# On the Integrability of Pfaffian Forms on ${\mathds{R}}^{n}$

Pedro F. da Silva Júnior  Universidade Federal de Pernambuco Caruaru, PE,
Brazil<EMAIL_ADDRESS>
###### Abstract.

This paper details the lesser known conditions on ${\mathds{R}}^{n}$ for the
integrability of pfaffian forms, or 1-forms. Emphasis is given to locality of
these conditions, and proofs in some additional detail are provided for
theorems due to Clairaut and Carathéodory. Considering the importance of the
integrability of pfaffian forms, in particular in mathematical-physics, this
paper shows that: there is a hidden content in Carathéodory’s theorem in the
direction of a global integrability.
Keywords: Pfaffian forms, integrability of pfaffian forms, Carathéodory’s
theorem.

##  1\. Introduction

Pfaffian forms are a particular case of differential forms: the 1-forms. The
treatment of this subject via differential forms, including the use of
exterior algebra and differentiable manifolds more general than
${\mathbb{R}}^{n}$, is beyond the scope of this paper111To the reader
interested in an extensive approach to the subject via differential forms, we
suggest consulting the book by Flanders [1], for a applied exposition in
physics, or the book by Morita [2], for one based in pure mathematics.. The
study of pfaffian forms has theoretical and practical relevance in itself,
particularly, in mathematical-physics [3, 4].

This paper has two main objectives. The first is to present to the reader, in
some detail, the lesser known conditions for the integrability of pfaffian
forms on ${\mathbb{R}}^{n}$, which have a local character and rarely appear in
textbooks of differential equations and differential forms. The second is to
discuss the possibility and obtainability of a global integrability criterion
for pfaffian forms. Frobenius Theorem does not fit the proposed roadmap for
this paper and thus will be omitted. Exception is a briefly comment in section
4.

Well, first studied by Clairaut, Fontaine, and Euler, according to Katz [5],
the pfaffian forms were so named in honor of Pfaff, who between 1814 and 1815,
treated the subject in greater detail [6]. Notable mathematicians later went
on to expand on Pfaff’s work, most notably Frobenius [7] and Cartan [8].

A suitable way to define pfaffian forms here is similar to how Morita [2]
does.

######  Definition 1.1.

(Pfaffian form). Let be a collection of n independent variables
$x_{1},x_{2},...,x_{n}$ in ${\mathbb{R}}^{n}$, and a collection of n functions
$F_{i}=F_{i}(x_{1},x_{2},...,x_{n})$ of class $C^{\infty}$ on an open set
$B\>{\subseteq}\>{\mathbb{R}}^{n}$. For the $\delta\xi$ objects,

$\displaystyle\delta{\xi}=\sum_{i=1}^{n}F_{i}(x_{1},x_{2},...,x_{n})dx_{i},$

defined in $B$, with $\delta\xi$ representing the infinitesimal of a certain
finite quantity $\xi$ in B, we call them pfaffian forms in n variables.

Henceforth when we refer to a generic pfaffian form we will be referring to a
pfaffian form $\delta\xi$, whose parameters are exemplified from Definition
1.1. Now, a pfaffian form may, or may not, refer to what we call the
differential of a function. When a pfaffian form $\delta\xi$ does not have the
functions $F_{i}$ identified as the partial derivatives of a function
$\xi=\xi(x_{1},x_{2},...,x_{n})$ with respect to the respective variables
$x_{i}$, ($F_{i}={\partial}{\xi}/{\partial}x_{i}$), then it does not represent
the differential of such a function $\xi$ and we call $\delta\xi$ an inexact
differential. That is, in this case, the quantity ${\delta}{\xi}$ represents
only the infinitesimal of a certain finite quantity ${\xi}$, and ${\xi}$ is
not a function of the $n$ independent variables $x_{i}$ in the sense of
${\xi}=\xi(x_{1},x_{2},...,x_{n})$.

Otherwise, if the pfaffian form ${\delta}\xi$ has the functions $F_{i}$
identified as the partial derivatives of a function
${\xi}=\xi(x_{1},x_{2},....,x_{n})$ with respect to the respective variables
$x_{i}$, ($F_{i}={\partial}{\xi}/{\partial}x_{i}$), then it is the
differential of such a function $\xi$, actually existing. Moreover, in this
case, we also call $\delta{\xi}$ an exact differential and replace, in the
symbolism of its designation, the symbol $\delta$ by the usual symbol $d$,
traditionally used to denote the infinitesimal of a quantity that is a usual
function.

It turns out that in some cases, even if the pfaffian form $\delta\xi$ is an
inexact differential, it can be written as the result of the product of a
function ${\mu}={\mu}(x_{1},x_{2},...,x_{n})$ with an exact differential
$d\psi$, where $d\psi$ is a function of the $n$ independent variables
$x_{1},x_{2},....,x_{n}$, that is, $\psi=\psi(x_{1},x_{2},...,x_{n})$. In
other words, there being $\mu$, and being
$\mu(x_{1},x_{2},...,x_{n})\;{\neq}\;0$, in an open set $A$, where
$A\>{\subseteq}\>B$, it follows that the quantity ${\delta}{\xi}/{\mu}$ will
be an exact differential $d\psi$ in $A$. When this happens we say that
$\delta\xi$ is integrable in $A$, and we call the function ${\mu}^{-1}$ the
integrating factor of $\delta\xi$. Furthermore, given the smoothness of
functions $F_{i}$ on all $B$, for the discussion of integrability we will also
assume that the studied pfaffian forms are always non-singular in $B$; i.e.,
not identically null on all $B$.

######  Definition 1.2.

(Integrable pfaffian form). Let $\delta\xi$ be a non-singular pfaffian form.
If there exist functions ${\mu}={\mu}(x_{1},x_{2},...,x_{n})$, with
$\mu(x_{1},x_{2},...,x_{n})\;{\neq}\;0$, and
$\psi=\psi(x_{1},x_{2},...,x_{n})$ such that

$\displaystyle\delta{\xi}=\mu{d\psi},$

on an open $A\>{\subseteq}\>B$, then $\delta\xi$ is said to be integrable on
$A$. Also, the function ${\mu}^{-1}$ is called the integrating factor of
$\delta\xi$.

Naturally, by Definition 1.2, every $\delta\xi$ such that $\delta\xi=d\xi$ is
integrable. To continue our discussion we need to mention the important
situation that occurs on paths in $B$ such that a pfaffian form nullifies,
where we get the so-called Pfaff equation associated with that pfaffian form.

######  Definition 1.3.

(Pfaff equation). The associated Pfaff equation for the pfaffian form
$\delta{\xi}$ is

$\displaystyle\delta{\xi}=0.$

It is very important to be said that a Pfaff equation not says that
$\delta{\xi}$ is identically null at $B$; that equation says at which paths of
$B$ the equation $\delta{\xi}=0$ has solution222This can be exemplified with
some uses of Pfaff equations in physics. In Analytic Mechanics, constraints
are often modeled by a Pfaff equation [9], and in Classical Thermodynamics the
usual condition for an adiabatic infinitesimal process
$\delta{\altmathcal{Q}}=0$ is precisely the Pfaff equation of the pfaffian
form heat, $\delta{\altmathcal{Q}}$ [10].. Next, to seek more familiarity with
the idea of integrable pfaffian forms, we will aim to analyze the solutions of
the Pfaff equations associated with them.

######  Definition 1.4.

(Pfaff Exact and Integrable Equation). The Pfaff equation associated with the
pfaffian form $\delta{\xi}$,

$\displaystyle\delta{\xi}=0,$

is called exact if, and only if, $\delta{\xi}$ is an exact differential,
$\delta{\xi}=d{\xi}$; if, and only if, the pfaffian form $\delta{\xi}$ is
integrable, the associated Pfaff equation is called integrable.

If $\delta{\xi}$ is a pfaffian form that constitutes an exact differential,
then $\delta{\xi}=d{\xi}$ and the associated Pfaff equation $d{\xi}=0$ has as
solution ${\xi}={\xi}(x_{1},x_{2},...,x_{n})$ = constant, which is
geometrically a hypersurface of $n-1$ dimension in $B$.

On the other hand, if $\delta{\xi}$ is a pfaffian form that is an inexact
differential but integrable, then $\delta{\xi}=0$ occurs in the same paths
where $d\psi=0$, i.e., where $\delta{\xi}=\mu{d\psi}$ holds, according to
Definition 1.2. In this situation, the solution of $\delta{\xi}=0$ is
$\psi=\psi(x_{1},x_{2},...,x_{n})$ = constant, which defines a hypersurface of
$n-1$ dimension, now, in $A$. Of course, if $\delta{\xi}$ is a non-integrable
Pfaffian form, then the solution of the equation $\delta{\xi}=0$ does not need
to define any geometric object restricted to $n-1$ dimensions as in the
previous cases.

This said, we can now ask the main question: in which situations is a pfaffian
form integrable? The sections 2 and 3 give us the answer.

##  2\. Local Integrability

This section deals with integrability conditions that have local character for
pfaffian forms; i.e., conditions that, when satisfied, are restricted to some
neighborhood $M$ contained in $B$, around some point $p$ of $B$. This will
become clearer in the section 3, where we will discuss conditions for global
integrability. For now, it is interesting that a definition for local
integrability be formalized.

######  Definition 2.1.

(Local integrability). If the non-singular pfaffian form $\delta{\xi}$ is
integrable restrictedly to some neighborhood $M\>{\subset}\>B$ of every point
$p\>{\in}\>B$, we say that $\delta{\xi}$ is locally integrable on $B$.

We will first discuss the simplest cases for local integrability: those of
pfaffian forms in two and three variables.

###  2.1. Pfaffian forms in two and three variables

To avoid unnecessary repetition, we will henceforth assume only non-singular
pfaffian forms. According to Definition 1.2, let be a pfaffian form
$\delta{\xi}$ in two variables, ${x_{1}}$ and ${x_{2}}$:

(1) $\delta{\xi}=F_{1}({x_{1}},{x_{2}})dx_{1}+F_{2}({x_{1}},{x_{2}})dx_{2}.$

The respective Pfaff equation associated with the pfaffian form of the
expression (1) is

(2) $F_{1}({x_{1}},{x_{2}})dx_{1}+F_{2}({x_{1}},{x_{2}})dx_{2}=0,$

which defines the following first-order ordinary differential equation,

(3)
$\frac{d{x_{2}}}{d{x_{1}}}=-\frac{F_{1}({x_{1}},{x_{2}})}{F_{2}({x_{1}},{x_{2}})}\,{\equiv}\,f({x_{1}},{x_{2}}),$

where $x_{2}=x_{2}(x_{1})$. Now, by the Existence and Uniqueness Theorem for
ordinary differential equations333Attention spent on formally stating this
theorem is redundant to the purpose of this paper. For more details of this
fundamental theorem we suggest reading the book by Coddington and Levinson
[11]. , if $f({x_{1}},{x_{2}})$ and
${\partial}f({x_{1}},{x_{2}})/{\partial}x_{2}$ are continuous on the open $B$,
then given some point
$p=({x_{1}}^{0},{x_{2}}^{0})\;{\in}B\;{\subseteq}\>{\mathbb{R}}^{2}$, there
then exists in $B$ a single curve ${\psi}({x_{1}},x_{2}(x_{1}))=$ constant,
parametrized by $x_{1}$, which provides the function ${x_{2}}=x_{2}({x_{1}})$
solution of the equation (3), such that it satisfies
${x_{2}}^{0}={x_{2}}({x_{1}}^{0})$ on some open interval $I$ containing
${x_{1}}^{0}$. We guarantee that the functions $F_{1}({x_{1}},{x_{2}})$ and
$F_{2}({x_{1}},{x_{2}})$ are $C^{\infty}$ on $B$ by definition, and we must
assume here that they are also, by construction, always non-null on $B$, given
the arbitrariness generated by setting up the equation (3) so that
$x_{2}=x_{2}(x_{1})$, instead of $x_{1}=x_{1}(x_{2})$. Otherwise, by this
arbitrariness, it could be $\delta{\xi}$ identically null, or indeterminate.
These collocations allow the use of this theorem for the equation (3). With
this in mind, we are able to deal with one of the most important theorems in
the theory of integrability of pfaffian forms.

