Propagation processes of correlations
of hard spheres

V. I. Gerasimenko111E-mail<EMAIL_ADDRESS>
Institute of Mathematics of the NAS of Ukraïne,
3, Tereshchenkivs’ka Str.,
01004, Kyïv, Ukraïne

I. V. Gapyak222E-mail<EMAIL_ADDRESS>
Taras Shevchenko National University of Kyïv,
Department of Mathematics and Mechanics,
4, Academician Glushkov ave.,
03127, Kyïv, Ukraïne

Abstract. The paper develops an approach to the description of the evolution
of correlations for many hard spheres based on a hierarchy of evolution
equations for the cumulants of the probability distribution function governed
by the Liouville equation. It is established that the constructed dynamics of
correlations underlies the description of the evolution of the states of many
hard spheres described by the BBGKY hierarchy for reduced distribution
functions or the hierarchy of nonlinear evolution equations for reduced
correlation functions. As an application of the developed approach to
describing the evolution of the state of many hard spheres within the
framework of dynamics of correlations, the challenges of the derivation of
kinetic equations are discussed.

Key words: Liouville hierarchy, BBGKY hierarchy, kinetic equation, cumulant,
correlation function.

2010 Mathematics Subject Classification: 82C22, 82C40, 35Q20.

math

###### Contents

  1. 1 Introduction
  2. 2 Dynamics of correlations of a hard-sphere system
    1. 2.1 Preliminaries: dynamics of finitely many hard spheres
    2. 2.2 Correlation functions
    3. 2.3 The Liouville hierarchy
  3. 3 Propagation of correlations in a hard-sphere system
    1. 3.1 Evolution of states described by the dynamics of correlations: reduced distribution functions
    2. 3.2 Evolution of states described by the dynamics of correlations: reduced correlation functions
  4. 4 The description of correlations by means of kinetic equations
    1. 4.1 The non-Markovian Enskog equation
    2. 4.2 On the low-density approximation of hard-sphere correlations
  5. 5 Conclusion

##  1 Introduction

Recently, mainly in connection with the problem of the rigorous derivation of
kinetic equations from the underlying particle dynamics [1]-[5], a number of
papers [6]-[16] have appeared discussing possible approaches to describing the
evolution of the state of many-particle systems, in particular, many hard
spheres.

As is known [1]-[3], the evolution of the state of finitely many hard spheres
is traditionally described by a probability distribution function governed by
the Liouville equation. This article is developed an alternative approach to
the description of the evolution of the state, which consists of the use of
functions defined as cumulants of the mentioned probability distribution
function. The cumulants of the probability distribution function are
interpreted as correlations of the states of hard spheres, and further, the
term correlation functions are used for them. The evolution of the correlation
functions is governed by the Liouville hierarchy for hard spheres constructed
in the article.

One more approach that allows describing the evolution of states of the
systems both from a finite and from an infinite number of particles [1]-[3] is
to describe the state by means of a sequence of so-called reduced distribution
functions (marginals [5]-[8]) governed by the BBGKY
(Bogolyubov–Born–Green–Kirkwood–Yvon) hierarchy [1]-[5]. Note that this method
is traditionally used in the rigorous derivation of the Boltzmann kinetic
equation from the dynamics of hard spheres [17]-[20]. An alternative of the
approach to such a description of a state is based on a sequence of functions
determined by the cluster expansions of reduced distribution functions that
are, by their cumulants. These functions are interpreted as the reduced
correlation functions and they are governed by the corresponding hierarchy of
the nonlinear evolution equations [21]-[Ch].

In this article, we develop an approach to the description of the evolution of
a state by means of reduced distribution functions and reduced correlation
functions, which is based on the dynamics of correlations in a system of hard
spheres. It should be emphasized that the generating operators of solution
expansions of the Cauchy problem of the corresponding hierarchies of evolution
equations are induced by the generating operators of an expansion of the
solution of the Cauchy problem of the Liouville hierarchy for correlation
functions which are represented by means of the corresponding-order cumulants
of the groups of operators of the Liouville equations [15],[22].

In the last section of the article, as an application of a developed approach
to describing the evolution of the state of many hard spheres within the
framework of the dynamics of correlations, we consider two issues related to
the problem of the rigorous derivation of kinetic equations. One of them is a
method for describing the evolution of the state of many hard spheres within
the framework of the evolution of the state of a typical particle, described
by a generalization of the Enskog equation [14], or, in other words, the
fundamentals of describing the evolution of correlations by kinetic equations
are discussed. We remark that the conventional approach to the mentioned
problem is based on the construction of the Boltzmann–Grad asymptotic behavior
[17]-[20] of a solution of the BBGKY hierarchy for reduced distribution
functions represented as the series of the perturbation theory for initial
data specified by a one-particle distribution function in the case of absence
of correlations (so-called initial chaos) [1]-[5]. In this connection, we also
consider a sketch of constructing the scaling asymptotics of the reduced
correlation functions in the low-density limit.

##  2 Dynamics of correlations of a hard-sphere system

###  2.1 Preliminaries: dynamics of finitely many hard spheres

At the initial instant in the space $\mathbb{R}^{3}$ the state of a hard-
sphere system of a non-fixed, i.e. arbitrary but finite average number of
identical particles, is described by the sequence
$D(0)=(1,D_{1}(0),\ldots,D_{n}(0),\ldots)$ of the probability distribution
functions $D_{n}(0),\,n\geq 1$ defined on the phase space
$\mathbb{R}^{3n}\times(\mathbb{R}^{3n}\setminus\mathbb{W}_{n})$ of $n$ hard
spheres. Each hard sphere of a diameter $\sigma>0$ is characterized by the
phase space coordinates $(q_{i},p_{i})\equiv
x_{i}\in\mathbb{R}^{3}\times\mathbb{R}^{3},\,i\geq 1$ and for configurations
in the space the following inequalities are satisfied:
$|q_{i}-q_{j}|\geq\sigma,$ $i\neq j\geq 1$, i.e. the set
$\mathbb{W}_{n}\equiv\big{\\{}(q_{1},\ldots,q_{n})\in\mathbb{R}^{3n}\big{|}|q_{i}-q_{j}|<\sigma$
for at least one pair $(i,j):\,i\neq j\in(1,\ldots,n)\big{\\}}$ is the
forbidden configuration set. The nonnegative functions
$D_{n}(0)=D_{n}(0,x_{1},\ldots,x_{n}),\,n\geq 1,$ that are symmetric with
respect to permutations of the arguments $x_{1},\ldots,x_{n}$, equal to zero
on the set of forbidden configurations $\mathbb{W}_{n}$ will be assumed to
belong to the space of integrable functions $L^{1}_{n}\equiv
L^{1}(\mathbb{R}^{3n}\times\mathbb{R}^{3n})$ equipped with the norm:
$\|f_{n}\|_{L^{1}_{n}}=\int_{\mathbb{R}^{3n}\times\mathbb{R}^{3n}}dx_{1}\ldots
dx_{n}|f_{n}(x_{1},\ldots,x_{n})|$.

As is known [1]-[4], the evolution of all possible states of a hard-sphere
system is described by the sequence $D(t)=(1,D_{1}(t),\ldots,D_{n}(t),\ldots)$
of the following probability distribution functions:

$\displaystyle\hskip-14.22636ptD_{n}(t,x_{1},\ldots,x_{n})=\begin{cases}D_{n}(0,X_{1}(-t,x_{1},\ldots,x_{n}),\ldots,X_{n}(-t,x_{1},\ldots,x_{n})),\\\
\hskip
128.0374pt\mathrm{if}\,(x_{1},\ldots,x_{n})\in(\mathbb{R}^{3n}\times(\mathbb{R}^{3n}\setminus\mathbb{W}_{n})),\\\
0,\hskip
116.6563pt\mathrm{if}\,(q_{1},\ldots,q_{n})\in\mathbb{W}_{n},\end{cases}$ (1)

where for $t\in\mathbb{R}$ the function $X_{i}(-t)$ is a phase space
trajectory of the $ith$ hard sphere constructed in [1], which is defined
almost everywhere on the phase space
$\mathbb{R}^{3n}\times(\mathbb{R}^{3n}\setminus\mathbb{W}_{n})$, namely,
beyond the set $\mathbb{M}_{n}^{0}$ of zero Lebesgue measure consisting of the
phase space points specified by such initial data that multiple collisions can
occur during the evolution, i.e. collisions of more than two particles, more
than one two-particle collision at the same instant and infinite number of
collisions within a finite time interval [1],[20].

On the space $L^{1}_{n}\equiv L^{1}(\mathbb{R}^{3n}\times\mathbb{R}^{3n})$
one-parameter mapping (1) generates an isometric strong continuous group of
operators

$\displaystyle S_{n}(-t,1,\ldots,n)D_{n}(0,x_{1},\ldots,x_{n})\doteq
D_{n}(t,x_{1},\ldots,x_{n}).$ (2)

On the subspace of continuously differentiable functions with compact supports
$L_{n,0}^{1}\subset L^{1}_{n}$ for the group of operators (2) the Duhamel
equation holds

$\displaystyle\hskip-14.22636ptS_{n}(-t,1,\ldots,n)=\prod\limits_{i=1}^{n}S_{1}(-t,i)+\int\limits_{0}^{t}d\tau\prod\limits_{i=1}^{n}S_{1}(\tau-t,i)\sum\limits_{j_{1}<j_{2}=1}^{n}\mathcal{L}_{\mathrm{int}}^{\ast}(j_{1},j_{2})S_{n}(-\tau,1,\ldots,n))=$
$\displaystyle\prod\limits_{i=1}^{n}S_{1}(-t,i)+\int\limits_{0}^{t}d\tau
S_{n}(\tau-t,1,\ldots,n)\sum\limits_{j_{1}<j_{2}=1}^{n}\mathcal{L}_{\mathrm{int}}^{\ast}(j_{1},j_{2})\prod\limits_{i=1}^{n}S_{1}(-\tau,i),$

where for $t>0$ the operator $\mathcal{L}_{\mathrm{int}}^{\ast}(j_{1},j_{2})$
is defined by the formula

$\displaystyle\hskip-14.22636pt\mathcal{L}_{\mathrm{int}}^{\ast}(j_{1},j_{2})f_{n}\doteq\sigma^{2}\int\limits_{\mathbb{S}_{+}^{2}}d\eta\langle\eta,(p_{j_{1}}-p_{j_{2}})\rangle
f_{n}(x_{1},\ldots,p_{j_{1}}^{*},q_{j_{1}},\ldots,$ (3) $\displaystyle
p_{j_{2}}^{*},q_{j_{2}},\ldots,x_{n})\delta(q_{j_{1}}-q_{j_{2}}+\sigma\eta)-f_{n}(x_{1},\ldots,x_{n})\delta(q_{j_{1}}-q_{j_{2}}-\sigma\eta)\big{)}.$

In definition (3) the symbol $\langle\cdot,\cdot\rangle$ means a scalar
product, the symbol $\delta$ denotes the Dirac measure,
$\mathbb{S}_{+}^{2}\doteq\\{\eta\in\mathbb{R}^{3}\big{|}\left|\eta\right|=1\langle\eta,(p_{1}-p_{2})\rangle>0\\}$
and the pre-collision momenta $p_{i}^{\ast},p_{j}^{\ast}$ are determined by
the equalities:

$\displaystyle p_{i}^{\ast}\doteq
p_{i}-\eta\left\langle\eta,\left(p_{i}-p_{j}\right)\right\rangle,$
$\displaystyle p_{j}^{\ast}\doteq
p_{j}+\eta\left\langle\eta,\left(p_{i}-p_{j}\right)\right\rangle.$

In the case $t<0$, the operator $\mathcal{L}_{\mathrm{int}}(j_{1},j_{2})$ is
defined by the corresponding expression [1].

Hence the infinitesimal generator $\mathcal{L}_{n}^{\ast}$ of the group of
operators $S_{n}(t)$ has the following structure

$\displaystyle\mathcal{L}_{n}^{\ast}(1,\ldots,n)f_{n}\doteq\sum\limits_{j=1}^{n}\mathcal{L}^{\ast}(j)f_{n}+\sum\limits_{j_{1}<j_{2}=1}^{n}\mathcal{L}_{\mathrm{int}}^{\ast}(j_{1},j_{2})f_{n},$
(4)

where the Liouville operator of free motion
$\mathcal{L}^{\ast}(j)\doteq-\langle p_{j},\frac{\partial}{\partial
q_{j}}\rangle$ defined on the subspace $L_{n,0}^{1}\subset L^{1}_{n}$, is
denoted by the symbol $\mathcal{L}^{\ast}(j)$.

