Content-Type: multipart/mixed; boundary="-------------1703091219308" This is a multi-part message in MIME format. ---------------1703091219308 Content-Type: text/plain; name="17-24.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="17-24.comments" 10 pages ---------------1703091219308 Content-Type: text/plain; name="17-24.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="17-24.keywords" marginal correlation operator, group of nonlinear operators, quantum kinetic equation ---------------1703091219308 Content-Type: application/x-tex; name="arxiv_mph.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="arxiv_mph.tex" \documentclass[letterpaper, 12pt]{article}[2000/05/19] \usepackage[english]{babel} \usepackage{amsfonts,amsmath,amssymb,amsthm,latexsym,amscd,mathrsfs} \usepackage{ifthen,cite} \usepackage[bookmarksnumbered=true]{hyperref} \hypersetup{pdfpagetransition={Split}} \newcommand{\ArticleLabel}{Article label} \newcommand{\evenhead}{Author \ name} \newcommand{\oddhead}{Article \ name} \newcommand{\theArticleName}{Article \ name} \newcommand{\Volume}{Vol.~1} \newcommand{\Paper}{Paper} \newcommand{\PaperNumber}{1} \newcommand{\PublicationYear}{2010} % Titlepage \newcommand{\FirstPageHeading}[1]{\thispagestyle{empty}% \noindent\raisebox{0pt}[0pt][0pt]{\makebox[\textwidth]{\protect\footnotesize \sf }}\par} \newcommand{\LastPageEnding}{\label{\ArticleLabel-lp}\newpage} \newcommand{\ArticleName}[1]{\renewcommand{\theArticleName}{#1}\vspace{-2mm}\par\noindent {\LARGE\bf #1\par}} \newcommand{\Author}[1]{\vspace{5mm}\par\noindent {\Large #1\par} \par\vspace{2mm}\par} \newcommand{\Address}[1]{\vspace{2mm}\par\noindent {\it #1} \par} \newcommand{\Email}[1]{\ifthenelse{\equal{#1}{}}{}{\par\noindent {\rm E-mail: }{\it #1} \par}} \newcommand{\URLaddress}[1]{\ifthenelse{\equal{#1}{}}{}{\par\noindent {\rm URL: }{\tt #1} \par}} \newcommand{\EmailD}[1]{\ifthenelse{\equal{#1}{}}{}{\par\noindent {$\phantom{\dag}$~\rm E-mail: }{\it #1} \par}} \newcommand{\URLaddressD}[1]{\ifthenelse{\equal{#1}{}}{}{\par\noindent {$\phantom{\dag}$~\rm URL: }{\tt #1} \par}} \newcommand{\ArticleDates}[1]{\vspace{2mm}\par\noindent {\small {\rm #1} \par}} \newcommand{\Abstract}[1]{\vspace{6mm}\par\noindent\hspace*{10mm} \parbox{140mm}{\small {\bf Abstract.} #1}\par} \newcommand{\Keywords}[1]{\vspace{3mm}\par\noindent\hspace*{10mm} \parbox{140mm}{\small {\bf Key words:} \rm #1}\par} \newcommand{\Classification}[1]{\vspace{3mm}\par\noindent\hspace*{10mm} \parbox{140mm}{\small {\it 2010 Mathematics Subject Classification:} \rm #1}\vspace{3mm}\par} \newcommand{\ShortArticleName}[1]{\renewcommand{\oddhead}{#1}} \newcommand{\AuthorNameForHeading}[1]{\renewcommand{\evenhead}{#1}} % Papersize \setlength{\textwidth}{175.0mm} \setlength{\textheight}{229.0mm} \setlength{\oddsidemargin}{0mm} \setlength{\evensidemargin}{0mm} \setlength{\topmargin}{-8mm} \setlength{\parindent}{5.0mm} \long\def\@makecaption#1#2{%\vskip\abovecaptionskip \sbox\@tempboxa{\small \textbf{#1.}\ \ #2}% \ifdim \wd\@tempboxa >\hsize {\small \textbf{#1.