Kinetic equations and hierarchies of evolution
equations of quantum systems

V.I. Gerasimenko∗111E-mail<EMAIL_ADDRESS>
∗Institute of Mathematics of the NAS of Ukraine,
3, Tereshchenkivs’ka Str.,
01601, Kyiv-4, Ukraine

Abstract. The article provides an overview of some advances in the
mathematical understanding of the nature of the kinetic equations of quantum
systems of many particles. The fundamental equations of modern mathematical
physics are studied, in particular, the hierarchies of evolution equations of
quantum systems and their asymptotic behavior described by kinetic nonlinear
equations.

Key words: kinetic equation, von Neumann hierarchy, BBGKY hierarchy, density
operator, correlation operator.

2010 Mathematics Subject Classification: 35Q20; 47J35.

math

###### Contents

  1. 1 A brief chronology of 150 years of the theory of kinetic equations
  2. 2 Hierarchies of evolution equations of quantum systems
    1. 2.1 Preliminaries: on evolution equations of quantum systems of many particles
    2. 2.2 Dynamics of correlations of particle states
    3. 2.3 The BBGKY hierarchies of evolution equations
  3. 3 On quantum kinetic equations
    1. 3.1 The origin of the kinetic evolution of a state
    2. 3.2 The generalized quantum kinetic equation
    3. 3.3 Scaling properties of state evolution
  4. 4 The kinetic evolution of observables
    1. 4.1 The hierarchy of kinetic equations for observables
    2. 4.2 The propagation of the initial chaos
    3. 4.3 Kinetic equations with initial correlations
  5. 5 Outlook

##  1 A brief chronology of 150 years of the theory of kinetic equations

Mathematically the collective properties of many-particle systems in kinetic
theory are described by kinetic equations, namely, nonlinear equations that
describe the evolution of the state of a system of many particles by means of
the evolution of the state of a typical particle. The generator of such
evolution equations consists of a term that describes the free-evolution
(motion by inertia) of a typical particle of the system and a nonlinear term
that simulates the self-interaction of a typical particle (collision
integral). The mean values of observables of the many-particle system are
determined by a solution of the kinetic equations. A well-known historical
example of a kinetic equation is the Boltzmann equation, which describes the
process of collision of particles in rarefied gases. Quantum kinetic equations
are the corresponding generalizations of the kinetic equations of classical
particle systems.

From a physical point of view, kinetic equations describe a certain stage of
the process of transition (relaxation) from a non-equilibrium state to the
state of thermodynamic equilibrium of the system of many particles. Indeed, in
the general case, in the process of relaxation, an arbitrary non-equilibrium
state of the particle system attracts to a state that can be fully described
in terms of a one-particle probability distribution function (for quantum
systems, it is a one-particle statistical operator, the kernel of which is
known as a one-particle density matrix), that is governed by the suitable
kinetic equation depending on the interaction potential of the particles. At
the next stage of relaxation, such a state of the particle system tends to a
state of local equilibrium, which is described by the Maxwell distribution,
characterized by hydrodynamic parameters that depend on a position in space.
The evolution of hydrodynamic parameters in turn is governed by the equations
of a continuous medium (hydrodynamics or diffusion equations).

One hundred and fifty-year history of kinetic equations is derived from the
works of J. C. Maxwell (1860, 1867) [54],[55] and L. E. Boltzmann (1872) [9].
In 1872 Ludwig Boltzmann published the paper [9] where the evolution equation
for a one-particle distribution function was formulated and was showed that
the Maxwell distribution describes only the equilibrium state of a gas. He
proved the so-called H-theorem (on the increase in entropy) about a property
of the solution of this equation, which explained the irreversibility of
macroscopic dynamics. Thus, from this time began a period of development of
the theory of kinetic equations, which was based on phenomenological models of
kinetic phenomena.

Later in order to generalize the Boltzmann equation to dense gases or liquids
for a system of many hard spheres, the Enskog kinetic equation was formulated
(D. Enskog, 1922) [14]. For a particle (Brownian) in a system of many
particles (an environment) was introduced the Fokker–Planck equation (A. D.
Fokker, 1914, M. Planck, 1917) [17],[63] and the Smoluchowski equation as its
partial case (M. von Smoluchowski, 1906) [70]. In this period of development
of kinetic theory were also formulated: the Leontovich equation for stochastic
dynamics of a system of many particles (M. A. Leontovich, 1935) [53], the
Landau equation (L. D. Landau, 1936) [50], the Vlasov equation (A. A. Vlasov,
1938) [68] and the Lenard–Balescu equation (A. Lenard, R. Balescu, R. L.
Guernsey, 1960) [1],[45],[52] for systems of many charged particles (ionized
gases, plasma).

With the beginning of the development of quantum theory the Uehling–Uhlenbeck
kinetic equation was formulated as a quantum analog of the Boltzmann equation
(L. W. Nordheim, 1928; E. A. Uehling, G. E. Uhlenbeck, 1933) [56],[67] and the
quantum Bogolyubov equation (M. M. Bogolyubov, K. P. Gurov, 1947) [7]. In the
mean-field approximation for pure states, the Hartree equation (D. R. Hartree,
1928) [46] or the Hartree–Fock equation for systems of fermions and bosons (V.
A. Fock, 1930) [16] were derived, and for mixed states, the quantum Vlasov
kinetic equation (A. A. Vlasov, 1947) [69]. For quantum many-particle systems
in condensed states, it was suggested the Bogolyubov kinetic equation (M. M.
Bogolyubov, 1947) [5] and later the Gross–Pitaevskii equation (E. P. Gross, L.
D. Pitaevskii, 1961) [44],[62].

From the second half of the 1940s, a new stage in the development of kinetic
theory began, namely, the creation of a formalized theory of kinetic
phenomena. In 1945 in Kyiv at the Institute of Mathematics in the famous
monograph [8], M. M. Bogolyubov formulated a consistent approach to deriving
kinetic equations from the dynamics of systems of many particles, namely, from
fundamental equations described the evolution of all possible states of many-
particle systems, they are now known as the BBGKY hierarchy
(Bogolyubov–Born–Green–Kirkwood–Yvon) [8],[10],[49],[71]. Using the methods of
perturbation theory, an approach was developed to construct a generalization
of the Boltzmann equation, known as the kinetic equation of Bogolyubov, as
well as the Vlasov and Landau kinetic equations. Thanks to this work, the
irreversibility mechanism of the evolution of systems of many particles in the
macroscopic scale, the dynamics of which are described reversible in time
equations of motion in a microscopic scale, became clear. Initially, the
mathematical theory of the BBGKY hierarchy was developed in works
[32],[57],[59] and see also in [12],[23],[58].

Later H. Grad (H. Grad, 1958) [43] formulated an approach to the derivation of
kinetic equations as the evolution equations, which describe the corresponding
scaling asymptotic of a solution of the BBGKY hierarchy. In recent decades,
this approach has been used as an accepted method for the rigorous derivation
of kinetic equations of complex systems of various nature. In general, the
problem of the rigorous derivation of kinetic equations from the dynamics of
systems of many particles remains an open problem of kinetic theory.

At the present stage of development of kinetic theory, the most advanced is
the mathematical theory of the nonlinear Boltzmann equation, which takes its
origin from the works of H. Poincaré (1906) [64], who drew the attention of
mathematicians to the need to substantiate kinetic theory and D. Hilbert
(1912) [48], who established the connection between a solution of the
Boltzmann equation and the hydrodynamics equations of (D. Hilbert’s sixth
problem was formulated at the International Congress of Mathematics in 1900
[47]), as well as the work of T. Carleman (1957) [11] on mathematical analysis
of the spatially homogeneous Boltzmann equation.

The mathematical theory of nonlinear kinetic equations began to develop
intensively in the early 80s of the XX century. One of the achievements of
this period was the rigorous derivation of the Boltzmann equation from the
dynamics of an infinite number of hard spheres in the Boltzmann–Grad limit
[51],[60]. Rigorous results of the theory of kinetic equations and their
justification at the end of the XX century were summarized in monographs
[12],[13],[66].

Over the last decade mathematical results on the derivation of the Boltzmann
kinetic equation for classical systems of particles with short-range
interaction potential have been summarized in the monograph [18] and new
methods have been developed for deriving the Boltzmann equation [20] and the
Enskog equation [30] from the collisional dynamics [31] (for details see the
references in the above works).

In the last two decades significant progress has also been observed in
deriving quantum kinetic equations in the scaling limits of the BBCKY
hierarchy solution constructed by methods of the perturbation theory [3],
[42], [58], in particular, of the quantum Boltzmann equation [2] in a weak
coupling limit, of the nonlinear Schrödinger equation [4], [61] and the
Gross–Pitaevskii equation [15] in the mean-field limit.

Below the fundamental equations that describe the nature of substance are
studies, namely the hierarchies of evolution equations of quantum systems of
many particles, and non-perturbative methods for constructing their solutions.
Based on this, two new approaches to the description of the kinetic evolution
of quantum systems [23] are considered. One of them is consists in the
description of the evolution of quantum systems in the mean-field scaling
limit of the hierarchy of evolution equations for the observables [19] and an
approach based on a non-Markovian generalization of quantum kinetic equations
[38].

##  2 Hierarchies of evolution equations of quantum systems

As is well-known, quantum systems are described by concepts such as
observables and states. The average value functional of the observables
(mathematical expectation) determines the duality of the observables and the
state. Consequently, there are two approaches to describing the evolution of a
quantum system of a finite number of particles, namely, in terms of
observables whose evolution is governed by the Heisenberg equation, or in
terms of a state whose evolution is governed by the von Neumann equation
(quantum Liouville equation) for the density operator (statistical operator)
which kernel is known as the density matrix, respectively.

An equivalent approach to describing the evolution of systems of many
particles, both of a finite and an infinite number of particles, is to
describe the evolution in terms of a sequence of reduced operators of
observables are governing by the dual BBGKY hierarchy
(Bogolyubov–Born–Green–Kirkwood–Yvon) [23],[28], or in terms of the state
describing by a sequence of the reduced density operators governed by the
BBGKY hierarchy [8].

An alternative method for describing the evolution of the state of a quantum
system of finitely many particles is to describe the state in terms of the
operators that are determined by cluster expansions of the density operator.
Such operators are interpreted as correlations of particle states, and their
evolution is governed by the von Neumann hierarchy for a sequence of
correlation operators [26],[34],[37].

###  2.1 Preliminaries: on evolution equations of quantum systems of many
particles

For generality, we will consider below quantum systems in space
$\mathbb{R}^{\nu},\,\nu\geq 1$of a non-fixed number of identical spinless
particles, i.e. of an arbitrary but finite average number of particles, which
obeying the Maxwell–Boltzmann statistics. We will use units where
$h={2\pi\hbar}=1$ is a Planck constant, $m=1$ is particle mass.

We denote the $n$-particle Hilbert space by
$\mathcal{H}_{n}=\mathcal{H}^{\otimes n}$ and $\mathcal{H}^{\otimes
0}=\mathbb{C}$. We denote the Fock space over the space $\mathcal{H}$ by
$\mathcal{F}_{\mathcal{H}}={\bigoplus\limits}_{n=0}^{\infty}\mathcal{H}_{n}$.
The self-adjoint operator $f_{n}$ defined on the space
$\mathcal{H}_{n}=\mathcal{H}^{\otimes n}$ further we will also denote by the
symbol $f_{n}(1,\ldots,n)$. Let $\mathfrak{L}(\mathcal{H}_{n})$ be a space of
bounded operators $g_{n}\equiv
g_{n}(1,\ldots,n)\in\mathfrak{L}(\mathcal{H}_{n})$ with operator norm
$\|.\|_{\mathfrak{L}(\mathcal{H}_{n})}$. Accordingly, let the space
$\mathfrak{L}^{1}(\mathcal{H}_{n})$ be the space of trace class operators
$f_{n}\equiv f_{n}(1,\ldots,n)\in\mathfrak{L}^{1}(\mathcal{H}_{n})$ with the
norm:
$\|f_{n}\|_{\mathfrak{L}^{1}(\mathcal{H}_{n})}=\mathrm{Tr}_{1,\ldots,n}|f_{n}(1,\ldots,n)|,$
where the symbol $\mathrm{Tr}_{1,\ldots,n}$ denotes partial traces of the
operator $f_{n}$. The subspace of finite sequences of degenerate operators
with infinitely differentiated kernels with compact supports is denoted by
$\mathfrak{L}^{1}_{0}(\mathcal{F}_{\mathcal{H}})$.

