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Cherkasy Univ. Bulletin: Phys. Math. Sci., No.1, pp. 3--12, 2017
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kinetic equation, correlation function, scaling limit, active soft matter
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\begin{document}

\FirstPageHeading{V.I. Gerasimenko}

\AuthorNameForHeading{V.I. Gerasimenko}

\ShortArticleName{Kinetic evolution of correlations}

\ArticleName{\textcolor{blue!50!black}{On Propagation of Initial Correlations\\ in Active Soft Condensed Matter}}

\Author{V.I. Gerasimenko$^\ast$\footnote{E-mail: \emph{gerasym@imath.kiev.ua}}}

\Address{$^\ast$Institute of Mathematics of the NAS of Ukraine,\\
                3,Tereshchenkivs'ka Str., 01601, Kyiv-4, \\ Ukraine}

\bigskip

\Abstract{For collisional dynamics, modeling the collective behavior of complex systems
of mathematical biology, the process of propagation of initial correlations is described.
The developed approach is based on the construction of a mean field limit for a solution
of the Cauchy problem of the nonlinear BBGKY hierarchy for marginal correlation functions.
}

\bigskip

\Keywords{kinetic equation, correlation function, scaling limit, active soft matter.}
\vspace{2pc}
\Classification{05.20.Dd; 05.90.+m; 87.10.-e; 02.30.Jr; 02.50.Ey; 87.23.Ge; 87.17.-d.}

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\newpage
%\vphantom{math}
\textcolor{blue!50!black}{\tableofcontents}
%\newpage
\vspace{0.5cm}

\textcolor{blue!50!black}{\section{Introduction}}

As well known a main consistent approach to the problem of the derivation of kinetic
equations from underlying large particle dynamics was formulated by M.M. Bogolyubov
\cite{B46}. The rigorous derivation of kinetic equations for many-particle systems
in condensed states is an open problem so far \cite{V},\cite{CGP97}.

In modern researches the main approach to the problem of the rigorous derivation of kinetic
equations consists in the construction of scaling asymptotics of a solution of evolution
equations which describe the evolution of states of large particle systems, for example,
a mean field limit of a solution of the BBGKY (Bogolyubov--Born--Green--Kirkwood--Yvon)
hierarchy constructed by methods of the perturbation theory \cite{V},\cite{CGP97}.

It should be noted modern wide applications of kinetic equations to the description of
collective processes of various nature, in particular, the collective behavior of complex
systems of biology. We emphasize that the considerable advance in modeling of the kinetic
evolution of systems of mathematical biology with a large number of constituents (entities),
for example, systems of large number of cells, is recently observed \cite{L08}-\cite{CDW}
(see also references cited therein).

In this paper we consider the problem of a rigorous description of the evolution of states within
the framework of marginal correlation functions governed by the nonlinear BBGKY hierarchy for a
large system of interacting stochastic processes of collisional kinetic theory \cite{L11}, modeling
the microscopic evolution of active soft condensed matter \cite{ALBA},\cite{MJRLPRAS}. The developed
approach to the derivation of kinetic equations is based on the construction of a mean field limit
of a nonperturbative solution of the Cauchy problem of the nonlinear BBGKY hierarchy. One of the
advantages of a such approach is the opportunity to describe the processes of propagation of initial
correlations in scaling limits, in particular, that can characterize the condensed states of soft matter.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\textcolor{blue!50!black}{\section{On collisional dynamics of active soft condensed matter}}

The many-constituent systems of active soft condensed matter \cite{ALBA},\cite{MJRLPRAS} are
dynamical systems displaying a collective behavior which differs from the statistical behavior
of usual many-particle systems \cite{CGP97}. To specify a nature of constituents (entities)
we consider the dynamical system suggested in papers \cite{L08},\cite{L11} which is based on
the Markov jump processes that must represent the intrinsic properties of living creatures
(self-propelled particles).

We consider a system of entities of various $M$ subpopulations introduced in paper \cite{L11}
in case of non-fixed, i.e. arbitrary, but finite average number of entities. Every $i$th
entity is characterized by: $\textbf{u}_i=(j_i,u_i)\in\mathcal{J}\times\mathcal{U}$, where
$j_i\in\mathcal{J}\equiv(1,\ldots,M)$ is a number of its subpopulation, and values
$u_i\in\mathcal{U}\subset\mathbb{R}^{d}$ is its microscopic characteristics.

