al.", "2009; §7).", "over the range of time for which the disk could be `readily' identified (see Sánchez-Blázquez et al.", "2009 for details pertaining to the `disk identification').", "In the upper-right panel of Fig 2, we decompose the infalling gas (black curve from the left panel, repeated again here, also in black) into polytropic/hot gas (what we label as `corona') in red, and non-polytropic/cooler gas (what we label as `streams') in blue.", "Within this simulation, (i) the infall from the corona is roughly constant in time, and (ii) at low-redshift, each component accounts for half of the current gas infall.", "In the lower-left panel of Fig 2, we show the infalling (outflowing) gas flux, again in black (magenta), through a cylinder of radius 25 kpc and height $\\pm $ 5 kpc; i.e., the sum of the fluxes shown here, plus those shown in the upper-left panel of Fig 2, correspond to the real/total accretion rate.", "For this simulation, the rate of gas infall/inflow entering the disk through the cylinder (i.e., `in-plane' infall/inflow) is fairly negligible.", "Finally, in the lower-right panel of Fig 2, we show the gas inflow rate through three small `holes' situated $\\pm $ 5 kpc from the mid-plane, at different radii.", "While noisy, due to the small sampling employed here, we can see the emergence of the fundamental tenet of inside-out disk growth: specifically, the lack of gas accretion at small galacto-centric radii at low-redshift (i.e., a trend for flatter infall profiles at larger radii).", "Figure: Upper-left: inflowing (outflowing) gas through slabs±\\pm 5 kpc from the mid-plane in black (magenta); Upper-right: inflowinghot/coronal (cooler/stream) gas through the same ±\\pm 5 kpc slabs inred (blue); Lower-left: inflowing (outflowing) gas through a±\\pm 5 kpc high cylinder of radius 25 kpc in black (magenta);Lower-right: inflowing gas through small `holes' ±\\pm 5 kpcfrom the mid-plane, at three different radii.", "See text for details." ], [ "Age Gradients", "The existence of U-shaped radial age profiles (inferred via radial colour profiles, in consort with stellar population modeling) in disk galaxies (particularly those with so-called Type II surface brightness profiles - i.e., those showing a `break' in the surface brightness) is now well-established (e.g.", "Bakos et al.", "2008; Sánchez-Blázquez et al.", "2011; Roediger et al.", "2012).", "Such troughs in age, near the break radius, were found in the exquisite models of Roskar et al.", "(2008), where the `up-bend' in the age profile in the outer disk was produced by stars that had migrated from the inner parts of the disk; in our cosmological simulation (Sánchez-Blázquez et al.", "2009), a similar trough/inflection in the age profile was found at the break radius, where the presence of a warp in the gas disk resulted in a decrease in the volume density and, hence, a `break' in the star formation density.", "The trough persists, regardless of the presence (or lack thereof) of radial migration (although migration clearly takes place and is critical!).", "Whether U-shaped age profiles are a natural byproduct within classical galactic chemical evolution models is less certain; to that end, we examined the spectrophotometric predictions associated with the same fiducial Milky Way models (Mollá & Díaz 2005; N=28) that were employed in our reecnt work on the temporal evolution of metallicity gradients in L$^\\star $ galaxies (Pilkington et al.", "2012a).", "In Fig 3, we show predicted present-day mass-weighted age gradients for a Milky Way analog, employing a range of star formation efficiencies (from high efficiency to low efficiency, in going from the top to the bottom curves at small galactocentric radii).", "We find that within these classical models, which by construct neglect radial migration, U-shaped age profiles are a natural outcome of the infall/star formation prescriptions.", "It is interesting to note that the position and depth of the trough is sensitive to the adopted star formation efficiency; high efficiencies drive the trough to be (i) positioned at larger galactocentric radii, and (ii) shallower (and vice versa for low star formation efficiencies).", "In the outer parts of the disk, beyond the minima of the age profiles, the high efficiency models show inverted age profiles, while the low efficiency models show declining age profiles.", "A more thorough investigation is clearly warranted.", "Figure: Radial, mass-weighted, age profiles for the fiducial Milky Waymodels of Mollá & Díaz (2005), for a range of star formationefficiencies.", "All models possess U-shaped age profiles, with the positionand depth of the trough being depending upon the adopted star formationefficiency.", "See text for details.Without the help of our collaborators, this work would not have been possible; we thank them all for their ongoing advice and guidance.", "BKG thanks both Monash and Saint Mary's Universities for their generous visitor support, and the organisers of what was an incredibly exciting, rewarding, and collegial School and Workshop.", "BKG, SC, MM and DC acknowledge the support of the UK's Science & Technology Facilities Council (ST/F002432/1 & ST/H00260X/1).", "SC acknowledges support from the the BINGO Project (ANR-08-BLAN-0316-01).", "Bakos, J., Trujillo, I., Pohlen, M. 2008, ApJ, 683, L103 Brook, C.B., Kawata, D., Gibson, B.K., et al.", "2004, 349, 52 Chiappini, C., Matteucci, F., Romano, D. 2001, ApJ, 554, 1044 Crain, R.A., Theuns, T., Dalla Vecchia, C., et al.", "2009, MNRAS, 399, 1773 Fenner, Y., Gibson, B.K.", "2003, PASA, 20, 189 Few, C.G., Courty, S., Gibson, B.K., et al.", "2012, MNRAS, submitted Gibson, B.K., Giroux, M.L., Penton, S.V., et al.", "2001, AJ, 122, 3280 Gibson, B.K., Courty, S., Sánchez-Blázquez, P., et al.", "2009, in The Galaxy Disk in Cosmological Context, J. Andersen et al.", "(ed.)", "Kawata, D., Gibson, B.K.", "2003, MNRAS, 346, 135 Lineweaver, C.H., Fenner, Y., Gibson, B.K.", "2004, Science, 303, 59 Mollá, M., Díaz, A.I.", "2005, MNRAS, 358, 521 Pilkington, K., Gibson, B.K., Calura, F., et al.", "2011, MNRAS, 417, 2891 Pilkington, K., Few, C.G., Gibson, B.K., et al.", "2012a, A&A, 540, A56 Pilkington, K., Gibson, B.K., Calura, F., et al.", "2012b, MNRAS, submitted Pisano, D.J., Barnes, D.G., Gibson, B.K., et al.", "2004, ApJ, 610, L17 Renda, A., Fenner, Y., Gibson, B.K., et al.", "2004, MNRAS, 354, 575 Roediger, J.C., Courteau, S., Sánchez-Blázquez, P. 2012, ApJ, submitted Roskar, R., Debattista, V.P., Quinn, T.R., et al.", "2008, ApJ, 684, L79 Sánchez-Blázquez, P., Courty, S., Gibson, B.K., et al.", "2009, MNRAS, 398, 591 Sánchez-Blázquez, P., Ocvirk, P., Gibson, B.K., et al.", "2011, MNRAS, 415, 709 Stinson, G., Brook, C., Prochaska, J.X., et al.", "2012, MNRAS, submitted Teyssier, R. 2002, A&A, 385, 337" ] ]

[ [ "Thermal modification of bottomonium spectra from QCD sum rules with the\n maximum entropy method" ], [ "Abstract The bottomonium spectral functions at finite temperature are analyzed by employing QCD sum rules with the maximum entropy method.", "This approach enables us to extract the spectral functions without any phenomenological parametrization, and thus to visualize deformation of the spectral functions due to temperature effects estimated from quenched lattice QCD data.", "As a result, it is found that \\Upsilon and \\eta_b survive in hot matter of temperature up to at least 2.3T_c and 2.1T_c, respectively, while \\chi_{b0} and \\chi_{b1} will disappear at T<2.5T_c.", "Furthermore, a detailed analysis of the vector channel shows that the spectral function in the region of the lowest peak at T=0 contains contributions from the excited states, \\Upsilon(2S) and \\Upsilon(3S), as well as the ground states \\Upsilon (1S).", "Our results at finite T are consistent with the picture that the excited states of bottomonia dissociate at lower temperature than that of the ground state.", "Assuming this picture, we find that \\Upsilon(2S) and \\Upsilon(3S) disappear at T=1.5-2.0T_c." ], [ "Introduction", "Properties of high density and temperature matter is one of the most exciting subjects of hadron physics.", "Quantum chromodynamics (QCD) predicts matter composed of quarks and gluons in the form of unconfined plasma phase, called quark gluon plasma (QGP).", "In such a phase of matter, production of quarkonium, a bound state of a heavy quark and a heavy antiquark, is expected to be suppressed.", "Especially, $J/\\psi $ suppression has traditionally been considered as a signal for the formation of QGP [1], [2] in high-energy heavy ion collisions.", "There, the charmonia including $J/\\psi $ may dissociate in the QGP due to temperature effects such as the color Debye screening.", "Such suppressions were observed in the heavy ion collisions at the Relativistic Heavy Ion Collider (RHIC).", "A similar phenomenon may occur in the bottomonium spectrum.", "Recent report has suggested significant modification of the bottomonium spectra from the comparison between the P-P and Pb-Pb collisions at the Large Hadron Collider (LHC).", "The data indicate larger suppression of the excited states than the ground state.", "One purpose of the present paper is to study this phenomenon by using QCD sum rules with MEM.", "It is thus interesting to see whether there are qualitative/quantitative similarities and differences between the behavior of the charmonia and bottomonia spectra at finite temperature.", "In order to quantify and predict when and how the quarkonium spectrum is modified, various theoretical approaches have been developed.", "They include Lattice QCD [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], QCD sum rules [14], [15], [16], [17], [18], AdS/QCD [19], [20], [21], [22], resummed perturbation theory [23], [24], and effective field theories [25], [26] as well as potential models [11], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36].", "Most of the previous studies were devoted to the charmonium spectra, while, for bottomonia, there are only a few theoretical results [12], [17], [27], [29], [33], [34], [35], [37], and it is important that they are examined by independent methods.", "In a previous work [18], we have investigated the behavior of charmonium spectral functions at finite temperature from QCD sum rules with the maximum entropy method (MEM).", "Our approach enables us to extract directly the shape of the spectral functions.", "Therefore, this is a suitable tool to study the deformation of the spectral functions upon the change of the temperature.", "In this paper, we will study the bottomonium spectrum by using the same method.", "We point out that the bottomonium spectral function is crucially different from that of charmonium in the sense that there are several excited states below the continuum threshold.", "For example, $\\Upsilon $ has the ground state ($1S$ ) and the excited states ($2S$ , $3S$ ).", "This paper is organized as follows: In Section , we discuss QCD sum rules for heavy quarkonia at finite temperature and demonstrate our method to extract spectral functions with MEM.", "In Section , we show the obtained spectral functions at zero and finite temperature.", "Here, we also investigate the behavior of the excited states of the bottomonium vector channel.", "Section is devoted to the summary and conclusion." ], [ "Analysis procedure", "This section overviews our analysis method of QCD sum rules with MEM.", "Historically, QCD sum rules for heavy quarkonia were introduced in [38], [39] and then elaborated in [40].", "Their Borel transformation were calculated in [41].", "Applying MEM to QCD sum rules was successfully performed for the $\\rho $ meson in [42] and the nucleon in [43], [44].", "We follow the same procedure as [18] for the QCD sum rules of heavy quarkonia at finite temperature." ], [ "QCD sum rules for bottomonia", "Let us start with the current correlation function $\\Pi ^{\\, J}(q) = i \\int d^4x e^{i q \\cdot x} \\langle T[j^{\\, J}(x) j^{\\, J \\dag }(0)] \\rangle , $ where $J$ stands for the pseudoscalar $(P)$ , vector$(V)$ , scalar $(S)$ , and axial-vector $(A)$ channel.", "Each current is defined as $j^{\\, P}=\\bar{b} \\gamma _5 b$ , $ j^{\\, V}_\\mu = \\bar{b} \\gamma _\\mu b$ , $ j^{\\, S} = \\bar{b}b$ , and $ j^{\\, A}_\\mu = (q_\\mu q_\\nu /q^2 -g_{\\mu \\nu })\\bar{b} \\gamma _5 \\gamma ^\\nu b$ with $b$ being the bottom quark operator.", "For the axial-vector current, the projection to the transverse components for the $\\chi _{b1}$ states is carried out.", "At finite temperature, the currents of the $V$ and the $A$ channels generally have two independent components.", "We assume spatial momentum of the bottomonia to be zero so that only one component becomes independent : $q^\\mu =(\\omega , \\bf {0})$ .", "Then, one can define the dimensionless correlation functions as $\\tilde{\\Pi }^{P, \\, S}(q^2) = \\Pi ^{P, \\, S}(q)/q^2 $ and $\\tilde{\\Pi }^{V, \\, A}(q^2) = \\Pi _\\mu ^{\\mu V, \\, A}(q)/(-3q^2) $ .", "Using the operator product expansion (OPE), one can expand the operator $j^{\\, J}(x) j^{\\, J \\dag }(0)$ of Eq.", "(REF ) as a series of local operators $O_n$ with mass dimension $n$ .", "Then, the dimensionless correlation functions are given as $\\tilde{\\Pi }^{\\, J}(q^2) = \\sum _n C_n^{\\, J} (q^2) \\langle O_n \\rangle .", "$ If the scale of the gluon condensates is smaller than the separation scale: $4m_b^2-q^2 \\gg \\langle G \\rangle \\sim (\\Lambda _{\\mathrm {Q}CD} +aT + b \\mu )^2$ , one can assume that all the temperature effects are included in the expectation values of the local operators $ \\langle O_n \\rangle $ [45], [17].", "Thus the Wilson coefficients $ C_n^{\\, J} (q^2)$ can be considered to be independent of $T$ , as long as the temperature is not too high.", "In order to improve the OPE convergence and suppress contributions of higher energy states, we perform the Borel transformation on Eq.", "(REF ).", "Then, the correlation function can be written down as ${\\mathcal {M}}^{\\, J}(\\nu ) = \\pi e^{-\\nu } A^J(\\nu ) [1+\\alpha _s(\\nu ) a^J(\\nu ) +b^J(\\nu ) \\phi _b(T)+c^J(\\nu ) \\phi _c(T)], $ where we use the dimensionless parameter $\\nu \\equiv 4m_b^2/M^2$ with $M$ standing for the usual Borel mass.", "Now the condition for the separation scale is given by $d!", "M^{2d} \\gg \\langle G^d \\rangle $ [17].", "The first term is the leading order of the OPE corresponding to the free current correlation function.", "The second term stands for the perturbative $\\alpha _s$ correction.", "The third and fourth terms include contributions of the scalar and twist-2 gluon condensates of mass dimension 4.", "Here, $\\phi _b$ and $\\phi _c$ are defined as $\\phi _b(T) = \\frac{4\\pi ^2}{9(4m_b^2)^2} G_0(T), $ $\\phi _c(T) = \\frac{4\\pi ^2}{3(4m_b^2)^2} G_2(T), $ where $G_0(T)$ and $G_2(T)$ are the scalar and twist-2 gluon condensates at finite temperature.", "$G_0$ and $G_2$ are defined as $G_0(T) = \\langle \\frac{\\alpha _s}{\\pi } G_{\\mu \\nu }^a G^{a\\mu \\nu } \\rangle $ and $( u^\\mu u^\\nu -\\frac{1}{4} g^{\\mu \\nu } ) G_2(T) = \\langle \\frac{\\alpha _s}{\\pi } G_\\rho ^{a\\mu } G^{a\\nu \\rho } \\rangle $ , where $u^\\mu $ is the four velocity of the medium.", "In our previous work on charmonium, we have added the scalar gluon condensate of mass dimension 6 which was found to be small.", "Thus, we can safely neglect it because its coefficient is strongly suppressed by the bottom quark mass.", "The detailed expressions of the Wilson coefficients $ A^J(\\nu )$ , $a^J(\\nu )$ , $b^J(\\nu )$ and $c^J(\\nu )$ are given in Ref.", "[17].", "The temperature dependences of the gluon condensates are obtained from the approach proposed in Refs.", "[14], [15], where the dimension-4 gluon condensates are related to the energy-momentum tensor, which can be expressed in terms of the energy density $\\epsilon $ , the pressure $p$ and the strong coupling constant $\\alpha _s$ .", "In concrete, $G_0(T)=G_0^{\\mathrm {v}ac} -\\frac{8}{11}[\\epsilon (T) - 3p(T)]$ and $G_2(T) = -\\frac{\\alpha _s(T)}{\\pi }[\\epsilon (T)+p(T)]$ .", "We then utilize the results of quenched lattice QCD [46], [47] to obtain the temperature dependence of $\\epsilon (T)$ , $p(T)$ and $\\alpha _s(T)$ .", "It should be noted here that, in the quenched approximation, the value of the critical temperature $T_c$ is about 260 MeV [46], while the cross-over temperature of full QCD is estimated to be in the region of 145-165 MeV [48], [49]Whereas the phase transition in pure SU(3) theory is a first order one, full QCD at physical quark masses exhibits a cross-over [50]..", "In order to make the predictions more realistic, full lattice QCD with physical quark masses [51] may be applied.", "We, however, note that the gluon condensates are not directly extracted from energy density and pressure in full QCD because of the light quark contribution.", "Therefore, in the present approach, we have to assume that $G_0(T)$ , $G_2(T)$ and $\\alpha _s(T)$ are functions of $T/T_c$ .", "For the scalar gluon condensate, this approximation is known to be a good around $T/T_c \\sim 1.0$ [52], [15].", "Thus this approach may provide a good approximation for charmonium systems.", "As we shall see, however, bottomonium systems receive significant modification at higher temperatures where not only $G_2(T)$ , unknown for full QCD, becomes more important [17] but also $G_0(T)$ in full QCD deviates from the scaling behavior.", "Therefore, resultant temperature dependence of the spectral functions, which will be discussed below, provides no more than qualitative guides for the full QCD case.", "Nevertheless, we emphasize that the results obtained in this paper, rather than being entirely applicable to full QCD, can provide useful information on the difference between the behavior of the S-wave and P-wave (or charmonium and bottomonium) states.", "The sum rule is constructed from the dispersion relation derived from the analytic properties of the correlation function of Eq.", "(REF ).", "For the spectral function $\\rho ^{\\, J}(\\omega )$ , we obtain ${\\mathcal {M}}^{\\, J}(\\nu ) = \\int _0^\\infty d\\omega ^2 \\, e^{- \\nu \\omega ^2 /4m_b^2} \\rho ^{\\, J}(\\omega ), $ where the left-hand sides are equal to Eq.", "(REF ).", "Note that, in the vector, scalar, and axial-vector channels, there is an additional constant term to Eq.", "(REF ) at finite temperature, which originates from a pole at $\\omega =0$ in $ \\rho ^{\\, J}(\\omega )$ and is called scattering term [53].", "Although the contribution of this term, proportional to $e^{-m_b/T}$ , should be much smaller than the charmonium case, we differentiate Eq.", "(REF ) with respect to $\\nu $ to eliminate this contribution: $\\frac{\\partial }{\\partial \\nu } {\\mathcal {M}}^{\\, J}(\\nu )= -\\frac{1}{4m_{b}^2} \\int _0^\\infty d\\omega ^2 \\, \\omega ^2 e^{- \\nu \\omega ^2 /4m_b^2} \\rho ^{\\, J}(\\omega ).", "$ To compare all the channels on the same basis, we also differentiate the sum rule of the pseudo-scalar channel.", "The validity of this procedure in the heavy quark sum rules was discussed in Ref.", "[54].", "Also, we have checked that the same results are obtained from both the original and differentiated sum rules for the pseudo-scalar channel." ], [ "MEM analysis of QCD sum rules", "In this subsection, we demonstrate our method to extract the spectral function $\\rho ^{\\, J}(\\omega )$ from Eq.", "(REF ).", "The conventional methods of analyzing QCD sum rules assume a particular form for the spectral function, the most popular one being the “pole + continuum” form.", "By contrast, such an assumption is not necessary in our method [42] as the shape of spectral functions is directly extracted from the MEM.", "Let us now briefly summarize the procedure of the MEM analysis.", "The basic idea is Bayes' theorem : $P[\\rho |\\mathcal {M} H] =\\frac{P[\\mathcal {M} |\\rho H] P[\\rho | H]}{ P [\\mathcal {M}|H]}, $ where $\\rho $ and $\\mathcal {M} $ are corresponding to the spectral function and the OPE in Eq.", "(REF ), respectively.", "$H$ denotes prior knowledge on $\\rho $ such as positivity and its asymptotic values.", "$P[\\rho |\\mathcal {M} H]$ represents the conditional probability of $\\rho $ given $\\mathcal {M}$ and $H$ .", "On the right-hand side, $P[\\mathcal {M} |\\rho H]$ is called the “likelihood function”, and $ P[\\rho | H]$ stands for the “prior probability”.", "$P [\\mathcal {M}|H]$ is only a normalization constant and can be ignored in later discussion since it does not depend on $\\rho $ .", "In order to maximize $P[\\rho |\\mathcal {M} H]$ , we estimate $P[\\mathcal {M} |\\rho H]$ and $ P[\\rho | H]$ .", "The likelihood function is written as $P[\\mathcal {M} |\\rho H] = e^{-L[\\rho ]}, \\\\$ $L[\\rho ] = \\frac{1}{2(\\nu _{\\mathrm {max}}-\\nu _{\\mathrm {min}})} \\int _{\\nu _{\\mathrm {min}}}^{\\nu _{\\mathrm {max}}} d\\nu \\frac{[\\mathcal {M} (\\nu ) - \\mathcal {M}_\\rho (\\nu )]^2}{\\sigma ^2(\\nu )}.", "$ Here, $\\mathcal {M} (\\nu )$ is obtained from the results of the OPE and corresponds to the left-hand side in Eq.", "(REF ), while $\\mathcal {M}_\\rho (\\nu )$ is defined as the right-hand one.", "$\\sigma (\\nu )$ stands for the uncertainty of $\\mathcal {M}(\\nu )$ (see Ref.", "[42]).", "On the other hand, the prior probability is written as $P[\\rho | H] = e^{\\alpha S[\\rho ]}, \\\\$ $S[\\rho ] = \\int _0^\\infty d\\omega \\left[ \\rho (\\omega ) - m(\\omega ) - \\rho (\\omega ) \\log \\left( \\frac{\\rho (\\omega )}{m(\\omega )} \\right) \\right], $ where $S[\\rho ]$ is known as the Shannon-Jaynes entropy and $\\alpha $ is introduced as a scaling factor.", "$m(\\omega )$ is called the default model and determines the spectral function when there is no information from the OPE.", "For the default model, we use a constant corresponding to the perturbative value of the spectral functions at high energy.", "Using Eq.", "(REF ) and Eqs.", "(REF ), one can rewrite Eq.", "(REF ) as $P[\\rho |\\mathcal {M} H] &\\propto & P[\\mathcal {M}|\\rho H] P[\\rho |H] \\nonumber \\\\&=& e^{Q[\\rho ]}, \\\\Q[\\rho ] &\\equiv & \\alpha S[\\rho ] - L[\\rho ].$ In order to determine the most probable $\\rho (\\omega )$ , we find the maximum of the functional $Q[\\rho ]$ by the Bryan algorithm [55].", "It can be proven that the maximum of $Q[\\rho ]$ is unique if it exists, as it is shown explicitly in [56].", "Before starting the MEM analysis, we have to determine the criterion for the range of the dimensionless parameter $\\nu $ which is used for the investigation.", "We determine the upper bound $\\nu _{\\mathrm {max}}$ from the criterion that the perturbative $\\alpha _s$ correction term contributes less than 30 % of the leading term as in [17].", "The reason why we use the perturbative part rather than the term of the highest gluonic condensate as a criterion of the OPE convergence, is that for bottomonium the terms proportional to the gluon condensates are strongly suppressed due to the large bottom quark mass, as seen in Eqs.", "(REF ) and (REF ).", "Therefore, there is a region of $\\nu $ , where the $\\alpha _s$ correction is forbiddingly large although the contributions of the condensates seem to converge.", "On the other hand, for the lower bound $\\nu _{\\mathrm {min}}$ , we do not have to impose such a criterion unlike in usual QCD sum rule analyses, since we are not assuming the pole dominance of the dispersion integral.", "Thus, we choose the value which obtains the highest peak as the criterion of $\\nu _{\\mathrm {min}}$ .", "The range of $\\nu $ (namely, Borel window) for each channel is summarized in Table REF .", "We have checked that the obtained results do not depend on the chosen values of these bounds.", "Table: Range of dimensionless parameter ν\\nu ." ], [ "Analysis of mock data", "In order to estimate the resolution of the MEM, we should carry out a test analysis by using mock data generated from the experimental information for the vector channel, delta functions for the bound states and a smooth step-like function for the continuum: $\\mathrm {Im} \\Pi ^V(s)=\\sum _{\\mathrm {res}} \\frac{27}{4} \\cdot \\Gamma _R (e^+ \\, e^-) \\frac{m_R}{\\alpha ^2} \\cdot \\delta (s - m_R^2) + \\mathrm {continuum}, $ where, the sum is taken for the these bottomonium states, $\\Upsilon (1S)$ , $\\Upsilon (2S)$ and $\\Upsilon (3S)$ .", "$m_R$ represents their masses, $\\Gamma _R (e^+ \\, e^-)$ is the corresponding decay width to an electron-positron pair, and $\\alpha $ is the fine structure constant (see Ref.", "[40]).", "We construct mock data by substituting the phenomenological spectral function into Eq.", "(REF ) and evaluating the integral numerically over $s$ .", "We also consider the mock data for the individual bottomonium states, where a single state + continuum is taken for Eq.", "(REF ).", "The results of the mock data for the individual bottomonium states are shown in Fig.", "REF as the dotted lines.", "The result for the full spectrum with all the bound states is shown in Fig.", "REF as the solid line.", "One sees that the individual peaks are not resolved in the full spectrum.", "Namely, they are combined into a single peak after the MEM analysis.", "Moreover, the residue of the single peaks is consistent with the sum of the residues of $\\Upsilon (1S)$ , $\\Upsilon (2S)$ and $\\Upsilon (3S)$ .", "It should be noted that the peak position of the combined spectrum is located at a higher energy than the $\\Upsilon (1S)$ peak.", "The values of the peak positions and the residues are summarized in Table REF .", "Later, we will use this property of the combined peak, when we extract the temperature dependence of the excited bottomonium states.", "Figure: Spectral functions extracted from mock data based on Eq.", "() with MEM.Table: Mass and residue values obtained from mock data based on Eq.", "() with MEM." ], [ "Bottomonia at zero temperature", "The results of the MEM for the spectral functions of the vector, pseudoscalar, scalar and axial-vector $b\\bar{b}$ systems at zero temperature are shown in Fig.", "REF .", "We employ the bottom quark mass $\\bar{m}_b(m_b) = 4.167 \\pm 0.013 \\mathrm {GeV}$ [57], the strong coupling constant $\\alpha _s(M_Z)=0.1184\\pm 0.0007$ [58] and the vacuum gluon condensate $G_0^\\mathrm {vac} = 0.012 \\pm 0.0036 \\mathrm {GeV}^4$ [38].", "Each spectral function shows a clear peak at around $\\omega \\sim 10\\mathrm {GeV}$ .", "The average peak height and the estimated error of the average height in the MEM procedure are given by the horizontal and vertical lines.", "The lines are drawn at the range of $\\omega $ where the average is taken.", "One sees that the lowest-energy peaks of the S-wave channels are statistically significant.", "On the other hand, the lowest P-wave peaks have heights of the same order as their error-bars, the scalar channel being a bit higher while the axial-vector channel is lower.", "This means that with the precision presently available, we cannot make any strong statement about the existence of these peaks and about their behavior at finite temperature.", "Therefore, the results about the lowest P-wave peaks obtained in this paper should not be considered to be fully conclusive.", "Furthermore, for all peaks appearing at higher-energy, their statistical significances are not good.", "The lowest peaks are located at $m_{\\Upsilon }=9.56$ GeV, $m_{\\eta _b}=9.51$ GeV, $m_{\\chi _{b0}}=10.15$ GeV and $m_{\\chi _{b1}}=10.42$ GeV, respectively.", "These values are somewhat higher than the experimentally observed masses ($m_{\\mathrm {exp.", "}}=9.460$ GeV, $9.389$ GeV, $9.859$ GeV, and $9.893$ GeV, respectively).", "In fact, they are consistent with our analysis of mock data since the obtained peaks contain contributions from both the ground and excited states and their positions are shifted to higher energies.", "Validity of this picture is confirmed by evaluating the residue of the peak for the vector channel and comparing it with the residue obtained from the leptonic decay width (the value obtained from the present analysis is 0.0476 GeV, which should be compared to Table REF ).", "We have checked that such contributions of the excited states are also present in analyses by usual QCD sum rules, where the “pole + continuum” assumption is used, i.e.", "ground state cannot be separated as a single pole." ], [ "Bottomonia at finite temperature", "The results of the spectral functions at finite temperatures are shown in Fig.", "REF .", "All the channels show the same qualitative behavior.", "First, the peak position undergoes a shift to a lower energy with increasing temperature.", "Next, the peak gradually lowers, becomes broader and simultaneously shifts to slightly higher energies.", "At the same time, a continuum-like structure grows in the low energy region, penetrates into the peak regions and moves downward.", "As a further point, let us mention the crucial difference between the behavior of S-wave and P-wave channels.", "Quantitatively, the vector and pseudoscalar $b\\bar{b}$ states, $\\Upsilon $ and $\\eta _b$ , remain as clear peaks up to $T/T_c=2.0$ and may still survive at $T/T_c=2.5$ .", "In the case of $\\Upsilon $ , one sees a bump even at $T/T_c=3.0$ .", "On the other hand, $\\chi _{b0}$ and $\\chi _{b1}$ seem to disappear at $T/T_c=2.0-2.5$ .", "Figure: The solid lines and their error bars show the spectral functions of bottomonia obtained from finite temperature OPE data.", "The dashed lines stand for spectral functions at zero temperature using error at corresponding finite temperature.", "Upper left : vector (Υ\\Upsilon ), upper right : pseudoscalar (η b \\eta _b), lower left : scalar (χ b0 \\chi _{b0}), lower right : axial-vector (χ b1 \\chi _{b1}).Since the uncertainties involved in the calculation are quite large, as indicated by the error bars in Fig.", "REF , we can presently not make statements about specific numbers for the melting temperatures of the investigated states, but can only restrict the ranges of temperatures within which the peaks disappear.", "Concretely, we define the range of the melting temperatures as follows.", "The upper limit is determined as the temperature where the bump (extremum) disappears, while the lower limit taken as the temperature where the error bar exceeds the lowest-energy peak height of the spectral function, namely when the peak ceases to be statistically significant.", "The error bars for each temperature are shown in Fig.", "REF .", "The resultant upper and lower limits of the dissociation temperatures are summarized in Table REF .", "If we were able to calculate the OPE data with better precision, the lower (upper) limit of the temperature ranges would be increased (decreased).", "Table: Upper and lower limits of the dissociation temperature ranges for the lowest bottomonium states.", "The precise definition of these limits is given in the text.In order to confirm that the obtained results are caused by genuine physical effects, we have to check possible contributions of MEM artifacts at finite temperature.", "First, as the contributions of the gluon condensates increase at finite temperature, their uncertainties magnify the OPE error.", "Therefore, it is expected that the resolution of the MEM is reduced and the peaks of the extracted spectral functions become broader.", "Thus, to investigate this effect, we reanalyze the spectral functions by using the OPE at $T=0$ with error at finite temperature.", "The result for each channel is shown as dashed lines in Fig.", "REF .", "As one can see, although the heights of the peaks are reduced partly due to the MEM artifact described above, the peaks are still present in the S-wave (P-wave) channels with the error of $T/T_c = 3.0$ ($T/T_c = 2.5)$ .", "We also stress that the MEM artifact does not shift the peak position.", "From this analysis, we conclude that the disappearance of the peaks at the finite temperatures is caused by physical effects and not due to an MEM artifact." ], [ "Excited states of bottomonia", "In Section 3.1, we showed that the spectral function extracted by the MEM contains contributions of the excited states in the lowest peak.", "In order to extract finite temperature effects on the excited states from the spectral functions, we analyze the residue of the lowest peak of the vector channel.", "However, one cannot naively integrate the spectral function in the region of the peak because the spectral function is contaminated by the continuum, which is not negligible particularly at high temperatures.", "In order to exclude the continuum contributions and to estimate the sum of the residues only of the ground and excited $b\\bar{b}$ states, we fit the obtained spectral functions by using a Breit-Wigner (or Gaussian) function for the peak, and the continuum parametrized in the form of the leading order perturbative result.", "Specifically, we take $f(x) = \\frac{|\\lambda |^2}{2\\pi } \\frac{ \\Gamma }{(x-m)^2 +\\Gamma ^2 /4} +\\frac{1}{8\\pi ^2} \\sqrt{1-\\frac{4a^2}{x^2}} \\left(2+\\frac{4a^2}{x^2} \\right), $ for fitting the MEM results.", "The four fitting parameters, $|\\lambda |^2$ , $m$ , $\\Gamma $ , and $a$ correspond to the residue, peak position, width, and continuum threshold, respectively.", "Note that $a$ coincides with $m_b$ in the perturbative calculation.", "These parameters are fitted by the Levenberg-Marquardt method [59], [60].", "Furthermore, in order to exclude a possible initial value dependence of the fitting procedure and to investigate the existence of local minima, we take 200 initial values generated randomly for the four fitting parameters at each temperature.", "The obtained residues of the $\\Upsilon $ peak, $|\\lambda |^2$ , with increasing $T/T_c$ are plotted in Fig.", "REF .", "The left panel shows the results of the fitting with the function Eq.", "(REF ).", "For each temperature, 200 results corresponding to initial values are plotted.", "For some temperature, we find multiple solutions, which are supposed to be local minima solutions of the least-square function $\\chi ^2$ in the L-M method.", "In the case of the Breit-Wigner + continuum fitting, we find that the local minimum form three clusters, top, middle, and bottom.", "For some $T/T_c$ , one sees that the clusters are diffused and solutions are scattering to interpolate two clusters.", "Then, the minimum valley of $\\chi ^2$ seems to become flat between the two minima.", "Figure: Temperature dependence of the residue for the vector channel peak fitted with a Breit-Wigner + continuum (left) and as Gaussian + continuum (right).", "Red points are corresponding to true minima, and gray points stand for local minima.", "Fit range is fixed to 7.0-12.0 GeV.The red point at each $T/T_c$ is the true minimum point, where $\\chi ^2$ hits the minimum.", "One sees that at low $T/T_c$ , the true minimum is located in the top cluster, while it moves down to the middle cluster at around $T/T_c \\sim 1.5-1.6$ , and further down to the bottom cluster at around $T/T_c \\sim 1.9-2.0$ .", "At $T/T_c \\ge 2.5$ , we have only one stable solution.", "The peak positions and the continuum thresholds from the fitting with Breit-Wigner + continuum form is shown in Fig.", "REF .", "Both the peak positions and continuum threshold undergo a transition towards smaller values at $T/T_c \\sim 1.5-2.0$ .", "Figure: Temperature dependence of the fitting parameters for the vector channel peak fitted with a Breit-Wigner + continuum.", "Left : peak position, right : continuum threshold.", "Red points are corresponding to true minima, and gray points stand for local minima.", "Fit range is fixed to 7.0-12.0 GeV.To make sure that this fitting analysis is valid, we have repeated the same procedure with an alternate fitting function, namely, the Gaussian + continuum form.", "The results are shown in the right panel of Fig.", "REF .", "The qualitative behaviors are the same as the previous case.", "It however shows only two clusters of local minima and the transition from the higher to the lower cluster at $T/T_c \\sim 1.8-1.9$ .", "Comparing these two fitting results, we conclude that the qualitative behavior of the residue is independent of fitting peak function form.", "It can be concluded from the above results that the residue of $\\Upsilon $ peak decreases gradually with increasing temperature and becomes a constant value at higher temperature.", "Especially, the rapid reduction of the residue is seen at $T/T_c = 1.5 -2.0$ .", "It should be noted that this behavior does not directly imply that the excited states, $\\Upsilon (2S)$ and $\\Upsilon (3S)$ , disappear at lower temperature than ground state $\\Upsilon (1S)$ because one cannot eliminate other possibilities such as simultaneous reduction of the ground and excited states.", "Nevertheless, if we assume the disappearance of the excited states at lower temperature than the ground state, our results suggest that $\\Upsilon (2S)$ and $\\Upsilon (3S)$ disappear at $T/T_c = 1.5 -2.0$ , while $\\Upsilon (1S)$ survives up to $T/T_c = 3.0$ .", "In summary, we have analyzed the bottomonium spectral functions at zero and finite temperature by using a newly developed analysis method of QCD sum rules.", "The Maximum Entropy Method (MEM) is adapted to extract the spectral function from the sum rule.", "At $T = 0$ , the lowest peak has been obtained for each channel corresponding to $\\Upsilon $ , $\\eta _b$ , $\\chi _{b0}$ , and $\\chi _{b1}$ .", "Although these mass spectra agree qualitatively with experimental values, their peak positions shift slightly to higher energies.", "By analyzing mock data for the vector channel and evaluating the obtained residue, we conclude that the disagreement is caused by the contribution of the excited states.", "Next, we have investigated the temperature dependence of the spectral functions.", "Temperature dependences are taken into account in the gluon condensates, which are estimated from the quenched lattice QCD data at finite temperature.", "As a result, we have found that the spectral functions of bottomonia are modified much slower as functions of $T/T_c$ than those of charmonia, in which the lowest peak disappears suddenly at the vicinity of $T_c$ .", "Using the definitions of the upper and lower limits of the melting temperature given in Section REF , we find that $\\Upsilon $ and $\\eta _b$ survive as a peak in the spectral functions up to some temperature restricted to the regions of $T/T_c > 2.3$ and $T/T_c > 2.1$ , while the dissociation temperatures of $\\chi _{b0}$ and $\\chi _{b1}$ are confined to $T/T_c = 1.3 - 2.5$ and $T/T_c < 2.5$ , respectively.", "It should be noted, however, that our definition inevitably contains some ambiguity due to the limitation of the OPE and MEM.", "Therefore, respective results on the melting temperature should be regarded as qualitative guides.", "Furthermore, both the P-wave peaks are not found to be fully significant statistically even at $T=0$ , which means that we cannot make a definite conclusion about the fate of these states at finite temperature.", "To obtain more conclusive results on their behavior, further studies are needed once more precise information on the OPE is available.", "The current prediction of the melting temperatures depends on the extracted temperature dependences of the gluon condensates.", "For these we have used the quenched lattice QCD data for the energy density and pressure.", "To go beyond the quenched approximation, a more detailed analysis will be required to include full QCD information on the gluon condensates, which will be the subject of a future investigation.", "Our results are qualitatively consistent with previous QCD sum rule analysis in conventional method [17].", "As mentioned above, however, it turns out that the lowest peaks contain the excited states as well as the ground state, so that a deformation of such a peak depends on the behavior of the excited states.", "Therefore, to extract more detailed information on the spectral function of the vector channel, we have investigated the temperature dependence of the residue of the lowest peak obtained from MEM for this channel.", "Then, we have observed that the residue decreases with increasing temperature.", "The results are consistent with a picture that the excited states, $\\Upsilon (2S)$ and $\\Upsilon (3S)$ , dissociate at lower temperatures than the ground state $\\Upsilon (1S)$ .", "Our results indicate that $\\Upsilon (2S)$ and $\\Upsilon (3S)$ disappear in the temperature region of $T/T_c=1.5-2.0$ ." ], [ "Acknowledgments", "This work is partly supported by the Grant-in-Aid for Scientific Research from MEXT (No.", "22105503).", "K.S.", "acknowledges the financial support from the Global Center of Excellence Program by MEXT, Japan through the “Nanoscience and Quantum Physics” Project of the Tokyo Institute of Technology.", "P.G.", "gratefully acknowledges the support by the Japan Society for the Promotion of Science for Young Scientists (contract No.", "21.8079).", "K.M.", "is supported by Yukawa International Program for Quark-Hadron Sciences at Kyoto University (No.", "24540271)." ] ]

[ [ "Adaptation and migration of a population between patches" ], [ "Abstract A Hamilton-Jacobi formulation has been established previously for phenotypically structured population models where the solution concentrates as Dirac masses in the limit of small diffusion.", "Is it possible to extend this approach to spatial models?", "Are the limiting solutions still in the form of sums of Dirac masses?", "Does the presence of several habitats lead to polymorphic situations?", "We study the stationary solutions of a structured population model, while the population is structured by continuous phenotypical traits and discrete positions in space.", "The growth term varies from one habitable zone to another, for instance because of a change in the temperature.", "The individuals can migrate from one zone to another with a constant rate.", "The mathematical modeling of this problem, considering mutations between phenotypical traits and competitive interaction of individuals within each zone via a single resource, leads to a system of coupled parabolic integro-differential equations.", "We study the asymptotic behavior of the stationary solutions to this model in the limit of small mutations.", "The limit, which is a sum of Dirac masses, can be described with the help of an effective Hamiltonian.", "The presence of migration can modify the dominant traits and lead to polymorphic situations." ], [ "Introduction", "Non-local Lotka-Volterra equations arise in models of adaptive evolution of phenotypically structured populations.", "These equations have the property that the solutions concentrate generally, in the limit of small diffusion, on several isolated points, corresponding to distinct traits.", "Can we generalize these models by adding a spatial structure?", "How do the dominant traits evolve if we introduce a new habitat?", "To understand the interaction of ecological and evolutionary processes in population dynamics, spatial structure of the communities and adaptation of species to the environmental changes, it is crucial to dispose mathematical models that describe them jointly.", "We refer to [19] and the references therein for general literature on the subject.", "In this manuscript we consider a model where several distinct favorable habitable zones are possible.", "Population dynamics models structured by spatial patches have been studied using both deterministic and probabilistic methods (see for instance [30], [1]).", "Our model, in the case of two patches, is indeed very close to the one studied in [30] where the authors use adaptive dynamics theory (adaptive dynamics is a theory, based on dynamical systems and their stability, to study population dynamics [11]).", "Here we model similar phenomena, by adding a spatial structure to an earlier known integro-differential model describing the darwinian evolution.", "Integro-differential models have the advantage that the mutations can be considered directly in the model without assuming a separation of time scales of evolution and ecology.", "The present work provides a general description of the asymptotic stationary solutions, in the general case where two or several patches are possible.", "We study the asymptotic behavior of solutions of a system of coupled elliptic integro-differential equations with small diffusion terms.", "These solutions are the stationary solutions to a parabolic system describing the dynamics of a population density.", "The individuals are characterized by phenotypical traits, that we denote by $x\\in \\mathbb {R}^d$ .", "They can move between two or several patches, which are favorable habitable zones, with constant rates (that we denote by $\\nu ^1$ and $\\nu ^2$ in the case of two patches).", "The mathematical modeling is based on the darwinian evolution and takes into account mutations and competition between the traits.", "There is a large literature for mathematical modeling and analysis on the subject of adaptive evolution, we refer the interested reader to [16], [15], [11], [12], [22], [10].", "Here, we represent the birth and death term by a net growth term $R^i(x,I^i)$ that is different in each patch, for instance because of a change in the temperature, and depends on the integral parameter $I^i$ , which corresponds to the the pressure exerted by the whole population within patch $i$ on the resource.", "To model the mutations, we use Laplace terms with a small rate $\\varepsilon $ that is introduced to consider only rare mutations.", "We study the asymptotic behavior of stationary solutions as the mutation rate $\\varepsilon $ goes to 0.", "The asymptotic solutions are generally concentrated on one or several Dirac masses.", "We describe the position and the weight of these Dirac masses using a Hamilton-Jacobi approach.", "The time-dependent model, in the case of two patches, is written as $\\left\\lbrace \\begin{array}{rlr}\\partial _t n_\\varepsilon ^1-\\varepsilon \\Delta n_\\varepsilon ^1&=\\frac{1}{\\varepsilon }n_\\varepsilon ^1 R^1(x,I_\\varepsilon ^1)+\\frac{1}{\\varepsilon }\\nu ^2 n_\\varepsilon ^2-\\frac{1}{\\varepsilon }\\nu ^1 n_\\varepsilon ^1,&\\\\&& \\quad x\\in \\mathbb {R}^d,\\\\\\partial _t n_\\varepsilon ^2-\\varepsilon \\Delta n_\\varepsilon ^2&=\\frac{1}{\\varepsilon }n_\\varepsilon ^2 R^2(x,I_\\varepsilon ^2)+\\frac{1}{\\varepsilon }\\nu ^1 n_\\varepsilon ^1-\\frac{1}{\\varepsilon }\\nu ^2 n_\\varepsilon ^2,&\\end{array}\\right.$ with $I_\\varepsilon ^i=\\int \\psi ^i(x)n_\\varepsilon ^i(x)dx,\\qquad \\text{for $i=1,2.$}$ Such models, without the structure in space, have been derived from stochastic individual based models in the limit of large populations (see [7], [6]).", "This manuscript follows earlier works on parabolic Lotka-Volterra type equations to study concentration effects in models of phenotypically structured populations, that are based on a Hamilton-Jacobi formulation (see [12], [25], [2], [20]).", "The novelty of our work is that we add a spatial structure to the model by considering a finite number of favorable habitable zones.", "We thus have a system instead of a single equation.", "A Hamilton-Jacobi approach in the case of systems has also been introduced in [5] for an age structured model.", "See also [4] for a study of stationary solutions of the latter system.", "The Hamilton-Jacobi approach can also be used in problems other than adaptive evolution to prove concentration phenomena.", "See for instance [27], [26], [24] where related methods have been used to study the motion of motor proteins.", "We are interested in the equilibria of REF limited to a bounded domain, that are given by solutions of the following system $\\left\\lbrace \\begin{array}{rll}-\\varepsilon ^2 \\Delta n_\\varepsilon ^1&=n_\\varepsilon ^1 R^1(x,I_\\varepsilon ^1)+\\nu ^2 n_\\varepsilon ^2-\\nu ^1 n_\\varepsilon ^1&\\quad \\text{in $B_L(0)$},\\\\\\\\-\\varepsilon ^2 \\Delta n_\\varepsilon ^2&=n_\\varepsilon ^2 R^2(x,I_\\varepsilon ^2)+\\nu ^1 n_\\varepsilon ^1-\\nu ^2 n_\\varepsilon ^2&\\quad \\text{in $B_L(0)$},\\\\\\\\\\nabla n_\\varepsilon ^i\\cdot \\vec{n}&=0&\\quad \\text{in $\\partial B_L(0)$ and for $i=1,\\,2$},\\end{array}\\right.$ where $B_L(p)$ is a ball of radius $L$ with center in $p$ and $\\vec{n}(x)$ is the unit normal vector, at the point $x\\in \\partial B_L(0)$ , to the boundary of $B_L(0)$ .", "The Neumann boundary condition is a way to express that mutants cannot be born in $\\mathbb {R}^d\\setminus B_L(0)$ .", "To formulate our results we introduce the assumptions we will be using throughout the paper.", "We assume that, there exist positive constants $a_m$ and $a_M$ such that $\\psi ^1=\\psi ^2=\\psi ,\\quad a_m\\le \\psi (x)\\le a_M, \\quad \\Vert \\psi (x)\\Vert _{W^{2,\\infty }}\\le A \\quad \\text{and}\\quad \\nabla \\psi \\cdot \\vec{n}=0 \\text{ in $\\partial B_L(0)$}.$ Moreover there exist positive constants $I_m$ , $I_M$ , $\\delta $ and $C$ such that, for all $x\\in B_L(0)$ and $i,j=1,\\,2$ , $\\delta \\le \\min \\left(R^i(x,\\frac{\\nu ^j}{\\nu ^i} I_m),R^i(x,I_m)\\right),\\quad \\max \\left(R^i(x,\\frac{\\nu ^j }{\\nu ^i} I_M),R^i(x,I_M)\\right)\\le -\\delta ,$ $-C\\le \\frac{\\partial R^i}{\\partial I}(x,I)\\le -\\frac{1}{C},$ $-D\\le R_{\\xi \\xi }^i(x,I),\\quad \\text{for $x\\in B_L(0)$, $I\\in [I_m,I_M]$, $\\xi \\in \\mathbb {R}^d$, $|\\xi |=1$ and $i=1,2$.", "}$ We use the Hopf-Cole transformation $ n_\\varepsilon ^i =\\exp (\\frac{u_\\varepsilon ^i}{\\varepsilon }) ,\\qquad \\text{for $i=1,2,$}$ and replace the latter in the system satisfied by $n_\\varepsilon ^i$ to obtain $\\left\\lbrace \\begin{array}{rll}-\\varepsilon \\Delta u_\\varepsilon ^1&=|\\nabla u_\\varepsilon ^1|^2+ R^1(x,I_\\varepsilon ^1)+\\nu ^2 \\exp (\\frac{u_\\varepsilon ^2-u_\\varepsilon ^1}{\\varepsilon })-\\nu ^1,&\\quad \\text{in $B_L(0)$},\\\\\\\\-\\varepsilon \\Delta u_\\varepsilon ^2&=|\\nabla u_\\varepsilon ^2|^2+R^2(x,I_\\varepsilon ^2)+\\nu ^1 \\exp (\\frac{u_\\varepsilon ^1-u_\\varepsilon ^2}{\\varepsilon }) -\\nu ^2&\\quad \\text{in $B_L(0)$},\\\\\\\\\\nabla u_\\varepsilon ^i\\cdot \\vec{n}&=0&\\quad \\text{in $\\partial B_L(0)$} \\\\&&\\text{and for $i=1,\\,2$}.\\end{array}\\right.$ We prove the following Theorem 1.1 Assume REF –REF .", "Then, as $\\varepsilon \\rightarrow 0$ along subsequences, both sequences $(u_\\varepsilon ^1)_\\varepsilon $ and $(u_\\varepsilon ^2)_\\varepsilon $ converge uniformly in $B_L(0)$ to a continuous function $u\\in \\mathrm {C}(B_L(0))$ and $(I_\\varepsilon ^1,I_\\varepsilon ^2)$ goes to $(I^1,I^2)$ , with $(u,I^1,I^2)$ such that $u$ is a viscosity solution to the following equation $\\left\\lbrace \\begin{array}{ll}-|\\nabla u|^2= H(x,I^1,I^2),&\\quad \\text{in $B_L(0)$},\\\\\\\\\\max _{x\\in B_L(0)}u(x)=0,\\end{array}\\right.$ with $\\begin{array}{c} \\text{$H(x,I^1,I^2)$ the largest eigenvalue of the matrix}\\\\\\displaystyle \\mathcal {A}=\\left(\\begin{array}{cc} R^1(x,I^1)-\\nu ^1&\\nu ^2\\\\\\nu ^1 & R^2(x,I^2)-\\nu ^2\\end{array}\\right).\\end{array}$ The function $H$ is indeed an effective Hamiltonian that contains information from the two patches and helps us in Theorem REF to describe the support of the weak limits of $(n_\\varepsilon ^1,n_\\varepsilon ^2)$ as $\\varepsilon \\rightarrow 0$ .", "We can interpret $H(x,I^1,I^2)$ as the fitness of the system in the limit of $\\varepsilon \\rightarrow 0$ (see [23] for the definition of fitness).", "The difficulty here is to find appropriate regularity estimates on $u_\\varepsilon ^i$ , that we obtain using the Harnack inequality [3] and the Bernstein method [9].", "To prove convergence to the Hamilton-Jacobi equation, we are inspired from the method of perturbed test functions in homogenization [14].", "The above information on the limit of $u_\\varepsilon ^i$ allows us to describe the limit of the densities $n_\\varepsilon ^i$ as $\\varepsilon $ vanishes.", "We prove Theorem 1.2 Assume REF –REF .", "Consider a subsequence such that $u_\\varepsilon ^1$ and $u_\\varepsilon ^2$ converge uniformly to $u\\in C\\left(B_L(0)\\right)$ and $(I_\\varepsilon ^1,I_\\varepsilon ^2)$ goes to $(I^1,I^2)$ , as $\\varepsilon \\rightarrow 0$ , with $(u,I^1,I^2)$ solution of REF .", "Let $n_\\varepsilon ^i$ , for $i=1,2$ , converge weakly in the sense of measures to $n^i$ along this subsequence.", "We have $ \\mathrm {supp } \\;n^i\\subset \\Omega \\cap \\Gamma ,\\quad \\text{for $i=1,2$,}$ with $\\begin{array}{c}\\Omega =\\lbrace x\\in B_L(0) \\,| \\, u(x)=0\\rbrace ,\\\\ \\Gamma =\\lbrace x\\in B_L(0)\\,| \\, H(x,I^1,I^2)=\\max _{x\\in B_L(0)}H(x,I^1,I^2)=0\\rbrace .\\end{array}$ Moreover, we have $\\left(R^1(x)-\\nu ^1\\right)n^1(x)+\\nu ^2n^2(x)=0,\\quad \\left(R^2(x)-\\nu ^2\\right)n^2(x)+\\nu ^1n^1(x)=0, \\quad \\text{in }B_L(0)$ in the sense of distributions.", "The above condition is coupled by $\\int _{B_L(0)} \\psi ^i(x)n^i(x)=I^i.$ Theorem REF provides us with a set of algebraic constraints on the limit, which allows us to describe the latter.", "In particular, if the support of $n^i$ , for $i=1,2$ , is a set of distinct points: $\\mathrm {supp}\\,n^i\\subset \\lbrace x_1,x_2,\\cdots ,x_k\\rbrace $ , REF implies that $n^i=\\sum _{j=1}^k\\rho ^i_j \\delta (x-x_j),\\qquad \\text{for $i=1,2$,}$ with $\\rho _j^2=\\rho _j^1\\left(\\frac{\\nu ^1-R^1(x_j,I^1)}{\\nu ^2}\\right)=\\rho _j^1\\left(\\frac{\\nu ^1}{\\nu ^2-R^2(x_j,I^2)}\\right).$ Furthermore, the weights $(\\rho ^i_1,\\cdots ,\\rho ^i_k)$ satisfy the normalization condition $\\sum _{j=1}^k \\psi ^i(x_j)\\rho ^i_j=I^i,\\qquad \\ \\text{for $i=1,2$.", "}$ Condition REF means that the vector $\\left(\\begin{array}{c} \\rho _j^1\\\\\\rho _j^2 \\end{array}\\right)$ is the eigenvector corresponding to the largest eigenvalue of the matrix $\\mathcal {A}$ at the point $x_j$ , which is 0.", "Thereby REF implies once again that $\\mathrm {supp}\\,n^i\\subset \\Gamma $ .", "We point out that since $n^i$ , for $i=1,2$ , is such that the fitness $H$ vanishes on the support of $n^i$ and is negative outside the support, we can interpret $n^i$ as evolutionary stable distribution of the model.", "In adaptive dynamics, evolutionary stable distribution (ESD) corresponds to a distribution that remains stable after introduction of small mutants (see [21], [13], [17] for a more detailed definition).", "See also [10], [28] for related works on stability and convergence to ESD for trait-structured integro-differential models.", "The set of assumptions in Theorem REF allows us to describe the asymptotics of the stationary solutions, in the limit of rare or small mutations.", "In Section we provide some examples where based on this information we can describe the asymptotics.", "In particular, we notice that the introduction of a new environment can lead to dimorphic situations.", "We refer to [8] for a related work using the Hamilton-Jacobi approach, where polymorphic situations can also appear in a model with multiple resources.", "The paper is organized as follows.", "In Section we prove some bounds on $I_\\varepsilon $ and some regularity properties on $u_\\varepsilon $ that allow us to pass to the limit as $\\varepsilon \\rightarrow 0$ and derive the Hamilton-Jacobi equation with constraint.", "Theorem REF is proved in Section .", "Using the results obtained on the asymptotic behavior of $(u_\\varepsilon ^i)_\\varepsilon $ we prove Theorem REF in Section .", "In Section we provide some examples where the information given by Theorem REF and Theorem REF allows us to describe the limit.", "The asymptotic behavior of the stationary solutions in a more general framework, where more than two habitable zones are considered, is given in Section .", "Finally in Section we present some numerical simulations for the time-dependent problem and compare them with the behavior of stationary solutions." ], [ "Regularity results", "Lemma 2.1 Under assumptions REF –REF we have, for $\\varepsilon \\le \\varepsilon _0$ chosen small enough, $I_m\\le I_\\varepsilon ^i\\le I_M, \\qquad \\text{for $i=1,\\,2$}.$ In particular, along subsequences, $(I_\\varepsilon ^1,I_\\varepsilon ^2)_\\varepsilon $ converges to $(I^1,I^2)$ , with $I_m\\le I^1,\\; I^2\\le I_M$ .", "Remark 1 This is the only part, where we use Assumption REF .", "If $(n_\\varepsilon ^1,n_\\varepsilon ^2)$ is a solution of REF such that REF is satisfied, then the results of Theorems REF and REF hold true without necessarily assuming REF .", "In particular, one can take $\\psi ^1\\lnot \\equiv \\psi ^2$ .", "Proof 1 We prove the result by contradiction.", "We suppose that $I_\\varepsilon ^1> I_M$ (the case with $I_\\varepsilon ^2>I_M$ , and the inequalities from below can be treated following similar arguments).", "We multiply the first equation in REF by $\\psi (x)$ , integrate, and use REF to obtain $-\\varepsilon ^2\\frac{A}{a_m}I_\\varepsilon ^1\\le \\int \\psi (x) n_\\varepsilon ^1(x)R^1(x,I_\\varepsilon ^1)dx+\\nu ^2 I_\\varepsilon ^2-\\nu ^1I_\\varepsilon ^1.$ Using now REF , REF and the fact that $I_\\varepsilon ^1>I_M$ we deduce that, for $\\varepsilon \\le \\varepsilon _0$ small enough, $0\\le \\left(\\delta -\\varepsilon ^2\\frac{A}{a_m}\\right)I_\\varepsilon ^1 \\le \\nu ^2 I_\\varepsilon ^2-\\nu ^1I_\\varepsilon ^1,$ and thus $\\frac{\\nu ^1}{\\nu ^2}\\,I_M\\le I_\\varepsilon ^2.$ Now we multiply the equations in REF by $\\psi (x)$ , integrate and add them and use REF to obtain $-\\varepsilon ^2\\frac{A}{a_m}(I_\\varepsilon ^1+I_\\varepsilon ^2) \\le \\int \\psi (x) n_\\varepsilon ^1(x)R^1(x,I_\\varepsilon ^1)dx+\\int \\psi (x) n_\\varepsilon ^2(x)R^2(x,I_\\varepsilon ^2)dx.$ From REF , REF and the above bounds on $I_\\varepsilon ^1$ and $I_\\varepsilon ^2$ it follows that $-\\varepsilon ^2\\frac{A}{a_m}(I_\\varepsilon ^1+I_\\varepsilon ^2) \\le -\\delta (I_\\varepsilon ^1+I_\\varepsilon ^2),$ which is not possible if $\\varepsilon $ is small enough.", "We conclude that $I_\\varepsilon ^1\\le I_M$ .", "Theorem 2.2 Assume REF –REF .", "Then (i) there exists a positive constant $D$ , such that for $\\varepsilon \\le \\varepsilon _0$ , $ |u_\\varepsilon ^i(x)-u_\\varepsilon ^j(y)|\\le D\\varepsilon ,\\qquad \\text{for all $x,\\,y\\in B_L(0)$, $|x-y|\\le \\varepsilon $ and $i,\\,j\\in \\lbrace 1,2\\rbrace $.", "}$ (ii) For $i=1,\\,2$ and all $\\varepsilon \\le \\varepsilon _0$ , the family $(u_\\varepsilon )_\\varepsilon $ is uniformly Lipschitz and uniformly bounded from below.", "(iii) For all $a>0$ , there exists $\\varepsilon _1=\\varepsilon _1(a)$ such that for all $\\varepsilon \\le \\varepsilon _1$ , $u_\\varepsilon ^i(x)\\le a,\\qquad \\text{for $x\\in B_L(0)$ and $i=1,\\,2$.", "}$ Proof 2 (i) We define $\\widetilde{n}_\\varepsilon ^i(y)=n_\\varepsilon ^i(\\varepsilon y),\\qquad \\text{for $i=1,\\,2$.", "}$ From REF we have $\\displaystyle \\left\\lbrace \\begin{array}{rll}- \\Delta \\widetilde{n}_\\varepsilon ^1&=\\widetilde{n}_\\varepsilon ^1 R^1(\\varepsilon x,I_\\varepsilon ^1)+\\nu ^2\\widetilde{n}_\\varepsilon ^2-\\nu ^1 \\widetilde{n}_\\varepsilon ^1&\\quad \\text{in $B_{\\frac{L}{\\varepsilon }}(0)$},\\\\\\\\- \\Delta \\widetilde{n}_\\varepsilon ^2&=\\widetilde{n}_\\varepsilon ^2 R^2(\\varepsilon x,I_\\varepsilon ^2)+\\nu ^1\\widetilde{n}_\\varepsilon ^1-\\nu ^2 \\widetilde{n}_\\varepsilon ^2&\\quad \\text{in $B_{\\frac{L}{\\varepsilon }}(0)$},\\end{array}\\right.$ Moreover, from REF , REF and REF we have, for $\\varepsilon \\le \\varepsilon _0$ , $\\delta -C(I_M-I_m)\\le R(\\varepsilon x,I_\\varepsilon )\\le -\\delta +C(I_M-I_m).$ Therefore the coefficients of the linear elliptic system REF are bounded uniformly in $\\varepsilon $ .", "It follows from the classical Harnack inequality ([3], Theorem 8.2) that there exists a constant $D=D(C,I_m,I_M,\\delta ,\\nu ^1,\\nu ^2)$ such that for all $y_0\\in B_{\\frac{L}{\\varepsilon }}(0)$ such that $B_1(y_0)\\subset B_{\\frac{L}{\\varepsilon }}(0)$ and for $i,j=1,2$ , $\\sup _{z\\in B_1(y_0)}\\widetilde{n}_\\varepsilon ^i(z)\\le D\\,\\inf _{z\\in B_1(y_0)}\\widetilde{n}_\\varepsilon ^j(z).$ Rewriting the latter in terms of $ n_\\varepsilon ^1$ and $ n_\\varepsilon ^2$ and replacing $(y_0,z)$ by $(\\frac{x}{\\varepsilon },\\frac{z^{\\prime }}{\\varepsilon })$ we obtain $\\sup _{z^{\\prime }\\in B_\\varepsilon (x)}n_\\varepsilon ^i(z^{\\prime })\\le D\\,\\inf _{z^{\\prime }\\in B_\\varepsilon (y_0)}n_\\varepsilon ^j(z^{\\prime }),$ and thus from REF we deduce REF .", "(ii) To prove the Lipschitz bounds, we use the Bernstein method (see [9]).", "We assume that $ \\max _{x\\in B_L(0)}(|\\nabla u_\\varepsilon ^1(x)|,|\\nabla u_\\varepsilon ^2(x)|)=|\\nabla u_\\varepsilon ^1(x_\\varepsilon )|,$ that is the maximum is achieved at a point $x_\\varepsilon \\in B_L(0)$ and for $i=1$ (the case where the maximum is achieved for $i=2$ can be treated by similar arguments).", "From the Neumann boundary condition in REF we know that $x_\\varepsilon $ is an interior point of $B_L(0)$ .", "We define $p=|\\nabla u_\\varepsilon ^1|^2$ and notice that $\\Delta p=2 \\mathrm {Tr}\\;(\\mathrm {Hess}\\;u_\\varepsilon ^1)^2+2\\nabla (\\Delta u_\\varepsilon ^1)\\cdot \\nabla u_\\varepsilon ^1.$ We now differentiate the first equation in REF with respect to $x$ and multiply it by $\\nabla u_\\varepsilon ^1$ to obtain $-\\varepsilon \\nabla (\\Delta u_\\varepsilon ^1)\\cdot \\nabla u_\\varepsilon ^1=\\nabla p\\cdot \\nabla u_\\varepsilon ^1+\\nabla R^1\\cdot \\nabla u_\\varepsilon ^1+\\nu ^2\\left(\\frac{\\nabla u_\\varepsilon ^2-\\nabla u_\\varepsilon ^1}{\\varepsilon }\\right)\\cdot \\nabla u_\\varepsilon ^1 \\exp (\\frac{u_\\varepsilon ^2-u_\\varepsilon ^1}{\\varepsilon }).$ From REF we have $\\left(\\nabla u_\\varepsilon ^2(x_\\varepsilon )-\\nabla u_\\varepsilon ^1(x_\\varepsilon )\\right)\\cdot \\nabla u_\\varepsilon ^1(x_\\varepsilon )\\le 0,$ and thus $-\\frac{\\varepsilon }{2}\\Delta p(x_\\varepsilon )+\\varepsilon \\mathrm {Tr}\\;(\\mathrm {Hess}\\;u_\\varepsilon ^1(x_\\varepsilon ))^2\\le \\nabla p(x_\\varepsilon )\\cdot \\nabla u_\\varepsilon ^1(x_\\varepsilon )+\\nabla R^1(x_\\varepsilon )\\cdot \\nabla u_\\varepsilon ^1(x_\\varepsilon ).$ Moreover from REF we have $\\nabla p(x_\\varepsilon )=0$ and $\\Delta p\\le 0$ .", "It follows that $\\varepsilon \\left(\\Delta u_\\varepsilon ^1(x_\\varepsilon )\\right)^2\\le \\varepsilon d\\, \\mathrm {Tr}\\;(\\mathrm {Hess}\\;u_\\varepsilon ^1(x_\\varepsilon ))^2\\le d\\nabla R^1(x_\\varepsilon )\\cdot \\nabla u_\\varepsilon ^1(x_\\varepsilon ).$ Using again REF we obtain $\\left( |\\nabla u_\\varepsilon ^1|^2+R^1(x_\\varepsilon ,I_\\varepsilon ^1)+\\nu ^2\\exp \\left(\\frac{u_\\varepsilon ^2-u_\\varepsilon ^1}{\\varepsilon }\\right)-\\nu ^1 \\right)^2\\le \\varepsilon d\\,\\nabla R^1(x_\\varepsilon ,I_\\varepsilon ^1)\\cdot \\nabla u_\\varepsilon ^1(x_\\varepsilon ).$ From REF , REF and REF we find that $(R^1(x,I_\\varepsilon ^1))_\\varepsilon $ is uniformly bounded for $\\varepsilon \\le \\varepsilon _0$ .", "We conclude that $(u_\\varepsilon ^1)_\\varepsilon $ is uniformly Lipschitz for $\\varepsilon \\le \\varepsilon _0$ .", "To prove uniform bounds from below, we notice from REF and REF that, for $i=1,2$ , there exists a point $\\overline{x}_i\\in B_L(0)$ such that $\\varepsilon \\ln \\left(\\frac{I_m}{a_M|B_L(0)|}\\right)\\le u_\\varepsilon ^i(\\overline{x}_i).$ From the latter and the Lipschitz bounds we obtain that $-2LC_1+\\varepsilon \\ln \\left(\\frac{I_m}{a_M|B_L(0)|}\\right)\\le u_\\varepsilon ^i,\\qquad \\text{in $B_L(0)$ and for $i=1,2$}.$ It follows that the families $(u_\\varepsilon ^i)_\\varepsilon $ are bounded from below for $\\varepsilon \\le \\varepsilon _0$ and $i=1,2$ .", "(iii) We prove REF for $i=1$ by contradiction.", "The proof for $i=2$ follows the same arguments.", "We assume that there exists a sequence $(\\varepsilon _k,x_k)$ such that $\\varepsilon _k\\rightarrow 0$ as $k\\rightarrow \\infty $ , $x_k\\in B_L(0)$ and $u_{\\varepsilon _k}^1(x_k)>a$ .", "Using the uniform Lipschitz bounds obtained in (ii) we have $n_{\\varepsilon _k}^1(x)>\\exp \\left(\\frac{a}{2\\varepsilon _k}\\right),\\qquad \\text{in }[x_k-\\frac{a}{2C_1},x_k+\\frac{a}{2C_1}]\\cap B_L(0).$ This is in contradiction with the bound from above in REF , for $\\varepsilon _k$ small enough.", "Therefore REF holds." ], [ "Convergence to the Hamilton-Jacobi equation", "In this section we prove Theorem REF .", "Proof 3 Convergence to the Hamilton-Jacobi equation: From (ii) and (iii) in Theorem REF we have that for $i=1,2$ , the families $(u_\\varepsilon ^i)_\\varepsilon $ are uniformly bounded and Lipschitz.", "Therefore, from the Arzela-Ascoli Theorem we obtain that, along subsequences, $(u_\\varepsilon ^1)_\\varepsilon $ and $(u_\\varepsilon ^2)_\\varepsilon $ converge locally uniformly to some continuous functions $u^i\\in \\mathrm {C}(B_L(0); \\mathbb {R})$ , with $i=1,2$ .", "Moreover, from (i) in Theorem REF we deduce that $u^1=u^2$ .", "Here we consider a subsequence of $(I_\\varepsilon ^1,I_\\varepsilon ^2,u_\\varepsilon ^1,u_\\varepsilon ^2)_\\varepsilon $ that converges to $(I^1,I^2,u,u)$ .", "Let $H(x,I_\\varepsilon ^1,I_\\varepsilon ^2)$ , be the largest eigenvalue of the matrix $\\mathcal {A}_\\varepsilon =\\left(\\begin{array}{cc} R^1(x,I_\\varepsilon ^1)-\\nu ^1&\\nu ^2\\\\\\nu ^1 & R^2(x,I_\\varepsilon ^2)-\\nu ^2\\end{array}\\right),$ and $\\left(\\begin{array}{c}\\chi _\\varepsilon ^1(x)\\\\\\chi _\\varepsilon ^2(x)\\end{array}\\right)$ be the corresponding eigenvector.", "Since the non-diagonal terms in $\\mathcal {A}_\\varepsilon $ are strictly positive, using the Perron-Frobinius Theorem, we know that such eigenvalue exist and that $\\chi _\\varepsilon ^1$ and $\\chi _\\varepsilon ^2$ are strictly positive.", "We write $\\phi _\\varepsilon ^i(x)=\\ln \\chi _\\varepsilon ^i(x),\\qquad \\text{for $i=1,2$.", "}$ We prove that $u$ is a viscosity solution of $ - |\\nabla u | ^2 = H(x,I^1,I^2), \\qquad \\text{in $B_L(0)$}.", "$ To this aim, suppose that $u-\\varphi $ has a maximum in $x\\in B_L(0)$ .", "Then, we consider a sequence $x_\\varepsilon \\in B_L(0)$ , such that as $\\varepsilon \\rightarrow 0$ , $x_\\varepsilon \\rightarrow x$ and $u_\\varepsilon ^1(x_\\varepsilon )-\\varphi (x_\\varepsilon )-\\varepsilon \\phi _\\varepsilon ^1(x_\\varepsilon )=\\max _{\\underset{i=1,2}{x\\in B_L(0)}} u_\\varepsilon ^i(x)-\\varphi (x)-\\varepsilon \\phi _\\varepsilon ^i(x)$ is attained at the point $x_\\varepsilon $ and for $i=1$ (The case with $i=2$ can be treated similarly).", "In this case, we have in particular that $u_\\varepsilon ^2(x_\\varepsilon )-u_\\varepsilon ^1(x_\\varepsilon )\\le \\varepsilon \\left(\\phi _\\varepsilon ^2(x_\\varepsilon )-\\phi _\\varepsilon ^1(x_\\varepsilon )\\right).$ Using the latter and the viscosity criterion for the first equation in REF we obtain that $\\begin{split}-\\varepsilon (\\Delta \\varphi (x_\\varepsilon )+\\varepsilon \\Delta \\phi _\\varepsilon ^1(x_\\varepsilon ))-|\\nabla \\varphi (x_\\varepsilon )+\\varepsilon \\nabla \\phi _\\varepsilon ^1(x_\\varepsilon )|^2- R^1(x,I_\\varepsilon ^1)&\\\\-\\nu ^2 \\exp \\left(\\phi _\\varepsilon ^2(x_\\varepsilon )-\\phi _\\varepsilon ^1(x_\\varepsilon )\\right)+\\nu ^1&\\le 0.\\end{split}$ We notice that, by definition of $\\phi _\\varepsilon ^1$ and $\\phi _\\varepsilon ^2$ , we have $- R^1(x,I_\\varepsilon ^1)-\\nu ^2 \\exp \\left(\\phi _\\varepsilon ^2(x_\\varepsilon )-\\phi _\\varepsilon ^1(x_\\varepsilon )\\right)+\\nu ^1=-H(x,I_\\varepsilon ^1,I_\\varepsilon ^2).$ From the latter and by letting $\\varepsilon \\rightarrow 0$ in REF we deduce that $-|\\nabla \\varphi (x)|^2\\le H(x,I^1,I^2),$ and thus $u$ is a subsolution of REF in the viscosity sense.", "The supersolution criterion can be proved in a similar way.", "The constraint on the limit ($\\max _{x\\in B_L(0)}u(x)=0$ ): From REF we obtain that $u(x)\\le 0$ .", "To prove that $0\\le \\max _{x\\in B_L(0)}u(x)$ , we use the lower bounds on $I_\\varepsilon ^i$ in REF .", "The proof of this property is classical and we refer to [2], [20] for a detailed proof." ], [ "Asymptotic behavior of stationary solutions", "In this section we prove Theorem REF .", "Proof 4 Support of $n^i$ : From REF , we deduce that, along subsequences and for $i=1,2$ , $(n_\\varepsilon ^i)_\\varepsilon $ converges weakly to a measure $n^i$ .", "The fact that $\\mathrm {supp } \\;n^i\\subset \\Omega ,\\text{ for $i=1,2$,}$ is a consequence of the Hopf-Cole transformation REF .", "To prove REF it is enough to prove $\\Omega \\subset \\Gamma $ .", "To this aim following the idea in [25] we first prove that, for $i=1,2$ , $u_\\varepsilon ^i$ are uniformly semi-convex.", "Recall that the smooth function $v$ is semiconvex with constant $C$ , if we have $v_{\\xi \\xi }\\ge -C,\\qquad \\text{for all $|\\xi |=1$}.$ Let $\\min \\lbrace u_{\\varepsilon ,\\xi \\xi }^i(x)\\,|\\, x\\in B_L(0),\\, i=1,2,\\, \\xi \\in \\mathbb {R}^d, \\, |\\xi |=1\\rbrace =u_{\\varepsilon ,\\eta \\eta }^1(x_\\varepsilon ).$ The case where the minimum is achieved for $i=2$ can be treated similarly.", "We differentiate twice the first equation in REF with respect to $\\eta $ and obtain $\\begin{split}-\\varepsilon \\Delta u_{\\varepsilon ,\\eta \\eta }^1&=2\\nabla u_\\varepsilon ^1\\cdot \\nabla u_{\\varepsilon ,\\eta \\eta }^1+2|\\nabla u_{\\varepsilon ,\\eta }^1|^2+R_{\\eta \\eta }^1\\\\&+\\nu ^2\\left(\\left(\\frac{u_{\\varepsilon ,\\eta }^2-u_{\\varepsilon ,\\eta }^1}{\\varepsilon }\\right)^2+\\frac{u_{\\varepsilon ,\\eta \\eta }^2-u_{\\varepsilon ,\\eta \\eta }^1}{\\varepsilon }\\right)\\exp \\left(\\frac{u_\\varepsilon ^2-u_\\varepsilon ^1}{\\varepsilon }\\right).\\end{split}$ From REF we obtain that $\\Delta u_{\\varepsilon ,\\eta \\eta }^1(x_\\varepsilon )\\ge 0$ , $\\nabla u_{\\varepsilon ,\\eta \\eta }^1(x_\\varepsilon )=0$ and $u_{\\varepsilon ,\\eta \\eta }^2(x_\\varepsilon )-u_{\\varepsilon ,\\eta \\eta }^1(x_\\varepsilon )\\ge 0$ .", "Using REF It follows that $|\\nabla u_{\\varepsilon ,\\eta }^1(x_\\varepsilon )|^2\\le \\frac{D}{2}.$ Since $u_{\\varepsilon ,\\eta \\eta }^1=\\nabla u_{\\varepsilon ,\\eta }^1\\cdot \\eta $ , we have $|u_{\\varepsilon ,\\eta \\eta }^1|\\le |\\nabla u_{\\varepsilon ,\\eta }^1|$ .", "We deduce that $|u_{\\varepsilon ,\\eta \\eta }^1(x_\\varepsilon )|^2\\le \\frac{D}{2},$ and thus $\\min \\lbrace u_{\\varepsilon ,\\xi \\xi }^i(x)\\,|\\, x\\in B_L(0),\\, i=1,2,\\, \\xi \\in \\mathbb {R}^d, \\, |\\xi |=1\\rbrace \\ge -\\sqrt{ \\frac{D}{2}}.$ This proves that $u_\\varepsilon ^i$ , for $i=1,2$ are semiconvex functions with constant $-\\sqrt{ \\frac{D}{2}}$ .", "By passing to the limit in $\\varepsilon \\rightarrow 0$ we obtain that $u$ is also semiconvex with the same constant.", "A semiconvex function is differentiable at its maximum points.", "Therefore $u$ is differentiable with $\\nabla u=0$ in the set $\\Omega $ .", "From REF , we deduce, that for all $x\\in \\Omega $ , $H(x,I^1,I^2)=0$ , and thus $\\Omega \\subset \\lbrace x\\in B_L(0)\\,|\\,H(x,I^1,I^2)=0\\rbrace $ .", "The fact that $\\max _{x\\in B_L(0)}H(x,I^1,I^2)=0$ is immediate from REF and the facts that $u$ is almost everywhere differentiable and $H(x,I^1,I^2)$ is a continuous function.", "Value of $n^i$ on the support: Let $\\xi \\in \\mathrm {C}^\\infty _\\mathrm {c}(B_L(0))$ , i.e.", "$\\xi $ is a smooth function with compact support in $B_L(0)$ .", "We multiply REF by $\\xi $ and integrate with respect to $x$ in $B_L(0)$ to obtain, for $ \\lbrace i,j\\rbrace =\\lbrace 1,2\\rbrace $ , $\\begin{split}- \\varepsilon ^2 \\int _{B_L(0)} n_\\varepsilon ^i(x) \\Delta \\xi (x) dx &=\\int _{B_L(0)} \\xi (x) n_\\varepsilon ^i(x) R^i(x,I_\\varepsilon ^i) dx\\\\& -\\nu ^i \\int _{B_L(0)} \\xi (x) n_\\varepsilon ^i(x)dx +\\nu ^j \\int _{B_L(0)} \\xi (x)n_\\varepsilon ^j(x) dx.", "\\end{split}$ Since $n_\\varepsilon ^l{\\quad } n^l$ weakly and $I_\\varepsilon ^l\\rightarrow I^l$ , for $l=1,2$ , as $\\varepsilon \\rightarrow 0$ , we obtain that, for $ \\lbrace i,j\\rbrace =\\lbrace 1,2\\rbrace $ , $\\int _{B_L(0)} \\xi (x) n^i(x) R^i(x,I^i) dx -\\nu ^i \\int _{B_L(0)} \\xi (x) n^i(x)dx +\\nu ^j \\int _{B_L(0)} \\xi (x)n^j(x) dx=0,$ and thus REF .", "Finally, REF follows from REF ." ], [ "Examples of application", "In REF –REF we give a description of $(n^1,n^2)$ , assuming that the support of $n^i$ , for $i=1,2$ , is a set of distinct points, i.e.", "$n^i$ is a sum of Dirac masses and does not have a continuous distribution.", "This is what we expect naturally in the models based on darwinian evolution.", "More precisely, from Volterra-GauseÕs competitive exclusion principle (see [18], [29]) it is known in theoretical biology that in a model with $K$ limiting factors (as nutrients or geographic parameters) at most $K$ distinct species can generally survive.", "Here we have two limiting factors, represented by $I^1$ and $I^2$ , that correspond to the environmental pressures in the two patches.", "We thus expect to observe only monomorphic or dimorphic situations.", "This is also the case in the numerical simulations represented in Section .", "From REF we know that the support of $n^i$ is included in the set of maximum points of $H(x,I^1,I^2)$ , $\\Gamma $ , with $(I^1,I^2)$ limits of $(I_\\varepsilon ^1,I_\\varepsilon ^2)$ .", "If now $H$ is such that, for fixed $(I^1,I^2)$ , the corresponding set $\\Gamma $ consists of isolated points, it follows that the supports of $n^1$ and $n^2$ consist also of isolated points.", "We give an example below where $H$ has clearly this property.", "Example 5.1 (monomorphism towards dimorphism) Consider a case with the following values for the parameters of the system $R^1(x,I)=a^1x^2+b^1x+c^1-d^1I,\\qquad R^2(x,I)=a^2x^2+b^2x+c^2-d^2I,$ with $a^i,b^i,c^i,d^i\\in \\mathbb {R}, \\qquad a^i<0<d^i, \\qquad \\text{for $i=1,2$.", "}$ Then the supports of $n^1$ and $n^2$ consist at most of two single points.", "We first notice that in the case where there is no migration between patches ($\\nu ^1=\\nu ^2=0$ ), from the results in [20], we know that in patch $i$ , the population concentrates in large time on the maximum points of $R^i(\\cdot ,I^i)$ with $I^i$ the limit of $I_\\varepsilon ^i$ .", "Since $R^i$ is a quadratic function in $x$ , it has a unique maximum and thus $n^i$ is a single Dirac mass on this maximum point.", "However, allowing migration by taking positive values for $\\nu ^1$ and $\\nu ^2$ the population can become dimorphic.", "In Section we give a numerical example where a dimorphic situation appears (see Figure REF ).", "This is in accordance with the competitive exclusion principle since we have introduced a new limiting factor, which is the choice of habitable zones.", "Next, we prove the result: Proof 5 (Proof of Example REF . )", "From REF we have that the stationary solutions concentrate asymptotically on the maximum points of $H$ defined as below $\\begin{array}{rl}H(x,I^1,I^2)&=\\frac{1}{2}F+\\frac{1}{2}\\sqrt{F^2-4G},\\end{array}$ with $\\begin{array}{c}F(x,I^1,I^2):=R^1(x,I^1)-\\nu ^1+R^2(x,I^2)-\\nu ^2,\\\\ G(x,I^1,I^2):=(R^1(x,I^1)-\\nu ^1)(R^2(x,I^2)-\\nu ^2)-\\nu ^1\\nu ^2,\\end{array}$ Since $\\max _{x\\in B_L(0)}H(x,I^1,I^2)=0$ , we deduce that $\\min _{x\\in B_L(0)}G(x,I^1,I^2)=0,$ and $\\Gamma =\\lbrace x\\in B_L(0)\\,|\\, H(x,I^1,I^2)=0\\rbrace =\\lbrace x\\in B_L(0)\\,|\\, G(x,I^1,I^2)=0\\rbrace .$ For fixed $(I^1,I^2)$ , $G(x,I^1,I^2)$ is a polynomial of order 4.", "Therefore it has at most two maximum points.", "It follows that $\\Gamma $ consists of one or two distinct points.", "Example 5.2 (An asymmetric case) We assume that the parameters are such that the support of $n^i$ , for $i=1,2$ , consists of isolated points, and we have $\\begin{split}R^i(x,I)=R^i(x)-cI,\\quad \\text{for $i=1,2$,} \\quad R^1(x)=R^2(\\tau (x)),\\quad \\text{for all $x\\in B_L(0)$}\\\\\\text{ and }\\quad \\nu ^1=\\nu ^2=\\nu ,\\end{split}$ with $\\tau :B_L(0)\\rightarrow B_L(0)$ such that $\\tau \\circ \\tau =\\rm {Id}$ .", "Let $(I^1,I^2)$ be a limit point of $(I_\\varepsilon ^1,I_\\varepsilon ^2)$ .", "We have $I^1=I^2=I$ , where $I$ is such that $\\max _x H(x,I,I)=\\min _x G(x,I,I)=0,$ with $H$ and $G$ defined respectively in REF and REF .", "In particular, if $\\bar{x}\\in \\Gamma $ then we have $\\tau (\\bar{x})\\in \\Gamma $ , with $\\Gamma $ defined in REF .", "Assumption REF covers the case where the growth terms have the following forms $R^1(x)=f(|x-a|),\\qquad R^2(x)=f(|x+a|),$ with $f:\\rm {B_L(0)}\\rightarrow \\mathbb {R}$ a function and $a$ constant (we consider the application $\\tau (x)=-x$ ).", "In this case the competition terms in the patches have a simple form: the fitness, in absence of migration, has a shift in traits from one zone to another, for instance due to a difference in the temperature.", "We can thus characterize the limit in this case.", "If moreover, we suppose that the growth terms satisfy REF , we conclude that in the limit while $\\varepsilon \\rightarrow 0$ , the population, is either monomorphic with a single Dirac mass at the origin, or it is dimorphic with two Dirac masses located on two symmetric points, one of the winning traits being more favorable for zone 1 and the other one being more favorable for zone 2.", "Proof 6 (Proof of Example REF .)", "We prove the claim by contradiction and we assume that $I^1\\ne I^2$ .", "Without loss of generality we suppose that $I^1< I^2$ .", "Let $\\bar{x}_j\\in \\rm {supp}\\;n^1$ .", "From REF and REF , we have that $G$ has a minimum in $\\bar{x}_j$ and in particular, $G(\\bar{x}_j)\\le G(\\tau (\\bar{x}_j))$ , namely, $(R^1(\\bar{x}_j)-I^1-\\nu )(R^2(\\bar{x}_j)-I^2-\\nu )\\le (R^1(\\tau (\\bar{x}_j))-I^1-\\nu )(R^2(\\tau (\\bar{x}_j))-I^2-\\nu ).$ It follows that $(R^1(\\bar{x}_j)-I^1-\\nu )(R^2(\\bar{x}_j)-I^2-\\nu )\\le (R^2(\\bar{x}_j)-I^1-\\nu )(R^1(\\bar{x}_j)-I^2-\\nu ).$ We deduce that $0\\le (I^1-I^2)\\left(R^2(\\bar{x}_j)-R^1(\\bar{x}_j)\\right), $ and thus $R^2(\\bar{x}_j)\\le R^1(\\bar{x}_j).$ From the latter, $I^1<I^2$ and REF we obtain that $\\rho _j^2<\\rho _j^1.", "$ Since this is true for all $\\bar{x}_j\\in \\rm {supp}\\; n^1= \\rm {supp}\\; n^2$ , we obtain from REF that $I^2<I^1$ .", "This is a contradiction and thus $I^1=I^2$ ." ], [ "The case with several patches", "The result can be extended to the case with more than two patches.", "The model for $K$ patches is written as $\\left\\lbrace \\begin{array}{rll}-\\varepsilon ^2 \\Delta n_\\varepsilon ^i&=n_\\varepsilon ^i R^i(x,I_\\varepsilon ^i)+\\sum _j \\nu ^{ij} n_\\varepsilon ^j-\\nu ^{ii} n_\\varepsilon ^i&\\quad \\text{in $B_L(0)$ and for $1\\le i \\le K$},\\\\\\\\\\nabla n_\\varepsilon ^i\\cdot \\vec{n}&=0&\\quad \\text{in $\\partial B_L(0)$ and for $1\\le i \\le K$},\\end{array}\\right.$ with $I_\\varepsilon ^i=\\int \\psi ^i(x)n_\\varepsilon ^i(x)dx,\\qquad \\text{for $i\\in \\lbrace 1, \\cdots ,K\\rbrace $}.$ We suppose that $(n_\\varepsilon ^1,\\cdots , n_\\varepsilon ^K)$ is a solution of REF –REF such that $\\min (I_\\varepsilon ^1,\\cdots , I_\\varepsilon ^K)\\le I_M,\\quad I_m\\le \\max (I_\\varepsilon ^1,\\cdots ,I_\\varepsilon ^K),\\quad (I_\\varepsilon ^1,\\cdots ,I_\\varepsilon ^K)\\underset{\\varepsilon \\rightarrow 0}{\\longrightarrow }(I^1,\\cdots ,I^K).$ We also replace assumption REF by $\\begin{array}{c}|R^i(x,I)|\\le C,\\quad -D\\le R_{\\xi \\xi }^i(x,I), \\\\\\text{for $x\\in B_L(0)$, $0\\le I$, } \\text{$\\xi \\in \\mathbb {R}^d$, $|\\xi |=1$ and $1\\le i\\le K$,}\\end{array}$ and we use again the Hopf-Cole transformation $n_\\varepsilon ^i=\\exp (\\frac{u_\\varepsilon ^i}{\\varepsilon }),\\qquad \\text{for $i=1,\\cdots ,K$.", "}$ To present the result we also introduce the following matrix $\\mathcal {B}=\\left(\\begin{array}{ccc} R^1(x,I^1)-\\nu ^{11}&\\cdots &\\nu ^{1K}\\\\\\vdots &\\ddots &\\vdots \\\\\\nu ^{K1} &\\cdots &R^K(x,I^K)-\\nu ^{KK}\\end{array}\\right),$ and as in the case with two patches we define $\\Omega =\\lbrace x\\in B_L(0) \\,| \\, u(x)=0\\rbrace ,$ and $\\Gamma =\\lbrace x\\in B_L(0)\\,| \\, H(x,I^1,\\cdots ,I^K)=\\max _{x\\in B_L(0)}H(x,I^1,\\cdots ,I^K)=0\\rbrace .$ We have Theorem 6.1 We assume that $(n_\\varepsilon ^1,\\cdots , n_\\varepsilon ^K)$ is a solution of REF –REF with REF , REF and REF .", "Then, after extraction of a subsequence, the sequences $(u_\\varepsilon ^i)_\\varepsilon $ , for $i=1,\\cdots ,K$ , converge to a continuous function $u\\in \\mathrm {C}(B_L(0))$ that is a viscosity solution to the following equation $\\left\\lbrace \\begin{array}{ll}-|\\nabla u|^2= H(x,I^1,\\cdots ,I^K),&\\quad \\text{in $B_L(0)$},\\\\\\\\\\max _{x\\in B_L(0)}u(x)=0,\\end{array}\\right.$ with $H(x,I^1,I^2)$ the largest eigenvalue of the matrix $\\mathcal {B}$ .", "Let $n^i$ , for $i=1,\\cdots ,K$ , be a weak limit of $n_\\varepsilon ^i$ along this subsequence.", "We have $\\mathrm {supp } \\;n^i\\subset \\Omega \\cap \\Gamma ,\\quad \\text{for $i=1,\\cdots ,K$}$ Moreover, if the support of $n^i$ , for $i=1,\\cdots ,K$ , is a set of distinct points: $\\mathrm {supp}\\,n^i\\subset \\lbrace x_1,x_2,\\cdots ,x_l\\rbrace $ , we then have $n^i=\\sum _{j=1}^l\\rho ^i_j \\delta (x-x_j),\\qquad \\text{for $i=1,\\cdots ,K$,}$ with $\\left(\\begin{array}{c}\\rho ^{1}_j\\\\\\vdots \\\\ \\rho ^K_j\\end{array}\\right)$ the eigenvector corresponding to the largest eigenvalue of $\\mathcal {B}$ at the point $x_j$ , which is 0, coupled by $\\sum _j \\rho _j^i\\psi ^i(x_j)=I^i.$ Proof 7 The proof of Theorem REF follows along the same lines as the one of Theorem REF and Theorem REF .", "The only difference is in the proof of lower and upper bounds on $u_\\varepsilon $ which are obtained using the uniform bounds on $I_\\varepsilon ^i$ .", "Indeed Assumption REF is slightly weaker than REF .", "To prove uniform bounds on $u_\\varepsilon ^i$ , with $i=1,\\cdots ,K$ , using REF we first prove that for an index $j\\in \\lbrace 1,\\cdots ,K\\rbrace $ which is such that the minimum (respectively the maximum) of $(I^1,\\cdots ,I^K)$ is attained for $I^j$ , $u_\\varepsilon ^j$ is uniformly bounded from above (respectively from below), then we use an estimate of type REF to obtain a uniform bound from above (respectively from below) on $u_\\varepsilon ^i$ for all $i\\in \\lbrace 1,\\cdots ,K\\rbrace $ ." ], [ "Time dependent problem and numerics", "How well the asymptotics of the solutions of REF (that are stationary solutions of REF ) approximate the large time behavior of the solution of the time-dependent problem REF , while $\\varepsilon $ vanishes ?", "In this section, using numerical simulations we try to answer to this question.", "Theoretical study of the time-dependent problem, which requires appropriate regularity estimates, is beyond the scope of the present paper and is left for future work.", "The numerical simulations for REF have been performed in Matlab using the following parameters $\\begin{array}{c}R^1(x,I)=3-(x+1)^2-I,\\quad R^2(x,I)=3-(x-1)^2-I,\\quad \\psi ^1(x)=\\psi ^2(x)=1,\\\\\\ \\nu ^1=\\nu ^2=2.5,\\quad {\\varepsilon =.001},\\quad L=2.\\end{array}$ We notice that these parameters verify the properties in both examples REF and REF .", "Therefore, we expect that the stationary solutions are concentrated on one or two Dirac masses that are symmetric with respect to the origin.", "As we observe in Figure REF , $n_\\varepsilon ^1$ and $n_\\varepsilon ^2$ , with $(n_\\varepsilon ^1,n_\\varepsilon ^2)$ solution of the time-dependent problem REF with the above parameters, concentrate in large time on a single Dirac mass at the origin, which is the mean value of the favorable traits in each zone in absence of migration.", "In this simulation, initially $n_\\varepsilon ^1$ is concentrated on $x=-0.3$ and $n_\\varepsilon ^2$ is concentrated on $x=0.3$ .", "Figure: Dynamics of the time-dependent problem with parameters given in .", "In both figures, horizontally is time tt and vertically is trait xx.", "The gray layers represent the value of n ε 1 n_\\varepsilon ^1(left) and n ε 2 n_\\varepsilon ^2(right).", "Initially n ε 1 n_\\varepsilon ^1 is concentrated on x=-0.3x=-0.3 and n ε 2 n_\\varepsilon ^2 is concentrated on x=0.3x=0.3.", "Due to migration both traits appear rapidly in the two patches, but in large time only one dominant trait persists.", "This point is the mean value of favorable traits in each patch in absence of migration.Depending on the parameters of the model, one can also observe stability in large time of dimorphic situations.", "For instance, if we vary the values of $\\nu ^1$ and $\\nu ^2$ in REF as follows $\\begin{array}{c}R^1(x,I)=3-(x+1)^2-I,\\quad R^2(x,I)=3-(x-1)^2-I,\\quad \\psi ^1(x)=\\psi ^2(x)=1\\\\\\nu ^1=\\nu ^2=1,\\quad {\\varepsilon =.001},\\quad L=2,\\end{array}$ then $n_\\varepsilon ^1$ and $n_\\varepsilon ^2$ , with $(n_\\varepsilon ^1,n_\\varepsilon ^2)$ solution of the time-dependent problem REF , concentrate in large time on two distinct Dirac masses, one of them more favorable to patch 1 and the second one more favorable to patch 2 (see Figure REF ).", "We note indeed that, in absence of migration, the local optimal trait in patch 1 is $x=-1$ and in patch 2 is $x=1$ .", "In presence of migration, the two initial traits appear immediately in the two patches and evolve to two points, one close to $x=-0.86$ and the other close to $x=0.86$ .", "Figure: Dynamics of the time-dependent problem with parameters given in .", "In both figures, horizontally is time tt and vertically is trait xx.", "The gray layers represent the value of n ε 1 n_\\varepsilon ^1(left) and n ε 2 n_\\varepsilon ^2(right).", "In absence of migration, the local optimal trait in patch 1 is x=-1x=-1 and in patch 2 is x=1x=1.", "Initially n ε 1 n_\\varepsilon ^1 is concentrated on x=-0.3x=-0.3 and n ε 2 n_\\varepsilon ^2 is concentrated on x=0.3x=0.3.", "Due to migration both traits appear rapidly in the two patches, and evolve to two points close to -0.86 -0.86 and 0.860.86.Does the above numerical solution converge in long time to the solution described by the algebraic constraints given in Theorem REF ?", "The values of $I_\\varepsilon ^1$ and $I_\\varepsilon ^2$ are depicted in Figure REF showing that both these quantities converge in long time to $2.25$ .", "We can also compute the value of $H$ at the final time step.", "As we observe in Figure REF , $\\max _x H=0$ and the maximum is attained at the points $x=-.86$ and $x=.86$ which correspond to the positions of the Dirac masses in Figure REF .", "We can also compute numerically the weights of the Dirac masses at the final time step, to obtain $n_\\varepsilon ^1(t=5)\\approx 1.77\\, \\delta (x+.86)+.48\\, \\delta (x-.86),\\quad n_\\varepsilon ^2(t=5)\\approx .48\\, \\delta (x+.86)+1.77\\, \\delta (x-.86).$ One can verify that the above weights satisfy REF –REF .", "Figure: Dynamics of the total populations: I ε 1 (t)I_\\varepsilon ^1(t)(left) and I ε 2 (t)I_\\varepsilon ^2(t)(right), using the parameters in .", "In both patches, the total population converges to a constant close to 2.252.25.Figure: The value of H(·,I ε 1 (t),I ε 2 (t)H(\\cdot ,I_\\varepsilon ^1(t),I_\\varepsilon ^2(t) defined in , at time t=5t=5." ], [ "Acknowledgments", "The author benefits from a 2 year \"Fondation Mathématique Jacques Hadamard\" (FMJH) postdoc scholarship.", "She would like to thank Ecole Polytechnique for its hospitality.", "She is also grateful to Clément Fabre for useful discussions." ] ]

[ [ "The Kumaraswamy Pareto distribution" ], [ "Abstract The modeling and analysis of lifetimes is an important aspect of statistical work in a wide variety of scientific and technological fields.", "For the first time, the called Kumaraswamy Pareto distribution is introduced and studied.", "The new distribution can have a decreasing and upside-down bathtub failure rate function depending on the values of its parameters.", "It includes as special sub-models the Pareto and exponentiated Pareto (Gupta et al.", "[12]) distributions.", "Some structural properties of the proposed distribution are studied including explicit expressions for the moments and generating function.", "We provide the density function of the order statistics and obtain their moments.", "The method of maximum likelihood is used for estimating the model parameters and the observed information matrix is derived.", "A real data set is used to compare the new model with widely known distributions." ], [ "Introduction", "Introduction The Pareto distribution is a very popular model named after a professor of economics: Vilfredo Pareto.", "The various forms of the Pareto distribution are very versatile and a variety of uncertainties can be usefully modelled by them.", "For instance, they arise as tractable `lifetime' models in actuarial sciences, economics, finance, life testing and climatology, where it usually describes the occurrence of extreme weather.", "The random variable $X$ has the Pareto distribution if its cumulative distribution function (cdf) for $x\\ge \\beta $ is given by $G(x;\\beta ,k) = 1-\\left(\\frac{\\beta }{x}\\right)^k,$ where $ \\beta >0$ is a scale parameter and $k > 0$ is a shape parameter.", "The probability density function (pdf) corresponding to (REF ) is $g(x;\\beta ,k) = \\frac{k\\,\\beta ^k}{x^{k+1}}.$ Several generalized forms of the Pareto distribution can be found in the literature.", "The term “generalized Pareto” (GP) distribution was first used by Pickands [22] when making statistical inferences about the upper tail of a distribution function.", "As expected, the Pareto distribution can be seen as a special case of the GP distribution.", "It can also be obtained as a special case of another extended form generated by compounding a heavy-tailed skewed conditional gamma density function with parameters $\\alpha $ and $\\beta ^{-1}$ , where the weighting function for $\\beta $ has a gamma distribution with parameters $k$ and $\\theta $ (Hogg et al. [13]).", "Gupta et al.", "[12] extended the Pareto distribution by raising (REF ) to a positive power.", "In this note, we refer to this extension as the exponentiated Pareto (EP) distribution.", "Recently, many authors have considered various exponentiated-type distributions based on some known distributions such as the exponential, Rayleigh, Weibull, gamma and Burr distributions; see, for example, Gupta and Kundu ([9], [10] and [11]), Surles and Padgett [26], Kundu and Raqab [17] and Silva et al. [25].", "The methods of moments and maximum likelihood have been used to fit these models.", "Further, Akinsete et al.", "[1] and Mahmoudi  [18] extended the Pareto and GP distributions by defining the beta Pareto (BP) and beta generalized Pareto (BGP) distributions, respectively, based on the class of generalized (so-called “beta-G\") distributions introduced by Eugene et al. [8].", "The generalized distributions are obtained by taking any parent G distribution in the cdf of a beta distribution with two additional shape parameters, whose role is to introduce skewness and to vary tail weight.", "Following the same idea, many beta-type distributions were introduced and studied, see, for example, Barreto-Souza et al.", "[2], Silva et al.", "[24] and Cordeiro et al. [6].", "In this context, we propose an extension of the Pareto distribution based on the family of Kumaraswamy generalized (denoted with the prefix “Kw-G\" for short) distributions introduced by Cordeiro and de Castro [4].", "Nadarajah et al.", "[20] studied some mathematical properties of this family.", "The Kumaraswamy (Kw) distribution is not very common among statisticians and has been little explored in the literature.", "Its cdf (for $0<x<1$ ) is $F(x) = 1 - (1-x^a)^b$ , where $a>0$ and $b>0$ are shape parameters, and the density function has a simple form $f(x)= a\\,b\\,x^{a-1}(1-x^a)^{b-1}$ , which can be unimodal, increasing, decreasing or constant, depending on the parameter values.", "It does not seem to be very familiar to statisticians and has not been investigated systematically in much detail before, nor has its relative interchangeability with the beta distribution been widely appreciated.", "However, in a very recent paper, Jones [14] explored the background and genesis of this distribution and, more importantly, made clear some similarities and differences between the beta and Kw distributions.", "In this note, we combine the works of Kumaraswamy [16] and Cordeiro and de Castro [4] to derive some mathematical properties of a new model, called the Kumaraswamy Pareto (Kw-P) distribution, which stems from the following general construction: if G denotes the baseline cumulative function of a random variable, then a generalized class of distributions can be defined by $F(x) = 1 - \\left[1 - G(x)^a\\right]^b,$ where $a > 0$ and $b > 0$ are two additional shape parameters which govern skewness and tail weights.", "Because of its tractable distribution function (REF ), the Kw-G distribution can be used quite effectively even if the data are censored.", "Correspondingly, its density function is distributions has a very simple form $f(x) = a\\,b\\,g(x)\\,G(x)^{a-1}\\left[1 - G(x)^a\\right]^{b-1}.$ The density family (REF ) has many of the same properties of the class of beta-G distributions (see Eugene et al.", "[8]), but has some advantages in terms of tractability, since it does not involve any special function such as the beta function.", "Equivalently, as occurs with the beta-G family of distributions, special Kw-G distributions can be generated as follows: the Kw-normal distribution is obtained by taking $G(x)$ in (REF ) to be the normal cumulative function.", "Analogously, the Kw-Weibull (Cordeiro et al.", "[5]), Kw-generalized gamma (Pascoa et al.", "[21]), Kw-Birnbaum-Saunders (Saulo et al.", "[23]) and Kw-Gumbel (Cordeiro et al.", "[7]) distributions are obtained by taking $G(x)$ to be the cdf of the Weibull, generalized gamma, Birnbaum-Saunders and Gumbel distributions, respectively, among several others.", "Hence, each new Kw-G distribution can be generated from a specified G distribution.", "This paper is outlined as follows.", "In section 2, we define the Kw-P distribution and provide expansions for its cumulative and density functions.", "In addition, we study the limit behavior of its pdf and hazard rate function.", "A range of mathematical properties of this distribution is considered in sections 3-7.", "These include quantile function, simulation, skewness and kurtosis, order statistics, generating and characteristic functions, incomplete moments, L-moments and mean deviations.", "The Rényi entropy is calculated in section 8.", "Maximum likelihood estimation is performed and the observed information matrix is determined in section 9.", "In section 10, we provide an application of the Kw-P distribution to a flood data set.", "Finally, some conclusions are addressed in section 11." ], [ "The Kw-P distribution", "The Kw-P distribution If $G(x;\\beta ,k)$ is the Pareto cumulative distribution with parameters $\\beta $ and $k$ , then equation (REF ) yields the Kw-P cumulative distribution (for $x \\ge \\beta $ ) $F(x;\\beta ,k,a,b) = 1 - \\left\\lbrace 1 -\\left[1-\\left(\\frac{\\beta }{x}\\right)^{k}\\right]^{a} \\right\\rbrace ^{b},$ where $\\beta >0$ is a scale parameter and the other positive parameters $k,a$ and $b$ are shape parameters.", "The corresponding pdf and hazard rate function are $f(x;\\beta ,k,a,b) = \\frac{a\\,b\\, k\\,\\beta ^k}{x^{k+1}}\\left[1-\\left(\\frac{\\beta }{x}\\right)^k\\right]^{a-1}\\left\\lbrace 1-\\left[1-\\left(\\frac{\\beta }{x}\\right)^k\\right]^{a}\\right\\rbrace ^{b-1},$ and $\\tau (x;\\beta ,k,a,b) = \\frac{a\\,b\\, k\\,\\beta ^k\\left[1-\\left(\\beta /x\\right)^k\\right]^{a-1}}{x^{k+1}\\,\\lbrace 1 -[1-(\\beta /x)^{k} ]^{a} \\rbrace },$ respectively.", "The Kw-P distribution is not in fact very tractable.", "However, its heavy tail can adjust skewed data that cannot be properly fitted by existing distributions.", "Furthermore, the cumulative and hazard rate functions are simple.", "In Figures REF and REF , we plot the density and failure rate functions of the Kw-P distribution for selected parameter values, respectively.", "We can verify that this distribution can have a decreasing and upside-down bathtub failure rate function depending on the values of its parameters." ], [ "Expansions for the cumulative and density functions", "Expansions for the cumulative and density functions Here, we give simple expansions for the Kw-P cumulative distribution.", "By using the generalized binomial theorem (for $0 < a < 1$ ) $(1+a)^\\nu = \\sum _{i=0}^{\\infty } \\binom{\\nu }{i}\\,a^i,$ where $\\binom{\\nu }{i}=\\frac{n(n-1)\\ldots (\\nu -i+1)}{i!", "},$ in equation (REF ), we can write $ \\nonumber F(x;\\beta ,k,a,b) &=& 1 -\\sum ^{\\infty }_{i=0}(-1)^i\\,\\binom{b}{i}\\left[1-\\left(\\frac{\\beta }{x}\\right)^k\\right]^{ai}\\\\&=& 1 - \\sum ^{\\infty }_{i=0}\\eta _i\\,H(x;\\beta ,k,ia),$ where $\\eta _i = (-1)^i\\binom{b}{i}$ and $H(x;\\beta , k, ia)$ denotes the EP cumulative distribution (with parameters $\\beta $ , $k$ and $ia$ ) given by $H(x;\\beta ,k,\\alpha )=\\left[ 1-\\left(\\frac{\\beta }{x}\\right)^k\\right]^\\alpha .$ Now, using the power series (REF ) in the last term of (REF ), we obtain $\\nonumber f(x;\\beta ,k,a,b) &=& \\frac{a\\, b\\, k\\,\\beta ^k}{x^{k+1}}\\sum ^{\\infty }_{i=0}(-1)^i\\,\\binom{b-1}{i}\\,\\left[1-\\left(\\frac{\\beta }{x}\\right)^k\\right]^{ a(i+1)-1}\\\\&=& \\sum _{j=0}^{\\infty }w_{j}\\, g(x; \\beta , k(j+1)),$ where $w_{j} = \\frac{a\\, b}{(j+1)}\\sum _{i=0}^{\\infty }(-1)^{i+j}\\binom{b-1}{i}\\binom{a(i+1)-1}{j} \\nonumber $ and $g(x; \\beta , k(j+1))$ denotes the Pareto density function with parameters $\\beta $ and $k(j+1)$ and cumulative distribution as in (REF ).", "Thus, the Kw-P density function can be expressed as an infinite linear combination of Pareto densities.", "Thus, some of its mathematical properties can be obtained directly from those properties of the Pareto distribution.", "For example, the ordinary, inverse and factorial moments, moment generating function (mgf) and characteristic function of the Kw-P distribution follow immediately from those quantities of the Pareto distribution." ], [ "Limiting behaviour of Kw-P density and hazard functions", "Limiting behaviour of Kw-P density and hazard functions Lemma 1.", "The limit of the Kw-P density function as $x\\rightarrow \\infty $ is 0 and the limit as $x\\rightarrow \\beta $ are $\\lim _{x\\rightarrow \\beta }f(x;\\beta ,k,a,b) = \\left\\lbrace \\begin{array}{ll}\\infty , & \\textrm {for} \\, \\, 0 < a < 1,\\\\\\dfrac{b k}{\\beta }, & \\textrm {for} \\, \\, a = 1,\\\\0, & \\textrm {for} \\, \\, a > 1.\\\\\\end{array} \\right.$ Proof.", "It is easy to demonstrate the result from the density function (REF ).", "Lemma 2.", "The limit of the Kw-P hazard function as $x\\rightarrow \\infty $ is 0 and the limit as $x\\rightarrow \\beta $ are $\\lim _{x\\rightarrow \\beta }\\tau (x;\\beta ,k,a,b) = \\left\\lbrace \\begin{array}{ll}\\infty , & \\textrm {for} \\, \\, 0 < a < 1,\\\\\\dfrac{b k}{\\beta }, & \\textrm {for} \\, \\, a = 1,\\\\0, & \\textrm {for} \\, \\, a > 1.\\\\\\end{array} \\right.$ Proof.", "It is straightforward to prove this result from (REF )." ], [ "Moments and generating function", "Moments and generating function Here and henceforth, let $X$ be a Kw-P random variable following (REF )." ], [ "Moments", "Moments The $r$ th moment of $X$ can be obtained from (REF ) as $ \\nonumber \\mbox{E}(X^r) &=&\\sum ^{\\infty }_{j=0} w_j \\int _{\\beta }^{\\infty }x^r \\, g(x; \\beta , k(j+1)) dx\\\\&=&k\\,\\beta ^r\\sum ^{\\infty }_{j=0}\\frac{(j+1)\\,w_{j}}{\\left[k(j+1)-r\\right]},$ for $ r < bk$ .", "In particular, setting $r = 1$ in (REF ), the mean of $X$ reduces to $\\mu = \\mbox{E}(X) = k\\,\\beta \\sum ^{\\infty }_{j=0}\\frac{(j+1)\\,w_{j}}{\\left[k(j+1)-1\\right]},\\quad \\text{for}\\ \\ bk > 1.$ Setting $a=b=1$ , we have $w_j = \\left\\lbrace \\begin{array}{ll}1, & \\textrm {for j=0},\\\\0, & \\textrm {for j \\ge 1\\,.", "}\\\\\\end{array} \\right.$ Then, equation (REF ) reduces to (for $k>1$ ) $\\mbox{E}(X) = \\frac{k\\beta }{k-1}, \\nonumber $ which is precisely the mean of the Pareto distribution." ], [ "Incomplete moments", "Incomplete moments If $Y$ is a random variable with a Pareto distribution with parameters $\\beta $ and $k$ , the $r$ th incomplete moment of $Y$ , for $r< k$ , is given by $M_r(z) = \\int ^{z}_{\\beta }y^r\\, g(y;\\beta ,k)dy =\\frac{k\\beta ^r}{(k-r)}\\,\\Biggl [1-\\biggl (\\frac{\\beta }{z}\\biggr )^{k-r}\\Biggr ]\\,.$ From this equation, we note that $M_r(z) \\rightarrow E(Y^r)$ when $z \\rightarrow \\infty $ , whenever $k>r$ .", "Let X $\\sim $ Kw-P($\\beta ,k,a,b$ ).", "The $r$ th incomplete moment of $X$ is then equal to $M_r(z) = \\int ^{z}_{\\beta }x^r f(x;\\beta ,k,a,b)dx = k\\,\\beta ^r\\sum ^{\\infty }_{j=0}\\frac{(j+1)\\,w_{j}}{[k(j+1)-r]}\\,\\Biggl [1-\\biggl (\\frac{\\beta }{z}\\biggr )^{k(j+1)-r}\\Biggr ]\\,,$ which provided that $r< bk$ ." ], [ "Generating function", "Generating function First, the mgf $M_Y(t)$ corresponding to a random variable $Y$ with Pareto distribution with parameters $\\beta $ and $k$ is only defined for non-positive values of $t$ .", "It is given by $ \\nonumber M_Y(t)=k\\,(-\\beta t)^k\\,\\Gamma (-k,-\\beta t)\\,,\\, &\\text{if} \\,\\, t<0,$ where $\\Gamma (\\cdot ,\\cdot )$ denotes the incomplete gamma function $\\Gamma (s,x)=\\int _x^\\infty t^{s-1}e^{-t}dt\\,.$ Thus, using $M_Y(t)$ and (REF ), we can write for $t<0$ $ \\nonumber M_{X}(t)&=& \\sum _{j=0}^{\\infty } w_j \\int _{\\beta }^{\\infty }\\mathrm {e}^{tx}\\,g(x; \\beta , k(j+1))dx \\nonumber \\\\&=& k\\sum _{j=0}^{\\infty } (j+1)(-\\beta t)^{k(j+1)}\\,w_j\\,\\Gamma (-k(j+1),-\\beta t).$ In the same way, the characteristic function of the Kw-P distribution becomes $\\phi _X(t)=M_X(it)$ , where $i=\\sqrt{-1}$ is the unit imaginary number." ], [ "Quantile function and simulation", "Quantile function and simulation We present a method for simulating from the Kw-P distribution (REF ).", "The quantile function corresponding to (REF ) is $Q(u) = F^{-1}(u) = \\frac{\\beta }{\\lbrace 1 - [1 - (1 - u)^{1/b}]^{1/a}\\rbrace ^{1/k}}.$ Simulating the Kw-P random variable is straightforward.", "Let $U$ be a uniform variate on the unit interval $(0,1)$ .", "Thus, by means of the inverse transformation method, we consider the random variable $X$ given by $X = \\frac{\\beta }{\\lbrace 1 - [1 - (1 - U)^{1/b}]^{1/a} \\rbrace ^{1/k}},$ which follows (REF ), i.e., $X \\sim \\operatorname{Kw-P}(\\beta ,k,a,b)$ .", "The plots comparing the exact Kw-P densities and histograms from two simulated data sets for some parameter values are given in Figure REF .", "These plots indicate that the simulated values are consistent with the Kw-P theoretical density function." ], [ "Skewness and Kurtosis", "Skewness and Kurtosis The shortcomings of the classical kurtosis measure are well-known.", "There are many heavy-tailed distributions for which this measure is infinite.", "So, it becomes uninformative precisely when it needs to be.", "Indeed, our motivation to use quantile-based measures stemmed from the non-existance of classical kurtosis for many of the Kw distributions.", "The Bowley's skewness (see Kenney and Keeping [15]) is based on quartiles: $B=\\frac{Q(3/4)-2Q(1/2)+Q(1/4)}{Q(3/4)-Q(1/4)}$ and the Moors' kurtosis (see Moors [19]) is based on octiles: $M=\\frac{Q(7/8)-Q(5/8)-Q(3/8)+Q(1/8)}{Q(6/8)-Q(2/8)},$ where $Q(\\cdot )$ represents the quantile function.", "Plots of the skewness and kurtosis for some choices of the parameter $b$ as function of $a$ , and for some choices of the parameter $a$ as function of $b$ , for $\\beta = 1.0$ and $k=1.5$ , are shown in Figure REF .", "These plots show that the skewness and kurtosis decrease when $b$ increases for fixed $a$ and when $a$ increases for fixed $b$ ." ], [ "Order statistics", "Order statistics Moments of order statistics play an important role in quality control testing and reliability, where a practitioner needs to predict the failure of future items based on the times of a few early failures.", "These predictors are often based on moments of order statistics.", "We now derive an explicit expression for the density function of the $i$ th order statistic $X_{i:n}$ , say $f_{i:n}(x)$ , in a random sample of size $n$ from the Kw-P distribution.", "We can write $f_{i:n}(x) =\\frac{n!}{(i-1)!(n-i)!", "}f(x)\\,F^{i-1}(x)[1-F(x)]^{n-i},\\nonumber $ where $f(\\cdot )$ and $F(\\cdot )$ are the pdf and cdf of the Kw-P distribution, respectively.", "From the above equation and using the series representation (REF ) repeatedly, we obtain a useful expression for $f_{i:n}(x)$ given by $f_{i:n}(x) = \\sum _{r=0}^{\\infty }c_{i:n}^{(r)}\\,\\,g(x; k(r+1), \\beta ),$ where $c_{i:n}^{(r)} =\\frac{n!\\,a\\,b}{(i-1)!(n-i)!", "}\\sum _{l=0}^{\\infty }\\sum _{m=0}^{\\infty }\\frac{(-1)^{l+m+r}}{r+1} \\binom{i-1}{l}\\,\\binom{b(n+l+1-i)-1}{m}\\,\\binom{a(m+1)-1}{r} \\nonumber $ and $g(x; k(r+1), \\beta )$ denotes the Pareto density function with parameters $k(r+1)$ and $\\beta $ .", "So, the density function of the order statistics is simply an infinite linear combination of Pareto densities.", "The pdf of the $i$ th order statistic from a random sample of the Pareto distribution comes by setting $a=b=1$ in (REF ).", "Evidently, equation (REF ) plays an important role in the derivation of the main properties of the Kw-P order statistics.", "For example, the $s$ th raw moment of $X_{i:n}$ can be expressed as $E(X^s_{i:n}) = k \\,\\beta ^s\\sum _{r=0}^{\\infty }\\frac{(r+1)c_{i:n}^{(r)}}{k(r+1)-s}.$ The L-moments are analogous to the ordinary moments but can be estimated by linear combinations of order statistics.", "They are linear functions of expected order statistics defined by $\\lambda _{m+1} = \\frac{1}{m+1} \\sum _{k=0}^{m}(-1)^k\\binom{m}{k}\\,\\operatorname{E}(X_{m+1-k:m+1}),\\,\\,m=0,1,\\ldots $ The first four L-moments are: $\\lambda _1=\\mbox{E}(X_{1:1})$ , $\\lambda _2=\\frac{1}{2}\\mbox{E}(X_{2:2}-X_{1:2})$ , $\\lambda _3=\\frac{1}{3}\\mbox{E}(X_{3:3}-2X_{2:3}+X_{1:3})$ and $\\lambda _4=\\frac{1}{4}\\mbox{E}(X_{4:4}-3X_{3:4}+ 3X_{2:4}-X_{1:4})$ .", "The L-moments have the advantage that they exist whenever the mean of the distribution exists, even though some higher moments may not exist, and are relatively robust to the effects of outliers.", "From equation (REF ) with $s=1$ , we can easily obtain explicit expressions for the L-moments of $X$ ." ], [ "Mean deviations", "Mean deviations The mean deviations about the mean and the median can be used as measures of spread in a population.", "Let $\\mu = E(X)$ and $m$ be the mean and the median of the Kw-P distribution, respectively.", "The mean deviations about the mean and about the median can be calculated as $D(\\mu ) = E(|X-\\mu |) = \\int _{\\beta }^{\\infty }|x-\\mu |\\,f(x)dx \\nonumber $ and $D(m) = E(|X-m|) = \\int _{\\beta }^{\\infty }|x-m|f(x)\\,dx\\,, \\nonumber $ respectively.", "We obtain $D(\\mu ) = \\int _{\\beta }^{\\infty }|x-\\mu |\\,f(x)dx = 2 \\mu F(\\mu ) - 2M_1(\\mu )\\,,$ where $M_1(\\mu )$ denotes the first incomplete moment calculated from (REF ) for $r=1$ .", "Similarly, the mean deviation about the median follows as $D(m) = \\int _{\\beta }^{\\infty }|x-m|f(x)dx = \\mu -2M_1(m)\\,.$" ], [ "Rényi entropy", "Rényi entropy The entropy of a random variable $X$ is a measure of uncertainty variation.", "The Rényi entropy is defined as $I_{R}(\\delta ) = \\frac{1}{1-\\delta }\\log \\left[I(\\delta )\\right],\\nonumber $ where $I(\\delta )=\\int _{\\mathbb {R}}{f^{\\delta }(x)dx}$ , $\\delta >0$ and $\\delta \\ne 1$ .", "We have $I(\\delta )= a^\\delta b^\\delta k^\\delta \\beta ^{k\\delta }\\int _{\\beta }^{\\infty }\\frac{1}{x^{\\delta (k+1)}}\\left[1-\\left(\\frac{\\beta }{x}\\right)^k\\right]^{\\delta (a-1)}\\left\\lbrace 1-\\left[1-\\left(\\frac{\\beta }{x}\\right)^k\\right]^{a}\\right\\rbrace ^{\\delta (b-1)}dx.\\nonumber $ By expanding the last term of the above integrand as in equation (REF ), we obtain $I(\\delta )= a^\\delta b^\\delta k^\\delta \\beta ^{k\\delta }\\sum _{j=0}^{\\infty }(-1)^{j}\\binom{\\delta (b-1)}{j}\\int _{\\beta }^{\\infty }\\frac{1}{x^{\\delta (k+1)}}\\left[1-\\left(\\frac{\\beta }{x}\\right)^k\\right]^{a(\\delta +j)-\\delta }dx.\\nonumber $ Transforming variables, this equation becomes $I(\\delta )= a^\\delta b^\\delta k^{\\delta -1} \\beta ^{\\delta +1}\\sum _{j=0}^{\\infty }(-1)^{j}\\,\\binom{\\delta (b-1)}{j}B\\left(a(\\delta +j)-\\delta +1,\\frac{\\delta (k+1)-1}{k}\\right), \\nonumber $ where $B(a,b) = \\int ^{1}_{0}t^{a-1} (1-t)^{b-1}dt$ denotes the beta function.", "Hence, the Rényi entropy reduces to $I_{R}(\\delta ) &=& \\frac{\\delta \\log (a\\,b)}{1-\\delta } - \\log k + \\log \\beta \\\\&+& \\frac{1}{1-\\delta }\\log \\sum _{j=0}^{\\infty }(-1)^{j}\\binom{\\delta (b-1)}{j}B\\left(a(\\delta +j)-\\delta +1,\\frac{\\delta (k+1)-1}{k}\\right).$" ], [ "Estimation and information matrix", "Estimation and information matrix In this section, we discuss maximum likelihood estimation and inference for the Kw-P distribution.", "Let $x_1, \\ldots , x_n$ be a random sample from $X \\sim \\mbox{Kw-P}(\\beta , k, a, b)$ and let $\\theta =(\\beta , k, a, b)^\\top $ be the vector of the model parameters.", "The log-likelihood function for $\\theta $ reduces to $ \\nonumber \\ell (\\theta ) &=& n \\log a + n \\log b + n\\log k + nk\\log \\beta -(k+1)\\sum _{i=1}^{n}\\log (x_i)\\\\&+& (a-1)\\sum _{i=1}^{n} \\log \\left[1-\\left(\\frac{\\beta }{x_i}\\right)^k\\right] + (b-1)\\sum _{i=1}^{n} \\log \\left\\lbrace 1-\\left[1-\\left(\\frac{\\beta }{x_i}\\right)^k\\right]^{a}\\right\\rbrace .$ The score vector is $U(\\theta )=(\\partial \\ell /\\partial k, \\partial \\ell /\\partial a, \\partial \\ell /\\partial b)^\\top $ , where the components corresponding to the model parameters are calculated by differentiating (REF ).", "By setting $z_i = 1- \\left(\\beta /x_i\\right)^k$ , we obtain $\\frac{\\partial \\ell }{\\partial k} &= \\frac{n}{k} +\\frac{1}{k}\\sum _{i=1}^{n}\\log (1-z_i) - \\frac{(a-1)}{k}\\sum _{i=1}^{n} \\frac{(1-z_i)\\log (1-z_i)}{z_i} \\\\&+ \\frac{a (b-1)}{k} \\sum _{i=1}^{n} \\frac{z_i^{a-1}(1-z_i)\\log (1-z_i)}{ (1-z_{i}^{a})}, \\\\\\frac{\\partial \\ell }{\\partial a} &= \\frac{n}{a} + \\sum _{i=1}^{n}\\log z_i - (b-1)\\sum _{i=1}^{n}\\frac{z_i^{a} \\log z_i}{1 - z_i^{a}}\\\\\\multicolumn{2}{l}{\\text{and}}\\\\\\frac{\\partial \\ell }{\\partial b} &= \\frac{n}{b} + \\sum _{i=1}^{n}\\log (1 - z_i^{a}).", "\\\\$ The maximum likelihood estimates (MLEs) of the parameters are the solutions of the nonlinear equations $\\nabla \\ell =0$ , which are solved iteratively.", "The observed information matrix given by $J_n(\\theta )=n\\left[\\begin{array}{ccc}J_{k k }&J_{k a}& J_{k b}\\\\J_{a k }&J_{a a}& J_{a b}\\\\J_{b k }&J_{b a}& J_{b b}\\\\\\end{array}\\right],$ whose elements are $J_{kk} &= -\\frac{n}{k^2} - \\frac{2(a-1)}{k}\\sum _{i=1}^{n}\\frac{(1-z_i)\\log (1-z_i)}{z_i^2} \\\\&+ \\frac{2a(b-1)}{k}\\sum _{i=1}^{n}\\frac{z_i^{a-1}(1-z_i)\\log (1-z_i)\\left[a-(a-1)(z_i^a+z_i^{-1})+(a-2)z_i^{a-1}\\right]}{(1-z_i^a)^2},\\\\J_{ka} &= \\frac{(b-1)}{k}\\sum _{i=1}^{n}\\frac{z_{i}^{a-1}(1-z_i)\\log (1-z_i)\\left[1-z_i^a - a\\log z_{i}\\right]}{(1-z_{i}^{a})^2}- \\frac{1}{k} \\sum _{i=1}^{n}\\frac{(1-z_i)\\log (1-z_i)}{z_i},\\\\J_{kb} &= \\frac{a}{k} \\sum _{i=1}^{n}\\frac{z_{i}^{a-1}(1-z_i)\\log (1-z_i)}{1-z_{i}^{a}}, \\quad J_{a a} = -\\frac{n}{a^2} - 2(b-1) \\sum _{i=1}^{n} \\frac{z_i^{a}\\log z_i}{(1-z_i^{a})^2}, \\\\J_{a b} &= -\\sum _{i=1}^{n} \\frac{z_i^{a}\\log z_i}{1-z_i^{a}} \\quad \\mbox{and} \\quad J_{bb} = -\\frac{n}{b^2}.\\\\$" ], [ "Simulation study and application", "Simulation study and application In this section, we illustrate the usefulness of the Kw-P distribution." ], [ "Simulation study", "Simulation study We conduct Monte Carlo simulation studies to assess on the finite sample behavior of the MLEs of $\\beta ,k, a$ and $b$ .", "All results were obtained from 1000 Monte Carlo replications and the simulations were carried out using the statistical software package R. In each replication, a random sample of size n is drawn from the Kw-P($\\beta ,k,a,b$ ) distribution and the BFGS method has been used by the authors for maximizing the total log-likelihood function $\\ell (\\theta )$ .", "The Kw-P random number generation was performed using the inversion method.", "The true parameter values used in the data generating processes are $\\beta = 1.5, k = 1.0, a = 0.5$ and $b = 2.5$ .", "Table REF lists the means of the MLEs of the four parameters that index the Kw-P distribution along with the respective biases for sample sizes $n = 30, n = 50$ and $n = 100$ .", "The figures in Table REF indicate that the biases of the MLEs of $\\beta , k, a,$ and $b$ decay toward zero as the sample size increases, as expected." ], [ "The Wheaton River data", "The Wheaton River data The data correspond to the exceedances of flood peaks (in $\\mbox{m}^3/$ s) of the Wheaton River near Carcross in Yukon Territory, Canada.", "The data consist of 72 exceedances for the years 1958–1984, rounded to one decimal place.", "They were analysed by Choulakian and Stephens [3] and are listed in Table REF .", "The distribution is highly skewed to the left.", "Recently, Akinsete et al.", "[1] and Mahmoudi [18] analysed these data using the BP and BGP distributions, respectively.", "We fit the Kw-P distribution to these data and compare the results with those by fitting some of its sub-models such as the EP and Pareto distributions, as well as the non-nested BP distribution.", "The required numerical evaluations are implemented using the SAS (PROCNLMIXED) and R softwares.", "Tables REF and REF provide some descriptive statistics and the MLEs (with corresponding standard errors in parentheses) of the model parameters.", "Since $x \\ge \\beta $ , the MLE of $\\beta $ is the first-order statistic $x_{(1)}$ , accordingly to Akinsete et al. [1].", "Since the values of the Akaike information criterion (AIC), Bayesian information criterion (BIC) and consistent Akaike information criterion (CAIC) are smaller for the Kw-P distribution compared with those values of the other models, the new distribution seems to be a very competitive model to these data.", "Plots of the estimated pdf and cdf of the Kw-P, BP, EP and Pareto models fitted to these data are given in Figure REF .", "They indicate that the Kw-P distribution is superior to the other distributions in terms of model fitting.", "Table REF gives the values of the Kolmogorov-Smirnov (K-S) statistic and of $-2\\ell (\\hat{\\theta })$ .", "From these figures, we conclude that the Kw-P distribution provides a better fit to these data than the BP, EP and Pareto models." ], [ "Concluding remarks", "Concluding remarks The well-known two-parameter Pareto distribution is extended by introducing two extra shape parameters, thus defining the Kumaraswamy Pareto (Kw-P) distribution having a broader class of hazard rate functions.", "This is achieved by taking (REF ) as the baseline cumulative distribution of the generalized class of Kumaraswamy distributions defined by Cordeiro and de Castro [4].", "A detailed study on the mathematical properties of the new distribution is presented.", "The new model includes as special sub-models the Pareto and exponentiated Pareto (EP) distributions (Gupta et al. [12]).", "We obtain the moment generating function, ordinary moments, order statistics and their moments and Rényi entropy.", "The estimation of the model parameters is approached by maximum likelihood and the observed information matrix is derived.", "An application to a real data set shows that the fit of the new model is superior to the fits of its main sub-models.", "We hope that the proposed model may attract wider applications in statistics.", "Acknowledgments We gratefully acknowledge the grants from CAPES and CNPq (Brazil)." ] ]

[ [ "Internal convection in thermoelectric generator models" ], [ "Abstract Coupling between heat and electrical currents is at the heart of thermoelectric processes.", "From a thermal viewpoint this may be seen as an additional thermal flux linked to the appearance of electrical current in a given thermoelectric system.", "Since this additional flux is associated to the global displacement of charge carriers in the system, it can be qualified as convective in opposition to the conductive part associated with both phonons transport and heat transport by electrons under open circuit condition, as, e.g., in the Wiedemann-Franz relation.", "In this article we demonstrate that considering the convective part of the thermal flux allows both new insight into the thermoelectric energy conversion and the derivation of the maximum power condition for generators with realistic thermal coupling." ], [ "Introduction", "The field of thermoelectricity was born in the 1820s after Seebeck reported his observation of a phenomenon he interpreted as thermomagnetism.", "Orsted, in the same years, gave a better interpretation of the phenomenon and termed it thermoelectricity: the appearance of an electrical current in a circuit subjected to a temperature difference.", "The name of Seebeck was later given to the coefficient linking the temperature difference $\\Delta T$ and the resulting electromotive force $\\Delta V$ : $\\alpha = - \\Delta V / \\Delta T$ .", "A few years after Seebeck's works were published [1], Peltier noticed that, conversely, an electrical current $I$ may absorb or release heat at a junction between two different materials [2].", "The flux of absorbed heat $I_Q$ is related to the electrical current $I$ through the difference of Peltier coefficients $\\Pi $ between the materials of the junction: $I_Q = (\\Pi _b - \\Pi _a) I$ (with $I$ positive when flowing from material $a$ to material $b$ ).", "The relation between these two effects was later clearly expressed by Thomson who established the connection between the Peltier and Seebeck coefficients: $\\Pi = \\alpha T$ , with $T$ the temperature of the considered material [3].", "This relation demonstrates that both effects have the same physical origin: the ability of charge carriers to carry heat along with electrical charge.", "As thermoelectric (TE) devices possess the particularity to strongly couple heat and electrical transports, they can be used in refrigeration or energy conversion applications [4].", "However to achieve good performances both TE materials and device design have to be optimized.", "For materials, optimization amounts to increasing the so-called figure of merit $ZT$ (see for example the recent review by Shakouri [5]).", "Yet, to make the best use of the materials it is necessary also to lead a reflection on the integration of the devices into an external environment.", "One thus needs a complete model of thermoelectric device to fully comprehend the interaction between the constitutive laws of the device and the laws related to the external exchanges.", "In this work we propose a simple model of thermoelectric generator (TEG) that accounts for the heat transported by electrical carriers; this results in the introduction of an effective thermal conductance associated with this phenomenon.", "As this heat propagation is associated with global movement of particles, by analogy with fluid mechanics, we call it convection.", "We then study and illustrate the consequences that this additional term may bear on both the physics of TE energy conversion and on the practical design and optimization of thermoelectric generators (TEGs)." ], [ "Thermoelectric modeling: Accounting for a convective term", "Let us consider a thermoelectric generator connected to two heat baths at temperature $T_{\\rm hot}$ and $T_{\\rm cold}$ ($T_{\\rm hot} > T_{\\rm cold}$ ) respectively with an average temperature $T = (T_{\\rm hot} + T_{\\rm cold}) / 2$ .", "Though a thermoelectric device may operate indifferently either as a cooler or as a generator, in this work, we restrict our analysis to the case of a generator to which an electrical load resistance $R_{load}$ is connected in order to extract power.", "Now that we have set the thermal and electrical environment, we state the phenomenological laws that govern the device by relating the electrical current $I$ and the average thermal current $I_Q$ to the voltage $\\Delta V$ and the temperature difference $\\Delta T$ across the generator.", "The equations follow directly from the force-flux formalism introduced by Onsager and then extended to thermoelectricity by Callen [6].", "For our purpose, we find convenient to use the macroscopic coefficients and potential differences, rather than local coefficients and gradients, and we get: $ I = \\frac{1}{R}\\Delta V + \\frac{\\alpha }{R} \\Delta T$ $ I_Q = \\frac{\\alpha T}{R} \\Delta V + \\left(\\frac{\\alpha ^2 T}{R} + K_{0}\\right) \\Delta T$ where $R$ is the electrical resistance, $K_0$ the thermal conductance under open-circuit condition and $\\alpha $ the Seebeck coefficient.", "The TEG is also characterized by its figure of merit $ZT$ obtained from these three parameters: $ZT = \\alpha ^2 T /R K_0$ .", "This dimensionless number gives a direct quantitative estimation of the generator's quality [7].", "The convention for the currents and potential differences, $\\Delta V = V_1 - V_0$ and $\\Delta T = T_{\\rm hot} - T_{\\rm cold}$ , are displayed in Fig.", "(REF ).", "Equation (REF ) gives a definition of the Seebeck coefficient under open-circuit condition ($I = 0$ ), namely $\\alpha = - \\Delta V / \\Delta T$ .", "If the Seebeck coefficient $\\alpha $ vanishes, thermoelectric effects become negligible, and one recovers the traditional macroscopic expressions of Ohm's law and Fourier's law, from Eqs.", "(REF ) and (REF ) respectively.", "The two previous equations may be combined to express the thermal flux as the sum of two terms: $I_Q = \\alpha T I + K_{0} \\Delta T$ Figure: Thermoelectric generator: Decomposition of the TEG thermal conductance into a conductive part K 0 K_0 and a convective part K conv K_{\\rm conv}.The first term is the contribution due to the heat transported within the electrical current, the so-called convective thermal flux [8], [9], and the second is the conductive part traditionally associated with Fourier's law.", "The quantity of energy transported by each carrier is $|\\alpha | T e$ , $e$ being the elementary electric charge.", "To compare these two terms we now rewrite Eq.", "(REF ) to obtain a form similar to Fourier's law: A relation of proportionality between the heat current and the temperature difference.", "For this purpose we express the electrical current through the resistive load as a function of $\\Delta T$ .", "From Ohm's law: $I = -\\Delta V / R_{\\rm load}$ , we get: $I = \\frac{\\Delta V + \\alpha \\Delta T}{R} = \\frac{\\alpha \\Delta T}{R_{\\rm load} + R}$ After substitution of the above expression for $I$ in Eq.", "(REF ), the average heat flux may be expressed as: $I_Q = \\left( \\underbrace{\\frac{\\alpha ^2 T}{R_{\\rm load} + R}}_{\\displaystyle K_{\\rm conv}} + K_{0} \\right)\\Delta T = K_{\\rm TEG}\\Delta T,$ where an additional term, $K_{\\rm conv}$ , for the total thermal conductance of the generator now appears clearly.", "This term is directly related to the electrical condition on the circuit since the load resistance $R_{\\rm load}$ is explicitly involved in its definition.", "The effective thermal conductance of the generator $K_{\\rm TEG}$ is thereby equivalent to the two thermal conductances $K_{\\rm conv}$ and $K_{0}$ in parallel as illustrated in Fig.", "(REF ).", "When $R_{\\rm load} \\rightarrow \\infty $ , i.e., in open-circuit condition, we recover, as expected, the fact that $K_{\\rm TEG}$ is equivalent to $K_{0}$ only, whereas when $R_{\\rm load} \\rightarrow 0$ , $K_{\\rm TEG} = \\alpha ^2 T / R + K_{0}$ .", "To derive Eq.", "(REF ) we made a few assumptions: We considered only an average value for the heat flux; we neglected the Joule effect and the thermal energy converted into electric energy along the device.", "These approximations are reasonable if the temperature difference $\\Delta T$ is kept small; a condition that must be satisfied to make use of constant coefficients to describe the TE generator.", "All these assumptions amount to stating that we only consider a linear framework.", "Second, we neglected the heat flowing through the electrical load.", "So, dealing with a true unileg generator as in Ref.", "[10], implies $K_{\\rm TEG} \\gg K_{\\rm load}$ .", "This configuration may also be related to a generator with two legs: The parameters then used are only equivalent ones [7], in which case the load sustains no temperature difference, so that no heat flux goes through it." ], [ "Illustrative examples of the effects of thermoelectric convection", "In this section we illustrate the impact that the convective part of the generator thermal conductance may have on the properties and operation of a TEG.", "We first look at the construction and meaning of the figure of merit $ZT$ ; next we discuss the question of optimization of thermoelectric generators when the connections to the heat reservoirs are not perfect; finally we examine the effective thermal conductance of two thermoelectric modules in parallel." ], [ "On the meaning of the figure of merit ZT", "To discuss the properties of thermoelectric systems, we find convenient to proceed by analogy using the classical heat transfer description in terms of convection and conduction.", "To determine whether heat transfer in a fluid is dominated either by conduction or by convection, one may introduce the Prandtl number $\\sigma _{\\rm p}$ defined as $\\sigma _{\\rm p} = \\frac{\\nu }{D}$ where $\\nu $ is the kinematic viscosity of this fluid, characterizing momentum diffusion, and $D$ its thermal diffusivity, characterizing heat diffusion [11].", "We retained this idea to define a thermoelectric Prandtl number $(\\sigma _{\\rm p})_{_{\\rm TE}}$ given by the ratio of the convective thermal flux and of the conductive thermal flux: $(\\sigma _{\\rm p})_{_{\\rm TE}} = \\frac{K_{\\rm conv} \\Delta T}{K_{0} \\Delta T} = \\frac{\\alpha ^2 T}{K_{0}(R + R_{\\rm load})} = \\frac{ZT}{1 + R_{\\rm load} / R}$ It appears that $(\\sigma _{\\rm p})_{_{\\rm TE}}$ is proportional $ZT$ .", "The proportionality coefficient depends on the ratio between the electrical resistances of the generator and the load.", "If the electrical circuit is shorted, i.e., if $R_{\\rm load} \\ll R$ , the thermoelectric Prandtl number $(\\sigma _{\\rm p})_{_{\\rm TE}}$ is then equal to $ZT$ , which is the maximum value of the ratio between the convective part and the conductive part of the thermal flux.", "It is consistent with the fact that $ZT$ must be increased to improve the TEG performances: The convective thermal flux is a direct consequence of the coupling between heat and electrical charge transport, and, as such, it is a necessary process to obtain energy conversion.", "Conversely, the conductive thermal flux amounts to pure loss effects only: So maximizing $ZT$ is equivalent to maximizing the useful part of the thermal flux with respect to the losses.", "The ratio $R_{\\rm load} / R$ controls the convective component of the heat flux.", "However, maximization of $(\\sigma _{\\rm p})_{_{\\rm TE}}$ does not lead to the best performances: To extract power from the heat that flows in the generator, one has to satisfy impedance matching too.", "If $R_{\\rm load}$ becomes too small, the electrical current, and hence the convective thermal current, will increase but the power will decrease.", "It appears that the optimal value of $(\\sigma _{\\rm p})_{_{\\rm TE}}$ associated to the maximum power output is not $ZT$ but $ZT /2$ .", "We can even extend this analysis to the maximization of the TEG efficiency: As shown by Ioffe [7], for a TEG with constant parameters the efficiency is maximized when $R_{\\rm load} / R = \\sqrt{1 + ZT}$ .", "Thereby, for the efficiency the optimal value of $(\\sigma _{\\rm p})_{_{\\rm TE}}$ is $ ZT / (1 + \\sqrt{1 + ZT})$ .", "This last result should be compared with the compatibility approach of thermoelectric optimization: Snyder and Ursell [12] demonstrated that in order to obtain the best efficiency, the relative current $u = J / (\\kappa \\nabla T)$ (with $J$ the electrical current density and $\\kappa $ the local thermal conductivity under open circuit-condition) has to be kept constant along the device and should be equal to an optimal value $s = (\\sqrt{1 + ZT} - 1) / \\alpha T$ .", "Expressed differently the condition for efficiency maximization reads: $\\alpha T J / (\\kappa \\nabla T) = (\\sqrt{1 + ZT} - 1)$ .", "As the term on the left hand side is a local definition of $(\\sigma _{\\rm p})_{_{\\rm TE}}$ , we recover exactly the same result as above.", "It is interesting to note that similar conditions were derived using the ratio of the total heat current to the convective thermal flux in Refs.", "[13], [14].", "Finally the notion of convective thermal conductance can also be useful to practically determine the figure of merit of a thermoelectric generator: In 1966, Lisker proposed to obtain $ZT$ by measuring the effective thermal conductance of the TE module for two different electrical load conditions instead of extracting from three different measurements the value of each parameter $R$ , $\\alpha $ and $K_0$ [15].", "The ratio $K_{\\Delta V = 0}/K_0$ is indeed equal to $ZT + 1$ ." ], [ "Thermal matching of thermoelectric generators", "Assume that the generator is not directly coupled to the heat reservoirs; the finite thermal conductance of the contacts must be taken into account.", "These are denoted $K_{\\rm hot}$ on the hot side and $K_{\\rm cold}$ on the cold side of the generator.", "The total equivalent conductance for both contacts is given by $K_{\\rm contact}^{-1} = K_{\\rm hot}^{-1} + K_{\\rm cold}^{-1}$ .", "The main effect of these imperfect contacts is that the actual temperature difference across the TEG is not $\\Delta T$ but $\\Delta T^{\\prime } = T_{\\rm hM} - T_{\\rm cM}$ as illustrated on the left side of Fig.", "(REF ).", "We first need to determine the relation between these two temperature differences.", "To do so we use an analogue to the potential divider formula for the temperature, assuming that the heat flux remains constant along the structure (this hypothesis is the same as that used to derive the expression of $K_{\\rm conv}$ ).", "We obtain the following relation: $\\Delta T^{\\prime } = T_{\\rm hM} - T_{\\rm cM} \\approx \\frac{K_{\\rm contact}}{K_{\\rm TEG}+K_{\\rm contact}}~\\Delta T$ Since the thermal conductance $K_{\\rm TEG}$ depends on the electrical load resistance through $K_{\\rm conv}$ , the temperature difference across the generator, $\\Delta T^{\\prime }$ , is modulated by the electrical load condition.", "The consequence of this modulation is that the electrical circuit modeling of the TEG as diplayed in Fig.", "(REF ) is no longer a Thévenin generator: In such model the voltage source must be independent of the load which is not the case when finite thermal contacts are introduced.", "To keep using a proper Thévenin generator we split the thermoelectric voltage $V_{\\rm oc} = \\alpha \\Delta T^{\\prime }$ into two terms: One is independent of the electrical load, and the other is linked to an internal resistance [16]: $V_{\\rm oc} =\\alpha \\Delta T \\frac{K_{\\rm contact}}{K_{0}+K_{\\rm contact}}-I \\frac{\\alpha ^2 T}{K_{_{I=0}}+K_{\\rm contact}}$ This equation is of the form: $V_{\\rm oc} = V_{\\rm oc}^{\\prime } - IR^{\\prime }$ with $R^{\\prime }= \\alpha ^2 T / (K_{_{I=0}}+K_{\\rm contact})$ .", "We thus obtain a rigorous Thévenin modeling of the electrical part of the TEG with the definitions of the open circuit voltage given by $V_{\\rm oc}^{\\prime }$ and the internal resistance is $R_{\\rm TEG} = R + R^{\\prime }$ : As for the effective thermal conductance, the effective electrical resistance of the generator shows an additional contribution directly related to the thermal conditions imposed on the generator through the thermoelectric coupling coefficient $\\alpha $ .", "Further, it is remarkable that the expressions of $R^{\\prime }$ and $K_{\\rm conv}$ present the same form.", "One can switch from one to the other by changing thermal conductances to electrical resistances and vice versa.", "Recently, the introduction of an extra electrical resistance depending on thermal conditions was also suggested by Spry [17].", "With our model, the power produced by the TEG may thus be simply expressed as: $P = \\Delta VI = \\frac{{V_{\\rm oc}^{\\prime }}^2 R_{\\rm load}}{(R_{\\rm TEG}+R_{\\rm load})^2}$ from which we deduce that power maximization is obtained for $R_{\\rm load} = R_{\\rm TEG}$ .", "Such a condition was already derived mathematically by Freunek and co-workers [18].", "Figure: Thermoelectric generator with dissipative thermal coupling (left) and maximum power output as a function of the ratio K 0 /K contact K_0 / K_{\\rm contact} for different values of ZT (right).As regards thermal matching, Stevens proposed that to maximize power production, the TEG should be thermally matched with $K_{0} = K_{\\rm contact}$ .", "Let us examine the situation using the equations we just derived.", "An expression for the maximum power may be obtained using Eq.", "(REF ) in the case of electrical matching and the definition of $R_{\\rm TEG}$ : $P_{\\rm max} = \\frac{(K_{\\rm contact}\\Delta T)^2}{4(K_{_{I=0}}+K_{\\rm contact})T}~\\frac{ZT}{1+ZT+K_{\\rm contact}/K_{_{I=0}}}$ For a given value of $ZT$ , $P_{\\rm max}$ shows a clear dependence on the ratio $K_{_{I=0}} / K_{\\rm contact}$ .", "If we consider a fixed value of $K_{\\rm contact}$ , due for example to technological constraints, the choice of the value for $K_{_{I=0}}$ will strongly impact the ability of the generator to produce power.", "On the right side of Fig.", "(REF ) we show the relation between $P_{\\rm max}$ and $K_{_{I=0}} / K_{\\rm contact}$ for three different values of $ZT$ (1, 3 and 10).", "Note that for each point electrical impedance matching is achieved.", "We see that this condition for power maximization through thermal impedance matching slightly differs from the one obtained by Stevens, and that it depends on $ZT$ .", "Actually the optimum value of $K_0$ corresponds to the thermal matching between $K_{\\rm contact}$ and $K_{\\rm TEG}$ : $K_{\\rm TEG} = K_{\\rm contact}$ [16].", "The dependence on $ZT$ is explained by the fact that the greater $ZT$ is, the greater the contribution of $K_{\\rm conv}$ to $K_{\\rm TEG}$ is.", "Thus for sufficiently high TEG efficiency, i.e., high $ZT$ , the discrepancy between these two conditions of thermal impedance matching is not negligible: See the difference in Fig.", "(REF ) between the maximum power and the value of power for the condition $K_{_{I=0}} = K_{\\rm contact}$ displayed by the vertical dot-dashed line.", "Finally we express the simultaneous conditions of thermal and electrical matching in terms of ratios $K_{0} / K_{\\rm contact}$ and $R_{\\rm load} / R$ (as $R$ and $K_{0}$ are more suitable quantities for device design than $K_{\\rm TEG}$ and $R_{\\rm TEG}$ ): $\\frac{K_{0}}{K_{\\rm contact}} = \\frac{1}{\\sqrt{1 + ZT}}$ $\\frac{R_{\\rm load}}{R} = \\sqrt{1 + ZT}$ Once again, we emphasize the symmetry that arises between thermal and electrical quantities.", "In a recent publication, Yazawa and Shakouri obtained the same relations from a more mathematical viewpoint [20]." ], [ "Effective thermal conduction increase in inhomogeneous material {{cite:41342c4be6c93a2b1dbc44c2f30e5919ec496646}}", "Our last example concerns the determination of the effective thermal conductance of two thermoelectric modules in parallel as represented in Fig.", "(REF ).", "We keep the same notations for modules' properties except that we add a subscript (1 or 2) to distinguish the two modules.", "First, we derive the total average thermal flux $I_{Q_{_{\\rm eq}}}$ inside the system: It is given by the sum of the average thermal flux for each module, with a conductive and a convective part for each.", "So we have: $I_{Q_{_{\\rm eq}}}=I_{Q_{_{1}}}+I_{Q_{_{2}}} = \\alpha _1 T I_1 + K_{0,1}\\Delta T + \\alpha _2 T I_2 + K_{0,2}\\Delta T$ Figure: Association of 2 thermoelectric generators in parallel.As we focus on the open electrical circuit condition to determine the equivalent thermal conductance $K_{0,{\\rm eq}}$ , the total electrical current is set to $I = 0$ .", "However this does not mean that the currents $I_1$ and $I_2$ vanish as well: The only consequence is that $I_1 = - I_2$ .", "A simple analysis of the electrical circuit in Fig.", "(REF ) leads to the following expression for the internal current: $I_1 = - I_2 = \\frac{(\\alpha _1 - \\alpha _2)\\Delta T}{R_1 + R_2}$ This internal current only exists if the electromotive forces, given by $\\alpha \\Delta T$ , do not compensate each other, i.e., if $\\alpha _1 \\ne \\alpha _2$ .", "In that case we have to consider the convective contribution to the thermal flux.", "Substituting Eq.", "(REF ) in Eq.", "(REF ) and simplifying by $\\Delta T$ we get the equivalent open circuit thermal conductance: $K_{0,{\\rm eq}} = K_{0,1} + K_{0,2} + \\frac{(\\alpha _1 - \\alpha _2)^2T}{R_1 + R_2}$ This last equation demonstrates that the convective component of the thermal flux has a significant impact on the effective thermal conductance of the two thermoelectric modules in parallel if they possess very dissimilar Seebeck coefficients.", "However this term is of limited interest if the electrical resistances of the modules are too high: the magnitude of electrical internal current is then small and, consequently, so is the convective thermal flux.", "Knowledge of the effective thermal conductance may be useful when dealing with superlattices for which the layers can be viewed as parallel modules [9].", "More surprisingly, Price obtained a similar expression for the effective thermal conductance of an ambipolar conductor where the subscripts 1 and 2 are associated with electrons and holes repectively: The two types of carriers are indeed considered as evolving in parallel inside the system [22]." ], [ "Conclusion", "Convective thermal flux will have more and more impact on the behavior of thermoelectric generator as the performance of materials will be increased.", "Indeed we have shown that the proportion of convective to conductive thermal current is directly related to the figure of merit $ZT$ which is expected to increase significantly in the future to allow a wider use of thermoelectric power production.", "In this paper we have demonstrated that this additional contribution to heat flow should not be neglected for device design as well as for materials research, especially those concerning composites where internal currents can develop and lead to a performance decrease." ], [ "Acknowledgments", "This work is part of the SYSPACTE projects funded by the Fonds Unifié Interministériel 7.", "Y.", "A. acknowledges financial support from the Ministère de l'Enseignement Supérieur et de la Recherche." ] ]

[ [ "Effect of dipolar interactions on optical nonlinearity of\n two-dimensional nanocomposites" ], [ "Abstract In this work, we calculate the contribution of dipole-dipole interactions to the optical nonlinearity of the two-dimensional random ensemble of nanoparticles that possess a set of exciton levels, for example, quantum dots.", "The analytical expressions for the contributions in the cases of TM and TE-polarized light waves propagating along the plane are obtained.", "It is shown that the optical nonlinearity, caused by the dipole-dipole interactions in the planar ensemble of the nanoparticles, is several times smaller than the similar nonlinearity of the bulk nanocomposite.", "This type of optical nonlinearity is expected to be observed at timescales much larger than the quantum dot exciton rise time.", "The proposed method may be applied to various types of the nanocomposite shapes." ], [ "Effect of dipolar interactions on optical nonlinearity of two-dimensional nanocomposites Andrey V. Panov$^{1,2}$ [email protected] $^1$ Institute of Automation and Control Processes, Far East Branch of Russian Academy of Sciences, 5, Radio st., Vladivostok, 690041, Russia $^2$ School of Natural Sciences, Far Eastern Federal University, 8, Sukhanova st., Vladivostok, 690950, Russia In this work, we calculate the contribution of dipole-dipole interactions to the optical nonlinearity of the two-dimensional random ensemble of nanoparticles that possess a set of exciton levels, for example, quantum dots.", "The analytical expressions for the contributions in the cases of TM and TE-polarized light waves propagating along the plane are obtained.", "It is shown that the optical nonlinearity, caused by the dipole-dipole interactions in the planar ensemble of the nanoparticles, is several times smaller than the similar nonlinearity of the bulk nanocomposite.", "This type of optical nonlinearity is expected to be observed at timescales much larger than the quantum dot exciton rise time.", "The proposed method may be applied to various types of the nanocomposite shapes.", "The final publication (Andrey V Panov, 2013, J. Opt.", "15, 055201) is available at http://iopscience.iop.org/2040-8986/15/5/055201, DOI: 10.1088/2040-8978/15/5/055201 78.67.Sc, 42.65.An, 42.65.Hw In recent years, much attention has been given to optical nanostructures containing, for instance, semiconductor quantum dots.", "Frequently, such nanocomposites are organized as two-dimensional structures [1], [2], [3], [4].", "Normally, these structures are dense-packed systems of nanoparticles.", "The high concentration of nanoparticles will require taking into account the contribution of interparticle interactions, specifically long-range dipole-dipole ones, to the nanocomposite optical properties.", "Usually, the dipole-dipole coupling in the system of nanoparticles is treated by numerical methods [5], [6].", "This approach is very popular for description of the systems of oscillating dipoles.", "Widely used approximations based on homogenization (effective medium theories) have restrictions on the size of inclusions [7], typical quantum dots do not fall in this range.", "In Ref.", "[8] mean-field theory was used in order to calculate the impact of dipole-dipole interactions in the materials doped with five-level nanoparticles.", "It is worth mentioning that the mean-field approaches are only suitable for the medium comprising a set of dipoles with some ordering and for the nonzero mean field.", "In this study, we obtain analytical expressions describing the effect of the dipolar interactions on the nonlinear dielectric susceptibility of quantum dot planar systems.", "In these nanocomposites, incident laser light can excite the electric dipole moment in the nanoparticles.", "Upon increasing the quantity of the dipolar excited nanoparticles, the dipole-dipole interactions in the nanocomposite become of particular importance.", "To calculate the contribution of the dipolar interactions to nonlinear optical characteristics of the two-dimensional system of randomly positioned cylindrical nanoparticles, the approach developed in Ref.", "[9] can be used.", "In this approximation, it is assumed that the mean value $p$ of the induced electric dipole moment of the particle is proportional to the electric field amplitude $E_{in}$ of the incident (external) radiation, $p = \\varepsilon _m \\alpha v E_{in},$ where $\\alpha $ is the dimensionless polarizability of the particle, $v$ is its volume, $\\varepsilon _m$ is the dielectric function of ambient medium.", "Here $p$ denotes the absolute value of the particle dipole moment $\\mathbf {p}$ which can be arbitrarily oriented in three dimensions.", "This continuous approximation can be justified for the quantum dots possessing a number of exciton levels.", "The experimentally measured values of the polarizabilities of the CdSe quantum dots in terahertz or static electric fields lie within the range from 0.4 (Ref.", "[10]) to 3.6 (Ref. [11]).", "The lifetime of the quantum dot excited state with dipole moment $\\mathbf {p}$ is several orders greater than the period of an optical field oscillation.", "After the period much larger than the exciton rise time, typically ranging from tenths of picosecond [12], [13] to tens of nanoseconds [14], a great number of the particles will be in the excited state and the system achieves a steady state.", "In the steady state, we suppose that the average quantity of the particles with the dipole moment does not vary and the system reaches the thermodynamic limit.", "Thus, on a timescale of the optical field oscillation, the polarized particles can be regarded as static dipoles.", "By this means, the quantum dots with the dipole moment will induce on the test particle, being located at the coordinate origin, additional random electric field $\\mathbf {E}=\\sum _l\\frac{3(\\mathbf {r}_l\\cdot \\mathbf {p}_l)\\mathbf {r}_l-\\mathbf {p}_lr_l^2}{\\varepsilon _m r_l^5},$ where $\\mathbf {r}_l$ is the position vector of the $l$ -th particle, $\\mathbf {p}_l$ is its dipole moment.", "In polar coordinates, $\\mathbf {r}_l$ has two components $(r_l,\\phi _l)$ .", "Hereinafter, we use the orientation of the test particle dipole moment as the selected direction.", "As a first approximation, we assume that the local field $\\mathbf {E}$ or induced nonlinearities do not change polarizability $\\alpha $ of the individual particles.", "This field is random since the quantum dots are polarized arbitrarily and placed haphazardly.", "After obtaining the probability distribution function $W(E)$ for the projection $E$ of the field $\\mathbf {E}$ onto the selected direction, the contribution of the dipole-dipole interactions to the nonlinear dielectric susceptibility of the nanocomposite can be calculated with the help of the statistical mechanics methods [9].", "Let us consider the monolayer of cylindrical particles that lie randomly in a plane.", "These particles are assumed to have the electric dipole polarization (REF ) after exposition to laser light and after the transition to the steady state.", "Due to the probabilistic nature of the excitation, the number of the dipoles with one direction is balanced by dipoles with opposite alignment.", "Thus, the ensemble lacks the ordering and the mean of the random field $\\mathbf {E}$ is zero.", "As shown in the Ref.", "[15] for such a system, $W(E)$ becomes the Gaussian distribution at high nanoparticle surface concentrations $c=N s/S>0.6$ , where $N$ is the number of the particles having been excited, $S$ is the area of the sample, $s$ is the area of the particle.", "Hence, we cannot apply directly the approach of Ref.", "[9] exploiting the Gaussian distribution for the calculation of the third-order dielectric susceptibility as such high surface nanoparticle concentrations are hardly achievable in practice.", "In order to obtain $W(E)$ at $c<0.6$ , making use of the negative cumulant expansion followed by the inverse Fourier transform was proposed [15]: $W(E) =\\frac{1}{2\\pi }\\int _{-\\infty }^\\infty \\exp \\left\\lbrace -\\frac{c}{s}\\left[ \\frac{1}{2}\\lambda _2\\rho ^2-\\frac{1}{4!", "}\\lambda _4\\rho ^4+\\frac{1}{6!", "}\\lambda _{6}\\rho ^{6}+\\ldots \\right]-i\\rho E \\right\\rbrace \\mathrm {d}\\rho ,$ where $\\rho $ is the Fourier-transform variable and the cumulants $\\lambda _n$ are defined as follows $\\lambda _n=\\frac{1}{p^n}\\int \\left\\lbrace \\frac{3(\\mathbf {r}\\cdot \\mathbf {p})\\mathbf {r}-\\mathbf {p}r^2}{r^5}\\right\\rbrace ^n \\tau (\\mathbf {p})\\mathrm {d} \\mathbf {p}\\, \\mathrm {d}\\mathbf {r},$ $\\tau \\left( \\mathbf {p} \\right)$ is the distribution function for dipole moment orientations.", "This approach is useful for the case of the identical functions $\\tau \\left( \\mathbf {p} \\right)$ for all the particles.", "Integration over radius $r$ in Eq.", "(REF ) begins from $2r_0$ which is the minimal possible distance between two centers of the cylindrical particles having a radius $r_0$ .", "It should be emphasized that in the lack of the ordering in the system, cumulant expansion in Eq.", "(REF ) contains only even-numbered $\\lambda _n$ [15].", "Here, the absence of the ordering means that the net dipolar moment of the system is zero.", "The free energy density of the ensemble is found by the standard method: $F=-k_B T \\frac{c}{v}\\ln \\int _{-\\infty }^\\infty \\exp (-p E /k_B T)W(E) \\mathrm {d}E,$ where $T$ is the temperature and $k_B$ is the Boltzmann constant, $v=sh$ , $h$ is the width of the monolayer (height of the particle).", "Projection $P$ of the sample macroscopic polarization onto the selected direction for monochromatic radiation and non-absorbing medium can be obtained using the thermodynamic relation [16] $P=-\\frac{\\partial \\langle F\\rangle _t}{\\partial E_{in}},$ where $\\langle \\rangle _t$ denotes time averaging.", "It is worth recalling that in this model the dipole of the test particle can possess two opposite orientations with equal probabilities.", "To perform the time averaging, we check our results for the independence on the change of the sign of the exponential argument in Eq.", "REF (Ref. [9]).", "In the case of linear or circular light polarization, expanding $P$ in a series $P=\\chi _1 E_{in} + \\chi _3 |E_{in}|^2 E_{in} + \\ldots ,$ it is possible to obtain cubic optical susceptibility of the system $\\chi _3$ as well as higher order nonlinear susceptibilities.", "Upon differentiating (REF ) and expanding $P$ in a series, we get $P=\\frac{c \\alpha ^2 v^2 \\varepsilon _m^2}{ k_B T s h I_{0,0}^2}\\Biggl \\lbrace \\left[\\frac{c}{s}\\lambda _2 (k_B T)^2 I_{2,0}-I_{0,2}\\right]I_{0,0}+I_{0,1}^2\\Biggr \\rbrace E_{in}+\\\\\\quad \\frac{c \\alpha ^3 v^3 \\varepsilon _m^3}{2 (k_B T)^2 s h I_{0,0}^3}\\Biggl \\lbrace \\left[ 3 \\frac{c}{s} (k_B T)^2I_{1,2}\\lambda _2-{I_{0,3}}\\right]I_{0,0}^2 - \\\\3\\left[ \\frac{c}{s}(k_B T)^2I_{0,2}\\lambda _2-{I_{0,2}}\\right]I_{0,1}I_{0,0} -2{I_{0,1}^3} \\Biggr \\rbrace E_{in}^2+\\\\\\frac{c^2 \\alpha ^4 v^4 \\varepsilon _m^4}{6 k_B T s^2 h I_{0,0}^4}\\Biggl \\lbrace -I_{0,0}^3\\biggl [\\left(\\lambda _4+3\\lambda _2^2\\frac{c}{s}\\right)(k_B T)^2 I_{4,0}-6I_{2,2}\\lambda _2+\\frac{s}{c(k_B T)^2}I_{0,4}\\biggr ]+ \\\\4 I_{0,0}^2 I_{0,1} \\left[\\frac{sI_{0,2}}{c(k_B T)^2} - 3I_{2,0}\\lambda _2\\right]+12 I_{0,0} I_{0,1}^2 \\left[I_{2,0}\\lambda _2 - \\frac{sI_{0,2}}{c(k_B T)^2}\\right]+\\\\3 I_{0,0}^2 \\frac{c}{s}\\left[I_{2,0}\\lambda _2 k_B T - \\frac{sI_{0,2}}{c k_B T}\\right]^2 + 6 I_{0,1}^4\\Biggr \\rbrace E_{in}^3+\\ldots ,$ where $I_{k,l}=\\int _{-\\infty }^\\infty \\int _{-\\infty }^\\infty \\exp (-i\\rho E)\\rho ^k E^l \\mathrm {d}\\rho \\mathrm {d} E$ To simplify these expressions, let us invoke the Fourier integral representation of the Dirac $\\delta $ -function [17] $\\delta ^{(n)}(E)=\\frac{1}{2\\pi }\\int (i\\rho )^n\\exp (i\\rho E) \\mathrm {d}\\rho $ and the definition of the Dirac $\\delta $ -function $n$ -th derivative [17] $\\int f(E)\\delta ^{(n)}(E)\\mathrm {d}E=(-1)^n f^{(n)}(0).$ Here $f$ is some function.", "The use of the above two formulas reveals that integrals (REF ) obtained after the expansion have nonzero values if $k=l$ and $k=0,2,4,\\ldots $ , namely $I_{0,0}=-2\\pi $ , $I_{2,2}=4\\pi $ .", "In the absence of the ordering, the first non-vanishing term in expansion (REF ) contains a third-order optical susceptibility $\\chi _3(\\omega ;\\omega ,\\omega ,-\\omega )$ (here $\\omega $ is the cyclic frequency of the incident light).", "After simplifications we derive formula for the self-induced Kerr nonlinear susceptibility: $\\chi _3 = -\\frac{ 2 c^2 \\alpha ^4 v^4 \\varepsilon _m^4 \\lambda _2}{k_B T s^2 h}.$ It should be noticed that the same relationship for $\\chi _3$ can be obtained using the Gaussian distribution as $W(E)$ .", "The next nonzero term in expansion (REF ) contains $\\chi _7$ : $\\chi _7 = -\\frac{ c^2 \\alpha ^8 v^8 \\varepsilon _m^8 \\lambda _4}{3(k_B T)^3 s^3 h}.$ It should be noted that the above formulas are derived under condition $\\varepsilon _m>0$ .", "In the above calculations we did not limit the number of the terms in the negative cumulant expansion in (REF ), so that the higher cumulants make contributions to higher-order susceptibilities.", "Further, let us consider some special dipole configurations that can occur if the monolayer of the quantum dots is integrated into a planar waveguide or a surface polariton propagates along an interface with the nanoparticles.", "When all the dipoles have in-plane alignment and they are collinear with two equiprobable opposite directions, the distribution function $\\tau \\left( \\mathbf {p} \\right)$ for dipole moment orientations is given by $\\tau (\\gamma )=\\frac{\\delta (\\gamma )+\\delta (\\gamma -\\pi )}{2},$ where the angle $\\gamma $ is measured from the selected direction which is determined by the dipole orientation.", "Then, $\\lambda _2 = \\frac{ 11 \\pi }{16 \\left( 2 r_0 \\right)^4 \\varepsilon _m^2}.$ For the case of the dipoles lying at the interface between two half-spaces with $\\varepsilon _1$ and $\\varepsilon _2$ [18], $\\varepsilon _m = \\frac{\\varepsilon _1 + \\varepsilon _2}{2}.$ For the in-plane oriented dipoles having arbitrary directions, $\\tau (\\gamma )=\\frac{1}{2\\pi },\\quad \\lambda _2 = \\frac{ 5\\pi }{8 \\left( 2 r_0 \\right)^4 \\varepsilon _m^2}.$ Both of the above cases can occur when a TM-polarized wave propagates along the plane containing the nanoparticles.", "Another situation arises when TE-polarized wave propagates along the interface.", "In this case, the dipoles are expected to be oriented perpendicular to the plane.", "So we utilize function (REF ) taking into account the fact that the polar axis is perpendicular to the plane.", "Then, $\\lambda _2 = \\frac{ \\pi }{2 \\left( 2 r_0 \\right)^4 \\varepsilon _m^2}.$ For the medium consisting of two half-spaces with $\\varepsilon _1$ and $\\varepsilon _2$ [18], $\\varepsilon _m = \\frac{2 \\varepsilon _1 \\varepsilon _2}{\\varepsilon _1 + \\varepsilon _2}.$ As one can see, all of these variants of $\\lambda _2$ differ little.", "By way of illustration, in the case of the dipoles with in-plain collinear polarizations, substituting $v=sh$ , $s=\\pi r_0^2$ , and (REF ) in (REF ) we get $\\chi _3 = -\\frac{ 11 c^2 \\alpha ^4 h^3 \\varepsilon _m^2}{128 k_B T}.$ It is obvious that the obtained expression for the Kerr nonlinear susceptibility does not depend explicitly on the particle radius.", "However, one should keep in mind that the polarizability of the particles strongly depends on their size.", "Provided $c=0.05$ , $\\alpha =0.4$ , $h=5$  nm, $\\varepsilon _m=2.3$ , $T=300$  K using Eq.", "(REF ) we arrive at $\\chi _3 \\approx -8\\times 10^{-11}$  esu.", "This value is of the same order as observed by experiment with nanosecond laser pulses for cadmium chalcogenide nanocomposites ($-10^{-11}$ –$-10^{-10}$  esu)[19], [20].", "As well as for the bulk nanocomposite, the contribution of the dipole-dipole interactions to the Kerr optical nonlinear susceptibility of the planar system has a negative sign.", "This is due to the disarranging effect of the random field of the induced dipoles.", "It is worth mentioning that the dipolar optical nonlinearity would become significant at the time intervals which exceed the exciton rise time many-fold.", "Comparison with the results of Ref.", "[9] shows that for the same $r$ and $h$ the contribution of the dipole-dipole interactions to the Kerr susceptibility of the two-dimensional nanocomposite is several times less than for the bulk nanocomposites.", "This is especially important for the nanoparticles which acquire the large values of the dipole moment, for instance, the II-VI semiconductor quantum dots [21].", "Often, under femtosecond laser pulses, these quantum dots exhibit the positive real part of the third-order susceptibility [22], [23], but at nanosecond pulse durations they show negative nonlinear refractive index [19], [20].", "This phenomenon may be attributed to the effect of the dipole-dipole interactions.", "If the dipolar optical nonlinearity is undesirable, the organization of the nanocomposites, which contain the nanoparticles with the great magnitudes of the induced dipole moment, as planar structures is justified.", "However, one would take into account that other types of optical nonlinearity also may decrease with reducing the number of dimensions.", "As one can see from Eq.", "(REF ), $\\chi _3$ is proportional to $c^2$ .", "It is the dependence on concentration that is the key feature of the contribution of the interparticle dipole-dipole interactions to the optical nonlinearity of the composite.", "This is reasonable since the number of interparticle couplings varies as the square of the quantity of the particles in the sample.", "As one could expect, other types of optical nonlinearities would linearly depend on the nanoparticle concentration.", "It should be underlined that the proposed here approach to the calculation of the dipole-dipole interaction contribution to the nonlinear susceptibility of nanocomposite can be applied not only to the two-dimensional geometry but also to other configurations.", "The method, based on the cumulant expansion of the characteristic function, can be also utilized for other types of interparticle interactions, such as the quadrupolar ones, since there is no need to determine probability distribution function $W(E)$ explicitly.", "In conclusion, the analytical expressions for Kerr optical nonlinearity $\\chi _3$ of the two-dimensional system of the randomly located quantum dots have been derived.", "It has been shown that for the different types of the light polarization, $\\chi _3$ changes insignificantly.", "The dipole-induced optical nonlinearity of the two-dimensional sample is several-fold smaller than for the bulk one so that this can be used in practice." ] ]

[ [ "Contributions of hyperon-hyperon scattering to subthreshold cascade\n production in heavy ion collisions" ], [ "Abstract Using a gauged flavor SU(3)-invariant hadronic Lagrangian, we calculate the cross sections for the strangeness-exchange reactions YY to N\\Xi (Y=\\Lambda, \\Sigma) in the Born approximation.", "These cross sections are then used in the Relativistic Vlasov-Uehling-Uhlenbeck (RVUU) transport model to study \\Xi production in Ar+KCl collisions at incident energy of 1.76A GeV and impact parameter b=3.5 fm.", "We find that including the contributions of hyperon-hyperon scattering channels strongly enhances the yield of \\Xi, leading to the abundance ratio \\Xi^{-}/(\\Lambda+\\Sigma^{0})=3.38E-3, which is essentially consistent with the recently measured value of $(5.6 \\pm 1.2_{-1.7}^{+1.8})\\times 10^{-3}$ by the HADES collaboration at GSI." ], [ "Introduction", "The study of particle production in heavy ion collisions at energies below their thresholds in nucleon-nucleon collisions was a topic of extensive studies during the 1990s [1], [2], [3], [4], [5].", "The main motivation for such study is that it offers the possibility of extracting information on the nuclear equation of state (EOS) at densities above that of normal nuclear matter.", "In particular, the yield of strange hadrons, such as the kaon, has been shown to be sensitive to the stiffness of the nuclear equation of state up to three times normal nuclear matter density, with a softer EOS giving a larger yield than a stiff EOS.", "Indeed, experimental results obtained by the KaoS Collaboration [6] at the Society for Heavy Ion Research (GSI) in Germany on the yield of kaons in heavy ion collisions at subthreshold energies have led to the conclusion that the nuclear equation of state at high densities is soft, consistent with an incompressibility of about 200 MeV extracted from the collective flow studies by the Plastic Ball [7] and EOS [8] Collaborations from Lawrence Berkeley Laboratory (LBL) and the E877 [9] and E895 [10] Collaborations at the Alternating Gradient Synchrotron (AGS) of Brookhaven National Laboratory (BNL).", "More recently, the doubly strange baryons $\\Xi $ from Ar+KCl collisions at 1.76A GeV, which is below the threshold energy of $3.74$ GeV in a nucleon-nucleon collision, was measured by the HADES Collaboration at GSI [11].", "The measured abundance ratio including the statistical and systematic errors is $\\Xi ^{-}/(\\Lambda +\\Sigma ^{0})=(5.6\\pm 1.2_{-1.7}^{+1.8})\\times 10^{-3}$ .", "This value is about 10-20 times larger than those given by the statistical model [12] and the relativistic transport model [13].", "Because of the very low collision energy, secondary reactions other than the direct reaction $NN\\rightarrow N\\Xi KK$ are expected to contribute significantantly to $\\Xi $ production in these collisions.", "In Ref.", "[13], the strangeness-exchange reaction $\\bar{K}Y\\rightarrow \\pi \\Xi $ $(Y=\\Lambda , \\Sigma )$ between antikaon and hyperon was introduced in the Vlasov-Uheling-Uhlenbeck (RVUU) transport model [14] to study $\\Xi $ production in heavy ion collisions.", "The cross sections used in Ref.", "[13] were taken from the coupled-channel calculation of Ref.", "[15] based on a gauged flavor SU(3)-invariant hadronic Lagrangian.", "Since there are more hyperons than anitkaons in heavy ion collisions at this energy, the strangeness-exchange reaction $YY\\rightarrow N\\Xi $ between two hyperons is expected to be important for $\\Xi $ production in these collisions.", "In the present study, we use the same hadronic Lagarangian as in Ref.", "[15] to evaluate the cross sections for the reaction $YY\\rightarrow N\\Xi $ .", "For an exploratory study, these cross sections are calculated in the Born approximation with the cutoff parameter in the form factors at interaction vertices fitted to the cross sections for the reactions $\\bar{K}Y\\rightarrow \\pi \\Xi $ obtained in Ref. [15].", "For completeness, we also include the reaction $\\bar{K}N\\rightarrow K\\Xi $ with its cross section taken from empirically available values.", "Our results show that the inclusion of the reaction $YY\\rightarrow N\\Xi $ significantly enhances the yield of $\\Xi $ in heavy ion collisions at subthreshold energies, resulting in the abundance ratio $\\Xi ^{-}/(\\Lambda +\\Sigma ^{0})=3.38\\times 10^{-3}$ in Ar+KCl collisions at 1.76A GeV and impact parameter $b=3.5$ fm, which is essentially consistent with the recently measured experimental value.", "We find, however, that the contribution of the reaction $\\bar{K}N\\rightarrow K\\Xi $ to the $\\Xi $ yield is negligible.", "The paper is organized as follows.", "In Sec.", "II, we describe the gauged flavor SU(3)-invariant hadronic Lagrangian [15], calculate the amplitudes for the reaction $YY \\rightarrow N\\Xi $ in the Born approximation, and parametrize the resulting cross sections.", "In Sec.", "III, we introduce the parametrization of the empirical cross section for the reaction $\\bar{K}N\\rightarrow K\\Xi $ as a function of the center of mass energy.", "We then briefly review in Sec.", "IV the RVUU transport model for high energy heavy ion collisions.", "Numerical results on the time evolution of the $\\Xi $ abundance in Ar+KCl collisions at 1.76A GeV and impact parameter $b=3.5$ fm are presented in Sec.", "V. Finally, we present some discussions in Sect.", "VI and a summary in Sec.", "VII." ], [ "The hadronic model", "Possible reactions for $\\Xi $ production from hyperon-hyperon collisions are $\\Lambda \\Lambda \\rightarrow N\\Xi $ , $\\Lambda \\Sigma \\rightarrow N\\Xi $ , and $\\Sigma \\Sigma \\rightarrow N\\Xi $ .", "Cross sections for these reactions can be evaluated using the same Lagrangian introduced in Ref [15] for studying $\\Xi $ production from the reactions ${\\bar{K}}\\Lambda \\rightarrow \\pi \\Xi $ and ${\\bar{K}}\\Sigma \\rightarrow \\pi \\Xi $ .", "This Lagrangian is based on the gauged SU(3) flavor symmetry but with empirical masses.", "The coupling constants are taken, if possible, from empirical information.", "Otherwise, the SU(3) relations are used to relate unknown coupling constants to known ones.", "Also, form factors are introduced at interaction vertices to take into account the finite size of hadrons." ], [ "The Lagrangian", "As in Ref.", "[15], we use the following flavor SU(3)-invariant hadronic Lagrangian for pseudoscalar mesons and baryons $\\mathcal {L}&=&i\\,\\mathtt {Tr}(\\bar{B}{\\partial \\hspace{-6.111pt}/}B)+\\mathtt {Tr}[(\\partial _{\\mu }P^{\\dagger }\\partial ^{\\mu }P)]\\nonumber \\\\&+&g^{\\prime }\\left\\lbrace \\mathtt {Tr}\\left[\\left( 2\\alpha -1\\right)\\bar{B}\\gamma ^{5}\\gamma ^{\\mu }B\\partial _{\\mu }P+\\bar{B}\\gamma ^{5}\\gamma ^{\\mu }\\left( \\partial _{\\mu }P\\right) B\\right]\\right\\rbrace ,\\nonumber \\\\$ where $B$ and $P$ denote, respectively, the baryon and pseudoscalar meson octets $B=\\left(\\begin{array}{ccc}\\frac{\\Sigma ^{0}}{\\sqrt{2}}+\\frac{\\Lambda }{\\sqrt{6}} & \\Sigma ^{+} & p \\\\\\Sigma ^{-} & -\\frac{\\Sigma ^{0}}{\\sqrt{2}}+\\frac{\\Lambda }{\\sqrt{6}} & n \\\\-\\Xi ^{-} & \\Xi ^{0} & -\\sqrt{\\frac{2}{3}}\\Lambda \\end{array}\\right)$ and $P=\\frac{1}{\\sqrt{2}}\\left(\\begin{array}{ccc}\\frac{\\pi ^{0}}{\\sqrt{2}}+\\frac{\\eta _{8}}{\\sqrt{6}}+\\frac{\\eta _{1}}{\\sqrt{3}} & \\pi ^{+} & K^{+} \\\\\\pi ^{-} & -\\frac{\\pi ^{0}}{\\sqrt{2}}+\\frac{\\eta _{8}}{\\sqrt{6}}+\\frac{\\eta _{1}}{\\sqrt{3}} & K^{0} \\\\K^{-} & \\bar{K}^{0} & -\\sqrt{\\frac{2}{3}}\\eta _{8}+\\frac{\\eta _{1}}{\\sqrt{3}}\\end{array}\\right),$ with $g^\\prime $ being a coupling constant and $\\alpha $ being a parameter.", "For the interactions of baryons and pseudoscalar mesons with the vector meson octet $V_\\mu $ , $V=\\frac{1}{\\sqrt{2}}\\left(\\begin{array}{ccc}\\frac{\\rho ^{0}}{\\sqrt{2}}+\\frac{\\omega }{\\sqrt{2}} & \\rho ^{+} & K^{\\ast +}\\\\\\rho ^{-} & -\\frac{\\rho ^{0}}{\\sqrt{2}}+\\frac{\\omega }{\\sqrt{2}} & K^{\\ast 0}\\\\K^{\\ast -} & \\bar{K}^{\\ast 0} & \\phi \\end{array}\\right),$ they are included by replacing the partial derivative $\\partial _{\\mu }$ in Eq.", "(REF ) with the covariant derivative $D_{\\mu }=\\partial _{\\mu }-\\frac{i}{2}g\\left[ V_{\\mu },\\right]$ , where $g$ is another coupling constant.", "We further include the tensor interactions between baryons and vector mesons via the interaction Lagrangian $\\mathcal {L}^{t}=\\frac{g^{t}}{2m}\\mathtt {Tr}\\left[ \\left( 2\\alpha -1\\right)\\bar{B}\\sigma ^{\\mu \\nu }B\\partial _{\\mu }V_{\\nu }+\\bar{B}\\sigma ^{\\mu \\nu }\\left( \\partial _{\\mu }V_{\\nu }\\right) B\\right],$ with $g^t$ being the tensor coupling constant." ], [ "Born approximation to the reactions $\\Lambda \\Lambda \\rightarrow N\\Xi $ ,\n{{formula:8b439f44-65af-4e20-9e57-e203ab7110d9}} , and {{formula:71dece67-948d-4f0a-a665-a86322bb53f5}}", "In the Born approximation, the reactions $\\Lambda \\Lambda \\rightarrow N\\Xi $ , $\\Lambda \\Sigma \\rightarrow N\\Xi $ , and $\\Sigma \\Sigma \\rightarrow N\\Xi $ are described by the tree-level $t$ -channel and $u$ -channel diagrams shown in Fig. 1.", "To evaluate their amplitudes requires the following interaction Lagrangian densities that are deduced from the hadronic Lagrangian in the previous subsection, i.e., ${\\cal L}_{KN\\Lambda }&=&\\frac{f_{KN\\Lambda }}{m_K}\\bar{N} \\gamma ^5\\gamma ^\\mu \\Lambda \\partial _\\mu K+{\\rm H.c.}, \\nonumber \\\\{\\cal L}_{KN\\Sigma }&=&\\frac{f_{KN\\Sigma }}{m_K}\\bar{N} \\gamma ^5\\gamma ^\\mu (\\vec{\\tau }\\cdot \\vec{\\Sigma }) \\partial _\\mu K+{\\rm H.c.}, \\nonumber \\\\{\\cal L}_{K\\Lambda \\Xi }&=&\\frac{f_{K\\Lambda \\Xi }}{m_K}\\bar{\\Xi }\\gamma ^5\\gamma ^\\mu \\Lambda \\partial _\\mu K^c+{\\rm H.c},\\nonumber \\\\{\\cal L}_{K\\Sigma \\Xi }&=&\\frac{f_{K\\Sigma \\Xi }}{m_K}\\bar{\\Xi }\\gamma ^5\\gamma ^\\mu (\\vec{\\tau }\\cdot \\vec{\\Sigma })\\partial _\\mu K^c+{\\rm H.c},\\nonumber \\\\{\\cal L}_{K^*N\\Lambda }&=&g_{K^*N\\Lambda }\\bar{N}\\left(\\gamma ^\\mu K^*_\\mu +\\frac{\\kappa _{K^*\\Lambda N}}{m_N+m_\\Lambda }\\sigma ^{\\mu \\nu }\\partial _\\mu K^*_\\nu \\right)\\Lambda \\nonumber \\\\&+&{\\rm H.c},\\nonumber \\\\{\\cal L}_{K^*N\\Sigma }&=&g_{K^* N\\Sigma }\\nonumber \\\\&\\times &\\vec{\\bar{\\Sigma }}\\cdot \\left(\\gamma ^\\mu \\vec{\\tau }K^*_\\mu +\\frac{\\kappa _{K^*N\\Sigma }}{m_N+m_\\Sigma }\\sigma ^{\\mu \\nu }\\vec{\\tau }\\partial _\\mu K^*_\\nu \\right)N\\nonumber \\\\&+&{\\rm H.c},\\nonumber \\\\{\\cal L}_{K^* \\Lambda \\Xi }&=&g_{K^* \\Lambda \\Xi }\\bar{\\Xi }\\left(\\gamma ^\\mu K^{*c}_\\mu +\\frac{\\kappa _{K^*\\Lambda \\Xi }}{m_\\Lambda +m_\\Xi }\\sigma ^{\\mu \\nu }\\partial _\\mu K^{*c}_\\nu \\right)\\Lambda \\nonumber \\\\&+&{\\rm H.c},\\nonumber \\\\{\\cal L}_{K^* \\Sigma \\Xi }&=&g_{K^* \\Sigma \\Xi }\\nonumber \\\\&\\times &\\vec{\\bar{\\Sigma }}\\cdot \\left(\\gamma ^\\mu \\vec{\\tau }K^{*c}_\\mu +\\frac{\\kappa _{K^*\\Sigma \\Xi }}{m_\\Sigma +m_\\Xi }\\sigma ^{\\mu \\nu }\\vec{\\tau }\\partial _\\mu K^{*c}_\\nu \\right)\\Xi \\nonumber \\\\&+&{\\rm H.c}.\\nonumber \\\\$ In the above, $\\vec{\\tau }$ are Pauli matrices; $\\vec{\\pi }$ , $\\vec{\\rho }$ , and $\\vec{\\Sigma }$ denote the pion, rho meson, and sigma hyperon isospin triplets, respectively; $K=(K^+,K^0)^T$ ($K^*=(K^{*+},K^{*0})^T$ ) and $K^c=(\\bar{K}^0,-K^-)^T$ ($K^{*c}=(\\bar{K}^{*0},-K^{*-})^T$ ) denote the pseudoscalar (vector) kaon and antikaon isospin doublets, respectively; and $\\Xi =(\\Xi ^0,\\Xi ^-)^T$ is the cascade hyperon isospin doublet.", "The coupling constants in above interaction Lagrangian densities are relate to those in Sec.", "REF by $\\frac{f_{KN\\Lambda }}{m_K}&=&\\frac{2\\alpha -3}{2\\sqrt{3}}g^\\prime ,\\ \\frac{f_{KN\\Sigma }}{m_K}=\\frac{2\\alpha -1}{2}g^\\prime ,\\nonumber \\\\\\frac{f_{K\\Lambda \\Xi }}{m_K}&=&\\frac{3-4\\alpha }{2\\sqrt{3}} g^\\prime ,\\ \\frac{f_{K\\Sigma \\Xi }}{m_K}=-\\frac{1}{2}g^\\prime ,\\nonumber \\\\g_{K^*N \\Lambda }&=&-g_{K^* \\Lambda \\Xi }=-\\frac{\\sqrt{3}}{4}g, \\nonumber \\\\\\ g_{K^*N\\Sigma }&=&g_{K^* \\Sigma \\Xi }=-\\frac{g}{4},\\nonumber \\\\\\kappa _{K^*\\Lambda N}&=&\\frac{g^t_{K^*\\Lambda N}}{g_{K^*\\Lambda N}},\\ \\kappa _{K^*N\\Sigma }=\\frac{g^t_{K^*N\\Sigma }}{g_{K^*N\\Sigma }},\\nonumber \\\\\\kappa _{K^*\\Lambda \\Xi }&=&\\frac{g^t_{K^*\\Lambda \\Xi }}{g_{K^*\\Lambda \\Xi }},\\ \\kappa _{K^*\\Sigma \\Xi }=\\frac{g^t_{K^*\\Sigma \\Xi }}{g_{K^*\\Sigma \\Xi }},\\nonumber \\\\\\frac{g^t_{K^*N\\Lambda }}{m_N+m_\\Lambda }&=&\\frac{2\\alpha -3}{2\\sqrt{3}}\\frac{g^t}{2m},\\ \\frac{g^t_{K^*N\\Sigma }}{m_N+m_\\Sigma }=\\frac{2\\alpha -1}{2}\\frac{g^t}{2m},\\nonumber \\\\\\frac{g^t_{K^*\\Lambda \\Xi }}{m_\\Lambda +m_\\Xi }&=&\\frac{3-4\\alpha }{2\\sqrt{3}}\\frac{g^t}{2m},\\ \\frac{g^t_{K^*\\Sigma \\Xi }}{m_\\Sigma +m_\\Xi }={\\sf \\color [rgb]{0,0,1}{-}}\\frac{g^t}{4m}.\\nonumber \\\\$ The cross sections for these reactions are then given by $\\sigma _{YY\\rightarrow N\\Xi }(s)=\\frac{1}{64\\pi sp_i^2}\\int dt\\overline{\\left|\\mathcal {M}\\right|^{2}},$ where $s=(p_1+p_2)^2$ and $t=(p_1-p_3)^2$ are the usual squared center of mass energy of colliding hyperons and the squared four momentum transfer in the reaction; and $p_i$ is the momentum of initial hyperons in their center of mass frame.", "The spin-isospin averaged amplitude $\\overline{\\left|\\mathcal {M}\\right|^{2}}$ in the above equation is given by $&&\\overline{\\left|\\mathcal {M}\\right|^{2}}=\\frac{1}{\\left( 2s_{1}+1\\right) \\left( 2s_{2}+1\\right) \\left( 2I_{1}+1\\right) \\left(2I_{2}+1\\right)}\\nonumber \\\\&&\\times \\sum _{s_{1}s_{2}s_{1}^{\\prime }s_{2}^{\\prime }}\\left[ \\eta _{tt}|\\mathcal {M}_{s_{1}s_{2}s_{1}^{\\prime }s_{2}^{\\prime }}^{t}|^2-\\eta _{tu}\\mathcal {M}_{s_{1}s_{2}s_{1}^{\\prime }s_{2}^{\\prime }}^{t}\\mathcal {M}_{s_{1}s_{2}s_{1}^{\\prime }s_{2}^{\\prime }}^{u\\ast }\\right.\\nonumber \\\\&&-\\left.\\eta _{ut}\\mathcal {M}_{s_{1}s_{2}s_{1}^{\\prime }s_{2}^{\\prime }}^{u}\\mathcal {M}_{s_{1}s_{2}s_{1}^{\\prime }s_{2}^{\\prime }}^{t\\ast }+\\eta _{uu}|\\mathcal {M}_{s_{1}s_{2}s_{1}^{\\prime }s_{2}^{\\prime }}^{u}|^2\\right],$ where $M^t_{s_1s_2s_1^\\prime s_2^\\prime }$ and $M^u_{s_1s_2s_1^\\prime s_2^\\prime }$ are the spin-dependent amplitudes for the two Born diagrams shown in Fig.", "REF and are given by $&&\\mathcal {M}^{t}_{s_1s_2s_1^\\prime s_2^\\prime }(s,t)=-\\frac{f_{KY_1\\Xi }f_{KNY_2}}{m_K^2}F^2({\\bf p}_1-{\\bf p}_3,\\Lambda )\\nonumber \\\\&&\\times \\left[ \\bar{\\Xi }\\left(p_3\\right)\\gamma ^{5}\\gamma ^{\\mu }Y_1\\left(p_1\\right) \\right]\\frac{t_{\\mu }t_{\\nu }}{t-m_{K}^{2}}\\left[ \\bar{N}\\left(p_4\\right) \\gamma ^{5}\\gamma ^{\\nu }Y_2\\left(p_2\\right)\\right]\\nonumber \\\\&&+g_{K^{\\ast }Y_1\\Xi }g_{K^{\\ast }NY_2}\\nonumber \\\\&&\\times \\left[\\bar{\\Xi }\\left(p_3\\right) \\left(\\left( 1+\\kappa _{K^{\\ast }Y_1\\Xi }\\right) \\gamma ^{\\mu }-\\kappa _{K^{\\ast }Y_1\\Xi }\\frac{\\left( p_3+p_1\\right) ^{\\mu }}{m_{Y_1}+m_{\\Xi }}\\right)\\right.\\nonumber \\\\&&\\times \\left.Y_1\\left(p_1\\right)\\right]\\frac{g_{\\mu \\nu }-t_{\\mu }t_{\\nu }/m_{K^{\\ast }}^{2}}{t-m_{K^*}^{2}}\\left[\\bar{N}\\left(p_4\\right)\\left(\\left(1+\\kappa _{K^{\\ast }NY_2}\\right)\\gamma ^{\\nu }\\right.\\right.\\nonumber \\\\&&\\left.\\left.+\\kappa _{K^{\\ast }NY_2}\\frac{\\left(p_3+p_1\\right)^{\\nu }}{m_{N}+m_{Y_2}}\\right)Y_2\\left(p_2\\right) \\right]$ and $\\mathcal {M}_{s_{1}s_{2}s_{1}^{\\prime }s_{2}^{\\prime }}^{u}(s,u)=\\mathcal {M}_{s_{1}s_{2}s_{1}^{\\prime }s_{2}^{\\prime }}^{t}(s,t),$ with $u=(p_1-p_4)^2$ .", "The form factor $F$ introduced at the interaction vertex because of the hardron structure is taken to have the monopole form, $F\\left( \\mathbf {q},\\Lambda \\right) =\\frac{\\Lambda ^{2}}{\\Lambda ^{2}+\\mathbf {q}^{2}},$ and depends on the three momentum transfer ${\\bf q}$ and the parameter $\\Lambda $ .", "The isospin factors $\\eta _{tt}$ , $\\eta _{tu}=\\eta _{ut}$ , and $\\eta _{uu}$ in Eq.", "(REF ), which are obtained from summing the isospins of initial and final particle, are 18, 10, and 18 for the reaction $\\Sigma \\Sigma \\rightarrow \\Xi N$ ; 6, 2, and 6 for the reaction $\\Lambda \\Sigma \\rightarrow \\Xi N$ , and 2, 2, and 2 for the reaction $\\Lambda \\Lambda \\rightarrow \\Xi N$ ." ], [ "Cross sections for the reactions $\\Lambda \\Lambda \\rightarrow N\\Xi $ ,\n{{formula:bc284201-92f0-47f7-841c-3a761197f778}} , and {{formula:0f2f7da3-ed08-4da0-ba4e-6ca62a76b1cb}}", "For numerical calculations of the cross sections, we use the coupling constants shown in Table I.", "These values are obtained from $g^\\prime =14.4~{\\rm GeV}^{-1}$ , $g=13.0$ , and $g^t/2m=19.8/m_N$ that are determined from the empirical values $f_{\\pi NN}=1.00$ , $g_{\\rho NN}=3.25$ , $g_{\\rho NN}^{t}=19.8$ [16], and $\\alpha =0.64$  [17] using relations based on the $SU(3)$ symmetry, i.e., $\\frac{f_{\\pi NN}}{m_\\pi }=\\frac{g^\\prime }{2},\\ g_{\\rho NN}=\\frac{g}{4},\\ \\frac{g^t_{\\rho NN}}{2m_N}=\\frac{g^t}{4m}.$ For the cutoff parameter $\\Lambda $ in the form factor, its value is taken to be $\\Lambda =0.7$ GeV in order to reproduce, as shown in Fig.", "REF , the cross sections for the reactions $\\bar{K}\\Lambda \\rightarrow \\pi \\Xi $ and $\\bar{K}\\Sigma \\rightarrow \\pi \\Xi $ that are obtained from the coupled-channel calculation based on the same hadronic Lagrangian [15].", "Figure: (Color online) Isospin-averaged cross sections for (a)K ¯Λ→πΞ\\bar{K}\\Lambda \\rightarrow \\pi \\Xi and (b) K ¯Σ→πΞ\\bar{K}\\Sigma \\rightarrow \\pi \\Xi .", "Solidlines are from the Born approximation with the cutoff parameterΛ=0.7 GeV \\Lambda =0.7 {\\rm GeV} in the form factor, and dashed lines arethose based on the coupled-channel calculation , .Figure: (Color online) Cross sections for (a) ΛΛ→NΞ\\Lambda \\Lambda \\rightarrow N\\Xi , (b) ΛΣ→NΞ\\Lambda \\Sigma \\rightarrow N\\Xi , (c) ΣΣ→NΞ\\Sigma \\Sigma \\rightarrow N\\Xi , (d)NΞ→ΛΛN\\Xi \\rightarrow \\Lambda \\Lambda , (e) NΞ→ΛΣN\\Xi \\rightarrow \\Lambda \\Sigma , and (f)NΞ→ΣΣN\\Xi \\rightarrow \\Sigma \\Sigma as functions of the center-of-mass energys\\sqrt{s} from the Born approximation with cutoff parametersΛ=0.5 GeV \\Lambda =0.5~ {\\rm GeV} (dashed lines), Λ=0.7 GeV \\Lambda =0.7~{\\rm GeV}(solid lines), and Λ=1 GeV \\Lambda =1~{\\rm GeV} (dotted lines).In Fig.", "REF , we show by solid lines the isospin-averaged cross sections for the reactions $\\Lambda \\Lambda \\rightarrow N\\Xi $ (panel (a)), $\\Lambda \\Sigma \\rightarrow N\\Xi $ (panel (b)), and $\\Sigma \\Sigma \\rightarrow N\\Xi $ (panel (c)) as functions of the center-of-mass energy $\\sqrt{s}$ , obtained with $\\Lambda =0.7~{\\rm GeV}$ .", "These cross sections can be parametrized as $\\sigma _{\\Lambda \\Lambda \\rightarrow N\\Xi }&=&37.15\\frac{p_{N}}{p_{\\Lambda }}(\\sqrt{s}-\\sqrt{s_{0}})^{-0.16}~\\mathrm {mb},\\nonumber \\\\\\sigma _{\\Lambda \\Sigma \\rightarrow N\\Xi }&=&25.12(\\sqrt{s}-\\sqrt{s_{0}})^{-0.42} ~\\mathrm {mb},\\nonumber \\\\\\sigma _{\\Sigma \\Sigma \\rightarrow N\\Xi }&=&8.51(\\sqrt{s}-\\sqrt{s_{0}})^{-0.395}~ \\mathrm {mb},$ where $p_{\\Lambda }$ and $p_{N}$ are initial $\\Lambda $ and final nucleon momenta in the center-of-mass frame.", "We note that the magnitude of our cross section for the reaction $\\Xi N\\rightarrow \\Lambda \\Lambda $ is similar to that of Ref.", "[18] obtained from the SU$_6$ quark model formulated in the resonance group method but is smaller than that extracted from the $(K^-,K^+)\\Xi ^-$ reactions in a nucleus [19].", "For comparisons, we also show in Fig.", "REF the cross sections for the reaction $YY\\rightarrow N\\Xi $ for the cutoff parameters $\\Lambda =0.5~{\\rm GeV}$ (dashed lines) and $\\Lambda =1~{\\rm GeV}$ (dotted lines).", "As expected, the cross sections are larger for a larger $\\Lambda $ .", "The cross sections for the inverse reactions $\\sigma _{N\\Xi \\rightarrow \\Lambda \\Lambda }$ , $\\sigma _{N\\Xi \\rightarrow \\Lambda \\Sigma }$ , and $\\sigma _{N\\Xi \\rightarrow \\Sigma \\Sigma }$ are related to above cross sections by the detailed balance relations: $\\sigma _{N\\Xi \\rightarrow \\Lambda \\Lambda }=\\frac{1}{4}\\left(\\frac{p_{\\Lambda }}{p_{N}}\\right)^{2}\\sigma _{\\Lambda \\Lambda \\rightarrow N\\Xi },\\nonumber \\\\\\sigma _{N\\Xi \\rightarrow \\Lambda \\Sigma }=\\frac{3}{4}\\left(\\frac{p_{\\Lambda }}{p_{N}}\\right)^{2}\\sigma _{\\Lambda \\Sigma \\rightarrow N\\Xi },\\nonumber \\\\\\sigma _{N\\Xi \\rightarrow \\Sigma \\Sigma }=\\frac{9}{4}\\left(\\frac{p_{\\Sigma }}{p_{N}}\\right)^{2}\\sigma _{\\Sigma \\Sigma \\rightarrow N\\Xi }.$ For completeness, we also include in the present study the reaction $\\bar{K}N\\rightarrow K\\Xi $ .", "Both the differential and total cross sections for this reaction were measured in 1960s and 70s [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], and they are shown in Fig.", "REF by solid squares for $K^-+p\\rightarrow K^+\\Xi ^-$ (panel (a)), $K^-+p\\rightarrow K^0\\Xi ^0$ (panel (b)), and $K^-+n\\rightarrow K^0\\Xi ^0$ (panel (c)).", "Recently, a phenomenological model was introduced in Ref.", "[30] to describe these reactions, and the results are shown by dashed lines in Fig.", "REF .", "In the present study, we use the following parametrization for these cross sections: $\\sigma _{K^{-}p\\rightarrow K^{+}\\Xi ^{-}} &=& 235.6 \\left(1-\\frac{\\sqrt{s_{0}}}{\\sqrt{s}} \\right)^{2.4} \\left(\\frac{\\sqrt{s_{0}}}{\\sqrt{s}} \\right)^{16.6} ~ \\mathrm {mb},\\nonumber \\\\\\sigma _{K^{-}p\\rightarrow K^{0}\\Xi ^{0}} &=& 7739.9 \\left(1-\\frac{\\sqrt{s_{0}}}{\\sqrt{s}} \\right)^{3.8} \\left(\\frac{\\sqrt{s_{0}}}{\\sqrt{s}} \\right)^{26.5} ~ \\mathrm {mb},\\nonumber \\\\\\sigma _{K^{-}n\\rightarrow K^{0}\\Xi ^{-}} &=& 235.6 \\left(1-\\frac{\\sqrt{s_{0}}}{\\sqrt{s}} \\right)^{2.4} \\left(\\frac{\\sqrt{s_{0}}}{\\sqrt{s}} \\right)^{16.6} ~ \\mathrm {mb}.\\nonumber \\\\$ In terms of these cross sections, the isospin averaged cross section for the reaction ${\\bar{K}}N\\rightarrow K\\Xi $ can be expressed as $\\sigma _{\\bar{K}N\\rightarrow K\\Xi }&=&0.5(\\sigma _{K^{-}p\\rightarrow K^{+}\\Xi ^{-}}+\\sigma _{K^{-}p\\rightarrow K^{0}\\Xi ^{0}}\\nonumber \\\\&+&\\sigma _{K^{-}n\\rightarrow K^{0}\\Xi ^{-}}).$ The detailed balance relation then allows us to express the cross section for the inverse reaction $K\\Xi \\rightarrow {\\bar{K}}N$ as $\\sigma _{K\\Xi \\rightarrow \\bar{K}N}=\\left(\\frac{p_{N}}{p_{\\Xi }}\\right)^{2}\\sigma _{\\bar{K}N\\rightarrow K\\Xi }$ where $p_N$ and $p_\\Xi $ are the 3-momenta of nucleon and $\\Xi $ in the center-of-mass frame." ], [ "the relativistic Vlasov-Uhling-Ulenbeck transport model", "To study $\\Xi $ production in heavy ion collisions at subthreshold energies, we generalize the RVUU transport model [14] to include the reactions $YY\\rightarrow N\\Xi $ and ${\\bar{K}}N\\rightarrow K\\Xi $ and their inverse reactions besides the reaction ${\\bar{K}}Y\\rightarrow \\pi \\Xi $ and its inverse reaction that were already included in Ref. [13].", "In addition to these reactions and other reactions involving nucleons, Delta resonances, hyperons, pions, kaons, and antikaons, the VUU model also includes the mean-field effect on the propagation of baryons, kaons, and antikaons.", "For nucleons and Delta resonances, their mean-field potentials are taken from the relativistic mean-field model via the scalar and vector potentials, so their motions are given by the following equations of motion: $\\dot{\\mathbf {x}}&=&\\frac{\\mathbf {p}^{*}}{E^{*}}\\nonumber \\\\\\dot{\\mathbf {p}}&=&-\\nabla _{x}(E^{*}+W_{0})$ where $m^{*}=m-\\Phi $ , $\\mathbf {p}^{*}=\\mathbf {p}-\\mathbf {W}$ , $E^{*}=\\sqrt{\\mathbf {\\mathrm {p}}^{*2}+m^{*2}}$ with $\\Phi $ and $W=(W_{0},\\mathbf {W})$ being the scalar and vector mean fields, respectively.", "These mean fields are calculated from the effective chiral Lagrangian of Ref.", "[31] with parameters determined from fitting the nuclear matter incompressibility $K_{0}=194\\mathrm {MeV}$ and the nucleon effective mass $m^{*}/m=0.6$ at normal nuclear matter density $\\rho _{0}=0.15~\\mathrm {fm^{3}}$ .", "For $\\Lambda $ and $\\Sigma $ hyperons, their mean-field potentials are taken to be 2/3 of the nucleon mean-field potential according to their light quark content.", "Similarly, the mean-field potential for $\\Xi $ is 1/3 of that of the nucleon .", "For kaons and antikaons, their mean-field potentials are derived, on the other hand, from the dispersion relation obtained in the chiral Lagrangian [32] $\\omega _{K,\\bar{K}}=\\left[m_{K,\\bar{K}}^{2}+\\mathbf {p}^{2}-\\frac{\\Sigma _{NK}}{f^{2}}\\rho _{s}+\\left(\\frac{3}{8}\\frac{\\rho _{N}}{f^{2}} \\right)^{2} \\right]^{1/2}\\pm \\frac{3}{8}\\frac{\\rho _{N}}{f^{2}},$ where $\\rho _{s}=\\langle \\bar{N}N\\rangle $ is the scalar density, $f=103$ MeV is the pion decay constant, and the $\\pm $ is taken as “+” for kaons and “-” for antikaons.", "The $KN$ and $\\bar{K}N$ sigma term $\\Sigma _{NK}$ in the above equation can in principle be calculated from the $SU(3)_{L}\\times SU(3)_{R}$ chiral Lagrangian but are taken to have the values $\\Sigma _{NK}/f^{2}=0.22~ \\mathrm { GeV^{2}fm^{3}}$ and $\\Sigma _{N\\bar{K}}/f^{2}=0.35~ \\mathrm { GeV^{2}fm^{3}}$ as in Ref.", "[13] from fitting the kaon and antikaon yields in heavy ion collisions.", "Besides affecting the propagation of particles, the mean-field potential also has effect on the threshold energy for particle production as a result of the potential difference between the initial and final states of a reaction.", "For example, this effect is important for understanding the enhanced production of antikaon through the reactions $BB\\leftrightarrow BBK\\bar{K}$ , $\\pi B\\leftrightarrow K\\bar{K}B$ , and $\\pi Y\\leftrightarrow \\bar{K}N$ in heavy ion collisions at subthresold energies.", "As a result, the contribution of the reaction ${\\bar{K}}Y\\rightarrow \\pi \\Xi $ to $\\Xi $ production in heavy ion collisions at subthreshold energies was found in Ref.", "[13] to be further enhanced.", "We note that in the RVUU model, kaons, antikaons, hyperons (lambdas and sigmas), and cascade particles are treated perturbatively by neglecting the effect of their production and annihilation on the collision dynamics, which is dominated by the more abundant nucleons, Delta resonances, and pions.", "In this approach, kaons, antikaons, and hyperson are produced from nucleon (Delta)-nucleon (Delta) and pion-nucleon (Delta) collisions whenever it is energetically allowed, and they are given probabilities that are determined by the ratios of their respective production cross sections to the total cross sections of the colliding particles.", "For $\\Xi $ production from antikaon collisions with nucleons or hyperons and from hyperon-hyperon collisions, it is similarly treated but the probability of the produced $\\Xi $ is reduced by the probabilities of colliding particles.", "The annihilation of these rare particles is treated in a similar way and leads to reductions of their probabilities.", "The present approach thus takes into account the small probability associated with the production of two rare particles in a subthrehold heavy ion collision that are involved in the production of a $\\Xi $ ." ], [ "results", "In this Section, we show the results for $^{40}\\mathrm {Ar}+\\mathrm {KCl}$ collisions at incident energy 1.76 AGeV, taking as an average of $^{40}\\mathrm {Ar}+\\mathrm {K}^{39}$ collisions and $^{40}\\mathrm {Ar}+\\mathrm {Cl}^{35}$ collisions, and compare them with the data from the HADES Collaboration at SIS.", "The HADES trigger (LVL1) selects approximately the most central $35\\%$ of the total reaction cross section [11].", "According to GEANT simulations [33] with the UrQMD [34], [35] transport approach as event generator, the average value and width of the corresponding impact parameter distribution amount to 3.5 and 1.5 fm, respectively.", "For simplicity, we take $b=3.5$ fm in the present study.", "Fig.", "REF (a) shows the time evolution of $\\pi $ and $\\Delta $ abundances (left scale) and the central baryon density (right scale).", "It is seen that the colliding system reaches its highest density of about $1.87\\rho _{0}$ at about 7 $\\mathrm {fm}/c$ when most particles are produced.", "The $\\pi $ abundance saturates at $10.3$ .", "Assuming isospin symmetry, the $\\pi ^-$ number is then 3.43 which is very close to the measured number of $3.9\\pm 0.1\\pm 0.1$ by the HADES Collaboration [36], [37].", "The time evolution for the abundances of $K$ , $\\bar{K}$ , $\\Lambda $ , and $\\Sigma $ are shown in Fig.", "REF (b), and they saturate at the values of $5.32\\times 10^{-2}$ , $1.15\\times 10^{-3}$ , $2.60\\times 10^{-2}$ , and $2.60\\times 10^{-2}$ , respectively.", "Assuming isospin symmetry gives $2.61\\times 10^{-2}$ for the $K^+$ number, $5.75\\times 10^{-4}$ for the $K^-$ number, and $3.47\\times 10^{-2}$ for the $\\Lambda +\\Sigma ^0$ number.", "These numbers are again close to corresponding measured numbers of $(2.8\\pm 0.2\\pm 0.1\\pm 0.1)\\times 10^{-2}$ , $(7.1\\pm 1.5\\pm 0.3\\pm 0.1)\\times 10^{-4}$ , and $(4.09\\pm 0.1\\pm 0.17)\\times 10^{-2}$ by the HADES Collaboration [38].", "For the time evolution of the $\\Xi $ abundance, it is shown by the solid curve in Fig.REF (c) and is seen to saturate at the value $2.34\\times 10^{-4}$ .", "Taking $\\Xi ^{-}$ as half of $\\Xi $ by assuming isospin symmetry, we obtained a $\\Xi ^-$ number of $1.17\\times 10^{-4}$ which is about half of the measured number of $(2.3\\pm 0.9)\\times 10^{-4}$ by the HADES Collaboration [39].", "Our results thus lead to an abundance ratio $\\Xi ^{-}/(\\Lambda +\\Sigma ^{0})=3.38\\times 10^{-3}$ , which is essentially consistent with the measured value of $(5.6\\pm 1.2_{-1.7}^{+1.8})\\times 10^{-3}$ by the HADES collaboration.", "The contributions to $\\Xi $ production from different reaction channels are also shown in Fig.REF (c).", "Dotted, dashed-dotted, and dash lines denote, respectively, the abundance of the $\\Xi $ particles from the reactions $YY\\rightarrow N\\Xi $ , $\\bar{K}Y\\rightarrow \\pi \\Xi $ , and $\\bar{K}N\\rightarrow K\\Xi $ .", "Compared to the total $\\Xi $ abundance, shown by the solid line in Fig.REF , the contributions are $97.5\\%$ , $2.40\\%$ , and $0.1\\%$ from the reactions $YY\\rightarrow N\\Xi $ , $\\bar{K}Y\\rightarrow \\pi \\Xi $ , and $\\bar{K}N\\rightarrow K\\Xi $ , respectively.", "So the $YY\\rightarrow N\\Xi $ channel dominates $\\Xi $ production in heavy ion collisions at subthreshold energies.", "This can be explained by the fact that the cross section for $YY\\rightarrow N\\Xi $ is almost 3-4 times the cross section for $\\bar{K}Y\\rightarrow \\pi \\Xi $ , and almost hundred times the cross section for $\\bar{K}N\\rightarrow K\\Xi $ .", "Also, the hyperon abundance in the system is almost 20 times the anti-kaon abundance.", "We note that the relative contributions to the $\\Xi $ yield from the reactions $\\Lambda \\Lambda \\rightarrow N\\Xi $ , $\\Lambda \\Sigma \\rightarrow N\\Xi $ and $\\Sigma \\Sigma \\rightarrow N\\Xi $ are about 1, 4 and 1." ], [ "discussions", "Our results are obtained without the consideration of the isospin asymmetry effect due to different proton and neutron numbers in the colliding nuclei, which is expected to increase the final abundance ratio $\\Xi ^{-}/(\\Lambda +\\Sigma ^{0})$ .", "If we assume that the abundance of $\\Xi $ has reached chemical equilibrium in heavy ion collisions, which is certainly questionable in view of the failure of the statistical model in describing the experimental data, this enhancement can be estimated using $\\Xi ^{-}/\\Xi ^{0}=\\mathrm {e}^{-\\mu _{C}/T}=\\Sigma ^{-}/\\Sigma ^{0}=\\Sigma ^{0}/\\Sigma ^{+}=N/Z$ , where $\\mu _{C}$ is the charge chemical potential and $T$ is the temperature of the system.", "With the value $N/Z\\sim 1.14$ for $\\mathrm {Ar}^{40}+\\mathrm {K}^{39}$ or $\\mathrm {Ar}^{40}+\\mathrm {Cl}^{35}$ , we have $\\Xi ^{-}=0.533~\\Xi $ and $\\Sigma ^{0}=0.3314~\\Sigma $ , leading to the ratio $\\Xi ^{-}/(\\Lambda +\\Sigma ^{0})=3.60\\times 10^{-3}$ that is $6.5\\%$ larger than that for an isospin symmetric system.", "Also, the nuclear EOS used in the transport model can affect the final $\\Xi $ abundance in heavy ion collisions.", "The results presented in the previous Section are based on a soft EOS.", "Using a stiff EOS, we find that the $\\Lambda $ , $\\Sigma $ , and $\\Xi $ abundances are reduced to $1.74\\times 10^{-2}$ , $1.77\\times 10^{-2}$ , and $1.46\\times 10^{-4}$ , respectively.", "The reason for this reduction in the hyperon abundances is that the energy density of the colliding system increases faster for a stiff EOS, thus making its expansion faster and reaction time short.", "However, the abundance ratio $\\Xi ^{-}/(\\Lambda +\\Sigma ^{0})=3.13\\times 10^{-3}$ for the stiff EOS is essentially the same as that for a soft EOS.", "Furthermore, the results presented here are for the impact parameter $b=3.5$ fm.", "A more realistic comparison with experimental data should include a distribution of impact parameters.", "We have checked that using different impact parameters, the ratio $\\Xi ^{-}/(\\Lambda +\\Sigma ^{0})$ remains, however, essentially unchanged, since both hyperons and cascade abundances change by almost the same factor.", "Finally, because of the very large $\\Xi $ production cross sections and the small size of the colliding system, the geometrical treatment of $\\Xi $ production from hyperon-hyperon scattering in terms of their scattering cross section as used in the RVUU transport model may become inaccurate.", "This can be seen from the dependence of final $\\Xi $ abundance on the value of the cutoff parameter $\\Lambda $ in the form factor used in evaluating the cross sections of the reactions $YY\\rightarrow N\\Xi $ .", "As shown in Fig.", "REF , these cross sections increase with increasing value of $\\Lambda $ .", "Results from our transport model study show, on the other hand, that the $\\Xi $ abundance increases with decreasing value of $\\Lambda $ .", "However, our conclusion in the present work is expected to remain unchanged since the $\\Xi $ abundance changes only by about $30\\%$ when the $\\Xi $ production cross sections change by more than a factor of 4.", "We note that a more accurate treatment of particle scattering may be achieved by using the stochastic method of Ref.", "[40] based on the transition probability, and we hope to purse such an improved study in the future." ], [ "Summary", "We have calculated the cross sections for the reaction $YY\\rightarrow N\\Xi $ ($Y=\\Lambda $ , $\\Sigma $ ) based on a gauged SU(3)-invariant hadronic Lagrangian in the Born approximation and found that these cross sections are almost 4 times the cross sections for the reaction $\\bar{K}Y\\rightarrow \\pi \\Xi $ that was considered in previous studies.", "We then used these cross sections to study $\\Xi $ production in $^{40}\\mathrm {Ar}+\\mathrm {KCl}$ collisions at the subthreshold energy of 1.76 AGeV within the frame work of a relativistic transport model that includes explicitly the nucleon, $\\Delta $ , pion, and perturbatively the kaon, antikaon, hyperons and $\\Xi $ .", "We found that the reaction $YY \\rightarrow N\\Xi $ would enhance the $\\Xi $ abundance by a factor of about 16 compared to that from the reaction ${\\bar{K}}Y\\rightarrow \\pi \\Xi $ , resulting in abundance ratio $\\Xi ^{-}/(\\Lambda +\\Sigma ^{0})=3.38\\times 10^{-3}$ that is essentially consistent with that measured by the HADES Collaboration at GSI.", "Our study has thus helped in resolving one of the puzzles in particle production from heavy ion collisions at subthrehold energies." ], [ "Acknowledgements", "This work was supported in part by the U.S. National Science Foundation under Grant Nos.", "PHY-0758115 and PHY-1068572, the Welch Foundation under Grant No.", "A-1358, the NNSF of China under Grant Nos.", "10975097 and 11135011, the Shanghai Rising-Star Program under grant No.", "11QH1401100, the \"Shu Guang\" project supported by Shanghai Municipal Education Commission and Shanghai Education Development Foundation, the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, the Science and Technology Commission of Shanghai Municipality (11DZ2260700), and the Korean Research Foundation under Grant No.", "KRF-2011-0020333." ] ]

[ [ "A Laser Frequency Comb System for Absolute Calibration of the VTT\n Echelle Spectrograph" ], [ "Abstract A wavelength calibration system based on a laser frequency comb (LFC) was developed in a co-operation between the Kiepenheuer-Institut f\\\"ur Sonnenphysik, Freiburg, Germany and the Max-Planck-Institut f\\\"ur Quantenoptik, Garching, Germany for permanent installation at the German Vacuum Tower Telescope (VTT) on Tenerife, Canary Islands.", "The system was installed successfully in October 2011.", "By simultaneously recording the spectra from the Sun and the LFC, for each exposure a calibration curve can be derived from the known frequencies of the comb modes that is suitable for absolute calibration at the meters per second level.", "We briefly summarize some topics in solar physics that benefit from absolute spectroscopy and point out the advantages of LFC compared to traditional calibration techniques.", "We also sketch the basic setup of the VTT calibration system and its integration with the existing echelle spectrograph." ], [ "The Need for High-Accuracy Absolute Calibration in Solar Spectroscopy", "The wavelength of spectral lines formed on the solar photosphere is influenced by many well-known processes.", "Some of the observed wavelength shifts are due to a true Doppler shift introduced by motion of the plasma on the solar photospheric surface, e.g., rotation (2000 m s-1), granulation (500 m s-1), and solar oscillations (500 m s-1).", "The convective blueshift and gravitational redshift introduce wavelength shifts on the order of 500 m s-1, while magnetic fields alter the magnitude of the convective blueshift itself, but these effects may not be interpreted as Doppler shifts.", "Also, the rapidly changing line-of-sight velocity between an observer on the Earth and the observed feature on the Sun amounts to roughly 1000 m s-1.", "To measure small-scale velocities of the plasma on the solar surface, all these effects must be separated carefully.", "Sensitive and stable instruments are needed to measure Doppler velocities of a few meters per second on top of the solar “noise” because long averaging times are required.", "While most processes can be characterized with spectroscopic measurements that are stable to an arbitrarily chosen zero point, precise absolute calibration is crucial to some measurements of small-scale Doppler velocities.", "We shall now summarize some of the topics in solar physics that would benefit from high-precision absolute wavelength measurements.", "The convective blueshift is a shift of spectral lines of up to 500 m s-1 caused by solar granulation [2].", "The blueshift is strongest at the disk center and shows a center-to-limb variation, which is also known as the “limb effect” [1].", "The limb effect further complicates Doppler measurements on the solar surface, as it cannot be averaged out by taking longer time series.", "Instead, a precise characterization of the limb effect depending on the position on the solar disk and the spectral line is required to correct measurements.", "However, so far, absolute measurements of the limb function have been limited to a few lines where accurate wavelength references are available [10] and model computations are employed for calibration purposes [7].", "For any precise spectroscopic measurement of the meridional flow as described below, a preceding determination of the limb function is a prerequisite, as the effect could also vary with the solar activity." ], [ "Meridional Motion", "The meridional motion is a weak, poleward directed flow on the solar surface with an am- plitude of about 20 m s-1.", "At high latitudes, the flow is expected to sink to deeper layers and turn equatorward, resulting in a circulation whose revolution time scale might be connected to the 11-year solar magnetic cycle [8].", "So far, most mea- surements of the meridional flow are derived from feature tracking or helioseismic methods.", "However, the velocity derived from feature tracking is different from the true plasma mo- tions, and helioseismology does not give reliable results at the surface due to noise from solar granulation.", "Thus, Doppler velocities derived from absolute calibrated high-resolution spectra would complement the existing measurements." ], [ "Sun-as-a-Star Radial Velocity vs. Solar Activity", "Stellar radial velocity measurements have now reached a precision well below the meters per second scale [15], and the required 5 cm s-1 sensitivity to detect Earth-like planets orbiting Sun-like stars seems to be feasible.", "In this radial velocity regime, Doppler noise from stellar activity usually dominates the detection limit, as it readily exceeds the Doppler signal from small orbiting planets and the instrument stability.", "Studying how the solar activity affects the radial velocity signal of the Sun thus is of great importance in detecting signatures of Earth-like exoplanets orbiting Sun-like stars.", "Detailed studies were readily carried out on this subject [12], [16], and high-resolution solar spectra with an absolute wavelength calibration will be useful to verify model calculations and measurements taken with different instruments." ], [ "Wavelength Calibration Techniques", "Several traditional calibration techniques commonly used in astronomy provide reference lines that are stable at the meters per second level.", "However, for absolute wavelength calibration, stability of the used reference lines is necessary but not sufficient, as their absolute laboratory wavelength must also be known with a corresponding accuracy.", "The ideal wavelength reference for spectrograph calibration provides a dense spectrum of equidistant emission lines of equal intensity with a line width smaller than the instrument's resolution.", "The frequencies of the reference lines should be derivable from fundamental physics so that the calibration source becomes exchangeable without affecting the calibration.", "Different calibration techniques were compared with frequency comb calibrators by [17] for their suitability for extremely stable night-time spectrograph calibration.", "They concluded that so far only frequency combs allow one to exploit the limits of existing and future high-precision spectrographs.", "While most of their findings are also applicable for solar spectroscopy, some aspects are different in this field.", "In night-time spectroscopy, thorium-argon lamps are commonly used for calibration.", "The Th-Ar spectrum covers the visible spectrum with thousands of sharp lines, many of which are known with an accuracy of about 10 m s-1 [14].", "But while the dim intensity of these lamps is sufficient for typical integration times of several minutes in night-time astronomy, solar spectroscopy usually deals with exposure times well below one second even at very high spectral resolution.", "The use of iodine cells is limited to the spectral range between 500 and 630 nm.", "The dense I2 spectrum blends with solar lines, and deconvolution techniques must be employed to recover the true solar spectrum from the measured data [11].", "While both Th-Ar and I2 calibrators provide very stable references with a reasonable number of calibration lines, a major drawback is the huge variation in intensity of their emission/absorption lines.", "Also, their atomic parameters are not known precisely enough for absolute calibration at a few meters per second.", "Recently, several frequency comb-based calibration systems were proposed for astro- nomical spectrographs [17], [5], [13], and a few test setups readily demonstrated their superior performance for wavelength calibration [19], [3], [22].", "[22] also showed that with grating spectrographs it is not sufficient to rely on a few calibration lines only, as aberrations of the spectrograph optics itself or even inter-pixel irregularities of the CCD could introduce deviations from the best-fit pixel-to-frequency curve that exceed several 10 m s-1.", "The feasibility of calibrating astronomical spectrographs with frequency combs was demonstrated for the first time by [19] using the Vacuum Tower Telescope (VTT) echelle spectrograph with a frequency comb operated in the infrared range.", "During the last two years, a laser frequency comb-based wavelength calibration system was developed for the VTT spectrograph in a cooperation between the Kiepenheuer-Institut für Sonnenphysik, Freiburg, Germany and the Max-Planck-Institut für Quantenoptik, Garching, Germany.", "The system was planned to cover a spectral range of at least $530\\,\\pm 50$  nm in the visible with a mode separation of about 5 pm.", "The VTT spectrograph has a spectral resolution of $\\frac{\\lambda }{\\delta \\lambda } > 10^6$ (0.5 pm @ 500 nm).", "For our application the rather large mode separation is required, because it easily allows us to unambiguously identify the frequencies of the comb lines by comparison with a nearby line in the solar spectrum.", "For the details of frequency comb generation and their application in astronomy we refer to [9], [17], [19] and [21].", "Here we merely describe the basic principle of our astro-comb setup and its integration with the spectrograph." ], [ "Optical Frequency Combs", "The spectrum of a mode-locked femtosecond laser forms a comb-like structure of equidistantly spaced modes with a frequency separation that is equal to the repetition rate $f_\\mathrm {rep}= 1/T$ of the laser, where $T$ is the round-trip time of the pulse in the laser resonator.", "Inequality of group and phase velocity in the resonator introduces a nonzero offset frequency $\\left|f_\\mathrm {0}\\right| < f_\\mathrm {rep}$ of the laser modes so that the frequency of the $n$ th mode, $f_n = f_\\mathrm {0}+ n f_\\mathrm {rep},$ is completely defined by the two radio frequencies (RFs) $f_\\mathrm {0}$ and $f_\\mathrm {rep}$ .", "Equation (REF ) establishes a direct link between the RFs and optical spectrum.", "Both frequencies can be measured and controlled very precisely, enabling the generation of optical frequencies with the same precision that previously was only available to RF electronics.", "While $f_\\mathrm {rep}$ can be measured straightforwardly with a fast photo-diode, determination of $f_\\mathrm {0}$ requires additional effort.", "One possibility is to use an $f$ :$2f$ -interferometer where the red part of the LFC spectrum is frequency doubled and the beat note with the blue part then yields $f_\\mathrm {0}$ .", "This requires the comb spectrum to span at least one octave and was first accomplished by spectral broadening of the comb in a photonic crystal fiber (PCF) [9].", "The equidistant spacing of the comb modes can be derived from theory, and no deviation from the equidistancy could be detected experimentally at a level of a few parts in 1016.", "While frequency combs are now routinely used in many fields in metrology and spectroscopy, their application in astronomy was delayed until recently for two reasons: 1) the fundamental repetition rate of the comb lasers (typically of the order of 100 MHz or approximately 0.1 pm @ 500 nm) is much to small to be resolved even by high-resolution astronomical spectrographs and, 2), the wavelength coverage of the comb spectrum is too narrow to be useful for multipurpose astronomical spectroscopy." ], [ "The VTT Astro-Comb", "There are several solutions to the problems mentioned above.", "The approach we follow is to filter the the fundamental repetition rate of the frequency comb with Fabry-Pérot cavities (FPCs) to a higher effective repetition rate $f_\\mathrm {rep, astrocomb} = m f_\\mathrm {rep}$ where $m$ is the integer ratio of the free spectral range (FSR) of the cavity and $f_\\mathrm {rep}$ .", "The filtered spectrum is amplified, converted to the visible spectrum, and finally broadened in a PCF.", "Figure: Schematic view of the VTT astro-comb.A schematic view of the setup is shown in Figure REF .", "The heart of the VTT astro-comb is a commercial frequency comb system (Menlo Systems FC1000).", "It consists of a ytterbium fiber laser operated at 1060 nm with a repetition rate of 250 MHz.", "The repetition rate and offset frequency of the laser are phase-locked to a GPS disciplined reference oscillator with a stability and accuracy of 10-12 @ 1 s. While with Ti:Sa lasers, a higher fundamental repetition rate of $\\approx 1$  GHz would be possible, fiber lasers have the big advantage of turnkey operation, allowing for unattended operation by non-experts.", "The comb is filtered with $m=22$ with two identical planoconcave FPCs with a free spectral range of 5.5 GHz and a finesse of approximately 500.", "By using two cavities we obtain a suppression of the nearest 250 MHz sidemodes by more than 60 dB [20].", "The plane mirrors of the cavities are piezo-driven for very fast adjustment of the resonator length.", "Both cavities are locked with the Pound-Drever-Hall method [4] to the transmission signal of a continuous wave (CW) laser with a polarization state that is orthogonal to that of the comb laser.", "The CW laser itself is locked to one of the comb modes transmitted by the cavities.", "Since the CW laser is comparably stable, the cavities stay locked even when the seed laser is switched off or blocked for a short time.", "As the cavities might drift out of the piezo range due to changes in pressure or temperature, a coarse tracking system with thermoelectrial heaters is implemented.", "To compensate for losses of the laser power in the filter cavities, the signal from the seed laser is amplified with core-pumped fiber amplifiers before passing the cavities.", "After filtering, a high-power double-clad fiber amplifier and a pulse compressor are inserted in the beam as the second-harmonic generation (SHG) requires high pulse powers to be efficient and enough power must also be available afterwards for spectral broadening in the PCF.", "We achieve good spectral broadening with an average power of about 1.5 W before the SHG stage.", "Up to the pulse compressor, all optical components are fiber coupled except for a free-space section in the FPCs.", "This makes the optical alignment of the system very robust; readjustment will probably only be necessary during the regular telescope maintenance periods.", "To enhance the non-linear effects that enable the spectral broadening, the PCF is tapered [18].", "The characteristics of the broadening strongly depend on the fiber which can not yet be manufactured with completely identical properties.", "With most of the PCFs we have tried so far, broad spectra ranging from approximately 480 to 640 nm could be obtained with enough power for short exposure times that are still limited by the flux from the sunlight." ], [ "Integration with the Spectrograph", "In solar physics spectroscopy usually also involves imaging of features on the solar surface.", "By scanning the spectrograph slit, images can be obtained with high spatial and spectral resolution (at the cost of temporal resolution).", "However, this further complicates the use of external wavelength calibration standards, as the calibration light must be combined with the sunlight so that identical illumination of the spectrograph grating from both light sources is ensured.", "This is probably the most critical issue of the complete setup.", "Night-time astronomy is different in this aspect.", "Stars are always unresolved point sources; the star light and calibration light can be fed to the spectrograph in two optical fibers which provide excellent stability and any offset between both fibers can easily be measured.", "For observations that rely on the spatial information provided by the slit spectrograph we are currently developing a slit illumination unit that will allow us to align the calibration light fed to the spectrograph in a singlemode fiber with respect to the optical axis of the telescope.", "Telescope and fiber pupil are imaged on a CCD camera, and the position of both can be tracked and aligned with a precision that translates to a few meters/second.", "While this would still limit our ability for absolute calibration, the stability of the calibration itself is not affected.", "To characterize the limits of the spectrograph calibration we are also preparing an experimental all-fiber setup where integrated sunlight from a full-disk telescope and the calibration light is fed to the spectrograph in the same single mode fiber.", "By using a single mode fiber, a perfect mode match between calibration light and sunlight can be guaranteed, eliminating any systematic effects related to unequal illumination of the grating with sunlight and calibration light.", "This setup then can also be used for high-precision Sun-as-a-Star spectroscopy." ], [ "Expected Performance", "The photon-noise limited uncertainty in velocity information that can be extracted from a single line can be estimated [6] by the relation $\\sigma _v = A \\frac{\\mathrm {FWHM}}{\\mathrm {SNR}\\sqrt{n}}$ where $A$ is a weighting factor that depends on the line shape (0.41 for a gaussian according to [17]), FWHM is the full width at half maximum of the line in meters/second, SNR is the peak signal-to-noise-ratio of the line profile, and $n$ is the number of pixels per FWHM at which the spectrum is sampled on the CCD.", "For the VTT spectrograph $\\sigma _v$ corresponds to 0.7 meters/second at a moderate SNR of 100.", "Figure: A part of the solar spectrum at about 530 nm simultaneouslyrecorded with the calibration spectrum from the frequency comb on 8 Oct2011 17:35 UTC near the disk center (upper panel).", "The lower panel showsa cut through the LFC spectrum in horizontal direction at the same imagescale.As the VTT spectrograph is neither temperature nor pressure stabilized, drifts of the order of 0.1 pm (50 m/s) per hour are common.", "While this drift seems rather large, at exposure times of one second it is still below the detection limit and can easily be tracked when the calibration spectrum is recorded simultaneously with the science data.", "Figure REF shows a test exposure taken during the installation campaign in October 2011.", "In the upper panel, a part of the solar spectrum at about 530 nm is shown with the calibration spectrum from the LFC at the lower edge of the CCD image.", "At 530 nm, the PCF introduces strong intensity variations that however quickly flatten with departure from the central wavelength of the frequency-doubled laser comb." ], [ "Conclusion and Outlook", "A frequency comb-based calibration system was developed for the VTT spectrograph and successfully installed at the telescope in October 2011.", "The astro-comb allows for absolute calibration of solar spectroscopic measurements in a broad wavelength range between approximately 480 and 640 nm.", "As many spectroscopic measurements in solar physics suffer from unreliable calibration, we see great potential for a very high-resolution spectrograph in combination with an absolute wavelength calibration system.", "The system is currently being tested and characterized thoroughly; the first scientific campaigns are scheduled for the next observing season.", "We also plan to make the system available for regular observation campaigns beginning with the 2013 observing season.", "Acknowledgements This project is in part funded by the Leibniz-Gemeinschaft within the “Pakt für Forschung und Innovation”.", "A. Fischer, K. Gerber, T. Sonner, and M. Weissschädel provided invaluable technical support for the setup at the VTT.", "0.0pt" ] ]

[ [ "Emergence: Key physical issues for deeper philosophical inquiries" ], [ "Abstract A sketch of three senses of emergence and a suggestive view on the emergence of time and the direction of time is presented.", "After trying to identify which issues philosophers interested in emergent phenomena in physics view as important I make several observations pertaining to the concepts, methodology and mechanisms required to understand emergence and describe a platform for its investigation.", "I then identify some key physical issues which I feel need be better appreciated by the philosophers in this pursuit.", "I end with some comments on one of these issues, that of coarse-graining and persistent structures." ], [ "Introduction", "In this talk I present some latest thoughts on three inter-related subjects: 1) Emergence: After describing three different senses of emergence, I point out that effective field theory (EFT) or renormalization group (RG) is a suitable, maybe even necessary, but not sufficient set of conceptual means for describing emergence.", "EFT or RG [A1] may suggest how different physics manifest at different scales, but one also needs to identify the mechanisms or processes whereby different levels of structures and the laws governing them, including the symmetry principles, emerge.", "That depends on deeper interplay of collectivity, complexity, stochasticity and self-organization.", "2) Emergent Gravity: There are at least two intimately related veins in viewing gravity as emergent: a) “General Relativity as Hydrodynamics?\"", "(This viewpoint which can be traced back to Sakharov (1968) [1] was first pronounced in this vein in [2].", "Other major proponents of this view are Volovik [3] and Wen [4].", ")– in the sense that gravity is an effective theory valid only at the long wavelength, low energy limit of some underlying theory (quantum gravity [5]) for the microscopic structures of spacetime and matter [A2][A3].", "b) Gravity as Thermodynamics , where such a view is often shaped by considering the effects of an event horizon on the quantum fluctuations of a field, shown by Bekenstein [10], Hawking [11], Unruh [12] and others, which underlies what is known today as the holography principle [13].", "This view is represented by the works of Jacobson [14], Padmanabhan [15] and Verlinde [16] [A4].", "3) Gravity and Thermodynamics: Since both gravity and thermodynamics are classical theories of macroscopic structures, if a deep connection exists, we should be able to see their direct relation at this level, without relying on arguments invoking the microscopic structure of matter (quantum fluctuations).", "This I first posed as a challenge in [A5], one which physicists in the 19c in principle may be able to resolve.", "If we can meet this challenge we may see the simpler and deeper connection between gravity and thermodynamics without invoking quantum mechanics.", "If we fail we will perhaps see more clearly the essential role of quantum physics in explaining gravity and the necessary implication that a) either the macroscopic world is fundamentally quantum.", "(For viewing spacetime as a condensate, see, e.g, [19]) or b) quantum mechanics is also emergent from a deeper structure [20], a representation of stochastic processes [21], or as a form of organizational rules like statistical mechanics [22].", "————- [A1] For a recent meeting on this topic, see, e.g., http://www.perimeterinstitute.ca/Events/ Emergence-and-Effective-Field-Theories/ Schedule/ [A2] For references to writings of the major proponents see, e.g., [6], [7].", "[A3] It is easier and more natural for string theorists to view gravity as emergent.", "See, e.g., [8], [9] [A4] For references to earlier work and a critique, see, e.g., [17] [A5] B. L. Hu, ¡°Gravity and Thermodynamics: What exactly do we want?¡± Invited talk at ESF Exploratory Workshop: Gravity and Thermodynamics, SISSA Sept 8, 2011.", "Further discussion can be found in [18].", "******* The above is the Abstract of my talk.", "In writing up my report, rather than a sketchy summary of all the points raised above, I thought perhaps it is more useful to select one topic and go deeper.", "Of these, since emergence is the overarching issue receiving increasing attention in physics and beyond, in particular, philosophy, I will focus on this subject matter.", "In keeping with the selected focus of this paper I will leave my Point 1) in the Abstract as is, pending future elaboration.", "Point 2) above has been discussed in recent meetings and the reader can find more details and other authors' related work in the references given in [A2]A5].", "On Point 3) about the challenge I posed: “Can we deduce a relation between Gravity and Thermodynamics without invoking any quantum consideration?\"", "either one succeeds or one fails to provide such a direct link without relying on quantum arguments, the implications for theoretical physics are equally significant.", "If one can meet this challenge , that thermodynamics rules can be used to understand or even derive gravity without appeal to the microphysical constituents of matter or spacetime and the physical laws governing them, it would be a very important step forward.", "If on the contrary this bridge between gravity and thermodynamics requires quantum mechanics to build, as is assumed from Bekenstein, Hawking onward, then it says something about the role of quantum mechanics in the macroscopic realm, including, and particularly important for spacetimes in this regard.", "This paper is organized as follows: In Sec.", "2 I give a sketch of the three senses of emergence These are thoughts I have toyed with on and off in the past two decades since the 1991 Huelva Workshop on The Physical Origin of Time-Asymmetry and the 1993 Santa Fe Institute Workshop on `Fluctuations and Forms'.", "which await further development.", "In Sec.", "3 I present some notes on essays written by philosophers trying to understand emergence in the physics context for the purpose of identifying what issues these philosophers view as important in their inquires.", "In Sec.", "4 I make several observations pertaining to the concepts, methodology and mechanisms required to understand emergence and describe a platform for its investigation, namely, nonequilibrium statistical mechanics (e.g., [23], [24], [25], [26]).", "In Sec.", "5 I try to identify some key physical issues which I feel need be better appreciated by the philosophers who wish to deepen their inquiries and bring themselves closer to practising scientists pursuing research topics bearing on this issue.", "I end with a brief description of the issue of coarse-graining and persistent structures." ], [ "Emergence: Processes and Mechanisms", "In this section I jot down some notes on the different senses of emergence which underlie different scientific disciplines, as well as a view of emergent time.", "These are merely sketches of ideas for an idea (as Wheeler liked to say), most likely not new, but which I find useful as nucleus for gathering amorphous threads of thoughts for further analysis." ], [ "Three Senses of Emergence", "1.", "Emergence in the sense of difference in manifestations– Role of coarse-graining: * Different manifestations at different levels of structures, hierarchical in form, and corresponding interactions.", "* Requires the identification of the range and precision of measurement, thus interfaces necessarily with an observer's probing ability and observation range.", "Effective field theory has this concern.", "* Stability of emergent structures depends on the degree of repeated specific coarse-graining.", "* Robustness of emergent structures against the variation of different coarse-graining measures.", "(Vaguely, in philosopher's language, the first two points bring out the issues of novelty, and the latter two touch on autonomy.", "More in the next two sections.)", "This aspect of emergence is sensitive to the conditions for the appearance of meta-stable structures.", "The key issues are embodied in studies of the emergence of quasi-classical domains [27] in environment-induced decoherence [28], [29] and decoherent or consistent histories [30], [31] formulation of quantum mechanics.", "Historically these studies focused more on the quantum aspects (transition to classicality), but I want to point out that issues in nonequilibrium statistical mechanics enter in a major way.", "See, e.g., Sec.", "5 of [32] and [33].", "2.", "Emergence in the sense of conditioning or control– Role of stochasticity: * Selection by the system's environment through their interactions.", "For quantum open systems and decoherence studies this would correspond loosely to the functionality of the “pointer basis\" [34].", "A particular set of bases is `selected' by the way a system interacts with its environment.", "It is more sensitive to the type of coupling than the magnitude of interaction.", "* Adaptation of a biological system to its environment is well-known in evolutionary theory.", "The emergence of new species is the outcome of both genetic (intrinsic) variation and environmental (extrinsic) selection, the former arising from mutation, a stochastic variable This is similar to the environment-induced decoherence of a quantum system although the stochastic element there is from the noise and fluctuations in an environment and the quantum to classical transition is on the whole not a stochastic process..", "Note, even if a reduced density matrix is almost diagonal with respect to some pointer basis, there is still the question of how its diagonal components, the classical probability or the weighing functions, are distributed, which influences the outcome of emergent entities and behavior.", "3.", "Emergence in the sense of dynamics ( sequencing, `updates', but not time evolution): Here I have in mind models from * Cellula automata (CA) [37]: Simple conjunctive rules between neighboring elements when iterated a large number of steps can lead to very different structures – many will terminate but some will keep evolving.", "There is no time, thus no dynamics in terms of time evolution, simply sequencing.", "Those which survive can be viewed as emergent structures since they possess both the qualities of being novel and autonomous.", "But note that in this sense, given the conjunctive rules of the cells, one can predict exactly the outcome of `evolution' (thus not in the biological sense 2 above) without any element of stochasticity involved, even though the outcome may appear irregular or seemingly unpredictable.", "* Growth and form in driven diffusive systems [38], self-organization: Different forms emerge arising from the interplay of stochasticity and nonlinearly.", "Role of nonlinearity in self-organization is widely known.", "Role of stochasticity (noise and fluctuations) in the genesis of new forms appear in new branches of nonequilibrium physics- based fields in the 80-90s such as soft matter physics.", "I want to add that the effect of memory is also important.", "(Protein-folding is a well-known example.)", "This calls for recognizing the importance of non-Markovian processes in emergent phenomena For example: every snowflake has its own distinct identity, being a fine record of the density, humidity, temperature as the condensate traverses the different layers of water vapor, growing into a flake.. Non-Markovian (or memory-laden) processes, or histories (usually connoted as with memories) do not have to refer to dynamics in the sense of time-evolution.", "It could also be understood in the sense of `sequencing' referred to above in the example of cellula automata." ], [ "Sense of Time in Emergence", "The essence or relevance of time is in its ordering function.", "Conceptually, let us consider three distinct elements: a) configurations in each step as basic entities, b) sequencing or iteration of steps (try to avoid using the word `update', since date is a marking of time).", "An example of a): Energetics of phase transition using the free energy density functional $F(T)$ describes critical phenomena, not critical dynamics.", "It is a one parameter (temperature $T$ ) family of curves, where the minima signify the existence of meta-stable states.", "One can discuss the probability of tunneling from one such state to another in this (assuming multi-dimensional space) landscape, but there is no dynamics involved.", "Using CA as an example of b), the conjunction rules produce one configuration in one step, then another after the second step, which make up a sequence.", "There are sequences which terminate after relatively few iterations, others sustain many iterations and begin to produce various forms.", "There is still no sense of time in the sense of dynamics yet.", "The third ingredient c) is the key.", "In simple cases like CA, iteration in increasing steps can be viewed as time, as the word `update' conjures.", "But it is more challenging to understand time in more complex systems where they can interact with various environments.", "Using biological systems as example, replication continues the species and provides the stability.", "This is like the sustaining sequences in CA – those long-living sequences as measured by the number of iterations.", "Sufficient stability is necessary for a notion of time to emerge.", "Mutations in the genes of a system provides the species a wider sampling pool to adapt to a changing environment.", "But this stochastic element also introduces a source of instability.", "Thus even though the sequencing function could facilitate a notion of micro-time this stochasticity and instability at the cellula level undermines the macro-time as we experience it which requires some sustained unidirectional development.", "Note at this stage there is no `arrow of time' yet.", "Survival of the species is the precondition for the notion of `direction of time' to enter.", "Fluctuation triggers instability in the system in varying degrees: the adverse ones could bring about extinction of the species, the `beneficial' ones enable the species to better adapt and continue to evolve.", "A sense of `direction of time' or a sense of (forward) history comes only when there is sufficient `success' as measured by the species' ability to survive, thrive and improve on itself.", "The word `advanced' in advanced civilization connotes a sense of positive direction of evolution.", "Thus it not only reflects the passive conditioning by the environment but it also allows for some degrees of regulation, control and adjustment by the system itself.", "This amounts to biasing a random process.", "Therefore, in summary, sustained sequencing with some degree of stability is the precondition for the notion of time to emerge and the direction of time emerges only with positive feedback in the sustained sequencing of biased random processes.", "The emergence of time and the sense of direction of time are explained above by using the paradigms of evolutionary biology.", "Placing these issues in the physics context, concepts and techniques in nonequilibrium statistical mechanics including open systems dynamics and stochastic processes enter in essential ways.", "This will be discussed in a later section.", "Suffice it here to say that time asymmetry in many physical processes is influenced by many factors: the way one stipulates the boundary conditions and initial states, the time scale of observation in comparison to the dynamical time scale, how one decides what the most suitable relevant variables are and how they are separated from the irrelevant ones, how the irrelevant variables are coarse-grained, and what assumptions one makes and what limits one takes in shaping a macroscopic picture from one's imperfect knowledge of the underlying microscopic structure and dynamics.", "More discussions follow in Sec.", "4, 5." ], [ "Philosophical Studies", "In this section I gather some notes in a naive attempt to extract the wisdom from philosophers trying to understand emergence in the physics context.", "I begin with the rudimentary exercise of finding out what the philosophers interpret the physicists (I mean by necessity a selected subset sampling) in their use of words like “constructivism\" “protectorate\" and the philosophers' terminology “microphysicalism\" “supervenience\" and key words used by both communities, such as “novelty\" “autonomy\" which characterize emergence.", "Of course these exercises are not just for clarifying the terminology, but for extracting the meaning behind these words.", "What I find in this preliminary exercise is that the philosophers are mostly interested in issues of their own, in metaphysics, in the philosophy of the mind, etc, not unexpectedly.", "From the questions they raised I also see some ambiguity (or not clear enough definitions) in the physicists' use of words like “fundamental\", “reductionism\", which need be clarified.", "The focusses of philosophers of science with a physics background or those who are attuned to physicists' language or way of thinking enable us to see what issues they consider as important to bring back to their own community.", "At the same time I also see the philosophers' choice of focus is somewhat partial, or skewed, missing some key issues which physicists consider as important in understanding emergent phenomena.", "This is likely a result of the scarcity of explanations by the practising physicists of their technical findings in the light of emergence phenomena, even scarcer in philosophical terms." ], [ "Sources and Tracks", "I shall first identify a few sources of literature on both sides so readers can see how my readings could be limited or biased and identify what I have missed out clearly.", "Classic papers in physics: Anderson [39], Laughlin and Pines [40] on the one side and Weinberg [41] on the other.", "Physicists (at PI workshop [A1]) who addressed key issues on emergence: Goldenfeld, Kadanoff Issues of emergence exemplified in quantum decoherence: Stamp Philosophers (at PI workshop) who raised key issues on emergence: Batterman [42], Morrison [43] Philosophers with physics training or attuned to physical issues; Bain [44], [45], Wilson [46].", "Useful introduction: Mainwood PhD thesis [47], with metaphysical emphasis." ], [ "Terminology, Contents, Meaning and Issues", "Physicists usage: `Fundamental', Reductionism, Constructivism, “protectorate\" [40] Emergent entities, properties, principles (Weinberg) Philosopher's usage: microphysicalism, supervenience Defining properties of emergence largely agreed-upon by philosophers: Novelty, Autonomy(reductive, predictive, causal and/or explanatory) A description rather than a definition is probably best to illustrate what novelty refers to: `Instead, at each level of complexity entirely new properties appear' (Anderson [39], p. 393).", "“When you put enough elementary units together, you get something that is more than the sum of these units.", "A substance made of a great number of molecules, for instance, has properties such as pressure and temperature that no one molecule possesses.", "It may be a solid or a liquid or a gas, although no single molecule is solid or liquid or gas.\"", "(Wheeler and Ford [48] p.341, quoted by Mainwood [47] p.18) Reductionism– reduction, prediction; derivable, deducible: “In most philosophical discussions the concept of emergence is intimately related to the following notions: antireductionism, unpredictability, and novelty.", "Emergence in these contexts is also typically associated with parts and wholes.", "The idea being that a phenomenon is emergent if its behavior is not reducible to some sort of sum of the behaviors of its parts, if its behavior is not predictable given full knowledge of the behaviors of its parts, and if it is somehow new – most typically this is taken to mean that emergent phenomenon displays causal powers not displayed by any of its parts.", "In addition to irreducibility, unpredictability, and novelty, it is often asserted that emergent phenomena are inexplicable – they defy explanation in terms of the behaviors of their components.", "And, as “explanation\" is typically understood to be explanation by a particular theory, this means that the behavior of the emergent whole is not fully explained by the theory that governs the behavior of the component parts.\"", "(Batterman [42] p.1-2) “Almost all conceptions of emergence characterise novelty by appealing to some conception of predictability or derivability, questioning whether it is possible to obtain one set of properties from another by some mechanical process of logical derivation.", "Kim's 1995 definition is typical from one concerned with the philosophy of mind: `a property of a complex system is said to be `emergent' just in case, although it arises out of the properties and relations characterizing simpler constituents, it is neither predictable from, nor reducible to, these lower-level characteristics.'", "The thought is that if it is possible to derive systemic properties from those of their components, these properties cannot be truly novel – for they were \"there all along\" in the parts, and it was merely a matter of careful analysis to make this fact manifest.", "This leads naturally to a distinction between this `prediction' or `reduction' being one that could be carried out in practice, and one that is possible only in principle.\"", "(Mainwood [47] p. 27) Here lies the distinction between ontological (in principle) vs epistemological (in practice) emergence.", "Reductionism- Constructivism “The workings of our minds and bodies, and of all the animate or inanimate matter of which we have any detailed knowledge, are assumed to be controlled by the same set of fundamental laws, which except under certain extreme conditions we feel we know pretty well.\"", "(Anderson [39] p. 393) “ ... the material composition of organisms is exactly the same as that found in the inorganic world.", "Further ... none of the events and processes encountered in the world of living organism is in any conflict with the physico-chemical phenomena at the level of atoms and molecules.\"", "(Mayr's “constitutive reductionism' in the context of biology [49] p.59 quoted by Mainwood [47] p. 16 “... the reductionist hypothesis does not by any means imply a constructionist one: The ability to reduce everything to simple fundamental laws does not imply the ability to start from those laws and reconstruct the universe.", "“The constructionist hypothesis breaks down when confronted with the twin difficulties of scale and complexity.", "(Anderson [39] p. 393) Microphysicalism and the `New Emergentists\" What Anderson and Weinberg refer to as `reductionism' is at odds with the metaphysicians' usage, who call it `microphysicalism'.", "“ In metaphysics, the claim endorsed by Anderson and Mayr is usually called microphysicalism.", "[M]icrophysicalism ... is the doctrine that actually (but not necessarily) everything non-microphysical is composed out of microphysical entities and is governed by microphysical laws.\"", "(Pettit [50]) “ Pettit also brings out the two components which constitute microphysicalism.", "The first is that everything in the empirical world – every particular in that world – is composed in some sense out of subatomic materials.", "And second is that everything that happens in the empirical world happens, ultimately, under the controlling influence of subatomic forces and laws.\"", "(Pettit [51] p.342-3) Mainwood refers to proponents and adherents of these tenets as the New Emergentists.", "On the issue of “in practice\" versus “in principle\" mentioned above, “.. the New Emergentists are accused of illegitimately citing evidence for epistemological emergence (a practical difficulty in deriving the properties of a large system) as support for claims of ontological emergence (there exist emergent properties that are “novel\" in a metaphysically important sense).", "(Mainwood [47] ) Supervenience – a key concept philosophers use for qualifying emergence.", "“The most promising approach to capturing the `determination' inherent in property physicalism is through a relation of supervenience.", "Take a set of objects O, let their physical properties form a set $P_A$ , and all their properties (both physical and ostensibly non-physical) form a set $P_B$ .", "To capture physicalism via supervenience, we demand that the extensions of elements of $P_B$ are fixed by specifying the extensions of the elements of $P_A$ .", "The simplest way of accommodating any such n-adic relations amongst objects $o_1, o_2, o_3$ is to treat it as a one-place property of the larger object composed of all of these.", "So we need to close the set of objects O under the operation of taking ordered n-tuples.", "A little more formally: the properties $P_B$ supervene on $P_A$ with respect to a set of objects O, on which both are defined, iff any two objects in O that match with respect to all properties in $P_A$ , also match with respect to all properties in $P_B$ .", "The central claim of property physicalism can then be expressed very simply: all properties supervene on physical properties.\"", "(Mainwood [47] p. 22) Using these technical terms in philosophy, Mainwood summarizes, “... systemic properties are novel, if and only if it is practically impossible to derive them from the microphysical properties mentioned in microphysical supervenience.\"", "([47] p. 30) Three proposals of novelty were examined by Mainwood [47] (Sec.", "1.5): 1) a failure of inter-theoretic reduction; 2) an impossibility of deducing the systemic properties from the properties of the parts; or 3) a failure of mereological supervenience.", "The approaches appeal to three entirely separate distinctions between the properties of parts and wholes; thus according to Mainwood, there are three entirely different sets of criteria for emergence.", "We leave it for the interested reader to follow his treatise but offer some insights on these issues in a later section." ], [ "Key words and concepts used by physicists", "Fundamental: “The modern theory of critical phenomena has interesting implications for our understanding of what constitutes \"fundamental\" physics.", "For many important problems, a fundamental under- standing of the physics involved does not necessarily lie in the science of the smallest available time or length scale.", "The extreme insensitivity of the hydrodynamics of fluids to the precise physics at high frequencies and short distances is highlighted when we remember that the Navier-Stokes equations were derived in the early nineteenth century, at a time when even the discrete atomistic nature of matter was in doubt.", "( [52] p.3, quoted by Batterman [42]] Protectorate: According to Laughlin and Pines (LP) [40] .. a (quantum) protectorate, (is) “a stable state of matter whose generic low-energy properties are determined by a higher organizing principle and nothing else.\"", "An example of LP's “higher organizing principle\", used also by Anderson earlier, is spontaneous symmetry breaking.", "“.. one does not need to prove the existence of sound in a solid, for it follows from the existence of elastic moduli at long length scales, which in turn follows from the spontaneous breaking of translational and rotation symmetry characteristic of the crystalline state.\"", "What is more telling is, “Conversely, one therefore learns little about the atomic structure of a crystalline solid by measuring its acoustics.\"", "Morrison [43]) “.. referred to them as `theoretical principles' in order to capture the idea that there is a dynamical process associated with these principles responsible for producing certain kinds of behavior.\"", "In the eyes of the theoretical physicists, `theoretical' is too general a term to identify these principles, and can become nondescriptive.", "These principles provide not only how the micro-parts are organized into the meso-`whole', and here we are referring two adjacent rungs in a hierarchy, but also the dynamical process which they come into being.", "This would correspond to the specification of – how to choose or constitute – the collective variables and how to describe their states and dynamics, the collective behavior, which is the critically important issue.", "Parts and Whole relation: I agree with these summary statements by Morrison below and by Batterman next: “Not only do we need to reorient our thinking about the role of natural kinds as a method for differentiating essential features of matter, but new ways of thinking about the part/whole relations involved in defining and describing emergent phenomena are required.\"", "“And perhaps most importantly, a re-evaluation of reductionist strategies for defining the relationship between different theoretical levels is crucial for making sense of emergence in physics.", "In connection with the ¡°levels¡± approach characteristic of this picture of emergence, it is important to note the differences between emergence and the fact that phenomena at different scales may obey different fundamental laws.\"", "(Morrison [43] p.886)" ], [ "Overall Assessment", " “A number of philosophers of science have imposed these philosophical conceptions upon physical theory in an attempt to address the issue of emergence in physics.", "While I believe there are some benefits to this methodological approach, on the whole I think it is better to turn the process on its head.", "We should look to physics and to “emergent relations\" between physical theories to get a better idea about what the nature of emergence really is.", "Trying to impose a conceptual framework designed primarily to deal with the problem of the mental's relation to the physical is by and large unhelpful.\"", "(Batterman [42] p.2) I think the same can be said about the relation of science and philosophy of science in general.", "If one wants to not just talk about the philosophy of a subject X (the layman's use of the word “philosophize on\" is even worse), but to provide philosophical insights to the key issues of X, in terms of asking better questions and seeing clearer directions, it is imperative that one should first have the technical mastery of X.", "E.g., one must have lived a life to talk about the philosophy of life.", "In this section I wish to make some observations on how philosophers can better communicate with, and of greater help to physicists, e.g., sorting out the complexity of issues in problems physicist wrestle with and probing deeper into specific key issues in these problems.", "Three aspects: 1) Conceptual Constructs: It is not enough to rely on a reductionist conceptual approach, such as what Mainard related to about Russell's impact on philosophers, that all mathematical statements can be reduced to set theory.", "Many physical phenomena defy simple descriptions or monolithic construction, some may even appear illogical (in reasoning, not in its intrinsic consistency).", "A new physical theory will become more logical and its conceptual construct more rational (think special relativity and quantum physics) only after it has been proposed, examined and checked against observations and experiments, not before.", "Physicists create models to capture the essence of the novel phenomena and they make approximations to test out characteristic behavior in different parameter regimes.", "What they are wrestling with is physical reality in whatever shape and form it be, raw and often unruly.", "Physicists approach truth from reality.", "This process may be aided by abstractions of theories with some sense of completeness and perfection but the final judgments come from experiments.", "Here we see the difference in attitude stemming from approaching truth in the epistemic rather than the ontological sense.", "For emergent phenomena, rather than assuming a linear progression of logical deduction or inference we need a conceptual scheme which can explain the emergence of a hierarchical construct, with a tower of levels of structure, the collective variables most suitable for each level, the laws governing them, and even more demanding, the inter-level meta-constituents and their activities.", "2) Methodology and Vehicles: It seems that there is still heavy reliance on the method of close scrutiny of the internal logical consistency of an argument – down to every sentence written or word used, to the extent that a slight detected crack in the logic of an argument immediately spells the demise of the whole theory and the triumph of the protagonist.", "This golden example set up by the classic philosophers before the advent of modern science is unfortunately dated and inadequate when applied to physical issues, because reality often defies logic, at least in the epistemic level.", "This is in contrast to mathematical constructs, especially when the challenge is to come up with a new paradigm to deal with new issues such as emergence.", "This methodology is not wrong, it works for disciplines based on pure logic.", "The mathematical tools physicists use to derive equations are constructed with rigorous logic.", "But there is a divide, often unnoticed or downplayed by non-physicists, between casting physical ideas to mathematical models and inferring physical meanings from mathematical results.", "It is just wrongly applied to arguments, and to the extent arguments cannot be represented rigorous by mathematical concepts, this approach can easily lead one to drawing sweeping or overgeneralized statements or/and missing the key issues, defeating the purpose of philosophy.", "3) Mechanisms and Processes For the study of emergence in physics, a modest way I can suggest which produces reliable and helpful results for philosophical inquiries is to examine existing clear-cut examples with valid physical theories at both the micro (basic or originating) and macro (effective or emergent) levels, examine the underlying physical issues, extract the commonalities and differences, identify notions, terminology and concepts which need further clarification.", "Examples are thermodynamics via kinetic theory, hydrodynamics via molecular dynamics, collective excitations in (atomic-based) condensed matter systems; quantum Hall effect [45], chiral-dynamics in QCD, dimensional reduction in Kaluza-Klein theories.", "In terms of constructing paradigms for emergence from a thorough understanding of the specifics, in addition to the modeling and approximation schemes physicist use all the time, here, what is particularly relevant are the mechanisms and processes whereby new phenomena emerge at a particular collective (or derived) level from a more basic (or elementary) level.", "Symmetry breaking mentioned by Anderson has been cited repeatedly by philosophers.", "There are more, which can be gleaned from the well-established examples mentioned above.", "We will dwell on a few of these in the following.", "A useful platform in physics to explore emergence is nonequilibrium statistical mechanics.", "It consists not just of the formalisms or theorems for physical systems out of equilibrium, not just finding real-time causal equations of motion for the evolution of such systems, but more importantly, for our purpose here, its conceptual constructs in the foundational issues, the physical meaning they convey, but also the methodology adapted to specific physical setups, and the specific mechanisms and processes.", "They all prove to be essential for the explanation of the emergence of forms and structures, even new symmetry principles and constituting laws.", "Just as equilibrium statistical mechanics, in particular Gibbs' ensemble theory, has acted as the base for condensed matter physics for almost a century, nonequilibrium statistical mechanics is the underpinning of disciplines from soft matter physics to chemical-biological systems to sociological- collective behavior to financial-economic modeling.", "It is in these complex systems where emergent phenomena manifest most vividly and strikingly.", "The key issues of nonequilibrium statistical mechanics, which includes stochastic mechanics, open system dynamics, etc, are: (i) collectivity, coarse-graining, correlations - coherence; (ii) noise, fluctuations, stochasticity; (iii) nonlinearity, nonMarkovianity, nonlocality.", "They all have a bearing on emergence in varying degrees.", "We can discuss only one issue here from this list, that of coarse-graining and persistent structures (“protectorates\").", "A description of various physical systems with reference to how certain information in the system is kept, neglected, lost or degraded is facilitated by the distinction between closed, open and effectively open systems.", "This aspect which underlies the thematic material of the next section has been discussed before (see, e.g, the last sections in [32], excerpts appeared in Chapter 1 of [26] with some reference updating).", "We learn that even the very first step in defining an open system or an effectively open but otherwise closed system, is non-trivial.", "In fact the judicious choice of an appropriate set of collective variables already provides half of the answer to the issues of emergence.", "(E.g., witness how simple yet powerful the Laughlin wave function is to capture the fractional quantum Hall effect.)", "The remaining half comes from deriving the dynamics of these collective variables in describing the physical phenomena at this level of structure and interactions (e.g., thermodynamics in terms of temperature, chemical potential, entropy, enthalpy)." ], [ "Key physical issues worthy of closer attention by philosophers: 3Cs and 3Ds", "The 3Cs are “Collectivity, Coarse-graining, Correlation- Coherence\".", "The 3Ds are: Details, Details, Details I'm not referring to what is contained in the title of [54], but to the details in a physical phenomenon and in a physical theory attempting to explain such phenomena.", ": minute details in the study of emergence could be important.", "We feel that in a new field which lacks a well-defined paradigm it is important to pay attention to all the details before drawing some general conclusion than concocting general principles from pure reasons in the belief that they can explain or even predict all the details A section meant to illustrate this point with the processes of environment-induced quantum decoherence (See, e.g., [35], [36]) in the emergence of the classical world is omitted here due to space limitation.", "What I wanted to demonstrate with this example is, even when a process or mechanism can be and has been identified, different parameter regimes of the factors involved (e.g., low temperature, super-Ohmic spectral density) can yield very different outcomes, see, e.g., [55].", "In the decoherent histories approach the existence of quasi-classical domains [27] is, contrary to what was initially conjured, a very complex issue.", "See, e.g, [33].. We shall focus on the issue of coarse-graining and persistent structures here.", "There is no space to include the second set of issues, namely, correlation, and for quantum systems, quantum coherence and entanglement.", "These are important issues for understanding how one level of structure connects to, or yields / reduces to another.", "They appear especially acutely for strongly correlated systems or systems with memories because they make the choice of coarse-graining measures particularly difficult From the projection operator [25] viewpoint, this corresponds to situations where, confronted with the integro-differential equation for one system (containing the dynamics of the other subsystem), one can find no easily justifiable or implementable approximations..", "Some discussions of this point can be found in [56] which addresses these aspects of macroscopic quantum phenomena with references." ], [ "Elimination of degrees of freedom (DOF) and emergence in effective field theory (EFT)", "To set the stage for our discussions I quote the Abstract of Bain [44] who used two examples in physics [45] to illustrate some issues of emergence in effective field theory (EFT) (see, e.g., [53]).", "As stated in the beginning, the more specific a physical theory or model a philosopher can relate to, the more helpful it is to physicists who want to see how related philosophical issues are being dealt with.", "The issue of the `elimination of degrees of freedom' raised there by Bain and by Wilson [46], as related by Bain, are relevant to our theses discussed in the next subsection, namely, on the robustness of coarse-graining measures and persistent structures.", "Bain [44] suggests that EFTs satisfy the following disiderata for a notion of emergence: (i) Emergence should involve microphysicalism, in the sense that the emergent system should ultimately be composed of microphysical systems that comprise the fundamental system and that obey the fundamental system's laws.", "(ii) Emergence should involve novelty, in the sense that the properties of the emergent system should not be deducible from the properties of the fundamental system.", "These disiderata are underwritten in an EFT by the elimination of degrees of freedom in its construction.", "Thus the properties of a system described by an effective Langrangian density Leff can be said to emerge from a fundamental system described by a high energy Langrangian density L in the following sense: (a) High-energy degrees of freedom are integrated out of L. This secures microphysicalism insofar as it entails that the degrees of freedom of $L_{eff}$ are exactly the low-energy degrees of freedom of L. (b) $L_{eff}$ is expanded in a local operator expansion (either to guarantee locality or as a means of approximating the functional integral in (a)).", "The result is dynamically distinct from L, and this secures novelty in the sense of the failure of lawlike deducibility from L of the properties described by $L_{eff}$ .", "According to Mainwood ([47] pp.", "107, 116), the mechanisms Anderson and Laughlin and Pines identify as underwriting (i) and (ii) are spontaneous symmetry breaking and universality .", "Neither of these is generally applicable to EFTs, as the Quantum Hall liquid example indicates.", "Elimination of DOF Mainwood ([47] p. 284) further suggests that a nontrivial notion of emergence requires the specification of a physical mechanism to underwrite (i) and (ii), and it might seem that the elimination of degrees of freedom in an EFT is a formal, as opposed to a physical, mechanism.", "As related to by Bain, Wilson [46] similarly identifies the elimination of degrees of freedom (DOF) as an essential characteristic of a notion of emergence.", "For Wilson, DOF elimination plays two roles.", "First it secures the physical acceptability of an emergent entity by securing the lawlike deducibility of the entity's behavior from its composing parts (p. 295), and such physical acceptability partially underwrites physicalism.", "Second, according to Wilson, DOF elimination entails that an emergent entity is characterized by different law-governed properties and behavior than those of its composing parts, and this suggests that the former cannot be reduced to the latter (p. 301).", "This failure of ontological reduction might charitably be associated with a notion of novelty (although Wilson's explicit goal is simply to establish peaceful coexistence between physicalism and non-reductivism).", "Bain's comment: “This might suggest similarity with the above account of emergence in EFTs.", "However, the type of DOF elimination involved in the construction of an EFT is distinct from Wilson's notion in two major respects.", "First, DOF elimination in an EFT is typically characterized by a failure of lawlike deducibility: The lawlike behavior of entities described by an EFT cannot, in general, be deduced from the lawlike behavior of the entities described by its high-energy theory.", "This failure, I suggested above, is what underwrites a notion of novelty.", "Second, in the presence of such failure, physicalism is preserved, insofar as, in DOF elimination in an EFT, the degrees of freedom of the EFT are exactly the low-energy degrees of freedom of the high-energy theory.\"", "Our comments: Emergence of a new (macro) level of structure does not obtain simply by the “elimination\" of (micro) degrees of freedom.", "This emerged level of structure usually comes from a new set of collective variables which bears no relation to the variables of the sub-level.", "An easy example is again thermodynamics, where one important collective variable is the temperature T. The DOF of the micro theory are the coordinates and momenta $(q_i, p_i)$ of N ($N_A$ Avagadro's number $10^{23}$ ) molecules.", "Elimination of this huge number of DOF to a few does not lead one any closer to seeing the foremost collective variable T. It is in fact the wrong way to think and to go.", "Temperature is related to the kinetic energy of the molecules, not extractable from the elimination of these variables.", "The judicious choice, or more often than not, the creative construction of a set of collective variables from a given set of micro-variables is required and usually poses the most critical challenge.", "The following two subsections excerpted from [32] reflect the complexity of this issue." ], [ "Measures of Coarse-graining", "Coarse graining in the most general sense refers to some information lost, removed, or degraded from a system.", "It could come about because these information is inaccessible to us, due to the limited accuracy in our observation or measurement.", "A drastic example is Planck scale physics, the details of which are mostly lost (hard to retrieve) because the world we live in today is an ultra-low energy construct.", "For this one needs to invoke ideas like effective field theory [53].", "Even when information is fully accessible to us in principle, in practice one may only be interested in some aspects of the system.", "We choose to ignore certain variables such as ignoring the higher order correlations in Boltzmann's kinetic theory, or ignoring the phase information in a quantum system by imposing a random phase approximation.", "We do this by `integrating over' or “projecting out' these `irrelevant' variables.", "Here are some examples of coarse-graining in action.", "We start with the familiar Boltzmann theory: implementation of the molecular chaos assumption (i.e., the 2 particle distribution function $f_2 = f_1f_1$ can be expressed schematically as a product of two 1-particle distribution functions $f_1$ ) entails performing a coarse-graining in the collision integral of space over the range of interaction and of time over the duration of a collision.", "Note also that coarse-graining is a necessary but not sufficient condition for entropy generation.", "It does not always produce a dissipative system.", "Truncation of the BBGKY hierarchy leads to a closed subsystem composed of $n$ - particle correlation functions whose dynamical equations are unitary.", "(An example mentioned before is the Vlasov equation describing particle interaction via long range forces.)", "In quantum field theory equations derived from a finite-loop effective action are also unitary – at one loop the effect of the quantum field on the particles manifests through the renormalize masses and charges (to be exact, the equations of motion derived from a finite-loop effective action are unitary if none of the relevant correlation functions are `slaved' See [26] Chapter 6, Sec.", "3 and Chapter 9 Sec.", "2.3 for a discussion of this concept.", "for $\\ell $ loops, one must keep the first $\\ell +1 $ th order correlations, otherwise dissipation in the sense defined above sets in - dissipation is absent only in very specific situations, such as a free theory or equilibrium initial conditions).", "That is perhaps why (if one limits one's attention to loop expansions) statistical mechanical concepts rarely came to the fore, until one starts asking questions of a distinct nature, such as how dissipative dynamics appears in an otherwise unitary system, and the origin and nature of noise in quantum field theory.", "A causal condition need be introduced to render the dynamics of the subsystem irreversible.", "This opens up another important theme: effective field theory viewed in the open system framework (See [57] for an example of how noise from the higher energy sector can serve as a measure of the validity of a low energy effective field theory.)" ], [ "Persistent structures in the physical world – `protectorates'", "We have seen from the above discussions that the appearance of irreversibility is often traced to the initial condition being special in some sense.", "The dynamics of the system and how it interacts with its environment also enter in determining whether the system exhibits mixing or dissipative behavior.", "For the sake of highlighting the contrast we could broadly divide the processes into two classes depending on how sensitive they are to the initial conditions versus the dynamics.", "One can say that the first class is a priori determined by the initial conditions, the other is a posteriori rather insensitive to the initial conditions.", "Of the examples we have seen, the first group includes divergent trajectories in molecular (micro) dynamics, Landau damping, vacuum particle creation, the second class includes gas (macro) or fluid dynamics, diffusion, particle creation with interaction, decoherence.", "Appearance of dissipation is accompanied by a degradation of information via coarse graining (such as the molecular chaos assumption in kinetic theory, restriction to one-particle distribution in particle creation with interaction, `integrating out' some class of histories in decoherence).", "Many perceived phenomena in the observable physical world, including the phenomenon of time-asymmetry, can be understood in the open-system viewpoint via the approximations introduced to the objective microscopic world by a macroscopic observer.", "[29], [30], [58] We have discussed the procedures such as the system-environment split and the coarse-graining of the environment which can bring about these results.", "However, a set of more important and challenging issues remain largely unexplored, i.e., under what conditions the outcomes become less subjective and less sensitive to these procedures.", "These procedures provide one with a viable prescription to get certain general qualitative results, but are still not specific and robust enough to explain how and why the variety of observed phenomena in the physical world arise and stay in their particular ways.", "To address these issues one should ask a different set of questions: 1) By what criteria are the system variables chosen?", "– collectivity and hierarchy of structure and interactions In a model problem, one picks out the system variables – be it the Brownian particle or the minisuperspace variables – by fiat.", "One defines one's system in a particular way because one wants to calculate the properties of that particular system.", "But in the real world, certain variables distinguish themselves from others because they possess a relatively well-defined, stable, and meaningful set of properties for which the observer can carry out measurements and derive meaningful results.", "Its meaningfulness is defined by the range of validity or degree of precision or the level of relevance to what the observer chooses to extract information from.", "In this sense, it clearly carries a certain degree of subjectivity– not in the sense of arbitrariness in the exercise of free will of the observer, but in the specification of the parameters of observation and measurement.", "For example, a thermodynamic variable like temperature provide excellent description of systems close to equilibrium; in other regimes one needs to describe the system in terms of kinetic-theoretical or molecular-dynamical variables where temperature carries no direct meaning.", "The level of relevance which defines one's system changes with the level of structure of matter and the relative importance of the forces at work at that level.", "The improvement of the Weinberg-Salam model with $W, Z$ intermediate bosons over the Fermi model of four-point interactions is what is needed in probing a deeper level of interaction and structure which puts the electromagnetic and weak forces on the same footing.", "Therefore, one needs to explore the rules for the formation of such relatively distinct and stable levels, before one can sensibly define one's system (and the environment) to carry out meaningful inquiries of a statistical nature.", "What is interesting here is that these levels of structures and interactions come in approximate hierarchical order (thus, e.g., one doesn't need QCD to calculate the rotational spectrum of a nucleus, and the manifold picture of spacetime will hopefully provide most of what we need in the post-Planckian era).", "One needs both some knowledge of the hierarchy of interactions and the way effective theories emerge from `integrating out' variables at very different energy scales in the hierarchical structure (e.g., ordinary gravity plus particle theory regarded as a low energy effective higher-dimension or Kaluza-Klein theory) The first part involves fundamental constituents and interactions and the second part the application of statistical methods.", "One should also keep in mind that what is viewed as fundamental at one level can be a composite or statistical mixture at a finer level.", "There are system-environment separation schemes which are designed to accommodate or reflect these more intricate structures, from the mean field-fluctuation field split to the multiple source or nPI formalism (see [26] Chapter 6) for the description of the dynamics of correlations and fluctuations.", "The validity of these approximations depends quite sensitively on where exactly one wants to probe in between any two levels of structure.", "Statistical properties of the system such as the appearance of dissipative effects and the associated irreversibility character of the dynamics in an open system certainly depend on this separation.", "2) How does the behavior of the subsystem depend on coarse-graining?", "– sensitivity and variability of coarse-graining, stability and robustness of emergent structure Does there exist a common asymptotic regime as the result of including successively higher order iterations in the same coarse-graining routine?", "This measures the sensitivity of the end result to a particular kind of coarse-graining.", "How well can different kinds of coarse-graining measure produce and preserve the same result?", "This is measured by its variability.", "Based on these properties of coarse-graining, one can discuss the relative stability of the behavior of the resultant open system after a sequence of coarse-grainings within the same routine, and its robustness with respect to changes to slightly different coarse-graining routines.", "Let us illustrate this problem with some simple examples.", "When we present a microscopic derivation of the transport coefficients (viscosity, heat conductivity, etc) in kinetic theory via the system-environment separation scheme, we usually get the same correct answer independent of the way the environment is chosen or coarse-grained.", "Why?", "It turns out that this is the case only if we operate in the linear-response regime.", "(See [59]).", "The linear coupling between the system and the environment makes this dependence simple.", "This is something we usually take for granted, but has some deeper meaning.", "For nonlinear coupling, the above problem becomes nontrivial.", "Another aspect of this problem can be brought out [60], [61] by comparing these two levels of structure and interaction, e.g., the hydrodynamic regime and the kinetic regime.", "Construct the relevant entropy from the one-particle classical distribution function $f_1$ , that gives us the kinetic theory entropy $S_{kt}$ which is simply $-k H_{B}$ , where $H_{B}$ is the Boltzmann's H-function.", "Now compare it with the hydrodynamic entropy function $S_{hd}$ given in terms of the hydrodynamic variables (in this case, the number and energy density), one sees that $S_{hd} > S_{kt}$ .", "A simple physical argument for this result is that the information contained in the correlations amongst the particles are not included in the hydrodynamic approximation.", "Even within the kinetic theory regime there exist intermediate stages described by suitably chosen variables [61].", "The entropy functions constructed therefrom will reflect how much fine-grained information is lost.", "In this sense $S_{hd}$ is a maximum in the sequence of different coarse-graining procedures.", "In the terminology we introduced above, by comparison with the other regimes, the hydrodynamic regime is more robust in its structure and interactions with respect to varying levels of coarse-graining.", "One way to account for this is, as we know, the hydrodynamic variables enter in the description of systems in equilibrium and they obey conservation laws [62], [63], [64].", "Further coarse-grainings on these systems is expected to produce the same results, i.e., the hydrodynamic regime is a limit point of sorts after the action from a sequence of coarse-grainings.", "Therefore, a kind of `maximal entropy principle' with respect to variability of coarse-graining is one way where thermodynamically robust systems can be located.", "While including successively higher orders of the same coarse-graining measure usually gives rise to quantitative differences (if there is a convergent result, that is, but this condition is not guaranteed, especially if a phase transition intervenes), coarse-graining of a different nature will in general result in very different behavior in the dynamics of the open system.", "Let us look further at the relation of variability of coarse-graining and robustness of structure.", "Sometimes the stability of a system with respect to the variability of coarse-graining is an implicit criterion behind the proper identification of a system.", "For example, Boltzmann's equation governing the one-particle distribution function which gives a very adequate depiction of the physical world is, as we have seen, only the lowest order equation in an infinite (BBGKY) hierarchy.", "If coarse-graining is by the order of the hierarchy – e.g., if the second and higher order correlations are ignored, then one can calculate without ambiguity the error introduced by such a truncation.", "The dynamics of the open system which includes dissipation effects and irreversible behavior will not change drastically if one uses a different (say more fine-grained) procedure, such as retaining the fourth order correlations (if the series converges, which is a non-trivial issue (see, e.g., [65]).", "Consider now a different approximation: For a binary gas of large mass discrepancy, if one considers the system as the heavy mass particles, ignore their mutual interactions and coarse-grain the effect of the light molecules on the heavy ones, the system now behaves like a Brownian particle motion described by a Fokker-Planck equation.", "We get a qualitatively very different result in the behavior of the system.", "In general the variability of different coarse-grainings in producing a qualitatively similar result is higher (more variations allowed) when the system one works with is closer to a stable level in the interaction range or in the hierarchical order of structure of matter.", "The result is more sensitive to different coarse-graining measures if it is far away from a stable structure, usually falling in between two stable levels.", "Only robust systems survive in nature and carry definite meaning in terms of their persistent structure and systematic evolution.", "This is where the relation of coarse-graining and persistent structures enters.", "So far we have only discussed the activity around one level of robust structure.", "To investigate the domain lying in-between two levels of structures (e.g., between nucleons and quark-gluons) one needs to first know the basic constituents and interactions of the two levels.", "This brings back our consideration of levels of structures above.", "Studies in the properties of coarse-graining can provide a useful guide to venture into the often nebulous and elusive area between the two levels and extract meaningful results pertaining to the collective behavior of the underlying structure.", "But one probably cannot gain new information about the fine structure and the new interactions from the old just by these statistical measures.", "(cf.", "the old bootstrapping idea in particle physics versus the quark model).", "I thank Gerhard Grossing for his invitation and hospitality.", "I am grateful to Professor Hua-Tung Nieh, Director of the Institute for Advanced Study at Tsing Hua University, Beijing, China for making the arrangements for my visit in April-May 2011, Professor Zheng-Yu Weng of IAS-THU for showing me how in his high-$T_c$ superconductivity theory emergence of collective variables and their dynamics comes about and Professor Lu Yu of the Institute of Physics, Chinese Academy of Natural Sciences for a discussion on the proper Chinese technical terminology for emergence.", "This kind of non mission-driven, non utilitarian work addressing purely intellectual issues is not expected to be supported by any U.S. grant agency." ] ]