For a quantum system of a non-fixed number of particles, the observables are
described by sequences $A=(A_{0},A_{1}(1),\ldots,A_{n}(1,\ldots,n),\ldots)$ of
the self-adjoint operators $A_{n}\in\mathfrak{L}(\mathcal{H}_{n})$. The
evolution of the observables
$A(t)=(A_{0},A_{1}(t,1),\ldots,A_{n}(t,1,\ldots,n),\ldots)$, where
$t\in\mathbb{R}$, is determined by the Cauchy problem for a sequence of the
Heisenberg equations:

$\displaystyle\frac{\partial}{\partial t}A(t)=\mathcal{N}A(t),$ (1)
$\displaystyle A(t)|_{t=0}=A(0),$ (2)

where $A(0)=(A_{0},A_{1}^{0}(1),\ldots,A_{n}^{0}(1,\ldots,n),\ldots)$ is the
initial observable, the generator
$\mathcal{N}=\oplus_{n=0}^{\infty}\mathcal{N}_{n}$ is defined by the formula

$\displaystyle\mathcal{N}_{n}g_{n}\doteq-i\,(g_{n}H_{n}-H_{n}g_{n}),$ (3)

and the self-adjoint operator
$H_{n}={\sum\limits}_{j=1}^{n}K(j)+\epsilon{\sum\limits}_{j_{1}<j_{2}=1}^{n}\Phi(j_{1},j_{2})$
is the Hamilton operator of a system of $n$ particles, that is, the operator
$K(j)$ is the kinetic energy operator of $j$ particle, $\Phi$ is the bounded
operator of the pair interaction potential, $\epsilon>0$ is the scaling
parameter.

If $A(0)\in\mathfrak{L}(\mathcal{F}_{\mathcal{H}})$, then for $t\in\mathbb{R}$
in the space $\mathfrak{L}(\mathcal{F}_{\mathcal{H}})$ the unique solution of
the Cauchy problem (1),(2) is represented by a one-parameter family of
mappings $\mathcal{G}(t)=\oplus_{n=0}^{\infty}\mathcal{G}_{n}(t)$:

$\displaystyle A(t)=\mathcal{G}(t)A(0),$

where the mapping $\mathcal{G}_{n}(t)$ is defined by the formula

$\displaystyle\mathbb{R}^{1}\ni t\mapsto\mathcal{G}_{n}(t)g_{n}\doteq
e^{itH_{n}}g_{n}e^{-itH_{n}}.$ (4)

In the space $\mathfrak{L}(\mathcal{H}_{n})$ one-parameter mapping (4) forms a
$\ast$-weakly continuous group of operators, the infinitesimal generator of
which coincides with operator (3) on the domain of its definition.

The average value (mathematical expectation) of the observable $A(t)$ at time
$t\in\mathbb{R}$ is determined by a continuous linear functional, which is
represented by such an expansion in a series [23]:

$\displaystyle\langle
A\rangle(t)=(I,D(0))^{-1}\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\mathrm{Tr}_{1,\ldots,n}\,A_{n}(t)\,D_{n}^{0},$
(5)

where the sequence $D(0)=(I,D_{1}^{0},\ldots,D_{n}^{0},\ldots)$ of positive
self-adjoint operators $D_{n}^{0}\in\mathfrak{L}^{1}(\mathcal{H}_{n})$ is the
sequence of density operators by which they are described all possible states
of the quantum system of a non-fixed number of particles at the initial
instant, the coefficient

$(I,D(0))={\sum\limits}_{n=0}^{\infty}\frac{1}{n!}\mathrm{Tr}_{1,\ldots,n}D_{n}^{0}$

is the normalizing factor. Functional (5), which determines the duality of
observables and a state, exists for
$D_{n}^{0}\in\mathfrak{L}^{1}(\mathcal{H}_{n})$ and
$A_{n}(t)\in\mathfrak{L}(\mathcal{H}_{n})$.

For the normalizing factor, the following equality is valid

$\displaystyle(I,D(0))=(I,\mathcal{G}^{\ast}(t)D(0)).$ (6)

On the space $\mathfrak{L}^{1}(\mathcal{F}_{\mathcal{H}})$ the one-parameter
mapping $\mathcal{G}^{\ast}(t)=\oplus_{n=0}^{\infty}\mathcal{G}^{\ast}_{n}(t)$
adjoint to the group (4) is defined by

$\displaystyle\mathbb{R}^{1}\ni t\mapsto\mathcal{G}^{\ast}_{n}(t)f_{n}\doteq
e^{-itH_{n}}f_{n}e^{itH_{n}},$ (7)

and forms a strongly continuous isometric group of operators, which preserves
positivity and self-adjointness of operators.

Due to equality (6) for functional (5) the following representation is valid:

$\displaystyle\hskip-34.1433pt(A(t),D(0))=(I,D(0))^{-1}\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\mathrm{Tr}_{1,\ldots,n}\,\mathcal{G}_{n}(t)A_{n}^{0}\,D_{n}^{0}=$
$\displaystyle(I,\mathcal{G}^{\ast}(t)D(0))^{-1}\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\mathrm{Tr}_{1,\ldots,n}\,A_{n}^{0}\,\mathcal{G}^{\ast}_{n}(t)D_{n}^{0}=$
$\displaystyle(I,D(t))^{-1}(A(0),D(t)),$

that is, the evolution of quantum systems of many particles in an equivalent
way can be described as the evolution of the state. Indeed, the evolution of
all possible states, i.e. the sequence
$D(t)=(1,D_{1}(t),\ldots,D_{n}(t),\ldots)\in\mathfrak{L}^{1}(\mathcal{F}_{\mathcal{H}})$
of the density operators $D_{n}(t),\,n\geq 1$, is described by the Cauchy
problem for a sequence of the von Neumann equations (quantum Liouville
equations):

$\displaystyle\frac{\partial}{\partial t}D(t)=\mathcal{N}^{\ast}D(t),$ (8)
$\displaystyle D(t)|_{t=0}=D(0),$ (9)

where the generator
$\mathcal{N}^{\ast}=\oplus^{\infty}_{n=0}\mathcal{N}^{\ast}_{n}$ of the von
Neumann equation (8) is adjoint operator to generator (3) of the Heisenberg
equation (1) and is defined by the formula

$\displaystyle\mathcal{N}^{\ast}_{n}f_{n}\doteq-i\,(H_{n}f_{n}-f_{n}H_{n}).$
(10)

Operator (10) has such structure:
$\mathcal{N}^{\ast}_{n}=\sum_{j=1}^{n}\mathcal{N}^{\ast}(j)+\epsilon\sum_{j_{1}<j_{2}=1}^{n}\mathcal{N}^{\ast}_{\mathrm{int}}(j_{1},j_{2})$,
where the operator $\mathcal{N}^{\ast}(j)$ is the generator of the von Neumann
equation of noninteracting particles and the operator
$\mathcal{N}^{\ast}_{\mathrm{int}}$ is defined is defined by the operator of
the particle interaction:
$\mathcal{N}^{\ast}_{\mathrm{int}}(j_{1},j_{2})f_{n}\doteq-i\,(\Phi(j_{1},j_{2})f_{n}-f_{n}\Phi(j_{1},j_{2}))$.

Thus, the unique solution of the Cauchy problem (8),(9) is representing by the
group of operators (4)

$\displaystyle D(t)=\mathcal{G}^{\ast}(t)D(0).$

Note that the density operator is represented by a convex linear combination
of projectors of the first rank. Density operator, which is a projector of the
first rank $D_{n}(t)=P_{\psi_{n}}(t),\,\psi_{n}\in\mathcal{H}_{n},$ is known
as the pure state, and the arbitrary state is interpreted as a mixed state. As
a consequence of the validity for the projector of such equality:
$P_{\psi_{n}}(t)=P_{\psi_{n}(t)}$, where $\psi_{n}(t)=e^{-itH_{n}}\psi_{n}$,
the evolution of the pure state can also be described by the Cauchy problem
for a sequence of the Schrödinger equations:

$\displaystyle i\frac{\partial}{\partial t}\psi_{n}(t)=H_{n}\psi_{n}(t),$
$\displaystyle\psi(t)_{n}\big{|}_{t=0}=\psi_{n}^{0},\quad n\geq 1,$

where the operator $H_{n}$ is the Hamiltonian of the system of $n$ particles.

###  2.2 Dynamics of correlations of particle states

An alternative approach to describing the state of a quantum system of a
finite average number of particles is to describe the state using cumulants of
density operators, which are interpreted as correlations of the states of
clusters of particles [27],[34],[37].

We introduce sequence $g(t)=(I,g_{1}(t,1),\ldots,g_{s}(t,1,\ldots,s),\ldots)$
of correlation operators by means of cluster expansions of the density
operators $D(t)=(I,D_{1}(t,1),\ldots,D_{n}(t,1,\ldots,n),\ldots)$:

$\displaystyle\hskip-34.1433ptD_{n}(t,1,\ldots,n)=g_{n}(t,1,\ldots,n)+\sum\limits_{\mbox{\scriptsize$\begin{array}[]{c}\mathrm{P}:(1,\ldots,n)=\bigcup_{i}X_{i},\\\
|\mathrm{P}|>1\end{array}$}}\prod_{X_{i}\subset\mathrm{P}}g_{|X_{i}|}(t,X_{i}),\quad
n\geq 1,$ (13)

where
${\sum\limits}_{\mathrm{P}:(1,\ldots,n)=\bigcup_{i}X_{i},\,|\mathrm{P}|>1}$ is
the sum over all possible partitions $\mathrm{P}$ of the set of indices
$(1,\ldots,n)$ on $|\mathrm{P}|>1$ non-empty subsets
$X_{i}\subset(1,\ldots,n)$, which are not mutually intersecting.

The solutions of recurrent relations (13) are determined by the following
expansions:

$\displaystyle\hskip-42.67912ptg_{s}(t,1,\ldots,s)=D_{s}(t,1,\ldots,s)+$ (14)
$\displaystyle\hskip-34.1433pt\sum\limits_{\mbox{\scriptsize$\begin{array}[]{c}\mathrm{P}:(1,\ldots,s)=\bigcup_{i}X_{i},\\\
|\mathrm{P}|>1\end{array}$}}(-1)^{|\mathrm{P}|-1}(|\mathrm{P}|-1)!\,\prod_{X_{i}\subset\mathrm{P}}D_{|X_{i}|}(t,X_{i}),\quad
s\geq 1.$ (17)

The structure of expansions (14) is such that correlation operators can be
interpreted as cumulants (semi-invariants) of density operators (2.1).

If $g_{s}^{0}\in\mathfrak{L}^{1}(\mathcal{H}_{s}),\,s\geq 1$, then for
$t\in\mathbb{R}$ sequence of correlation operators (14) is the unique solution
of the Cauchy problem of the von Neumann hierarchy [34],[37]:

$\displaystyle\hskip-42.67912pt\frac{\partial}{\partial
t}g_{s}(t,1,\ldots,s)=\mathcal{N}^{\ast}_{s}g_{s}(t,1,\ldots,s)+$ (18)
$\displaystyle\hskip-34.1433pt\sum\limits_{\mathrm{P}:\,(1,\ldots,s)=X_{1}\bigcup
X_{2}}\,\sum\limits_{i_{1}\in X_{1}}\sum\limits_{i_{2}\in
X_{2}}\epsilon\,\mathcal{N}_{\mathrm{int}}^{\ast}(i_{1},i_{2})g_{|X_{1}|}(t,X_{1})g_{|X_{2}|}(t,X_{2}),$
$\displaystyle\hskip-42.67912ptg_{s}(t,1,\ldots,s)\big{|}_{t=0}=g_{s}^{0}(1,\ldots,s),\quad
s\geq 1,$ (19)

where the symbol ${\sum\limits}_{\mathrm{P}:\,(1,\ldots,s)=X_{1}\bigcup
X_{2}}$ means the sum over all possible partitions $\mathrm{P}$ of the set
$(1,\ldots,s)$ on two nonempty subsets $X_{1}$ and $X_{2}$, which are not
mutually intersect, and the operators $\mathcal{N}^{\ast}_{s}$,
$\mathcal{N}_{\mathrm{int}}^{\ast}(i_{1},i_{2})$ are defined by formulas (10).
We emphasize that the von Neumann hierarchy (18) is a set of recurrent
evolution equations.