If $D_{n}(0)\in L^{1}_{n},\,n\geq 1$, the sequence of distribution functions
defined by formula (1) is a unique solution of the Cauchy problem of the weak
formulation of a sequence of evolution equations for the state known as the
Liouville equation:

$\displaystyle\frac{\partial}{\partial
t}D_{n}(t)=\mathcal{L}_{n}^{\ast}(1,\ldots,n)D_{n}(t),$ (5) $\displaystyle
D_{n}(t)|_{t=0}=D_{n}(0),\quad n\geq 1.$ (6)

Thus, the traditional approach to the description of the evolution of all
possible states of finitely many hard spheres is specified by the Cauchy
problem for the sequence of Liouville equations.

###  2.2 Correlation functions

An alternative approach to the description of states of a hard-sphere system
of finitely many particles is given by means of functions determined by the
cluster expansions of the probability distribution functions. They are called
here as correlation functions (the cumulants of probability distribution
functions).

We introduce the sequence of correlation functions
$g(t)=(1,g_{1}(t,x_{1}),\ldots,g_{s}(t,x_{1},\ldots,x_{s}),\ldots)$ by means
of cluster expansions of the probability distribution functions
$D(t)=(1,D_{1}(t,x_{1}),\ldots,$ $D_{n}(t,x_{1},\ldots,x_{n}),\ldots)$,
defined on the set of allowed configurations
$\mathbb{R}^{3n}\setminus\mathbb{W}_{n}$ as follows:

$\displaystyle\hskip-19.91692ptD_{n}(t,x_{1},\ldots,x_{n})=g_{n}(t,x_{1},\ldots,x_{n})+\sum\limits_{\mbox{\scriptsize$\begin{array}[]{c}\mathrm{P}:(x_{1},\ldots,x_{n})=\bigcup_{i}X_{i},\\\
|\mathrm{P}|>1\end{array}$}}\prod_{X_{i}\subset\mathrm{P}}g_{|X_{i}|}(t,X_{i}),\,\,\,\,n\geq
1,$ (9)

where
${\sum\limits}_{\mathrm{P}:(x_{1},\ldots,x_{n})=\bigcup_{i}X_{i},\,|\mathrm{P}|>1}$
is the sum over all possible partitions $\mathrm{P}$ of the set of the
arguments $(x_{1},\ldots,x_{n})$ into $|\mathrm{P}|>1$ nonempty mutually
disjoint subsets $X_{i}\subset(x_{1},\ldots,x_{n})$.

On the set $\mathbb{R}^{3n}\setminus\mathbb{W}_{n}$ solutions of recursion
relations (9) are given by the following expansions:

$\displaystyle\hskip-19.91692ptg_{s}(t,x_{1},\ldots,x_{s})=D_{s}(t,x_{1},\ldots,x_{s})+$
(10)
$\displaystyle\sum\limits_{\mbox{\scriptsize$\begin{array}[]{c}\mathrm{P}:(x_{1},\ldots,x_{s})=\bigcup_{i}X_{i},\\\
|\mathrm{P}|>1\end{array}$}}(-1)^{|\mathrm{P}|-1}(|\mathrm{P}|-1)!\,\prod_{X_{i}\subset\mathrm{P}}D_{|X_{i}|}(t,X_{i}),\quad
s\geq 1.$ (13)

The structure of expansions (10) is such that the correlation functions can be
treated as cumulants (semi-invariants) of the probability distribution
functions (1).

Thus, correlation functions (10) are to enable to describe of the evolution of
states of finitely many hard spheres by the equivalent method in comparison
with probability distribution functions (1), namely within the framework of
dynamics of correlations.

If the initial state is specified by the sequence
$g(0)=(1,g_{1}^{0}(x_{1}),\ldots,g_{n}^{0}(x_{1},\ldots,$ $x_{n}),\ldots)$, of
correlation functions $g_{n}^{0}\in L^{1}_{n},\,n\geq 1,$ then the evolution
of all possible states, i.e. the sequence $g(t)=(1,$
$g_{1}(t,x_{1}),\ldots,g_{s}(t,x_{1},\ldots,x_{s}),\ldots)$ of the correlation
functions $g_{s}(t),\,s\geq 1$, is determined by the following expansions:

$\displaystyle\hskip-22.76219ptg_{s}(t,x_{1},\ldots,x_{s})=\sum\limits_{\mathrm{P}:\,(x_{1},\ldots,x_{s})=\bigcup_{j}X_{j}}\mathfrak{A}_{|\mathrm{P}|}(t,\\{\widehat{X}_{1}\\},\ldots,\\{\widehat{X}_{|\mathrm{P}|}\\})\prod_{X_{j}\subset\mathrm{P}}g_{|X_{j}|}^{0}(X_{j}),\quad
s\geq 1,$ (14)

where the symbol $\sum_{\mathrm{P}:\,(x_{1},\ldots,x_{s})=\bigcup_{j}X_{j}}$
is the sum over all possible partitions $\mathrm{P}$ of the set
$(x_{1},\ldots,x_{s})$ into $|\mathrm{P}|$ nonempty mutually disjoint subsets
$X_{j}$, the symbol $\widehat{X}$ means the set of indexes of the set $X$ of
phase space coordinates and the set
$(\\{\widehat{X}_{1}\\},\ldots,\\{\widehat{X}_{|\mathrm{P}|}\\})$ consists of
elements that are subsets $\widehat{X}_{j}\subset(1,\ldots,s)$, i.e.
$|(\\{\widehat{X}_{1}\\},\ldots,\\{\widehat{X}_{|\mathrm{P}|}\\})|=|\mathrm{P}|$.
The generating operator $\mathfrak{A}_{|\mathrm{P}|}(t)$ in expansions (14) is
the $|\mathrm{P}|th$-order cumulant of the groups of operators (2) which is
defined by the expansion

$\displaystyle\hskip-19.91692pt\mathfrak{A}_{|\mathrm{P}|}(t,\\{\widehat{X}_{1}\\},\ldots,\\{\widehat{X}_{|\mathrm{P}|}\\})\doteq$
(15)
$\displaystyle\sum\limits_{\mathrm{P}^{{}^{\prime}}:\,(\\{\widehat{X}_{1}\\},\ldots,\\{\widehat{X}_{|\mathrm{P}|}\\})=\bigcup_{k}Z_{k}}(-1)^{|\mathrm{P}^{{}^{\prime}}|-1}({|\mathrm{P}^{{}^{\prime}}|-1})!\prod\limits_{Z_{k}\subset\mathrm{P}^{{}^{\prime}}}S_{|\theta(Z_{k})|}(-t,\theta(Z_{k})),$

where $\theta$ is the declusterization mapping:
$\theta(\\{\widehat{X}_{1}\\},\ldots,\\{\widehat{X}_{|\mathrm{P}|}\\})\doteq(1,\ldots,s)$.
The simplest examples of correlation functions (14) are given as follows:

$\displaystyle g_{1}(t,x_{1})=\mathfrak{A}_{1}(t,1)g_{1}^{0}(x_{1}),$
$\displaystyle
g_{2}(t,x_{1},x_{2})=\mathfrak{A}_{1}(t,\\{1,2\\})g_{2}^{0}(x_{1},x_{2})+\mathfrak{A}_{2}(t,1,2)g_{1}^{0}(x_{1})g_{1}^{0}(x_{2}).$

The structure of expansions (14) is established as a result of the permutation
of the terms of cumulant expansions (10) for correlation functions and cluster
expansions (9) for initial probability distribution functions (2). Thus, the
cumulant origin of correlation functions induces the cumulant structure of
their dynamics (14).

In particular, in the absence of correlations between hard spheres at the
initial moment (initial state satisfying the chaos condition [1],[3]) the
sequence of the initial correlation functions on allowed configurations has
the form $g^{(c)}(0)=(0,g_{1}^{0}(x_{1}),0,\ldots,0,\ldots)$. In terms of a
sequence of the probability distribution functions the chaos condition means
that we start from an initial datum of the form
$D^{(c)}(0)=(1,D_{1}^{0}(x_{1}),D_{1}^{0}(x_{1})D_{1}^{0}(x_{2})\mathcal{X}_{\mathbb{R}^{6}\setminus\mathbb{W}_{2}},\ldots,$
$\prod^{n}_{i=1}D_{1}^{0}(x_{i})\mathcal{X}_{\mathbb{R}^{3n}\setminus\mathbb{W}_{n}},\ldots)$,
where the function $\mathcal{X}_{\mathbb{R}^{3n}\setminus\mathbb{W}_{n}}$ is
the Heaviside step function of allowed configurations of $n$ hard spheres. In
this case for
$(x_{1},\ldots,x_{s})\in\mathbb{R}^{3s}\times(\mathbb{R}^{3s}\setminus\mathbb{W}_{s})$
expansions (14) are represented as follows:

$\displaystyle
g_{s}(t,x_{1},\ldots,x_{s})=\mathfrak{A}_{s}(t,1,\ldots,s)\,\prod\limits_{i=1}^{s}g_{1}^{0}(x_{i})\mathcal{X}_{\mathbb{R}^{3s}\setminus\mathbb{W}_{s}},\quad
s\geq 1,$ (16)

where the generating operator $\mathfrak{A}_{s}(t)$ of this expansion is the
$sth$-order cumulant of groups of operators (2) defined by the expansion

$\displaystyle\hskip-22.76219pt\mathfrak{A}_{s}(t,1,\ldots,s)=\sum\limits_{\mathrm{P}:\,(1,\ldots,s)=\bigcup_{i}X_{i}}(-1)^{|\mathrm{P}|-1}({|\mathrm{P}|-1})!\prod\limits_{X_{i}\subset\mathrm{P}}S_{|X_{i}|}(-t,X_{i}),$
(17)

with notations accepted in formula (14). From the structure of series (16) it
is clear that in case of the absence of correlations at the initial instant
the correlations generated by the dynamics of a system of hard spheres are
completely determined by cumulants (17) of the groups of operators of the
Liouville equation (5).

We note that in the case of initial data $g^{(c)}(0)$ expansions (16) can be
rewritten in another representation that explains their physical meaning.
Indeed, for $n=1$ we have

$\displaystyle
g_{1}(t,x_{1})=\mathfrak{A}_{1}(t,1)g_{1}^{0}(x_{1})=g_{1}^{0}(p_{1},q_{1}-p_{1}t).$

Then, according to formula (16) and the definition of the first-order cumulant
$\mathfrak{A}_{1}(t)=S_{1}(-t)$, and its inverse group of operators
$S_{1}^{-1}(-t)=S_{1}(t)$, we express the correlation functions $g_{s}(t),$
$s\geq 2$, in terms of the one-particle correlation function $g_{1}(t)$.
Therefore, for $s\geq 2$ expansions (16) are represented in the following
form:

$\displaystyle
g_{s}(t,x_{1},\ldots,x_{s})=\widehat{\mathfrak{A}}_{s}(t,1,\ldots,s)\,\prod_{i=1}^{s}\,g_{1}(t,x_{i}),\quad
s\geq 2,$

where $\widehat{\mathfrak{A}}_{s}(t,1,\ldots,s)$ is the $s$-order cumulant
(17) of the scattering operators

$\displaystyle\widehat{S}_{n}(t,1,\ldots,n)\doteq
S_{n}(-t,1,\ldots,n)\mathcal{X}_{\mathbb{R}^{3n}\setminus\mathbb{W}_{n}}\prod_{i=1}^{n}S_{1}(t,i),\quad
n\geq 1.$

On subspace $L^{1}_{0,n}\subset L^{1}_{n}$ the generator of the scattering
operator $\widehat{S}_{n}(t,1,\ldots,n)$ is determined by the operator:
$\frac{d}{dt}\widehat{S}_{n}(t,1,\ldots,n)\mid_{t=0}=\sum_{j_{1}<j_{2}=1}^{n}\mathcal{L}_{\mathrm{int}}^{\ast}(j_{1},j_{2}),$
where for $t\geq 0$ the operator
$\mathcal{L}_{\mathrm{int}}^{\ast}(j_{1},j_{2})$ is defined according to
formula (3).

