# To the kinetic theory of dense gases and liquids. Calculation of quasi-
equilibrium particle distribution functions by the method of collective
variables

Tokarchuk M<EMAIL_ADDRESS>Institute for Condensed Matter Physics of the
National Academy of Sciences of Ukraine,
1 Svientsitskii str., 79011, Lviv, Ukraine

###### Abstract

Based on a chain of BBGKI equations with a modified boundary condition that
takes into account multiparticle correlations, kinetic equations in the
approximate “pairs” collisions and in the polarization approximation, taking
into account the interaction through the third particle, obtained. The
specifics of the model representation of the pair potential of particle
interaction through short-range and long-range parts were taken into account.
In the case of the short-range potential in the form of the potential of solid
spheres, the contribution of Enskog’s revised theory to the complete
integration of the collision of the kinetic equation is obtained. The
collision integrals include paired quasi-equilibrium distribution functions
that depend on the nonequilibrium mean values of the particle number density
and the inverse temperature. The method of collective variables Yukhnovskii is
applied for the calculation of pair quasi-equilibrium distribution function
with an allocation of short-range and long-range parts in the potential of the
interaction of particles. In this case, the system with short-range
interaction is considered as a frame of reference.

###### keywords:

American Chemical Society, LaTeX

Lviv Polytechnic National University, 12 S. Bandera str., 79013, Lviv, Ukraine
IR,NMR,UV

##  1 Introduction

Studies of the role of repulsive and attractive forces between atoms,
molecules of liquids and their influence on thermodynamic, structural, and
dynamic properties have a history, starting with Van der Waals 1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 16, 18, 19, 20, 21, 22, 23, 24, 25,
26. And this is probably not all the work in which the problem is primarily
the correct definition of the nature of forces, the construction of model
capacity within the first coordination sphere of the liquid state. Everything
that happens with the nature of the interaction of particles in the first
coordination sphere of liquids is essential because, for highly correlated
liquids, the contribution of long-range attraction will not be significant.
For complex fluids: molecular, polar, and magnetic, when the orientational
degrees of freedom are also necessary, the structure of the first coordination
sphere will obviously determine the nature of the forces of interaction, the
emergence of certain cluster formations, collective hydrodynamic sound,
thermal modes, and nonhydrodynamic modes 27, 28, 29, 30. On a spatial and
temporal scale, nonequilibrium processes, which are the main within the first
coordination sphere of simple fluids, belong to the kinetic stage of the
temporal evolution of transfer processes.

In the nonequilibrium theory of dense gases and liquids, the approach of
kinetic equations makes it possible to obtain the transfer equation for the
hydrodynamic densities of the number of particles, their momentum, and energy,
and the expressions for the transfer coefficients as a function of the
particle parameters and external parameters 31, 32, 33, 34, 35. However,
constructing a kinetic theory for ordinary potentials at high densities is
significantly complicated (there are no small parameters). For dense gases and
liquids, time intervals $\tau_{f}$ — time of free run and $\tau_{hyd}$ —
characteristic time of change of sizes which are locally stored — densities
mass, momentum, and energy have the same order of magnitude. Therefore in the
process of “collision” of two particles, an important role is played by other
system particles. This means that the kinetics of dense systems must consider
the collective effects that are characteristic of the hydrodynamic approach.
In other words, this means that for dense gases and liquids, kinetics and
hydrodynamics are closely related and should be considered simultaneously.
There are only a few model systems for which it is possible to introduce a
proper kinetic equation (and then in the approximation of paired collisions)
and obtain formulas for the transfer coefficients. There are only a few model
systems for which it is possible to introduce a proper kinetic equation (and
then in the approximation of paired collisions) and obtain formulas for the
transfer coefficients. This is a system of solid spheres 36, 32, 33, 37, 38,
34, 39, 40, 41, a system with a square-well potential(SWP) 42, 43, 45, 44, 46,
47, 48, 49, 50, 51 and with a multistep potential (MSP) 52, 53, 54, 55, 56,
57, 58, 59 as a generalization of SWP. The first model system replaces all the
real potential by the effective diameter of solid spheres. It is a sure recipe
for calculating the contact value of the pair distribution function. The
latter two are unique in that they explicitly consider the far-reaching part
of the potential due to the corresponding irreversible integral of the
collisions.

In other models, Enskog’s kinetic theory or its revised variant (RET) 37, 38
is mainly used as a basis, and long-range interaction is taken into account
indirectly. In particular, Rice and Allnatt 60 added a Fokker–Planck
contribution to the integral of Enskog collisions: the particles between
collisions on a solid core seem to undergo Brownian motion due to the far-
reaching part of the potential. Due to this approximation, the numerical
calculations for shear viscosity and thermal conductivity differ significantly
61 from the data of molecular dynamics. Based on the entropy maximization
approach with some elms, for the potential “solid spheres + smooth tail”,
several kinetic theories of the mean-field type (KTMF) have been proposed 62,
63, 64, which are parallel and indirect, and take into account the attractive
interaction. In 65, the kinetic theory of the mean-field is applied to
mixtures, and the result is obtained within the Katz boundary. In 66, the
calculations of KTMF for the coefficients of self-diffusion, shear, and bulk
viscosity are compared with the results of other indirect approaches —
stochastic kinetic theory and renormalized Kirkwood theory. Other works that
use semi-phenomenological methods to account for attractive interactions
include 67, where an averaged scattering cross-section for viscosity is
introduced. The work 68 also proposes a kinetic theory for a system of
particles with the potential “hard spheres + truncated tail”, which combines a
kinetic theory for SWP systems and KTMF. This theory gives a good agreement
between the numerical calculations for the transfer coefficients and the
results of computer simulations and experiments.

The multistage potential has some advantage over a square-well because it
allows us to approximate real interaction better. It should note that in the
calculations of the transfer coefficients for argon 56, we have to some extent
seen the role of the first coordination sphere in the optimal choice of the
number of attraction walls in the multistage potential, which simulates the
Lennard–Jones potential. In fact, in these calculations, there is a problem of
optimal choice of the number of repulsive and attractive walls, which is
primarily manifested in the pair function of particle distribution, the
behavior of which determines the first coordination area. In fact, due to
these pairwise particle distribution functions, the liquid argon transfer
coefficients were calculated. It was shown that after the self-consistent
optimal number of attraction walls, the increase in their number did not
significantly affect the calculation results. It is obvious that within the
first coordination sphere in simple, complex liquids, there are short-acting
repulsive and attractive forces, the true nature of which is connected with
the internal nuclear-electronic structure of atoms and molecules, and has a
quantum-mechanical nature (regardless of temperature) in a certain way we
model one or another potential of interaction at the classical level of
description. The kinetic theory of mixtures with a multistage interaction
potential is considered in 59. The boundary transition to a smooth continuous
potential for the kinetic equation and the balanced equation for the potential
energy density is analyzed. When the limit potential is chosen in the form of
“solid spheres + smooth tail”, the kinetic equation is reduced to the equation
of kinetic variation theory (KVT). Still, the limit equation of balance for
the kinetic energy density differs from the corresponding equation KVT.

To highlight the contribution of short-range repulsive and attracting
particles within the formation of the short order (first coordination sphere)
in the theory of nonequilibrium processes, we consider a chain of BBGKI
equations for nonequilibrium distribution functions of interacting particles
of a classical system. We use the approach proposed in 69 based on the
modification of the boundary conditions for the weakening of correlations to
the chain of BBGKI equations. This approach was further developed in the works
70, 71, 72, 73, 55, 74, 75, 76, 56, 77, 78, 79, 30. It considers both the
nonequilibrium of the one-particle distribution function and the local laws of
conservation of mass, momentum and total energy, which form the basis of the
hydrodynamic description of the system’s evolution. Based on the modification
of the boundary condition of attenuation of correlations to the Liouville
equation, a system of coupled generalized equations for the nonequilibrium
one-particle distribution function and the average total energy density was
obtained, which are valid for describing nonequilibrium states of both far and
near equilibrium. An important achievement of this approach is that the
formulated modified boundary condition, which takes into account local
conservation laws to the chain BBGKI, made it possible for the first time in
71 to consistently derive the kinetic equation of the revised Enskog theory
37, 38, 34 for the system of solid spheres. Based on this result, we obtained
a kinetic equation for a system with a model multistep interparticle
interaction potential 55 (in particular, for this kinetic equation was proved
$H$-theorem in 55), as well as the kinetic equation Landau 70, 71, 80 for a
system of charged solid spheres. For the new kinetic equations obtained in
this way, normal solutions were found by the Chapman–Enskog method, which was
used to perform numerical calculations of the transfer coefficients of bulk,
shear viscosities, and thermal conductivity for systems that simulate argon
80, 56, once ionized argon 74.

An important problem in applying the obtained kinetic equations with integrals
of collisions in approximations of pair collisions or polarization is the
calculation of the pair quasi-equilibrium distribution function
$g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)$ as functions of nonequilibrium
values: of average density $n(\vec{r};t)$ and inverse temperature
$\beta(\vec{r};t)$. In the works 64, 39, 41, when considering the
corresponding models of the collision integral for this distribution function,
we used the generalization of the virial decomposition by the density chosen
over time.

In the second section, we consider a chain of BBGKI equations with a modified
boundary condition and with the selection of short-range repulsive and
attractive interactions between particles of the classical system. In the
third section, we consider a chain of BBGKI equations with a modified boundary
condition to approximate “paired ” collisions and obtain the corresponding
kinetic equation. In this section, we obtain the kinetic equation for the
nonequilibrium one-particle distribution function with the collision integral
in the polarization approximation when the interaction of two particles is
taken into account through the interaction of the third particle. In the
fourth section, we present one of the ways to calculate the pair quasi-
equilibrium distribution function, which is included in the collision
integrals of the corresponding kinetic equations.

##  2 Chain of BBGKI equations for nonequilibrium particle distribution
functions of a simple liquid

As a model of a simple fluid, we will consider a system of classical
interacting particles with the following Hamiltonian:

$H=\sum_{j=1}^{N}\frac{\vec{p}_{j}}{2m}+\frac{1}{2}\sum_{j,l}^{N}\Phi(\vec{r}_{j},\vec{r}_{l})$

where the indices $j$, $l$ number the particles with mass $m$ with vector
pulses $\vec{p}_{j}$ and vectors with coordinates $\vec{r}_{j}$ simple liquid,
$N$ is the total number of particles in the volume $V$;

$\Phi(\vec{r}_{j},\vec{r}_{l})=\Phi_{rep}(\vec{r}_{j},\vec{r}_{l})+\Phi_{attr}(\vec{r}_{j},\vec{r}_{l})=\Phi_{rep}^{sr}(\vec{r}_{j},\vec{r}_{l})+\Phi_{attr}^{sr}(\vec{r}_{j},\vec{r}_{l})+\Phi_{attr}^{lr}(\vec{r}_{j},\vec{r}_{l}),$
(1)

there is a pair potential for interaction between particles, consisting of a
short-acting repulsive (for example, a model of solid spheres)
$\Phi_{rep}^{sr}(\vec{r}_{j},\vec{r}_{l})$, short-range attractive
$\Phi_{attr}^{sr}(\vec{r}_{j},\vec{r}_{l})$ and long-range attractive parts
$\Phi_{attr}^{lr}(\vec{r}_{j},\vec{r}_{l})$. The full short-term interaction
is equal to

$\Phi_{tot}^{sr}(\vec{r}_{j},\vec{r}_{l})=\Phi_{rep}^{sr}(\vec{r}_{j},\vec{r}_{l})+\Phi_{attr}^{sr}(\vec{r}_{j},\vec{r}_{l}).$

The nonequilibrium state of the system is completely described by the
nonequilibrium distribution function of all particles
$\rho(x_{1},\ldots,x_{N};t)=\rho(x^{N};t)$ (where the notation
$x^{N}=x_{1},\ldots,x_{N}$, $x_{j}=\vec{p}_{j}\vec{r}_{j}$), whose evolution
in time is given Liouville equation

$\frac{\partial}{\partial t}\rho(x^{N};t)+iL_{N}\rho(x^{N};t)=0.$

with the Liouville operator

$iL_{N}=\sum_{j=1}^{N}\frac{\vec{p}_{j}}{m}\cdot\frac{\partial}{\partial\vec{r}_{j}}-\sum_{j,l}^{N}\frac{\partial}{\partial\vec{r}_{j}}\Phi(\vec{r}_{j},\vec{r}_{l})\left(\frac{\partial}{\partial\vec{p}_{j}}-\frac{\partial}{\partial\vec{p}_{l}}\right).$