If the initial state is described by a sequence of correlation operators
$g(0)=(I,g_{1}^{0}(1),\ldots,$
$g_{n}^{0}(1,\ldots,n),\ldots)\in\oplus_{n=0}^{\infty}\mathfrak{L}^{1}(\mathcal{H}_{n})$,
then the evolution of all possible states of a quantum system of many
particles, i.e., the sequence
$g(t)=(I,g_{1}(t,1),\ldots,g_{s}(t,1,\ldots,s),\ldots)$ of correlation
operators $g_{s}(t),\,s\geq 1$, is determined by such a group of nonlinear
operators [34]:

$\displaystyle\hskip-42.67912ptg_{s}(t,1,\ldots,s)=\mathcal{G}(t;1,\ldots,s\mid
g(0))\doteq$ (20)
$\displaystyle\hskip-28.45274pt\sum\limits_{\mathrm{P}:\,(1,\ldots,s)=\bigcup_{j}X_{j}}\mathfrak{A}_{|\mathrm{P}|}(t,\\{X_{1}\\},\ldots,\\{X_{|\mathrm{P}|}\\})\prod_{X_{j}\subset\mathrm{P}}g_{|X_{j}|}^{0}(X_{j}),\quad
s\geq 1,$

where $\sum_{\mathrm{P}:\,(1,\ldots,s)=\bigcup_{j}X_{j}}$ is the sum over all
possible partitions $\mathrm{P}$ of the set of indices $(1,\ldots,n)$ on
$|\mathrm{P}|>1$ non-empty subsets $X_{j}$, which are not mutually
intersecting, the set $(\\{X_{1}\\},\ldots,\\{X_{|\mathrm{P}|}\\})$ consists
of elements that 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 (20) is the
$|\mathrm{P}|$-th order cumulant of groups of operators (7), which is
determined by such expansion

$\displaystyle\hskip-42.67912pt\mathfrak{A}_{|\mathrm{P}|}(t,\\{X_{1}\\},\ldots,\\{X_{|\mathrm{P}|}\\})\doteq$
(21)
$\displaystyle\hskip-14.22636pt\sum\limits_{\mathrm{P}^{{}^{\prime}}:\,(\\{X_{1}\\},\ldots,\\{X_{|\mathrm{P}|}\\})=\bigcup_{k}Z_{k}}(-1)^{|\mathrm{P}^{{}^{\prime}}|-1}({|\mathrm{P}^{{}^{\prime}}|-1})!\prod\limits_{Z_{k}\subset\mathrm{P}^{{}^{\prime}}}\mathcal{G}^{\ast}_{|\theta(Z_{k})|}(t,\theta(Z_{k})),$

where $\theta$ is the declusterization mapping:
$\theta(\\{X_{1}\\},\ldots,\\{X_{|\mathrm{P}|}\\})\doteq(1,\ldots,s)$.

In the absence of correlations between the particles at the initial instant
(the initial state satisfies the chaos condition [3],[66]), that is, of the
sequence of initial correlation operators
$g^{(c)}(0)=(0,g_{1}^{0}(1),0,\ldots,0,\ldots)$ (in the case of the
Maxwell–Boltzmann statistics in terms of density operators this condition
means that
$D^{(c)}(0)=(I,D_{1}^{0}(1),\ldots,\prod^{n}_{i=1}D_{1}^{0}(i),\ldots)$),
expansions (20) take the form:

$\displaystyle
g_{s}(t,1,\ldots,s)=\mathfrak{A}_{s}(t,1,\ldots,s)\,\prod\limits_{i=1}^{s}g_{1}^{0}(i),\quad
s\geq 1,$

where the $s$-th order cumulant $\mathfrak{A}_{s}(t)$ of the groups of
operators (7) is defined by the following expansion

$\displaystyle\hskip-34.1433pt\mathfrak{A}_{s}(t,1,\ldots,s)=\sum\limits_{\mathrm{P}:\,(1,\ldots,s)=\bigcup_{i}X_{i}}(-1)^{|\mathrm{P}|-1}({|\mathrm{P}|-1})!\prod\limits_{X_{i}\subset\mathrm{P}}\mathcal{G}^{\ast}_{|X_{i}|}(t,X_{i}),$
(22)

and where it was used notations accepted in formula (20).

Thus, the cumulant origin of correlation operators (14) induces a cumulant
structure of one-parameter mapping (20).

We emphasize that the dynamics of correlations, i.e. the hierarchy of
fundamental equations (18) which describes the evolution of state correlations
can be used as a basis for describing the evolution of states both a system of
a finite and infinite number of particles instead of the von Neumann equations
(8) for density operators.

###  2.3 The BBGKY hierarchies of evolution equations

To describe the evolution of quantum systems of both finite and infinite
numbers of particles, another approach to the description of observables and a
state is used, which equivalent to the approach formulated above in the case
of systems of a finite average number of particles. [6],[12].

Indeed, the mean value functional of observables (5) can be else represented
in the following form

$\displaystyle\hskip-34.1433pt\langle
A\rangle(t)=(I,D(0))^{-1}\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\mathrm{Tr}_{1,\ldots,n}\,A_{n}(t)\,D_{n}^{0}=$
(23)
$\displaystyle\sum\limits_{s=0}^{\infty}\frac{1}{s!}\,\mathrm{Tr}_{1,\ldots,s}\,B_{s}(t,1,\ldots,s)\,F_{s}^{0}(1,\ldots,s),$

where to describe the observables and the state introduced sequences of the
reduced ($s$-particle) operators of observables
$B(t)=(B_{0},B_{1}(t,1),\ldots,B_{s}(t,1,\ldots,s),\ldots)$ and the reduced
($s$-particle) density operators
$F(0)=(I,F_{1}^{0}(1),\ldots,F_{s}^{0}(1,\ldots,s),\ldots)$, respectively
[6],[66]. Therefore, the reduced observables are defined in terms of the
observables by such expansions [28]:

$\displaystyle\hskip-34.1433ptB_{s}(t,1,\ldots,s)=\sum_{n=0}^{s}\,\frac{(-1)^{n}}{n!}\sum_{j_{1}\neq\ldots\neq
j_{n}=1}^{s}A_{s-n}(t,(1,\ldots,s)\setminus(j_{1},\ldots,j_{n})),\quad s\geq
1,$ (24)

and reduced density operators are defined by density operators as follows
[12]:

$\displaystyle\hskip-34.1433ptF_{s}^{0}(1,\ldots,s)=(I,D)^{-1}\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\mathrm{Tr}_{s+1,\ldots,s+n}\,D_{s+n}^{0}(1,\ldots,s+n),\quad
s\geq 1.$ (25)

We emphasize that the possibility of describing observables and a state with
the help of corresponding reduced operators naturally arises as a result of
dividing the series in the expression (5) by the series of the normalization
factor, i.e. in consequence of redefining the representation of mean value
functional (23).

If at the initial moment an observable is determined by the sequence of
reduced observables
$B(0)=(B_{0},B_{1}^{0}(1),\ldots,B_{s}^{0}(1,\ldots,s),\ldots)\in\mathfrak{L}(\mathcal{F}_{\mathcal{H}})$,
then for arbitrary $t\in\mathbb{R}$ the sequence
$B(t)=(B_{0},B_{1}(t,1),\ldots,B_{s}(t,1,\ldots,s),\ldots)$ of reduced
observables (24) satisfies the Cauchy problem of the quantum dual BBGKY
hierarchy [21],[23],[19]:

$\displaystyle\hskip-34.1433pt\frac{\partial}{\partial
t}B_{s}(t,1,\ldots,s)=\big{(}\sum\limits_{j=1}^{s}\mathcal{N}(j)+\epsilon\hskip-5.69054pt\sum\limits_{j_{1}<j_{2}=1}^{s}\mathcal{N}_{\mathrm{int}}(j_{1},j_{2})\big{)}B_{s}(t,1,\ldots,s)+$
(26) $\displaystyle\hskip-14.22636pt+\epsilon\,\sum_{j_{1}\neq
j_{2}=1}^{s}\mathcal{N}_{\mathrm{int}}(j_{1},j_{2})B_{s-1}(t,1,\ldots,j_{1}-1,j_{1}+1,\ldots,s),$
$\displaystyle\hskip-34.1433ptB_{s}(t,1,\ldots,s)_{\mid
t=0}=B_{s}^{0}(1,\ldots,s),\quad s\geq 1,$ (27)

where the notation accepted in formula (3) was used.

We remark that the hierarchy of equations (26) has the structure of recurrent
evolution equations, for example,

$\displaystyle\hskip-34.1433pt\frac{\partial}{\partial
t}B_{1}(t,1)=\mathcal{N}(1)B_{1}(t,1),$
$\displaystyle\hskip-34.1433pt\frac{\partial}{\partial
t}B_{2}(t,1,2)=\big{(}\sum\limits_{j=1}^{2}\mathcal{N}(j)+\epsilon\,\mathcal{N}_{\mathrm{int}}(1,2)\big{)}B_{2}(t,1,2)+\epsilon\,\mathcal{N}_{\mathrm{int}}(1,2)\big{(}B_{1}(t,1)+B_{1}(t,2)\big{)}.$

The solution of the Cauchy problem (26),(27) is represented by the following
expansions [28],[19]:

$\displaystyle\hskip-34.1433ptB_{s}(t,1,\ldots,s)=\sum_{n=0}^{s}\,\frac{1}{n!}\sum_{j_{1}\neq\ldots\neq
j_{n}=1}^{s}\mathfrak{A}_{1+n}\big{(}t,\\{(1,\ldots,s)\setminus(j_{1},\ldots,j_{n})\\},$
(28)
$\displaystyle(j_{1},\ldots,j_{n})\big{)}\,B_{s-n}^{0}(1,\ldots,j_{1}-1,j_{1}+1,\ldots,j_{n}-1,j_{n}+1,\ldots,s),\quad
s\geq 1,$

where the generating operator of this expansion is the $(1+n)$-th order
cumulant of the groups of operators (4):

$\displaystyle\hskip-34.1433pt\mathfrak{A}_{1+n}(t,\\{(1,\ldots,s)\setminus(j_{1},\ldots,j_{n})\\},(j_{1},\ldots,j_{n}))\doteq$
$\displaystyle\sum\limits_{\mathrm{P}:\,(\\{(1,\ldots,s)\setminus(j_{1},\ldots,j_{n})\\},(j_{1},\ldots,j_{n}))={\bigcup}_{i}X_{i}}(-1)^{\mathrm{|P|}-1}({\mathrm{|P|}-1})!\prod_{X_{i}\subset\mathrm{P}}\mathcal{G}_{|\theta(X_{i})|}(t,\theta(X_{i})),\quad
n\geq 0,$

and used notation similar to the formula (21).

We note that expansion (28) for the solution can be represented as an
iteration series (perturbation theory series) of recurrent evolution equations
(26) in the result of the application of analogues of the Duhamel equation to
the generating operators, i.e. to the cumulants of groups of operators (4).

According to the definition of additive type observables (24) correspond to
the one-component sequences $B^{(1)}(0)=(0,b_{1}^{0}(1),0,\ldots)$ of reduced
observables, and the sequences $B^{(k)}(0)=(0,\ldots,$
$0,b_{k}^{0}(1,\ldots,k),0,\ldots)$ correspond to $k$-ary (non-additive) type
observables. Then for a certain type of observables, the structure of
expansion (28) takes the appropriate form

$\displaystyle
B_{s}^{(1)}(t,1,\ldots,s)=\mathfrak{A}_{s}(t,1,\ldots,s)\sum_{j=1}^{s}b_{1}^{0}(j),\quad
s\geq 1,$ (29)

where the generating operator $\mathfrak{A}_{s}(t)$ is the $s$-th order
cumulant of the groups of operators (4), and, if $s\geq k$,

$\displaystyle\hskip-34.1433ptB_{s}^{(k)}(t,1,\ldots,s)=\frac{1}{(s-k)!}\sum_{j_{1}\neq\ldots\neq
j_{s-k}=1}^{s}\mathfrak{A}_{1+s-k}\big{(}t,\\{(1,\ldots,s)\setminus(j_{1},\ldots,j_{s-k})\\},j_{1},\ldots,$
(30) $\displaystyle
j_{s-k}\big{)}\,b_{k}^{0}(1,\ldots,j_{1}-1,j_{1}+1,\ldots,j_{s}-k-1,j_{s}-k+1,\ldots,s),$

and, if $1\leq s<k$, we have: $B_{s}^{(k)}(t)=0$.

Traditionally, the evolution of many-particle systems is described within the
framework of the evolution of the state governed by the BBGKY hierarchy for
reduced density operators [3],[6],[12],[66].

Indeed, for functional (23) the following representation is hold

$\displaystyle(B(t),F(0))=(B(0),F(t)),$

that is, the evolution of quantum systems of many particles in an equivalent
way can be described as the evolution of the state using reduced density
operators (25).

