# Kinetic description of ion transport in the system “ionic solution – porous
environment”

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

###### Abstract

A kinetic approach based on a modified chain of BBGKI equations for
nonequilibrium particle distribution functions was used to describe the ion
transfer processes in the ionic solution – porous medium system. A generalized
kinetic equation of the revised Enskog–Vlasov–Landau theory for the
nonequilibrium ion distribution function in the model of charged solid spheres
is obtained, taking into account attractive short-range interactions for the
ionic solution – porous medium system.

###### keywords:

American Chemical Society, LaTeX

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

##  1 Introduction

Studies of particle transfer processes (atoms, molecules, ions) in porous
media (soils, nano heterogeneous composite materials – electrodes, polymer,
and biological membranes, etc.) are relevant in terms of modern technologies
in the creation of nano batteries, supercapacitors, membrane structures for
separation purification of aqueous solutions, etc. The importance and
justification of such studies are noted in many works, including 1, 2, 3, 4,
5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19. The study of the
mechanisms of anomalous dispersion of rheological fluid flows in inhomogeneous
porous media 11, 16, 17, 18 is relevant from the point of view of practical
applications. Another aspect of research is related to the ionic conductivity
of ionic solutions in porous and layered structures, which is important in
connection with the anomalous behavior of ion diffusion and polarization
effects 20, 2, 21, 22, 23, 14. In particular, theoretical studies of
electrodiffusion processes of ion transfer in systems are relevant
“electrolyte – electrode” 4, 5, 14, 29, 30, 24, 25, 26, 27, 28, 31 and
associated with the need to describe nonequilibrium processes of
intercalation–deintercalation of ions, and with the need for a theory suitable
for practical application for forecasting and management these processes.
Problems in the description of electrode processes are connected, first of
all, as with the superficial phenomena at the interface “electrolyte –
electrode”, where complex processes of adsorption, desorption, diffusion, and
in the middle of the electrodes (porous, layered in structure), where complex
processes of association, dissociation between ions, their anomalous (sub or
super) diffusion due to complex interaction with the structure of the
electrode, which in turn are associated with problems of charge accumulation
on electrodes in batteries. Therefore, it is very important to take into
account, to some extent, the change in microstructure electrode material, in
particular, due to its polarization properties and porosity. In the vast
majority of studies to describe electrodiffusion processes of ion transfer in
systems “electrolyte – electrode” equations of nonequilibrium thermodynamics
24 with constant diffusion coefficients are used.

It is interesting to study the processes of self-diffusion of ions in charged
nanoporous media (particles which are frozen) by computer simulation methods
32, 33. At the same time, an important feature of these systems is their
significant spatial heterogeneity, when the diffusion coefficients are
functions of spatial coordinates and time, that is, temporal correlation
functions “stream – stream”
$\langle\hat{\vec{j}}(\vec{r}_{l};t)\hat{\vec{j}}(\vec{r}_{l^{\prime}};t^{\prime})\rangle$
in each of the phases and between phases. It should be noted that some way of
calculating the diffusion coefficients of ions depending on the coordinates
for the systems “electrolyte solution – membrane”, “electrolyte solution –
vitreous fuel-containing materials”, “electrolyte solution – soil” was
proposed in 34, 35, 36. In 14, 27, a statistical theory was proposed to
describe electrodiffusion ion transfer processes in the system “electrolyte –
electrode ” taking into account the spatial inhomogeneities and memory effects
using the nonequilibrium method statistical operator (NSO). In 14, 28, 29, 31,
experimental and theoretical study of subdiffusion impedance for multilayer
system GaSe with encapsulated $\beta$-cyclodextrin having a porous fractal
structure was carried out. From the point of view of theoretical research, the
generalized equations of electrodiffusion of Cattaneo type in fractional
derivatives 31 were applied. However, in most of the mentioned works, except
for 34, 35, 36, in the corresponding transfer equations, the transfer
coefficients: diffusion, viscosity, and thermal conductivity, which determine
the main mechanisms of transfer processes, are constants at the corresponding
temperatures. Obviously, to elucidate such mechanisms, it is very important to
apply such theoretical approaches that link the transfer coefficients with the
characteristic potentials of the interaction of particles and their
distribution functions.

In this paper, we will apply the kinetic approach to the description of ion
transfer processes in the system ionic solution – porous medium. The second
section analyzes the modified chain of BBGKI equations for nonequilibrium ion
distribution functions in the ionic solution – porous medium system, depending
on whether the two subsystems interacted at the initial time. The third
section considers the chain of BBGKI equations for nonequilibrium ion
distribution functions in the pairwise collision approximation for the case
when the ionic solution and porous matrix subsystems are interacting. For this
case, a generalized kinetic equation of the revised Enskog–Vlasov–Landau
theory for the nonequilibrium ion distribution function in the model of
charged solid spheres will be obtained, taking into account attractive short-
range interactions for the ionic solution-porous medium system. To calculate
the paired quasi-equilibrium coordinate distribution functions for ions and
particles of the porous matrix, inhomogeneous, time-dependent Ornstein–Zernike
equations will be proposed for the corresponding complete correlation
functions.

##  2 The chain of BBGKI equations for the ionic solution system in a porous
medium

We consider a system of ionic solution that interacts with a porous medium by
diffusing at its place. Positively and negatively charged ions of the solution
can penetrate into the structure of the porous medium (matrix) and move in it,
interacting with its particles. We assume the whole volume of the system to be
equal to $V=V_{l}+V_{s}$, where $V_{l}$ is the volume occupied by the ionic
solution, and $V_{s}$ is the true volume porous matrix. Entering the volume
$V_{por}$ of the porous space of the matrix, we can determine its porosity:
$\psi=1-\frac{V_{por}}{V}$.

The ionic solution will be considered with specific dielectric properties
without explicit consideration molecular subsystem, and the porous matrix
formed by moving particles (atoms, molecules) varieties $\xi$ subsystem,
kinetic energy, which is much less than its potential energy. We believe that
the particles of the porous matrix perform oscillating motions, and its
dynamics can be described by interacting phonons, for example, in the case of
porous silicon, and other porous materials 40. When interacting with an ionic
solution, both the structure of the porous matrix and the dynamic and
structural properties of the ionic solution itself can change. The Hamiltonian
of such a system is an ionic solution – a porous matrix can be represented as:

$\displaystyle H$
$\displaystyle=\sum_{\alpha,j=1}^{N_{\alpha}}\frac{\big{(}\vec{p}_{j}-\frac{Z_{\alpha}e}{c}\vec{A}(\vec{r}_{j};t)\big{)}^{2}}{2m_{\alpha}}+\sum_{\alpha,\gamma}\sum_{j,l}^{N_{\alpha},N_{\gamma}}\Phi_{\alpha\gamma}(\vec{r}_{j},\vec{r}_{l})$
$\displaystyle\quad+\sum_{\alpha,\xi}\sum_{j,l}^{N_{\alpha},N_{\xi}}\Phi_{\alpha\xi}(\vec{r}_{j},\vec{R}_{l})+\sum_{\alpha}\sum_{j}^{N_{\alpha}}Z_{\alpha}e\varphi(\vec{r}_{j};t)+H_{s},$

where the indices $j$, $l$ number the ions of the solution of varieties
$\alpha$, $\gamma$, with masses $m_{\alpha}$, $m_{\gamma}$ and vector–momentum
$\vec{p}_{j}$, $\vec{p}_{l}$, $\vec{r}_{j}$, $\vec{R}_{l}$ are the coordinate
vectors of ions and particles, respectively porous matrix, $N_{\alpha}$ is the
total number of ions of the variety $\alpha$;
$\Phi_{\alpha\gamma}(\vec{r}_{j},\vec{r}_{l})=\Phi^{sh}_{\alpha\gamma}(\vec{r}_{j},\vec{r}_{l})+\Phi^{l}_{\alpha\gamma}(\vec{r}_{j},\vec{r}_{l})$
is an even potential for interaction between ions of the grade $\alpha$,
$\gamma$,
$\Phi_{\alpha\xi}(\vec{r}_{j},\vec{R}_{l})=\Phi^{sh}_{\alpha\xi}(\vec{r}_{j},\vec{R}_{l})+\Phi^{l}_{\alpha\xi}(\vec{r}_{j},\vec{R}_{l})$
is the pairwise potential of ion interaction with porous matrix particles that
have short-range and long-range (attractive) contributions.