If $g^{0}_{n}\in L^{1}_{n},\,n\geq 1$, one-parameter mapping (14) generates
strong continuous group of nonlinear operators

$\displaystyle\mathcal{G}(t;1,\ldots,s\mid g(0))\doteq
g_{s}(t,x_{1},\ldots,x_{s}),$ (18)

and it is bounded, and the following estimate holds:
$\big{\|}\mathcal{G}(t;1,\ldots,s\mid g)\big{\|}_{L^{1}_{s}}\leq s!c^{s},$
where $c\equiv\max(1,$
$\max_{\mathrm{P}:\,(1,\ldots,s)=\bigcup_{i}X_{i}}\|g_{|X_{i}|}\|_{L^{1}_{|X_{i}|}})$.
For $g_{n}\in L^{1}_{n,0},\,n\geq 1$, the infinitesimal generator of this
group of nonlinear operators has the following structure

$\displaystyle\hskip-14.22636pt\mathcal{L}(1,\ldots,s\mid
g)\doteq\mathcal{L}^{\ast}_{s}(1,\ldots,s)g_{s}(x_{1},\ldots,x_{s})+$ (19)
$\displaystyle\sum\limits_{\mathrm{P}:\,(x_{1},\ldots,x_{s})=X_{1}\bigcup
X_{2}}\,\sum\limits_{i_{1}\in\widehat{X}_{1}}\sum\limits_{i_{2}\in\widehat{X}_{2}}\mathcal{L}_{\mathrm{int}}^{\ast}(i_{1},i_{2})g_{|X_{1}|}(X_{1})g_{|X_{2}|}(X_{2}),$

where we used the notation adopted above in expansions (14).

###  2.3 The Liouville hierarchy

If $g_{s}^{0}\in L^{1}_{s},\,s\geq 1,$ then for $t\in\mathbb{R}$ the sequence
of correlation functions (14) defined on the set of allowed configurations is
a unique solution of the Cauchy problem of the weak formulation of the
Liouville hierarchy:

$\displaystyle\hskip-22.76219pt\frac{\partial}{\partial
t}g_{s}(t,x_{1},\ldots,x_{s})=\mathcal{L}^{\ast}_{s}(1,\ldots,s)g_{s}(t,x_{1},\ldots,x_{s})+$
(20) $\displaystyle\sum\limits_{\mathrm{P}:\,(x_{1},\ldots,x_{s})=X_{1}\bigcup
X_{2}}\,\sum\limits_{i_{1}\in\widehat{X}_{1}}\sum\limits_{i_{2}\in\widehat{X}_{2}}\mathcal{L}_{\mathrm{int}}^{\ast}(i_{1},i_{2})g_{|X_{1}|}(t,X_{1})g_{|X_{2}|}(t,X_{2}),$
$\displaystyle\hskip-22.76219ptg_{s}(t,x_{1},\ldots,x_{s})\big{|}_{t=0}=g_{s}^{0}(x_{1},\ldots,x_{s}),\quad
s\geq 1,$ (21)

where ${\sum\limits}_{\mathrm{P}:\,(x_{1},\ldots,x_{s})=X_{1}\bigcup X_{2}}$
is the sum over all possible partitions $\mathrm{P}$ of the set
$(x_{1},\ldots,x_{s})$ into two nonempty mutually disjoint subsets $X_{1}$ and
$X_{2}$, the symbol $\widehat{X}_{i}$ means the set of indexes of the set
$X_{i}$ of phase space coordinates and the operator $\mathcal{L}^{\ast}_{s}$
is defined on the subspace $L^{1}_{0}\subset L^{1}$ by formula (4). It should
be noted that the Liouville hierarchy (20) is the evolution recurrence
equations set.

For $t\geq 0$ we give a few examples of recurrence equations set (20) for a
system of hard spheres:

$\displaystyle\hskip-19.91692pt\frac{\partial}{\partial
t}g_{1}(t,x_{1})=-\langle p_{1},\frac{\partial}{\partial q_{1}}\rangle
g_{1}(t,x_{1}),$ $\displaystyle\hskip-19.91692pt\frac{\partial}{\partial
t}g_{2}(t,x_{1},x_{2})=-\sum\limits_{j=1}^{2}\langle
p_{j},\frac{\partial}{\partial q_{j}}\rangle g_{2}(t,x_{1},x_{2})+$
$\displaystyle\sigma^{2}\int\limits_{\mathbb{S}_{+}^{2}}d\eta\langle\eta,(p_{1}-p_{2})\rangle\big{(}g_{2}(t,q_{1},p_{1}^{\ast},q_{2},p_{2}^{\ast})\delta(q_{1}-q_{2}+\sigma\eta)-$
$\displaystyle g_{2}(t,x_{1},x_{2})\delta(q_{1}-q_{2}-\sigma\eta)\big{)}+$
$\displaystyle\sigma^{2}\int\limits_{\mathbb{S}_{+}^{2}}d\eta\langle\eta,(p_{1}-p_{2})\rangle\big{(}g_{1}(t,q_{1},p_{1}^{\ast})g_{1}(t,q_{2},p_{2}^{\ast})\delta(q_{1}-q_{2}+\sigma\eta)-$
$\displaystyle
g_{1}(t,x_{1})g_{1}(t,x_{2})\delta(q_{1}-q_{2}-\sigma\eta)\big{)},$

where it was used notations accepted above in definition (3).

We note that because the Liouville hierarchy (20) is the recurrence evolution
equations set, we can construct a solution of the Cauchy problem (20),(21),
integrating each equation of the hierarchy as the inhomogeneous Liouville
equation. For example, as a result of the integration of the first two
equations of the Liouville hierarchy (20), we obtain the following equalities:

$\displaystyle\hskip-14.22636ptg_{1}(t,x_{1})=S_{1}(-t,1)g_{1}^{0}(x_{1}),$
$\displaystyle\hskip-14.22636ptg_{2}(t,1,2)=S_{2}(-t,1,2)g_{2}^{0}(x_{1},x_{2})+$
$\displaystyle\int\limits_{0}^{t}dt_{1}S_{2}(t_{1}-t,1,2)\mathcal{L}^{\ast}_{\mathrm{int}}(1,2)S_{1}(-t_{1},1)S_{1}(-t_{1},2)g_{1}^{0}(x_{1})g_{1}^{0}(x_{2}).$

Then for the corresponding term on the right-hand side of the second equality,
an analog of the Duhamel equation holds

$\displaystyle\int\limits_{0}^{t}dt_{1}S_{2}(t_{1}-t,1,2)\mathcal{L}^{\ast}_{\mathrm{int}}(1,2)S_{1}(-t_{1},1)S_{1}(-t_{1},2)=$
$\displaystyle-\int\limits_{0}^{t}dt_{1}\frac{d}{dt_{1}}\big{(}S_{2}(t_{1}-t,1,2)S_{1}(-t_{1},1)S_{1}(-t_{1},2)\big{)}=$
$\displaystyle S_{2}(-t,1,2)-S_{1}(-t,1)S_{1}(-t,2)=\mathfrak{A}_{2}(t,1,2),$

where $\mathfrak{A}_{2}(t)$ is the second-order cumulant (17) of groups of
operators (2). As a result of similar transformations for $s>2$, the solution
of the Cauchy problem (20),(21), constructed by an iterative procedure, is
represented in the form of expansions (14).

The following statement is true.

For $t\in\mathbb{R}$ a unique solution of the Cauchy problem of the Liouville
hierarchy (20),(21) is represented by a sequence of expansions (14). For
$g_{n}^{0}\in L^{1}_{n,0}\subset L^{1}_{n},\,n\geq 1$, a sequence of functions
(14) is a classical solution and for arbitrary initial data $g_{n}^{0}\in
L^{1}_{n},\,n\geq 1$, one has a generalized solution.

The proof of the theorem is similar to the proof of the existence theorem for
the BBGKY hierarchy in the space of sequences of integrable functions
[1],[22]. Indeed, if the initial data is $g_{n}^{0}\in L^{1}_{n,0},\,n\geq 1$,
then the infinitesimal generator of the group of nonlinear operators (18)
coincides with the operator (19) and hence the Cauchy problem (20),(21) has a
classical (strong) solution (14).

We remark that a steady solution of the Liouville hierarchy (20) is a sequence
of the Ursell functions on the allowed configurations of hard spheres, i.e. it
is the sequence $g^{(eq)}=(0,e^{-\beta\frac{p^{2}_{1}}{2}},0,\ldots,)$, where
$\beta$ is a parameter inversely proportional to temperature.

Finally, we emphasize that the dynamics of correlations, that is, the
fundamental equations (20) describing the evolution of correlations of states
of hard spheres, can be used as a basis for describing the evolution of the
state of both a finite and an infinite number of hard spheres instead of the
Liouville equations for the state (5). Further, we establish that the
constructed dynamics of correlation underlies the description of the dynamics
of infinitely many hard-spheres governed by the BBGKY hierarchies for reduced
distribution functions or reduced correlation functions.

##  3 Propagation of correlations in a hard-sphere system

###  3.1 Evolution of states described by the dynamics of correlations:
reduced distribution functions

As is known, an equivalent approach adapted to describing the evolution of
states of systems of both finite and infinite number of hard spheres is to
describe a state by means of a sequence of so-called reduced distribution
functions (marginals) governed by the BBGKY hierarchy [1]. In what follows, we
outline an approach to describing the evolution of a state using both reduced
distribution functions and reduced correlation functions, based on the
dynamics of correlations in a system of hard spheres governed by the Liouville
hierarchy for correlation functions (20).

For a hard-sphere system of a non-fixed, i.e. arbitrary but finite average
number of identical particles, the reduced distribution functions are defined
by means of probability distribution functions as follows [1]:

$\displaystyle\hskip-34.1433ptF_{s}(t,x_{1},\ldots,x_{s})\doteq(I,D(t))^{-1}\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\int\limits_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{n}}dx_{s+1}\ldots
dx_{s+n}\,D_{s+n}(t,x_{1},\ldots,x_{s+n}),$
$\displaystyle\hskip-34.1433pts\geq 1,$

where the normalizing factor
$(I,D(t))\doteq\sum_{n=0}^{\infty}\frac{1}{n!}\int_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{n}}dx_{1}\ldots
dx_{n}D_{n}(t,x_{1},\ldots,x_{n})$ is a grand canonical partition function.
The possibility of redefining of the reduced distribution functions naturally
arises as a result of dividing the series in expression (3.1) by the series of
the normalization factor.

A definition of reduced distribution functions equivalent to definition (3.1)
is formulated on the basis of correlation functions (14) of a system of hard
spheres by means of the following series expansion:

$\displaystyle\hskip-34.1433ptF_{s}(t,x_{1},\ldots,x_{s})\doteq\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\int\limits_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{n}}dx_{s+1}\ldots
dx_{s+n}\,g_{1+n}(t,\\{x_{1},\ldots,x_{s}\\},x_{s+1},\ldots,x_{s+n}),$ (23)
$\displaystyle\hskip-34.1433pts\geq 1,$

where on the set of allowed configurations
$\mathbb{R}^{3(s+n)}\setminus\mathbb{W}_{s+n}$ the correlation functions of
clusters of hard spheres $g_{1+n}(t),n\geq 0,$ are determined by the
expansions:

$\displaystyle\hskip-34.1433ptg_{1+n}(t,\\{x_{1},\ldots,x_{s}\\},x_{s+1},\ldots,x_{s+n})=$
(24)
$\displaystyle\sum_{\mbox{\scriptsize$\begin{array}[]{c}\mathrm{P}:(\\{x_{1},\ldots,x_{s}\\},\\\
x_{s+1},\ldots,x_{s+n})=\bigcup_{i}X_{i}\end{array}$}}\mathfrak{A}_{|\mathrm{P}|}\big{(}t,\\{\theta(\widehat{X}_{1})\\},\ldots,\\{\theta(\widehat{X}_{|\mathrm{P}|})\\}\big{)}\prod_{X_{i}\subset\mathrm{P}}g_{|X_{i}|}^{0}(X_{i}),\quad
n\geq 0.$ (27)

We remind that in expansions (24) the symbol
$\sum_{\mathrm{P}:(\\{x_{1},\ldots,x_{s}\\},x_{s+1},\ldots,x_{s+n})=\bigcup_{i}X_{i}}$
means the sum over all possible partitions $\mathrm{P}$ of the set
$(\\{x_{1},\ldots,x_{s}\\},x_{s+1},\ldots,$ $x_{s+n})$ into nonempty mutually
disjoint subsets $X_{i}$, and the generating operator
$\mathfrak{A}_{|\mathrm{P}|}(t)$ is the $|\mathrm{P}|th$-order cumulant (15)
of the groups of operators (2).