The stochastic dynamics of $n$ entities of various subpopulations is described by
the semigroup $e^{t\Lambda_n^\ast}$ of the Markov jump processes defined on the space
$L^{1}_{n}$ of the integrable functions $f_n(\textbf{u}_1,\ldots,\textbf{u}_n)$
defined on $(\mathcal{J}\times\mathcal{U})^n$, that are symmetric with respect to
permutations of the arguments $\textbf{u}_1,\ldots,\textbf{u}_n$, and equipped with
the norm: $$\|f_n\|_{L^{1}_n}=\sum_{j_1\in\mathcal{J}}\ldots\sum_{j_n\in\mathcal{J}}\,
\int_{\mathcal{U}^n}du_1\ldots du_{n}|f_n(\textbf{u}_1,\ldots,\textbf{u}_n)|.$$
The infi\-nitesimal generator $\Lambda_n^\ast$ of this semigroup (the generator of the
Kolmogorov backward equation for states of $n$ entities) is defined on the space
$L^{1}_{n}$ and it has the following structure \cite{L11}:
\begin{eqnarray}\label{gen_state_gener}
   &&\hskip-7mm (\Lambda^\ast_n f_n)(\textbf{u}_1,\ldots,\textbf{u}_n)=
         \sum_{m=1}^M \varepsilon^{m-1}\sum_{i_1\neq\ldots\neq i_m=1}^n
        (\Lambda^{\ast[m]}(i_1,\ldots,i_m)f_n)(\textbf{u}_1,\ldots,\textbf{u}_n)\doteq\\
   &&\hskip-7mm \sum_{m=1}^M\varepsilon^{m-1}\sum_{i_1\neq\ldots\neq i_m=1}^n
     \big(\int_{\mathcal{J}\times\mathcal{U}}
     A^{[m]}(\textbf{u}_{i_1};\textbf{v},\textbf{u}_{i_2},\ldots,\textbf{u}_{i_m})a^{[m]}(\textbf{v},\nonumber\\
   &&\hskip-7mm\textbf{u}_{i_2},\ldots,\textbf{u}_{i_m})f_n(\textbf{u}_1,\ldots,
     \textbf{u}_{{i_1}-1},\textbf{v},\textbf{u}_{{i_1}+1},\ldots,\textbf{u}_n)d\textbf{v}-
     a^{[m]}(\textbf{u}_{i_1},\ldots,\textbf{u}_{i_m})f_n(\textbf{u}_1,\ldots,\textbf{u}_n)\big)\nonumber,
\end{eqnarray}
where $\varepsilon>0$ is a scaling parameter \cite{BL14}, the functions
$a^{[m]}(\textbf{u}_{i_1},\ldots,\textbf{u}_{i_m}),\,m\geq1,$ characterize the interaction
between entities, in particular, in case of $m=1$ it is the interaction of entities with an
external environment. These functions are measurable positive bounded functions on
$(\mathcal{J}\times\mathcal{U})^n$ such that:
$0\leq a^{[m]}(\textbf{u}_{i_1},\ldots,\textbf{u}_{i_m})\leq a^{[m]}_*$,
where $a^{[m]}_*$ is some constant. The functions
$A^{[m]}(\textbf{v};\textbf{u}_{i_1},\ldots,\textbf{u}_{i_m}),\,m\geq1$, are measurable positive
integrable functions which describe the probability of the transition of the $i_1$ entity in the
microscopic state $u_{i_1}$ to the state $v$ as a result of the interaction with entities in the
states $u_{i_2},\ldots,u_{i_m}$ (in case of $m=1$ it is the interaction with an external environment).
The functions $A^{[m]}(\textbf{v};\textbf{u}_{i_1},\ldots,\textbf{u}_{i_m}),\,m\geq1$, satisfy
the conditions: $ \int_{\mathcal{J}\times\mathcal{U}}A^{[m]}(\textbf{v};\textbf{u}_{i_1},\ldots,
\textbf{u}_{i_m})d\textbf{v}=1, \quad m\geq1$. We refer to paper \cite{L11}, where examples of the
functions $a^{[m]}$ and $A^{[m]}$ are given in the context of biological systems.