We will describe the nonequilibrium state of the system with the help of chain
of BBGKI equations for nonequilibrium fluid particle distribution functions.
To do this, we use the approach proposed in 71, 75, where a modified chain of
equations BBGKI is built, taking into account the concept of a consistent
description of kinetics and hydrodynamics of nonequilibrium processes of the
system of interacting particles in the method of nonequilibrium statistical
operator Zubarev, based on the Liouville equation with the source:

$\frac{\partial}{\partial
t}\rho(x^{N};t)+iL_{N}\rho(x^{N};t)=-\varepsilon(\rho(x^{N};t)-\varrho_{rel}(x^{N};t)),$
(2)

which selects the delayed ($\varepsilon\rightarrow+0$, after thermodynamic
transition) solutions of the Liouville equation under given initial
conditions:

$\rho(x^{N};t)_{t=t_{0}}=\varrho_{rel}(x^{N};t_{0}),$

$\rho_{rel}(x^{N_{\alpha}};t)$ is the relevant (quasi-equilibrium) particle
distribution function obtained according to 71, 75 from the condition of the
maximum of the Gibbs entropy functional with preserved normalization
conditions for the distribution and the set parameters of the abbreviated
description of the nonequilibrium state of the liquid:
$\langle\hat{n}(x)\rangle^{t}=f_{1}(x;t)$ is the nonequilibrium one-particle
particle distribution function of liquid and
$\langle\hat{\varepsilon}_{int}(\vec{r})\rangle^{t}$ is the nonequilibrium
average energy of the interaction of liquid particles, which has the following
structure:

$\rho_{rel}(x^{N};t)=\exp\left\\{-\Phi(t)-\int
d\vec{r}\,\beta(\vec{r},t)\hat{\varepsilon}_{int}(\vec{r})-\int
dx\,a(x,t)\hat{n}(x)\right\\},$ (3)

where $\Phi(t)$ is the Masier–Planck functional

$\displaystyle\Phi(t)$ $\displaystyle=\int d\Gamma_{N}(x)\exp\left\\{-\int
d\vec{r}\,\beta(\vec{r},t)\hat{\varepsilon}_{int}(\vec{r})-\int
dx\,a(x,t)\hat{n}(x)\right\\}=\ln Z_{rel}(t),$ $\displaystyle Z_{rel}(t)$
$\displaystyle=\int d\Gamma_{N}(x)\exp\left\\{-\int
d\vec{r}\,\beta(\vec{r},t)\hat{\varepsilon}_{int}(\vec{r})-\int
dx\,a(x,t)\hat{n}(x)\right\\}$

is the statistical sum of the quasi-equilibrium distribution
$\rho_{rel}(x^{N};t)$, in which

$\hat{n}(x)=\sum_{j=1}^{N}\delta(x-x_{j})=\sum_{j=1}^{N}\delta(\vec{r}-\vec{r}_{j})\delta(\vec{p}-\vec{p}_{j})$

is the microscopic phase density of the number of particles and

$\hat{\varepsilon}_{int}(\vec{r})=\frac{1}{2}\sum_{j,l}^{N}\Phi(\vec{r}_{j},\vec{r}_{l})\delta(\vec{r}-\vec{r}_{j})=\hat{\varepsilon}_{rep}^{sr}(\vec{r})+\hat{\varepsilon}_{attr}^{sr}(\vec{r})+\hat{\varepsilon}_{attr}^{lr}(\vec{r})$

is the microscopic energy density of the interaction of particles with the
selected contributions of repulsive and attractive interactions;
$d\Gamma=(dx)^{N}/N!$, $dx=d\vec{p}\,d\vec{r}$. Moreover, the sum of the first
two $\hat{\varepsilon}_{tot}^{sr}(\vec{r})$ form the microscopic energy
density of the interaction of particles in the first coordination sphere. The
Lagrange parameters $\beta(\vec{r},t)$ (inverse of the nonequilibrium
temperature of the liquid), $a(x,t)$ are determined from the conditions of
self-agreement:

$\langle\hat{\varepsilon}_{int}(\vec{r})\rangle^{t}=\langle\hat{\varepsilon}_{int}(\vec{r})\rangle^{t}_{rel},\qquad\langle\hat{n}(x)\rangle^{t}=\langle\hat{n}(x)\rangle^{t}_{rel}.$
(4)

Taking into account, the structure $\rho_{rel}(x^{N};t)$ and the approach 71,
75, integrating the Liouville equation with the source (2) by the
corresponding coordinates and momentum of the particles, we obtain chain of
BBGKI equations with modified boundary conditions (taking into account space-
time interparticle correlations) for a simple fluid:

$\displaystyle\left(\frac{\partial}{\partial
t}+iL(1)\right)f_{1}(x_{1};t)+\int dx_{2}\,iL(1,2)f_{2}(x_{1},x_{2};t)=0,$ (5)
$\displaystyle\left(\frac{\partial}{\partial
t}+iL(1)+iL(2)+iL(1,2)\right)f_{2}(x_{1},x_{2};t)+\int
dx_{3}\big{(}iL(1,3)+iL(2,3)\big{)}f_{3}(x_{1},x_{2},x_{3};t)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad=-\varepsilon\big{(}f_{2}(x_{1},x_{2};t)-g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)f_{1}(x_{1};t)f_{2}(x_{2};t)\big{)},$
(6)

where $\varepsilon\rightarrow+0$ after the limit thermodynamic transition,

$\displaystyle iL(j)$
$\displaystyle=\frac{\vec{p}_{j}}{m}\cdot\frac{\partial}{\partial\vec{r}_{j}},$
$\displaystyle iL(j,l)$
$\displaystyle=-\frac{\partial}{\partial\vec{r}_{j}}\Phi(\vec{r}_{j},\vec{r}_{l})\cdot\left(\frac{\partial}{\partial\vec{p}_{j}}-\frac{\partial}{\partial\vec{p}_{l}}\right)$
$\displaystyle=-\frac{\partial}{\partial\vec{r}_{j}}\left(\Phi_{rep}^{sr}(\vec{r}_{j},\vec{r}_{l})+\Phi_{attr}^{sr}(\vec{r}_{j},\vec{r}_{l})+\Phi_{attr}^{lr}(\vec{r}_{j},\vec{r}_{l})\right)\cdot\left(\frac{\partial}{\partial\vec{p}_{j}}-\frac{\partial}{\partial\vec{p}_{l}}\right),$

there are one-part and two-part parts of the Liouville operator,
$g_{2}(\vec{r}_{1},\vec{r}_{2}|n;t)$ is a pair of quasi-equilibrium coordinate
functions of the distribution of fluid particles

$g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)=\frac{1}{n(\vec{r}_{1};t)n(\vec{r}_{2};t)}\int
d\Gamma_{N}(x)\,\hat{n}(\vec{r}_{1})\,\hat{n}(\vec{r}_{2})\,\rho_{rel}(x^{N};t).$
(7)

The distribution functions of the three particles satisfy the following
equations of the chain of equations BBGKI, which include nonequilibrium
functions of the distribution of four particles. It is important to note that
one- and two-particle nonequilibrium distribution functions determine the
behavior of hydrodynamic variables: average nonequilibrium values densities of
the number of particles $n(\vec{r};t)$, their momentum $\vec{p}(\vec{r};t)$,
kinetic energy $\varepsilon^{kin}(\vec{r};t)$, as well as the potential energy
$\varepsilon^{int}(\vec{r};t)$:

$\displaystyle n(\vec{r};t)$ $\displaystyle=\int
d\vec{p}\,f_{1}(\vec{r},\vec{p};t),\qquad\vec{p}(\vec{r};t)=\int
d\vec{p}\,f_{1}(\vec{r},\vec{p};t)\vec{p},$
$\displaystyle\varepsilon^{kin}(\vec{r};t)$ $\displaystyle=\int
d\vec{p}\,\frac{p^{2}}{2m}f_{1}(\vec{r},\vec{p};t),$
$\displaystyle\varepsilon^{int}(\vec{r};t)$ $\displaystyle=\int d\vec{p}\int
d\vec{p}^{\prime}\int
d\vec{r}^{\prime}\,\Phi(\vec{r},\vec{r}^{\prime})f_{2}(\vec{r},\vec{p},\vec{r}^{\prime},\vec{p}^{\prime};t),$

which satisfy the corresponding conservation laws of the average
nonequilibrium values of the number of particles of a simple liquid
$n(\vec{r};t)$, their momentum $\vec{p}(\vec{r};t)$ and full of energy

$\varepsilon(\vec{r};t)=\varepsilon^{kin}(\vec{r};t)+\varepsilon^{int}(\vec{r};t),$

underlying the hydrodynamic description of nonequilibrium processes of a
simple fluid. In addition, the pair is a quasi-equilibrium coordinate
distribution function $g_{2}(\vec{r},\vec{r}^{\prime}|n,\beta;t)$ and higher
coordinate quasi-equilibrium distribution functions
$g_{3}(\vec{r},\vec{r}^{\prime},\vec{r}^{\prime\prime}|n,\beta;t)$, …,
$g_{s}(\vec{r}^{s}|n,\beta;t)$ in the following equations BBGKI chains
describe multiparticle correlations and are functions of nonequilibrium
densities $n(\vec{r};t)$ and temperature $\beta(\vec{r};t)$. The structure of
the data of correlation functions should reflect the features of the selected
model for the pair potential of particle interaction (1). One of the ways to
calculate them will be considered in the last section.

In the next section, we consider the chain of BBGKI equations in the
approximation of pairwise collisions between particles.

##  3 The chain of BBGKI equations in the approximation of pairwise collisions
of particles and in the polarization approximation

In the approximation of pairwise collisions between particles, when the
distribution functions of the three particles are not taken into account, for
nonequilibrium two-particle distribution functions, we obtain 71, 75 the
following equation:

$\bigg{(}\frac{\partial}{\partial
t}+iL(1)+iL(2)+iL(1,2)\\!\bigg{)}\\!f_{2}(x_{1},x_{2};t)=-\varepsilon\big{(}f_{2}(x_{1},x_{2};t)-g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)f_{1}(x_{1};t)f_{1}(x_{2};t)\big{)}.$
(8)

The solution of this equation (8) can be represented as

$f_{2}(x_{1},x_{2};t)=\varepsilon\int_{-\infty}^{0}d\tau\,e^{(\varepsilon+i\bar{L}(1,2))\tau}g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t+\tau)f_{1}(x_{1};t+\tau)f_{1}(x_{2};t+\tau),$

where

$i\bar{L}(1,2)=iL(1)+iL(2)+iL(1,2).$

Substituting this solution into the equation (5), we obtain a non-Markov
kinetic equation for nonequilibrium one-particle simple fluid distribution
function:

$\left(\frac{\partial}{\partial
t}+iL(1)\\!\right)\\!f_{1}(x_{1};t)=-\int\\!\\!dx_{2}\,iL(1,2)\varepsilon\int_{-\infty}^{0}\\!\\!\\!\\!d\tau\,e^{(\varepsilon+i\bar{L}(1,2))\tau}g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t+\tau)f_{1}(x_{1};t+\tau)f_{1}(x_{2};t+\tau).$

The obtained kinetic equation for the nonequilibrium one-particle particle
distribution function is taken into account also the spatial heterogeneity of
the system. According to the nature of the interaction of particles (1), the
integral of the collisions in the right part is given in the form:

$\displaystyle I(x;t)$ $\displaystyle=-\int
dx_{2}\,iL(1,2)\varepsilon\int_{-\infty}^{0}d\tau\,e^{(\varepsilon+i\bar{L}(1,2))\tau}g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t+\tau)f_{1}(x_{1};t+\tau)f_{1}(x_{2};t+\tau)$
(9) $\displaystyle=-\int
d\vec{p}_{2}\bigg{(}\int_{0}^{\sigma}iL_{rep}^{sr}(1,2)\,\varepsilon\int_{-\infty}^{0}d\tau\,e^{(\varepsilon+i\bar{L}_{rep}^{sr}(1,2))\tau}+\int_{\sigma}^{r_{min}}iL_{attr}^{sr}(1,2)\,\varepsilon\int_{-\infty}^{0}d\tau
e^{(\varepsilon+i\bar{L}_{attr}^{sr}(1,2))\tau}$
$\displaystyle+\int_{r_{min}}^{\infty}iL_{attr}^{lr}(1,2)\,\varepsilon\int_{-\infty}^{0}d\tau\,e^{(\varepsilon+i\bar{L}_{attr}^{lr}(1,2))\tau}\bigg{)}g_{2}(\vec{r}_{1},\vec{r}_{2}|n\beta;t+\tau)f_{1}(x_{1};t+\tau)f_{1}(x_{2};t+\tau)\,d\vec{r}_{2},$

where the first term of the collision integral describes the repulsive short-
range processes of particle scattering, and the second represents the
attractive short-range scattering processes in the approximate pair
interactions. They form the kinetics of the middle order (the first
coordination sphere) in a simple fluid. The third term in the collision
integral describes the attractive long-range processes of particle scattering.
It forms a long-range order with a consistent description of the kinetics and
hydrodynamics of a simple fluid. In the future, we will choose the repulsive
short-range interaction potential in the form of a model of solid spheres:

$\Phi_{rep}^{sr}(|\vec{r}_{jl}|)=\Phi^{h.s.}(|\vec{r}_{jl}|)=\lim_{\epsilon\rightarrow\infty}\Phi^{\epsilon}(|\vec{r}_{jl}|),$

where

$\Phi^{\epsilon}(|\vec{r}_{jl}|)=\left\\{\begin{array}[]{ll}\epsilon,&|\vec{r}_{jl}|<\sigma;\\\
0,&|\vec{r}_{jl}|\geq\sigma.\\\ \end{array}\right.$

$\sigma$ is the diameter of a solid sphere. The potential
$\Phi^{h.s.}(|\vec{r}_{jl}|)$ for solid spheres is strongly singular (in
particular, the operator Liouville $iL(j,l)$ is poorly defined). That’s why we
will consider the potential $\Phi^{\epsilon}(|\vec{r}_{jl}|)$ and only in the
final expressions we put $\lim_{\epsilon\rightarrow\infty}$. Note the
characteristics of the potential $\Phi^{h.s.}(|\vec{r}_{jl}|)$,
$\Phi^{\epsilon}(|\vec{r}_{jl}|)$. First, since the radius of their
interaction is $r_{0}=\sigma^{+}$, then the area of interaction $\Delta
r_{0}\rightarrow+0$, and, secondly, time interaction $\tau_{0}\rightarrow+0$,
which is due to the singular the nature of the potential. Then, according to
the approach 71, 75, this part of the collision integral can be represented in
the form of the collision integral in the revised theory of Enskog:

$\displaystyle I_{rep}^{sr}(x;t)$
$\displaystyle=-\\!\\!\int\\!\\!d\vec{p}_{2}\int_{0}^{\sigma}\\!\\!iL_{rep}^{sr}(1,2)\,\varepsilon\int_{-\infty}^{0}\\!\\!\\!\\!\\!d\tau
e^{(\varepsilon+i\bar{L}_{rep}^{sr}(1,2))\tau}g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t+\tau)f_{1}(x_{1};t+\tau)f_{1}(x_{2};t+\tau)d\vec{r}_{2}$
$\displaystyle=-n\int
dx_{2}\,{\hat{T}}(1,2)g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)f_{1}(x_{l};t)f_{1}(x_{2};t),$
(10)

where

${\hat{T}}(1,2)=\sigma^{2}\int
d\hat{\vec{\sigma}}\,\Theta(\hat{\vec{\sigma}}\cdot\vec{g})(\hat{\vec{\sigma}}\cdot\vec{g})\left[\delta(\vec{r}_{1}-\vec{r}_{2}+\sigma^{+}\hat{\vec{\sigma}}){\hat{B}}(\vec{\sigma})-\delta(\vec{r}_{1}+\vec{r}_{2}-\sigma^{+}\hat{\vec{\sigma}})\right]$

is the operator of collisions of solid elastic spheres of Enskog, where

$\vec{v}^{\prime}_{1}=\vec{v}_{1}+\hat{\vec{\sigma}}(\hat{\vec{\sigma}}\cdot\vec{g}),\qquad\vec{v}^{\prime}_{2}=\vec{v}_{2}-\hat{\vec{\sigma}}(\hat{\vec{\sigma}}\cdot\vec{g}),$
(11)

$\vec{v}_{1}$, $\vec{v}_{2}$ are vectors — the velocities of particles to the
interaction, and $\vec{v}^{\prime}_{1}$, $\vec{v}^{\prime}_{2}$ are vectors —
their velocities after interaction with the relative velocity
$\vec{g}=\vec{v}_{2}-\vec{v}_{1}$,
${\hat{B}}(\hat{\sigma})\varphi(\vec{v}_{1},\vec{v}_{2})=\varphi(\vec{v}^{\prime}_{1},\vec{v}^{\prime}_{2})$
is the speed shift operator defined ratios (11). Further, if in the second and
third terms of the collision integral (9) perform the integration in parts and
take into account (3), the collision integral can be represented as:

$\displaystyle I(x;t)$ $\displaystyle=-n\int
dx_{2}\,{\hat{T}}(1,2)\,g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)\,f_{1}(x_{l};t)\,f_{1}(x_{2};t)$
$\displaystyle\quad-\int
d\vec{p}_{2}\int_{\sigma}^{r_{\min}}iL_{attr}^{sr}(1,2)\,g_{2}(\vec{r}_{1},\vec{r}_{2}|n;t)\,f_{1}(x_{1};t)\,f_{1}(x_{2};t)d\vec{r}_{2}$
$\displaystyle\quad+\int
d\vec{p}_{2}\int_{\sigma}^{r_{\min}}iL_{attr}^{sr}(1,2)\int_{-\infty}^{0}d\tau\,e^{(\varepsilon+i\bar{L}_{attr}^{sr}(1,2))\tau}\left(\frac{\partial}{\partial
t}+i\bar{L}_{attr}^{sr}(1,2)\right)$ $\displaystyle\qquad\qquad\times
g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t+\tau)f_{1}(x_{1};t+\tau)f_{1}(x_{2};t+\tau)\,d\vec{r}_{2}$
$\displaystyle\quad-\int
d\vec{p}_{2}\int_{r_{\min}}^{\infty}iL_{attr}^{lr}(1,2)\,g_{2}(\vec{r}_{1},\vec{r}_{2}|n;t)\,f_{1}(x_{1};t)\,f_{1}(x_{2};t)\,d\vec{r}_{2}$
$\displaystyle\quad+\int
d\vec{p}_{2}\int_{r_{\min}}^{\infty}iL_{attr}^{lr}(1,2)\int_{-\infty}^{0}d\tau\,e^{(\varepsilon+i\bar{L}_{attr}^{lr}(1,2))\tau}\left(\frac{\partial}{\partial
t}+i\bar{L}_{attr}^{lr}(1,2)\right)$ $\displaystyle\qquad\qquad\times
g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t+\tau)\,f_{1}(x_{1};t+\tau)\,f_{1}(x_{2};t+\tau)\,d\vec{r}_{2},$
(12)

in which the second and fourth terms in the right part correspond to the
generalized middle fields with potentials of short-range and long-range
attraction. The third and fifth terms of the collision integral are of the
second order in terms of interactions taking into account non-Markov processes
(time delay).

The kinetic equation with the collision integral (3) is derived from the BBGKI
chain of equations in approaching “pair” collisions, although a significant
part of the coordinates of dynamic correlations is already taken into account
in the quasi-equilibrium distribution functions
$g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)$.

For analysis of chain connections of BBGKI equations (5)–(2) in higher
approximations by interparticle correlations it is convenient to apply the
concept of group compositions 81, 82, 83. Group deposits were applied to the
chain of equations of BBGKI in many works 81, 82, 84, 85, 86, 87, 88, 89, 83,
92 with a boundary condition corresponding to the attenuation condition
correlations by Bogolyubov, in the works of Zubarev and Novikov 83, 90, 91,
which he was developed diagram method for constructing connections of the
chain of BBGKI equations. For the connection of the chain of equations (5)–(2)
similarly as in the works 83, 90, 91, and even more so in the works 81, 82,
84, let’s move from nonequilibrium of distribution functions $f_{s}(x^{s};t)$
to irreducible distribution functions $F_{s}(x^{s};t)$, which are introduced
by the quantities 81, 83, but in in our case with some modification:

$\displaystyle f_{1}(x_{1};t)$ $\displaystyle=F_{1}(x_{1};t),$ (13)
$\displaystyle f_{2}(x_{1},x_{2};t)$
$\displaystyle=F_{2}(x_{1},x_{2};t)+g_{2}(\vec{r}_{1},\vec{r}_{2};t)F_{1}(x_{1};t)F_{1}(x_{2};t),$
$\displaystyle f_{3}(x_{1},x_{2},x_{3};t)$
$\displaystyle=F_{3}(x_{1},x_{2},x_{3};t)$
$\displaystyle\quad+\sum\nolimits_{P}F_{2}(x_{1},x_{2};t)F_{1}(x_{3};t)+g_{3}(\vec{r}_{1},\vec{r}_{2},\vec{r}_{3};t)F_{1}(x_{1};t)F_{1}(x_{2};t)F_{1}(x_{3};t),\quad\ldots,$

in which the coordinate quasi-equilibrium distribution functions
$g_{2}(\vec{r}_{1},\vec{r}_{2};t)$,
$g_{3}(\vec{r}_{1},\vec{r}_{2},\vec{r}_{3};t)$, $g_{s}(\vec{r}^{s};t)$ are
determined by the relations (7) and

$g_{s}(\vec{r}^{s}|n,\beta;t)=\frac{1}{n(\vec{r}_{1};t)\ldots
n(\vec{r}_{s};t)}\int
d\Gamma_{N}(x)\hat{n}(\vec{r}_{1})\ldots\hat{n}(\vec{r}_{s})\rho_{rel}(x^{N};t).$

The modification of group compositions (13) is that a significant part of
simple correlations is taken into account over time quasi-equilibrium
functions $g_{s}(\vec{r}^{s};t)$. When $g_{s}(\vec{r}^{s};t)=1$ for
$s=2,3,\ldots$ these group clauses coincide with group clauses 81, 82, 84.
Since each line of equivalences (13) introduces exactly one new one function
$F_{s}(\vec{r}^{s};t)$, $s=1,2,3,\ldots$, then the data of the equation can be
related relatively of irreducible separation functions and write:

$\displaystyle F_{1}(x_{1};t)$ $\displaystyle=f_{1}(x_{1};t),$ (14)
$\displaystyle F_{2}(x_{1},x_{2};t)$
$\displaystyle=f_{2}(x_{1},x_{2};t)-g_{2}(\vec{r}_{1},\vec{r}_{2};t)f_{1}(x_{1};t)f_{1}(x_{2};t),$
$\displaystyle F_{3}(x_{1},x_{2},x_{3};t)$
$\displaystyle=f_{3}(x_{1},x_{2},x_{3};t)-\sum\nolimits_{P}f_{2}(x_{1},x_{2};t)f_{1}(x_{3};t)$
$\displaystyle\quad-
h_{3}(\vec{r}_{1},\vec{r}_{2},\vec{r}_{3};t)f_{1}(x_{1};t)f_{1}(x_{2};t)f_{1}(x_{3};t),\quad\ldots.$
(15)

Here and in (13) $\sum_{P}$ denotes the sum of all different permutations the
coordinate of three or more particles,

$h_{3}(\vec{r}_{1},\vec{r}_{2},\vec{r}_{3};t)=g_{3}(\vec{r}_{1},\vec{r}_{2},\vec{r}_{3};t)-g_{2}(\vec{r}_{1},\vec{r}_{2};t)-g_{2}(\vec{r}_{1},\vec{r}_{3};t)-g_{2}(\vec{r}_{2},\vec{r}_{3};t)$
(16)

is a three-particle quasi-equilibrium correlation function. Now we write down
the chain of equations BBGKI (5)–(2) for irreducible distribution functions
$F_{s}(x^{s};t)$, the first two comparison. The first equation of the chain
has the form:

$\left(\frac{\partial}{\partial t}+iL(1)\right)F_{1}(x_{1};t)+\int
dx_{2}\,iL(1,2)\,g_{2}(\vec{r}_{1},\vec{r}_{2};t)\,F_{1}(x_{1};t)\,F_{1}(x_{2};t)\\\
+\int dx_{2}\,iL(1,2)\,F_{2}(x_{1},x_{2};t)=0.$ (17)

Differentiating by time the expression for $F_{2}(x_{1},x_{2};t)$ in (14) and
using the second equation of the BBGKI chain (2) for the function
$f_{2}(x_{1},x_{2};t)$, for $F_{2}(x_{1},x_{2};t)$, the following equation:

$\left(\frac{\partial}{\partial
t}+iL_{2}+\varepsilon\right)F_{2}(x_{1},x_{2};t)=-\left(\frac{\partial}{\partial
t}+iL_{2}\right)g_{2}(\vec{r}_{1},\vec{r}_{2};t)F_{1}(x_{1};t)F_{1}(x_{2};t)\\\
-\int
dx_{3}\Big{\\{}iL(1,3)+iL(2,3)\Big{\\}}\Big{\\{}F_{3}(x_{1},x_{2},x_{3};t)+\sum\nolimits_{P}F_{2}(x_{1},x_{2};t)F_{1}(x_{3};t)\\\
+g_{3}(\vec{r}_{1},\vec{r}_{2},\vec{r}_{3};t)F_{1}(x_{1};t)F_{1}(x_{2};t)F_{1}(x_{3};t)\Big{\\}}.$
(18)

In a similar way, you can get an equation for a three-particle of the
irreducible function $F_{3}(x_{1},x_{2},x_{3};t)$ and higher functions
$F_{s}(x^{s};t)$ of particle distribution. Recall that the emergence of quasi-
equilibrium functions distribution $g_{2}(\vec{r}_{1},\vec{r}_{2};t)$,
$g_{3}(\vec{r}_{1},\vec{r}_{2},\vec{r}_{3};t)$ and $g_{s}(\vec{r}^{s};t)$ in
the chain of equations (17), (18) due to the fact that the boundary condition
for the binding of the equation Liuville (2) is considered an imbalance one-
particle distribution functions and local security laws, which corresponds to
the agreed description of kinetics and hydrodynamics systems 69, 70, 71, 72,
75, 78, 79. Because in this work we will analyze only the first two equations
(17), (18), then the following we will not write down the chain. It is
necessary note that if put formally $g_{s}(\vec{r}^{s};t)=1$, $s=2,3,\ldots$
in (17), (18), then we get the first two equations of the BBGKI chain of
equations for irreducible distribution functions $F_{1}(x_{1};t)$,
$F_{2}(x_{1},x_{2};t)$, taken in the work 90, 91. A characteristic feature
systems of equations (17), (18) is the first term in the right part of the
equation (18), ie the term with the time derivative from the double quasi-
equilibrium distribution function $g_{2}(\vec{r}_{1},\vec{r}_{2};t)$. As we
already are marked according to (7), the quasi-equilibrium pair the
distribution function is a functional of local values temperatures
$\beta(\vec{r};t)$ and average particle density $n(\vec{r};t)$. Therefore,
derivatives in time from
$g_{2}\big{(}\vec{r}_{1},\vec{r}_{2}|\beta(t),n(t)\big{)}$ will apply to
$\beta(\vec{r};t)$ and $n(\vec{r};t)$, which in turn under the terms of self-
agreement (4) will be to be expressed through the average values of energy in
the monitored system counting $\langle\hat{\cal
E}^{\prime}(\vec{r})\rangle^{t}$ and $\langle\hat{n}(\vec{r})\rangle^{t}$,
which form the basis of the hydrodynamic description nonequilibrium state of
the system.

Now consider the system of equations (17), (18) in approximation, when the
second equation (18) does not take into account three-particle irreducible
distribution functions $F_{3}(x_{1},x_{2},x_{3};t)$,
$h_{3}(\vec{r}_{1},\vec{r}_{2},\vec{r}_{3};t)$, which is analogous
polarization approximation in the case of retrieval of the
Bogolyubov–Lenard–Balescu kinetic equation 31, 93, 94, 95 for Coulomb plasma
in the same case. Taking into account (16), and that
$F_{3}(x_{1},x_{2},x_{3};t)\equiv 0$, and also
$h_{3}(\vec{r}_{1},\vec{r}_{2},\vec{r}_{3};t)\equiv 0$. The equation (18) is
written as follows:

$\left(\frac{\partial}{\partial
t}+iL_{2}+\varepsilon\right)F_{2}(x_{1},x_{2};t)=\left(\frac{\partial}{\partial
t}+iL_{2}\right)g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)F_{1}(x_{1};t)F_{1}(x_{2};t)\\\
-\int
dx_{3}\,iL(1,3)\left\\{F_{2}(x_{1},x_{2};t)F_{1}(x_{3};t)+F_{2}(x_{2},x_{3};t)F_{1}(x_{1};t)\right\\}\\\
-\int
dx_{3}\,iL(2,3)\left\\{F_{2}(x_{1},x_{2};t)F_{1}(x_{3};t)+F_{2}(x_{1},x_{3};t)F_{1}(x_{2};t)\right\\}\\\
-\int
dx_{3}\left\\{iL(1,3)+iL(2,3)\right\\}\left\\{g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)+g_{2}(\vec{r}_{1},\vec{r}_{3}|n,\beta;t)+g_{2}(\vec{r}_{2},\vec{r}_{3}|n,\beta;t)\right\\}\\\
\times F_{1}(x_{1};t)F_{1}(x_{2};t)F_{1}(x_{3};t).$ (19)

Next, we introduce the operator that can be obtained by varying of the mean
field collision operator near the nonequilibrium distribution
$F_{1}(x_{1};t)$:

$\delta\left(\int dx_{3}\,iL(1,3)\,F_{1}(x_{3};t)\,F_{1}(x_{1};t)\right)=\int
dx_{3}\,iL(1,3)\,F_{1}(x_{3};t)\,\delta F_{1}(x_{1};t)=L(x_{1};t)\,\delta
F_{1}(x_{1};t).$

Then the equation (19) using the operator $L(x_{1};t)$ is represented in in
the form of

$\left(\frac{\partial}{\partial
t}+iL_{2}+L(x_{1},x_{2};t)+\varepsilon\right)F_{2}(x_{1},x_{2};t)\\\
=-\left(\frac{\partial}{\partial
t}+iL_{2}+L(x_{1},x_{2};t)\right)g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)F_{1}(x_{1};t)F_{1}(x_{2};t),$

where we find the formal solution for the fundamental two-particle function
distribution:

$F_{2}(x_{1},x_{2};t)=-\int_{-\infty}^{t}dt^{\prime}e^{\varepsilon(t^{\prime}-t)}U(t,t^{\prime})\\\\[-2.84526pt]
\times\left(\frac{\partial}{\partial
t^{\prime}}+iL_{2}+L(x_{1},x_{2};t^{\prime})\right)g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t^{\prime})F_{1}(x_{1};t^{\prime})F_{1}(x_{2};t^{\prime}),$
(20)

where $U(t,t^{\prime})$ is an evolution operator:

$\displaystyle U(t,t^{\prime})$
$\displaystyle=\exp_{+}\left(-\int_{t^{\prime}}^{t}dt^{\prime\prime}\left(iL_{2}+L(x_{1},x_{2};t^{\prime\prime})\right)\right),$
$\displaystyle L(x_{1},x_{2};t)=L(x_{1};t)+L(x_{2};t).$

As a result, we obtain the expression for the irreducible quasi-equilibrium
two-particle distribution functions $F_{2}(x_{1},x_{2};t)$ in generalized
polarization approximation. Substitute this expression (20) into the first
equation of the chain (17), then we get:

$\left(\frac{\partial}{\partial t}+iL(1)\right)F_{1}(x_{1};t)+\int
dx_{2}\,iL(1,2)\,g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)\,F_{1}(x_{1};t)\,F_{1}(x_{2};t)\\\
=\int
dx_{2}\int_{-\infty}^{t}dt^{\prime}e^{\varepsilon(t^{\prime}-t)}iL(1,2)U(t,t^{\prime})\\\
\times\left(\frac{\partial}{\partial
t^{\prime}}+iL_{2}+L(x_{1},x_{2};t^{\prime})\right)g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t^{\prime})F_{1}(x_{1};t^{\prime})F_{1}(x_{2};t^{\prime})$
(21)

is a generalized kinetic equation for a nonequilibrium one-particle function
distribution with the non-Markov collision integral in the generalized
polarization approximation. It should be noted that the presence in the
integral collisions (21) of the mean field collision operator
$L(x_{1},x_{2};t)$ indicates that it takes into account the collective
effects. In general, the analysis of the integral of collisions in (21) is a
rather complex mathematical task. Obviously, for each physical model of a
system of particles, or of the nonequilibrium state, the integral of
collisions in (21), or the expression for $F_{2}(x_{1},x_{2};t)$ (20) can be
significantly simplified. In addition, for further application of the obtained
kinetic equations, in particular, their solution by the Chapman–Enskog method
and construction of the corresponding equations of hydrodynamics, it is
necessary to set the algorithm for calculating the pair coordinate quasi-
equilibrium distribution function $g_{2}(\vec{r},\vec{r}^{\prime}|n,\beta;t)$.

In the next section, the method of collective variables Yukhnovskii 96, 97,
98, 99, 100 will be used to calculate the pair quasi-equilibrium distribution
function and higher-order distribution functions.

##  4 Calculation of the statistical sum of the quasi-equilibrium distribution
and coordinate functions of the distribution $g_{s}((\vec{r})^{s}|n,\beta;t)$
by the method of collective variables

For calculations of the even coordinate quasi-equilibrium distribution
function $g_{2}(\vec{r},\vec{r}^{\prime}|n,\beta;t)$ and higher correlation
functions $g_{s}((\vec{r})^{s}|n,\beta;t)$ use the statistical sum of the
quasi-equilibrium distribution:

$Z_{rel}(t)=\int d\Gamma_{N}(x)\,\exp\left\\{-\int
d\vec{r}\,\beta(\vec{r},t)\,\hat{\varepsilon}_{int}(\vec{r})-\int
dx\,a(x,t)\,\hat{n}(x)\right\\}.$

Correlation functions $g_{s}((\vec{r})^{s}|n,\beta;t)$ are determined by
variational derivatives on the parameter $a(x,t)$ of the corresponding order
from the statistical sum $Z_{rel}(t)$:

$g_{s}((\vec{r})^{s}|n,\beta;t)=\frac{1}{n(\vec{r}_{1},t)\ldots
n(\vec{r}_{s},t)}\int d\vec{p}_{1}\ldots\int
d\vec{p}_{s}\,\frac{\delta}{\delta
a(\vec{p}_{1},\vec{r}_{1},t)}\ldots\frac{\delta}{\delta
a(\vec{p}_{s},\vec{r}_{s},t)}\ln Z_{rel}(t).$

The calculation of the statistical sum of the quasi-equilibrium distribution
will be performed using the method of collective variables 96, 97, 98, 99,
100. In doing so, we will take into account the nature of short-range and
long-range interactions of particles, therefore $Z_{rel}(t)$ is given in the
form:

$Z_{rel}(t)=\int d\Gamma_{N}(x)\,\exp\left\\{-\int
d\vec{r}\,\beta(\vec{r},t)\,\hat{\varepsilon}_{int}^{sr}(\vec{r})-\int
d\vec{r}\,\beta(\vec{r},t)\,\hat{\varepsilon}_{int}^{lr}(\vec{r})-\int
dx\,a(x,t)\,\hat{n}(x)\right\\}.$

Next, assuming that for the far-reaching part of the potential
$\Phi_{attr}^{lr}(|{\bf r}_{lj}|)$ there is a Fourier image $\nu(\vec{k})$ in
the space of wave vectors $\vec{k}$, the far-reaching part of $\int
d\vec{r}\,\beta(\vec{r},t)\,\hat{\varepsilon}_{int}^{lr}(\vec{r})$ will be
presented in the form of

$\int
d\vec{r}\,\beta(\vec{r},t)\,\hat{\varepsilon}_{int}^{lr}(\vec{r})=\frac{1}{2V^{2}}\sum_{\vec{q}}\sum_{\vec{k}}\beta_{-\vec{q}}(t)\,\nu(\vec{k})\left(\hat{\rho}_{\vec{q}+\vec{k}}\,\hat{\rho}_{-\vec{k}}-\hat{\rho}_{\vec{q}}\right),$
(22)

where

$\hat{\rho}_{\vec{k}}=\sum_{j=1}^{N}\mbox{e}^{-i{\vec{k}}\vec{r}_{j}}$

is the Fourier component of the particle number density. Taking into account
(22), the statistical sum of the quasi-equilibrium distribution is given in
the form:

$\displaystyle Z_{rel}(t)$ $\displaystyle=\int d\Gamma_{N}(x)\exp\left\\{-\int
d\vec{r}\,\beta(\vec{r},t)\,\hat{\varepsilon}_{int}^{sr}(\vec{r})-\int
dx\,a(x,t)\,\hat{n}(x)\right\\}$
$\displaystyle\quad\times\exp\left\\{-\frac{1}{2V^{2}}\sum_{\vec{q}}\sum_{\vec{k}}\beta_{-\vec{q}}(t)\,\nu(\vec{k})\left(\hat{\rho}_{\vec{q}+\vec{k}}\hat{\rho}_{-\vec{k}}-\hat{\rho}_{\vec{q}}\right)\right\\}.$
(23)