[ [ "Equivalence of interest rate models and lattice gases" ], [ "Abstract We consider the class of short rate interest rate models for which the short rate is proportional to the exponential of a Gaussian Markov process x(t) in the terminal measure r(t) = a(t) exp(x(t)).", "These models include the Black, Derman, Toy and Black, Karasinski models in the terminal measure.", "We show that such interest rate models are equivalent with lattice gases with attractive two-body interaction V(t1,t2)= -Cov(x(t1),x(t2)).", "We consider in some detail the Black, Karasinski model with x(t) an Ornstein, Uhlenbeck process, and show that it is similar with a lattice gas model considered by Kac and Helfand, with attractive long-range two-body interactions V(x,y) = -\\alpha (e^{-\\gamma |x - y|} - e^{-\\gamma (x + y)}).", "An explicit solution for the model is given as a sum over the states of the lattice gas, which is used to show that the model has a phase transition similar to that found previously in the Black, Derman, Toy model in the terminal measure." ], [ "Introduction", "We consider in this paper the class of one-factor interest rate models with log-normally distributed short rate in the terminal measure.", "In these models the short rate is driven by one Gaussian Markov process $x(t)$ .", "Such a process is defined by two conditions: i) for any set of times $t_1<t_2< \\cdots < t_k$ , the values $(x(t_1),x(t_2),\\cdots , x(t_k))$ have a joint normal distribution; ii) the evolution of $x(s)$ for all $s>t$ depends only on $x(t)$ .", "It can be shown that the most general process of this type is a time-changed Brownian motion, and includes the Ornstein-Uhlenbeck process as a particular case [1].", "This class of models includes the Black, Derman, Toy (BDT) [2] model, and the Black-Karasinski (BK) [3] model, formulated in the terminal measure.", "The terminal measure is sometimes used in practice for these models [4], as opposed to the spot measure in which the models were originally formulated, due to the ease of calibration and simulation.", "Such models have been also proposed as approximations to the Libor market model [5], [6], and as particular parametric realizations of Markov functional models [7], [4].", "A choice of measure amounts to a distributional assumption for the dynamical variables of the model.", "See [8] for a readable introduction to the related concepts of martingales and measure for stochastic processes, and their relation to arbitrage pricing theory.", "In this paper we show that these interest rate models are equivalent with lattice gases with attractive two-body interaction $V(t_1,t_2)=-\\mbox{Cov}(x(t_1),x(t_2))$ , placed in an external potential.", "The solution of the models can be expressed explicitly as an expression for the one-step zero coupon bond given by a sum over occupation numbers in the lattice gas.", "The expectation values required for the simulation of the model correspond to thermodynamical potentials in the lattice gas model.", "We discuss in some detail the Black, Karasinski model with constant mean reversion $\\gamma $ , which is equivalent to a lattice gas with attractive two-body interaction $V(x,y) = -\\alpha (e^{-\\gamma |x - y|} - e^{-\\gamma (x + y)})$ .", "This is similar to a lattice gas model considered by Kac [9], Kac, Uhlenbeck, Hemmer [10] and Kac, Helfand [11], [12].", "This model generalizes the BDT model in the terminal measure, which corresponds to $\\gamma =0$ , and is equivalent with a Coulomb lattice gas with attractive two-body interactions.", "The latter model was studied in Ref.", "[13], where it was shown that it displays discontinuous behaviour in volatility, which is similar to a phase transition in condensed matter physics [14], [15].", "The equivalence with the lattice gas models suggests alternative simulation methods for these interest rate models, which express expectation values as sums over the states of the lattice gas.", "For small lattices this can be done by explicit summation over the lattice gas states, while for bigger lattices efficient numerical methods are available from statistical mechanics, such as Gibbs sampling and the Metropolis algorithm.", "We illustrate this approach by a numerical study of the BK model, which shows that the volatility phase transition observed in the BDT model in Ref.", "[13] persists also for this model." ], [ "The interest rate model", "We consider a short rate interest rate model in discrete time.", "The model is defined on a finite set of dates $0 = t_0 < t_1 < \\cdots < t_n$ For simplicity we will assume that $t_i$ are equally spaced, and denote $\\tau = t_{i+1} - t_i$ with $i=0,1,\\cdots , n-1$ .", "The fundamental dynamical quantities of the model are the zero coupon bonds $P_{i,j}\\equiv P_{t_i,t_j}$ .", "They are defined as the price at time $t_i$ of a payment of 1 made at time $t_j$ .", "They are stochastic quantities, and can be expressed as functions of an one-dimensional Markov process $x(t)$ .", "For definiteness we consider in the following that $x(t)$ is an Ornstein-Uhlenbeck process with zero mean reversion level $dx(t) = -\\gamma x(t) dt + \\sigma dW(t)\\,.$ The mean and variance of $x(t)$ conditional on $x(0)=0$ are $&& \\mathbb {E}[x(t)|x(0)=0]=0 \\\\&& \\mathbb {E}[x^2(t)|x(0)=0] = \\frac{\\sigma ^2}{2\\gamma }(1 - e^{-2\\gamma t})\\equiv G(t)\\,.$ The arguments of this paper can be easily extended to the more general case of $x(t)$ an arbitrary Gaussian Markov process.", "By the Doob's representation, the most general Gaussian Markov process can be represented as a time-modified Brownian motion [1] $x(t) = f(t) \\int _0^t g(s) dW(s)$ with $f(t),g(t)$ deterministic functions of time, and $W(t)$ a Brownian motion.", "We define the Libor rate (or simply Libor) for the $(t_i,t_{i+1})$ period as $L_{i} = \\tau ^{-1} \\Big ( \\frac{1}{P_{i,i+1}} - 1 \\Big )\\,.$ The model is defined by specifying the functional dependence of the Libor rate $L_i$ on the Markov driver $x(t_i)$ $L_i = \\tilde{L}_i \\exp \\Big ( x(t_i) - \\frac{1}{2} G(t_i) \\Big )$ where $\\tilde{L}_i$ are constants to be chosen such that the initial yield curve $P_{0,t}$ is correctly reproduced.", "This implies that the Libors $L_i$ are log-normally distributed in the terminal measure.", "This model is similar with the Black-Karasinski model [2], [3], up to the difference that the latter is usually formulated in the risk-neutral measure, while in the model considered here the short rate $L_i$ is expressed in terms of $x(t)$ defined in the terminal measure.", "In the limit when the time step is taken to zero $\\tau \\rightarrow 0$ , this model becomes a continuous time short rate model, and the short rate $r(t) = \\lim _{\\tau \\rightarrow 0} L_{t/\\tau }(t)$ satisfies the stochastic differential equation $\\frac{dr(t)}{r(t)} = (a(t) - \\gamma \\ln r(t)) dt + \\sigma dW(t)$ with $a(t)$ a function depending on $\\tilde{L}_i$ and $\\sigma $ .", "We recognize this as the short rate evolution in the Black-Karasinski model [3]." ], [ "Explicit solution of the model", "According to the fundamental theorem of arbitrage pricing theory [8], the price of a financial asset $V(t)$ expressed in units of a simpler asset $N(t)$ (called numeraire) is a martingale.", "The mathematical statement of this result is expressed as $V(t)/N(t) = \\mathbb {E}[V(T)/N(T)|{\\cal F}_t]\\,,$ for any $t<T$ .", "This holds under fairly general assumptions, among which market completeness is the most important one.", "Speaking loosely this means that the model contains sufficiently many tradeable instruments to allow any possible payout to be reproduced as a combination thereof.", "The choice of the numeraire $N(t)$ is not unique, and any particular choice defines a measure for the stochastic process followed by the discounted asset prices $V(t)/N(t)$ .", "Two particular choices are most common in the context discussed here.", "The spot measure, or the risk-neutral measure, takes $N(t)$ to be the money market account at time $t$ , while the terminal measure (or $t_n$ -forward measure) takes $N(t)=P_{t,n}$ to be the zero coupon bond maturing at time $t_n$ .", "Once the condition (REF ) is imposed, different measure choices produce different observable distributional properties of the dynamical quantities of the model (rates and bonds), and thus effectively correspond to different models.", "We will work in the terminal measure in the following.", "It is convenient to introduce the zero coupon bond prices divided by the numeraire $P_{t,n}$ , which will be denoted as $\\hat{P}_{i,j} = P_{i,j}/P_{i,n}$ .", "They are martingales in the terminal measure, and thus satisfy the condition (REF ), which reads explicitly $\\hat{P}_{i,j} = \\mathbb {E}\\Big [\\frac{P_{k,j}}{P_{k,n}} | {\\cal F}_i\\Big ]$ for all $i < k < j \\le n$ .", "The one-step discounted zero bond $\\hat{P}_{i,i+1}(x_i)$ will play an important role in writing the analytical solution of this model.", "It satisfies a few conditions, following from the martingale condition (REF ).", "First, its expectation value is known in terms of the initial yield curve $\\mathbb {E}[\\hat{P}_{i,i+1}(x_i)] = \\hat{P}_{0,i+1} \\,.$ It also satisfies the two conditions $&& \\hat{P}_{i,i+1}(x_i) = \\mathbb {E}[ \\hat{P}_{i+1,i+2}(x_{i+1})(1 + L_{i+1}(x_{i+1})\\tau ) | {\\cal F}_i] \\nonumber \\\\&& \\\\&& \\hat{P}_{0,i} = \\mathbb {E}[\\hat{P}_{i,i+1}(x_i) (1 + L_i(x_i)\\tau ) ]$ The first condition (REF ) determines recursively the functional form of $\\hat{P}_{i,i+1}(x_i)$ , starting with $\\hat{P}_{n-1,n}=1$ and proceeding backwards in time.", "This is given explicitly as a conditional expectation value $\\hat{P}_{i,i+1}(x_i) &=& \\mathbb {E}\\Big [ \\prod _{k=i+1}^{n-1} (1 + \\tilde{L}_k\\tau e^{x_k - \\frac{1}{2} G_k}) | {\\cal F}_i\\Big ] \\,.$ The second condition () can be used to determine $\\tilde{L}_i$ also recursively, once $\\hat{P}_{i,i+1}(x_i)$ has been determined, using the relation $\\tilde{L}_i = \\frac{\\hat{P}_{0,i} - \\hat{P}_{0,i+1}}{\\mathbb {E}[\\hat{P}_{i,i+1}\\exp (x_i - \\frac{1}{2} G_i)] \\tau } \\,.$ For simplicity we denote the value of the Markov driver at time $t_i$ as $x_i \\equiv x(t_i)$ , and its variance as $G(t_i) = G_i$ .", "We will state in the following the closed form of the solution of this model.", "The solution expresses the discounted one-step zero coupon bonds $\\hat{P}_{i,i+1}(x_i)$ as a sum of terms containing $0,1,2,\\cdots , n-i-1$ $\\tilde{L}_j$ factors.", "Writing the first few terms explicitly this is given by $&& \\hat{P}_{i,i+1}(x_i) = 1 + \\sum _{j=i+1}^{n-1}\\tilde{L}_j \\tau \\exp ( w^{j-i} x_i - \\frac{1}{2} w^{2(j-i)} G_i ) \\nonumber \\\\&& + \\sum _{j>k=i+1}^{n-1}\\tilde{L}_j\\tilde{L}_k \\tau ^2\\exp \\Big ( (w^{j-i} + w^{k-i}) x_i- \\frac{1}{2} (w^{j-i} + w^{k-i})^2 G_i + X_{jk}\\Big ) + \\cdots \\\\&& + \\sum _{k\\le n-i-1} \\sum _{S_k \\in T_i}\\tilde{L}_{j_1}\\tilde{L}_{j_2}\\cdots \\tilde{L}_{j_k} \\tau ^k\\exp \\Big ( \\sum _{a=1}^k w^{j_a-i} x_i -\\frac{1}{2} (\\sum _{a=1}^k w^{j_a-i})^2 G_i +\\sum _{1<a<b<k} X_{j_a, j_b} \\Big ) \\nonumber \\,.$ We denoted here the weight $w=\\exp (-\\gamma \\tau )$ , and the auto-covariance of the Markov process $x(t)$ as $X_{jk} &=& \\mbox{Cov}(x(t_j), x(t_k))\\\\&=& \\frac{\\sigma ^2}{2\\gamma }(e^{-\\gamma |t_j - t_k|} - e^{-\\gamma (t_j + t_k)})\\,.\\nonumber $ The general term in Eq.", "(REF ) containing $k\\le n-i-1$ factors of $\\tilde{L}_j$ is given by a sum over all subsets $S_k = \\lbrace j_1, j_2, \\cdots , j_k\\rbrace $ of $k$ indices chosen from the $n-i-1$ indices $T_i \\equiv \\lbrace i+1,i+2, \\cdots , n-1\\rbrace $ .", "In the limit of zero mean reversion $\\gamma \\rightarrow 0$ , we have $w=1$ and $G(t) = \\sigma ^2 t$ , and the expression (REF ) simplifies drastically.", "In this limit all terms with the same number of $\\tilde{L}_j$ factors have the same functional dependence of $x_i$ , and we recover the simple form obtained in Ref.", "[13] $\\hat{P}_{i,i+1}(x) = \\sum _{j=0}^{n-1} c_j^{(i)} e^{j x_i -\\frac{1}{2} j^2 G_i}$ where the coefficients $c_j^{(i)}$ are given by $c_k^{(i)} = \\sum _{S_k}\\tilde{L}_{j_1}\\tilde{L}_{j_1}\\cdots \\tilde{L}_{j_k} \\tau ^k\\exp ( \\sum _{1<a<b<k} X_{j_a, j_b} )$ where $X_{j,k} = \\sigma ^2 \\mbox{min}(t_j, t_k)$ .", "In [13] these coefficients were determined recursively from a recursion relation, see Eq.", "(12) in Ref. [13].", "Equation (REF ) gives an explicit solution of this recursion relation.", "An important role is played in this model by the expectation values of the form $&& N_i(\\phi ) = \\mathbb {E}[\\hat{P}_{i,i+1} e^{\\phi x_i - \\frac{1}{2} \\phi ^2 G_i}]\\\\&& =1 + \\sum _{j=i+1}^{n-1}\\tilde{L}_j \\tau \\exp (\\phi w^{j-i} G_i) + \\cdots \\nonumber \\\\&& \\qquad + \\sum _{k\\le n-i-1}\\sum _{S_k}\\tilde{L}_{j_1}\\tilde{L}_{j_2}\\cdots \\tilde{L}_{j_k} \\tau ^k \\nonumber \\\\&& \\times \\exp ( \\phi G_i \\sum _{a=1}^k w^{j_a-i} + \\sum _{1<a<b<k} X_{j_a, j_b} )\\nonumber \\,.$ We enumerate in the following the applications of these expectation values with $\\phi = 0,1,\\cdots $ .", "The expectation value of $\\hat{P}_{i,i+1}$ (corresponding to $\\phi =0$ ) is constrained by the requirement that the initial yield curve $P_{0,i}$ is correctly reproduced, see (REF ).", "$&& \\mathbb {E}[\\hat{P}_{i,i+1}] = \\hat{P}_{0,i+1} =1 + \\sum _{j=i+1}^{n-1} \\tilde{L}_j \\tau + \\cdots \\\\&& + \\sum _{S_k}\\tilde{L}_{j_1}\\tilde{L}_{j_1}\\cdots \\tilde{L}_{j_k} \\tau ^k\\exp ( \\sum _{1<a<b<k} X_{j_a, j_b} ) + \\cdots \\nonumber $ The sum on the right-hand side is linear in $\\tilde{L}_{i+1}$ and thus can be used to solve explicitly for this constant, provided that all $\\tilde{L}_j$ with $j=i+2, \\cdots , n-1$ are already known.", "This is given in Eq.", "(REF ) in a form more convenient for practical calculation.", "The $\\phi =1$ expectation value appears in the calculation of the convexity-adjusted Libors $\\tilde{L}_i$ Eq.", "(REF ), which can be written equivalently as $\\tilde{L}_i = \\hat{P}_{0,i+1} L_i^{\\rm fwd} \\frac{1}{N_i(1)}\\,.$ Finally, $N_i(j)$ with $j\\in \\mathbb {Z}_+, j>1$ determines the $j-$ th moment of the Libor distribution in its natural (forward) measure according to the relation [16] $\\mathbb {E}_{i+1}[(L_i)^j] &=& \\frac{1}{\\hat{P}_{0,i+1}} (\\tilde{L}_i)^j\\mathbb {E}_n [ \\hat{P}_{i,i+1} e^{j x_i - \\frac{1}{2} j G_i} ] \\nonumber \\\\&=&\\frac{1}{\\hat{P}_{0,i+1}} (\\tilde{L}_i)^j e^{-\\frac{1}{2} (j-j^2) G_i} N_i(j)$ In the limit of zero mean-reversion $\\gamma \\rightarrow 0$ the above expectation values are given by simple expressions [13] $N_i(\\phi ) = \\mathbb {E}[\\hat{P}_{i,i+1} e^{\\phi x_i -\\frac{1}{2}\\phi ^2 G_i}] =\\sum _{j=0}^{n-i-1} c_j^{(i)} e^{j \\phi ^2 \\sigma ^2 t_i}\\,.$ For sufficiently small volatility $\\sigma $ , the expectation values $N_i(\\phi )$ given in Eq.", "(REF ) can be computed in an expansion of the small parameter $\\tilde{L}_i\\tau \\ll 1$ , and keeping only the terms linear in this parameter is sufficient for most applications.", "In this approximation we have $N_i(\\phi ) = 1 + \\sum _{j=i+1}^{n-1} L_j^{\\rm fwd} \\tau e^{\\phi w^{j-i} G_i}+ O((L_k^{\\rm fwd}\\tau )^2)$ The distribution of the Libors in their natural measure is approximatively log-normal and the ATM caplet volatility is $\\sigma _{\\rm LN}^2 = \\frac{G(t_i)}{t_i}\\,.$ In the model with zero mean reversion $\\gamma =0$ , it was noted in Ref.", "[13] that for volatility $\\sigma $ above some critical value, the higher order terms in the expansion (REF ) become comparable to the linear terms of $O(\\tilde{L}_i \\tau )$ .", "The actual expansion parameter becomes $\\tilde{L}_i \\tau \\exp (\\sigma ^2 t_i)$ and terms of all orders in $L_i^{\\rm fwd} \\tau $ become important.", "This leads to a discontinuity in the first derivative of the expectation value $N_i(\\phi )$ with respect to the volatility $\\sigma $ , which is similar to a phase transition in condensed matter physics [14], [15].", "In the next section we express the expectation values (REF ) as averages over the grand canonical ensemble in an equivalent lattice gas model.", "This is used to show the existence of a phase transition also in this model, using a numerical simulation." ], [ "Proof", "The result (REF ) can be proven using the following basic identity.", "For any numbers $n_k=0,1$ associated with the ordered sequence of times $t \\equiv t_0 \\le t_1 \\le t_2 \\cdots < t_N$ , the following expectation value with $x(t)$ the Ornstein-Uhlenbeck process (REF ) is given by $&& \\mathbb {E}\\Big [\\exp \\Big (\\sum _{k=1}^N n_k (x_k -\\frac{1}{2}G_k ) \\Big ) | {\\cal F}_t\\Big ] \\\\&& = \\exp \\Big (x_t\\sum _{k=1}^N n_k e^{-\\gamma t_k}-\\frac{1}{2} G_t (\\sum _{k=1}^N n_k e^{-\\gamma t_k})^2 \\nonumber \\\\&& \\qquad + \\frac{1}{2} \\sum _{j\\ne k=1}^N X_{j,k} n_j n_k\\Big ) \\nonumber $ where $X_{j,k}$ is the covariance of the process $x(t)$ given above in Eq.", "(REF ).", "This is a slight generalization of an identity used in Ref.", "[9], [11] to compute the partition function of a lattice gas with exponential interaction.", "It can be easily generalized to the case of a general Gaussian Markov process $x(t)$ .", "The discounted one-step bond $\\hat{P}_{i,i+1}(x_i)$ is given by the conditional expectation (REF ).", "Expanding out the product yields terms with $0, 1, 2, \\cdots $ factors of $\\tilde{L}_k \\tau $ , up to $n-i-1$ factors.", "There are $\\binom{n-i-1}{N}$ terms containing $N$ such factors, and they are given by a sum over all subsets $\\lbrace n_k \\rbrace =\\lbrace n_{k_1}, n_{k_2}, \\cdots , n_{k_N}\\rbrace $ of $N$ indices out of the total of $n-i-1$ indices.", "A generic term has the form $&& \\sum _{\\lbrace n_k \\rbrace }\\Pi _{j=1}^{N} (\\tilde{L}_{k_j} \\tau )\\mathbb {E}[\\exp \\Big ( \\sum _{j=1}^{N} n_{k_j} (x_{k_j} - \\frac{1}{2} G_{k_j} )\\Big )|{\\cal F}_i] \\nonumber \\\\&& = \\sum _{\\lbrace n_k \\rbrace }\\Pi _{j=1}^{N} (\\tilde{L}_{k_j} \\tau ) \\\\&& \\times \\exp \\Big (x_i\\sum _{k=1}^N e^{-\\gamma t_{k_j}} n_{k_j} -\\frac{1}{2} G_i (\\sum _{k=1}^N n_{k_j} e^{-\\gamma t_{k_j}})^2\\nonumber \\\\&& \\qquad +\\sum _{k_j < k_l} X_{k_j,k_l} n_{k_j} n_{k_l} \\Big )\\nonumber $ where the expectation value was computed using the identity (REF ).", "This reproduces the terms containing $N$ factors of $\\tilde{L}_k\\tau $ in Eq.", "(REF ).", "This completes the proof of (REF )." ], [ "The lattice gas model", "The interest rate model considered in the previous section is equivalent with a one-dimensional lattice gas with attractive long-range potential $V(x,y) = -\\alpha (e^{-\\gamma |x - y|} - e^{-\\gamma (x + y)})$ The particles of the lattice gas are constrained to sit at positions $x_i = \\tau i$ , with $i=1,2,\\cdots , n-1$ .", "The $n$ sites of the lattice gas are labeled as $j = 0, 1, \\cdots , n-1$ .", "The sites $j$ are in one-to-one correspondence with the discrete set of simulation times $\\lbrace t_j \\rbrace $ of the interest rate model.", "At each site at most one particle can be present.", "We define $n_j$ the occupation number of the site $j$ .", "It can take values 0 or 1, depending on whether the site $j$ is vacant or occupied.", "The Hamiltonian of the lattice gas model is $H = \\sum _{j=1}^{n-1} \\varepsilon _j n_j +\\sum _{j>k=1}^{n-1} \\varepsilon _{jk} n_j n_k$ The two-body interaction is $\\varepsilon _{jk} =-\\alpha (e^{-\\gamma \\tau |j-k|} - e^{-\\gamma \\tau (j + k)})$ and the single-site energies are $\\varepsilon _j = - \\beta ^{-1} \\ln (\\tilde{L}_j \\tau )\\,.$ For the application to the interest rate model we are interested not only in the entire lattice system, but also in the subsystem $\\mathbb {T}_i$ of the lattice consisting of the sites $\\mathbb {T}_i: \\lbrace i+1, \\cdots , n-1\\rbrace $ , in total $n_f=n-i-1$ sites.", "Assume that the subsystem $\\mathbb {T}_i$ of the lattice gas is placed in a position-dependent chemical potential $\\mu ^{(i)}(t) = \\mu G_i e^{-\\gamma (t-t_i)}$ The grand partition function of the subsystem $\\mathbb {T}_i$ of the lattice gas with the Hamiltonian (REF ) and placed in the chemical potential (REF ) is given by ${\\cal Z}_i(\\mu ,T) = \\sum _{N=0}^{n-i-1} \\sum _{S_N}\\exp \\Big (-\\beta H + \\beta \\sum _{j\\in S_N}\\mu ^{(i)}(t_j)\\Big )$ The sum over the number of particles $N$ runs from 0 to $n-i-1$ , the number of lattice sites in the subsystem $\\mathbb {T}_i$ .", "For each $N$ the sum runs over all configurations $S_N$ of $N$ occupied sites, which are subsets of $N$ sites of the $n-i-1$ sites in the system $\\mathbb {T}_i$ .", "The correspondence of this lattice gas model with the interest rate model is realized through the following relation between the grand partition function ${\\cal Z}_i(\\mu ,T)$ and the expectation value (REF ) $N_i(\\phi ) = {\\cal Z}_i(\\mu ,T) \\,,$ provided that the parameters of the lattice gas are related to those of the interest rate model as $\\alpha \\beta = \\frac{\\sigma ^2}{2\\gamma }\\,,\\qquad \\phi = \\beta \\mu $ This system is similar to the one-dimensional gas considered by Kac [9] and by Kac, Uhlenbeck, Hemmer [10].", "A lattice version of the gas model, very similar to that considered here, was examined by Kac and Helfand in Ref.", "[11], [12].", "More precisely, the latter papers consider a lattice gas, where the particles occupy a lattice with $N$ nodes and lattice spacing 1, and interact by two-body attractive potentials $V(|x-y|) = -\\alpha \\gamma e^{-\\gamma |x-y|}$ .", "This model has a phase transition in the so-called van der Waals limit, which is obtained by first going to the thermodynamical limit of large $N$ , followed by the infinite range limit $\\gamma \\rightarrow 0$ .", "In the van der Waals limit the lattice gas model has a liquid-gas phase transition with critical temperature $\\beta _c\\alpha = \\frac{1}{2}$ , and the equation of state is given by the van der Waals equation supplemented by the equal area rule [10].", "At this point it may be useful to recall a few well-known facts about phase transitions in one-dimensional systems [17].", "Although a phase transition does not exist in a one-dimensional system with short range interactions [18], it is possible for such a system to have a phase transition provided that the interaction is sufficiently long range.", "Sufficient conditions which have to be satisfied by the interaction in order for a phase transition to exist in a one-dimensional system were given in [19].", "The papers [10] provided the first instance of phase transition in a one-dimensional system, and showed explicitly that this can occur in a system with long-range interactions.", "The results of [10] have been extended to more general interactions and higher dimensional systems in Ref. [20].", "The zero mean-reversion limit of the interest rate model $\\gamma =0$ is the Black, Derman, Toy model in the terminal measure [13], and is equivalent with a lattice gas model with attractive Coulomb two-body interactions, placed into an external potential.", "This can be seen by writing the covariance of the Markov driver for this case as $&& -V(t_1,t_2) = \\mbox{Cov}(x(t_1), x(t_2))_{\\gamma =0} =\\sigma ^2 \\mbox{min} (t_1,t_2) \\nonumber \\\\&& \\qquad = \\frac{1}{2} \\sigma ^2 (|t_1 - t_2| - (t_1+t_2))\\,.$ The first term describes an attractive linear interaction between the pair of particles at sites $t_1,t_2$ , while the second term can be represented as their interactions with the repulsive external field of a static charge placed at the site $i=0$ .", "The one-dimensional gas with Coulomb interaction between several types of charges was studied, using methods very similar to those employed here, by Edwards and Lenard [21].", "Our Coulomb lattice gas is different from a usual Coulomb gas in that all particles attract each other.", "The thermodynamics of a one-dimensional system with linear attractive potentials was considered in Ref.", "[22], although periodic boundary conditions were imposed such that the resulting form of the interaction is different from that considered here.", "A connection between stochastic processes and the (two-dimensional) Coulomb gas was realized in a different context in Ref. [23].", "Figure: (Color online) Plots of lnN i (1)\\ln N_i(1) vs σ\\sigma for several values of themean-reversion parameter γ\\gamma , with i=30i=30 in a simulation withn=40n=40 quarterly time steps τ=0.25\\tau =0.25.", "The black curve (leftmost)corresponds toγ=0\\gamma =0 and is obtained using the method used in Ref.", ".The other curves (from left to right) are obtained by explicit summation overthe occupation numbers of the lattice gas as explained in the text:γ=0.1%\\gamma =0.1\\% (blue), 1%1\\% (red), 2%2\\% (green), 5%5\\% (orange).The lattice gas with non-zero mean-reversion considered here differs from that studied by Kac and Helfand [11], [12] in several respects, due to the peculiarities of the interest rate model.", "1.", "The presence of the $\\tilde{L}_j$ factors requires the introduction of single-site energies $\\varepsilon _j$ associated with the lattice sites.", "These energies are different and thus the space homogeneity of the system is lost.", "This space homogeneity was crucial for the analytical solution of the model in the thermodynamical limit [9], [10], [11].", "A similar approach is unlikely in this case for this reason.", "The single-site energies $\\varepsilon _j$ are constrained by the condition (REF ) such that the initial yield curve $P_{0,i}$ is correctly reproduced.", "According to this relation, $\\varepsilon _j$ depends on the properties of the subsystem $T_{j-1}$ of the lattice gas, and must be determined by a recursive procedure starting with the smallest subsystem $T_{n-2}$ and adding one lattice site at a time.", "2.", "The two-body interaction in the lattice gas (REF ) contains a second exponential term $\\exp (-\\gamma (t_i + t_j))$ , which is not present in Refs.", "[11], [12].", "This is due to the fact that the expectation values (REF ) are conditional on $x(0)=0$ , while [11], [12] integrate over $x(0)$ .", "While the new term does not have the typical form of a two-body interaction, its inclusion does not present any problem of principle.", "Also, this term becomes vanishingly small if the subsystem $\\mathbb {T}_i$ is chosen such that $\\gamma t_i \\gg 1$ , and the simple exponential Kac interaction is recovered in this limit.", "The equivalence of these interest rate models with lattice gases suggests an alternative way of calibrating and simulating such models.", "The expectation values $N_i(\\phi )$ are usually [4], [7] computed by evaluating the nested integrations over the values of the Markov driver $x(t)$ at the simulation times, using numerical approaches such as finite difference or Monte Carlo methods.", "The results (REF ) and (REF ) suggest that the expectation values $N_i(\\phi )$ can be also computed as averages over the grand canonical ensemble in the lattice gas.", "For small lattices, this can be done by explicit summation over all possible occupation numbers ($2^n$ configurations for a lattice with $n$ sites), while for larger lattices alternative methods familiar from statistical mechanics can be used, such as Gibbs sampling and the Metropolis-Hastings algorithm [24], [25].", "As an illustration of this approach, we show in Fig.", "REF the results of a simulation of the BK model in the terminal measure performed by summing over the occupation numbers of the lattice gas.", "These plots show the multiplicative convexity adjustment $\\ln N_i(1)$ for $i=30$ as function of $\\sigma $ for several values of the mean-reversion parameter $\\gamma $ .", "The simulation assumed $n=40$ quarterly time steps $\\tau = 0.25$ , for a total simulation time $t_n=10$ years.", "The forward yield curve is flat with $L_i^{\\rm fwd}=5\\%$ .", "The $\\gamma =0$ curve is obtained using the recurrence method of [13], and the remaining curves were obtained by computing $N_i(1)$ using (REF ) by explicit summation over the $2^{n-i-1}=512$ states of the subsystem $\\mathbb {T}_{30}$ of the lattice gas.", "These results show that the transition observed in Ref.", "[13] persists also in the model considered here.", "The mean-reversion $\\gamma $ allows one to control the range of the two-body interaction in the lattice gas.", "In the $\\gamma \\rightarrow 0$ limit the lattice model particles attract each other with Coulomb potentials, while for $\\gamma \\ne 0$ the potential becomes exponential and is given in Eq.", "(REF ).", "In the $\\gamma \\rightarrow 0$ limit the results of [13] are recovered: the convexity adjustment factor increases suddenly above the critical volatility $\\sigma _{\\rm cr} \\simeq 32\\%$ .", "As the mean reversion $\\gamma $ is increased from zero, the transition persists, and the critical volatility increases from its $\\gamma =0$ value.", "The $\\gamma \\rightarrow 0$ limit is well-behaved, as expected for a finite size lattice.", "The study of the $\\gamma =0$ limit of this model presented in Ref.", "[13] showed that the phase transition is not visible under usual simulation methods used in practice for such interest rates models, such as finite difference or Monte Carlo methods.", "This is due to the fact that these methods effectively truncate the range of values of the Markovian driver $x(t)$ to a few ($\\sim 5$ ) multiples of $\\sigma \\sqrt{t}$ .", "Such a truncation omits the contributions to the expectation values $N_i(\\phi )$ which are responsible for the phase transition.", "The alternative method proposed here offers a possible way to study the properties of these models, free of these limitations." ], [ "Conclusions", "We presented in this paper the exact solution of a class of interest rates models with log-normally distributed short rates in the terminal measure.", "The solution is formulated naturally in terms of a lattice gas with sites corresponding to the simulation times of the model $t_i$ .", "At each site only one particle can be present, and the particles interact by attractive two-body potentials $V_{ij}$ which are determined by the stochastic process followed by the short rate.", "The analogy with the lattice gas models simplifies very much the simulation of these models, as many of the important expectation values in the interest rate model can be written in closed form as averages over the grand canonical ensemble in the corresponding lattice gas.", "The numerical evaluation of these averages is straightforward for small lattices (few simulation times in the interest rate model), while for larger lattices the number of configurations ($2^n$ for a lattice with $n$ sites) becomes too large for direct evaluation, and approximation methods familiar from statistical mechanics may have to be used [24], [25].", "We used the exact lattice gas solution to study numerically the Black, Karasinski model in the terminal measure with constant mean-reversion and volatility.", "This showed the appearance of a phase transition in the convexity adjustments of single-period interest rates, similar to that noted in the Black, Derman, Toy model in the terminal measure in Ref. [13].", "This adds further support to the suggestion made in Ref.", "[13] that the presence of such a transition is generic for all interest rate models with log-normally distributed rates in the terminal measure.", "Although the present numerical study considered only the version of the model with constant parameters, the method can be extended without any major difficulty also to the more general case of time-dependent model parameters.", "This is in contrast to the method of the recursion relations used in [13] to solve the $\\gamma =0$ limit of the model with uniform volatility, which does not appear to be easily extended beyond this case due to the unmanageable complexity of the resulting expressions.", "The equivalence of the interest rates models considered with interacting lattice gases shows that the former have a rich dynamics which has not been fully explored.", "Physical intuition about the lattice gas equivalent should give further insight into the dynamics of the interest rate models.", "In particular, one natural question is whether a phase transition similar to that studied in Ref.", "[10] is present also in the lattice gas considered here, and if it is observed also for a finite size lattice.", "The analog of the van der Waals limit for this case corresponds to simultaneously scaling the volatility as $\\sigma = \\sigma _0 \\gamma $ as the mean reversion is taken to zero $\\gamma \\rightarrow 0$ .", "It would be interesting to see if the behaviour of the system in this limit has implications also for the practically relevant case of non-zero volatility.", "Finally, it would be interesting to investigate whether the exact solution presented here can be extended also to other interest rate models, with more general distributional properties.", "Hopefully the lattice gas analogy will remain useful also for more general interest rate models." ] ]

[ [ "Operator entanglement of two-qubit joint unitary operations revisited:\n Schmidt number approach" ], [ "Abstract Operator entanglement of two-qubit joint unitary operations is revisited.", "Schmidt number is an important attribute of a two-qubit unitary operation, and may have connection with the entanglement measure of the unitary operator.", "We found the entanglement measure of two-qubit unitary operators is classified by the Schmidt number of the unitary operators.", "The exact relation between the operator entanglement and the parameters of the unitary operator is clarified too." ], [ "Introduction", "Unitary operations have been placed in a very important position in the quantum communication and entanglement manipulating, such as, quantum cryptography[1], teleportation[2], entanglement swapping[3], quantum states purification[4], entanglement production[5] and so on.", "In quantum teleportation, to transfer the unknown quantum state to the remote user, the sender must apply an joint unitary operator on the unknown state particle and one of the entangled particles.", "In quantum entanglement swapping, a joint unitary transformation on two particles(they are from two different entangled pairs) will let two remote particles entangled without direct interaction.", "In entanglement purification process, joint unitary operations and measurements can transfer the entanglement from many partially entangled pairs to few near perfect entangled pairs.", "In entanglement generation, the joint unitary operations and single qubit operations can let the initially product particles entangled.", "From the above applications we can see that, it is the nonlocal attribute of the bipartite joint unitary transformation that plays the most important role.", "The nonlocal attribute of a bipartite joint unitary operator has been studied from different aspects, such as entangling power[6], operator entanglement[7], [8], and entanglement-changing power[9].", "Entangling power is the mean entanglement(linear entropy) produced by acting with U on a given distribution of pure product states[6].", "Because a quantum operator belongs to a Hilbert-Schmidt space, one can consider the entanglement of the operator itself, which is named as Operator entanglement[7].", "It is a natural extension of the entanglement measures of quantum states[10], [11], [12], [13] to the level of general quantum evolutions.", "Up to now, several methods have been proposed to quantify the entanglement of an unitary bipartite operator, such as linear entropy[7], von Neumann entropy[8], concurrence[14] and Schmidt strength[15] etc.", "The relations between the entangling power and these operator entanglement measures have also been discussed recently[8], [7], [16].", "In general, the entangling power of an unitary operator is related to those operator entanglement measures in complicated or indirect ways, so does the relation between the different operator entanglement measures.", "Clarifying the exact relation between the entangling power and different operator entanglement measures, and the relation between different operator entanglement measures will be very helpful for us to understand the nonlocal attributes and entanglement capacity of a joint unitary operator.", "After getting the whole nonlocal features of joint unitary operators, we can choose the optimal unitary operator to produce the specific entangled state as we want it to be, and the quantum communication protocols(such as teleportation, entanglement swapping etc) can be realized in an optimal way by introducing the appropriate joint operations[17].", "For two-qubit unitary operators, the two operator entanglement measures Schmidt strength and linear entropy are shown to have a one-to-one relation between them for the Schmidt number 2 case, but no such relation exists for the Schmidt number 4 case[16].", "This result also shows that, the Schmidt number is a very important parameter of an unitary operator when the entangling power and operator entanglement of it is concerned.", "In this paper, we are going to study the operator entanglement of joint two-qubit unitary operators with different Schmidt numbers.", "In this paper, we use the linear entropy as the entanglement measure of joint two-qubit unitary operator[7], [8], and study the Schmidt number and the entanglement measure of any unitary operator in four-dimensional Hilbert-Schmidt space.", "The Schmidt number of two-qubit unitary operators has the following three possible situations: 1, 2 or 4[15].", "We will show that the entanglement measure of two-qubit unitary operators is classified by the Schmidt number of the unitary operator.", "In the light of the numerical analysis, we can get the extreme value of the operator entanglement for the two-qubit unitary operators.", "Further, the relation between the operator entanglement and the parameters of the unitary operator will be clarified here." ], [ "Operator entanglement and Schmidt number of two-qubit joint unitary operations", "There exist local unitary operators $U_{A}$ ,$U_{B}$ ,$V_{A}$ ,$V_{B}$ and a two-qubit unitary operator $U_{d}$ , so that arbitrary two-qubit unitary operator $U_{AB}$ can be canonically decomposed as[18], [19]: $U_{AB}=(U_{A}\\otimes U_{B})\\cdot U_{d}\\cdot (V_{A}\\otimes V_{B}),$ where $U_{d}=\\exp [-i\\vec{\\sigma }_{A}^{T}d \\vec{\\sigma }_{B}]$ , and $d$ is a diagonal matrix.", "In the light of this theory, any bipartite unitary operator can be decomposed as the form above.", "Moreover, the entanglement measure of a unitary operator's must be invariant under the local unitary transformations[14].", "So, the entanglement measure of any bipartite unitary operator can be simplified into the entanglement measure of operator $U_{d}$ .", "In the standard computational basis, we have[20]: $U_{d}=\\left(\\begin{array}{cccc}e^{-i{c}_{3}}c^{-} & 0 & 0 & {-i}e^{-i{c}_{3}}s^{-} \\\\0 & e^{i{c}_{3}}c^{+} & {-i}e^{i{c}_{3}}s^{+} & 0 \\\\0 & {-i}e^{i{c}_{3}}s^{+} & e^{i{c}_{3}}c^{+} & 0 \\\\{-i}e^{{-i}{c}_{3}}s^{-} & 0 & 0 & e^{{-i}{c}_{3}}c^{-} \\\\\\end{array}\\right),$ where $c^{\\pm }=\\cos ({c}_{1}\\pm {c}_{2})$ , $s^{\\pm }=\\sin ({c}_{1}\\pm {c}_{2})$ , and one can always restrict oneself to the region $\\frac{\\pi }{4}\\ge {c}_{1}\\ge {c}_{2}\\ge |{c}_{3}|$ , which is the so-called Weyl chamber[19].", "Any operator $U$ acting on the systems $A$ and $B$ can be written in the operator-Schmidt decomposition[21]: $U=\\sum _{l}{s}_{l}{A}_{l}\\otimes {B}_{l},$ where ${s}_{l}$ are the Schmidt coefficients with the positive value and ${A}_{l}$ , ${B}_{l}$ are orthonormal operator bases for $A$ and $B$ , respectively.", "To calculate the operator entanglement of the unitary operator $U_{AB}$ , we only need to make the Schmidt decomposition of the unitary operator ${U}_{d}$ .", "From the Ref.", "[8], entanglement measure of a unitary operator can be expressed as: $E(U)=1-\\sum _{l}\\frac{s_{l}^{4}}{d_{1}^{2}d_{2}^{2}},$ where ${d}_{1}$ and ${d}_{2}$ are dimensions of $A$ and $B$ , respectively.", "So we can get the entanglement measure for the unitary operator ${U}_{d}$ as follows: $E({U}_{d})&=&1-\\frac{1}{4}\\lbrace 1-\\sin ^{2}({c}_{1}+{c}_{2})\\cos ^{2}({c}_{1}+{c}_{2}) -\\sin ^{2}({c}_{1}-{c}_{2})\\cos ^{2}({c}_{1}-{c}_{2})\\\\ \\nonumber &&+[1+2\\cos ^{2}(2{c}_{3})]\\sin ^{2}({c}_{1}+{c}_{2})\\sin ^{2}({c}_{1} -{c}_{2})\\\\ \\nonumber &&+[1+2\\cos ^{2}(2{c}_{3})]\\cos ^{2}({c}_{1}+{c}_{2})\\cos ^{2}({c}_{1}-{c}_{2})\\rbrace .$ The Schmidt number[10], [15] is the number of non-zero coefficients $s_{l}$ .", "For the unitary operator $U_{d}$ , the Schmidt coefficients $s_{l}$ are as follows: ${s}_{1}=[\\cos ^{2}(c_{1}+c_{2})+\\cos ^{2}(c_{1}-c_{2})+2\\cos (2c_{3})\\cos (c_{1}+c_{2})\\cos (c_{1}-c_{2})]^{1/2},$ ${s}_{2}=[\\sin ^{2}(c_{1}+c_{2})+\\sin ^{2}(c_{1}-c_{2})+2\\cos (2c_{3})\\sin (c_{1}+c_{2})\\sin (c_{1}-c_{2})]^{1/2},$ ${s}_{3}=[\\sin ^{2}(c_{1}+c_{2})+\\sin ^{2}(c_{1}-c_{2})-2\\cos (2c_{3})\\sin (c_{1}+c_{2})\\sin (c_{1}-c_{2})]^{1/2},$ ${s}_{4}=[\\cos ^{2}(c_{1}+c_{2})+\\cos ^{2}(c_{1}-c_{2})-2\\cos (2c_{3})\\cos (c_{1}+c_{2})\\cos (c_{1}-c_{2})]^{1/2}.$ We made numerical analysis for the Schmidt number and the entanglement measure of the unitary operator, and got the relation between the Schmidt number and the entanglement measure of the unitary operator(shown in Table.", "REF ).", "Table: The Schmidt number versus theentanglement measure of the unitary operator U d {U}_{d} .For the Schmidt number 4 case, the first plot in Fig.REF shows how the entanglement measure of $U_{d}$ depends on the parameters $c_{1}$ and $c_{2}$ for $c_{3}=0$ when the Schmidt number of $U_{d}$ is 4.", "As the parameters $c_{1}$ and $c_{2}$ approach to 0, which represents a unit matrix, the entanglement measure of the unitary operator approaches to 0.", "As the parameters $c_{1}$ and $c_{2}$ are equal to $\\frac{\\pi }{4}$ , which represents the SWAP gate, the entanglement measure of the unitary operator is equal to the maximum value $\\frac{3}{4}$ .", "As the parameters ${c}_{1}$ and ${c}_{2}$ are increasing, the entanglement measure of unitary operator $U_{d}$ is increasing too.", "If $c_{3}\\ne 0$ , the changing pattern of the operator entanglement is similar to that of the $c_{3}=0$ case.", "From the other three plots in Fig.REF , we can see that the minimum entanglement for the Schmidt-4 operator is oscillating with $c_{3}$ , and the maximum value and the period of the oscillation are $0.5$ and $\\frac{\\pi }{2}$ , respectively.", "When $c_{3}=\\frac{\\pi }{4}$ , the minimum entanglement reaches its maximum $0.5$ .", "The maximum value of operator entanglement is still $\\frac{3}{4}$ .", "Figure: Entanglement measure of unitary operator U d U_{d}versus the parameters c 1 c_{1}, c 2 c_{2} for parameterc 3 =0,π/16,π/8,π/4,c_{3}=0,\\pi /16, \\pi /8,\\pi /4, respectively, when the Schmidt number is 4.For the Schmidt number 2 case, if $c_{3}\\ne 0 $ , then $c_{1},c_{2}$ must be zero, so the operator entanglement can be expressed in a very simple form $E({U}_{d})=\\frac{1}{2}\\sin ^{2}(2c_{3})$ .", "Fig.REF shows how the entanglement measure of $U_{d}$ depends on the parameters $c_{1}$ and $c_{2}$ for $c_{3}=0$ when the Schmidt number of $U_{d}$ is 2.", "This curve is the boundary line of the first plot in Fig.REF .", "As the parameters $c_{1}$ and $c_{2}$ approach to 0, the entanglement measure of unitary operator $U_{d}$ approaches to 0.", "The entanglement measure of unitary operator will increase as the parameters $c_{1}$ or $c_{2}$ increase.", "The entanglement measure of the unitary operator with Schmidt number 2 can get to the extreme value $\\frac{1}{2}$ .", "Figure: Entanglement measure of unitary operator U d U_{d}versus the parameters c 1 c_{1}, c 2 c_{2} for parameterc 3 =0{c}_{3}=0 when the Schmidt number is 2.From the two figures we can see that, if we want to design an operation so that it has a specific operator entanglement(or entangling power), we have infinite design schemes(i.e.", "$c_{1},c_{2},c_{3}$ ) for the Schmidt-number-4 type operations.", "But we only have two design schemes for the Schmidt-number-2 type operations.", "That is to say, Schmidt-number-4 type operations have a variety of design schemes rather than the only two design schemes of the Schmidt-number-2 type operations.", "So, the Schmidt-number-4 type operations are superior to the Schmidt-number-2 type operations.", "In addition, the maximum operator entanglement of the Schmidt-number-4 type operations can reach $\\frac{3}{4}$ , while the maximum operator entanglement of the Schmidt-number-2 type operations only reaches $\\frac{1}{2}$ .", "So, in practice, we will prefer the Schmidt-number-4 type operations." ], [ "An example in Cavity QED system", "To demonstrate the abstract relation between the operator entanglement and the parameters of the unitary operator, we will take the following detailed example.", "Consider two two-level atoms(1, 2) trapped in a single-mode optical cavity, and the two atoms are coupled to the cavity mode with the same coupling constant $g$ .", "The excited state$|e\\rangle _{i}$ and the ground state$|g\\rangle _{i}$ , $(i=1,2)$ are the two levels used to encode quantum information.", "The two atoms have different transition frequencies $\\omega _{1}$ , $\\omega _{2}$ , and $\\omega _{1}\\ne \\omega _{2}$ .", "The frequency of the cavity mode is denoted by $\\omega _{0}$ .", "The atom 1 is resonantly driven by an external classical field with coupling constant $\\Omega $ .", "Suppose the cavity mode is initially prepared in vacuum state, under the large detuning condition $\\delta _{1}=\\omega _{1}-\\omega _{0}\\gg g$ , $\\delta _{2}=\\omega _{2}-\\omega _{0}\\gg g$ and in the strong driving regime $\\Omega \\gg \\frac{g^{2}}{\\delta _{1}}$ , the effective Hamiltonian of the total system can be expressed as[22]: $H_{eff}=\\frac{\\lambda }{2}\\sigma _{1}^{x}\\sigma _{2}^{x},$ where $\\lambda =\\frac{g^{2}}{\\delta _{1}}$ is the effective coupling constant between atoms 1 and 2, and $\\sigma _{i}^{x}$ is the Pauli operator of the $i$ th atom.", "The unitary transformation induced by this effective Hamiltonian can be expressed as: $ U_{eff}=\\left(\\begin{array}{cccc}cos(\\frac{\\lambda t}{2}) & 0 & 0 & {-i}sin(\\frac{\\lambda t}{2}) \\\\0 & cos(\\frac{\\lambda t}{2}) & {-i}sin(\\frac{\\lambda t}{2}) & 0 \\\\0 & {-i}sin(\\frac{\\lambda t}{2}) & cos(\\frac{\\lambda t}{2}) & 0 \\\\{-i}sin(\\frac{\\lambda t}{2}) & 0 & 0 & cos(\\frac{\\lambda t}{2}) \\\\\\end{array}\\right).$ If we set $c_{1}=\\frac{\\lambda t}{2}$ , ${c}_{2}=0$ , ${c}_{3}=0$ in Eq.", "(REF ), it is just the joint unitary operator in Eq.", "(REF ).", "That is to say, the above mentioned physical process is just a physical realization of the joint unitary operation (REF ).", "The Schmidt number of the operator (REF ) is 2, and the operator entanglement measure of it can be expressed as $E(U_{eff})=\\frac{1}{2}\\sin ^{2}(\\lambda t)$ .", "The relationship between the operator's entanglement measure and the effective interaction time $\\lambda t$ between the two atoms is depicted in Fig.REF .", "From this figure we can easily see that the maximum operator entanglement is $\\frac{1}{2}$ with $Sch=2$ .", "Figure: Entanglement measure of unitary operatorU eff U_{eff} versus the effective interaction time λt\\lambda t between the two atoms.", "Here the Schmidt number of U eff U_{eff} is 2." ], [ "Conclusion", "In this paper, the linear entropy and the Schmidt number of an arbitrary two-qubit unitary operator is discussed.", "The results have shown that the Schmidt number is related with the entanglement measure of unitary operators closely.", "For the same operator entanglement within the range $(0,\\frac{1}{2}]$ , there exist infinite unitary operators with Schmidt number 4 but only 2 unitary operators with Schmidt number 2.", "In this sense, we can say that the unitary operators with Schmidt number 4 can be realized more easily than the unitary operators with Schmidt number 2 if the same operator entanglement is required.", "In addition, for the unitary operators with Schmidt number 4, the range for the operator entanglement is $(0,\\frac{3}{4}]$ .", "But, for the unitary operators with Schmidt number 2, the range decline to $(0,\\frac{1}{2}]$ .", "There must be some requirement of entanglement which can be available for the unitary operator with Schmidt number 4 only." ], [ "Acknowledgments", "This work is supported by National Natural Science Foundation of China (NSFC) under Grants No.", "10704001, No.", "61073048, No.10905024 and 11005029, the Specialized Research Fund for the Doctoral Program of Higher Education(20113401110002), the Key Project of Chinese Ministry of Education.", "(No.210092), the China Postdoctoral Science Foundation under Grant No.", "20110490825, Anhui Provincial Natural Science Foundation under Grants No.", "11040606M16 and 10040606Q51, the Key Program of the Education Department of Anhui Province under Grants No.", "KJ2012A020, No.", "KJ2012A244, No.", "KJ2010A287, No.", "KJ2012B075 and No.", "2010SQRL153ZD, the `211' Project of Anhui University, the Talent Foundation of Anhui University under Grant No.33190019, the personnel department of Anhui province, and Anhui Key Laboratory of Information Materials and Devices (Anhui University)." ] ]

[ [ "Certifying the restricted isometry property is hard" ], [ "Abstract This paper is concerned with an important matrix condition in compressed sensing known as the restricted isometry property (RIP).", "We demonstrate that testing whether a matrix satisfies RIP is NP-hard.", "As a consequence of our result, it is impossible to efficiently test for RIP provided P \\neq NP." ], [ "Introduction", "It is now well known that compressed sensing offers a method of taking few sensing measurements of high-dimensional sparse vectors, while at the same time enabling efficient and stable reconstruction [1].", "In this field, the restricted isometry property is arguably the most popular condition to impose on the sensing matrix in order to acquire state-of-the-art reconstruction guarantees: Definition 1 We say a matrix $\\Phi $ satisfies the $(K,\\delta )$ -restricted isometry property (RIP) if $(1-\\delta )\\Vert x\\Vert ^2\\le \\Vert \\Phi x\\Vert ^2\\le (1+\\delta )\\Vert x\\Vert ^2$ for every vector $x$ with at most $K$ nonzero entries.", "To date, RIP-based reconstruction guarantees exist for Basis Pursuit [2], CoSaMP [3] and Iterative Hard Thresholding [4], and the ubiquitous utility of RIP has made the construction of RIP matrices a subject of active research [5]–[7].", "Here, random matrices have found much more success than deterministic constructions [5], but this success is with high probability, meaning there is some (small) chance of failure in the construction.", "Furthermore, RIP is a statement about the conditioning of all $\\binom{N}{K}$ submatrices of an $M\\times N$ sensing matrix, and so it seems computationally intractable to check whether a given instance of a random matrix fails to satisfy RIP; it is widely conjectured that certifying RIP for an arbitrary matrix is $$ -hard.", "In the present paper, we prove this conjecture.", "Problem 2 Given a matrix $\\Phi $ , a positive integer $K$ , and some $\\delta \\in (0,1)$ , does $\\Phi $ satisfy the $(K,\\delta )$ -restricted isometry property?", "In short, we show that any efficient method of solving Problem REF can be called in an algorithm that efficiently solves the $$ -complete subset sum problem.", "As a consequence of our result, there is no method by which one can efficiently test for RIP provided $¶\\ne $ .", "This contrasts with previous work [8], in which the reported hardness results are based on less-established assumptions on the complexity of dense subgraph problems.", "In the next section, we review the basic concepts we will use from computational complexity, and Section 3 contains our main result." ], [ "A brief review of computational complexity", "In complexity theory, problems are categorized into complexity classes according to the amount of resources required to solve them.", "For example, the complexity class $¶$ contains all problems which can be solved in polynomial time, while problems in $$ may require as much as exponential time.", "Problems in $$ have the defining quality that solutions can be verified in polynomial time given a certificate for the answer.", "As an example, the graph isomorphism problem is in $$ because, given an isomorphism between graphs (a certificate), one can verify that the isomorphism is legitimate in polynomial time.", "Clearly, $¶\\subseteq $ , since we can ignore the certificate and still solve the problem in polynomial time.", "While problem categories provide one way to describe complexity, another important tool is the polynomial-time reduction, which allows one to show that a given problem is “more complex” than another.", "To be precise, a polynomial-time reduction from problem $A$ to problem $B$ is a polynomial-time algorithm that solves problem $A$ by exploiting an oracle which solves problem $B$ ; the reduction indicates that solving problem $A$ is no harder than solving problem $B$ (up to polynomial factors in time), and we say “$A$ reduces to $B$ ,” or $A\\le B$ .", "Such reductions lead to some of the most popular definitions in complexity theory: We say a problem $B$ is called $$ -hard if every problem $A$ in $$ reduces to $B$ , and a problem is called $$ -complete if it is both $$ -hard and in $$ .", "In plain speak, $$ -hard problems are harder than every problem in $$ , while $$ -complete problems are the hardest of problems in $$ .", "Contrary to popular intuition, $$ -hard problems are not merely problems that seem to require a lot of computation to solve.", "Of course, $$ -hard problems have this quality, as an $$ -hard problem can be solved in polynomial time only if $¶=$ ; this is an open problem, but it is widely believed that $¶\\ne $  [9].", "However, there are other problems which seem hard but are not known to be $$ -hard (e.g., the graph isomorphism problem).", "As such, while testing for RIP in the general case seems to be computationally intensive, it is not obvious whether the problem is actually $$ -hard.", "Indeed, by the definition of $$ -hard, one must compare its complexity to the complexity of every problem in $$ .", "To this end, notice that $A\\le B$ and $B\\le C$ together imply $A\\le C$ , and so to demonstrate that a problem $C$ is $$ -hard, it suffices to show that $B\\le C$ for some $$ -hard problem $B$ .", "In the present paper, we demonstrate the hardness of certifying RIP by reducing from the following problem: Problem 3 Given a matrix $\\Psi $ and some positive integer $K$ , do there exist $K$ columns of $\\Psi $ which are linearly dependent?", "Problem REF has a brief history in computational complexity.", "First, McCormick [10] demonstrated that the analogous problem of testing the girth of a transversal matroid is $$ -complete, and so by invoking the randomized matroid representation of Marx [11], Problem REF is hard for $$ under randomized reductions [12].", "Next, Khachiyan [13] showed that the problem is $$ -hard by focusing on the case where $K$ equals the number of rows of $\\Psi $ ; using a particular matrix construction with Vandermonde components, he reduced this instance of the problem to the subset sum problem.", "Recently, Tillmann and Pfetsch [14] used ideas similar to McCormick's to strengthen Khachiyan's result: they prove Problem REF is $$ -hard without focusing on such a specific instance of the problem.", "Each of these complexity results use $M\\times N$ matrices with integer entries whose binary representations take $\\le p(M,N)$ bits for some polynomial $p$ ; we will exploit this feature in our proof." ], [ "Main result", "Theorem 4 Problem REF is $$ -hard.", "Reducing from Problem REF , suppose we are given a matrix $\\Psi $ with integer entries.", "Letting $\\mathrm {Spark}(\\Psi )$ denote the size of the smallest collection of linearly dependent columns of $\\Psi $ , we wish to determine whether $\\mathrm {Spark}(\\Psi )\\le K$ .", "To this end, we take $P\\le 2^{p(M,N)}$ to be the size of the largest entry in $\\Psi $ , and define $C=2^{\\lceil \\log _2 \\sqrt{MN}P\\rceil }$ and $\\Phi =\\frac{1}{C}\\Psi $ ; note that we choose $C$ to be of this form instead of $\\sqrt{MN}P$ to ensure that the entries of $\\Phi $ can be expressed in $(M,N)$ bits without truncation.", "Of course, linear dependence between columns is not affected by scaling, and so testing $\\Phi $ is equivalent to testing $\\Psi $ .", "In fact, since we plan to appeal to an RIP oracle, it is better to test $\\Phi $ since the right-hand inequality of Definition REF is already satisfied for every $\\delta >0$ : $\\Vert \\Phi \\Vert _2\\le \\sqrt{MN}\\Vert \\Phi \\Vert _{\\mathrm {max}}=\\sqrt{MN}\\frac{P}{C}\\le 1\\le \\sqrt{1+\\delta }.$ We are now ready to state the remainder of our reduction: For some value of $\\delta $ (which we will determine later), ask the oracle if $\\Phi $ is $(K,\\delta )$ -RIP; then $\\begin{array}{rcll}\\mbox{(i)} & \\mbox{$\\Phi $ is $(K,\\delta )$-RIP} & \\Longrightarrow & \\mathrm {Spark}(\\Psi )>K,\\\\\\mbox{(ii)} & \\mbox{$\\Phi $ is not $(K,\\delta )$-RIP} & \\Longrightarrow & \\mathrm {Spark}(\\Psi )\\le K.\\end{array}$ The remainder of this proof will demonstrate (i) and (ii).", "Note that (i) immediately holds for all choices of $\\delta \\in (0,1)$ by the contrapositive.", "Indeed, $\\mathrm {Spark}(\\Psi )\\le K$ implies the existence of a nonzero vector $x$ in the nullspace of $\\Phi $ with $\\le K$ nonzero entries, and $\\Vert \\Phi x\\Vert ^2=0<(1-\\delta )\\Vert x\\Vert ^2$ violates the left-hand inequality of Definition REF .", "For (ii), we also consider the contrapositive.", "When $\\mathrm {Spark}(\\Psi )>K$ , we have that every size-$K$ subcollection of $\\Psi $ 's columns is linearly independent.", "Letting $\\Psi _\\mathcal {K}$ denote the submatrix of columns indexed by a size-$K$ subset $\\mathcal {K}\\subseteq \\lbrace 1,\\ldots ,N\\rbrace $ , this implies that $\\lambda _\\mathrm {min}(\\Psi _\\mathcal {K}^*\\Psi _\\mathcal {K}^{})>0$ , and so $\\det (\\Psi _\\mathcal {K}^*\\Psi _\\mathcal {K}^{})>0$ .", "Since the entries of $\\Psi $ lie in $\\lbrace -P,\\ldots ,P\\rbrace $ , we know the entries of $\\Psi _\\mathcal {K}^*\\Psi _\\mathcal {K}^{}$ lie in $\\lbrace -MP^2,\\ldots ,MP^2\\rbrace $ , and since $\\Psi _\\mathcal {K}^*\\Psi _\\mathcal {K}^{}$ is integral with positive determinant, we must have $\\det (\\Psi _\\mathcal {K}^*\\Psi _\\mathcal {K}^{})\\ge 1$ .", "In fact, $1&\\le \\det (\\Psi _\\mathcal {K}^*\\Psi _\\mathcal {K}^{})\\\\&=\\prod _{k=1}^K\\lambda _k(\\Psi _\\mathcal {K}^*\\Psi _\\mathcal {K}^{})\\\\&\\le \\lambda _\\mathrm {min}(\\Psi _\\mathcal {K}^*\\Psi _\\mathcal {K}^{})\\cdot \\lambda _\\mathrm {max}(\\Psi _\\mathcal {K}^*\\Psi _\\mathcal {K}^{})^{K-1}\\\\&\\le \\lambda _\\mathrm {min}(\\Psi _\\mathcal {K}^*\\Psi _\\mathcal {K}^{})\\cdot \\big (K\\Vert \\Psi _\\mathcal {K}^*\\Psi _\\mathcal {K}^{}\\Vert _{\\mathrm {max}}\\big )^{K-1},$ and so we can rearrange to get $\\lambda _\\mathrm {min}(\\Phi _\\mathcal {K}^*\\Phi _\\mathcal {K}^{})&=\\frac{1}{C^2}\\lambda _\\mathrm {min}(\\Psi _\\mathcal {K}^*\\Psi _\\mathcal {K}^{})\\\\&\\ge \\frac{1}{C^2(KMP^2)^{K-1}}\\ge 2^{-5MNp(M,N)},$ where the last inequality follows from $K\\le M\\le N$ and other coarse bounds.", "Therefore, if we pick $\\delta :=1-2^{-5MNp(M,N)}$ , then since our choice for $\\mathcal {K}$ was arbitrary, we conclude that $\\Phi $ is $(K,\\delta )$ -RIP whenever $\\mathrm {Spark}(\\Psi )>K$ , as desired.", "Moreover, since $\\delta $ can be expressed in the standard representation using $(M,N)$ bits, we can ask the oracle our question in polynomial time.", "It is important to note that Theorem REF is a statement about testing for RIP in the worst case; this result does not rule out the existence of matrices for which RIP is easily verified (e.g., using coherence in conjunction with the Gershgorin circle theorem for small values of $K$  [5])." ] ]

[ [ "Effects of magnetic dipole-dipole interactions in atomic Bose-Einstein\n condensates with tunable s-wave interactions" ], [ "Abstract The s-wave interaction is usually the dominant form of interactions in atomic Bose-Einstein condensates (BECs).", "Recently, Feshbach resonances have been employed to reduce the strength of the s-wave interaction in many atomic speicies.", "This opens the possibilities to study magnetic dipole-dipole interactions (MDDI) in BECs, where the novel physics resulting from long-range and anisotropic dipolar interactions can be explored.", "Using a variational method, we study the effect of MDDI on the statics and dynamics of atomic BECs with tunable s-wave interactions.", "We benchmark our calculation against previously observed MDDI effects in $^{52}$Cr with excellent agreement, and predict new effects that should be promising to observe experimentally.", "A parameter of magnetic Feshbach resonances, $\\epsilon_{dd,\\text{max}}$, is used to quantitatively indicate the feasibility of experimentally observing MDDI effects in different atomic species.", "We find that strong MDDI effects should be observable in both in-trap and time-of-flight behaviors for the alkali BECs of $^{7}$Li, $^{39}$K, and $^{133}$Cs.", "Our results provide a helpful guide for experimentalists to realize and study atomic dipolar quantum gases." ], [ "Introduction", "The physics of ultracold dipolar quantum gases is a rich and promising area of research.", "There is great interest in the physical behaviors that result from dipole-dipole interactions [1], [2].", "An ultracold atomic gas of atoms that posses a magnetic moment will have a magnetic dipole-dipole interaction (MDDI).", "However, it is often difficult to observe the effects of MDDI in many atomic species (particularly alkalis), where the MDDI is typically much weaker than the isotropic, s-wave interactions.", "For atomic species with magnetic Feshbach resonances, however, the s-wave scattering length, $a_s$ , can be tuned via magnetic fields [3].", "Employing Feshbach resonances has led to fruitful and impressive developments in ultracold atom research.", "Of interest here is that it allows for the exploration of MDDI in Bose-Einstein condensates (BECs).", "By tuning $a_s$ to near zero, MDDI can become the strongest interaction in the BEC.", "The effect of MDDI in ultracold atomic gases was first observed using a BEC of $^{52}$ Cr atoms, which posses a strong magnetic moment, $\\mu $ , of 6 Bohr magnetons ($\\mu _B$ ) [4].", "The MDDI were observed to affect the aspect ratio of the $^{52}$ Cr BEC in time-of-flight (TOF) free expansion.", "The MDDI effect on the stability of a BEC was also experimentally studied in $^{52}$ Cr, where for various trap configurations the MDDI either made the BEC more or less stable against collapse [5].", "More recently, in $^{7}$ Li the MDDI effect was seen when comparing the axial length of the BEC near $a_s = 0$ for two different trapping geometries [6].", "Effects of MDDI have also been observed on the decoherence rate in a $^{39}$ K BEC atomic interferometer [7], and on the spin domains of a spinor $^{87}$ Rb Bose gas [8].", "Very recently, a BEC has been realized with $^{164}$ Dy, with a strong magnetic moment of 10$\\mu _B$ , exhibiting dipolar effects even with no tuning of the scattering length [9].", "Many further effects due to MDDI have been predicted such as the excitation of collective modes by tuning the dipolar interaction [10], the emergence of a biconcave structure with local collapse [11] [12], and the modification of the phase diagram of dipolar spin-1 BECs [13] [14] [15], of the soliton stability in 1D BECs [16], and of vortices in BECs [17].", "We refer the reader to recent reviews for a further discussion of the multiplicity of MDDI effects [2], [1].", "Motivated by such rich physics, we theoretically model the effects of MDDI and possibility of experimentally detecting such effects in BECs of $^{52}$ Cr and all the alkalis.", "In Section II, we explain the variational method we employ to model MDDI using a cylindrically-symmetric, Gaussian Ansatz for the BEC wave-function [18].", "In Section III, we start by discussing the relevant parameters for our simulations.", "We also introduce a key quantity used in this paper, the ratio of s-wave scattering length, $a_s$ , to a length defined for MDDI, $a_{dd}$ .", "Next in that section we present simulations of the effects of MDDI in $^{52}$ Cr, including those reported in Refs.", "[5] and [19], and predict several additional effects that should be readily observable.", "We also present simulations for the effects of MDDI in $^{7}$ Li, $^{39}$ K and $^{133}$ Cs, the three alkali BECs we identified as promising for observing MDDI effects.", "In Section IV, we conclude." ], [ "Variational Method", "While a few other methods exist to model dipole-dipole interaction effects in BECs [20], [21], [22], [23], [1], the variational method we employ has shown great utility because its simple, analytic solutions are valid over a wide range of experimental parameters [24], [18], [25].", "Two types of interactions are considered.", "The s-wave interaction is characterized by the s-wave scattering length, $a_s$ , and for dipole-dipole interactions a parameter defined as $a_{dd}$ is used.", "MDDI can be characterized by $a_{dd} = \\mu _0 \\mu ^2 M / (12 \\pi \\hbar ^2)$ , where $\\mu $ is the magnetic moment of the atom, $M$ its mass, and $\\mu _0$ is the permeability of free space Electronic dipole-dipole interactions could be similarly characterized but are not considered further in this paper..", "The atom-atom interaction potential thus has two terms, one for each interaction: $V_{atom-atom}(\\vec{R}) = a_s\\frac{4 \\pi \\hbar ^2}{M} \\delta (\\vec{R}) + a_{dd} \\frac{3 \\hbar ^2}{M}\\frac{1-3cos^2 \\theta }{R^3}$ Using that interaction term, the Gross-Pitaevskii equation for a BEC takes the form (in dimensionless units): $\\begin{split}i \\frac{\\partial \\psi (\\vec{r})}{\\partial t} = -\\frac{1}{2}\\nabla ^2 \\psi (\\vec{r}) + V_{ext}(\\vec{r}) \\psi (\\vec{r}) + \\frac{4 \\pi N a_s}{a_{r}} |\\psi (\\vec{r})|^2 \\psi (\\vec{r}) \\\\ +\\frac{N a_{dd}}{3 a_{r}} \\int \\frac{1-3 cos^2 \\theta }{R^3} |\\psi (\\vec{r^{\\prime }})|^2 d\\vec{r^{\\prime }} \\psi (\\vec{r})\\end{split}$ where the length unit is $a_r = \\sqrt{\\hbar / m\\omega _r}$ , the trap frequencies are $\\omega _r$ and $\\omega _z$ for the respective radial and axial direction, the time unit $ t=2\\pi /\\omega _r$ , $\\psi $ is normalized to unity ($|\\psi |^2 = 1$ ), and $V_{ext}(\\vec{r}) = \\frac{1}{2} (x^2 + y^2 + \\lambda z^2)/2 $ is the trap potential where the trap aspect ratio is $ \\lambda = \\omega _z / \\omega _r$ .", "For this paper, we adopt the nomenclature for trap shapes that is common to experiments: the symmetrical a.k.a.", "spherical ($\\lambda = 1$ ), the cigar a.k.a.", "prolate ($\\lambda < 1$ ), and the pancake a.k.a.", "oblate ($\\lambda > 1$ ) shapes [5]; we also assume that the magnetic field is applied along the axial direction.", "Using a cylindrical-symmetric, Gaussian Ansatz for the BEC wave-function We note that there are limitations to the Gaussian anzatz.", "For example, it does not account for biconcave structure predicted to occur in pancake traps.", "However, this is an issue more specific to the pancake traps and only in a certain region of interactions[11].", "For most trap geometries (especially cigar traps) and properties studied in this paper, the Gaussian ansatz provides a good model to guide the experimental investigation of MDDI effects.", "$\\psi \\left(\\vec{r}\\right)= \\exp \\left(-\\frac{x^2}{2 q_r^2}\\right) \\exp \\left(-\\frac{y^2}{2 q_r^2}\\right) \\exp \\left(-\\frac{z^2}{2 q_z^2}\\right)$ the variational method results in two differential equations that describe the mean axial, $q_z$ , and radial, $q_r$ , lengths of the BEC (detailed solution in Ref.", "[18], noting Typo in [18], [25] corrected.", "(S. Yi, Private Communications)): $\\ddot{q_r} + q_r & = & \\frac{1}{q^3_r} - \\sqrt{\\frac{2}{\\pi }}\\frac{1}{q_r^3 q_z} \\frac{N}{a_{\\rho }} [a_{dd} f(\\kappa ) - a_s] \\\\\\ddot{q_z} +\\lambda ^2 q_z & = & \\frac{1}{q_z^3} - \\sqrt{\\frac{2}{\\pi }}\\frac{1}{q_r^2 q_z^2}\\frac{N}{a_{r}} [a_{dd} g(\\kappa ) - a_s] $ where $q_r & \\equiv & \\sqrt{\\frac{<x^2>}{2}} \\equiv \\sqrt{\\frac{<y^2>}{2}} \\\\q_z &\\equiv & \\sqrt{\\frac{<z^2>}{2}} \\\\\\kappa & \\equiv & q_r / q_z \\mbox{, the BEC aspect ratio} \\\\f(\\kappa ) &\\equiv & \\frac{[-4 \\kappa ^4-7\\kappa ^2+2+9\\kappa ^4 H(\\kappa )]}{2(\\kappa ^2-1)^2} \\\\g(\\kappa ) &\\equiv & \\frac{[-2\\kappa ^4+10\\kappa ^2 + 1-9\\kappa ^2 H(\\kappa )]}{(\\kappa ^2-1)^2} \\\\H(\\kappa ) &\\equiv & \\frac{\\tanh ^{-1} \\left( \\sqrt{1-\\kappa ^2}\\right)}{ \\sqrt{1-\\kappa ^2}}$ These coupled differential equations model BEC behavior with (keeping $a_{dd}$ ) and without (setting $a_{dd}=0$ ) MDDI effects, and they can be numerically solved to model three experimentally relevant situations: The static, in-situ sizes for a trapped BEC are found by setting the time-dependent components, $\\ddot{q_r}$ and $\\ddot{q_z}$ , to zero.", "In-trap dynamics are modeled by keeping all terms.", "Time-of-flight (TOF) free expansion behavior is modeled by removing the the terms $q_r$ and $\\lambda ^2 q_z$ , which represent the trapping potential, on the left sides of Eqns.", "." ], [ "Results", "In this section, we employ the above method to solve for both the in-trap and TOF behaviors of BECs with MDDI Code developed for this were written in Matlab, and are available at www.physics.purdue.edu/quantum/MDDI.html and www.abeolson.com/physics/MDDI.html.", "We also find the threshold $a_s$ , denoted $a_s^{\\mathrm {threshold}}$ , below which the BEC is unstable and collapses.", "We benchmark our variational calculations against available experimental results in $^{52}$ Cr and find good agreement (Section III.B).", "We also make predictions of MDDI effects in the alkalis, and find that $^7$ Li, $^{39}$ K, and $^{133}$ Cs are the species most favorable for the exploration of MDDI effects." ], [ "Parameters", "The input parameters used in our simulation are the number of atoms in the trap ($N$ ), the magnetic dipole moment of the atom ($\\mu $ ), the mass of the atom ($M$ ), the axial ($f_z$ ) and radial ($f_r$ ) frequencies of the trap ($2 \\pi f_{r,z} = \\omega _{r,z}$ ), and the s-wave scattering length ($a_s$ ).", "An applied magnetic field can tune $a_s$ via a Feshbach resonance with an analytic approximation given by $a_{s} (B)=a_{bg}\\left(1-\\frac{\\Delta }{B-B_{\\infty }}\\right)$ , where $\\Delta $ is the width and $B_{\\infty }$ is the location of the Feshbach resonance, and $a_{bg}$ is the scattering length far from any resonances.", "An experimental limit for reaching small $a_s$ is the precision of control over the magnetic fields.", "In typical ultracold atom experiments an experimental precision $\\delta B/B$ of approximately $10^{-5}$ can be realized [30], [31], [32].", "As $\\mu $ depends on the strength of the magnetic field, we calculate $\\mu $ at $B_0=B_{\\infty }+\\Delta $ (where $a_s=0$ ), denoted $\\mu _{cross}$ .", "For reference, some parameters for known Feshbach resonances from the literature are listed along with the calculated values of $\\mu _{cross}$ in Appendix  Table REF .", "To compare the potential for experimentally observing MDDI effects in the various atomic species of interest, we employ the dimensionless ratio $\\epsilon _{dd}=a_{dd}/a_s$ used in Ref.", "[19], focusing on its maximal value that can be achieved experimentally: $\\epsilon _{dd\\text{, max}} \\equiv \\frac{a_{dd}}{a_{s,\\text{min}}}=\\frac{\\mu _0 \\mu ^2 m}{12 \\pi \\hbar ^2 a_{bg}(\\delta B / \\Delta )}$ where $a_{s,\\text{min}}\\approx a_{bg} \\left(\\delta B / \\Delta \\right)$ is the minimal $a_s$ that can be achieved given a typical experimental magnetic field stability (assumed to be $\\delta B/B_0 \\approx 10^{-5}$ ).", "The alkali species that are the best candidates for observing and studying MDDI effects in BECs are clearly $^7$ Li and $^{39}$ K and $^{133}$ Cs (see Table REF ).", "We note that $^{52}$ Cr, while having a much larger $a_{dd}$ , does not have a broad Feshbach resonance that allows for as precise tuning of $a_s$ as in some of the alkalis.", "Figure: The calculated in-trap BEC aspect ratio (a), radial length (b), and axial length (c) of a 52 ^{52}Cr BEC with (solid, blue) and without (dashed, black) MDDI for a range of trap aspect ratios, λ\\lambda .", "In this simulation, the f avg =700f_{avg} = 700 Hz, the atom number N=2×10 4 N= 2\\times 10^4, and a s =15a 0 a_s=15 a_0.", "The magnetic field is, as for all simulations in this paper, aligned along the axial direction.Table: Parameters for variational computations, with the maximum value of ϵ dd \\epsilon _{dd} as calculated from Eqn.", ".As an initial verification of our model, we compute $\\kappa $ , in-situ ($t=0$ ) and in the TOF asymptotic limit ($t\\rightarrow \\infty $ ) assuming only s-wave interactions, for both strongly interaction ($a_s \\rightarrow \\text{Large}$ ) and non-interacting ($a_s=0$ ) cases.", "The expected in-trap and TOF behaviors of a BEC in such limiting cases are (see Refs.", "[33], [34]): If $a_s = 0$ and $t=0$ , expect $\\kappa =\\sqrt{\\frac{\\omega _a}{\\omega _r}} = \\lambda ^{1/2}$ .", "If $a_s = \\text{Large}$ , $t=0$ , expect $\\kappa =\\frac{\\omega _a}{\\omega _r} = \\lambda $ .", "If $a_s = 0$ , $t\\rightarrow \\infty $ , expect $\\kappa =\\sqrt{\\frac{\\omega _r}{\\omega _a}} = \\lambda ^{-1/2}$ .", "If $\\lambda \\ll 1$ or $\\lambda \\gg 1$ , and if $a_s = \\text{Large}$ with $t\\rightarrow \\infty $ , expect $\\kappa =\\frac{2}{\\pi } \\frac{\\omega _r}{\\omega _a} = 2 \\lambda ^{-1} /\\pi $ .", "Our variational calculation (performed with $\\mu =0$ in these cases) does reproduce these expected values for $\\kappa $ ." ], [ "Effects of MDDI in $^{52}$ Cr BEC", "The first observation of MDDI in a BEC was with $^{52}$ Cr.", "As $^{52}$ Cr is the most studied atomic species so far for MDDI effects in BECs, we present and benchmark our calculation and results for $^{52}$ Cr before discussing the alkalis.", "We used $^{52}$ Cr to demonstrate four characteristic MDDI effects, discussed in detail below." ], [ "Effect of MDDI on in-situ aspect ratio of a BEC", "A calculated result of a $^{52}$ Cr BEC trapped in a harmonic trap is shown in Fig.", "REF .", "For a nearly symmetrical trap ($\\lambda \\approx 1$ , which is the case for the experiment in Ref.", "[5]) the MDDI increase the axial length of the BEC and reduce its radial length compared to a BEC with no MDDI, leading to a decreased aspect ratio.", "However, if the trap is very prolate, $\\lambda \\ll 1$ (very oblate, $\\lambda \\gg 1$ ), we find that the BEC will shrink (expand) in both the axial and radial directions, in a way that leads to an increased aspect ratio.", "There are two values of $\\lambda $ , one for when the trap is oblate and one for when the trap is prolate, where the MDDI do not change the aspect ratio of the BEC.", "Similar in-situ effects of MDDI on $\\kappa $ were found in all simulations of stable BECs for the alkali as well.", "Figure: Stability diagram and aspect ratio of a 52 ^{52}Cr BEC, with parameters (f avg =700f_{avg}= 700 Hz, and N=2×10 4 N=2\\times 10^4) chosen to resemble those in the experiment Ref.", ".", "The BEC aspect ratio, κ\\kappa , solved using our method, is plotted in the color map as functions of a s a_s and λ\\lambda in the stable regime (log 10 scale).", "The energy variational solution and experimental data from Ref.", "for a s threshold a_s^{\\mathrm {threshold}} are included for comparison.", "This shows the effectiveness of our method in solving not just the a s threshold a_s^{\\mathrm {threshold}} where the BEC collapses but also the BEC size in the stable regime.Figure: The aspect ratio of a 52 ^{52}Cr BEC vs the time-of-flight duration,t TOF t_{TOF}, for different scattering lengths (a-d) with parameters chosen to resemble those in Ref. .", "We additionally show simulated TOF behavior where MDDI nearly (e) and actually (f) causes BEC collapse.", "Upon the release of the BEC, the scattering length is tuned to a s =10a 0 a_s =10 \\, a_0 in (e), and a s =5a 0 a_s =5 \\, a_0 in (f).", "For (a-f), N=3×10 4 N=3\\times 10^4, f r =600f_r = 600 Hz, and f z =370f_z = 370 Hz." ], [ "Effect of MDDI on stability of a trapped BEC", "For BECs with only s-wave interactions, it is shown that if $a_s<0$ (attractive interactions) the BEC will collapse if $N\\gtrsim 0.55 a_{ho} / |a_s|$ , where $a_{ho} = \\sqrt{\\hbar / m\\bar{\\omega }}$ and $\\bar{\\omega }$ is the average trap frequency [35], [36], [34].", "For any purely s-wave interacting BECs held in a three dimensional harmonic trap, the BEC will not collapse if $a_s>0$ .", "However, MDDI can destabilize or stabilize a BEC that would otherwise be stable or unstable.", "The effect of MDDI on the stability of a trapped BEC was first observed in $^{52}$ Cr [5].", "For cigar traps with the B-field aligned along the axial direction, MDDI can lead to BEC collapse at a larger value of $a_s$ (in some cases, even $a_s>0$ ) than in an otherwise identical BEC with no MDDI.", "For pancake traps, the effect is opposite, and the MDDI stabilize the BEC.", "Koch et al.", "performed an experimental study of this MDDI effect [5].", "To model their results, Koch et al.", "employed an energy argument based on a variational Gaussian Ansatz for the BEC density distribution.", "Their method relied on finding when a minimum in the energy vanishes as $a_s$ is reduced.", "Different from their method, our variational calculation solves Eqns.", "REF and  over a range of $a_s$ , and find the threshold $a_s$ when those equations do not have stable numerical solutions, which indicates BEC collapse.", "Our method allows not only the determination of the threshold $a_s$ for collapse (which agrees with those obtained in Ref.", "[5]) but also the axial and radial BEC lengths over the entire range of $a_s$ where the BEC is stable.", "We employed our method, using $N$ and $f_{avg}$ identical to those used in Fig.", "3 of Ref.", "[5], to solve for the threshold $a_s$ and also $\\kappa $ for $a_s>a_s^{\\mathrm {threshold}}$ over a range of $\\lambda $ (see Fig.", "REF ).", "We find excellent agreement with both Koch et al's calculation and their experimental data.", "Our calculation shows, as observed in the original experiment [5], that the anisotropic dipole-dipole interactions cause the threshold $a_s$ to depend strongly on the trap geometry.", "A similar dependence of $a_s^{\\mathrm {threshold}}$ on $\\lambda $ is also seen later in this paper for alkali BECs.", "Figure: The aspect ratio,(a), the axial length, (b), and the radial length,(c), for a 52 ^{52}Cr BEC evolving in trap with an initial radial length twice the static value and the initial axial length half the static value.", "(d) and (e) show the free expansion for the BEC if it were released at times from the trap when the aspect ratio is at a minimum and maximum values respectively.", "The release points are indicated with arrows in (a).", "The parameters used in (a-e) are f z =f r =700Hzf_z=f_r=700 Hz, N=2×10 4 N=2\\times 10^4, and a s =14a 0 a_s=14 a_0." ], [ "MDDI effect in time-of-flight behavior", "MDDI affects the behavior of a BEC released from a trap, and such an effect was first observed in an experiment on $^{52}$ Cr, where the BEC's aspect ratio in TOF changed when the applied magnetic field was along the axial verses radial direction [4] [19].", "We have simulated the experiment in Ref.", "[19] (specifically the data shown in their Fig.", "4) and find good agreement.", "We assume cylindrically-symmetric ($f_r = 600$ Hz and $f_z = 370$ Hz) trap for ease of calculation with parameters approximating their “trap 2”, which had frequencies $f_x$ ,$f_y$ ,$f_z$ =660,540,370 Hz, and N$=3\\times 10^4$ .", "The values of $a_s$ (112 $a_0$ , 96 $a_0$ , 30.5$a_0$ , 20.5$a_0$ ) and resulting $\\epsilon _{dd}$ are chosen to match the values in Fig.", "4 a-d of [19] ($\\epsilon _{dd} =$ 0.14, 0.16, 0.5, 0.75).", "Even with the cylindrical trap approximation, the simulation results agree with the observed data quite well, and clearly shows the effect of MDDI to reduce the aspect ratio of the BEC in TOF from this trap configuration, as shown in Fig.", "REF .", "We further show that in the case of small $a_s$ , MDDI-induced collapse could also be observed after a BEC is released from a trap if the $a_s$ is tuned to a small value upon release.", "Fig.", "REF also shows the MDDI-induced near-collapse (e) and the collapse (f) of atoms in free-expansion.We have verified that typical experimental limitations in changing $a_s$ will not limit the observation of this effect, as collapse still occurs even when $a_s$ is changed after a short (a few 100's of $\\mu $ s for this simulation) TOF." ], [ "Effect of MDDI on in-trap dynamics", "Our model can also solve the in-trap dynamics of BECs with MDDI.", "A simulation of the aspect ratio, radial size ($q_r$ ), and axial size ($q_a$ ) of a trapped $^{52}$ Cr BEC—initially perturbed from its static state—evolving with time is shown in Fig.", "REF , revealing an oscillatory behavior.", "One effect of the MDDI in this situation is to reduce amplitude of the oscillations.", "The oscillations may be difficult to observe in-situ due to the small condensate size, but would become easier to observe in TOF measurements taken from different instants of the oscillations.", "As seen in Fig.", "REF (d) and (e), the aspect ratio in TOF changes by a factor of about 2 because of the MDDI.", "An in-depth treatment of the modes of oscillatory, in-trap dynamics of a BEC with MDDI can be found in Refs.", "[38] and [39].", "Figure: Simulation of a 52 ^{52}Cr BEC frequency spectra of the in-trap oscillations.", "Frequency spectra are obtained by taking the Fourier transform of the in-trap oscillations of the aspect ratio κ\\kappa (a,d), radial length q r q_r (b,e), and the axial length q z q_z (c,f) that occur after changing the axial size to q a,i q_{a,i} from its in-trap equilibrium value, q a,0 q_{a,0}.", "The colorbar gives the amplitude of the FFT for each plot.", "Note the shifts in both amplitude and frequency of oscillations between the simulations including (a-c) and not including (d-f) MDDI.", "The parameters assumed a spherical trap with f r =f a =700f_r=f_a=700Hz, N=2×10 4 2\\times 10^4, and a s =14a 0 a_s=14 a_0.MDDI effects have been observed in collective oscillations of $^{52}$ Cr BECs [40].", "While the results of Ref.", "[40] cannot be directly simulated by our method because their magnetic field was not aligned on the axial direction of the BEC, we are motivated by this experiment to study the collective oscillations of BECs with MDDI.", "By taking the Fourier transform of the time-dependent, in-trap oscillations, we obtain the frequency spectra of the collective oscillations of a BEC with MDDI, exemplified in Fig REF .", "The three lowest modes of oscillation were observed, and shifts due to MDDI were seen in both the amplitude and frequency of the modes." ], [ "Highlight of MDDI effects in Alkali BECs: stability", "A central point of our paper is that MDDI effects are also possible to observe in BECs of the alkalis, even with a much smaller $\\mu $ than $^{52}$ Cr.", "To show this, we compare dipolar collapse for BECs of various species in Fig.", "REF .", "If the MDDI are not included in the model, then the BECs are stable for any positive value of $a_s$ .", "However, with MDDI the BEC aspect ratio decreases more substantially as $a_s$ is reduced towards zero.", "The BEC collapses beyond a $a_s^{\\mathrm {threshold}}$ , indicated by the heavy dot.", "The value of $a_s^{\\mathrm {threshold}}$ can be compared with the $a_{s,\\text{min}}$ of Table REF , indicated by the colored bars in the figure.", "This comparison clearly indicates whether the observation of collapse due to MDDI is feasible with current experimental abilities and reveals $^{7}$ Li, $^{39}$ K, and $^{133}$ Cs as promising alkali species for such an observation." ], [ "$^{7}$ Li, {{formula:b9e1c9c7-4bf0-44b9-89b6-8f04a97773b6}} K, and {{formula:1bfedd55-1214-4339-affc-093fd0fa417e}} Cs", "We find that $^{7}$ Li, $^{39}$ K, and $^{133}$ Cs possess the greatest potential among alkali BECs for exploring MDDI effects.", "Several examples of such effects are shown in the simulations below.", "We describe $^7$ Li in detail, and similar results are presented for $^{39}$ K and $^{133}$ Cs.", "The $|1,+1>$ state of $^{7}$ Li is excellent for studying MDDI effects as it has the the widest known Feshbach resonance of the alkali atoms; the slope at zero-crossing is only $\\approx 0.1 a_0/G$ [6].", "Using our variational method, we find that each of the MDDI effects discussed so far in $^{52}$ Cr should be observable within current experimental capabilities with $^7$ Li as well.", "Figure: The in-trap aspect ratio for a BEC near a s =0a_s=0.", "The colored, vertical bars indicate the minimum scattering length currently achievable in typical experiments, a s,min a_{s,\\text{min}}.", "Calculation for each species assumes a cigar trap with λ=100\\lambda =100 where f z =20f_z=20 Hz, f r =2000f_r = 2000 Hz, and 10 6 10^6 atoms in the BEC.", "A thick dot at the end of each solid curve indicates the a s threshold a_s^{\\mathrm {threshold}}, which depends both on the mass and the magnetic moment of the atom.", "The dotted line, shown only for 52 ^{52}Cr as a demonstrative example, indicates the numerically unstable solutions for a s <a s threshold a_s < a_s^{\\mathrm {threshold}} and is used to determine the threshold a s a_s.", "Collapse threshold values from the smallest to largest are (0.01, 0.02, 0.20, 0.24, 0.54, 13.50)×a 0 \\times \\,a_0 for ( 7 ^7Li, 23 ^{23}Na, 85 ^{85}Rb, 39 ^{39}K, 133 ^{133}Cs, 52 ^{52}Cr) respectively.In Fig.", "REF , (a) shows the effect of $\\lambda $ and $a_s$ on the stability of a BEC with MDDI.", "The line for each simulated BEC atom numbers $N$ indicates the boundary between the stable (above) and unstable (below) regimes.", "Similar to the $^{52}$ Cr case (Fig.", "REF ), the effect of MDDI is to stabilize a BEC in a pancake trap, and to destabilize a BEC in a cigar trap.", "(b) shows $\\kappa $ over a range of $\\lambda $ , with the collapsed regime indicated by the dotted faint line.", "(c) shows plots of $\\kappa $ after release from a trap.", "The in-trap starts with $a_s=0.01 a_0$ (in the stable regime), and upon release (at $t=0$ in the simulation) $a_s$ is tuned to $0.001 a_0$ to induce collapse in free expansion.", "Such rapid modulation of $a_s$ has already been performed experimentally in $^7$ Li BECs [41], and here could be used to reveal MDDI effects.", "(d) shows the in-trap evolution of a BEC initially at $a_s=0.01 a_0$ and then tuned to $0.001 a_0$ at $t=0$ to induce in-trap collapse due to MDDI.", "Neglecting MDDI would result in a stable, oscillating BEC, as seen in Fig.", "REF (d).", "The results in Fig.", "REF offer four methods for detecting MDDI in $^7$ Li BECs.", "Similar calculations are provided for $^{39}$ K and $^{133}$ Cs, as seen in Fig.", "REF and REF respectively.", "We have shown that the MDDI can have substantial impact on the shape and stability for each of these alkali BECs.", "Figure: Effect of MDDI in a 7 ^7Li BEC.", "(a) the calculated threshold a s a_s between stable and unstable regimes for a range of λ\\lambda and three representative atom numbers.", "(b) the BEC aspect ratio over a range of λ\\lambda , with the collapse regime indicated by the faint dotted line, for which there is no stable solution found for Eqns.", "and .", "(c) Evolution of the aspect ratio of the BEC in TOF expansion after release from a trap.", "In the trap a s =0.1a 0 a_s = 0.1 a_0 (where the BEC is stable), and upon release a s a_s is tuned to 0.001a 0 0.001 a_0 to induce collapse in free expansion.", "(d) the in-trap evolution of a BEC similar to (c) with a s a_s tuned to 0.001a 0 0.001 a_0 at t=0t=0 to induce in-trap dipolar collapse.", "Parameters used in the simulation: (a) and (b) f avg =700f_{avg}=700 Hz; (b), N=5×10 6 N=5\\times 10^6 and a s =0.001a 0 a_s = 0.001 a_0; (c) and (d), f z =200f_z=200 Hz and f r =2000f_r=2000 Hz with N=5×10 6 N=5\\times 10^6.Figure: Effect of MDDI in a 39 ^{39}K BEC.", "(a) the calculated threshold a s a_s between stable and unstable regimes for a range of λ\\lambda and three representative atom numbers.", "(b) the BEC aspect ratio over a range of λ\\lambda , with the collapse regime indicated by the faint dotted line, for which there is no stable solution found for Eqns.", "and .", "(c) Evolution of the aspect ratio of the BEC in TOF expansion after release from a trap.", "In the trap a s =0.5a 0 a_s = 0.5 a_0 (where the BEC is stable), and upon release a s a_s is tuned to 0.05a 0 0.05 a_0 to induce collapse in free expansion.", "(d) the in-trap evolution of a BEC similar to (c) with a s a_s tuned to 0.05a 0 0.05 a_0 at t=0t=0 to induce in-trap dipolar collapse.", "Parameters used in the simulation: (a) and (b), f avg =700f_{avg}=700 Hz; (b), N=5×10 5 N=5\\times 10^5 and a s =0.05a 0 a_s = 0.05 a_0; (c) and (d), f z =200f_z=200 Hz and f r =2000f_r=2000 Hz with N=5×10 5 N=5\\times 10^5.Figure: Effect of MDDI in a 133 ^{133}Cs BEC.", "(a) the calculated threshold a s a_s between stable and unstable regimes for a range of λ\\lambda and three representative atom numbers.", "(b) the BEC aspect ratio over a range of λ\\lambda , with the collapse regime indicated by the faint dotted line, for which there is no stable solution found for Eqns.", "and .", "(c) Evolution of the aspect ratio of the BEC in TOF expansion after release from a trap.", "In the trap a s =5a 0 a_s = 5 a_0 (where the BEC is stable), and upon release a s a_s is tuned to 0.1a 0 0.1 a_0 to induce collapse in free expansion.", "(d) the in trap evolution of a BEC similar to (c) with a s a_s tuned to 0.1a 0 0.1 a_0 at t=0t=0 to induce in-trap dipolar collapse.", "Parameters used in the simulation: (a) and (b), f avg =700f_{avg}=700 Hz; (b), N=2×10 6 N=2\\times 10^6 and a s =0.1a 0 a_s = 0.1 a_0; (c) and (d), f z =200f_z=200 Hz and f r =2000f_r=2000 Hz with N=2×10 6 N=2\\times 10^6." ], [ "Other Alkalis", "While some of the other alkalis have potential for observing the effects of MDDI, the effects are usually small.", "For example, in $^{23}$ Na and $^{85}$ Rb BECs with atom number and trapping frequencies similar to current experiments (see Table REF ), a calculation including MDDI makes a five to ten percent difference in TOF aspect ratio from a calculation where MDDI are not included.", "Thus, though small, the MDDI effect lies within the bounds of possible experimental observations.", "For $^{41}$ K and $^{87}$ Rb, however, the Feshbach resonance is too narrow to provide the precision control of $a_s$ to carry out the type of experiments proposed here, and the MDDI effect is less than a one percent perturbation on the aspect ratio." ], [ "Results for $^{164}$ Dy and {{formula:2465f89c-9b1b-4dda-89b2-52c3566894d8}} Er", "Due to the much higher dipole moments in $^{164}$ Dy and $^{168}$ Er ($10 \\mu _B$ [9] and $7 \\mu _B$ [42], respectively) the effects of MDDI will be strongly apparent with larger values of $a_s$ than the alkalis.", "In Fig.", "REF we present similar calculations to those of the alkali figures for these highly magnetic moment species, with the collapse dynamics occuring at higher values of $a_s$ .", "Figure: Effects of MDDI in 164 ^{164}Dy and 168 ^{168}Er.", "The left column (a-c) contains results for 164 ^{164}Dy and the righ column (a-c) for 168 ^{168}Er.", "(a,d) the BEC aspect ratio over a range of λ\\lambda , with the collapse regime indicated by the faint dotted line, for which there is no stable solution found for Eqns.", "and .", "(b,e) Evolution of the aspect ratio of the BEC in TOF expansion after release from a trap.", "In the trap a s =150a 0 a_s = 150 a_0 (where the BEC is stable for both species), and upon release a s a_s is tuned to 75a 0 75 a_0 (50a 0 50 a_0) for 164 ^{164}Dy ( 168 ^{168}Er) to induce collapse in free expansion.", "(c,f) the in trap evolution of a BEC similar to (b) with a s a_s tuned to 75a 0 75 a_0 (50a 0 50 a_0) for 164 ^{164}Dy ( 168 ^{168}Er) at t=0t=0 to induce in-trap dipolar collapse.", "For all 164 ^{164}Dy calculations N=1.5×10 4 N = 1.5 \\times 10^4, and for all 168 ^{168}Er N=7×10 4 N = 7 \\times 10^4.", "For (a,d), f avg =170f_{avg}=170 Hz.", "For (b,c), f z =100f_z=100 Hz and f r =200f_r=200 Hz, and for (e,f) f z =70f_z=70 Hz and f r =250f_r=250 Hz.We showed that the variational method provides a useful and simple tool to simulate the effects of MDDI in BECs and presented various results for $^{52}$ Cr and the alkalis.", "For example, examining the the aspect ratio of freely expanding BECs should be sufficient for detecting the effects of MDDI in many species, and we suggest the investigation of $^{7}$ Li, $^{39}$ K , and $^{133}$ Cs as favorable to detect such effects.", "We mention that future investigation of MDDI among non-alkali species looks promising as well.", "The achievement of BECs with $^{164}$ Dy ($\\mu =10\\mu _B$ )[9] and $^{168}$ Er ($\\mu =7\\mu _B$ ) [43] is quite exciting, as the $a_{dd}$ for $^{168}$ Er and $^{164}$ Dy are 66.3 $a_0$ and 131.5$a_0$ , much greater than even that of $^{52}$ Cr.", "The ability to tune $a_s$ by Feshbach resonance could make these species unparalleled for the observation of strong MDDI effects.", "The method presented here is applicable to both species.", "To close, we highlight that the effects of MDDI on the BEC shape provide a clear and intuitive picture of MDDI in BECs.", "The examination of the BEC aspect ratio, for example, can be used as a sensitive measurement of the $a_s$ value in situ, and may prove a helpful calibration method for future studies of other—perhaps more exotic—MDDI effects.", "We also note that while our discussion is limited to magnetic dipole-dipole interactions in this paper, the variational method we present is general for all dipolar BECs and may be employed in calculating effects of electric dipole-dipole interactions (e.g.", "polar molecular BECs) as well [18]." ], [ "Supplemental Experimental Parameters", "Typical atom numbers and trap frequencies currently employed in experimental studies of alkali BECs from the literature are listed below to show the experimental feasibility of the simulations provided here (Table REF ).", "Table: An example list of representative parameters for alkali BECs from the literature." ], [ "Calculating the Magnetic Moment", "A key parameter for the variational simulation is the value of the magnetic dipole moment, as $a_{dd} \\propto \\mu ^2$ .", "In low fields, the magnetic moment is found from the Zeeman effect, where $\\Delta E_{|F m_F>}=\\mu _B g_F m_F B_z$ In such a case, $\\mu = \\mu _B g_F m_F$ .", "For higher fields, however, the Breit-Rabi formula is used for the ground states of alkali atoms [49]: $E=\\frac{-\\Delta E_{hfs}}{2(2 I +1)} + g_I \\mu _B m_F B \\pm \\frac{\\Delta E_{hfs}}{2}\\left(1+\\frac{2 m_F}{I+1/2} x + x^2\\right)^{1/2}$ where $m_F=m_I \\pm 1/2$ , $\\Delta E_{hfs} = A_{hfs} (I+1/2)$ is the hyperfine splitting, $x=\\frac{g \\mu _B B}{\\Delta E_{hfs}}$ , and $g=g_J - g_I$ .", "The magnetic dipole moment is simply the derivative of the energy with respect to the magnetic field, resulting in $\\mu (B) = g_I \\mu _B m_F \\pm \\frac{\\frac{2 m_F}{2 I+1}+x}{2\\left(1+\\frac{4 m_F}{2 I +1} x + x^2\\right)^{1/2}} g \\mu _B$ This is used to calculate the magnetic moment, $\\mu _{cross}$ , at $B$ where $a_s=0$ of all the alkali species (see Table REF ).", "Table: Various parameters for the atomic species discussed in this paper.", "Values for g I g_I are from Ref. .", "Values for A hfs _{hfs} are from .", "We use g J =2.0023193043622(15)g_J=2.002 319 304 3622(15) from .", "Other values are from references listed in the table and .", "* ^{\\ast } from Ref.", ".The authors thank R.M.", "Wilson, R.G.", "Hulet, Han Pu, Yi Su and Li You for discussions, and T. Lahaye and T. Koch for providing the $^{52}$ Cr data.", "YPC acknowledges the support of the Miller Family Foundation and NSF Grant No.", "CCF-0829918.", "AJO was supported by the Department of Defense through the National Defense Science and Engineering Graduate (NDSEG) Fellowship Program." ] ]

[ [ "\"TNOs are Cool\": A survey of the trans-Neptunian region -- VII. Size and\n surface characteristics of (90377) Sedna and 2010 EK139" ], [ "Abstract We present estimates of the basic physical properties (size and albedo) of (90377) Sedna, a prominent member of the detached trans-Neptunian object population and the recently discovered scattered disk object 2010 EK139, based on the recent observations acquired with the Herschel Space Observatory, within the \"TNOs are Cool!\"", "key programme.", "Our modeling of the thermal measurements shows that both objects have larger albedos and smaller sizes than the previous expectations, thus their surfaces might be covered by ices in a significantly larger fraction.", "The derived diameter of Sedna and 2010 EK139 are 995 +/- 80 km and 470 +35/-10 km, while the respective geometric albedos are pV 0.32 +/- 0.06 and 0.25 +0.02/-0.05.", "These estimates are based on thermophysical model techniques." ], [ "Introduction", "The Herschel Space Observatory [21] allows the detection of thermal radiation from several trans-Neptunian objects (TNOs) at the precision level of $<1\\,{\\rm mJy}$ .", "Since the expected fluxes around the peak of the spectral energy distribution (SED) significantly exceed this precision, Herschel provides a great opportunity to characterize TNOs and obtain basic thermophysical information.", "In this work, we present recent measurements of the prominent objects (90377) Sedna and 2010 EK$_{139}$ using the Photodetector Array Camera and Spectrometer instrument [22] on board the Herschel Space Observatory.", "These observations are part of the “TNOs are Cool!", ": a survey of the trans-Neptunian region” Open Time Key Program [17], [18], [14], [15].", "Sedna is a prominent member of the detached objects, that is often classified as an inner Oort-cloud object.", "Until now, no accurate measurements of the diameter and albedo have been available for this object.", "Both direct imaging [3] and upper limits to the thermal radiation using the Spitzer Space Telescope [28] have yielded an upper limit of $\\approx 1670\\,{\\rm km}$ for its diameter (within 97% confidence).", "2010 EK$_{139}$ has been discovered in 2010 by [27] in the course of a southern Galactic plane survey.", "Prediscovery observations date back to 2002, allowing for a relatively accurate orbit determination.", "This places 2010 EK$_{139}$ among the scattered disk objects.", "2010 EK$_{139}$ orbits the Sun on an eccentric orbit ($e\\approx 0.53$ ) and has a perihelion distance of $q\\approx 32.5\\,{\\rm AU}$ .", "In addition, 2010 EK$_{139}$ is a suspected member of the 2:7 resonance grouphttp://boulder.swri.edu/buie/kbo/astrom/10EK139.html.", "We note that a more complete sample of SDOs/detached objects observed with Herschel/PACS is presented by [25].", "Table: Summary of Herschel observations of Sednaand 2010 EK 139 _{139}.", "The columns are:i) observation identifier,ii) date and time,iii) scan angle direction with respect to the detector array, andiv) filter configuration." ], [ "Observations, data reduction and photometry", "Sedna was observed by Herschel/PACS in two visits: the first started on 2010 August 6, 10:55:17 UTC and a follow-up started on 2010 August 9, 08:11:37 UTC, both taking place during the Routine Science Phase observation series of the “TNO's are Cool!” key programme [17].", "2010 EK$_{139}$ was also observed by Herschel/PACS in two visits, the first started on 2010 December 23, 07:04:30 UTC, and a follow-up started the same day, 19:58:27 UTC.", "Herschel/PACS observed Sedna and 2010 EK$_{139}$ for $\\approx 3.14$ and $\\approx 1.26$ hours, respectively.", "For both objects, we used both the blue/red ($70/160\\,\\mu {\\rm m}$ ) and green/red ($100/160\\,\\mu {\\rm m}$ ) channel combinations.", "The actual details of these observations are summarized in Table REF .", "Raw observational data were reduced using the Herschel Interactive Processing Environment (HIPEData presented in this paper were analyzed using “HIPE”, a joint development by the Herschel Science Ground Segment Consortium, consisting of ESA, the NASA Herschel Science Center, and the HIFI, PACS and SPIRE consortia members, see http://herschel.esac.esa.int/DpHipeContributors.shtml., see also Ott, 2010) and the processing scripts are similar to the ones employed in [16], [25], or [29].", "For each observation, these scripts create a pair of maps, one for the blue or green channel and one for the red channel.", "The maps have an effective pixel size of $1.\\!\\!^{\\prime \\prime }1$ , $1.\\!\\!^{\\prime \\prime }4$ , and $2.\\!\\!^{\\prime \\prime }1$ , for the blue, green, and red filters, respectively: these pixel sizes are set to sample the respective point spread functions (PSFs) properly.", "Data frames were selected by the actual scan speed ($10^{\\prime \\prime }/{\\rm sec}\\le {\\rm speed}\\le 25^{\\prime \\prime }/{\\rm sec}$ ) of the spacecraft, which maximized the effective usage of the detector and yielded significantly higher signal-to-noise (S/N) ratios than the standard setting (approximately $20^{\\prime \\prime }/{\\rm sec}$ ).", "Since the apparent motion of Sedna and 2010 EK$_{139}$ between the two visits ($15-35$ map pixels, depending on the actual filter) is relatively large compared to the PSF but small compared to the detector size, the location of the target in the first visit can simply be used as a background area on the maps of the second visit and vice versa.", "Owing to the satellite pointing uncertainty that is about a few arcsec [22], we derived the true map-center displacements using the red channel maps – on which the background confusion is the strongest – as follows.", "By varying the proper motion vector between the two visits, we computed the cross-correlation residuals for each trial vector.", "By minimizing the residuals, we obtained a more precise value for the shift between the visits and the photometry of combined maps was found to be more reliable.", "Since simple averaging the registered maps does not cancel the background confusion noise, we employed background removal techniques as it is described in [25] or [16].", "The maps on which the photometry was then performed are shown in Fig.", "REF .", "Figure: Image stamps showing the combined maps of Sedna (upper panels) and2010 EK 139 _{139} (lower panels) in the 70μm70\\,\\mu {\\rm m} (blue),100μm100\\,\\mu {\\rm m} (green), and 160μm160\\,\\mu {\\rm m} (red) channels.Each stamp covers an area of 56 '' ×56 '' 56^{\\prime \\prime }\\times 56^{\\prime \\prime }, whilethe tick marks on the axes show therelative positions in pixels.", "The effective beam size (i.e.", "the circle with adiameter corresponding to the full width at half magnitude)is also displayed in the lower-left cornersof the stamps.Table: Thermal fluxes of Sedna and2010 EK 139 _{139} derived from our Herschel measurements.Table: Orbital and optical data for Sednaand 2010 EK 139 _{139} at the time of the Herschel observations.", "The parameters rr andΔ\\Delta denote the heliocentric distance and thedistance from Herschel, α\\alpha is the phase angle,and H V H_{\\rm V} is the absolute visual magnitude, which is availablefrom the literature.Regardless of the background structures, in the subtracted and combined maps the only expected source is the TNO itself and this source can be treated as an isolated point source.", "We estimated the fluxes and their uncertainties using (1) a single aperture that maximizes the expected S/N ratio; (2) the aperture growth curve method and implanted artificial sources in a Monte Carlo fashion [25]; and (3) we also checked the individual (non-combined) maps on which they had sufficient S/N ratio.", "For a more detailed description, we refer to [16] and [25].", "Figure: Spectral energy distribution of Sedna (left)and 2010 EK 139 _{139} (right) in the far-infrared region, based onHerschel/PACS measurements.", "Superimposedare the best-fit TPM (solid lines) andSTM curves with floating beaming parameter (dashed lines).We found that all three methods yielded the same fluxes and uncertainties for each channel.", "The individual analysis of maps for 2010 EK$_{139}$ also showed consistent results.", "Therefore, we accepted the means of all measurements per object and filter as final fluxes (see Table REF ) and used them for thermal modeling.", "We note here that the color corrections provided in [22] are negligible: it is nearly or less than 1 per cent for 2010 EK$_{139}$ and for Sedna, it is 6 per cent in the $70\\,\\mu {\\rm m}$ channel and less than 3 per cent in the other longer wavelength channels, so almost less by a magnitude than the relative photometric uncertainties in all of these cases." ], [ "Thermal properties", "The basic physical properties of Sedna and 2010 EK$_{139}$ were estimated by a hybrid standard thermal model [13], [28] in which the beaming parameter is adjustable and the asteroid thermophysical model [10], [11], [12].", "The absolute magnitudes of the reflected sunlight from these TNOs are available from the literature [23], [27].", "In addition for 2010 EK$_{139}$ , we conservatively increased the formal uncertainty ($0.03$  mag, based on MPC data) up to $0.10$  mag: we took into account the possible omission of the phase angle corrections and also added quadratically an average TNO lightcurve amplitude of $0.088$  mag [6].", "The employed geometric parameters and absolute magnitudes are summarized in Table REF .", "The hybrid STM predicts thermal fluxes from the geometric albedo, diameter, and beaming parameter and these fluxes can be computed for arbitrary solar and geocentric distances.", "Hybrid STM provides reliable estimates only for small phase angles (via a simple form of phase angle corrections), although owing to the distances of these objects, this estimate is fairly sufficient in our cases.", "To estimate the physical parameters and their respective uncertainties, we used a Monte Carlo (MC) approach by varying the fit input around their mean with the standard deviation equal to their respective uncertainty.", "For both targets, we used the fixed-$\\eta $ approach for the beaming parameter, i.e.", "taking $\\eta =1.14\\pm 0.15$ for each MC step.", "This mean value of and scatter in the beaming factor are taken from [25] and seem to be an acceptable approach for TNOs.", "To estimate the phase integral $q$ , i.e.", "the ratio of the Bond to geometric albedo (i.e.", "$A=qp_{\\rm V}$ ), we employed an iterative approach.", "First, the phase integral is computed for unity slope parameter ($G=1$ , i.e.", "$q=0.29+0.68G$ ), and then refined using eq.", "1 of [5] until convergence.", "This procedure applied for hybrid STM yielded the diameter, geometric albedo, and slope parameter of $D=1060\\pm 100$  km, $p_{\\rm V}=0.290\\pm 0.061$ , and $G=0.42\\pm 0.04$ for Sedna and $D=535\\pm 30$  km, $p_{\\rm V}=0.187\\pm 0.027$ , and $G=0.37\\pm 0.03$ for 2010 EK$_{139}$ , respectively.", "We repeated the similar procedure by allowing the beaming parameter $\\eta $ to vary.", "This analysis yielded $\\eta =0.95\\pm 0.43$ for Sedna, with the corresponding diameter and albedo of $D=990\\pm 95$  km and $p_{\\rm V}=0.336\\pm 0.072$ .", "For 2010 EK$_{139}$ , the best-fit value of the beaming parameter is somewhat smaller, $0.70\\pm 0.31$ , while the diameter and albedo values are $D=450\\pm 35$  km and $p_{\\rm V}=0.261\\pm 0.047$ .", "In the case of Sedna, we note that the linear phase coefficient $\\beta =0.151\\pm 0.033$ [23] would imply a phase integral of $q=0.89^{+0.55}_{-0.29}$ , assuming the same phase behavior over the whole phase angle range.", "Although the phase curve is known for very small domains [23], this value broadly agrees, within a nearly 1-$\\sigma $ difference from the phase integral of $q=0.59\\pm 0.03$ as implied by the radiometric albedo and the Brucker formula.", "The results of the TPM estimates were the following.", "For Sedna, we used the rotation period of $\\approx 10.27$  hours [9] and assumed an equator-on rotation and the most favorable solution for the thermal inertia was found to be $0.2\\,{\\rm J}\\,{\\rm m}^{-2}\\,{\\rm K}^{-1}\\,{\\rm s}^{-1/2}$ , which corresponds to the diameter of $D=995 \\pm 80\\,{\\rm km}$ and the geometric albedo of $p_{\\rm V}=0.32 \\pm 0.06$ .", "For 2010 EK$_{139}$ , we assumed a period of 12 hours and equator-on rotation and found that this object also requires a very low thermal inertia, $0.1\\,{\\rm J}\\,{\\rm m}^{-2}\\,{\\rm K}^{-1}\\,{\\rm s}^{-1/2}$ .", "This may change slightly if the rotation period and the spin vector orientation were very different, although all feasible solutions put the thermal inertia below $1.0\\,{\\rm J}\\,{\\rm m}^{-2}\\,{\\rm K}^{-1}\\,{\\rm s}^{-1/2}$ .", "Our best fit yielded a diameter of $D=470^{+35}_{-10}\\,{\\rm km}$ and a geometric albedo of $p_{\\rm V}=0.25^{+0.02}_{-0.05}$ , which do not differ significantly from the hybrid STM model results in which the beaming parameter was also varied.", "We note that here the uncertainties include both the statistical errors and the ambiguities in the rotation parameters [19].", "In Fig.", "REF , we displayed the far-infrared SEDs for these two objects as it is estimated from the hybrid STM and TPM fitting and our best-fit data and the floating $\\eta $ values.", "We also note that the rotation period of Sedna found by [9] corresponds to a peak-to-peak amplitude of $0.02$  mag.", "The small amplitude does not change the reliability of the thermal modeling and the corresponding shape effects are not relevant to the size determination." ], [ "Discussion", "We have estimated the sizes and surface albedos for the trans-Neptunian objects Sedna and 2010 EK$_{139}$ using recent observations of their thermal emission at $70/100/160\\,\\mu {\\rm m}$ with Herschel/PACS.", "On the basis of earlier Spitzer measurements, Sedna had already only an upper limit to its size estimate [28].", "Our analysis has shown that for Sedna and 2010 EK$_{139}$ , the respective geometric albedos are $p_{\\rm V}=0.32\\pm 0.06$ and $p_{\\rm V}=0.25^{+0.02}_{-0.05}$ , thus both objects have brighter surfaces than the average TNO population [28] or SDOs/detached population [25].", "We note that the albedos of Sedna and 2010 EK$_{139}$ closely match those of detached objects in fig.", "4a ($p_{\\rm V}$ vs. diameter) in [25].", "According to [26] and [4], even with these newly derived parameters, Sedna lies in the region in which volatiles are expected to be retained in the surface (see Fig.", "1 in that paper for an equivalent temperature of $20\\pm 2$  K), hence one can also expect a brighter surface [2], [8].", "Sedna is currently approaching its perihelion.", "Thus, if the brightness of the surface were changing owing to the ongoing sublimation of ices, it might be detectable in the variation in the absolute magnitude on a timescale of decades.", "In contrast, 2010 EK$_{139}$ falls in the region in which volatiles should have been lost (using $48\\pm 3$  K for equivalent temperature).", "However, objects of this size can have such a high albedo if water ice is present on the surface [1], [24], [7].", "The presence of water could be tested by measuring the intrinsic color to see whether it is bluish [3].", "As they lack known satellites, we do not know the masses and hence the surface gravity and escape velocities of these objects that could place tighter constraints on the surface properties.", "We gratefully thank the valuable comments and suggestions of the referee, Josh Emery.", "The work of A. P., Cs.", "K., and N. Sz.", "has been supported by the ESA grant PECS 98073.", "A. P. and Cs. K.", "were also supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.", "Cs. K.", "thanks the support of the OTKA grant K101393.", "Part of this work was supported by the German DLR projects number 50 OR 1108, 50 OR 0903, and 50 OR 0904.", "M. R. acknowledges support from the German Deutsches Zentrum für Luft- und Raumfahrt, DLR project number 50OFO 0903.", "M. M. acknowledges support through the DFG Special Priority Program 1385.", "P. S.-S. would like to acknowledge financial support by the Centre National de la Recherche Scientifique (CNRS).", "J.-L. O. acknowledges the Spanish grants AYA2008-06202-C03-01, AYA2011-30106-C02-01 and 2007-FQM2998." ] ]

[ [ "Numerical study of the conductivity of graphene monolayer within the\n effective field theory approach" ], [ "Abstract We report on the direct numerical measurements of the conductivity of graphene monolayer.", "Our numerical simulations are performed in the effective lattice field theory with noncompact 3 + 1-dimensional Abelian lattice gauge fields and 2 + 1-dimensional staggered lattice fermions.", "The conductivity is obtained from the Green-Kubo relations using the Maximum Entropy Method.", "We find that in a phase with spontaneously broken sublattice symmetry the conductivity rapidly decreases.", "For the largest value of the coupling constant used in our simulations g = 4.5, the DC conductivity is less than the DC conductivity in the weak-coupling phase (at g < 3.5) by at least three orders of magnitude." ], [ "Introduction", "Graphene, a single layer of carbon atoms which form a two-dimensional honeycomb lattice, is probably the most widely discussed material in modern condensed matter physics.", "A peculiar feature of charge carriers in graphene is that their energy spectrum near the Fermi point is similar to that of the free $2+1$ -dimensional massless Dirac fermions.", "This explains the unusual transport properties of graphene, such as Klein tunneling or novel types of the quantum Hall effect [1], [2], [3].", "The four spinor components of these Dirac fermions correspond to charge carriers which are localized on one of the two elementary rhombic sublattices of the honeycomb lattice and which are close to one of the two distinct Fermi points in the Brillouin zone of graphene.", "In this low-energy description two components of the non-relativistic spin are treated as two independent fermionic flavors.", "Interactions between fermions are mediated by electromagnetic fields which propagate freely in $3+1$ -dimensional space.", "The strength of electromagnetic interactions can be controlled by placing graphene layers on substrates with different dielectric permittivities.", "Since charge carriers in graphene propagate with speed $v_F \\approx c/300$ , the effective coupling constant for electromagnetic interactions turns out to be quite large, $\\alpha = \\alpha _0/v_F = 300/137 \\approx 2$ for suspended graphene [1], [2], [3].", "In this case the non-perturbative effects could play an important role.", "Theoretical considerations suggest that such strong interaction between charge carriers could result in the insulator-semimetal phase transition in graphene [4], [5], [6], [7], [8], [9].", "However, due to the large value of the effective coupling constant there are no reliable analytical methods which allow to study this phase transition from the first principles, and one has to use numerical simulations.", "The effective field theory of graphene can be efficiently simulated using lattice staggered fermions [10], [11], [12], [13], [14], [15], [16], [17].", "A single flavor of staggered fermions on $2 + 1$ -dimensional square lattice corresponds to two independent flavors of continuum Dirac fermions [18], [19], [20], which exactly reproduces the number of fermion flavors in the graphene effective field theory.", "In papers [10], [11], [12], [13], [14], [15], [16], [17] the insulator-semimetal phase transition was studied numerically by considering the fermionic“chiral” condensate $\\langle \\bar{\\psi } \\psi \\rangle $ .", "Within the effective field theory of Dirac quasiparticles, nonzero condensate signals the opening of a gap in the quasiparticle spectrum, thus it plays the role of the order parameter for the semimetal-insulator phase transition.", "In [10], [11], [12], [13], [14] Coulomb interactions between fermions were modeled by a non-compact $3 + 1$ -dimensional Abelian lattice gauge field.", "It was found that the condensate is formed at a critical coupling constant of the non-compact gauge field $\\beta \\sim 0.1$ .", "Motivated by the theoretical considerations of [9], [21], the authors of [15], [16], [17] have also studied a similar theory with a contact interaction instead of Coulomb potential.", "They have also found that fermionic condensate is formed at sufficiently strong coupling.", "However, the value of the conductivity has not yet been directly measured in numerical simulations.", "In this paper we report on direct numerical measurements of AC and DC conductivities of graphene within the effective field theory of Dirac quasiparticles.", "Our lattice regularization of this effective field theory is similar to the one used in [10], [11], [12], [13], [14].", "The conductivity is obtained from the Green-Kubo dispersion relations for the ground-state correlators of electromagnetic currents.", "These relations are inverted with the help of the Maximum Entropy Method [22], [23].", "In agreement with the predictions of [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17] we find that when a nonzero fermionic condensate is formed, the DC conductivity rapidly decreases.", "For the maximal value of the coupling constant used in our simulations, which corresponds to substrate dielectric permittivity $\\epsilon = 1.75$ , the DC conductivity is smaller than the DC conductivity in the weak-coupling limit by at least three orders of magnitude.", "The paper is organized as follows: in Section we briefly review the effective field theory of graphene and discuss its lattice regularization, as well as suitable simulation algorithms.", "In Section we present and discuss our numerical results for the graphene conductivity and the fermionic condensate $ \\langle \\, \\bar{\\psi } \\psi \\, \\rangle $ .", "Section contains some concluding remarks and the discussion of the obtained results.", "We start from the Euclidean path integral representation of the partition function of the effective field theory of graphene [1], [2], [3]: $\\mathcal {Z} =\\int \\mathcal {D}\\bar{\\psi }_f\\mathcal {D}\\psi _f \\,\\mathcal {D}A_{\\mu } \\,\\exp \\left(- \\frac{1}{2}\\int d^4x \\, \\left( \\partial _{\\left[\\mu \\right.}", "A_{\\left.\\nu \\right]} \\right) ^2- \\right.", "\\nonumber \\\\ -\\int d^3x \\,\\bar{\\psi }_f \\, \\Gamma _{0} \\, \\left( \\partial _{0} - i e A_{0} \\right) \\, \\psi _f- \\nonumber \\\\ \\left.", "-\\sum \\limits _{i=1,2} \\int d^3x \\,\\bar{\\psi }_f \\, \\Gamma _{i} \\, \\left( \\partial _{i} - i e \\, v_F \\, A_{i} \\right) \\, \\psi _f\\right) , $ where $A_{\\mu }$ , $\\mu = 0 \\ldots 3$ is the vector potential of the $3 + 1$ electromagnetic field, $\\Gamma _{\\mu }$ are Euclidean gamma-matrices and $\\psi _f$ , $f = 1, 2$ are two flavours of Dirac fermions which correspond to the two spin components of the non-relativistic electrons in graphene.", "We have also taken into account that Dirac fermions propagate in $2 + 1$ -dimensional subspace at $x^3 = 0$ with speed $v_F \\approx 1/300$ .", "After rescaling of the coordinates and the vector potential $x^0 \\rightarrow x^0/v_F,\\quad A^0 \\rightarrow \\sqrt{v_F} \\, A^0,\\quad A_i \\rightarrow \\frac{1}{\\sqrt{v_F}} \\, A_i \\quad .$ we conclude that the fluctuations of the spatial components $A_i$ of the vector potential are suppressed by a factor $1/v_F$ and we can set $A_i = 0$ in practical calculations.", "We thus arrive at the following partition function: $\\mathcal {Z} =\\int \\mathcal {D}\\bar{\\psi } \\mathcal {D}\\psi \\mathcal {D}A_0\\exp \\left( -\\frac{1}{2}\\int d^4x \\left( \\partial _{i} A_{0} \\right) ^2- \\right.", "\\nonumber \\\\ \\left.", "-\\int d^3x \\, \\bar{\\psi }_f \\, \\left( \\Gamma _0 \\, \\left( \\partial _0 - i g A_0 \\right) - \\sum \\limits _{i=1,2}\\Gamma _i \\partial _i \\right) \\psi _f\\right) ,$ where the effective coupling constant $g^2 = e^2/v_F \\approx 300/137 \\sim 2$ .", "Finite temperature $T$ can be introduced by imposing periodic boundary conditions in Euclidean time $x^0$ with the period $\\frac{v_F}{k T}$ .", "By virtue of the commutation relations $[O_a,\\Gamma _0 \\Gamma _i] = 0$ with $O_a = 1, \\Gamma ^3, \\, \\Gamma ^5, \\, i \\Gamma ^3 \\Gamma ^5$ the action of the effective field theory (REF ) has the global $U(4)$ symmetry $\\psi _f \\rightarrow \\exp {\\left( i O_a \\otimes \\tau _b \\, \\alpha ^{ab} \\right)} \\, \\psi _f ,$ where $\\tau _0 = 1$ and $\\tau _i$ - are spin Pauli matrices which act on flavor index $f$ .", "Finite temperature and chemical potential do not break this global symmetry on the level of the Lagrangian, however, it might be broken spontaneously due to sufficiently strong Coulomb interactions [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]." ], [ "Lattice action", "Following [10], [11], [12], [13], [14], [15], [16], [17] we use staggered fermions [19], [20] in order to discretize the fermionic part of the action in (REF ).", "One flavor of staggered fermions in $2 + 1$ dimensions corresponds to two flavors of continuum Dirac fermions [18], [19], [20], which makes them especially suitable for simulations of the graphene effective field theory.", "The action for staggered fermions coupled to Abelian lattice gauge field is $S_{\\Psi } \\left[ \\bar{\\Psi }_x, \\Psi _x, \\theta _{x, \\, \\mu } \\right] =\\sum \\limits _{x, y} \\bar{\\Psi }_x \\, D_{x, y} \\left[ \\theta _{x, \\, \\mu } \\right] \\, \\Psi _y= \\nonumber \\\\ =\\frac{1}{2} \\, \\sum \\limits _{x} \\, \\delta _{x_3, \\, 0} \\, \\left( \\sum \\limits _{\\mu =0, 1, 2}\\bar{\\Psi }_x \\alpha _{x, \\mu } e^{i \\theta _{x, \\, \\mu }} \\Psi _{x+\\hat{\\mu }}- \\right.", "\\nonumber \\\\ \\left.", "-\\sum _{\\mu =0, 1, 2}\\bar{\\Psi }_x \\alpha _{x, \\mu } e^{-i \\theta _{x, \\, \\mu }} \\Psi _{x-\\hat{\\mu }}+ m {\\bar{\\Psi }}_x \\Psi _x \\right) ,$ where the lattice coordinates $x$ take integer values $x^{\\mu } = 0 \\ldots L_{\\mu }-1$ and $x^3$ is restricted to $x^3 = 0$ , $\\bar{\\Psi }_x$ is a single-component Grassman-valued field, $\\alpha _{x, \\mu } = (-1)^{x_0 + \\ldots + x_{\\mu -1}}$ , and $\\theta _{x, \\, \\mu }$ are the link variables which are the lattice counterpart of the vector potential $A_{\\mu } \\left( x \\right) $ .", "For further convenience, we have also introduced the matrix elements $D_{x, y}$ of the staggered Dirac operator.", "The fields $\\bar{\\Psi }_x$ , $\\Psi _x$ satisfy periodic boundary conditions in spatial directions and anti-periodic boundary conditions in the Euclidean time direction.", "To account for the latter, we make a shift $\\theta _{x, 0} \\rightarrow \\theta _{x, 0} + \\pi $ in (REF ) at the time slice with $x^0 = 0$ .", "In order to recover the original spinor and flavor indices of the continuum Dirac fermions in (REF ), the lattice should be subdivided into the cubic blocks consisting of $2 \\times 2 \\times 2$ elementary lattice cells.", "Now the coordinates of all lattice sites can be enumerated as $x_{\\mu } = 2 y_{\\mu } + \\eta _{\\mu }$ , where $\\eta _{\\mu } = 0, 1$ .", "We define the new fields on the lattice of $y$ coordinates [19], [20], [11]: $ \\left[ \\Phi _y \\right] ^{\\alpha }_f = \\frac{1}{4 \\sqrt{2}} \\sum _{\\eta } \\left[ \\Gamma _0^{\\eta _0} \\Gamma _1^{\\eta _1} \\Gamma _2^{\\eta _2} \\right] ^{\\alpha }_fW_{y, \\, \\eta }\\Psi _{2 y + \\eta } ,$ where $W_{y, \\, \\eta }$ is the product of $e^{i \\theta _{x, \\mu }}$ along the the path which connects lattice sites with coordinates $2 y$ and $2 y + \\eta $ , $\\alpha = 1, 2, 3, 4$ is the Dirac spinor index and $f = 1, ..., 4$ is the flavor index.", "It can be shown that in terms of these new fields defined on the lattice with double lattice spacing the staggered fermion action (REF ) reproduces the naive discretization of the continuum fermionic action in (REF ).", "However, there are additional terms which explicitly break the global $U \\left( 4 \\right) $ symmetry of the continuum action in (REF ) down to its $U \\left( 1 \\right) \\otimes U \\left( 1 \\right) $ subgroup and which decouple only in the long-wavelength limit [19], [20], [11].", "Since the fermion action (REF ) is restricted to the $2 + 1$ -dimensional subspace with $x^3 = 0$ , not all components of $ \\left[ \\Phi _y \\right] ^{\\alpha }_f$ are independent.", "Due to the absence of $\\Gamma ^3$ in (REF ) it satisfies the constraint $\\Gamma _3 \\, \\Gamma _5 \\, \\Phi _y \\, \\Gamma _5 \\, \\Gamma _3 = \\Phi _y .$ It is easy to check that in a representation of Euclidean gamma-matrices with $\\Gamma _3 \\, \\Gamma _5 = {\\rm diag} \\, \\left( 1, 1, -1, -1 \\right) $ this constraint implies the following block-diagonal form of the matrices $\\Phi _y$ : $\\Phi = \\left(\\begin{array}{cc}A & 0\\\\0 & B\\end{array}\\right) ,$ which is equivalent to two flavors of 4-component Dirac spinors.", "Now let us consider lattice discretization of the action of the electromagnetic field in (REF ).", "There exist two basic formulations of the $U \\left( 1 \\right) $ lattice gauge theory: compact and non-compact.", "In order to exclude non-physical confining phase of the compact $U \\left( 1 \\right) $ gauge theory [24] here we use the non-compact action for the gauge fields: $S_g \\left[ \\theta _{x, \\, \\mu } \\right] = \\frac{\\beta }{2} \\, \\sum \\limits _x \\sum \\limits ^{3}_{i=1} \\left( \\theta _{x, \\, 0} - \\theta _{x + \\hat{i}, \\, 0} \\right) ^2 ,$ where summation over $x$ now goes over the whole four-dimensional lattice.", "As discussed above, the fluctuations of the spatial components of the vector potential $A_i \\left( x \\right) $ are suppressed in the effective field theory of graphene (REF ).", "Correspondingly, we also set to zero the spatial link variables $\\theta _{x, \\, i}$ .", "In continuous space, the inverse lattice coupling constant $\\beta $ is related to the substrate dielectric permittivity $\\epsilon $ as $\\beta \\equiv \\frac{1}{g^2} = \\frac{v_F}{4 \\pi e^2} \\, \\frac{\\epsilon + 1}{2} ,$ where the factor $\\frac{\\epsilon + 1}{2}$ takes into account the screening of the electrostatic interactions by the substrate.", "However, this relation can be modified due to finite lattice spacing effects such as the flavor symmetry breaking for staggered fermions [25], [26].", "Generally, such effects tend to shift the phase transition towards the weak-coupling region [25], [26].", "We leave the study of such finite-spacing artifacts for future work.", "We note also that although the gauge field action (REF ) is non-compact, the fermionic action (REF ) is still “compact”, that is, periodic in the variables $\\theta _{\\mu } \\left( x \\right) $ .", "In general, it is impossible to couple the gauge field to lattice fermions in a non-compact way while preserving the gauge invariance of the theory.", "Since the fermionic action (REF ) is bilinear in the fermion fields, they can be integrated out in the partition function (REF ): $\\mathcal {Z} =\\int \\mathcal {D}\\bar{\\Psi }_x \\, \\mathcal {D}\\Psi _x \\, \\mathcal {D}\\theta _{x, \\, 0}\\,\\nonumber \\\\\\exp \\left( - S_g \\left[ \\theta _{x, \\, 0} \\right] - S_{\\Psi } \\left[ \\bar{\\Psi }_x, \\Psi _x, \\theta _{x, \\, 0} \\right] \\right) = \\nonumber \\\\ =\\int \\mathcal {D}\\theta _{x, \\, 0}\\,\\det \\left( D \\left[ \\theta _{x, \\, 0} \\right] \\right) \\exp {\\left( -S_g \\left[ \\theta _{x, \\, 0} \\right] \\right)} .$ Thus we deal with the effective action $S_{eff} \\left[ \\theta _{x, \\, 0} \\right] = S_g \\left[ \\theta _{x, \\, 0} \\right] - \\ln \\det \\left( D \\left[ \\theta _{x, \\, 0} \\right] \\right) $ which includes the determinant $\\det \\left( D \\left[ \\theta _{x, \\, \\mu } \\right] \\right) $ of the staggered Dirac operator $D_{x, y} \\left[ \\theta _{x, \\, \\mu } \\right] $ introduced in (REF )." ], [ "Simulation algorithm", "We use the standard Hybrid Monte-Carlo Method for generation of the configurations of the field $\\theta _{x, \\, 0}$ with the statistical weight $ \\exp {\\left( -S_{eff} \\left[ \\theta _{x, \\, 0} \\right] \\right)} $ [19], [20], [11].", "In order to calculate the determinant of the staggered Dirac operator in (REF ), we take into account that in the basis of even and odd lattice sites (which are defined as lattice sites with even or odd sum of all coordinates $x^0 + x^1 + x^2$ ) it takes the form [19], [20], [11] $D \\left[ \\theta _{x, \\, 0} \\right] = \\left(\\begin{matrix}m & D_{eo} \\\\D_{oe} & m\\end{matrix}\\right)$ with $D_{eo}^{\\dag } = - D_{oe}$ .", "The determinant of $D$ is thus equal to $\\det \\left( D \\right) = \\det \\left( m^2 + D^{\\dag }_{eo} \\, D_{eo} \\right) ,$ which is a manifestly positive quantity.", "Note that the operator $m^2 + D^{\\dag }_{eo} \\, D_{eo}$ acts only on the subspace of even lattice sites.", "We use the $\\Phi $ -algorithm in our simulations [19], [20], in which the determinant (REF ) is represented in terms of a Gaussian integral over the pseudo-fermion field $\\phi _x$ : $\\det \\left( m^2 + D^{\\dag }_{eo} \\, D_{eo} \\right) = \\int \\mathcal {D}\\bar{\\phi }_x \\, \\mathcal {D}\\phi _x\\nonumber \\\\ \\exp {\\left( - \\sum \\limits _{x, y} \\bar{\\phi }_x \\, \\left( m^2 + D^{\\dag }_{eo} \\, D_{eo} \\right) ^{-1}_{x, y}\\, \\phi _y \\right)} ,$ where the sum over $x, y$ goes only over even lattice sites.", "The field $\\phi _x$ is then stochastically sampled with the weight (REF ).", "To this end we generate the random field $\\xi _x$ according to the Gaussian distribution $P \\left[ \\xi _x \\right] \\sim \\exp {\\left( - \\sum \\limits _{x} \\bar{\\xi }_x \\, \\xi _x \\right)} $ and then calculate $\\phi _{x} = \\sum \\limits _{y} \\left( m^2 + D^{\\dag }_{eo} \\, D_{eo} \\right) ^{-1}_{x, y} \\, \\xi _y$ at the beginning of each Molecular Dynamics trajectory [19], [20].", "Nonzero mass term in (REF ) and (REF ) is necessary in order to ensure the invertibility of the staggered Dirac operator.", "Numerical results in the physical limit of zero mass were obtained by performing simulations at several nonzero values of $m$ and by extrapolating the expectation values of physical observables to $m \\rightarrow 0$ .", "In order to speed up the simulations we also perform local heatbath updates of the gauge field outside of the graphene plane (at $x^3 \\ne 0$ ) between Hybrid Monte-Carlo updates.", "Both algorithms satisfy the detailed balance condition for the weight (REF ) [19], [20].", "Successive application of these algorithms does not, in general, have this property.", "Nevertheless, by using the composition rule for transition probabilities it is easy to demonstrate that the path integral weight (REF ) is still the stationary probability distribution for such a combination of both algorithms.", "While local heatbath updates are computationally very cheap, they significantly decrease the autocorrelation time of the algorithm." ], [ "Physical observables on the lattice", "The main goal of this paper is to measure the electric conductivity of graphene, that is, a linear response of the electric current density $J_i \\left( x \\right) = \\bar{\\psi } \\left( x \\right) \\, \\gamma _i \\, \\psi \\left( x \\right) $ to the applied homogeneous electric field $E_j \\left( t \\right) $ (where $t$ is the real Minkowski time).", "It is convenient to introduce the AC conductivity $\\sigma _{ij} \\left( w \\right) $ , so that $\\tilde{J}_i \\left( w \\right) = \\sigma _{ij} \\left( w \\right) \\, \\tilde{E}_j \\left( w \\right) $ , where $\\tilde{J}_i \\left( w \\right) = \\int dt \\, e^{-i w t} J_i \\left( t \\right) $ and $\\tilde{E}_j \\left( w \\right) = \\int dt \\, e^{-i w t} E_j \\left( t \\right) $ .", "Due to rotational symmetry of the effective field theory (REF ), $\\sigma _{ij} \\left( w \\right) $ should have the form $\\sigma _{ij} \\left( w \\right) = \\delta _{ij} \\, \\sigma \\left( w \\right) $ .", "Correspondingly, the DC conductivity is equal to the value of $\\sigma \\left( w \\right) $ at $w \\rightarrow 0$ .", "By virtue of the Green-Kubo dispersion relations [27], [22], [23], the Euclidean current-current correlators $G \\left( \\tau \\right) = \\frac{1}{2} \\, \\sum \\limits _{i=1,2} \\, \\int dx^1 \\, dx^2 \\, \\langle J_i \\left( 0 \\right) \\, J_i \\left( x \\right) \\rangle $ can be expressed in terms of $\\sigma \\left( w \\right) $ as $G \\left( \\tau \\right) = \\int \\limits ^{\\infty }_{0}\\frac{dw}{2 \\pi } \\, K \\left( w, \\tau \\right) \\, \\sigma \\left( w \\right) ,$ where the thermal kernel $K \\left( w, \\tau \\right) $ is $K \\left( w, \\tau \\right) = \\frac{w \\cosh \\left( w \\left( \\tau - \\frac{1}{2T} \\right) \\right) }{\\sinh \\left( \\frac{w}{2T} \\right) }$ and $\\tau \\equiv x^0$ is the Euclidean time.", "We use here a nonstandard definition of the kernel (REF ) from [23], which is more convenient for numerical analysis.", "Note that the current density in graphene is the charge which flows through the unit length in unit time and thus has the dimensionality of $L^{-2}$ (where $L$ stands for length) in units with $\\hbar = c = 1$ .", "Correspondingly, the current density in lattice units is the charge which flows through a link of the dual lattice of length $a$ in time $a/v_F$ .", "Thus in order to express the current-current correlator (REF ) in physical units, one should multiply the result obtained on the lattice by $a^2 \\, v_F^2/a^4$ , where an additional factor of $a^2$ comes from integration over $x^1$ , $x^2$ in (REF ).", "With the Euclidean time $\\tau $ in (REF ), (REF ) and (REF ) being expressed in units of lattice spacing in temporal direction $a/v_F$ , integration over $w$ in (REF ) also includes a factor $v_F^2/a^2$ .", "We thus conclude that the AC conductivity $\\sigma \\left( w \\right) $ is dimensionless.", "Moreover, the DC conductivity $\\sigma \\left( 0 \\right) $ is a universal quantity which does not depend on the lattice spacing or on the ratio of lattice spacings in temporal and spatial directions.", "For conversion to the SI system of units, it should be multiplied by $e^2/ \\left( 2 \\pi h \\right) $ .", "In numerical simulations $G \\left( \\tau \\right) $ is measured for several ($\\sim 10^1$ ) discrete values of $\\tau $ .", "A commonly used method to invert the relation (REF ) and to extract the continuum function $\\sigma \\left( w \\right) $ from the lattice discretization of $G \\left( \\tau \\right) $ is the Maximum Entropy Method [22], [23].", "For staggered fermions the electric current $J_i \\left( y \\right) $ can be expressed in terms of the fields $\\Psi _x$ as [20]: $J_i \\left( y \\right) =\\frac{1}{8} \\sum \\limits _{\\eta } \\,\\delta _{\\eta _3, \\, 0} \\, \\delta _{\\eta _i, \\, 0} \\,\\left(\\bar{\\Psi }_{2 y + \\eta } \\alpha _{\\eta , \\, i} \\Psi _{2 y + \\eta + \\hat{i}}+ \\right.", "\\nonumber \\\\ \\left.", "+\\bar{\\Psi }_{2 y + \\eta + \\hat{i}}\\alpha _{\\eta , \\, i} \\Psi _{2 y + \\eta }\\right) ,$ where we have taken into account that the spatial link variables $\\theta _{x, i}$ are effectively equal to zero.", "Since the current (REF ) is defined on the lattice with double lattice spacing, we calculate the Euclidean current-current correlator (REF ) only on time slices with even $\\tau $ .", "In order to make sure that we reproduce the results of [10], [11], [12], [13], [14], we have also calculated the fermionic chiral condensate.", "In terms of staggered fermions it can be written as $ \\langle \\, \\bar{\\psi } \\, \\psi \\, \\rangle =\\frac{1}{8 \\, L_0 \\, L_1 \\, L_2} \\, \\sum \\limits _{x} \\, \\langle \\, \\bar{\\Psi }_x \\Psi _x \\, \\rangle .$ After the fermions in the partition function are integrated out, the current-current correlator (REF ) and the chiral condensate (REF ) can be expressed in terms of expectation values of certain combinations of the staggered fermion propagator $D^{-1}_{x, y} \\left[ \\theta _{x, \\mu } \\right] $ with respect to the weight (REF ).", "We give the explicit expressions for these combinations in Appendix ." ], [ "Numerical results", "Using the algorithm described in Subsection REF , we have generated 400 statistically independent gauge field configurations on the $20^4$ lattice for each point in the space of lattice parameters $\\beta $ and $m$ .", "For each value of $\\beta $ in the range $\\beta = 0.05 \\ldots 0.025$ (which corresponds to substrate dielectric permittivities $\\epsilon = 1.75 \\ldots 12.75$ according to (REF )) the measurements were performed at three different values of mass $m = 0.01, \\, 0.02, \\, 0.03$ .", "For the smallest mass $m = 0.005$ , for which the simulations are most expensive, $\\beta $ took values in the range $\\beta = 0.05 \\ldots 0.15$ ($\\epsilon = 1.75 \\ldots 7.25$ ).", "In order to estimate the finite-volume effects, we have also generated 100 gauge field configurations on the $28^4$ lattice with $\\beta = 0.05 \\ldots 0.21$ and $m = 0.01$ .", "Figure: Fermionic condensate 〈ψ ¯ψ〉 \\langle \\, \\bar{\\psi } \\, \\psi \\, \\rangle as a function of inverse lattice coupling constants β\\beta at different values of mass mmand extrapolation to the limit m→0m \\rightarrow 0.Solid line is the fit of the extrapolated data with the function〈ψ ¯ψ〉∼β c -β γ \\langle \\, \\bar{\\psi } \\, \\psi \\, \\rangle \\sim \\left( \\beta _c - \\beta \\right) ^\\gamma with β c =0.0908±0.0018\\beta _c = 0.0908 \\pm 0.0018 and γ=1.0±0.16\\gamma = 1.0 \\pm 0.16.To check our simulation algorithms and to make sure that we reproduce the results of [10], [11], we first consider the fermionic chiral condensate $ \\langle \\, \\bar{\\psi } \\, \\psi \\, \\rangle $ .", "It is plotted on Fig.", "REF for different values of the mass $m$ .", "The condensate rapidly decreases as the inverse lattice coupling constant $\\beta $ (or, equivalently, the substrate dielectric permittivity $\\epsilon $ (REF )) is increased.", "For each value of $\\beta $ , we have fitted the mass dependence of the condensate with a quadratic polynomial and used this fit to extrapolate the data to the limit $m \\rightarrow 0$ .", "The result of such extrapolation is also shown on Fig.", "REF .", "It suggests that there is a critical value of $\\beta $ in the range $0.08 < \\beta _c < 0.09$ (which corresponds to $3.4 < \\epsilon _c < 4.0$ ) such that the condensate is zero for $\\beta > \\beta _c$ .", "A fit of the extrapolated data of the form $ \\langle \\, \\bar{\\psi } \\, \\psi \\, \\rangle = b \\left( \\beta _c - \\beta \\right) ^\\gamma $ for $\\beta < \\beta _c$ yields $\\beta _c = 0.091 \\pm 0.002$ ($\\epsilon _c = 4.0 \\pm 0.1$ ) and $\\gamma =1.0 \\pm 0.16$ .", "These values are in agreement with the results of [10], [11], where the critical inverse coupling constant $\\beta _c$ and the critical index $\\gamma $ were estimated as $0.071 \\ge \\beta _c \\le 0.091$ and $\\gamma \\backsimeq 1$ .", "Figure: Current - current correlators () for m=0.01m = 0.01.", "Solid lines are the fits obtained using the Maximum Entropy Method.Figure: AC conductivity σw\\sigma \\left( w \\right) in () for m=0.01m=0.01 at β\\beta close to β c \\beta _c (on the left) and at β>β c \\beta > \\beta _c (on the right).Our measurements of the conductivity of graphene start from the calculation of the current-current correlators (REF ), which are shown on Fig.", "REF for $m = 0.01$ and for different values of inverse coupling constant $\\beta $ .", "We note that the correlators decay significantly faster for smaller values of $\\beta $ .", "The AC conductivity $\\sigma \\left( w \\right) $ is extracted from the correlators using the Maximum Entropy Method with four basis eigenfunctions of the kernel (REF ) and a constant model function $\\sigma _0 \\left( w \\right) = 0.1$ [22], [23].", "The profiles of $\\sigma \\left( w \\right) $ are plotted on Fig.", "REF and the corresponding fits of the correlators are shown on Fig.", "REF with solid lines.", "It is important to note at this point that on Fig.", "REF the angular frequency $w$ is given in lattice units.", "For qualitative comparison with experimental data one can assume that the lattice spacing $a$ for the spatial directions of the cubic lattice used in our simulations is comparable with the lattice spacing $a = 0.246 \\, {\\rm nm}$ of the hexagonal lattice in graphene.", "After the rescaling (REF ) the discretization step for the Euclidean time $\\tau $ should be of order ${\\Delta \\tau } \\sim a/v_F$ .", "The temperature $T$ in (REF ) and (REF ) is then equal to $k T = \\hbar / \\left( L_0 \\, {\\Delta \\tau } \\right) \\sim 0.1 \\, {\\rm eV}$ , which is much smaller than the characteristic binding energy in graphene $\\sim 1 \\, {\\rm eV}$ .", "Thus our simulation results should correspond to sufficiently low physical temperatures as compared to characteristic excitation energies.", "For the inverse coupling constant $\\beta $ below approximately $0.08$ ($\\epsilon <3.4$ ), $\\sigma \\left( w \\right) $ has one very broad peak around $w \\approx 1.2$ , and the DC conductivity $\\sigma \\left( 0 \\right) $ has some small nonzero value.", "As $\\beta $ increases towards $\\beta _c$ , the second peak emerges at $w = 0$ and both peaks become narrower and higher (see Fig.", "REF , left plot).", "The emergence of the second peak results in the rapid growth of the DC conductivity.", "At $\\beta > \\beta _c$ , the two peaks continue to grow, and their widths become comparable to the temperature $T = L_0^{-1}$ in lattice units (see Fig.", "REF , right plot).", "In order to understand such peak structure in the weak-coupling limit, remember that for free Dirac fermions the AC conductivity $\\sigma \\left( w \\right) $ has a delta-function singularity at $w = 0$ [30], [31], which is a manifestation of the absence of scattering of charge carriers.", "When the interactions are turned on, this peak is smeared, which results in a large but finite value of the DC conductivity $\\sigma \\left( 0 \\right) $ .", "The second peak practically does not move as the mass $m$ is changed.", "We conjecture that this second peak corresponds to a saddle point in the dispersion relation of staggered fermions which is situated in the middle of a straight line which connects the two distinct Fermi points [28].", "The position of this peak should thus depend on the lattice regularization of the effective field theory (REF ) and should correspond to the optical frequency range for real graphene.", "Figure: DC conductivity per spin per valley σ0/4\\sigma \\left( 0 \\right) /4 in units of e 2 /he^2/h as a function of inverse lattice coupling constants at different values of mass mm.", "The result of extrapolation of the data to the limit m→0m \\rightarrow 0 is plotted with black dots and solid lines.The DC conductivity $\\sigma \\left( 0 \\right) $ is shown on Fig.", "REF as a function of inverse coupling constant $\\beta $ at different values of the mass $m$ .", "We normalize the conductivity to a single spin component and a single Dirac point, thus on Fig.", "REF we plot $\\sigma \\left( 0 \\right) /4$ rather than $\\sigma \\left( 0 \\right) $ .", "$\\sigma \\left( 0 \\right) $ quickly decreases as both $m$ and $\\beta $ become smaller.", "One can also note two distinct discontinuities in the dependence of $\\sigma \\left( 0 \\right) $ on $\\beta $ .", "For example, at $m=0.01$ the first discontinuity is situated between $\\beta =0.07$ ($\\epsilon = 3$ ) and $\\beta =0.09$ ($\\epsilon = 4$ ), and the second one - between $\\beta =0.12$ ($\\epsilon =5.6$ ) and $\\beta =0.13$ ($\\epsilon =6.2$ ).", "The position of the first discontinuity depends only weakly on the mass $m$ and roughly corresponds to the critical inverse coupling constant $\\beta _c$ obtained from the analysis of the chiral condensate (see Fig.", "REF ).", "The second discontinuity shifts to smaller $\\beta $ and becomes somewhat weaker as $m$ decreases.", "Linear extrapolation of the position of this discontinuity to the limit of zero mass (see Fig.", "REF ) suggests that at $m = 0$ both discontinuities coincide.", "We also note that the profile of the AC conductivity $\\sigma \\left( w \\right) $ practically does not change across this second discontinuity.", "Thus there seems to be a single phase transition in the chiral limit $m \\rightarrow 0$ , in agreement with the results of [10], [11], [12].", "The corresponding critical value of the inverse coupling constant can be estimated to lie in the range $0.07 \\le \\beta _c \\le 0.09$ ($3 \\le \\epsilon _c \\le 4$ ).", "For each value of $\\beta $ between the two discontinuities, we also perform the quadratic fit of the mass dependence of the conductivity and use it to extrapolate the data to $m \\rightarrow 0$ .", "This extrapolation is shown on Fig.", "REF with black dots and solid lines.", "On Fig.", "REF we compare the DC conductivity at $m = 0.01$ on $20^4$ and $28^4$ lattices.", "The dependence of $\\sigma \\left( 0 \\right) $ on the inverse coupling constant $\\beta $ is qualitatively the same for both lattices, in particular, the positions of the discontinuities practically coincide.", "However, the actual values of the DC conductivity differ beyond the error bars, especially in the strong-coupling phase.", "This suggests that finite-volume and finite-temperature effects could be quite large for our lattice parameters.", "Indeed, quite large finite-temperature effects have been reported in a recent Monte-Carlo study of the tight-binding model on the hexagonal lattice [28], where the values of lattice parameters were quite close to those used in this work.", "We leave the detailed study of finite-temperature and finite-volume effects as a direction for further investigations.", "Figure: Linear extrapolation of the position of the second discontinuity of the DC conductivity to the limit m→0m\\rightarrow 0.Figure: DC conductivity per spin per valley σ0/4\\sigma \\left( 0 \\right) /4 in units of e 2 /he^2/h as a function of inverse lattice coupling constant β\\beta for 20 4 20^4 and 28 4 28^4 lattices with m=0.01m=0.01." ], [ "Conclusions", "In this paper we have numerically studied the AC and DC conductivities of graphene by using lattice Monte-Carlo simulations with $2 + 1$ -dimensional staggered fermions which interact with $3 + 1$ -dimensional non-compact Abelian lattice gauge field.", "We have found that in a phase with spontaneously broken chiral symmetry (which corresponds to sublattice symmetry of the original hexagonal lattice) the DC conductivity rapidly decreases as the substrate dielectric permittivity becomes smaller.", "The estimates of the corresponding critical values $0.07 \\le \\beta _c \\le 0.09$ ($3 \\le \\epsilon _c \\le 4$ ) obtained both from the measurements of the chiral condensate and the conductivity agree with each other and with the results of [10], [11], [12], [14].", "This supports the existence of a single insulator-semimetal phase transition in graphene.", "Interestingly, our estimate of $\\epsilon _c$ is close to the dielectric permittivity of silicon dioxide $\\epsilon _{SiO_2} = 3.9$ , which is often used as a substrate for graphene.", "According to the data presented on Fig.", "REF , for the largest value of the coupling constant used in our simulations ($\\beta = 1/g^2 = 0.05$ ) the DC conductivity turns out to be smaller than the DC conductivity in the semimetal phase (at $\\beta > \\beta _c$ ) by a factor of order of $10^3$ .", "Finally, we note that our value of the DC conductivity in the semimetal phase is significantly larger than the conductivity of non-interacting quasiparticles in the ideal monolayer graphene ($\\sigma \\left( 0 \\right) \\sim 4 e^2/h$ ) obtained in [29], [30], [2] using the Landauer approach.", "However, as was stressed in [30], in Landauer approach a finite value of the conductivity of free Dirac fermions is determined solely by scattering on the boundaries of the sample.", "In the absence of boundaries (for instance, on the lattice with torus topology) the AC conductivity $\\sigma \\left( w \\right) $ has a delta-function singularity at $w = 0$ , and thus the DC conductivity $\\sigma \\left( 0 \\right) $ is formally infinite.", "In an interacting theory, this singularity is smeared out, and the DC conductivity takes some finite (but large) value.", "The authors are much obliged to Dr. Timo Lahde who was the first to draw their attention to graphene field theory.", "The authors are grateful to Prof. Mikhail Zubkov for interesting and useful discussions.", "The work was supported by Grant RFBR-11-02-01227-a and by the Russian Ministry of Science and Education, under contract No.", "07.514.12.4028.", "Numerical calculations were performed at the ITEP system Stakan (authors are much obliged to A.V.", "Barylov, A.A. Golubev, V.A.", "Kolosov, I.E.", "Korolko, M.M.", "Sokolov for the help), the MVS 100K at Moscow Joint Supercomputer Center and at Supercomputing Center of the Moscow State University." ], [ "Calculation of fermionic observables", "Here we give explicit expressions for the vacuum expectation values of fermionic observables used in our simulations.", "Fermionic chiral condensate corresponds to the diagonal elements of the staggered fermion propagator: $ \\langle \\, \\bar{\\psi } \\, \\psi \\, \\rangle =\\frac{1}{8 \\, L_0 \\, L_1 \\, L_2} \\, \\sum \\limits _{x} \\, \\langle \\, D^{-1}_{x, x} \\, \\rangle $ where $ \\langle \\, \\ldots \\, \\rangle $ on the right-hand side denotes averaging over lattice gauge field $\\theta _{x, \\mu }$ with the weight (REF ).", "We calculate this trace using the stochastic estimator [19], [20].", "Current-current correlator is a sum of connected and disconnected parts: $G \\left( y^0 \\right) = C \\left( y^0 \\right) - D \\left( y^0 \\right) .$ The connected contributions can be expressed in terms of staggered fermion propagator as [20]: $C \\left( y^0 \\right) = \\frac{1}{64} \\, \\sum \\limits _{y_1, y_2}\\sum \\limits _{\\eta , \\eta ^{\\prime }}\\alpha _{\\eta , i} \\alpha _{\\eta ^{\\prime }, i}\\times \\nonumber \\\\ \\times \\left( \\langle \\, S \\left( 2 y + \\eta , \\hat{i} + \\eta ^{\\prime } \\right) S \\left( \\eta ^{\\prime }, 2 y + \\hat{i} + \\eta \\right) \\, \\rangle \\right.", "+ \\nonumber \\\\ + \\langle \\, S \\left( 2 y + \\eta , \\eta ^{\\prime } \\right) S \\left( \\hat{i} + \\eta ^{\\prime }, 2 y + \\hat{i} + \\eta \\right) \\, \\rangle + \\nonumber \\\\ + \\langle \\, S \\left( 2 y + \\hat{i} + \\eta , \\hat{i} + \\eta ^{\\prime } \\right) S \\left( \\eta ^{\\prime }, 2 y + \\eta \\right) \\, \\rangle + \\nonumber \\\\ + \\left.", "\\langle \\, S \\left( 2 y + \\hat{i} + \\eta , \\eta ^{\\prime } \\right) S \\left( \\hat{i} + \\eta ^{\\prime }, 2 y + \\eta \\right) \\, \\rangle \\right) ,$ where $S \\left( x, y \\right) \\equiv D^{-1}_{x, y}$ .", "For the calculation of the connected part of the correlator (REF ) we take into account that the solution of the linear equation $\\chi _y = D_{y, z} \\psi _z$ with $\\chi _y = \\delta _{x, y}$ yields the staggered fermion propagator $D^{-1}_{x, y}$ for all $y$ .", "The disconnected part takes the following form: $D \\left( y^0 \\right) =\\frac{1}{64} \\sum \\limits _{y_1, y_2}\\sum \\limits _{\\eta , \\eta ^{\\prime }}\\alpha _{\\eta , i} \\, \\alpha _{\\eta ^{\\prime }, i}\\nonumber \\\\\\langle \\left( S \\left( \\eta ^{\\prime }, \\hat{i} + \\eta ^{\\prime } \\right) +S \\left( \\hat{i} + \\eta ^{\\prime }, \\eta ^{\\prime } \\right) \\right) \\times \\nonumber \\\\ \\times \\left( S \\left( 2 y + \\hat{i}, 2 y + \\eta \\right) +S \\left( 2 y + \\eta , 2 y + \\hat{i} + \\eta \\right) \\right) \\rangle .$ In practice the disconnected part of the correlator (REF ) is calculated using stochastic estimators [19], [20], similarly to the chiral condensate.", "In our simulations we have found that the disconnected part of the correlator is much smaller and much noisier than the connected one.", "Therefore we have neglected it in our measurements of the conductivity." ] ]

[ [ "Set It and Forget It: Approximating the Set Once Strip Cover Problem" ], [ "Abstract We consider the Set Once Strip Cover problem, in which n wireless sensors are deployed over a one-dimensional region.", "Each sensor has a fixed battery that drains in inverse proportion to a radius that can be set just once, but activated at any time.", "The problem is to find an assignment of radii and activation times that maximizes the length of time during which the entire region is covered.", "We show that this problem is NP-hard.", "Second, we show that RoundRobin, the algorithm in which the sensors simply take turns covering the entire region, has a tight approximation guarantee of 3/2 in both Set Once Strip Cover and the more general Strip Cover problem, in which each radius may be set finitely-many times.", "Moreover, we show that the more general class of duty cycle algorithms, in which groups of sensors take turns covering the entire region, can do no better.", "Finally, we give an optimal O(n^2 log n)-time algorithm for the related Set Radius Strip Cover problem, in which all sensors must be activated immediately." ], [ "Introduction", "Suppose that $n$ sensors are deployed over a one-dimensional region that they are to cover with a wireless network.", "Each sensor is equipped with a finite battery charge that drains in inverse proportion to the sensing radius that is assigned to it, and each sensor can be activated only once.", "In the Set Once Strip Cover (OnceSC) problem, the goal is to find an assignment of radii and activation times that maximizes the lifetime of the network, namely the length of time during which the entire region is covered.", "Formally, we are given as input the locations $x \\in [0,1]^n$ and battery charges $b \\in \\mathbb {Q}^n$ for each of $n$ sensors.", "While we cannot move the sensors, we do have the ability to set the sensing radius $\\rho _i$ of each sensor and the time $\\tau _i$ when it should become active.", "Since each sensor's battery drains in inverse proportion to the radius we set (but cannot subsequently change), each sensor covers the region $[x_i - \\rho _i, x_i + \\rho _i]$ for $b_i/\\rho _i$ time units.", "Our task is to devise an algorithm that finds a schedule $S = (\\rho , \\tau ) \\in [0,1]^n \\times [0,\\infty )^n$ for any input $(x,b)$ , such that $[0,1]$ is completely covered for as long as possible." ], [ "Scheduling problems of this ilk arise in many applications, particularly when the goal is barrier coverage (see [8], [21] for surveys, or [13] for motivation).", "Suppose that we have a highway, supply line, or fence in territory that is either hostile or difficult to navigate.", "While we want to monitor activity along this line, conditions on the ground make it impossible to systematically place wireless sensors at specific locations.", "However, it is feasible and inexpensive to deploy adjustable range sensors along this line by, say, dropping them from an airplane flying overhead (e.g.", "[7], [18], [20]).", "Once deployed, the sensors send us their location via GPS, and we wish to send a single radius-time pair to each sensor as an assignment.", "Replacing the battery in any sensor is infeasible.", "How do we construct an assignment that will keep this vital supply line completely monitored for as long as possible?", "While the focus of this paper is the OnceSC problem, we touch upon three closely related problems.", "In each problem the location and battery of each sensor are fixed, and a solution can be viewed as a finite set of radius-time pairs.", "In OnceSC, both the radii and the activation times are variable, but can be set only once.", "In the more general Strip Cover problem, the radius and activation time of each sensor can be set finitely many times.", "On the other hand, if the radius of each sensor is fixed and given as part of the input, then we call the problem of assigning an activation time to each sensor so as to maximize network lifetime Set Time Strip Cover (TimeSC).", "Set Radius Strip Cover (RadSC) is another variant of OnceSC in which all of the sensors are scheduled to activate immediately, and the problem is to find the optimal radial assignment.", "Figure REF summarizes the important differences between related problems and illustrates their relationship to one another.", "Figure: Relationship of Problem Variants.TimeSC, which is known as Restricted Strip Covering, was shown to be NP-hard by Buchsbaum et al.", "[6], who also gave an $O(\\log \\log n)$ -approximation algorithm.", "Later, a constant factor approximation algorithm was discovered by Gibson and Varadarajan [12].", "Close variants of RadSC have been the subject of previous work.", "Whereas RadSC requires area coverage, Peleg and Lev-Tov [14] studied target coverage.", "In this problem the input is a set of $n$ sensors and a finite set of $m$ points on the line that are to be covered, and the goal is to find the radial assignments with the minimum sum of radii.", "They used dynamic programming to devise a polynomial time alorithm.", "Bar-Noy et al.", "[5] improved the running time to $O(n+m)$ .", "Recently, Bar-Noy et al.", "considered an extension of RadSC in which sensors are mobile.", "Strip Cover was first considered by Bar-Noy and Baumer [3], who gave a $\\frac{3}{2}$ lower bound on the performance of RoundRobin, the algorithm in which the sensors take turns covering the entire region (see Observation REF ), but were only able to show a corresponding upper bound of $1.82$ .", "The similar Connected Range Assignment (CRA) problem, in which radii are assigned to points in the plane in order to obtain a connected disk graph, was studied by Chambers et al. [11].", "They showed that the best one circle solution to CRA also yields a $\\frac{3}{2}$ -approximation guarantee, and in fact, the instance that produces their lower bound is simply a translation of the instance used in Observation REF .", "The notion of duty cycling as a mean to maximize network lifetime was also considered in the literature of discrete geometry.", "In this context, maximizing the number of covers $t$ serves as a proxy for maximizing the actual network lifetime.", "Pach [15] began the study of decomposability of multiple coverings.", "Pach and Tóth [16] showed that a $t$ -fold cover of translates of a centrally-symmetric open convex polygon can be decomposed into $\\Omega (\\sqrt{t})$ covers.", "This was later improved to the optimal $\\Omega (t)$ covers by Aloupis et al.", "[2], while Gibson and Varadarajan [12] showed the same result without the centrally-symmetric restriction.", "Motivated by prior invocations of duty cycling [19], [17], [1], [7], [9], [10], Bar-Noy et al.", "[4] studied a duty cycle variant of OnceSC with unit batteries in which sensors must be grouped into shifts of size at most $k$ that take turns covering $[0,1]$ .", "(RoundRobin is the only possible algorithm when $k=1$ .)", "They presented a polynomial-time algorithm for $k=2$ and showed that the approximation ratio of this algorithm is $\\frac{35}{24}$ for $k>2$ .", "It was also shown that its approximation ratio is at least $\\frac{15}{11}$ , for $k \\ge 4$ , and $\\frac{6}{5}$ , for $k = 3$ .", "A fault-tolerance model, in which smaller shifts are more robust, was also proposed.", "We introduce the Set Once model that corresponds to the case where the scheduler does not have the ability to vary the sensor's radius once it has been activated.", "We show that OnceSC is NP-hard (Section ) and that RoundRobin is a $\\frac{3}{2}$ -approximation algorithm for both OnceSC and Strip Cover (Section ).", "This closes a gap between the best previously known lower and upper bounds ($\\frac{3}{2}$ and $1.82$ , resp.)", "on the performance of this algorithm.", "Our analysis of RoundRobin is based on the following approach: We slice an optimal schedule into strips in which the set of active sensors is fixed.", "For each such strip we construct an instance with unit batteries and compare the performance of RoundRobin to the RadSC optimum of this instance.", "In Section  we show that the class of duty cycle algorithms cannot improve on this $\\frac{3}{2}$ guarantee.", "In Section , we provide an $O(n^2 \\log {n})$ -time algorithm for RadSC.", "We note that the same approach would work for the case where, for every sensor $i$ , the $i$ th battery is drained in inverse proportion to $\\rho _i^\\alpha $ , for some $\\alpha >0$ .", "The Set Once Strip Cover (abbreviated OnceSC) is defined as follows.", "Let $U \\stackrel{\\scriptscriptstyle \\triangle }{=}[0,1]$ be the interval that we wish to cover.", "Given is a vector $x = (x_1,\\ldots ,x_n) \\in U^n$ of $n$ sensor locations, and a corresponding vector $b = (b_1,\\ldots ,b_n) \\in \\mathbb {Q}_+^n$ of battery charges, with $b_i \\ge 0$ for all $i$ .", "We assume that $x_i \\le x_{i+1}$ for every $i \\in \\lbrace 1,\\ldots ,n-1\\rbrace $ .", "We sometimes abuse notation by treating $x$ as a set.", "An instance of the problem thus consists of a pair $I = (x,b)$ , and a solution is an assignment of radii and activation times to sensors.", "More specifically a solution (or schedule) is a pair $S =(\\rho ,\\tau )$ where $\\rho _i$ is the radius of sensor $i$ and $\\tau _i$ is the activation time of $i$ .", "Since the radius of each sensor cannot be reset, this means that sensor $i$ becomes active at time $\\tau _i$ , covers the range $[x_i - \\rho _i, x_i +\\rho _i]$ for $b_i/\\rho _i$ time units, and then becomes inactive since it has exhausted its entire battery.", "Any schedule can be visualized by a space-time diagram in which each coverage assignment can be represented by a rectangle.", "It is customary in such diagrams to view the sensor locations as forming the horizontal axis, with time extending upwards vertically.", "In this case, the coverage of a sensor located at $x_i$ and assigned the radius $\\rho _i$ beginning at time $\\tau _i$ is depicted by a rectangle with lower-left corner $(x_i - \\rho _i, \\tau _i)$ and upper-right corner $(x_i + \\rho _i, \\tau _i + b_i/\\rho _i)$ .", "Let the set of all points contained in this rectangle be denoted as $Rect(\\rho _i, \\tau _i)$ .", "A point $(u,t)$ in space-time is covered by a schedule $(\\rho ,\\tau )$ if $(u,t) \\in \\bigcup _{i} Rect(\\rho _i, \\tau _i)$ .", "The lifetime of the network in a solution $S = (\\rho ,\\tau )$ is the maximum value $T$ such that every point $(u,t) \\in U \\times [0,T]$ is covered.", "Graphical depictions of two schedules are shown below in Figure REF .", "In OnceSC our goal is to find a schedule $S=(\\rho ,\\tau )$ that maximizes the lifetime $T$ .", "Given an instance $I = (x,b)$ , the optimal lifetime is denoted by $\\textsc {Opt} (x,b)$ .", "(We sometimes use $\\textsc {Opt} $ , when the instance is clear from the context.)", "The Set Radius Strip Cover (RadSC) problem is a variant of OnceSC in which $\\tau _i=0$ , for every $i$ .", "Hence, a solution is simply a radial assignment $\\rho $ .", "Set Time Strip Cover (TimeSC) is another variant in which the radii are given in the input, and a solution is an assignment of activation times to sensors.", "Strip Cover is a generalization of OnceSC in which a sensor's radius may be changed finitely many times.", "In this case a solution is a vector of piece-wise constant functions $\\rho (t)$ , where $\\rho _i(t)$ is the sensing radius of sensor $i$ at time $t$ .", "The solution is feasible if $U$ is covered for all $t \\in [0,T]$ , and if $\\int _0^\\infty \\rho _i(t)\\, dt \\le b_i$ , for every $i$ .", "The segment $[0,1]$ is covered at time $t$ , if $[0,1] \\subseteq \\bigcup _i [x_i-\\rho _i(t),x_i+\\rho _i(t)]$ .", "The best possible lifetime of an instance $(x,b)$ is $2\\sum _i b_i$ .", "We state this formally for OnceSC, but the same holds for the other variants.", "Observation 1 The lifetime of a OnceSC instance $(x,b)$ is at most $2\\sum _i b_i$ .", "Consider an optimal solution $(\\rho ,\\tau )$ for $(x,b)$ with lifetime $T$ .", "A sensor $i$ covers an interval of length $2\\rho _i$ for $\\frac{b_i}{\\rho _i}$ time.", "The lifetime $T$ is at most the total area of space-time covered by the sensors, which is at most $\\sum _i 2 \\rho _i \\cdot b_i/\\rho _i$ .", "We focus on a simple algorithm we call RoundRobin.", "The RoundRobin algorithm forces the sensors to take turns covering $U$ , namely it assigns, for every $i$ , $\\rho _i = r_i \\stackrel{\\scriptscriptstyle \\triangle }{=}\\max \\lbrace x_i,1-x_i\\rbrace $ and $\\tau _i =\\sum _{j=1}^{i-1} b_j/\\rho _j$ .", "The lifetime of RoundRobin is thus $\\textstyle \\textsc {RR} (x,b) \\stackrel{\\scriptscriptstyle \\triangle }{=}\\sum _{i=i}^n b_i/r_i~.$ Notice that Observation REF implies an upper bound of 2 on the approximation ratio of RoundRobin, since $r_i \\le 1$ , for every $i$ .", "A lower bound of $\\frac{3}{2}$ on the approximation guarantee of RoundRobin was given in [3] using the two sensor instance $x = (\\frac{1}{4},\\frac{3}{4})$ , $b = (1,1)$ .", "The relevant schedules are depicted graphically in Figure REF .", "Observation 2 ([3]) The approximation ratio of RoundRobin is at least $\\frac{3}{2}$ .", "Figure: RoundRobin vs. Opt with x=(1 4,3 4)x = (\\frac{1}{4},\\frac{3}{4}) andb=(1,1)b = (1,1).", "The sensors are indicated by (red) dots.", "Each of the(blue) rectangles represents the active coverage region for onesensor.", "The dashed gray arrow helps to clarify which sensor is activeat a particular point in time.Given an instance $(x,b)$ of OnceSC, let $B \\stackrel{\\scriptscriptstyle \\triangle }{=}\\sum _i b_i$ be the total battery charge of the system and $\\overline{r} = \\sum _i\\frac{b_i}{B} \\cdot r_i$ be the average of the $r_i$ 's, weighted by their respective battery charge.", "We define the following lower bound on $\\textsc {RR} (x,b)$ : $\\textsc {RR}^{\\prime }(x,b) \\stackrel{\\scriptscriptstyle \\triangle }{=}B/\\overline{r}~.$ Lemma 3 $\\textsc {RR}^{\\prime }(x,b) \\le \\textsc {RR} (x,b)$ , for every OnceSC instance $(x,b)$ .", "We have that $\\textsc {RR} (x,b)= \\sum _{i=1}^n \\frac{b_i}{r_i}= \\sum _{i=1}^n \\frac{b_i^2}{b_i r_i}\\ge \\frac{(\\sum _{i=1}^n b_i)^2}{\\sum _{i=1}^n b_i r_i}= \\frac{\\sum _{i=1}^n b_i}{\\overline{r}}= \\textsc {RR}^{\\prime }(x,b)~,$ where the inequality is due to an implication of the Cauchy-Schwarz Inequality: $\\sum _j\\frac{c_j^2}{d_j} \\ge \\frac{(\\sum _j c_j)^2}{\\sum _j d_j}$ , for any positive $c,d \\in \\mathbb {R}^n$ ." ], [ "Set Once Hardness Result", "In this section we show that OnceSC is NP-hard.", "This is done using a reduction from Partition.", "Theorem 1 OnceSC is NP-hard.", "Let $Y = \\lbrace y_1,\\ldots ,y_n\\rbrace $ be a given instance of Partition, and define $B = \\frac{1}{2} \\sum _{i=1}^n y_i$ .", "We create an instance of OnceSC by placing $n$ sensors with battery $y_i$ at $\\frac{1}{2}$ , and two additional sensors equipped with battery $B$ at $\\frac{1}{6}$ and $\\frac{5}{6}$ , respectively.", "That is, the instance of OnceSC consists of sensor locations $x = (\\frac{1}{6}, \\underbrace{\\textstyle \\frac{1}{2},\\ldots , \\frac{1}{2}}_n, \\frac{5}{6} )$ and batteries $b = (B,y_1,\\ldots ,y_n,B)$ .", "We show that $Y \\in \\textsc {Partition} $ if and only if the maximum possible lifetime of $8B$ is achievable for the OnceSC instance.", "First, suppose $Y \\in \\textsc {Partition} $ , hence there exist two non-empty disjoint subsets $Y_0, Y_1 \\subseteq Y$ , such that $Y_0 \\cup Y_1 = Y$ , and $\\sum _{y \\in Y_0} y = B = \\sum _{y \\in Y_1} y$ .", "Schedule the sensors in $Y_0$ to iteratively cover the region $[\\frac{1}{3},\\frac{2}{3}]$ .", "Since all of these sensors are located at $\\frac{1}{2}$ , this requires that each sensor's radius be set to $\\frac{1}{6}$ , i.e.", "$\\rho _{i+1} = \\frac{1}{6}$ , for every $i \\in Y_0$ .", "Since the sum of their batteries is $B$ , this region can be covered for exactly $6B$ time units.", "With the help of the additional sensors located at $\\frac{1}{6}$ and $\\frac{5}{6}$ , whose radii are also set to $\\rho _1 = \\rho _{n+2} = \\frac{1}{6}$ , the sensors in $Y_0$ can thus cover $[0,1]$ for $6B$ time units (see Figure REF for an example).", "Next, the sensors in $Y_1$ can cover $[0,1]$ for an additional $2B$ time units, since they all require a radius of $\\rho _{i+1} = \\frac{1}{2}$ , for every $i \\in Y_1$ .", "Thus, the total lifetime is $8B$ .", "Figure: Proof of NP-hardness.", "Y={1,2,3,4}Y = \\lbrace 1,2,3,4\\rbrace is a given instanceof Partition, and (x,b)=(1 6,1 2,...,1 2,5 6),(5,1,2,3,4,5)(x,b) = \\left( (\\frac{1}{6}, \\frac{1}{2}, \\ldots ,\\frac{1}{2}, \\frac{5}{6}), (5,1,2,3,4,5) \\right) is the translated OnceSC instance.Now suppose that for such a OnceSC instance, the lifetime of $8B$ is achievable.", "Since the maximum possible lifetime is achievable, no coverage can be wasted in the optimal schedule.", "In this case the radii of the sensors at $\\frac{1}{6}$ and $\\frac{5}{6}$ must be exactly $\\frac{1}{6}$ , since otherwise, they would either not reach the endpoints $\\lbrace 0,1\\rbrace $ , or extend beyond them.", "Moreover, due the fact that all of the other sensors are located at $\\frac{1}{2}$ , and their coverage is thus symmetric with respect to $\\frac{1}{2}$ , it cannot be the case that sensor 1 and sensor $n+2$ are active at different times.", "Thus, the solution requires a partition of the sensors located at $\\frac{1}{2}$ into two groups: the first of which must work alongside sensors 1 and $n+2$ with a radius of $\\frac{1}{6}$ and a combined lifetime of $6B$ ; and the second of which must implement RoundRobin for a lifetime of $2B$ .", "The batteries of these two partitions form a solution to Partition." ], [ "Round Robin", "We show in Appendix  that OnceSC is NP-hard, so we turn our attention to approximation algorithms.", "While RoundRobin is among the simplest possible algorithms (note that its running time is exactly $n$ ), the precise value of its approximation ratio is not obvious (although it is not hard to see that 2 is an upper bound).", "In [3] an upper bound of $1.82$ and a lower bound of $\\frac{3}{2}$ were shown.", "In this section, we show that the approximation ratio of RoundRobin in OnceSC is exactly $\\frac{3}{2}$ .", "The structure of the proof is as follows.", "We start with an optimal schedule $S$ , and cut it into disjoint time intervals, or strips, such that the same set of sensors is active within each time interval.", "Each strip induces a RadSC instance $I_j$ and a corresponding solution $S_j$ .", "Next, we show that for any such instance $I_j$ , there exists a unit-battery instance $I_j^{\\prime }$ with the same optimum lifetime.", "Finally, we prove a lower bound on the performance of RoundRobin on such unit battery instances.", "By combining these results, we prove that $\\textsc {RR} (x,b) \\ge \\frac{2}{3} T$ ." ], [ "Cutting the Schedule into Strips", "Given an instance $I = (x,b)$ , and a solution $S = (\\rho ,\\tau )$ with lifetime $T$ , let $\\Omega $ be the set of times until $T$ in which a sensor was turned on or off, namely $\\Omega = \\bigcup _i \\lbrace \\tau _i,\\tau _i+b_i/\\rho _i\\rbrace \\cap [0,T]$ .", "Let $\\Omega = \\left\\lbrace 0=\\omega _0,\\ldots ,\\omega _\\ell = T \\right\\rbrace $ , where $\\omega _j <\\omega _{j+1}$ , for every $j$ .", "Next, we partition the time interval $[0,T]$ into the sub-intervals $[\\omega _j,\\omega _{j+1}]$ , for every $j \\in \\lbrace 0,\\ldots ,\\ell -1\\rbrace $ .", "Next, we define a new instance for every sub-interval.", "For every $j\\in \\lbrace 0,\\ldots ,\\ell -1\\rbrace $ , let $x^j \\subseteq x$ be the set of sensors that participate in covering $[0,1]$ during the $j$ th sub-interval of time, i.e., $x^j =\\left\\lbrace x_i : [\\omega _j,\\omega _{j+1}] \\subseteq [\\tau _i, \\tau _i+b_i/ \\rho _i] \\right\\rbrace $ .", "Also, let $T_j = \\omega _{j+1} - \\omega _j$ , and let $b^j_i$ be the energy that was consumed by sensor $i$ during the $j$ th sub-interval, i.e., $b^j_i = \\rho _i \\cdot T_j$ .", "Observe that $I_j = (x^j,b^j)$ is a valid instance of RadSC, for which $\\rho ^j$ , where $\\rho ^j_i =\\rho _i$ for every sensor $i$ such that $x_i \\in x^j$ , is a solution that achieves a lifetime of exactly $T_j$ .", "Figure REF provides an illustration of this procedure.", "Figure: Cutting an optimal schedule into strips.", "Note that coverageoverlaps may occur in both the horizontal and vertical directions inthe optimal schedule, but only horizontally in a strip.We further modify the instance $I_j=(x^j,b^j)$ and the solution $\\rho ^j$ as follows: Starting with $i=1$ , remove sensor $i$ from the instance, if the interval $[0,1]$ is covered during $[\\omega _j,\\omega _{j+1}]$ without $i$ .", "Decrease the battery and the radius of the left-most sensor as much as possible, and also decrease the battery and the radius of the right-most sensor as much as possible.", "Observation 4 Let sensors 1 and $m$ be the leftmost and rightmost sensors in $x^j$ .", "Then, either $\\rho ^j_1 = x^j_1$ or the interval $[0,x^j_1+\\rho ^j_1)$ is only covered by sensor 1.", "Also, either $\\rho ^j_m = 1-x^j_m$ or the interval $(x^j_m-\\rho ^j_m,1]$ is only covered by sensor $m$ .", "For now, it is important to note only that $\\textsc {RR} (x^j,b^j) = \\sum _{x_i\\in x^j} \\frac{b^j_i}{r_i}$ is the RoundRobin lifetime of the $j$ th strip, which is a specific RadSC instance $I_j$ with the properties outlined above.", "Reduction to Set Radius Strip Cover with Uniform Batteries Given the RadSC instance $I_j = (x^j,b^j)$ and a solution $\\rho ^j$ , we construct an instance $I^{\\prime }_j = (y^j,\\mathbf {1})$ with unit size batteries and a RadSC solution $\\sigma ^j$ , such that the lifetime of $\\sigma ^j$ is $T_j$ .", "That is, the optimal lifetime of $I_j^{\\prime }$ is exactly the same as for $I_j$ , but it uses only unit batteries.", "Let $\\textsc {Opt} _0$ denote the optimal RadSC lifetime.", "We assume that $b^j_i\\in \\mathbb {N}$ and $b^j_i \\ge 3$ for every $i$ , since [(i)] $b^j_i \\in \\mathbb {Q}$ for every $i$ , $\\textsc {Opt} _0(x,\\beta b) = \\beta \\cdot \\textsc {Opt} _0(x,b)$ , and $\\textsc {RR} (x, \\beta b) = \\beta \\cdot \\textsc {RR} (x,b)$ .", "The instance $I^{\\prime }_j$ is constructed as follows.", "We replace each sensor $i$ such that $x^j_i \\in x^j$ with $b^j_i$ unit battery sensors whose average location is $x_i$ .", "These unit battery sensors are called the children of $i$ .", "To do this, we divide the interval $[x^j_i -\\rho ^j_i, x^j_i + \\rho ^j_i]$ into $b^j_i$ equal sub-intervals, and place a unit battery sensor in the middle of each sub-interval.", "Observe that child sensors may be placed outside $[0,1]$ , namely to the left of 0 or to the right of 1.", "The solution $\\sigma ^j$ is defined as follows.", "For any child $k$ of a sensor $i$ in $I_j$ , we set $\\sigma ^j_k = \\rho ^j_i/b^j_i$ .", "An example is shown in Figure REF .", "Figure: Reduction of a non-uniform battery strip I j I_j to a uniformbattery instance I j ' I_j^{\\prime }: At the top, I j =(1 4,19 24),(3,4)I_j = \\left( (\\frac{1}{4},\\frac{19}{24}), (3,4) \\right), while at the bottom, I j ' =(1 12,3 12,5 12,13 24,17 24,21 24,25 24),1I_j^{\\prime } = \\left((\\frac{1}{12}, \\frac{3}{12}, \\frac{5}{12}, \\frac{13}{24},\\frac{17}{24}, \\frac{21}{24}, \\frac{25}{24} ), \\mathbf {1}\\right).Lemma 5 The lifetime of $\\sigma ^j$ is $T_j$ .", "First, the $b^j_i$ children of a sensor $i$ in $I_j$ cover the interval $[x^j_i-\\rho ^j_i, x^j_i + \\rho ^j_i]$ .", "Also, a child $k$ of $i$ survives $1/\\sigma ^j_k = b^j_i/\\rho ^j_i = T_j$ time units.", "Next, we prove that the lower bound on the performance of RoundRobin may only decrease.", "Lemma 6 $\\textsc {RR}^{\\prime }(y^j,\\mathbf {1}) \\le \\textsc {RR}^{\\prime }(x^j,b^j)$ .", "Let $p^j$ be the RoundRobin radii of $y^j$ .", "Observe that if $x^j_i \\le \\frac{1}{2}$ , it follows that $\\sum _{k:k \\text{ correspond to } i} \\!\\!\\!\\!\\!\\!", "p^j_k= \\sum _{k:k \\text{ correspond to } i} \\!\\!\\!\\!\\!\\!", "\\max \\left\\lbrace y^j_k,1-y^j_k \\right\\rbrace \\ge \\sum _{k:k \\text{ correspond to } i} \\!\\!\\!\\!\\!\\!", "(1-y^j_k)~=~ b^j_i (1-x^j_i)~=~ b^j_i r^j_i~,$ and that if $x^j_i \\ge \\frac{1}{2}$ , we have that $\\sum _{k:k \\text{ correspond to } i} \\!\\!\\!\\!\\!\\!", "p^j_k= \\sum _{k:k \\text{ correspond to } i} \\!\\!\\!\\!\\!\\!", "\\max \\left\\lbrace y^j_k,1-y^j_k \\right\\rbrace \\ge \\sum _{k:k \\text{ correspond to } i} \\!\\!\\!\\!\\!\\!", "y^j_k~=~ b^j_i x^j_i~=~ b^j_i r^j_i~.$ Hence, $\\textsc {RR}^{\\prime }(y^j, \\mathbf {1})= \\frac{\\sum _i b^j_i}{\\overline{p^j}}= \\frac{B^j}{\\frac{1}{B^j} \\sum _{k} p_k^j}\\le \\frac{B^j}{\\frac{1}{B^j} \\sum _{i} b^j_i r^j_i}= \\frac{B^j}{\\overline{r^j}}= \\textsc {RR}^{\\prime }(x^j, b^j)~,$ and the lemma follows.", "Analysis of Round Robin for Unit Batteries For the remainder of this section, we assume that we are given a unit battery instance $x$ that corresponds to the $j$ th strip.", "(We drop the subscript $j$ and go back to $x$ for readability.)", "Recall that $x\\cap [0,1]$ is not necessarily equal to $x$ , since some children could have been created outside $[0,1]$ in the previous step.", "We show that $\\textsc {RR}^{\\prime }(x) \\ge \\frac{2}{3} \\textsc {Opt} _0(x)$ .", "Let $i_0 = \\min \\left\\lbrace i : x_i \\ge 0 \\right\\rbrace $ and let $i_1 = \\max \\left\\lbrace i : x_i\\le 1 \\right\\rbrace $ be the indices of the left-most and right-most sensors in $[0,1]$ , respectively.", "Lemma 7 $\\max _{i \\in \\lbrace i_0,\\ldots ,i_1-1\\rbrace } \\lbrace x_{i+1}-x_i\\rbrace = \\max _{i \\in \\lbrace 1,\\ldots ,n-1\\rbrace } \\lbrace x_{i+1}-x_i\\rbrace $ .", "By Observation REF either $\\rho _1 = x_1$ and hence none of its children are located to the left of 0, or the points to the left of $x_1+\\rho _1$ are only covered by sensor 1, which means that the gaps between 1's children to the left of zero also appears between its children within $[0,1]$ .", "(Recall that $b^j_i \\ge 3$ , for all $i$ .)", "The same argument can be used for the right-most sensor.", "As illustrated in Figure REF , we define $\\Delta _0 &\\stackrel{\\scriptscriptstyle \\triangle }{=}{\\left\\lbrace \\begin{array}{ll}x_{i_0}-x_{i_0-1} & i_0 > 1, -x_{i_0-1} < x_{i_0}, \\\\2x_{i_0} & \\text{otherwise},\\end{array}\\right.", "}$ $\\Delta _1 & \\stackrel{\\scriptscriptstyle \\triangle }{=}{\\left\\lbrace \\begin{array}{ll}x_{i_1+1}-x_{i_1} & i_1 < n, x_{i_1+1}-1 < 1-x_{i_1}, \\\\2(1-x_{i_1}) & \\text{otherwise,}\\end{array}\\right.", "}$ and $\\Delta \\stackrel{\\scriptscriptstyle \\triangle }{=}\\max \\left\\lbrace \\Delta _0, \\Delta _1, \\max _{i \\in \\lbrace i_0,\\ldots ,i_1-1\\rbrace } \\lbrace x_{i+1}-x_i\\rbrace \\right\\rbrace ~.$ Figure: Illustration of the gaps in a unit battery instance xx.Note that i 0 =2i_0 = 2 and i 1 =8i_1 = 8.", "Δ 0 =d 1 \\Delta _0 = d_1, since sensor 1is closer to 0 than sensor 2.", "Also, Δ 1 =2(1-x 8 )\\Delta _1 = 2(1-x_8).", "Hence,Δ=maxd 4 ,d 1 ,2(1-x 8 )\\Delta = \\max \\left\\lbrace d_4,d_1,2(1-x_8) \\right\\rbrace .We describe the optimal RadSC lifetime in terms of $\\Delta $ .", "Lemma 8 The optimum lifetime of $x$ is $\\frac{2}{\\Delta }$ .", "To verify that $2/\\Delta $ can be achieved, consider the solution in which $\\rho _i = \\Delta /2$ for all $i$ .", "Clearly, $[0,1]$ is covered, and all sensors die after $2/\\Delta $ time units.", "Now suppose that a solution $\\rho $ exists with lifetime strictly greater than $2/\\Delta $ .", "Hence $\\max _i \\lbrace \\rho _i\\rbrace < \\Delta /2$ .", "By definition, $\\Delta $ must equal $\\Delta _0$ , $\\Delta _1$ , or the maximum internal gap.", "If the latter, then there exists a point $u \\in [0,1]$ between the two sensors forming the maximum internal gap that is uncovered.", "On the other hand, if $\\Delta = \\Delta _0$ , then if $\\Delta _0 = 2x_{i_0}$ , 0 is uncovered, and otherwise, there is a point in $[0,x_{i_0}]$ that is uncovered.", "A similar argument holds if $\\Delta = \\Delta _1$ .", "In the next definition we transform $x$ into an instance $x^{\\prime }$ by pushing sensors away from $\\frac{1}{2}$ , so that each internal gap between sensors is of equal width.", "See Figure REF for an illustration.", "Definition 1 For a given instance $x$ , let $k$ be a sensor whose location is closest to $1/2$ .", "Then we define the stretched instance $x^{\\prime }$ of $x$ as follows: $x_i^{\\prime } ={\\left\\lbrace \\begin{array}{ll}(1-r_k) - (\\left\\lceil {n/2} \\right\\rceil -i) \\Delta & i \\le \\left\\lceil {n/2} \\right\\rceil , \\\\(1-r_k) + (i-\\left\\lceil {n/2} \\right\\rceil ) \\Delta & i > \\left\\lceil {n/2} \\right\\rceil .", "\\\\\\end{array}\\right.", "}$ Figure: Transformation of instance xx to stretched instance x ' x^{\\prime }.", "The sensor closest to 1 2\\frac{1}{2} (x 3 x_3) remains in place, while the other sensors are placed at increasing intervals of Δ\\Delta away from x 3 x_3.", "The RoundRobin lifetime of a sensor is shown as a continuous function of its location xx.Observation 9 Let $x^{\\prime }$ be a stretched instance of $x$ .", "Then $|\\left\\lbrace i : x^{\\prime }_i \\le \\frac{1}{2} \\right\\rbrace | = \\left\\lceil {\\frac{n}{2}} \\right\\rceil $ and $|\\left\\lbrace i : x^{\\prime }_i > \\frac{1}{2} \\right\\rbrace | =\\left\\lfloor {\\frac{n}{2}} \\right\\rfloor $ .", "Lemma 10 Let $x^{\\prime }$ be the stretched instance of $x$ .", "Then, $\\textsc {Opt} _0(x^{\\prime }) =\\textsc {Opt} _0(x)$ and $\\textsc {RR}^{\\prime }(x^{\\prime }) \\le \\textsc {RR}^{\\prime }(x)$ .", "First, by construction, the internal gaps in $x^{\\prime }$ are of length $\\Delta $ and $\\Delta ^{\\prime }_0,\\Delta ^{\\prime }_1 \\le \\Delta $ .", "Thus, by Lemma REF , $\\textsc {Opt} _0(x^{\\prime }) = \\textsc {Opt} _0(x)$ .", "By Lemma REF we know that the sensors moved away from $\\frac{1}{2}$ , hence $\\sum _i r^{\\prime }_i \\ge \\sum _i r_i$ and $\\textsc {RR}^{\\prime }(x^{\\prime }) \\le \\textsc {RR}^{\\prime }(x)$ .", "Now we are ready to bound $\\textsc {RR} (x)$ .", "Lemma 11 $\\textsc {RR}^{\\prime }(x) \\ge \\frac{2}{3} \\textsc {Opt} _0(x)$ , for every instance $I =(x,\\mathbf {1})$ of RadSC, where sensors may be located outside $[0,1]$ .", "By Lemma REF we may assume that the instance is stretched.", "First, suppose that $n$ is even.", "Since $x$ is a stretched instance, it must be the case that exactly half of the sensors lie to the left of $1/2$ , and exactly half lie to the right.", "Hence, $\\overline{r}\\stackrel{\\scriptscriptstyle \\triangle }{=}\\frac{1}{n} \\sum _{i=1}^n r_i& = \\frac{1}{n}\\left[ \\sum _{j=0}^{n/2-1} (r_{n/2} + j \\Delta ) +\\sum _{j=0}^{n/2-1} (r_{n/2+1} + j \\Delta )\\right] \\\\& = \\frac{1}{n}\\left[ \\frac{n}{2} \\cdot r_{n/2} + \\Delta \\binom{n/2}{2} +\\frac{n}{2} \\cdot r_{n/2+1} + \\Delta \\binom{n/2}{2}\\right] \\\\& = \\frac{r_{n/2} + r_{n/2+1}}{2} + \\frac{2 \\Delta }{n} \\binom{n/2}{2} \\\\& = \\frac{1 + \\Delta }{2} + \\frac{\\Delta (n-2)}{4} \\\\& = \\frac{1}{2} + \\frac{n\\Delta }{4}~,$ where we have used the fact that since the sequence is stretched $r_{n/2} + r_{n/2+1} = 1 + \\Delta $ .", "Furthermore, since $n \\Delta \\ge 1$ , it now follows that $\\frac{\\textsc {RR}^{\\prime }(x)}{\\textsc {Opt} _0(x)}= \\frac{n/\\overline{r}}{2/\\Delta }= \\frac{n \\Delta }{1 + n \\Delta /2}= \\frac{1}{\\frac{1}{n\\Delta } + \\frac{1}{2}}\\ge \\frac{2}{3} ~.$ If $n$ is odd, then w.l.o.g.", "there are $\\frac{n+1}{2}$ sensors to the left of $1/2$ , and $\\frac{n-1}{2}$ to the right.", "Then $\\overline{r}& = \\frac{1}{n}\\left[ \\sum _{j=0}^{(n-1)/2} (r_{(n+1)/2} + j \\Delta ) +\\sum _{j=0}^{(n-3)/2} (r_{(n+3)/2} + j \\Delta )\\right] \\\\& = \\frac{1}{n}\\left[ \\frac{n+1}{2} \\cdot r_{(n+1)/2} +\\Delta \\binom{(n+1)/2}{2} +\\frac{n-1}{2} \\cdot r_{(n+3)/2} +\\Delta \\binom{(n-1)/2}{2}\\right] \\\\& = \\frac{r_{(n+1)/2} + r_{(n+3)/2}}{2} +\\frac{r_{(n+1)/2} - r_{(n+3)/2}}{2n} +\\frac{\\Delta }{n} \\cdot \\frac{(n-1)^2}{4} \\\\& \\le \\frac{1+\\Delta }{2} + \\frac{\\Delta }{n} \\cdot \\frac{(n-1)^2}{4} \\\\& = \\frac{1}{2} + \\Delta \\frac{n^2 + 1}{4n}~.$ We have two cases.", "If $r_1 \\ge 1$ , then there are $n-1$ gaps of size $\\Delta $ , as well as one gap of size at most $\\Delta /2$ .", "Since the gaps cover the entire interval, we have that $(n-1)\\Delta + \\frac{\\Delta }{2} \\ge 1$ .", "It follows that $n\\Delta \\ge \\frac{2n}{2n-1}$ .", "Thus, we can demonstrate the same bound, since $\\frac{\\textsc {RR}^{\\prime }(x)}{\\textsc {Opt} _0(x)}= \\frac{n/\\overline{r}}{2/\\Delta }\\ge \\frac{n \\Delta }{1 + \\frac{(n^2+1) \\Delta }{2n}}= \\frac{1}{\\frac{1}{n\\Delta } + \\frac{1}{2} + \\frac{1}{2n^2}}\\ge \\frac{2n^2}{3n^2 -n + 1}> \\frac{2}{3}~.$ Finally, we consider the case where $r_1 < 1$ .", "For some $\\epsilon \\in (0,\\Delta /2]$ , we can set $r_{(n+1)/2} = \\frac{1}{2} + \\epsilon $ .", "Since sensors $(n+1)/2$ and $(n+3)/2$ are of distance $\\Delta $ from one another, it follows that $r_{\\frac{n+3}{2}} - r_{\\frac{n+1}{2}}= \\left( 1/2 + \\Delta - \\epsilon \\right) - \\left( 1/2 + \\epsilon \\right)= \\Delta - 2 \\epsilon ~.$ Moreover, we will show that $\\epsilon \\le \\Delta /4$ , and thus $r_{(n+3)/2} - r_{(n+1)/2} \\ge \\Delta /2$ .", "To see this, note first that it follows from the definition of a stretch sequence and the assumption that $r_1 < 1$ that $r_1 = r_{(n+1)/2} + \\Delta (n-1)/2$ and $r_2 = r_{(n+3)/2} - \\Delta (n-3)/2$ .", "Hence their difference is $\\textstyle r_1 - r_n= ( r_{\\frac{n+1}{2}} + \\frac{1}{2}\\Delta (n-1) ) -( r_{\\frac{n+3}{2}} + \\frac{1}{2}\\Delta (n-3) )= r_{\\frac{n+1}{2}} - r_{\\frac{n+3}{2}} + \\Delta = 2\\epsilon ~.$ However since $1-\\Delta /2 \\le r_n \\le r_1 < 1$ , it must be the case that $r_1-r_n \\le \\Delta /2$ , and this implies that $\\epsilon \\le \\Delta /4$ .", "Finally, a computation similar to the one above reveals that $\\overline{r}\\le \\frac{r_{\\frac{n+1}{2}} + r_{\\frac{n+3}{2}}}{2} +\\frac{r_{\\frac{n+1}{2}} - r_{\\frac{n+3}{2}}}{2n} +\\frac{\\Delta }{n} \\frac{(n-1)^2}{4}\\le \\frac{1+\\Delta }{2} - \\frac{\\Delta }{4n} +\\frac{\\Delta }{n} \\frac{(n-1)^2}{4}= \\frac{1}{2} + \\frac{n \\Delta }{4}~.$ As this is the same bound that we obtained in the even case, we similarly achieve the same $2/3$ bound.", "Putting It All Together It remains only to connect the pieces we have accumulated in the previous three sections.", "Lemma 12 $\\textsc {RR}^{\\prime }(x^j,b^j) \\ge \\frac{2}{3} T_j$ , for every strip $j$ .", "The result follows immediately from Lemmas REF , REF , and  REF .", "Our main result now follows from our construction.", "Theorem 2 RoundRobin is a $\\frac{3}{2}$ -approximation algorithm for OnceSC.", "First, observe that $\\sum _j \\textsc {RR} (x^j,b^j)= \\sum _j \\sum _{x_i \\in x^j} \\frac{b^j_i}{r_i}= \\sum _i \\frac{1}{r_i} \\sum _{j : x_i \\in x^j} b^j_i\\le \\sum _i \\frac{1}{r_i} b_i= \\textsc {RR} (x,b)~.$ By Lemmas REF and REF we have that $\\textsc {RR} (x,b)\\ge \\sum _j \\textsc {RR} (x^j,b^j)\\ge \\sum _j \\textsc {RR}^{\\prime }(x^j,b^j)\\ge \\sum _j \\frac{2}{3} T_j= \\frac{2}{3} T= \\frac{2}{3} \\textsc {Opt} (x,b)~,$ and we are done.", "Strip Cover Theorem REF readily extends to the Strip Cover problem.", "Theorem 3 RoundRobin is a $\\frac{3}{2}$ -approximation algorithm for Strip Cover.", "Duty Cycle Algorithms In this paper we analyzed the RoundRobin algorithm in which each sensor works alone.", "One may consider a more general version of this approach, where a schedule induces a partition of the sensors into sets, or shifts, and each shift works by itself.", "In RoundRobin each shift consists of one active sensor.", "We refer to such an algorithm as a duty cycle algorithm.", "In this section we show that, in the worst case, no duty cycle algorithm outperforms RoundRobin.", "More specifically, we show that the approximation ratio of any duty cycle algorithm is at least $\\frac{3}{2}$ .", "Lemma 13 The approximation ratio of any duty cycle algorithm is at least $\\frac{3}{2}$ for both OnceSC and Strip Cover.", "Consider an instance where $x = (\\frac{1}{4}, \\frac{3}{4},\\frac{3}{4})$ and $b = (2,1,1)$ .", "An optimal solution is obtained by assigning $\\rho _1 = \\rho _2 = \\rho _3 = \\frac{1}{4}$ , $\\tau _1 = \\tau _2 =0$ and $\\tau _3 = 4$ .", "That is, sensor 1 covers the interval $[0,0.5]$ for 8 time units, sensors 2 covers $[0.5,1]$ until time 4, and sensors 3 covers $[0.5,1]$ from time 4 to 8.", "This solution is optimal in that it achieves the maximum possible lifetime of $8 = 2 \\sum _i b_i$ .", "Figure: Best schedule vs. best duty cycle schedule.Here x=(1 4,3 4,3 4)x = (\\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}) and b=(2,1,1)b = (2,1,1).On the other hand, the best duty cycle algorithm is RoundRobin, which achieves a lifetime of $16/3$ time units.", "(The shifts $\\lbrace 1,2\\rbrace $ and $\\lbrace 3\\rbrace $ would also result in a lifetime of $16/3$ time units.)", "Both schedules are shown in Figure REF .", "Set Radius Strip Cover In this section we present an optimal $O(n^2 \\log n)$ -time algorithm for the RadSC problem.", "Recall that in RadSC we may only set the radii of the sensors since all the activation times must be set to 0.", "More specifically, we assign non-zero radii to a subset of the sensors which we call active, while the rest of the sensors get $\\rho _i=0$ and do not participate in the cover.", "Given an instance $(x,b)$ , a radial assignment $\\rho $ is called proper if the following conditions hold: Every sensor is either inactive, or exhausts its battery by time $T$ , where $T$ is the lifetime of $\\rho $ .", "That is, $\\rho _i\\in \\lbrace 0,b_i/T\\rbrace $ , No sensor's coverage is superfluous.", "That is, for every active sensor $i$ there is a point $u_i \\in [0,1]$ such that $u_i \\in [x_i-\\rho _i,x_i+\\rho _i]$ and $u_i \\notin [x_k-\\rho _k,x_k+\\rho _k]$ , for every active $k \\ne i$ .", "Lemma 14 There is a proper optimal assignment for every RadSC instance.", "Let $I=(x,b)$ be a RadSC instance, and let $\\rho $ be an optimal assignment for $I$ with lifetime $T$ .", "We first define the assignment $\\rho ^{\\prime } = b/T$ and show that it is feasible.", "Since $\\rho $ has lifetime $T$ , any point $u \\in [0,1]$ is covered by some sensor $i$ throughout the time interval $[0,T]$ .", "It follows that $\\rho _i \\le b_i/T =\\rho ^{\\prime }_i$ .", "Hence, $u \\in [x_i-\\rho ^{\\prime }_i,x_i+\\rho ^{\\prime }_i]$ , and thus $\\rho ^{\\prime }$ has lifetime $T$ .", "Next, we construct an assignment $\\rho ^{\\prime \\prime }$ .", "Initially, $\\rho ^{\\prime \\prime }=\\rho ^{\\prime }$ .", "Then starting with $i=1$ , we set $\\rho ^{\\prime \\prime }_i=0$ as long as $\\rho ^{\\prime \\prime }$ remains feasible.", "Clearly, $\\rho ^{\\prime \\prime }_i \\in \\lbrace 0,b_i/T\\rbrace $ .", "Furthermore, for every sensor $i$ there must be a point $u_i \\in [x_i-\\rho ^{\\prime \\prime }_i,x_i+\\rho ^{\\prime \\prime }_i]$ such that $u_i \\notin [x_k-\\rho ^{\\prime \\prime }_k,x_k+\\rho ^{\\prime \\prime }_k]$ , for every active $k \\ne i$ , since otherwise $i$ would have been deactivated.", "Hence, $\\rho ^{\\prime \\prime }$ is a proper assignment with lifetime $T$ , and is thus optimal.", "Given a proper optimal solution, we add two dummy sensors, denoted 0 and $n+1$ , with zero radii and zero batteries at 0 and at 1, respectively.", "The dummy sensors are considered active.", "We show that the optimal lifetime of a given instance is determined by at most two active sensors.", "Lemma 15 Let $T$ be the optimal lifetime of a given RadSC instance $I=(x,b)$ .", "There exist two sensors $i, k \\in \\lbrace 0,\\ldots ,n+1\\rbrace $ , where $i < k$ , such that $T = \\frac{b_k+b_i}{x_k - x_i}$ .", "Let $\\rho $ be the proper optimal assignment, whose existence is guaranteed by Lemma REF .", "We claim that there exist two neighboring active sensors $i$ and $k$ , where $i<k$ , such that $\\rho _i+ \\rho _k = x_k - x_i$ .", "The lemma follows, since $\\rho _i = b_i/T$ and $\\rho _k = b_k/T$ .", "Observe that if $\\rho _i + \\rho _k < x_k - x_i$ , for two neighboring active sensors $i$ and $k$ , then there is a point in the interval $(x_i,x_k)$ that is covered by neither $i$ and $k$ , but is covered by another sensor.", "This means that either $i$ or $k$ is redundant, in contradiction to $\\rho $ being proper.", "Hence, $\\rho _i + \\rho _k \\ge x_k- x_i$ , for every two neighboring active sensors $i$ and $k$ .", "Let $\\alpha = \\min \\left\\lbrace \\frac{\\rho _k + \\rho _i}{x_k - x_i} : i, k\\text{ are active} \\right\\rbrace $ .", "If $\\alpha = 1$ , then we are done.", "Otherwise, we define the assignment $\\rho ^{\\prime } = \\rho /\\alpha $ .", "$\\rho ^{\\prime }$ is feasible since $\\rho ^{\\prime }_i + \\rho ^{\\prime }_k = \\frac{1}{\\alpha } (\\rho _i + \\rho _k) \\ge x_k- x_i$ , for every two neighboring active sensors $i$ and $k$ .", "Furthermore, the lifetime of $\\rho ^{\\prime }$ is $\\alpha T$ , in contradiction to the optimality of $\\rho $ .", "Lemma REF implies that there are $O(n^2)$ possible lifetimes.", "This leads to an algorithm for solving RadSC.", "Theorem 4 There exists an $O(n^2 \\log n)$ -time algorithm for solving RadSC.", "First if $n = 1$ , then $\\rho _1 \\leftarrow r_1 \\stackrel{\\scriptscriptstyle \\triangle }{=}\\max (x_1,1-x_1)$ and we are done.", "Otherwise, let $T_{ik} \\leftarrow \\frac{b_k + b_i}{x_k-x_i}$ , for every $i, k\\in \\lbrace 0,\\ldots ,n+1\\rbrace $ such that $i<k$ .", "After sorting the set $\\left\\lbrace T_{ik} : i<k \\right\\rbrace $ , perform a binary search to find the largest potentially feasible lifetime.", "The feasibility of candidate $T_{ik}$ can be checked using the assignment $\\rho ^{ik}_\\ell \\leftarrow b_\\ell /T_{ik}$ , for every sensor $\\ell $ .", "There are $O(n^2)$ candidates, each takes $O(1)$ to compute, and sorting takes $O(n^2 \\log n)$ time.", "Checking the feasibility of a candidate takes $O(n)$ time, and thus the binary search takes $O(n\\log n)$ .", "Hence, the overall running time is $O(n^2 \\log n)$ .", "Discussion and Open Problems We have shown that RoundRobin, which is perhaps the simplest possible algorithm, has a tight approximation ratio of $\\frac{3}{2}$ for both OnceSC and Strip Cover.", "We have also shown that OnceSC is NP-hard, but it remains to be seen whether the same is true for Strip Cover.", "Future work may include finding algorithms with better approximation ratios for either problem.", "However, we have eliminated duty cycle algorithms as candidates.", "Observe that both OnceSC and TimeSC are NP-hard, while RadSC can be solved in polynomial time.", "This suggests that hardness comes from setting the activation times.", "We have assumed that the battery charges dissipate in direct inverse proportion to the assigned sensing radius (e.g.", "$\\tau =b/\\rho $ ).", "It is natural to suppose that an exponent could factor into this relationship, so that, say, the radius drains in quadratic inverse proportion to the sensing radius (e.g.", "$\\tau = b/\\rho ^2$ ).", "One could expand the scope of the problem to higher dimensions.", "Before moving both the sensor locations and the region being covered to the plane, one might consider moving one but not the other.", "This yields two different problems: 1) covering the line with sensors located in the plane; and 2) covering a region of the plane with sensors located on a line." ], [ "Duty Cycle Algorithms", "In this paper we analyzed the RoundRobin algorithm in which each sensor works alone.", "One may consider a more general version of this approach, where a schedule induces a partition of the sensors into sets, or shifts, and each shift works by itself.", "In RoundRobin each shift consists of one active sensor.", "We refer to such an algorithm as a duty cycle algorithm.", "In this section we show that, in the worst case, no duty cycle algorithm outperforms RoundRobin.", "More specifically, we show that the approximation ratio of any duty cycle algorithm is at least $\\frac{3}{2}$ .", "Lemma 13 The approximation ratio of any duty cycle algorithm is at least $\\frac{3}{2}$ for both OnceSC and Strip Cover.", "Consider an instance where $x = (\\frac{1}{4}, \\frac{3}{4},\\frac{3}{4})$ and $b = (2,1,1)$ .", "An optimal solution is obtained by assigning $\\rho _1 = \\rho _2 = \\rho _3 = \\frac{1}{4}$ , $\\tau _1 = \\tau _2 =0$ and $\\tau _3 = 4$ .", "That is, sensor 1 covers the interval $[0,0.5]$ for 8 time units, sensors 2 covers $[0.5,1]$ until time 4, and sensors 3 covers $[0.5,1]$ from time 4 to 8.", "This solution is optimal in that it achieves the maximum possible lifetime of $8 = 2 \\sum _i b_i$ .", "Figure: Best schedule vs. best duty cycle schedule.Here x=(1 4,3 4,3 4)x = (\\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}) and b=(2,1,1)b = (2,1,1).On the other hand, the best duty cycle algorithm is RoundRobin, which achieves a lifetime of $16/3$ time units.", "(The shifts $\\lbrace 1,2\\rbrace $ and $\\lbrace 3\\rbrace $ would also result in a lifetime of $16/3$ time units.)", "Both schedules are shown in Figure REF ." ], [ "Set Radius Strip Cover", "In this section we present an optimal $O(n^2 \\log n)$ -time algorithm for the RadSC problem.", "Recall that in RadSC we may only set the radii of the sensors since all the activation times must be set to 0.", "More specifically, we assign non-zero radii to a subset of the sensors which we call active, while the rest of the sensors get $\\rho _i=0$ and do not participate in the cover.", "Given an instance $(x,b)$ , a radial assignment $\\rho $ is called proper if the following conditions hold: Every sensor is either inactive, or exhausts its battery by time $T$ , where $T$ is the lifetime of $\\rho $ .", "That is, $\\rho _i\\in \\lbrace 0,b_i/T\\rbrace $ , No sensor's coverage is superfluous.", "That is, for every active sensor $i$ there is a point $u_i \\in [0,1]$ such that $u_i \\in [x_i-\\rho _i,x_i+\\rho _i]$ and $u_i \\notin [x_k-\\rho _k,x_k+\\rho _k]$ , for every active $k \\ne i$ .", "Lemma 14 There is a proper optimal assignment for every RadSC instance.", "Let $I=(x,b)$ be a RadSC instance, and let $\\rho $ be an optimal assignment for $I$ with lifetime $T$ .", "We first define the assignment $\\rho ^{\\prime } = b/T$ and show that it is feasible.", "Since $\\rho $ has lifetime $T$ , any point $u \\in [0,1]$ is covered by some sensor $i$ throughout the time interval $[0,T]$ .", "It follows that $\\rho _i \\le b_i/T =\\rho ^{\\prime }_i$ .", "Hence, $u \\in [x_i-\\rho ^{\\prime }_i,x_i+\\rho ^{\\prime }_i]$ , and thus $\\rho ^{\\prime }$ has lifetime $T$ .", "Next, we construct an assignment $\\rho ^{\\prime \\prime }$ .", "Initially, $\\rho ^{\\prime \\prime }=\\rho ^{\\prime }$ .", "Then starting with $i=1$ , we set $\\rho ^{\\prime \\prime }_i=0$ as long as $\\rho ^{\\prime \\prime }$ remains feasible.", "Clearly, $\\rho ^{\\prime \\prime }_i \\in \\lbrace 0,b_i/T\\rbrace $ .", "Furthermore, for every sensor $i$ there must be a point $u_i \\in [x_i-\\rho ^{\\prime \\prime }_i,x_i+\\rho ^{\\prime \\prime }_i]$ such that $u_i \\notin [x_k-\\rho ^{\\prime \\prime }_k,x_k+\\rho ^{\\prime \\prime }_k]$ , for every active $k \\ne i$ , since otherwise $i$ would have been deactivated.", "Hence, $\\rho ^{\\prime \\prime }$ is a proper assignment with lifetime $T$ , and is thus optimal.", "Given a proper optimal solution, we add two dummy sensors, denoted 0 and $n+1$ , with zero radii and zero batteries at 0 and at 1, respectively.", "The dummy sensors are considered active.", "We show that the optimal lifetime of a given instance is determined by at most two active sensors.", "Lemma 15 Let $T$ be the optimal lifetime of a given RadSC instance $I=(x,b)$ .", "There exist two sensors $i, k \\in \\lbrace 0,\\ldots ,n+1\\rbrace $ , where $i < k$ , such that $T = \\frac{b_k+b_i}{x_k - x_i}$ .", "Let $\\rho $ be the proper optimal assignment, whose existence is guaranteed by Lemma REF .", "We claim that there exist two neighboring active sensors $i$ and $k$ , where $i<k$ , such that $\\rho _i+ \\rho _k = x_k - x_i$ .", "The lemma follows, since $\\rho _i = b_i/T$ and $\\rho _k = b_k/T$ .", "Observe that if $\\rho _i + \\rho _k < x_k - x_i$ , for two neighboring active sensors $i$ and $k$ , then there is a point in the interval $(x_i,x_k)$ that is covered by neither $i$ and $k$ , but is covered by another sensor.", "This means that either $i$ or $k$ is redundant, in contradiction to $\\rho $ being proper.", "Hence, $\\rho _i + \\rho _k \\ge x_k- x_i$ , for every two neighboring active sensors $i$ and $k$ .", "Let $\\alpha = \\min \\left\\lbrace \\frac{\\rho _k + \\rho _i}{x_k - x_i} : i, k\\text{ are active} \\right\\rbrace $ .", "If $\\alpha = 1$ , then we are done.", "Otherwise, we define the assignment $\\rho ^{\\prime } = \\rho /\\alpha $ .", "$\\rho ^{\\prime }$ is feasible since $\\rho ^{\\prime }_i + \\rho ^{\\prime }_k = \\frac{1}{\\alpha } (\\rho _i + \\rho _k) \\ge x_k- x_i$ , for every two neighboring active sensors $i$ and $k$ .", "Furthermore, the lifetime of $\\rho ^{\\prime }$ is $\\alpha T$ , in contradiction to the optimality of $\\rho $ .", "Lemma REF implies that there are $O(n^2)$ possible lifetimes.", "This leads to an algorithm for solving RadSC.", "Theorem 4 There exists an $O(n^2 \\log n)$ -time algorithm for solving RadSC.", "First if $n = 1$ , then $\\rho _1 \\leftarrow r_1 \\stackrel{\\scriptscriptstyle \\triangle }{=}\\max (x_1,1-x_1)$ and we are done.", "Otherwise, let $T_{ik} \\leftarrow \\frac{b_k + b_i}{x_k-x_i}$ , for every $i, k\\in \\lbrace 0,\\ldots ,n+1\\rbrace $ such that $i<k$ .", "After sorting the set $\\left\\lbrace T_{ik} : i<k \\right\\rbrace $ , perform a binary search to find the largest potentially feasible lifetime.", "The feasibility of candidate $T_{ik}$ can be checked using the assignment $\\rho ^{ik}_\\ell \\leftarrow b_\\ell /T_{ik}$ , for every sensor $\\ell $ .", "There are $O(n^2)$ candidates, each takes $O(1)$ to compute, and sorting takes $O(n^2 \\log n)$ time.", "Checking the feasibility of a candidate takes $O(n)$ time, and thus the binary search takes $O(n\\log n)$ .", "Hence, the overall running time is $O(n^2 \\log n)$ .", "Discussion and Open Problems We have shown that RoundRobin, which is perhaps the simplest possible algorithm, has a tight approximation ratio of $\\frac{3}{2}$ for both OnceSC and Strip Cover.", "We have also shown that OnceSC is NP-hard, but it remains to be seen whether the same is true for Strip Cover.", "Future work may include finding algorithms with better approximation ratios for either problem.", "However, we have eliminated duty cycle algorithms as candidates.", "Observe that both OnceSC and TimeSC are NP-hard, while RadSC can be solved in polynomial time.", "This suggests that hardness comes from setting the activation times.", "We have assumed that the battery charges dissipate in direct inverse proportion to the assigned sensing radius (e.g.", "$\\tau =b/\\rho $ ).", "It is natural to suppose that an exponent could factor into this relationship, so that, say, the radius drains in quadratic inverse proportion to the sensing radius (e.g.", "$\\tau = b/\\rho ^2$ ).", "One could expand the scope of the problem to higher dimensions.", "Before moving both the sensor locations and the region being covered to the plane, one might consider moving one but not the other.", "This yields two different problems: 1) covering the line with sensors located in the plane; and 2) covering a region of the plane with sensors located on a line." ], [ "Discussion and Open Problems", "We have shown that RoundRobin, which is perhaps the simplest possible algorithm, has a tight approximation ratio of $\\frac{3}{2}$ for both OnceSC and Strip Cover.", "We have also shown that OnceSC is NP-hard, but it remains to be seen whether the same is true for Strip Cover.", "Future work may include finding algorithms with better approximation ratios for either problem.", "However, we have eliminated duty cycle algorithms as candidates.", "Observe that both OnceSC and TimeSC are NP-hard, while RadSC can be solved in polynomial time.", "This suggests that hardness comes from setting the activation times.", "We have assumed that the battery charges dissipate in direct inverse proportion to the assigned sensing radius (e.g.", "$\\tau =b/\\rho $ ).", "It is natural to suppose that an exponent could factor into this relationship, so that, say, the radius drains in quadratic inverse proportion to the sensing radius (e.g.", "$\\tau = b/\\rho ^2$ ).", "One could expand the scope of the problem to higher dimensions.", "Before moving both the sensor locations and the region being covered to the plane, one might consider moving one but not the other.", "This yields two different problems: 1) covering the line with sensors located in the plane; and 2) covering a region of the plane with sensors located on a line." ] ]