In case of $M=1$ generator (\ref{gen_state_gener}) has the form $\sum_{i_1=1}^n\Lambda^{[1]}_n(i_1)$
and it describes the free stochastic evolution of entities, i.e. the evolution of self-propelled particles.
The case of $M=m\geq2$ corresponds to a system with the $m$-body interaction of entities in the sense
accepted in kinetic theory \cite{CGP97}. The $m$-body interaction of entities is the distinctive property
of biological systems in comparison with many-particle systems, for example, gases of atoms with a pair
interaction potential.

On the space $L^{1}_{n}$ the one-parameter mapping $e^{t\Lambda^\ast_n}$ is a bounded strong continuous
semigroup of operators.

Further we restrict ourself by the case of $M=2$ subpopulations to simplify the cumbersome formulas.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\textcolor{blue!50!black}{\section{Dynamics of correlations}}

The evolution of all possible states of a system of non-fixed number of the Markov jump processes
within the framework of dynamics of correlations \cite{Pri},\cite{GP} is described by means of
the sequence $g(t)=(1,g_1(t,\textbf{u}_1),\ldots,g_{n}(t,\textbf{u}_1,\ldots,\textbf{u}_n),\ldots)$
of correlation functions governed by the following Liouville hierarchy (hierarchy of the Kolmogorov
backward equations) \cite{GP},\cite{G12}:
\begin{eqnarray}\label{KE2}
       &&\frac{\partial}{\partial t}g_{n}(t,\textbf{u}_1,\ldots,\textbf{u}_n)
          =(\Lambda^\ast_ng_n(t))(\textbf{u}_1,\ldots,\textbf{u}_n)+\\
       &&+\varepsilon\sum\limits_{\mathrm{P}:\,(1,\ldots,n)=X_{1}\bigcup X_2}\,
         \sum\limits_{i_{1}\in X_{1}}\sum\limits_{i_{2}\in X_{2}}\Lambda^{\ast[2]}(i_1,i_2)
         g_{|X_{1}|}(t,\textbf{U}_{1})g_{|X_{2}|}(t,\textbf{U}_{2}),\quad n\geq1,\nonumber
\end{eqnarray}
where $\sum_{\mathrm{P}:\,(1,\ldots,n)=X_{1}\bigcup X_2}$ is the sum over all possible partitions
$\mathrm{P}$ of the set of indexes $(1,\ldots,n)$ into two nonempty mutually disjoint subsets $X_1, X_2$
and the arguments $\textbf{U}_{i}$ of the function $g_{|X_{i}|}$ correspond to indexes from the set $X_i$.

For initial states $g(0)=(g_0,g_{1}^{0,\varepsilon},\ldots,g_{n}^{0,\varepsilon},\ldots)$ from the space
$L^{1}=\oplus^{\infty}_{n=0}L^{1}_{n}$ a nonperturbative solution of the Cauchy problem of hierarchy
(\ref{KE2}) is represented as follows:
\begin{eqnarray}\label{ghs}
   &&g(t,\textbf{u}_1,\ldots,\textbf{u}_n)=\mathcal{G}(t;1,\ldots,n\mid g(0)), \quad n\geq1,
\end{eqnarray}
where on the space $L^{1}_n$ the nonlinear one-parameter mapping $\mathcal{G}(t;1,\ldots,n\mid \bullet)$
is defined by the expansion \cite{GP}
\begin{eqnarray}\label{rozvNh}
    &&\mathcal{G}(t;1,\ldots,n\mid f)\doteq\sum\limits_{\mathrm{P}:\,(1,\ldots,n)=\bigcup_j X_j}
      \mathfrak{A}_{|\mathrm{P}|}(t,\{X_1\},\ldots,\{X_{|\mathrm{P}|}\})
      \prod_{X_j\subset \mathrm{P}}f_{|X_j|}(\textbf{U}_j),\quad n\geq1,
\end{eqnarray}
in which the generating operator is the $|\mathrm{P}|th$-order cumulant of groups of operators
$e^{t\Lambda^\ast_n},\, n\geq1,$
\begin{eqnarray} \label{cumulantP}
   &&\hskip-12mm \mathfrak{A}_{|\mathrm{P}|}(t,\{X_1\},\ldots,\{X_{|\mathrm{P}|}\})\doteq
   \sum\limits_{\mathrm{P}^{'}:\,(\{X_1\},\ldots,\{X_{|\mathrm{P}|}\})=\bigcup_k Z_k}
       (-1)^{|\mathrm{P}^{'}|-1}({|\mathrm{P}^{'}|-1})!
      \prod\limits_{Z_k\subset\mathrm{P}^{'}}e^{t\Lambda^\ast_{|\theta(Z_{k})|}},
\end{eqnarray}
the symbol $\sum_{\mathrm{P}:\,(1,\ldots,n)=\bigcup_j X_j}$ means the sum over all
possible partitions $\mathrm{P}$ of the set $(1,\ldots,n)$ into $|\mathrm{P}|$ nonempty
mutually disjoint subsets $X_j$, the set $(\{X_1\},\ldots,\{X_{|\mathrm{P}|}\})$ consists
from elements of which are subsets $X_j\subset (1,\ldots,s)$ and the declusterization
mapping $\theta$ is defined by equality:
$\theta(\{X_1\},\ldots,\{X_{|\mathrm{P}|}\})\doteq(1,\ldots,n)$.