Next, enter the Jacobian transition

$\hat{J}(\rho)=\delta(\hat{\rho}-\rho)=\prod_{\vec{k}}\delta(\hat{\rho}_{\vec{k}}-\rho_{\vec{k}})$

to the collective variables $\rho_{\vec{k}}$, the statistical sum (4) is
written as:

$Z_{rel}(t)=Z_{rel}^{sh}(t)\int
d\rho\,\langle\hat{J}(\rho)\rangle^{t}_{rel,sh}\exp\left\\{-\frac{1}{2V^{2}}\sum_{\vec{q}}\sum_{\vec{k}}\beta_{-\vec{q}}(t)\,\nu(\vec{k})\left(\rho_{\vec{q}+\vec{k}}\rho_{-\vec{k}}-\rho_{\vec{q}}\right)\right\\},$

where $d\rho=\prod_{\vec{k}}d\rho_{\vec{k}}$ and

$\langle\hat{J}(\rho)\rangle^{t}_{rel,sh}=\int
d\Gamma_{N}(x)\delta(\hat{\rho}-\rho)\,\rho_{rel,sh}(x^{N};t)$

is the average Jacobian transition to collective variables by quasi-
equilibrium distribution, which takes into account only short-range repulsive
and attractive interactions between particles

$\rho_{rel,sh}(x^{N};t)=\frac{1}{Z_{rel}^{sh}(t)}\exp\left\\{-\int
d\vec{r}\,\beta(\vec{r},t)\,\hat{\varepsilon}_{int}^{sr}(\vec{r})-\int
dx\,a(x,t)\,\hat{n}(x)\right\\},$ (24)

the statistical sum of which has the following structure:

$Z_{rel}^{sh}(t)=\int d\Gamma_{N}(x)\exp\left\\{-\int
d\vec{r}\,\beta(\vec{r},t)\,\hat{\varepsilon}_{int}^{sr}(\vec{r})-\int
dx\,a(x,t)\,\hat{n}(x)\right\\}.$

Next, we apply the integral representation for the $\delta$-function, then
$\hat{J}(\rho)$ is given as

$\hat{J}(\rho)=\int
d\omega\exp\Big{\\{}-i\pi\sum_{\vec{k}}\omega_{\vec{k}}(\hat{\rho}_{\vec{k}}-\rho_{\vec{k}})\Big{\\}}.$

Using the cumulative schedule 98, 99, 100 for
$\langle\hat{J}(\rho)\rangle^{t}_{rel,sh}=W(\rho;t)$, we get:

$\displaystyle W(\rho;t)$
$\displaystyle=\langle\hat{J}(\rho)\rangle^{t}_{rel,sh}=\int
d\Gamma_{N}\rho_{rel,sh}(x^{N};t)\hat{J}(\rho)$ $\displaystyle=\int
d\omega\exp\bigg{\\{}-i\pi\sum_{\vec{k}}\omega_{\vec{k}}\bar{\rho}_{\vec{k}}-\frac{\pi^{2}}{2}\sum_{{\vec{k}}_{1},{\vec{k}}_{2}}\mathfrak{M}_{2}({\vec{k}}_{1},{\vec{k}}_{2};t)\omega_{{\vec{k}}_{1}}\omega_{{\vec{k}}_{2}}\bigg{\\}}\exp\bigg{\\{}\sum_{n\geq
3}D_{n}(\omega;t)\bigg{\\}},$ (25)

where

$\displaystyle\bar{\rho}_{\vec{k}}$
$\displaystyle=\rho_{\vec{k}}-\langle\hat{\rho}_{\vec{k}}\rangle_{rel,sh}^{t},$
$\displaystyle
d\omega=\Pi_{\vec{k}}d\omega_{\vec{k}}^{r}d\omega_{\vec{k}}^{s},$
$\displaystyle\omega_{\vec{k}}$
$\displaystyle=\omega_{\vec{k}}^{r}-i\omega_{\vec{k}}^{s},$
$\displaystyle\omega_{-\vec{k}}=\omega_{\vec{k}}^{*},$
$D_{n}(\omega;t)=\frac{(-i\pi)^{n}}{n!}\sum_{{\vec{k}}_{1},\ldots,{\vec{k}}_{n}}\mathfrak{M}_{n}({\vec{k}}_{1},\ldots,{\vec{k}}_{n};t)\,\omega_{{\vec{k}}_{1}}\ldots\omega_{{\vec{k}}_{n}}.$

Quasi-equilibrium cumulative averages of $n$-order

$\mathfrak{M}_{n}({\vec{k}}_{1},\ldots,{\vec{k}}_{n};t)=\langle\hat{\rho}_{{\vec{k}}_{1}}\ldots\hat{\rho}_{{\vec{k}}_{n}}\rangle_{rel,sh}^{t,c}$

calculated with the relevant distribution function with short-range
interparticle interaction (24), where the superscript $c$ means the cumulative
average. It is important to note that in (3) we have separated the
contributions from the short-term ones and long-range interactions. Short-term
interactions are taken into account in the relevant distribution (24) (which
can be considered as basic), and long-range interactions are presented through
collective variables.

For further calculation, the structural function $W(\rho;t)$ is given in the
form:

$\displaystyle W(\rho;t)$ $\displaystyle=\int
d\omega\exp\bigg{\\{}-i\pi\sum_{\vec{k}}\omega_{\vec{k}}\bar{\rho}_{\vec{k}}-\frac{\pi^{2}}{2}\sum_{{\vec{k}}_{1},{\vec{k}}_{2}}\mathfrak{M}_{2}({\vec{k}}_{1},{\vec{k}}_{2};t)\omega_{{\vec{k}}_{1}}\omega_{{\vec{k}}_{2}}\bigg{\\}}$
$\displaystyle\quad\times\left(1+B+\frac{1}{2!}B^{2}+\frac{1}{3!}B^{3}+\ldots+\frac{1}{n!}B^{n}+\dots\right),$
(26)

where $B=\sum_{n\geq 3}D_{n}(\omega;t)$. If in the schedule in a number of
exponents (4), ie $\exp\\{\sum_{n\geq 3}D_{n}(\omega;t)\\}$, save only the
first term, equal to one, we obtain the Gaussian approximation for
$W(\rho;t)$:

$W^{G}(\rho;t)=\int
d\omega\exp\bigg{\\{}i\pi\sum_{\vec{k}}\omega_{\vec{k}}\bar{\rho}_{\vec{k}}-\frac{\pi^{2}}{2}\sum_{{\vec{k}}_{1},{\vec{k}}_{2}}\mathfrak{M}_{2}({\vec{k}}_{1},{\vec{k}}_{2};t)\,\omega_{{\vec{k}}_{1}}\omega_{{\vec{k}}_{2}}\bigg{\\}},$
(27)

where the quasi-equilibrium cumulative average density–density has the form:

$\mathfrak{M}_{2}({\vec{k}}_{1},{\vec{k}}_{2};t)=\langle\hat{\rho}_{{\vec{k}}_{1}}\hat{\rho}_{{\vec{k}}_{2}}\rangle_{rel,sh}^{t,c}=\langle\hat{\rho}_{\vec{k}}\hat{\rho}_{-\vec{k}}\rangle_{rel,sh}^{t}-\langle\hat{\rho}_{\vec{k}}\rangle_{rel,sh}^{t}\langle\hat{\rho}_{-\vec{k}}\rangle_{rel,sh}^{t}.$

To integrate for $d\omega$ in (27) it is necessary to give expression in the
exponent to the quadratic diagonal form for $\omega_{\vec{k}}$. In this
regard, you need to find your values by solving the equation:

$\det\big{|}\tilde{\mathfrak{M}}_{2}({\vec{k}}_{1},{\vec{k}}_{2};t)-\tilde{E}({\vec{k}}_{1},{\vec{k}}_{2};t)\big{|}=0,$

$\tilde{E}({\vec{k}}_{1},{\vec{k}}_{2};t)$ is a diagonal matrix. With this in
mind, we obtain:

$W^{G}(\rho;t)=\int
d\omega\,\exp\bigg{\\{}-i\pi\sum_{\vec{k}}\bar{\rho}_{\vec{k}}\omega_{\vec{k}}-\frac{\pi^{2}}{2}\sum_{{\vec{k}}}E({\vec{k}};t)\,\omega_{\vec{k}}\omega_{-\vec{k}}\bigg{\\}}.$
(28)

The subintegral expression in (28) is a quadratic function
$\tilde{\omega}_{\vec{k}}$, therefore, performing integration for
$d\omega_{\vec{k}}$, for a structural function in the Gaussian approximation
$W^{G}(a;t)$ we obtain:

$W^{G}(\rho;t)=Z(t)\exp\left\\{-\frac{1}{2}\sum_{\vec{k}}E^{-1}({\vec{k}};t)\,\bar{\rho}_{{\vec{k}}}\,\bar{\rho}_{-{\vec{k}}}\right\\},$

where

$Z(t)=\exp\left\\{-\frac{1}{2}\sum_{\vec{k}}\ln\pi\det\tilde{E}({\vec{k}};t)\right\\}.$

The structural function $W^{G}(\rho;t)$ in the Gaussian approximation makes it
possible to calculate full structural function (4) in higher approximations by
Gaussian moments 99, 100:

$W(\rho;t)=\bar{W}^{G}(\rho;t)\exp\left\\{\sum_{n\geq
3}\langle\tilde{D}_{n}(\rho;t)\rangle_{G}\right\\},$

where

$\bar{W}^{G}(\rho;t)=Z(t)\exp\left\\{-\frac{1}{2}\sum_{\vec{k}}E^{-1}({\vec{k}};t)\,\bar{\rho}_{{\vec{k}}}\,\bar{\rho}_{-{\vec{k}}}\right\\}$

and $\langle\tilde{D}_{n}(\rho;t)\rangle_{G}$ approximate as follows:

$\displaystyle\langle\tilde{D}_{3}(\rho;t)\rangle_{G}$
$\displaystyle=\langle\bar{D}_{3}(\rho;t)\rangle_{G},$
$\displaystyle\langle\tilde{D}_{4}(\rho;t)\rangle_{G}$
$\displaystyle=\langle\bar{D}_{4}(\rho;t)\rangle_{G},$
$\displaystyle\langle\tilde{D}_{6}(\rho;t)\rangle_{G}$
$\displaystyle=\langle\bar{D}_{6}(\rho;t)\rangle_{G}-\frac{1}{2}\langle\bar{D}_{3}(\rho;t)\rangle_{G}^{2},$
$\displaystyle\langle\tilde{D}_{8}(\rho;t)\rangle_{G}$
$\displaystyle=\langle\bar{D}_{8}(\rho;t)\rangle_{G}-\langle\bar{D}_{3}(\rho;t)\rangle_{G}\langle\bar{D}_{5}(\rho;t)\rangle_{G}-\frac{1}{2}\langle\bar{D}_{4}(\rho;t)\rangle_{G}^{2},$
$\displaystyle\langle\tilde{D}_{n}(\rho;t)\rangle_{G}$
$\displaystyle=\frac{1}{\bar{W}^{G}(\rho;t)}\sum_{{\vec{k}}_{1},\ldots,{\vec{k}}_{n}}\bar{\mathfrak{M}}_{n}({\vec{k}}_{1},\ldots,{\vec{k}}_{n};t)\frac{1}{(i\pi)^{n}}\frac{\delta^{n}}{\delta\bar{\rho}_{{\vec{k}}_{1}}\ldots\delta\bar{\rho}_{{\vec{k}}_{n}}}\bar{W}^{G}(\rho;t).$

$\langle\tilde{D}_{n}(\rho;t)\rangle_{G}$ are renormalized $n$-quasi-
equilibrium cumulative averages for variables $\bar{\rho}_{\vec{k}}$ higher
orders.