}\ \ #2}\par \else \global \@minipagefalse \hb@xt@\hsize{\hfil\box\@tempboxa\hfil}% \fi \vskip\belowcaptionskip} % Defines the \numberwithin command from AMS-LaTeX \def\Re{\vbox{\hbox to8.9pt{I\hskip-2.1pt R\hfil}}} \def\numberwithin#1#2{\@ifundefined{c@#1}{\@nocounterr{#1}}{% \@ifundefined{c@#2}{\@nocnterr{#2}}{% \@addtoreset{#1}{#2}% \toks@\@xp\@xp\@xp{\csname the#1\endcsname}% \@xp\xdef\csname the#1\endcsname {\@xp\@nx\csname the#2\endcsname.\the\toks@}}}} \def\E^#1{{\buildrel #1 \over\vee}} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \newtheorem{corollary}{Corollary} \newtheorem{proposition}{Proposition} {\theoremstyle{definition} \newtheorem{definition}{Definition} \newtheorem{example}{Example} \newtheorem{remark}{Remark} } \usepackage{tikz} \begin{document} \FirstPageHeading{V.I. Gerasimenko} \ShortArticleName{Towards rigorous description of correlations of quantum states} \AuthorNameForHeading{V.I. Gerasimenko} \ArticleName{\textcolor{blue!50!black}{Towards rigorous description of correlations \\ of quantum states by means of kinetic equations}} \Author{V.I. Gerasimenko\footnote{E-mail: \emph{gerasym@imath.kiev.ua}}} \Address{\hspace*{2mm} Institute of Mathematics of the NAS of Ukraine,\\ \hspace*{2mm}3, Tereshchenkivs'ka Str.,\\ \hspace*{2mm}01004, Kyiv, Ukraine} \bigskip \Abstract{ We develop an approach to the description of processes of the creation of correlations and the propagation of initial correlations in large particle quantum systems by means of a one-particle density operator that is a solution of the generalized quantum kinetic equation with initial correlations. Moreover, a mean field asymptotic behavior of the constructed correlation operators of the quantum states is established. } \bigskip \Keywords{marginal correlation operator, group of nonlinear operators, quantum kinetic equation.} \vspace{2pc} \Classification{35Q40, 82C40, 82C10.} \makeatletter \renewcommand{\@evenhead}{ \hspace*{-3pt}\raisebox{-15pt}[\headheight][0pt]{\vbox{\hbox to \textwidth {\thepage \hfil \evenhead}\vskip4pt \hrule}}} \renewcommand{\@oddhead}{ \hspace*{-3pt}\raisebox{-15pt}[\headheight][0pt]{\vbox{\hbox to \textwidth {\oddhead \hfil \thepage}\vskip4pt\hrule}}} \renewcommand{\@evenfoot}{} \renewcommand{\@oddfoot}{} \makeatother \newpage \vphantom{math} %\protect\tableofcontents \protect\textcolor{blue!50!black}{\tableofcontents} %\newpage %\vspace{0.5cm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \textcolor{blue!50!black}{\section{Introduction}} As known, the marginal correlation operators give an equivalent approach to the description of the evolution of states of large particle quantum systems in comparison with marginal density operators. The physical interpretation of marginal correlation operators is that the macroscopic characteristics of fluctuations of mean values of observables are determined by them on the microscopic level \cite{BQ},\cite{GP13}. Traditionally marginal correlation operators are introduced by means of the cluster expansions of the marginal density operators \cite{G12}. In article \cite{G16} we developed an approach based on the definition of the marginal correlation operators within the framework of dynamics of correlations governed by the von Neumann hierarchy \cite{GP11}. As a result of which it is established that the marginal