F^{(c)}\equiv\big{(}1,F_{1}^{\epsilon,0}(x_{1}),\ldots,\prod_{i=1}^{s}F_{1}^{\epsilon,0}(x_{i})\mathcal{X}_{\mathbb{R}^{3s}\setminus\mathbb{W}_{s}},\ldots\big{)},$
(3.19)

where $\mathcal{X}_{\mathbb{R}^{3s}\setminus\mathbb{W}_{s}}$ is the Heaviside
step function of the allowed configurations. This assumption about initial
state is intrinsic for the kinetic theory, because in this case all possible
states of gases are described by means of a one-particle distribution
function.

Let $F_{1}^{0,\epsilon}\in
L^{\infty}_{\xi}(\mathbb{R}^{3}\times\mathbb{R}^{3})$, i.e., the following
inequality holds:
$|F_{1}^{0,\epsilon}(x_{i})|\leq\xi\exp(-\beta\frac{p^{2}_{i}}{2})$, where
$\xi>0,\beta\geq 0$ are parameters. We assume that the Boltzmann–Grad limit of
the initial one-particle (reduced) distribution function
$F_{1}^{0,\epsilon}\in L^{\infty}_{\xi}(\mathbb{R}^{3}\times\mathbb{R}^{3})$
exists in the sense of a weak convergence of the space
$L^{\infty}_{\xi}(\mathbb{R}^{3}\times\mathbb{R}^{3})$, namely,

$\displaystyle\mathrm{w-}\lim_{\epsilon\rightarrow
0}(\epsilon^{2}\,F_{1}^{0,\epsilon}-f_{1}^{0})=0,$ (3.20)

then the Boltzmann–Grad limit of the initial state (3.19) satisfies a chaos
property too, i.e.,
$f^{(c)}\equiv\big{(}1,f_{1}^{0}(x_{1}),\ldots,\prod_{i=1}^{s}f_{1}^{0}(x_{i}),\ldots\big{)}$.

We note that assumption (3.20) with respect to the Boltzmann–Grad limit of
initial states holds true for the equilibrium state [60].

If $b(t)\in\mathcal{C}_{\gamma}$ and
$|f_{1}^{0}(x_{i})|\leq\xi\exp(-\beta\frac{p^{2}_{i}}{2})$, then the
Boltzmann–Grad limit of mean value functional $\big{(}B(t),F^{(c)}\big{)}$
exists under the condition that [83]:
$t<t_{0}\equiv\big{(}\text{const}(\xi,\beta)\|f_{1}^{0}\|_{L^{\infty}_{\xi}(\mathbb{R}^{3}\times\mathbb{R}^{3})}\big{)}^{-1}$,
and it is determined by the following series expansion:

$\displaystyle\hskip-22.76219pt\big{(}b(t),f^{(c)}\big{)}=\sum\limits_{s=0}^{\infty}\,\frac{1}{s!}\,\int\limits_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{s}}dx_{1}\ldots
dx_{s}\,b_{s}(t,x_{1},\ldots,x_{s})\prod\limits_{i=1}^{s}f_{1}^{0}(x_{i}).$

For the limit of additive-type reduced observables (3.9) the following
equality holds [51]:

$\displaystyle\hskip-34.1433pt\big{(}b^{(1)}(t),f^{(c)}\big{)}=\sum\limits_{s=0}^{\infty}\,\frac{1}{s!}\,\int\limits_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{s}}dx_{1}\ldots
dx_{s}\,b_{s}^{(1)}(t,x_{1},\ldots,x_{s})\prod\limits_{i=1}^{s}f_{1}^{0}(x_{i})=$
$\displaystyle\int\limits_{\mathbb{R}^{3}\times\mathbb{R}^{3}}dx_{1}\,b_{1}^{0}(x_{1})f_{1}(t,x_{1}),$

where the function $b_{s}^{(1)}(t)$ is given by expansion (3.9) and the
distribution function $f_{1}(t,x_{1})$ is represented by the series

$\displaystyle\hskip-34.1433ptf_{1}(t,x_{1})=\sum\limits_{n=0}^{\infty}\int\limits_{0}^{t}dt_{1}\ldots\int\limits_{0}^{t_{n-1}}dt_{n}\,\int\limits_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{n}}dx_{2}\ldots
dx_{n+1}\,S_{1}^{\ast}(t-t_{1},1)\mathcal{L}_{\mathrm{int}}^{0,\ast}(1,2)\times$
(3.21)
$\displaystyle\hskip-14.22636pt\prod\limits_{j_{1}=1}^{2}S_{1}^{\ast}(t_{1}-t_{2},j_{1})\ldots\prod\limits_{i_{n}=1}^{n}S_{1}^{\ast}(t_{n-1}-t_{n},i_{n})\sum\limits_{k_{n}=1}^{n}\mathcal{L}_{\mathrm{int}}^{0,\ast}(k_{n},n+1)\times$
$\displaystyle\hskip-14.22636pt\prod\limits_{j_{n}=1}^{n+1}S_{1}^{\ast}(t_{n},j_{n})\prod\limits_{i=1}^{n+1}f_{1}^{0}(x_{i}),$

and the following operator was introduced:

$\displaystyle\hskip-34.1433pt\int\limits_{\mathbb{R}^{3}\times\mathbb{R}^{3}}dx_{n+1}\mathcal{L}_{\mathrm{int}}^{0,\ast}(i,n+1)f_{n+1}(x_{1},\ldots,x_{n+1})\equiv$
(3.22)
$\displaystyle\int\limits_{\mathbb{R}^{3}\times\mathbb{S}_{+}^{2}}dp_{n+1}d\eta\,\langle\eta,(p_{i}-p_{n+1})\rangle\big{(}f_{n+1}(x_{1},\ldots,q_{i},p_{i}^{*},\ldots,x_{s},q_{i},p_{n+1}^{*})-$
$\displaystyle f_{n+1}(x_{1},\ldots,x_{s},q_{i},p_{n+1})\big{)}.$

A one-particle distribution function represented as a series (3.21) is a
solution of the Cauchy problem of the Boltzmann kinetic equation:

$\displaystyle\hskip-34.1433pt\frac{\partial}{\partial
t}f_{1}(t,x_{1})=-\langle p_{1},\frac{\partial}{\partial q_{1}}\rangle
f_{1}(t,x_{1})+$ (3.23)
$\displaystyle\int\limits_{\mathbb{R}^{3}\times\mathbb{S}^{2}_{+}}dp_{2}\,d\eta\,\langle\eta,(p_{1}-p_{2})\rangle\big{(}f_{1}(t,q_{1},p_{1}^{*})f_{1}(t,q_{1},p_{2}^{*})-f_{1}(t,x_{1})f_{1}(t,q_{1},p_{2})\big{)},$
$\displaystyle\hskip-34.1433ptf_{1}(t,x_{1})_{\mid t=0}=f_{1}^{0}(x_{1}).$
(3.24)

Thus, we establish that the dual Boltzmann hierarchy (3.17) for additive-type
reduced observables and initial state (3.20) describe the evolution of hard
sphere systems just as the Boltzmann kinetic equation (3.23).

We remark that in a one-dimensional space, the collision integral of the
Boltzmann equation with elastic hard sphere collisions identically equals
zero. In a one-dimensional space, the Boltzmann–Grad limit is not trivial in
the case of hard sphere dynamics with inelastic collisions. In the paper [43]
for one-dimensional granular gas, the process of the creation and propagation
of correlations in the Boltzmann–Grad scaling limit was also described (see
also section 5.1).

Correspondingly, if the initial state of hard spheres is given by a sequence
of reduced distribution functions (3.19), then in the Boltzmann–Grad limit,
the property of the propagation of initial chaos holds [51]. It is a result of
the validity of the following equality for the limit $k$-ary reduced
observables, i.e., for the sequences
$b^{(k)}(0)=(0,\ldots,b_{k}^{0}(x_{1},\ldots,x_{k}),0,\ldots)$,

$\displaystyle\hskip-34.1433pt\big{(}b^{(k)}(t),f^{(c)}\big{)}=\sum\limits_{s=0}^{\infty}\,\frac{1}{s!}\,\int\limits_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{s}}dx_{1}\ldots
dx_{s}\,b_{s}^{(k)}(t,x_{1},\ldots,x_{s})\prod\limits_{i=1}^{s}f_{1}^{0}(x_{i})=$
(3.25)
$\displaystyle\hskip-14.22636pt\frac{1}{k!}\int\limits_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{k}}dx_{1}\ldots
dx_{k}\,b_{k}^{0}(x_{1},\ldots,x_{k})\prod\limits_{i=1}^{k}f_{1}(t,x_{i}),\quad
k\geq 2,$

where the limit one-particle reduced distribution function $f_{1}(t)$ is
defined by expansion (3.21) and therefore it is governed by the Cauchy problem
of the Boltzmann kinetic equation (3.23),(3.24).