On allowed configurations the correlation functions of particle clusters in
series (23), i.e. the functions $g_{1+n}(t,\\{x_{1},\ldots,x_{s}\\},$
$x_{s+1},\ldots,x_{s+n}),\,n\geq 0$, are defined as solutions of generalized
cluster expansions of a sequence of solutions of the Liouville equations (5):

$\displaystyle\hskip-19.91692ptD_{s+n}(t,x_{1},\ldots,x_{s+n})=\sum_{\mbox{\scriptsize$\begin{array}[]{c}\mathrm{P}:(\\{x_{1},\ldots,x_{s}\\},\\\
x_{s+1},\ldots,x_{s+n})=\bigcup_{i}X_{i}\end{array}$}}\prod_{X_{i}\subset\mathrm{P}}g_{|X_{i}|}(t,X_{i}),\quad
s\geq 1,\,n\geq 0,$ (30)

namely,

$\displaystyle\hskip-19.91692ptg_{1+n}(t,\\{x_{1},\ldots,x_{s}\\},x_{s+1},\ldots,x_{s+n})=$
$\displaystyle\sum_{\mbox{\scriptsize$\begin{array}[]{c}\mathrm{P}:(\\{x_{1},\ldots,x_{s}\\},\\\
x_{s+1},\ldots,x_{s+n})=\bigcup_{i}X_{i}\end{array}$}}(-1)^{|\mathrm{P}|-1}(|\mathrm{P}|-1)!\,\prod_{X_{i}\subset\mathrm{P}}D_{|\theta(X_{i})|}(t,\theta(X_{i})),\quad
s\geq 1,\,n\geq 0,$

where $\theta$ is the declusterization mapping defined in formula (15), the
probability distribution function $D_{|\theta(X_{i})|}(t,\theta(X_{i}))$ is
solution (1) of the Liouville equations (5).

The correlation functions of particle clusters satisfy the Liouville hierarchy
of evolution equations with the following generator

$\displaystyle\hskip-34.1433pt\mathcal{L}(\\{1,\ldots,s\\},s+1,\ldots,s+n\mid\mathfrak{d}_{\\{Y\\}}g(t))\doteq$
(32) $\displaystyle\hskip
14.22636pt\mathcal{L}^{\ast}_{s+n}(1,\ldots,s+n)g_{1+n}(t,X)+$
$\displaystyle\hskip 14.22636pt\sum\limits_{\mathrm{P}:\,X=X_{1}\bigcup
X_{2}}\,\sum\limits_{i_{1}\in\theta(\widehat{X}_{1})}\sum\limits_{i_{2}\in\theta(\widehat{X}_{2})}\mathcal{L}_{\mathrm{int}}^{\ast}(i_{1},i_{2})g_{|X_{1}|}(t,X_{1})g_{|X_{2}|}(t,X_{2}),\quad
n\geq 0,$

where
$X\equiv(\\{Y\\},x_{s+1},\ldots,x_{s+n})\equiv(\\{x_{1},\ldots,x_{s}\\},x_{s+1},\ldots,x_{s+n})$,
the sequence of solutions of generalized cluster expansions (30) is denoted by
means of the mapping

$\displaystyle\hskip-14.22636pt(\mathfrak{d}_{\\{Y\\}}g)_{n}(x_{1},\ldots,x_{n})\doteq
g_{1+n}(\\{x_{1},\ldots,x_{s}\\},x_{s+1},\ldots,x_{s+n}),\quad n\geq 0,$

and we also used the notations adopted above in expansion (14).

We note that on the allowed configurations correlation functions of hard-
sphere clusters can be expressed through correlation functions of hard spheres
(14) by the following relations:

$\displaystyle\hskip-34.1433ptg_{1+n}(t,\\{x_{1},\ldots,x_{s}\\},x_{s+1},\ldots,x_{s+n})=\sum_{\mbox{\scriptsize$\begin{array}[]{c}\mathrm{P}:(\\{x_{1},\ldots,x_{s}\\},\\\
x_{s+1},\ldots,x_{s+n})=\bigcup_{i}X_{i}\end{array}$}}(-1)^{|\mathrm{P}|-1}(|\mathrm{P}|-1)!\times$
(35)
$\displaystyle\prod_{X_{i}\subset\mathrm{P}}\,\sum\limits_{\mathrm{P^{\prime}}:\,\theta(X_{i})=\bigcup_{j_{i}}Z_{j_{i}}}\,\prod_{Z_{j_{i}}\subset\mathrm{P^{\prime}}}g_{|Z_{j_{i}}|}(t,Z_{j_{i}}),\quad
n\geq 0.$

In particular case $n=0$, i.e. the correlation function of a cluster of the
$s$ hard spheres, these relations take the form

$\displaystyle
g_{1+0}(t,\\{x_{1},\ldots,x_{s}\\})=\sum\limits_{\mathrm{P}:\,\theta(\\{x_{1},\ldots,x_{s}\\})=\bigcup_{i}X_{i}}\,\prod_{X_{i}\subset\mathrm{P}}g_{|X_{i}|}(t,X_{i}).$

As a consequence of these relations, for the initial state satisfying the
chaos condition, from (24) the following generalization of expansions (16)
holds:

$\displaystyle\hskip-34.1433ptg_{s+n}(t,\\{x_{1},\ldots,x_{s}\\},x_{s+1},\ldots,x_{s+n})=$
(36)
$\displaystyle\mathfrak{A}_{1+n}(t,\\{1,\ldots,s\\},s+1,\ldots,s+n)\,\prod\limits_{i=1}^{s+n}g_{1}^{0}(x_{i})\mathcal{X}_{\mathbb{R}^{3(s+n)}\setminus\mathbb{W}_{s+n}},\quad
s\geq 1,\,n\geq 0.$

As we noted above, the possibility of the description of the evolution of a
state based on the dynamics of correlations (23) occurs naturally in
consequence of dividing the series of expressions (3.1) by the series of the
normalizing factor. To provide evidence of this statement, we will introduce
the necessary concepts and prove the validity of some auxiliary equalities.

On sequences of functions $f,\widetilde{f}\in
L^{1}\oplus_{n=0}^{\infty}L^{1}_{n}$ we define the following $\ast$-product
[23]

$\displaystyle(f\ast\widetilde{f})_{s}(x_{1},\ldots,x_{s})=\sum\limits_{Z\subset(x_{1},\ldots,x_{s})}\,f_{|Z|}(Z)\,\widetilde{f}_{s-|Z|}((x_{1},\ldots,x_{s})\setminus
Z),$ (37)

where $\sum_{Z\subset(x_{1},\ldots,x_{s})}$ is the sum over all subsets $Z$ of
the set $(x_{1},\ldots,x_{s})$. Using the definition of the $\ast$-product
(37), we introduce the mapping ${\mathbb{E}}\mathrm{xp}_{\ast}$ and the
inverse mapping ${\mathbb{L}}\mathrm{n}_{\ast}$ on sequences
$h=(0,h_{1}(x_{1}),\ldots,h_{n}(x_{1},\ldots,x_{n}),\ldots)$ of functions
$h_{n}\in L^{1}_{n}$ by the expansions:

$\displaystyle\hskip-19.91692pt({\mathbb{E}}\mathrm{xp}_{\ast}\,h)_{s}(x_{1},\ldots,x_{s})=\big{(}\mathbb{I}+\sum\limits_{n=1}^{\infty}\frac{h^{\ast
n}}{n!}\big{)}_{s}(x_{1},\ldots,x_{s})=$ (38) $\displaystyle\hskip
34.1433pt\delta_{s,0}+\sum\limits_{\mathrm{P}:\,(x_{1},\ldots,x_{s})=\bigcup_{i}X_{i}}\,\prod_{X_{i}\subset\mathrm{P}}h_{|X_{i}|}(X_{i}),$

where we used the notations accepted in formula (9),
$\mathbb{I}=(1,0,\ldots,0,\ldots)$ and $\delta_{s,0}$ is the Kronecker symbol,
and respectively,

$\displaystyle\hskip-19.91692pt({\mathbb{L}}\mathrm{n}_{\ast}(\mathbb{I}+h))_{s}(x_{1},\ldots,x_{s})=\big{(}\sum\limits_{n=1}^{\infty}(-1)^{n-1}\,\frac{h^{\ast
n}}{n}\big{)}_{s}(x_{1},\ldots,x_{s})=$ (39) $\displaystyle\hskip
34.1433pt\sum\limits_{\mathrm{P}:\,(x_{1},\ldots,x_{s})=\bigcup_{i}X_{i}}(-1)^{|\mathrm{P}|-1}(|\mathrm{P}|-1)!\,\prod_{X_{i}\subset\mathrm{P}}h_{|X_{i}|}(X_{i}).$

Therefore in terms of sequences of functions recursion relations (9) are
rewritten in the form

$\displaystyle\hskip-19.91692ptD(t)={\mathbb{E}}\mathrm{xp}_{\ast}\,\,g(t),$

where
$D(t)=\mathbb{I}+(0,D_{1}(t,x_{1}),\ldots,D_{n}(t,x_{1},\ldots,x_{n}),\ldots)$.
As a result, we get

$\displaystyle\hskip-19.91692ptg(t)={\mathbb{L}}\mathrm{n}_{\ast}\,\,D(t).$

Thus, according to definition (37) of the $\ast$-product and mapping (39), in
the component-wise form solutions of recursion relations (9) are represented
by expansions (10).

For arbitrary $f=(f_{0},f_{1},\ldots,f_{n},\ldots)\in L^{1}$ and the set
$Y\equiv(x_{1},\ldots,x_{s})$ we define the linear mapping
$\mathfrak{d}_{Y}:f\rightarrow\mathfrak{d}_{Y}f$, by the formula

$\displaystyle(\mathfrak{d}_{Y}f)_{n}(x_{1},\ldots,x_{n})\doteq
f_{s+n}(x_{1},\ldots,x_{s},x_{s+1},\ldots,x_{s+n}),\quad n\geq 0.$ (40)

For the set $\\{Y\\}$ consisting of the one element $Y=(x_{1},\ldots,x_{s})$,
we have, respectively

$\displaystyle\hskip-14.22636pt(\mathfrak{d}_{\\{Y\\}}f)_{n}(x_{1},\ldots,x_{n})\doteq
f_{1+n}(\\{x_{1},\ldots,x_{s}\\},x_{s+1},\ldots,x_{s+n}),\quad n\geq 0.$ (41)

On sequences $\mathfrak{d}_{Y}f$ and $\mathfrak{d}_{Y^{\prime}}\widetilde{f}$
we introduce the $\ast$-product

$\displaystyle(\mathfrak{d}_{Y}f\ast\mathfrak{d}_{Y^{\prime}}\widetilde{f})_{|X|}(X)\doteq\sum\limits_{Z\subset
X}f_{|Z|+|Y|}(Y,Z)\,\widetilde{f}_{|X\backslash
Z|+|Y^{\prime}|}(Y^{\prime},X\setminus Z),$

where $X,Y,Y^{\prime}$ are the sets, which characterize clusters of hard
spheres, and $\sum_{Z\subset X}$ is the sum over all subsets $Z$ of the set
$X$. In particular case $Y=\emptyset,\,Y^{\prime}=\emptyset$, this definition
reduces to definition of $\ast$-product (37).

Let us establish some properties of introduced mappings (38) and (41).