######  Theorem 1.

Every pfaffian form in two variables on an open $B$ is locally integrable on
$B$.

Proof. Every Pfaff equation of a pfaffian form in two variables, whether this
equation is exact or not, defines an first-order ordinary differential
equation as the equation (3) that ensures, by the Existence and Uniqueness
Theorem, at least locally, the existence of a unique solution curve
${\psi}({x_{1}},x_{2}(x_{1}))=$ constant. Consider $d\psi$ in an open
$B\>{\subseteq}\>{\mathbb{R}}^{2}$:

(4)
$d\psi=\frac{{\partial}\psi}{{\partial}x_{1}}d{x_{1}}+\frac{{\partial}\psi}{{\partial}x_{2}}d{x_{2}}=0.$

Notice that equation (4) describes the same curves $x_{2}=x_{2}(x_{1})$ as
equation (3). By then substituting equation (3) into equation (4), we have:

(5)
$d\psi=\frac{{\partial}\psi}{{\partial}x_{1}}d{x_{1}}-\frac{F_{1}({x_{1}},{x_{2}})}{F_{2}({x_{1}},{x_{2}})}\frac{{\partial}\psi}{{\partial}x_{2}}d{x_{1}}=0.$

Rearranging the equation (5), and in view of Definition 1.2, we are invited to
define the function ${\mu}={\mu}(x_{1},x_{2})$, clearly non-zero, given by,

(6)
${{\mu}(x_{1},x_{2})}^{-1}\,{\equiv}\,\frac{1}{F_{1}({x_{1}},{x_{2}})}\frac{{\partial}\psi}{{\partial}x_{1}}=\frac{1}{F_{2}({x_{1}},{x_{2}})}\frac{{\partial}\psi}{{\partial}x_{2}},$

from which it immediately follows that:

(7)
${\mu}d{\psi}=F_{1}({x_{1}},{x_{2}})dx_{1}+F_{2}({x_{1}},{x_{2}})dx_{2}=\delta{\xi}.$

$\square$

Another proof for Theorem 1 can be found in [12]. In studying the
integrability of a pfaffian form $\delta{\xi}$ in three variables, $x_{1}$,
$x_{2}$ and $x_{3}$,

(8)
$\delta{\xi}=F_{1}({x_{1}},{x_{2}},{x_{3}})dx_{1}+F_{2}({x_{1}},{x_{2}},{x_{3}})dx_{2}+F_{3}({x_{1}},{x_{2}},{x_{3}})dx_{3},$

it is more pertinent to use the vector notation:
$\textbf{F}\,{\equiv}\,(F_{1},F_{2},F_{3})$,
$d\textbf{r}\,{\equiv}\,(d{x_{1}},d{x_{2}},d{x_{3}})$. Thus, $\delta{\xi}$ and
its associated Pfaff equation are represented by, respectively:
$\delta{\xi}=\textbf{F}\,{\cdot}\,d\textbf{r}$ and
$\textbf{F}\,{\cdot}\,d\textbf{r}=0$. In an open set, verification of the
following equation is necessary and sufficient for the integrability of the
pfaffian form in question:

(9) $\textbf{F}\,{\cdot}\,\nabla{\times}\textbf{F}=0.$

The preceding statement is a theorem. The proof of this result using the means
discussed so far is long, in particular as to whether the equation (9) is
sufficient for integrability, and so will be omitted in this paper. Later,
when we deal with the integrability of pfaffian forms in any number of
variables, the demonstration of the condition (9) for the case of three
variables will be immediately retrieved. A fact to be pointed out now is that
in the complete demonstration of the integrability condition (9) use is made
of the Theorem 1 for the conclusion of the integrability of pfaffian forms in
three variables [13]. As previously discussed, Theorem 1 has exclusively local
character. The result of this is that the condition (9) guarantees the
integrability of pfaffian forms in three variables also only locally, for an
appropriate open $B\>{\subseteq}\>{\mathbb{R}}^{3}$.

######  Theorem 2.

A pfaffian form in three variables on an open $B$ is locally integrable on $B$
if, and only if, $\textbf{F}\;{\cdot}\,\nabla{\times}\textbf{F}=0$ on $B$.

Indication of proof. Noting the Theorem 1, see Chapter 1 of Sneddon’s book
[13]. $\square$

However, it is not difficult to see that the equation (9) is a necessary
condition for the integrability of a pfaffian form in three variables. Let us
see, with the vector notation presented earlier, from the vector calculus we
take that if $\nabla{\times}\textbf{F}\neq\textbf{0}$, then $\delta{\xi}$ is
an inexact differential, because of Schwarz’s Theorem. In order for
$\delta{\xi}$ to be integrable, then there must exist a non-identically null
function $\mu$ such that, $\nabla{\times}({\mu}^{-1}\textbf{F})=\textbf{0}$.
In other words,

(10)
${\mu}^{-1}\nabla{\times}\textbf{F}+\nabla({\mu}^{-1}){\times}\textbf{F}=\textbf{0}.$

By the scalar multiplication of (10) by F, we obtain that the condition sought
is:

(11) $\textbf{F}\,{\cdot}\,\nabla{\times}\textbf{F}=0.$

For pfaffian forms in $n$ variables, we start by finding a necessary condition
for integrability that generalizes (11).

###  2.2. Pfaffian forms in $n$ variables

The important first results that follow were initially [5] obtained by
Clairaut.

######  Lemma 1.

A necessary condition for the integrability of a pfaffian form in n variables
is that $\mathfrak{R}_{ijk}$:

$\displaystyle\mathfrak{R}_{ijk}\>{\equiv}\>F_{i}\Bigg{[}\frac{{\partial}F_{k}}{{\partial}x_{j}}-\frac{{\partial}F_{j}}{{\partial}x_{k}}\Bigg{]}+F_{j}\Bigg{[}\frac{{\partial}F_{i}}{{\partial}x_{k}}-\frac{{\partial}F_{k}}{{\partial}x_{i}}\Bigg{]}+F_{k}\Bigg{[}\frac{{\partial}F_{j}}{{\partial}x_{i}}-\frac{{\partial}F_{i}}{{\partial}x_{j}}\Bigg{]},$

be annulled, for any $i$, $j$, $k$.

Proof. We start with writing a pfaffian form $\delta{\xi}$ in $n$ variables,
$x_{1},x_{2},...,x_{n}$:

(12) $\delta{\xi}=\sum_{i=1}^{n}F_{i}(x_{1},x_{2},...,x_{n})dx_{i}.$

From Definition 1.2, if $\delta{\xi}$ is integrable, then there exist
functions $\mu$ and $\psi$ which, under the appropriate conditions, satisfy
$\delta{\xi}=\mu{d{\psi}}$. It follows that, for each $i$:

(13) $\frac{{\partial}\psi}{{\partial}x_{i}}=\frac{1}{\mu}F_{i}.$

Now, if we derive the equation (13) with respect to some other variable,
namely $x_{j}$, we will have,

(14)
$\frac{{\partial}^{2}\psi}{{\partial}x_{j}{\partial}x_{i}}=\frac{{\partial}{\mu}^{-1}}{{\partial}x_{j}}F_{i}+{\mu}^{-1}\frac{{\partial}F_{i}}{{\partial}x_{j}}.$

By Schwarz’s Theorem,
${{\partial}^{2}\psi}/{{\partial}x_{j}{\partial}x_{i}}={{\partial}^{2}\psi}/{{\partial}x_{i}{\partial}x_{j}}$,
so,

(15)
$\frac{{\partial}{\mu}^{-1}}{{\partial}x_{i}}F_{j}+{\mu}^{-1}\frac{{\partial}F_{j}}{{\partial}x_{i}}=\frac{{\partial}{\mu}^{-1}}{{\partial}x_{j}}F_{i}+{\mu}^{-1}\frac{{\partial}F_{i}}{{\partial}x_{j}}$

Regrouping (15) and multiplying the whole equation by $\mu$,

(16)
$\frac{{\partial}F_{j}}{{\partial}x_{i}}-\frac{{\partial}F_{i}}{{\partial}x_{j}}={\mu}F_{i}\frac{{\partial}{\mu}^{-1}}{{\partial}x_{j}}-{\mu}F_{j}\frac{{\partial}{\mu}^{-1}}{{\partial}x_{i}}.$

Compared to the condition (11) for three variables, the left-hand side of (16)
invites us to look for ways to nullify it and thus obtain an integrability
condition that depends only on the derivatives of the functions $F_{i}$. This
occurs if we multiply (16) by another function $F_{k}$, and then cyclically
add terms analogous to
$F_{k}[{{\partial}F_{j}}/{{\partial}x_{i}}-{{\partial}F_{i}}/{{\partial}x_{j}}]$
so that,

(17)
$\mathfrak{R}_{ijk}\,{\equiv}\,F_{i}\Bigg{[}\frac{{\partial}F_{k}}{{\partial}x_{j}}-\frac{{\partial}F_{j}}{{\partial}x_{k}}\Bigg{]}+F_{j}\Bigg{[}\frac{{\partial}F_{i}}{{\partial}x_{k}}-\frac{{\partial}F_{k}}{{\partial}x_{i}}\Bigg{]}+F_{k}\Bigg{[}\frac{{\partial}F_{j}}{{\partial}x_{i}}-\frac{{\partial}F_{i}}{{\partial}x_{j}}\Bigg{]}=0,$

because the terms on the right-hand side of (16) cancel out with the analogous
terms when we add them up. $\square$

Immediately one sees the recovery of condition (11) by setting
$(i,j,k)=(1,2,3)$ on (17). The reciprocal of the Lemma 1 is valid, at least
locally. To show this, we first need to observe that: if the quantity
$\mathfrak{R}_{ijk}$ is null in a collection of variables, it will remain null
by a change of those variables.

######  Lemma 2.

The nullity of $\mathfrak{R}_{ijk}$ is invariant by a change of variables.