Thus, in the Boltzmann–Grad scaling limit, an equivalent approach to the
description of the kinetic evolution of hard spheres in terms of the Cauchy
problem of the Boltzmann kinetic equation (3.23),(3.24) is given by the Cauchy
problem of the dual Boltzmann hierarchy with hard sphere collisions
(3.17),(3.18) for the additive-type reduced observables. In the case of non-
additive-type reduced observables, a solution of the dual Boltzmann hierarchy
with hard sphere collisions (3.17) is equivalent to the property of the
propagation of initial chaos in the sense of equality (3.25).

###  3.4 The Boltzmann kinetic equation with initial correlations

We now consider the case of the more general initial state of a hard sphere
system specified by the one-particle reduced distribution function
$F_{1}^{0,\epsilon}\in L^{\infty}_{\xi}(\mathbb{R}^{3}\times\mathbb{R}^{3})$
in the presence of correlations, i.e., the initial state that is specified by
the following sequence of reduced distribution functions:

$\displaystyle\hskip-22.76219ptF^{(cc)}=\big{(}1,F_{1}^{0,\epsilon}(x_{1}),g_{2}^{\epsilon}\prod_{i=1}^{2}F_{1}^{0,\epsilon}(x_{i}),\ldots,g_{n}^{\epsilon}\prod_{i=1}^{n}F_{1}^{0,\epsilon}(x_{i}),\ldots\big{)},$
(3.26)

where the functions $g_{n}^{\epsilon}(x_{1},\ldots,x_{n})\equiv
g_{n}^{\epsilon}\in
C_{n}(\mathbb{R}^{3n}\times(\mathbb{R}^{3n}\setminus\mathbb{W}_{n})),\,n\geq
2$, specify the initial correlations. Since many-particle systems in condensed
states are characterized by correlations, sequence (3.26) describes the
initial state of the kinetic evolution of hard sphere fluids.

We assume that the Boltzmann–Grad limit of the initial one-particle reduced
distribution function $F_{1}^{0,\epsilon}\in
L^{\infty}_{\xi}(\mathbb{R}^{3}\times\mathbb{R}^{3})$ exists in the sense as
above, i.e., in the sense of a weak convergence the equality holds:
$\lim_{\epsilon\rightarrow 0}(\epsilon^{2}\,F_{1}^{0,\epsilon}-f_{1}^{0})=0$,
and in the case of correlation functions, let be: $\lim_{\epsilon\rightarrow
0}(g_{n}^{\epsilon}-g_{n})=0,\,n\geq 2$, then in the Boltzmann–Grad limit,
initial state (3.26) is specified by the following sequence of the limit
reduced distribution functions:

$\displaystyle\hskip-22.76219ptf^{(cc)}=\big{(}1,f_{1}^{0}(x_{1}),g_{2}\prod_{i=1}^{2}f_{1}^{0}(x_{i}),\ldots,g_{n}\prod_{i=1}^{n}f_{1}^{0}(x_{i}),\ldots\big{)}.$
(3.27)

We now consider relationships between the constructed Boltzmann–Grad
asymptotic behavior of reduced observables and the nonlinear Boltzmann-type
kinetic equation in the case of initial state specified by sequence (3.27).

For the limit additive-type reduced observables (3.9) and initial state (3.27)
the following equality is true:

$\displaystyle\hskip-34.1433pt\big{(}b^{(1)}(t),f^{(cc)}\big{)}=\sum\limits_{s=0}^{\infty}\,\frac{1}{s!}\,\int\limits_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{s}}dx_{1}\ldots
dx_{s}\,b_{s}^{(1)}(t,x_{1},\ldots,x_{s})g_{s}(x_{1},\ldots,x_{s})\prod_{i=1}^{s}f_{1}^{0}(x_{i})=$
$\displaystyle\int\limits_{\mathbb{R}^{3}\times\mathbb{R}^{3}}dx_{1}\,b_{1}^{0}(x_{1})f_{1}(t,x_{1}),$

where the functions $b_{s}^{(1)}(t)$ are represented by expansions (3.9) and
the limit reduced distribution function $f_{1}(t)$ is represented by the
following series expansion

$\displaystyle\hskip-34.1433ptf_{1}(t,x_{1})=\sum\limits_{n=0}^{\infty}\,\int\limits_{0}^{t}dt_{1}\ldots\int\limits_{0}^{t_{n-1}}dt_{n}\,\int\limits_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{n}}dx_{2}\ldots
dx_{n+1}S_{1}^{\ast}(t-t_{1},1)\times$ (3.28)
$\displaystyle\hskip-14.22636pt\mathcal{L}_{\mathrm{int}}^{0,\ast}(1,2)S_{1}^{\ast}(t_{1}-t_{2},j_{1})\ldots\prod\limits_{i_{n}=1}^{n}S_{1}^{\ast}(t_{n}-t_{n},i_{n})\sum\limits_{k_{n}=1}^{n}\mathcal{L}_{\mathrm{int}}^{0,\ast}(k_{n},n+1)\times$
$\displaystyle\hskip-14.22636pt\prod\limits_{j_{n}=1}^{n+1}S_{1}^{\ast}(t_{n},j_{n})g_{1+n}(x_{1},\ldots,x_{n+1})\prod\limits_{i=1}^{n+1}f_{1}^{0}(x_{i}).$

Series (3.28) is uniformly convergent for a finite time interval under the
condition as above.

The function $f_{1}(t)$ represented by series (3.28) is a weak solution of the
Cauchy problem of the Boltzmann kinetic equation with initial correlations
[30],[55]:

$\displaystyle\hskip-34.1433pt\frac{\partial}{\partial
t}f_{1}(t,x_{1})=-\langle p_{1},\frac{\partial}{\partial q_{1}}\rangle
f_{1}(t,x_{1})+$ (3.29)
$\displaystyle\hskip-14.22636pt\int\limits_{\mathbb{R}^{3}\times\mathbb{S}^{2}_{+}}dp_{2}\,d\eta\,\langle\eta,(p_{1}-p_{2})\rangle\Big{(}g_{2}(q_{1}-p_{1}^{*}t,p_{1}^{*},q_{2}-p_{2}^{*}t,p_{2}^{*})f_{1}(t,q_{1},p_{1}^{*})f_{1}(t,q_{1},p_{2}^{*})-$
$\displaystyle\hskip-14.22636ptg_{2}(q_{1}-p_{1}t,p_{1},q_{2}-p_{2}t,p_{2})f_{1}(t,x_{1})f_{1}(t,q_{1},p_{2})\Big{)},$
$\displaystyle\hskip-34.1433ptf_{1}(t,x_{1})\big{|}_{t=0}=f_{1}^{0}(x_{1}).$
(3.30)

This fact is proved similarly to the case of a perturbative solution of the
BBGKY hierarchy for hard spheres represented by the iteration series
[14],[58].

Thus, in the case of initial states specified by a one-particle reduced
distribution function (3.27) we establish that the dual Boltzmann hierarchy
with hard sphere collisions (3.17) for additive-type reduced observables
describes the evolution of a hard sphere system just as the Boltzmann kinetic
equation with initial correlations (3.29).