We note that in case of the absence of correlations between the Markov jump processes 
at initial time, i.e. initial states
$g(0)=(0,g_{1}^{0,\varepsilon},0,\ldots,0,\ldots)$, satisfying a chaos condition \cite{CGP97},
solution (\ref{ghs}) of the Cauchy problem of the Liouville hierarchy (\ref{KE2})
is represented by the following expansions:
\begin{eqnarray*}\label{gth}
   &&g_{s}(t,\textbf{u}_1,\ldots,\textbf{u}_s)=\mathfrak{A}_{s}(t,1,\ldots,s)\,
               \prod\limits_{i=1}^{s}g_{1}^{0,\epsilon}(\textbf{u}_i),\quad s\geq1,
\end{eqnarray*}
where the $sth$-order cumulant of groups of operators $e^{t\Lambda^\ast_n},\, n\geq1,$ 
is determined by the expansion
\begin{eqnarray*}\label{cumcp}
   &&\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}}e^{t\Lambda^\ast_{|X_i|}},
\end{eqnarray*}
and notations accepted in formula (\ref{cumulantP}) is used.

The evolution of states of large particle systems can be also described within the
framework of marginal ($s$-particle) correlation functions governed by the fundamental
evolution equations known as the nonlinear BBGKY hierarchy \cite{G12},\cite{G17}.
We note that macroscopic characteristics of fluctuations of mean values of observables
are determined by means of marginal correlation functions \cite{B46}.

The marginal correlation functions are defined within the framework of solution (\ref{ghs})
of the Cauchy problem of hierarchy (\ref{KE2}) by the following series expansions:
\begin{eqnarray}\label{Gexpsg}
   &&G_{s}(t,\textbf{u}_1,\ldots,\textbf{u}_s)\doteq\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,
      \int\limits_{(\mathcal{J}\times\mathcal{U})^n}d\textbf{u}_{s+1}\ldots
      d\textbf{u}_{s+n}\,\mathcal{G}(t;1,\ldots,s+n\mid g(0)),\quad s\geq1.
\end{eqnarray}

According to the estimate: $$\big\|\mathcal{G}(t;1,\ldots,s\mid f)\big\|_{L^{1}_{s}}\leq s!e^{2s}c^{s},$$
where $c\equiv e^{3}\max(1,\max_{\mathrm{P}:\,(1,\ldots,s)=\bigcup_iX_i}\|f_{|X_{i}|}\|_{L^{1}_{|X_{i}|}})$,
series (\ref{Gexpsg}) exists and the following estimate holds:
$\big\|G_s(t)\big\|_{L^{1}_{s}}\leq s!(2e^2)^s\emph{c}^s\sum_{n=0}^{\infty}(2e^2)^n\emph{c}^n$.