Moreover, short-term contributions can have repulsive and attractive
components $\Phi^{sh}_{\alpha\gamma}(\vec{r}_{j},\vec{r}_{l})=\Phi^{sh-
rep}_{\alpha\gamma}(\vec{r}_{j},\vec{r}_{l})+\Phi^{sh-
att}_{\alpha\gamma}(\vec{r}_{j},\vec{r}_{l})$, i.e. repulsive and attractive
forces can act between ions at small distances. In particular, attractive
short-range interactions can describe the processes of association between
ions. $H_{s}$ is a Hamiltonian of a porous matrix, the structure of which will
not be specified at this stage of modeling. $\vec{A}(\vec{r}_{j};t)$,
$\varphi(\vec{r}_{j};t)$ are the total vector and scalar potentials of the
electromagnetic field generated by valence ions $Z_{\alpha}$ and external
field, $e$ is the electron charge and $c$ is the speed of light. In what
follows, we will not consider vortex electromagnetic processes of order
$\frac{e}{c}$, but only potential contributions from the scalar potential
$\varphi(\vec{r}_{j};t)$. In addition, at this stage of research we do not
take into account the possible influence of the electromagnetic field on the
particles of the porous matrix, which may be associated with the processes of
polarization, changes in its dielectric properties. Although in reality, in
many cases, in particular for electrodes, biological membranes, such
consideration is important.

The nonequilibrium state of the ionic solution when interacting with the
porous matrix is completely described by the Liouville equation for
nonequilibrium distribution function of all particles
$\rho(x_{1},\ldots,x_{N_{\alpha}}|\vec{R}_{1},\ldots,\vec{R}_{N_{\xi}};t)=\rho(x^{N_{\alpha}},R^{N_{\xi}};t)$
(where the notation is entered: $x^{N_{\alpha}}=x_{1},\ldots,x_{N_{\alpha}}$,
$x_{j}=\vec{p}_{j}\vec{r}_{j}$ is the coordinate and momentum of $j$-th ion of
the solution, $R^{N_{\xi}}=\vec{R}_{1},\ldots,\vec{R}_{N_{\xi}}$,
$\vec{R}_{l}$ is the coordinate of the $l$-th particle of the porous matrix)

$\frac{\partial}{\partial
t}\rho(x^{N_{\alpha}},R^{N_{\xi}};t)+iL_{N}\rho(x^{N_{\alpha}},R^{N_{\xi}};t)=0$

with the Liouville operator

$\displaystyle iL_{N}$
$\displaystyle=\sum_{\alpha,j=1}^{N_{\alpha}}\frac{\vec{p}_{j}}{m_{\alpha}}\cdot\frac{\partial}{\partial\vec{r}_{j}}+\sum_{\xi,l=1}^{N_{\xi}}\frac{\vec{P}_{l}}{m_{\xi}}\cdot\frac{\partial}{\partial\vec{R}_{l}}-\sum_{\alpha,\gamma}\sum_{j,l}^{N_{\alpha}N_{\gamma}}\frac{\partial}{\partial\vec{r}_{j}}\Phi_{\alpha\gamma}(\vec{r}_{j},\vec{r}_{l})\left(\frac{\partial}{\partial\vec{p}_{j}}-\frac{\partial}{\partial\vec{p}_{l}}\right)$
$\displaystyle\quad-\sum_{\alpha,\xi}\sum_{j,l}^{N_{\alpha},N_{\xi}}\frac{\partial}{\partial\vec{r}_{j}}\left(\Phi_{\alpha\xi}(\vec{r}_{j},\vec{R}_{l})+Z_{\alpha}e\varphi(\vec{r}_{j};t)\right)\cdot\frac{\partial}{\partial\vec{p}_{j}},$

at the same time in the Liouville operator we do not consider terms
$-\sum_{\alpha,\xi}\sum_{j,l}^{N_{\alpha},N_{\xi}}\frac{\partial}{\partial\vec{R}_{l}}\Phi_{\alpha\xi}(\vec{r}_{j},\vec{R}_{l})\cdot\frac{\partial}{\partial\vec{P}_{l}},$
since we believe that the microscopic forces acting on the atoms of the porous
matrix on the ion of the solution do not change the momentum of the particles
of the matrix.

The nonequilibrium state of the system ionic solution – porous medium will be
described using a modified chain of BBGKI equations 37, 38, 39, 41, 42 for
partial nonequilibrium distribution functions of ions and particles of a
porous matrix. To do this, we use the approach proposed in the works 37, 38,
39, 41, 42, where the 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:

$\frac{\partial}{\partial
t}\rho\left(x^{N_{\alpha}},R^{N_{\xi}};t\right)+iL_{N}\rho\left(x^{N_{\alpha}},R^{N_{\xi}};t\right)=-\varepsilon\left(\rho\left(x^{N_{\alpha}},R^{N_{\xi}};t\right)-\varrho_{rel}\left(x^{N_{\alpha}},R^{N_{\xi}};t\right)\right),$
(1)

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

In this case, the initial condition for solving the Liouville equation (Cauchy
problem) can be considered two options: first

$\rho\left(x^{N_{\alpha}},R^{N_{\xi}};t\right)\big{|}_{t=t_{0}}=\varrho_{\rm
rel}\left(x^{N_{\alpha}},R^{N_{\xi}};t_{0}\right)=\rho_{\rm rel}^{\rm
liq}\left(x^{N_{\alpha}};t_{0}\right)\rho_{s}\left(R^{N_{\xi}};t_{0}\right),$

this means that at the initial moment of time $t_{0}$ the ionic solution and
the porous matrix are considered independent; other

$\rho\left(x^{N_{\alpha}},R^{N_{\xi}};t\right)\big{|}_{t=t_{0}}=\rho_{\rm
rel}\left(x^{N_{\alpha}},R^{N_{\xi}};t_{0}\right),$

when the ionic solution and the porous matrix are considered at the initial
moment of time by interacting subsystems.

In the first case, $\rho_{\rm rel}^{\rm liq}(x^{N_{\alpha}};t)$ is the
relevant ion distribution function obtained by 37, 38, 39, 41, 42 from the
condition of the maximum of the Gibbs entropy functional while preserving the
normalization conditions for the distribution and the given parameters of the
abbreviated description of the nonequilibrium state of the ionic solution:
$\langle\hat{n}_{\alpha}(x)\rangle^{t}=f_{\alpha}(x;t)$ is the nonequilibrium
one-particle ion distribution function of the variety $\alpha$ and
$\langle\hat{\varepsilon}_{\rm int}(\vec{r})\rangle^{t}$ is the
nonequilibrium. The average interaction energy of the ions of the solution has
the following structure:

$\rho_{\rm rel}^{liq}(x^{N_{\alpha}};t)=\exp\left\\{-\Phi_{\rm liq}(t)-\int
d\vec{r}\,\beta_{\rm liq}(\vec{r},t)\hat{\varepsilon}_{\rm
int}(\vec{r})-\sum_{\alpha}\int
dx\,a_{\alpha}(x,t)\hat{n}_{\alpha}(x)\right\\},$

where $\Phi_{\rm liq}(t)$ is the Masier–Planck functional

$\Phi_{\rm liq}(t)=\int d\Gamma_{N_{\rm liq}}(x)\,\exp\left\\{-\int
d\vec{r}\,\beta_{\rm liq}(\vec{r},t)\hat{\varepsilon}_{\rm
int}(\vec{r})-\sum_{\alpha}\int
dx\,a_{\alpha}(x,t)\hat{n}_{\alpha}(x)\right\\},$

in which

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

is the microscopic phase density of the number of ions of the variety $\alpha$
and

$\hat{\varepsilon}_{\rm
int}(\vec{r})=\frac{1}{2}\sum_{\alpha,\gamma}\sum_{j,l}^{N_{\alpha},N_{\gamma}}\Phi_{\alpha\gamma}(\vec{r}_{j},\vec{r}_{l})\delta(\vec{r}-\vec{r}_{j})$

is the microscopic energy density of the interaction of the ions of the
solution. The Lagrange parameters $\beta_{\rm liq}(\vec{r},t)$ (inverse of the
nonequilibrium temperature of the ionic solution), $a_{\alpha}(x,t)$ are
determined from the conditions of self-consistent:

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

In this case, the ionic solution has a temperature $\beta_{\rm
liq}(\vec{r},t)$, and the porous matrix $\beta_{s}$.