The property of the propagation of initial correlations is a consequence of
the validity of the following equality for the mean value functional of the
limit $k$-ary reduced observables:

$\displaystyle\hskip-34.1433pt\big{(}b^{(k)}(t),f^{(cc)}\big{)}=\sum\limits_{s=0}^{\infty}\,\frac{1}{s!}\,\int\limits_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{s}}dx_{1}\ldots
dx_{s}\,b_{s}^{(k)}(t,x_{1},\ldots,x_{s})g_{s}(x_{1},\ldots,x_{s})\prod\limits_{j=1}^{s}f_{1}^{0}(x_{j})=$
(3.31)
$\displaystyle\hskip-14.22636pt\frac{1}{k!}\int\limits_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{k}}dx_{1}\ldots
dx_{k}\,b_{k}^{0}(x_{1},\ldots,x_{k})\prod_{i_{1}=1}^{k}S_{1}^{\ast}(t,i_{1})g_{k}(x_{1},\ldots,x_{k})\times$
$\displaystyle\hskip-14.22636pt\prod_{i_{2}=1}^{k}(S_{1}^{\ast})^{-1}(t,i_{2})\prod\limits_{j=1}^{k}f_{1}(t,x_{j}),\quad
k\geq 2,$

where the one-particle reduced distribution function $f_{1}(t,x_{j})$ is
solution (3.28) of the Cauchy problem of the Boltzmann kinetic equation with
initial correlations (3.29),(3.30), and the inverse group to the group of
operators $S_{1}^{\ast}(t)$ we denote by
$(S_{1}^{\ast})^{-1}(t)=S_{1}^{\ast}(-t)=S_{1}(t)$.

This fact is proved similarly to the proof of a property on the propagation of
initial chaos (3.25).

We note that, according to equality (3.31), in the Boltzmann–Grad limit, the
reduced correlation functions are defined as cluster expansions of reduced
distribution functions, namely,

$\displaystyle\hskip-34.1433ptf_{s}(t,x_{1},\ldots,x_{s})=\sum\limits_{\mbox{\scriptsize$\begin{array}[]{c}\mathrm{P}:(x_{1},\ldots,x_{s})=\bigcup_{i}X_{i}\end{array}$}}{\prod\limits}_{X_{i}\subset\mathrm{P}}\,g_{|X_{i}|}(t,X_{i}),\quad
s\geq 1,$

and they have the explicit form:

$\displaystyle\hskip-34.1433ptg_{1}(t,x_{1})=f_{1}(t,x_{1}),$ (3.33)
$\displaystyle\hskip-34.1433ptg_{s}(t,x_{1},\ldots,x_{s})=\tilde{g}_{s}(q_{1}-p_{1}t,p_{1},\ldots,q_{s}-p_{s}t,p_{s})\prod\limits_{j=1}^{s}f_{1}(t,x_{j}),\quad
s\geq 2,$

where for initial correlation functions (3.27) it is used the following
notations:

$\displaystyle\hskip-34.1433pt\tilde{g}_{s}(x_{1},\ldots,x_{s})=\sum\limits_{\mbox{\scriptsize$\begin{array}[]{c}\mathrm{P}:(x_{1},\ldots,x_{s})=\bigcup_{i}X_{i}\end{array}$}}\prod_{X_{i}\subset\mathrm{P}}g_{|X_{i}|}(X_{i}),$

the symbol $\sum_{\mathrm{P}}$ means the sum over possible partitions
$\mathrm{P}$ of the set of arguments $(x_{1},\ldots,x_{s})$ on $|\mathrm{P}|$
nonempty subsets $X_{i}$, and the one-particle reduced distribution function
$f_{1}(t)$ is a solution of the Cauchy problem of the Boltzmann kinetic
equation with initial correlations (3.29),(3.30).

Thus, in the case of the limit $k$-ary reduced observables, a solution of the
dual Boltzmann hierarchy with hard sphere collisions (3.17) is equivalent to a
property of the propagation of initial correlations for the $k$-particle
reduced distribution function in the sense of equality (3.31) or in other
words, the Boltzmann–Grad scaling dynamics does not create new correlations.

##  4 Origin of kinetic equations

One of the challenges of kinetic theory, as mentioned above, is understanding
the nature of the possibility of describing the evolution of the state of a
system of many hard spheres by means of the state of a typical hard sphere.
More precisely, we further focus on the problem of the origin of the
description of the evolution of the state of hard spheres by the Enskog-type
kinetic equation.

In the circumstances where the initial state is specified by a one-particle
reduced distribution function, for the mean value functional of observables at
an arbitrary instant, the representation is also valid in terms of a one-
particle reduced distribution function, the evolution of which is governed by
a non-Markovian nonlinear evolution equation. In other words, for such initial
data, the Cauchy problem of the BBGKY hierarchy for hard spheres is equivalent
to the nonlinear Enskog-type kinetic equation and a sequence of reduced
functionals determined by the solution of this evolution equation.

###  4.1 The generalized Enskog kinetic equation

In the case of initial state (3.19) the dual picture of the evolution to the
picture described by employing observables governed by the dual BBGKY
hierarchy (2.1) for hard spheres is the picture of the evolution of a state
described by means of the non-Markovian Enskog kinetic equation and by a
sequence of explicitly defined functionals of the solution of such a kinetic
equation that describe the evolution of all possible correlations in a system
of hard spheres [46],[47].

In view of the fact that the initial state is completely specified by a one-
particle reduced distribution function on allowed configurations (3.19), for
mean value functional (1.11) the following representation holds [47]:

$\displaystyle\big{(}B(t),F^{(c)}\big{)}=\big{(}B(0),F(t\mid
F_{1}(t))\big{)},$ (4.1)

where $F^{(c)}$ is the sequence of initial reduced distribution functions
(3.19), and the sequence $F(t\mid F_{1}(t))=\big{(}1,F_{1}(t),F_{2}(t\mid
F_{1}(t)),\ldots,F_{s}(t\mid F_{1}(t)),\ldots\big{)}$ is a sequence of the
reduced functionals of the state $F_{s}(t,x_{1},\ldots,x_{s}\mid F_{1}(t)),$
$s\geq 2,$ represented by the series expansions over the products of the one-
particle distribution function $F_{1}(t)$, namely

$\displaystyle\hskip-34.1433ptF_{s}(t,x_{1},\ldots,x_{s}\mid F_{1}(t))\doteq$
(4.2)
$\displaystyle\hskip-19.91692pt\sum_{n=0}^{\infty}\frac{1}{n!}\,\int\limits_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{n}}dx_{s+1}\ldots
dx_{s+n}\,\mathfrak{V}_{1+n}(t,\\{1,\ldots,s\\},s+1,\ldots,s+n)\prod_{i=1}^{s+n}F_{1}(t,x_{i}),$
$\displaystyle\hskip-34.1433pts\geq 2.$

where the generating operators of series (4.2) are the $(1+n)th$-order
operators $\mathfrak{V}_{1+n}(t),\,n\geq 0$, determined by the following
expansions [47]:

$\displaystyle\hskip-34.1433pt\mathfrak{V}_{1+n}\bigl{(}t,\\{1,\ldots,s\\},s+1,\ldots,s+n\bigr{)}=n!\,\sum_{k=0}^{n}\,(-1)^{k}\,\sum_{n_{1}=1}^{n}\ldots$
(4.3) $\displaystyle\sum_{n_{k}=1}^{n-n_{1}-\ldots-
n_{k-1}}\frac{1}{(n-n_{1}-\ldots-
n_{k})!}\,\widehat{\mathfrak{A}}_{1+n-n_{1}-\ldots-n_{k}}(t,\\{1,\ldots,s\\},$
$\displaystyle s+1,\ldots,s+n-n_{1}-\ldots-
n_{k})\prod_{j=1}^{k}\,\sum\limits_{\mbox{\scriptsize$\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}$}}\frac{1}{|\mathrm{D}_{j}|!}\times$ (4.6)
$\displaystyle\sum_{i_{1}\neq\ldots\neq
i_{|\mathrm{D}_{j}|}=1}^{s+n-n_{1}-\ldots-
n_{j}}\,\prod_{X_{l_{j}}\subset\mathrm{D}_{j}}\,\frac{1}{|X_{l_{j}}|!}\,\widehat{\mathfrak{A}}_{1+|X_{l_{j}}|}(t,i_{l_{j}},X_{l_{j}}),$

In expansion (4.3) the symbol
$\sum_{\mathrm{D}_{j}:Z_{j}=\bigcup_{l_{j}}X_{l_{j}}}$ means 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, and the
$(1+n)th$-order scattering cumulant we denoted by the operator:

$\displaystyle\hskip-34.1433pt\widehat{\mathfrak{A}}_{1+n}(t,\\{1,\ldots,s\\},s+1,\ldots,s+n)\doteq$
(4.7)
$\displaystyle\mathfrak{A}_{1+n}^{\ast}(t,\\{1,\ldots,s\\},s+1,\ldots,s+n)\mathcal{X}_{\mathbb{R}^{3(s+n)}\setminus\mathbb{W}_{s+n}}\prod_{i=1}^{s+n}\mathfrak{A}_{1}^{\ast}(t,i)^{-1},$

where the operator $\mathfrak{A}_{1+n}^{\ast}(t)$ is the $(1+n)th$-order
cumulant of the groups of operators (1.6) of hard spheres.