[ [ "The AMBRE Project: Stellar parameterisation of the ESO:FEROS archived\n spectra" ], [ "Abstract The AMBRE Project is a collaboration between the European Southern Observatory (ESO) and the Observatoire de la Cote d'Azur (OCA) that has been established in order to carry out the determination of stellar atmospheric parameters for the archived spectra of four ESO spectrographs.", "The analysis of the FEROS archived spectra for their stellar parameters (effective temperatures, surface gravities, global metallicities, alpha element to iron ratios and radial velocities) has been completed in the first phase of the AMBRE Project.", "From the complete ESO:FEROS archive dataset that was received, a total of 21551 scientific spectra have been identified, covering the period 2005 to 2010.", "These spectra correspond to ~6285 stars.", "The determination of the stellar parameters was carried out using the stellar parameterisation algorithm, MATISSE (MATrix Inversion for Spectral SynthEsis), which has been developed at OCA to be used in the analysis of large scale spectroscopic studies in galactic archaeology.", "An analysis pipeline has been constructed that integrates spectral reduction and radial velocity correction procedures with MATISSE in order to automatically determine the stellar parameters of the FEROS spectra.", "Stellar atmospheric parameters (Teff, log g, [M/H] and [alpha/Fe]) were determined for 6508 (30.2%) of the FEROS archived spectra (~3087 stars).", "Radial velocities were determined for 11963 (56%) of the archived spectra.", "2370 (11%) spectra could not be analysed within the pipeline.", "12673 spectra (58.8%) were analysed in the pipeline but their parameters were discarded based on quality criteria and error analysis determined within the automated process.", "The majority of these rejected spectra were found to have broad spectral features indicating that they may be hot and/or fast rotating stars, which are not considered within the adopted reference synthetic spectra grid of FGKM stars." ], [ "Introduction", "Astronomy has entered an era of large scale astronomical surveys, the scientific goals of which have the potential to considerably expand our understanding of the formation and evolution of the Universe.", "In particular current and future large scale spectroscopic surveys of the Milky Way will allow astronomers to trace in incredible detail the chemical and kinematic history of our Galaxy.", "These surveys are being undertaken over a range of resolutions.", "Low-resolution surveys (R = 2,000 to 8,000) are now widespread, for example RAVE [60] and SEGUE [64], while high-resolution surveys are a more recent endeavour such as APOGEE [44]) and the Gaia-ESO Survey (P.I.s: Gerry Gilmore & Sofia Randich).", "The science goals of these ground-based surveys will significantly contribute to the studies of galactic archaeology as well as provide complementary information to the upcoming astrometric survey, the European Space Agency (ESA) Gaia Mission.", "The ESA Gaia satellite will observe approximately a billion stars in the Milky Way for which distances will be determined to milliarcsecond accuracies.", "Its Radial Velocity Spectrometer (RVS) will observe spectra at medium-resolution (R$\\simeq $ 7000 - 11,500) which will be used to obtain radial velocities for all the targets, as well as to determine stellar atmospheric parameters and chemical abundances for some ten's of millions of stars.", "The multi-object instruments at high-resolution (R$\\sim $ 20,000; MIKE on Magellan and FLAMES on the Very Large Telescope), medium-resolution (R$\\sim $ 8,000; AAOmega on the Australian Astronomical Telescope (AAT)) and low-resolution (R$\\le $ 2,200; Fibre Multi Object Spectrograph (FMOS) on the Subaru Telescope) are also being used to address science goals specific to the field of galactic archaeology.", "One key instrument that is currently being built and has been designed primarily for galactic archaeology research is the High Efficiency and Resolution Multi-Element Spectrograph [7] on the AAT.", "This ongoing accumulation of large spectral datasets from surveys and individual spectrographs has compelled the development of automated stellar parameterisation algorithms that can reliably and effectively analyse every spectrum.", "The stellar parameterisation algorithm, MATISSE, has been developed at the Observatoire de la Côte d'Azur (OCA) [58] to be included in the automated pipeline for the analysis and parameterisation of the Gaia-RVS stellar spectra.", "MATISSE has also been developed as a standalone java application for use in a wide variety of projects [23], [34].", "The AMBRE Project team at OCA, which oversees the development of MATISSE, is connected to the Gaia Data Processing Consortium (DPAC) under the Generalized Stellar Parametrizer-spectroscopy (GSP-spec) Top Level Work Package which is overseen by Coordination Unit 8 (CU8).", "With such large datasets soon to be available a crucial aspect of the analysis is to be able to compare the results from the different surveys.", "To do, this a comprehensive set of standard objects is required that can be used to calibrate the datasets.", "Preparation for this is in the form of the development of spectral libraries.", "These are datasets of spectra with homogeneously determined characteristics, such as stellar parameters, which can be used as calibration stars for these surveys.", "In particular standard star lists are being developed for Gaia for both the radial velocity [12] and stellar parameter measurements [59].", "The work carried out for the AMBRE Project in effect converts the spectra in the European Southern Observatory (ESO) archive into a comprehensive spectral library of homogeneously determined stellar parameters.", "There are three primary objectives for the AMBRE Project: To rigorously test MATISSE on large spectral datasets over a range of wavelengths and resolutions, including those for the Gaia RVS, To provide ESO with a database of stellar temperatures, gravities, metallicities, alpha to iron ratios and radial velocities for the associated archived spectra that will then be made available to the international scientific community via the ESO Archive, To create a chemical map of the Galaxy from the combined ESO archived sample upon which stellar and galactic formation and evolution archaeological analysis can be carried out.", "The first phase of the AMBRE Project was the analysis of the FEROS archived spectra.", "From the complete ESO:FEROS archive dataset that was delivered to OCA, a total of 21551 scientific spectra have been identified, covering the period 2005 to 2010.", "These spectra correspond to 6285 different stars based on a coordinate matching calculation with a radius of 10”.", "The structure of this paper is as follows: Section  introduces the AMBRE Project; Section  introduces the MATISSE algorithm and the synthetic spectra grid; Section  describes the analysis pipeline that has been built around MATISSE for the analysis of the archived spectra; Section  discusses the internal errors that have been calculated for the pipeline; Section  discusses the external errors analysis based on a reference sample of stars; Section  presents the application of key rejection criteria to the spectral dataset; Section  presents the stellar parameter results for FEROS and finally Section  concludes the paper." ], [ "The AMBRE Project", "Under a contract with ESO the archived spectra of four ESO spectrographs are being analysed using the automated parameterisation programme MATISSE in a project overseen by the AMBRE Project team.", "Table REF lists the main characteristics of the four spectrographs: the Fiberfed Extended Range Optical Spectrograph (FEROS) [33]; the High Accuracy Radial velocity Planet Searcher (HARPS) [46]; the Ultraviolet and Visual Echelle Spectrograph [14]; and the Fibre Large Array Multi Element Spectrograph/GIRAFFE (FLAMES/GIRAFFE) [55].", "The atmospheric stellar parameters of effective temperature ($T_{\\textrm {eff}}$ ), surface gravity ($\\log g$ ), global metallicity ([M/H]), $\\alpha $ element to iron ratio ([$\\alpha $ /Fe]) and radial velocity (V$_{rad}$ ) will be derived for each of the archived stellar spectra.", "These will be delivered to ESO for inclusion in the ESO database and then made available to the astronomical community via the ESO archive.", "It is intended that the availability of stellar parameters for each spectra will encourage further use of the archived spectra.", "Previously unconsidered samples can be found through searches on the parameters, for examples extracting all the spectra in a particular metallicity range, or with very similar temperatures and gravities.", "Table: Details of the four ESO spectrographs and their publicly available archived spectra sample that are part of the AMBRE Project.The analysis of the archived spectra of these four spectrographs presents a unique opportunity to test the performance of MATISSE on large datasets of real spectra.", "In particular key instrument configurations of FLAMES/GIRAFFE cover the Gaia RVS wavelength domain and resolutions.", "Rigorous testing of MATISSE is necessary in order to optimise its performance in the Gaia-RVS analysis pipeline that is being compiled at the Centre National d’Etudes Spatiales (CNES).", "As such the AMBRE Project has been formally designated as a sub-work package under GSP-spec.", "The stars analysed by AMBRE will also be available for use as standard or calibration stars for the Gaia-ESO survey, and as secondary standards for the Gaia Mission.", "The first phase of the AMBRE Project, the analysis of the FEROS archived spectra, is now complete.", "FEROS is a state-of-the-art bench-mounted high resolution spectrograph that was built by a consortiumhttp://www.lsw.uni-heidelberg.de/projects/instrumentation/Feros/ of four astronomical institutes: Landessternwarte Heidelberg, Astronomical Observatory Copenhagen, Institut d'Astrophysique de Paris and Observatoire de Paris/Meudon.", "In 1998 FEROS was installed and commissioned on the ESO 1.52 m telescope at La Silla, Chile.", "In 2002 FEROS was moved to the MPG/ESO 2.2 m telescope where it permanently resideshttp://www.eso.org/sci/facilities/lasilla/instruments/feros/index.html.", "The high resolution and full coverage of the optical domain means that FEROS is suitable for a wide range of astronomical projects from radial velocities and variability studies to chemical abundances in stellar populations.", "In 2009-10 the FEROS spectra from October 2005 to December 2009 were reanalysed by the ESO archivists using an improved reduction pipeline.", "These reduced spectra were the sample that was delivered to OCA for analysis in AMBRE.", "Figure: Number of FEROS observations per year for analysis in AMBRE.", "The 2005 observations begin in October.", "The number of distinct objects observed and the number with repeated observations are as indicated in the key.Figure REF shows a histogram of the number of FEROS observations per year that were received by OCA, the number for 2005 representing only 3 months of observations.", "The number counts for the number of distinct objects that have been observed by FEROS in this timeframe and the number with repeated observations are also shown.", "The analysis procedure and results for these FEROS spectra are presented in this paper.", "This analysis has been the testbed for producing many of the tools that will also be used in the analysis of the UVES, HARPS and FLAMES/GIRAFFE archived spectra.", "These tools have been integrated into an analysis pipeline that feeds the processed spectra into MATISSE for derivation of the spectral parameters." ], [ "MATISSE & the synthetic spectra grid", "MATISSE (MATrix Inversion for Spectral SynthEsis) is an automated stellar parameterisation algorithm based on a local multi-linear regression method.", "It derives stellar parameters ($\\theta $ = $T_{\\textrm {eff}}$ , $\\log \\ g$ , [M/H], individual chemical abundances) by the projection of an input observed spectrum on a vector function $B_{\\theta }(\\lambda )$ , $\\hspace{0.0pt} \\hat{\\theta }_{i} = \\sum \\limits _{\\lambda } \\ B_{\\theta }(\\lambda ) \\cdot O_{i}(\\lambda )$ with $\\hat{\\theta }$ being the derived value.", "The $B_{\\theta }(\\lambda )$ vector function is an optimal linear combination of theoretical spectra, $S_{i}(\\lambda )$ , calculated from a synthetic spectra grid.", "$\\hspace{0.0pt} B_{\\theta _{i}}(\\lambda ) = \\sum \\limits _{j} \\ \\alpha _{ij} \\ S_{j}(\\lambda )$ Key features in the observed spectrum due to a particular $\\theta $ are reflected in the corresponding $B_{\\theta }(\\lambda )$ vector indicating the particular regions which are sensitive to $\\theta $ [58], [9]).", "MATISSE also generates synthetic spectra interpolated to each set of output stellar parameters.", "For each set of parameters, the synthetic spectrum on the grid, $S_{0}(\\lambda )$ , with the closest stellar parameters is identified along with the associated $B_{\\theta }(\\lambda )$ function.", "The $B_{\\theta }(\\lambda )$ functions located locally about this point are used to estimate the variations in the flux between the synthetic spectra at the corresponding grid points and the final interpolated synthetic spectrum corresponding to the required stellar parameters ($\\theta _{k}$ ) [34].", "This interpolated spectrum is then used to calculate a $\\chi ^2$ between the interpolated and input normalised spectrum.", "This provides a measure of the goodness of the fit of the derived stellar parameters to the observed spectrum.", "Stellar parameters, with $\\log \\chi ^2$ , and corresponding interpolated synthetic spectra are generated for every spectrum that is analysed in MATISSE." ], [ "Grid of synthetic spectra for the classification algorithm", "Within the AMBRE Project, the adopted procedure for the automatic classification of stellar spectra relies on a library of reference spectra in order to derive their atmospheric parameters and their chemical abundances.", "Due to the lack of a library of observed spectra that covers a large range of atmospheric parameters and chemical abundances over a very large spectral domain and resolution, the only solution was to compute large grids of synthetic spectra.", "For the AMBRE application, the computed grid of theoretical stellar spectra has to cover the whole optical spectral range at very high resolution in order to be used for the analysis of the majority of the spectroscopic data.", "For that reason, we have computed a synthetic spectra grid covering the wavelength range between 300 and 1 200 nm with a wavelength step of 0.001 nm (900,000 pixels in total) [13].", "Since this project is mostly devoted to the analysis of FGKM stars, this grid is based on the latest generation of MARCS model atmospheres presented in [30].", "An extension to hotter effective temperatures with Kurucz stellar atmosphere models [39] is planned for the near future.", "The considered parameters of the spectral grid are the effective temperature ($T_{\\textrm {eff}}$ in K), the stellar surface gravity ($\\log g$ in dex), the mean metallicity ([M/H] in dex) and the enrichment in $\\alpha $ -elements with respect to iron ([$\\alpha $ /Fe] in dex).", "The metallicity definition here is different to the classical use of [Fe/H] to denote the metallicity of a star.", "For [Fe/H] the metallicity is defined using the derived abundances from Fe lines only.", "MATISSE provides the opportunity to use all the available metal lines (those atoms heavier than He) to define a metallicity providing a global metallicity designated as [M/H].", "The opportunity also exists with MATISSE to derive a global $\\alpha $ element abundance using as many $\\alpha $ element spectral lines as possible, where we assumed that in the generation of the synthetic spectra the abundances of the different $\\alpha $ elements vary in lockstep.", "Throughout the AMBRE Project, the following chemical species are assumed to be $\\alpha $ -elements: O, Ne, Mg, Si, S, Ar, Ca and Ti, although for any selected wavelength region spectral features for all of these elements may not necessarily be present.", "From the selected MARCS model atmospheres, the synthetic spectra were computed with the turbospectrum code ([5], and further improvements by Plez) in plane-parallel and spherical geometry assuming hydrostatic and local thermodynamic equilibrium.", "Atomic lines have been recovered from the Vienna Atomic Line Database (in August 2009; [38]).", "The molecular line list has been provided by B. Plez.", "It includes transitions from ZrO, TiO, VO, OH, CN, C2, CH, SiH, CaH, MgH and FeH with their corresponding isotopic variations (see [30], for a list of references).", "This grid covers the following ranges of atmospheric parameters: $T_{\\textrm {eff}}$ between 2 500 K and 8 000 K, $\\log g$ from $-0.5$ to $+5.5$  dex, and [Fe/H] from $-5.0$ to $+1.0$  dex, although not all combinations of the parameters are available within the grid.", "The selected MARCS models have [$\\alpha $ /Fe]=$0.0$ for [M/H] $\\ge $ $0.0$ , [$\\alpha $ /Fe]=$+0.4$ for [M/H] $\\le $ $-1.0$ and, in between, [$\\alpha $ /Fe]=$-0.25$ x[M/H].", "For the spectra computation from each of these MARCS models, we considered an [$\\alpha $ /Fe] enrichment from $-0.4$ to $+0.4$  dex with respect to the canonical values that correspond to the original abundances of the MARCS models.", "The final AMBRE synthetic spectra grid consists of 16783 flux normalized spectra [13].", "The microturbulence ($\\xi $ ) is not a free parameter in this synthetic spectra grid.", "For the atmospheric models with high $\\log g$ ($+3.5 \\le \\log g \\le +5.5$ ) $\\xi $ is set at 1.0 kms$^{-1}$ .", "For low $\\log g$ ($\\log g < +3.0$ ) $\\xi $ is set at 2.0 kms$^{-1}$ , being typical values for dwarfs and giants respectively.", "These values reflect the MARCS model atmospheres configurations, and further details on the model selection is given in [13].", "Due to the microturbulence being hardwired into the synthetic grid it was not possible to carry out tests on the effects of variations in $\\xi $ on the stellar parameter determination across the whole synthetic spectra grid.", "However the effects of changes in $\\xi $ are most prominent for strong lines.", "By using the global metallicity [M/H] rather than [Fe/H] we expect the contribution from strong lines on the [M/H] to be negligible due to the significantly larger quantity of non-$\\xi $ sensitive small metallic lines.", "Section  describes the extensive testing of the pipeline that was carried out using observed spectra from which an external error on the resulting parameters is derived by comparison to literature values.", "To gauge the effect of microturbulence on the derived stellar parameters a key sample of 66 Main Sequence (MS) stars [8], and two key samples of 16 Red Giant Branch (RGB) stars [47], [31] were considered from within the test sample.", "The most noticeable effect within the MS sample was an underestimation of the $T_{\\textrm {eff}}$ for the hotter MS stars, for which the constant $\\xi $ of 1.0 kms$^{-1}$ underestimates the accepted value.", "Hence at $T_{\\textrm {eff}} \\sim 6500$  K, where $\\Delta \\xi \\sim +0.8$ kms$^{-1}$ the effect was $\\Delta T_{\\textrm {eff}} \\sim -100$  K. This is well within the derived $T_{\\textrm {eff}}$ external error of 120 K (see Section ).", "The most noticeably effect in the RGB sample was on the derivation of $\\log g$ .", "At the base of the RGB, where the constant $\\xi $ of 2.0 kms$^{-1}$ is an overestimation, the derived $\\log g$ was observed to be overestimated by $\\sim 0.3$  dex for $\\Delta \\xi \\sim +0.7$ kms$^{-1}$ .", "At the RGB tip, where $\\xi = 2.0$  kms$^{-1}$ is an underestimation, $\\log g$ was observed to be underestimated by $\\sim 0.3$  dex for $\\Delta \\xi \\sim -0.7$  kms$^{-1}$ .", "These variations are within the derived $\\log g$ external error of 0.37 dex.", "Hence based on these samples, variations of $\\xi $ from the assumed constant values do have an effect on the derived parameters that varies in magnitude with stellar evolutionary stage.", "However the external error derived from the entire reference sample takes account of them in a global sense (see Section )." ], [ "AMBRE:FEROS subgrid", "The FEROS spectra cover most of the optical wavelengths but only a subset of these wavelengths was required for the MATISSE analysis.", "In order to carry out the MATISSE training phase, where the $B_{\\theta }(\\lambda )$ vectors are generated, for the FEROS analysis the optimum resolution, wavelength regions, and sampling of the FEROS spectra were determined.", "The full AMBRE synthetic spectra grid was then adapted to the same specifications creating the AMBRE:FEROS subgrid that was used in the training phase to generate the AMBRE:FEROS $B_{\\theta }(\\lambda )$ vectors.", "In this analysis, only $B_{\\theta }(\\lambda )$ functions computed from the direct numerical inversion of the correlation matrix [58] were considered.", "These functions are thus not optimized for very low SNR spectra.", "However, due to the large number of spectral features available in the selected spectral domains, it was not necessary to use the approximated $B_{\\theta }(\\lambda )$ functions that are computed with the Landweber algorithm as in [34].", "As will be shown in Section  the estimated internal errors based on the AMBRE:FEROS grid are indeed already very small (see Figure REF )." ], [ "Selection of wavelength regions", "A detailed analysis of the full wavelength range of the FEROS spectra was undertaken in order to select wavelength regions that would provide the greatest amount of information for each of the derived parameters while also minimising the number of pixels in order to reduce computing time.", "Figure: As for Figure  but for the corresponding B θ (λ)B_{\\theta }(\\lambda ) vectors of the AMBRE:FEROS synthetic spectra grid.", "The y-axis is scaled down compared to Figure .FEROS disperses the spectra into 39 orders of varying wavelength interval from $\\sim $ 3500 Å to $\\sim $ 9200 Å at high resolution (R$\\sim $ 48,000).", "The optimum spectral regions for the determination of the stellar parameters were selected taking into account the need to avoid excessive computing time and to avoid low spectral information region.", "To this end, the signal-to-noise (SNR) per pixel (0.03 or 0.06 Å per pixel for FEROS) as a function of the wavelength was derived for each archived spectra by determining the SNR profile of each spectral order and thus providing a SNR profile over the entire spectral domain.", "Wavelengths regions with low SNR were rejected, typically the start and end of each order.", "The wavelengths regions affected by sky absorption and telluric features were also rejected, and also regions where continuum placement proved too difficult owing to wide spectral features in the region.", "Some key spectral features, such as H$_{\\alpha }$ and the Ca II H & K lines, were also discarded as these features were found to be poorly synthesised based on our current understanding of stellar atmospheres as well as being difficult to normalise automatically.", "The remaining spectral regions were then investigated for their intrinsic sensitivity to three of the four stellar parameters to be determined by MATISSE: $T_{\\textrm {eff}}$ ; $\\log g$ ; and [M/H].", "This was carried out using preliminary $B_{\\theta }(\\lambda )$ vectors that covered the full optical domain at very high resolution generated using a reduced synthetic grid.", "Due to the underlying equation for calculating the $B_{\\theta }(\\lambda )$ vectors, the high resolution $B_{\\theta }(\\lambda )$ vectors could not be simply degraded to lower resolutions to investigate the sensitivity, as this would not accurately reflect the information at that resolution.", "However the high resolution $B_{\\theta }(\\lambda )$ vectors are useful to give a general sense of the sensitivity.", "For stellar parameters at: $T_{\\textrm {eff}}$ =6000 K, $\\log g$ =4.0 dex, [M/H]=0.0 dex (Sun); $T_{\\textrm {eff}}$ =6000 K, $\\log g$ =2.0 dex, [M/H]=0.0 dex; $T_{\\textrm {eff}}$ =4500 K, $\\log g$ =2.0 dex, [M/H]=–0.5 dex (Arcturus); $T_{\\textrm {eff}}$ =4500 K, $\\log g$ =4.0 dex, [M/H]=–0.5 dex, the $B_{\\theta }(\\lambda )$ vectors at the same $T_{\\textrm {eff}}$ , $\\log g$ , and [M/H] have been analysed for their extrema distribution.", "This selection explores the sensitivity of both dwarfs and giants that are similar to the two standard stars, the Sun and Arcturus.", "The panels in Figure REF show how the spectral domain changes in sensitivity with the different stellar parameters.", "The standard deviation ($\\sigma $ ) for each $B_{\\theta }(\\lambda )$ vector was calculated as the spread of the pixel values in the $B_{\\theta }(\\lambda )$ vector.", "Then for each $B_{\\theta }(\\lambda )$ vector the pixels outside 2 $\\sigma $ were binned in $\\sim $ 200 Å bins over the wavelength range from $\\sim $ 4000 Å to $\\sim $ 9500 Å.", "Hence the bars in Figure REF show the number of pixels per bin which have a high sensitivity to the respective $\\theta $ .", "Clearly the bluer wavelengths show the greatest sensitivity to all three parameters, reflecting the greater quantity of spectral features in the blue.", "Table: FEROS échelle order, starting wavelength and finishing wavelength for each region used in the analysis of the FEROS spectra.This sensitivity to $\\theta $ for different ranges in $\\theta $ was used to select the wavelengths regions in the FEROS spectra to be used in the AMBRE analysis.", "As the greatest sensitivity was located towards the blue, regions were selected, where possible, to capture this sensitivity.", "The grey regions in Figure REF represent the final wavelength regions selected for AMBRE:FEROS which are listed in Table REF .", "The wavelengths include key spectral features such as the magnesium triplet at $\\sim $ 5160 Å, H$_{\\beta }$ (4861 Å), H$_{\\gamma }$ (4340 Å), H$_{\\delta }$ (4101 Å), a CH bandhead (4305 Å) and a CN bandhead (4142 Å) as well as the spectral features for many atomic elements.", "This wealth of information provided sufficient sensitivity over the range of required stellar parameters in anticipation of the range of spectral types that were potentially included within the FEROS archived sample.", "For confirmation of the $B_{\\theta }(\\lambda )$ sensitivity, Figure REF was generated by the same process as Figure REF but using the corresponding $B_{\\theta }(\\lambda )$ vectors for the final AMBRE:FEROS grid.", "Therefore Figure REF reflects the sensitivity of the $B_{\\theta }(\\lambda )$ vectors at the actual resolution of the AMBRE:FEROS analysis.", "Due to the reduced number of points in sampling and resolution the 2 $\\sigma $ limit is significantly reduced.", "While some regions show there are subtle differences in the sensitivity of some regions, overall the distribution of greater sensitivity in the blue is in good agreement with the sensitivity at high resolution." ], [ "Convolution & sampling of synthetic spectra grid", "At the high resolution of R$\\sim $ 48,000, the FEROS spectra have typical wavelength sampling ($\\Delta \\lambda $ ) of 0.03 Å (1$\\times $ 1 binning) or 0.06 Å (1$\\times $ 2 binning).", "The final selected wavelengths corresponded to a total of $\\sim $ 1500 Å and at this sampling this translated to a (maximum) total of $\\sim $ 50,000 pixels.", "Creating the FEROS synthetic spectra subgrid of $\\sim $ 16,000 spectra at this resolution and sampling would result in excessively large memory and computing requirements when creating the $B_{\\theta }(\\lambda )$ vectors in the training phase, and also when loading these vectors during the MATISSSE analysis.", "It was necessary to reduce the resolution and sampling to a more reasonable pixel total, but without sacrificing the key spectral information.", "The most reasonable number of pixels per spectra was determined to be $\\sim $ 15,000 in order to ease the use of the available computing power.", "By optimising the memory and spectral information requirements, the final specifications for the FEROS synthetic spectra subgrid was determined to be 11890 pixels with R$\\sim $ 15,000 at $\\lambda \\sim 4500$  Å with a sampling of 0.1 Å over $\\sim $ 1500 Å.", "Hence the full AMBRE synthetic spectra grid [13] was then sliced to the wavelengths specified in Table REF , convolved by a Gaussian kernel with constant $\\sigma $ and resampled to the optimised FEROS pixels.", "The training phase was then carried out whereby the $B_{\\theta }(\\lambda )$ vectors were generated for use in the MATISSE analysis of the FEROS archived spectra." ], [ "FEROS analysis pipeline", "A complex analysis pipeline (written in shell, Java and IDL) has been built to wavelength slice, radial velocity correct, normalise and convolve the FEROS spectra and then feed them into MATISSE for the determination of their stellar parameters.", "Figure REF shows a flowchart of the key stages in the analysis pipeline.", "The pipeline has been developed so that it can be easily adapted to the remaining three instrument datasets of ESO archived spectra.", "The following sections outline the procedures carried out for each of the key stages in the pipeline.", "Figure: The FEROS Analysis Pipeline: The key stages are displayed in order of analysis.", "Spectral Processing A and Spectral Processing B carry out testing of spectra quality and preliminary parameter determination, including calculation of the radial velocity and spectral FWHM.", "In Spectral Processing C robust iterative procedures are carried out resulting in the final stellar parameters and normalised spectra.", "See text for detailed explanation of each stage." ], [ "Spectral Processing A: Spectra processing for radial velocity determination", "Spectral Processing A (SPA) is the initial stage that prepares the FEROS spectra in order to determine the radial velocities.", "The procedures used here are also used in the later stages of the pipeline.", "Key flags were defined within the pipeline to be attached to any spectra that satisfied the conditions of the flag.", "The flags were for `Faulty Spectra' (e.g.", "missing spectral orders), `Extreme Emission Features' (spectral emission features in the blue with width greater than 50 Å), `Excessive Noise' (extremely noisy spectra with the number of negative flux pixels $> 25\\%$ ) and `Poor Normalisation' (spectra with a total flux above $1.2 > 10\\%$ ).", "A spectrum may satisfy multiple flags, particularly in the case of excessive noise which naturally led to poor normalisation.", "Such types of spectra were unlikely to produce valid results in the MATISSE analysis and as such were rejected from the full analysis.", "Flags were also implemented to identify `Large Emission Features' (width greater than 1.2 Å) and the presence of odd features that were potentially `Instrumental Relics'.", "However these flags alone were not enough to reject a spectrum from the full analysis.", "All of the quality flags were accounted for in the generation of the final dataset of results for ESO, which is explained in Section .", "The first reduction process extracts the required wavelengths (Table REF ) from the spectra and then normalises them.", "However no convolution is carried out so the original resolution is maintained.", "A hot star spectrum (almost featureless) in the FEROS sample was identified and used to determine a continuum profile for each wavelength region.", "This profile is divided out of the spectra removing the residual instrument profile and performing a first rough normalisation to place the continuum near unity.", "At this stage the spectra must necessarily be considered as unknown in stellar parameters and hence the normalisation process must be carried out assuming that key pre-selected continuum regions need to be normalised to unity in intensity.", "This assumption may not in fact correctly normalise each spectrum.", "However in these initial stages it is the best, albeit crude, estimate available.", "Up to three continuum regions for each wavelength region were pre-selected by an investigation of synthetic spectra and a sample of FEROS spectra covering a range of stellar parameters.", "For the normalisation, a simple linear profile is constructed for each wavelength region by determination of the mean signal in the respective continuum regions.", "This profile is divided out of the wavelength region and then the region undergoes cleaning for cosmic rays.", "The region is scanned for emission features greater than a factor of 1.5 above the noise on the continuum (unity).", "These features are tested for width and those less than 1.2 Å (40 pixels) are cleaned to unity.", "Features larger than 1.2 Å (40 pixels) in width are flagged as large emission features and are not cleaned.", "This limit of 1.2 Å (40 pixels) was determined by an examination of many examples of cosmic rays in the FEROS spectra to get a sense of the number of pixels over which they could be dispersed.", "Each wavelength region is treated individually for each spectrum, and the regions are then combined into one vector.", "This merged spectrum is tested for the goodness of the normalisation and flagged if the normalisation is poor.", "The resulting spectrum is used in the determination of the radial velocity.", "At this stage the original resolution is retained.", "None of the FEROS spectra were rejected at this stage in case later testing and processing recovered a previously rejected spectra (i.e.", "mis-calculated radial velocity is corrected)." ], [ "Radial velocity determination", "A completely automatic radial velocity (V$_{rad}$ ) programme has been established that can analyse spectra across a very wide range of stellar parameters and is based on a cross-correlation algorithm that compares the normalised observed spectrum to binary masks [48].", "To complement this programme a procedure was developed at OCA that computes a set of binary masks to match the observations in terms of wavelength range and resolution, and are used as input to the radial velocity programme.", "Masks covering the parameter space of the AMBRE:FEROS analysis were computed using synthetic spectra taken from the high resolution, full optical domain, AMBRE synthetic spectra grid (see Section REF ).", "Six masks for stars hotter that 8000 K were also computed using synthetic spectra from the POLLUX database [54].", "For the AMBRE:FEROS analysis 51 masks were computed at the stellar parameters given in Table REF .", "Also listed are 5 standard masks for dwarf stars that were supplied with the radial velocity programme (C. Melo, private communication) and used in the AMBRE:FEROS analysis.", "Table: List of the stellar parameters for the binary masks used in the AMBRE:FEROS radial velocity determination that were built from the full AMBRE synthetic grid (45 masks) and the POLLUX database (6 masks).", "Five standard masks for dwarf stars provided by C. Melo are also listed.For each spectrum a cross-correlation function (CCF) ranging from $-500$ to $+500$  kms$^{-1}$ was calculated for each of the 56 masks.", "The CCF was calculated in radial velocity steps ($\\Delta $ V$_{rad}$ ) based on the specified wavelength step ($\\Delta \\lambda $ ) of the spectrum.", "For spectra with $\\Delta \\lambda = 0.03 Å$ the CCF step was calculated to be $\\Delta $ V$_{rad} \\sim 1.8$  kms$^{-1}$ , and for $\\Delta \\lambda = 0.06 Å$ the CCF step was $\\Delta $ V$_{rad} \\sim 3.6$  kms$^{-1}$ .", "For each CCF two separate gaussian fits were made in order to determine the minimum of the profile, and hence the radial velocity.", "The first fit used the majority of the profile, while the second fit used a section of the profile centred near the mininum with a width equal to the full-width-at-half-maximum (FWHM) of the profile.", "Figure: CCFs produced for calculating radial velocities.", "a) Noisy CCF with an asymmetric profile.", "b) Core of profile showing skewed full profile gaussian fit (black).", "The secondary gaussian fit (red) provides a better estimate.", "c) CCF of the spectroscopic binary, HD135728, showing two profiles.Figure REF a. shows an example of a noisy CCF although the key profile is distinct.", "However, as seen in Figure REF b., the fit to the full profile (black) provides a minimum skewed away from the actual minimum due to asymmetries of the full profile.", "The use of the central core of the profile provides a more accurate fit (red).", "Figure REF c. shows an example of a CCF for which two profiles are observed.", "This is the spectroscopic binary HD135728.", "The pipeline selects the most prominent of the two profiles in the determination of the radial velocity.", "The CCF seems the most likely tool with which to be able to identify the spectra of spectroscopic binaries.", "However at this stage we have not developed such a routine and so binaries are not specifically detected in the pipeline.", "The stellar parameters for such spectra are likely to be poorly determined and so will be at least identified as having large associated errors from the radial velocity and stellar parameter analysis.", "Quality control flags, such as the $\\chi ^2$ , may also be used to help identify spectroscopic binaries.", "Figure REF shows the CCF corresponding to the best fit mask for three FEROS spectra at different stellar parameters.", "The full CCF for each spectrum is shown in the first row, while the second row shows more clearly the profile used to calculate the radial velocity.", "The full CCF can show varying degrees of noise and gradient depending on the quality of the spectra as is shown in each of these three examples.", "The respective profiles used to determine the radial velocity reflect the characteristics of the different spectra.", "The two cooler stars have very prominent inverted peaks, reflecting the numerous spectral lines available with which to make the cross-correlation.", "The hotter star, with fewer spectral lines, has a much less pronounced profile.", "These differences can be quantified by the measurement of the contrast of the profile.", "Figure: Examples of cross-correlation functions of the best fit masks for three of the FEROS spectra.", "The first row shows the full CCFs while the second row shows the profiles from which the radial velocities were determined.", "The final AMBRE:FEROS stellar parameters (T eff ,loggT_{\\textrm {eff}}, \\log \\ g, [M/H], [α\\alpha /Fe]) and radial velocities are stated for each spectrum.", "The red lines are used to calculate the contrast and FWHM for each profile.The contrast was defined here as the amplitude of the profile multiplied by 100 (hence a percentage) where the amplitude of the profile is the length of the vertical red line as shown in each example profile in Figure REF .", "The amplitude of the CCF, the continuum placement of the CCF and their associated errors are calculated automatically when using the GAUSSFIT routine in IDL.", "The sign of the amplitude does change depending on whether the CCF has an absorption or an emission profile.", "The expected profile in this analysis is an absorption profile.", "The two conditions that were required for a profile to have been well fitted by a gausssian were first: the contrast must be less than zero ensuring an absorption profile; and second, the error on the amplitude must be less than 20$\\%$ of the amplitude ensuring that the CCF profile dominates above the noise.", "CCFs that satisfied these conditions were considered to be well-defined.", "Finally the error on the radial velocity ($\\sigma _{Vrad}$ ) was calculated using the prescription outlined in [61].", "This prescription makes use of the relative heights of the primary and secondary peaks in the CCF profile hence $\\sigma _{Vrad}$ reflects the goodness of the definition of the primary peak.", "The radial velocity programme carried out the calculations for each of the 56 masks, resulting in 56 determinations of the radial velocity for each spectrum.", "The errors derived from the quality of the gaussian fit and $\\sigma _{Vrad}$ were used to select the best fit radial velocity.", "Typically the majority of these determinations were in good agreement at the correct radial velocity.", "However for masks with parameters far from the true parameters of the star, a radial velocity would be determined that was incorrect but sometimes had sufficiently small errors such that it would be incorrectly selected as the best fit, if the smallest error was the only selection criteria.", "To avoid this a binning procedure was implemented as follows that discarded such outliers.", "For a single spectrum, all of the V$_{rad}$ determinations with well-defined CCFs were binned by V$_{rad}$ into a maximum of 3 bins in correlation with the overall spread of the V$_{rad}$ values.", "The bin size was set in each instance by the difference in the maximum and minimum V$_{rad}$ values divided by the number of bins.", "The bin with the largest number of members was retained and the outlying bins were discarded.", "The V$_{rad}$ within accepted bin with the smallest $\\sigma _{Vrad}$ was selected as the final V$_{rad}$ .", "In the case of an equal number of members within 2 or more bins, the bin with the smallest spread in V$_{rad}$ values (so best agreement) was selected.", "Typically, most of the radial velocities for a spectrum were in good agreement, so the maximum bin was easily identified.", "Otherwise the radial velocities were highly dispersed, and this would be reflected in the $\\sigma _{Vrad}$ of the final selected value.", "The errors on the gaussian fit of the CCF and the $\\sigma _{Vrad}$ values were used later in the analysis pipeline to identify problematic spectra and select alternate radial velocities where necessary, and also as quality criteria for the final set of results to be delivered to ESO.", "The S$^4$ N [2] and [12] libraries provided good samples of stars with which to validate the radial velocities determined in the AMBRE:FEROS pipeline.", "The S$^4$ N library comprises of 118 F and G dwarf stars for which a detailed analysis was carried out using high resolution high SNR spectra.", "All four parameters ($T_{\\textrm {eff}}, \\log \\ g$ , [Fe/H] and [$\\alpha $ /Fe]) as well as radial velocities are available for each of the S$^4$ N stars.", "[12] is the preliminary list of 1420 candidate standard stars to be used to calibrate the radial velocities obtained with Gaia RVS.", "Section  gives further discussion on the use of these libraries to validate the AMBRE:FEROS results.", "Within the FEROS archived dataset 30 stars (338 spectra) were found that are also within the S$^4$ N library, and 183 stars (411 spectra) were found that are also within [12].", "Figure REF a and b compare the S$^4$ N and [12] radial velocities values with those determined for AMBRE:FEROS.", "The mean and standard deviation of the differences between the two sets for each sample are also shown.", "There is a small offset in both sets with the S$^4$ N having a greater offset of 1.8$\\pm $ 1.4 kms$^{-1}$ compared to [12] of 0.52$\\pm $ 0.35 kms$^{-1}$ .", "The FEROS spectra were sampled at either 0.03 Å or 0.06 Å from the ESO:FEROS reduction pipeline corresponding to expected V$_{rad}$ accuracies of 1.6 kms$^{-1}$ and 3.3 kms$^{-1}$ respectively.", "Hence these offsets indicate very good agreement for both samples, particularly for the [12] values.", "Figure: Comparison of AMBRE:FEROS radial velocities: a) S 4 ^4N library for 29 stars (338 spectra); and b) for 158 stars (318 spectra).Figure: Histogram of the radial velocity values calculated for all of the FEROS archived spectra defined as good spectra with well-defined CCFs: a) V rad _{rad}, and b) FWHM of the V rad _{rad} CCF.All of the FEROS spectra were analysed for their radial velocity.", "However for 4217 spectra ($19.6\\%$ ) poorly defined or non-standard cross-correlation functions lead to unreliable estimates of the radial velocity.", "This is most likely due to some peculiar or non-stellar properties of the spectra (i.e.", "nova).", "Of these 4217 spectra, 549 also failed the conditions of the rejection flags as defined in Section REF .", "Figure REF a is a histogram of the radial velocities calculated for the FEROS spectra defined as good quality with well-defined CCF (15513 spectra).", "The majority of the spectra have radial velocities between $-100$ and $+50$  kms$^{-1}$ .", "Figure REF b is a histogram of the FWHM of the CCF, calculated from the full CCF profile, for the same sample of spectra.", "It shows that the majority of spectra returned a FWHM of less than 50 kms$^{-1}$ .", "The effects of the CCF on the selection of the final dataset will be discussed in Section ." ], [ "Spectral Processing B: Spectra reduction, convolution and first parameter estimation", "Spectral Processing B (SPB) proceeds under two iterations through the normalisation procedure and the MATISSE analysis.", "These are necessary as key tests on the normalisation and radial velocity correction are carried out between the two iterations allowing alterations to be made that provide better first estimates of the stellar parameters for the remaining pipeline procedures.", "For both iterations the original archived spectra are re-analysed using the same procedures as outlined in SPA in terms of wavelength selection, cosmic ray cleaning and normalisation.", "However in both iterations of SPB the measured V$_{rad}$ correction is automatically applied to each spectrum prior to normalisation.", "This allows each spectrum to be shifted to the laboratory rest frame at which the synthetic grid, and so the $B_{\\theta }(\\lambda )$ vector functions, have been calculated.", "The AMBRE:FEROS synthetic grid was set to a resolution lower than the resolution of the observed FEROS spectra.", "In order to convolve the observed spectra to the appropriate resolution three factors needed to be taken into account.", "First, the observed spectra are observed with constant resolving power, which corresponds to increasing full-width-at-half-maximum (FWHM) of the spectral features with wavelength.", "Second, the synthetic spectra grid was convolved using a gaussian profile with constant FWHM hence the synthetic spectra have constant spectral FWHM with wavelength, not a constant resolution.", "Third, the construction of the full AMBRE synthetic spectra grid did not include astrophysical broadening such as those due to $Vsini$ and $\\xi $ [13].", "Hence a straight-forward convolution using the nominal instrument FWHM and the FWHM applied to obtain the AMBRE:FEROS synthetic spectra grid would result in convolved observed spectra that had increasing FWHM with wavelength in disagreement with the synthetic spectra.", "Also for stars with astrophysical broadening greater than the broadening due to the resolving power of the instrument the spectra would be too broad for the appropriate parameter range of synthetic spectra.", "Therefore a procedure was implemented that measured the spectral FWHM for each of the 17 selected wavelength regions (see Table REF ) for each spectrum and then applied an appropriate gaussian profile to smooth each wavelength region in order that the convolution resulted in observed spectra that had spectral FWHM as close as possible to the constant FWHM of the synthetic spectra grid.", "The FWHM values were measured as part of SPB and this was found to significantly increase the processing time of this stage of the pipeline.", "Hence, due to it being primarily a testing phase, for the first iteration of SPB the observed spectra are convolved using predetermined `mean' spectral FWHM values that were determined for each of the 17 wavelength regions.", "These default spectral FWHM values were determined by a statistical analysis of a subsample of 384 of the FEROS archived spectra, whereby the FWHM for as many weak and medium strength spectral lines as possible were measured in each of the sample spectra.", "The FWHM was measured by fitting a gaussian function to each spectral feature and calculating the width of the gaussian profile at the midpoint of the depth of the profile.", "For each spectrum the FWHM values increased with wavelength hence assuming one FWHM for the entire spectrum was not reasonable across the wavelength domain of the AMBRE:FEROS analysis.", "Hence a mean FWHM value for each of the 17 wavelength sections was determined and these were used in the convolution of the corresponding wavelength section.", "The convolution was carried out using a transformation gaussian profile for which the standard deviation ($\\sigma _X$ ) was defined as: $\\sigma _X &= \\sqrt{\\frac{FWHM_{grid}}{\\sigma _{FWHM}}^2 - \\frac{FWHM_{obs}}{\\sigma _{FWHM}}^2} \\\\\\sigma _{FWHM} &= \\sqrt{2.0 \\ln (2.0)}$ where FWHM$_{obs}$ is the default FWHM for the region, FWHM$_{grid}$ = 0.33 mÅ and $\\sigma _{FWHM}$ transforms the FWHMs to $\\sigma $ s. Therefore each region of the observed spectrum was convolved such that the FWHM of the convolved spectrum matched the FWHM of the synthetic spectrum.", "Part of the convolution process is to resample the observed spectra to the same wavelength bins as the FEROS synthetic spectra grid.", "Also for SPB, the same quality tests outlined in SPA are used but again no spectra were rejected based on these tests in this first iteration.", "The convolved FEROS spectra were then analysed using MATISSE to obtain the first estimate of the stellar atmospheric parameters ($T_{\\textrm {eff}}$ , $\\log g$ , [M/H] and [$\\alpha $ /Fe]).", "MATISSE also outputs synthetic spectra interpolated to the derived stellar parameters.", "Within the MATISSE algorithm the determination of the stellar parameters undergoes 10 iterations during which the algorithm converges on the final parameters for each spectrum.", "This number is set empirically and the majority of spectra converge to their parameters in much less than 10 iterations.", "After the MATISSE analysis is complete each spectrum is tested for whether convergence on the final $B_{\\theta }(\\lambda )$ function occurred within the 10 iterations and also the spectrum is tested as to the goodness of the fit between the normalised and the interpolated synthetic spectra in a $\\log (\\chi ^2)$ calculation.", "In the case of non-convergence within MATISSE, the parameters determined at the ninth and tenth iterations are compared and the solution with the lowest $\\log (\\chi ^2)$ is selected as the final set of parameters.", "If there is no convergence and/or a high $\\log (\\chi ^2)$ , the spectra are flagged for further investigation.", "The majority of these instances were attributable to incorrect selection of the radial velocity.", "A separate routine was developed that carries out an automated visual inspection of these spectra in order to select, where possible, an improved radial velocity correction from the full list of radial velocity masks.", "In some instances the normalisation of the spectra was poor due to noisy spectra, strange spectral features or an invalid radial velocity correction.", "In the latter case the adjustment to the radial velocity would correct the poor normalisation.", "In the former cases these spectra were captured by quality/rejection flags in the normalisation process or in the construction of the final ESO dataset.", "After these adjustments were made the spectra were reprocessed in the second iteration of SPB.", "This process is exactly the same as the first iteration except the mean spectral FWHM of the absorption lines for each wavelength section is now measured directly for each individual spectrum.", "This second iteration of SPB provides the first scientific estimate of the parameters, and hence convolving the spectrum with spectral FWHM values measured specifically for each spectrum provides a more robust analysis.", "The measured spectral FWHM values were used to convolve each region of each spectrum to match the FWHM of the AMBRE:FEROS synthetic spectra grid.", "Therefore in Equation REF the FWHM$_{obs}$ is the measured spectral FWHM for each wavelength section for each spectrum.", "Extensive testing was carried out which compared spectral lines that were in common between the observed and synthetic spectra both before and after convolution in order to ensure that the transformation calculation resulted in appropriately convolved observed spectra.", "This included testing the procedure on the original high resolution synthetic spectra to ensure the AMBRE:FEROS resolution was obtained after convolution by this procedure.", "The FWHM$_{grid}$ (0.33 mÅ) was calculated based on a sample of the AMBRE:FEROS synthetic spectra for which the FWHM values were determined by the same method as for the observed spectrum and were found to be constant with wavelength as expected due to how the synthetic spectra were generated.", "This ensured consistency in the calculation and application of the FWHM in the convolution process.", "Part of the calculation of the spectral FWHM was to classify each spectrum using the number of spectral lines and the widths that were measured.", "Three categories were used: `Weak' for less than 25 lines measured which were typically found to have FWHM $<0.11$ (FWHM$_{weak}$ ) and were generally noisy spectra; `Medium' for more than 25 lines measured, where the line depths of the measured lines were between 0.5 and 0.95 in normalised intensity, and no more than 1 strong line was identified (FWHM$_{medium}$ ); and `Strong' for more than 1 strong line identified where the line depth of the `Strong' line was between 0.35 and 0.5 of the normalised intensity (FWHM$_{strong}$ ).", "The spectral lines were identified by locating the minima and maxima in the spectra, fitting a gaussian at each minima and then discarding those lines which failed the quality tests of the gaussian fit.", "For a spectrum typically classified as FWHM$_{medium}$ 1000 to 2000 lines were identified which were reduced to $\\sim $ 400 lines with good gaussian fits.", "The `Strong' classification was an attempt to identify the spectra with large, broad spectral features as this provided another estimate of the spectral classification of the star.", "This was accomplished by applying a high FWHM smoothing function to each spectrum and then testing for which spectral features still remained.", "A spectrum without `Strong' lines would be reduced to a line near the continuum, while `Strong' features would otherwise still be prominent.", "This method had to be tailored to the FEROS wavelength region in order to place relevant limits on the possible number of `Strong' lines and their location within the wavelength regions.", "If, by this method, more than 1 `Strong' line were found then the spectrum was classifed as FWHM$_{strong}$ and the mean spectral FWHM of the `Strong' lines was recorded.", "In the cases where the spectrum was designated as FWHM$_{weak}$ or FWHM$_{strong}$ , then default spectral FWHM values were used instead.", "The spectral FWHM values for each spectrum that were used in the convolution, measured or mean values, were saved to an external file to be used in the next stage of the pipeline.", "Extensive testing was carried out using corresponding spectral lines in the original, convolved and synthetic spectra to confirm that the spectral FWHM procedure correctly convolved the archived spectra to the resolution of the synthetic grid across the wavelength range.", "At this stage the spectra which failed the tests identifying un-analysable spectra (extreme emission features, extreme noise-dominated spectra etc) were rejected from the analysis process.", "This reduced list was analysed once more in MATISSE in order to determine the first estimate of the stellar parameters to be used in the next stage of the analysis pipeline.", "The convergence and $\\log \\chi ^2$ were tested again to catch any further mis-identified radial velocity corrections.", "At the end of the second iteration of SPB 2370 FEROS spectra were rejected, 11% of the total number of FEROS archived spectra.", "Figure: Normalisation procedure for a two stars with T eff ∼4600T_{\\textrm {eff}} \\sim 4600 K (LEFT) and T eff ∼5300T_{\\textrm {eff}} \\sim 5300 K (RIGHT) shown by the wavelength region about two lines of the magnesium triplet.", "TOP: the second iteration of SPB, MIDDLE: first iteration of SPC, and BOTTOM: final solution from SPC.", "The stellar parameters (T eff T_{\\textrm {eff}}, logg\\log g, [M/H] and [α\\alpha /Fe]) and goodness of fit over all the AMBRE:FEROS wavelengths (logχ 2 \\log \\chi ^2) are shown.", "The synthetic spectrum interpolated to these parameters is shown in red, while the normalised observed spectrum is shown in black.", "The difference between observed and synthetic spectra is also shown at 0.0." ], [ "Spectral Processing C: Iterative spectra normalisation & parameterisation", "The final number of FEROS spectra that were passed to the final stage of the analysis pipeline, Spectral Processing C (SPC), was 19181.", "SPC again uses the same procedures as SPA and SPB with two differences.", "The first difference is that the spectra are convolved using the spectral FWHM values determined in the second iteration of SPB rather than re-measuring the FWHM.", "The second difference is the use of a more robust method for the normalisation of the spectra.", "Previously, in SPA and SPB, a `rough' normalisation process was carried out that used pre-selected continuum regions to normalise the spectra to unity.", "But this is an invalid assumption for many stars, in particular for cool stars which have depressed continuum regions due to molecular bandheads that should not be normalised to unity.", "However at the beginning of SPC we have an estimate of the stellar parameters from the second iteration of SPB that gives us some valid information about each spectra that we did not previously possess.", "We take advantage of this information by using the interpolated synthetic spectra generated for each set of stellar parameters as a better estimate of the continuum placement for the normalisation of the observed spectra.", "In an iterative process that discards absorption and emission features by polynomial fitting and sigma clipping to leave only `continuum' regions, the synthetic spectrum is divided out of the corresponding observed spectrum leaving a residual of continuum points.", "A continuum profile is fitted to this residual which is then divided out of the observed spectrum, normalising it to a pseudo-continuum that better represents the parameters of the star.", "This process overrides any residual curvature that remains after, or was introduced by, the normalisation to the hot star continuum.", "These pseudo-continuum normalised spectra are reanalysed in MATISSE to obtain a second estimate of the stellar parameters and another set of synthetic spectra.", "The `normalisation treatment & MATISSE analysis' cycle is repeated 9 times, which had been determined to be a sufficient number of iterations within which the normalised spectra and stellar parameters could converge to their optimum state.", "This typically occurred within five iterations.", "This approach provides a robust incremental adjustment of the parameters and normalisation process that hones in on a realistic estimate of the stellar parameters of each spectra, a process which is not possible using the normalisation method of SPA and SPB.", "Figure REF shows the normalisation procedure for two FEROS spectra in the wavelength region about two of the magnesium triplet at 5160Å as an example of depressed continuum regions.", "The two stars correspond to temperatures of $T_{\\textrm {eff}} \\sim 4600$  K (SNR = 115) and $T_{\\textrm {eff}} \\sim 5300$  K (SNR = 204) respectively.", "For each spectrum the normalisation by SPB, the first iteration of SPC and the final solution that was found in SPC are shown.", "The stellar parameters and $\\log \\chi ^2$ (which is calculated over all the AMBRE:FEROS wavelengths) determined at each stage are shown as well as the synthetic spectrum interpolated to the stellar parameters.", "The difference between the observed and synthetic spectra is also included to show how it converges.", "For the cooler star this rough normalisation of SPB is not a good fit as the observed spectrum is set too high compared to the synthetic spectrum generated at the corresponding stellar parameters.", "This shows how inadequate normalisation to unity is for cool star spectra.", "However the rough normalisation does provides a reasonable fit to the synthetic spectrum for the hot star although the placement of the continuum is still not ideal.", "At the initial iteration of SPC the spectrum undergoes its first normalisation to the previous solution synthetic spectrum.", "For the cooler star the observed spectrum now sits below unity in better agreement with the new synthetic spectrum, but there are still mismatches in the relative placement of the two spectra and the line depths.", "For the hotter star there is very good agreement between the observed and synthetic spectra.", "In both cases the improvement in the match are shown by the lower $\\log \\chi ^2$ values.", "The final solution for the cooler star ($i=6$ ) shows excellent agreement between the observed and the synthetic spectra, which is reflected in the much lower $\\log \\chi ^2$ value.", "Indeed the visual inspection of the spectra show that they are in good agreement in this section of wavelength.", "For the hotter star the solution ($i=7$ ) is very close to the solution determined in the first iteration of SPC, indeed the $\\log \\chi ^2$ does not change.", "Visually this good fit is obvious between the observed and synthetic spectra.", "The comparison of these two stars illustrates the greater difficulty there is in the normalisation and parameterisation of cool stars.", "However the procedure developed for the AMBRE pipeline can successfully manage both cases and the normalisation procedure in SPC is particularly effective in the case of cool stars where the continuum regions are more heavily obscured by spectral features.", "Typically convergence occured by the fourth or fifth iteration of the normalisation and stellar parameter determination cycle.", "When convergence has occurred the final set of parameters are selected as those from the converged solutions with the lowest $\\log (\\chi ^2)$ .", "In the case of non-convergence within the 9 iterations of SPC, it is assumed that for the final six iterations any bias introduced by the initial rough normalisation has been erased.", "Of these final six iterations the solution with the lowest $\\log (\\chi ^2)$ is selected as the final stellar parameters for the spectra along with the corresponding normalised and synthetic spectra.", "Hence the radial velocity, stellar parameters ($T_{\\textrm {eff}}$ , $\\log g$ , [M/H] and [$\\alpha $ /Fe]), normalised spectra and corresponding inteprolated synthetic spectra were determined in SPC for 19181 of the 21551 FEROS spectra.", "Extensive testing of the FEROS analysis pipeline was carried out in order to optimise the procedures and to validate the MATISSE results by making comparison to literature values.", "The following sections describe the different tests and validations that were carried out." ], [ "AMBRE:FEROS internal error analysis", "The internal errors can be used to determine how well MATISSE will derive the stellar parameters of similar stellar types which have different levels of noise and different uncertainties in radial velocity.", "In order to test for noise and radial velocity effects a sample of 500 synthetic spectra were generated at random stellar parameters covering the entire stellar parameter range, spectral resolution and wavelength domains of the AMBRE:FEROS synthetic spectra grid." ], [ "Internal errors: SNR", "To test the effects of noise the sample of synthetic spectra was reproduced five times by adding differing levels of noise per pixel (0.1 Å per pixel for AMBRE:FEROS synthetic grid) at SNR = 10, 20, 50, 100.", "Each sample was then re-analysed in MATISSE and the derived stellar parameters were compared to the original values.", "Figure: Internal error for each parameter with changes in SNR.", "a) The ΔT eff \\Delta T_{\\textrm {eff}} that 70% of the synthetic sample were less than or equal to when calculating the difference between the nominal T eff T_{\\textrm {eff}} and the T eff T_{\\textrm {eff}} determined for the respective SNR.", "b) As for a) but for logg\\log \\ g. c) As for a) but for [M/H].", "d) As for a) but for [α\\alpha /Fe].Figure: Internal error for each parameter with changes in the V rad _{rad} uncertainty (kms -1 ^{-1}).", "a) The ΔT eff \\Delta T_{\\textrm {eff}} that 70% of the synthetic sample were less than or equal to when calculating the difference between the nominal T eff T_{\\textrm {eff}} and the T eff T_{\\textrm {eff}} determined for the respective V rad _{rad} uncertainty.", "b) As for a) but for logg\\log \\ g. c) As for a) but for [M/H].", "d) As for a) but for [α\\alpha /Fe].Figures REF a. to d. show how the difference in stellar parameters changes with increasing SNR for $T_{\\textrm {eff}}$ , $\\log g$ , [M/H] and [$\\alpha $ /Fe] respectively.", "For each noise-added sample the difference between the original and derived parameters was calculated and then the difference value at the 70th percentile ($\\sim 1 \\sigma $ ) was determined.", "For example, in Figure REF a. at a SNR of 10 (the noisiest), 70% of the synthetic spectra returned $T_{\\textrm {eff}}$ values within 14 K of the original $T_{\\textrm {eff}}$ value.", "At a SNR of 100, 70% of the sample returned values within 2 K of the original $T_{\\textrm {eff}}$ value.", "Similarly small errors were also found for $\\log g$ , [M/H] and [$\\alpha $ /Fe] that diminished to negligible levels at SNR of 100 in all cases.", "This analysis shows that for two synthetic spectra with very similar intrinsic stellar parameters, where the differences in the spectra are only attributable to noise, MATISSE will determine close to the same stellar parameters indicating that the internal errors of MATISSE are very small.", "This can be attributed to the large wavelength coverage and high resolution of the FEROS spectra which contain a great deal of information on the stellar parameters in terms of spectral lines, even at low SNR, of which MATISSE takes advantage.", "Overall this analysis shows that the internal error due to SNR is negligible in the FEROS results.", "The internal ($int$ ) error on each parameter due to SNR ($\\sigma _{int,snr}$ ) is calculated as part of the MATISSSE program.", "It was used to calculate the total internal error as explained at the end of this section." ], [ "Internal errors: V$_{rad}$ & {{formula:f0c0225a-19eb-45e7-a80b-4d0c5bf6560e}}", "A similar investigation was carried out in order to determine the effect of uncertainties in radial velocity on the determination of the stellar parameters by MATISSE.", "The sample of interpolated spectra were reproduced this time undergoing radial velocity shifts of $\\Delta $ V$_{rad}$ = 0, 0.5, 1.0, 5 and 10 kms$^{-1}$ with an SNR of 100.", "Figures REF a to d show how the difference in stellar parameters changes with increasing radial velocity uncertainty for $T_{\\textrm {eff}}$ , $\\log g$ , [M/H] and [$\\alpha $ /Fe] respectively.", "Again the difference between the original and derived stellar parameters was obtained and the difference value at the 70th percentile for each sample was calculated.", "At a radial velocity uncertainty of 5 kms$^{-1}$ ($\\sim $  0.09 pixels) the error in stellar parameter determination becomes significant ($>$ 1$\\%$ ) for each of the stellar parameters.", "However if the uncertainty in the radial velocity is less than 5 kms$^{-1}$ the MATISSE parameters compare well to the true values.", "The internal uncertainty in each stellar parameter due to radial velocity was calculated using the equations that connect the points in Figure REF .", "The error on the radial velocity, $\\sigma (\\textrm {V}_{rad})$ , determined in the radial velocity programme was used as the input to determine the internal error due to the V$_{rad}$ uncertainty ($\\sigma _{int,\\textrm {V}_{rad}}$ ) for each parameter for each spectrum according to the following conditions: ${\\footnotesize \\sigma (T_{eff})_{int,\\textrm {V}_{rad}}={\\left\\lbrace \\begin{array}{ll} 5.2 & \\text{if $\\sigma (\\textrm {V}_{rad}) < 0.5$,}\\\\9.60 \\sigma (\\textrm {V}_{rad}) + 0.40 &\\text{if $0.5 \\le \\sigma (\\textrm {V}_{rad}) < 1.0 $,}\\\\10.05 \\sigma (\\textrm {V}_{rad}) - 0.05 &\\text{if $1.0 \\le \\sigma (\\textrm {V}_{rad}) < 5.0$,}\\\\13.24 \\sigma (\\textrm {V}_{rad}) - 16.0 &\\text{if $5.0 \\le \\sigma (\\textrm {V}_{rad}) \\le 10.0$.}\\end{array}\\right.}", "}$ ${\\footnotesize \\sigma (\\log \\ g)_{int,\\textrm {V}_{rad}}={\\left\\lbrace \\begin{array}{ll} 0.015 & \\text{if $\\sigma (\\textrm {V}_{rad}) < 0.5$,}\\\\0.028 \\sigma (\\textrm {V}_{rad}) + 0.001 &\\text{if $0.5 \\le \\sigma (\\textrm {V}_{rad}) < 1.0 $,}\\\\0.0287 \\sigma (\\textrm {V}_{rad}) + 0.00025 &\\text{if $1.0 \\le \\sigma (\\textrm {V}_{rad}) < 5.0$,}\\\\0.0558 \\sigma (\\textrm {V}_{rad}) + 0.135 &\\text{if $5.0 \\le \\sigma (\\textrm {V}_{rad}) \\le 10.0$.}\\end{array}\\right.}", "}$ ${\\footnotesize \\sigma ([\\text{M/H}])_{int,\\textrm {V}_{rad}}={\\left\\lbrace \\begin{array}{ll} 0.008 & \\text{if $\\sigma (\\textrm {V}_{rad}) < 0.5$,}\\\\0.012 \\sigma (\\textrm {V}_{rad}) + 0.002 &\\text{if $0.5 \\le \\sigma (\\textrm {V}_{rad}) < 1.0 $,}\\\\0.0213 \\sigma (\\textrm {V}_{rad}) - 0.0073 &\\text{if $1.0 \\le \\sigma (\\textrm {V}_{rad}) < 5.0$,}\\\\0.0510 \\sigma (\\textrm {V}_{rad}) - 0.156 &\\text{if $5.0 \\le \\sigma (\\textrm {V}_{rad}) \\le 10.0$.}\\end{array}\\right.}", "}$ ${\\footnotesize \\sigma ([\\alpha \\text{/Fe}])_{int,\\textrm {V}_{rad}}={\\left\\lbrace \\begin{array}{ll} 0.006 & \\text{if $\\sigma (\\textrm {V}_{rad}) < 0.5$,}\\\\0.010 \\sigma (\\textrm {V}_{rad}) + 0.001 &\\text{if $0.5 \\le \\sigma (\\textrm {V}_{rad}) < 1.0 $,}\\\\0.0128 \\sigma (\\textrm {V}_{rad}) - 0.0018 &\\text{if $1.0 \\le \\sigma (\\textrm {V}_{rad}) < 5.0$,}\\\\0.0204 \\sigma (\\textrm {V}_{rad}) - 0.0400 &\\text{if $5.0 \\le \\sigma (\\textrm {V}_{rad}) \\le 10.0$.}\\end{array}\\right.}", "}\\vspace{14.22636pt}$ The rotational velocity ($Vsini$ ) of a star can have an impact the broadening of the observed spectral features depending on the magnitude of the rotational velocity and the resolution of the instrument.", "The AMBRE synthetic spectra grid was generated with no variations in $Vsini$ , assuming all stars to be slow rotators.", "However it is important to consider whether variations in $Vsini$ will have an effect on the determined stellar parameters.", "[23] carried out a robust investigation into the $Vsini$ limits of reliable stellar parameter determination using MATISSE and found that spectra with $Vsini < 11$  kms$^{-1}$ produced good results in the MATISSE analysis of a sample of FGK dwarfs.", "Based on this, [23] accepted all CCF FWHM $\\le 20$  kms$^{-1}$ for spectra at a resolution of R$\\sim $ 26,000.", "The FEROS sample and the AMBRE synthetic spectra grid are of much broader range in stellar parameters and wavelength, and of lower resolution (R$\\sim $ 15,000) than the [23] study.", "Hence the effects of $Vsini$ on the range of stars for which the synthetic grid has been optimised, would be well-masked for the AMBRE:FEROS configuration.", "In Section REF we examine the FEROS sample in the context of the CCF FWHM to identify spectra which are not well-represented by the AMBRE synthetic spectra grid." ], [ "Internal errors: Normalisation", "The effects of normalisation on the MATISSE determination were measured using the iterative normalisation process (SPC) of the FEROS analysis pipeline.", "Using a test sample of 384 FEROS spectra ($\\langle $ SNR$\\rangle $ =100$\\pm $ 45) the changes in normalisation between iterations of SPC was explored.", "Figure REF shows the progression of the goodness of fit between the reconstructed spectra and the normalised spectra at each iteration of SPC as a Box & Whisker graph.", "Assuming that by the 9th iteration convergence has occured, the absolute difference in the $\\log \\chi ^2$ between the 9th and ith iteration was calculated for each of the 384 test spectra.", "The spread in values is largest at $i=0$ and there is a large decrease in the spread for $i=1$ .", "The following iterations show a more sedate decrease in spread and appear to have converged from $i=6$ .", "Figure: Box & Whisker graph of the absolute difference in the logχ 2 \\log \\chi ^2 of the reconstructed to the normalised spectra for the 384 test spectra between the 9th SPC iteration and each of the previous iterations (|logχ 2 (9)-logχ 2 (i)|| \\log \\chi ^2(9) - \\log \\chi ^2(i) |).", "The box is constrained by the 25th and 75th percentiles, and the red lines are the medians.", "The whiskers extend to the furthest data points not considered to be outliers.Figure: Internal error with changes in Normalisation for a) T eff T_{\\textrm {eff}}; b) logg\\log g; c) [M/H]; and d) [α\\alpha /Fe].", "The red lines are determined as the 2σ2 \\sigma uncertainty on values in two SNR bins (Δ\\Delta SNR ≈30\\approx 30) centred at SNR = 65 & 100 and constant uncertainty for SNR >> 125.To measure the internal error associated with the normalisation process we investigated the difference in stellar parameters between the 9th and 7th iteration for the test sample.", "Hence we assumed that the solutions have converged from at least the 7th iteration and the subsequent variations in the stellar parameters are due to minor adjustment of the normalisation of the observed spectra.", "The variation from convergence ($i=7$ ) to the final iteration ($i=9$ ) was used to give the internal error due to the normalisation process.", "Figure REF shows the difference in each stellar parameter between the 9th and 7th iterations against SNR for the 384 test spectra as black points.", "These values were binned into two SNR bins ($\\Delta $ SNR $\\approx 30$ ) centred at SNR = 65 & 100, and a single bin for all points with SNR $>$ 125.", "The 2 $\\sigma $ uncertainty for each bin is shown as red squares at the bin centre.", "Red lines connect the bin uncertainty values.", "For $T_{\\textrm {eff}}$ the spread is constant for each bin and so a constant value for the internal uncertainty due to normalisation was adopted.", "For $\\log g$ , [M/H] and [$\\alpha $ /Fe] the spread diminishes with increased SNR until SNR = 125 after which a constant spread is found.", "The adopted internal errors for these three parameters were therefore a piecewise function of SNR.", "The following equations define the internal errors due to normalisation for each parameter ($\\sigma (\\theta )_{int,norm}$ ): ${\\footnotesize \\sigma (T_{\\textrm {eff}})_{int,norm}= 11.0 \\text{~K}} \\\\$ ${\\footnotesize \\sigma (\\log \\ g)_{int,norm}= {\\left\\lbrace \\begin{array}{ll}-0.0017 \\text{~SNR} + 0.2414 \\text{~dex} & \\text{if SNR $< 100$,}\\\\-0.0018 \\text{~SNR} + 0.2500 \\text{~dex} & \\text{if $100 \\le $ SNR $< 125$,}\\\\-0.025 \\text{~dex} & \\text{if SNR $\\ge 125$,}\\\\\\end{array}\\right.", "}\\\\}$ ${\\footnotesize \\sigma (\\text{[M/H]})_{int,norm}= {\\left\\lbrace \\begin{array}{ll}-0.0007 \\text{~SNR} + 0.0964 \\text{~dex} & \\text{if SNR $< 100$,}\\\\-0.0004 \\text{~SNR} + 0.0690 \\text{~dex} & \\text{if $100 \\le $ SNR $< 125$,}\\\\-0.014 \\text{~dex} & \\text{if SNR $\\ge 125$,}\\\\\\end{array}\\right.", "}\\\\}$ ${\\footnotesize \\sigma ([\\alpha /\\text{Fe]})_{int,norm}= {\\left\\lbrace \\begin{array}{ll}-0.0006 \\text{~SNR} + 0.0671 \\text{~dex} & \\text{if SNR $< 100$,}\\\\-0.0002 \\text{~SNR} + 0.0260 \\text{~dex} & \\text{if $100 \\le $ SNR $< 125$,}\\\\-0.006 \\text{~dex} & \\text{if SNR $\\ge 125$,}\\\\\\end{array}\\right.", "}\\\\}$" ], [ "Total internal error", "The internal errors due to the SNR, V$_{rad}$ and Normalisation calculated above were combined in quadrature to give the total internal error for each parameter ($\\sigma (\\theta )_{int}$ ) for each spectrum.", "This was calculated as follows: $\\sigma (\\theta )_{int} = \\sqrt{\\sigma ^2(\\theta )_{int,snr} + \\sigma ^2(\\theta )_{int,\\textrm {V}_{rad}} + \\sigma ^2(\\theta )_{int,norm}}$ These values were reported for each spectrum in the ESO dataset as described in Section ." ], [ "AMBRE:FEROS external error analysis", "The external error was quantified by comparing the AMBRE:FEROS stellar parameter values to literature values for key reference stars that exist within the AMBRE:FEROS dataset.", "A list of stars that had been observed with FEROS but were also part of other high quality spectroscopic studies were identified and this reference sample was used to provide quality criteria for each derived stellar parameter.", "Unfortunately, reference stars covering the whole range of possible parameters could not be found in the literature.", "In several cases, estimates of the mean stellar metallicity were lacking and the situation was even worse for the [$\\alpha $ /Fe] chemical contents since too few reference stars with known [$\\alpha $ /Fe] content could be found in the literature." ], [ "Building the FEROS reference sample", "In order to compare the MATISSE stellar atmospheric parameters to the literature and to define some quality criteria, a list of reference stars was built using key spectral atlases and libraries.", "Table REF lists the standard stars and the corresponding stellar atlases that were used to test the FEROS analysis pipeline.", "Of these atlases only the S$^4$ N solar atlas was observed using FEROS.", "However the adaptability of the pipeline meant that the use of spectra from other instruments presented no difficulty as long as the required wavelength ranges were available.", "The accepted stellar parameters for each standard star are listed in bold in Table REF .", "The final stellar parameters that have been derived in the AMBRE:FEROS analysis with bias corrections (see end of this section) for each atlas are also given.", "Table: List of standard stars and the corresponding spectral atlases used to test the FEROS analysis pipeline.", "The accepted parameters for the standard stars are listed (bold) as well as the final stellar parameters with bias corrections derived for each atlas in the AMBRE:FEROS analysis.These atlases were combined in a test sample that included high resolution synthetic spectra for the Sun and Arcturus, as well as synthetic spectra for each grid point in the AMBRE:FEROS synthetic grid corresponding to the Sun and Arcturus.", "These were used to check the internal consistency of the pipeline in terms of normalisation and convolution of the input spectra.", "The results for the Sun are in very good agreement with the accepted values.", "For Arcturus we over-estimate the $T_{\\textrm {eff}}$ and $\\log g$ values however we are within 2 $\\sigma $ of the accepted values for $T_{\\textrm {eff}}$ .", "For Procyon there is good agreement in $T_{\\textrm {eff}}$ and $\\log g$ , although the $T_{\\textrm {eff}}$ is slightly under–estimated within the stated external errors.", "However the metallicity is under-estimated by –0.25 dex which is outside the 2 $\\sigma $ limit.", "The under–estimation of the $T_{\\textrm {eff}}$ may account for some of this discrepancy in metallicity.", "However further investigation into understanding the variation in the stellar parameters for these standard stars is ongoing.", "List of spectral libraries used to select comparison stars from the FEROS archived spectra.", "The number of stars from each library and the corresponding number of FEROS spectra are also listed.", "Table: NO_CAPTION" ], [ "Spectral libraries reference sample", "Preliminary investigations showed that merely making comparison between the MATISSE results and all of the stellar parameter values that were available in SIMBAD for the FEROS stars resulted in a great deal of disparity.", "There was no quality determination on the values with which we made the comparison.", "Homogeneously analysed large sample datasets were necessary in order to ensure that a valid comparison was being made.", "To this end key spectral libraries were investigated in order to locate within the FEROS archive dataset stars that had been analysed in high quality studies.", "This list of spectral libraries is given in Table REF .", "The primary spectral library that was used was the PASTEL database [59]http://pastel.obs.u-bordeaux1.fr, which provides a selection of key papers that carried out detailed spectroscopic studies at high resolution and high SNR to derive high quality stellar parameters.", "From these papers we were able to identify a total of 148 stars corresponding to 618 FEROS archived spectra (Table REF ).", "There was significant crossover of stars between the studies which provided another level of comparison for this work.", "From the PASTEL sample $\\sim $ 120 stars ($\\sim $ 300 spectra) had values for three stellar parameters, $T_{\\textrm {eff}}, \\log g$ and [Fe/H], while the remaining stars had $T_{\\textrm {eff}}$ values only.", "In the following comparison we assume the literature values to be correct.", "However we note that the literature values, while they are high quality studies, were determined using a range of techniques [59], in particular, several studies determined photometric $T_{\\textrm {eff}}$ values using the Infra-Red Flux Method [3], [4], [57], [24].", "Figure REF a. to b compare the $T_{\\textrm {eff}}$ , $\\log g$ , [M/H] and [$\\alpha $ /Fe] values determined by AMBRE:FEROS with the literature values of the reference sample and stellar atlases.", "The legend in each provides the bias and $\\sigma $ for each subsample.", "The axes represent the limits of the final accepted parameters and clearly the reference sample does not cover the entire grid as would be ideal.", "Despite this limitation the reference sample was sufficient to define bias corrections which were necessary to apply to the AMBRE:FEROS stellar parameters.", "Figure: Comparison of stellar parameters determined in AMBRE:FEROS (A:F) and the reference sample (RS) from PASTEL: a) T eff T_{\\textrm {eff}}, b) logg\\log g, c) [M/H] compared with [Fe/H], d) [α\\alpha /Fe].", "The legend identifies the relevant individual papers and the associated sample bias±σ\\pm \\sigma .", "The stellar atlases are also shown.", "The dashed line indicates a 1:1 agreement.", "The axis limits represent the limits on the final accepted parameters." ], [ "Reference sample: $T_{\\textrm {eff}}$", "Figure REF a. compares the $T_{\\textrm {eff}}$ for each star.", "It is important to note that many of the literature $T_{\\textrm {eff}}$ values were not determined spectroscopically, as stated above.", "Despite the differences in measurement technique, overall there is excellent agreement between the AMBRE:FEROS results and the PASTEL values, and also good agreement for the stellar atlases.", "There is a slight turning over of the distribution from $T_{\\textrm {eff}}\\gtrsim \\sim 5750$  K for which a small bias correction was applied that will be quantified at the end of this section." ], [ "Reference sample: $\\log g$", "Figure REF b. shows the comparision of the $\\log g$ values for which there is reasonably good agreement.", "However the majority of the sample are dwarfs (high $\\log g$ ), and the giants (low $\\log g$ ) are under-represented.", "There is a slight gradient within the sample which resulted in a small bias correction being applied at high $\\log g$ for the final results (see end of the section).", "The lack of a significant sample of giants made it difficult to accurately define a bias at low $\\log g$ , hence the correction was only applied at high $\\log g$ .", "However the uncertainty on the giants is definitely higher." ], [ "Reference sample: [M/H]", "Figure REF c. shows the comparison of the AMBRE:FEROS [M/H] values with the literature [Fe/H] values.", "The difference in definition between [M/H] and [Fe/H], as discussed previously, means that this is not an accurate comparison, as so many more elements (albeit of lesser contribution) than just Fe are included in the MATISSE metallicity.", "The comparison shows an overall systematic offset between the AMBRE:FEROS and literature values reflected in the subsample biases.", "The sources of this bias are difficult to quantify so we assumed a direct comparison between [Fe/H] and [M/H] then made a systematic bias correction to the final [M/H] results as outlined at the end of this section." ], [ "Reference sample: [$\\alpha $ /Fe]", "Unfortunately very few comparison stars with published values of their [$\\alpha $ /Fe] exist in the literature.", "The stellar parameters in the S$^4$ N library [2] also include $\\alpha $ element abundances where possible, which provided some comparisons with the AMBRE:FEROS [$\\alpha $ /Fe] results.", "The S$^4$ N $T_{\\textrm {eff}}$ values were determined using photometric calibrations while the $\\log g$ were determined from Hipparcos parallaxes.", "The chemical abundances were determined using $\\chi ^2$ minimisation of the spectral line profile between the observed spectrum and a grid of synthetic spectra [2].", "A key study that was investigated specifically in order to validate the AMBRE:FEROS [$\\alpha $ /Fe] values was [8].", "This study was a detailed analysis of 66 F and G dwarf stars in the galactic disk, and accurate abundances were determined for the $\\alpha $ elements, Mg, Si, Ca, Ti.", "The mean of these abundances was taken as the value for the global [$\\alpha $ /Fe] with which to compare the AMBRE:FEROS [$\\alpha $ /Fe] values.", "This study was particularly useful because the spectra analysed in [8] were observed with FEROS.", "However the observations took place in 2000 and 2001 and so the spectra were not part of the archived sample delivered to AMBRE.", "The original spectra were obtained (Bensby, private communication) and analysed in the AMBRE:FEROS pipeline.", "Hence a direct comparison between AMBRE:FEROS and [8] could be made for all four parameters (See Figures REF a to c).", "Figure REF d. shows the comparison of the AMBRE:FEROS [$\\alpha $ /Fe] results with the values from the S$^4$ N library, [8] and the stellar atlases.", "The biases are also listed for each sample.", "There is reasonably good agreement with these samples.", "In particular there is excellent agreement in the results for the [8] stars, which as stated above, is a study where the $\\alpha $ element abundances were carefully determined, thereby providing an excellent validation of the AMBRE:FEROS [$\\alpha $ /Fe] results." ], [ "Bias corrections", "The sample of reference stars and the spectral atlases provided a crucial comparison at all stages of the development of the AMBRE:FEROS analysis pipeline.", "Although it was not possible to cover the entire parameter space, and the reference sample was ultimately biased towards metal-rich dwarfs over a small temperature range, the results provided sufficient information with which to identify biases within the analysis, assuming the literature values to be correct.", "The following corrections were made in order to remove these biases: ${\\footnotesize T_{\\textrm {eff}}(cor)={\\left\\lbrace \\begin{array}{ll} T_{\\textrm {eff}} - 35 & \\text{if $T_{\\textrm {eff}} < 5300$~K,}\\\\T_{\\textrm {eff}} + 0.21 \\times T_{\\textrm {eff}} - 1141.4 &\\text{if $5300 \\le T_{\\textrm {eff}} \\le 6000$~K,}\\\\T_{\\textrm {eff}} + 110 &\\text{if $T_{\\textrm {eff}} > 6000$~K.}\\end{array}\\right.}", "}$ ${\\footnotesize \\log g(cor)={\\left\\lbrace \\begin{array}{ll} \\log \\ g - 0.296 \\times \\log g + 1.388 & \\text{if $\\log g \\ge 4.0$,}\\\\\\log g + 0.204 &\\text{if $3.75 \\le \\log g < 4.0$,}\\\\\\log g + 0.817 \\times \\log g - 2.860 &\\text{if $3.5 \\le \\log g < 3.75$,}\\\\\\log g &\\text{if $\\log g < 3.5$.}\\end{array}\\right.}", "}$ $\\text{[M/H]}(cor) &= \\text{[M/H]} + 0.15 \\\\[\\alpha \\text{/Fe}](cor) &= [\\alpha \\text{/Fe}]$ There are several potential sources of the bias corrections for the AMBRE:FEROS analysis.", "First, the differing techniques, as well as differing spectral domains and resolutions, used in the determination of the stellar parameters by the reference studies compared with AMBRE:FEROS analysis may make a significant contribution to the difference in derived parameters.", "Second, while the normalisation process within the AMBRE analysis pipeline was designed to be as robust as possible, the normalisation of stellar spectra is an inherently complex problem, particularly over large wavelength ranges with many spectral features.", "For spectra of particular spectral types (e.g.", "cool, low gravity, metal-rich) the normalisation procedure may not be as robust as for less detailed spectra.", "Third, due to the length of the atomic and molecular linelist that was required to synthesise the AMBRE:FEROS wavelength regions, it was inefficient to carry out a calibration of the linelist to standard stars (i.e.", "the Sun).", "It was assumed that the number of lines was statistically sufficient to dampen the noise from ill-fitted spectral features but non-calibration of the line list may play some role in the degree of the bias corrections.", "Individually, and in combination, these are the most likely sources of the bias corrections for the AMBRE:FEROS stellar parameters." ], [ "External error", "The external error for each stellar parameter was determined from the above reference sample analysis.", "For each individual study the mean difference and spread in differences ($\\sigma $ ) between the literature and AMBRE:FEROS values was calculated.", "The mean of these $\\sigma $ values for each stellar parameter was taken as the global external error ($\\sigma (\\theta )_{ext}$ ) for all of the spectra as follows: $\\sigma (T_{\\textrm {eff}})_{ext} &= 120 \\text{~K} \\\\\\sigma (\\log \\ g)_{ext} &={\\left\\lbrace \\begin{array}{ll}0.20 \\text{~dex} &\\text{if $\\log g \\ge 3.2$,}\\\\0.37 \\text{~dex} &\\text{if $\\log g < 3.2$,}\\\\\\end{array}\\right.", "}\\\\\\sigma (\\text{[M/H]})_{ext} &= 0.10 \\text{~dex} \\\\\\sigma ([\\alpha /\\text{Fe]})_{ext} &= 0.10 \\text{~dex} \\\\$" ], [ "ESO Table: Rejection criteria", "The construction of the ESO Table of stellar parameters that would be delivered to the ESO Archive was the final stage of the pipeline.", "At this stage the quality flags and tests included throughout the pipeline were drawn together to provide the final set of stellar parameters.", "The columns headings, their definitions, range of values, null values and rejection conditions used to construct the ESO Table are listed in Table .", "There were two phases of rejecting spectra in the AMBRE:FEROS analysis.", "First, prior to analysis in SPC spectra were rejected due to spectral quality issues.", "This will be described in the Section REF .", "Second, the remaining spectra were all analysed in SPC for their stellar parameters and then another set of rejection criteria were applied to construct the final table.", "These criteria are described and discussed in Sections REF to REF ." ], [ "Pre-SPC: Non-standard spectra", "Prior to SPC, 11% of the FEROS archived spectra were rejected as being non-analysable primarily due to being non-standard spectra.", "The rejection flags, as defined in Section REF , were for `Faulty Spectra', `Extreme Emission Features',`Poor Normalisation' and `Excessive Noise'.", "Often the nature of a spectrum rejected at this stage meant the conditions for two or more of these flags were met.", "For instance, an extremely noisy spectrum was also consequently poorly normalised.", "Quality flags were also attached for `Large Emission Features' and `Instrumental Relics' but these spectra were not rejected based solely on these flags.", "The number of spectra identified for each rejection flag and for each quality flag are listed in Table REF .", "Table: The number of spectra rejected (Rej) prior to SPC based on the defined rejection flags and the number of spectra accepted (Acc) for SPC but identified by the quality flags." ], [ "Rejection criteria post-SPC", "The remaining 89% of the FEROS spectra were analysed in SPC and so stellar parameters were obtained for each of these spectra.", "Figure REF a shows the HR diagram for all the stellar parameters obtained in SPC.", "The giant branch and main sequence can be observed but there is a great deal of mis-classification at hot temperatures for $\\log g \\gtrsim 2$  dex, and an overdensity at $T_{\\textrm {eff}} \\approx 3000$  K and $\\log g \\approx 2$  dex.", "Most noticeable is that there are many spectra (8947) for which the derived parameters lie outside the $T_{\\textrm {eff}}$ and $\\log g$ grid limits which are indicated in red.", "In the construction of the ESO Table it would have been very straightforward to just reject all spectra whose parameters lay outside the limits of the synthetic grid and so only report to ESO those spectra with parameters inside the grid.", "However to have the synthetic grid limits as the sole reason to reject so many spectra ($\\sim 47\\%$ ) without further explanation seemed inadequate.", "Thus we further investigated the observational and astrophysical characteristics (indicators) of the spectra in order to better understand the dataset and to be able to impose more robust quality control criteria on the final dataset to be reported to ESO.", "The SNR and $\\log \\chi ^2$ were investigated as potential indicators but both proved to be functions of spectral type (the more spectral features the lower the SNR, the poorer the observed to synthetic spectra fit) and the criteria rejected RGB stars rather than the high temperature mis-classifications.", "Indicators relating to the measurement of the spectral FWHM and the radial velocity proved to be the least affected by spectral type and better identified the mis-classifications.", "The key indicators that were explored were: the V$_{rad}$ CCF contrast, the error on the amplitude and the error on continuum of the V$_{rad}$ CCF; the V$_{rad}$ error; the FWHM of the V$_{rad}$ CCF; the spectral FWHM; and the AMBRE:FEROS synthetic grid boundaries in $T_{\\textrm {eff}}$ , $\\log g$ , [M/H], [$\\alpha $ /Fe].", "Table REF summarises the number of spectra that satisfied the conditions of each rejection criteria for the spectra rejected before and after SPC.", "The post-SPC rejection criteria were applied to the spectra rejected before SPC to further characterise that sample.", "To illustrate the effects of the rejection criteria on the sample of spectra analysed in SPC, Figure REF shows progressively the HR diagram of the AMBRE:FEROS stellar parameters as each criteria is applied.", "The limits of the synthetic grid in $T_{\\textrm {eff}}$ and $\\log g$ are also shown.", "This process is discussed in the following sections.", "Table: Summary of the number of spectra flagged as satisfying the conditions of the post-SPC rejection criteria.", "The spectra that were rejected prior to the SPC analysis were rejected due to other quality issues (see Section ).", "The spectra analysed in SPC were accepted or rejected based on the listed criteria." ], [ "CCF contrast, amplitude & continuum", "As described in Section REF , the radial velocity was determined using binary masks to calculate a CCF.", "The contrast, the depth (amplitude) of the CCF, the continuum placement of the CCF and their associated errors (as calculated in the IDL:GAUSSFIT routine) are measures of the quality of the CCF.", "The rejection criteria were derived from these quantities as follows: A negative contrast means that the CCF profile is inverted, which is contrary to the expected result, hence all spectra with a negative contrast were rejected.", "The relative error on the amplitude of the CCF ($\\frac{\\sigma _{amp}}{Amp}$ ) is mainly a measure of how much noise there is in the CCF.", "If the $\\frac{\\sigma _{Amp}}{Amp} \\le 0.2$ then the noise makes up less than 20% of the profile depth.", "We rejected spectra with $\\frac{\\sigma _{Amp}}{Amp} > 0.2$ .", "The placement of the continuum is also an indication of how well the CCF was defined.", "The relative error on the continuum ($\\frac{\\sigma _{Cont}}{Cont}$ ) gives a measure of the noise in the continuum placement.", "We rejected all spectra with $\\frac{\\sigma _{Cont}}{Cont} > 0.1$ .", "These three criteria identify spectra with poorly-defined CCFs.", "Combined they are the first criteria to be applied to the SPC dataset resulting in the rejection of a further 3668 spectra from the analysis.", "Figure REF b shows the resulting HR diagram after these spectra were removed.", "The hot star mis-classifications and overdensity at $T_{\\textrm {eff}} \\approx 3000$  K are diminished.", "Figure REF a and b show histograms of the SNR per pixel (0.03 or 0.06 Å per pixel) for the full sample of FEROS archived spectra.", "The SNR has been calculated within the analysis pipeline using sigma clipping on either the predefined continuum regions in SPB or the pseudo-continuum residual in SPC.", "The highest SNR values do not necessarily mean optimum spectra and are identified by the quality flags within the pipeline as required.", "Figure REF a is the histogram of the spectra that were accepted as being of good quality and with a well-defined CCF, whereas Figure REF b is the histogram of the spectra rejected as being of poor quality or with poorly-defined CCF.", "Note that the y-axis scale is different between the figures.", "The histogram in Figure REF a peaks between 50 and 100 SNR.", "There are significantly fewer spectra with SNR$<$ 50 in Figure REF a compared with Figure REF b, indicating the cleaning thus far has indeed identified low SNR spectra.", "However, as Figure REF a to d shows, even at SNR of 50 the internal error is negligible.", "Figure: Histogram of the measured SNR values of the FEROS archived spectra in bins of Δ\\Delta SNR =50=50: a) Spectra defined as good quality and with well-defined CCFs (N=15513); b) Spectra rejected as poor quality and/or poorly defined CCFS (N=6038).", "The y-axes are to different scale to better display each sample." ], [ "V$_{rad}$ error", "The next criterion to be applied was the V$_{rad}$ error ($\\sigma _{\\textrm {V}_{rad}}$ ) which had been calculated using the prescription in [61].", "Figure REF shows histograms of the $\\sigma _{\\textrm {V}_{rad}}$ determined for each of the FEROS archived spectra of good quality with a well-defined CCF.", "Figure REF a is a histogram of V$_{rad}$ error $<10$  kms$^{-1}$ in bins of 1 kms$^{-1}$ , and Figure REF b is a histogram of $\\sigma _{\\textrm {V}_{rad}}$  $>10$  kms$^{-1}$ in bins of 10 kms$^{-1}$ .", "The majority of the spectra have a low $\\sigma _{\\textrm {V}_{rad}}$ ($<0.5$  kms$^{-1}$ ), and based on the synthetic spectra analysis in Section REF , Figure REF shows that $\\Delta $ V$_{rad} \\le 5$  kms$^{-1}$ correspond to reasonable variations in the stellar parameters ($\\theta $ ).", "However there are a significant number of spectra with $\\sigma _{\\textrm {V}_{rad}}$ greater than 10 kms$^{-1}$ (see Figure REF b) which correspond to much larger uncertainties in the $\\theta $ determination.", "Figure: Histogram of measured V rad _{rad} uncertainty for each of the FEROS archived spectra of good quality with a well-defined CCF.", "a) The distribution of the sample with σ V rad <10\\sigma _{\\textrm {V}_{rad}} < 10 kms -1 ^{-1}.", "b) The distribution of the sample with σ V rad >10\\sigma _{\\textrm {V}_{rad}} > 10 kms -1 ^{-1}.Hence we decided to reject all spectra with $\\sigma _{\\textrm {V}_{rad}} > 10$  kms$^{-1}$ .", "This resulted in a further 3550 spectra being rejected from the SPC dataset.", "The resulting HR diagram is shown in Figure REF c. Again more of the mis-classifications at hot $T_{\\textrm {eff}}$ , particular for high $\\log g$ , have been removed and the overdensity at $T_{\\textrm {eff}} \\approx 3000$  K is considerably diminished.", "Figure: T eff T_{\\textrm {eff}} vs FWHM of the CCF for a) Giants (logg<3.5\\log g < 3.5 dex), b) Dwarfs (logg≥3.5\\log g \\ge 3.5 dex).", "The nominal temperature values for spectral types A0, F0, G0, K0 are shown as cyan lines.", "A CCF FWHM of 40 kms -1 ^{-1} is indicated in red." ], [ "FWHM of V$_{rad}$ CCF", "Astrophysically, the FWHM of the V$_{rad}$ CCF can be used to calculate the rotational velocity ($Vsini$ ) of a star.", "That particular calculation was not carried out here, for the purposes of expediency, but the FWHM of the CCF was used to identify spectra with broadened spectral features that would not be well represented by the synthetic grid.", "As discussed in Section REF , the synthetic spectra grid was generated with no variations in $Vsini$ , assuming all stars to be slow rotators.", "The analysis in [23] found that for a sample of FGK dwarfs $Vsini < 11$  kms$^{-1}$ produced good results in the stellar parameter determination by MATISSE.", "At a resolution of R$\\sim $ 26,000 an upper limit of CCF FWHM $= 20$  kms$^{-1}$ was set.", "Due to the lower resolution (R$\\sim $ 15,000), and hence greater masking of the effects of $Vsini$ , of the AMBRE:FEROS analysis we sought to relax that criterion.", "The comparison of spectral type with $Vsini$ is used to show the increase in the number of fast rotators with hotter spectral type for both dwarfs and giants [29].", "For dwarfs, fast rotators begin to appear at approximately F2 in spectral type, for giants at approximately G2.", "Using the stellar parameters derived in SPC and the spectral dataset that remained after the application of the $\\sigma _{\\textrm {V}_{rad}}$ rejection criterion, we replicated Spectral Type vs $Vsini$ using $T_{\\textrm {eff}}$ and the CCF FWHM as proxies.", "The spectra were separated into subsamples of dwarfs and giants using the derived values for $\\log g$ .", "The comparisons are shown in Figure REF a and b.", "The location of the Spectral Types in temperature are indicated (cyan lines).", "There is a clear dense grouping of spectra between 4000 and 6000 K with CCF FWHM less than 20 for both giants and dwarfs.", "Both samples have a great deal of scatter, particularly for $T_{\\textrm {eff}}$ greater than 7000 K. Above the expected spectral type of F2 for the giants there is more dense scatter at higher CCF FWHMs.", "For the dwarfs the scatter becomes denser around 10000 K. A CCF FWHM of 40 kms$^{-1}$ is indicated in both cases (red line).", "This threshold was selected as being a factor of two greater than the threshold in [23], as the resolution of the AMBRE:FEROS synthetic grid is approximately a factor of two lower in resolution than that study.", "Hence the astrophysical broadening is masked to at least a factor of 2 in the convolved FEROS spectra.", "Above the 40 kms$^{-1}$ threshold the distribution of the points is sporadic while below this value the distribution is more coherent.", "However the majority of the points lie below 20 kms$^{-1}$ in line with [23].", "Ultimately we decided to reject all points with a CCF FWHM greater than 40 kms$^{-1}$ .", "The effects of this cleaning on the HR diagram are shown in Figure REF d. A significant number of mis-classifications with $T_{\\textrm {eff}} > 7000$  K have been removed as expected from the threshold shown in Figure REF .", "These rejected stars should be analysed with a hotter synthetic spectra grid with variations in $Vsini$ ." ], [ "FWHM of the stellar spectrum", "The calculation of the spectral FWHM was necessary in order to convolve the observed spectra to the resolution of the synthetic grid.", "However the FWHM values also provide an extra degree of information with which to understand the spectral dataset.", "The FWHM measurements were separated into FWHM$_{weak}$ , FWHM$_{strong}$ and FWHM$_{medium}$ classifications (Section REF ).", "In particular the FWHM$_{strong}$ classification allowed us to identify spectra with broad features, independantly of the FWHM of the V$_{rad}$ CCF.", "For each FWHM$_{strong}$ spectra, the FWHM of the medium strength lines was measured (where possible) as well as the FWHM of the identified strong lines.", "Figure REF a, b and c show the relationships between FWHM$_{medium}$ and CCF FWHM, FWHM$_{medium}$ and $T_{\\textrm {eff}}$ , and FWHM$_{strong}$ and $T_{\\textrm {eff}}$ respectively for the dataset as of the application of the $\\sigma _{\\textrm {V}_{rad}}$ criterion.", "Figure: Comparison of a) FWHM medium _{medium} with the CCF FWHM, b) T eff T_{\\textrm {eff}} for the dataset with CCF FWHM << 40 kms -1 ^{-1} compared to the FWHM medium _{medium}, and c) as for b) but for FWHM strong _{strong}.", "Thresholds at CCF FWHM = 40 kms -1 ^{-1}, FWHM medium _{medium} = 0.8 mÅ, FWHM strong _{strong} = 8 mÅ, and T eff =7000T_{\\textrm {eff}}=7000 K are also shown.In Figure REF a there is a clear trend of increasing CCF FWHM with FWHM$_{medium}$ .", "The line at FWHM$_{medium}$ = 0.8 mÅ intersects the trend at the threshold of CCF FWHM = 40 kms$^{-1}$ where the trend becomes less dense.", "This provided the first criterion that we applied such that any spectra with a measured FWHM$_{medium}$ greater than 0.8 mÅ were rejected.", "Figure REF e shows the HR diagram after the removal of spectra based on this criterion.", "There is some improvement such that the density of the hot mis-classifications is decreased.", "However there are still a significant number of mis-classifications at temperatures greater than 7000 K. Further investigation showed that these remaining mis-classifications were connected to the `strong' FWHM classification (FWHM$_{strong}$ ).", "There was no clear trend between the CCF FWHM and the FWHM$_{strong}$ with which to derive a threshold of rejection.", "Instead a comparison was made to $T_{\\textrm {eff}}$ .", "Figure REF b compares the $T_{\\textrm {eff}}$ with the corresponding FWHM$_{medium}$ .", "The lower boundary of possible values of $T_{\\textrm {eff}}$ does increase with FWHM$_{medium}$ and at the derived threshold of FWHM$_{medium}$ = 0.8 mÅ this trend intersects with a $T_{\\textrm {eff}} = 7000$  K. For greater FWHM$_{medium}$ values the trend of the lowest $T_{\\textrm {eff}}$ disappears into greater scatter, essentially attributing a temperature value to the FWHM$_{medium}$ threshold.", "This temperature value was used to infer a FWHM$_{medium}$ threshold for the strong spectral lines.", "Figure REF c shows the measured FWHM$_{strong}$ (where possible) with the $T_{\\textrm {eff}}$ .", "There is distinct trend of increasing temperature with increasing FWHM$_{strong}$ .", "The nominal temperature threshold at 7000 K intersects with this trend at a FWHM$_{strong}$ of 8 mÅ.", "At greater FWHM$_{strong}$ there is an increase in scatter and the trend becomes more dispersed.", "The greater spread in values for spectra with FWHM$_{strong}$ $>$ 8 mÅ, and for FWHM$_{medium}$ $>$ 0.8 mÅ, may indicate a greater uncertainty in the parameterisation as the boundary of the grid is approached.", "Based on this we can assume that spectra with a FWHM$_{strong}$ not more than 8 mÅ have stellar parameters that fall within the synthetic grid and thus are well-defined.", "Hence this provided the second criterion to apply such that spectra with a FWHM$_{strong}$ greater than 8 mÅ were rejected.", "Figure REF f shows that indeed the mis-classifications between 7000 K and 8000 K are significantly reduced after the application of this criterion." ], [ "Understanding the rejected spectra", "Figure REF explores the types of stars to which the 11971 rejected spectra correspond.", "Figures REF a compares $T_{\\textrm {eff}}$ with $\\sigma _{\\textrm {V}_{rad}}$ for the rejected sample.", "It is clearly bimodal with the primary peak at 8190$\\pm $ 924 K (10068 spectra) and the secondary peak at 3351$\\pm $ 679 K (1903 spectra).", "The datasets corresponding to the key rejection criteria are also indicated: FWHM$_{medium}$ rejections (pink), FWHM$_{strong}$ rejections (yellow) and CCF FWHM rejections (cyan).", "The spectra rejected on the basis of FWHM$_{medium}$ and FWHM$_{strong}$ are concentrated in the two peaks.", "This argues that at least 84% of these spectra with broad features have returned parameters greater than the 8000 K temperature limit of the synthetic spectra grid.", "Figure: Comparison of the T eff T_{\\textrm {eff}} of the 11971 rejected spectra (black) with a) σ V rad \\sigma _{\\textrm {V}_{rad}}, b) logg\\log g, c) [M/H] and d) [α\\alpha /Fe].", "The datasets for the FWHM medium _{medium} rejections (pink), FWHM strong _{strong} rejections (yellow) and CCF FWHM rejections (cyan) are shown.Figure REF b is the HR diagram of the rejected spectra.", "The majority have $T_{\\textrm {eff}} > 6000$  K, but also with $\\log g > 2$  dex implying these are sub-giants and dwarf stars.", "It is possible that these spectra have parameters that lie outside the available synthetic grid parameter space and MATISSE has compensated by varying one or all of the stellar parameters away from the true values in order to converge on a solution.", "Combined with the high number of these spectra that have been measured to have broadened spectral features, it is possible that these are in fact hot and/or fast rotating stars for which the synthetic grid has not been designed.", "In Figure REF a 16% of the rejected sample are located near the cool temperature limit of the synthetic grid (3351$\\pm $ 679 K).", "Are these truly such cool stars?", "Figure REF c compares the $T_{\\textrm {eff}}$ with [M/H].", "The cool temperature limit spectra show an interesting trend of decreasing [M/H] with decreasing temperature.", "For spectra with $T_{\\textrm {eff}} < 3000$  K the [M/H] extends significantly outside the grid range.", "A possible answer is that these are hot or fast rotating stars (based on the FWHM measurements) for which MATISSE has converged to a significantly cooler temperature with significantly decreased metallicity.", "A similar but less dramatic argument can be made for the spectra rejected at the hot temperature limit.", "These lie within the [M/H] range for the synthetic grid although many are metal-poor to $-$ 3.5 dex.", "Combined with the high gravity values of these spectra the low metallicity emphasises the possiblity that these stars have been mis-classified due to their parameters lying outside the grid parameter space.", "The comparison of [$\\alpha $ /Fe] to $T_{\\textrm {eff}}$ in Figure REF d adds weight to this, as the cool temperature limit group have very high [$\\alpha $ /Fe] values.", "For the hot temperature limit group the [$\\alpha $ /Fe] are typically within the range of the synthetic grid.", "From this examination of the key variables, the main argument that can be drawn as to why $\\sim $ 66.1% of the spectra analysed in SPC have been rejected by these very strict criteria is that they are most likely hot and/or fast rotating stars that are not well-represented by the current synthetic grid.", "Hence MATISSE has converged on solutions which mis-classify the spectra in the absence of a grid covering the required parameter range.", "This is not an unreasonable conclusion given that FEROS was designed and built by researchers within the hot star community [33].", "A visual inspection was carried out on large random samples of the spectra that were rejected based on each of the rejection criteria.", "This inspection supported the appropriate application of the criteria as the rejected spectra corresponded to noisy, peculiar or strong-lined spectra, and there was an over-representation of spectra comprising solely of strong balmer lines.", "However to confirm this conclusion these spectra can be re-analysed at a later date when a synthetic grid with $T_{\\textrm {eff}} > 8000$  K and a synthetic grid with a range in $Vsini$ values will be available." ], [ "Synthetic grid boundaries", "The above rejection criteria succeeded in cleaning the stellar parameter sample without losing the well-defined branches of stellar evolution that are observed in the HR diagram.", "Finally, the stellar parameters determined near the boundaries of the synthetic grid are inherently less reliable due to there being less synthetic spectra present to aid in the definition of the parameters.", "Hence the final rejection criteria that were applied are based on the stellar parameters themselves.", "Taking into account boundary effects in the learning phase of MATISSE, the limits on the accepted parameters are defined as follows: $3000 \\ \\le \\ &T_{\\textrm {eff}} \\ \\le 7625 \\\\1 \\ \\le \\ &\\log g \\ \\le \\ 5 \\\\-3.5 \\ \\le \\ &\\text{[M/H]} \\ \\le \\ 1 \\\\-0.4 \\ \\le \\ &[\\alpha /\\text{Fe}] \\ \\le \\ 0.4 \\hspace{14.22636pt} \\text{if [M/H] $\\ge 0.0$} \\\\-0.4 \\ \\le \\ &[\\alpha /\\text{Fe}] \\ \\le \\ 0.8 \\hspace{14.22636pt} \\text{if $-1.0 < [M/H] < 0$} \\\\0.0 \\ \\le \\ &[\\alpha /\\text{Fe}] \\ \\le \\ 0.8 \\hspace{14.22636pt} \\text{if $[M/H] \\le -1$} \\\\$ Three categories were established using these definitions: TON: $T_{\\textrm {eff}}$ values only within the accepted limits; TGM: $T_{\\textrm {eff}}$ , $\\log g$ & [M/H] values within accepted limits; TGMA: $T_{\\textrm {eff}}$ , $\\log g$ , [M/H] & [$\\alpha $ /Fe] within accepted limits.", "The final count of spectra for these three categories were: TON = 394 spectra ($\\sim $ 292 stars), TGM = 97 spectra ($\\sim $ 87 stars), TGMA = 6017 spectra ($\\sim $ 2780 stars).", "In total 12673 spectra were rejected from the SPC analysis.", "The final set of the stellar parameters accepted for delivery to ESO was comprised of 6508 spectra ($\\sim $ 3087 stars).", "For the radial velocity analysis 11963 spectra ($\\sim $ 4505 stars) were determined to have radial velocities with $\\sigma _{\\textrm {V}_{rad}} \\le 10$  kms$^{-1}$ and these values were also accepted for delivery to ESO.", "The full breakdown of the spectral number count is given in Table REF .", "Table: Number counts and percentages of spectra accepted (Acc.)", "or rejected (Rej.)", "during the AMBRE:FEROS analysis." ], [ "Repeated observations", "The FEROS dataset contains many stars with repeated observations.", "Figure REF shows histograms of the number of stars per number of observations for the TGMA sample.", "Figure REF a shows the histogram for the number of stars with observations less than or equal to 10, while Figure REF b shows the number of stars with observations greater than 10.", "There is a significant number of stars with 2 or more observations, and some stars have been observed a great many times.", "The repeated observations were used to assess whether the AMBRE analysis derived consistent stellar parameters for the same star from different spectra.", "Figure: Histogram of the humber of repeat observations in the TGMA sample for stars with: a) the number of observations less than or equal to 10; b) the number of observations greater than 10.All objects with 2 or more observations were extracted from the TGMA sample in order to explore the agreement in parameters for a single object over a number of observations of similar SNR.", "We also extracted from SIMBAD a list of objects within the FEROS dataset (based on a 10” coordinate search) that were designated as a binary and/or variable star.", "We found 17 binary stars and 111 variable stars in the final sample which we eliminated from the sample of repeat observations.", "Hence we removed those stars with possible astrophysical variations in the parameters to be left with variations that are solely due to the analysis process.", "(It must be noted that, as the list in SIMBAD is not exhaustive, it is likely that there are still stars for which the parameters vary astrophysically within the repeated observations sample.)", "The repeated observations sample was further constrained such that each set of repeats had a mean SNR less than 200, as the observations above SNR=200 were sparcely sampled.", "Due to the prior application of the rejection criteria there were no spectra present with SNR $<$  40.", "Finally, the sample of repeats comprised of 584 stars equating to 3438 spectra.", "The maximum number of repeats was 73 for a single star in the repeats sample.", "This sample represents the internal error of the AMBRE analysis hence we compared it to the internal error ($\\sigma _{int}$ ) that we calculated for each spectra based on the SNR, V$_{rad}$ and Normalisation (Section ).", "Figure: Change in θ\\theta with SNR (in bins of 20) for the repeat observations within TGMA (black symbols) compared with the σ(θ) int \\sigma (\\theta )_{int} values (Section ) for TGMA (grey points).", "a) The ΔT eff \\Delta T_{\\textrm {eff}} that 70% of the repeat sample were less than or equal to at each SNR (black squares).", "The trend with SNR is shown by the black lines.", "b) As for a) but for variations in logg\\log g (black stars).", "c) As for a) but for variations in [M/H] (black circles).", "d) As for a) but for variations in [α\\alpha /Fe] (black diamonds).Figure: The final FEROS stellar parameters for TGM (red) and TGMA (black) samples for a) The Hertzsrung-Russell diagram; b) [M/H] vs T eff T_{\\textrm {eff}}; c) [α\\alpha /Fe] vs T eff T_{\\textrm {eff}}; d) [M/H] vs logg\\log g; e) [α\\alpha /Fe] vs logg\\log g; f) [M/H] vs [α\\alpha /Fe].Figure REF replicates the relationships in Figure REF but for the repeats sample.", "For each set of repeats the mean and standard deviation were calculated for the SNR and stellar parameters.", "The standard deviation of each of the parameters was used to represent the change in the parameters ($\\Delta \\theta $ ) for each set.", "The mean SNR values were binned (binsize = 25) and for each bin the 70th percentile of the $\\Delta \\theta $ was calculated.", "These values are shown in black.", "Also included in grey for each parameter are the internal errors calculated for the TGMA spectra ($\\sigma (\\theta )_{int}$ , see Section ).", "In Figure REF a the variations in $T_{\\textrm {eff}}$ with SNR for the repeats sample is greater than the calculated internal error distribution and almost traces out the upper limit on the internal error.", "However, in general these two independant measurements of the internal error are on the same order of magnitude confirming the internal consistency within the AMBRE:FEROS analysis.", "As defined in Section  the internal errors for $\\log g$ , [M/H] and [$\\alpha $ /Fe] are calculated as a piecewise function of SNR and this is reflected in the distribution of the internal errors for TGMA as shown in Figure REF b to d. The lower limit of the internal errors for each of these three parameters decreases with increasing SNR to the defined constant value at SNR = 125.", "This distribution is replicated in the variation of each parameter with SNR for the repeats sample.", "At low SNR there is an offset in $\\log g$ and [M/H] between the two sets of values where the internal errors are of greater value.", "This implies that the internal errors may be overestimated at low SNR in light of the repeats sample analysis.", "For [$\\alpha $ /Fe] the variations due to the repeats sample trace the internal errors very well across the range of SNR values.", "But overall, as for $T_{\\textrm {eff}}$ , these two independant measurements of the internal error for each parameter are in very good agreement.", "The distribution of the variation of the parameters with SNR for the repeats sample in Figure REF also replicates the analysis of the internal error due to SNR that was carried out using the synthetic spectra sample as shown in Figure REF .", "For the lowest SNR bin for each parameter the $\\Delta \\theta $ are only 2 or 3 times the equivalent value in the synthetic case which is reasonable, and as the SNR increases the $\\Delta \\theta $ decreases significantly also replicating the synthetic case.", "The comparison between the two measures of the internal error from the observed spectra and with the synthetic spectra analysis shows that the analysis carried out within the pipeline is consistent such that closely comparable parameters are found for different spectra of the same star, and so for different stars of similar stellar parameters, at a particular SNR value." ], [ "The AMBRE:FEROS stellar parameters", "The final FEROS stellar parameters are shown in Figure REF in six combinations with which to examine the distribution of the parameters.", "Figure REF a shows the Hertzsprung-Russell (HR) diagram of the TGM (red) and TGMA (black) samples.", "The branches of stellar evolution are distinct although broad in width with some small scatter evident throughout the figure.", "Figure REF b compares [M/H] with $T_{\\textrm {eff}}$ and the majority of the spectra show near solar metallicities ($-1.0 <$  [M/H] $< 0.5$  dex) consistently across the temperatures.", "Figure REF c compares [$\\alpha $ /Fe] with $T_{\\textrm {eff}}$ and shows an interesting distribution of low ($< 0.0$  dex) and high ($> 0.2$  dex) [$\\alpha $ /Fe] at low temperatures but an even distribution of solar [$\\alpha $ /Fe] across all temperatures.", "Figure REF d compares [M/H] with $\\log g$ and the separation between the giants and dwarfs is evident.", "There is larger scatter in [M/H] at low $\\log g$ (giants).", "Figure REF e compares [$\\alpha $ /Fe] with $\\log g$ and again the scatter is larger at low $\\log g$ (giants).", "Figure REF f compares [$\\alpha $ /Fe] with [M/H] for which there is a clear trend of enhanced [$\\alpha $ /Fe] ($\\sim 0.35$  dex) at low [M/H] ($<-1.0$  dex).", "For [M/H] of solar and above the [$\\alpha $ /Fe] decreases to solar and depleted values.", "Combining three parameters in one graph provides another perspective with which to further explore this dataset.", "Figure REF shows the HR diagram of the TGM and TGMA samples binned by colour in [M/H].", "The near solar [M/H] bins contain the majority of the sample as expected from Figure REF across the Main Sequence and RGB.", "The samples within the more metal-poor bins lie to the left of the stellar evolution branches which agrees with stellar evolutionary tracks.", "Interestingly the majority of the Horizontal Branch (HB) are metal-poor ([M/H] $=-1.62$  dex).", "Figure: The HR diagram of the TGM and TGMA FEROS stellar parameter samples binned by colour in [M/H].", "The centre of each bin are listed in the legend." ], [ "Conclusions & future work", "The stellar parameterisation of the FEROS archive completes the first phase of the AMBRE Project.", "Of the 21551 FEROS object spectra that were delivered to OCA, 19181 could be analysed for their stellar parameters.", "The quality flags and error analysis resulted in a final total of 6508 spectra ($\\sim $ 3087 stars) with stellar parameters to be delivered to ESO.", "Also delivered to ESO are the radial velocity values for 11963 spectra ($\\sim $ 4505 stars) Approximately 70% of the spectra were rejected from the stellar parameterisation analysis.", "While 28% were rejected due to the quality of the spectra being insufficient for analysis, 42% were rejected due to astrophysical reasons.", "From this analysis it seems very likely that hot and/or fast rotating stars are the favoured observational object for FEROS.", "As the current synthetic grid is defined for stars cooler than 8000 K only then the high number of rejections was inevitable.", "In general the AMBRE synthetic spectra grid is currently configured for slow-rotating FGK stars.", "Non-standard stars, such as binaries and chemically peculiar stars, within the FEROS sample may still pollute the final accepted dataset as we were unable to identify such stars at this stage.", "However, stellar parameter extensions to the synthetic spectra grid are under development and identification tools are also being developed such that a larger range of spectra may be reliably analysed at a later date.", "In summary the work carried out in the analysis of the FEROS archived spectra has resulted in: the development of a complex and robust analysis pipeline for the determination of stellar parameters of large spectral datasets through automated iterative spectral reduction and MATISSE analysis; the establishment of tools with which to exploit spectral atlases and libraries for the comprehensive testing and validation of the results from the AMBRE analysis pipeline; the determination of stellar parameters ($T_{\\textrm {eff}}$ , $\\log g$ , [M/H], [$\\alpha $ /Fe]) for 6508 of 21551 FEROS archived spectra to be made available in the ESO Archive, including quality flags.", "the determination of radial velocities (V$_{rad}$ ) for 11963 of 21551 FEROS archived spectra to be made available in the ESO Archive.", "The AMBRE pipeline is currently being used to determine stellar parameters for the UVES and HARPS archived spectra, the next two phases of analysis in the AMBRE Project.", "The analysis of the FLAMES/GIRAFFE archived spectra is expected to commence in early 2012.", "The AMBRE Project team members would like to thank ESO, OCA and CNES for their financial support of this project.", "We would also like to thank O.", "Begin, Y. Vernisse, S. Rousset and F. Guitton for their work on the development of the AMBRE analysis pipeline as well as the OCA Mesocentre members.", "We would like to thank C. Soubiran and T. Bensby for proving stellar parameters and spectra for testing the pipeline.", "We would like to thank B. Plez for the use of Turbospectrum and the molecular linelists, and also C. Melo for the use of the radial velocity programme.", "Thanks also to L. Pasquini for initiating the project, and to M. Romaniello and J. Melnich for their help within ESO.", "This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France, as well as the NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration." ], [ "Description of ESO Table for AMBRE:FEROS", " Description of columns in the table of FEROS stellar parameters delivered to ESO.", "Table: NO_CAPTION" ] ]

[ [ "Implications of the UHECRs penetration depth measurements" ], [ "Abstract The simple interpretation of PAO's UHECRs' penetration depth measurements suggests a transition at the energy range $1.1 - 35 \\cdot 10^{18} $ eV from protons to heavier nuclei.", "A detailed comparison of this data with air shower simulations reveals strong restrictions on the amount of light nuclei (protons and He) in the observed flux.", "We find a robust upper bound on the observed proton fraction of the UHECRs flux and we rule out a composition dominated by protons and He.", "Acceleration and propagation effects lead to an observed composition that is different from the one at the source.", "Using a simple toy model that take into account these effects, we show that the observations requires an extreme metallicity at the sources with metals to protons mass ratio of 1:1, a ratio that is larger by a factor of a hundred than the solar abundance.", "This composition imposes an almost impossible constraint on all current astrophysical models for UHECRs accelerators.", "This may provide a first hint towards new physics that emerges at $\\sim 100$ TeV and leads to a larger proton cross section at these energies." ], [ "Implications of the UHECRs penetration depth measurements Nimrod Shaham Tsvi Piran The Racah institute of Physics, The Hebrew university of Jerusalem The simple interpretation of PAO's UHECRs' penetration depth measurements suggests a transition at the energy range $1.1 - 35 \\cdot 10^{18} $ eV from protons to heavier nuclei.", "A detailed comparison of this data with air shower simulations reveals strong restrictions on the amount of light nuclei (protons and He) in the observed flux.", "We find a robust upper bound on the observed proton fraction of the UHECRs flux and we rule out a composition dominated by protons and He.", "Acceleration and propagation effects lead to an observed composition that is different from the one at the source.", "Using a simple toy model that take into account these effects, we show that the observations requires an extreme metallicity at the sources with metals to protons mass ratio of 1:1, a ratio that is larger by a factor of a hundred than the solar abundance.", "This composition imposes an almost impossible constraint on all current astrophysical models for UHECRs accelerators.", "This may provide a first hint towards new physics that emerges at $\\sim 100$ TeV and leads to a larger proton cross section at these energies.", "Among the most interesting results of the Pierre Auger Observatory (PAO) are the penetration depth measurements [1], [2], [3]: the observed mean depth where maximal number of secondaries are generated, $\\langle X_{max} \\rangle (E)$ , and the fluctuations of this quantity $RMS(X_{max})(E)\\equiv \\sigma (X_{max}))$ .", "When compared with extensive shower simulations [4], [5], [6], [7] both $\\langle X_{max} \\rangle $ and $\\sigma (X_{max})$ are consistent with a protonic composition around $1~EeV$ .", "However, at higher energies, $\\langle X_{max} \\rangle $ and $\\sigma (X_{max})$ fall below the protonic simulated values, towards values estimates for intermediate mass (e.g.", "iron) nucleiThe interactions between UHECRs and atmospheric particles takes place at CM energies of $\\sim 100$ TeV.", "This is about a hundred times higher than energies in which cross sections have been measured.", ".", "At first sight this suggests a transition, around 10 EeV, from protons to intermediate mass nuclei [1]While PAO has the best statistics, the High resolution fly's eye (HiRes) and the telescope array (TA) observatories [8], [9] composition results are consistent with proton dominated composition all the way up to the high end of the spectrum.", "This disagreement among the different observatories is still to be settled.. We explore here the implications of the penetration depth data combined with the observed energy spectrum [10], [11], [12] .", "Consider a cosmic ray flux composed of N species each with a fraction $f_j$ , a mean penetration depth, $\\langle X_{max} \\rangle _j$ , and RMS variation $\\sigma _j $ .", "At each energy these are related to the measured mean and RMS values as: $\\sum \\limits _{j=1}^N f_j \\langle X_{max} \\rangle _j = \\langle X_{max} \\rangle , $ $\\sum \\limits _{j=1}^N f_j (\\sigma _j^2+\\langle X_{max} \\rangle _j^2) - \\langle X_{max} \\rangle ^2 = \\sigma (X_{max})^2 .", "$ In the following we examine several possible solutions of these equations for different compositions.", "Consider, first, a mixture of protons (denoted p) and another arbitrary component, denoted 0.", "This arbitrary component may be a single species or a combination of a few.", "The condition $ \\sigma _0^2>0$ yields an upper bound on $f_p(E)$ : $& f_p \\le \\frac{1}{2} \\biggl \\lbrace 1+ \\frac{\\sigma (X_{max})^2+(\\langle X_{max} \\rangle -\\langle X_{max} \\rangle _p )^2}{\\sigma _p^2} - \\\\\\nonumber &\\sqrt{\\bigl [1+\\frac{\\sigma (X_{max})^2+(\\langle X_{max} \\rangle -\\langle X_{max} \\rangle _p )^2}{\\sigma _p^2}\\bigr ]^2 -\\frac{4\\sigma (X_{max})^2}{\\sigma _p^2} }~ \\biggr \\rbrace .$ Fig.", "REF depicts the upper limit obtained using the observed values of $\\langle X_{max} \\rangle $ and $\\sigma (X_{max})$ and the simulated values of $\\langle X_{max} \\rangle _p$ and $\\sigma _p$ .", "The maximal proton fraction is smaller than $50 \\%$ at $E>10^{19}$ $eV$ and it decreases below 30% at higher energies.", "This upper bound depends only on the observed PAO data and on the shower simulation results for protons (it does not even depend on the shower simulations for nuclei).", "It is independent of the acceleration or propagation of the UHECRs.", "When we replace the hypothetical ingredient with any composition of He, N, Si and Fe the upper bound on $f_p$ at the two highest energy bins is smaller by $\\approx 10\\%$ than the upper limit derived using $\\sigma _0=0$ .", "Figure: An upper bound on the proton fraction of the observed flux as a function of energy, calculated using the PAO data and different extensive air shower simulations: QGSJET01 , QGSJET-II , Sibyll2.1 and EPOSv1.99 (see legend).Protons and Fe survives best the interactions with the cosmic photon field while propagating from the sources to Earth [13].", "As such they are the most natural UHECR ingredients.", "However, with just two components the system of eqs.", "(REF )-(REF ) is overdetermined.", "Wilk and Włodarczyk [14] have shown that there is no consistent solution for protons and Fe within the error bars of the PAO data and the shower simulations.", "Eqs.", "(REF )-(REF ) have a marginal ($\\chi ^2 / dof \\approx 1$ ) solution for a mixture of protons and He (see also [15]).", "However, when propagation effects are taken into account this composition can be ruled out.", "He nuclei photodisintegrate rapidly on their way from the sources to Earth.", "For simplicity we neglect redshift effects and assuming a uniform distribution of sources.", "Under this approximations the fraction of He nuclei with energy E surviving and reaching earth is $F_{{GZK}_{He}}(E)= l_{HE}/l_h \\ll 1$ , where $l_{HE}$ is the mean free path for He photodisintegration and $l_h$ is the horizon size.", "For a given observed He flux, $(1-f_p(E)) J(E)$ , the flux at the source is $(1-f_p(E)) J(E)/F_{{GZK}_{He}}(E)$ .", "Since $F_{{GZK}_{He}}(E)\\ll 1 $ most of the He nuclei disintegrate producing secondary protons with energy $E/4$ .", "The resulting secondary proton flux, $J_{p,sec}(E)$ is(i) The protons GZK distance is comparable to the horizon distance at these energies.", "(ii) We have overestimated here the GZK distances of $^3He$ and $^2H$ as equal to the GZK distance of $^4He$ .", ": $J_{p,sec}(E) \\approx 4(1- F_{{GZK}_{He}}(4E))^3 \\frac{(1-f_p(4E)) J(4E)}{ F_{{GZK}_{He}}(4E)}$ Using the upper bound on $f_p(E)$ obtained earlier (eq.", "(REF )) we find (see fig.", "REF ) that this secondary proton flux is larger than the maximal proton flux allowed.", "Figure: The total observed total flux (blue dotted line with diamonds) and the calculated secondary proton flux arising from He disintegration (eq.", "()) (red dashed line with circles), for a composition with a maximal number of primary protons(satisfying eq.", "())and He (color online).", "At high energies the number of secondary protons exceeds the observed flux.Kampert and Unger [15] have shown that a mixture of protons, He and intermediate elements like N, Si and Fe can provide a solution of eqs.", "(REF ) and (REF ) for the observed composition which is within the uncertainties of the observed data and the simulations.", "However, as we have seen for p and He, to determine the sources' composition we need to take propagation effects into account.", "Moreover, since different species are accelerated differently within a given accelerator, acceleration should also be considered.", "To examine these effects we consider a toy model based on only two components: protons and Fe.", "As mentioned earlier, this composition cannot satisfy equations (REF ) and (REF ) and other intermediate elements in addition to Fe are needed.", "Therefore, when we examine a proton and Fe composition we consider only the overall spectra and $\\langle X_{max} \\rangle $ and we ignore, for simplicity, the RMS data.", "This simple example is sufficient for demonstrating the nature of the problem.", "Any electromagnetic acceleration process that accelerates protons to energy $E$ accelerates nuclei (with charge $Z$ ) to energy In the diffusive shock acceleration, a proton and a nucleus, with an atomic weight A, crossing the shock front will have the same Lorentz factor, and therefore a nucleus will be A times more energetic.", "This suggest that we need to compare nuclei at energy E with protons with energy $E/A$ and not $E/Z$ .", "This will add a factor of $ (A/Z)^{\\alpha -1} \\sim 3 $ to the composition ratio.", "This factor doesn't change qualitatively the results $Z E$ .", "This suggests a natural [15], [15] explanation for the transition in composition: the source accelerates protons to a power law energy distribution, $E^{-\\alpha }$ up to some maximal energy where a gradual cutoff begins.", "The same source accelerates nuclei to the same power law but up to an energy that is $Z$ times larger.", "This naturally produces a heavier observed composition than the one at the source and may suggest that the drop at very high energies in the UHECR flux is not necessarily due to a GZK effect but simply due to lack of available accelerators that can accelerate UHECRs to extremely large energies [15].", "We characterized the accelerator's cutoff by an unknown function of the rigidity, $g({E}/{Z})$ , with $0 \\le g(E) \\le 1$ and $g(E)=1$ at low energies.", "However, before adopting this model propagation effects should also be taken into account.", "Like acceleration, propagation in the IGM magnetic field depends on the rigidity.", "On the other hand GZK attenuation depends on the nucleus at hand.", "In this energy range (1.1-35 EeV), it is negligible for protons but it is significant for all nuclei.", "We characterize the propagation effects using $F_{{GZK}_{Fe}}=l_{Fe}/l_h$ [15].", "Under these assumptions the total observed UHECRs flux is: $J(E)=c_p g(E) E^{-\\alpha }+c_{Fe} F_{{GZK}_{Fe}}(E) g(\\frac{E}{26}) \\frac{E^{-\\alpha }}{26^{1-\\alpha }}$ Where $c_{p},c_{Fe}$ are normalization factors for the proton and Fe nuclei fluxes respectively.", "$c_p$ is obtained by the condition $g(E)=1$ at the minimal energy.", "The normalization of $c_{Fe}$ is such that $c_{Fe}/c_{p}$ equals the Fe nuclei to protons number ratio at the source (before acceleration)Within this acceleration model the number of Fe nuclei accelerated to energies larger than 26E equals to $c_{Fe}/c_{p}$ times the number of protons accelerated to energies larger than E (Note that [15] use somewhat different notations).", "Using the proton fraction $ f_{p}(E) \\equiv {c_p g(E) E^{-\\alpha }}/{J(E)}$ and $g(E/26)=1$ , which is valid over the relevant energy range (1.1-35 EeV) eq.", "(REF ) becomes: $\\frac{J(E)(1-f_{p}(E))}{F_{{GZK}_{Fe}}(E)} = \\frac{c_{Fe}}{26^{1-\\alpha }} E^{-\\alpha } ,$ We solve eq.", "(REF ) for $f_p(E)$ using the measured $\\langle X_{max} \\rangle $ and the simulated values for protons and Fe.", "Now that all the quantities at the l.h.s of (REF ) are known we fit a power law ($E^{-\\alpha }$ ) to the l.h.s to to obtain $\\alpha $ and $c_{Fe}$ .", "The best fit results are: $\\alpha = 2.1\\pm 0.1$ , $c_{Fe}/c_p = (2.2 \\pm 0.6) \\cdot 10^{-2} $ , with ${\\chi ^2}/{dof}=0.21$ .", "Figure: The l.h.s of eq.", "() ( with errorbars, blue circles for Fe, green squares for Si) and the best fit for the r.h.s of eq.", "() (green solid line for Si, blue dashed line for Fe, color online).", "The Si results are shiftedto the right for clarity.", "The best fit results for Fe and protons: α=2.1±0.1\\alpha = 2.1\\pm 0.1 , c Fe /c p =(2.2±0.6)·10 -2 c_{Fe}/ c_p = (2.2 \\pm 0.6) \\cdot 10^{-2} , with χ 2 /dof=0.21{\\chi ^2}/{dof}=0.21 and for Si and protons:α=1.9±0.1\\alpha = 1.9 \\pm 0.1, c Si /c p =(5±1)·10 -2 c_{Si}/ c_p = (5 \\pm 1) \\cdot 10^{-2} , with χ 2 /dof=0.4{\\chi ^2}/{dof}=0.4.The number ratio we obtained corresponds to a mass ratio of $ \\approx 1:1$ .", "A similar analysis with Si instead of Fe yields $\\alpha = 1.9 \\pm 0.1$ , $c_{Si}/ c_p(E) = (5 \\pm 1) \\cdot 10^{-2} $ , with ${\\chi ^2}/{dof}=0.4$ .", "The mass ratio is again $ \\approx 1:1$ .", "As both Fe and Si show almost the same trend a composite solution that will satisfy both the $\\langle X_{max}\\rangle $ and $\\sigma (X_{max})$ data [15] will have similar features and a $\\sim 1:1$ mass ratio between the protons and the metals.", "Note that we have neglected secondary protons arising from photodisintegration.", "Taking those into account would have resulted in even higher metallicity.", "Shifting the observed energies by about 20%, as suggested by comparison of the PAO spectra with the spectra observed in other main UHECRs observatories [16], does not change qualitatively our results.", "Interestingly, this extremely heavy composition at the source is comparable to the upper limit obtained using the angular distribution [15] of these UHECRs.", "These ratios of ${N_{Fe}(>26E)} / {N_p(>E)}$ are $>0.072$ for the VCV catalogue and $ >0.084$ for correlations with Cen A.", "Finally, we note that the spectral index, found here, is much harder than what observed in lower energies, $\\alpha =3$ [15].", "This arises from the GZK attenuation affecting nuclei at these energies.", "Such hard spectra were obtained in detailed propagation simulations [15], [15], [15].", "We have shown that the PAO penetration depth measurements and the penetration depth numerical simulations yield a robust upper limit on the observed proton fraction of the UHECRs flux.", "This limit drops below $50\\%$ at energies higher than 10 EeV and below 30% at higher energies.", "These measurements are inconsistent with the composition of 75% protons and 24% He, that is common in the Universe and they require a significant fraction of intermediate mass nuclei.", "The conversion of the observed composition to the composition at the source depends on the acceleration and propagation.", "Using a simple toy model we find that the protons to metals mass ratio at the source should be about $1:1$ .", "This metallicity is larger by a factor of a hundred than the solar metal abundance of $\\approx 1$ %, which reflects typical metallicity in the Universe.", "This high metallically puts a new severe constraint on the sources, since objects dominated by nuclei heavier than He are rare in the astrophysical landscape.", "Active galactic nuclei (AGNs) are natural UHECRs accelerators [15], [15].", "Most AGNs can accelerate metals to $\\sim 10^{20}$ eV and protons to an order of magnitude lower.", "This would produce, naturally, the observed composition transition [15], [15] as well as the cutoff in the spectrum at higher energies [15].", "However this would require a very heavy composition, whereas AGNs typically show solar-like metallicities [15].", "Gamma ray bursts (GRBs) are another natural UHECRs accelerators [15] (see however [15], [15]).", "If UHECRs are produced by GRBs' internal shocks, they are composed of the original material of the jets.", "One has to invoke, in this case, a very efficient nucleosynthesis within the jets and survival of the produced nuclei during the acceleration [15], [15].", "It is not clear that this can be achieved generically.", "Recall that the UHECR output of GRBs should be comparable or larger than their $\\gamma $ -rays output.", "Alternatively if UHECRs are produced by external shocks they will be composed by the circum-burst winds surrounding the star.", "GRBs' progenitors, Wolf - Rayet stars, have He, C and O dominated wind whose composition is too light.", "A variation on this theme was proposed by Liu and Wang [15] who invoke, instead of regular GRBs, low-luminosity GRBs which they describe as “semi relativistic core collapse supernovae that involve an engine activity\" and call “hypernovae\".", "These are also based on Wolf - Rayet progenitors.", "However, low-luminosity GRBs jets do not penetrate the stellar envelope and the observed emission is produced by a shock breakout [17].", "Thus, it is not clear how could these bursts accelerate UHECRs in the first place.", "Furthermore, as sources of heavy UHECRs they suffer from all problems mentioned earlier concerning regular GRBs.", "A rapidly rotating young pulsar with a strong but reasonable magnetic field ($\\sim 10^{13}$ G) can accelerate Fe to UHECR energies [18].", "Fang et al.", "[19] suggested that young pulsars are UHECR sources and the origin of the heavy composition is the Fe rich crust.", "However, X-ray observations of neutron stars suggest the existence of an atmosphere composed of proton and light elements above the crust (see [18] and citations therein).", "Acceleration of this atmospheric component will result in a light composition.", "This poses a serious doubt concerning this model.", "Overall, while astrophysical heavy UHECRs sources cannot be ruled out with absolute certainty, the strict constraint on the composition obtained here (that constrain both the protonic and the He components) makes the UHECRs sources puzzle even harder to solve.", "In particular it rules out the most natural sources, AGNs.", "There is no single clear model that naturally produces UHECRs with such a composition.", "One may search for acceleration processes that are not rigidity dependent and favor heavy nuclei over light ones, however such a process is not readily available.", "Given this situation one can consider the following alternatives to the heavy composition.", "First, the observational data might be incorrect or somehow dominated by poor statistics: these results are based on about 1500 events at the lowest energy bin and on only about 50 at the highest one.", "The possibility of a miscalculation in the shower simulations is unlikely.", "Different simulations [4], [5], [6], [7] obtain comparable results.", "However, the simulations depend on the extrapolations of the proton's cross section from the measured energies of a few TeV to the range of an UHECR - atmospheric nuclei collision, which are factor of 100 higher in energy.", "Is it possible that this extrapolation breaks down?", "A larger cross section than the one extrapolated can explain the shorter penetration depth.", "If so these findings might provide hints of a new physics that set in at energies of several dozen TeV[20].", "This research was partially supported by an ERC Advanced research grant and by the ISF center for High Energy Astrophysics." ] ]

[ [ "Rare earth substitution in lattice-tuned Sr0.3Ca0.7Fe2As2 solid\n solutions" ], [ "Abstract The effects of aliovalent rare earth substitution on the physical properties of Sr0.3Ca0.7Fe2As2 solid solutions are explored.", "Electrical transport, magnetic susceptibility and structural characterization data as a function of La substitution into (Sr_1-y_Ca_y)_1-x_La_x_Fe2As2 single crystals confirm the ability to suppress the antiferromagnetic ordering temperature from 200 K in the undoped compound down to 100 K approaching the solubility limit of La.", "Despite up to ~30% La substitution, the persistence of magnetic order and lack of any signature of superconductivity above 10 K present a contrasting phase diagram to that of Ca1-xLaxFe2As2, indicating that the suppression of magnetic order is necessary to induce the high-temperature superconducting phase observed in Ca1-xLaxFe2As2." ], [ "Introduction: ", "The iron-pnictide superconductors have garnered much attention since the first observation [1] that chemical manipulation of these materials could lead to high-temperature superconductivity [2].", "Particular interest has fallen on the structural “122” family with the ThCr$_2$ Si$_2$ crystal structure, because of their intermetallic nature and relative ease of synthesis.", "Superconducting transition temperatures ($T_c$ ) as high as 25 K have been observed in transition metal substituted BaFe$_2$ As$_2$  [3], while $T_c$  up to 38 K has been seen when hole doping on the alkaline earth site in Ba$_{1-x}$ K$_x$ Fe$_2$ As$_2$[4].", "Recently, superconductivity with $T_c$  values reaching as high as 47 K was observed to occur via aliovalent rare earth substitution into the alkaline earth site of CaFe$_2$ As$_2$  [5], approaching the highest $T_c$  values of all iron-based superconductors ($\\sim 55$  K in SmO$_{1-x}$ F$_x$ FeAs  [6]).", "Stabilized by the suppression of antiferromagnetic order via a combination of both chemical pressure and electron doping, the superconductivity observed in the Ca$_{1-x}$ R$_x$ Fe$_2$ As$_2$  series (R = La, Ce, Pr, Nd) arises in a highly tunable system where the choice of rare earth species allows for structural tuning toward the collapsed tetragonal phase  [7], such as shown in the Pr- and Nd-doped series, or more simply electron doping with minimal change in unit cell size as shown for La substitution [5].", "The close ionic radius match of the light rare earths with Ca in the CaFe$_2$ As$_2$ system makes for an ideal system with which to study the interplay between superconductivity and structural instability.", "However, the observations of partial volume-fraction diamagnetic screening in low-concentration rare earth-substituted CaFe$_2$ As$_2$ [5], [9], [8], where the solubility limit of La is 30%, poses a challenge to understanding the reasons for a lack of bulk-phase superconductivity.", "One promising route is through the inclusion of higher concentrations of rare earth substitution.", "Previous studies of La substitution into the SrFe$_2$ As$_2$  system [10], with $T_c$ values up to  $\\sim 22$  K, have shown a significant increase in the superconducting volume fraction when La content reaches 40% La for Sr, at which point the volume fraction jumps to nearly 70%.", "The introduction of higher concentrations of rare earth in the CaFe$_2$ As$_2$ series is thus a promising route to achieving bulk superconductivity with high $T_c$ values.", "While it has been shown that the application of pressure during synthesis can provide the desired results [10], a clue to another possible route is provided by the materials themselves (i.e., in the case of the Ca$_{1-x}$ La$_x$ Fe$_2$ As$_2$ series, where the $c$ -axis remains constant but the $a$ -axis increases with increasing rare-earth concentration up to the solubility limit of $\\sim 30\\%$ [5].)", "It follows that a larger unit cell would likely be able to accomodate a larger concentration of rare earths.", "It has also been shown previously that substituting Sr in the place of Ca leads to a controlled expansion of the unit cell in accordance with Vegard's Law [11]; therefore, such a solid solution may serve as the base formula into which rare earth atoms can be further doped.", "In this article, we examine the feasibility of increasing the solubility limit of rare earth substitution by doping La into the (Sr,Ca)Fe$_2$ As$_2$ solid solution system.", "The synthesis of such single crystals with La substitution shows widely ranging chemical compositions and suggest a competition between Sr and La.", "We detail the effects of increasing La content using systematic X-ray, electrical transport, and magnetization measurements, and compare these effects with those observed when La is doped into the CaFe$_2$ As$_2$  parent material, specifically tracking the suppression of the AFM ordering temperature and signs of superconductivity in the system.", "Figure: Results of energy dispersive spectroscopic (EDS) analysis of the elements occupying the alkaline earth site.", "a) Measured (actual) La content vs. Nominal (pre-reaction) La content.", "Black close circles represent samples with pre-reaction stoichiometries of (Sr 0.3 _{0.3}Ca 0.7 _{0.7}) 1-x _{1-x}La x _xFe 2 _2As 2 _2.", "Colored symbols represent samples with pre-reaction stoichiometries of (Sr 1-y _{1-y}Ca y _y) 1-x _{1-x}La x _xFe 2 _2As 2 _2, where yy is denoted in the legend.b) Measured La content vs. measured Sr content in the same samples shown in a).", "All values are taken from EDS analysis.", "The black solid line serves as the upper bound for the solubility of La." ], [ "Experiment", "Single crystals were grown via a self-flux method using elemental stoichiometries of $(1-y)(1-x)$ :$y(1-x)$ :$x$ :4 for Sr:Ca:La:FeAs according to the formula (Sr$_{1-y}$ Ca$_y$ )$_{1-x}$ La$_x$ Fe$_2$ As$_2$  with FeAs flux [12].", "Starting materials were placed inside alumina crucibles and sealed in quartz tubes under partial atmospheric pressure of Ar.", "The growths were heated and allowed to slow cool, resulting in crystals with typical dimensions of ($5.0 \\times 5.0 \\times 0.10$ ) mm$^3$ , which were mechanically separated from the frozen flux.", "Chemical analysis was obtained using both energy-dispersive (EDS) and wavelength-dispersive (WDS) X-ray spectroscopy, showing 1:2:2 stoichiometry between (Sr,Ca,La), Fe, and As concentrations.", "EDS was conducted on a large number of samples in order to determine general concentration trends, while WDS was used to determine very accurately the concentrations of elements for samples used in X-ray, resistivity, and magnetization measurements.", "Single crystal X-ray diffraction was performed on a Bruker Smart Apex2 diffractometer equipped with a CCD detector, graphite monochromator, and monocap collimator.", "Crystal structures were refined (SHELXL-97 package) using the I4/mmm space group against 113 and 106 independent reflections measured at 250 K and corrected for absorption using the integration method based on face indexing (SADABS software).", "Because three different atoms were occupying the same crystallographic site, refinement of chemical compositions was not possible, and refinement instead was focused on obtaining unit cell parameters.", "Resistivity measurements were performed using the standard four-probe ac method, via gold wire and In/Sn solder contacts with typical contact resistance of $\\sim 0.5~\\Omega $ at room temperature, using up to 1 mA excitation currents at low temperatures.", "Magnetic susceptibility was measured using a commercial superconducting quantum interference device magnetometer." ], [ "Results and Discussion", "EDS of the actual concentration of La in the (Sr$_{1-y}$ Ca$_y$ )$_{1-x}$ La$_x$ Fe$_2$ As$_2$  series reveals that at low La values, the actual La content is higher than the nominal content, rising to a limit of $\\sim 30$ % for growths with nominal La higher than 50% as shown in FIG.", "1a.", "While it is illustrated that increasing the starting ratio of Ca:Sr leads to higher La concentrations in the final materials, a stronger correlation between the Sr, Ca, and La concentrations can be found by plotting the measured La content against the measured Sr content as shown in FIG.", "1b.", "It is evident here that La and Sr are inversely correlated in this material and increasing the Sr concentration seems to strongly limit the amount of La that is able to dope into the sample.", "Figure: Resistivity of (Sr,Ca,La)Fe 2 _2As 2 _2 samples as a function of temperature, normalized to 300 K and then vertically shifted for clarity; the Ca-rich samples are presented in (a), while the Sr-rich samples are presented in (b).Single crystal x-ray analysis allows us to analyze the progression of the lattice parameters as a function of the concentrations of Sr, Ca, and La in each sample.", "In FIG.", "2, we plot the lattice parameters of the samples used in this study alongside the lattice parameters observed for solid solutions of the parent compounds SrFe$_2$ As$_2$ and CaFe$_2$ As$_2$  [11].", "Previously, it was shown that doping La for Ca in CaFe$_2$ As$_2$  results in a $c$ -axis lattice parameter that does not change, despite expansion of the $a$ -axis [5].", "Taking this into account, we have plotted these points as Sr$_{1-x}$ (Ca,La)$_x$ – the Ca and La values are taken together in order to determine the composition $x$ , which places our points in good agreement with the $c$ -axis values from the Sr$_{1-x}$ Ca$_x$ Fe$_2$ As$_2$  study (FIG.", "2b;, however, the $a$ -axis values diverge as La content increases (FIG.", "2a).", "This implies that by selecting the proper Sr, Ca, and La content, we can tune the $a$ -axis and $c$ -axis parameters nearly independently.", "This is in striking contrast to most doping studies on these materials, which show a strong coupling between $a$ and $c$ -axes lattice parameters [13].", "As seen in similar doping studies of iron-pnictides [5], [10], [14], it is expected that increasing the La content in these samples will be manifest in resistivity data as a systematic decrease in the Neél ordering temperature $T_N$ .", "Electrical resistivity data of these samples (shown in FIG.", "2) roughly resembles the expected behavior, as it is obvious that $T_N$  is suppressed upon increased doping of La into the system.", "A key difference here lies in the ranging chemical compositions obtained using WDS; subtraction of the La content leaves two classes of samples, i.e.", "the Ca-rich (shown in FIG.", "3a) and the Sr-rich (shown in FIG.", "3b).", "In the Sr-rich case, no sample was found to contain less than $\\sim 10$ % La or more than $\\sim 22$ % La, whereas in the Ca-rich case, a much wider range of La concentrations can be found (up to $\\sim 30$ % La).", "In the Sr-rich samples, $T_N$  is gradually suppressed down to $\\sim $ 130 K and no superconductivity is found to exist due to La substitution.", "In the Ca-rich case, $T_N$  is suppressed to a slightly lower value of $\\sim $ 100 K, but no trace of a superconducting phase similar to that seen in the Ca$_{1-x}$ La$_x$ Fe$_2$ As$_2$  case [5], where high-$T_c$  values in the range 30-47 K are indicative of rare earth doping-induced superconductivity, was found.", "Note that the Ca-rich samples do exhibit traces of a superconductivity onset near $T^*\\sim $ 10 K, which we attribute to the strain-induced phase often observed under non-hydrostatic pressure conditions [15], [16] and posited to nucleate at AFM domain walls [17].", "It is interesting to highlight the fact that this “10 K” phase appears predominantly in Ca-rich samples, suggesting its stability is tied strictly to the CaFe$_2$ As$_2$ magnetic and/or crystallographic structure.", "Figure: Magnetic susceptibility (χ\\chi ) vs. temperature for several La doped samples.", "The data are shifted along the vertical axis for clarity.", "The kink in each curve indicates the temperature of the antiferromagnetic transition, T N T_N .", "The inset shows the low field magnetic susceptibility of the 29%-La doped sample at low temperatures, where superconductivity below T c T_c ∼10\\sim 10 K is seen; the estimated superconducting volume fraction is ∼11\\sim 11%.Temperature dependence of magnetic susceptibility $\\chi $ ($T$ ) data for Ca-rich samples (shown in FIG.", "4) corroborates the picture drawn by electrical transport data.", "As expected from previous studies, $T_N$  is revealed as an antiferromagnetic ordering temperature traced by a kink in $\\chi $ ($T$ ).", "The suppression of $T_N$ occurs at the same rate observed in transport data, with ordering at $\\sim $ 100 K still present for samples which show superconductivity at $T^*\\sim $ 10 K (inset of FIG.", "4) at low field.", "The Meissner screening fraction of the $\\sim $ 10 K superconductivity of this sample is still seen to be relatively small, of the order of 10%.", "A slight Curie tail is observed in the highly-doped La samples at low temperatures, similar to that observed in Ca$_{1-x}$ La$_x$ Fe$_2$ As$_2$ samples [5], which may arise from paramagnetism associated with the FeAs lattice.", "Figure: (a) The phase diagram for the Ca 1-x _{1-x}La x _xFe 2 _2As 2 _2 system , where high T c T_c  superconductivity is induced on the border of AFM order and coexists with the T * ∼T^*\\sim 10 K superconducting phase.", "(b) Suggested phase diagram for the (Sr 1-y _{1-y}Ca y _y) 1-x _{1-x}La x _xFe 2 _2As 2 _2 system.", "The solid and broken lines are guides to the eye.Figure 5 presents a proposed phase diagram for the (Sr$_{1-y}$ Ca$_y$ )$_{1-x}$ La$_x$ Fe$_2$ As$_2$  system in comparison with that of the Ca$_{1-x}$ La$_x$ Fe$_2$ As$_2$  system [5].", "As shown in FIG.", "5b, a key observation in (Sr$_{1-y}$ Ca$_y$ )$_{1-x}$ La$_x$ Fe$_2$ As$_2$  is the absence of the high $T_c$   superconducting phase, which is observed ubiquitously in the rare-earth doped CaFe$_2$ As$_2$  materials upon suppression of the AFM phase, despite a similar electron doping scheme.", "The lack of a high-$T_c$ superconducting phase in (Sr$_{1-y}$ Ca$_y$ )$_{1-x}$ La$_x$ Fe$_2$ As$_2$ samples with La concentrations more than sufficient to induce superconductivity in Ca$_{1-x}$ La$_x$ Fe$_2$ As$_2$ (FIG.", "5a) suggests that a scenario where the superconducting phase arises solely from the presence of sufficient rare earth atoms that presumably cluster or percolate in some manner is improbable.", "The persistence of AFM order up to high concentrations of La may play a role here, as it seems as though high $T_c$  superconductivity competes with AFM order and does not emerge until the complete suppression of magnetic ordering.", "Indeed, in every other rare-earth substituted 122 system, 30+ K superconductivity and antiferromagnetism are never found to coexist [5].", "This agrees with the occurrence of the highest $T_c$ in the 1111 iron-pnictide family [6], [2], indicating that the highest $T_c$ superconducting phase and magnetic ordering may be mutually exclusive.", "Of course, further investigation will be necessary to bear out such a result.", "However, the conspicuous absence here of the high $T_c$  phase, which has been thought to be an impurity phase of ReOFeAs, despite similar growth techniques and materials, lends credence to the idea that it is in fact intrinsic to the rare-earth substituted CaFe$_2$ As$_2$  system." ], [ "Summary", "In summary, we have studied the effect of electron doping by La substitution on (Sr$_{1-y}$ Ca$_y$ )$_{1-x}$ La$_x$ Fe$_2$ As$_2$  solid solutions by growing single crystals.", "We have constructed a phase diagram based on transport, magnetic susceptibility and structural characterization.", "Chemical analysis indicates a strong inverse correlation between Sr and La.", "Nonetheless, independent tunability of the $a$ - and $c$ -axis lattice parameters can be achieved.", "The Sr-rich and Ca-rich regions show differing behavior; in Ca-rich samples, antiferromagnetic ordering is found to coexist with superconductivity with $T^*\\sim $ 10 K with a volume fraction $\\sim 10$ %.", "But, in contrast to CaFe$_2$ As$_2$ , in neither case is $T_N$  fully suppressed and no high-$T_c$   superconducting phase is observed, placing the constraint that complete suppression of AFM order is a necessary condition for the latter phase, which may provide an important clue for the superconducting pairing in the new iron-superconductors.", "ACKNOWLEDGEMENTS This work was supported by AFOSR-MURI Grant FA9550-09-1-0603." ] ]

[ [ "Chondrule Formation in Bow Shocks around Eccentric Planetary Embryos" ], [ "Abstract Recent isotopic studies of Martian meteorites by Dauphas & Pourmond (2011) have established that large (~ 3000 km radius) planetary embryos existed in the solar nebula at the same time that chondrules - millimeter-sized igneous inclusions found in meteorites - were forming.", "We model the formation of chondrules by passage through bow shocks around such a planetary embryo on an eccentric orbit.", "We numerically model the hydrodynamics of the flow, and find that such large bodies retain an atmosphere, with Kelvin-Helmholtz instabilities allowing mixing of this atmosphere with the gas and particles flowing past the embryo.", "We calculate the trajectories of chondrules flowing past the body, and find that they are not accreted by the protoplanet, but may instead flow through volatiles outgassed from the planet's magma ocean.", "In contrast, chondrules are accreted onto smaller planetesimals.", "We calculate the thermal histories of chondrules passing through the bow shock.", "We find that peak temperatures and cooling rates are consistent with the formation of the dominant, porphyritic texture of most chondrules, assuming a modest enhancement above the likely solar nebula average value of chondrule densities (by a factor of 10), attributable to settling of chondrule precursors to the midplane of the disk or turbulent concentration.", "We calculate the rate at which a planetary embryo's eccentricity is damped and conclude that a single planetary embryo scattered into an eccentric orbit can, over ~ 10e5 years, produce ~ 10e24 g of chondrules.", "In principle, a small number (1-10) of eccentric planetary embryos can melt the observed mass of chondrules in a manner consistent with all known constraints." ], [ "Introduction", "Chondrules are submillimeter to millimeter-sized, mostly Fe-Mg silicate spherules found within chondritic meteorites.", "Chondrule precursors were heated to high temperatures and melted, and then slowly cooled and crystallized to form chondrules, while they were independent, free-floating objects in the early solar nebula.", "A wealth of elemental and isotopic compositional measurements, as well as mineralogical and petrological information, potentially place important constraints on the melting and cooling of chondrules.", "The physical conditions and processes acting in the early solar nebula could be deciphered, if the chondrule formation mechanism could be understood.", "Various chondrule formation models have been proposed.", "These are assessed against their ability to match the following meteoritic constraints: (1) Models must explain why most chondrules appear to have formed only millions of years after the oldest solids formed in the solar nebula (calcium-rich, aluminum-rich inclusions, otherwise known as CAIs).", "Radiometric (Al-Mg and Pb-Pb) dating shows that nearly all extant chondrules formed $\\approx 1.5 -4 \\, {\\rm Myr}$ after CAIs, with the majority forming 2-3 Myr after CAIs (Kurahashi et al.", "2008; Villeneuve et al.", "2009).", "(2) Models must explain the high number density of chondrules in the formation region.", "About 5% of chondrules are compound chondrules, chondrules that fused together while still partially molten (Gooding & Keil 1981; Ciesla et al.", "2004a).", "Based on this frequency and an assumed relative velocity $100 \\, {\\rm cm} \\, {\\rm s}^{-1}$ , a number density of chondrules $\\sim 1 - 10 \\, {\\rm m}^{-3}$ has been inferred (Gooding & Keil 1981; Ciesla et al.", "2004a).", "Cuzzi & Alexander (2006) also require a density $> 10 \\, {\\rm m}^{-3}$ so that the partial pressure of volatiles following chondrule melting remains high enough to suppress substantial loss of further volatiles.", "This density is significantly higher than the average density of chondrules in the solar nebula.", "Based on a silicate-to-gas mass fraction $0.5\\%$ (Lodders 2003) and the observation that chondrules comprise 75% of the mass of an ordinary chondrite (e.g., Grossman 1988), we expect the chondrules-to-gas ratio to be $0.375\\%$ (although this value may be lower if a large percentage of the available solids are locked up in planetary embryos).", "Assuming a chondrule mass $m_{\\rm c} = 3.7 \\times 10^{-4} \\, {\\rm g}$ , consistent with a chondrule of radius $300 \\, \\mu {\\rm m}$ and internal density $3.3 \\, {\\rm g} \\, {\\rm cm}^{-3}$ , and a nebula gas density $1 \\times 10^{-9} \\, {\\rm g} \\, {\\rm cm}^{-3}$ (Desch 2007), we would expect a chondrule number density $0.01 \\, {\\cal C} \\, {\\rm m}^{-3}$ , where the chondrules-to-gas mass ratio is defined to be $3.75 \\times 10^{-3} \\, {\\cal C}$ , with ${\\cal C}$ representing an additional concentration factor.", "Here, the chondrule precursor radius of $300\\mu $ m is chosen to be consistent with chondrules found in most chondrites.", "It is reasonable, however, to assume that there was a size distribution between matrix-sized grains and large chondrule precursors in the solar nebula prior to a chondrule-forming event.", "Morris & Desch (2012) find that cooling rates are increased slightly for simple size distributions, but not enough to affect the general results of the shock model.", "We therefore consider only a single size for simplicity.", "In the chondrule-forming region, chondrules appear to have been concentrated by large factors, up to 100 times their background density (for a relative velocity of $100 \\, {\\rm cm} \\, {\\rm s}^{-1}$ ).", "(3) Models must allow for simultaneous chondrule formation over large regions ($> 10^3 \\, {\\rm km}$ in size) to limit diffusion of volatiles away from chondrules, in order to keep the partial pressures of the volatiles high enough to suppress isotopic fractionation of these volatiles (Cuzzi & Alexander 2006).", "Likewise, the existence of compound chondrules argues for regions larger than the mean free path of a chondrule through a cloud of chondrules, which is 100 - 1000 km for the densities quoted above.", "(4) Models must explain the relatively slow cooling of chondrules over hours.", "Comparison of chondrule petrologic textures with experimental analogs show that the 80% of chondrules in ordinary chondrites that have porphyritic textures (Gooding & Keil 1981) cooled from their peak temperatures at rates $\\sim 10 - 10^{3} \\, {\\rm K} \\, {\\rm hr}^{-1}$ and crystallized in a matter of hours (Desch & Connolly 2002; Hewins et al.", "2005; Connolly et al.", "2006; Lauretta et al.", "2006).", "(5) The partial pressure of Na may have been considerably higher than is easily explained by volatile loss from chondrules.", "Alexander et al.", "(2008) recently reported high mass fractions $\\approx 0.010 - 0.015 \\; {\\rm wt}\\%$ of Na$_2$ O in the cores of olivine phenocrysts of Semarkona chondrules, indicating that the olivine melt contained substantial Na.", "The implied partial pressures of Na vapor in the chondrule-forming region are $\\sim 10^{-5} \\, {\\rm bar}$ .", "If this vapor is supplied by the chondrules themselves, chondrule concentrations ${\\cal C} > 10^5$ would be required.", "Such a high chondrule concentration would have significant and even incredible physical consequences, but it has been difficult to explain the high Na concentrations in the chondrule olivine by any other method (Alexander et al.", "2008).", "Many additional constraints on chondrule formation exist as well (see Desch et al.", "2012, accepted).", "The model that appears most consistent with the constraints on chondrule formation is the nebular shock model, in which chondrules are melted by passage through shock waves in the solar nebula gas (Desch et al.", "2005, 2010, 2012).", "Of the sources of shocks considered to date, two were found to be most plausible by Desch et al.", "(2005): large-scale shocks driven by gravitational instabilities (e.g., Boss & Durisen 2005; Boley & Durisen 2008), and bow shocks driven by planetesimals on eccentric orbits (e.g., Hood 1998; Ciesla et al.", "2004b; Hood et al.", "2009; Hood & Weidenschilling 2011).", "A planetesimal or other body on an eccentric orbit will inevitably drive a bow shock in front of it, because its orbital velocity necessarily differs from that of the gas, which remains on circular orbits with nearly the Keplerian velocity $v_{\\rm K}$ .", "A body with eccentricity $e$ will have a purely azimuthal velocity at aphelion or perihelion, but one that differs from the local Keplerian velocity by an amount $\\sim e v_{\\rm K} / 2$ .", "At other locations in its orbit the protoplanet also has a radial velocity.", "If the body's orbit is inclined, this also enhances the relative velocity.", "The combined effect is that the body's velocity relative to the gas can remain a substantial fraction of the Keplerian velocity ($v_{\\rm K} \\sim 20 \\, {\\rm km} \\, {\\rm s}^{-1}$ at 2 AU) over its entire orbit.", "As one plausible example, which we discuss further in section 6.1, we show in Figure 1 the velocity difference $V_{\\rm s}$ between the gas and solid body on an eccentric orbit with semi-major axis $a = 1.25 \\, {\\rm AU}$ , eccentricity $e = 0.2$ and inclination $i = 15^{\\circ }$ .", "The relative velocity is seen to vary over the orbit but it remains $> 3 \\, {\\rm km} \\, {\\rm s}^{-1}$ at all times.", "As the typical sound speed in the gas is $< 1 \\, {\\rm km} \\, {\\rm s}^{-1}$ , a bow shock is the inevitable result of an eccentric planetary body.", "The authors above (Hood 1998; Ciesla et al.", "2004b; Hood et al.", "2009; Hood & Weidenschilling 2011) have argued that planetesimals, bodies up to several $\\times 10^2 \\, {\\rm km}$ in radius, on eccentric orbits, would be common enough to drive sufficient shocks to melt the observed mass of chondrules, which is $2 \\times 10^{-4} \\, M_{\\oplus }$ (Grossman 1988).", "Despite their presumed ubiquity, planetesimal bow shocks have not been favored as the chondrule-forming mechanism.", "Because chondrules are associated with asteroids and formation in the asteroid belt, only planetesimals $\\raisebox {-0.6ex}{\\, \\stackrel{\\raisebox {-.2ex}{\\textstyle <}}{\\sim }\\,}10^{3} \\, {\\rm km}$ in diameter have been considered so far.", "For such bodies, the size of the heated region, which is necessarily comparable to the size of the planetesimal itself, is probably too small to satisfy the above constraints on the chondrule-forming region.", "More importantly, the small size of the heated region means chondrules are never far from cold, unshocked gas to which they can radiate and cool.", "The optically thin geometry is expected to lead to very fast cooling rates, and matching the cooling rates necessary to reproduce the dominant porphyritic texture in particular, $\\raisebox {-0.6ex}{\\, \\stackrel{\\raisebox {-.2ex}{\\textstyle <}}{\\sim }\\,}10^3 \\, {\\rm K} \\, {\\rm hr}^{-1}$ , requires physical conditions that appear extreme (e.g., Ciesla et al.", "2004b; Morris et al.", "2010a,b).", "Intuitively, suppression of the radiation that cools the chondrules requires requires the optical depth $\\tau $ between the chondrule-forming region and the unshocked gas to be $\\raisebox {-0.6ex}{\\, \\stackrel{\\raisebox {-.2ex}{\\textstyle >}}{\\sim }\\,}1$ .", "In their treatment of the problem, Morris et al.", "(2010a,b) found that optical depths $\\tau \\raisebox {-0.6ex}{\\, \\stackrel{\\raisebox {-.2ex}{\\textstyle >}}{\\sim }\\,}0.1$ led to cooling rates $\\sim 10^3 \\, {\\rm K} \\, {\\rm hr}^{-1}$ , just barely consistent with the constraints.", "Our treatment of chondrule cooling here (see §4) suggests slightly slower cooling, but the cooling rates $\\sim 100 \\, {\\rm K} \\, {\\rm hr}^{-1}$ consistent with porphyritic textures require optical depths $\\tau \\sim 0.1$ .", "The optical depth is a function of the opacity and the size of the region.", "Dust grains that might otherwise increase the opacity are found to vaporize in shocks strong enough to melt chondrules (Morris & Desch 2010).", "The opacity of chondrules themselves, at their background levels, does not yield $\\tau > 0.1$ unless the size of the region is $l \\gg 3 \\times 10^{4} \\, {\\rm km}$ .", "Even if chondrules are concentrated by the considerable factor of 100 times their background level, so that $n_{\\rm c} = 1 \\, {\\rm m}^{-3}$ , planetesimals at least several hundred kilometers in radius are required to yield marginally consistent cooling rates (Ciesla et al.", "2004b; Morris et al.", "2010a,b).", "To the extent that mass is locked up in large planetary bodies rather than dust and chondrule-sized objects, even higher concentrations of chondrules above background levels are required.", "Another difficulty of the planetesimal bow shock model, one that has not been discussed previously, is that due to the small size of the planetesimal, chondrule precursors passing close enough to the body to be melted as chondrules are likely to be accreted by the planetesimal.", "Unlike the gas, whose velocity is immediately altered at the shock front to flow around the body, chondrules have sufficient inertia that they retain their pre-shock velocity until they have collided with their own mass of gas.", "Chondrules can be deflected around the body driving the bow shock only by moving a minimum length through the gas past the shock front.", "For typical parameters this length is $\\approx 300 \\, {\\rm km}$ .", "The stand-off distance between the bow shock and a 500 km-radius planetesimal driving it is typically $\\raisebox {-0.6ex}{\\, \\stackrel{\\raisebox {-.2ex}{\\textstyle <}}{\\sim }\\,}100 \\, {\\rm km}$ (see §2), meaning that accretion onto the planetesimal is probable.", "This is problematic because chondrules are expected to mix with cold nebular components before accretion, and a given chondrite will contain chondrules with a wide variety of ages.", "Chondrules melted in bow shocks around bodies larger than planetesimals may conform better to the constraints on chondrule cooling rates and the size of the chondrule forming region.", "The terrestrial planets have long been recognized to have formed from planetary embryos, bodies $> 1000 \\, {\\rm km}$ in diameter or even much larger, and it has been recognized that these embryos could have formed in a few Myr or less (Wetherill & Stewart 1993; Weidenschilling 1997).", "Mars itself is considered a starved planetary embryo (Chambers & Wetherill 1998), which allows the timing of embryo formation to be fixed.", "Previous Hf-W isotopic analyses had suggested Mars had differentiated within 10 Myr of CAI formation (Nimmo & Kleine 2007).", "More recently, the analysis of Hf and W, in conjunction with Th, by Dauphas & Pourmond (2011) suggests that Mars accreted 50% of its mass (and 80% of its radius) within 2 Myr of CAIs.", "Formation of the terrestrial planets requires dozens of such embryos.", "The timing of Mars's formation establishes that the solar nebula contained not just planetesimals, but protoplanets, during the epoch of chondrule formation.", "Moreover, some of these planetary embryos quite plausibly found themselves on eccentric orbits at least some of the time.", "Mutual scattering events are common in N-body simulations (Chambers 2001; Raymond et.", "al 2006).", "One recent model for the origin of Mars in particular suggests that it suffered close encounters with other bodies in an annulus interior to 1 AU, and was scattered into an inclined, eccentric orbit with aphelion at 1.5 AU (Hansen 2009).", "Alternatively, Walsh et al.", "(2011) have simulated the early migration of Jupiter and shown it leads to a truncated inner disk (consistent with the Hansen 2009 model) and an excess of dynamically excited planetary embryos (at least 5 in number) at 1.0 AU, from which they propose Mars formed.", "The true extent to which embryos were scattered is not well known, but is a common feature of planet formation models.", "The recent confirmation by Dauphas & Pourmand (2011) that large planetary embryos did exist at the time of chondrule formation, the expectation that such planetary embryos may be scattered onto eccentric orbits, and the larger size of the chondrule forming region around planetary embryos strongly motivate us to consider chondrule formation in bow shocks around planetary embryos several 1000 km in radius.", "Despite the obvious similarities to chondrule formation in planetesimal bow shocks, chondrule formation in planetary embryo bow shocks differs in several significant ways.", "First, as the planetary radius and size of the chondrule forming region increase above a threshold of about 1000 km (for the embryo radius), the region of heated chondrules becomes optically thick, fundamentally altering the chondrule cooling rates.", "Second, chondrules flowing around bodies larger than about 1000 km in radius can dynamically recouple with the gas and avoid hitting the embryo's surface.", "Chondrules in planetesimal bow shocks, if their impact parameters are less than the planetesimal radius, are likely to be accreted by the planetesimal.", "A third fundamental difference is that planetary embryos larger than about 1000 km in radius are able to accrete a primary atmosphere from the nebula and retain secondary outgassed atmospheres.", "The escape velocity from a body with radius $R_{\\rm p}$ and Mars-like density ($\\rho $ = 3.94 g cm$^{-3}$ ) is $1.5 \\, (R_{\\rm p} / 1000 \\, {\\rm km})^{1/2} \\, {\\rm km} \\, {\\rm s}^{-1}$ , compared to a typical sound speed $0.7 \\, {\\rm km} \\, {\\rm s}^{-1}$ (at 150 K).", "The atmosphere retained by the larger planetary embryo will affect the trajectories of gas and chondrules flowing past it, and will also alter the chemistry of the chondrule formation environment.", "We are motivated to study chondrule formation in planetary embryo bow shocks because of these fundamental differences.", "Here we present new calculations of the melting of chondrules in bow shocks around large protoplanets on eccentric orbits.", "We show that for reasonable parameters, chondrule formation in planetary embryo bow shocks is consistent with the major constraints listed above.", "We show that for the largest eccentricities and inclinations we consider plausible the shocks speeds of $6 - 8 \\, {\\rm km} \\, {\\rm s}^{-1}$ needed for chondrule formation (Morris et al.", "2010a,b; §4) are sustained for much of the protoplanet's orbit.", "For the first time, we follow the trajectories of chondrules formed in bow shocks and demonstrate that they are not accreted by the protoplanet, whereas chondrules formed in bow shocks around planetesimals are accreted.", "Based on the spatial distribution of chondrules, we calculate the optical depths and calculate chondrule cooling rates.", "We show that the large scale of planetary embryo bow shocks yields chondrule cooling rates consistent with meteoritic constraints.", "We show that chondrules formed in planetary embryo bow shocks do pass through what is essentially the planet's upper atmosphere, and therefore may be exposed to high vapor pressures of species outgassed from the protoplanet's magma ocean, including ${\\rm H}_{2}{\\rm O}$ and Na vapor.", "We compute the rate at which a protoplanet's eccentricity and inclination are damped, and show that the protoplanet's orbit remains eccentric and can drive shocks for $\\sim 10^5 \\, {\\rm yr}$ .", "We find that a single scattered planetary embryo can process up to $10^{24} \\, {\\rm g}$ of chondrules." ], [ "Hydrodynamics of Planetary Embryo Bow Shocks", "We have numerically investigated the hydrodynamics of a bow shock around a planetary embryo using the adaptive-mesh, Eulerian hydrodynamics code, FLASH (Fryxell et al.", "2000).", "We consider a stationary protoplanet on a cylindrical (axisymmetric) 2-D grid.", "We simulate the relative motion of the protoplanet through the gas by imposing a supersonic inflow of gas along the symmetry axis, originating at the $-z$ boundary.", "Our assumed boundary conditions allow for inflow on the $-z$ boundary, reflection on the symmetry axis, and diode (zero-gradient) boundary conditions on the other boundaries.", "We consider two physical cases: that of a planetesimal and one of a planetary embryo.", "In the first case of a planetesimal, we consider a body with mass $M_{\\rm p} = 2 \\times 10^{24} \\, {\\rm g}$ , a radius of $R_{\\rm p} = 500 \\, {\\rm km}$ , yielding a density of $3.80 \\, {\\rm g} \\, {\\rm cm}^{-3}$ (similar to Mars's density).", "The computational domain has radius 16,000 km and vertical extent 32,000 km.", "The simulation is run with 7 levels of refinement, resulting in our best grid resolution of 32 km = $X_{\\rm size}/[N_{X_{\\rm block}}\\cdot 2^{(\\rm levels\\;of\\;refinement - 1)}]$ , where $N_{X_{\\rm block}}$ = 8.", "In the second case of a planetary embryo, motivated by the results of Dauphas & Pourmand (2011), we initialize our simulations with a protoplanet of mass $M_{\\rm p} = 3.20 \\times 10^{26} \\, {\\rm g}$ (half of Mars's mass) and a radius $R_{\\rm p} = 2720 \\, {\\rm km}$ (80% of Mars's radius), with density $3.80 \\, {\\rm g} \\, {\\rm cm}^{-3}$ .", "Once again, the computational domain has radius 16,000 km and vertical extent 32,000 km, and the simulation is run with 8 levels of refinement.", "In both cases the body is embedded in a non-radiating ${\\rm H}_{2}$ / He gas with uniform initial density $\\rho _{0} = 1.0 \\times 10^{-9} \\, {\\rm g} \\, {\\rm cm}^{-3}$ , pressure $P_{0} = 5.3 \\times 10^{-6} \\, {\\rm atm}$ , temperature $T_{0} = 150 \\, {\\rm K}$ , and assumed adiabatic index $\\gamma = 7/5$ , appropriate for the solar nebula at around 2 AU (Desch 2007).", "The gravitational pull of the protoplanet was simulated using a point-mass gravity module.", "Before we initialize the inflow, we allow the planetary embryo to acquire a primary atmosphere of ${\\rm H}_{2}$ and He.", "It is straightforward to show that accretion of adiabatic gas will lead to a spherically symmetric profile $\\rho (r) = \\rho _{0} \\left[1 + \\frac{ r_{\\rm g} }{ r } \\right]^{1 / (\\gamma -1)},$ where $r$ is the distance from the protoplanet center, $r_{\\rm g} = (\\gamma -1) (G M_{\\rm p} \\rho _{0} / P_{0}) / \\gamma $ is akin to a scale height.", "For the case of the planetesimal, $r_{\\rm g} \\approx 71 \\, {\\rm km}$ , and the atmosphere that develops is very thin and hardly resolveable by our best grid resolution of 32 $\\, {\\rm km}$ .", "Physically it behaves as part of the planetesimal surface in our runs.", "For the case of the planetary embryo, $r_{\\rm g} \\approx 11,500 \\, {\\rm km}$ .", "At the protoplanet surface the density is expected to reach $6.2 \\times 10^{-8} \\, {\\rm g} \\, {\\rm cm}^{-3}$ , the pressure to reach $1.74 \\times 10^{-3} \\, {\\rm bar}$ , and the temperature to reach $780 \\, {\\rm K}$ , values that are 62.5, 330.0, and 5.2 times their background values, respectively.", "The effective scale height near the protoplanet surface is $\\approx 1000 \\, {\\rm km}$ , and is easily resolved by our best grid resolution of 32 km.", "The primary atmosphere acquired by the planetary embryo in our simulations matched this solution within a few percent.", "The mass of the atmosphere inside 1000 km of the protoplanet surface is very nearly $3 \\times 10^{18} \\, {\\rm g}$ , also in accord with analytical estimates.", "We do not directly simulate the effects of a secondary atmosphere of species outgassed from the protoplanet's magma ocean, which would be orders of magnitude denser, but we expect it would behave very similarly dynamically.", "We begin our simulation of the bow shock with the protoplanet and primary atmosphere, and introduce gas streaming up from the lower $z$ boundary with density $\\rho _{0} = 1 \\times 10^{-9} \\, {\\rm g} \\, {\\rm cm}^{-3}$ and upward velocity $V_{\\rm s} = 8 \\, {\\rm km} \\, {\\rm s}^{-1}$ .", "We chose this density to represent conditions of the chondrule-forming region, presumed to be at 2-3 AU (but see §6).", "However, disk models differ on gas densities in the protoplanetary disk.", "For example, the density $\\rho _{0} = 1 \\times 10^{-9} \\, {\\rm g} \\, {\\rm cm}^{-3}$ is achieved in a minimum-mass solar nebula model (Hayashi 1981) at 1.25 AU (depending on the scale height), or at several (up to 5) AU in more massive disk models (Desch 2007).", "We chose the relative velocity of $8 \\, {\\rm km} \\, {\\rm s}^{-1}$ based on the results of Morris & Desch (2010), who showed that this speed was necessary to achieve chondrule melting.", "Such a large relative velocity may appear to be extreme, but such values are expected to be met episodically throughout planet formation, even though the typical relative velocity is expected to be much lower.", "A velocity difference of $8 \\, {\\rm km} \\, {\\rm s}^{-1}$ between the body and the gas can occur for a range of eccentricities and inclinations, which will be explored in more detail in Section 6.", "Here, consider the example of a body on an orbit with a semi-major axis $a = 1.25$ AU and $e = 0.29$ .", "This body will produce a $\\sim 8$ km/s bow shock at quadrature ($a\\sim 1.1$ AU), which is a plausible site for the formation of some chondrules.", "Compared with the total population of embryos, such high relative velocities may be rare, but when they do occur, they can be very significant, as we will show.", "In both the planetesimal and planetary embryo cases, the evolution of the gas is marked by a reverse shock bouncing off the body surface and multiple shocks reflecting off each other at the symmetry axis.", "Within a few crossing times (i.e., a few $\\times 10^3 \\, {\\rm s}$ ) for the planetesimal case, and a few $\\times \\sim 10^4 \\, {\\rm s}$ for the planetary embryo case), a quasi steady-state bow shock structure is reached.", "An animation showing the case for the protoplanet is shown in Figure REF .", "Contour plots of gas densities $4 \\times 10^{4} \\, {\\rm s}$ after the introduction of supersonic material on the grid is shown in Figures REF a and REF b.", "In both panels the gas density is depicted in blue, with the darkest levels referring to densities of $2.5 \\times 10^{-10} \\, {\\rm g} \\, {\\rm cm}^{-3}$ and the brightest levels to $2.5 \\times 10^{-8} \\, {\\rm g} \\, {\\rm cm}^{-3}$ .", "At the bow shock itself the density increases by a factor of 6 at the shock front, from $1 \\times 10^{-9} \\, {\\rm g} \\, {\\rm cm}^{-3}$ to $6 \\times 10^{-9} \\, {\\rm g} \\, {\\rm cm}^{-3}$ , as expected for a $\\gamma = 7/5$ gas, in a strong shock.", "We observed that the bow shock around the planetesimal quickly achieved steady state, but as is evident from Figure REF b, steady-state is not formally achieved around the protoplanet.", "This is observed by the varying density in the wake of the protoplanet, and the slightly irregular structure of the bow shock (e.g., the kink at $x = 6000 \\, {\\rm km}$ ).", "We attribute this to the boundary layer that exists between the primary atmosphere and the post-shock flow of gas past the protoplanet.", "Inside this boundary, evident at a distance $\\approx 1000 \\, {\\rm km}$ from the protoplanet surface, the densities are considerable higher $(> 10^{-8} \\, {\\rm cm}^{-3}$ ) than the densities $\\approx 6 \\times 10^{-9} \\, {\\rm g} \\, {\\rm cm}^{-3}$ between this boundary and the bow shock.", "This boundary is unstable to the Kelvin-Helmholtz (KH) instability, and we observed frequent shedding of vortices and rolls of material at this interface; these rolls are evident in Figure REF b at regular ($\\approx 2000 \\, {\\rm km}$ ) intervals.", "Reflections of sound waves off of these rolls is responsible for the irregular structure at the bow shock.", "Eventually these KH rolls will strip away the atmosphere surrounding the protoplanet.", "Numerical simulations of the stripping of gravitationally bound gas by KH rolls at the surfaces of protoplanetary disks by Ouellette et al.", "(2007) suggest that the efficiency of stripping is about 1%, meaning that the mass of gas that is stripped is about 1% of the mass of the impacting gas.", "Based on the inflow of gas $\\rho _{0} \\, \\pi R_{\\rm p}^2 \\, V_{\\rm s}$ , we estimate a stripping rate $2 \\times 10^{12} \\, {\\rm g} \\, {\\rm s}^{-1}$ .", "This suggests that loss of the primary atmosphere would take $> 10^6 \\, {\\rm s}$ , hundreds of times longer than the duration of our simulation.", "We were able to quantify this stripping rate via a “dye\" calculation in which we kept track separately of material accreted before the supersonic flow of gas on the grid.", "We found that after an initial loss of gas, the primary atmosphere was being lost on timescales $\\sim 10^{5} \\, {\\rm s}$ , suggesting a stripping rate $> 10^{13} \\, {\\rm g} \\, {\\rm s}^{-1}$ .", "The existence of this interface between an atmosphere and the post-bow shock flow is a fundamental difference between the planetesimal and protoplanet cases." ], [ "Chondrule Trajectories in Planetary Embryo Bow Shocks", "Having established the dynamics of the gas and the structure of the bow shock surrounding the planetary embryo, we now turn our attention to the trajectories of chondrules and their precursors in the flow.", "In general, the particles are dynamically coupled to the gas, except in the vicinity of the bow shock, when the gas suddenly changes velocity over a few molecular mean free paths (i.e., over a few meters).", "Particles will dynamically recouple with the gas on a timescale comparable to the aerodynamic stopping time, $t_{\\rm stop} = \\frac{ \\rho _{\\rm s} a_{\\rm c} }{ \\rho _{\\rm g} c_{\\rm s} },$ where $\\rho _{\\rm s} \\approx 3 \\, {\\rm g} \\, {\\rm cm}^{-3}$ is the particle internal density, $a_{\\rm c} \\approx 300 \\, \\mu {\\rm m}$ is its radius, $\\rho _{\\rm g}$ the post-shock gas density, and $c_{\\rm s}$ the post-shock gas sound speed.", "For $\\rho _{\\rm g} = 6 \\times 10^{-9} \\, {\\rm g} \\, {\\rm cm}^{-3}$ and $T_{\\rm g} \\approx 2500 \\, {\\rm K}$ , the stopping time is 50 seconds.", "If the planetary body is traveling through the gas at a speed $V_{\\rm s} = 8 \\, {\\rm km} \\, {\\rm s}^{-1}$ , particles will move past the shock front several hundred km before they dynamically couple to the gas.", "Based on the numerical simulations above, this lengthscale is less than or comparable to the distance between the shock front and the planetary bodies themselves.", "The expectation is that particles passing through a bow shock will be accreted onto a planetesimal and will flow around a planetary embryo, although clearly a careful calculation of the trajectories is required to determine their fate.", "We are also motivated to calculate the trajectories of particles carefully to determine if they will melt and cool in a manner consistent with chondrules.", "The maximum heating of the chondrules is set by their velocity relative to the gas after passing through the shock front, which sets their degree of melting.", "Their cooling after melting is set by the optical depth between the particles and the unshocked gas, which in turn is set by the geometry and spatial distribution of the particles.", "These must be calculated numerically.", "We have computed the trajectories of particles by integrating the particles' motions through the gas, assuming the gas flow around the planetary body is in a steady state.", "We assume the particles are tracers only, with no dynamical effect on the gas.", "Particles are accelerated by the gravity of the planetary embryo (with mass $M_{\\rm p}$ ) and by the drag force due to gas of density $\\rho _{\\rm g}$ : $m_{\\rm c} \\frac{d \\mbox{$\\mbox{$V$}_{\\rm c}$}}{dt}= -\\frac{G M_{\\rm p} m_{\\rm c}}{r^3} \\, \\mbox{$r$}-\\pi a_{\\rm c}^{2} \\, \\left( \\frac{ C_{\\rm D} }{2} \\right) \\,\\rho _{\\rm g} \\,\\left| \\mbox{$\\mbox{$V$}_{\\rm c}$}- \\mbox{$V_{\\rm g}$}\\right| \\, \\left( \\mbox{$\\mbox{$V$}_{\\rm c}$}- \\mbox{$V_{\\rm g}$}\\right),$ where $\\mbox{$r$}$ is the position vector of the chondrule (with respect to the planet center), and $\\mbox{$\\mbox{$V$}_{\\rm c}$}$ and $\\mbox{$V_{\\rm g}$}$ are the chondrule and gas velocities.", "Particles are in the Epstein limit (much smaller than the mean free paths of gas molecules), and the drag force is given by $C_{\\rm D} = \\frac{2}{3 s} \\left( \\frac{\\pi T_{\\rm c}}{T_{\\rm g}} \\right)^{1/2}+ \\frac{2 s^2 + 1}{\\pi ^{1/2} s^3} \\, \\exp \\left( -s^2 \\right)+ \\frac{4 s^4 + 4 s^2 - 1}{2 s^4} \\, {\\rm erf}(s),$ where we will assume the particle temperature $T_{\\rm c}$ locally equals the gas temperature $T_{\\rm g}$ , and where $s = \\frac{ \\left| \\mbox{$\\mbox{$V$}_{\\rm c}$}- \\mbox{$V_{\\rm g}$}\\right| }{ (2 k T_{\\rm g} / \\bar{m})^{1/2} },$ where $\\bar{m} = 2.33 m_{\\rm H}$ is the mean molecular weight of the gas and $k$ is Boltzmann's constant (Probstein 1968).", "Gas velocities $\\mbox{$V_{\\rm g}$}$ , temperatures $T_{\\rm g}$ and densities $\\rho _{\\rm g}$ are found from the output of the hydrodynamic simulations, using bilinear interpolation on a grid with spatial resolution of 32 km.", "Particle velocities are integrated using a simple explicit scheme that is first-order in time, with timesteps of 0.05 s. Particles move no more than 0.4 km during each timestep, which is significantly smaller than the spatial resolution of the hydrodynamics simulations.", "The trajectories of chondrules around both the planetesimal and the planetary embryo are depicted in Figures REF a and REF b.", "These are seen to be significantly affected by the size of the region between the shock front and the body.", "At the shock front the gas undergoes a nearly instantaneous shift in velocity.", "Chondrules regain dynamical equilibrium with the gas only after they move through the gas by a stopping length $V_{\\rm rel} t_{\\rm stop}$ , where $t_{\\rm stop}$ is the stopping time (Equation 2) and $V_{\\rm rel}$ is the velocity of a chondrule with respect to the gas.", "This quantity depends only on the gas density $\\rho _{\\rm g}$ , not the radius of the planetary body.", "For both the planetesimal and planetary embryo cases, the stopping time $t_{\\rm stop} \\approx \\, 50 \\, {\\rm s}$ (assuming a gas density $\\rho _{\\rm g} = 1 \\times 10^{-9} \\, {\\rm cm}^{-3}$ ), over which time the chondrules slow from a velocity (relative to the planet) of $8 \\, {\\rm km} \\, {\\rm s}^{-1}$ to about $2 \\, {\\rm km} \\, {\\rm s}^{-1}$ .", "Assuming an average velocity $\\approx 5 \\, {\\rm km} \\, {\\rm s}^{-1}$ , chondrules will typically move about 250 km past the shock front.", "This stopping length is to be compared to the size of the post-shock region, specifically the standoff distance between the shock front and the planetary body itself.", "For the $R_{\\rm p} = 500 \\, {\\rm km}$ planetesimal case, this standoff distance between the shock front and the planetesimal is $\\approx 110 \\, {\\rm km} \\approx 0.2 \\, R_{\\rm p}$ .", "It is therefore not surprising that all chondrules with impact parameter $b < 460 \\, {\\rm km}$ are accreted onto the planetesimal.", "In contrast, for the $R_{\\rm p} = 2720 \\, {\\rm km}$ planetary embryo case, the standoff distance to the planet is $\\approx 2000 \\, {\\rm km} \\approx 0.8 \\, R_{\\rm p}$ , or $\\approx 1400 \\, {\\rm km} \\approx 0.5 \\, R_{\\rm p}$ to the atmosphere, much greater than the stopping length.", "Thus, the vast majority of chondrules dynamically recouple with the gas before hitting the planet.", "Only those chondrules with $b \\raisebox {-0.6ex}{\\, \\stackrel{\\raisebox {-.2ex}{\\textstyle <}}{\\sim }\\,}400 \\, {\\rm km}$ might be accreted, and chondrules that escape accretion flow around the body.", "Along the way, their trajectories are affected by the KH instabilities between the shocked gas and the primary atmosphere.", "This makes it difficult to predict their exact trajectories, but it is clear that they will escape accretion onto the body.", "Moving forward it is important to distinguish between particles that pass through the shock front more directly and are melted as chondrules, and those particles that pass through the shock front far from the body and are not melted as chondrules.", "Only particles that pass through the shock front with sufficient velocity $V_{\\rm rel}$ relative to the gas will melt as chondrules.", "The heating rate of chondrules after they pass through the shock front scales as $V_{\\rm rel}^{3}$ (up to the liquidus temperature).", "As we discuss in §4, a shock speed of $6 \\, {\\rm km} \\, {\\rm s}^{-1}$ is required to ensure melting of chondrules, which is equivalent to $V_{\\rm rel} \\raisebox {-0.6ex}{\\, \\stackrel{\\raisebox {-.2ex}{\\textstyle >}}{\\sim }\\,}5 \\, {\\rm km} \\, {\\rm s}^{-1}$ .", "The relative velocity between chondrules and gas is a function of the impact parameter $b$ , and is easily calculated for the case $b = 0$ km (chondrules and gas impact the body directly).", "Just past the shock front, chondrules retain their velocity but gas is immediately slowed by a factor of $(\\gamma +1)/(\\gamma -1) \\approx 6$ .", "The velocity difference between gas and chondrules is thus $\\approx (5/6) (8 \\, {\\rm km} \\, {\\rm s}^{-1}$ ) $\\approx 6.7 \\, {\\rm km} \\, {\\rm s}^{-1}$ .", "As the impact parameter $b$ is increased, this relative velocity decreases.", "This is because only the component of the velocity normal to the shock front is reduced, whereas the tangential component of velocity is conserved.", "As $b$ increases and the shock becomes increasingly oblique, the relative velocity is decreased, and therefore the chondrule heating rate is decreased.", "We therefore anticipate a maximum impact parameter for precursors to be melted as chondrules.", "This variation of relative velocity with impact parameter is indeed seen in our numerical simulations, as illustrated in Figures REF a (planetesimal case) and REF b (planetary embryo case).", "In both cases, the maximum relative velocity at $b = 0 \\, {\\rm km}$ approaches $\\approx 6.7 \\, {\\rm km} \\, {\\rm s}^{-1}$ , and decreases with increasing impact parameter.", "In the $R_{\\rm p} = 500 \\, {\\rm km}$ case, the relative velocity drops below the critical threshold for melting, $5 \\, {\\rm km} \\, {\\rm s}^{-1}$ , for impact parameters $\\raisebox {-0.6ex}{\\, \\stackrel{\\raisebox {-.2ex}{\\textstyle >}}{\\sim }\\,}400 \\, {\\rm km}$ .", "For the $R_{\\rm p} = 2720 \\, {\\rm km}$ planetary case, on the other hand, particles experience $V_{\\rm rel} > 5 \\, {\\rm km} \\, {\\rm s}^{-1}$ for impact parameters as high as $b \\approx 4300 \\, {\\rm km}$ .", "If the planetary embryo had even higher speed with respect to the gas, this maximum impact parameter would be expanded.", "Because of the slightly irregular shape of the shock front, some regions are not as oblique as in others.", "In the snapshot of Figure REF b, for example, particles with $b \\approx 6000 \\, {\\rm km}$ encounter the shock more directly and also may experience thermal histories consistent with chondrules.", "Note that even particles with $b = 10,000$ km will be significantly heated (by a shock with speed $2.5 \\, {\\rm km} \\, {\\rm s}^{-1}$ ), but will not be melted as chondrules.", "Comparing the cross sections, the vast majority ($> 80\\%)$ of particles encountered by the bow shock will be heated, but not melted.", "There is generally a maximum impact parameter for which particles are melted as chondrules, but a minimum impact parameter for which particles escape accretion.", "For a planetesimal with radius $R_{\\rm p} = 500 \\, {\\rm km}$ , only particles with $b < 400 \\, {\\rm km}$ are heated strongly enough to melt as chondrules.", "However, only particles with $b > 460 \\, {\\rm km}$ escape accretion onto the planetesimal.", "From this we see that all particles melted as chondrules in a planetesimal bow shock are accreted onto the planetesimal.", "For a planetary embryo with radius $R_{\\rm p} = 2720 \\, {\\rm km}$ , on the other hand, all particles with $b < 4300 \\, {\\rm km}$ (possibly up to 6000 km) are melted as chondrules, whereas only particles with $b < 400 \\, {\\rm km}$ are possibly (but not necessarily) accreted.", "More than $99\\%$ of all particles melted as chondrules in a planetary embryo bow shock escape accretion by the body.", "Larger bodies will present a larger shock front to incoming particles, increasing the impact parameter out to which chondrules form, and will produce shocks that stand off farther from the body, decreasing the fraction of particles accreted.", "One important caveat to the results stated above is that the chondrule trajectories were calculated under the assumption that the gas flow was in steady state.", "This assumption is valid for gas as it flows to the shock, and even in the immediate post-shock region, but is manifestly not valid at the boundary where the post-shock gas flows past the planetary atmosphere, since this boundary is strongly affected by KH rolls.", "Figure REF b makes clear that where chondrules intercept a KH roll (e.g., at impact parameters 2000 km and 5000 km), the gas flow drives chondrules out of the KH roll.", "Likewise, at impact parameter $\\approx 4000$ km, characterized by the absence of KH rolls, the gas is flowing from the shock front inward, driving chondrules into the void.", "Thus, there is a tendency for chondrules to “skim\" the boundary surface between the atmosphere and the post-shock gas, although it is difficult to confirm this tendency under the assumption of a steady-state flow.", "Chondrules in the vicinity of the flow-atmosphere boundary have a range of velocities $\\approx 1.8 - 5.6 \\, {\\rm km} \\, {\\rm s}^{-1}$ (see below), while the KH rolls themselves have velocities (pattern speeds) on the order of $3.5 \\, {\\rm km} \\, {\\rm s}^{-1}$ , so that chondrules will probably move in and out of KH rolls, but not at the rates they do so in this calculation, in which the gas flow is kept in steady state.", "The trajectories of chondrules determine their density in the chondrule-forming region.", "In what follows, we specifically focus on particles that achieved temperatures high enough to melt as chondrules.", "To be specific, we consider the particles processed by the planetary embryo bow shock, in Figure 2b, which will pass through a slice with $Y = +3000 \\, {\\rm km}$ near the shock front, with $X \\approx 7000 - 10000 \\, {\\rm km}$ .", "Considering just the subset of those particles that melted as chondrules, they have impact parameters $b = 400 - 4300 \\, {\\rm km}$ , and pass through $X = 7180 - 8060 \\, {\\rm km}$ .", "Particles with the lower end of impact parameter ($b = 400 \\, {\\rm km}$ ) move most slowly, at speeds $\\approx 1.8 \\, {\\rm km} \\, {\\rm s}^{-1}$ , and take 2-4 hours to reach this region.", "Chondrules with the higher impact parameter ($b = 4300 \\, {\\rm km}$ ) tend to have speeds $\\approx 5.6 \\, {\\rm km} \\, {\\rm s}^{-1}$ and take just under 1 hour to reach this region.", "On average, chondrules' velocities are reduced from their pre-shock values ($8 \\, {\\rm km} \\, {\\rm s}^{-1}$ ) by factors $\\approx 2.3$ , implying that chondrule densities will increase by a factor 2.3 as they slow.", "In addition, the chondrules are also seen to be geometrically concentrated as they are funneled into a smaller conical volume in past the bow shock.", "This increases their density by a factor $(4300^2 - 400^2) / (8060^2 - 7180^2) = 1.37$ .", "Combined, the chondrule density is raised above the background value by a factor of 3.2.", "A similar calculation for those particles with impact parameters $4300 \\, {\\rm km} < b < 10000 \\, {\\rm km}$ shows the density of these particles is increased by a similar factor.", "These particles will not have achieved the same peak temperatures as those particles with smaller impact parameters, but they will be almost as hot.", "For example, if particles with $b = 0$ km achieved peak temperatures of 2000 K, and peak temperature scales as $V_{\\rm rel}^{3/4}$ , then Figure 3b suggests that particles with $b = 8000 \\, {\\rm km}$ will have peak temperatures $\\approx 1400 \\, {\\rm K}$ .", "The chondrule densities derived above bear on the frequency of compound chondrules.", "The mean free path of a chondrule in a sea of chondrules is $l_{\\rm mfp} = (n_{\\rm c} \\, 4 \\pi a_{\\rm c}^{2} )^{-1}$ .", "For $a_{\\rm c} = 300 \\, \\mu {\\rm m}$ and assuming a compression of a factor of 3.2 in the post-shock region, we find $l_{\\rm mfp} \\approx 2.7 \\times 10^{4} \\, {\\cal C}^{-1} \\, {\\rm km}$ .", "For ${\\cal C} = 10$ , the mean free path is 2700 km (about the radius of the planet).", "Within the part of the flow containing chondrules, their velocities range between 1.8 and $5.6 \\, {\\rm km} \\, {\\rm s}^{-1}$ over a distance 800 km, shorter than the mean free path.", "Chondrules therefore will collide with other chondrules at relative velocities $\\Delta V$ up to and above that needed to shatter chondrules, which has been estimated to lie in the range $1 \\, {\\rm m} \\, {\\rm s}^{-1}$ (Kring 1991) to $100 \\, {\\rm m} \\, {\\rm s}^{-1}$ (Gooding & Keil 1981).", "We assume a median threshold for shattering of $10 \\, {\\rm m} \\, {\\rm s}^{-1}$ , assert that collisions with chondrules up to (and beyond) this speed will occur, and that these will dominate the production of compound chondrules.", "The compound chondrule frequency $f$ we predict is $f = (\\Delta V) t_{\\rm plast} l_{\\rm mfp}^{-1}= 0.04 \\,\\left( \\frac{\\Delta V}{10 \\, {\\rm m} \\, {\\rm s}^{-1}} \\right) \\,\\left( \\frac{ t_{\\rm plast} }{ 10^4 \\, {\\rm s} } \\right) \\,\\left( \\frac{ {\\cal C} }{ 10 } \\right),$ where $t_{\\rm plast}$ is the time for which chondrules remain plastic enough to stick after a collision.", "The observed frequency of compound chondrules is reproduced in a planetary embryo bow shock assuming chondrules are concentrated by factors ${\\cal C} \\approx 10$ , although the exact percentage will depend on the maximum speed at which chondrules can collide and stick without shattering.", "Finally, the density of chondrules in the post-shock region will also bear on the optical depths between chondrules and the cool, unshocked gas, which we estimate as follows.", "The optical depth is $\\tau = n_{\\rm c} \\, (\\epsilon \\pi a_{\\rm c}^{2}) \\, l$ , where $l$ is the distance between the locations of chondrules and the bow shock, and $\\epsilon \\, \\pi a_{\\rm c}^{2}$ is the wavelength-integrated absorption cross-section, with $\\epsilon \\approx 0.8$ in the near-infrared (Li & Greenberg 1997).", "Dust evaporates at the shock front, so the opacity is due solely to chondrules (Morris & Desch 2010).", "The number density of chondrules in the background nebular gas is $n_{\\rm c,0} = \\frac{ (\\rho _{\\rm c} / \\rho _{\\rm g}) \\rho _{\\rm g} }{ 4\\pi \\rho _{\\rm s} a_{\\rm c}^3 / 3} \\approx 1 \\times 10^{-8} \\, {\\cal C} \\, {\\rm cm}^{-3},$ where we have assumed a chondrule radius $a_{\\rm c} = 300 \\, \\mu {\\rm m}$ , an internal density $\\rho _{\\rm s} = 3.3 \\, {\\rm g} \\, {\\rm cm}^{-3}$ , and defined ${\\cal C} \\equiv (\\rho _{\\rm c} / \\rho _{\\rm g}) / (3.75 \\times 10^{-3})$ .", "We estimate $l$ by considering those chondrules (objects with $b < 4300 \\, {\\rm km}$ ) with most probable impact parameter, $b \\approx 3000 \\, {\\rm km}$ .", "These chondrules pass $Y = +3000 \\, {\\rm km}$ at $X \\approx 7800 \\, {\\rm km}$ .", "The particles with the largest impact parameter to be chondrules ($b = 4300 \\, {\\rm km}$ ) pass through $Y = +3000 \\, {\\rm km}$ at $X = 8060 \\, {\\rm km}$ .", "Importantly, though, even particles with $b = 8000 \\, {\\rm km}$ will achieve near-chondrule-like temperatures of 1400 K, and pass through $Y = +3000 \\, {\\rm km}$ at about $X = 9000 \\, {\\rm km}$ .", "The unshocked gas lies at $X > 10000 \\, {\\rm km}$ at $Y = +3000 \\, {\\rm km}$ .", "Thus the typical chondrule lies 250 km from the edge of the cloud of particles shocked enough to melt as chondrules, but 2100 km from the bow shock (after correcting for the tilt of shock front).", "We assume chondrules lie 2200 km from the bow shock and consider chondrules to be concentrated above their background value by a factor of 3.2, to find $\\tau = 0.16 \\, \\left( \\frac{ {\\cal C} }{10} \\right) \\,\\left( \\frac{ l }{ 2200 \\, {\\rm km} } \\right).$ The gradual spatial transition from shocked chondrules to unshocked gas makes it difficult to define $l$ more exactly.", "Nevertheless, this calculation demonstrates that if chondrules are concentrated by factors of about 10, then $\\tau \\sim 0.16$ for most chondrules melted by planetary embryo bow shocks." ], [ "Thermal Histories of Chondrules in Planetary Embryo Bow Shocks", "Having calculated the trajectories of chondrules past the planetary embryo, an important goal is to calculate their thermal histories.", "These are influenced strongly by the absorption of infrared radiation emitted by nearby chondrules, and by the proximity of a cooler region into which the chondrules can radiate (without receiving a return of radiation).", "In the previously studied cases of large-scale shocks driven by disk-wide gravitational instabilities (e.g., Morris & Desch 2010), the shock front is planar and the geometry is 1-D.", "The cooler region into which chondrules radiate is the pre-shock gas (ahead of a radiation front).", "Chondrule trajectories are parallel to each other and normal to the shock front, and chondrule properties do not vary in the lateral direction.", "In contrast, following melting by a bow shock around a planetesimal or planetary embryo, the cooler region to which chondrules radiate is in the lateral direction, and the shock clearly has a 2-D geometry.", "The only correct way to calculate chondrule thermal histories is to compute the fully 2-D radiation field in cylindrical geometry, but the chondrule trajectories displayed in Figure REF b suggest that a perturbation to the 1-D chondrule thermal histories may yield an adequate solution.", "First, the particles are seen to flow in a more or less laminar fashion, without crossing paths.", "This suggests that chondrule thermal histories are a function of distance past the shock front, as in the 1-D example.", "Second, chondrules on parallel trajectories do not differ from each other significantly.", "As stated above, it takes about 3 hours for chondrules with the lowest initial impact parameter ($b = 400 \\, {\\rm km}$ ), and about 1 hour for chondrules with the highest impact parameter ($b = 4300 \\, {\\rm km}$ ), to reach a certain representative point downstream ($Y = +3000 \\, {\\rm km}$ ).", "We show below that typical chondrule cooling rates are roughly $60 \\, {\\rm K} \\, {\\rm hr}^{-1}$ , so that at this location all chondrules are within about 100 K of each other.", "Compared to the typical chondrule temperature at this location ($\\approx 1600 \\, {\\rm K}$ ), this difference is not significant.", "Third, these properties hold even as the particles move downstream by roughly 20,000 km (not shown in Figure REF b).", "Further downstream, rarefaction of the shocked gas is presumed to take place.", "Indeed, some hints of this are seen in the slightly diverging chondrule trajectories in Figure REF b, but this does not affect the gross geometry.", "Certainly the flow remains basically the same for at least the 5 hours it takes for chondrules in our simulations to cool below their solidus temperatures.", "These facts suggest that a simple alteration to the 1-D code of Morris & Desch (2010), to allow for loss of energy by radiation into the unshocked gas, may allow a useful estimate of chondrule thermal histories.", "We accommodate for radiative losses in an approximate way using the formulation of Morris et al.", "(2010a,b).", "Specifically, we conduct a 1-D simulation as in Morris & Desch (2010), based on Desch & Connolly (2002), including emission, absorption, and transfer of continuum radiation from solids, and the molecular radiation from ${\\rm H}_{2}{\\rm O}$ , dissociation and recombination of ${\\rm H}_{2}$ molecules, and evaporation of small particles.", "However, we replace the 1-D radiation field $J$ seen by chondrules with one that is a mixture of the local 1-D radiation field and the radiation field they see from the cool gas on the other side of the bow shock.", "To calculate this mixed radiation field, we consider chondrules to lie in one of two semi-infinite spaces separated by a plane, which represents the bow shock.", "As seen in Figure REF b, chondrules tend to move very close to, and parallel to, the shock front.", "We assume they all remain at a fixed optical depth $\\tau $ from the bow shock.", "On the unshocked side of the bow shock, we assume the temperature is that of the background nebula, $T_{\\rm bkgrnd} \\approx 150 \\, {\\rm K}$ .", "On the other side lies heated chondrules, and the hot ($T \\sim 10^{3} \\, {\\rm K}$ ) gas in the planetary embryo's wake.", "The column density across the wake region is $\\sim 10 \\, {\\rm g} \\, {\\rm cm}^{-2}$ , which for a solar-composition gas yields a column density of water molecules $N_{\\rm H2O} \\sim 10^{21} \\, {\\rm cm}^{-2}$ , which is optically thick in the infrared (Morris et al.", "2009).", "In addition, gas that might contain dust is mixed into the wake region, especially at $Y > +8000 \\, {\\rm km}$ .", "This justifies the assumption of a semi-infinite space at high temperature, which we assume is the local temperature of chondrules, $T_{\\rm ch}$ .", "Making these assumptions, it is straightforward to show that chondrules see a radiation field with mean intensity $J(\\tau ) = B(T_{\\rm ch}) + \\frac{1}{2} \\left[ B(T_{\\rm bkgrnd}) - B(T_{\\rm ch}) \\right] \\, {\\rm E}_{2}(\\tau ),$ where $B$ is the Planck radiation function and ${\\rm E}_{2}$ is the second exponential integral.", "We replace the radiation field seen by chondrules with this quantity, where $\\tau $ is an input parameter describing the optical depth between the chondrules and the gas.", "We note that the previous work of Morris et al.", "(2010a,b) differed in two ways.", "In that work, ${\\rm E}_{2}(\\tau )$ was approximated by $\\exp (-\\tau )$ , which is a reasonably good assumption.", "That work also assumed the wake of the planetesimal was not optically thick, so that $\\exp (-\\tau )$ was used instead of the more appropriate ${\\rm E}_{2}(\\tau )$ used here.", "Because of these differences, cooling rates are roughly an order of magnitude slower here than found by Morris et al.", "(2010a,b).", "As discussed above in §3, we consider $\\tau = 0.16$ to be a typical value for those particles that melt as chondrules (impact parameters from 400 to 4300 km), provided ${\\cal C} \\approx 10$ .", "Based on this, ${\\rm E}_{2}(0.16) / 2 = 0.313 = \\exp (-1.16)$ , meaning that the radiation field seen by chondrules is roughly 69% the radiation field they would normally see in the 1-D shock, mixed with 31% of the background radiation field.", "Chondrules are warmed by the radiation field from nearby chondrules, but not as much as they would be in a 1-D shock.", "We now consider the possible effects of H$_2$ O line cooling in this case.", "Morris & Desch (2010) found that molecular line cooling due to H$_2$ O was negligible in large scale shocks, due to the combined effects of high column densities of water seconds after the shock, backwarming, and, most importantly, ${\\rm H}_{2}$ recombinations.", "However, we must consider whether line cooling would become significant in the geometry of a bow shock.", "Recall from above that the column density of water molecules across the wake region is $N_{\\rm H2O} \\sim 10^{21} \\, {\\rm cm}^{-2}$ , so the region is optically thick to line photons shortly after the shock (Morris et al.", "2009).", "Not only is the wake region optically thick to line photons, but the high column density of water also tends to block continuum radiation escaping through the wake region.", "Although the effects of backwarming may be reduced somewhat due to the bow shock geometry, buffering by ${\\rm H}_{2}$ recombinations will be unaffected.", "The combination of high column densities of water and buffering by ${\\rm H}_{2}$ recombinations will dominate over any reduction in the efficiency of backwarming due to the geometry.", "Therefore, we still consider H$_2$ O line cooling negligible in bow shocks.", "In Figure 5, the thermal histories of chondrules are shown for four possible values of the shock speed: $V_{\\rm s} = 5 \\, {\\rm km} \\, {\\rm s}^{-1}$ , $6 \\, {\\rm km} \\, {\\rm s}^{-1}$ , $7 \\, {\\rm km} \\, {\\rm s}^{-1}$ and $8 \\, {\\rm km} \\, {\\rm s}^{-1}$ .", "In each case chondrules are heated by absorbing infrared radiation emitted by other chondrules in the vicinity, by thermal exchange with the gas in the post-shock region, and by frictional heating in the minute or so past the shock where the chondrules move at supersonic speeds $V_{\\rm rel}$ , with respect to the gas.", "As described above, the radiation field seen by chondrules has been artifically lowered from the radiation field they would normally see, using the factor $\\tau = 0.1$ ($\\tau ^{\\prime } = 1$ ).", "The peak temperature is sensitive to the supersonic heating rate, which scales as $V_{\\rm rel}^{3}$ (up to the liquidus temperature), and strongly correlates with shock speed.", "We find $T_{\\rm peak} \\approx 1430 \\, {\\rm K}$ for $V_{\\rm s} = 5 \\, {\\rm km} \\, {\\rm s}^{-1}$ , $T_{\\rm peak} \\approx 1800 \\, {\\rm K}$ for $V_{\\rm s} = 6 \\, {\\rm km} \\, {\\rm s}^{-1}$ , $T_{\\rm peak} \\approx 1820 \\, {\\rm K}$ for $V_{\\rm s} = 7 \\, {\\rm km} \\, {\\rm s}^{-1}$ , and $T_{\\rm peak} \\approx 1820 \\, {\\rm K}$ for $V_{\\rm s} = 8 \\, {\\rm km} \\, {\\rm s}^{-1}$ .", "Note that thermal histories become increasingly uncertain for times $> 5$ hours (i.e., for temperatures $< 1400$ K) because the assumption of a 1-D geometry becomes increasingly less valid.", "Chondrule peak temperatures must exceed 1770 - 2120 K to produce porphyritic textures (Hewins & Connolly 1996; Desch & Connolly 2002).", "Formation of chondrules therefore requires $V_{\\rm s} \\raisebox {-0.6ex}{\\, \\stackrel{\\raisebox {-.2ex}{\\textstyle >}}{\\sim }\\,}6 \\, {\\rm km} \\, {\\rm s}^{-1}$ , or $V_{\\rm rel} \\raisebox {-0.6ex}{\\, \\stackrel{\\raisebox {-.2ex}{\\textstyle >}}{\\sim }\\,}5 \\, {\\rm km} \\, {\\rm s}^{-1}$ .", "For the runs where peak temperatures were sufficient to melt the chondrules, the cooling rates through the crystallization temperature range ($\\approx 1400 - 1820 \\, {\\rm K}$ , Desch & Connolly 2002) were $< 10^2 \\, {\\rm K} \\, {\\rm hr}^{-1}$ (the cooling rate for chondrules in the shock with $V_{\\rm s} = 6 \\, {\\rm km} \\, {\\rm s}^{-1}$ did not begin to cool slowly until their temperatures dropped below $1400 \\, {\\rm K}$ ).", "If we repeat our calculations with ${\\cal C} \\raisebox {-0.6ex}{\\, \\stackrel{\\raisebox {-.2ex}{\\textstyle >}}{\\sim }\\,}100$ , we recover the optically thick, 1-D limit for which cooling rates are $< 10^2 \\, {\\rm K} \\, {\\rm hr}^{-1}$ .", "If we reduce ${\\cal C}$ below 10, the chondrule cooling rates increase.", "They may not increase significantly if the wake region remains optically thick, but if it does not, then ${\\cal C} \\sim 1$ leads to cooling rates $\\sim 10^{3} \\, {\\rm K} \\, {\\rm hr}^{-1}$ (Morris et al.", "2009).", "From all this, we conclude that both peak temperatures and cooling rates are consistent with formation of porphyritic textures of chondrules, provided shocks speeds are $V_{\\rm s} > 6 \\, {\\rm km} \\, {\\rm s}^{-1}$ ($V_{\\rm rel} > 5 \\, {\\rm km} \\, {\\rm s}^{-1}$ ), and ${\\cal C} > 10$ .", "Lower particle concentrations may also be consistent with cooling rates $< 10^3 \\, {\\rm K} \\, {\\rm hr}^{-1}$ , although we have not tested this.", "The required shock speeds are obtained so long as the planetary embryo has sufficient orbital eccentricity and/or inclination.", "Because cooling rates are sensitive to chondrule concentrations ${\\cal C}$ , it is worth considering likely values of ${\\cal C}$ as well as the lengthscales over which such concentrations are achieved.", "One mechanism that may lead to a high concentration of chondrules is settling to the midplane, which is inferred observationally in certain young protoplanetary disks (Furlan et al.", "2011).", "In this scenario, the level of turbulence at the midplane is low enough that chondrules have a scale height $h_{\\rm c}$ significantly smaller than the scale height $H$ of the gas.", "Dubrulle et al.", "(1995) calculated the scale height of particles as a function of the level of turbulence, as parameterized by $\\alpha $ .", "This can be written in terms of ${\\cal S} = \\alpha / (\\Omega \\; t_{\\rm stop})$ , where $\\Omega $ is the Keplerian orbital frequency, $t_{\\rm stop} = \\rho _{\\rm s} a_{\\rm c} / \\rho _{\\rm g} c_s$ is the chondrules' aerodynamic stopping time as defined in Section 3, $\\nu = \\alpha c_s H$ is the turbulent viscosity, $c_s$ is the sound speed of the gas, and $H = c_s / \\Omega $ .", "The chondrule scale height is $h_{\\rm c} = h \\left[ 1 + \\left( \\frac{h}{H} \\right)^2 \\right]^{-1/2},$ where $h / H = (\\gamma +1)^{-1/4} {\\cal S}^{1/2}$ .", "If $\\alpha \\ll \\Omega \\; t_{\\rm stop}$ , then $h \\ll H$ and chondrules may be concentrated at the disk midplane.", "Assuming $\\rho _{\\rm g} = 1 \\times 10^{-9} \\, {\\rm g} \\, {\\rm cm}^{-3}$ , $C = 0.7 \\, {\\rm km} \\, {\\rm s}^{-1}$ ($T = 150 \\, {\\rm K}$ ) and $\\Omega = 7 \\times 10^{-8} \\, {\\rm s}^{-1}$ ($r = 2 \\, {\\rm AU}$ ), $t_{\\rm stop} = 1400 \\, {\\rm s}$ and $\\Omega \\; t_{\\rm stop} = 1 \\times 10^{-4}$ .", "If $\\alpha \\ll 10^{-4}$ then chondrules will concentrate to the midplane.", "If the turbulent viscosity of the disk is dominated by the magnetorotational instability, it is possible, but not certain, that $\\alpha $ may be low enough at the midplane to allow chondrules to settle.", "Even low values of $\\alpha $ are insufficient to allow chondrules to settle: for $\\alpha = 1 \\times 10^{-4}$ , ${\\cal S} = 1$ , $h / H = 0.8$ (for $\\gamma = 1.4$ ) and $h_{\\rm c} = 0.8 H$ , signifying relatively good mixing of chondrules with the gas.", "To obtain ${\\cal C} = 10$ requires $\\alpha < 1.6 \\times 10^{-6}$ , so that ${\\cal S} < 1.6 \\times 10^{-2}$ , $h / H < 0.101$ and $h_{\\rm c} < 0.100 H$ , implying significant settling to the midplane.", "Chondrule concentrations would then be 10 times their background nebular values, over a thickness $\\raisebox {-0.6ex}{\\, \\stackrel{\\raisebox {-.2ex}{\\textstyle <}}{\\sim }\\,}H / 10$ .", "This analysis is complicated by the possibility that a significant fraction of the solids mass may be locked up in large planetesimals.", "If half of the solids mass is locked up in planetesimals, the same optical depths in the chondrule-forming region require chondrules to be concentrated by a factor of 20 instead of 10, which would require $\\alpha < 3.9 \\times 10^{-7}$ so that $h / H < 2.5 \\times 10^{-3}$ and $h_{\\rm c} < 0.050 H$ .", "If 90% of the solids mass is locked up in planetesimals, the same optical depths would require a concentration by a factor of 100 instead of 10, which would require $\\alpha < 1.6 \\times 10^{-8}$ so that $h_{\\rm c} < 0.010 H$ .", "It is not clear what value of $\\alpha $ is appropriate at the midplanes of protoplanetary disks in which the magnetorotational instability cannot act at the midplane.", "In principle, lower values are possible (e.g., Turner et al.", "2010), but values $\\alpha \\approx 10^{-4}$ are thought to be appropriate (Fleming & Stone 2003), which would make it difficult for significant concentrations of chondrules to be achieved.", "Turbulence itself provides a second mechanism for concentrating chondrules.", "Cuzzi et al.", "(2001) demonstrated that particles with an aerodynamic stopping time equal to the turnover time of the Kolmogorov-scale eddies, can be concentrated in the stagnant zones between turbulent eddies.", "Particles with this stopping time have a size $a_{\\rm c} = \\frac{ \\rho _{\\rm g} c_s \\nu _{\\rm m}^{1/2}}{ \\rho _{\\rm s} \\Omega (\\alpha c_s H)^{1/2} }.$ The molecular viscosity $\\nu _{\\rm m} = \\eta / \\rho _{\\rm g}$ , where $\\eta $ is given by Sutherland's formula.", "Following the discussion in Desch (2007), and assuming a solar composition gas with $T = 150 \\, {\\rm K}$ and the parameters above, we calculate $\\eta = 59 \\, \\mu {\\rm P}$ and $\\nu _{\\rm m} = 5.9 \\times 10^{4} \\, {\\rm cm}^2 \\, {\\rm s}^{-1}$ .", "For $\\alpha = 10^{-4}$ we determine the size of particles necessary for concentration is $a_{\\rm c} = 280 \\, \\mu {\\rm m}$ .", "Chondrule precursors with radius $300 \\, \\mu {\\rm m}$ , as we have assumed in our model, have almost the exact size to be optimally concentrated.", "Particle concentrations ${\\cal C} \\sim 10$ are easily achieved (Cuzzi & Hogan 2003).", "Even if a significant fraction of the mass of solids is locked up in large planetesimals, there will be many regions in the nebula that have these concentrations of chondrule precursors.", "For example, in nebular gas with $\\alpha = 10^{-4}$ , nearly 80% of all chondrule precursors are in regions where their densities are concentrated by factors of 10 above background values, 70% are in regions with concentrations 20 times their background density, and 20% are in regions with concentrations 100 times their background density (Cuzzi et al.", "2001).", "Of course, only small fractions of the volume of the nebula see concentrations this high, but these regions are where most chondrule precursors reside.", "The lengthscale over which a concentration ${\\cal C}$ is achieved roughly scales as $(10^6 \\, {\\rm km}) / {\\cal C}$ (Cuzzi & Hogan 2003), so even regions with ${\\cal C} = 100$ are $> 10^4 \\, {\\rm km}$ in extent and thus wider than the planetary embryo bow shock itself.", "We are justified in assuming a uniform value of ${\\cal C}$ when computing chondrule thermal histories.", "Note that while a planetary embryo will pass through clumps with ${\\cal C} = 10 - 100$ , with sizes $10^{5} - 10^{4} \\, {\\rm km}$ , in only 3 - 0.3 hours, the newly-formed chondrules remain in the vicinity of each other.", "To fix ideas, we assume that half of the solids mass is locked in large planetesimals, so that the optical depths assumed in the thermal histories above requires concentrations by factors of 20, which are achieved over regions 50,000 km in extent, which contain 70% of all chondrule precursors .", "If the planetary embryo is on an orbit eccentric and/or inclined enough to yield shock speeds $> 6 \\, {\\rm km} \\, {\\rm s}^{-1}$ , the chondrule precursors in these clumps would be melted and would cool in a manner consistent with the dominant porphyritic textures." ], [ "Chemical Environment of Planetary Embryo Bow Shocks", "We now consider the formation environment experienced by chondrules in bow shocks around planetary embryos, which we expect to be quite different from the environment in planetesimal bow shocks.", "Dauphas & Pourmand (2011) have suggested that Mars accreted roughly half its mass within $1.8^{+0.9}_{-1.0} \\, {\\rm Myr}$ after the formation of CAIs.", "It therefore must have contained abundant live ${}^{26}{\\rm Al}$ when it formed, sufficient to melt the planet (Dauphas & Pourmand 2011; Grimm & McSween 1993).", "It is presumed that such a body would also form a magma ocean, would convect, and would rapidly outgas volatile species.", "Zahnle et al.", "(2007) have considered the outgassing of volatiles from the Earth following the Moon-forming impact, when the Earth was in its magma ocean state, and concluded that an atmosphere in chemical equilibrium with the magma could rapidly develop in $< 10^3 \\, {\\rm yr}$ .", "Elkins-Tanton (2008) showed that the mantle of an Earth-sized planet takes 5 Myr to become 98% solidified.", "This suggests that planetary embryos could maintain a magma ocean and outgas volatile species for several Myr, perhaps aided by the continuous decay of ${}^{26}{\\rm Al}$ .", "The composition of the planetary embryo's atmosphere probably was similar to the Earth's outgassed atmosphere during its magma ocean stage.", "Zahnle et al.", "(2007) considered the atmosphere outgassed from the proto-Earth's magma ocean.", "They calculated that a fraction of the Earth's ${\\rm CO}_{2}$ would be outgassed, equivalent to a partial pressure $P_{\\rm CO2} \\approx 100 \\, {\\rm bar}$ , and that a substantial fraction of the ${\\rm H}_{2}{\\rm O}$ would be outgassed too, equivalent to a partial pressure $P_{\\rm H2O} \\approx 100 \\, {\\rm bar}$ .", "Other moderately volatile species such as S, Na, Zn, Cl and K, are also expected to be outgassed during the magma ocean stage (Zahnle et al.", "2007; Holland 1984).", "We estimate the abundances of these species in the atmosphere of the planetary embryo by assuming that while outgassing may continue for many Myr, at any one instant in time the magma ocean is in chemical equilibrium with the atmosphere.", "The mass fraction of ${\\rm CO}_{2}$ dissolved in the magma scales linearly with the partial pressure in the atmosphere, as $x_{\\rm CO2} = 4.4 \\times 10^{-7} \\, \\left( \\frac{ P_{\\rm CO2} }{1 \\, {\\rm bar}} \\right)$ (Stolper & Holloway 1988).", "It is straightforward to show that if the planetary embryo has the same total abundance of ${\\rm CO}_{2}$ (C) that $P_{\\rm CO2}$ will scale as $g^2$ , where $g$ is the gravitational acceleration at the planet surface.", "We accordingly estimate $P_{\\rm CO2} \\approx 10 \\, {\\rm bar}$ .", "The solubility behavior of ${\\rm N}_{2}$ is similar to that of ${\\rm CO}_{2}$ (Fricker & Reynolds 1968) and we assume $P_{\\rm N2} \\sim 0.1 \\, {\\rm bar}$ .", "The mass fraction of water dissolved in the magma, $x_{\\rm H2O}$ , is related to the partial pressure of water vapor in the atmosphere, $P_{\\rm H2O}$ , by the relation $x_{\\rm H2O} = 6 \\times 10^{-7} \\, \\left( \\frac{ P_{\\rm H2O} }{ 1 \\, {\\rm dyn} \\, {\\rm cm}^{-2} } \\right)^{0.54}$ (Fricker & Reynolds 1968).", "Analyses of Martian meteorites have led to estimates of Mars's bulk water content in a range of 140-250 ppm (McCubbin et al.", "2011), or 0.5wt% (Craddock & Greeley 2009) to $\\approx 1.4 - 1.8$ wt% (McSween & Harvey 1993; McSween et al.", "2001; Dann et al.", "2001).", "We take a median value, $x_{\\rm H2O} = 0.2$ wt%, as a representative value (a mass fraction equivalent to the Earth having 8 oceans, in line with current estimates by Mottl et al.", "2007), so that the total mass of ${\\rm H}_{2}{\\rm O}$ in the planetary embryo is $6.6 \\times 10^{23} \\, {\\rm g}$ , and $P_{\\rm H2O} = 3.3 \\, {\\rm bar}$ .", "Assuming $M_{\\rm p} = 3.31 \\times 10^{26} \\, {\\rm g}$ , $R_{\\rm p} = 2720 \\, {\\rm km}$ , and a gravitational acceleration $g = 300 \\, {\\rm cm} \\, {\\rm s}^{-2}$ , we determine an equilibrium mass of ${\\rm H}_{2}{\\rm O}$ in the atmosphere of $1.0 \\times 10^{22} \\, {\\rm g}$ , which is a small fraction of the total mass.", "The equilibrium abundance of Na in the atmosphere is complicated by the fact that it may dissolve in the magma either as ${\\rm Na}_{2}{\\rm O}$ or as ${\\rm NaOH}$ , in which case its solubility is affected by the water content.", "van Limpt et al.", "(2006) considered Na solubility for silicate glasses and found the following relationship to hold when Na dissolves as NaOH (i.e.", "in the presence of H$_2$ O): $P_{\\rm NaOH} \\approx 1.2 \\times 10^{-3} \\, \\left( \\frac{ P_{\\rm H2O} }{ 1 {\\rm bar} } \\right)^{0.5} \\, {\\rm bar}.$ For $P_{\\rm H2O} = 3.3 \\, {\\rm bar}$ , we find $P_{\\rm NaOH} \\approx 2 \\times 10^{-3} \\, {\\rm bar}$ .", "Altogther, we infer that if the bulk abundance of water on the planetary embryo is 0.2%, then the planetary embryo's atmosphere during the magma ocean stage is mostly (88wt%) ${\\rm CO}_{2}$ with partial pressure 10 bar, some (12wt%) ${\\rm H}_{2}{\\rm O}$ with partial pressure 3.3 bar, and alkalis and volatiles being trace species, with Na having a partial presure $\\approx 2 \\times 10^{-3} \\, {\\rm bar}$ .", "The total mass of the atmosphere is $\\approx 4 \\times 10^{22} \\, {\\rm g}$ .", "This atmosphere will be continuously stripped by KH instabilities at the interface between the atmosphere and the post-shock gas streaming by the planet.", "We did not include such an outgassed atmosphere, but the primary atmosphere accreted by the planetary embryo in our simulations plays the same role.", "The stripping by KH rolls at the interface between the planetary atmosphere and the post-shock gas, in particular, is expected to follow the same general trends.", "We find using dye calculations (see §2) that the atmosphere is lost at a rate $\\approx 2 \\times 10^{12} \\, {\\rm g} \\, {\\rm s}^{-1}$ , about 1% of the rate at which the planetary embryo intercepts nebular gas.", "This is in line with similar rates of KH stripping in other examples of strongly bound gas (e.g., around protoplanetary disks; Ouellette et al.", "2007).", "This is sufficiently rapid to deplete the atmosphere after about 600 years, but outgassing can continue to replenish the atmosphere as the magma ocean convects (on shorter timescales) and bring new volatiles to the surface.", "The planetary inventory of water, in particular, in this example could persist for $8 \\times 10^4 \\, {\\rm yr}$ .", "The reservoirs of Na and other species would persist for many Myr.", "Not only are gases from the planetary embryo's atmosphere stripped by KH rolls at the interface with the shocked gas, as seen in Figure REF ; chondrules also make excursions in and out of these rolls and this planetary gas.", "Chondrules will be exposed, at least some of the time, to volatiles such as ${\\rm H}_{2}{\\rm O}$ and Na outgassed from the planetary embryo's magma ocean.", "The scale height of the atmosphere proper (which has been shocked to $\\approx 2500 \\, {\\rm K}$ ) is approximately 185 km, so even at the 800 km distance between the planetary surface and the boundary layer at the top of the atmosphere, gas pressures should be $\\sim 1\\%$ of their values at the surface.", "We estimate $P_{\\rm H2O} \\sim 3 \\times 10^{-2} \\, {\\rm bar}$ and $P_{\\rm Na} \\sim 2 \\times 10^{-5} \\, {\\rm bar}$ .", "Chondrules passing through KH rolls will experience approximately these partial pressures of volatile species.", "It is important to note that when we calculated approximate chondrule thermal histories in §4, we assumed the only gas to which chondrules were exposed was the shocked nebula gas, with pressure $\\sim 10^{-3} \\, {\\rm bar}$ .", "If chondrules pass in and out of the plumes of planetary atmosphere gas in KH rolls, they will see much higher presssures, $\\sim 10^{-1} \\ {\\rm bar}$ .", "This could potentially affect chondrule thermal histories in two ways: during the first minute past the shock front, higher gas densities can more rapidly decelerate the chondrule and heat it by friction; and thermal exchange with the hot gas will become more efficient.", "Neither of these effects is likely to drastically alter the thermal histories of the chondrules considered here, because nearly all of the chondrule deceleration takes place in the shocked nebula gas during the first minute, outside of the planetary atmosphere (see Figure REF b).", "Also, the denser plumes of gas are not likely to be at significantly different temperatures, and chondrule temperatures will be dominated by radiative effects.", "A full treatment of chondrule thermal histories in future work must more fully consider these effects.", "Elevated partial pressures of volatiles may explain puzzling features of the chondrule formation environment.", "An enhanced abundance of water vapor is potentially consistent with indicators of elevated oxygen fugacity during chondrule formation (Krot et al.", "2000; Fedkin & Grossman 2006; Grossman et al.", "2011).", "Partial pressures $P_{\\rm H2O} > 10^{-3} \\, {\\rm bar}$ have been suggested to explain the fayalite content of chondrule olivine (Fedkin & Grossman 2006).", "Likewise, the high partial pressure of Na vapor may potentially explain the finding by Alexander et al.", "(2008) that olivine phenocrysts in chondrules from Semarkona contain $\\sim 0.010 - 0.015 {\\rm wt}\\%$ ${\\rm Na}_{2}{\\rm O}$ , even at their cores, strongly implying that Na was dissolved in the melt as the olivine crystallized.", "Alexander et al.", "(2008) expressed the needed partial pressures in terms of a density of solids, assuming that the Na derived from the chondrule melts and that 10% of this Na entered the gas phase.", "Their inferred chondrule densities ranged from low values up to at least $\\approx 400 \\, {\\rm g} \\, {\\rm m}^{-3}$ .", "Assuming the chondrule is 0.01wt% ${\\rm Na}_{2}{\\rm O}$ , this implies a Na vapor density $3 \\times 10^{-9} \\, {\\rm g} \\, {\\rm cm}^{-3}$ and a partial pressure of Na $\\approx 2 \\times 10^{-5} \\, {\\rm bar}$ .", "Remarkably, the volatiles escaping from the planetary embryo's magma ocean and mixing with the post-shock gas may provide the partial pressures of ${\\rm H}_{2}{\\rm O}$ and Na assumed to have been experienced by chondrules.", "We have not modeled in detail the production of a secondary atmosphere, nor quantified in detail how that atmosphere is stripped, or how often chondrules encounter this stripped gas.", "We have demonstrated that chondrules do make excursions into gas that is strongly bound to the planet, and that this gas may contain elevated abundance of species outgassed during a planet's magma ocean stage, including ${\\rm H}_{2}{\\rm O}$ , ${\\rm CO}_{2}$ , Na, and K vapor, in particular.", "Our best estimates of the partial pressures of water vapor and Na vapor are consistent with values needed to provide an environment with elevated oxygen fugacity and the ability to suppress evaporation of alkalis, although our results depend on the assumption $x_{\\rm H2O} = 0.2$ wt%.", "Further tests are required to better test these ideas quantitatively." ], [ "Masses of Chondrules Formed by Planetary Embryo Bow Shocks", "A very large number of chondrules can be produced by a single planetary embryo on an eccentric and/or inclined orbit.", "Chondrules will be produced as long as the velocity difference between the body and the gas is $V_{\\rm s} \\raisebox {-0.6ex}{\\, \\stackrel{\\raisebox {-.2ex}{\\textstyle >}}{\\sim }\\,}6 \\, {\\rm km} \\, {\\rm s}^{-1}$ , which requires the eccentricity $e$ and inclination $i$ of the body to be sufficiently large.", "Over time, however, the interaction of a planetary embryo with nebular gas will cause its eccentricity and inclination to damp to low values.", "The total number of chondrules produced depends on the $e$ and $i$ damping timescales.", "In this section we construct a model for estimating the orbital conditions, duration, and mass processed during a single scattering event.", "The model includes $e$ and $i$ damping due to gas drag and disk torques, as well as semi-major axis evolution due to energy dissipation from drag.", "Migration due to torques is ignored for convenience and to reflect the fact that the magnitude and direction of migration can differ depending on the equation of state (e.g., Paardekooper & Mellema 2006) and on whether radiative transfer is included or not (e.g., Bitsch & Kley 2010).", "Below, we discuss the methods for including torques and drag, in turn, and then combine them to estimate the orbital evolution and mass of chondrules processed." ], [ "Eccentricity Damping of Planetary Embryos", "We first consider eccentricity damping by torques.", "Recent simulations by Cresswell et al.", "(2007) and Bitsch & Kley (2010) demonstrate that eccentric planets in radiatively cooling disks follow analytic limits.", "When the eccentricity $e$ is high, $de/dt\\vert _h = -K_e \\, e^{-2}$ (Papaloizou & Larwood 2000), where $K_e$ is a constant.", "When $e$ is low, $de/dt$ follows an exponential decay that is well characterized by $de/dt\\vert _\\ell = -e / t_{\\rm ecc} = -e \\, 0.78 \\, q \\, \\left( \\Sigma a^2/M_*\\right) h^{-4} \\, \\Omega $ , where $q$ is the planet-to-star mass ratio, $a$ is the planet's semi-major axis, $M_*$ is the mass of the star, $h=H/a$ for scaleheight $H(a)$ at heliocentric distance $a$ , $\\Sigma (a)$ is the total surface density at heliocentric distance $a$ , and $\\Omega (a)$ is the average angular speed of the planet (Tanaka & Ward 2004).", "The low-$e$ limit is valid for $e \\lesssim 0.1$ .", "Using these results, we explore parameter space without using costly hydrodynamics simulations by making the ansatz that the total rate is the harmonic sum of the limits, such that $de/dt = -K_{e} e/(K_{e} t_{\\rm ecc}+e^3)$ .", "We select a family of solutions by varying only $K_{e}$ and assuming that $K_{e} t_{ecc}$ is a constant over the parameter space of interest.", "Based on Cresswell et al.", "(2007) we set $K_e t_{ecc} = 0.00253$ , which allows us to match their eccentricity evolutions.", "For the inclination evolution, the rates follow the same limiting behavior as in eccentricity damping, with $di/dt\\vert _\\ell = -i / t_{\\rm inc} = -i \\, 0.544 \\, q \\, \\left( \\Sigma a^2/M_*\\right) h^{-4} \\, \\Omega $ (Tanaka & Ward 2004) and $di/dt\\vert _h = -K_i \\, i^{-2}$ (Bitsch & Kley 2011).", "Based on the simulations of Bitsch & Kley (2011), we make the same assumptions for the total inclination damping as we have made for the eccentricity damping, and we vary $K_{i}$ while keeping constant $K_i t_{\\rm inc} = 0.00075$ .", "Strictly speaking, the inclination and eccentricity damping are coupled, but they are treated independently for the estimates we present here.", "For disk torques, we neglect dynamical friction between the embryo and a sea of planetesimals, which is given by $de/dt \\vert _{\\rm DF} = - \\ln (1+\\Lambda ^2) G \\Sigma _p q /(\\sqrt{2} \\Omega a e^3)$ (Ford & Chiang 2007).", "Here, $\\Sigma _p$ is the surface density of planetesimals and the natural log term is the Coulomb parameter.", "The ratio $(de/dt\\vert _\\ell ) / (de/dt\\vert _{\\rm DF}) \\approx (e/h)^4 \\Sigma /\\Sigma _p (\\ln (1+\\Lambda ^2))^{-1}$ .", "For typical parameters ($\\Sigma /\\Sigma _p$ $\\sim $ 100, $\\ln (1+\\Lambda ^2)\\sim 10$ , and h $\\sim $ 0.05) dynamical friction is unimportant until $e \\sim 0.03$ .", "We show below that chondrule-producing shocks are only achieved for $e\\gtrsim 0.1$ .", "Therefore, gas disk torques will dominate over dynamical friction.", "Now we consider the effects of gas drag on large solids.", "Planetesimals and planets are much larger than the mean free path of gas molecules $\\lambda $ , so the drag is in the Stokes regime, i.e., the Knudsen number ${\\rm Kn} \\equiv 2 R_{\\rm p} / \\lambda \\ll 1$ .", "In this regime the gas drag takes the form $d(\\delta v) / dt = -3 \\zeta \\, {\\rm Kn} \\, k_d \\, \\delta v$ , where $\\delta v$ is the velocity difference, $k_d$ is a coupling coefficient, and $\\zeta = (8/\\pi )^{1/2} \\, (\\rho _{\\rm g} c_{\\rm a}) / (\\rho _{\\rm s} R_{\\rm p})$ $= (8\\gamma / \\pi )^{1/2} \\, t_{\\rm stop}^{-1}$ , where $c_{\\rm a}$ is the adiabatic sound speed and $t_{\\rm stop}$ and other quantitites are defined as before.", "The coupling coefficient $k_d$ depends on the Reynolds number ${\\rm Re} = 3 \\, (\\pi /8)^{1/2} \\, \\mathcal {M} / {\\rm Kn}$ , where $\\mathcal {M} = \\delta v / c_{\\rm s}$ .", "In the regime of interest (${\\rm Re} > 1500$ ), $k_d = 0.11 \\, {\\rm Re}$ (Paardekooper 2007).", "This yields $d(\\delta v) / dt \\approx -(\\delta v) / t_{\\rm stop}$ , in accord with Adachi et al.", "(1976).", "The evolution of the planetary embryo's orbit can now be determined if $\\delta v$ can be expressed as a function of $e$ and $i$ .", "In general $\\delta v$ will vary along the body's orbit, depending on the value of the true anomaly $\\phi $ .", "We model the damping by solving Kepler's equation for following the orbit of a planet, and by using a predictor-corrector for modifying the eccentricity and inclinations over a timestep.", "The time derivatives of the radial and tangential velocity components for a Keplerian orbit are $\\frac{dv_{\\phi }}{dt} & = & \\left(\\frac{\\mu }{a(1-e^2)}\\right)^{1/2}\\left[ \\left( \\frac{e}{1-e^2}\\left(1+e\\cos {\\phi }\\right)+\\cos \\phi \\right) \\frac{de}{dt}-e\\sin \\phi \\frac{d \\phi }{dt}\\right] \\\\\\frac{dv_r}{dt} & = & \\left(\\frac{\\mu }{a(1-e^2)}\\right)^{1/2}\\left[ \\left( \\frac{e^2}{1-e^2}+1\\right) \\frac{de}{dt}\\sin \\phi +e\\cos \\phi \\frac{d \\phi }{dt}\\right],$ where $\\mu =G(M_*+M_p)$ .", "We find an expression for the change in eccentricity by keeping only the terms with $de/dt$ , as the other terms represent the unaltered Keplerian orbit.", "This gives $\\left(\\frac{ d \\delta v}{dt}\\right)^2 = \\frac{\\mu }{a(1-e^2)}\\left(\\frac{d e}{dt}\\right)^2\\left[\\left(\\frac{e}{1-e^2}\\left(1+e\\cos \\phi \\right)+\\cos \\phi \\right)^2+\\left(\\left( \\frac{e^2}{1-e^2}+1\\right) \\sin \\phi \\right)^2 \\right].$ The change in eccentricity over time can thus be expressed in terms of the change of the relative velocity over a computational time step.", "A similar approach can be taken with the inclination, where $d(\\delta v_z)/dt \\approx (\\mu / r_{\\rm cyl})^{1/2} \\, d i/dt$ , where $r_{\\rm cyl}$ is the projected radial separation.", "We have assumed small $i$ here because drag predominantly takes place within one scale height of the disk, and we in fact only include drag when $\\vert z\\vert < H$ .", "Finally, we use the energy gained or dissipated in a timestep, $\\delta E$ , to computed an updated semi-major axis $a = -(\\mu /2) (E + \\delta E)^{-1}$ .", "We now apply these formulae to conditions in protoplanetary disks.", "In our orbital damping calculations we use a disk model with gas surface density $\\Sigma (r) = 149 \\, (r / 5.2 \\, {\\rm AU})^{-1.5} \\, {\\rm g} \\, {\\rm cm}^{-2}$ , a gas scale height $H = 0.05 r$ , a midplane density $\\rho _0 = \\Sigma / (2 H)$ , and a stellar mass of $1 \\, M_{\\odot }$ .", "To illustrate which damping terms are most important, we plot in Figure REF the eccentricity damping timescale $e \\, (de/dt)^{-1}$ as a function of planetary mass $M_{\\rm p}$ , for our model conditions and with $e = 0.4$ and $i = 0^{\\circ }$ .", "The internal density in this example was set to be $\\rho _{\\rm s} = 5.7 \\, {\\rm g} \\, {\\rm cm}^{-3}$ , appropriate for Earth-like bodies but slightly denser than our proto-Mars planetary embryo.", "Each curve represents damping at pericenter for different $a$ .", "In Figure REF it is seen that bodies with small mass ($M_{\\rm p} < 0.1 M_{\\oplus }$ ) are in the drag-dominated regime, with damping timescales that increase with increasing mass as $M_{\\rm p}^{1/3}$ .", "Bodies with large mass ($M_{\\rm p} > 1 M_{\\oplus }$ ) are in the disk torque regime, with damping timescales that decrease as the planet's mass is increased, roughly as $M_{\\rm p}^{-1}$ .", "The transition between these regimes, and the maximum damping timescales $\\sim 10^{5} \\, {\\rm yr}$ , occurs for bodies of mass approximately $0.1 - 0.2 M_{\\oplus }$ , somewhat larger than the planetary embryo we consider ($= 0.055 M_{\\oplus }$ ).", "In Figure REF we show the eccentricity evolution of a scattered planetary embryo with mass $M_{\\rm p} = 0.055 \\, M_{\\oplus }$ , and $R_{\\rm p} = 2720 \\, {\\rm km}$ .", "The semi-major axis $a$ is set to 2.5 AU, but various starting eccentricities $e_0$ and inclinations $i_0$ for strong scattering events are explored.", "These plots demonstrate that for plausible scattering scenarios the planetary embryo retains high eccentricity for a few $\\times 10^5 \\, {\\rm yr}$ .", "More relevantly, the maximum shock speeds are shown to exceed the critical value ($\\approx 6 \\, {\\rm km} \\, {\\rm s}^{-1}$ ) for $1 - 4 \\times 10^{5} \\, {\\rm yr}$ , if $e_0 = 0.4$ , and for a shorter time if $e_0 = 0.2$ .", "Although the large eccentricities and inclinations explored here may not be common, they do represent values that could be met repeatedly during the lifetime of a disk (e.g., Chambers 2001; Hansen 2009)." ], [ "Mass of Chondrules Produced Per Embryo", "We now calculate the mass of chondrules, $M_{\\rm c}$ , produced by a single planetary embryo.", "The rate at which chondrules are processed over time is $d M_{\\rm c} / dt = \\sigma \\, V_{\\rm s} \\, \\rho _{\\rm c}$ , where $\\sigma $ is the cross section, which we set to $\\pi (4300 \\, {\\rm km})^{2}$ for the planetary embryo described in §2.", "The density of chondrule (precursors) is defined to be $\\rho _{\\rm c} = f \\, {\\cal C} \\rho _{\\rm g}$ , with $f = 3.75 \\times 10^{-3}$ .", "The concentration factor of chondrules, ${\\cal C}$ , is a local value that can vary spatially throughout the disk due to settling or turbulence.", "The instantaneous velocity difference between the planetary embryo and the gas is $V_{\\rm s}$ , which is also the shock speed.", "If $V_{\\rm s} < 6 \\, {\\rm km} \\, {\\rm s}^{-1}$ (or greater than $8 \\, {\\rm km} \\, {\\rm s}^{-1}$ ) it is assumed that chondrules are not melted (or are completely vaporized), and we set $d M_{\\rm c} / dt = 0$ for that part of the protoplanet's orbit.", "We calculate the total mass produced by integrating $d M_{\\rm c} / dt$ over individual orbits, and over the entire damping history of the protoplanet.", "Tables 1-6 list the masses of chondrules produced in bow shocks around planetary embryos with a variety of initial semi-major axes $a$ , eccentricities $e$ and inclinations, $i$ .", "Based on the need to have ${\\cal C} \\raisebox {-0.6ex}{\\, \\stackrel{\\raisebox {-.2ex}{\\textstyle >}}{\\sim }\\,}10$ to match chondrule cooling rates (see §4), we consider two situations: one in which chondrules are locally concentrated by turbulence, and one in which chondrules have settled to the midplane.", "We first consider the case where chondrules are concentrated via turbulence.", "As discussed in §4, chondrules more often than not find themselves in regions of enhanced chondrule density (${\\cal C} > 10$ ).", "As seen from the planetary embryo, which randomly samples all volumes of the disk, the average density of chondrules is ${\\cal C} = 1$ .", "Tables 1-3 list the masses of chondrules produced assuming ${\\cal C} = 1$ .", "We next consider the case where chondrules have settled to the midplane, by setting ${\\cal C} = 10$ if $\\vert z\\vert < H / 10$ , and ${\\cal C} = 0$ for $z$ farther from the midplane.", "Tables 4-6 list the masses of chondrules produced assuming this form of ${\\cal C}$ .", "It is important to note that these masses will be reduced if a large fraction of the solids mass is locked up in large planetary bodies.", "A lower solids-to-gas ratio would also result in higher chondrule cooling rates, due to lower optical depth, but probably not significantly so; we expect them to remain within the range consistent with chondrule formation ($10 - 10^3 \\, {\\rm K} \\, {\\rm hr}^{-1}$ ).", "As Tables 1-6 demonstrate, a single planetary embryo can produce a mass of chondrules close to the inferred present-day mass of chondrules, $\\approx 2 \\times 10^{-4} \\, M_{\\oplus }$ (Grossman 1988).", "Higher initial eccentricities and lower initial inclinations favor greater chondrule formation in both concentration scenarios.", "Note that when planetary bodies are scattered out of the plane of the disk, the potential chondrule-forming duration is greatly extended because gas drag is only effective near the orbital nodes; however, chondrules would only be produced at the orbital nodes, so the longer duration does not necessarily equate to more mass processed.", "Comparing the cases where chondrules are concentrated by turbulence vs. settled to the midplane, the total mass of chondrules is comparable when the planetary embryo has high inclination, whereas if the inclination is low ($\\raisebox {-0.6ex}{\\, \\stackrel{\\raisebox {-.2ex}{\\textstyle <}}{\\sim }\\,}10^{\\circ }$ ), a greater mass of chondrules is produced when chondrules settle to the midplane.", "For $a = 1.5 \\, {\\rm AU}$ , $e_0 = 0.4$ and $i = 0^{\\circ }$ , $9 \\times 10^{-5} \\, M_{\\oplus }$ of chondrules are produced over 64,000 years for the turbulent concentration case.", "For the case where chondrules settle to the midplane, a factor of 10 more chondrules are produced.", "Tables 1-6 assume a planetary embryo mass $0.055 \\, M_{\\oplus }$ .", "This may be close to the optimal mass for producing chondrules.", "Provided the body is large enough to actually produce chondrules (i.e., it is not a planetesimal for which chondrules are accreted, but is a large planetary embryo), the cross section $\\sigma $ scales roughly as $R_{\\rm p}^{2}$ .", "The product of this cross section and $t_{\\rm damp}$ yields the total mass of chondrules produced.", "For small bodies whose damping is dominated by gas drag, the mass of chondrules that is produced scales linearly with $M_{\\rm p}$ .", "For large bodies whose damping is dominated by disk torques, the mass of chondrules that is produced scales roughly as $M_{\\rm p}^{-1/3}$ .", "In fact, the mass of chondrules that can be produced per embryo is optimal for Mars-sized bodies with $M_{\\rm p} \\sim 0.1 \\, M_{\\oplus }$ .", "Nevertheless, as long as the body is larger than about 1000 km in radius, a comparable amount (to within factors of 3) of chondrules will be produced.", "The chondrule masses produced per embryo are intriguingly close to the mass $2 \\times 10^{-4} \\, M_{\\oplus }$ , and in principle a single scattered planetary embryo could produce the entire present-day inventory of chondrules.", "Nevertheless, to produce the observed mass of chondrules several planetary embryos probably would be required.", "This is because some of the solids mass is locked up in large bodies, making fewer chondrule precursors available for processing.", "Also, the present-day asteroid belt may have been depleted since it formed, meaning there were more chondrules originally, potentially by a factor of 100 (Weidenschilling 1977).", "On the other hand, eccentric planetary embryos may be a common event.", "Even for the minimum-mass solar nebula mass distribution assumed above, the annulus between 1.5 and 2.5 AU would have contained $\\approx 1.5 \\, M_{\\oplus }$ of solids.", "If half of the mass was in small (chondrule-like) particles and half locked up in large planetary embryos, potentially $\\approx 25$ planetary embryos could exist in that one annulus alone.", "Our calculations show that, given the right orbital parameters, chondrules can form over a range of heliocentric distances 1.0 - 2.5 AU.", "Chondrules embedded in a disk with even low levels of turbulence may be spread over lengthscales $\\Delta r = 1 \\, {\\rm AU}$ in $\\lesssim 10^{5} \\, {\\rm yr}$ , assuming an alpha viscosity law $\\nu = \\alpha H^2 \\Omega $ (where $H$ is the gas scale height, $\\Omega $ the Keplerian orbital frequency and $\\alpha $ a dimensionless constant $\\approx 10^{-2}$ ; Hartmann et al.", "1998).", "Chondrules produced at 1.0 - 2.5 AU will relatively quickly be spread throughout the present-day asteroid belt to be incorporated into chondrite parent bodies." ], [ "Discussion and Summary", "Following the recent discovery that a planetary embryo, the half-formed Mars, existed at the same time chondrules were forming (Dauphas & Pourmond 2011), we have been motivated to study chondrule formation in the bow shocks of planetary embryos on eccentric orbits.", "We have calculated the hydrodynamics of gas flowing through the bow shock of a large (radius 2720 km) planetary embryo.", "In contrast to bow shocks around smaller planetesimals, an embryo acquires a substantial primary atmosphere from the solar nebula.", "A boundary layer exists between this atmosphere and the shocked gas flowing past the planetary embryo, and this layer is unstable to KH instabilities.", "The planetary embryo is likely to have possess a magma ocean (Dauphas & Pourmand 2011), and the secondary atmosphere outgassed from the magma ocean would behave similarly dynamically.", "We have calculated the trajectories of chondrules through the planetary embryo bow shock, subject to gravitational and gas drag forces.", "We find that the relative velocity between chondrules and gas is sufficient to melt chondrules ($> 5 \\, {\\rm km} \\, {\\rm s}^{-1}$ , equivalent to shocks with speed $> 6 \\, {\\rm km} \\, {\\rm s}^{-1}$ ) only for chondrules with impact parameter $b < 400 \\, {\\rm km}$ around a 500 km radius planetesimal, or impact parameter $b < 4300 \\, {\\rm km}$ around a 2720 km radius planetary embryo.", "We find that the dynamical coupling between chondrules and gas is not acheived until the chondrules have passed about 400 km past the bow shock.", "For a small planetesimal the bow shock stands only $\\approx 100$ km from the body, and chondrules are not well coupled to the gas.", "All chondrules with $b < 460 \\, {\\rm km}$ , including all those particles achieving peak temperatures of chondrules and melted as chondrules, are accreted onto the planetesimal.", "For a larger planetary embryo, the bow shock stands $\\approx 1000 \\, {\\rm km}$ off of the boundary with the atmosphere, and very few chondrules are accreted by the protoplanet, only those with $b < 400 \\, {\\rm km}$ ($< 1\\%$ of the chondrules formed in the bow shock).", "We have calculated the thermal histories of chondrules in a planetary embryo bow shock.", "Our calculation of thermal histories builds on the 1-D algorithm of Morris & Desch (2010) with the substitution of Morris et al.", "(2010a,b) for the radiation field, to allow the chondrules to radiate to the cold, unshocked gas.", "We find that shock speeds $> 6 \\, {\\rm km} \\, {\\rm s}^{-1}$ and optical depths between chondrules and the unshocked gas $> 0.1$ are sufficient to melt chondrules and yield cooling rates through the crystallization temperature range $< 100 \\, {\\rm K} \\, {\\rm hr}^{-1}$ , consistent with formation of porphyritic textures.", "These optical depths, we calculate, require concentrations of chondrules ${\\cal C} > 10$ above a presumed background density $3.75 \\times 10^{-3} \\, \\rho _{\\rm g}$ for a gas density $\\rho _{\\rm g} = 10^{-9} \\, {\\rm g} \\, {\\rm cm}^{-3}$ .", "These concentrations are achievable by chondrules settling to the midplane or turbulent concentration.", "Moreover, a concentration factor of ${\\cal C} = 10$ yields a compound chondrule frequency in accord with observations.", "To achieve the same optical depths in bow shocks around planetesimals requires much higher concentrations of chondrules.", "We find that chondrules formed in a planetary embryo bow shock pass through the KH rolls at the boundary between the shocked gas and the planetary atmosphere, and will be exposed to the atmosphere gas.", "This atmosphere is composed of nebular gas but will be dominated by volatiles outgassed from the protoplanet's magma ocean.", "We have estimated the composition of the secondary atmosphere outgassed from the protoplanet to be roughly 10 bars of ${\\rm CO}_{2}$ , 3 bars of ${\\rm H}_{2}{\\rm O}$ , and $2 \\times 10^{-3}$ bars of Na vapor, along with other trace species.", "We estimate a partial pressure of water vapor $\\sim 3 \\times 10^{-2}$ bar and a partial pressure of Na vapor $\\sim 2 \\times 10^{-5}$ bar at the boundary region through which chondrules traverse.", "The partial pressure of water is consistent with the abundance of fayalite in chondrules (Fedkin et al.", "2006) and indicators of variable oxygen fugacity in the chondrule formation environment (Krot et al.", "2000).", "The high partial pressures of Na may resolve the mystery of how olivines crystallized form chondrule melts were able to incorporate substantial Na (Alexander et al.", "2008).", "We find that for plausibleinitial eccentricities ($e_0\\gtrsim 0.2$ ) and inclinations ($i_0 \\approx 0 $ to $ 15^{\\circ }$ ) following a scattering event, the velocity of the planetary embryo with respect to the gas, and the speed of the bow shock, are sufficient to melt chondrules and to continue to melt chondrules over the eccentricity damping timescale.", "For typical chondrule-forming parameters we estimate damping timescales $\\sim 10^5$ years.", "Chondrules may be a byproduct of planet formation, not a necessary step toward planet formation.", "Chondrule formation is likely to be delayed, perhaps by 2 Myr, until the first large planetary embryos are formed and scattered.", "Once such bodies form, several planetary embryos can potentially melt chondrules over a span of several Myr, each one for $\\sim 0.1 \\ {\\rm Myr}$ at a time.", "The mass of chondrules processed by a single eccentric planetary embryo is potentially as high as the entire mass of chondrules inferred to exist in the asteroid belt today, $\\sim 2 \\times 10^{24} \\, {\\rm g}$ (Grossman 1988).", "A small number of planetary embryos forming chondrules intermittently over several Myr would be consistent with the ages of chondrules (Kurahashi et al.", "2008; Villeneuve et al.", "2009).", "A diversity of chondrule sizes, compositions and textural types could result from the evolution of the solar nebula between chondrule-forming events involving protoplanets with different $a$ , $e_{0}$ and $i_{0}$ .", "Yet another consideration is what type of object might be produced outside of the impact parameter necessary to produce chondrules.", "It is likely that solids will be fully or partially melted in these regions, and experience rapid cooling rates, resulting in objects that do not fit the classical definition of chondrules.", "For example, Miura et al.", "(2010) proposed that silicate crystals found in meteorites could be produced by supercooling of a silicate vapor in bow shocks.", "Alexander et al.", "(2007) found that the microstructure of pyroxene in interplanetary dust particles (IDPs) indicates that it formed at temperatures $>$ 1258 K and cooled relatively rapidly ($\\sim $ 1000 K hr$^{-1}$ ), suggesting some type of shock heating.", "Scott & Krot (2005) proposed that most matrix silicates likely formed as condensates close to chondrules in transient heating events, cooling below 1300 K at $\\sim $ 1000 K hr$^{-1}$ .", "Rubin & Wasson (2003) suggest that ferroan olivine overgrowths on chondrules and chondrule fragments are produced by cooling rates orders of magnitude greater than those experienced by the chondrules themselves.", "Formation outside the chondrule-forming impact parameter in a planetary embryo bow shock may explain objects such as the silicates found in meteorites (Scott & Krot 2005; Miura et al.", "2010), pyroxene in IDPs (Alexander et al.", "2007), and overgrowths on chondrules (Wasson & Rubin 2003; Wasson et al.", "2003).", "Further detailed modeling of the thermal histories of objects formed in bow shocks around planetary embryos is necessary to address this intriguing possibility.", "Finally, we speculate that there is also likely to be a critical size for a body that produces chondrules with the proper thermal histories, that are not subsequently accreted.", "Based on the standoff distances found for the two bodies considered in this study (§3), we anticipate that this critical size is $\\sim $ 1000 km in radius.", "In future work, we plan a parameter study of bodies ranging in size from radii of 500 - 2720 km in order to test this hypothesis.", "In this study, we have found that chondrule formation in planetary embryo bow shocks is fundamentally different from formation in planetesimal bow shocks, but on the basis of our preliminary calculations this model appears to fit all the known constraints on chondrule formation.", "We intend to further develop the model, especially to include fully 2-D radiative transfer in determining chondrule thermal histories, to better test whether chondrules could form in planetary bow shocks and what other types of object may form along with them.", "We intend to determine the critical size of a body that is necessary for chondrule formation in bow shocks.", "At the same time, we do not exclude other formation mechanisms for chondrules.", "Chondrules in CH/CB chondrites, in particular, appear to have formed from an impact plume (Krot et al.", "2005).", "Large-scale shocks driven by gravitational instabilities also remain a viable option for chondrule formation (Morris & Desch 2010).", "Nevertheless, although there are some details that need to be explored further, the overall ability of the model to match chondrule constraints — especially the elevated partial pressures of volatiles — makes the planetary embryo bow shock model a very promising mechanism for chondrule formation.", "We gratefully acknowledge support from NASA Origins of Solar Systems grant NNHX10AH34G (PI SJD).", "ACB's support was provided by a contract with the California Institute of Technology (Caltech) funded by NASA through the Sagan Fellowship Program.", "Table: Masses of chondrules produced by planetary embryo bow shock over the orbital evolution of thebody, assuming an initial semi-major axis of a=1.0 AU a = 1.0 \\, {\\rm AU} and different initial valueseccentricity ee and inclination ii.We assume chondrule formation occurs only when the instantaneous velocity difference V s V_{\\rm s}between the embryo and the gas is in the range 6-8 km s -1 6 - 8 \\, {\\rm km} \\, {\\rm s}^{-1}.Max V s V_{\\rm s} is the maximum shock velocity during the orbital evolution.The duration signifies the time period beginning with the first episode of chondrule formationand ending with the last episode of chondrule formation before damping of the embryo's ee and ii.A concentration factor 𝒞=1{\\cal C} = 1 (implying chondrules-to-gas mass ratio 0.00375) was assumed.Only material that is within H/CH/C, for disk gas scale height HH is included in M s M_s.The notation a(-b)a(-b) means a×10 -b a\\times 10^{-b}.Table: Same as Table 1, but for a=1.5 AU a = 1.5 \\, {\\rm AU}.Table: Same as Table 1, but for a=2.5 AU a = 2.5 \\, {\\rm AU}.Table: Same as Table 1, but assuming a concentration of chondrules𝒞=10{\\cal C} = 10 within a height |z|<H/10\\vert z\\vert < H/10 of the midplane.Table: Same as Table 4, but with a=1.5 AU a = 1.5 \\, {\\rm AU}.Table: Same as Table 4, but with a=2.5 AU a = 2.5 \\, {\\rm AU}.Figure: Velocity difference V s V_{\\rm s} between nebular gas (on a circular Keplerian orbit)and a planetary embryo on an eccentric orbit with semi-major axis a=1.25 AU a = 1.25 \\, {\\rm AU}, eccentricitye=0.2e = 0.2 and inclination i=15 ∘ i = 15^{\\circ }.The radial, azimuthal and vertical components, and the total velocity difference V s V_{\\rm s}(equal to the speed of the bow shock) are shown as functions of time over the orbit, where t=0t = 0signifies perihelion passage.In this example the planet passes through the ascending node at t≈1.1t \\approx 1.1 years.V s V_{\\rm s} is supersonic and bow shocks are driven throughout the orbit.Figure: Animation showing the evolution of a bow shock around a planetary embryo with radius 2720 km (i.e.", "half a Mars mass), on an inclined and eccentric orbit.Figure: Left: Gas density (blue contours) and velocities (white arrows),and trajectories of chondrule precursors / chondrules (gold streamlines),in the vicinity of the bow shock surrounding a 500-km radius planetesimal.Velocities are measured in the frame of the planetesimal, moving at8 km s -1 8 \\, {\\rm km} \\, {\\rm s}^{-1} with respect to the local gas.", "Chondruleswith impact parameters ≲\\lesssim 460 km are accreted.Right: The same plot repeated for a planetary embryo with radius2720 km (i.e., half a Mars mass).", "Note the change in scale.", "In thiscase, particles with impact parameter ≳\\gtrsim 400 km arenot accreted.Figure: Left: Maximum relative velocity between chondrules and gas,achieved just after passing through the bow shock, as a function ofthe impact parameter bb, for the R p =500 km R_{\\rm p} = 500 \\, {\\rm km} case.Values of bb that lead to the chondrules being accreted are plottedwith a dotted line instead of a solid line.", "The minimum relativevelocity needed for chondrules to melt (=5 km s -1 = 5 \\, {\\rm km} \\, {\\rm s}^{-1})is shown.Right: The same but for the R p =2720 km R_{\\rm p} = 2720 \\, {\\rm km} case.Only particles with b≲400 km b \\lesssim 400 \\, {\\rm km} might be accreted.Figure: Thermal histories of chondrules formed in bow shocks around planetary embryos, assuming shock speedsof 5 km s -1 5 \\, {\\rm km} \\, {\\rm s}^{-1} (top left), 6 km s -1 6 \\, {\\rm km} \\, {\\rm s}^{-1} (top right), 7 km s -1 7 \\, {\\rm km} \\, {\\rm s}^{-1} (bottom left), and8 km s -1 8 \\, {\\rm km} \\, {\\rm s}^{-1} (bottom right).", "Chondrule precursors pass through the shock front at t=0t = 0.See text for details.Figure: Eccentricity damping times as a function of planetary embryo mass.", "Damping for smallbodies (M<0.1M ⊕ M < 0.1 \\, M_{\\oplus }) is dominated by gas drag, and damping of large bodies(M>1M ⊕ M > 1 \\, M_{\\oplus }) is dominated by disk torques.", "For heliocentric distances 1 - 3 AU,Mars-sized bodies are in the transition regime that is least damped.Figure: Left: Eccentricity evolution for three scattered-planet scenarios.Each embryo has mass 0.055M ⊕ 0.055 \\, M_{\\oplus }, radius 2720 km, and semi-major axis 2.5 AU.Curves correspond to starting conditions e 0 =0.4e_0 = 0.4, i 0 =0 ∘ i_0 = 0^{\\circ } (solid triangles),e 0 =0.2e_0 = 0.2, i 0 =15 ∘ i_0 = 15^{\\circ } (solid circles)e 0 =0.4e_0 = 0.4, i 0 =15 ∘ i_0 = 15^{\\circ } (solid and open squares).Right: Maximum shock velocities during an orbit, as a function of time, for the same three cases." ] ]

[ [ "Approximate Analytical Solutions of a Two-Term Diatomic Molecular\n Potential with Centrifugal Barrier" ], [ "Abstract Approximate analytical bound state solutions of the radial Schr\\\"odinger equation are studied for a two-term diatomic molecular potential in terms of the hypergeometric functions for the cases where $q\\geq1$ and $q=0$.", "The energy eigenvalues and the corresponding normalized wave functions of the Manning-Rosen potential, the 'standard' Hulth\\'{e}n potential and the generalized Morse potential are briefly studied as special cases.", "It is observed that our analytical results are the same with the ones obtained before." ], [ "Introduction", "In this letter, we study the bound state solutions of a diatomic molecular potential having the form $V(r,\\beta ,q)=-V_{0}\\,\\frac{e^{-\\beta r}}{1-qe^{-\\beta r}}+V_{1}\\,\\frac{e^{-2\\beta r}}{(1-qe^{-\\beta r})^2}\\,,$ which has been firstly presented by Sun to fit experimental data of some diatomic molecular systems [1].", "Analytical study of the above potential could be interesting since it involves several potential forms (for example, $q=0$ gives the Morse potential, $q>0$ corresponds to the 'generalized' Hulthén potential, etc.)", "meaning that we can simply extend the solutions to the ones of these special cases.", "The above potential is one of the central potentials which are a powerful ground for experimental and theoretical computations in different areas of physics such as in high energy physics where they were used to describe hadrons as bound states [2], in atomic physics where some important subjects such as binding energy and inclusive momentum distributions are studied by using of central potentials [3], in theoretical molecular dynamics model to study the intramolecular and intermolecular interactions and atomic pair correlation functions [4].", "Moreover, the central potentials have been used in an important quantum mechanical problem which is also related with quantum information theory, the Fisher uncertainty relation and applied to the hydrogen atom and isotropic harmonic oscillator [5] and also for some theoretical calculations within the information theory to study some statistical quantities such as the Boltzmann-Shannon entropy [6].", "The construction of an algorithm could be an interesting problem where the aim is to solve the radial Schrödinger equation (SE) for a given central potential $V(r)$ numerically [7].", "To our knowledge, the potential under consideration has been studied within the supersymmetric quantum mechanics [8] and in terms of Green's function [9] in non-relativistic domain.", "We search the bound state spectrum and the wave functions of the above potential by using an approximation instead of the centrifugal term in the same domain.", "We find an analytical expression for the energy spectrum and obtain the normalization constant by using some properties of the hypergeometric functions.", "Throughout this work, we restrict ourself to the cases where $q\\ge 1$ and $q=0$ and give our numerical results for two diatomic molecules for different values of quantum number pair ($n, \\ell $ )." ], [ "Energy Spectrum and Wave Functions", "The radial Schrödinger equation is written [10] $\\frac{d^{2}R(r)}{dr^2}+\\left[\\frac{2m}{\\hbar ^2}\\left[E_{n\\ell }-V(r)\\right]-\\frac{\\ell (\\ell +1)}{r^2}\\right]R(r)=0\\,,$ where $\\ell $ is the angular momentum quantum number, $m$ is the particle mass, $V(r)$ is the central potential and $E_{n\\ell }$ is the non-relativistic energy.", "Inserting Eq.", "(1) into Eq.", "(2) gives $\\frac{d^{2}R(r)}{dr^2}+\\left[-\\frac{2mV_{1}}{\\hbar ^2}\\,\\frac{1}{(e^{\\beta r}-q)^2}+\\frac{2mV_{0}}{\\hbar ^2}\\,\\frac{1}{e^{\\beta r}-q}+\\frac{2mE_{n\\ell }}{\\hbar ^2}-\\frac{\\ell (\\ell +1)}{r^2}\\right]R(r)=0\\,.$ where $V_{0}, V_{1}, \\beta $ and $q$ are real parameters defined by $V_{1}=D_{0}(e^{\\mu }-q)$ , $V_{0}=2V_{1}$ , $\\beta =\\mu /r_{0}$ , where $D_{0}$ is the depth of the potential, $r_{0}$ is the equilibrium of the molecule and $q$ is the shape parameter.", "We use the following approximation [11] instead of the centrifugal term among the others [12-15] to obtain an analytical solution of Eq.", "(3) $\\frac{1}{r^2} \\approx \\beta ^2\\,\\frac{e^{\\beta r}}{(e^{\\beta r}-q)^2}\\,,$ Defining a new variable $z=qe^{-\\beta r}$ and taking a trial function as $R(z)=z^{A_{1}}(1-z)^{A_{2}}\\phi (z)$ and with the help of Eq.", "(4), Eq.", "(3) turns into $z(1-z)\\frac{d^2\\phi (z)}{dz^2}&+&\\left[1+2A_{1}-(1+2A_{1}+2A_{2})z\\right]\\frac{d\\phi (z)}{dz}\\nonumber \\\\&+&\\left[-2A_{1}A_{2}-A^2_{2}+\\frac{2mV_{1}}{\\beta ^2\\hbar ^2}+\\frac{2mV_{0}}{q\\beta ^2\\hbar ^2}\\right]\\phi (z)=0\\,,$ where we set the parameters $&A^2_{1}=-\\frac{2m E_{n\\ell }}{\\hbar ^2}\\,,\\\\&A_{2}(A_{2}-1)=\\frac{1}{q}\\,\\ell (\\ell +1)+\\frac{2mV_{1}}{\\beta ^2\\hbar ^2}\\,.$ By using the abbreviations $&c=1+2A_{1}\\,,\\\\&b=A_{1}+A_{2}+\\sqrt{A^2_{1}+\\frac{2m}{\\beta ^2\\hbar ^2}\\left(V_{1}+\\frac{V_{0}}{q}\\right)\\,}\\,,\\\\&a=A_{1}+A_{2}-\\sqrt{A^2_{1}+\\frac{2m}{\\beta ^2\\hbar ^2}\\left(V_{1}+\\frac{V_{0}}{q}\\right)\\,}\\,,$ Eq.", "(5) becomes an equation having the form of the hypergeometric-type equation [16] $z(1-z)\\phi ^{\\prime \\prime }(z)+[c-(a+b+1)z]\\phi ^{\\prime }(z)-ab\\phi (z)=0\\,,$ whose solution is the hypergeometric functions $\\phi (z) \\sim \\, _{2}F_{1}(a, b; c; z)\\,.$ When either $a$ or $b$ equals to a negative integer $-n$ , the hypergeometric function $\\phi (z)$ can be reduced to a finite solution.", "This gives us a polynomial of degree $n$ in Eq.", "(9) and the following quantum condition $A_{1}+A_{2}-\\sqrt{A^2_{1}+\\frac{2m}{\\beta ^2\\hbar ^2}\\left(V_{1}+\\frac{V_{0}}{q}\\right)\\,}=-n\\,,$ which gives the energy values of the two-term potential for any $\\ell $ -values $E_{n\\ell }=-\\frac{\\beta ^2\\hbar ^2}{2m}\\left[\\frac{n^2+(2n+1)\\left(A^{\\prime }_{2}+\\frac{1}{2}\\right)+\\frac{1}{q}\\left[\\ell (\\ell +1)-\\frac{2mV_{0}}{\\beta ^2\\hbar ^2}\\right]}{2n+1+2A^{\\prime }_{2}}\\right]^2\\,,$ where $A^{\\prime }_{2}=\\sqrt{\\frac{1}{4}+\\frac{\\ell (\\ell +1)}{q}+\\frac{2mV_{1}}{\\beta ^2\\hbar ^2}\\,}\\,.$ By using Eq.", "(10) we obtain the total wave functions $R(z)=Nz^{A_{1}}(1-z)^{A_{2}}\\,_{2}F_{1}(-n,n+2A_{1}+2A_{2};1+2A_{1};z)\\,.$ where $N$ is the normalization constant and will be derived in Appendix A.", "We summarize our numerical results in Table 1 and 2 where the computations are made for two diatomic molecules, namely $H_{2}$ and $LiH$ .", "The values of potential parameters we used for these molecules are as follows [13]: $D_{0}=4.744600$ eV, $r_{0}=0.741600\\,Å$ , $m=0.503910$ amu, $\\mu =1.440558$ and $E_{0}=\\hbar ^2/(mr^2_{0})=1.508343932 \\times 10^{-2}$ eV for $H_{2}$ molecule and $D_{0}=2.515287$ eV, $r_{0}=1.595600\\,Å$ , $m=0.8801221$ amu, $\\mu =1.7998368$ and $E_{0}=1.865528199 \\times 10^{-3}$ eV for $LiH$ molecule [17].", "It is seen that the energy values decrease while the values of the quantum numbers increase and the energy eigenvalues are also inversely proportional with the shape parameter for each of the molecules.", "Now we intend briefly to study some special cases whose energy eigenvalue equation obtained from Eq.", "(11) by suitable choices of the potential parameters." ], [ "Manning-Rosen Potential", "The Manning-Rosen potential can be written as [18] $V(r)=-\\frac{A\\hbar ^2}{2mb^2}\\,\\frac{1}{e^{r/b}-1}+\\frac{\\alpha (\\alpha -1)\\hbar ^2}{2mb^2}\\,\\frac{1}{(e^{r/b}-1)^2}\\,,$ If we write our parameters as $V_{0}=\\frac{A}{2b^2}$ ; $V_{1}=\\frac{\\alpha (\\alpha -1)}{2b^2}$ ;$\\beta =\\frac{1}{b}$ and $q=1$ then we obtain the energy eigenvalues of the Manning-Rosen potential $E_{n\\ell }=-\\frac{1}{2b^2}\\left[\\frac{n^2+(2n+1)\\left(\\frac{1}{2}+\\sqrt{\\frac{1}{4}+\\ell (\\ell +1)+\\alpha (\\alpha -1)\\,}\\,\\right)+\\ell (\\ell +1)-A}{2n+1+2\\sqrt{\\frac{1}{4}+\\ell (\\ell +1)+\\alpha (\\alpha -1)\\,}}\\right]^2\\,.$ which is the same result obtained in Ref.", "[19].", "The normalization constant in Eq.", "(13) is obtained from Eq.", "(A8) as $N=\\left[\\frac{1}{g(A_{1}^{(1)},A_{2}^{(1)},k)g(A_{1}^{(1)},A_{2}^{(1)},l)\\,_{2}F_{1}(-2A_{2}^{(1)},1+2A_{1}^{(1)}+k+l;2+2A_{2}^{(1)}+k+l;1)}\\right]^{1/2}\\,.\\nonumber \\\\$ where $A_{1}^{(1)}=\\sqrt{-2mE_{n\\ell }b^2/\\hbar ^2\\,}$ and $A_{2}^{(1)}=(1/2)\\left(1+\\sqrt{1+4\\ell (\\ell +1)+4m\\alpha (\\alpha -1)/\\hbar ^2\\,}\\right)$ .", "We summarize our numerical results obtained from Eq.", "(15) in Table 3 where we set the parameters as $A=2b$ and $\\alpha =0.75$ to compare the results with the ones given in Ref.", "[19].", "Please note that the parameter $D_{0}$ used in Ref.", "[19] is zero in the present work since our approximation used for the centrifugal term is different from the one used in Ref.", "[19] where energy eigenvalues are computed in atomic units." ], [ "Standard Hulthén Potential", "Eq.", "(1) gives the standard Hulthén potential for $V_{1}=0$ and $q=1$ $V(r)=-V_{0}\\,\\frac{e^{-\\beta r}}{1-e^{-\\beta r}}\\,,$ and we obtain the energy eigenvalues from Eq.", "(11) $E_{n\\ell }=-\\frac{\\beta ^2\\hbar ^2}{2m}\\left[\\frac{(n+\\ell )(n+\\ell +2)+1-\\frac{2mV_{0}}{\\beta ^2\\hbar ^2}}{2(n+1+\\ell )}\\right]^2\\,.$ and the normalization constant of the corresponding wave functions from Eq.", "(A8) $N=\\left[\\frac{\\Gamma (2+4A_{2}^{(2)}+k+\\ell )}{g(A_{1},A_{2}^{(2)},k)g(A_{1},A_{2}^{(2)},\\ell )(1+2A_{2}^{(2)}+k+\\ell )\\Gamma (1-2A_{1}+4A_{2}^{(2)})}\\right]^{1/2}$ where $A_{2}^{(2)}=1+\\ell $  .", "Choosing the parameters as $\\beta =\\frac{1}{a}$ and $V_{0}=\\alpha $ gives the following expression ($m=\\hbar =1$ ) $E_{n\\ell }=-\\frac{1}{2a^2}\\left[\\frac{(n+\\ell )(n+\\ell +2)+1-2\\alpha a^2}{2(n+1+\\ell )}\\right]^2\\,.$ which is the same result given in Ref.", "[20].", "The standard Hulthén potential in Eq.", "(16) could gives the Coulomb potential for $\\beta r \\ll 1$ $V(r)=-\\frac{Ze^2}{r}\\,,$ where we set $V_{0}=Ze^2\\beta $  .", "We obtain the energy spectrum of the Coulomb potential from Eq.", "(18) ($m=\\hbar =e=1$ ) $E_{n\\ell }=-\\frac{Z^2}{2(n+1+\\ell )^2}\\,.$ which is the same result obtained in Ref.", "[19].", "The normalization constant of the corresponding wave functions is given with the help of Eq.", "(19) under the above assumptions.", "The numerical energy values of the Hulthén potential obtained from Eq.", "(18) are placed in second part of Table 3 for different quantum number pair ($n, \\ell $ ) in atomic units.", "We choose the parameters as $V_{0}=\\beta =\\delta $ as in Ref [21]." ], [ "Generalized Morse Potential", "We obtain the generalized Morse potential for the limit $q\\rightarrow 0$ in Eq.", "(1) $V(r)=V_{1}e^{-2\\beta r}-V_{0}e^{-\\beta r}\\,,$ which gives the following equation for $s$ -waves $\\left(\\frac{d^2}{dr^2}-\\frac{2mV_{1}}{\\hbar ^2}\\,e^{-\\beta r}+\\frac{2mV_{0}}{\\hbar ^2}\\,e^{-2\\beta r}+\\frac{2mE_{n\\ell }}{\\hbar ^2}\\right)R(r)=0\\,,$ Defining a new variable $z=e^{-\\beta r}$ and taking the wave function of the form $R(z)=e^{-B_{1}z/2}z^{B_{2}/2}\\phi (z)$ , we obtain $z\\frac{d^2\\phi (z)}{dz^2}+\\left(1+B_{2}-B_{1}z\\right)\\frac{d\\phi (z)}{dz}+\\left[-\\frac{B_{1}B_{2}}{2}-\\frac{B_{1}}{2}+\\frac{2mV_{0}}{\\beta ^2\\hbar ^2}\\right]\\phi (z)=0\\,,$ where $B^2_{1}=\\frac{8mV_{0}}{\\beta ^2\\hbar ^2}$ and $B^2_{2}=-\\frac{8mE}{\\beta ^2\\hbar ^2}$ .", "Using a new variable $y=B_{1}z$ gives $y\\frac{d^2\\phi (y)}{dy^2}+\\left(1+B_{2}-y\\right)\\frac{d\\phi (z)}{dz}+\\left[-\\frac{B_{2}}{2}-\\frac{1}{2}+\\frac{2mV_{0}}{B_{1}\\beta ^2\\hbar ^2}\\right]\\phi (y)=0\\,,$ which is the Laguerre differential equation $xy^{\\prime \\prime }+(\\alpha +1-y)y^{\\prime }+ny=0\\,.$ where the factor $n$ should be zero or a positive integer to get a polynomial solution [22].", "So, the solution of Eq.", "(25) are given in terms of the Laguerre polynomials as $\\phi (y) \\sim L_{n}^{\\overline{\\sigma }}(y)\\,,$ where $\\overline{\\sigma }=B_{2}$ and $n=-\\frac{B_{2}}{2}-\\frac{1}{2}+\\frac{2mV_{0}}{B_{1}\\beta ^2\\hbar ^2}$  .", "We get the total eigenfunctions of the Morse potential $R(z)=Ne^{-B_{1}z/2}z^{B_{2}/2}L_{n}^{\\overline{\\sigma }}(B_{1}z)\\,.$ and the energy eigenvalues $E_{n\\ell }=-\\frac{\\beta ^2\\hbar ^2}{8m}\\left\\lbrace 2n+1-\\frac{V_{0}}{\\beta \\hbar }\\sqrt{\\frac{2m}{V_{1}}\\,}\\right\\rbrace ^2\\,.$ We present the numerical energy values of the Morse potential obtained from the above equation in Table 3.", "We give the results for $H_{2}$ molecule (in $eV$ ) by taking the same parameter values to obtain the results given in Table 1 and by setting the potential parameters as $V_{0}=2D_{0}$ , $V_{1}=D_{0}$ .", "Using the following representation of the Laguerre polynomials [22] $L_{n}^{\\overline{\\sigma }}(x)=\\sum _{k=0}^{n}(-1)^{k}\\Bigg (\\begin{array}{c}n+\\overline{\\sigma } \\\\n-k\\end{array}\\Bigg )\\frac{x^{k}}{k!", "}\\,,$ the normalization condition is written as $\\left|N\\right|^{2}g(n)g(m)\\int _{0}^{1}e^{B_{1}z}z^{B_{2}+2k}dz=1\\,,$ where $g(n)=\\sum _{k=0}^{n}(-1)^{k}\\Bigg (\\begin{array}{c}n+B_{2} \\\\n-k\\end{array}\\Bigg )\\frac{B_{1}^{k}}{k!", "}\\,\\,;g(m)=\\sum _{k=0}^{n}(-1)^{k}\\Bigg (\\begin{array}{c}m+B_{2} \\\\m-k\\end{array}\\Bigg )\\frac{B_{1}^{k}}{k!", "}\\,.$ Changing the variable $t=B_{1}z$ in Eq.", "(32) gives $\\left|N\\right|^{2}g(n)g(m)B_{1}^{-(1+B_{2}+2k)}\\int _{0}^{1}e^{-t}t^{B_{2}+2k}dt=1\\,,$ which includes the incomplete Gamma function defined as [22] $\\gamma (a,x)\\equiv \\int _{0}^{x}t^{a-1}e^{-t}dt=\\frac{1}{a}x^{a}e^{-x}\\,_{1}F_{1}(1;1+a;x)\\,,$ Finally the normalization constant is obtained as $N=\\left[\\frac{e\\Omega B_{1}^{\\Omega }}{g(n)g(m)\\,_{1}F_{1}(1;1+\\Omega ;1)}\\right]^{1/2}\\,,\\,\\,\\,\\Omega =1+B_{2}+2k\\,.$ We have studied the approximate bound state solutions of the radial SE equation for a two-term potential.", "We have obtained the energy eigenvalues and the corresponding normalized wave functions approximately in terms of the hypergeometric functions.", "We have presented our numerical results of the energy eigenvalues of two diatomic molecules in Tables 1 and 2.", "We have also studied the analytical bound state solutions of the Manning-Rosen potential, the 'standard' Hulthén potential and the generalized Morse potential as special cases.", "We have observed that our all analytical results are the same with the ones obtained in the literature.", "We have also summarized some numerical results of the energy eigenvalues of the above three potentials in Table 3 and observed that our results are good agreement with the ones obtained before." ], [ "Acknowledgments", "This research was partially supported by the Scientific and Technical Research Council of Turkey." ], [ "Normalization Constant", "The wave functions in Eq.", "(13) is $R(z)=Nz^{A_{1}}(1-z)^{A_{2}}\\,_{2}F_{1}(-n,n+2A_{1}+2A_{2};1+2A_{1};z)\\,,$ which is written in terms of the new variable $z=q\\xi $ ($0\\le \\xi \\le 1$ ) $R(q\\xi )=Nq^{A_{1}}\\xi ^{A_{1}}(1-q\\xi )^{A_{2}}\\,_{2}F_{1}(-n,n+2A_{1}+2A_{2};1+2A_{1};q\\xi )\\,,$ The normalization condition $\\int _{0}^{1}\\left|R(q\\xi )\\right|^2d\\xi =1$ gives $\\left|N\\right|^2q^{1+2A_{1}}\\int _{0}^{1}\\xi ^{2A_{1}}(1-q\\xi )^{2A_{2}}\\left[\\,_{2}F_{1}(-n,n+2A_{1}+2A_{2};1+2A_{1};q\\xi )\\right]^2dx=1\\,.$ Using the representation of the hypergeometric functions [22] $\\,_{2}F_{1}(-n,b;c;z)=\\sum _{k=0}^{n}\\frac{(-n)_{k}(b)_{k}}{(c)_{k}k!", "}z^{k}\\,,$ Eq.", "(A3) becomes $\\left|N\\right|^2q^{1+2A_{1}}g(A_{1},A_{2},k)g(A_{1},A_{2},l)\\int _{0}^{1}\\xi ^{2A_{1}+k+l}(1-q\\xi )^{2A_{2}}d\\xi =1\\,,$ where $(-n)_{k}=(-1)^{k}(n-k+1)_{k}=(-1)^{k}\\,\\frac{\\Gamma (n+1)}{\\Gamma (n-k+1)}$ and $g(A_{1},A_{2},k)=\\sum _{k=0}^{n}\\frac{(-n)_{k}(n+2A_{1}+2A_{2})_{k}}{(1+2A_{1})_{k}k!", "}z^{k}\\,.$ and $g(A_{1},A_{2},l)=g(A_{1},A_{2},k \\rightarrow l)$  .", "Using the following identity for the hypergeometric functions [22] $\\,_{2}F_{1}(\\alpha ^{\\prime },\\beta ^{\\prime };\\delta ^{\\prime };z)=\\frac{\\Gamma (\\delta ^{\\prime })}{\\Gamma (\\beta ^{\\prime })\\Gamma (\\delta ^{\\prime }-\\beta ^{\\prime })}\\,\\int _{0}^{1}t^{\\beta ^{\\prime }-1}(1-t)^{\\delta ^{\\prime }-\\beta ^{\\prime }-1}(1-tz)^{-\\alpha ^{\\prime }}dt\\,,$ we obtain the normalization constant from Eq.", "(A5) $N=\\left[\\frac{1}{q^{1+2A_{1}}g(A_{1},A_{2},k)g(A_{1},A_{2},l)\\,_{2}F_{1}(-2A_{2},1+2A_{1}+k+l;2+2A_{2}+k+l;q)}\\right]^{1/2}\\,.\\nonumber \\\\$" ] ]