If $f_{n}\in L^{1}_{n},\,n\geq 1$ for the sequences
$f=(0,f_{1},\ldots,f_{n},\ldots)$, according to definitions of mappings (38)
and (41), the following equality holds

$\displaystyle\mathfrak{d}_{\\{Y\\}}\mathbb{E}\mathrm{xp}_{\ast}f=\mathbb{E}\mathrm{xp}_{\ast}f\ast\mathfrak{d}_{\\{Y\\}}f,$
(42)

and for mapping (40) respectively

$\displaystyle\mathfrak{d}_{Y}\mathbb{E}\mathrm{xp}_{\ast}f=\mathbb{E}\mathrm{xp}_{\ast}f\ast\sum\limits_{\mathrm{P}:\,Y=\bigcup_{i}X_{i}}\mathfrak{d}_{X_{1}}f\ast\ldots\ast\mathfrak{d}_{X_{|\mathrm{P}|}}f,$

where ${\sum\limits}_{\mathrm{P}:\,Y=\bigcup_{i}X_{i}}$ is the sum over all
possible partitions $\mathrm{P}$ of the set $Y\equiv(x_{1},\ldots,x_{s})$ into
$|\mathrm{P}|$ nonempty mutually disjoint subsets $X_{i}\subset Y$.

Hence in terms of mappings (40) and (41) generalized cluster expansions (30)
take the form

$\displaystyle\mathfrak{d}_{Y}D(t)=\mathfrak{d}_{\\{Y\\}}{\mathbb{E}}\mathrm{xp}_{\ast}\,\,g(t).$
(43)

On sequences of functions $f\in L^{1}=\oplus_{n=0}^{\infty}L^{1}_{n}$ we also
define the analogue of the annihilation operator

$\displaystyle(\mathfrak{a}f)_{n}(x_{1},\ldots,x_{n})=\int\limits_{\mathbb{R}^{3}\times\mathbb{R}^{3}}dx_{n+1}f_{n+1}(x_{1},\ldots,x_{n},x_{n+1}).$
(44)

Then for sequences $f,\widetilde{f}\in L^{1}$, the following equality holds

$\displaystyle(e^{\mathfrak{a}}f\ast\widetilde{f})_{0}=(e^{\mathfrak{a}}f)_{0}(e^{\mathfrak{a}}\widetilde{f})_{0},$
(45)

where such a notation was used

$\displaystyle(e^{\mathfrak{a}}f)_{0}=\sum\limits_{n=0}^{\infty}\frac{1}{n!}\int\limits_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{n}}dx_{1}\ldots
dx_{n}\,f_{n}(x_{1},\ldots,x_{n}).$ (46)

Now let us prove the equivalence of definition (3.1) of the reduced
distribution functions and their definition (23) within the framework of the
dynamics of correlations.

In terms of mapping (40) and notation (46) the definition of reduced
distribution functions (3.1) is written as follows

$\displaystyle
F_{s}(t,x_{1},\ldots,x_{s})=(e^{\mathfrak{a}}D(t))^{-1}_{0}(e^{\mathfrak{a}}\mathfrak{d}_{Y}D(t))_{0}.$

Using generalized cluster expansions (43), and as a consequence of equalities
(42) and (45), we find

$\displaystyle(e^{\mathfrak{a}}\mathfrak{d}_{Y}D(t))_{0}=(e^{\mathfrak{a}}\mathfrak{d}_{\\{Y\\}}{\mathbb{E}}\mathrm{xp}_{\ast}\,g(t))_{0}=$
$\displaystyle(e^{\mathfrak{a}}\mathbb{E}\mathrm{xp}_{\ast}g(t)\ast\mathfrak{d}_{\\{Y\\}}g(t))_{0}=(e^{\mathfrak{a}}\mathbb{E}\mathrm{xp}_{\ast}g(t))_{0}(e^{\mathfrak{a}}\mathfrak{d}_{\\{Y\\}}g(t))_{0}.$

Taking into account that, according to the particular case $Y=\emptyset,$ of
cluster expansions (30), the equality holds

$\displaystyle(e^{\mathfrak{a}}\mathbb{E}\mathrm{xp}_{\ast}g(t))_{0}=(e^{\mathfrak{a}}D(t))_{0},$

as a result, we establish the following representation for the reduced
distribution functions

$\displaystyle
F_{s}(t,x_{1},\ldots,x_{s})=(e^{\mathfrak{a}}\mathfrak{d}_{\\{Y\\}}g(t))_{0}.$

Therefore, in componentwise-form we obtain relation (23).

Since the correlation functions $g_{1+n}(t),\,n\geq 0,$ are governed by the
corresponding Liouville hierarchy for the cluster of hard spheres and hard
spheres, the reduced distribution functions (23) are governed by the BBGKY
hierarchy for hard spheres

$\displaystyle\frac{\partial}{\partial
t}F(t)=e^{\mathfrak{a}}\mathcal{L}(\\{\cdot\\},\cdot\mid
e^{-\mathfrak{a}}F(t)),$ (47)

where the operator $\mathcal{L}(\\{\cdot\\},\cdot\mid f)$ is generator (32) of
the Liouville hierarchy for a cluster of hard spheres and hard spheres. For a
generator of this hierarchy of evolution equations takes place the following
representation:

$\displaystyle e^{\mathfrak{a}}\mathcal{L}(\\{\cdot\\},\cdot\mid
e^{-\mathfrak{a}}F(t))=e^{\mathfrak{a}}\mathcal{L}^{\ast}e^{-\mathfrak{a}}F(t),$

where the operator
$\mathcal{L}^{\ast}=\oplus_{n=0}^{\infty}\mathcal{L}_{n}^{\ast}$ is a direct
sum of the Liouville operators (4) and the operator $\mathfrak{a}$ is defined
by formula (44). Due to the fact that pairwise collisions occur during the
evolution, a generator of this hierarchy is reduced to the operator of such a
structure [1]

$\displaystyle
e^{\mathfrak{a}}\mathcal{L}^{\ast}e^{-\mathfrak{a}}=\mathcal{L}^{\ast}+[\mathfrak{a},\mathcal{L}^{\ast}],$

where the bracket $[\cdot,\cdot]$ is the commutator of operators. For $t\geq
0$ the collision part $[\mathfrak{a},\mathcal{L}^{\ast}]$ of a generator of
the BBGKY hierarchy for hard spheres has the following explicit form

$\displaystyle\hskip-34.1433pt\big{(}[\mathfrak{a},\mathcal{L}^{\ast}]F(t)\big{)}_{s}(x_{1},\ldots,x_{s})=\sigma^{2}\sum\limits_{i=1}^{s}\int\limits_{\mathbb{R}^{3}\times\mathbb{S}_{+}^{2}}dp_{s+1}d\eta\,\langle\eta,(p_{i}-p_{s+1})\rangle\big{(}F_{s+1}(t,x_{1},\ldots,$
$\displaystyle
q_{i},p_{i}^{*},\ldots,x_{s},q_{i}-\sigma\eta,p_{s+1}^{*})-F_{s+1}(t,x_{1},\ldots,x_{s},q_{i}+\sigma\eta,p_{s+1})\big{)},$

where the notations adopted above were used.

We note that the BBGKY hierarchy for hard spheres (47) was mathematically
justified in paper [20] (see also [1]).

In consequence of definition (23) and the cumulant structure of representation
of a solution (14) for the Liouville hierarchy (20), if initial state
specified by the sequence of reduced distribution functions
$F(0)=(1,F_{1}^{0}(x_{1}),\ldots,$ $F_{n}^{0}(x_{1},\ldots,x_{n}),\ldots)$,
then the evolution of all possible states, i.e. the sequence of the reduced
distribution functions $F_{s}(t),\,s\geq 1$, is determined by the following
series expansions [22]:

$\displaystyle\hskip-34.1433ptF_{s}(t,x_{1},\ldots,x_{s})=\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\int\limits_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{n}}dx_{s+1}\ldots
dx_{s+n}\,\mathfrak{A}_{1+n}(t,\\{1,\ldots,s\\},$ (48) $\displaystyle\hskip
22.76219pts+1,\ldots,{s+n})F_{s+n}^{0}(x_{1},\ldots,x_{s+n}),\quad s\geq 1,$

where the generating operator of these series

$\displaystyle\hskip-34.1433pt\mathfrak{A}_{1+n}(t,\\{1,\ldots,s\\},s+1,\ldots,{s+n})=$
(49)
$\displaystyle\sum_{\mbox{\scriptsize$\begin{array}[]{c}\mathrm{P}:(\\{x_{1},\ldots,x_{s}\\},\\\
x_{s+1},\ldots,x_{s+n})=\bigcup_{i}X_{i}\end{array}$}}(-1)^{|\mathrm{P}|-1}(|\mathrm{P}|-1)!\prod_{X_{i}\subset\mathrm{P}}S_{|\theta(X_{i})|}(-t,\theta(X_{i}))$
(52)

is the $(1+n)th$-order cumulant (15) of the groups of operators (2) and the
notations adopted above was used.

We remark that the representation (48) is directly established for the initial
states satisfying the chaos condition due to the validity in this case of the
representation (36) for the correlation functions of the hard-sphere cluster
and of the hard spheres.

Consequently, as follows from the above, the cumulant structure of generating
operators of expansions for correlation functions (14) or (24) induces the
cumulant structure (49) of generating operators of series expansions for
reduced distribution functions (48) or in other words, the evolution of the
state of a system of an infinite number of hard spheres is governed by the
dynamics of correlations on a microscopic scale.

Thus, we have established relation (23) between the reduced distribution
functions and correlation functions.

###  3.2 Evolution of states described by the dynamics of correlations:
reduced correlation functions

As is known, on a microscopic scale, the macroscopic characteristics of
fluctuations of observables are directly determined by means of the reduced
correlation functions (marginal or $s$-particle correlation functions [4], or
cumulants of marginals [7],[8]). Assuming as a basis an alternative approach
to the description of the evolution of states of a hard-sphere system within
the framework of correlation functions (14), then the reduced correlation
functions are defined by means of a solution of the Cauchy problem of the
Liouville hierarchy (20),(21) as follows:

$\displaystyle\hskip-34.1433ptG_{s}(t,x_{1},\ldots,x_{s})\doteq\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\int\limits_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{n}}dx_{s+1}\ldots
dx_{s+n}\,g_{s+n}(t,x_{1},\ldots,x_{s+n}),\quad s\geq 1,$ (53)

where the generating function $g_{s+n}(t,x_{1},\ldots,x_{s+n})$ is defined by
expansion (14), or in terms of mapping (40) and notation (46) this definition
takes the form

$\displaystyle
G_{s}(t,x_{1},\ldots,x_{s})=(e^{\mathfrak{a}}\mathfrak{d}_{Y}g(t))_{0},$

or in terms of sequences of functions this expression has the form

$\displaystyle G(t)=e^{\mathfrak{a}}g(t).$

We emphasize that $nth$ term of expansions (53) of the reduced correlation
functions are determined by the $(s+n)th$-particle correlation function (14)
in contrast with the expansions of reduced distribution functions (23) which
are determined by the $(1+n)th$-particle correlation function of clusters of
hard spheres (24).

Such a representation for reduced correlation functions (53) can be derived as
a result of the fact that the reduced correlation functions are cumulants of
reduced distribution functions (23). Indeed, traditionally reduced correlation
functions are introduced by means of the cluster expansions of the reduced
distribution functions similar to the cluster expansions of the probability
distribution functions (9) and on the set of allowed configurations
$\mathbb{R}^{3n}\setminus\mathbb{W}_{n}$ they have the form:

$\displaystyle\hskip-22.76219ptF_{s}(t,x_{1},\ldots,x_{s})=\sum\limits_{\mbox{\scriptsize$\begin{array}[]{c}\mathrm{P}:(x_{1},\ldots,x_{s})=\bigcup_{i}X_{i}\end{array}$}}\prod_{X_{i}\subset\mathrm{P}}G_{|X_{i}|}(t,X_{i}),\quad
s\geq 1,$ (55)

where as above the symbol
${\sum\limits}_{\mathrm{P}:(x_{1},\ldots,x_{s})=\bigcup_{i}X_{i}}$ is the sum
over all possible partitions $\mathrm{P}$ of the set $(x_{1},\ldots,x_{s})$
into $|\mathrm{P}|$ nonempty mutually disjoint subsets
$X_{i}\subset(x_{1},\ldots,x_{s})$. As a consequence of this, the solution of
recurrence relations (55) are represented through reduced distribution
functions as follows:

$\displaystyle\hskip-22.76219ptG_{s}(t,x_{1},\ldots,x_{s})=\sum\limits_{\mbox{\scriptsize$\begin{array}[]{c}\mathrm{P}:(x_{1},\ldots,x_{s})=\bigcup_{i}X_{i}\end{array}$}}(-1)^{|\mathrm{P}|-1}(|\mathrm{P}|-1)!\prod_{X_{i}\subset\mathrm{P}}F_{|X_{i}|}(t,X_{i}),\quad
s\geq 1.$ (57)

Functions (57) are interpreted as the functions which describe the
correlations of hard-sphere states. The structure of expansions (57) is such
that the reduced correlation functions are cumulants (semi-invariants) of the
reduced distribution functions (48).