Proof. Let be a pfaffian form $\delta{\xi}$ in $n$ variables
$x_{1},x_{2},...,x_{n}$ such that it undergoes a change of variables to new
$n$ variables ${\bar{x}_{1}},{\bar{x}_{2}},...,{\bar{x}_{n}}$. The
differential of a variable
$x_{i}=x_{i}({\bar{x}_{1}},{\bar{x}_{2}},...,{\bar{x}_{n}})$ is then:

(18)
$d{x_{i}}=\sum_{j=1}^{n}\frac{{\partial}x_{i}}{{\partial}{\bar{x}}_{j}}d{\bar{x}}_{j}.$

The pfaffian form $\delta{\xi}$ can be represented in both collections of
variables, with their associated functions. Hence,

(19)
$\delta{\xi}=\sum_{i=1}^{n}{F_{i}}({x}_{1},{x}_{2},...,{x}_{n})d{x}_{i}=\sum_{j=1}^{n}{\bar{F}}_{j}({\bar{x}_{1}},{\bar{x}_{2}},...,{\bar{x}_{n}})d{\bar{x}}_{j},$

where, substituting (18) into (19), we obtain:

(20)
${\bar{F}}_{j}({\bar{x}_{1}},{\bar{x}_{2}},...,{\bar{x}_{n}})=\sum_{i=1}^{n}\frac{{\partial}x_{i}}{{\partial}{\bar{x}}_{j}}{F_{i}}({x}_{1},{x}_{2},...,{x}_{n}).$

Suppressing the explicit dependence to variables, by (17), in the collection
of new variables, the quantity ${\bar{\mathfrak{R}}}_{ijk}$ is:

(21)
${\bar{F}}_{i}\Bigg{[}\frac{{\partial}{\bar{F}}_{k}}{{\partial}{\bar{x}}_{j}}-\frac{{\partial}{\bar{F}}_{j}}{{\partial}{\bar{x}}_{k}}\Bigg{]}+{\bar{F}}_{j}\Bigg{[}\frac{{\partial}{\bar{F}}_{i}}{{\partial}{\bar{x}}_{k}}-\frac{{\partial}{\bar{F}}_{k}}{{\partial}{\bar{x}}_{i}}\Bigg{]}+{\bar{F}}_{k}\Bigg{[}\frac{{\partial}{\bar{F}}_{j}}{{\partial}{\bar{x}}_{i}}-\frac{{\partial}{\bar{F}}_{i}}{{\partial}{\bar{x}}_{j}}\Bigg{]}.$

We will initially analyze only the second term of (21), thus appropriately
substituting the new functions given in (20). By doing so,

(22)
$\sum_{i=1}^{n}\frac{{\partial}x_{i}}{{\partial}{\bar{x}}_{j}}{F_{i}}\Bigg{[}\sum_{k=1}^{n}\sum_{j=1}^{n}\frac{{\partial}x_{k}}{{\partial}{\bar{x}}_{i}}\frac{{\partial}{F}_{k}}{{\partial}{x}_{j}}\frac{{\partial}x_{j}}{{\partial}{\bar{x}}_{k}}-\sum_{j=1}^{n}\sum_{k=1}^{n}\frac{{\partial}x_{j}}{{\partial}{\bar{x}}_{k}}\frac{{\partial}{F}_{j}}{{\partial}{x}_{k}}\frac{{\partial}x_{k}}{{\partial}{\bar{x}}_{i}}\Bigg{]},$

we see that we obtain a term proportional to the first term of
${\mathfrak{R}}_{ijk}$, since the second order partial derivatives in the
variables vanish, by Schwarz’s Theorem. Repeating the same for the remaining
terms of (21), we have that:

(23)
${\bar{\mathfrak{R}}}_{ijk}=\Bigg{[}\sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{k=1}^{n}\frac{{\partial}x_{i}}{{\partial}{\bar{x}}_{j}}\frac{{\partial}x_{j}}{{\partial}{\bar{x}}_{k}}\frac{{\partial}x_{k}}{{\partial}{\bar{x}}_{i}}\Bigg{]}{\mathfrak{R}}_{ijk}.$

Therefore, if ${\mathfrak{R}}_{ijk}=0$, then ${\bar{\mathfrak{R}}}_{ijk}=0$.
$\square$

######  Theorem 3.

A sufficient condition for the local integrability of a pfaffian form in n
variables, in an open B, is that $\mathfrak{R}_{ijk}$ annuls, for any $i$,
$j$, $k$.

Proof. We will present a proof via finite induction which initially seeks to
show that the condition $\mathfrak{R}_{ijk}=0$, for any $i$, $j$, $k$, is
sufficient for integrability. From this it will follow that the most we can
say about such a condition is that, in fact, it is sufficient to local
integrability, only.

For a pfaffian form in one variable, $x_{1}$, by construction, it is clear
that $\delta{\xi}$ is always integrable. Whereas for a pfaffian form in $n$
variables,

(24) $\delta{\xi}=\sum_{i=1}^{n}F_{i}(x_{1},x_{2},...,x_{n})dx_{i},$

we assume that ${\mathfrak{R}}_{ijk}=0$, for any $i,j,k$. Next, we choose to
examine $\delta{\xi}$ for a path in an open $B\;{\subseteq}\;{\mathbb{R}}^{n}$
such that $d{x_{n}}=0$. The pfaffian form that results from fixing $x_{n}$ at
(24), is,

(25) ${\delta}\eta=\sum_{i=1}^{n-1}F_{i}(x_{1},x_{2},...,x_{n})dx_{i},$

so that naturally ${\mathfrak{R}}_{ijk}=0$ remains unchanged in (25), as the
induction hypothesis, since the nullity of this quantity does not change
because we fix a variable. We then assume that ${\delta}\eta$ is integrable,
under the circumstance ${\mathfrak{R}}_{ijk}=0$, for any $i,j,k$ different
from $n$ in $B$. On account of this, there must exist functions $\lambda$,
with $\lambda(x_{1},x_{2},...,x_{n-1})\;{\neq}\;0$, and
${\sigma}={\sigma}(x_{1},x_{2},...,x_{n-1})$, in some open
$A\>{\subseteq}\>B$, such that:

(26)
${\delta}\eta=\lambda{d\sigma}=\lambda\sum_{i=1}^{n-1}\frac{{\partial}\sigma}{{\partial}{x_{i}}}d{x_{i}}.$

Now, by letting $x_{n}$ vary, we can rewrite ${\delta}\xi$ as a function of
${\delta}\eta$, with ${\delta}\eta$ integrable by hypothesis, as we put it.
That is,

(27) ${\delta}\xi=\lambda{d\sigma}+{F_{n}}d{x_{n}},$

where, since ${\delta}\xi$ is a pfaffian form in $n$ variables, writing (27)
is equivalent to a change of variables in ${\delta}\xi$, from the variables
$x_{1},x_{2},...,x_{n}$, to certain new variables
${\bar{x}_{1}},{\bar{x}_{2}},...,{\bar{x}_{n-2}},{\sigma},{x_{n}}$. In this
new collection of variables it occurs that ${{\bar{F}}_{i}}=0$, for all
$i=\\{1,2,...,n-2\\}$. From the Lemma 2, the hypothesis of the nullity of
${\mathfrak{R}}_{ijk}$ holds for the new collection of variables. Again, this
relationship is preserved when examining only the collection of variables
${\bar{x}_{1}},{\bar{x}_{2}},...,{\bar{x}_{n-2}}$. Explicitly,

(28)
${\bar{\mathfrak{R}}}_{ijk}={\lambda}\frac{{\partial}{F_{n}}}{{\partial}{\bar{x}}_{i}}-{F_{n}}\frac{{\partial}\lambda}{{\partial}{\bar{x}}_{i}}=0,$

and we obtain that, on the variables
${\bar{x}_{1}},{\bar{x}_{2}},...,{\bar{x}_{n-2}},{\sigma},{x_{n}}$, the
quotient ${F_{n}}/\lambda$ must depend solely on $\sigma$ and $x_{n}$. With
$\lambda\;{\neq}\;0$, we can rewrite (27) as:

(29)
${\delta}\xi=\lambda\Bigg{(}{d\sigma}+\frac{F_{n}}{\lambda}d{x_{n}}\Bigg{)},$

The term in parentheses in (29) is a pfaffian form in two variables, which is,
by Theorem 1, locally integrable. Therefore, there exist functions $\mu$ and
$\psi$ such that,

(30) ${\delta}\xi=\lambda\mu{d\psi},$

under the appropriate conditions, and so ${\delta}\xi$ is locally integrable
on $B$. $\square$

We will now present one last result. Originally obtained in the formalization
of Classical Thermodynamics by C. Carathéodory in 1909 [14], and probably
figuring as the criterion for integrability of pfaffian forms most absent from
differential equations textbooks since then. This is a verification that
provides the local integrability of a pfaffian form from a topological
condition of the set $B$ to which the pfaffian form in question resides. The
proof of this Carathéodory’s Theorem is presented here in a little more detail
than in the works that first investigated it [15, 16], after Carathéodory’s
original proof [14].

######  Theorem 4.

(Carathéodory’s Theorem) A necessary and sufficient condition for the local
integrability of a pfaffian form ${\delta}{\xi}$ in n variables in an open B,
is that in every neighborhood $M\>{\subset}{\>}B$ arbitrarily close to any
point $p\>{\in}{\>}B$ there exist points unreachable from $p$ by curves such
that ${\delta}{\xi}=0$.

Proof. [16] Let be the pfaffian form ${\delta}\xi$ in $n$ variables, in an
open $B\>{\subseteq}\>{\mathbb{R}}^{n}$, and
$p=({x_{1}}^{0},{x_{2}}^{0},...,{x_{n}}^{0})$,
$q=({x_{1}}^{*},{x_{2}}^{*},...,{x_{n}}^{*})$,
$r=({x_{1}}^{**},{x_{2}}^{**},...,{x_{n}}^{**})$ points of $B$. Let be the
curves ${\gamma}_{1}$ and ${\gamma}_{2}$ in $B$, smooth, parametrized by a
real parameter $t$,

(31a)
$\displaystyle{\gamma}_{1}(t)=(f_{1}(t),f_{2}(t),...,f_{n}(t))\,{\equiv}{\,}(f_{i}(t)),$
(31b)
$\displaystyle{\gamma}_{2}(t)=(f_{1}(t)+{\nu}g_{1}(t),f_{2}(t)+{\nu}g_{2}(t),...,f_{n}(t)+{\nu}g_{n}(t))\,{\equiv}{\,}(f_{i}(t)+{\nu}g_{i}(t)),$

with ${\nu}$ real, such that ${\delta}{\xi}=0$, with the following conditions,
respectively,

(32a) $\displaystyle{\gamma}_{1}(t_{0})=$
$\displaystyle{\,}p,\quad{\gamma}_{1}(t_{*})=q,$ (32b)
$\displaystyle{\gamma}_{2}(t_{0})=$
$\displaystyle{\,}p,\quad{\gamma}_{2}(t_{**})=r,$

where $t_{0}<t_{*}<t_{**}$, with $|t_{*}-t_{0}|<\epsilon_{1}$ and
$|t_{**}-t_{*}|<\epsilon_{2}$, for arbitrarily small $\epsilon_{1}$ and
$\epsilon_{2}$. With a sufficiently small $\nu$, our goal is to examine the
situation $\epsilon_{2}\;{\rightarrow}\;0$. The Pfaff equation to which
${\gamma}_{2}$ is solution is given by,

(33)
$\displaystyle\sum_{i=1}^{n}F_{i}(f_{l}(t)+{\nu}g_{l}(t))d(f_{i}(t)+{\nu}g_{i}(t))=0$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}F_{i}(f_{l}(t)+{\nu}g_{l}(t))[\dot{f}_{i}(t)+{\nu}\dot{g}_{i}(t)],$

with $d{f_{i}}(t)/dt\,{\equiv}{\,}\dot{f}_{i}(t)$,
$d{g_{i}}(t)/dt\,{\equiv}{\,}\dot{g}_{i}(t)$. Deriving (33) by $\nu$, at
$\nu=0$,