If $F(0)\in\mathfrak{L}^{1}(\mathcal{F}_{\mathcal{H}})$, the for arbitrary
$t\in\mathbb{R}$ the sequence $F(t)=(I,F_{1}(t,1),\ldots,F_{s}(t,$
$1,\ldots,s),\ldots)$ of reduced density operators (25) satisfies the Cauchy
problem of the quantum BBGKY hierarchy [6],[12]:

$\displaystyle\hskip-34.1433pt\frac{\partial}{\partial
t}F_{s}(t,1,\ldots,s)=\mathcal{N}^{\ast}_{s}F_{s}(t,1,\ldots,s)+\epsilon\,\sum\limits_{j=1}^{s}\mathrm{Tr}_{s+1}\mathcal{N}^{\ast}_{\mathrm{int}}(j,s+1)F_{s+1}(t,1,\ldots,s,s+1),$
(31)
$\displaystyle\hskip-34.1433ptF_{s}(t,1,\ldots,s)\mid_{t=0}=F_{s}^{0}(1,\ldots,s),\quad
s\geq 1,$ (32)

where the notation from formula (10) is used.

The solution of the Cauchy problem (31),(32) is represented by the following
series [35],[36]:

$\displaystyle\hskip-34.1433ptF_{s}(t,1,\ldots,s)=$ (33)
$\displaystyle\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\mathrm{Tr}_{s+1,\ldots,{s+n}}\,\mathfrak{A}_{1+n}(t,\\{1,\ldots,s\\},s+1,\ldots,{s+n})F_{s+n}^{0}(1,\ldots,{s+n}),\quad
s\geq 1,$

where the generating operator of this series

$\displaystyle\hskip-42.67912pt\mathfrak{A}_{1+n}(t,\\{1,\ldots,s\\},s+1,\ldots,{s+n})=$
(34)
$\displaystyle\sum\limits_{\mathrm{P}\,:(\\{1,\ldots,s\\},s+1,\ldots,{s+n})={\bigcup\limits}_{i}X_{i}}(-1)^{|\mathrm{P}|-1}(|\mathrm{P}|-1)!\prod_{X_{i}\subset\mathrm{P}}\mathcal{G}^{\ast}_{|\theta(X_{i})|}(t,\theta(X_{i}))$

is the $(1+n)$-th order cumulant of the groups of operators (7). In expansion
(34) symbol ${\sum\limits}_{\mathrm{P}}$ means the sum over all possible
partitions $\mathrm{P}$ of the set $(\\{1,\ldots,s\\},s+1,\ldots,{s+n})$ on
$|\mathrm{P}|$ nonampty subsets
$X_{i}\subset(\\{1,\ldots,s\\},s+1,\ldots,{s+n})$, which do not mutually
intersect, and the notations introduced in (20) are used.

We remark that one of the methods for constructing solutions (33) and (28) is
based on the application of cluster expansions [23],[36] to groups of
operators (7) and (4), which are the generating operators of the series (25)
for the reduced density operators and the expansions (24) for the reduced
observables ones, respectively.

We note that a common form of representation of the solution of the Cauchy
problem of the BBGKY hierarchy is its representation as a perturbation theory
series (iteration series of the BBGKY hierarchy) [6],[57] (see also [3] [12]
and therein references):

$\displaystyle\hskip-34.1433ptF_{s}(t,1,\ldots,s)=$ (35)
$\displaystyle\sum\limits_{n=0}^{\infty}\,\int\limits_{0}^{t}dt_{1}\ldots\int\limits_{0}^{t_{n-1}}dt_{n}\mathrm{Tr}_{s+1,\ldots,s+n}\mathcal{G}^{\ast}_{s}(t-t_{1})\sum\limits_{j_{1}=1}^{s}\mathcal{N}^{\ast}_{\mathrm{int}}(j_{1},s+1))\mathcal{G}^{\ast}_{s+1}(t_{1}-t_{2})\ldots$
$\displaystyle\mathcal{G}^{\ast}_{s+n-1}(t_{n-1}-t_{n})\sum\limits_{j_{n}=1}^{s+n-1}\mathcal{N}^{\ast}_{\mathrm{int}}(j_{n},s+n))\mathcal{G}^{\ast}_{s+n}(t_{n})F_{s+n}^{0}(1,\ldots,s+n),\quad
s\geq 1,$

where notations from the expression (10) are used. The representation of the
solution by the series (33) is equivalent to series (35) due to the validity
under the appropriate conditions for the initial data and the interaction
potential of particles of analogs of the Duhamel equation for generating
operators (34) of series (33) that is, for cumulants of groups of operators
(4).

Thus, there are two approaches to describing the evolution of quantum systems
of many particles, namely, in terms of observables whose evolution is governed
by the dual BBGKY hierarchy (26), or in terms of a state whose evolution is
governed by the BBGKY hierarchy (31). For systems of finitely many particles,
these hierarchies of evolution equations are equivalent to the Heisenberg
equation (1) and the von Neumann equation (8), respectively.

In paper [23], these hierarchies of evolution equations are generalized for
many-particle systems with multiparticle interaction potentials.

An alternative approach, as noted above, to the description of the evolution
of the state of quantum systems, consisting of both finitely and infinitely
many particles, can be formulated by means of operators defined by cluster
expansions of the reduced density operators, namely:

$\displaystyle\hskip-42.67912ptG_{s}(t,1,\ldots,s)=\sum\limits_{\mbox{\scriptsize$\begin{array}[]{c}\mathrm{P}:(1,\ldots,s)=\bigcup_{i}X_{i}\end{array}$}}(-1)^{|\mathrm{P}|-1}(|\mathrm{P}|-1)!\,\prod_{X_{i}\subset\mathrm{P}}F_{|X_{i}|}(t,X_{i}),\quad
s\geq 1,$ (37)

where the notation from formula (14) was used. Such cumulants of reduced
density operators are interpreted as reduced correlations of the state [6].

On a microscopic scale, the macroscopic characteristics of the fluctuations of
the observables are directly determined by the reduced correlation operators,
for example, the functional of dispersion of the additive type observables,
that is, of such sequences
$A^{(1)}=(0,a_{1}(1),\ldots,\sum_{i=1}^{n}a_{1}(i),\ldots)$, is represented by
the following formula

$\displaystyle\hskip-34.1433pt\langle(A^{(1)}-\langle
A^{(1)}\rangle)^{2}\rangle(t)=\mathrm{Tr}_{1}\,(a_{1}^{2}(1)-\langle
A^{(1)}\rangle^{2}(t))G_{1}(t,1)+\mathrm{Tr}_{1,2}\,a_{1}(1)a_{1}(2)G_{2}(t,1,2),$

where the mean value functional of observables of an additive type we denoted
by $\langle A^{(1)}\rangle(t)=\mathrm{Tr}_{1}\,a_{1}(1)G_{1}(t,1)$.

If $G(0)\in\in\mathfrak{L}^{1}(\mathcal{F}_{\mathcal{H}})$, then for arbitrary
$t\in\mathbb{R}$ the sequence of reduced correlation operators (37) satisfies
the Cauchy problem for the hierarchy of nonlinear evolution equations (quantun
nonlinear BBGKY hierarchy) [6]:

$\displaystyle\hskip-34.1433pt\frac{\partial}{\partial
t}G_{s}(t,1,\ldots,s)=\mathcal{N}^{\ast}_{s}G_{s}(t,1,\ldots,s)+$ (38)
$\displaystyle\sum\limits_{\mathrm{P}:\,(1,\ldots,s)=X_{1}\bigcup
X_{2}}\,\sum\limits_{i_{1}\in X_{1}}\sum\limits_{i_{2}\in
X_{2}}\epsilon\,\mathcal{N}_{\mathrm{int}}^{\ast}(i_{1},i_{2})G_{|X_{1}|}(t,X_{1})G_{|X_{2}|}(t,X_{2}))+$
$\displaystyle\mathrm{Tr}_{s+1}\sum_{i\in(1,\ldots,s)}\epsilon\,\mathcal{N}^{\ast}_{\mathrm{int}}(i,s+1)\big{(}G_{s+1}(t,1,\ldots,s+1)+$
$\displaystyle\sum_{\mbox{\scriptsize$\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}$}}G_{|X_{1}|}(t,X_{1})G_{|X_{2}|}(t,X_{2})\big{)},$ (41)
$\displaystyle\hskip-34.1433ptG_{s}(t,1,\ldots,s)\big{|}_{t=0}=G_{s}^{0}(,1,\ldots,s),\quad
s\geq 1,$ (42)

where the notations for the hierarchy of equations (18) is used.

In the case of the initial state in the absence of correlations between the
particles, i.e. the state described by the sequence
$G^{(c)}=(0,G_{1}^{0},0,\ldots,0,\ldots)$, the solution of the Cauchy problem
(38),(42) is represented by the following series:

$\displaystyle\hskip-42.67912ptG_{s}(t,1,\ldots,s)=\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\mathrm{Tr}_{s+1,\ldots,s+n}\,\mathfrak{A}_{s+n}(t;1,\ldots,s+n)\prod_{i=1}^{s+n}G_{1}^{0}(i),\quad
s\geq 1,$ (43)

where the generating operator $\mathfrak{A}_{s+n}(t)$ is the $(s+n)$-th order
cumulant (22) of groups of operators (7). We note that for the specified
initial state, the expressions for the reduced density operators (33) and the
reduced correlation operators (43) differ only in the order of the generating
operator for the corresponding terms of the series representing these
operators.

For an arbitrary initial state (42) a solution of the Cauchy problem for the
nonlinear BBGKY hierarchy (38) was constructed in the paper [25] (see also
[27]).

Note that the description of the evolution of the state of quantum systems of
many particles by means of both reduced density operators and reduced
correlation operators can be based on an approach founded on the dynamics of
correlations, which is governed by the von Neumann hierarchy (18) for
correlation operators. Within the framework of this approach, the reduced
density operators are defined by the following series:

$\displaystyle\hskip-34.1433ptF_{s}(t,1,\ldots,s)\doteq\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\mathrm{Tr}_{s+1,\ldots,s+n}\,\,g_{1+n}(t,\\{1,\ldots,s\\},s+1,\ldots,s+n),\quad
s\geq 1,$ (44)

where the correlation operators $g_{1+n}(t),\,n\geq 0,$ of particle cluster
and particles are represented by the following expansions:

$\displaystyle\hskip-34.1433ptg_{1+n}(t,\\{1,\ldots,s\\},s+1,\ldots,s+n)=$
(45)
$\displaystyle\sum\limits_{\mathrm{P}:\,(\\{1,\ldots,s\\},\,s+1,\ldots,s+n)=\bigcup_{i}X_{i}}\mathfrak{A}_{|\mathrm{P}|}\big{(}t,\\{\theta(X_{1})\\},\ldots,\\{\theta(X_{|\mathrm{P}|})\\}\big{)}\prod_{X_{i}\subset\mathrm{P}}g_{|X_{i}|}^{0}(X_{i}),\quad
n\geq 0,$

in which the notations from formula (20) are used. The reduced correlation
operators are defined by the corresponding series:

$\displaystyle\hskip-22.76219ptG_{s}(t,1,\ldots,s)\doteq\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\mathrm{Tr}_{s+1,\ldots,s+n}\,\,g_{s+n}(t,1,\ldots,s+n),\quad
s\geq 1,$ (46)

where the correlation operators $g_{s+n}(t),\,n\geq 0,$ are represented by
expansions (20).

Thus, as a result of definitions (44) and (46), we establish that the cumulant
structure of generating operators of expansions for correlation operators
(20), or more general expansions (45), induces a cumulant structure of series
for reduced density operators (33) and reduced correlation operators (43),
i.e. in fact, the evolution of a system of infinitely many particles is
generated by the dynamics of correlations.

Note also that in the article [29] the hierarchies of evolution equations
(18),(26),(31),(38) are studied as evolution equations in functional
derivatives.

##  3 On quantum kinetic equations

This section describes an approach to describing the evolution of a state
using the state of a typical particle of a quantum system of many particles,
or, in other words, discusses the origin of describing the evolution of a
state by means of quantum kinetic equations [G09],[33],[38].

###  3.1 The origin of the kinetic evolution of a state

We consider a system of many particles that satisfy the Maxwell-Boltzmann
statistics, in the absence of correlations between particles at the initial
time, i.e. the initial state of which is determined by a one-particle density
operator, namely, such a sequence of reduced density operators
$F^{(c)}=(0,F_{1}^{0}(1),\ldots,\prod^{n}_{i=1}F_{1}^{0}(i),\ldots)$. We
emphasize that the formulated assumption about the initial state is inherent
in the kinetic theory of systems of many particles [3],[23],[43].