Thus, taking into account representation (23) of the reduced distribution
functions, in consequence of the validity of relations (35) we derive
representation (53) of the reduced correlation functions through correlation
functions

$\displaystyle\hskip-22.76219ptG_{s}(t,x_{1},\ldots,x_{s})=\sum\limits_{\mathrm{P}:(x_{1},\ldots,x_{s})=\bigcup_{i}X_{i}}(-1)^{|\mathrm{P}|-1}(|\mathrm{P}|-1)!\,\prod_{X_{i}\subset\mathrm{P}}(e^{\mathfrak{a}}\mathfrak{d}_{\\{X_{i}\\}}g(t))=(e^{\mathfrak{a}}\mathfrak{d}_{Y}g(t))_{0}.$

Since the correlation functions $g_{s+n}(t),\,n\geq 0,$ are governed by the
Liouville hierarchy for hard spheres (20), the reduced correlation functions
defined as (53) are governed by the hierarchy of nonlinear equations for hard
spheres (the nonlinear BBGKY hierarchy):

$\displaystyle\hskip-34.1433pt\frac{\partial}{\partial
t}G_{s}(t,x_{1},\ldots,x_{s})=\mathcal{L}^{\ast}_{s}G_{s}(t,x_{1},\ldots,x_{s})+$
(58) $\displaystyle\sum\limits_{\mathrm{P}:\,(x_{1},\ldots,x_{s})=X_{1}\bigcup
X_{2}}\,\sum\limits_{i_{1}\in\widehat{X}_{1}}\sum\limits_{i_{2}\in\widehat{X}_{2}}\mathcal{L}_{\mathrm{int}}^{\ast}(i_{1},i_{2})G_{|X_{1}|}(t,X_{1})G_{|X_{2}|}(t,X_{2}))+$
$\displaystyle\int\limits_{\mathbb{R}^{3}\times\mathbb{R}^{3}}dx_{s+1}\big{(}\sum_{i=1}^{s}\mathcal{L}^{\ast}_{\mathrm{int}}(i,s+1)G_{s+1}(t,x_{1},\ldots,x_{s+1})+$
$\displaystyle\sum\limits_{\mathrm{P}:\,(x_{1},\ldots,x_{s+1})=X_{1}\bigcup
X_{2}}\sum_{i\in\widehat{X}_{1};s+1\in\widehat{X}_{2}}\mathcal{L}^{\ast}_{\mathrm{int}}(i,s+1)G_{|X_{1}|}(t,X_{1})G_{|X_{2}|}(t,X_{2})\big{)},$
$\displaystyle\hskip-34.1433ptG_{s}(t,x_{1},\ldots,x_{s})\big{|}_{t=0}=G_{s}^{0}(x_{1},\ldots,x_{s}),\quad
s\geq 1,$ (59)

where the symbol
$\sum_{\mbox{\scriptsize$\begin{array}[]{c}\mathrm{P}:(x_{1},\ldots,x_{s+1})=X_{1}\bigcup
X_{2},\end{array}$}}$ means the sum over all possible partitions of the set
$(x_{1},\ldots,x_{s+1})$ into two mutually disjoint subsets $X_{1}$ and
$X_{2}$, the sum over the index $i$ which takes values from the subset
$\widehat{X}_{1}$ provided that the index $s+1$ belongs to the subset
$\widehat{X}_{2}$ is denoted by
$\sum_{\mbox{\scriptsize$\begin{array}[]{c}i\in\widehat{X}_{1};s+1\in\widehat{X}_{2}\end{array}$}}$
and notations accepted in the Liouville hierarchy (20) are used.

A generator of this hierarchy of nonlinear evolution equations has the
following structure:

$\displaystyle\frac{\partial}{\partial
t}G(t)=e^{\mathfrak{a}}\mathcal{L}(\cdot\mid e^{-\mathfrak{a}}G(t)),$

where the operator $\mathcal{L}(\cdot\mid
f)=\oplus_{n=0}^{\infty}\mathcal{L}(1,\ldots,n\mid f)$ is a direct sum of
generators (19) of the Liouville hierarchy (20). Here are some component-wise
examples of hierarchy (58):

$\displaystyle\hskip-34.1433pt\frac{\partial}{\partial
t}G_{1}(t,x_{1})=\mathcal{L}^{\ast}_{1}(1)G_{1}(t,x_{1})+$
$\displaystyle\int\limits_{\mathbb{R}^{3}\times\mathbb{R}^{3}}dx_{2}\mathcal{L}^{\ast}_{\mathrm{int}}(1,2)\big{(}G_{2}(t,x_{1},x_{2})+G_{1}(t,x_{1})G_{1}(t,x_{2})\big{)},$
$\displaystyle\hskip-34.1433pt\frac{\partial}{\partial
t}G_{2}(t,x_{1},x_{2})=\mathcal{L}^{\ast}_{2}(1,2)G_{2}(t,x_{1},x_{2})+\mathcal{L}_{\mathrm{int}}^{\ast}(1,2)G_{1}(t,x_{1})G_{1}(t,x_{2})+$
$\displaystyle\int\limits_{\mathbb{R}^{3}\times\mathbb{R}^{3}}dx_{3}\Big{(}\sum_{i=1}^{2}\mathcal{L}^{\ast}_{\mathrm{int}}(i,3)\big{(}G_{3}(t,x_{1},x_{2},x_{3})+G_{2}(t,x_{1},x_{2})G_{1}(t,x_{3})\big{)}+$
$\displaystyle\mathcal{L}^{\ast}_{\mathrm{int}}(2,3)G_{2}(t,x_{1},x_{3})G_{1}(t,x_{2})+\mathcal{L}^{\ast}_{\mathrm{int}}(1,3)G_{2}(t,x_{2},x_{3})G_{1}(t,x_{1})\Big{)},$

where it was used notations accepted above in definition (3).

If $G(0)=(1,G_{1}^{0}(x_{1}),\ldots,G_{s}^{0}(x_{1},\ldots,x_{s}),\ldots)$ is
a sequence of reduced correlation functions at initial instant, then by means
of mappings (18) the evolution of all possible states, i.e. the sequence of
the reduced correlation functions $G_{s}(t),\,s\geq 1$, is determined by the
following series expansions:

$\displaystyle\hskip-34.1433ptG_{s}(t,x_{1},\ldots,x_{s})=$ (60)
$\displaystyle\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\int\limits_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{n}}dx_{s+1}\ldots
dx_{s+n}\,\,\mathfrak{A}_{1+n}(t;\\{1,\ldots,s\\},s+1,\ldots,s+n\mid
G(0)),\quad s\geq 1,$

where the generating operator
$\mathfrak{A}_{1+n}(t;\\{1,\ldots,s\\},s+1,\ldots,s+n\mid G(0))$ of this
series is the $(1+n)th$-order cumulant of groups of nonlinear operators (14):

$\displaystyle\hskip-22.76219pt\mathfrak{A}_{1+n}(t;\\{1,\ldots,s\\},s+1,\ldots,s+n\mid
G(0))\doteq$
$\displaystyle\sum\limits_{\mathrm{P}:\,(\\{1,\ldots,s\\},s+1,\ldots,s+n)=\bigcup_{k}X_{k}}(-1)^{|\mathrm{P}|-1}({|\mathrm{P}|-1})!\mathcal{G}(t;\theta(X_{1})\mid\ldots\mathcal{G}(t;\theta(X_{|\mathrm{P}|})\mid
G(0))\ldots),$ $\displaystyle\hskip-22.76219ptn\geq 0,$

and where the composition of mappings (14) of the corresponding noninteracting
groups of particles was denoted by
$\mathcal{G}(t;\theta(X_{1})\mid\ldots\mathcal{G}(t;\theta(X_{|\mathrm{P}|})\mid
G(0))\ldots)$, for example,

$\displaystyle\hskip-14.22636pt\mathcal{G}\big{(}t;1\mid\mathcal{G}(t;2\mid
G(0))\big{)}=\mathfrak{A}_{1}(t,1)\mathfrak{A}_{1}(t,2)G^{0}_{2}(x_{1},x_{2}),$
$\displaystyle\hskip-14.22636pt\mathcal{G}\big{(}t;1,2\mid\mathcal{G}(t;3\mid
G(0))\big{)}=\mathfrak{A}_{1}(t,\\{1,2\\})\mathfrak{A}_{1}(t,3)G^{0}_{3}(x_{1},x_{2},x_{3})+$
$\displaystyle\hskip
14.22636pt\mathfrak{A}_{2}(t,1,2)\mathfrak{A}_{1}(t,3)\big{(}G^{0}_{1}(x_{1})G^{0}_{2}(x_{2},x_{3})+G^{0}_{1}(x_{2})G^{0}_{2}(x_{1},x_{3})\big{)}.$

We will adduce examples of expansions (3.2). The first order cumulant of the
groups of nonlinear operators (14) is the group of these nonlinear operators

$\displaystyle\mathfrak{A}_{1}(t;\\{1,\ldots,s\\}\mid
G(0))=\mathcal{G}(t;1,\ldots,s\mid G(0)).$

In case of $s=2$ the second order cumulant of nonlinear operators (14) has the
structure

$\displaystyle\hskip-14.22636pt\mathfrak{A}_{1+1}(t;\\{1,2\\},3\mid
G(0))=\mathcal{G}(t;1,2,3\mid
G(0))-\mathcal{G}\big{(}t;1,2\mid\mathcal{G}(t;3\mid G(0))\big{)}=$
$\displaystyle\mathfrak{A}_{1+1}(t,\\{1,2\\},3)G^{0}_{3}(1,2,3)+$
$\displaystyle\big{(}\mathfrak{A}_{1+1}(t,\\{1,2\\},3)-\mathfrak{A}_{2}(t,2,3)\mathfrak{A}_{1}(t,1)\big{)}G^{0}_{1}(x_{1})G^{0}_{2}(x_{2},x_{3})+$
$\displaystyle\big{(}\mathfrak{A}_{1+1}(t,\\{1,2\\},3)-\mathfrak{A}_{2}(t,1,3)\mathfrak{A}_{1}(t,2)\big{)}G^{0}_{1}(x_{2})G^{0}_{2}(x_{1},x_{3})+$
$\displaystyle\mathfrak{A}_{1+1}(t,\\{1,2\\},3)G^{0}_{1}(x_{3})G^{0}_{2}(x_{1},x_{2})+\mathfrak{A}_{3}(t,1,2,3)G^{0}_{1}(x_{1})G^{0}_{1}(x_{2})G^{0}_{1}(x_{3}),$

where the operator

$\mathfrak{A}_{3}(t,1,2,3)=\mathfrak{A}_{1+1}(t,\\{1,2\\},3)-\mathfrak{A}_{2}(t,2,3)\mathfrak{A}_{1}(t,1)-\mathfrak{A}_{2}(t,1,3)\mathfrak{A}_{1}(t,2)$

is the third-order cumulant (17) of groups of operators (2) of a system of
hard spheres.

If $G(0)\in\oplus_{n=0}^{\infty}L^{1}_{n}$, then provided that $\max_{n\geq
1}\big{\|}G_{n}^{0}\big{\|}_{L^{1}_{n}}<(2e^{3})^{-1}$, for $t\in\mathbb{R}$
the sequence of reduced correlation functions (60) is a unique solution of the
Cauchy problem of the nonlinear BBGKY hierarchy (58),(59) for hard spheres.