correlation operators governed by the hierarchy of nonlinear evolution equations, known as the quantum nonlinear BBGKY (Bogolyubov--Born--Green--Kirkwood--Yvon) hierarchy, are represented in the form of series expansions over the number of particles of subsystems which generating operators are the corresponding-order cumulants of the groups of nonlinear operators of the von Neumann hierarchy for a sequence of correlation operators \cite{GP11}. In this paper we consider the problem of the rigorous description of the evolution of states of large particle quantum systems within the framework of a one-particle (marginal) density operator that is a solution of the generalized quantum kinetic equation with initial correlations. We remark that initial states specified by correlations are typical for the condensed states of many-particle systems in contrast to their gaseous state \cite{BQ},\cite{S-R}. Moreover, in the paper mean field asymptotic behavior of processes of the creation of correlations and the propagation of initial correlations in large particle quantum systems is established. We note that the conventional approach to the problem of the description of the propagation of initial chaos \cite{CGP}, i.e. in case of initial states specified by a one-particle density operator without correlation operators, is based on the consideration of an asymptotic behavior of a solution of the quantum BBGKY hierarchy for marginal density operators constructed within the framework of the perturbation theory \cite{Go13},\cite{ESchY2},\cite{CG}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \textcolor{blue!50!black}{\section{Preliminaries: marginal correlation operators}} Let the space $\mathcal{H}$ be a one-particle Hilbert space, then the $n$-particle space $\mathcal{H}_n=\mathcal{H}^{\otimes n}$ is a tensor product of $n$ Hilbert spaces $\mathcal{H}$. We adopt the usual convention that $\mathcal{H}^{\otimes 0}=\mathbb{C}$. The Fock space over the Hilbert space $\mathcal{H}$ we denote by $\mathcal{F}_{\mathcal{H}}={\bigoplus\limits}_{n=0}^{\infty}\mathcal{H}_{n}$. A self adjoint operator $f_{n}$ defined on the $n$-particle Hilbert space $\mathcal{H}_{n}=\mathcal{H}^{\otimes n}$ will be also denoted by the symbol $f_{n}(1,\ldots,n)$. Let $\mathfrak{L}^{1}(\mathcal{H}_{n})$ be the space of trace class operators $f_{n}\equiv f_{n}(1,\ldots,n)\in\mathfrak{L}^{1}(\mathcal{H}_{n})$ that satisfy the symmetry condition: $f_{n}(1,\ldots,n)=f_{n}(i_{1},\ldots,i_{n})$ for arbitrary $(i_{1},\ldots,i_{n})\in(1,\ldots,n)$, and equipped with the norm \begin{eqnarray*} &&\|f_{n}\|_{\mathfrak{L}^{1}(\mathcal{H}_{n})}=\mathrm{Tr}_{1,\ldots,n}|f_{n}(1,\ldots,n)|, \end{eqnarray*} where $\mathrm{Tr}_{1,\ldots,n}$ are partial traces over $1,\ldots,n$ particles. We denote by $\mathfrak{L}^{1}_0(\mathcal{H}_{n})$ the everywhere dense set of finite sequences of degenerate operators with infinitely differentiable kernels with compact supports. On the space $\mathfrak{L}^{1}(\mathcal{F}_\mathcal{H})=\oplus_{n=0}^{\infty}\mathfrak{L}^{1}(\mathcal{H}_{n})$ of sequences $f=(f_0,f_{1},\ldots,$ $f_{n},\ldots)$ of trace class operators $f_{n}\in\mathfrak{L}^{1}(\mathcal{H}_{n})$ and $f_0\in\mathbb{C}$ it is defined the following nonlinear one-parameter mapping: \begin{eqnarray}\label{rozvNh} &&\hskip-8mm \mathcal{G}(t;1,\ldots,s\mid f)\doteq\sum\limits_{\mathrm{P}:\,(1,\ldots,s)=\bigcup_j X_j} \mathfrak{A}_{|\mathrm{P}|}(t,\{X_1\},\ldots,\{X_{|\mathrm{P}|}\}) \prod_{X_j\subset \mathrm{P}}f_{|X_j|}(X_j),\quad s\geq1, \end{eqnarray} where the symbol $\sum_{\mathrm{P}:\,(1,\ldots,s)=\bigcup_j X_j}$ means the sum over all possible partitions $\mathrm{P}$ of the set $(1,\ldots,s)$ into $|\mathrm{P}|$ nonempty mutually disjoint subsets $X_j$, the set $(\{X_1\},\ldots,\{X_{|\mathrm{P}|}\})$ consists from elements of which are subsets $X_j\subset (1,\ldots,s)$, i.e., $|(\{X_1\},\ldots,\{X_{|\mathrm{P}|}\})|=|\mathrm{P}|$. The generating operator $\mathfrak{A}_{|\mathrm{P}|}(t)$ of expansion (\ref{rozvNh}) is the $|\mathrm{P}|th$-order cumulant of the groups of operators defined by the following expansion: \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}^{'}}\mathcal{G}^{\ast}_{|\theta(Z_{k})|}(t,\theta(Z_{k})), \end{eqnarray} where $\theta$ is the declusterization mapping: $\theta(\{X_1\},\ldots,\{X_{|\mathrm{P}|}\})\doteq(1,\ldots,s)$, and on the space $\mathfrak{L}^1(\mathcal{H}_{n})$ the one-parameter mapping $\mathcal{G}^{\ast}_n(t)$ is defined by the formula \begin{eqnarray}\label{grG} &&\mathbb{R}^1\ni t\mapsto\mathcal{G}^{\ast}_n(t)f_n\doteq e^{-itH_{n}}f_n e^{itH_{n}}. \end{eqnarray} In (\ref{grG}) the operator $H_{n}$ is the Hamiltonian of a system of $n$ particles, obeying Maxwell--Boltzmann statistics, and we use units where $h={2\pi\hbar}=1$ is a Planck constant and $m=1$ is the mass of particles. The inverse group to the group $\mathcal{G}_{n}^{\ast}(t)$ we denote by $(\mathcal{G}_{n}^{\ast})^{-1}(t)=\mathcal{G}_{n}^{\ast}(-t)$. On its domain of the definition the infinitesimal generator $\mathcal{N}^{\ast}_{n}$ of the group of operators (\ref{grG}) is determined in the sense of the strong convergence of the space $\mathfrak{L}^1(\mathcal{H}_{n})$ by the operator \begin{eqnarray}\label{infOper1} &&\lim\limits_{t\rightarrow 0}\frac{1}{t}\big(\mathcal{G}^{\ast}_n(t)f_n-f_n \big) =-i\,(H_n f_n - f_n H_n)\doteq\mathcal{N}^{\ast}_n f_n, \end{eqnarray} that has the structure: $\mathcal{N}^{\ast}_n=\sum_{j=1}^{n}\mathcal{N}^{\ast}(j)+ \epsilon\sum_{j_{1}0$. The evolution of all possible states of large particle quantum systems, obeying the Maxwell--Boltzmann statistics, can be described by means of the sequence $G(t)=(I,G_1(t),G_2(t),\ldots,$ $G_s(t),\ldots)\in\mathfrak{L}^{1}(\mathcal{F}_\mathcal{H})$ of marginal correlation operators governed by the hierarchy of nonlinear evolution equations known as the quantum nonlinear BBGKY hierarchy \cite{BQ}. If $G(0)=(I,G_1^{0,\epsilon}(1),\ldots,G_s^{0,\epsilon}(1,\ldots,s),\ldots)$ is a sequence of initial marginal correlation operators, then a nonperturbative solution of the Cauchy problem of the quantum nonlinear BBGKY hierarchy is represented by a sequence of the following operators \cite{G16}: \begin{eqnarray}\label{sss} &&\hskip-12mm G_{s}(t,1,\ldots,s)=\sum\limits_{n=0}^{\infty}\frac{1}{n!