We provide some examples of expressions for the generating operators of series
(4.2) for reduced functionals of the state:

$\displaystyle\hskip-34.1433pt\mathfrak{V}_{1}(t,\\{1,\ldots,s\\})=\widehat{\mathfrak{A}}_{1}(t,\\{1,\ldots,s\\})\doteq
S_{s}^{\ast}(t,1,\ldots,s)\mathcal{X}_{\mathbb{R}^{3(s)}\setminus\mathbb{W}_{s}}\prod_{i=1}^{s}S_{1}^{\ast}(t,i)^{-1},$
$\displaystyle\hskip-34.1433pt\mathfrak{V}_{2}(t,\\{1,\ldots,s\\},s+1)=\widehat{\mathfrak{A}}_{2}(t,\\{1,\ldots,s\\},s+1)-\widehat{\mathfrak{A}}_{1}(t,\\{1,\ldots,s\\})\sum_{i_{1}=1}^{s}\widehat{\mathfrak{A}}_{2}(t,i_{1},s+1).$

The method of constructing reduced state functionals (4.2) is based on the
application of the variation of cluster expansions, the so-called kinetic
cluster expansions, to generating operators (2.21) of series representing
solutions of hierarchies of evolution equations [47].

The one-particle distribution function $F_{1}(t)$, i.e., the first element of
the sequence $F(t\mid F_{1}(t))$, is determined by series (2.19), namely

$\displaystyle\hskip-34.1433ptF_{1}(t,x_{1})=\sum\limits_{n=0}^{\infty}\frac{1}{n!}\int\limits_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{n}}dx_{2}\ldots
dx_{n+1}\,\mathfrak{A}_{1+n}^{\ast}(t,1,\ldots,n+1)\prod_{i=1}^{n+1}F_{1}^{\epsilon,0}(x_{i})\mathcal{X}_{\mathbb{R}^{3(1+n)}\setminus\mathbb{W}_{1+n}},$
(4.8)

where the generating operator $\mathfrak{A}_{1+n}^{\ast}(t)$ is the $(1+n)$th-
order cumulant of groups of operators (1.6).

Let us note that in particular case of initial data (2.2) specified by the
additive-type reduced observables, according to solution expansion (2.10),
equality (4.1) takes the form

$\displaystyle\hskip-22.76219pt\big{(}B^{(1)}(t),F(0)\big{)}=\int\limits_{\mathbb{R}^{3}\times\mathbb{R}^{3}}dx_{1}\,b_{1}^{\epsilon}(x_{1})F_{1}(t,x_{1}),$

where the one-particle distribution function $F_{1}(t)$ is determined by
series (4.8). In the case of initial data (2.2) specified by the $s$-ary
reduced observable $s\geq 2$, equality (4.1) has the form

$\displaystyle\hskip-34.1433pt\big{(}B^{(s)}(t),F(0)\big{)}=\frac{1}{s!}\int\limits_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{s}}dx_{1}\ldots
dx_{s}\,b_{s}^{\epsilon}(x_{1},\ldots,x_{s})F_{s}(t,x_{1},\ldots,x_{s}\mid
F_{1}(t)),$

where the reduced functionals of the state $F_{s}(t,x_{1},\ldots,x_{s}\mid
F_{1}(t))$ are determined by series (4.2).

Thus, for the initial state specified by a one-particle distribution function,
the evolution of all possible states of a system of many hard spheres can be
described by means of the state of a typical particle without any scaling
approximations. We emphasize that reduced functionals of the state(4.2)
describe all possible correlations created during the evolution of many hard
spheres in terms of the state of a typical hard sphere.

For $t\geq 0$ the one-particle distribution function (4.8) is a solution of
the following Cauchy problem of the non-Markovian generalized Enskog kinetic
equation [53],[47]:

$\displaystyle\hskip-34.1433pt\frac{\partial}{\partial
t}F_{1}(t,q_{1},p_{1})=-\langle p_{1},\frac{\partial}{\partial q_{1}}\rangle
F_{1}(t,q_{1},p_{1})+$ (4.9)
$\displaystyle\hskip-14.22636pt\epsilon^{2}\int\limits_{\mathbb{R}^{3}\times\mathbb{S}^{2}_{+}}dp_{2}d\eta\langle\eta,(p_{1}-p_{2})\rangle\Big{(}F_{2}(t,q_{1},p_{1}^{*},q_{1}-\epsilon\eta,p_{2}^{*}\mid
F_{1}(t))-$
$\displaystyle\hskip-14.22636ptF_{2}(t,q_{1},p_{1},q_{1}+\epsilon\eta,p_{2}\mid
F_{1}(t))\Big{)},$
$\displaystyle\hskip-34.1433ptF_{1}(t)\big{|}_{t=0}=F_{1}^{\epsilon,0},$
(4.10)

where the collision integral is determined by the reduced functional of the
state (4.2) in the case of $s=2$ and the expressions $p_{1}^{*}$ and
$p_{2}^{*}$ are the pre-collision momenta of hard spheres (1.4). The series on
the right-hand side of this equation converges under the condition:
$\|F_{1}(t)\|_{L^{1}(\mathbb{R}\times\mathbb{R})}<e^{-8}$ .

Hence in the case of the additive-type reduced observables the generalized
Enskog kinetic equation (4.9) is dual to the dual BBGKY hierarchy of hard
spheres (2.1) with respect to bilinear form (1.11).

We observe that the structure of the collision integral of the generalized
Enskog equation (4.9) is such that the first term of its expansion is the
collision integral of the Boltzman–Enskog kinetic equation, and the next terms
describe all possible correlations that are created by the dynamics of hard
spheres and by the propagation of initial correlations connected with the
forbidden configurations, indeed

$\displaystyle\hskip-34.1433pt\frac{\partial}{\partial
t}F_{1}(t,x_{1})=-\langle p_{1},\frac{\partial}{\partial q_{1}}\rangle
F_{1}(t,x_{1})+\mathcal{I}_{GEE},$

where the collision integral is determined by the following series expansion:

$\displaystyle\hskip-34.1433pt\mathcal{I}_{GEE}\doteq\epsilon^{2}\sum_{n=0}^{\infty}\frac{1}{n!}\int\limits_{\mathbb{R}^{3}\times\mathbb{S}^{2}_{+}}dp_{2}d\eta\int\limits_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{n}}dx_{3}\ldots
dx_{n+2}\langle\eta,(p_{1}-p_{2})\rangle\times$
$\displaystyle\hskip-14.22636pt\big{(}\mathfrak{V}_{1+n}(t,\\{1^{*},2^{*}_{-}\\},3,\ldots,n+2)F_{1}(t,q_{1},p_{1}^{*})F_{1}(t,q_{1}-\epsilon\eta,p_{2}^{*})\prod_{i=3}^{n+2}F_{1}(t,x_{i})-$
$\displaystyle\hskip-14.22636pt\mathfrak{V}_{1+n}(t,\\{1,2_{+}\\},3,\ldots,n+2)F_{1}(t,x_{1})F_{1}(t,q_{1}+\epsilon\eta,p_{2})\prod_{i=3}^{n+2}F_{1}(t,x_{i})\big{)},$

and the notations adopted for the conventional notation of the Enskog
collision integral were used: indices $(1^{\sharp},2^{\sharp}_{\pm})$ denote
that the evolution operator $\mathfrak{V}_{1+n}(t)$ acts on the corresponding
phase points $(q_{1},p_{1}^{\sharp})$ and
$(q_{1}\pm\epsilon\eta,p_{2}^{\sharp})$, and the $(n+1)$th-order evolution
operator $\mathfrak{V}_{1+n}(t),\,n\geq 0$, is determined by expansion (4.3)
in the case of $s=2$.

We note that in the work [47] for the initial-value problem (4.9),(4.10) the
existence theorem was proved in the space of integrable functions. The
accordance of the generalized Enskog equation (4.9) and of the Markovian
Enskog-type kinetic equation was also established there. By the point, we
remark that in the paper [97] the explicit soliton-like solutions of kinetic
equation (4.9) were found.