[ [ "Computing Functionals of Multidimensional Diffusions via Monte Carlo\n Methods" ], [ "Abstract We discuss suitable classes of diffusion processes, for which functionals relevant to finance can be computed via Monte Carlo methods.", "In particular, we construct exact simulation schemes for processes from this class.", "However, should the finance problem under consideration require e.g.", "continuous monitoring of the processes, the simulation algorithm can easily be embedded in a multilevel Monte Carlo scheme.", "We choose to introduce the finance problems under the benchmark approach, and find that this approach allows us to exploit conveniently the analytical tractability of these diffusion processes." ], [ "Introduction", "In mathematical finance, the pricing of financial derivatives can under suitable conditions be shown to amount to the computation of an expected value, see e.g.", "[53], [56].", "Depending on the financial derivative and the model under consideration, it might not be possible to compute the expected value explicitly, however, numerical methods have to be invoked.", "A candidate for the computation of such expectations is the Monte Carlo method, see e.g.", "[11], [30], and [44].", "Applying the Monte Carlo method typically entails the sampling of the distribution of the relevant financial state variables, e.g.", "an equity index, a short rate, or a commodity price.", "It is then, of course, desirable to have at one's disposal a recipe for drawing samples from the relevant distributions.", "In case these distributions are known, one refers to exact simulation schemes, see e.g.", "[55], but also [7], [8], [9], and [16], for further references on exact simulation schemes.", "If exact simulation schemes are not applicable, discrete time approximations, as analyzed in [44] and [55] become relevant.", "In recent years, it has been shown under certain assumptions that using the multilevel Monte Carlo method, see [29] and also [38], [39], the standard Monte Carlo convergence rate, achieved by exact simulation schemes, can be recovered.", "For modeling financial quantities of interest, it is important to know a priori if exact simulation schemes exist, so that financial derivatives can be priced, even if expected values cannot be computed explicitly.", "In this paper, we discuss classes of stochastic processes for which this is the case.", "For one-dimensional diffusions, Lie symmetry analysis, see [10], and [54] turns out to be a useful tool.", "Besides allowing one to discover transition densities, see [21], it also allows us to compute Laplace transforms of important multidimensional functionals, see e.g.", "[20].", "In particular, we find that squared Bessel processes fall into the class of diffusions that can be handled well via Lie symmetry methods.", "The Wishart process, [13], is the multidimensional extension of the squared Bessel process.", "It turns out, see [33] and [34], that Wishart processes are affine processes, i.e.", "their characteristic function is exponentially affine in the state variables.", "We point out that in [33], and [34] the concept of an affine process was generalized from real-valued processes to matrix-valued processes, where the latter category covers Wishart processes.", "Furthermore, the characteristic function can be computed explicitly, see [33], and [34].", "Finally, we remark that in [1] an exact simulation scheme for Wishart processes was presented.", "Modeling financial quantities, one aims for models which provide an accurate reflection of reality, whilst at the same time retaining analytical tractability.", "The benchmark approach, see [56], offers a unified framework to derivative pricing, risk management, and portfolio optimization.", "It allows us to use a much wider range of empirically supported models than under the classical no-arbitrage approach.", "At the heart of the benchmark approach sits the growth optimal portfolio (GOP).", "It is the portfolio which maximizes expected log-utility from terminal wealth.", "In particular, the benchmark approach uses the GOP as numéraire and the real world probability for taking expectations.", "We find that the class of processes for which exact simulation is possible is easily accommodated under the benchmark approach, which we illustrate using examples.", "The remaining structure of the paper is as follows: In Section we introduce the benchmark approach using a particular model for illustration, the minimal market model (MMM), see [56].", "Section introduces Lie symmetry methods and discusses how they can be used in the context of the benchmark approach.", "Section presents Wishart processes and shows how they can be used to extend the MMM.", "Section concludes the paper." ], [ "Benchmark Approach", "The GOP plays a pivotal role as benchmark and numéraire under the benchmark approach.", "It also enjoys a prominent position in the finance literature, see [43], but also [12], [45], [42], [49], [50], and [58].", "The benchmark approach uses the GOP as the numéraire.", "Since the GOP is the numéraire portfolio, see [49], contingent claims are priced under the real world probability measure.", "This avoids the restrictive assumption on the existence of an equivalent risk-neutral probability measure.", "We remark, it is argued in [56] that the existence of such a measure may not be a realistic assumption.", "Finally, we emphasize that the benchmark approach can be seen as a generalization of risk-neutral pricing, as well as other pricing approaches, such as actuarial pricing, see [56].", "To fix ideas in a simple manner, we model a well-diversified index, which we interpret as the GOP, using the stylized version of the MMM, see [56].", "Though parsimonious, this model is able to capture important empirical characteristics of well-diversified indices.", "It has subsequently been extended in several ways, see e.g.", "[56], and also [4].", "To be precise, consider a filtered probability space $(\\Omega ,{\\mathcal {A}},\\underline{\\mathcal {A}},P)$ , where the filtration $\\underline{\\mathcal {A}}=({\\mathcal {A}}_t)_{\\,t \\in [0,\\infty )}$ is assumed to satisfy the usual conditions, which carries for simplicity one source of uncertainty, a standard Brownian motion $W=\\lbrace W_t,{\\,t \\in [0,\\infty )}\\rbrace $ .", "The deterministic savings account is modeled using the differential equation $dS^0_t = r\\,S^0_t\\,dt \\, ,$ for ${\\,t \\in [0,\\infty )}$ with $S^0_0=1$ , where $r$ denotes the constant short rate.", "Next, we introduce the model for the well diversified index, the GOP $S^{{{\\delta _*}}}_t$ , which is given by the expression $S^{{\\delta _*}}_t = S^0_t\\,{\\bar{S}}^{{\\delta _*}}_t = S^0_t\\,Y_t\\,\\alpha ^{{\\delta _*}}_t \\, .$ Here $Y_t = \\frac{\\alpha ^{{{\\delta _*}}}_t}{{\\bar{S}}^{{\\delta _*}}_t}$ is a square-root process of dimension four, satisfying the stochastic differential equation (SDE) $dY_t = (1-\\eta \\,Y_t)\\,dt + \\sqrt{Y_t}\\,dW_t \\, ,$ for ${\\,t \\in [0,\\infty )}$ with initial value $Y_0 > 0$ and net growth rate $\\eta > 0$ .", "The deterministic function of time $\\alpha ^{{\\delta _*}}_t $ is given by the exponential function $\\alpha ^{{{\\delta _*}}}_t = \\alpha _0 \\exp \\left\\lbrace \\eta t \\right\\rbrace \\, ,$ with scaling parameter $\\alpha _0 > 0$ .", "Furthermore, it can be shown by the Itô formula that $\\alpha ^{{{\\delta _*}}}_t$ is the drift at time $t$ of the discounted GOP ${\\bar{S}}^{{\\delta _*}}_t := \\frac{S^{{{\\delta _*}}}_t}{S^0_t} \\, ,$ so that the parameters of the model are $S^{{{\\delta _*}}}_0$ , $\\alpha _0$ , $\\eta $ , and $r$ .", "We note that one obtains for the GOP the SDE $d S^{{{\\delta _*}}}_t = S^{{{\\delta _*}}}_t \\left( \\left( r + \\frac{1}{Y_t} \\right) dt + \\sqrt{\\frac{1}{Y_t}} d W_t \\right) \\, ,$ which illustrates the well-observed leverage effect, since as the index $S^{{{\\delta _*}}}_t$ decreases, its volatility $\\frac{1}{\\sqrt{Y_t}}=\\sqrt{\\frac{\\alpha ^{{{\\delta _*}}}_t}{{\\bar{S}}^{{\\delta _*}}_t}}$ increases and vice versa.", "It is useful to define the transformed time $\\varphi (t)$ as $\\varphi (t) = \\varphi (0) + \\frac{1}{4} \\int ^t_0 \\alpha ^{{{\\delta _*}}}_s ds \\, .$ Setting $X_{\\varphi (t)} = {\\bar{S}}^{{\\delta _*}}_t \\, ,$ we obtain the SDE $d X_{\\varphi (t)} = 4 d \\varphi (t) + 2 \\sqrt{X_{\\varphi (t)}} d W_{\\varphi (t)} \\, ,$ where $d W_{\\varphi (t)} = \\sqrt{\\frac{\\alpha ^{{{\\delta _*}}}_t}{4} } d W_t$ for $t \\in [0,\\infty )$ .", "This shows that $\\,X=\\lbrace X_\\varphi ,\\,\\varphi \\in [\\varphi (0),\\infty )\\rbrace $ is a time transformed squared Bessel process of dimension four and $\\,W=\\lbrace W_\\varphi ,\\,\\varphi \\in [\\varphi (0),\\infty )\\rbrace $ is a Wiener process in the transformed $\\varphi $ -time $\\varphi (t) \\in [\\varphi (0), \\infty )$ , see [57].", "The merit of the dynamics given by (REF ) is that transition densities of squared Bessel processes are well studied; in fact we derive them in Section using Lie symmetry methods.", "We remark that the MMM does not admit a risk-neutral probability measure because the Radon-Nikodym derivative $\\Lambda _t = \\frac{{\\bar{S}}^{{\\delta _*}}_0}{{\\bar{S}}^{{\\delta _*}}_t}$ of the putative risk-neutral measure, which is the inverse of a time transformed squared Bessel process of dimension four, is a strict local martingale and not a martingale, see [57].", "On the other hand, $S^{{\\delta _*}}$ , is the numéraire portfolio, and thus, when used as numéraire to denominate any nonnegative portfolio, yields a supermartingale under the real-world probability measure $P$ .", "This implies that the financial market under consideration is free of those arbitrage opportunities that are economically meaningful in the sense that they would allow to create strictly positive wealth out of zero initial wealth via a nonnegative portfolio, that is, under limited liability, see [48] and [56].", "This also means that we can price contingent claims under $P$ employing $S^{{{\\delta _*}}}$ as the numéraire.", "This pricing concept is referred to as real-world pricing, which we now recall, see [56]: for a nonnegative contingent claim with payoff $H$ at maturity $T$ , where $H$ is ${\\mathcal {A}}_T$ -measurable, and $E \\left( \\frac{ H }{S^{{\\delta _*}}_T} \\right) < \\infty $ , we define the value process at time $\\, t \\in [0,T]$ by $ V_t := S^{{{\\delta _*}}}_t E \\left( \\frac{H}{S^{{{\\delta _*}}}_T} \\,\\bigg |\\,{\\mathcal {A}}_t \\right) \\, .$ Note that since $V_T=H$ , the benchmarked price process $\\frac{V_t}{S^{{{\\delta _*}}}_t}$ is an $\\left(\\underline{\\mathcal {A}}, P\\right)$ -martingale.", "Formula (REF ) represents the real-world pricing formula, which provides the minimal possible price and will be used in this paper to price derivatives.", "If the expectation in equation (REF ) cannot be computed explicitly, one can resort to Monte Carlo methods.", "In that case, it is particularly convenient, if the relevant financial quantities, such as $S^{{{\\delta _*}}}_T$ can be simulated exactly.", "In the next section, we derive the transition density of $S^{{{\\delta _*}}}$ via Lie symmetry methods, which then allows us to simulate $S^{{{\\delta _*}}}_T$ exactly.", "Note, in Section , we generalize the MMM to a multidimensional setting and present a suitable exact simulation algorithm." ], [ "Lie Symmetry Methods", "The aim of this section is to present Lie symmetry methods as an effective tool for designing tractable models in mathematical finance.", "Tractable models are, in particular, useful for the evaluation of derivatives and risk measures in mathematical finance.", "We point out that in the literature, Lie symmetry methods have been used to solve mathematical finance problems explicitly, see e.g.", "[19], and [40].", "Within the current paper we want to demonstrate that they can also be used to design efficient Monte Carlo algorithms for complex multidimensional functionals.", "The advantage of the use of Lie symmetry methods is that it is straightforward to check whether the method is applicable or not.", "If the method is applicable, then the relevant solution or its Laplace transform has usually already been obtained in the literature or can be systematically derived.", "We will demonstrate this in finance applications using the benchmark approach for pricing.", "We now follow [20], and recall that if the solution of the Cauchy problem $u_t &=& b x^{\\gamma } u_{x x} + f(x) u_x - g(x) u \\, , \\, x > 0 \\, , \\, t \\ge 0 \\, , \\\\ u(x,0) &=& \\varphi (x) \\, , \\, x \\in \\Omega =[0, \\infty ) \\, ,$ is unique, then by using the Feynman-Kac formula it is given by the expectation $u(x,t) = E \\left( \\exp \\left( - \\int ^t_0 g(X_s) ds \\right) \\varphi (X_t) \\right) \\, ,$ where $X_0=x$ , and the stochastic process $X= \\left\\lbrace X_t \\, , \\, t \\ge 0 \\right\\rbrace $ satisfies the SDE $d X_t = f(X_t) dt + \\sqrt{2 b X^{\\gamma }_t} dW_t \\, .$ We now briefly describe the intuition behind the application of Lie Symmetry methods to problems from mathematical finance, in particular, the integral transform method developed in [47], and the types of results this approach can produce.", "Lie's method allows us to find vector fields $\\mathbf {v} = \\xi (x,y,u) \\partial _x + \\tau (x,t,u) \\partial _t + \\phi (x,t,u) \\partial _u \\, ,$ which generate one parameter Lie groups that preserve solutions of (REF ).", "It is standard to denote the action of $\\mathbf {v}$ on solutions $u(x,t)$ of (REF ) by $ \\rho ( \\exp \\epsilon \\mathbf {v} ) u(x,t) = \\sigma (x,t; \\epsilon ) u (a_1(x,t; \\epsilon ) , a_2 (x,t; \\epsilon ))$ for some functions $\\sigma $ , $a_1$ , and $a_2$ , where $\\epsilon $ is the parameter of the group, $\\sigma $ is referred to as the multiplier, and $a_1$ and $a_2$ are changes of variables of the symmetry.", "For the applications we have in mind, $\\epsilon $ and $\\sigma $ are of crucial importance, $\\epsilon $ will play the role of the transform parameter of the Fourier or Laplace transform and $\\sigma $ will usually be the Fourier or Laplace transform of the transition density.", "Following [19], we assume that (REF ) has a fundamental solution $p(t,x,y)$ .", "For this paper, it suffices to recall that we can express a solution $u(x,t)$ of the PDE (REF ) subject to the initial condition $u(x,0)=f(x)$ in the form $u(x,t) = \\int _{\\Omega } f(y) p(t,x,y) dy \\, ,$ where $p(t,x,y)$ is a fundamental solution of (REF ).", "The key idea of the transform method is to connect (REF ) and (REF ).", "Now consider a stationary, i.e.", "a time-independent solution, say $u_0(x)$ .", "Of course, (REF ) yields $\\rho \\left( \\exp \\epsilon \\mathbf {v} \\right) u_0(x) = \\sigma \\left( x , t ; \\epsilon \\right) u_0 \\left( a_1 (x,t ; \\varepsilon ) \\right) \\, ,$ which also solves the initial value problem.", "We now set $t=0$ and use (REF ) and (REF ) to obtain $\\int _{\\Omega } \\sigma (y,0,\\epsilon ) u_0 \\left( a_1 \\left( y , 0, ; \\epsilon \\right) \\right) p \\left( t,x,y \\right) dy = \\sigma \\left( x,t ; \\epsilon \\right) u_0 \\left( a_1 \\left( x, t; \\epsilon \\right) \\right) \\, .$ Since $\\sigma $ , $u_0$ , and $a_1$ are known functions, we have a family of integral equations for $p(t,x,y)$ .", "To illustrate this idea using an example, we consider the one-dimensional heat equation $u_t = \\frac{1}{2} g^2 u_{xx} \\, .$ We will show that if $u(x,t)$ solves (REF ), then for $\\epsilon $ sufficiently small, so does $\\tilde{u} (t,z) = \\exp \\left\\lbrace \\frac{\\epsilon t^2}{2 g^2} - \\frac{z \\epsilon }{g^2} \\right\\rbrace u \\left( z - t \\epsilon , t \\right) \\, .$ Taking $u_0=1$ , (REF ) gives $\\int ^{\\infty }_{-\\infty } \\exp \\left\\lbrace - \\frac{y \\epsilon }{g^2} \\right\\rbrace p(t,x,y) dy = \\exp \\left\\lbrace \\frac{ t \\epsilon ^2 }{2 g^2} - \\frac{x \\epsilon }{g^2} \\right\\rbrace \\, .$ Setting $a=-\\frac{\\epsilon }{g^2}$ , we get $\\int ^{\\infty }_{-\\infty } \\exp \\lbrace a y \\rbrace p(t,x,y) dy = \\exp \\left\\lbrace \\frac{a^2 g^2 t}{2} + a x \\right\\rbrace \\, .$ We recognize that (REF ) is the moment generating function of the Gaussian distribution, so $p(t,x,y)$ is the Gaussian density with mean $x$ and variance $g^2 t$ .", "We alert the reader to the fact that $\\epsilon $ plays the role of the transform parameter and $\\sigma $ corresponds to the moment generating function.", "Finally, we recall a remark from [17], namely the fact that Laplace and Fourier transforms can be readily obtained through Lie algebra computations, which suggests a deep relationship between Lie symmetry analysis and harmonic analysis.", "Lastly, we remark that in order to apply the approach, we require the PDE (REF ) to have nontrivial symmetries.", "The approach developed by Craddock and collaborators, see [17], [18], [19], [20], and [21], provides us with the following: A statement confirming if nontrivial symmetries exist and an expression stemming from (REF ), which one only needs to invert to obtain $p(t,x,y)$ .", "We first present theoretical results, and then apply these to the case of the MMM.", "Now we discuss the question whether the PDE (REF ) has nontrivial symmetries, see [20], Proposition 2.1.", "If $\\gamma \\ne 2$ , then the PDE $u_t = b x^{\\gamma } u_{x x} + f(x) u_x - g(x) u \\, , \\quad x \\ge 0 \\, , b > 0$ has a nontrivial Lie symmetry group if and only if $h$ satisfies one of the following families of drift equations $ b x h^{\\prime } - b h + \\frac{1}{2} h^2 + 2 b x^{2 - \\gamma } g(x) &= 2 b A x^{2-\\gamma } + B \\, , \\\\ b x h^{\\prime } - b h + \\frac{1}{2} h^2 + 2 b x^{2 - \\gamma } g(x) &= \\frac{A x^{4-2 \\gamma }}{2 \\left( 2 - \\gamma \\right)^2} + \\frac{B x^{2- \\gamma }}{2 - \\gamma } +C \\, , \\\\ b x h^{\\prime } - b h + \\frac{1}{2} h^2 + 2 b x^{2 - \\gamma } g(x) &= \\frac{A x^{4 - 2 \\gamma }}{2 \\left( 2 - \\gamma \\right)^2} + \\frac{B x^{3 - \\frac{3}{2} \\gamma }}{3 - \\frac{3}{2} \\gamma } + \\frac{C x^{2- \\gamma }}{2 - \\gamma } - \\kappa \\, ,$ with $\\kappa = \\frac{\\gamma }{8} \\left( \\gamma - 4 \\right) b^2$ and $h(x)=x^{1- \\gamma } f(x)$ .", "For the case $\\gamma =2$ , a similar result was obtained in [20], Proposition 2.1.", "Regarding the first Ricatti equation, (REF ), the following result was described in [20], Theorem 3.1: Suppose $\\gamma \\ne 2$ and $h(x) = x^{1- \\gamma } f(x)$ is a solution of the Ricatti equation $b x h^{\\prime } - b h + \\frac{1}{2} h^2 + 2 b x^{2 - \\gamma } g(x) = 2 b A x^{2 - \\gamma } + B \\, .$ Then the PDE (REF ) has a symmetry of the form $ \\overline{U}_{\\varepsilon }(x,t) &= \\frac{1}{\\left( 1 + 4 \\varepsilon t \\right)^{\\frac{1- \\gamma }{2 - \\gamma }}} \\exp \\left\\lbrace \\frac{- 4 \\varepsilon \\left( x^{2 - \\gamma } + A b \\left( 2 - \\gamma \\right)^2 t^2 \\right) }{b \\left( 2 - \\gamma \\right)^2 \\left( 1 + 4 \\varepsilon t \\right)} \\right\\rbrace \\\\ &\\quad \\exp \\left\\lbrace \\frac{1}{2 b} \\left( F \\left( \\frac{x}{ \\left( 1+ 4 \\varepsilon t \\right)^{\\frac{2}{2-\\gamma }}} \\right) - F \\left( x \\right) \\right) \\right\\rbrace \\\\ &\\quad u \\left( \\frac{x}{\\left( 1 + 4 \\varepsilon t \\right)^{\\frac{2}{2 - \\gamma }}} , \\frac{t}{1+ 4 \\varepsilon t} \\right) \\, ,$ where $F^{\\prime }(x) = f(x) / x^{\\gamma }$ and $u$ is a solution of the respective PDE.", "That is, for $\\varepsilon $ sufficiently small, $U_{\\varepsilon }$ is a solution of (REF ) whenever $u$ is.", "If $u(x,t)=u_0(x)$ with $u_0$ an analytic, stationary solution there is a fundamental solution $p(t,x,y)$ of (REF ) such that $\\int ^{\\infty }_0 \\exp \\lbrace - \\lambda y^{2-\\gamma } \\rbrace u_0 \\left( y \\right) p \\left( t,x,y \\right) dy = U_{\\lambda }(x,t) \\, .$ Here $U_{\\lambda }(x,t) = \\overline{U}_{\\frac{1}{4} b \\left( 2 - \\gamma \\right)^2 \\lambda }$ .", "Further, if $u_0=1$ , then $\\int ^{\\infty }_0 p(t,x,y)dy=1$ .", "For the remaining two Ricatti equations, () and (), we refer the reader to Theorems 2.5 and 2.8 in [17].", "We would now like to illustrate how the method can be used.", "Consider a squared Bessel process of dimension $\\delta $ , where $\\delta \\ge 2$ , $d X_t = \\delta dt + 2 \\sqrt{X_t} dW_t \\, ,$ where $X_0 = x >0$ .", "The drift $f(x)=\\delta $ satisfies equation (REF ) with $A=0$ .", "Consequently, using Theorem with $A=0$ and $u(x,t)=1$ , we obtain $\\overline{U}_{\\varepsilon }(x,t) = \\exp \\left\\lbrace - \\frac{4 \\epsilon x}{b \\left( 1 + 4 \\varepsilon t\\right)} \\right\\rbrace \\left( 1 + 4 \\varepsilon t \\right)^{- \\frac{\\delta }{b}} \\, ,$ where $b=2$ .", "Setting $\\varepsilon = \\frac{b \\lambda }{4}$ , we obtain the Laplace transform $U_{\\lambda }(x,t) &=& \\int ^{\\infty }_0 \\exp \\left\\lbrace - \\lambda y \\right\\rbrace p(t,x,y) dy\\\\ &=& \\exp \\left\\lbrace - \\frac{x \\lambda }{1+ 2 \\lambda t} \\right\\rbrace \\left(1 + 2 \\lambda t \\right)^{- \\frac{\\delta }{2}} \\, ,$ which is easily inverted to yield $ p(t,x,y) = \\frac{1}{2 t} \\left( \\frac{x}{y} \\right)^{\\frac{\\nu }{2}} I_{\\nu } \\left( \\frac{\\sqrt{x y}}{t} \\right) \\exp \\left\\lbrace - \\frac{(x+y)}{2 t} \\right\\rbrace \\, ,$ where $\\nu = \\frac{\\delta }{2}-1$ denotes the index of the squared Bessel process.", "Equation (REF ) shows the transition density of a squared Bessel process started at time 0 in $x$ for being at time $t$ in $y$ .", "Recall that $I_{\\nu }$ denotes the modified Bessel function of the first kind.", "This result, together with the real world pricing formula, (REF ), allows us to price a wide range of European style and path-dependent derivatives with payoffs of the type $H=f(S^*_{t_1}, S^*_{t_2}, \\dots , S^*_{t_d} )$ , where $d \\ge 1$ and $t_1 , t_2 , \\dots , t_d$ are given deterministic times.", "By exploiting the tractability of the underlying processes, Lie symmetry methods allow us to design efficient Monte Carlo algorithms, as the following example from [2] and [3] shows.", "We now consider the problem of pricing derivatives on realized variance.", "Here we define realized variance to be the quadratic variation of the log-index, and we formally compute the quadratic variation of the log-index in the form, $\\left[ \\log (S^{{{\\delta _*}}}_{\\cdot } ) \\right]_T = \\int ^T_0 \\frac{dt}{Y_t} \\, .$ Recall from Section that $Y= \\left\\lbrace Y_t \\, , \\, t \\ge 0 \\right\\rbrace $ is a square-root process whose dynamics are given in equation (REF ).", "In particular, we focus on put options on volatility, where volatility is defined to be the square-root of realized variance.", "We remark that call options on volatility can be obtained via the put-call parity relation in Lemma 4.1 in [2].", "The real-world pricing formula (REF ) yields the following price for put options on volatility $ S^{{{\\delta _*}}}_t E \\left( \\frac{(K - \\sqrt{\\frac{1}{T} \\int ^T_0 \\frac{ds}{Y_s} })^+}{S^{{{\\delta _*}}}_T} \\bigg | {\\mathcal {A}}_t \\right) \\, .$ For computing the expectation in (REF ) via Monte Carlo methods, one first needs to have access to the joint density of $(S^{{{\\delta _*}}}_T , \\int ^T_0 \\frac{ds}{Y_s} )$ and subsequently perform the Monte Carlo simulation.", "Before presenting the relevant result, we recall that $S^{{{\\delta _*}}}_T = S^0_T \\alpha ^{{{\\delta _*}}}_T Y_T$ , i.e.", "it suffices to have access to the joint distribution of $(Y_T , \\int ^T_0 \\frac{dt}{Y_t} )$ .", "We remark that if we have access to the Laplace transform of $(Y_T , \\int ^T_0 \\frac{dt}{Y_t} )$ , i.e.", "$ E \\left( \\exp \\left( - \\lambda Y_T - \\mu \\int ^T_0 \\frac{dt}{Y_t} \\right) \\right) \\, ,$ then we have, in principle, solved the problem.", "From the point of view of implementation though, inverting a two-dimensional Laplace transform numerically is expensive.", "The following result from [20], see Corollaries 5.8 - 5.9, goes further: In fact the fundamental solution corresponds to inverting the expression in (REF ) with respect to $\\lambda $ , which significantly reduces the computational complexity.", "The joint Laplace transform of $Y_T$ and $\\int ^T_0 \\frac{dt}{Y_t} $ is given by ${E \\left( \\exp \\left( - \\lambda Y_T - \\mu \\int ^T_0 \\frac{1}{Y_t} dt \\right) \\right)}\\\\ &=& \\frac{\\Gamma (3/2 + \\nu /2)}{\\Gamma (\\nu +1)} \\beta x^{-1} \\exp \\left( \\eta \\left( T+ x - \\frac{x}{\\tanh \\left( \\eta T /2 \\right) } \\right) \\right)\\\\ && \\frac{1}{\\beta \\alpha } \\exp \\left( \\beta ^2 / ( 2 \\alpha ) \\right) M_{-k, \\nu /2} \\left( \\frac{\\beta ^2}{\\alpha } \\right) \\, ,$ where $\\alpha = \\eta \\left( 1 + \\coth (\\frac{\\eta t}{2}) \\right) +\\lambda $ , $\\beta = \\frac{\\eta \\sqrt{x}}{\\sinh \\left(\\frac{\\eta t}{2} \\right)}$ , $\\nu =2\\sqrt{\\frac{1}{4}+2 \\mu }$ , and $M_{s,r}(z)$ denotes the Whittaker function of the first kind.", "In [20], the inverse with respect to $\\lambda $ was already performed explicitly and is given as $ \\nonumber p(T,x,y) &=& \\frac{\\eta }{\\sinh \\left( \\eta T /2 \\right)} \\left( \\frac{y}{x} \\right)^{1/2}\\\\ && \\exp \\left( \\eta \\left( T + x-y - \\frac{x+y}{\\tanh (\\eta T/2)} \\right) \\right) I_{\\nu } \\left( \\frac{2 \\eta \\sqrt{x y}}{\\sinh \\left( \\eta T/2 \\right)} \\right) \\, .$ Consequently, to recover the joint density of $(Y_T, \\int ^T_0 \\frac{dt}{Y_t} )$ , one only needs to invert a one-dimensional Laplace transform.", "For further details, we refer the interested reader to [3].", "By gaining access to the relevant joint densities, this example demonstrates that Lie symmetry methods allow us to design efficient Monte Carlo algorithms for challenging finance problems." ], [ "Wishart Processes", "Very tractable and highly relevant to finance are models that generalize the previously mentioned MMM.", "Along these lines, in this section we discuss Wishart processes with a view towards exact simulation.", "As demonstrated in [13], Wishart processes turn out to be the multidimensional extensions of squared Bessel processes.", "However, they also turn out to be affine, see [33], and [34].", "Prior to the latter two contributions, the literature was focused on affine processes taking values in the Euclidean space, see e.g.", "[27], and [28].", "Subsequently, matrix-valued affine processes were studied, see e.g.", "[22], and [35].", "Since [33], and [34], it has been more widely known that Wishart processes are analytically tractable, since their characteristic function is available in closed form; see also [31].", "In this section, we exploit this fact when we discuss exact simulation of Wishart processes.", "Firstly, we fix notation and present an existence result.", "Wishart processes are $S^+_d$ or $\\overline{S^+_d}$ valued, i.e.", "they assume values in the set of positive definite or positive semidefinite matrices, respectively.", "This makes them natural candidates for the modeling of covariance matrices, as noted in [33].", "Starting with [33] and [34], there is now a substantial body of literature applying Wishart processes to problems in finance, see [14], [15], [23], [24], [25], [26], and [32].", "In the current paper we study Wishart processes in a pure diffusion setting.", "For completeness, we mention that matrix valued processes incorporating jumps have been studied, see e.g.", "in [5], and [46].", "These processes are all contained in the affine framework introduced in [22], where we direct the reader interested in affine matrix valued processes.", "In the following, we introduce the Wishart process as described in the work of Grasselli and collaborators; see [25] and [35].", "For ${{\\mbox{$x$}}}\\in \\overline{S^+_d}$ , we introduce the $\\overline{S^+_d}$ valued Wishart process ${{\\mbox{$X$}}}^{{{\\mbox{\\scriptsize {$x$}}}}} = {{\\mbox{$X$}}}= \\left\\lbrace {{\\mbox{$X$}}}_t \\, , \\, t \\ge 0 \\right\\rbrace $ , which satisfies the SDE $ d {{\\mbox{$X$}}}_t = \\left( \\alpha {{\\mbox{$a$}}}^{\\top } {{\\mbox{$a$}}}+ {{\\mbox{$b$}}}{{\\mbox{$X$}}}_t + {{\\mbox{$X$}}}_t {{\\mbox{$b$}}}^{\\top } \\right) dt + \\left( \\sqrt{{{\\mbox{$X$}}}_t} d {{\\mbox{$W$}}}_t {{\\mbox{$a$}}}+ {{\\mbox{$a$}}}^{\\top } d {{\\mbox{$W$}}}^{\\top }_t \\sqrt{{{\\mbox{$X$}}}_t} \\right) \\, ,$ where $\\alpha \\ge 0$ , ${{\\mbox{$b$}}}\\in \\mathcal {M}_d$ , ${{\\mbox{$a$}}}\\in \\mathcal {M}_d$ .", "Here $\\mathcal {M}_d$ denotes the set of $d \\times d$ matrices taking values in $\\Re $ .", "An obvious question to ask is whether equation (REF ) admits a solution, and, furthermore, if such a solution is unique and strong.", "For results on weak solutions we refer the reader to [22], and for results on strong solutions to [51].", "We now present a summary of results, which in this form also appeared in [1]; see Theorem 1 in [1].", "Assume that ${{\\mbox{$x$}}}\\in \\overline{S^+_d}$ , and $\\alpha \\ge d-1$ , then equation (REF ) admits a unique weak solution.", "If ${{\\mbox{$x$}}}\\in S^+_d$ and $\\alpha \\ge d+1$ , then this solution is strong.", "In this paper, we are interested in exact simulation schemes to be used in Monte Carlo methods.", "Hence weak solutions suffice for our purposes and we assume that $\\alpha >d-1$ , so that the weak solution is unique.", "As in [1], we use $WIS_d({{\\mbox{$x$}}},\\alpha ,{{\\mbox{$b$}}},{{\\mbox{$a$}}})$ to denote a Wishart process and $WIS_d({{\\mbox{$x$}}},\\alpha ,{{\\mbox{$b$}}},{{\\mbox{$a$}}};t)$ for the value of the process at the time point $t$ .", "We begin with the study of some special cases, which includes an extension of the MMM to the multidimensional case.", "We use ${{\\mbox{$B$}}}_t$ to denote an $n \\times d$ Brownian motion and set $ {{\\mbox{$X$}}}_t = {{\\mbox{$B$}}}^{\\top }_t {{\\mbox{$B$}}}_t \\, .$ Then it can be shown that ${{\\mbox{$X$}}}= \\left\\lbrace {{\\mbox{$X$}}}_t \\, , \\, t \\ge 0 \\right\\rbrace $ satisfies the SDE $d {{\\mbox{$X$}}}_t = n {{\\mbox{$I$}}}_d dt + \\sqrt{{{\\mbox{$X$}}}_t} d{{\\mbox{$W$}}}_t + d {{\\mbox{$W$}}}^{\\top }_t \\sqrt{{{\\mbox{$X$}}}_t} \\, , $ where ${{\\mbox{$W$}}}_t$ is a $d \\times d$ Brownian motion, and ${{\\mbox{$I$}}}_d$ denotes the $d \\times d$ identity matrix.", "This corresponds to the case where we set ${{\\mbox{$a$}}}= {{\\mbox{$I$}}}_d \\, , \\, {{\\mbox{$b$}}}= {{\\mbox{$0$}}}\\, , \\, \\alpha = n \\, .$ We now provide the analogous scalar result, showing that Wishart processes generalize squared Bessel processes: Let $\\delta \\in \\mathcal {N}$ , and set $x= \\sum ^{\\delta }_{k=1} (w^k)^2 \\, .$ Now we set $ X_t = \\sum ^{\\delta }_{k=1} (W^k_t + w^k)^2 \\, .$ Then $X$ can be shown to satisfy the SDE $d X_t = \\delta dt + 2 \\sqrt{X_t} d B_t \\, ,$ where $B= \\left\\lbrace B_t\\, , \\, t \\ge 0 \\right\\rbrace $ is a scalar Brownian motion.", "This shows that (REF ) is the generalization of (REF ).", "Furthermore, it is also clear how to simulate (REF ).", "Next, we illustrate how Wishart processes can be used to extend the MMM from Section .", "We recall some results pertaining to matrix-valued random variables, see e.g.", "[36], and [52].", "We introduce some auxialiary notation.", "We denote by $\\mathcal {M}_{m,n} (\\Re )$ the set of all $m \\times n$ matrices with entries in $\\Re $ .", "Next, we present a one-to-one relationship between vectors and matrices.", "Let ${{\\mbox{$A$}}}\\in \\mathcal {M}_{m,n}(\\Re )$ with columns ${{\\mbox{$a$}}}_i \\in \\Re ^m$ , $i=1,\\dots ,n$ , and define the function $vec: \\mathcal {M}_{m,n} (\\Re ) \\rightarrow \\Re ^{m n }$ via $vec({{\\mbox{$A$}}}) = \\left( \\begin{array}{c} {{\\mbox{$a$}}}_1 \\\\ \\vdots \\\\ {{\\mbox{$a$}}}_n \\end{array} \\right) \\, .$ We can now define the matrix variate normal distribution.", "A $p \\times n$ random matrix is said to have a matrix variate normal distribution with mean ${{\\mbox{$M$}}}\\in \\mathcal {M}_{p,n}(\\Re )$ and covariance ${{\\mbox{$\\Sigma $}}}\\otimes {{\\mbox{$\\Psi $}}}$ , where ${{\\mbox{$\\Sigma $}}}\\in \\mathcal {S}^+_p$ , ${{\\mbox{$\\Psi $}}}\\in \\mathcal {S}^+_n$ , if $vec({{{\\mbox{$X$}}}}^{\\top }) \\sim \\mathcal {N}_{p n} ( vec({{{\\mbox{$M$}}}}^{\\top }), {{\\mbox{$\\Sigma $}}}\\otimes {{\\mbox{$\\Psi $}}})$ , where $\\mathcal {N}_{p n}$ denotes the multivariate normal distribution on $\\Re ^{p n}$ with mean $vec({{{\\mbox{$M$}}}}^{\\top })$ and covariance ${{\\mbox{$\\Sigma $}}}\\otimes {{\\mbox{$\\Psi $}}}$ .", "We will use the notation ${{\\mbox{$X$}}}\\sim \\mathcal {N}_{p,n}({{\\mbox{$M$}}}, {{\\mbox{$\\Sigma $}}}\\otimes {{\\mbox{$\\Psi $}}})$ .", "Next, we introduce the Wishart distribution, which we link in the subsequent theorem to the normal distribution.", "A $p \\times p$ -random matrix ${{\\mbox{$X$}}}$ in $\\mathcal {S}^+_p$ is said to have a noncentral Wishart distribution with parameters $p \\in {\\mathcal {N}}$ , $n \\ge p$ , ${{\\mbox{$\\Sigma $}}}\\in \\mathcal {S}^+_p$ and ${{\\mbox{$\\Theta $}}}\\in \\mathcal {M}_p(\\Re )$ , if its probability density function is of the form $ {f_{{{\\mbox{$X$}}}}({{\\mbox{$S$}}})}\\\\ &=& \\left( 2^{\\frac{1}{2} n p} \\Gamma _p ( \\frac{n}{2} ) det ({{\\mbox{$\\Sigma $}}})^{ \\frac{n}{2}} \\right)^{-1} etr \\left( - \\frac{1}{2} ( {{\\mbox{$\\Theta $}}}+ {{\\mbox{$\\Sigma $}}}^{-1} {{\\mbox{$S$}}}) \\right)\\\\ && det({{\\mbox{$S$}}})^{ \\frac{1}{2} ( n - p -1) } \\phantom{i}_0 F_1 \\left( \\frac{n}{2} ; \\frac{1}{4} {{\\mbox{$\\Theta $}}}{{\\mbox{$\\Sigma $}}}^{-1} {{\\mbox{$S$}}}\\right)$ where ${{\\mbox{$S$}}}\\in \\mathcal {S}^+_p$ and $\\phantom{i}_0 F_1$ is the matrix-valued hypergeometric function, see [36], and [52] for a definition.", "We write ${{\\mbox{$X$}}}~\\sim ~\\mathcal {W}_p(n, {{\\mbox{$\\Sigma $}}}, {{\\mbox{$\\Theta $}}}) \\, .$ Before stating the next result, recall that scalar non-central chi-squared random variables of integer degrees of freedom, can be constructed via sums of normal random variables; see e.g.", "[41].", "The following result presents the matrix variate analogy.", "Let ${{\\mbox{$X$}}}\\sim \\mathcal {N}_{p,n}({{\\mbox{$M$}}}, {{\\mbox{$\\Sigma $}}}\\otimes {{\\mbox{$I$}}}_n)$ , $n \\in \\left\\lbrace p, p+1, \\dots \\right\\rbrace $ .", "Then ${{\\mbox{$X$}}}{{{\\mbox{$X$}}}}^{\\top } \\sim \\mathcal {W}_p ( n , {{\\mbox{$\\Sigma $}}}, {{\\mbox{$\\Sigma $}}}^{-1} {{\\mbox{$M$}}}{{{\\mbox{$M$}}}}^{\\top } ) \\, .$" ], [ "Bivariate MMM", "Theorem is now employed to extend the MMM to a bivariate case.", "We consider exchange rate options, and follow the ideas from [37].", "The GOP denominated in units of the domestic currency is denoted by $S^a$ , and the GOP denominated in the foreign currency by $S^b$ .", "An exchange rate at time $t$ can be expressed in terms of a ratio of two GOP denominations.", "Then one would pay at time $t$ , $\\frac{S^a_t}{S^b_t}$ units of currency $a$ to obtain one unit of the foreign currency $b$ .", "As the domestic currency is indexed by $a$ , the price of, say, a call option with maturity $T$ on the exchange rate can be expressed via the real world pricing formula (REF ) as: $ S^a_0 E \\left( \\frac{\\left( \\frac{S^a_T}{S^b_T} - K \\right)^+}{S^a_T} \\right) \\, .$ We now discuss a bivariate extension of the MMM from Section , which is still tractable, as we can employ the non-central Wishart distribution to compute (REF ).", "For $k \\in \\left\\lbrace a , b \\right\\rbrace $ , we set $S^k_t = S^{0,k}_t {\\bar{S}}^k_t \\, ,$ where $S^{0,k}_t = \\exp \\lbrace r_k t \\rbrace $ , $S^{0,k}_0 = 1$ , so $S^{0,k}$ denotes the savings account in currency $k$ , which for simplicity is assumed to be a deterministic exponential function of time.", "As for the stylized MMM, we model the discounted GOP, $\\bar{S}^k_t$ , denominated in units of the $k$ th savings account, $S^{0,k}_t$ , as a time-changed squared Bessel process of dimension four.", "We introduce the $2 \\times 4$ matrix process ${{\\mbox{$X$}}}=\\left\\lbrace {{\\mbox{$X$}}}_t \\, , \\, t \\ge 0 \\right\\rbrace $ via ${{\\mbox{$X$}}}_t = \\left[ \\begin{array}{cccc} \\left( W^{1,1}_{\\varphi ^1(t)} + w^{1,1} \\right) & \\left( W^{2,1}_{\\varphi ^1(t)} + w^{2,1} \\right) & \\left( W^{3,1}_{\\varphi ^1(t)} + w^{3,1} \\right) & \\left( W^{4,1}_{\\varphi ^1(t)} + w^{4,1} \\right) \\\\ \\left( W^{1,2}_{\\varphi ^2(t)} + w^{1,2} \\right) & \\left( W^{2,2}_{\\varphi ^2(t)} + w^{2,2} \\right) & \\left( W^{3,2}_{\\varphi ^2(t)} + w^{3,2} \\right) & \\left( W^{4,2}_{\\varphi ^2(t)} + w^{4,2} \\right) \\end{array} \\right] \\, .$ The processes $W^{i,1}_{\\varphi ^1}$ , $i=1,\\dots ,4$ , denote independent Brownian motions, subject to the deterministic time-change $\\varphi ^1(t) = \\frac{\\alpha ^1_0}{4 \\eta ^1} \\left( \\exp \\lbrace \\eta ^1 t \\rbrace - 1 \\right) = \\frac{1}{4} \\int ^t_0 \\alpha ^1_s ds \\, ,$ c.f.", "Section .", "Similarly, also $W^{i,2}_{\\varphi ^2}$ , $i=1,\\dots ,4$ , denote independent Brownian motions, subject to the deterministic time change $\\varphi ^2(t) = \\frac{\\alpha ^2_0}{4 \\eta ^2} \\left( \\exp \\lbrace \\eta ^2 t \\rbrace -1 \\right) = \\frac{1}{4} \\int ^t_0 \\alpha ^2_s ds \\, .$ Now, consider the process ${{\\mbox{$Y$}}}=\\left\\lbrace {{\\mbox{$Y$}}}_t \\, , \\, t \\ge 0 \\right\\rbrace $ , which assumes values in $S^+_2$ , and is given by ${{\\mbox{$Y$}}}_t := {{\\mbox{$X$}}}_t {{{\\mbox{$X$}}}}^{\\top }_t \\, , \\, t \\ge 0 \\, ,$ which yields ${{{\\mbox{$Y$}}}_t =}\\\\& \\left[ \\begin{array}{cc} \\sum ^4_{i=1} \\left( W^{i,1}_{\\varphi ^1(t)} + w^{i,1} \\right)^2 & \\sum ^4_{i=1} \\sum ^2_{j=1} \\left( W^{i,j}_{\\varphi ^j(t)} + w^{i,j} \\right) \\\\ \\sum ^4_{i=1} \\sum ^2_{j=1} \\left( W^{i,j}_{\\varphi ^j(t)} + w^{i,j} \\right) & \\sum ^4_{i=1} \\left( W^{i,2}_{\\varphi ^2(t)} + w^{i,2} \\right)^2 \\end{array} \\right] \\, .$ We set ${\\bar{S}}^{a}_t = Y^{1,1}_t \\, ,$ and ${\\bar{S}}^b_t = Y^{2,2}_t \\, ,$ so we use the diagonal elements of ${{\\mbox{$Y$}}}_t$ to model the GOP in different currency denominations.", "Next, we introduce the following dependence structure: The Brownian motions $W^{i,1}$ and $W^{i,2}$ , $i=1,\\dots ,4$ , covary as follows, $\\langle W^{i,1}_{\\varphi ^1 (\\cdot )} , W^{i,2}_{\\varphi ^2 (\\cdot )} \\rangle _t = \\frac{\\varrho }{4} \\int ^t_0 \\sqrt{ \\alpha ^1_s \\alpha ^2_0 } ds, i=1,\\dots ,4 \\, ,$ where $-1 < \\varrho <1$ .", "The specification (REF ) allows us to employ the non-central Wishart distribution; we work through this example in detail, as it illustrates how to extend the stylized MMM to allow for a non-trivial dependence structure, but still exploit the tractability of the Wishart distribution.", "We recall that $vec({{{\\mbox{$X$}}}}^{\\top }_T)$ stacks the two columns of ${{{\\mbox{$X$}}}}^{\\top }_T$ , hence $vec( {{{\\mbox{$X$}}}}^{\\top }_T ) = \\left[ \\begin{array}{c} \\left( W^{1,1}_{\\varphi ^1(T)} + w^{1,1} \\right) \\\\ \\vdots \\\\ \\left( W^{4,1}_{\\varphi ^1(T)} + w^{4,1} \\right) \\\\ \\left( W^{1,2}_{\\varphi ^2(T)} + w^{1,2} \\right) \\\\ \\vdots \\\\ \\left( W^{4,2}_{\\varphi ^2(T)} + w^{4,2} \\right) \\end{array} \\right] \\, .$ It is easily seen that the mean matrix ${{\\mbox{$M$}}}$ of $vec(X^{\\top }_T)$ satisfies $ vec \\left( {{{\\mbox{$M$}}}}^{\\top } \\right) = \\left[ \\begin{array}{c} w^{1,1} \\\\ \\vdots \\\\ w^{4,1} \\\\ w^{1,2} \\\\ \\vdots \\\\ w^{4,2} \\end{array} \\right]$ and the covariance matrix of $vec({{{\\mbox{$X$}}}}^{\\top }_T)$ is given by $ {{\\mbox{$\\Sigma $}}}\\otimes {{\\mbox{$I$}}}_4 = \\left[ \\begin{array}{cc} \\Sigma ^{1,1} {{\\mbox{$I$}}}_4 & \\Sigma ^{1,2} {{\\mbox{$I$}}}_4 \\\\ \\Sigma ^{2,1} {{\\mbox{$I$}}}_4 & \\Sigma ^{2,2} {{\\mbox{$I$}}}_4 \\end{array} \\right] \\, ,$ where ${{\\mbox{$\\Sigma $}}}$ is a $2 \\times 2$ matrix with $\\Sigma ^{1,1} = \\varphi ^1(T)$ , $\\Sigma ^{2,2} = \\varphi ^2(T)$ , and $\\Sigma ^{1,2} = \\Sigma ^{2,1} = \\frac{\\varrho }{4}\\int ^t_0 \\sqrt{ \\alpha ^1_s \\alpha ^2_s }ds \\, .$ We remark that assuming $-1 < \\varrho < 1$ results in ${{\\mbox{$\\Sigma $}}}$ being positive definite.", "It now immediately follows from Theorem that ${{\\mbox{$X$}}}_T {{{\\mbox{$X$}}}}^{\\top }_T \\sim W_2 \\left( 4 , {{\\mbox{$\\Sigma $}}}, {{\\mbox{$\\Sigma $}}}^{-1} {{\\mbox{$M$}}}{{\\mbox{$M$}}}^{\\top } \\right) \\, ,$ where ${{\\mbox{$M$}}}$ and ${{\\mbox{$\\Sigma $}}}$ are given in equations (REF ) and (REF ), respectively.", "Recall that we set ${{\\mbox{$Y$}}}_t &=& {{\\mbox{$X$}}}_t {{{\\mbox{$X$}}}}^{\\top }_t \\, ,\\\\ {\\bar{S}}^a_t &=& Y^{1,1}_t \\, ,\\\\ {\\bar{S}}^b_t &=& Y^{2,2}_t \\, ,$ hence we can compute (REF ) using $E \\left( f ({{\\mbox{$Y$}}}_T) \\right) \\, ,$ where $f : S^+_2 \\rightarrow \\Re $ is given by $f (y) = \\frac{\\left( \\frac{\\exp \\lbrace r_1 T \\rbrace y^{1,1} }{ \\exp \\lbrace r_2 T \\rbrace y^{2,2} } - K \\right)^+}{\\exp \\lbrace r_1 T \\rbrace y^{1,1}} \\, ,$ for $y \\in S^+_2$ , and $y^{i,i}$ , $i=1,2$ , are the diagonal elements of $y$ , and the probability density function of ${{\\mbox{$Y$}}}_T$ is given in Definition .", "We now discuss further exact simulation schemes for Wishart processes, where we rely on [1] and [6].", "For integer valued parameters $\\alpha $ in (REF ), we have the following exact simulation scheme, which generalizes a well-known result from the scalar case, linking Ornstein-Uhlenbeck and square-root processes.", "In particular, this lemma shows that, in principle, certain square-root processes can be simulated using Ornstein-Uhlenbeck processes.", "Let $A>0$ , $Q>0$ , and define the SDEs $dX^i_t = - A X^i_t dt + Q dW^i_t \\, ,$ for $i= 1, \\dots , \\beta $ , where $\\beta \\in \\mathcal {N}$ , $W^1, W^2, \\dots , W^{\\beta }$ are independent Brownian motions.", "Then $Z_t = \\sum ^{\\beta }_{i=1} (X^i_t)^2$ is a square-root process of dimension $\\beta $ , whose dynamics are characterised by an SDE $d Z_t = ( \\beta Q^2 - 2 A Z_t ) dt + 2 Q \\sqrt{Z_t} dB_t \\, ,$ where $B$ is a resulting Brownian motion.", "The proof follows immediately from the Itô-formula.", "This result is easily extended to the Wishart case, for integer valued $\\alpha $ , see Section 1.2.2 in [6].", "We define $ {{\\mbox{$V$}}}_t = \\sum ^{\\beta }_{k=1} {{\\mbox{$X$}}}_{k,t} {{\\mbox{$X$}}}^{\\top }_{k,t} \\, ,$ where $ d {{\\mbox{$X$}}}_{k,t} = A {{\\mbox{$X$}}}_{k,t} dt + {{\\mbox{$Q$}}}^{\\top } d {{\\mbox{$W$}}}_{k,t} \\, , k=1, \\dots , \\beta \\, ,$ where $A \\in \\mathcal {M}_d$ , ${{\\mbox{$X$}}}_t \\in \\Re ^d$ , ${{\\mbox{$Q$}}}\\in \\mathcal {M}_d$ , ${{\\mbox{$W$}}}_k \\in \\Re ^d$ , so that ${{\\mbox{$V$}}}_t \\in \\mathcal {M}_d$ .", "The following lemma gives the dynamics of ${{\\mbox{$V$}}}= \\left\\lbrace {{\\mbox{$V$}}}_t \\, , \\, t \\ge 0 \\right\\rbrace $ .", "Assume that ${{\\mbox{$V$}}}_t$ is given by equation (REF ), where ${{\\mbox{$X$}}}_t$ satisfies equation (REF ).", "Then $d {{\\mbox{$V$}}}_t = \\left( \\beta {{\\mbox{$Q$}}}^{\\top } {{\\mbox{$Q$}}}+ A {{\\mbox{$V$}}}_t + {{\\mbox{$V$}}}_t A^{\\top } \\right) dt + \\sqrt{{{\\mbox{$V$}}}_t} d {{\\mbox{$W$}}}_t {{\\mbox{$Q$}}}+ {{\\mbox{$Q$}}}^{\\top } d{{\\mbox{$W$}}}^{\\top }_t \\sqrt{{{\\mbox{$V$}}}_t} \\, ,$ where ${{\\mbox{$W$}}}= \\left\\lbrace {{\\mbox{$W$}}}_t \\, , \\, t \\ge 0 \\right\\rbrace $ is a $d \\times d$ matrix valued Brownian motion that is determined by $\\sqrt{{{\\mbox{$V$}}}_t} d {{\\mbox{$W$}}}_t = \\sum ^{\\beta }_{k=1} {{\\mbox{$X$}}}_{k,t} d {{\\mbox{$W$}}}^{\\top }_{t,k} \\, .$ Finally, we remind the reader that vector-valued Ornstein-Uhlenbeck processes can be simulated exactly, see e.g.", "Chapter 2 in [55].", "For the general case, we refer the reader to [1].", "In that paper, a remarkable splitting property of the infinitesimal generator of the Wishart process was employed to come up with an exact simulation scheme for Wishart processes without any restriction on the parameters.", "Furthermore, in [1] higher-order discretization schemes for Wishart processes and second-order schemes for general affine diffusions on positive semidefinite matrices were presented.", "These results emphasize that Wishart processes are suitable candidates for financial models, since exact simulation schemes are readily available." ], [ "Conclusion", "In this paper, we discussed classes of stochastic processes for which exact simulation schemes are available.", "In the one-dimensional case, our first theorem gives access to explicit transition densities via Lie symmetry group results.", "In the multidimensional case the probability law of Wishart processes is described explicitly.", "When considering applications in finance, one needs a framework that can accommodate these processes as asset prices, in particular, when they generate strict local martingales.", "We demonstrated that the benchmark approach is a suitable framework for these processes and allows to systematically exploit the tractability of the models described.", "For long dated contracts in finance, insurance and for pensions the accuracy of the proposed simulation methods is extremely important." ] ]

[ [ "Efficient $N$-particle $W$ state concentration with different parity\n check gates" ], [ "Abstract We present an universal way to concentrate an arbitrary $N$-particle less-entangled $W$ state into a maximally entangled $W$ state with different parity check gates.", "It comprises two protocols.", "The first protocol is based on the linear optical elements say the partial parity check gate and the second one uses the quantum nondemolition (QND) to construct the complete parity check gate.", "Both of which can achieve the concentration task.", "These protocols have several advantages.", "First, it can obtain a maximally entangled W state only with the help of some single photons, which greatly reduces the number of entanglement resources.", "Second, in the first protocol, only linear optical elements are required which is feasible with current techniques.", "Third, in the second protocol, it can be repeated to perform the concentration step and get a higher success probability.", "All these advantages make it be useful in current quantum communication and computation applications." ], [ "Introduction", "Entanglement is the important quantum resource in both quantum communication and computation [1], [2].", "The applications of entanglement information processings such as quantum teleportation [3], [4], quantum key distribution (QKD) [5], [6], [7], quantum dense coding [8], [9], quantum secret sharing [10], [11], [12] and quantum secure direct communication (QSDC) [13], [14], [15] all resort the entanglement for setting up the quantum channel between long distance locations.", "Unfortunately, during the practical transmission, an entangled quantum system can not avoid the channel noise that comes from the environment, which will degrade the entanglement.", "It will make a maximally entangled state system become a mixed one or a partially entangled one.", "Therefore, these nonmaximally entangled systems will decrease the security of a QKD protocol if it is used to set up the quantum channel.", "Moreover, they also will decrease the fidelity of quantum dense coding and quantum teleportation.", "Entanglement purification is a powerful tool for parties to improve the fidelity of the entangled state from a mixed entangled ensembles [16], [17], [20], [22], , [24], [19], [23], [26], [27], [25], [21], [28].", "On the other hand, the entanglement concentration protocol (ECP) is focused on the pure less-entangled system, which can be used to recover a pure less-entangled state into a pure maximally entangled state with only local operation and classical communications [29], [30], [31], [32], [33], [34], [35], [37], [36], [43], [39], [40], [41], [42], [38].", "Most of the ECPs such as the Schmidt decomposition protocol proposed by Bennett et al.", "[29], the ECPs based on entanglement swapping [30], [31], linear optics [33], [34], [32], and cross-Kerr nonlinearity [35], [36] are all focused on the Bell states and multi-partite Greenberger-Horne-Zeilinger (GHZ) states.", "Because all the ECPs for Bell stats can be easily extended to the GHZ states.", "On the other hand, the $W$ state, which has the different entanglement structure and can not be convert to the GHZ state directly with only local operation and classical communication, has began to receive attention both in theory and experiment [44], [45], [46], [47], [48], [49].", "Agrawal and Pati presented a perfect teleportation and superdense coding with $W$ states in 2006 [45].", "In 2010, Tamaryan et al.", "discussed the universal behavior of the geometric entanglement measure of many-qubit $W$ states [47].", "Eibl et al.", "also realized a three-qubit entangled $W$ state in experiment [48].", "Several ECPs for less-entangled $W$ state were also proposed [39], [40], [41], [42], [43].", "In 2003, Cao and Yang has discussed the $W$ state concentration with the help of joint unitary transformation[39].", "In 2007, a $W$ state ECP based on the Bell-state measurement has been proposed [40].", "Then in 2010, Wang et al.", "have proposed an ECP which focuses on a special kind of W state [41].", "Recently, Yildiz proposed an optimal distillation of three-qubit asymmetric $W$ states [42].", "We also have proposed an ECP with both linear optics and cross-Kerr for three-particle $W$ state [43].", "Unfortunately, these ECPs described above all focus on the three-particle $W$ state and they are mostly to concentrate some $W$ states with the special structures.", "In this paper, we will present an ECP for arbitrary multi-partite polarized $W$ entangled systems.", "We will describe this protocol in two different ways.", "First, we use the partial parity check (PPC) gate constructed by linear optics to perform this protocol.", "Second, we introduce the complete parity check (CPC) gate to achieve this task.", "Compared with other conventional ECPs, we only resort the single photon as an auxiliary which largely reduce the consumed quantum resources.", "Moreover, with the help of CPC gate, this protocol can be repeated and get a higher success probability.", "This paper is organized as follows: in Sec.", "II, we first briefly explain our PPC gate and CPC gate.", "in Sec.", "III, we describe our ECP with both PPC and CPC gates respectively.", "In Sec.", "IV, we make a discussion and summary." ], [ "Parity check gate", "Before we start to explain this protocol, we first briefly describe the parity check gate.", "Parity check gate is the basic element in quantum communication and computation.", "It can be used to construct the controlled-not (CNOT) gate [50], [51].", "It also can be used to perform the entanglement purification [20], [22] and concentration protocol [33], [32]." ], [ "partial parity check gate", "There are two different kinds of parity check gates.", "One is the partial parity check (PPC) gate and the other is the complete parity check (CPC) gate .", "In optical system, a polarization beam splitter (PBS) is essentially a good candidate for PPC gate as shown in Fig.1.", "Suppose that two polarized photons of the form $|\\varphi _{1}\\rangle =\\alpha |H\\rangle +\\beta |V\\rangle ,|\\varphi _{2}\\rangle =\\gamma |H\\rangle +\\delta |V\\rangle ,$ entrance into the PBS from different spatial modes.", "Here $|\\alpha |^{2}+|\\beta |^{2}=1$ , and $|\\gamma |^{2}+|\\delta |^{2}=1$ .", "$|H\\rangle $ and $|V\\rangle $ represent the horizonal and the vertical polarization of the photons, respectively.", "Figure: A schematic drawing of our PPC gate.", "It is constructed by apolarization beam splitter(PBS).", "It is used to transfer a|H〉|H\\rangle polarization photon and to reflect a |V〉|V\\rangle polarization photon.Let $|\\varphi _{1}\\rangle $ be in the spatial mode $a_{1}$ and $|\\varphi _{2}\\rangle $ be in the spatial mode $a_{2}$ .", "The whole system can be described as $|\\varphi _{1}\\rangle &\\otimes &|\\varphi _{2}\\rangle =(\\alpha |H\\rangle _{a_{1}}+\\beta |V\\rangle _{a_{1}})\\otimes (\\gamma |H\\rangle _{a_{2}}+\\delta |V\\rangle _{a_{2}})\\nonumber \\\\&=&\\alpha \\gamma |H\\rangle _{a_{1}}|H\\rangle _{a_{2}}+\\beta \\delta |V\\rangle _{a_{1}}|V\\rangle _{a_{2}}\\nonumber \\\\&+&\\alpha \\delta |H\\rangle _{a_{1}}|V\\rangle _{a_{2}}+\\beta \\gamma |V\\rangle _{a_{1}}|H\\rangle _{a_{2}}$ Then after passing through the PBS, it evolves as $&\\rightarrow &\\alpha \\gamma |H\\rangle _{b_{1}}|H\\rangle _{b_{1}}+\\beta \\delta |V\\rangle _{b_{1}}|V\\rangle _{b_{2}}\\nonumber \\\\&+&\\alpha \\delta |H\\rangle _{b_{1}}|V\\rangle _{b_{1}}+\\beta \\gamma |V\\rangle _{b_{2}}|H\\rangle _{b_{2}}.$ From above description, items $|H\\rangle _{b_{1}}|H\\rangle _{b_{1}}$ and $|V\\rangle _{b_{1}}|V\\rangle _{b_{2}}$ , say the even parity states will lead the output modes $b_{1}$ and $b_{2}$ both exactly contain only one photon.", "But items $|H\\rangle _{b_{1}}|V\\rangle _{b_{1}}$ and $|V\\rangle _{b_{2}}|H\\rangle _{b_{2}}$ will lead the two photons be in the same output mode, which cannot be distinguished.", "Based on the post selection principle, only the even parity state is the successful case.", "Therefore, the total success probability is $|\\alpha \\gamma |^{2}+|\\beta \\delta |^{2}<1$ .", "This is the reason that we call it PPC gate." ], [ "complete parity check gate", "Another parity check gate say CPC gate is shown in Fig.", "2.", "We adopt the cross-Kerr nonlinearity to construct the CPC gate.", "Cross-Kerr nonlinearity has been widely used in quantum information processing, [54], [55], .", "In general, the Hamiltonian of the cross-Kerr nonlinearity is described as $H=\\hbar \\chi \\hat{n_{a}}\\hat{n_{b}}$ , where the $\\hbar \\chi $ is the coupling strength of the nonlinearity.", "It is decided by the material of cross-Kerr.", "The $ \\hat{n_{a}}(\\hat{n_{b}})$ are the number operator for mode $a(b)$[51], [52].", "Now we reconsider the two photon system $|\\varphi _{1}\\rangle \\otimes |\\varphi _{2}\\rangle $ coupled with the coherent state $|\\alpha \\rangle $ .", "From Fig.", "2, the whole system evolves as $|\\varphi _{1}\\rangle &\\otimes |&\\varphi _{2}\\rangle \\otimes |\\alpha \\rangle =(\\alpha \\gamma |H\\rangle _{a_{1}}|H\\rangle _{a_{2}}+\\beta \\delta |V\\rangle _{a_{1}}|V\\rangle _{a_{2}}\\nonumber \\\\&+&\\alpha \\delta |H\\rangle _{a_{1}}|V\\rangle _{a_{2}}+\\beta \\gamma |V\\rangle _{a_{1}}|H\\rangle _{a_{2}})\\otimes |\\alpha \\rangle \\nonumber \\\\&\\rightarrow &(\\alpha \\gamma |H\\rangle _{b_{1}}|H\\rangle _{b_{1}}+\\beta \\delta |V\\rangle _{b_{1}}|V\\rangle _{b_{2}})|\\alpha \\rangle \\nonumber \\\\&+&\\alpha \\delta |H\\rangle _{b_{1}}|V\\rangle _{b_{1}}|\\alpha e^{-i2\\theta }\\rangle +\\beta \\gamma |V\\rangle _{b_{2}}|H\\rangle _{b_{2}}|\\alpha e^{i2\\theta }\\rangle .\\nonumber \\\\$ Figure: A schematic drawing of our CPC gate.It is obvious to see that the even parity states make the coherent state $|\\alpha \\rangle $ pick up no phase shift, but the odd parity state $|H\\rangle _{b_{1}}|V\\rangle _{b_{2}}$ makes the coherent state pick up the phase shift $-2\\theta $ .", "The other odd parity state $|V\\rangle _{b_{1}}|H\\rangle _{b_{2}}$ make the coherent state pick up the phase shift with $2\\theta $ .", "With a general homodyne-heterodyne measurement, the phase shift $2\\theta $ and $-2\\theta $ can not be distinguished[51].", "Then one can distinguish the different parity state according to their different phase shifts.", "So the success probability of the initial states collapsing to the even and odd parity state is $|\\alpha \\gamma |^{2}+|\\beta \\delta |^{2}+|\\alpha \\delta |^{2}+|\\beta \\gamma |^{2}=1$ , in principle.", "So we call it CPC gate.", "Compared with the PPC gate, the success probability for CPC gate can reach the max value 1 but the PPC gate cannot reach 1.", "Another advantage of the CPC gate is that we get the both even and odd parity state by measuring the phase shift of the coherent state.", "That is to say, we do not need to measure the two photons directly.", "So after the measurement, the two photons can be remained.", "It is so called quantum nondemolition(QND) measurement.", "But in PPC gate, we should use the post selection principle to detect the two photons being in the different spatial modes by coincidence counting.", "After both detectors register the photons with a success case, the photons are destroyed and cannot be used further more." ], [ "$N$ -particle less-entangled {{formula:4a9ec4fb-058c-4d81-8d1d-bbc801f79261}} state concentration with PPC gate", "In this section, we start to describe our $N$ -particle ECP with PPC gate.", "An $N$ -particle $W$ state can be described as $|\\Psi \\rangle _{N}&=&\\alpha _{1}|V\\rangle _{1}|H\\rangle _{2}|H\\rangle _{3}+\\cdots +|H\\rangle _{N-1}|H\\rangle _{N}\\nonumber \\\\&+&\\alpha _{2}|H\\rangle _{1}|V\\rangle _{2}|H\\rangle _{3}+\\cdots +|H\\rangle _{N-1}|H\\rangle _{N}\\nonumber \\\\&+&\\cdots +\\alpha _{N}|H\\rangle _{1}|H\\rangle _{2}+\\cdots +|H\\rangle _{N-1}|V\\rangle _{N}\\nonumber \\\\&=&\\alpha _{1}|V\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}+\\alpha _{2}|H\\rangle _{1}|V\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}\\nonumber \\\\&+&\\cdots +\\alpha _{N}|H\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-3}|V\\rangle _{N}.$ where $|\\alpha _{1}|^{2}+|\\alpha _{2}|^{2}+\\cdots +|\\alpha _{N}|^{2}=1$ .", "In order to explain this ECP clearly simply, we let $\\alpha _{1}$ , $\\alpha _{2}$ , $\\cdots $ be real.", "Certainly, this ECP is also suitable for the case of $\\alpha _{1}$ , $\\alpha _{2}$ , $\\cdots $ being complex.", "$|\\widetilde{H}\\rangle ^{N-2}$ means that the $N-2$ photons say $|H\\rangle _{3}|H\\rangle _{4}\\cdots |H\\rangle _{N}$ are all in the $|H\\rangle $ polarization.", "Figure: A schematic drawing of our ECP with PPCgate.", "Each parties except Bob2 own the PPC gate and performthe parity check.", "If they pick up the even parity state, it is successful, otherwise, it is a failure.From Fig.", "3, the $N$ -photon less-entangled $W$ state of the form Eq.", "(REF ) is distributed to $N$ parties, saies Bob1, Bob2, $\\cdots $ , Bob$N$ .", "Bob1 receives the photon of number 1 in the spatial mode $a_{1}$ .", "Bob2 receives the number 2 in the spatial mode $a_{2}$ , and Bob$N$ receives the photon number $N$ in the spatial mode $a_{N}$ .", "That is to say, each of the parties owns one photon.", "The principle of our ECP with PPC gate is shown in Fig.", "3.", "The basic idea of realizing the concentration is to use the local operation and classical communication to make each coefficients on each items of Eq.", "(REF ) all equal to $\\alpha _{2}$ .", "If all coefficients are equal, they can be regarded as a common factor and can be neglected.", "The remained state is essentially the maximally entangled $W$ state.", "Thus, the whole process can be briefly described as follows: we first divide the whole procedure into $N-1$ steps.", "In each step, each party say Bob$K$ should first prepare a single photon.", "In Fig.", "3, the single-photon sources $S_{1}$ , $S_{3}$ , $\\cdots $ , $S_{K}$ , $\\cdots $ , $S_{N}$ are used to prepare the single photons locally.", "Then he performs a parity check measurement for his two photons.", "The one comes from the single photon he prepared, and the other is the photon from the less-entangled $W$ state.", "If the parity check measurement is successful, then he asks the other to perform the further operation.", "Bob1 first perform the parity check on the photon of number 1 and the prepared single photon.", "The single-photon resource $S_{1}$ for Bob1 prepares a single photon in the spatial mode $b_{1}$ of the form $|\\Phi \\rangle _{1}=\\frac{\\alpha _{1}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|H\\rangle +\\frac{\\alpha _{2}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|V\\rangle .$ Then the initial less-entangled $W$ state $|\\Psi \\rangle _{N}$ combined with $|\\Phi \\rangle _{1}$ can be described as $&&|\\Psi \\rangle _{N+1}=|\\Psi \\rangle _{N}\\otimes |\\Phi \\rangle _{1}=(\\alpha _{1}|V\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}\\nonumber \\\\&+&\\alpha _{2}|H\\rangle _{1}|V\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}+\\cdots +\\alpha _{N}|H\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-3}|V\\rangle _{N})\\nonumber \\\\&\\otimes &(\\frac{\\alpha _{1}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|H\\rangle +\\frac{\\alpha _{2}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|V\\rangle )\\nonumber \\\\&=&\\frac{\\alpha _{1}^{2}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|H\\rangle |V\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}\\nonumber \\\\&+&\\frac{\\alpha _{2}^{2}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|V\\rangle |H\\rangle _{1}|V\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}\\nonumber \\\\&+&\\frac{\\alpha _{1}\\alpha _{2}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|H\\rangle |H\\rangle _{1}|V\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}\\nonumber \\\\&+&\\frac{\\alpha _{1}\\alpha _{3}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|H\\rangle |H\\rangle _{1}|H\\rangle _{2}|V\\rangle _{3}|\\widetilde{H}\\rangle ^{N-3}\\nonumber \\\\&+&\\cdots \\nonumber \\\\&+&\\frac{\\alpha _{1}\\alpha _{N}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|H\\rangle |H\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-3}|V\\rangle _{N}\\nonumber \\\\&+&\\frac{\\alpha _{1}\\alpha _{2}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|V\\rangle |V\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}\\nonumber \\\\&+&\\frac{\\alpha _{2}\\alpha _{3}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|V\\rangle |H\\rangle _{1}|H\\rangle _{2}|V\\rangle _{3}|\\widetilde{H}\\rangle ^{N-3}\\nonumber \\\\&+&\\cdots \\nonumber \\\\&+&\\frac{\\alpha _{2}\\alpha _{N}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|V\\rangle |H\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-3}|V\\rangle _{N}.$ After passing through the PPC gate in Bob1's location, Bob1 only picks up the even parity state in the spatial mode $c_{1}$ and $d_{1}$ .", "Therefore, the above state collapses to $|\\Psi \\rangle ^{\\prime }_{N+1}&=&\\frac{\\alpha _{1}\\alpha _{2}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|H\\rangle |H\\rangle _{1}|V\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}\\nonumber \\\\&+&\\frac{\\alpha _{1}\\alpha _{3}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|H\\rangle |H\\rangle _{1}|H\\rangle _{2}|V\\rangle _{3}|\\widetilde{H}\\rangle ^{N-3}\\nonumber \\\\&+&\\cdots \\nonumber \\\\&+&\\frac{\\alpha _{1}\\alpha _{N}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|H\\rangle |H\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-3}|V\\rangle _{N}\\nonumber \\\\&+&\\frac{\\alpha _{1}\\alpha _{2}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|V\\rangle |V\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}.$ It can be rewritten as $&&|\\Psi \\rangle ^{\\prime \\prime }_{N+1}=\\frac{\\alpha _{2}}{\\sqrt{2\\alpha _{2}^{2}+\\alpha _{3}^{2}+\\cdots +\\alpha _{N}^{2}}}|H\\rangle |H\\rangle _{1}|V\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}\\nonumber \\\\&+&\\frac{\\alpha _{2}}{\\sqrt{2\\alpha _{2}^{2}+\\alpha _{3}^{2}+\\cdots +\\alpha _{N}^{2}}}|V\\rangle |V\\rangle _{1}|V\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}\\nonumber \\\\&+&\\frac{\\alpha _{3}}{\\sqrt{2\\alpha _{2}^{2}+\\alpha _{3}^{2}+\\cdots +\\alpha _{N}^{2}}}|H\\rangle |H\\rangle _{1}|H\\rangle _{2}|V\\rangle _{3}|\\widetilde{H}\\rangle ^{N-3}\\nonumber \\\\&+&\\cdots \\nonumber \\\\&+&\\frac{\\alpha _{N}}{\\sqrt{2\\alpha _{2}^{2}+\\alpha _{3}^{2}+\\cdots +\\alpha _{N}^{2}}}|H\\rangle |H\\rangle _{1}|\\widetilde{H}\\rangle ^{N-2}|V\\rangle _{N}.$ Finally, Bob1 measures the photon in the spatial mode $d_{1}$ (the first photon in Eq.", "(REF )) in the basis $|\\pm \\rangle $ , with $|\\pm \\rangle =\\frac{1}{\\sqrt{2}}(|H\\rangle \\pm |V\\rangle )$ .", "Then they will get $|\\Psi ^{\\pm }\\rangle ^{1}_{N}&=&\\pm \\frac{\\alpha _{2}}{\\sqrt{2\\alpha _{2}^{2}+\\alpha _{3}^{2}+\\cdots +\\alpha _{N}^{2}}}|V\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}\\nonumber \\\\&+&\\frac{\\alpha _{2}}{\\sqrt{2\\alpha _{2}^{2}+\\alpha _{3}^{2}+\\cdots +\\alpha _{N}^{2}}}|H\\rangle _{1}|V\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}\\nonumber \\\\&+&\\frac{\\alpha _{3}}{\\sqrt{2\\alpha _{2}^{2}+\\alpha _{3}^{2}+\\cdots +\\alpha _{N}^{2}}}|H\\rangle _{1}|H\\rangle _{2}|V\\rangle _{3}|\\widetilde{H}\\rangle ^{N-3}\\nonumber \\\\&+&\\cdots \\nonumber \\\\&+&\\frac{\\alpha _{N}}{\\sqrt{2\\alpha _{2}^{2}+\\alpha _{3}^{2}+\\cdots +\\alpha _{N}^{2}}}|H\\rangle _{1}|\\widetilde{H}\\rangle ^{N-2}|V\\rangle _{N}.\\nonumber \\\\$ The superscription 1 means that they perform the concentration on the first particle.", "If the measurement result is $|+\\rangle $ , they will get $|\\Psi ^{+}\\rangle ^{1}_{N}$ .", "If the result is $|-\\rangle $ , they will get $|\\Psi ^{-}\\rangle ^{1}_{N}$ .", "In order to get $|\\Psi ^{+}\\rangle ^{1}_{N}$ , one of the parties, Bob1, Bob2, $\\cdots $ should perform a local operation of phase rotation on his photon.", "The total success probability is $P^{1}&=&\\frac{2\\alpha ^{2}_{1}\\alpha ^{2}_{2}+\\alpha ^{2}_{1}(\\alpha ^{2}_{3}+\\alpha ^{2}_{4}+\\cdots +\\alpha ^{2}_{N})}{\\alpha ^{2}_{1}+\\alpha ^{2}_{2}}\\nonumber \\\\&=&\\frac{\\alpha ^{2}_{1}(2\\alpha ^{2}_{2}+\\alpha ^{2}_{3}+\\alpha ^{2}_{4}+\\cdots +\\alpha ^{2}_{N})}{\\alpha ^{2}_{1}+\\alpha ^{2}_{2}}.$ Compared with Eq.", "(), the coefficient of $\\alpha _{1}$ has disappeared in the state of Eq.", "(REF ).", "The next step is to prepare another single photon in single-photon source $S_{3}$ in the spatial mode $b_{3}$ of the form $|\\Phi \\rangle _{3}=\\frac{\\alpha _{2}}{\\sqrt{\\alpha _{2}^{2}+\\alpha _{3}^{2}}}|V\\rangle +\\frac{\\alpha _{3}}{\\sqrt{\\alpha _{2}^{2}+\\alpha _{3}^{2}}}|H\\rangle .$ Following the same principle described above, Bob3 lets the photon of number 3 in $|\\Psi ^{+}\\rangle ^{1}_{N}$ in the spatial mode $a_{3}$ combined with the single photon $|\\Phi \\rangle _{3}$ in the spatial mode $b_{3}$ pass through his PPC gate.", "Then the whole system evolves to $|\\Psi ^{+}\\rangle ^{1}_{N}\\otimes |\\Phi \\rangle _{3}&\\rightarrow &\\frac{\\alpha _{2}\\alpha _{3}}{\\sqrt{2\\alpha _{2}^{2}+\\alpha _{3}^{2}+\\cdots +\\alpha _{N}^{2}}\\sqrt{\\alpha _{2}^{2}+\\alpha _{3}^{2}}}|V\\rangle _{1}|H\\rangle _{2}|H\\rangle _{3}|H\\rangle |\\widetilde{H}\\rangle ^{N-3}\\nonumber \\\\&+&\\frac{\\alpha _{2}\\alpha _{3}}{\\sqrt{2\\alpha _{2}^{2}+\\alpha _{3}^{2}+\\cdots +\\alpha _{N}^{2}}\\sqrt{\\alpha _{2}^{2}+\\alpha _{3}^{2}}}|H\\rangle _{1}|V\\rangle _{2}|H\\rangle _{3}|H\\rangle |\\widetilde{H}\\rangle ^{N-3}\\nonumber \\\\&+&\\frac{\\alpha _{2}\\alpha _{3}}{\\sqrt{2\\alpha _{2}^{2}+\\alpha _{3}^{2}+\\cdots +\\alpha _{N}^{2}}\\sqrt{\\alpha _{2}^{2}+\\alpha _{3}^{2}}}|H\\rangle _{1}|H\\rangle _{2}|V\\rangle _{3}|V\\rangle |\\widetilde{H}\\rangle ^{N-3}\\nonumber \\\\&+&\\cdots \\nonumber \\\\&+&\\frac{\\alpha _{3}\\alpha _{N}}{\\sqrt{2\\alpha _{2}^{2}+\\alpha _{3}^{2}+\\cdots +\\alpha _{N}^{2}}\\sqrt{\\alpha _{2}^{2}+\\alpha _{3}^{2}}}|H\\rangle _{1}|H\\rangle _{2}|H\\rangle _{3}|H\\rangle |\\widetilde{H}\\rangle ^{N-3}|V\\rangle _{N},$ If he picks up the even parity state, then Bob3 measures the photon in the spatial mode $d_{3}$ in the basis $|\\pm \\rangle $ .", "They will get $&&|\\Psi ^{\\pm }\\rangle ^{3}_{N}=\\frac{\\alpha _{2}}{\\sqrt{3\\alpha ^{2}_{2}+\\alpha ^{2}_{4}+\\cdots \\alpha ^{2}_{N}}}|V\\rangle _{1}|H\\rangle _{2}|H\\rangle _{3}|\\widetilde{H}\\rangle ^{N-3}\\nonumber \\\\&&+\\frac{\\alpha _{2}}{\\sqrt{3\\alpha ^{2}_{2}+\\alpha ^{2}_{4}+\\cdots \\alpha ^{2}_{N}}}|H\\rangle _{1}|V\\rangle _{2}|H\\rangle _{3}|H\\rangle |\\widetilde{H}\\rangle ^{N-3}\\nonumber \\\\&&\\pm \\frac{\\alpha _{2}}{\\sqrt{3\\alpha ^{2}_{2}+\\alpha ^{2}_{4}+\\cdots \\alpha ^{2}_{N}}}|H\\rangle _{1}|H\\rangle _{2}|V\\rangle _{3}|\\widetilde{H}\\rangle ^{N-3}\\nonumber \\\\&&+\\frac{\\alpha _{4}}{\\sqrt{3\\alpha ^{2}_{2}+\\alpha ^{2}_{4}+\\cdots \\alpha ^{2}_{N}}}|H\\rangle _{1}|H\\rangle _{2}|H\\rangle _{3}|V\\rangle _{4}|\\widetilde{H}\\rangle ^{N-4}\\nonumber \\\\&&+\\cdots \\nonumber \\\\&&+\\frac{\\alpha _{N}}{\\sqrt{3\\alpha ^{2}_{2}+\\alpha ^{2}_{4}+\\cdots \\alpha ^{2}_{N}}}|H\\rangle _{1}|H\\rangle _{2}|H\\rangle _{3}|\\widetilde{H}\\rangle ^{N-4}|V\\rangle _{N}.\\nonumber \\\\$ The total success probability is $P^{3}=\\frac{3\\alpha ^{2}_{2}\\alpha ^{2}_{3}+\\alpha ^{2}_{3}(\\alpha ^{2}_{4}+\\cdots +\\alpha ^{2}_{N})}{(2\\alpha ^{2}_{2}+\\alpha ^{2}_{3}+\\alpha ^{2}_{4}+\\cdots +\\alpha ^{2}_{N})(\\alpha _{2}^{2}+\\alpha _{3}^{2})}.$ $P^{3}$ essentially contains two parts.", "The first one is the success probability to get $|\\Psi ^{\\pm }\\rangle ^{1}_{N}$ , and the second one is the success probability for Bob3 to pick up the even parity state.", "Interestingly, from Eq.", "(REF ), the coefficient $\\alpha _{3}$ has also disappeared.", "The following concentration steps are similar to the above description.", "That is each one performs a parity check measurement and picks up the even parity state.", "For instance, in the $Kth$ step, Bob$K$ first prepares a single photon of the form $|\\Phi \\rangle _{K}=\\frac{\\alpha _{2}}{\\sqrt{\\alpha _{2}^{2}+\\alpha _{K}^{2}}}|V\\rangle +\\frac{\\alpha _{K}}{\\sqrt{\\alpha _{2}^{2}+\\alpha _{K}^{2}}}|H\\rangle .$ After he performing the parity check measurement and picks up the even parity state, the original less-entangled $W$ state becomes $&&|\\Psi ^{\\pm }\\rangle ^{K}_{N}=\\frac{\\alpha _{2}}{\\sqrt{K\\alpha ^{2}_{2}+\\alpha ^{2}_{K+1}+\\cdots \\alpha ^{2}_{N}}}|V\\rangle _{1}|\\widetilde{H}\\rangle ^{N-1}\\nonumber \\\\&&+\\frac{\\alpha _{2}}{\\sqrt{K\\alpha ^{2}_{2}+\\alpha ^{2}_{4}+\\cdots \\alpha ^{2}_{N}}}|H\\rangle _{1}|V\\rangle _{2}|H\\rangle _{3}|\\widetilde{H}\\rangle ^{N-3}\\nonumber \\\\&&+\\cdots \\nonumber \\\\&&\\pm \\frac{\\alpha _{2}}{\\sqrt{K\\alpha ^{2}_{2}+\\alpha ^{2}_{K+1}+\\cdots \\alpha ^{2}_{N}}}|\\widetilde{H}\\rangle ^{K-1}|V\\rangle _{K}|\\widetilde{H}\\rangle ^{N-K}\\nonumber \\\\&&+\\frac{\\alpha _{K+1}}{\\sqrt{K\\alpha ^{2}_{2}+\\alpha ^{2}_{K+1}+\\cdots \\alpha ^{2}_{N}}}|\\widetilde{H}\\rangle ^{K}|V\\rangle _{K+1}|\\widetilde{H}\\rangle ^{N-K-1}\\nonumber \\\\&&+\\cdots \\nonumber \\\\&&+\\frac{\\alpha _{N}}{\\sqrt{K\\alpha ^{2}_{2}+\\alpha ^{2}_{K+1}+\\cdots \\alpha ^{2}_{N}}}|\\widetilde{H}\\rangle ^{N-1}|V\\rangle _{N}.$ The success probability can be written as $P^{K}=\\frac{K\\alpha ^{2}_{2}\\alpha ^{2}_{K}+\\alpha ^{2}_{K}(\\alpha ^{2}_{K+1}+\\alpha ^{2}_{K+2}\\cdots +\\alpha ^{2}_{N})}{((K-1)\\alpha ^{2}_{2}+\\alpha ^{2}_{K}+\\alpha ^{2}_{K+1}+\\cdots +\\alpha ^{2}_{N})(\\alpha _{2}^{2}+\\alpha _{K}^{2})}.\\nonumber \\\\$ If $K=N$ , then they will get the maximally entangled $W$ state, with the probability of $P^{N}=\\frac{N\\alpha ^{2}_{2}\\alpha ^{2}_{N}}{((N-1)\\alpha ^{2}_{2}+\\alpha ^{2}_{N})(\\alpha _{2}^{2}+\\alpha _{N}^{2})}.$ Therefore, the total success probability to get the maximally entangled $W$ state from Eq.", "() is $P_{T}=P^{1}P^{3}\\cdots P^{N}=\\frac{N\\alpha ^{2}_{1}\\alpha ^{2}_{2}\\alpha ^{2}_{3}\\cdots \\alpha ^{2}_{N}}{(\\alpha ^{2}_{2}+\\alpha ^{2}_{1})(\\alpha ^{2}_{2}+\\alpha ^{2}_{3})\\cdots (\\alpha ^{2}_{2}+\\alpha ^{2}_{N})}.\\nonumber \\\\$ Interestingly, if $N=2$ , it is the concentration of two-particle Bell state with $P_{T}=2\\alpha ^{2}_{1}\\alpha ^{2}_{2}$ .", "It is equal to the success probability in Refs.", "[33], [35], [36], [38].", "By far, we have fully explained our ECP with PPC gate.", "During the whole process, we require $N-1$ single photons to achieve this task with the success probability of $P_{T}$ .", "Except Bob2, each parties needs to perform a parity check.", "If the parity check measurement result is even, it is successful and he asks others to retain their photons.", "From Sec.", "II, the PPC gate essentially is based on linear optics and we should resort the post selection principle.", "That is to say, the detection will destroy their photons.", "This disadvantage will greatly limits its practical application, because it has to require all of the parties to perform the parity check simultaneously.", "On the other hand, the total success probability is extremely low.", "Because they should ensure all $N-1$ parity checks be successful.", "If any of parity check in Bob$K$ is fail, then the whole ECP is fail.", "It is quite different from the ECP of $N$ -particle GHZ state[33], [35], [36], due to the same entanglement structure with Bell state.", "The ECP of Bell state is suitable to the $N$ -particle GHZ state with the same success probability $2\\alpha ^{2}_{1}\\alpha ^{2}_{2}$ with linear optics[33].", "That is to say, the success probability does not change with the particle number $N$ .", "However, in this ECP, we find that the $P_{T}$ changes when $N$ changes." ], [ "N-particle $W$ state concentration with CPC gate", "From above description, we show that the PPC gate can be used to achieve this concentration task.", "However, it is not an economical one and the success probability is extremely low.", "The reason is that we only pick up the even parity state and discard the odd one.", "In this section, we will adopt the PPC gate to redescribe this ECP.", "The basic principle of our ECP is shown in Fig.", "4.", "We use the CPC gates to substitute the PPC gates.", "Figure: A schematic drawing of our ECP with CPCgate.", "Compared with Fig.", "3, we use the CPC gate shown in Fig.", "2 to substitutethe PPC gate.", "By using the CPC gate, the odd parity state can also be reused to improve the success probability and the concentrated state can also be retained.In the first step, the initial state $|\\Psi \\rangle _{N}$ and $|\\Phi \\rangle _{1}$ combined with the coherent state $|\\alpha \\rangle $ evolve as $&&|\\Psi \\rangle _{N+1}\\otimes |\\alpha \\rangle =|\\Psi \\rangle _{N}\\otimes |\\Phi \\rangle _{1}=(\\alpha _{1}|V\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}\\nonumber \\\\&+&\\alpha _{2}|H\\rangle _{1}|V\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}+\\cdots +\\alpha _{N}|H\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-3}|V\\rangle _{N})\\nonumber \\\\&\\otimes &(\\frac{\\alpha _{1}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|H\\rangle +\\frac{\\alpha _{2}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|V\\rangle )\\otimes |\\alpha \\rangle \\nonumber \\\\&\\rightarrow &\\frac{\\alpha _{1}^{2}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|H\\rangle |V\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}\\otimes |\\alpha e^{-i2\\theta }\\rangle \\nonumber \\\\&+&\\frac{\\alpha _{2}^{2}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|V\\rangle |H\\rangle _{1}|V\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}\\otimes |\\alpha e^{i2\\theta }\\rangle \\nonumber \\\\&+&\\frac{\\alpha _{1}\\alpha _{2}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|H\\rangle |H\\rangle _{1}|V\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}\\otimes |\\alpha \\rangle \\nonumber \\\\&+&\\frac{\\alpha _{1}\\alpha _{3}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|H\\rangle |H\\rangle _{1}|H\\rangle _{2}|V\\rangle _{3}|\\widetilde{H}\\rangle ^{N-3}\\otimes |\\alpha \\rangle \\nonumber \\\\&+&\\cdots \\nonumber \\\\&+&\\frac{\\alpha _{1}\\alpha _{N}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|H\\rangle |H\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-3}|V\\rangle _{N}\\otimes |\\alpha \\rangle \\nonumber \\\\&+&\\frac{\\alpha _{1}\\alpha _{2}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|V\\rangle |V\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}\\otimes |\\alpha \\rangle \\nonumber \\\\&+&\\frac{\\alpha _{2}\\alpha _{3}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|V\\rangle |H\\rangle _{1}|H\\rangle _{2}|V\\rangle _{3}|\\widetilde{H}\\rangle ^{N-3}\\otimes |\\alpha e^{-i2\\theta }\\rangle \\nonumber \\\\&+&\\cdots \\nonumber \\\\&+&\\frac{\\alpha _{2}\\alpha _{N}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|V\\rangle |H\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-3}|V\\rangle _{N}\\otimes |\\alpha e^{^{-i2\\theta }} \\rangle .$ Obviously, if the coherent state $|\\alpha \\rangle $ picks up no phase shift, the original state will collapse to the even state similar to $|\\Psi \\rangle ^{\\prime }_{N+1}$ in Eq.", "(REF ).", "It can also be rewritten as $|\\Psi \\rangle ^{\\prime \\prime }_{N+1}$ with the probability of $P^{1}$ .", "In this way, they can also obtain the same state $|\\Psi ^{\\pm }\\rangle ^{1}_{N}$ in Eq.", "(REF ) and can be used to start the next concentration step on the number 3 photon performed by Bob3.", "On the other hand, there is the probability of $1-P^{1}$ that the original state will collapse to the odd state, if the phase shift of coherent state is $2\\theta $ .", "Therefore, it can be written as $|\\Psi _{^{\\bot }}\\rangle ^{\\prime }_{N+1}&=&\\frac{\\alpha _{1}^{2}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|H\\rangle |V\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}\\nonumber \\\\&+&\\frac{\\alpha _{2}^{2}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|V\\rangle |H\\rangle _{1}|V\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}\\nonumber \\\\&+&\\frac{\\alpha _{2}\\alpha _{3}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|V\\rangle |H\\rangle _{1}|H\\rangle _{2}|V\\rangle _{3}|\\widetilde{H}\\rangle ^{N-3}\\rangle \\nonumber \\\\&+&\\cdots \\nonumber \\\\&+&\\frac{\\alpha _{2}\\alpha _{N}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|V\\rangle |H\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-3}|V\\rangle _{N}.\\nonumber \\\\$ After measuring the photon in $d_{1}$ mode in the basis $|\\pm \\rangle $ , above state becomes $|\\Psi ^{\\pm }_{^{\\bot }}\\rangle ^{\\prime }_{N}&=&\\pm \\frac{\\alpha _{1}^{2}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|V\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}\\nonumber \\\\&+&\\frac{\\alpha _{2}^{2}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|H\\rangle _{1}|V\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}\\nonumber \\\\&+&\\frac{\\alpha _{2}\\alpha _{3}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|H\\rangle _{1}|H\\rangle _{2}|V\\rangle _{3}|\\widetilde{H}\\rangle ^{N-3}\\rangle \\nonumber \\\\&+&\\cdots \\nonumber \\\\&+&\\frac{\\alpha _{2}\\alpha _{N}}{\\sqrt{\\alpha _{1}^{2}+\\alpha _{2}^{2}}}|H\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-3}|V\\rangle _{N}.\\nonumber \\\\$ If the measurement result is $|+\\rangle $ , they will get $|\\Psi ^{+}_{^{\\bot }}\\rangle ^{\\prime }_{N}$ , otherwise, they will get $|\\Psi ^{-}_{^{\\bot }}\\rangle ^{\\prime }_{N}$ .", "Above equation can be rewritten as $|\\Psi ^{\\pm }_{^{\\bot }}\\rangle ^{\\prime \\prime }_{N}&=&\\pm \\frac{\\alpha _{1}^{2}}{T}|V\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}\\nonumber \\\\&+&\\frac{\\alpha _{2}^{2}}{T}|H\\rangle _{1}|V\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}\\nonumber \\\\&+&\\frac{\\alpha _{2}\\alpha _{3}}{T}|H\\rangle _{1}|H\\rangle _{2}|V\\rangle _{3}|\\widetilde{H}\\rangle ^{N-3}\\rangle \\nonumber \\\\&+&\\cdots \\nonumber \\\\&+&\\frac{\\alpha _{2}\\alpha _{N}}{T}|H\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-3}|V\\rangle _{N}.$ $T=\\sqrt{\\alpha ^{4}_{1}+\\alpha ^{2}_{2}(\\alpha ^{2}_{2}+\\alpha ^{2}_{3}+\\cdots +\\alpha ^{2}_{N})}$ .", "Interestingly, the state of Eq.", "(REF ) essentially is a lesser-entangled $W$ state.", "It can be reconcentrated with another single photon on the number 1 photon.", "The another single photon is written as $|\\Phi \\rangle ^{\\prime }_{1}=\\frac{\\alpha ^{2}_{1}}{\\sqrt{\\alpha _{1}^{4}+\\alpha _{2}^{4}}}|H\\rangle +\\frac{\\alpha ^{2}_{2}}{\\sqrt{\\alpha _{1}^{4}+\\alpha _{2}^{4}}}|V\\rangle .$ So Bob1 can restart this ECP with the help of a second single photon $|\\Phi \\rangle ^{\\prime }_{1}$ .", "The state $|\\Psi ^{+}_{^{\\bot }}\\rangle ^{\\prime \\prime }_{N}$ and $|\\Phi \\rangle ^{\\prime }_{1}$ combined with the coherent state $|\\alpha \\rangle $ evolves as $&&|\\Psi ^{+}_{^{\\bot }}\\rangle ^{\\prime \\prime }_{N}\\otimes |\\Phi \\rangle ^{\\prime }_{1}\\otimes |\\alpha \\rangle =\\frac{\\alpha _{1}^{2}}{T}|V\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}\\nonumber \\\\&+&\\frac{\\alpha _{2}^{2}}{T}|H\\rangle _{1}|V\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}\\nonumber \\\\&+&\\frac{\\alpha _{2}\\alpha _{3}}{T}|H\\rangle _{1}|H\\rangle _{2}|V\\rangle _{3}|\\widetilde{H}\\rangle ^{N-3}\\rangle \\nonumber \\\\&+&\\cdots \\nonumber \\\\&+&\\frac{\\alpha _{2}\\alpha _{N}}{T}|H\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-3}|V\\rangle _{N}\\nonumber \\\\&\\otimes &(\\frac{\\alpha ^{2}_{1}}{\\sqrt{\\alpha _{1}^{4}+\\alpha _{2}^{4}}}|H\\rangle +\\frac{\\alpha ^{2}_{2}}{\\sqrt{\\alpha _{1}^{4}+\\alpha _{2}^{4}}}|V\\rangle )\\otimes |\\alpha \\rangle \\nonumber \\\\&\\rightarrow &\\frac{\\alpha ^{4}_{1}}{T\\sqrt{\\alpha _{1}^{4}+\\alpha _{2}^{4}}}|H\\rangle |V\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}|\\alpha e^{-i2\\theta }\\rangle \\nonumber \\\\&+&\\frac{\\alpha ^{2}_{1}\\alpha ^{2}_{2}}{T\\sqrt{\\alpha _{1}^{4}+\\alpha _{2}^{4}}}|V\\rangle |V\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}|\\alpha \\rangle \\nonumber \\\\&+&\\frac{\\alpha ^{2}_{1}\\alpha ^{2}_{2}}{T\\sqrt{\\alpha _{1}^{4}+\\alpha _{2}^{4}}}|H\\rangle |H\\rangle _{1}|V\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}|\\alpha \\rangle \\nonumber \\\\&+&\\frac{\\alpha ^{4}_{1}}{T\\sqrt{\\alpha _{1}^{4}+\\alpha _{2}^{4}}}|V\\rangle |H\\rangle _{1}|V\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}|\\alpha e^{i2\\theta }\\rangle \\nonumber \\\\&+&\\frac{\\alpha _{1}^{2}\\alpha _{2}\\alpha _{3}}{T\\sqrt{\\alpha _{1}^{4}+\\alpha _{2}^{4}}}|H\\rangle |H\\rangle _{1}|H\\rangle _{2}|V\\rangle _{3}|\\widetilde{H}\\rangle ^{N-3}\\rangle |\\alpha \\rangle \\nonumber \\\\&+&\\cdots \\nonumber \\\\&+&\\frac{\\alpha _{1}^{2}\\alpha _{2}\\alpha _{N}}{T\\sqrt{\\alpha _{1}^{4}+\\alpha _{2}^{4}}}|H\\rangle |H\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-3}|V\\rangle _{N}|\\alpha \\rangle \\nonumber \\\\&+&\\frac{\\alpha _{2}^{3}\\alpha _{3}\\alpha _{N}}{T\\sqrt{\\alpha _{1}^{4}+\\alpha _{2}^{4}}}|V\\rangle |H\\rangle _{1}|H\\rangle _{2}|V\\rangle _{3}|\\widetilde{H}\\rangle ^{N-3}|\\alpha e^{i2\\theta }\\rangle \\nonumber \\\\&+&\\cdots \\nonumber \\\\&+&\\frac{\\alpha _{2}^{3}\\alpha _{N}\\alpha _{N}}{T\\sqrt{\\alpha _{1}^{4}+\\alpha _{2}^{4}}}|V\\rangle |H\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-3}|V\\rangle _{N}|\\alpha e^{i2\\theta }\\rangle \\nonumber \\\\$ Figure: The success probability of concentration of each photon ineach step is altered with the iteration number MM.", "Here we choosethe five-photon less entangled WW state withα 1 =α 2 =α 3 =0.5\\alpha _{1}=\\alpha _{2}=\\alpha _{3}=0.5, α 4 =0.3\\alpha _{4}=0.3,α 5 =0.4\\alpha _{5}=0.4.", "Curve B: the success probability of concentrationthe number 1 and 3 photons according to α 1 =α 3 =0.5\\alpha _{1}=\\alpha _{3}=0.5.Curve C: the success probability of concentration the number 4photon according to α 4 =0.3\\alpha _{4}=0.3.", "Curve D: the successprobability of concentration the number 4 photon according toα 5 =0.4\\alpha _{5}=0.4.Obviously, if Bob1 picks up no phase shift, above equation collapses to $&&|\\Psi _\\perp \\rangle ^{\\prime }_{N+1}=\\frac{\\alpha ^{2}_{1}\\alpha ^{2}_{2}}{T\\sqrt{\\alpha _{1}^{4}+\\alpha _{2}^{4}}}|H\\rangle |H\\rangle _{1}|V\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}\\nonumber \\\\&+&\\frac{\\alpha ^{2}_{1}\\alpha ^{2}_{2}}{T\\sqrt{\\alpha _{1}^{4}+\\alpha _{2}^{4}}}|H\\rangle |H\\rangle _{1}|V\\rangle _{2}|\\widetilde{H}\\rangle ^{N-2}\\nonumber \\\\&+&\\frac{\\alpha _{1}^{2}\\alpha _{2}\\alpha _{3}}{T\\sqrt{\\alpha _{1}^{4}+\\alpha _{2}^{4}}}|H\\rangle |H\\rangle _{1}|H\\rangle _{2}|V\\rangle _{3}|\\widetilde{H}\\rangle ^{N-3}\\rangle |\\alpha \\rangle \\nonumber \\\\&+&\\cdots \\nonumber \\\\&+&\\frac{\\alpha _{1}^{2}\\alpha _{2}\\alpha _{N}}{T\\sqrt{\\alpha _{1}^{4}+\\alpha _{2}^{4}}}|H\\rangle |H\\rangle _{1}|H\\rangle _{2}|\\widetilde{H}\\rangle ^{N-3}|V\\rangle _{N}.$ Figure: The total success probability of our ECPaltered with iteration number MM with CPC gate.", "We also letα 1 =α 2 =α 3 =0.5\\alpha _{1}=\\alpha _{2}=\\alpha _{3}=0.5, α 4 =0.3\\alpha _{4}=0.3,α 5 =0.4\\alpha _{5}=0.4.Interestingly, above state essentially is the state $|\\Psi \\rangle ^{\\prime \\prime }_{N+1}$ in Eq.", "(REF ), if it is normalized.", "Then it can be used to concentrate the number 3 photon with the same CPC gate like above.", "The success probability is $P_{2}^{1}=\\frac{2\\alpha ^{4}_{1}\\alpha ^{4}_{2}+\\alpha ^{4}_{1}\\alpha ^{2}_{2}(\\alpha ^{2}_{3}+\\alpha ^{2}_{4}+\\cdots +\\alpha ^{2}_{N})}{(\\alpha ^{2}_{1}+\\alpha ^{2}_{2})(\\alpha ^{4}_{1}+\\alpha ^{4}_{2})}.$ Following the same principle, Bob1 can repeat this ECP for $M$ times and they can get the success probability in each step as $P_{3}^{1}&=&\\frac{2\\alpha ^{8}_{1}\\alpha ^{8}_{2}+\\alpha ^{8}_{1}\\alpha ^{6}_{2}(\\alpha ^{2}_{3}+\\alpha ^{2}_{4}+\\cdots +\\alpha ^{2}_{N})}{(\\alpha ^{2}_{1}+\\alpha ^{2}_{2})(\\alpha ^{4}_{1}+\\alpha ^{4}_{2})(\\alpha ^{8}_{1}+\\alpha ^{8}_{2})},\\nonumber \\\\&\\cdots &\\nonumber \\\\P_{M}^{1}&=&\\frac{2\\alpha ^{2^{M}}_{1}\\alpha ^{2^{M}}_{2}+\\alpha ^{2^{M}}_{1}\\alpha ^{2^{M}-2}_{2}(\\alpha ^{2}_{3}+\\alpha ^{2}_{4}+\\cdots +\\alpha ^{2}_{N})}{(\\alpha ^{2}_{1}+\\alpha ^{2}_{2})(\\alpha ^{2^{2}}_{1}+\\alpha ^{2^{2}}_{2})\\cdots (\\alpha ^{2^{M}}_{1}+\\alpha ^{2^{M}}_{2})}.\\nonumber \\\\$ Here the superscription 1 means that concentration on the number 1 photon.", "The subscription $M$ means that the ECP is performed $M$ times.", "After performing the concentration ECP on the number 1 photon, they will have a total success probability with $P_{total}^{1}=P^{1}_{1}+P^{1}_{2}+\\cdots +P^{1}_{M}=\\sum ^{\\infty }_{M=1}P^{1}_{M}$ to obtain $|\\Psi ^{\\pm }\\rangle ^{1}_{N}$ , which can be used to performing the concentration scheme on the number 3 photon.", "So far, we have explained our ECP performed on the number 1 photon with CPC gate.", "Different from the scheme described with PPC gate, it can be repeated to get a high success probability.", "Following the same principle, they can also use this way to concentrating each photons.", "If they perform this ECP on the $Kth$ ($K\\ne 2$ ) photon with $M$ times, they can get the success probability $P^{K}_{M}$ $P^{K}_{M}=\\frac{K\\alpha ^{2^{M}}_{2}\\alpha ^{2^{M}}_{K}+\\alpha ^{2^{M}}_{K}\\alpha ^{2^{M}-2}_{2}(\\alpha ^{2}_{K+1}+\\alpha ^{2}_{K+2}+\\cdots +\\alpha ^{2}_{N})}{[(K-1)\\alpha ^{2}_{2}+\\alpha ^{2}_{K}+\\alpha ^{2}_{K+1}+\\cdots \\alpha ^{2}_{N}][(\\alpha ^{2}_{2}+\\alpha ^{2}_{K})(\\alpha ^{2^{2}}_{2}+\\alpha ^{2^{2}}_{K})\\cdots (\\alpha ^{2^{M}}_{2}+\\alpha ^{2^{M}}_{K})]}.$ If $K=N$ , they can get $P^{N}_{M}=\\frac{N\\alpha ^{2^{M}}_{2}\\alpha ^{2^{M}}_{N}}{[(N-1)\\alpha ^{2}_{2}+\\alpha ^{2}_{N}][(\\alpha ^{2}_{2}+\\alpha ^{2}_{N})(\\alpha ^{2^{2}}_{2}+\\alpha ^{2^{2}}_{N})\\cdots (\\alpha ^{2^{M}}_{2}+\\alpha ^{2^{M}}_{N})]}.$ Therefore, if we use the CPC gate to perform the EPC, each parties can repeat this ECP to increase the success probability.", "Suppose each one all perform this ECP for $M$ times, the success probability of get a maximally entangled $W$ state from the initial state in Eq.", "() can be described as $P&=&P^{1}_{total}P^{3}_{total}P^{4}_{total}\\cdots P^{N}_{total}\\nonumber \\\\&=&(P^{1}_{1}+P^{1}_{2}+\\cdots +P^{1}_{M})(P^{3}_{1}+P^{3}_{2}\\nonumber \\\\&+&\\cdots +P^{3}_{M})\\cdots (P^{N}_{1}+P^{N}_{2}+\\cdots +P^{N}_{M})\\nonumber \\\\&=&\\prod ^{N}_{K=1,K\\ne 2}(\\sum ^{\\infty }_{M=1}P^{K}_{M}).$ Compared with the ECP with PPC gate, the success probability in Eq.", "(REF ) is the case of $M=1$ in Eq.", "(REF ).", "In Fig.", "5, we show that the success probability of concentration of each photon altered with the iteration number $M$ .", "We take the five-photon less-entangled $W$ state as an example.", "We let $\\alpha _{1}=\\alpha _{2}=\\alpha _{3}=0.5$ , $\\alpha _{4}=0.3$ and $\\alpha _{5}=0.4$ .", "Interestingly, if $\\alpha _{1}=\\alpha _{2}=\\alpha _{3}=0.5$ , the success probability of concentration number 1 photon $P_{M}^{1}$ is equal to $P_{M}^{3}$ , shown in Curve B.", "We calculated the total success probability of our protocol with CPC gate shown in Fig.6.", "It is shown that, if we use the PPC gate, the success probability is the case of $M=1$ , that is 0.03228.", "But if we use the CPC gate and iterate it for eight times, the success probability can be increased to 0.28575.", "It is about nine times greater than the success probability of using PPC gate." ], [ "discussion and summary", "So far, we have fully described our ECP for N-particle less-entangled $W$ state.", "We explain this ECP with two different methods.", "The first one is to use the PPC gates and the second one is to use the CPC gates.", "In our ECP, after successfully performing this parity check, all coefficients in the initial state are equal to $\\alpha _{2}$ .", "In fact, this is not the unique way to achieve this task.", "We can also choose $\\alpha _{1}$ and make all coefficients be equal to $\\alpha _{1}$ after performing this ECP.", "Choosing different coefficients do not change the basic principle of this ECP, but it will change the total success probability.", "In detail, we take four-particle less-entangled $W$ state and five-particle less-entangled $W$ state for example.", "Fig.", "7 shows the success probability altering with the iteration number $M$ for case of four-particle.", "Figure: The total success probability of our ECPfor four-partite WW state altered with iteration number MM with CPCgate.", "Curve A: α 1 =1 6\\alpha _{1}=\\frac{1}{\\sqrt{6}},α 2 =1 12\\alpha _{2}=\\frac{1}{\\sqrt{12}}, α 3 =1 2\\alpha _{3}=\\frac{1}{\\sqrt{2}},α 4 =1 2\\alpha _{4}=\\frac{1}{2}.", "Curve B: α 1 =1 2\\alpha _{1}=\\frac{1}{2},α 2 =1 6\\alpha _{2}=\\frac{1}{\\sqrt{6}}, α 3 =1 2\\alpha _{3}=\\frac{1}{\\sqrt{2}},α 4 =1 2\\alpha _{4}=\\frac{1}{2}.", "Curve C: α 1 =1 2\\alpha _{1}=\\frac{1}{\\sqrt{2}},α 2 =1 2\\alpha _{2}=\\frac{1}{2}, α 3 =1 6\\alpha _{3}=\\frac{1}{\\sqrt{6}},α 4 =1 12\\alpha _{4}=\\frac{1}{\\sqrt{12}}.", "Curve D:α 1 =1 12\\alpha _{1}=\\frac{1}{\\sqrt{12}}, α 2 =1 2\\alpha _{2}=\\frac{1}{\\sqrt{2}},α 3 =1 2\\alpha _{3}=\\frac{1}{2}, α 1 =1 6\\alpha _{1}=\\frac{1}{\\sqrt{6}}.In Fig.", "7, the less-entangled $W$ states corresponding to different curves essentially have the same entanglement.", "Because they can change to each other with local operations.", "However, it is shown that the same initial entanglement have the different success probabilities if we choose different $\\alpha _{2}$ .", "In Fig.", "8, we also calculate the similar case of five-particle less-entangled $W$ state.", "One can see that choosing different $\\alpha _{2}$ leads different total success probability.", "That is $\\alpha _{2}$ smaller, the total success probability is greater.", "Figure: The total success probability of our ECPfor five-particle WW state altered with iteration number MM with CPCgate.", "Curve A: α 1 =0.4\\alpha _{1}=0.4, α 2 =0.3\\alpha _{2}=0.3,α 3 =α 4 =α 5 =0.5\\alpha _{3}=\\alpha _{4}=\\alpha _{5}=0.5.", "Curve B: α 1 =0.5\\alpha _{1}=0.5,α 2 =0.4\\alpha _{2}=0.4, α 3 =0.3\\alpha _{3}=0.3, α 4 =α 5 =0.5\\alpha _{4}=\\alpha _{5}=0.5.Curve C: α 1 =α 2 =α 3 =0.5\\alpha _{1}=\\alpha _{2}=\\alpha _{3}=0.5, α 4 =0.3\\alpha _{4}=0.3,α 5 =0.4\\alpha _{5}=0.4.We can explain this result from Eq.", "(REF ) and Eq.", "(REF ).", "In Eq.", "(REF ), if $\\alpha _{1},\\alpha _{2},\\cdots \\alpha _{N}$ are given, the numerator is a constant.", "But the value of denominator is decided by $\\alpha _{2}$ .", "Therefore, choosing the smallest value of $\\alpha _{2}$ will get the highest success probability.", "This result provide us an useful way to performing this ECP.", "If $\\alpha _{2}$ is not the smallest one, then we can rotate the polarization of each photon with half-wave plate until to obtain the smallest $\\alpha _{2}$ .", "Certainly, we should point out that the ECP with PPC gate and with CPC gate are quite different form each other in the practical manipulation.", "Because the PPC gate is equipped with the linear optical elements and we should resort the pose selection principle to achieve this task.", "That is, after successfully performed this ECP, the maximally entangled $W$ state is destroyed by the sophisticated single photon detectors.", "This condition greatly limit its practical application.", "In addition, in Sec.", "III A, we explain it by dividing the whole ECP into $N-1$ steps.", "In each step, one of the parties prepares one single photon and makes a parity check measurement.", "Practically, each parties except Bob2 should perform the parity check measurement simultaneously due to the post selection principle.", "If all parity check measurements are even parities, then by classical communication, they ask each to retain their photons, and it is a successful case.", "On the other hand, if we adopt CPC gate to perform this ECP, each parties can operate his photons independently.", "That is, each one can repeat to perform concentration until it is successful.", "The most fundamental reason is that QND is only to check the phase shift of the coherent state and it does not destroy the photon after measurement.", "In our ECP, after performing the parity check measurement using CPC gate, the next operation is decided by the measurement result.", "If it is even parity, it is successful, otherwise, each one can restart to concentrate his photon with another single photon.", "This strategy makes the total concentration efficiency be greatly improved.", "Finally, let us discuss the key element of our ECP, that is the cross-Kerr nonlinearity.", "In Ref.", "[35] and [37], they also adopt the cross-Kerr nonlinearity to construct the parity check gate to achieve the concentration tasks.", "Unfortunately, in order to increase the efficiency of the protocol, they should resort the coherent state to obtain $\\pi $ phase shift.", "Although there are several strategies to increase the phase shift, such as increasing the strength of the coherent state, controlling the coupling time of the coherent state and the Kerr media, and choosing the suitable Kerr media, to get giant phase shift is still difficult in current technology [57], [58].", "Meanwhile, cross-Kerr nonlinearity is also a controversial topic.", "The focus of the argument is still that one cannot get giant phase shift on the single-photon level.", "This conclusion is agree with the results of Shapiro, Razavi, and Gea-Banacloche[59], [60], [61].", "On the other hand, Hofmann pointed out that with a single two-level atom in a one-sided cavity, a large phase-shift of $\\pi $ can be achieved[62].", "Current research showed that it is possible to amplify a cross-Kerr-phase-shift to an observable value by using weak measurements, which is much larger than the intrinsic magnitude of the single-photon-level nonlinearity[63].", "Zhu and Huang also discussed the possibility of obtain the giant cross-Kerr nonlinearities using a double-quantum-well structure with a four-lever, double-type configuration[64].", "Fortunately, we do not require the coherent state to get $\\pi $ phase shift.", "It is an improvement of Refs.", "[35], [37], [36].", "This kind of parity check gate is first used to perform the entanglement purification in Ref.", "[24].", "Then Guo et al.", "developed this idea, and used it to perform the Bell-state analyzer, prepare the cluster-state and so on [56].", "As discussed by Guo et al., compared with the previous parity check gate[35], [37], [36], it has several advantages: first, it is an effective simplification by removing two PBSs and several mirrors; second, it has a lower error rate.", "Third, it does not require the $\\pi $ phase shift which is more suitable in current experimental conditions.", "In summary, we have present an universal way to concentrate an N-particle less-entangled $W$ state into a maximally entangled $W$ state with both PPC gate and CPC gate.", "In the former, we require the linear optical elements and post selection principle.", "In the later, we use cross-Kerr nonlinearity to construct the QND.", "Different from other concentration protocols, we only need single photon as an auxiliary to achieve the task.", "Then this ECP does not largely consume the less-entangled photon systems.", "Especially, with the help of QND, each parties can operated independently and this ECP can be repeated to get a higher success probability.", "These advantages may make this ECP more useful in practical application in current quantum information processing." ], [ "ACKNOWLEDGEMENTS", "This work is supported by the National Natural Science Foundation of China under Grant No.", "11104159, Scientific Research Foundation of Nanjing University of Posts and Telecommunications under Grant No.", "NY211008, University Natural Science Research Foundation of JiangSu Province under Grant No.", "11KJA510002, and the open research fund of Key Lab of Broadband Wireless Communication and Sensor Network Technology (Nanjing University of Posts and Telecommunications), Ministry of Education, China, and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions." ] ]

[ [ "Densities and entropies in cellular automata" ], [ "Abstract Following work by Hochman and Meyerovitch on multidimensional SFT, we give computability-theoretic characterizations of the real numbers that can appear as the topological entropies of one-dimensional and two-dimensional cellular automata." ], [ "In this section, we construct a south-deterministic SFT with null entropies which letter-factors onto $*_S$ .", "In the SFT, there is a special layer which consists exactly in $S^*$ : from Lemma , we can a priori assume that all configurations of this layers are in ${A^}^*$ , by implicitly having a layer in $$ .", "We will now add a layer whose purpose is to check that if $x\\in \\alpha ^*$ is read from this layer, with $\\alpha \\in $ , then $\\alpha $ is really in the wanted set $S$ , by simulating the application of a machine $$ corresponding to the machine $\\tilde{}$ that rejects any configuration that is not in $S$ (see Lemma ).", "A naive simulation of the machine for an infinite time would create invalid limit configurations.", "A solution to this problem is to build the additional layer in a self-similar way, in the fashion of , , : we build a family of south-deterministic SFT $(Y_n)$ such that $Y_n$ simulates the TM for $n$ steps, and also simulates $Y_{n+1}$ with some parameters $B_n,T_n$ .", "That way, if $n$ was not enough to figure out that the input had to be rejected, then a higher level will notice it.", "More precisely, $Y_n$ will be able to apply the TM over the input $x{B_{n+1}0{B_n}+j}$ for some $j\\in 0{B_{n+1}}$ .", "The simulation of $Y_{n+1}$ , as defined previously, consists in dividing naturally every valid configuration of $Y_n$ into rectangles of size $B_n \\times T_n$ called the $Y_n$ -macrotiles.", "An important feature is that this family admits a uniform description: one single SFT is actually described.", "Each configuration is conscious of the level $Y_n$ it belongs to, and will check that it simulates a configuration of the next one.", "The details of the construction ensuring these conditions can be found in .", "The following lemma applies machine $$ from Lemma to finite configurations composed of some arithmetic progressions in lines of the SFT, that are still in $\\alpha $ .", "Null entropies come from the self-simulation.", "If $S\\subset $ is an effectively closed set, then $S^*$ is S0-sofic.", "Finally, let us see how Lemma REF can be used to prove Theorem : it simply independently splits each letter 1 into two letters, so that its density is transformed into entropy.", "[of Theorem ] One direction corresponds to Theorem .", "Let us prove the converse.", "Should we make the product with the shift over $2^{\\alpha }$ symbols, whose entropy is $\\alpha $ , we can assume that $\\alpha \\in [0,1[$ .", "Let $F$ be the shift composed with the CA corresponding to the deterministic SFT given by Lemma REF for the effectively closed set $S$ consisting of binary representations of real numbers from the interval $[0,\\alpha ]$ , $A$ its alphabet, and $\\pi :A\\rightarrow $ be the corresponding letter projection.", "Let $\\tilde{F}$ be the CA over alphabet $(A \\times \\lbrace 0\\rbrace )\\sqcup (\\pi ^{-1}(1)\\times \\lbrace 1\\rbrace )$ such that the first component performs $F$ and the second one performs the shift.", "in the first component we can see the 0-entropy $F$ and, in the second one the one-dimensional subshift: $S^\\nabla ={(y_i)_{i\\in }}{^}{\\exists (x_i)_{i\\in }\\in S,\\forall i\\in , \\text{ if } x_i=0, \\text{ then } y_i=0}.$ It is known that the entropy of a product is the sum of the entropies, hence the entropy of $\\tilde{F}$ is that of $S^\\nabla $ .", "$U(S^\\nabla )=\\sum _{u\\in _U(S)}2^{u1}$ can be bounded by $U(S)2^{\\sup _{u\\in _U(S)}u1}$ .", "Hence, the entropy $S^\\nabla $ is: $S^\\nabla =\\lim _{r\\rightarrow \\infty }\\frac{\\log {r}(S^\\nabla )}{{r}}\\le {S}+\\lim _{r\\rightarrow \\infty }\\sup _{u\\in _{r}(S)}\\frac{u1}{{r}}.$ However, since ${S} = 0$ , $S^\\nabla $ is not more than the maximal density $\\alpha $ of configurations of $S$ .", "Conversely, if $x\\in \\alpha \\subset S$ , then $_{r}(S^\\nabla )\\supset {(x_i,y_i)_{i<r}}{\\forall i\\in ,y_i\\in \\lbrace x_i,2x_i\\rbrace }$ ; hence ${r}(S)\\ge 2^{{x_{r}}1}$ and $S^\\nabla \\ge \\limsup _{r\\rightarrow \\infty }{x_{r}}1=\\alpha $ .", "Therefore, ${\\tilde{F}}=S^\\nabla =\\alpha $ ." ] ]

[ [ "Stability properties of multiplicative representations of free groups" ], [ "Abstract We extend the construction of multiplicative representations for a free group G introduced by Kuhn and Steger (Isr.", "J., (144) 2004) in such a way that the new class Mult(G) so defined is stable under taking the finite direct sum, under changes of generators (and hence is Aut(G)-invariant), under restriction to and induction from a subgroup of finite index.", "The main tool is the detailed study of the properties of the action of a free group on its Cayley graph with respect to a change of generators, as well as the relative properties of the action of a subgroup of finite index after the choice of a \"nice\" fundamental domain.", "These stability properties of Mult(G) are essential in the construction of a new class of representations for a virtually free group (Iozzi-Kuhn-Steger, arXiv:1112.4709v1)" ], [ "Introduction", "Let $\\Gamma $ be a finitely generated non-abelian free group.", "We shall say that a unitary representation $(\\pi ,\\mathcal {H})$ of a group $G$ is tempered if it is weakly contained in the regular representation.", "In [4], the second and the third author introduced a new family of tempered unitary representations of $\\Gamma $ .", "This class is large enough to include all known representations that are obtained by embedding $\\Gamma $ into the automorphism group of its Cayley graph.", "Beside being rather exhaustive, these representations have interesting properties in their own right, such as for example beeing representations of the crossed product $C^*$ -algebra $\\Gamma \\ltimes \\mathcal {C}(\\partial \\Gamma )$ where $\\mathcal {C}(\\partial \\Gamma )$ is the $C^*$ -algebra of continuos functions on the boundary $\\partial \\Gamma $ of $\\Gamma $ (see the discussion after Theorem REF ).", "The definition of these representations requires a set of data, called matrix system with inner product, consisting of a (complex) vector space and a positive definite sesquilinear form for each generator, as well as linear maps between any two pairs of vector spaces, all subject to some compatibility condition (recalled in § ).", "We generalize in this paper the construction in [4] by releasing the condition that the matrix system with inner product be irreducible (see Definition REF ).", "The irreducibility in [4] insured that, except in sporadic and well understood special cases, the unitary representations so constructed would be irreducible.", "The starting point in this paper is the following result, according to which irreducibility of the matrix system is not essential: representations arising from non-irreducible matrix systems are anyway finitely reducible in the following sense: Theorem 1 Every representation $(\\pi ,\\mathcal {H})$ constructed from a matrix system with inner products $(V_a,H_{ba},B_a)$ decomposes into the orthogonal direct sum with respect to $\\mathcal {B}=(B_a)$ of a finite number of representations constructed from irreducible matrix systems.", "We call such a representation multiplicative and we denote by $\\mathbf {Mult}(\\Gamma )$ the class of representations that are unitarily equivalent to a multiplicative representation (see the end of §  for the precise definition).", "That we are allowed to drop the dependence of the set of free generators follows from the following important result: Theorem 2 Let $A$ and $A^{\\prime }$ be two symmetric sets of free generators of a free group $\\Gamma $ , and let us denote by $\\mathbb {F}_A$ and $\\mathbb {F}_{A^{\\prime }}$ the group $\\Gamma $ as generated respectively by $A$ and $A^{\\prime }$ .", "Then for every $\\pi \\in \\mathbf {Mult}{\\mathbb {F}_{A^{\\prime }}}$ there exists a matrix system with inner product indexed on $A$ , such that $\\pi \\in \\mathbf {Mult}{\\mathbb {F}_A}$ .", "In particular the class $\\mathbf {Mult}(\\Gamma )$ is $\\operatorname{Aut}(\\Gamma )$ -invariant.", "In [4] the authors give an explicit realization of several known representations, such as for example the spherical series of Figà-Talamanca and Picardello [1], as multiplicative representations with respect to scalar matrices acting on one dimentional spaces.", "At the same time in [6] it is shown that if $\\pi _s$ and $\\Pi _s$ are spherical series representations corresponding to different generating sets, say $A^{\\prime }$ and $A$ , then they cannot be equivalent unless $A$ is obtainable by $A^{\\prime }$ by an automorphism of the Cayley graph associated to the generating set $A^{\\prime }$ .", "The above theorem insures that, when we think of a spherical representation as a multiplicative representation this pathology disappears, in the sense that a spherical representation $\\pi _s$ corresponding to a given generating set $A^{\\prime }$ will be realized as a multiplicative representation with respect to another generating set $A$ (although in this case the new matrices will fail to be scalars, as on can see in Example REF ).", "The class $\\mathbf {Mult}(\\Gamma )$ allows us to define a new class of representations for virtually free groups $\\Lambda $ (see [2]): $\\mathbf {Mult}(\\Lambda )$ is defined as the class of representations obtained by inducing to $\\Lambda $ a multiplicative representation of a free subgroup of finite index.", "The proof that the class $\\mathbf {Mult}(\\Lambda )$ is independent of the choice of the free subgroup depends on the following further interesting stability property of the class $\\mathbf {Mult}(\\Gamma )$ .", "Theorem 3 Assume that $\\Gamma $ is a finitely generated non-abelian free group and let $\\Gamma ^{\\prime }<\\Gamma $ be a subgroup of finite index.", "If $\\pi \\in \\mathbf {Mult}(\\Gamma )$ , then the restriction of $\\pi $ to $\\Gamma ^{\\prime }$ belongs to $\\mathbf {Mult}(\\Gamma ^{\\prime })$ .", "If $\\pi \\in \\mathbf {Mult}(\\Gamma ^{\\prime })$ , then the induction of $\\pi $ to $\\Gamma $ belongs to $\\mathbf {Mult}(\\Gamma )$ .", "Since representations of the class $\\mathbf {Mult}(\\Gamma )$ are tempered, the same is true for those of the class $\\mathbf {Mult}(\\Lambda )$ .", "The representations in the class $\\mathbf {Mult}(\\Gamma )$ appear also in a natural way as boundary representations, that is representations of the cross product $C^\\ast $ -algebra $\\Gamma \\ltimes \\mathcal {C}(\\partial \\Gamma )$ , where $\\mathcal {C}(\\partial \\Gamma )$ is the $C^\\ast $ -algebra of the continuous functions on the boundary $\\partial \\Gamma $ of $\\Gamma $ .", "Boundary representations are exactly those which admit a boundary realization, that is, a relization as a direct integral over $\\partial \\Gamma $ with respect to some quasi-invariant measure.", "As boundary representations as well, the representations in the class $\\mathbf {Mult}(\\Gamma )$ enjoy all of the above properties and this is again an essential ingredient in the proof that every representation in the class $\\mathbf {Mult}(\\Lambda )$ extends to a representation of $\\Lambda \\ltimes \\mathcal {C}(\\partial \\Gamma )$ and hence admits a boundary realization after identifying the two boundaries $\\partial \\Lambda $ and $\\partial \\Gamma $ .", "Incidentally, it is proved in [2] that every tempered representation of a torsion-free not almost cyclic Gromov hyperbolic group has a boundary realization.", "However, while the existence of such a boundary realization for a representation of a Gromov hyperbolic group follows from general $C^\\ast $ -algebra inclusions as well extension properties using Hanh–Banach theorem, and is hence highly non-constructive, for representations in the class $\\mathbf {Mult}(\\Gamma )$ the boundary realization is more accessible and sometimes very concrete.", "Its uniqueness is also studied in details in the scalar case in [3], but remains in general an open question." ], [ "Multiplicative Representations of the Free Group", "Fix a symmetric set $A$ of free generators for $\\mathbb {F}_A$ , $A=A^{-1}$ .", "Throughout, when we use $a, b, c,d,a_j$ , for $j\\in \\mathbf {N}$ , for elements of $\\mathbb {F}_A$ , it is intended that they are elements of $A$ .", "There is a unique reduced word for every $x\\in \\mathbb {F}_A$ : $x=a_1a_2\\dots a_n \\qquad \\text{where for all $j$, $a_j\\in A$ and $a_ja_{j+1}\\ne e$.", "}$ The Cayley graph of $\\mathbb {F}_A$ has as vertices $\\mathcal {V}$ the elements of $\\mathbb {F}_A$ and as undirected edges the couples $\\lbrace x,xa\\rbrace $ for $x\\in \\mathbb {F}_A$ , $a\\in A$ .", "This is a tree $\\mathcal {T}$ with $\\#A$ edges attached to each vertex and the action of $\\mathbb {F}_A$ on itself by left translation preserves the tree structure.", "Since the set of vertices $\\mathcal {V}$ is independent of the generating set, whenever we need to emphasize this independence, we identify elements of the free group with vertices of its associated Cayley graph.", "A sequence $(x_0,x_1,\\dots ,x_n)$ of vertices in the tree is a geodesic segment if for all $j$ , $x_{j+1}$  is adjacent to $x_j$ and $x_{j+2}\\ne x_j$ .", "We denote such geodesic segment joining $x_0$ with $x_n$ with $[x_0,x_1,\\dots ,x_n]\\qquad \\text{ or }\\qquad [x_0,x_n]\\,,$ whenever the intermediate vertices are not important.", "If the vertex $z\\in \\mathcal {V}$ is on the geodesic from $x_0$ to $x_n$ , we write $z\\in [x_0,x_n]$ .", "We define the distance between two vertices of the tree as the number of edges in the path joining them.", "This gives $d(e,x)=|x|$ , $d(x,y)=|x^{-1}y|$ .", "Definition 2.1 A matrix system or simply system $(V_a,H_{ba})$ is obtained by choosing a complex finite dimensional vector space $V_a$ for each $a\\in A$ , and a linear map $H_{ba}:V_a\\rightarrow V_b$ for each pair $a,b\\in A$ , where $H_{ba}=0$ whenever $ab=e$ .", "Definition 2.2 A tuple of linear subspaces $W_a\\subseteq V_a$ is called an invariant subsystem of $(V_a,H_{ba})$ if $H_{ba}W_a\\subseteq W_b\\qquad \\text{ for all $a$, $b$.", "}$ For any given invariant subsystem $(W_a,H_{ba})$ of $(V_a,H_{ba})$ the quotient system $(\\widetilde{V}_a,\\widetilde{H}_{ba})$ is defined on $\\widetilde{V}_a=V_a/W_a$ in the obvious way: $\\widetilde{H}_{ba}\\widetilde{v}_a:=\\widetilde{H_{ba}v_a} \\qquad \\text{where $v_a$ is any representativefor $\\widetilde{v}_a$.", "}$ The system $(V_a,H_{ba})$ is called irreducible if it is nonzero and if it admits no invariant subsystems except for itself and the zero subsystem.", "Definition 2.3 A map from the system $(V_a,H_{ba})$ to the system $(V^{\\prime }_a,H^{\\prime }_{ba})$ is a tuple $(J_a)$ where $J_a: V_a\\rightarrow V^{\\prime }_a$ is a linear map and $H^{\\prime }_{ab}J_b=J_a H_{ab}\\;.$ The tuple $(J_a)$ is called an equivalence if each $J_a$  is a bijection.", "Two systems are called equivalent if there is an equivalence between them.", "Remark 2.4 A map $(J_a)$ between irreducible systems $(V_a,H_{ba})$ and $(V^{\\prime }_a,H^{\\prime }_{ba})$ is either 0 or an equivalence.", "This is because the kernels (respectively, the images) of the maps $J_a$ constitute an invariant subsystem.", "For $x\\in \\mathcal {V}$ we set once and for all $\\begin{aligned}E(x)&:=\\lbrace y\\in \\mathcal {V}:\\;\\text{the reduced word for }y \\text{ ends with }x\\rbrace \\\\C(x)&:=\\lbrace y\\in \\mathcal {V}:\\;\\text{the reduced word for }y \\text{ starts with }x\\rbrace \\\\&\\hphantom{:}=\\lbrace y\\in \\mathcal {V}:\\;x\\in [e,y]\\rbrace \\,.\\end{aligned}$ Definition 2.5 A (vector-valued) multiplicative function is a function $f:\\mathbb {F}_A~\\rightarrow ~\\coprod _{a\\in A}~V_a$ for which there exists $N=N(f)\\ge 0$ such that for every $x\\in \\mathcal {V}$ , with $|x|\\ge N$ $\\begin{alignedat}{3}&f(x)\\in V_a &\\quad &\\text{if } &&x\\in E(a) \\\\&f(xb)= H_{ba}f(x)&\\quad &\\text{if } && x\\in E(a) \\text{ and}\\;|xb| =|x|+1\\,.\\end{alignedat}$ We denote by $\\mathcal {H}_0^\\infty (V_a,H_{ba})$ (or $\\mathcal {H}_0^\\infty $ is there is no risk of confusion) the space of multiplicative functions with respect to the system $(V_a,H_{ba})$ .", "Note that if $f$ satisfies (REF ) for some $N=N_0$ , it also satisfies (REF ) for all $N\\ge N_0$ .", "We define two multiplicative functions $f$ and $g$ to be equivalent, $f\\sim g$ , if $f(x)=g(x)$ for all but finitely many elements of $ \\mathcal {V}$ and $\\mathcal {H}^\\infty $ is defined as the quotient of the space of multiplicative functions with respect to this equivalence relation $\\mathcal {H}^\\infty :=\\mathcal {H}^\\infty _0/\\sim $ .", "The vector space structure on $\\mathcal {H}^\\infty $ is given by pointwise multiplication by scalars and pointwise addition, where we choose an arbitrary value for $(f_1+f_2)(x)$ for those finitely many $x$ for which $f_1(x)$ and $f_2(x)$ do not belong to the same space $V_a$ .", "In the following we will need a particular type of multiplicative function which we now define.", "Definition 2.6 Let $x$ be a reduced word in $E(a)$ and $v_a\\in V_a$ .", "A shadow $\\mu [x,v_a]$ is (the equivalence class of) a multiplicative function supported on the cone $C(x)$ , such that $N\\big (\\mu [x,v_a]\\big )=|x|\\;\\text{ and }\\mu [x,v_a](x):=v_a\\,.$ It is clear that every multiplicative function can be written as the sum of a finite number of shadows.", "For each $a\\in A$ choose a positive definite sesquilinear form $B_a$ on $V_a\\times V_a$ and set $\\langle f_1,f_2\\rangle :=\\sum _{|x|=N}\\;\\;\\sum _{ \\begin{array}{c} \\;a\\\\ |xa|=|x|+1\\end{array}}B_a\\big (f_1(xa),f_2(xa)\\big )$ where $N$ is large enough so that both $f_i$ satisfy (REF ).", "It is easy to verify that for the definition to be independent of $N$ the $B_a^{\\prime }s$ must satisfy the condition $B_a(v_a,v_a)=\\sum _{b}B_b(H_{ba}v_a,H_{ba}v_a)$ , for all $a\\in A$ and $v_a\\in V_a$ .", "Definition 2.7 The triple $(V_a,H_{ba},B_a)$ is a system with inner products if $(V_a,H_{ba})$ is a matrix system, $B_a$  is a positive definite sesquilinear form on $V_a$ for each $a\\in A$ and for $v_a\\in V_a$ $B_a(v_a,v_a)=\\sum _{b\\in A}B_b(H_{ba}v_a,H_{ba}v_a)\\,.$ We refer to (REF ) as to a compatibility condition.", "Assuming that such a family exists define a unitary representation $\\pi $ of $\\mathbb {F}_A$ on $\\mathcal {H}^\\infty $ by the rule $(\\pi (x)f)(y)=f({x}^{-1} y)\\,.$ The existence of a family of sesquilinear forms satisfying the compatibility condition was shown in [4] as follows.", "Definition 2.8 For each $a\\in A$ , let $S_a$ be the real vector space of symmetric sesquilinear forms on $V_a\\times V_a$ .", "Let $\\mathcal {S}=\\bigoplus _{a\\in A}S_a$ .", "We say that a tuple $\\mathcal {B}=(B_a)\\in \\mathcal {S}$ is positive definite (resp.", "positive semi-definite) if each of its components is positive definite (resp.", "positive semi-definite), in which case we write $\\mathcal {B}>0$ (resp.", "$\\mathcal {B}\\ge 0$ ).", "Let $\\mathcal {K}\\subseteq \\mathcal {S}$ denote the solid cone consisting of positive semi-definite tuples.", "Define a linear map $\\mathcal {L}:\\mathcal {S}\\rightarrow \\mathcal {S}$ by the rule $(\\mathcal {L}\\mathcal {B})_a(v_a,v_a)=\\sum _{b} B_b( H_{ba}v_a,H_{ba}v_a)\\,,$ where $\\mathcal {B}=(B_a)$ , and observe that $\\mathcal {L}(\\mathcal {K})\\subseteq \\mathcal {K}$ .", "The existence of a tuple $(B_a)_{a\\in A}$ compatible with $(V_a,H_{ba})$ depends on eigenvalues of $\\mathcal {L}$ .", "The following lemma summarizes the results of [4]: Lemma 2.9 ([4]) For any given matrix system $(V_a,H_{ba})$ , there exists a positive number $\\rho $ and a tuple of positive semi-definite sesquilinear forms $(B_a)$ on $V_a$ such that $\\sum _{b}B_b(H_{ba}v_a,H_{ba}v_a)=\\rho B_a(v_a,v_a)\\,.$ If $\\lambda $ is any other number such that $\\sum _{b}B_b(H_{ba}v_a,H_{ba}v_a)=\\lambda B_a(v_a,v_a)$ then $|\\lambda |\\le \\rho $ .", "If the matrix system is irreducible then each $B_a$ is strictly positive definite and, up to multiple scalars, there exists a unique tuple satisfying (REF ).", "We shall refer to $\\rho $ as the Perron–Frobenius eigenvalue of the system $(V_a,H_{ba})$ .", "As a consequence of the above lemma, it follows that, up to a normalization of the matrices $H_{ba}$ , every matrix system becomes a system with inner products.", "Complete now $\\mathcal {H}^\\infty $ to $\\mathcal {H}=\\mathcal {H}(V_a,H_{ab},B_a)$ with respect to the norm defined in (REF ) (where, again, we shall drop the dependence from $(V_a,H_{ab},B_a)$ unless necessary) and extend the representation $\\pi $ defined in (REF ) to a unitary representation on $\\mathcal {H}$ .", "Two equivalent systems $(V_a,H_{ba},B_a)$ and $(V^{\\prime }_a,H^{\\prime }_{ba},B^{\\prime }_a)$ give rise to equivalent representations $\\pi $ and $\\pi ^{\\prime }$ on $\\mathcal {H}=\\mathcal {H}(V_a,H_{ab},B_a)$ and $\\mathcal {H}=\\mathcal {H}(V^{\\prime }_a,H^{\\prime }_{ab},B^{\\prime }_a)$ .", "In fact, if the tuple $(J_a)$ gives the equivalence of the two systems in Definition REF , the operator defined by $U\\big (\\mu [x,v_a]\\big ):=\\mu [x,J_av_a]$ for $v_a\\in V_a$ extends to a unitary equivalence between $(\\pi ,\\mathcal {H}(V_a,H_{ab},B_a))$ and $(\\pi ^{\\prime },\\mathcal {H}(V^{\\prime }_a,H^{\\prime }_{ab},B^{\\prime }_a))$ .", "Notice that the converse is not true, namely non-equivalent systems can give rise to equivalent representations: the simplest example is given by any spherical representation of the principal series of Figà-Talamanca and Picardello corresponding to a non-real parameter $q^{-\\frac{1}{2}+is}$ [4].", "The irreducibility condition in the last statement in Lemma REF is only sufficient.", "In fact, even if the matrix system is reducible, we can always assume that the $B_a^{\\prime }s$ are strictly positive definite by passing to an appropriate quotient, as the following shows: Lemma 2.10 Let $(V_a, H_{ba},B_a)$ be a matrix system with inner product and let $\\pi $ a multiplicative representation on $\\mathcal {H}(V_a,H_{ba},B_a)$ .", "Then there exist a matrix system with inner product $(\\widetilde{V}_a,\\widetilde{H}_{ba},\\widetilde{B}_a)$ and a representation $\\widetilde{\\pi }$ on $\\widetilde{\\mathcal {H}}(\\widetilde{V}_a,\\widetilde{H}_{ab},\\widetilde{B}_a)$ equivalent to $\\pi $ such that $\\widetilde{\\mathcal {B}}=(\\widetilde{B}_a)>0$ .", "If $(B_a)$ is not strictly positive definite, then for some $a\\in A$ , $W_a:=\\lbrace w_a\\in V_a\\setminus \\lbrace 0\\rbrace :\\;B_a(w_a,w_a)=0\\rbrace \\ne \\emptyset \\,.$ Since for $w_a\\in W_a$ $0=B_a(w_a,w_a)=\\sum _b B_b(H_{ba}w_a,H_{ba}w_a)$ and all the terms $B_b(H_{ba}w_a,H_{ba}w_a)$ on the right are non-negative, each of these must be zero.", "Thus, $H_{ba}w_a\\in W_b $ and we conclude that $(W_a)$ is a nontrivial invariant subsystem.", "Let $(\\widetilde{V}_a,\\widetilde{H}_{ba})$ be the quotient system.", "The tuple $(\\widetilde{B}_a)$ given by $\\widetilde{B}_a(\\widetilde{v}_a,\\widetilde{v}_a)=B_a(v_a,v_a)\\qquad \\text{for some $v_a\\in \\widetilde{v}_a$}$ is well-defined and strictly positive on $(\\widetilde{V}_a)$ .", "In the representation space $\\mathcal {H}^\\infty (V_a,H_{ba})$ define the invariant subspace $\\begin{aligned}\\mathcal {H}_W^\\infty =\\lbrace f\\in \\mathcal {H}^\\infty (V_a,H_{ba}):\\;f(xa)\\in W_a\\text{ for all }a\\in A\\text{ and for all }&\\\\x\\in \\mathbb {F}_A\\text{ with }|x|\\ge N(f)\\text{ and }|xa|=|x|+1&\\rbrace \\,.\\end{aligned}$ and consider the quotient representation $\\pi _W$ on $\\mathcal {H}^\\infty (V_a,H_{ba})/\\mathcal {H}^\\infty _W$ .", "Then the representation space $\\mathcal {H}^\\infty (V_a,H_{ba})/\\mathcal {H}^\\infty _W$ may be identified with the space $\\mathcal {H}^\\infty (\\widetilde{V}_a,\\widetilde{H}_{ba})$ of vector-valued multiplicative functions taking values in $\\bigoplus _{a\\in A} \\widetilde{V}_a$ and, after the appropriate completion, $\\pi $ is equivalent to $\\pi _W$ .", "We conclude this section with the definition of the class of representations whose stability properties are the subject of study of this paper.", "Definition 2.11 Given a free group $\\mathbb {F}_A$ on a symmetric set of generators $A$ , we say that a representation $(\\rho ,H)$ belongs to the class $\\mathbf {Mult}{\\mathbb {F}_A}$ if there exists a system with inner products $(V_a,H_{ba},B_a)$ , a dense subspace $M\\subseteq H$ and a unitary operator $U:H\\rightarrow \\mathcal {H}=\\mathcal {H}(V_a,H_{ba},B_a)$ such that $U$ is an isomorphism between $M$ and the space $\\mathcal {H}^\\infty (V_a,H_{ba},B_a)$ of vector-valued multiplicative functions.", "$U(\\rho (x)m)=\\pi (x)(Um)$ for every $m\\in M$ and $x\\in \\mathbb {F}_A$ ." ], [ "The Compatibility Condition and the Norm of a Multiplicative Function", "Let $f$ be a function multiplicative for $|x|\\ge N$ .", "Fix any vertex $x$ such that $d(e,x)\\ge N$ and denote by $t(x)$ the last letter in the reduced word for $x$ .", "Then the compatibility condition can be rewritten as $B_{t(x)}\\big (f(x),f(x)\\big )=\\sum _{\\begin{array}{c}\\; y\\\\ |y|=|x|+1\\end{array}}B_{t(y)}\\big (f(y),f(y)\\big )\\,,$ so that, from (REF ), $\\Vert f\\Vert _{\\mathcal {H}}^2=\\sum _{|x|=N}\\Vert f(x)\\Vert ^2\\,,$ where $\\Vert f(x)\\Vert ^2:=B_{t(x)}(f(x),f(x))\\,.$ The hypothesis of compatibility (REF ) has further consequences in the computation of the norm of a function, that we illustrate now.", "We start with some definitions and notation.", "Definition 3.1 Let $\\mathcal {T}$ be a tree of degree $q+1$ and $\\mathcal {X}$ a finite subtree.", "We say that $\\mathcal {X}$ is non-elementary if it contains at least two vertices.", "If $x$ is a vertex of $\\mathcal {X}$ , its degree relative to $\\mathcal {X}$ is the number of neighborhoods of $x$ that lie in $\\mathcal {X}$ .", "A finite subtree $\\mathcal {X}$ is called complete if all its vertices have relative degree equal either to 1 or to $q+1$ .", "The vertices having degree 1 are called terminal while the others are called interior.", "The set of terminal vertices is denoted by $T(\\mathcal {X})$ .", "If $\\mathcal {X}$ is a complete nonelementary subtree not containing $e$ as an interior vertex, we denote by $\\bar{x}_e$ the unique vertex of $\\mathcal {X}$ which minimizes the distance from $e$ and $x_e$ the unique vertex of $\\mathcal {X}$ connected to $\\bar{x}_e$ (which exists since $\\bar{x}_e\\in T(\\mathcal {X})$ ).", "We call $\\mathcal {X}$ a complete (nonelementary) subtree based at $x_e$.", "We set moreover $T_e(\\mathcal {X}):=T(\\mathcal {X})\\setminus \\lbrace \\bar{x}_e\\rbrace $ and denote by $B(x,N) =\\lbrace y\\in \\mathcal {T}:\\;d(x,y)\\le N\\rbrace $ the (closed) ball of radius $N$ centered at $x\\in \\mathcal {T}$ .", "Lemma 3.2 Let $\\mathcal {X}$ be any complete nonelementary subtree not containing $e$ as an interior vertex.", "With the above notation, assume that $f$ is a function multiplicative outside the ball $B\\big (e,|x_e|\\big )$ .", "Then $\\Vert f(x_e)\\Vert ^2=\\sum _{t\\in T_e(\\mathcal {X})}\\Vert f(t)\\Vert ^2\\,.$ Let $n=\\sup _{x\\in \\mathcal {X}} d(x_e,x)\\,.$ The statement can be easily proved by induction on $n$ .", "When $n=1$ the subtree $\\mathcal {X}$ must be exactly $B\\big (x_e,1\\big )$ and (REF ) reduces to (REF ).", "Assume now that (REF ) is true for $n$ and pick any $y_1$ such that $d(x_e,y_1)=n+1=\\sup _{x\\in \\mathcal {X}} d(x_e,x)\\,.$ Denote by $[x_e,\\dots , \\bar{y}_1,y_1]$ the geodesic joining $x_e$ to $y_1$ .", "By construction $y_1$ is a terminal vertex while $\\bar{y}_1$ is an interior vertex.", "Let $\\mathcal {X}_1$ be the subtree obtained from $\\mathcal {X}$ by removing all the $q$ neighbors of $\\bar{y}_1$ at distance $n+1$ from $x_e$ .", "Now $\\bar{y}_1$ is a terminal vertex of $\\mathcal {X}_1$ .", "If the supremum over all the vertices of the new complete subtree $\\mathcal {X}_1$ of the distances $d(x_e,x)$ is $n$ use induction, otherwise, if $n+1=\\sup _{x\\in \\bar{\\mathcal {X}}} d(x_e,x)\\;;$ pick any $y_2$ such that $n+1=d(x_e,y_2)$ and proceed as before.", "In a finite number of steps we shall end with a finite complete subtree $\\mathcal {X}_k$ satisfying $n=\\sup _{x\\in \\mathcal {X}_k} d(x_e,x)$ for which (REF ) holds.", "Since by inductive hypothesis $\\mathcal {X}$ can be obtained from $\\mathcal {X}_k$ by adding all the $q$ neighbors of each point $\\bar{y}_i$ which are at distance $n+1$ from $x_e$ , $i=1,\\dots ,k$ , again (REF ) follows from (REF ).", "We saw that the norm of a multiplicative function can be computed as the sum of the values of $\\Vert f(x)\\Vert ^2$ , where $x$ ranges over all terminal vertices in $B(e,N)$ for $N$ large enough; building on the previous lemma, the next result asserts that branching off in some direction along a complete subtree and considering again all terminal vertices does not change the norm.", "Lemma 3.3 Let $\\mathcal {X}$ be any finite complete subtree containing $B(e,N)$ and let $f$ be multiplicative for $|x|\\ge N$ .", "Then $\\Vert f\\Vert _{\\mathcal {H}}^2 = \\sum _{x\\in T(\\mathcal {X})}\\Vert f(x)\\Vert ^2\\,.$ Let $L\\ge N$ be the radius of the largest ball $B(e,L)$ completely contained in $\\mathcal {X}$ , so that $\\Vert f\\Vert _{\\mathcal {H}}^2=\\sum _{|x|=L}\\Vert f(x)\\Vert ^2$ .", "If $B(e,L)\\ne \\mathcal {X}$ , the set of points $I:=\\big \\lbrace x\\in \\mathcal {X}:\\,d(e,x)=L\\text{ and }x\\notin T(\\mathcal {X})\\big \\rbrace $ is not empty.", "Apply now Lemma REF to the complete subtree $\\mathcal {X}_x$ of $\\mathcal {X}$ based at $x$ for all $x\\in I$ ." ], [ "The Perron–Frobenius Eigenvalue", "Before we conclude this section we prove the following two lemmas, which shed some light on the possible values of the Perron–Frobenius eigenvalue of a given matrix system.", "Both lemmas, together with Lemma REF , will be necessary in the proof of Theorem REF .", "Lemma 3.4 Let $(V_a, H_{ba}, B_a)$ be a matrix system with inner product, $(W_a, H_{ba})$ an invariant subsystem.", "Let $\\pi $ be the multiplicative representation on $\\mathcal {H}(V_a, H_{ba}, B_a)$ and let $\\pi _W$ be the restriction of $\\pi $ to a multiplicative representation on $\\mathcal {H}(W_a,H_{ba},B_a)$ .", "Assume that the quotient system $(\\widetilde{V}_a,\\widetilde{H}_{ba})$ is irreducible.", "If the Perron–Frobenius eigenvalue $\\rho $ of the quotient system $(\\widetilde{V}_a,\\widetilde{H}_{ba})$ is less than 1 then the representations $\\pi $ and $\\pi _W$ are equivalent.", "By Lemma REF we may assume that the $B_a$ 's are strictly positive definite.", "For each $a$ let $W_a^\\perp :=\\lbrace v_a\\in V_a:\\;B_a(w_a,v_a)=0\\text{ for all }w_a\\in W_a\\rbrace $ be the orthogonal complement (with respect to $B_a$ ) of $W_a$ in $V_a$ .", "Let $\\varphi _a:V_a\\rightarrow \\widetilde{V}_a$ , respectively $P_a:V_a\\rightarrow W_a^\\perp $ , denote the projection of $V_a$ onto $\\widetilde{V}_a$ and the orthogonal projection of $V_a$ onto $W_a^\\perp $ .", "Set $H^\\perp _{ba}~:=~P_bH_{ba}P_a$ .", "The following diagram ${V_a[rr]^{\\varphi _a}& &\\widetilde{V}_a\\\\V_a[u]^=[rr]_{P_a}& &W_a^\\perp [u]_{\\varphi _a|_{W_a^\\perp }}}$ is commutative, so that the system $(W_a^\\perp ,H_{ba}^\\perp )$ may be viewed as an invariant subsystem of the quotient system $(\\widetilde{V}_a,\\widetilde{H}_{ba})$ .", "Since the dimensions are the same, the two systems must be equivalent.", "Denote by $\\rho $ the Perron-Frobenius eigenvalue of the system $(\\widetilde{V}_a,\\widetilde{H}_a)$ .", "By Lemma REF there exists an essentially unique tuple $\\widetilde{B}_a$ of sesquilinear forms on $\\widetilde{V}_a$ such that $\\sum _{b\\in A}\\widetilde{B}_b(\\widetilde{H}_{ba}\\widetilde{v}_a,\\widetilde{H}_{ba}\\widetilde{v}_a)=\\rho \\widetilde{B}_a(\\widetilde{v}_a,\\widetilde{v}_a)\\,,$ which can be chosen to be positive definite since the system $(\\widetilde{V}_a,\\widetilde{B}_a)$ is irreducible.", "By identifying the finite dimensional subspaces $ W_a^\\perp $ and $\\widetilde{V}_a$ , the norms induced on $W_a^\\perp $ by $B_a$ and on $\\widetilde{V}_a$ by $\\widetilde{B}_a$ are equivalent and there exists a constant $K$ so that $B_a\\big (P_a(v_a),P_a(v_a)\\big )\\le K\\widetilde{B}_a\\big (\\varphi (v_a),\\varphi (v_a)\\big )$ for all $a\\in A$ .", "Define, as in Lemma REF , $\\begin{aligned}\\mathcal {H}_W^\\infty =\\lbrace f\\in \\mathcal {H}^\\infty (V_a,H_{ba}):\\;f(xa)\\in W_a\\text{ for all }a\\in A\\text{ and for all }&\\\\x\\in \\mathbb {F}_A \\text{ with }|x|\\ge N(f)\\text{ and }|xa|=|x|+1&\\rbrace \\,.\\end{aligned}$ Under the assumption that $\\rho <1$ , we shall prove that $\\mathcal {H}^\\infty _W$ is dense in $\\mathcal {H}^\\infty (V_a,H_{ba})$ with respect to the norm induced by the $B_a$ 's, from which the assertion will follow.", "Choose $f$ in $\\mathcal {H}^\\infty (V_a,H_{ba})$ and $\\epsilon >0$ .", "Let $N=N(f)$ be such that $f$ is multiplicative for $n\\ge N$ and let us fix $x\\in \\mathbb {F}_A$ and $a\\in A$ such that $|x|\\ge N$ and $|xa|=|x|+1$ .", "Write $f(xa)=w_a+w_a^\\perp $ , where $w_a\\in W_a$ and $w^\\perp _a\\in W^\\perp _a$ , and observe that $\\begin{aligned}P_b\\big (f(xab)\\big )&=P_b\\big (H_{ba}f(xa)\\big )=P_b\\big (H_{ba}(w_a+w^\\perp _a)\\big )\\\\&=P_bH_{ba}w^\\perp _a=H^\\perp _{ba}w_a^\\perp \\,.\\end{aligned}$ Define now $g_0:=\\sum _{b:\\;\\; ab\\ne e}\\mu [xab,f(xab)-P_b(f(xab))]$ and compute $\\Vert {f-g_0}\\Vert _{\\mathcal {H}}^2\\;&=\\sum _{ \\begin{array}{c} \\;b\\\\ |xab|=|x|+2\\end{array}}B_b\\big (f(xab)-g_0(xab),f(xab)-g_0(xab)\\big ) \\\\&=\\sum _{ \\begin{array}{c} \\;b\\\\ |xab|=|x|+2\\end{array}}B_b(H^\\perp _{ba}w_a^\\perp ,H^\\perp _{ba}w_a^\\perp )\\\\&\\le K\\sum _{\\begin{array}{c}\\; b\\\\ |xab|=|x|+2\\end{array}}\\widetilde{B}_b(H^\\perp _{ba}w_a^\\perp ,H^\\perp _{ba}w_a^\\perp )\\\\&=K\\rho \\widetilde{B}_a(w_a^\\perp ,w_a^\\perp )\\,.$ Let $n$ be large enough so that $K\\rho ^n\\widetilde{B}_a(w_a^\\perp ,w_a^\\perp )<\\epsilon \\,.$ Let $z:=a_1\\dots a_n$ a reduced word of length $n$ so that $y=xazb$ has length $|y|=|x|+2+n$ .", "Define $H^\\perp _y=H^\\perp _{ba_n}\\dots H^\\perp _{a_1a}$ and use induction and (REF ) to see that $P_b(f(y))=H^\\perp _yw_a^\\perp \\,.$ A repeated application of (REF ) yields $\\sum _{b\\in A}\\;\\sum _{\\begin{array}{c}y\\in {C}(xa)\\cap E(b)\\\\ |y|=|x|+2+n\\end{array}}\\widetilde{B}_b(H^\\perp _yw_a^\\perp ,H^\\perp _yw_a^\\perp )\\;=\\;\\rho ^{n+1}\\widetilde{B}_a(w_a^\\perp ,w_a^\\perp )\\,.$ If we set, as before, $g_n:=\\sum _{b\\in A}\\;\\;\\sum _{\\begin{array}{c}y\\in {C}(xa)\\cap E(b)\\\\ |y|=|x|+2+n\\end{array}}\\mu [y,f(y)-P_b(y)]\\,,$ then $\\begin{aligned}\\Vert {f-g_n}\\Vert _{\\mathcal {H}}^2&=\\sum _{b\\in A}\\;\\;\\sum _{\\begin{array}{c}y\\in {C}(xa)\\cap E(b)\\\\ |y|=|x|+2+n\\end{array}}B_b\\big (P_b(f(y),P_bf(y)\\big )\\\\&\\le K\\sum _{b\\in A}\\;\\sum _{\\begin{array}{c}y\\in {C}(xa)\\cap E(b)\\\\ |y|=|x|+2+n\\end{array}}\\widetilde{B}_b(H^\\perp _yw_a^\\perp ,H^\\perp _yw_a^\\perp )\\\\&=K\\rho ^{n+1}\\widetilde{B}_a(w_a^\\perp ,w_a^\\perp )\\,,\\end{aligned}$ and hence $\\Vert {f-g_n}\\Vert _{\\mathcal {H}}^2\\le K\\rho ^{n+1}\\widetilde{B}_a(w_a^\\perp ,w_a^\\perp )<\\epsilon \\,.$ Since $g_n$ belongs to $\\mathcal {H}_W$ this concludes the proof.", "Lemma 3.5 Let $(V_a,H_{ba},B_a)$ be a matrix system with inner products and $(W_a,H_{ba})$ a maximal nontrivial invariant subsystem with quotient $(\\widetilde{V}_a,\\widetilde{H}_{ba})$ .", "Then there exists a tuple of strictly positive definite forms on $\\widetilde{V}_a$ with Perron–Frobenius eigenvalue $\\rho =1$ .", "We may assume that $\\mathcal {B}:=(B_a)>0$ .", "The maximality of $(W_a,H_{ba})$ implies that the quotient system $(\\widetilde{V}_a,\\widetilde{H}_{ba})$ is irreducible, hence by Lemma REF there exists a tuple of strictly positive definite forms $(\\widetilde{B}_a)$ satisfying $\\sum _{b}\\widetilde{B}_b(\\widetilde{H}_{ba}\\widetilde{v}_a,\\widetilde{H}_{ba}\\widetilde{v}_a)=\\rho \\widetilde{B}_a(\\widetilde{v}_a,\\widetilde{v}_a)$ for some positive $\\rho $ .", "If the Perron–Frobenius eigenvalue $\\rho $ relative to $(\\widetilde{V}_a,\\widetilde{H}_{ba})$ were strictly smaller than one, by Lemma REF the representations $\\pi $ on $\\mathcal {H}(V_a,H_{ba},B_a)$ and $\\pi _W$ on $\\mathcal {H}(W_a,H_{ba},B_a)$ would be equivalent and we could restrict ourselves to the new system $(W_a,H_{ba},B_a)$ of strictly smaller dimension.", "We may assume therefore that $\\rho \\ge 1$ .", "Assume, by way of contradiction, that $\\rho >1$ .", "Lift the $\\widetilde{B}_a$ to a positive semi-definite form on $V_a$ by setting it equal to zero on $W_a$ .", "Rewrite our conditions in terms of the operator $\\mathcal {L}$ defined in (REF ): $\\mathcal {L}\\mathcal {B}=\\mathcal {B}\\quad \\text{ and} \\quad \\mathcal {L}\\widetilde{\\mathcal {B}}=\\rho \\widetilde{\\mathcal {B}}$ where $\\mathcal {B}=(B_a)_{a\\in A}$ and $\\widetilde{\\mathcal {B}}=(\\widetilde{B}_a)_{a\\in A}$ .", "Since all the $B_a$ are strictly positive definite, there exists a positive number $k$ such that $k B_a-\\widetilde{B}_a$ is strictly positive definite on $V_a$ for each $a\\in A$ .", "Hence for every integer $n$ $\\mathcal {L}^n(k\\mathcal {B}-\\widetilde{\\mathcal {B}})=k\\mathcal {L}^n(\\mathcal {B})-\\mathcal {L}^n(\\widetilde{\\mathcal {B}})=k\\mathcal {B}-\\rho ^n\\widetilde{\\mathcal {B}}\\ge 0$ Choose now $v_a\\in V_a$ so that $\\widetilde{B}_a(v_a,v_a)\\ne 0$ and $n$ large enough to get a contradiction." ], [ "Stability Under Orthogonal Decomposition ", "A representation that arises from an irreducible matrix system with inner product is in most of the cases irreducible or, in some special cases, sum of two irreducible ones.", "As mentioned already, this is the situation considered in [4].", "In this section we analyze representations arising from non-irreducible matrix systems showing that they are still well behaved as the following theorem shows.", "Theorem 4.1 Every representation $(\\pi ,\\mathcal {H})$ constructed from a matrix system with inner products $(V_a,H_{ba},B_a)$ decomposes into the orthogonal direct sum with respect to $\\mathcal {B}=(B_a)$ of a finite number of representations constructed from irreducible matrix systems.", "Let $(V_a, H_{ba},B_a)$ be a matrix system with inner products and assume that that $\\mathcal {B}=(B_a)>0$ (see Lemma REF ).", "Let $(W_a,H_{ba})$ be a maximal nontrivial invariant subsystem with irreducible quotient $(\\widetilde{V}_a, \\widetilde{H}_{ba})$ and let $(\\widetilde{B}_a)$ be a tuple of strictly positive definite forms with Perron–Frobenius eigenvalue $\\rho =1$ , whose existence follows from Lemma REF .", "Pull back the forms $(\\widetilde{B}_a)$ to obtain a tuple of positive semi-definite forms on $V_a$ which have $W_a$ as the kernel and which we still denote by $\\widetilde{B}_a$ .", "Define $\\lambda _0=\\sup \\lbrace \\lambda >0:\\;B_a-\\lambda \\widetilde{B}_a\\ge 0\\text{ for all }a\\in A\\rbrace $ Since $(B_a)$ are strictly positive $\\lambda _0$ is finite.", "Moreover, for such $\\lambda _0$ , $B_a-\\lambda _0\\widetilde{B}_a$ is not strictly positive for some $a$ and hence, for these $a$ 's $W^0_a:= \\lbrace v_a\\in V_a:\\;(B_a-\\lambda _0\\widetilde{B}_a) (v_a,v_a)=0\\rbrace \\ne \\lbrace 0\\rbrace \\,.$ Set $(\\mathcal {B}^0)_a:=B_a-\\lambda _0\\widetilde{B}_a$ and observe that $\\begin{aligned}\\mathcal {B}^0&=\\mathcal {B}-\\lambda _0\\widetilde{\\mathcal {B}}\\ge 0\\\\\\mathcal {L}(\\mathcal {B}-\\lambda _0\\widetilde{\\mathcal {B}})&=\\mathcal {L}\\mathcal {B}-\\lambda _0\\mathcal {L}\\widetilde{\\mathcal {B}}=\\mathcal {B}-\\lambda _0\\widetilde{\\mathcal {B}}\\,.\\end{aligned}$ Arguing as in Lemma REF one can see that also the $(W^0_a)$ , and hence the ($W_a+W^0_a$ ), constitute an invariant subsystem.", "We claim that $V_a=W_a\\oplus W^0_a$ .", "In fact, since $\\widetilde{B}_a|_{W_a}\\equiv 0$ , then $W_a\\cap W^0_a=0$ for all $a$ .", "Moreover, if $\\varphi _a:V_a\\rightarrow \\widetilde{V}_a$ denotes the projection, the system $\\varphi _a(W_a\\oplus W^0_a)$ would be invariant and hence, by irreducibility of $(\\widetilde{V}_a)$ , the image $\\varphi _a(W_a\\oplus W^0_a)$ has to be all of $\\widetilde{V}_a$ , that is to say $V_a=W_a\\oplus W^0_a$ for all $a$ .", "Moreover $B_a=B^0_a+\\lambda _0\\widetilde{B}_a$ is the sum of two orthogonal forms.", "The representation $(\\pi ,\\mathcal {H})$ constructed from the system $(V_a,H_{ba},B_a)$ decomposes as the sum of the two sub-representations corresponding to the systems $(W_a,H_{ba},B^0_a)$ and $(W^0_a,H_{ba},\\widetilde{B}_a)$ where the latter is an irreducible system.", "To complete the proof repeat the above argument for the system $(W_a,H_{ba},B^0_a)$ : since all the $V_a$ are finite dimensional, this reduction process will stop with an irreducible subsystem." ], [ "Stability Under Change of Generators", "Let $A, A^{\\prime }$ denote two symmetric set of free generators for the free group and write $a_i, b_i,c_i$ , and $\\alpha _j, \\beta _j,\\gamma _j$ , for generic elements of $A$ or $A^{\\prime }$ , respectively.", "Denote by $\\mathcal {T}$ and $\\mathcal {T}^{\\prime }$ the tree relative to the generating set $A$ and $A^{\\prime }$ , and by $|x|$ , $|x|^{\\prime }$ the tree distance of $x$ from $e$ in $\\mathcal {T}$ and $\\mathcal {T}^{\\prime }$ .", "The aim of this section is to prove the following: Theorem 5.1 Let $\\pi \\in \\mathbf {Mult}(\\mathbb {F}_{A^{\\prime }})$ be a multiplicative representation with respect to the set $A^{\\prime }$ of generators.", "Then there exists a matrix system with inner product $(V_a,H_{ab},B_a)$ indexed on the set of generators $A$ , such that $\\pi \\in \\mathbf {Mult}(\\mathbb {F}_A)$ .", "This allows us to refine the definition of the class of multiplicative representations.", "Definition 5.2 Given a non abelian finitely generated free group $\\Gamma $ , we say that a representation $\\pi $ belongs to the class $\\mathbf {Mult}(\\Gamma _{})$ if there exists a symmetric set of generators $A$ such that $\\pi \\in \\mathbf {Mult}(\\mathbb {F}_A)$ .", "Observe that the property of being invariant under a change of generators is enjoyed by the class $\\mathbf {Mult}(\\Gamma _{})$ , but not by single representations, as will be shown in the Example REF at the end of this section.", "We begin with some definitions.", "Every element has a unique expression as a reduced word in both alphabets and we shall write $z=a_1\\dots a_n$ or $z=\\alpha _1\\dots \\alpha _k$ .", "If $\\ell (A,A^{\\prime })$ denotes the maximum length of the elements of $A$ with respect to the elements of $A^{\\prime }$ , then $|z|^{\\prime }\\le \\ell (A,A^{\\prime })|z|\\,.$ We recall from (REF ) that $C(z)=\\lbrace y\\in \\mathcal {V}:\\;z\\in [e,y]\\rbrace $ and we define analogously $C^{\\prime }(z)=\\lbrace y\\in \\mathcal {V}:\\;z\\in [e,y]^{\\prime }\\rbrace \\,,$ where $[e,y]^{\\prime }$ denotes the geodesic joining $e$ and $y$ in the tree $\\mathcal {T}^{\\prime }$ .", "Hence, if $z=\\alpha _1\\dots \\alpha _k\\in \\mathbb {F}_{A^{\\prime }}$ and $z=a_1\\dots a_n\\in \\mathbb {F}_A$ , $C^{\\prime }(z)$ consists of all reduced words in the alphabet $A^{\\prime }$ of the form $y=\\alpha _1\\dots \\alpha _k s$ with $|y|=k+|s|$ while $C(z)$ consists of all reduced words in the alphabet $A$ of the form $y=a_1\\dots a_n s$ with $|y|=n+|s|$ .", "We remark that, for $xy\\ne e$ , in general we have that $C(xy)\\subseteq xC(y)\\,,$ as $xC(y)$ might contain the identity and hence need not be a cone.", "The following lemma gives conditions under which there is, in fact, equality.", "Lemma 5.3 Let $x,y\\in \\mathcal {V}$ .", "$xC(y)=C(xy)$ if and only if $y$ does not belong to the geodesic fom $e$ to ${x}^{-1}$ in $\\mathcal {T}$ .", "Let $a\\in A$ be such that $|xa|=|x|+1$ and assume that $C^{\\prime }(y)\\subseteq C(a)$ .", "Then $xC^{\\prime }(y)=C^{\\prime }(xy)$ .", "The identity is not in $xC(y)$ if and only if $x$ does not cancel $y$ , that is, if and only if $y\\notin [e,{x}^{-1}]$ .", "To prove the second assertion, observe that, since $|xa|=|x|+1$ , the element ${x}^{-1}$ does not belong to $C(a)$ and, a fortiori to $C^{\\prime }(y)$ by hypothesis.", "Hence $y$ does not belong the geodesic $[e,{x}^{-1}]^{\\prime }$ in $\\mathcal {T}^{\\prime }$ , which, by (i) is equivalent to saying that $xC^{\\prime }(y)=C^{\\prime }(xy)$ .", "The following easy lemma will be useful in the definition of the matrices and the proof of their compatibility.", "Lemma 5.4 Let $a\\in A$ and $z\\in \\mathcal {V}$ such that $C^{\\prime }(z)\\subseteq C(a)$ .", "Then for every $b\\in A$ , $ab\\ne e$ , the last letter of $z$ and of $bz$ in the alphabet $A^{\\prime }$ coincide.", "If not, multiplication by $b$ on the left would delete $z$ , that is the reduced expression in the alphabet $A^{\\prime }$ of the generator $b\\in A$ would be $b=\\alpha _1\\dots \\alpha _t {z}^{-1}$ .", "Taking the inverses one would have ${b}^{-1}=z{\\alpha _t}^{-1}\\dots {\\alpha _1}^{-1}$ , thus contradicting the hypothesis that $C^{\\prime }(z)\\subseteq C(a)$ .", "We have seen in the last two lemmas the first consequences of the inclusion of cones with respect to the two different sets of generators.", "Analogous inclusions follow from the fact that, given two generating systems $A$ and $A^{\\prime }$ , for every $k\\ge 0$ there exists an integer $N=N(k)$ such that the first $N(k)$ letters of a word $z$ in the alphabet $A^{\\prime }$ determine the first $k$ letters of $z$ in the alphabet $A$ .", "In other words, for any given $z\\in \\mathcal {V}$ there exists $N(|z|)$ and $y$ with $|y|^{\\prime }\\le N(|z|)$ so that $C^{\\prime }(y)\\subseteq C(z)\\,.$ The set of $y\\in \\mathcal {V}$ with this property is not necessarily unique.", "To refine the study of the consequences of this cone inclusion, we need to consider, among the $y$ that satisfy (REF ), those that are the “shortest” with this property, in the appropriate sense.", "To make this precise, we use the following notation: $\\begin{aligned}\\bar{y}&\\text{ is the last vertex before }y\\text{ in the geodesic }[e,\\dots ,\\bar{y},y]^{\\prime }\\subset \\mathcal {T}^{\\prime }\\\\\\widetilde{y}_z&\\text{ is the first vertex in the geodesic }[e,y]^{\\prime }\\text{ such that }C^{\\prime }(\\widetilde{y}_z)\\subseteq C(z)\\,.\\end{aligned}$ (For ease of notation, we will remove the subscript $z$ whenever this does not cause any confusion.)", "For any $z\\in \\mathcal {V}$ we then define $\\begin{aligned}Y(z)=&\\left\\lbrace y\\in \\mathcal {V}:\\;C^{\\prime }(y)\\subseteq C(z)\\text{ and }C^{\\prime }(\\bar{y})\\nsubseteq C(z)\\right\\rbrace \\\\=&\\left\\lbrace y\\in \\mathcal {V}:\\;C^{\\prime }(y)\\subseteq C(z)\\text{ and }y=\\widetilde{y_z}\\right\\rbrace \\end{aligned}$ Then we have the following analogue of Lemma REF : Corollary 5.5 For every $a,b\\in A$ , $ab\\ne e$ , we have $aY(b)=Y(ab)\\,.$ Let $y\\in Y(b)$ .", "By Lemma REF (ii) , $\\overline{ay}=a\\bar{y}$ .", "Since $C^{\\prime }(y)\\subseteq C^{\\prime }(\\bar{y})\\nsubseteq C(b)$ and $C^{\\prime }(\\bar{y})\\supseteq C^{\\prime }(y)$ there exists a reduced word $\\bar{y} t$ in the alphabet $A^{\\prime }$ so that $\\bar{y} t\\in C(d)$ for some $d\\in A$ with $d\\ne b$ .", "Hence the element $a\\bar{y} t$ will not be contained in $C(ab)$ .", "For any given $\\pi ^{\\prime }$ in $\\mathbf {Mult}(\\mathbf {F}_{A^{\\prime }})$ we shall now construct $\\pi $ in $\\mathbf {Mult}(\\mathbb {F}_A)$ so that $\\pi ^{\\prime }$ is either a subrepresentation or a quotient of $\\pi $ .", "Namely, if we are given a matrix system with inner products $(V^{\\prime }_\\alpha ,H^{\\prime }_{\\beta \\alpha },B^{\\prime }_\\alpha )$ , we need to define a new system $(V_a, H_{ba}, B_a)$ in such a way that the original system appears as a quotient or as a subsystem of the new one.", "Definition 5.6 Let $z=\\alpha _1\\dots \\alpha _{k-1}\\alpha _k\\in \\mathbb {F}_{A^{\\prime }}$ and define $V^{\\prime }_z=V^{\\prime }_{\\alpha _k}\\quad B^{\\prime }_z=B^{\\prime }_{\\alpha _k}\\,.$ We set $V_a=\\bigoplus _{z\\in Y(a)} V^{\\prime }_z \\qquad B_a=\\bigoplus _{z\\in Y(a)} B^{\\prime }_z$ We need now to define the new matrices $H_{ba}:V_a\\rightarrow V_b$ , for $b\\ne {a}^{-1}$ .", "To this extent, take $z\\in Y(b)$ .", "Since $b\\ne {a}^{-1}$ , then $az\\in C(a)$ and hence, by definition, $\\widetilde{(az)}_a\\in Y(a)$ .", "Then we have two cases: either $az=\\widetilde{(az)}_a$ and hence $az\\in Y(a)$ ; or $az=\\widetilde{(az)}_a\\,x$ with $x\\ne e$ .", "In this case, if the reduced expression for $x$ in the alphabet $A^{\\prime }$ is $x=\\alpha _1\\dots \\alpha _n$ and $\\alpha \\ne {\\alpha }^{-1}_1$ is the last letter (in $A^{\\prime }$ ) of $\\widetilde{(az)}_a\\,$ , define $H^{\\prime }_{az,\\widetilde{az}}:=H^{\\prime }_{\\alpha _n\\alpha _{n-1}}\\dots H^{\\prime }_{\\alpha _1\\alpha }$ where we wrote ${az,\\widetilde{az}}$ for ${az,\\widetilde{(az)}_a}$ for ease of notation.", "The new matrices $H_{ba}:V_a\\rightarrow V_b$ can hence be defined to be block matrices indexed by pairs $(z,w)$ , with $z\\in Y(b)$ and $w\\in Y(a)$ , as follows: $(H_{ba})_{z,w}:={\\left\\lbrace \\begin{array}{ll}\\operatorname{Id}&\\text{ if }w=az=\\widetilde{(az)_a}\\\\H^{\\prime }_{az,\\widetilde{az}}&\\text{ if }w=\\widetilde{(az)_a}\\ne az\\end{array}\\right.", "}$ and $(H_{ba})_{z,w}=0$ for all other $w\\in Y(a)$ with $w\\ne \\widetilde{(az)_a}$ .", "In the course of the definition we have shown that $\\bigcup _{\\begin{array}{c}z\\in Y(b)\\\\ b\\ne {a}^{-1}\\end{array}}\\widetilde{(az)}_a\\subseteq Y(a)\\,,$ but to show that the matrices so defined give a compatible matrix system we need to show that the above inclusion is in fact an equality, namely: Proposition 5.7 We have that $Y(a)=\\bigcup _{\\begin{array}{c}z\\in Y(b)\\\\ b\\ne {a}^{-1}\\end{array}}\\widetilde{(az)}_a$ Take any $w\\in Y(a)$ so that $C^{\\prime }(w)\\subseteq C(a)$ .", "Hence either there exists $b\\ne {a}^{-1}$ such that $C^{\\prime }(w)~\\subseteq ~C(ab)$ , in which case $w\\in Y(ab)$ , or $C^{\\prime }(w)~\\nsubseteq ~C(ab)$ for all $b\\ne {a}^{-1}$ .", "In this case, according to the discussion after Lemma REF , there exists $b\\ne {a}^{-1}$ and $t_b\\in \\mathcal {V}$ with the following properties: $|wt_b|^{\\prime }=|w|^{\\prime }+|t_b|^{\\prime }$ ; $C^{\\prime }(wt_b)\\subseteq C(ab)$ ; $t_b$ is minimal with the above properties, that is $C^{\\prime }(w\\overline{t}_b)\\nsubseteq C(ab)$ .", "In the last case one has, by definition, $wt_b\\in Y(ab)$ .", "By Corollary REF $Y(ab)=aY(b)$ , so that either $w=az$ or $wt_b=az$ for some $z\\in Y(b)$ .", "Since $w\\in Y(a)$ , it is obvious that $w=\\widetilde{(az)}_a$ when $w=az$ .", "To finish we must show that $w=\\widetilde{(wt_b)}_a$ when $wt_b=az$ .", "By definition $\\widetilde{(az)}_a$ is the first vertex in the geodesic $[e,wt_b]^{\\prime }=[e,az]^{\\prime }$ such that $C^{\\prime }(\\widetilde{(az})_a)\\subset C(a)$ .", "But by hypothesis $w\\in Y(a)$ , that is $C^{\\prime }(w)\\subset C(a)$ and $C^{\\prime }(\\overline{w})\\nsubseteq C(a)$ .", "Thus $\\widetilde{(az)}_a=w$ .", "[scale=0.8] [very thick] (-1.14,-1.63) – (0,0) – (10,0) (2,0) – ++(75:5) (2,0) – ++(285:5) (4,0) – ++(40:5) (4,0) – ++(320:5) (2.5,-1.7) – ++(335:4.5) (6,0) – ++(45:2) – (7.5,1.5) – (10.5,1.5); (-.2,.2) node $e$ (1.8,.3) node $a^{-1}$ (3.5,.3) node $a^{-2}=\\overline{w}$ (7.1,-.3) node $a^{-3}=w\\in Y_0(a^{-2})$ (10,.3) node $a^{-4}=w\\alpha =wt_{a^{-1}}\\in Y_0(a^{-2})$ (2.9,-1.414) node $a^{-1}b=w\\beta =wt_b\\in Y_0(a^{-1}b)$ (-1.8,-1.8) node $\\overline{w}\\beta =b$ (6,-3.9) node ${C(a^{-1}b)}$ (4.5,2) node $C(a^{-1})$ (8.3,2.4) node $C(a^{-2})$ (7,1.8) node $a^{-3}b^{-1}$ (8.8,1.8) node $a^{-3}b^{-1}a^{-1}$ (13,1.8) node $a^{-3}b^{-1}a^{-2}=w\\beta =wt_{a^{-1}}\\in Y_0(a^{-2})$; [thick, red] (-1.14,-1.63) – (4,0) – (10,0) (2.5,-1.7) – (6,0) – (10.5,1.5) (2.5,-1.7) – ++(300:3.5) (2.5,-1.7) – ++(320:4); [red] (4.8,-4.3) node ${C^{\\prime }(a^{-1}b)}$; [ultra thick] (0,0) circle [radius=.05] (-1.14,-1.63) circle [radius=.05] (2,0) circle [radius=.05] (4,0) circle [radius=.05] (6,0) circle [radius=.05] (8,0) circle [radius=.05] (2.5,-1.7) circle [radius=.05] (7.5,1.5) circle [radius=.05] (9,1.5) circle [radius=.05] (10.5,1.5) circle [radius=.05] ; Figure 1: The trees $\\mathcal {T}$ (in black) and $\\mathcal {T}^{\\prime }$ (in red) associated respectively to $\\mathbb {F}_A$ and $\\mathbb {F}_{A^{\\prime }}$ , where $A=\\lbrace a,b,a^{-1}, b^{-1}\\rbrace $ and $A^{\\prime }$ is obtained with the change of generators $a\\mapsto \\alpha $ and $b\\mapsto \\beta =a^2b$ .", "In the course of the proof of the above proposition we have distinguished two types of elements of $Y(a)$ , and we can consequently conclude the following: Corollary 5.8 We have $Y(a)=Y_0(a)\\sqcup Y_1(a)\\,,$ where $\\begin{aligned}Y_1(a):&= \\bigcup _{b\\ne {a}^{-1}} \\left(Y(a)\\cap Y(ab)\\right)\\\\&=\\big \\lbrace w\\in Y(a):\\;\\text{there exists }b\\ne {a}^{-1}\\text{ and }z\\in Y(b),\\text{ such that}\\\\&\\hphantom{hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh}w={az}= \\widetilde{(az)}_a\\big \\rbrace \\\\\\end{aligned}$ and $\\begin{aligned}Y_0(a):&=\\big \\lbrace w\\in Y(a):\\;\\text{for all }b\\ne {a}^{-1},\\,C^{\\prime }(w)\\nsubseteq C(ab)\\big \\rbrace \\\\&=\\big \\lbrace w\\in Y(a):\\;\\text{for some }b\\ne {a}^{-1}\\text{ there exists }\\,z\\in Y(b), \\text{ such}\\\\&\\hphantom{hhhhhhhhhhhhhhhh}\\text{that }w=\\widetilde{(az)}_a\\;\\text{ and }az=w\\,x,\\text{ with }x\\ne e\\big \\rbrace \\,.\\end{aligned}$ To prove the compatibility condition we will make use of Lemma REF , so that we need to construct an appropriate finite complete subtree in $\\mathcal {T}^{\\prime }$ .", "Notice that for all $w\\in \\mathcal {V}$ , the set $\\overline{w}\\cup C^{\\prime }(w)$ is a complete subtree, but infinite.", "To \"prune\" it so that it will be finite and still complete, consider an element $w\\in Y_0(a)$ and the following decomposition $\\begin{aligned}C^{\\prime }(w)=&\\big \\lbrace y\\in C^{\\prime }(w):\\;\\, C^{\\prime }(y)\\nsubseteq C(ab)\\text{ for all } b\\ne {a}^{-1}\\big \\rbrace \\\\&\\cup \\big \\lbrace y\\in C^{\\prime }(w):\\;\\, C^{\\prime }(y)\\subseteq C(ab)\\text{ for some }b\\ne {a}^{-1}\\big \\rbrace \\\\=&I^{\\prime }_w\\cup \\bigcup _{b\\ne {a}^{-1}} \\big \\lbrace y\\in C^{\\prime }(w):\\;C^{\\prime }(y)\\subseteq C(ab)\\big \\rbrace \\,,\\end{aligned}$ where we have set $I^{\\prime }_w:=\\big \\lbrace y\\in C^{\\prime }(w):\\;\\, C^{\\prime }(y)\\nsubseteq C(ab)\\text{ for all }b\\ne {a}^{-1}\\big \\rbrace \\,.$ Since the set $I^{\\prime }_w$ is finite and $w\\in I^{\\prime }_w$ , we need to prune the other set.", "Proposition 5.9 Let $w\\in Y_0(a)$ and define $\\begin{aligned}T^{\\prime }_w:=&\\bigcup _{b\\ne {a}^{-1}} \\big \\lbrace y\\in C^{\\prime }(w):\\;C^{\\prime }(y)\\subseteq C(ab),\\,C^{\\prime }(\\overline{y})\\nsubseteq C(ab)\\big \\rbrace \\\\=&\\bigcup _{b\\ne {a}^{-1}}\\big (C^{\\prime }(w)\\cap Y(ab)\\big )\\,.\\end{aligned}$ The set $\\mathcal {X}^{\\prime }_w:=\\lbrace \\overline{w}\\rbrace \\cup I^{\\prime }_w\\cup T^{\\prime }_w$ is a finite complete subtree in $\\mathcal {T}^{\\prime }$ whose terminal vertices are $\\overline{w}$ and $T^{\\prime }_w$ .", "Before proceeding to the proof, we remark that this kind of construction will be performed also in other parts of the paper, whenever we need to construct a finite complete subtree (see for example Lemmas REF , REF and REF in § REF ).", "By definition if $y\\in I^{\\prime }_w\\setminus \\lbrace w\\rbrace $ , then $\\overline{y}\\in I^{\\prime }_w$ and if $y\\in T^{\\prime }_w$ , then $\\overline{y}\\in I^{\\prime }_w$ .", "This shows in particular that $T^{\\prime }_w\\subset T(\\mathcal {X}^{\\prime }_w)$ .", "To see that the set of terminal vertices consists of $\\lbrace \\overline{w}\\rbrace \\cup T^{\\prime }_w$ , observe that if $y\\in I^{\\prime }_w$ and $y\\alpha \\in \\mathcal {T}^{\\prime }$ is such that $|y\\alpha |^{\\prime }=|y|^{\\prime }+1$ , then by construction either $y\\alpha \\in I^{\\prime }_w$ or $y\\alpha \\in T^{\\prime }_w$ .", "We are now finally ready to prove the compatibility condition.", "Proposition 5.10 The system $(V_a,B_a, H_{ba})$ is a compatible matrix system in the sense of (REF ).", "We need to show that if $v_a\\in V_a$ , then $B_a(v_a,v_a)=\\sum _{b\\ne {a}^{-1}}B_b(H_{ba}v_a, H_{ba}v_a)\\,.$ As in (REF ) write $Y(a)=\\bigcup _{\\begin{array}{c}z\\in Y(b)\\\\ b\\ne {a}^{-1}\\end{array}}\\widetilde{(az)}_a=Y_0(a)\\bigcup Y_1(a)\\,.$ By definition of $B_a$ and by Corollary REF we can write the left hand side as $B_a(v_a,v_a)=\\sum _{w\\in Y_0(a)}B^{\\prime }_w(v^{\\prime }_w,v^{\\prime }_w)+\\sum _{w\\in Y_1(a)}B^{\\prime }_w(v^{\\prime }_w,v^{\\prime }_w)$ and, likewise the right hand side as $\\begin{aligned}& \\sum _{b\\ne {a}^{-1}}B_b(H_{ba}v_a, H_{ba}v_a)=\\sum _{b\\ne {a}^{-1}}\\sum _{z\\in Y(b)}\\sum _{\\begin{array}{c}w=\\widetilde{az}\\end{array}}B^{\\prime }_z(H^{\\prime }_{az,\\widetilde{az}}\\,v^{\\prime }_w,H^{\\prime }_{az,\\widetilde{az}}\\,v^{\\prime }_w)=\\\\&\\sum _{b\\ne {a}^{-1}}\\sum _{z\\in Y(b)}\\sum _{\\begin{array}{c}w=\\widetilde{az}\\ne az\\\\w\\in Y_0(a)\\end{array}}B^{\\prime }_z(H^{\\prime }_{az,\\widetilde{az}}\\,v^{\\prime }_w,H^{\\prime }_{az,\\widetilde{az}}\\,v^{\\prime }_w)\\\\&+\\sum _{b\\ne {a}^{-1}}\\sum _{z\\in Y(b)}\\sum _{\\begin{array}{c}w=\\widetilde{az}=az\\\\w\\in Y_1(a)\\end{array}}B^{\\prime }_z(v^{\\prime }_w,v^{\\prime }_w) \\;,\\end{aligned}$ where we used the definition of the $H_{ba}$ (REF ).", "Write $Y_1(a)=\\coprod _{b:\\;b\\ne {a}^{-1}}(Y(a)\\cap Y(ab))$ , a disjoint union.", "Since, for every $b\\ne {a}^{-1}$ , the set $Y(a)\\cap Y(ab)$ consists of those elements $w$ of the form $w=az=\\widetilde{az}$ for some $z\\in Y(b)$ , using Lemma REF we get $\\sum _{w\\in Y_1(a)}B^{\\prime }_w(v^{\\prime }_w,v^{\\prime }_w)=\\sum _{b\\ne {a}^{-1}}\\sum _{z\\in Y(b)}\\sum _{w=az\\in Y_1(a)} B^{\\prime }_{az}(v^{\\prime }_w,v^{\\prime }_w)\\,,$ so that showing (REF ) reduces to showing that $\\sum _{w\\in Y_0(a)}B^{\\prime }_w(v^{\\prime }_w,v^{\\prime }_w)=\\sum _{b\\ne {a}^{-1}}\\sum _{z\\in Y(b)}\\sum _{w=\\widetilde{az}\\in Y_0(a)}B^{\\prime }_z(H^{\\prime }_{az,\\widetilde{az}}\\,v^{\\prime }_w,H^{\\prime }_{az,\\widetilde{az}}\\,v^{\\prime }_w)\\,.$ To this purpose, observe that, for any element $w\\in Y_0(a)$ there exists a geodesic $[w,wt_b]^{\\prime }$ which starts at the vertex $w$ and ends up in the cone $C(ab)$ for some $b\\ne {a}^{-1}$ (see Proposition REF and Figure 1).", "This geodesic is \"minimal\" in the sense that $C^{\\prime }(w\\bar{t_b})$ would fail to be in the cone $C(ab)$ .", "The endpoints $wt_b$ of these geodesics, for all possible $b$ , are exactly the terminal points $T^{\\prime }_w$ of the tree $\\mathcal {X}^{\\prime }_w$ .", "Hence, for each $w\\in Y_0(a)$ , by Lemma REF applied to the shadow $\\mu [w,v^{\\prime }_w]$ at the point $w$ and the tree $\\mathcal {X}^{\\prime }_w$ ,one has $B^{\\prime }_w(v^{\\prime }_w,v^{\\prime }_w)=\\sum _{b\\ne {a}^{-1}}B^{\\prime }_{wt_b}(v^{\\prime }_{wt_b},v^{\\prime }_{wt_b})\\;.$ We need now to compare the two quantities $B^{\\prime }_{wt_b}(v^{\\prime }_{wt_b},v^{\\prime }_{wt_b})$ and $B^{\\prime }_{z}(H_{az,\\widetilde{az}}v^{\\prime }_{w},H_{az,\\widetilde{az}}v^{\\prime }_{w})$ .", "By Proposition REF we have seen that such terminal vertices can be written as $wt_b=az$ for some $z\\in Y(b)$ and that $\\widetilde{az}_a=w$ .", "By definition of $H_{ba}$ one has $B^{\\prime }_{wt_b}(v^{\\prime }_{wt_b},v^{\\prime }_{wt_b})=B^{\\prime }_{z}(H_{az,\\widetilde{az}}v^{\\prime }_{w},H_{az,\\widetilde{az}}v^{\\prime }_{w})$ where we have used again Lemma REF .", "Summing over $w\\in Y_0(a)$ (or, that is the same, over $az\\in Y_0(a)$ ), we obtain the desired assertion.", "Let now $\\pi $ be the left regular action of $\\mathbb {F}_a$ on $\\mathcal {H}^\\infty (V_a,H_{ba})$ and let $\\mathcal {H}(V_a, H_{ba}, B_a)$ be the completion of $\\mathcal {H}^\\infty (V_a,H_{ba})$ with respect to the norm induced by the $(B_a)$ .", "We define now the intertwining operator $U:\\mathcal {H}^\\infty (V^{\\prime }_\\alpha ,H^{\\prime }_{\\beta \\alpha },B^{\\prime }_\\alpha )\\rightarrow \\mathcal {H}^\\infty (V_a,H_{ba},B_a)\\,.$ For every $f\\in \\mathcal {H}^\\infty (V^{\\prime }_\\alpha ,H^{\\prime }_{\\beta \\alpha })$ and a reduced word $xa$ in the alphabet $A$ we set $(U f)(xa):=\\sum _{y\\in Y(xa)}f(y)\\,.$ To see that $U$ intertwines $\\pi ^{\\prime }$ to $\\pi $ fix any $y\\in \\mathcal {V}$ and assume that $|y|\\le |x|+1$ .", "For any such $x$ and $y$ one has $\\pi (y)U f(xa)&=U f({y}^{-1}xa)=\\sum _{z\\in Y({y}^{-1}xa)}f(z)=\\sum _{z\\in {y}^{-1} Y(xa)}f(z)\\\\&=\\sum _{u\\in Y(xa)}f({y}^{-1}u)=U\\big (\\pi ^{\\prime }(y) f\\big )(xa)$ since $Y({y}^{-1}xa)={y}^{-1}Y(xa)$ if $|y|\\le |x|+1$ .", "It follows that $U \\pi ^{\\prime }(y)f (xa)$ and $\\pi (y)U f(xa)$ differ only for a finite set of values of $x$ , and hence are equal in $\\mathcal {H}^\\infty (V_a,H_{ba})$ .", "We conclude with the following Theorem 5.11 $U$ is unitary.", "Assume that $f\\in \\mathcal {H}^\\infty (V^{\\prime }_\\alpha ,H^{\\prime }_{\\beta \\alpha })$ is multiplicative for $|y|^{\\prime }\\ge N$ .", "We may also assume that $f$ is zero if $|y^{\\prime }|\\le N-1$ .", "By the discussion after Lemma REF there exists an integer $k$ such that $|y|\\le k$ whenever $|y|^{\\prime }\\le N$ .", "Define $S^0_k=\\lbrace z\\in \\mathbb {F}:\\;C^{\\prime }(z)\\nsubseteq C(x)\\text{ for all } x \\text{ with } |x|=k\\rbrace $ and $\\mathcal {S}^{\\prime }(k)=\\lbrace e\\rbrace \\cup S^0_k\\cup \\bigcup _{\\begin{array}{c}x\\in \\mathcal {T}\\\\ |x|=k\\end{array}}Y(x)\\,.$ Arguing as in the proof of Proposition REF one can show that $\\mathcal {S}^{\\prime }(k)$ is a finite complete subtree in $\\mathcal {T}^{\\prime }$ whose terminal vertices are the elements of $Y(x)$ for all $x$ with $|x|=k$ .", "Since every $y$ belongs to $C^{\\prime }(y)$ , we see that $\\mathcal {S}^{\\prime }(k)$ contains the ball of radius $N$ about the origin in $\\mathcal {T}^{\\prime }$ .", "Use now Lemma REF to conclude the proof.", "We conclude this section with an example illustrating the effect of a nontrivial change of generators on a given multiplicative representation.", "Example 5.12 Let $\\Gamma =\\mathbb {F}_A$ , where $A=\\lbrace a,b,a^{-1}, b^{-1}\\rbrace $ .", "Consider the change of generators given by $\\alpha =a$ and $\\beta =ab$ and let $\\pi _s$ be the spherical series representation of Figà–Talamanca and Picardello [1] constructed from the set of generators $A^{\\prime }=\\lbrace \\alpha ,{\\alpha }^{-1},\\beta {\\beta }^{-1}\\rbrace $ .", "Denote by $a^{\\prime }$ , $b^{\\prime }$ the generic elements of $A^{\\prime }$ .", "In [4] it is shown that $\\pi _s$ can be realized as a multiplicative representation with respect to the following matrix system: $V_{a^{\\prime }}&=\\mathbf {C}&\\forall a^{\\prime }\\in A^{\\prime }\\\\H_{b^{\\prime }a^{\\prime }}&= 3^{-\\frac{1}{2}+is} &\\forall a^{\\prime },b^{\\prime }\\in A^{\\prime }\\\\B_{a^{\\prime }}(v,v)&=\\frac{|v|^2}{4}=:\\lambda &\\,.$ Moreover, in [6] it is also shown that it is impossible to realize $\\pi _s$ as any spherical representation arising from the generators $a$ and $b$ .", "We show here that it is however possible to realize $\\pi _s$ as a multiplicative representation with respect to the other generators $a$ and $b$ .", "In fact one can verify that $&Y(a)=\\lbrace \\alpha ,\\beta \\rbrace \\\\&Y(b)=\\lbrace {\\alpha }^{-1}\\beta \\rbrace \\\\&Y({{a}^{-1}})=\\lbrace \\alpha ^{-2},{\\alpha }^{-1}{\\beta }^{-1}\\rbrace \\\\&Y({b}^{-1})=\\lbrace {\\beta }^{-1}\\rbrace $ According to Definition REF the spaces $V_a$ and $V_{{a}^{-1}}$ are two dimensional while $V_b=V_{{b}^{-1}}=\\mathbf {C}$ .", "The matrices appearing in REF are: $&H_{aa} =H_{{a}^{-1}{a}^{-1}}=\\left(\\begin{array}{cc}\\lambda & 0\\\\\\lambda & 0\\\\\\end{array}\\right)\\\\& H_{b{a}^{-1}}=H_{{b}^{-1} a}=\\left(\\begin{array}{cc}\\lambda & 0\\end{array}\\right)\\\\&H_{ba}= H_{{b}^{-1}{a}^{-1}}=\\left(\\begin{array}{cc}0 & 1\\end{array}\\right)\\\\&H_{bb}=H_{{b}^{-1}{b}^{-1}}=\\lambda ^2&&\\\\&H_{ab}=H_{{a}^{-1}{b}^{-1}}=\\left(\\begin{array}{c}\\lambda \\\\\\lambda \\end{array}\\right)\\\\&H_{a{b}^{-1}} = H_{{a}^{-1} b}=\\left(\\begin{array}{c}\\lambda ^2\\\\\\lambda ^2\\end{array}\\right)\\\\ \\,.$ Let $W_a$ (respectively $W_{{a}^{-1}}$ ) denote the subspace of $V_a$ (respectively $V_{{a}^{-1}}$ ) generated by the vector $(1,1)$ .", "The reader can verify that the subspaces $W_a$ , $W_{{a}^{-1}}$ , $W_b=V_b=\\mathbf {C}$ and $W_{{b}^{-1}}=V_{{b}^{-1}}=\\mathbf {C}$ constitute an invariant subsystem and that the quotient system has Perron–Frobenius eigenvalue zero.", "According to Lemma REF the representation $\\pi _s$ is equivalent to the multiplicative representation constructed from the subsystem $W$ ." ], [ "Stability Under Restriction and Unitary Induction", "In this section the set $A$ of generators for $\\Gamma $ is fixed once and for all.", "As before, we write $\\bar{x}$ for the (reduced) word obtained from $x$ by deleting the last letter of the reduced expression for $x$ .", "Set also $\\bar{a}=e$ if $a$ belongs to $A$ .", "Definition 6.1 A Schreier system $S$ in $\\Gamma $ is a nonempty subset of $\\Gamma $ satisfying the following conditions: $e\\in S$ ; if $x\\in S$ , then $\\bar{x}\\in S$ .", "Assume that $\\Gamma ^{\\prime }$ is a subgroup of finite index in $\\Gamma $ .", "Essential in the following will be a choice of an appropriate fundamental domain $D$ for the action of $\\Gamma ^{\\prime }$ on the Cayley graph of $\\Gamma $ with respect to a symmetric set of generators $A$ .", "It is well known (see for example [5]) that one can choose in $\\Gamma $ a set $S^{\\prime }$ of representatives for the left cosets $\\Gamma ^{\\prime }\\gamma $ which is also a Schreier set.", "Identifying $S^{\\prime }$ with an appropriate set of vertices $D$ of $\\mathcal {T}$ , it turns out that $D$ has the following properties: $D$ is a finite subtree containing $e$ .", "$D$ is a fundamental domain with respect to the left action on the vertices of $\\mathcal {T}$ in the sense that the set of vertices of $\\mathcal {T}$ is the disjoint union of the subtrees $x^{\\prime } D$ with $x^{\\prime }\\in \\Gamma ^{\\prime }$ .", "We shall refer to every such $D$ as to a fundamental subtree.", "Corresponding to that choice of $D$ one has also a natural choice of generators for $\\Gamma ^{\\prime }$ , namely one can prove that $\\Gamma ^{\\prime }$ is generated by the set $A^{\\prime }:=\\big \\lbrace a_j^{\\prime }\\in \\Gamma :\\;d(D,a_j^{\\prime }D)=1\\big \\rbrace \\,.$ We shall assume in this Section that $D$ is a fixed fundamental subtree and that $A^{\\prime }$ is the corresponding generating set defined as in REF .", "We write $a^{\\prime },b^{\\prime },\\dots $ to denote a generic element of $A^{\\prime }$ .", "The following lemma summarizes the properties of the translates of $D$ which will be used in several occasions to build finite complete subtrees.", "Lemma 6.2 Let $\\gamma ^{\\prime }a^{\\prime }\\ne e$ be a reduced word in $\\Gamma ^{\\prime }$ .", "There exists $x\\in \\Gamma $ such that $\\gamma ^{\\prime }a^{\\prime }D\\subset C(x)$ but $\\gamma ^{\\prime }D\\lnot \\subset C(x)$ .", "Moreover $\\gamma ^{\\prime }a^{\\prime }b^{\\prime }D\\subset C(x)$ for all $b^{\\prime }$ such that $a^{\\prime }b^{\\prime }\\ne e$ .", "The geodesic in $\\mathcal {T}$ connecting $\\gamma ^{\\prime }a^{\\prime }D$ and $e$ crosses $\\gamma ^{\\prime }D$ .", "Let $a^{\\prime }\\in A^{\\prime }$ be a generator of $\\Gamma ^{\\prime }$ and $D$ a fundamental subtree.", "Let $x(a^{\\prime })\\in a^{\\prime }D$ be the vertex of $a^{\\prime }D$ closest to $D$ .", "Since the distance between $D$ and $a^{\\prime }D$ is one, there exists a unique edge $(x,x(a^{\\prime }))$ such that $x\\in D$ and $x(a^{\\prime })\\in a^{\\prime }D$ .", "We claim that $a^{\\prime }D\\subset C(x(a^{\\prime }))$ .", "Assume the contrary: namely assume that there exists $v\\in a^{\\prime }D$ whose reduced word does not start with $x(a^{\\prime })$ .", "Since $a^{\\prime }D$ is a subtree it must contain the geodesic $[v,x(a^{\\prime })]$ connecting $v$ to $x(a^{\\prime })$ , but this is impossible since $x\\in [v,x(a^{\\prime })]$ .", "Let $b^{\\prime }\\in A^{\\prime }$ be such that $a^{\\prime }b^{\\prime }\\ne e$ .", "Denote by $(w,w^{\\prime })$ ($w\\in a^{\\prime }D$ , $w^{\\prime }\\in a^{\\prime }b^{\\prime }D$ ) the unique edge connecting $a^{\\prime }b^{\\prime }D$ to $a^{\\prime }D$ .", "If $a^{\\prime }b^{\\prime }D\\lnot \\subset C(x(a^{\\prime }))$ it must be $w=x(a^{\\prime })$ and $w^{\\prime }=x$ , which is impossible.", "By induction one has $a^{\\prime }\\gamma ^{\\prime }D\\subset C(x(a^{\\prime }))$ for every $\\gamma ^{\\prime }$ so that $a^{\\prime }\\gamma ^{\\prime }=1+|\\gamma ^{\\prime }|$ .", "Let now $\\gamma ^{\\prime }a^{\\prime }$ be a reduced word in $\\Gamma ^{\\prime }$ and let $x(\\gamma ^{\\prime }a^{\\prime })$ denote the vertex of $\\gamma ^{\\prime }a^{\\prime }D$ closest to $D$ .", "Translating the picture by ${\\gamma ^{\\prime }}^{-1}$ one can see that ${\\gamma ^{\\prime }}^{-1} x(\\gamma ^{\\prime }a^{\\prime })=x(a^{\\prime })$ , that is $x(\\gamma ^{\\prime }a^{\\prime })=\\gamma ^{\\prime }x(a^{\\prime })\\,.$ Since we have $\\gamma ^{\\prime }a^{\\prime }D\\subset \\gamma ^{\\prime }C(x(a^{\\prime })$ (1) will be proved as soon as we show that $\\gamma ^{\\prime }C(x(a^{\\prime })=C(\\gamma ^{\\prime }x(a^{\\prime }))$ .", "Let ${d^{\\prime }}^{-1}$ denote the last letter of $\\gamma ^{\\prime }$ , so that ${d^{\\prime }}^{-1}\\ne {a^{\\prime }}^{-1}$ .", "Since the two subtrees $d^{\\prime }D$ and $a^{\\prime }D$ are both at distance one from $D$ they cannot be contained in the same cone: so that neither $x(a^{\\prime })$ is the first part of $x(d^{\\prime })$ nor the converse.", "In particular $x(a^{\\prime })$ does not belong to the geodesic, in $\\mathcal {T}$ , $[e,{\\gamma ^{\\prime }}^{-1}]$ so that, by Lemma REF , $\\gamma ^{\\prime }C(x(a^{\\prime })=C(\\gamma ^{\\prime }x(a^{\\prime }))$ .", "To complete the proof observe that, since $a^{\\prime }D\\subset C(x(a^{\\prime }))$ and $e\\in D$ , the geodesic connecting $D$ and $a^{\\prime }b^{\\prime }D$ must cross $x(a^{\\prime })$" ], [ "Stability Under Restriction", "The goal of this section is to prove the following: Theorem 6.3 Assume that $\\Gamma $ is a finitely generated free group and $\\Gamma ^{\\prime }\\subseteq \\Gamma $ is a subgroup of finite index.", "If $\\pi \\in \\mathbf {Mult}(\\Gamma _{ })$ , then the restriction of $\\pi $ to $\\Gamma ^{\\prime }$ belongs to $\\mathbf {Mult}(\\Gamma ^{\\prime })$ .", "Choose $D$ and $A^{\\prime }$ as in Definition REF .", "Although $D$ is a finite subtree, it is not complete.", "The strategy of the proof consists of completing $D$ to a complete subtree $D^{\\prime }$ , then translating $D^{\\prime }$ by a generator of $\\Gamma ^{\\prime }$ , so that most of it (in fact, all of it with the exception of the unique edge closer to the identity) is contained in a cone at distance one from $D$ .", "A wise definition of $(V_{a^{\\prime }},H_{b^{\\prime }a^{\\prime }})$ and $B_{a^{\\prime }}$ , together with the help of a shadow supported on the cone, will provide the construction of a matrix system with inner product for the subgroup $\\Gamma ^{\\prime }$ .", "Let, as in the proof of Lemma REF , denote by $x(a^{\\prime })$ the vertex of $a^{\\prime }D$ closest to $D$ .", "Let $D^{\\prime }$ be the subtree obtained by adding to $D$ the vertices $x(a^{\\prime })$ (and the relative edges) corresponding to all $a^{\\prime }\\in A^{\\prime }$ .", "Write $x(a^{\\prime })$ in the generators of $\\Gamma $ and denote by $q(a^{\\prime })$ the last letter of its reduced expressions, that, with the notation used in (REF ), we have that $q(a^{\\prime })=t(x(a^{\\prime }))$ .", "Lemma 6.4 Let $D$ , $D^{\\prime }$ , $x(a^{\\prime })$ as above.", "The subtree $D^{\\prime }$ is complete and its terminal vertices consist of exactly all the $x(a^{\\prime })_{a^{\\prime }\\in A^{\\prime }}$ .", "For every $a^{\\prime },b^{\\prime }\\in A^{\\prime }$ , the vertex of $a^{\\prime }b^{\\prime }D$ closest to $a^{\\prime }D$ is $a^{\\prime }x(b^{\\prime })$ .", "Assume that $a^{\\prime }b^{\\prime }\\ne e$ .", "Then the geodesic joining $e$ and $a^{\\prime }x(b^{\\prime })$ crosses $x(a^{\\prime })$ and the last letter of $a^{\\prime }x(b^{\\prime })$ is $q(b^{\\prime })$ .", "(1) Let $v\\in D$ and assume that $v^{\\prime }$ is a neighbor of $v$ .", "If $v^{\\prime }\\notin D$ there exists $x^{\\prime }\\in \\Gamma ^{\\prime }$ and $u\\in D$ such that $v^{\\prime }=x^{\\prime } u$ .", "Hence the distance between $D$ and $x^{\\prime }D$ is one: this implies that $x^{\\prime }=a^{\\prime }$ for some $a^{\\prime }\\in A^{\\prime }$ and $v^{\\prime }=x(a^{\\prime })$ .", "This proves that every vertex of $D$ is an interior vertex of $D^{\\prime }$ .", "Choose now any $x(a^{\\prime })$ and consider its $q+1$ neighbors: one of them belongs to $D$ and the others, being at distance two from $D$ , cannot be in $D^{\\prime }$ .", "This proves that $D^{\\prime }$ is complete with terminal vertices $x(a^{\\prime })_{a^{\\prime }\\in A^{\\prime }}$ .", "(2) follows immediately from (REF ).", "In particular the vertex of $a^{\\prime }b^{\\prime }D$ closest to $a^{\\prime }D$ is $a^{\\prime }x(b^{\\prime })=x(a^{\\prime }b^{\\prime })$ .", "(3) By Lemma REF , the geodesic joining $e$ and $x(a^{\\prime }b^{\\prime })$ , crosses $x(a^{\\prime })$ .", "In term of the generators of $\\Gamma $ this means that $x(a^{\\prime })$ is the first piece of the reduced word for $a^{\\prime }x(b^{\\prime })$ and, in particular, passing from $x(a^{\\prime })$ to $a^{\\prime }x(b^{\\prime })$ , the last letter of $x(a^{\\prime })$ is not canceled.", "To prove the second assertion, observe that $e$ does not belong to ${x(b^{\\prime })}^{-1}{(a^{\\prime })}^{-1}D$ .", "In fact, if it did, one would have $e={x(b^{\\prime })}^{-1}{(a^{\\prime })}^{-1}\\xi _0$ for some $\\xi _0\\in D$ : but since we also have $x(b^{\\prime })=b^{\\prime }\\xi _1$ this would imply that $\\xi _0=\\xi _1$ and $b^{\\prime }={(a^{\\prime })}^{-1}$ .", "Hence the subtree ${x(b^{\\prime })}^{-1}{(a^{\\prime })}^{-1}D$ is contained in the cone $C(c)$ for some $c\\in A$ .", "Since $d({x(b^{\\prime })}^{-1}D,{x(b^{\\prime })}^{-1}{(a^{\\prime })}^{-1}D)=d(D,{(a^{\\prime })}^{-1}D)=1\\,,$ the subtree ${x(b^{\\prime })}^{-1}D$ is at distance one from ${x(b^{\\prime })}^{-1}{(a^{\\prime })}^{-1}D$ .", "This is possible only in two ways: either ${x(b^{\\prime })}^{-1}D$ is contained in $C(c)$ or ${x(b^{\\prime })}^{-1}D$ contains the identity $e$ .", "The second possibility is ruled out because $x(b^{\\prime })\\notin D$ .", "This implies that the last letter of $x(b^{\\prime })$ is the same as the last letter of $a^{\\prime }x(b^{\\prime })$ .", "We collect here the results as they will be needed later.", "Corollary 6.5 With the above notation the subtree $a^{\\prime }D^{\\prime }$ is a non-elementary tree based at $x(a^{\\prime })$ whose terminal vertices are $T(a^{\\prime }D^{\\prime })=\\lbrace a^{\\prime }x(b^{\\prime }):\\,b^{\\prime }\\in A^{\\prime }\\rbrace $ .", "The terminal vertex closest to $e$ is $a^{\\prime }x(a^{\\prime -1})$ , so that $T_e(a^{\\prime }D^{\\prime })=\\lbrace a^{\\prime }x(b^{\\prime }):\\,b^{\\prime }\\in A^{\\prime },\\,a^{\\prime }b^{\\prime }\\ne e\\rbrace $ and $a^{\\prime }x(b^{\\prime })=x(a^{\\prime })a_1a_2\\dots a_kt(b^{\\prime })=a^{\\prime }x(a^{\\prime -1})q(a^{\\prime })a_1a_2\\dots a_kq(b^{\\prime })$ is the reduced expression of $a^{\\prime }x(b^{\\prime })$ in the alphabet $A$ .", "We are now ready to define the matrix system $(V_{a^{\\prime }},H_{b^{\\prime }a^{\\prime }})$ .", "Definition 6.6 With (REF ) in mind, we set $\\begin{aligned}&V_{a^{\\prime }}:=V_{q(a^{\\prime })}\\,,\\quad \\text{and}\\\\&H_{b^{\\prime }a^{\\prime }}:={\\left\\lbrace \\begin{array}{ll}H_{q(b^{\\prime })a_k}\\dots H_{a_2a_1}H_{a_1q(a^{\\prime })}&\\text{ if }b^{\\prime }a^{\\prime }\\ne e\\\\0&\\text{ if }b^{\\prime }a^{\\prime }=e\\,.\\end{array}\\right.", "}\\end{aligned}$ Lemma 6.7 The tuple $(B_{a^{\\prime }})_{a^{\\prime }\\in A^{\\prime }}$ defined by $B_{a^{\\prime }}:=B_{q(a^{\\prime })}$ is compatible with the matrix system $(V_{a^{\\prime }},H_{b^{\\prime }a^{\\prime }})$ .", "We need to prove that, for every $v_{a^{\\prime }}\\in V_{a^{\\prime }}$ $B_{a^{\\prime }}(v_{a^{\\prime }},v_{a^{\\prime }})=\\sum _{b^{\\prime }:\\;a^{\\prime }b^{\\prime }\\ne e}B_{b^{\\prime }}(H_{b^{\\prime }a^{\\prime }}v_{a^{\\prime }},H_{b^{\\prime }a^{\\prime }}v_{a^{\\prime }})\\,.$ Let $\\mu [x(a^{\\prime }),v_{a^{\\prime }}]$ be the shadow as in Definition REF .", "Since by definition $B_{a^{\\prime }}(v_{a^{\\prime }},v_{a^{\\prime }})=\\big \\Vert \\mu [x(a^{\\prime }),v_{a^{\\prime }}](x(a^{\\prime }))\\big \\Vert ^2\\,,$ showing (REF ) is equivalent to showing that $\\big \\Vert \\mu [x(a^{\\prime }),v_{a^{\\prime }}](x(a^{\\prime }))\\big \\Vert ^2=\\sum _{b^{\\prime }:\\,a^{\\prime }b^{\\prime }\\ne e}\\big \\Vert H_{b^{\\prime }a^{\\prime }}\\mu [x(a^{\\prime }),v_{a^{\\prime }}](x(a^{\\prime }))\\big \\Vert ^2\\,.$ Moreover, since $\\mu [x(a^{\\prime }),v_{a^{\\prime }}]$ is multiplicative, according to the definition of $H_{b^{\\prime }a^{\\prime }}$ we have $\\mu [x(a^{\\prime }),v_{a^{\\prime }}](a^{\\prime }x(b^{\\prime }))=H_{b^{\\prime }a^{\\prime }}\\mu [x(a^{\\prime }),v_{a^{\\prime }}](x(a^{\\prime }))\\,.$ By Lemma REF , Corollary REF and (REF ) it follows that $\\begin{aligned}\\big \\Vert \\mu [x(a^{\\prime }),v_{a^{\\prime }}](x(a^{\\prime }))\\big \\Vert ^2&=\\sum _{t\\in T_e(a^{\\prime }D^{\\prime })}\\big \\Vert \\mu [x(a^{\\prime }),v_{a^{\\prime }}](t)\\big \\Vert ^2\\\\&=\\sum _{b^{\\prime }:\\,b^{\\prime }a^{\\prime }\\ne e}\\big \\Vert \\mu [x(a^{\\prime }),v_{a^{\\prime }}](a^{\\prime }x(b^{\\prime }))\\big \\Vert ^2\\\\&=\\sum _{b^{\\prime }:\\,a^{\\prime }b^{\\prime }\\ne e}\\big \\Vert H_{b^{\\prime }a^{\\prime }}\\mu [x(a^{\\prime }),v_{a^{\\prime }}](x(a^{\\prime }))\\big \\Vert ^2\\,,\\end{aligned}$ which completes the proof.", "We need to define now the intertwining operator between the restriction $\\pi |_{\\Gamma ^{\\prime }}$ to $\\Gamma ^{\\prime }$ of the representation $\\pi $ on $\\mathcal {H}(V_a,H_{ba},B_a)$ and the representation $\\rho $ of $\\Gamma ^{\\prime }$ on $\\mathcal {H}(V_{a^{\\prime }},H_{b^{\\prime }a^{\\prime }},B_{a^{\\prime }})$ defined by $\\rho (x^{\\prime })f(y^{\\prime }):=f({x^{\\prime }}^{-1}y^{\\prime })\\,,$ for $x^{\\prime },y^{\\prime }\\in \\Gamma ^{\\prime }$ and $f\\in \\mathcal {H}(V_{a^{\\prime }},H_{b^{\\prime }a^{\\prime }},B_{a^{\\prime }})$ .", "Definition 6.8 Let $f\\in \\mathcal {H}^\\infty (V_a,H_{ba},B_a)$ .", "If $x^{\\prime }=y^{\\prime }a^{\\prime }\\in \\Gamma ^{\\prime }$ with $a^{\\prime }\\in A^{\\prime }$ and $|x^{\\prime }|_{\\Gamma ^{\\prime }}=|y^{\\prime }|_{\\Gamma ^{\\prime }}+1$ (in the word metric with respect to the generators $A^{\\prime }$ ), define $(Uf)(x^{\\prime }):=f\\big (y^{\\prime }x(a^{\\prime })\\big )\\,.$ It is easy to check that the operator $U$ maps the restriction to $\\Gamma ^{\\prime }$ of multiplicative functions in $\\mathcal {H}^\\infty (V_a,H_{ba},B_a)$ to multiplicative functions in $\\mathcal {H}^\\infty (V_{a^{\\prime }},H_{b^{\\prime }a^{\\prime }},B_{a^{\\prime }})$ .", "In fact, if $x^{\\prime }=y^{\\prime }a^{\\prime }\\in \\Gamma $ with $a^{\\prime }\\in \\Gamma ^{\\prime }$ and $|x^{\\prime }|_{\\Gamma ^{\\prime }}=|y^{\\prime }|_{\\Gamma ^{\\prime }}+1$ , then $(Uf)(x^{\\prime })=f\\big (y^{\\prime }x(a^{\\prime })\\big )\\in V_{t(x(a^{\\prime }))}=V_{q(a^{\\prime })}\\,.$ Moreover, if $y^{\\prime }a^{\\prime }b^{\\prime }\\in \\Gamma ^{\\prime }$ with $a^{\\prime },b^{\\prime }\\in A^{\\prime }$ and $|y^{\\prime }a^{\\prime }b^{\\prime }|_{\\Gamma ^{\\prime }}=|y^{\\prime }|_{\\Gamma ^{\\prime }}+2$ , then $(Uf)(y^{\\prime }a^{\\prime }b^{\\prime })=f\\big (y^{\\prime }a^{\\prime }x(b^{\\prime })\\big )=H_{b^{\\prime }a^{\\prime }}\\big (f(y^{\\prime }a^{\\prime })\\big )\\,.$ Furthermore, it is straightforward to check that $U\\big (\\pi |_{\\Gamma ^{\\prime }}(x^{\\prime })f\\big )=\\rho (x^{\\prime })(Uf)\\,,$ thus completing the proof." ], [ "Stability Under Unitary Induction", "The goal of this section is to prove the following Theorem 6.9 Assume that $\\Gamma $ is a finitely generated free group and $\\Gamma ^{\\prime }\\le \\Gamma $ is a subgroup of finite index.", "If $\\pi ^{\\prime }\\in \\mathbf {Mult}(\\Gamma ^{\\prime })$ then $\\operatorname{Ind}_{\\Gamma ^{\\prime }}^\\Gamma {(\\pi ^{\\prime })}$ is in the class $\\mathbf {Mult}(\\Gamma _{ })$ .", "Let $\\Gamma ^{\\prime }\\le \\Gamma $ be a subgroup of finite index and let $D$ be a fundamental subtree for the action of $\\Gamma ^{\\prime }$ on $\\mathcal {T}$ .", "By Theorem REF we may assume that $A^{\\prime }$ is the generating set of $\\Gamma ^{\\prime }$ corresponding to $D$ as in (REF ).", "Suppose that we are given a matrix system with inner products $(V_{a^{\\prime }}, H_{b^{\\prime }a^{\\prime }},B_{a^{\\prime }})$ relative to $\\Gamma ^{\\prime }$ and hence a representation $\\pi ^{\\prime }$ of the class $\\mathbf {Mult}(\\Gamma ^{\\prime })$ acting on $\\mathcal {H}_s:=\\mathcal {H}(V_{a^{\\prime }},H_{b^{\\prime }a^{\\prime }},B_{a^{\\prime }})$ .", "Because of Theorem REF we may always assume that the system is irreducible.", "Let $\\operatorname{Ind}_{\\Gamma ^{\\prime }}^\\Gamma {(\\pi ^{\\prime })}$ denote the induced representation acting on $\\operatorname{Ind}_{\\Gamma ^{\\prime }}^\\Gamma {(\\mathcal {H}_s)}$ .", "We recall that $\\operatorname{Ind}_{\\Gamma ^{\\prime }}^\\Gamma {(\\mathcal {H}_s)}:=\\big \\lbrace f:\\Gamma \\rightarrow \\mathcal {H}_s:\\,\\pi ^{\\prime }(h)f(g)=f(g{h}^{-1}),\\text{ for all }h\\in \\Gamma ^{\\prime },g\\in \\Gamma \\big \\rbrace \\,,$ on which $\\Gamma $ acts by $\\big (\\operatorname{Ind}_{\\Gamma ^{\\prime }}^\\Gamma {(\\pi ^{\\prime })}(g_0)f\\big )(g):=f({g_0}^{-1}g)\\,,$ for all $g_0,g\\in \\Gamma $ .", "In particular $f(g)$ is uniquely determined by its values on a set of representatives for the right cosets of $\\Gamma ^{\\prime }$ in $\\Gamma $ , which, with our choice of generators of $\\Gamma ^{\\prime }$ , can also be taken to be the fundamental subtree $D$ .", "Denote by $\\mathcal {H}_s^\\infty :=\\mathcal {H}^\\infty (V_{a^{\\prime }},H_{b^{\\prime }a^{\\prime }},B_{a^{\\prime }})$ the dense subspace $\\mathcal {H}_s$ consisting of multiplicative functions and define, with a slight abuse of notation, the dense subset $\\begin{aligned}\\operatorname{Ind}_{\\Gamma ^{\\prime }}^\\Gamma {(\\mathcal {H}_s^\\infty )}:=\\big \\lbrace f:\\Gamma \\rightarrow \\mathcal {H}^\\infty (V_{a^{\\prime }},H_{b^{\\prime }a^{\\prime }},B_{a^{\\prime }}):\\,\\pi ^{\\prime }(h)f(g)=f(g{h}^{-1}),&\\\\\\text{ for all }h\\in \\Gamma ^{\\prime },g\\in \\Gamma \\big \\rbrace &\\end{aligned}$ which, by definition of $\\mathcal {H}_s^\\infty $ , can be identified with $\\begin{aligned}\\operatorname{Ind}_{\\Gamma ^{\\prime }}^\\Gamma {(\\mathcal {H}_s^\\infty )}&\\cong \\big \\lbrace \\varphi :D\\cdot \\Gamma ^{\\prime }\\rightarrow \\coprod _{a^{\\prime }\\in A^{\\prime }}V_{a^{\\prime }}:\\,\\pi ^{\\prime }(h)\\varphi (g)=\\varphi (g{h}^{-1}),\\\\\\text{ for all }&h\\in \\Gamma ^{\\prime },g\\in \\Gamma \\text{ and }\\varphi \\text{ is multiplicative as a function of }\\Gamma ^{\\prime }\\big \\rbrace \\end{aligned}$ via the map $f\\mapsto \\Phi (f)$ , where $\\Phi (f)(x):=f(u)(h)$ , for $x=uh$ , with $h\\in \\Gamma ^{\\prime }$ and $u\\in D$ .", "The invariance property of functions in $\\operatorname{Ind}_{\\Gamma ^{\\prime }}^\\Gamma {(\\mathcal {H}_s^\\infty )}$ imply that $\\Phi (f)$ is well defined.", "We want to show that there exists a matrix system with inner product $(V_a,H_{ba},B_a)$ on $\\Gamma $ so that $\\operatorname{Ind}_{\\Gamma ^{\\prime }}^\\Gamma {(\\pi ^{\\prime })}$ is equivalent to a multiplicative representation $\\pi $ on $\\mathcal {H}(V_a,H_{ba},B_a)$ .", "The vector spaces $V_a$ will be direct sums of possibly multiple copies of the vector spaces $V_{a^{\\prime }}$ , according to some appropriately chosen \"coordinates\" on subsets of the cones $C(a)$ .", "To this purpose, let us define for any generator $a$ of $\\Gamma $ , the set $P(a)=({D}^{-1}\\cdot A^{\\prime })\\cap C(a)\\,,$ where ${D}^{-1}=\\lbrace {u}^{-1}:\\;u\\in D\\rbrace $ .", "The following lemma is technical, but only specifies the multiplicative property of the chosen coordinates.", "Lemma 6.10 Let us fix $a\\in A$ and $v\\in D$ .", "Assume that $va^{-1}\\in D$ and let $c^{\\prime }\\in A^{\\prime }$ be any generator.", "Then $av^{-1}c^{\\prime }\\in P(a)$ if and only if $v^{-1}c^{\\prime }\\in P(b)$ for some $b\\in A$ with $ab\\ne e$ .", "Assume that $va^{-1}\\notin D$ .", "Then there exists $c^{\\prime }\\in A^{\\prime }$ and $u\\in D$ such that $av^{-1}=u^{-1}c^{\\prime }\\in P(a)$ ; furthermore for every $d^{\\prime }\\in A^{\\prime }$ such that $c^{\\prime }d^{\\prime }\\ne e$ , there exists a unique $b\\in A$ with $ab\\ne e$ such that $v^{-1}d^{\\prime }\\in P(b)$ .", "(1) Let $b\\in A$ be such that $v^{-1}c^{\\prime }\\in P(b)$ .", "Then in particular $v^{-1}c^{\\prime }$ starts with $b$ and hence $av^{-1}c^{\\prime }\\in C(a)$ if $ab\\ne e$ .", "Since by hypothesis $va^{-1}\\in D$ , it follows that $av^{-1}c^{\\prime }\\in P(a)$ .", "Conversely, let $b\\in A$ be such that $v^{-1}c^{\\prime }\\in C(b)$ .", "Since $av^{-1}c^{\\prime }\\in P(a)$ , it follows that $ab\\ne e$ .", "Moreover, since $v\\in D$ , we have that $v^{-1}c^{\\prime }\\in P(b)$ .", "(2a) Since $v\\in D$ but $va^{-1}\\notin D$ and $D$ is a Schreier system, then $|va^{-1}|=|v|+1$ , that is $d(va^{-1},D)=1$ .", "By (REF ), there exist $u\\in D$ and $(c^{\\prime })^{-1}\\in A^{\\prime }$ such that $va^{-1}=(c^{\\prime })^{-1}u$ , from which it follows that $av^{-1}=u^{-1}c^{\\prime }\\in P(a)$ .", "(2b) Choose $d^{\\prime }\\in A^{\\prime }$ .", "By (REF ), $D$ and $d^{\\prime }D$ are disjoint subtrees at distance one from each other.", "We claim that if $d^{\\prime }\\ne (c^{\\prime })^{-1}$ , neither of their translates $av^{-1}D$ and $av^{-1}d^{\\prime }D$ contains the identity $e$ .", "In fact, if $e$ were to belong to $av^{-1}D$ , we would have that $va^{-1}\\in D$ , which is excluded by hypothesis.", "If on the other hand $e$ were to belong to $av^{-1}d^{\\prime }D$ , then we would have that for some $u_0\\in D$ , $av^{-1}=u_0^{-1}(d^{\\prime })^{-1}$ .", "But by (2a) we know that $av^{-1}=u^{-1}c^{\\prime }$ , so that, by uniqueness of the decomposition, one would conclude that $c^{\\prime }=(d^{\\prime })^{-1}$ , which is also excluded by hypothesis.", "Hence both subtrees are contained in some cone $C(b)$ , where $b\\in A$ and, since they are at distance one from each other, this cone must be the same for both.", "But since $v\\in D$ , then $a\\in av^{-1}D$ , so that $av^{-1}D$ , and hence $av^{-1}d^{\\prime }D$ , are contained in $C(a)$ .", "Since $e\\in D$ , this means in particular that $av^{-1}d^{\\prime }\\in C(a)$ , so that $v^{-1}d^{\\prime }\\in C(b)$ for some $b$ such that $ab\\ne e$ .", "Hence $v^{-1}d^{\\prime }\\in P(b)$ .", "We are now ready to define the matrix system $(V_a,H_{ba})$ .", "Definition 6.11 For every $u\\in D$ and $a$ in $A$ let $V_{u,a}$ be the direct sum of the spaces $V_{c^{\\prime }}$ for all $c^{\\prime }$ such that ${u}^{-1}c^{\\prime }$ belongs to $P(a)$ , namely $V_{u,a}:=\\bigoplus \\big \\lbrace V_{c^{\\prime }}:\\;\\,\\, c^{\\prime }\\in A^{\\prime }\\text{ and } {u}^{-1}c^{\\prime }\\in P(a)\\big \\rbrace \\,,$ and set $V_a:=\\bigoplus _{u\\in D} V_{u,a}=\\bigoplus \\big \\lbrace V_{c^{\\prime }}:\\,\\,u\\in D,\\,c^{\\prime }\\in A^{\\prime }\\text{ and }u^{-1}c^{\\prime }\\in P(a)\\big \\rbrace \\,.$ In other words, we can think of the $V_a$ 's as consisting of blocks, corresponding to elements $u\\in D$ each of them containing a copy of $V_{c^{\\prime }}$ whenever $u^{-1}c^{\\prime }\\in P(a)$ .", "With this definition of the $V_a$ 's, we can now define a map $U:\\operatorname{Ind}_{\\Gamma ^{\\prime }}^\\Gamma {(\\mathcal {H}_s^\\infty (V_{a^{\\prime }},H_{b^{\\prime }a^{\\prime }}))}\\rightarrow \\big \\lbrace \\Gamma \\rightarrow \\bigoplus _{a\\in A}V_a\\big \\rbrace $ with the idea in mind that the target will have to be the space of multiplicative functions on some matrix system with inner product $(V_a,H_{ba},B_a)$ .", "Fix $a\\in A$ and let $u^{-1}c^{\\prime }\\in P(a)$ .", "Then for all $x\\in \\Gamma $ such that $|xa|=|x|+1$ and for $f\\in \\operatorname{Ind}_{\\Gamma ^{\\prime }}^\\Gamma {(\\mathcal {H}_s^\\infty (V_{a^{\\prime }},H_{b^{\\prime }a^{\\prime }}))}$ , we define $Uf(xa)$ to be the vector whose $(u,c^{\\prime })$ -component is given by $Uf(xa)_{{u},c^{\\prime }}:=\\Phi (f)(x{u}^{-1}c^{\\prime })$ or, equivalently, $Uf(xa)=\\bigoplus _{(u,c^{\\prime })}f(x{u}^{-1})(c^{\\prime })$ It is not difficult to convince oneself on how to construct the linear maps $H_{ba}$ so that the functions $Uf$ will be multiplicative: we give here an explanation, and one can find the formula in (REF ).", "Since the functions $Uf$ will have to be multiplicative, if $|xab|=|x|+2$ they will have to satisfy $f(xav^{-1}d^{\\prime })=(Uf)(xab)_{v,d^{\\prime }}=\\big (H_{ba}(Uf)(xa)\\big )_{v,d^{\\prime }}$ whenever $v^{-1}d^{\\prime }\\in P(b)$ for some $H_{ba}:V_a\\rightarrow V_b$ to be specified.", "Thinking of the \"block decomposition\" alluded to above, the linear maps $H_{ba}$ will also be block matrices that will perform three kinds of operations on a vector $w_a\\in V_a$ with coordinates $w_a=(w_{u,c})_{u^{-1}c^{\\prime }\\in P(a)}$ .", "If there exists $d^{\\prime }\\in A^{\\prime }$ such that for some $v\\in D$ , $a^{-1}vd^{\\prime }\\in P(a)$ and $v^{-1}d^{\\prime }\\in P(b)$ , (see Lemma REF (1)), then $H_{ba}$ will just move the $(va^{-1},d^{\\prime })$ -component of $w_a$ to the $(v^{\\prime },d^{\\prime })$ -component of $H_{ba}w_a$ .", "According to Lemma REF (1) this happens precisely when $va^{-1}\\in D$ .", "If on the other hand for $u,v\\in D$ , $u^{-1}c^{\\prime }\\in P(a)$ and $v^{-1}d^{\\prime }\\in P(b)$ , then $c^{\\prime }d^{\\prime }\\ne e$ (cf.", "Lemma REF (2)) and $H_{ba}|_{V_{u,c^{\\prime }}}:V_{u,c^{\\prime }}\\rightarrow V_{v,d^{\\prime }}$ will be nothing but $H_{d^{\\prime }c^{\\prime }}$ .", "In all other cases $H_{ba}$ will be set equal to zero .", "More precisely we define $(H_{ba}w_a)_{{v},d^{\\prime }}:={\\left\\lbrace \\begin{array}{ll}\\begin{aligned}&(w_a)_{v{a}^{-1},d^{\\prime }}&\\qquad \\text{if }v{a}^{-1}\\in D\\\\&H_{d^{\\prime }c^{\\prime }}(w_a)_{u,c^{\\prime }} &\\hphantom{XX}\\text{if } v{a}^{-1}\\notin D \\text{ and }a^{-1}v=u^{-1}c^{\\prime }\\\\&0&\\text{otherwise}\\,.\\end{aligned}\\end{array}\\right.", "}$ That this makes sense follows directly from Lemma REF as we explained above.", "The definition of a tuple of positive definite forms is now obvious, namely the $(u,c^{\\prime })$ -component of $B_{a}$ is given by the following $({B_a})_{{u},c^{\\prime }}:=B_{c^{\\prime }}\\qquad \\text{where ${u}^{-1}c^{\\prime }\\in P(a)$}$ Proposition 6.12 The tuple $(B_a)_{a\\in A}$ is compatible with the system $H_{ba}$ defined in (REF ).", "We must check that, for every $ w_a\\in V{a}$ one has ${B_a}(w_a,w_a)=\\sum _{b:\\;ab\\ne e}{B_b}(H_{ba}w_a,H_{ba} w_a)\\,.$ Remembering that, by definition of $V_a$ and $B_a$ ${B_a}(w_a, w_a)=\\sum _{u\\in F}\\;\\;\\sum _{{u}^{-1}c^{\\prime }\\in P(a)} B_{c^{\\prime }}\\big ((w_a)_{u,c^{\\prime }},(w_a)_{u,c^{\\prime }}\\big )\\,,$ we must prove that $\\begin{aligned}&\\sum _{u\\in F}\\sum _{{u}^{-1}c^{\\prime }\\in P(a)} B_{c^{\\prime }}\\big ((w_a)_{u,c^{\\prime }},(w_a)_{u,c^{\\prime }}\\big )=\\\\&\\sum _{b:\\;ab\\ne e}\\;\\;\\sum _{v\\in F}\\;\\;\\sum _{{v}^{-1}d^{\\prime }\\in P(b)} B_{d^{\\prime }}\\big ((H_{ba} w_a)_{v,d^{\\prime }},(H_{ba} w_a)_{v,d^{\\prime }}\\big )\\,.\\end{aligned}$ Fix $a$ in $A$ and define $D_a=\\lbrace u\\in D:\\;u= v{a}^{-1}\\;\\; \\text{for some $v\\in D$}\\rbrace \\,,$ so that ${D_a\\cdot a}=\\lbrace v\\in D:\\;v=ua\\;\\; \\text{for some $u\\in D_a$}\\rbrace $ is in bijective correspondence with $D_a$ .", "Split the sums on each side of (REF ) to obtain $\\begin{aligned}&\\sum _{u\\in D_a}\\;\\;\\sum _{{u}^{-1}c^{\\prime }\\in P(a)} B_{c^{\\prime }}\\big ((w_a)_{u,c^{\\prime }},(w_a)_{u,c^{\\prime }}\\big )\\\\&+\\sum _{u\\in D\\setminus D_a}\\;\\;\\sum _{{u}^{-1}c^{\\prime }\\in P(a)} B_{c^{\\prime }}\\big ((w_a)_{u,c^{\\prime }},(w_a)_{u,c^{\\prime }}\\big )\\\\&=\\sum _{v\\in {D_a\\cdot a}}\\;\\;\\sum _{b:\\;ab\\ne e}\\;\\;\\sum _{{v}^{-1}d^{\\prime }\\in P(b)} B_{d^{\\prime }}\\big ((H_{ba} w_a)_{v,d^{\\prime }},(H_{ba} w_a)_{v,d^{\\prime }}\\big )\\\\&+\\sum _{v\\in D\\setminus {D_a\\cdot a}}\\;\\;\\sum _{b:\\;ab\\ne e}\\;\\;\\sum _{{v}^{-1}d^{\\prime }\\in P(b)} B_{d^{\\prime }}\\big ((H_{ba} w_a)_{v,d^{\\prime }},(H_{ba} w_a)_{v,d^{\\prime }}\\big )\\,.\\end{aligned}$ We will show the equality $\\begin{aligned}&\\sum _{{u}^{-1}c^{\\prime }\\in P(a)} B_{c^{\\prime }}\\big ((w_a)_{u,c^{\\prime }},(w_a)_{u,c^{\\prime }}\\big )\\\\&=\\sum _{b:\\;ab\\ne e}\\;\\;\\sum _{{v}^{-1}d^{\\prime }\\in P(b)} B_{d^{\\prime }}\\big ((H_{ba} w_a)_{v,d^{\\prime }},(H_{ba} w_a)_{v,d^{\\prime }}\\big )\\\\\\end{aligned}$ in the two cases $u\\in D_a$ and $v=ua\\in D_a\\cdot a$ , $u\\notin D_a$ and $v=ua\\notin D_a\\cdot a$ .", "Then (REF ) will follow by summing (REF ) once over $D_a$ and once over $D\\setminus D_a$ and adding the resulting equations.", "(1) Let $u\\in D_a$ and $v\\in D_a\\cdot a$ .", "Then for a fixed $c^{\\prime }\\in A^{\\prime }$ with ${u}^{-1}c^{\\prime }\\in P(a)$ , Lemma REF (1) implies that $\\begin{aligned}&\\sum _{c^{\\prime }:\\;a{v}^{-1}c^{\\prime }\\in P(a)} B_{c^{\\prime }}\\big ((w_a)_{v{a}^{-1},c^{\\prime }},(w_a)_{v{a}^{-1},c^{\\prime }}\\big )\\\\&=\\sum _{b:\\;ab\\ne e}\\;\\;\\sum _{c^{\\prime }:\\;{v}^{-1}c^{\\prime }\\in P(b)} B_{c^{\\prime }}\\big ((w_a)_{v{a}^{-1},c^{\\prime }},(w_a)_{v{a}^{-1},c^{\\prime }}\\big )\\,,\\end{aligned}$ so that $\\begin{aligned}&\\sum _{c^{\\prime }:\\;{u}^{-1}c^{\\prime }\\in P(a)} B_{c^{\\prime }}\\big ((w_a)_{u,c^{\\prime }},(w_a)_{u,c^{\\prime }}\\big )\\\\&=\\sum _{c^{\\prime }:\\;a{v}^{-1}c^{\\prime }\\in P(a)} B_{c^{\\prime }}\\big ((w_a)_{v{a}^{-1},c^{\\prime }},(w_a)_{v{a}^{-1},c^{\\prime }}\\big )\\\\&=\\sum _{b:\\;ab\\ne e}\\;\\;\\sum _{c^{\\prime }:\\;{v}^{-1}c^{\\prime }\\in P(b)} B_{c^{\\prime }}\\big ((w_a)_{v{a}^{-1},c^{\\prime }},(w_a)_{v{a}^{-1},c^{\\prime }}\\big )\\\\&=\\sum _{b:\\;ab\\ne e}\\;\\;\\sum _{c^{\\prime }:\\;{v}^{-1}c^{\\prime }\\in P(b)} B_{c^{\\prime }}\\big ((H_{ba} w_a)_{v,c^{\\prime }},(H_{ba} w_a)_{v,c^{\\prime }}\\big )\\\\&=\\sum _{b:\\;ab\\ne e}\\;\\;\\sum _{d^{\\prime }:\\;{v}^{-1}d^{\\prime }\\in P(b)} B_{d^{\\prime }}\\big ((H_{ba} w_a)_{v,d^{\\prime }},(H_{ba} w_a)_{v,d^{\\prime }}\\big )\\,,\\end{aligned}$ where the next to the last equation comes from the definition of the $H_a$ and the last just from renaming the variable.", "(2) Fix now any $v$ in $D\\setminus D_a\\cdot a$ and write $a{v}^{-1}={u}^{-1}c^{\\prime }$ (Lemma REF (2a)).", "Choose any $d^{\\prime }$ with $c^{\\prime }d^{\\prime }\\ne e$ and let $b\\in A$ with $ab\\ne e$ be the unique $b$ such that ${v}^{-1}d^{\\prime }\\in P(b)$ (Lemma REF (2b)) By definition of $B_a$ $(H_{ba}w_a)_{v,d^{\\prime }}= H_{d^{\\prime }c^{\\prime }}(w_a)_{u,c^{\\prime }}\\,.$ To every $b$ corresponds a subset $A^{\\prime }_b$ of $A^{\\prime }$ consisting of all $d^{\\prime }$ such that ${v}^{-1}d^{\\prime }$ belongs to $P(b)$ and we observed before that $\\bigcup _b A^{\\prime }_b=A^{\\prime }\\setminus {(c^{\\prime })}^{-1}$ .", "Hence $\\begin{aligned}&\\sum _{b:\\;ab\\ne e}\\;\\;\\sum _{{v}^{-1}d^{\\prime }\\in P(b)} B_{d^{\\prime }}\\big ((H_{ba} w_a)_{v,d^{\\prime }},(H_{ba} w_a)_{v,d^{\\prime }}\\big )=\\\\&\\sum _{b:\\;ab\\ne e}\\;\\;\\sum _{d^{\\prime }\\in A^{\\prime }_b} B_{d^{\\prime }}(H_{d^{\\prime }c^{\\prime }}(w_a)_{u,c^{\\prime }},H_{d^{\\prime }c^{\\prime }}(w_a)_{u,c^{\\prime }})=\\\\&\\sum _{d^{\\prime }\\in A^{\\prime }\\setminus (c^{\\prime })^{-1}}B_{d^{\\prime }}(H_{d^{\\prime }c^{\\prime }}(w_a)_{u,c^{\\prime }},H_{d^{\\prime }c^{\\prime }}(w_a)_{u,c^{\\prime }})=B_{c^{\\prime }}((w_a)_{u,c^{\\prime }},(w_a)_{u,c^{\\prime }})\\,,\\end{aligned}$ where the last equality is nothing but the compatibility of the $(B_{a})$ .", "In particular to every $v$ in $D\\setminus {D_a\\cdot a}$ corresponds a unique $u$ in $D\\setminus D_a$ and a unique $c^{\\prime }\\in A^{\\prime }$ such that ${u}^{-1}c^{\\prime }\\in P(a)$ and $\\begin{aligned}\\sum _{u\\in D\\setminus D_a}\\sum _{b:\\;ab\\ne e}\\sum _{{v}^{-1}d^{\\prime }\\in P(b)} B_{d^{\\prime }}\\big ((H_{ba} w_a)_{v,d^{\\prime }},(H_{ba} w_a)_{v,d^{\\prime }}\\big )\\\\=\\sum _{v\\in D\\setminus D_a\\cdot a}B_{c^{\\prime }}\\big ((w_a)_{u,c^{\\prime }},(w_a)_{u,c^{\\prime }}\\big )\\,.\\end{aligned}$ The upshot of the above discussion is that we have shown that the map $U$ takes values in the space of multiplicative functions.", "We still need to show that $U$ is an unitary operator and hence it extends to a unitary equivalence between $\\operatorname{Ind}_{\\Gamma ^{\\prime }}^\\Gamma {(\\mathcal {H}(V_{a^{\\prime }},H_{b^{\\prime }a^{\\prime }},B_{a^{\\prime }}))}$ and $\\mathcal {H}(V_a,H_{ba},B_a)$ .", "The following theorem will complete the proof.", "Theorem 6.13 Let $V_a$ , $H_{ba}$ and $B_a$ be as in (REF ), (REF ) and (REF ) and let $U:\\operatorname{Ind}_{\\Gamma ^{\\prime }}^\\Gamma {(\\mathcal {H}^\\infty (V_{a^{\\prime }},H_{b^{\\prime }a^{\\prime }},B_{a^{\\prime }}))}\\rightarrow \\mathcal {H}^\\infty (V_a,H_{ba},B_a)$ be as in (REF ).", "Then $U$ is an unitary operator and hence it extends to a unitary equivalence $U:\\operatorname{Ind}_{\\Gamma ^{\\prime }}^\\Gamma {(\\mathcal {H}(V_{a^{\\prime }},H_{b^{\\prime }a^{\\prime }},B_{a^{\\prime }}))}\\rightarrow \\mathcal {H}(V_a,H_{ba},B_a)\\,.$ Let us simply write as before $\\mathcal {H}^\\infty _s$ for $\\mathcal {H}^\\infty (V_{a^{\\prime }},H_{b^{\\prime }a^{\\prime }}, B_{a^{\\prime }})$ and $\\mathcal {H}^\\infty $ for $\\mathcal {H}^\\infty (V_{a},H_{ba}, B_{a})$ .", "For every $f\\in \\operatorname{Ind}_{\\Gamma ^{\\prime }}^\\Gamma {(\\mathcal {H}_s^\\infty )}$ we have by definition of the induced norm that $\\Vert {f}\\Vert ^2_{\\operatorname{Ind}_{\\Gamma ^{\\prime }}^\\Gamma {(\\mathcal {H}_s^\\infty )}} =\\sum _{u\\in D}\\Vert {f(u)}\\Vert _{\\mathcal {H}_s^\\infty }^2\\,,$ and, since the above sum is orthogonal, we may assume that $f$ is supported on $z\\cdot \\Gamma ^{\\prime }$ for some $z\\in D$ .", "For such an $f$ it will be hence enough to show that $\\Vert {Uf}\\Vert ^2_{\\mathcal {H}^\\infty }=\\Vert {f(z)}\\Vert ^2_{\\mathcal {H}_s^\\infty }\\,.$ Using the definition of the norm in (REF ) as well as the definitons of $U$ in (REF ) and of $B_a$ in (REF ) we obtain that for $N$ large enough $\\begin{aligned}\\Vert {Uf}\\Vert ^2_{\\mathcal {H}^\\infty }&=\\sum _{a\\in A}\\sum _{\\begin{array}{c}|x|=N\\\\|xa|=|x|+1\\end{array}} B_a\\big (Uf(xa),Uf(xa)\\big )\\\\&=\\sum _{a\\in A}\\sum _{\\begin{array}{c}|x|=N\\\\|xa|=|x|+1\\end{array}} \\sum _{{u}^{-1}c^{\\prime }\\in P(a)} B_{c^{\\prime }}\\big (f(x{u}^{-1})(c^{\\prime }),f(x{u}^{-1})(c^{\\prime })\\big )\\,.\\end{aligned}$ Since $f(z)\\in \\mathcal {H}_s^\\infty $ , there exists $M>0$ such that $f(z)$ is multiplicative outside the ball $B^{\\prime }(e,M)$ in $\\mathcal {T}^{\\prime }$ of radius $M$ .", "To complete the proof it will be hence enough to show the following Lemma 6.14 There exists a finite complete subtree $\\mathcal {S}^{\\prime }\\subset \\mathcal {T}^{\\prime }$ containing $B^{\\prime }(e,M)$ whose terminal elements are $T(\\mathcal {S}^{\\prime })=\\lbrace \\gamma ^{\\prime }={z}^{-1}xy\\in \\Gamma ^{\\prime }:\\,|x|=N,\\,|xa|=N+1,\\, y\\in P(a)\\rbrace $ Observe that since, according to the above lemma, $\\gamma ^{\\prime }\\in T(\\mathcal {S}^{\\prime })$ has the form $\\gamma ^{\\prime }={z}^{-1}x{u}^{-1}c^{\\prime }$ with $u\\in D$ and $c^{\\prime }\\in A^{\\prime }$ , the invariance property of $f$ translates into the equality $f(z)(\\gamma ^{\\prime })=f(x{u}^{-1})(c^{\\prime })\\,.$ From this in fact, using Lemma REF and denoting $\\overline{\\gamma ^{\\prime }}$ to be as before the reduced word obtained by deleting the last letter (in $\\Gamma ^{\\prime }$ ) of $\\gamma ^{\\prime }$ , we deduce that $\\begin{aligned}\\Vert {f(z)}\\Vert ^2_{\\mathcal {H}_s^\\infty }&=\\sum _{\\begin{array}{c}\\gamma ^{\\prime }\\in T(\\mathcal {S}^{\\prime })\\\\ \\gamma ^{\\prime }=\\overline{\\gamma ^{\\prime }}c^{\\prime }\\end{array}}B_{c^{\\prime }}\\big (f(z)(\\gamma ^{\\prime }),f(z)(\\gamma ^{\\prime })\\big )\\\\&=\\sum _{a\\in A}\\sum _{\\begin{array}{c}|x|=N\\\\|xa|=|x|+1\\end{array}} \\sum _{{u}^{-1}c^{\\prime }\\in P(a)} B_{c^{\\prime }}\\big (f(x{u}^{-1})(c^{\\prime }),f(x{u}^{-1})(c^{\\prime })\\big )\\,,\\end{aligned}$ thus concluding the proof.", "We need now to show Lemma REF .", "We start recording the following obvious fact, which follows immediately from the observation that left translates of $D$ are subtrees (hence convex) and that cones are disjoint and convex.", "Lemma 6.15 Let $\\Gamma ^{\\prime }\\le \\Gamma $ be a subgroup of a free group with associated trees $\\mathcal {T}^{\\prime }\\subset \\mathcal {T}$ and let $D$ a fundamental subtree in $\\mathcal {T}$ .", "Then for any $w\\in \\Gamma $ we can write $\\mathcal {T}=w\\,B(e,N+1)\\sqcup \\bigsqcup _{\\begin{array}{c}|x|=N\\\\ |xa|=N+1\\end{array}}w\\,C(xa)$ and $\\begin{aligned}\\mathcal {T}^{\\prime }=\\big \\lbrace \\gamma ^{\\prime }\\in \\Gamma ^{\\prime }:\\,\\gamma ^{\\prime }D\\,\\cap \\, w\\,B(e,N+1)\\ne \\emptyset \\big \\rbrace \\sqcup &\\\\\\sqcup \\bigsqcup _{\\begin{array}{c}|x|=N\\\\ |xa|=N+1\\end{array}}\\big \\lbrace \\gamma ^{\\prime }\\in \\Gamma ^{\\prime }:\\,\\gamma ^{\\prime }D\\subseteq w\\, C(xa)\\big \\rbrace &\\,.\\end{aligned}$ Clearly there are finitely many $\\gamma ^{\\prime }\\in \\Gamma ^{\\prime }$ such that $\\gamma ^{\\prime }D\\,\\cap \\,w\\ B(e,N+1)\\ne \\emptyset $ , but infinitely many $\\gamma ^{\\prime }\\in \\Gamma ^{\\prime }$ such that $\\gamma ^{\\prime }D\\subseteq w\\, C(xa)$ for some fixed $x$ and $a$ .", "The right finiteness condition is imposed in the following lemma.", "Lemma 6.16 Fix any $z\\in \\Gamma $ and choose $N>|z|$ large enough so that $\\gamma ^{\\prime } D\\cap {z}^{-1}B(e,N+1)\\ne \\emptyset $ for all $|\\gamma ^{\\prime }|\\le M$ .", "Define $\\begin{aligned}S^{\\prime }_0 &:= \\lbrace \\gamma ^{\\prime }\\in \\Gamma ^{\\prime }:\\;\\gamma ^{\\prime }D\\cap {z}^{-1}B(e,N+1)\\ne \\emptyset \\rbrace \\,,\\\\S^{\\prime }_t &:=\\lbrace \\gamma ^{\\prime }\\in \\Gamma ^{\\prime }:\\;\\gamma ^{\\prime }D\\subseteq {z}^{-1} C(xa)\\text{ for some }x,\\,a\\text{ with } |xa|=N+1\\\\&\\hphantom{hhhhhhhhhhh} \\text{ and } \\overline{\\gamma ^{\\prime }}D\\nsubseteq {z}^{-1}C(xa)\\rbrace \\\\\\mathcal {S}^{\\prime }&:=S^{\\prime }_0\\sqcup S^{\\prime }_t\\,.\\end{aligned}$ Then $\\mathcal {S}^{\\prime }$ is a finite complete subtree (containing $B^{\\prime }(e,M)$ ), whose terminal vertices are $T(\\mathcal {S}^{\\prime })=S^{\\prime }_t$ and can be characterized as follows $T(\\mathcal {S}^{\\prime })=\\lbrace \\gamma ^{\\prime }={z}^{-1}xy\\in \\Gamma ^{\\prime }:\\,|x|=N,\\,|xa|=N+1,\\, y\\in P(a)\\rbrace \\,.$ We shall prove a sequence of simple claims.", "Notice that since $|z|< N$ , then for all $x\\in \\Gamma $ and $a\\in A$ such that $|xa|=|x|+1$ , $xa$ does not belong to the geodesic between $e$ and $z$ and hence, according to Lemma REF , ${z}^{-1}C(xa)=C({z}^{-1}xa)$ .", "Claim 1.", "If $\\gamma ^{\\prime }\\in S^{\\prime }_0$ , then $\\overline{\\gamma ^{\\prime }}\\in S^{\\prime }_0$ and hence the set $S^{\\prime }_0$ is a subtree.", "Proof: Let $v\\in \\gamma ^{\\prime }D\\cap {z}^{-1}\\,B(e,N+1)$ be a vertex and let $x_0=v,x_1,\\dots ,x_r=e$ be a sequence of vertices of the unique geodesic in $\\mathcal {T}$ from $x_0=v$ to $x_r=e$ .", "By convexity of ${z}^{-1}B(e,N+1)$ , $x_j\\in {z}^{-1}B(e,N+1)$ for all $0\\le j\\le r$ .", "Since $\\gamma ^{\\prime }D$ is a subtree, the set $\\lbrace i:\\;0\\le i\\le r,\\,x_i\\in \\gamma ^{\\prime }D\\rbrace $ is an interval, say $[0,i_0]\\cap \\mathbf {Z}$ .", "Let $\\gamma ^{\\prime \\prime }\\in \\Gamma ^{\\prime }$ be (the unique element) such that $x_{i_0+1}\\in \\gamma ^{\\prime \\prime }D$ .", "Then by construction $d(\\gamma ^{\\prime }D,\\gamma ^{\\prime \\prime }D)=1$ so that $\\gamma ^{\\prime \\prime }=\\overline{\\gamma ^{\\prime }}$ and $\\overline{\\gamma ^{\\prime }}\\,D\\cap {z}^{-1}B(e,N+1)\\ne \\emptyset $ , thus showing that $\\overline{\\gamma ^{\\prime }}\\in S^{\\prime }_0$ .", "dist$d(\\gamma ^{\\prime }D,\\overline{\\gamma ^{\\prime }}D)=1$ g'd$\\gamma ^{\\prime }D$ g'bd$\\overline{\\gamma ^{\\prime }}D$ x$x$ xa$xa$ Figure: NO_CAPTION Figure 2: $\\gamma ^{\\prime }\\in S^{\\prime }_t$ and $\\overline{\\gamma ^{\\prime }}D\\in S^{\\prime }_0$ .", "Claim 2.", "If $\\gamma ^{\\prime }\\in S^{\\prime }_t$ , then $\\overline{\\gamma ^{\\prime }}\\in S^{\\prime }_0$ and hence the set $\\mathcal {S}^{\\prime }$ is a subtree and $S^{\\prime }_t\\subseteq T(\\mathcal {S}^{\\prime })$ .", "Proof: Let $\\gamma ^{\\prime }\\in S^{\\prime }_t$ and let $\\gamma ^{\\prime }D\\subset {z}^{-1}C(xa)$ with $\\overline{\\gamma ^{\\prime }}D\\notin {z}^{-1}C(xa)$ .", "Lemma REF implies then immediately that $\\overline{\\gamma ^{\\prime }}D\\cap {z}^{-1}\\,B(e,N+1)\\ne \\emptyset $ and hence $\\overline{\\gamma ^{\\prime }}\\in S^{\\prime }_0$ .", "Claim 3.", "The tree $\\mathcal {S}^{\\prime }$ is complete and $S^{\\prime }_t= T(\\mathcal {S}^{\\prime })$ .", "Proof: Let $\\gamma ^{\\prime }\\in S^{\\prime }_0$ and let $a^{\\prime }\\in A\"$ so that $|\\gamma ^{\\prime }a^{\\prime }|^{\\prime }=|\\gamma ^{\\prime }|^{\\prime }+1$ .", "If $\\gamma ^{\\prime }a^{\\prime }\\notin S^{\\prime }_0$ , then, by Lemma REF , $\\gamma ^{\\prime }a^{\\prime }D\\in {z}^{-1}C(xa)$ for some $|x|=N$ and $|xa|=N+1$ .", "On the other hand $\\overline{\\gamma ^{\\prime }a^{\\prime }}D=\\gamma ^{\\prime }D\\notin {z}^{-1}C(xa)$ and hence $\\gamma ^{\\prime }\\in S^{\\prime }_t$ .", "Claim 4.", "$T(\\mathcal {S}^{\\prime })=\\lbrace \\gamma ^{\\prime }={z}^{-1}xy\\in \\Gamma ^{\\prime }:\\,|x|=N,\\,|xa|=N+1,\\, y\\in P(a)\\rbrace $ .", "Proof: By definition if $\\gamma ^{\\prime }\\in S^{\\prime }_t$ , then $\\gamma ^{\\prime }D\\subseteq {z}^{-1} C(xa)$ and hence $\\gamma ^{\\prime }={z}^{-1} xay$ , for some $y\\in \\Gamma $ .", "However, since we have also that $\\overline{\\gamma ^{\\prime }}D\\nsubseteq {z}^{-1} C(xa)$ , then ${z}^{-1}x\\in \\overline{\\gamma ^{\\prime }}D$ .", "Thus there exists $u\\in D$ such that $\\overline{\\gamma ^{\\prime }}={z}^{-1}x{u}^{-1}$ .", "The assertion now follows by completing $\\gamma ^{\\prime }$ with its last letter $c^{\\prime }\\in A^{\\prime }$ in the reduced expression." ] ]

[ [ "The first results from the Herschel-HIFI mission" ], [ "Abstract This paper contains a summary of the results from the first years of observations with the HIFI instrument onboard ESA's Herschel space observatory.", "The paper starts by outlining the goals and possibilities of far-infrared and submillimeter astronomy, the limitations of the Earth's atmosphere, and the scientific scope of the Herschel-HIFI mission.", "The presentation of science results from the mission follows the life cycle of gas in galaxies as grouped into five themes: Structure of the interstellar medium, First steps in interstellar chemistry, Formation of stars and planets, Solar system results and Evolved stellar envelopes.", "The HIFI observations paint a picture where the interstellar medium in galaxies has a mixed, rather than a layered structure; the same conclusion may hold for protoplanetary disks.", "In addition, the HIFI data show that exchange of matter between comets and asteroids with planets and moons plays a large role.", "The paper concludes with an outlook to future instrumentation in the far-infrared and submillimeter wavelength ranges." ], [ "Introduction", "The Herschel space observatory is a facility to study celestial objects in far-infrared light, which is the third most common kind of radiation in the UniverseSee http://astro.ucla.edu/$\\sim $ wright/CIBR/.", "As measured by energy density, the most common type of radiation are photons from the Cosmic Microwave Background, which were emitted in the early Universe when protons and electrons first (re)combined to form hydrogen atoms; studying this type of light is studying our past.", "The second most common type of radiation ($\\approx $ 6% of the CMB intensity) is optical and near-infrared light, which is emitted by the current generation of stars; studying this light is studying our present.", "At $\\approx $ 3% of the CMB intensity, far-infrared light is about half as common as optical light, and at least as important: this light is emitted by cold clouds of gas and dust where new stars and planets are forming (Figure REF ).", "Studying far-infrared light from the local Universe is therefore studying our future.", "The far-infrared and submillimeter part of the spectrum is well suited to study the origin of galaxies, stars and planets, because gas and dust clouds with temperatures of 30–100 K emit the bulk of their radiation in this range.", "In particular, continuum observations readily probe the mass and the temperature of the clouds.", "Here, the advantage over mid-infrared or shorter-wavelength observations is that the radiation is mostly optically thin, so that it traces the entire volume of the clouds rather than just their surfaces.", "Furthermore, the large number of atomic fine structure and molecular rotational transitions spanning a wide range of excitation energies (from $\\sim $ 1 to $\\sim $ 1000 K) and radiative decay rates provides a powerful tool to measure the densities, temperatures and masses of interstellar clouds (Figure REF ).", "In addition, observations at high spectral resolution enable to study the kinematics of clouds in detail, polarization measurements give information about magnetic fields, and the chemical composition of clouds contains signatures of otherwise hidden (energetic) radiation fields.", "The goal of far-infrared and submillimeter astronomy is therefore a basic understanding of the physics and chemistry of interstellar clouds, star-forming regions, protoplanetary disks, the envelopes of evolved stars, planetary atmospheres, active galactic nuclei, and starburst galaxies.", "Figure: The HIFI spectrum of the Orion-KL region shows a multitude of lines from which many physical and chemical properties of the gas can be inferred.", "Red numbers denote isolated methanol transitions, blue text denotes transitions from other molecules and methanol transitions which are blended, green text and ÒBÓ denotes methanol lines which are blended with different parity states, and ÒUÓ denotes unidentified lines.", "The chemical compounds seen in this spectrum were known from ground-based observations, but these lines are seen with HIFI for the first time.", "From Wang et al (2011).Ground-based telescopes can only partly reach the above goals, because the Earth's atmosphere is opaque for a large fraction of the submillimeter range and all of the far-infrared.", "Due to pressure broadening, absorption by atmospheric gases not only blocks our view of abundant atmospheric constituents such as H$_2$ O and O$_2$ , but also of many other species and of entire frequency ranges, including the entire far-infrared at frequencies above $\\sim $ 1000 GHz, including the fine structure lines of C$^+$ and O at 1901 and 4745 GHz which require airborne observatories such as SOFIA.", "Below $\\sim $ 1000 GHz, the transmission at some frequencies is so low that ground-based observations are limited to bright sources and small regions on the sky; an important example are the fine structure lines of C at 492 and 809 GHz, which can only under favourable conditions be observed from high-altitude observatories such as Mauna Kea (JCMT, CSO, SMA) and Chajnantor (APEX, ALMA).", "Following successful missions such as ISO and KAO, the Heterodyne Instrument for the Far Infrared (HIFI) onboard ESA's Herschel space observatory has been designed to give astronomers near-complete coverage of the submillimeter waveband.", "The orbit of Herschel around the L2 point and the passively cooled mirror provide superior sensitivity and stability compared with ground-based observatories.", "The spectral coverage of HIFI ranges from 480 to 1250 GHz in five bands and from 1410 to 1910 GHz in two additional bands, or in other words from the fine structure lines of C to that of C$^+$ with a gap around the p-H$_2$ D$^+$ ground state line.", "By mixing the sky signal with a locally generated frequency standard, HIFI achieves a spectral resolution of $\\approx $ 0.1 MHz or a resolving power of $\\sim $ 10$^7$ .", "The angular resolution is set by the diffraction limit of Herschel's 3.5-m mirror and ranges from 39$^{\\prime \\prime }$ at the lowest to 13$^{\\prime \\prime }$ at the highest frequencies.", "The instrument measures a single beam on the sky, but maps can be made by scanning the telescope and combining the data afterward.", "See [68] for a description of the Herschel telescope, [16] for a description of the HIFI instrument, and [71] for its in-orbit calibration and performance.", "This paper reviews the results from the first half of the Herschel-HIFI mission, grouped into five themes: Structure of the interstellar medium (§ ), First steps in interstellar chemistry (§ ), Formation of stars and planets (§ ), Solar system results (§ ), and Evolved stellar envelopes (§ ).", "Results from Herschel's other two instruments, PACS and SPIRE, will only be mentioned to provide context for HIFI results; space limitations prevent a more complete coverage of PACS and SPIRE results.", "Also, the science cases for each of the selected HIFI results will necessarily be brief.", "For a general introduction in the field of interstellar medium physics, see the book by [21].", "The formation of stars is discussed extensively in the book by [74].", "An up-to-date comprehensive review of interstellar chemistry does not exist currently, but the proceedings of IAU Symposium 280 provide a good starting point.", "The envelopes of evolved stars are reviewed by [34], while the book by [18] covers Solar system science." ], [ "Structure of the interstellar medium", "The space between the stars is far from empty: about 10% of the baryonic mass in galaxies is contained in the interstellar medium (ISM).", "This medium consists of several phases: hot and warm ionized gas, warm and cold neutral gas, and molecular clouds, which are in approximate pressure equilibrium.", "While the warm diffuse ionized and atomic phases occupy most of the ISM volume, the cold dense molecular phase contain most of the mass.", "In this latter phase, pressure equilibrium breaks down and self-gravity becomes important, which leads to gravitational collapse and the formation of new generations of stars and planets.", "The cold dense ISM of galaxies is a key science area for Herschel, because many questions about its nature can only be addressed with far-infrared observations which are impossible from the ground.", "In particular, HIFI observations are instrumental to study the physical conditions, the geometrical structure, and the chemical composition of the ISM, as detailed in the following subsections." ], [ "A new population of interstellar clouds", "A key objective of the HIFI mission are observations of the fine-structure line of C$^+$ at 158 $\\mu $ m, which is the primary coolant of diffuse interstellar gas clouds.", "Measurements with the COBE, BICE and KAO observatories have revealed the large-scale distribution of C$^+$ emission, but HIFI offers superior angular and spectral resolution which is necessary to disentangle close neighbours on the sky and along the line of sight.", "Figure REF shows the most surprising result of the C$^+$ observations with HIFI: that significant C$^+$ emission is observed from interstellar clouds where most hydrogen is in the form of H$_2$ but carbon is not locked up in CO.", "This regime of physical conditions ($T \\approx 100$  K, $n_H \\approx 300$  cm$^{-3}$ , $A_V = 0.1 - 1.3$  mag) is referred to as \"warm dark gas\" [43].", "The amount of gas under these conditions is much larger than previously assumed from $\\gamma $ -ray data (e.g., [33]): as much as $\\approx $ 25% of the total H$_2$ in these `warm dark' clouds may be in H$_2$ layers which are not traced by CO [80].", "This fraction may be much higher in dwarf galaxies, where $L$ [CII]/$L$ (CO) (${_>\\atop {^\\sim }}$ 10$^4$ ) is at least an order of magnitude greater than in the most metal-rich starburst galaxies [47].", "The power of HIFI for understanding extragalactic C$^+$ emission is nicely illustrated by [56], who present a study of the gas around the H$^+$ region BCLMP 302 in the nearby spiral M33.", "The widths of the [CII] line profiles are found to be intermediate between those of the CO and HI 21cm lines, and the spatial correlation of C$^+$ with both CO and H is found to be rather poor.", "The authors estimate that about $\\approx $ 25% of the C$^+$ emission has an origin in ionized gas." ], [ "Thermal balance of dense interstellar clouds", "The cooling of dense molecular clouds is a long-standing question of astrophysics.", "Models of pure gas-phase chemistry predict large abundances of O$_2$ and H$_2$ O, which would make these molecules the major carriers of oxygen and major coolants of gas at densities ${_>\\atop {^\\sim }}$ 10$^4$  cm$^{-3}$ [30], [5].", "Thermal emission from these species can only be observed from space, and the first tests of those predictions were made by the SWAS and Odin missions around the year 2000.", "On the pc-sized scales of interstellar clouds, the measured H$_2$ O abundance was found to be orders of magnitude below the gas-phase predictions, probably due to depletion of oxygen on grain surfaces [50].", "The O$_2$ molecule was only detected toward one position [44]; this result and upper limits toward many other sources also indicate a low abundance.", "The HIFI instrument is much better suited to search for interstellar O$_2$ than SWAS and Odin for three reasons.", "Its lower system temperature implies a much better sensitivity, its larger frequency coverage enables observation of three transitions instead of just one, and its 3–10 times smaller beam size is better coupled to the expected sizes of the emitting regions.", "The result is the first clear detection of interstellar O$_2$ toward Orion [31] and the confirmation of the Odin detection toward $\\rho $  Oph [46].", "Upper limits have been obtained toward many other sources, which indicates that the formation of O$_2$ is very sensitive to local conditions.", "Clearly, the role of O$_2$ as oxygen carrier and gas coolant is very limited.", "Like for O$_2$ , HIFI is well suited to characterize the distribution and excitation of interstellar H$_2$ O, since many transitions (including the isotopologs H$_2^{18}$ O,H$_2^{17}$ O, and HDO) lie within its frequency range.", "Many Herschel programs include observations of H$_2$ O, as a probe of physical conditions, kinematics, or chemistry.", "The basic result of these studies is that while H$_2$ O may locally reach high abundances, particularly in shocks (§ REF ), the bulk of interstellar gas contains very little H$_2$ O.", "The typical abundance of H$_2$ O in non-shocked gas is $\\sim $ 10$^{-9}$ , which is limited by freeze-out on grain surfaces in dense clouds, and by photodissociation at low densities (e.g., [76]).", "This result implies that in the general interstellar medium, H$_2$ O is not a major oxygen carrier, nor a major coolant, confirming earlier results from SWAS [6]." ], [ "The transition between diffuse and dense interstellar clouds", "In cold dense interstellar gas clouds, the bulk of the hydrogen is in the form of H$_2$ , which is basically unobservable at low temperatures.", "The common proxy for H$_2$ is the CO molecule, which is abundant and chemically stable, but which is prone to photodissociation at $N$ (H$_2$ )${_<\\atop {^\\sim }}$ 1.5$\\times $ 10$^{21}$  cm$^{-2}$ [81].", "A good alternative for this regime is HF, which is also chemically very stable.", "The reaction of F with H$_2$ leading to HF is exothermic, so that HF is the main carrier of gas-phase fluorine, especially at low (column) densities.", "Observations with ISO have confirmed these predictions, although they only probe excited states of HF [62].", "The HIFI instrument gives our first access to the rotational ground state of HF, and observations of widespread absorption in the $J$ =1–0 line indicate an HF abundance of (1–2)$\\times $ 10$^{-8}$ in diffuse clouds, close to the interstellar fluorine abundance [73], [54].", "Towards dense clouds, the inferred abundance is $\\sim $ 100 times lower [67], suggesting that depletion on grain surfaces or excitation effects play a role.", "The large dipole moment of HF and the high frequency of its 1–0 line imply a rapid radiative decay rate, which explains why the line usually appears in absorption.", "Only in very dense environments such as the inner envelopes of late-type stars, the line appears in emisison [1].", "The only detection of HF emission from the Galactic interstellar medium so far is toward the Orion Bar [77], which is a surprise because the H$_2$ density in this region is not high enough to excite the line.", "Instead, collisional excitation by electrons appears to dominate, whereas non-thermal excitation mechanisms such as infrared pumping or chemical pumping appear unlikely [75].", "Emission lines of HF thus appear to trace regions with a high electron density, caused by strong ultraviolet irradiation of dense molecular gas.", "The high electron density in the Orion Bar may also apply to the active nucleus Mrk 231 where the SPIRE spectrum shows HF emission [78].", "Other active galactic nuclei such as the Cloverleaf quasar[55] show HF in absorption, suggesting that these nuclei have low electron densities, while yet others such as Arp 220 [70] show P Cygni profiles, suggesting the presence of HF in a wind with a high electron density." ], [ "Mixing of atomic and molecular phases", "Models of the structure of interstellar gas clouds predict that at low densities, hydrogen is primarily in atomic form, while at higher densities, the molecular form predominates.", "It is therefore common practice to divide such clouds in two classes: atomic clouds and molecular clouds (e.g., [72]).", "Observations of ionized water species with HIFI have however changed this picture.", "The formation of H$_2$ O in cold gas starts with O$^+$ , reacting with H$_2$ to first form OH$^+$ , then H$_2$ O$^+$ , and then H$_3$ O$^+$ , which recombines with an electron to form H$_2$ O.", "These reactions have no barriers and proceed fast, so that no significant amounts of the intermediate products OH$^+$ and H$_2$ O$^+$ are expected if all hydrogen is in the form of H$_2$ .", "Observations with HIFI have revealed large column densities of both OH$^+$ and H$_2$ O$^+$ , which is inconsistent with the above predictions [64], [28].", "Before the Herschel mission, H$_2$ O$^+$ was only known in comets [39] and only a tentative detection of OH$^+$ with ISO was reported toward by [32].", "The implication of widespread interstellar OH$^+$ and H$_2$ O$^+$ is that `mixed' clouds appear to be common, where hydrogen is partly H and partly H$_2$ [60].", "The ratio of H$_2$ O$^+$ to H$_2$ O is observed to vary, probably due to variations in the density of the clouds [87].", "HIFI has also observed ionized water in nearby starburst galaxies such as M82, NGC 253, and NGC 4945 (Figure REF ).", "The observed column density of H$_2$ O$^+$ in M82 is almost as high as that of H$_2$ O [84], probably because H$_2$ O suffers from rapid photodissociation by ultraviolet radiation from the young stellar population, while H$_2$ O$^+$ is much more robust against ultraviolet light.", "Even more extreme conditions occur in the active galactic nucleus of Mrk 231, where lines of OH$^+$ and H$_2$ O$^+$ appear in emission [78].", "This unusual phenomenon may be the result of rapid formation via X-rays or cosmic rays, along with strong excitation by electron collisions.", "Figure: Spectrum of H 2 _2O and H 2 _2O + ^+ lines toward the active nucleus of M82.", "The high H 2 _2O + ^+/H 2 _2O ratio indicates fast photodissociation of H 2 _2O by the strong UV radiation field from the starburst.", "From Weiß et al (2010)." ], [ "First steps in interstellar chemistry", "The chemical composition of interstellar clouds and star-forming regions gives valuable clues to physical processes which are not directly visible.", "Spectral surveys at (sub)millimeter wavelengths from the ground have shown some of this potential but are limited to atmospheric windows.", "The HIFI instrument gives us the first full view of the molecular line spectra in the 500–2000 GHz range.", "Besides the major oxygen carriers H$_2$ O and O$_2$ discussed above, the HIFI range includes several key species to improve our understanding of the physics of interstellar clouds.", "This section reviews the areas where HIFI has already contributed to this understanding." ], [ "Turbulent chemistry", "The large abundance of interstellar CH$^+$ has been a puzzle for decades since its discovery in 1941.", "Steady-state models where ultraviolet irradiation from stars drives the chemistry in the clouds do not nearly produce the observed CH$^+$ abundances.", "The ingredients C$^+$ and H$_2$ are readily available in diffuse clouds, but their direct reaction is highly endothermic so that at $T \\sim 100$  K, the production of CH$^+$ must proceed via a slow radiative association reaction.", "Non-thermal formation mechanisms have been proposed, using turbulence as an energy source to overcome the barrier of the reaction of C$^+$ with H$_2$ [29].", "These models could only be tested indirectly so far because from the ground, only optical absorption lines can be observed at limited velocity resolution.", "The rotational transitions of CH$^+$ are inaccessible from the ground, although the $^{13}$ CH$^+$ $J$ =1–0 line has been detected from Mauna Kea [26].", "With HIFI, both the 1–0 and the 2–1 transitions of CH$^+$ have been observed toward a number of sources [25], [9].", "The strong absorption lines of CH$^+$ and $^{13}$ CH$^+$ confirm the earlier abundance estimates of CH$^+$ /H$_2$ = $\\sim $ 3$\\times $ 10$^{-8}$ from the optical spectra.", "Models of so-called turbulent dissipation regions (TDRs) where the dissipation of turbulence through shocks acts as energy source, reproduce these abundances and also the observed CH$^+$ /HCO$^+$ ratios toward interstellar clouds [24].", "In the case of high-mass protostellar envelopes, the lines of CH$^+$ appear in emission, which suggests a different formation mechanism.", "The models presented by [9] explain the observed abundance and excitation of CH and CH$^+$ toward the source AFGL 2591 with a scenario of irradiated outflow walls, where a cavity etched out by the outflow allows protostellar far-ultraviolet photons to irradiate and heat the envelope to large distances driving the chemical reactions that produce these molecules." ], [ "Chlorine chemistry", "The dominant type of chemical reaction in the gas phase at low temperatures (${_<\\atop {^\\sim }}$ 100 K) are ion-molecule reactions, which usually have no activation barrier and proceed at the Langevin rate.", "The chemistry of interstellar chlorine is a good test case of this ion-molecule reaction scheme.", "The dominant form of chlorine in diffuse interstellar clouds is Cl$^+$ , which reacts exothermically with H$_2$ .", "The product HCl$^+$ reacts again with H$_2$ , and the resulting H$_2$ Cl$^+$ produces HCl upon dissociative recombination with an electron.", "This scheme is similar to the chemistry of carbon, with the difference that all reactions are exothermic.", "However, of the chlorine hydrides, only HCl is observable from the ground [8], [65].", "By giving access to HCl$^+$ and H$_2$ Cl$^+$ , HIFI provides the first test of the above reaction scheme.", "Observations with HIFI have resulted in the interstellar detections of both HCl$^+$ [17] and H$_2$ Cl$^+$ [45].", "The abundance ratios of HCl, HCl$^+$ and H$_2$ Cl$^+$ are in good agreement with expectations, but the absolute abundances are $\\sim $ 10$\\times $ higher than the models predict.", "The origin of this discrepancy is not understood; further observations may help to clarify the picture [63].", "The chlorine isotopic ratio $^{35}$ Cl/$^{37}$ Cl of $\\approx $ 2 measured toward the massive star-forming region W3 A is somewhat below the Solar value of $\\approx $ 3, but the difference is probably not significant [12]." ], [ "Hydrides as radiation diagnostics", "One way for young stars to disperse their natal envelopes is by energetic radiation (ultraviolet and X-rays), but the efficiency of this process is not well known because the envelopes are opaque to such short-wavelength radiation.", "Chemical signatures may help to determine the amount and nature of hidden energetic radiation, and the abundances of certain hydride molecules appear to be particularly good diagnostics.", "The small reduced masses of such molecules imply high line frequencies, so that HIFI offers the first opportunity of their observation.", "A number of hydride lines have been observed towards the high-mass protostars AFGL 2591 [9] and W3 IRS5 [2].", "Detected species are CH, CH$^+$ , OH$^+$ , H$_2$ O$^+$ , H$_3$ O$^+$ , NH and SH$^+$ , while upper limits were obtained for SH and NH$^+$ .", "The lines are narrow, which combined with the estimated abundances indicates an origin in the walls of outflow cavities which are being irradiated by protostellar far-ultraviolet radiation.", "The ground state lines of OH$^+$ and H$_2$ O$^+$ show pure absorption profiles suggesting an origin in the outer envelopes of the young stars or in foreground clouds, depending on their velocity shift.", "Excited state lines of CH$^+$ , OH$^+$ and H$_2$ O$^+$ show P Cygni profiles which indicate an origin in a wind, as also suggested by source-to-source variations in H$_2$ O/H$_2$ O$^+$ line ratios [87].", "Some emission lines, in particular those of H$_2$ O, OH and OH$^+$ , also exhibit a broad component ($\\Delta $V$\\approx $ 30 km s$^{-1}$ ) which likely originates in shocks [82].", "In conclusion, the hydride observations indicate that the interaction of young high-mass stars with their environment is dominated by far-UV radiation and shocks, whereas X-rays do not seem to play a role." ], [ "Formation of stars and planets", "The stars that make up the bulk of the mass and the luminosity of galaxies have not always existed and will not remain forever.", "The formation of stars and planets is a central question of modern astrophysics, and a prerequisite to understand the evolution of galaxies.", "All three instruments onboard the Herschel telescope have star formation as a high scientific priority.", "The high spectral resolution of HIFI makes it especially suitable to disentangle the rich forest-like line spectra of star-forming regions and infer physical conditions from the myriad of molecular lines, for example through modeling of line ratios.", "In addition, only HIFI can spectrally resolve the line profiles so that gas motions can be studied in exquisite detail.", "Finally, the chemical composition of star-forming regions is a powerful tool to study physical processes which are not directly observable, such as ionization and dissociation by cosmic rays and energetic radiation fields.", "The next subsections give highlights of what HIFI has been able to achieve in these areas so far." ], [ "Structure of star-forming regions", "Understanding the formation of stars requires accurate descriptions of the distribution of physical conditions in star-forming regions.", "Observations of molecular rotational emission lines are an excellent tool to constrain these conditions, as the emitted line spectrum is sensitive to basic parameters such as kinetic temperature and volume density.", "A good example is the CH$_3$ OH molecule, which due to its asymmetric structure has many of its lines grouped together in bands.", "Such bands give access to a large range in energy levels within a narrow frequency range, thereby enabling accurate measurements of gas temperatures.", "Figure: Rotation diagram analysis of CH 3 _3OH lines toward the Orion-KL region, showing how gas temperatures may be inferred from HIFI multi-line spectra.", "From Wang et al (2011).Many of these CH$_3$ OH bands, in particular the so-called Q-branches (sets of lines with $\\Delta $J=0) are unobservable from the ground, and HIFI is the first opportunity to use their diagnostic value.", "Examples are the studies of the OMC2-FIR4 core, which is found to be heated from the inside [41], and of the Compact Ridge in Orion, where external heating appears to dominate [83]; see Figure REF .", "The capability of HIFI to measure line spectra over a broad wavelength range is also useful to constrain physical conditions in star-forming regions.", "For example, [88] use high-$J$ lines of CO to measure the temperature profiles of low-mass star-forming cores, and [69] use C$^{18}$ O lines from $J$ =5–4 to 17–16 to measure the total column density of CO towards Orion-KL directly in all rotational states, independent of excitation.", "The populations of the energy levels of C$^{18}$ O are found to actually follow thermal distributions, at temperatures consistent with previous estimates for the various components of the Orion-KL region." ], [ "Shocks and hot cores", "The H$_2$ O molecule is a useful tracer of physical conditions in star-forming regions in several ways.", "Its line ratios can be used as tracers of the kinetic temperature and the volume density of the gas, and its line intensities are a measure of its abundance, which gives information about chemical formation and destruction processes.", "Towards low-mass star-forming regions, a third method turns out to be particularly powerful: H$_2$ O line profiles are found to be sensitive tracers of gas dynamics.", "[42] has found `bullets' in HIFI spectra of the outflow of the young protostar L1448 (Figure REF ).", "These high-velocity features in the line profiles were known from ground-based CO observations, but they are much brighter in H$_2$ O [7].", "Kristensen et al derive high temperatures ($\\approx $ 150 K) and densities ($\\approx $ 10$^5$  cm$^{-3}$ ) for the outflow shocks where the bullet features are thought to arise.", "The H$_2$ O and CO cooling appears to be similar in magnitude, and the high H$_2$ O abundance indicates that the hydrogen in the shock is mostly in the form of H$_2$ .", "Figure: HIFI spectra of H 2 _2O lines toward the low-mass protostar L1448 show high-velocity 'bullets' at V lsr V_{\\rm lsr} ≈\\approx –50 and +60 km s -1 ^{-1}, which demonstrate that H 2 _2O is a good tracer of gas motions.", "From Kristensen et al (2011).Observations of gas-phase molecules toward dense interstellar clouds indicate reduced abundances of all but the lightest species, presumably as a result of depletion onto dust grains.", "The bulk material of the grain surfaces is H$_2$ O ice, as shown by the strong and broad mid-infrared absorption features in the spectra of any cloud with $A_V > 3$ [86], [85].", "When newly formed stars heat up their surroundings, the ice mantles will evaporate and enrich the gas phase with molecules synthesized on the grain surfaces.", "Evidence that this process occurs are observations of complex organic molecules in the spectra of young stars which have heated up their surroundings, the so-called hot cores [37].", "Contrary to expectation, however, Herschel-HIFI finds that the H$_2$ O abundance in hot cores is $\\sim $ 10$^{-6}$ which is only $\\sim $ 1% of gas-phase oxygen and much less than the H$_2$ O abundance in the ice mantles.", "The most accurate measurements are toward high-mass hot cores (Figure REF ), which are bright enough to see excited state lines and rare isotopic lines of H$_2$ O [48], [13], [23] but the result holds for lower-mass objects as well [79].", "In some cases, observations of high-$J$ lines of rare H$_2$ O isotopologs reveal higher abundances [52], [38], indicating that the low-$J$ lines of H$_2$ O itself are not sensitive to the evaporated grain mantle material, but how general this result is remains to be seen.", "Figure: Observations with HIFI toward the high-mass protostar NGC 6334I show H 2 _2O lines from several isotopes and from a wide range of energy levels.", "The line profiles show a mix of emission and absorption from the protostellar envelope, the outflow, and several foreground clouds.", "From Emprechtinger et al (2010)." ], [ "Protoplanetary disks", "The star formation process leads to a disk of gas and dust which surrounds its parent star for $\\sim $ 10 Myr after its formation.", "The structure of such disks is of great interest since they are the likely birthplaces of exoplanets.", "The gross structure of protoplanetary disks is well known: the density is high in the midplane, where molecules freeze out onto dust grains, and low in the disk atmosphere, where stellar ultraviolet radiation first desorbs and then dissociates the molecules.", "The radial structure of the disk is usually described by decreasing (power-law) functions of temperature and density with increasing radius.", "However, key parameters of the disks are not well understood, such as the amount of mixing between the various parts of the disk, and the amount of settling of dust grains as a function of time [3].", "Observations of H$_2$ O lines with HIFI may improve our understanding of protoplanetary disks, but the expected signals are weak so that only few objects can be observed.", "The spectrum of DM Tau does not show H$_2$ O lines at all which gives a firm upper limit on the H$_2$ O abundance [4].", "The implication of this observation is that the dust in this disk has already settled to the midplane and probably started coagulating, which makes photodesorption less effective.", "The HIFI data thus imply that planet formation is underway in this object.", "Observations of a second disk, TW Hya, have resulted in the detection of both ground-state lines of H$_2$ O, as shown in Figure REF [40].", "The lines are likely optically thick, but radiative transfer calculations suggest that not only the abundance of H$_2$ O in this object is low, but also its ortho/para ratio.", "In particular, the o/p ratio of H$_2$ O in the TW Hya disk is lower than that in comets in our Solar System, which is a surprise because comets are thought to originate in the outer protosolar nebula, which is the part that the HIFI data probe.", "This result suggests that some processing of pre-cometary material occurred in the inner protosolar nebula, for example as a result of mixing of the inner and outer parts of protoplanetary disks." ], [ "Solar system results", "Planets form in circumstellar disks which are a natural byproduct of star formation.", "As realized by Kant and Laplace in the eighteenth century, the fact that the planets of the Solar system share one orbital plane and revolve around the Sun in the same direction suggests their formation from a protoplanetary disk, and argues against a scenario where the planets were captured one by one.", "However, the diversity of objects in our Solar system is large, and no two planets are alike, which implies that the formation of planetary systems is far from a simple process.", "The origin of this diversity and the likelihood of habitable planets around other stars are currently major science topics.", "Today, most young stars are known to be surrounded by disks [22] and a sizeable fraction of mature stars (up to 50%) are known to host planets [49].", "Studying the disks and planets around other stars remains a challenge, because of the high demands that it sets to instrumentation.", "Our Solar system covers only a fraction of parameter space: in particular, hot Jupiters and super-Earths do not occur around the Sun.", "Nevertheless, studies of the local planets, moons, comets and asteroids give a first impression of the possible outcomes of the planet formation process.", "This section describes key results from HIFI concerning physical processes and the chemical composition of Solar system objects." ], [ "The source of H$_2$ O to the early Earth", "A key result of HIFI concerns the D/H ratio of cometary water.", "The origin of water on Earth is a long-standing question because the early Earth was too warm to retain such volatile species.", "Delivery by comets during the Great Bombardment is an attractive solution, except that the isotopic composition, in particular the D/H ratio, of water in well-known comets such as Halley, Hale-Bopp and Hyakutake is at least twice the value of 1.56$\\times $ 10$^{-4}$ for the Earth's oceans (Figure REF ).", "Therefore, the Nice model for the evolution of the Solar system has most of the volatiles, including H$_2$ O, coming from asteroids [57].", "However, the measured comets all stem from the Oort cloud, where they were expelled by gravitational interaction with the giant planets.", "The D/H ratios of such long-period comets thus may not reflect pristine conditions but be affected by processing in the inner Solar system.", "With HIFI, the first measurement has been obtained of the D/H ratio of water in a short-period comet from the Kuiper belt, which should reflect pristine conditions in the outer Solar system.", "The measured ratio of (1.61 $\\pm $ 0.24)$\\times $ 10$^{-4}$ [36] is consistent with the value for terrestrial ocean water, implying that comets may have delivered at least some of the water to the early Earth.", "The relative importance of comets and asteroids as sources of volatiles is a subject for future investigation." ], [ "External supply of H$_2$ O to the giant planets", "The far-infrared line profiles of H$_2$ O seen with ISO toward Saturn show broad absorption from tropospheric H$_2$ O, as well as narrower emission due to H$_2$ O in the stratosphere [27].", "While H$_2$ O is expected to be the main oxygen carrier in the troposphere, the stratospheric H$_2$ O must have an external origin.", "The cryovolcanic activity on Enceladus has been a candidate supplier since its discovery by the Cassini spacecraft, but measurements with its mass spectrometer are inconclusive about the amount of ejected H$_2$ O, which is also thought to produce the E ring [66].", "Observations of H$_2$ O lines with HIFI have resulted in the first accurate measurement of Enceladus' H$_2$ O production rate [35].", "The torus is found to have an H$_2$ O column density of 4$\\times $ 10$^{13}$  cm$^{-2}$ and a scale height of 50,000 km, which is sufficient to be the major source of H$_2$ O for Saturn's upper atmosphere, but not for that of Titan.", "Similarly, HIFI observations of H$_2$ O toward Jupiter argue for an origin of its stratospheric H$_2$ O in cometary impacts such as that of Shoemaker-Levy 9 in July 1994 [11].", "The ISO satellite also found H$_2$ O in the atmosphere of Titan, but its low spectral resolution precluded a determination of the vertical profile [15].", "Observations with HIFI reveal emission lines of H$_2$ O and HNC which given the narrow line width must originate in the upper atmosphere, at altitudes above 300 km [58].", "Preliminary analysis suggests that ablation of micrometeorites is the likely source, although cometary impacts and local ring/satellite source may also play a role.", "The first detection of HNC on Titan allows to determine the HCN/HNC ratio at high altitudes ($\\sim $ 700 km), which helps to constrain photochemical models of Titan's atmosphere [59]." ], [ "The envelopes of evolved stars", "After leaving the main sequence, stars start losing significant amounts of mass and swell up to become red giants of several types.", "The mass loss reaches its peak during the AGB phase, after which the star traverses the Hertzsprung gap and starts cooling down as a white dwarf.", "The ejecta form the so-called AGB remnant, which contains up to half the stellar mass.", "Young white dwarfs have high surface temperatures and ionize their surroundings, which become visible as planetary nebulae.", "As these nebulae expand, they disappear from view and the material disperses into the interstellar medium.", "These late stages of stellar evolution are a key phase in the lifecycle of gas and dust, when enriched material returns to the ISM, from which future generations of stars may form.", "In addition, the winds of AGB stars have suitable conditions (high density, low temperature) for the formation of dust grains, which play a key role in the physics and chemistry of interstellar matter.", "The Herschel-HIFI mission has made several key observations of the material surrounding evolved stars, which have improved our understanding of this part of the life cycle of gas and dust.", "One crucial observation concerns the origin of H$_2$ O in circumstellar envelopes.", "The SWAS mission has discovered gas-phase H$_2$ O in the envelopes of C-rich AGB stars, which was attributed to the evaporation of comet-like bodies [51].", "The observations of a single spectral line did however not constrain the temperature of the H$_2$ O, which was a major uncertainty in this interpretation.", "Observations with the PACS and HIFI instruments onboard Herschel have now revealed about a dozen H$_2$ O lines and show that the temperature is at least several 100 K [19], [20].", "The initial observations of the prototype star IRC 10216 are confirmed by a survey of 8 C-rich AGB stars [61].", "The high H$_2$ O temperature rules out comet evaporation; instead, a possible origin of the observed H$_2$ O is the penetration of ultraviolet photons deep into a clumpy circumstellar envelope [19].", "This mechanism would also trigger the formation of other molecules such as NH$_3$ , whose observed abundances are much higher than thermal equilibrium models predict [53].", "Alternatively, non-equilibrium chemistry such as shock processing of the ejecta at the base of the wind, just above the photosphere, predicts H$_2$ O abundances which are very similar to the observed values [14].", "The HIFI instrument has also been useful to measure the temperatures of the winds in protoplanetary nebulae and young planetary nebulae.", "[10] present observations of CO, H$_2$ O, and other species toward ten such objects and find a surprisingly large range of wind temperatures: from $\\approx $ 30 K to ${_>\\atop {^\\sim }}$ 200 K. These differences may be due to cooling in shocks and hence reflect the ages of the winds which range from $\\sim $ 100 yr for the youngest and warmest winds to $\\sim $ 1000 yr for the oldest and coolest ones." ], [ "Conclusions and outlook", "The Herschel mission is still ongoing, and current estimates of its lifetime suggest that liquid helium will not run out before February 2013.", "Nevertheless, it is already possible to draw some first conclusions from the HIFI results obtained so far.", "First, it appears that interstellar clouds have a mixed instead of a layered structure.", "This conclusion follows in particular from the observations of C$^+$ which trace `warm dark gas', but also from the ubiquitous nature and high abundances of interstellar OH$^+$ and H$_2$ O$^+$ .", "In all these cases, the formerly common classification of interstellar clouds into diffuse clouds where hydrogen is largely atomic and dense clouds where it is mostly molecular must be replaced by a new view that the atomic and molecular phases of interstellar clouds are mixed together.", "The situation is less clear for protoplanetary disks, where the observations of ortho- and para-H$_2$ O toward TW Hya suggest that comets are the mixing products of the entire disk, while the observed D/H ratio in comet Hartley indicates distinct conditions for the inner and outer parts of the disks.", "In this area, more observations are needed.", "A second major conclusion is that neither H$_2$ O nor O$_2$ is a major carrier of interstellar oxygen, and that both species play only a minor role in the cooling of interstellar clouds.", "This conclusion strengthens earlier findings from the SWAS and Odin missions.", "Only in exceptional situations, such as strong shocks, does H$_2$ O cooling become important.", "In the case of Solar system objects, the main conclusion from the HIFI results is that planetary bodies do not lead solitary lives.", "Interaction plays an important role, as shown by the observations of exchange of H$_2$ O and other species between comets and asteroids with planets and their moons.", "The measurement of the HDO/H$_2$ O ratio in a Jupiter family comet which shows that comets were a major supplier of H$_2$ O to the early Earth is an example of such interactions in the past of the Solar system.", "In the near future, the newly commissioned ALMA interferometer will break new ground on each of these science topics.", "In particular, the high angular resolution of ALMA will be crucial to understand the structure of small objects such as protoplanetary disks.", "The good sensitivity of ALMA will be especially important to study distant processes such as star formation in the early Universe.", "In the more distant future, several planned telescopes and space missions will build on the legacy of Herschel and HIFI.", "In particular, the CCAT telescope will make deep wide-field images of the submillimeter sky at high ($\\sim $ 10$^{\\prime \\prime }$ ) resolution, both in continuum and spectral lines.", "From $\\approx $ 2015 on, the GUSSTO balloon experiment will survey the Southern sky in the fine structure lines of O, C$^+$ and N$^+$ at high spectral resolution.", "The SAFARI instrument onboard the JAXA-led SPICA mission, to be launched around 2020, will make very deep images of spectral line emission in the far-infrared.", "Finally, the FIRI interferometer will bring high angular resolution to the far-infrared regime in the year $\\approx $ 2030." ], [ "Acknowledgements", "The author thanks Frank Helmich, Xander Tielens and Ted Bergin for useful discussions and comments on the manuscript.", "HIFI has been designed and built by a consortium of institutes and university departments from across Europe, Canada and the US under the leadership of SRON Netherlands Institute for Space Research, Groningen, The Netherlands with major contributions from Germany, France and the US.", "Consortium members are: Canada: CSA, U.Waterloo; France: CESR, LAB, LERMA, IRAM; Germany: KOSMA, MPIfR, MPS; Ireland, NUI Maynooth; Italy: ASI, IFSI-INAF, Arcetri-INAF; Netherlands: SRON, TUD; Poland: CAMK, CBK; Spain: Observatorio Astronómico Nacional (IGN), Centro de Astrobiología (CSIC-INTA); Sweden: Chalmers University of Technology - MC2, RSS & GARD, Onsala Space Observatory, Swedish National Space Board, Stockholm University - Stockholm Observatory; Switzerland: ETH Zürich, FHNW; USA: Caltech, JPL, NHSC." ] ]