Then, according to definition (\ref{Gexpsg}), the evolution of all possible states of a system
of non-fixed number of stochastic processes defined above can be described by means of the sequence
$G(t)=(1,G_1(t),G_2(t),\ldots,G_s(t),\ldots)\in L^{1}=\oplus_{n=0}^{\infty}L^{1}_n$
of marginal correlation functions governed by the Cauchy problem of the nonlinear BBGKY
hierarchy: \cite{G12},\cite{G17}:
\begin{eqnarray}
 \label{gBigfromDFBa}
   &&\hskip-5mm\frac{\partial}{\partial t}G_s(t,\textbf{u}_1,\ldots,\textbf{u}_s)=
   (\Lambda^\ast_s G_s(t))(\textbf{u}_1,\ldots,\textbf{u}_s)+\\
   &&+\varepsilon\sum\limits_{\mathrm{P}:\,(1,\ldots,s)=X_{1}\bigcup X_2}\,
         \sum\limits_{i_{1}\in X_{1}}\sum\limits_{i_{2}\in X_{2}}\Lambda^{\ast[2]}(i_1,i_2)
         G_{|X_{1}|}(t,\textbf{U}_{1})G_{|X_{2}|}(t,\textbf{U}_{2})+\nonumber\\
     &&\varepsilon\int\limits_{\mathcal{J}\times\mathcal{U}}d\textbf{u}_{s+1}
            \sum_{i=1}^{s}\Lambda^{\ast[2]}(i,s+1)\big(G_{s+1}(t,\textbf{u}_1,\ldots,\textbf{u}_{s+1})+\nonumber\\
   &&\sum_{\mbox{\scriptsize$\begin{array}{c}\mathrm{P}:(1,\ldots,s+1)=X_1\bigcup X_2,\\i\in
      X_1;s+1\in X_2\end{array}$}}G_{|X_1|}(t,\textbf{U}_{1})G_{|X_2|}(t,\textbf{U}_{2})\big),\nonumber\\
 \label{gBigfromDFBai}
   &&\hskip-5mmG_{s}(t)\big|_{t=0}=G_{s}^{0,\varepsilon}, \quad s\geq1,
\end{eqnarray}
where $\varepsilon>0$ is a scaling parameter and we use accepted in hierarchy (\ref{KE2}) notations.

A nonperturbative solution of the Cauchy problem (\ref{gBigfromDFBa}),(\ref{gBigfromDFBai})
is represented by a sequence of marginal correlation functions:
\begin{eqnarray}\label{sss}
    &&G_{s}(t,\textbf{u}_1,\ldots,\textbf{u}_s)=\\
    &&\sum\limits_{n=0}^{\infty}\frac{1}{n!}
        \,\int\limits_{(\mathcal{J}\times\mathcal{U})^n}d\textbf{u}_{s+1}\ldots
      d\textbf{u}_{s+n}\,\mathfrak{A}_{1+n}(t;\{1,\ldots,s\},s+1,\ldots,s+n\mid G(0)),\quad s\geq1,\nonumber
\end{eqnarray}
where we denote by $G(0)=(1,G_1^{0,\varepsilon},\ldots,G_s^{0,\varepsilon},\ldots)$
a sequence of initial marginal correlation functions. The generating operator $\mathfrak{A}_{1+n}(t\mid \bullet)$
of series expansion (\ref{sss}) is the $(1+n)th$-order cumulant of groups of nonlinear operators (\ref{rozvNh})
of the Liouville hierarchy (\ref{KE2})
\begin{eqnarray} \label{cc}
   &&\hskip-8mm \mathfrak{A}_{1+n}(t;\{1,\ldots,s\},s+1,\ldots,s+n\mid G(0))\doteq\\
   &&\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),\nonumber
\end{eqnarray}
where the composition of mappings (\ref{rozvNh}) of corresponding noninteracting groups of interacting
stochastic processes is denoted by
$\mathcal{G}(t;\theta(X_1)\mid \ldots\mathcal{G}(t;\theta(X_{|\mathrm{P}|})\mid G(0))\ldots)$ \cite{G17}.

Series expansion (\ref{sss}) exists under the condition that:
$\max_{n\geq1}\big\|G_n^{0,\varepsilon}\big\|_{L^{1}_{n}}<(2e^{3})^{-1}$. If
$G_{n}^{0,\varepsilon}\in L^{1}_{0,n}\subset L^{1}_{n}$, it is a strong solution and for
arbitrary initial data $G_{n}^{0,\varepsilon}\in\ L^{1}_{n}$ it is a weak solution of the
Cauchy problem (\ref{gBigfromDFBa}),(\ref{gBigfromDFBai}).