Thus, if the initial state is specified by a one-particle distribution
function on allowed configurations, then the evolution of many hard spheres
governed by the dual BBGKY hierarchy (2.1) for reduced observables can be
completely described by the generalized Enskog kinetic equation (4.9) and by a
sequence of reduced functionals of the state (4.2).

We remark also that in the case of the initial state that involves
correlations (3.26) considered approach permits to take into consideration the
initial correlations in the kinetic equations [52].

Further, we sketch out the Boltzmann–Grad scaling behavior of the non-
Markovian Enskog kinetic equation (4.9) and reduced state functional (4.2).

Taking into account the validity of assumption (3.20) for the initial one-
particle distribution function (3.19), in that case for a finite time
interval, the Boltzmann–Grad limit of dimensionless solution (4.8) of the
Cauchy problem of the non-Markovian Enskog kinetic equation (4.9),(4.10)
exists in the same sense, namely

$\displaystyle\lim_{\epsilon\rightarrow
0}\big{(}\epsilon^{2}F_{1}(t,x_{1})-f_{1}(t,x_{1})\big{)}=0,$

where the limit one-particle distribution function $f_{1}(t)$ is a weak
solution of the Cauchy problem of the Boltzmann kinetic equation
(3.23),(3.24).

Taking into consideration the fact of the existence of the Boltzmann–Grad
scaling limit of a solution of the non-Markovian Enskog kinetic equation
(4.9), for reduced functionals of the state (4.2) the following statement
holds [47]:

$\displaystyle\lim\limits_{\epsilon\rightarrow
0}\big{(}\epsilon^{2s}F_{s}\big{(}t,x_{1},\ldots,x_{s}\mid
F_{1}(t)\big{)}-\prod\limits_{j=1}^{s}f_{1}(t,x_{j})\big{)}=0,$

where the limit one-particle distribution function $f_{1}(t)$ is governed by
the Boltzmann kinetic equation with hard sphere collisions (3.23). Because all
possible correlations of many hard spheres with elastic collisions are
described by reduced functionals of the state (4.2), as noted above, this
property means the propagation of the initial chaos in the Boltzmann–Grad
limit.

The proof of these statements is based on the properties of cumulants of
asymptotically perturbed groups of operators (1.6) and the explicit structure
(4.3) of the generating operators of series expansions (4.2) for reduced
functional of state and of series (4.8).

###  4.2 Dynamics of correlations governed by kinetic equations

Let the initial state be specified by a one-particle reduced correlation
function, namely, the initial state specified by a sequence of reduced
correlation functions satisfying the chaos property stated above, i.e., by the
sequence $G^{(c)}=(G_{0},G_{1}^{0},0,\ldots,0,\ldots)$. We note that such an
assumption about initial states is intrinsic to the contemporary kinetic
theory of many-particle systems [14],[15].

Since the initial data $G^{(c)}$ is completely specified only by a one-
particle correlation function, the Cauchy problem (2.75),(2.76) of the
nonlinear hierarchy for hard spheres is not a completely well-defined Cauchy
problem because the initial data is not independent for every unknown function
determined of the hierarchy of mentioned evolution equations. As a
consequence, 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 for the kinetic equation.

We formulate such a restated Cauchy problem and the sequence of the suitable
functionals. 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 (2.79), i.e.,

$\displaystyle\hskip-34.1433ptG_{1}(t,x_{1})=\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\int\limits_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{n}}dx_{2}\ldots
dx_{1+n}\,\mathfrak{A}^{\ast}_{1+n}(t,1,\ldots,n+1)\prod_{i=1}^{n+1}G_{1}^{0}(x_{i})\mathcal{X}_{\mathbb{R}^{3(n+1)}\setminus\mathbb{W}_{n+1}},$
(4.11)

as the following series expansions:

$\displaystyle\hskip-34.1433ptG_{s}\bigl{(}t,x_{1},\ldots,x_{s}\mid
G_{1}(t)\bigr{)}=$ (4.12)
$\displaystyle\hskip-14.22636pt\sum_{n=0}^{\infty}\frac{1}{n!}\,\int\limits_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{n}}dx_{s+1}\ldots
dx_{s+n}\,\mathfrak{V}_{s+n}\bigl{(}t,1,\ldots,s+n\bigr{)}\prod_{i=1}^{s+n}G_{1}(t,x_{i}),\quad
s\geq 2.$

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:

$\displaystyle\hskip-34.1433pt\mathfrak{V}_{s+n}\bigl{(}t,1,\ldots,s,s+1,\ldots,s+n\bigr{)}=n!\,\sum_{k=0}^{n}\,(-1)^{k}\,\sum_{n_{1}=1}^{n}\ldots$
(4.13) $\displaystyle\hskip-22.76219pt\sum_{n_{k}=1}^{n-n_{1}-\ldots-
n_{k-1}}\frac{1}{(n-n_{1}-\ldots-
n_{k})!}\,\hat{\mathfrak{A}}_{s+n-n_{1}-\ldots-
n_{k}}(t,1,\ldots,s+n-n_{1}-\ldots-n_{k})\times$
$\displaystyle\hskip-22.76219pt\prod_{j=1}^{k}\,\sum\limits_{\mbox{\scriptsize$\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}$}}\frac{1}{|\mathrm{D}_{j}|!}\sum_{i_{1}\neq\ldots\neq
i_{|\mathrm{D}_{j}|}=1}^{s+n-n_{1}-\ldots-
n_{j}}\,\prod_{X_{l_{j}}\subset\mathrm{D}_{j}}\,\frac{1}{|X_{l_{j}}|!}\hat{\mathfrak{A}}_{1+|X_{l_{j}}|}(t,i_{l_{j}},X_{l_{j}}),$
(4.16)

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

$\displaystyle\hskip-34.1433pt\hat{\mathfrak{A}}_{s+n}(t,1,\ldots,s+n)\doteq\mathfrak{A}^{\ast}_{s+n}(t,1,\ldots,s+n)\mathcal{X}_{\mathbb{R}^{3(s+n)}\setminus\mathbb{W}_{s+n}}\prod_{i=1}^{s+n}(\mathfrak{A}_{1}^{\ast})^{-1}(t,i),$

and notations accepted above were used.

We adduce simplest examples of generating operators (4.13):

$\displaystyle\hskip-34.1433pt\mathfrak{V}_{s}(t,1,\ldots,s)=\mathfrak{A}_{s}(t,1,\ldots,s)\mathcal{X}_{\mathbb{R}^{3s}\setminus\mathbb{W}_{s}}\prod_{i=1}^{s}(\mathfrak{A}_{1}^{\ast})^{-1}(t,i),$
$\displaystyle\hskip-34.1433pt\mathfrak{V}_{s+1}(t,1,\ldots,s,s+1)=\mathfrak{A}_{s+1}(t,1,\ldots,s+1)\mathcal{X}_{\mathbb{R}^{3(s+1)}\setminus\mathbb{W}_{s+1}}\prod_{i=1}^{s+1}(\mathfrak{A}_{1}^{\ast})^{-1}(t,i)-$
$\displaystyle\mathfrak{A}_{s}(t,1,\ldots,s)\mathcal{X}_{\mathbb{R}^{3s}\setminus\mathbb{W}_{s}}\prod_{i=1}^{s}(\mathfrak{A}_{1}^{\ast})^{-1}(t,i)\sum_{j=1}^{s}\mathfrak{A}_{2}(t,j,s+1)\times$
$\displaystyle\mathcal{X}_{\mathbb{R}^{6}\setminus\mathbb{W}_{2}}(\mathfrak{A}_{1}^{\ast})^{-1}(t,j)(\mathfrak{A}_{1}^{\ast})^{-1}(t,s+1).$

If $\|G_{1}(t)\|_{L^{1}(\mathbb{R}^{3}\times\mathbb{R}^{3})}<e^{-(3s+2)}$, for
arbitrary $t\in\mathbb{R}$ series (4.12) converges in the norm of the space
$L^{1}_{s}$ [47].

We note that in the case of initial state specified by a one-particle
correlation function the reduced correlation functionals (4.12) describe all
possible correlations generated by the dynamics of many hard spheres in terms
of a one-particle correlation function.

Thus, according to the representation (2.79) of reduced correlation functions,
the cumulant structure of their generating operators induces a generalized
cumulant structure of the generating operators for series (4.12) of reduced
correlation functionals.

In this case, the method of constructing reduced reduced correlation
functionals (4.12) is based on the application of the variation of cluster
expansions, the so-called kinetic cluster expansions [47], to generating
operators (2.40) of series representing reduced correlation functions (2.79).