[ [ "Probing quasiparticle excitations in a hybrid single electron transistor" ], [ "Abstract We investigate the behavior of quasiparticles in a hybrid electron turnstile with the aim of improving its performance as a metrological current source.", "The device is used to directly probe the density of quasiparticles and monitor their relaxation into normal metal traps.", "We compare different trap geometries and reach quasiparticle densities below 3um^-3 for pumping frequencies of 20 MHz.", "Our data show that quasiparticles are excited both by the device operation itself and by the electromagnetic environment of the sample.", "Our observations can be modelled on a quantitative level with a sequential tunneling model and a simple diffusion equation." ], [ "Probing quasiparticle excitations in a hybrid single electron transistor H. S. Knowles Low Temperature Laboratory (OVLL), Aalto University, P.O.", "Box 15100, FI-00076 AALTO, Finland Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom V. F. Maisi Low Temperature Laboratory (OVLL), Aalto University, P.O.", "Box 15100, FI-00076 AALTO, Finland Centre for Metrology and Accreditation (MIKES), P.O.", "Box 9, FI-02151 Espoo, FinlandJ.", "P. Pekola Low Temperature Laboratory (OVLL), Aalto University, P.O.", "Box 15100, FI-00076 AALTO, Finland We investigate the behavior of quasiparticles in a hybrid electron turnstile with the aim of improving its performance as a metrological current source.", "The device is used to directly probe the density of quasiparticles and monitor their relaxation into normal metal traps.", "We compare different trap geometries and reach quasiparticle densities below 3 $$ m$^{-3}$ for pumping frequencies of 20 MHz.", "Our data show that quasiparticles are excited both by the device operation itself and by the electromagnetic environment of the sample.", "Our observations can be modelled on a quantitative level with a sequential tunneling model and a simple diffusion equation.", "Applications of superconductors generally rely on the fact that electronic excitations can be generated only if energy higher or equal to the gap energy $\\Delta $ is available.", "Hence the number of the excitations is ideally exponentially small at low temperatures and the properties specific for superconductors appear.", "If the number of excitations, typically characterized by the density of quasiparticles, increases, the superconducting features degrade.", "Such an effect has been studied in several devices such as superconducting qubits [1], [2], [3], [4], superconductor-insulator-normal metal-insulator-superconductor (SINIS) microcoolers [5], [6], [7], [8] and kinetic inductance detectors [9], [10].", "In this letter we focus on the effects of quasiparticles on a hybrid single-electron transistor.", "We use the Coulomb blockaded transistor for direct and simple probing of the quasiparticle excitation density.", "The device operates as a charge pump and is a promising candidate for the realization of a metrological current source [11].", "We show experimentally that the quasiparticle excitations limit the current quantization.", "By optimizing the quasiparticle relaxation, however, we estimate that it is possible to reach metrological accuracy.", "Figure: (a) Scanning electron micrograph of the l=20l = 20 m type A SINIS20 turnstile showing the superconducting Al leads connected to the Cu metal island via AlO 2 _2 barrier junctions and the gate electrode used to regulate the potential of the island.", "(b) shows the type B SINISopen geometry sample where the normal metal traps are within 200 nm of the junctions and the leads open up straight from the junction.", "In (c) the full length ll of the superconducting line of sample SINIS20 between the junction and the trap is visible as well as the quasiparticle traps formed by overlapping Cu and Al shadows.", "A sketch of the basic measurement circuitry is depicted and (d) shows the IV characteristics of the turnstile.In order to observe how quasiparticles influence the performance of the turnstile we designed samples (type A) where the quasiparticle relaxation in normal metal traps was purposefully delayed by extending the bare superconducting lines that connect the junctions to the traps.", "The beginning of this isolated superconducting line can be seen in the scanning electron microscope image of the turnstile in Fig.", "REF .", "(a) where it connects to the normal metal island via oxide junctions that appear as lighter areas.", "Its extension is visible in (c) all the way through to the wide traps of overlapping superconductor and normal metal separated by an oxide layer.", "The sample shown in (b) (type B) has wide leads with normal metal traps close to the junctions to enable efficient quasiparticle evacuation.", "The samples were fabricated with the standard electron-beam lithography and shadow mask technique [12].", "We compare the behaviour of quasiparticles in SINIS turnstile samples with different geometries.", "The length of the isolated superconducting line (given by the separation of the transistor junction from the trap) was varied between $l = 200$ nm and $l = 20$ $$ m. Measurements were performed in a dilution refrigerator at a base temperature of approximately 60 mK.", "Figure REF (d) shows the current-voltage characteristics of the turnstile sample presented in (a) and (c).", "The black line shows the current through the turnstile when the bias voltage V$_b$ is swept across the superconducting gap and the gate potential V$_g$ is varied between the gate open ($n_g = 0.5$ ) and the gate closed ($n_g = 0$ ) states.", "Simulations based on sequential tunnelling are fitted to the data in order to extract the parameters of the superconducting gap $\\Delta = 216\\ \\mathrm {eV}$ , the charging energy of the island $E_c = 0.74 \\Delta $ and the tunnelling resistance of the junction $R_T = 91$ k$\\Omega $ , specific to each sample (see Table 1 for parameters of all samples measured).", "These simulations are shown in blue for the gate open and red for the gate closed state.", "On this coarse level, the heating of the superconducting leads does not make a significant contribution and can be neglected.", "The overheating of the normal metallic island (thickness 30 nm) is taken into account by considering the electron-phonon coupling with material parameter value of $\\Sigma = 2 \\cdot 10^9\\ \\mathrm {WK^{-5}m^{-3}}$ which is consistent with values obtained in previous experiments [13], [14].", "In the subgap and pumping experiments we have the opposite situation: the island does not heat since the signals are much smaller but as we look at small differences, the overheating of the superconductor starts to have an effect.", "Figure: (a) and (b) show the normalized current through the oxide trap turnstile of geometry l=20l = 20 m smoothed over 5 data points as a function of V b V_b for gate voltages varying between n g =0n_g = 0 and n g =0.5n_g = 0.5.", "The charge stability measurement plotted in (a) was performed in a sample stage with a single cover; (b) shows measurements of the same sample in an indium sealed two-cover stage.", "In (c) and (d) we plot the normalized current computed with a simulation based on sequential tunneling for the situations of (a) and (b) respectively.", "The inset of (d) shows the quasiparticle density n qp n_{qp} inferred from the temperature reproducing the measured current inside the superconductor gap as a function of the length ll of the isolated superconducting line.We now investigate the current of the turnstile more closely for bias voltages within the superconducting gap.", "These measurements allow the direct probing of the quasiparticle density.", "This is not possible with voltage biased NIS junctions that probe the excitations more indirectly [15], [5], [7], [6], [8] and correspond to the gate open case in these experiments.", "Figure REF  (a) shows the current as a function of $V_b$ and $V_g$ .", "With no quasiparticles present we would expect zero current for low biases.", "However, we observe a current pattern periodic in $V_g$ within the gap with currents rising up to 10 fA.", "We find that we can reproduce the measured current pattern with a simulation shown in panel (c) using a high superconductor temperature $T_S$ that gives rise to a quasiparticle population above the gap.", "We compute the average current through the left junction via $I = e \\sum _{n}(\\Gamma _{LI} (n) - \\Gamma _{IL} (n)) P(n,t)$ where $\\Gamma _{LI(IL)}$ is sequential tunnelling rate to (from) the island through the left junction and $P(n,t)$ is the probability of the system being in the charge state $n$ of the island.", "The temperatures used to fit these data are $T_S = 205$ mK and the normal metal temperature $T_N = 92$  mK.", "The current-voltage characteristics are surprising: At degeneracy (half integer $n_g$ ) no net current flows, whereas in Coulomb blockade (integer $n_g$ ) we obtain a finite current.", "The simulations give us an insight into the on-going processes: at degeneracy the hot quasiparticle excitations lying at high energies are able to tunnel in both directions equally, hence there is no net current.", "In Coulomb blockade, the tunneling of a quasiparticle excitation is followed by a fast relaxation to the lowest lying charge state.", "The relaxation always happens in the forward direction given by $V_b$ and leads to a net current through the device.", "These features are a strong indication of having quasiparticle excitations as the source of the sub-gap current in the device.", "We measured the same sample in two different sample stages.", "Instead of having an enclosed stage with one metallic cover as in the measurement displayed in Fig.", "REF (a) we also used an indium sealed double hermetic metallic sample stage.", "In the latter case, the radio frequancy (RF) line had an additional 20 cm thermocoax cable to enhance the sample shielding.", "This was not included in the wiring of the single cover stage.", "However, as the RF line is not electrically connected to the turnstile, we do not expect this to influence the direct heat conduction to the sample.", "The dc wiring of the two setups was similar, made of approximately two meters of thermocoax cable.", "The two sample stages are thermalized to the cryostat base temperature in an identical way and the sole purpose of this sealing is to create an RF shield to the sample.", "The result is shown in panel (b).", "In this case no sub-gap current can be resolved.", "This behavior was fitted with temperatures of $T_S \\le 167$  mK and $T_N = 72$  mK which we present in Fig.", "2 (d).", "The inset in Fig.", "REF  (d) displays the quasiparticle density $n_{qp}$ inferred from the relation $n_{qp} = 2D(E_F) \\int _0^\\infty n_S(E) e^{-\\beta E} \\,dE = \\sqrt{2\\pi }\\ D(E_F) \\Delta \\sqrt{k_BT/\\Delta }\\ e^{-\\Delta /k_B T}$ which is valid at low temperatures $k_BT_S << \\Delta $ for three different samples with varying distance to the trap and measured with the poorly filtered sample stage.", "We use $D(E_F) = 1.45 \\times 10^{47} m^{-3}J^{-1}$ as the normal state density of states at the Fermi energy[16] .", "We observe a monotonous increase in $n_{qp}$ with increasing distance of the trap from the junction.", "We deduce that the presence of environmentally excited quasiparticles is determined by both the trap relaxation rate and the diffusion rate through the superconducting line.", "Next we turn to the dynamic case of the turnstile operation.", "In addition to environmental excitation, quasiparticles are now injected to the superconducting leads once in every pump cycle.", "The pumping frequency thus allows us to control the injected power and the number of quasiparticles.", "Figure REF shows current plateaus measured on type A SINIS20 sample.", "Three bias voltages were chosen around the optimum operation voltage of $eV_b = \\Delta $ [17] ranging from 0.8$\\Delta /e$ to 1.6$\\Delta /e$ .", "Two main effects were observed: a slight overshoot for the highest bias voltage at the beginning of the plateau and a spreading of the plateau value for different $V_b$ .", "Simulations including single electron and two-electron Andreev processes were used to model the process.", "The peak at the beginning of the plateau can be reproduced by allowing for an Andreev current with a conduction channel area of 30 nm$^2$ .", "This value is taken from previous experiments with similar samples [18], [19].", "The spreading of the plateaus in $V_b$ is caused by quasiparticle excitations and follows the expected behaviour with respect to variations in geometry and pumping operation as described below.", "The spread increases with the pumping frequency $f$ from $\\Delta I = 8$ fA for 1 MHz (panel (a) of Fig.", "REF ) to $\\Delta I = 15$ fA for 10 MHz (panel (b)).", "The injected power $P_{inj}$ increases with $f$ and we model this increase in mean number of quasiparticles in the superconductor by raising its temperature $T_S$ .", "The bias-dependence of current on the plateau is not influenced by Andreev currents and we can thus fit it simply by ascribing it to the quasiparticle number.", "The black lines correspond to simulations with quasiparticle densities of $n_{qp} = 41.1\\ $ m$^{-3}$ at 1 MHz and $n_{qp} = 82.6\\ $ m$^{-3}$ at 10 MHz.", "Figure: Current plateaus under pumping.", "Current through the turnstile as a function of the gate modulation amplitude measured on the type A SINIS20 sample at (a) 1 MHz, (b) 10 MHz and (c) 20 MHz sinusoidal V g V_g modulation frequency and on the type B sample SINISopen in (d) at 20 MHz.", "The pump was operated at bias voltage values of eV b =0.8ΔeV_b = 0.8 \\Delta (blue dots), eV b =1.2ΔeV_b = 1.2\\Delta (green dots) and eV b =1.6ΔeV_b = 1.6 \\Delta (red dots).", "Fits from simulations including two-electron Andreev processes are displayed as solid black lines, dashed black lines are simulations only including single electron processes.To demonstrate that the quantization can be improved by enhancing the quasiparticle relaxation we measured the pumping plateaus of a sample with broader leads, see Fig.", "REF  (b).", "Panels (c) and (d) in Fig.", "REF show the comparison of poor and good quasiparticle trapping respectively.", "The current quantization improves by two orders of magnitude when using leads with enhanced quasiparticle relaxation.", "To model the relaxation of the quasiparticle excitations we consider their diffusion in a thin superconducting line.", "The heat diffusion equation is $\\nabla \\cdot \\left(- \\kappa _S \\nabla T \\right) = -p_{\\mathrm {trap}},$ where the thermal conductivity of the superconductor is $\\kappa _S = \\frac{6}{\\pi ^2} \\left( \\frac{\\Delta }{k_BT}\\right) ^2 e^{-\\Delta /k_BT} L_0T/\\rho _n$ with the Lorentz number $L_0$ and the normal state resistivity $\\rho _n$ [20], [21].", "The heat is removed from the superconductor by a normal metallic trap to which it is tunnel coupled.", "The trap removes the heat $p_{\\mathrm {trap}}~=~\\frac{2 \\sigma _T}{e^2 d} \\int _0^\\infty E n_S(E) (f_N(E) - f_S(E)) \\,dE$ per unit area where $\\sigma _T$ is the conductance of the trap per unit area and $d = 22$ nm is the thickness of the superconducting film.", "The power $P_{inj}$ injected into the line during turnstile operation sets a boundary condition to the beginning of the line: $P_{inj} = A \\cdot (-\\kappa _S \\nabla T)$ , where $A = wd$ is the cross-sectional area of the lead with width $w$ .", "We rewrite Eq.", "(REF ) in terms of quasiparticle density $n_{qp}$ by considering only the strong exponential dependencies on $T$ to obtain $\\nabla ^2 n_{qp} = \\lambda ^{-2}(n_{qp}-n_{qp0}),$ where $\\lambda ^2 = \\frac{\\sqrt{2}d}{\\sqrt{\\pi }\\rho _n \\sigma _T} \\left(\\frac{k_BT}{\\Delta }\\right)^{1/2}$ and $n_{qp0}$ is the quasiparticle density of the superconductor when it is fully thermalised to the normal metal.", "Next we solve Eq.", "(REF ) for three different sections of the bias line for type A samples.", "In the first part we treat the bare aluminium line of length $l$ and constant width $w_1$ with no quasiparticle trapping ($p_{trap} = 0$ ).", "The next part deals with the widening of the lead from $w_1$ to $w_2$ where we also neglect quasiparticle trapping since its contribution in this section of the lead is small for our samples.", "Then, in the last part, the lead continues with a constant cross section of $w_2$ and is in contact with the quasiparticle trap: $p_{trap} \\ne 0$ .", "As a result, we obtain the quasiparticle density at the junction to be $n_{qp} = \\frac{\\sqrt{\\pi }e^2 D(E_F) \\rho _n}{\\sqrt{2\\Delta k_B T}} \\frac{P_{inj}}{wd} \\left(l + w_1 \\log \\left( \\frac{w_1}{w_2}\\right) + \\lambda \\frac{w_1}{w_2} \\right).$ The first term is the diffusion in the bare aluminium wire, the second term is the spreading to the wider line and the last term arises from the relaxation to the trap.", "Similarly, we can solve the diffusion for the type B sample with opening bias lines.", "Here we assume that the line starts at radius $r_0$ and the injected power is distributed evenly to all directions.", "We take $r_0 = 70\\ \\mathrm {nm}$ so that the area the power is injected into, $\\frac{\\pi }{2} r_0 d = (50\\ \\mathrm {nm)^2}$ , matches the junction area of the sample.", "The quasipartice density at the junction is then $n_{qp} = \\frac{\\sqrt{\\pi }e^2 D(E_F) \\rho _n}{\\sqrt{2\\Delta k_B T}}\\frac{P_{inj}}{\\theta r_0 d} \\frac{K_0(r_0/\\lambda )}{K_1(r_0/\\lambda )},$ where $K_n$ is the modified Bessel function of second kind and $\\theta $ the opening angle of the line.", "We now compare the quasiparticle relaxation in different sample geometries.", "From fits to measurements similar to those shown in Fig.", "REF we extract $n_{qp}$ as a function of $f$ .", "These values are displayed as dots in Fig.", "REF for the SINIS20 (blue), the SINIS5 (green) and the SINISopen samples (red).", "Using the diffusion model described above we calculate $n_{qp}$ as a function of $f$ corresponding to an injection power on the plateau given by $P_{inj} = ef\\Delta $ (solid lines).", "The injected heat calculated from the simulations deviated less than 10 % from this power even at the highest frequencies measured.", "We used $\\rho _n = 31\\ \\mathrm {n\\Omega m}$ as the normal state resistivity for all samples.", "Individual sample parameters used in the simulations are listed in Table 1.", "The measured densities clearly show a linear behavior with increasing $f$ as predicted by the model (see Eqs.", "(3) and (4)).", "As indicated by the slopes for increasing length $l$ of the isolated line, the further the quasiparticle trap lies from the oxide junction and the thinner the connecting line is, the slower the relaxation process is and the more the superconductor is heated.", "For the samples SINIS20 and SINIS5 a finite quasiparticle density is observed even in the absence of injected power.", "This density is of the same order of magnitude as the leakage currents shown in Fig.", "REF and it points towards interactions with electromagnetic radiation from the environment.", "The SINISopen sample clearly shows the most efficient quasiparticle relaxation with only a few quasiparticles per $$ m$^{-3}$ at 50 MHz driving frequency.", "Figure: Quasiparticle density as a function of the driving gate voltage ff.", "Fits to measurements including first and second order tunnelling processes are displayed as dots and quasiparticle densities derived from the diffusion model are displayed as solid lines.", "We compare two type A samples with delayed relaxation, SINIS20 (blue) and SINIS5 (green) and one type B sample SINISopen (red).", "Diagrams of the sample geometries are displayed in the upper part of the figure, dark colours representing quasiparticle traps, lighter areas the isolated superconducting lines and the small black bulks the normal metal islands.", "Below, a diagram shows the two pathways to quasiparticle excitation, during pumping via the normal metal island and through interactions with the electromagnetic environment.Table: Sample parametersWe have investigated the process of quasiparticle relaxation in hybrid SINIS turnstiles for various geometries.", "Both direct current and gate-driven pumping measurements can be well understood and simulated using models based on first and second order tunnelling processes.", "We were also able to model the relaxation of the excitations by a simple diffusion model.", "We find that the main sources for quasiparticles in the single electron transistor are those injected above the superconducting band gap during turnstile operation and those excited by radiation from a hot environment.", "In the best structure studied in this Letter we reached an accuracy $\\delta I /I$ of the order of $1 \\cdot 10^{-4}$ .", "We estimate that accuracy better than $10^{-6}$ at $50\\ \\mathrm {MHz}$ would be obtained with the following improvements: The aluminum should be made an order of magnitude thicker and the trap ten times more transparent.", "Alternatively, the bias leads can be extended to the third dimension.", "The resistance of the junctions should be increased by a factor of four and charging energy to $E_c> 2 \\Delta $ .", "Finally, the density of environmentally activated quasiparticles needs to be reduced to a level $n_{qp} \\ll 0.1\\ \\mathrm {\\mu m^{-3}}$ which has been demonstrated experimentally in a recent work [22].", "With these realistic modifications we expect to reach metrological accuracy in turnstile operation.", "We thank A. Kemppinen and O.-P. Saira for useful discussions and M. Meschke for technical assistance.", "The work has been supported partially by the National Doctoral Programme in Nanoscience (NGS-NANO) and the European Community's FP7 Programme under Grant Agreements No.", "228464 (MICROKELVIN, Capacities Specific Programme) and No.", "218783 (SCOPE)." ] ]

[ [ "Exploiting Channel Correlation and PU Traffic Memory for Opportunistic\n Spectrum Scheduling" ], [ "Abstract We consider a cognitive radio network with multiple primary users (PUs) and one secondary user (SU), where a spectrum server is utilized for spectrum sensing and scheduling the SU to transmit over one of the PU channels opportunistically.", "One practical yet challenging scenario is when \\textit{both} the PU occupancy and the channel fading vary over time and exhibit temporal correlations.", "Little work has been done for exploiting such temporal memory in the channel fading and the PU occupancy simultaneously for opportunistic spectrum scheduling.", "A main goal of this work is to understand the intricate tradeoffs resulting from the interactions of the two sets of system states - the channel fading and the PU occupancy, by casting the problem as a partially observable Markov decision process.", "We first show that a simple greedy policy is optimal in some special cases.", "To build a clear understanding of the tradeoffs, we then introduce a full-observation genie-aided system, where the spectrum server collects channel fading states from all PU channels.", "The genie-aided system is used to decompose the tradeoffs in the original system into multiple tiers, which are examined progressively.", "Numerical examples indicate that the optimal scheduler in the original system, with observation on the scheduled channel only, achieves a performance very close to the genie-aided system.", "Further, as expected, the optimal policy in the original system significantly outperforms randomized scheduling, pointing to the merit of exploiting the temporal correlation structure in both channel fading and PU occupancy." ], [ "Introduction", "Over the past decade, cognitive radio (CR) has been identified as one promising solution to ease the “spectrum scarcity” associated with the traditional static spectrum allocation [1], [2], [3], [4].", "Going beyond the fixed and licensed spectrum allocation, a secondary user (SU) can opportunistically access the spectrum owned by the primary users (PUs) in a CR network.", "This paradigm shift from static to dynamic spectrum allocation has been shown to bring significant improvement in the spectrum utilization, and hence the system's overall performance.", "A fundamental principle enabling the cognitive capability is built upon the SU's dynamic adaptation of its operation parameters (such as power, frequency, etc.", "), according to the environmental variations over time.", "One such variation is the channel fading.", "Often times an i.i.d.", "flat fading model is used in abstracting fading channels, which fails to capture the temporal channel memory observed in realistic scenarios [5].", "An alternative model, namely the Gilbert-Elliot (GE) model [6], has been widely used (see, e.g., [7], [8], [9]) to capture the temporal correlation in the fading process.", "Specifically, the GE model uses a first-order Markov chain with two states: one representing a “good” channel where the user experiences error-free transmissions, and the other representing a “bad” channel with unsuccessful transmissions.", "Another variation to be considered is the PU's activity on the channels.", "Note that in a CR network, the SUs have a strictly lower priority in the spectrum usage, and can only access the channels when the PUs are absent [1], [2], [3], [4].", "This unique spectrum usage structure necessitates the inclusion of the channel's PU occupancy state in determining the channel's accessability by an SU.", "In many of the existing works (see, e.g., [7], [8], [10], [11], [12]), only one set of the system states – either the channel fading, or the PU occupancy – has been taken into consideration in developing spectrum access strategies by the SU.", "In this work, we take a step forward, and explore the utility of both the states for opportunistic channel access.", "Specifically, we consider a CR network consisting of multiple PU channels and one SU opportunistically accessing one of the PU channels at a time.", "A spectrum server is utilized to periodically schedule the SU to one of the channels for transmission.", "Worth noting is that the usage of the spectrum server is consistent with the recent FCC ruling on the use of a spectrum database in CR network operations [13].", "Further, the spectrum server facilitates spectrum management, and enhances the scalability of the network [14].", "Dynamic spectrum access in the presence of the temporal variations can be cast as a sequential control problem.", "We formulate this sequential control problem as a partially observable Markov decision process [15].", "In this context, the spectrum server makes scheduling decisions in terms of allocating a PU channel to the SU, based on the channel's PU occupancy state and fading state.", "We model the channel fading by using a two-state first-order Markov chain, i.e., the GE model.", "On the other hand, since the PU activity may possess a long temporal memory (see, e.g., [16], [17]), we develop an “age” model to capture the temporal correlation structure of the PU occupancy state.", "Building on the above model, we examine the intricate tradeoffs resulting from the dynamic interaction of the system states.", "Our main contributions can be summarized as follows: We study opportunistic spectrum scheduling by exploiting the temporal correlation structure in both the channel fading and the PU occupancy states.", "This, to the best of our knowledge, has not been addressed systematically in the literature so far.", "We show that the optimal scheduling involves a multi-tier “exploitation vs. exploration” tradeoff.", "For certain special cases, we establish the optimality of a simple greedy policy, and examine the intricacy of the fundamental tradeoffs.", "To gain a better understanding of the tradeoffs for the general case, we introduce a full-observation genie-aided system, where the spectrum server collects channel fading states from all the PU channels.", "Using the genie-aided system, we decompose the multiple tiers of the tradeoffs, and examine them progressively.", "The rest of the paper is organized as follows.", "Section  introduces the basic setting and problem formulation in detail.", "In Section , we identify the fundamental tradeoffs and illustrate them via special cases.", "Section  further examines the tradeoffs by developing a genie-aided system that isolates the impact of channel fading and PU occupancy on the optimal reward.", "In Section , numerical results are presented where we evaluate and compare the performance of the optimal policy in the original system with baseline cases.", "We also study the impact of the memory in the channel fading and PU occupancy on the relative performances of various baseline cases.", "This is followed by concluding remarks in Section .", "We consider a CR network with one SU and $N$ PUsEach user is assumed to be a pair of transmitter and receiver.. Each PU is licensed to one of $N$ independent channels, henceforth identified as PU channels.", "A PU generates packets according to a stationary process, transmits over its channel if there are backlogged packets, and leaves upon the completion of the transmissions.", "The PU traffic activity is assumed to be identical and independent across channels.", "The SU, on the other hand, is backlogged with packets and opportunistically transmits these packets over the PU channels with the help of a spectrum server.", "Time is divided into two timescales: mini-slots and the control slots each constituting $K$ mini-slots, as illustrated in Fig.", "REF .", "The length of each mini-slot is normalized to fit the transmission of one data packet of the PU or the SU.", "At the beginning of each control slot, the spectrum server schedules the SU to the “best” PU channel that is expected to yield the highest average throughput for the SU.", "The SU then transmits packets in the scheduled channel, until it detects the return of a PUThis can be accomplished by incorporating collision detection by the SU at the mini-slot timescale.", "We also assume that PU arrivals coincide with the mini-slot boundaries..", "Upon such an event, the SU suspends transmissions until the beginning of the next control slot, when the spectrum server re-schedules the SU to a PU channel based on most recent observations.", "At the end of each mini-slot when the SU transmitted a packet, it sends accurate feedback on the channel fading state (of the PU channel, as seen by the SU) corresponding to that mini-slot, to the spectrum server.", "The spectrum server uses this channel fading feedback, the PU traffic observations, along with the memory inherent in these processes to perform informed scheduling decisions at the beginning of the next control slot.", "We discuss the system model and the scheduling problem formulation in more detail in the following.", "Figure: A sketch of the two timescale model." ], [ "Problem Formulation", "The opportunistic spectrum access at hand can be viewed as a sequential control problem, which we formulate as a partially observable Markov decision process.", "In the following, we introduce and elaborate the entities involved in the formulation.", "Channel occupancy: The usage pattern on each of the PU channels can be modeled as an ON-OFF process at the mini-slot timescale, with ON denoting the busy state where the PU transmits data over the channel, and OFF the idle state where the PU is absent.", "Channel occupancy is the idle or busy state of the PU channels.", "Let $o_{t,k}(n)$ be a binary random variable, denoting whether PU channel $n$ , for $n\\in \\lbrace 1,\\ldots ,N\\rbrace $ , is idle ($o_{t,k}(n)=0$ ), or not ($o_{t,k}(n)=1$ ), in the $k$ th mini-slot of control slot $t$ .", "The corresponding idle probability is denoted by $\\pi _{t,k}^o(n) \\triangleq \\textrm {Pr}(o_{t,k}(n)=0)$ .", "Idle/Busy age: The PU traffic is temporally correlated, i.e., the current occupancy state on each of the channel depends on the history of the channel occupancy.", "We introduce the notion of “age,” defined as follows, to characterize the occupancy history: The age of a PU channel is the number of consecutive mini-slots immediately preceding the current mini-slot, during which the channel is in the same occupancy state as in the current mini-slot.", "The age is denoted as “idle age” if the channel is in idle state in the current mini-slot and “busy age” otherwise.", "We use $x_t(n)$ to denote the age of channel $n$ at the beginning of control slot $t$ .", "As noted earlier, we assume long memory in the PU occupancy state.", "Specifically, with the definition of age in place, we adopt a family of functions monotonically decreasing in age, to denote the conditional probability that a channel will be idle (or busy), given that it has been idle (or busy) for $x \\ge 1$ mini-slots: $ P_I(x) &=& \\frac{1}{x^u + C_I}, \\nonumber \\\\P_B(x) &=& \\frac{1}{x^u + C_B}, ~~u = 1,2,...,$ where $C_I$ and $C_B$ are normalizing constants taking positive values.", "Our occupancy model essentially imposes the following realistic correlation structures: [1)] the occupancy memory weakens with time, i.e., the impact of past occupancy events on the current occupancy state diminishes since the said event happened; the conditional probability that the PU channel is busy or idle now, is purely a function of the length of time the channel has been in the most recent state, and is independent of the channel occupancy history before the time of the latest transition to the most recent state.", "In sight of this, the quantities $P_I$ and $P_B$ defined in (REF ) are sufficient for capturing the temporal correlation in the channels' PU occupancy state.", "Channel fading model: At the end of each mini-slot after transmitting a packet, the SU measures the channel fading between its transmitter and receiver on the scheduled channel, and feeds back this information to the spectrum server.", "Inspired by recent works [10], [8], [7], we capture the memory in the fading (of the PU channel) between the SU's transmitter and receiver using a two-state, first-order Markov chain, with state variations occurring at the mini-slot timescale.", "The Markov chain model is i.i.d.", "across the PU channels.", "Each state of the Markov chain corresponds to the degree of decodability of the data sent through the channel, where state 1 denotes full decodability and state 0 denotes zero decodability.", "Note that the states can also be interpreted as a quantized representation of the underlying channel fading, in the sense that state 1 corresponds to “good” channel fading, while state 0 corresponds to “bad” fading.", "The probability transition matrix of this Markov chain is given as: $ \\textbf {P}:= \\left[\\begin{array}{cc}1 - r & r \\\\1 - p & p\\end{array}\\right],$ where $p$ is the conditional probability that the channel fading is good, given that it was good in the previous mini-slot; and $r$ is the conditional probability that the channel fading is good, given that it was bad in the previous mini-slot.", "Throughout the paper, we will focus on the case when the fading channels are positively correlated, i.e., $p>r$ .", "Belief of channel fading state: Denote by $\\pi _{t,k}^s(n)$ the belief of channel fading state in the $k$ th mini-slot of control slot $t$ on channel $n$ .", "As is standard [15], [7], the fading state belief is a sufficient statistic that characterizes the current channel fading state as perceived by the SU.", "Further, let $f_{t,k}(a_t)$ be a binary random variable denoting the fading state feedback obtained at the end of the $k$ th mini-slot in control slot $t$ on the scheduled channel $a_t$ .", "Also, define $\\mathrm {T}^L(\\cdot )$ , for $L \\in \\lbrace 0,1,\\ldots \\rbrace $ , as the $L$ th step belief evolution operator, taking the form: for $\\gamma \\in (0,1)$ , $ \\mathrm {T}^L(\\gamma ) = \\mathrm {T}(\\mathrm {T}^{L-1}(\\gamma )),$ with $\\mathrm {T}^0(\\gamma ) = \\gamma $ and $\\mathrm {T}(\\gamma ) = \\gamma p + (1-\\gamma )r$ .", "Now, the update of the fading state belief is governed by the underlying Markov chain model, and any new information obtained on the channel fading, i.e.", ": $ \\pi _{t,k+1}^s(n) =\\left\\lbrace \\begin{array}{ll}p, ~~~& a_t = n, f_{t,k}(a_t)=1, \\\\r, ~~~& a_t = n, f_{t,k}(a_t)=0, \\\\\\mathrm {T}(\\pi _{t,k}^s(n)),~~~& a_t \\ne n.\\end{array}\\right.$ Action space: This refers to the set of channels that the scheduling decision is made from.", "The spectrum server selects channels only from those that are currently idleThis is a policy level constraint to protect the PU's priority in spectrum access., and the action space $\\mathcal {A}_t$ in control slot $t$ can thus be written as: $\\mathcal {A}_t = \\lbrace n:o_{t,1}(n) = 0\\rbrace .$ State: At the beginning of each control slot, the spectrum server makes the scheduling decision based on three factors: For each of the PU channels, [1)] the idle/busy state at the moment; the length of time the channel has been in the current occupancy state (i.e., age); and the fading state belief value.", "That is, the state of each PU channel $n$ , is represented by a three dimensional vector: $S_t(n) = [o_{t,1}(n), x_t(n), \\pi _{t,1}^s(n)]$ .", "Accordingly, the state of the system at the beginning of current control slot $t$ is described by a $N \\times 3$ matrix $\\textbf {S}_t$ : $\\textbf {S}_t \\hspace{-1.42262pt}:= \\hspace{-1.42262pt}[S_t(1);\\ldots ;S_t(N)] \\hspace{-2.84526pt}=\\hspace{-2.84526pt} \\left[\\hspace{-2.84526pt}\\begin{array}{ccc}o_{t,1}(1) & x_t(1) & \\pi _{t,1}^s(1) \\\\o_{t,1}(2) & x_t(2) & \\pi _{t,1}^s(2) \\\\\\vdots & \\vdots & \\vdots \\\\o_{t,1}(N) & x_t(N) & \\pi _{t,1}^s(N)\\end{array}\\hspace{-2.84526pt}\\right].$ Horizon: The horizon is the number of consecutive control slots over which scheduling is performed.", "We index the control slots in a decreasing order with control slot 1 being the end of the horizonFor the mini-slots, we use the conventional increasing time indexing..", "Throughout the paper, we denote the length of the horizon by $m$ , i.e., the scheduling process begins at control slot $m$ .", "Stationary scheduling policy: A stationary scheduling policy $\\mathcal {P}$ establishes a stationary mapping from the current state $\\textbf {S}_t$ to an action $a_t$ in each control slot $t$ .", "Expected immediate reward: The expected immediate reward is the reward accrued by the SU within the current control slot.", "Specifically, the SU collects one unit of reward in each mini-slot, if the channel is idle and has good channel fading (i.e., conditions that indicate successful transmission by SU).", "Since the scheduled channel must be idle in the first mini-slot of the current control slot, the expected immediate reward can be calculated as: $R_t(\\textbf {S}_t,a_t)=\\sum _{k=2}^K \\pi _{t,k}^o(a_t) \\pi _{t,k}^s(a_t) + \\pi _{t,1}^s(a_t).$ Total discounted reward: Given a scheduling policy $\\mathcal {P}$ , the total discounted reward, accumulated from the current control slot $t$ , until the horizon, can be written asIn the subsequent sections, we may drop $\\mathcal {P}$ and the tiers of expectation to simplify the notation.", "$V_t(\\textbf {S}_t; \\mathcal {P}) = R_t(\\textbf {S}_t,a_t) + \\beta E_{{\\mathbf {\\pi }}_{t-1}^s} E_{\\textbf {O}_{t-1}} V_{t-1}\\left(\\textbf {S}_{t-1}; \\mathcal {P}\\right),$ where $\\beta \\in (0,1)$ is the discount factor, facilitating relative weighing between the immediate and future rewards, and the expectation is taken with respect to fading state belief: $\\mathbf {\\pi }_{t-1}^s = \\lbrace \\pi _{t-1,1}^s(1),\\ldots ,\\pi _{t-1,1}^s(N)\\rbrace $ , and PU occupancy: $\\textbf {O}_{t-1} = \\lbrace o_{t-1,1}(1),\\ldots ,o_{t-1,1}(N)\\rbrace $ .", "Objective function: The objective of the scheduling problem is to maximize the SU's throughput, i.e., SU's total discounted reward.", "A scheduling policy $\\mathcal {P}^*$ is optimal if and only if the following optimality equation is satisfied: $V^(\\textbf {S}_t; \\mathcal {P}^*) \\hspace{-2.84526pt} = \\hspace{-2.84526pt} \\max _{a_t \\in \\mathcal {A}_t}\\Bigg \\lbrace R_t(\\textbf {S}_t, a_t)+ \\beta E_{{\\mathbf {\\pi }}_{t-1}^s} E_{\\textbf {O}_{t-1}} V^(\\textbf {S}_{t-1}; \\mathcal {P}^*)\\Bigg \\rbrace .$ The function $V^*(\\textbf {S}_t) := V^(\\textbf {S}_t; \\mathcal {P}^*)$ is the objective function of the scheduling problem.", "Fundamental Tradeoffs The decision on opportunistic spectrum scheduling is made based on two sets of system states: the PU occupancy on the channel and the channel fading perceived by the SU.", "On one hand, PUs may return in the middle of a control slot and hinder further transmissions of the SU, leading to a decreased reward for the SU.", "The temporal memory resident in the PU occupancy suggests that the past history of channel's occupancy, measured by the age, influences the occupancy state of the channel in the future.", "On the other hand, the PU channels may suffer from “bad” channel fading in the middle of a control slot, even if a PU does not return to hinder SU's transmissions..", "Similar to the PU occupancy, the historic observation on the fading process would help determine the expected channel fading in the future.", "Note that by way of the channel feedback arrangement, an observation of a PU channel fading is made only when that channel is scheduled to the SU.", "Thus scheduling is inherently tied to channel fading learning.", "Roughly speaking, to maximize the SU's reward, the spectrum server must schedule a channel such that the combination of the perceived channel occupancy and channel fading strikes a “perfect” balance between the immediate gains and channel learning for future gains.", "We discuss this intricate tradeoff in the following.", "Classic “Exploitation vs. Exploration” Tradeoff In the existing literature (e.g., [7], [8], [10], [11], [12]), focus has been cast on considering only one of the factors: either channel fading or channel occupancy, along with the associated temporal correlation.", "The optimal decision is a mapping that best balances the tradeoff of “exploitation” and “exploration” on the single factor being considered.", "The exploitation side lets the scheduler choose the channel with the best perceived channel fading (or occupancy state) at the moment, corresponding to immediate gains; while the exploration side tends to favor the channel with the least learnt information so far, probing which can contribute to the overall understanding of the channel fading (or occupancy state) in the network, and thus better opportunistic scheduling decisions in the future.", "“Exploitation vs. Exploration” Tradeoff in Dynamics of Both Channel Fading and PU Occupancy States In contrast to the existing works, we examine the tradeoffs when the temporal correlation in both the channel fading and PU occupancy are considered.", "While the classic tradeoff described above apparently exists, additional tradeoffs arise in our context due to the interactions between the two sets of system states.", "In particular, the long temporal memory in the PU occupancy state adds a new layer to the tradeoffs inherent in the problem.", "For instance, note that the SU can only transmit on idle channels.", "To carry out exploration, the channel being favored in the traditional sense, i.e., the one on which the least information is available, may no longer be the preferred choice if this channel is perceived to be unavailable (i.e., busy) for a prolonged duration in the near future.", "In other words, it may not be worth learning the channel as the SU cannot utilize the learned knowledge in the near future.", "Fig.", "REF is a pictorial illustration of the impact from the occupancy state on the SU's expected reward.", "The history of occupancy, represented by the idle ages $x_t(1)$ and $x_2(t)$ , affects both the immediate and future rewards of the SU.", "Specifically, as the idle age increases, the temporal memory in the PU's occupancy pushes the channel to transit to busy sooner (i.e., time point $b$ comes earlier than $a$ in the figure).", "Therefore, the average availability on the PU channel in the current control slot decreases, which leads to a smaller immediate reward for the SU.", "Further, note that the latest mini-slot for which the spectrum server receives channel fading feedback is also the last mini-slot before the PU returns, i.e., time points $a$ and $b$ respectively for the two cases in Fig.", "REF .", "The duration $d_1$ (likewise, $d_2$ ) in Fig.", "REF is an indication of how “fresh” the channel fading information is for the scheduling decision at the beginning of the next control slot, i.e., $t-1$ .", "With $d_1 < d_2$ , channel feedback is more fresh in the former case, with a lower idle age $x_t(1)$ .", "Thus, age, through its effect on the freshness of feedback, and the availability of the PU channels in the future slots, adds another layer to the tradeoffs, thereby influencing the optimal scheduling decision.", "Figure: An illustration of the impact of age on the SU's reward.To better perceive the intricate tradeoffs in the system, we proceed, in what follows, with a number of break-down results that aim at illustrating each tier of the tradeoffs progressively.", "Tradeoffs Inherent in Immediate Reward Consider the PU channel scheduled in the current control slot $t$ .", "Let $k_0$ denote the latest mini-slot before the PU of the scheduled channel returns in the current control slot.", "Clearly, $k_0$ is a random variable, taking values in $k_0 \\in \\lbrace 1,\\ldots ,K\\rbrace $ .", "Let $p_z \\triangleq \\textrm {Pr}(k_0 = z)$ .", "With $x$ denoting the idle age of the scheduled channel, $k_0$ is distributed as follows: For $K=2$ : $p_1 = 1 - P_I(x+1),~~~p_2 = P_I(x+1),$ and for $K \\ge 3$ : $ p_1 &= 1 - P_I(x+1), \\nonumber \\\\p_z &= \\prod _{j=1}^{z-1} P_I(x+j) (1-P_I(x+z)),z=2,\\ldots ,K-1,\\nonumber \\\\p_K &= \\prod _{j=1}^{K-1} P_I(x+j).$ In the following lemma, we establish the structure of the distribution of $k_0$ .", "Lemma 1 The distribution of $k_0$ is monotonically decreasing in the idle age, $x_t$ , for $z = 2,\\ldots , K$ , and monotonically increasing in $x_t$ , for $z=1$ .", "For $K=2$ , it is straightforward to establish the conclusion.", "We next focus on $K\\ge 3$ .", "First, it is easy to show that for $z=1$ , $p_1$ is monotonically increasing in $x$ since $P_I(x)$ is monotonically decreasing.", "Further, note that if two positively valued functions, $f_1(x) > 0$ and $f_2(x) > 0$ , are both monotonically decreasing in $x$ , i.e., for any positive integer $\\Delta \\ge 1$ , $f_1(x) > f_1(x + \\Delta ),~~~f_2(x) > f_2(x+\\Delta ),$ then the product of the two, $f_1(x)f_2(x)$ , is also a decreasing function since $f_1(x) f_2(x) > f_1(x + \\Delta ) f_2(x+\\Delta ).$ Therefore, $p_K$ is monotonically decreasing in $x$ .", "Next, we show that $p_z,z=2,\\ldots ,K-1$ are monotonically decreasing in $x$ .", "Based on the above argument, it is sufficient to show that for any $z = 2,\\ldots , K-1$ , the following function, $g(x) \\triangleq P_I(x+z-1) (1-P_I(x+z)),$ is monotonically decreasing.", "We have the following simplifications: For $\\Delta \\ge 1$ , a positive integer, $\\frac{g(x)}{g(x+\\Delta )} \\hspace{-8.53581pt}&=&\\hspace{-8.53581pt} \\frac{P_I(x+z-1) (1-P_I(x+z))}{P_I(x+\\Delta +z-1) (1-P_I(x+\\Delta +z))} \\nonumber \\\\\\hspace{-8.53581pt}&=&\\hspace{-8.53581pt} \\frac{(x+\\Delta +z-1)^u + C_I}{(x+z)^u + C_I} \\hspace{-2.84526pt}\\cdot \\hspace{-2.84526pt} \\frac{(x+\\Delta +z)^u + C_I}{(x+\\Delta +z)^u + C_I-1}\\cdot \\frac{(x+z)^u + C_I-1}{(x+z-1)^u + C_I}\\nonumber \\\\&\\triangleq & {B}_1 {B}_2 {B}_3.$ Since $x \\ge 0, C_I > 0, u\\ge 1$ and $\\Delta \\ge 1$ , we immediately have $B_1 \\ge 1$ , $B_2 > 1$ .", "Further, using the binomial theorem, we obtain $(x+z)^u= \\sum _{i=0}^u {u\\atopwithdelims ()i} (x+z-1)^i \\ge (x+z-1)^u + 1,$ and hence $B_3 \\ge 1$ .", "As a result, $\\frac{g(x)}{g(x+\\Delta )} > 1$ and $p_2, \\ldots , p_{K-1}$ are monotonically decreasing in $x$ .", "This concludes the proof.", "We next present a result that demonstrates the tradeoff inherent in the immediate reward with respect to age.", "Proposition 1 The immediate reward on the scheduled channel is monotonically decreasing in the idle age.", "As the system model implies, we can rewrite the immediate reward as the following weighted sum: $ R_t(\\textbf {S}_t,a_t) = \\sum _{z = 1}^K \\sum _{k=1}^z \\pi _{t,k}^s(a_t)p_z,$ where $p_z = \\textrm {Pr}(k_0 = z)$ is given by (REF ).", "When $K=2$ , with idle age on $a_t$ being $x_t$ , we have $R_t(\\textbf {S}_t,a_t) = \\pi _{t,1}^s(a_t) + P_I(x_t+1) \\pi _{t,2}^s(a_t)$ .", "Apparently, it increases as $x_t$ decreases.", "For $K\\ge 3$ , denote by $\\theta _z \\triangleq \\sum _{k=1}^z \\pi _{t,k}^s(a_t), z = 1,\\ldots ,K$ .", "It is clear that $\\theta _1 < \\theta _2 < \\cdots < \\theta _K,$ and $\\lbrace \\theta _z\\rbrace $ 's are constants in the idle age $x_t$ .", "To emphasize the role of the argument $x_t$ , we rewrite $R_t(\\textbf {S}_t,a_t)$ as $R_t(x_t)$ , and $p_z$ as $p_z(x_t)$ .", "Utilizing the result of Lemma REF , and noting that for any positive integer $\\Delta \\ge 1$ , $|p_1(x_t) - p_1(x_t + \\Delta )| = \\sum _{z=2}^K (p_z(x_t) - p_z(x_t + \\Delta ))$ , we obtain: $R_t(x_t) -R_t(x_t+\\Delta ) =\\sum _{z=1}^K \\theta _z \\left(p_z(x_t) - p_z(x_t + \\Delta )\\right)> \\theta _1(p_1(x_t) - p_1(x_t + \\Delta )) + \\theta _2 \\sum _{z=2}^K (p_z(x_t) - p_z(x_t + \\Delta ))> 0,$ i.e., $R_t(x_t)$ is monotonically decreasing in the idle age $x_t$ , and this establishes the proposition.", "The above result can be readily extended to the following corollary.", "Corollary 1 When all the PU channels have equal fading state beliefs, the immediate reward is maximized by scheduling the SU to the channel with the lowest idle age.", "Next, recall that the channel fading is modeled by a positively-correlated Markov chain.", "Hence, if $\\pi _{t,1}^s(a_t) > \\pi _{t,1}^{s^{\\prime }}(a_t)$ , then the inequality $\\pi _{t,k}^s(a_t) > \\pi _{t,k}^{s^{\\prime }}(a_t)$ holds, for all $k =2,\\ldots ,K$ .", "We present the following proposition without further proof.", "Proposition 2 The immediate reward on the scheduled channel is monotonically increasing in its fading state belief at the moment.", "Further, given equal idle ages across all PU channels, the immediate reward is maximized by scheduling the SU to the channel with the largest fading state belief value at the moment.", "Tradeoffs Inherent in Total Reward In this subsection, we illustrate the tradeoffs inherent in the total reward by examining a special case with two channels $N=2$ and number of mini-slots $K=1$ .", "In particular, we show that under these conditions, a simple greedy scheduling policy is optimal.", "The greedy policy is formally defined as follows: In any control slot, the greedy decision maximizes the immediate reward, ignoring the future rewards, i.e., $\\hat{a}_t = \\max _{a_t\\in {\\mathcal {A}_t}} \\lbrace R_t(\\textbf {S}_t,a_t)\\rbrace .$ We now formally record the result on greedy policy optimality in the following proposition.", "Proposition 3 The greedy policy is optimal when $K=1$ and $N=2$ .", "To prove the proposition, we begin with the following induction hypothesis: Induction Hypothesis: With the length of the horizon denoted by $m, m \\ge 2$ , assume that greedy policy is optimal in all the control slots $t\\in \\lbrace m-1, \\ldots ,1\\rbrace $ .", "The proof proceeds in two steps: First, we fix a sequence of scheduling decisions $\\vec{l} := \\lbrace a_{m},\\ldots ,a_{t+1}\\rbrace $ , and show that the expected immediate reward in control slot $t$ , under the greedy policy, is independent of the scheduling decisions $\\vec{l}$ .", "Then, we provide induction based arguments to validate the induction hypothesis and hence establish that the greedy policy is optimal in all the control slots.", "Let $U_t^{(\\vec{l})}$ denote the expected immediate reward in slot $t \\in \\lbrace m-1,\\ldots ,1\\rbrace $ , given the scheduling decisions $\\vec{l}$ .", "$U_t^{(\\vec{l})}$ can be calculated as: $ U_t^{(\\vec{l})} = \\sum _{\\lbrace o_t(1),o_t(2)\\rbrace } U_t^{(\\vec{l})}(o_t(1),o_t(2)) \\textrm {Pr}(o_t(1),o_t(2)),$ where $o_t(1)$ (likewise, $o_t(2)$ ) is the binary indicator of whether channel 1 (or 2) is idle ($o_t(1)=0$ ) or not ($o_t(1)=1$ ) in the $t$ th control slot, and $\\textrm {Pr}(o_t(1),o_t(2))$ denotes the joint probability of both channels' availability status (idle or busy).", "There exist four realizations of the vector $(o_t(1),o_t(2))$ , namely $\\lbrace (0,0),(0,1),(1,0),(1,1)\\rbrace $ .", "In what follows, we show that the value of $U_t^{(\\vec{l})}(o_t(1),o_t(2))$ calculated under each of the realizations is independent of the scheduling decisions $\\vec{l}$ .", "Case 1: When $(o_t(1),o_t(2)) = (0,0)$ .", "In this case, both channels are idle in control slot $t$ .", "Let $\\pi _{t+1}^s(n)$ be the fading state belief on channel $n$ in control slot $t+1$ .", "The expected immediate reward in control slot $t$ , under the greedy policy, can be calculated as $U_t^{(\\vec{l})}(0,0) \\hspace{-2.84526pt}=\\hspace{-2.84526pt} \\pi _{t+1}^s(a_{t+1})p+(1-\\pi _{t+1}^s(a_{t+1}))\\mathrm {T}(\\pi _{t+1}^s(\\tilde{a}_{t+1})),$ where $\\tilde{a}_{t+1} = \\lbrace 1,2\\rbrace \\backslash a_{t+1}$ .", "For notational convenience, we write $\\alpha \\triangleq \\pi _{t+1}^s(a_{t+1})$ and $\\tilde{\\alpha } \\triangleq \\pi _{t+1}^s(\\tilde{a}_{t+1})$ .", "The reward $U_t^{(\\vec{l})}(0,0)$ can now be further expressed as $U_t^{(\\vec{l})}(0,0) &=& p \\alpha + (1-\\alpha ) \\mathrm {T}(\\tilde{\\alpha }) \\nonumber \\\\&=& p \\alpha + (1-\\alpha ) (\\alpha p + (1-\\alpha )r) \\nonumber \\\\&=& p P_1 + r P_2,$ where $P_1 \\triangleq \\alpha + \\tilde{\\alpha } - \\alpha \\tilde{\\alpha }, ~~~P_2 \\triangleq (1-\\alpha )(1-\\tilde{\\alpha }).$ That is, $P_1$ is the probability that at least one of the channels experiences good channel fading in the previous control slot $t+1$ , while $P_2$ is the probability that both channels see bad channel fading.", "It is noted that these probabilities are controlled by the underlying Markov dynamics only, and thus $P_1$ and $P_2$ are independent of the scheduling decisions $\\vec{l}$ .", "Therefore, $U_t^{(\\vec{l})}(0,0)$ is independent of $\\vec{l}$ .", "Case 2: When $(o_t(1),o_t(2)) = (0,1)$ .", "In this case, only channel 1 is idle and can be scheduled.", "The reward $U_t^{(\\vec{l})}(0,1)$ is obtained as $U_t^{(\\vec{l})}(0,1) =\\left\\lbrace \\begin{array}{ll}p \\pi _{t+1}^s(1) + r(1-\\pi _{t+1}^s(1)), &\\hspace{-7.11317pt}\\textrm {if}~ a_{t+1} = 1,\\\\\\mathrm {T}(\\pi _{t+1}^s(1)), &\\hspace{-7.11317pt}\\textrm {if}~ a_{t+1} = 2.\\end{array}\\right.", "\\hspace{-5.69054pt}$ It follows from (REF ) that $U_t^{(\\vec{l})}(0,1)|_{a_{t+1} = 1} = U_t^{(\\vec{l})}(0,1)|_{a_{t+1} = 2},$ i.e., $U_t^{(\\vec{l})}(0,1)$ is independent of $\\vec{l}$ .", "Case 3: When $(o_t(1),o_t(2)) = (1,0)$ .", "Similar to Case 2, only channel 2 can be scheduled in this case, and we have: $U_t^{(\\vec{l})}(1,0)=\\left\\lbrace \\begin{array}{ll}\\mathrm {T}(\\pi _{t+1}^s(2)), &\\hspace{-7.11317pt}\\textrm {if}~ a_{t+1} = 1, \\\\p \\pi _{t+1}^s(2) + r(1-\\pi _{t+1}^s(2)), &\\hspace{-7.11317pt}\\textrm {if}~ a_{t+1} = 2.\\end{array}\\right.", "\\hspace{-5.69054pt}$ Again, this indicates that $U_t^{(\\vec{l})}(1,0)|_{a_{t+1} = 1} = U_t^{(\\vec{l})}(1,0)|_{a_{t+1} = 2},$ i.e., $U_t^{(\\vec{l})}(1,0)$ is independent of $\\vec{l}$ .", "Case 4: When $(o_t(1),o_t(2)) = (1,1)$ .", "In this case, both channels are busy, and it follows immediately that $U_t^{(\\vec{l})}(1,1) = 0.$ Clearly, $U_t^{(\\vec{l})}(1,1)$ is independent of $\\vec{l}$ as well.", "Next, note that $\\textrm {Pr}(o_t(1),o_t(2))$ is a function of the ages $(x_t(1),x_t(2))$ only, which evolve independently from the scheduling decisions $\\vec{l}$ .", "Thereby, we conclude that $\\textrm {Pr}(o_t(1),o_t(2))$ is independent of the scheduling decisions $\\vec{l}$ , and so is the expected immediate reward in control slot $t$ , i.e, $U_t^{\\vec{l}} = U_t$ .", "Now, by extension, we have that the total reward collected from control slot $t$ till the horizon is independent of $\\vec{l}$ , i.e., $\\sum _{t^{\\prime }= t}^1 U_{t^{\\prime }}^{\\vec{l}} = \\sum _{t^{\\prime }=t}^1 U_{t^{\\prime }}.$ Thus, the greedy policy is optimal in control slot $t+1$ as well.", "Since $t \\in \\lbrace m-1,\\ldots \\rbrace $ is arbitrary, the greedy policy is optimal in every control slot $\\lbrace m,\\ldots ,1\\rbrace $ under the induction hypothesis.", "Finally, as the greedy policy is trivially optimal at the horizon, i.e., $t=1$ , using backward induction, we validate the induction hypothesis, and arrive at the conclusion that greedy is optimal in all control slots $t \\in \\lbrace m,\\ldots ,1\\rbrace $ .", "This establishes the proposition.", "Remarks: Note that the tradeoffs inherent in the special case considered above, i.e., $K=1, N=2$ , is more intricate than those observed in related recent works (e.g., [7], [8]), where a control slot coincides with a mini-slot and only one of the states: channel fading or PU occupancy, is considered.", "This is because, despite $K=1$ , the question of “whether to learn a channel that may not be available for scheduling in the near future due to channel occupancy state” still exists.", "Thus the tradeoffs discussed in the preceding subsections are retained in this special case, essentially adding value to our result on greedy optimality.", "In the subsequent section, we proceed to further understand the tradeoffs in the original system by introducing a conceptual “genie-aided system.” Multi-tier Tradeoffs: A Closer Look via A Genie-Aided System In the previous section, we partially showed the interaction between various state elements by examining the immediate reward and certain special cases.", "In order to obtain a more complete understanding of the inherent dynamics, we next introduce a full-observation genie-aided system that helps decompose and characterize the various tiers of the multi-dimensional tradeoffs.", "A Genie-Aided System The genie-aided system is a variant of the original system with the following modification: The spectrum server receives channel fading feedbacks from all the channels and not only the scheduled channel.", "These feedbacks are collected at the same times as those of the feedback from the scheduled channel.", "Thus when the PU returns on the scheduled channel, the feedback from all the channels stop at once.", "Note that this is a conceptual system, without practical significance, which as we will see, is helpful in better understanding the complicated tradeoffs inherent in the original system.", "Tradeoffs Associated with Channel Fading Proposition 4 When the idle ages are the same across all PU channels, it is optimal to schedule the SU to the channel with the highest fading state belief at the moment, i.e., $a_t^* = \\arg \\max _n \\lbrace \\pi _{t,1}^s(n)\\rbrace .$ First, from Proposition REF , the immediate reward is maximized when scheduling the channel with the highest fading state belief at the moment.", "Now, we focus on showing that the future reward is independent of the action in the current control slot and therefore establishing the proposition.", "Specifically, at the current control slot $t$ , $t \\ge 2$ , consider an arbitrary control slot in the future, $t_0 \\in \\lbrace t-1,\\ldots ,1\\rbrace $ .", "In the following, we show that the expected immediate reward in this control slot, denoted by $U_{t_0}$ , is independent of the current action $a_t$ , and thus the future reward is independent of $a_t$ .", "Let $k_0^{\\prime }$ denote the latest idle mini-slot in control slot $t_0 + 1$ before the PU returns.", "The reward $U_{t_0}$ is then calculated as: $ U_{t_0} = \\sum _{k_0^{\\prime } = 1}^K U_{t_0}(k_0^{\\prime }) \\textrm {Pr}(k_0^{\\prime }),$ where $\\textrm {Pr}(k_0^{\\prime })$ is the distribution of $k_0^{\\prime }$ , identical to that of $k_0$ as given in (REF ), and $U_{t_0}(k_0^{\\prime })$ is the expected reward in control slot $t_0$ for a given $k_0^{\\prime }$ .", "In the genie-aided system, the spectrum server obtains the feedbacks of the fading states from all $N$ channels simultaneously, i.e., at the end of mini-slot $k_0^{\\prime }$ .", "Place the channels on which good channel fading is observed at $k_0^{\\prime }$ in the set $\\mathcal {C}_1$ , and the rest in another set $\\mathcal {C}_0$ .", "The characterization of the reward $U_{t_0}(k_0^{\\prime })$ can be further divided into the following cases.", "Case 1: $\\mathcal {C}_0 = \\emptyset $ .", "This corresponds to the case where $f_{t_0+1,k_0^{\\prime }}(n) = 1, \\forall n = 1,\\ldots ,N$ .", "Let $W_{t_0}^{(\\textrm {case} 1)}(n)$ be the expected reward in control slot $t_0$ on channel $n$ in this case.", "We have $ W_{t_0}^{(\\textrm {case} 1)}(n) = \\pi _{t_0,1}^o(n) \\sum _{z=1}^K \\sum _{k=1}^z \\pi _{t_0,k}^s(n) p_z(x_{t_0}(n)),$ where $\\pi _{t_0,1}^s(n) = \\mathrm {T}^{K-k_0^{\\prime }}(p), \\forall n \\in \\mathcal {C}_1$ .", "It follows that $\\pi _{t_0,k}^s(1) = \\ldots = \\pi _{t_0,k}^s(N), \\forall k = 1,\\ldots ,K$ .", "Further, since $x_t(1) = \\ldots =x_t(N)$ , we have $p_z(x_{t_0}(1)) = \\ldots = p_z(x_{t_0}(N))$ , and $\\pi _{t_0,1}^o(1) = \\ldots = \\pi _{t_0,1}^o(N)$ , and therefore, scheduling any of the idle channels in $\\mathcal {C}_1$ achieves the same expected reward in control slot $t_0$ , i.e., $U_{t_0}(k_0^{\\prime })|_{\\textrm {case} 1} = W_{t_0}^{(\\textrm {case} 1)}(n), \\forall n \\in \\mathcal {C}_1, o_{t_0,1}(n)=0.$ Now, note that given equal idle ages on all $N$ channels, the distribution of $k_0^{\\prime }$ is identical across channels.", "Therefore, for all $n=1,\\ldots ,N$ , the values of $\\pi _{t_0,k}^s(n), k=1,\\ldots ,K$ , $\\pi _{t_0,1}^o(n)$ and $p_z(x_{t_0}(n))$ all stay unchanged if a different channel $a_t \\ne a_t^{\\prime }$ is scheduled in the current control slot $t$ .", "This implies that $U_{t_0}(k_0^{\\prime })|_{\\textrm {case} 1}$ is unchanged, and is thus independent of $a_t$ .", "Case 2: $\\mathcal {C}_1 = \\emptyset $ .", "In this case, $f_{t_0+1,k_0^{\\prime }}(n) = 0, \\forall n = 1,\\ldots ,N$ .", "The expected reward collected in control slot $t_0$ on channel $n$ , denoted by $W_{t_0}^{(\\textrm {case} 2)}(n)$ , can be expressed the same as (REF ), where the channel strength belief $\\pi _{t_0,1}^s(n) = \\mathrm {T}^{K-k_0^{\\prime }}(r)$ , for all $n\\in \\mathcal {C}_0$ .", "Then, using the similar reasoning as in Case 1, we obtain that scheduling any of the idle channels in $\\mathcal {C}_0$ achieves the same expected reward in control slot $t_0$ , i.e., $U_{t_0}(k_0^{\\prime })|_{\\textrm {case} 2} = W_{t_0}^{(\\textrm {case} 2)}(n), \\forall n \\in \\mathcal {C}_0, o_{t_0,1}(n)=0.$ Further, the expected reward achieved in this case, $U_{t_0}(k_0^{\\prime })|_{\\textrm {case} 2}$ , does not change when the action in current control slot $t$ varies, i.e., $U_{t_0}(k_0^{\\prime })|_{\\textrm {case} 2}$ is independent of $a_t$ .", "Case 3: $\\mathcal {C}_0 \\ne \\emptyset , \\mathcal {C}_1 \\ne \\emptyset $ .", "In this case, we first show that the maximum expected reward in control slot $t_0$ is achieved by scheduling any of the idle channels in the set $\\mathcal {C}_1$ , which are perceived to have better channel fading state in the subsequent control slot $t_0$ than the channels in set $\\mathcal {C}_0$ .", "Specifically, picking any one of the channels from each of the set, $n_1 \\in \\mathcal {C}_1$ and $n_0 \\in \\mathcal {C}_0$ , we have $W_{t_0}^{(\\textrm {case} 3)}(n_1) \\hspace{-5.69054pt} &=&\\hspace{-5.69054pt} \\sum _{z=1}^K \\sum _{k=1}^z \\pi _{t_0,k}^s(n_1) p_z(x_{t_0}(n_1)), \\nonumber \\\\W_{t_0}^{(\\textrm {case} 3)}(n_0) \\hspace{-5.69054pt} &=&\\hspace{-5.69054pt} \\sum _{z=1}^K \\sum _{k=1}^z \\pi _{t_0,k}^s(n_0) p_z(x_{t_0}(n_0)),$ where $\\pi _{t_0,k}^s(n_1) &=& \\mathrm {T}^{K-k_0^{\\prime }+k-1}(p),\\nonumber \\\\\\pi _{t_0,k}^s(n_0) &=& \\mathrm {T}^{K-k_0^{\\prime }+k-1}(r).$ Based on (REF ) and the property of the positively-correlated Markov chain, the following inequality holds: For all $k = 1,\\ldots ,K$ , $\\pi _{t_0^{\\prime },k}^s(n_1) = \\mathrm {T}^{K-k_0^{\\prime }+k-1}(p) \\ge \\mathrm {T}^{K-k_0^{\\prime }+k-1}(r) = \\pi _{t_0,k}^s(n_0),$ with the equality achieved when $K \\rightarrow \\infty $ .", "Further, applying similar argument as before, we obtain $W_{t_0}^{(\\textrm {case} 3)}(n_1) > W_{t_0}^{(\\textrm {case} 3)}(n_0).$ Now, since $n_1 \\in \\mathcal {C}_1$ (likewise, $n_0 \\in \\mathcal {C}_0$ ) is arbitrary, from the conclusion drawn in Case 1, the maximal expected reward in control slot $t$ under this case is achieved by scheduling the SU to any of the idle channels in the set $\\mathcal {C}_1$ , i.e., $U_{t_0}(k_0^{\\prime })|_{\\textrm {case} 3} = W_{t_0}^{(\\textrm {case} 3)}(n_1), \\forall n_1 \\in \\mathcal {C}_1, o_{t_0,1}(n_1)=0.$ Next, based on the similar reasoning as in the previous cases, we know that $U_{t_0}(k_0^{\\prime })|_{\\textrm {case} 3}$ is independent of $a_t$ .", "Finally, note that $U_{t_0}(k_0^{\\prime })$ can be written as: $U_{t_0}(k_0^{\\prime }) = \\sum _{i=1}^3 U_{t_0}(k_0^{\\prime })|_{\\textrm {case}~i} \\textrm {Pr} (\\textrm {Case}~ i),$ where $ \\textrm {Pr} (\\textrm {Case} 1) &=& \\prod _{n \\in \\mathcal {C}_1} \\pi _{t_0+1,k_0^{\\prime }}^s(n), \\nonumber \\\\\\textrm {Pr} (\\textrm {Case} 2) &=& \\prod _{n \\in \\mathcal {C}_0} (1-\\pi _{t_0+1,k_0^{\\prime }}^s(n)), \\nonumber \\\\\\textrm {Pr} (\\textrm {Case} 3) &=& \\prod _{n \\in \\mathcal {C}_1} \\pi _{t_0+1,k_0^{\\prime }}^s(n) \\prod _{n^{\\prime }\\in \\mathcal {C}_0} (1-\\pi _{t_0+1,k_0^{\\prime }}^s(n^{\\prime })),$ are quantities dependent on $k_0^{\\prime }$ only.", "Based on the fact that the idle age at the moment are identical across the channels, the probabilities $\\textrm {Pr} (\\textrm {Case}~i), i=1,2,3$ , and the distribution $\\textrm {Pr}(k_0^{\\prime })$ , are the same across the channels as well.", "Thus $U_{t_0}$ is independent of $a_t$ .", "Since $t_0 \\in \\lbrace t-1,\\ldots ,1\\rbrace $ is arbitrary, by extension, we have that the total reward collected from control slot $t_0$ till the horizon, i.e., $\\sum _{t^{\\prime } = t_0}^1 U_{t^{\\prime }}$ , which is the future reward of current control slot $t$ , is independent from $a_t$ .", "This concludes the proof and establishes the proposition.", "Remarks: Proposition REF illustrates the effect of fading state belief on the optimal decisions in the genie-aided system.", "With ages equalized across the PU channels and the classic “exploitation vs. exploration” tradeoff neutralized (by definition of the genie-aided system), we saw that, higher fading belief favors the immediate reward and that the future reward is, in fact, independent of the current decision.", "In the following, we study the effect of PU occupancy and age on the optimal decisions in the genie-aided system and, in turn, its impact on the original system.", "Tradeoffs Associated with PU Occupancy The following proposition identifies the effect of channel occupancy state on the optimal scheduling decisions.", "Proposition 5 When the fading state beliefs are the same across all PU channels, it is optimal to schedule the SU to the channel with the lowest idle age at the moment, i.e., $a_t^* = \\arg \\min _n \\lbrace x_t(n)\\rbrace .$ We prove the proposition by showing that scheduling the channel with the lowest idle age favors: [1)] the immediate reward $R_t(\\textbf {S}_t,a_t)$ ; and the optimal future reward $V_{t-1}^*(\\textbf {S}_{t-1})$ .", "The first part can be readily shown by appealing to Proposition REF and Corollary REF .", "To show the second part, we adopt the following induction hypothesis: Induction Hypothesis: The optimal future reward is convex in the fading state belief.", "When channel $a_t$ is scheduled in the current control slot $t$ , the expected future reward can be evaluated as: $V_{t-1}^*(\\mathbf {S}_{t-1})|_{a_t}=\\pi _{t,k_0}^s(a_t) V_{t-1}^*(p) + (1-\\pi _{t,k_0}^s(a_t))V_{t-1}^*(r),$ where $V_{t-1}^*(p)$ and $V_{t-1}^*(r)$ represent the future reward calculated when the channel fading state observed in the $k_0$ th mini-slot of control slot $t$ is good or bad, respectively.", "More specifically, $V_{t-1}^*(p) \\triangleq V_{t-1}^*\\left(\\pi _{t-1,1}^s(a_t) = \\mathrm {T}^{K-k_0}(p)\\right), \\nonumber \\\\V_{t-1}^*(r) \\triangleq V_{t-1}^*\\left(\\pi _{t-1,1}^s(a_t) = \\mathrm {T}^{K-k_0}(r)\\right).$ Based on (REF ), we have, for $\\gamma \\in (0,1)$ , $\\mathrm {T}^L(\\gamma ) = (p-r)^L\\gamma + r\\frac{1-(p-r)^L}{1-(p-r)}, L = 0,1,\\ldots ,$ and $\\lim _{L \\rightarrow \\infty } \\mathrm {T}^L(\\gamma ) = \\frac{r}{1-p+r} \\triangleq \\pi _{s|s}$ , where $\\pi _{s|s}$ denotes the steady-state probability of perceiving good channel fading on any of the PU channels.", "This indicates that, a smaller $k_0$ , associated with a higher idle age at the moment (recall discussions in Section ), results in a larger $K-k_0$ and thus a value of $\\pi _{t-1,1}^s(a_t)$ closer to $\\pi _{s|s}$ , in which case, $V_{t-1}^*(\\mathbf {S}_{t-1})|_{a_t} \\hspace{-9.95845pt}&\\rightarrow &\\hspace{-9.95845pt} \\pi _{t,k_0}^s(a_t) V_{t-1}^*(\\pi _{s|s}) \\hspace{-3.41432pt}+\\hspace{-3.41432pt} (1-\\pi _{t,k_0}^s(a_t))V_{t-1}^*(\\pi _{s|s})\\triangleq V_{t-1}^*(a_t,E_{\\pi _{s|s}}).$ On the contrary, as $k_0$ becomes larger because of a lower idle age, the value of $\\pi _{t-1,1}^s(a_t)$ deviates further away from $\\pi _{s|s}$ , but is closer to $p$ (or $r$ ).", "Also, $\\pi _{t,k_0}^s(a_t)$ gets closer to $\\pi _{s|s}$ .", "Therefore, $V_{t-1}^*(\\mathbf {S}_{t-1})|_{a_t} \\hspace{-9.95845pt}&\\rightarrow &\\hspace{-9.95845pt} \\pi _{s|s} V_{t-1}^*(p) \\hspace{-3.1298pt}+\\hspace{-3.1298pt} (1-\\pi _{s|s}) V_{t-1}^*(r) \\triangleq E_{\\pi _{s|s}} V_t^*(a_t).$ Now, appealing to the induction hypothesis and the Jensen's inequality [18], we have that $E_{\\pi _{s|s}} V_t^*(a_t) > V_{t-1}^*(a_t,E_{\\pi _{s|s}})$ , and therefore the future reward is maximized by scheduling the channel with the lowest idle age, i.e., $a_t^* = \\min _{a_t}\\lbrace x_t(a_t)\\rbrace ~~s.t.~~ V_{t-1}^*(\\textbf {S}_{t-1})|_{a_t^*} = \\max _{a_t}\\lbrace V_{t-1}^*(\\textbf {S}_{t-1})|_{a_t}\\rbrace .$ Finally, we proceed to verify the induction hypothesis.", "At $t=2$ , the optimal future reward equals the optimal immediate reward at the horizon $t=1$ , i.e., $V_1^*(\\textbf {S}_1) = R_1(\\textbf {S}_1, a_1^*) := \\max _{a_1\\in \\mathcal {A}_1}\\lbrace R_1(\\textbf {S}_1,a_1)\\rbrace .$ Since $R_1(\\textbf {S}_1,a_1)$ is linear in the strength belief, using the property of convex function [18], we have that $V_1^*(\\textbf {S}_1)$ is convex in the strength beliefs, which validates the induction hypothesis.", "Then, using backward induction, we establish the proposition.", "Remarks: Proposition REF explicitly illustrates the effect of the PU occupancy and age on the optimal decisions.", "With fading state beliefs equalized and the classic “exploitation vs. exploration” tradeoff neutralized (by definition of the genie-aided system), we saw that: [1)] lower idle age on the PU channel favors the immediate reward by allowing more idle time on the channel; and lower idle age also favors the future reward by way of better freshness of the channel fading feedback.", "Thus by studying the full-observation genie-aided system, via the results in Propositions REF and REF , we have decomposed the tradeoffs associated with the channel occupancy and the fading state beliefs in the original system.", "Indeed, the results in Propositions REF and REF rigorously support the understanding we developed earlier in Section  on the tradeoffs in the original system.", "Numerical Results & Further Discussions In this section, we evaluate and compare the optimal rewards of the original system (denoted as $V_{ori}^*$ ) and the genie-aided system ($V_{genie}^*$ ).", "Also, the optimal policy in the original system is compared to a randomized scheduling policy (with reward denoted as $V_{random}^*$ ), where the spectrum server chooses a channel, among all the idle ones, randomly and uniformly, and allocates it to the SU for data transmission.", "The numerical results are collected for a two channel system with $K=2$ , horizon length $m=6$ , and discounted factor $\\beta = 0.9$ .", "For notational convenience, denote $\\Delta _{{ga}-{ori}} = V_{genie}^* - V_{ori}^*$ and $\\Delta _{{ori}-{rnd}} = V_{ori}^* - V_{random}^*$ .", "Table REF records the rewards obtained under various baseline cases, for various values of $\\delta \\triangleq p-r$ , which broadly captures the temporal memory in the channel fading.", "In particular, as $\\delta $ decreases, the channel fading memory fades and the difference between the baselines, which are primarily differentiated by the degree to which they exploit the memory in the system, tends to decrease.", "This is observed from Table REF .", "Table: Comparison of rewards when the channel fading memory varies.", "System parameters used: u=1,C I =1,C B =2,x t (1)=10,x t (2)=5,π t,1 s (1)=0.4,π t,1 s (2)=0.7u=1,C_I = 1, C_B = 2, x_t(1) = 10, x_t(2) = 5, \\pi _{t,1}^s(1) = 0.4, \\pi _{t,1}^s(2) = 0.7Next, in Table REF , we compare the baseline rewards under different values of the power exponent $u$ , used in the definitions of $P_I$ and $P_B$ in (REF ).", "To build a better understanding of the trend reflected in these numerical results, we consider an arbitrary mini-slot, denoted as $k^{\\prime }$ , as the current mini-slot, and the following two exhaustive sets of PU occupancy histories: [1)] the set of histories $h_x^I, x=1,2,\\ldots $ , corresponds to the case when there are exactly $x$ mini-slots between the current mini-slot $k^{\\prime }$ and the most recent mini-slot (preceding $k^{\\prime }$ ) when the channel was in busy state, i.e., the idle age of mini-slot $k^{\\prime }$ is $x$ ; and the set of histories $h_x^B, x=1,2,\\ldots $ , corresponds to the case when there are exactly $x$ mini-slots between the current mini-slot $k^{\\prime }$ and the most recent mini-slot (preceding $k^{\\prime }$ ) when the channel was in idle state, i.e., the busy age of mini-slot $k^{\\prime }$ is $x$ .", "In Fig.", "REF , we plot the two sets of occupancy histories.", "As has been pointed out in Section REF , the idle/busy age is a sufficient statistic for capturing the memory in the PU channels' occupancy states.", "Thereby, with the construction of $h_x^I$ and $h_x^B$ , we can examine the effect of the temporal correlation of PU occupancy on the system performance.", "Specifically, the conditional idle probability in mini-slot $k^{\\prime }$ , given occupancy history, can be obtained as: $\\pi _{k^{\\prime }}^o |_{h_x^I} \\hspace{-5.69054pt}&=&\\hspace{-5.69054pt} P_I(x) = \\frac{1}{x^u+C_I},\\nonumber \\\\\\pi _{k^{\\prime }}^o |_{h_x^B} \\hspace{-5.69054pt}&=&\\hspace{-5.69054pt} 1 - P_B(x) = 1 - \\frac{1}{x^u+C_B}.$ As an example, we plot $\\pi _{k^{\\prime }}^o |_{h_x^I}$ in Fig.", "REF .", "It is clear that as $u$ increases, the conditional probability curves become steeper.", "Define the threshold point, $x_0$ , such that for all $x>x_0$ , the difference in $\\pi _{k^{\\prime }}^o |_{h_x^I}$ is insignificant (below $10^{-2}$ ).", "Now, note from the figure that the threshold $x_0$ decreases with increasing $u$ , i.e., $x_0^{(u=5)} < x_0^{(u=3)} < x_0^{(u=1)}$ .", "This indicates that the impact of different occupancy histories on the current PU occupancy state diminishes with increasing $u$ and thus a decreased memory in the PU occupancy.", "Similar argument holds when considering $h_x^B$ .", "In short, the exponent $u$ broadly captures the PU occupancy memory, and as its value increases, the memory decreases and thus the rewards under various cases, as expected, come closer with increasing $u$ .", "This is illustrated in Table REF .", "Figure: Occupancy histories & the conditional idle probability.Table: Comparison of rewards when the occupancy memory varies.", "System parameters used: p=0.9,r=0.1,C I =1,C B =2,x t (1)=0,x t (2)=1,π t,1 s (1)=0.4,π t,1 s (2)=0.7p = 0.9, r = 0.1, C_I = 1, C_B = 2, x_t(1) = 0, x_t(2) = 1, \\pi _{t,1}^s(1) = 0.4, \\pi _{t,1}^s(2) = 0.7Finally, as can be seen from both tables, the original system performs very close to the genie-aided system, while the cost of measuring and sending the channel fading feedback is only $\\frac{1}{N}$ of the latter.", "Also, the optimal policy significantly outperforms the randomized policy, indicating that the temporal correlation structure in the channel fading and PU occupancy can greatly benefit the opportunistic spectrum scheduling, when appropriately exploited.", "Conclusions In this work, we studied opportunistic spectrum access for a single SU in a CR network with multiple PU channels.", "We formulated the problem as a partially observable Markov decision process, and examined the intricate tradeoffs in the optimal scheduling process, when incorporating the temporal correlation in both the channel fading and PU occupancy states.", "We modeled the channel fading variation with a two-state first-order Markov chain.", "The temporal correlation of PU traffic was modeled using age and a class of monotonically decreasing functions, which can have a long memory.", "The optimality of the simple greedy policy was established under certain conditions.", "For the general case, we individually studied the tradeoffs in the immediate reward, and the total reward.", "Further, by developing a genie-aided system with full observation of the channel fading feedbacks, we decomposed and characterized the multiple tiers of the intricate tradeoffs in the original system.", "Finally, we numerically studied the performance of the two systems and showed that the original system achieved an optimal total reward very close (within $1\\%$ ) to that of the genie-aided system.", "Further, the optimal policy in the original system significantly outperformed randomized scheduling, highlighting the merit of exploiting the temporal correlation in the system states.", "We believe that our formulation and the insights we have obtained open up new horizons in better understanding spectrum allocation in cognitive radio networks, with problem settings that go beyond the traditional ones." ], [ "Fundamental Tradeoffs", "The decision on opportunistic spectrum scheduling is made based on two sets of system states: the PU occupancy on the channel and the channel fading perceived by the SU.", "On one hand, PUs may return in the middle of a control slot and hinder further transmissions of the SU, leading to a decreased reward for the SU.", "The temporal memory resident in the PU occupancy suggests that the past history of channel's occupancy, measured by the age, influences the occupancy state of the channel in the future.", "On the other hand, the PU channels may suffer from “bad” channel fading in the middle of a control slot, even if a PU does not return to hinder SU's transmissions..", "Similar to the PU occupancy, the historic observation on the fading process would help determine the expected channel fading in the future.", "Note that by way of the channel feedback arrangement, an observation of a PU channel fading is made only when that channel is scheduled to the SU.", "Thus scheduling is inherently tied to channel fading learning.", "Roughly speaking, to maximize the SU's reward, the spectrum server must schedule a channel such that the combination of the perceived channel occupancy and channel fading strikes a “perfect” balance between the immediate gains and channel learning for future gains.", "We discuss this intricate tradeoff in the following." ], [ "Classic “Exploitation vs. Exploration” Tradeoff", "In the existing literature (e.g., [7], [8], [10], [11], [12]), focus has been cast on considering only one of the factors: either channel fading or channel occupancy, along with the associated temporal correlation.", "The optimal decision is a mapping that best balances the tradeoff of “exploitation” and “exploration” on the single factor being considered.", "The exploitation side lets the scheduler choose the channel with the best perceived channel fading (or occupancy state) at the moment, corresponding to immediate gains; while the exploration side tends to favor the channel with the least learnt information so far, probing which can contribute to the overall understanding of the channel fading (or occupancy state) in the network, and thus better opportunistic scheduling decisions in the future." ], [ "“Exploitation vs. Exploration” Tradeoff in Dynamics of Both Channel Fading and PU Occupancy States", "In contrast to the existing works, we examine the tradeoffs when the temporal correlation in both the channel fading and PU occupancy are considered.", "While the classic tradeoff described above apparently exists, additional tradeoffs arise in our context due to the interactions between the two sets of system states.", "In particular, the long temporal memory in the PU occupancy state adds a new layer to the tradeoffs inherent in the problem.", "For instance, note that the SU can only transmit on idle channels.", "To carry out exploration, the channel being favored in the traditional sense, i.e., the one on which the least information is available, may no longer be the preferred choice if this channel is perceived to be unavailable (i.e., busy) for a prolonged duration in the near future.", "In other words, it may not be worth learning the channel as the SU cannot utilize the learned knowledge in the near future.", "Fig.", "REF is a pictorial illustration of the impact from the occupancy state on the SU's expected reward.", "The history of occupancy, represented by the idle ages $x_t(1)$ and $x_2(t)$ , affects both the immediate and future rewards of the SU.", "Specifically, as the idle age increases, the temporal memory in the PU's occupancy pushes the channel to transit to busy sooner (i.e., time point $b$ comes earlier than $a$ in the figure).", "Therefore, the average availability on the PU channel in the current control slot decreases, which leads to a smaller immediate reward for the SU.", "Further, note that the latest mini-slot for which the spectrum server receives channel fading feedback is also the last mini-slot before the PU returns, i.e., time points $a$ and $b$ respectively for the two cases in Fig.", "REF .", "The duration $d_1$ (likewise, $d_2$ ) in Fig.", "REF is an indication of how “fresh” the channel fading information is for the scheduling decision at the beginning of the next control slot, i.e., $t-1$ .", "With $d_1 < d_2$ , channel feedback is more fresh in the former case, with a lower idle age $x_t(1)$ .", "Thus, age, through its effect on the freshness of feedback, and the availability of the PU channels in the future slots, adds another layer to the tradeoffs, thereby influencing the optimal scheduling decision.", "Figure: An illustration of the impact of age on the SU's reward.To better perceive the intricate tradeoffs in the system, we proceed, in what follows, with a number of break-down results that aim at illustrating each tier of the tradeoffs progressively." ], [ "Tradeoffs Inherent in Immediate Reward", "Consider the PU channel scheduled in the current control slot $t$ .", "Let $k_0$ denote the latest mini-slot before the PU of the scheduled channel returns in the current control slot.", "Clearly, $k_0$ is a random variable, taking values in $k_0 \\in \\lbrace 1,\\ldots ,K\\rbrace $ .", "Let $p_z \\triangleq \\textrm {Pr}(k_0 = z)$ .", "With $x$ denoting the idle age of the scheduled channel, $k_0$ is distributed as follows: For $K=2$ : $p_1 = 1 - P_I(x+1),~~~p_2 = P_I(x+1),$ and for $K \\ge 3$ : $ p_1 &= 1 - P_I(x+1), \\nonumber \\\\p_z &= \\prod _{j=1}^{z-1} P_I(x+j) (1-P_I(x+z)),z=2,\\ldots ,K-1,\\nonumber \\\\p_K &= \\prod _{j=1}^{K-1} P_I(x+j).$ In the following lemma, we establish the structure of the distribution of $k_0$ .", "Lemma 1 The distribution of $k_0$ is monotonically decreasing in the idle age, $x_t$ , for $z = 2,\\ldots , K$ , and monotonically increasing in $x_t$ , for $z=1$ .", "For $K=2$ , it is straightforward to establish the conclusion.", "We next focus on $K\\ge 3$ .", "First, it is easy to show that for $z=1$ , $p_1$ is monotonically increasing in $x$ since $P_I(x)$ is monotonically decreasing.", "Further, note that if two positively valued functions, $f_1(x) > 0$ and $f_2(x) > 0$ , are both monotonically decreasing in $x$ , i.e., for any positive integer $\\Delta \\ge 1$ , $f_1(x) > f_1(x + \\Delta ),~~~f_2(x) > f_2(x+\\Delta ),$ then the product of the two, $f_1(x)f_2(x)$ , is also a decreasing function since $f_1(x) f_2(x) > f_1(x + \\Delta ) f_2(x+\\Delta ).$ Therefore, $p_K$ is monotonically decreasing in $x$ .", "Next, we show that $p_z,z=2,\\ldots ,K-1$ are monotonically decreasing in $x$ .", "Based on the above argument, it is sufficient to show that for any $z = 2,\\ldots , K-1$ , the following function, $g(x) \\triangleq P_I(x+z-1) (1-P_I(x+z)),$ is monotonically decreasing.", "We have the following simplifications: For $\\Delta \\ge 1$ , a positive integer, $\\frac{g(x)}{g(x+\\Delta )} \\hspace{-8.53581pt}&=&\\hspace{-8.53581pt} \\frac{P_I(x+z-1) (1-P_I(x+z))}{P_I(x+\\Delta +z-1) (1-P_I(x+\\Delta +z))} \\nonumber \\\\\\hspace{-8.53581pt}&=&\\hspace{-8.53581pt} \\frac{(x+\\Delta +z-1)^u + C_I}{(x+z)^u + C_I} \\hspace{-2.84526pt}\\cdot \\hspace{-2.84526pt} \\frac{(x+\\Delta +z)^u + C_I}{(x+\\Delta +z)^u + C_I-1}\\cdot \\frac{(x+z)^u + C_I-1}{(x+z-1)^u + C_I}\\nonumber \\\\&\\triangleq & {B}_1 {B}_2 {B}_3.$ Since $x \\ge 0, C_I > 0, u\\ge 1$ and $\\Delta \\ge 1$ , we immediately have $B_1 \\ge 1$ , $B_2 > 1$ .", "Further, using the binomial theorem, we obtain $(x+z)^u= \\sum _{i=0}^u {u\\atopwithdelims ()i} (x+z-1)^i \\ge (x+z-1)^u + 1,$ and hence $B_3 \\ge 1$ .", "As a result, $\\frac{g(x)}{g(x+\\Delta )} > 1$ and $p_2, \\ldots , p_{K-1}$ are monotonically decreasing in $x$ .", "This concludes the proof.", "We next present a result that demonstrates the tradeoff inherent in the immediate reward with respect to age.", "Proposition 1 The immediate reward on the scheduled channel is monotonically decreasing in the idle age.", "As the system model implies, we can rewrite the immediate reward as the following weighted sum: $ R_t(\\textbf {S}_t,a_t) = \\sum _{z = 1}^K \\sum _{k=1}^z \\pi _{t,k}^s(a_t)p_z,$ where $p_z = \\textrm {Pr}(k_0 = z)$ is given by (REF ).", "When $K=2$ , with idle age on $a_t$ being $x_t$ , we have $R_t(\\textbf {S}_t,a_t) = \\pi _{t,1}^s(a_t) + P_I(x_t+1) \\pi _{t,2}^s(a_t)$ .", "Apparently, it increases as $x_t$ decreases.", "For $K\\ge 3$ , denote by $\\theta _z \\triangleq \\sum _{k=1}^z \\pi _{t,k}^s(a_t), z = 1,\\ldots ,K$ .", "It is clear that $\\theta _1 < \\theta _2 < \\cdots < \\theta _K,$ and $\\lbrace \\theta _z\\rbrace $ 's are constants in the idle age $x_t$ .", "To emphasize the role of the argument $x_t$ , we rewrite $R_t(\\textbf {S}_t,a_t)$ as $R_t(x_t)$ , and $p_z$ as $p_z(x_t)$ .", "Utilizing the result of Lemma REF , and noting that for any positive integer $\\Delta \\ge 1$ , $|p_1(x_t) - p_1(x_t + \\Delta )| = \\sum _{z=2}^K (p_z(x_t) - p_z(x_t + \\Delta ))$ , we obtain: $R_t(x_t) -R_t(x_t+\\Delta ) =\\sum _{z=1}^K \\theta _z \\left(p_z(x_t) - p_z(x_t + \\Delta )\\right)> \\theta _1(p_1(x_t) - p_1(x_t + \\Delta )) + \\theta _2 \\sum _{z=2}^K (p_z(x_t) - p_z(x_t + \\Delta ))> 0,$ i.e., $R_t(x_t)$ is monotonically decreasing in the idle age $x_t$ , and this establishes the proposition.", "The above result can be readily extended to the following corollary.", "Corollary 1 When all the PU channels have equal fading state beliefs, the immediate reward is maximized by scheduling the SU to the channel with the lowest idle age.", "Next, recall that the channel fading is modeled by a positively-correlated Markov chain.", "Hence, if $\\pi _{t,1}^s(a_t) > \\pi _{t,1}^{s^{\\prime }}(a_t)$ , then the inequality $\\pi _{t,k}^s(a_t) > \\pi _{t,k}^{s^{\\prime }}(a_t)$ holds, for all $k =2,\\ldots ,K$ .", "We present the following proposition without further proof.", "Proposition 2 The immediate reward on the scheduled channel is monotonically increasing in its fading state belief at the moment.", "Further, given equal idle ages across all PU channels, the immediate reward is maximized by scheduling the SU to the channel with the largest fading state belief value at the moment." ], [ "Tradeoffs Inherent in Total Reward", "In this subsection, we illustrate the tradeoffs inherent in the total reward by examining a special case with two channels $N=2$ and number of mini-slots $K=1$ .", "In particular, we show that under these conditions, a simple greedy scheduling policy is optimal.", "The greedy policy is formally defined as follows: In any control slot, the greedy decision maximizes the immediate reward, ignoring the future rewards, i.e., $\\hat{a}_t = \\max _{a_t\\in {\\mathcal {A}_t}} \\lbrace R_t(\\textbf {S}_t,a_t)\\rbrace .$ We now formally record the result on greedy policy optimality in the following proposition.", "Proposition 3 The greedy policy is optimal when $K=1$ and $N=2$ .", "To prove the proposition, we begin with the following induction hypothesis: Induction Hypothesis: With the length of the horizon denoted by $m, m \\ge 2$ , assume that greedy policy is optimal in all the control slots $t\\in \\lbrace m-1, \\ldots ,1\\rbrace $ .", "The proof proceeds in two steps: First, we fix a sequence of scheduling decisions $\\vec{l} := \\lbrace a_{m},\\ldots ,a_{t+1}\\rbrace $ , and show that the expected immediate reward in control slot $t$ , under the greedy policy, is independent of the scheduling decisions $\\vec{l}$ .", "Then, we provide induction based arguments to validate the induction hypothesis and hence establish that the greedy policy is optimal in all the control slots.", "Let $U_t^{(\\vec{l})}$ denote the expected immediate reward in slot $t \\in \\lbrace m-1,\\ldots ,1\\rbrace $ , given the scheduling decisions $\\vec{l}$ .", "$U_t^{(\\vec{l})}$ can be calculated as: $ U_t^{(\\vec{l})} = \\sum _{\\lbrace o_t(1),o_t(2)\\rbrace } U_t^{(\\vec{l})}(o_t(1),o_t(2)) \\textrm {Pr}(o_t(1),o_t(2)),$ where $o_t(1)$ (likewise, $o_t(2)$ ) is the binary indicator of whether channel 1 (or 2) is idle ($o_t(1)=0$ ) or not ($o_t(1)=1$ ) in the $t$ th control slot, and $\\textrm {Pr}(o_t(1),o_t(2))$ denotes the joint probability of both channels' availability status (idle or busy).", "There exist four realizations of the vector $(o_t(1),o_t(2))$ , namely $\\lbrace (0,0),(0,1),(1,0),(1,1)\\rbrace $ .", "In what follows, we show that the value of $U_t^{(\\vec{l})}(o_t(1),o_t(2))$ calculated under each of the realizations is independent of the scheduling decisions $\\vec{l}$ .", "Case 1: When $(o_t(1),o_t(2)) = (0,0)$ .", "In this case, both channels are idle in control slot $t$ .", "Let $\\pi _{t+1}^s(n)$ be the fading state belief on channel $n$ in control slot $t+1$ .", "The expected immediate reward in control slot $t$ , under the greedy policy, can be calculated as $U_t^{(\\vec{l})}(0,0) \\hspace{-2.84526pt}=\\hspace{-2.84526pt} \\pi _{t+1}^s(a_{t+1})p+(1-\\pi _{t+1}^s(a_{t+1}))\\mathrm {T}(\\pi _{t+1}^s(\\tilde{a}_{t+1})),$ where $\\tilde{a}_{t+1} = \\lbrace 1,2\\rbrace \\backslash a_{t+1}$ .", "For notational convenience, we write $\\alpha \\triangleq \\pi _{t+1}^s(a_{t+1})$ and $\\tilde{\\alpha } \\triangleq \\pi _{t+1}^s(\\tilde{a}_{t+1})$ .", "The reward $U_t^{(\\vec{l})}(0,0)$ can now be further expressed as $U_t^{(\\vec{l})}(0,0) &=& p \\alpha + (1-\\alpha ) \\mathrm {T}(\\tilde{\\alpha }) \\nonumber \\\\&=& p \\alpha + (1-\\alpha ) (\\alpha p + (1-\\alpha )r) \\nonumber \\\\&=& p P_1 + r P_2,$ where $P_1 \\triangleq \\alpha + \\tilde{\\alpha } - \\alpha \\tilde{\\alpha }, ~~~P_2 \\triangleq (1-\\alpha )(1-\\tilde{\\alpha }).$ That is, $P_1$ is the probability that at least one of the channels experiences good channel fading in the previous control slot $t+1$ , while $P_2$ is the probability that both channels see bad channel fading.", "It is noted that these probabilities are controlled by the underlying Markov dynamics only, and thus $P_1$ and $P_2$ are independent of the scheduling decisions $\\vec{l}$ .", "Therefore, $U_t^{(\\vec{l})}(0,0)$ is independent of $\\vec{l}$ .", "Case 2: When $(o_t(1),o_t(2)) = (0,1)$ .", "In this case, only channel 1 is idle and can be scheduled.", "The reward $U_t^{(\\vec{l})}(0,1)$ is obtained as $U_t^{(\\vec{l})}(0,1) =\\left\\lbrace \\begin{array}{ll}p \\pi _{t+1}^s(1) + r(1-\\pi _{t+1}^s(1)), &\\hspace{-7.11317pt}\\textrm {if}~ a_{t+1} = 1,\\\\\\mathrm {T}(\\pi _{t+1}^s(1)), &\\hspace{-7.11317pt}\\textrm {if}~ a_{t+1} = 2.\\end{array}\\right.", "\\hspace{-5.69054pt}$ It follows from (REF ) that $U_t^{(\\vec{l})}(0,1)|_{a_{t+1} = 1} = U_t^{(\\vec{l})}(0,1)|_{a_{t+1} = 2},$ i.e., $U_t^{(\\vec{l})}(0,1)$ is independent of $\\vec{l}$ .", "Case 3: When $(o_t(1),o_t(2)) = (1,0)$ .", "Similar to Case 2, only channel 2 can be scheduled in this case, and we have: $U_t^{(\\vec{l})}(1,0)=\\left\\lbrace \\begin{array}{ll}\\mathrm {T}(\\pi _{t+1}^s(2)), &\\hspace{-7.11317pt}\\textrm {if}~ a_{t+1} = 1, \\\\p \\pi _{t+1}^s(2) + r(1-\\pi _{t+1}^s(2)), &\\hspace{-7.11317pt}\\textrm {if}~ a_{t+1} = 2.\\end{array}\\right.", "\\hspace{-5.69054pt}$ Again, this indicates that $U_t^{(\\vec{l})}(1,0)|_{a_{t+1} = 1} = U_t^{(\\vec{l})}(1,0)|_{a_{t+1} = 2},$ i.e., $U_t^{(\\vec{l})}(1,0)$ is independent of $\\vec{l}$ .", "Case 4: When $(o_t(1),o_t(2)) = (1,1)$ .", "In this case, both channels are busy, and it follows immediately that $U_t^{(\\vec{l})}(1,1) = 0.$ Clearly, $U_t^{(\\vec{l})}(1,1)$ is independent of $\\vec{l}$ as well.", "Next, note that $\\textrm {Pr}(o_t(1),o_t(2))$ is a function of the ages $(x_t(1),x_t(2))$ only, which evolve independently from the scheduling decisions $\\vec{l}$ .", "Thereby, we conclude that $\\textrm {Pr}(o_t(1),o_t(2))$ is independent of the scheduling decisions $\\vec{l}$ , and so is the expected immediate reward in control slot $t$ , i.e, $U_t^{\\vec{l}} = U_t$ .", "Now, by extension, we have that the total reward collected from control slot $t$ till the horizon is independent of $\\vec{l}$ , i.e., $\\sum _{t^{\\prime }= t}^1 U_{t^{\\prime }}^{\\vec{l}} = \\sum _{t^{\\prime }=t}^1 U_{t^{\\prime }}.$ Thus, the greedy policy is optimal in control slot $t+1$ as well.", "Since $t \\in \\lbrace m-1,\\ldots \\rbrace $ is arbitrary, the greedy policy is optimal in every control slot $\\lbrace m,\\ldots ,1\\rbrace $ under the induction hypothesis.", "Finally, as the greedy policy is trivially optimal at the horizon, i.e., $t=1$ , using backward induction, we validate the induction hypothesis, and arrive at the conclusion that greedy is optimal in all control slots $t \\in \\lbrace m,\\ldots ,1\\rbrace $ .", "This establishes the proposition.", "Remarks: Note that the tradeoffs inherent in the special case considered above, i.e., $K=1, N=2$ , is more intricate than those observed in related recent works (e.g., [7], [8]), where a control slot coincides with a mini-slot and only one of the states: channel fading or PU occupancy, is considered.", "This is because, despite $K=1$ , the question of “whether to learn a channel that may not be available for scheduling in the near future due to channel occupancy state” still exists.", "Thus the tradeoffs discussed in the preceding subsections are retained in this special case, essentially adding value to our result on greedy optimality.", "In the subsequent section, we proceed to further understand the tradeoffs in the original system by introducing a conceptual “genie-aided system.” In the previous section, we partially showed the interaction between various state elements by examining the immediate reward and certain special cases.", "In order to obtain a more complete understanding of the inherent dynamics, we next introduce a full-observation genie-aided system that helps decompose and characterize the various tiers of the multi-dimensional tradeoffs." ], [ "A Genie-Aided System", "The genie-aided system is a variant of the original system with the following modification: The spectrum server receives channel fading feedbacks from all the channels and not only the scheduled channel.", "These feedbacks are collected at the same times as those of the feedback from the scheduled channel.", "Thus when the PU returns on the scheduled channel, the feedback from all the channels stop at once.", "Note that this is a conceptual system, without practical significance, which as we will see, is helpful in better understanding the complicated tradeoffs inherent in the original system." ], [ "Tradeoffs Associated with Channel Fading", "Proposition 4 When the idle ages are the same across all PU channels, it is optimal to schedule the SU to the channel with the highest fading state belief at the moment, i.e., $a_t^* = \\arg \\max _n \\lbrace \\pi _{t,1}^s(n)\\rbrace .$ First, from Proposition REF , the immediate reward is maximized when scheduling the channel with the highest fading state belief at the moment.", "Now, we focus on showing that the future reward is independent of the action in the current control slot and therefore establishing the proposition.", "Specifically, at the current control slot $t$ , $t \\ge 2$ , consider an arbitrary control slot in the future, $t_0 \\in \\lbrace t-1,\\ldots ,1\\rbrace $ .", "In the following, we show that the expected immediate reward in this control slot, denoted by $U_{t_0}$ , is independent of the current action $a_t$ , and thus the future reward is independent of $a_t$ .", "Let $k_0^{\\prime }$ denote the latest idle mini-slot in control slot $t_0 + 1$ before the PU returns.", "The reward $U_{t_0}$ is then calculated as: $ U_{t_0} = \\sum _{k_0^{\\prime } = 1}^K U_{t_0}(k_0^{\\prime }) \\textrm {Pr}(k_0^{\\prime }),$ where $\\textrm {Pr}(k_0^{\\prime })$ is the distribution of $k_0^{\\prime }$ , identical to that of $k_0$ as given in (REF ), and $U_{t_0}(k_0^{\\prime })$ is the expected reward in control slot $t_0$ for a given $k_0^{\\prime }$ .", "In the genie-aided system, the spectrum server obtains the feedbacks of the fading states from all $N$ channels simultaneously, i.e., at the end of mini-slot $k_0^{\\prime }$ .", "Place the channels on which good channel fading is observed at $k_0^{\\prime }$ in the set $\\mathcal {C}_1$ , and the rest in another set $\\mathcal {C}_0$ .", "The characterization of the reward $U_{t_0}(k_0^{\\prime })$ can be further divided into the following cases.", "Case 1: $\\mathcal {C}_0 = \\emptyset $ .", "This corresponds to the case where $f_{t_0+1,k_0^{\\prime }}(n) = 1, \\forall n = 1,\\ldots ,N$ .", "Let $W_{t_0}^{(\\textrm {case} 1)}(n)$ be the expected reward in control slot $t_0$ on channel $n$ in this case.", "We have $ W_{t_0}^{(\\textrm {case} 1)}(n) = \\pi _{t_0,1}^o(n) \\sum _{z=1}^K \\sum _{k=1}^z \\pi _{t_0,k}^s(n) p_z(x_{t_0}(n)),$ where $\\pi _{t_0,1}^s(n) = \\mathrm {T}^{K-k_0^{\\prime }}(p), \\forall n \\in \\mathcal {C}_1$ .", "It follows that $\\pi _{t_0,k}^s(1) = \\ldots = \\pi _{t_0,k}^s(N), \\forall k = 1,\\ldots ,K$ .", "Further, since $x_t(1) = \\ldots =x_t(N)$ , we have $p_z(x_{t_0}(1)) = \\ldots = p_z(x_{t_0}(N))$ , and $\\pi _{t_0,1}^o(1) = \\ldots = \\pi _{t_0,1}^o(N)$ , and therefore, scheduling any of the idle channels in $\\mathcal {C}_1$ achieves the same expected reward in control slot $t_0$ , i.e., $U_{t_0}(k_0^{\\prime })|_{\\textrm {case} 1} = W_{t_0}^{(\\textrm {case} 1)}(n), \\forall n \\in \\mathcal {C}_1, o_{t_0,1}(n)=0.$ Now, note that given equal idle ages on all $N$ channels, the distribution of $k_0^{\\prime }$ is identical across channels.", "Therefore, for all $n=1,\\ldots ,N$ , the values of $\\pi _{t_0,k}^s(n), k=1,\\ldots ,K$ , $\\pi _{t_0,1}^o(n)$ and $p_z(x_{t_0}(n))$ all stay unchanged if a different channel $a_t \\ne a_t^{\\prime }$ is scheduled in the current control slot $t$ .", "This implies that $U_{t_0}(k_0^{\\prime })|_{\\textrm {case} 1}$ is unchanged, and is thus independent of $a_t$ .", "Case 2: $\\mathcal {C}_1 = \\emptyset $ .", "In this case, $f_{t_0+1,k_0^{\\prime }}(n) = 0, \\forall n = 1,\\ldots ,N$ .", "The expected reward collected in control slot $t_0$ on channel $n$ , denoted by $W_{t_0}^{(\\textrm {case} 2)}(n)$ , can be expressed the same as (REF ), where the channel strength belief $\\pi _{t_0,1}^s(n) = \\mathrm {T}^{K-k_0^{\\prime }}(r)$ , for all $n\\in \\mathcal {C}_0$ .", "Then, using the similar reasoning as in Case 1, we obtain that scheduling any of the idle channels in $\\mathcal {C}_0$ achieves the same expected reward in control slot $t_0$ , i.e., $U_{t_0}(k_0^{\\prime })|_{\\textrm {case} 2} = W_{t_0}^{(\\textrm {case} 2)}(n), \\forall n \\in \\mathcal {C}_0, o_{t_0,1}(n)=0.$ Further, the expected reward achieved in this case, $U_{t_0}(k_0^{\\prime })|_{\\textrm {case} 2}$ , does not change when the action in current control slot $t$ varies, i.e., $U_{t_0}(k_0^{\\prime })|_{\\textrm {case} 2}$ is independent of $a_t$ .", "Case 3: $\\mathcal {C}_0 \\ne \\emptyset , \\mathcal {C}_1 \\ne \\emptyset $ .", "In this case, we first show that the maximum expected reward in control slot $t_0$ is achieved by scheduling any of the idle channels in the set $\\mathcal {C}_1$ , which are perceived to have better channel fading state in the subsequent control slot $t_0$ than the channels in set $\\mathcal {C}_0$ .", "Specifically, picking any one of the channels from each of the set, $n_1 \\in \\mathcal {C}_1$ and $n_0 \\in \\mathcal {C}_0$ , we have $W_{t_0}^{(\\textrm {case} 3)}(n_1) \\hspace{-5.69054pt} &=&\\hspace{-5.69054pt} \\sum _{z=1}^K \\sum _{k=1}^z \\pi _{t_0,k}^s(n_1) p_z(x_{t_0}(n_1)), \\nonumber \\\\W_{t_0}^{(\\textrm {case} 3)}(n_0) \\hspace{-5.69054pt} &=&\\hspace{-5.69054pt} \\sum _{z=1}^K \\sum _{k=1}^z \\pi _{t_0,k}^s(n_0) p_z(x_{t_0}(n_0)),$ where $\\pi _{t_0,k}^s(n_1) &=& \\mathrm {T}^{K-k_0^{\\prime }+k-1}(p),\\nonumber \\\\\\pi _{t_0,k}^s(n_0) &=& \\mathrm {T}^{K-k_0^{\\prime }+k-1}(r).$ Based on (REF ) and the property of the positively-correlated Markov chain, the following inequality holds: For all $k = 1,\\ldots ,K$ , $\\pi _{t_0^{\\prime },k}^s(n_1) = \\mathrm {T}^{K-k_0^{\\prime }+k-1}(p) \\ge \\mathrm {T}^{K-k_0^{\\prime }+k-1}(r) = \\pi _{t_0,k}^s(n_0),$ with the equality achieved when $K \\rightarrow \\infty $ .", "Further, applying similar argument as before, we obtain $W_{t_0}^{(\\textrm {case} 3)}(n_1) > W_{t_0}^{(\\textrm {case} 3)}(n_0).$ Now, since $n_1 \\in \\mathcal {C}_1$ (likewise, $n_0 \\in \\mathcal {C}_0$ ) is arbitrary, from the conclusion drawn in Case 1, the maximal expected reward in control slot $t$ under this case is achieved by scheduling the SU to any of the idle channels in the set $\\mathcal {C}_1$ , i.e., $U_{t_0}(k_0^{\\prime })|_{\\textrm {case} 3} = W_{t_0}^{(\\textrm {case} 3)}(n_1), \\forall n_1 \\in \\mathcal {C}_1, o_{t_0,1}(n_1)=0.$ Next, based on the similar reasoning as in the previous cases, we know that $U_{t_0}(k_0^{\\prime })|_{\\textrm {case} 3}$ is independent of $a_t$ .", "Finally, note that $U_{t_0}(k_0^{\\prime })$ can be written as: $U_{t_0}(k_0^{\\prime }) = \\sum _{i=1}^3 U_{t_0}(k_0^{\\prime })|_{\\textrm {case}~i} \\textrm {Pr} (\\textrm {Case}~ i),$ where $ \\textrm {Pr} (\\textrm {Case} 1) &=& \\prod _{n \\in \\mathcal {C}_1} \\pi _{t_0+1,k_0^{\\prime }}^s(n), \\nonumber \\\\\\textrm {Pr} (\\textrm {Case} 2) &=& \\prod _{n \\in \\mathcal {C}_0} (1-\\pi _{t_0+1,k_0^{\\prime }}^s(n)), \\nonumber \\\\\\textrm {Pr} (\\textrm {Case} 3) &=& \\prod _{n \\in \\mathcal {C}_1} \\pi _{t_0+1,k_0^{\\prime }}^s(n) \\prod _{n^{\\prime }\\in \\mathcal {C}_0} (1-\\pi _{t_0+1,k_0^{\\prime }}^s(n^{\\prime })),$ are quantities dependent on $k_0^{\\prime }$ only.", "Based on the fact that the idle age at the moment are identical across the channels, the probabilities $\\textrm {Pr} (\\textrm {Case}~i), i=1,2,3$ , and the distribution $\\textrm {Pr}(k_0^{\\prime })$ , are the same across the channels as well.", "Thus $U_{t_0}$ is independent of $a_t$ .", "Since $t_0 \\in \\lbrace t-1,\\ldots ,1\\rbrace $ is arbitrary, by extension, we have that the total reward collected from control slot $t_0$ till the horizon, i.e., $\\sum _{t^{\\prime } = t_0}^1 U_{t^{\\prime }}$ , which is the future reward of current control slot $t$ , is independent from $a_t$ .", "This concludes the proof and establishes the proposition.", "Remarks: Proposition REF illustrates the effect of fading state belief on the optimal decisions in the genie-aided system.", "With ages equalized across the PU channels and the classic “exploitation vs. exploration” tradeoff neutralized (by definition of the genie-aided system), we saw that, higher fading belief favors the immediate reward and that the future reward is, in fact, independent of the current decision.", "In the following, we study the effect of PU occupancy and age on the optimal decisions in the genie-aided system and, in turn, its impact on the original system." ], [ "Tradeoffs Associated with PU Occupancy", "The following proposition identifies the effect of channel occupancy state on the optimal scheduling decisions.", "Proposition 5 When the fading state beliefs are the same across all PU channels, it is optimal to schedule the SU to the channel with the lowest idle age at the moment, i.e., $a_t^* = \\arg \\min _n \\lbrace x_t(n)\\rbrace .$ We prove the proposition by showing that scheduling the channel with the lowest idle age favors: [1)] the immediate reward $R_t(\\textbf {S}_t,a_t)$ ; and the optimal future reward $V_{t-1}^*(\\textbf {S}_{t-1})$ .", "The first part can be readily shown by appealing to Proposition REF and Corollary REF .", "To show the second part, we adopt the following induction hypothesis: Induction Hypothesis: The optimal future reward is convex in the fading state belief.", "When channel $a_t$ is scheduled in the current control slot $t$ , the expected future reward can be evaluated as: $V_{t-1}^*(\\mathbf {S}_{t-1})|_{a_t}=\\pi _{t,k_0}^s(a_t) V_{t-1}^*(p) + (1-\\pi _{t,k_0}^s(a_t))V_{t-1}^*(r),$ where $V_{t-1}^*(p)$ and $V_{t-1}^*(r)$ represent the future reward calculated when the channel fading state observed in the $k_0$ th mini-slot of control slot $t$ is good or bad, respectively.", "More specifically, $V_{t-1}^*(p) \\triangleq V_{t-1}^*\\left(\\pi _{t-1,1}^s(a_t) = \\mathrm {T}^{K-k_0}(p)\\right), \\nonumber \\\\V_{t-1}^*(r) \\triangleq V_{t-1}^*\\left(\\pi _{t-1,1}^s(a_t) = \\mathrm {T}^{K-k_0}(r)\\right).$ Based on (REF ), we have, for $\\gamma \\in (0,1)$ , $\\mathrm {T}^L(\\gamma ) = (p-r)^L\\gamma + r\\frac{1-(p-r)^L}{1-(p-r)}, L = 0,1,\\ldots ,$ and $\\lim _{L \\rightarrow \\infty } \\mathrm {T}^L(\\gamma ) = \\frac{r}{1-p+r} \\triangleq \\pi _{s|s}$ , where $\\pi _{s|s}$ denotes the steady-state probability of perceiving good channel fading on any of the PU channels.", "This indicates that, a smaller $k_0$ , associated with a higher idle age at the moment (recall discussions in Section ), results in a larger $K-k_0$ and thus a value of $\\pi _{t-1,1}^s(a_t)$ closer to $\\pi _{s|s}$ , in which case, $V_{t-1}^*(\\mathbf {S}_{t-1})|_{a_t} \\hspace{-9.95845pt}&\\rightarrow &\\hspace{-9.95845pt} \\pi _{t,k_0}^s(a_t) V_{t-1}^*(\\pi _{s|s}) \\hspace{-3.41432pt}+\\hspace{-3.41432pt} (1-\\pi _{t,k_0}^s(a_t))V_{t-1}^*(\\pi _{s|s})\\triangleq V_{t-1}^*(a_t,E_{\\pi _{s|s}}).$ On the contrary, as $k_0$ becomes larger because of a lower idle age, the value of $\\pi _{t-1,1}^s(a_t)$ deviates further away from $\\pi _{s|s}$ , but is closer to $p$ (or $r$ ).", "Also, $\\pi _{t,k_0}^s(a_t)$ gets closer to $\\pi _{s|s}$ .", "Therefore, $V_{t-1}^*(\\mathbf {S}_{t-1})|_{a_t} \\hspace{-9.95845pt}&\\rightarrow &\\hspace{-9.95845pt} \\pi _{s|s} V_{t-1}^*(p) \\hspace{-3.1298pt}+\\hspace{-3.1298pt} (1-\\pi _{s|s}) V_{t-1}^*(r) \\triangleq E_{\\pi _{s|s}} V_t^*(a_t).$ Now, appealing to the induction hypothesis and the Jensen's inequality [18], we have that $E_{\\pi _{s|s}} V_t^*(a_t) > V_{t-1}^*(a_t,E_{\\pi _{s|s}})$ , and therefore the future reward is maximized by scheduling the channel with the lowest idle age, i.e., $a_t^* = \\min _{a_t}\\lbrace x_t(a_t)\\rbrace ~~s.t.~~ V_{t-1}^*(\\textbf {S}_{t-1})|_{a_t^*} = \\max _{a_t}\\lbrace V_{t-1}^*(\\textbf {S}_{t-1})|_{a_t}\\rbrace .$ Finally, we proceed to verify the induction hypothesis.", "At $t=2$ , the optimal future reward equals the optimal immediate reward at the horizon $t=1$ , i.e., $V_1^*(\\textbf {S}_1) = R_1(\\textbf {S}_1, a_1^*) := \\max _{a_1\\in \\mathcal {A}_1}\\lbrace R_1(\\textbf {S}_1,a_1)\\rbrace .$ Since $R_1(\\textbf {S}_1,a_1)$ is linear in the strength belief, using the property of convex function [18], we have that $V_1^*(\\textbf {S}_1)$ is convex in the strength beliefs, which validates the induction hypothesis.", "Then, using backward induction, we establish the proposition.", "Remarks: Proposition REF explicitly illustrates the effect of the PU occupancy and age on the optimal decisions.", "With fading state beliefs equalized and the classic “exploitation vs. exploration” tradeoff neutralized (by definition of the genie-aided system), we saw that: [1)] lower idle age on the PU channel favors the immediate reward by allowing more idle time on the channel; and lower idle age also favors the future reward by way of better freshness of the channel fading feedback.", "Thus by studying the full-observation genie-aided system, via the results in Propositions REF and REF , we have decomposed the tradeoffs associated with the channel occupancy and the fading state beliefs in the original system.", "Indeed, the results in Propositions REF and REF rigorously support the understanding we developed earlier in Section  on the tradeoffs in the original system.", "Numerical Results & Further Discussions In this section, we evaluate and compare the optimal rewards of the original system (denoted as $V_{ori}^*$ ) and the genie-aided system ($V_{genie}^*$ ).", "Also, the optimal policy in the original system is compared to a randomized scheduling policy (with reward denoted as $V_{random}^*$ ), where the spectrum server chooses a channel, among all the idle ones, randomly and uniformly, and allocates it to the SU for data transmission.", "The numerical results are collected for a two channel system with $K=2$ , horizon length $m=6$ , and discounted factor $\\beta = 0.9$ .", "For notational convenience, denote $\\Delta _{{ga}-{ori}} = V_{genie}^* - V_{ori}^*$ and $\\Delta _{{ori}-{rnd}} = V_{ori}^* - V_{random}^*$ .", "Table REF records the rewards obtained under various baseline cases, for various values of $\\delta \\triangleq p-r$ , which broadly captures the temporal memory in the channel fading.", "In particular, as $\\delta $ decreases, the channel fading memory fades and the difference between the baselines, which are primarily differentiated by the degree to which they exploit the memory in the system, tends to decrease.", "This is observed from Table REF .", "Table: Comparison of rewards when the channel fading memory varies.", "System parameters used: u=1,C I =1,C B =2,x t (1)=10,x t (2)=5,π t,1 s (1)=0.4,π t,1 s (2)=0.7u=1,C_I = 1, C_B = 2, x_t(1) = 10, x_t(2) = 5, \\pi _{t,1}^s(1) = 0.4, \\pi _{t,1}^s(2) = 0.7Next, in Table REF , we compare the baseline rewards under different values of the power exponent $u$ , used in the definitions of $P_I$ and $P_B$ in (REF ).", "To build a better understanding of the trend reflected in these numerical results, we consider an arbitrary mini-slot, denoted as $k^{\\prime }$ , as the current mini-slot, and the following two exhaustive sets of PU occupancy histories: [1)] the set of histories $h_x^I, x=1,2,\\ldots $ , corresponds to the case when there are exactly $x$ mini-slots between the current mini-slot $k^{\\prime }$ and the most recent mini-slot (preceding $k^{\\prime }$ ) when the channel was in busy state, i.e., the idle age of mini-slot $k^{\\prime }$ is $x$ ; and the set of histories $h_x^B, x=1,2,\\ldots $ , corresponds to the case when there are exactly $x$ mini-slots between the current mini-slot $k^{\\prime }$ and the most recent mini-slot (preceding $k^{\\prime }$ ) when the channel was in idle state, i.e., the busy age of mini-slot $k^{\\prime }$ is $x$ .", "In Fig.", "REF , we plot the two sets of occupancy histories.", "As has been pointed out in Section REF , the idle/busy age is a sufficient statistic for capturing the memory in the PU channels' occupancy states.", "Thereby, with the construction of $h_x^I$ and $h_x^B$ , we can examine the effect of the temporal correlation of PU occupancy on the system performance.", "Specifically, the conditional idle probability in mini-slot $k^{\\prime }$ , given occupancy history, can be obtained as: $\\pi _{k^{\\prime }}^o |_{h_x^I} \\hspace{-5.69054pt}&=&\\hspace{-5.69054pt} P_I(x) = \\frac{1}{x^u+C_I},\\nonumber \\\\\\pi _{k^{\\prime }}^o |_{h_x^B} \\hspace{-5.69054pt}&=&\\hspace{-5.69054pt} 1 - P_B(x) = 1 - \\frac{1}{x^u+C_B}.$ As an example, we plot $\\pi _{k^{\\prime }}^o |_{h_x^I}$ in Fig.", "REF .", "It is clear that as $u$ increases, the conditional probability curves become steeper.", "Define the threshold point, $x_0$ , such that for all $x>x_0$ , the difference in $\\pi _{k^{\\prime }}^o |_{h_x^I}$ is insignificant (below $10^{-2}$ ).", "Now, note from the figure that the threshold $x_0$ decreases with increasing $u$ , i.e., $x_0^{(u=5)} < x_0^{(u=3)} < x_0^{(u=1)}$ .", "This indicates that the impact of different occupancy histories on the current PU occupancy state diminishes with increasing $u$ and thus a decreased memory in the PU occupancy.", "Similar argument holds when considering $h_x^B$ .", "In short, the exponent $u$ broadly captures the PU occupancy memory, and as its value increases, the memory decreases and thus the rewards under various cases, as expected, come closer with increasing $u$ .", "This is illustrated in Table REF .", "Figure: Occupancy histories & the conditional idle probability.Table: Comparison of rewards when the occupancy memory varies.", "System parameters used: p=0.9,r=0.1,C I =1,C B =2,x t (1)=0,x t (2)=1,π t,1 s (1)=0.4,π t,1 s (2)=0.7p = 0.9, r = 0.1, C_I = 1, C_B = 2, x_t(1) = 0, x_t(2) = 1, \\pi _{t,1}^s(1) = 0.4, \\pi _{t,1}^s(2) = 0.7Finally, as can be seen from both tables, the original system performs very close to the genie-aided system, while the cost of measuring and sending the channel fading feedback is only $\\frac{1}{N}$ of the latter.", "Also, the optimal policy significantly outperforms the randomized policy, indicating that the temporal correlation structure in the channel fading and PU occupancy can greatly benefit the opportunistic spectrum scheduling, when appropriately exploited.", "Conclusions In this work, we studied opportunistic spectrum access for a single SU in a CR network with multiple PU channels.", "We formulated the problem as a partially observable Markov decision process, and examined the intricate tradeoffs in the optimal scheduling process, when incorporating the temporal correlation in both the channel fading and PU occupancy states.", "We modeled the channel fading variation with a two-state first-order Markov chain.", "The temporal correlation of PU traffic was modeled using age and a class of monotonically decreasing functions, which can have a long memory.", "The optimality of the simple greedy policy was established under certain conditions.", "For the general case, we individually studied the tradeoffs in the immediate reward, and the total reward.", "Further, by developing a genie-aided system with full observation of the channel fading feedbacks, we decomposed and characterized the multiple tiers of the intricate tradeoffs in the original system.", "Finally, we numerically studied the performance of the two systems and showed that the original system achieved an optimal total reward very close (within $1\\%$ ) to that of the genie-aided system.", "Further, the optimal policy in the original system significantly outperformed randomized scheduling, highlighting the merit of exploiting the temporal correlation in the system states.", "We believe that our formulation and the insights we have obtained open up new horizons in better understanding spectrum allocation in cognitive radio networks, with problem settings that go beyond the traditional ones." ], [ "Numerical Results & Further Discussions", "In this section, we evaluate and compare the optimal rewards of the original system (denoted as $V_{ori}^*$ ) and the genie-aided system ($V_{genie}^*$ ).", "Also, the optimal policy in the original system is compared to a randomized scheduling policy (with reward denoted as $V_{random}^*$ ), where the spectrum server chooses a channel, among all the idle ones, randomly and uniformly, and allocates it to the SU for data transmission.", "The numerical results are collected for a two channel system with $K=2$ , horizon length $m=6$ , and discounted factor $\\beta = 0.9$ .", "For notational convenience, denote $\\Delta _{{ga}-{ori}} = V_{genie}^* - V_{ori}^*$ and $\\Delta _{{ori}-{rnd}} = V_{ori}^* - V_{random}^*$ .", "Table REF records the rewards obtained under various baseline cases, for various values of $\\delta \\triangleq p-r$ , which broadly captures the temporal memory in the channel fading.", "In particular, as $\\delta $ decreases, the channel fading memory fades and the difference between the baselines, which are primarily differentiated by the degree to which they exploit the memory in the system, tends to decrease.", "This is observed from Table REF .", "Table: Comparison of rewards when the channel fading memory varies.", "System parameters used: u=1,C I =1,C B =2,x t (1)=10,x t (2)=5,π t,1 s (1)=0.4,π t,1 s (2)=0.7u=1,C_I = 1, C_B = 2, x_t(1) = 10, x_t(2) = 5, \\pi _{t,1}^s(1) = 0.4, \\pi _{t,1}^s(2) = 0.7Next, in Table REF , we compare the baseline rewards under different values of the power exponent $u$ , used in the definitions of $P_I$ and $P_B$ in (REF ).", "To build a better understanding of the trend reflected in these numerical results, we consider an arbitrary mini-slot, denoted as $k^{\\prime }$ , as the current mini-slot, and the following two exhaustive sets of PU occupancy histories: [1)] the set of histories $h_x^I, x=1,2,\\ldots $ , corresponds to the case when there are exactly $x$ mini-slots between the current mini-slot $k^{\\prime }$ and the most recent mini-slot (preceding $k^{\\prime }$ ) when the channel was in busy state, i.e., the idle age of mini-slot $k^{\\prime }$ is $x$ ; and the set of histories $h_x^B, x=1,2,\\ldots $ , corresponds to the case when there are exactly $x$ mini-slots between the current mini-slot $k^{\\prime }$ and the most recent mini-slot (preceding $k^{\\prime }$ ) when the channel was in idle state, i.e., the busy age of mini-slot $k^{\\prime }$ is $x$ .", "In Fig.", "REF , we plot the two sets of occupancy histories.", "As has been pointed out in Section REF , the idle/busy age is a sufficient statistic for capturing the memory in the PU channels' occupancy states.", "Thereby, with the construction of $h_x^I$ and $h_x^B$ , we can examine the effect of the temporal correlation of PU occupancy on the system performance.", "Specifically, the conditional idle probability in mini-slot $k^{\\prime }$ , given occupancy history, can be obtained as: $\\pi _{k^{\\prime }}^o |_{h_x^I} \\hspace{-5.69054pt}&=&\\hspace{-5.69054pt} P_I(x) = \\frac{1}{x^u+C_I},\\nonumber \\\\\\pi _{k^{\\prime }}^o |_{h_x^B} \\hspace{-5.69054pt}&=&\\hspace{-5.69054pt} 1 - P_B(x) = 1 - \\frac{1}{x^u+C_B}.$ As an example, we plot $\\pi _{k^{\\prime }}^o |_{h_x^I}$ in Fig.", "REF .", "It is clear that as $u$ increases, the conditional probability curves become steeper.", "Define the threshold point, $x_0$ , such that for all $x>x_0$ , the difference in $\\pi _{k^{\\prime }}^o |_{h_x^I}$ is insignificant (below $10^{-2}$ ).", "Now, note from the figure that the threshold $x_0$ decreases with increasing $u$ , i.e., $x_0^{(u=5)} < x_0^{(u=3)} < x_0^{(u=1)}$ .", "This indicates that the impact of different occupancy histories on the current PU occupancy state diminishes with increasing $u$ and thus a decreased memory in the PU occupancy.", "Similar argument holds when considering $h_x^B$ .", "In short, the exponent $u$ broadly captures the PU occupancy memory, and as its value increases, the memory decreases and thus the rewards under various cases, as expected, come closer with increasing $u$ .", "This is illustrated in Table REF .", "Figure: Occupancy histories & the conditional idle probability.Table: Comparison of rewards when the occupancy memory varies.", "System parameters used: p=0.9,r=0.1,C I =1,C B =2,x t (1)=0,x t (2)=1,π t,1 s (1)=0.4,π t,1 s (2)=0.7p = 0.9, r = 0.1, C_I = 1, C_B = 2, x_t(1) = 0, x_t(2) = 1, \\pi _{t,1}^s(1) = 0.4, \\pi _{t,1}^s(2) = 0.7Finally, as can be seen from both tables, the original system performs very close to the genie-aided system, while the cost of measuring and sending the channel fading feedback is only $\\frac{1}{N}$ of the latter.", "Also, the optimal policy significantly outperforms the randomized policy, indicating that the temporal correlation structure in the channel fading and PU occupancy can greatly benefit the opportunistic spectrum scheduling, when appropriately exploited.", "Conclusions In this work, we studied opportunistic spectrum access for a single SU in a CR network with multiple PU channels.", "We formulated the problem as a partially observable Markov decision process, and examined the intricate tradeoffs in the optimal scheduling process, when incorporating the temporal correlation in both the channel fading and PU occupancy states.", "We modeled the channel fading variation with a two-state first-order Markov chain.", "The temporal correlation of PU traffic was modeled using age and a class of monotonically decreasing functions, which can have a long memory.", "The optimality of the simple greedy policy was established under certain conditions.", "For the general case, we individually studied the tradeoffs in the immediate reward, and the total reward.", "Further, by developing a genie-aided system with full observation of the channel fading feedbacks, we decomposed and characterized the multiple tiers of the intricate tradeoffs in the original system.", "Finally, we numerically studied the performance of the two systems and showed that the original system achieved an optimal total reward very close (within $1\\%$ ) to that of the genie-aided system.", "Further, the optimal policy in the original system significantly outperformed randomized scheduling, highlighting the merit of exploiting the temporal correlation in the system states.", "We believe that our formulation and the insights we have obtained open up new horizons in better understanding spectrum allocation in cognitive radio networks, with problem settings that go beyond the traditional ones." ], [ "Conclusions", "In this work, we studied opportunistic spectrum access for a single SU in a CR network with multiple PU channels.", "We formulated the problem as a partially observable Markov decision process, and examined the intricate tradeoffs in the optimal scheduling process, when incorporating the temporal correlation in both the channel fading and PU occupancy states.", "We modeled the channel fading variation with a two-state first-order Markov chain.", "The temporal correlation of PU traffic was modeled using age and a class of monotonically decreasing functions, which can have a long memory.", "The optimality of the simple greedy policy was established under certain conditions.", "For the general case, we individually studied the tradeoffs in the immediate reward, and the total reward.", "Further, by developing a genie-aided system with full observation of the channel fading feedbacks, we decomposed and characterized the multiple tiers of the intricate tradeoffs in the original system.", "Finally, we numerically studied the performance of the two systems and showed that the original system achieved an optimal total reward very close (within $1\\%$ ) to that of the genie-aided system.", "Further, the optimal policy in the original system significantly outperformed randomized scheduling, highlighting the merit of exploiting the temporal correlation in the system states.", "We believe that our formulation and the insights we have obtained open up new horizons in better understanding spectrum allocation in cognitive radio networks, with problem settings that go beyond the traditional ones." ] ]

[ [ "Experimental phase-space-based optical amplification of scar modes" ], [ "Abstract Waves billiard which are chaotic in the geometrical limit are known to support non-generic spatially localized modes called scar modes.", "The interaction of the scar modes with gain has been recently investigated in optics in micro-cavity lasers and vertically-cavity surface-emitting lasers.", "Exploiting the localization properties of scar modes in their wave analogous phase space representation, we report experimental results of scar modes selection by gain in a doped D-shaped optical fiber." ], [ "Experimental phase-space-based optical amplification of scar modes C. Michel$^1$ , S. Tascu$^2$ , V. Doya$^1$ , P. Aschiéri$^1$ , W. Blanc$^1$ , O. Legrand$^1$ and F. Mortessagne$^1$ $^1$ Laboratoire de Physique de la Matière Condensée, Université Nice-Sophia Antipolis, CNRS, UMR 7336, France $^2$ Research Center on Advanced Materials and Technologies, Faculty of Physics, Alexandru Ioan Cuza University of Iasi, Blvd.", "Carol I, nr.", "11, 700506, Iasi, Romania Waves billiard which are chaotic in the geometrical limit are known to support non-generic spatially localized modes called scar modes.", "The interaction of the scar modes with gain has been recently investigated in optics in micro-cavity lasers and vertically-cavity surface-emitting lasers.", "Exploiting the localization properties of scar modes in their wave analogous phase space representation, we report experimental results of scar modes selection by gain in a doped D-shaped optical fiber.", "Wave chaos is investigating the wave behavior of bounded systems with chaotic classical dynamics.", "The common feature of chaotic modes in a linear (passive) medium is ergodicity [1]: modes of a chaotic cavity exhibit a generic statistically uniform field distribution [2].", "Scar modes that result from constructive wave interferences along the least unstable classical periodic orbits break the ergodicity property [3] but are rather scarce events.", "Even though scar phenomenon has been intensively investigated since their numerical discovery by Mc Donald and Kaufman [4], their localization feature still arouses interest as in the profilic context of graphene investigations [5].", "Interaction between spatially localized scar modes and nonlinear effects is of fundamental interest and constitutes a recent evolution of Wave Chaos.", "The subtle interplay of gain with wave chaos and specifically scar modes has been widely examined in optics in micro-cavity lasers [6], [7], [8] and in vertically-cavity surface-emetting lasers [9].", "The problem of the interaction between the lasing medium and the light field in a two-dimensional (2D) laser cavity is still a vivid research area [10].", "In the presence of a saturable nonlinear effect, which does not involve any phase change, scar modes take a crucial part on the improvement of the lasing efficiency.", "Thanks to their low-loss properties compared with generic ergodic modes, scar modes are commonly selected during the lasing process leading to enhanced emission directivity and lowered lasing threshold [11].", "In recent studies, the use of gain has also proved to favor the selection of scar modes in guided optics: numerical investigations revealed that the interaction of the chaotic modes with a localized active medium in a D-shaped optical fiber amplifier led to a selective amplification of scar modes [12].", "In this letter, we report the first, to our knowledge, experimental demonstration of a selective amplification of scar modes through gain in a highly multimode chaotic system.", "This amplification is based upon the realization of a D-shaped optical fiber doped with a spatially localized gain.", "The amplified scar modes are built on the 2-bounce periodic orbit (2-PO) (see inset of Fig.REF ) of the associated D-shaped billard.", "Using the specific features of the scar modes in their wave-analogous phase space representation and a spatially coherent amplification process, we show that the selection mechanism of scars is strong enough to enhance scar modes and make them take over ergodic modes.", "Indeed, we can establish that certain non-scarred modes, displaying a good spatial overlap with the gain area, remain unamplified while the neighboring scar modes are clearly amplified, essentially because those non-scarred modes are not phase-space localized in the way the 2-PO scar modes are.", "Our experimental system is a non standard Ytterbium doped multimode optical fiber with a core diameter of 121 $\\mu $ m, truncated at its half radius (see inset in Fig.REF ).", "Figure: Experimental setup for the amplification of scar modes in the double clad Ytterbium doped fiber amplifier.", "L 1 _1, lens (15-mm working distance), L 2 _2, lens (10-mm working distance), L 3 _3, lens (2.5-cm focal length f), DM, dichroic mirror (T=90%\\%@1064 nm, R=90%\\%@978 nm), P 1 _1, P 2 _2, 3-D positioning supports, F 1 _1, bandpass filter (3 nm @1064 nm), BC, beam collimator, BE, beam expander, M, mirror.", "Right inset, transverse cross section of the D-shaped fiber amplifier with the off-centered doped area (circle) and the 2-PO along the x-axis.", "The polymer cladding is not represented.", "Color onlineWe fabricate the fiber preform by a standard MCVD (modified chemical vapor deposition) method.", "The core is made of pure silica including a 15-$\\mu m$ -diameter doped area of Ytterbium, free from any other co-dopant.", "A low refractive index polymer surrounds the silica core and acts as an optical cladding.", "The doped area is off-centered and located in a specific position deduced from numerical investigations [12] which corresponds to the maximum of intensity for most of the 2-PO scar modes.", "The preform has been cut and shaped before being stretched for this purpose.", "The optical index mismatch between the doped area and the D-shaped core is induced by the ions concentration and takes the value $\\Delta n=5\\times 10^{-4}$ ($[\\text{Yb}^{3+}]=900$ ppm).", "This value of $\\Delta n$ , coupled with a triangular index profile, prohibits any light guidance in the doped area.", "Both sides of the fiber are optically polished to avoid either at the input or at the output any diffraction induced by defects.", "The signal is given by a YAG laser ($P_s$ =400 mW) at $\\lambda _s$ =1064 nm and expanded thanks to a couple of microscope objectives ($\\times 4$ and $\\times 10$ ) that play the role of a beam expander (BE).", "The resulting beam, of 5 mm of diameter, is sent through a 15-mm working distance lens (L$_1$ ) and is transmitted by a dichroic mirror (DM, with 90 $\\%$ of transmission at $\\lambda _s$ ).", "The pump ($P_p$ =10 W) is provided by a cw laser diode (coupled in an angle-polished fiber of 105-$\\mu $ m diameter) at $\\lambda _p$ =974 nm.", "The pump is focused thanks to a 10-mm working distance lens (L$_2$ ) on the input of the D-shaped optical fiber through the dichroic mirror (90 $\\%$ of reflexion at $\\lambda _p$ ).", "For an optimization of the pump absorption (excitation of a maximized number of modes), the beam is strongly focused into the multimode core [13], [14].", "The pump absorption coefficient has been measured under low pump power to prevent any population inversion and takes the value $\\alpha _p=6.5$ dB/m.", "The absorption and emission cross sections of the pump and the signal are respectively $\\sigma _{pa}=2.65\\times 10^{-24}$ m$^2$ , $\\sigma _{pe}=2.65\\times 10^{-24}$ m$^2$ , $\\sigma _{sa}=5.56\\times 10^{-26}$ m$^2$ and $\\sigma _{se}=6.00\\times 10^{-25}$ m$^2$ .", "The gain of the signal has been measured along a 20 m-length fiber, and the maximum value $G= 4.5$ dB (with an output signal to noise ratio of 27 dB) is obtained at the optimal length $L_{\\text{opt}}=15$ m. This gain value, which is rather low as compared to standards of telecom optical amplifiers, serves our purpose in demonstrating that amplification is really at work in our highly multimode fiber.", "At the output of the fiber, the residual pump is filtered by a bandpass filter $F_1$ of 3 nm-width around $\\lambda _s$ and a CCD camera is placed in the focal plane of a 2.5 cm-focal length lens to image the far-field intensity pattern.", "As we want to observe strong signatures of waves localization associated with scar modes, we choose to vizualize the signal intensity in the far-field configuration.", "Indeed, the square modulus of the spatial Fourier transform of the field distribution gives some information about the directions and modulus of the outgoing wave vectors.", "At the fiber entrance, we control the direction of the transverse wave vectors propagating in the fiber.", "A selection of the injected transverse wave vectors is obtained by a $(\\hat{x},\\hat{y})$ -translation of the polished input of the optical fiber in the divergent part of the focused signal beam.", "The figure REF (a) presents the typical far field intensity of the signal obtained at the output of the multimode 15 m-length D-shaped fiber amplifier when pump is off.", "Even if the injection of the signal is optimized to excite a scar mode among few ergodic modes, the bending of the fiber tends to distribute the signal intensity among numerous other modes.", "The observed statistically uniform distribution of intensity reveals the superposition of a large number of modes in the fiber.", "Figure REF (b) represents a cut along the $\\kappa _x$ -axis for $\\kappa _y=0$ .", "It confirms the broad distribution of transverse wavenumbers.", "For the same signal configuration, but with pump turned on, the figure REF (c) presents the far field intensity pattern of the outgoing signal.", "A pair of high intensity spots symmetrically distributed on both sides along the $\\kappa _x$ -axis appears in the center of the far field.", "These two peaks demonstrate that the signal propagates along the fiber with two privileged transverse directions given by $\\kappa $ and $-\\kappa $ along the $x$ -direction.", "Those directions are associated with those of the 2-PO.", "The observation of two symetrical peaks along the 2-PO direction is the fingerprint of a scar mode in the far-field intensity representation as can be seen in the $\\kappa _x$ -cut of the far-field intensity pattern of a typical calculated scar mode (Fig.", "REF (f)).", "Figure: Far field intensity observations (a), (c) and their transverse horizontal cut at κ y =0\\kappa _y=0 (b), (d) for the unpumped and pumped configuration respectively for κ p =7\\kappa _p=7.", "Far field intensity pattern and transverse horizontal cut superimposed for the pumped configuration with κ p =6\\kappa _p=6 (e).", "Horizontal cut of the far field intensity of a calculated scar mode (f).", "Color onlineThe transverse wavenumber corresponding to the observed experimental peaks can be estimated from figure REF (d).", "The calibration of the far field representation gives one pixel = (2.405$\\pm $ 0.005)$\\times 10^{-3}$ $\\mu $ m$^{-1}$ and then, the transverse wavenumber associated to the peaks is deduced.", "We get $\\kappa =(0.279\\pm 0.003)$ $\\mu $ m$^{-1}=(17.0\\pm 0.2)/R$ .", "This value has to be compared with the theoretical value deduced from a condition of phase coherence of waves that bounce back and forth along the 2-PO in the D-shaped billiard: $\\kappa _p\\mathcal {L}+\\Delta \\phi +\\frac{\\pi }{2}=2\\pi p$ where $\\mathcal {L}=3R$ is the 2-PO-length, $\\Delta \\phi $ is the phaseshift induced by the reflexion at the core/cladding interface, the additional $\\pi /2$ phaseshift is a consequence of the single self-focal point of the 2-PO, and $p$ is an integer that gives the order of the scar mode.", "For $p=7$ , we get a value $\\kappa _{p=7}=17.08/R$ which is in perfect agreement with the measured value: the scar mode of order 7 is selected thanks to the localized gain.", "We change the signal launching condition by translating the fiber input in the $(\\hat{x},\\hat{y})$ plane (Fig.", "REF ) in order to seed the amplification process on another scar mode.", "Figure REF (e) reports the enhancement of two symetrical peaks along the $\\kappa _x$ -direction for this new input illumination.", "The measured value of the transverse wavenumber $\\kappa =(15.0\\pm 0.2)/R$ has to be compared to the calculated value (Eq.", "REF ) $\\kappa _{p=6}=15.0/R$ .", "The agreement is again remarkable and one can deduce that the scar mode of order $p=6$ is enhanced thanks to amplification.", "These two examples are the proof that, by controlling the initial illumination condition, one can select a given scar mode along the 2-PO thanks to the optical amplification process.", "These preliminary experimental results demonstrate our ability to selectively amplify scar modes in a highly multimode fiber thanks to a localized gain.", "In agreement with common fiber amplification processes, the spatial overlap of the modes with the active area is usually the main control parameter of the amplification efficiency.", "We propose a complementary interpretation of the scar modes enhancement by gain based on the physical nature of scars.", "According to E. J. Heller, “scars manifest themselves as enhanced probability in phase space, as measured by overlap with coherent states placed on the periodic orbits” [3].", "Therefore, scar modes, viewed as fingerprints of classical orbits in the near-field intensity pattern, also show strongly privileged wave directions as highlighted through their far-field intensity.", "Both spatial and directional features of scar modes can be most efficiently combined in the so-called Husimi representation [15].", "The Husimi representation is commonly used as a wave equivalent to the classical Poincaré surface of section.", "Thus, it establishes a strong correspondence between some particular modes and their associated classical trajectories in the semiclassical regime.", "In the case of 2D cavities, the Husimi function measures the normal derivative of the eigenfunction on the boundaries of the cavity [16], [17], [18], [19].", "Fig.", "REF (b-d-f) present the Husimi representation of three different calculated modes of the chaotic optical fiber with metallic boundary in the 2-dimensional space ($\\theta ,\\kappa \\sin \\chi )$ with $\\theta $ the curved abscissa, $\\kappa $ the transverse wavenumber of the modes and $\\kappa \\sin \\chi $ the projection of the transverse wavevector along the tangential direction to the boundary (Fig.", "REF (a)).", "The Husimi representation (Fig.", "REF (b)) of a generic ergodic mode (Fig.", "REF (a)) generally explores a large part of the available space: no specific transverse wavevector directions appear to be prevailing.", "This observation is consistent with the hypothesis of generic modes built upon an ergodic semiclassical behavior [2], [20].", "Ergodic modes result from the superposition of a large number of plane waves with fixed transverse wavenumber $\\kappa $ but random directions $\\chi $ .", "On the contrary, the Husimi representation (Fig.", "REF (d)) of a scar mode associated to the 2-PO (Fig.", "REF (c)) exhibits a strong localization of the field around the angular abscissa $\\theta _0=0$ and the exclusive direction of the 2-PO, that is, $\\chi _0=0$ .", "All the 2-PO scar modes present such a similar localization attribute in their Husimi representation.", "Figure: Field intensity of specific modes and their respective Husimi representations.", "For a generic speckle mode (a),(b); for a scar mode along the 2-PO (c), (d); and for the mode with κ NS \\kappa _{\\mathrm {NS}} (e),(f).", "Color onlineThe spatial and directional extensions of the Husimi pattern define a range of angular abscissa $\\Delta \\theta $ around $\\theta _0$ and of transverse wavevector directions $\\Delta \\chi $ derived from $\\Delta (\\kappa \\sin \\chi )$ around $\\chi _0$ .", "Thus, a scar mode is built upon few highly directional plane waves, ensuring a strong spatial coherence.", "In a geometrical approach in terms of rays in the D-shaped billiard, this span of initial conditions extracted from the Husimi distribution determines an area in the billiard plane where rays are converging.", "This area is located in the vicinity of the so-called self-focal point of the 2-PO.", "This location of the active ions guarantees an efficient round-trip of rays through the active medium, as well as a spatially coherent amplification and also a good overlap with the field amplitude of the scar modes.", "We stress that all these conditions should be fulfilled to assure a scar mode amplification.", "Indeed, we observe that non-scarred (NS) modes can be insensitive to amplification despite a good spatial overlap with the gain medium.", "For instance, the field amplitude of the NS mode of Fig.REF (e) has a good spatial overlap with the doped area but is not selected during the amplification process [21].", "Using a Beam Propagation Method algorithm [22], we simulate the propagation of a gaussian-like initial signal injected into the fiber with a transverse wavenumber $\\kappa _{\\mathrm {NS}}=17.07/R$ that corresponds to the specific NS mode.", "The modulus of the transverse wavevector is fixed, but its direction is chosen so as to avoid the 2-PO direction and thus a 2-PO scar mode excitation.", "Figure: Intensity spectrum at the input of the fiber amplifier (dashed) and after amplification at the output of the fiber (gray) associated to a gaussian signal launched into the fiber with κ NS \\kappa _{\\mathrm {NS}}.", "Color onlineFig.REF presents the intensity spectrum at the output of the D-shaped fiber amplifier.", "The targeted NS mode is not amplified whereas the scar mode of order $p=7$ is strongly amplified despite its reduced contribution in the initial spectrum: the spatial field overlap is not able to clarify this favored amplification of the scar mode.", "The explanation is supplied in the analysis of the Husimi representation (Fig.", "REF (f)).", "Indeed, the NS mode presents a broad distribution of transverse wavevectors directions in the Husimi representation that prohibits a coherent amplification mechanism.", "As a consequence, a strong localization in the Husimi representation around the signature of the 2-PO is essential to ensure the efficiency of modes amplification.", "In this paper, we have presented an experimental observation of scar modes amplification in the D-shaped optical fiber with localized gain.", "An active medium with a spatial location controlled by the phase space signature of scars is a propitious system to selectively amplify scar modes.", "Investigations about scar modes in nonlinear systems are opening new attractive issues.", "For instance, having a scar amplifier may offer new perspectives for optical communications.", "The intensity spectrum of a scar modes amplifier has a response similar to the spectral response of a Fabry-Perot cavity.", "Each scar mode appears well-separated with a spacing between adjacent transverse wavenumbers equal to $2\\pi /\\mathcal {L^{\\prime }}$ with $\\kappa \\mathcal {L^{\\prime }}=2\\pi p$ .", "These well-resolved modes injected in a multimode fiber would be less sensitive to crosstalk and could be employed as independent transmission channels in a process of mode division multiplexing [23].", "We express grateful thanks to S. Trzesien and M. Ude for their priceless involvement in the preform and fiber manufacture." ] ]

[ [ "The Minimal Non-Koszul A(Gamma)" ], [ "Abstract The algebras $A(\\Gamma)$, where $\\Gamma$ is a directed layered graph, were first constructed by I. Gelfand, S. Serconek, V. Retakh and R. Wilson.", "These algebras are generalizations of the algebras $Q_n$, which are related to factorizations of non-commutative polynomials.", "It was conjectured that these algebras were Koszul.", "In 2008, T.Cassidy and B.Shelton found a counterexample to this claim, a non-Koszul $A(\\Gamma)$ corresponding to a graph $\\Gamma$ with 18 edges and 11 vertices.", "We produce an example of a directed layered graph $\\Gamma$ with 13 edges and 9 vertices which produces a non-Koszul $A(\\Gamma)$.", "We also show this is the minimal example with this property." ], [ "Introduction", "The relationship between the factorizations and coefficients of a non-commutative polynomial is described in terms of pseudo-roots by the non-commutative version of Vieta's theorem [4].", "The algebra $Q_n$ of the pseudo-roots for a polynomial of degree $n$ has been described and studied by Gelfand, Retakh and Wilson in [5].", "These algebras are quadratic and Koszul, and their corresponding dual algebras $Q_n^!$ have finite dimension.", "The generators of this algebra correspond to elements of $B_n$ , the boolean lattice of all subsets of an $n-$ element set.", "This construction was then generalized to form the class of algebras $A(\\Gamma )$ , each of which is determined by a layered graph $\\Gamma $ .", "The algebras $Q_n$ are simply the algebras $A(\\Gamma )$ , where $\\Gamma $ equals $B_n$ .", "Depending on a condition called uniformity of the graph $\\Gamma $ , the algebra $A(\\Gamma )$ may be quadratic [8], thus leading to the question of which of these algebras are Koszul.", "It was discovered that the algebras were Koszul for the boolean lattice $B_n$ , simplicial complexes and complete layered graphs with arbitrary numbers of vertices at each level [9], [10].", "It was also conjectured that all algebras $A(\\Gamma )$ were Koszul.", "This was shown not to be the case when Cassidy and Shelton found the first example of a non-Koszul $A(\\Gamma )$ [3]; see Figure REF .", "Figure: A Non-Koszul A(Γ)A(\\Gamma ) with Eleven VerticesThis led to the question of what the smallest $\\Gamma $ with a quadratic non-Koszul $A(\\Gamma )$ might be.", "In 2010, Retakh, Serconek, and Wilson found that when one edge is removed from a particular layer, we retain both uniformity and non-Koszulity of $A(\\Gamma )$ [7] (see Figure REF .)", "Figure: Another Non-Koszul A(Γ)A(\\Gamma ) with Eleven VerticesBy computer, we found that we could extend the example further, removing one vertex from the second highest layer though this was not minimal; see Figure REF .", "Figure: A Non-Koszul A(Γ)A(\\Gamma ) with Ten VerticesIn this paper we now produce the smallest non-Koszul example.", "We named this graph $H$ after HiLGA,The name HiLGA comes from “Hilbert Series of Layered Graph Algebras”.", "the program we wrote to study algebras of the form $A(\\Gamma ).$ Computers are not needed for showing either the minimality in terms of number of vertices, or the non-Koszulity of $H.$" ], [ "Theory", "Let $\\Gamma =(V,E)$ be a directed graph, where $V = \\coprod _{i=0}^{N}V_i$ and $V_i \\ne \\emptyset $ for $i \\in \\lbrace 0, 1, \\cdots , N\\rbrace .$ We call these $V_i$ the layers of $\\Gamma $ .", "For each vertex $v$ , we write $|v|=i$ , and say that $v$ has height $i$ if $v \\in V_i$ .", "For each edge, let $t(e)$ denote the tail of $e$ and $h(e)$ denote the head of $e$ .", "We say that $\\Gamma $ is a layered graph of height $N$ if $|t(e)| = |h(e)| + 1$ for all $e \\in E.$ Given a layered graph, we construct an algebra $A(\\Gamma )$ as follows: Begin with the free algebra $T(E)$ generated by the edges of $\\Gamma $ .", "Let $\\pi $ and $\\pi ^{\\prime }$ be any two paths $\\pi = (e_1,e_2, \\cdots e_k)$ and $\\pi ^{\\prime } = (f_1, f_2, \\cdots f_k)$ with the same starting and finishing vertices.", "For every such pair $\\pi , \\pi ^{\\prime }$ , we impose the relation $(t-e_1)(t-e_2) \\cdots (t-e_k) = (t-f_1)(t-f_2) \\cdots (t-f_k)$ where $t$ is a formal parameter commuting with all edges in $E.$ Matching the coefficients of $t$ gives us a collection of relations.", "Let $I$ be the collection of all of these relations for any two paths meeting our conditions.", "Then $A(\\Gamma )$ is $T(E)/I.$ The standard definition of Koszulity requires a quadratic algebra.", "Not all algebras of the form $A(\\Gamma )$ are quadratic, but there is a condition on $\\Gamma $ which guarantees $A(\\Gamma )$ is quadratic.", "Two vertices $v,v^{\\prime }$ of the same layer $l$ are connected by an up-down sequence if there is a sequence $v=v_0, v_1, \\cdots ,v_k=v^{\\prime }$ of vertices of level $l$ with the following property: For each $i$ in $\\lbrace 1, \\cdots , k\\rbrace $ there are edges $e$ and $f$ so $t(e)=v_{i-1}$ , $t(f)=v_i$ and $h(e)=h(f).$ A layered graph $\\Gamma $ is said to be uniform if for any pair of edges $e,e^{\\prime }$ with a common tail, their heads are connected by an up down sequence $v_0, v_1, \\cdots v_k$ with $v_i$ adjacent to $t(e)$ for $i\\in \\lbrace 1, \\cdots , k\\rbrace .$ It has been shown in [8] that if $\\Gamma $ is a uniform layered graph then $A(\\Gamma )$ is a quadratic algebra.", "If $\\Gamma $ has a unique minimal vertex of layer zero then there is a construction giving us a new algebra we call $B(\\Gamma ).$ The algebras $B(\\Gamma )$ can be presented by generators $u \\in V^+ =\\coprod _{i=1}^{N} V_i$ and relations $u \\cdot w = 0$ if there is no edge from $u$ to $w$ $u \\cdot \\sum _{w \\in S(u)} w = 0$ where $S(u)$ is the collection of vertices $w$ for which there is an edge from $u$ to $w$ .", "The algebras $B(\\Gamma )$ are the dual algebras of the associated graded algebras of $A(\\Gamma )$ under a certain filtration.", "They have the same Hilbert series as the actual dual of $A(\\Gamma )$ and when it exists, $B(\\Gamma )$ is Koszul if and only if $A(\\Gamma )$ is [7]." ], [ "A Layered Graph $\\Gamma $ on Nine Vertices with non-Koszul A({{formula:9ba23ee7-462a-417c-874b-5cc336fb47d6}} )", "We introduce a layered graph $H$ on nine vertices; see Figure REF .", "Figure: The Poset HHTheorem 3.1 The algebra $A(\\Gamma )$ is not Koszul for $\\Gamma = H$ .", "We use the numerical Koszulity test from theorem 4.2.1 in [7].", "Considering the $i=4$ case we know that $A(\\Gamma )$ is not numerically Koszul if $\\dim (H^{-1}(\\Gamma _{a,4})) -\\dim (H^{0}(\\Gamma _{a,4})) + \\dim (H^{1}(\\Gamma _{a,4})) \\ne 0$ where the layer of $a$ is greater than or equal to 4.", "Thus $a$ must be the maximal vertex in $H$ so $\\Gamma _{a,4}$ is the graph in Figure REF .", "Figure: The Graph Γ a,4 \\Gamma _{a,4}By the Euler-Pointcare formula, we know $\\sum (-1)^i \\dim (H^i) =\\sum (-1)^j \\dim (C^j).$ Thus we have $\\dim (H^{-1}(\\Gamma _{a,4})) - \\dim (H^{0}(\\Gamma _{a,4})) +\\dim (H^{1}(\\Gamma _{a,4})) =$ $\\dim (C^{-1}(\\Gamma _{a,4})) - \\dim (C^{0}(\\Gamma _{a,4})) +\\dim (C^{1}(\\Gamma _{a,4})) - \\dim (C^{2}(\\Gamma _{a,4})) +\\dim (H^{2}(\\Gamma _{a,4})) \\ge $ $\\dim (C^{-1}(\\Gamma _{a,4})) - \\dim (C^{0}(\\Gamma _{a,4})) +\\dim (C^{1}(\\Gamma _{a,4})) - \\dim (C^{2}(\\Gamma _{a,4}))$ $= 1 - 7 + 13 - 6 = 1.$ This shows that $H$ is not numerically Koszul.", "Since Koszulity implies numerical Koszulity, $H$ is not Koszul as well." ], [ "The Minimal non-Koszul A($\\Gamma $ )", "We now wish to show that $H$ is the unique minimal example.", "We must restrict ourselves to uniform $\\Gamma $ in order to guarantee that the algebra is quadratic There is still room, however, to generalize here using one of the different definitions of Koszulity for non-quadratic algebras.. We will need part of a proposition of Polishchuk and Positselksi found in [6] which we summarize here for clarity.", "Proposition 4.1 Let $W$ be a vector space and $X_1, \\cdots , X_N \\subset W$ be a collection of its subspaces.", "Then the following are equivalent: The collection $X_1, \\cdots , X_N$ is distributive.", "There exists a direct sum decomposition $W = \\bigoplus _{\\eta \\in H} W_\\eta $ of the vector space $W$ such that each of the subspaces $X_i$ is the sum of a set of subspaces $W_\\eta .$ Lemma 4.1 Suppose the vector subspaces $W_1, W_2, \\cdots , W_M$ and $X_1, X_2,\\cdots , X_N$ generate distributive lattices in Y and Z respectively.", "Then the subspaces $W_1 \\otimes Z, W_2 \\otimes Z, \\cdots , W_M\\otimes Z, Y \\otimes X_1, Y \\otimes X_2, \\cdots , Y \\otimes X_N$ generate a distributive lattice in $Y \\otimes Z.$ By proposition REF , there exist index sets $\\alpha , \\beta $ and decompositions $Y = \\oplus _{i \\in \\alpha } T_i$ and $Z = \\oplus _{i\\in \\beta } U_i$ so that for every $i \\in \\lbrace 1, \\cdots , M\\rbrace $ and $j\\in \\lbrace 1, \\cdots , N\\rbrace $ there are sets $\\alpha _i \\subseteq \\alpha $ and $\\beta _j \\subseteq \\beta $ so that $W_i = \\oplus _{k \\in \\alpha _i}T_k$ and $X_j = \\oplus _{k \\in \\beta _i} U_k.$ We can construct the direct sum decomposition $Y \\otimes Z = \\oplus _{i \\in \\alpha , j \\in \\beta } (T_i \\otimes U_j).$ Then for any $i \\in \\lbrace 1, \\cdots , M\\rbrace $ , $W_i \\otimes Z = \\oplus _{j \\in \\alpha _i, k \\in \\beta } (T_j \\otimes U_k)$ and $Y \\otimes X_i = \\oplus _{j \\in \\alpha , k \\in \\beta _i} (T_j\\otimes U_k)$ .", "Referring to proposition REF once more completes the proof.", "In order to show a quadratic algebra is Koszul, we need to show that for any positive integer $n$ , the collection of subspaces $V^iRV^{n-i-2}$ generates a distributive lattice in $V^n.$ We repeat the following lemma from Serconek and Wilson's paper [10] which will allow us to reduce the problem to one where we check for distributivity inside a much smaller vector space.", "Lemma 4.2 Let $V = \\sum _{i \\in I} V_{[i]}$ be a graded vector space and $\\lbrace X_j | j \\in J \\rbrace $ be a collection of subspaces of $V$ .", "Assume that each $X_j$ is graded, $X_j = \\sum _{i \\in I} X_{j,[i]},$ and $X_{j,[i]} = X_j \\cap V_{[i]}.$ Then $\\lbrace X_j | j \\in J \\rbrace $ generates a distributive lattice in $V$ if and only if, for every $i\\in I, \\lbrace X_{j,[i]} | j \\in J \\rbrace $ generates a distributive lattice in $V_{[i]}.$ Lemma 4.3 Let $A$ be a quadratic algebra where the generators are partitioned into the disjoint spaces $V_1, V_2, \\cdots , V_N.$ Suppose every relation is contained inside the space $V_{i+1}V_i$ for some $i \\in \\lbrace 1, \\cdots , n-1\\rbrace $ .", "Consider the $\\mathbb {Z}^n$ grading of $V^n$ where $x_1x_2 \\cdots x_n$ is in $V^n_{[(z_1, z_2, \\cdots , z_n)]}$ if and only if $x_i \\in V_i$ for all $i \\in \\lbrace 1, \\cdots , n\\rbrace $ .", "The collection $\\lbrace RV^{n-2}, \\cdots , V^{n-2}R\\rbrace $ generates a distributive lattice in $V^n$ if and only if $\\lbrace RV^{n-2}_{[(n,n-1,\\cdots , 1)]},$ $VRV^{n-3}_{[(n,n-1,\\cdots , 1)]},$ $\\cdots ,$ $V^{n-2}R_{[(n,n-1,\\cdots , 1)]} \\rbrace $ generates a distributive lattice in $V^n_{[(n,n-1,\\cdots , 1)]}.$ By lemma REF we know that $\\lbrace RV^{n-2}, \\cdots , V^{n-2}R\\rbrace $ generates a distributive lattice in $V^n$ if and only if $\\lbrace RV^{n-2}_{[(z_n,z_{n-1},\\cdots , z_1)]},$ $VRV^{n-3}_{[(z_n,z_{n-1},\\cdots , z_1)]},$ $\\cdots ,$ $V^{n-2}R_{[(z_n,z_{n-1},\\cdots , z_1)]}$ generates a distributive lattice in $V^n_{[(z_n,z_{n-1},\\cdots , z_1)]} $ for every $(z_n,z_{n-1},\\cdots , z_1).$ Thus we only need to show that if $\\lbrace RV^{n-2}_{[(n,n-1,\\cdots , 1)]},$ $VRV^{n-3}_{[(n,n-1,\\cdots , 1)]},$ $\\cdots ,$ $V^{n-2}R_{[(n,n-1,\\cdots , 1)]} \\rbrace $ generates a distributive lattice in $V^n_{[(n,n-1,\\cdots , 1)]}$ then $\\lbrace RV^{n-2}_{[(z_n,z_{n-1}, \\cdots , z_1)]},$ $VRV^{n-3}_{[(z_n,z_{n-1},\\cdots , z_1)]},$ $\\cdots ,$ $V^{n-2}R_{[(z_n,z_{n-1}, \\cdots , z_1)]}\\rbrace $ generates a distributive lattice in $V^n_{[(z_n,z_{n-1}, \\cdots ,z_1)]}$ for every $(z_n,z_{n-1},\\cdots , z_1).$ Consider the collection $\\lbrace V_{z_n}V_{z_{n-1}}V_{z_{n-2}} \\cdots V_{z_{b+1}} W_i V_{z_{a-1}} V_{z_{a-2}} \\cdots V_{z_{1}} \\rbrace _{i \\in \\lbrace a+1, \\cdots , b \\rbrace } $ where $W_i =$ $V_b V_{b-1}$ $\\cdots $ $V_{i+1} R_i V_{i-2} V_{i-3}$ $\\cdots $ $V_{a+1} V_a$ .", "It is enough to show that this collection is distributive in $V^n_{[(z_n,z_{n-1}, \\cdots , z_2, z_1)]}$ where $z_{i+1} = z_{i} + 1$ for $i \\in \\lbrace a, \\cdots , b-1\\rbrace $ because then we can string these increasing runs together using lemma REF to complete the proof.", "By our assumption, we do know that the collection $\\lbrace V_{n} V_{n-1}V_{n-2} \\cdots V_{b+1} W_i V_{a-1} V_{a-2} \\cdots V_{1} \\rbrace _{i \\in \\lbrace a+1, \\cdots , b \\rbrace }$ is distributive in $V^n_{[(n, n-1, \\cdots , 2,1)]}$ which implies the collection $\\lbrace W_i \\rbrace _{i \\in \\lbrace a+1, \\cdots ,b \\rbrace } $ is distributive in $V^{b-a+1}_{[(b, b-1, \\cdots , a+1,a)]}$ .", "This implies the distributivity of $\\lbrace V_{z_n}V_{z_{n-1}}V_{z_{n-2}} \\cdots V_{z_{b+1}} W_i V_{z_{a-1}}V_{z_{a-2}} \\cdots V_{z_{1}} \\rbrace _{i \\in \\lbrace a+1, \\cdots , b\\rbrace } $ in $V^n_{[(z_n, z_{n-1}, \\cdots , z_2, z_1)]}$ completing the proof.", "Theorem 4.1 Suppose $\\Gamma $ is a uniform layered graph on $V = \\cup _{i=0}^NV_i$ with exactly one vertex at layer $k$ .", "Let $\\Gamma _0$ be the induced subgraph of $\\Gamma $ on $\\cup _{i=0}^k V_i$ and $\\Gamma _1$ be the induced subgraph of $\\Gamma $ on $\\cup _{i=k}^N V_i$ .", "Then $A(\\Gamma )$ is Koszul if $A(\\Gamma _0)$ and $A(\\Gamma _1)$ are Koszul.", "We work in terms of the associated graded algebra of $A(\\Gamma )$ (equal to $B(\\Gamma )^!$ ) so that the relations meet the hypotheses of lemma REF .", "The lemma allows us to assume the height decreases at each step.", "As $B(\\Gamma _1)^!$ is Koszul we know that $\\lbrace (\\otimes _{i=j+1}^{N}V_i) \\otimes R_j \\otimes (\\otimes _{i=k+1}^{j-2} V_i) \\rbrace _{j \\in \\lbrace k+2, \\cdots , N \\rbrace }$ is distributive in $Y = \\otimes _{i=k+1}^{N}V_i.$ As $B(\\Gamma _0)^!$ is Koszul we know that $\\lbrace (\\otimes _{i=j+1}^{k} V_i) \\otimes R_j \\otimes (\\otimes _{i=1}^{j-2} V_i) \\rbrace _{j \\in \\lbrace 2, \\cdots , k \\rbrace }$ is distributive in $Z = \\otimes _{i=1}^{k} V_i.$ Applying lemma $\\ref {YZ}$ gives us the distributivity of the collection $\\lbrace (\\otimes _{i=j+1}^{N} V_i) \\otimes R_j \\otimes (\\otimes _{i=1}^{j-2} V_i) \\rbrace _{j \\in \\lbrace 2, \\cdots , k \\rbrace \\cup \\lbrace k+2,\\cdots , N \\rbrace }$ in $\\otimes _{i=1}^{N} V_i$ .", "Since $R_{k+1} = {0}$ this implies distributivity for $j \\in \\lbrace 2, \\cdots , N \\rbrace $ thus showing $B(\\Gamma )^!$ and thus $A(\\Gamma )$ is Koszul.", "We can now use this to reduce the number of cases we need to consider.", "Say that a layered graph $\\Gamma $ is a $[z_0,z_1, \\cdots , z_N]$ -graph if $|V_i| = z_i$ for each $i$ , where $V_i$ is the collection of vertices of layer $i.$ Proposition 4.2 Any uniform layered graph with unique minimal and maximal elements and non-Koszul $A(\\Gamma )$ has at least nine vertices.", "Suppose the $[1,z_1,z_2, \\cdots , z_{n-1}, 1]$ -graph is minimal in number of vertices amongst graphs with non-Koszul $A(\\Gamma )$ .", "Then $z_i> 1$ for all $0 < i < N.$ Otherwise we could use theorem REF to find a smaller example.", "We also know from [7] that no example of a non-Koszul $A(\\Gamma )$ exists for graphs of height three or less.", "This leaves one possibility for a non-Koszul $A(\\Gamma )$ with under nine vertices: that of a [1,2,2,2,1]-graph.", "There are only ten such graphs with unique maximal and minimal vertices [2], and of those only five [1] are uniform; see Figure REF .", "Figure: The Five Uniform [1,2,2,2,1]-graphsIt is easy to check these five cases, by either hand or computer to see these are all Koszul.", "Theorem 4.2 Consider the collection of all uniform layered graphs with unique maximal and minimal elements.", "$\\Gamma = H$ is the minimal example producing a non-Koszul $A(\\Gamma ).$ To clarify, by minimal we mean in terms of number of vertices though it has been checked by computer that this is also the edge-minimal example.", "Using proposition REF we only need to show that $H$ is the unique example with nine vertices producing a non-Koszul $A(\\Gamma )$ .", "To show this, we need to check the ten uniform [1,3,2,2,1]-graphs, ten uniform [1,2,2,3,1]-graphs, and twenty-three uniform [1,2,3,2,1]-graphs (see Figures REF , REF , and REF .)", "Figure: The Ten Uniform [1,3,2,2,1]-graphsFigure: The Ten Uniform [1,2,2,3,1]-graphsFigure: The Twenty-three Uniform [1,2,3,2,1]-graphsThis was done by computer, though these 43 cases individually can still each be done by hand.", "The graph $H$ is minimal under more general conditions as well.", "We can first drop the requirement that the graph must have a unique maximal element, but still require that maximal elements be of maximal rank.", "This changes very little, but does include the possibility of a [1,2,2,2,2]-graph with non-Koszul $A(\\Gamma )$ .", "There are thirty-five of these, twenty one of these being uniform, also not too time consuming for us to check.", "To generalize further, we drop the last condition and allow maximal elements at every level.", "Now that we include graphs like the one shown in Figure REF , Figure: An Arbitrary Uniform [1,2,3,2,1]-graphwe have many more examples to consider.", "For ruling out [1,2,2,2,1]-graphs, there are thirty-three cases needing to be checked.", "To show $H$ is minimal amongst graphs with nine vertices we must check 83 [1,3,2,2,1]-graphs, 170 [1,2,3,2,1]-graphs, 93 [1,2,2,3,1]-graphs, and 65 [1,2,2,2,2]-graphs.", "This is a bit much to check by hand, but it can and has been done by computer.", "With this, the following has been shown: Theorem 4.3 Consider the collection of all uniform layered graphs with a unique minimal element.", "$\\Gamma = H$ is the minimal example producing a non-Koszul $A(\\Gamma ).$" ], [ "Acknowledgements", "The authors would like to thank Robert Wilson for his guidance and advice.", "We would also like to thank programmer John Yeung for introducing and mentoring us in Python, the language HiLGA was written in, which led us to the discovery of the poset $H$ ." ] ]

[ [ "Dynamical systems of eternal inflation: A possible solution to the\n problems of entropy, measure, observables and initial conditions" ], [ "Abstract There are two main approaches to non-equlibrium statistical mechanics: one using stochastic processes and the other using dynamical systems.", "To model the dynamics during inflation one usually adopts a stochastic description, which is known to suffer from serious conceptual problems.", "To overcome the problems and/or to gain more insight, we develop a dynamical systems approach.", "A key assumption that goes into analysis is the chaotic hypothesis, which is a natural generalization of the ergodic hypothesis to non-Hamiltonian systems.", "The unfamiliar feature for gravitational systems is that the local phase space trajectories can either reproduce or escape due to the presence of cosmological and black hole horizons.", "We argue that the effect of horizons can be studied using dynamical systems and apply the so-called thermodynamic formalism to derive the equilibrium (or Sinai-Ruelle-Bowen) measure given by a variational principle.", "We show that the only physical measure is not the Liouville measure (i.e.", "no entropy problem), but the equilibrium measure (i.e.", "no measure problem) defined over local trajectories (i.e.", "no problem of observables) and supported on only infinite trajectories (i.e.", "no problem of initial conditions).", "Phenomenological aspects of the fluctuation theorem are discussed." ], [ "Introduction", "How to make testable and sensible prediction is one of the most important unresolved problems in contemporary cosmology.", "A number of interesting, but controversial, ideas had been put forward (e.g.", "quantum cosmology [1], [2], holographic cosmology [3], [4], [5], [6], [7]), but by far the most popular approach is realized in the context of eternal inflation [8], [9], [10], where the problem of making predictions is known as the measure problem [11], [12], [13].", "In recent years the idea of eternal inflation has gained a renewed interest due to a possible unification of inflationary cosmology and string theory in the context of a huge landscape of vacua [14].", "It is also argued that the unified framework may simultaneously help us to solve the cosmological constant problem using either a non-anthropic solution [15] or an anthropic solution [16] with very mild assumptions about an underlying probability measure.", "However, to declare a victory one has to derive the measure from first principles which has proven to be a very difficult task.", "So, the main question is: can the measure problem in eternal inflation be really solved?", "The answer, perhaps, depends crucially on how one defines inflation.", "So far, most of the attempts to tackle the problem were using stochastic description which can be modeled, for example, by diffusion in a configuration space.", "Given the stochastic model one can start asking probabilistic questions, but, as it turned out, the answer always depends on either initial conditions [17], [18], [19] (i.e.", "problem of initial conditions) or on a cut-off procedure [11], [12], [13] (i.e.", "measure problem).", "This would be a real pity if one had to postulate an additional rule such as initial conditions or a probability measure to determine observables in a system which seems to have an attractor dynamics (e.g.", "cosmic inflation).", "Moreover, many otherwise phenomenologically acceptable stochastic measures (e.g.", "causal patch measure [19] or scale factor measure [20]) give rise to very counterintuitive and somewhat paradoxical predictions [24], [22], [23].", "In other words, it is not always clear how to define a probability space of observables without violating the basic principles of the probability theory.", "We will refer to it as the problem of defining cosmological observables or simply the problem of observables.", "At this point one might start worrying whether the stochastic description, which is at most an approximation to the underlying microscopic dynamics, is a good mathematical model of eternal inflation.", "The objective of this paper is to construct an alternative mathematical model of inflation using dynamical systems, but before we proceed, let us briefly review another related problem - the entropy problem [25].", "Consider a finite Hamiltonian system.", "For such systems the most physical time-invariant measure is given by the Liouville measure, according to which a typical observer should find himself in a highly entropic state.", "Cosmology for such observers (often called Boltzmann observers [27]) would be very boring in a sharp contrast to what we actually observe.", "This is the so-called entropy problem.", "On the other hand, one can certainly define other non-invariant measures on the surface of initial conditions of a given observer (e.g.", "geocentric measures [24]), but it might be more desirable to have a dynamical mechanism to explain cosmological observations.Indeed, in its origin the geocentric approach adopts a Bayesian (or subjective) interpretation of probabilities when a frequentists (or objective) interpretation is often easier digested by physicists.", "So, another relevant question is: can the entropy problem in cosmology be really solved?", "There are at least two approaches that one might take.", "Roughly speaking, we need to violate the Liouville theorem by providing a mechanism to either add or remove phase space trajectories.", "Clearly, a global description of gravitational systems provides a natural mechanism to accommodate both phenomena.", "For example, the eternally inflating space-time constantly “adds” new local trajectories (i.e.", "more and more local observers fall out of causal contact with each other) and the constantly forming black holes “remove” old local trajectories when the observers hit singularities.", "Since the phase space volume for such observers is no longer conserved, the Liouville measure is not very useful, but one might still wonder whether there are any good time-invariant measures.", "As we will argue below, the space of time-invariant measures for a generic dynamical system is very large, but the so-called equilibrium measure is often a unique measure given by a variational principle.", "So, it appears that the entropy and measure problems can be simultaneously avoided, if not solved, in the context of more general dynamical systems.", "In addition, the equilibrium measure is defined on a space of local trajectories with support on only infinite trajectories.Although the precise mathematical definition of equilibrium measures involves an infinite time limit, for all practical purposes it is sufficient to follow the system for a very long but finite time.", "Moreover, for the systems which only allow finite trajectories the phrase “infinite trajectories” should be read throughout the paper as “very long trajectories”.", "This provides a possible resolution to the problem of defining relevant cosmological observables (i.e.", "observables are the trajectories) as well as to the problem of initial conditions (i.e.", "for infinite trajectories the problem is irrelevant).", "The paper is organized as follows.", "In Section we review the stochastic approach and problems associated with it.", "In Section we introduce the dynamical systems approach with an emphasis to a variational principle and a fluctuation theorem.", "In Section we construct a dynamical system of eternal inflation and derive its equilibrium measures.", "In Section we summaries the main results." ], [ "Stochastic approach", "Consider a deterministic dynamical system whose evolution is defined by a velocity flow ${\\bf v}({\\bf x}) \\equiv \\frac{d \\bf x}{d t}$ and the system at time $t$ is described by a state vector ${\\bf x}(t) \\in X$ .", "If the system does not have any absorbing states, then the evolution of an arbitrary distribution function $\\mu ({\\bf x},t)$ can be followed in time using the continuity equation: $\\frac{\\partial \\mu ({\\bf x},t)}{\\partial t} = - \\frac{\\partial }{\\partial {\\bf x}} \\cdot {\\bf v}({\\bf x}) \\mu ({\\bf x},t).$ The main challenge in the stochastic approach is to solve the continuity equation for a given model of the velocity flow ${\\bf v}({\\bf x})$ .", "In what follows we will consider three models of the flow all of which lead to yet unresolved cosmological problems." ], [ "Entropy problem", "Perhaps the most studied dynamical systems are Hamiltonian systems.", "For such systems the components of a state vector ${\\bf x}$ come in conjugate pairs ${\\bf x} =\\lbrace {\\bf p, q}\\rbrace $ corresponding to momentum ${\\bf p}$ and position ${\\bf q}$ coordinates and the Hamiltonian equations of motion imply that the velocity flow is not compressible $ \\frac{\\partial }{\\partial {\\bf x}} \\cdot {\\bf v} = 0$ .", "Under the incompressibility assumption Eq.", "(REF ) becomes the classical Liouville equation: $\\frac{\\partial \\mu ({\\bf x},t)}{\\partial t} = - {\\bf v}({\\bf x}) \\cdot \\frac{\\partial }{\\partial {\\bf x}} \\mu ({\\bf x},t),$ which has a trivial time-independent solution $\\mu _L = \\Gamma ^{-1}$ known as the Liouville measure, where $\\Gamma $ is the volume of phase space $X$ .", "This measure is known to be very useful for describing large thermodynamic system, but it is not very useful for describing the universe.", "In particular, the Liouville measure gives rise to the entropy problem - entropy of the observable universe is much smaller than what one would naively expect [25].", "It also follows immediately that for infinite ($\\Gamma =\\infty $ ) Hamiltonian systems the Liouville measure does not exist, which is the main source of a measure problem to be discussed below.", "But what is really the size of the phase space of eternal inflation: finite or infinite?", "In a flat slicing of de Sitter space, $d s^2 = dt^2 - e^{2 H t} dx^2,$ the new degrees of freedom constantly come from under the Planck scale, but as the modes are eventually stretched out to super-horizon scales they can no longer be observed.", "This is a global picture that suggests that the total number of degrees of freedom is infinite, which is misleading if one wants to count only the states accessible to a local observer.", "From a local viewpoint, the evolution is most conveniently described in static coordinates, $d s^2 = (1-(r H)^2) dt^2 - \\frac{d r^2}{1-(r H)^2} - r^2 d \\Omega ,$ where the amount of information accessible to a local observer is only finite (when a cut-off is imposed at the Planck scale).", "Note that in a quasi-de Sitter space, with possible transitions between different vacua, the size of the phase space should be set by an exponent of the entropy of a vacua with with the smallest positive energy density.", "One could still ran into problems with Minkowski vacua, but this will turn out not to be the case for the time-invariant measures discussed in Section .", "The phase space might still be huge but, what is more important, it is finite." ], [ "Measure problem", "If we try to model a finite dynamical system, consisting of only degrees of freedom accessible to a local observer, then to solve the entropy problem one should abandon the idea of a Hamiltonian description whose predictions are in conflict with observations.", "The appearance of a non-Hamiltonian dynamics is not entirely new and happens all the time whenever some of the degrees of freedom of a larger Hamiltonian system are ignored.", "This is exactly the situation in a local description of gravitational systems with horizons.", "Once we integrate over the degrees of freedom unaccessible to a local observer, the local dynamical system should start to behave as a non-Hamiltonian system.", "In a global stochastic description of eternal inflation one usually models the dynamics with three non-Hamiltonian ingredients [11].", "First of all, the quantum (or thermal) effects are modeled by a compressible flow in configuration space (e.g.", "$\\frac{1}{8 \\pi ^2} \\partial _{\\bf q} H({\\bf q})^{\\gamma } \\partial _{\\bf q} H({\\bf q})^{3-\\gamma } \\mu ({\\bf q},t)$ during slow-roll inflation).", "Moreover, the constant addition and removal of local trajectories are modeled with reproducing (e.g.", "$3 H({\\bf q})^\\alpha \\mu ({\\bf q},t)$ ) and absorbing states (e.g.", "$\\mu ({\\bf q}) = 0$ for ${\\bf q} \\in \\partial X$ ).", "Then, Eq.", "(REF ) becomes a branching-diffusion equation with escape: $\\frac{\\partial \\mu ({\\bf q},t)}{\\partial t} = \\frac{1}{8 \\pi ^2} \\frac{\\partial }{\\partial {\\bf q}} H({\\bf q})^{\\gamma } \\frac{\\partial }{\\partial {\\bf q}} {H({\\bf q})^{3-\\gamma }} \\mu ({\\bf q},t) - {\\bf v}({\\bf q}) \\cdot \\frac{\\partial }{\\partial {\\bf q}} \\mu ({\\bf q},t) + 3 H({\\bf q})^\\alpha \\mu ({\\bf q},t),$ where $H({\\bf q})$ is the Hubble scale.", "Although the stochastic eternal inflation avoids the entropy problem, it immediately introduces a well known measure problem, which one can think of as a counterpart of the entropy problem for more general dynamical systems.", "In other words, what time coordinate (or $\\alpha $ ) should we use for calculating probabilities?", "It is well known that different choices can lead to very different answers, and the most popular choice $\\alpha =0$ , corresponding to the scale factor measure, is often chosen on purely phenomenological grounds [20].", "This might be acceptable phenomenologically, but it is not acceptable from the theoretical viewpoint where one wants to derive the measure from first principles." ], [ "Problem of initial conditions", "Since the main source of the measure problem was due to the presence of reproducing states one might wonder what would happen if we ignore the reproduction term, $3 H({\\bf x})^\\alpha \\mu ({\\bf x},t)$ .", "In fact, this is a well known limit, corresponding to a local description of eternal inflation, where one concentrates on the evolution of comoving distributions, i.e.", "$\\frac{\\partial \\mu ({\\bf q},t)}{\\partial t} = \\frac{1}{8 \\pi ^2} \\frac{\\partial }{\\partial {\\bf q}} H({\\bf q})^{\\gamma }\\frac{\\partial }{\\partial {\\bf q}} {H({\\bf q})^{3-\\gamma }} \\mu ({\\bf q},t) - {\\bf v}({\\bf q}) \\cdot \\frac{\\partial }{\\partial {\\bf q}} \\mu ({\\bf q},t).$ The local stochastic approach was originally proposed to study the effects of quantum fluctuations during inflation [9], [17], but later it was adopted to study the landscape models of eternal inflation [18].", "Due to the presence of absorbing states the answer always depends on the initial conditions and even unequal weighting (e.g.", "entropic or anthropic) of states would not cure the problem.", "In other words, even if a given local measure (e.g.", "causal patch measure [19]) gives phenomenologically acceptable results for some ranges of initial conditions and some ranges of parameters of a model, it does not solve the problem of initial conditions.", "One can certainly take a point of view that any physical problem must involve the knowledge of initial conditions.", "This was an attitude in the early days of quantum cosmology [1] as well as very recently in the geocentric approach [24].", "But then it seems unnecessary complicated to postulate a measure in addition to postulating initial conditions.", "If the only role of cosmology is to assign probabilities to local observations, then it is always only a problem of initial conditions and instead of fighting it we should learn how to construct a theory of initial conditions [1], [24].", "This would have been an acceptable “solution” if we did not have examples where, in the long run, the system completely forgets its initial state.", "For example, a large Hamiltonian system close to a thermal equilibrium is a system for which one can study its macroscopic properties without the knowledge of initial conditions.", "Of course, as we have argued above, the universe is not in a thermal state, but one might still hope that a similar phenomena would occur for more general, and perhaps, non-Hamiltonian systems.", "In constructing a system which eventually forgets its initial state we should be careful not to introduce any other problems as it was in the case of a global description of eternal inflation discussed above.", "However, in our opinion, the best possible solution to the problem of initial conditions would be if the initial conditions did not exist.", "In other words, if the universe would be infinite to the past (as well as to the future), then the question of initial condition would be irrelevant.", "For some time it was believed that eternal inflation might provide a possible framework to accomplish this task, until a no-go theorem was proved which states that the eternally inflating space-times are not past complete [26].", "Of course, to prove any no-go theorem one makes certain assumptions which often turn out to be false and finding such “loopholes” is one of the biggest challenges for theoretical physics.", "In fact, the conclusions of Ref.", "[26] do not apply to infinite trajectories, that play a central role in the dynamical systems approach developed in this paper, even when their Liouville measure is zero." ], [ "Problem of observables", "Another problem associated with stochastic descriptions of eternal inflation is related to the problem of defining relevant cosmological observables.", "More precisely, the problem is to define a measurable space of observables on which the cosmological measures are to be constructed.", "A priori, there is a lot of freedom in choosing a relevant measurable space and some popular choices include a space of local states, a space of states of local observers, or even a space of states of local brains.", "However, it turned out to be a very non-trivial task to define a measurable space which avoids paradoxes [11], [21], [24], [22], [23].", "For example, if one applies the (global or local) stochastic measures to laboratory experiments, then even the most popular phenomenological choices (scale-factor measure [20] and causal patch measure [19]) are not free of logical inconsistencies [24], [22], [23].", "The problems arise due to an exponential growth of the distribution $\\mu ({\\bf x},t)$ defined on a measurable space of local states.", "In such exponentially growing models one can construct paradoxical situations where the probabilities of past events change with time [22].", "This is, perhaps, an indication that the measure $\\mu ({\\bf x},t)$ on a space of local states might not be suitable for describing inflationary systems.", "For Hamiltonian systems the measure on states was certainly very useful for calculating macroscopic observables using microcanonical, canonical or grand canonical ensembles, but it does not have to be appropriate for more general dynamical systems.", "The measure $\\mu ({\\bf x},t)$ contains only a very limited amount of information about the dynamics which was sufficient for equilibrium statistical mechanics, but might be insufficient for describing eternal inflation.", "For example, if the relevant distributions are to be defined on a space of trajectories then such distributions would contain much more information than any distribution on states.", "Evidently, one can easily calculate a measure on states from a measure on trajectories by using a sequential cutoff measure (see Ref.", "[22] for details), but not the other way around." ], [ "Dynamical systems approach", "A stochastic approach to statistical mechanics was originated over a century ago by Boltzmann, while a dynamical systems approach was proposed by Ruelle only forty years ago and later developed into a consistent mathematical framework [28].", "Although most of the precise results are known only for mathematically “simple” systems such as Anosov (or hyperbolic) systems, the more complicated dynamical systems are usually analyzed under the so-called chaotic hypothesis.", "It says that for computing macroscopic observables, any chaotic dynamical system can be considered as an Anosov system.", "In contrast to non-chaotic (or integrable) systems, the chaotic systems allow us to define time averages independent of initial conditions which is a desired property if one wants to solve the cosmological problem of initial conditions.", "The chaotic hypothesis can be viewed as a generalization of the ergodic hypothesis to more general non-Hamiltonian systems." ], [ "Equilibrium measures", "The problems of interest in the measure theoretic discussions of non-Hamiltonian systems involve finding the most physical measure $\\mu \\in {\\cal M}$ , where ${\\cal M}$ is the space of all time-invariant measures (i.e.", "measures which are invariant under the time evolution).", "Although the space is very large, there is often a unique measure $\\mu _+$ defined as a late time attractor starting from an arbitrary (continuous with respect to $\\mu _L$ ) distribution.For Hamiltonian systems $\\mu _+ = \\mu _L$ , but for non-Hamiltonian systems the two measures need not be the same.", "More precisely, if ${\\cal O}({\\bf x})$ is some observable (continuous with respect to $\\mu _L$ ), then $\\int {\\cal O}({\\bf y}) \\mu _+({\\bf y}) d{\\bf y} = \\lim _{t\\rightarrow \\infty } \\frac{1}{t} \\int _{0}^{t} {\\cal O}({\\bf x}(\\tau )) d\\tau $ should be satisfied almost surely for all but a measure zero of initial states ${\\bf x}(0)$ with respect to $\\mu _+$ .", "These measures were originally proposed by Sinai [29], Ruelle [30] and Bowen [31] and go by the name of SRB measures [28].", "In this article we will refer to them as the equilibrium measures with respect to a certain energy-like function $E$ to be defined below.", "Recently, the equilibrium measures were proved to be related to steady states in thermostated systems.For example, a Gaussian thermostat [32] is a collection of particles subject to a non-conservative force $\\bf F$ and a constraint $\\alpha = \\frac{\\sum _j \\left( {\\bf F}({\\bf q}_j) - \\frac{\\partial V({\\bf q}_i)}{\\partial {\\bf q}_i}\\right) \\cdot {\\bf p}_j}{ \\sum _j {\\bf p}_j^2}$ with non-Hamiltonian equations of motions given by $\\frac{d{\\bf p}_i}{dt} = - \\frac{\\partial V({\\bf q}_i)}{\\partial {\\bf q}_i}+{\\bf F}({\\bf q}_i) - \\alpha {\\bf p}_i\\;\\;\\;\\; \\text{and}\\;\\;\\;\\;\\frac{d{\\bf q}_i}{dt} = \\frac{{\\bf p}_i}{m},$ such that the total kinetic energy remains constant $\\sum _j {\\bf p}_j^2=\\text{const}$ .", "Although the equilibrium measure seems to be the most physical it is by no means unique.", "The situation is completely analogous to the equilibrium statistical mechanics where the Gibbs measure is defined only once the energy function is specified.", "This is also not a unique choice and under certain circumstances other measures with respect to other constraints can be more physically relevant (e.g.", "grand canonical ensemble).", "The general rule for finding an appropriate measure is given by the MaxEnt (maximal entropy) principle proposed by Jaynes [33].", "It says that for any given set of constraints on a system or for a given knowledge about the system, the probability measure, which best represents the state of knowledge, is the one with largest entropy.", "What is, however, unique about the equilibrium measure $\\mu _+$ is that it is the only measure which is a zero-noise limit of small perturbations around deterministic trajectories [28].", "In other words, if we slightly perturb our deterministic evolution and take the perturbation to zero, then the equilibrium measure is the only measure which converges to itself.", "Thus, if we are to construct a measure which respects the quantum-classical correspondence principle then $\\mu _+$ might be the only choice within a framework of dynamical systems.A possible connection of the equilibrium measures to quantum gravity was expressed in Ref.", "[34].", "To make the above statement more precise a full quantization of the cosmological systems must be carried out which proved to be a difficult task, although a number of recent attempts have been made to advance our understanding of quantum mechanics on cosmological scales[35]." ], [ "Variational principle", "To study the statistical properties of non-Hamiltonian systems, a thermodynamic formalism was developed with many ideas borrowed from the equilibrium statistical mechanics, but one very important difference.", "In the conventional statistical mechanics we are usually interested in states, when in dynamical system the key role is played by a time-ordered collection of states or by trajectories.", "Thus, it is convenient to think of time as a thermodynamical volume which is a conjugate variable to the so-called topological pressure.", "The topological pressure of a given energy-like function $E({\\bf x})$ is defined as $p(\\beta E) = \\lim _{t\\rightarrow \\infty } \\frac{1}{t} \\log {\\cal Z},$ where ${\\cal Z} = \\int d{\\bf x}(0) e^{ -\\beta \\int _0^t E({\\bf x}(\\tau ))d\\tau }$ is a dynamical partition function.There is no $1/\\beta $ in the definition of the topological pressure which might cause some confusion whenever $\\beta \\ne 1$ , but it seems to be a standard convention in the dynamical systems literature.", "For $\\beta E=0$ the topological pressure (known as topological entropy) is equal to the rate of growth of the number of topologically distinguishable trajectories and for $\\beta E \\ne 0$ these trajectories are also weighted by $\\exp (-\\beta E)$ .", "The energy-like function $E_+$ which corresponds to the forward equilibrium measures $\\mu _+$ is given by a sum of local Lyapunov exponents $\\chi _i$ (defined as local rates of separation of nearby trajectories) over directions corresponding to only positive (global) Lyapunov exponents $\\lambda _i$ (defined as rates of separation of nearby trajectories in the limit of infinite times), i.e.", "$E_+({\\bf x}) = \\sum _{\\lambda _i>0} \\chi _i({\\bf x}).$ Similarly, the energy-like function $E_-$ , which corresponds to the backward equilibrium measure $\\mu _-$ , is given by a sum over negative Lyapunov exponents or by a sum over positive Lyapunov exponents of a time-reversed system, $E_-({\\bf x}) = - \\sum _{\\lambda _i<0} \\chi _i({\\bf x}).$ For Hamiltonian systems the Lyapunov exponents come in conjugate pairs (i.e.", "$E_+({\\bf x}) = E_-({\\bf x})$ ) and the two equilibrium measures (backward and forward) are identical, i.e.", "$\\mu _+=\\mu _-$ .", "Another important quantity is the Kolmogorov-Sinai entropy defined for any given time-invariant measure $\\mu ({\\bf x})$ using the Shannon entropy formula on a space of trajectories.", "For a discrete time dynamical system $T: X \\rightarrow X$ the entropy is defined as${\\cal C} =\\lbrace C_1, C_2,....,C_n\\rbrace $ is a finite partition of $X$ if $X = \\cup _{i=1}^n C_i$ and $C_j \\cap C_k = \\emptyset $ for $j\\ne k$ .", "$S_\\mu \\equiv \\text{sup}\\lbrace \\lim _{t\\rightarrow \\infty } \\frac{1}{t} S_\\mu (\\vee _{k=0}^{t-1} T^{-k}({\\cal C}) ) : {\\cal C}\\; \\text{is a finite partition of}\\; {X} \\rbrace $ where $S_{\\mu }({\\cal C}) = - \\sum _{i=1}^n \\mu (C_i) \\log (\\mu (C_i)).$ The union of two partitions (${\\cal C} =\\lbrace C_1, C_2,....,C_n\\rbrace $ and ${\\cal D} =\\lbrace D_1, D_2,....,D_m\\rbrace $ ) is defined as ${\\cal C} \\vee {\\cal D} \\equiv \\lbrace C_i \\cap D_j : C_i \\in {\\cal C}, D_j \\in {\\cal D} \\rbrace $ .", "The Kolmogorov-Sinai entropy (pre unit time) can also be defined for a continues time process [28].", "Intuitively $S_\\mu $ quantifies the significance of long periodic orbits with respect to a given measure $\\mu $ , and should not be confused with a thermodynamic entropy on states.", "We are now ready to state one of the two most important results of the thermodynamic formalism - a variational principle.The variation principle is analogous to the Gibbs variational principle which defines the equilibrium state of a system by minimizing its free energy.", "Thus, it might be helpful to think of $\\int E_+({\\bf x}) \\mu ({\\bf x}) d{\\bf x}- S_\\mu /\\beta $ as a dynamical free energy per unit time.", "It says that $p(\\beta E_+) = \\sup \\lbrace S_\\mu - \\beta \\int E_+({\\bf x}) \\mu ({\\bf x}) d{\\bf x} : \\mu \\in {\\cal M} \\rbrace .$ where the extremum is realized for $\\mu =\\mu _+$ .", "The variational principle allows us to calculate the topological pressure $p(\\beta E_+) = S_{\\mu _+} - \\beta \\int E_+({\\bf x}) \\mu _+({\\bf x}) d{\\bf x}$ as well as the equilibrium measure, corresponding to $\\beta =1$ , $\\mu _+({\\bf x}(0)) \\propto \\exp \\left(-\\int _0^t E_+({{\\bf x}(\\tau )}) d\\tau \\right)$ from a spectrum of Lyapunov exponents.", "For closed systems without any absorbing sates the topological pressure vanishes, $p(E_+) =0$ and for open systems $p(E_+) =-\\gamma $ , where $\\gamma $ is the escape rate of trajectories." ], [ "Absorbing states", "In the context of eternal inflation it will be useful to study dynamical systems whose phase space trajectories can either reproduce or escape.", "The escape of trajectories is easier to understand when a system with absorbing states is followed forward in time.", "Similarly, if the system with escape is followed backward in time then it would seem as though the forward trajectories reproduce.", "Thus, it is convenient to define the reproducing states as absorbing states of a time-reversed system.", "Of course, if we are only interested in the time-invariant measures, then none of the forward nor backward trajectories would ever escape and the existence of absorbing and reproducing states is not directly observable.", "The simplest dynamical systems with absrobing states are called cookie-cutters.", "Cookie-cutters are defined by a discrete map $T$ from a union of disjoint subsets $A_i \\subset [0,1]$ to the entire unit interval such that $[0,1] - \\cup A_i \\ne \\emptyset $ .", "For example, $T(x) = {\\left\\lbrace \\begin{array}{ll} 3x\\;\\;\\;&\\text{if}\\;\\;\\; x\\in [0,1/3] \\\\2 x- 1\\;\\;\\;&\\text{if}\\;\\;\\; x\\in [1/2, 1].\\end{array}\\right.", "}$ On each iteration the map of an open interval $(1/3,1/2)$ is undetermined which represents a terminal or absorbing state.", "The set of all points that are never mapped to $(1/3,1/2)$ is $A\\equiv \\sup \\lbrace Y | Y=T(Y)\\rbrace $ .", "The set $A$ is a fractal whose Liouville (or more precisely Lebesgue) measure is zero (i.e.", "$\\mu _L(A)=0$ ), but one could still ask whether it is possible to construct an equilibrium time-invariant measure on $A$ .", "The space of all time-invariant measures is very large (e.g.", "$\\mu (x) = \\delta (x-0)$ , $\\mu (x) = \\delta (x-1/5)/2+\\delta (x-3/5)/2$ , etc.", "), but the equilibrium measure $\\mu _+$ is a unique late time limit of an arbitrary (continuous with respect to $\\mu _L$ ) distribution, i.e.", "$\\mu _+(x) \\propto \\exp \\left(- \\beta \\sum _{i=0}^{t-1} E_+(T^i(x)) \\right),$ where $E_+(x) = \\log \\left| \\left[\\frac{dT}{dy}\\right]_{x} \\right|$ is the only positive Lapunov exponents at $x$ .", "Although $\\mu _+(x)$ is defined precisely for infinite $t$ one can study a coarse-grained partition function for a fixed $t$ by summing over periodic orbits: ${\\cal Z} = \\sum _{T^t(x)=x} \\exp \\left(- \\beta \\sum _{i=0}^{t-1} \\log \\left| \\left[\\frac{dT}{dx}\\right]_{T^i(x)} \\right| \\right).$ Then, the topological pressure is $p = \\lim _{t\\rightarrow \\infty } \\frac{1}{t} \\log {\\cal Z} = \\lim _{t\\rightarrow \\infty } \\frac{1}{t} \\log (3^{-\\beta } + 2^{-\\beta })^t= \\log (3^{-\\beta } + 2^{-\\beta }).$ According to Bowen [31], the vanishing topological pressure $p=0$ implies that $\\beta \\approx 0.7879$ is the fractal dimension of $A$ , but the most physical choice is given by the equilibrium measure with $\\beta =1$ ." ], [ "Fluctuation theorem", "In dynamical systems literature, the entropy production rate had been identified with (minus) the phase space contraction rate, $e({\\bf x}) = - \\frac{\\partial }{\\partial {\\bf x}} \\cdot {\\bf v}({\\bf x}).$ whose average with respect to the equilibrium measures $\\mu _+$ is non-negative, i.e.", "$- \\int \\frac{\\partial }{\\partial {\\bf x}} \\cdot {\\bf v}({\\bf x}) d\\mu _+({\\bf x}) \\ge 0,$ and is strictly positive for dissipating systems.", "In our notations the local entropy production rate is given by a sum of all local Lyapunov exponents: $e({\\bf x}) \\equiv \\sum _i \\chi _i({\\bf x}) = E_+({\\bf x}) - E_-({\\bf x}),$ where the expressions for both $E_+({\\bf x})$ and $E_-({\\bf x})$ in the context of eternal inflation will be derived in the following section.", "Alternative calculations of the entropy production during inflation using stochastic methods is described in Refs.", "[39].", "Since any physical system is usually observed only for a finite period of time, one might want to define a finite time average as $e_T({\\bf x}(0)) \\equiv \\frac{1}{T}\\int _{0}^{T}e({\\bf x}(t)) dt$ and study the properties of the probability distribution $P(e_T)$ .", "This problem was analyzed in the context of Anosov systems, where it was found that $\\log \\left(\\frac{P(e_T)}{P(-e_T)}\\right) = e_T T$ is exactly linear with no higher order terms for an arbitrarily large $e_T$ .", "This relation is the second of the two most important results of the thermodynamic formalism known as the fluctuation theorem [40], [41].", "The symmetry of the distribution $P(e_T)$ (sometimes called Gallavotti-Cohen symmetry) involves the statistics of very atypical fluctuations and was first discovered in numerical simulations [40].", "Later, the fluctuation theorem was also proved analytically for Anosov systems [41].", "However, if we assume that the chaotic hypothesis holds for a dynamical system of eternal inflation (i.e.", "eternal inflation can be regarded as an Anosov system), then we should also expect to see the symmetry described by Eq.", "(REF ) in primordial fluctuations.", "Note, that this symmetry is more than just Gaussianity, but involves the statistics of very improbable events.", "On a positive side, the Cosmic Microwave Background experiments allow us to retrieve information about separate (i.e.", "causally disconnected) trajectories by simply looking at different directions on the sky.", "This is certainly an advantage compared to other non-equilibrium systems such as thermostats [32] where the fluctuation relation is being tested.", "For example, one can divide the CMB sky into $N$ equal regions and use the CMB data to estimate what could have been the entropy production $e({\\bf x})$ during inflation on each of the corresponding trajectories separately.", "Then, according to the fluctuation theorem, the distribution $P(e_T)$ must have a symmetry described by Eq.", "(REF ) which can always be verified for a sufficiently large $N$ and sufficiently small $T$ ." ], [ "Eternal inflation", "The main objective of this section is to construct the time-invariant equilibrium measures of eternal inflation using dynamical systems, but before we proceed it is instructive to highlight the main properties of such measures within a more familiar stochastic approach.", "Since all time-invariant measures, $\\mu \\in {\\cal M}$ , are defined on a space of infinite (or very large) trajectories, the relevant observables are the trajectories.A possibility of defining measures on trajectories was already expressed in Refs.", "[36].", "As was emphasized above, this might potentially solve the problem of defining the relevant cosmological observables.", "For example, one can show that various paradoxes [24], [22], [23] can be resolved whenever the probabilities of cosmological observations are defined on a space of trajectories.", "In addition, the time-invariant measures provide a simple solution to the problem of initial conditions which does not exist for infinite trajectories.The measures on only non-singular infinite trajectories were also discussed in Ref.", "[37] in the context of homogeneous cosmologies.", "In eternal inflation literature such trajectories are usually neglected on the grounds of zero measure (with respect to comoving volume), but because of their infinite lengths one might also argue that any infinite trajectory is infinitely more probable than any finite trajectory.", "Clearly, there is an order of limits issue that we are going to discuss next." ], [ "Order of limits", "Consider a stochastic mode of eternal inflation with absorbing (or terminal) states.", "The only relevant, for our considerations, parameter is the decay rate per unit time $\\gamma $ to one of the absorbing states.", "Our task is to define a measure $\\mu (T)$ on a measurable space of trajectories parametrized by their duration $T$ before the final transition to an absorbing state.", "There are two factors that might go into $\\mu (T) \\propto w_{\\text{evolution}}(T) w_{\\text{observation}}(T)$ .", "First of all, the trajectories could be weighted by their probabilities with respect to some Markovian evolution operator, $w_{\\text{evolution}}(T)$ .", "In addition, the trajectories could also be weighted by some monotonic function $w_{\\text{observation}}(T)$ such that $w_{\\text{observation}}(\\infty )=\\infty $ .", "The first factor can be argued for using, for example, semiclassical methods [17], [18], and the second factor can be argued for using, for example, anthropic principle [16] since the longer trajectories intersect more observers that could observe them.", "Now, if we compare forward trajectories of length $T$ , with trajectories of infinite length, then from the point of view of evolution $\\frac{w_{\\text{evolution}}(T)}{w_{\\text{evolution}}(\\infty )} = \\frac{\\exp (-\\gamma T)}{\\exp (-\\gamma \\infty )} = \\infty ,$ but from the point of view of observations $\\frac{w_{\\text{observation}}(T)}{w_{\\text{observation}}(\\infty )} = 0,$ where the exact form of $w_{\\text{observation}}(T)$ is not important.", "To make sense of $\\frac{\\mu (T)}{\\mu (\\infty )} = \\infty \\times 0$ we can introduce two separate cut-offs, $\\frac{\\mu (T)}{\\mu (\\infty )} = \\lim _{b \\rightarrow \\infty } \\lim _{a \\rightarrow \\infty } \\frac{w_{\\text{evolution}}(T)}{w_{\\text{evolution}}(b)} \\frac{w_{\\text{observation}}(T)}{w_{\\text{observation}}(a)},$ where the order of limits is not specified a priori.", "The standard choice is to first take $b \\rightarrow \\infty $ (or to take both limits simultaneously $a=b \\rightarrow \\infty $ ), which usually gives a divergent answer, i.e.", "$\\frac{\\mu (T)}{\\mu (\\infty )} = \\infty $ , or a zero measure for infinite trajectories.", "Such ordering of limits is known to give answers that depend on ether initial conditions (for local stochastic measures), or cutoffs (for global stochastic measures).", "Another alternative is to first take $a \\rightarrow \\infty $ which would yield a completely different answer, i.e.", "$\\frac{\\mu (T)}{\\mu (\\infty )} = 0$ .", "According to the dynamical systems approach the latter ordering procedure is a lot more natural and corresponds to the time-invariant measures.", "The situation is very similar in a negative time direction.", "It is well known that all but a measure-zero (with respect to comoving volume) of trajectories are past-incomplete [26] and that is why all past-complete trajectories are usually neglected.", "However, if the relevant measurable space is a space of trajectories, then the measure on trajectories depends on the order of limits as in Eq.", "(REF ) and the infinite past-complete trajectories must not be overlooked.", "In fact, such trajectories are the only trajectories which acquire a finite weight with respect to the ordering procedure suggested by the dynamical systems approach.", "However, in the models, where the past-complete infinite trajectories are strictly forbidden the time-invariant measures would still be supported on only future-infinite trajectories.", "In a stochastic picture the time-invariant measures might be thought of as fractal measures defined on only eternal geodesics [18] in the limit when the cut-off $a$ is taken to infinity and $b$ remains large, but finite.", "Of course, for any finite value of $a \\gg b$ the measure assigns a non-vanishing weight to all trajectories within a small neighborhood around each eternal geodesic.", "It might be tempting to conclude that the probability to observe a terminal vacua (i.e.", "AdS vacua) is extremely small, if not identically zero.", "This is a very nice prediction which is in agreement with the observed positive value of the cosmological constant.", "On the other hand, if there would be a mechanism (perhaps quantum) to resolve the black-hole singularities by recycling local trajectories back to eternal inflation, then the system would not contain any absorbing states and the above conclusion would certainly change." ], [ "Backward measure", "We are now ready to switch to a more formal measure-theoretic discussion of eternal inflation.", "To warm up, we start with a construction of a backward (in time) equilibrium measure for a single scalar field inflation.", "The objective is to describe the dynamics of the field from the point of view of a local observer moving along a time-like geodesic.", "The background equations of motion are given by $\\frac{d{\\pi }}{dt} = - \\partial _\\varphi V + 3H {\\pi }$ and $\\frac{d{\\varphi }}{dt}= \\pi $ where $\\varphi $ and $\\pi $ are the position and momentum coordinates.", "In Minkowski space $H=0$ and the dynamics is described by a time-independent Hamiltonian leading to an incompressible flow of the nearby phase space trajectories, but during inflation $H\\ne 0$ and the flow becomes compressible.", "More precisely, the Hubble friction introduces a single negative local Lyapunov exponent $\\chi _i$ of a forward evolution or a single positive local Lyapunov exponent of a backward evolution.", "If this exponent has the same sign as the corresponding (global) Lyapunov exponent $\\lambda _i$ , then $E_-({\\bf x}) = - \\sum _{\\lambda _i<0} \\chi _i({\\bf x}) \\approx 3H({\\bf x}),$ where ${\\bf x}(t)\\equiv \\lbrace \\pi (t), \\varphi (t)\\rbrace $ .", "In fact, every field contributing to a quasi de Sitter expansion by forming a condensate is likely to contribute to a sum of the negative local Lyapunov exponents, but, for the time being, we assume that there is only a single scalar field that drives inflation.", "Note that all non-inflating fields would also have negative local Lyapunov exponents $\\chi _i$ during inflation, but their relation to signs of the corresponding (global) Lyapunov exponents $\\lambda _i$ is not direct, and we will assume that on average they do not contribute to the compressibility of the flow.", "Since the Hubble friction introduces only negative Lyapunov exponents it plays a central role for constructing the equilibrium measure on backward trajectories, i.e.", "$\\mu _-({\\bf x}(0)) \\propto \\exp \\left( {- \\int _0^t E_-({\\bf x}(\\tau )) d\\tau }\\right) = \\exp \\left( {- \\int _0^t 3H({\\bf x}(\\tau )) d\\tau }\\right),$ but has no effect on the equlibrium measure of forward trajectories.", "This implies that $E_+({\\bf x})\\ne E_-({\\bf x})$ (at the level of background dynamics) and $\\mu _+ \\ne \\mu _-$ which should not be too surprising.", "The equilibrium measures on forward and backward trajectories of a non-Hamiltonian system usually differ even though they are defined on the same space of infinite trajectories.", "In fact this will turn out to be the case for the dynamical system of eternal inflation even when perturbations are taken into account." ], [ "Forward measure", "At the level of background dynamics the nearby phase space trajectories do not expand and one must go beyond the homogeneous limit in order to derive the equilibrium measures on forward trajectories.", "In this limit the quantum effects cannot be ignored and one often employes the semiclassical tools to study quantum fluctuation generated during inflation.", "It is well known that the semiclassical analysis gives rise to a stochastic picture which is sufficient for modeling inflation using diffusion Eqs.", "(REF ) and (REF ), but for the methods developed in the previous section to be useful we should also learn how to extract the microscopic properties.", "In particular, it is desired to map the microscopic parameters, such as Lyapunov exponents, to the macroscopic parameters, such as transport coefficients.", "This would enable us to estimate the equilibrium measures on forward trajectories and to study their properties.", "A number of different ideas had been put forward to address this issue, but perhaps the simplest of all is the escape-rate formalism [38].", "The idea is to express the rate of escape $\\gamma $ from a given phase space neighborhood of size $L$ using thermodynamic formalism and then to equate it to the escape rate calculated using a diffusion equation, i.e.", "$\\gamma = - p(E_+)=\\int E_+({\\bf x}) \\mu _+({\\bf x}) d{\\bf x} - S_{\\mu _+} = \\left(\\frac{\\pi }{L}\\right)^2 D$ where $S_{\\mu _+}$ is the Kolmogorov-Sinai entropy of the neighborhood.", "The semiclassical analysis [17] suggests that the evolution is described by Eq.", "(REF ) with diffusion coefficient given by $D = \\frac{H^3}{8 \\pi ^2},$ where we ignore the problem of factor ordering.", "Then, according to Eqs.", "(REF ) and (REF ), the sum of positive local Lyapunov exponents is likely to scale linearly with diffusion coefficient, i.e.", "$E_+({\\bf x}) \\propto H({\\bf x})^3.$ This is our best guess of what the underlying microscopic properties of the system should be based on the semiclassical methods [17] and on the escape-rate formalism [38].", "Of course, one would want to go beyond the semiclassical theory to confirm (or disprove) the linear dependence of microscopic Lyapunov exponents on macroscopic diffusion coefficients.", "In addition to diffusion, the fluctuations are constantly stretched by cosmological expansion which gradually reduces the effect of any particular mode (along a given local trajectory) on the Lyapunov spectrum.", "This can be captured by an additional factor $\\propto a({\\bf x})^3$ in the expression for $E_+({\\bf x})$ , where $a({\\bf x})$ is a local scale factor which describes the local FRW geometry.", "Then the final expression for the sum of positive Lyapunov exponents is $E_+({\\bf x}) \\propto H({\\bf x})^3 a({\\bf x})^3$ and the corresponding equilibrium measure is $\\mu _+({\\bf x}(0)) \\propto \\exp \\left( {- \\int _0^t E_+({\\bf x}(\\tau )) d\\tau }\\right) = \\exp \\left( {- \\beta \\int _0^t H({\\bf x}(\\tau ))^{3} {a}({\\bf x}(\\tau ))^3 d\\tau }\\right),$ where $\\beta $ is yet undetermined constant.", "We would like to stress that the above equation does not contain all of the quantum effects (e.g.", "quantum tunnelings), but only provides an estimate of the effect of linear inflationary perturbations on the equilibrium measure of forward trajectories due to positive Lyapunov exponents." ], [ "Effect of horizons", "In the dynamical systems of inflation discussed so far the local phase space trajectories did not escape nor reproduce, but a generic dynamical system may contain both absorbing and reproducing (i.e.", "absorbing states of a time-reversed system) states.", "For such systems the rate of escape of forward trajectories is usually given by the sum of positive Lyapunov exponents (REF ).", "Then the contribution to the escape rate from $N$ scalar fields is given by $E^{\\text{phase-space}}_+({\\bf x}) \\approx \\sum _{i=1}^{N} \\beta _i H({\\bf x})^{3} a({\\bf x})^3.$ Roughly speaking, the larger the rate of a phase space expansion the easier it is for a given trajectory to escape or to hit an absorbing state if such a state exists.", "Similarly, the rate of reproduction of local trajectories is given by the rate of escape of backward trajectories or by minus the sum of negative Lyapunov exponents (REF ).", "For inflation driven by $N$ scalar field the rate is $E^{\\text{phase-space}}_-({\\bf x}) \\approx \\sum _{i=1}^{N} 3 H({\\bf x}) = 3 N H({\\bf x}).$ This is a phase-space picture.", "At the same time, eternal inflation can be described from a physical-space point of view where the local trajectories reproduce or escape due to the presence of cosmological and black hole horizons.", "According to the physical-space picture the local trajectories reproduce with the following rate $\\frac{d}{dt} \\log \\left(\\frac{V_0 a^3}{H^{-3}}\\right) = 3 \\;H({\\bf x}) + 3\\frac{\\dot{H}({\\bf x})}{H({\\bf x})}.$ where $V_0 a^3/H^{-3}$ is the number of independent local trajectories at time $t$ inside of comoving volume $V_0$ .", "The second term of Eq.", "(REF ), averaged over periodic orbits, is exactly zero, but the first term gives a non-zero contribution to the reproduction rate, $E^{\\text{physical-space}}_-({\\bf x}) = 3 \\;H({\\bf x}).$ Roughly speaking, this is the rate with which local observers fall out of causal contact with each other and start to follow their own local trajectories.", "In addition to reproduction, the physical-space picture suggests that the local trajectories might eventually escape into black hole singularities whenever the energy density contrast during inflation is of order one or larger.", "This is usually the case for inflationary perturbations generated above the self-reprodcution scale, if they are not stretched out by dark energy, but might occasionally happen on any scale if the classical drift of a scalar field is of the same order or smaller than a given quantum jump, i.e $\\delta \\varphi _{\\text{class}} \\lesssim \\delta \\varphi _{\\text{quant}}$ .", "For Gaussian fluctuations (below self-reproduction scale) one finds that the corresponding rate of escape is, $E^{\\text{physical-space}}_+({\\bf x}) \\sim \\frac{H({\\bf x})^3}{\\partial _\\varphi H({\\bf x})}.$ Of course, the above equation is only valid towards the end of inflation so that the fluctuations have time to reenter horizon and the local observers have time to escape into the singularity before the cosmological constant starts to dominate.", "To apply the methods developed in this paper to an arbitrary model of eternal inflation one might want to describe the effects of horizons described by Eqs.", "(REF ) and (REF ) in a language of dynamical systems.", "This can be accomplished by generalizing the energy-like functions $E_-$ and $E_+$ to include both the phase space and physical space contributions, i.e.", "$E_-({\\bf x}) = E^{\\text{phase-space}}_-({\\bf x})+E^{\\text{physical-space}}_-({\\bf x})$ and $E_+({\\bf x}) =E^{\\text{phase-space}}_+({\\bf x})+E^{\\text{physical-space}}_+({\\bf x}).$ Depending on a model of eternal inflation the horizons may or may not change significantly the corresponding equilibrium measures of Eqs.", "(REF ) and (REF )." ], [ "Summary of results", "The dynamical systems approach to non-equlbrium statistical mechanics is a viable alternative to the stochastic approaches with a wide range of applications.", "For cosmological systems the approach is introduced here for the first time, but it is based on a well known mathematics developed over the last few decades.", "The main motivation to study something new was to solve the cosmological problems of entropy, measure, observables and initial conditions.", "We do not claim that the solution presented here is unique, but we do claim that none of the previously proposed scenarios solve all of these problems simultaneously.", "Below we list the most popular frameworksOther promising holographic approaches which are currently under development include dS/CFT [3], FRW/CFT [4], dS/dS [5], HST [6] and Holographic Multiverse [7].", "and their problems:“No” means that the respective problem can be solved or avoided at least for some systems, and “Yes” means that, to our knowledge, the problem cannot be solved nor avoided within a given framework.", "Table: NO_CAPTIONPerhaps the most interesting result about the new approach is that, if correct, it gives us a hope to derive the cosmological predictions from the dynamics itself without a need to postulate any additional rules (e.g.", "measure, initial conditions, space of obervables etc.).", "This would be a truly dynamical solution to the existing cosmological problems.", "We conclude with a critical summary of the main results: 1) Problem of observables.", "Many cosmological measures, defined over a space of local states, are known to suffer from serious logical inconsistencies.", "Although, the probability spaces over local states are natural within a stochastic approach, it is not the case for the dynamical systems approach developed in this paper.", "The new framework suggests that the relevant observables should be local trajectories and the measures should be defined over the space of trajectories instead of states.", "Without going into details we mention that one might still run into philosophical issues (e.g.", "terminal states have zero probability) whose implications remain to be better understood.", "2) Entropy problem.", "The Liouville measure is the most physical measure for a finite Hamiltonian system which is known to suffer from the entropy problem.", "A solution proposed here involves a generalization of a purely Hamiltonian dynamics to include absorbing and reproducing states.", "For such systems the most physical measures are not the Liouville measures, but more general equilibrium measures.", "This does not mean that a general dynamical system would never be in a conflict with observations, but it does mean that such problems can be avoided.", "3) Measure problem.", "There are many “solutions” to the entropy problem using stochastic processes (i.e.", "local, global, stationary), but their major drawback is that they always introduce new problems such as the problem of measure or the problem of initial conditions.", "This is not the case for the dynamical systems approach whose equilibrium measures are uniquely defined by a variational principle.", "Of course, for some dynamical systems close to a dynamical phase transition the equilibrium measure might still be degenerate and such critical systems certainly deserve a closer examination.The dynamical phase transitions usually occurs in non-hyperbolic dynamical systems and should not be confused with phase transitions in statistical mechanics.", "4) Problem of initial conditions.", "The equilibrium measures are only supported on infinite trajectories to the past as well as to the future.", "For such measures the initial conditions are irrelevant by construction.", "However, even if one demands to start with a distribution (continuous with respect to the Liouville measure) the dynamical system would eventually forget its initial state, similarly to what happens in finite Hamiltonian systems.", "5) Fluctuation theorem.", "The dynamical systems approach would not be very useful without the chaotic hypothesis.", "The hypothesis is a natural generalization of the ergodic hypothesis and is often assumed for analysis of sufficiently chaotic systems.", "One of the results that follows immediately is a symmetry described by the fluctuation theorem which could in principle be observable in the Cosmic Microwave Background radiation where the nearby local trajectories can be compared side by side.", "This involves the analysis of very improbable fluctuations which is a challenging task.", "Acknowledgments.", "The author is grateful to Alan Guth, Andrei Linde, Mahdiyar Noorbala and Alexander Vilenkin for very useful discussions and comments on the manuscript.", "The work was supported in part by NSF grant PHY-0756174." ] ]

[ [ "Online submodular welfare maximization: Greedy is optimal" ], [ "Abstract We prove that no online algorithm (even randomized, against an oblivious adversary) is better than 1/2-competitive for welfare maximization with coverage valuations, unless $NP = RP$.", "Since the Greedy algorithm is known to be 1/2-competitive for monotone submodular valuations, of which coverage is a special case, this proves that Greedy provides the optimal competitive ratio.", "On the other hand, we prove that Greedy in a stochastic setting with i.i.d.items and valuations satisfying diminishing returns is $(1-1/e)$-competitive, which is optimal even for coverage valuations, unless $NP=RP$.", "For online budget-additive allocation, we prove that no algorithm can be 0.612-competitive with respect to a natural LP which has been used previously for this problem." ], [ "Introduction", "We study an online variant of the welfare maximization problem in combinatorial auctions: $m$ items are arriving online, and each item should be allocated upon arrival to one of $n$ agents whose interest in different subsets of items is expressed by valuation functions $w_i: 2^{[m]} \\rightarrow {\\mathbb {R}}_+$ .", "The goal is to maximize $\\sum _{i=1}^{n} w_i(S_i)$ where $S_i$ is the set of items allocated to agent $i$ .", "Variants of the problem arise by considering different classes of valuation functions $w_i$ and different models (adversarial/stochastic) for the arrival ordering of the items.", "We remark that in this work we do not consider any game-theoretic aspects of this problem.", "The origin of this line of work can be traced back to a seminal paper of Karp, Vazirani and Vazirani [15] on online bipartite matching.", "This can be viewed as a welfare maximization problem where one side of the bipartite graph represents agents and the other side items; each agent $i$ is interested in the items $N(i)$ joined to $i$ by an edge, and he is completely satisfied by 1 item, meaning the valuation function can be written as $w_i(S) = \\min \\lbrace |S \\cap N(i)|, 1 \\rbrace $ .", "Karp, Vazirani and Vazirani gave an elegant $(1-1/e)$ -competitive randomized algorithm, which improves a greedy $1/2$ -approximation and is optimal in this setting.", "Recent interest in online allocation problems arises from applications in online advertising, where the items represent ad slots associated with search queries, and agents are advertisers interested in having their ad displayed in connection with certain queries.", "A popular model in this context is the budget-additive framework [13], [19], [3] where valuations have the form $w_i(S) = \\min \\lbrace \\sum _{j \\in S} b_{ij}, B_i \\rbrace $ .", "More generally, combinatorial auctions [17] form a setting where multiple items are sold to multiple agents with valuation functions $w_i$ .", "Again, practical restrictions often require that the decision about each item needs to be made immediately, rather than after seeing the entire pool of items.", "Hence the online model, which we study in this paper.", "The baseline algorithm in this setting is the greedy algorithm, due to Fisher, Nemhauser and Wolsey, who initiated the study of problems involving maximization of submodular functions [22], [12], [21].", "The greedy algorithm simply allocates each incoming item to the agent who gains from it the most and is $1/2$ -competitive whenever the valuation functions of the agents are monotone submodular [12], [17].", "This is in fact the most general setting known where a constant-factor approximation can be achieved even for the offline welfare maximization problem (using value queries; more general classes of valuations can be handled when more powerful queries are available [8]).", "Thus the basic question in most variants of this problem is whether the factor of $1/2$ is optimal or can be improved.", "For the offline welfare maximization problem with monotone submodular valuations, a $(1-1/e)$ -approximation has been found [23], and this is optimal [16].", "At the other end of the spectrum is the above-mentioned bipartite matching problem, which can be viewed as a welfare maximization problem with valuation functions of the form $w_i(S) = \\min \\lbrace |S \\cap N(i)|, 1\\rbrace $ (a very special case of a submodular function).", "The $(1-1/e)$ -competitive algorithm of [15] is optimal in the adversarial online setting; several improvements have been obtained in various stochastic settings [14], [11], [2], [18], [20].", "Factor $(1-1/e)$ -competitive algorithms have been also found in two adversarial budget-additive settings, the small-bids case, $w_i(S) = \\min \\lbrace \\sum _{j \\in S} b_{ij}, B_i\\rbrace $ where $b_{ij} \\ll B_i$ [19], and the single-bids case, $w_i(S) = \\min \\lbrace \\sum _{j \\in S} b_{ij}, B_i\\rbrace $ where $b_{ij} \\in \\lbrace 0,b_i\\rbrace $ for some $b_i$ independent of $j$ [1].", "A unifying generalization of these $(1-1/e)$ -competitive algorithms to the budget-additive setting, $w_i(S) = \\min \\lbrace \\sum _{j \\in S} b_{ij}, B_i\\rbrace $ , has been conjectured but still remains open.", "Prior to this work, it was conceivable that a $(1-1/e)$ -competitive algorithm might exist for arbitrary monotone submodular valuations, but the best known online algorithm gave only an $o(1)$ improvement over $1/2$ [6]." ], [ "Our results.", "We prove that: In the online setting with submodular valuations, the factor of $1/2$ cannot be improved unless $NP=RP$ (even by randomized algorithms against an oblivious adversary).", "Hence, the greedy $1/2$ -competitive algorithm is optimal up to lower-order terms.", "This holds in fact for the special case of coverage valuations (see Section ).", "In the online setting with budget-additive valuations, we prove that no (randomized) algorithm is $0.612$ -competitive with respect to a natural LP, which has been used successfully in the special case of small bids [3].", "Thus, a $(1-1/e)$ -competitive algorithm would need to use a different approach (see Section ).", "In a stochastic setting with items arriving i.i.d.", "from an unknown distribution, the greedy algorithm is $(1-1/e)$ -competitive for valuations with the property of diminishing returns (a natural extension of submodularity to multisets which we define in Section REF ).", "This is optimal even for coverage valuations and a known (uniform) distribution, unless $NP=RP$ (see Section )." ], [ "Our techniques.", "Our hardness result for online algorithms in the adversarial setting relies on a combination of two sources of hardness: (1) the inapproximability of Max $k$ -cover due to Feige [7], and (2) the lack of information arising from the unknown online ordering.", "A careful combination of these two ingredients gives an optimal hardness result ($1/2+\\epsilon $ ) for online algorithms under coverage valuations.", "Our hardness result in the i.i.d.", "stochastic setting also relies on the hardness of Max $k$ -cover.", "A consequence of our use of the computational hardness of (offline) Max $k$ -cover is that our results rely on a complexity-theoretic assumption ($NP=RP$ ), which is somewhat unusual in the context of online algorithms.", "Our negative result for budget-additive valuations is based on an integrality-gap example for the natural LP and does not rely on any complexity-theoretic assumption.", "This result does not rule out, e.g., a $(1-1/e)$ -competitive algorithm for budget-additive valuations, but we consider it instructive, considering recent efforts to develop online algorithms in the primal-dual framework.", "Our result points to the fact that perhaps the natural LP is too weak for the general budget-additive setting, and stronger LPs such as the Configuration LP should be considered (also, see a discussion in [4]).", "Finally, our positive result for the greedy algorithm in the i.i.d.", "stochastic setting is an extension of a similar analysis for budget-additive valuations [5].", "Here, we want to point out the definition of valuation functions satisfying the property of diminishing returns (Section REF ).", "This is a generalization of submodularity to functions on multisets.", "We remark that for set functions, the properties of diminishing returns and submodularity coincide, but this is not the case for functions on multisets.", "We believe that our generalization is a natural one, considering the original motivation for submodularity in the context of combinatorial auctions, and we wish to highlight this definition for possible future work.", "In the following, we state our results more formally and present the proofs." ], [ "Welfare maximization.", "In the welfare maximization problem (sometimes also referred to as the “allocation problem\" or “combinatorial auctions\"), the goal is to allocate $|M| = m$ items to $n$ agents with valuation functions $w_i:2^M \\rightarrow {\\mathbb {R}}_+$ in a way that maximizes $\\sum _{i=1}^{n} w_i(S_i)$ , where $S_i$ is the set of items allocated to agent $i$ (satisfying $S_i \\cap S_j = \\emptyset $ for $i \\ne j$ )." ], [ "Online welfare maximization.", "In the online version of the problem, items arrive one by one and we have to allocate each item when it arrives, knowing only the agents' valuations on the items that have arrived so far.", "An algorithm is $c$ -competitive if, for any ordering of the incoming items, it achieves at least a $c$ -fraction of the (offline) optimal welfare.", "A randomized algorithm is $c$ -competitive against an oblivious adversary if, for any ordering of the incoming items (fixed before running the algorithm), it achieves at least a $c$ -fraction of the optimal welfare in expectation." ], [ "Coverage valuations.", "A valuation function $w:2^M \\rightarrow {\\mathbb {R}}_+$ is called a coverage valuation if there is a set system $\\lbrace A_j: j \\in M\\rbrace $ such that $w(S) = |\\bigcup _{j \\in S} A_j|$ for all $S \\subseteq M$ ." ], [ "Submodular valuations.", "A valuation function $w:2^M \\rightarrow {\\mathbb {R}}_+$ is called submodular if $w(S \\cup T) + w(S \\cap T) \\le w(S) + w(T)$ .", "It is called monotone if $w(S) \\le w(T)$ whenever $S \\subseteq T$ ." ], [ "Succinct representation and oracles.", "For complexity-theoretic considerations, it is important how the valuation functions are presented on the input.", "In this paper, we assume that coverage valuations are presented explicitly, by a succinct representation of size polynomial in $|M|$ .", "Submodular valuations are presented by means of a value oracle, which can answer queries in the form “What is the value of $w_i(S)$ ?\"", "Our main result for online welfare maximization is as follows.", "Theorem 2.1 Unless $NP = RP$ , there is no $(1/2+\\delta )$ -competitive polynomial-time algorithm (even randomized, against an oblivious adversary) for the online welfare maximization problem with coverage valuations and constant $\\delta >0$ .", "Our main tool is Feige's hardness reduction, which proves the optimality of $(1-1/e)$ -approximation for Max $k$ -cover [7].", "We also require some additional properties of this reduction, which have been described in [9], [10].", "We summarize the properties that we need as follows:" ], [ "Hardness of Max $k$ -cover.", "For any fixed $c_0>0$ and $\\epsilon >0$ , it is NP-hard to distinguish between the following two cases for a given collection of sets ${\\cal S}\\subset 2^U$ , partitioned into groups ${\\cal S}_1,\\ldots ,{\\cal S}_k$ : YES case: There are $k$ disjoint sets, 1 from each group ${\\cal S}_i$ , whose union is the universe $U$ .", "NO case: For any choice of $\\ell \\le c_0 k$ sets, their union covers at most a $(1 - (1-1/k)^\\ell + \\epsilon )$ -fraction of the elements of $U$ .", "This holds even for set systems with the following properties: every set has the same (constant) size $s$ ; and each group contains the same (constant) number of sets $n$ .", "As we show in Section REF , this reduction also gives hard instances for welfare maximization with coverage valuations, proving that any (offline) $(1-1/e+\\delta )$ -approximation would imply $P=NP$ .", "In Section REF , we prove our result, Theorem REF ." ], [ "Our approach.", "We produce instances of online welfare maximization by taking multiple copies of a hard instance ${\\cal I}$ for offline welfare maximization and repeating them with certain (random) agents gradually dropping out of the system.", "We prove that an online algorithm faces two obstacles: it cannot solve the offline instance ${\\cal I}$ optimally (in fact it already loses a factor of $1-1/e$ there), and in addition it does not know in advance which agents will drop out at what time.", "A careful analysis of these two obstacles in combination gives the optimal hardness of $(1/2+\\delta )$ -approximation for online algorithms." ], [ "Warm-up: hardness of offline welfare maximization", "First, we show how Feige's reduction implies the hardness $(1-1/e+\\delta )$ -approximation for welfare maximization with coverage valuations.", "This was previously proved by a more involved technique in [16].", "The result of [16] has the additional property that it holds even when all agents have the same coverage valuation; in our reduction the valuations are different." ], [ "Reduction.", "Consider a set system that forms a hard instance of Max $k$ -cover as described above.", "We produce an instance of welfare maximization with $n$ agents and $m = kn$ items (where $n$ is the number of sets in each group, and $k$ is the number of groups).", "Each agent will have a valuation associated with this set system.", "However, the way items are associated with sets will be different for each agent.", "Let the $kn$ items be described by pairs $(j_1,j_2) \\in [k] \\times [n]$ , and let the sets in the set system be denoted by $A_{j_1,j_2}$ where $(j_1,j_2) \\in [k] \\times [n]$ .", "Then, the item $(j_1,j_2)$ for agent $i$ is associated with the set $A_{j_1,j_2+i \\pmod {n}}$ .", "In other words, the value of a set of items $S$ for agent $i$ is $ w_i(S) = \\Big | \\bigcup _{(j_1,j_2) \\in S} A_{j_1,{(j_2+i \\bmod n)}} \\Big |.$ Now consider the two cases: YES case: There are $k$ sets, one from each group, covering the universe.", "Denote these sets by $A_{j,\\pi (j)}$ for some function $\\pi :[k] \\rightarrow [n]$ .", "Then, there is an allocation where agent $i$ receives the set of items $S_i = \\lbrace (j, \\pi (j)-i \\bmod n): j \\in [k] \\rbrace $ .", "Note that these sets of items are disjoint, due to the cyclic shift depending on $i$ .", "Also, each agent is perfectly satisfied, since the union of the sets associated with her items is $\\bigcup _{j \\in [k]} A_{j, \\pi (j)} = U$ .", "Hence $w_i(S_i) = |U|$ for all $i$ .", "NO case: For each choice of $\\ell \\le c_0 k$ sets, they cover at most a $(1 - (1-1/k)^\\ell + \\epsilon )$ -fraction of the universe.", "(We choose $c_0$ to be a large constant.)", "In other words, any agent who receives $\\ell \\le c_0 k$ sets gets value at most $w_i(S_i) \\le (1 - (1-1/k)^\\ell + \\epsilon ) |U|$ .", "Here, it does not matter who receives which items, as we have a bound depending solely on the number of items received.", "Since this bound is a concave function, the best possible welfare is achieved when each agent receives exactly $k$ items, and this yields welfare $(1 - (1-1/k)^k + \\epsilon ) |U|$ per agent.", "By choosing $k$ arbitrarily large and $\\epsilon >0$ arbitrarily small, we obtain welfare arbitrarily close to $(1-1/e) |U|$ per agent.", "In the following, we will use this hard instance of welfare maximization with coverage valuation as a black box." ], [ "Hardness of online welfare maximization", "Here we prove Theorem REF .", "We produce a reduction from Max $k$ -cover to online welfare maximization as follows." ], [ "The hard online instances.", "Let ${\\cal I}$ be an instance of welfare maximization with coverage valuations, obtained from a hard instance of Max $k$ -cover (as in Section REF ), with $n$ agents and $m = kn$ items.", "For a parameter $t \\ge 1$ , we produce the following instance ${\\cal I}^{(t)}$ of online welfare maximization, with $tn$ agents and $tm$ items, proceeding in $t$ stages: In the first stage, we have $t$ copies of each agent of the instance ${\\cal I}$ , with exactly the same valuation function.", "The valuation function for each agent is determined by the set system of ${\\cal I}$ .", "The $m$ items of instance ${\\cal I}$ arrive in an arbitrary order.", "After each stage, one copy of each agent is effectively “deactivated\", in the sense that all subsequent items have zero value for her.", "The copy of each agent that disappears is chosen by an adversary.", "In stage $t^{\\prime } \\in \\lbrace 1,\\ldots ,t\\rbrace $ , we have $(t-t^{\\prime }+1)n$ “active agents\" remaining, who are still interested in the remaining items.", "In each stage, $m$ items of the original instance arrive in an arbitrary order, but now they are valuable only for the remaining active agents.", "For these agents, the items are effectively copies of the items that arrived in previous stages, and they are represented by the same sets.", "These instances were inspired by the $1-1/e$ lower bound for online matching [15].", "Essentially, we take an instance of bipartite matching that is hard for online algorithms and expand each incoming vertex into an entire instance of welfare maximization with coverage valuations, to impose the additional difficulty of approximating an APX-hard problem at each stage.", "We analyze this instance in a series of claims.", "Claim 2.2 The offline optimum in the YES case is $t n |U|$ .", "The offline optimum allocates all items in stage $j$ to those agents who will be deactivated at the end of this stage.", "Since these are $n$ agents whose valuations of the items of this stage correspond exactly to the instance ${\\cal I}$ , in the YES case they can obtain optimal value $n |U|$ (since every agent can cover the universe $U$ ).", "Adding up over all stages, the total value collected by all agents is $t n |U|$ .", "Claim 2.3 Let the adversary choose a copy of each agent to be deactivated after each stage independently and uniformly at random from the remaining active copies.", "Then the expected total number of items allocated to the agents deactivated at the end of stage $j$ is at most $m \\ln \\frac{t}{t-j}$ .", "Let $A_j$ denote the agents deactivated right after stage $j$ ; $A_j$ contains exactly 1 copy of each agent.", "Consider $i \\le j$ and condition on the set of agents active in stage $i$ .", "The choice of which agents will appear in $A_j$ will be made after stage $j$ , independently of what the algorithm does in stage $i$ .", "Since the choice of $A_j$ is uniform in each stage $j$ , each of the $t-i+1$ copies of a given agent active in stage $i$ has the same probability $(\\frac{1}{t-i+1})$ of appearing in $A_j$ .", "The number of items allocated in each stage is $m$ , hence the expected number of items allocated to $A_j$ in stage $i$ is $\\frac{m}{t-i+1}$ .", "By linearity of expectation, the number of items allocated to $A_j$ between stages 1 and $j$ is $\\sum _{i=1}^{j} \\frac{m}{t-i+1} \\le \\int _0^j \\frac{m}{t-x} dx = m \\ln \\frac{t}{t-j}.", "$ Claim 2.4 For every $\\epsilon ^{\\prime }>0$ , there are $\\epsilon ,c_0>0$ and a constant lower bound on $k$ (parameters of the Max $k$ -cover reduction) such that for every $j \\le (1-\\epsilon ^{\\prime }) t$ , the expected value collected in the NO case by the agents deactivated at the end of stage $j$ is at most $(j/t + \\epsilon ^{\\prime }) n |U|$ .", "Denote again by $A_j$ the agents deactivated at the end of stage $j$ .", "By Claim REF , the expected number of items allocated to $A_j$ is at most $m \\ln \\frac{t}{t-j}$ .", "Let $\\mu $ denote the expected number of items allocated per agent in $A_j$ : we get $\\mu \\le \\frac{m}{n} \\ln \\frac{t}{t-j} = k \\ln \\frac{t}{t-j}$ .", "Assuming that $j \\le (1-\\epsilon ^{\\prime }) t$ , we have $\\mu \\le k \\ln \\frac{1}{\\epsilon ^{\\prime }}$ .", "Let us set $c_0 = 1 + \\ln \\frac{1}{\\epsilon ^{\\prime }}$ , and let us denote by $\\nu (\\ell )$ the largest value that an agent can possibly obtain from $\\ell $ items.", "By properties of the NO case, we know that for $\\ell \\le c_0 k$ , we have $\\nu (\\ell ) \\le (1 - (1-1/k)^\\ell + \\epsilon ) |U|$ .", "For $\\epsilon = \\frac{1}{2} \\epsilon ^{\\prime }$ and $k$ lower-bounded by some sufficiently large constant, we can replace this bound by $\\nu (\\ell ) \\le (1 - e^{-\\ell / k} + \\epsilon ^{\\prime }) |U|$ .", "A technical point here is that this bound holds only for $\\ell \\le c_0 k$ , while the actual number of allocated items is random and could be much larger.", "However, we can deal with this issue as follows.", "Let us define $\\phi (x) = (1 - e^{-x / k} + \\epsilon ^{\\prime }) |U|$ .", "The derivative of $\\phi $ at $\\mu $ is $\\phi ^{\\prime }(\\mu ) = \\frac{1}{k} e^{-\\mu /k} |U|$ .", "Therefore, since the function $\\phi (x)$ is concave, we have $\\phi (x) \\le \\phi (\\mu ) + \\phi ^{\\prime }(\\mu ) (x-\\mu )$ , for $\\ell \\in [0, c_0 k]$ .", "Thus we obtain a (weaker) linear bound: $\\nu (\\ell ) \\le \\phi (\\mu ) + \\phi ^{\\prime }(\\mu ) (\\ell -\\mu )= (1 - e^{-\\mu /k} + \\epsilon ^{\\prime } + \\frac{1}{k} e^{-\\mu /k} (\\ell -\\mu )) |U|.$ Furthermore, we always have the trivial bound $\\nu (\\ell ) \\le |U|$ , for any $\\ell $ .", "This bound is anyway stronger than the one above for $\\ell \\ge c_0 k$ , because we have $c_0 k \\ge \\mu + k$ .", "Therefore, we obtain the following bound for all $\\ell \\ge 0$ : $ \\nu (\\ell ) \\le \\min \\lbrace |U|, (1 - e^{-\\mu /k} + \\epsilon ^{\\prime } + \\frac{1}{k} e^{-\\mu /k} (\\ell - \\mu )) |U|\\rbrace ,$ and this (piecewise linear) bound is still concave.", "Since the expected number of items per player is ${\\bf E}[\\ell ] = \\mu $ , the worst case is that each agent in $A_j$ indeed receives $\\mu $ items (deterministically), and her value is $\\nu (\\mu ) \\le (1 - e^{-\\mu /k} + \\epsilon ^{\\prime }) |U|$ .", "Using our bound $\\mu \\le k \\ln \\frac{t}{t-j}$ , we obtain that the expected value collected per agent in $A_j$ is at most $(1 - \\frac{t-j}{t} + \\epsilon ^{\\prime }) |U| = (\\frac{j}{t} + \\epsilon ^{\\prime }) |U|$ .", "[Theorem REF ] Let us assume now that there is a $(\\frac{1}{2} + \\delta )$ -competitive algorithm for online welfare maximization with coverage valuations.", "We set $\\epsilon ^{\\prime } = \\delta / 4$ and the parameters $c_0, \\epsilon $ accordingly to this value of $\\epsilon ^{\\prime }$ (see Claim REF ).", "Given an instance ${\\cal I}$ of Max $k$ -cover, we can also assume that $k$ is sufficiently large as required by Claim REF ; otherwise all parameters of the Max $k$ -cover instance are constant, and we can solve it by exhaustive search.", "If $k$ is sufficiently large, we run the presumed online algorithm on the random instance ${\\cal I}^{(t)}$ that we constructed above.", "In the NO case, denote by $V_j$ the expected value collected by agents deactivated after stage $j$ .", "By Claim REF , we have $V_j \\le (\\frac{j}{t} + \\epsilon ^{\\prime }) n |U|$ for $j \\le (1-\\epsilon ^{\\prime }) t$ .", "The value collected by the agents deactivated in each of the last $\\epsilon ^{\\prime } t$ stages is $V_j \\le n |U|$ , because every agent can possibly get value at most $|U|$ .", "Adding up the values of agents over all stages, we obtain that the online algorithm returns a solution of expected value $\\sum _{j=1}^{t} V_j \\le \\sum _{j=1}^{(1-\\epsilon ^{\\prime })t} \\left(\\frac{j}{t} + \\epsilon ^{\\prime }\\right) n |U| + \\epsilon ^{\\prime } t n |U|\\le \\frac{t}{2} n |U| + 2 \\epsilon ^{\\prime } t n |U| = \\left(\\frac{1}{2}+2\\epsilon ^{\\prime }\\right) t n |U|.$ In contrast, the offline optimum in the YES case is $t n |U|$ (by Claim REF ) and hence the $(\\frac{1}{2}+\\delta )$ -competitive algorithm must return expected value at least $(\\frac{1}{2} + \\delta ) t n |U| = (\\frac{1}{2} + 4 \\epsilon ^{\\prime }) t n |U|$ , a constant fraction better than the NO case.", "Since the possible values returned by the algorithm are in the range $[0, t n |U|]$ , we can distinguish the two cases with constant two-sided error.", "In fact, we can make the error one-sided as follows.", "If some agent receives $\\ell \\le c_0 k$ items ($c_0$ as in the proof of Claim REF ) whose value is more than $(1 - (1-1/k)^\\ell + \\epsilon ) |U|$ , we answer YES, otherwise we answer NO.", "Note that by the proof of Claim REF , in the YES case, we will answer YES with probability $\\Omega (1)$ , because otherwise the solution is almost always bounded by the same analysis as in the NO case, and the expected value of the solution would be less than $(\\frac{1}{2} + 4 \\epsilon ^{\\prime }) t n |U|$ , which cannot be the case.", "In the NO case, we always answer NO, because there are no $\\ell \\le c_0 k$ items of value more than $(1 - (1-1/k)^\\ell + \\epsilon ) |U|$ .", "Thus we can solve the Max $k$ -cover decision problem with constant one-sided error, which implies $NP=RP$ ." ], [ "Online budget-additive allocation", "In this section we prove that no online algorithm can obtain a better than $0.612$ approximation with respect to the standard LP in the budget-additive case.", "We now define the budgeted allocation problem[4].", "Definition 3.1 Let $Q$ be a set of $m$ indivisible items and $A$ a set of $n$ agents, respectively, where agent $a$ is willing to pay $b_{ai}$ for item $i$ .", "Each agent $a$ has a budget constraint $B_a$ , and on receiving a set $S\\subseteq Q$ of items pays $\\min \\lbrace B_a, \\sum _{i\\in S} b_{ai} \\rbrace $ .", "An allocation $\\Gamma :A \\rightarrow 2^{Q}$ is a partitioning of the items $Q$ into disjoint subsets $\\Gamma (1),\\ldots , \\Gamma (n)$ .", "The maximum budgeted allocation problem, or simply MBA, is to find the allocation which maximizes the total revenue, that is $\\sum _{a\\in A}\\min \\left\\lbrace B_a,\\sum _{i\\in \\Gamma (a)}b_{ai}\\right\\rbrace $ .", "Note that one can assume without loss of generality that $b_{ai}\\le B_a$ , $\\forall a\\in A, i\\in Q$ .", "Indeed, if bids are larger than budget, decreasing them to the budget does not change the value of any allocation.", "We now introduce the standard LP relaxation of the maximum budgeted allocation problem [4]: $\\begin{split}& \\max \\sum _{a\\in A, i\\in Q} b_{ai} x_{ai}: \\\\&\\forall a\\in A, \\sum _{i\\in Q} b_{ai}x_{ai}\\le B_a; \\\\& \\forall i\\in Q, \\sum _{a\\in A} x_{ai}\\le 1;\\\\&\\forall a\\in A, i\\in Q, x_{ai}\\ge 0.\\end{split}$ It was shown in [4] that the integrality gap of LP (REF ) is exactly $3/4$ .", "We now show that no online algorithm can obtain value better than a factor $0.612$ of this LP.", "Thus, if a $(1-1/e)$ -competitive algorithm exists, it has to use other techniques, perhaps a stronger LP relaxation.", "Our basic building block will be an instance with agents $A=\\lbrace a_1, a_2\\rbrace $ with budgets $B_{a_1}=B_{a_2}=3$ and items $I=\\lbrace i_1,i_2, i_3\\rbrace $ such that $b_{a_j, i_k}=2$ for all $j=1,2$ and $k=1, 2, 3$ .", "Note that the value of the standard LP on this instance is 6, while the maximum allocation is 5 since an agent that gets two items can only pay 1 for the second item that is allocated to him.", "We now use the small instance that we just described to construct an online instance similarly to Section .", "Denote the set of agents in the system by $A$ .", "We will have $|A|=2t$ and $B_a=3$ for all $a\\in A$ .", "Items will arrive in $t$ stages.", "It will be convenient to use a partition $A=\\bigcup _{s=1}^t A^{(s)}$ of $A$ into disjoint sets of size 2.", "The agents will gradually drop out of the system, i.e., agents who drop out at time $j$ will not be interested in items that arrive after $j$ .", "As before, for each $j=1,\\ldots ,t$ we refer to the set of agents that did not drop out before time $j$ as active at time $j$, and refer to the other sets of agents as deactivated at time $j$ .", "Initially all sets $A^{(s)}, s=1,\\ldots , t$ are active.", "After each stage, $A^{(s)}$ for $s$ uniformly random among the remaining active sets is deactivated.", "We denote the set deactivated after stage $j$ by $A_{j}$ .", "In each stage $j=1,\\ldots , t$ a set of items $I_{j}$ arrives, where $|I_{j}|=3$ and $b_{ai}=2$ for all $a\\in A$ that are active in stage $j$ .", "Note that the value of the standard LP for our instance is $3t$ : for each $j=1,\\ldots , t$ allocate $2/3$ of each item in $I_j$ to each agent in the set $A_j$ .", "We now upper bound the value of any allocation that an online algorithm can obtain.", "Let $g(x)=\\left\\lbrace \\begin{array}{cc}2x,&\\text{~if~}x\\le 1\\\\x+1&\\text{~if~}1\\le x\\le 2\\\\3&\\text{o.w.", "}\\end{array}\\right.$ We first prove: Claim 3.2 Let $a\\in A$ denote an agent and let $X$ denote the (random) number of items allocated to $a$ .", "Then the expected value obtained by $a$ for these items is upper bounded by $g({\\bf \\mbox{\\bf E}}[X])$ , where $g(\\cdot )$ is given by (REF ).", "This follows from concavity of $\\min \\left\\lbrace B_a, \\sum _{i\\in \\Gamma (a)} b_{ai}\\right\\rbrace $ .", "Let $x={\\bf \\mbox{\\bf E}}[X]$ .", "Then if $x<1$ , the maximum is achieved if exactly one item is allocated to $a$ with probability $x$ , yielding value $2x$ .", "If $1\\le x\\le 2$ , then the maximum is achieved if 1 item is always allocated to $a$ at the price of 2, and then a second item is allocated with probability $x$ at the price of 1, yielding value $2+(x-1)=x+1$ .", "Otherwise if $x\\ge 2$ , the payoff cannot be larger than the budget of $a$ , i.e. 3.", "We can now prove: Theorem 3.3 No online (randomized) algorithm for the budgeted allocation problem can achieve (in expectation) more than a $0.612$ -fraction of the optimal value of the linear program (REF ).", "We first upper bound the expected number of items allocated to agents $A_j$ (recall that $A_j$ is the set of agents deactivated after stage $j$ ).", "Let $X^1_{j}, X^2_j$ denote the (random) number of items allocated to the two agents in $A_j$ .", "By the same argument as in the proof of Claim REF , which we do not repeat here, we have that each agent that is active at time $i=1,\\ldots , j$ appears in $A_j$ with probability $\\frac{1}{t-i+1}$ .", "Since three items arrive in each stage, we have ${\\bf \\mbox{\\bf E}}[X^1_j+X^2_j] \\le \\sum _{i=1}^j\\frac{3}{t-i+1}\\le 3\\int _0^j\\frac{1}{t-x}dx=3\\ln \\left(\\frac{t}{t-j}\\right).$ Now by Claim REF together with convexity of the function $g(\\cdot )$ we get that the value obtained by the online algorithm is upper bounded by $\\sum _{j=1}^t \\left[g({\\bf \\mbox{\\bf E}}[X^1_j])+g({\\bf \\mbox{\\bf E}}[X^2_j])\\right]\\le \\sum _{j=1}^{t} 2g\\left(\\frac{3}{2} \\ln \\left(\\frac{t}{t-j}\\right)\\right)\\le t\\int _0^{1}2g\\left(\\frac{3}{2} \\ln \\left(\\frac{1}{1-x}\\right)\\right)dx.$ We now split the interval $[0, 1]$ as $[0, 1]=[0, x_1]\\cup [x_1, x_2]\\cup [x_2, 1]$ , where $\\frac{3}{2} \\ln \\left(\\frac{1}{1-x_1}\\right)=1$ and $\\frac{3}{2} \\ln \\left(\\frac{1}{1-x_2}\\right)=2$ , i.e., $x_1=1-e^{-2/3}$ and $x_2=1-e^{4/3}$ .", "We get by (REF ) that the RHS of (REF ) is equal to $t\\int _0^{1-e^{-2/3}}3 \\ln \\left(\\frac{1}{1-x}\\right)dx+t\\int _{1-e^{-2/3}}^{1-e^{-4/3}}\\left[\\frac{3}{2} \\ln \\left(\\frac{1}{1-x}\\right)+1\\right]dx+3t\\cdot e^{-4/3}< 0.612\\cdot 3t.$ Recalling that the value of the standard LP on our instance is $3t$ completes the proof." ], [ "The i.i.d. stochastic model.", "Here we consider a model where items arrive from some (possibly known or unknown) distribution ${\\cal D}$ over a fixed collection of items $M$ .", "In each step, an item is drawn independently at random from ${\\cal D}$ and we must allocate it irrevocably to an agent.", "The total number of items can be either known or unknown.", "In this model, we compare to the expected offline optimum, $OPT = {\\bf E}[OPT(M)]$ where $M$ is the random multiset of items that appear on the input.", "We say that an algorithm is $c$ -competitive if it achieves at least $c \\cdot OPT$ in expectation over the random inputs (and possibly its own randomness)." ], [ "Diminishing returns on multisets", "In this section, we would like to consider the class of submodular valuations and its extension to multisets.", "Submodular valuations on $\\lbrace 0,1\\rbrace ^m$ express the property of diminishing returns, and this has indeed been the primary motivation for their modeling power as valuation functions.", "However, considering the stochastic setting with i.i.d.", "samples, we should clarify how we deal with possible multiple copies of an item.", "In other words, we need to consider valuation functions $f:{\\mathbb {Z}}_+^m \\rightarrow {\\mathbb {R}}$ .", "An extension of submodularity to the ${\\mathbb {Z}}_+^m$ lattice that has been used in the literature is the following condition: $f(x \\vee y) + f(x \\wedge y) \\le f(x) + f(y)$ , where $\\vee $ and $\\wedge $ are the coordinate-wise max/min operations.", "Unfortunately, this condition does not quite capture the property of diminishing returns as it does in the case of $\\lbrace 0,1\\rbrace ^m$ : note that in particular it does not impose any restrictions on $f(x)$ if the domain is 1-dimensional, $x \\in {\\mathbb {Z}}_+$ .", "Considering the property of diminishing returns, we would like the condition to imply that $f$ is concave in this 1-dimensional setting.", "Therefore, we define the following property.", "Definition 4.1 A function $f:{\\mathbb {Z}}_+^m \\rightarrow {\\mathbb {R}}$ has the property of diminishing returns, if for any $x \\le y$ (coordinate-wise) and any unit basis vector $e_i = (0,\\ldots ,0,1,0,\\ldots ,0)$ , $ i \\in [m]$ , $ f(x+e_i) - f(x) \\ge f(y+e_i) - f(y).$ Note that when restricted to $\\lbrace 0,1\\rbrace ^m$ , this property is equivalent to submodularity.", "Also, note that a simple way to extend a monotone submodular function $f:\\lbrace 0,1\\rbrace ^m \\rightarrow {\\mathbb {R}}$ to $\\tilde{f}: {\\mathbb {Z}}_+^m \\rightarrow {\\mathbb {R}}$ , by declaring that additional copies of any item bring zero marginal value (i.e.", "$\\tilde{f}(x) = f(x \\wedge {\\bf 1})$ ), satisfies the property of diminishing returns.", "In particular, coverage valuations on multisets interpreted in a natural way (multiple copies of the same set do not cover any new elements), have the property of diminishing returns.", "In some sense, we believe that this is the “right extension\" of submodularity to multisets, at least for applications related to combinatorial auctions and welfare maximization.", "We also consider the following natural notion of monotonicity.", "Definition 4.2 A function $f:{\\mathbb {Z}}_+^m \\rightarrow {\\mathbb {R}}$ is monotone, if $f(x) \\le f(y)$ whenever $x \\le y$ ." ], [ "Our results", "We prove that in the i.i.d.", "stochastic model with valuations satisfying the property of diminishing returns, the best one can achieve is a $(1-1/e)$ -competitive algorithm.", "In fact, the factor of $1-1/e$ is achieved by the same greedy algorithm that gives a $1/2$ -approximation in the adversarial online model [12], [17]." ], [ "Greedy algorithm:", "Suppose the multisets assigned to the $n$ agents before item $j$ arrives are $(T_1,\\ldots ,T_n)$ .", "Then assign item $j$ to the agent who maximizes $w_i(T_i+j) - w_i(T_i)$ .", "We remark that this algorithm obviously does not need to know the distribution or the number of items in advance.", "Theorem 4.3 The greedy algorithm is $(1-1/e)$ -competitive for welfare maximization with valuations satisfying the property of diminishing returns in the stochastic i.i.d. model.", "Theorem 4.4 Unless $NP = RP$ , there is no $(1-1/e+\\delta )$ -competitive polynomial-time algorithm for welfare maximization with coverage valuations in the i.i.d.", "stochastic model, for fixed $\\delta >0$ .", "Since coverage valuations satisfy the property of diminishing returns, we conclude that $1-1/e$ is the optimal factor in the stochastic i.i.d.", "model for coverage valuations as well as any valuations satisfying diminishing returns." ], [ "Analysis of the greedy algorithm for stochastic input", "Here we prove Theorem REF .", "Our proof is a relatively straightforward extension of the analysis of [5] in the budget-additive case.", "First, we need a bound on the expected optimum.", "Lemma 4.5 The expected optimum in the stochastic model where $m$ items arrive independently, item $j$ with probability $p_j$ , is bounded by $LP & = & \\max \\sum _{i,S} x_{i,S} w_i(S): \\\\& & \\forall j; \\sum _{i,S} x_{i,S} c_j(S) \\le p_j m; \\\\& & \\forall i; \\sum _S x_{i,S} = 1; \\\\& & \\forall i,S; x_{i,S} \\ge 0$ where $w_i$ is the valuation of agent $i$ , $S$ runs over all multisets of at most $m$ items, and $c_j(S) \\ge 0$ denotes the number of copies of $j$ contained in $S$ .", "Consider the optimal (offline) solution $OPT(M)$ for each realization of the random multiset $M$ of arriving items.", "Let $x_{i,S}$ denote the probability that the multiset allocated to agent $i$ in the optimal solution is $S$ .", "Then the expected value of the optimum is ${\\bf E}[OPT(M)] = \\sum _{i,S} x_{i,S} w_i(S)$ .", "Also, each multiset $S$ contains $c_j(S)$ copies of item $j$ , so the expected number of allocated copies of item $j$ is $\\sum _{i,S} x_{i,S} c_j(S)$ .", "On the other hand, this cannot be more than the expected number of copies of $j$ in $M$ , which is ${\\bf E}[c_j(M)] = p_j m$ .", "Therefore, $x_{i,S}$ is a feasible solution of value $OPT = {\\bf E}[OPT(M)]$ .", "Lemma 4.6 Assume that $w_i$ are monotone valuations with the property of diminishing returns.", "Condition on the partial allocation at some point being $(T_1,\\ldots ,T_n)$ .", "Then the expected gain from allocating the next random item is at least $\\frac{1}{m} (LP - \\sum _i w_i(T_i))$ .", "Let $x_{i,S}$ be any feasible LP solution and let $y_{ij} = \\sum _S x_{i,S} c_j(S)$ .", "Recall that $c_j(S)$ denotes the number of copies of $j$ contained in $S$ .", "Note that by the LP constraints, $\\sum _i y_{ij} \\le p_j m$ .", "We use the notation $T_i+S$ to denote the union of multisets (adding up the multiplicities of each item).", "By the property of diminishing returns, we have $ w_i(T_i+S) - w_i(T_i) \\le \\sum _j c_j(S) (w_i(T_i+j) - w_i(T_i)).$ Adding up these inequalities multiplied by $x_{i,S} \\ge 0$ , we get $\\sum _{i,S} x_{i,S} (w_i(T_i+S) - w_i(T_i))\\le \\sum _{i,j,S} x_{i,S} c_j(S) (w_i(T_i+j) - w_i(T_i))= \\sum _{i,j} y_{ij} (w_i(T_i+j) - w_i(T_i)).$ Since $\\sum _S x_{i,S} = 1$ by the LP constraints, and $w_i(T_i+S) \\ge w_i(S)$ by monotonicity, we obtain $\\sum _{i,S} x_{i,S} w_i(S) - \\sum _i w_i(T_i) \\le \\sum _{i,j} y_{ij} (w_i(T_i+j) - w_i(T_i)).$ Now consider a hypothetical allocation rule (depending on the fractional solution): If the incoming item is $j$ , we allocate it to agent $i$ with probability $\\frac{y_{ij}}{p_j m}$ .", "(By the LP constraints, these probabilities for a fixed $j$ add up to at most 1.)", "Since item $j$ appears with probability $p_j$ , overall we allocate item $j$ to agent $i$ with probability $\\frac{y_{ij}}{m}$ .", "By (REF ), the expected gain of this randomized allocation rule is ${\\bf E}[\\mbox{random gain}] = \\sum _{i,j} \\frac{y_{ij}}{m} (w_i(T_i+j) - w_i(T_i))\\ge \\frac{1}{m} \\left(\\sum _{i,S} x_{i,S} w_i(S) - \\sum _i w_i(T_i) \\right).$ However, the greedy allocation rule gives each item to the agent maximizing her gain.", "Therefore, the greedy rule gains at least as much as the randomized allocation rule, for any feasible solution $x_{i,S}$ .", "This implies ${\\bf E}[\\mbox{greedy gain}] \\ge \\max {\\bf E}[\\mbox{random gain}]\\ge \\frac{1}{m} \\left(LP - \\sum _i w_i(T_i)\\right).", "$ Now we can prove Theorem REF .", "[Proof of Theorem REF ] Denote the allocation obtained after allocating $t$ items $(T^{(t)}_1,\\ldots ,T^{(t)}_n)$ .", "Lemma REF states that conditioned on $(T^{(t)}_1,\\ldots ,T^{(t)}_n)$ , the expected value after allocating 1 random item will be ${\\bf E}\\Big [\\sum _i w_i(T^{(t+1)}_i) \\mid T^{(t)}_1,\\ldots ,T^{(t)}_n\\Big ]\\ge \\sum _i w_i(T^{(t)}_i) + \\frac{1}{m} \\left(LP - \\sum _i w_i(T^{(t)}_i)\\right).$ Taking an expectation over the partial allocation $(T^{(t)}_1,\\ldots ,T^{(t)}_n)$ , we obtain ${\\bf E}\\Big [\\sum _i w_i(T^{(t+1)}_i)\\Big ]\\ge {\\bf E}\\Big [\\sum _i w_i(T^{(t)}_i)\\Big ] + \\frac{1}{m} {\\bf E}\\Big [LP - \\sum _i w_i(T^{(t)}_i)\\Big ].$ Let us denote $W(t) = {\\bf E}[\\sum _i w_i(T^{(t)}_i)]$ .", "The last inequality states $W(t+1) \\ge W(t) + \\frac{1}{m} (LP - W(t))$ , or equivalently $LP - W(t+1) \\le (1 - \\frac{1}{m}) (LP - W(t))$ .", "By induction, we obtain $ LP - W(t) \\le \\left(1-\\frac{1}{m}\\right)^t (LP - W(0)) \\le e^{-t/m} LP.$ The expected value of the solution found by the greedy algorithm after $m$ items is $W(m) = {\\bf E}[\\sum _i w_i(T^{(m)}_i)]$ ; we conclude that $W(m) \\ge (1 - 1/e) LP \\ge (1-1/e) OPT$ ." ], [ "Optimality of $1-1/e$ in the stochastic i.i.d. model", "Here we prove Theorem REF .", "We prove essentially that the stochastic online problem cannot be easier than the offline problem.", "However, the reduction is not quite black-box and we need some properties of the hard coverage instances that we discussed in Section .", "Recall the instance ${\\cal I}$ of welfare maximization with coverage valuations (Section ), for which it is NP-hard to achieve approximation better than $1-1/e$ .", "We transform it into an instance ${\\cal I}^{[t]}$ in the stochastic i.i.d.", "model as follows.", "We pick a parameter $t = poly(m)$ and produce $t$ identical copies of each agent in ${\\cal I}$ .", "If the number of items in ${\\cal I}$ is $m$ , we let $tm$ i.i.d.", "items arrive from the uniform distribution on the $m$ items of ${\\cal I}$ .", "By Chernoff bounds, with high probability the number of copies of each item on the input will be $t \\pm O(\\sqrt{t \\log m}) = t \\pm O(\\sqrt{t \\log t})$ .", "Consider the YES case.", "The items of ${\\cal I}$ can be allocated so that each of the $n$ agents covers the universe.", "Since we have at least $t - O(\\sqrt{t \\log t}) = (1-o(1)) t$ copies of each item with high probability, they can be allocated to $(1-o(1)) t n$ agents of the instance ${\\cal I}^{[t]}$ so that these agents get full value $|U|$ .", "Thus the expected offline optimum is at least $(1 - o(1)) t n |U|$ .", "On the other hand, in the NO case, any agent in ${\\cal I}$ who gets $\\ell \\le c_0 m / n$ items has value at most $(1 - (1 - n/m)^\\ell + \\epsilon ) |U|$ .", "Since the total number of items on the input of ${\\cal I}^{[t]}$ is $t m$ and the number of agents is $t n$ , an agent can only get $m / n$ items on average.", "As the bound on the value as a function of the number of items is concave (and we can deal with the fact that this bound works only up to $\\ell \\le c_0 m / n$ , similarly to Section ), the optimum value is achieved if each agent receives $m/n$ items.", "Then the total value collected is $(1 - (1-n/m)^{m/n} + \\epsilon ) t n |U|$ , which can be made arbitrarily close to $(1-1/e) t n |U|$ .", "Note that this holds with probability 1, irrespective of the randomness on the input.", "If we had a $(1-1/e+\\delta )$ -competitive algorithm in the stochastic i.i.d.", "model, we could distinguish these two cases with constant one-sided error, which would imply $NP = RP$ .", "[1]BibliographyMyBibliography" ] ]