The sequence of marginal correlation functions (\ref{sss}) describes the processes of the creation and
the propagation of correlations in a large system of interacting stochastic processes, modeling the
microscopic evolution of active soft condensed matter.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\textcolor{blue!50!black}{\section{A mean field asymptotic behavior of marginal correlation functions}}

We describe the processes of the creation and the propagation of correlations in a mean field
limit, namely, we establish the mean field asymptotic behavior \cite{G13} of constructed marginal
correlation functions (\ref{sss}) in case of the initial state specified by the one-particle
marginal correlation function with correlations
\begin{eqnarray}\label{insc}
   &&\hskip-8mm G^{(cc)}=\big(1,G_1^{0,\varepsilon}(\textbf{u}_1),g_{2}^{\varepsilon}(\textbf{u}_1,\textbf{u}_2)
      \prod_{i=1}^{2}G_1^{0,\varepsilon}(\textbf{u}_i),\ldots,
      g_{n}^{\varepsilon}(\textbf{u}_1,\ldots,\textbf{u}_n)\prod_{i=1}^{n}G_1^{0,\varepsilon}(\textbf{u}_i),\ldots\big),
\end{eqnarray}
where functions
$g_{n}^{\varepsilon}(\textbf{u}_1,\ldots,\textbf{u}_n)\equiv g_{n}^{\varepsilon},\,n\geq2$,
are specified the initial correlations. We note that a such assumption about initial states is
intrinsic for the kinetic description of many-particle systems \cite{CGP97}, on the other hand,
initial states (\ref{insc}) are typical for soft condensed matter.

We assume the existence of a mean field limit of initial one-particle marginal correlation
function in the following sense
\begin{eqnarray}\label{asic1}
   &&\hskip-8mm\lim\limits_{\varepsilon\rightarrow 0}\big\|\varepsilon G_{1}^{0,\varepsilon}-
      g_{1}^{0}\big\|_{L^{1}_{1}}=0,
\end{eqnarray}
and for initial correlations, respectively,
\begin{eqnarray}\label{asic}
   &&\lim\limits_{\varepsilon\rightarrow 0}\big\|g_{n}^{\varepsilon}-
      g_{n}\big\|_{L^{1}_n}=0, \quad n\geq2.
\end{eqnarray}
Hence a mean field limit of initial state (\ref{insc}) is specified by the sequence
of the limit marginal correlation functions
\begin{eqnarray}\label{lins}
   &&\hskip-5mm g^{(cc)}=\big(1,g_1^{0}(\textbf{u}_1),g_{2}(\textbf{u}_1,\textbf{u}_2)
      \prod_{i=1}^{2}g_1^{0}(\textbf{u}_i),\ldots,
        g_{n}(\textbf{u}_1,\ldots,\textbf{u}_n)\prod_{i=1}^{n}g_1^{0}(\textbf{u}_i),\ldots\big).
\end{eqnarray}