Due to the fact that the initial state is determined by a one-particle density
operator, for the mean value functional of observables (23) the following
representation is valid

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

that is, the evolution of all possible states is described by a sequence
$F(t\mid F_{1}(t))=\big{(}I,F_{1}(t),F_{2}(t\mid F_{1}(t)),\ldots,F_{s}(t\mid
F_{1}(t)),\ldots\big{)}$ of reduced functionals of the state
$F_{s}(t,1,\ldots,s\mid F_{1}(t)),\,s\geq 2,$ which are represented by the
series:

$\displaystyle\hskip-34.1433ptF_{s}\bigl{(}t,1,\ldots,s\mid F_{1}(t)\bigr{)}=$
(47)
$\displaystyle\sum_{n=0}^{\infty}\frac{1}{n!}\,\mathrm{Tr}_{s+1,\ldots,s+n}\,\mathfrak{V}_{1+n}\bigl{(}t,\\{1,\ldots,s\\},s+1,\ldots,s+n\bigr{)}\prod_{i=1}^{s+n}F_{1}(t,i),\quad
s\geq 2,$

with respect of a one-particle density operator

$\displaystyle\hskip-34.1433ptF_{1}(t,1)=\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\mathrm{Tr}_{2,\ldots,{1+n}}\,\mathfrak{A}_{1+n}(t,1,\ldots,n+1)\prod_{i=1}^{n+1}F_{1}^{0}(i),$
(48)

which describes the evolution of the state of a typical particle of a quantum
system of many particles. The generating operators of the series (48) are the
cumulants (34) of the corresponding order of the groups of operators (7).

The reduced functionals of the state $F_{s}(t\mid F_{1}(t)),\,s\geq 2$,
describe all possible correlations that created in the process of evolution of
the quantum system of many particles in terms of the state of a typical
particle. The $(1+n)$-th order generating operator of the series (47) is
determined by the following expansion [38]

$\displaystyle\hskip-25.60747pt\mathfrak{V}_{1+n}\bigl{(}t,\\{1,\ldots,s\\},s+1,\ldots,s+n\bigr{)}=n!\,\sum_{k=0}^{n}\,(-1)^{k}\,\sum_{n_{1}=1}^{n}\ldots$
(49) $\displaystyle\hskip-19.91692pt\sum_{n_{k}=1}^{n-n_{1}-\ldots-
n_{k-1}}\frac{1}{(n-n_{1}-\ldots-
n_{k})!}\,\widehat{\mathfrak{A}}_{1+n-n_{1}-\ldots-
n_{k}}(t,\\{1,\ldots,s\\},s+1,\ldots,s+n-$
$\displaystyle\hskip-19.91692ptn_{1}-\ldots-
n_{k})\prod_{j=1}^{k}\,\sum\limits_{\mbox{\scriptsize$\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}$}}\frac{1}{|\mathrm{D}_{j}|!}\sum_{i_{1}\neq\ldots\neq
i_{|\mathrm{D}_{j}|}=1}^{s+n-n_{1}-\ldots-
n_{j}}\,\prod_{X_{l_{j}}\subset\mathrm{D}_{j}}\,\frac{1}{|X_{l_{j}}|!}\,\widehat{\mathfrak{A}}_{1+|X_{l_{j}}|}(t,i_{l_{j}},X_{l_{j}}),$
(52)

where 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 generating
operator of this expansion $\widehat{\mathfrak{A}}_{1+n}(t)$ is the $(1+n)$-th
order cumulant (34) of the groups of scattering operators

$\displaystyle\widehat{\mathcal{G}}_{n}(t)\doteq\mathcal{G}_{n}^{\ast}(t,1,\ldots,n)\prod_{i=1}^{n}(\mathcal{G}_{1}^{\ast}(t,i))^{-1},\quad
n\geq 1,$

We give the simplest examples of generating operators of expansion (49):

$\displaystyle\hskip-22.76219pt\mathfrak{V}_{1}(t,\\{1,\ldots,s\\})=\widehat{\mathfrak{A}}_{1}(t,\\{1,\ldots,s\\}),$
$\displaystyle\hskip-22.76219pt\mathfrak{V}_{1+1}(t,\\{1,\ldots,s\\},s+1)=\widehat{\mathfrak{A}}_{2}(t,\\{1,\ldots,s\\},s+1)-\widehat{\mathfrak{A}}_{1}(t,\\{1,\ldots,s\\})\sum_{i=1}^{s}\widehat{\mathfrak{A}}_{2}(t,i,s+1).$

Thus, according to the definition of (44), the cumulant structure of
generating operators of expansions for correlation operators (45) induces a
generalized cumulant structure of generating operators for series of reduced
functionals of the state (47).

Noticing that for the initial state specified by the one-particle density
operator $F^{(c)}$, the Cauchy problem (38),(42) is not correctly well-
defined, because the initial data are not are independent for each unknown
reduced density operator of the BBGKY hierarchy. As a consequence, such a
Cauchy problem can be reformulated as a new Cauchy problem for a one-particle
density operator with an independent initial condition and a sequence of
explicitly defined functionals of the solution of the Cauchy problem of the
evolution equation for a one-particle density operator (kinetic equation). In
this case, the method of constructing reduced state functionals (47) is based
on the application of the species of cluster expansions, the so-called kinetic
cluster expansions [38], to generating operators (34) of series representing
reduced density operators (33).

###  3.2 The generalized quantum kinetic equation

If $F_{1}^{0}\in\mathfrak{L}^{1}(\mathcal{H})$, then for an arbitrary
$t\in\mathbb{R}$ a one-particle density operator (48) satisfies the Cauchy
problem for the generalized quantum kinetic equation [38]:

$\displaystyle\hskip-22.76219pt\frac{\partial}{\partial
t}F_{1}(t,1)=\mathcal{N}^{\ast}(1)F_{1}(t,1)+\epsilon\,\mathrm{Tr}_{2}\,\mathcal{N}_{\mathrm{int}}^{\ast}(1,2)F_{2}\bigl{(}t,1,2\mid
F_{1}(t)\bigr{)},$ (53)
$\displaystyle\hskip-22.76219ptF_{1}(t,1)\big{|}_{t=0}=F_{1}^{0}(1),$ (54)

where the collision integral is determined by the two-particle functional of
the state (47) and the notation is used (10).

In [38], the existence theorem of the Cauchy problem (53), (54) in the space
of the trace class operators is proved.

Thus, for the initial state in the absence of correlations between particles,
i.e. of the state specified by the one-particle density operator, the
evolution of all possible states of the quantum system of many particles can
be described without any approximations by means of a one-particle density
operator (53) and of a sequence of functionals (47) of this operator (48).

###  3.3 Scaling properties of state evolution

The generally accepted philosophy of describing evolution by kinetic equations
is as follows [8],[43]. If the initial state is determined by the state of a
typical particle of the system, i.e. at the initial moment there are no
correlations between particles (chaos condition), then in a certain scaling
approximation [3],[15],[23],[42] the evolution of the state of a system of
many particles can be effectively described within the state of a typical
particle, i.e. by a one-particle density operator, which is governed by the
corresponding nonlinear kinetic equation.

Further, the scaling asymptotic behavior of the reduced functionals of the
state $F(t\mid F_{1}(t))$ in the specific case of the mean-field limit is
considered [40].

Let there exists the mean-field limit of the initial one-particle density
operator in the following sense

$\displaystyle\lim\limits_{\epsilon\rightarrow
0}\big{\|}\epsilon\,F_{1}^{0}-f_{1}^{0}\big{\|}_{\mathfrak{L}^{1}(\mathcal{H})}=0,$
(55)

where $\epsilon>0$ is a scaling parameter.

Since for an arbitrary finite time interval for an asymptotically perturbed
first-order cumulant of groups of operators (7), i.e. for a strongly
continuous group (7), such equality holds

$\displaystyle\lim\limits_{\epsilon\rightarrow
0}\Big{\|}\mathcal{G}^{\ast}_{s}(t,1,\ldots,s)f_{s}-\prod\limits_{j=1}^{s}\mathcal{G}^{\ast}_{1}(t,j)f_{s}\Big{\|}_{\mathfrak{L}^{1}(\mathcal{H}_{s})}=0.$

then for the $(1+n)$-th order cumulant of asymptotically perturbed groups of
operators (7) the equality valid

$\displaystyle\hskip-34.1433pt\lim\limits_{\epsilon\rightarrow
0}\Big{\|}\frac{1}{\epsilon^{n}}\,\frac{1}{n!}\mathfrak{A}_{1+n}(t,\\{1,\ldots,s\\},s+1,\ldots,s+n)f_{s+n}-$
(56)
$\displaystyle\hskip-5.69054pt\int\limits_{0}^{t}dt_{1}\ldots\int\limits_{0}^{t_{n-1}}dt_{n}\prod\limits_{j=1}^{s}\mathcal{G}_{1}^{\ast}(t-t_{1},j)\sum\limits_{i_{1}=1}^{s}\mathcal{N}^{\ast}_{\mathrm{int}}(i_{1},s+1)\prod\limits_{j_{1}=1}^{s+1}\mathcal{G}_{1}^{\ast}(t_{1}-t_{2},j_{1})\ldots$
$\displaystyle\hskip-5.69054pt\prod\limits_{j_{n-1}=1}^{s+n-1}\mathcal{G}^{\ast}_{1}(t_{n-1}-t_{n},j_{n-1})\sum\limits_{i_{n}=1}^{s+n-1}\mathcal{N}^{\ast}_{\mathrm{int}}(i_{n},s+n)\prod\limits_{j_{n}=1}^{s+n}\mathcal{G}^{\ast}_{1}(t_{n},j_{n})f_{s+n}\Big{\|}_{\mathfrak{L}^{1}(\mathcal{H}_{s+n})}=0,\quad
n\geq 1.$

The last equations are a consequence of the validity for the bounded
interaction potential of analogues of the Duhamel equation for cumulants (34)
of the groups of operators (7).

According to equalities (56) for a one-particle density operator (48) the
following limit theorem holds.

If condition (55) is satisfied, then for series (48) the equality is valid

$\displaystyle\lim\limits_{\epsilon\rightarrow 0}\big{\|}\epsilon
F_{1}(t)-f_{1}(t)\big{\|}_{\mathfrak{L}^{1}(\mathcal{H})}=0,$

where for an arbitrary finite time interval the limiting one-particle density
operator $f_{1}(t)$ is determined by convergent series over the norm of space
$\mathfrak{L}^{1}(\mathcal{H})$

$\displaystyle\hskip-34.1433ptf_{1}(t,1)=\sum\limits_{n=0}^{\infty}\,\int\limits_{0}^{t}dt_{1}\ldots\int\limits_{0}^{t_{n-1}}dt_{n}\,\mathrm{Tr}_{2,\ldots,n+1}\mathcal{G}^{\ast}_{1}(t-t_{1},1)\mathcal{N}^{\ast}_{\mathrm{int}}(1,2)\prod\limits_{j_{1}=1}^{2}\mathcal{G}^{\ast}_{1}(t_{1}-t_{2},j_{1})\ldots$
(57)
$\displaystyle\prod\limits_{i_{n}=1}^{n}\mathcal{G}^{\ast}_{1}(t_{n}-t_{n},i_{n})\sum\limits_{k_{n}=1}^{n}\mathcal{N}^{\ast}_{\mathrm{int}}(k_{n},n+1)\prod\limits_{j_{n}=1}^{n+1}\mathcal{G}^{\ast}_{1}(t_{n},j_{n})\prod\limits_{i=1}^{n+1}f_{1}^{0}(i).$

For bounded interaction potentials, the series (57)the series converges in the
norm of space $\mathfrak{L}^{1}(\mathcal{H})$ provided that:
$t<t_{0}\equiv(2\,\|\Phi\|_{\mathfrak{L}(\mathcal{H}_{2})}\|f_{1}^{0}\|_{\mathfrak{L}^{1}(\mathcal{H})})^{-1}$.