[ [ "Particle filtering in high-dimensional chaotic systems" ], [ "Abstract We present an efficient particle filtering algorithm for multiscale systems, that is adapted for simple atmospheric dynamics models which are inherently chaotic.", "Particle filters represent the posterior conditional distribution of the state variables by a collection of particles, which evolves and adapts recursively as new information becomes available.", "The difference between the estimated state and the true state of the system constitutes the error in specifying or forecasting the state, which is amplified in chaotic systems that have a number of positive Lyapunov exponents.", "The purpose of the present paper is to show that the homogenization method developed in Imkeller et al.", "(2011), which is applicable to high dimensional multi-scale filtering problems, along with important sampling and control methods can be used as a basic and flexible tool for the construction of the proposal density inherent in particle filtering.", "Finally, we apply the general homogenized particle filtering algorithm developed here to the Lorenz'96 atmospheric model that mimics mid-latitude atmospheric dynamics with microscopic convective processes." ], [ "Introduction", "The main goal of filtering is to obtain, recursively in time, the best statistical estimate of a natural or physical system based on noisy partial observations of the same.", "More precisely, filtering problems consist of an unobservable signal process and an observation process that is a function of the signal corrupted by noise.", "This is given by the conditional distribution of the signal given the observation process.", "This paper deals with real time filtering of chaotic signals from atmospheric models involving many degrees of freedom.", "Since small errors in our estimate of the current state of a chaotic system can grow to have a major impact on the subsequent forecast, better estimate of the signal is needed to make accurate predictions of the future state.", "This is obviously a problem with significant importance in real time filtering and prediction of weather and climate which involves coupled atmosphere-ocean system as well as the spread of hazardous plumes or pollutants governed by extremely complex flows.", "A complete study of these complex systems with practical impact, involves models of extremely unstable, chaotic dynamical systems with several million degrees of freedom.", "Proper and accurate climate models can only be obtained by combining the models with data.", "Lower dimensional climate models require first the identification of slowly evolving climate modes and fast evolving non-climate modes.", "Contrary to standard initial value problems, we do not have access to the initial state of these dynamical systems.", "Instead, we have a known initial probability density function for the initial state and treat the model states as realizations of a random variable.", "Lower dimensional climate models also contain noise terms that account for the interaction of the resolved climate modes with the neglected non-climate modes.", "Hence these problems require efficient new algorithms to estimate the present and future state of the climate models, based upon corrupted, distorted, and possibly partial observations of the climate modes and fast evolving non-climate modes.", "While perfect determination of the state is impossible under these noisy observations, it may still be desirable to obtain probabilistic estimates of the state conditioned on the information that is available.", "The sheer number of calculations required in directly solving large-scale random dynamical systems becomes computationally overwhelming.", "Hence, we consider the Lorenz'96 model of type II ([17]) with two time-scale simplified ordinary differential equation describing advection, damping and forcing of some (slow) resolved atmospheric variables being coupled to some (fast) sub-scale variables as a nontrivial example of an atmospheric dynamics model.", "Though the model considered in this paper is a simple chaotic “toy model\" of the atmosphere, which is an excellent test-bed for schemes that will be developed in Section , it has a more realistic dimension than the multitude of “Lorenz 1963\"-type studies (i.e., 360 dimensions rather than 3 dimensions).", "Even though this simple model is still a long way from the Primitive Equations that we need to eventually study, it is hoped that the filtering methods presented here can be ultimately adapted for the realistic models that need to be computationally solved for weather and climate predictions.", "In the past decade, vast amount of data has been produced via various sources from network-connected sensor arrays to satellites, all with a wide range of scale separations.", "To deal with such an onslaught of data, it is necessary to have a new framework capable of harnessing and processing these data with multiscale models.", "Hence, a data assimilation scheme that can handle chaotic systems that are sensitive to initial conditions (characterized by positive Lyapunov exponents) and noisy multiscale observations is needed.", "The theory of nonlinear filtering forms the framework in our study for the assimilation of data into multiscale models.", "Particle filters represent the posterior conditional distribution of the state variables by a collection of particles, which evolves and adapts recursively as new information becomes available.", "Hence, particle filtering provides a recursive procedure for estimating an evolving state from a noisy observation process.", "Despite the general applicability and rigorous convergence results, particle filters have not been extensively used in state estimation of high dimensional problems, for example, weather prediction, as pointed out in [25].", "This is due to the fact that a large number of particles is required and the particle filters suffer from particle degeneracy (see, for example, [3], [25]) in high dimensional systems.", "In this paper, we combine our study of stochastic dimensional reduction and nonlinear filtering to provide a rigorous framework for developing an algorithm for computing lower dimensional particle filters which are specifically adapted to the complexities of the underlying multiscale signal.", "When the rates of change of different model variables differ by orders of magnitude, efficient data assimilation can be accomplished by constructing a particle filter approach for nonlinear filtering equations for the coarse-grained signal.", "In moderate dimensional problems, particle filters are an attractive alternative to numerical approximation of the stochastic partial differential equations (SPDEs) by finite difference or finite element methods.", "Our approach described in this paper, not only reduces the computational burden for real time applications but also helps solve the problem of particle degeneracy.", "First objective is to predict the self-contained description of the coarse-grained dynamics without fully resolving the dynamics described in fast scales.", "In these problems, extracting coarse-grained dynamics is at the heart a problem of weak convergence of stochastic processes, or more precisely weak convergence of the laws of Markov processes.", "We begin in Section  by presenting the general formulation of the multiscale nonlinear filtering problem.", "Here we introduce the homogenized equations that were derived in [11] for the reduced dimension unnormalized filter.", "In Section  we present the model that was originally suggested by Lorenz (Lorenz'96), which incorporated a pattern with convective scales and introduced a crude model of $K$ slow variables plus $JK$ fast variables that varies with two distinct time scales.", "This is an excellent test-bed for data assimilation schemes that are developed in this paper.", "Section outlines a sequential particle filtering algorithm (the Sequential Importance Sampling method) and a numerical algorithm for dimensional reduction (the Heterogeneous Multiscale Method of [27], [5]).", "These two methods are combined to give an efficient algorithm for particle filtering in a multiscale environment.", "Finally, we present in Section , the results from several data assimilation experiments on the Lorenz'96 model and discuss future research directions based on “adaptive” or “targeted” observations and sensor placement." ], [ "Formulation of multiscale nonlinear filtering problems", "The results presented here are set within the context of slow-fast dynamical systems, where the rates of change of different variables differ by orders of magnitude.", "The effects of the multiscale signal and observation processes via the study of lower dimensional Zakai equations in a canonical way was presented in [10], [11], where the convergence of the optimal filter to the homogenized filter is shown using backward stochastic differential equations (BSDEs) and asymptotic techniques.", "This paper provided rigorous mathematical results that support the numerical algorithms based on the idea that stochastically averaged models provide qualitatively useful results which are potentially helpful in developing inexpensive lower-dimensional filtering.", "For the reduced nonlinear model an appropriate form of particle filter can be a viable and useful scheme.", "Hence, we present the numerical solution of the lower dimensional stochastic partial differential equation derived here, as it is applied to several higher dimensional multiscale applications.", "We will present the main result of [11] here.", "Let $(\\Omega , \\mathcal {F}, (\\mathcal {F}_t), \\mathbb {Q})$ be a filtered probability space that supports a standard Brownian motion $(V, W, B)\\in \\mathbb {R}^{k\\times l\\times d}$ .", "We consider the signal process that is the solution of the two time scale stochastic differential equations (SDEs) driven by $(W,V)$ : $dX^\\varepsilon _t & = b(X^\\varepsilon _t, Z^\\varepsilon _t) dt + \\sigma (X^\\varepsilon _t, Z^\\varepsilon _t) dV_t, \\quad X^\\varepsilon _0= \\xi \\in \\mathbb {R}^m \\\\ dZ^\\varepsilon _t & = \\frac{1}{\\varepsilon }f(X^\\varepsilon _t, Z^\\varepsilon _t)dt + \\frac{1}{\\sqrt{\\varepsilon }}g(X^\\varepsilon _t, Z^\\varepsilon _t)dW_t, \\quad Z^\\varepsilon _0 = \\eta \\in \\mathbb {R}^n.$ Here, $\\varepsilon << 1$ is the timescale separation parameter, so $X^\\varepsilon $ is the slow component and $Z^\\varepsilon $ is the fast component.", "$W$ , $V$ and $B$ are independent of each other as well as the random initial conditions $\\xi $ and $\\eta $ .", "The functions $f$ , $g$ , $b$ , $\\sigma $ are assumed to be Borel-measurable.", "Associated to the signal is the $d$ -dimensional observation $Y^\\varepsilon $ that is perturbed by the Brownian motion $B$ , given by $Y^\\varepsilon _t = \\int _0^t h(X^\\varepsilon _s, Z^\\varepsilon _s) ds + B_t,$ where $h$ is a Borel-measurable function.", "Define the sigma algebra generated by the observation, $\\mathcal {Y}^\\varepsilon _t = \\sigma (Y^\\varepsilon _s: 0 \\le s \\le t)\\vee \\mathcal {N}$ , where $\\mathcal {N}$ are the $\\mathbb {Q}$ -negligible sets.", "The main objecive of filtering theory is to obtain the best estimate of the signal process based on information from the observation.", "The best estimate, called the optimal filter (the Zakai equation), is a conditional expectation that satisfies a recursive equation driven by the observation ([29]).", "For a formal definition, denote the optimal filter for the multiscale system by $\\pi ^\\varepsilon $ , a finite measure on $\\mathbb {R}^{m+n}$ .", "The goal is to calculate the (normalized) observation-dependent filter $\\pi ^\\varepsilon _t(\\varphi ) \\,\\text{\\tiny def}\\over {=}\\,\\mathbb {E}_\\mathbb {Q}[\\varphi (X^\\varepsilon _t, Z^\\varepsilon _t) | \\mathcal {Y}^\\varepsilon _t]$ , where $\\varphi $ is a bounded measurable function on $\\mathbb {R}^{m+n}$ .", "In order to calculated the specified conditional expectation, it is easier to work on a new probability space $\\mathbb {P}^\\varepsilon $ , on which the observation $Y^\\varepsilon $ is a Brownian motion independent of the signal noise $(W,V)$ .", "$\\mathbb {P}^\\varepsilon $ is related to the original probability space $\\mathbb {Q}$ by the Girsanov transform [2] $D^\\varepsilon _t \\,\\text{\\tiny def}\\over {=}\\,\\left.\\frac{d\\mathbb {P}^\\varepsilon }{d\\mathbb {Q}}\\right|_{\\mathcal {F}_t} = \\exp \\left( - \\int _0^t h(X^\\varepsilon _s, Z^\\varepsilon _s)^T dB_s - \\frac{1}{2} \\int _0^t |h(X^\\varepsilon _s, Z^\\varepsilon _s)|^2 ds\\right),$ which effectively removes the drift $h$ from the observation equation.", "Define an unnormalized filter as $\\rho ^\\varepsilon _t(\\varphi ) \\,\\text{\\tiny def}\\over {=}\\,\\mathbb {E}_{\\mathbb {P}^\\varepsilon } [ \\varphi (X^\\varepsilon _t, Z^\\varepsilon _t)\\left(D^\\varepsilon _t\\right)^{-1} |\\mathcal {Y}^\\varepsilon _t ]$ .", "The normalized and unnormalized filters are related using the Girsanov transform: $\\pi ^\\varepsilon _t(\\varphi ) = \\mathbb {E}_\\mathbb {Q}[\\varphi (X^\\varepsilon _t, Z^\\varepsilon _t) | \\mathcal {Y}^\\varepsilon _t]= \\frac{\\mathbb {E}_{\\mathbb {P}^\\varepsilon }[\\varphi (X^\\varepsilon _t, Z^\\varepsilon _t) (D^\\varepsilon _t)^{-1} | \\mathcal {Y}^\\varepsilon _t]}{\\mathbb {E}_{\\mathbb {P}^\\varepsilon }[(D^\\varepsilon _t)^{-1} | \\mathcal {Y}^\\varepsilon _t]} = \\frac{\\rho ^\\varepsilon _t(\\varphi )}{\\rho ^\\varepsilon (1)}.$ The unnormalized filter $\\rho ^\\varepsilon $ satisfies the Zakai equation (see, for example, [2]): $d\\rho ^\\varepsilon _t(\\varphi ) = \\rho ^\\varepsilon _t(\\mathcal {L}^\\varepsilon \\varphi ) dt + \\rho ^\\varepsilon _t (h \\varphi ) dY^\\varepsilon _t, \\quad \\rho ^\\varepsilon _0(\\varphi ) = \\mathbb {E}_\\mathbb {Q}[\\varphi (X^\\varepsilon _0, Z^\\varepsilon _0)].$ Here, $\\mathcal {L}^\\varepsilon = \\frac{1}{\\varepsilon }\\mathcal {L}_F + \\mathcal {L}_S$ is the differential operator associated to $(X^\\varepsilon , Z^\\varepsilon )$ , with $\\mathcal {L}_F & = \\sum _{i=1}^n f_i(x,z) \\frac{\\partial }{\\partial z_i } + \\frac{1}{2} \\sum _{i,j=1}^n (g g^T)_{ij}(x,z) \\frac{\\partial ^2}{\\partial z_i \\partial z_j } \\\\\\mathcal {L}_S & = \\sum _{i=1}^m b_i(x,z) \\frac{\\partial }{\\partial x_i} + \\frac{1}{2} \\sum _{i,j=1}^m (\\sigma \\sigma ^T)_{ij}(x,z) \\frac{\\partial ^2}{\\partial x_i \\partial x_j}$ where $\\cdot ^T$ denotes the transpose of a matrix or a vector.", "Next, we briefly present a result from stochastic averaging theory for the multiscale setting of the problem considered.", "We assume that for every $x\\in \\mathbb {R}^m$ , the solution $Z^x$ of () with $X^\\varepsilon =x$ fixed is ergodic and converges rapidly to its unique stationary distribution $p_\\infty (x:\\cdot )$ .", "In this case, it is well known that $X^\\varepsilon $ converges in distribution to a diffusion $X^0$ governed by an SDE $dX^0_t = \\bar{b}(X^0_t) dt + \\bar{\\sigma }(X^0_t) dV_t$ for appropriately averaged $\\bar{b}$ and $\\bar{\\sigma }$ .", "In other words, a stochastically averaged model provides a qualitatively useful approximation to the actual multiscale system.", "Hence, if we are only interested in estimating the slow process, or the “coarse-grained” dynamics, then we should make use of the homogenized diffusion $X^0$ .", "Specifically, if we are only interested in the $x$ -marginal, $\\pi ^{\\varepsilon ,x}$ , of the optimal $\\pi ^\\varepsilon $ , then this $X^0$ can be used to construct an averaged, or homogenized, filter $\\pi ^0$ that can appropriately replace $\\pi ^{\\varepsilon , x}$ .", "In high-dimensional problems, $\\pi ^0$ would be preferable to $\\pi ^{\\varepsilon ,x}$ since calculation of $\\pi ^0$ using $X^0$ does not directly involve calculations of the fast process.", "By making use of $X^0$ , we would like to find a homogenized (unnnormalized) filter $\\rho ^0$ that satisfies $d\\rho ^0_t(\\varphi ) = \\rho ^0_t(\\bar{\\mathcal {L}} \\varphi ) dt + \\rho ^0_t(\\bar{h}\\varphi ) dY^\\varepsilon _t, \\quad \\rho ^0_0(\\varphi ) = \\mathbb {E}_\\mathbb {Q}[\\varphi (X^0_0)],$ such that for small $\\varepsilon $ , the $x$ -marginal of $\\rho ^\\varepsilon $ , $\\rho ^{\\varepsilon , x}$ , is close to $\\rho ^0$ .", "We let the generator $\\bar{\\mathcal {L}}$ of $X^0$ be defined as $\\bar{\\mathcal {L}} = \\sum _{i=1}^m \\bar{b}_i(x) \\frac{\\partial }{\\partial x_i} + \\frac{1}{2} \\sum _{i,j=1}^m \\bar{a}_{ij}(x,z) \\frac{\\partial ^2}{\\partial x_i \\partial x_j}$ where $\\bar{b}(x) = \\int b(x,z) p_\\infty (x,dz)$ and $\\bar{a} = \\int (\\sigma \\sigma ^T)(x,z) p_\\infty (x,dz)$ .", "Also, define $\\bar{h}(x) = \\int h(x,z) p_\\infty (x,dz)$ .", "Note that the homogenized filter is still driven by the real observation $Y^\\varepsilon $ and not by a “homogenized observation”, which is practical for implementation of the homogenized filter in applications since such avearaged observation is usually not available.", "However, even if such homogenized observation is available, using it would lead to loss of information for estimating the signal compared to using the actual observation.", "Now define the measure-valued processes $\\pi ^0$ and $\\pi ^{\\varepsilon , x}$ in terms of $\\rho ^0$ and $\\rho ^{\\varepsilon , x}$ as $\\pi ^\\varepsilon $ was in terms of $\\rho ^\\varepsilon $ : $\\pi ^0_t (\\varphi ) = \\frac{\\rho ^0_t(\\varphi )}{\\rho ^0_t(1)} \\qquad \\text{and} \\qquad \\pi ^{\\varepsilon , x}_t(\\varphi ) = \\frac{\\rho ^{\\varepsilon , x}_t(\\varphi )}{\\rho ^{\\varepsilon ,x}_t(1)}.$ The main result of [11] is that under appropriate assumptions on the coefficients of (REF ), () and $h$ , there exists a metric $d$ on the space of probability measures such that for every $T \\ge 0$ there exists $C>0$ such that $\\mathbb {E}_\\mathbb {Q}\\left[ d(\\pi ^{\\varepsilon ,x}_T, \\pi ^0_T)\\right]\\le \\sqrt{\\varepsilon } C, ~\\textrm {i.e.", "}~ \\lim _{\\varepsilon \\rightarrow 0} \\mathbb {E}_\\mathbb {Q}\\left[ d(\\pi ^{\\varepsilon ,x}_T, \\pi ^0_T)\\right] = 0 ~\\textrm {for any}~ T>0.$ In other words, the $x$ -marginal of the optimal filter $\\pi ^\\varepsilon $ converges to the averaged filter $\\pi ^0$ in the space of probability measures as the separation parameter $\\varepsilon $ goes to 0.", "Hence, the homogenized filter is an appropriate measure to use in place of the actual $\\pi ^{\\varepsilon ,x}$ in estimating the “coarse-grained” dynamics $X^\\varepsilon $ in a setting with wide timescale separation.", "In terms of filtering applications, $\\pi ^0$ presents the advantage of not requiring exact knowledge of the fast dynamics for the purpose of estimating the “coarse-grained” dynamics.", "Only knowledge of the invariant measure of $Z^x$ is required, hence by applying appropriate multiscale averaging numerical schemes, computation and information storage for the fast dynamics can be reduced.", "The convergence of the normalized filters, $\\pi ^{\\varepsilon ,x}\\rightarrow \\pi ^0$ , was shown by first obtaining the convergence of the unnormalized filters, $\\rho ^{\\varepsilon ,x}$ to $\\rho ^0$ .", "The dual representations of $\\rho ^{\\varepsilon ,x}_T(\\varphi )$ and $\\rho ^{0}_T(\\varphi )$ were introduced as in [21]: $v^{\\varepsilon ,T,\\varphi }_t(x,z) \\,\\text{\\tiny def}\\over {=}\\,\\mathbb {E}_{\\mathbb {P}^\\varepsilon _{t,x,z}}[\\varphi (X^\\varepsilon _T) \\left(D^\\varepsilon _{t,T}\\right)^{-1} | \\mathcal {Y}^\\varepsilon _{t,T}], \\quad \\textrm {and} \\quad v^{0,T,\\varphi }_t(x) = \\mathbb {E}_{\\mathbb {P}^0_{t,x}}[\\varphi (X^0_T) \\left(D^0_{t,T}\\right)^{-1} | \\mathcal {Y}^\\varepsilon _{t,T}].$ $\\mathbb {P}_{t,x,z}^\\varepsilon $ and $\\mathbb {P}^0_{t,x}$ are the respective measures under which $(X^\\varepsilon ,Z^\\varepsilon )$ and $X^0$ are governed by the same dynamics as under $\\mathbb {P}^\\varepsilon $ and $P^0$ , but $(X^\\varepsilon , Z^\\varepsilon )$ and $X^0$ stays in $(x,z)$ and $x$ until time $t$ .", "$D^\\varepsilon _{t,T}$ and $D^0_{t,T}$ were defined as the Girsanov transform (REF ), but with moving limit of integration $t$ , $D^\\varepsilon _t \\,\\text{\\tiny def}\\over {=}\\,\\left.\\frac{d\\mathbb {P}^\\varepsilon }{d\\mathbb {Q}}\\right|_{\\mathcal {F}_t} = \\exp \\left( - \\int _t^T h(X^\\varepsilon _s, Z^\\varepsilon _s)^T dB_s - \\frac{1}{2} \\int _t^T |h(X^\\varepsilon _s, Z^\\varepsilon _s)|^2 ds\\right),$ $D^0_{t,T} = \\exp \\left( - \\int _t^T \\bar{h}(X^0_r)^T dY^\\varepsilon _r + \\frac{1}{2} \\int _t^T |\\bar{h}(X^0_r)|^2dr \\right),$ and $\\mathcal {Y}^\\varepsilon _{t,T} = \\sigma (Y^\\varepsilon _r - Y^\\varepsilon _t : t \\le r \\le T) \\vee \\mathcal {N}$ is the filtration generated by the observation over $[t,T]$ , minus observation history from 0 to $t$ .", "The Markov property of $(X^\\varepsilon , Z^\\varepsilon , X^0)$ results gives the relations between the unnormalized filters and the respective duals: $\\rho ^{\\varepsilon ,x}_T(\\varphi ) = \\int v^{\\varepsilon , T, \\varphi }_0 (x,z) \\mathbb {Q}_{(X^\\varepsilon _0,Z^\\varepsilon _0)}(dx,dz) \\quad \\textrm {and} \\quad \\rho ^0_T(\\varphi ) = \\int v^{0, T, \\varphi }_0 (x) \\mathbb {Q}_{X^0_0}(dx).$ Note that $\\mathbb {Q}_{X^0_0} = \\mathbb {Q}_{X^\\varepsilon _0}$ , because the homogenized process has the same starting distribution as the un-homogenized one.", "For fixed $T$ and $\\varphi \\in C^2_b(\\mathbb {R}^m, \\mathbb {R})$ , we will write $v^\\varepsilon _t = v^{\\varepsilon , T, \\varphi }_t$ and $v_t^0 = v^{0, T, \\varphi }_t$ .", "The dual process $v_t^\\varepsilon $ essentially represents the conditional expectation $\\rho _T^{\\varepsilon ,x}$ by an alternate conditional expectation that is run backwards in time from $T$ to 0.", "For fixed starting point $(x,z)$ at $t=0$ , this backward in time conditional expectation is constructed using processes $(X_T^\\varepsilon ,Z_T^\\varepsilon )$ that started at $(x,z)$ and ran backwards to $t=0$ .", "By integrating $v_0^\\varepsilon $ over $\\mathbb {Q}_{(X_0^\\varepsilon ,Z_0^\\varepsilon )}$ , we are integrating over all possible starting points $(x,z)$ , hence giving $\\rho _T^\\varepsilon $ .", "This interpretation is similar for $v^0$ of $\\rho ^0$ .", "From (REF ), we have $\\mathbb {E}[|\\rho ^{\\varepsilon ,x}_T(\\varphi ) - \\rho ^0_T(\\varphi )|^p] \\le \\int \\mathbb {E}[ | v^\\varepsilon _0(x,z) - v_0^0(x)|^p] \\mathbb {Q}_{(X^\\varepsilon _0, Z^\\varepsilon _0)}(dx, dz).$ So, if $| v^\\varepsilon _0(x,z) - v_0^0(x)|$ is small, then $|\\rho ^{\\varepsilon ,x}_T(\\varphi ) - \\rho ^0_T(\\varphi )|$ will also be small as long as $\\mathbb {Q}_{(X^\\varepsilon _0, Z^\\varepsilon _0)}$ is well behaved and (REF ) will lead to the normalized filter convergence result.", "Therefore, the aim is to show that for nice test functions $\\varphi $ , $\\mathbb {E}[|v^\\varepsilon _0(x,z) - v^0_0(x)|^p]$ is small.", "The key point is that $v^\\varepsilon $ and $v^0$ solve backward SPDEs.", "We formally expand $v^\\varepsilon $ as $v^\\varepsilon _t(x, z) = \\underbrace{u^0_t(x,z)}_{v^0(t,x)} + \\underbrace{\\varepsilon u^1_{t/\\varepsilon } \\left( x, z \\right)}_{\\psi (t,x,z)} + \\underbrace{\\varepsilon ^2 u^2_{t/\\varepsilon }(x, z)}_{R(t,x,z)}.$ Then, $v^0$ , $\\psi $ and $R$ satisfy linear partial differential equations with appropriate terminal conditions.", "By existence and uniqueness of the solutions to these linear equations, we can apply superposition to obtain that indeed $v^\\varepsilon _t(x,z) = v^0_t(x) + \\psi _t(x,z) + R_t(x,z),$ where $\\psi $ and $R$ are the corrector and remainder terms, respectively.", "Based on the expansion, the problem of showing $L^p$ -convergence of $v^\\varepsilon $ to $v^0$ reduces to showing $L^p$ -convergence of $\\psi $ and $R$ to 0.", "Details of the proof are provided in [11].", "The outline of the method of proof is as follows: The backwards SPDEs were converted to their respective proabilistic representations, which are backward doubly stochastic differential equations (BDSDEs).", "The diffusion operators were replaced by the associated diffusions and explicit estimates for the finite dimensional BDSDEs in terms of the transition density function of the fast process were able to be obtained.", "[22] proved very precise estimates for this transition function, and were used to obtain the desired bounds on $\\psi $ and $R$ ." ], [ "The Lorenz'96 System", "The systematic strategy for the identification of slowly evolving climate modes and fast evolving non-climate modes requires a lengthy analysis.", "In this paper, we consider a simple atmospheric model, which nonetheless exhibits many of the difficulties arising in realistic models, to gain insight into predictability and data assimilation.", "The Lorenz'96 model was originally introduced (in [17]) to mimic multiscale mid-latitude weather, considering an unspecified scalar meteorological quantity at $K$ equidistant grid points along a latitude circle: $&\\dot{X}_t^k = -X_t^{k-1}(X_t^{k-2}-X_t^{k+1}) - X_t^k + F_x +\\frac{h_x}{J}\\sum _{j=1}^{J}Z_t^{k,j}, \\quad k = 1,\\ldots ,K, \\\\&\\dot{Z}_t^{k,j} = \\frac{1}{\\varepsilon }\\left\\lbrace -Z_t^{k,j+1}(Z_t^{k,j+2}-Z_t^{k,j-1}) - Z_t^{k,j} + h_zX_t^k\\right\\rbrace , \\quad j = 1,\\ldots ,J.$ Equation (REF ) describes the dynamics of some atmospheric quantity $X$ , and $X_t^k$ can represent the value of this variable at time $t$ , in the $k^{\\rm th}$ sector defined over a latitude circle in the mid-latitude region (Note: We use superscripts $k$ and $j$ to conform with the typical spatial indexing notation used for the Lorenz '96 model.", "In sections that follow, subscripts $k$ and $j$ will be used as discrete time indices, not to be confused with the spatial indices of the Lorenz model).", "The latitude circle is divided into $K$ sectors (with values of $K$ ranging from $K=4$ to $K=36$ ).", "Each $X_t^k$ is coupled to its neighbors $X_t^{k+1}$ , $X_t^{k-1}$ , and $X_t^{k-2}$ by (REF ).", "(REF ) applies for all values of $k$ by letting $X_t^{k+K} = X_t^{k-K} = X_t^{k}$ , so, for example for $k=1$ , $X_t^{k+1} = X_t^{2}$ , $X_t^{k-1} = X_t^{36}$ , and $X_t^{k-2} = X_t^{35}$ .", "The system was extended to study the influence of multiple spatio-temporal scales on the predictability of atmospheric flow by the dividing each segment $k$ into $J$ subsectors ($J=4$ to $J=10$ ), and introducing a fast variable, $Z_t^{k,j}$ given by (), associated with each subsector.", "Thus, each $X_t^{k}$ represents a slowly-varying, large amplitude atmospheric quantity, with $J$ fast-varying, low amplitude, similarly coupled quantities, $Z_t^{k,j}$ , associated with it.", "In the context of climate modeling, the slow component is also known as the resolved climate modes while the rapidly-varying component is known as the unresolved non-climate modes.", "The coupling terms between neighbors model advection between sectors and subsectors, while the coupling between each sector and its subsectors model damping within the model.", "The model is subjected to linear external forcing, $F_x$ , on the slow timescale.", "The dynamics of the unresolved modes can be modified to include nonlinear self-interaction effects by adding forcing in the form of stochastic terms (see, for example, [18], [19]).", "The use of stochastic terms to represent nonlinear self-interaction effects at short timescales in the unresolved modes is appropriate if we are only interested in the coarse-grained dynamics occuring in the long timescale.", "This is called stochastic consistency in [19].", "Considering (), where only quadratic nonlinearity is present, the motivation behind adding stochastic forcing is thus to model higher order self-interaction effects.", "This can be done using a mean-zero Ornstein-Uhlenback process, specifically a process with amplitude $\\frac{1}{\\sqrt{\\varepsilon }}$ , that is, $&\\dot{Z}_t^{k,j} = \\frac{1}{\\varepsilon }\\left\\lbrace -Z_t^{k,j+1}(Z_t^{k,j+2}-Z_t^{k,j-1}) - Z_t^{k,j} + h_zX_t^k\\right\\rbrace + \\frac{1}{\\sqrt{\\varepsilon }} \\zeta (t).$ This is done more explicitly in a general context in Section .", "The two-scale Lorenz'96 model was also used extensively by [28] to study stochastic parametrization and by [16] for analyzing targeted observations, and by [9] and several others for analyzing the influence of large-scale spatial patterns on the growth of small perturbations." ], [ "Homogenized Hybrid Particle Filter (HHPF)", "Numerical simulation of multiscale dynamical systems is quite problematic because of the wide separation in the time-scales involved.", "Based on the results presented in Section , we combine our study of stochastic dimensional reduction and nonlinear filtering to provide a rigorous framework for developing an algorithm for lower dimensional particle filters specifically adapted to the complexities of underlying multiscale signals.", "The proposed particle filter algorithm is adapted to a system with time-scales separation by incorporating a homogenization scheme in the nonlinear filter, in conjunction with the stochastic dimensional reduction results of Section .", "In order to apply stochastic dimensional reduction in the problem presented in Section , we introduce a stochastic forcing term in () to represent external forcing effects on the system due to unresolved dynamics.", "To illustrate the homogenization scheme, we consider a general system of stochastic differential equations (SDEs) that corresponds with the problem of Section , with stochastic forcing in the fast component: $\\dot{X}^\\varepsilon _t &= b(X^\\varepsilon _t,Z^\\varepsilon _t), \\quad X^\\varepsilon _0 \\sim \\mathcal {N}(\\mu ^x_0, \\sigma ^x_0) \\\\ \\nonumber \\dot{Z}^\\varepsilon _t &= \\varepsilon ^{-1}f(X^\\varepsilon _t,Z^\\varepsilon _t)+\\varepsilon ^{-1/2} g(X^\\varepsilon _t,Z^\\varepsilon _t)\\dot{W}_t, \\quad Z^\\varepsilon _0 \\sim \\mathcal {N}(\\mu ^z_0, \\sigma ^z_0).$ In the following two subsections we explain the Heterogeneous Multiscale Method and Sequential Importance Sampling.", "These techniques are then combined in the Homogenized Hybrid Particle Filter." ], [ "Multiscale numerical integration", "For numerical simulations of (REF ), the timestep $\\delta t$ required for the forward integration of the fast process $Z^\\varepsilon $ needs to be smaller than $\\varepsilon $ for stability of the integration scheme.", "However, with such timestep, significant changes in the slow variable can only be seen on the time-scale of $\\mathcal {O}(1)$ , i.e., much of the computational resources is wasted in solving for an almost stationary evolution of the slow process $X^\\varepsilon $ .", "Based on the results presented in Section , we can solve this problem by adopting a reduced system through stochastic averaging, that is, $\\dot{\\bar{X}}_t = \\bar{b}(\\bar{X}_t),$ where $\\bar{b}(x) = \\int b(x,z) \\mu _x(dz).$ By adopting (REF ) with (REF ), we are under the assumption that (REF ) is exponentially mixing, and the Doeblin condition is satisfied.", "Stated informally: With the slow process fixed at $X_t^\\varepsilon = x$ , the corresponding fast process $Z^{x}$ has a unique invariant measure $\\mu _x(dz)$ , i.e.", "the transition probability measure $P^{x}(z,t)$ of $Z_t^{x}$ converges exponentially to $\\mu _x(dz)$ in the weak sense as $t\\rightarrow \\infty $ , locally uniformly in $x$ and $z$ .", "This implies that for any test function $\\varphi \\in C_b(\\mathbb {R}^m)$ $\\mathbb {E}_z\\left[ \\varphi (Z_t^{x}) \\right] \\rightarrow \\int \\varphi (z)\\mu _x(dz) \\quad \\rm {as} \\quad t\\rightarrow \\infty ,$ uniformly in $x$ and $z$ in any compact set.", "However, the evaluation of the high dimensional integration in (REF ) is nontrivial, and usually it is impossible to obtain an invariant distribution of the fast variable analytically.", "To determine the invariant distribution numerically, we adopt the Heterogeneous Multiscale Method (HMM) introduced in [27].", "The HMM is a method of determining the effective dynamics (REF ) through numerical approximation of the invariant distribution of the fast process.", "It is based on the observation that the fast variable $Z^\\varepsilon $ reaches its invariant distribution (equilibrium) on a time-scale much smaller than the time-scale needed to evolve the slow variable $X^\\varepsilon $ (the Doeblin condition).", "This implies that we could use a much larger timestep value $\\Delta t$ for the evolution of the slow process while keeping the timestep for the fast process small for stability.", "Following the procedure presented in [27], we describe the HMM.", "The evolution of the averaged equation (REF ) is approximated numerically using a forward integration scheme.", "For simplicity, we will use the Euler and Euler-Maruyama schemes in the following description, although higher order schemes are also applicable (see, for example, [13]).", "Consider the interval $[0,T]$ discretized into timesteps of size $\\Delta t = \\left\\lfloor \\frac{T}{N} \\right\\rfloor $ .", "Let $t_k \\,\\text{\\tiny def}\\over {=}\\,k\\Delta t$ and write $X_{t_k}$ as $X_k$ .", "$\\bar{X}_{k+1} = \\bar{X}_k + \\tilde{b}(\\bar{X}_k) \\Delta t$ is the macro-solver with $\\Delta t$ being the macro-timestep (Note: As mentioned in Section , subcript $k$ here indicates time index, different from the spatial index superscript $k$ in the Lorenz '96 model).", "The value of $\\tilde{b}$ , which is an approximation of the averaged coefficient $\\bar{b}$ in (REF ), can be calculated as $\\tilde{b}(\\bar{X}_k) =\\frac{1}{MN_m} \\sum _{r=1}^M \\sum _{j = n_T}^{n_T+N_m} b(\\bar{X}_k,Z^{\\varepsilon ,r}_{k,j})$ where $M$ is the number of replicas of the fast process $Z$ for spatial averaging, $N_m$ is the number of micro-timesteps $\\delta t$ for time averaging, and $n_T$ is the number of micro-timesteps skipped to eliminate transient effects.", "The evolution of $Z^{\\varepsilon ,r}_{k,j}$ is governed by the following micro-solver $Z^{\\varepsilon ,r}_{k,j+1} = Z^{\\varepsilon ,r}_{k,j} + \\frac{1}{\\varepsilon } f(\\bar{X}_k,Z^{\\varepsilon ,r}_{k,j}) \\delta t + \\frac{1}{\\sqrt{\\varepsilon }} g(\\bar{X}_k,Z^{\\varepsilon ,r}_{k,j}) \\delta W.$ Note that a combination of spatial and temporal averaging is used in (REF ) but it is shown in [27] that the combinations of $M$ , $N_m$ , and $\\delta t$ can be chosen based on error analysis such that no spatial averaging or no spatial and temporal averaging is required.", "For more detailed explanation and error analysis, refer to the references [27], [7]." ], [ "Homogenized Hybrid Particle Filter (HHPF)", "The algorithm for the continuous-time HHPF is presented in [23].", "As in standard particle filtering methods, the continuous-time equations can be discretized and the filtering can be done on the resulting discrete-time models.", "Here, we present the discrete-time version of the HHPF using the sequential importance sampling (SIS) algorithm, which is also commonly known as bootstrap filtering (see, for example, [1], [8]).", "We will first provide a brief overview of the idea behind importance sampling and then illustrate how it is applied sequentially in particle filters.", "Then we present how the HHPF uses the SIS algorithm." ], [ "Importance Sampling", "In particle filtering, there is always the need to represent some distribution using a collection of particles.", "When it is difficult to sample from a given distribution, the idea is to sample from another distribution that is more tractable to sample from, and properly use that sample to represent the distribution of interest.", "Importance sampling is a technique for approximating integrals with respect to one probability distribution using a collection of samples from another.", "Let $p$ be the target distribution of interest over space $\\mathbb {X}$ and $q \\gg p$ ($q$ is absolutely continuous with respect to $p$ ) be the distribution from which sampling is done ($q$ is also called the proposal distribution).", "Denote by $\\mathbb {E}_p [.", "]$ and $\\mathbb {E}_q [.", "]$ the expectation with respect to the distributions $p$ and $q$ , respectively.", "For any integrable function $ \\varphi :\\mathbb {X}\\rightarrow \\mathbb {R}$ , we have $\\mathbb {E}_p\\left[ \\varphi (X) \\right] & = \\int _\\mathbb {X} \\varphi (x)p(dx) = \\int _\\mathbb {X} \\varphi (x)\\frac{dp}{dq}(x) q(dx) \\\\ \\nonumber & = \\int _\\mathbb {X} \\varphi (x)w(x)q(dx) = \\mathbb {E}_q\\left[ w(X)\\varphi (X) \\right],$ where $w\\,\\text{\\tiny def}\\over {=}\\,\\frac{dp}{dq}$ .", "A collection $\\lbrace x^i\\rbrace _{i=1}^{N_s}$ of $N_s$ particles can be sampled from $q$ and the particles can be weighted according to $w^i\\propto \\frac{dp}{dq}(x^i)$ to represent the target distribution $p$ i.e.", "$p(x)\\approx \\sum _{i=1}^{N_s}{w}^i\\delta (x-x^i).$ The weights ${w}^i$ are normalized such that $\\sum {w}^i=1$ .", "The strong law of large numbers can be employed to verify that the empirical average of $\\varphi $ with respect to the weighted sample from $q$ converges as $N_s\\rightarrow \\infty $ , with probability 1, to the expected value of $\\varphi $ under the target distribution $p$ , i.e.", "$\\int \\left[\\sum _{i=1}^{N_s}{w}^i\\delta (x-x^i)\\right]\\varphi (x)dx=\\sum _{i=1}^{N_s}{w}^i\\varphi (x^i)\\rightarrow \\mathbb {E}_p\\left[ \\varphi (X) \\right].$" ], [ "Sequential Importance Sampling (SIS)", "The SIS algorithm is a technique of using Monte-Carlo simulations for Bayesian filtering through the incorporation of importance sampling.", "Consider a discrete-time signal $X_{t_k}$ and a discrete-time observation $Y_{t_k}$ and let $p\\left(\\left.", "x_{0:k}\\right| y_{0:k}\\right)$ be the density of the target posterior distribution at timestep $t_k$ .", "The SIS algorithm approximates the target distribution using appropriately weighted samples from a proposal density $q\\left( \\left.", "x_{0:k} \\right| y_{0:k}\\right)$ .", "Suppose we represent $p\\left(\\left.", "x_{0:k}\\right| y_{0:k}\\right)$ using a collection $\\lbrace x^i_{0:k}\\rbrace _{i=1}^{N_s}$ of $N_s$ particles sampled according to $q\\left( \\left.", "x_{0:k} \\right| y_{0:k}\\right)$ as $p\\left( \\left.", "x_{0:k} \\right| y_{0:k}\\right) \\approx \\sum _{i=1}^{N_s} w^i_k \\delta (x - x^i_{0:k}),$ where $w^i_k \\propto \\frac{p\\left( \\left.", "x^i_{0:k} \\right| y_{0:k}\\right)}{q\\left( \\left.", "x^i_{0:k} \\right| y_{0:k}\\right)}$ , with $\\sum _i w^i_k=1$ .", "We choose to arrive at proposal densities sequentially $q\\left( \\left.", "x_{0:k} \\right| y_{0:k}\\right) & = q\\left( \\left.", "x_k \\right| x_{k-1},y_{k}\\right) q\\left( \\left.", "x_{0:k-1} \\right| y_{0:k-1}\\right).$ Making use of the identity (see, for example, [1], equation (45)) $p\\left( \\left.", "x_{0:k} \\right| y_{0:k}\\right) & = \\frac{ p\\left( \\left.", "y_k \\right| x_k\\right) p\\left( \\left.", "x_k \\right| x_{k-1}\\right) p\\left( \\left.", "x_{0:k-1} \\right| y_{0:k-1}\\right) }{ p\\left( \\left.", "y_k \\right| y_{0:k-1}\\right) },$ we write (after removing $p\\left( \\left.", "y_k \\right| y_{0:k-1}\\right)$ because it is same for all the particles) $w^i_k & \\propto \\frac{p\\left( \\left.", "y_k \\right| x_k^i\\right) p\\left( \\left.", "x^i_k \\right| x^i_{k-1}\\right) }{q\\left( \\left.", "x^i_k \\right| x^i_{k-1},y_{k} \\right) } \\frac{p\\left( \\left.", "x^i_{0:k-1} \\right| y_{0:k-1}\\right)}{q\\left( \\left.", "x^i_{0:k-1} \\right| y_{0:k-1}\\right)} \\\\ \\nonumber & \\propto \\frac{p\\left( \\left.", "y_k \\right| x_k^i\\right) p\\left( \\left.", "x^i_k \\right| x^i_{k-1}\\right) }{q\\left( \\left.", "x^i_k \\right| x^i_{k-1},y_{k} \\right) } w^i_{k-1}.$ Hence we have a sequential formula for computing the unnormalized particle weights.", "Because our choice of $q$ is done sequentially, the proposal density depends on the location of the particle only at the previous time step.", "Only $x^i_{k-1}$ need be stored and the rest of the path $x^i_{0:k}$ can be discarded.", "We then have $p\\left( \\left.", "x_k \\right| y_{0:k}\\right) \\approx \\sum _{i=1}^{N_s} {w}^i_k \\delta (x - x^i_{k})$ with weights updated according to (REF ) and normalized by $\\sum _i w^i_k=1$ .", "So, the SIS algorithm is this: At step $k-1$ the location of particles $\\lbrace x^i_{k-1}\\rbrace _{i=1}^{N_s}$ is known.", "New observation $y_k$ is recorded.", "Choose a form for the proposal $q(\\cdot | x^i_{k-1},y_k)$ .", "Sample a particle according to this proposal density and get the new location $x^i_k$ .", "Evaluate the number $q(x^i_k | x^i_{k-1},y_k)$ , and then evaluate the quantities in the numerator of equation (REF ), using the sensor dynamics and the signal dynamics.", "Then update the weights according to equation (REF ).", "In filtering with a fixed number of particles, it is crucial to keep the variance of the weights to minimum: if a collection of particles is such that the weight is concentrated in a small number of particles, this collection represents the distribution poorly.", "One can better it by selecting a collection which has many particles near the region of high concentration and the particles sharing nearly equal weights.", "As can be seen from the weight update equation (REF ), the particle weights depend crucially on the choice of the proposal distribution $q$ .", "So, $q$ should be choosen with an aim of minimizing the variance of particle weights.", "It can be shown [1] that the proposal density which keeps the variance of the weights to a minimum is $q^{\\mathrm {opt}} (x_{k}| x_{k-1},y_{k}) = \\frac{p\\left( \\left.", "y_k \\right| x_k\\right) p\\left( \\left.", "x_k \\right| x_{k-1}\\right)}{\\int p\\left( \\left.", "y_k \\right| x_k\\right) p\\left( \\left.", "x_k \\right| x_{k-1}\\right) dx_k}.$ Employing this in weight update equation REF we have that, if we choose the optimal proposal density, $w^i_k & \\propto w^i_{k-1}\\int p\\left( \\left.", "y_k \\right| x_k\\right) p\\left( \\left.", "x_k \\right| x_{k-1}^i\\right) dx_k.$ In general case, it is difficult to sample from $q^{\\mathrm {opt}}$ .", "However, it is easy when both the likelihood $p\\left( \\left.", "y_k \\right| x_k\\right)$ and the conditional prior $p\\left( \\left.", "x_k \\right| x_{k-1}\\right)$ are Gaussians, cf.", "[1].", "Consider the system dynamics and observation equation: $X_{k+1}&=F(X_k) + \\sigma _{X} W_{k+1} \\\\Y_{k+1}&=HX_{k+1}+\\sigma _{Y}V_{k+1}$ where $W_k$ and $V_k$ are independent centered Gaussian increments with variance one and $Q \\,\\text{\\tiny def}\\over {=}\\,\\sigma _X\\sigma _X^T$ , $R\\,\\text{\\tiny def}\\over {=}\\,\\sigma _Y\\sigma _Y^T$ are strictly positive definite.", "Since this subsection on SIS and the next one on optimal control discuss a general method, we have used $F$ to denote the vector field instead of $b$ and $f$ as in the previous sections.", "We then have that the likelihood $p(y_k|x_k)=\\mathcal {N}(Hx_k,R)$ and the conditional prior $p\\left( \\left.", "x_k \\right| x_{k-1}\\right)=\\mathcal {N}(F(X_{k-1}),Q)$ .", "Using (REF ), we have $q^{\\mathrm {opt}}(x_{k}| x_{k-1},y_{k})&=\\mathcal {N}(F(x_{k-1})+\\alpha (x_{k-1},y_k),\\hat{Q}), \\nonumber \\\\\\hat{Q}&=\\left(Q^{-1}+H^TR^{-1}H\\right)^{-1}, \\nonumber \\\\\\alpha (x_{k-1},y_k)&=\\hat{Q}H^TR^{-1}(y_k-HF(x_{k-1})).$ This $q^{\\mathrm {opt}}$ is a Gaussian.", "Once we have particle locations $\\lbrace x^i_{k-1}\\rbrace _{i=1}^{N_s}$ representing the posterior at time $k-1$ , and the observation $y_k$ is recorded, we can evaluate $\\alpha (x_{k-1},y_k)$ .", "We can then sample a particle $x^i_{k}$ from the above Gaussian $\\mathcal {N}(F(x_{k-1}^i)+\\alpha (x_{k-1}^i,y_k),\\hat{Q})$ and arrive at a collection of particles $\\lbrace x^i_{k}\\rbrace $ .", "Alternatively, the particles can be evolved according to $X_{k+1}&=F(X_k) + \\alpha (X_k,y_{k+1}) + \\hat{\\sigma }_{X} W_{k+1}$ where $\\hat{\\sigma }_{X}$ is such that $\\hat{\\sigma }_{X}\\hat{\\sigma }_{X}^T = \\hat{Q}$ .", "Then $X_{k}^i$ behaves like a particle sampled from $q^{\\mathrm {opt}} (\\cdot | x_{k-1}^i,y_k).$ The weights are updated according to (REF ): we will have $w^i_k\\,\\,&\\propto \\,\\, w^i_{k-1}\\exp \\left\\lbrace -\\frac{1}{2} (y_k-HF(x_{k-1}^i))^T\\hat{R}^{-1}(y_k-HF(x_{k-1}^i))\\right\\rbrace , \\nonumber \\\\\\hat{R}^{-1}&=R^{-1}\\left(\\mathbf {1}-H\\hat{Q}H^TR^{-1}\\right).$" ], [ "Stochastic optimal control approach", "We consider the same discrete-time nonlinear system with linear observation (REF ), ().", "The particle method presented in this section consists of control terms in the “prognostic\" equations, that nudge the particles toward the observations.", "We nudge the particles by applying a control $u_k(y,x)$ according to $X_{k+1}&=F(X_k) + u_k(y_{k+1},X_k)+ \\sigma _{X} W_{k+1}.$ The technique for determining the nudging term $u_k(y_{k+1},x)$ is by method of stochastic optimal control by minimizing a quadratic cost: $J := \\mathbb {E}_{k,x}\\frac{1}{2}\\left[ u_k^T(y,x)Q^{-1} u_k(y,x) + (y - h(X^{(k,x)}_{k+1}))^T R^{-1} (y-h(X^{(k,x)}_{k+1}))\\right],$ where $Q$ , $R$ are the signal and observation noise covariance matrices and by $X^{(k,x)}$ we mean that the process $X$ started at time $k$ at the value $x$ .", "Again, we assume the linear sensor function with observation available at every timestep, given by ().", "The purpose of the present section is to show that control methods can be used as a basic and flexible tool for the construction of the proposal density inherent in particle filtering.", "In this framework, $u$ can be interpreted as the optimal control for minimizing the cost $J$ .", "The first term in (REF ) represents the control energy and if we allow $u$ to become too big, then heuristically all the particles will coincide with the observation.", "Then the particles will be a sample from a Dirac distribution, whereas the conditional distribution that we try to simulate is absolutely continuous.", "The second term in (REF ) represents the distance between $HX_{k+1}$ and the observation that we want minimized.", "Covariance matrices $Q$ and $R$ in the quadratic terms indicate that dimensions of the signal and observation that have larger noise variance are penalized less by the control.", "This means that in directions where noise amplitude is large, we allow for more correction by taking $Q^{-1}$ , which puts less penalty on the size of the control in the dimensions with large noise amplitude.", "Similarly, the terminal cost given by the second term in (REF ) incurs a penalty for being far away from the actual signal based on observation, but in directions where the quality of the observation is not very good, we allow our particle to be further away from the observation, hence $R^{-1}$ .", "Define the value function $V^{opt}(k,x) := \\inf _{u_k(y,x)} \\mathbb {E}_{k,x}\\frac{1}{2}\\left[ u_k^T(y,x)Q^{-1} u_k(y,x) + (y - h(X^{(k,x)}_{k+1}))^T R^{-1} (y-h(X^{(k,x)}_{k+1}))\\right].$ Using (REF ) and (), and substituting for $X^{(k,x)}_{k+1}$ in the second expression in the value function, we have $& \\left(y - HX^{(k,x)}_{k+1}\\right)^TR^{-1}\\left(y - HX^{(k,x)}_{k+1}\\right) \\\\& = \\left(y - H\\tilde{F}(x,u_k)\\right)^TR^{-1}\\left(y - H\\tilde{F}(x,u_k)\\right) - 2\\left(y - H\\tilde{F}(x,u_k)\\right)^TR^{-1}\\left(H{\\sigma }_XW_{k+1}\\right) \\\\& \\quad + \\left(H{\\sigma }_XW_{k+1}\\right)^TR^{-1}\\left(H{\\sigma }_XW_{k+1}\\right),$ where $\\tilde{F}(x,u_k) = F(x) + u_k(y,x)$ .", "Because $u_k$ depends only on $y$ and $x$ , both of which are given, we have $\\mathbb {E}_{k,x}\\left[u_kQ^{-1}u_k\\right]=u_kQ^{-1}u_k$ , and similarly $\\mathbb {E}_{k,x}\\left( \\left(y - H\\tilde{F}(x,u_k)\\right)^TR^{-1}\\left(y - H\\tilde{F}(x,u_k)\\right) \\right) = \\left(y - H\\tilde{F}(x,u_k)\\right)^TR^{-1}\\left(y - H\\tilde{F}(x,u_k)\\right).$ $W_{k+1}$ is a standard Gaussian random variable independent of $X_k=x$ , so $\\mathbb {E}_{k,x}\\left[ \\left(y - H\\tilde{F}(x,u_k)\\right)^TR^{-1}\\left(H{\\sigma }_XW_{k+1}\\right) \\right] & = \\left(y - H\\tilde{F}(x,u_k)\\right)^TR^{-1}H{\\sigma }_X \\mathbb {E}\\left[ W_{k+1}\\right]=0,$ $\\mathbb {E}_{k,x}\\left[ \\left(H{\\sigma }_XW_{k+1}\\right)^TR^{-1}\\left(H{\\sigma }_XW_{k+1}\\right)\\right] & = \\operatorname{tr}\\left(\\left(H{\\sigma }_X\\right)^TR^{-1}\\left(H{\\sigma }_X\\right)\\right)$ Therefore, $V(k,x) = \\frac{1}{2}\\inf _{u_k}\\left\\lbrace u_k^TQ^{-1}u_k + \\left(y - H\\tilde{F}(x,u_k)\\right)^TR^{-1}\\left(y - H\\tilde{F}(x,u_k)\\right) + \\operatorname{tr}\\left(\\left(H{\\sigma }_X\\right)^TR^{-1}\\left(H{\\sigma }_X\\right)\\right) \\right\\rbrace $ and $\\frac{\\partial }{\\partial u_k}V(k,x) & = {Q}^{-1}u_k + H^TR^{-1}H(F(x)+u_k) - H^TR^{-1}y \\\\& = ({Q}^{-1} + H^TR^{-1}H)u_k - H^TR^{-1}\\left(y-HF(x)\\right),$ hence the optimal control is $ u_k^{opt} = ({Q}^{-1} + H^TR^{-1}H)^{-1}H^TR^{-1}(y-HF(x)), $ which is similar to the solution from the previous approach, with the difference being the noise term, $\\hat{\\sigma }_X$ in (REF ) and ${\\sigma }_X$ in (REF ).", "This section provided the results related to the control design of the particles (prior to updating the weights) that is needed to nudge the particle solutions toward the observations.", "This procedure consists of adding, forcing terms to the “prognostic\" equations for the construction of the proposal density inherent in particle filtering.", "However, it is possible that the nudging terms may become too large and destroy the balance between the effects of the noise and the control terms.", "The stochastic optimal control approach presented in this section is similar to the derivation of the 4D-VAR method that is used in geophysical data assimilation (see, for example, [12]).", "The 4D-VAR method considers the problem of determining the best initial condition at time $t_0$ for the forward integration of the model PDEs based on discrete observations collected, up to a finite time $t_K$ , in the future of $t_0$ .", "In the 4D-VAR method, the cost function to be minimized with respect to the initial condition $x(t_0)$ is $J(x(t_0)) &=\\frac{1}{2}[x(t_0)-x^b(t_0)]^TB_0^{-1}[x(t_0)-x^b(t_0)] \\\\&\\quad + \\frac{1}{2}[H(x(t_K))-y(t_K)]^TR^{-1}[H(x(t_K))-y(t_K)],$ where $x^b(t_0)$ was predicted using the model equations from time before $t_0$ and $x(t_K)$ is obtained by integration of the model PDEs using $x(t_0)$ as initial condition.", "From this point of view, the stochastic optimal control approach presented here can be viewed as determining the optimal initial condition at every discrete time $t_k$ using the next available observation at $t_{k+1}$ .", "The optimal control $u_k^{opt}$ is the correction made to the state $x_k$ predicted from $t_{k-1}$ .", "The SIS presented in Section REF allows the modification of the drift terms as well as the stochastic coefficients in the “prognostic\" equations and SIS is an easy approach to implement numerically, therefore we will use SIS in conjunction with HHPF throughout this paper.", "However, when dealing with sparse data, a proposal density based on stochastic control theory presented in this section is essential, as can be shown in [15]." ], [ "HHPF", "In the following we describe how the HHPF uses the SIS algorithm.", "For now, we would choose the proposal density $q\\left( \\left.", "\\cdot \\right| x^i_{k-1},y_{k} \\right)=p(\\cdot |x^i_{k-1})$ i.e.", "the conditional prior which can be obtained from the signal dynamics.", "A particle can be sampled from this $q$ by propagating the location of the particle at $k-1$ using the signal dynamics.", "We note that by choosing such a $q$ we are being blind to the observation $y_k$ .", "We address a better choice in Section REF .", "The HHPF is developed based on the results presented in Section .", "It incorporates the HMM described in Section REF in the particle evolution step of a particle filtering algorithm.", "Although the problem presented in Section has continuous-time signal, the HHPF is applied on the associated discretized model.", "Consider the time interval $[0,T]$ , discretized into macro-timesteps of size $\\Delta t = \\left\\lfloor \\frac{T}{N} \\right\\rfloor $ and denote the micro-timestep by $\\delta t << \\Delta t$ , where $\\delta t$ is chosen small enough compared to $\\varepsilon $ for numerical stability.", "In the context of the HMM, the intervals $[k \\Delta t, (k+1) \\Delta t]$ , $k = 1,\\ldots ,N-1$ , are the macro-timestep intervals over which evolution of the slow process occurs.", "The fast process is evolved over micro-timestep intervals $[j \\delta t, (j+1) \\delta t]$ , $j=1,\\ldots ,n_T + N_m$ within each macro-timestep interval.", "The discretized version of the equation (REF ) governing the coarse-grained dynamics $\\bar{X}_t$ , is used for propagation of particles $\\left\\lbrace \\bar{x}^i_k \\right\\rbrace _{i=1}^{N_s}$ over macro-timesteps: $X_{k} & = X_{k-1} + \\tilde{b}\\left(X_{k-1}\\right)\\Delta t + \\sigma _x \\Delta W_{k},$ where $\\tilde{b}$ is defined in (REF ).", "Note that (REF ) is the same as (REF ) but with additive noise, which is introduced to regularize the system, for the purpose of weight calculation.", "Instead of continuous observations, we assume that observations are available at discrete macro-timesteps.", "Write $Y^\\varepsilon _k \\,\\text{\\tiny def}\\over {=}\\,Y^\\varepsilon _{t_k}$ .", "The discrete-time observations are $Y^\\varepsilon _{k} = h\\left( X^\\varepsilon _{k}, Z^\\varepsilon _{k,\\left\\lfloor \\frac{\\Delta t}{\\delta t} \\right\\rfloor } \\right) + \\sigma _y V_{k}.$ Observations $Y_k^\\varepsilon $ are used to update the sample by altering particle weights and resampling the system of particles.", "For notational consistency, we denote the particles representing the averaged $\\bar{X}_k$ by $\\bar{x}^i_k$ .", "The evolution of the particle system via the HHPF is given by the following steps with a graphical illustration of each step shown in Figure REF .", "Figure: HHPF illustrative scheme initial condition: Samples of $N_s$ particles for the slow process, $\\bar{X}_k$ , and $N_s \\times M$ particles for the fast process, $Z_{k,j}^\\varepsilon $ (subscripts $k$ and $j$ are the macro-step and micro-step indices, respectively, not to be confused with the spatial indices, superscripts $k$ and $j$ , in Section ), are drawn from the initial distribution of the state variables.", "The initial distribution of the slow process, $\\pi ^0_{0}$ , is approximated by a sample of $N_s$ particles $\\lbrace \\bar{x}^i_0\\rbrace _{i=1}^{N_s}$ drawn from $\\pi ^0_0$ , each of mass $\\frac{1}{N_s}$ , i.e.", "$\\pi ^0_{0} \\approx \\frac{1}{N_s} \\sum _{i=1}^{N_s} \\delta (x - \\bar{x}_0^i).$ Similarly, the initial distribution of the fast process is approximated by a collection of $M$ particles $\\left\\lbrace z_0^{\\varepsilon ,i,r}\\right\\rbrace _{i,r=1}^{N_s,M}$ which are equally weighted ($z_0^{\\varepsilon ,i,r} \\in \\mathbb {R}^n$ is the position of the $r^{th}$ particle).", "Note that we are only interested in the coarse-grained dynamics $\\dot{\\bar{X}}_t$ of the system.", "The fast process, assumed to be exponentially mixing, is spatially and temporally averaged through the implementation of the HMM.", "Thus, under the assumption of ergodicity, we can set $M<< N_s$ or even $M= 1$ without loss of accuracy in approximating the distribution of $\\bar{X}_k$ .", "prediction and update: The prediction and update step is where HHPF differs from regular particle filters through the incorporation of dimensional reduction and homogenization techniques.", "In a regular particle filter, particles evolve independently according to the discretized signal equations which govern the behaviour over a micro-step $\\delta t$ .", "Recall that $\\delta t$ is small compared to $\\varepsilon $ .", "So, the regular filters simulate the slow process also every $\\delta t$ , even though it does not change much in such a small time, thus wasting the computation resources.", "The HHPF utilizes multiscale scheme of the HMM to propagate the sample forward in time.", "Particles $\\left\\lbrace \\bar{x}_k^i \\right\\rbrace _{i=1}^{N_s}$ are propagated independently over macro-timesteps $\\Delta t$ using the macro-solver (REF ), where the averaged drift is approximated using (REF ).", "Particle propagations in a macro-timestep interval are illustrated in the “predict” segment of Figure REF .", "Within each macro-timestep interval, the particles representing the fast process, $\\left\\lbrace z_{k,j}^{\\varepsilon ,i,r}: k~{\\rm fixed}\\right\\rbrace _{i,r=1}^{N_s,M}$ , evolve according to the micro-solver (REF ) over micro-timestep intervals $[j\\delta t,(j+1)\\delta t]$ , $j = 1,\\ldots ,N_m-1$ , while the slow process, $\\bar{X}_k = x$ is fixed throughout the macro-timestep interval.", "Note that in the HMM scheme, $N_m<\\frac{\\Delta t}{\\delta t}$ ; $N_m$ only needs to be sufficiently large for the fast process to attain an invariant measure within $[k\\Delta t,(k+1)\\Delta t]$ .", "Also, the particles representing $z_{k,j}^{\\varepsilon ,i,r}$ that are propagated over the micro-timesteps are in total $N_s\\times M$ , but $M$ can be set to be 1 by appropriately adjusting the value of $N_m$ , thus the number of fast particles that need to be propagated can be $N_s$ , even with implementation of the HMM scheme.", "The coefficient $\\tilde{b}$ in (REF ) is computed by averaging over the particle locations $\\left\\lbrace z_{k,j}^{\\varepsilon ,i,r}: j=1,\\ldots ,n_T+N_m \\right\\rbrace _{i,r=1}^{N_s,M}$ .", "Averaging over $z_{k,j}^{\\varepsilon ,i,r}$ s is performed spatially over the sample of $M$ particles and temporally over the micro-timesteps interval $[n_T\\delta t,(n_T+N_m)\\delta t]$ .", "The prediction step is given by (REF ).", "The particles are updated at time-steps when observations are available, as illustrated in the update segment of Figure REF .", "Only the sample of fast process $\\left\\lbrace \\bar{x}_k^i \\right\\rbrace _{i=1}^{N_s}$ is updated, since, in the multiscale scheme, we are interested only in the dynamics of the homogenized $\\bar{X}_k$ .", "The SIS algorithm is used for updating the sample.", "In propagating the particles $\\lbrace \\bar{x}^i_k\\rbrace $ , as described above, according to the homogenized signal dynamics over $\\Delta t$ , we arrive at new location of the particles $\\lbrace \\bar{x}^i_{k+1}\\rbrace $ .", "These new locations can be thought of as a sample drawn from the proposal density $q(\\cdot |x_{k})$ which is same as the prior $p(\\cdot |x_{k})$ of the homogenized dynamics.", "Particle weights are calculated sequentially according to (REF ) and because of our assumption of the proposal, we have $w^i_k = p\\left( \\left.", "y^\\varepsilon _k \\right| \\bar{x}^i_k\\right) w^i_{k-1}.$ We would like to point out another difference of the HHPF compared to a regular SIS filter, that arises due to homogenization.", "Considering independent standard Gaussian noise increments for the sensor noise, i.e.", "$V_k \\sim \\mathcal {N}(0,\\mathbb {I})$ , in (REF ), the likelihood function is Gaussian: $p\\left(\\left.", "y^\\varepsilon _k \\right| \\bar{x}_k\\right) \\propto \\exp \\left\\lbrace -\\frac{1}{2}\\left(y^\\varepsilon _k - \\bar{h}\\left(\\bar{x}_k\\right)\\right)^T \\left(\\sigma _y \\sigma _y^T\\right)^{-1} \\left( y^\\varepsilon _k - \\bar{h}\\left(\\bar{x}_k\\right)\\right)\\right\\rbrace .$ Instead of the sensor function $h\\left(X^\\varepsilon _k,Z^\\varepsilon _{k,\\left\\lfloor \\frac{\\Delta t}{\\delta t}\\right\\rfloor }\\right)$ , the averaged sensor function $\\bar{h}(x) = \\int _{\\mathbb {R}^n} h(x,z) \\mu _x(dz)$ is used, since we are dealing with the coarse-grained dynamics.", "As with the homogenized drift, the averaged sensor function is approximated by $\\tilde{h}$ via the HMM.", "$\\tilde{h}$ has the form $\\tilde{h}(X_k) = \\frac{1}{MN_m} \\sum _{r=1}^M \\sum _{j = n_T}^{n_T+N_m} h(\\bar{X}_k,Z^\\varepsilon _{k,j,r}).$ It is worthwhile to note that the actual available observation $Y_k^\\varepsilon $ is used instead of a fictitious averaged $\\bar{Y}_k$ in calculating the weights.", "Thus the homogenized system (REF ) is combined with the actual observation $Y^\\varepsilon $ , from which the name “homogenized hybrid” was derived.", "After a few time steps, all the weights may tend to concentrate on a very few particles, which drastically reduces the effective sample size.", "This issue is addressed in the following step.", "resampling: The nature of the sequential importance sampling algorithm is such that the variance of the unnormalized weights increases with each iteration (see, for example, Prop.", "3, p. 7 in [4], Theorem, p. 285 in [14]).", "Due to the gradual increase in weights variance over time, weights will tend to concentrate on a few particles, causing the issue of sample degeneracy.", "Sample degeneracy decreases the ability of the weighted sample to properly represent the target posterior distribution since only a limited number of particles will have significant weights.", "Additionally, it incurs the cost of propagating and storing particles with insignificant weights, which effectively do not contribute to representing the target distribution.", "One method of addressing this issue is to perform occasional resampling when the sample degeneracy level reaches a certain threshold.", "Resampling does not overcome the issue of weights degeneracy; it only serves to rejuvenate the sample by eliminating particles with insignificant weights and multiplying those with weights that significantly contribute to approximating the posterior.", "The measure of sample degeneracy can be determined by the effective sample size $N^{\\mathrm {eff}}$ (see, for example, [1]): $N^{\\mathrm {eff}}_k \\,\\text{\\tiny def}\\over {=}\\,\\frac{N_s}{1+\\textrm {Var}\\left(\\bar{w^*}_k\\right)}, \\qquad \\bar{w^*}^i_k \\,\\text{\\tiny def}\\over {=}\\,\\frac{p\\left( \\left.", "\\bar{x}^i_k \\right| y^\\varepsilon _{0:k} \\right)}{q\\left( \\left.", "\\bar{x}^i_k \\right| \\bar{x}^i_{k-1}, y^\\varepsilon _{k} \\right)},$ where $\\bar{w^*}^i$ is the “true” weight.", "The exact values of $\\bar{w^*}^i$ cannot be evaluated since $p( \\bar{x}^i_k | y^\\varepsilon _{0:k} )$ , the actual posterior, is not known, so the effective sample size is approximated numerically using the normalized weights by [1] $\\tilde{N}^{\\mathrm {eff}}_k = \\frac{1}{\\sum _{i=1}^M (\\bar{w}^i_k)^2}.$ $\\tilde{N}^{\\mathrm {eff}}_k \\le N_s$ represents the effective sample size at timestep $k$ , and a resampling procedure is carried out whenever $\\tilde{N}^{\\mathrm {eff}}_k$ falls below a set threshold.", "One resampling procedure is the systematic resampling procedure, as described in [1], [4].", "The resampling procedure involves sampling with replacement on the current sample, multiplying particles with significant weights and discarding those with insignificant weights.", "The new sample is reinitialized with uniform weights $\\frac{1}{N_s}$ for each particle.", "In addition to resampling, the issue of sample degeneracy can also be addressed by choosing a good importance sampling density $q$ .", "As described so far, the importance density $q\\left(\\left.", "x_k \\right| x_{k-1} \\right)$ is based on the propagation of the sample from the previous estimation step.", "i.e.", "the observation $Y_k$ was not involved in the importance density at time $k$ .", "In Section REF below, we propose a method of specifying an importance density $q\\left(\\left.", "x_k \\right| x_{k-1}, y_k \\right)$ at time $k$ that makes use of the observation at time $k$ .", "Algorithm REF shows a pseudo code for the HHPF.", "Overall, the advantages of Algorithm REF are: The number of fast samples evaluations is greatly reduced (if the number of fast sample replicas is set as $M=1$ by choosing $N_m$ appropriately), since $N_m<\\lfloor \\Delta t/\\delta t \\rfloor $ .", "The total number of timesteps is decreased due to the relatively large macro-timesteps.", "The number of function evaluations are also decreased accordingly.", "Even though there are additional function evaluations (as in equation (REF )) in incorporating the HMM, these are negligible compared to the original weight calculations of the regular branching particle method, which have been reduced in the HHPF through (REF ).", "Although the HHPF has been adapted for multiscale computations, another issue arises in its application on complex problems with inherent chaotic nature such as that described in Section .", "As shown in Section , we are able to perform state estimation for a chaotic system using the HHPF as described so far, the procedure can be made more efficient by importance sampling.", "However, as mentioned earlier, a key aspect of the importance sampling algorithm is the choice of the sampling density $q$ .", "In the following section, we present a method of constructing a good sampling density by introducing a forcing term in the particle evolution of (REF ).", "[H] HHPF ([24]) Draw samples from initial distribution: $\\lbrace \\bar{x}^{i}_{k=1}\\rbrace _{i=1}^{N_s}$ ,$\\lbrace z^{\\varepsilon ,i,r}_{k=1,j=1}\\rbrace _{i,r=1}^{N_s,M}$ k = 1:number of macro-timesteps ($K$ ) r = 1:number of replicas ($M$ ) j = 1:number of micro-timesteps ($N_m$ ) Solve micro-solver (REF ): $z^{\\varepsilon ,i,r}_{k,j+1}$ Perform averaging (REF ) and (REF ): $\\lbrace \\tilde{f}(\\bar{x}^i_k), \\tilde{h}(\\bar{x}^i_k)\\rbrace _{i=1}^{N_s}$ Solve macro-solver (REF ): $\\lbrace \\bar{x}^i_{k+1}\\rbrace _{i=1}^{N_s}$ Compute weights: $\\lbrace \\bar{w}_{k+1}^i\\rbrace _{i=1}^{N_s}$ Compute effective sample size: $\\tilde{N}_{\\mathrm {eff}}^{k+1}$ Resample using choice of resampling algorithm if $\\tilde{N}_{\\mathrm {eff}}^{k+1}<N_{\\mathrm {thres}}$ Reinitialize: $z^{\\varepsilon , i,r}_{k+1,j=1}=z^{\\varepsilon , i,r}_{k,N_m}$" ], [ "HHPF using the optimal proposal", "In the event that the averaged sensor function (REF ) is expressible as $\\bar{h}(x)=Hx+C_k$ , where $C_k$ is a vector which can change with time $k$ , then one can use the optimal proposal which keeps the variance of weights to minimum, as explained in REF .", "Note that $\\bar{h}$ is of this form if the observation function $h$ is linear and only depends on the slow components.", "Only the following changes need to be made to the algorithm: Instead of propagating the particles $\\bar{x}^i_{k+1}$ by using (REF ), we propagate according to $X_{k} & = X_{k-1} + \\tilde{b}\\left(X_{k-1}\\right)\\Delta t + \\alpha (X_k,y_k-C_k)+ \\hat{\\sigma }_x \\Delta W_{k},$ where $\\hat{\\sigma }_x$ is such that $\\hat{Q}\\,\\text{\\tiny def}\\over {=}\\,(\\hat{\\sigma }_x\\hat{\\sigma }_x^T)=((\\sigma _x\\sigma _x^T)^{-1}+H^T(\\sigma _y\\sigma _y^T)^{-1}H)^{-1}$ , and $\\alpha (X_k,y_k)=\\hat{Q}H^T R^{-1}(y_k-Hf(X_{k-1}))$ , where $f(x)=(x + \\tilde{b}\\left(x\\right)\\Delta t)$ .", "The weights are updated according to $w^i_k\\,\\,&\\propto \\,\\, w^i_{k-1}\\exp \\left\\lbrace -\\frac{1}{2} (y_k-C_k-Hf(x_{k-1}^i))^T\\hat{R}^{-1}(y_k-C_k-Hf(x_{k-1}^i))\\right\\rbrace , \\nonumber \\\\\\hat{R}^{-1}&=R^{-1}\\left(\\mathbf {1}-H\\hat{Q}H^TR^{-1}\\right).$" ], [ "Application", "Based on the results of homogenization and optimal importance sampling, we have developed a new lower-dimensional particle filter, the HHPF, for state estimation in nonlinear multi-scale systems.", "In this section, we illustrate the HHPF's potential for state estimation in a high-dimensional complex problem by applying the HHPF algorithm to the Lorenz '96 ([17]) atmospheric model (see Section ) to estimate the slow variables.", "The HHPF algorithm is implemented in discrete time, using SIS, as presented in Section REF .", "The model parameters for application of the HHPF are as follows: $K = 36$ $J = 10$ , $F_x = 10$ , $h_x = -0.8$ , $h_z = 1$ , $\\varepsilon = 1/128$ .", "The model was first simulated for 40960 timesteps of size $2^{-11}$ starting from arbitrary initial conditions to represent the “true” signal $X^\\varepsilon $ that was to be estimated.", "The timestep value of $2^{-11}$ was picked such that it is small enough compared to the separation parameter $\\varepsilon $ to ensure numerical stability.", "The Lorenz '96 model given by (REF ), (REF ) was integrated using a split timestepping scheme: fourth order Runge-Kutta scheme for the deterministic drift and the Euler-Maruyama scheme for the stochastic parts.", "These schemes were selected for simplicity of implementation in the numerical experiments, but of course, coarser timesteps may be used by implementing higher order integration schemes.", "Observations generated from the true states were of the form assumed in Section REF : $Y^\\varepsilon _t = HX^\\varepsilon _t + B_t$ , i.e.", "depends linearly on $X^\\varepsilon $ , perturbed by a standard Gaussian noise.", "Observations were generated every 128 timesteps, i.e.", "at every timestep of size $2^{-4}$ .", "This is the size of the macrotimestep $\\Delta t$ that we select for the HMM integration scheme.", "This means that we are considering the case where observations are available sequentially (at every timestep) at the timescale that we had chosen for the numerical integration of the slow process.", "In [17], the timestep chosen for numerical integration of the multiscale system with $\\varepsilon = 0.01$ was $0.05$ , which corresponds to 36 minutes in real time.", "We follow [7] in selecting $\\Delta t = 2^{-4}$ and $\\delta t = 2^{-11}$ for the HMM scheme for $\\varepsilon = 2^{-7}$ .", "Based on [17]'s scale, these timesteps approximately correspond to 45 minutes and 35 seconds, respectively, in real time and the duration of the model simulation and data assimilation experiments correspond to 10 days (14400 minutes).", "On the real time scale, the assumption of sequentially available observation data corresponds to observation data collection every 45 minutes, which is not an unrealistic assumption.", "We consider two cases for the linear observation matrix $H$ .", "The first is $H=I_{K\\times K}$ , where $I$ is the identity matrix, i.e.", "all slow states are observed (non-sparse observations).", "The second is $H_{ij}=1$ if $i=j$ and $i$ odd, and otherwise $H_{ij}=0$ , i.e.", "only the odd-indexed slow dimensions are observed (sparse observations).", "Observation noise covariance matrix is assumed to be the identity matrix, i.e.", "observation noise in each dimension is independent of the rest.", "We chose the slow and fast signal noise covariance matrices to be 1 on the diagonals and $0.5$ on the sub- and super-diagonals as in [26], with the fast noise scaled by $\\varepsilon ^{-1/2}$ .", "Initial conditions for the true signal are arbitrarily chosen from mean zero normal distributions with variances 3 and 5 for the slow and fast processes, respectively.", "The filtering objective is to estimate the slow states using sequentially available observations.", "We discuss the numerical experiments in the following.", "The numerical experiments were performed using MATLAB v. 7.11.0.584 (R2010b), without explicitly employing parallel processing capabilities, on an Intel Xeon DP Hexacore X5675 3.07 GHz processor with 12$\\times $ 4 Gb RAM." ], [ "Optimized HHPF with linear observations", "In the first set of numerical experiments, we consider the case of non-sparse observations and consider two variants of the HHPF to study the effects of the optimal particle propagation for linear observation functions discussed in Section REF .", "The SIS algorithm is used in both variants of the HHPF.", "In one variant, the proposal density used for the importance sampling procedure is generated using particles propagated using the optimal drift and diffusion as given in (REF ), with weights updated accordingly.", "In the other variant, proposal density is the prior density generated by propagating particles directly according to (REF ).", "We call the first one the optimized HHPF and the second one the direct HHPF.", "For the HMM scheme, the HMM window was set at $m_N = 64$ microtimesteps, and number of microtimesteps skipped to ignore transient effects is $m_T = 32$ .", "The number of replicas is set at $m_r = 1$ .", "Numerical experiments were performed using both HHPFs with varying sample sizes starting from $N_s = 2$ to $N_s = 400$ .", "Both filters achieved better accuracy with increasing $N_s$ , as they should, and Figures REF and REF each show the comparison of one dimension of the true state with its estimates using the HHPFs and the corresponding estimation errors.", "In both upper figures, blue curve represents the true state, broken red curve represents the estimate using the optimized HHPF, and green curve represents that of the direct HHPF.", "In the lower figures, blue curve represents the observation error, broken red curve and blue curves represent the estimation error of the optimized and direct HHPFs, respectively.", "The error shown is the absolute error over all 36 slow dimensions, i.e.", "${{\\rm error}}_t = \\sqrt{\\sum _{k=1}^{36} (\\hat{X}^{k}_t-X^{k, \\varepsilon }_t)^2}$ .", "Figure REF shows the HHPFs using just 2 particles and the optimized HHPF was seen to perform fairly well in estimating the true state, and the error plot showed that the estimate was only slightly worse than the observation error.", "This is not surprising, since all the slow states were observed and the particles were driven towards those states as indicated by the observations.", "Estimation error was contributed to in part by the stochasticity in the system and the observation noise, as well as the error due to approximation by numerical homogenization.", "The direct HHPF however fails to estimate the truth with $N_s=2$ .", "The estimate initially converged to the truth but diverges at around $t=2.5$ and fails to recapture it after that.", "Both the optimized and direct HHPFs took just 5 seconds to run over the entire interval of 320 macrotimesteps, which is a good performance for the optimized HHPF.", "But considering that the estimates were at best only as good as the noisy observations, the most computationally efficient choice would be to just use observations directly if observations of all states are available.", "By increasing the number of particles used, we see in the error plot of Figure REF that the optimized HHPF was able to provide a better estimate of the truth than what was known directly from the observations.", "The run time for the experiment presented in the figure was 135 seconds, which is the typical time recorded for the optimized HHPF with $N_s=100$ .", "The estimate from the direct HHPF improved but was at best as good as the observation with $N_s=100$ .", "Increasing $N_s$ up to 400 improved both filters' estimates but not significantly for the optimized HHPF.", "The direct HHPF was seen to be able to match the optimized HHPF with $N_s=100$ by using 600 particles but the run time required was 639 seconds.", "The second and third columns of Table REF show the run times for different sample sizes for a typical experiment using the HHPFs.", "For each fixed $N_s$ , run times for the optimized and direct HHPFs were observed to be within the same range, which was expected, since the only additional function evaluations in the optimized version is the calculation of the “nudging” correction that is just a linear combination of observation vector with particle locations.", "Occasional drastic variation in run times between the optimized and direct HHPFs, for example for $N_s=100$ and $N_s=400$ , can be explained based on the number of times the resampling procedure were required to be performed in each filter.", "In each case, resampling may have been required more frequently due to the initial sample being significantly far from the truth, hence more resampling had to be performed in the beginning of the run, or at some point in time, a large number of particles were forced significantly far from the true state by the stochastic forcing.", "The second scenario is less likely to happen since the signal noise amplitude is small in comparison with that of the states.", "Resampling, however, will not drastically increase for the optimized HHPF because particles are simultanesously driven towards the truth based on the observations through the optimal propagation procedure.", "Figure: Non-sparse observations, optimized and direct HHPFs, N s =2N_s=2.", "The optimized HHPF estimated the truth fairly well but was at best as good as observations.", "Increasing N s N_s led to estimates that were better than obervation data.Figure: Non-sparse observations, optimized and direct HHPFs, N s =100N_s=100.", "The optimized HHPF estimate was better than observation data but the direct HHPF was still not as good.", "The direct HHPF was able to match the optmized one with N s =600N_s=600 at the cost of longer computation time.The same numerical experiments were also performed for the case of sparse observations (i.e.", "observing only the dimensions with odd index), to study the performance of the optimized HHPF in estimating hidden states.", "The same trend as for the case of non-sparse observations was observed in comparing the estimates using the optimized and direct HHPFs.", "For fixed $N_s$ , the optimized HHPF achieved better accuracy than the direct version, although both estimates displayed higher estimation errors compared to the non-sparse observation case at the same $N_s$ due to the presence of hidden states.", "For this case, the optimized HHPF estimate was able to match the observation data of the observed states using 20 particles, and estimate the unobserved states as well.", "Figure REF shows the comparisons of the estimates from the HHPFs, using 100 particles for the optimized and 400 particles for the direct, with the truth.", "The upper plot shows the estimates of an observed state and the lower shows those of an unobserved state.", "The optimized HHPF (broken red curve) provided good estimates of all the states, including the unobserved ones and the corresponding error plot in Figure REF shows that the estimation was as good as the observation (of course, observations only observed half the states and the error shown only compares error in the observed dimensions; the HHPF estimates included the unobserved dimensions, shown to capture the truth in the lower plot in Figure REF ).", "The direct HHPF, was also able to capture the shape of the true fluctuations in the observed and unobserved dimensions, but in the observed dimensions, the estimates were poorer than the observations, even with 400 particles.", "Increasing sample size further up to $N_s=800$ did not lead to significant improvement.", "The run times and the trend of increase in run time with sample size for the sparse observations experiments were the same as that of the non-sparse observations experiment shown in Table REF .", "The homogenized filters comparison experiments showed that the optimized HHPF displayed significant estimation performance over the direct version for a fixed sample size.", "This indirectly led to improvement in computation time in comparison with the direct HHPF, in the sense that for the same or even better level of estimation error, the optimized HHPF could be implemented using smaller sample size than the direct version, hense reducing computational costs.", "In the discussion here and in Table REF , we have only considered the comparisons of the HHPFs with $N_s$ up to 400, beyond which the run time advantage of the HHPF over other unhomogenized nonlinear filters, for the same level of accuracy, is lost.", "In the next set of experiments, we compare the optimized HHPF with two other nonlinear filters.", "Figure: Sparse observations, optimized (N s =100N_s=100) and direct (N s =400N_s=400) HHPFs.", "The upper plot shows the estimates of an observed state, the lower plot shows that of an unobserved state.", "The unobserved state was estimated well by the optimzied HHPF with N s =100N_s=100.", "Even with N s =400N_s=400, The direct HHPF captures the fluctuations in the truth but did not follow the trajectory well.Figure: Sparse observations, estimation error of the observed states for the optimized (N s =100N_s=100) and direct (N s =400N_s=400) HHPFs compared with observation error.", "The optimized HHPF estimates were as good as observations (but the optimized HHPF also provided estimates of the unobserved states so more information is gained by using the filter instead of just observation)." ], [ "Comparison of the optimized HHPF with other nonlinear filtering schemes", "In the second set of numerical experiments, we compared the optimized HHPF with two other nonlinear filtering schemes: the ensemble Kalman filter (enKF) and a particle filter without homogenization.", "The aim was to compare the run time reduction in the HHPF, due to the implementation of a homogenization scheme, relative to unhomogenized nonlinear filters and assess the trade-off in estimation accuracy due to homogenization.", "We do not expect the optimized HHPF to outperform an unhomogenized filter in terms of estimation accuracy, but the optimized HHPF should possess comparable estimation capabilities to unhomogenized filters, with the advantage of the HHPF being that it requires shorter run times.", "Since observations were generated at macrotimesteps of size $2^{-4}$ , we have observations being sparse in time as well for the unhomogenized filters, which use the same timestep size of $2^{-11}$ as the model simulation.", "The enKF [6] was implemented directly without modifications, by propagating the ensemble forward in time according to the model dynamics and performing information updates at timesteps when observations are available.", "For the particle filter, we implement a modified SIS particle filter algorithm developed by [26] that is designed to accommodate for observations that are sparse in time.", "This particle filter is similar to the optimized HHPF presented here, in the sense that it uses the presently available observation to construct a better proposal density at a present time by driving particles in between observation timesteps using a time exponential function that is proportional to the model noise covariance and the distance of the intermediate particle locations from the observed state.", "For details and better insight to this particle filter, see [26], and from here on, we will denote this particle filter as just PF, but it is implied that it is the particle filter adapted for sparse-in-time observations.", "Similar numerical experiments as for the comparison of the optimized and direct HHPFs were performed for the enKF and the PF and the estimation results were compared with those of the HHPFs.", "We will first discuss the case of non-sparse (spatially) observations, i.e.", "the case $H=I_{K\\times K}$ .", "The PF was able to provide good estimates of the truth with a sample size of just 2.", "Estimation error decreased as $N_s$ was increased to 20 and was not seen to further decrease significantly as $N_s$ was increased from 20.", "Figure REF shows the estimates of one dimension of the true state using the EnKF, the optimized HHPF, and the PF with $N_s=20$ , and their corresponding estimation errors.", "The blue curve is for the truth, the black for the EnKF, red for the optimized HHPF, and green for the PF.", "The optimized HHPF displayed the highest estimation error, its estimate being as good as the observation.", "The EnKF and PF estimates are better than the observation, with estimation errors of equal magnitude.", "However, when considering the run times, the optimized HHPF took 30 seconds while the EnKF and PF took 540 and 1757 seconds, respectively.", "Even with $N_s=2$ , the PF took 1727 seconds due to the timestep size and the functional evaluations required for particle weight calculations, as well as the resampling procedures.", "Additionally, based on the error plot in Figure REF , the estimation error of the optimized HHPF was not much worse than those of the EnKF and PF, and the estimate trajectory followed the truth very well apart from slight over- and under-shoots at local maxima and minima.", "This indicates that the optimized HHPF required less run time compared to the unhomogenized filters at a comparable level of estimation error.", "Increasing sample size showed that the optimized HHPF was almost as good as the PF at $N_s=100$ , as shown in Figure REF , with the EnKF being slightly better than both particle filters.", "The key point is that using 100 particles, the optimized HHPF took 134 seconds while the EnKF and PF took 2169 and 920 seconds, respectively.", "Considering that the levels of estimation errors are almost equal, the optimized HHPF provided significant advantage in terms of computation time.", "Further increasing the sample size to 400 enabled the optimized HHPF to match the PF.", "Beyond $N_s=400$ however, the optimized HHPF lost its computation time advantage, for the EnKF could be implemented using $N_s=50$ or 100 with about the same level of estimation accuracy and computation time.", "Figure: Non-sparse observations, EnKF, PF and optimized HHPF comparison at fixed N s =20N_s=20.", "The estimate of the optimized HHPF is comparable to those of the EnKF and the PF, but is obtained in a shorter run time.Figure: Non-sparse observations, estimation error comparison the EnKF, PF and optimized HHPF comaprison at fixed N s =100N_s=100.", "The optimized HHPF could match the PF, withthe EnKF being slightly better, but the optimized HHPF required the least computation time.Similar numerical experiments were again performed for the case of sparse observations (i.e.", "observing dimensions with odd index) and Figures REF and REF show the comparisons of the estimates of the observed and unobserved states from the filters using 20 and 100 particles, respectively.", "With 20 particles, the EnKF and PF provided slightly better estimates of the observed states than the observation, but the optimized HHPF performed rather poorly in comparison.", "However, at $N_s=100$ , the optimized HHPF's performance improved significantly, performing as well as the PF.", "Figure REF shows the estimation error over the observed dimensions and, to compare the filter estimate over all slow dimensions, Figure REF shows the estimation error over all slow dimensions for each filter.", "The PF estimate converged to the truth faster but the optimized HHPF estimate eventually became as good as the PF's, with the EnKF's being the best.", "The key point is again that the optimized HHPF took 134 seconds while the EnKF and PF took 920 and 2169 seconds respectively.", "Even if we considered using the EnKF with $N_s=20$ (540 seconds) and the PF with $N_s=2$ (1727 seconds), both of which provided relatively low-error estimates, the optimized HHPF with $N_s=100$ is still faster.", "So, even in the sparse observations case, the optimized HHPF could be implemented in shorter time with estimation error comparable or equal to the EnKF and the PF.", "Figure: Sparse observations, EnKF, PF and optimized HHPF comparison at fixed N s =20N_s=20.", "The optimized HHPF did not perform as well as the EnKF or PF.Figure: Sparse observations, EnKF, PF and optimized HHPF comparison at fixed N s =100N_s=100.", "The optimized HHPF performed as well as the PF, but in shorter time than both the EnKF and PF.Figure: Sparse observations, comparison of estimation errors of the EnKF, PF and optimized HHPF with observation error.Figure: Sparse observations, comparison of estimation errors of the EnKF, PF and optimized HHPF with observation error.", "Estimation error of the optimized HHPF and the PF are of the same magnitude at N s =100N_s=100.", "The optimized HHPF still required less computation time than the EnKF and PF even with the EnKF and PF being implemented using minimum N s N_s possible.We do not claim that the HHPF is better than the EnKF or the PF; as shown, the unhomogenized filters provided lower estimation error than the homogenized filter at low fixed $N_s$ .", "As mentioned in the discussion of the previous set of experiments, the accuracy of the optimzied HHPF could be increased by increasing $N_s$ , but beyond $N_s=400$ , it loses computation time advantage over the EnKF.", "However, in terms of computation time and cost of storage, the optimized HHPF displayed advantage over the unhomogenized filters, with comparable level of accuracy.", "The EnKF would have been the better choice of filter if the lowest possible estimation error was required over computational time.", "Additionally, the PF was designed to accommodate temporally sparse observations, as it has been shown to do in [26] and here.", "This capability still needs to be incorporated in the HHPF as presented here, because in most real time applications, the time interval for availability of observation data can be greater than the 45 minute-interval assumed here.", "In comparing the computational times of the numerical experiments using the EnKF, the PF and the optimized HHPF so far, we have not taken into account the cost of computing the homogenized observation function $\\bar{H}$ .", "In addition to $H$ being a constant matrix, we also assumed that the sensor observed only the slow states, hence the observation function was independent of the fast process.", "So, the homogenized observation function is the same as the unhomogenized one.", "However, we do not expect the cost of evaluating the homogenized observation function to drastically affect the computational time advantage of the optimized HHPF over other unhomogenized nonlinear filters.", "Table: Typical computation times (in sec.)", "for different sample sizes for the different nonlinear filters, in the case of non-sparse observations.", "We see that for a fixed N s N_s, the HHPF required less computational time.", "Circled are the computation times corresponding to the sample sizes that led to the same levels of estimation accuracy for the three different filters compared in Section .", "For the same level of accuracy, the optimized HHPF required a larger sample size but still performed in less time.", "Times for the case of sparse observations are of the same magnitudes and display the same trend." ], [ "Future directions", "In chaotic systems, such as the one studied in this paper, the transients become irrelevant from the dynamical systems point of view and the motion of the solution settles typically near a subset of the state space, called an attractor.", "However, in the data assimilation problem that is of interest in this paper, we are interested in the transients and, in particular, in directions that are stretched by the transient dynamics.", "This sensitivity to initial conditions are characterized by the finite time Lyapunov exponents, which are determined by the behavior of two neighboring orbits or the two point motion of the nonlinear systems.", "We are mostly interested in filtering deterministic chaotic systems, and the particle filtering methods developed above won't work without the addition of noise, because knowing the initial conditions $X_0$ , the distribution of $X_t$ is a Dirac measure.", "Therefore we add Gaussian noise artificially.", "Hence, the finite time Lyapunov exponents may be handy in deciding how we choose the magnitude of the noise.", "The second question then arises as to the property of the sensor function, in the linear case, the span of the observation matrix.", "As pointed out by Lorenz [16] and [20], who have coined the word “adaptive” or “targeted” observations, the sensors should be deployed at any given time, if the data that they gather are to be most effective in improving the analysis and forecasts.", "The results presented in this paper assume that the observations are available every 45 minutes.", "The extension of this work that deals with every combination of the spatial and temporal sparsities, performing intermittent in time sparse data assimilation that mimics a global weather model is presented in [15].", "The particle method presented in [15] consists of control terms in the “prognostic\" equations, that nudge the particles toward the observations, specially in the sparsest situation of $\\frac{K}{4}$ observations in every 48 hours, and shows that control methods can be used as a basic and flexible tool for the construction of the proposal density inherent in particle filtering.", "Acknowledgement.", "Nishanth Lingala, N. Sri Namachchivaya, and Hoong C. Yeong are supported by the National Science Foundation under grant number EFRI 10-24772 and by AFOSR under grant number FA9550-08-1-0206.", "Nicolas Perkowski is supported by a Ph.D. scholarship of the Berlin Mathematical School.", "Part of this research was carried out while Nicolas Perkowski was visiting the Department of Aerospace Engineering of University of Illinois at Urbana-Champaign.", "He is grateful for the hospitality at UIUC.", "The visit of Nicolas Perkowski was funded by NSF grant number EFRI 10-24772 and by the Berlin Mathematical School.", "Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation." ] ]

[ [ "Hysteresis behavior in current-driven stationary resonance induced by\n nonlinearity in the coupled sine-Gordon equation" ], [ "Abstract Recently novel current-driven resonant states characterized by the $\\pi$-phase kinks were proposed in the coupled sine-Gordon equation.", "In these states hysteresis behavior is observed with respect to the application process of current, and such behavior is due to nonlinearity in the sine term.", "Varying strength of the sine term, there exists a critical strength for the hysteresis behavior and the amplitude of the sine term coincides with the applied current at the critical strength." ], [ "Hysteresis behavior in current-driven stationary resonance induced by nonlinearity in the coupled sine-Gordon equation Yoshihiko Nonomura [email protected] Computational Materials Science Unit, National Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan Recently novel current-driven resonant states characterized by the $\\pi $ -phase kinks were proposed in the coupled sine-Gordon equation.", "In these states hysteresis behavior is observed with respect to the application process of current, and such behavior is due to nonlinearity in the sine term.", "Varying strength of the sine term, there exists a critical strength for the hysteresis behavior and the amplitude of the sine term coincides with the applied current at the critical strength.", "05.45.Xt, 74.50.+r, 85.25.Cp Introduction.", "Recently the coupled sine-Gordon equation has been intensively studied numerically and analytically as a model of THz electromagnetic wave emission from intrinsic Josephson junctions (IJJs).", "[1], [2] In current-driven emission without external magnetic field, novel resonant states characterized by the $\\pi $ -phase kinks were proposed for fairly large surface impedance $Z$ , [3], [4] and the present author showed [5] that such states were stationary for any $Z$ with large enough current.", "In these resonant states hysteresis behavior is observed.", "Strong emission which clearly breaks the Ohm's law only takes place in the current-increasing process, and the law almost holds in the current-decreasing process.", "Such dynamical behavior is due to nonlinearity of the system, namely the sine term in the present equation.", "The main question of the present Letter is how such nonlinearity affects on hysteresis behavior in the current-driven resonance.", "May infinitesimal nonlinearity cause such behavior, or may there exist a critical value of it?", "In order to resolve this question, we introduce an artificial parameter $\\gamma $ in the equation and verify the strength of nonlinearlity continuously.", "As the parameter $\\gamma $ decreases from the value in the original equation ($\\gamma =1$ ), width of hysteresis also decreases, and there seems to exist a nonvanishing critical value $\\gamma _{\\rm c}$ with vanishing hysteresis.", "Model and formulation.", "When the capacitive coupling is not taken into account, IJJs are described by the coupled sine-Gordon equation, [6] $\\partial _{x^{\\prime }}^{2}\\psi _{l}=(1-\\zeta \\Delta ^{(2)})\\left( \\partial ^{2}_{t^{\\prime }}\\psi _{l}+\\beta \\partial _{t^{\\prime }}\\psi _{l}+\\sin \\psi _{l}-J^{\\prime } \\right),$ with the layer index $l$ and the operator $\\Delta ^{(2)}$ defined in $\\Delta ^{(2)}X_{l} \\equiv X_{l+1}-2X_{l}+X_{l-1}$ .", "Quantities are scaled as $x^{\\prime }=x/\\lambda _{c},\\ t^{\\prime }=\\omega _{\\rm p}t,\\ J^{\\prime }=J/J_{\\rm c};\\ \\omega _{\\rm p}=c/\\left(\\sqrt{\\epsilon _{\\rm c}}\\lambda _{c}\\right),$ with the penetration depth along the $c$ axis $\\lambda _{c}$ , the plasma frequency in each layer $\\omega _{\\rm p}$ , and the critical current $J_{\\rm c}$ .", "Using material parameters of Bi$_{2}$ Sr$_{2}$ CaCu$_{2}$ O$_{8}$ given in Ref.", "[7], we result in a large inductive coupling $\\zeta =4.4\\times 10^{5}$ , and $\\epsilon _{c}=10$ and $\\beta =0.02$ are taken.", "Neglecting temperature fluctuations and assuming homogeneity along the $y$ axis, we have the two-dimensional formula (REF ).", "Following our previous study, [5] width of the junction is chosen as $L_{x}=86\\mu $ m, and the periodic boundary condition (PBC) along the $c$ axis is considered.", "Then, plasma velocity of the stationary state automatically coincides with that of light in IJJs, which corresponds to the case with infinite number of junctions, though actual number of junctions $N$ still affects on physical properties even in the PBC, especially on response to the in-plane magnetic field.", "[8] Here we concentrate on the simplest case $N=4$ , which takes spatial inhomogeneity of the superconducting phase into account.", "Since direct evaluation of electromagnetic wave emission from edges of a thin sample to vacuum is quite complicated, we use a simplified version [9] of the dynamical boundary condition, [10] where effects outside of the sample are only included in the relation between dynamical parts of the rescaled electric and magnetic fields, $\\tilde{E}^{\\prime }_{l}=\\mp Z \\tilde{B}^{\\prime }_{l}$ , with rescaled quantities $E^{\\prime }_{l}$ and $B^{\\prime }_{l}$ related with $\\psi _{l}$ as $\\partial _{t^{\\prime }}\\psi _{l}=E^{\\prime }_{l}$ and $\\partial _{x^{\\prime }} \\psi _{l}=(1-\\zeta \\Delta ^{(2)})B^{\\prime }_{l}$ , respectively.", "Here we take $Z=30$ , which gives strong enough emission close to the optimal value.", "[5] The sample along the $x$ axis is divided into 80 numerical grids, and calculations are based on the RADAU5 ODE solver.", "[11] In order to investigate effect of nonlinearity in the sine term, Eq.", "(REF ) is slightly modified as $\\partial _{x^{\\prime }}^{2}\\psi _{l}=(1-\\zeta \\Delta ^{(2)})\\left(\\partial ^{2}_{t^{\\prime }}\\psi _{l}+\\beta \\partial _{t^{\\prime }}\\psi _{l}+\\gamma \\sin \\psi _{l}-J^{\\prime }\\right),$ and the parameter $\\gamma $ is controlled hereafter.", "Hysteresis in the original equation.", "First, the $I$ -$V$ curve of the original equation (REF ) is displayed in Figs.", "REF (a) and REF (b).", "As long as the fundamental and first harmonic modes are observed, large hysteresis is quite apparent.", "In the current-increasing process (Fig.", "REF (a)), the current rapidly increases as the voltage approaches the value corresponding to the cavity resonance point given by the ac Josephson relation (broken lines), $V=\\phi _{0}f=\\phi _{0}\\frac{c}{\\sqrt{\\epsilon _{c}}}\\frac{n}{2L_{x}}\\approx 1.14\\ n\\ [{\\rm mV}],$ with the number of nodes $n$ ($=1$ : the fundamental mode).", "In the vicinity of emission peaks, the voltage exceeds the value given by Eq.", "(REF ), which assumes perfect cavity resonance.", "In experiments about 10% of frequency looks tunable in a single resonant branch by varying the current, [12], [13] which is consistent with this result.", "When a much larger value of $Z$ is taken, the range of voltage becomes much smaller [3] during varying the current in the region of strong emission.", "Figure: II-VV curve (solid line) for the(a) current-increasing and (b) current-decreasing processes.Dotted line stands for the curve in another figure, and thesolid-dotted lines denote the voltages corresponding tothe cavity resonance points ().Figure: Emission intensity versus current for the(a) current-increasing and (b) current-decreasing processes.Dotted line stands for the curve in another figure.", "Circles andsquares represent the emission peak in each node and in thecurrent-increasing and current-decreasing processes, respectively.Emission intensity corresponding to the situations for Figs.", "REF (a) and REF (b) is plotted versus current in Figs.", "REF (a) and REF (b), respectively.", "Strong emission observed in the current-increasing process is quite reduced in the current-decreasing process.", "Especially in the fundamental mode, the maximum intensity in the current-decreasing process is almost one order smaller than that in the current-increasing process.", "In such a case emission is expected to be invisible when experimental noise is overloaded in the reverse (current-decreasing) process, which may represent “irreversible\" emission in experiments.", "Hysteresis in the equations with varying nonlinearity.", "Next, the $I$ -$V$ curve of the modified equation (REF ) is investigated.", "Here we concentrate on hysteresis behavior around the emission peaks in the fundamental ($n=1$ ) mode, and only observe the $n=1$ curves in the current-increasing process near the upper edges and the $n=2$ curves in the current-decreasing process near the lower edges.", "In Fig.", "REF (a), parameter dependence of the upper and lower edges is visualized by various symbols for a wide range of parameters between $\\gamma =0.8$ and $0.1$ .", "As $\\gamma $ decreases, the width of hysteresis, namely the difference of the voltages at the both edges in the current-varying processes, becomes smaller and smaller.", "Hysteresis is observed up to $\\gamma =0.2$ , while it is invisible at $\\gamma =0.1$ .", "Then, more precise measurement is made between $\\gamma =0.2$ and $0.1$ as shown in Fig.", "REF (b).", "Although the upper edges of the $n=1$ curve decrease monotonically, the lower edges of the $n=2$ curve exhibit non-monotonic behavior.", "The lower edges decrease up to $\\gamma =0.17$ , increase up to $\\gamma =0.15$ , and decrease again for smaller $\\gamma $ .", "Hysteresis behavior is observed up to $\\gamma =0.16$ , and from $\\gamma =0.15$ current at the upper and lower edges is the same, or the current-varying process becomes reversible.", "Figure: II-VV curve for various values of γ\\gamma for (a) γ=0.8\\gamma =0.8 to 0.10.1 and (b) γ=0.2\\gamma =0.2 to 0.10.1(expanded figure around the irreversible-reversible boundary).Edges of the n=1n=1 curve in the current-increasing process andthe n=2n=2 curve in the current-decreasing process are visualizedby various symbols.Discussions.", "These results suggest that the boundary between the irreversible and reversible behavior locates between $\\gamma =0.16$ and $0.15$ , but further precise calculation in search for the next digit of $\\gamma $ may not be productive.", "Instead, we point out that the current at $\\gamma =0.15$ is $J^{\\prime }=0.150$ on the edges, or that the amplitude of the sine term coincides with the constant term at $\\gamma =0.15$ in the equation (REF ).", "This fact strongly suggests that $\\gamma =0.15$ is the irreversible-reversible boundary.", "Quite recently reversible THz wave emission from IJJs was reported, [14] which may be related with the present finding.", "Introduction of the parameter $\\gamma $ can be regarded as modification of the critical current, namely $J_{\\rm c} \\rightarrow \\gamma J_{\\rm c}$ .", "Then, $\\gamma =J^{\\prime }$ means that the current $J$ is equal to the modified critical current $\\gamma J_{\\rm c}$ , where superconductivity breaks down and the present model is not justified anymore.", "It is natural that hysteresis behavior of the model may also change there.", "In the original equation (REF ), the situation $J>J_{\\rm c}$ occurs in higher harmonic modes and similar behavior may also be observed.", "Numerical study along this direction is now in progress.", "[15] Frequency of resonance $f$ in Eq.", "(REF ) is derived from the linearized version of Eq.", "(REF ), [16] and the role of nonlinearity is to generate the $\\pi $ -phase kinks [3] and hysteresis behavior.", "The present result shows that the hysteresis behavior is also nontrivial.", "We consider this behavior is general and can be regarded as a prototype of hysteresis in harmonic oscillations induced by nonlinearity.", "Summary.", "In the present Letter we numerically investigate characteristic behavior of novel current-driven resonant states in the coupled sine-Gordon equation.", "These states were proposed theoretically as strong emission states in THz electromagnetic wave emission from intrinsic Josephson junctions, and such emission depends on the application process of current.", "That is, hysteresis is observed in emission behavior driven by applied current.", "Such irreversible behavior is due to nonlinearity of the equation, and we vary the strength of the sine term.", "As long as the fundamental resonant mode is observed, the maximum intensity of emission (and consequently the maximum value of the current) decreases monotonically as the strength of nonlinearity decreases, and at the nonvanishing critical value hysteresis behavior disappears and the amplitude of the sine term coincides with that of the applied rescaled current at the emission peak.", "From a physical point of view, superconductivity and hysteresis behavior vanish at the same time.", "Acknowledgments.", "The present work was partially supported by Grant-in-Aids for Scientific Research (C) No.", "20510121 from JSPS." ] ]

[ [ "Computing covariant vectors, Lyapunov vectors, Oseledets vectors, and\n dichotomy projectors: a comparative numerical study" ], [ "Abstract Covariant vectors, Lyapunov vectors, or Oseledets vectors are increasingly being used for a variety of model analyses in areas such as partial differential equations, nonautonomous differentiable dynamical systems, and random dynamical systems.", "These vectors identify spatially varying directions of specific asymptotic growth rates and obey equivariance principles.", "In recent years new computational methods for approximating Oseledets vectors have been developed, motivated by increasing model complexity and greater demands for accuracy.", "In this numerical study we introduce two new approaches based on singular value decomposition and exponential dichotomies and comparatively review and improve two recent popular approaches of Ginelli et al.", "(2007) and Wolfe and Samelson (2007).", "We compare the performance of the four approaches via three case studies with very different dynamics in terms of symmetry, spectral separation, and dimension.", "We also investigate which methods perform well with limited data." ], [ "Introduction", "The asymptotic behaviour of a linear ODE $\\dot{x}(t)=Ax(t)$ , $x(t)\\in \\mathbb {R}^d$ is completely determined by the spectral properties of the $d\\times d$ matrix $A$ .", "Similarly, the long-term behaviour of a nonlinear ODE $\\dot{x}(t)=f(x(t))$ in a small neighbourhood of a fixed point $x_0$ , for which $f(x_0)=0$ , is completely determined by the spectral properties of the linearisation of $f$ at $x_0$ .", "Well-known extensions of these facts can be constructed when $x_0$ is periodic via Floquet theory.", "However, for general time-dependent linear ODEs $\\dot{x}(t)=A(t)x(t)$ , the eigenvalues of $A(t)$ contain no useful information about the asymptotic behaviour as the simple example of [6] illustrates.", "On the other hand, if the $A(t)$ are generated by a process with well-defined statistics, there is a good spectral theory for the system $\\dot{x}(t)=A(t)x(t)$ , and this is the content of the celebrated Oseledets Multiplicative Ergodic Theorem (MET) ([23], see also Arnold [1] for a thorough treatment), which we state and explain shortly.", "The “well-defined statistics” are often generated by some underlying (typically ergodic) dynamical system.", "For clarity of exposition, we will discuss discrete-time dynamics; it is trivial to convert a continuous-time system to a discrete-time system by creating eg.", "time-1 maps flowing from time $t$ to time $t+1$ .", "Let $X$ denote our base space, the space on which the underlying process that controls the time-dependence of the matrices $A$ occurs.", "As we will place a probability measure on $X$ , we formally need a $\\sigma $ -algebra $\\mathfrak {X}$ of sets that we can measurefor example, if $X$ is a topological space, we can set $\\mathfrak {X}$ to be the standard Borel $\\sigma $ -algebra generated by open sets on $X$ .. We denote the underlying process on $X$ by $T:X\\circlearrowleft $ and assume that $T$ is invertible.", "One formally requires that $T$ is measurableif $X$ is a topological space, and $T$ is continuous, then $T$ is measurable with respect to the standard Borel $\\sigma $ -algebra generated by open sets.", "with respect to $\\mathfrak {X}$ .", "The “well-defined statistics” are captured by a $T$ -invariant probability measure $\\mu $ on $X$ ; that is, $\\mu =\\mu \\circ T^{-1}$ , and we say that $T$ preserves $\\mu $ .", "Finally, it is common to assume that the underlying process is ergodic, which means that any subsets $X^{\\prime }\\in \\mathfrak {X}$ of $X$ that are invariant ($T^{-1}(X^{\\prime })=X^{\\prime }$ , implying that trajectories beginning in $X^{\\prime }$ stay in $X^{\\prime }$ forever in forward and backward time) have either $\\mu $ -measure 0 (they are trivial), or $\\mu $ -measure 1 (up to sets of $\\mu $ -measure 0 they are all of $X$ ).", "Now we come to the matrices $A$ , which are generated by a measurable matrix-valued function $A:X\\rightarrow M_d(\\mathbb {R})$ , where $M_d(\\mathbb {R})$ is the space of $d\\times d$ real matrices.", "We choose an initial $x\\in X$ and begin iterating $T$ to produce an orbit $x,Tx,T^2x,\\ldots $ .", "Concurrently, we multiply $\\cdots A(T^2x)\\cdot A(Tx)\\cdot A(x)$ , and we are interested in the asymptotic behaviour of this matrix product.", "In particular, we are interested in (i) the growth rates $\\lambda (x,v):=\\lim _{n\\rightarrow \\infty } \\frac{1}{n}\\log \\Vert A(T^{n-1}x)\\cdots A(Tx)\\cdot A(x)v\\Vert $ as $v$ varies in $\\mathbb {R}^d$ and (ii) the subspaces $W(x)\\subset \\mathbb {R}^d$ on which the various growth rates occur.", "Throughout, $\\left\\Vert \\cdot \\right\\Vert $ denotes the standard Euclidean vector norm or the associated matrix operator norm $\\left\\Vert A \\right\\Vert = \\max _{\\left\\Vert v \\right\\Vert =1}\\left\\Vert Av \\right\\Vert $ ; whether $\\left\\Vert \\cdot \\right\\Vert $ is a vector or matrix norm will be clear from the context.", "Surprisingly, the “well-defined statistics” and ergodicity ensures that these limits exist, and that there are at most $d$ different values $\\lambda _1>\\lambda _2>\\cdots >\\lambda _\\ell \\ge -\\infty $ that $\\lambda (x,v)$ can take, as $v$ varies over $\\mathbb {R}^d$ and $x$ varies over $\\mu $ -almost all of $X$ (note we allow $\\lambda _\\ell =-\\infty $ to include the case of non-invertible $A$ ).", "We can also decompose $\\mathbb {R}^d$ pointwise in $X$ as $\\mathbb {R}^d=\\bigoplus _{i=1}^\\ell W_i(x)$ , where for all $v\\in W_i(x)\\setminus \\lbrace 0\\rbrace $ , one has $\\lim _{n\\rightarrow \\infty }\\frac{1}{n} \\log \\Vert A(T^{n-1}x)\\cdots A(x)v\\Vert =\\lambda _i.$ The subspaces $W_i$ are equivariant (or covariant) with respect to $A$ over $T$ ; that is, they satisfy $W_i(Tx)=A(x)W_i(x)$ for $1\\le i<\\ell $ .", "We use the following stronger version of the MET, which guarantees an Oseledets splitting even when the matrices $A$ are non-invertible.", "Theorem 1.1 ([14], Theorem 4.1) Let $T$ be an invertible ergodic measure-preserving transformation of the probability space $(X,\\mathfrak {X},\\mu )$ .", "Let $A:X\\rightarrow M_d(\\mathbb {R})$ be a measurable family of matrices satisfying $\\int \\log ^+\\Vert A(x)\\Vert \\ d\\mu (x)<\\infty .$ Then there exist $\\lambda _1>\\lambda _2>\\cdots >\\lambda _\\ell \\ge -\\infty $ and dimensions $m_1,\\ldots ,m_\\ell $ with $m_1+\\cdots +m_\\ell =d$ , and a measurable family of subspaces $W_i(x)\\subset \\mathbb {R}^d$ such that for $\\mu $ -almost every $x\\in X$ , the following hold.", "$\\dim W_i(x)=m_i$ , $\\mathbb {R}^d=\\bigoplus _{i=1}^\\ell W_i(x)$ , $A(x)W_i(x)\\subset W_i(Tx)\\mbox{ with equality if $\\lambda _i>-\\infty $,}$ For all $v\\in W_i(x)\\setminus \\lbrace 0\\rbrace $ , one has $\\lim _{n\\rightarrow \\infty }\\frac{1}{n} \\log \\Vert A(T^{n-1}x)\\cdots A(x)v\\Vert =\\lambda _i.$ The range of applications of the MET to the analysis of dynamical systems is vast.", "Below, we mention just of few of the settings in which the MET is used.", "Example 1.2     Differentiable dynamics: One of the first applications of the MET was to differentiable dynamical systems $T:X\\circlearrowleft $ on smooth $d$ -dimensional compact manifolds.", "The matrix function $A$ is the spatial derivative of $T$ , denoted $DT$ .", "The space $\\mathbb {R}^d$ is associated with the tangent space of $X$ and the equivariance condition becomes $W_i(Tx)=DT(x)\\cdot W_i(x)$ .", "If $T$ is uniformly hyperbolic, $\\bigoplus _{i:\\lambda _i>0}W_i(x)=W^u(x)$ , the unstable subspace at $x\\in X$ and $\\bigoplus _{i:\\lambda _i<0}W_i(x)=W^s(x)$ , the stable subspace at $x$ .", "The spaces $W_i(x)$ provide a refinement of $W^u(x)$ and $W^s(x)$ into subspaces with different growth rates.", "Hard disk system: Consider a fixed number $N$ of hard disks in a region $L_{x} \\times L_{y}$ moving freely between collisions.", "In each collision a pair of disks change their velocities [22].", "The region may be finite (hard walls) or periodic (toroidal) in either coordinate direction.", "The quasi-one-dimensional system studied here is a two-dimensional system with $L_{y}$ less than twice the particle diameter so that the disks remain ordered in the $x$ direction.", "Here $X = ([0,L_x]\\times [0,L_y])^N \\times \\mathbb {R}^{2N}$ (with the appropriate equivalence classes depending on the choice of hard wall or toroidal boundary conditions) is the collection of $4N$ -tuples containing all the coordinates and momenta of the $N$ particles.", "The map $T:X\\rightarrow X, x\\mapsto \\mathcal {C} \\circ \\mathcal {F}^{\\tau (x)}(x)$ is the composition of a free-flow map $\\mathcal {F}^{\\tau (x)}$ and a collision map $\\mathcal {C}$ .", "The free-flow map moves the disks in straight lines according to their momentum while none of the disks are colliding.", "The time between collisions is the free-flow time $\\tau (x)$ which depends on the initial condition $x\\in X$ .", "Collisions occur when the boundary of two disks (or one disk and a wall) touch, and the collision map exchanges velocities along the direction of collision (since all disks are of equal mass).", "Again, the matrix function $A$ is the spatial derivative of $T$ , so that $A(x) = DT(x) = D\\left(\\mathcal {C}\\circ \\mathcal {F}^{\\tau (x)}\\right)(x)$ .", "Precise details may be found in [5].", "PDE: The Kuramoto-Sivashinski equation is a model for weakly turbulent fluids and flame fronts $\\eta _{t} = (\\eta ^{2})_{x} - \\eta _{xx} - \\nu \\eta _{xxxx},$ where $\\nu $ is a damping coefficient.", "Another familiar example is the complex Ginzburg-Landau equation $\\eta _{t}=\\eta -(1+i\\beta ) |\\eta |^{2} \\eta + (1+i\\alpha ) \\eta _{xx}$ where $\\eta (x,t)$ is complex and $\\alpha $ and $\\beta $ are parameters.", "In both of these cases it is possible to approximate solutions of the partial differential equations using Fourier spectral methods (see [8] for details).", "For instance, in the case of the 1-dimensional Kuramoto-Sivashinski PDE we look for solutions of the form $\\eta (x,t) = \\sum _{k=-\\infty }^\\infty a_k(t)e^{ikx/\\tilde{L}},$ where $\\tilde{L}$ is a unitless length parameter, then solve the following system of ODEs for the Fourier coefficients $a_k(t)$ : $\\dot{a}_k = (q_k^2 - q_k^4)a_k-i\\frac{q_k}{2}\\sum _{m=-\\infty }^\\infty a_ma_{k-m},$ where $q_k = k/\\tilde{L}$ .", "Since the $a_k$ decrease rapidly with $k$ , truncating the above system of ODEs is justified.", "In the setting of this review we treat $X$ as the space of Fourier coefficients $(a_1,\\ldots ,a_d)$ of the truncated PDE, and consider the transformation $T:X\\rightarrow X$ defined by choosing some $\\tau >0$ and letting $T\\left((a_1,\\ldots , a_d)\\right) = (a_1(\\tau ),\\ldots ,a_d(\\tau ))$ where the $a_k(t)$ are solutions to the system of ODEs with initial conditions $a_k(0) = a_k$ .", "The matrix function $A$ is again the spatial derivative of $T$ so that $ A(x) = DT(x) = \\begin{pmatrix} \\frac{\\partial a_1(\\tau )}{\\partial a_1} & \\frac{\\partial a_1(\\tau )}{\\partial a_2} & \\cdots \\\\ \\frac{\\partial a_2(\\tau )}{\\partial a_1} & \\frac{\\partial a_2(\\tau )}{\\partial a_2} & \\\\ \\vdots & & \\ddots \\end{pmatrix}.$ Nonautonomous ODEs and transfer operators: Consider an autonomous ODE $\\dot{x}(t)=f(x(t))$ on $X$ (for example, the Lorenz flow on $X=\\mathbb {R}^3$ ), and its flow map $\\xi (\\tau ,x)$ which flows the points $x$ forward $\\tau $ time units.", "We think of the coordinates $x$ as a “generalised time” and the ODE $\\dot{x}(t)=f(x(t))$ is our base system.", "We use this base ODE (the driving system) to construct a nonautonomous ODE or skew product ODE as $\\dot{y}(t)=F(\\xi (t,x),y(t))$ .", "Given an initial time $t$ and a flow time $\\tau $ , one may construct finite-rank approximations $P^{(\\tau )}_x(t)$ of the Perron-Frobenius operator $\\mathcal {P}^{(\\tau )}(x(t))$ that track the evolution of densities from base “time” $x(t)$ to $x(t+\\tau )$ ; see [15] for details.", "The matrices $P^{(\\tau )}_x(t)$ form a cocycle and Oseledets subspace computations enable the extraction of coherent sets in the nonautonomous flow (see [15]).", "Coherent sets are time-dependent analogues of almost-invariant sets for autonomous systems; see [9], [13], [12].", "Finite-time constructions for coherent sets are described in [16].", "In the setting of this review, $T:X\\rightarrow X$ is defined as $T(x)=\\xi (\\tau ,x)$ , and $A(x)=P^{(\\tau )}_x(t)$ .", "From now on, we denote $A(T^{n-1}x)\\cdots A(x)$ as $A(x,n)$ .", "The proof of the classical MET [23] proves that the matrix limit $\\Psi (x) = \\lim _{n\\rightarrow \\infty } \\left(\\left(A(x,n)\\right)^* A(x,n)\\right)^{1/2n} $ exists for $\\mu $ -almost all $x\\in X$ .", "The matrix $\\Psi (x)$ is symmetric, depends measurably on $x$ , and its eigenvalues are $e^{\\lambda _1(x)}>\\cdots >e^{\\lambda _\\ell (x)}$ .", "The corresponding eigenspaces are denoted $U_1(x),\\ldots ,U_\\ell (x)$ and one has $V_i(x):=\\bigoplus _{j=i}^\\ell W_j(x)=\\bigoplus _{j=i}^\\ell U_j(x).$ Thus $V_i(x)$ captures growth rates from $\\lambda _i(x)$ down to $\\lambda _\\ell (x)$ ; the “$U$ ”decomposition of $V_i(x)$ is orthogonal, while the “$W$ ” decomposition (the Oseledets splitting) is equivariant (or covariant).", "An alternative notion of stability for non-autonomous systems is the so called Sacker-Sell spectrum, cf.", "[28].", "It is based on exponential dichotomies, cf.", "[7], [24] which we briefly introduce for linear difference equations of the form $w_{n+1} = A_n w_n,\\quad n \\in \\mathbb {Z},\\quad A_n\\in M_d(\\mathbb {R}) \\text{ invertible}.$ In the current context we associate the sequence of matrices $\\lbrace A_n\\rbrace _{n\\in \\mathbb {Z}}$ with an invertible matrix cocycle over a single orbit, e.g.", "for some $x\\in X$ let $A_n = A(T^nx)$ .", "We restrict the introduction of exponential dichotomies to invertible systems only, and note that a justification of our algorithm for computing dichotomy projectors strongly depends on this assumption.", "Theory defining exponential dichotomies for non-invertible matrices is contained in eg.", "[2].", "Numerical experiments indicate that Algorithms REF and REF also apply in the non-invertible case, however, the corresponding analysis is a topic of future research.", "We denote by $\\Phi $ the solution operator of (REF ), defined as $\\Phi (n,m) :={\\left\\lbrace \\begin{array}{ll}A_{n-1} \\ldots A_m,\\quad & \\text{for }n>m,\\\\I, &\\text{for } n = m,\\\\A_n^{-1} \\ldots A_{m-1}^{-1},\\quad &\\text{for } n < m.\\end{array}\\right.", "}$ Definition 1.3 The linear difference equation (REF ) has an exponential dichotomy with data $(K,\\alpha _s,\\alpha _u,P_n^{s},P_n^u)$ on $J\\subset \\mathbb {Z}$ , if there exist two families of projectors $P_n^{s}$ and $P_n^u = I-P_n^{s}$ and constants $K, \\alpha _s, \\alpha _u >0$ , such that the following statements hold: $P_n^s \\Phi (n,m) = \\Phi (n,m)P_m^s \\quad \\forall n,m \\in J,$ $\\begin{array}{ccl}\\left\\Vert \\Phi (n,m)P_m^{s} \\right\\Vert & \\le Ke^{-\\alpha _s(n-m)}\\vspace{5.69054pt}\\\\\\left\\Vert \\Phi (m,n)P_n^u \\right\\Vert & \\le K e^{-\\alpha _u(n-m)}\\end{array}\\quad \\forall n \\ge m,\\ n,m\\in J.$ Consider the scaled equation $w_{n+1} = {e^{-\\lambda }} A_n w_n,\\quad n \\in \\mathbb {Z}.$ Definition 1.4 The Sacker-Sell or dichotomy spectrum is defined as $\\sigma _{\\text{ED}} := \\lbrace {\\lambda \\in \\mathbb {R}}: \\text{ (\\ref {i5}) has noexponential dichotomy on }\\mathbb {Z}\\rbrace .$ The complementary set $\\mathbb {R}\\setminus \\sigma _{\\text{ED}}$ is called the resolvent set.", "The Sacker-Sell spectrum consists of at most $d$ disjoint, closed intervals, where $d$ denotes the dimension of the space, cf.", "[28], i.e.", "there exists an $\\ell < d$ such that $\\sigma _{\\text{ED}} = \\bigcup _{i=1}^\\ell {[\\lambda _i^-,\\lambda _i^+]},\\quad \\text{where } {\\lambda _{i+1}^+ < \\lambda _i^-}\\quad \\text{for } i=1,\\dots ,\\ell -1.$ It is well known that the Lyapunov spectrum, when it exists, is a subset of the Sacker-Sell spectrum, see [10].", "While the Lyapunov spectrum provides information on bounded solutions of $\\Phi (n,0)$ , $n\\ge 0$ , the Sacker-Sell spectrum answers this question for $\\Phi (n,m)$ , $n\\ge m$ .", "These answers may be different for different initial $n$ because, in contrast to the MET setting, there is no a priori stationarity assumption on a base dynamical system generating the matrix cocycle.", "Note that for ${\\lambda \\in \\mathbb {R}\\setminus \\sigma _{\\text{ED}}}$ it follows from [24] that the inhomogeneous equation $w_{n+1} = {e^{-\\lambda }} A_n w_n + r_n$ has for every bounded sequence $r_\\mathbb {Z}$ a unique bounded solution on $\\mathbb {Z}$ .", "Dichotomy projectors of the scaled equation (REF ) are constant in resolvent intervals $R_i := (\\lambda _{i}^+,\\lambda _{i-1}^-)$ , $i=1,\\dots ,\\ell +1$ , where $\\lambda _0^- = \\infty $ and $\\lambda _{\\ell +1}^+ =-\\infty $ , see Figure REF .", "We denote these families of projectors by $(P_n^{i,s}, P_n^{i,u})$ .", "Figure: Spectral setup.In analogy to the MET we obtain the family of subspaces $W_n^{i} = \\mathcal {R}(P_n^{i,s}) \\cap \\mathcal {R}(P_n^{i+1,u}),\\quad n\\in \\mathbb {Z},\\quad i =1,\\dots ,\\ell $ that decompose $\\mathbb {R}^d$ for each $n\\in \\mathbb {Z}$ $\\mathbb {R}^d = \\bigoplus _{i=1}^\\ell W^i_n,$ and using the cocycle property (REF ) it follows for all $i=1,\\dots ,\\ell $ that $A_n W_n^i = W_{n+1}^i,\\quad n\\in \\mathbb {Z}.$ Furthermore, for each $w \\in W_m^i$ , there exists a constant $K = K(w)>0$ such that the following equations hold $\\left\\Vert \\Phi (n,m)w \\right\\Vert & = K e^{\\left(\\lambda _i^+ + r^+_i(n-m)\\right)(n-m)}, \\quad \\text{for } n \\ge m, \\quad \\text{where } \\limsup _{n\\rightarrow \\infty } r^+_i(n) = 0, \\\\\\left\\Vert \\Phi (n,m)w \\right\\Vert & = K e^{\\left(\\lambda _i^- +r_i^-(n-m)\\right)(n-m)},\\quad \\text{for } n<m.\\quad \\text{where }\\limsup _{n\\rightarrow \\infty }r_i^-(n) = 0.$ When working with data over a finite time interval, one has access only to a finite sequence $A_0,A_1,\\ldots ,A_{n-1}$ .", "In this case, one either assumes there is an underlying ergodic process generating the sequence $A_0,A_1,\\ldots ,A_{n-1}$ or one considers exponential dichotomies.", "An outline of the paper is as follows.", "In Sections 2 and 3, we introduce two new methods for computing Oseledets vectors.", "The first method is based on the proof of the generalised MET in [14] and is particularly simple to implement and fast to execute.", "The second method is an adaptation of an approach to compute dichotomy projectors [18].", "In Section 4 we review the approaches by Ginelli et al [17] and Wolfe and Samelson [34].", "In Sections 2, 3, and 4 we provide MATLAB code snippets to implement the algorithms presented.", "Section 5 contains numerical comparisons of the performance of the four methods on three dynamical systems.", "The first case study is a dynamical systems formed via composition of a sequence of $8\\times 8$ matrices constructed so that all Oseledets vectors are known at time 0; we thus compare the accuracy of the methods exactly in this case study.", "The second case study is an eight-dimensional system generated by two hard disks in a quasi-one-dimensional box.", "The third case study is a nonlinear model of time-dependent fluid flow in a cylinder; the matrices are generated by finite-rank approximations of the corresponding time-dependent transfer operators.", "The three case studies have been chosen to represent a cross-section of a variety of features of systems that either help or hinder the computation of Oseledets vectors, and we draw out the advantages and disadvantages of each of the four methods considered." ], [ "An SVD-based approach", "The approach outlined in this section is simple to execute and exhibits quick convergence.", "However, as the length of the sample orbit becomes too large this approach fails.", "In [14] it is proven that the limit $\\lim _{N \\rightarrow \\infty }A(T^{-N}x,N)U_i(T^{-N}x)$ exists and is equal to the $i$ th Oseledets subspace $W_i(x)$ .", "That is, if one computes $U_i$ in the far past and pushes forward to the present, the result is a subspace close to $W_i(x)$ .", "Thus, the strategy in [14] is to first estimate $U_i$ in the past and push forward.", "The numerical method of approximating $W_j(x)$ , $x\\in X$ , is implemented in the following steps: Algorithm 2.1 (To estimate $W_j(x)$ )     Choose $M,N >0$ and form the matrix $\\Psi ^{(M)}(T^{-N}x) = \\left(A(T^{-N}x,M)^* A(T^{-N}x,M)\\right)^{1/2M}$ as an approximation of (REF ) at $T^{-N}x \\in X$ .", "Compute $U_j^{(M)}(T^{-N}x)$ , the $j$ th orthonormal eigenspace of $\\Psi ^{(M)}(T^{-N}x)$ as an approximation of $U_j(T^{-N}x)$ .", "Define $W_j^{(M,N)}(x) = A(T^{-N}x,N)U_j^{(M)}(T^{-N}x)$ , approximating the Oseledets subspace $W_j(x)$ .", "Listing shows part of a MATLAB implementation of Algorithm REF .", "The array A$ = \\left[A(T^{-N}x) \\ | \\ A(T^{-N+1}x) \\ | \\cdots | \\ A(T^{M-1}x)\\right]$ contains the $d\\times d$ matrices which generate the cocycle $A:X\\times \\mathbb {Z}^+ \\rightarrow M_d(\\mathbb {R})$ , and the matrix Psi is formed by multiplying the matrices contained in A.", "Step REF of Algorithm REF is performed prior to the code in Listing , Step REF is performed in lines 1-3 and lines 4-7 perform Step REF .", "The function returns Wj as its estimate to $W_j(x)$ .", "Figure: NO_CAPTIONThe values of $M$ and $N$ can be chosen with relative freedom and in our examples that follow we have chosen $M= 2N$ to compute over a time window centred on $x$ , from $T^{-N}x$ to $T^Nx$ .", "Unfortunately, we cannot choose $M$ and $N$ arbitrarily large and expect accurate results.", "If $A(T^{-N}x,M)$ is constructed via the product $A(T^{-N}x,M) = A(T^{M-N-1}x) \\cdots A(T^{-N+1}x)A(T^{-N}x) $ then with larger $M$ the numerical inaccuracies of matrix multiplication compound and this product becomes more singular and thus a poorer approximation of $A(T^{-N}x,M)$ .", "Because of this, $\\Psi ^{(M)}(T^{-N}x)$ cannot be expected to accurately approximate $\\Psi (T^{-N}x)$ for large $M$ .", "However, even if we suppose $\\Psi ^{(M)}(T^{-N}x)$ accurately approximates $\\Psi (T^{-N}x)$ , the small, but non-zero, difference in $U^{(M)}_j(T^{-N}x)$ and $U_j(T^{-N}x)$ grows roughly as $O\\left(e^{N(\\lambda _1-\\lambda _j)}\\right)$ during the push-forward in step REF above.", "For these reasons $M$ and $N$ must be chosen carefully." ], [ "Improving the basic SVD-based approach", "We present a simple improvement that can overcome one of the sources of numerical instability, namely the push-forward process in step REF above.", "Figure: Schematic of the re-orthogonalisation described in Section .", "The black line represents the orbit centred at x∈Xx\\in X and the points T -N k xT^{-N_k}x are those points at which we ensure orthogonality with the subspaces V j (T -N k x) ⊥ V_j(T^{-N_k}x)^\\perp .", "To do this we use the (blue) approximations Ψ (M) (T -N k x)\\Psi ^{(M)}(T^{-N_k}x) to approximate V j (T -N k x) ⊥ V_j(T^{-N_k}x)^\\perp and perform the (red) push-forward and orthoganlisation steps starting with U j (M) (T -N 1 x)U_j^{(M)}(T^{-N_1}x) and ending with W j (M,0) (x)W_j^{(M,0)}(x) (see Algorithm ).Recall that the subspace $V_j(x) = U_j(x) \\oplus \\cdots \\oplus U_\\ell (x)=\\left(U_1(x)\\oplus \\cdots \\oplus U_{j-1}(x)\\right)^\\perp $ is $A$ -invariant and that for $v\\in V_j(x) \\backslash V_{j+1}(x)$ (with $V_{\\ell +1}(x) = \\lbrace 0\\rbrace $ ) we have $\\lambda _j(x) = \\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\left\\Vert A(x,n)v \\right\\Vert .", "$ The subspace $V_j(x)$ contains $W_j(x), W_{j+1}(x),\\ldots , W_\\ell (x)$ , and so the Oseledets subspace $W_j(x)$ is necessarily perpendicular to all $U_1(x),\\ldots ,U_{j-1}(x)$ .", "To solve the numerical instability of step REF we enforce this condition periodically.", "The amended algorithm is implemented as follows: Algorithm 2.2 (To estimate $W_j(x)$ )     Choose $M,N_1>N_2>\\cdots > N_n=0$ and form the matrices $\\Psi ^{(M)}(T^{-N_k}x) = \\left(A(T^{-N_k}x,M)^* A(T^{-N_k}x,M)\\right)^{1/2M},\\quad k=1,\\ldots ,n.$ Compute all the orthonormal eigenspaces $U^{(M)}_i(T^{-N_k}x)$ , $i=1,\\ldots ,j-1$ of (REF ) (replacing $N$ with $N_k$ in (REF )) and the eigenspace $U^{(M)}_j(T^{-N_1}x)$ .", "Let $\\operatorname{proj}_V:\\mathbb {R}^d \\rightarrow \\mathbb {R}^d$ be the orthogonal projection onto the subspace $V$ so that $\\ker \\left(\\operatorname{proj}_V\\right) = V^\\perp $ and $V^{(M)}_j(x) = \\left(U^{(M)}_1(x) \\oplus \\cdots \\oplus U^{(M)}_{j-1}(x)\\right)^\\perp $ .", "Define $W^{(M,N_1)}_j(T^{-N_1}x) = U_j^{(M)}(T^{-N_1}x)$ , and then define iteratively by pushing forward and taking orthogonal projections: $W^{(M,N_{k+1})}_j\\left(T^{-N_{k+1}}x\\right) = \\operatorname{proj}_{V_j^{(M)}\\left(T^{-N_{k+1}}x\\right)} \\left(A\\left(T^{-N_k}x, N_{k+1} - N_k\\right) W^{(M,N_k)}_j\\left(T^{-N_k}x\\right)\\right) $ $W^{(M,N_n)}_j(x)=W^{(M,0)}_j(x)$ is our approximation of $W_j(x)$ .", "Listing shows an example implementation of Algorithm REF in MATLAB.", "Lines 1-18 are responsible for performing Steps REF and REF , whilst the push forward procedure of Step REF is performed in lines 20-30.", "Again, the matrix cocycle is stored in A$=\\left[ A(T^{-N}x) \\ \\right|\\left.", "\\ A(T^{N-1}x) \\ \\right|\\left.", "\\cdots \\right|\\left.", "\\ A(T^{M-1}x)\\right]$ and the function returns Wj as its approximation of $W_j(x)$ .", "The variable Nk is a one-dimensional array containing the elements of $\\lbrace N_k\\rbrace $ and is counted by k. Figure: NO_CAPTIONRemark 1 Unfortunately, some numerical issues with this approach remain.", "They stem primarily from the long multiplication involved in building the variable Psi of Listings and .", "This results in Psi becoming too singular and hence $U^{(M)}_j(T^{-n}x)$ ($j\\ne 1$ ) poorly approximates $U_j(T^{-n}x)$ .", "As can be seen in Section , Algorithm REF works superbly for $W_2(x)$ as $U^{(M)}_1(T^{-n}x)$ is well approximated for large $n$ .", "However when estimating $W_j(x)$ , $j>2$ , a good estimate of $U^{(M)}_{j-1}(x)$ is required for an accurate projection $\\operatorname{proj}_{V^{(M)}_j}$ and for large $n$ such an estimate becomes unreliable." ], [ "A dichotomy projector approach", "We derive an approach for the computation of a vector $w^j_n \\in W^j_n = \\mathcal {R}(P^{j,s}_n)\\cap \\mathcal {R}(P^{j+1,u}_n)$ .", "For this task, we first need a guess of ${\\Lambda ^\\text{right} \\in R_{i}}$ and of ${\\Lambda ^\\text{left} \\in R_{i+1}}$ in two neighbouring resolvent intervals that lie close to the common spectral interval, see Figure REF .", "Figure: Choice of Λ left/right \\Lambda ^\\text{left/right} in case i=2i = 2.Numerical experiments indicate that we get the best results by choosing $\\Lambda ^\\text{right}$ and $\\Lambda ^\\text{left}$ close to (but outside) the second Sacker-Sell interval.", "This conclusion is supported by theoretical estimates on the approximation error for Algorithm REF , discussed at the end of Section 3.", "The following observation from [18], [19] allows the computation of dichotomy projectors by solving $w_{n+1}^i = A_n w_n^i + \\delta _{n,m-1} e_i,\\quad n \\in \\mathbb {Z},\\quad e_i\\ i\\text{-th unit vector.", "}$ With Green's function, cf.", "[24], the unique bounded solution $w_\\mathbb {Z}^i$ of (REF ) has the explicit form $w_{n}^i = G(n,m)e_i,\\ n \\in \\mathbb {Z}, \\quad \\text{where} \\quad G(n,m) ={\\left\\lbrace \\begin{array}{ll}\\Phi (n,m) P_m^s,\\quad & n \\ge m,\\\\-\\Phi (n,m) P_m^u,\\quad & n < m,\\end{array}\\right.", "}$ and consequently $P^s_m =\\begin{pmatrix}| & & | \\\\w^1_m & \\cdots & w^d_m \\\\| & & |\\end{pmatrix}.$ Numerically, we approximate the unique bounded solution on $\\mathbb {Z}$ by the least squares solution of (REF ) on a sufficiently long interval.", "For an error analysis of this approximation process, we refer to [19].", "The algorithms that we propose in this section compute a vector $w \\in W_0^j$ in analogy to $W_j(x)$ in the previous sections.", "For simplicity, we restrict the representation to the case $j=2$ and assume that $W_n^1$ and $W_n^2$ are one-dimensional subspaces.", "In the absence of information about the dichotomy intervals, one may proceed as follows.", "Given a finite sequence of matrices, one can estimate a point in the spectral interval ${[\\lambda _q^-,\\lambda _q^+]}, q=1,2,3$ by computing the (logarithmic) growth rates of one-, two-, and three-dimensional subspaces using direct multiplication; these growth rates should approximate $\\lambda _1, \\lambda _1+\\lambda _2,$ and $\\lambda _1+\\lambda _2+\\lambda _3$ , respectively.", "By taking differences to obtain estimates ${\\hat{\\lambda }_q}, q=1,2,3$ , (the caret indicating estimated quantities) one should obtain values in the interior of ${[\\lambda _q^-,\\lambda _q^+]}, q=1,2,3$ .", "We then estimate ${\\Lambda ^\\text{left}=\\hat{\\lambda }_2-(\\hat{\\lambda }_2-\\hat{\\lambda }_3)/10\\lessapprox \\lambda _2^-}$ and ${\\Lambda ^\\text{right}=\\hat{\\lambda }_2-(\\hat{\\lambda }_2-\\hat{\\lambda }_1)/10\\gtrapprox \\lambda _2^+}$ .", "In the first step of our first algorithm, we compute a basis of the two-dimensional space $\\mathcal {R}(P_0^{3,u})$ .", "Then, in the second step, we search for the direction $w$ in this subspace that additionally lies in $\\mathcal {R}(P_0^{2,s})$ and assure in this way that $w \\in \\mathcal {R}(P_0^{2,s}) \\cap \\mathcal {R}(P_0^{3,u}) = W_0^2$ .", "Algorithm 3.1 (A Dichotomy Projector approach to estimate $W_2(x)$ by computing $W_0^2$ )     Suppose $N \\in \\mathbb {N}$ and consider $n\\in [-N,N]\\cap \\mathbb {Z}$ .", "Let $A_n = A(T^nx)$ .", "Solve the least squares problem $\\tilde{w}^i_{n+1} &= {e^{-\\Lambda ^\\text{left}}} A_n \\tilde{w}_n^i + \\delta _{n,-1}r^i, \\quad n = -N,\\ldots ,N-1, \\ i=1,2 \\\\& \\text{such that} \\ \\Vert (\\tilde{w}^i_{-N},\\ldots ,\\tilde{w}^i_N)\\Vert _2 \\ \\text{is minimised,}$ where the $r^i$ are chosen at random and $\\left\\Vert \\cdot \\right\\Vert _2$ is the $\\ell ^2$ -norm.", "Define $p^i: = A_{-1}\\tilde{w}^{i}_{-1}$ , $i = 1,2$ .", "Solve for $\\tilde{w}_{[0,N]}$ and $\\kappa $ the least squares problem $\\tilde{w}_{n+1} &= {e^{-\\Lambda ^\\text{right}}} A_n \\tilde{w}_n,\\quad n =0,\\dots ,N-1,\\\\\\tilde{w}_0 + {\\kappa } p^1 + p^2 &= 0,\\\\& \\text{such that} \\ \\left\\Vert (\\tilde{w}_0,\\ldots , \\tilde{w}_{N},{\\kappa }) \\right\\Vert _2 \\ \\text{is minimised}.$ Then $\\tilde{w}_0$ is our approximation of $w_2(x) \\in W_2(x)$ .", "The unique bounded solutions on $\\mathbb {Z}$ of these two steps satisfy $p^1,p^2 \\in \\mathcal {R}(P_0^{3,u})$ and these vectors are generically linear independent.", "Furthermore $\\tilde{w}_0 \\in \\mathcal {R}(P_0^{2,s})$ due to (REF ) and $\\tilde{w}_0\\in \\mathcal {R}(P_0^{3,u})$ due to ().", "Thus $\\tilde{w}_0 \\in \\mathcal {R}(P_0^{3,u}) \\cap \\mathcal {R}(P_0^{2,s}) = W_0^2$ .", "Note that (REF ) has the form $B \\tilde{w} = r, \\quad \\text{with } B \\in M_{2dN,d(2N+1)}(\\mathbb {R}), \\ r \\in \\mathbb {R}^{2dN},$ where $B=\\begin{pmatrix}-{e^{-\\Lambda ^\\text{left}}} A_{-N} & I \\\\& \\ddots & \\ddots \\\\&& -{e^{-\\Lambda ^\\text{left}}}A_{N} & I\\\\\\end{pmatrix},\\ \\tilde{w} = \\begin{pmatrix} \\tilde{w}_{-N} \\\\ \\vdots \\\\ \\tilde{w}_{N-1}\\end{pmatrix},$ and the $n$ th entry of $r$ is the vector $\\delta _{n,-1}r^i \\in \\mathbb {R}^d$ for $i=1,2$ .", "The least squares solution can be obtained, using the Moore-Penrose inverse: $\\tilde{w} = B^+ r,\\quad \\text{where } B^+ = B^T(BB^T)^{-1},$ cf.", "[30].", "Numerically, we find $\\tilde{w}$ by solving the linear system $BB^T y = r$ ; then $\\tilde{w} = B^T y$ .", "Figure: NO_CAPTIONNote that in the unlikely case where $p^1 \\in W^2_0$ , Algorithm REF fails.", "An alternative approach for computing vectors in $W_0^2$ that avoids this problem is introduced in Algorithm REF .", "The main idea of this algorithm is to take a random vector $r$ , project it first to $\\mathcal {R}(P_0^{3,u})$ and then eliminate components in the wrong subspaces, by projecting with $P_0^{2,s}$ .", "Algorithm 3.2 (An alternate Dichotomy Projector approach)     Again, suppose $N\\in \\mathbb {N}$ and consider $n\\in [-N,N]\\cap \\mathbb {Z}$ and let $A_n = A(T^nx)$ as above.", "Solve the least squares problem $\\tilde{w}_{n+1} &= {e^{-\\Lambda ^\\text{left}}}A_n \\tilde{w}_n +\\delta _{n,-1} r,\\quad n =-N,\\dots ,N-1,\\\\&\\text{such that} \\ \\Vert (\\tilde{w}_{-N},\\ldots ,\\tilde{w}_N)\\Vert _2 \\ \\text{is minimised},$ where $r$ is chosen at random, and define $r^{\\prime } = A_{-1} \\tilde{w}_{-1}$ .", "Solve the least squares problem $\\tilde{w}^{\\prime }_{n+1} &= {e^{-\\Lambda ^\\text{right}}} A_n \\tilde{w}^{\\prime }_n + \\delta _{n,-1} r^{\\prime },\\quad n =-N,\\dots ,N-1,\\\\&\\text{such that} \\ \\Vert (\\tilde{w}^{\\prime }_{-N},\\ldots ,\\tilde{w}^{\\prime }_N)\\Vert _2 \\ \\text{is minimised}.$ Then $\\tilde{w}^{\\prime }_0$ is our approximation of $w_2(x) \\in W_2(x)$ .", "The solution $w_0$ on $\\mathbb {Z}$ of these two steps satisfies $w_0 = P_0^{2,s} P_0^{3,u} r \\in \\mathcal {R}(P_0^{2,s}) \\cap \\mathcal {R}(P_0^{3,u}) = W^2_0$ .", "Figure: NO_CAPTION" ], [ "Error estimate", "We give an error estimate for the solution of Algorithm REF for a finite choice of $N$ .", "Details on deriving this estimate are postponed to a forthcoming publication.", "For ${\\Lambda ^\\text{left}}$ and ${\\Lambda ^\\text{right}}$ close to the boundary of the second Sacker-Sell spectral interval, we denote the dichotomy rates of $w_{n+1} = {e^{-\\Lambda ^\\text{left}}} A_n w_n,\\quad w_{n+1} = {e^{-\\Lambda ^\\text{right}}} A_n w_n,\\quad n \\in \\mathbb {Z}$ by $(\\alpha ^{\\ell ,s}, \\alpha ^{\\ell ,u})$ and $(\\alpha ^{r,s}, \\alpha ^{r,u})$ , respectively.", "Let $w_0$ be the solution of Algorithm REF on $\\mathbb {Z}$ and let $\\tilde{w}_0$ be its approximation for a finite choice of $N$ .", "Careful estimates show that the approximation error in the “wrong subspace\" $\\mathcal {R}(Q)$ , with $Q :=I-P_0^{2,s} P_0^{3,u}$ is given as $\\Vert Q(w_0 - \\tilde{w}_0)\\Vert \\le C N (e^{-\\alpha ^{\\ell ,s} N} + e^{-\\alpha ^{r,u}N}),$ where the constant $C>0$ does not depend on $N$ .", "The exponential dichotomy rates $\\alpha ^{\\ell ,s}$ and $\\alpha ^{r,u}$ of the difference equations (REF ) depend on the choice of $\\Lambda ^\\text{left}$ and $\\Lambda ^\\text{right}$ in the following way: for $\\Lambda ^\\text{left}$ in the resolvent set $R_3=[\\lambda _3^+,\\lambda _2^-]$ the difference equation $ w_{n+1} = e^{-\\Lambda ^\\text{left}} A_n w_n, \\quad n\\in \\mathbb {Z}$ has an exponential dichotomy with stable dichotomy rate $\\alpha ^{\\ell ,s}$ for all $\\alpha ^{\\ell ,s}$ with $0 < \\alpha ^{\\ell ,s} < \\Lambda ^\\text{left} - \\lambda ^+_3.", "$ Similarly, for $\\Lambda ^\\text{right}$ in the resolvent set $R_2 = [\\lambda _2^+,\\lambda _1^-]$ the difference equation $ w_{n+1} = e^{-\\Lambda ^\\text{right}}A_n w_n, \\quad n \\in \\mathbb {Z}$ has an exponential dichotomy with unstable dichotomy rate $\\alpha ^{r,u}$ for all $\\alpha ^{r,u}$ with $ 0 < \\alpha ^{r,u} < \\lambda _1^- - \\Lambda ^\\text{right}.", "$ Note that both of the above inequalities are strict.", "Inspecting equation (REF ), we get the best (smallest) maximal error if we choose $\\Lambda ^\\text{left} \\in R_3$ and $\\Lambda ^\\text{right}\\in R^2$ so as to maximise $\\alpha ^{\\ell ,s}$ and $\\alpha ^{r,u}$ .", "Consequently, we get the best numerical approximations, if ${\\Lambda ^\\text{left}}$ and ${\\Lambda ^\\text{right}}$ are chosen close to, but not equal to, the boundary of the common spectral interval ${[\\lambda _2^-,\\lambda _2^+]}$ ." ], [ "The Ginelli scheme", "The Ginelli Scheme was first presented by Ginelli et al.", "in [17] as a method for accurately computing the covariant Lyapunov vectors of an orbit of an invertible differentiable dynamical system where the $A(x)=DT(x)$ are the Jacobian matrices of the flow or map.", "Estimates of the $W_j(x)$ are found by constructing equivariant subspaces $S_j(x)=W_1(x)\\oplus \\cdots \\oplus W_j(x)$ and filtering the invariant directions contained therein using a power method on the inverse system restricted to the subspaces $S_j(x)$ .", "To construct the subspaces $S_j(x)$ we utilise the notion of the stationary Lyapunov basis [11].", "Choose $j$ orthonormal vectors $s_1(T^{-n}x), s_2(T^{-n}x),\\ldots , s_j(T^{-n}x)$ , $n\\ge 1$ , such that $s_i(T^{-n}x) \\notin V_{j+1}(T^{-n}x)$ for $1\\le i\\le j$ and construct $\\tilde{s}^{(n)}_i(x) = A(T^{-n}x,n)s_i(T^{-n}x), \\quad i=1,\\ldots ,j.$ Using the Gram-Schmidt procedure, construct the orthonormal basis $\\lbrace s_1^{(n)}(x),\\ldots , s_j^{(n)}(x)\\rbrace $ from $\\lbrace \\tilde{s}_1^{(n)}(x), \\ldots , \\tilde{s}_j^{(n)}(x)\\rbrace $ , that is, $s_1^{(n)}(x) & = \\frac{1}{\\left\\Vert \\tilde{s}_1^{(n)}(x) \\right\\Vert }\\tilde{s}_1^{(n)}(x), \\\\s_2^{(n)}(x) & = \\frac{1}{\\left\\Vert \\left(\\tilde{s}_2^{(n)}(x) - \\left(\\tilde{s}_2^{(n)}(x)\\cdot s_1^{(n)}(x)\\right)s_1^{(n)}(x)\\right) \\right\\Vert }\\left(\\tilde{s}_2^{(n)}(x) - \\left(\\tilde{s}_2^{(n)}(x)\\cdot s_1^{(n)}(x)\\right)s_1^{(n)}(x) \\right), \\\\& \\vdots $ Then as $n\\rightarrow \\infty $ the basis $\\lbrace s_1^{(n)}(x) , \\ldots , s_j^{(n)}(x)\\rbrace $ converges to a set of orthonormal vectors $\\lbrace s_1^{(\\infty )}(x),\\ldots ,s_j^{(\\infty )}(x)\\rbrace $ which span the $j$ fastest expanding directions of the cocycle $A$ [11], that is, if the multiplicities $m_1 = \\cdots = m_j = 1$ $S_j(x):=\\operatorname{span}\\left\\lbrace s^{(\\infty )}_1(x),\\ldots ,s^{(\\infty )}_j(x)\\right\\rbrace & = W_1(x)\\oplus \\cdots \\oplus W_j(x) \\\\& = V_1(x) \\backslash V_{j+1}(x).", "$ If the Oseledets subspaces are not all one-dimensional, that is the Lyapunov spectrum is degenerate, then we choose $S_j(x)$ only for those $j$ which are the sum of the first $k$ multiplicities, i.e., $j= m_1 + \\cdots +m_k$ .", "Then $S_j(x) & = W_1(x) \\oplus \\cdots \\oplus W_k(x) \\\\& = V_1(x)\\backslash V_{k+1}(x).", "$ In the interest of readability we assume the Oseledets subspaces are one-dimensional but note that the approach may be extended to the multi-dimensional case.", "Note that the $S_j(x)$ are equivariant by construction: $A(x,n) S_j(x) = S_j(T^nx) $ provided $j \\le m_1 + \\cdots + m_{\\ell -1}$ if $\\lambda _\\ell = -\\infty $ .", "We describe the Ginelli approach to finding $W_2(x)$ .", "Suppose $\\dim W_1(x) = \\dim W_2(x) = 1$ and $\\lambda _1 > \\lambda _2 > -\\infty $ and that the basis $\\lbrace s_1^{(\\infty )}(x),s_2^{(\\infty )}(x)\\rbrace $ is known at $x\\in X$ .", "Note first that $\\operatorname{span}\\left\\lbrace s_1^{(\\infty )}(x)\\right\\rbrace = W_1(x)$ .", "Let $c(x) \\in \\mathbb {R}^2$ denote the coefficients of $w_2(x)\\in W_2(x)$ in the basis $\\lbrace s_1^{(\\infty )}(x), s_2^{(\\infty )}(x)\\rbrace $ (recall that the orthogonal projection of $w_2(x)$ onto $s_i^{(\\infty )}(x)$ is zero for $i=3,4,\\ldots ,d$ ) then $w_2(x) = c_1(x)s_1^{(\\infty )}(x) + c_2(x) s_2^{(\\infty )}(x).", "$ Lemma 4.1 Let $Q(x)$ denote the $d\\times 2$ matrix whose $i$ th column is $s_i^{(\\infty )}(x)$ .", "Then for each $n \\ge 0$ there exists an upper triangular, $2 \\times 2$ matrix $R(x,n)$ satisfying $A(x,n)Q(x) = Q(T^nx)R(x,n).", "$ Note that $A(x,n)Q(x) & = A(x,n) \\begin{pmatrix} | & | \\\\ s_1^{(\\infty )}(x) & s_2^{(\\infty )}(x) \\\\ | & | \\end{pmatrix} \\\\& = \\begin{pmatrix} | & | \\\\ A(x,n)s_1^{(\\infty )}(x) & A(x,n)s_2^{(\\infty )}(x) \\\\ | & | \\end{pmatrix} \\\\& = Q(T^nx) R(x,n) $ where $Q(T^nx) & =\\begin{pmatrix}| & | \\\\s_1^{(\\infty )}(T^nx) & s_2^{(\\infty )}(T^nx) \\\\| & |\\end{pmatrix} $ and $R(x,n) & =\\begin{pmatrix}\\left\\Vert A(x,n)s_1^{(\\infty )}(x) \\right\\Vert & \\left\\langle s_1^{(\\infty )}(T^nx),A(x,n)s_2^{(\\infty )}(x)\\right\\rangle \\\\0 & \\left\\Vert A(x,n)s_2^{(\\infty )}(x) \\right\\Vert \\end{pmatrix}, $ using the equivariance of $S_1(x) = \\operatorname{span}\\lbrace s_1^{(\\infty )}(x)\\rbrace $ and $S_2(x) = \\operatorname{span}\\lbrace s_1^{(\\infty )}(x),s_2^{(\\infty )}(x)\\rbrace $ .", "Thus, the QR-decomposition of Lemma REF is equivalent to the Gram-Schmidt orthonormalisation that defines the stationary Lyapunov bases.", "The columns of $Q(T^nx)$ form the stationary Lyapunov basis at $T^nx$ .", "We have chosen the above notation $R(x,n)$ specifically since, defined in this way, $R$ forms a cocycle which is the restriction of $A$ to the invariant subspaces $S_j$ .", "Lemma 4.2 The matrix $R(x,n)$ defined above forms a cocycle over $T$ .", "Let $n,m \\ge 0$ then $A(x,n+m)Q(x) & = Q(T^{n+m}x)R(x,n+m) $ by Lemma REF .", "Since $A(x,n+m) = A(T^nx,m)A(x,n)$ , $A(x,n+m)Q(x) & = A(T^nx,m)A(x,n)Q(x) \\\\& = A(T^nx,m)Q(T^nx)R(x,n) \\\\& = Q(T^{n+m}x)R(T^nx,m)R(x,n).", "$ Equating (REF ) and (REF ) gives $R(T^nx,m)R(x,n) = R(x,n+m), $ as $Q(T^{n+m}x)$ is left-invertible.", "Since $c(x)$ is the vector of coefficients of the second Oseledets vector of the cocycle $A$ , it is the second Oseledets vector of the cocycle $R$ .", "To see this, recall $w_2(x) = Q(x)c(x) \\in W_2(x)$ so that $\\lambda _2 = \\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\left\\Vert A(x,n)Q(x)c(x) \\right\\Vert $ which, due to (REF ), becomes $\\lambda _2 &= \\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\left\\Vert Q(T^nx)R(x,n)c(x) \\right\\Vert \\\\& = \\lim _{n\\rightarrow \\infty } \\frac{1}{n}\\log \\left\\Vert R(x,n)c(x) \\right\\Vert $ since the columns of $Q(T^nx)$ are orthonormal.", "We may approximate $c(x)$ numerically using a simple power method on the inverse cocycle $R^{-1}$ (which exists since $\\lambda _1 > \\lambda _2 > -\\infty $ ).", "The Ginelli method can be summarised by the following steps: Algorithm 4.3 (Ginelli method of approximating $w_2(x)\\in W_2(x)$ )     Choose $x\\in X$ and $M>0$ and form $\\lbrace s_1^{(M)}(x),s_2^{(M)}(x)\\rbrace $ by first randomly selecting two orthonormal vectors $\\lbrace s_1(T^{-M}x),s_2(T^{-M}x)\\rbrace $ then performing the push-forward/Gram-Schmidt procedure given by (REF ).", "That is, define $ \\tilde{s}_i^{(M)}(x) = A(T^{-M}x,M)s_i(T^{-M}x), \\quad i = 1,2,$ followed by setting $s_1^{(M)}(x) &= \\mathcal {N}\\left(\\tilde{s}_1^{(M)}(x)\\right), \\\\s_2^{(M)}(x) & = \\mathcal {N} \\left(\\tilde{s}_2^{(M)}(x) - \\left(\\tilde{s}_2^{(M)}(x)\\cdot s_1^{(M)}(x)\\right)s_1^{(M)}(x)\\right),$ where $\\mathcal {N}:v\\mapsto v/\\left\\Vert v \\right\\Vert $ .", "The vectors $\\lbrace s_1^{(M)}(x),s_2^{(M)}(x)\\rbrace $ form an approximation to the stationary Lyapunov basis $\\lbrace s_1^{(\\infty )}(x),s_2^{(\\infty )}(x)\\rbrace $ .", "Choose $N>0$ and using the approximate basis $\\lbrace s_1^{(M)}(x),s_2^{(M)}(x)\\rbrace $ in (REF ), form an approximation to $R(x,N)$ , denoted by $R^{(M)}(x,N)$ .", "Choose $c^{\\prime }\\in \\mathbb {R}^2$ either at random, or by some guess at the second Oseledets vector of $R$ at $T^N(x)\\in X$ , in this review we found $c^{\\prime }=(0,1)$ to work well.", "Use the inverse iteration method to approximate $c(x)$ , that is, define our approximation to $c(x)$ as $c^{(M,N)}(x)& = R^{(M)}(x,N)^{-1}c^{\\prime } \\\\& = R^{(M)}(T^Nx,-N)c^{\\prime } $ Then $w_2^{(M,N)}(x)=\\begin{pmatrix}| & |\\\\s_1^{(M)}(x) & s_2^{(M)}(x)\\\\| & |\\end{pmatrix}c^{(M,N)}(x)$ is our approximation to $w_2(x)\\in W_2(x)$ .", "As before, there is some freedom of choice of both $M$ and $N$ as well as of the initial orthonormal basis $\\lbrace s_1(T^{-M}x),s_2(T^{-M}x)\\rbrace $ , used to approximate $S_2(x)$ , and of the 2-tuple $c^{\\prime }$ .", "The larger $M$ and $N$ are chosen, the more accurate $w^{(M,N)}_2(x)$ will be, provided $s_2(T^{-M}x) \\notin W_1(T^{-M}x)\\cup V_3(T^{-M}x)$ and $c^{\\prime } \\notin E_1(T^Mx)$ where $E_1$ is the Oseledets subspace of $R$ with Lyapunov exponent $\\lambda _1$ .", "Figure: NO_CAPTIONListing shows an example implementation of Algorithm REF in MATLAB which approximates $w_j(x) \\in W_j(x)$ using $M = N$ and $s_1(T^{-M}x),\\ldots ,s_j(T^{-M}x)$ are chosen at random and $c^{\\prime }=(\\underbrace{0,\\ldots ,0}_{j-1 \\text{ entries}},1)$ .", "Lines 2 through 6 construct the approximation of the stationary Lyapunov basis, $\\left\\lbrace s_1^{(M)}(x),\\ldots ,s_j^{(M)}(x)\\right\\rbrace $ which is stored as columns of the matrix $Q(x)$ represented as the variable Q0, while lines 7 through 11 construct the cocycle $R$ stored in AllR as $[ R(x) \\ | \\ R(Tx) \\ | \\cdots | \\ R(T^Nx)]$ .", "Lines 12 through 17 perform a simple power method on $R(x,N)^{-1}$ to find the coefficient vector $c$ , which represents the approximation of $w_j(x)$ in the basis $\\left\\lbrace s_1^{(M)}(x),\\ldots ,s_j^{(M)}(x)\\right\\rbrace $ .", "Thus, the approximation is given by $Q(x)c$ .", "Although Algorithm REF is specific to the case where $j=2$ , Listing is applicable to any $j$ for which $R(x,N)^{-1}$ exists.", "It can be shown that, in this case where the top Lyapunov exponent has multiplicity 1, $E_1 = \\operatorname{span}\\lbrace (1,0)^T\\rbrace $ .", "Lemma 4.4 If the first Lyapunov exponent has multiplicity 1, the dominant Oseledets subspace of the cocycle $R$ is $E_1 = \\operatorname{span}\\lbrace (1,0)^T\\rbrace $ .", "Recall that $s_1^{(\\infty )}(x) \\in W_1(x)$ since $\\operatorname{span}\\left\\lbrace s_1^{(\\infty )}(x)\\right\\rbrace = S_1(x) = V_1(x) \\backslash V_2(x)$ (from (REF )) and for all $s \\in V_1(x)\\backslash V_2(x)$ $\\lambda _1 & = \\lim _{n\\rightarrow \\infty } \\frac{1}{n} \\log \\left\\Vert A(x,n)s \\right\\Vert .", "$ We may write $s = Q(x)(a,0)^T$ for some $a\\in \\mathbb {R}$ then $\\lambda _1 & = \\lim _{n\\rightarrow \\infty }\\frac{1}{n} \\log \\left\\Vert A(x,n)Q(x)(a,0)^T \\right\\Vert \\\\& = \\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\left\\Vert Q(T^nx)R(x,n)(a,0)^T \\right\\Vert $ and since $\\left\\Vert Q(T^nx)s \\right\\Vert = \\left\\Vert s \\right\\Vert $ (the columns of $Q$ are orthonormal) $\\lambda _1 & = \\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\left\\Vert R(x,n)(a,0)^T \\right\\Vert .", "$ Since $S_1$ is $A$ -invariant, $\\operatorname{span}\\lbrace (1,0)^T\\rbrace $ is $R$ -invariant and the proof is complete." ], [ "Limited Data Scenario", "In the case where convergence isn't satisfactory because the amount of cocycle data available is too small (for any $M$ and $N$ to be small), the approximations from Algorithm REF can be improved by using better guesses at $s_1(T^{-M}x),s_2(T^{-M}x)$ and $c^{\\prime }$ .", "Note that $s_1^{(\\infty )}(x)$ and $s_2^{(\\infty )}(x)$ are two orthonormal vectors optimised for maximal growth over the time interval $[-\\infty ,0]\\cap \\mathbb {Z}$ .", "In the situation where the values of $M$ and $N$ are limited, one can choose those two vectors that are optimised for growth over the shorter time interval $[-M,0]\\cap \\mathbb {Z}$ .", "In [34] this is achieved by computing the left singular vectors of $A(T^{-M}x,M)$ .", "This approach works well for small $M$ but can become inaccurate for very large $M$ for the reasons in Remark REF .", "In practice, we have observed that a combination of Step 1 in Algorithm REF and the above provides the most robust method of accurately approximating $s_1^{(\\infty )}(x)$ and $s_2^{(\\infty )}(x)$ .", "As an alternative to Algorithm REF the following may be used: For $M\\ge M^{\\prime }$ , compute vectors optimised for growth from $-M$ to $-M + M^{\\prime }$ , then push-forward these vectors from $-M+M^{\\prime }$ to 0.", "Algorithm 4.5 ( Improved Algorithm REF )     Choose $x\\in X$ and $M \\ge M^{\\prime } > 0$ .", "Compute the two left singular vectors of $A(T^{-M}x,M^{\\prime })$ corresponding to the two largest singular values and call them $\\tilde{s}_1(T^{-M+M^{\\prime }}x)$ and $\\tilde{s}_2(T^{-M+M^{\\prime }}x)$ .", "Now define $\\left\\lbrace s_1^{(M,M^{\\prime })}(x),s_2^{(M,M^{\\prime })}(x)\\right\\rbrace $ as an approximation to $\\left\\lbrace s_1^{(\\infty )}(x),s_2^{(\\infty )}(x)\\right\\rbrace $ by the Gram-Schmidt orthonormalisation of $A(T^{-M+M^{\\prime }}x,M-M^{\\prime })s_1(T^{-M+M^{\\prime }}x)$ and $A(T^{-M+M^{\\prime }}x,M-M^{\\prime })s_2(T^{-M+M^{\\prime }}x)$ as in (REF ).", "Steps 2–4 as in Algorithm REF .", "In practice, one should choose $M^{\\prime }$ large enough so that enough data is sampled, but not so large that $A(T^{-M}x,M^{\\prime })$ is too singular." ], [ "The Wolfe scheme", "The approach followed by Wolfe et al.", "[34] directly computes the subspace splitting as the intersection of two sets of invariant subspaces.", "The description of the numerical construction of the subspaces $S_j(x)$ featured below differs slightly from [34], however, the essential features of the approach are retained.", "In fact, the constructions featured here improve upon those in [34] in terms of accuracy versus amount of cocycle data used – in the notation of Algorithm REF below, if $M_1$ is made larger, $w_2^{(M_1,M_1^{\\prime },M_2)}(x)$ is more accurate, which is not the case in [34] for the same reasons discussed in Remark REF .", "Recall the eigenspace decomposition $U_j(x)$ of the limiting matrix $\\Psi (x)$ presented in Section and define $V_j(x) = U_j(x)\\oplus \\cdots \\oplus U_\\ell (x)$ .", "Recall that $V_j(x) \\supset W_j(x),W_{j+1}(x),\\ldots ,W_\\ell (x)$ .", "Also recall from the previous section that $S_j(x) \\supset W_1(x),W_2(x),\\ldots , W_j(x)$ .", "Thus $W_j(x) = V_j(x) \\cap S_j(x).$ Again, in the interest of readability we assume the Oseledets subspaces $W_j(x)$ , and the eigenspaces $U_j(x)$ , are one-dimensional.", "As in the case of the previous section, the ideas here may be extended to the case in which the Oseledets subspaces are not one-dimensional.", "Let $u_j(x)$ be the singular vector spanning $U_j(x)$ and let $s_j(x)=s^{(\\infty )}_j(x)$ be the $j$ th element of the stationary Lyapunov basis as in the previous section.", "Then note $w_j(x) & = \\sum _{i=1}^j \\left\\langle w_j(x),s_i(x)\\right\\rangle s_i(x), \\\\w_j(x) & = \\sum _{i=j}^d \\left\\langle w_j(x),u_i(x)\\right\\rangle u_i(x).", "$ Taking inner products with $u_k(x)$ and $s_k(x)$ respectively gives $\\left\\langle w_j(x),u_k(x)\\right\\rangle & = \\sum _{i=1}^j\\left\\langle w_j(x),s_i(x)\\right\\rangle \\left\\langle s_i(x),u_k(x)\\right\\rangle \\text{for $k\\ge j$}, \\\\\\left\\langle w_j(x),s_k(x)\\right\\rangle & = \\sum _{i=j}^d\\left\\langle w_j(x),u_i(x)\\right\\rangle \\left\\langle u_i(x),s_k(x)\\right\\rangle \\text{for $k\\le j$}.", "$ Substituting (REF ) into () and rearranging gives $\\left\\langle w_j(x),s_k(x)\\right\\rangle & = \\sum _{i=1}^j \\left(\\sum _{h=j}^d \\left\\langle s_k(x),u_h(x)\\right\\rangle \\left\\langle u_h(x),s_i(x)\\right\\rangle \\right)\\left\\langle s_i(x),w_j(x)\\right\\rangle .", "$ Note that $\\sum _{h=1}^d\\left\\langle s_k(x),u_h(x)\\right\\rangle \\left\\langle u_h(x),s_i(x)\\right\\rangle = \\delta _{ki}$ so $\\sum _{h=j}^d \\left\\langle s_k(x),u_h(x)\\right\\rangle \\left\\langle u_h(x),s_i(x)\\right\\rangle = \\delta _{ki} - \\sum _{h=1}^{j-1}\\left\\langle s_k(x),u_h(x)\\right\\rangle \\left\\langle u_h(x),s_i(x)\\right\\rangle .", "$ Then (REF ) becomes $\\left\\langle w_j(x),s_k(x)\\right\\rangle & = \\sum _{i=1}^j \\delta _{ki}\\left\\langle s_i(x),w_j(x)\\right\\rangle - \\sum _{i=1}^j \\sum _{h=1}^{j-1}\\left\\langle s_k(x),u_h(x)\\right\\rangle \\left\\langle u_h(x),s_i(x)\\right\\rangle \\left\\langle s_i(x),w_j(x)\\right\\rangle \\\\& = \\left\\langle s_k(x),w_j(x)\\right\\rangle - \\sum _{i=1}^j \\sum _{h=1}^{j-1}\\left\\langle s_k(x),u_h(x)\\right\\rangle \\left\\langle u_h(x),s_i(x)\\right\\rangle \\left\\langle s_i(x),w_j(x)\\right\\rangle \\\\0 & = \\sum _{i=1}^j \\sum _{h=1}^{j-1}\\left\\langle s_k(x),u_h(x)\\right\\rangle \\left\\langle u_h(x),s_i(x)\\right\\rangle \\left\\langle s_i(x),w_j(x)\\right\\rangle .", "$ Equation (REF ) may be considered as a $j\\times j$ homogeneous linear equation by defining a matrix entry-wise as $D_{ki} = \\sum _{h=1}^{j-1}\\left\\langle s_k(x),u_h(x)\\right\\rangle \\left\\langle u_h(x),s_i(x)\\right\\rangle $ and solving $Dy = 0, $ where $y_i = \\left\\langle s_i(x),w_j(x)\\right\\rangle $ .", "The entries of $y$ are then the coefficients of $w_j(x)$ with respect to the basis $s_1(x)\\ldots ,s_j(x)$ .", "The Wolfe approach may be implemented as follows: Algorithm 4.6 (Improved Wolfe approach to approximating $w_j(x) \\in W_j(x)$ )     Choose $x\\in X$ and $M_1 \\ge M_1^{\\prime } > 0$ and construct $\\lbrace s_1^{(M_1,M_1^{\\prime })}(x),\\ldots ,s_j^{(M_1,M_1^{\\prime })}(x)\\rbrace $ as an approximation of the stationary Lyapunov basis vectors $\\lbrace s_1^{(\\infty )}(x),\\ldots ,s_j^{(\\infty )}(x)\\rbrace $ using the methods outlined in Step 1 of Algorithm REF , that is, compute the left singular vectors of $A(T^{-M_1}x,M_1^{\\prime })$ corresponding to the $j-1$ largest singular values and call them $\\tilde{s}_i(T^{-M_1+M_1^{\\prime }}x)$ , $i=1,\\ldots ,j-1$ .", "Then form the $s_i^{(M_1,M_1^{\\prime })}(x)$ by the Gram-Schmidt orthornormalisation of $A(T^{-M_1+M_1^{\\prime }}x,M_1-M_1^{\\prime })\\tilde{s}_i(T^{-M_1 + M_1^{\\prime }}x)$ for $i=1,\\ldots ,j-1$ .", "Choose $M_2>0$ and construct the one-dimensional eigenspaces $U_1^{(M_2)}(x),\\ldots ,U_{j-1}^{(M_2)}(x)$ as approximations to the eigenspaces $U_1(x),\\ldots , U_{j-1}(x)$ as in Step 1 of Algorithm REF , that is, construct $\\Psi ^{(M_2)}(x) = \\left(A(x,M_2)^* A(x,M_2)\\right)^{1/2M_2},$ and let $U^{(M_2)}_i(x)$ be the $i$ th orthonormal eigenspace of $\\Psi ^{(M_2)}(x)$ .", "Define $u_i^{(M_2)}(x) \\in U_i^{(M_2)}(x)$ , $i=1,\\ldots , j-1$ .", "Form the matrix $D$ as above: $D_{ki} = \\sum _{h=1}^{j-1}\\left\\langle s_k^{(M_1,M_1^{\\prime })}(x),u_h^{(M_2)}(x)\\right\\rangle \\left\\langle u_h^{(M_2)}(x),s_i^{(M_1,M_1^{\\prime })}(x)\\right\\rangle .", "$ Solve the homogeneous linear equation $Dy=0$ .", "Then $w_j^{(M_1,M_1^{\\prime },M_2)} = \\sum _{i=1}^j y_i s_i(x)$ forms our approximation of $w_j(x)\\in W_j(x)$ .", "This approach suffers from the same numerical stability issue of Algorithm REF of Section REF .", "Namely, the vector spaces $U_1^{(M)}(x),\\ldots ,U_{j-1}^{(M)}(x)$ may only poorly approximate $U_1(x),\\ldots ,U_{j-1}(x)$ for $M$ too large (see the final paragraph of Section REF ).", "A recent paper [20] provides alternative descriptions of both the Ginelli et al.", "and Wolfe and Samelson methods, well-suited to those familiar with the QR-decomposition based numerical method for estimating Lyapunov exponents due to Benettin et al.", "[3], [4] and Shimada and Nagashima [29].", "The discussion in [20] is restricted to invertible cocycles generated by the Jacobian matrices of a dynamical system.", "Although this assumption allows stable numerical methods to be constructed, i.e., better convergence obtained for larger data sets, it means some important examples in which the matrix cocycle is non-invertible are overlooked, for example the case study of Section REF .", "While the memory footprint of the implementations discussed in [20] is estimated, there is no discussion of convergence rates or accuracy with respect to the amount of cocycle data available.", "Finally, while the examples featured in [20] explain the methods presented in the context of differentiable dynamical systems the case studies of Section in the present paper focus on comparative performance of the methods presented, via a broad range of possible applications." ], [ "Numerical comparisons of the four approaches ", "We present three detailed case studies, comparing the four approaches for calculating Oseledets subspaces.", "The first case study is a nontrivial model for which we know the Oseledets subspaces exactly and can therefore precisely measure the accuracy of the methods.", "The second case study produces a relatively low-dimensional matrix cocycle, while the third case study generates a very high-dimensional matrix cocycle; in these case studies we use two fundamental properties of Oseledets subspaces to assess the accuracy of the four approaches." ], [ "Case Study 1: An exact model", "In general the Oseledets subspaces cannot be found analytically which makes the task of determining the efficacy of the above approaches difficult.", "However, the exact model described below allows us to compare the numerical approximations with the exact solution by building a cocycle in which the subspaces are known a priori.", "We generate a system with simple Lyapunov spectrum $\\lambda _1 > \\lambda _2 > \\cdots > \\lambda _d > -\\infty $ .", "We form a diagonal matrix $R ={\\begin{pmatrix}e^{\\lambda _1} & & &0 \\\\& e^{\\lambda _2} & & \\\\& & \\ddots & \\\\0 & & & e^{\\lambda _d} \\\\\\end{pmatrix}}$ and generate the cocycle by the sequence of matrices $\\lbrace A_n\\rbrace $ where $A_n & = S_n R S_{n-1}^{-1} \\\\S_n & ={\\left\\lbrace \\begin{array}{ll}I + \\epsilon Z, & \\text{ for $n\\ne -1, n\\in [-N,N]\\cap \\mathbb {Z}$,}\\\\I +\\begin{pmatrix}0 & & & & \\\\z_2 & 0 & & \\\\& \\ddots & \\ddots & \\\\& & z_d & 0\\end{pmatrix},& \\text{ for $n=-1$.}\\end{array}\\right.}", "$ The entries of $Z$ and the numbers $z_2,\\ldots , z_d$ are uniformly randomly generated from the interval $[0,1]$ .", "By construction, the columns of $S_{n-1}$ span the Oseledets subspaces at time $n\\in [-N,N]\\cap \\mathbb {Z}$ .", "We compare the exact result at time $n=0$ with the approximations computed by the various algorithms for $d=8$ , $\\lbrace \\lambda _1,\\ldots ,\\lambda _8\\rbrace = \\lbrace \\log 8, \\log 7, \\log 6,\\ldots ,\\log 1\\rbrace $ and $\\epsilon = 0.1$ for varying amounts of cocycle data $\\left\\lbrace A(T^{-N}x),\\ldots , A(T^Nx)\\right\\rbrace $ .", "The exact model has a well separated spectrum, is generated using invertible matrices, and is of relatively low dimension.", "For this model we use the following choice of parameters to execute the algorithms: Algorithm REF : $M = N$ and $\\lbrace N_k\\rbrace = \\lbrace 1,6,\\ldots ,5k-4,\\ldots ,5K-4,N\\rbrace $ where $5K-4 < N \\le 5K+1$ .", "Algorithm REF : We estimate the three largest Lyapunov exponents $\\lambda _1 > \\lambda _2 >\\lambda _3$ and set $\\Lambda ^\\text{right} = {\\lambda _2 + 0.1(\\lambda _1 - \\lambda _2)}$ and $\\Lambda ^\\text{left} = {\\lambda _2 - 0.1(\\lambda _2 - \\lambda _3})$ .", "Algorithm REF : As for Algorithm REF .", "Algorithm REF : $M = N$ , $M^{\\prime } = 5$ , and $c^{\\prime }= (0,1)$ .", "Algorithm REF : Let $M_1 = N$ , $M_1^{\\prime } = 5$ and $M_2 = N$ .", "Figure: Comparing the approximations of the second Oseledets subspace W 2 (N) (x)W^{(N)}_2(x) with the exact solution W 2 (x)W_2(x) which is known a priori.", "Each “NN-approximation” is computed using cocycle data {A(T -N x),A(T -N+1 x),...,A(T N x)}\\lbrace A(T^{-N}x), A(T^{-N+1}x), \\ldots , A(T^Nx)\\rbrace .", "The comparison is simply the Euclidean norm of the separation of the two unit vectors w 2 (N) (x)∈W 2 (N) (x)w_2^{(N)}(x)\\in W_2^{(N)}(x) and w 2 (x)∈W 2 (x)w_2(x) \\in W_2(x).Figure REF compares the approximations yielded from the various approaches outlined in Sections , and with the known solution of Equation (REF ).", "Each algorithm exhibits approximately exponential convergence with respect to the length of the sample cocycle up to (almost) machine accuracy of about $10^{-16}$ .", "Algorithm REF is notably erratic whereas the other algorithms converge smoothly, which suggests that in the limited data scenario (small $N$ ) it represents the less satisfactory choice.", "Algorithm REF is slightly more accurate than the other algorithms for large $N$ , while there is a limit to the accuracy of Algorithms REF and REF .", "It is worth noting that Algorithms REF and REF do not perform as well when approximating Oseledets subspaces corresponding to Lyapunov exponents $\\lambda _3, \\ldots , \\lambda _\\ell $ .", "Whilst they reach machine accuracy with ease for $W_1$ and $W_2$ , forming $A(x,n) = A(T^{n-1}x)\\cdots A(x)$ via numerical matrix multiplication produces greater inaccuracies for subspaces $W_3$ through $W_\\ell $ (see Remark REF ).", "On the other hand, Algorithms REF , REF and REF do not suffer from the same issue because they do not need to form $A(x,n)$ but use only the generating matrices $A(x)$ .", "As such, they still reach machine accuracy, although a greater amount of data (larger $N$ ) is required.", "Figure: Comparing the approximation of the second Oseledets vector with the exact solution for the two SVD based approaches, demonstrating the numerical instability which is overcome in the adapted SVD approach.Figure REF is similar to Figure REF except that it compares only Algorithms REF and REF .", "In doing so, it highlights the result of one of the numerical instabilities of Algorithm REF , namely the pushing forward of $U_j^{(M)}(T^{-N}x)$ in Step REF .", "Figure: Comparing the execution time τ\\tau of the various algorithms using MATLAB's timing functionality.", "Each algorithm is executed using the cocycle data A(T -N x),...,A(T N x)\\left\\lbrace A(T^{-N}x),\\ldots , A(T^Nx)\\right\\rbrace .Finally, Figure REF shows the execution times of Algorithms REF , REF , REF , REF and REF , which were timed using MATLAB's timing functionality.", "The most time-consuming step in Algorithm REF is the SVD performed as part of the alterations from Section REF .", "Algorithm REF must perform two SVDs and Algorithm REF must perform many more, which accounts for their longer execution times." ], [ "Case Study 2: Particle dynamics - two disks in a quasi-one-dimensional box ", "We consider the quasi-one-dimensional heat system studied extensively by Morriss et al.", "[5], [22], [27], [33], [31], [32] which consists of two disks of diameter $\\sigma =1$ in a rectangular box, $[0,L_x]\\times [0,L_y]$ , in which the shorter side, has length $L_y < 2\\sigma $ so that the disks may not change order.", "The two disks interact elastically with each other and the short walls, but periodic boundary conditions are enforced in the $y$ -direction.", "The phase space of the system is then the set $X\\subset \\mathbb {R}^8$ $X = \\left(\\left([0,L_x]\\times [0,L_y]\\right) / \\sim \\right)^2\\times \\mathbb {R}^2 \\times \\mathbb {R}^2 $ where $\\sim $ is the equivalence class associated with periodic boundary conditions, that is, $(x_1,y_1),(x_2,y_2)\\in [0,L_x]\\times [0,L_y]$ have $(x_1,y_1) \\sim (x_2,y_2)$ if $y_1 = y_2 \\mod {L}_y$ and $x_1=x_2$ .", "The flow $\\phi ^\\tau :X\\rightarrow X$ consists of free-flight maps of time $\\tau $ , $\\mathcal {F}^\\tau :X\\rightarrow X$ , and collision maps $\\mathcal {C}:X\\rightarrow X$ so that $\\phi ^\\tau (x) = \\mathcal {C} \\circ \\mathcal {F}^{\\tau _n}\\circ \\cdots \\circ \\mathcal {F}^{\\tau _2}\\circ \\mathcal {C} \\circ \\mathcal {F}^{\\tau _1}(x)$ where $\\tau _1 + \\cdots + \\tau _n = \\tau $ .", "We consider a discrete-time version of the system by mapping from the instant after collision to the instant after the next collision, that is $x\\mapsto \\mathcal {C} \\circ \\mathcal {F}^{\\tau (x)}(x)$ where $\\tau (x)$ is the free-flight time in the continuous system.", "The matrix cocycle is generated by the $8\\times 8$ Jacobian matrices or the derivative of the flow evaluated instantly after each collision (i.e.", "$A(x) = D\\left(\\mathcal {C} \\circ \\mathcal {F}^{\\tau (x)}\\right)(x)$ , see [5] for details).", "Due to a number of dynamic symmetries the system has Lyapunov exponents $\\lambda _1 > \\lambda _2 > 0 > -\\lambda _2 > -\\lambda _1$ with multiplicities $1,1,4,1$ and 1 respectively.", "This system has some symmetry, a high variation in expansion rates from iteration to iteration, a well separated spectrum, invertible Jacobian matrices, and relatively low dimension.", "Numerical integration of an orbit consisting of 4646 collisions yielded a sequence of Jacobian matrices $\\left\\lbrace A(T^{-2323}x),\\ldots ,A(T^{2322}x)\\right\\rbrace $ which generate the cocycle $A$ .", "For this model we use the same choice of parameters to execute the algorithms as with the previous model: Algorithm REF : $M = N$ and $\\lbrace N_k\\rbrace = \\lbrace 1,6,\\ldots ,5k-4,\\ldots ,5K-4,N\\rbrace $ where $5K-4 < N \\le 5K+1$ .", "Algorithm REF : We estimate the three largest Lyapunov exponents $\\lambda _1 > \\lambda _2 >\\lambda _3$ and set $\\Lambda ^\\text{right} = {\\lambda _2 + 0.1(\\lambda _1 - \\lambda _2)}$ and $\\Lambda ^\\text{left} = {\\lambda _2 - 0.1(\\lambda _2 - \\lambda _3})$ .", "Algorithm REF : As for Algorithm REF .", "Algorithm REF : $M = N$ , $M^{\\prime } = 5$ , and $c^{\\prime }= (0,1)$ .", "Algorithm REF : Let $M_1 = N$ , $M_1^{\\prime } = 5$ and $M_2 = N$ ." ], [ "Criteria to assess the accuracy of estimated Oseledets spaces", "Since the Oseledets subspaces for this model are unknown, we test the approximations for two properties of Oseledets subspaces, namely their equivariance and the expansion rate, which defines the corresponding Lyapunov exponent." ], [ "Equivariance:", "To test for equivariance, we approximate the second Oseledets vector, $w_2^{(N)}(T^nx)$ , at each time $n=0,1,\\ldots ,30$ .", "We then compute $\\left\\Vert \\mathcal {N}\\left(A(x,n)w_2^{(N)}(x)\\right) - w_2^{(N)}(T^nx) \\right\\Vert $ and plot the result, where $v \\smash{\\mathop {\\mapsto }\\limits ^{\\mathcal {N}}} v/\\left\\Vert v \\right\\Vert $ .", "If the approximations are equivariant this value would be zero." ], [ "Expansion Rate:", "To test the expansion rate, each approach is used to compute the second Oseledets vector, $w_2^{(N)}(x) \\in W_2^{(N)}(x)$ , at time $n=0$ and we plot $\\frac{1}{m}\\log \\left\\Vert A(x,m)w_2^{(N)}(x) \\right\\Vert $ versus $m$ .", "If $W^{(N)}_2(x)$ is accurate, elements of $W^{(N)}_2(x)$ should grow at the correct rate: $\\lambda _2$ .", "Whilst the Oseledets vector $w_2(x)$ must satisfy the above two properties, we must be careful when examining the results of these numerical experiments.", "For instance, (i) it is possible to choose vectors that are equivariant despite not being contained in any single Oseledets subspace, and (ii) any element of $V_2(x)\\backslash V_3(x) W_2(x)$ (a much larger set than $W_2(x)$ ) has Lyapunov exponent $\\lambda _2$ ." ], [ "Numerical Results", "Figure REF shows the results of the equivariance test for the quasi-one-dimensional two disk model.", "At the lower end of cocycle data length ($N=75$ ) all Algorithms except REF display reasonable equivariance, although Algorithm REF remains equivariant for only a handful of steps.", "For $N=150$ and $N=225$ all approaches appear to produce close to equivariant results (note the changing scales in the vertical direction), with Algorithms REF and REF lagging behind when $N=225$ .", "Figure: The equivariance test for the various algorithms on the quasi-one-dimensional two disk model.", "Each approach is used to approximate the second Oseledets vector, w 2 (N) (T n x)∈W 2 (N) (T n x)w_2^{(N)}(T^nx)\\in W_2^{(N)}(T^nx) using cocycle data {A(T -N x),...,A(T N x)}\\lbrace A(T^{-N}x),\\ldots , A(T^Nx)\\rbrace , at each time n=0,1,...,30n=0,1,\\ldots ,30.", "We then compute 𝒩A(x,n)w 2 (N) (x)-w 2 (N) (T n x)\\left\\Vert \\mathcal {N}A(x,n)w_2^{(N)}(x) - w_2^{(N)}(T^nx) \\right\\Vert and plot the result.", "Note the different scales on each vertical axis.", "The plots shown are for N=75N=75 (top left), N=150N = 150 (top right) and N=225N = 225 (bottom).Figure REF shows the results of the expansion rate test for the quasi-one-dimensional two disk model for various amounts of cocycle data $\\left\\lbrace A(T^{-N}x),\\ldots , A(T^{N}x)\\right\\rbrace $ .", "As expected, when there is a limited amount of data available ($N$ small) the approximations either expand at the higher rate of $\\lambda _1$ or only expand at the rate of $\\lambda _2$ for a brief time before the error grows too large.", "As $N$ is increased, the approximations expand at $\\lambda _2$ for longer periods, suggesting that they more accurately represent $w_2(x)$ .", "Figure: The expansion rate test for the various approaches on the quasi-one-dimensional two disk model.", "The second Oseledets vector, w 2 (N) (x)∈W 2 (N) (x)w_2^{(N)}(x)\\in W_2^{(N)}(x), is approximated using cocycle data {A(T -N x),...,A(T N x)}\\lbrace A(T^{-N}x),\\ldots ,A(T^Nx)\\rbrace and we plot 1 mlogA(x,m)w 2 (N) (x)\\frac{1}{m}\\log \\left\\Vert A(x,m)w_2^{(N)}(x) \\right\\Vert versus mm.", "If the approximation is accurate this quantity should tend to the value of λ 2 ≈0.210\\lambda _2 \\approx 0.210, otherwise it would tend to the value of λ 1 ≈0.325\\lambda _1 \\approx 0.325 both of which are shown in blue.", "The plots shown are for N=25N=25 (top left), N=75N=75 (top right) and N=150N=150 (bottom).Most algorithms perform similarly regarding expansion rate.", "Note that the amount of cocycle data (size of $N$ ) needed to perform well in the Expansion Rate test is less than that needed to perform well in the Equivariance test - this demonstrates the importance of good performance in both tests in order to assess whether or not the algorithms are performing well." ], [ "Case Study 3: Time-dependent fluid flow in a cylinder; a transfer operator description", "An important emerging application for Oseledets subspaces is the detection of strange eigenmodes, persistent patterns, and coherent sets for aperiodic time-dependent fluid flows.", "In the periodic setting strange eigenmodes have been found as eigenfunctions of a Perron-Frobenius operator via classical Floquet theory; [25], [21], [26].", "However, in the aperiodic time-dependent setting, Floquet theory cannot be applied.", "An extension to aperiodically driven flows was derived in [15], based on the new multiplicative ergodic theory of [14].", "Discrete approximations of a Perron-Frobenius cocycle representing the aperiodic flow are constructed and in this aperiodic setting the leading sub-dominant Oseledets subspaces play the role of the leading sub-dominant eigenfunctions in the periodic forcing case.", "We review the four methods of approximating Oseledets subspaces with the aperiodically driven cylinder flow from [15].", "The flow domain is $Y=[0,2\\pi ]\\times [0,\\pi ]$ , $t\\in \\mathbb {R}^+$ and the flow is defined by the following forced ODE: $\\begin{split}\\dot{x} &= c-\\tilde{A}(\\tilde{z}(t))\\sin (x-\\nu \\tilde{z}(t))\\cos (y)+\\varepsilon G(g(x,y,\\tilde{z}(t)))\\sin (\\tilde{z}(t)/2)\\qquad \\mod {2\\pi }\\\\\\dot{y} &= \\tilde{A}(\\tilde{z}(t))\\cos (x-\\nu \\tilde{z}(t))\\sin (y).\\\\\\end{split}$ Here, $\\tilde{z}(t)=6.6685z_1(t)$ , where $z_1(t)$ is generated by the standard Lorenz flow, $\\tilde{A}(\\tilde{z}(t))=1+0.125\\sin (\\sqrt{5}\\tilde{z}(t))$ , $G(\\psi ):=1/{(\\psi ^2+1)}^2$ and the parameter function $\\psi =g(x,y,\\tilde{z}(t)):=\\sin (x-\\nu \\tilde{z}(t))\\sin (y)+y/2-\\pi /4$ vanishes at the level set of the streamfunction of the unperturbed ($\\varepsilon =0$ ) flow at instantaneous time $t=0$ , i.e., $s(x,y,0)=\\pi /4$ , which divides the phase space in half.", "Figure: The second Oseledets subspace as determined by (a) Algorithm , (b) Algorithm , (c) Algorithm , (d) Algorithm and (e) Algorithm .We set $\\varepsilon =1$ as this value is sufficiently large to ensure no KAM tori remain in the jet regime, but sufficiently small to maintain islands originating from the nested periodic orbits around the elliptic points of the unperturbed system.", "We construct the discretised Perron-Frobenius matrices $P_x^{(\\tau )}(t)=:A(x)$ as described in Section 3 of [15], and briefly recapped in Example REF , using a uniform grid of $120\\times 60$ boxes, $\\tau =8$ and $-32\\le t \\le 32$ .", "In total, we generate 8 such matrices of dimension $7200\\times 7200$ .", "Thus, in this case study we have a limited amount of data, no symmetry, high dimension, and the matrices are non-invertible and sparse.", "In order to obtain reasonable results we executed Algorithms REF , REF , REF , REF and REF with the following parameters: Algorithm REF : $M=N=4$ and $\\lbrace N_k\\rbrace = \\lbrace 2,4\\rbrace $ .", "Algorithm REF : We estimate the three largest Lyapunov exponents $\\lambda _1 > \\lambda _2 >\\lambda _3$ and set $\\Lambda ^\\text{right} = {\\lambda _2 + 0.1(\\lambda _1 - \\lambda _2)}$ and $\\Lambda ^\\text{left} = {\\lambda _2 - 0.1(\\lambda _2 - \\lambda _3})$ .", "Algorithm REF : As for Algorithm REF .", "Algorithm REF : $M^{\\prime } = M = 4$ (so that only an SVD is used, and no push-forward step), $N=4$ and $c^{\\prime } = (0,1)$ .", "Algorithm REF : $M_1=M_1^{\\prime }=4$ and $M_2 = 4$ .", "Figure: Comparing the approximations of the second Oseledets vector w 2 (Tx)w_2(Tx) at time t=8t=8 with the push-forward of the approximations at time t=0t=0.", "Those labelled (a) are the push-forwards A(x,1)w 2 (4) (x)A(x,1)w_2^{(4)}(x) whilst those labelled (b) are independently computed approximations w 2 (4) (Tx)w_2^{(4)}(Tx) of w 2 (Tx)w_2(Tx).", "The algorithms used are as follows: (1) Algorithm , (2) Algorithm , (3) Algorithm , (4) Algorithm and (5) Algorithm .The results of these numerical experiments are shown in Figures REF and REF .", "Recall that in this setting, the cocycle $A(x,n)$ is a cocycle of discretised Perron-Frobenius operators acting on piecewise constant functions defined on $Y$ ; we identify these piecewise constant functions (with 7200 pieces) with vectors in $\\mathbb {R}^{7200}$ .", "Figure REF first shows the approximations of the second Oseledets vector $w_2(x)$ at time $t=0$ .", "In this setting the Oseledets vectors locate coherent structures: Figure REF compares the push-forward of the approximations in Figure REF with independently computed approximations of $w_2(Tx)$ - the second Oseledets vector at time $t=8$ .", "In this study the data sample is insufficiently long for Algorithm REF to work effectively, but the other algorithms produce similar results.", "A visual inspection of Figure REF shows that the highlighted structures are approximately equivariant/coherent." ], [ "Conclusion", "We introduced two new methods for computing Oseledets subspaces: one based on singular value decompositions and the other based on dichotomy projectors.", "We also reviewed recent methods by Ginelli et al.", "[17] and Wolfe and Samelson [34], and presented an improvement to both of these approaches that intelligently selected initial bases when only short time series were available to compute with.", "Finally, we carried out a comparative numerical investigation involving all four methods.", "Generally speaking, we found that Algorithms REF , REF , and REF outperformed the dichotomy projector methods (Algorithms REF and REF ) when limited to moderate amounts of data were available, however, the dichotomy projector methods performed very well when long time series of matrices were available.", "The Ginelli approach (in particular the improved Algorithm REF ) also worked very well with long time series.", "The improvements made to Algorithm REF in Section REF (namely the orthogonalisation step in Algorithm REF ) produced an algorithm that could take advantage of longer matrix sequences and return very accurate results.", "Of course, for each Algorithm one must choose the associated parameters sensibly to ensure good results.", "When only a short to moderate time series was available, we found mixed results in terms of the best algorithm.", "The improved SVD approach (Algorithm REF ) was best for low to moderate length time series in the exact Toy model, while the improved Ginelli (Algorithm REF ) and improved Wolfe (Algorithm REF ) were marginally best in terms of equivariance and expansion rate, respectively for the 2-disk model.", "Each of these three algorithms produced similar results in the fluid-flow system.", "Choosing appropriate parameters for a particular application can be difficult.", "In the present review, good values were chosen by educated experimentation.", "On the other hand, the dichotomy projector methods, Algorithms REF and REF , use parameters ($\\Lambda ^\\text{right}$ and $\\Lambda ^\\text{left}$ ) which can be chosen in a deterministic manner - by estimating Lyapunov exponents, which is a reasonably robust numerical procedure.", "Furthermore, a rigorous error approximation exists for Algorithm REF , a feature currently lacking for Algorithms REF , REF and REF .", "The memory footprint of each approach scales quite differently with dimension.", "In Section REF , Algorithms REF and REF could take advantage of the sparseness of the $d\\times d$ generating matrices of the cocycle.", "However, since $A(x,n)$ is formed by matrix multiplication, for large $n$ the matrix $A(x,n)$ becomes dense and may require memory of the order of $d^2$ floating point numbers.", "The dichotomy projector Algorithms REF and REF , need to form an $Nd \\times (N+1)d$ matrix, but with sparse generating matrices, this requires memory much less than of the order of $d^2$ floating point numbers.", "Algorithm REF has the most conservative memory footprint, but depends on its initialisation parameter $M^{\\prime }$ and the Oselelets subspace number $j$ .", "If $M^{\\prime }$ is large, then $A(T^{-M}x,M^{\\prime })$ in Step 1 can become dense and require ${ \\mathcal {O} (d^2)}$ floating point numbers.", "On the other hand, the stationary Lyapunov basis requires $jd$ floating point numbers to be stored, so if $j\\approx d$ this can becomes comparable to $d^2$ .", "Section REF involves non-invertible generating matrices and apart from Algorithm REF , each approach succeeded in producing a reasonable solution, showing that the Algorithms can perform well in the non-invertible setting.", "Continuing with the non-invertible situation, if one wishes to approximate Oseledets subspaces corresponding to negative numbers with very large magnitudes ($\\lambda _j \\approx -\\infty $ ), then Algorithms REF and REF may struggle as rapidly contracting directions (relative to the dominant direction corresponding to $\\lambda _1$ ) are quickly squashed during the matrix multiplication used to approximate $A(x,n)$ leading to inaccurate numerical representation of $A(x,n)$ .", "The dichotomy projector approaches of Algorithms REF and REF are able to compute Oseledets subspaces corresponding to smaller, sub-dominant Lyapunov exponents $\\lambda _3,\\lambda _4, \\ldots $ provided larger amounts of cocycle data is available.", "However, if $\\lambda _j \\approx -\\infty $ , we are forced to choose $\\Lambda ^\\text{right}$ or $\\Lambda ^\\text{left} \\approx -\\infty $ which means either problem (REF ) or (REF ) (in Algorithm REF which also feature in Algorithm REF ) are ill-conditioned and fail.", "The same problem manifests itself in Algorithm REF , even though it is able to compute Oseledets subspaces corresponding to smaller, sub-dominant Lyapunov exponents.", "The sum of the logarithm of the diagonal entries of the $j\\times j$ generating matrices of the cocycle $R(x,n)$ average to the logarithmic expansion rate of the $j$ -parallelepiped formed at $x$ by the stationary Lyapunov vectors $s_1^{(\\infty )}(x),\\ldots ,s_j^{(\\infty )}(x)$ as it is pushed-forward.", "Thus, the logarithm of the $i$ th diagonal entry of the generating matrices of $R(x,n)$ has a time average of $\\lambda _i$ [4] and if $\\lambda _j \\approx -\\infty $ , $R(x,n)$ will feature diagonal entries close to, or equal to zero and $R(x,n)^{-1}$ won't exist.", "In summary, Algorithms REF and REF are best suited to situations with limited cocycle data when one of the most dominant Oseledets subspaces is desired.", "Algorithm REF can be applied to both limited and high data situations by choosing $M^{\\prime }$ appropriately, and can compute most Oseledets subspaces provided their Lyapunov exponents are well-conditioned.", "If ample data is available and information regarding the system is lacking (making the choice of parameters for the other approaches difficult), the approaches of Algorithms REF and REF may be preferred for their relatively deterministic parameter selection." ] ]

[ [ "Message Passing for Dynamic Network Energy Management" ], [ "Abstract We consider a network of devices, such as generators, fixed loads, deferrable loads, and storage devices, each with its own dynamic constraints and objective, connected by lossy capacitated lines.", "The problem is to minimize the total network objective subject to the device and line constraints, over a given time horizon.", "This is a large optimization problem, with variables for consumption or generation in each time period for each device.", "In this paper we develop a decentralized method for solving this problem.", "The method is iterative: At each step, each device exchanges simple messages with its neighbors in the network and then solves its own optimization problem, minimizing its own objective function, augmented by a term determined by the messages it has received.", "We show that this message passing method converges to a solution when the device objective and constraints are convex.", "The method is completely decentralized, and needs no global coordination other than synchronizing iterations; the problems to be solved by each device can typically be solved extremely efficiently and in parallel.", "The method is fast enough that even a serial implementation can solve substantial problems in reasonable time frames.", "We report results for several numerical experiments, demonstrating the method's speed and scaling, including the solution of a problem instance with over 30 million variables in 52 minutes for a serial implementation; with decentralized computing, the solve time would be less than one second." ], [ "Introduction", "A traditional power grid is operated by solving a number of optimization problems.", "At the transmission level, these problems include unit commitment, economic dispatch, optimal power flow (OPF), and security-constrained OPF (SCOPF).", "At the distribution level, these problems include loss minimization and reactive power compensation.", "With the exception of the SCOPF, these optimization problems are static with a modest number of variables (often less than 10000), and are solved on time scales of 5 minutes or more.", "However, the operation of next generation electric grids (i.e., smart grid) will rely critically on solving large-scale, dynamic optimization problems involving hundreds of thousands of devices jointly optimizing tens to hundreds of millions of variables, on the order of seconds rather than minutes [13], [24].", "More precisely, the distribution level of a smart grid will include various types of active dynamic devices, such as distributed generators based on solar and wind, batteries, deferrable loads, curtailable loads, and electric vehicles, whose control and scheduling amount to a very complex power management problem [35], [8].", "In this paper, we consider a general problem, which we call the optimal power scheduling problem (OPSP), in which a network of dynamic devices are connected by lossy capacitated lines, and the goal is to jointly minimize a network objective subject to local constraints on the devices and lines.", "The network objective is the sum of the objective functions of each device.", "These objective functions extend over a given time horizon and encode operating costs such as fuel consumption and constraints such as limits on power generation or consumption.", "In addition, the objective functions encode dynamic objectives and constraints such as limits on ramp rates or charging limits.", "The variables for each device consist of its consumption or generation in each time period and can also include local variables which represent internal states of the device over time, such as the state of charge of a storage device.", "When all device objective functions and line constraints are convex, the OPSP is a convex optimization problem, which can in principle be solved efficiently [6].", "If not all device objective functions are convex, we can solve a relaxed form of the OPSP which can be used to find good, local solutions to the OPSP.", "The optimal value of the relaxed OPSP also gives a lower bound for the optimal value of the OPSP which can be used to evaluate the suboptimality of a local solution.", "For any network, the corresponding OPSP contains at least as many variables as the number of devices multiplied by the length of the time horizon.", "For large networks with hundreds of thousands of devices and a time horizon with tens or hundreds of time periods, the extremely large number of variables present in the corresponding OPSP makes solving it in a centralized fashion computationally impractical, even when all device objective functions are convex.", "We propose a decentralized optimization method which efficiently solves the OPSP by distributing computation across every device in the network.", "This method, which we call prox-average message passing, is iterative: At each iteration, every device passes simple messages to its network neighbors and then solves an optimization problem that minimizes the sum of its own objective function and a simple regularization term that only depends on the messages it received from its network neighbors in the previous iteration.", "As a result, the only non-local coordination needed between devices for prox-average message passing is synchronizing iterations.", "When all device objective functions are convex, we show that prox-average message passing converges to a solution of the OPSP.", "Our algorithm can be used several ways.", "It can be implemented in a traditional way on a single computer or cluster of computers, by collecting all the device constraints and objectives.", "We will demonstrate this use with an implementation that runs on a single 8-core computer.", "A more interesting use is in a peer-to-peer architecture.", "In this architecture, each device contains its own processor, which carries out the required local dynamic optimization and exchanges messages with its neighbors on the network.", "In this setting, the devices do not need to divulge their objectives or constraints; they must only support a simple protocol for interacting with the neighbors.", "Our algorithm ensures that the network power flows will converge to their optimal values, even though each device has very little information about the rest of the network, and only exchanges limited messages with its immediate neighbors.", "Due to recent advances in convex optimization [37], [28], [29], in many cases the optimization problems that each device solves in each iteration of prox-average message passing can be executed at millisecond or even microsecond time-scales on inexpensive, embedded processors.", "Since this execution can happen in parallel across all devices, the entire network can execute prox-average message passing at kilohertz rates.", "We present a series of numerical examples to illustrate this fact by using prox-average message passing to solve instances of the OPSP with up to 10 million variables serially in 17 minutes.", "(This is on an 8-core computer; with 64 cores, the time would be around 2 minutes.)", "Using decentralized computing, the solve time would be essentially independent of the size of the network and measured in fractions of a second.", "We note that although the primary application for our method is power management, it can easily be adapted to more general resource allocation and graph-structured optimization problems [31]." ], [ "Related work.", "The use of optimization in power systems dates back to the 1920s and has traditionally concerned the optimal dispatch problem [18], which aims to find the lowest cost method for generating and delivering power to consumers, subject to physical generator constraints.", "With the advent of computer and communication networks, many different ways to numerically solve this problem have been proposed [38] and more sophisticated variants of optimal dispatch have been introduced, such as OPF, economic dispatch, and dynamic dispatch [10], which extend optimal dispatch to include various reliability and dynamic constraints.", "For reviews of optimal and economic dispatch as well as general power systems, we direct the reader to [5] and the book and review papers cited above.", "When modeling AC power flow, the OPSP is a dynamic version of the OPF [7], extending the latter to include many more types of devices such as storage units.", "The OPSP also introduces a time horizon with coupling constraints between variables across time periods.", "The OPF has been a fundamental problem in power systems for over 50 years and is known to be non-convex.", "Recently, it was shown that the OPF can be solved exactly in certain circumstances by recasting it as a semidefinite program and solving its dual problem [22].", "Distributed optimization methods are naturally applied to power networks given the graph-structured nature of the transmission and distribution networks.", "There is an extensive literature on distributed optimization methods, dating back to the early 1960’s.", "The prototypical example is dual decomposition [11], [14], which is based on solving the dual problem by a gradient method.", "In each iteration, all devices optimize their local (primal) variables based on current prices (dual variables).", "Then the dual variables are updated to account for imbalances in supply and demand, with the goal being to determine prices for which supply equals demand.", "Unfortunately, dual decomposition methods are not robust, requiring many technical conditions, such as strict convexity and finiteness of all local cost functions, for both theoretical and practical convergence.", "One way to loosen these technical conditions is to use an augmented Lagrangian [19], [30], [3], resulting in the method of multipliers.", "This subtle change allows the method of multipliers to converge under mild technical conditions, even when the local cost functions are not strictly convex or necessarily finite.", "However, this method has the disadvantage of no longer being separable across subsystems.", "To achieve both separability and robustness for distributed optimization, we can instead use the alternating direction method of multipliers (ADMM) [16], [17], [12], [4].", "ADMM is very closely related to many other algorithms, and is identical to Douglas-Rachford operator splitting; see, e.g., the discussion in [4].", "Building on the work of [22], a distributed algorithm was recently proposed [25] to solve the dual OPF using a standard dual decomposition on subsystems that are maximal cliques of the power network.", "Augmented Lagrangian methods (including ADMM) have previously been applied to the study of power systems with static, single period objective functions on a small number of distributed subsystems, each representing regional power generation and consumption [20].", "For an overview of related decomposition methods applied to power flow problems, we direct the reader to [21], [1] and the references therein." ], [ "Outline.", "The rest of this paper is organized as follows.", "In § we give the formal definition of our network model.", "In § we give examples of how to model specific devices such as generators, deferrable loads and energy storage systems in our formal framework.", "In §, we describe the role that convexity plays in the OPSP and introduce the idea of convex relaxations as a tool to find solutions to the OPSP in the presence of non-convex device objective functions.", "In § we derive the prox-average message passing equations.", "In § we present a series of numerical examples, and in § we discuss how our framework can be easily extended to include use cases we do not explicitly cover in this paper." ], [ "Formal definition", "A network consists of a finite set of terminals $\\mathcal {T}$ , a finite set of devices $\\mathcal {D}$ , and a finite set of nets $\\mathcal {N}$ .", "The sets $\\mathcal {D}$ and $\\mathcal {N}$ are both partitions of $\\mathcal {T}$ .", "Thus, each terminal is associated with exactly one device and exactly one net.", "Equivalently, a network can be defined as a bipartite graph with one set of vertices given by devices, the other set of vertices given by nets, and edges given by terminals.", "Each terminal $t\\in \\mathcal {T}$ has an associated power schedule $p_t = (p_t(1), \\ldots , p_t(T)) \\in {\\mbox{\\bf R}}^T$ , where $T$ is a given time horizon.", "Here $p_t(\\tau )$ is the amount of energy consumed by device $d$ through terminal $t$ in time period $\\tau $ , where $t\\in d$ (i.e., terminal $t$ is associated with device $d$ ).", "When $p_t(\\tau ) < 0$ , $-p_t(\\tau )$ is the energy generated by device $d$ through terminal $t$ in time period $\\tau $ .", "The set of all terminal power schedules is denoted $p$ .", "This is a function from $\\mathcal {T}$ (the set of terminals) into ${\\mbox{\\bf R}}^T$ (time periods); we can identify $p$ with a $|\\mathcal {T}| \\times T$ matrix, whose rows are the terminal power schedules.", "For each device $d\\in \\mathcal {D}$ , $p_d$ consists of the set of power schedules of terminals associated with $d$ , which we identify with a $|d| \\times T$ matrix whose rows are taken from the rows of $p$ corresponding to the terminals in $d$ .", "Each device $d$ has an associated objective function $f_d:{\\mbox{\\bf R}}^{|d|\\times T} \\rightarrow {\\mbox{\\bf R}}\\cup \\lbrace +\\infty \\rbrace $ , where we set $f_d(p_d) = \\infty $ to encode constraints on the power schedules for the device.", "When $f_d(p_d)<\\infty $ , we say that $p_d$ is a set of realizable power schedules for device $d$ , and we interpret $f_d(p_d)$ as the cost (or revenue, if negative) to device $d$ for the power schedules $p_d$ .", "Similarly, for each net $n \\in \\mathcal {N}$ , $p_n$ consists of the set of power schedules of terminals associated with $n$ , which we identify with a $|n| \\times T$ matrix whose rows are taken from the rows of $p$ corresponding to the terminals in $n$ .", "Each net $n$ is a lossless energy carrier (commonly referred to as a bus in power systems literature), which is represented by the constraint $ \\sum _{t \\in n} p_t(\\tau )=0, \\quad \\tau =1, \\ldots ,T. $ In other words, in each time period the power flows on each net balance.", "For any terminal, we define the average net power imbalance $\\bar{p}:\\mathcal {T}\\rightarrow {\\mbox{\\bf R}}^T$ , as $ \\bar{p}_t = \\frac{1}{|n|}\\sum _{t^{\\prime } \\in n} p_{t^{\\prime }},$ where $t\\in n$ , i.e., terminal $t$ is associated with net $n$ .", "In other words, $\\bar{p}_t(\\tau )$ is the average power schedule of all terminals associated with the same net as terminal $t$ at time $\\tau $ .", "We overload this notation for devices by defining $\\bar{p}_d = (\\bar{p}_{i_1}, \\ldots , \\bar{p}_{i_{|d|}})$ , where device $d$ is associated with terminals $i_1, \\ldots , i_{|d|}$ .", "Using an identical notation for nets, we can see that $\\bar{p}_n$ simply contains $|n|$ copies of the average net power imbalance for net $n$ .", "The net power balance constraint for all terminals can be equivalently expressed as $\\bar{p} =0$ ." ], [ "Optimal power scheduling problem.", "We say that a set of power schedules $p: \\mathcal {T}\\rightarrow {\\mbox{\\bf R}}^T$ is feasible if $f_d(p_d) < \\infty $ for all $d \\in \\mathcal {D}$ (i.e., all devices' power schedules are realizable), and $\\bar{p} = 0$ (i.e., power balance holds across all nets).", "We define the network objective as $f(p) = \\sum _{d \\in \\mathcal {D}}f_d(p_d)$ .", "The optimal power scheduling problem (OPSP) is $\\begin{array}{ll}\\mbox{minimize} & f(p)\\\\\\mbox{subject to} & \\bar{p} = 0,\\end{array}$ with variable $p: \\mathcal {T}\\rightarrow {\\mbox{\\bf R}}^T$ .", "We refer to $p$ as optimal if it solves (REF ), i.e., globally minimizes the objective among all feasible $p$ .", "We refer to $p$ as locally optimal if it is a locally optimal point for (REF )." ], [ "Dual variables and locational marginal prices.", "Suppose $p^0$ is a set of optimal power schedules, that also minimizes the Lagrangian $f(p) + \\sum _{t\\in \\mathcal {T}} \\sum _{\\tau =1}^T y_t^0(\\tau ) \\bar{p}_t(\\tau ),$ with no power balance constraint, where $y^0: \\mathcal {T}\\rightarrow {\\mbox{\\bf R}}^T$ .", "In this case we call $y^0$ a set of optimal Lagrange multipliers or dual variables.", "When $p^0$ is a locally optimal point, which also locally minimizes the Lagrangian, then we refer to $y^0$ as a set of locally optimal Lagrange multipliers.", "The dual variables $y^0$ are related to the traditional concept of locational marginal prices $\\mathcal {L}^0:\\mathcal {T}\\rightarrow {\\mbox{\\bf R}}^T$ by rescaling the dual variables associated with each terminal according to the size of its associated net, i.e., $\\mathcal {L}_t^0 = y_t^0/|n|$ , where $t\\in n$ .", "This rescaling is due to the fact that locational marginal prices are the dual variables associated with the constraints in (REF ) rather than their scaled form used in (REF ) [15]." ], [ "Discussion", "We now describe our model in a more intuitive, less formal manner.", "Devices include generators, loads, energy storage systems, and other power sources, sinks, and converters.", "Terminals are ports on a device through which power flows, either into or out of the device (or both, at different times, as in a storage device).", "Nets are used to model ideal lossless uncapacitated connections between terminals over which power is transmitted; losses, capacities, and more general connection constraints between a set of terminals can be modeled with the addition of a device and individual nets which connect each terminal to the new device.", "Our network model does not specify nor require a specific type of energy transport mechanism (e.g., DC, single or 3-phase AC), but rather can abstractly model arbitrary heterogeneous energy transport and exchange mechanisms.", "The objective function of a device is used to measure the cost (which can be negative, representing revenue) associated with a particular mode of operation, such as a given level of consumption or generation of power.", "This cost can include the actual direct cost of operating according to the given power schedules, such as a fuel cost, as well as other costs such as $\\mathrm {CO}_2$ generation, or costs associated with increased maintenance and decreased system lifetime due to structural fatigue.", "The objective function can also include local variables other than power schedules, such as the state of charge of a storage device.", "Constraints on the power schedules and internal variables for a device are encoded by setting the objective function to $+\\infty $ for power schedules that violate the constraints.", "In many cases, a device's objective function will only take on the values 0 and $+\\infty $ , indicating no local preference among feasible power schedules." ], [ "Example transformation to abstract network model", "We illustrate how a traditional power network can be recast into our network model in figure REF .", "The original power network, shown on the left, contains 2 loads, 3 buses, 2 generators, and a single battery storage system.", "We can transform this small power grid into our model by representing it as a network with 11 terminals, 8 devices (3 of them transmission lines), and 3 nets, shown on the right of figure REF .", "Terminals are shown as small filled circles.", "Single terminal devices, which are used to model loads, generators, and the battery, are shown as boxes.", "The transmission lines are two terminal devices represented by solid lines.", "The nets are shown as dashed rounded boxes.", "Terminals are associated with the device they touch and the net in which they are contained.", "Figure: A simple network (left),and its transformation into standard form (right).The set of terminals can be partitioned by either the devices they are associated with, or the nets in which they are contained.", "Figure REF shows the network in figure REF as a bipartite graph, with devices on the left and nets on the right.", "Figure: The network in figure represented as a bipartitegraph.", "Devices (boxes) are shown on the left with their associated terminals (dots).The terminals are connected to their corresponding nets (solid boxes) on the right." ], [ "Device examples", "We present a few examples of how common devices can be modeled in our framework.", "We note that these examples are intentionally kept simple, but could easily be extended to more refined objectives and constraints.", "In these examples, it is easier to discuss operational costs and constraints for each device separately.", "A device's objective function is equal to the device's cost function unless any constraint is violated, in which case we set the objective value to $+\\infty $ ." ], [ "Generator.", "A generator is a single-terminal device with power schedule $p_\\mathrm {gen}$ , which generates power over a range, $P^\\mathrm {min} \\le -p_\\mathrm {gen} \\le P^\\mathrm {max}$ , and has ramp-rate constraints $R^\\mathrm {min} \\le -Dp_\\mathrm {gen} \\le R^\\mathrm {max},$ which limit the change of power levels from one period to the next.", "Here, the operator $D\\in {\\mbox{\\bf R}}^{(T-1)\\times T}$ is the forward difference operator, defined as $(Dx)(\\tau ) = x(\\tau +1)-x(\\tau ), \\quad \\tau = 1,\\ldots ,T-1.$ The cost function for a generator has the separable form $\\psi _\\mathrm {gen}(p_\\mathrm {gen}) =\\sum _{\\tau =1}^T \\phi _\\mathrm {gen}(-p_\\mathrm {gen}(\\tau )),$ where $\\phi : {\\mbox{\\bf R}}\\rightarrow {\\mbox{\\bf R}}$ gives the cost of operating the generator at a given power level over a single time period.", "This function is typically, but not always, convex and increasing.", "It could be piecewise linear, or, for example, quadratic: $\\phi _\\mathrm {gen}(u) = \\alpha u^2 + \\beta u,$ where $\\alpha , \\beta > 0$ .", "More sophisticated models of generators allow for them to be switched on or off, with an associated cost each time they are turned on or off.", "When switched on, the generator operates as described above.", "When the generator is turned off, it generates no power but can still incur costs for other activities such as idling." ], [ "Transmission line.", "A transmission line is a two terminal device that transmits power across some distance with power schedules $p_1$ and $p_2$ and zero cost function.", "The sum $p_1+p_2$ represents the loss in the line and is always nonnegative.", "The difference $p_1-p_2$ can be interpreted as twice the power flow from terminal one to terminal two.", "A line has a maximum flow capacity, which is given by $|p_1 - p_2| \\le C^\\mathrm {max},$ as well as a loss function, $\\ell (p_1,p_2):{\\mbox{\\bf R}}^{2\\times T} \\rightarrow {\\mbox{\\bf R}}^T_+$ , which defines the constraint $p_1 + p_2 + \\ell (p_1,p_2) = 0.$ In many cases, $\\ell $ is a convex function with $\\ell (0,0)=0$ .", "Under a simple resistive model, $\\ell $ is a convex quadratic function of $p_1$ and $p_2$ .", "Under a model for AC power transmission, the feasible region defined by power loss is given by an ellipse [23]." ], [ "Battery.", "A battery is a single terminal energy storage device with power schedule $p_\\mathrm {bat}$ , which can take in or deliver energy, depending on whether it is charging or discharging.", "The charging and discharging rates are limited by the constraints $-D^\\mathrm {max} \\le p_\\mathrm {bat} \\le C^\\mathrm {max}$ , where $C^\\mathrm {max}\\in {\\mbox{\\bf R}}^T$ and $D^\\mathrm {max}\\in {\\mbox{\\bf R}}^T$ are the maximum charging and discharging rates.", "At time $t$ , the charge level of the battery is given by local variables $q(\\tau ) = q^\\mathrm {init}+\\sum _{t=1}^\\tau p_\\mathrm {bat}(t),\\quad \\tau = 1,\\ldots ,T,$ where $q^\\mathrm {init}$ is the initial charge.", "It has zero cost function and the charge level must not exceed the battery capacity, i.e., $0 \\le q(\\tau ) \\le Q^\\mathrm {max}$ , $\\tau = 1,\\ldots ,T$ .", "It is common to constrain the terminal battery charge $q(T)$ to be some specified value or to match the initial charge $q^\\mathrm {init}$ .", "More sophisticated battery models include (possibly state-dependent) charging and discharging inefficiencies as well as charge leakage.", "In addition, they can include costs which penalize excessive charge-discharge cycling." ], [ "Fixed load.", "A fixed energy load is a single terminal device with zero cost function which consists of a desired consumption profile, $l\\in {\\mbox{\\bf R}}^T$ .", "This consumption profile must be satisfied in each period, i.e., we have the constraint $p_\\mathrm {load} = l$ ." ], [ "Thermal load.", "A thermal load is a single terminal device with power schedule $p_\\mathrm {therm}$ which consists of a heat store (room, cooled water reservoir, refrigerator), with temperature profile $\\theta \\in {\\mbox{\\bf R}}^T$ , which must be kept within minimum and maximum temperature limits, $\\theta ^\\mathrm {min}\\in {\\mbox{\\bf R}}^T$ and $\\theta ^\\mathrm {max}\\in {\\mbox{\\bf R}}^T$ .", "The temperature of the heat store evolves according to $\\theta (\\tau +1) = \\theta (\\tau )+(\\mu /c)(\\theta ^\\mathrm {amb}(\\tau )-\\theta (\\tau ))-(\\eta /c) p_\\mathrm {therm}(\\tau ),\\quad \\tau = 1,\\ldots ,T-1,\\qquad \\theta (1) = \\theta ^\\mathrm {init},$ where $0 \\le p_\\mathrm {therm} \\le H^\\mathrm {max}$ is the cooling power consumption profile, $H^\\mathrm {max}\\in {\\mbox{\\bf R}}^T$ is the maximum cooling power, $\\mu $ is the ambient conduction coefficient, $\\eta $ is the heating/cooling efficiency, $c$ is the heat capacity of the heat store, $\\theta ^\\mathrm {amb}\\in {\\mbox{\\bf R}}^T$ is the ambient temperature profile, and $\\theta ^\\mathrm {init}$ is the initial temperature of the heat store.", "A thermal load has zero cost function.", "More sophisticated models include temperature-dependent cooling and heating efficiencies for heat pumps, more complex dynamics of the system whose temperature is being controlled, and additional additive terms in the thermal dynamics, to represent occupancy or other heat sources." ], [ "Deferrable load.", "A deferrable load is a single terminal device with zero cost function that must consume a minimum amount of power over a given interval of time, which is characterized by the constraint $\\sum _{\\tau =A}^{D} p_\\mathrm {load}(\\tau ) \\ge E$ , where $E$ is the minimum total consumption for the time interval $\\tau = A,\\ldots ,D$ .", "The energy consumption in each time period is constrained by $0 \\le p_\\mathrm {load} \\le L^\\mathrm {max}$ .", "In some cases, the load can only be turned on or off in each time period, i.e., $p_\\mathrm {load}(\\tau ) \\in \\lbrace 0, L^\\mathrm {max}\\rbrace $ for $\\tau =A, \\ldots , D$ ." ], [ "Curtailable load.", "A curtailable load is a single terminal device which does not impose hard constraints on its power requirements, but instead penalizes the shortfall between a desired load profile $l\\in {\\mbox{\\bf R}}^T$ and delivered power.", "In the case of a linear penalty, its cost function is given by $\\alpha (l - p_\\mathrm {load})_+,$ where $(z)_+ = \\max (0,z)$ , $p_\\mathrm {load} \\in {\\mbox{\\bf R}}^T$ is the amount of electricity delivered to the device, and $\\alpha > 0$ is a penalty parameter." ], [ "Electric vehicle.", "An electric vehicle is a single terminal device with power schedule $p_\\mathrm {ev}$ which has a desired charging profile $c^\\mathrm {des} \\in {\\mbox{\\bf R}}^T$ and can be charged within a time interval $\\tau = A,\\ldots ,D$ .", "To avoid excessive charge cycling, we assume that the electric vehicle battery cannot be discharged back into the grid (in more sophisticated vehicle-to-grid models, this assumption is relaxed), so we have the constraints $0 \\le p_\\mathrm {ev} \\le C^\\mathrm {max}$ , where $C^\\mathrm {max}\\in {\\mbox{\\bf R}}^T$ is the maximum charging rate.", "We assume that $c^\\mathrm {des}(\\tau ) = 0$ for $\\tau \\notin \\lbrace A,\\ldots ,D\\rbrace $ , and for $\\tau = A,\\ldots ,D$ , the charge level is given by $q(\\tau ) = q^\\mathrm {init}+\\sum _{t = A}^\\tau p_\\mathrm {ev}(t),$ where $q^\\mathrm {init}$ is the initial charge of the vehicle when it is plugged in at time $\\tau = A$ .", "We can model electric vehicle charging as a deferrable load, where we require a given charge level to be achieved at some time.", "A more realistic model is as a combination of a deferrable and curtailable load, with cost function $\\alpha \\sum _{\\tau =A}^D(c^\\mathrm {des}(\\tau )-q(\\tau ))_+,$ where $\\alpha > 0$ is a penalty parameter.", "Here $c^\\mathrm {des}(\\tau )$ is the desired charge level at time $\\tau $ , and $c^\\mathrm {des}(\\tau )-q(\\tau ))_+$ is the shortfall." ], [ "External tie with transaction cost.", "An external tie is a connection to an external source of power.", "We represent this as a single terminal device with power schedule $p_\\mathrm {ex}$ .", "In this case, $p_\\mathrm {ex}(\\tau )_- =\\max \\lbrace -p_\\mathrm {ex}(\\tau ),0\\rbrace $ is the amount of energy pulled from the source, and $p_\\mathrm {ex}(\\tau )_+ = \\max \\lbrace p_\\mathrm {ex}(\\tau ),0\\rbrace $ is the amount of energy delivered to the source, at time $\\tau $ .", "We have the constraint $|p_\\mathrm {ex}(\\tau )| \\le E^\\mathrm {max}(\\tau )$ , where $E^\\mathrm {max}\\in {\\mbox{\\bf R}}^T$ is the transaction limit.", "We suppose that the prices for buying and selling energy are given by $c \\pm \\gamma $ respectively, where $c(\\tau )$ is the midpoint price, and $\\gamma (\\tau )>0$ is the difference between the price for buying and selling (i.e., the transaction cost).", "The cost function is then $-(c-\\gamma )^T(p_\\mathrm {ex})_++(c+\\gamma )^T(p_\\mathrm {ex})_- =-c^Tp_\\mathrm {ex}+\\gamma ^T|p_\\mathrm {ex}|,$ where $|p_\\mathrm {ex}|$ , $(p_\\mathrm {ex})_+$ , and $(p_\\mathrm {ex})_-$ are all interpreted elementwise." ], [ "Devices", "We call a device convex if its objective function is a convex function.", "A network is convex if all of its devices are convex.", "For convex networks, the OPSP is a convex optimization problem, which means that in principle we can efficiently find a global solution [6].", "When the network is not convex, even finding a feasible solution for the OPSP can become difficult, and finding and certifying a globally optimal solution to the OPSP is generally intractable.", "However, special structure in many practical power distribution problems allows us to guarantee optimality in certain cases.", "In the examples from §, the battery, fixed load, thermal load, curtailable load, electric vehicle, and external tie are all convex devices using the constraints and objective functions given.", "A deferrable load is convex if we drop the constraint that it can only be turned on or off.", "We discuss the convexity properties of the generator and transmission line in the following section." ], [ "Relaxations", "One technique to deal with non-convex networks is to use convex relaxations.", "We use the notation $g^\\mathrm {env}$ to denote the convex envelope [32] of the function $g$ .", "There are many equivalent definitions for the convex envelope, for example, $g^\\mathrm {env}$ = $(g^*)^*$ , where $g^*$ denotes the convex conjugate of the function $g$ .", "We can equivalently define $g^\\mathrm {env}$ to be the largest convex lower bound of $g$ .", "If $g$ is a convex, closed, proper (CCP) function, then $g=g^\\mathrm {env}$ .", "We define the relaxed optimal power scheduling problem (rOPSP) as $\\begin{array}{ll}\\mbox{minimize} & f^\\mathrm {env}(p)\\\\\\mbox{subject to} & \\bar{p} = 0,\\end{array}$ with variable $p:\\mathcal {T}\\rightarrow {\\mbox{\\bf R}}^T$ .", "This is a convex optimization problem, whose optimal value can in principle be computed efficiently, and whose optimal objective value is a lower bound for the optimal objective value of the OPSP.", "In some cases, we can guarantee a priori that a solution to the rOPSP will also be a solution to the OPSP based on a property of the network objective such as monotonicity.", "Even when the relaxed solution does not satisfy all of the constraints in the unrelaxed problem, it can be used as a starting point to help construct good, local solutions to the unrelaxed problem.", "The suboptimality of these local solutions can then be bounded by the gap between their network objective and the lower bound provided by the solution to the rOPSP.", "If this gap is small for a given local solution, we can guarantee that it is nearly optimal." ], [ "Generator.", "When a generator is modeled as in § and is always powered on, it is a convex device.", "However, when given the ability to be powered on and off, the generator is no longer convex.", "In this case, we can relax the generator objective function so that its cost for power production in each time period, given in figure REF , is a convex function.", "This allows the generator to produce power in the interval $[0,P^\\mathrm {min}]$ ." ], [ "Transmission line.", "In a lossless transmission line, we have $\\ell (p_1,p_2) = 0$ , and thus the set of feasible power schedules is the line segment $L = \\lbrace (p_1,p_2) \\mid p_1=-p_2, \\quad p_2 \\in [-C^\\mathrm {max}/2, C^\\mathrm {max}/2]\\rbrace ,$ as shown in figure REF in black.", "When the transmission line has losses, in most cases the loss function $\\ell $ is a convex function of the input and output powers, which leads to a feasible region like the grey arc in figure REF .", "For example, using a lumped $\\Pi $ model and under the common assumption that the voltage magnitude is fixed [5], a transmission line with series admittance $g - ib$ gives the quadratic loss $ \\ell (p_1, p_2) = -(g/4)((p_1 + p_2)^2/g^2 + (p_1 -p_2)^2/b^2).$ The feasible set of a relaxed transmission line is given by the convex hull of the original transmission line's constraints.", "The right side of figure REF shows examples of this for both lossless and lossy transmission lines.", "Physically, this relaxation gives lossy transmission lines the ability to discard some additional power beyond what is simply lost to heat.", "Since electricity is generally a valuable commodity in power networks, the transmission lines will generally not throw away any additional power in the optimal solution to the rOPSP, leading to the power line constraints in the rOPSP being tight and thus also satisfying the unrelaxed power line constraints in the original OPSP.", "As was shown in [23], when the network is a tree, this relaxation is always tight.", "In addition, when all locational marginal prices are positive and no other non-convexities exist in the network, the tightness of the line constraints in the rOPSP can be guaranteed in the case of networks that have separate phase shifters on each loop in the networks whose shift parameter can be freely chosen [34]." ], [ "Decentralized method", "We begin this section by deriving the prox-average message passing equations assuming that all the device objective functions are convex closed proper (CCP) functions.", "We then compare the computational and communication requirements of prox-average message passing with a centralized solver for the OPSP.", "The additional requirements that the functions are closed and proper are technical conditions that are in practice satisfied by any convex function used to model devices.", "We note that we do not require either finiteness or strict convexity of any device objective function, and that all results apply to networks with arbitrary topology.", "Whenever we have a set of variables that maps terminals to time periods, $x:\\mathcal {T}\\rightarrow {\\mbox{\\bf R}}^T$ (which we can also associate with a $|\\mathcal {T}| \\times T$ matrix), we will use the same index and over-line notation for the variables $x$ as we do for power schedules $p$ .", "For example, $x_t \\in {\\mbox{\\bf R}}^T$ consists of the time period vector of values of $x$ associated with terminal $t$ and $\\bar{x}_t =(1/|n|)\\sum _{t^{\\prime }\\in n} x_{t^{\\prime }}$ , where $t \\in n$ , with similar notation for indexing $x$ by devices and nets." ], [ "Prox-average message passing", "We derive the prox-average message passing equations by reformulating the OPSP using the alternating direction method of multipliers (ADMM) and then simplifying the resulting equations.", "We refer the reader to [4] for a thorough overview of ADMM.", "We first rewrite the OPSP as $\\begin{array}{ll}\\mbox{minimize} & \\sum _{d\\in \\mathcal {D}} f_d(p_d) +\\sum _{n\\in \\mathcal {N}} g_n(z_n) \\\\\\mbox{subject to} & p = z,\\end{array}$ with variables $p,z: \\mathcal {T}\\rightarrow {\\mbox{\\bf R}}^T$ , where $g_n(z_n)$ is the indicator function on the set $\\lbrace z_n: \\bar{z}_n = 0\\rbrace $ .", "We use the notation from [4] and, ignoring a constant, form the augmented Lagrangian $L_\\rho (p,z,u) = \\sum _{d\\in \\mathcal {D}} f_d(p_d) + \\sum _{n\\in \\mathcal {N}} g_n(z_n)+ (\\rho /2)\\Vert p - z + u\\Vert _2^2,$ with the scaled dual variables $u = y/\\rho : \\mathcal {T}\\rightarrow {\\mbox{\\bf R}}^T$ , which we associate with a $|\\mathcal {T}| \\times T$ matrix.", "Because devices and nets are each partitions of the terminals, the last term of (REF ) can be split across either devices or nets, i.e., $(\\rho /2)\\Vert p - z + u\\Vert _2^2 =\\sum _{d\\in \\mathcal {D}} (\\rho /2)\\Vert p_d - z_d + u_d\\Vert _2^2 =\\sum _{n\\in \\mathcal {N}} (\\rho /2)\\Vert p_n - z_n + u_n\\Vert _2^2.$ The resulting ADMM algorithm is then given by the iterations $p_d^{k+1} &:=& \\mathop {\\rm argmin}_{p_d} \\left( f_d(p_d) + (\\rho /2)\\Vert p_d-z_d^k + u_d^k\\Vert ^2_2\\right), \\quad d \\in \\mathcal {D}, \\\\z_n^{k+1} &:=& \\mathop {\\rm argmin}_{z_n} \\left( g(z_n) + (\\rho /2)\\Vert z_n-u_n^k-p_n^{k+1}\\Vert ^2_2\\right), \\quad n \\in \\mathcal {N}, \\\\u_n^{k+1} &:=& u_n^k + (p_n^{k+1}-z_n^{k+1}), \\quad n \\in \\mathcal {N},$ where the first step is carried out in parallel by all devices, and then the second and third steps are carried out in parallel by all nets.", "Since $g_n(z_n)$ is simply an indicator function for each net $n$ , the second step of the algorithm can be computed analytically and is given by $z^{k+1}_n &:=& u^k_n + p^{k+1}_n - \\bar{u}^k_n - \\bar{p}^{k+1}_n.$ By plugging this quantity into the $u$ update step, the algorithm can be simplified further to yield the prox-average message passing algorithm: Proximal power schedule update.", "$p_d^{k+1} := \\mathbf {prox}_{f_d, \\rho }(p_d^k - \\bar{p}_d^k - u_d^k),\\quad d\\in \\mathcal {D}.$ Scaled price update.", "$u_n^{k+1} := u_n^k + \\bar{p}_n^{k+1}, \\quad n \\in \\mathcal {N}.$ The proximal function for a function $f$ is given by $ \\mathbf {prox}_{f,\\rho }(x) = \\mathop {\\rm argmin}_{y}\\left(f(y)+(\\rho /2)\\Vert x-y\\Vert _2^2 \\right),$ which is guaranteed to exist when $f$ is CCP [32].", "We can now see where the name prox-average message passing comes from.", "In each iteration, every device computes the proximal function of its objective function, with an argument that depends on messages passed to it through its terminals by its neighboring nets in the previous iteration ($\\bar{p}_d^k$ and $u_d^k$ ).", "Then, every devices passes to its terminals the newly computed power schedules, $p_d^{k+1}$ , which are then passed to the terminals' associated nets.", "Every net computes its new average power imbalance, $\\bar{p}_n^{k+1}$ , updates its dual variables, $u_n^{k+1}$ , and broadcasts these values to its associated terminals' devices.", "Since $\\bar{p}_n^k$ is simply $|n|$ copies of the same vector for all $k$ , we can see that all terminals connected to the same net have the same value for their dual variables throughout the algorithm, i.e., for all values of $k$ , $u_t^k = u_{t^{\\prime }}^k$ whenever $t,t^{\\prime } \\in n$ for any $n \\in \\mathcal {N}$ .", "As an example, consider the network represented by figures REF and REF .", "The prox-average algorithm performs the power schedule update on the devices (the boxes on the left in figure REF ).", "The devices share the respective power profiles via the terminals, and the nets (the solid boxes on the right) compute the scaled price update.", "For any network, the prox-average algorithm can be thought of as alternating between devices (on the left) and nets (on the right)." ], [ "Convergence.", "We make a few comments about the convergence of prox-average message passing.", "Since prox-average message passing is a version of ADMM, all convergence results that apply to ADMM also apply to prox-average message passing.", "In particular, when all devices have closed, convex, proper (CCP) objective functions and a feasible solution to the OPSP exists, the following hold.", "Residual convergence.", "$\\bar{p}^k \\rightarrow 0$ as $k\\rightarrow \\infty $ , Objective convergence.", "$\\sum _{d\\in \\mathcal {D}} f_d(p_d^k)+\\sum _{n\\in \\mathcal {N}}g_n(p_n^k)\\rightarrow f^\\star $ as $k\\rightarrow \\infty $ , Dual variable convergence.", "$\\rho u^k = y^k \\rightarrow y^\\star $ as $k\\rightarrow \\infty $ , where $f^\\star $ is the optimal value for the OPSP, and $y^\\star $ are the optimal dual variables.", "The proof of these conditions can be found in [4].", "As a result of the third condition, the optimal locational marginal prices $\\mathcal {L}^\\star $ can be found for each net $n\\in \\mathcal {N}$ by setting $\\mathcal {L}_n^\\star = y_n^\\star /|n|$ ." ], [ "Stopping criterion.", "Following [4], we can define primal and dual residuals, which for prox-average message passing simplify to $r^k = \\bar{p}^k, \\quad s^k = \\rho ((p^k-\\bar{p}^k)-(p^{k-1}-\\bar{p}^{k-1})).$ We give a simple interpretation of each residual.", "The primal residual is simply the net power imbalance across all nets in the network, which is the original measure of primal feasibility in the OPSP.", "The dual residual is equal to the difference between the current and previous iterations of the difference between power schedules and their average net power.", "The locational marginal price at each net is determined by the deviation of all associated terminals' power schedule from the average power on that net.", "As the change in these deviations approaches zero, the corresponding locational marginal prices converge to their optimal values.", "We can define a simple criterion for terminating prox-average message passing when $\\Vert r^k\\Vert _2 \\le \\epsilon ^\\mathrm {pri},\\quad \\Vert s^k\\Vert _2 \\le \\epsilon ^\\mathrm {dual},$ where $\\epsilon ^\\mathrm {pri}$ and $\\epsilon ^\\mathrm {dual}$ are, respectively, primal and dual tolerances.", "We can normalize both of these quantities to network size by the relation $\\epsilon ^\\mathrm {pri} =\\epsilon ^\\mathrm {dual} =\\epsilon ^\\mathrm {abs}\\sqrt{|\\mathcal {T}|T},$ for some absolute tolerance $\\epsilon ^\\mathrm {abs} > 0$ ." ], [ "Choosing a value of $\\rho $ .", "Numerous examples show that the value of $\\rho $ can have a dramatic effect on the rate of convergence of ADMM and prox-average message passing.", "Many good methods for picking $\\rho $ in both offline and online fashions are discussed in [4].", "We note that unlike other versions of ADMM, the scaling parameter $\\rho $ enters very simply into the prox-average equations and can thus be modified online without incurring any additional computational penalties, such as having to re-factorize a matrix.", "We can modify the prox-average message passing algorithm with the addition of a third step Parameter update and price rescaling.", "$\\rho ^{k+1} &:=& h(\\rho ^k, r^k, s^k), \\\\u^{k+1} &:=& \\frac{\\rho ^k}{\\rho ^{k+1}} u^{k+1},$ for some function $h$ .", "We desire to pick an $h$ such that the primal and dual residuals are of similar size throughout the algorithm, i.e., $\\rho ^k \\Vert r^k\\Vert _2 \\approx \\Vert s^k\\Vert _2$ for all $k$ .", "To accomplish this task, we use a simple proportional-derivative controller to update $\\rho $ , choosing $h$ to be $h(\\rho ^k) = \\rho ^k \\exp (\\lambda v^k + \\mu (v^k - v^{k-1})),$ where $v^k = \\rho ^k \\Vert r^k\\Vert _2/\\Vert s^k\\Vert _2 - 1$ and $\\lambda $ and $\\mu $ are nonnegative parameters chosen to control the rate of convergence.", "Typical values of $\\lambda $ and $\\mu $ are between $10^{-3}$ and $10^{-1}$ .", "When $\\rho $ is updated in such a manner, convergence is sped up in many examples, sometimes dramatically.", "Although it can be difficult to prove convergence of the resulting algorithm, a standard trick is to assume that $\\rho $ is changed only for a large but bounded number of iterations, at which point it is held constant for the remainder of the algorithm, thus guaranteeing convergence." ], [ "Non-convex case.", "When one or more of the device objective functions is non-convex, we can no longer guarantee that prox-average message passing converges to the optimal value of the OPSP or even that it converges at all (i.e., reaches a fixed point).", "Prox functions for non-convex devices must be carefully defined as the set of minimizers in (REF ) is no longer necessarily a singleton.", "Even when they can be defined, prox functions of non-convex functions are intractable to compute in many cases.", "One solution to these issues is to use prox-average message passing to solve the rOPSP.", "It is easy to show that $f^\\mathrm {env}(p) = \\sum _{d\\in \\mathcal {D}} f^\\mathrm {env}_d(p_d)$ .", "As a result, we can run prox-average message passing using the proximal functions of the relaxed device objective functions.", "Since $f_d^\\mathrm {env}$ is a CCP function for all $d\\in \\mathcal {D}$ , prox-average message passing in this case is guaranteed to converge to the optimal value of the rOPSP and yield the optimal relaxed locational marginal prices." ], [ "Discussion", "In order to compute the prox-average messages, devices and nets only require knowledge of who their network neighbors are, the ability to send small vectors of numbers to those neighbors in each iteration, and the ability to store small amounts of state information and efficiently compute prox functions (devices) or averages (nets).", "As all communication is local and peer-to-peer, prox-average message passing supports the ad hoc formation of power networks, such as micro grids, and is robust to device failure and unexpected network topology changes.", "Due to recent advances in convex optimization [37], [28], [29], many of the prox function calculations that devices must perform can be very efficiently executed at millisecond or microsecond time-scales on inexpensive, embedded processors.", "Since all devices and all nets can each perform their computations in parallel, the time to execute a single, network wide prox-average message passing iteration (ignoring communication overhead) is equal to the sum of the maximum computation time over all devices and the maximum computation time of all nets in the network.", "As a result, the computation time per iteration is small and essentially independent of the size of the network.", "In contrast, solving the OPSP in a centralized fashion requires complete knowledge of the network topology, sufficient communication bandwidth to centrally aggregate all devices objective function data, and sufficient centralized computational resources to solve the resulting OPSP.", "In large, real-world networks, such as the smart grid, all three of these requirements are generally unattainable.", "Having accurate and timely information on the global connectivity of all devices is infeasible for all but the smallest of dynamic networks.", "Centrally aggregating all device objective functions would require not only infeasible bandwidth and data storage requirements at the aggregation site, but also the willingness of all devices to expose what could be proprietary function parameters in their objective functions.", "Finally, a centralized solution to the OPSP requires solving an optimization problem with $\\Omega (|\\mathcal {T}|T)$ variables, which leads to an identical lower bound on the time scaling for a centralized solver, even if problem structure is exploited.", "As a result, the centralized solver cannot scale to solve the OPSP on very large networks." ], [ "Numerical examples", "We illustrate the speed and scaling of prox-average message passing with a range of numerical examples.", "In the first two sections, we describe how we generate network instances for our examples.", "We then describe our implementation, showing how multithreading can exploit problem parallelism and how our method would scale in a fully peer-to-peer implementation.", "Lastly, we present our results, and demonstrate how the number of prox-average iterations needed for convergence is essentially independent of network size and also significantly decreases when the algorithm is seeded with a reasonable warm-start." ], [ "Network topology", "We generate a network instance by first picking the number of nets $N$ .", "We generate the locations $x_i \\in {\\mbox{\\bf R}}^2$ , $i=1,\\ldots , N$ by drawing them uniformly at random from $[0,\\sqrt{N}]^2$ .", "(These locations will be used to determine network topology.)", "Next, we introduce transmission lines into the network as follows.", "We first connect a transmission line between all pairs of nets $i$ and $j$ independently and with probability $\\gamma (i,j) = \\alpha \\min (1,d^2/\\Vert x_i - x_j\\Vert _2^2).$ In this way, when the distance between $i$ and $j$ is smaller than $d$ , they are connected with a fixed probability $\\alpha > 0$ , and when they are located farther than distance $d$ apart, the probability decays as $1/\\Vert x_i-x_j\\Vert _2^2$ .", "After this process, we add a transmission line between any isolated net and its nearest neighbor.", "We then introduce transmission lines between distinct connected components by selecting two connected components uniformly at random and then selecting two nets, one inside each component, uniformly at random and connecting them by a transmission line.", "We continue this process until the network is connected.", "For the network instances we present, we chose parameter values $d =0.15$ and $\\alpha = 0.8$ as the parameters for generating our network.", "This results in networks with an average degree of $2.3$ .", "Using these parameters, we generated networks with 100 to 100000 nets, which resulted in optimization problems with approximately 30 thousand to 30 million variables.", "Figure: A sample random network.", "Devices are color-coded:generators are in green, batteries are in blue, and loads are inred.", "Edges represent transmission lines." ], [ "Devices", "After we generate the network topology described above, we randomly attach a single (one-terminal) device to each net according to the distribution in table REF .", "The models for each device in the network are identical to the ones given in section , with model parameters chosen in a manner we describe below.", "Figure REF shows an example network for $N = 100$ (30 thousand variables) generated in this fashion.", "For simplicity, our examples only include networks with the devices listed below.", "For all devices, the time horizon was chosen to be $T=96$ , indicating 15 minute intervals for a 24 hour power schedule, with the time period $\\tau =1$ corresponding to midnight.", "Table: Fraction of one-terminal devices present in the generated networks." ], [ "Generator.", "Generators have the quadratic cost functions given in § and are divided into three types: small, medium, and large.", "In each case, we allow the generator to be turned on and off by setting $P_\\mathrm {min} = 0$ .", "Small generators have the smallest maximum power output, but the largest ramp rates, while large generators have the largest maximum power output, but the slowest ramp rates.", "Medium generators lie in between.", "Large generators are generally more efficient than small and medium generators which is reflected in their cost function by having smaller values of $\\alpha $ and $\\beta $ .", "Whenever a generator is placed into a network, its type is selected uniformly at random, and its parameters are taken from the appropriate row in table REF .", "Table: Generator parameters." ], [ "Battery.", "For a given instance of a battery, its parameters are generated by setting $q^\\mathrm {init} = 0$ and selecting $Q^\\mathrm {max}$ uniformly at random from the interval $[20,50]$ .", "The charging and discharging rates are selected to be equal (i.e., $C^\\mathrm {max} = D^\\mathrm {max}$ ) and drawn uniformly at random from the interval $[5,10]$ ." ], [ "Fixed load.", "The load profile for a fixed load instance is a sinusoid, $l(\\tau ) = c + a\\sin (2\\pi (\\tau - \\phi _0)/T), \\quad \\tau =1, \\ldots , T,$ with the amplitude $a$ chosen uniformly at random from the interval $[1,5]$ , and the DC term $c$ chosen so that $c = a + u$ , where $u$ is chosen uniformly at random from the interval $[0,0.5]$ , which ensures that the load profile remains elementwise positive.", "The phase shift $\\phi _0$ is chosen uniformly at random from the interval $[60,72]$ , ensuring that the load profile peaks between the hours of 3pm and 6pm." ], [ "Deferrable load.", "For an instance of a deferrable load, we choose $E$ uniformly at random from the interval $[500,1000]$ .", "The start time index $A$ is chosen uniformly at random from the discrete set $\\lbrace 1,\\ldots , T-7\\rbrace $ .", "The end time index $D$ is then chosen uniformly at random over the set $\\lbrace A+7,\\ldots ,T\\rbrace $ , so that the minimum time window to satisfy the load is 8 time periods (2 hours).", "We set the maximum power so that it is possible to satisfy the total energy constraint by only operating in half of the available time periods, i.e., $L^\\mathrm {max} = 2E/(D-A)$ ." ], [ "Curtailable loads.", "For an instance of a curtailable load, the desired load $l$ is constant over all time periods with a magnitude chosen uniformly at random from the interval $[5, 15]$ .", "The penalty parameter $\\alpha $ is chosen uniformly at random from the interval $[1,2]$ ." ], [ "Transmission line.", "For an instance of a transmission line, we choose its parameters by first solving the OPSP with lossless, uncapacitated lines, where we add a small quadratic cost function $\\epsilon (p_1^2 + p_2^2)$ , with $\\epsilon = 10^{-3}$ , to each transmission line in order to help spread power flow throughout the network.", "Using the flow values given by the solution to that problem, we set $C^\\mathrm {max} = \\max (10, 4 F^\\mathrm {max})$ for each line, where $F^\\mathrm {max}$ is equal to the maximum flow (from the solution to the lossless problem) along that line over all time periods.", "We use the loss function for transmission lines with a series admittance $g - ib$ given by (REF ).", "We choose $g$ and $b$ such that $b = \\gamma g$ , where $\\gamma $ is chosen uniformly at random from the interval $[4.5,5.5]$ ; on average, the susceptance is 5 times larger than the conductance.", "After we pick $C^\\mathrm {max}$ and $\\gamma $ , the values of $g$ and $b$ are chosen such that the loss when transmitting power at maximum capacity is uniformly at random between 5 to 15 percent of $C^\\mathrm {max}$ ." ], [ "Serial multithreaded implementation", "Our OPSP solver is implemented in C++, with the core prox-average message passing equations occupying fewer than 25 lines of C++ (excluding problem setup and classes).", "The code is compiled with gcc $4.4.5$ on an 8-core, $3.4$ GHz Intel Xeon processor with 16GB of RAM running the Debian OS.", "We used the compiler option -O3 to leverage full code optimization.", "To approximate a fully distributed implementation, we use gcc's implementation of OpenMP (version $3.0$ ) and multithreading to parallelize the computation of the prox functions for the devices.", "We use 8 threads (one per core) to solve each example.", "Assuming perfect load balancing, this means that 8 prox functions are being evaluated in parallel.", "Effectively, we evaluate the prox functions by stepping serially through the devices in blocks of size 8.", "We do not, however, parallelize the computation of the dual update over the nets since the overhead of spawning threads dominates the vector operation itself.", "The prox functions for fixed loads and curtailable loads are separable over $\\tau $ and can be computed analytically.", "For more complex devices, such as a generator, battery, or deferrable load, we compute the prox function using CVXGEN [28].", "The prox function for a transmission line is computed by projecting onto the convex hull of the line constraints.", "For a given network, we solve the associated OPSP with an absolute tolerance $\\epsilon ^\\mathrm {abs} = 10^{-3}$ .", "This translates to three digits of accuracy in the solution.", "The CVXGEN solvers used to evaluate the prox operators for some devices have an absolute tolerance of $10^{-8}$ .", "For our $\\rho $ -update function, $h$ , we use the parameter values $\\lambda = 0.005$ and $\\mu = 0.01$ and clip our values of $\\rho $ to be between $\\epsilon ^\\mathrm {abs}$ and $1/\\epsilon ^\\mathrm {abs}$ to prevent roundoff error." ], [ "Peer-to-peer implementation", "We have not yet created a fully peer-to-peer, bulk synchronous parallel [36], [26] implementation of prox-average message passing, but have carefully tracked prox-average solve times in our serial implementation in order to facilitate a first order analysis of such a system.", "In a peer-to-peer implementation, the proximal power schedule updates occur in parallel across all devices followed by (scaled) price updates occurring in parallel across all nets.", "As previously mentioned, the computation time per prox-average iteration is thus the maximum time, over all devices, to evaluate the proximal function of their objective, added to the maximum time across all nets to average their terminal power schedules and add that quantity to their existing price vector.", "Since evaluating the prox function for some devices requires solving a convex optimization problem, whereas the price update only requires a small number of vector operations that can be performed as a handful of SIMD instructions, the compute time for the price update is negligible in comparison to the proximal power schedule update.", "The determining factor in solve time, then, is in evaluating the prox functions for the power schedule update.", "In our examples, the maximum time taken to evaluate any prox function is 1 ms. To solve a problem with $N=100000$ nets (30 million variables) with approximately 500 iterations of our prox-average algorithm then takes only 500 ms.", "In practice, the actual solve time would clearly be dominated by network communication latencies and actual runtime performance will be determined by how quickly and reliably packets can be delivered.", "As a result, in a true peer-to-peer implementation, a negligible amount of time is actually spent on computation.", "However, it goes without saying that many other issues must be addressed with a peer-to-peer protocol, including handling network delays and security." ], [ "Results", "We first consider a single example: a network instance with $N=3000$ (1 million variables).", "Figure REF shows that after fewer than 500 iterations of prox-average message passing, both the relative suboptimality and the average net power imbalance are both less than $10^{-3}$ .", "The convergence rates for other network instances over the range of sizes we simulated are similar.", "Figure: The relative suboptimality (left) and primalinfeasibility (right) of prox-average message passing on anetwork instance with N=3000N=3000 nets (1 million variables).", "The dashedline shows when the stopping criterion is satisfied.In figure REF , we present average timing results for solving the OPSP for a family of examples, with networks of size $N=100$ , 300, 1000, 3000, 10000, 30000, and 100000.", "For each network size, we generated and solved 20 network instances to generate average solve times and confidence intervals around those averages.", "For network instances with $N=100000$ nets, the problem has approximately 30 million variables, which we solve serially using prox-average message passing in 52 minutes on average.", "For a peer-to-peer implementation, the runtime of prox-average message passing should be essentially constant, and in particular independent of the size of the network.", "For our multithreaded implementation, with bounded computation, this would be reflected by a runtime that only increases linearly with the number of nets in a network instance.", "By fitting a line to figure REF , we find that our parallel implementation scales as $O(N^{0.923})$ .", "The slight discrepancy between this and the ideal exponent of 1 is accounted for by implementation details such as operating system background processes consuming some compute cycles and slightly imperfect load balancing across all 8 cores in our system, especially for smaller values of $N$ .", "Figure: Average prox-average execution times for a family ofnetworks on 8 cores.", "Error bars show 95%95\\% confidence bounds.", "Thedotted line shows the least-squares fit to the data on a log-logscale, resulting in an exponent of 0.9230.923.We note that figure REF shows cold start runtimes for solving the OPSP.", "If we have access to a good estimate of the power schedules and locational marginal prices for each terminal, we can use these estimates to warm start our OPSP solver.", "To show the effect of warm-starting, we solve a specific problem instance with $N=3000$ nets (1 million variables).", "We define $K^\\mathrm {cold}$ to be the number of iterations needed to solve an instance of this problem.", "We then uniformly scale the load profiles of each device by separate and independent lognormal random variables.", "The new load profiles, $\\hat{l}$ , are obtained from the original load profiles $l$ according to $\\hat{l} = l \\exp (\\sigma X),$ where $X \\sim \\mathcal {N}(0,1)$ , and $\\sigma >0$ is given.", "Using the solution of the original problem to warm start our solver, we solve the perturbed problem and report the number of iterations $K^\\mathrm {warm}$ needed to solve it.", "Figure REF shows the ratio $K^\\mathrm {warm}/K^\\mathrm {cold}$ as we vary $\\sigma $ , and indicates the significant computational savings that warm-starting can achieve, even under relatively large perturbations.", "Figure: Number of warm start iterations needed to converge for various perturbations of load profiles." ], [ "Receding horizon control.", "The speed with which prox-average message passing converges on very large networks shows its applicability in coordinating real time decisions across massive networks of devices.", "A direct extension of our work to real-time network operation can be achieved using receding horizon control (RHC) [27], [2].", "In RHC, we solve the OPSP at each time step to determine conditional consumption and generation profiles for every device over the next $T$ time periods.", "We then execute the first step of these profiles and resolve the OPSP using new information and measurements that have become available.", "RHC has been successfully applied to a wide range of areas, including power systems, and allows us to take advantage of warm starting our algorithm, which we have shown to significantly decrease the number of iterations needed for convergence." ], [ "Hierarchical models.", "The power gird has a natural hierarchy, with generation and transmission occurring at the highest level and residential consumption and distribution occurring at the most granular.", "Prox-average message passing can be easily extended into hierarchical interactions by scheduling messages on different time scales and between systems at the similar levels of the hierarchy [9].", "We can recursively apply prox-average message passing at each level of the hierarchy.", "At the highest level, all regional systems exchange their proximal updates once they have computed their own prox-function.", "It can be shown that computing this function for a given region is equivalent to computing a partial minimization over the sum of the objective functions of devices located inside that region, subject to intra-region power balance.", "This too can be computed using prox-average message passing.", "This process can be continued down to the individual device level, at which point the device must solve its own prox function directly as the base case." ], [ "Local stopping criteria and $\\rho $ updates.", "The stopping criterion and the algorithm we propose to update $\\rho $ in § both currently require global device coordination — specifically the global values of the primal and dual residuals at each iteration.", "In principle, these could be computed in a decentralized fashion across the network by gossip algorithms [33], but that would require many rounds of gossip in between each iteration of prox-average message passing, significantly increasing runtime.", "We are currently investigating methods by which both the stopping criterion and different values of $\\rho $ can be independently chosen by individual devices or even individual terminals, all based only on local information, such as the primal and dual residuals of a given device and its neighboring nets." ], [ "Conclusion", "We have presented a fully decentralized method for dynamic network energy management based on message passing between devices.", "Prox-average message passing is simple and highly extensible, relying solely on peer to peer communication between devices that exchange energy.", "When the resulting network optimization problem is convex, prox-average message passing converges to the optimal value and gives optimal locational marginal prices.", "We have presented a parallel implementation that shows the time per iteration and the number of iterations needed for convergence of prox-average message passing are essentially independent of the size of the network.", "As a result, prox-average message passing can scale to extremely large networks with almost no increase in solve time." ], [ "Acknowledgments", "The authors thank Yang Wang for extensive discussions on the problem formulation as well as ADMM methods; Yang Wang and Brendan O'Donoghue for help with the $\\rho $ update method; and Ram Rajagopal and Trudie Wang for helpful comments.", "This research was supported in part by Precourt 1140458-1-WPIAE, by AFOSR grant FA9550-09-1-0704, by AFOSR grant FA9550-09-0130, and by NASA grant NNX07AEIIA." ] ]