} \,\mathrm{Tr}_{s+1,\ldots,s+n}\,\mathfrak{A}_{1+n}(t;\{1,\ldots,s\},s+1,\ldots,s+n\mid G(0)),\quad s\geq1, \end{eqnarray} where the generating operator $\mathfrak{A}_{1+n}(t;\{1,\ldots,s\},s+1,\ldots,s+n\mid G(0))$ of series expansion (\ref{sss}) is the $(1+n)th$-order cumulant of groups of nonlinear operators (\ref{rozvNh}) of the von Neumann hierarchy for correlation operators \begin{eqnarray} \label{cc} &&\hskip-12mm\mathfrak{A}_{1+n}(t;\{1,\ldots,s\},s+1,\ldots,s+n\mid G(0))\doteq\\ &&\hskip-5mm \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\\ &&\hskip-12mm n\geq0,\nonumber \end{eqnarray} and the composition of mappings (\ref{rozvNh}) of the corresponding noninteracting groups of particles we denote by the symbol $\mathcal{G}(t;\theta(X_1)\mid \ldots\mathcal{G}(t;\theta(X_{|\mathrm{P}|})\mid G(0))\ldots)$ \cite{G16}. We remark that nonperturbative solution (\ref{sss}) of the quantum nonlinear BBGKY hierarchy is transformed to the solution represented be perturbation (iteration) series as a result of the application of analogs of the Duhamel equation to cumulants (\ref{cumulantP}) of the groups of operators (\ref{grG}). In case of initial states specified in terms of a one-particle (marginal) density operator and correlation operators the evolution of all possible states of large particle quantum systems can be described in an equivalent way within the framework of a one-particle density operator governed by the kinetic equation, i.e. without any approximations. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 \textcolor{blue!50!black}{\section{A main result: marginal correlation functionals of the state}} We shall consider the case of initial states specified by a one-particle marginal density operator with correlations, namely, initial states specified by the following sequence of marginal correlation operators: \begin{eqnarray}\label{insc} &&\hskip-12mm G^{(c)}=\big(I,G_1^{0,\epsilon}(1),g_{2}^{\epsilon}(1,2)\prod_{i=1}^{2}G_1^{0,\epsilon}(i), \ldots,g_{n}^{\epsilon}(1,\ldots,n)\prod_{i=1}^{n}G_1^{0,\epsilon}(i),\ldots\big), \end{eqnarray} where the operators $g_{n}^{\epsilon}(1,\ldots,n)\equiv g_{n}^{\epsilon}\in\mathfrak{L}^{1}_0(\mathcal{H}_n),\,n\geq2$, are specified the initial correlations. We remark that such assumption about initial states is intrinsic for the kinetic description of many-particle systems. On the other hand, initial data (\ref{insc}) is typical for the condensed states of large particle quantum systems, for example, the equilibrium state of the Bose condensate satisfies the weakening of correlation condition with the correlations which characterize the condensed state \cite{BQ},\cite{S-R}. For initial states specified in terms of a one-particle density operator and correlation operators (\ref{insc}) the evolution of states given in the framework of the sequence $G(t)=(I,G_1(t),\ldots,G_s(t),\ldots)$ of marginal correlation operators (\ref{sss}) can be described by means of the sequence $G(t\mid G_{1}(t))=(I,G_1(t),G_2(t\mid G_{1}(t)),\ldots,G_s(t\mid G_{1}(t)),\ldots)$ of marginal correlation functionals: $G_s(t,1,\ldots,s\mid G_{1}(t)),\,s\geq2$, with respect to the one-particle correlation operator $G_1(t)$ governed by the kinetic