$Z_{rel}(t)=Z_{rel}^{sh}(t)\int
d\rho\,W_{\beta}(\rho;t)\,\bar{W}^{G}(\rho;t)\exp\bigg{\\{}\sum_{n\geq
3}\langle\tilde{D}_{n}(\rho;t)\rangle_{G}\bigg{\\}},$

where

$W_{\beta}(\rho;t)=\exp\left(-\frac{1}{2V^{2}}\sum_{\vec{q}}\sum_{\vec{k}}\beta_{-{\vec{q}}}(t)\nu({\vec{k}})\big{(}\rho_{{\vec{q}}+{\vec{k}}}\rho_{-{\vec{k}}}-\rho_{{\vec{q}}}\big{)}\right).$

In the Gaussian approximation of the collective variables for
$W_{\beta}(\rho;t)$ quasi-equilibrium distribution function will look like
this:

$Z_{rel}^{G}(t)=Z_{rel}^{sh}(t)\int(d\rho)\,W_{\beta}(\rho;t)\,\bar{W}^{G}(\rho;t).$

In this approximation for the pair quasi-equilibrium function of particles, we
obtain:

$g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)=\frac{1}{n(\vec{r}_{1},t)\,n(\vec{r}_{2},t)}\int
d\vec{p}_{1}\int d\vec{p}_{2}\,\frac{\delta}{\delta
a(\vec{p}_{1},\vec{r}_{1},t)}\frac{\delta}{\delta
a(\vec{p},\vec{r}_{2},t)}\ln\bigg{(}Z_{rel}^{sh}(t)$ (29)
$\times\exp\bigg{\\{}-\frac{1}{2}\sum_{\vec{k}}\ln\pi\det\tilde{E}({\vec{k}};t)\bigg{\\}}V(\beta,n;t)\bigg{)},$

where

$V(\beta,n;t)=\int(d\rho)\exp\bigg{\\{}-\frac{1}{2}\sum_{\vec{k}}E^{-1}({\vec{k}};t)\,\bar{\rho}_{{\vec{k}}}\,\bar{\rho}_{-{\vec{k}}}\bigg{\\}}\exp\bigg{(}\\!-\frac{1}{2V^{2}}\sum_{\vec{q}}\sum_{\vec{k}}\beta_{-{\vec{q}}}(t)\,\nu({\vec{k}})\big{(}\rho_{{\vec{q}}+{\vec{k}}}\rho_{-{\vec{k}}}-\rho_{{\vec{q}}}\big{)}\bigg{)}.$

To integrate the collective variables in $V(\beta,n;t)$ in the first exponent
under the integral we write
$\bar{\rho}_{{\vec{k}}}=\rho_{{\vec{k}}}-\langle\hat{\rho}_{{\vec{k}}}\rangle_{rel,sh}^{t}$.
Then we get

$-\frac{1}{2}\sum_{\vec{k}}E^{-1}({\vec{k}};t)\bar{\rho}_{{\vec{k}}}\bar{\rho}_{-{\vec{k}}}=-\frac{1}{2}\sum_{\vec{k}}E^{-1}({\vec{k}};t)\rho_{{\vec{k}}}\rho_{-{\vec{k}}}\\\\[-2.84526pt]
+\sum_{\vec{k}}E^{-1}({\vec{k}};t)\rho_{{\vec{k}}}\langle\hat{\rho}_{-{\vec{k}}}\rangle_{rel,sh}^{t}-\frac{1}{2}\sum_{\vec{k}}E^{-1}({\vec{k}};t)\langle\hat{\rho}_{{\vec{k}}}\rangle_{rel,sh}^{t}\langle\hat{\rho}_{-{\vec{k}}}\rangle_{rel,sh}^{t}.$

Then in the second exponent under the integral, you need the first term to the
quadratic form:

$-\frac{1}{2V^{2}}\sum_{\vec{q}}\sum_{\vec{k}}\beta_{-{\vec{q}}}(t)\,\nu({\vec{k}})\big{(}\rho_{{\vec{q}}+{\vec{k}}}\rho_{-{\vec{k}}}-\rho_{{\vec{q}}}\big{)}=-\frac{1}{2V^{2}}\sum_{\vec{k}}\lambda(\vec{k},t)\,\rho_{{\vec{k}}}\,\rho_{-{\vec{k}}}+\frac{1}{2V^{2}}\sum_{\vec{k}^{\prime}}\nu({\vec{k}^{\prime}})\sum_{\vec{k}}\beta_{-{\vec{k}}}(t)\,\rho_{{\vec{k}}},$

where $\lambda({\vec{k}}_{1},{\vec{k}}_{2};t)$ there are eigenvalues from the
equation:

$\det\big{|}\tilde{\mathfrak{B}}({\vec{k}}_{1},{\vec{k}}_{2};t)-\tilde{\lambda}({\vec{k}}_{1},{\vec{k}}_{2};t)\big{|}=0,$

in which
$\mathfrak{B}({\vec{k}}_{1},{\vec{k}}_{2};t)=\beta_{{\vec{k}}_{1}-{\vec{k}}_{2}}(t)\nu({\vec{k}}_{1})$.

Given the transformation data in the exponents of $V(\beta,n;t)$, you can
submit it as

$V(\beta,n;t)=\int(d\rho)\exp\bigg{\\{}-\frac{1}{2}\sum_{\vec{k}}G({\vec{k}};t)\Big{(}\rho_{{\vec{k}}}-\frac{1}{2}R({\vec{k}};t)\Big{)}^{2}\bigg{\\}}\\\\[-2.84526pt]
\times\exp\bigg{\\{}-\frac{1}{2}\sum_{\vec{k}}\Big{(}E^{-1}({\vec{k}};t)\langle\hat{\rho}_{{\vec{k}}}\rangle_{rel,sh}^{t}\langle\hat{\rho}_{-{\vec{k}}}\rangle_{rel,sh}^{t}-\frac{1}{4}G({\vec{k}};t)R^{2}({\vec{k}};t)\Big{)}\bigg{\\}}$

where

$\displaystyle G({\vec{k}};t)$
$\displaystyle=E^{-1}({\vec{k}};t)+\frac{1}{2V^{2}}\lambda(\vec{k},t),$
$\displaystyle R({\vec{k}};t)$
$\displaystyle=\frac{1}{1+\frac{1}{2V^{2}}\lambda(\vec{k},t)E({\vec{k}};t)}\bigg{(}\langle\hat{\rho}_{{\vec{k}}}\rangle_{rel,sh}^{t}+\frac{1}{2V^{2}}\sum_{\vec{k}^{\prime}}\nu({\vec{k}^{\prime}})\beta_{{\vec{k}}}(t)\bigg{)}.$

After integration by collective variables, we finally get

$V(\beta,n;t)=\exp\bigg{\\{}-\frac{1}{2}\sum_{\vec{k}}\ln\,\det\tilde{G}({\vec{k}};t)\bigg{\\}}\\\\[-2.84526pt]
\times\exp\bigg{\\{}-\frac{1}{2}\sum_{\vec{k}}\bigg{(}E^{-1}({\vec{k}};t)\langle\hat{\rho}_{{\vec{k}}}\rangle_{rel,sh}^{t}\langle\hat{\rho}_{-{\vec{k}}}\rangle_{rel,sh}^{t}-\frac{1}{4}G({\vec{k}};t)R^{2}({\vec{k}};t)\bigg{)}\bigg{\\}}.$

After substituting $V(\beta,n;t)$ in (29), we get:

$\displaystyle g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)$
$\displaystyle=g_{2}^{sh}(\vec{r}_{1}\,\vec{r}_{2}|n,\beta;t)-\frac{1}{n(\vec{r}_{1},t),n(\vec{r}_{2},t)}$
$\displaystyle\quad\times\int\\!\\!d\vec{p}_{1}\int\\!\\!d\vec{p}_{2}\,\frac{\delta}{\delta
a(\vec{p}_{1},\vec{r}_{1},t)}\,\frac{\delta}{\delta
a(\vec{p}_{2},\vec{r}_{2},t)}\bigg{(}\frac{1}{2}\sum_{\vec{k}}\ln\pi\det\tilde{E}({\vec{k}};t)+\frac{1}{2}\sum_{\vec{k}}\ln\det\tilde{G}({\vec{k}};t)$
$\displaystyle\qquad\qquad+\frac{1}{2}\sum_{\vec{k}}\bigg{(}E^{-1}({\vec{k}};t)\langle\hat{\rho}_{{\vec{k}}}\rangle_{rel,sh}^{t}\langle\hat{\rho}_{-{\vec{k}}}\rangle_{rel,sh}^{t}-\frac{1}{4}G({\vec{k}};t)R^{2}({\vec{k}};t)\bigg{)}\bigg{)},$

where $g_{2}^{sh}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)$ is an even quasi-
equilibrium coordinate function of particle distribution with short-range
interaction. The following terms are complex in structure and are related to
the functional derivatives $\frac{\delta}{\delta
a(\vec{p}_{1},\vec{r}_{1},t)}\frac{\delta}{\delta a(\vec{p},\vec{r}_{2},t)}$,
acting on $\langle\hat{\rho}_{{\vec{k}}}\rangle_{rel,sh}^{t}$ and
$\mathfrak{M}_{2}({\vec{k}}_{1},{\vec{k}}_{2};t)$, through which the functions
$E({\vec{k}};t)$, $G({\vec{k}};t)$ , $R({\vec{k}};t)$. The results are as
follows:

$\displaystyle\frac{\delta}{\delta a(\vec{p}_{2},\vec{r}_{2},t)}$
$\displaystyle\langle\hat{\rho}_{{\vec{k}}}\rangle_{rel,sh}^{t}=\langle\hat{\rho}_{{\vec{k}}}\rangle_{rel,sh}^{t}\langle\hat{n}(x_{2})\rangle_{rel,sh}^{t}-\langle\hat{\rho}_{{\vec{k}}}\hat{n}(x_{2})\rangle_{rel,sh}^{t},$
$\displaystyle\frac{\delta}{\delta a(\vec{p}_{1},\vec{r}_{1},t)}$
$\displaystyle\frac{\delta}{\delta
a(\vec{p}_{2},\vec{r}_{2},t)}\langle\hat{\rho}_{{\vec{k}}}\rangle_{rel,sh}^{t}=2\langle\hat{\rho}_{{\vec{k}}}\rangle_{rel,sh}^{t}\langle\hat{n}(x_{1})\rangle_{rel,sh}^{t}\langle\hat{n}(x_{2})\rangle_{rel,sh}^{t}-\langle\hat{n}(x_{1})\rangle_{rel,sh}^{t}\langle\hat{\rho}_{{\vec{k}}}\hat{n}(x_{2})\rangle_{rel,sh}^{t}$
$\displaystyle-\langle\hat{\rho}_{{\vec{k}}}\rangle_{rel,sh}^{t}\langle\hat{n}(x_{1})\hat{n}(x_{2})\rangle_{rel,sh}^{t}-\langle\hat{n}(x_{2})\rangle_{rel,sh}^{t}\langle\hat{\rho}_{{\vec{k}}}\hat{n}(x_{1})\rangle_{rel,sh}^{t}+\langle\hat{\rho}_{{\vec{k}}}\hat{n}(x_{1})\hat{n}(x_{2})\rangle_{rel,sh}^{t},$
$\displaystyle\frac{\delta}{\delta a(\vec{p}_{2},\vec{r}_{2},t)}$
$\displaystyle\langle\hat{\rho}_{{\vec{k}}}\hat{\rho}_{-{\vec{k}}}\rangle_{rel,sh}^{t}=\langle\hat{\rho}_{{\vec{k}}}\hat{\rho}_{-{\vec{k}}}\rangle_{rel,sh}^{t}\langle\hat{n}(x_{2})\rangle_{rel,sh}^{t}-\langle\hat{\rho}_{{\vec{k}}}\hat{\rho}_{-{\vec{k}}}\hat{n}(x_{2})\rangle_{rel,sh}^{t},$
$\displaystyle\frac{\delta}{\delta a(\vec{p}_{1},\vec{r}_{1},t)}$
$\displaystyle\frac{\delta}{\delta
a(\vec{p}_{2},\vec{r}_{2},t)}\langle\hat{\rho}_{{\vec{k}}}\hat{\rho}_{-{\vec{k}}}\rangle_{rel,sh}^{t}=2\langle\hat{\rho}_{{\vec{k}}}\hat{\rho}_{-{\vec{k}}}\rangle_{rel,sh}^{t}\langle\hat{n}(x_{1})\rangle_{rel,sh}^{t}\langle\hat{n}(x_{2})\rangle_{rel,sh}^{t}$
$\displaystyle-\langle\hat{\rho}_{{\vec{k}}}\hat{\rho}_{-{\vec{k}}}\rangle_{rel,sh}^{t}\langle\hat{n}(x_{1})\hat{n}(x_{2})\rangle_{rel,sh}^{t}-\langle\hat{n}(x_{1})\rangle_{rel,sh}^{t}\langle\hat{\rho}_{{\vec{k}}}\hat{\rho}_{-{\vec{k}}}\hat{n}(x_{2})\rangle_{rel,sh}^{t}$
$\displaystyle-\langle\hat{n}(x_{2})\rangle_{rel,sh}^{t}\langle\hat{\rho}_{{\vec{k}}}\hat{\rho}_{-{\vec{k}}}\hat{n}(x_{1})\rangle_{rel,sh}^{t}+\langle\hat{\rho}_{{\vec{k}}}\hat{\rho}_{-{\vec{k}}}\hat{n}(x_{1})\hat{n}(x_{2})\rangle_{rel,sh}^{t}.$

After integrating by impulses $\int d\vec{p}_{1}\int d\vec{p}_{2}$ we get the
expression for $g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)$ through one-, two-,
three-, four-particle coordinate quasi-equilibrium distribution functions for
particles with short-range interaction. In addition,
$g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)$ also depends on the long-range part
of the interaction $\nu({\vec{k}})$ due to certain renormalizations
$G({\vec{k}};t)$, and is also a function of the local inverse temperature
$\beta({\vec{r}};t)$. Thus, we obtain the expression for the quasi-equilibrium
pair coordinate function of the particle distribution in the Gaussian
approximation $W^{G}(\rho;t)$ (chaotic phase type approximation). Higher
approximations will be related to quasi-equilibrium renormalized cumulatives
$\langle\tilde{D}_{n}(\rho;t)\rangle_{G}$, which will also be expressed in
terms of one-, two-, three-, four- and higher-order particle coordinate quasi-
equilibrium distribution functions for particles with short-range interaction.
And so we come to the problem of calculating one-, two-, three-, four- and
higher-order particle coordinate quasi-equilibrium distribution functions for
particles with short-range interaction.