Under conditions (\ref{asic1}),(\ref{asic}) on initial state (\ref{insc}) there exists
a mean field limit of marginal correlation functions (\ref{sss}) in the following sense:
\begin{eqnarray*}\label{asymp}
   &&\lim\limits_{\varepsilon\rightarrow 0}\big\|\varepsilon^{s}G_{s}(t)-
      g_{s}(t)\big\|_{L^{1}_s}=0, \quad s\geq1,
\end{eqnarray*}
where for $s\geq2$ the limit marginal ($s$-particle) correlation function $g_s(t)$
is determined by the formula
\begin{eqnarray}\label{pc}
   &&\hskip-12mm g_s(t,\textbf{u}_1,\ldots,\textbf{u}_s)=
       \prod_{i_1=1}^{s}e^{t\Lambda^{\ast[1]}(i_1)}
       \sum\limits_{\mbox{\scriptsize $\begin{array}{c}\mathrm{P}:(1,\ldots,s)=\bigcup_{i}X_{i}\end{array}$}}
        \prod_{X_i\subset \mathrm{P}}g_{|X_i|}(\textbf{U}_i)
       \prod_{i_2=1}^{s}e^{-t\Lambda^{\ast[1]}(i_2)}\prod\limits_{j=1}^{s}g_{1}(t,\textbf{u}_j),
\end{eqnarray}
and, respectively, the limit one-particle correlation function $g_1(t)$ is represented
by the following series expansion:
\begin{eqnarray}\label{viterc}
   &&\hskip-12mm g_{1}(t,\textbf{u}_1)=\sum\limits_{n=0}^{\infty}\,\int\limits_0^tdt_{1}\ldots\int\limits_0^{t_{n-1}}dt_{n}
        \int_{(\mathcal{J}\times\mathcal{U})^n}d \textbf{u}_{2}\ldots d \textbf{u}_{n+1}\,
      e^{(t-t_{1})\Lambda^{\ast[1]}(1)}\times\\
    &&\hskip-7mm \times\Lambda^{\ast[2]}(1,2)
      \prod\limits_{j_1=1}^{2}e^{(t_{1}-t_{2})\Lambda^{\ast[1]}(j_1)}\ldots
      \prod\limits_{j_{n-1}=1}^{n}e^{(t_{n-1}-t_{n})\Lambda^{\ast[1]}(j_{n-1})}\times\nonumber\\
    &&\hskip-7mm \times\sum\limits_{i_{n}=1}^{n}\Lambda^{\ast[2]}(i_{n},n+1)
      \prod\limits_{j_n=1}^{n+1}e^{t_{n}\Lambda^{\ast[1]}(j_{n})}
      \sum\limits_{\mbox{\scriptsize $\begin{array}{c}\mathrm{P}:(1,\ldots,n+1)=\bigcup_{i}X_{i}\end{array}$}}
        \prod_{X_i\subset \mathrm{P}}g_{|X_i|}(\textbf{U}_i)\prod\limits_{i=1}^{n+1}g_1^0(\textbf{u}_i),\nonumber
\end{eqnarray}
where it is used notations accepted above.

The operator $g_{1}(t)$ represented by series (\ref{viterc}) is a solution of the Cauchy
problem of the Vlasov-type kinetic equation with initial correlations:
\begin{eqnarray}\label{Vlasov1}
    &&\hskip-5mm\frac{\partial}{\partial t}g_{1}(t,\textbf{u}_1)= \Lambda^{\ast[1]}(1)g_{1}(t,\textbf{u}_1)+\\
    &&+\int_{\mathcal{J}\times\mathcal{U}}d\textbf{u}_{2}
       \Lambda^{\ast[2]}(1,2)\prod_{i_1=1}^{2}e^{t\Lambda^{\ast[1]}(i_1)}\big(1+g_{2}(\textbf{u}_1,\textbf{u}_2)\big)
     \prod_{i_2=1}^{2}e^{-t\Lambda^{\ast[1]}(i_2)}
     g_{1}(t,\textbf{u}_1)g_{1}(t,\textbf{u}_2),\nonumber\\ \nonumber\\
  \label{Vlasov2}
    &&\hskip-5mmg_{1}(t,\textbf{u}_1)|_{t=0}=g_1^0(\textbf{u}_1),
\end{eqnarray}
where the operators $\Lambda^{\ast[1]}(1)$ and $\Lambda^{\ast[2]}(1,2)$ are defined according
to formula (\ref{gen_state_gener}). We point out that derived kinetic equation for a many-constituent
system modeling collective behavior of active soft condensed matter (\ref{Vlasov1}) is the non-Markovian
evolution equation.

We remark that in case of initial states of statistically independent interacting stochastic processes
specified by a one-particle marginal distribution function the kinetic evolution is governed by the
Vlasov kinetic equation and equality (\ref{pc}) means the property of the propagation of initial chaos \cite{G13}.

The proof of stated results is based on the corresponding formulas for cumulants of asymptotically
perturbed semigroups of operators (\ref{cumulantP}).