[ [ "A search for 21 cm HI absorption in AT20G compact radio galaxies" ], [ "Abstract We present results from a search for 21 cm associated HI absorption in a sample of 29 radio sources selected from the Australia Telescope 20 GHz survey.", "Observations were conducted using the Australia Telescope Compact Array Broadband Backend, with which we can simultaneously look for 21 cm absorption in a redshift range of 0.04 < z < 0.08, with a velocity resolution of 7 km/s .", "In preparation for future large-scale H I absorption surveys we test a spectral-line finding method based on Bayesian inference.", "We use this to assign significance to our detections and to determine the best-fitting number of spectral-line components.", "We find that the automated spectral-line search is limited by residuals in the continuum, both from the band-pass calibration and spectral-ripple subtraction, at spectral-line widths of \\Deltav_FWHM > 103 km/s .", "Using this technique we detect two new absorbers and a third, previously known, yielding a 10 per cent detection rate.", "Of the detections, the spectral-line profiles are consistent with the theory that we are seeing different orientations of the absorbing gas, in both the host galaxy and circumnuclear disc, with respect to our line-of-sight to the source.", "In order to spatially resolve the spectral-line components in the two new detections, and so verify this conclusion, we require further high-resolution 21 cm observations (~0.01 arcsec) using very long baseline interferometry." ], [ "Introduction", "Observations of the 21 cm H i line of neutral atomic hydrogen in absorption against bright radio continuum sources can provide a unique probe of the distribution and kinematics of gas in the innermost regions of radio galaxies.", "For a fixed background continuum source brightness the detection limit for H i and molecular absorption lines is independent of the redshift of the absorber.", "Therefore measurements of associated H i absorption in active galaxies are sensitive to relatively small amounts of neutral gas, provided the central radio continuum source is sufficiently strong and compact.", "H i absorption-line measurements can detect both circumnuclear gas discs and large scale gas, including inflows and outflows related to to Active Galactic Nuclei (AGN) fuelling and feedback (e.g.", "; ; ).", "Recent studies by Morganti and colleagues have found broad, shallow absorption-line components arising from the large-scale gas outflows in radio galaxies (; ), which provide some of the most direct evidence for AGN-driven feedback in massive galaxies .", "Previous work may suggest that the detection rate of associated H i absorption is highest in the most compact radio galaxies, searched for H i absorption in 23 nearby ($z<0.22$ ) radio galaxies from the southern 2 Jy sample , from which they obtained five detections (an overall detection rate of 22 per cent).", "They found that the H i detection rate in compact radio galaxies was much higher than in classical FR-i and broad emission-line FR-ii systems.", "Furthermore detected H i absorption in 19 (33 per cent) of a sample of 57 compact radio galaxies at $z<0.85$ .", "show that the integrated optical depth for 21 cm H i absorption increases with decreasing source size, suggesting that the probability of detection might be highest for the more compact sources.", "However show that this relationship might break down for the most compact (high-frequency peaker) sources.", "Recent searches for associated H i absorption in nearby galaxies have also found relatively high numbers of detections by targetting compact radio sources (e.g.", "; ).", "In the work presented here we use the recently-completed Australia Telescope 20 GHz survey to extend these studies to a sample of nearby ($z<0.08$ ) compact sources, which are selected at high frequency and so are expected to be the youngest and most recently-triggered radio AGN in the local Universe.", "This provides an important complement to earlier studies, allowing us to improve our knowledge of the local population of associated H i absorption-line systems.", "tested an automated spectral-line finding method, based on Bayesian inference and using simulated data from the Australian Square Kilometre Array Pathfinder , as part of the preparation for the First Large Absorption Survey in H i (FLASH)http://www.physics.usyd.edu.au/sifa/Main/FLASH.", "The data obtained here, from the Compact Array Broadband Backend on the Australia Telescope Compact Array (ATCA), provide another test of this spectral-line finding method.", "We use these data to characterize any limitations that can arise from sources of systematic error, such as imperfect band-pass calibration and continuum subtraction.", "Throughout this paper we adopt a flat $\\Lambda $ CDM cosmology with $\\Omega _\\mathrm {M} = 0.27$ , $\\Omega _{\\Lambda } = 0.73$ and $\\mathrm {H}_{0} = 71$  km s$^{-1}$  Mpc$^{-1}$ .", "All uncertainties refer to the 68.3 per cent confidence interval, unless otherwise stated." ] ]

[ [ "Similar zone-center gaps in the low-energy spin-wave spectra of NaFeAs\n and BaFe2As2" ], [ "Abstract We report results of inelastic-neutron-scattering measurements of low-energy spin-wave excitations in two structurally distinct families of iron-pnictide parent compounds: Na(1-{\\delta})FeAs and BaFe2As2.", "Despite their very different values of the ordered magnetic moment and N\\'eel temperatures, T_N, in the antiferromagnetic state both compounds exhibit similar spin gaps of the order of 10 meV at the magnetic Brillouin-zone center.", "The gap opens sharply below T_N, with no signatures of a precursor gap at temperatures between the orthorhombic and magnetic phase transitions in Na(1-{\\delta})FeAs.", "We also find a relatively weak dispersion of the spin-wave gap in BaFe2As2 along the out-of-plane momentum component, q_z.", "At the magnetic zone boundary (q_z = 0), spin excitations in the ordered state persist down to 20 meV, which implies a much smaller value of the effective out-of-plane exchange interaction, J_c, as compared to previous estimates based on fitting the high-energy spin-wave dispersion to a Heisenberg-type model." ], [ "=1 Similar zone-center gaps in the low-energy spin-wave spectra of Na$_{1-\\delta }$ FeAs and BaFe$_2$ As$_2$ J. T.", "Park$^{\\!,}$ [FirstAuth]$^*$ G. Friemel Max-Planck-Institut für Festkörperforschung, Heisenbergstraße 1, 70569 Stuttgart, Germany $^{\\!,}$ [FirstAuth]$^*$ T. Loew Max-Planck-Institut für Festkörperforschung, Heisenbergstraße 1, 70569 Stuttgart, Germany V. Hinkov Quantum Matter Institute, University of British Columbia, Vancouver, B.C.", "V6T 1Z1, Canada Max-Planck-Institut für Festkörperforschung, Heisenbergstraße 1, 70569 Stuttgart, Germany Yuan Li Max-Planck-Institut für Festkörperforschung, Heisenbergstraße 1, 70569 Stuttgart, Germany International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China B. H. Min Department of Emerging Materials Science, DGIST, Daegu 711-873, Republic of Korea D. L. Sun Max-Planck-Institut für Festkörperforschung, Heisenbergstraße 1, 70569 Stuttgart, Germany A. Ivanov Institut Laue-Langevin, 6 rue Jules Horowitz, 38042 Grenoble Cedex 9, France A. Piovano Institut Laue-Langevin, 6 rue Jules Horowitz, 38042 Grenoble Cedex 9, France C. T. Lin Max-Planck-Institut für Festkörperforschung, Heisenbergstraße 1, 70569 Stuttgart, Germany B. Keimer Max-Planck-Institut für Festkörperforschung, Heisenbergstraße 1, 70569 Stuttgart, Germany Y. S. Kwon$^{4,\\hspace{0.5pt}}$ [CorrAuthor1]$^\\dagger $ D. S. Inosov$^{1,\\,}$ [CorrAuthor2]$^\\ddagger $ We report results of inelastic-neutron-scattering measurements of low-energy spin-wave excitations in two structurally distinct families of iron-pnictide parent compounds: Na$_{1-\\delta }$ FeAs and BaFe$_2$ As$_2$ .", "Despite their very different values of the ordered magnetic moment and Néel temperatures, $T_{\\rm N}$ , in the antiferromagnetic state both compounds exhibit similar spin gaps of the order of 10 meV at the magnetic Brillouin-zone center.", "The gap opens sharply below $T_{\\rm N}$ , with no signatures of a precursor gap at temperatures between the orthorhombic and magnetic phase transitions in Na$_{1-\\delta }$ FeAs.", "We also find a relatively weak dispersion of the spin-wave gap in BaFe$_2$ As$_2$ along the out-of-plane momentum component, $q_z$ .", "At the magnetic zone boundary ($q_z=0$ ), spin excitations in the ordered state persist down to $\\sim $  20 meV, which implies a much smaller value of the effective out-of-plane exchange interaction, $J_{\\rm c}$ , as compared to previous estimates based on fitting the high-energy spin-wave dispersion to a Heisenberg-type model.", "spin waves, magnetic excitations, antiferromagnetism, anisotropy gap, inelastic neutron scattering, iron pnictide superconductors 75.30.Ds 74.70.Xa 78.70.Nx The discovery of unconventional superconductivity in iron-pnictide compounds [1] with critical temperatures, $T_{\\rm c}$ , as high as 56 K has fostered a tremendous interest in these materials in recent years [2].", "There are several structurally distinct families of iron-based superconductors with similar phase diagrams [2], governed by an interplay of antiferromagnetism, persistent in pure compounds under ambient pressure, and superconductivity that can be induced by pressure or chemical doping [3].", "Although the highest values of $T_{\\rm c}$ are usually found in doped compounds with a nonstoichiometric composition, our physical understanding of these systems undoubtedly depends on the detailed knowledge of magnetic properties in the respective parent compounds, which are also easier to treat theoretically due to their simple crystal structure with no substitutional disorder.", "Among the variety of such stoichiometric materials serving as parent phases for numerous iron-based superconductors, only a few have so far received proper experimental treatment, especially by inelastic neutron scattering (INS), due to miscellaneous reasons related to the availability of sizeable single crystals or their chemical stability.", "For instance, to the best of our knowledge, direct measurements of spin-wave excitations in the antiferromagnetic (AFM) state of iron pnictides have so far remained limited to a few members of the so-called `122' family with the ThCr$_2$ Si$_2$ -type structure, whose single crystals are typically stable in air and are readily available in the large sizes necessary for INS experiments.", "In particular, high-energy spin-wave modes have been mapped out in CaFe$_2$ As$_2$ [4], [5] and BaFe$_2$ As$_2$ [6] using time-of-flight (TOF) neutron spectroscopy, which enabled estimations of the effective magnetic exchange interactions in the framework of a localized Heisenberg-type $J_{1a}$  - $J_{1b}$  - $J_{2}$  - $J_{c}$ model.", "These results are complemented by INS measurements at lower energies, performed on polycrystalline BaFe$_2$ As$_2$ [7] and on single crystals of SrFe$_2$ As$_2$ [8], CaFe$_2$ As$_2$ [9], [10], and BaFe$_2$ As$_2$ [11].", "All of these measurements have unequivocally demonstrated the existence of a large anisotropy gap in the spin-wave dispersion at the magnetic Brillouin-zone center, which varies from 6.5 – 7.0 meV in SrFe$_2$ As$_2$ and CaFe$_2$ As$_2$ [8], [9], [10] to almost 10 meV in BaFe$_2$ As$_2$ [7], [11].", "A more recent polarized INS experiment has revealed two distinct components of this gap in BaFe$_2$ As$_2$ , characterized by the out-of-plane and in-plane polarizations [12], with the onsets at 10 meV and 16 meV, respectively.", "At present, first-principles calculations are faced with apparent difficulties in reproducing these energy scales in the spin-wave spectra even on a qualitative level [13], [12].", "Moreover, the identical crystal structure of all measured compounds, distinct from the structures of other families, precludes generalizations to all iron pnictides, making it difficult to relate the measured gaps to such microscopic structural and magnetic properties of the material as the ordered magnetic moment, exchange interactions, or crystallographic parameters.", "Therefore, in the present study we have performed INS measurements of the low-energy spin-wave spectrum in Na$_{1-\\delta }$ FeAs [14] with the `111'-type structure, which we compare to that of BaFe$_2$ As$_2$ .", "A single crystal of Na$_{1-\\delta }$ FeAs with a small Na deficiency, $\\delta \\approx 0.1$ , as estimated by energy-dispersive x-ray analysis, and a mass of $\\sim $  0.5 g was grown by the vertical Bridgman method [15].", "Pure FeAs$_{1.17}$ precursors were first synthesized from the reaction of high-purity Fe (powder, 99.999%) and As (chips, 99.999%) in sealed quartz containers at 1050$^\\circ $ C. A Na$_{1-\\delta }$ FeAs single crystal was then grown from a mixture of Na and precursor FeAs$_{1.17}$ with a molar ratio of 2:1 in a sealed molybdenum crucible.", "Higher molar concentrations of Na and As were necessary because of their high vapor pressures.", "During growth, the center of the furnace was heated to 1450$^\\circ $ C. Because of the extreme chemical sensitivity of Na$_{1-\\delta }$ FeAs to oxygen and air moisture [16], meticulous care had to be taken to exclude any contact with air while handling the crystal over the entire process of sample preparation and measurements.", "Prior to the INS experiment, the crystal had been sealed inside an aluminum can in helium atmosphere and oriented using a 4-circle neutron diffractometer Morpheus of the Paul Scherrer Institute (PSI), Switzerland.", "These preliminary measurements indicated that the sample consists predominantly of one single-crystalline grain with a mosaicity $<0.5^\\circ $ .", "Our second sample was a mosaic of BaFe$_2$ As$_2$ single crystals with a total mass of $\\sim $  0.9 g, grown by the self-flux method as described elsewhere [17], coaligned on a silicon wafer using x-ray Laue diffraction to a mosaicity $\\lesssim $  1.0$^\\circ $ .3pt A remarkable property of the Na$_{1-\\delta }$ FeAs compounds is the large splitting in temperature between the magnetic and structural phase transitions, which is present even in the parent phase [18], [19], [20], [21], whereas in the `122' family of iron pnictides the two transition temperatures usually merge together upon the reduction of doping [22], [24], [23], [25].", "This makes it possible to study the narrow temperature window between the two transitions, which is typically associated with a so-called “electronic liquid crystal” (or “spin nematic”) phase with a spontaneously broken rotational symmetry of the electronic eigenstates.", "In our Na$_{1-\\delta }$ FeAs sample, the AFM and orthorhombic phase transitions appear as anomalies in the temperature derivative of the magnetization [Fig.", "REF  (a)] and in the order-parameter-like dependencies of the magnetic and nuclear Bragg peak intensities [Fig.", "REF  (b, c)], respectively.", "From these measurements, the corresponding transition temperatures, $T_{\\rm N}=45$  K and $T_{\\rm s}=57$  K, could be determined, which are in agreement with literature values for samples of similar composition [18], [19], [20], [21].", "Although the orthorhombic distortion is too weak to be directly resolved as the splitting of Bragg reflections in our experiment, the abrupt change in the (200) nuclear Bragg intensity at $T_{\\rm s}$ , seen in Fig.", "REF  (c), is explained by the extinction release associated with a minor change in the sample's mosaicity across the orthorhombic transition, and has been previously used by several authors as a convenient and highly sensitive probe of the weak structural distortion [24], [25], [26].", "In addition, our Na$_{1-\\delta }$ FeAs sample exhibits a weak superconducting (SC) transition at $T_{\\rm c,\\,onset}\\approx 8$  K, but the volume fraction of the SC phase is below 10%, according to magnetization measurements [Fig.", "REF  (a), inset], and can be therefore neglected for the purpose of the present study.", "In BaFe$_2$ As$_2$ , both neutron diffraction and magnetization measurements (not shown) have revealed the AFM ordering below $T_{\\rm N}=137$  K, in agreement with previous reports [22].", "Figure: Determination of the structural and AFM transition temperatures in Na 0.9 _{0.9}FeAs from (a) magnetization measurements, (b) magnetic Bragg intensity, and (c) nuclear Bragg intensity.", "Magnetic susceptibility data in the inset of panel (a) show the SC transition.Figure: Typical unprocessed constant-energy scans for Na 0.9 _{0.9}FeAs (left) and BaFe 2 _2As 2 _2 (right), measured above and below T N T_{\\rm N} along the |𝐐|=const|\\mathbf {Q}|=\\text{const} trajectories in momentum space, centered at (1 0.8pt20L)(\\frac{1}{\\protect \\raisebox {0.8pt}{\\scriptsize 2}}0 L).", "The values of LL at the center of the scan and the energy transfer are indicated in every panel.Unprocessed INS data shown in Fig.", "REF illustrate several representative momentum scans along the $|\\mathbf {Q}|=\\text{const}$ trajectories in the $(H0L)$ plane, measured on both samples above and below $T_{\\rm N}$ in the spin-gap region.", "Here and henceforth, we are using the unfolded reciprocal-space notation of the Fe sublattice because of its simplicity and the natural correspondence to the symmetry of the observed signal [27].", "The wave vector $\\mathbf {Q}=(HKL)$ is given in reciprocal-lattice units (r.l.u.", "), i.e.", "in units of the reciprocal-lattice vectors of the Fe sublattice, in which the AFM ordering vector is $\\mathbf {Q}_{\\rm AFM}=(\\frac{1}{\\protect \\raisebox {0.8pt}{\\scriptsize 2}}\\,0\\,\\frac{1}{\\protect \\raisebox {0.8pt}{\\scriptsize 2}})$ .", "These measurements were done with a fixed final neutron momentum, $k_{\\rm f}=2.662$  Å$^{-1}$ , and with a pyrolytic graphite filter to eliminate higher-order reflections from the analyzer.", "In the paramagnetic state (Fig.", "REF , circles), a commensurate peak is observed at the ordering vector down to the lowest energies, indicating the presence of gapless paramagnon excitations.", "Below the AFM ordering temperature, its intensity vanishes in the low-energy region, below the spin gap energy [panels (a) and (c)], or gets partially suppressed at intermediate energies close to the gap onset [panel (b)].", "For BaFe$_2$ As$_2$ , we also show the corresponding scan centered at $L=2$ [at the magnetic zone boundary, panel (d)], where the situation is similar with the exception of a reduced intensity of the signal.", "The presence of a relatively strong paramagnetic intensity on the $(\\frac{1}{\\protect \\raisebox {0.8pt}{\\scriptsize 2}}\\,0\\,L)$ line even far away from the ordering wave vector has been already pointed out in earlier works [11], [10].", "It can be well understood in terms of Fermi surface nesting, which is maximized near $\\mathbf {Q}_{\\rm AFM}$ , but still remains substantial for all values of $L$ , according to band structure calculations [27].", "The detailed temperature dependence of the low-energy INS signals in both samples is illustrated by Fig.", "REF .", "Again, one can appreciate the similarity between the magnetic response measured at $\\mathbf {Q}_{\\rm AFM}$ (half-integer $L$ ) and at the magnetic zone boundary (integer $L$ ) in BaFe$_2$ As$_2$ at an energy transfer of 6 meV [Fig.", "REF  (b)].", "Despite the somewhat lower amplitude of the signal at $(\\frac{1}{\\protect \\raisebox {0.8pt}{\\scriptsize 2}}\\,0\\,2)$ as compared to $(\\frac{1}{\\protect \\raisebox {0.8pt}{\\scriptsize 2}}\\,0\\,\\frac{3}{\\protect \\raisebox {0.8pt}{\\scriptsize 2}})$ , both increase monotonically upon cooling towards $T_{\\rm N}$ , where they exhibit a sharp kink due to the onset of the spin gap in the AFM ordered state.", "This anomaly is much sharper at $\\mathbf {Q}_{\\rm AFM}$ due to the critical scattering intensity at the ordering wave vector around $T_{\\rm N}$ .", "It is remarkable that despite the presence of two well separated phase transitions in Na$_{1-\\delta }$ FeAs, the low-energy magnetic spectrum is only sensitive to the AFM transition and exhibits no pronounced anomaly at $T_{\\rm s}$ outside of our experimental uncertainty [Fig.", "REF  (a)].", "It is interesting to consider this observation in the framework of various models describing the electronic “nematic” state that was claimed responsible for the orthorhombic distortion in the narrow temperature range $T_{\\rm N} < T < T_{\\rm s}$ [28].", "Since a comprehensive theory of such a “nematic” electronic state in iron arsenides still awaits development, no definitive predictions about its magnetic excitation spectrum have been proposed.", "It has been recently suggested, however, that with the onset of a preemptive nematic order, the magnetic correlation length should discontinuously increase below $T_{\\rm s}$ , leading to a possible spectral weight redistribution and a consequent formation of a pseudogap [29].", "This could possibly result in a partial precursor gapping of the low-energy spin excitations already below $T_{\\rm s}$ , which is not confirmed by our measurements.", "Figure: Temperature dependence of the background-subtracted low-energy INS signal in Na 0.9 _{0.9}FeAs (left) and BaFe 2 _2As 2 _2 (right), indicating the abrupt spin-gap opening below T N T_{\\rm N}.", "Solid lines are guides to the eyes.", "Note the absence of any pronounced anomaly at the structural phase transition in Na 0.9 _{0.9}FeAs (T s =57T_{\\rm s}=57 K).In Fig.", "REF , we plot the energy dependence of the background-subtracted magnetic intensity, obtained from Gaussian fits of the momentum scans similar to those shown in Fig.", "REF (larger symbols) or from 3-point measurements, in which the background intensity was obtained at two points on both sides of the peak (smaller symbols).", "Measurements with $k_{\\rm f}=2.662$  Å$^{-1}$ , $k_{\\rm f}=3.837$  Å$^{-1}$ and $k_{\\rm f}=4.1$  Å$^{-1}$ are shown with different symbols.", "In the paramagnetic state, a gapless spectrum of spin fluctuations with nearly energy-independent intensity is observed both at integer and half-integer $L$ .", "As the temperature is decreased below the AFM transition, the low-energy spectral weight is transferred to higher energies, resulting in a clear spin gap in the magnetic excitation spectrum (see also Fig.", "REF ).", "At $\\mathbf {Q}_{\\rm AFM}$ (half-integer $L$ ), the onset of magnetic intensity at $T=1.5$  K is observed at approximately $\\sim $  10 meV both in the Na$_{0.9}$ FeAs and BaFe$_2$ As$_2$ compounds, so that the low-temperature spectra in Figs.", "REF  (a) and (b) are nearly indistinguishable within the experimental uncertainty.", "Based on the recent polarized INS measurements [12], we can ascribe this onset to the smaller out-of-plane anisotropy gap.", "The onset of the in-plane scattering that occurs at a slightly higher energy can not be resolved as a separate step in our unpolarized data.", "Figure: Energy dependence of the background-subtracted magnetic INS intensity for Na 0.9 _{0.9}FeAs (left) and BaFe 2 _2As 2 _2 (right), measured above and below T N T_{\\rm N}, (a, b) at the ordering wave vector (L=3/2L=3/2) and (c) at the magnetic zone boundary (L=1,2L=1,\\,2).At the magnetic zone boundary (integer $L$ ), the gap in BaFe$_2$ As$_2$ is only twice larger than at the zone center and amounts to approximately 20 meV, in agreement with the assumptions of Ref. HarrigerSchneidewind09.", "This clearly refutes the commonly accepted viewpoint that zone-boundary spin waves in BaFe$_2$ As$_2$ are limited to high energies of the order of 50 – 100 meV [6], [12].", "In terms of the Heisenberg-type exchange interaction constants, our result suggests a much weaker out-of-plane exchange coupling, $J_{\\rm c}$ , than previously estimated from fitting high-energy TOF data to the anisotropic Heisenberg model [6].", "Indeed, assuming the in-plane exchange interactions ($J_{\\rm 1a}$ , $J_{\\rm 1b}$ , and $J_{\\rm 2}$ ) of Ref.", "HarrigerLuo11 to be unchanged, we can use the zone-center and zone-boundary gap magnitudes from our present study (10 and 20 meV, respectively) to re-estimate the two remaining parameters in the model: the effective out-of-plane exchange energy, $SJ_{\\rm c}=0.22$  meV, and the single ion anisotropy constant, $SJ_{\\rm s}=0.14$  meV.", "This reevaluated value of $J_{\\rm c}$ is almost one order of magnitude smaller than previously reported [6], leading to a better agreement with the spin-wave velocities estimated from the nuclear-magnetic-resonance data [31].", "Table: Comparison of the Néel temperatures (T N T_{\\rm N}), values of the ordered magnetic moment (μ Fe \\mu _{\\rm Fe}), and zone-center gap energies (Δ 𝐐 AFM \\Delta _{\\mathbf {Q}_{\\rm AFM}}) in several iron-arsenide compounds.In Table REF , we have summarized several parameters of the AFM state, such as the Néel temperature ($T_{\\rm N}$ ), value of the ordered magnetic moment ($\\mu _{\\rm Fe}$ ), and zone-center gap energy ($\\Delta _{\\mathbf {Q}_{\\rm AFM}}$ ) for various iron-arsenide compounds.", "First, we observe that despite the tenfold difference in the ordered magnetic moment and a much lower ordering temperature, Na$_{1-\\delta }$ FeAs exhibits an anisotropy gap of the same order of magnitude as the materials of the `122' family.", "This is highly unusual, as theory predicts the anisotropy gap in spin-density-wave compounds to increase monotonically with the value of the sublattice magnetization, following a simple power law [35], whereas our observations point to an anomalous behavior of the magneto-crystalline anisotropy in iron pnictides that is inconsistent with this general trend.", "Second, we note that in `122'-compounds the spin gap is rapidly reduced upon doping Ni or Co into the Fe plane.", "As a result, in the doping range, where superconductivity coexists with static AFM order, spin fluctuations can already be observed below $T_{\\rm N}$ at energies as low as 2–3 meV [30], [36].", "Whenever these fluctuations extend below the energy of the SC gap, $2\\Delta $ , they can possibly serve as a source of spectral weight for the formation of a magnetic resonant mode below $T_{\\rm c}$ , which has been reported even in strongly underdoped `122' samples with $T_{\\rm c}$ as low as 11 K [36].", "In contrast to this scenario, superconductivity in the phase diagram of doped NaFeAs is found in the immediate vicinity of the parent phase [20], [21], where we have shown the anisotropy gap to be much larger than $2\\Delta $ .", "Indeed, for our sample with $T_{\\rm c}=8$  K, even the exceptionally high ratio of $2\\Delta /k_{\\rm B}T_{\\rm c} \\approx 8$ reported by Liu et al.", "[37] would result in $2\\Delta $ of only 5.5 meV, whereas the weak-coupling ratio of $2\\Delta /k_{\\rm B}T_{\\rm c} = 3.53$ yields $2\\Delta = 2.4$  meV, which is 4 times smaller than the magnetic anisotropy gap.", "Therefore, if the conventional scenario for the formation of the magnetic resonant mode also holds in the Na-111 family of superconductors, only 1–2% of Co or Ni would have to induce a substantial rearrangement of the low-energy magnetic spectral weight in NaFeAs, in order to reduce the zone-center gap and lead to a finite magnetic intensity below $2\\Delta $ .", "Further INS experiments on doped samples are required to explore the interplay between magnetism and superconductivity in this system.", "The authors are grateful to G. Khaliullin, A. Yaresko, N. Shannon, and A. Smerald for fruitful discussions and thank J.", "White and C. Busch for the assistance in sample characterization.", "All presented INS data were collected using the triple-axis thermal-neutron spectrometer IN8 at the Institut Laue-Langevin (ILL) in Grenoble.", "This work has been supported, in part, by the DFG within the Schwerpunktprogramm 1458, under Grant No.", "BO3537/1-1, and by the MPI – UBC Center for Quantum Materials.", "B. H. M. and Y. S. K. acknowledge support from the Basic Science Research Program (Grant No.", "2010-0007487) and the Mid-career Researcher Program (Grant No.", "2010-0007487) through NRF funded by the Ministry of Education, Science and Technology of Korea.", "Y. L. acknowledges support from the Alexander von Humboldt Foundation." ] ]

[ [ "Optical and X-ray Transients from Planet-Star Mergers" ], [ "Abstract We evaluate the prompt observational signatures of the merger between a massive close-in planet (a `hot Jupiter') and its host star, events with an estimated Galactic rate of ~0.1-1/yr.", "Depending on the ratio of the mean density of the planet rho_p to that of the star rho_star, a merger results in three possible outcomes.", "If rho_p/rho_star > 5, then the planet directly plunges below the stellar atmosphere before being disrupted by tidal forces.", "Dissipation of orbital energy creates a hot wake behind the planet, producing a EUV/soft X-ray transient as the planet sinks below the stellar surface.", "The peak luminosity L_X ~ 1e36 erg/s is achieved weeks-months prior to merger, after which the stellar surface is enshrouded by an outflow.", "The final inspiral is accompanied by an optical transient powered by the recombination of hydrogen in the outflow, which peaks at L~1e37-38 erg/s on a timescale ~days.", "If instead rho_planet/rho_star < 5, then Roche Lobe overflow occurs above the stellar surface.", "For rho_p/rho_star < 1, mass transfer is stable, resulting the planet being accreted on a relatively slow timescale.", "However, for 1 < rho_p/rho_star < 5, mass transfer may instead be unstable, resulting in the planet being dynamically disrupted into an accretion disk around the star.", "Super-Eddington outflows from the disk power an optical transient with L~1e37-38 erg/s and characteristic duration ~week-months.", "The disk itself becomes visible once the accretion rate become sub-Eddington, resulting in a bolometric brightening and spectral shift to the UV.", "Optical transients from planet merger events may resemble classical novae, but are distinguished by lower ejecta mass and velocity ~100s km/s, and by hard pre- and post-cursor emission, respectively.", "Promising search strategies include combined optical, UV, and X-ray surveys of nearby massive galaxies with cadences from days to months." ], [ "Introduction", "Theoretical models of planet formation suggest that gas giants form beyond the ice line at radii $\\mathrel {\\hbox{\\unknown.", "{\\hbox{$\\sim $}}}\\hbox{$>$}}$$ several AU from their central star (e.g.~\\cite {Inaba+03}).", "It is thus a mystery how some exoplanets, colloquially known as `hot Jupiters^{\\prime }, are transported to their current locations at radii $$\\sim $$<$ 0.5$ AU (e.g.~\\cite {Ida&Lin04}).", "The misalignment between the orbital planes of some hot Jupiters and the spin axes of their host stars (\\cite {Triaud+10}; \\cite {Schlaufman10}; \\cite {Winn+10}) suggests that not all migrate via interaction with the proto-stellar disk alone (although see \\cite {Foucart&Lai11}).", "At least some hot Jupiters appear to have migrated via other mechanisms, such as planet-planet scattering (e.g.~\\cite {Rasio&Ford96}; \\cite {Weidenschilling&Marzari96}; \\cite {Juric&Tremaine08}) or the \\cite {Kozai62} mechanism (e.g.~\\cite {Takeda&Rasio05}; \\cite {Fabrycky&Tremaine07}; \\cite {Socrates+11}).", "In both cases the planet initially approaches small radii on an eccentric trajectory, before tidal dissipation circularizes the orbit.$, on the other hand, argue that the current locations of many close-in exoplanets are inconsistent with migration resulting solely due to the inward scattering from original orbits exterior to the ice line, since the small pericenter distances implied by such trajectories would have resulted either in tidal-disruption of the planet or have completely ejected it from the system.", "One interpretation of their result is that some planets have migrated further after their orbit is circularized, due to tidal dissipation within the star.", "The angular momentum of most hot Jupiters is sufficiently low that no state of tidal equilibrium exists (the system is ` unstable'; ; ), suggesting that the ultimate `end state' of tidal dissipation is a merger between the planet and its host star (e.g. ).", "Depending on the quality factor of tidal dissipation within the star $Q^{\\prime }_{\\star } \\sim 10^{6}$ , the semi-major axes of several known hot Jupiters are sufficiently small that a merger will indeed occur on a relatively short timescale $\\mathrel {\\hbox{\\unknown.", "{\\hbox{$\\sim $}}}\\hbox{$<$}}$ 10-100$ Myr (e.g.~\\cite {Li+10}).", "The lack of old planetary systems with very short orbital periods hints that mergers indeed result from tidal orbital decay (\\cite {Jackson+09}).$ One might expect that since $\\sim 1$ per cent of stars host hot Jupiters (e.g.", "), and since the Galactic star-formation rate is $\\sim 1-10$ yr $^{-1}$ , then the Galactic rate of planet-star mergers should be $\\mathrel {\\hbox{\\unknown.", "{\\hbox{$\\sim $}}}\\hbox{$<$}}$ 0.1$ yr$ -1$.", "However, the actual merger rate of planets with main sequence stars can in principle be significantly higher $ 1$ yr$ -1$, depending on the value of $ Q'$ and the rate at which hot Jupiters are replenished by migration from the outer stellar system (e.g.~\\cite {Socrates+11}; see $ §$).", "A comparable or greater number of mergers may occur in evolved stellar systems, since the timescale for tidal migration $ plunge R-5$ (eq.~[\\ref {eq:plungetime}]) depends sensitively on stellar radius $ R$, which increases as the star evolves off the main sequence.", "In fact, irrespective of the efficiency of tidal dissipation, it seems inevitable that almost all planets with main-sequence orbital separation $$\\sim $$<$ 1$ AU eventually merger with their stars, since stars expand dramatically during the red giant and asymptotic giant branch phases of evolution (although post-main-sequence mergers are significantly less energetic events than their main sequence brethren).$ In this paper we explore the direct observational consequences of the merger of gas giant planets with their host stars.", "Depending on the properties of the planet-star system, we show that the merger event can be violent, resulting a bright and long-lived electromagnetic signature.", "Given the rapid pace of recent advances in our knowledge of exoplanets, and the advent of sensitive wide-field surveys across the electromagnetic spectrum, now is an optimal time to evaluate the transient signatures of planet-star mergers and how best to go about detecting them.", "This paper is organized as follows.", "In $§\\ref {sec:freq}$ we summarize the properties of known hot Jupiters and use them to estimate the rate of planet-star mergers in galaxies similar to the Milky Way.", "In § we identify three qualitatively different merger outcomes, which are distinguished in part by whether the planet fills its Roche radius at an orbital separation $a_{\\rm t}$ which is larger or smaller, respectively, than the stellar radius $R_{\\star }$ .", "If $a_{t} \\mathrel {\\hbox{\\unknown.", "{\\hbox{$\\sim $}}}\\hbox{$<$}}$ R$ then the planet is not disrupted until it has fallen below the stellar photosphere.", "In $ §$ we describe the dynamics of the planetary inspiral and argue that such `direct-impact^{\\prime } merger events are accompanied by a extreme ultraviolet (EUV)/soft X-ray transient that originates from the hot gas created behind the planet as it penetrates the stellar atmosphere ($ §REF $).", "This emission, which ceases weeks$ -$months prior to merger, is immediately followed by rising optical emission powered by the recombination of hydrogen in an outflow from the stellar surface, which peaks on a timescale $$days after the planet plunges below the stellar surface ($ §REF $).$ If $a_{t} \\mathrel {\\hbox{\\unknown.", "{\\hbox{$\\sim $}}}\\hbox{$>$}}$ R$, then the planet instead overflows its Roche radius above the stellar surface.", "If mass transfer is stable, then the planet is consumed on the relatively long timescale set by the rate of tidal dissipation; this results in a relatively short-lived mass-transferring binary, but is unlikely to produce a bright electromagnetic display.", "If, on the other hand, mass transfer is unstable (or if the initial planet trajectory is eccentric), then the planet is disrupted into an accretion disk just outside the stellar surface (`tidal-disruption event^{\\prime }).", "Thermal radiation from the disk, and from super-Eddington outflows from its surface, powers a bright optical-UV transient ($ §$).", "In $ §$ we discuss our results, including a concise summary of the predicted transients from planet-star mergers ($ §REF $); detection prospects and strategies, in particular with surveys of nearby galaxies at optical, UV, and X-ray wavelengths ($ §REF $); and a brief conclusion, which focuses on unresolved theoretical issues and future work ($ §REF $).$ Our work here builds on, and is broadly consistent with, past theoretical work into the observational signatures of binary stellar mergers (e.g.", "; ), events which refer to as `mergebursts' (see also )." ], [ "Galactic rate of Planet-Star Mergers", "Dissipation of tides raised on a planet and on its host star can act to change their orbital separation.", "Two important categories of tidal interactions can be termed “circularization tides” and “synchronization tides.” Dissipation of the former kind acts to damp eccentricity at constant orbital angular momentum $\\mathcal {L}$ .", "Since $\\mathcal {L} \\propto \\sqrt{a(1-e^2)}$ , where $a$ is semi-major axis and $e$ is eccentricity, circularization of the orbit ($e \\rightarrow 0$ ) at constant $\\mathcal {L}$ shrinks the binary separation.", "Dissipation of synchronization tides instead occurs due to asynchronous rotation of the bodies with respect to the orbital mean motion, even in the absence of eccentricity.", "In particular, if a planet orbits faster than its host star rotates, then tidal torques transfer angular momentum from the planet to the star, leading to orbital decay (; ; ).", "The characteristic timescale for this process (the tidal `plunging time' $\\tau _{\\rm plunge}$ ) is found by integrating the tidal dissipation equations assuming a slowly rotating star (; ), which can be estimated as (see also Appendix ) $\\tau _{\\rm plunge} & \\sim & \\frac{4}{117} \\frac{Q^{\\prime }_\\star }{n} \\left( \\frac{a}{R_\\star } \\right)^5 \\frac{M_\\star }{M_{\\rm p}} \\\\\\nonumber & \\approx & 1.5\\times 10^{5}{\\rm \\,yr}\\left(\\frac{Q^{\\prime }_\\star }{10^6} \\right)\\left(\\frac{M_{\\star }}{M_{\\odot }}\\right)^{1/2}\\left(\\frac{a}{2R_{\\odot }}\\right)^{13/2} \\left( \\frac{R_\\star }{R_\\odot } \\right)^{-5}\\left( \\frac{M_{\\rm p}}{M_{\\rm J}} \\right)^{-1},$ where $Q^{\\prime }_{\\star } \\equiv 3Q_{\\star }/k_2$ is the modified tidal quality factor of the star ($Q_{\\star }$ is the specific tidal dissipation function and $k_2$ is the Love number); $n \\equiv 2\\pi / P \\simeq (GM_{\\star }/a^{3})^{1/2}$ is the orbital mean motion; $P$ is the orbital period; $a$ is the orbital semi-major axis; $R_{\\star }$ is the stellar radius; and $M_\\star $ and $M_{\\rm p}$ are the stellar and planetary masses, respectively, where the latter is normalized to the mass of Jupiter $M_{\\rm J}$ .", "Figure REF shows the cumulative number of planets as a function of tidal plunge time, calculated by applying equation (REF ) to the distribution of $\\sim 160$ known transiting extrasolar planets ().http://exoplanet.eu This $\\tau _{\\rm plunge}$ distribution can be used to estimate the rate of planet-star mergers if one can extrapolate the known sample of exoplanets to those of the galaxy as a whole.", "In order to make this conversion, we assume that the observational selection function is given simply by the geometric probability of transit ($R_\\star /a$ ).", "We also assume that a fraction $\\sim 10^{-5}$ of all stars in the Milky Way have been searched by transit surveys (G. Bakos, private communication), such that each transiting planet used in Figure REF “counts” as $10^5(a/R_{\\star })$ planets in the Galaxy.", "Dashed lines in Figure REF represent the expected plunge-time distribution if one assumes that mergers occur with a steady-state rate of $10^6/Q^{\\prime }_{\\star }$ yr$^{-1}$ and $0.5(10^6/Q^{\\prime }_{\\star })$ yr$^{-1}$ , respectively.", "A comparison of the observed distribution to these models indicate that, at low values of plunge time $\\sim 10^{6-7}(Q^{\\prime }_{\\star }/10^6)$  yrs, roughly one merger occurs in the Galaxy every $2(Q^{\\prime }_{\\star }/10^6)$ years.", "Obvious deviations of the cumulative rate from the model predictions occur at large values $\\tau _{\\rm plunge}$ .", "These indicate the onset of additional effects not included in our simple model, such as a dependence of $Q^{\\prime }_{\\star }$ on orbital period (e.g.", "); the onset of additional selection effects; or a break-down of the steady-state assumption.", "The latter could result from the ongoing injection of planets from larger radii (e.g.", "), or by insitu planet formation following a binary star merger (e.g. ).", "Although our estimate of the merger rate from Figure REF is uncertain by at least an order of magnitude, the fiducial range of values $\\sim 0.1-1$ yr$^{-1}$ that we find are still sufficiently high that transient surveys with duration comparable to this mean interval $\\sim $ year—decade could in principle detect such events within the Milky Way or nearby galaxies, provided that the resulting emission is sufficiently bright.", "As mentioned in the Introduction, the rate of planet-star mergers could be enhanced as a result of post main sequence evolution.", "After the star evolves off of the main sequence, its larger radius $R_\\star $ significantly decreases the plunging timescale $\\tau _{\\rm plunge} \\propto R_{\\star }^{-5}$ (eq.", "[REF ]; even in the subgiant phase, for which $R_{\\star } \\sim $ few $R_{\\odot }$ ), more than compensating for the shorter duration of post main sequence evolution.", "This suggests that a sizable fraction of all stars with semi-major axes within $a \\mathrel {\\hbox{\\unknown.", "{\\hbox{$\\sim $}}}\\hbox{$<$}}$$ 0.1AU may be consumed (\\cite {Sato+08}; \\cite {Hansen10}; \\cite {Lloyd11}).", "Since the fraction of stars with planets in this radius range approaches unity \\cite {Mayor+11}, the Galactic rate of planet-star mergers may approach a significant fraction of the star-formation rate $ 1-10$ yr$ -1$.$ Mergers are likely to be even more common during the giant phase when the stellar radius expands to $R_{\\star } \\sim $ AU (e.g.", ", ; ).", "The low density of the stellar envelope in this case would almost certainly result in what we define as a direct-impact merger ($§\\ref {sec:interior}$ ).", "However, the much lower gravitational energy released near the stellar surface in such an event as compared to the merger with a dwarf or subgiant results in much fainter transient emission that is more challenging to detect (although these events may have other observable consequences; ; ; ).", "For this reason we focus on planet mergers with compact stars, even though their total rate may be somewhat lower.", "Figure: Cumulative number of planet-star mergers in the Galaxy (solid blue line) as a function of plunging time τ plunge \\tau _{\\rm plunge} (eq.", "[]), the latter normalized to a fiducial value Q ☆ ' ∼10 6 Q^{\\prime }_{\\star } \\sim 10^{6} for the rate of tidal dissipation in the star.", "Overlaid dashed lines indicate the expected distributions for steady-state merger rates of (10 6 /Q ☆ ' )(10^6/Q^{\\prime }_{\\star }) yr -1 ^{-1} (red) and 0.5(10 6 /Q ☆ ' )0.5(10^6/Q^{\\prime }_{\\star }) yr -1 ^{-1} (green), respectively.", "Deviations of the model predictions from the observed distribution at larger values of τ plunge \\tau _{\\rm plunge} indicate the presence of additional effects not included in our model, such as a dependence of Q ☆ ' Q^{\\prime }_{\\star } on orbital period or a break-down of the steady state assumption (see text for further details)." ], [ "Three Possible Merger Outcomes", "So long as no mass transfer occurs between the planet and the star, tidal forces control the rate at which the orbit decays.", "As the planet inspirals towards the star, its fate depends on whether it fills its Roche-lobe above or below the stellar surface.", "In the latter case the planet plunges directly into the stellar atmosphere.", "This produces an event that we define as a `direct-impact' merger event, the result of which we describe in detail in $§\\ref {sec:interior}$ .", "If Roche-lobe overflow instead begins above the stellar surface, then the planet is either dynamically disrupted into a compact disk around the star (`tidal-disruption' merger event; $§\\ref {sec:exterior}$ ), or it gradually transfers its mass over a much longer timescale (stable mass transfer).", "The volume of the Roche-lobe of the planet in a synchronous orbit about a star of much greater mass is approximately given by $V_{\\rm RL}\\simeq 0.5 (M_{\\rm p}/M_{\\star })a^3$ .", "Roche-lobe overflow thus begins when the orbital semi-major axis reaches the value $a_t\\simeq 2R_{\\star }(\\bar{\\rho }_{\\star }/\\bar{\\rho }_{\\rm p})^{1/3},$ where $\\bar{\\rho }_{\\star } = 3M_{\\star }/4\\pi R_{\\star }^{3}$ and $\\bar{\\rho }_{\\rm p} = 3M_{\\rm p}/4\\pi R_{\\rm p}^{3}$ are the mean density of the star and planet, respectively.", "At the orbital separation given by equation (REF ) the L1 Lagrange point is located at a radial distance $X_t \\simeq 0.7 (M_{\\rm p}/M_{\\star })^{1/3}a_t\\simeq 1.4 R_{\\rm p}$ from the center of mass of the planet.", "The condition that Roche-lobe overflow occurs entirely below the stellar surface is thus given by $a_t<R_{\\star }+X_t$ , or $2 (\\bar{\\rho }_{\\star }/\\bar{\\rho }_{\\rm p})^{1/3}<1+1.4(R_{\\rm p}/R_{\\star }).$ If equation (REF ) is satisfied, then a direct-impact merger occurs ($§\\ref {sec:interior}$ ).", "If Roche-lobe overflow instead occurs outside the star (i.e.", "if eq.", "[REF ] is not satisfied), then the evolution of the system is more subtle.", "In this case mass transfer is stable(unstable) when mass loss from the planet results in the Roche-lobe growing faster(slower) than the radius of the planet.", "Here we describe the qualitative distinction between the system evolution in these two cases, the details of which are provided in Appendix .", "Table: Outcomes of planet-star mergers based on known transiting planetsIf overflow occurs when the orbital separation is sufficiently large, then material crossing L1 has sufficient specific angular momentum to form a disk well above the stellar surface.", "Viscous redistribution of angular momentum within the disk causes most of its mass to accrete onto the star, whereas the majority of the angular momentum is transported to large radii where it can be transferred back into the orbit via tides exerted by the planet.", "Since mass is transferred from the less massive planet to the more massive star, this (approximate) conservation of orbital angular momentum acts to widen the orbital separation.", "Since the Roche-lobe volume $V_{\\rm RL}\\propto a^3$ increases rapidly as the orbit expands, mass transfer is temporarily halted.", "Orbital decay due to tidal dissipation then again drives the system together, resulting in a series of overflow episodes that slowly exhaust the planetary material on the tidal decay timescale $\\tau _{\\rm plunge}(a = a_{t} \\mathrel {\\hbox{\\unknown.", "{\\hbox{$\\sim $}}}\\hbox{$>$}}$ R) $\\sim $$>$ 103$ yr (eq.~[\\ref {eq:plungetime}]).", "This is the stable mass transfer case.$ If, on the other hand, Roche-lobe overflow occurs only at a sufficiently small planet-star separation, then material crossing L1 plunges directly onto, or circularizes just above, the stellar surface.", "In this case the bulk of the mass and angular momentum of the accreting material is deposited into the stellar envelope, which is capable of accepting it due to its considerable inertia.", "Since orbital angular momentum is lost to the star, the semi-major axis of the orbit (and hence the Roche volume of the planet) remains unchanged, whereas the planet expands upon mass loss (Appendix ).", "This results in unstable mass transfer and hence to dynamical tidal-disruption of the planet on a characteristic timescale of several orbital periods $\\sim $ hours ($§\\ref {sec:exterior}$ ).", "Figure REF shows the sample of transiting planets used in Figure REF in the space of mean stellar density $\\bar{\\rho }_{\\star }$ versus mean planetary density $\\bar{\\rho }_{\\rm p}$ .", "Each planet is marked according to both the value of its tidal plunging time (eq.", "[REF ]) and by our best estimate of its ultimate fate upon merger (`direct-impact'; `stable mass transfer'; or `tidal-disruption').", "The criteria for the latter are evaluated assuming that the current masses and radii of the planets and their host stars are identical to those at the time of merger (the validity of which we discuss in $§\\ref {sec:conclusions}$ ).", "We find that if the density ratio is sufficiently high ($\\bar{\\rho }_{\\rm p}/\\bar{\\rho }_{\\star }\\mathbin {\\unknown.", "{>}}$ 5$), then the the planet is disrupted below the stellar surface (direct-impact mergers).", "Planets somewhat denser than their host star ($ 1" ] ]

[ [ "Stochastic finite differences and multilevel Monte Carlo for a class of\n SPDEs in finance" ], [ "Abstract In this article, we propose a Milstein finite difference scheme for a stochastic partial differential equation (SPDE) describing a large particle system.", "We show, by means of Fourier analysis, that the discretisation on an unbounded domain is convergent of first order in the timestep and second order in the spatial grid size, and that the discretisation is stable with respect to boundary data.", "Numerical experiments clearly indicate that the same convergence order also holds for boundary-value problems.", "Multilevel path simulation, previously used for SDEs, is shown to give substantial complexity gains compared to a standard discretisation of the SPDE or direct simulation of the particle system.", "We derive complexity bounds and illustrate the results by an application to basket credit derivatives." ], [ "Introduction", "Various stochastic partial differential equations (SPDEs) have emerged over the last two decades in different areas of mathematical finance.", "A classical example is the Heath-Jarrow-Morton interest rate model [19] of the form ${\\rm d} r(x,t) &=& \\frac{\\partial }{\\partial x}\\left(r(t,x) +\\frac{1}{2} \\left| \\textstyle \\int _t^{t+x}\\sigma (t,u){\\,\\rm d}u\\right|^2\\right) {\\,\\rm d}t + \\sigma (t,t+x) {\\,\\rm d}W_t,$ where $r(x,t)$ is the forward rate of tenor $x$ at time $t$ and $\\sigma (t,t+x)$ its instantaneous volatility.", "In [3], a similar equation has been proposed more recently to model electricity forwards.", "Most of the SPDEs studied share with (REF ) the property that the derivatives of the solution only appear in the drift term; in the case of (REF ) the volatility of the Brownian driver does not depend on the solution $r$ at all.", "Numerical methods for hyperbolic SPDEs of the type (REF ) have been studied, for example, in [29].", "This article, in contrast, considers the parabolic SPDE $\\, {\\rm d} v= -\\mu \\, \\frac{\\partial v}{\\partial x} \\, {\\rm d} t+ \\frac{1}{2} \\, \\frac{\\partial ^2 v }{\\partial x^2} \\, {\\rm d} t- \\sqrt{\\rho }\\, \\frac{\\partial v}{\\partial x} \\, {\\rm d} M_t,$ where $M$ is a standard Brownian motion, and $\\mu $ and $0\\le \\rho \\le 1$ are real-valued parameters.", "It is clear that the behaviour of this equation is fundamentally different from those with additive or multiplicative noise.", "The significance of (REF ) for the following applications is that it describes the limiting density of a large system of exchangeable particles.", "Specifically, if we consider the system of SDEs $\\, {\\rm d} X_t^i = \\mu \\, {\\rm d} t+ \\sqrt{1 \\!-\\!", "\\rho }\\, \\, {\\rm d} W_t^i + \\sqrt{\\rho }\\, \\, {\\rm d} M_t,$ for $1\\le i \\le N$ , with $\\langle \\, {\\rm d} W_t^i, \\, {\\rm d} W_t^j\\rangle = \\delta _{ij}$ and $\\langle \\, {\\rm d} W_t^i, \\, {\\rm d} M_t\\rangle = 0$ , where $X_0^i$ are assumed i.i.d.", "with finite second moment, the empirical measure $\\nu _t^N = \\frac{1}{N} \\sum _{i=1}^N \\delta (\\cdot -X_t^i)$ has a limit $\\nu _t$ for $N\\rightarrow \\infty $ , whose density $v$ satisfies (REF ) in a weak sense.", "For a derivation of this result in the more general context of quasi-linear PDEs see [24].", "While the motivation in [24] is to use a large particle system (REF ) to approximate the solution to the SPDE (REF ), our view point is to use (REF ) as an approximate model for a large particle system, and we will argue later the (computational) advantages of this approach in situations when the number of particles is large.", "As a first possible application, one may consider $X^i$ as the log price processes of a basket of equities, which have idiosyncratic components $W^i$ and share a common driver $M$ (the “market”).", "If the size of the basket is large enough, the solution to the SPDE can be used to find the values of basket derivatives.", "In this paper, we study an application of a similar model to basket credit derivatives.", "We mention in passing that equations of the form (REF ) arise also in stochastic filtering.", "To be precise, (REF ) is the Zakai equation for the distribution of a signal $X$ given observation of $M$ , see e.g.", "[1].", "It is interesting to note that the solution to the SPDE (REF ) without boundary conditions can be written as the solution of the PDE $\\frac{\\partial u}{\\partial t} = \\frac{1}{2} (1\\!-\\!\\rho )\\, \\frac{\\partial ^2 u}{\\partial x^2}- \\mu \\, \\frac{\\partial u}{\\partial x},$ shifted by the current value of the Brownian driver, $v(t,x) = u(t, x\\!-\\!\\sqrt{\\rho }\\,M_t).$ In particular, if $v(0,x)=\\delta (x_0\\!-\\!x)$ , then $v(T,x) = \\frac{1}{\\sqrt{2\\pi \\, (1\\!-\\!\\rho )\\,T}}\\ \\exp \\left( -\\ \\frac{(\\rule {0in}{0.16in}x - x_0 - \\mu \\, T - \\sqrt{\\rho } \\, M_T)^2}{2\\,(1\\!-\\!\\rho )\\,T} \\right).$ The intuitive interpretation of this result is that the independent Brownian motions $W^i$ have averaged into a deterministic diffusion in the infinite particle limit, whereas the common factor $M$ , which moves all processes in parallel, shifts the whole profile (and also adds to the diffusion, via the Itô term).", "In [8], the analysis of the large particle system is extended to cases with absorption at the boundary ($x=0$ ), $\\nonumber X_t^i &=& 0,\\quad t\\ge T_0^i, \\\\T_0^i &=& \\inf \\lbrace t: X_t^i=0\\rbrace .$ It is shown that there is still a limit measure $\\nu _t$ , which may now be decomposed as $\\nu _t = \\nu _t^+ + L_t \\delta _0,$ where $L_t = \\lim _{N\\rightarrow \\infty } L_t^N =\\lim _{N\\rightarrow \\infty } \\frac{1}{N} \\sum _{i=1}^N 1_{T_0^i\\le t}$ is the proportion of absorbed particles (the “loss function”), and the density $v$ of $\\nu _t^+$ satisfies (REF ) in $(0,\\infty )$ with absorbing boundary condition $v(t,0) = 0.$ [8] consider applications to basket credit derivatives.", "For the market pricing examples, they consider a simplified model, where defaults are monitored only at a discrete set of dates.", "Between these times, the default barrier is inactive and (REF ) is solved on the real line by using (REF ) and (REF ).", "For the initial-boundary value problem (REF ), (REF ), however, such a semi-analytic solution strategy is no longer possible and an efficient numerical method is needed.", "Moreover, there is a loss of regularity at the boundary in this case, such that $x u_{xx} \\in L_2$ but not $u_{xx}$ , as is documented in [23].", "Recent papers on the numerical solution of SPDEs deal with cases relevant to ours, yet structurally crucially different.", "A comprehensive analysis of finite difference and finite element discretisations of the stochastic heat equation with multiplicative white noise and non-linear driving term is given in [18], [17] and [33], respectively.", "[25] shows a Lax equivalence theorem for the SDE ${\\rm d} X_t = A X_t \\, {\\rm d} t+ G(X_t) \\, {\\rm d} M_t,$ in a Hilbert space, driven by a process $M$ from a class including Brownian motion, where $A$ is a suitable (e.g.", "elliptic differential) operator and $G$ a Lipschitz function.", "In this paper, we propose a Milstein finite difference discretisation for (REF ) and analyse its stability and convergence in the mean-square sense by Fourier analysis.", "A main consideration of this paper is the computational complexity of the proposed methods, and we will demonstrate that a multilevel approach achieves a cost for the SPDE simulation no larger than that of direct Monte Carlo sampling from a known univariate distribution, $O(\\varepsilon ^{-2})$ for r.m.s.", "accuracy $\\epsilon $ , and is in that sense optimal.", "Multilevel Monte Carlo path simulation, first introduced in [13], is an efficient technique for computing expected values of path-dependent payoffs arising from the solution of SDEs.", "It is based on a multilevel decomposition of Brownian paths, similar to a Brownian Bridge construction.", "The complexity gain can be explained by the observation that the variance of high-level corrections – involving a large number of timesteps – is typically small, and consequently only a relatively small number Monte Carlo samples is required to estimate these contributions to an acceptable accuracy.", "Overall, for SDEs, if a r.m.s.", "accuracy of $\\varepsilon $ is required, the standard Monte Carlo method requires $O(\\varepsilon ^{-3})$ operations, whereas the multilevel method based on the Milstein discretisation [14] requires $O(\\varepsilon ^{-2})$ operations.", "The first extension of the multilevel approach to SPDEs was for parabolic PDEs with a multiplicative noise term [16].", "There have also been recent extensions to elliptic PDEs with random coefficients [2], [11].", "Our approach, for a rather different parabolic SPDE, is similar to the previous work on SDEs and SPDEs in that the solution is decomposed into a hierarchy with increasing resolution in both time and space.", "Provided the variance of the multilevel corrections decreases at a sufficiently high rate as one moves to higher levels of refinement, the number of fine grid Monte Carlo simulations which is required is greatly reduced.", "Indeed, the total cost is only $O(\\varepsilon ^{-2})$ to achieve a r.m.s.", "accuracy of $\\varepsilon $ compared to an $O(\\varepsilon ^{-7/2})$ cost for the standard approach which combines a finite difference discretisation of the spatial derivative terms and a Milstein discretisation of the stochastic integrals.", "The rest of the paper is structured as follows.", "Section outlines the finite difference scheme used, and analyses its accuracy and stability in the standard Monte Carlo approach.", "Section presents the modification to multilevel path simulation of functionals of the solution.", "Numerical experiments for a CDO tranche pricing application are given in section , providing empirical support for the postulated properties of the scheme and demonstrating the computational gains achieved.", "Section discusses the benefits over standard Monte Carlo simulation of particle systems and outlines extensions." ], [ "Milstein finite differences", "Integrating (REF ) over the time interval $[t,t\\!+\\!k]$ gives $v(t\\!+\\!k,x) = v(t,x) + \\int _t^{t+k} \\left( -\\, \\mu \\frac{\\partial v}{\\partial x}+ \\frac{1}{2}\\, \\frac{\\partial ^2 v }{\\partial x^2}\\right) \\, {\\rm d} s-\\int _t^{t+k} \\!\\!\\sqrt{\\rho }\\ \\frac{\\partial v}{\\partial x}\\, \\, {\\rm d} M_s.$ Making the approximation $v(s,x) \\!\\approx \\!", "v(t,x)$ for $t\\!<\\!s\\!<\\!t\\!+\\!k$ in the first integral and $v(s,x) \\approx v(t,x) - \\sqrt{\\rho } \\left.\\frac{\\partial v}{\\partial x}\\right|_{(t,x)} (M_s\\!-\\!M_t)$ in the second, and noting the standard Itô calculus result that $\\int _t^{t+k} (M_s\\!-\\!M_t)\\, \\, {\\rm d} M_s = \\frac{1}{2} \\left(\\rule {0in}{0.16in}(\\Delta M^n)^2 \\!-\\!", "k\\right),$ where $\\Delta M^n \\equiv M_{t+k}-M_t = \\sqrt{k}\\, Z_n$ with $Z_n \\sim N(0,1)$ , [15], [22], leads to the Milstein semi-discrete approximation $v^{n+1}(x) = v^n(x) - (\\mu \\, k +\\sqrt{\\rho \\, k}\\, Z_n)\\, \\frac{\\partial v^n}{\\partial x}+ \\frac{1}{2} \\left(\\rule {0in}{0.16in} (1\\!-\\!\\rho )\\, k + \\rho \\, k\\, Z_n^2\\right)\\frac{\\partial ^2 v^n }{\\partial x^2}.$ Using a spatial grid with uniform spacing $h$ , standard central difference approximations to the spatial derivatives [28] then give the finite difference equation $v_j^{n+1} &=& v_j^n\\ -\\ \\frac{\\mu \\, k + \\sqrt{\\rho \\, k}\\, Z_n}{2h} \\left(v_{j+1}^n - v_{j-1}^n\\right)\\nonumber \\\\&&~~~~ +\\ \\frac{(1\\!-\\!\\rho )\\, k + \\rho \\, k\\, Z_n^2}{2h^2}\\left(v_{j+1}^n - 2 v_j^n + v_{j-1}^n\\right),$ in which $v_j^n$ is an approximation to $v(nk,jh)$ .", "The spatial domain is truncated by introducing an upper boundary at $x_{max} \\!=\\!", "J\\,h$ and using the boundary condition $v_J^n\\!=\\!0$ .", "Since the initial distribution will be assumed localised, both the localisation error for a given path $M$ , and the expected error of functionals of the solution, can be made as small as needed by a large enough choice $x_{max}>0$ .", "The system of SDEs can then be written in matrix-vector form $V_{n+1} &=& V_n\\ -\\ \\frac{\\mu \\, k + \\sqrt{\\rho \\, k}\\, Z_n}{2h} D_1 V_n+\\ \\frac{(1\\!-\\!\\rho )\\, k + \\rho \\, k\\, Z_n^2}{2 h^2} D_2 V_n,$ where $V_n$ is the vector with elements $v_j^n, j=1, \\ldots , J\\!-\\!1$ and $D_1$ and $D_2$ are the matrices corresponding to first and second central differences as explicitly given in Appendix .", "Remark 2.1 An alternative discretisation arises if the spatial derivatives are discretised first, and the Milstein scheme is applied to the resulting system of SDEs.", "The practical implication is that the Itô term then contains a second finite difference with twice the step size, $D_1^2$ instead of $D_2$ , resulting in pentadiagonal discretisation matrices instead of tridiagonal ones, specifically $V_{n+1} = V_n - \\frac{\\mu \\, k + \\sqrt{\\rho \\, k}\\, Z_n}{2h} D_1 V_n+ \\frac{k}{2 h^2} D_2 V_n+ \\frac{(\\rho \\, k\\, (Z_n^2-1)}{2 h^2} D_1^2 V_n.$ The accuracy and stability of the two schemes are similar, hence we will not go into details.", "To approximate the initial condition on the grid, we apply the initial measure $\\nu ^N_0$ to a basis of `hat functions' $\\left\\langle \\Psi _j\\right\\rangle _{0<j<J}$ , where $\\Psi _j(x) = \\frac{1}{h}\\max \\left( h-|x-x_j|,0 \\right),$ giving $v_j^0 = \\langle \\Psi _j, v_0 \\rangle =\\int _{-\\infty }^{\\infty }\\Psi _j(x)\\ v_0(x) {\\, \\rm d}x.$ For the corresponding PDE ($\\rho =0$ ), this is known to result in $O(h^2)$ convergence provided $k$ satisfies a certain stability limit, even when the initial data are not smooth [27], [10].", "We will see that an extension of this analysis holds for SPDEs." ], [ "Fourier stability analysis", "Finite difference Fourier stability analysis ignores the boundary conditions and considers Fourier modes of the form $v_j^n = g_n \\exp (i j \\theta ), \\quad |\\theta | \\le \\pi ,$ which satisfy (REF ) provided $g_{n+1} = \\left( a(\\theta ) + b(\\theta )\\, Z_n + c(\\theta )\\, Z_n^2 \\right) g_n,$ where $a(\\theta ) &=& 1 - \\frac{i \\, \\mu \\, k}{h}\\, \\sin \\theta - \\frac{2\\, (1\\!-\\!\\rho )\\, k}{h^2} \\sin ^2 { \\textstyle \\frac{\\theta }{2} }, \\\\[0.05in]b(\\theta ) &=& -\\, \\frac{i \\sqrt{\\rho \\, k}}{h}\\, \\sin \\theta , \\\\[0.05in]c(\\theta ) &=& -\\, \\frac{2\\, \\rho \\, k}{h^2} \\sin ^2 { \\textstyle \\frac{\\theta }{2} }.$ Following the approach of mean-square stability analysis from [20], [31], we obtain $\\mathbb {E}[\\, |g_{n+1}|^2 ]&=& \\mathbb {E}\\left[ (a + b\\, Z_n + c\\, Z_n^2)(a^* + b^* Z_n + c^* Z_n^2)\\ |g_n|^2 \\right] \\\\[0.1in]&=& \\left( \\, |a\\!+\\!c|^2 + |b|^2 + 2 |c|^2 \\,\\right)\\ \\mathbb {E}\\left[\\, |g_n|^2 \\right],$ where $a^*$ denotes the complex conjugate of $a$ .", "Mean-square stability therefore requires $&& |a|^2 + |b|^2 + 3 |c|^2 + a\\, c^* + a^* c \\\\&& =\\ 1- 4 \\sin ^2 { \\textstyle \\frac{\\theta }{2} } \\left\\lbrace \\frac{k}{h^2}- (1 + 2 \\rho ^2) \\left(\\frac{k}{h^2}\\right)^2 \\sin ^2{ \\textstyle \\frac{\\theta }{2} }- \\left( \\left(\\frac{\\mu \\,k}{h}\\right)^2 + \\frac{\\rho \\, k}{h^2} \\right) \\cos ^2{ \\textstyle \\frac{\\theta }{2} }\\right\\rbrace \\\\&& \\le \\ 1,$ and enforcing this for all $\\theta $ leads to the two conditions summarised in the following theorem.", "Theorem 2.1 The Milstein finite difference scheme (REF ) is stable in the mean-square sense provided $\\mu ^2 k &\\le & 1 - \\rho , \\\\\\frac{k}{h^2} &\\le & (1+2\\rho ^2)^{-1}.$ The analysis in Appendix combines mean-square and matrix stability analysis to prove that in the limit $k,h\\rightarrow 0$ the condition $k/h^2 \\le (1+2\\rho ^2)^{-1}$ is also a sufficient condition for mean-square stability of the initial-boundary value problem with the boundary conditions at $j\\!=\\!0$ and $j\\!=\\!J$ ." ], [ "Fourier analysis of accuracy", "Fourier analysis can also be used to examine the accuracy of the finite difference approximation in the absence of boundary conditions.", "Considering a Fourier mode of the form $g(t) \\exp ( i \\kappa x),$ the PDE (REF ) reveals that $g(t) = g(0) \\ \\exp \\left(\\rule {0in}{0.16in}\\!", "-{ \\textstyle \\frac{1}{2} } (1\\!-\\!\\rho )\\, \\kappa ^2\\, t- i \\, \\kappa \\, (\\mu \\, t + \\sqrt{\\rho }\\, M_t) \\right),$ and therefore $g(t_{n+1}) = g(t_n) \\ \\exp \\left(\\rule {0in}{0.16in}\\!", "-{ \\textstyle \\frac{1}{2} } (1\\!-\\!\\rho )\\, \\kappa ^2 \\, k- i \\,\\kappa \\, (\\mu \\, k \\!+\\!", "\\sqrt{\\rho \\,k}\\, Z_n) \\right),$ where $M_{t_{n+1}}- M_{t_n} = \\sqrt{k}\\, Z_n$ .", "Fourier analysis of the discretisation gives $g_{n+1} = \\left( a(\\kappa h) + b(\\kappa h)\\, Z_n + c(\\kappa h)\\, Z_n^2\\right) g_n,$ where $a(\\theta )$ , $b(\\theta )$ , $c(\\theta )$ are as defined before.", "Writing $a(\\kappa h) + b(\\kappa h)\\, Z_n + c(\\kappa h)\\, Z_n^2 =\\exp \\left(\\rule {0in}{0.16in}\\!", "-{ \\textstyle \\frac{1}{2} } (1\\!-\\!\\rho )\\, \\kappa ^2\\, t- i \\, \\kappa \\, (\\mu \\, k \\!+\\!", "\\sqrt{\\rho \\,k}\\, Z_n) + e_n \\right)$ we obtain, after performing lengthy expansions using MATLAB's Symbolic Toolbox, $e_n = \\sqrt{\\rho \\, k} \\ \\kappa ^2 \\mu \\, k\\, Z_n- { \\textstyle \\frac{1}{6} } i\\, \\sqrt{\\rho \\, k} \\ \\kappa ^3 Z_n\\left(\\rule {0in}{0.16in}3\\, (1\\!-\\!\\rho )\\, k + \\rho \\, k\\, Z_n^2 - h^2\\right)+ r_n,$ where, for all $p\\ge 1$ , $\\mathbb {E}[|r_n|^p] \\le c(\\kappa ,p) \\left(k^2 + k h^2\\right)^p.$ When summing over $T/k$ timesteps, $\\sum Z_n$ and $\\sum Z^3_n$ are both $O(k^{-1/2})$ since they have zero expectation and $O(k^{-1})$ variance.", "Hence, it follows that $\\left( \\mathbb {E}\\left[ \\left( \\sum _n e_n \\right)^2 \\right]\\right)^{1/2}= O(k, h^2),$ so the RMS error in the Fourier mode over the full simulation interval is $O(k, h^2)$ .", "Following the method of analysis in [10], it can be deduced from this that the RMS $L_2$ and $L_\\infty $ errors for $V_n$ are both $O(k,h^2)$ .", "This is consistent with the usual $O(k,h^2)$ accuracy of the forward-time central space discretisation of a parabolic PDE [28], and the $O(k)$ strong accuracy of the Milstein discretisation of an SDE, see e.g.", "[22]." ], [ "Convergence tests", "We now test the accuracy and stability of the scheme numerically.", "The chosen parameters for (REF ) are $\\sigma = 0.22$ , $\\rho = 0.2$ , $r=0.042$ , $\\mu =(r-\\sigma ^2/2)/\\sigma $ , $v(0,x)\\!=\\!\\delta (x\\!-\\!x_0)$ with $x_0 = 5$ .", "These values are typical for the applications later on.", "The upper boundary for the computation is $x_{max}=16$ , and chosen to ensure that the use of zero Dirichlet data has negligible influence on the solution.", "We approximate the initial value problem without boundary conditions on $[-16/3,16]$ (note $x_0=5$ is roughly in the centre), and also the initial-boundary value problem with zero Dirichlet conditions on $[0,16]$ .", "Figure REF plots several solutions to the latter problem at $T=5$ , each corresponding to a different Brownian path $M_t$ .", "Figure: Some solutions v(T,x)v(T,x) for different drivingBrownian motions W t W_t.It can be seen that for those realisations for which $M_t$ has been largely positive, the boundary at $x\\!=\\!0$ has had negligible influence and so the solution is approximately equal to the displaced Normal distribution given by (REF ).", "For the unbounded case, we approximate the mean-square $L_2$ -error by $\\nonumber E(h,k)^2 &=& \\mathbb {E}\\left[\\sum _{j=0}^J (v_j^N(\\omega )-v(N k,j h;\\omega ))^2 \\, h\\right] \\\\&\\approx & \\frac{1}{M} \\sum _{m=1}^M\\sum _{j=0}^J (v_j^N(\\omega _m)-v(N k,j h;\\omega _m))^2 \\, h,$ where $J$ is the number of grid intervals, $N$ the number of timesteps, and the expectation is taken over Brownian paths $\\omega $ , of which $\\omega _m$ are $M$ samples.", "To study the convergence, we introduce decreasing grid sizes $h_l = h_0\\, 2^{-l}$ and timesteps $k_l = k_0\\, 4^{-l}$ , motivated by the second order convergence in $h$ and first order convergence in $k$ as well as the stability limit for the explicit scheme, and denote $E_l = E(h_l,k_l)$ the mean-square $L_2$ -error at level $l$ .", "For the initial-boundary value problem, no analytical solution is known, but we can compute error indicators via the difference between a fine grid solution $f$ with mesh parameters $k$ and $h$ , and a coarse solution $c$ with mesh parameters $4 k$ and $2 h$ , $\\nonumber e(h,k)^2 &=& \\mathbb {E}\\left[\\sum _{j=0}^{J/2} (f_{2j}^{N}(\\omega )-c_j^{N/4}(\\omega ))^2 \\, h\\right]\\\\&\\approx & \\frac{1}{M} \\sum _{m=1}^M\\sum _{j=0}^{J/2} (f_{2j}^N(\\omega _m)-c_j^{N/4}(\\omega _m))^2 \\, h,$ and define $e_l = e(h_l,k_l)$ .", "On the coarsest level, $h_0=4/3$ , $k_0=1/4$ , such that $x_0$ does not coincide with a grid point.", "Fig.", "REF shows both the computed values of $E_l^2$ and $e_l^2$ for the unbounded case, and $e_l^2$ for the bounded case.", "Figure: Mean-square error estimators for theMilstein scheme.For the unbounded case, L 2 L_2 difference between true and numerical solution,E L E_L as in (),and coarse and fine grid solutions, e l e_l as in ();for the bounded case, difference between coarse and fine grid solutionse l e_l.The results confirm the theoretical $O(k,h^2)$ convergence in the unbounded case, and support the conjecture that the convergence order is unchanged in the bounded case.", "We now formalise this conjecture about the error due to the SPDE discretisation.", "Denote $U_T = u(T,\\cdot )$ the solution to the initial boundary value problem at time $T$ for a given Brownian path and $\\widehat{U}_T$ its numerical approximation with grid size $h$ and timestep $k\\propto h^2$ .", "Conjecture 2.1 The error in the solution at time $T$ satisfies the strong error estimate in the $L_2$ norm $\\Vert \\cdot \\Vert $ $\\sqrt{ \\mathbb {E}[ \\Vert \\widehat{U}_T - U_T \\Vert ^2 ] }= O(h^{2}).$ Corollary 2.1 A Lipschitz payoff function $P(U_T)$ has a similar strong error bound $\\sqrt{\\mathbb {E}[ (P(\\widehat{U}_T) - P(U_T))^2 ]} = O(h^{2}).$ We know Conjecture REF to be true for the initial value problem on $\\mathbb {R}$ , and it conjectures that the introduction of the boundary condition at $x\\!=\\!0$ does not affect the (weak and) strong error, which is supported by the numerical results.", "The conjecture assumes $x_{max}\\!=\\!\\infty $ ; in practice, a finite value for $x_{max}$ will introduce an additional truncation error which will decay exponentially in $x_{max}$ .", "Numerical tests with different parameters, not reproduced here, indicate that the error in the bounded case increases for values of $x_0$ close to 0, in which case more paths contribute solutions with large higher derivatives close to the zero boundary, even if the asymptotic convergence order is still $O(k,h^2)$ .", "We analyse this loss of regularity further in Appendix ." ], [ "Multilevel Monte Carlo simulation", "We now consider estimating the expectation of a scalar functional of the SPDE solution, an expected “payoff” $P$ in a computational finance context.", "Defining $\\widehat{P}$ to be the payoff arising from a single SPDE approximation, the standard Monte Carlo approach is to average the payoff from $N$ independent SPDE simulations, each one using an independent vector of $N(0,1)$ Normal variables $Z = (Z^0, Z^1, \\ldots , Z^{n-1})$ .", "Thus the estimator for the expected value is $\\widehat{Y}= \\frac{1}{N} \\sum _{i=1}^N \\widehat{P}^{(i)}.$ The mean square error for this estimator can be expressed as the sum of two contributions, one due to the variance of the estimator and the other due to the error in its expectation, $\\mathbb {E}\\left[ \\left( \\widehat{Y}- \\mathbb {E}[P] \\right)^2 \\right]= N^{-1} \\mathbb {V}[\\widehat{P}] + \\left( \\mathbb {E}[\\widehat{P}] \\!-\\!", "\\mathbb {E}[P] \\right)^2.$ To achieve a r.m.s.", "error of $\\varepsilon $ requires that both of these terms are $O(\\varepsilon ^2)$ .", "This in turn requires that $N\\!=\\!O(\\varepsilon ^{-2})$ , $k\\!=\\!O(\\varepsilon )$ and $h\\!=\\!O(\\varepsilon ^{1/2})$ , based on the conjecture that the weak error $\\mathbb {E}[\\widehat{P}] \\!-\\!", "\\mathbb {E}[P]$ is $O(k,h^2)$ .", "Since the computational cost is proportional to $N k^{-1} h^{-1}$ this implies an overall cost which is $O(\\varepsilon ^{-7/2})$ .", "The aim of the multilevel Monte Carlo simulation is to reduce this complexity to $O(\\varepsilon ^{-2})$ .", "Consider Monte Carlo simulations with different levels of refinement, $l = 0, 1, \\ldots , L$ , with $l=0$ being the coarsest level, (i.e.", "the largest values for $k$ and $h$ ) and level $L$ being the finest level corresponding to that used by the standard Monte Carlo method.", "Let $\\widehat{P}_l$ denote an approximation to payoff $P$ using a numerical discretisation with parameters $k_l$ and $h_l$ .", "Because of the linearity of the expectation operator, it is clearly true that $\\mathbb {E}[\\widehat{P}_L] = \\mathbb {E}[\\widehat{P}_0] + \\sum _{l=1}^L \\mathbb {E}[\\widehat{P}_l \\!-\\!", "\\widehat{P}_{l-1}].$ This expresses the expectation on the finest level as being equal to the expectation on the coarsest level plus a sum of corrections which give the difference in expectation between simulations using different numbers of timesteps.", "The multilevel idea is to independently estimate each of the expectations on the right-hand side in a way which minimises the overall variance for a given computational cost.", "Let $\\widehat{Y}_0$ be an estimator for $\\mathbb {E}[\\widehat{P}_0]$ using $N_0$ samples, and let $\\widehat{Y}_l$ for $l\\!>\\!0$ be an estimator for $\\mathbb {E}[\\widehat{P}_l \\!-\\!", "\\widehat{P}_{l-1}]$ using $N_l$ samples.", "Each estimator is an average of $N_l$ independent samples, which for $l\\!>\\!0$ is $\\widehat{Y}_l = N_l^{-1} \\sum _{i=1}^{N_l} \\left( \\widehat{P}_l^{(i)} \\!-\\!", "\\widehat{P}_{l-1}^{(i)} \\right).$ The key point here is that the quantity $\\widehat{P}_l^{(i)} \\!-\\!", "\\widehat{P}_{l-1}^{(i)}$ comes from two discrete approximations using the same Brownian path.", "The variance of this simple estimator is $\\displaystyle \\mathbb {V}[\\widehat{Y}_l] = N_l^{-1} V_l$ where $V_l$ is the variance of a single sample.", "Combining this with independent estimators for each of the other levels, the variance of the combined estimator $\\sum _{l=0}^L \\widehat{Y}_l$ is $\\sum _{l=0}^L N_l^{-1} V_l.$ The corresponding computational cost is $\\sum _{l=0}^L N_l\\, C_l$ where $C_l$ represents the cost of a single sample on level $l$ .", "Treating the $N_l$ as continuous variables, the variance is minimised for a fixed computational cost by choosing $N_l$ to be proportional to $\\displaystyle \\sqrt{V_l / C_l}$ , with the constant of proportionality chosen so that the overall variance is $O(\\varepsilon ^{-2})$ .", "The total cost on level $l$ is proportional to $\\sqrt{V_l \\, C_l}$ .", "If the variance $V_l$ decays more rapidly with level than the cost $C_l$ increases, the dominant cost is on level 0.", "The number of samples on that level will be $O(\\varepsilon ^{-2})$ and the cost savings compared to standard Monte Carlo will be approximately $C_0/C_L$ , reflecting the different costs of samples on level 0 compared to level $L$ .", "On the other hand, if the variance $V_l$ decays more slowly than the cost $C_l$ increases, the dominant cost will be on the finest level $L$ , and the cost savings compared to standard Monte Carlo will be approximately $V_L/V_0$ , reflecting the difference between the variance of the finest grid correction compared to the variance of the standard Monte Carlo estimator, which is similar to $V_0$ .", "This outline analysis is made more precise in the following theorem: Theorem 3.1 Let $P$ denote a functional of the solution of an SPDE for a given Brownian path $M_t$ , and let $\\widehat{P}_l$ denote the corresponding level $l$ numerical approximation.", "If there exist independent estimators $\\widehat{Y}_l$ based on $N_l$ Monte Carlo samples, and positive constants $\\alpha , \\beta , \\gamma , c_1, c_2, c_3$ such that $\\alpha \\!\\ge \\!", "{ \\textstyle \\frac{1}{2} }\\,\\gamma $ and $\\displaystyle \\left| \\mathbb {E}[\\widehat{P}_l \\!-\\!", "P] \\right| \\le c_1\\, 2^{-\\alpha \\, l}$ $\\displaystyle \\mathbb {E}[\\widehat{Y}_l] = \\left\\lbrace \\begin{array}{ll}\\mathbb {E}[\\widehat{P}_0], & l=0 \\\\[0.1in]\\mathbb {E}[\\widehat{P}_l \\!-\\!", "\\widehat{P}_{l-1}], & l>0\\end{array}\\right.$ $\\displaystyle \\mathbb {V}[\\widehat{Y}_l] \\le c_2\\, N_l^{-1} 2^{-\\beta \\, l}$ $\\displaystyle C_l \\le c_3\\, N_l\\, 2^{\\gamma \\, l},$ where $C_l$ is the computational complexity of $\\widehat{Y}_l$ then there exists a positive constant $c_4$ such that for any $\\varepsilon \\!<\\!", "e^{-1}$ there are values $L$ and $N_l$ for which the multilevel estimator $\\widehat{Y}= \\sum _{l=0}^L \\widehat{Y}_l,$ has a mean-square-error with bound $MSE \\equiv \\mathbb {E}\\left[ \\left(\\widehat{Y}- E[P]\\right)^2\\right] < \\varepsilon ^2$ with a computational complexity $C$ with bound $C \\le \\left\\lbrace \\begin{array}{ll}c_4\\, \\varepsilon ^{-2} , & \\beta >\\gamma , \\\\[0.1in]c_4\\, \\varepsilon ^{-2} (\\log \\varepsilon )^2, & \\beta =\\gamma , \\\\[0.1in]c_4\\, \\varepsilon ^{-2-(\\gamma \\!-\\!\\beta )/\\alpha }, & 0<\\beta <\\gamma .\\end{array}\\right.$ Proof The proof is a slight generalisation of the proof in [13].", "$\\Box $ In our application, we choose $h_l \\propto 2^{-l}$ and $k_l\\propto 4^{-l}$ so that the ratio $k_l/h_l^2$ is held fixed to satisfy the finite difference stability condition.", "The computational cost increases by factor 8 in moving from level $l$ to $l\\!+\\!1$ , so $\\gamma \\!=\\!3$ .", "Given that the payoff is a Lipschitz function of the loss approximations at various dates, Conjecture REF implies that the weak error is also $O(h^2)$ and so $\\alpha \\!=\\!", "2 > { \\textstyle \\frac{1}{2} }\\,\\gamma $ .", "Also, due to the triangle inequality $\\sqrt{\\mathbb {V}[ \\widehat{P}_l \\!-\\!", "\\widehat{P}_{l-1} ]}&\\le &\\sqrt{\\mathbb {V}[ \\widehat{P}_l \\!-\\!", "P ]} +\\sqrt{\\mathbb {V}[ \\widehat{P}_{l-1} \\!-\\!", "P ]}\\\\ &\\le &\\sqrt{\\mathbb {E}[ (\\widehat{P}_l \\!-\\!", "P)^2 ]} +\\sqrt{\\mathbb {E}[ (\\widehat{P}_{l-1} \\!-\\!", "P)^2 ]} ,$ Conjecture REF and its corollary give $\\beta \\!=\\!", "4$ .", "Consequently, the computational cost to achieve a r.m.s.", "error of $\\varepsilon $ is $O(\\varepsilon ^{-2})$ ." ], [ "Numerical experiments", "In this section we study the numerical performance of the algorithms presented earlier on the example of CDO tranche pricing in a large basket limit." ], [ "CDO pricing in the structural credit model", "Basket credit derivatives provide protection against the default of a certain segment (`tranche') of a basket of firms.", "A typical example is that of a collateralized debt obligation where the protection buyer receives a notional amount, minus some recovery proportion $0\\le R\\le 1$ , if firms in a specified tranche of the basket default, and in return pays a regular spread until the default event occurs.", "The arbitrage-free spread depends crucially on the (risk-neutral, when hedged with defaultable bonds) probability of joint defaults.", "For a tranche with attachment point $0\\le a < 1$ and detachment point $1\\ge d>a$ , the outstanding tranche notional $P(L_t) = \\max (d-L_t,0) - \\max (a-L_t,0),$ where the loss variable $L_t$ is the proportion of losses in the basket up to time $t$ , determines the spread and default payments related to that tranche.", "The risk-neutral value of the tranche spread can be derived as $s = \\frac{\\sum _{i=1}^{n}\\, {\\rm e}^{-r T_{i}} \\mathbb {E}^{\\mathbb {Q}}[P(L_{T_{i-1}}) - P(L_{T_{i}})]}{\\delta \\sum _{i=1}^{n} \\, {\\rm e}^{-r T_i} \\mathbb {E}^{\\mathbb {Q}}[P(L_{T_{i}})]},$ see, e.g., [8].", "Here, $n$ is the maximum number of spread payments, $T$ the expiry, $T_i$ the payment dates for $1\\le i\\le n$ , $\\delta = 0.25$ the interval between payments.", "Spreads are quoted as an annual payment, as a ratio of the notional, but assumed to be paid quarterly.", "There is a variation for the equity tranche, and recently sometimes the mezzanine tranche, but we do not go into details here.", "For an extensive survey of credit derivatives and pricing models we refer the reader to [32], and note only that they typically fall in one of two classes: so-called reduced-form models, which model default times of firms directly as (dependent) random variables; structural models, which capture the evolution of the firms' values, and model default events as the first passage of a lower default barrier.", "We will consider the latter here.", "In a structural model in the spirit of the classical works of [26] and [4], as extended to multiple firms e.g.", "in the work of [21], a company $i$ 's firm value, $1\\le i\\le N$ , is assumed to follow a model of the type $\\frac{\\, {\\rm d} A_t^i}{A_t^i} = r \\, {\\rm d} t+ \\sqrt{1 \\!-\\!", "\\rho _i}\\ \\sigma _i \\, {\\rm d} W_t^i + \\sqrt{\\rho _i}\\ \\sigma _i\\, {\\rm d} M_t, \\qquad A_0^i = a^i,$ where $0\\le \\rho _i\\le 1$ is a correlation parameter, $r$ is the risk-free interest rate, $\\sigma _i$ the volatility, and $M$ , $W^i$ are standard Brownian motions.", "Here, the individual firms are correlated through a common `market' factor $M$ , but independent conditionally on $M$ .", "We make in the following the assumption that the firms are exchangeable in the sense that their dynamics is governed by the same set of parameters, specifically $\\rho =\\rho _i$ , $\\sigma =\\sigma _i$ .", "Their initial values $a_i$ are not necessarily identical which allows for different default probabilities for individual firms, consistent with their CDS spreads.", "In this framework, the default time $T_0^i$ of the $i$ -th firm is modelled as the first hitting time of a default barrier $B^i$ , for simplicity constant, and the distance-to-default $X_t^i = \\frac{1}{\\sigma }\\left(\\,\\log A_t^i - \\log B^i \\,\\right),$ evolves according to (REF ), where $\\mu =\\left(r- \\frac{1}{2} \\sigma ^2 \\right)/\\sigma $ , $x^i= \\left(\\,\\log a^i - \\log B^i \\,\\right)/\\sigma $ .", "$T^i_0$ as in (REF ) is precisely the default time.", "In the majority of applications, Monte Carlo simulation of the firm value processes is used for the estimation of tranche spreads, see e.g.", "[21] or [12], [9].", "This is largely enforced by the size of $N$ , for instance in the case of index tranches $N=125$ .", "This, however, puts the model precisely in the realm of the large basket approximation (REF ).", "See [7] for a numerical study of this large basket approximation.", "The loss functional is thereby approximated by $\\widehat{L}_{T_i} = (1-R) \\left(1- h \\sum _{j=1}^{J-1} v_j^{T_i/k}\\right)$ in terms of the numerical SPDE solution $v_j^{T_i/k}$ at time $T_i$ , which feeds into the estimator for the outstanding tranche notional (REF ) and subsequently the tranche spreads (REF )." ], [ "Pricing results", "All following results are for a representative set of parameter values, taken from a calibration performed in [8] to 2007 market data, $\\sigma = 0.22$ , $\\rho = 0.2$ , $r=0.042$ , $\\mu =(r-\\sigma ^2/2)/\\sigma $ .", "While the initial distribution used in [8] is somewhat spread out to match individual CDS spreads of obligors, most of the mass is around $x_0 = 5$ and for simplicity we centre all firms there for the numerical tests, $v(0,x)\\!=\\!\\delta (x\\!-\\!x_0)$ .", "The upper boundary is chosen as $x_{max}=16$ .", "We consider a maturity $T=5$ and tranches $[a,d] = [0,0.03]$ , $[0.03,0.06]$ , $[0.06,0.09]$ , $[0.09,0.12]$ , $[0.12,0.22]$ , $[0.22,1]$ .", "Figure REF shows the multilevel results for the expected protection payment from the first tranche, $\\sum _{i=1}^{n} \\, {\\rm e}^{-r T_{i}} \\mathbb {E}^{\\mathbb {Q}}[P(\\widehat{L}_{T_{i-1}}) - P(\\widehat{L}_{T_{i}})],$ expressed as a fraction of the initial tranche notional.", "We pick mesh sizes $h_l = h_0 \\, 2^{-l}$ and $k_l = k_0 \\, 4^{-l}$ for $l>0$ and $h_0=8/5$ , $k_0=1/4$ .", "This is motivated by the second and first order consistency in $h$ and $k$ respectively, and within the stability limit (REF ).", "Figure: Multilevel results for the expected loss from the first tranche.The top-left plot shows the convergence of the variance $\\mathbb {V}[\\widehat{P}_l \\!-\\!", "\\widehat{P}_{l-1}]$ as well as the variance of the standard single level estimate, while the top-right plot shows the convergence of the expectation $\\mathbb {E}[\\widehat{P}_l \\!-\\!", "\\widehat{P}_{l-1}]$ .", "The bottom-left plot shows results for different multilevel calculations, each with a different user-specified accuracy requirement.", "For each value of the accuracy $\\varepsilon $ , the multilevel algorithm determines (by formula (12) from [13]) the numbers of levels of refinement which are needed to ensure that the contribution to the Mean Square Error (MSE) due to the weak error on the finest grid is less than ${ \\textstyle \\frac{1}{2} } \\varepsilon ^2$ , and it determines (by formula (10) from [13]) the optimal number of samples on each level so that the combined variance of the multilevel estimate is also less than ${ \\textstyle \\frac{1}{2} } \\varepsilon ^2$ , and hence the MSE is less than $\\varepsilon ^2$ .", "Treating a single finite difference calculation on the coarsest level as having a unit cost, the bottom-right plot compares the total cost of the standard and multilevel algorithms.", "Since the objective is to achieve a computational cost which is approximately proportional to $\\varepsilon ^{-2}$ , it is the cost $C$ multiplied by $\\varepsilon ^2$ which is plotted versus $\\varepsilon $ .", "The results confirm that $\\varepsilon ^2 C$ does not vary much as $\\varepsilon \\rightarrow 0$ for the multilevel method, in line with the prediction that $\\varepsilon ^2 C \\propto 1$ , whereas it grows significantly for the standard method since $\\varepsilon ^2 C \\propto 8^L$ , where $L$ is the index of the finest level.", "Hence, there is a big jump in the cost of the standard method each time it is necessary to switch to a finer level to ensure the weak error is less than $\\varepsilon /\\sqrt{2}$ , whereas the jump is minimal for the multilevel method.", "In practice, default is not monitored continuously but only at a discrete set of times, and for pricing often the simplifying assumption is made that default is only determined at the spread payment dates $T_l$ .", "This can be incorporated as follows.", "In the time intervals $(T_{l-1},T_l)$ , we solve (REF ) on a sufficiently large domain (e.g.", "$[-4,16]$ ), and apply the following interface conditions at $T_l$ : $\\lim _{t\\downarrow T_l} v(t,x) =\\left\\lbrace \\begin{array}{rl}0, & x\\le 0, \\\\\\lim _{t\\uparrow T_l} v(t,x), & x > 0.\\end{array}\\right.$ To maintain quadratic grid convergence in spite of the discontinuity introduced by (REF ), we choose the mesh such that a grid point coincides with the 0 boundary (e.g.", "by setting $h_0=2$ in the above example), and set the numerical solution after default monitoring to 0 for grid coordinates below 0 and to $1/2$ its previous value at 0, see e.g. [27].", "It is seen in Fig.", "REF that convergence is very similar to the previous case.", "Figure: Multilevel results for the expected loss from the first tranche,with discrete monitoring." ], [ "Complexity and cost", "Here we discuss the computational complexity of the multilevel solution of the SPDE compared with the alternative use of the multilevel method for solving the SDEs which arise from directly simulating a large number of SDEs.", "We have already explained that to achieve a r.m.s.", "accuracy of $\\varepsilon $ requires $O(\\varepsilon ^{-7/2})$ work when solving the SPDE by the standard Monte Carlo method, but the cost is $O(\\varepsilon ^{-2})$ using the multilevel method, provided Conjecture REF is correct.", "Consider now the alternative of using a finite number of particles (firms), $M$ , to estimate the tranche loss in the limit of an infinite number of particles (firms).", "In this case, empirical results suggest that there is an additional $O(M^{-1})$ error (see also [7]), and the proof of this convergence order is the subject of current research.", "Taking this to be the case, the optimal choice of $M$ to minimise the computational complexity to achieve an r.m.s.", "error of $\\varepsilon $ is $O(\\varepsilon ^{-1})$ .", "Using the standard Monte Carlo method, the optimal timestep is $O(\\varepsilon )$ , and the optimal number of paths is $O(\\varepsilon ^{-2})$ , so the overall cost is $O(\\varepsilon ^{-4})$ .", "Using the multilevel method for the SDEs reduces the cost per company to $O(\\varepsilon ^{-2})$ , so the total cost is $O(\\varepsilon ^{-3})$ .", "This complexity information is summarised in Table REF .", "There is also a practical implementation aspect to note.", "The computational cost per grid point in the finite difference approximation of the SPDE is minimal, requiring just three floating point multiply-add operations if equation (REF ) is re-cast as $v_j^{n+1} = a v_{j-1}^n + b v_{j}^n + c v_{j+1}^n$ with the coefficients $a, b, c$ computed once per timestep, for all $j$ .", "If we let $C$ be the cost of generating all of the Gaussian random numbers $Z_n$ for a single SPDE simulation, then the cost of the rest of the finite difference calculation with 20 points in $x$ (as used on the coarsest level of our multilevel calculations) is probably similar, giving a total cost of $2C$ for each SPDE.", "On the other hand, each SDE needs its own Gaussian random numbers for the idiosyncratic risk, and so the cost of simulating each SDE is approximately $C$ , roughly half of the cost of the SPDEs on the coarsest level of approximation, giving a total cost of $M \\, C$ .", "Table: Comparison of the complexity of the SDE and SPDE models usingthe standard and multilevel Monte Carlo methods" ], [ "Further work", "We have shown that stochastic finite differences combined with a multilevel simulation approach achieve optimal complexity for the computation of expected payoffs of an SPDE model.", "In the case of an absorbing boundary, the complexity estimate is a conjecture in so far it relies on the convergence order of the finite difference scheme, which does not follow from the Fourier analysis of the unbounded case.", "The matrix stability analysis in Appendix could form part of a rigorous analysis if a Lax equivalence theorem could be proved.", "In the case of multiplicative white noise this is shown in [25].", "One difficulty in the present case of an SPDE with stochastic drift is the loss of regularity towards the boundary, which may be accounted for by weighted Sobolev norms of the solution, but even then it is not clear that convergence of the functionals of interest follows.", "There are several possible extensions of the present basic model as discussed in [7], ranging from stochastic volatility and jump-diffusion to contagion models.", "The methods developed in this paper should be of use there also, building for example on multilevel versions of jump-adapted discretisations for jump-diffusion SDEs [5], [34]." ], [ "Mean-square matrix stability analysis", "If $V_n$ is the vector with elements $v_j^n, j=1, \\ldots , J\\!-\\!1$ then the finite difference equation can be expressed as $V_{n+1} &=& (A + B\\, Z_n + C\\,Z_n^2)\\ V_n, \\\\[0.1in]A &=& I - \\frac{\\mu \\, k}{2h}\\, D_1 + \\frac{(1\\!-\\!\\rho )\\, k}{2h^2}\\, D_2, \\\\[0.05in]B &=& -\\, \\frac{\\sqrt{\\rho \\, k}}{2h}\\, D_1, \\\\[0.05in]C &=& \\frac{\\rho \\, k}{2h^2}\\, D_2,$ where $I$ is the identity matrix and $D_1$ and $D_2$ are the matrices corresponding to central first and second differences, which for $J=6$ are $D_1 =\\left(\\begin{array}{rrrrr}0&1&&& \\\\-1&0&1&& \\\\&-1&0&1& \\\\&&-1&0&1 \\\\&&&-1&\\ \\ 0\\end{array}\\right), \\quad D_2 =\\left(\\begin{array}{rrrrr}-2&1&&& \\\\1&-2&1&& \\\\&1&-2&1& \\\\&&1&-2&1 \\\\&&&1&-2\\end{array}\\right).$ From the recurrence relation we get $\\mathbb {E}[\\, V_{n+1}^T V_{n+1} ]&=& \\mathbb {E}\\left[ V_n^T (A^T + B^T\\, Z_n + C^T\\, Z_n^2)(A + B\\, Z_n + C\\, Z_n^2)\\ V_n \\right] \\\\[0.1in]&=& \\mathbb {E}\\left[ V_n^T \\left( (A\\!+\\!C)^T (A\\!+\\!C) + B^T B + 2\\, C^T C \\,\\right) V_n \\right].$ Noting that $D_1$ is anti-symmetric and $D_2$ is symmetric, and that $D_1 D_2 - D_2 D_1 = E_1 - E_2, \\quad D_1^2 = D_3 + E_1 + E_2,$ where $D_3$ corresponds to a central second difference with twice the usual span, $D_3 =\\left(\\begin{array}{rrrrr}-3&0&1&& \\\\0 &-2&0&1& \\\\1&0&-2&0&1 \\\\&1&0&-2&0 \\\\&&1&0&-3\\end{array}\\right)$ (with the end values of $-3$ being chosen to correspond to $V_{-1}\\equiv - V_1$ and $V_{J+1}\\equiv - V_{J-1}$ ), and $E_1$ and $E_2$ are each entirely zero apart from one corner element, $E_1 =\\left(\\begin{array}{rrrrr}2&&&& \\\\&&&& \\\\&&&& \\\\&&&& \\\\&&&&\\end{array}\\right), \\qquad E_2 =\\left(\\begin{array}{rrrrr}&&&& \\\\&&&& \\\\&&&& \\\\&&&& \\\\&&&& 2\\end{array}\\right),$ then after some lengthy algebra we get ${ \\mathbb {E}\\left[ V_n^T \\left( (A\\!+\\!C)^T (A\\!+\\!C) + B^T B + 2\\, C^T C\\,\\right) V_n \\right]} \\\\& =& \\mathbb {E}\\left[ V_n^T M V_n \\right]- \\left( e_1 + e_2 \\right) \\mathbb {E}[(v_1^n)^2]- \\left( e_1 - e_2 \\right) \\mathbb {E}[(v_{J-1}^n)^2],$ where $M = I - \\frac{k}{h^2}\\, D_2 + \\frac{k^2}{4\\,h^4}\\, D_2^2- \\left( \\frac{\\rho k}{4\\,h^2} + \\frac{\\mu ^2 k^2}{4\\,h^2} \\right) D_3,$ and $e_1 = \\frac{\\rho k}{2\\,h^2} + \\frac{\\mu ^2 k^2}{2\\,h^2}, \\qquad e_2 = \\frac{\\mu k^2}{2\\,h^3}.$ It can be verified that the $m^{th}$ eigenvector of $M$ has elements $\\sin j \\theta _m$ for $\\theta _m = m\\, \\pi / J$ , and the associated eigenvalue is $|a(\\theta _m)\\!+\\!c(\\theta _m)|^2 + |b(\\theta _m)|^2 + 2 |c(\\theta _m)|^2,$ where $a(\\theta ), b(\\theta ), c(\\theta )$ are the same functions as defined in the mean-square Fourier analysis.", "In addition, in the limit $h, k/h\\rightarrow 0$ , $e_1 \\!\\pm \\!", "e_2 > 0$ , and therefore in this limit the Fourier stability condition $\\sup _\\theta \\left\\lbrace |a(\\theta )\\!+\\!c(\\theta )|^2 + |b(\\theta )|^2 + 2 |c(\\theta )|^2 \\right\\rbrace \\le 1$ is also a sufficient condition for mean-square matrix stability." ], [ "Regularity considerations", "Figure REF shows the convergence behaviour as the computational grid is refined.", "Level 0 has $h \\!=\\!", "1/4, k \\!=\\!", "T/4$ ; $h$ is reduced by factor 2 and $k$ by factor 4 in moving to finer levels.", "Figure: Convergence plots as a function of grid levelThe top-left plot shows the convergence of $\\mathbb {E}[\\widehat{P}_{l}-\\widehat{P}_{l-1}]$ , with notation as in Sections and REF ; the top-right plot shows the convergence of $\\mathbb {V}[\\widehat{P}_{l}-\\widehat{P}_{l-1}]$ .", "The bottom two plots show the behaviour of $\\partial ^2 v/\\partial x^2(T,0)$ , which is estimated on each grid level using the standard second difference $\\frac{\\partial ^2 v}{\\partial x^2}(0) \\approx \\frac{v_2 - 2v_1 + v_0}{h^2}.$ The left plot indicates that the mean of this quantity is well behaved, but the right plot indicates a singular behaviour of its variance, with the value increasing rapidly with increased grid resolution.", "This is in accordance with the result shown in [23], that for the unique solution $v$ to the SPDE, $x\\, v_{xx}$ is square integrable, but $v_{xx}$ has a singularity at 0." ] ]

[ [ "Energy Loss Signals in the ALICE TRD" ], [ "Abstract We present the energy loss measurements with the ALICE TRD in the $\\beta\\gamma$ range 1--10$^{4}$, where $\\beta=v/c$ and $\\gamma=1/\\sqrt{1-\\beta^2}$.", "The measurements are conducted in three different scenarios: 1) with pions and electrons from testbeams; 2) with protons, pions and electrons in proton-proton collisions at center-of-mass energy 7 TeV; 3) with muons detected in ALICE cosmic runs.", "In the testbeam and cosmic ray measurements, ionization energy loss (dE/dx) signal as well as ionization energy loss plus transition radiation (dE/dx+TR) signal are measured.", "With cosmic muons the onset of TR is observed.", "Signals from TeV cosmic muons are consistent with those from GeV electrons in the other measurements.", "Numerical descriptions of the signal spectra and the $\\beta\\gamma$-dependence of the most probable signals are also presented." ], [ "Introduction", "The TRD [1] of the ALICE experiment [2] at the LHC is devoted to electron identification [3] and tracking of charged particles.", "It provides also Level-1 triggering on electrons and jets [4].", "It is a cylindrical detector system located in radius between 2.9 and 3.7 meters from the beamline and segmented in 6 layers.", "It has a 2$\\pi $ azimuthal coverage in 18 super-modules and a polar coverage between 45$^{\\circ }$ and 135$^{\\circ }$ in 5 stacks (Fig.", "REF ).", "Figure: Layout of the ALICE TRD.Individual TRD chamber consists of a 4.8 cm thick layer of fibres/foam sandwich radiator and a drift chamber filled with Xe, CO$_{2}$ (15%).", "The depths of the drift and amplification regions are 3 cm and 0.7 cm respectively.", "The induced charges are readout by cathode pads which have a typical size of 0.7$\\times $ 8.8 cm$^{2}$  (Fig.", "REF ) every 100 ns.", "A charged particle loses energy in primary collisions with gas atoms by ionization (dE/dx).", "Energetic ones with $\\gamma $ above $10^3$ in addition emit TR photons when passing through the radiator.", "The radial positions of the primary clusters are reconstructed from the drift time.", "Therefore besides the energy loss measurement, the TRD is also used for momentum determination.", "The TRD signals presented in the following sections are the integrated charge over all drift time corrected for the non-perpendicular incident angle.", "Figure: Schematic cross-sectional view of a ALICE TRD chamber." ], [ "TRD Signals from Testbeam Measurement and Proton-Proton Collisions", "The testbeam measurement was carried out at CERN's PS in 2004 [5] with secondary beams of pions and electrons of momenta from 1 to 10 GeV/c.", "The setup is shown in Fig.", "REF .", "Prototype chambers without (with) radiators were used for the dE/dx (dE/dx+TR) measurement.", "Figure: Beam line setup of TRD testbeam measurement .", "Scintillator S1–3 are trigger detectors.", "Silicon strip detector Si1–2 locate the beams.", "Cherenkov counter 1–2 and lead glass calorimeter are used to identify pions and electrons.In Figure REF the TRD signal distributions from 3 GeV/c testbeam pions and electrons are shown.", "Both are fit with the following modified Landau-Gaussian convolution: $(\\textrm {Exponential}\\times \\textrm {Landau})*\\textrm {Gaussian},$ where the Landau distribution is weighted by an exponential damping $\\textrm {Landau}(x)\\rightarrow e^{-\\kappa x}\\textrm {Landau}(x).$ This function is used also for distributions from proton-proton collisions (Fig.", "REF ) and cosmic runs (Fig.", "REF ) to extract the most probable energy loss.", "Figure: TRD signals from 3 GeV/c pions (upper) and electrons (lower).Figure: TRD signals measured at βγ=1\\beta \\gamma =1 from identified protons in proton-proton collisions.Figure: TRD signals measured at βγ=10 3 \\beta \\gamma =10^3 from cosmic runs.", "(Upper: dE/dx, lower: dE/dx+TR, see Section  for details.", ")Since the end of March in 2010, ALICE has collected data from proton-proton collisions at center-of-mass energy 7 TeV.", "The TRD signals are measured for protons, pions and electrons in minimum bias events.", "More details can be found in Ref.", "[7]." ], [ "TRD Signals from Cosmic Muons", "The TRD energy loss measurements in the testbeam and proton-proton runs described in Section  do not cover the $\\beta \\gamma $ range 10$^2$ – 10$^3$ which can be filled in by cosmic muons.", "Because the ALICE detector is situated underground with 28 meters of material above, cosmic rays which leave long trajectories in the ALICE Time Projection Chamber (TPC) [8] are predominantly muons.", "Because the TR photons are emitted in direction of the passage, they enter the drift section and are absorbed by the heavy gas when the particle traverses the radiator first.", "Contrarily, only the dE/dx signal is measured when the particle traverses the drift section and then the radiator.", "Due to the cylindrical placement of the TRD layers, dE/dx signals are associated with the in-coming passages of the cosmic muons and dE/dx+TR signals with the out-going ones (Fig.", "REF ).", "Figure: One cosmic event in ALICE TPC and TRD (3D view along the beam pipe).In order to measure precisely the momenta of the muons, a specific track fitting program was developed.", "Compared to the standard tracking which only fits half of the muon trajectory in the TPC, the combined track fit uses all $\\mathrm {TPC}$ clusters (Fig.", "REF ) and therefore achieves a 10 times better momentum resolution: the $1/p_t$ -resolution is 8.1$\\times 10^{-4}$ c/GeV at momentum 1 TeV/c (integrated for all cosmic ray geometries in the TPC).", "Figure REF shows the most probable TRD dE/dx+TR signals measured in the testbeam and cosmic runs.", "At $\\beta \\gamma $ above 10$^3$ , the cosmic ray signal by the standard TPC tracking flattens as a result of the limited momentum resolution.", "With the improvement provided by the combined track fit, the cosmic-ray and testbeam results are consistent up to $\\beta \\gamma =10^4$ , beyond which the statistics is limited.", "Figure: Most probable TRD dE/dx+TR signals in testbeam and cosmic ray measurements.The standard ALICE 0.5 T solenoidal magnetic field is too strong for sub-GeV muons to leave long trajectories in the TPC.", "In order to measure the muon minimum ionization signals in the TRD, cosmic runs with a magnetic field of 0.1 T were conducted.", "The momentum is determined by the standard TPC-TRD tracking, which in addition to the standard TPC tracking also takes into account the space points in the TRD.", "This is the first ALICE running at 0.1 T. The most probable TRD signals from cosmic muons are shown in Fig.", "REF .", "By comparing the dE/dx and dE/dx+TR signals, the TR onset is observed at $\\beta \\gamma >700$ .", "Figure: Most probable TRD signals from cosmic muons.", "Horizontal error bars are obtained from the estimated momentum resolution." ], [ "Results", "Combining the testbeam, proton-proton collisions and cosmic ray measurements we obtained the TRD energy loss signals in the $\\beta \\gamma $ range 1 – 10$^4$  (Fig.", "REF ).", "Results from different measurements are consistent.", "Figure: Most probable TRD signals from testbeam, proton-proton collisions and cosmic ray measurements.The most probable (mp) TRD dE/dx signal in unit of minimum ionization is well described by the ALEPH [9], [10] parameterization (Fig.", "REF ): $\\frac{\\mathrm {d}E}{\\mathrm {d}x}_\\mathrm {mp}=0.20\\times \\frac{4.4-\\beta ^{2.26}-\\ln \\left[0.004+\\frac{1}{\\left(\\beta \\gamma \\right)^{0.95}}\\right]}{\\beta ^{2.26}}.$ From the fit result two numbers of practical interest can be deduced: $\\beta \\gamma $ of minimum ionizing particle: 3.5, dE/dx in the relativistic limit: 1.7 times minimum ionization.", "Figure: Compilation of TRD dE/dx data from testbeam, proton-proton collisions and cosmic ray measurements.Fixing the dE/dx contribution according to the ALEPH parameterization, the most probable dE/dx+TR signal in unit of minimum ionization is well described by including an additional Logistic function (Fig.", "REF ): $\\mathrm {TR_{mp}}=\\frac{0.706}{1+\\exp ^{-1.85\\times (\\ln \\gamma -7.80)}}.$ The following practical TR properties can be deduced: saturated TR yield in the relativistic limit: 0.7 times minimum ionization, $\\beta \\gamma $ for half saturation: $2.4\\times 10^3$ .", "Figure: Compilation of TRD dE/dx+TR data from testbeam, proton-proton collisions and cosmic ray measurements.", "The ALEPH component of the fit is fixed with the dE/dx results in Fig.", "." ], [ "Summary", "We have presented an overview of the energy loss $\\mathrm {signals}$ in the ALICE TRD, measured with prototypes in a testbeam and with the actual detector in proton-proton collisions and cosmic runs in the ALICE setup at the LHC.", "In the cosmic ray measurement we exploit the geometry of the detector to measure separately the dE/dx and dE/dx+TR signals in the TRD.", "In addition, a combined track fit within the ALICE TPC was developed to exploit the full length of the track, leading to a $1/p_t$ -resolution of 8.1$\\times 10^{-4}$ c/GeV at momentum 1 TeV/c.", "The TR from TeV cosmic muons is unambiguously observed.", "We have shown numerical descriptions of the signal spectra and of the dependence of the most probable energy loss on $\\beta \\gamma $ , which will allow to establish reference distributions for particle identification with TRD over a broad momentum range." ] ]

[ [ "The chiral phase transition in charge ordered 1T-TiSe2" ], [ "Abstract It was recently discovered that the low temperature, charge ordered phase of 1T-TiSe2 has a chiral character.", "This unexpected chirality in a system described by a scalar order parameter could be explained in a model where the emergence of relative phase shifts between three charge density wave components breaks the inversion symmetry of the lattice.", "Here, we present experimental evidence for the sequence of phase transitions predicted by that theory, going from disorder to non-chiral and finally to chiral charge order.", "Employing X-ray diffraction, specific heat, and electrical transport measurements, we find that a novel phase transition occurs ~7 K below the main charge ordering transition in TiSe2, in agreement with the predicted hierarchy of charge ordered phases." ], [ "The chiral phase transition in charge ordered 1T-TiSe$_2$ John-Paul Castellan$^1$ Stephan Rosenkranz$^1$ Ray Osborn$^1$ Qing'an Li$^1$ K.E.", "Gray$^1$ X. Luo$^1$ U. Welp$^1$ Goran Karapetrov$^{1,2}$ J.P.C.", "Ruff$^{3,4}$ Jasper van Wezel$^{1,5}$ [email protected] $\\phantom{\\text{Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA.", "}}$ $^1$ Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA.", "$^2$ Physics Department, Drexel University, Philadelphia, PA 19104, USA.", "$^3$ Advanced Photon Source, Argonne National Laboratory, Argonne, IL 60439, USA.", "$^4$ CHESS, Cornell University, Ithaca, NY 14853, USA.", "$^5$ H.H.", "Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, U.K.", "It was recently discovered that the low temperature, charge ordered phase of 1T-TiSe$_2$ has a chiral character.", "This unexpected chirality in a system described by a scalar order parameter could be explained in a model where the emergence of relative phase shifts between three charge density wave components breaks the inversion symmetry of the lattice.", "Here, we present experimental evidence for the sequence of phase transitions predicted by that theory, going from disorder to non-chiral and finally to chiral charge order.", "Employing X-ray diffraction, specific heat, and electrical transport measurements, we find that a novel phase transition occurs $\\sim 7$  K below the main charge ordering transition in TiSe$_2$ , in agreement with the predicted hierarchy of charge ordered phases.", "71.45.Lr,64.60.Ej,11.30.Rd Introduction.–The modulation of electronic density which emerges in charge ordered materials reduces the translational symmetry of the underlying lattice.", "Additional symmetries may be broken through the coupling to atomic displacements, and these are often key in understanding the material properties of charge ordered systems.", "The broken rotational symmetry in 2H-TaSe$_2$ , for example, yields a reentrant phase transition under pressure [1], while the broken inversion symmetry in rare-earth nickelates RNiO$_3$ , renders them multiferroic [2].", "Recently, it was discovered that the breakdown of inversion symmetry in the charge ordered phase of 1T-TiSe$_2$ leads to the presence of a chiral structure at low temperatures [3], [4], [5], [6], [7], [8], [9].", "In this phase, a helical charge density distribution arises from a rotation of the dominant charge density wave component as one progresses through consecutive atomic layers.", "While helical phases are common among spin-density waves, with vectorial order parameters, 1T-TiSe$_2$ is one of only very few materials so far in which a chiral charge ordered phase, with a scalar order parameter, has been suggested to exist [10], [11].", "A mechanism for the formation of this scalar chirality in TiSe$_2$ was recently proposed, in which the chiral phase is interpreted to be simultaneously charge and orbital ordered [5].", "The scanning tunneling microscopy experiments in which the chirality of TiSe$_2$ was initially revealed, were performed well below the onset temperature of charge order in this material [3].", "There is currently no experimental information on the nature of the transition between the chiral and the normal state.", "Here, we present experimental evidence for a sequence of two transitions, the well-known onset of charge order at $\\sim 190$  K and a novel transition at $\\sim 183$  K. The temperature dependence of X-ray superlattice reflections unnoticed in previous studies, combined with an analysis of their structure factors, and the temperature dependences of the specific heat and resistance anisotropy, strongly suggests a scenario in which the transition at $\\sim 183$  K indicates the emergence of chiral charge order out of the non-chiral charge density wave state.", "The existence of this hierarchy of transitions, as well as the observed experimental signatures, are in agreement with recent theoretical predictions [5], [6].", "Chiral Charge Order.—1T-TiSe$_2$ is a quasi-two dimensional material, in which hexagonal layers of Ti are sandwiched between similar layers of Se, and individual sandwiches are separated from each other by a large van der Waals gap (see Fig.", "REF a).", "It is observed to undergo a phase transition into a commensurate charge density wave state, accompanied by periodic lattice distortions.", "The mechanism underlying the transition is heavily debated [12], [13], [14], [15], [16].", "Regardless of what drives its formation, however, it is clear that the charge density modulation consists of three components, each involving a charge transfer process between one particular Ti-$3d$ orbital and two Se-$4p$ orbitals.", "One such component is depicted in Fig.", "REF b.", "A non-zero phase shift of this component corresponds to more charge being transferred along the red upper than along the green lower bonds, and a corresponding translation of the displacement wave.", "If the three components are superposed without any relative phase differences, the 2x2x2 non-chiral charge density wave state originally proposed for TiSe$_2$ is produced [17].", "Superposing three components with different phase shifts result in a chiral 2x2x2 lattice distortion [5].", "Whether the chiral or the non-chiral state is energetically favorable, is determined by minimizing the Landau free energy.", "Writing the order parameter as the sum of three complex parameters $\\psi _j = \\psi _0 e^{i \\vec{q}_j \\cdot \\vec{x} + \\varphi _j}$ with equal amplitudes, the free energy may be written as: $F = & \\psi _0^2 [ a_0 (T/T_{\\text{CDW}}-1) + 2 a_1 \\sum _j \\cos ^2(\\varphi _j) ] \\\\+ & \\psi _0^4 [ b_0 + 2 b_1 \\sum _j \\cos ^2(\\varphi _j - \\varphi _{j+1}) ].$ Here $a_0$ and $b_0$ represent the combined effects of Coulomb interaction and the competition between charge density wave components, while the terms $a_1$ and $b_1$ are the leading order Umklapp terms which signify the coupling between the electronic order parameter and the atomic lattice [5].", "Minimizing the free energy with respect to the phase variables yields two possible solutions.", "Coming from high temperatures, the non-chiral state with $\\varphi _1=\\varphi _2=\\varphi _3$ is first realized at $T=T_{\\text{CDW}}$ , when the amplitude $\\psi _0$ becomes non-zero.", "At the lower temperature $T_{\\text{Chiral}} = T_{\\text{CDW}} ( 1 - \\frac{2 a_1}{3 a_0} [6 + \\frac{b_0}{b_1} ] )$ , the differences between the three phase variables $\\varphi _j$ become non-zero, and the chiral charge order sets in at a second-order phase transition.", "Since the contribution from Umklapp effects is generally weaker than that of the direct interactions, $1 - T_{\\text{Chiral}} / T_{\\text{CDW}} \\propto \\frac{a_1}{a_0}$ may be expected to be small [5].", "X-ray diffraction.—Single crystals of TiSe$_2$ were prepared using the iodine vapor transport method [18].", "A sample of 3x3x$0.05$  mm$^3$ was mounted on the cold finger of a closed cycle displex and aligned on a Huber six-circle diffractometer.", "X-ray measurements utilizing area detectors on the sector 6-ID-D high energy station of the APS were performed in transmission geometry, using an incident photon energy of 80 keV.", "Detailed order parameter measurements of the charge ordered phase were made using the superlattice reflection [$\\frac{3}{2}$ ,$\\frac{3}{2}$ ,$\\frac{1}{2}$ ], which is indicative of a doubling of the unit cell in all crystallographic directions.", "The three-dimensional integrated intensity of these scans is presented in Fig.", "REF a.", "Notice that the transition temperature $T_{\\text{CDW}}=190$  K is slightly lower than the optimal transition temperature reported for TiSe$_2$ , indicating a slight deviation from stoichiometry [19].", "We then used area detectors to map out large volumes of reciprocal space to search for evidence of the emerging chiral order below $T_{\\text{CDW}}$ .", "We discovered extra peaks in the low temperature maps at wave vectors of the form [$H+\\frac{1}{2}$ ,$K$ ,0], with intensities $\\sim 50$ times weaker than the primary charge density wave reflections.", "We measured one of these peaks, located at the [$\\frac{5}{2}$ ,1,0] superlattice reflection, as a function of temperature.", "Notice that although this reflection is not forbidden by symmetry in the non-chiral space group, it has a low intensity due to its small structure factor $S({\\bf q})$ .", "Up to an element-specific form factor, the contribution of each species of atom to the structure factor is determined by the Fourier transform of their positions in the crystal lattice.", "Using the predicted displacements for each of the charge density wave components (as shown in Fig.", "REF b), it becomes clear that, for any given amplitude of the distortion, the contributions to the structure factor for the peak at [$\\frac{5}{2}$ ,1,0] are significantly greater in the chiral configuration than in the corresponding non-chiral charge ordered state: $S^{Ti}_{Chiral}([\\frac{5}{2},1,0]) / S^{Ti}_{CDW}([\\frac{5}{2},1,0]) \\simeq 1.86$ while $S^{Se}_{Chiral}([\\frac{5}{2},1,0]) / S^{Se}_{CDW}([\\frac{5}{2},1,0]) \\simeq 64$ , which implies a pronounced enhancement of the diffraction intensity at $[\\frac{5}{2},1,0]$ as the chirality sets in, as seen in Fig.", "REF a.", "Figure: (Color online) a) The integrated intensities of the superlattice reflections at [3 2\\frac{3}{2},3 2\\frac{3}{2},1 2\\frac{1}{2}] (upper, red curve) and [5 2\\frac{5}{2},1,0] (lower, blue curve; amplified by a factor of 50) .", "inset) Schematic depiction of a single layer of TiSe 2 _2.", "The plane of Ti atoms (large and blue) is indicated along with the crystallographic axes aa, bb and cc.", "The orbital basis instead uses xx, yy and zz axes, which connect the Ti and Se atoms (smaller, red and green).", "b) The same data on a log-log scale, as a function of 1-T/T CDW 1-T/T_{\\text{CDW}} (red data) or 1-T/T Chiral 1-T/T_{\\text{Chiral}} (blue), along with power law fits.", "The error bars for the red curve do not exceed the size of the symbols.", "In both panels, the open symbols form a fluctuation tail, extending beyond T C T_C.", "inset) Depiction of one of the individual charge density wave components.The evolutions of the two peaks under decreasing tem- perature depicted in Fig.", "REF a show that the intensity at [$\\frac{3}{2}$ ,$\\frac{3}{2}$ ,$\\frac{1}{2}$ ] first becomes non-zero at $T_{\\text{CDW}}=190$  K. The intensity of this peak is expected to be proportional to the squared amplitude of the charge density wave order parameter.", "The intensity of the reflection at [$\\frac{5}{2}$ ,1,0] on the other hand, remains zero below the onset temperature $T_{\\text{CDW}}$ , and begins to gain intensity only at the lower temperature $T_{\\text{Chiral}}=183$  K, which we identify as a second phase transition.", "The evolution of the two peak intensities with decreasing temperature can be clearly seen in Fig.", "REF b to be significantly different.", "The intensity of the main charge density wave peak close to $T_{\\text{CDW}}$ is best fitted by the power law $I=I_0 (1-T/T_{\\text{CDW}})^{2 \\beta }$ , with $2 \\beta \\simeq 0.68$ .", "The second peak on the other hand, is a second harmonic since it necessarily involves two components of the charge density wave.", "Its intensity therefore scales with the fourth power of $\\beta $ , and is best described by a power law with $4 \\beta \\simeq 1.0$ .", "The difference between the best fits for these critical exponents is suggestive of the onset of a different type of order at $T_{\\text{Chiral}}$ , or $\\sim 7$ K below the initial onset of charge order at $T_{\\text{CDW}}$ .", "Signatures of this second phase transition are also seen in specific heat and resistivity measurements, as described below.", "Specific Heat.— The temperature dependence of the specific heat of a 165x165x7 $\\mu $ m$^3$ TiSe$_2$ crystal cut from the sample used in the X-ray diffraction and electric transport experiments, is shown in Fig.", "REF .", "We performed calorimetric measurements using a membrane-based steady-state ac-micro-calorimeter [20], [21], [22], with a thermocouple composed of Au-$1.7\\%$ Co and Cu films deposited onto a 150 nm thick Si$_3$ N$_4$ -membrane as a thermometer.", "This technique enables precise measurements of changes in specific heat.", "The absolute accuracy of our data was checked against measurements on gold samples of similar size as the samples studied here.", "The upper, red curve in the inset of Fig.", "REF displays the bare specific heat data.", "An anomaly around $\\sim 190$  K indicating the charge density wave transition is clearly seen.", "Plotting the data with respect to a linear extrapolation of the high-temperature background (the red dashed line) yields a detailed presentation of the specific heat anomaly (lower, blue curve in the inset).", "The height of the anomaly of $\\sim 0.5$  J/mol K and the extended high-temperature tail are similar to previous reports [23].", "This tail induces some uncertainty into the determination of the transition temperature $T_{\\text{CDW}}$ .", "The inflection point yields $T_{\\text{CDW}} \\sim 191$  K whereas the specific heat peak is at $\\sim 189$  K. Remarkably, the specific heat is linear in temperature below the peak position, as highlighted by the blue dashed line, followed by a kink near $\\sim 182$  K. This kink is clearly seen in the main panel, which presents the data plotted with respect to the blue dashed line.", "The high-temperature tail of the charge density wave transition indicates the presence of fluctuations or a distribution of transition temperatures.", "At present, these effects cannot be separated from the mean-field anomaly of the specific heat, and a quantitative analysis based on the free energy expression in Eq.", "(REF ) is not possible.", "However, the break in the slope of the specific heat signals a change in the thermodynamic state of the sample, which we identify with the transition into the chiral phase.", "This is supported by the mean-field behavior predictions of Eq.", "(REF ).", "Taking the second derivative with respect to temperature for the lowest energy solutions yields linear temperature dependences for both $T_{\\text{Chiral}} < T < T_{\\text{CDW}}$ and for $T < T_{\\text{Chiral}}$ , but with different slopes: $C_{V} = \\left\\lbrace \\begin{array}{lcr} \\frac{T}{T_{\\text{CDW}}^2} \\frac{a_0^2}{2 b_0+12 b_1}, & \\ & T_{\\text{Chiral}} < T < T_{\\text{CDW}} \\\\ \\frac{T}{T_{\\text{CDW}}^2} \\frac{a_0^2}{2 b_0+3 b_1}, & & T < T_{\\text{Chiral}} \\end{array} \\right.$ We anticipate that this break in slope is apparent in the data even when fluctuations and inhomogeneous broadening are superimposed.", "The additional step-like discontinuity expected at the chiral transition is not clearly resolved in the data, either because our present experimental resolution is insufficient, or because it is masked by broadening due to the fluctuations surrounding the transition, or due to an increased sample inhomogeneity resulting from the formation of domain walls in the chiral charge ordered state.", "We note that the high-temperature tails above $T_{\\text{CDW}}$ as well as $T_{\\text{Chiral}}$ are also apparent in the X-ray diffraction data, and that the two transition temperatures identified in the specific heat data closely match those suggested by the X-ray measurements.", "Figure: (Color online) The specific heat as a function of temperature.", "inset) Bare specific heat before (red, upper curve) and after (blue lower curve) subtracting the high temperature background (red dashed line).", "The blue dashed line emphasizes the linear temperature dependence below the initial onset of charge order at T CDW T_{\\text{CDW}}, as well as the break in slope which occurs at the lower temperature T Chiral T_{\\text{Chiral}}.", "This is shown more clearly in the main panel, which displays the specific heat after subtracting the linear fit below T CDW T_{\\text{CDW}}.Figure: (Color online) a) The principal components of resistivity as a function of temperature.", "Notice that the scales for the two curves differ by a factor of ∼575\\sim 575. b) The derivatives of both curves, displaying sharp minima at T CDW T_{\\text{CDW}}, the initial onset temperature of (non-chiral) charge order.", "c) The resistivity anisotropy ρ c /ρ ab \\rho _c / \\rho _{ab} as a function of temperature.", "The peak at 183 K, defining the second transition temperature T Chiral T_{\\text{Chiral}}, can be clearly identified.", "The broad feature centered at 161 K follows from the presence of broad maxima in ρ c (T)\\rho _c(T) and ρ ab (T)\\rho _{ab}(T), and does not signify an additional transition.", "It can be modeled by fitting the broad peaks in panel a to parabolic functions, resulting in the dotted line.Electrical Transport.—In a layered material like TiSe$_2$ , traditional four-terminal methods to determine the resistivity along the $c$ -axis, $\\rho _c$ , and in the $ab$ -plane, $\\rho _{ab}$ , may be unreliable for single crystals, especially if the conductance anisotropy is large.", "This problem is overcome by a six-terminal method that uses a single crystal in a rectangular shape [24].", "We employed four parallel Au stripes for electrical contacts, that were sputtered onto each $c$ -axis-normal surface, perpendicular to the longest in-plane dimension of the crystal.", "The current was injected through the outermost contacts of one surface and voltages were measured across the innermost contacts of each surface.", "The Laplace equation was then solved and inverted to find $\\rho _{ab}$ and $\\rho _c$ [24], [25].", "This method also allows a test of sample homogeneity by permuting the electrodes used for current and voltage [26].", "Fig.", "REF a shows the principle components of the resistivity for the same TiSe$_2$ crystal used in the X-ray diffraction studies.", "The maximum variation upon permuting the electrodes is less than 5 %, implying a high degree of homogeneity.", "The temperature profiles of $\\rho _{ab}$ and $\\rho _c$ are also almost identical, although their absolute values differ by an anisotropy factor of $\\sim 575$ .", "Each resistivity component exhibits a maximum close to 168 K, which has previously been suggested to arise from an initial decrease in the density of available carriers, caused by the opening of a gap in the charge ordered phase, which is overtaken at lower temperatures by both the decrease of scattering channels due to the developing order, and an increase in density of states due to the downward shift of the conduction band minimum below $T_{\\text{CDW}}$ [27].", "The position of the maximum thus does not coincide with any charge ordering transition.", "Instead, Di Salvo et al.", "found that a sharp minimum in $d\\rho /dT$ matches the onset of the charge ordered superlattice as determined by neutron scattering [17].", "This is confirmed by our data in Fig.", "REF b, which show $d\\rho _c/dT$ and $d\\rho _{ab}/dT$ to be virtually coincident after scaling by the anisotropy, with a minimum at 191 K that closely matches the onset of charge order as determined by X-ray diffraction (see Fig.", "REF a).", "While $\\rho (T)$ in Fig.", "REF a varies by more than a factor of three, the temperature variation of the resistivity anisotropy $\\rho _c / \\rho _{ab}$ shown in Fig.", "REF c, is significantly smaller at only $\\sim 1.5$  %.", "This observation agrees with the suggestion that the overall temperature dependence of $\\rho (T)$ is determined primarily by the evolution of the carrier density [27], while the variations in anisotropy are determined by the temperature dependence of the scattering, which evidently plays a much smaller role.", "The anisotropy exhibits a sharp peak at 183 K and a relatively broad maximum centered at 161 K. The latter feature is caused by slight differences in position and height of the maxima in $\\rho _c(T)$ and $\\rho _{ab}(T)$ , and does not signify a qualitative change in physics.", "It can be modeled by parabolic fits of the maxima in Fig.", "REF a, which results in the anisotropy indicated by the dotted curve in Fig.", "REF c. The sharp feature at 183 K on the other hand, closely matches the second transition temperature identified in our X-ray diffraction and specific heat experiments.", "The onset of chiral order will be accompanied by the formation of right and left-handed domains.", "These have been shown to occur in both pristine and Cu-intercalated TiSe$_2$ , using (surface sensitive) scanning tunneling microscopy [3], [7], [9], as well as (bulk) reflectivity measurements [3].", "The typical sizes of the observed domains are of the order of tens of nanometers [9].", "The corresponding presence of domain walls within the $a,b$ -plane may be expected to lead to a sharp increase of electronic scattering within the plane, without affecting electrical transport along the $c$ axis, thus causing a decrease of the resistivity anisotropy $\\rho _c / \\rho _{ab}$ with decreasing temperature below $T_{\\text{Chiral}}$ .", "The feature at 183 K in the anisotropy in Fig.", "REF is thus consistent with the expected increase of $\\rho _{ab}$ below $T_{\\text{Chiral}}$ resulting from increased scattering off chiral domain walls.", "Conclusions.—We conclude that the low temperature charge ordered state in 1T-TiSe$_2$ arises in a sequence of two closely separated phase transitions.", "The first transition, at $T_{\\text{CDW}} \\simeq 190$  K, is well known to indicate the onset of a 2x2x2 charge ordered state.", "The second transition, which we identify to occur at $T_{\\text{Chiral}} \\simeq 183$  K, is characterized by the emergence of previously unobserved X-ray diffraction peaks, a sudden change in slope of the specific heat, and a sharp peak in the resistivity anisotropy.", "The temperature dependence of the X-ray superlattice reflections, combined with an analysis of their structure factors, and the temperature dependence of the specific heat and resistivity, strongly suggests a scenario in which the transition at $\\sim 183$  K indicates the emergence of chiral charge order out of the non-chiral charge density wave state.", "This sequence of two transitions leading first from the normal state to the non-chiral charge density wave, and only then to the chiral charge ordered state, agrees with the predictions arising from the theoretical model of Ref.", "[5], which describes the recently discovered chiral state of 1T-TiSe$_2$ in terms of combined charge and orbital order.", "Work at the Advanced Photon Source and Material Science Division of Argonne National Laboratory were supported by the U.S. DOE-BES under Contract No.", "NE-AC02-06CH11357." ] ]

[ [ "Hyperbolicity and Stability for Hamiltonian flows" ], [ "Abstract We prove that a Hamiltonian star system, defined on a 2d-dimensional symplectic manifold M, is Anosov.", "As a consequence we obtain the proof of the stability conjecture for Hamiltonians.", "This generalizes the 4-dimensional results in [6]." ], [ "Structural stability and hyperbolicity", "Let $\\mathcal {S}$ be a dynamical system defined on a closed manifold.", "The concept of structural stability was introduced in the mid 1930s by Andronov and Pontrjagin ([1]).", "Roughly speaking it means that under small perturbations the dynamics are topologically equivalent: a dynamical system is $C^r$ -structurally stable if it is topologically conjugated to any other system in a $C^r$ neighbourhood.", "These conjugations are often defined in sets where the dynamics is relevant, usually in its non-wandering set, $\\Omega (\\mathcal {S})$ , and the system is said to be $\\Omega $ -stable.", "We recall that $\\Omega (\\mathcal {S})$ is the set of points in the manifold such that, for every neighbourhood $U$ , there exists an iterate $n$ satisfying $\\mathcal {S}^n(U)\\cap U\\ne \\emptyset $ .", "Smale's program in the early 1960s aimed to prove the (topological) genericity of structurally stable systems.", "Although Smale's program was proved to be wrong one decade later, it played a fundamental role in the development of the theory of dynamical systems.", "It led to the construction of Hyperbolic theory, studying uniform hyperbolicity, and characterizing structural stability as being essentially equivalent to uniform hyperbolicity.", "In the attempt to unify several classes of structurally stable systems, e.g., Morse-Smale systems, the horseshoe and Anosov's systems, Smale conceived Axiom A: a system $\\mathcal {S}$ is said to satisfy the Axiom A property if the closure of its closed orbits is equal to $\\Omega (\\mathcal {S})$ and, moreover, this set is hyperbolic.", "It turned out to be one of the most challenging problems in the modern theory of dynamical systems to know if a $C^r$ -structurally stable system satisfies the Axiom A property.", "A cornerstone to this program was the remarkable proof done by Mañé of the stability conjecture for the case of $C^1$ -dissipative diffeomorphisms ([17]).", "The proof of Mañé essentially uses the property, that holds for stable diffeomorphisms, that all periodic orbits are robustly hyperbolic.", "Therefore, one could ask if there exists a weaker property than stability that guarantees Axiom A.", "This remounts to another old problem attributed to Liao and Mañé (see, e.g.", "[15]) that asks wether for a system to loose the $\\Omega $ -stability it must undergo a bifurcation in a critical element.", "In other words, must a system robustly free of any critical-element-bifurcation be $\\Omega $ -stable?" ], [ "The star systems", "Back to the early 1980s, Mañé defined a set $\\mathcal {F}^1$ , of dissipative diffeomorphisms having a $C^1$ -neighbourhood $\\mathcal {U}$ such that every diffeomorphism inside $\\mathcal {U}$ has all periodic orbits of hyperbolic type.", "Therefore, a diffeomorfism in $\\mathcal {F}^1$ is a system that has robustly no critical-element-bifurcation.", "Given that being in $\\mathcal {F}^1$ concerns only to critical points and that the hyperbolicity on critical points is merely orbit-wise, but not uniform, this property looks, a priori, quite weak.", "Indeed, the Axiom A plus the no-cycle property, which is necessary and sufficient for a system to be $\\Omega $ -stable looks much stronger.", "Recall that, by the spectral decomposition of an Axiom A system $\\mathcal {S}$ , we have that $\\Omega (\\mathcal {S})=\\cup _{i=1}^k \\Lambda _i$ where each $\\Lambda _i$ is a basic piece.", "We define an order relation by $\\Lambda _i\\prec \\Lambda _j$ if there exists $x$ (outside $\\Lambda _i\\cup \\Lambda _j$ ) such that $\\alpha (x)\\subset \\Lambda _i$ and $\\omega (x)\\subset \\Lambda _j$ .", "We say that $\\mathcal {S}$ has a cycle if there exists a cycle with respect to $\\prec $ (see  [24] for details).", "Thus, the above conjecture of Liao and Mañé can be stated as follows: does every system robustly free of non-hyperbolic critical elements satisfy Axiom A and the no-cycle property?", "For diffeomorphisms the answer is affirmative.", "In  [18], Mañé proved that every surface dissipative diffeomorphism of $\\mathcal {F}^1$ satisfies the Axiom A property.", "Hayashi ([16]) extended this result for higher dimensions.", "In fact, the mentioned results by Mañé and Hayashi guarantee that diffeomorphisms in $\\mathcal {F}^1$ satisfy the Axiom A and the no-cycle properties (see also a result by Aoki [2]).", "We point out that classic results imply that being in $\\mathcal {F}^1$ is a necessary condition to satisfy the Axiom A and the no-cycle condition (see [17] and the references wherein).", "In the conservative setting we refer the seminal paper of Newhouse [20] where it was proved that any symplectomorphism robustly free of non-hyperbolic periodic orbits is Anosov.", "Recently, an analogous result was obtained by Arbieto and Catalan [3] for volume-preserving diffeomorphisms.", "For the continuous-time case the analogous to the set $\\mathcal {F}^1$ is traditionally denoted by $\\mathcal {G}^1$ , and a flow in it is called a star flow.", "Obviously, in this setting, the hyperbolicity of the equilibrium points (singularities of the vector field) is also imposed.", "It is well known that the dissipative star flow defined by the Lorenz differential equations (see e.g.", "[25]) belongs to $\\mathcal {G}^1$ .", "However, the hyperbolic saddle-type singularity is accumulated by (hyperbolic) closed orbits and they are contained in the non-wandering set preventing the flow to be Axiom A.", "The problem of Liao and Mañé of knowing if every (nonsingular) dissipative star flow satisfies the Axiom A and the no-cycle condition remained unsolved for almost 20 years, in part due to the technical difficulties specific of the flow setting.", "This central result was proved by Gan and Wen ([15]).", "If we consider flows that are divergence-free and restrict the definition of $\\mathcal {G}^1$ to this setting, which means that the star property is satisfied when one restricts to the conservative setting (but possibly not in the broader space of dissipative flows), using a completely different approach, based in conservative-type seminal ideas of Mañé, two of the authors (see [10]) proved recently that any divergence-free star vector field defined in a closed three-dimensional manifold does not have singularities and moreover it is Anosov (the manifold is uniformly hyperbolic).", "This result was recently generalized in [14] for a $d$ -dimensional closed manifold, $d \\ge 4$ .", "We point out that the proof in [10] could not be trivially adapted to higher dimensions.", "Indeed, in dimension 3, the normal bundle is splitted in two 1-dimensional subbundles.", "Consequently, using volume-preserving arguments the authors were able to prove the existence of a dominated splitting for the linear Poincaré flow and then the hyperbolicity.", "The main novelties of the proof in [14] are the use of a new strategy to prove the absence of singularities and the adaptation of an argument of Mañé in [18] to obtain hyperbolicity from a dominated splitting, which follows easily in dimension 3.", "The key ingredient in the proof is the following dichotomy for $C^1$ -divergence-free vector fields: a periodic orbit of large period either admits a dominated splitting of a prescribed strengh or can be turned into a parabolic one by a $C^1$ -small perturbation along the orbit.", "This dichotomy is a consequence of an adaptation ([11]) to the conservative setting of a dichotomy by Bonatti, Gourmelon and Vivier ([13]).", "In the context of Hamiltonian flows, and following the strategy described in [10], it was obtained in [6] an affirmative answer to the problem of Liao and Mañé: any Hamiltonian star system defined on a 4-dimensional symplectic manifold is Anosov.", "We remark that the proof makes use of some results that are only available in dimension four (see [7], [8]).", "In this paper we consider the setting of Hamiltonian flows defined on a $2d$ -dimensional compact symplectic manifold $(M,\\omega )$ ($d \\ge 2$ ).", "Here, we generalize the results in [6] to higher dimensions and we prove that any Hamiltonian star system defined on $2d$ -dimensional compact symplectic manifold is Anosov.", "As a consequence we obtain the proof of the stability conjecture for Hamiltonians.", "A key ingredient is a Hamiltonian version of the previously mentioned dichotomy of Bonatti, Gourmelon and Vivier which will be developed in Section §REF ." ], [ "Hamiltonians and tangent map structures", "A Hamiltonian is a real-valued $C^r$ function on a Riemannian symplectic manifold $M$ , $2\\le r\\le \\infty $ , equipped with a symplectic form $\\omega $ , whose set is denoted by $C^r(M,\\mathbb {R})$ .", "Associated to $H$ , we have the Hamiltonian vector field $X_H$ which generates the Hamiltonian flow $X_H^t$ .", "Observe that $H$ is $C^2$ if and only if $X_H$ is $C^1$ and that, since $H$ is continuous and $M$ is compact, $Sing(X_H)\\ne \\emptyset $ , where $Sing(X_H)$ denotes the singularities of $X_H$ or, in other words, the critical points of $H$ or the equilibria of $X_H^t$ .", "Let $\\mathcal {R}(H)=M\\setminus Sing(X_H)$ stands for the regular points.", "A scalar $e\\in H(M)\\subset \\mathbb {R}$ is called an energy of $H$ .", "An energy hypersurface $\\mathcal {E}_{H,e}$ is a connected component of $H^{-1}(\\left\\lbrace e\\right\\rbrace )$ and it is regular if it does not contain singularities.", "If $H^{-1}(\\left\\lbrace e\\right\\rbrace )$ is regular, then $H^{-1}(\\left\\lbrace e\\right\\rbrace )$ is the union of a finite number of energy hypersurfaces.", "Definition 2.1 A Hamiltonian system is a triple $(H,e, \\mathcal {E}_{H,e})$ , where $H$ is a Hamiltonian, $e$ is an energy and $\\mathcal {E}_{H,e}$ is a regular connected component of $H^{-1}(\\lbrace e\\rbrace )$ .", "Fixing a small neighbourhood $\\mathcal {W}$ of a regular $\\mathcal {E}_{H,e}$ , there exist a small neighbourhood $\\mathcal {U}$ of $H$ and $\\epsilon >0$ such that, for all $\\tilde{H} \\in \\mathcal {U}$ and $\\tilde{e} \\in (e-\\epsilon ,e+\\epsilon )$ , $\\tilde{H}^{-1}(\\lbrace \\tilde{e}\\rbrace )\\cap \\mathcal {W}=\\mathcal {E}_{\\tilde{H},\\tilde{e}}$ .", "We call $\\mathcal {E}_{\\tilde{H},\\tilde{e}}$ the analytic continuation of $\\mathcal {E}_{H,e}$ .", "In the space of Hamiltonian systems we consider the topology generated by a fundamental systems of neighbourhoods.", "Definition 2.2 Given a Hamiltonian system $(H,e, \\mathcal {E}_{H,e})$ we say that $\\mathcal {V}(\\mathcal {U},\\epsilon )$ is a neighbourhood of $(H,e, \\mathcal {E}_{H,e})$ if there exist a small neighbourhood $\\mathcal {U}$ of $H$ and $\\epsilon >0$ such that for all $\\tilde{H} \\in \\mathcal {U}$ and $\\tilde{e} \\in (e-\\epsilon ,e+\\epsilon )$ one has that the analytic continuation $\\mathcal {E}_{\\tilde{H},\\tilde{e}}$ of $\\mathcal {E}_{H,e}$ is well-defined.", "For each $x$ in a regular energy hypersurface take the orthogonal splitting $T_xM=\\mathbb {R} X_H(x)\\oplus N_x$ , where $N_x=(\\mathbb {R} X_H(x))^\\perp $ is the normal fiber at $x$ .", "Consider the automorphism of vector bundles $DX^{t}_{H}\\colon T_{\\mathcal {R}}M \\rightarrow T_{\\mathcal {R}}M$ defined by $DX^{t}_{H}(x,v) = (X^{t}_{H}(x),DX_{H}^{t}(x)\\, v)$ .", "Of course that, in general, the subbundle $N_{\\mathcal {R}}$ is not $DX_{H}^{t}$ -invariant.", "So we relate to the $DX^{t}_{H}$ -invariant quotient space $\\widetilde{N}_{\\mathcal {R}}=T_{\\mathcal {R}}M / \\mathbb {R}X_{H}(\\mathcal {R})$ with an isomorphism $\\phi _{1}\\colon N_{\\mathcal {R}}\\rightarrow \\widetilde{N}_{\\mathcal {R}}$ .", "The unique map $P_{H}^{t}\\colon N_{\\mathcal {R}}\\rightarrow N_{\\mathcal {R}}$ such that $\\phi _{1}\\circ P_{H}^{t}=DX^{t}_{H}\\circ \\phi _{1}$ is called the linear Poincaré flow for $H$ .", "Let $\\Pi _{x}\\colon T_xM\\rightarrow N_x$ be the canonical orthogonal projection, so the linear Poincaré flow $P^{t}_{H}(x)\\colon N_{x}\\rightarrow N_{X^{t}_{H}(x)}$ is defined by $P^{t}_{H}(x)\\, v=\\Pi _{X^{t}_{H}(x)}\\circ DX^{t}_{H}(x)\\, v$ .", "We now consider $\\mathcal {N}_x=N_x\\cap T_x H^{-1}(e),$ where $T_xH^{-1}(e)=\\ker dH(x)$ is the tangent space to the energy level set with $e=H(x)$ .", "Thus, $\\mathcal {N}_\\mathcal {R}$ is invariant under $P^{t}_{H}$ .", "So we define the map $\\Phi _{H}^{t}\\colon \\mathcal {N}_{\\mathcal {R}}\\rightarrow \\mathcal {N}_{\\mathcal {R}},\\qquad \\Phi _{H}^{t}=P^{t}_{H}|_{\\mathcal {N}_\\mathcal {R}},$ called the transversal linear Poincaré flow for $H$ such that $\\Phi ^{t}_{H}(x)\\colon \\mathcal {N}_{x}\\rightarrow \\mathcal {N}_{X^{t}_{H}(x)},\\quad \\Phi ^{t}_{H}(x)\\, v=\\Pi _{X^{t}_{H}(x)}\\circ DX^{t}_{H}(x)\\, v$ is a linear symplectomorphism for the symplectic form induced on $\\mathcal {N}_\\mathcal {R}$ by $\\omega $ ." ], [ "Invariant splittings and hyperbolicity", "Given $x\\in \\mathcal {R}(H)$ , we say that $x$ is a periodic point if $X^t_H(x)=x$ for some $t$ .", "The smallest $t>0$ is called period of $x$ and we denote it by $\\pi (x)$ .", "A period point is said to be hyperbolic if there exist $\\theta \\in (0,1)$ and a splitting of the normal subbundle $\\mathcal {N}$ along the orbit of $x$ , $\\mathcal {N}=E^s\\oplus E^u$ , such that $\\Vert \\Phi _H^t(y)|_{E^s_y}\\Vert <\\theta ^t$ and $\\Vert (\\Phi _H^t(y)|_{E^u_y})^{-1}\\Vert <\\theta ^t$ for all $y$ in the orbit of $x$ .", "In an analogous way, given a compact and $X^t_H$ -invariant set $\\Lambda \\subset \\mathcal {R}(H)$ , we say that $\\Lambda $ is a (uniformly) hyperbolic set if there exist $\\theta \\in (0,1)$ and a $\\Phi ^t_H$ -invariant splitting of the normal subbundle $\\mathcal {N}_\\Lambda =E^s_\\Lambda \\oplus E^u_\\Lambda $ such that $\\Vert \\Phi _H^t(x)|_{E^s_x}\\Vert <\\theta ^t$ and $\\Vert (\\Phi _H^t(x)|_{E^u_x})^{-1}\\Vert <\\theta ^t$ for all $x\\in \\Lambda $ .", "Observe that changing the Riemannian metric the constant of hyperbolicity $\\theta $ can be taken equal to $1/2$ .", "Once the metric is fixed, as $\\theta $ approximates to 1 the hyperbolicity gets weaker.", "Now, consider a $\\Phi _H^t$ -invariant splitting $\\mathcal {N}=\\mathcal {N}^1\\oplus \\cdots \\oplus \\mathcal {N}^{k}$ over a compact, $X_H^t$ -invariant and regular set $\\Lambda $ .", "Assume that, for $1\\le k\\le \\dim (M)-2$ , all these subbundles have constant dimension.", "This splitting is $\\ell $ -dominated if there exists ${\\ell }>0$ such that, for any $0\\le i<j\\le k$ , ${\\Vert \\Phi _H^{\\ell }(x)|_{{\\mathcal {N}}_x^i}\\Vert }\\cdot {\\Vert \\Phi _H^{-\\ell }(X^{\\ell }(x))|_{{\\mathcal {N}}_{X^{\\ell }(x)}^j}\\Vert }\\le 1/2, \\: \\: \\forall \\: x\\in \\Lambda .$ The $\\Phi _H^t$ -invariant splitting $\\mathcal {N}=\\mathcal {N}^u\\oplus \\mathcal {N}^c \\oplus \\mathcal {N}^{s}$ over $\\Lambda $ is said to be partially hyperbolic if there exists ${\\ell }>0$ such that, $\\mathcal {N}^u$ is uniformly hyperbolic and expanding with constant of hyperbolicity $1/2$ ; $\\mathcal {N}^s$ is uniformly hyperbolic and contracting with constant of hyperbolicity $1/2$ and $\\mathcal {N}^u$ $\\ell $ -dominates $\\mathcal {N}^c$ and $\\mathcal {N}^c$ $\\ell $ -dominates $\\mathcal {N}^s$ .", "We introduce the notion of Hamiltonian star system.", "Definition 2.3 A Hamiltonian system $(H,e, \\mathcal {E}_{H,e})$ is a Hamiltonian star system if there exists a neighbourhood $\\mathcal {V}$ of $(H,e, \\mathcal {E}_{H,e})$ such that, for any $(\\tilde{H},\\tilde{e}, \\mathcal {E}_{\\tilde{H},\\tilde{e}})\\in \\mathcal {V}$ , the correspondent regular energy hypersurface $\\mathcal {E}_{\\tilde{H},\\tilde{e}}$ has all the closed orbits hyperbolic.", "We denote by $\\mathcal {E}^{\\star }_{H,e}$ the regular energy hypersurface with the previous property and by $\\mathcal {G}^{2}(M)$ the set of all Hamiltonian star systems defined on a $2d$ -dimensional symplectic manifold, $d\\ge 2$ .", "The next definition states when a Hamiltonian system is Anosov.", "Definition 2.4 A Hamiltonian system $(H,e, \\mathcal {E}_{H,e})$ is said to be Anosov if $\\mathcal {E}_{H,e}$ is uniformly hyperbolic for the Hamiltonian flow $X_H^t$ associated to $H$ .", "For $d\\ge 2$ , let $\\mathcal {A}(M)$ denote the set of all Anosov Hamiltonian systems, defined on a $2d$ -dimensional symplectic manifold." ], [ "Statement of the results", "Our main result states that a Hamiltonian star system, defined on a $2d$ -dimensional symplectic manifold, is an Anosov Hamiltonian system.", "Theorem 1 If $(H,e, \\mathcal {E}^{\\star }_{H,e})\\in \\mathcal {G}$$^{2}(M)$ then $(H,e, \\mathcal {E}^{\\star }_{H,e})\\in \\mathcal {A}(M)$ .", "We say that a Hamiltonian system $(H,e, \\mathcal {E}_{H,e})$ is isolated in the boundary of Anosov Hamiltonian systems if given a neighbourhood $\\mathcal {V}$ of $(H,e, \\mathcal {E}_{H,e})$ and $(\\tilde{H},\\tilde{e}, \\mathcal {E}_{\\tilde{H},\\tilde{e}})\\in \\mathcal {V}$ the correspondent energy hypersurface $\\mathcal {E}_{\\tilde{H},\\tilde{e}}$ is uniformly hyperbolic but $\\mathcal {E}_{H,e}$ is not.", "As a consequence of Theorem REF , we have the following result.", "Corollary 1 The boundary of $\\mathcal {A}(M)$ has no isolated points.", "Definition 2.5 We say that the Hamiltonian system $(H,e,\\mathcal {E}_{H,e})$ is structurally stable if there exists a homeomorphism $h_{\\tilde{H},\\tilde{e}}$ between $\\mathcal {E}_{H,e}$ and $\\mathcal {E}_{\\tilde{H},\\tilde{e}}$ , preserving orbits and their orientations.", "Moreover, $h_{\\tilde{H},\\tilde{e}}$ is continuous on the parameters $\\tilde{H}$ and $\\tilde{e}$ , and converges to $id$ when $\\tilde{H}$ $C^2$ -converges to $H$ and $\\tilde{e}$ converges to $e$ .", "Accordingly with these definitions, we show that structurally stable Hamiltonian systems, defined on a $2d$ -dimensional symplectic manifold, are Anosov.", "Theorem 2 If $(H,e, \\mathcal {E}_{H,e})$ is a structurally stable Hamiltonian system, then $(H,e, \\mathcal {E}_{H,e})$ is Anosov." ], [ "A dichotomy for Hamiltonian periodic linear differential systems", "In this section we intend to contextualize the results in [13] for the Hamiltonian scenario.", "Actually, in [13] it is studied the abstract setting of linear bounded cocycles over sets of periodic orbits of large period and it is proved, in brief terms, that a dichotomy between uniform dominated splitting or else one-point spectrum holds (see [13]).", "In Theorem REF we provide a version of this result adapted to Hamiltonians.", "We denote by $Sp(2d,\\mathbb {R})$ ($d\\ge 1$ ), the symplectic Lie group of $2d\\times 2d$ matrices $A$ and with real entries satisfying $A^TJA=J$ , where $J=\\begin{pmatrix}0 & -\\textbf {1}_{d}\\\\\\textbf {1}_{d} &0\\end{pmatrix}$ denotes the skew-symmetric matrix, $\\textbf {1}_{d}$ is the $d$ -dimensional identity matrix and $A^T$ the transpose matrix of $A$ .", "Let $\\mathbb {R}^{2d}$ be a symplectic vector space equipped with a symplectic form $\\omega $ .", "Let $S$ be a two-dimensional subspace of $\\mathbb {R}^{2d}$ .", "We denote the $\\omega $ -orthogonal complement of $S$ by $S^{\\perp }$ which is defined by those vectors $u\\in \\mathbb {R}^{2d}$ such that $\\omega (u,v) = 0$ , for all $v\\in S$ .", "Clearly $\\dim (S^{\\perp })=2d-\\dim (S)$ .", "When, for a given subspace $S\\subset \\mathbb {R}^{2d}$ , we have that $\\omega |_{S\\times S}$ is non-degenerate (say $S^{\\perp }\\cap S=\\lbrace 0\\rbrace $ ) then $S$ is said to be a symplectic subspace.", "We say that the basis $\\lbrace e_1,...,e_{d},e_{\\hat{1}},...e_{\\hat{d}}\\rbrace $ is a symplectic base of $\\mathbb {R}^{2d}$ if $\\omega (e_i,e_j)=0$ , for all $j\\ne \\hat{i}$ and $\\omega (e_i,e_{\\hat{i}})=1$ .", "Let $\\mathfrak {sp}(2d,\\mathbb {R})$ denote the symplectic $2d$ -dimensional Lie algebra of matrices $H$ such that $JH+H^TJ=0$ , $\\Sigma $ a set of (infinite) periodic orbits and $C^{0}(\\Sigma , \\mathfrak {sp}(2d,\\mathbb {R}))$ denote the space of continuous maps (infinitesimal generators) with values on the Lie algebra $\\mathfrak {sp}(2d,\\mathbb {R})$ over a Hamiltonian flow $\\varphi ^t\\colon \\Sigma \\rightarrow \\Sigma $ .", "We endow $C^{0}(\\Sigma ,\\mathfrak {sp}(2d,\\mathbb {R}))$ with the uniform convergence topology defined by $\\Vert H_0-H_1\\Vert _{0}={\\underset{x\\in {\\Sigma }}{\\text{max}}}\\Vert H_0(x)-H_1(x)\\Vert ,$ for any $H_0,H_1\\in C^{0}(\\Sigma ,\\mathfrak {sp}(2d,\\mathbb {R}))$ .", "Given $H \\in C^{0}(\\Sigma , \\mathfrak {sp}(2d,\\mathbb {R}))$ , for each $x\\in \\Sigma $ we consider the non-autonomous linear differential equation $u^\\prime (s)\\Big |_{s=t}=H(\\varphi ^{t}(x))\\cdot u(t),$ known as linear variational equation.", "Fixing the initial condition $u(0)=\\textbf {1}_{2d}$ the unique solution of (REF ) is called the fundamental solution related to the system $H$ .", "The solution of  (REF ) is a linear flow $\\Phi _H^t\\colon \\mathbb {R}^{2d}_x\\rightarrow \\mathbb {R}^{2d}_{\\varphi ^{t}(x)}$ , where $\\Phi ^t_H\\in Sp(2d,\\mathbb {R}))$ which may be seen as the skew-product flow $\\begin{array}{cccc}\\Phi _H^t\\colon & \\Sigma \\times \\mathbb {R}^{2d} & \\longrightarrow & \\Sigma \\times \\mathbb {R}^{2d} \\\\& (x,v) & \\longrightarrow & (\\varphi ^{t}(x),\\Phi ^{t}_{H}(x)\\cdot v).\\end{array}$ Furthermore, we have the cocycle identity $\\Phi ^{t+s}_{H}(x)=\\Phi ^{s}_{H}(\\varphi ^{t}(x))\\Phi _{H}^{t}(x)$ for all $x\\in \\Sigma $ and $t,s\\in \\mathbb {R}$ .", "Moreover, $H$ satisfies the differential equation $H(x)=\\frac{d}{dt}\\Phi ^{t}_{H}(x)|_{t=0}$ for all $x\\in \\Sigma $ .", "We call $H$ the infinitesimal generator associated to $\\Phi _{H}^{t}$ .", "Given $H\\in \\mathfrak {sp}(2d,\\mathbb {R})$ , $\\xi >0$ and $P\\in \\mathfrak {sp}(2d,\\mathbb {R})$ satisfying $\\Vert P\\Vert _0<\\xi $ we say that $H+P$ is a $\\xi $ -$C^0$ -perturbation of $H$ .", "The Hamiltonian dynamics induced by $H+P$ is given by the solution of $u^\\prime (s)\\Big |_{s=t}=(H+P)(\\varphi ^{t}(x))\\cdot u(t).$ We begin by proving a basic perturbation lemma which will be the main tool for obtaining the results from [13] to our Hamiltonian context.", "Roughly, we would like to change a little bit the action of the cocycle on a certain two-dimensional symplectic subspace in time-one.", "Lemma 3.1 Given $H\\in \\mathfrak {sp}(2d,\\mathbb {R})$ and $\\epsilon >0$ , there exists $\\xi _{0}>0$ (depending on $H$ and $\\epsilon $ ), such that given any $\\xi \\in (0,\\xi _{0})$ , any $p\\in {\\Sigma }$ (with period larger than 1), any 2-dimensional symplectic subspace $S_p \\subset \\mathbb {R}^{2d}_{p}$ and any $R_\\xi \\in Sp(2d,\\mathbb {R})$ which is $\\xi $ -$C^0$ -close to $id$ and $R_\\xi |_{W_p}=id$ (where $W_{p}$ is the orthogonal symplectic complement of $S_p$ in $\\mathbb {R}^{2d}_{p}$ ), there exists $P\\in \\mathfrak {sp}(2d,\\mathbb {R})$ (depending on $\\xi $ and $p$ ) such that: $\\Vert P\\Vert _0<\\epsilon $ ; $P$ is supported in $\\varphi ^{t}(p)$ for $t\\in [0,1]$ ; $\\Phi ^{t}_{H+P}(p)=\\Phi ^{t}_{H}(p)$ on $W_p$ ; $\\Phi ^{1}_{H+P}(p)\\cdot v=\\Phi ^{1}_{H}(p) R_{\\xi } \\cdot v$ , $\\forall v \\in S_p$ .", "Take $K:=\\underset{p\\in \\Sigma }{\\text{max}}\\Vert \\Phi _{H}^{\\pm t}(p)\\Vert $ for $t\\in [0,1]$ .", "We claim that it is sufficient to take $\\xi _{0}>0$ such that: $\\xi _{0}\\le \\frac{\\epsilon }{4 K^{2}}.$ Let $\\alpha \\colon \\mathbb {R} \\rightarrow [0,1]$ be any $C^{\\infty }$ function such that $\\alpha (t)=0$ for $t\\le 0$ , $\\alpha (t)=1$ for $t\\ge 1$ , and $0\\le \\alpha ^{\\prime }(t)\\le 2$ , for all $t$ .", "We define the 1-parameter family of symplectic linear maps $\\Psi ^{t}(p)\\colon \\mathbb {R}^{2d}_{p} \\rightarrow \\mathbb {R}^{2d}_{p}$ for $t\\in [0,1]$ as follows; we fix two symplectic basis $\\lbrace e_1,e_{\\hat{1}}\\rbrace $ of $S_p$ and $\\lbrace e_2,e_3,...,e_d,e_{\\hat{2}},e_{\\hat{3}},...,e_{\\hat{d}}\\rbrace $ of $W_p$ .", "Take $\\xi \\in (0,\\xi _{0})$ and let $R_\\xi \\in Sp(2d,\\mathbb {R})$ taken $\\xi $ -$C^0$ -close to $id$ and $R_\\xi |_{W_p}=id$ .", "Since $S_p \\oplus W_p=\\mathbb {R}^{2d}_{p}$ , given any $u\\in \\mathbb {R}^{2d}_{p}$ we decompose $u=u_S+u_W$ , where $u_S\\in S_p$ and $u_W \\in W_p$ .", "Let $R_{t}\\colon S_p\\oplus W_p\\rightarrow S_p\\oplus W_p$ be an isotopy of symplectic linear maps from $id$ to $R_\\xi $ such that: $R_{t}=\\alpha (t)R_\\xi +(1-\\alpha (t)id)$ and $R_{t}$ and $R_t^{-1}$ is $\\xi $ -$C^0$ -close to $id$ for any $t\\in \\mathbb {R}$ .", "Finally, we consider the 1-parameter family of linear maps $\\Psi ^{t}(p)\\colon \\mathbb {R}^{2d}_{p} \\rightarrow \\mathbb {R}^{2d}_{\\varphi ^{t}(p)}$ where $\\Psi ^{t}(p):=\\Phi _{H}^{t}(p) R_{t}$ .", "We take time derivatives and we obtain: $(\\Psi ^{t}(p))^{\\prime }&=& (\\Phi _{H}^{t}(p))^{\\prime }R_{t}+\\Phi _{H}^{t}(p)(R_{t})^{\\prime }=\\\\&=& H(\\varphi ^{t}(p))\\Phi _{H}^{t}(p)R_{t}+\\Phi _{H}^{t}(p)(R_{t})^{\\prime }=\\\\&=& H(\\varphi ^{t}(p))\\Psi ^{t}(p)+\\Phi _{H}^{t}(p)(R_{t})^{\\prime }(\\Psi ^{t}(p))^{-1}\\Psi ^{t}(p)=\\\\&=& \\left[H(\\varphi ^{t}(p))+P(\\varphi ^{t}(p))\\right]\\cdot \\Psi ^{t}(p).$ Hence we define the perturbation $P$ by, $P(\\varphi ^{t}(p))=\\Phi _{H}^{t}(p)(R_{t})^{\\prime }(R_{t})^{-1}(\\Phi _{H}^{t}(p))^{-1}.$ Let us now show that $P\\in \\mathfrak {sp}(2d,\\mathbb {R})$ , that is $JP+P^TJ=0$ holds.", "Recall the symplectic identities: for any $\\Phi \\in Sp(2d,\\mathbb {R})$ ; $J^{-1}=J^T=-J$ , $\\Phi ^TJ\\Phi =J$ and $\\Phi ^{-1}=J^{-1}\\Phi ^TJ$ .", "$JP+P^TJ&=& J\\Phi _{H}^{t}(p)(R_{t})^{\\prime }(R_{t})^{-1}(\\Phi _{H}^{t}(p))^{-1}+[\\Phi _{H}^{t}(p)(R_{t})^{\\prime }(R_{t})^{-1}(\\Phi _{H}^{t}(p))^{-1}]^TJ\\\\&=& (\\Phi _{H}^{-t}(p))^TJ(R_{t})^{\\prime }(R_{t})^{-1}(\\Phi _{H}^{t}(p))^{-1}+(\\Phi _{H}^{-t}(p))^T((R_{t})^{-1})^T((R_{t})^{\\prime })^T(\\Phi _{H}^{t}(p))^TJ\\\\&=& (\\Phi _{H}^{-t}(p))^TJ(R_{t})^{\\prime }(R_{t})^{-1}\\Phi _{H}^{-t}(p)+(\\Phi _{H}^{-t}(p))^T((R_{t})^{-1})^T((R_{t})^{\\prime })^TJ\\Phi _{H}^{-t}(p)\\\\&=& (\\Phi _{H}^{-t}(p))^T[J(R_{t})^{\\prime }(R_{t})^{-1}+((R_{t})^{-1})^T((R_{t})^{\\prime })^TJ]\\Phi _{H}^{-t}(p)\\\\&=& (\\Phi _{H}^{-t}(p))^TJ[-(R_{t})^{\\prime }(R_{t})^{-1}J-J((R_{t})^{-1})^T((R_{t})^{\\prime })^T]J\\Phi _{H}^{-t}(p)\\\\&=& (\\Phi _{H}^{-t}(p))^TJ[(R_{t})^{\\prime }J^{-1}(R_{t})^T+R_t J^{-1}((R_{t})^{\\prime })^T]J\\Phi _{H}^{-t}(p)\\\\&=& (\\Phi _{H}^{-t}(p))^TJ[(R_{t})J^{-1}(R_{t})^T]^{\\prime }J\\Phi _{H}^{-t}(p)\\\\&=& (\\Phi _{H}^{-t}(p))^TJ[J^{-1}]^{\\prime }J\\Phi _{H}^{-t}(p)=0.$ Now to prove (1) we compute the $C^0$ -norm of $P$ : $\\Vert P(\\varphi ^{t}(p))\\Vert _0&=&\\Vert \\Phi _{H}^{t}(p)(R_{t})^{\\prime }(R_{t})^{-1}(\\Phi _{H}^{t}(p))^{-1}\\Vert _0\\\\&\\le &K^{2} \\Vert (R_{t})^{\\prime }(R_{t})^{-1}\\Vert _0\\\\&\\le &K^{2} \\Vert (R_{t})^{\\prime }\\Vert _0\\Vert (R_{t})^{-1}\\Vert _0\\\\&\\le & 2K^{2} \\Vert (\\alpha ^\\prime (t)R_\\xi +(1-\\alpha (t)id))^{\\prime }\\Vert _0\\\\&\\le & 2K^{2} \\Vert \\alpha ^\\prime (t)(R_\\xi -id)\\Vert _0\\\\&\\le &4K^{2} \\Vert R_\\xi -id\\Vert _0\\le 4K^2\\xi <\\epsilon .$ Moreover, by our choice of $\\alpha $ , we have that $\\text{Supp}(P)$ is $\\varphi ^{t}(p)$ for $t\\in [0,1]$ and (2) is proved.", "Observe that the perturbed system $H+P$ generates the linear flow $\\Phi _{H+P}^{t}(p)$ which is the same as $\\Psi ^{t}$ , hence given $u\\in W_{p}$ we have, since $u=u_S+u_W$ (where $u_S=0$ ): $\\Phi _{H+P}^{t}(p)\\cdot u =\\Psi ^{t}(p)\\cdot u=\\Phi _{H}^{t}(p)[R_{t}(u_S)+u_W]=\\Phi _{H}^{t}(p)\\cdot u_W=\\Phi _{H}^{t}(p)\\cdot u,$ and (3) follows.", "At last, to prove (4), taking $u\\in S_{p}$ we obtain, $\\Phi _{H+P}^{1}(p)\\cdot u &=&\\Psi ^{1}(p)\\cdot u=\\Phi _{H}^{1}(p) R_{1}\\cdot u=\\Phi _{H}^{1}(p)[R_{\\xi }(u_S)+u_W]=\\\\&=& \\Phi _{H}^{1}(p) R_{\\xi }(u_S)=\\Phi _{H}^{1}(p) R_{\\xi }\\cdot u,$ and Lemma REF is proved.", "Now, we borrow the arguments in  [13] and use Lemma REF , when perturbations are needed, in order to obtain the following result which can be seen as the Hamiltonian version of  [13].", "In fact, the perturbations that are used in [13] are mainly directional homotheties (contractions and expansions with the same factor) and rotations which are clearly also symplectic.", "Theorem 3.2 Given any dimension $2d$ ($d\\ge 1$ ) and any $\\epsilon >0$ , there exist $m,n\\in \\mathbb {N}$ such that any $H\\in \\mathfrak {sp}(2d,\\mathbb {R})$ over any periodic orbit $p\\in \\Sigma $ with period $\\pi (p)>n$ satisfies one of the following two assertions: either $\\Phi ^t_H(p)$ admits an $m$ -dominated splitting; or there exists an $\\epsilon $ -$C^0$ -perturbation $H+P$ of $H$ such that $\\Phi ^{\\pi (p)}_{H+P}(p)$ has all eigenvalues with modulus equal to 1.", "Once we have done the work in the abstract setting of Hamiltonian periodic linear differential systems we would like to consider the $(2d-2)$ -linear differential system which is given by the tangent map to the (Hamiltonian) vector field associated to a Hamiltonian defined in a symplectic manifold of dimension $2d$ but ignoring the flow direction and restricted to an energy level.", "We call this linear differential system the dynamical linear differential system.", "Since we are interested in perturb along closed orbits the framework developed in previous section is the adequated one.", "Next, we present a result which is a version of Franks' lemma for Hamiltonians (see [26]).", "Roughly, it says that we can realize a Hamiltonian corresponding to a given perturbation of the transversal linear Poincaré flow.", "This lemma is the piece that makes possible the connection between abstract linear differential systems and the dynamical one.", "Lemma 3.3 Take $H \\in C^2(M,\\mathbb {R})$ , $\\epsilon $ , $\\tau >0$ and $x\\in M$ .", "Then, there exists $\\delta >0$ such that for any flowbox $V$ of an injective arc of orbit $X_{H}^{[0,t]}(x)$ , $t\\ge \\tau $ , and a transversal symplectic $\\delta $ -perturbation $F$ of $\\Phi _{{H}}^t(x)$ , there is $H_0\\in C^2(M,\\mathbb {R})$ satisfying: $H_0$ is $\\epsilon $ -$C^2$ -close to $H$ ; $\\Phi _{{H_0}}^t(x)=F$ ; $H=H_0$ on $X_{H}^{[0,t]}(x)\\cup (M\\backslash V)$ .", "Using Lemma REF and Theorem REF we obtain the following result which will be very useful in the sequel.", "Theorem 3.4 Let $H\\in C^2(M,\\mathbb {R})$ and $\\mathcal {U}$ be a neighborhood of $H$ in the $C^2$ -topology.", "Then for any $\\epsilon >0$ there are $m,n\\in \\mathbb {N}$ such that, for any $H_0\\in \\mathcal {U}$ and for any periodic point $p$ of period $\\pi (p)\\ge n$ : either $\\Phi ^t_{H_0}(p)$ admits an $m$ -dominated splitting along the orbit of $p$ ; or, for any tubular flowbox neighborhood $\\mathcal {T}$ of the orbit of $p$ , there exists an $\\epsilon $ -$C^2$ -perturbation $H_1$ coinciding with $H_0$ outside $\\mathcal {T}$ and whose transversal linear Poincaré flow $\\Phi ^{\\pi (p)}_{H_1}(p)$ has all eigenvalues with modulus equal to 1." ], [ "Global hyperbolicity", "Lemma 3.5 If $(H,e, \\mathcal {E}^{\\star }_{H,e})\\in \\mathcal {G}$$^{2}(M)$ , then $\\Phi ^t_H$ admits a dominated splitting on $\\mathcal {E}^{\\star }_{H,e}$ .", "Consider $(H,e,\\mathcal {E}^{\\star }_{H,e})\\in \\mathcal {G}$$^{2}(M)$ and a $C^2$ -neighbourhood $\\mathcal {V}(\\mathcal {U},\\epsilon )$ of $(H,e,\\mathcal {E}^{\\star }_{H,e})$ such that, for any $H_0 \\in \\mathcal {U}$ and any $e_0 \\in (e-\\epsilon ,e+\\epsilon )$ , the analytic continuation $\\mathcal {E}^{\\star }_{H_0,e_0}$ of $\\mathcal {E}^{\\star }_{H,e}$ also has all the closed orbits hyperbolic, and such that the dicothomy in Theorem REF holds.", "Therefore, by Theorem  REF , there exist positive constants $m$ and $n$ such that $\\Phi ^t_{H_0}$ admits an $m$ -dominated splitting along the $X^t_{H_0}$ -orbit of any periodic point $p$ in $\\mathcal {E}^{\\star }_{H_0,e_0}$ with period $\\pi (p) \\ge n$ .", "Observe that, since any periodic point in $\\mathcal {E}^{\\star }_{H_0,e_0}$ is hyperbolic, we have the following $X_{H_0}^t$ -invariant splitting $\\mathcal {N}_p=\\mathcal {N}^u_p \\oplus \\mathcal {N}^s_p$ such that any subbundle has constant dimensionWe observe that, in the symplectic context, the index of hyperbolic orbits is always equal to $d$ .. We claim that this splitting is $m$ -dominated for any periodic point $p$ with period $\\pi (p) \\ge n$ .", "If this claim is not true, there is a periodic point $q$ with period $\\pi (q) \\ge n$ such that the angle between $\\mathcal {N}^u_q$ and $\\mathcal {N}^s_q$ is arbitrarily close to 0 or such that $q$ is wealy hyperbolic.", "In these situations, it is straightfoward to see that, applying the Franks' lemma for Hamiltonians (Lemma REF ) several times, we can $C^2$ -perturb $H_0$ in $\\mathcal {U}$ in order to have $H_1$ such that $q$ is a parabolic closed orbit of $H_1$ in the correspondent energy hypersurface $\\mathcal {E}^{\\star }_{H_1,e_1}$ .", "But this is a contradiction, since $(H_1,e_1,\\mathcal {E}^{\\star }_{H_1,e_1}) \\in \\mathcal {G}$$^{2}(M)$ .", "Therefore, any periodic point $p$ with period $\\pi (p) \\ge n$ admits the $m$ -dominated splitting $\\mathcal {N}_p=\\mathcal {N}^u_p \\oplus \\mathcal {N}^s_p$ .", "Recall that a dominated splitting can be continuously extended to the closure of a set.", "Thus, the $m$ -dominated splitting over the set of periodic points $p$ in $\\mathcal {E}^{\\star }_{H_0,e_0}$ , with period $\\pi (p) \\ge n$ , can be continuously extended to its closure.", "Furthermore, we observe that, since $(H_0,e_0,\\mathcal {E}^{\\star }_{H_0,e_0}) \\in \\mathcal {G}$$^{2}(M)$ , the set of periodic points $p$ in $\\mathcal {E}^{\\star }_{H_0,e_0}$ with period $\\pi (p) < n$ has a finite number of elements.", "Hence, the closure of the set of periodic points $p$ in $\\mathcal {E}^{\\star }_{H_0,e_0}$ with period $\\pi (p) \\ge n$ coincides with the set of periodic points in $\\mathcal {E}^{\\star }_{H_0,e_0}$ .", "So, we have just shown that any Hamiltonian $(H_0,e_0,\\mathcal {E}^{\\star }_{H_0,e_0})$ in $\\mathcal {V}(\\mathcal {U},\\epsilon )$ admits a dominated splitting on the closure of the set of periodic points in $\\mathcal {E}^{\\star }_{H_0,e_0}$ .", "Now, let $x$ be any point in $\\mathcal {E}^{\\star }_{H,e}$ .", "Clearly, by the Poincaré recurrence theorem, $x$ is a non-wandering point.", "Furthermore, by the Hamiltonian version of the ergodic closing lemma (see  [5]), there exist $H_n \\in \\mathcal {U}$ , $C^2$ -converging to $H$ , and periodic points $p_n$ of $H_n$ converging to $x$ .", "Thus, it follows from above that $x$ can be approximated by periodic points that admit a dominated splitting.", "Since the dominated splitting can be continuously extended to the closure of a set, we obtain that $\\Phi ^t_H$ admits a dominated splitting on $\\mathcal {E}^{\\star }_{H,e}$ .", "Remark 3.1 Observe that the previous lemma remains valid if we assume that $(H,e, \\mathcal {E}_{H,e})$ is an isolated point in the boundary of $\\mathcal {A}(M)$ .", "In fact, to prove Lemma REF , we use the fact that $(H,e, \\mathcal {E}^{\\star }_{H,e})\\in \\mathcal {G}$$^{2}(M)$ to ensure the existence of a dominated splitting over a periodic orbit $p$ , with arbitrarially large period $\\pi $ , for a Hamiltonian $H_0$ , $C^2$ -close to $H$ , given by Theorem REF .", "Therefore, if we start the proof by assuming that $(H,e, \\mathcal {E}_{H,e})$ is an isolated point in the boundary of $\\mathcal {A}(M)$ , we must obtain the same conclusion, because any $C^2$ -perturbation $H_1$ of $H$ must be Anosov, and so it cannot display a periodic orbit $q$ with period $\\pi $ such that $\\Phi ^{\\pi (p)}_{H_1}(p)$ has all eigenvalues with modulus equal to 1.", "The following auxiliary result asserts that, for a star Hamiltonian $(H,e, \\mathcal {E}^{\\star }_{H,e})$ , any closed orbit is uniformly hyperbolic in the period.", "This is a crucial step to derive, from Lemma REF , uniform hyperbolicity on $\\mathcal {E}^{\\star }_{H,e}$ .", "Lemma 3.6 Let $(H,e, \\mathcal {E}^{\\star }_{H,e})\\in \\mathcal {G}$$^{2}(M)$ .", "There exist a $C^2$ -neighbourhood $\\mathcal {V}$ of $(H,e, \\mathcal {E}^{\\star }_{H,e})$ and a constant $\\theta \\in (0,1)$ such that, for any $(H_0,e_0, \\mathcal {E}^{\\star }_{H_0,e_0}) \\in \\mathcal {V}$ , if $p$ is a periodic point in $\\mathcal {E}^{\\star }_{H_0,e_0}$ with period $\\pi (p)$ and has the hyperbolic splitting $\\mathcal {N}_p=\\mathcal {N}^s_p \\oplus \\mathcal {N}_p^u$ then: $\\Vert \\Phi ^{\\pi (p)}_{H_0} |_{\\mathcal {N}_p^s}\\Vert < \\theta ^{\\pi (p)}$ and $\\Vert \\Phi ^{-\\pi (p)}_{H_0} |_{\\mathcal {N}_p^u}\\Vert < \\theta ^{\\pi (p)}$ .", "Given that $(H,e, \\mathcal {E}^{\\star }_{H,e})\\in \\mathcal {G}$$^{2}(M)$ , there exists a $C^2$ -neighbourhood $\\mathcal {V}(\\mathcal {U},\\epsilon )$ of $(H,e, \\mathcal {E}^{\\star }_{H,e})$ such that, for any $H_0 \\in \\mathcal {U}$ and any $e_0 \\in (e-\\epsilon ,e+\\epsilon )$ , the analytic continuation $\\mathcal {E}^{\\star }_{H_0,e_0}$ of $\\mathcal {E}^{\\star }_{H,e}$ also has all the closed orbits hyperbolic.", "This means that for any periodic point $p$ in $\\mathcal {E}^{\\star }_{H_0,e_0}$ , with period $\\pi (p)$ , we have that $\\mathcal {N}_p=\\mathcal {N}^s_p \\oplus \\mathcal {N}_p^u$ and there is a constant $\\theta _p \\in (0,1)$ such that $\\Vert \\Phi ^{\\pi (p)}_{H_0}(p) |_{\\mathcal {N}_p^s}\\Vert < \\theta _p^{\\pi (p)}$ and $\\Vert \\Phi ^{-\\pi (p)}_{H_0}(p) |_{\\mathcal {N}_p^u}\\Vert < \\theta _p^{\\pi (p)}$ .", "However, we want to prove that, in fact, we can choose $\\theta _p$ not depending on $p$ .", "Let us prove $(a)$ .", "Suppose, by contradiction, that given $\\theta =1-2\\delta $ , with $\\delta >0$ small, there exist $H_0 \\in \\mathcal {U}$ , $e_0 \\in (e-\\epsilon ,e+\\epsilon )$ and a periodic point $p \\in \\mathcal {E}^{\\star }_{H_0,e_0}$ , with period $\\pi (p)$ , hyperbolic by hypothesis, such that $(1-2\\delta )^{\\pi (p)} \\le \\Vert \\Phi ^{\\pi (p)}_{H_0}(p) |_{\\mathcal {N}_p^s}\\Vert .$ Let $A_t$ , $0 \\le t \\le \\pi (p)$ , be the one-parameter family of linear perturbations of $\\Phi _{H_0}^t(p)$ given by $A_t=\\Phi _{H_0}^{t}(p)(1-2\\delta )^{-t}.$ Observe that $\\Vert A_t-\\Phi _{H_0}^{t}(p)\\Vert $ can be made arbitrarily close to 0, taking $\\delta $ small enough.", "Take $\\tilde{\\epsilon }>0$ such that any $\\tilde{\\epsilon }$ -$C^2$ -perturbation $H_1$ of $H_0$ belongs to $\\mathcal {U}$ and take $0<\\tau \\le \\pi (p)$ .", "It follows from Franks' lemma for Hamiltonians (Lemma REF ) that there exists $\\delta >0$ such that for any flowbox $V$ of an injective arc of orbit $X_{H_0}^{[0,\\pi (p)]}(p)$ , there exists $H_1$ $\\tilde{\\epsilon }$ -$C^2$ -close to $H_0$ coinciding with $H_0$ outside $V$ and such that $H_1^{\\pi (p)}(p)=p$ and $\\Phi ^{\\pi (p)}_{H_1}(p) = A_{\\pi (p)}$ .", "But, by construction, we get that $\\Vert \\Phi ^{\\pi (p)}_{H_1}(p) |_{\\mathcal {N}_p^s}\\Vert =\\Vert \\Phi _{H_0}^{\\pi (p)}(p)|_{\\mathcal {N}_p^s}\\Vert (1-2\\delta )^{-\\pi (p)}\\ =1.$ This is a contradiction because $p$ is an hyperbolic periodic point of $H_1$ .", "Then $(a)$ must hold.", "Item $(b)$ is obtained using a similar argument.", "The following lemma is proved in [12] and, in brief terms, says that in the symplectic world, the existence of a dominated splitting implies partial hyperbolicity.", "Lemma 3.7 If $\\mathcal {N}^u\\oplus \\mathcal {N}^2$ is a dominated splitting for a symplectic linear map $\\Phi _H^t$ , with $\\dim \\mathcal {N}^u\\le \\dim \\mathcal {N}^2$ , then $\\mathcal {N}^2$ splits invariantly as $\\mathcal {N}^2 = \\mathcal {N}^c \\oplus \\mathcal {N}^s$ , with $\\dim \\mathcal {N}^s = \\dim \\mathcal {N}^u$ .", "Furthermore, the splitting $\\mathcal {N}^u \\oplus \\mathcal {N}^c \\oplus \\mathcal {N}^s$ is dominated, $\\mathcal {N}^u$ is uniformly expanding, and $\\mathcal {N}^s$ is uniformly contracting.", "In conclusion, $\\mathcal {N}^u\\oplus \\mathcal {N}^c \\oplus \\mathcal {N}^s$ is partially hyperbolic.", "Now, by Lemma REF , we handle with the last step of the proof of Theorem REF .", "Lemma 3.8 If $(H,e, \\mathcal {E}^{\\star }_{H,e})\\in \\mathcal {G}$$^{2}(M)$ and $\\mathcal {N}^1\\oplus \\mathcal {N}^2$ is a dominated splitting, then this splitting is hyperbolic.", "Since by Lemma REF we know that the splitting is partially hyperbolic it remains to prove that the central subbundle is trivial.", "The following arguments are borrowed by the ones of Mañé  [18].", "We begin by stating, cf.", "[18], the following useful result concerning a dominated splitting $\\mathcal {N}^1\\oplus \\mathcal {N}^2$ .", "Claim 3.1 If $\\displaystyle \\liminf _{t\\rightarrow \\infty }\\Vert \\Phi _H^t(x)|_{\\mathcal {N}^2_x}\\Vert =0$ and $\\displaystyle \\liminf _{t\\rightarrow \\infty }\\Vert \\Phi _H^{-t}(x)|_{\\mathcal {N}^1_x}\\Vert =0$ , for all $x\\in \\mathcal {E}^{\\star }_{H,e}$ , then $\\mathcal {E}^{\\star }_{H,e}$ is Anosov.", "We shall prove that $\\Phi _H^t|_{\\mathcal {N}^2}$ is uniformly contracting on $ \\mathcal {E}^{\\star }_{H,e}$ .", "That $\\Phi _H^t|_{\\mathcal {N}^1}$ is uniformly expanding on $\\mathcal {E}^{\\star }_{H,e}$ is analog and we leave it to the reader.", "By Claim  REF , we just have to show that $\\displaystyle \\liminf _{t\\rightarrow \\infty }\\Vert \\Phi _H^t(x)|_{\\mathcal {N}^2_x}\\Vert =0,\\:\\forall \\:x\\in \\mathcal {E}^{\\star }_{H,e}.$ By contradiction, assume that there is $x\\in \\mathcal {E}^{\\star }_{H,e}$ such that $\\displaystyle \\liminf _{t\\rightarrow \\infty }\\Vert \\Phi _H^t(x)|_{\\mathcal {N}^2_x}\\Vert >0.$ Take a subsequence $t_n\\underset{n\\rightarrow \\infty }{\\rightarrow }\\infty $ such that $\\displaystyle \\lim _{n\\rightarrow \\infty }\\dfrac{1}{t_n}\\log \\Vert \\Phi _H^{t_n}(x)|_{\\mathcal {N}^2_x}\\Vert \\ge 0.$ Now, define $\\begin{array}{cccc}\\Psi _n\\colon & C^0(\\mathcal {E}^{\\star }_{H,e}) & \\longrightarrow & \\mathbb {R}\\\\& f & \\longrightarrow & \\frac{1}{s_n}\\int _{0}^{s_n} f(X^{s}_H(x))ds\\end{array}$ where $C^0(\\mathcal {E}^{\\star }_{H,e})$ stands for the set of continuous functions on $\\mathcal {E}^{\\star }_{H,e}$ equipped with the $C^0$ -topology.", "Take a subsequence of $\\Psi _n$ converging to $\\Psi \\colon C^0(\\mathcal {E}^{\\star }_{H,e}) \\rightarrow \\mathbb {R}$ .", "By the Riesz representation theorem, there exists a $X_H^t$ -invariant Borel probability measure $\\mu $ defined on $\\mathcal {E}^{\\star }_{H,e}$ such that, for any continuous observable $f$ on $\\mathcal {E}^{\\star }_{H,e}$ we have, $\\int _{\\mathcal {E}^{\\star }_{H,e}}f(x)d\\mu (x)=\\underset{n\\rightarrow \\infty }{\\lim }\\frac{1}{s_n}\\int _{0}^{s_n} f(X^{s}_H(x))ds=\\Psi (f).$ Define a continuous observable $f_H\\colon \\mathcal {E}^{\\star }_{H,e}\\rightarrow \\mathbb {R}$ by $f_H(x)=\\partial _h(\\log \\Vert \\Phi _H^h(x)|_{\\mathcal {N}_x^2}\\Vert )_{h=0}=\\underset{h\\rightarrow 0}{\\lim }\\frac{1}{h}\\log \\Vert \\Phi _H^h(x)|_{\\mathcal {N}_x^2}\\Vert .$ $\\int _{ \\mathcal {E}^{\\star }_{H,e}}f_H(x)\\:d\\mu (x)&=\\lim _{n\\rightarrow +\\infty }\\frac{1}{t_n}\\int _0^{t_n}f_H(X_H^s(x))\\:ds\\nonumber \\\\&=\\lim _{n\\rightarrow +\\infty } \\frac{1}{t_n}\\int _0^{t_n}\\partial _h(\\log \\Vert \\Phi _H^h(X_H^s(x))|_{\\mathcal {N}_{X_H^s(x)}^2}\\Vert )_{h=0}\\:ds\\nonumber \\\\&=\\lim _{n\\rightarrow +\\infty } \\frac{1}{t_n} \\log \\Vert \\Phi _H^{t_n}(x)|_{\\mathcal {N}_{x}^2}\\Vert \\overset{(\\ref {exp1})}{\\ge }0.\\nonumber $ As a direct consequence of Birkhoff's ergodic theorem we get, $\\int _{ \\mathcal {E}^{\\star }_{H,e}}f_H(x) \\:d\\mu (x)=\\int _{\\mathcal {E}^{\\star }_{H,e}}\\lim _{t\\rightarrow +\\infty }\\frac{1}{t}\\int _0^{t}f_H(X_H^s(x))\\:dsd\\mu (x)\\ge 0.$ Now, let $\\Sigma (\\mathcal {E}^{\\star }_{H,e})$ be the set of points $x\\in \\mathcal {E}^{\\star }_{H,e}$ such that, for any $C^2$ -neighbourhood $\\mathcal {U}$ of $H$ and $\\delta >0$ , there exist $H_0\\in \\mathcal {U}$ and a $X^t_{H_0}$ -closed orbit $y\\in \\mathcal {E}^{\\star }_{H,e}$ of period $\\pi $ such that $H=H_0$ except on the $\\delta $ -neighborhood of the $X_{H_0}^t$ -orbit of $y$ , and that $d(X_{H_0}^t(y),X_H^t(x))<\\delta $ , for $0\\le t\\le \\pi $ .", "By the Hamiltonian version of the ergodic closing lemma (see  [5]), given a $X_H^t$ -invariant Borel probability measure $\\mu $ , $\\mu (\\Sigma (\\mathcal {E}^{\\star }_{H,e}))=1$ .", "So, there is $x\\in \\Sigma (\\mathcal {E}^{\\star }_{H,e})$ such that $\\lim _{t\\rightarrow +\\infty }\\frac{1}{t}\\int _0^{t}f_H(X^s(x))\\:ds=\\lim _{t\\rightarrow +\\infty }\\frac{1}{t}\\log \\Vert \\Phi _H^{t}(x)|_{\\mathcal {N}_{x}^2}\\Vert \\ge 0.$ Fix $\\theta \\in (0,1)$ given by Lemma REF and depending on the neighborhood $\\mathcal {U}$ of $H$ .", "Take an arbitrary small $\\delta <0$ such that $\\log \\theta <\\delta $ .", "Thus, there is $t_{\\delta }$ such that, for $t\\ge t_{\\delta }$ , $\\frac{1}{t}\\log \\Vert \\Phi _H^{t}(x)|_{\\mathcal {N}_{x}^2}\\Vert \\ge \\delta .\\nonumber $ Since $x\\in \\Sigma (\\mathcal {E}^{\\star }_{H,e})$ , there are $H_n\\in \\mathcal {U}$ , $C^2$ -converging to $H$ , and periodic points $p_n$ of $H_n$ with period $\\pi _n$ .", "Notice that $\\pi _n\\rightarrow +\\infty $ as $n\\rightarrow \\infty $ , otherwise, $x$ would be a periodic point of $H$ with period $\\pi $ and the properties of dominated splitting, conservativeness and (REF ) contradict the hypothesis that $H$ has the star property.", "So, assuming that $\\pi _n>t_{\\delta }$ for every $n$ , by the continuity of the dominated splitting we have that, for $n$ large enough, $\\Vert \\Phi _{H_n}^{\\pi _n}(p_n)|_{\\mathcal {N}_{p_n}^2}\\Vert \\ge \\exp (\\delta \\pi _n)>\\theta ^{\\pi _n}.$ But this contradicts (a) in Lemma REF , because $H_n\\in \\mathcal {U}$ .", "So, $\\Phi _H^t|_{\\mathcal {N}^2}$ is uniformly contracting and the Lemma is proved.", "As a consequence of Theorem REF we have the following result.", "Corollary 3.9 If $(H,e, \\mathcal {E}^{\\star }_{H,e})\\in \\mathcal {G}$$^{2}(M)$ then the closure of the set of periodic orbits of $\\mathcal {E}^{\\star }_{H,e}$ is dense in $\\mathcal {E}^{\\star }_{H,e}$ .", "Let $(H,e, \\mathcal {E}^{\\star }_{H,e})\\in \\mathcal {G}$$^{2}(M)$ .", "By Theorem REF , we have that $(H,e, \\mathcal {E}^{\\star }_{H,e})$ is an Anosov Hamiltonian system.", "Now, we use Anosov closing lemma to obtain the conclusion of the corollary.", "We end this section with the proof of Corollary REF .", "(of Corollary REF ) By contradiction, assume there exists a Hamiltonian $(H,e, \\mathcal {E}_{H,e})$ isolated on the boundary of the set $\\mathcal {A}(M)$ .", "By Remark REF , $\\Phi _H^t$ admits a dominated splitting over $\\mathcal {E}_{H,e}$ .", "Therefore, we just have to follow the proof of Theorem REF , in order to conclude that $(H,e, \\mathcal {E}_{H,e}) \\in \\mathcal {A}(M)$ , which is a contradiction.", "So, the boundary of the set $\\mathcal {A}(M)$ cannot have isolated points." ], [ "Stability conjecture for Hamiltonians - proof of Theorem  ", "In this section we prove that structurally stable Hamiltonian systems are Anosov.", "Actually, the Hamiltonian version of the structural stability conjecture is implicitly treated in [20].", "See also [20] and the paragraph before it.", "However, we cannot find a formal statement and a proof in the literature and for that reason we fill here this gap although using a different approach from the one implicit in [20].", "(of Theorem REF ) Let us fix a $C^2$ -structurally stable Hamiltonian $(H,e, \\mathcal {E}_{H,e})$ and choose a $C^2$ -neighbourhood $\\mathcal {U}$ of $H$ whose elements are topologically equivalent to $H$ .", "If $H \\notin \\mathcal {A}(M)=\\mathcal {G}$$^{2}(M)$ , then there exists $\\tilde{H}_0 \\in \\mathcal {U}$ such that $\\tilde{H}_0$ has a non-hyperbolic periodic orbit.", "Since, by Robinson's version of the Kupka-Smale theorem (see [23]), a $C^2$ -generic Hamiltonian has all closed orbits of hyperbolic or elliptic type, there exists $H_0$ , close to $\\tilde{H}_0$ , such that $H_0$ has a $k$ -elliptic periodic orbit $p$ of period $\\tilde{\\pi }$ (recall that a periodic point $p$ of period $\\tilde{\\pi }$ is $k$ -elliptic, $1 \\le k \\le d-1$ , if $\\Phi ^{\\tilde{\\pi }}_{H_0}(p)$ has $2k$ non-real eigenvalues of norm one, and its remaining eigenvalues have norm different from one).", "Therefore, there exists a splitting of the normal subbundle $\\mathcal {N}^c$ along the orbit of $p$ into $k$ -subspaces $\\mathcal {N}^c_j$ , $1 \\le j \\le k$ , of dimension 2, such that $\\mathcal {N}^c=\\mathcal {N}^c_1 \\oplus \\ldots \\oplus \\mathcal {N}^c_k$ .", "Let $\\theta _1, \\ldots , \\theta _k \\in [0,2\\pi [$ be such that each $\\rho _j=\\mbox{\\rm exp}(\\theta _j i)$ is an eigenvalue of $\\Phi _{H_0}^{\\tilde{\\pi }}(p)|_{\\mathcal {N}^c_j}$ .", "Let $R_{\\theta }$ be the rotation matrix of angle $\\theta $ .", "The Poincaré map near $p$ , $f_{H_0}$ , associated to $\\Phi _{H_0}$ , is a map from a $(2d-1)$ -dimensional manifold to itself such that when it is restricted to a $(2d-2)$ -energy hypersurface it is a local symplectomorphism close to $R_{\\theta _j}$ in each subspace $\\mathcal {N}^c_j$ .", "Applying Theorem 3 in [9] to $f_{H_0}$ , we have that there exists ${H}_1 \\in \\mathcal {U}$ such that each appropriate restriction of $f_{{H}_1}$ is conjugated to the rotation $R_{\\theta _j}$ defined in $\\mathcal {N}^c_j$ .", "We can suppose that each $\\theta _j \\in \\mathbb {Q}$ , i.e., $\\theta _j=p_j/q_j$ .", "Otherwise, we slightly perturb each rotation and then apply  [9] to obtain a Hamiltonian whose Poincaré map restricted to a two-dimensional submanifold $\\Sigma _j^c$ is conjugated to a rational rotation, defined in $\\mathcal {N}^c_j$ , and close to $R_{\\theta _j}$ , $1 \\le j \\le k$ .", "Take $\\ell =\\Pi _{j=1}^k q_j$ .", "Now, $f_{{H}_1}^{\\ell }(q)=q$ , for any $q \\in \\cup _{j=1}^k\\Sigma ^c_j$ .", "Then, each $q \\in \\cup _{j=1}^k\\Sigma ^c_j$ is a periodic point whose period divides $\\ell $ .", "However, as shown by Robinson in [23], $C^2$ -generically there are not non-trivial resonance relations.", "In particular, $C^2$ -generically the periodic orbits are isolated.", "So, ${H}_1$ must be conjugated to a Hamiltonian which has only a finite number of closed orbits with period is limited by $\\max \\lbrace \\tilde{\\pi },\\ell \\rbrace $ .", "As, by the definition of structural stability, the conjugation is close to the $id$ , this leads to a contradiction." ], [ "Acknowledgements", "MB was partially supported by National Funds through FCT - “Fundação para a Ciência e a Tecnologia\", project PEst-OE/MAT/UI0212/2011.", "MJT was partially financed by FEDER Funds through “Programa Operacional Factores de Competitividade - COMPETE” and by Portuguese Funds through FCT, within the Project PEst-C/MAT/UI0013/2011.", "JR was partially funded by the European Regional Development Fund through the programme COMPETE and by the Portuguese Government through the FCT under the project PEst - C/MAT/UI0144/2011.", "MB and JR partially supported by the FCT, project PTDC/MAT/099493/2008." ] ]

[ [ "Bio-Polymer Hairpin Loops Sustained by Polarons" ], [ "Abstract We show that polarons can sustain loop-like configurations in flexible bio-polymers and that the size of the loops depend on both the flexural rigidity of the polymer and the electron-phonon coupling constant.", "In particular we show that for single stranded DNA (ssDNA) such loops can have as little as 10 base pairs.", "For polyacetylene the shortest loop must have at least 12 nodes.", "We also show that these configurations are very stable under thermal fluctuations and can facilitate the formation of hairpin-loops of ssDNA." ], [ "Introduction", "Conformational transitions of biopolymers as a result of the coupling between the electronic and elastic degrees of freedom are important for understanding native states of globular proteins and secondary structures of biopolymers such as DNA and RNA.", "In an attempt to understand toroidal states of DNA the globule-coil transition for semi-flexible polymers in poor solvents has been explored using Brownian dynamics simulations [1], [2].", "The intermediate states arising in these systems have also been classified [1], [2].", "However the collapse transition in polymers induced by polarons has been less explored [3].", "Polarons are the result on the interaction between a free electron in the conducting band of a polymer chain and the phonons of that chain.", "They were discovered by Davydov [4], [5], [6] who proposed them as a mechanism to explain how energy can be transported along alpha-helices in living cells.", "In this paper we explore the possibility of polaron induced polymer loop formation and stabilisation arising in semi-flexible chains.", "Our model is similar to the model proposed by Mingaleev et al.", "[3] who generalised the original model of Davydov [4], [5] by incorporating long ranged electron-phonon interactions.", "In their work Mingaleev et al.", "showed that at zero temperature polarons can induce a spontaneous bend in a straight chain if the bending modulus is less than a critical threshold.", "A careful examination of the model however reveals that realistic polymers e.g.", "DNA and polyacetylene are more rigid having their bending modulus twice and twenty times above the threshold value respectively.", "Therefore though interesting from a theoretical point of view, the spontaneous bending on polymers induced by polarons is limited in scope when applied to physical systems.", "In a recent paper [7] we showed that the Mingaleev et al.", "model can explain spontaneous polaron transport on a chain having a bending gradient, e.g.", "alpha-helices of light harvesting proteins.", "In this case, the bending of the chain is generated by the natural folding of the protein which can induce a spontaneous polaron displacement.", "We showed that with the polymer configuration frozen in, the polaron spontaneously accelerates along the bending gradient, and gets reflected across sharply kinked junctions.", "Further we showed that at finite temperatures the polaron undergoes a biased random walk to a region of high curvature.", "While polarons are not able to induce spontaneous conformational transitions in DNA and polyacetylene, on account of their rigidity, they might sustain a folded configuration that might have been formed by other means e.g.", "thermal fluctuations, or mechanical stress.", "This is particularly true for ssDNA whose bending modulus is only twice as large as the threshold value for spontaneous bending.", "This is what we are investigating in detail in this paper which is organised as follows.", "In section we review the Mingaleev et al.", "[3] model .", "In section we study loop configurations in which the last two nodes of the chains are held together by a polaron.", "We extend this analysis to study hairpin-loop configurations in for which the two opposite ends of the chain run parallel to each other, while the loop links the parallel strands together.", "Finally we show that one can estimate analytically the value of the parameters for which loops can be formed in section.", "In the last two sections and we look in some detail at loop and hairpin-loop configurations for both single stranded DNA and polyacetylene and we show that the polarons, in these two systems, are very stable and that they can facilitate the formation of hairpin-loops." ], [ "Model", "The model proposed by Mingaleev [3] is described by the Hamiltonian $H = \\sum _{n} \\left[\\frac{\\hat{M}}{2}\\left(\\frac{d {\\vec{R}_{n}}}{d\\tau }\\right)^2+ \\hat{U}_n({\\vec{R}_{}}) - \\frac{1}{2} \\Delta |\\phi _n|^4\\right.\\left.+ W\\left(2|{\\phi _n}|^2 - \\sum _{m\\ne n}J_{nm}\\phi _n^*\\phi _m \\right)\\right],$ where ${\\vec{R}_{n}}$ describes the position of each chain node, $\\hat{M}$ is the node mass, $W$ is the linear excitation transfer energy and $\\Delta $ the non-linear self-trapping interaction.", "The excitation transfer coefficients $J_{n,m}$ are of the form: $J_{n,m} = J(|{\\vec{R}_{n}}-{\\vec{R}_{m}}|) = (e^\\alpha -1)\\,e^{-\\alpha |{\\vec{R}_{n}}-{\\vec{R}_{m}}|/\\hat{a}},$ where $\\alpha ^{-1}$ sets the relative length scale over which the interaction decreases, in units of $\\hat{a}$ , where $\\hat{a}$ is the rest distance between two adjacent sites.", "The function $J_{n,m}$ describes the long range interaction between the electron field at different lattice sites $n$ and $m$ ; its value decreases exponentially with the distance between them.", "Notice that when $\\alpha $ is large and $|{\\vec{R}_{n}}-{\\vec{R}_{m}}|\\approx \\hat{a} $ , this corresponds to a nearest neighbour interaction with $J_{n,m}\\approx \\delta _{n,m\\pm 1}(1+\\alpha (1-\\frac{|{\\vec{R}_{n}}-{\\vec{R}_{m}}|}{\\hat{a}}))$ .", "In our formulation of the model, the normalisation of the electron field is preserved i.e.", "$\\sum _n |\\phi _n|^2 = 1.$ The phonon potential $\\hat{U}_n$ consists of three terms: $\\hat{U}_n({\\vec{R}_{}})=\\frac{\\hat{\\sigma }}{2} (|{\\vec{R}_{n}}-{\\vec{R}_{n-1}}|-\\hat{a})^2 +\\frac{\\hat{k}}{2} \\frac{\\theta _n^2}{\\left[1-(\\theta _n/\\theta _{max})^2\\right]}\\nonumber \\\\+\\frac{\\hat{\\delta }}{2} \\sum _{m\\ne n} (\\hat{d}-|{\\vec{R}_{n}}-{\\vec{R}_{m}}|)^2\\Theta (\\hat{d} -|{\\vec{R}_{n}}-{\\vec{R}_{m}}|),$ where the Heaviside function is defined as $\\Theta (x) = 1$ for $x > 1$ and $\\Theta (x) = 0$ for $x < 1$ .", "The first two terms in $\\hat{U}_n$ describe the elastic and the bending energy of the chain respectively.", "$\\hat{a}$ is the equilibrium separation between nodes and $\\theta _n$ is the angle between ${\\vec{R}_{n}}-{\\vec{R}_{n-1}}$ and ${\\vec{R}_{n+1}}-{\\vec{R}_{n}}$ .", "Finally $\\theta _{max}$ is the largest angle allowed between adjacent links.", "The term proportional to $\\hat{\\delta }$ in $\\hat{U}_n$ , models hard-core repulsion between the atoms of the chain.", "$\\hat{\\delta }$ should always be larger than $\\hat{\\sigma }$ and $\\hat{d}$ will correspond to the minimum distance allowed between nodes.", "For convenience, the symbols denoted by an overhead carat sign e.g.", "$\\hat{M}$ , $\\hat{\\sigma }$ etc.", "correspond to physical variables carrying units and dimensions while those without it correspond to dimensionless variables and parameters described below, except $H$ , $\\Delta $ and $W$ which are dimensional quantities.", "We also use the symbol $\\vec{R}$ for position of the nodes in physical units and $\\vec{r}$ in dimensionless units.", "First we define the time scale $\\tau _0=\\hbar \\Delta /W^2$ and use the lattice spacing $\\hat{a}$ as the length scale.", "We can then define the dimensionless time $t$ , position $r$ and coupling constant $g$ as $t &=& \\frac{\\tau }{\\tau _0}\\qquad \\qquad g = \\frac{\\Delta }{W}\\qquad \\qquad {\\vec{r}_{}} = \\frac{{\\vec{R}_{}}}{\\hat{a}}.$ In terms of these variables the Hamiltonian takes the form $H &=& \\frac{W^2 }{\\Delta } \\sum _{n}\\left[ \\frac{M}{2}\\left(\\frac{d{\\vec{r}_{n}}}{dt}\\right)^2+ U_n({\\vec{r}_{}})\\right.\\nonumber \\\\&&\\left.+ g\\left(2|{\\phi _n}|^2 - \\sum _{m\\ne n}J_{nm}\\phi _n^*\\phi _m \\right)- \\frac{g^2}{2} |\\phi _n|^4\\right],$ where $U_n({\\vec{r}_{}})&=& \\frac{\\sigma }{2} (|{\\vec{r}_{n}}-{\\vec{r}_{n-1}}|-a)^2 +\\frac{k}{2} \\frac{\\theta _n^2}{\\left[1-(\\theta _n/\\theta _{max})^2\\right]}\\nonumber \\\\&&+ \\frac{\\delta }{2} \\sum _{m\\ne n} (d-|{\\vec{r}_{n}}-{\\vec{r}_{m}}|)^2\\Theta (d -|{\\vec{r}_{n}}-{\\vec{r}_{m}}|)$ with $M &=& \\hat{M} \\frac{\\hat{a}^2 W^2}{\\hbar ^2 \\Delta }\\qquad \\qquad \\sigma = \\hat{\\sigma }\\frac{\\hat{a}^2\\Delta }{W^2}\\qquad \\qquad \\delta = \\hat{\\delta }\\frac{\\hat{a}^2\\Delta }{W^2}\\nonumber \\\\k &=& \\hat{k} \\frac{\\Delta }{W^2}\\qquad \\qquad a = 1\\qquad \\qquad d = \\frac{\\hat{d}}{\\hat{a}}.$ Writing ${\\vec{r}_{n}}=(x_{1,n},x_{2,n},x_{3,n},)$ we can derive the equation of motion for $x_{i,n}$ from the Hamiltonian (REF ): $M\\frac{d^2x_{i,n}}{dt^2} + \\Gamma \\frac{d x_{i,n}}{dt} + F(t)+\\sum _m \\frac{dU_m}{dx_{i,n}} &&\\nonumber \\\\-g\\sum _k \\sum _{m < k}\\frac{d J_{km}}{dx_{i,n}}(\\phi _k^*\\phi _m+\\phi _m^*\\phi _k) &=& 0\\nonumber \\\\i\\frac{d\\phi _n}{dt}-2 \\phi _n + \\sum _{m\\ne n} J_{nm}\\phi _m+ g |{\\phi _n}|^2\\phi _n&=& 0,$ where the force $F(t)$ and the friction term $\\Gamma \\,d x_{i,n}/dt $ , were added by hand to incorporate thermal fluctuations and $F(t)$ was chosen as a delta correlated white noise satisfying $<F(0) F(s)> = 2 \\Gamma k_{B} T \\delta (s)$ where $k_{B} T = \\hat{k}_B \\hat{T} \\frac{\\hat{W}^2}{\\hat{\\Delta }}= \\hat{k}_B\\hat{T} \\hat{W} g.$ As the equation for $x_i$ is expressed in units of $\\hat{W}^2/(\\hat{\\Delta } \\hat{a})$ , we have $\\Gamma = \\hat{\\Gamma } \\hat{a}^2/\\hbar $ .", "The friction coefficient $\\hat{\\Gamma }$ can be evaluated from $\\hat{\\Gamma }\\approx 6\\pi \\mu R_0$ where $\\mu = 0.001 \\mbox{Pa\\, s}$ for water, and up to 4 times that value for the cytoplasm, where $R_0$ is the average radius of a single molecule of the lattice.", "Notice also that the electron field $\\phi _n$ is coupled to the phonon field $x_{i,n}$ through the function $J_{nm}$ .", "In what follows we are primarily interested in stationary configurations.", "To compute such solutions numerically we choose an initial lattice configuration with a loop structure and localised the electron so that it overlapped with both tails of the loop.", "We achieved this by using an approximation for the polaron electron field and distributing it over a few nodes spread between the two ends of chain.", "This way the polaron was able to bind the loop extremities together.", "We then relax the electron field keeping the lattice configuration unchanged and then evolved the entire system with an absorption term until it relaxed to a static configuration.", "This was achieved by solving equation (REF ) without thermal noise.", "In all our simulations we started from a very small value of $k$ , typically $k=0.005$ , so that the lattice offered very little resistance to bending.", "We then increased the value of $k$ in small increments using the relaxed conformation obtained for the previous $k$ value as the initial configuration.", "We then equilibrated the system for the new value of $k$ .", "By repeating the procedure for each value of $g$ we have determined the critical value $k_{crit}(g)$ up to which the given configuration can be sustained by the polaron.", "Unless otherwise stated, we have used the following parameter values: $\\delta =10000$ , $\\sigma =1000$ and $M=0.5$ .", "For stationary solutions the mass term does not affect the results and $\\delta $ was chosen so that the repulsion potential is close to that of a hard shell.", "Finally for all the computed configurations, nothing prevents the nodes from being very close to their equilibrium distance and hence we have selected a relatively large value for $\\sigma $ to approximate stiff cross-node links.", "Following Mingaleev et al, we have also considered mainly the case $\\alpha =2$ and $d=0.6$ .", "Finally we have also considered the effect of varying the values of these two parameters.", "To solve equation (REF ) we used a 4th order Runge-Kutta method with a time step $dt=0.0001$ in dimensionless units.", "To compute static configurations, we took $T=0$ , i.e no thermal noise, setting $\\Gamma =1$ and then integrating equation (REF ) until the system relaxed to a stationary solution.", "To study the thermal stability of the configurations for DNA and polyacetylene, we solved equation (REF ), taking $T=300K$ and estimated $\\Gamma $ from the radius of the molecules as described above.", "For those simulations we started from the static configuration for which we wanted to evaluate the stability and let the system thermalise itself.", "The time needed for this thermalisation was always orders of magnitude smaller than the average life time of the configurations we considered and so we did not need to resort to sophisticated thermalisation procedure as we did in [7]." ], [ "Plain Loop Configurations", "Our first investigation involved considering a simple loop configurations for which all the nodes lie more or less on a circle with the two end points close to each other (separated by a distance $d$ ).", "When $k$ is very small, the favoured configuration is one similar to the one presented in figure REF .a.", "In this figures, the electron probability density is represented by the colour of the node.", "A dark colour corresponds to a null value while a light value corresponds to a higher probability density.", "The node at which the polaron field has its maximum value is close to the last 2 points at the opposite ends of the chain.", "This allows the electron field to be distributed on 2 nearby nodes rather than a single one and, as $k$ is small, the deformation of the chain does not prevent this from taking place.", "As $k$ increases, such a localisation becomes energetically expensive and the configuration assumes the shape of a horse-shoe, as presented in figures REF .b.", "and REF .c.", "As one increases $k$ further, there is a point at which the stretching energy is too large and the polaron is not able to sustain the loop anymore.", "Figure: Loop configuration for N=9N=9 nodes fora) g=2.5g=2.5, k=0.5k=0.5, |φ 0 | 2 =0.407|\\phi _0|^2=0.407, |φ 8 | 2 =0.221|\\phi _8|^2=0.221,|φ 7 | 2 =0.249|\\phi _7|^2=0.249;b) g=2.5g=2.5 k=4k=4, |φ 0 | 2 =|φ 8 | 2 =0.397|\\phi _0|^2= |\\phi _8|^2=0.397;c) g=5g=5 k=5k=5, |φ 0 | 2 =0.642|\\phi _0|^2=0.642, |φ 8 | 2 =0.272|\\phi _8|^2=0.272.The difference between figures REF .b.", "and REF .c is that in the former the electron is localised equally on the 2 end nodes while in the later it is localised mostly on a single node.", "The difference is dictated by the value of $g$ : for small $g$ , the polaron is wide and the electron spreads itself nearly equally between the two end points of the chain (Fig REF .b).", "As $g$ increases, the polaron becomes more localised and the electron becomes localised, more asymmetrically, on a single node (Fig REF .c).", "The critical value of $k$ as a function of $g$ is presented in Figure REF .a for loops consisting of 9 to 14 nodes.", "It is interesting to note that when $g$ is small, the critical value of $k$ is small.", "This can be explained by the fact that the coupling parameter $g$ is small but also by the fact that the polaron is delocalised and hence the fraction of the electron close to the end point is smaller than for larger values of $g$ .", "The maximum value of $k_{crit}$ is reached for $g\\approx 5$ .", "For very large values of $g$ , the electron is nearly fully localised on a single lattice point, but the attraction exerted by the polaron, surprisingly decreases very slowly.", "Having followed [3] and taken the values $\\alpha =2$ and $d=0.6$ for the results presented so far, it is worth checking how these two parameters affect the results that we have obtained.", "We started by varying $\\alpha $ , which controls, through $J_{n,m}$ , how fast the coupling between nodes decreases with the distance separating them.", "The results are presented in Figure REF .b where we see that $k_{crit}$ , contrary to what one might expect, increases with $\\alpha $ .", "This is easily explained: having chosen $d=0.6$ , increasing $\\alpha $ , not only reduces the long distance interaction between nodes but it also increases exponentially the binding energy of nodes that are very close to each other.", "The binding energy of the end nodes, which are separated by a distance $d < 1$ , thus increases with $\\alpha $ .", "For this reason, we have decided to take $d=a=1$ when we consider single stranded DNA and polyacetylene later in the paper.", "In figure REF .c we show how the critical value of $k$ varies with $d$ .", "As the parameter $d$ sets the minimal distance allowed between 2 nodes and given that $J_{n,m}$ decreases with the distance between nodes, it is not surprising that $k_{crit}$ decreases when $d$ becomes larger, but loop configurations can still be held by the polaron.", "Figure: Critical value of kk for the existence of a loop configuration.a) α=2\\alpha =2 and N=9N=9 to 14 nodes.b) N=9N=9 nodes α=1,1.5,2,25,3\\alpha =1,1.5,2,25,3 and d=0.6d=0.6." ], [ "Hairpin-Loop Configurations", "Now we consider a hairpin-loop configuration as presented in Figure REF similar to the structure that single stranded DNA can form and which is potentially more relevant to long chains.", "As for the plain loops, we generated these configurations for a small $k$ and then slowly increased its value until the number of links, $L$ , making the loop increased by one unit.", "This gave us the critical value, $k_{crit}$ , for which the hairpin-loop configuration of a given size can be sustained by the polaron.", "The results are presented in figure REF where we can see a sharp transition around $g=10$ .", "Below this value the hairpin-loop is only viable for relatively small values of $k$ but above it, they are sustainable for much more rigid chains.", "This is due to the fact that for small values of $g$ , the polaron is always distributed over the handle of the hairpin-loop while when $g>10$ , it is localised mostly on one lattice site, at the base of the loop.", "In that case the interaction is stronger and supports loops for larger values of $k$ .", "To make sure this was not an artefact of our procedure, we have tried to construct solutions using various initial conditions.", "We also used solutions obtained for $g>10$ as initial conditions and then slowly decreased the value of $g$ .", "Regardless of the procedure we used, we always obtained the curve of Figure REF .a.", "As expected, the configurations of figure REF are harder to sustain than a simple loop as the chain needs to be bent near the stem of the hairpin-loop.", "Figure: Hairpin-loop configuration for N=18N=18 nodes.The brightness of the nodes is proportional to |φ| 2 |\\phi |^2.a) g=1.5g=1.5 and k=0.5k=0.5 (max |φ| 2 =0.188|\\phi |^2= 0.188),b) g=11g=11 and k=10k=10 (max |φ| 2 =0.887|\\phi |^2=0.887 )Figure: Critical value of kk for the existence of an hairpin-loopconfiguration.a) α=2\\alpha =2 and N=7N=7 to 11 node loops.b) N=9N=9 nodes and α=1,1.5,2,2.5,3\\alpha =1,1.5,2,2.5,3." ], [ "Analytic Approximation", "Having computed numerically the critical value $k_{crit}(g)$ for which the polaron is able to sustain a loop of a given size, we now try to estimate this value analytically.", "To do this, we consider a circular configuration of radius $R$ made out of $N$ segments, one of length $b$ and $N-1$ of length $a$ , as depicted in figure REF .", "Note that the two nodes separated by the distance $b$ are not linked to each other.", "If $\\xi $ and $\\mu $ are the angles opposite $a$ and $b$ , respectively, we have $\\begin{array}{lll}(n-1)\\xi + \\mu = 2 \\pi \\qquad &\\sin (\\frac{\\xi }{2}) = \\frac{a}{ 2R}\\qquad &\\sin (\\frac{\\mu }{2}) = \\frac{b}{ 2R}\\end{array}$ and so $\\sin (\\frac{\\xi }{2}) = \\frac{a}{b} \\sin (\\frac{\\mu }{2})= \\frac{a}{b} \\sin (\\frac{(n-1)\\xi }{2}).$ Figure: Schematic representation of a polymer-loop with NN nodes.N-1N-1 bonds with rest distance aa subtend an angle ξ\\xi at the centre.The two end nodes, on which the electron is localised, are separated bya distance bb and span an angle μ\\mu at the centre.Choosing specific values $a=1$ , $b=b_0$ , this transcendental equation can be solved numerically to obtain the corresponding value $\\xi =\\xi _0$ .", "We can then perform a first order expansions around this solution: $b = b_0 +\\delta b, \\xi = \\xi _0+\\delta \\xi $ and obtain $\\delta \\xi = \\frac{ \\sin (\\frac{\\xi }{2})}{\\frac{N-1}{4}\\cos ((N-1)\\frac{\\xi }{2})-\\frac{b_0}{4}\\cos (\\frac{\\xi }{2})}\\delta b$ To determine the critical value of $g$ and $k$ for which a loop configuration can exist, we have to minimise the Hamiltonian and, for each value of $g$ and $b$ , determine the value of $k$ for which this Hamiltonian has a minimum.", "For each $g$ , we then select the value of $b$ for which $k$ is the largest.", "Let us assume that the loop is symmetric, so that the $N-2$ bending terms are all identical and are functions of $\\xi $ .", "The elastic terms are then also equal, but as they do not depend on $\\xi $ , they are constant and can thus be ignored for the minimisation.", "The repulsion term proportional to $\\delta $ can also be ignored if $b> d$ .", "When $b<d$ , the repulsion term leads to a very large energy increase and we can thus consider than $b=d$ is the smallest value we should consider.", "To evaluate the electron field, we take the continuum limit of equation REF for stationary solutions: $\\frac{ d^2 \\phi _c}{dx^2} + \\hat{g} |\\phi _c|^2\\phi _c -\\lambda \\phi _c =0,$ which is the well known non-linear Schoedinger equation, where $\\hat{g} = g/(1-e^{-\\alpha })$ , and which admits the following solution: $\\phi _c(x) = \\sqrt{\\frac{\\hat{g}}{8}}{\\cosh (\\frac{\\hat{g}x}{4})}.$ Note that $\\int |\\phi (x)|^2 dx = 1$ and $\\lambda = -\\hat{g}^2/16$ .", "From the numerical solutions, we know that the wave function is centred on one of the two end lattice points and we can thus take $\\phi _0=\\phi _c(0)$ and $\\phi _{N-1}=\\phi _c(b_0)$ .", "The Hamiltonian can then be approximated by the following function of $b_0$ and $\\xi _0$ $H &\\approx & -2\\,g\\,(e^\\alpha -1)e^{-\\alpha b_0/a}\\phi _0\\phi _{N-1}+\\frac{k}{2} \\frac{(N-2)\\xi _0^2}{\\left[1-\\left(\\frac{\\xi _0}{\\theta _{max}}\\right)^2\\right]}\\,.$ Next we compute the variation of $H$ with respect to $b$ and $\\xi $ $\\delta H =2 g \\alpha \\phi _0\\phi _{N-1} (e^\\alpha -1)e^{-\\alpha b_0}\\delta b+ \\frac{k (N-2) \\xi _0}{\\left[1-\\left(\\frac{\\xi _0}{\\theta _{max}}\\right)^2\\right]}(1-\\xi _0^2(1-\\frac{1}{\\theta _{max}^2}))\\delta \\xi $ Using equation (REF ) and imposing $\\delta H =0$ we get the condition $k_{crit} &=& \\frac{g}{D} \\phi _0\\phi _{N-1}$ where $D &=&\\frac{1}{2\\alpha (e^\\alpha -1)e^{-\\alpha b_0}}\\frac{(N-2) \\xi _0}{\\left[1-\\left(\\frac{\\xi _0}{\\theta _{max}}\\right)^2\\right]}(1-\\xi _0^2(1-\\frac{1}{\\theta _{max}^2}))\\nonumber \\\\&& \\frac{ -4\\sin (\\frac{\\xi _0}{2})}{(N-1)\\cos ((N-1)\\frac{\\xi _0}{2})-b_0\\cos (\\frac{\\xi _0}{2})}$ which depends on $g$ but not on $k$ .", "From equations (REF ) and (REF ) it is clear that $k_{crit}$ increases as $b_0$ decreases and so we have to choose the smallest possible value for $b_0$ i.e.", "$b_0=d$ .", "Figure: Theoretical estimation of the critical value of kk for theexistence of a loop configuration as a function of α\\alpha and NN.a) α=2\\alpha =2 and N=9N=9 to 14 nodes.b) N=9N=9 nodes and α=1,1.5,2,25,3\\alpha =1,1.5,2,25,3Table: Value of ξ 0 \\xi _0 for various number of nodes NN.Taking $a=1$ , $b_0=d=0.6$ , we can solve equation (REF ) to obtain the values $\\xi _0$ listed in Table 1. which we can use to estimate $k_{crit}$ .", "The results are presented in figure REF from which we see that our evaluation reproduces the gross features of the results obtained numerically (figure REF ): $k_{crit}$ is small when $g$ is very small, and increases with $g$ until a maximum is reached.", "Then, as $g$ increases further $k_{crit}$ slowly decreases.", "The maximum value obtained for $k_{crit}$ is slightly smaller than the numerical value obtained before.", "The biggest discrepancy between the numerical and analytic results are for large $g$ , but this is to be expected as this is the limit where the polaron is strongly localised and so is less well approximated by equation (REF )." ], [ "Single Stranded DNA", "Having considered the Mingaleev et al.", "polaron model in general, we now consider two explicit cases: DNA and polyacetylene, both of which have parameters allowing the polaron to sustain loops.", "The parameters of our model were obtained from several sources.", "First of all, $\\hat{k}$ can be determined from the flexural rigidity, $\\hat{k} = \\lambda \\hat{k}_B\\hat{T}/R_0$ where $R_0$ is the radius of the DNA strand.", "We do not have experimental values of $\\hat{\\delta }$ , but its actual value does not play an important role except that it must be large enough to mimic a hard shell repulsion.", "In practice, we chose a value larger than $\\hat{\\sigma }$ .", "For single stranded DNA we have $R_0\\approx 0.33 nm$  [8] $\\Delta \\approx 0.4eV$ , $W\\approx 0.3eV$  [3], $\\hat{k}\\approx 0.11eV$  [8] and $\\hat{\\sigma }\\approx 1.5eV/A^2$  [9].", "This leads to the following dimensionless values: $\\begin{array}{llll}g \\approx 1.33 &\\sigma \\approx 72.51 & k\\approx 0.487 & M\\approx 2.5\\times 10^5\\nonumber \\\\k_BT = \\hat{k_BT} \\frac{\\hat{\\Delta }}{\\hat{W}^2}\\approx 0.115 &\\Gamma \\approx 3210 &\\tau _0\\approx 2.92\\times 10^{-15}s &\\end{array}$ We thus see that single stranded DNA sits at the bottom left region of figures REF , REF and REF .", "For our simulations, we have chosen $\\alpha =2$ , $\\delta =100$ and $d=1$ , the later parameter was taken as the worst case we could consider.", "We then found that DNA can easily sustain loops of 10 segments and hairpin-loops with 11 segments.", "We then studied the thermal stability of the configurations that we have obtained at $T=300K$ .", "To achieve this, we started from a static configuration that we had obtained for DNA.", "We then solved equation (REF ), including the thermal noise, and ran 100 simulations for an extended period of time.", "We started by running 100 simulations for a loop made out of 10 nodes and we measured an average unfolding time of $1ns$ .", "We also observed that a chain made out of 11 nodes is much more stable, and experiences a very slow unfolding of the loops with an average decay time of approximately $1.3\\mu s$ .", "We have also performed thermal simulations for an hairpin-loop configuration of 18 nodes and $L=11$ and did not observe a single unfolding of the chain in $20\\mu s$ .", "As the integration time steps was approximately $3\\, 10^{-17}s$ this required 100 simultaneous simulations each performing around $10^{10}$ integration steps and we decided not to run it any longer as the stability of the loop was sufficiently well established.", "Under thermal noise, the stem, made out of the the two parallel ends of the chain, deforms itself and the chain takes the shape of a loop where the polaron links the two opposite ends of the chain around a couple of nodes, as presented in figure REF .", "In our simulations we observed that as the polaron moves along the chain, the size of the loop that it formed fluctuated constantly in time but it never unfolded.", "We can thus conclude that the DNA polarons loops are very stable.", "Figure: Thermalised, T=300KT=300K, hairpin-loop DNA configuration for N=18N=18 nodes.The size of the disk are an exaggerated indication of their depth in thedirection transverse to the plane of view.", "(The two nodes close to thecrossing point are separated by the same distance as two neighbour points).The brightness of the nodes is proportional to |φ| 2 |\\phi |^2.This loop configuration could play an important role in the formation of single stranded DNA hairpin-loops in vivo.", "The formation of such configurations depends on the likelihood of complementary DNA bases to face each other before they can bind by hydrogen bonding and this likelihood decreases rapidly as the length of the chain increases [10] and partially matching DNA base sequences are less stable than perfectly matched ones [11].", "While homogeneous sequences of DNA bases can bind quite rapidly like the one used in [10] and [11] for example, irregular sequences, like $ATGCAGTC ... GACTGCAT$ are less likely to match purely randomly.", "As the polaron folds the ssDNA into a loop and moves along the chain, the DNA bases on the opposite segments of the loop slide relative to each other, under the action of thermal excitation, increasing the probability for a complementary sequence of bases to face each other and bind.", "In vitro, the reaction time of hairpin-loops has been determined to be several $\\mu s$  [10], a length of time that, as we have shown, DNA-polaron can outlive easily.", "Hence, we can conclude that DNA polarons can increase the rate of formation of hairpin-loops.", "In our model, we have not taken into account the effect of water on the polaron.", "Recent studies [12], [13] have suggested that its effect would be to reduce the polaron size, which, as can be seen from equation (REF ), corresponds to increasing the value of $g$ .", "The net effect would thus be to move DNA to a parameter region where the polaron is stronger, as can be seen in figures REF and REF .", "As a result the polaron would then be able to sustain smaller loops." ], [ "Polyacetylene", "For polyacetylene, the physical value of the parameters are given by $R_0\\approx 0.24nm$ , $W\\approx 2.5eV$ , $\\hat{\\sigma }\\approx 21eV/A^2$  [14], $\\hat{k}\\approx 3.7eV/$  [15] and so $\\begin{array}{llll}g \\approx 2.56 &\\sigma \\approx 128 & k\\approx 3.79 &M\\approx 3.2\\times 10^4\\nonumber \\\\k_BT\\approx 0.026 &\\Gamma \\approx 1316 &\\tau _0\\approx 6.74\\times 10^{-16}s&\\end{array}$ Once again, we took $d=1$ and $\\alpha =2$ to avoid the potentially spurious effects induced at close distance by $J_{n,m}$ .", "In this case, we were able to make loops out of 12 nodes and hairpin-loops of 12 nodes too.", "Under thermal fluctuation, both were very stable.", "In this case, the integration time step was approximately $7\\,10^{-18}s$ and our attempt to evaluate the configuration average life time was achieved by running 100 simultaneous each performing over $10^{10}$ integration steps and we did not observe a single unfolding of the chain in $13\\mu s$ .", "We also ran simulations for an hairpin-loop configuration of $N=18$ nodes and $L=12$ and also did not observe a single unfolding of the chain in over $10\\mu s$ ." ], [ "Conclusions", "In this paper we have studied the possibility of a polaron to sustain loops and hairpin-loop configurations.", "In these configurations the polaron was localised over lattice nodes that were well separated along the chain backbone but spatially close to each other because of the bending of the chain.", "The polaron then acted as a linker between the two regions of the chain and so could sustain the loop configuration if the chain was not too rigid.", "The Mingaleev model we have used to describe this property takes into account the long distance interactions between the electron and the phonon field, with a strength decreasing with the distance.", "For the configurations we have studied, the most important contribution comes from lattice nodes that are spatially close to each other and the energy contribution from next to nearest neighbour is not essential, unlike in our study of spontaneous polaron displacements [7] where the next to nearest neighbour terms were essential for the polaron to move along the bending gradient of the chain.", "We have determined the critical value of the chain rigidity $k_{crit}$ as a function of the polaron coupling constant $g$ and we have shown that polarons are able to sustain relatively small loops for a wide range of parameters values.", "We have then shown that both DNA and polyacetylene are flexible enough for a polaron to sustain hairpin-loop configurations.", "Moreover, we have also shown that these hairpin-loop configurations are very stable under thermal excitations, with average live times exceeding $10\\mu $ s, and that they can facilitate the formation of hairpin-loops of single stranded DNA." ], [ "Acknowledgements", "BC was partially supported by EPSRC grant EP/I013377/1." ] ]

[ [ "Geometry of Weak Stability Boundaries" ], [ "Abstract The notion of a weak stability boundary has been successfully used to design low energy trajectories from the Earth to the Moon.", "The structure of this boundary has been investigated in a number of studies, where partial results have been obtained.", "We propose a generalization of the weak stability boundary.", "We prove analytically that, in the context of the planar circular restricted three-body problem, under certain conditions on the mass ratio of the primaries and on the energy, the weak stability boundary about the heavier primary coincides with a branch of the global stable manifold of the Lyapunov orbit about one of the Lagrange points." ], [ "Introduction", "We consider the planar circular restricted three-body problem for a small mass ratio of the primaries.", "We give a general definition of the weak stability boundary set in the region of the heavier primary.", "We consider the global stable manifold of the Lyapunov orbit about the Lagrange point located between the primaries.", "We prove analytically that, under restrictions on the energy, the weak stability boundary coincides with the branch of the global stable manifold in the region of the heavier primary.", "The concept of WSB was introduced in [1], [2] to design low energy transfers from Earth to Moon, and subsequently applied to the rescue of the Japanese mission Hiten in 1991.The GRAIL mission of NASA, arriving at the Moon on January 1, 2012, is using the same transfer as Hiten [8].", "(See also [3].)", "A particular feature of the `WSB method' useful for applications is that it allows the capture of a spacecraft into an elliptic orbit about the Moon, with specified eccentricity of the ellipse, and with specified true anomaly at the capture.", "There has been considerable work devoted to understand the concept of WSB from the point of view of dynamical systems, and to enhance its applicability (see, e.g., [6], [13], [4], [12]).", "A remarkable property of the WSB is that, in the context of the planar circular restricted three-body problem, for some range of energies, and under some topological conditions on the hyperbolic invariant manifolds associated to the libration points, the weak stability boundary points coincide with the points on the stable manifolds satisfying some additional conditions.", "This has been observed numerically in [6], and argued geometrically in [4].", "The classical definition of the WSB is as follows: for each radial segment emanating from the Moon, we consider trajectories that leave that segment at the periapsis of an osculating ellipse whose semi-major axis is a part of the radial segment; a trajectory is called weakly $n$ -stable if it makes $n$ full turns around the Moon without going around the Earth, and it has negative Kepler energy when it returns to the radial segment; if the trajectory is weakly $(n-1)$ -stable but fails to be weakly $n$ -stable, it is called weakly $n$ -unstable; the points that make the transition from the weakly $n$ -stable regime to the weakly $n$ -unstable regime are by definition the points of the WSB of order $n$ .", "We note that WSB points lie on different Hamiltonian energy levels.", "Also, the WSB is not an invariant set for the Hamiltonian flow.", "We remark that, since the stability/instability criteria, as described above, are concerned with the behavior of trajectories for finite time, they inherently introduce `artifacts', i.e., points with very similar trajectories that are categorized differently with respect to these criteria.", "See [4], [11].", "In the present note, we propose a more general definition of the WSB.", "We remove the condition that the infinitesimal mass leaves the radial segment at the periapsis of an osculating ellipse whose semi-major axis is a part of the radial segment.", "We remove the condition on negative Kepler energy at the return.", "We define a point on the radial segment as being weakly $n$ -stable provided that it makes $n$ turns around the primary, such that the distance from the infinitesimal mass to the primary measured along the trajectory does not get bigger than some critical distance.", "Otherwise the point is redeemed as unstable.", "(Some of these ideas are also suggested in [11].)", "The main result of this paper is that the WSB points, which make the transition from the weakly stable to the weakly unstable regime, are the points on the stable manifold of the Lyapunov orbit for the corresponding energy level.", "The argument for the main result is analytical, relying on topological arguments and estimates from [5], [10], [9].", "For this reason, we deal with the WSB set about the heavier primary (unlike in the WSB original setting).", "An interesting aspect of the WSB method is that it uses `local' information on the dynamics, namely the return of trajectories to a surface of section about one of the primaries, to infer some `global' information on the dynamics, namely the existence of trajectories that execute transfers from one primary to the other." ], [ "The planar circular restricted three-body problem", "We consider the planar circular restricted three-body problem (PCRTBP) with the mass ratio of the primaries sufficiently small.", "The system consists of two mass points $P_1,P_2$ , called primaries, of masses $m_1>m_2>0$ , respectively, that move under mutual Newtonian gravity on circular orbits about their barycenter, and a third point $P_3$ , of infinitesimal mass, that moves in the same plane as the primaries under their gravitational influence, but without exerting any influence on them.", "Let $\\mu = m_2/(m_1 + m_2)$ be the relative mass ratio of $m_2$ .", "In the sequel, we will assume that $0<\\mu <1$ is very small, which will be made precise later.", "It is customary to study the motion of the infinitesimal mass in a co-rotating system of coordinates $(x,y)$ that rotates with the primaries.", "Relative to this system, $P_1$ is positioned at $(\\mu ,0)$ and $P_2$ is positioned at $(-1+\\mu ,0)$ .", "After some rescaling, the equations of motions are given by $ \\ddot{x} -2\\dot{y} = \\displaystyle \\frac{\\partial \\omega }{\\partial x},\\qquad \\ddot{y} +2\\dot{x} =\\displaystyle \\frac{\\partial \\omega }{\\partial y},$ where the effective potential $\\omega $ is given by $ \\omega (x,y) = \\frac{1}{2}(x^2+y^2) +\\frac{1-\\mu }{r_1} + \\frac{\\mu }{r_2} + \\frac{1}{2}\\mu (1-\\mu ),$ with $r_{1} =((x-\\mu )^2 + y^2)^{1/2}$ , $r_{2} = ((x+1-\\mu )^2 + y^2)^{1/2}$ .", "The equations of motion can be described by a Hamiltonian system given by the following Hamiltonian (energy function): $ H(x,y,p_x,p_y) = \\frac{1}{2}((p_x+y)^2+(p_y-x)^2)-\\omega (x,y),$ where $\\dot{x}=p_x+y$ and $\\dot{y}=p_y-x$ .", "For each fixed value $H$ of the Hamiltonian, the energy hypersurface $M_H$ is a non-compact 3-dimensional manifold in the 4-dimensional phase space.", "The projection of the energy hypersurface onto the configuration space $(x,y)$ is called a Hill's region, and its boundary is a zero velocity curve.", "See Fig.", "REF .", "Every trajectory is confined to the Hill's region corresponding to the energy level of that trajectory.", "Figure: A Hill's region, H∈(H(L 1 ),H(L 2 ))H\\in (H(L_1),H(L_2)).The equilibrium points of the differential equations (REF ) are given by the critical points of $\\omega $ .", "There are five equilibrium points for this problem: three of them, $L_1$ , $L_2$ and $L_3$ , are collinear with the primaries (where $L_1$ is between $L_2$ and $L_3$ ), while the other two, $L_4, L_5$ , form equilateral triangles with the primaries.", "The distance from $L_1$ to $P_2$ is given by the only positive solution $x_+$ to Euler's quintic equation $x^5-(3-\\mu )x^4+(3-2\\mu )x^3-\\mu x^2+2\\mu x-\\mu =0,$ and so the distance from $L_1$ to $P_1$ is $1-x_+$ .", "The values $H(L_i)$ of the Hamiltonian (REF ) at the points $L_i$ , $i=1,\\ldots ,5$ , satisfy $H(L_5)=H(L_4)>H(L_3)>H(L_2)>H(L_1)$ .", "For $H<H(L_1)$ , the Hill's region has three components: two bounded components, one about $P_1$ and the other about $P_2$ , and a third component which is unbounded.", "For $H\\in (H(L_1),H(L_2))$ , the Hill's region has two components, one bounded, which is topologically equivalent to the connected sum of the two bounded components from the case $H<H(L_1)$ , and the other one unbounded (Fig.", "REF ).", "The linearized stability of the equilibrium point $L_1$ is of saddle-center type, with the linearized equations possessing a pair of non-zero real eigenvalues $\\pm \\lambda $ , and a pair of complex conjugate, purely imaginary eigenvalues $\\pm i\\nu $ .", "For each $H \\gtrapprox H(L_1)$ , near the equilibrium point $L_1$ there exists a unique hyperbolic periodic orbit $\\gamma _H$ , referred as a Lyapunov orbit.", "This orbit has 2-dimensional stable and unstable manifolds $W^s(\\gamma _H)$ , $W^u(\\gamma _H)$ , respectively, that are locally diffeomorphic to 2-dimensional cylinders.", "These manifolds have the following separatrix property: when restricted to a compact neighborhood $B_H(a,b)$ of $\\gamma _H$ in the energy hypersurface $M_H$ , of the type $B_H(a,b)=\\lbrace a\\le x\\le b\\rbrace $ , with $a<x_{L_1}<b$ sufficiently close to $x_{L_1}$ , each of the manifolds $W^s(\\gamma _H), W^u(\\gamma _H)$ separates $B_H(a,b)$ into two connected components." ], [ "Conley's isolating block", "Let $\\phi :M\\times \\mathbb {R}\\rightarrow M$ be a $C^1$ -flow on a $C^1$ -differentiable manifold $M$ .", "Given a compact submanifold with boundary $B\\subseteq M$ , with $\\dim (B)=\\dim (M)$ , we define $B^-=\\lbrace p\\in \\partial B \\,|\\, \\exists \\varepsilon >0 \\textrm { s.t. }", "\\phi _{(0, \\varepsilon )}(p)\\cap B=\\emptyset \\rbrace , \\\\B^+=\\lbrace p\\in \\partial B \\,|\\, \\exists \\varepsilon >0 \\textrm { s.t. }", "\\phi _{(-\\varepsilon ,0)}(p)\\cap B=\\emptyset \\rbrace ,\\\\B^0=\\lbrace p\\in \\partial B \\,|\\, \\phi _t \\textrm { is tangent to }\\partial B \\textrm { at } p \\rbrace .$ We obviously have $\\partial B=B^0\\cup B^-\\cup B^+$ .", "We call $B^-$ the exit set and $B^+$ the entry set of $B$ .", "An open set $V$ is called an isolating neighborhood for the flow if $\\partial V$ contains no orbit of $\\phi $ .", "An invariant set $S$ for the flow $\\phi $ is an isolated invariant set if there exists an isolating neighborhood $V$ for the flow such that $S$ is the maximal invariant set in $V$ .", "The compact submanifold $B$ is called an isolating block for the flow $\\phi $ provided that: (i) $B^-\\cap B^+=B^0$ , (ii) $B^0$ is a smooth submanifold of $\\partial B$ of codimension 1, and, consequently, $B^-,B^+$ are submanifolds with common boundary $B^0$ .", "The interior of an isolating block is an isolating neighborhood and so determines an isolated invariant set, possibly empty.", "In the PCRTBP, Conley has constructed an isolating block around $L_1$ that can be used to study the nearby dynamics.", "Consider the part of the Hill's region which satisfies $a\\le x\\le b$ , where $(a, b)$ contains the $x$ -coordinate $x_{L_1}$ of $L_1$ .", "This set determines a “dynamical channel” which allows for the transit of trajectories between the $P_1$ and $P_2$ regions.", "The lift $B_H=B_H(a,b)$ of this set to the energy hypersurface, where $a,b$ are chosen close to $x_{L_1}$ , is Conley's isolating block.", "Geometrically, this is a 3-dimensional manifold with boundary $\\partial B_H$ consisting of the set of points in the energy hypersurface that projects onto $x = a$ and $x=b$ in the configuration space.", "It is diffeomorphic to the product of a line segment with a two sphere, $B_H \\approx [a,b]\\times S^2$ , and its boundary $\\partial B_H$ is diffeomorphic to the union of two 2-spheres, $\\partial B_H= B_{H,a}\\cup B_{H,b}\\approx (\\lbrace a\\rbrace \\times S^2)\\cup (\\lbrace b\\rbrace \\times S^2)$ .", "The isolating block conditions in this case are that every trajectory intersecting $\\partial B$ tangentially must lie outside of $B_H$ both before and after the intersection, that is, if $x(t) = a$ and $\\dot{x} (t) = 0$ then $\\ddot{x}(t) < 0$ and if $x(t) = b$ , and $\\dot{x} (t) = 0$ then $\\ddot{x}(t) > 0$ .", "So we have $B_H^0 &=& \\lbrace (x,y,\\dot{x},\\dot{y})\\in \\partial B_H\\,|\\, x(t) = a \\textrm { or } x(t) = b \\textrm { and } \\dot{x} (t) = 0\\rbrace , \\\\B_H^-&=&\\lbrace (x,y,\\dot{x},\\dot{y})\\in \\partial B_H\\,|\\, x(t) = a \\textrm { and } \\dot{x} (t) < 0, \\textrm { or } x(t) = b \\textrm { and } \\dot{x} (t) > 0\\rbrace , \\\\B_H^+&=&\\lbrace (x,y,\\dot{x},\\dot{y})\\in \\partial B_H\\,|\\, x(t) = a \\textrm { and } \\dot{x} (t) > 0, \\textrm { or } x(t) = b \\textrm { and } \\dot{x} (t) < 0\\rbrace .$ For each component of $\\partial B_H$ , the exit and entry sets determine a pair of disjoint open 2-dimensional topological disks, which we denote as follows: $B_{H,a}^{-}$ , $B_{H,b}^{-}$ are the exit sets of the boundary components $B_{H,a}$ , $B_{H,b}$ , respectively, and $B_{H,a}^{+}$ , $B_{H,b}^{+}$ are the entry sets of the boundary components $B_{H,a}$ , $B_{H,b}$ , respectively.", "The complement in $B_{H,b} $ of $B^{-}_{H,b}\\cup B^{+}_{H,b}$ is the set $B_{H,b}^{0}=B_H^0\\cap \\lbrace x=b\\rbrace $ .", "A similar statement holds for $B_{H,a}$ .", "The exit and entry sets are further broken up into components with dynamical roles.", "The set $B_{H,b}^{+}$ is the union of three sets, a spherical cap $B_{H,b}^{+,a} $ , corresponding to trajectories that enter the block $B_H$ through the entry part of $B_{H,b}$ and later leave the block through the exit part of $B_{H,a}$ , a spherical zone $B_{H,b}^{+,b} $ , corresponding to trajectories that enter the block $B_H$ through the entry part of $B_{H,b}$ and leave the block through the exit part of $B_{H,b}$ , and a topological circle separating them, corresponding to the intersection of $W^s(\\gamma _H)$ with $B_{H,b}$ .", "Similarly, $B_{H,b}^{-}=B_{H,b}^{-,a} \\cup B_{H,b}^{-,b} \\cup (B_{H,b} \\cap W^u(\\gamma _H))$ , where the notation is analogous to the above.", "There is a similar decomposition for the entry and exit set components of $B_{H,a}$ .", "See Fig.", "REF .", "Later in the paper, we will use the following fact, which is a consequence of the above discussion.", "There are three possible behaviors for trajectories that start from the $P_1$ -region and enter the isolating block: (i) Trajectories enter the block through $B_{H,b}^{+,a} $ , exit the block through $B_{H,a}^{-,b} $ , and so they execute a transfer from the $P_1$ -region to the $P_2$ -region.", "(ii) Trajectories enter the block through $B_{H,b}^{+,b}$ , exit the block through $B_{H,b}^{-,b} $ , and so they do not transfer to the $P_2$ -region.", "(iii) Trajectories enter the block through $B_{H,b}\\cap W^s(\\gamma _H)$ and are forward asymptotic to $\\gamma _H$ , and so they never leave the block.", "For further details on this subsection, see [5].", "Figure: (a) Projection of Conley's isolating block onto configuration space.", "(b) Schematic representation of the dynamics across Conley's isolating block." ], [ "Hyperbolic invariant manifolds", "The geometry of the hyperbolic invariant manifolds can be described analytically inside the $P_1$ -region, for some range of energies and mass ratios, following some results from [10], [9].", "First, there exists an open set $O_1$ in the $(\\mu ,H)$ -parameter plane, with $0<\\mu \\ll 1$ and $H \\gtrapprox H(L_1)$ such that, for $(\\mu ,H)\\in O_1$ , the following hold: (i) The energy hypersurface $M_{H}$ contains an invariant 2-torus $\\mathcal {T}_H$ separating $P_1$ from $L_1$ .", "(ii) There exist $a<x_{L_1}<b$ such that the flow inside the isolating block $B_H=B_H(a,b)$ is conjugate to the linearized flow.", "(iii) In the region $\\mathcal {N}_H$ in $M_H$ bounded by $\\mathcal {T}_H$ and $B_{H,b}$ , the longitudinal angular coordinate $\\theta $ is increasing along trajectories.", "Second, for all $0<\\mu \\ll 1$ sufficiently small, the $(x,y)$ -projections of the branches of $W^u(L_1),W^s(L_1)$ inside the $P_1$ -region have the following properties: (iv) The distance $d$ to the zero velocity curve, and the angular coordinate $\\theta $ satisfy the following estimates: $ d&=\\mu ^{1/3}\\left(\\frac{2}{3}N-3^{1/6} +M\\cos t + o(1) \\right),\\\\ \\theta &=-\\pi +\\mu ^{1/3}\\left(Nt+2M\\sin t+o(1) \\right),$ where $M,N$ are constants, the parameter $t$ means the physical time measured from a suitable origin, and $o(1)\\rightarrow 0$ when $\\mu \\rightarrow 0$ uniformly in $t$ as $t=O(\\mu ^{-1/3})$ .", "These expressions hold true outside $B_H$ .", "(v) There exists an open set $O_2\\subseteq O_1$ in the $(\\mu ,H)$ -parameter plane, with $0<\\mu \\ll 1$ and $H \\gtrapprox H(L_1)$ such that, for $(\\mu ,H)\\in O_2$ , the $(x,y)$ -projections of the branches of $W^u(\\gamma _H), W^s(\\gamma _H)$ inside the $P_1$ -region satisfy estimates similar to (REF ) and ().", "That is, these invariant manifolds turn around $P_1$ in the region $\\mathcal {N}_H$ bounded by the torus $\\mathcal {T}_H$ and the boundary component $B_{H,b}$ of the isolating block $B_H$ .", "Moreover, there exists a sequence of mass ratios $\\mu _k$ for which $W^u(\\gamma _H)$ and $W^s(\\gamma _H)$ have symmetric transverse intersections, provided $(\\mu _k,H)\\in O_2$ .", "The geometry of the hyperbolic invariant manifolds for the range of parameters considered above allows to extend the separatrix property of these manifolds from the local case, as described in Subsection REF , to the global case.", "For as long as the stable and unstable manifolds do not intersect each other, the cuts of these manifolds with a surface of section are topological circles.", "If a point is inside the $i$ -th cut $\\Gamma _{\\theta _0,i}^s(\\gamma _H)$ made by the stable manifold $W^s(\\gamma _H)$ with the surface of section $S_{\\theta _0}$ , which is assumed to be a topological circle, then the forward trajectory of that point stays inside the cylinder bounded by $W^s(\\gamma _H)$ in $M_H$ for $i$ -turns and transfers from the $P_1$ -region to the $P_2$ -region afterwards.", "If a point in $S_{\\theta _0}$ is outside the $i$ -th cut $\\Gamma _{\\theta _0,i}^s(\\gamma _H)$ , then its forward trajectory stays inside the $P_1$ -region for at least $(i+1)$ -turns.", "A similar statement holds for the cuts made by the unstable manifold and backwards trajectories.", "If the stable and unstable manifolds intersect, say $\\Gamma _{\\theta _0,i}^s(\\gamma _H)$ intersects $\\Gamma _{\\theta _0,j}^u(\\gamma _H)$ , then the intersection points are homoclinic points that make $(i+j)$ -turns about $P_1$ , and some future cuts of the invariant manifolds cease to be topological circles.", "For example, $\\Gamma _{\\theta _0,i+j}^s(\\gamma _H)$ is a finite union of open curve segments whose endpoints wind asymptotically towards $\\Gamma _{\\theta _0,i}^s(\\gamma _H)$ .", "Due to the asymptotic behavior of the endpoints, each of these open curves divides $S_{\\theta _0}$ into transfer and non-transfer orbits.", "Thus, the separatrix property extends to the case when the cuts of the hyperbolic invariant manifolds cease being topological circles.", "See [7], [4].", "There are no analogues of the above analytical results for the $P_2$ -region about the lighter mass.", "Figure: Projection of McGehee's separating torus onto configuration space, and trajectory near the zero velocity curve." ], [ "Equations of motion relative to polar coordinates", "We recall the relations between the motion of the infinitesimal mass $P_3$ relative relative to the barycentric rotating coordinates $(x, y,\\dot{x}, \\dot{y})$ , relative to the polar coordinates $(r,\\theta , \\dot{r},\\dot{\\theta })$ about $P_1$ , and relative to the classical orbital elements $(a, e, \\phi , \\tau )$ about $P_1$ .", "The relation between barycentric and polar coordinate is $r=((x-\\mu )^2+y^2)^{1/2}$ and $\\tan \\theta =y/(x-\\mu )$ .", "The orbital elements are characterized by the semi-major axis $a$ of an ellipse with a focus at $P_1$ , the ellipse eccentricity $e\\in [0,1)$ , the argument of the periapsisis $\\phi \\in [0,2\\pi ]$ , and the true anomaly $\\tau \\in [0,2\\pi ]$ .", "We have the following coordinate transformations $ \\begin{split}x & = r \\cos (\\phi +\\tau ) + \\mu , \\\\y & = r \\sin (\\phi +\\tau ), \\\\\\dot{x} &= \\dot{r}\\cos (\\phi +\\tau ) - r\\dot{\\tau }\\sin (\\phi +\\tau ) + r \\sin (\\phi +\\tau ), \\\\ \\dot{y} & = \\dot{r}\\sin (\\phi +\\tau ) + r\\dot{\\tau }\\cos (\\phi + \\tau ) - r\\cos (\\phi +\\tau ),\\end{split}$ and the following formulas $ \\begin{split}r & = \\frac{a(1 - e^2)}{1+ e \\cos \\tau }, \\\\\\dot{r} & = \\frac{a e (1-e^2)\\dot{\\tau }\\sin \\tau }{(1+e\\cos \\tau )^2},\\\\\\theta &=\\phi +\\tau ,\\\\\\dot{\\theta }& =\\dot{\\tau }= \\frac{\\sqrt{a(1-e^2)(1-\\mu )}}{r^2}.\\end{split}$ The Hamiltonian function in polar coordinates is given by $H(r,p_r,\\theta ,p_\\theta )= \\frac{1}{2}(p_r^2+\\frac{1}{r^2}\\theta ^2)-p_\\theta +\\mu r\\cos \\theta +\\omega (r,p_r,\\theta ,p_\\theta ),$ where the canonical momenta are given by $p_r=\\dot{r},\\, p_\\theta =r^2(\\dot{\\theta }+1).$ Note that the conservation of energy implies that the initial position $(r,\\theta )$ relative to $P_1$ and the initial radial velocity $\\dot{r}$ uniquely determine a trajectory, up to a choice of a sign for $\\dot{\\theta }$ .", "Suppose that we know the initial data $(r,\\dot{r},\\theta )$ on a trajectory.", "Using (REF ), the eccentricity of the osculating ellipse to this trajectory at the initial point uniquely determines the trajectory, and hence its energy.", "This implicitly defines $\\phi $ and $\\tau $ .", "Conversely, if we have a trajectory for which the initial angle coordinate $\\theta $ , the initial angular velocity $\\dot{r}$ , and the eccentricity of the osculating ellipse $e$ at the initial condition are fixed, then the energy level $H$ of the trajectory uniquely determines its initial value of $r$ ." ], [ "Weak Stability Boundary", "We consider the system of polar coordinates $(r,\\theta )$ about $P_1$ as above, and we let $H(r,\\dot{r},\\theta ,\\dot{\\theta })$ be the Hamiltonian relative to this coordinate system.", "As discussed above, the energy is also uniquely determined by the $(r,\\dot{r},\\theta , e)$ -data, where $e$ is the eccentricity of the osculating ellipse at the initial point.", "We consider a Poincaré section through $P_1$ that makes an angle $\\theta _0$ with the $x$ -axis, which is given by $S_{\\theta _0}=\\lbrace (r,\\dot{r},\\theta ,\\dot{\\theta })\\,|\\,\\theta =\\theta _0,\\, \\dot{\\theta }>0\\rbrace .$ Let $l_{\\theta _0}$ denote the radial segment obtained as the intersection of $S_{\\theta _0}$ with the $(x,y)$ -space.", "Any trajectory that meets $S_{\\theta _0}$ transversally is uniquely determined by the $(r,\\dot{r})$ -coordinates of the intersection point, as the $\\theta $ -coordinate equals $\\theta _0$ in this section, and the $\\dot{\\theta }$ -coordinate can be solved uniquely from the energy condition $H(r,\\dot{r},\\theta ,\\dot{\\theta })=H$ , provided $\\dot{\\theta }>0$ .", "Consider a trajectory with the initial condition $z_0=z_0(r_0, \\dot{r}_0,\\theta _0,e_0)$ with initial position $r(0)=r_0$ , $\\theta (0)=\\theta _0$ , initial radial velocity $\\dot{r}(0)=\\dot{r}_0$ , and $\\dot{\\theta }(0)>0$ , for which the osculating ellipse at the initial point has eccentricity $e_0$ .", "We keep the values of $\\dot{r}_0, \\theta _0, e_0$ fixed and investigate the change of behavior of the trajectories when $r_0$ changes.", "Note that different initial values of $r_0$ yield different energies $H_0$ .", "Fix a value of $\\mu $ sufficiently small for which there exists an open range of energies $(H(L_1),H^*)$ with $(\\mu , H)\\in O_2$ for each $H\\in (H(L_1),H^*)$ , as in Subsection REF .", "For this range of energies the estimates (REF ) are valid.", "Fix $a<x_{L_1}<b$ such that $B_H(a,b)$ is an isolating block for all $H\\in (H(L_1),H^*)$ .", "Let $y_b$ be the supremum of the $y$ -coordinates on the segment $x=b$ inside the Hill's regions for $H\\in (H(L_1),H^*)$ .", "Define $\\theta _1=\\arctan ({y_b}/(\\mu -b))$ .", "Let $D_1$ be the distance from $P_1$ to $x=a$ , that is $D_1=\\mu -a$ .", "Fix $H\\in (H(L_1),H^*)$ and consider the projection $\\textrm {pr}_{(x,y)}(\\mathcal {N}_H)$ of $\\mathcal {N}_H$ onto the $(x,y)$ -configuration plane.", "For each angle coordinate $\\theta \\in [0,2\\pi ]$ , there exists a well defined interval $(r_1(H,\\theta ), r_2(H,\\theta ))$ such that $(r,\\theta )\\in \\textrm {pr}_{(x,y)}(\\mathcal {N}_H)$ if and only if $r\\in (r_1(H,\\theta ), r_2(H,\\theta ))$ .", "For each trajectory point $(r,\\theta ) \\in \\textrm {pr}_{(x,y)}(\\mathcal {N}_H)$ there exists a set of admissible values of the radial velocity $\\dot{r}$ and of the eccentricity of the osculating ellipse $e$ corresponding to the trajectory at that point.", "When we let $H$ vary in $(H(L_1),H^*)$ , then for each $\\theta \\in [0,2\\pi ]$ , we obtain an open set of admissible values of $(\\dot{r},e)$ corresponding to all trajectories for all of these energy levels.", "We fix an angle $\\theta _0$ and a pair of admissible values $(\\dot{r}_0,e_0)$ .", "Since the energy $H$ is uniquely determined by the data $(r_0,\\dot{r}_0,\\theta _0,e_0)$ , there exists an open set $\\mathcal {R}(\\dot{r}_0,\\theta _0,e_0)\\subseteq (r_1(H,\\theta ), r_2(H,\\theta ))$ of $r_0$ -values such that ${H_0}=H(r_0,\\dot{r}_0,\\theta _0,e_0)\\in (H(L_1),H^*)$ provided $r_0\\in \\mathcal {R}(\\dot{r}_0,\\theta _0,e_0)$ .", "In the next definition, we will consider trajectories with initial points $z_0$ lying on the radial segment $l_{\\theta _0}$ .", "We will restrict to values of $r_0$ in the set $\\mathcal {R}(\\dot{r}_0,\\theta _0,e_0)$ .", "Definition 3.1 We say that a forward trajectory with initial point $z_0=z_0(r_0,\\theta _0)$ in $l_{\\theta _0}$ , initial radial velocity $\\dot{r}_0$ and initial eccentricity of the osculating ellipse $e_0$ , is weakly $n$ -stable provided that it turns $n$ -times around $P_1$ , with all intersections with $l_{\\theta _0}$ being transverse, and such that the distance to $P_1$ is always less than $D_1$ .", "If the trajectory is weakly $(n-1)$ -stable but fails to be weakly $n$ -stable, we say that the trajectory is weakly $n$ -unstable.", "The conditions on the parameters assumed for the Definition REF are imposed in order to define the critical distance $D_1$ in a consistent way for the whole range of energy values $H\\in (H(L_1),H^*)$ .", "We recall that in the classical definition of the WSB, a trajectory is called $n$ -stable if it turns $n$ -times around $P_1$ , without turning around $P_2$ ; in that case one can consider the distance from $P_1$ to $P_2$ as the critical distance.", "We note that the transversality requirement in Definition REF , on the intersections of the trajectory of the infinitesimal mass with $l_{\\theta _0}$ , implies that weak $n$ -stability is an open condition, that is, if a trajectory starting at some $z_0=(r_0,\\dot{r}_0, \\theta _0,e_0)$ is weakly $n$ -stable, then all trajectory starting inside some domain of the type $(r,\\dot{r}, \\theta , e)\\in (r_0-\\varepsilon ,r_0+\\varepsilon )\\times (\\dot{r}_0-\\varepsilon ,\\dot{r}_0+\\varepsilon )\\times (\\theta _0-\\varepsilon ,\\theta _0+\\varepsilon )\\times (e_0-\\varepsilon ,e_0+\\varepsilon )$ with $\\varepsilon >0$ sufficiently small, are also weakly $n$ -stable.", "Thus we obtain the following set of weakly stable points in the phase space $ {\\mathcal {W}}_n=\\lbrace z_0(r_0,\\dot{r}_0,\\theta _0,e_0)\\,|\\, z_0\\textrm { is weakly $ $-stable relative to } l_{\\theta _0},\\, \\theta _0\\in [0,2\\pi ]\\rbrace .$$$ Due to the open conditions on the $n$ -stable trajectories, the set ${\\mathcal {W}}_n$ is an open set of points in the phase space.", "If we fix the parameters $\\dot{r}_0$ , $\\theta _0$ and $e_0$ , then we obtain an open set ${\\mathcal {W}}_n(\\dot{r}_0,\\theta _0,e_0)$ in $l_{\\theta _0}$ , which is a countable union of disjoint open intervals ${\\mathcal {W}}_n(\\dot{r}_0,\\theta _0,e_0)=\\bigcup _{k\\ge 1}(r_{2k-1},r_{2k}).$ The points of the type $r_{2k-1},r_{2k}$ at the ends of these intervals are weakly $n$ -unstable.", "Definition 3.2 The WSB of order $n$ , denoted $\\mathcal {W}^*_n$ , is the set of all points $r^*(r_0,\\dot{r}_0,\\theta _0,e_0)$ that are at the boundary of the set of the weakly $n$ -stable points, i.e., $\\mathcal {W}^*_n =\\partial {\\mathcal {W}}_n.$ We also denote by $\\mathcal {W}^*_n(\\dot{r}_0,\\theta _0,e_0)$ the set of WSB points on the radial segment $l_{\\theta _0}$ of fixed parameters $\\dot{r}_0$ and $e_0$ .", "Thus, the WSB set $\\mathcal {W}^*_n(\\dot{r}_0,\\theta _0,e_0)$ contains the closure of the set of all points of the type $r_{2k-1},r_{2k}$ , which are the endpoints of the intervals of weakly $n$ -stable points within each radial segment $l_{\\theta _0}$ as in (REF ).", "The main result of the paper says that, if we restrict to some angle range of $\\theta _0$ outside the angle sector $[\\pi -\\theta _1,\\pi +\\theta _1]$ , where $\\theta _1$ is defined as above, then the WSB set is completely determined by the stable manifolds of Lyapunov orbits.", "To state this result, we have to adopt a convention on how to count the number of cuts made by the stable manifold with a surface of section $S_{\\theta _0}$ .", "We label a cut made by the stable manifold $W^s(\\gamma _{H_0})$ with $S_{\\theta _0}$ as the $i$ -th cut provided that the net change $\\Delta \\theta $ of the angle $\\theta $ along all trajectories starting from $S_{\\theta _0}$ and ending asymptotically at $\\gamma _{H_0}$ satisfies $2i\\pi \\le \\Delta \\theta <2(i+1)\\pi $ .", "Note that as long as $\\theta _0 \\notin [\\pi -\\theta _1,\\pi +\\theta _1]$ there is no ambiguity about the labeling of the cuts with the section $S_{\\theta _0}$ .", "Theorem 3.3 Fix a pair of admissible values $(\\dot{r}_0, e_0)$ as defined above.", "Assume $\\theta _0\\in (-\\pi +\\theta _1,\\pi -\\theta _1)$ , where $\\theta _1$ is defined as above.", "Then a point $z_0=z_0(r_0,\\dot{r}_0,\\theta _0,e_0)$ , with $r_0\\in \\mathcal {R}(\\dot{r}_0,\\theta _0,e_0)$ , is in $\\mathcal {W}^*_n(\\dot{r}_0,\\theta _0,e_0)$ if and only if $z_0$ lies on the $(n-1)$ -st cut $\\Gamma ^s_{\\theta _0,n-1}(\\gamma _{H_0})$ of the stable manifold $W^s(\\gamma _{H_0})$ with the surface of section $\\mathcal {S}_{\\theta _0}$ , where ${H_0}$ is the energy level corresponding to $z_0$ .", "The restrictions imposed on the parameters in Theorem REF are needed to apply the analytical arguments from Subsection REF .", "It is nevertheless shown in [4] that the WSB overlaps with some subset of the stable manifold of the Lyapunov orbit under much weaker conditions, provided that the hyperbolic invariant manifolds satisfy some topological condition (they turn around the primaries for a long enough time, without colliding with the primaries).", "Moreover, in [4] a wider energy range is considered, in which case the WSB is identified with a subset of the union of the stable manifolds of the Lyapunov orbits about $L_1$ and about $L_2$ .", "The situation described by Theorem REF is just a special case when the required topological conditions can be verified analytically.", "Now we explain the relation between WSB and hyperbolic invariant manifolds in a more concrete way.", "Assume that we fix some energy level ${H_0}\\in (H(L_1),H^*)$ .", "We generate the stable manifold of the Lyapunov orbit $\\gamma _{H_0}$ , and we count the successive cuts made by the stable manifold with some Poincaré surface of section $S_{\\theta _0}$ .", "Let $z_0$ be a point on the $(n-1)$ -st cut $\\Gamma ^s_{\\theta _0,n-1}(\\gamma _{H_0})$ of $W^s(\\gamma _{H_0})$ with $\\mathcal {S}_{\\theta _0}$ .", "Let $\\dot{r}_0$ be the radial velocity at $z_0$ , and $e_0$ the eccentricity of the osculating ellipse at $z_0$ .", "Then the point $z_0$ is in the WSB set $\\mathcal {W}^*_n(\\dot{r}_0,\\theta _0,e_0)$ .", "Moreover, every WSB point can be obtained in this way." ], [ "Proof of the main result", "Due to the angle restriction $\\theta _0\\in (-\\pi +\\theta _1,\\pi -\\theta _1)$ , in Theorem REF , we restrict to the following set of weakly $n$ -stable points $\\tilde{\\mathcal {W}}_n= \\lbrace z_0(r_0,\\dot{r}_0,\\theta _0,e_0)\\,|\\, z_0\\textrm { is weakly $ $-stable relative to } l_{\\theta _0},\\, \\theta _0\\in (-\\pi +\\theta _1,\\pi -\\theta _1)\\rbrace .$$$ Since the $n$ -stability is an open condition and the angle range $(-\\pi +\\theta _1,\\pi -\\theta _1)$ is also open, the set $\\tilde{\\mathcal {W}}_n$ is an open set in the phase space.", "We prove that a point $z_0=z_0(r_0,\\dot{r}_0,\\theta _0,e_0)$ is in $\\tilde{\\mathcal {W}}^*_n$ if and only if it is in the $(n-1)$ -st cut $\\Gamma ^s_{\\theta _0,n-1}(\\gamma _{H_0})$ made by the stable manifold $W^s(\\gamma _{H_0})$ with $\\mathcal {S}_{\\theta _0}$ , where ${H_0}$ is the energy level corresponding to $z_0$ .", "For this, we first show that $z_0$ is a weakly $n$ -stable point on $l_{\\theta _0}$ if and only if it is outside the domain in $S_{\\theta _0}$ bounded by $\\Gamma ^s_{\\theta _0,n-1}(\\gamma _{H_0})$ , and is weakly $n$ -unstable if and only if it is inside the domain in $S_{\\theta _0}$ bounded by $\\Gamma ^s_{\\theta _0,n-1}(\\gamma _{H_0})$ .", "First, we show that the points inside the cylinder bounded by the stable manifold are weakly unstable.", "Let $z_0=z_0(r_0,\\dot{r}_0,\\theta _0,e_0)$ be a point in $l_{\\theta _0}$ .", "Then (REF ) gives the value ${H_0}$ of the energy of the trajectory with initial condition $z_0$ .", "Assume that $z_0$ is inside the domain in $S_{\\theta _0}$ bounded by $\\Gamma ^s_{\\theta _0,n-1}(\\gamma _{H_0})$ .", "By the separatrix property from Subsection REF the trajectory turns counterclockwise precisely $(n-1)$ -times inside the domain $\\mathcal {N}_{H_0}$ , while staying inside the region of the cylinder bounded by $W^s(\\gamma _{H_0})$ , enters the isolating block $B_{H_0}$ through the entry set region $B_{{H_0},b}^{+,a}$ , crosses the block and exits it through the exit set region $B_{{H_0},a}^{-,b}$ .", "When the trajectory leaves the block $B_{H_0}$ , the distance from $P_1$ is bigger than $D_1$ .", "Since the trajectory achieves a distance to $P_1$ bigger than the threshold value prior to completing an $n$ -th turn around $P_1$ , the trajectory is weakly $n$ -unstable.", "Second, we show that the points outside the cylinder bounded by the stable manifold are weakly stable.", "Assume that $z_0$ is outside the domain in $S_{\\theta _0}$ bounded by $\\Gamma ^s_{\\theta _0,n-1}(\\gamma _{H_0})$ .", "By the separatrix property from Subsection REF the trajectory will turn counterclockwise inside the domain $\\mathcal {N}_{H_0}$ and will keep staying outside the region of the cylinder bounded by $W^s(\\gamma _{H_0})$ for at least $n$ turns.", "If the trajectory leaves the domain $\\mathcal {N}_{H_0}$ , it has to meet the block $B_{H_0}$ at $B^0_{H_0}$ or at $B_{{H_0},b}^{+,b}$ .", "In the first case, the trajectory bounces back to the domain $\\mathcal {N}_{H_0}$ and it continues its counterclockwise motion about $P_1$ .", "In the second case, it cannot leave the block $B_{H_0}$ through $B_{{H_0},a}$ , since only the points that are inside the cylinder bounded by $W^s(\\gamma _{H_0})$ can do that; it cannot remain inside the block $B_{H_0}$ for all future times since only the points on $W^s(\\gamma _{H_0})$ have this property; hence, it has to leave $B_{H_0}$ through the exit set region $B_{{H_0},b}^{-,b}$ , and to go back to the domain $\\mathcal {N}_{H_0}$ .", "The time spent by the trajectory inside the block $B_{H_0}$ does not affect the count of turns about $P_1$ .", "Since $z_0$ is outside the cut $\\Gamma ^s_{\\theta _0,n-1}(\\gamma _{H_0})$ , the trajectory cannot leave the $P_1$ -region after only $(n-1)$ -turns, so it turns around $P_1$ for at least $n$ -turns.", "Thus the trajectory is weakly $n$ -stable.", "Now, we prove the statement of the main theorem.", "First, assume that $z_0$ is in $\\Gamma ^s_{\\theta _0,n-1}(\\gamma _{H_0})$ .", "Its forward trajectory turns $(n-1)$ -times around $P_1$ and then approaches asymptotically $\\gamma _{H_0}$ .", "Thus the trajectory is weakly $n$ -unstable.", "To show that $z_0$ is an WSB point it is sufficient to prove that there exists a sequence $(z_0^k)_{k\\ge 1}$ with $z_0^k\\in \\tilde{\\mathcal {W}}_n$ and $z_0^k\\rightarrow z_0$ as $k\\rightarrow \\infty $ .", "Take a small 4-dimensional open ball $\\mathcal {U}$ around $z_0$ in the phase space.", "Let $T>0$ be such that the time-$T$ map $\\phi _T$ of the Hamiltonian flow takes $z_0$ to a point in $B_{H_0,b}$ , where $H_0=H(z_0)$ .", "The image $\\phi _T(\\mathcal {U})$ of $\\mathcal {U}$ by $\\phi _T$ is a 4-dimensional open topological ball about $\\phi _{T}(z_0)$ .", "We intersect $\\phi _T(\\mathcal {U})$ with the 4-dimensional submanifold with boundary $\\bigcup _{H\\in (H(L_1),H^*)}B_{H}$ .", "The ball $\\phi _T(\\mathcal {U})$ has non-empty intersection with $\\bigcup _{H\\in (H(L_1),H^*)}B^{+,b}_{H,b}$ .", "These intersection points yield weakly $n$ -stable trajectories.", "Thus $\\phi _T(\\mathcal {U})\\cap \\bigcup _{H\\in (H(L_1),H^*)}B_{H}$ contains a 4-dimensional open, topological ball $\\mathcal {V}$ , which contains $\\phi _{T}(z_0)$ on its boundary, consisting of points that correspond to weakly $n$ -stable trajectories, i.e., those trajectories that return to the $P_1$ region for at least one extra turn about $P_1$ .", "Now consider the set $\\phi _{-T}(\\mathcal {V})$ .", "This is a 4-dimensional open, topological ball in $\\mathcal {U}$ that contains $z_0$ on its boundary.", "There exist $\\theta ^{\\prime }$ arbitrarily close to $\\theta _0$ such that the intersection $\\phi _{-T}(\\mathcal {V})\\cap S_{\\theta ^{\\prime }}$ is a non-empty open set.", "All points $z^{\\prime }\\in \\phi _{-T}(\\mathcal {V})\\cap {S}_{\\theta ^{\\prime }}$ are weakly $n$ -stable points.", "We note that these points may not lie on $l_{\\theta _0}$ , nor on the same energy level as $z_0$ ; they can also have the eccentricity of the osculating ellipse different from $e_0$ .", "Thus, arbitrarily near $z_0$ one can always find weakly $n$ -stable points, and since $z_0$ itself is weakly $n$ -unstable, it follows that $z_0\\in \\partial \\tilde{\\mathcal {W}}_n=\\tilde{\\mathcal {W}}^*_n$ .", "Second, assume that $z_0\\in \\tilde{\\mathcal {W}}^*_n(\\dot{r}_0,\\theta _0, e_0)$ .", "Then there exists a sequence of points $(z_0^k)_{k\\ge 1}$ on $l(\\theta _0)$ such that $z_0^k$ is weakly $n$ -stable and $z_0^k\\rightarrow z_0$ as $k\\rightarrow \\infty $ .", "From the above, we know that the weakly $n$ -stable points are those inside the cylinder bounded by the stable manifold.", "Thus, there exists a corresponding sequence of stable manifold cuts $\\Gamma ^{s}_{\\theta _0,n-1}(\\gamma _{H_k})$ where $H_k=H(z_0^k)$ , such that $z_0^k$ is inside the region in $S_{\\theta _0}$ bounded by $\\Gamma ^{s}_{\\theta _0,n-1}(\\gamma _{H_k})$ .", "Since $z_k^0\\rightarrow z_0$ it follows that $H(z_0^k)\\rightarrow H(z_0)=H_0$ as $k\\rightarrow \\infty $ .", "The stable manifold cuts also depend continuously on the energy, so $\\Gamma ^{s}_{\\theta _0,n-1}(\\gamma _{H_k})$ approaches $\\Gamma ^{s}_{\\theta _0,n-1}(\\gamma _{{H_0}})$ as $k\\rightarrow \\infty $ .", "Hence $z_0\\in \\Gamma ^{s}_{\\theta _0,n-1}(\\gamma _{{H_0}})$ .", "Through double inclusion, we conclude that $z_0\\in \\tilde{\\mathcal {W}}^*_n(\\dot{r}_0,\\theta _0, e_0)\\textrm { if and only if } z_0\\in \\Gamma ^{s}_{\\theta _0,n-1}(\\gamma _{H_0}).$" ], [ "Concluding remarks", "We compare the invariant manifold method with the WSB method.", "The invariant manifold method is based on identifying geometric objects that serve as building blocks that organize the global dynamics: equilibrium points, periodic orbits, and their stable and unstable invariant manifolds, if they exist.", "The WSB method is a local method for deciding whether the trajectories about one of the primaries exhibit some kind of stability in terms of the return to a surface of section.", "The conclusion of this paper, corroborated with the results in [6], [4], is that in simple models the two methods overlap for a substantial range of parameters.", "One can think of some other kinds of indicators that mark the passage between the weakly $n$ -stable and the weakly $n$ -unstable regimes.", "One such a possible indicator is the continuity of the Poincaré return map.", "The $n$ -th return map to $S_{\\theta _0}$ is continuous at all weakly $n$ -stable points.", "At the WSB points, the return map exhibits essential discontinuities of infinite type.", "Thus, the set of points where the return map fails to be continuous contains the WSB points." ] ]