(34)
$\sum_{i=1}^{n}F_{i}(f_{l}(t))\dot{g}_{i}(t)+\sum_{i=1}^{n}\sum_{j=1}^{n}\frac{{\partial}F_{i}(f_{l}(t))}{{\partial}{x_{j}}}\dot{f}_{i}(t){g}_{j}(t)=0,$

or

(35)
$\sum_{i=1}^{n}F_{i}(f_{l}(t))\dot{g}_{i}(t)=-\sum_{i=1}^{n}\sum_{j=1}^{n}\frac{{\partial}F_{i}(f_{l}(t))}{{\partial}{x_{j}}}\dot{f}_{i}(t){g}_{j}(t).$

Equation (35) is equivalent to us choosing $n-1$ of the functions ${g}_{j}(t)$
in an arbitrary manner and the $n$-th one we ensure obeys (35). Let then be
that $n$-th function ${g}_{k}(t)$, so that, isolating the $k$-index terms, we
have:

(36) $\displaystyle F_{k}(f_{l}(t))\dot{g}_{k}(t)+$
$\displaystyle\sum_{\begin{subarray}{c}i=1\\\ i\neq
k\end{subarray}}^{n}\frac{{\partial}F_{i}(f_{l}(t))}{{\partial}{x_{k}}}\dot{f}_{i}(t){g}_{k}(t)=$
$\displaystyle-$ $\displaystyle\sum_{\begin{subarray}{c}j=1\\\ j\neq
k\end{subarray}}^{n}F_{j}(f_{l}(t))\dot{g}_{j}(t)-\sum_{\begin{subarray}{c}i=1\\\
i\neq k\end{subarray}}^{n}\sum_{\begin{subarray}{c}j=1\\\ j\neq
k\end{subarray}}^{n}\frac{{\partial}F_{i}(f_{l}(t))}{{\partial}{x_{j}}}\dot{f}_{i}(t){g}_{j}(t).$

The function ${g}_{k}(t)$ then becomes the object to be studied if we make
$\epsilon_{2}\;{\rightarrow}\;0$. The left-hand side of (36) is a first-order
linear ordinary differential equation in ${g}_{k}(t)$, so, by Leibniz’s method
of the integrating factor, let be the function ${\eta}={\eta}(t)$, non-null,
such that:

(37)
$\frac{d({\eta}(t)F_{k}(f_{l}(t)){g}_{k}(t))}{dt}={\eta}(t)\Bigg{[}F_{k}(f_{l}(t))\dot{g}_{k}(t)+\sum_{\begin{subarray}{c}i=1\\\
i\neq
k\end{subarray}}^{n}\frac{{\partial}F_{i}(f_{l}(t))}{{\partial}{x_{k}}}\dot{f}_{i}(t){g}_{k}(t)\Bigg{]}.$

Developing the left-hand side of (37),

(38) $\displaystyle\frac{d({\eta}(t)F_{k}(f_{l}(t)){g}_{k}(t))}{dt}$
$\displaystyle={\eta}(t)F_{k}(f_{l}(t))\dot{g}_{k}(t)+$
$\displaystyle{\eta}(t)\sum_{\begin{subarray}{c}i=1\\\ i\neq
k\end{subarray}}^{n}\frac{{\partial}F_{k}(f_{l}(t))}{{\partial}{x_{i}}}\dot{f}_{i}(t){g}_{k}(t)+\dot{{\eta}}(t)F_{k}(f_{l}(t)){g}_{k}(t),$

and with the comparison between (37) and (38) comes,

(39)
$\dot{{\eta}}(t)F_{k}(f_{l}(t))={\eta}(t)\dot{f}_{i}(t)\sum_{\begin{subarray}{c}i=1\\\
i\neq
k\end{subarray}}^{n}\Bigg{[}\frac{{\partial}F_{i}(f_{l}(t))}{{\partial}{x_{k}}}-\frac{{\partial}F_{k}(f_{l}(t))}{{\partial}{x_{i}}}\Bigg{]}.$

Now, perpetuating Leibniz’s method and multiplying the two sides of (36) by
${\eta}(t)$, using (38) and (39), we can isolate ${g}_{k}(t)$ by integrating
the left-hand side of (37), which turns out to be equal to the right-hand side
of (36) when we multiply the latter by ${\eta}(t)$. We then have:

(40)
${\eta}(t^{\prime})F_{k}(f_{l}(t^{\prime})){g}_{k}(t^{\prime})=-\int_{t_{0}}^{t^{\prime}}{\eta}(t)\Bigg{[}\sum_{\begin{subarray}{c}j=1\\\
j\neq
k\end{subarray}}^{n}F_{j}(f_{l}(t))\dot{g}_{j}(t)+\sum_{\begin{subarray}{c}j=1\\\
j\neq k\end{subarray}}^{n}\sum_{\begin{subarray}{c}i=1\\\ i\neq
k\end{subarray}}^{n}\frac{{\partial}F_{i}(f_{l}(t))}{{\partial}{x_{j}}}\dot{f}_{i}(t){g}_{j}(t)\Bigg{]}dt,$

where $g_{i}(t_{0})=0$ for all $i$, by the conditions (32). Integrating by
parts the first term in the integrand of (40), using (39) and again that
$g_{i}(t_{0})=0$, we obtain directly:

(41)
$\displaystyle-\int_{t_{0}}^{t^{\prime}}{\eta}(t)\sum_{\begin{subarray}{c}j=1\\\
j\neq
k\end{subarray}}^{n}F_{j}(f_{l}(t))\dot{g}_{j}(t)dt=-{\eta}(t^{\prime})\sum_{\begin{subarray}{c}j=1\\\
j\neq k\end{subarray}}^{n}F_{j}(f_{l}(t^{\prime})){g}_{j}(t^{\prime})$
$\displaystyle+\int_{t_{0}}^{t^{\prime}}{\eta}(t)\sum_{\begin{subarray}{c}i=1\\\
i\neq k\end{subarray}}^{n}\sum_{\begin{subarray}{c}j=1\\\ j\neq
k\end{subarray}}^{n}\dot{f}_{i}(t){g}_{j}(t)\Bigg{(}\frac{F_{j}(f_{l}(t))}{F_{k}(f_{l}(t))}\Bigg{[}\frac{{\partial}F_{i}(f_{l}(t))}{{\partial}{x_{k}}}-\frac{{\partial}F_{k}(f_{l}(t))}{{\partial}{x_{i}}}\Bigg{]}+\frac{{\partial}F_{j}(f_{l}(t))}{{\partial}{x_{i}}}\Bigg{)}dt.$

Substituting (41) into (40), and putting the function $F_{k}(f_{l}(t))$ in
evidence on the integrand, we have:

(42)
$\displaystyle{\eta}(t^{\prime})F_{k}(f_{l}(t^{\prime})){g}_{k}(t^{\prime})=$
$\displaystyle-{\eta}(t^{\prime})\sum_{\begin{subarray}{c}j=1\\\ j\neq
k\end{subarray}}^{n}F_{j}(f_{l}(t^{\prime})){g}_{j}(t^{\prime})$
$\displaystyle+\sum_{\begin{subarray}{c}i=1\\\ i\neq
k\end{subarray}}^{n}\sum_{\begin{subarray}{c}j=1\\\ j\neq
k\end{subarray}}^{n}\int_{t_{0}}^{t^{\prime}}\frac{{\eta}(t)\dot{f}_{i}(t){g}_{j}(t)}{F_{k}(f_{l}(t))}\Bigg{(}F_{j}(f_{l}(t))\Bigg{[}\frac{{\partial}F_{i}(f_{l}(t))}{{\partial}{x_{k}}}-\frac{{\partial}F_{k}(f_{l}(t))}{{\partial}{x_{i}}}\Bigg{]}$
$\displaystyle+F_{k}(f_{l}(t))\Bigg{[}\frac{{\partial}F_{j}(f_{l}(t))}{{\partial}{x_{i}}}-\frac{{\partial}F_{i}(f_{l}(t))}{{\partial}{x_{j}}}\Bigg{]}\Bigg{)}dt.$

Since, by the curve $\gamma_{1}$ it is true that,

(43) $\sum_{i=1}^{n}F_{i}(f_{l}(t))\dot{f}_{i}(t)=0,$

then,

(44) $\sum_{\begin{subarray}{c}i=1\\\ i\neq
k\end{subarray}}^{n}\sum_{\begin{subarray}{c}j=1\\\ j\neq
k\end{subarray}}^{n}\int_{t_{0}}^{t^{\prime}}\frac{{\eta}(t)\dot{f}_{i}(t){g}_{j}(t)}{F_{k}(f_{l}(t))}F_{i}(f_{l}(t))\Bigg{[}\frac{{\partial}F_{k}(f_{l}(t))}{{\partial}{x_{j}}}-\frac{{\partial}F_{j}(f_{l}(t))}{{\partial}{x_{k}}}\Bigg{]}dt=0.$

Substituting (44) into (42), isolating ${g}_{k}(t^{\prime})$ in the process,
and identifying ${\mathfrak{R}}_{ijk}$ in the integrand, we have:

(45)
$\displaystyle{g}_{k}(t^{\prime})=\frac{1}{{\eta}(t^{\prime})F_{k}(f_{l}(t^{\prime}))}\Bigg{\\{}$
$\displaystyle-{\eta}(t^{\prime})\sum_{\begin{subarray}{c}j=1\\\ j\neq
k\end{subarray}}^{n}F_{j}(f_{l}(t^{\prime})){g}_{j}(t^{\prime})$
$\displaystyle+\sum_{\begin{subarray}{c}i=1\\\ i\neq
k\end{subarray}}^{n}\sum_{\begin{subarray}{c}j=1\\\ j\neq
k\end{subarray}}^{n}\int_{t_{0}}^{t^{\prime}}\frac{{\eta}(t)\dot{f}_{i}(t){g}_{j}(t)}{F_{k}(f_{l}(t))}{\mathfrak{R}}_{ijk}dt\Bigg{\\}}.$

If $t^{\prime}{\rightarrow}{t_{*}}$, with $\epsilon_{1}$ arbitrarily small,
where $|t_{*}-t_{0}|<\epsilon_{1}$, as well as $\epsilon_{2}$, where
$|t_{**}-t_{*}|<\epsilon_{2}$, then, in the limit
$\epsilon_{2}\;{\rightarrow}{\;}0$, we will have the formation of a
neighborhood $M{\>}{\subset}{\>}B$ around $p$, arbitrarily close to $p$, such
that there are points reachable from $p$ by curves on which ${\delta}{\xi}=0$.
The only exception to this is if every function ${g}_{k}(t)$ is identically
null, for all $t$. In the integrand of (45), this requires that,

(46) $\sum_{\begin{subarray}{c}i=1\\\ i\neq
k\end{subarray}}^{n}\dot{f}_{i}(t){\mathfrak{R}}_{ijk}=0,$

since, by construction, ${\eta}={\eta}(t){\>}{\neq}{\>}0$, the ${g_{j}}(t)$
cannot be fixed as zero and, obviously,
${F_{k}(f_{l}(t))}^{-1}{\>}{\neq}{\>}0$, for all $k$. Since the
$\dot{f}_{i}(t)$ can be arbitrary, except $\dot{f}_{k}(t)$, which does not
appear in (46), we conclude that ${\mathfrak{R}}_{ijk}=0$, for any $i$, $j$,
$k$. By the Lemma 1 and the Theorem 3 this proof ends. $\square$

##  3\. Global Integrability

The Carathéodory’s Theorem 4 has historically been placed under debate for
guaranteeing only a local integrability for pfaffian forms [17]. However, its
use in Classical Thermodynamics provides clues that the local nature of the
Carathéodory’s Theorem lies in the generality of its premise in topological
terms. More than that, when analyzed according to the descriptive needs of
Classical Thermodynamics [10], this theorem appears to ask for more than it
needs to obtain an integrating factor, by presuming a non-connection relation
between points in space (in the present context, the ${\mathbb{R}}^{n}$) valid
for any, arbitrarily small, neighborhood in this space.