For the initial state $f_{1}^{0}\in\mathfrak{L}^{1}(\mathcal{H})$ limiting
operator (57) satisfies the Cauchy problem for the quantum Vlasov kinetic
equation

$\displaystyle\hskip-14.22636pt\frac{\partial}{\partial
t}f_{1}(t,1)=\mathcal{N}^{\ast}_{1}(1)f_{1}(t,1)+\mathrm{Tr}_{2}\,\mathcal{N}^{\ast}_{\mathrm{int}}(1,2)f_{1}(t,1)f_{1}(t,2),$
(58) $\displaystyle\hskip-14.22636ptf_{1}(t)|_{t=0}=f_{1}^{0}.$ (59)

We note that for pure states, i.e. $f_{1}(t)=|\psi(t)\rangle\langle\psi(t)|$
or in terms of the kernel of this operator
$f_{1}(t,q,q^{\prime})=\psi(t,q)\psi^{\ast}(t,q^{\prime})$, the quantum Vlasov
kinetic (58) is reduced to the Hartree equation

$\displaystyle i\frac{\partial}{\partial
t}\psi(t,q)=-\frac{1}{2}\Delta_{q}\psi(t,q)+\int
dq^{\prime}\Phi(q-q^{\prime})|\psi(q^{\prime})|^{2}\psi(t,q).$

In particular, if the kernel of the interaction potential $\Phi(q)=\delta(q)$
is a Dirac measure, then the Hartree kinetic equation turns into the nonlinear
Schrödinger equation with cubic nonlinearity

$\displaystyle i\frac{\partial}{\partial
t}\psi(t,q)=-\frac{1}{2}\Delta_{q}\psi(t,q)+|\psi(t,q)|^{2}\psi(t,q).$

According to property (56) for cumulants of asymptotically perturbed groups of
operators for generating operators (49) the equalities true:

$\displaystyle\hskip-14.22636pt\lim\limits_{\epsilon\rightarrow
0}\Big{\|}\big{(}\mathfrak{V}_{1}(t,\\{1,\ldots,s\\})-I\big{)}f_{s}\Big{\|}_{\mathfrak{L}^{1}(\mathcal{H}_{s})}=0,$

and for $n\geq 1$

$\displaystyle\hskip-14.22636pt\lim\limits_{\epsilon\rightarrow
0}\Big{\|}\frac{1}{\epsilon^{n}}\,\mathfrak{V}_{1+n}(t,\\{1,\ldots,s\\},s+1,\ldots,s+n)f_{s+n}\Big{\|}_{\mathfrak{L}^{1}(\mathcal{H}_{s+n})}=0,$

respectively.

Due to such equalities and convergence in the mean-field limit of the solution
of the Cauchy problem of the generalized quantum kinetic equation (53),(54) to
the solution of the Cauchy problem of the quantum Vlasov kinetic equation
(58),(59) for reduced functionals of the state (47) the following equalities
are valid:

$\displaystyle\lim\limits_{\epsilon\rightarrow
0}\Big{\|}\epsilon^{s}F_{s}\big{(}t,1,\ldots,s\mid
F_{1}(t)\big{)}-\prod\limits_{j=1}^{s}f_{1}(t,j)\Big{\|}_{\mathfrak{L}^{1}(\mathcal{H}_{s})}=0,\quad
s\geq 2,$

where the limit operator $f_{1}(t)$ is determined by series (57), which
represents the solution of the quantum Vlasov equation (58).

The last statement describes the process of propagation of the initial chaos
in the mean-field limit, i.e., if at the initial moment there are no
correlations in the system, then in the process of evolution correlations are
not created in this approximation.

Note that the traditional approach to the problem of the propagation of
initial chaos is based on the construction of the asymptotic behavior of a
solution of the quantum BBGKY hierarchy for reduced density operators within
the perturbation theory [3],[42].

##  4 The kinetic evolution of observables

In this section, we consider the scaling asymptotic behavior of a solution of
the Cauchy problem (28) for the dual BBGKY hierarchy (26),(27) in the case of
the mean-field limit [19], or, in other words, we consider the foundations of
the description of the kinetic evolution of quantum systems of many particles
within the framework of observables.

Note that one of the advantages of such an approach to the description of
kinetic evolution is the possibility to describe the propagation of initial
correlations in scaling limits.

###  4.1 The hierarchy of kinetic equations for observables

Suppose that at the initial moment of time there exists the mean-field limit
of reduced observables (27) in the sense of $\ast$-weak convergence of the
space of bounded operators $\mathfrak{L}(\mathcal{H}_{s})$

$\displaystyle\mathrm{w^{\ast}-}\lim\limits_{\epsilon\rightarrow
0}\big{(}\epsilon^{-s}B_{s}^{\epsilon,0}-b_{s}^{0}\big{)}=0,$ (60)

where $\epsilon>0$ is a scaling parameter.

Then the following limit theorem holds for reduced observables (28), which are
the solution of the dual BBGKY hierarchy (26).

If the condition (60) is satisfied, then for an arbitrary finite time interval
there is a mean-field limit for the sequence of reduced observables (28) in
the same sense [24],[19]

$\displaystyle\mathrm{w^{\ast}-}\lim\limits_{\epsilon\rightarrow
0}\big{(}\epsilon^{-s}B_{s}(t)-b_{s}(t)\big{)}=0,$ (61)

where the reduced observables $b_{s}(t),\,s\geq 1$, are determined by the
following expansions:

$\displaystyle\hskip-34.1433ptb_{s}(t,1,\ldots,s)=\sum\limits_{n=0}^{s-1}\,\int\limits_{0}^{t}dt_{1}\ldots\int\limits_{0}^{t_{n-1}}dt_{n}\,\mathcal{G}_{s}^{0}(t-t_{1})\sum\limits_{i_{1}\neq
j_{1}=1}^{s}\mathcal{N}_{\mathrm{int}}(i_{1},j_{1})\mathcal{G}_{s-1}^{0}(t_{1}-t_{2})\ldots$
(62)
$\displaystyle\hskip-34.1433pt\mathcal{G}_{s-n+1}^{0}(t_{n-1}-t_{n})\sum\limits^{s}_{\mbox{\scriptsize$\begin{array}[]{c}i_{n}\neq
j_{n}=1,\\\
i_{n},j_{n}\neq(j_{1},\ldots,j_{n-1})\end{array}$}}\mathcal{N}_{\mathrm{int}}(i_{n},j_{n})\mathcal{G}_{s-n}^{0}(t_{n})b_{s-n}^{0}((1,\ldots,s)\setminus({j_{1}},\ldots,{j_{n}})),$
(65) $\displaystyle\hskip-34.1433pts\geq 1,$

and for the group of operators of non-interacting particles, the notation was
used

$\displaystyle\mathcal{G}_{s-n+1}^{0}(t_{n-1}-t_{n})\equiv\prod\limits_{j\in(1,\ldots,s)\setminus(j_{1},\ldots,j_{n-1})}\mathcal{G}_{1}(t_{n-1}-t_{n},j).$

For a certain type of observables, the structure of expansion (62) takes a
particular form, for example, in the case of the $k$-ary type of observables
we have:

$\displaystyle\hskip-34.1433ptb_{s}^{(k)}(t,1,\ldots,s)=\int\limits_{0}^{t}dt_{1}\ldots\int\limits_{0}^{t_{s-k-1}}dt_{s-k}\,\mathcal{G}_{s}^{0}(t-t_{1})\sum\limits_{i_{1}\neq
j_{1}=1}^{s}\mathcal{N}_{\mathrm{int}}(i_{1},j_{1})\mathcal{G}_{s-1}^{0}(t_{1}-t_{2})\ldots$
(66)
$\displaystyle\mathcal{G}_{s-n+1}^{0}(t_{s-k-1}-t_{s-k})\sum\limits^{s}_{\mbox{\scriptsize$\begin{array}[]{c}i_{s-k}\neq
j_{s-k}=1,\\\
i_{s-k},j_{s-k}\neq(j_{1},\ldots,j_{s-k-1})\end{array}$}}\mathcal{N}_{\mathrm{int}}(i_{s-k},j_{s-k})\mathcal{G}_{s-n}^{0}(t_{s-k})\times$
(69) $\displaystyle
b_{k}^{0}((1,\ldots,s)\setminus({j_{1}},\ldots,{j_{s-k}})),\quad 1\leq s\leq
k.$

If $b^{0}\in\mathfrak{L}(\mathcal{F}_{\mathcal{H}})$, then the sequence
$b(t)=(b_{0},b_{1}(t),\ldots,b_{s}(t),\ldots)$ of limit observables (62)
satisfies the Cauchy problem of the dual Vlasov hierarchy:

$\displaystyle\hskip-34.1433pt\frac{\partial}{\partial
t}b_{s}(t,1,\ldots,s)=\sum\limits_{j=1}^{s}\mathcal{N}(j)\,b_{s}(t,1,\ldots,s)+\sum_{j_{1}\neq
j_{2}=1}^{s}\mathcal{N}_{\mathrm{int}}(j_{1},j_{2})\,b_{s-1}(t,(1,\ldots,s)\setminus(j_{1})),$
(70)
$\displaystyle\hskip-34.1433ptb_{s}(t,1,\ldots,s)\mid_{t=0}=b_{s}^{0}(1,\ldots,s),\quad
s\geq 1.$ (71)

We give some examples of the dual Vlasov hierarchy (70) in terms of kernels of
the operators for the reduced observables:

$\displaystyle\hskip-22.76219pti\,\frac{\partial}{\partial
t}b_{1}(t,q_{1};q^{\prime}_{1})=-\frac{1}{2}(-\Delta_{q_{1}}+\Delta_{q^{\prime}_{1}})b_{1}(t,q_{1};q^{\prime}_{1}),$
$\displaystyle\hskip-22.76219pti\,\frac{\partial}{\partial
t}b_{2}(t,q_{1},q_{2};q^{\prime}_{1},q^{\prime}_{2})=\big{(}-\frac{1}{2}\sum\limits_{i=1}^{2}(-\Delta_{q_{i}}+\Delta_{q^{\prime}_{i}})+$
$\displaystyle\hskip
28.45274pt(\Phi(q^{\prime}_{1}-q^{\prime}_{2})-\Phi(q_{1}-q_{2}))\big{)}b_{2}(t,q_{1},q_{2};q^{\prime}_{1},q^{\prime}_{2}))+$
$\displaystyle\hskip
28.45274pt\big{(}\Phi(q^{\prime}_{1}-q^{\prime}_{2})-\Phi(q_{1}-q_{2})\big{)}\big{(}b_{1}(t,q_{1};q^{\prime}_{1})+b_{1}(t,q_{2};q^{\prime}_{2})\big{)}.$

We note that the sequence of evolution equations (70) has the structure of
recursive equations.

Thus, in the mean-field limit, the collective behavior (kinetic evolution) of
quantum systems of many particles is described in terms of the sequence of
limiting reduced observables (62), in particular, by sequence (66) whose
evolution is governed by the Cauchy problem of the dual Vlasov hierarchy
(70),(71).

###  4.2 The propagation of the initial chaos

Let us consider the relationship of collective behavior within the mean-field
approximation of a quantum system of many particles, which is described by the
dual Vlasov hierarchy (70) for the limiting reduced observables, and by the
Vlasov kinetic equation (58) for the state of a typical particle of a system.

Let the initial state satisfy a chaos condition, i.e. at the initial instant
there are no correlations between particles (the statistically independent
particles), namely, in the case of the Maxwell–Boltzmann statistics by such a
sequence of limiting reduced density operators

$\displaystyle
f^{(c)}\equiv\big{(}I,f_{1}^{0}(1),\ldots,\prod_{i=1}^{s}f_{1}^{0}(i),\ldots\big{)}.$
(72)

As mentioned above, this assumption regarding the initial state is
characteristic of the kinetic description of the gas, because in this case the
state is completely determined using a one-particle density operator.

For mean value functional (23) of additive type limit observables (66) and
initial state (72) on a finite time interval, the equality holds

$\displaystyle\hskip-22.76219pt\big{(}b^{(1)}(t),f^{(c)}\big{)}=\sum\limits_{s=0}^{\infty}\frac{1}{s!}\,\mathrm{Tr}_{1,\ldots,s}\,b_{s}^{(1)}(t,1,\ldots,s)\prod\limits_{i=1}^{s}f_{1}^{0}(i)=$
$\displaystyle\hskip 19.91692pt\mathrm{Tr}_{1}\,b_{1}^{0}(1)f_{1}(t,1),$

where a one-particle density operator is represented by such a series

$\displaystyle\hskip-34.1433ptf_{1}(t,1)=\sum\limits_{n=0}^{\infty}\int\limits_{0}^{t}dt_{1}\ldots\int\limits_{0}^{t_{n-1}}dt_{n}\,\mathrm{Tr}_{2,\ldots,n+1}\prod\limits_{i_{1}=1}^{1}\mathcal{G}^{\ast}_{1}(t-t_{1},i_{1})\mathcal{N}^{\ast}_{\mathrm{int}}(1,2)\prod\limits_{j_{1}=1}^{2}\mathcal{G}^{\ast}_{1}(t_{1}-t_{2},j_{1})\ldots$
(73)
$\displaystyle\prod\limits_{i_{n}=1}^{n}\mathcal{G}^{\ast}_{1}(t_{n}-t_{n},i_{n})\sum\limits_{k_{n}=1}^{n}\mathcal{N}^{\ast}_{\mathrm{int}}(k_{n},n+1)\prod\limits_{j_{n}=1}^{n+1}\mathcal{G}^{\ast}_{1}(t_{n},j_{n})\prod\limits_{i=1}^{n+1}f_{1}^{0}(i).$

The one-particle density operator (73) satisfies the Cauchy problem of the
quantum Vlasov kinetic equation (58),(59).