[ [ "Building Grassmann Numbers from PI-Algebras" ], [ "Abstract This works deals with the formal mathematical structure of so called Grassmann Numbers applied to Theoretical Physics, which is a basic concept on Berezin integration.", "To achieve this purpose we make use of some constructions from relative modern Polynomial Identity Algebras (PI-Algebras) applied to the special case of the Grassmann algebra." ], [ "The ideas of H. Grassmann in Mathematics and later F. Berezin in Physics[1] are a good example of how science fields required Mathematics to obtain new results.", "But as any other field, Mathematics evolves.", "Follow on this philosophy, we make use of some kind of new Mathematics to examine the Grassmann Numbers, now using the concepts of Polynomial Identity Theory (PI-Theory), that is part of non-commutative algebras.", "The misunderstanding arises in the following polynomial on Berezin's work: $p(\\theta )=a+b\\theta ,$ that according to usual polynomial definition over some field $K$ , the coefficients $a,b$ must belong to the same field, this means that $a,b\\in K$ .", "But Berezin's work states that $a\\in \\mathbb {C}$ (complex number) and $b$ is a kind of C-number, since anti-commutation is required for $b$ and $\\theta $ .", "This apparent paradox is the central motivation of this work, which is a result of discussions of the authors over some earlier works[2].", "We divided this work as: Next section we make a brief review of PI-Algebras concepts.", "After a section on Mathematical background, we start a new section on Physics background, where we will describe how the Berezin's ideas appears on the context of Supersymmetry[3].", "The next section we will joins the ideas discussed in previous sections focusing on the above polynomial.", "The final section contains our conclusions." ], [ "In this section we will present some introductory ideas and the basic Mathematics we required to tackle this problem.", "This sections is written in order Theoretical Physicists can understand the main ideas of PI-Algebras.", "We will started with some historical motivation and with basics definitions on non-commutative algebra, after that we will give some examples of polynomial identities, and finally give, as an example, the case of Grassmann algebras from this point of view.", "In 1987 Kemer [4] showed that T-prime algebras (i.e., algebras with T-ideals that are T-prime) that are non-trivial (in characteristic zero) are of type $M_n(K)$ , $M_n(\\mathbb {G})$ and $M_{a,b}(\\mathbb {G})$ , where $\\mathbb {G}$ is the finite dimensional Grassmann algebra over a field $K$ .", "After a short introductory, now we are going to give some definitions [5].", "The main structure to study is an algebra, then it is defined as Definition 1 (Algebra) A vector space $A$ ($a,b,c\\in A$ ) over a field $K$ ($\\alpha \\in K$ ) with a binary operation on $A$ defines an algebra if and only if: $(a+b)c=ac+bc$ , $a(b+c)=ab+bc$ , and $\\alpha (ab)=(\\alpha a)b=a(\\alpha b)$ .", "Some times this kind of algebra is also called K-Algebra Now lets introduce some non-commutativity.", "Let $X=\\lbrace x_1,x_2,\\cdots ,x_n\\rbrace $ be a set, where each $x_i$ must obey in general the property $x_ix_j\\ne x_jx_i,\\ \\forall i,j:i\\ne j$ , and of course this last condition implies there exist some binary operation among the $x_i$ 's.", "Let's introduce another binary operation, and denote this by $(+)$ , altogether with the previous one, allows us to built up elements of type, for example, $x_ix_jx_k+x_jx_kx_i+x_kx_ix_j,$ $x_ix_j+x_jx_k,$ etc., that according the mandatory definition of binary operation, must also belongs to the main set $X$ .", "Using the field $K$ and the elements of $X$ we can make the following construction, $B=\\lbrace 1,x_{i_1}\\cdots x_{i_k}:\\ i_l=1,2\\cdots \\rbrace ,$ then we have a vector space $V$ generated by $B$ over $K$ , and also $K\\langle X\\rangle $ is the associative algebra generated by the vector space $V$ where the the binary operation is now defined as: $(x_{i_1}\\cdots x{i_k})(x_{j_1}\\cdots x{j_l})=x_{i_1}\\cdots x{i_k}x_{j_1}\\cdots x{j_l}.$ We will call the elements of $K\\langle x\\rangle $ as polynomials.", "Definition 2 (right-ideal) Let $A$ be an algebra, and $I$ a sub-algebra of $A$ , then $I$ represents an right-ideal if and only if for all $a\\in A$ and $i\\in I$ we have $ai\\in I$ In the same form we define a left-ideal.", "Definition 3 (2-ideal) An ideal $I$ of an algebra $A$ is said a two-ideal if it is both, right-ideal and also left-ideal.", "Definition 4 (homomorphism) A linear transformation $\\varphi :A\\rightarrow B$ between the algebras $A$ and $B$ is said to be an homomorphism is and only if $\\varphi (ab)=\\varphi (a)\\varphi (b),$ in few words, operations are preserved.", "If the above definition is applied over the same set, the homomorphism is called endomorphism.", "Definition 5 (T-ideal) An ideal $I$ of an algebra $A$ is defined as a T-ideal if $I$ is invariant under all endomorphisms on $A$ , i.e., $\\Phi (I)\\subseteq I$ for all $\\Phi :A\\rightarrow A$ .", "Our construction requires another important definition: Definition 6 (Quotient Algebra) Let $I$ be an 2-Ideal of $K\\langle x\\rangle $ and $f,g\\in K\\langle x\\rangle $ , then we define the quotient algebra over $I$ as $K\\langle x\\rangle /I$ where as usual $(f+I)+(g+I)=(f+g)+I$ , $(f+I)(g+I)=fg+I$ .", "Notice that $\\bar{f}=f+I\\in K\\langle x\\rangle /I$ Now we are ready to define the Grassmann algebra $\\mathbb {G}$ [6]: Definition 7 (Grassmann Algebra) Let be a field $K$ , a non-commutative algebra $A$ over $K$ , and the ideal $I=I=\\langle x_ix_j+x_jx_i\\rangle ,\\ i,j\\ge 1$ , the quotient algebra $\\mathbb {G}=A/I.$ According to this definition it is not hard to see the Grassmann algebra can be written as: $\\mathbb {G}=alg_K\\lbrace 1,x_{i_1}x_{i_2}\\cdots x_{i_n};\\ i_1<i_2<\\cdots <i_n\\rbrace .$ Another important definition is concerned with the commutative sub-algebra of an algebra $A$ that is called the center of the algebra and is defined as: Definition 8 (Center of an Algebra) Let $A$ be a non-commutative algebra, then the sub-set $Z[A]=\\lbrace x\\in A:\\forall g\\in A,\\ gxg^{-1}=x\\rbrace $ is defined as the center of the algebra $A$ .", "Using this last definition, we can obtain two sub-spaces of $\\mathbb {G}$ : $\\mathbb {G}_0=alg_K\\lbrace 1,x_{i_1}x_{i_2}\\cdots x_{i_n};\\ i_1<i_2<\\cdots <i_n,\\ n\\ even\\rbrace $ $\\mathbb {G}_1=alg_K\\lbrace x_{i_1}x_{i_2}\\cdots x_{i_n};\\ i_1<i_2<\\cdots <i_n,\\ n\\ odd\\rbrace $ Where $\\mathbb {G}_0$ is the center of $\\mathbb {G}$ , this means $\\mathbb {G}=\\mathbb {G}_0\\oplus \\mathbb {G}_1$ .", "Let us to give one more definition: Definition 9 (Graded Algebra) Let $A$ have the structure of a non-commutative algebra, if $A=\\oplus _{g\\in G}A_g$ , where $A_g$ is a subspace of $A$ ($\\forall g\\in G$ ), and ($\\forall g,h\\in G$ ), $A_gA_h\\subseteq A_{g+h}$ , then $A$ is called a G-graded algebra.", "According this definition, the Grassmann algebra is also a 2-Graded algebra.", "Here we stop our succinct description of PI-Theory." ], [ "Trying to understand the fundamental structure of Nature is not an easy task.", "For many years researchers have been working hard to obtain a better understanding of this problem, and Mathematics, a trust discipline, was (and is) a mandatory machine giving assistance on this kind of work for many years, this is because, in a few words, Physics is a science that makes use of Mathematics trying to describe Nature.", "Fist we will describe in few lines, the main physical ideas, in order mathematicians figure out the Physics behind these concepts.", "It is not our aim to give a full review on this topic, for that purpose there are good works on it [7].", "One interesting fact in Nature is that Nature itself is split in two main disjoint sets (mathematically speaking, we are saying two equivalence classes): Fermions, obeying the anti-commuting algebra: $\\lbrace b,b^\\dag \\rbrace =1,$ and Bosons, that observe the commuting algebra: $[a,a^\\dag ]=1.$ This historical fact (associated with statistical distributions) also gives support to understand main concepts in Elementary Particle Physics.", "In our opinion, a good candidate for describing the fundamental interactions of nature is superstring theory, that uses the ideas of super-symmetry[3], where the Grassmann algebra and Berezin integration[1] are important and make use polynomials of type: $p(\\theta )=a+b\\theta .$ As states above, according the basics text of algebra, the coefficients $a$ and $b$ must belongs to the same field, say it $K$ , but according to Theoretical Physics, $a$ belongs to the complex field and $b$ to the so called Grassmann Numbers.", "On the following section we will work on this apparent paradox.", "Summing up, the particle physics classifications as understand it, today from a (quantum) statistical distribution point of view is: we have some particles called Bosons (in memory of S. E. Bose and A. Einstein), and some other particles called Fermions (in memory of E. Fermi and P. A. M. Dirac).", "It is at this point that Grassmann algebra arises." ], [ "Working with the previous ideas, we can see that for case $n=1$ in Grassmann algebra we have: $\\mathbb {G}=alg_k\\lbrace 1,x_k\\rbrace $ that implies the polynomial expansion: $p(x_1)=a+bx_1;\\ \\ \\ a,b\\in K.$ Also for $n=2$ : $\\mathbb {G}=alg_k\\lbrace 1,x_1,x_2,x_1x_2\\rbrace $ that also implies: $p(x_1,x_2)=a+bx_1+cx_2+dx_1x_2+ex_2x_1;\\ \\ \\ a,b,c,d,e\\in K.$ Observe, since $\\lbrace x_1,x_2\\rbrace =0$ , polynomial $p(x_1,x_2)$ can be written as: $p(x_1,x_2)=a+bx_1+cx_2+(d-e)x_1x_2;$ this makes the coefficient of $x_1x_2$ behave like a non-commuting number, but in fact it is made of two commuting numbers that belongs to the field $K$ .", "The main tangle here was due considering the polynomial $p(\\theta )=a+b\\theta $ as part of mathematical structure, but as described above, this type of polynomial does not have a foundation on Mathematics.", "One interesting observation is that since the Grassmann algebra $\\mathbb {G}=\\mathbb {G}_0\\oplus \\mathbb {G}_1$ , the polynomial $p(x_1,x_2)\\in \\mathbb {G}$ can also be split as $p(x_1,x_2)=p_1(x_1,x_2)+p_0(x_1,x_2)$ where $p_1(x_1,x_2)\\in \\mathbb {G}_1$ takes the form: $p_1(x_1,x_2)=bx_1+cx_2,$ also $p_0(x_1,x_2)\\in \\mathbb {G}_0$ takes the form: $p_0(x_1,x_2)=a+(d-e)x_1x_2.$ It is this last equation (REF ) resembles the very initial equation (REF ) used in supersymmetry works, but now making the identifications: $a&\\rightarrow &a\\\\b&\\rightarrow &(d-e)\\\\\\theta &\\rightarrow &x_1x_2.$ where $\\theta ^2=x_1x_2x_1x_2=-x_1x_1x_2x_2=-0.0=0$ , that agrees with usual interpretations of Berezin's and supersymmetry." ], [ "We saw that mathematical consistency requires the use of polynomial expansions as we have seen in eq.", "(REF ) or eq.", "(REF ) instead of that used in eq.", "(REF ).", "But as we knew from previous section, the polynomial that gives the Berezin's polynomial expansion does not have any supporting mathematical structure but is in fact only part of polynomial (REF ).", "One question arises: what is the meaning of the missing terms?", "This is not an easy question to answer since we are dealing with one supposed well know fact form supersymmetry and it will causes some discomfort to realize this misinterpretation happened, but Mathematical construction is very clear and for its formal developments we can trust on them.", "Perhaps the following lemma will give some advice: Lemma 1 Let $p(x_1,x_2)=a+bx_1+cx_2+(d-e)x_1x_2$ be a polynomial for the Grassmann algebra $\\mathbb {G}=\\langle 1,x_1,x_2\\rangle $ , then $(p-a)^k=0$ ($\\forall k>1$ ).", "Proof Is suffices to prove the $k=2$ case.", "We have: $(p(x_1,x_2)-a)^2=(bx_1+cx_2+(d-e)x_1x_2)^2$ that is of the form $(a+b+c)^2=a^2+b^2+c^2+ab+ba+ac+ca+bc+cb,$ observe that for this case, each squared term vanishes.", "This is true also for the crossed terms with $(d-e)x_1x_2$ since them will generate factors of $x_1^2$ or $x_2^2$ , the only remaining terms will be $bcx_1x_2+bcx_2x_1$ that clearly vanish.", "So proof is completed.", "$\\rule {2mm}{2mm}$ The lemma have a form $Q^2=0$ , that resembles the definition of nilpotent operators, used both, in Mathematics (as it is done in Operator Theory) and in Physics (e.g.", "Quantum Field Theory).", "This is, to us, the main counsel or advice, that a better understanding of non-commutative algebras (with the proper attention on PI-Theory new results) for this kind of manipulations will improve advantageous and effective results in Supersymmetry and in general in Theoretical Physics.", "This partially clarifies the initial questions of this work about the $b\\theta $ term of the foremost Berezin's polynomial $p=a+b\\theta ,$ wide discussed on this work, now observing coefficient $b$ not being a C-number, but constructed from a difference of elements of some field $K$ over the Grassman algebra is built.", "But this also opens new questions to be worked out.", "Acknowledge: Both authors are very grateful to professor German I. Gomero (at Universidade Estadual de Santa Cruz) for discussions on ideas concerning this work and also to Universidade Estadual de Santa Cruz for facilities when this work was prepared." ] ]

[ [ "Complete control of a matter qubit using a single picosecond laser pulse" ], [ "Abstract We demonstrate for the first time that a matter physical two level system, a qubit, can be fully controlled using one ultrafast step.", "We show that the spin state of an optically excited electron, an exciton, confined in a quantum dot, can be rotated by any desired angle, about any desired axis, during such a step.", "For this we use a single, resonantly tuned, picosecond long, polarized optical pulse.", "The polarization of the pulse defines the rotation axis, while the pulse detuning from a non-degenerate absorption resonance, defines the magnitude of the rotation angle.", "We thereby achieve a high fidelity, universal gate operation, applicable to other spin systems, using only one short optical pulse.", "The operation duration equals the pulse temporal width, orders of magnitude shorter than the qubit evolution life and coherence times." ], [ "Sample description", "The sample used in this work was grown by molecular-beam epitaxy on a (001) oriented GaAs substrate.", "One layer of strain-induced $\\rm In_xGa_{1-x}As$ quantum dots (QDs) was deposited in the center of a one-wavelength microcavity formed by two unequal stacks of alternating quarter-wavelength layers of AlAs and GaAs, respectively.", "The height and composition of the QDs were controlled by partially covering the InAs QDs with a 3 nm layer of GaAs and subsequent growth interruption.", "To improve photon collection efficiency, the microcavity was designed to have a cavity mode which matches the QD emission due to ground-state e-h pair recombinations.", "During the growth of the QD layer the sample was not rotated, resulting in a gradient in the density of the formed QDs.", "The estimated QD density in the sample areas that were measured is $\\rm 10^8 cm^{-2}$ ; however, the density of QDs that emit in resonance with the microcavity mode is more than two orders of magnitude lower [1].", "Thus, single QDs separated by few tens of micrometers were easily located by scanning the sample surface during micro-photoluminescence (PL) measurements.", "Strong antibunching in intensity autocorrelation measurements was then used to verify that the isolated QDs are single ones and that they form single-photon sources." ], [ "Experimental setup", "The experimental setup that we used for the optical measurements is described in Fig.", "REF .", "The sample was placed inside a sealed metal tube immersed in liquid helium, maintaining a temperature of 4.2 K. A $\\times $ 60 microscope objective with numerical aperture of 0.85 was placed above the sample and used to focus the light beams on the sample surface and to collect the emitted PL.", "In the measurements described in Figs.", "3-5 of the Letter we used two dye lasers (Styryl 13), synchronously pumped by the same frequency-doubled Nd:YVO$_4$ (Spectra Physics-Vanguard™) laser for generating the resonantly tuned optical pulses, as described in the figure.", "The repetition rate of the setup was 76 MHz, corresponding to a pulse separation of about 13 nsec.", "The lasers' emission energy could have been continuously tuned using coordinated rotations of two plate birefringent filters and a thin etalon.", "The temporal width of the dye lasers' pulses was about 9 psec and their spectral width about 100 $\\mu $ eV.", "A third, Ti:Sapphire pulsed laser (Spectra Physics, Tsunami™) was locked to the clock of the Vanguard™laser.", "Its pulse duration was about 2 psec and spectral width of about 500 $\\mu $ eV.", "The delay between the pulses was controlled by 2 retroreflectors on translation stages.", "The polarizations of the pulses were independently adjusted using a polarized beam splitter (PBS) and two pairs of computer-controlled liquid crystal variable retarders (LCVRs).", "The polarization of the emitted PL was analyzed by the same LCVRs and PBS.", "The PL was spectrally analyzed by a 1-meter monochromator and detected by either a silicon avalanche photodetector (using lock-in detection) or by a cooled charged coupled array detector.", "Figure: Schematic description of the experimental setup.", "The delay between the three pulses is controlled by 2 computer controlled motorized retroreflectors.", "(P)BS stands for (polarizing) beam splitter, VBS for a variable beam splitter, BC for beam combiner, and LCVR for a liquid crystal variable retarder." ], [ "Rotation by a detuned pulse to a non- degenerate biexcitonic transition.", "A coherent exciton spin state may be described as a vector on the Bloch sphere: $|X(\\theta , \\phi )\\rangle = \\cos \\left( \\frac{\\theta }{2} \\right) |H\\rangle +ie^{i \\phi } \\sin \\left( \\frac{\\theta }{2} \\right) |V\\rangle \\\\= \\alpha (\\theta ) |H\\rangle + \\beta (\\theta , \\phi ) |V\\rangle .$ Here $|H\\rangle $ and $|V\\rangle $ are the two exciton's eigenstates $\\rm 1/\\sqrt{2} \\left[(1e^1)_{-1/2} (1h^1)_{3/2} + (1e^1)_{1/2} (1h^1)_{-3/2} \\right]$ and $\\rm 1/\\sqrt{2} \\left[(1e^1)_{-1/2} (1h^1)_{3/2} - (1e^1)_{1/2} (1h^1)_{-3/2}\\right]$ respectively.", "Here the number denotes the energetic order of the confined single carrier level, the letter stands for the carrier type (e-for electron and h for heavy hole), the superscript (either 1 or 2) for the level's occupation and the subscript for the spin projection of same charge carriers [2].", "These non-degenerate eigenstates form the poles of the Bloch sphere (Fig.", "2 of the Letter).", "Such an exciton is photogenerated by a short optical pulse (whose bandwidth is larger than the energy difference between the two eigen-energies) resonantly tuned to an excitonic transition provided that the pulse polarization is given by: $ \\vec{P}_X(\\theta , \\phi ) = \\cos \\left( \\frac{\\theta }{2} \\right) \\hat{H} + ie^{i\\phi } \\sin \\left( \\frac{\\theta }{2} \\right) \\hat{V},$ where $\\hat{H}$ and $\\hat{V}$ represent linear polarizations parallel to the major and minor axis of the QD, respectively [2], [3].", "This exciton is excited by a second pulse which is resonantly tuned to a non-degenerate biexciton resonance.", "Here the resonance is $\\rm (1e^2) (1h^1 4h^1)_{T_0}$ , in which the two electrons form a spin singlet in the ground state and the holes, one in the ground s-like level and one in the $\\rm d_{HH}$ - like 4$\\rm ^{th}$ level, form a total spin projection zero triplet [2].", "Maximal absorption occurs if the polarization of the second pulse is crossed-polarized with the exciton spin (and therefore with the polarization of the first pulse, if both pulses are simultaneous), $\\vec{P}_{XX}(\\pi -\\theta , \\pi +\\phi ) = \\sin \\left( \\frac{\\theta }{2} \\right) \\hat{H} - ie^{i\\phi } \\cos \\left( \\frac{\\theta }{2} \\right) \\hat{V},$ as described in Refs.", "twophoton, writeread.", "The cross-polarized exciton state: $|\\bar{X} (\\theta , \\phi )\\rangle = |X (\\pi - \\theta , \\pi + \\phi )\\rangle = \\\\= \\sin \\left( \\frac{\\theta }{2} \\right) |H\\rangle - ie^{i \\phi } \\cos \\left( \\frac{\\theta }{2} \\right) |V\\rangle = \\\\= \\alpha (\\theta ) |H\\rangle + \\beta (\\theta , \\phi ) |V\\rangle .$ is completely unaffected by resonant pulses with this polarization, but maximally coupled to \"cross polarized\" pulses with polarization described by Eq.", "(REF ).", "When a control, 2$\\pi $ -pulse with $\\vec{P}_{XX}$ polarization is applied to an arbitrary coherent excitonic state such as $|X (\\theta ^{\\prime }, \\phi ^{\\prime })\\rangle $ $|X (\\theta ^{\\prime }, \\phi ^{\\prime })\\rangle = \\cos \\left( \\frac{\\theta ^{\\prime }}{2} \\right) |H\\rangle +ie^{i \\phi ^{\\prime }} \\sin \\left( \\frac{\\theta ^{\\prime }}{2} \\right) |V\\rangle = \\\\= \\alpha (\\theta ^{\\prime }) |H\\rangle + \\beta (\\theta ^{\\prime }, \\phi ^{\\prime }) |V\\rangle .$ which can be conveniently expressed also in terms of the coherent states $|X(\\theta , \\phi )\\rangle $ and $|\\bar{X}(\\theta , \\phi )\\rangle $ as: $|X (\\theta ^{\\prime }, \\phi ^{\\prime })\\rangle = \\alpha (\\theta ^p) |X(\\theta , \\phi )\\rangle + \\beta (\\theta ^p, \\phi ^p) |\\bar{X}(\\theta , \\phi )\\rangle $ In Eq.", "(REF ) spherical symmetry considerations are used and the angles $\\theta ^p$ and $\\phi ^p$ are measured relative to the pulse's polarization direction.", "Since the $\\vec{P}_{XX}(\\theta , \\phi )$ polarized 2$\\pi $ pulse couples only to the $|\\bar{X}(\\theta , \\phi )\\rangle $ part of the exciton wavefunction, this part acquires a geometrical phase of $\\delta $ (defined by Eq.", "(1) in the Letter) relative to the $|X(\\theta , \\phi )\\rangle $ part of the exciton.", "Therefore after the control 2$\\pi $ -pulse ends, the new exciton state is given by: $|X (\\theta ^{\\prime \\prime }, \\phi ^{\\prime \\prime })\\rangle = \\\\= \\alpha (\\theta ^p) |X(\\theta , \\phi )\\rangle + e^{-i\\delta } \\beta (\\theta ^p, \\phi ^p) |\\bar{X}(\\theta , \\phi )\\rangle = \\\\\\alpha (\\theta ^p) |X(\\theta , \\phi )\\rangle + \\beta (\\theta ^p, \\phi ^p - \\delta ) |\\bar{X}(\\theta , \\phi )\\rangle .$ Thus, the geometrical description of the control pulse action is a clockwise rotation by an angle $\\delta $ about an axis connecting the states $|X(\\theta , \\phi )\\rangle $ and $|\\bar{X}(\\theta , \\phi )\\rangle $ , parallel to the polarization direction of the control pulse." ], [ "Description of the action of the polarized 2$\\pi $ control pulse as a universal rotation", "A unit vector in the polarization direction of an exciton $|X(\\theta , \\phi )\\rangle $ , which is coupled to a polarized 2$\\pi $ control pulse is given by: $\\hat{n} = \\left( n_x, n_y, n_z \\right) = \\left( \\cos \\phi \\sin \\theta , \\sin \\phi \\sin \\theta , \\cos \\theta \\right),$ where the Cartesian axes are chosen such that: $\\hat{x} \\equiv |R\\rangle = 1/\\sqrt{2} \\left( |H\\rangle + i |V\\rangle \\right)$ , $\\hat{y} \\equiv |\\bar{D}\\rangle = 1/\\sqrt{2} \\left( |H\\rangle - i |V\\rangle \\right)$ , $\\hat{z} \\equiv |H\\rangle $ .", "As discussed above, the effect of the control pulse on the exciton wavefunction can be simply viewed as a rotation about $\\hat{n}$ by the angle -$\\delta $ .", "Such a rotation in the eigenstates base is described by the operator: $R_{\\hat{n}}\\left( \\delta \\right) = \\exp \\left( i \\vec{\\sigma }\\cdot \\hat{n} \\frac{\\delta }{2} \\right),$ where $\\vec{\\sigma }\\equiv \\left( \\sigma _x, \\sigma _y, \\sigma _z \\right)$ is the Pauli matrix vector.", "When this operator is applied to an exciton state such as $|X (\\theta ^{\\prime }, \\phi ^{\\prime })\\rangle $ (Eq.", "(REF )), the freedom in choosing$\\theta , \\phi $ and $\\delta $ constructs a universal rotation of the exciton spin polarization." ], [ "Estimation of the phase shift and visibility of the control pulse action.", "Each one of the measurements displayed in Figs.", "2-3 of the Letter, was fitted to a functional of the form: $C \\cdot e^{-\\frac{t}{\\tau }} \\left[ 1 - V \\cdot \\cos \\left( \\frac{2 \\pi }{T} \\cdot \\Delta t + \\Delta \\phi \\right) \\right],$ where $\\tau $ , T and C are the exciton lifetime, its precession period and an overall normalization factor.", "The parameters $\\tau $ and T are accurately evaluated independently.", "The parameters V and $\\Delta \\phi $ are the visibility and the phase shift of the signal, respectively.", "These are extracted from the fits of Eq.", "(REF ) to each one of the traces in the three pulse experiments presented in Figs.", "3-5, for $\\Delta t > T$ .", "Here $\\Delta \\phi $ is measured relative to $\\phi = 0$ in the two-pulse experiments (lowest trace in Figs.", "3-5).", "The visibility of the control pulse is then normalized by the visibility of the two-pulse experiments (without the control pulse), which is slightly less than one [3]." ] ]

[ [ "Probing Colored Particles with Photons, Leptons, and Jets" ], [ "Abstract If pairs of new colored particles are produced at the Large Hadron Collider, determining their quantum numbers, and even discovering them, can be non-trivial.", "We suggest that valuable information can be obtained by measuring the resonant signals of their near-threshold QCD bound states.", "If the particles are charged, the resulting signatures include photons and leptons and are sufficiently rich for unambiguously determining their various quantum numbers, including the charge, color representation and spin, and obtaining a precise mass measurement.", "These signals provide well-motivated benchmark models for resonance searches in the dijet, photon+jet, diphoton and dilepton channels.", "While these measurements require that the lifetime of the new particles be not too short, the resulting limits, unlike those from direct searches for pair production above threshold, do not depend on the particles' decay modes.", "These limits may be competitive with more direct searches if the particles decay in an obscure way." ], [ "Introduction and Motivation", "The high collision energy available at the Large Hadron Collider (LHC) permits the exploration of a vast region of particle physics territory that was previous inaccessible.", "To assure sensitivity to the rich array of signatures that may arise from physics beyond the Standard Model, a diverse set of analysis techniques is needed.", "Already, limits on some particles have reached as high as 2 TeV and beyond.", "But in some cases, the limits are much weaker.", "New colored particles even as light as 200 GeV, which would have large production cross sections, are still not excluded, because of difficult backgrounds, trigger limitations, or the need for dedicated analysis methods.", "In this paper we will show that pair-produced colored particles from many Beyond-the-Standard-Model scenarios are constrained, in a way that is largely model-independent, by the non-observation of dijet (or other) resonances arising from their QCD bound states.", "In some cases the constraints obtained by this method would be stronger than the ones from the more standard searches.", "Furthermore, if new pair-produced colored particles are discovered, by any method, their bound state signals can provide a uniquely powerful tool for determining their quantum numbers, including charge, spin, color representation and multiplicity.", "With such quantum numbers determined, the bound state resonances then also provide a high-precision mass measurement.", "This approach is especially useful for particles in color representations higher than the triplet (so that the bound states have a sizeable cross-section) that also carry electric charge (so that multiple resonant signals may be observable, including $\\gamma \\gamma $ , $\\gamma $ +jet and $\\ell ^+\\ell ^-$ ).", "These ideas are hardly new, of course.", "Bound states were crucial in understanding charm and bottom quarks, and had the top quark been lighter than 130 GeV, measurements of toponium at $e^+e^-$ and hadron colliders (see, for example, [1], [2], [3], [4]) could have provided very detailed information about its properties.", "But it has been some time since these ideas were current, and so they need to be dusted off and shined for use at the LHC.", "This has been done recently in [5] and further in [6], where the focus was on bound states of particles that appear in theories of supersymmetry or extra dimensions, though other cases were briefly discussed.", "Utilizing bound state annihilation signals at the LHC has been considered also in [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18].", "In this work we explore these ideas in a much more general context and confront the potential signals with data from the 7 TeV LHC.", "The only essential model-dependent assumption that we make about the new particle $X$ is that its life is slightly prolonged, so that the $X\\overline{X}$ bound state (“$X$ -onium”) decays not by the disintegration of the $X$ itself (as in toponium) but by the annihilation of the $X$ with the $\\overline{X}$ (as in charmonium).", "This condition is satisfied relatively easily as long as the particle does not have any unsuppressed 2-body decays.", "For particles in higher representations of color this situation becomes even more generic because, as we will see, the annihilation rates are proportional to many powers of color factors and can be enhanced by orders of magnitude.", "Particles in the triplet and octet representations of color appear in many common scenarios and our results will be relevant to many of them.", "We will also consider particles in more exotic representations, for some of which the signals we discuss can be unusually large.", "(Other aspects of the phenomenology of massive vectorlike matter in exotic representations have been considered in the past in [19], [20], [21], [22], [23], [24], [25], [26], [27], [28].)", "Our motivations for discussing such representations are several.", "First, searches for resonances will be done anyway, and can be interpreted in terms of any representations, so it makes sense to understand their implications for all available representations without imposing theoretical assumptions.", "Second, massive vector-like matter in other representations often arises both in perturbative string theory, our only top-down model framework, and quantum field theory models with strong dynamics (or their dual warped-extra-dimensional cousins).", "In fact, string theory typically predicts extra matter that, though chiral at high energies, becomes vector-like after symmetry breaking which may well occur at or just above the weak scale.", "The types of matter commonly predicted by string theory are biased by the fact that strings have two ends, and each end carries an index under a gauge or flavor group.", "This means that matter in perturbative string theory tends to appear in a limited set of representations with two indices, namely adjoints, symmetric and antisymmetric tensors, bifundamentals, and fundamentals.", "At the least, therefore, one should consider the possibility of massive sextet representations of color.", "Meanwhile, if non-perturbative quantum field theory has a role to play near the weak scale, composite particles in more complicated representations may arise through new confining dynamics.", "In QCD we encounter massive octet and decuplet states of flavor-$SU(3)$ (with $U(1)$ -baryon charge) around the scale of confinement; in a similar vein, we might easily find technibaryons or other confined states near the TeV scale in the octet or decuplet of color, and carrying $U(1)$ hypercharge.", "Furthermore, the two-index tensor representations expected in perturbative string theory might be combined through strong dynamics into more elaborate massive bound states.", "This can occur in confining supersymmetric quantum field theories, where massless states with high representations of global symmetries (into which $SU(3)$ -color may be embedded) sometimes arise.", "In section  we will set the stage for our study by discussing the various colored particles that we will consider and the degree to which they are constrained by current data.", "In section  we will review the basics of the bound state formalism.", "In section  we will analyze the bound state annihilation signals in the $\\gamma \\gamma $ , $\\gamma $ +jet, dijet and $\\ell ^+\\ell ^-$ channels.", "In section  we will comment on the widths of the resonances.", "In section  we will present a strategy that combines the information from the various channels for determining the properties of the new particles.", "We will summarize and give final remarks in section ." ], [ "New Colored Particles: Preliminary Considerations", "Since the properties of the bound state signals are commonly model-independent, we will not restrict ourselves to a particular new physics scenario.", "We will simply assume that nature contains a new pair-produced particle $X$ of spin $j=0$ , $\\frac{1}{2}$ or 1, electric charge $Q$ , and color representation $R$ (details of various color representations are given in table REF ).", "For simplicity and brevity, we will assume that $X$ is a singlet of the electroweak $SU(2)$ , though our methods apply also to other $SU(2)$ representations.", "We also assume that the only interactions that contribute to production and annihilation of $X\\overline{X}$ pairs, and the $X\\overline{X}$ binding potential, are the Standard Model gauge interactions.Our results can be significantly modified for particles which have large Yukawa-type interactions, such as gluinos (when the squarks are not much heavier) [5], [6] or fourth-generation quarks [29].", "Note, however, that some of the representations that we study here cannot have any renormalizable non-gauge interactions with Standard Model particles, simply because their quantum numbers do not allow it.", "Even when other renormalizable couplings are permitted, any subsequent effects are competing against the large value of $\\alpha _s$ , and so our formulas for production and decay rates will often be accurate even in this case.", "Then, at least in the case of $j=0$ or $\\frac{1}{2}$ particles, everything is fixed by gauge invariance.", "The $j=1$ case is a bit more subtle and discussed in appendix .", "Table: Properties of SU(3)SU(3) representations , , ordered by their quadratic Casimir C R C_R.", "Also shown are the dimension D R D_R, the index T R T_R, the anomaly coefficient A R A_R, and the triality t R t_R.", "The last column shows the representations obtained by combining the representation RR with its complex conjugate (where representations that give rise to non-binding potential between the particles are enclosed in parentheses).", "Our conventions and useful SU(3)SU(3) identities are given in appendix ." ], [ "Constraints on quantum numbers", "New stable colored particles would form exotic atoms that are not observed [33], so we will focus on particles $X$ that are able to decay down to known particles, possibly along with invisible new particles.", "This restricts the charge $Q$ to be an integer times $1/3$ and imposes a correlation between $Q$ and the triality $t_R$ of $R$ [which can be computed from (REF )].", "For any representation $R$ in the decomposition of $R_1\\otimes R_2$ , $t_R = \\left(t_{R_1} + t_{R_2}\\right)\\,\\mbox{mod}\\,3$ and thus triality is conserved mod 3.", "Quarks ($R={\\bf 3}$ ; $Q=-1/3$ or $+2/3$ ), antiquarks ($R={\\bf \\overline{3}}$ ; $Q=+1/3$ or $-2/3$ ) and gluons ($R={\\bf 8}$ ; $Q=0$ ) all have charge and triality that satisfy $t_R = (-3Q)\\,\\mbox{mod}\\,3$ as do the colorless particles of the Standard Model, and any new invisible particles.", "Consequently, for $X$ to be able to decay, it must also satisfy (REF ).", "Particles $X$ in representations with vanishing triality (including $\\mathbf {8}$ , $\\mathbf {10}$ , and $\\mathbf {27}$ ) therefore must have integral charges, while particles in other representations (including $\\mathbf {3}$ , $\\mathbf {6}$ , $\\mathbf {15}$ , $\\mathbf {24}$ , $\\mathbf {15^{\\prime }}$ ) must have fractional charges which are multiples of $1/3$ .", "Spin $j$ is unconstrained by these considerations, since three quarks can have the quantum numbers of a neutron ($j=1/2$ ; $R={\\bf 1}$ ; $Q=0$ ) allowing a change of spin by a half-integer with no change in charge or color.", "One might consider constraining the list of particles by requiring that the running of the Standard Model gauge couplings should not be too severely affected.", "High representations in $j$ and $R$ can have a huge impact on the running of $\\alpha _s$ , and (for $Q\\ne 0$ ) electroweak couplings.", "Requiring asymptotic freedom in $SU(3)$ would preclude some of the representations we consider.", "However, if $X$ were composite, then above its confinement scale the beta functions would change again, and asymptotic freedom might be restored.", "Alternatively, in a sufficiently exotic scenario with a whole set of new particles, including vectors as well as scalars or fermions, their effects on the beta function may partially cancel.", "Similar remarks apply to the potential contributions of the particles to loops affecting precision measurements.", "Lacking a clear-cut criterion, we leave these considerations to the reader's judgement, and impose no constraint of our own." ], [ "Restrictions from collider searches", "Let us now consider the direct constraints on new colored particles from various collider measurements.", "These may be especially stringent for particles in high representations in $j$ and $R$ , which have large pair-production cross-sections,Since we assume $X$ has a small width, resonant production of $X$ is too rare to be important.", "as shown in figures REF –REF (based on the expressions in appendix ).We assumed the fields $X$ to be complex.", "Electrically-neutral fields (of any spin) in the $\\mathbf {8}$ or $\\mathbf {27}$ can be real, in which case the cross sections would be smaller by a factor of 2.", "However, existing limits depend very significantly on the particles' decay channels, while our aim is to remain as model-independent as possible.", "We will discuss several examples but will be cautious in applying constraints except where obviously necessary.", "Figure: Tevatron pair-production cross sections for particles of mass mm and spin j=0j=0 (solid black), 1 2\\frac{1}{2} (dashed blue) or 1 (dotted red) in color representations RR indicated in the figure.", "The 95% CL exclusion limit on particles decaying to 3 jets is the thick blue curve (CDF , , , 3.23.2 fb -1 ^{-1}).Figure: 7 TeV LHC pair production cross sections for particles of mass mm and spin j=0j=0 (solid black), 1 2\\frac{1}{2} (dashed blue) or 1 (dotted red) in color representations RR indicated in the figure.", "The 95% CL exclusion limits on particles decaying to 2 jets are the thick red curve (ATLAS with 4.6 fb -1 ^{-1}  for m>150m > 150 GeV, 34 pb -1 ^{-1}  for lower masses) and the thick pink curve (CMS with 2.2 fb -1 ^{-1} ).", "The limit on particles decaying to 3 jets is the thick blue curve (CMS with 5 fb -1 ^{-1} , for m>280m > 280 GeV, 35 pb -1 ^{-1}  for lower masses).Table: The minimal number of jets to which a spin-jj particle in a particular color representation can decay, accounting for color, charge and angular momentum conservation.", "As discussed in the text, the 8\\mathbf {8}, 10\\mathbf {10} and 27\\mathbf {27} must have integer electric charges while the others' must be fractional.", "For particular values of the charge, the minimal number of jets will sometimes be higher than shown in the table.", "For the 10\\mathbf {10} or 27\\mathbf {27}, j=0j=0, any non-zero charge requires the particle to decay to at least three jets, as indicated in parentheses.One signature particularly relevant to particles in high representations of color is decays to jets only.", "In table REF is shown the minimal number of quarks, antiquarks and/or gluons to which a particle $X$ with spin $j$ and color representation $R$ could decay, consistent with conservation of color, angular momentum and charge.", "Notice that a high-dimensional operator is sometimes required for the corresponding decay, automatically suppressing decays with easier signatures (except possibly for decays involving top quarks), and at the same time giving a sufficiently long lifetime for bound states to annihilate.If the lifetime of $X$ is very large, then when produced above threshold it may travel a macroscopic distance in the detector (within an R-hadron).", "While our results would still apply, the direct observation of such particles would typically make discovery and measurement of quantum numbers much easier (see, e.g., [43] and references therein) and constraints on them are already very powerful [44], [45].", "Note that this is neither the best nor the worst case as far as sensitivity to the new particles goes.", "For instance, if the final state contains invisible particles and consequently a lot of missing transverse momentum, discovery would likely have occurred through searches aimed at supersymmetry, while if the new particles decay to jets plus a new colorless particle that itself typically decays to three jets, it is not obvious whether any searches would be sensitive other than those seeking black holes and similar phenomena.", "Signatures with jets only, and no missing energy, have been traditionally considered hard due to the large QCD background.", "Nevertheless, several successful search techniques for pair-produced hadronically decaying particles have been developed recently, starting with searches for particles decaying to 3 jets by CDF [34], [35], [36] ($3.2$  fb$^{-1}$ ) and CMS [42] (35 pb$^{-1}$ , recently updated to 5 fb$^{-1}$  [40]The article [40] presents the limit curve only up to $m = 1000$  GeV, but a more complete curve (up to 1600 GeV) has been presented in [41].", "), and followed by searches for particles decaying to 2 jets by ATLAS [38] (34 pb$^{-1}$ , recently updated to $4.6$  fb$^{-1}$  [37]) and CMS [39] ($2.2$  fb$^{-1}$ ).", "Estimates of their 95% CL exclusion limits are shownFor presenting the results of [39], we assumed the acceptance to be the same as for the coloron examined in [39].", "We determined the acceptance by dividing the acceptance-times-cross-section curve from [39] by the cross section of the model used in that study [46].", "This is consistent with the two values quoted in [39] ($3\\%$ for $m = 300$  GeV and $13\\%$ for $m = 1000$  GeV).", "as thick curves in figures REF –REF .", "These limits are likely to be conservative since the cross sections we plot do not include higher-order QCD corrections (which for many cases are not yet available in the literature) which would typically make them bigger.One correction that will be especially important for the high color representations is the Sommerfeld enhancement which, like the bound states, originates from the attractive QCD potential between the two particles (see, e.g., [47]).", "We have estimated that typically (e.g., for 1 TeV particles at the 7 TeV LHC or 200 GeV particles at the Tevatron) the enhancement factors relative to the leading-order cross sections would be $\\sim 1$ -2 for $X$ in the $\\mathbf {6}$ and $\\mathbf {8}$ representations, $\\sim 3$ for $\\mathbf {10}$ and $\\mathbf {15}$ , and $\\sim 5$ for $\\mathbf {27}$ , $\\mathbf {24}$ and $\\mathbf {15^{\\prime }}$ .", "However, since the searches for hadronically decaying particles typically require the particles to be somewhat boosted, so as to reduce QCD multijet background, they are not sensitive to pairs produced near the threshold, where the effect of the Sommerfeld enhancement is most pronounced.", "As a result, the number of events passing the cuts will be enhanced by smaller factors than those stated above.", "Addressing this in more detail requires a separate study.", "Meanwhile, the limits on spin-1 particles are conservative also, because we did not include the contribution from the $q\\overline{q}$ channel in that case since it is necessarily model-dependent and typically subdominant, as discussed in more detail in appendix .", "We should also note that the limits inferred from figures REF –REF do not take into account that the angular and invariant mass distributions of the pairs depend on the spin and color representation, so the acceptance varies somewhat between the different cases.", "For some of the representations the cross sections are so large that they are likely to be highly constrained even without dedicated searches, by the absence of anomalies in high-multiplicity events.", "Those have been explored recently in the CMS black holes search [48] (with 4.7 fb$^{-1}$ ).", "While a careful interpretation of the limits from this search would require a detailed analysis, an examination of the backgrounds suggests that such an analysis would likely exclude certain regions, roughly within the 1000–1500 GeV mass range, for all the particles from table REF decaying to 4 jets (spin-$\\frac{1}{2}$ particles in the $\\mathbf {27}$ , and spin-0 and 1 particles in the $\\mathbf {15^{\\prime }}$ ) and for some of the particles decaying to 3 jets (spin-1 particles in the $\\mathbf {15}$ , $\\mathbf {10}$ , $\\mathbf {27}$ , and $\\mathbf {24}$ , and spin-$\\frac{1}{2}$ particles in the $\\mathbf {24}$ and $\\mathbf {15^{\\prime }}$ ).", "Note from figure REF that the resulting limits on these 3-jet cases would be complementary to those from the dedicated search for particles that decay this way.", "The dedicated 3-jet search [40], [41], which is motivated by models with low signal-to-background ratio, needs to use hard cuts for reducing the background, resulting in signal acceptance of 2-3%.", "However, for signals that are larger than the background, or at least comparable to the uncertainty on the background, such cuts are no longer needed and limits can be set by inclusive searches like [48] even when the number of signal events is small.", "Also, for sufficiently large representations, the new particles will alter the overall dijet spectrum through large loop corrections.", "But the most prominent effect of these loops is likely to be below threshold, where they are enhanced by Coulomb effects and manifest themselves through the production and annihilation of bound states.", "Since the contribution from the bound states is resonant, and benefits from higher parton luminosity, it is probably the source of the most powerful limits arising from the dijet spectrum.", "Limits of this type will be discussed below.", "Overall, we see that the direct limits on hadronically decaying particles are not extremely strong, sometimes even for particles in high representations, and apply only in certain mass ranges for each specific decay mode.", "We will see that bound state signals may sometimes set limits of comparable strength which are less model-dependent.", "We also note that the direct searches for particles decaying to 3 jets suffer from uncertainties due to the very hard cuts needed for reducing the QCD background, leading to strong sensitivity to the far tails of the kinematic distributions which are difficult to simulate reliably.For example, the cross section of a hadronically decaying top which can be inferred from the CDF search for particles decaying to 3 jets [34], [35], [36] appears larger than expectation by a factor of $\\sim 10$ .", "While the excess can be accounted for by a $2\\sigma $ fluctuation or indicate a new physics source of boosted top quarks, the discrepancy may also be attributed to underestimating the systematic uncertainty on the very small signal efficiency ($\\sim 10^{-4}$ ).", "By contrast, the bound state signals have $\\mathcal {O}(1)$ efficiencies." ], [ "Bound State Formalism", "Any new colored particle will produce a rich spectrum of bound states, similarly to the charmonium system.", "Often there will be even more variety, since particles in any representation other than the triplet can form bound states in more than one representation of color, and spin-1 particles provide additional possibilities for the spins of the bound states.", "However, for each of our signals, just one or a few of the bound states will contribute significantly to observable LHC signals.", "In particular, the probability to produce a bound state depends on the size of its wavefunction at the origin, $\\psi (\\mathbf {0})$ , which is non-vanishing only for S waves.", "For any particular annihilation channel, the spins and the color representations of the bound states that need to be considered are further constrained.", "Luckily, as we will see, in most cases the contribution from the relevant S waves is not suppressed in any way, so we will not need to take into account higher angular momentum states or radiative transitions between states.", "Only the dilepton channel will require a more detailed analysis.", "For simplicity, in the rest of this introductory section we will focus on the case of directly produced S-wave bound states, and discuss the additional states relevant to the dilepton channel when we come to it.", "We will start by describing the leading-order expressions that we will be using and then comment on higher-order corrections, which become more and more important as we go higher in the color representation $R$ ." ], [ "Leading order", "For particles of mass $m \\gg \\Lambda _{\\rm QCD}$ , and as long as the Bohr radius of the relevant bound state is much smaller than the QCD scale and the velocity of its constituents is non-relativistic, we can obtain reasonable estimates using a standard modified-hydrogenic approximation.", "For a particle $X$ in representation $R$ , the potential between $X$ and $\\overline{X}$ depends on the color representation ${\\cal R}$ of the $X\\overline{X}$ pair through the quadratic Casimirs of $R$ and ${\\cal R}$ as $V(r) = -C\\frac{\\overline{\\alpha }_s}{r}\\,, \\qquad \\quad C = C_R - \\frac{1}{2} C_{{\\cal R}}$ Here $\\overline{\\alpha }_s \\equiv \\alpha _s(r_{\\rm rms})$ is defined as the running coupling at the scale of the average distance between the two particles in the corresponding hydrogenic state, $r_{\\rm rms} \\equiv \\sqrt{\\langle r^2\\rangle }$ , which is of the order of the Bohr radiusMore precisely, $r_{\\rm rms} = \\sqrt{3}\\,a_0$ for the S-wave ground state and $\\sqrt{30}\\,a_0$ for the lowest P-wave state.", "Since $a_0$ itself depends on $\\overline{\\alpha }_s$ , we determine $r_{\\rm rms}$ numerically using a self-consistency condition.", "$a_0 = 2/(C\\overline{\\alpha }_s m)$ .", "(The symbol $\\alpha _s$ without a bar will be reserved for its value at the scale $m$ .)", "The binding energies and the wavefunctions at the origin for the ground state $(n=1)$ and its radial excitations ($n=2,3,\\ldots $ ) are given by $E_b = -\\frac{1}{4n^2}\\, C^2\\overline{\\alpha }_s^2 m \\,,\\qquad \\left|\\psi (\\mathbf {0})\\right|^2 \\equiv \\frac{1}{4\\pi }\\left|R(0)\\right|^2 = \\frac{C^3 \\overline{\\alpha }_s^3 m^3}{8\\pi \\,n^3}$ and the cross-section for the bound state $\\mathcal {B}$ to be produced by initial-state partons $a$ and $b$ is $\\hat{\\sigma }_{ab\\rightarrow \\mathcal {B}}(\\hat{s})= \\frac{8\\pi }{m}\\,\\frac{\\hat{\\sigma }_{ab\\rightarrow X\\overline{X}}^{\\rm free}(\\hat{s})}{\\beta (\\hat{s})}\\, |\\psi (\\mathbf {0})|^2\\,2\\pi \\,\\delta (\\hat{s}-M^2)$ where $M = 2m + E_b$ is the mass of the bound state, $\\hat{\\sigma }_{ab\\rightarrow X\\overline{X}}^{\\rm free}(\\hat{s})$ is the production cross section for a free pair at threshold (i.e., for $\\beta (\\hat{s}) \\rightarrow 0$ , where $\\beta (\\hat{s})$ is the velocity of $X$ or $\\overline{X}$ in their center of mass frame).", "(If $X$ is real, then $\\overline{X}=X$ , and eq.", "(REF ) still holds with $\\psi (\\mathbf {0})$ defined through the expression (REF ), rather than by the appropriately symmetrized wave-function.)", "Meanwhile one may show that the production cross section of any narrow resonance $\\mathcal {B}$ of mass $M$ and spin $J$ from $a$ and $b$ , and the decay rate back to $a$ and $b$ , are related by $\\hat{\\sigma }_{ab\\rightarrow \\mathcal {B}}(\\hat{s}) = \\frac{2\\pi \\left(2J+1\\right) D_{\\mathcal {B}}}{D_a D_b}\\,\\frac{\\Gamma _{\\mathcal {B}\\rightarrow ab}}{M}\\;2\\pi \\,\\delta (\\hat{s} - M^2)\\qquad \\left(\\,\\times 2\\;\\mbox{ for } a = b \\,\\right)$ where $D_p$ denotes the dimension of the color representation of particle $p$ .", "From (REF ), we may see that the binding energies would typically be small relative to the bound state mass.", "Even for high representations like the $\\mathbf {15}$ and $\\mathbf {10}$ , the binding energy for a color-singlet S-wave bound state in its ground state is only $E_b \\sim -0.05 M$ .", "It is even smaller for bound states with non-zero orbital angular momentum and/or color.", "Therefore observing the splittings between the various states would generally be difficult at the LHC due to resolution limitations, except in the case of very high representations.", "For $R = \\mathbf {10}$ and $\\mathbf {15}$ , observation of radially excited states distinctly from the ground state might barely be possible in diphoton and dielectron channels, but the rates for the excited states are very low and a very large data set would be required.", "Furthermore, annihilation rates fall like $1/n^3$ , making radially excited states more vulnerable to decays of the constituent $X$ particles themselves, which could eliminate their annihilation signals.", "For these reasons, we will assume in this paper that no information can be gleaned from the presence of several distinct states in the same channel.", "When combined with detector resolution, the effect of the radially excited states, whose production rate falls as $1/n^3$ , will be a small distortion on the high-energy side of the observed resonance shape, increasing the area under the observed resonance by only $\\zeta (3)=\\sum _{n=1}^\\infty \\frac{1}{n^3} \\approx 1.2$ Since this effect is small and, as mentioned above, the radial excitations are more vulnerable to constituent decays, we will take the conservative approach of not including the factor (REF ) in the cross sections." ], [ "The importance of higher order corrections", "The computations in this paper will all be at the leading order both in the short-distance production and annihilation amplitudes and in the non-relativistic and Coulomb-like treatment of the bound states,We note that higher order results have been computed in the context of supersymmetric theories for bound states of color-triplet scalars (squarks) [49], [50] and color-octet fermions (gluinos) [51], [17].", "with the exception of the running of $\\alpha _s$ which we do account for to a degree, as described above, and of our use of NLO parton distribution functions (PDFs).", "We must therefore emphasize that our results will be rather imprecise.", "Our aim in this paper is not high precision, but rather to show that bound state searches may serve as a useful tool at the LHC, and to motivate future calculations of greater precision that will allow maximal information to be extracted from those searches.", "We will also demonstrate, however, that high precision is often not required for determining discrete quantum numbers.", "The leading order approximation becomes less valid as we go up in the representations.", "In particular, the particle velocity in the ground state of a color-singlet S-wave bound state is $v \\approx C_R\\overline{\\alpha }_s/2$ , where $C_R$ is the second Casimir of the representation $R$ , and thus for too high representations the bound states cannot be treated non-relativistically.", "The relativistic corrections contribute at the next-to-next-to-leading order in the $\\overline{\\alpha }_s$ expansion.", "Other corrections that depend on the combination $C_R\\overline{\\alpha }_s$ also start appearing at that order.", "For more details, see [17], which studied the effect of such corrections on a color-singlet S-wave bound state of $R=\\mathbf {8}$ fermions (gluinos).", "In their case, the corrections make $|\\psi (\\mathbf {0})|^2$ larger than the leading-order expression by a factor that varies from $1.8$ for $m=300$  GeV to $1.5$ for $m=1500$  GeV.", "Based on the values of $C_R$ from table REF , we expect a comparable effect for $R = \\mathbf {6}$ and a larger effect for $R = \\mathbf {15}$ and $\\mathbf {10}$ (where $C_R$ is about twice as large).", "For $R = \\mathbf {27}$ , $\\mathbf {24}$ and $\\mathbf {15^{\\prime }}$ , $C_R\\overline{\\alpha }_s \\approx 1$ , so we cannot obtain reliable results for these (or higher) representations.", "Therefore in the following sections we will restrict our attention to $R = \\mathbf {3}$ , $\\mathbf {8}$ , $\\mathbf {6}$ , $\\mathbf {15}$ and $\\mathbf {10}$ .", "We will briefly describe what the leading-order expressions give for higher representations in section REF ." ], [ "Signals", "In this section we will identify the channels in which the bound state resonances would be most easily measurable and compute the corresponding cross sections as a function of the mass, color representation, charge and spin of the constituent particles.", "The strong and electroweak couplings of the particles allow them to annihilate to a pair of gauge bosons, each of which can be a gluon, photon or $Z$ .", "They can also annihilate to a pair of quarks through an $s$ -channel gluon or any pair of fermions through an $s$ -channel $\\gamma /Z$ .", "Since for simplicity we assumed the particles to be $SU(2)$ singlets, there are no processes involving $W$ bosons.", "The promising final states that we will analyze are diphoton ($\\gamma \\gamma $ ), photon+jet ($\\gamma g$ ), dijet ($gg$ and $q\\overline{q}$ ), and dilepton ($\\ell ^+\\ell ^-$ ).", "Annihilation to $\\gamma \\gamma $ , $\\gamma g$ or $gg$ is possible only for bound states with total angular momentum $J = 0$ or 2.", "Such bound states (color-singlet for $\\gamma \\gamma $ , color-octet for $\\gamma g$ , and various representations for $gg$ ) can only be produced in the $gg$ channel.", "Annihilation to $q\\overline{q}$ or $\\ell ^+\\ell ^-$ (through an $s$ -channel gauge boson) is possible only for $J = 1$ bound states.", "Color-octet $J = 1$ bound states which will annihilate to $q\\overline{q}$ will be produced in the $q\\overline{q}$ channel.", "On the other hand, there is no leading-order strong-interaction process that can produce the color-singlet $J = 1$ bound states which can annihilate to $\\ell ^+\\ell ^-$ .", "Therefore, we will have to take into account various subleading processes which will lead to a measurable $\\ell ^+\\ell ^-$ signal.", "The photon+jet, diphoton and dilepton signals will unfortunately be absent if the particles are neutral.", "Note though that according to the arguments in section REF , for many representations ($\\mathbf {3}$ , $\\mathbf {6}$ , $\\mathbf {15}$ , etc.)", "neutral particles would be exactly stable and are likely already excluded.", "In this section we will also present approximate current limits based on recent LHC searches.", "For simplicity, since the signal efficiencies (including acceptance) are $\\mathcal {O}(1)$ , and since the limits are still changing rapidly as the analyses are being updated with more data, we will only display one representative limit curve per search, without simulating the signal efficiency relevant to each of our cases separately.", "We will present the cross sections as a function of half the bound state mass, $M/2$ , which is roughly the mass $m$ of the constituent particle.In the dijet channel where the contribution comes from several bound states with different binding energies, we took $M$ to correspond to the lightest one.", "However, all the states are included in the computation of the cross section, each with its own mass, since they will usually contribute to the same peak within the experimental resolution." ], [ "$\\gamma \\gamma $ channel", "Any particle with charge and color, no matter what its spin, can be produced in pairs (in $gg$ collisions) in an S-wave $J=0$ color-singlet bound state that can then decay as a typically narrow $\\gamma \\gamma $ resonance.", "For spin-1 particles, S-wave $J=2$ color-singlet bound states contribute as well.", "Even though the dominant decay in all cases is back to a $gg$ pair, the $\\gamma \\gamma $ signal benefits from smaller backgrounds and better resolution.", "The cross section of the $\\gamma \\gamma $ signal due to a spin-$J$ bound state isThe $\\gamma \\gamma $ annihilation signal has been computed in the past for bound states of color triplets (see, e.g., [15], [49], [50], [6] for scalars, [4], [29], [6] for fermions, and [6] for vectors) and color octets (see, e.g., [16] for scalars).", "$\\sigma _{\\gamma \\gamma } \\simeq \\sigma _{gg\\rightarrow \\mathcal {B}}\\,\\frac{\\Gamma _{\\mathcal {B}\\rightarrow \\gamma \\gamma }}{\\Gamma _{\\mathcal {B}\\rightarrow gg}}= \\frac{\\left(2J+1\\right)C_R^3}{2^{14}}\\,\\overline{\\alpha }_s^3\\, \\frac{\\mathcal {L}_{gg}(M^2)}{M^2}\\int _0^\\pi d\\theta \\,\\sin \\theta \\,\\sum _{\\varepsilon _1,\\varepsilon _2}\\left|\\mathcal {M}_{X\\overline{X} \\leftrightarrow \\gamma \\gamma }(\\theta )\\right|^2$ Here ${\\cal M}_{X\\overline{X}\\leftrightarrow \\gamma \\gamma }$ is the matrix element for free $X$ and $\\overline{X}$ to convert to a pair of photons and the sum is over the photon polarizations.", "Also appearing in this expression is the parton luminosity for a pair of partons $a$ and $b$ , defined as $\\mathcal {L}_{ab}(\\hat{s}) = \\frac{\\hat{s}}{s}\\,\\int _{\\hat{s}/s}^1 \\frac{dx}{x}\\, f_{a/p}(x)\\, f_{b/p}\\left(\\frac{\\hat{s}}{xs}\\right) \\ ,$ where $\\sqrt{s}$ is the collider center-of-mass energy.", "The resulting $\\gamma \\gamma $ signal from the bound states is $\\sigma _{\\gamma \\gamma } = \\frac{Q^4 C_R^3 D_R}{64}\\,\\pi ^2\\alpha ^2 \\overline{\\alpha }_s^3\\, \\frac{\\mathcal {L}_{gg}(M^2)}{M^2}\\times \\left\\lbrace 1,\\, 2,\\, 19\\right\\rbrace \\qquad \\mbox{for $j=\\left\\lbrace 0, \\frac{1}{2}, 1 \\right\\rbrace $}$ We have evaluatedHere and in the following, we are using the NLO MSTW 2008 PDFs [52] evaluated at the scale $m$ .", "this for the LHC in figure REF , which also shows our estimate of the exclusion limit from ATLAS [53] and CMS [54], [55].Unfortunately the high-luminosity ATLAS and CMS searches [53], [55] have not presented limits for $M < 450$  GeV or $M < 500$  GeV, respectively (even though good data seems to be available down to at least $M \\approx 200$  GeV in both cases; and of course Higgs boson searches work at even lower masses).", "As a result, we could only present limits from the much earlier 36 pb$^{-1}$ CMS search [54] in that range of masses.", "We also could not make use of the CMS search with 2 fb$^{-1}$  [56] since it has not presented limits on the cross section times branching ratio.", "Figure: γγ\\gamma \\gamma signal cross sections at the 7 TeV (left) and 14 TeV (right) LHC for bound states of particles with spin j=0j=0 (solid black), 1 2\\frac{1}{2} (dashed blue) or 1 (dotted red) and charge Q=1Q = 1.", "For other values of QQ they need to be multiplied by Q 4 Q^4.", "Our estimates of the current 95%95\\% CL exclusion limits are the thick red curve (ATLAS, 4.9 fb -1 ^{-1} ), and the thick blue curve (CMS, 1.14 fb -1 ^{-1} ) which is supplemented for M/2<250M/2 < 250 GeV by a limit from a lower luminosity search (CMS, 36 pb -1 ^{-1} ).For $j=0$ or $\\frac{1}{2}$ , the signal is coming from $J=0$ bound states and is therefore isotropic.", "For $j=1$ , both $J=0$ and $J=2$ bound states contribute.", "The signal from $J=2$ bound states, which contributes $16/19$ of the cross section, will have the angular distribution $\\frac{1}{\\Gamma _{\\mathcal {B}\\rightarrow \\gamma \\gamma }}\\,\\frac{d\\Gamma _{\\mathcal {B}\\rightarrow \\gamma \\gamma }}{\\sin \\theta \\,d\\theta } = \\frac{5}{4}\\left(\\cos ^8\\frac{\\theta }{2} + \\sin ^8\\frac{\\theta }{2}\\right)$ where $\\theta $ specifies the direction of motion of the photons relative to the beam axis in the center of mass frame.", "This angular dependence should allow us to distinguish between bound states of spin-1 particles and those of spin-0 or spin-$1/2$ particles." ], [ "$\\gamma +$ jet channel", "For particles in color representations higher than the triplet, the potential is attractive also in the color-octet state.", "Color-octet bound states can decay to a photon and a gluon, producing a photon-jet resonance.", "Perhaps surprisingly, we find that these states can often be observable.", "For spin-0 and spin-1 constituents, these states are especially important, as di-lepton resonances are rarely produced, as we will see.", "Happily, measuring the rates for the two-photon and photon-jet resonances provides complementary information about the charge $Q$ and the representation $R$ .", "An octet state can be constructed from a pair of particles in any representation $R$ by using the color wavefunction $\\frac{1}{\\sqrt{T_R}} \\left(T_R^a\\right)^j_i$ where $\\left(T_R^a\\right)^j_i$ are the generators and $T_R$ in the prefactor is the index of the representation.", "For some representations there is more than one way to form an octet (e.g., for $R= \\mathbf {8}$ , one can use $d^{abc}$ as well as $f^{abc} \\propto \\left(T_R^a\\right)^{bc}$ ).", "However, the $\\gamma g$ matrix element turns out to be proportional to $\\left(T_R^a\\right)^j_i$ , so color octets formed with wavefunctions other than (REF ) do not contribute to the $\\gamma +$ jet signal.", "The matrix elements for production from $gg$ (for particles of any spin $j$ ) are proportional to $\\left\\lbrace T_R^b,T_R^c\\right\\rbrace _j^i$ which will be combined in a color trace with (REF ), so such color-octet states will be produced with a rate proportional (at leading order) to the square of the anomaly coefficient $A_R$ .", "Unless $A_R$ vanishes (as happens for $R = \\mathbf {8}$ ), the results from the previous section, for singlet states annihilating to two photons, carry over directly to this section, with very small adjustments.", "Our result (REF ) is simply modified by the replacements $\\ C_R^3 \\rightarrow 8 \\left(C_R - \\frac{3}{2}\\right)^3 \\ ,\\qquad Q^4\\, D_R\\, \\alpha ^2 \\rightarrow Q^2\\, T_R\\, \\alpha \\, \\alpha _s \\ ,$ The first step in the replacement stems from the change in the potential, which in the singlet state is proportional to $C_R$ and in the octet state to $C_R-\\frac{3}{2}$ ; see eq.", "(REF ).", "This propagates, cubed, into the squared wave function at the origin, $|\\psi (\\mathbf {0})|^2$ .", "Finally there is a factor of 8 due to the octet multiplicity.", "The second part of the replacement rule arises from the differences in the matrix elements (contracted with the color wavefunctions), which for $\\gamma \\gamma $ are proportional to $e^2 Q^2\\, \\delta ^i_j \\cdot \\frac{\\delta ^j_i}{\\sqrt{D_R}} = e^2 Q^2 \\sqrt{D_R}$ and for $\\gamma g$ to $e Q\\, g_s \\left(T_R^b\\right)^i_j \\cdot \\frac{1}{\\sqrt{T_R}}\\left(T_R^a\\right)^j_i = eQ\\, g_s \\sqrt{T_R}\\,\\delta ^{ab}$ We thus get $\\sigma _{\\gamma g} = \\frac{Q^2 \\left(C_R-\\frac{3}{2}\\right)^3 T_R}{8}\\,\\pi ^2 \\alpha \\, \\alpha _s\\, \\overline{\\alpha }_s^3\\, \\frac{\\mathcal {L}_{gg}(M^2)}{M^2} \\times \\left\\lbrace 1,\\, 2,\\, 19\\right\\rbrace \\qquad \\mbox{for $j=\\left\\lbrace 0, \\frac{1}{2}, 1 \\right\\rbrace $}$ As before, the factor of 19 involves 3 from the $J=0$ state and 16 from the $J=2$ state.", "The resulting LHC cross sections are shown in figure REF , which also shows our estimate of the exclusion limit from ATLAS [57].No limits on narrow $\\gamma g$ resonances have been presented as yet.", "Such resonances can be simulated by using FeynRules [58] to create a UFO model [59] which can be provided as an input to MadGraph 5 [60].", "To simulate the search [57], we showered the events in Pythia 8 [61], applied the photon isolation criterion, jet algorithm, photon reconstruction efficiency of $85\\%$ , and all the cuts.", "To obtain the exclusion curve, we used the limit of [57] on generic Gaussian-shape peaks following the procedure explained in the appendix of the analogous dijet search [62], including cutting the non-Gaussian tail, adjusting the mass, and determining the width of the effective Gaussian.", "We found the acceptance (times efficiency) to be $\\approx 33\\%$ and the width (dominated by showering, but including also the detector resolution from [57]) to vary between 8% at low masses and 7% at high masses.", "We used the limit from [57] which assumes 7% width.", "The angular distributions of the annihilation products are the same as in the $\\gamma \\gamma $ channel.", "Figure: γ+\\gamma +jet signal cross sections at the 7 TeV (left) and 14 TeV (right) LHC for bound states of particles with spin j=0j=0 (solid black), 1 2\\frac{1}{2} (dashed blue) or 1 (dotted red) and charge Q=1Q = 1.", "For other values of QQ they need to be multiplied by Q 2 Q^2.", "The thick red curve is our estimate of the 95%95\\% CL exclusion limit from ATLAS  (with 2.11 fb -1 ^{-1})." ], [ "Dijet channel", "For particles with any spin and in any color representation, there are S-wave bound states (with $J=0$ and, for $j=1$ , also $J=2$ ) produced via $gg \\rightarrow \\mathcal {B}$ and annihilating mostly back to $gg$ .", "These include the color-singlet and color-octet bound states discussed in the previous two subsections, as well as bound states in the $\\mathbf {27}$ .", "The result is a resonant dijet signal.", "For $j=\\frac{1}{2}$ there is also a comparable contribution from S-wave $J=1$ color-octet bound states produced via $q\\overline{q} \\rightarrow \\mathcal {B}$ and annihilating to $q\\overline{q}$ .", "The annihilation produces all quark flavors equally (unless limited by phase space), so $1/6$ of the cases are actually $t\\overline{t}$ rather than ordinary jets.", "The squared matrix element for pair production of $X\\overline{X}$ from $gg$ at the threshold is proportional to $|\\mathcal {M}|^2 \\propto \\mbox{Tr}\\left(\\left\\lbrace T_R^a, T_R^b\\right\\rbrace \\left\\lbrace T_R^c, T_R^d\\right\\rbrace \\right)$ where $a,b$ and $c,d$ are the color indices of the gluons from $\\mathcal {M}$ and $\\mathcal {M}^\\ast $ , respectively, and as usual $R$ is the representation of $X$ .", "The $X\\overline{X}$ bound state in a representation ${\\cal R}$ can only be produced when the gluon pairs are each in that representation.", "We may project the gluon pairs onto the representation ${\\cal R}$ using projection operators given in eq.", "(B.15) of [63].", "Direct calculation yields $P_\\mathbf {1}^{abcd}\\,\\mbox{Tr}\\left(\\left\\lbrace T_R^a, T_R^b\\right\\rbrace \\left\\lbrace T_R^c, T_R^d\\right\\rbrace \\right)&=& \\frac{1}{2}C_R^2 D_R \\\\P_\\mathbf {8_S}^{abcd}\\mbox{Tr}\\left(\\left\\lbrace T_R^a, T_R^b\\right\\rbrace \\left\\lbrace T_R^c, T_R^d\\right\\rbrace \\right)&=& \\frac{4}{5}\\left(C_R + \\frac{3}{4}\\right)C_R D_R \\\\P_\\mathbf {27}^{abcd}\\,\\mbox{Tr}\\left(\\left\\lbrace T_R^a, T_R^b\\right\\rbrace \\left\\lbrace T_R^c, T_R^d\\right\\rbrace \\right)&=& \\frac{27}{10}\\left(C_R - \\frac{4}{3}\\right)C_R D_R$ with projection onto $\\mathbf {8_A}, \\mathbf {10}, \\mathbf {\\overline{10}}$ giving zero.", "Figure: Dijet signal cross section at the Tevatron (top), 7 TeV LHC (bottom left) and 14 TeV LHC (bottom right) for bound states of particles with spin j=0j=0 (solid black), 1 2\\frac{1}{2} (dashed blue) or 1 (dotted red).", "For the Tevatron, the thick blue curve is the limit from CDF  (with 1.13 fb -1 ^{-1}).", "For the 7 TeV LHC, the thick red curve is the limit from ATLAS  (with 4.8 fb -1 ^{-1}), supplemented for M/2<650M/2 < 650 GeV by limits from earlier ATLAS searches with lower luminosity , (with 1 fb -1 ^{-1} and 36 pb -1 ^{-1}, respectively), and the thick blue curve is the limit from CMS  (with 5 fb -1 ^{-1} for M/2>500M/2 > 500 GeV, and 0.13 fb -1 ^{-1} analyzed with a special technique for lower masses).From this and table REF we see that there will be color-singlet bound states for particles in any representation $R$ , color-octet bound states for all representations except for $\\mathbf {3}$ , and color-$\\mathbf {27}$ bound states for all $R$ except for $\\mathbf {3}$ , $\\mathbf {8}$ and $\\mathbf {6}$ .", "Their production cross sections are all proportional to the easily-obtained production rate for the color singlet, and thus we havePrevious results for the $gg$ channel covered bound states of color triplets (see, e.g., [15], [49], [50], [6] for $j=0$ and [4], [29], [6] for $j=\\frac{1}{2}$ ) and color octets (see, e.g., [16] for $j=0$ ).", "$\\sigma ^{gg}_{jj,\\mathbf {1}} = \\frac{D_R C_R^5}{512}\\,\\pi ^2\\alpha _s^2 \\overline{\\alpha }_s^3\\, \\frac{\\mathcal {L}_{gg}(M^2)}{M^2}\\times \\left\\lbrace 1,\\, 2,\\, 19\\right\\rbrace \\quad \\mbox{for $j=\\left\\lbrace 0, \\frac{1}{2}, 1 \\right\\rbrace $}$ $\\sigma ^{gg}_{jj,\\mathbf {8}} = \\frac{D_R C_R\\left(C_R + \\frac{3}{4}\\right)\\left(C_R - \\frac{3}{2}\\right)^3}{320}\\,\\pi ^2\\alpha _s^2 \\overline{\\alpha }_s^3\\, \\frac{\\mathcal {L}_{gg}(M^2)}{M^2}\\times \\left\\lbrace 1,\\, 2,\\, 19\\right\\rbrace \\quad \\mbox{for $j=\\left\\lbrace 0, \\frac{1}{2}, 1 \\right\\rbrace $}$ $\\sigma ^{gg}_{jj,\\mathbf {27}} = \\frac{27 D_R C_R\\left(C_R - \\frac{4}{3}\\right)\\left(C_R - 4\\right)^3}{2560}\\,\\pi ^2\\alpha _s^2 \\overline{\\alpha }_s^3\\, \\frac{\\mathcal {L}_{gg}(M^2)}{M^2}\\times \\left\\lbrace 1,\\, 2,\\, 19\\right\\rbrace \\quad \\mbox{for $j=\\left\\lbrace 0, \\frac{1}{2}, 1 \\right\\rbrace $}$ For $j=\\frac{1}{2}$ and any $R$ except for $\\mathbf {3}$ , there is an additional contribution to the cross section from the $q\\overline{q}$ channel: $\\sigma ^{q\\overline{q}}_{jj,\\mathbf {8}} = \\frac{D_R C_R\\left(C_R - \\frac{3}{2}\\right)^3}{9}\\,\\pi ^2\\alpha _s^2 \\overline{\\alpha }_s^3\\, \\frac{\\sum _q\\mathcal {L}_{q\\overline{q}}(M^2)}{M^2}$ The total dijet cross sections (excluding the $t\\overline{t}$ contribution) for the 7 and 14 TeV LHC (as well as Tevatron) are shown in figure REF , which also shows our estimates of the exclusion limits from CDF [64], ATLAS [65], [62], [66] and CMS [67].For the Tevatron, we used the RS graviton limit of [64].", "For the 7 TeV ATLAS data we used the limit of [65] on a color-octet scalar resonance decaying to gluons.", "Since the mass range of this 4.8 fb$^{-1}$ search is limited from below by the triggers, we used the analogous limit from the earlier 1 fb$^{-1}$ search [62] for $M/2 < 650$  GeV, and the yet earlier 36 pb$^{-1}$ search [66] for $M/2 < 500$  GeV.", "In [66], a limit on a color-octet scalar was not available and we used the limit on signals which appear as Gaussians, following the instructions from the appendix of [62].", "For CMS [67], we used the limit on a resonance decaying to $gg$ (for $M/2 < 500$  GeV, the data were analyzed with a special approach adapted to the high event rate at low masses, using however only the last 0.13 fb$^{-1}$ ).", "For this CMS search, we used their quoted acceptance of ${\\cal A} \\approx 60\\%$ (for isotropic decays), while for the other searches we simulated a scalar $gg$ resonance in Pythia.", "For [65] and [62] we get ${\\cal A} \\approx 60\\%$ .", "For [66], ${\\cal A} \\approx 30\\%$ , the mean mass of the effective Gaussian is shifted down by $\\approx 8.5\\%$ relative to the real mass (in our figure, $M$ is the real mass), and its width (dominated by showering, but including detector resolution as well) is $\\approx 9\\%$ (so we used the limit of [66] on 10%-width resonances).", "For [64], ${\\cal A}$ varies between 50% at low masses and 75% at high masses.", "It would be beneficial to keep improving the LHC reach in the low-mass region (which is constrained very weakly by the Tevatron), which would involve working in a regime where the trigger is not fully efficient, using a prescaled trigger and/or storing reduced event content as has been done by CMS in [67].", "Note that unlike all the other signals studied in this paper, the dijet signal is present even if the particles are not charged.", "While we were assuming the fields $X$ to be complex, if $Q=0$ and the color representation is real (such as $\\mathbf {8}$ ), the field $X$ may be real.", "In such case the cross sections (REF )–(REF ) would be half as big.The resulting expressions would apply, for example, to bound states of gluinos, see e.g.", "[11], [6], or KK gluons [6], in the limit that the squarks or the KK quarks are heavy.", "The angular distributions of the annihilation products are the same as in the $\\gamma \\gamma $ channel, except for $j=\\frac{1}{2}$ particles where there is an additional contribution from the spin-1 bound states of the $q\\overline{q}$ channel with the angular distribution $\\frac{1}{\\Gamma _{\\mathcal {B}\\rightarrow q\\overline{q}}}\\,\\frac{d\\Gamma _{\\mathcal {B}\\rightarrow q\\overline{q}}}{\\sin \\theta \\,d\\theta } = \\frac{3}{8}\\left(1 + \\cos ^2\\theta \\right)$" ], [ "$\\ell ^+\\ell ^-$ channel", "Spin-$\\frac{1}{2}$ particles in any color representation can form color-singlet $J=1$ S-wave bound states analogous to the $J/\\psi $ and $\\Upsilon $ .", "The interesting signal of these bound states is their annihilation into a pair of leptons (via an $s$ -channel photon or $Z$ ).", "They cannot be produced directly from a pair of gluons or quarks (except by electroweak interactions), so we need to consider the various subdominant production processes.", "Besides the electroweak production from $q\\overline{q}$ , these include production from $gg$ in association with a gluon (or a photon or $Z$ ) and production via a radiative transition of a color-singlet or color-octet P-wave bound state (along with a soft photon or gluon).", "The P-wave bound states themselves are produced directly, although with a suppressed rate, from a pair of gluons.", "It turns out that any of these processes can be important, depending on the representation, charge and collider energy.", "This is illustrated in table REF for the case of $m = 1$  TeV.", "The ratios between the processes also vary strongly as a function $m$ , primarily because the different processes depend on different PDFs.", "And of course one must keep in mind that, like everything in this paper, the ratios are calculated at leading order and may change significantly upon including higher order corrections.", "The $p_T$ distribution of the bound state is an interesting observable, and it is sensitive to the relative contributions of the different production modes, which will have different $p_T$ spectra.", "In particular, the $gg\\rightarrow {\\cal B}g$ process provides a hard gluon against which the bound state can recoil, unlike the electroweak production or the radiative processes where a hard recoiling gluon can come only from initial state radiation.", "Consequently, a measurement of the $p_T$ distribution is sensitive to the quantum numbers of the state.", "But methods much more sophisticated than ours would be needed to predict this observable.", "We will now analyze the production mechanisms one by one, then compute the dilepton branching ratio of the bound state and plot the resulting signal, and at the end comment on the much smaller signals expected from bound states of spin-0 or spin-1 particles.", "Table: The fraction (in %\\%) contributed by each of the production mechanisms for color-singlet J=1J=1 S-wave bound states (ℬ\\mathcal {B}) of spin-1 2\\frac{1}{2} particles at the 7 TeV LHC (top table) and 14 TeV LHC (bottom table).", "The numbers are presented for bound states of SU(2)SU(2) singlets with Q=2 3Q = \\frac{2}{3} (left) or 2 (right) and m=1m = 1 TeV.", "Note these numbers change significantly as a function of mm as well as RR and QQ." ], [ "Electroweak production from $q\\overline{q}$", "Given that we mostly consider $X\\overline{X}$ states with mass far above the $Z$ , we can largely ignore the $Z$ mass to our level of approximation, and write the electroweak production cross section via an $s$ -channel photon or $Z$ as $\\sigma = \\frac{\\pi ^2}{108}\\, D_R C_R^3 Q^2\\,\\frac{\\alpha ^2 \\overline{\\alpha }_s^3}{\\cos ^4\\theta _W} \\left(17\\sum _{q=u,c} + 5\\sum _{q=d,s,b}\\right)\\frac{\\mathcal {L}_{q\\overline{q}}(M^2)}{M^2}$" ], [ "Production in association with a gauge boson", "Generalizing the results for $J/\\psi $ and $\\Upsilon $ (see, e.g., [68], [69]), the cross section for production in association with a gluon, $gg \\rightarrow \\mathcal {B}\\,g$ , is $\\sigma = \\frac{5\\pi }{192\\,m^2}\\,\\frac{A_R^2 C_R^3}{D_R}\\,\\alpha _s^3\\overline{\\alpha }_s^3 \\int _0^1 dx_1 \\int _0^1 dx_2\\; f_{g/p}(x_1)\\, f_{g/p}(x_2)\\; I\\left(\\frac{x_1 x_2 s}{M^2}\\right)$ where $I(x) = \\theta (x-1)\\left[\\frac{2}{x^2}\\left(\\frac{x+1}{x-1} - \\frac{2x\\ln x}{(x-1)^2}\\right) + \\frac{2(x-1)}{x(x+1)^2} + \\frac{4\\ln x}{(x+1)^3}\\right]$ Production in association with a photon, $gg \\rightarrow \\mathcal {B}\\,\\gamma $ , is described by the same expression with the replacement $\\frac{A_R^2}{D_R}\\,\\alpha _s\\rightarrow \\frac{3}{20}\\,D_R C_R^2\\, Q^2\\alpha $ Production in association with a $Z$ is given by the expression for the photon times $\\tan ^2\\theta _W$ ." ], [ "Production via color-singlet P-wave states", "Here we consider the possibility of production through the radiative decays of the lowest color-singlet P states, $\\,^3P_J^{(\\mathbf {1})}$ (a.k.a.", "$\\chi _J$ ) with $J = 0, 2$ : $gg \\rightarrow \\,^3P_J^{(\\mathbf {1})} \\rightarrow \\mathcal {B}\\,\\gamma $ The binding energy of these P waves is $E_b = -C_R^2\\,\\overline{\\alpha }_s^2 m/16$ , which is about 4 times smaller than of the S wave of interest.", "The rate of the radiative transitions from a P wave to an S wave is (see, e.g., [70]) $\\Gamma (\\,^3P_J^{(\\mathbf {1})} \\rightarrow \\mathcal {B}\\gamma ) = \\frac{4}{9}\\,Q^2\\overline{\\alpha }\\, E_\\gamma ^3\\left|\\langle R_S|r|R_P\\rangle \\right|^2\\sim \\frac{128}{6561}\\,Q^2 C_R^4\\, \\overline{\\alpha }\\,\\overline{\\alpha }_s^4 m$ where the last expression is very approximate, and presented only to illustrate parametric dependence.", "The reason is that different values of $\\overline{\\alpha }_s$ arise.", "In our plots below we evaluate the photon energy $E_\\gamma = E_{b,P} - E_{b,S}$ using $\\overline{\\alpha }_s$ defined self-consistently at the average radii of the P-wave and S-wave states, respectively (see footnote REF ).", "Meanwhile, we estimate the transition amplitude $\\left|\\langle R_S|r|R_P\\rangle \\right|^2 = \\frac{2^{17}}{3^9\\, C_R^2\\,\\overline{\\alpha }_s^2\\, m^2}$ using the average $\\overline{\\alpha }_s$ of the two states.", "The radiative transitions must compete with annihilation of the P-wave states to $gg$ , which have the rates $\\Gamma (\\,^3P_0^{(\\mathbf {1})} \\rightarrow gg) = \\frac{9}{8}D_R C_R^2\\,\\alpha _s^2\\frac{|R^{\\prime }_P(0)|^2}{m^4}= \\frac{3}{2048} D_R C_R^7\\,\\alpha _s^2\\overline{\\alpha }_s^5 m$ $\\Gamma (\\,^3P_2^{(\\mathbf {1})} \\rightarrow gg) = \\frac{3}{10}D_R C_R^2\\,\\alpha _s^2\\frac{|R^{\\prime }_P(0)|^2}{m^4}= \\frac{1}{2560} D_R C_R^7\\,\\alpha _s^2\\overline{\\alpha }_s^5 m$ We obtained these expressions from those of quarkonia (see, e.g., [70]) with appropriately generalized color factors.", "Since the P wavefunction vanishes at the origin, the matrix elements are proportional to its derivative, so the rates are suppressed by two additional powers of $\\overline{\\alpha }_s$ compared with the S-wave processes.", "For $Q \\approx 1$ , the branching ratios for the radiative transitions of the color-singlet P waves are $\\mathcal {O}(1)$ for $R = \\mathbf {3}$ , $\\mathbf {6}$ , and $\\mathbf {8}$ , but become much smaller for higher representations because the annihilation rates are proportional to more powers of the color factors.", "Similarly, the production cross sections of $^3P_J^{(\\mathbf {1})}$ are $\\sigma = \\frac{D_R C_R^7}{2^{13}}\\,\\pi ^2\\alpha _s^2\\overline{\\alpha }_s^5\\, \\frac{\\mathcal {L}_{gg}(M^2)}{M^2} \\times \\left\\lbrace \\frac{3}{4}\\,,\\, 1\\right\\rbrace \\quad \\mbox{for}\\quad J=\\left\\lbrace 0,\\, 2\\right\\rbrace $" ], [ "Production via color-octet P-wave states", "Finally we would like to consider the process $gg \\rightarrow \\,^3P_J^{(\\mathbf {8})} \\rightarrow \\mathcal {B}\\,g$ The binding energy of these color-octet P waves is $E_b = -(C_R - \\frac{3}{2})^2\\,\\overline{\\alpha }_s^2 m/16$ , which is more than 4 times smaller than for the color-singlet S wave.", "We need to derive the rate for the chromoelectric dipole transition of an octet to a singlet with the emission of a soft gluon.", "We can start with the rate of electric dipole transitions with the emission of a photon (see, e.g., [70]) $\\Gamma (i\\rightarrow f\\gamma ) = \\frac{4}{3}\\,Q^2\\,\\overline{\\alpha }\\, E_\\gamma ^3\\left|\\langle F|\\mathbf {r}|I\\rangle \\right|^2$ where $|I\\rangle $ and $|F\\rangle $ are the spatial wavefunctions of the initial state $i$ and the final state $f$ .", "We then make the replacement $eQ\\langle F|\\mathbf {r}|I\\rangle \\;\\rightarrow \\;g_s\\mbox{Tr}\\left(\\chi _F\\,T^a_R\\,\\chi _I^a\\right) \\langle F|\\mathbf {r}|I\\rangle $ where $\\chi _I^a$ and $\\chi _F$ are the color wavefunctions of the initial and final state and there is no summation over the color index of the initial state, $a$ .", "Substituting $\\chi _F = \\delta _{jk}/\\sqrt{D_R}$ (where $j,k$ are indices in the representation $R$ ) we have $\\Gamma (i\\rightarrow f g) = \\frac{4}{3}\\,\\overline{\\alpha }_s\\, E_g^3\\, \\frac{\\left|\\mbox{Tr}\\left(\\chi _I^a\\, T_R^a\\right)\\right|^2}{D_R}\\left|\\langle F|\\mathbf {r}|I\\rangle \\right|^2$ For the case of $\\chi _I^a \\propto T_R^a$ as in (REF ), we have $\\Gamma (i\\rightarrow f g) = \\frac{1}{6}\\,C_R\\,\\overline{\\alpha }_s\\, E_g^3\\left|\\langle F|\\mathbf {r}|I\\rangle \\right|^2$ while for any other possible color-octet wavefunction the result vanishes when $\\chi _I^a$ is contracted with $T_R^a$ .", "Specializing to $\\,^3P_J^{(\\mathbf {8})}$ (with $J = 0$ or 2) we get $\\Gamma (\\,^3P_J^{(\\mathbf {8})} \\rightarrow \\mathcal {B}g) = \\frac{1}{18}\\,C_R\\,\\overline{\\alpha }_s E_g^3\\, \\left|\\langle R_S^{(\\mathbf {1})}|r|R_P^{(\\mathbf {8})}\\rangle \\right|^2\\sim \\frac{16}{6561}\\, C_R^4 \\frac{\\left(C_R+\\frac{3}{2}\\right)^3\\left(C_R-\\frac{3}{2}\\right)^5}{\\left(C_R-\\frac{1}{2}\\right)^7}\\, \\overline{\\alpha }_s^5\\, m$ where again, as in (REF ), the last expression is very approximate.", "As in the discussion following (REF ), we evaluate the gluon energy $E_g$ using the values of $\\overline{\\alpha }_s$ self-consistently determined at the radii of the P-wave and S-wave states (see footnote REF ).", "For the explicit power of $\\overline{\\alpha }_s$ in the middle expression in (REF ) and for the transition amplitude involving $R_P$ and $R_S$ (the P and S radial wavefunctions) $\\left|\\langle R_S^{(\\mathbf {1})}|r|R_P^{(\\mathbf {8})}\\rangle \\right|^2 = \\frac{2^{17}\\,C_R^3\\left(C_R-\\frac{3}{2}\\right)^5}{3^9 \\left(C_R-\\frac{1}{2}\\right)^{10}\\overline{\\alpha }_s^2\\, m^2}$ we use the average $\\overline{\\alpha }_s$ of the two states.", "The annihilation rates of the P waves to $gg$ , which compete with their radiative transitions, can be obtained by appropriately modifying the various color factors in (REF )–(REF ): $\\Gamma (\\,^3P_0^{(\\mathbf {8})} \\rightarrow gg) = \\frac{5}{512} \\frac{A_R^2 \\left(C_R - \\frac{3}{2}\\right)^5}{D_R C_R}\\,\\alpha _s^2\\overline{\\alpha }_s^5 m$ $\\Gamma (\\,^3P_2^{(\\mathbf {8})} \\rightarrow gg) = \\frac{1}{384} \\frac{A_R^2 \\left(C_R - \\frac{3}{2}\\right)^5}{D_R C_R}\\,\\alpha _s^2\\overline{\\alpha }_s^5 m$ The branching ratios for the radiative transitions of the P waves are typically very close to 1 for $R = \\mathbf {8}$ , $\\mathbf {6}$ and $\\mathbf {15}$ , and somewhat smaller for $R = \\mathbf {10}$ , where the annihilation rates are comparable to the transition rate.", "The production cross sections of $^3P_J^{(\\mathbf {8})}$ , based on (REF )–(REF ), are $\\sigma = \\frac{5}{768} \\frac{A_R^2 \\left(C_R - \\frac{3}{2}\\right)^5}{D_R C_R}\\,\\pi ^2\\alpha _s^2\\overline{\\alpha }_s^5\\, \\frac{\\mathcal {L}_{gg}(M^2)}{M^2} \\times \\left\\lbrace \\frac{3}{4}\\,,\\, 1\\right\\rbrace \\quad \\mbox{for}\\quad J=\\left\\lbrace 0,\\, 2\\right\\rbrace $" ], [ "Dilepton branching ratio", "To determine the observable rate for the dilepton resonance, we need to know the branching fraction of the S-wave $J=1$ bound state into Standard Model fermions.", "For a $J=1$ bound state of fermions $X$ with charge $Q$ and hypercharge $Y$ , annihilation can proceed through a photon or $Z$ .", "Neglecting the mass of the $Z$ relative to the bound state mass, we can write the annihilation rate using $SU(2)\\times U(1)$ states, giving for any flavor of fermions $f_L, f_R$ $\\Gamma _{\\mathcal {B}\\rightarrow f\\overline{f}} = \\frac{n_c}{12}\\,D_R C_R^3\\sum _{\\sigma =R,L}\\left(\\frac{Y_{f_\\sigma } Y}{\\cos ^2\\theta _W} + \\frac{\\left(Q_{f_\\sigma }-Y_{f_\\sigma }\\right)\\left(Q-Y\\right)}{\\sin ^2\\theta _W}\\right)^2\\alpha ^2\\overline{\\alpha }_s^3 m$ where $n_c = 1$ for leptons and 3 for quarks.", "In this paper we restrict ourselves to $SU(2)$ singlets, so $Y = Q$ and only the first term appears.", "To compute the branching fractions, we must also compute the partial widths for annihilation to $ggg$ , $\\gamma gg$ , and $Zgg$ ; we believe there are no other decays with considerable rates.The bound state can annihilate to $Z$ +Higgs via a diagram with an $s$ -channel $Z$ .", "However, the rate is small.", "For example, for the Standard Model Higgs, in the limit $M \\gg m_H, m_Z$ , the rate to $ZH$ is equal to the total rate to fermion pairs times $3/80$ .", "By generalizing the quarkonium result (see, e.g., [73]), we obtain $\\Gamma _{\\mathcal {B}\\rightarrow ggg} = \\frac{5\\left(\\pi ^2-9\\right)}{27\\pi }\\,\\frac{A_R^2 C_R^3}{D_R}\\,\\alpha _s^3\\overline{\\alpha }_s^3 m$ The rate for $\\mathcal {B}\\rightarrow \\gamma gg$ is described by the same expression with the replacement $\\frac{A_R^2}{D_R}\\,\\alpha _s\\rightarrow \\frac{9}{20}\\,D_R C_R^2\\, Q^2\\alpha $ The rate for $\\mathcal {B}\\rightarrow Zgg$ is given by the expression for $\\gamma gg$ times $\\tan ^2\\theta _W$ .", "The branching ratio to any single flavor of leptons, which with just (REF ) would always be $12.5\\%$ , remains $\\sim 10\\%$ for $R = \\mathbf {3}$ and $\\mathbf {8}$ but becomes smaller for the other representations, sometimes even below $1\\%$ , depending on the charge, representation and mass.", "The resulting $\\ell ^+\\ell ^-$ signal (for any single flavor of leptons) is shown in figure REF , which also shows our estimates of the exclusion limits from ATLAS [71] and CMS [72].We plot the ATLAS limit on a narrow-width RS graviton ($k/\\overline{M}_{\\rm Pl} = 0.1$ ) and the CMS limit on a $Z^{\\prime }_{\\rm SSM}$ (whose width is $\\sim 3\\%$ ).", "In both cases we use the combined limit from $e^+e^-$ and $\\mu ^+\\mu ^-$ channels." ], [ "The cases of spin-0 and spin-1 particles", "Bound states of spin-0 or spin-1 particles are very unlikely to have a dilepton signal comparable to that of spin-$\\frac{1}{2}$ particles.", "Spin-0 particles cannot form S-wave spin-1 bound states, while the color-singlet S-wave spin-1 bound state of spin-1 particles has the quantum numbers $J^{PC} = 1^{+-}$ so it cannot decay to $\\ell ^+\\ell ^-$ (nor can it be easily produced).", "As a result, the $\\ell ^+\\ell ^-$ signal, primarily from spin-1 P waves, is likely to be several orders of magnitude smaller than for spin-$\\frac{1}{2}$ particles.", "First, the P-wave states will preferentially transition into S waves rather than annihilate to leptons (or anything else), in part because their annihilation rates are suppressed by the vanishing wavefunction at the origin.", "Second, the P waves have no large production modes.", "Their direct production cross section will be suppressed (relative to the S waves of the $j=\\frac{1}{2}$ case) because of their vanishing wavefunction at the origin.", "Meanwhile, production of P-wave states via a radiative transition is also suppressed.", "Radiation from a D wave is suppressed because the direct production of D waves is very rare, as the derivative of their wavefunction also vanishes at the origin.", "Meanwhile, although excited S waves are easier to produce and may also transition to the P waves, their annihilation rates are large, making radiative transitions rare.", "In summary, we do not expect to be able to see $\\ell ^+\\ell ^-$ resonances if $j=0$ or 1." ], [ "Widths", "Earlier, in section , we considered the splittings between the various states relative to the detector resolution.", "Here we will briefly consider the effects of the states' intrinsic widths.", "It is important to check that the widths $\\Gamma $ of the bound states are smaller than the binding energies $E_b$ , since otherwise the bound state's lifetime would be short compared to its orbital time, and rather than appearing as a distinct resonance in, for example, the $\\gamma \\gamma $ spectrum, it would merely distort the continuum enhancement of $\\gamma \\gamma $ production by $X\\overline{X}$ loops.", "It is also important experimentally to know whether the states are narrower or wider than the intrinsic resolution of the detectors.", "Besides the possibility that the constituents are short-lived (which we will not discuss here since the lifetime is very model-dependent), there is an irreducible contribution to the bound state width from the annihilation processes, which can be significant for bound states of particles in high color representations.", "The expressions for the widths of all the relevant bound states are derived in appendix .", "We find that $\\Gamma /|E_b| \\ll 1$ in all cases that we considered in the previous section, so we indeed have well-defined bound states below threshold.", "Now let's turn to the experimental resolutions.", "In the $\\gamma \\gamma $ channel, where the signal is coming from ${\\cal R}=\\mathbf {1}$ , $J=0,2$ bound states and the resolution in invariant mass is $\\sim 1\\%$ , the widths are negligible for the low representations, but become comparable to the resolution for $R=\\mathbf {10}$ and $\\mathbf {15}$ .", "In the $\\gamma $ +jet channel, where the signal is coming from ${\\cal R}=\\mathbf {8}$ , $J=0,2$ bound states and the experimental resolution is $\\sim 3\\%$ , the widths are negligible.", "In the dijet channel, where the signal is coming from all the possible representations $\\cal R$ and the resolution is $\\sim 5\\%$ , the widths are again negligible.", "In the $\\ell ^+\\ell ^-$ channel, where the signal is coming from ${\\cal R}=\\mathbf {1}$ , $J=1$ bound states (a fraction of which are produced from P waves), the widths are negligible relative to the resolution, which is $\\sim 1\\%$ for $e^+e^-$ and larger for $\\mu ^+\\mu ^-$ .", "In many cases the bound state is long-lived on the QCD scale, $\\Gamma \\lesssim \\Lambda _{\\rm QCD}$ .", "If this happens to colored bound states, they will hadronize with light quarks and gluons before annihilating.", "However, the $X\\overline{X}$ core of the state, which is generally much smaller than the QCD length scale, will not be much affected by this “brown muck”, whose size is of order $1/\\Lambda _{\\rm QCD}$ , and in particular our wave-function and annihilation calculations will be largely unaffected.", "There is one exception to this statement.", "For exceptionally long bound-state lifetimes, namely when $\\Gamma \\lesssim \\Lambda _{\\rm QCD}^2/m$ , and if the bound state has spin, its polarization may start oscillating if the resulting hadron is not in its total spin eigenstate, as has been discussed recently in more detail in [74].", "This effect would change the angular distribution of the annihilation products.", "However, since this can only happen to colored bound states, the angular distribution (REF ) in the $\\gamma \\gamma $ channel is always unaffected.", "Moreover, a simple comparison of the annihilation rates to $\\Lambda _{\\rm QCD}^2/m$ indicates that the dijet and $\\gamma $ +jet channels are unlikely to be affected even for the lowest masses that we considered in this paper." ], [ "Strategy", "In this section we will show how the bound state signals can be used for determining the various quantum numbers of the new particles.", "As discussed in section REF , unstable particles $X$ must have integral charges for $R = \\mathbf {8}$ and $\\mathbf {10}$ and fractional charges which are multiples of $\\frac{1}{3}$ for $R=\\mathbf {3}$ , $\\mathbf {6}$ and $\\mathbf {15}$ .", "We will appeal to this fact when needed.", "In our general discussion we will assume the particle to be charged and will dedicate a separate subsection to neutral particles, for which only the dijet signal is present.", "We will also devote a subsection to representations higher than the $\\mathbf {10}$ and $\\mathbf {15}$ , for which some qualitative and semi-quantitative statements are merited despite the lack of any trustworthy quantitative technique." ], [ "Determining spin", "One can identify spin-1 particles based on the angular distribution in the diphoton (or photon+jet) channel.", "For bound states of spin-0 or spin-$\\frac{1}{2}$ particles it will be isotropic, while for spin-1 particles the dominant contribution will be coming from $J=2$ bound states with the angular distribution (REF ).", "For distinguishing between spin-0 and spin-$\\frac{1}{2}$ particles, three methods are available.", "One striking indication that the constituent particles have spin $\\frac{1}{2}$ would be the presence of an $\\ell ^+\\ell ^-$ signal.", "This is a good method as long as the charge and the color representation are not too small for the observation of this signal within a reasonable amount of time.", "Another method is to use the angular distribution in the dijet channel.", "For spin-0 particles, the distribution would be isotropic, while for spin-$\\frac{1}{2}$ particles the contribution of the $J=1$ bound states produced in the $q\\overline{q}$ channel (which exists for $R \\ne \\mathbf {3}$ ) would follow (REF ).", "This method is useful as long as the $q\\overline{q}$ channel contribution to the cross section is comparable to or larger than the $gg$ contribution.", "This condition is satisfied for $m \\gtrsim 500$  GeV in the case of the 7 TeV LHC and $m \\gtrsim 1000$  GeV in the case of the 14 TeV LHC.", "The third method is to use a precise measurement of the cross section of one of the signals.", "In the dijet channel, the cross sections for bound states of spin-$\\frac{1}{2}$ particles are more than twice as large as those of spin-0 particles in the same representation $R$ .", "In the photon+jet and diphoton channels, they are exactly twice as large, for equal $R$ and $Q$ .", "(Since there is only a discrete number of possibilities for $R$ and $Q$ , the fact that the signal depends on these variables as well, to be explored in more detail in the next subsection, will not generically prevent us from determining the spin.)", "However, these differences are not large compared with the uncertainty of our results, so this method will only become viable once a sufficiently precise higher-order calculation is available." ], [ "Determining charge and color representation", "The cross sections in the dijet and $\\gamma $ +jet channels are likely to be measured first and they alone will be sufficient for determining the representation and the charge in many cases, as demonstrated in figure REF  (left) for bound states of spin-0 or spin-1 particles and in figure REF  (left) for bound states of spin-$\\frac{1}{2}$ particles.", "It will still be impossible to know the charge of particles in the $\\mathbf {3}$ and $\\mathbf {8}$ representations, which do not have a $\\gamma $ +jet signal, and some ambiguity may exist in some of the other cases due to the unknown $K$ -factors.", "Most of these issues will be resolved once the cross section in the diphoton channel is measured too, as demonstrated in figure REF  (right).", "In the case of spin-$\\frac{1}{2}$ particles, the dilepton channel can be used as well, as shown in figure REF  (right)." ], [ "Determining mass", "The mass $m$ of the constituent particles is immediately known to within a few percent from the location of the resonance peak (since $M \\approx 2m$ ).", "An even more accurate measurement of the mass is possible by using $M = 2m + E_b$ since the binding energy $E_b$ is calculable once $j$ , $R$ and $Q$ are all known (although the uncertainty in leading-order expressions such as (REF ) is large and a higher order calculation is essential)." ], [ "More general scenarios", "Note that if all signals are present, the problem we have posed is over-determined.", "The full set of quantum numbers can be extracted from the cross sections in the $\\gamma $ +jet and diphoton channels (see figure REF , right), the angular distribution in one of those channels which is anisotropic for $j=1$ , and the observation of a dilepton signal which may be needed to break a degeneracy between $j=0$ and $j=\\frac{1}{2}$ .", "This leaves several other observables, including the cross section in the dijet channel, certain angular distributions of the annihilation products, and (for spin-$\\frac{1}{2}$ particles) the cross section and $p_T$ distribution of the dilepton resonance.", "One may also try to measure the $t\\overline{t}$ signal, which is available via (REF ) for spin-$\\frac{1}{2}$ particles (we have not discussed this channel in detail because we found that with currently available searches it is inferior to the dijet channel).", "Besides that, we have not utilized the signal shape, i.e., the resonance widths and the mass differences between the various resonances contributing to a given scenario, which for a few cases (see sections  and ) may be larger than the experimental resolution in the diphoton and dielectron channels.", "These extra observables may be used to relax some of our assumptions.", "For example, if the decay rate of the constituent particles is non-negligible relative to the annihilation rate, all the signals will be scaled down by a fixed branching ratio, and the dijet signal provides the additional measurement that can be used for determining it.", "The additional observables may be helpful in resolving ambiguities introduced by the possibility of large $K$ -factors, or in testing whether the particles have other large couplings besides Standard Model gauge interactions (such as discussed for spin-1 particles in appendix ).", "And they allow us to generalize our strategy to include cases where $X$ is part of a roughly degenerate multiplet, either of electroweak $SU(2)$ or of some as yet unknown approximate global symmetry.", "For $SU(2)$ multiplets, our results for the $\\gamma \\gamma $ , $\\gamma $ +jet and dijet signals would still apply for each member of the multiplet separately, while determining the dilepton signal would require small modifications to our analysis." ], [ "The case of neutral particles", "For $R=\\mathbf {8}$ and $\\mathbf {10}$ , the particles may be neutral, and then only the dijet signal will be present.", "In principle, as demonstrated in figure REF , the size of the signal allows one to determine the representation and the spin.", "In cases where the cross sections differ by not much more than a factor of 2 some ambiguity may remain due to unknown $K$ -factors before a higher order calculation is available.", "In such situations one can use the angular distribution to obtain further information.", "We have already discussed this in section REF for $j=0$ vs. $j=\\frac{1}{2}$ .", "These cases can also be distinguished from $j=1$ which will be dominated by the angular distribution for the $J=2$ state [which will be the same as in (REF )], with only a small isotropic component." ], [ "Even higher color representations", "Following the discussion in section REF , we have been restricting our attention to the first five color representations from table REF since for the $\\mathbf {27}$ , $\\mathbf {24}$ , $\\mathbf {15^{\\prime }}$ (and higher representations) our leading-order methods for computing the bound states are inadequate.", "Still, such particles may in principle exist in nature and give rise to bound state annihilation signals despite the fact that we cannot compute them precisely.", "Let us therefore describe the leading-order predictions for such signals anyway, at least to get an idea about some of their qualitative features.", "The signals for the $\\mathbf {27}$ , $\\mathbf {24}$ and $\\mathbf {15^{\\prime }}$ are typically significantly larger than those of the $\\mathbf {15}$ and $\\mathbf {10}$ (for same spin and charge) due to their larger color factors.", "The $\\mathbf {27}$ has vanishing anomaly coefficient and therefore, like the $\\mathbf {8}$ , does not have a $\\gamma $ +jet signal at the leading order.", "One qualitatively new feature of these high representations is that the annihilation rates for the color-singlet $J=0$ and $J=2$ bound states (which would contribute to the diphoton and dijet resonances) are so large that the bound state widths are comparable to or even somewhat larger than the binding energy, especially for $j = \\frac{1}{2}$ or 1.", "As a result, the signatures of these states will be much less pronounced.", "However, their radial excitations (see eq.", "(REF )) would still appear as narrow peaks since their annihilation rates are suppressed by $1/n^3$ .", "In the dijet channel, all the colored bound states are still sufficiently narrow so their contributions remain intact.", "As a result, even without the overly broad states, the diphoton and dijet signals for $R = \\mathbf {27}$ , $\\mathbf {24}$ , and $\\mathbf {15^{\\prime }}$ are still somewhat larger than those of the $\\mathbf {15}$ and $\\mathbf {10}$ .", "We also find for these representations that in some cases the bound state widths become comparable to (but not much larger than) the experimental resolution in various channels (for details, see appendix ).", "The determination of quantum numbers proceeds along the same general lines as we have discussed for the lower representations.", "A difficulty arises though in distinguishing between the $\\mathbf {24}$ and $\\mathbf {15^{\\prime }}$ because their color factors are similar and the allowed charges are fractional in both cases.", "For $j=0$ or 1, information from pair production above the threshold can help.", "According to table REF , the $\\mathbf {15^{\\prime }}$ would decay to at least 4 jets, while for the $\\mathbf {24}$ , decays to 3 jets are possible.", "Finally, it is interesting to note that the binding energies for the $\\mathbf {27}$ , $\\mathbf {24}$ , and $\\mathbf {15^{\\prime }}$ become large compared to the bound state mass, as large as $\\sim 0.1 M$ for S-wave color-singlet bound states.", "As a result, the splittings between some of the states may exceed the detector resolution so that several distinct peaks will be observable.", "The dielectron channel (available in the $j=\\frac{1}{2}$ case) seems to be especially promising, while the diphoton channel (available for any $j$ ) is somewhat less so because the ground states of the bound states contributing to that channel are very broad, as we have just discussed.", "Unfortunately, it will not be straightforward to extract much quantitative information from the observed splittings, since it will be difficult to compute properties of bound states of the $\\mathbf {27}$ , $\\mathbf {24}$ , and $\\mathbf {15^{\\prime }}$ reliably." ], [ "Summary and Discussion", "Bound state annihilation signals provide a uniquely powerful and largely model-independent probe that may be utilized at the LHC to study new colored pair-produced particles.", "One may use them to investigate hadronically-decaying particles using photons and leptons, measure precisely and unambiguously the masses of particles with partly-invisible decays, and in especially difficult cases even exclude or discover particles more effectively than is possible with more direct methods.", "Specifically, we have argued that if new colored and charged pair-produced particles are present, then as long as their decays are not too rapid, their bound states will generically give rise to dijet, $\\gamma $ +jet, $\\gamma \\gamma $ , and sometimes $\\ell ^+\\ell ^-$ resonances.", "We have computed the signals in the various channels and demonstrated that they typically allow unambiguous determination of the constituent particles' spin, color representation, and electric charge.", "In fact, only a subset of the signals is needed.", "There are more observables than unknowns, so many of our assumptions can be relaxed, extending the framework to multiplets of electroweak-$SU(2)$ or other as-yet unknown symmetries, and to situations in which the constituent decays compete with the bound state annihilation rates.", "The non-observation of new resonances due to bound state annihilation imposes constraints on models containing new pair-produced colored particles.", "For large representations, the bound state cross sections (dominated by the dijet channel, figure REF ) are very large, sometimes of the order of the pair production cross sections (figure REF ).", "This is perhaps surprising, but easy to understand.", "First, the usual suppression of bound state formation by $|\\psi (\\mathbf {0})|^2 \\propto C^3\\overline{\\alpha }_s^3$ is much reduced for higher representations.", "Second, the parton luminosities fall quickly with energy, and the bound states, which are slightly below the threshold, benefit from much higher luminosities than the free pairs produced well above the threshold.", "Happily, the model dependence of the bound state signals is usually orthogonal to that of the pair production and decay signatures.", "The limits from bound states depend only on a relatively mild condition — that the particle's decay rate is not too fast — and not at all on its decay modes.", "As a result, in cases with obscure decays the bound state searches can be competitive with, and thereby complementary to, direct searches.", "For example, with the full 7 TeV LHC dataset, the mass limit on pair-produced fermions in the $\\mathbf {10}$ representation (which would decay to 3 quarks) is only 610 GeV from the direct search (see figure REF ), but 780 GeV from the dijet resonance search (figure REF ).", "If these particles are charged with $Q=2$ , all the resonances that we studied set better limits, with the strongest limit of 860 GeV coming from the diphoton channel (figure REF ).", "Color-octet fermions with $Q=2$ (which would also decay to 3 quarks) are excluded up to 650 GeV by the diphoton search, while the limit from the direct search is only 500 GeV.", "For color-sextet fermions with $Q=4/3$ or $5/3$ , the diphoton channel sets a better limit as well (550 or 590 GeV, respectively, vs. 500 GeV).", "We have been describing the bound states, as well as the short-distance parts of their production and annihilation processes, at the leading order in $\\alpha _s$ (although we did distinguish between $\\alpha _s$ at the scale of the hard process and $\\overline{\\alpha }_s$ at the scale of the Bohr radius).", "This is approximately valid in the limit that $C\\overline{\\alpha }_s$ is small.", "For the range of masses we considered, higher-order corrections can be large ($K$ -factors of $\\sim 2$ or even larger for production, large corrections to annihilation rates, and substantial corrections to the bound states themselves) and will need to be computed, especially if such signals are observed.", "Note however that since the possible values for the spin, color representation and charge are discrete, our methods for determining the quantum numbers will still be useful even in the presence of large theoretical uncertainties.", "Also, since in most of the known cases the corrections are positive, the reach of the bound state signals will probably be greater than what we have presented.", "Since the existence of new colored and charged particles is a very generic possibility, and bound state signals are substantially model-independent and appear even for particles that decay in obscure ways, we suggest including the bound state signals as benchmark models for resonance searches.", "In particular, regardless of the original motivation for a particular resonance search, there is clearly value in presenting model-independent limits on the cross sections of narrow resonances.", "Some of our examples emphasize the need to keep improving the exclusion limits even at relatively low masses.", "In some cases this will require analyzing data in regimes where the triggers are not fully efficient or collecting data with prescaled triggers.", "We hope this work will help motivate such efforts." ], [ "Acknowledgments", "We would like to thank Bogdan Dobrescu, Yuri Gershtein, Yuval Grossman, Eva Halkiadakis, Maxim Perelstein, Aaron Pierce, Gavin Salam, Torbjörn Sjöstrand, and Kai Yi for discussions and communications related to various aspects of this project.", "The research of YK and MJS is supported in part by DOE grant DE-FG02-96ER40959.", "The research of MJS is also supported in part by NSF grant PHY-0904069." ], [ "Group theory identities for $SU(N)$", "In the following summary of our conventions and useful identities, $T_R^a$ denote the generators in representation $R$ of $SU(N)$ , $D_R$ is the dimension of the representation, $C_R$ is the quadratic Casimir, $T_R$ is the index, and $A_R$ is the anomaly coefficient (which are listed for various representations of $SU(3)$ in table REF ): $[T_R^a,T_R^b] = if^{abc}\\,T_R^c \\;, \\qquad \\left(T_R^a\\right)_{ij}\\left(T_R^a\\right)_{jl} = C_R\\, \\delta _{il}$ $\\mbox{Tr}\\, T_R^a = 0 \\;, \\quad \\mbox{Tr}\\left(T_R^a T_R^b\\right) = T_R\\, \\delta ^{ab} \\;, \\quad T_R = \\frac{D_R}{D_A} C_R$ $\\mbox{Tr}\\left(T_R^a\\left[T_R^b,T_R^c\\right]\\right) = i f^{abc}\\, T_R \\;, \\quad \\mbox{Tr}\\left(T_R^a\\left\\lbrace T_R^b,T_R^c\\right\\rbrace \\right) = \\frac{1}{2} d^{abc} A_R$ $f^{abc}f^{abd} = N\\delta ^{cd} \\;,\\quad d^{abc}d^{abd} = \\frac{N^2-4}{N}\\,\\delta ^{cd} \\;,\\quad d^{aac} = 0$ $f^{abe}f^{ecd} + f^{cbe}f^{aed} + f^{dbe}f^{ace} = 0 \\;, \\qquad d^{ecd}f^{abe} + d^{aed}f^{cbe} + d^{ace}f^{dbe} = 0$ $d^{abc}f^{adg}f^{beg} = \\frac{N}{2}\\,d^{cde} \\;, \\qquad f^{abc}f^{adg}f^{beg} = \\frac{N}{2}\\,f^{cde}$ $d^{abe}d^{cde} = \\frac{1}{3}\\left(\\delta ^{ac}\\delta ^{bd} + \\delta ^{ad}\\delta ^{bc} - \\delta ^{ab}\\delta ^{cd} + f^{ace}f^{bde} + f^{ade}f^{bce}\\right)$ For any representation $R = \\left(a,b\\right)$ of $SU(3)$ , $D_R = \\frac{1}{2}\\left(a+1\\right)\\left(b+1\\right)\\left(a+b+2\\right)\\,,\\quad C_R = \\frac{1}{3}\\left(a^2 + 3a + ab + 3b + b^2\\right)$ and the triality is $t_R = (a + 2b)\\,\\mbox{mod}\\,3$" ], [ "Interactions of spin-1 particles", "For spin-1 particles $X^\\mu $ , gauge invariance alone does not fix the interactions with the gauge bosons completely, and we must specify the effective Lagrangian: $\\mathcal {L}= -\\frac{1}{2}X_{\\mu \\nu }^\\ast X^{\\mu \\nu } - i g_s X_\\mu ^\\ast T_R^a X_\\nu G^{\\mu \\nu a} - i e\\, Q\\, X_\\mu ^\\ast X_\\nu F^{\\mu \\nu } + m^2 X_\\mu ^\\ast X^\\mu $ where $T^a_R$ is the $SU(3)$ generator in the representation $R$ , $X_{\\mu \\nu } = D_\\mu X_\\nu - D_\\nu X_\\mu $ with $D_\\mu = \\partial _\\mu - ig_s T_R^a A_\\mu ^a - ieQ A_\\mu $ , $G^a_{\\mu \\nu }$ is the gluon field strength, and $F^{\\mu \\nu }$ is the photon field strength.", "The coefficients of the second and third terms in (REF ) were chosen such that the theory preserves tree-level unitarity at high energy (a special case being when $X^\\mu $ is a gauge boson of an extended gauge group) [22].", "The Feynman rules are similar to those of vector leptoquarks (with vanishing anomalous couplings) [75], [76].", "Another subtlety is in the interaction with quarks.", "Pairs of particles of any spin can be produced from $q\\overline{q}$ through a diagram with an $s$ -channel gluon.", "However, as mentioned in appendix , for spin-1 particles this diagram violates unitarity at high energies, so any consistent theory must contain additional $q\\overline{q}$ diagrams involving some new interactions.", "For instance there may be diagrams with the exchange of some new particles in the $t$ channel.", "Therefore, for spin-1 particles, we have neglected the $q\\overline{q}$ channel contribution to the pair production cross sections in figures REF –REF because it is model-dependent.", "We have checked that even if we used the diverging diagram (whose eventual contribution is finite because of the falling PDFs), the cross sections for vectors in the range of masses presented in figures REF –REF would still be dominated by the $gg$ channel, except in the case of particles in the $\\mathbf {3}$ representation, where the contributions of the $gg$ and $q\\overline{q}$ channels may be comparable.", "Now let's discuss the possible effects on bound states.", "The unitarity-violating $q\\overline{q}$ diagram would not produce S-wave bound states by itself because it vanishes at the threshold.", "However, the additional interactions that must be present in the theory may affect bound state processes.", "As an example, bound states of KK gluons in a theory of universal extra dimensions, studied in [6], are affected by $q\\overline{q}$ diagrams with KK quarks, which contribute to the production and annihilation of $J=2$ bound states.", "By examining that example (and generalizing it to particles in other color representations) we notice that the effects of the $q\\overline{q}$ channel will typically be small (although sometimes not completely negligible) relative to the $gg$ channel as long as the KK quarks are at least twice as heavy as the KK gluons.", "This suggests that in typical models there are regimes of parameter space where the bound state processes are dominated by the $gg$ channel, and this motivates us to assume, for simplicity and model-independence, that the contributions to bound states of spin-1 particles from the $q\\overline{q}$ channel are small." ], [ "Pair production cross sections", "This appendix contains the expressions for the $X\\overline{X}$ pair production cross sections, relevant to figures REF –REF (where $\\alpha _s$ and the NLO PDFs [52] were evaluated at the scale $m$ ).", "For scalars [22], [24], [77], [25]: $\\hat{\\sigma }(gg\\rightarrow X\\overline{X}) &=& \\frac{T_R}{16}\\,\\frac{\\pi \\alpha _s^2}{\\hat{s}}\\left[4C_R\\left(2-\\beta ^2\\right)\\beta + \\left(3-5\\beta ^2\\right)\\beta \\frac{}{}\\right.\\nonumber \\\\&&\\qquad \\qquad \\left.", "- \\left(2C_R(1-\\beta ^4) + 3\\left(1-\\beta ^2\\right)^2\\right)\\ln \\frac{1+\\beta }{1-\\beta }\\right] \\\\\\hat{\\sigma }(q\\overline{q} \\rightarrow X\\overline{X}) &=& \\frac{4}{27} T_R \\frac{\\pi \\alpha _s^2}{\\hat{s}}\\beta ^3$ where $\\beta = \\sqrt{1-4m^2/\\hat{s}}$ .", "These expressions agree with those for squarks [78] for $R = \\mathbf {3}$ .The result for scalars in the $gg$ channel in [22] is incorrect.", "It does not reduce to the expression for squarks.", "Ref.", "[75] has also found that they disagree with the result of [22] for scalars.", "For fermions [22], [23], [25]: $\\hat{\\sigma }(gg\\rightarrow X\\overline{X}) &=& \\frac{T_R}{8}\\,\\frac{\\pi \\alpha _s^2}{\\hat{s}}\\left[ - 4C_R\\left(2-\\beta ^2\\right)\\beta - \\left(9 - 5\\beta ^2\\right)\\beta \\frac{}{}\\right.\\nonumber \\\\&&\\qquad \\qquad \\left.", "+ \\left(2 C_R\\left(3 - \\beta ^4\\right) + 3\\left(1-\\beta ^2\\right)^2\\right) \\ln \\frac{1+\\beta }{1-\\beta }\\right] \\\\\\hat{\\sigma }(q\\overline{q} \\rightarrow X\\overline{X}) &=& \\frac{8}{27}T_R\\frac{\\pi \\alpha _s^2}{\\hat{s}}\\beta \\left(3-\\beta ^2\\right)$ These results agree with those for heavy quarks [79] for $R = \\mathbf {3}$ .", "For vectors [22]: $\\hat{\\sigma }(gg\\rightarrow X\\overline{X}) &=& \\frac{T_R}{64}\\,\\frac{\\pi \\alpha _s^2}{m^2}\\left[4C_R\\left(22 - 9\\beta ^2 + 3\\beta ^4\\right)\\beta + 3\\left(19 - 24\\beta ^2 + 5\\beta ^4\\right)\\beta \\frac{}{}\\right.", "\\nonumber \\\\&&\\left.\\qquad \\qquad \\; - \\frac{12m^2}{\\hat{s}}\\left(2C_R\\left(1-\\beta ^4\\right) + 19 - 6\\beta ^2 + 3\\beta ^4\\right) \\ln \\frac{1+\\beta }{1-\\beta }\\right] \\\\\\hat{\\sigma }(q\\overline{q}\\rightarrow X\\overline{X}) &=& \\frac{T_R}{108}\\,\\pi \\alpha _s^2\\,\\frac{\\hat{s}}{m^4}\\,\\left(27-26\\beta ^2+3\\beta ^4\\right)\\beta ^3$ Note that in the $gg$ channel in the limit $\\hat{s}\\rightarrow \\infty $ $\\hat{\\sigma }(gg\\rightarrow X\\overline{X}) \\rightarrow T_R C_R\\,\\frac{\\pi \\alpha _s^2}{m^2}$ which is consistent with unitarity.", "For $R = \\mathbf {3}$ , eq.", "(REF ) reduces to the result obtained for vector leptoquarks (with no anomalous couplings) in [75] and for $R = \\mathbf {8}$ to twice that of the coloron of [80], [81] (because the coloron field is real).", "On the other hand, the expression for the $q\\overline{q}$ channel, eq.", "(), where only the $s$ -channel gluon diagram has been included, diverges with $\\hat{s}$ and thus violates unitarity, as has been already noted in [22].", "A UV completion is needed in this case, and the resulting contribution from the $q\\overline{q}$ channel will be model-dependent.", "We discuss this issue further in appendix ." ], [ "Widths", "In this appendix we provide details about the bound state widths (due to the annihilation processes alone), as discussed in section  (and for the very high representations in section REF ).", "For $J=0$ and $J=2$ bound states contributing in the $\\gamma \\gamma $ , $\\gamma $ +jet and dijet channels, the dominant annihilation process is to $gg$ .", "For constituents of a given spin $j$ and color representation $R$ , the annihilation rates depend on the spin $J$ and color representation ${\\cal R}$ of the bound state.", "We are interested in bound states with ${\\cal R}=\\mathbf {1}$ , $\\mathbf {8}$ and $\\mathbf {27}$ .", "For constituent particles in representations $R=\\mathbf {8}$ , $\\mathbf {15}$ , $\\mathbf {27}$ , $\\mathbf {24}$ there are two ways to form a bound state with ${\\cal R}=\\mathbf {8}$ , as indicated in table REF , one of which is universal and described by the wavefunction (REF ).", "In the following, we will denote it by $\\mathbf {8}$ and the second possibility by $\\mathbf {\\tilde{8}}$ .", "Similarly, for $R = \\mathbf {15}$ , $\\mathbf {27}$ , $\\mathbf {24}$ there are two or three kinds of ${\\cal R}=\\mathbf {27}$ bound states.", "For bound states of spin-0 particles we find $\\Gamma _{j=0}\\left(\\mathcal {B}^{{\\cal R}=\\mathbf {1}}_{J=0}\\rightarrow gg\\right) = \\frac{D_R C_R^5}{32}\\,\\alpha _s^2\\,\\overline{\\alpha }_s^3\\,m$ $\\Gamma _{j=0}\\left(\\mathcal {B}^{{\\cal R}=\\mathbf {8}}_{J=0}\\rightarrow gg\\right) = \\frac{5\\,A_R^2\\left(C_R - \\frac{3}{2}\\right)^3}{192\\,T_R}\\,\\alpha _s^2\\,\\overline{\\alpha }_s^3\\,m$ $\\Gamma _{j=0}\\left(\\mathcal {B}^{{\\cal R}=\\mathbf {\\tilde{8}}}_{J=0}\\rightarrow gg\\right) = \\frac{1}{32}\\left(\\frac{D_R C_R\\left(C_R+\\frac{3}{4}\\right)}{5} - \\frac{5\\,A_R^2}{6\\,T_R}\\right)\\left(C_R-\\frac{3}{2}\\right)^3\\alpha _s^2\\,\\overline{\\alpha }_s^3\\,m$ $\\sum _{\\mathbf {27}}\\Gamma _{j=0}\\left(\\mathcal {B}^{{\\cal R}=\\mathbf {27}}_{J=0}\\rightarrow gg\\right) = \\frac{D_R C_R\\left(C_R-\\frac{4}{3}\\right)\\left(C_R-4\\right)^3}{160}\\,\\alpha _s^2\\,\\overline{\\alpha }_s^3\\,m$ where the sum in (REF ) is over all the possible ways to form an ${\\cal R} = \\mathbf {27}$ bound state (which is easier to obtain than computing the separate rates).", "For bound states of spin-$\\frac{1}{2}$ particles, the widths are given by the same expressions multiplied by a factor of 2: $\\Gamma _{j=1/2}\\left(\\mathcal {B}^{{\\cal R}}_{J=0}\\rightarrow gg\\right) = 2\\,\\Gamma _{j=0}\\left(\\mathcal {B}^{{\\cal R}}_{J=0}\\rightarrow gg\\right)$ and for bound states of spin-1 particles $\\Gamma _{j=1}\\left(\\mathcal {B}^{{\\cal R}}_{J=0}\\rightarrow gg\\right) = 3\\,\\Gamma _{j=0}\\left(\\mathcal {B}^{{\\cal R}}_{J=0}\\rightarrow gg\\right)$ $\\Gamma _{j=1}\\left(\\mathcal {B}^{{\\cal R}}_{J=2}\\rightarrow gg\\right) = \\frac{16}{5}\\,\\Gamma _{j=0}\\left(\\mathcal {B}^{{\\cal R}}_{J=0}\\rightarrow gg\\right)$ Table REF compares these rates with the binding energies (REF ), and table REF (top) presents the same widths as a fraction of the bound state mass $M$ , which is useful for the discussion of the experimental resolution.", "Table: Annihilation widths as a fraction of the bound state mass, Γ/M\\Gamma /M.", "Top table: j=0,J=0j=0, J=0.", "Same comments as in table  apply.", "For R=27,24,15 ' R = \\mathbf {27},\\mathbf {24},\\mathbf {15^{\\prime }} with ℛ=1{\\cal R} = \\mathbf {1} we give the width for the first radial excitation (since the ground state is too broad).", "Middle table: j=1 2,J=1,ℛ=8j=\\frac{1}{2}, J=1, {\\cal R} = \\mathbf {8}.", "Bottom table: j=1 2,J=1,ℛ=1j=\\frac{1}{2}, J=1, {\\cal R} = \\mathbf {1} for two values of the charge QQ.For the dijet channel in the case of $j=\\frac{1}{2}$ we also need the annihilation rate of $J=1$ color-octet bound states: $\\Gamma _{j=1/2}\\left(\\mathcal {B}^{{\\cal R}=\\mathbf {8}}_{J=1}\\rightarrow q\\overline{q}\\right) = \\frac{D_R C_R\\left(C_R - \\frac{3}{2}\\right)^3}{16}\\,\\alpha _s^2\\,\\overline{\\alpha }_s^3\\,m$ This rate is compared with the binding energy in table REF and with the mass in table REF (middle).", "For the $\\ell ^+\\ell ^-$ channel, we are interested in color-singlet spin-1 bound states of spin-$\\frac{1}{2}$ particles.", "They cannot annihilate to $gg$ or $q\\overline{q}$ via the strong interaction, so the width is determined by subleading processes, which are the annihilation to fermion pairs via a photon or $Z$ , and the annihilation to three gauge bosons.", "Based on the expressions in section REF , the total rate is $&&\\Gamma _{j=1/2}\\left(\\mathcal {B}^{{\\cal R}=\\mathbf {1}}_{J=1} \\rightarrow f\\overline{f},ggg,\\gamma gg,Zgg\\right) = \\nonumber \\\\&&\\quad \\frac{5}{3}\\left[\\frac{1}{2} D_R\\, Q^2\\frac{\\alpha ^2}{\\cos ^4\\theta _W}+ \\frac{\\pi ^2-9}{9\\pi }\\left(\\frac{A_R^2}{D_R}\\,\\alpha _s + \\frac{9}{20}\\,D_R C_R^2\\, Q^2\\frac{\\alpha }{\\cos ^2\\theta _W}\\right)\\alpha _s^2\\right] C_R^3\\,\\overline{\\alpha }_s^3\\,m$ These bound states are very narrow: $\\Gamma /|E_b| \\ll 1$ for any $R$ .", "In the context of experimental resolution, the widths are presented as a fraction of $M$ in table REF (bottom)." ] ]

[ [ "The AGN content in luminous IR galaxies at z\\sim2 from a global SED\n analysis including Herschel data" ], [ "Abstract We use Herschel-PACS far-infrared data, combined with previous multi-band information and mid-IR spectra, to properly account for the presence of an active nucleus and constrain its energetic contribution in luminous infrared (IR) sources at z\\sim2.", "The sample is composed of 24 sources in the GOODS-South field, with typical IR luminosity of 10^{12} Lo.", "Data from the 4 Ms Chandra X-ray imaging in this field are also used to identify and characterize AGN emission.", "We reproduce the observed spectral energy distribution (SED), decomposed into a host-galaxy and an AGN component.", "A smooth-torus model for circum-nuclear dust is used to account for the direct and re-processed contribution from the AGN.", "We confirm that galaxies with typical L_{8-1000um}\\sim10^{12}Lo at z\\sim2 are powered predominantly by star-formation.", "An AGN component is present in nine objects (\\sim35% of the sample) at the 3sigma confidence level, but its contribution to the 8-1000 um emission accounts for only \\sim5% of the energy budget.", "The AGN contribution rises to \\sim23% over the 5-30 um range (in agreement with Spitzer IRS results) and to \\sim60% over the narrow 2-6 um range.", "The presence of an AGN is confirmed by X-ray data for 3 (out of nine) sources, with X-ray spectral analysis indicating the presence of significant absorption, i.e.", "NH\\sim10^{23}-10^{24} cm^{-2}.", "An additional source shows indications of obscured AGN emission from X-ray data.", "The comparison between the mid-IR--derived X-ray luminosities and those obtained from X-ray data suggests that obscuration is likely present also in the remaining six sources that harbour an AGN according to the SED-fitting analysis." ], [ "Introduction", "Ultra-luminous infrared galaxies (ULIRGs, defined as having $L_{IR}>10^{12}L_{\\odot }$ ; [55]) are among the most luminous objects of the Universe, radiating most of their energy in the infrared (IR) band.", "In the local Universe, ULIRGs are rare objects and, although very luminous, account for only ${\\sim }5\\%$ of the total integrated IR luminosity density [54].", "Moving to higher redshift, the contribution of ULIRGs to the energy budget increases, as clearly illustrated by infrared luminosity function studies ([28]; [9]).", "[35], deriving the infrared luminosity function up to $z{\\sim }2.3$ from deep 24 and 70 ${\\mu }$ m Spitzer data, obtained a contribution of ${\\sim }17 \\%$ from ULIRGs to the IR luminosity density at $z{\\sim }2$ .", "This result has been recently confirmed also by the Herschel Science Demonstration Phase (SDP) preliminary results; [22], by deriving the total IR luminosity function up to $z\\sim {3}$ , estimated that ULIRGs account for ${\\sim }30 \\%$ to the IR luminosity density at $z{\\sim }2$ .", "At high redshift ($z{\\sim }2$ ), a key question regards the nature of the sources that power these ultra-luminous objects (i.e., star-formation vs. accretion).", "Locally, thanks to Infrared Space observatory (ISO) mid-IR spectroscopy, ULIRGs were proven to be powered predominantly by star formation in the mid-IR, with the fraction of AGN-powered objects increasing with luminosity (from ${\\sim }$ 15% at $L_{IR}<2{\\times }10^{12}~L_{\\odot }$ to about 50% at higher luminosity; [33]).", "A limited percentage (15–20%) of mid-IR light dominated by accretion processes has also been found using $L$ -band (3-4 $\\mu $ m) observations of local ULIRGs with $L_{IR}\\sim 10^{12}L_{\\odot }$ by [47].", "Interestingly, these authors show that, although their sources are powered mostly by starburst processes, at least 60% of them contain an active nucleus.", "Using accurate optical spectral line diagnostics applied to a sample of 70 $\\mu $ m selected luminous sources at $z\\sim $ 1, [58] found that only 20–30% of their objects may host an active nucleus.", "However, such AGN are never bolometrically dominant.", "In the same paper, a discussion on how the low AGN incidence can be partially due to selection effects (i.e.", "the 70 $\\mu $ m band sampling the starburst 50 $\\mu $ m rest-frame far-IR bump) is also presented.", "At $z\\sim 2$ , our understanding of the AGN content in ULIRGs is much more uncertain.", "On the one hand, sources selected with a 24 ${\\mu }$ m flux density $$ > $$$0.9--1~mJy mainlysample the bright end of the ULIRG population.", "The analysis of the IRSspectra of a color-selected sample of sources withS$ 24m$\\; > \\over \\sim \\;$ 0.9$~mJy and at $ z>1.5$($ LIR71012L$) shows that the majority of thesesources are AGN-dominated ($ 75%$; \\cite {2008ApJ...683..659S}).On the other hand, sub-millimeter selected galaxies (SMGs) at $ z2$,falling in the ULIRG regime, appear largely starburst-dominated objects(e.g., \\cite {2007ApJ...660.1060V}; \\cite {2008ApJ...675.1171P}).$ In this paper, we aim at providing a better understanding of luminous sources at $z\\sim $ 2 with typical IR luminosities of $10^{12}L_{\\odot }$ , i.e.", "sampling the knee of the IR luminosity function instead of the bright end.", "We will re-analyse the sample from [18], where an estimate of the AGN contribution in ULIRGs has already been presented, based on Spitzer IRS data, Chandra 2 Ms observations, optical/near-IR multi-band properties, and ACS morphological properties.", "In this work, we will extend the analysis to far-IR data, obtained recently by the Herschel satellite as part of the guaranteed survey “PACS Evolutionary Probe” (PEP; [32]), and to the recently published Chandra 4 Ms data ([63]).", "We note that an analysis of the F10 sample (including the new far-IR data) has been already presented by [40] [hereafter N12], where the general properties of the mid-to-far IR spectral energy distributions (SEDs) of 0.7$<z<$ 2.5 galaxies have been investigated.", "The authors confirm the early Herschel results (e.g., [16]), i.e.", "the star-formation rates at $z\\sim 2$ are over-estimated if derived from 24 ${\\mu }$ m flux densities, and discuss how this effect can be due to enhanced PAH emission with respect to local templates (see also [16]; [15]).", "In N12, the SED library from [11] was used, longward of 6 $\\mu $ m rest-frame, and the IRS spectra data were stacked to improve the faint signal of sources at $z\\sim {2}$ .", "In this paper we adopt a different method to study the properties of $z\\sim 2$ IR galaxies, as described in $§$; a comparison of our results vs. those obtained by N12 is presented in $§$REF .", "Hereafter, we adopt the concordance cosmology ($H_{0}=70$  km sec$^{-1}$  Mpc$^{-1}$ , $\\Omega _{m}$ =0.3, and $\\Omega _{\\Lambda }$ =0.7, [56]).", "Magnitudes are expressed in the Vega system." ], [ "The data", "The sample is composed of 24 luminous sources at $z{\\sim }$ 2, selected by F10 in the Chandra Deep Field South (CDF-S), with 24 ${\\mu }$ m flux densities $S(24~{\\mu }m){\\sim }0.14-0.55$ mJy and at $z=1.75-2.40$ .", "The sample can be considered luminosity-selected, since all of the sources satisfying these selection criteria are considered.", "Given the redshift of the sources, their 24 $\\mu $ m flux densities translate roughly into infrared luminosity around $10^{12}L_{\\odot }$ .", "We excluded two objects, originally defined as luminous IR galaxies (LIRGs; $L_{IR}>10^{11}L_{\\odot }$ ) by F10 (L5511 and L6211) and subsequently included in the ULIRG class thanks to the new IRS redshift measurements, since they would be the only two sources outside the HST ACS area (L5511 is also outside the Herschel area).", "The sample benefits from a large amount of information available for the CDF-S, from multi-band photometry to the recent Spitzer IRS spectroscopy.", "The IRS observations were performed at low-resolution in the observed 14-35 $\\mu $ m wavelength regime, i.e.", "sampling at $z\\sim 2$ the important rest-frame PAH features at 6.2 $\\mu $ m and 7.7 $\\mu $ m, and the 9.7 $\\mu $ m silicate feature (see F10).", "As reported above, in this work we also use recent far-IR data obtained from the Herschel satellite, which consists of PACS data at 70, 100 and 160 ${\\mu }$ m from the PEP survey [32].", "We use the PACS blind catalogue v1.3 down to 3$\\sigma $ limits of 1.2, 1.2 and 2.0 mJy at 70, 110 and 160 ${\\mu }$ m, respectively.", "Herschel data have been matched to 24 $\\mu $ m Spitzer MIPS sources [34] using the likelihood ratio technique ([57]; [12]), as described in [5].", "Shorter wavelength information has been included by matching the 24 $\\mu $ m sources to the multi-band (from UV to Spitzer IRAC bands) GOODS-MUSIC photometric catalog [51].", "In particular, we cross-correlated the 24 $\\mu $ m selected ULIRGs with the PEP catalogue, considering the positions of the 24 $\\mu $ m sources already associated to the PEP ones.", "Among the 24 ULIRGs, 21 sources have counterparts in PACS.", "For the 3 sources undetected by Herschel (U5050, U5152, and U5153), we consider the conservative 5$\\sigma $ upper limits (2.2, 2.0 and 3.0 mJy at 70, 100 and 160 $\\mu $ m, respectively).", "For PACS detections, the typical separation between 24 $\\mu $ m and PEP sources is less than 1, with the exception of U5805 and U16526 (${\\sim }$ 4).", "By visually checking the far-IR images, these sources appear as blends of more than one 24 ${\\mu }$ m source.", "In these cases, the PACS flux densities are considered as upper limits.", "Photometry at 16 $\\mu $ m, obtained by F10 from IRS data, has also been included.", "Finally, we consider the recently published Chandra 4 Ms point-like source catalogue in the CDF-S ([63]).", "Eight of the 24 sources have an X-ray counterpart within 1.", "Table REF lists the source names, redshifts from optical and IRS spectroscopy (see F10), source IDs in the GOODS-MUSIC catalogue, flux densities and associated errors from mid-IR (16 $\\mu $ m) to far-IR (160 ${\\mu }$ m).", "Table: The sample: multi-band information" ], [ "SED decomposition", "The IR energy budget of a galaxy can be mainly ascribed to stellar photosphere and star-formation emissions and accretion processes; to estimate the relative importance of these three processes, a proper SED decomposition should be carried out.", "Disentangling the different contributions to the total SED is becoming more and more effective with the advent of the Spitzer and Herschel satellites.", "In this regard, many studies have been performed to compare the full range of observed photometric data with expectations from a host-galaxy component and models for the circum-nuclear dust emission (e.g., [44]; [23]; [62]; [46]).", "Here, together with the full-band photometric SED, we benefit also from the IRS spectra, that we combine with the photometric datapoints (Sec.", "REF ; see also [37], [2] for examples of combined photometric/spectroscopic data analysis).", "The IRS spectra provide an important diagnostic, sampling the rest-frame mid-IR spectral range (${\\sim }5-12~\\mu $ m for our sample), where the difference between starburst and AGN is strongest.", "In this wavelength range, starburst galaxies are generally characterized by prominent polycyclic aromatic (PAH) features and weak 10 $\\mu $ m continuum, whereas AGN display weak or no PAH features plus a strong continuum (e.g.", "[27])." ], [ "AGN and stellar components", "We have decomposed the observed SEDs using three distinct components: stars, having the bulk of the emission in the optical/near-IR; hot dust, mainly heated by UV/optical emission due to gas accreting onto the super-massive black hole (SMBH) and whose emission peaks between a few and a few tens of microns; cold dust, principally heated by star formation (we refer to [46]; but see also [23] for a detailed description of the properties of the AGN and host-galaxy (stars$+$ cold dust) components).", "Here we report only the most important issues concerning this analysis, with the AGN component being the main focus.", "The stellar component has been modelled as the sum of simple stellar population (SSP) models of solar metallicity and ages ranging from ≈1 Myr to 2.3 Gyr, which corresponds to the time elapsed between $z=6$ (the redshift assumed for the initial star formation stars to form) and $z\\sim {2}$ in the adopted cosmology.", "A [50] initial mass function (IMF), with mass in the range 0.15–120 M$_{\\odot }$ , is assumed.", "The SSP spectra have been weighted by a Schmidt-like law of star formation (see [4]): $SFR(t)=\\frac{T_{G}-t}{T_{G}}{\\times }\\exp \\left({-\\frac{T_{G}-t}{T_{G}{\\tau }_{sf} }}\\right)$ where $T_{G}$ is the age of the galaxy (i.e.", "of the oldest SSP) and $\\tau _{sf}$ is the duration of the burst in units of the oldest SSP.", "A common value of extinction is applied to stars of all ages, and a [8]) attenuation law has been adopted ($R_{V }$ =4.05).", "To account for emission above 24 $\\mu $ m, a component coming from colder, diffuse dust, likely heated by star-formation processes, has been included in the fitting procedure.", "It is represented by templates of well-studied starburst galaxies (i.e.", "Arp 220, M82, M83, NGC 1482, NGC 4102, NGC 5253 and NGC 7714) and five additional host-galaxy average templates obtained recently by [37] from the starburst templates of [7].", "This set of five templates have been included since they properly reproduce the relative PAH strengths in the average IRS spectra of our ULIRG sample (see F10).", "Regarding the AGN component, we have used the radiative transfer code of [20].", "This model follows the formalism developed by different authors (e.g., [43]; [21]; [14]), where the IR emission in AGN originates in dusty gas around the SMBH with a “flared disk”, “smooth distribution”.", "Recently, this model has been widely used and found to successfully reproduce the photometric data, including the 9.7 $\\mu $ m silicate feature in emission observed for type-I AGN (e.g., [52]).", "Recent high-resolution, interferometric mid-IR observations of nearby AGN (e.g., [25]) have confirmed the presence of a geometrically thick, torus-like dust distribution on pc-scales; this torus is likely irregular or “clumpy”.", "Over the last decade, many codes have been developed to deal with clumpy dust distributions (e.g., [38]; [39]; [24]).", "According to [13], the two models (smooth and clumpy) do not differ significantly in reproducing sparse photometric datapoints (see also [17]).", "The main difference is in the strength of the silicate feature observed in absorption in objects seen edge-on, which is, on average, weaker for clumpy models with the same global torus parameters.", "The comparison between the two models has been applied only to few samples including IRS data and the results are not conclusive.", "Recently, [61] made a comparison of the smooth vs. clumpy models for the matter responsible for reprocessing the nuclear component of a hyper-luminous absorbed X-ray quasar at $z{\\sim }0.442$ (IRAS 09104$+$ 4109), for which both photometric and spectroscopic (IRS) data were available.", "While smooth solutions (the F06 model) are able to reproduce the complete dataset, clumpy models ([39]) have problems in reproducing the source photometry and spectroscopy at the same time.", "In [37], smooth vs. clumpy models are tested against a sample of ${\\sim }$ 10 local Swift/BAT AGN with prominent emission at IRAS wavelengths.", "In this case, the authors claim that clumpy solutions reproduce the data better; smooth model parameters produce a much wider range of SED solutions, i.e.", "this model seems to produce too degenerate solutions.", "A further, more complete and extensive comparison of smooth vs. clumpy solutions is presented in Feltre et al.", "(submitted), where the theoretical SED shapes and the detailed spectral features of the two classes of models (i.e.", "F06 for the “smooth distribution” and [39] for the “clumpy distribution”) are compared using a large compilation of AGN with IRS spectroscopic data.", "Overall, similar results to those obtained by [13] are derived, i.e.", "SED fitting applied to both photometric and spectroscopic data is not a sufficiently reliable tool to discriminate between the smooth and the clumpy distributions.", "We remind the reader that in the present paper we are focusing on the torus “global” energy output (i.e., the relevance of accretion-related emission with respect to the total source SED), not on the details of the torus structure and geometry, so the choice of the adopted model does not critically influence the results; as shown by Feltre et al.", "(submitted), the two models provide consistent results in terms of energetics.", "Figure: a) Rest-frame broad-band datapoints (red dots) compared withthe best-fit model obtained as the sum (solid black line) of a stellar (reddotted line), an AGN (blue dashed line) and a starburst component(green dot-dashed line).", "IRS spectra are shown as magenta lines.The area filled with diagonal lines represents the AGN solutions at the3σ\\sigma confidence level.Figure: b) As in Fig.", "a.Figure: Rest-frame SEDs and datapoints, as in Fig.", ",where a zoom in the 2–20 μ\\mu m wavelength range is shown for thenine sources where an AGN component is required at the 3 sigma confidencelevel from the SED-fitting analysis." ], [ "SED-fitting procedure", "In most previous work using the F06 code, only photometric datapoints were used, and the quality of the fitting solutions was estimated using a standard $\\chi ^{2}$ minimization technique, where the observed values are the photometric flux densities (from optical-to-mid-IR/far-IR) and the model values are the “synthetic” flux densities obtained by convolving the sum of stars, AGN, and starburst components through the filter response curves (see [23]).", "In [61], the spectroscopic information was taken into account a posteriori in order to discriminate among different photometric best-fitting solutions.", "Here we propose a first attempt to simultaneously fit the photometric and spectroscopic datapoints using smooth torus models by transforming the mid-IR 14–35 $\\mu $ m observed-frame spectra into “narrow-band” photometric points of 1 $\\mu $ m band-width in the observed frame (i.e., subdividing the spectral transmission curve in sub-units), and estimating the corresponding fluxes and uncertainties using ordinary procedure of filter convolution and error propagation.", "The “new” filters have been chosen to achieve, in each wavelength bin, a sufficient signal-to-noise ratio without losing too much in terms of spectral resolution, which is needed to reproduce the 9.7 ${\\mu }$ m feature, when present.", "To take into account slit loss effects, IRS data have been normalized to the 24 $\\mu $ m flux density, deriving normalization factors between $\\sim $ 1.2 and 1.8.", "We would like to remark here that the F06 code does not necessarily impose the presence of an AGN component, i.e., solutions with only stellar emission are possible, as described in $§$REF .", "Furthermore, the relative normalization of the optical/near-IR component and the far-IR emission of the host galaxy is free, given the extremely complex physical relation of the two (i.e.", "[3]).", "Overall, the SED-fitting procedure ends with 11 free parameters: six are related to the AGN, two to the stellar component, and one to the starburst.", "The further two free parameters are the normalizations of the stellar and of the starburst components; the torus normalization is estimated by difference, i.e., it represents the scaling factor of the torus model capable to account for the data-to-model residuals once the stellar components have already been included.", "Here we briefly recall the parameters involved in our SED fitting analysis, and refer to F06 and Feltre et al.", "(submitted) for a detailed description of the AGN model parameter.", "The six parameters related to the torus are: the ratio $R_{max}/R_{min}$ between the outer and the inner radius of the torus (the latter being defined by the sublimation temperature of the dust grains); the torus opening angle $\\Theta $ ; the optical depth $\\tau $ at 9.7 $\\mu $ m ($\\tau _{9.7{\\mu }m}$ ); the line of sight $\\theta $ with respect to the equatorial plane; two further parameters, $\\gamma $ and $\\alpha $ , describe the law for the spatial distribution of the dust.", "In the currently adopted version (see Feltre et al., submitted), the “smooth” torus database contains 2368 AGN models.", "In [46] (see also [23]), the degeneracies related to the six torus parameters are extensively described: in fact, various combinations of parameter values are equally able to provide good results in reproducing a set of observed data points.", "In particular, the optical depth $\\tau _{9.7{\\mu }m}$ has the largest effect on the fit.", "The infrared luminosity, provided by the SED-fitting code, is robustly determined (within $\\sim $ 0.1 dex) and appears “solid” against parameter degeneracies.", "Concerning the parameters associated with the other components, two are related to the stellar emission: $\\tau _{sf}$ , i.e.", "the parameter of the Schmidt-like law for the star formation, and the reddening $E(B-V)$ .", "Regarding the starburst component, the free parameter is related to the choice of the best-fitting template among the starburst library.", "The given number of free parameters means that the acceptable solutions, within 1(3)$\\sigma $ confidence levels, are derived, for each source, by considering the parameter regions encompassing $\\chi ^{2}_{min}$ +(12.65, 28.5), respectively, in presence of 11 degrees of freedom (d.o.f.", "; see [26]).", "Figure: Fractional contribution due to the AGN component in the2-6 μ\\mu m (top panel), 5–30 μ\\mu m (middle panel), and8–1000 μ\\mu m (bottom panel) range.", "The error barsaccount for the AGN model dispersion at the 1σ\\sigma confidence level.Red points indicate the sources where an AGN component is detected atthe 3σ\\sigma confidence level from the SED fitting, and the trianglesthose with X-ray emission pointing clearly towards an AGN classification." ], [ "SED-fitting results", "In Fig.1a,b the observed UV–160 $\\mu $ m datapoints (filled red points) are reported along with the best-fitting solutions (black lines) and the range of AGN models within the 3$\\sigma $ confidence level (filled region).", "All the sources need a host galaxy (red dotted-line) and a starburst component (green dot-dashed line).", "The host galaxy dominates the UV–8 $\\mu $ m photometry (at $z\\sim 2$ , the IRAC 8 $\\mu $ m band samples the 2.7 $\\mu $ m rest-frame), while the starburst component dominates at longer wavelengths.", "For the three sources with no PACS detection (U5050, U5152, and U5153), the starburst component is required by the SED-fitting procedure to reproduce the mid-IR spectral data, although its shape is not well constrained.", "Regarding the AGN component, our goal is to check whether its presence is required by the data and, in this case, to estimate its contribution to the IR luminosity.", "The SED-fitting procedure found that for all but one source (U428) the presence of an AGN is consistent with the photometry, although for only nine sources (${\\sim }35\\%$ of the sample: U4639, U4950, U4958, U5150, U5152, U5153, U5652, U5805, and U5877) its presence is significant at the 3$\\sigma $ confidence level (i.e., solutions with no torus emission have ${\\chi }^{2}{\\ge }{\\chi }^{2}_{min}$ +28.5).", "However, in these sources as well, the AGN component far from dominates the whole spectral range, but emerges only in the narrow 2–10 $\\mu $ m range, where the stellar emission from the host galaxy has a minimum while the warm dust heated by the AGN manifests itself (e.g., [27]; [22]).", "In Fig.", "REF we report a zoom of the SED over the 2–20 $\\mu $ m range in order to visualize the AGN emission for all of the nine sources where such component is required at the 3$\\sigma $ confidence level.", "At a visual inspection, the presence of a nuclear component can be inferred for sources with a power-law SED (i.e., U4950 and U5877), for sources where the stellar component alone is not able to reproduce the datapoints around 3 $\\mu $ m (i.e., U4958, U5153) or for sources where the starburst component, normalized to the far-IR datapoints, has already declined around 5–6 $\\mu $ m (i.e.", "U5152, U5153, and U5805).", "Finally, there are few cases (i.e., U4639) where visual inspection of the SED decomposition is not strongly suggesting an AGN component, which is however required by the SED-fitting analysis.", "The likely AGN in these sources is either obscured or of low-luminosity, although a combination of both effects is plausible as well.", "While our analysis is able to place constraints on the presence of an AGN, we cannot draw any firm conclusion on either the obscured or the low-luminosity hypothesis for the AGN emission from mid-IR data.", "Despite the uncertainties affecting the estimate of the gas column density $N_{H}$ derived from the dust optical depths (e.g., [36]), what we observe is that for all of the sources with an AGN component, a certain level of obscurationThe column density has been derived from the the optical depth at 9.7$\\mu $ m, using the Galactic extinction law ([10]) and dust-to-gas ratio ([6]).", "($10^{22}$$\\; < \\over \\sim \\;$$N_{H}$$\\; < \\over \\sim \\;$$10^{24}$  cm$^{-2}$ ) is required.", "In particular, the three sources where the presence of an AGN is particularly evident in the mid-IR regime from our SED decomposition (U4950, U4958, and U5877) – as already pointed out by F10 on the basis of the presence of a powerlaw mid-IR SED (U4950 and U5877), optical emission lines (U4958, with an apparently broad C iii] feature) and optically unresolved nucleus (U4950 and U5877) – also show a relatively bright X-ray counterpart (see Sec.", "REF ).", "The optical depth at 9.7 $\\mu $ m of the best-fitting solutions corresponds to $N_{H}\\sim {(2-7)}{\\times }10^{22}$  cm$^{-2}$ (i.e., in the Compton-thin regime; $<N_{H}>=6{\\times }10^{22}$  cm$^{-2}$ is obtained once the solutions at the 3$\\sigma $ level are considered).", "Of the remaining six sources, one (U4639) has an association with a relatively weak X-ray source, while for the others only an upper limit to the X-ray emission can be placed (see Table REF ).", "For these six sources, the SED-fitting procedure indicates an obscuration still in the Compton-thin regime, but higher (up to $4{\\times }10^{23}$  cm$^{-2}$ , with $<N_{H}>=2{\\times }10^{23}$  cm$^{-2}$ when the solutions at the 3$\\sigma $ level are taken into account) than for the previous three sources.", "Further insights on the properties of these sources, hence on the nature of their broad-band emission, will be discussed using X-ray diagnostics (see Sec.", "REF ).", "Turning now to source energetics, for the nine objects with an AGN component from the mid-IR, we computed the total and nuclear luminosity in three different spectral ranges: the whole (8–1000 $\\mu $ m) IR range, the mid-IR (5–30 $\\mu $ m) range (partially sampled by IRS), and the narrow 2–6 $\\mu $ m wavelength interval.", "We find that the 8–1000 $\\mu $ m luminosity is always completely dominated by star-formation emission, the AGN nuclear contribution being $$ < $$ 5%$ (see Fig.~\\ref {figure_agn_contr}, {\\it bottom panel}),and that in only one source (U4950) out of the nine is the nuclear componentcontribution significant ($ 20%$).Our finding that starburst processes dominate the 8--1000 $$memission is consistent with the F10 and N12 conclusions.$ In the 5–30 $\\mu $ m range, where the re-processed emission from the dust surrounding the nuclear source peaks (e.g., [53]), we find a larger AGN contribution (i.e.", "${\\sim }$ 25%; see Fig.", "REF , middle panel).", "As shown in Fig.1a,b, at ${\\lambda }{\\sim }10$  ${\\mu }$ m the nuclear component typically starts being overwhelmed by the starburst emission.", "We note that the mid-IR AGN/starburst relative contribution is also discussed by F10, adopting a completely different method than ours (i.e., scaling the SED of Mrk 231 to the 5.8 $\\mu $ m continuum and fitting the residual SED with the average starburst from [7]).", "Excluding from their analysis the three sources with the strongest AGN evidence from either X-ray or optical data (U4950, U4958, and U5877), F10 found a nuclear fraction of $\\sim $ 20% (see their Fig.", "22, top panel), which is consistent with our results.", "The only wavelength range where we find that the AGN overcomes the galaxy emission and contributes to ${\\sim }$ 60% of the emission is the narrow 2–6 ${\\mu }$ m interval (see Fig.", "REF , top panel).", "The power of the near-IR spectral range to detect obscured AGN emission confirms previous results (e.g., [48] using $L$ -band spectroscopy, but see also [29], where a weak near-IR excess continuum emission, detected in disk galaxies thanks to ISOPHOT spectral observations, was ascribed to interstellar dust emission at temperatures of ${\\sim }10^{3}$ K).", "The dominance of the AGN emission in the narrow 2–6 ${\\mu }$ m interval is not in contrast with N12 conclusions, i.e.", "that these sources in the mid-IR are dominated by the PAH features, once the different spectral range is taken into account.", "In Table REF , we report the total IR luminosities (8–1000 $\\mu $ m), the AGN fractions over the three wavelengths discussed above (for the nine sources with an AGN emission detected at least at the 3$\\sigma $ confidence level), and both observed and predicted 2–10 keV luminosities for our sample.", "The latter luminosities have been derived from the 5.8 $\\mu $ m luminosity, using the [19] correlation (see Sec.", "REF for further details), and only for the nine sources with an AGN component detected at least at 3$\\sigma $ confidence (last column of Table REF ).", "The integration of the SED over the 8–1000 $\\mu $ m range confirms the ULIRG classification ($L_{IR}>10^{12}L_{\\odot }$ ), inferred by F10 on the basis of the 24 $\\mu $ m flux densities and redshifts, for 14 out of the 24 sources, the remaining 10 sources showing slightly lower IR luminosities, between 0.5${\\times }10^{12}$ and $10^{12}~L_{\\odot }$ .", "The fact that 24 ${\\mu }$ m–based measurements tend to over-estimate the $L_{IR}$ for $z{\\sim }2$ sources is consistent with other works based on Herschel-PEP data (e.g., [16]; N12) and stacking methods (e.g., [41]).", "The final IR luminosity range of our sample is 0.5–2.8${\\times }10^{12}~L_{\\odot }$ ($<L_{IR}>=1.4{\\times }10^{12}~L_{\\odot }$ , with a dispersion $\\sigma =7{\\times }10^{11}~L_{\\odot }$ )." ], [ "X-ray results", "The 4 Ms X-ray source catalog in the CDF-S ([63]) provides additional information for the sub-sample of eight matched sources (see Table REF ).", "The depth of the Chandra mosaic in the field, though variable across its area, provides constraints down to very faint flux limits ($\\sim 10^{-17}$   ${\\rm erg~cm}^{-2}~{\\rm s}^{-1}$ in the 0.5–2 keV band).", "In the following, we use the X-ray information to provide an independent estimate of the AGN presence in our sample of IR-luminous galaxies and, for sources with most counts, characterize their X-ray emission.", "We also provide upper limits to the X-ray luminosity of the sources which are not detected in the 4 Ms CDF-S image.", "A basic source classification is reported by [63] (AGN/galaxy/star), where five different classification criteria are adopted to separate AGN from galaxiesThe criteria to classify a source as an AGN are: high X-ray luminosity (above $3\\times 10^{42}$  erg s$^{-1}$ ); flat effective photon index ($\\Gamma \\le 1.0$ ); X-ray-to-optical flux ratio log(f$_{\\rm X}$ /f$_{\\rm R}$ )$>-1$ ; an X-ray luminosity at least three times higher than that possibly due to star formation (see [1]); presence of either broad or high-ionisation emission lines in the optical spectra.", "See $§$ 4.4 of [63] for details.", "; at least one of these criteria should be satisfied.", "According to such classification, only two sources of the present sample are classified as galaxies, U428 and U4639, the remaining six being AGN.", "However, in the following we will provide indications for a “revision” of this classification using all of the available X-ray information (e.g., spectral properties, luminosities, count distribution).", "In Table REF the source classifications from X-ray data ([63] and current work, respectively), along with the results from the SED-fitting analysis (see Sec.", "REF ), are presented.", "X-ray luminosities, coupled with X-ray spectral analysis (see below), indicate the presence of an AGN in the three sources (U4950, U4958, and U5877) for which the SED fitting analysis and F10 already suggested an AGN.", "Also U4642 has an X-ray luminosity typical of AGN emission.", "In comparison with the 2 Ms Chandra data ([30]) used by F10, U4958 represents a new AGN detection, although the presence of an active nucleus was already inferred by the mid-IR featureless spectra and the optical spectrum, showing a broad C iii] line and strong N v and C iv emission lines.", "All of the X-ray matched sources have a full-band (0.5–8 keV) detection.", "One source (U4958) has an upper limit in the soft band (0.5–2 keV); this result is suggestive of heavy absorption, since Chandra has its highest effective area, and hence best sensitivity, in this energy range.", "Basic X-ray spectral analysis (using an absorbed powerlaw with photon index fixed to 1.8, as typically observed in AGN; [42]), actually confirms the presence of strong, possibly Compton-thick obscurationA source is called Compton-thick if its column density $N_{\\rm H}>1/\\sigma _{\\rm T}\\sim 1.5\\times 10^{24}$  cm$^{-2}$ , where $\\sigma _{\\rm T}$ is the Thomson cross section.", "towards this source (N$_{\\rm H}=1.9^{+1.4}_{-0.7}\\times 10^{24}$  cm$^{-2}$ ).", "Four sources have hard-band (2–8 keV) upper limits: U428, U4639, U5632, and U5775, all of which being characterized by limited counting statistics (12–26 counts in the full band).", "Their X-ray luminosity (see Table REF )For the four sources with 2–8 keV upper limits, the 2–10 keV luminosities are reported in Table REF as these sources were detected in the hard band.", "We note, in fact, that the values reported in the table are derived from the 0.5–8 keV fluxes, where all of these sources are detected.", "is consistent with star-formation activity, although the presence of a low-luminosity AGN cannot be excluded.", "U4950 and U5877 are characterized by $\\sim $ 1420 and $\\sim $ 360 counts in the 0.5–8 keV band, which allow a moderate-quality X-ray spectral analysis.", "Both sources can be fitted using a powerlaw model, but the flat photon index ($-1$ and $+$ 1, respectively) is indicative of absorption, which has been estimated to be 1.2$^{+0.3}_{-0.2}\\times 10^{23}$  cm$^{-2}$ and 6.0$^{+2.0}_{-1.5}\\times 10^{23}$  cm$^{-2}$ , respectively, after fixing the photon index to 1.8.", "The same model applied to U4642 data provides good results, although the derived column density is poorly constrained ($N_{\\rm H}=4.5^{+6.1}_{-4.3}\\times 10^{22}$  cm$^{-2}$ ).", "Apart from U4950, U4958, U5877 and U4642, for the remaining X-ray matched sources, the limited counting statistics prevent us from placing constraints on any possible obscuration.", "For the sources which were not detected in the 4 Ms CDF-S observations, we derived rest-frame 2–10 keV luminosity upper limits by converting the full-band sensitivity map (weighted over a region surrounding the source of interest of 4 radius to minimize possible spurious fluctuations in the map) using a powerlaw with $\\Gamma $ =1.4 (see Table REF ).", "The assumption of $\\Gamma $ =1.8 would imply luminosities higher by $\\sim 0.1-0.2$ dex.", "The derived X-ray upper limits place strong constraints on the further presence of luminous AGN in our sample, unless they are heavily obscured.", "Following F10, for the nine sources with an AGN from the SED fitting, we estimated the rest-frame X-ray emission using the mid-IR emission of the AGN component as a proxy of the nuclear power (Table REF ).", "In particular, we use the $L_{2-10keV}$ –5.8 $\\mu $ m relation ([19]; $\\log L_{2-10keV}=43.57 + 0.72\\times (\\log L_{5.8\\hbox{$\\mu $m}}-44.2)$ , with luminosities expressed in erg s$^{-1}$ ; see also [31]), which was calibrated using unobscured AGN in the CDF-S and COSMOS surveys.", "In Fig.", "REF the predicted and measured X-ray luminosities are compared.", "For the three sources with clear AGN signatures (also from X-ray data), the luminosities have been corrected for the absorption (see Table REF and above), while for the remaining sources, no correction has been applied, due to the lack of constraints to any possible column density from the low-count X-ray spectra.", "Overall, it appears that the three sources showing clear AGN signatures in the mid-IR band and in X-rays (U4950, U4958 and U5877) are broadly consistent with the 1:1 relation, considering the dispersion in the mid-IR/X-ray relation, the observational errors, and the uncertainties in deriving the intrinsic 2–10 keV luminosity properly.", "The correction accounting for the absorption seems to be correct also for the most heavily obscured of our X-ray sources, U4958.", "For the other six sources, the expected 2–10 keV luminosity is a factor of ${\\sim }$ 10–100 higher than the measured X-ray luminosity.", "Although caution is obviously needed in all the cases where mid-IR vs. X-ray luminosity correlations are used (see, e.g., the discussion in [60]), the obscuration derived from the SED fitting for the remaining six sources ($\\sim 2{\\times }$ 10$^{23}$  cm$^{-2}$ ) and the ratio between the measured and predicted luminosities suggest the presence of significant obscuration in these sources.", "Our conclusion is robust against the possible presence of absorption affecting also the $L_{5.8\\hbox{$\\mu $m}}$ values, since in this case the predicted X-ray luminosities should be considered as lower limits.", "Sensitive X-ray data over a larger bandpass would be needed to test the hypothesis of heavy obscuration in these sources.", "Figure: Predicted vs. measured 2–10 keV luminosities for the ninesources where an AGN component is inferred from the SED fitting.", "Thetriangles indicate the three sources with clear AGN X-ray emission;for these three sources only, the 2–10 keV luminosities have beencorrected for absorption derived from X-ray spectral analysisdata (see Table ).The dashed diagonal lines indicate ratios of 1:1, 10:1 and 100:1 between thepredicted and the measured X-ray luminosity (from right to left).Table: IR and X-ray luminosities, and AGN fractions for the sampleof z∼2z\\sim 2 IR-luminous galaxies" ], [ "Summary", "In this paper we have studied the multi-band properties of a sample of IR luminous sources at $z{\\sim }2$ in order to estimate the AGN contribution to their mid-IR and far-IR emission.", "The sample was selected by F10 at faint 24 $\\mu $ m flux densities ($S(24{\\mu }m){\\sim }0.14-0.55$  mJy) and $z=1.75-2.40$ to specifically target luminosities around $10^{12}~L_{\\odot }$ , i.e.", "sampling the knee of the IR luminosity function.", "We have extended the previous analysis by taking advantage of new far-IR data recently obtained by the Herschel satellite as part of the guaranteed survey “PACS Evolutionary Probe” (PEP, [32]), and of the recently published 4 Ms Chandra data ([63]).", "The available photometry, coupled with IRS mid-IR data, have been used to reconstruct the broad-band SEDs of our IR-luminous galaxies.", "These SEDs have been modeled using a SED-fitting technique with three components, namely a stellar, an AGN, and a starburst component.", "The most up-to-date smooth torus model by F06 have been adopted for the AGN emission.", "The major results of the work can be summarized as follows.", "SED fitting indicates that emission from the host galaxy in the optical/near-IR and star formation in the mid-IR/far-IR is required for all of the sources.", "Presence of an AGN component is consistent with the data for all but one source (U428), although only for 9 out of the 24 galaxies (${\\sim }35\\%$ of the sample) is this emission significant at least at the 3$\\sigma $ level.", "Of the sub-sample of nine sources that likely harbour an AGN according to the SED fitting, we find that their total (8–1000 ${\\mu }$ m) and mid-IR (5–30 $\\mu $ m) emissions are dominated by starburst processes, with the AGN-powered emission accounting for only $\\sim 5\\%$ and $\\sim 23\\%$ of the energy budget in these wavelength ranges, respectively.", "We find that the AGN radiation overcomes the stellar + starburst components only in the narrow 2–6 $\\mu $ m range, where it accounts for $\\sim $ 60% of the energy budget.", "In this wavelength range, stellar emission has significantly declined and emission from PAH features and starburst emission is not prominent yet.", "For this same sub-sample, the gas column densities (derived by converting the dust optical depth at 9.7 $\\mu $ m obtained from the SED fitting) are indicative of a significant level of obscuration.", "In particular, three sources, also detected as relatively bright X-ray sources (U4950, U4958, and U5877, see below), have $<N_{H}>=6{\\times }10^{22}$  cm$^{-2}$ (considering all the solutions at the 3$\\sigma $ confidence level), while the remaining six sources have $<N_{H}>={2}{\\times }10^{23}$  cm$^{-2}$ .", "X-ray analysis confirm that three sources (U4950, U4958, and U5877) are actually powered by an AGN at short wavelengths, and that this AGN varies from being moderately (U4950 and U5877) to heavily obscured, possibly Compton thick (U4958).", "The X-ray luminosity of U4642 is also suggestive of moderately obscured AGN emission.", "The remaining four sources detected by Chandra have X-ray emission consistent with star-formation processes.", "For the six sources where the AGN is required only by SED fitting (i.e., no strong AGN emission is observed in X-ray), we estimate an intrinsic X-ray nuclear luminosity from the AGN continuum at 5.8 ${\\mu }$ m. The ratio (from 10 up to 100) between the predicted and the measured luminosities suggests the presence of significant obscuration in these sources." ], [ "acknowledgements", "The authors thank the referee for his/her useful comments.", "FP and CV thank the Sterrenkundig Observatorium (Universiteit Gent), in particular Prof. M. Baes and Dr. J. Fritz, for their kind hospitality.", "The authors are grateful to F.E.", "Bauer and F. Vito for their help with Chandra spectra.", "CV acknowledges partial support from the Italian Space Agency (contract ASI/INAF/I/009/10/0).", "PACS has been developed by a consortium of institutes led by MPE (Germany) and including UVIE (Austria); KU Leuven, CSL, IMEC (Belgium); CEA, LAM (France); MPIA (Germany); INAF- IFSI/OAA/OAP/OAT, LENS, SISSA (Italy); IAC (Spain).", "This development has been supported by the funding agencies BMVIT (Austria), ESA-PRODEX), CEA/CNES (France), DLR (Germany), ASI/INAF (Italy), and CICYT/MCYT (Spain)." ] ]

[ [ "Quantum discord evolution of three-qubit states under noisy channels" ], [ "Abstract We investigated the dissipative dynamics of quantum discord for correlated qubits under Markovian environments.", "The basic idea in the present scheme is that quantum discord is more general, and possibly more robust and fundamental, than entanglement.", "We provide three initially correlated qubits in pure Greenberger-Horne-Zeilinger (GHZ) or W state and analyse the time evolution of the quantum discord under various dissipative channels such as: Pauli channels $\\sigma_{x}$, $\\sigma_{y}$, and $\\sigma_{z}$, as well as depolarising channels.", "Surprisingly, we find that under the action of Pauli channel $\\sigma_{x}$, the quantum discord of GHZ state is not affected by decoherence.", "For the remaining dissipative channels, the W state is more robust than the GHZ state against decoherence.", "Moreover, we compare the dynamics of entanglement with that of the quantum discord under the conditions in which disentanglement occurs and show that quantum discord is more robust than entanglement except for phase flip coupling of the three qubits system to the environment." ], [ "Introduction", "One of the most remarkable properties of quantum mechanics is represented by the quantum correlation.", "Entanglement, which is a prominent feature of quantum correlation plays an important role in quantum computing and informational processing [1], [2], [3], [4].", "Recently, it has been perceived that entanglement is not the only kind of quantum correlation.", "In this context, others suitable measures of quantum correlations such as quantum discord [5], quantum deficit [6], [7], [8], quantumness of correlations [9] and quantum dissonance [10] have been proposed.", "Among them, quantum discord as a measure that is based on the difference between two quantum extensions of classically equivalent concepts has received considerable attention.", "This measure which quantifies the all nonclassical correlations between parts of a quantum system, actually supplements the measure of entanglement.", "For pure entangled states quantum discord coincides with the entropy of entanglement.", "It can also be nonzero for some mixed separable state.", "It is worth mentioning that quantum discord is related to other concepts such as Maxwell's demons [11], [12], quantum phase transitions [13], [14], completely positive maps [11], and relative entropy [15].", "Moreover, the characteristics of quantum discord have been studied in some physical models and information processing.", "It was shown that quantum discord can be considered as a more universal quantum resource than quantum entanglement in some sense.", "It offers new prospects for quantum information processing [16], [17], [18], [19], [20].", "Studying of the quantum discord evolution exposed to noisy environments has led to the surprising result that it may obviously differ from entanglement evolution.", "In fact, the unavoidable interaction of the systems with their environment, based on completely positive quantum dynamical semi groups, can be modelled by means of the noisy quantum channels.", "Quantum channels are completely positive and trace preserving maps between spaces of operators.", "The study of those can be broadly divided into Markovian and non-Markovian channels depend on the interaction of the system and environment.", "Markovian channels describes memoryless environments.", "The prototype of it is given by a quantum dynamical semi group, that is by solving a master equation for the reduced density matrix with Lindblad structure [21], [22], [23].", "For non-Markovian channels, environmental memory plays an important role, so the master equations describing their dynamics are often complicated integro-differential equations which are rarely exactly solvable [22], [23].", "When a system of qubits with quantum correlation is exposed to noisy channels disentanglement can occur suddenly, but the quantum discord mostly decays in the asymptotic time [24], [25], [28], [26], [27].", "This points to a controversial fact that quantum discord may be more robust against decoherence than entanglement.", "Hence quantum algorithms that are based only on the quantum discord correlations can be more robust than those based on entanglement.", "The aim of this paper is to illustrate the mentioned subject for a system of three-qubit which is initially prepared in pure state by Greenberger-Horne-Zeilinger (GHZ) [29] or W [30] state as $ |\\psi ^{GHZ}\\rangle &=&\\frac{1}{\\sqrt{2}}(|000\\rangle +|111\\rangle )\\cr |\\psi ^{W}\\rangle &=&\\frac{1}{2}(|100\\rangle +|010\\rangle +\\sqrt{2}|001\\rangle ).$ Jung et al.", "[31] showed that these pure states will be mixed due to transmission through some of the common channels for qubits.", "Moreover, it was shown that under these noisy channels the sudden death of entanglement of three qubits occurs in a finite time [32].", "The question is, what happens to quantum discord under the same conditions in which disentanglement can occur?", "In reply to this question we lead to remarkable results.", "The decoherence induced by the Pauli channel $\\sigma _{x}$ , can not affect the quantum discord of GHZ state contrary to the W state.", "We also show that the W state lose less quantum discord than the GHZ state due to transmission through the Pauli channels $\\sigma _{y}$ , $\\sigma _{z}$ and the depolarising channel.", "Comparison of entanglement and quantum discord demonstrates that for the Pauli channels $\\sigma _{x}$ and $\\sigma _{y}$ and the depolarising channel, quantum discord is more robust against decoherence than entanglement.", "The organisation of this paper is as follows: Section 2 is devoted to the necessary theoretical background to describe the time evolution of the system and introduces the global quantum discord.", "Evolution of quantum discord in transmission through the Pauli and the depolarising channels for the GHZ and W states is calculated in section 3 and 4, respectively.", "Finally, we summarise our results.", "In an open quantum system the prototype of a Markov process is given by a quantum dynamical semigroup of a completely positive and trace preserving map $\\Phi (t)=\\exp [£t]$ , $t\\ge 0$ .", "In this case, quantum evolution of the system is given by the solution of a Markovian master equation with Lindblad structure for the reduced density matrix [21], $ \\frac{d}{dt}\\rho (t)&=&£\\rho (t)\\cr &=& -i[H_{s},\\rho (t)]+\\sum _{i}L_{i}\\rho (t)L_{i}^{\\dag }-\\frac{1}{2}\\lbrace L_{i}^{\\dag }L_{i},\\rho (t)\\rbrace .$ For any Pauli channel $\\sigma _{\\alpha }$ ($\\alpha =x,y,z$ ), the decoherence dynamic is described by Lindblad operators $L_{A_{j},\\alpha }\\equiv \\sqrt{\\kappa _{A_{j},\\alpha }}\\,\\sigma _{\\alpha }^{A_{j}}$ , $j=1,2,3$ , which act independently upon the $j$ -th qubit.", "In these operators, $\\sigma _{\\alpha }^{A_{j}}$ denote the Pauli matrices of the $j$ -th qubit and the constants $\\kappa _{A_{j},\\alpha }$ are relaxation rates.", "For depolarising channel nine of these operators are needed.", "Here we assume that the Hamiltonian of the system is zero $H_{s}=0$ , and the strength of the coupling between each of the qubits and channels is equal.", "Recently, Jung et al.", "[31] analysed time evolution of three qubit GHZ and W states in the presence of noisy channels.", "Here we use their results." ], [ "Quantum discord", "For a bipartite system AB quantum discord is given by [5] $ \\textit {D}(\\rho ^{AB})=\\textit {I}(\\rho ^{AB})- \\textit {C}(\\rho ^{AB}),$ where $\\textit {I}(\\rho ^{AB})=S(\\rho ^{A})+S(\\rho ^{B})-S(\\rho ^{AB})$ , is quantum mutual information which includes the total correlation between A and B.", "The last section on the right represents classical correlation $\\textit {C}(\\rho ^{AB})=max_{\\lbrace \\Pi _{k}\\rbrace }[S(\\rho ^{A})-S(\\rho ^{AB}|\\lbrace \\Pi _{k}\\rbrace )]$ with $S(\\rho )=-Tr[\\rho \\log _{2}\\rho ]$ as the von-Neumann entropy.", "Notice that the maximum is taken over the set of projective measurements $\\lbrace \\Pi _{k}\\rbrace $ [34].", "By definition the conditional density operator $\\rho ^{AB}_{k}=\\frac{1}{p_{k}}\\lbrace (I^{A}\\otimes \\Pi _{k}^{B})\\rho ^{AB}(I^{A}\\otimes \\Pi _{k}^{B})\\rbrace $ with $p_{k}=Tr[(I^{A}\\otimes \\Pi _{k}^{B})\\rho ^{AB}]$ as the probability of obtaining the outcome $k$ .", "We can define the conditional entropy of $A$ as $S(\\rho ^{AB}|\\lbrace \\Pi _{k}\\rbrace )=\\sum _{k}p_{k}S(\\rho _{k}^{A})$ .", "This entropy includes the knowledge of subsystem $B$ , with $\\rho _{k}^{A}=Tr_{B}[\\rho _{k}^{AB}]$ and $S(\\rho _{k}^{A})=S(\\rho _{k}^{AB})$ .", "It has been shown that $\\textit {D}(\\rho ^{AB})\\ge 0$ with the equal sign, only for classical correlation [35].", "Very recently, Rulli et al.", "[33] have proposed a global measure of quantum discord based on a systematic extension of the bipartite quantum discord.", "Global quantum discord (GQD) which satisfy the basic requirements of a correlation function, for an arbitrary multipartite state $\\rho ^{A_{1}...A_{N}}$ under a set of local measurement $\\lbrace \\Pi _{j}^{A_{1}}\\otimes ... \\otimes \\Pi _{j}^{A_{N}}\\rbrace $ is defined as $ \\textit {D}(\\rho ^{A_{1}...A_{N}})=\\min _{\\lbrace \\Pi _{k}\\rbrace }\\,[S(\\rho ^{A_{1}...A_{N}}\\Vert \\Phi (\\rho ^{A_{1}...A_{N}}))-\\sum _{j=1}^{N}S(\\rho ^{A_{j}}\\Vert \\Phi _{j}(\\rho ^{A_{j}}))].$ Where $\\Phi _{j}(\\rho ^{A_{j}})=\\sum _{i}\\Pi _{i}^{A_{j}}\\rho ^{A_{j}}\\Pi _{i}^{A_{j}}$ and $\\Phi (\\rho ^{A_{1}...A_{N}})=\\sum _{k}\\Pi _{k}\\rho ^{A_{1}...A_{N}}\\Pi _{k}$ with $\\Pi _{k}=\\Pi _{j_{1}}^{A_{1}}\\otimes ... \\otimes \\Pi _{j_{N}}^{A_{N}}$ and $k$ denoting the index string ($j_{1}...j_{N}$ ).", "We could eliminate dependence on measurement by minimising the set of projectors $\\lbrace \\Pi _{j_{1}}^{A_{1}}, ... ,\\Pi _{j_{N}}^{A_{N}}\\rbrace $ .", "With these remarks about the global quantum discord (4), one can describe the time evolution of the quantum discord for three-qubit GHZ and W states when they are passed through a noisy channel.", "By selecting a set of von-Neumann measurements as $\\Pi _{1}^{A_{j}}=\\left({\\cos ^{2}(\\frac{\\theta _{j}}{2})& \\,\\,\\,\\,\\,\\,\\,\\,\\,e^{i\\varphi _{j}}\\cos (\\frac{\\theta _{j}}{2})\\sin (\\frac{\\theta _{j}}{2})\\cr e^{-i\\varphi _{j}}\\cos (\\frac{\\theta _{j}}{2})\\sin (\\frac{\\theta _{j}}{2})& \\,\\,\\,\\,\\,\\,\\,\\,\\,\\sin ^{2}(\\frac{\\theta _{j}}{2})}\\right), \\cr \\cr \\cr \\Pi _{2}^{A_{j}}=\\left({\\sin ^{2}(\\frac{\\theta _{j}}{2})& -e^{-i\\varphi _{j}}\\cos (\\frac{\\theta _{j}}{2})\\sin (\\frac{\\theta _{j}}{2})\\cr -e^{i\\varphi _{j}}\\cos (\\frac{\\theta _{j}}{2})\\sin (\\frac{\\theta _{j}}{2})&\\cos ^{2}(\\frac{\\theta _{j}}{2})}\\right),$ the quantum discord for $\\rho =\\rho ^{GHZ}(t)$ or $\\rho ^{W}(t)$ can be obtained from $ \\textit {D}(\\rho )=\\min _{\\lbrace \\theta _{j},\\varphi _{j}\\rbrace }\\,[S(\\rho \\Vert \\Phi (\\rho ))-\\sum _{j=1}^{3}S(\\rho ^{A_{j}}\\Vert \\Phi _{j}(\\rho ^{A_{j}}))].$ Where $\\theta _{j}\\in [0,\\pi )$ and $\\varphi _{j}\\in [0,2\\pi )$ , for $j=1,2,3$ , are azimuthal and polar angles, respectively." ], [ "Pauli channel $\\sigma _{x}$", "When a three-qubit system with initial GHZ state is coupled to a shift-flip noise channel, in which each qubit is coupled to its own channel, the time evolution is obtained by the solution of the master equation (2) with Lindbald operators $L_{j,x}\\equiv \\sqrt{\\kappa _{j,x}}\\,\\sigma _{x}^{A_{j}}, (j=1,2,3$ ).", "For this coupling, the master equation reduces to 8 diagonal and 28 off-diagonal coupled linear differential equations.", "So after transmission of the GHZ through the Pauli channel $\\sigma _{x}$ the density matrix is given by [31] $\\rho _{x}^{GHZ}(t)=\\frac{1}{8}\\left({\\alpha _{+}&0&0&0&0&0&0&\\alpha _{+}\\cr 0&\\alpha _{-}&0&0&0&0&\\alpha _{-}&0 \\cr 0&0&\\alpha _{-}&0&0&\\alpha _{-}&0&0 \\cr 0&0&0&\\alpha _{-}&\\alpha _{-}&0&0&0 \\cr 0&0&0&\\alpha _{-}&\\alpha _{-}&0&0&0 \\cr 0&0&\\alpha _{-}&0&0&\\alpha _{-}&0&0 \\cr 0&\\alpha _{-}&0&0&0&0&\\alpha _{-}&0 \\cr \\alpha _{+}&0&0&0&0&0&0&\\alpha _{+}}\\right),$ with $\\alpha _{+}&\\equiv & 1+3e^{-4\\kappa t},\\cr \\alpha _{-}&\\equiv & 1-e^{-4\\kappa t}.$ For this case, the lower bound to the concurrence is [32] $\\tau _{3}(\\rho _{x}^{GHZ}(t))=e^{-4\\kappa t}.$ In order to find the analytical expression for the quantum discord with $\\rho _{x}^{GHZ}(t)$ we must consider equation (6).", "By tracing out two qubits, the one qubit density matrices representing the individual subsystems are proportional to the identity operator $ \\rho _{x}^{(A_{1})}(t)=\\rho _{x}^{(A_{2})}(t)=\\rho _{x}^{(A_{3})}(t)=\\frac{\\alpha _{+}+3\\alpha _{-}}{8}I.$ Therefore, we have $S(\\rho _{x}^{(j)}(t)\\Vert \\Phi _{j}(\\rho _{x}^{(j)}(t)))=0$ $(j=1,2,3)$ and the equation(6) reduces to $ \\textit {D}(\\rho _{x}^{GHZ}(t))&=&\\min _{\\lbrace \\theta _{j},\\varphi _{j}\\rbrace }\\lbrace S(\\rho _{x}^{GHZ}(t)\\Vert \\Phi (\\rho _{x}^{GHZ}(t)))\\rbrace \\cr &=&\\min _{\\lbrace \\theta _{j},\\varphi _{j}\\rbrace }\\lbrace S(\\Phi (\\rho _{x}^{GHZ}(t)))-S(\\rho _{x}^{GHZ}(t))\\rbrace .$ The entropy $S(\\rho _{x}^{GHZ}(t))$ can be obtained as $ S(\\rho _{x}^{GHZ}(t))=2-\\frac{3\\alpha _{-}}{4}\\log _{2}\\alpha _{-}-\\frac{\\alpha _{+}}{4}\\log _{2}\\alpha _{+}.$ In order to obtain the maximum classical correlation among the parts of $\\rho _{x}^{GHZ}(t)$ , by varying the angles $\\theta _{j}$ and $\\varphi _{j}$ , we must find the measurement bases that minimise $\\textit {D}(\\rho _{x}^{GHZ}(t))$ .", "After some calculation we have perceived that, by adopting local measurements in the $\\sigma _{z}$ eigenbases for each particle, the value of quantum discord will be minimised.", "So the von-Neumann entropy, after completing such measurements, can be written as $ S(\\Phi (\\rho _{x}^{GHZ}(t)))=3-\\frac{3\\alpha _{-}}{4}\\log _{2}\\alpha _{-}-\\frac{\\alpha _{+}}{4}\\log _{2}\\alpha _{+}.$ By substituting equations (12) and (13) into equation (11) quantum discord is readily found to be $ \\emph {D}(\\rho _{x}^{GHZ}(t))=1.$ As can be seen, the Pauli channel $\\sigma _{x}$ can not change the quantum discord of the three qubit systems with the initial GHZ state.", "We have plotted the dynamic evolution of the quantum discord for density matrix (7) versus the dimensionless scaled time $\\kappa t$ in Fig.", "1 (dashed violet )." ], [ "Pauli channel $\\sigma _{y}$", "If the three-qubit GHZ state are transmitted through the Pauli channel $\\sigma _{y}$ , its density matrix takes the following form [31] $\\hspace{-19.91692pt}\\rho _{y}^{GHZ}(t)=\\frac{1}{8}\\left({\\alpha _{+}&0&0&0&0&0&0&\\beta _{1}\\cr 0&\\alpha _{-}&0&0&0&0&-\\beta _{2}&0 \\cr 0&0&\\alpha _{-}&0&0&-\\beta _{2}&0&0 \\cr 0&0&0&\\alpha _{-}&-\\beta _{2}&0&0&0 \\cr 0&0&0&-\\beta _{2}&\\alpha _{-}&0&0&0 \\cr 0&0&-\\beta _{2}&0&0&\\alpha _{-}&0&0 \\cr 0&-\\beta _{2}&0&0&0&0&\\alpha _{-}&0 \\cr \\beta _{1}&0&0&0&0&0&0&\\alpha _{+}}\\right),$ where $\\alpha _{\\pm }$ are given in equation (8) and, $\\beta _{1}$ and $\\beta _{2}$ are defined as $\\beta _{1}&\\equiv &3e^{-2\\kappa t}+e^{-6\\kappa t},\\cr \\beta _{2}&\\equiv &e^{-2\\kappa t}-e^{-6\\kappa t}.$ For this matrix, the entanglement vanishes after some finite time due to the condition [32] $\\tau _{3}(\\rho _{y}^{GHZ}(t))=max\\lbrace 0,\\frac{1}{4}(3e^{-2\\kappa t}+e^{-4\\kappa t}+e^{-6\\kappa t}-1)\\rbrace .$ The reduced density matrices of subsystem of (15), is the same as equation (10).", "Hence, the quantum discord becomes $ \\textit {D}(\\rho _{y}^{GHZ}(t))=\\min _{\\lbrace \\theta _{j},\\varphi _{j}\\rbrace }\\lbrace S(\\Phi (\\rho _{y}^{GHZ}(t)))-S(\\rho _{y}^{GHZ}(t))\\rbrace .$ When the projective measurement in the eigenprojectors of $\\sigma _{z}$ is performed on any one of the remaining qubits of $\\rho _{y}^{GHZ}(t)$ , the minimum quantum discord is obtained as $ \\textit {D}(\\rho _{y}^{GHZ}(t))&=&\\frac{(\\alpha _{+}-\\beta _{1})}{8}\\log _{2}(\\alpha _{+}-\\beta _{1})+\\frac{(\\alpha _{+}+\\beta _{1})}{8}\\log _{2}(\\alpha _{+}+\\beta _{1})\\cr &+&\\frac{3(\\alpha _{-}-\\beta _{2})}{8}\\log _{2}(\\alpha _{-}-\\beta _{2})+\\frac{3(\\alpha _{-}+\\beta _{2})}{8}\\log _{2}(\\alpha _{-}+\\beta _{2})\\cr &-&\\frac{\\alpha _{+}}{4}\\log _{2}(\\alpha _{+})-\\frac{3\\alpha _{-}}{4}\\log _{2}(\\alpha _{-}).$ Quantum discord is reduced due to the Pauli noisy channel $\\sigma _{y}$ and disappears with low speed, as shown in Fig.", "1 (solid blue )." ], [ "Pauli channel $\\sigma _{z}$", "Transmission of the GHZ state through the Pauli channel $\\sigma _{z}$ result in the master equation (2) reduces to 8 diagonal and 28 off-diagonal first order differential equations with a simply trivial solution.", "So $\\rho ^{GHZ}(0)=|GHZ \\rangle \\langle GHZ|$ is evolved into [31] $ \\hspace{-25.60747pt}\\rho _{z}^{GHZ}(t)=\\frac{1}{2}(\\vert 000 \\rangle \\langle 000 \\vert + \\vert 111 \\rangle \\langle 111 \\vert ) +\\frac{1}{2}e^{-6\\kappa t}(\\vert 000 \\rangle \\langle 111 \\vert + \\vert 000 \\rangle \\langle 111 \\vert ).", "\\nonumber \\\\$ For this mixed state, the lower bound to concurrence is a mono-exponential function of time $ \\tau _{3}(\\rho _{z}^{GHZ}(t))=e^{-6\\kappa t}.$ Let us now focus on the quantum discord for $\\rho _{z}^{GHZ}(t)$ as defined by (6).", "After tracing out two qubits, the three reduce density matrices are equal, given by $\\rho _{z}^{(A_{1})}(t)=\\rho _{z}^{(A_{2})}(t)=\\rho _{z}^{(A_{3})}(t)=\\frac{I}{2}$ .", "Hence, we have $S(\\rho _{z}^{(A_{j})}(t)\\Vert \\Phi _{j}(\\rho _{z}^{(A_{j})}(t)))=0$ ($j=1,2,3$ ) and the equation (6) reduces to $ \\textit {D}(\\rho _{z}^{GHZ}(t))=\\min _{\\lbrace \\theta _{j},\\varphi _{j}\\rbrace }\\lbrace S(\\Phi (\\rho _{z}^{GHZ}(t)))-S(\\rho _{z}^{GHZ}(t))\\rbrace .$ The von-Neumann entropy of $\\rho _{z}^{GHZ}(t)$ is $ \\hspace{-28.45274pt}S(\\rho _{z}^{GHZ}(t))=\\lbrace 1-\\frac{1+e^{-6\\kappa t}}{2}\\log _{2}(1+e^{-6\\kappa t})-\\frac{1-e^{-6\\kappa t}}{2}\\log _{2}(1-e^{-6\\kappa t})\\rbrace .\\nonumber \\\\$ We have found that for the density matrix (20), measurements in the eigenprojectors of $\\sigma _{z}^{j}$ , that is boundary values $\\varphi _{j}=0$ and $\\theta _{j}=0$ , minimise $\\Phi (\\rho _{z}^{GHZ}(t))$ as $ \\Phi (\\rho _{z}^{GHZ}(t))=\\frac{1}{2}\\lbrace |+++\\rangle \\langle +++|+ |---\\rangle \\langle ---|\\rbrace ,$ which cause $S(\\Phi (\\rho _{z}^{GHZ}(t)))=1$ .", "In these circumstances, the quantum discord is expressed as $ \\textit {D}(\\rho _{z}^{GHZ}(t))=\\frac{1+e^{-6\\kappa t}}{2}\\log _{2}(1+e^{-6\\kappa t})+\\frac{1-e^{-6\\kappa t}}{2}\\log _{2}(1-e^{-6\\kappa t}).$ In Fig.", "1, we have plotted the dynamic evolution of the quantum discord for the density matrix (20) as a function of $\\kappa t$ (solid red).", "It can be seen that the quantum discord decreases from its maximal value $\\textit {D}(\\rho _{z}^{GHZ}(t))=1$ and vanishes after some finite time." ], [ "Depolarising channel ", "For depolarising noise which is described by nine Lindblad operators, $L_{j,z}$ , $L_{j,x}$ and $L_{j,y}$ ($j=1,2,3$ ) the state of three-qubit system that were initially described by the GHZ state replaces $\\rho _{d}^{GHZ}(t)=\\frac{1}{8}\\left({\\tilde{\\alpha }_{+}&0&0&0&0&0&0&\\gamma \\cr 0&\\tilde{\\alpha }_{-}&0&0&0&0&0&0 \\cr 0&0&\\tilde{\\alpha }_{-}&0&0&0&0&0 \\cr 0&0&0&\\tilde{\\alpha }_{-}&0&0&0&0 \\cr 0&0&0&0&\\tilde{\\alpha }_{-}&0&0&0 \\cr 0&0&0&0&0&\\tilde{\\alpha }_{-}&0&0 \\cr 0&0&0&0&0&0&\\tilde{\\alpha }_{-}&0 \\cr \\gamma &0&0&0&0&0&0&\\tilde{\\alpha }_{+}}\\right),$ where $\\tilde{\\alpha }_{+}&\\equiv & 1+3e^{-8\\kappa t}\\cr \\tilde{\\alpha }_{-}&\\equiv & 1-e^{-8\\kappa t}\\cr \\gamma &\\equiv &4e^{-12\\kappa t}.$ Since $\\rho _{d}^{(A_{j})}(t)=\\frac{\\tilde{\\alpha }_{+}+3\\tilde{\\alpha }_{-}}{8}I$ , then $S(\\rho _{d}^{(A_{j})}(t)\\Vert \\Phi _{j}(\\rho _{d}^{(A_{j})}(t)))=0$ .", "The quantum discord, due to the condition of depolarizing channel may be written as $ \\textit {D}(\\rho _{d}^{GHZ}(t))=\\min _{\\theta _{j},\\varphi _{j}}\\lbrace S(\\Phi (\\rho _{d}^{GHZ}(t)))-S(\\rho _{d}^{GHZ}(t))\\rbrace .$ We find that measurements in the eigenprojectors of $\\sigma _{z}^{j}$ minimise the quantum discord.", "After some algebraic calculation it is found that $ \\textit {D}(\\rho _{d}^{GHZ}(t))&=&\\frac{(\\alpha _{+}+\\gamma )}{8}\\log _{2}(\\alpha _{+}+\\gamma )\\cr &+&\\frac{(\\alpha _{+}-\\gamma )}{8}\\log _{2}(\\alpha _{+}-\\gamma )-\\frac{\\alpha _{+}}{4}\\log _{2}(\\alpha _{+}).$ The time evolution of quantum discord (6) as a function of $\\kappa t$ in the case of the depolarising channel is plotted in Fig.", "1 (solid black).", "Due to the noisy channel, quantum discord decays from its maximum value $\\textit {D}(\\rho _{z}^{GHZ}(0))=1$ and disappears after some finite time.", "As we have pointed out previously, transmission of the GHZ state through the Pauli channel $\\sigma _{x}$ contrary to other channels, could not disturb the quantum discord.", "We also observed that the quantum discord of the initial state due to transmission of the GHZ state through the Pauli channel $\\sigma _{y}$ decayed less quickly than $\\sigma _{z}$ and the depolarising channel, as shown in Fig.", "1.", "In order to comparse quantum discord and entanglement we have used the results of reference [32].", "In Fig.", "2, the time dependent entanglement of the GHZ state under the noisy channels is shown.", "In all cases, one can see decrease of the entanglement due to the noise of the channels.", "Clearly, under the dissipative dynamics considered here, except for phase flip, coupling of the three qubits system to the environment, quantum discord is more robust than entanglement.", "In the next section we will discuss the effects of noisy channels when we prepare three qubits with initial W state." ], [ "Pauli channels $\\sigma _{x}$ and {{formula:8b9d9891-aad4-4614-bf7a-dfb2637afcf5}}", "If the three-qubit system is initially prepared in the W state, a similar analysis as in the previous section to compute the quantum discord can be made.", "Under the bit flip or bit-phase flip coupling to environment the density matrix of three-qubit W state at time t has the following analytical expression [31] $\\hspace{-71.13188pt}\\rho _{\\pm }^{W}(t)=\\frac{1}{16}\\left({2\\alpha _{2}&0&0&\\pm \\sqrt{2}\\alpha _{2}&0&\\pm \\sqrt{2}\\alpha _{2}&\\pm \\alpha _{2}&0 \\cr 0&2\\alpha _{1}&\\sqrt{2}\\alpha _{1}&0&\\sqrt{2}\\alpha _{1}&0&0&\\pm \\alpha _{3} \\cr 0&\\sqrt{2}\\alpha _{1}&2\\beta _{+}&0&\\alpha _{1}&0&0&\\pm \\sqrt{2}\\alpha _{3} \\cr \\pm \\sqrt{2}\\alpha _{2}&0&0&2\\beta _{-}&0&\\alpha _{4}&\\sqrt{2}\\alpha _{4} &0 \\cr 0&\\sqrt{2}\\alpha _{1}&\\alpha _{1}&0&2\\beta _{+}&0&0&\\pm \\sqrt{2}\\alpha _{3} \\cr \\pm \\sqrt{2}\\alpha _{2}&0&0&\\alpha _{4}&0&2\\beta _{-}&\\sqrt{2}\\alpha _{4}&0 \\cr \\pm \\alpha _{2}&0&0&\\sqrt{2}\\alpha _{4}&0&\\sqrt{2}\\alpha _{4}&2\\alpha _{4}&0 \\cr 0&\\pm \\alpha _{3}&\\pm \\sqrt{2}\\alpha _{3}&0&\\pm \\sqrt{2}\\alpha _{3}&0&0&2\\alpha _{3}}\\right),\\cr $ where the $+$ sign refers to the $\\sigma _{x}$ and $-$ to the $\\sigma _{y}$ channel, respectively.", "The density matrix elements are given by $\\alpha _{1}&=&1+e^{-2\\kappa t}+ e^{-4\\kappa t}+e^{-6\\kappa t}\\cr \\alpha _{2}&=&1+e^{-2\\kappa t}- e^{-4\\kappa t}-e^{-6\\kappa t}\\cr \\alpha _{3}&=&1-e^{-2\\kappa t}- e^{-4\\kappa t}+e^{-6\\kappa t}\\cr \\alpha _{4}&=&1-e^{-2\\kappa t}+ e^{-4\\kappa t}-e^{-6\\kappa t}\\cr \\beta _{\\pm }&=&1\\pm e^{-6\\kappa t}.$ It worth mentioning that since two density matrices $\\rho _{\\pm }^{W}(t)$ have the same structure of matrix elements, we expect that the quantum discord dynamic of these density matrices coincide.", "After tracing out any two qubits, the reduced density matrices are given by $ \\rho _{\\pm }^{(A_{j})}(t)=\\frac{1}{4}\\left({2+e^{-2\\kappa t}&0\\cr 0&2-e^{-2\\kappa t}}\\right) \\,\\,\\,\\,\\, j=1,2,3.$ To find the measurement bases that minimise quantum discord, we used the same procedure as in the previous section.", "We perceived that the maximum classical correlation for any reduced density matrix (32) was obtained by completing the projective measurements in the eigenprojectors of $\\sigma _{z}$ .", "It leads to $S(\\rho _{\\pm }^{(A_{j})}(t)\\Vert \\Phi _{j}(\\rho _{\\pm }^{(A_{j})}(t)))=0$ .", "The minimum of $\\textit {D}(\\rho _{x}^{W}(t))$ and $\\textit {D}(\\rho _{y}^{W}(t))$ is obtained after the two project measurements in the eigneprojectors of $\\sigma _{x}$ and $\\sigma _{z}$ on $\\rho _{x}^{W}(t)$ and $\\rho _{y}^{W}(t)$ , respectively.", "We do not show the achieved analytic expressions for the quantum discord here because its form are not compact.", "We have plotted the quantum discord for density matrix (30) as a function of the dimensionless scaled time $\\kappa t$ at Fig.", "3 (dashed violet).", "Here, we can see that quantum discord due to the transmission through $\\sigma _{x}$ and $\\sigma _{y}$ channels, as expected, coincide and decrease at a low rate of speed from its initial value $\\textit {D}(\\rho _{\\pm }^{W}(0))=1.5$ and the asymptotic limit select the exact value $0.813$ ." ], [ "Pauli channel $\\sigma _{z}$", "After transmission through the Pauli channel $\\sigma _{z}$ , the time evolution of the W state is described by [31] $\\rho _{z}^{W}(t)=\\frac{1}{4}\\left({0&0&0&0&0&0&0&0\\cr 0&2&\\sqrt{2}e^{-4\\kappa t}&0&\\sqrt{2}e^{-4\\kappa t}&0&0&0 \\cr 0&\\sqrt{2}e^{-4\\kappa t}&1&0&e^{-4\\kappa t}&0&0&0 \\cr 0&0&0&0&0&0&0&0 \\cr 0&\\sqrt{2}e^{-4\\kappa t}&e^{-4\\kappa t}&0&1&0&0&0 \\cr 0&0&0&0&0&0&0&0 \\cr 0&0&0&0&0&0&0&0 \\cr 0&0&0&0&0&0&0&0}\\right).$ The density matrices representing each of the one qubit subsystem of (33) are given by $\\rho _{z}^{(A_{j})}(t)=\\frac{1}{4}\\lbrace 3|0\\rangle \\langle 0|+|1\\rangle \\langle 1|\\rbrace $ ($j=1,2,3$ ).", "By using the same procedure as above to find the measurement bases that minimise $\\textit {D}(\\rho _{z}^{W}(t))$ , we find that by performing the projective measurement in the eigenprojectors of $\\sigma _{z}^{j}$ quantum discord reaches its lowest point.", "Thus one can verify that $S(\\rho _{z}^{(A_{j})}(t)\\Vert \\Phi _{j}(\\rho _{z}^{(A_{j})}(t)))=0$ and $S(\\Phi (\\rho _{z}^{W}(t)))=\\frac{3}{2}$ .", "The von-Neumann entropy of $\\rho _{z}^{W}(t)$ density matrix is found as $ S(\\rho _{z}^{W}(t))&=&\\frac{1}{4}(11+e^{-4\\kappa t})-\\frac{1}{4}(1-e^{-4\\kappa t})\\log _{2}(1-e^{-4\\kappa t})\\cr &-&\\frac{1}{8}\\lbrace (3+e^{-4\\kappa t}-\\sqrt{1-2e^{-4\\kappa t}+17e^{-8\\kappa t}})\\cr &&\\log _{2}(3+e^{-4\\kappa t}-\\sqrt{1-2e^{-4\\kappa t}+17e^{-8\\kappa t}})\\cr &+&(3+e^{-4\\kappa t}+\\sqrt{1-2e^{-4\\kappa t}+17e^{-8\\kappa t}})\\cr && \\log _{2}(3+e^{-4\\kappa t}+\\sqrt{1-2e^{-4\\kappa t}+17e^{-8\\kappa t}})\\rbrace .$ Therefore, the expression for quantum discord $\\textit {D}(\\rho _{z}^{W}(t))$ is given by $ \\textit {D}(\\rho _{z}^{W}(t))&=&\\frac{-1}{4}(5+e^{-4\\kappa t})+\\frac{1}{4}(1-e^{-4\\kappa t})\\log _{2}(1-e^{-4\\kappa t})\\cr &+&\\frac{1}{8}\\lbrace (3+e^{-4\\kappa t}-\\sqrt{1-2e^{-4\\kappa t}+17e^{-8\\kappa t}})\\cr &&\\log _{2}(3+e^{-4\\kappa t}-\\sqrt{1-2e^{-4\\kappa t}+17e^{-8\\kappa t}})\\cr &+&(3+e^{-4\\kappa t}+\\sqrt{1-2e^{-4\\kappa t}+17e^{-8\\kappa t}})\\cr &&\\log _{2}(3+e^{-4\\kappa t}+\\sqrt{1-2e^{-4\\kappa t}+17e^{-8\\kappa t}})\\rbrace .$ With the above channels, quantum discord of the selected system with the initial W state disappear under Pauli channel $\\sigma _{z}$ after a finite time.", "This is displayed in Fig.", "3 (solid red)." ], [ "Depolarising channel", "When three-qubite W state is transmitted through the depolarising channel, its time evolution is described by [31] $\\hspace{-42.67912pt}\\rho _{d}^{W}(t)=\\frac{1}{8}\\left({\\tilde{\\alpha }_{2}&0&0&0&0&0&0&0 \\cr 0&\\tilde{\\alpha }_{1}&\\sqrt{2}\\tilde{\\gamma }_{+}&0&\\sqrt{2}\\tilde{\\gamma }_{+}&0&0&0 \\cr 0&\\sqrt{2}\\tilde{\\gamma }_{+}&\\tilde{\\beta }_{+}&0&\\tilde{\\gamma }_{+}&0&0&0 \\cr 0&0&0&\\tilde{\\beta }_{-}&0&\\tilde{\\gamma }_{-}&\\sqrt{2}\\tilde{\\gamma }_{-} &0 \\cr 0&\\sqrt{2}\\tilde{\\gamma }_{+}&\\tilde{\\gamma }_{+}&0&\\tilde{\\beta }_{+}&0&0&0 \\cr 0&0&0&\\tilde{\\gamma }_{-}&0&\\tilde{\\beta }_{-}&\\sqrt{2}\\tilde{\\gamma }_{-}&0 \\cr 0&0&0&\\sqrt{2}\\tilde{\\gamma }_{-}&0&\\sqrt{2}\\tilde{\\gamma }_{-}&\\tilde{\\alpha }_{4}&0 \\cr 0&0&0&0&0&0&0&\\tilde{\\alpha }_{3}}\\right),\\cr $ with $\\tilde{\\alpha }_{1}&=&1+e^{-4\\kappa t}+ e^{-8\\kappa t}+e^{-12\\kappa t}\\cr \\tilde{\\alpha }_{2}&=&1+e^{-4\\kappa t}- e^{-8\\kappa t}-e^{-12\\kappa t}\\cr \\tilde{\\alpha }_{3}&=&1-e^{-4\\kappa t}- e^{-8\\kappa t}+e^{-12\\kappa t}\\cr \\tilde{\\alpha }_{4}&=&1-e^{-4\\kappa t}+ e^{-8\\kappa t}-e^{-12\\kappa t}\\cr \\tilde{\\beta }_{\\pm }&=&1\\pm e^{-12\\kappa t}\\cr \\tilde{\\gamma }_{\\pm }&=&e^{-8\\kappa t}\\pm e^{-12\\kappa t}.$ For density matrix (36), the single qubit density matrix is given by $ \\rho _{d}^{(A_{j})}(t)=\\frac{1}{4}\\left({2+e^{-4\\kappa t}&0\\cr 0&2-e^{-4\\kappa t}}\\right) \\,\\,\\, j=1,2,3.$ Our results show that the minimum quantum discord for the depolarising coupling of the three-qubit W state is attained at boundary values $\\varphi _{j}=0$ and $\\theta _{j}=0$ , that is for a local measurement along the eigenstates of Pauli matrix $\\sigma _{z}$ .", "The state of the single qubit is not induced by such measurement, therefore $S(\\rho _{d}^{(A_{j})}(t)\\Vert \\Phi _{j}(\\rho _{d}^{(A_{j})}(t)))=0$ .", "The von-Neumann entropy of $\\rho _{d}^{W}(t)$ after measurement is given explicitly by $ S(\\Phi (\\rho _{d}^{W}(t)))&=&\\frac{1}{4}\\lbrace 18-\\tilde{\\alpha }_{1}\\log _{2}\\tilde{\\alpha }_{1}-\\tilde{\\alpha }_{2}\\log _{2}\\tilde{\\alpha }_{2}\\cr &-&\\tilde{\\alpha }_{3}\\log _{2}\\tilde{\\alpha }_{3}-\\tilde{\\alpha }_{4}\\log _{2}\\tilde{\\alpha }_{4}-\\tilde{\\beta }_{+}\\log _{2}\\tilde{\\beta }_{+}-\\tilde{\\beta }_{-}\\log _{2}\\tilde{\\beta }_{-}\\rbrace .$ In this case, the time evolution of quantum discord is obtained as $ \\hspace{-42.67912pt}\\textit {D}(\\rho _{d}^{W}(t))&=&\\frac{1}{2}(3-e^{-8\\kappa t})-\\frac{1}{4}\\lbrace \\tilde{\\beta }_{+}\\log _{2}\\tilde{\\beta }_{+}+\\tilde{\\beta }_{-}\\log _{2}\\tilde{\\beta }_{-}\\cr &+&\\tilde{\\alpha }_{1}\\log _{2}\\tilde{\\alpha }_{1}+2\\tilde{\\alpha }_{2}\\log _{2}\\tilde{\\alpha }_{2} +2\\tilde{\\alpha }_{3}\\log _{2}\\tilde{\\alpha }_{3}+\\tilde{\\alpha }_{4}\\log _{2}\\tilde{\\alpha }_{4}\\rbrace \\cr &+&\\frac{1}{8}\\lbrace (\\tilde{\\beta }_{-}-\\tilde{\\gamma }_{-})\\log _{2}(\\tilde{\\beta }_{-}-\\tilde{\\gamma }_{-})+(\\tilde{\\beta }_{+}-\\tilde{\\gamma }_{+})\\log _{2}(\\tilde{\\beta }_{+}-\\tilde{\\gamma }_{+})\\rbrace \\cr &+&\\frac{1}{16}\\lbrace (\\tilde{\\beta }_{+}+\\tilde{\\gamma }_{+}+\\tilde{\\alpha }_{1}-\\sqrt{\\tilde{\\beta }_{+}^{2}+\\tilde{\\gamma }_{+}(2\\tilde{\\beta }_{+}+17\\tilde{\\gamma }_{+})-2\\tilde{\\alpha }_{1}(\\tilde{\\beta }_{+}+\\tilde{\\gamma }_{+})+\\tilde{\\alpha }_{1}^{2}})\\cr &&\\log _{2}(\\tilde{\\beta }_{+}+\\tilde{\\gamma }_{+}+\\tilde{\\alpha }_{1}-\\sqrt{\\tilde{\\beta }_{+}^{2}+\\tilde{\\gamma }_{+}(2\\tilde{\\beta }_{+}+17\\tilde{\\gamma }_{+})-2\\tilde{\\alpha }_{1}(\\tilde{\\beta }_{+}+\\tilde{\\gamma }_{+})+\\tilde{\\alpha }_{1}^{2}})\\cr &+&(\\tilde{\\beta }_{+}+\\tilde{\\gamma }_{+}+\\tilde{\\alpha }_{1}+\\sqrt{\\tilde{\\beta }_{+}^{2}+\\tilde{\\gamma }_{+}(2\\tilde{\\beta }_{+}+17\\tilde{\\gamma }_{+})-2\\tilde{\\alpha }_{1}(\\tilde{\\beta }_{+}+\\tilde{\\gamma }_{+})+\\tilde{\\alpha }_{1}^{2}})\\cr &&\\log _{2}(\\tilde{\\beta }_{+}+\\tilde{\\gamma }_{+}+\\tilde{\\alpha }_{1}+\\sqrt{\\tilde{\\beta }_{+}^{2}+\\tilde{\\gamma }_{+}(2\\tilde{\\beta }_{+}+17\\tilde{\\gamma }_{+})-2\\tilde{\\alpha }_{1}(\\tilde{\\beta }_{+}+\\tilde{\\gamma }_{+})+\\tilde{\\alpha }_{1}^{2}})\\cr &+&(\\tilde{\\beta }_{-}+\\tilde{\\gamma }_{-}+\\tilde{\\alpha }_{4}-\\sqrt{\\tilde{\\beta }_{-}^{2}+\\tilde{\\gamma }_{-}(2\\tilde{\\beta }_{-}+17\\tilde{\\gamma }_{-})-2\\tilde{\\alpha }_{4}(\\tilde{\\beta }_{-}+\\tilde{\\gamma }_{-})+\\tilde{\\alpha }_{4}^{2}})\\cr &&\\log _{2}(\\tilde{\\beta }_{-}+\\tilde{\\gamma }_{-}+\\tilde{\\alpha }_{4}-\\sqrt{\\tilde{\\beta }_{-}^{2}+\\tilde{\\gamma }_{-}(2\\tilde{\\beta }_{-}+17\\tilde{\\gamma }_{-})-2\\tilde{\\alpha }_{4}(\\tilde{\\beta }_{-}+\\tilde{\\gamma }_{-})+\\tilde{\\alpha }_{4}^{2}})\\cr &+&(\\tilde{\\beta }_{-}+\\tilde{\\gamma }_{-}+\\tilde{\\alpha }_{4}+\\sqrt{\\tilde{\\beta }_{-}^{2}+\\tilde{\\gamma }_{-}(2\\tilde{\\beta }_{-}+17\\tilde{\\gamma }_{-})-2\\tilde{\\alpha }_{4}(\\tilde{\\beta }_{-}+\\tilde{\\gamma }_{-})+\\tilde{\\alpha }_{4}^{2}})\\cr &&\\log _{2}(\\tilde{\\beta }_{-}+\\tilde{\\gamma }_{-}+\\tilde{\\alpha }_{4}+\\sqrt{\\tilde{\\beta }_{-}^{2}+\\tilde{\\gamma }_{-}(2\\tilde{\\beta }_{-}+17\\tilde{\\gamma }_{-})-2\\tilde{\\alpha }_{4}(\\tilde{\\beta }_{-}+\\tilde{\\gamma }_{-})+\\tilde{\\alpha }_{4}^{2}})\\rbrace .", "\\nonumber \\\\$ The responses of quantum discord with the initial W state to the different noises as functions of $\\kappa t$ are listed at Fig.", "3.", "As in the case of the GHZ state, the quantum discord for the W state decays due to the noisy channels.", "When a three-qubit system with initial W or GHZ state is coupled to a depolarising noise channel decay of the quantum discord occurs so fast.", "This means that the depolarising channel has the most destructive influence on the quantum discord.", "However, the quantum discord decreases due to the transmission through the Pauli channels $\\sigma _{x}$ or $\\sigma _{y}$ , but it catches the exact value $0.813$ in the long term as shown in Fig.", "3.", "This figure also exhibits the vanishing of the quantum discord for the initial W state under the phase flip channel.", "These results show that GHZ-type quantum discord is more fragile under certain types of environment coupling which can be modelled by means of the Pauli channels $\\sigma _{y}$ and $\\sigma _{z }$ as well as the depolarising channel as compared to the quantum discord of the W state.", "However, in the Pauli channel $\\sigma _{x}$ the situation becomes completely reversed.", "In order to show the difference between the entanglement and quantum discord under the various dissipative channels, by using the results of [32] we have plotted the evolution of entanglement of the W state in Fig.", "4.", "As in the case of the GHZ state, the entanglement for the W state decays exponentially due to the noisy channels.", "Observe that under the dissipative dynamics considered, except for the phase flip coupling of the three qubits system to the environment, discord is more robust than entanglement." ], [ "SUMMARY AND CONCLUSION", "To sum up, we prepared the three-qubit system with the initial state formed by GHZ and W states.", "We then studied the dynamics of quantum discord under interaction with independent Markovian environments which can be modelled by means of the various noisy channels, namely, the Pauli channels $\\sigma _{x}$ , $\\sigma _{y}$ and $\\sigma _{z}$ as well as the depolarising channel.", "It was clarified that coupling with the bit flip channel could not disorder the quantum discord of three-qubit GHZ state unlike in the W state.", "In other words, for this noisy channel, GHZ-type quantum discord is always more robust than quantum discord of the W state.", "In the Pauli channels $\\sigma _{y}$ , $\\sigma _{z}$ and depolarising channel, the W state preserved more quantum discord than the GHZ state in the long term.", "Also, we observed that under the dissipative Markovian dynamics considered, except for the Pauli channel $\\sigma _{z}$ quantum discord is more robust than entanglement.", "This points to the fact that quantum discord is another kind of quantum correlation different from entanglement and the absence of entanglement does not necessarily indicate the absence of quantum correlations." ], [ "References", "Figure Caption FIG.", "1.", "(Colour online) The quantum discord (6) for the three-qubit system with the initial GHZ state as a function of $\\kappa t$ , if transmitted through Markovian channels: Pauli channel $\\sigma _{x}$ (dashed violet), $\\sigma _{y}$ (solid blue), $\\sigma _{z}$ (solid red) and depolarising channel (solid black).", "FIG.", "2.", "(Colour online) The entanglement for the three-qubit system with the initial GHZ state as a function of $\\kappa t$ , if transmitted through Markovian channels: Pauli channel $\\sigma _{x}$ (dashed violet), $\\sigma _{y}$ (solid blue), $\\sigma _{z}$ (solid red) and depolarising channel (solid black).", "FIG.", "3.", "(Colour online) The quantum discord (6) for the three-qubit system with the initial W state as a function of $\\kappa t$ , if transmitted through Markovian channels: Pauli channel $\\sigma _{x}$ and $\\sigma _{y}$ (dashed violet), $\\sigma _{z}$ (solid red) and depolarising channel (solid black).", "FIG.", "4.", "(Colour online) The entanglement for the three-qubit system with the initial W state as a function of $\\kappa t$ , if transmitted through Markovian channels: Pauli channel $\\sigma _{x}$ (dashed violet), $\\sigma _{y}$ (solid blue), $\\sigma _{z}$ (solid red) and depolarising channel (solid black)." ] ]

[ [ "A relativistic non-relativistic Goldstone theorem: gapped Goldstones at\n finite charge density" ], [ "Abstract We adapt the Goldstone theorem to study spontaneous symmetry breaking in relativistic theo- ries at finite charge density.", "It is customary to treat systems at finite density via non-relativistic Hamiltonians.", "Here we highlight the importance of the underlying relativistic dynamics.", "This leads to seemingly new results whenever the charge in question is spontaneously broken and does not commute with other broken charges.", "We find that that the latter interpolate gapped excitations.", "In contrast, all existing versions of the Goldstone theorem predict the existence of gapless modes.", "We derive exact non-perturbative expressions for their gaps, in terms of the chemical potential and of the symmetry algebra." ], [ "A relativistic non-relativistic Goldstone theorem: gapped Goldstones at finite charge density Alberto Nicolis Physics Department and Institute for Strings, Cosmology, and Astroparticle Physics, Columbia University, New York, NY 10027, USA Federico Piazza Paris Center for Cosmological Physics and Laboratoire APC, Université Paris 7, 75205 Paris, France We adapt the Goldstone theorem to study spontaneous symmetry breaking in relativistic theories at finite charge density.", "It is customary to treat systems at finite density via non-relativistic Hamiltonians.", "Here we highlight the importance of the underlying relativistic dynamics.", "This leads to seemingly new results whenever the charge in question is spontaneously broken and does not commute with other broken charges.", "We find that that the latter interpolate gapped excitations.", "In contrast, all existing versions of the Goldstone theorem predict the existence of gapless modes.", "We derive exact non-perturbative expressions for their gaps, in terms of the chemical potential and of the symmetry algebra.", "Preliminary considerations.", "Non-relativistic Goldstone theorems [1], [2], [3], [4], [5], [6] display interesting twists compared to the standard relativistic one [7], [8].", "First, the non-relativistic version is notoriously less powerful.", "For instance, it guarantees the existence of gapless zero-momentum excitations, but says nothing about their properties, like e.g.", "stability, at finite momenta.", "A physically relevant example is phonons in superfluid helium-4, which are unstable with a decay rate $\\Gamma \\sim k^5$ —that is, they are not really eigenstates of the Hamiltonian.", "Second, as far as counting is concerned, a classic result by Nielsen and Chadha [2] states that for non-relativistic systems, the number of gapless Goldstone excitations equals the number of spontaneously broken generators, provided one counts the Goldstone excitations with even dispersion law (e.g.", "$\\omega \\sim k^2 $ ) twice.", "However, to the best of our knowledge, all fundamental interactions are described by relativistic field equations.", "This means that for the so-called non-relativistic systems in the real world, Lorentz invariance is broken only spontaneously, i.e.", "by the state of the system, rather than at the level of the dynamics.", "It is thus tempting to ask whether the very constrained framework of relativistic field theories can give non-trivial insights into physical systems that are effectively non-relativistic—and in particular, whether it can be used to sharpen or correct the non-relativistic versions of the Goldstone theorem.", "An immediate reaction to this idea is that in no way can relativistic effects be relevant for systems that, like condensed matter systems in the lab, have a very non-relativistic equation of state, a very small speed of sound compared to that of light, and so on.", "But there is more to relativity than just the so-called “relativistic effects”, which are weighed by $(v/c)^2$ .", "First, there is the statement of relativity itself—that all inertial frames are equivalent—which is valid, and powerful, even in the $c \\rightarrow \\infty $ limit, corresponding formally to the Galilean limit of Lorentz invariance.", "Then, there are properties of relativistic field theories that are not directly statements of symmetry, but that are nevertheless crucial for the consistency of the theory.", "For instance, we will see below that the vanishing of commutators for space-like separated local operators can be used to remove an assumption of the Nielsen-Chadha theorem.", "Finally—and this will be the useful aspect for our purposes—the fundamental relativistic viewpoint offers an unambiguous starting point to analyze the pattern of spontaneous symmetry breaking, for spacetime symmetries and internal ones.", "To clarify this last statement with an example, let's consider directly the system we want to focus on for the rest of paper: a relativistic theory with Hamiltonian $H$ and a group of internal symmetries, at finite density for one of the corresponding charges, $Q$ .", "The ground state $| \\mu \\rangle $ (i.e.", "the state of minimal energy for given average charge density) can be found by the method of Lagrange multipliers as the state minimizing the modified Hamiltonian $\\tilde{H} = H - \\mu Q$ , where $\\mu $ is the chemical potentialIn general, the r.h.s.", "should read $\\lambda | \\mu \\rangle $ , with non-vanishing $\\lambda $ .", "From applying the familiar thermodynamic relation $E+ PV = T S + \\mu Q$ to our zero-temperature system, we see that $\\lambda $ is related to the pressure.", "Such a charge density breaks Lorentz invariance, because it transforms like the time component of a four-vector.", "In the following we will drop $\\lambda $ from our formulae, to make them less cluttered, although keeping track of it is straightforward.", "Formally, for any given $\\mu $ , we can set $\\lambda $ to zero by adjusting the cosmological constant., $\\tilde{H} | \\mu \\rangle = \\big ( H - \\mu Q \\big ) |\\mu \\rangle = 0 \\; .$ It is standard practice to use the non-relativistic Hamiltonian $\\tilde{H}$ to study the system at finite density [9].", "However, it is clear from the outset that we are in the presence of a spontaneous—rather than explicit—breaking of Lorentz symmetry.", "Introducing $\\tilde{H}$ is purely a mathematical tool to find a state with the desired properties.", "The Hamiltonian of the system is still $H$ .", "Heisenberg picture operators evolve in time as dictated by $H$ .", "In order to better understand the role of $\\tilde{H}$ , consider the case in which also the (internal) symmetry generated by $Q$ is spontaneously broken, i.e.", "$\\langle \\mu | [Q,A(x)] | \\mu \\rangle \\ne 0$ for some order parameter $A(x)$ .", "Then $| \\mu \\rangle $ cannot be an eigenstate of $Q$ , because this would be inconsistent with (REF ).", "But since $| \\mu \\rangle $ obeys eq.", "(REF ), it cannot be an eigenstate of $H$ either.", "We conclude that, at finite density for $Q$ , if $Q$ is spontaneously broken, so is $H$ [10].", "This means that if $Q$ is broken, we cannot classify the states of the system—including our ground state $| \\mu \\rangle $ —as eigenstates of the fundamental Hamiltonian $H$ .", "The best we can do is trying to diagonalize the unbroken combination $\\tilde{H} = H - \\mu Q$ .", "$| \\mu \\rangle $ is the eigenstate with lowest eigenvalue.", "The excitations of the system—including the Goldstone bosons—will correspond to higher eigenstates.", "Since the Nielsen-Chadha theorem implicitly assumes that Heisenberg picture operators evolve in time with the same (non-relativistic) Hamiltonian that is minimized by the ground state, we conclude that that theorem does not apply to systems that are `non-relativistic' because of a finite density for a charge that is spontaneously broken.", "Indeed, for such systems it is not even clear how one would study the spontaneous breaking of $Q$ in a setup that is non-relativistic from the start.", "There are systems of this sort where $Q$ is broken `before' (i.e., at higher energy scales) Lorentz invariance is: any relativistic theory with ordinary spontaneous symmetry breaking in its Poincaré-invariant vacuum, can be put in a state of arbitrarily low density for the broken charge [10].", "When there are local operators obeying (REF ), an alternative viewpoint suggests itself.", "By eq.", "(REF ), the action of the Hamiltonian on $|\\mu \\rangle $ is proportional to that of the symmetry generator.", "We are thus in the presence of a state that evolves in time along a symmetry direction, at `speed' $\\mu $ .", "Any field $\\phi _j(x)$ that transforms non-trivially under the symmetry can then feature a spacially homogeneous, time-dependent expectation value, obeying $\\frac{d}{dt}\\langle {\\dot{\\phi }}_j \\rangle = \\mu \\langle \\delta \\phi _j \\rangle \\; ,$ where $\\phi _j \\rightarrow \\phi _j + \\delta \\phi _j$ is the action of the symmetry in field space.", "In [10], we dubbed this situation `spontaneous symmetry probing' (SSP)—there, we were using `$c$ ' in place of `$\\mu $ '.", "This viewpoint is particularly useful in the semiclassical limit, where we can think of time evolution in terms of classical trajectories in field space.", "We refer the reader to [10] for details.", "Assumptions and formalism.", "We now want to study the low-energy spectrum of the system at finite density, by considering the Goldstone states associated with $Q$ and with other broken generators.", "Let's assume that the theory enjoys a Lie group of internal symmetries, with generators $Q_1, Q_2, \\dots , Q_N$ .", "Without loss of generality we can set the first generator to be our $Q$ , $Q=Q_1$ .", "The remarks we made above lead to the following hypotheses: (a) The Heisenberg-picture currents evolve in time as dictated by the original Hamiltonian, i.e.", "$J^\\mu _a(t, {\\vec{x}}) = e^{i (H t - \\vec{P} \\cdot \\vec{x})} J^\\mu _a(0) e^{-i (H t - \\vec{P} \\cdot \\vec{x})}$ , where $a = 1,\\dots , N$ , and $\\vec{P}$ is the total momentum operator.", "(b) The state $|\\mu \\rangle $ is the ground state of $\\tilde{H} = H - \\mu \\, Q$ .", "The crucially different role played by $H$ and $\\tilde{H}$ is the origin of the discrepancy between our results and the existing literature on non-relativistic Goldstone theorems, e.g., [2], [3], [4], [5].", "Consider then the case in which the first $n$ of the $Q_a$ 's—including $Q_1$ —are spontaneously broken.", "By definition, for each spontaneously broken $Q_a$ , there must exist a local operator $ A_I (x)$ —an `order parameter'—that makes the matrix element $\\kappa _{aI} \\equiv \\langle \\mu |[Q_a(t), A_I (0)] | \\mu \\rangle $ nonzero.", "The index $I=1, \\dots , m \\le n$ in general runs over fewer values than the number of broken generators, simply because the same $A_I(x)$ typically serves as an order parameter for two or more symmetries.", "In the matrix element above, $A_I(x)$ is evaluated at the (space-time) origin.", "From now on, to simplify the notation, whenever a local operator is evaluated at the origin, we will drop its argument, ${\\cal O}(0) \\rightarrow {\\cal O}$ .", "$Q_a$ is formally evaluated—in Heisenberg picture—at time $t$ .", "But since $Q_a$ commutes with the Hamiltonian, it is constant in time, and so is the $\\kappa _{aI}$ matrix element.", "To convince oneself that this is true even though spontaneously broken charges are not completely well-defined operators, one can use the local conservation of the current: $\\int \\!", "d^3 x \\langle \\mu | [\\dot{J}_a^0 (\\vec{x},t) , A_I] |\\mu \\rangle + \\!", "\\int \\!d^3 x \\langle \\mu | [\\partial _i J_a^i (\\vec{x},t) , A_I] |\\mu \\rangle = 0 \\; .\\nonumber $ The first term is the time-derivative of our $\\kappa _{aI}$ , while the second is a boundary term that only receives contributions from spacial infinity.", "Since for relativistic QFTs the commutator of space-like separated local operators vanishes—and we are breaking Lorentz symmetry only spontaneously—such a term is guaranteed to vanishThis is the removal of one of the Nielsen-Chadha assmputions we alluded to above.. We conclude that $\\kappa _{aI}$ is constant in time.", "We now use assumptions (a) and (b) above to `pull out' of $J^0_a$ its $\\vec{x}$ - and $t$ -dependence: $ \\kappa _{aI} \\!&=& \\!", "\\int \\!", "d^3 x \\langle \\mu | J_a^0 (\\vec{x},t) A_I |\\mu \\rangle - {\\rm c.c.}", "\\\\\\!&=& \\!", "\\int \\!", "d^3 x \\langle \\mu | e^{i \\mu Q t} \\, J_a^0 \\, e^{-i (H t - P\\cdot \\vec{x})} A_I |\\mu \\rangle - {\\rm c.c.", "}\\nonumber \\, ,$ where we used that spacial translations are not spontaneously broken, $\\vec{P} | \\mu \\rangle = 0$ , and we are assuming that both $J^0_a$ and $A_I$ are hermitian operators.", "Inside the matrix element, we now insert a complete set of intermediate momentum eigenstates $|n,\\vec{p} \\, \\rangle $ , where $n$ labels other quantities that characterize these states.", "Schematically, $ & & \\langle \\mu | e^{i \\mu Q t} \\, J^0_a \\, e^{-i (H t-P\\cdot \\vec{x} )} A_I | \\mu \\rangle \\\\& = & \\sum _{n,p} e^{i \\vec{p} \\cdot \\vec{x}} \\langle \\mu | e^{i \\mu Q t} J^0_a e^{-i \\mu Q t} e^{- i \\tilde{H} t} |n,\\vec{p} \\, \\rangle \\langle n,\\vec{p} \\, | \\, A_I |\\mu \\rangle \\nonumber $ where we rewrote $H$ in terms of $\\tilde{H}$ and $Q$ .", "The Hamiltonian $\\tilde{H}$ commutes with the spatial momentum, because $H$ and $Q$ do.", "We can then choose the $|n, \\vec{p} \\rangle $ states to be eigenstates of $\\tilde{H}$ as well, with eigenvalues $E_n(\\vec{p} \\, )$ .", "The integral in $d^3 x $ projects onto the zero momentum states, and we are left with $\\kappa _{aI} = \\sum _n e^{- i E_n(0) t} \\langle \\mu | e^{i \\mu Q t} J^0_a e^{-i \\mu Q t} |n,0 \\rangle \\langle n,0 | A_I |\\mu \\rangle - {\\rm c.c.}", "$ The constancy in time of the above expression gives important information about the spectrum of the theory in the limit of zero spatial momentum.", "There are two distinct cases to consider, depending on whether $Q_a$ commutes with $Q \\equiv Q_1$ or not.", "For brevity, let us call these two classes of generators `C' and `NC', short for `commuting' and `non-commuting'.", "We will show that C-generators interpolate states $|n,\\vec{p} \\, \\rangle $ that have zero energy in the limit of zero momentum, $E_n(0) = 0$ .", "Vice versa, NC generators interpolate states that are gapped, in the sense that $E_n(0) \\ne 0$ .", "C-Generators: gapless modes.", "Eq.", "(REF ) should be compared with eq.", "(6) of Nielsen and Chadha's paper [2].", "They find that the quantity that is constant in time is, in our notation, $\\kappa _{aI} = \\sum _n e^{- i E_n(0) t} \\langle \\mu | J^0_a |n,0 \\rangle \\langle n,0 | A_I |\\mu \\rangle - {\\rm c.c.", "}$ The implicit assumption leading to (REF ) is that all currents evolve in time with the non-relativistic Hamiltonian of which $|\\mu \\rangle $ is the ground state ($\\tilde{H}$ ).", "This assumption is violated by our system.", "However, our eq.", "(REF ) does reduce to (REF ) whenever $Q_a$ commutes with $Q$ , since in this case $e^{i \\mu Q t} J^0_a e^{-i \\mu Q t} = J^0_a$ .Once the algebra for the charges is given, the charge-current commutators are uniquely determined up to possible contact terms that vanish at zero momentum, i.e., total spacial derivatives.", "Since we took the $\\vec{p} \\rightarrow 0$ limit, such possible extra terms will not matter for us.", "Note that for $|n,0\\rangle = |\\mu \\rangle $ the combination (REF ) just gives zero, because $J^0_a$ and $A_I$ are hermitian operators.", "In order for $\\kappa _{aI}$ to be time-independent and different from zero, there must exist a Goldstone state $| \\pi , \\vec{p} \\, \\rangle $ other than $|\\mu \\rangle $ , whose energy goes to zero in the zero-momentum limit, $E_\\pi (0) = 0$ .", "Moreover, the matrix elements $\\langle \\mu | J^0_a \\, |\\pi , \\vec{p}\\, \\rangle $ and $\\langle \\pi , \\vec{p} \\, | \\, A_I |\\mu \\rangle $ should be nonzero for such a state: both the broken current and the order parameter $A_I$ have to interpolate the Goldstone excitation.", "Beyond this basic argument, the detailed analysis of the number and nature of gapless Goldstone bosons follows closely that of [2] and features all the subtleties considered therein (see also [3], [6], [5] for more recent refinements).", "We have nothing to add to the existing analyses, other than emphasize that they apply here to broken generators of the C-type only.", "Notice that among the C-generators we have $Q$ itself.", "We devoted the bulk of [10] to studying the physical properties of the associated Goldstone boson.", "NC-Generators: the gap.", "Let us now consider the case where $Q_a$ does not commute with $Q=Q_1$ .", "For our purposes, it is useful to write the commutation relations in hybrid form with charges and currentsfootnote algebra: $[Q_a, J^0_b(x)] = i f_{ab}^c \\, J_c^0 (x) \\; ,$ where $f_{ab}^c$ are the group's structure constants, which are real for any Lie group.", "We can now go back to eq.", "(REF ).", "After expanding the exponentials on both sides of the current, using recursively (REF ) to eliminate all $Q$ 's, and re-exponentiating the result, we find $e^{i \\mu \\, Q t} J^0_a e^{-i \\mu \\, Q t} = (e^{-f_1 \\, \\mu t} )^b_a J^0_b \\; , $ where $f_1$ is a matrix with entries $f_{1a}^b$ .", "That is, $i f_1$ is the adjoint representation of $Q_1$ , $(Q_1^A)^b_a = i f_{1a}^b$ .", "Eq.", "(REF ) above is nothing but the usual statement—applied to the currents—that the generators of a group live in the adjoint representation of that group.", "The exponential acting on the current is now a finite dimensional matrix that `mixes' in a time-dependent fashion the different currents of the group: $\\kappa _{aI} = \\sum _n e^{- i E_n (0) t} \\, (e^{-f_1 \\, \\mu t} )^b_a \\, \\langle \\mu | J^0_b |n,0 \\rangle \\langle n,0| A_I |\\mu \\rangle - {\\rm c.c.", "}$ We will now assume—as usual—that the symmetry group under consideration is the direct product of simple compact Lie groups ($SU(n)$ , $SO(n)$ , etc.", "), and of $U(1)$ factors.", "In this case the structure constants $f_{ab}^c$ can be taken to be totally antisymmetric—see e.g. [8].", "Since $f_{1a}^b$ is real and antisymmetric, its eigenvalues are either zero or pure imaginary.", "The imaginary eigenvalues come in pairs $(+i q_a, -i q_a)$ , with corresponding hermitian-conjugate pairs of eigenvectors, defining a (non-hermitian) basis of generators in which $f_{1a}^b = i \\cdot {\\rm diag}(0, \\dots , 0, q_1 , - q_1, q_2, -q_2, \\dots ) \\, .$ By acting separately on each $\\pm q_a$ block , it is straightforward to define instead an hermitian basis, in which $ f_{1a}^b $ is real and block-diagonal, with $2\\times 2$ blocks of the form $\\left( \\begin{array}{cc}0 & + q_a \\\\- q_a & 0 \\end{array}\\right)\\, .$ Let's assume that we started with eq.", "(REF ) directly in this hermitian basis where $ f_{1a}^b $ is block-diagonal, and let's restrict our analysis to the spontaneously broken generators, which yield non-vanishing $\\kappa _{aI}$ .", "If $J^0_a$ corresponds to a vanishing eigenvalue of $Q_1^A = i f_1$ —i.e., if it commutes with $Q_1$ —then we go back to the C-generator case and we conclude that $J^0_a$ interpolates a gapless particles.", "If on the other hand $J^0_a$ is either of the paired currents acted upon by a $2\\times 2$ block of the form (REF ), then the exponential in (REF ) mixes its matrix elements with its companion's, with frequency $\\mu q_a$ .", "In this case, the time-independence of $\\kappa _{aI}$ implies a non-zero value for $E_n(0)$ .", "This is most easily seen by using a complex notation: $J^0_a$ can be expressed as $J^0_a = \\hat{J}^0_a + \\hat{J}^0_a {}^\\dagger $ —where $\\hat{J}^0_a$ and $\\hat{J}^0_a {}^\\dagger $ are an hermitian-conjugate pair of non-hermitian generators that diagonalize $f_1$ , as in (REF )—and eq.", "(REF ) simply becomes aI = n e- i (En(0) - qa )t   | J0a |n,0n,0| AI | + e- i (En(0) + qa )t   | J0a |n,0 n,0| AI | - c.c.", "Notice that we are not implicitly summing over $a$ .", "Without loss of generality, we will assume that $\\mu \\, q_a$ is positive and $- \\mu \\, q_a$ negative.", "By assumption, our $| \\mu \\rangle $ state is the ground state of the unbroken $\\tilde{H}$ Hamiltonian.", "Therefore, there cannot be states with negative $E_n(\\vec{p})$ .", "So, for the combination (REF ) to be constant in time and different from zero, two conditions have to be met: (i) There exists a state $| \\pi _a, \\vec{p} \\, \\rangle $ whose energy in the zero momentum limit is $E_a (0)= \\mu \\, q_a \\; .$ This is our main result: the gap for the Goldstone excitations associated with the broken NC-generators.", "(ii) Both $\\hat{J}^0_a $ and $A_I$ are interpolating fields for such a state, in the sense that $\\langle \\mu | \\hat{J}^0_a |\\pi _a, \\vec{p} \\, \\rangle \\ne 0$ and $\\langle \\mu | A_I |\\pi _a, \\vec{p} \\, \\rangle \\ne 0 $ (recall that $A_I$ is hermitian.)", "Notice that we have one gapped Goldstone mode for each pair of broken NC-generators.", "This is reminiscent of the Nielsen-Chadha counting [2], [3].", "In fact, we still find that for even dispersion relations the number of broken generators is twice the number of Goldstones, just in the broader sense that gapped states have an `even' dispersion relation, i.e.", "$E(p) \\sim |\\vec{p}\\, |^0 + {\\cal O}(p^2)$ .", "An intuitive picture of what is going on is provided by the following example.", "The linear $SO(3)$ sigma model.", "Consider a Lagrangian with internal symmetry $SO(3)$ , linearly realized on a scalar triplet $\\vec{\\phi }$ : $ {\\cal L} = -{\\textstyle \\frac{1}{2}} \\partial _\\mu \\vec{\\phi }\\cdot \\partial ^\\mu \\vec{\\phi }-{\\textstyle \\frac{1}{2}} m^2 |{\\vec{\\phi }} \\, |^{2} - {\\textstyle \\frac{1}{4}} \\lambda |{\\vec{\\phi }} \\, |^4 \\, .$ Let's pick one of the generators of $SO(3)$ , say $\\tau = \\tau _3$ , where $\\tau _i$ generates rotations about the $\\phi _i$ axis, and let's consider the theory at finite charge density for the corresponding charge $Q$ .", "The standard way to build the non-relativistic Lagrangian at finite density—see e.g.", "[9], [11]—is to introduce a constant non-dynamical gauge field pointing in the time direction, which in our case amounts to the replacement $- \\partial _\\mu \\Phi ^\\dagger \\partial ^\\mu \\Phi \\rightarrow (\\partial _0 - i \\mu ) \\Phi ^\\dagger (\\partial _0 + i\\mu ) \\Phi - \\partial _j \\Phi ^\\dagger \\partial _j \\Phi \\, ,$ where $\\Phi $ is the complex combination $\\Phi = \\phi _1+i \\phi _2$ .", "After going to polar coordinates, $\\Phi = \\sigma e^{i \\theta }$ , we find L()= -12 -12 2 - 12 3 3 - 12 m2   (2 +32) - 14   (2 +32)2 +   2 + 12 2 2 The ground state at finite chemical potential corresponds to constant field solutions of this Lagrangian.", "While we always have $\\langle \\phi _3 \\rangle = 0$ , there are cases when $\\langle \\sigma \\rangle \\ne 0$ , which spontaneously breaks the symmetry generated by $Q$ .", "This happens when the configuration $\\vec{\\phi }=0$ was unstable to begin with ($m^2<0$ ), or when the chemical potential exceeds a critical value ($\\mu ^2 > m^2$ ).", "In the broken phase, the VEV of the radial field is $\\langle \\sigma \\rangle ^2 = \\frac{1}{\\lambda }(\\mu ^2 - m^2)$ .", "Expanded to second order about this configuration, the Lagrangian (REF ) reads L() -12 -12 2 - 12 3 3 + 2 -(2 - m2) 2 - 12 2 32   .", "Let's assume that $\\langle \\vec{\\phi }\\rangle $ points in the $\\phi _1$ direction.", "Of the whole $SO(3)$ symmetry group, the only residual symmetry is that associated with $\\tau _1$ —rotations about $\\phi _1$ .", "This symmetry breaking pattern would normally be associated with two massless Goldstone bosons, one for each broken generator.", "Instead, here we see that the would-be Goldstone field $\\phi _3$ associated with $\\tau _2$ has acquired a mass $m_3= \\mu $ , in agreement with our general result, since $\\tau _2$ does not commute with $\\tau _3$ .", "The angular field ($\\theta $ ) is massless, and can be identified with the Goldstone boson associated with $\\tau _3$ , which obviously commutes with itself.", "In the alternative (but equivalent) SSP language [10], one starts from the relativistic Lagrangian (REF ), and looks for a time-dependent background solution that rotates about the $\\phi _3$ axis, with $\\dot{\\vec{\\phi }} = \\mu \\, \\tau _3 \\cdot \\vec{\\phi }$ .", "This just corresponds to a constant speed in the angular field, $\\theta (x) = \\mu \\, t$ .", "Such solutions are allowed only for $m^2<0$ or for $\\mu ^2 > m^2$ , in agreement with what we found above.", "After expanding in $\\theta $ fluctuations about such a solution, $\\theta \\rightarrow \\mu \\, t + \\theta $ , one finds precisely the Lagrangian (REF ), and its quadratic approximation (REF ), with the same excitation spectrum as above.", "However, the SSP language is closer to the viewpoint we emphasized in this paper, because it makes manifest that having a finite charge density for a spontaneously broken charge, necessarily implies a spontaneous breakdown of time-translations as well.", "Moreover, it stresses that Lorentz-invariance is broken spontaneously, by the field configuration one considers, rather than at the level of the Lagrangian.", "And finally, it gets the breaking pattern for internal symmetries right: all $SO(3)$ generators are broken, albeit in a time-dependent fashion, in the sense that $\\langle [\\tau _1, \\vec{\\phi }(x)] \\rangle $ and $\\langle [\\tau _2, \\vec{\\phi }(x)] \\rangle $ depend on time.", "This is what one would discover from our general analysis above, if one evaluated the order parameters $A_I$ at generic positions $x$ rather than at the origin.", "Concluding remarks.", "We conclude by stressing that our derivation of eq.", "(REF ) involved no approximation.", "As a result, our expression for the gap is an exact non-perturbative prediction.", "Our results evade the Nielsen-Chadha theorem, because one of its (implicit) assumptions is violated, namely that $Q$ commute with all other broken charges.", "This crucial difference should be taken into consideration also when comparing our results with the literature, like for instance the “kaon condensation\" model of refs.", "[4], [5].", "On the other hand, our results agree with the theorem of [5], which states—among other things—that there are no subtleties in counting the gapless Golstones for relativistic theories as long as all charge densities vanish.", "Acknowledgements.", "The work of AN is supported by the DOE (DE-FG02-92-ER40699) and by NASA ATP (09-ATP09-0049)." ] ]