Indeed, if $f_{s}\in L^{1}_{s}$, then for arbitrary finite time interval for the strongly continuous
semigroup $e^{t\Lambda^\ast_s}$ the following equality is valid:
\begin{eqnarray*}
    &&\lim\limits_{\varepsilon\rightarrow 0}\Big\|e^{t\Lambda^\ast_n}f_{s}-
        \prod\limits_{j=1}^{s}e^{t\Lambda^{\ast[1]}(j)}f_{s}\Big\|_{L^{1}_{s}}=0.
\end{eqnarray*}
As a result of this equality for the $(s+n)th$-order cumulants of semigroups of operators
(\ref{cumulantP}) the following equalities are true:
\begin{eqnarray*}\label{apg}
   &&\lim\limits_{\varepsilon\rightarrow0}\Big\|\frac{1}{\varepsilon^{n}}\,
     \mathfrak{A}_{s+n}(t,1,\ldots,s+n)f_{s+n}\Big\|_{L^{1}_{s}}=0,\quad s\geq2.
\end{eqnarray*}

In consequence of the validity of this equalities the representation of the limit one-particle correlation
function $g_1(t)$ by series expansion (\ref{viterc}) is also obtained directly in view that series expansion
(\ref{sss}) in the case of initial data (\ref{insc}) takes the form
\begin{eqnarray*}\label{ske}
   &&\hskip-7mm G_{1}(t,\textbf{u}_1)= \sum\limits_{n=0}^{\infty}\frac{1}{n!}\,
   \int_{(\mathcal{J}\times\mathcal{U})^n}d\textbf{u}_{2}\ldots d\textbf{u}_{n+1}\,\mathfrak{A}_{1+n}(t,1,\ldots,\\
   && n+1)\sum\limits_{\mbox{\scriptsize $\begin{array}{c}\mathrm{P}:(1,\ldots,n+1)=\bigcup_{i}X_{i}\end{array}$}}
        \prod_{X_i\subset \mathrm{P}}g^{\varepsilon}_{|X_i|}(\textbf{U}_i)\prod_{i=1}^{n+1}G_{1}^{0,\varepsilon}(\textbf{u}_{i}),
\end{eqnarray*}
where it is used notations accepted above.

We note that the sequence of limit marginal correlation functions (\ref{viterc}) and (\ref{pc})
is a solution of the nonlinear Vlasov hierarchy which describe the evolution of marginal correlation
functions in a mean field limit for arbitrary initial states \cite{G12},\cite{G13}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\textcolor{blue!50!black}{\section{Conclusion}}

For a large system of interacting stochastic processes of collisional kinetic theory \cite{L11},
modeling the microscopic evolution of active soft condensed matter, a mean field scaling asymptotics
of nonperturbative solution (\ref{sss}) of the Cauchy problem of the nonlinear BBGKY hierarchy
(\ref{gBigfromDFBa}),(\ref{gBigfromDFBai}) for marginal correlation functions was constructed.

The marginal correlation functions give an equivalent approach to the description of the evolution
of states of large particle systems in comparison with marginal distribution functions governed by
the BBGKY hierarchy \cite{CGP97}. The macroscopic characteristics of fluctuations of observables are
directly determined by marginal correlation functions (\ref{sss}) on the microscopic level \cite{B46},\cite{G12}.

In case of initial states specified by correlation functions (\ref{lins}), which can
characterize the analogs of condensed states of many-particle systems of statistical mechanics
for interacting entities of complex biological systems, a mean field asymptotic behavior of the
processes of the creation and the propagation of correlations were described (\ref{pc}). It was
established that mean field dynamics does not create new correlations except of those that
generating by initial correlations. It was also proved the property known as a propagation
of initial chaos, which underlies in mathematical derivation of effective evolution equations
of complex systems \cite{CGP97},\cite{G13}.

We note that the developed approach is related to the problem of a rigorous derivation
of the non-Markovian kinetic-type equations from underlying many-constituent dynamics
which make it possible to describe the memory effects of collective dynamics of complex
systems modeling active soft condensed matter.

We remark also that in papers \cite{GF13},\cite{GF14} (see also \cite{G18}) it was developed
two other approaches to the description of the process of the propagation of initial correlations
in a mean field limit. In paper \cite{GF13} the process of the propagation of initial chaos was
established within the framework of the evolution of marginal observables (in \cite{GF15} this
result was generalized on case of initial states with correlations) and in paper \cite{GF14} the
property of the propagation of initial correlations was proved within the framework of the description
of the evolution of states by means of a one-particle (marginal) distribution function governed
by the generalized kinetic equation.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{0.5cm}
\addcontentsline{toc}{section}{{References}}
\renewcommand{\refname}{\textcolor{blue!50!black}{References}}
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\end{document}
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