In the second case, $\rho_{\rm rel}(x^{N_{\alpha}},R^{N_{\xi}};t)$ is the
relevant function of the distribution of ions and particles of the porous
matrix obtained respectively 37, 38, 39, 41, 42 from the condition of the
maximum of the Gibbs entropy functional with preserved normalization
conditions for the distribution and set parameters of the abbreviated
description of the nonequilibrium state of the ionic solution:
$\langle\hat{n}_{\alpha}(x)\rangle^{t}=f_{\alpha}(x;t)$ is nonequilibrium one-
particle ion distribution function of $\alpha$,
$\langle\hat{E}_{int}(\vec{r})\rangle^{t}$ is nonequilibrium average ion
interaction energy solution and particles of a porous matrix,
$\langle\hat{n}_{\xi}(\vec{R})\rangle^{t}=n_{\xi}(\vec{R};t)$ is average the
particle density of the porous matrix of the variety $\xi$ has the following
structure:

$\rho_{\rm rel}(x^{N_{\alpha}},R^{N_{\xi}};t)\\\
=\exp\left\\{-\Phi(t)-\int\\!\\!d\vec{r}\,\beta(\vec{r},t)\big{(}\hat{E}_{int}(\vec{r})+H_{s}\big{)}-\sum_{\alpha}\int\\!\\!dx\,a_{\alpha}(x,t)\hat{n}_{\alpha}(x)-\sum_{\xi}\int\\!\\!d\vec{R}\,\mu_{\xi}(\vec{R})\hat{n}_{\xi}(\vec{R})\right\\},$

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

$\Phi(t)=\int d\Gamma(x,R)\\\
\times\exp\left\\{-\int\\!\\!d\vec{r}\,\beta(\vec{r},t)\big{(}\hat{E}_{int}(\vec{r})+H_{s}\big{)}-\sum_{\alpha}\int\\!\\!dx\,a_{\alpha}(x,t)\hat{n}_{\alpha}(x)-\sum_{\xi}\int\\!\\!d\vec{R}\,\mu_{\xi}(\vec{R})\hat{n}_{\xi}(\vec{R})\right\\},$

in which $\hat{n}_{\alpha}(x)$ is a microscopic phase density of the number of
ions of the variety $\alpha$ and

$\hat{E}_{\rm
int}(\vec{r})=\frac{1}{2}\sum_{\alpha,\gamma}\sum_{j,l}^{N_{\alpha},N_{\gamma}}\Phi_{\alpha\gamma}(\vec{r}_{j},\vec{r}_{l})\delta(\vec{r}-\vec{r}_{j})+\sum_{\alpha,\xi}\sum_{j,l}^{N_{\alpha},N_{\xi}}\Phi_{\alpha\xi}(\vec{r}_{j},\vec{r}_{l})\delta(\vec{r}-\vec{r}_{j})$

is the microscopic energy density of the interaction of solution ions and
particles of the porous matrix, which can perform oscillating motions. The
Lagrange parameters $\beta(\vec{r},t)$ (inverse of the nonequilibrium
temperature of the ionic solution system – porous matrix), $a_{\alpha}(x,t)$
are determined from the conditions of self-agreement:

$\big{\langle}(\hat{E}_{\rm
int}(\vec{r})+H_{s})\big{\rangle}^{t}=\big{\langle}(\hat{E}_{\rm
int}(\vec{r})+H_{s})\big{\rangle}^{t}_{\rm
rel},\quad\big{\langle}\hat{n}_{\alpha}(x)\big{\rangle}^{t}=\big{\langle}\hat{n}_{\alpha}(x)\big{\rangle}^{t}_{\rm
rel}.$

In this case, the ionic solution and the porous matrix have a temperature of
$\beta(\vec{r},t)$.

###  2.1 Kinetic equations with initial condition of independent subsystems:
ionic solution and porous matrix

Given the structure $\rho_{\rm rel}^{\rm liq}(x^{N_{\alpha}};t)$ and the
approach 37, 38, 39, 41, 42, 43, 44, integrating the Liouville equation with
the source (1) by the corresponding coordinates and momentum of the ions of
the solution and the coordinates of the particles of the porous matrix, we
obtain a chain of equations BBGKI with modified boundary conditions (taking
into account spatiotemporal interparticle correlations) for the system ionic
solution – porous matrix:

$\left(\frac{\partial}{\partial
t}+iL_{\alpha}(1)\right)f_{\alpha}(x_{1};t)+\sum_{\gamma}\int\\!\\!dx_{2}\,iL_{\alpha\gamma}(1,2)f_{\alpha\gamma}(x_{1},x_{2};t)+\int\\!\\!d\vec{R}_{s}\,iL_{\alpha
s}(1,s)f_{\alpha s}(x_{1},\vec{R}_{s};t)=0,$ (2)
$\left(\frac{\partial}{\partial
t}+iL_{\alpha}(1)+iL_{\gamma}(2)+iL_{\alpha\gamma}(1,2)\right)f_{\alpha\gamma}(x_{1},x_{2};t)\\\
+\sum_{\nu}\int\\!\\!dx_{3}\big{(}iL_{\alpha\xi}(1,3)+iL_{\gamma\xi}(2,3)\big{)}f_{\alpha\gamma\nu}(x_{1},x_{2},x_{3};t)+\int\\!\\!d\vec{R}_{s}\big{(}iL_{\alpha
s}(1,s)+iL_{\gamma s}(2,s)\big{)}f_{\alpha\gamma
s}(x_{1},x_{2},\vec{R}_{s};t)\\\
=-\varepsilon\big{(}f_{\alpha\gamma}(x_{1},x_{2};t)-g_{\alpha\gamma}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)f_{\alpha}(x_{1};t)f_{\gamma}(x_{2};t)\big{)},$
(3)
$\left(\frac{\partial}{\partial t}+iL_{\alpha}(1)+iL_{\alpha
s}(1,s)\right)f_{\alpha s}(x_{1},\vec{R}_{s};t)\\\ +\sum_{\gamma}\int
dx_{3}\big{(}iL_{\alpha\gamma}(1,3)+iL_{s\gamma}(s,3)\big{)}f_{\alpha
s\gamma}(x_{1},\vec{R}_{s},x_{3};t)+\int d\vec{R}_{s^{\prime}}iL_{\alpha
s^{\prime}}(1,s^{\prime})f_{\alpha
ss^{\prime}}(x_{1},\vec{R}_{s},\vec{R}_{s^{\prime}};t)\\\
=-\varepsilon\big{(}f_{\alpha
s}(x_{1},\vec{R}_{s};t)-f_{\alpha}(x_{1};t)n(\vec{R}_{s};t)\big{)},$ (4)