Thus, the hierarchy of evolutionary equations (70) for limit reduced
observables of additive type and initial state (72) describes the evolution of
quantum many-particle systems in an equivalent way compared to the Vlasov
equation (58).

Accordingly, for the mean value functional of the limit observables of
nonadditive type (66) and the initial state (72) on a finite time interval,
equations valid:

$\displaystyle\hskip-19.91692pt\big{(}b^{(k)}(t),f^{(c)}\big{)}=\sum\limits_{s=0}^{\infty}\frac{1}{s!}\,\mathrm{Tr}_{1,\ldots,s}\,b_{s}^{(k)}(t,1,\ldots,s)\prod\limits_{i=1}^{s}f_{1}^{0}(i)=$
(74)
$\displaystyle\frac{1}{k!}\mathrm{Tr}_{1,\ldots,k}\,b_{k}^{0}(1,\ldots,k)\prod\limits_{i=1}^{k}f_{1}(t,i),\quad
k\geq 2,$

where the one-particle density operator is represented by series (73), i.e.
satisfies the Cauchy problem for the quantum Vlasov kinetic equation
(58),(59).

Equality (74) describes the process of the propagation of initial chaos (72)
in the mean-field limit by means of solution (73) of the quantum Vlasov
kinetic eqation (58), namely, for an arbitrary finite time interval, the state
is represented by such a sequence of limit reduced operators:

$\displaystyle f_{k}(t,1,\ldots,k)=\prod\limits_{i=1}^{k}f_{1}(t,i),\quad
k\geq 2,$ (75)

that is, if at the initial moment of time in the system of particles there are
no correlations, then in this approximation in the process of evolution
correlations of particle states are not create.

###  4.3 Kinetic equations with initial correlations

We note that the above approach to the derivation of kinetic equations allows
us to formulate kinetic equations in the case of more general initial states,
which describe not only the gases of quantum particles (72) but also systems
of many particles in condensed states.

Further, we consider the initial states of quantum systems of many particles,
which are determined by a one-particle density operator and correlation
operators (Maxwell - Boltzmann statistics) [24]

$\displaystyle\hskip-34.1433ptf^{(cc)}=\big{(}I,f_{1}^{0}(1),g_{2}^{0}(1,2)\prod_{i=1}^{2}f_{1}^{0}(i),\ldots,g_{n}^{0}(1,\ldots,n)\prod_{i=1}^{n}f_{1}^{0}(i),\ldots\big{)},$
(76)

where the correlations of the initial states of the particles are determined
by the operators $g_{n}^{0}(1,\ldots,n)\equiv
g_{n}^{0}\in\mathfrak{L}^{1}_{0}(\mathcal{H}_{n}),\,n\geq 2$. We emphasize
that this assumption (76) with respect to the initial state is typical for the
kinetic description of systems of many particles in condensed states, which
are characterized by correlations, for example, such as in particle flows
[6],[65].

Then, using the method of derivation of kinetic equations based on the
hierarchy of kinetic equations for the observables, for an arbitrary finite
time interval we establish that the state is described by the sequence
$f(t)=\big{(}I,f_{1}(t),\ldots,f_{n}(t,1,\ldots,n),\ldots\big{)}$ of limiting
reduced density operators where a one-particle density operator is represented
by such a series

$\displaystyle\hskip-34.1433ptf_{1}(t,1)=\sum\limits_{n=0}^{\infty}\int\limits_{0}^{t}dt_{1}\ldots\int\limits_{0}^{t_{n-1}}dt_{n}\,\mathrm{Tr}_{2,\ldots,n+1}\prod\limits_{i_{1}=1}^{1}\mathcal{G}^{\ast}_{1}(t-t_{1},i_{1})\mathcal{N}^{\ast}_{\mathrm{int}}(1,2)\prod\limits_{j_{1}=1}^{2}\mathcal{G}^{\ast}_{1}(t_{1}-t_{2},j_{1})\ldots$
(77)
$\displaystyle\prod\limits_{i_{n}=1}^{n}\mathcal{G}^{\ast}_{1}(t_{n}-t_{n},i_{n})\sum\limits_{k_{n}=1}^{n}\mathcal{N}^{\ast}_{\mathrm{int}}(k_{n},n+1)\prod\limits_{j_{n}=1}^{n+1}\mathcal{G}^{\ast}_{1}(t_{n},j_{n})g_{n+1}^{0}(1,\ldots,n+1)\prod\limits_{i=1}^{n+1}f_{1}^{0}(i),$

and the limiting density operators $f_{k}(t,1,\ldots,k),\,k\geq 2,$ are
determined by the following expressions [41]:

$\displaystyle\hskip-34.1433ptf_{k}(t,1,\ldots,k)=\prod_{i_{1}=1}^{k}\mathcal{G}^{\ast}_{1}(t,i_{1})g_{k}^{0}(1,\ldots,k)\prod_{i_{2}=1}^{k}(\mathcal{G}_{1}^{\ast})^{-1}(t,i_{2})\prod\limits_{j=1}^{k}f_{1}(t,j),\quad
k\geq 2.$ (78)

Indeed, for the mean value functionals of the limiting observables (66) in the
case of the initial state (76) for a finite time interval, the equalities are
valid:

$\displaystyle\hskip-19.91692pt\big{(}b^{(k)}(t),f^{(cc)}\big{)}=\frac{1}{k!}\mathrm{Tr}_{1,\ldots,k}\,b_{k}^{0}(1,\ldots,k)f_{k}(t,1,\ldots,k),\quad
k\geq 1,$

where the sequence $f(t)$ is determined by expansions (77) and (78).

In [41], a similar result was obtained using a generalized quantum kinetic
equation with initial correlations.

In the case of initial states specified by a sequence of limiting correlation
operators (76), the process of propagation of initial correlations is
described by the following sequence of correlation operators (37):

$\displaystyle\hskip-34.1433ptg_{n}(t,1,\ldots,n)=\prod_{i_{1}=1}^{n}\mathcal{G}_{1}^{\ast}(t,i_{1})\sum\limits_{\mbox{\scriptsize$\begin{array}[]{c}\mathrm{P}:(1,\ldots,n)=\bigcup_{i}X_{i}\end{array}$}}(-1)^{|\mathrm{P}|-1}(|\mathrm{P}|-1)!\,\prod_{X_{i}\subset\mathrm{P}}g_{|X_{i}|}^{0}(X_{i})\times$
$\displaystyle\prod_{i_{2}=1}^{n}(\mathcal{G}_{1}^{\ast})^{-1}(t,i_{2})\prod\limits_{j=1}^{n}f_{1}(t,j),\quad
n\geq 2.$

Limiting one-particle density operator (77) is governed by the quantum Vlasov
kinetic equation with initial correlations [41]:

$\displaystyle\hskip-34.1433pt\frac{\partial}{\partial
t}f_{1}(t,1)=\mathcal{N}^{\ast}(1)f_{1}(t,1)+$ (80)
$\displaystyle\mathrm{Tr}_{2}\,\mathcal{N}^{\ast}_{\mathrm{int}}(1,2)\prod_{i_{1}=1}^{2}\mathcal{G}^{\ast}_{1}(t,i_{1})g_{2}^{0}(1,2)\prod_{i_{2}=1}^{2}(\mathcal{G}^{\ast}_{1})^{-1}(t,i_{2})f_{1}(t,1)f_{1}(t,2),$
$\displaystyle\hskip-34.1433ptf_{1}(t)|_{t=0}=f_{1}^{0},$ (81)

where the notations (10) is used and the inverse group of operators to group
(4) is denoted by $(\mathcal{G}^{\ast}_{1})^{-1}(t)$.

We note that the kinetic equation (80) is a non-Markov kinetic equation. For
pure states, equation (80) reduces to the Hartree kinetic equation with
initial correlations. For the initial states of the system of statistically
independent particles, the kinetic equation (80) coincides with the quantum
Vlasov equation (58), and reduced density operators (78) describe the process
of propagation of the initial chaos (75).

We remark also that in the case of arbitrary initial states in a mean-field
limit, a sequence of reduced density operators, as well as sequence (77),
(78), is the solution of the Cauchy problem of the Vlasov hierarchy:

$\displaystyle\hskip-34.1433pt\frac{\partial}{\partial
t}f_{s}(t,1,\ldots,s)=\sum\limits_{i=1}^{s}\mathcal{N}^{\ast}(i)f_{s}(t,1,\ldots,s)+\sum\limits_{i=1}^{s}\mathrm{Tr}_{s+1}\mathcal{N}^{\ast}_{\mathrm{int}}(i,s+1)f_{s+1}(t,1,\ldots,s+1),$
$\displaystyle\hskip-34.1433ptf_{s}(t)_{|t=0}=f_{s}^{0},\quad s\geq 1.$

Thus, in the case of initial states given by a one-particle density operator
and correlation operators (76), the dual Vlasov hierarchy (70) for the
additive type reduced observables describes the evolution of quantum systems
of many particles, just as non-Markov quantum Vlasov kinetic equation with
initial correlations (80). In the case of reduced nonadditive-type
observables, the dual Vlasov hierarchy (70) describes in an equivalent way in
the sense of equality (78) the process of propagation of initial correlations
in terms of reduced density operators (78). In other words, the alternative
method of describing the evolution of states of quantum systems of many
particles in the mean-field approximation is based on the non-Markovian Vlasov
kinetic equation with initial correlations (80).

##  5 Outlook

The possible approaches to the description of the evolution of quantum systems
of many particles are considered above, namely, in terms of the observables
ones, the evolution of which is described by the dual BBGKY hierarchy (26), or
in terms of the state whose evolution is described by the BBGKY hierarchy
(31). For systems of a finite number of particles, these hierarchies of
evolution equations are equivalent to the Heisenberg equation (1) and the von
Neumann equation (8), respectively. In particular, possible methods of
describing the evolution of the state by means of the hierarchy of fundamental
equations (18), which describes the evolution of state correlations, are
considered.

It was established that the concept of cumulant (22) of groups of operators
forms the basis of expansions for solutions of fundamental evolution
equations, which describe the evolution of quantum systems of many particles,
namely: in the case of the groups of operators (4) for the dual BBGKY
hierarchy for reduced observables; in the case of groups of operators (7) for
the von Neumann hierarchy (18) for correlation operators, for the BBGKY
hierarchy (31) for reduced density operators and for the BBGKY hierarchy of
nonlinear equations (38) for reduced correlation operators, respectively, as
well as for the basis of the kinetic description of infinitely many particles
systems (47),(53).

We emphasize that the structure of expansions for correlation operators (45),
in which the generating operators are of the appropriate order cumulants (21)
of the groups of operators (7), induces the cumulant structure of expansions
in a series for reduced density operators (33), reduced correlation operators
(46) and reduced functionals of the state (47). Thus, the dynamics of systems
of infinitely many particles are generated by the dynamics of correlations.

The article also was considered two new approaches to describing the kinetic
evolution of quantum systems of many particles. One of them consists in the
description of the kinetic evolution of the quantum system of particles by
means of the reduced observables in mean-field scaling limit (70). Another
approach is based on the non-Markovian generalization of quantum kinetic
equations (53).

One of the advantages of the considered approaches is connected with the
possibility of construction of kinetic equations in scaling approximations
taking into account correlations of particles at the initial instant which
characterize the condensed states of systems of many particles. Another
advantage is related to the problem of the rigorous derivation of kinetic
equations of the non-Markovian type based on the dynamics of quantum many-
particle systems, which allow us to describe the effects of memory in
nanostructures.

We remark that the approach to the derivation of the quantum Vlasov kinetic
equation from the dynamics of many-particle systems, which is based on the
generalized quantum kinetic equation (53), also allows the construction of
higher-order corrections to the approximation of mean-field for quantum
kinetic equations.

The above results can be extended to systems of many bosons or fermions
[34],[39].