In the particular case of the initial state specified by the sequence of
reduced correlation functions $G^{(c)}=(0,G_{1}^{0},0,\ldots,0,\ldots)$ on the
allowed configurations, that is, in the absence of correlations between hard
spheres at the initial moment of time [1],[19], according to definition (3.2)
of the generating operators, reduced correlation functions (60) are
represented by the following series expansions:

$\displaystyle\hskip-34.1433ptG_{s}(t,x_{1},\ldots,x_{s})=$ (62)
$\displaystyle\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\int\limits_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{n}}dx_{s+1}\ldots
dx_{s+n}\,\mathfrak{A}_{s+n}(t;1,\ldots,s+n)\prod_{i=1}^{s+n}G_{1}^{0}(x_{i})\mathcal{X}_{\mathbb{R}^{3(s+n)}\setminus\mathbb{W}_{s+n}},\quad
s\geq 1,$

where the generating operator $\mathfrak{A}_{s+n}(t)$ is the $(s+n)th$-order
cumulant (17) of the groups of operators (2) of the Liouville equations (5).

We emphasize that in the absence of correlations of states of hard spheres on
allowed configurations at the initial moment of time, the generators of
expansions into a series of reduced correlation functions (62) and reduced
distribution functions (48) differ only in the order of cumulants of groups of
operators of hard spheres. Therefore, by means of such reduced distribution
functions or reduced correlation functions, the process of creating
correlations in a system of hard spheres is described.

We note that the reduced correlation functions give an equivalent approach to
the description of the evolution of states of many hard spheres along with the
reduced distribution functions. Indeed, the macroscopic characteristics of
fluctuations of observables are directly determined by the reduced correlation
functions on the microscopic scale [21] for example, the functional of the
dispersion of an additive-type observable, i.e. the sequence
$A^{(1)}=(0,a_{1}(x_{1}),\ldots,\sum_{i_{1}=1}^{n}a_{1}(x_{i_{1}}),\ldots)$,
is represented by the formula

$\displaystyle\hskip-22.76219pt\langle(A^{(1)}-\langle
A^{(1)}\rangle)^{2}\rangle(t)=$ $\displaystyle\hskip
19.91692pt\int\limits_{\mathbb{R}^{3}\times\mathbb{R}^{3}}dx_{1}\,(a_{1}^{2}(x_{1})-\langle
A^{(1)}\rangle^{2}(t))G_{1}(t,x_{1})+\int\limits_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{2}}dx_{1}dx_{2}\,a_{1}(x_{1})a_{1}(x_{2})G_{2}(t,x_{1},x_{2}),$

where

$\displaystyle\hskip-22.76219pt\langle
A^{(1)}\rangle(t)=\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}dx_{1}\,a_{1}(x_{1})G_{1}(t,x_{1})$

is the mean value functional of an additive-type observable.

##  4 The description of correlations by means of kinetic equations

###  4.1 The non-Markovian Enskog equation

Now we will consider an approach to the description of the state evolution by
means of the state of a typical particle of a system of many hard spheres, or
in other words, foundations are overviewed of describing the evolution of a
state by kinetic equations.

Let the initial state specified by a one-particle reduced correlation
function, namely, the initial state specified by a sequence of reduced
correlation functions satisfying a chaos property stated above, i.e. by the
sequence $G^{(c)}=(G_{0},G_{1}^{0},0,\ldots,0,\ldots)$ on the allowed
configurations. We remark that such an assumption about initial states is
intrinsic in the contemporary kinetic theory of many-particle systems [1],[2].

Since the initial data $G^{(c)}$ is completely specified by a one-particle
correlation function, the Cauchy problem of the nonlinear BBGKY hierarchy
(58),(59) for hard spheres is not a completely well-defined Cauchy problem,
because the initial data is not independent for every unknown function
governed by the hierarchy of mentioned evolution equations. As a result, it
becomes possible to reformulate such a Cauchy problem as a new Cauchy problem
for a one-particle correlation function with independent initial data and
explicitly defined functionals of the solution of this Cauchy problem.

We formulate such a restated Cauchy problem and the sequence of the suitable
functionals. The following statement is true.

In the case of the initial state $G^{(c)}$ specified by a one-particle
correlation function the evolution that described by means of the sequence
$G(t)=\left(G_{0},G_{1}(t),\ldots,G_{s}(t),\ldots\right)$ of reduced
correlation functions (60), is also be described by the sequence $G(t\mid
G_{1}(t))=(G_{0},G_{1}(t),\ldots,G_{s}(t\mid G_{1}(t)),\ldots)$ of the reduced
(marginal) correlation functionals: $G_{s}(t,x_{1},\ldots,x_{s}\mid
G_{1}(t)),\,s\geq 2$, with respect to the one-particle correlation function
$G_{1}(t)$ governed by the non-Markovian Enskog kinetic equation.

Indeed, in the case under consideration the reduced correlation functionals
$G_{s}(t\mid G_{1}(t)),\,s\geq 2$, are represented with respect to the one-
particle correlation function (62), i.e.

$\displaystyle\hskip-22.76219ptG_{1}(t,x_{1})=\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\int\limits_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{n}}dx_{2}\ldots
dx_{1+n}\,\mathfrak{A}_{1+n}(t,1,\ldots,n+1)\prod_{i=1}^{n+1}G_{1}^{0}(x_{i})\mathcal{X}_{\mathbb{R}^{3(n+1)}\setminus\mathbb{W}_{n+1}},$
(63)

by the following series expansions:

$\displaystyle\hskip-22.76219ptG_{s}\bigl{(}t,x_{1},\ldots,x_{s}\mid
G_{1}(t)\bigr{)}=$ (64)
$\displaystyle\sum_{n=0}^{\infty}\frac{1}{n!}\,\int\limits_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{n}}dx_{s+1}\ldots
dx_{s+n}\,\mathfrak{V}_{s+n}\bigl{(}t,1,\ldots,s+n\bigr{)}\prod_{i=1}^{s+n}G_{1}(t,x_{i}),\quad
s\geq 2.$

The generating operator $\mathfrak{V}_{s+n}(t),\,n\geq 0$, of the
$(s+n)th$-order of this series is determined by the following expansion:

$\displaystyle\hskip-34.1433pt\mathfrak{V}_{s+n}\bigl{(}t,1,\ldots,s,s+1,\ldots,s+n\bigr{)}=n!\,\sum_{k=0}^{n}\,(-1)^{k}\,\sum_{n_{1}=1}^{n}\ldots$
(65) $\displaystyle\hskip-28.45274pt\sum_{n_{k}=1}^{n-n_{1}-\ldots-
n_{k-1}}\frac{1}{(n-n_{1}-\ldots-n_{k})!}\hat{\mathfrak{A}}_{s+n-n_{1}-\ldots-
n_{k}}(t,$ $\displaystyle\hskip-28.45274pt1,\ldots,s+n-n_{1}-\ldots-
n_{k})\prod_{j=1}^{k}\,\sum\limits_{\mbox{\scriptsize$\begin{array}[]{c}\mathrm{D}_{j}:Z_{j}=\bigcup_{l_{j}}X_{l_{j}},\\\
|\mathrm{D}_{j}|\leq s+n-n_{1}-\dots-
n_{j}\end{array}$}}\frac{1}{|\mathrm{D}_{j}|!}\times$ (68)
$\displaystyle\hskip-28.45274pt\sum_{i_{1}\neq\ldots\neq
i_{|\mathrm{D}_{j}|}=1}^{s+n-n_{1}-\ldots-
n_{j}}\,\prod_{X_{l_{j}}\subset\mathrm{D}_{j}}\,\frac{1}{|X_{l_{j}}|!}\hat{\mathfrak{A}}_{1+|X_{l_{j}}|}(t,i_{l_{j}},X_{l_{j}}),$

where $\sum_{\mathrm{D}_{j}:Z_{j}=\bigcup_{l_{j}}X_{l_{j}}}$ is the sum over
all possible dissections of the linearly ordered set
$Z_{j}\equiv(s+n-n_{1}-\ldots-n_{j}+1,\ldots,s+n-n_{1}-\ldots-n_{j-1})$ on no
more than $s+n-n_{1}-\ldots-n_{j}$ linearly ordered subsets, the
$(s+n)th$-order scattering cumulant is defined by the formula

$\displaystyle\hskip-22.76219pt\hat{\mathfrak{A}}_{s+n}(t,1,\ldots,s+n)\doteq\mathfrak{A}_{s+n}(t,1,\ldots,s+n)\mathcal{X}_{\mathbb{R}^{3(s+n)}\setminus\mathbb{W}_{s+n}}\prod_{i=1}^{s+n}\mathfrak{A}_{1}^{-1}(t,i),$

and notations accepted above were used.

We adduce simplest examples of generating operators (65):

$\displaystyle\hskip-19.91692pt\mathfrak{V}_{s}(t,1,\ldots,s)=\mathfrak{A}_{s}(t,1,\ldots,s)\mathcal{X}_{\mathbb{R}^{3s}\setminus\mathbb{W}_{s}}\prod_{i=1}^{s}\mathfrak{A}_{1}^{-1}(t,i),$
$\displaystyle\hskip-19.91692pt\mathfrak{V}_{s+1}(t,1,\ldots,s,s+1)=\mathfrak{A}_{s+1}(t,1,\ldots,s+1)\mathcal{X}_{\mathbb{R}^{3(s+1)}\setminus\mathbb{W}_{s+1}}\prod_{i=1}^{s+1}\mathfrak{A}_{1}^{-1}(t,i)-$
$\displaystyle\mathfrak{A}_{s}(t,1,\ldots,s)\mathcal{X}_{\mathbb{R}^{3s}\setminus\mathbb{W}_{s}}\prod_{i=1}^{s}\mathfrak{A}_{1}^{-1}(t,i)\sum_{j=1}^{s}\mathfrak{A}_{2}(t,j,s+1)\mathcal{X}_{\mathbb{R}^{6}\setminus\mathbb{W}_{2}}\mathfrak{A}_{1}^{-1}(t,j)\mathfrak{A}_{1}^{-1}(t,s+1).$

A method of the construction of reduced correlation functionals (64) is based
on the application of the so-called kinetic cluster expansions [14] to
generating operators (17) of series (62). If
$\|G_{1}(t)\|_{L^{1}(\mathbb{R}^{3}\times\mathbb{R}^{3})}<e^{-(3s+2)}$, for
arbitrary $t\in\mathbb{R}$ series (64) converges in the norm of the space
$L^{1}_{s}$ [14].

We note that in the case of initial state specified by a one-particle
correlation function the reduced correlation functionals (64) describe all
possible correlations generated by the dynamics of many hard spheres in terms
of a one-particle correlation function.

If initial data $G_{1}^{0}\in L^{1}_{1}$, then for arbitrary $t\in\mathbb{R}$
one-particle correlation function (63) is a weak solution of the Cauchy
problem of the non-Markovian Enskog kinetic equation

$\displaystyle\hskip-34.1433pt\frac{\partial}{\partial
t}G_{1}(t,x_{1})=\mathcal{L}^{\ast}(1)G_{1}(t,x_{1})+\int\limits_{\mathbb{R}^{3}\times\mathbb{R}^{3}}dx_{2}\,\mathcal{L}_{\mathrm{int}}^{\ast}(1,2)G_{1}(t,x_{1})G_{1}(t,x_{2})+$
(69) $\displaystyle\hskip
25.60747pt\int\limits_{\mathbb{R}^{3}\times\mathbb{R}^{3}}dx_{2}\,\mathcal{L}_{\mathrm{int}}^{\ast}(1,2)G_{2}\bigl{(}t,x_{1},x_{2}\mid
G_{1}(t)\bigr{)},$
$\displaystyle\hskip-34.1433ptG_{1}(t,x_{1})\big{|}_{t=0}=G_{1}^{0}(x_{1}),$
(70)

where the first part of the collision integral in equation (69) has the
Boltzmann–Enskog structure, and the second part of the collision integral is
determined in terms of the two-particle correlation functional represented by
series expansion (64) and it describes all possible correlations which are
created by hard-sphere dynamics and by the propagation of initial correlations
related to the forbidden configurations.