This indicates the possibility of a hidden content in Carathéodory’s Theorem
that can be revealed by appropriate modification of the notion of
neighborhood. We will do this next, and obtain as a result a sufficient
condition for the integrability of pfaffian forms on all $B$, except for a set
of measure zero contained in $B$; what we will call here global integrability
on $B$.

######  Definition 3.1.

(Surrounding line). Given a point
$p=({x_{1}}^{0},{x_{2}}^{0},...,{x_{n}}^{0})\>{\in}{\>}B\>{\subseteq}{\>}{\mathbb{R}}^{n}$,
for the line ${\Pi}(x_{i})$ of the points
$({x_{1}}^{0},{x_{2}}^{0},...,{x_{i}},...,{x_{n}}^{0})$, with the arbitrary
variable $x_{i}$ called the free variable, we call the surrounding line to $p$
associated with $x_{i}$.

Notice that the union $\bigcup_{i=1}^{n}{\Pi}(x_{i})$ of the $n$ possible
surrounding lines ${\Pi}(x_{i})$ to a point
$p\>{\in}{\>}B\>{\subseteq}{\>}{\mathbb{R}}^{n}$ does not constitute
neighborhood $M$ of $p$.

######  Theorem 5.

A sufficient condition for the global integrability of a pfaffian form
${\delta}\xi$ in n variables in an open B is that on the surrounding line
${\Pi}(x_{i})$ of any point $p\>{\in}{\>}B$, for some free variable $x_{i}$,
there exist points arbitrarily close to $p$ unreachable from $p$ by curves
such that ${\delta}\xi=0$.

Proof. Let be the pfaffian form ${\delta}\xi$ in $n$ variables, on an open
$B\>{\subseteq}\>{\mathbb{R}}^{n}$, and
$p=({x_{1}}^{0},{x_{2}}^{0},...,{x_{n}}^{0})$ an arbitrary point of $B$. Let
$q=({x_{1}}^{0},{x_{2}}^{0},...,{x_{n}}^{*})$ also be a point on the
surrounding line to $p$ associated with the variable $x_{n}$, ${\Pi}(x_{n})$,
and ${\gamma}$ a curve in $B$ such that:

(47) $\sum_{i=1}^{n}{F_{i}}({\gamma})d{x_{i}}({\gamma})=0.$

With $|{x_{n}}^{*}-{x_{n}}^{0}|<{\varepsilon}$, let us assume that $\gamma$
passes through $p$ but does not pass through $q$, for any arbitrarily small
$\varepsilon$. It follows from this that the equation,

(48)
$d{x_{n}}({x_{1}},{x_{2}},...,{x_{n-1}})=-\sum_{i=1}^{n-1}\frac{{F_{i}}({x_{1}},{x_{2}},...,{x_{n}})}{{F_{n}}({x_{1}},{x_{2}},...,{x_{n}})}d{x_{i}},$

arising from (47), denotes $d{x_{n}}$ as the differential of a function
${x_{n}}={x_{n}}({x_{1}},{x_{2}},...,{x_{n-1}})$, so since, as we know, the
function $F_{n}({x_{1}},{x_{2}},...,{x_{n}})$ is not identically null. We
obtain ${x_{n}}={x_{n}}({x_{1}},{x_{2}},...,{x_{n-1}})$ explicitly by
integrating (48),

(49)
${x_{n}}({x_{1}},{x_{2}},...,{x_{n-1}})={x_{n}}^{0}-\int_{({x_{1}}^{0},{x_{2}}^{0},...,{x_{n-1}}^{0})}^{({x_{1}},{x_{2}},...,{x_{n-1}})}\sum_{i=1}^{n-1}\frac{{F_{i}}({x_{1}},{x_{2}},...,{x_{n}})}{{F_{n}}({x_{1}},{x_{2}},...,{x_{n}})}d{x_{i}},$

where ${x_{n}}^{0}={x_{n}}({x_{1}}^{0},{x_{2}}^{0},...,{x_{n-1}}^{0})$. Now,
if we make the quantities ${x_{1}}^{0},{x_{2}}^{0},...,{x_{n-1}}^{0}$ vary, we
get ${x_{1}},{x_{2}},...,{x_{n}}^{0}$ as the new independent variables on
which the function ${x_{n}}$, now
${x_{n}}={x_{n}}({x_{1}},{x_{2}},...,{x_{n}}^{0})$, depends. The function
${x_{n}}$ is continuous with respect to the variables
${x_{1}},{x_{2}},...,{x_{n-1}},{x_{n}}^{0}$, and differentiable with respect
to the variables ${x_{1}},{x_{2}},...,{x_{n-1}}$, by (48). Hence:

(50) $\frac{{\partial}{x_{n}}}{{\partial}{x_{i}}}=-\frac{F_{i}}{F_{n}}.$

Furthermore, by equation (49), the quotients ${F_{i}}/{F_{n}}$ might depend on
${x_{n}}^{0}$ in some way. However, if we fix the other variables
${x_{1}},{x_{2}},...,{x_{n-1}}$, at (49), we get that
${x_{n}}={x_{n}}({x_{1}},{x_{2}},...,{x_{n}}^{0})$ is a monotone function of
${x_{n}}^{0}$. This does not change for any closed interval contained in
${\Pi}(x_{n})$ on which we can make the same assumptions that have been
posited so far. As a consequence, by Lebesgue’s Differentiation Theorem [18],
${x_{n}}={x_{n}}({x_{1}},{x_{2}},...,{x_{n}}^{0})$ is a differentiable
function of ${x_{n}}^{0}$ on all $B$, except for a set of measure zero
contained in $B$. Thus, we observe that the differential of the function
${x_{n}}={x_{n}}({x_{1}},{x_{2}},...,{x_{n}}^{0})$,

(51)
$d{x_{n}}({x_{1}},{x_{2}},...,{x_{n}}^{0})=\sum_{i=1}^{n-1}\frac{{\partial}{x_{n}}}{{\partial}{x_{i}}}d{x_{i}}+\frac{{\partial}{x_{n}}}{{\partial}{x_{n}}^{0}}d{x_{n}}^{0},$

in the same way refers to all $B$, except for the same set of measure zero
contained in $B$. Retaking (47) explicitly, and using (51) and (50), we have:

(52) $\displaystyle{\delta}{\xi}=$
$\displaystyle\sum_{i=1}^{n-1}{F_{i}}d{x_{i}}+{F_{n}}d{x_{n}}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n-1}{F_{i}}d{x_{i}}+{F_{n}}\Bigg{(}\sum_{i=1}^{n-1}\frac{{\partial}{x_{n}}}{{\partial}{x_{i}}}d{x_{i}}+\frac{{\partial}{x_{n}}}{{\partial}{x_{n}}^{0}}d{x_{n}}^{0}\Bigg{)}$
$\displaystyle=$
$\displaystyle{F_{n}}\frac{{\partial}{x_{n}}}{{\partial}{x_{n}}^{0}}d{x_{n}}^{0}.$

Therefore, ${\delta}{\xi}$ is integrable over almost everywhere on $B$.
$\square$

##  4\. Applications and concluding remarks

In this paper we introduce the reader to the integrability conditions of
pfaffian forms on ${\mathbb{R}}^{n}$, with except for the well-known Frobenius
Theorem. We divide our discussion between local aspects, in section 2, and
global aspects, in section 3, with respect to integrability. Inspired by
Carathéodory’s Theorem, and its use in Classical Thermodynamics [10], an
integrability criterion of global character, namely, the Theorem 5, was
obtained in section 3.

The Theorem 5 when applied on Classical Thermodynamics may generate a very
important result: the construction of a differentiable almost everywhere
entropy function, so that the regions which the differentiability vanish are,
by the experimental justification of the theory, the points on thermodynamic
space of states related to phase transitions. Indeed, introducing the zeroth
law of thermodynamics and consequently the concept of a empirical temperature
$\vartheta$, this quantity can be now identified how the free variable in the
context of Theorem 5; i.e., we assume that $\vartheta=x_{n}$. Next, in
according with [16], by identifying also ${\delta}{\xi}$ as the quantity of
heat ${\delta}{\altmathcal{Q}}$, ${F_{n}}={F_{\vartheta}}$, and
${x_{n}}^{0}={\vartheta}^{0}$, the last expression in (52) can be rewritten if
one observes the deduction of the premise of Theorem 5 from Kelvin’s, or
Clausius’s, statement of the second law of thermodynamics [10]. Then taking a
change for new variables, ${\mu}$ and $\sigma$, one obtains,

${\delta}{\altmathcal{Q}}={\mu}d{\sigma},$

where it is not difficult to check that $\sigma$ is an differentiable almost
everywhere function of appropriate variables under consideration. With a
little more thermodynamics arguments [16], one can conclude that

${\delta}{\altmathcal{Q}}=Td{\altmathcal{S}},$

being $T$ the absolute temperature, and ${\altmathcal{S}}$ the absolute
entropy. This absolute entropy function, or just entropy function, have the
following properties: a) additivity (over thermodynamics systems); b)
extremization (maximization or minimization444For the reader with familiarity
in physics, this possibility of maximization or minimization, in addition with
the absence of a property of monotone increasing variation with respect the
internal energy, actually are the blessings of Carathéodory’s thermodynamics;
there is a wide set of experimental evidences that supports this apparent gaps
in the theory [19].) on a irreversible adiabatic process (the premise of
Theorem 5 can be generalized from the thermodynamic point of view of
irreversible processes); c) differentiability almost everywhere. With the
experimental based assumption of local bounded variation of the thermodynamics
quantities [20], we can assume that the entropy function have also its first-
derivatives with the local bounded variation property. This, together with the
property of differentiability almost everywhere, generates a final property
for the entropy function in question: d) local Lipschitz continuity. Actually,
due to Rademacher’s Theorem, the properties of this entropy function can be
summarized in: additivity, extremization, and local Lipschitz continuity.