##  5 Conclusions

Within the concept of the coordinated description of kinetic and hydrodynamic
processes, generalized kinetic equations are obtained for the nonequilibrium
one-particle distribution function in the approximation of “paired” collisions
and in the polarization approximation when the interaction through the third
particle is taken into account. In this case, the contributions from the
short-range and long-range nature of the interaction of particles are
distinguished in the collision integrals. The resulting collision integrals
include a quasi-equilibrium coordinate distribution function
$g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)$ as a function of nonequilibrium
densities of the number of particles $n({\vec{r}};t)$ and inverse temperature
$\beta({\vec{r}};t)$ and describes multiparticle correlations.
$g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)$ is of independent interest, as it
may be related to the corresponding dynamic structural factor quasi-balanced
state of the system. The calculation of
$g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)$ is one of the important problems
64, 39, 41. In this paper, the calculation of
$g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)$ is applied to the method of
collective Yukhnovskii variables, with the allocation of the frame of
reference, which is a system of particles with a short-range nature of the
interaction. In addition, it is important to note that it is promising to use
the Ornstein–Zernike equation, which depends on the time 101, 102, 103, 104 to
calculate $g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)$.

In the following, we consider in detail the kinetic equation for a
nonequilibrium one-particle distribution function with collision integral in
the pair collision integral with a pair quasi-equilibrium coordinate
distribution function in the Gaussian approximation (chaotic phases) in the
Yukhnovskii method of collective variables.