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

$\displaystyle iL_{\alpha}(j)$
$\displaystyle=\frac{\vec{p}_{j}}{m_{\alpha}}\cdot\frac{\partial}{\partial\vec{r}_{j}}-\frac{\partial}{\partial\vec{r}_{j}}Z_{\alpha}e\varphi(\vec{r}_{j};t)\cdot\frac{\partial}{\partial\vec{p}_{j}},$
(5) $\displaystyle iL_{\alpha\gamma}(j,l)$
$\displaystyle=-\frac{\partial}{\partial\vec{r}_{j}}\Phi_{\alpha\gamma}(\vec{r}_{j},\vec{r}_{l})\cdot\left(\frac{\partial}{\partial\vec{p}_{j}}-\frac{\partial}{\partial\vec{p}_{l}}\right)$

there are one-part and two-part parts of the Liouville operator,

$iL_{\alpha s}(j,s)=-\frac{\partial}{\partial\vec{r}_{j}}\Phi_{\alpha
s}(\vec{r}_{j},\vec{R}_{s})\cdot\frac{\partial}{\partial\vec{p}_{j}}$

is a two-particle Liouville operator of particles of liquid subsystem and
porous subsystem. $g_{\alpha\gamma}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)$ are
quasi-equilibrium pair coordinate functions of ion distribution of varieties
$\alpha,\gamma$ solution subsystems

$\displaystyle g_{\alpha\gamma}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)$
$\displaystyle=\frac{1}{n_{\alpha}(\vec{r}_{1};t)n_{\gamma}(\vec{r}_{2};t)}\int
d\Gamma_{N_{\rm
liq}}(x)\,\hat{n}_{\alpha}(\vec{r}_{1})\hat{n}_{\gamma}(\vec{r}_{2})\rho_{\rm
rel}^{\rm liq}(x^{N_{\alpha}};t),$ $\displaystyle n_{\alpha}(\vec{r};t)$
$\displaystyle=\int d\vec{p}\,f_{\alpha}(x;t)$

is the nonequilibrium average value of the density of the number of ions of
the variety $\alpha$, $n(\vec{r}_{s};t)$ is a nonequilibrium average density
of the number of particles of the porous subsystem, $f_{\alpha}(x_{1};t)$,
$f_{\alpha\gamma}(x_{1},x_{2};t)$, $f_{\alpha\gamma\nu}(x_{1},x_{2},x_{3};t)$
are one-, two- and three-ionic nonequilibrium distribution functions,
$f_{\alpha\gamma s}(x_{1},x_{2},\vec{R}_{s};t)$, $f_{\alpha
s}(x_{1},\vec{R}_{s};t)$, $f_{\alpha
ss^{\prime}}(x_{1},\vec{R}_{s},\vec{R}_{s^{\prime}};t)$ are nonequilibrium
distribution functions ions and particles of the porous medium.

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

$\displaystyle\vec{p}_{\alpha}(\vec{r};t)$
$\displaystyle=\int\\!\\!d\vec{p}\,f_{\alpha}(\vec{r},\vec{p};t)\vec{p},$
$\displaystyle\varepsilon^{\rm kin}_{\alpha}(\vec{r};t)$
$\displaystyle=\int\\!\\!d\vec{p}\,\frac{p^{2}}{2m_{\alpha}}f_{\alpha}(\vec{r},\vec{p};t),$
$\displaystyle\varepsilon^{\rm int}_{\alpha}(\vec{r};t)$
$\displaystyle=\sum_{\gamma}\int\\!\\!d\vec{p}\int\\!\\!d\vec{p}^{\prime}\int\\!\\!d\vec{r}^{\prime}\,\Phi_{\alpha\gamma}(\vec{r},\vec{r}^{\prime})f_{\alpha\gamma}(\vec{r},\vec{p},\vec{r}^{\prime},\vec{p}^{\prime};t)+\int\\!\\!d\vec{p}\int\\!\\!d\vec{R}_{s}\Phi_{\alpha
s}(\vec{r},\vec{R}_{s})f_{\alpha s}(\vec{r},\vec{p},\vec{R}_{s};t),$

which satisfy the corresponding laws of conservation of average nonequilibrium
values of the number of ions $n_{\alpha}(\vec{r};t)$, full momentum
$\vec{p}(\vec{r};t)$

$\vec{p}(\vec{r};t)=\sum_{\alpha}\vec{p}_{\alpha}(\vec{r};t)$

and full of energy

$\varepsilon(\vec{r};t)=\sum_{\alpha}\big{(}\varepsilon^{\rm
kin}_{\alpha}(\vec{r};t)+\varepsilon^{\rm int}_{\alpha}(\vec{r};t)\big{)},$

underlying the hydrodynamic description of nonequilibrium processes in the
system ionic solution – porous medium. The system of equations (2)–(4) for
nonequilibrium ion distribution functions includes paired quasi-equilibrium
coordinate functions ion distribution (positively and negatively charged ions)
$g_{\alpha\gamma}(\vec{r},\vec{r}^{\prime}|n,\beta;t)$ ($g_{++}$, $g_{+-}$,
$g_{--}$), which are functions of nonequilibrium ion densities
$n_{\alpha}(\vec{r};t)$ and inverse temperature $\beta(\vec{r};t)$. They
describe multiparticle correlations and may be of independent interest, as
they may be related to the corresponding partial dynamic structural factors of
the corresponding quasi-equilibrium state.

The calculation of paired quasi-equilibrium distribution functions is one of
the important problems. In particular, in the case of simple fluids in the
works 45, 46, 47, 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 for the time-dependent density.
Another way is suggested in the recent work 44, in which the paired quasi-
equilibrium coordinate distribution function of a simple liquid is calculated
from the statistical sum of the corresponding quasi-equilibrium particle
distribution in the method of collective variables 48, 49, 50, 51, 52. In
addition, it is important to note that the use of the Ornstein–Zernicke
equation, which depends on the time 53, 54, 55, 56, is promising to calculate
$g_{2}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)$. In the case of equilibrium
spatially homogeneous and inhomogeneous ionic and ionic-molecular systems, the
methods for solving the Ornstein–Zernike equations are perfectly developed 57,
58, 59, 60.

In our case, the quasi-equilibrium distribution functions are even
$g_{\alpha\gamma}(\vec{r},\vec{r}^{\prime}|n,\beta;t)$ can be related to full
correlation functions $h_{\alpha\gamma}(\vec{r},\vec{r}^{\prime}|n,\beta;t)$
($h_{++}$, $h_{+-}$, $h_{--}$) that satisfy inhomogeneous
Ornstein–Zernike–type equations that depend on time:

$h_{\alpha\gamma}(\vec{r},\vec{r}^{\prime};t)=c_{\alpha\gamma}(\vec{r},\vec{r}^{\prime};t)+\sum_{\nu}\int
d\vec{r}^{\prime\prime}c_{\alpha\nu}(\vec{r},\vec{r}^{\prime\prime};t)n_{\nu}(\vec{r}^{\prime\prime};t)h_{\nu\gamma}(\vec{r}^{\prime\prime},\vec{r}^{\prime};t),$
(6)

where $c_{\alpha\gamma}(\vec{r},\vec{r}^{\prime};t)$ are direct correlation
functions of solution ion distribution. The system of equations (6) includes
time-dependent ion densities $n_{\nu}(\vec{r}^{\prime\prime};t)$ ($n_{+}$,
$n_{-}$), equations for which it is necessary to find from the corresponding
kinetic equation for $f_{\alpha}(\vec{p},\vec{r};t)$ after integration by
impulses $d\vec{p}$.

It is important to note that the boundary condition in the equation (3) takes
into account spatially inhomogeneous correlations between solution particles.
In contrast, the boundary condition in the equation (5) does not take into
account spatially inhomogeneous correlations between solution particles and a
porous matrix, which corresponds to the principle of complete weakening of
Bogolyubov correlations and, in this case, is a consequence of the
independence of the ionic solution from the porous matrix at the initial time.