equation. In this case the marginal correlation functionals $G_s(t\mid G_{1}(t)),\,s\geq2$, are defined with respect to the one-particle (marginal) density operator \begin{eqnarray}\label{ske} &&\hskip-12mm G_{1}(t,1)=\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\mathrm{Tr}_{2,\ldots,{1+n}}\, \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_{|X_i|}^{\epsilon}(X_i)\prod_{i=1}^{n+1}G_{1}^{0,\epsilon}(i), \end{eqnarray} where the generating operator $\mathfrak{A}_{1+n}(t)$ is the $(1+n)th$-order cumulant (\ref{cumulantP}) of the groups of operators (\ref{grG}), and these functionals are represented by the series expansions \begin{eqnarray}\label{f} &&\hskip-12mm G_{s}(t,1,\ldots,s\mid G_{1}(t))=\sum _{n=0}^{\infty }\frac{1}{n!}\,\mathrm{Tr}_{s+1,\ldots,{s+n}}\, \mathfrak{G}_{s+n}(t,\theta(\{1,\ldots,s\}),s+1,\ldots,s+n)\prod_{i=1}^{s+n}G_{1}(t,i),\\ &&\hskip-12mm s\geq2,\nonumber \end{eqnarray} where the $(s+n)th$-order generating operator $\mathfrak{G}_{s+n}(t),\,n\geq0$, of this series is determined by the following expansion: \begin{eqnarray}\label{skrrc} &&\hskip-12mm\mathfrak{G}_{s+n}(t,\theta(\{1,\ldots,s\}),s+1,\ldots,s+n)=\\ &&\hskip-5mm n!\,\sum_{k=0}^{n}\,(-1)^k\,\sum_{n_1=1}^{n}\ldots \sum_{n_k=1}^{n-n_1-\ldots-n_{k-1}}\frac{1}{(n-n_1-\ldots-n_k)!}\times\nonumber\\ &&\hskip-5mm \breve{\mathfrak{A}}_{s+n-n_1-\ldots-n_k}(t,\theta(\{1,\ldots,s\}),s+1,\ldots, s+n-n_1-\ldots-n_k)\times\nonumber\\ &&\hskip-5mm \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}|!}\breve{\mathfrak{A}}_{1+|X_{l_j}|}(t,i_{l_j},X_{l_j}).\nonumber \end{eqnarray} In formula (\ref{skrrc}) the sum over all possible dissections \cite{GT} 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 we denote by $\sum_{\mathrm{D}_{j}:Z_j=\bigcup_{l_j} X_{l_j}}$ and the $(s+n)th$-order scattering cumulant is defined by the formula \begin{eqnarray*} &&\hskip-5mm\breve{\mathfrak{A}}_{s+n}(t,\theta(\{1,\ldots,s\}),s+1,\ldots,s+n)= \mathfrak{A}_{s+n}(t,1,\ldots,s+n)g_{s+n}^{\epsilon}(1,\ldots,s+n) \prod_{i=1}^{s+n}\mathfrak{A}_{1}^{-1}(t,i), \end{eqnarray*} where the operator $g_{s+n}^{\epsilon}(1,\ldots,s+n)$ is specified initial correlations (\ref{insc}), and notations accepted above were used. We adduce simplest examples of generating operators (\ref{skrrc}) \begin{eqnarray*} &&\hskip-12mm\mathfrak{G}_{s}(t,\theta(\{1,\ldots,s\}))=\breve{\mathfrak{A}}_{s}(t,\theta(\{1,\ldots,s\}))=\\ &&\hskip-12mm\mathfrak{A}_{s}(t,1,\ldots,s))g_{s}^{\epsilon}(1,\ldots,s) \prod_{i=1}^{s}\mathfrak{A}_{1}^{-1}(t,i),\mathfrak{G}_{s+1}(t,\theta(\{1,\ldots,s\}),s+1)=\\ &&\hskip-5mm\mathfrak{A}_{s+1}(t,1,\ldots,s+1) g_{s+1}^{\epsilon}(1,\ldots,s+1)\prod_{i=1}^{s+1}\mathfrak{A}_{1}^{-1}(t,i)-\\ &&\hskip-5mm\mathfrak{A}_{s}(t,1,\ldots,s)g_{s}^{\epsilon}(1,\ldots,s) \prod_{i=1}^{s}\mathfrak{A}_{1}^{-1}(t,i) \sum_{j=1}^s\mathfrak{A}_{2}(t,j,s+1)g_{2}^{\epsilon}(j,s+1)\mathfrak{A}_{1}^{-1}(t,j)\mathfrak{A}_{1}^{-1}(t,s+1). \end{eqnarray*} A method of the construction of marginal correlation functionals (\ref{f}) is based on the application of kinetic cluster expansions \cite{G12} to the generating operators of series (\ref{sss}). If $\|G_{1}(t)\|_{\mathfrak{L}^{1}(\mathcal{H})}