Indeed, taking into account relations of kinetic cluster expansions for
scattering cumulants (4.7):

$\displaystyle\hskip-34.1433pt\widehat{\mathfrak{A}}_{s+n}(t,1,\ldots,s+n)=\sum_{n_{1}=0}^{n}\frac{n!}{(n-n_{1})!}\mathfrak{V}_{s+n-n_{1}}\big{(}t,1,\ldots,s+n-n_{1}\big{)}\times$
$\displaystyle\sum\limits_{\mbox{\scriptsize$\begin{array}[]{c}\mathrm{D}:Z=\bigcup_{l}X_{l},\\\
|\mathrm{D}|\leq
s+n-n_{1}\end{array}$}}\frac{1}{|\mathrm{D}|!}\sum_{i_{1}\neq\ldots\neq
i_{|\mathrm{D}|}=1}^{s+n-n_{1}}\,\,\prod_{X_{l}\subset\mathrm{D}}\,\frac{1}{|X_{l}|!}\,\,\widehat{\mathfrak{A}}_{1+|X_{l}|}(t,i_{l},X_{l}),$

where $\sum_{\mathrm{D}:Z=\bigcup_{l}X_{l},\,|\mathrm{D}|\leq s+n-n_{1}}$ is
the sum over all possible dissections $\mathrm{D}$ of the linearly ordered set
$Z\equiv(s+n-n_{1}+1,\ldots,s+n)$ on no more than $s+n-n_{1}$ linearly ordered
subsets, we derive the expansions of the reduced correlation functionals
$G_{s}(t,x_{1},\ldots,x_{s}\mid G_{1}(t)),\,s\geq 2$, on the basis of solution
expansions (2.79) of the hierarchy of nonlinear evolution equations (2.75).

Note that the structure of kinetic cluster expansions of scattering cumulants
of the groups of operators is similar to the structure of virial expansions of
equilibrium distribution functions, i.e., as a power series in the density.

If initial data $G_{1}^{0}\in L^{1}_{1}$, then for arbitrary $t\in\mathbb{R}$
one-particle correlation function (4.11) is a weak solution of the Cauchy
problem of the non-Markovian Enskog kinetic equation

$\displaystyle\hskip-34.1433pt\frac{\partial}{\partial
t}G_{1}(t,x_{1})=\mathcal{L}^{\ast}(1)G_{1}(t,x_{1})+\int\limits_{\mathbb{R}^{3}\times\mathbb{R}^{3}}dx_{2}\,\mathcal{L}_{\mathrm{int}}^{\ast}(1,2)G_{1}(t,x_{1})G_{1}(t,x_{2})+$
(4.18)
$\displaystyle\int\limits_{\mathbb{R}^{3}\times\mathbb{R}^{3}}dx_{2}\,\mathcal{L}_{\mathrm{int}}^{\ast}(1,2)G_{2}\bigl{(}t,x_{1},x_{2}\mid
G_{1}(t)\bigr{)},$
$\displaystyle\hskip-34.1433ptG_{1}(t,x_{1})\big{|}_{t=0}=G_{1}^{0}(x_{1}),$
(4.19)

where the first part of the collision integral in equation (4.18) 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 (4.12) which describes all possible correlations that are
created by hard-sphere dynamics and by the propagation of initial correlations
related to the forbidden configurations.

In the paper [47], similar statements were proved for the state evolution 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 (4.12) of the reduced correlation functionals are determined by the
$(s+n)th$-order generating operator (4.3) in contradistinction to the
expansions of reduced distribution functionals of the state constructed in
[47] which are determined by the $(1+n)th$-order generating operator (4.3).

Thus, for the initial state specified by a one-particle correlation function,
the evolution of all possible states of the system of hard spheres can be
described without any approximations within the framework of a one-particle
correlation function governed by the non-Markovian Enskog-type kinetic
equation (4.18), and by a sequence of explicitly defined functionals (4.12) of
its solution.

##  5 Conclusion

In conclusion, the challenges of the evolution of many hard spheres with
inelastic collisions will be reviewed, as will some applications of the
methods outlined above to complex systems of various natures.

###  5.1 On the dynamics of inelastic collisions

According to contemporary concept [96],[99] on a microscopic scale, the
characteristic properties of granular media are determined by dissipative
collisional dynamics and can be described as the evolution of a system of many
hard spheres with inelastic collisions.

Since the characteristic features of the collective behavior of inelastically
colliding particles in one-dimensional space reflect the main properties of
granular gases, an approach to the rigorous derivation of the Boltzmann-type
equation for one-dimensional granular gases will be presented below. We note
that, in contrast to the system of hard rods with inelastic collisions, in
one-dimensional space the evolution of hard rods with elastic collisions is
trivial in the Boltzmann–Grad scaling limit; it is known as so-called free
molecular motion or the Knudsen flow [14].

In the case of a one-dimensional granular gas for $t\geq 0$ in dimensionless
form the Cauchy problem of the non-Markovian generalized Enskog kinetic
equation (4.9),(4.10) takes the form [37],[42],[43]:

$\displaystyle\hskip-34.1433pt\frac{\partial}{\partial
t}F_{1}(t,q_{1},p_{1})=-p_{1}\frac{\partial}{\partial
q_{1}}F_{1}(t,q_{1},p_{1})+$ (5.1)
$\displaystyle\int\limits_{0}^{\infty}dP\,P\big{(}\frac{1}{(1-2\varepsilon)^{2}}\,F_{2}(t,q_{1},p_{1}^{\diamond}(p_{1},P),q_{1}-\epsilon,p_{2}^{\diamond}(p_{1},P)\mid
F_{1}(t))-$ $\displaystyle F_{2}(t,q_{1},p_{1},q_{1}-\epsilon,p_{1}+P\mid
F_{1}(t))\big{)}+$
$\displaystyle\int\limits_{0}^{\infty}dP\,P\big{(}\frac{1}{(1-2\varepsilon)^{2}}\,F_{2}(t,q_{1},\tilde{p}_{1}^{\diamond}(p_{1},P),q_{1}+\epsilon,\tilde{p}_{2}^{\diamond}(p_{1},P)\mid
F_{1}(t))-$ $\displaystyle F_{2}(t,q_{1},p_{1},q_{1}+\epsilon,p_{1}-P\mid
F_{1}(t))\big{)},$
$\displaystyle\hskip-34.1433ptF_{1}(t)|_{t=0}=F_{1}^{\epsilon,0},$ (5.2)

where $\varepsilon=\frac{1-e}{2}\in[0,\frac{1}{2})$ and $e\in(0,1]$ is a
restitution coefficient, $\epsilon>0$ is a scaling parameter (the ratio of a
hard sphere diameter (the length) $\sigma>0$ to the mean free path), the
collision integral is determined by reduced functional (4.2) of the state
$F_{1}(t)$ in the case of $s=2$ and the expressions:

$\displaystyle
p_{1}^{\diamond}(p_{1},P)=p_{1}-P+\frac{\varepsilon}{2\varepsilon-1}\,P,$
$\displaystyle
p_{2}^{\diamond}(p_{1},P)=p_{1}-\frac{\varepsilon}{2\varepsilon-1}\,P$

and the expressions

$\displaystyle\tilde{p}_{1}^{\diamond}(p_{1},P)=p_{1}+P-\frac{\varepsilon}{2\varepsilon-1}\,P,$
$\displaystyle\tilde{p}_{2}^{\diamond}(p_{1},P)=p_{1}+\frac{\varepsilon}{2\varepsilon-1}\,P,$

are transformed pre-collision momenta of inelastically colliding particles in
a one-dimensional space.