###  2.2 Kinetic equations with the initial condition of interacting
subsystems:
ionic solution and porous matrix

In the case when at the initial moment of time, the ionic solution and the
porous matrix are considered by interacting subsystems, the first equations of
the BBGKI chain have the following form:

$\left(\frac{\partial}{\partial
t}+iL_{\alpha}(1)\right)f_{\alpha}(x_{1};t)+\sum_{\gamma}\int\\!\\!dx_{2}\,iL_{\alpha\gamma}(1,2)f_{\alpha\gamma}(x_{1},x_{2};t)+\int\\!\\!d\vec{R}_{s}\,iL_{\alpha
s}(1,s)f_{\alpha s}(x_{1},\vec{R}_{s};t)=0,$
$\left(\frac{\partial}{\partial
t}+iL_{\alpha}(1)+iL_{\gamma}(2)+iL_{\alpha\gamma}(1,2)\right)f_{\alpha\gamma}(x_{1},x_{2};t)\\\
+\sum_{\xi}\int\\!\\!dx_{3}\big{(}iL_{\alpha\xi}(1,3)+iL_{\gamma\xi}(2,3)\big{)}f_{\alpha\gamma\xi}(x_{1},x_{2},x_{3};t)+\int\\!\\!d\vec{R}_{s}\big{(}iL_{\alpha
s}(1,s)+iL_{\gamma s}(2,s)\big{)}f_{\alpha\gamma
s}(x_{1},x_{2},\vec{R}_{s};t)\\\
=-\varepsilon\big{(}f_{\alpha\gamma}(x_{1},x_{2};t)-g_{\alpha\gamma}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)f_{\alpha}(x_{1};t)f_{\gamma}(x_{2};t)\big{)},$
$\left(\frac{\partial}{\partial t}+iL_{\alpha}(1)+iL_{\alpha
s}(1,s)\right)f_{\alpha s}(x_{1},\vec{R}_{s};t)\\\ +\sum_{\gamma}\int
dx_{3}\Big{(}iL_{\alpha\gamma}(1,3)+iL_{s\gamma}(s,3)\Big{)}f_{\alpha
s\gamma}(x_{1},\vec{R}_{s},x_{3};t)+\int\\!\\!d\vec{R}_{s^{\prime}}iL_{\alpha
s^{\prime}}(1,s^{\prime})f_{\alpha
ss^{\prime}}(x_{1},\vec{R}_{s},\vec{R}_{s^{\prime}};t)\\\
=-\varepsilon\big{(}f_{\alpha s}(x_{1},\vec{R}_{s};t)-g_{\alpha
s}(\vec{r}_{1},\vec{R}_{s}|n,\beta;t)f_{\alpha}(x_{1};t)n(\vec{R}_{s};t)\big{)},$

where $g_{\alpha\gamma}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)$, $g_{\alpha
s}(\vec{r}_{1},\vec{R}_{s}|n,\beta;t)$ are quasi-equilibrium pair ion
distribution functions of $\alpha,\gamma$ varieties solution subsystems and
solution – porous matrix, $n(\vec{r}_{s};t)$ is a nonequilibrium density
(unary function distribution) of the porous subsystem particles,
$f_{\alpha}(x_{1};t)$, $f_{\alpha\gamma}(x_{1},x_{2};t)$,
$f_{\alpha\gamma\xi}(x_{1},x_{2},x_{3};t)$ are one-, two- and three-ion
nonequilibrium distribution functions, $f_{\alpha\gamma
s}(x_{1},x_{2},\vec{R}_{s};t)$, $f_{\alpha s}(x_{1},\vec{R}_{s};t)$,
$f_{\alpha ss^{\prime}}(x_{1},\vec{R}_{s},\vec{R}_{s^{\prime}};t)$ are
nonequilibrium distribution functions ions and particles of the porous medium.
Paired quasi-equilibrium distribution functions
$g_{\alpha\gamma}(\vec{r},\vec{r}^{\prime}|n,\beta;t)$, $g_{\alpha
s}(\vec{r},\vec{R}_{s}|n,\beta;t)$ in the chain of equations BBGKI as in the
previous case can be associated with the corresponding complete correlation
functions $h_{\alpha\gamma}(\vec{r},\vec{r}^{\prime}|n,\beta;t)$ ($h_{++}$,
$h_{-+}$, $h_{--}$), $h_{\alpha s}(\vec{r},\vec{R}_{s}|n,\beta;t)$ ($h_{+s}$,
$h_{-s}$), which satisfy inhomogeneous equations Ornstein–Zernike, which
depend on time:

$h_{\alpha\gamma}(\vec{r},\vec{r}^{\prime};t)=c_{\alpha\gamma}(\vec{r},\vec{r}^{\prime};t)+\sum_{\xi}\int
d\vec{r}^{\prime\prime}c_{\alpha\xi}(\vec{r},\vec{r}^{\prime\prime};t)n_{\xi}(\vec{r}^{\prime\prime};t)h_{\xi\gamma}(\vec{r}^{\prime\prime},\vec{r}^{\prime};t)\\\
+\int d\vec{R}_{s}\,c_{\alpha
s}(\vec{r},\vec{R}_{s};t)n_{s}(\vec{R}_{s};t)h_{s\gamma}(\vec{R}_{s},\vec{r}^{\prime};t),$
$h_{\alpha s}(\vec{r},\vec{R}_{s};t)=c_{\alpha
s}(\vec{r},\vec{R}_{s};t)+\sum_{\xi}\int
d\vec{r}^{\prime\prime}c_{\alpha\xi}(\vec{r},\vec{r}^{\prime\prime};t)n_{\xi}(\vec{r}^{\prime\prime};t)h_{\xi
s}(\vec{r}^{\prime\prime},\vec{R}_{s};t)\\\ +\int
d\vec{R}_{s^{\prime}}\,c_{\alpha
s}(\vec{r},\vec{R}_{s};t)n_{s^{\prime}}(\vec{R}_{s^{\prime}};t)h_{s^{\prime}s}(\vec{R}_{s^{\prime}},\vec{R}_{s}),$
$h_{s\alpha}(\vec{R}_{s},\vec{r};t)=c_{s\alpha}(\vec{R}_{s},\vec{r};t)+\sum_{\xi}\int
d\vec{r}^{\prime\prime}c_{s\xi}(\vec{R}_{s},\vec{r}^{\prime\prime};t)n_{\xi}(\vec{r}^{\prime\prime};t)h_{\xi\alpha}(\vec{r}^{\prime\prime},\vec{r};t)\\\
+\int
d\vec{R}_{s^{\prime}}\,c_{ss^{\prime}}(\vec{R}_{s},\vec{R}_{s^{\prime}})n_{s^{\prime}}(\vec{R}_{s^{\prime}};t)h_{s^{\prime}\alpha}(\vec{R}_{s^{\prime}},\vec{r};t),$
$h_{ss}(\vec{R}_{s},\vec{R}_{s^{\prime}};t)=c_{ss}(\vec{R}_{s},\vec{R}_{s^{\prime}};t)+\int
d\vec{R}_{s^{\prime\prime}}c_{ss}(\vec{R}_{s},\vec{R}_{s^{\prime\prime}};t)n_{s}(\vec{R}_{s^{\prime\prime}};t)h_{ss}(\vec{R}_{s^{\prime\prime}},\vec{R}_{s^{\prime}};t),$

where $c_{\alpha\gamma}(\vec{r},\vec{r}^{\prime};t)$, $c_{\alpha
s}(\vec{r},\vec{R}_{s};t)$, $c_{ss}(\vec{R}_{s},\vec{R}_{s^{\prime}};t)$ are
direct correlation functions of ions and particles of the porous medium. As we
can see, the structure is a connected system of equations for finding the
correlation functions $h_{\alpha\gamma}(\vec{r},\vec{r}^{\prime}|n,\beta;t)$,
$h_{\alpha s}(\vec{r},\vec{R}_{s}|n,\beta;t)$. It includes the nonequilibrium
density $n_{s}(\vec{R}_{s^{\prime\prime}};t)$,
$h_{ss}(\vec{R}_{s},\vec{R}_{s^{\prime}};t)$,
$c_{ss}(\vec{R}_{s},\vec{R}_{s^{\prime}};t)$ particles of a porous matrix.
This raises the problem of developing methods for solving time-dependent
inhomogeneous Ornstein–Zernike equations.