## References

  * 1 Van der Waals J. D. On the Continuity of the Gas and Liquid State. Ph.D. Thesis, University of Leiden, Leiden (1873).
  * 2 Zwanzig R. W. High Temperature Equation of State by a Perturbation Method. I. Nonpolar Gases. J. Chem. Phys. 22 (8), 1420–1426 (1954). doi: 10.1063/1.1740409
  * 3 Rowlinson J. S., Widom B. Molecular Theory of Capillarity. Clarendon, Oxford, Chap. 1.6. (1982).
  * 4 Widom B. Intermolecular Forces and the Nature of the Liquid State. Science. 157 (3787), 375–382 (1967). doi: 10.1126/science.157.3787.375
  * 5 Fisher M. E. Correlation Functions and the Critical Region of Simple Fluids. J. Math. Phys. 5, 944–962 (1964). doi: 10.1063/1.1704197
  * 6 Widom B. Equation of State in the Neighborhood of the Critical Point. J. Chem. Phys. 43, 3898–3905 (1965). doi: 10.1063/1.1696618
  * 7 Longuet-Higgens H. C., Widom B. A rigid sphere model for the melting of argon. Mol. Phys. 8 (6), 549–556 (1964). doi: 10.1080/00268976400100611
  * 8 Barker J. A., Henderson D. Perturbation Theory and Equation of State for Fluids. II. A Successful Theory of Liquids. J. Chem. Phys. 47, 4714–4721 (1967). doi: 10.1063/1.1701689
  * 9 Weeks J. D., Chandler D., Andersen H. C. Role of Repulsive Forces in Determining the Equilibrium Structure of Simple Liquids. J. Chem. Phys. 54, 5237–5247 (1971). doi: 10.1063/1.1674820
  * 10 Kushickt J., Berne B. J. Role of attractive forces in self-diffusion in dense Lennard–Jones fluids. J. Chem. Phys. 59, 3732–3736 (1973). doi: 10.1063/1.1680543
  * 11 Chandler D. Equilibrium Structure and Molecular Motion in Liquids. Acc. Chem. Res. 7, 246–251 (1974). doi: 10.1021/ar50080a002
  * 12 Barker J. A., Henderson D. What is “liquid”? Understanding the states of matter. Reviews of Modern Physics. 48 (4), 587–671 (1976). doi: 10.1103/RevModPhys.48.587
  * 13 Dean D. P., Kushick J. N. On the role of attractive forces in the cage effect in simple liquids. J. Chem. Phys. 76 (1), 619–621 (1982). doi: 10.1063/1.442712
  * 14 Sung W., Dahler J. S. Theory of transport processes in dense fluids. J. Chem Phys. 80 (7), 3025–3037 (1984). doi: 10.1063/1.447137
  * 15 Barrat J. L., Hansen J. P. Basic Concepts for Simple and Complex Liquids. Cambridge University Press, Cambridge (2003). doi: 10.1017/CBO9780511606533
  * 16 Hansen J. P., McDonald J. R. Theory of Simple Liquids. Academic, New York (2005).
  * 17 Chandler D. Interfaces and the driving force of hydrophobic assembly. Nature. 437, 640–646 (2005). doi: 10.1038/nature04162
  * 18 Kirchner B. Theory of Complicated Liquids: Investigation of Liquids, Solvents and Solvent Effects with Modern Theoretical Methods. Phys. Rep. 440 (1–3), 1–111 (2007). doi: 10.1016/j.physrep.2006.11.005
  * 19 Berthier L., Tarjus G. Nonperturbative Effect of Attractive Forces in Viscous Liquids. Phys. Rev. Lett. 103, 170601 (2009). doi: 10.1103/PhysRevLett.103.170601
  * 20 Melnyk R., Nezbeda I., Henderson D., Trokhymchuk A. On the role of the reference system in perturbation theory: An augmented van der Waals theory of simple fluids. Fluid Phase Equilibria. 279 (1), 1–10 (2009). doi: 10.1016/j.fluid.2008.12.004
  * 21 Pedersen U. R., Schroder T. B., Dyre J. C. Repulsive Reference Potential Reproducing the Dynamics of a Liquid with Attractions. Phys. Rev. Lett. 105 (15), 157801 (2010). doi: 10.1103/PhysRevLett.105.157801
  * 22 Toxvaerd S., Dyre J. C. Communication: Shifted forces in molecular dynamics. J. Chem. Phys. 134 (8), 081102 (2011). doi: 10.1063/1.3558787
  * 23 Toxvaerd S., Dyre J. C. Role of the first coordination shell in determining the equilibrium structure and dynamics of simple liquids. J. Chem. Phys. 135 (13), 134501 (2011). doi: 10.1063/1.3643123
  * 24 Berthier L., Tarjus G. The role of attractive forces in viscous liquids. J. Chem. Phys. 134 (21), 214503 (2011). doi: 10.1063/1.3592709
  * 25 Ingebrigtsen T. S., Schrøder T. B., Dyre J. C. What Is a Simple Liquid? Phys. Rev. X. 2 (1), 011011 (2012). doi: 10.1103/PhysRevX.2.011011
  * 26 Hordiichuk V., Trokhymchuk A., Skvara J., Nezbed I. Excluded volume interaction in a Lennard–Jones fluid. Preprint of the Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, ICMP-18-07E (2018).
  * 27 Mryglod I. M., Omelyan I. P., Tokarchuk M. V. Generalized Collective Modes for the Lennard–Jones fluid. Mol. Phys. 84 (2), 235–259 (1995). doi: 10.1080/00268979500100181
  * 28 Mryglod I. M., Folk R., Tokarchuk M. V. On the hydrodynamic theory of a magnetic liquid. I. General description. Physica A. 220 (3–4), 325–348 (1995). doi: 10.1016/0378-4371(95)00232-V
  * 29 Mryglod I. M., Omelyan I. P., Tokarchuk M. V. Wavevector- and frequency-dependent shear viscosity of water: the modified collective mode approach and molecular dynamics calculations. Condens. Matter Phys. 8 (1), 25–46 (2005). doi: 10.5488/CMP.8.1.25
  * 30 Markiv B., Omelyan I., Tokarchuk M. Consistent description of kinetics and hydrodynamics of weakly nonequilibrium processes in simple liquids. J. Stat. Phys. 155, 843–866 (2014). doi: 10.1007/s10955-014-0980-4
  * 31 Bogolubov N. N. Problems of a dynamical theory in statistical physics. In: Studies in statistical mechanics. Vol. 1. Amsterdam, North-Holland (1962).
  * 32 Chapman S., Cowling T. The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction and diffusion in gases. Cambridge, University Press, 3rd ed. (1970).
  * 33 Ferziger J. H., Kaper H. G. Mathematical Theory of Transport Processes in Gases. North-Holland, Amsterdam (1972).
  * 34 Resibois P., de Leener M. Classical Kinetic Theory of Fluids. New York, John Willey and Sons (1977).
  * 35 Humenyuk Y. A., Tokarchuk M. V. Extension of Hydrodynamic Balance Equations for Simple Fluids. J. Stat. Phys. 142, 1052–1084 (2011). doi: 10.1007/s10955-011-0141-y
  * 36 Enskog D. Kinetische Theorie der Warmeleitung Reibung und Selbstdiffusion in gevissen verdichtetten Gasen und Flussigkeiten. Svenska Vetenskapsaked. Handl. 63, 4 (1922).
  * 37 Van Beijeren H., Ernst M. H. The modified Enskog equation. Physica. 68 (3), 437–456 (1973). doi: 10.1016/0031-8914(73)90372-8
  * 38 Van Beijeren H., Ernst M. H. The modified Enskog equation for mixtures. Physica. 70 (2), 225–242 (1973). doi: 10.1016/0031-8914(73)90247-4
  * 39 Polewczak J., Stell G. Transport Coefficients in Some Stochastic Models of the Revised Enskog Equation. J. Stat. Phys. 109, 569–590 (2002). doi: 10.1023/A:1020406413636
  * 40 Polewczak J. A review of the kinetic modelings for non-reactive and reactive dense fluids: Questions and problems. Rivista di Matematica della Universitat di Parma. 4, 23–55 (2001).
  * 41 Polewczak J. Hard-sphere kinetic models for inert and reactive mixtures. J. Phys.: Condens. Matter. 28, 414022 (2016). doi: 10.1088/0953-8984/28/41/414022
  * 42 Davis H. T., Rice S. A., Sengers J. V. On the kinetic theory of dense fluids. IX. The fluid of rigid spheres with a square-well attraction. J. Chem. Phys. 35 (6), 2210–2233 (1961). doi: 10.1063/1.1732234
  * 43 Karkheck J., van Beijeren H., de Schepper I., Stell G. Kinetic theory and H theorem for a dense square-well fluid. Phys. Rev. A. 32 (4), 2517–2520 (1985). doi: 10.1103/PhysRevA.32.2517
  * 44 Wilbertz H., Micheis J., van Beijeren H., Leegwater J. A. Self-Diffusion of Particles Interacting Through a Square-Well or Square-Shoulder Potential. J. Stat. Phys. 53 (5/6), 1155–1177 (1988). doi: 10.1007/BF01023862
  * 45 Van Beijeren H., Karkheck J., Sengers J. V. Nonequilibrium temperature and bulk viscosity for a dense fluid of square-well molecules. Phys. Rev. A. 37 (6), 2247–2250 (1988). doi: 10.1103/PhysRevA.37.2247
  * 46 Leegwater J. A., van Beijeren H., Michels P. J. Linear kinetic theory of the square-well fliud. J. Phys.: Condens. Matter. 1 (1), 237–255 (1989). doi: 10.1088/0953-8984/1/1/020
  * 47 Dufty J. W., Mo K. C., Gubbins K. E. Models for self-diffusion in the square well fluid. J. Chem. Phys. 94 (4), 3132–3140 (1991). doi: 10.1063/1.459783
  * 48 Garland G. E., Dufty J. W. Bound state contributions to transport coefficients. J. Chem. Phys. 95 (4), 2702–2716 (1991). doi: 10.1063/1.460922
  * 49 Yu Y.-X., Han M.-H., Gao G.-H. Self-diffusion in a Nuid of square-well spheres. Phys. Chem. Chem. Phys. 3, 437–443 (2001). doi: 10.1039/b006807l
  * 50 Dwivedee D. K., Srivastava R., Khanna K. N. Transport coefficients of square-well fluids. Fluid Phase Equilibria. 263 (2), 199–204 (2008). doi: 10.1016/j.fluid.2007.09.027
  * 51 Takada S., Saitoh K., Hayakawa H. Kinetic theory for dilute cohesive granular gases with a square-well potential. Phys. Rev. E. 94 (1), 012906 (2016). doi: 10.1103/PhysRevE.94.012906
  * 52 Tokarchuk M. V., Omelyan I. P. To the kinetic theory of processes transfer in dense gases. Preprint of the Institute for Theoretical Physics of the Academy of Sciences of Ukraine, ITP-87-152R (1987), (in Russian).
  * 53 Tokarchuk M. V., Omelyan I. P. The kinetic equation for the systems with multistage interparticle potential. H-theorem. Preprint of the Institute for Theoretical Physics of the Academy of Sciences of Ukraine, ITP-89-49R (1989).
  * 54 Tokarchuk M. V., Omelyan I. P. Model kinetic equations for dense gases and liquids. Ukr. Phys. J. 35 (8), 1255–1261 (1990).
  * 55 Omelyan I. P., Tokarchuk M. V. Kinetic equation for liquids with a multistep potential of interaction. H-theorem. Physica A. 234 (1–2), 89–107 (1996). doi: 10.1016/S0378-4371(96)00286-5
  * 56 Tokarchuk M. V., Omelyan I. P., Kobryn O. E. Kinetic equation for liquids ith a multistep potential of interaction: Calculation of transport coefficient. Phys. Rev. E. 62 (6), 8021–8036 (2000). doi: 10.1103/PhysRevE.62.8021
  * 57 Tokarchuk M. V., Humenyuk Y. A. Hydrodynamic equations for dense fluid mixtures with multistep interaction between particles. Condens. Matter Phys. 10 (2), 151–163 (2007). doi: 10.5488/CMP.10.2.151
  * 58 Tokarchuk M. V., Humenyuk Y. A. Derivation of hydrodynamic equations for dense gaseous mixtures with multi-step interaction between particles. Preprint of the Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, ICMP-04-09U (2004), (in Ukrainian).
  * 59 Humenyuk Y. A., Tokarchuk M. V. Limit behavior of kinetic theory for a system with multistage potential. Ukr. Phys. J. 55 (4), 453–460 (1999), (in Ukrainian).
  * 60 Rice S. A., Allnatt A. R. On the kinetic theory of dense fluids. VI. Singlet distribution function for rigid spheres with an attractive potential. J. Chem. Phys. 34 (6), 2144–2155 (1961). doi: 10.1063/1.1731836
  * 61 Vogelsang R., Hoheisel C. Rice-Allnatt theory: Comparison of calculated shear-viscosity and thermal-conductivity coefficients with molecular-dynamic results. Phys. Rev. A. 39 (12), 6391–6396 (1989). doi: 10.1103/PhysRevA.39.6391
  * 62 Karkheck J., Stell G. Kinetic mean-field theories. J. Chem. Phys. 75 (3), 1475–1487 (1981). doi: 10.1063/1.442154
  * 63 Stell G., Karkheck J., van Beijeren H. Kinetic mean-field theories: Results of energy constraint in maximizing entropy. J. Chem. Phys. 79 (6), 3166–3167 (1983). doi: 10.1063/1.446151
  * 64 Karkheck J., Stell G. Maximization of entropy, kinetic equations, and irreversible thermodynamics. Phys. Rev. A. 25 (6), 3302–3327 (1982). doi: 10.1103/PhysRevA.25.3302
  * 65 Karkheck J., Martina E., Stell G. Kinetic variational theory for mixtures: Kac-tail limit. Phys. Rev. A. 25 (6), 3328–3334 (1982). doi: 10.1103/PhysRevA.25.3328
  * 66 Dyer K. M., Pettitt B. M., Stell G. Systematic investigation of theories of transport in the Lennard–Jones fluid. J. Chem. Phys. 126, 034502 (2007). doi: 10.1063/1.2424714
  * 67 Morioka S. Phenomenological viscous factor in the nonequilibrium distribution function for liquids. Phys. Rev. E. 72 (5), 051203 (2005). doi: 10.1103/PhysRevE.72.051203
  * 68 Karkheck J., Stell G., Xu J. Transport theory for the Lennard–Jones dense fluid. J. Chem. Phys. 89 (9), 5829–5833 (1988). doi: 10.1063/1.455533
  * 69 Zubarev D. N., Morozov V. G. Formulation of boundary conditions for the BBGKY hierarchy with allowance for local conservation laws. Theoretical and Mathematical Physics. 60, 814–820 (1984). doi: 10.1007/BF01018982
  * 70 Zubarev D. N., Morozov V. G. Omelyan I. P., Tokarchuk M. V. Unification of kinetics and hydrodynamics in theory of transport phenomena. In: Collection of scientific works of Institute for Theoretical and Applied Mechanics of Siberian Branch of USSR Academy of Sciences. Models of mechanics of continuous media. Novosibirsk, 34–51 (1989), (in Russian).
  * 71 Zubarev D. N., Morozov V. G., Omelyan I. P., Tokarchuk M. V. Kinetic equations for dense gases and liquids. Theoretical and Mathematical Physics. 87 (1), 113–129 (1991). doi: 10.1007/BF01016582
  * 72 Zubarev D. N., Morozov V. G., Omelyan I. P., Tokarchuk M. V. Unification of the kinetic and hydrodynamic approaches in the theory of dense gases and liquids. Theoretical and Mathematical Physics. 96 (3), 997–1012 (1993). doi: 10.1007/BF01019063
  * 73 Morozov V. G., Kobryn A. E., Tokarchuk M. V. Modified kinetic theory with consideration for slow hydrodynamical processes. Condens. Matter Phys. 4, 117–127 (1994). doi: 10.5488/CMP.4.117
  * 74 Kobryn A. E., Morozov V. G., Omelyan I. P., Tokarchuk M. V. Enskog–Landau kinetic equation. Calculation of the transport coefficients for charged hard spheres. Physica A. 230 (1–2), 189–201 (1996). doi: 10.1016/0378-4371(96)00044-1
  * 75 Tokarchuk M. V., Omelyan I. P., Kobryn A. E. A consistent description of kinetics and hydrodynamics of systems of interacting particles by means of the nonequilibrium statistical operator method. Condens. Matter Phys. 1 (4), 687–751 (1998). doi: 10.5488/CMP.1.4.687
  * 76 Kobryn A. E., Omelyan I. P., Tokarchuk M. V. The modified group expansions for constructions of solutions to the BBGKY hierarchy. J. Stat. Phys. 92 (5/6), 973–994 (1998). doi: 10.1023/A:1023044610690
  * 77 Hlushak P. A., Tokarchuk M. V. A consistent description of kinetics and hydrodynamics of quantum Bose-systems. Condens. Matter Phys. 7 (3), 639–660 (2004). doi: 10.5488/CMP.7.3.639
  * 78 Markiv B. B., Omelyan I. P., Tokarchuk M. V. On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids. Condens. Matter Phys. 13 (2), 23005 (2010). doi: 10.5488/CMP.13.23005
  * 79 Markiv B. B., Omelyan I. P., Tokarchuk M. V. On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids: Collective excitations spectrum. Condens. Matter Phys. 15 (1), 14001 (2012). doi: 10.5488/CMP.15.14001
  * 80 Zubarev D. N., Morozov V. G., Omelyan I. P., Tokarchuk M. V. The Enskog–Landau kinetic equation for charged hard spheres. In: Problems of atomic science and technique. Series: Nuclear physics investigations (theory and experiment). 3 (24), 60–65 (1992), (in Russian).
  * 81 Green H. S. The Molecular Theory of Gases. Amsterdam, North Holland (1952).
  * 82 Cohen E. G. D. Cluster expansion and hierarchy. Physica. 28 (10), 1045–1059 (1962). doi: 10.1016/0031-8914(62)90009-5
  * 83 Zubarev D. N., Novikov M. Yu. Generalized formulation of the boundary condition for the Liouville equation and for BBGKY hierarchy. Theoretical and Mathematical Physics. 13 (3), 406–420 (1972), (in Russian). doi: 10.1007/BF01036148
  * 84 Cohen E. G. D. On the kinetic theory of dense gases. J. Math. Phys. 4 (2), 183–189 (1963). doi: 10.1063/1.1703942
  * 85 Prigogine I. Non-Equilibrium Statistical Mechanics. J. Wiley and Sons, New York, Publie (1962)
  * 86 Uhlenbeck G. E., Ford G. W. Lectures on Statistical Mechanics. (ed. M. Kac). American Mathematical Society, Providence RI (1963).
  * 87 Gurov K. P. Fundamentals of kinetic theory. Moscow, Nauka (1966), (in Russian).
  * 88 Vlasov A. A. Statistical distribution functions. Moscow, Nauka (1966), (in Russian).
  * 89 Silin V. P. Introduction to the kinetic theory of gases. Moscow, Nauka (1971), (in Russian).
  * 90 Zubarev D. N., Novikov M. Yu., Röpke G. Die Boltzmangleichung and mögliche Wege zur Entwicklung dynamischer Methoden in der Kinetishe Theorie. Fortschr. Physik. 21 (12), 703–734 (1973). doi: 10.1002/prop.19730211204
  * 91 Zubarev D. N., Novikov M. Yu. Renormalized kinetic equations for a system with weak interaction and for a low-density gas. Theoretical and Mathematical Physics. 19 (2), 237–251 (1974), (in Russian). doi: 10.1007/BF01035949
  * 92 Liboff R. L. Introduction to the Theory of Kinetic Equations. New York, John Willey and Sons (1969). doi: 10.1119/1.1975335
  * 93 Balescu R. Equilibrium and nonequilibrium statistical mechanics. Research supported by EURATOM. New York, Wiley-Interscience (1975).
  * 94 Balescu R. Statistical Mechanics of Charged Particles. Interscience Publishers (1963).
  * 95 Ecker G. Theory of completely ionized plasma. Acad. Press New York and London (1972).
  * 96 Yukhnovskii I. R., Holovko M. F. Statistical Theory of Classical Equilibrium Systems. Naukova Dumka, Kiev (1980), (in Russian).
  * 97 Zubarev D. N. Statistical thermodynammics of turbulent transport processes. Theoretical and Mathematical Physics. 53 (1), 1004–1014 (1982). doi: 10.1007/BF01014797
  * 98 Idzyk I. M., Ighatyuk V. V., Tokarchuk M. V. Fokker–Planck equation for nonequilibrium distribution function collective variables. I. Calculation of statistical weight, entropy, hydrodynamic speeds. Ukr. J. Phys. 41 (5–6), 596–604 (1996), (in Ukrainian).
  * 99 Hlushak P., Tokarchuk M. Chain of kinetic equations for the distribution functions of particles in simple liquid taking into account nonlinear hydrodynamic fluctuations. Physica A. 443, 231–245 (2016). doi: 10.1016/j.physa.2015.09.059
  * 100 Yukhnovskii I. R., Hlushak P. A., Tokarchuk M. V. BBGKY chain of kinetic equations, nonequilibrium statistical operator method and collective variable method in the statistical theory of nonequilibrium liquids. Condens. Matter Phys. 19, 43705 (2016). doi: 10.5488/CMP.19.43705
  * 101 Ramshaw J. D. Time-dependent direct correlashion function. Phys. Rev. A. 24 (3), 1567–1570 (1981). doi: 10.1103/PhysRevA.24.1567
  * 102 Eu B. C. Dynamic Ornstein-Zernike Equation. In: Transport Coefficients of Fluids. Chemical physics. Vol. 82, 221–240 (2006).
  * 103 Gan H. H., Eu B. C. Theory of the nonequilibrium structure of dense simple fluids: Effects of shearing. Phys. Rev. A. 45 (6), 3670–3686 (1992). doi: 10.1103/PhysRevA.45.3670
  * 104 Brader J. M., Schmidt M. Nonequilibrium Ornstein–Zernike relation for Brownian many-body dynamics. J. Chem. Phys. 139 (10), 104108 (2013). doi: 10.1063/1.4820399