The solution of the Cauchy problem (5.1),(5.2) is represented by the following
series:

$\displaystyle\hskip-34.1433ptF_{1}^{\epsilon}(t,x_{1})=\sum\limits_{n=0}^{\infty}\frac{1}{n!}\int\limits_{\mathbb{R}^{n}\times\mathbb{R}^{n}}dx_{2}\ldots
dx_{n+1}\,\mathfrak{A}_{1+n}^{\ast}(t)\prod_{i=1}^{n+1}F_{1}^{\epsilon,0}(x_{i})\mathcal{X}_{\mathbb{R}^{(1+n)}\setminus\mathbb{W}_{1+n}},$
(5.3)

where the generating operator $\mathfrak{A}_{1+n}^{\ast}(t)$ is the $(1+n)$th-
order cumulant (2.21) of the semigroups of operators (1.6) of inelastically
colliding hard rods in a one-dimensional space. Let the initial one-particle
distribution function satisfy the following condition:
$|F_{1}^{\epsilon,0}(x_{1})|\leq Ce^{\textstyle-\frac{\beta}{2}{p^{2}_{1}}},$
where $\textstyle{\beta}>0$ is a parameter and $C<\infty$ is some constant.
Then every term of series (5.3) exists; for a finite time interval it is the
uniformly convergent series with respect to $x_{1}$ from an arbitrary compact,
and function (5.3) is a weak solution of the Cauchy problem (5.1),(5.2) of the
non-Markovian Enskog-type equation with inelastic collisions.

We assume that, in the sense of weak convergence, there exists a limit

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

Then, for a finite time interval, the Boltzmann–Grad limit of solution (5.3)
of the Cauchy problem of the non-Markovian Enskog-type equation for a one-
dimensional granular gas (5.1) exists in the sense of a weak convergence

$\displaystyle\mathrm{w-}\lim\limits_{\epsilon\rightarrow
0}\big{(}F_{1}^{\epsilon}(t,x_{1})-f_{1}(t,x_{1})\big{)}=0,$ (5.4)

where the limit of one-particle distribution function (5.3) is determined by
the following series uniformly convergent on an arbitrary compact set

$\displaystyle\hskip-25.60747ptf_{1}(t,x_{1})=\sum\limits_{n=0}^{\infty}\frac{1}{n!}\int\limits_{\mathbb{R}^{n}\times\mathbb{R}^{n}}dx_{2}\ldots
dx_{n+1}\,\mathfrak{A}_{1+n}^{0}(t)\prod_{i=1}^{n+1}f_{1}^{0}(x_{i}),$ (5.5)

and the generating operator
$\mathfrak{A}_{1+n}^{0}(t)\equiv\mathfrak{A}_{1+n}^{0}(t,1,\ldots,n+1)$ is the
$(n+1)th$-order cumulant of semigroups (1.6) of point particles with inelastic
collisions. For $t\geq 0$ an infinitesimal generator of this semigroup of
operators is determined by the operator:

$\displaystyle\hskip-14.22636pt(\mathcal{L}_{n}^{\ast,0}f_{n})(x_{1},\ldots,x_{n})=-\sum_{j=1}^{n}p_{j}\,\frac{\partial}{\partial
q_{j}}f_{n}(x_{1},\ldots,x_{n})+$
$\displaystyle\sum_{j_{1}<j_{2}=1}^{n}|p_{j_{2}}-p_{j_{1}}|\big{(}\frac{1}{(1-2\varepsilon)^{2}}f_{n}(x_{1},\ldots,x^{\diamond}_{j_{1}},\ldots,x^{\diamond}_{j_{2}},\ldots,x_{n})-f_{n}(x_{1},\ldots,x_{n})\big{)}\delta(q_{j_{1}}-q_{j_{2}}),$

where $x_{j}^{\diamond}\equiv(q_{j},p_{j}^{\diamond})$ and the pre-collision
momenta $p^{\diamond}_{j_{1}},\,p^{\diamond}_{j_{2}}$ of inelastically
colliding particles are determined by the following expressions:

$\displaystyle
p^{\diamond}_{j_{1}}=p_{j_{2}}+\frac{\varepsilon}{2\varepsilon-1}\,(p_{j_{1}}-p_{j_{2}}),$
$\displaystyle
p_{j_{2}}^{\diamond}=p_{j_{1}}-\frac{\varepsilon}{2\varepsilon-1}\,(p_{j_{1}}-p_{j_{2}}).$

For $t\geq 0$ the limit one-particle distribution function represented by
series (5.5) is a weak solution of the Cauchy problem of the Boltzmann-type
kinetic equation of point particles with inelastic collisions [43]

$\displaystyle\hskip-34.1433pt\frac{\partial}{\partial
t}f_{1}(t,q,p)=-p\,\frac{\partial}{\partial q}f_{1}(t,q,p)+$ (5.6)
$\displaystyle\hskip-19.91692pt\int\limits_{-\infty}^{+\infty}dp_{1}\,|p-p_{1}|\big{(}\frac{1}{(1-2\varepsilon)^{2}}\,f_{1}(t,q,p^{\diamond})\,f_{1}(t,q,p^{\diamond}_{1})-f_{1}(t,q,p)\,f_{1}(t,q,p_{1})\big{)}+\sum_{n=1}^{\infty}\mathcal{I}^{(n)}_{0}.$

In kinetic equation (5.6) the remainder
$\sum_{n=1}^{\infty}\mathcal{I}^{(n)}_{0}$ of the collision integral is
determined by the following expressions:

$\displaystyle\hskip-34.1433pt\mathcal{I}^{(n)}_{0}\equiv\frac{1}{n!}\int\limits_{0}^{\infty}dP\,P\,\int\limits_{\mathbb{R}^{n}\times\mathbb{R}^{n}}dq_{3}dp_{3}\ldots$
$\displaystyle
dq_{n+2}dp_{n+2}\mathfrak{V}_{1+n}(t)\big{(}\frac{1}{(1-2\varepsilon)^{2}}f_{1}(t,q,p_{1}^{\diamond}(p,P))f_{1}(t,q,p_{2}^{\diamond}(p,P))-$
$\displaystyle
f_{1}(t,q,p)f_{1}(t,q,p+P)\big{)}\prod_{i=3}^{n+2}f_{1}(t,q_{i},p_{i})+$
$\displaystyle\int\limits_{0}^{\infty}dP\,P\,\int\limits_{\mathbb{R}^{n}\times\mathbb{R}^{n}}dq_{3}dp_{3}\ldots$
$\displaystyle
dq_{n+2}dp_{n+2}\mathfrak{V}_{1+n}(t)\big{(}\frac{1}{(1-2\varepsilon)^{2}}f_{1}(t,q,\tilde{p}_{1}^{\diamond}(p,P))f_{1}(t,q,\tilde{p}_{2}^{\diamond}(p,P))-$
$\displaystyle
f_{1}(t,q,p)f_{1}(t,q,p-P)\big{)}\prod_{i=3}^{n+2}F_{1}(t,q_{i},p_{i}),$

where the generating operators
$\mathfrak{V}_{1+n}(t)\equiv\mathfrak{V}_{1+n}(t,\\{1,2\\},3,\ldots,n+2),\,n\geq
0,$ of the series for a collision integral are represented by expansions (4.3)
with respect to the cumulants of semigroups of scattering operators of point
hard rods with inelastic collisions in a one-dimensional space

$\displaystyle\widehat{S}_{n}^{0}(t,1,\ldots,n)\doteq
S_{n}^{\ast,0}(t,1,\ldots,s)\prod_{i=1}^{n}(S_{1}^{\ast,0})^{-1}(t,i).$

In fact, series expansions for the collision integral of the non-Markovian
Enskog equation for a granular gas (5.6) or solution (5.3) are represented as
the power series over the density, so that the terms
$\mathcal{I}^{(n)}_{0},\,n\geq 1,$ of the collision integral in kinetic
equation (5.6) are corrections with respect to the density to the Boltzmann
collision integral of one-dimensional granular gases formulated in the paper
[96].

Since the scattering operator of point hard rods is an identity operator in
the approximation of elastic collisions, namely, in the limit
$\varepsilon\rightarrow 0$, the collision integral of the Boltzmann kinetic
equation (5.6) in a one-dimensional space is identical to zero. In the quasi-
elastic approximation [96] the limit one-particle distribution function (5.5)

$\displaystyle\lim_{\varepsilon\rightarrow 0}\varepsilon
f_{1}(t,q,p)=f^{0}(t,q,p),$

satisfies the nonlinear friction kinetic equation for one-dimensional granular
gases [96]:

$\displaystyle\hskip-34.1433pt\frac{\partial}{\partial
t}f^{0}(t,q,p)=-p\,\frac{\partial}{\partial
q}f^{0}(t,q,p)+\frac{\partial}{\partial
p}\int\limits_{-\infty}^{\infty}dp_{1}\,|p_{1}-p|\,(p_{1}-p)\,f^{0}(t,q,p_{1})f^{0}(t,q,p).$

Taking into consideration the result (5.4) on the Boltzmann–Grad asymptotic
behavior of the non-Markovian Enskog equation (5.1), for reduced functionals
of the state (4.2) in a one-dimensional space, the following statement is true
[43]:

$\displaystyle\hskip-34.1433pt\mathrm{w-}\lim\limits_{\epsilon\rightarrow
0}\big{(}F_{s}\big{(}t,x_{1},\ldots,x_{s}\mid
F_{1}^{\epsilon}(t)\big{)}-f_{s}\big{(}t,x_{1},\ldots,x_{s}\mid
f_{1}(t)\big{)}\big{)}=0,\quad s\geq 2,$ (5.7)

where in equality (5.7) the limit reduced functionals of the limit one-
particle distribution function (5.5) are determined by the series expansions
with a structure similar to series (4.2) and the generating operators
represented by expansions (4.3) over the cumulants of semigroups of scattering
operators of point hard rods with inelastic collisions in a one-dimensional
space.