## References

  * [1] R. Balescu, _Irreversible processes in ionized gases_ , Phys. of Fluids, 3, (1), (1960), 52–63.
  * [2] D. Benedetto, F. Castella, R. Esposito and M. Pulvirenti, _A short review on the derivation of the nonlinear quantum Boltzmann equations_. Commun. Math. Sci., 5, (2007), 55–71.
  * [3] N. Benedikter, M. Porta and B. Schlein, _Effective Evolution Equations from Quantum Dynamics_ , SpringerBriefs in Mathematical Physics, 2016.
  * [4] Ch. Boccato, S. Cenatiempo and B. Schlein, _Quantum many-body fluctuations around nonlinear Schrödinger dynamics_. Ann. Henri Poincaré, 18, (1), 2017, 113–191.
  * [5] M. M. Bogolyubov, _On the theory of superfluidity_. Proc. Inst. Mathematics of Acad. Sci. USSR (in Ukrainian), No. 9, (1947), 89–103.
  * [6] M. M. Bogolyubov, _Lectures on Quantum Statistics. Problems of Statistical Mechanics of Quantum Systems_. K.: Rad. Shkola, 1949 (in Ukrainian).
  * [7] M. M. Bogolyubov and K. P. Gurov, _Kinetic equations in quantum mechanics_. JETP (in Russian), 17, (7), (1947), 614—-628.
  * [8] M. M. Bogolyubov, _Problems of the dynamical theory in statistical physics_ , M.: Gostechizdat, 1946 (in Russian).
  * [9] L. E. Boltzmann, _Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen_. Sitzungsber. Akad. Wiss., 66, 1872, 275–-370.
  * [10] M. Born and H. S. Green, _A general kinetic theory of liquids. I. The molecular distribution functions_. Proceedings of the Royal Soc. of London. A, 188, (1012), (1946), 10–18.
  * [11] T. Carleman, _Probleèmes mathématiques dans la théorie cinétique des gaz_. Publ. Sci. Inst. Mittag-Leffler, 2 Almqvist & Wiksells Boktryckeri Ab, Uppsala, 1957.
  * [12] C. Cercignani, V.I. Gerasimenko and D.Ya. Petrina, _Many-Particle Dynamics and Kinetic Equations_. the Netherlands: Springer, 2012.
  * [13] C. Cercignani, R. Illner and M. Pulvirenti, _The mathematical theory of dilute gases_. Springer-Verlag, New York, 1994
  * [14] D. Enskog, _Kinetische Theorie der Wärmeleitung, Reibung und Selbstdiffusion in gewissen verdichteten Gasen und Flüssigkeiten_. Kungl. Sv. Vetenskapsakademiens Handl., 63, (4), (1922), 3–44.
  * [15] L. Erdős, B. Schlein, and H.-T. Yau, _Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate_. Ann. of Math., 172, (1), (2010), 291–370.
  * [16] V. Fock, _Näherungsmethode zur Lösung des quantenmechanischen Mehrkörperproblems_. Zeitschrift für Physik, 61, (1930), 126–148.
  * [17] A. D. Fokker, _Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld_ , Annalen der Physik, 348, (43), (1914), 810–-820.
  * [18] I. Gallagher, L. Saint-Raymond and B. Texier, _From Newton to Boltzmann: hard spheres and short-range potentials_ , EMS, Zürich, 2013.
  * [19] V.I. Gerasimenko, _Heisenberg picture of quantum kinetic evolution in mean-field limit_. Kinet. Relat. Models, 4, (1), (2011), 385–399.
  * [20] V. Gerasimenko and I. Gapyak, _Low-density asymptotic behavior of observables of hard sphere fluids_. Advances in Math. Phys., 2018, (2018), 1–11.
  * [21] V. Gerasimenko, V. Shtyk and A. Zagorodny, _Hydrodynamic equations for microscopic phase densities_ , Cent. Eur. J. Phys., 9, (1), (2011), 71–77.
  * [22] V. I. Gerasimenko, _Approaches to derivation of quantum kinetic equations_. Ukrainian J. Phys. 54, (8-9), (2009), 834–846.
  * [23] V. I. Gerasimenko, _Hierarchies of quantum evolution equations and dynamics of many-particle correlations_. Int. J. Evol. Equ., 7, (2), (2012), 109–163.
  * [24] V. I. Gerasimenko, _New approach to derivation of quantum kinetic equations with initial correlations_. Carpathian Math. Publ., 7, (1), (2015), 38–48.
  * [25] V. I. Gerasimenko, _Evolution of correlation operators of large particle quantum systems_. Methods Funct. Anal. Topology, 23, (2), (2017), 123–132.
  * [26] V. I. Gerasimenko, _Description of evolution of states in terms of operators originating by density matrix_ , In: Understanding Density Matrices, N.Y.: Nova Science Publ., Inc., 2019, 229–250.
  * [27] V. I. Gerasimenko, _Processes of Creation and Propagation of Correlations in Large Quantum Particle System_. In: Panorama of Contemporary Quantum Mechanics - Concepts and Applications, IntechOpen, 2019.
  * [28] V. I. Gerasimenko and G. Borgioli, _Initial-value problem of the quantum dual BBGKY hierarchy_. Il Nuovo Cimento C, 33, (1), (2010), 71–78.
  * [29] V. I. Gerasimenko and Yu. Yu. Fedchun, _Nonperturbative solution expansions of hierarchies of evolution equations in functional derivatives_. Proceedings Inst. Math. NASU, 9, (2), (2012), 347–375.
  * [30] V. I. Gerasimenko and I. V. Gapyak, _Hard sphere dynamics and the Enskog equation_ , Kinetic & Related Models, 5, (3), (2012), 459–484.
  * [31] V. I. Gerasimenko and I. V. Gapyak, _Boltzmann–Grad asymptotic behavior of collisional dynamics_. Reviews in Math. Phys., 33, (2), (2021), 2130001.
  * [32] V. I. Gerasimenko and D. Ya. Petrina, _Evolution of states of infinite systems in classical statistical mechanics_. In: Sov. Sci. Rev., sect. C: Math. Phys. Rev., v.5, N.Y.: Harwood Acad. Publ., 1985, 1–52.
  * [33] V. I. Gerasimenko and D. Ya. Petrina, _The generalized kinetic equation generated by the BBGKY hierarchy_. Ukr. J. Phys., 43, (6/7), (1998), 697–702.
  * [34] V. I. Gerasimenko and D. O. Polishchuk, _Dynamics of correlations of Bose and Fermi particles_. Math. Meth. Appl. Sci. 34, (1), (2011), 76–93.
  * [35] V. I. Gerasimenko and T. V. Ryabukha and M. O. Stashenko, _On the structure of expansions for the BBGKY hierarchy solutions_. J. Phys. A: Math. and General, 37, (42), (2004), 9861–9872.
  * [36] V.I. Gerasimenko and V. O. Shtyk, _Initial-value problem of the Bogolyubov hierarchy for quantum systems of particles_. Ukrain. Math. J. 58, (9), (2006), 1175–1191.
  * [37] V. I. Gerasimenko and V. O. Shtyk, _Evolution of correlations of quantum many-particle systems_. J. Stat. Mech.: Theory and Experiment, 2008, (03), (2008), P03007.
  * [38] V. I. Gerasimenko and Zh. A. Tsvir, _A description of the evolution of quantum states by means of the kinetic equation_ , J. Phys. A.: Math. and Theor., 43, (48), (2010), 485203.
  * [39] V. I. Gerasimenko and Zh. A. Tsvir, _The generalized quantum kinetic equation for interacting particles with quantum statistics_. Math. Bulletin Sh. Sci. Soc., 7, (2010), 351–367.
  * [40] V. I. Gerasimenko and Zh. A. Tsvir, _Mean field asymptotics of generalized quantum kinetic equation_ , Rep. Math. Phys., 70, (2), (2012), 135–147.
  * [41] V. I. Gerasimenko and Zh. A. Tsvir, _On quantum kinetic equations of many-particle systems in condensed states_. Physica A: Stat. Mech. and its Appl., 391, (24), (2012), 6362–6366.
  * [42] F. Golse, _On the dynamics of large particle systems in the mean field limit_ , In: Macroscopic and large scale phenomena: coarse graining, mean field limits and ergodicity, Lect. Notes Appl. Math. Mech., Springer, 2016, 3, 1–144.
  * [43] H. Grad, _Principles of the kinetic theory of gases_. In: Handbuch der Physik (herausgegeben von S. Flügge), Bd. 12, Thermodynamik der Gase, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958, 205–294.
  * [44] E. P. Gross, _Structure of a quantized vortex in boson systems_. Nuovo Cim., 20, (1961), 454–-477.
  * [45] R. L. Guernsey, _Kinetic equation for a completely ionized gas_. Phys. of Fluids, 5, (3), (1962), 322–328.
  * [46] D. R. Hartree, _The Wave Mechanics of an Atom with a non-Coulomb Central Field. Part III. Term Values and Intensities in Series in Optical Spectra_. Math. Proc. of the Cambridge Phil. Soc., 24, (3), (1928), 426–437.
  * [47] D. Hilbert, _Principles of the kinetic theory of gases_. In: Sur les problèmes futurs des mathématiques, Compte-Rendu du 2ème Congrés International de Mathématiques, tenuà Paris en 1900, Paris: Gauthier-Villars, 1902, 58–114.
  * [48] D. Hilbert, _Begründung der kinetischen Gastheorie_. Math. Ann., 72, (1912), 562–-577.
  * [49] J. G. Kirkwood, _The statistical mechanical theory of transport processes II. Transport in gases_. J. Chem. Phys., 15, (1), (1947), 72–76.
  * [50] L. D. Landau, _The Transport Equation in the Case of Coulomb Interactions_. Phys. Z. Sowjet., 10, (1936), 154–-168.
  * [51] O. E. Lanford, III, _Time evolution of large classical systems_. In: Dynamical systems, theory and applications, Lecture Notes in Phys., 38, 1975, 1–111.
  * [52] A. Lenard, _On Bogoliubov’s kinetic equation for a spatially homogeneous plasma_. Ann. Phys. (USA), 10, (3), (1960), 390–400.
  * [53] M. A. Leontovich, _Basic equations of the kinetic theory of gases from the point of view of random processes_. J. Exp. Theor. Phys., 5, (1935), 211–-214.
  * [54] J. C. Maxwell, _Illustrations of the Dynamical Theory of Gases_. Phil. Trans. of the Royal Soc. of London, 19, 20, (1860), 19–-32; 21–-37.
  * [55] J. C. Maxwell, _On the dynamical theory of gases_. Phil. Trans. of the Royal Soc. of London, 157, (1867), 49–88.
  * [56] L. W. Nordhiem, _Transport Phenomena in Einstein-Bose and Fermi-Dirac Gases. I_ , R. Soc. Lond. A, 119, (1928), 689–-698.
  * [57] D. Ya. Petrina, _On solutions of the Bogolyubov kinetic equations. Quantum statistics_. Theor. Math. Phys., 13, (3), (1972), 391–405.
  * [58] D. Ya. Petrina, _Mathematical Foundations of Quantum Statistical Mechanics. Continuous Systems_. Kluwer, 1995.
  * [59] D. Ya. Petrina and V. I. Gerasimenko, _A mathematical description of the evolution of the state of infinite systems of classical statistical mechanics_. Russian Math. Surveys, 38, (5), (1983), 1–61.
  * [60] D. Ya. Petrina and V. I. Gerasimenko, _Mathematical problems of statistical mechanics of a system of elastic balls_. Russian Math. Surveys, 45, (3), (1990), 153–211.
  * [61] F. Pezzotti and M. Pulvirenti, _Mean-field limit and semiclassical expansion of a quantum particle system_. Ann. Henri Poincaré, 10, (1), (2009), 145–187.
  * [62] L. P. Pitaevskii, _Vortex Lines in an Imperfect Bose Gas_. J. Exp. Theor. Phys., 13, (2), (1961), 451–-454.
  * [63] M. Planck, _Über einen Satz der statistischen Dynamik und seine Erweiterung in der Quantentheorie_. Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin, 24, (1917), 324–341.
  * [64] H. Poincaré, _The Milky Way and the theory of gases_ , Popular Astronomy, 14, (1906), 475–488.
  * [65] L. Saint-Raymond, _Kinetic models for superfluids: a review of mathematical results_ , Comptes Rendus Physique, 5, (1), (2004), 65–75.
  * [66] H. Spohn, _Large Scale Dynamics of Interacting Particles_. Springer, Berlin, 1991.
  * [67] E. A. Uehling and G. E. Uhlenbeck, _Transport Phenomena in Einstein–Bose and Fermi–Dirac Gases. I_. Phys. Rev., 43, (7), (1933), 552–561.
  * [68] A. A. Vlasov, _On Vibration Properties of Electron Gas_. J. Exp. Theor. Phys., 8, (3), (1938), 291–-318.
  * [69] A. A. Vlasov, _Many-particle theory and its application to plasma_. N. Y., Gordon and Breach, 1961.
  * [70] M. von Smoluchowski, _Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen_. Annalen der Physik, 326, (14), (1906), 756–780.
  * [71] J. Yvon, _La théorie statistique des fluides et l’équation d’état_. Hermann, no 203, 1935.