In Analytical Mechanics as well, the application of a integrability criterion
for pfaffian forms is the procedure which leads to the verification of whether
non-holonomic constraints are, or are not, integrable. Constraints are
relations between the mechanical generalized coordinates, generalized
velocities, and eventually the time coordinate; when a constraint is a
relation between only the generalized coordinates and time, it is called
holonomic, otherwise it is called non-holonomic. A set of $n$ non-holonomic
constraints imposed to a mechanical system are often represented by $n$
correspondents Pfaff equations: $\delta{{\xi}_{1}}=0,...,\delta{{\xi}_{n}}=0$.
There are a practical importance for Analytical Mechanics in the verification
of the integrability of non-holonomic constrains: the constraints can be
applied directly in the lagrangian function of the mechanical system, thus
making it easier to obtain the equations of motion by solving the Euler-
Lagrange equations [21]. Besides that, the traditional method to do the
integrability test is by using the Frobenius Theorem, which calls for handling
with exterior algebra and the analytical knowledge of the Pfaff equations in
question.

On the other hand, in terms of original Carathéodory’s Theorem the test of
integrability for non-holonomic constraints can be performed without more
mathematical features, instead making use of physical inferences in the phase
space of the mechanical system and its respective constraints. For example,
the traditional problem of a perfect cylinder rolling without sliding on an
inclined plane can be easily visualized from the perspective that there are
states in the phase space of the system that are not accessible, hence the
respective constraint must be integrable. In this point of view, the original
Carathéodory’s Theorem show itself to be more physically substantial than
Frobenius Theorem, although the latter have a greater rigour and be most
useful, specially with more complicated constraints. However, the application
and respective descriptive consequences of a specie of Carathéodory’s Theorem,
like Theorem 5, in the integrability of non-holonomic constraint must be
better investigate.

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# On the statistical description of classical open systems with integer
variables by the Lindblad equation

E. D. Vol<EMAIL_ADDRESS>B. Verkin Institute for Low Temperature Physics
and Engineering of the National Academy of Sciences of Ukraine 47, Lenin Ave.,
Kharkov 61103, Ukraine.

###### Abstract

We propose the consistent statistical approach to consider a wide class of
classical open systems whose states are specified by a set of positive
integers(occupation numbers).Such systems are often encountered in physics,
chemistry, ecology, economics and other sciences.Our statistical method based
on ideas of quantum theory of open systems takes into account both
discreteness of the system variables and their time fluctuations - two effects
which are ignored in usual mean field dynamical approach.The method let one to
calculate the distribution function and (or)all moments of the system of
interest at any instant.As descriptive examples illustrating the effectiveness
of the method we consider some simple models:one relating to nonlinear
mechanics,and others taken from population biology .In all this examples the
results obtained by the method for large occupation numbers coincide with
results of purely dynamical approach but for small numbers interesting
differences and new effects arise.The possible observable effects connected
with discreteness and fluctuations in such systems are discussed.

###### pacs:

03.65.Ta, 05.40.-a

##  I Introduction

Among the vast number of classical open dynamical systems under consideration
in physics, chemistry, biology, economics and other sciences there are many
such whose states in accordance with the sense of the problem are specified by
a set of integer variables $\left(\left\\{n_{i}\right\\}\right)$ , where $i$ =
0,1,2,..N (N-number of degrees of freedom). For example in physics $n_{i}$\-
are occupation numbers of cell states in phase space, in chemistry - numbers
of molecules of reactive elements, in ecology -numbers of individuals in
populations which live in the area and interact with other populations, in
economics the number of companies operating on the market. Since all these
systems are classical their dynamics as a rule is described by a system of
differential equations of the form

$\frac{dn_{i}}{dt}=F_{i}\left(\left\\{n_{\alpha}\right\\}\right),$ (1)

where $F_{i}(\left\\{n_{\alpha}\right\\})$ -some nonlinear functions,
depending on concrete problem. Obviously notation Eq. (1) implies that
variables $n_{i}$ in system Eq. (1) -are considered as continuous. In the case
when all $n_{i}\gg 1$ such approach can be easy justified. From the physical
point of view the system of equations (1) corresponds to mean field
approximation and $n_{i}$ are occupation numbers averaged over some
appropriate statistical ensemble. In the case when all $n_{i}\gtrsim 1$,
dynamical description becomes inadequate and the question naturally arises: is
there consistent statistical approach which takes into account both
discreteness of variables $n_{i\text{ }}$ and their time fluctuations that may
be not small. In addition it is naturally to demand that such approach gave
the same results as dynamical description in the large $n_{i\text{ }}$ limit.
In this paper we propose such approach based on the ideas of quantum theory of
open systems (QTOS) and consider some examples that demonstrate its
effectiveness. The rest of the article organized as follows. In the Sect.2 we
briefly describe minimal information from (QTOS) which is necessary for
understanding of the method used and present main steps of our method. In
Sect.3 we consider simple model of nonlinear autonomous oscillator with soft
exciting mode and give its statistical description on the basis of the method
proposed. All the main features of the method clearly come to light already in
this representative example. In Sect.4 with the help of our approach we
consider some problems from population dynamics relating to evolution of two
interaction populations living in a certain area..We show that in the case of
small populations statistical description leads to a number of differences
from the ordinary dynamical picture. On the other hand in the case of large
occupation numbers both descriptions are virtually identical. In conclusion we
discuss some generalizations of the method and its possible experimental
verification.

##  II Description of the method

In this section we briefly remind the main points of the method proposed by
author earlier 1a  which allows one to make the transition from known
dynamical equations of classical open system to the master equation for its
quantum analogue. The method based on the correspondence that can be set
between quantum master equation in the Lindblad form and the Liouville
equation for distribution function in phase space of the classical system of
interest. This correspondence allows one using classical equations of motion
to restore the form of all operators involved in the Lindblad equation.
Thereby we can apply the procedure of quantization at least in semiclassical
approximation in the case of a large class of nonhamiltonian dynamical
systems. In short (all details see in 1a  ) the recipe of quantization
proposed consists of three consecutive steps.

Step1: The input classical dynamical equations should be presented in the form
allowed the quantization (FAQ). For the purposes of present paper the most
convenient form is complex representation of equations of motion:

$\frac{dz_{i}}{dt}=-i\cdot\frac{dH}{dz_{i}^{\ast}}+\mathop{\displaystyle\sum}\limits_{\alpha}\left(\overline{R_{\alpha}}\frac{dR_{\alpha}}{dz_{i}^{\ast}}-R_{\alpha}\frac{d\overline{R_{\alpha}}}{dz_{i}^{\ast}}\right),$
(2)

where $z_{i}=\frac{x_{i}+iy_{i}}{\sqrt{2}},$
$z_{i}^{\ast}=\frac{x_{i}-iy_{i}}{\sqrt{2}}$ are complex dynamical coordinates
of the system of interest, $H$, $R_{\alpha}$, $\overline{R_{\alpha}}$ are
functions of $z_{i}$, $z_{i}^{\ast}$ ( $H$ is real function, and $R_{\alpha}$,
$\overline{R_{\alpha}}$ are complex, $\overline{R_{\alpha}}$ means function
which conjugate to $R_{\alpha}$). It is necessary to emphasize that it is the
most delicate step of the method because it is difficult exactly to formalize
this point.

Step2. Having in hands representation Eq. (2) we can use classical function
$H$, $R_{\alpha}$, $\overline{R}_{\alpha}$ and with their help determine their
quantum analogues - operators $\widehat{H}$, $\widehat{R_{\alpha}}$,
$\widehat{R^{+}}$. For this purpose the variables $z_{i}$, $z_{i}^{\ast\text{
}}$must be replaced by the correspondence Bose operators $\widehat{a}_{i}$ and
$\widehat{a}_{i}^{+}$ with usual commutation rules:
$\left[\widehat{a}_{i},\widehat{a}_{j}^{+}\right]=\delta_{ij}$,
$\left[\widehat{a}_{i},\widehat{a}_{j}\right]=\left[\widehat{a}_{i}^{+},\widehat{a}_{j}^{+}\right]=0$.

Step3. The operators $\widehat{H}$, $\widehat{R_{\alpha}}$,
$\widehat{R_{\alpha}^{+}}$ found in this manner should be substituted into the
quantum Lindblad equation for the evolution of density matrix of the system:

$\frac{d\widehat{\rho}}{dt}=-i\left[\widehat{H},\widehat{\rho}\right]+\mathop{\displaystyle\sum}\limits_{\alpha}\left[\widehat{R_{\alpha}}\cdot\widehat{\rho},\widehat{R_{\alpha}^{+}}\right]+\left[\widehat{R_{\alpha}},\widehat{\rho}\cdot\widehat{R_{\alpha}^{+}}\right].$
(3)

The correspondence principle guarantees us that such approach will give
correct description of evolution of quantum open system at least with the
accuracy up to the first order in h .

##  III Representative model. Autonomous nonlinear oscillator with self
exciting mode

It is convenient to demonstrate on concrete example all the features of the
approach proposed. We consider an oscillator with nonlinear damping in
situation when its equilibrium point loses stability and small fluctuations
switch the system to the new stationary state that corresponds to the closed
trajectory (limit cycle). The following system of equations gives the correct
mathematical description of the behavior of the oscillator in the vicinity of
bifurcation point (see 2a ):

$\displaystyle\frac{dx}{dt}$ $\displaystyle=$ $\displaystyle\omega y+\mu
x-x\left(x^{2}+y^{2}\right),$ (4) $\displaystyle\frac{dy}{dt}$
$\displaystyle=$ $\displaystyle-\omega x+\mu y-y\left(x^{2}+y^{2}\right),$

where $x$, $y$ are coordinates of oscillator in phase space and $\omega$ -its
frequency. The system of the equations of motion Eq. (4) can be written in the
complex form as one equation

$\frac{dz}{dt}=-i\omega z+\mu z-2z\left|z\right|^{2},$ (5)

where $z=\frac{x+iy}{\sqrt{2}}$. It is easy to show that equation Eq. (5) can
be represented in the FAC. For this purpose we introduce the functions
$H=\omega z^{\ast}z$, $R_{1}=\sqrt{\mu}z^{\ast}$ and $R_{2}=z^{2}$. One can
verify by direct checking that r.h.s. of Eq. (5) can be rewritten in the form

$\frac{dz}{dt}=-i\frac{\partial H}{\partial
z^{\ast}}+\left(\overline{R_{1}}\frac{\partial R_{1}}{\partial
z^{\ast}}-R_{1}\frac{\partial\overline{R_{1}}}{\partial
z^{\ast}}\right)+\left(\overline{R_{2}}\frac{\partial R_{2}}{\partial
z^{\ast}}-R_{2}\frac{\partial\overline{R_{2}}}{\partial z^{\ast}}\right).$ (6)

According to the recipe of quantization the Lindblad equation for quantum
analog of the system Eq. (4) can be written in the form

$\frac{d\rho}{dt}=-i\left[H,\rho\right]+\left[R_{1}\rho,R_{1}^{+}\right]+\left[R_{2}\rho,R_{2}^{+}\right]+h.c,$
(7)

where $\widehat{H}=\omega\widehat{a}^{+}\widehat{a}$,
$\widehat{R}_{1}=\sqrt{\mu}\widehat{a}^{+}$ and
$\widehat{R}_{2}=\widehat{a}^{2}$.