As mentioned above, in the case of a system of hard rods with elastic
collisions, the limit reduced functionals of the state are the product of the
limit one-particle distribution functions, describing the free motion of point
particles.

Thus, the Boltzmann–Grad asymptotic behavior of solution (5.3) of the non-
Markovian Enskog equation (5.1) is governed by the Boltzmann kinetic equation
(5.6) for a one-dimensional granular gas.

We emphasize that the Boltzmann-type equation (5.6) describes the memory
effects in a one-dimensional granular gas. In addition, the limits
$f_{s}\big{(}t,x_{1},\ldots,x_{s}\mid f_{1}(t)\big{)},\,s\geq 2,$ of reduced
functionals of the state which are defined above, describe the process of the
propagation of initial chaos in a one-dimensional granular gas, or,in other
words, the process of creating correlations in a system of hard rods with
inelastic collisions.

It should be noted that the Boltzmann–Grad asymptotic behavior of the non-
Markovian Enskog equation with inelastic collisions in a multidimensional
space is analogous to the Boltzmann–Grad asymptotic behavior of a hard sphere
system with the elastic collisions [43], i.e., it is governed by the Boltzmann
equation for a granular gas [98],[99], and the asymptotic behavior of the
reduced functionals of the state (4.2) is described by the product of one-
particle distribution functions of its solution, i.e., describes the
propagation of initial chaos.

###  5.2 Some bibliographic notes on collisional dynamics

Above, it was studied systems of identical colliding particles, which are
described by means of functions of observables and distribution functions,
which are symmetrical with respect to arbitrary permutations of their
arguments. In papers [32],[33],[67],[70],[82], the theory of the hierarchies
of evolution equations for systems of many colliding particles described by
non-symmetric functions was developed. An example of such a system is a one-
dimensional system of particles interacting with their nearest neighbors, so-
called non-symmetric systems of particles [32].

As is known, many-entity systems of active soft condensed matter are dynamic
systems exhibiting a collective behavior that differs from the statistical
behavior of ordinary gases. To describe the nature of entities (or self-
propelled particles), in the paper [78], collision dynamics based on Markov
jump processes, which should reflect the internal properties of living
creatures, were proposed. In works [36],[45] an approach was developed to
describe the collective behavior of complex systems of mathematical biology
within the framework of the evolution of observables of many colliding
stochastic processes, and the dual Vlasov hierarchy was constructed in the
mean field approximation. This representation of the kinetic evolution seems,
in fact, to be the direct mathematically fully consistent formulation modeling
the kinetic evolution of biological systems, since the notion of the state is
more subtle and is an implicit characteristic of populations of living
creatures. In the paper [36] the processes of creation of correlations
generated by the dynamics of active soft matter and propagation of initial
correlations have also been described by means of the non-Markovian
generalized kinetic equation with initial correlations, and, in particular, in
the mean-field scaling approximation, the Vlasov-type kinetic equation for
many colliding stochastic processes was constructed.

The study of systems of colliding particles in interaction with the
environment, the so-called open systems, involves a number of unsolved
fundamental problems. One of them is related to the challenge of the origin of
stochastic behavior in dynamical systems of many particles. In papers
[48],[49],[50], based on the approaches to the derivation of kinetic equations
outlined above, a generalization of the Fokker–Planck equation for open
systems of colliding particles was justified.

In previous decades, a lot of work has been performed on discrete-velocity
models of the Boltzmann equation, which are of significant conceptual interest
for the kinetic theory of gases and, at the same time, represent a fascinating
mathematical subject [84]. In connection with this topic of research, we note
the works [39],[40],[41],[54], in which the discrete-velocity model was
studied, related to the problem of deriving a model of the Enskog discrete-
velocity kinetic equation.

An overview of some modern applications of kinetic equations to the
description of non-equilibrium processes in complex systems of various natures
is presented in the monograph [23].

###  5.3 Outlook

The purpose of this review was to analyze the development and current advances
of the theory of evolution equations for systems of many colliding particles,
in particular, kinetic equations and their relations to the fundamental
equations that describe the laws of nature.

The problem of constructing a solution to the Cauchy problem for hierarchies
of evolution equations of observables (2.1) and the state (2.16) of a system
of hard spheres with elastic collisions for initial data belonging to some
functional spaces is considered. As was established, solutions of hierarchies
of evolution equations are determined by groups of operators, which are
represented by expansions over the groups of particles whose evolution is
described by cumulants of the corresponding order of the groups of operators
of the Liouville equations. Due to the fact that the cumulants of the groups
of operators are determined by cluster expansions of the groups of operators
of the Liouville equations, in the corresponding function spaces there are
different representations for solutions to the hierarchies of evolution
equations. These cluster expansions of the groups of operators underlie the
classification of possible solution representations to the Cauchy problem for
the hierarchies of evolution equations of many colliding particles.

To describe the evolution of the state of a many-particle system, there is an
alternative approach that is based on the dynamics of correlations. In this
approach, a state of finitely many hard spheres is described with the
employment of functions determined by the cluster expansions of the
probability distribution functions that are governed by the so-called
Liouville hierarchy (2.35). It was above established that the constructed
dynamics of correlation underlie the description of the dynamics of infinitely
many hard spheres governed by the BBGKY hierarchy for reduced distribution
functions (2.16) or the hierarchy of nonlinear evolution equations for reduced
correlation functions (2.75), i.e., of the cumulants of reduced distribution
functions. We emphasize the importance of the mathematical description of the
processes of the creation and propagation of correlations, in particular, for
numerous applications [2],[3],[86].

To describe the evolution of many hard spheres within the framework of the
evolution of states for an initial state close to ”kinetic,” i.e., a state
described in terms of the state of a typical particle, there is another
possibility: by means of the so-called non-Markovian Enskog kinetic equation
(4.9). In other words, the origin of the collective behavior of a hard-sphere
system on a microscopic scale was examined above. As already mentioned, one of
the advantages of such an approach to the derivation of kinetic equations from
underlying collisional dynamics is the opportunity to construct the kinetic
equations with initial correlations, which makes it possible to describe the
creation of correlations and propagation of initial correlations. Another
advantage of this approach is related to the rigorous derivation of the
Boltzmann equation (3.23) with higher-order corrections to the canonical term
of the collision integral.

Thus, the concept of cumulants of the groups of operators of Liouville
equations underlies non-perturbative expansions of solutions to hierarchies of
fundamental evolution equations that describe the evolution of observables and
a state of many colliding particles, as well as underlies the kinetic
description of their collective behavior. We note that for quantum many-
particle systems the concept of cumulants of groups of operators is considered
in review [38].

In the paper, possible approaches to the rigorous derivation [35] and
justification [62],[63] of the kinetic equations for many colliding particles
were considered. One of them is an approach to the description of the kinetic
evolution within the framework of the evolution of the observables of many
colliding particles [51]. The advances of the method based on the dual
Boltzmann hierarchy (3.17) are the opportunity to construct kinetic equations
(3.29), taking into account the correlations of particles of the initial
state, and the description of the process of propagation of initial
correlations in scaling approximations (3.33).

The paper [69] considered the challenge of deriving hydrodynamic equations
from the dual BBGKY hierarchy for reduced observed microscopic phase
densities. We notice that the rigorous derivation of hydrodynamic equations
from the dynamics of many colliding particles is still an open problem.
Regarding the classical problem of rigorous derivation of the hydrodynamic
equations from the Boltzmann kinetic equation in scaling limits, we refer to
the books [72],[93].

Acknowledgements. Glory to Ukraïne!

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