By virtue of definitions (3),(4) of the generator of the non-Markovian Enskog
equation (69), for $t>0$ the kinetic equation gets such explicit form

$\displaystyle\hskip-19.91692pt\frac{\partial}{\partial
t}G_{1}(t,x_{1})=-\langle p_{1},\frac{\partial}{\partial q_{1}}\rangle
G_{1}(t,x_{1})+$
$\displaystyle\sigma^{2}\int\limits_{\mathbb{R}^{3}\times\mathbb{S}_{+}^{2}}dp_{2}d\eta\,\langle\eta,(p_{1}-p_{2})\rangle\big{(}G_{1}(t,p_{1}^{\ast},q_{1})G_{1}(t,p_{2}^{\ast},q_{1}-\sigma\eta,)-G_{1}(t,x_{1})G_{1}(t,p_{2},q_{1}+\sigma\eta)\big{)}+$
$\displaystyle\sigma^{2}\int\limits_{\mathbb{R}^{3}\times\mathbb{S}_{+}^{2}}dp_{2}d\eta\,\langle\eta,(p_{1}-p_{2})\rangle\big{(}G_{2}\bigl{(}t,p_{1}^{\ast},q_{1},p_{2}^{\ast},q_{1}-\sigma\eta\mid
G_{1}(t)\bigr{)}-G_{2}\bigl{(}t,x_{1},p_{2},q_{1}+\sigma\eta\mid
G_{1}(t)\bigr{)}\big{)}.$

In the paper [14], similar statements were proved for the evolution of the
state of a hard-sphere system described in terms of reduced distribution
functions governed by the BBGKY hierarchy. We emphasize that the $nth$ term of
expansions (64) of the reduced correlation functionals are determined by the
$(s+n)th$-order generating operator (65) in contradistinction to the
expansions of reduced distribution functionals of the state constructed in
[14] which are determined by the $(1+n)th$-order generating operator (65).

Thus, for the initial state specified by a one-particle correlation function
all possible states of a system of hard spheres can be described without any
approximations within the framework of a one-particle correlation function
governed by the non-Markovian kinetic equation (69), and a sequence of
explicitly defined functionals (64) of its solution (63).

###  4.2 On the low-density approximation of hard-sphere correlations

The conventional philosophy of the description of kinetic evolution consists
in the following. If the initial state is specified by a one-particle
distribution function, then the evolution of the state can be effectively
described in a suitable scaling limit [4],[17] by means of a one-particle
distribution function governed by the nonlinear kinetic equation.

In the last decade, the Boltzmann–Grad limit (low-density scaling limit)
[17],[18] of the reduced distribution functions constructed by means of the
theory of perturbations was rigorously discussed in several papers, see for
example [7],[11],[15] and references therein. In this subsection, we consider
a scheme of constructing the scaling asymptotic behavior of reduced
correlation functions (60) in the Boltzmann–Grad limit for initial data
satisfying a chaos property stated above, namely, for the sequence of reduced
correlation functions (62).

Let for $t\geq 0$ the operator $\mathcal{L}^{\ast}_{\mathrm{int}}$ in the
dimensionless form of the hierarchy of evolution nonlinear equations (58) be
scaled in such a way that

$\displaystyle\hskip-19.91692pt\mathcal{L}_{\mathrm{int}}^{\ast}(j_{1},j_{2})f_{n}=\epsilon^{2}\int\limits_{\mathbb{S}_{+}^{2}}d\eta\langle\eta,(p_{j_{1}}-p_{j_{2}})\rangle\big{(}f_{n}(x_{1},\ldots,p_{j_{1}}^{*},q_{j_{1}},\ldots,$
$\displaystyle
p_{j_{2}}^{*},q_{j_{2}},\ldots,x_{n})\delta(q_{j_{1}}-q_{j_{2}}+\epsilon\eta)-f_{n}(x_{1},\ldots,x_{n})\delta(q_{j_{1}}-q_{j_{2}}-\epsilon\eta)\big{)},$

where $\epsilon>0$ is a scaling parameter (the ratio of the diameter
$\sigma>0$ to the mean free path of hard spheres) and the notations similar to
definition (3) are used. We will consider initial states of a hard-sphere
system specified by the scaled one-particle correlation function
$G_{1}^{0,\epsilon},$ such that: $|G_{1}^{0,\epsilon}(x_{1})|\leq
ce^{\textstyle-\frac{\beta}{2}{p^{2}_{1}}},$ where $\beta>0$ is a parameter
and $c<\infty$ is some constant. We emphasize that the states of a system of
infinitely many hard spheres are described by sequences of functions bounded
with respect to the configuration variables [1] as is assumed above. Regarding
the technical details of the scheme presented below for constructing the
Boltzmann–Grad asymptotics of reduced correlation functions (62), we refer to
our article [20].

Let us assume that the Boltzmann–Grad limit of the initial one-particle
correlation function $G_{1}^{0,\epsilon}$ exists in the sense of the weak
convergence [20]

$\displaystyle\mathrm{w-}\lim_{\epsilon\rightarrow
0}\big{(}\epsilon^{2}G_{1}^{0,\epsilon}(x_{1})-f_{1}^{0}(x_{1})\big{)}=0.$
(71)

If equality (71) holds for the initial one-particle correlation function, then
for series expansion (62) the following equality is true

$\displaystyle\mathrm{w-}\lim\limits_{\epsilon\rightarrow
0}\big{(}\epsilon^{2}G_{1}(t,x_{1})-f_{1}(t,x_{1})\big{)}=0,$

where for some finite time interval the limit one-particle correlation
function $f_{1}(t,x_{1})$ is represented by the series expansion

$\displaystyle\hskip-22.76219ptf_{1}(t,x_{1})=\sum\limits_{n=0}^{\infty}\int\limits_{0}^{t}dt_{1}\ldots\int\limits_{0}^{t_{n-1}}dt_{n}\,\int\limits_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{n}}dx_{2}\ldots
dx_{1+n}\,S_{1}(t_{1}-t,1)\times$ (72)
$\displaystyle\mathcal{L}^{\ast,0}_{\mathrm{int}}(1,2)\prod\limits_{j_{1}=1}^{2}S_{1}(t_{2}-t_{1},j_{1})\ldots\prod\limits_{i_{n}=1}^{n}S_{1}(t_{n}-t_{n-1},i_{n})\times$
$\displaystyle\sum\limits_{k_{n}=1}^{n}\mathcal{L}^{\ast,0}_{\mathrm{int}}(k_{n},n+1)\prod\limits_{j_{n}=1}^{n+1}S_{1}(-t_{n},j_{n})\prod\limits_{i=1}^{n+1}f_{1}^{0}(x_{i}).$

In this series expansion for $t\geq 0$ the operator
$\mathcal{L}_{\mathrm{int}}^{\ast,0}(j_{1},j_{2})$ is defined by the formula

$\displaystyle\hskip-19.91692pt\mathcal{L}_{\mathrm{int}}^{\ast,0}(j_{1},j_{2})f_{n}\doteq\int\limits_{\mathbb{S}_{+}^{2}}d\eta\langle\eta,(p_{j_{1}}-p_{j_{2}})\rangle\big{(}f_{n}(x_{1},\ldots,p_{j_{1}}^{*},q_{j_{1}},\ldots,$
$\displaystyle
p_{j_{2}}^{*},q_{j_{2}},\ldots,x_{n})-f_{n}(x_{1},\ldots,x_{n})\big{)}\delta(q_{j_{1}}-q_{j_{2}}),$

where notations adopted in formula (3) are used.

The function $f_{1}(t)$ represented by series (72) is a weak solution of the
Cauchy problem of the Boltzmann kinetic equation for hard spheres

$\displaystyle\hskip-19.91692pt\frac{\partial}{\partial
t}f_{1}(t,x_{1})=-\langle p_{1},\frac{\partial}{\partial q_{1}}\rangle
f_{1}(t,x_{1})+$ (73) $\displaystyle\hskip
14.22636pt\int_{\mathbb{R}^{3}\times\mathbb{S}^{2}_{+}}dp_{2}\,d\eta\,\langle\eta,(p_{1}-p_{2})\rangle(f_{1}(t,p_{1}^{*},q_{1})f_{1}(t,p_{2}^{*},q_{1})-f_{1}(t,p_{1},q_{1})f_{1}(t,p_{2},q_{1})),$
$\displaystyle\hskip-19.91692ptf_{1}(t,x_{1})\big{|}_{t=0}=f_{1}^{0}(x_{1}).$
(74)

The property of propagation of initial chaos in the of low-density limit is a
consequence of the following equalities for reduced correlation functions
(62):

$\displaystyle\mathrm{w-}\lim\limits_{\epsilon\rightarrow
0}\epsilon^{2s}G_{s}\bigl{(}t,x_{1},\ldots,x_{s}\bigr{)}=0,\quad s\geq 2.$
(75)

The proof of these statements (72) and (75) is based on the validity of the
limit theorems for cumulants of asymptotically perturbed groups of operators
(2) and the explicit structure of the generating operators of series
expansions of reduced correlation functions (62).

Indeed, if $|f_{s}|\leq
ce^{\textstyle-\frac{\beta}{2}{\sum^{s}_{i=1}p^{2}_{i}}}$, then for arbitrary
finite time interval for asymptotically perturbed first-order cumulant (17) of
the groups of operators (2) the following equality takes place [1],[20]

$\displaystyle\mathrm{w-}\lim\limits_{\epsilon\rightarrow
0}\big{(}S_{s}(-t,1,\ldots,s)f_{s}-\prod\limits_{j=1}^{s}S_{1}(-t,j)f_{s}\big{)}=0.$

Therefore, for the $(s+n)th$-order cumulant of asymptotically perturbed groups
of operators (2) the equalities are true:

$\displaystyle\mathrm{w-}\lim\limits_{\epsilon\rightarrow
0}\frac{1}{\epsilon^{2n}}\,\mathfrak{A}_{s+n}(t,1,\ldots,s+n)f_{s+n}=0,\quad
s\geq 2.$

In conclusion, we note that in the articles [16],[15] some other approaches to
the derivation of kinetic equations were developed, including the kinetic
equations for the states with correlations at the initial instant. In
particular, a new method of the description of the kinetic evolution of a
system with hard-sphere collisions has been suggested in the paper [16].
Formalism is based on the description of evolution within the framework of
observables which are governed by the dual Boltzmann hierarchy in the low-
density limit.

##  5 Conclusion

This article dealt with the mathematical problems of the description of the
evolution of many hard spheres based on various ways of describing their
state, namely by means of functions describing the propagation of
correlations. One of these approaches allows one to describe the evolution of
both a finite and an infinite average number of hard spheres using reduced
distribution functions (48) or reduced correlation functions (60), which are
determined by the dynamics of correlations (14) of a hard-sphere system.

We note the importance of the description of the processes of the creation and
propagation of correlations [24], in particular, it is related to the problem
of the description of the memory effects in many-particle systems with
collision dynamics.

It was established that the notion of cumulants (15) of the groups of
operators (2) underlies non-perturbative expansions of solutions for the
fundamental evolution equations describing the state evolution of a hard-
sphere system, namely, of the Liouville hierarchy (20) for correlation
functions, of the BBGKY hierarchy for reduced distribution functions and of
the nonlinear BBGKY hierarchy (58) for reduced correlation functions, as well
as it underlies the kinetic description of infinitely many hard spheres (64).
We remark that for quantum many-particle systems the concept of cumulants of
groups of operators is considered in the papers [25]-[27].

We emphasize that the structure of expansions for correlation functions (14),
in which the generating operators are the cumulants of the corresponding order
(15) of the groups of operators (2) of hard spheres, induces the cumulant
structure of series expansions for reduced distribution functions (48),
reduced correlation functions (60) and reduced correlation functionals (64).
Thus, in fact, the dynamics of systems of infinitely many hard spheres is
generated by the dynamics of correlations.

Above, one more approach to the description of the state evolution of a system
of many hard spheres in terms of the state evolution of a typical particle was
also studied. In other words, the origin of the collective behavior of a hard-
sphere system on a microscopic scale was described by means of a one-particle
correlation function that is determined by the non-Markovian Enskog kinetic
equation (69). As already mentioned, one of the advantages of such an approach
to the derivation of kinetic equations from underlying collisional dynamics
consists of an opportunity to construct the kinetic equations with initial
correlations, which makes it possible to describe the propagation of initial
correlations in the Boltzmann–Grad limit [15]. Another advantage of this
approach is related to the rigorous derivation of the Boltzmann equation with
higher-order corrections to the main term of the Boltzmann–Grad asymptotics of
collisional dynamics.

Acknowledgements. Glory to Ukraïne!

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