We are interesting in the stationary solution of Eq. 7 and from physical
considerations we will seek it in the form:
$\widehat{\rho}=\widehat{\rho}(\widehat{N})=\sum\left|n\right\rangle\rho_{n}\left\langle
n\right|,$ where $\left|n\right\rangle$ are the eigenfunctions of the operator
$\widehat{N}=\widehat{a}^{+}\widehat{a}$. It is convenient to introduce the
generating function $G(u)$ for the coefficients $\rho_{n}$. By definition
$G(u)=\sum\limits_{n=0}^{n=\infty}\rho_{n}\cdot u^{n}.$One can obtain from Eq.
7 in stationary case the following equation for the function $G(u)$:

$(1+u)\frac{d^{2}G}{du^{2}}-\mu u\frac{dG}{du}-\mu G(u)=0.$ (8)

The solution of Eq. 8 that satisfies all conditions of the problem may be
represented as

$G(u)=\frac{\Phi(1,\mu,\mu(1+u))}{\Phi(1,\mu,2\mu)},$ (9)

where $\Phi(a,c,x)$ is well known confluent hypergeometric function (see 3a )
- $\Phi(a,c,x)=1+\frac{ax}{c}+\frac{a(a+1)}{2!c(c+1)}x^{2}+...$. Note that
condition $G(u)=1$ corresponds to normalization of $\ \widehat{\rho_{st}}(n)-$
namely $\sum\limits_{n}\rho_{n}=1$. Having in hands the expression Eq. 9 for
the generation function one can find the average values for any physical
quantity of interest in the stationary state that is all moments of the
distribution $\rho(n)$. For example moments of first and second order are
determined by relations:
$\overline{n}=\sum\limits_{n}n\cdot\rho_{n}=\frac{dG}{du}$,
$\overline{n^{2}}-\overline{n}=\frac{d^{2}G}{du^{2}}$. (all derivatives are
taken at point $u=1$). Let us consider now the behavior of our system in two
limiting cases: $\mu\gg 1$ and $\ \mu$ $\ll 1$. When $\mu\gg 1$ using the
asymptotic formula for $\Phi(a,c,x)$ (see 3a )we obtain for $G\left(u\right)$
the next expression

$G(u)\approx e^{\mu(u-1)}\cdot\left(\frac{1+u}{2}\right)^{1-\mu},$ (10)

The average number of quanta in stationary state $\ \overline{n}$ and their
dispersion $\sigma=\overline{n^{2}}-\overline{n}^{2}$ in this case are:
$\overline{n}=\frac{\mu+1}{2}$ and $\ \sigma=\frac{3\mu+1}{4}$. The relative
fluctuation of quanta generated in stationary state is
$\frac{\sqrt{\sigma}}{\overline{n}}=\frac{\sqrt{3\mu+1}}{\mu+1}$ tends to zero
when $\mu\gg 1$. The obtained result testifies validity of deterministic
approach in this case since in classical case
$n_{cl}=\left|z\right|^{2}=\frac{\mu}{2}$ when the system is moving along the
limit cycle. Now let us consider the opposite case when $\mu\ll 1$. Using the
series expansion for $\ \Phi(a,c,x)$ 3a  we obtained for $G(u)$ the next
expression (up to the first order in $\mu$)

$G(u)\backsimeq\frac{(2+u)(1+\mu)+\mu(1+u)^{2}}{3+7\mu}.$ (11)

The relation Eq. 11 implies that $G(u)$ tends to $G_{0}(u)=\frac{2+u}{3}$ when
$\mu$ tends to zero. The first two moments of the distribution $\rho(n)$ are:
$\overline{n}=\frac{1}{3}$ and $\sigma=\frac{2}{9}$. The relative fluctuation
of $n=\frac{\sqrt{\sigma}}{\overline{n}}$ is equal to $\sqrt{2\text{ }}$ in
this case. We see that in contrast to classical situation $\overline{n}\neq 0$
when $\mu$ tends to zero moreover fluctuations of occupation number turn out
to be large and very essential.

It is necessary to note that in classical system Eq. 4 in addition to variable
$\left|z\right|^{2}$ we have phase variable $\varphi$ that satisfies to
equation $\frac{d\varphi}{dt}=\omega$. However phase dynamics determines only
velocity along limit cycle but does not influence on stationary state itself.
In principle we can turn $\omega$ to zero (which implies that $H=0$) and all
foregoing results do not change. It is the point which explains why the
Lindblad equation can be successfully applied for statistical description of
classical open systems with integer variables. The reason is that in the case
when $H=0$ in Eq. 4, h may be eliminated from the Lindblad equation and wave
properties of quantum system turn out to be irrelevant. After this crucial
remark we can apply our approach to different classical systems with integer
variables in particular population dynamics models.

##  IV Statistical description of population dynamics models.

We begin our consideration with well known Lotka-Volterra model (LVM) (see 4a
) describing ”the interaction” between two populations: prays and predators
(for example hares and lynxes) living in the same territory. Let $n_{1}(t)$
and $n_{2}(t)$ the current numbers of preys and predators correspondently.
Then input dynamical equations of the LVM can be written as

$\displaystyle\frac{dn_{1}}{dt}$ $\displaystyle=$ $\displaystyle
n_{1}\left(a-2n_{2}\right),$ (12) $\displaystyle\frac{dn_{2}}{dt}$
$\displaystyle=$ $\displaystyle-n_{2}\left(b-2n_{1}\right),$

where coefficients $a$, $b$ are positive and have clear ecological meaning.
Note that for convenience we have chosen time unit thus to put coefficient of
the term $n_{1}n_{2}$ in Eq. 12 equal to 2. Now let us demonstrate that system
Eq. 12 can be represented in FAQ. For this purpose we introduce two auxiliary
complex variables $z_{1}$ and $z_{2}$ so that $\
n_{1}=\left|z_{1}\right|^{2}$and $n_{2}=\left|z_{2}\right|^{2}$. Let us assume
that evolution $z_{1}$, $z_{2}$ in time is governed by the following system of
equations:

$\displaystyle\ \frac{dz_{1}}{dt}$ $\displaystyle=$
$\displaystyle\lambda_{1}^{2}z_{1}-z_{1}\left|z_{2}\right|^{2},$ (13)
$\displaystyle\frac{dz_{2}}{dt}$ $\displaystyle=$
$\displaystyle-\lambda_{2}^{2}z_{2}+z_{2}\left|z_{1}\right|^{2}.$

It is easy to see that system Eq. 13 implies the following equations for
$n_{1}$, $n_{2}$

$\displaystyle\frac{dn_{1}}{dt}$ $\displaystyle=$ $\displaystyle
2\lambda_{1}^{2}n_{1}-2n_{1}n_{2},$ (14) $\displaystyle\frac{dn_{2}}{dt}$
$\displaystyle=$ $\displaystyle-2\lambda_{2}^{2}n_{2}+2n_{1}n_{2},$

which is exactly coincides with Eq. 12 if one lets $a=2\lambda_{1}^{2}$,
$b=2\lambda_{2}^{2}$.

Now if we introduce three functions $R_{1}=\lambda_{1}z_{1}^{\ast}$,
$R_{2}=\lambda_{2}z_{2}$, and $R_{3}=z_{1}z_{2}^{\ast}$ it is easy to verify
that Eq. 13 can be represented in the form

$\displaystyle\frac{dz_{1}}{dt}$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{3}\left(\overline{R_{i}}\frac{\partial
R_{i}}{\partial z_{1}^{\ast}}-R_{i}\frac{\partial\overline{R_{i}}}{\partial
z_{1}^{\ast}}\right),$ (15) $\displaystyle\frac{dz_{2}}{dt}$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{3}\left(\overline{R_{i}}\ \frac{\partial
R_{i}}{\partial z_{2}^{\ast}}-R_{i}\frac{\partial\overline{R_{i}}}{\partial
z_{2}^{\ast}}\right).$

Using the foregoing recipe of quantization we maintain that description of LVM
which takes into account the discreteness of variables $n_{1}$ and $n_{2}$ and
their time fluctuations is given by the next master equation

$\frac{d\rho}{dt}=\sum\limits_{i=1}^{\
3}\left[\widehat{R}_{i}\rho,\widehat{R}_{i}^{+}\right]+h.c.,$ (16)

where $\widehat{R}_{1}=\lambda_{1}\widehat{a}_{1}^{+}$,
$\widehat{R}_{2}=\lambda_{2}\widehat{a}_{2}$ and
$\widehat{R}_{3}=\widehat{a}_{1}\widehat{a}_{2}^{+}$. Again we will
interesting in only the solutions of Eq. 16 which have the form
$\widehat{\rho}=\sum\limits_{n_{1,n_{2}}}\left|n_{1}n_{2}\right\rangle\rho_{n_{1,}n_{2}}\left\langle
n_{1}n_{2}\right|$. In this case for the coefficients of the expansion
$\rho_{n_{1}n_{2}}$ we obtain the next general equation

$\frac{d\rho_{n_{1}n_{2}}}{dt}=2\lambda_{1}^{2}\left[n_{1}\rho_{n_{1}-1,n_{2}}-\left(n_{1}+1\right)\rho_{n_{1}n_{2}}\right]+2\lambda_{2}^{2}\left[\left(n_{2}+1\right)\rho_{n_{1,}n_{2}+1}-n_{2}\rho_{n_{1}n_{2}}\right]+2\left[\left(n_{1}+1\right)n_{2}\rho_{n_{1}+1,n_{2}-1}-\left(n_{2}+1\right)n_{1}\rho_{n_{1}n_{2}}\right]$
(17)

. It is convenient to introduce the generating function
$G(u,v,t)=\sum\limits_{n_{1}n_{2}}\rho_{n_{1}n_{2}}\cdot u^{n_{1}}\cdot
v^{n_{2}}.$Then after the simple algebra we find that Eq. 17 implies the next
equation for $G(u,v,t)$:

$\frac{\partial G}{\partial t}=2\lambda_{1}^{2}(u-1)\frac{\partial}{\partial
u}(uG)+2\lambda_{2}^{2}(1-v)\frac{\partial G}{\partial
v}+2(v-u)\frac{\partial^{2}}{\partial u\partial v}(vG).$ (18)

The Eq. 17 and Eq. 18 in principle give us all the necessary information about
statistical behaviour of LVM. Now let us consider concrete results following
from these equations. Note that even in stationary case it is difficult to
find exact analytical solutions of Eq. 18 for arbitrary $\lambda_{1}$and
$\lambda_{2}$. But in the special case when
$\lambda_{2}^{2}=1+\lambda_{1}^{2}$ such solution easily can be found and have
the form $G(u,v)=\frac{\left(1-\varkappa\right)^{2}}{\left(1-\varkappa
u\right)\left(1-\varkappa v\right)}$ where
$\varkappa=\frac{\lambda_{1}^{2}}{1+\lambda_{1}^{2}}$. Since $G(u,v)=g(u)\cdot
g(v)$ it is clear that $n_{1}$and $n_{2}$ are independent variables in this
case and
$\overline{n_{1}}=\overline{n_{2}}=\frac{\varkappa}{1-\varkappa}=\lambda_{1}^{2}.$The
dispersion $\sigma_{1}^{2}$ in this case is equal to
$\lambda_{1}^{2}+\lambda_{1}^{4}$ and if we calculate relative fluctuation of
$n_{1}$ in stationary state we get the result: $\delta
n_{1}=\frac{\sqrt{\sigma_{1}}}{\overline{n_{1}}}=\sqrt{1+\frac{1}{\lambda_{1}^{2}}}$.
Thereby we see this quantity is not small and can be easily measured. More
detailed analysis of statistical behaviour of LVM following from Eq. 17 for
arbitrary $\lambda_{1\text{ }}$and $\lambda_{2}$ will be carried out
elsewhere. Here we show only that using Eq. 18 one can easy obtain a
collection of explicit relations connecting different moments of distribution
$\rho(n_{1},n_{2})$. For example if we differentiate stationary Eq. 18 with
respect to $u$ and after that put $u=v=1$ we obtain the simple relation
between moments of first and second order which reads as:

$\lambda_{1}^{2}\left(1+\overline{n_{1}}\right)-n_{1}-\overline{n_{1}n_{2}}=0.$
(19)

In a similar manner by differentiating of Eq. 18 with respect to v we obtain
the second relation:

$-\lambda_{2}^{2}\overline{n_{2}}\ +\overline{n_{1}}+\overline{n_{1}n_{2}}=0\
$ (20)

Relations Eq. 19 and Eq. 20 imply the helpful equation connecting
$\overline{n_{1}}$ and $\overline{n_{2}}$ namely
$\frac{\overline{n_{2}}}{1+\overline{n_{1}}}=\frac{\lambda_{1}^{2}}{\lambda_{2}^{2}}$.
It is worth to note that in classical LVM Eq. 14 similar relation exists:
$\frac{\overline{n_{2}}}{\overline{n_{1}}}=\frac{\lambda_{1}^{2}}{\lambda_{2}^{2}}$,
so we conclude that when $\overline{n_{1}}\gg 1$, $\overline{n_{2}}\gg 1$ the
results of statistical description completely coincide with pure dynamical
consideration. But in the case of small numbers $n_{1}$, $n_{2}$ the
difference between them may be essential. To demonstrate this distinction and
also to compare the approach proposed in the present paper with usual
Markovian description of such systems proposed in well known article of
Nicolis and Prigogine 5a  and expanded in their later book 6a  it is
appropriate to consider the truncated case of LVM, with
$\lambda_{1}=\lambda_{2}=0$. It is obvious that total number of individuals
N$=n_{1}+n_{2}$ in this model will be constant .This fact greatly simplifies
finding and analysis of solutions Eq. 18 which in this case takes the form:

$\frac{\partial G}{\partial t}=(v-u)\frac{\partial^{2}}{\partial u\partial
v}(vG)\ $ (21)

It is easy to see that Eq. 21 has solutions for any integer N in the form of
homogeneous polynomial in $u$ and $v$ of degree N , namely $G_{N\text{
}}(u,v,t)=\sum\limits_{k}A_{k}\left(t\right)u^{k}v^{N-k}$ where coefficients
$A_{k}$ satisfy the normalization condition $\sum\limits_{k}A_{k}=1$. Thereby
Eq. 21 is reduced to the linear system of equations of N+1order for
coefficients $A_{k}$ of the form$\ \frac{dA_{k}}{dt}=L_{km}A_{m}$, where
matrix elements $L_{km}$ can directly be found from Eq. 21 for any N. For
example in simplest case when N=2 the matrix $L_{km\text{ }}$has the
form$\begin{pmatrix}-2&0&0\\\ 2&-2&0\\\ 0&2&0\end{pmatrix}$.

Let us consider now this case more detail. Let
$G_{2}\left(u,v,t\right)=au^{2}+buv+cv^{2}$ is generating function of the
model. Then Eq. 21 implies the next system of equations for evolution of
coefficients $a,b,c$.

$\frac{da}{dt}=-2a,\frac{db}{dt}=2a-2b,\frac{dc}{dt}=2b.$ (22)

Together with normalization condition $a+b+c=1$ system Eq. 22 allows one to
give statistical description of the model at any time. In particular Eq. 22
implies that when t tends to infinity $\overline{n_{1}}$ tends to zero and
$\overline{n_{2\text{ }}}$ tends to 2.This completely agrees with solutions of
dynamical equations in this case. From the other hand let us consider now the
equation for generating function of this model obtained by Nicolis and
Prigogine (see eq .10.66 in their book 6a ) which in our notation has the form

$\frac{dG}{dt}=(v-u)v\frac{\partial^{2}G}{\partial u\partial v}.$ (23)

In the case $N=2$ Eq. 23 implies the system of equations for coefficients of
expansion $G_{2}=au^{2}+buv+cv^{2}$ which differs from 22 namely

$\frac{da}{dt}=0,\frac{db}{dt}=-b,\frac{dc}{dt}=b.$ (24)

Eq. (24) imply that when t tends to infinity both quantities
$\overline{n_{1}}$, $\overline{n_{2}}$ tend to nonzero values depending on
initial conditions what obviously does not agree with dynamical equations. But
when $N\gg 1$ it is easy to see that asymptotic behaviour solutions obtained
from equations Eq. (21) and Eq. (23) are virtually identical. Thus in this
example we see the benefits of approach proposed over the standard methods
which do not allow to consider explicitly the discreteness of variables of the
problem.

Finally in the last part of the paper using one concrete model of the
population dynamics we want to demonstrate that approach proposed let one to
give statistical description of the systems for which dynamical description in
the framework of mean field theory looks as oversimplified. For this purpose
we consider two ”competing” kins of cannibals eating each other so that
voracity of individuals in both kins is assumed to be distinctive. As one
knows cannibalism is widespread in living nature and plays important role in
evolution processes 7a . Besides many species possess special mechanisms which
let them to recognize relatives and to avoid of their eating 8a . Let us
assume that mutual eating is the major factor of changes of the number of
individuals in both kins. Then evolution of the number of individuals
$n_{1}$and $n_{2}$ in such model can be represented of simple system of
equation of the form

$\displaystyle\frac{dn_{1}}{dt}$ $\displaystyle=$ $\displaystyle
an_{1}n_{2}-bn_{1}n_{2},$ (25) $\displaystyle\frac{dn_{2}}{dt}$
$\displaystyle=$ $\displaystyle-an_{1}n_{2}+bn_{1}n_{2}.$

It is easy to see that total number of individuals in this model
N$=n_{1}+n_{2}$ conserves. But dynamical description of the system with the
help of Eq. (25) seems to be oversimplified. In particular it implies that for
any N when $a>b$ and t tends to infinity $n_{1}$tends to N , and $n_{2}$ tends
to zero. Now following the spirit of our method we will describe this system
by the help of two operators
$\widehat{R_{1}}=\lambda_{1}\widehat{a_{1}}\widehat{a_{2}^{+}}$ and
$\widehat{R_{2}}=\lambda_{2}\widehat{a_{1}^{+}}\widehat{a_{2}}$. Show that
such statistical version is completely consistent. Actually acting as in
previous examples we can write the master equation for the density matrix of
the system as

$\
\frac{d\widehat{\rho}}{dt}=\left[\widehat{R_{1}}\widehat{\rho,}\widehat{R_{1}^{+}}\right]+\left[\widehat{R_{2}}\widehat{\rho},\widehat{R_{2}^{+}}\right]+h.c..$
(26)

If again we are interesting in by solutions of the Eq. (26) of the form
$\widehat{\rho}=\sum\limits_{n_{1}n_{2}}\left|n_{1}n_{2}\right\rangle\rho_{n_{n}n_{2}}\left\langle
n_{1}n_{2}\right|$ then for the generating function of the problem
$G(u,v,t)=\sum\limits_{n_{1}n_{2}}\rho_{n_{1}n_{2}}u^{n_{1}}v^{n_{2}}$ we
obtain the equation

$\
\frac{dG}{dt}=a\left(u-v\right)\frac{d^{2}}{dudv}\left(uG\right)+b\left(v-u\right)\frac{d^{2}}{dudv}\left(vG\right),$
(27)

where $a=2\lambda_{2}^{2}$ and $b=2\lambda_{1}^{2}$. The equation Eq. (27)
implies that generating function of stationary state of this model in the case
when total number of individuals is equal to N can be represented in the form

$G_{st}\left(u,v\right)=C_{N}\left[\frac{\left(bv\right)^{N+1}-\left(au\right)^{N+1}}{bv-
au}\right]=C_{N}\left[\left(bv\right)^{N}+\left(bv\right)^{N-1}\left(au\right)+...\right]$
(28)

where $C_{N}$ is the normalization factor,
$C_{N}=\left(b^{N}+ab^{N-1}+...\right)^{-1}$ Having in hands expression Eq.
(28) for the generation function one can find all statistical characteristics
of the model for any N. In particular for N=2 for the average values of
individuals in both kins we obtain
$\overline{n_{1}}=\frac{2a^{2}+ab}{b^{2}+ab+a^{2}}$, and $\
\overline{n_{2}}=\frac{2b^{2}+ab}{b^{2}+ab+a^{2}}$. Now we want to show that
when N tends to infinity the properties of our statistical model will be
similar to ones of the dynamical model Eq. (25).Let us assume that $\ b>a$ and
let $\ \ \ \ \ \ \ \varkappa=\frac{a}{b},\left(\varkappa<1\right).$ Let us
calculate now the ratio $\frac{\overline{n_{1}}}{\overline{n_{2}}}$ in this
model. Using Eq. (28) we obtain

$\frac{\overline{n_{1}}}{\overleftarrow{n_{2}}}=\frac{Na^{N}+\left(N-1\right)a^{N-1}b+...}{Nb^{N}+\left(N-1\right)b^{N-1}a+...}=\frac{N\varkappa^{N}+\left(N-1\right)\varkappa^{N-1}+...\varkappa}{N\text{
}+\left(N-1\right)\varkappa+....\varkappa^{N-1}}$ (29)

. Expression Eq. (29) can be represented in the next convenient form

$\frac{\overline{n_{1}}}{\overline{n_{2}}}=\frac{\varkappa+\varkappa^{2}\frac{\partial
Lnf_{N}\left(\varkappa\right)}{\partial\varkappa}}{N-\varkappa\frac{\partial
Lnf_{N}\left(\varkappa\right)}{\partial\varkappa}},$ (30)

where $f_{N}\left(\varkappa\right)=\frac{1-\varkappa^{N}}{1-\varkappa}$. It is
easy to see that
$\frac{\partial\left(Lnf_{N}\right)}{\partial\varkappa}=\frac{\left(N-1\right)\varkappa^{N}-N\varkappa^{N-1}+1}{\left(1-\varkappa\right)\left(1-\varkappa^{N}\right)}$
. Obviously this expression tends to $\frac{1}{1-\varkappa}$ when
$\varkappa<1$ and N tends to infinity .Thus Eq. (30) implies that ratio
$\frac{\overline{n_{1}}}{\overline{n_{2}}}$ tends to zero but this case is
realized only for infinite population. From the other hand for any finite
population of cannibals all its statistical properties can be well described
by the approach proposed in our paper. In conclusion let us briefly summing up
our consideration. We propose the statistical method of describing of various
systems in physics,chemistry,ecology whose states can be represented by
integers. Although the method proposed is based on quantum Lindblad equation
nevertheless we have shown that it can be successfully used also for
description of classical open systems with integer variables (at least in the
cases when ”Hamiltonian” of the open system is equal to zero) .All examples
considered in present paper demonstrate that approach proposed results in
consistent conclusions and allows one in several cases to eliminate essential
inaccuracies of preceding considerations.

##  V Acknowledgement

The author acknowledges I.V. Krive and S.N. Dolya for the discussions of the
results of the paper and valuable comments.

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