In the next section, we consider the approximation of pair collisions between
particles for the case when the subsystems ionic solution and the porous
matrix at the initial time are considered interacting.

##  3 Approaching pairwise collisions

In the approximate pair collisions between particles, when three-particle
distribution functions are not taken into account, for nonequilibrium two-
particle distribution functions, we obtain 37, 39 the following equations:

$\left(\frac{\partial}{\partial
t}+iL_{\alpha}(1)+iL_{\gamma}(2)+iL_{\alpha\gamma}(1,2)\right)f_{\alpha\gamma}(x_{1},x_{2};t)\\\
=-\varepsilon\big{(}f_{\alpha\gamma}(x_{1},x_{2};t)-g_{\alpha\gamma}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)f_{\alpha}(x_{1};t)f_{\gamma}(x_{2};t)\big{)},$
(7)
$\left(\frac{\partial}{\partial t}+iL_{\alpha}(1)+iL_{\alpha
s}(1,s)\right)f_{\alpha s}(x_{1},\vec{R}_{s};t)\\\
=-\varepsilon\big{(}f_{\alpha s}(x_{1},\vec{R}_{s};t)-g_{\alpha
s}(\vec{r}_{1},\vec{R}_{s}|n,\beta;t)f_{\alpha}(x_{1};t)n(\vec{R}_{s};t)\big{)}.$
(8)

Solutions of these equations (7), (8) can be given as:

$\displaystyle f_{\alpha\gamma}(x_{1},x_{2};t)$
$\displaystyle=\varepsilon\int_{-\infty}^{0}\\!\\!d\tau\,e^{\big{(}\varepsilon+iL^{(2)}_{\alpha\gamma}(1,2)\big{)}\tau}g_{\alpha\gamma}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t+\tau)f_{\alpha}(x_{1};t+\tau)f_{\gamma}(x_{2};t+\tau),$
$\displaystyle f_{\alpha s}(x_{1},\vec{R}_{s};t)$
$\displaystyle=\varepsilon\int_{-\infty}^{0}\\!\\!d\tau\,e^{\big{(}\varepsilon+iL^{(2)}_{\alpha
s}(1,s)\big{)}\tau}g_{\alpha
s}(\vec{r}_{1},\vec{R}_{s}|n,\beta;t+\tau)f_{\alpha}(x_{1};t+\tau)n(\vec{R}_{s};t+\tau),$

where

$\displaystyle iL^{(2)}_{\alpha\gamma}(1,2)$
$\displaystyle=iL_{\alpha}(1)+iL_{\gamma}(2)+iL_{\alpha\gamma}(1,2),$
$\displaystyle iL^{(2)}_{\alpha s}(1,s)$
$\displaystyle=iL_{\alpha}(1)+iL_{\alpha s}(1,s).$

Substituting these solutions into the equation (2), we obtain a non-Markov
kinetic equation for nonequilibrium one-particle ion distribution function in
the system ionic solution – porous matrix:

$\left(\frac{\partial}{\partial
t}+iL_{\alpha}(1)\right)f_{\alpha}(x_{1};t)=-\sum_{\gamma}\int
dx_{2}\,iL_{\alpha\gamma}(1,2)\\\
\times\varepsilon\int_{-\infty}^{0}\\!\\!d\tau\,e^{\big{(}\varepsilon+iL^{(2)}_{\alpha\gamma}(1,2)\big{)}\tau}g_{\alpha\gamma}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t+\tau)\,f_{\alpha}(x_{1};t+\tau)f_{\gamma}(x_{2};t+\tau)\\\
-\int
d\vec{R}_{s}\,iL(1,s)\,\varepsilon\int_{-\infty}^{0}\\!\\!d\tau\,e^{\big{(}\varepsilon+iL^{(2)}_{\alpha
s}(1,s)\big{)}\tau}g_{\alpha
s}(\vec{r}_{1},\vec{R}_{s}|n,\beta;t+\tau)\,f_{\alpha}(x_{1};t+\tau)n(\vec{R}_{s};t+\tau).$
(9)

The obtained kinetic equation for the nonequilibrium one-particle ion
distribution function also takes into account the spatial heterogeneity of the
system. If this equation formally put $g_{\alpha
s}(\vec{r}_{1},\vec{R}_{s}|n,\beta;t+\tau)=1$, then we obtain the kinetic
equation for the nonequilibrium one-particle distribution function ions, which
corresponds to the initial condition of non-interacting subsystems of ionic
solution and porous matrix. When obtaining the kinetic equation (9), we did
not specify the paired potentials of interaction between ions and particles of
the porous matrix.

In the next section, we consider the model of charged solid spheres to
describe the ionic solution in the presence of a porous matrix.

##  4 Model of charged solid spheres. Kinetic equation of the revised
Enskog–Vlasov–Landau theory for the ionic solution-porous matrix system

Consider the model of charged solid spheres for an ionic subsystem when the
interaction potential can be submitted as an amount 37, 39, 43:

$\Phi_{\alpha\gamma}(\vec{r},\vec{r}^{\prime})=\Phi^{sh}_{\alpha\gamma}(\vec{r},\vec{r}^{\prime})+\Phi^{l}_{\alpha\gamma}(\vec{r},\vec{r}^{\prime}),$

where $\Phi^{\rm sh}_{\alpha\gamma}(\vec{r},\vec{r}^{\prime})$ is the short-
range interaction potential between ions, which will be modeled by the sum of
$\Phi^{\text{sh-rep}}_{\alpha\gamma}(\vec{r},\vec{r}^{\prime})$ is the
potential of solid spheres and $\Phi^{\text{sh-
att}}_{\alpha\gamma}(\vec{r},\vec{r}^{\prime})$ is short-range attraction
potential, which describes possible associative connections between ions;
$\Phi^{l}_{\alpha\gamma}(\vec{r},\vec{r}^{\prime})$ is a long-range potential
for interaction between ions, in particular the potential of the Yukawa type.
In addition, the interaction of ions and particles of the porous medium will
be described by short-term potential of solid spheres $\Phi^{\text{sh-
rep}}_{\alpha s}(\vec{r},\vec{R}_{s})$ and some attractive potential
$\Phi^{\text{sh-att}}_{\alpha s}(\vec{r},\vec{R}_{s})$ with effective range
$r_{eff}$. Based on the works 37, 39 in the case of the model of solid spheres
for the liquid subsystem from (9) we get:

$\left(\frac{\partial}{\partial
t}+iL_{\alpha}(1)\right)f_{\alpha}(x_{1};t)=-\sum_{\gamma}\int_{0}^{\sigma_{\gamma}}\\!\\!\\!d\vec{r}_{2}\int
d\vec{p}_{2}\,iL_{\alpha\gamma}^{\text{sh-rep}}(12)\\\\[-2.84526pt]
\times\varepsilon\int_{-\infty}^{0}\\!\\!\\!\\!\\!\\!d\tau\,e^{\big{(}\varepsilon+iL_{\alpha\gamma}^{0}(12)+iL_{\alpha\gamma}^{\text{sh-
rep}}(12)\big{)}\tau}g_{\alpha\gamma}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t+\tau)\,f_{\alpha}(x_{1};t+\tau)f_{\gamma}(x_{2};t+\tau)\\\\[-2.84526pt]
-\sum_{\gamma}\int_{\sigma_{\gamma}}^{r_{\rm
eff}}\\!\\!\\!\\!\\!\\!d\vec{r}_{2}\int\\!\\!d\vec{p}_{2}\,iL_{\alpha\gamma}^{\text{sh-
att}}(12)\,\varepsilon\int_{-\infty}^{0}\\!\\!\\!\\!\\!\\!d\tau\,e^{\big{(}\varepsilon+iL_{\alpha\gamma}^{0}(12)+iL_{\alpha\gamma}^{\text{sh-
att}}(12)\big{)}\tau}\\\ \times
g_{\alpha\gamma}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t+\tau)f_{\alpha}(x_{1};t+\tau)f_{\gamma}(x_{2};t+\tau)\\\
-\sum_{\gamma}\int_{r_{\rm
eff}}^{\infty}\\!\\!\\!\\!\\!d\vec{r}_{2}\int\\!\\!d\vec{p}_{2}\,iL_{\alpha\gamma}^{l}(12)\,\varepsilon\\!\\!\int_{-\infty}^{0}\\!\\!\\!\\!\\!\\!d\tau\,e^{\big{(}\varepsilon+iL_{\alpha\gamma}^{0}(12)+iL_{\alpha\gamma}^{l}(12)\big{)}\tau}g_{\alpha\gamma}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t+\tau)f_{\alpha}(x_{1};t+\tau)f_{\gamma}(x_{2};t+\tau)\\\\[-2.84526pt]
-\int_{0}^{\sigma_{s}}\\!\\!\\!\\!d\vec{R}_{s}\,iL_{\alpha s}^{\text{sh-
rep}}(1s)\,\varepsilon\int_{-\infty}^{0}\\!\\!\\!\\!\\!\\!d\tau\,e^{\big{(}\varepsilon+iL_{\alpha}(1)+iL_{\alpha
s}^{\text{sh-rep}}(1s)\big{)}\tau}g_{\alpha
s}(\vec{r}_{1},\vec{R}_{s}|n,\beta;t+\tau)f_{\alpha}(x_{1};t+\tau)n_{s}(\vec{R}_{s};t+\tau)\\\\[-2.84526pt]
-\int_{\sigma_{s}}^{r_{\rm eff}}\\!\\!\\!\\!d\vec{R}_{s}\,iL_{\alpha
s}^{\text{sh-
att}}(1s)\,\varepsilon\int_{-\infty}^{0}\\!\\!\\!\\!\\!\\!d\tau\,e^{\big{(}\varepsilon+iL_{\alpha}(1)+iL_{\alpha
s}^{\text{sh-att}}(1s)\big{)}\tau}g_{\alpha
s}(\vec{r}_{1},\vec{R}_{s}|n,\beta;t+\tau)f_{\alpha}(x_{1};t+\tau)n_{s}(\vec{R}_{s};t+\tau)$

is a kinetic equation for the nonequilibrium one-particle ion distribution
function, taking into account areas of short-range (solid-sphere) and long-
range potentials. Given that in the field of action of the potential of solid
spheres, the interaction time $\tau\rightarrow+0$ and detailed calculations
37, 39, 43, the equation can be given as:

$\left(\frac{\partial}{\partial
t}+iL_{\alpha}(1)\right)f_{\alpha}(x_{1};t)=-\sum_{\gamma}\int
dx_{2}\,\hat{T}_{\alpha\gamma}(12)\,g_{\alpha\gamma}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)\,f_{\alpha}(x_{1};t)f_{\gamma}(x_{2};t)\\\\[-2.84526pt]
-\int d\vec{R}_{s}\,\hat{T}_{\alpha s}(1s)\,g_{\alpha
s}(\vec{r}_{1},\vec{R}_{s}|n,\beta;t)f_{\alpha}(x_{1};t)n_{s}(\vec{R}_{s};t)\\\\[-2.84526pt]
-\sum_{\gamma}\int_{\sigma_{\gamma}}^{r_{\rm
eff}}\\!\\!\\!\\!\\!\\!d\vec{r}_{2}\int\\!\\!d\vec{p}_{2}\,iL_{\alpha\gamma}^{\text{sh-
att}}(12)\,\varepsilon\\!\int_{-\infty}^{0}\\!\\!\\!\\!\\!\\!d\tau\,e^{(\varepsilon+iL_{\alpha\gamma}^{0}(12)+iL_{\alpha\gamma}^{\text{sh-
att}}(12))\tau}\\\\[-2.84526pt]
\times\,\,g_{\alpha\gamma}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t+\tau)f_{\alpha}(x_{1};t+\tau)f_{\gamma}(x_{2};t+\tau)\\\\[-2.84526pt]
-\sum_{\gamma}\int_{r_{\rm
eff}}^{\infty}\\!\\!\\!\\!\\!\\!d\vec{r}_{2}\int\\!\\!d\vec{p}_{2}\,iL_{\alpha\gamma}^{l}(12)\,\varepsilon\\!\int_{-\infty}^{0}\\!\\!\\!\\!\\!\\!d\tau
e^{(\varepsilon+iL_{\alpha\gamma}^{0}(12)+iL_{\alpha\gamma}^{l}(12))\tau}g_{\alpha\gamma}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t+\tau)f_{\alpha}(x_{1};t+\tau)f_{\gamma}(x_{2};t+\tau)\\\\[-2.84526pt]
-\int_{\sigma_{s}}^{r_{\rm eff}}\\!\\!\\!\\!d\vec{R}_{s}\,iL_{\alpha
s}^{\text{sh-
att}}(1s)\,\varepsilon\int_{-\infty}^{0}\\!\\!\\!\\!\\!\\!d\tau\,e^{(\varepsilon+iL_{\alpha}(1)+iL_{\alpha
s}^{\text{sh-att}}(1s))\tau}g_{\alpha
s}(\vec{r}_{1},\vec{R}_{s}|n,\beta;t+\tau)f_{\alpha}(x_{1};t+\tau)n_{s}(\vec{R}_{s};t+\tau),$

where $\hat{T}_{\alpha\gamma}(12)$ is the Enskog collision operator for
charged solid spheres (ions) 37, $\hat{T}_{\alpha s}(1s)$ is the operator
Enskog collision for charged solid spheres and solid spheres, describing the
porous medium. Next, if in the long-range part of the collision integral to
perform integration by parts, we obtain the following kinetic equation:

$\left(\frac{\partial}{\partial
t}+iL_{\alpha}(1)\right)f_{\alpha}(x_{1};t)=-\sum_{\gamma}\int
dx_{2}\,\hat{T}_{\alpha\gamma}(12)\,g_{\alpha\gamma}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)\,f_{\alpha}(x_{1};t)\,f_{\gamma}(x_{2};t)\\\
-\int d\vec{R}_{s}\,\hat{T}_{\alpha s}(1s)\,g_{\alpha
s}(\vec{r}_{1},\vec{R}_{s}|n,\beta;t)\,f_{\alpha}(x_{1};t)\,n_{s}(\vec{R}_{s};t)\\\
-\sum_{\gamma}\int_{\sigma_{\gamma}}^{r_{\rm eff}}d\vec{r}_{2}\int
d\vec{p}_{2}\,iL_{\alpha\gamma}^{\text{sh-
att}}(12)\,g_{\alpha\gamma}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)\,f_{\alpha}(x_{1};t)\,f_{\gamma}(x_{2};t)\\\
+\sum_{\gamma}\int_{\sigma_{\gamma}}^{r_{\rm eff}}d\vec{r}_{2}\int
d\vec{p}_{2}\,iL_{\alpha\gamma}^{\text{sh-
att}}(12)\int_{-\infty}^{0}d\tau\,e^{(\varepsilon+iL_{\alpha\gamma}^{0}(12)+iL_{\alpha\gamma}^{\text{sh-
att}}(12))\tau}\\\
\times\left(\frac{\partial}{\partial\tau}+iL_{\alpha\gamma}^{0}(12)+iL_{\alpha\gamma}^{\text{sh-
att}}(12)\right)g_{\alpha\gamma}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t+\tau)\,f_{\alpha}(x_{1};t+\tau)\,f_{\gamma}(x_{2};t+\tau)\\\
-\sum_{\gamma}\int_{r_{\rm eff}}^{\infty}d\vec{r}_{2}\int
d\vec{p}_{2}\,iL_{\alpha\gamma}^{l}(12)\,g_{\alpha\gamma}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t)\,f_{\alpha}(x_{1};t)\,f_{\gamma}(x_{2};t)\\\
+\sum_{\gamma}\int_{r_{\rm eff}}^{\infty}d\vec{r}_{2}\int
d\vec{p}_{2}\,iL_{\alpha\gamma}^{l}(12)\int_{-\infty}^{0}d\tau\,e^{(\varepsilon+iL_{\alpha\gamma}^{0}(12)+iL_{\alpha\gamma}^{l}(12))\tau}\\\
\times\left(\frac{\partial}{\partial\tau}+iL_{\alpha\gamma}^{0}(12)+iL_{\alpha\gamma}^{l}(12)\right)g_{\alpha\gamma}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t+\tau)\,f_{\alpha}(x_{1};t+\tau)\,f_{\gamma}(x_{2};t+\tau)\\\
-\int_{\sigma_{s}}^{r_{\rm eff}}d\vec{R}_{s}\,iL_{\alpha s}^{\text{sh-
att}}(1s)g_{\alpha
s}(\vec{r}_{1},\vec{R}_{s}|n,\beta;t)\,f_{\alpha}(x_{1};t)\,n_{s}(\vec{R}_{s};t)\\\
+\int_{\sigma_{s}}^{r_{\rm eff}}d\vec{R}_{s}\,iL_{\alpha s}^{\text{sh-
att}}(1s)\int_{-\infty}^{0}d\tau\,e^{(\varepsilon+iL_{\alpha}(1)+iL_{\alpha
s}^{\text{sh-att}}(1s))\tau}\\\
\times\left(\frac{\partial}{\partial\tau}+iL_{\alpha}(1)+iL_{\alpha
s}^{\text{sh-att}}(1s)\right)g_{\alpha
s}(\vec{r}_{1},\vec{R}_{s}|n,\beta;t+\tau)\,f_{\alpha}(x_{1};t+\tau)\,n_{s}(\vec{R}_{s};t+\tau),$

where the third, fifth, and seventh terms in the right part are generalized
integrals of the Vlasov collision – generalized middle fields, and the fourth,
sixth, and eighth terms are generalized Landau–type collision integrals
between ions and ions and porous matrix particles, taking into account memory
effects.

Revealing the action of the operator Enskog in the right part, in a spatially
inhomogeneous case, (up to linear values by gradients) and without taking into
account the effects of memory, we obtain:

$\left(\frac{\partial}{\partial
t}+iL_{\alpha}(1)\right)f_{\alpha}(x_{1};t)=I^{(0)}_{\alpha
E}(x_{1};t)+I^{(1)}_{\alpha E}(x_{1};t)+I^{(1)}_{\alpha
MF}(x_{1};t)+I^{(1)}_{\alpha L}(x_{1};t),$ (10)

where the terms on the right are the integrals of collisions, due to the
contribution of an certain type of interparticle interaction. The first and
second of them are Enskog–type collision integrals of the RET theory 37:

$I^{(0)}_{\alpha E}(x_{1};t)=\sum_{\gamma}\int d\vec{v}_{2}\int
d\varepsilon\int
b\,db\,g(12)g_{\alpha\gamma}(\sigma_{\alpha\gamma}|n,\beta;t)\\\\[-5.69054pt]
\times\big{(}f_{\alpha}(\vec{r}_{1},\vec{v}^{\prime}_{1};t)f_{\gamma}(\vec{r}_{2},\vec{v}^{\prime}_{2};t)-f_{1}(\vec{r}_{\alpha},\vec{v}_{1};t)f_{\gamma}(\vec{r}_{2},\vec{v}_{2};t)\big{)},$
$I^{(1)}_{\alpha E}(x_{1};t)=\sum_{\gamma}\sigma_{\alpha\gamma}^{3}\int
d\hat{\vec{r}}_{12}\int
d\vec{v}_{2}\,\Theta\big{(}\hat{\vec{r}}_{12}\cdot\vec{g}(12)\big{)}\big{(}\hat{\vec{r}}_{12}\cdot\vec{g}(12)\big{)}\\\\[-2.84526pt]
\times\Big{(}g_{\alpha\gamma}(\vec{r}_{12}|n;t)\vec{r}_{12}\cdot[f_{\alpha}(\vec{r}_{1},\vec{v^{\prime}}_{1};t)\vec{\nabla}_{2}f_{\gamma}(\vec{r}_{2},\vec{v^{\prime}}_{2};t)-f_{\alpha}(\vec{r}_{1},\vec{v}_{1};t)\vec{\nabla}_{2}f_{\gamma}(\vec{r}_{2},\vec{v}_{2};t)]\\\\[-2.84526pt]
+\frac{1}{2}\big{(}\hat{\vec{r}}_{12}\cdot\vec{\nabla}_{2}g_{\alpha\gamma}(\vec{r}_{12}|n,\beta;t)\big{)}\big{[}f_{\alpha}(\vec{r}_{1},\vec{v^{\prime}}_{1};t)f_{\gamma}(\vec{r}_{2},\vec{v^{\prime}}_{2};t)-f_{\alpha}(\vec{r}_{1},\vec{v}_{1};t)f_{\gamma}(\vec{r}_{2},\vec{v}_{2};t)\big{]}\Big{)},$

where $b$ is the aiming parameter,
$g_{\alpha\gamma}(\sigma_{\alpha\gamma}|n;t)$ is the contact value of the
paired quasi-equilibrium distribution function,
$\hat{\vec{r}}_{12}=\frac{\vec{r}_{12}}{|\vec{r}_{12}|}$ is a single vector,
$\vec{v^{\prime}}_{1}=\vec{v}_{1}+\hat{\vec{r}}_{12}(\hat{\vec{r}}_{12}\cdot\vec{g}(12))$,
$\vec{v^{\prime}}_{2}=\vec{v}_{2}-\hat{\vec{r}}_{12}(\hat{\vec{r}}_{12}\cdot\vec{g}(12))$
is the value of particle velocities $1$, $2$ after the collision, while
$\vec{v}_{1}$, $\vec{v}_{2}$ is the value of their velocities before the
collision, where $\vec{g}(12)=\vec{v}_{2}-\vec{v}_{1}$ is the relative
velocity.

The next term is the contribution of the mean-field theory KMFT 37:

$I^{(1)}_{\alpha
MF}(x_{1};t)=\frac{1}{m_{\alpha}}\sum_{\gamma}\int_{\sigma_{\gamma}}^{r_{\rm
eff}}d\vec{r}_{2}\,\frac{\partial}{\partial\vec{r}_{1}}\Phi^{\text{sh-
att}}_{\alpha\gamma}(\vec{r}_{12})\cdot\frac{\partial}{\partial\vec{v}_{1}}g_{\alpha\gamma}(\vec{r}_{12}|n,\beta;t)f_{\alpha}(\vec{r}_{1},\vec{v}_{1};t)n_{\gamma}(\vec{r}_{2};t)\\\
+\frac{1}{m_{\alpha}}\sum_{\gamma}\int_{r_{\rm
eff}}^{\infty}d\vec{r}_{2}\,\frac{\partial}{\partial\vec{r}_{1}}\Phi^{l}_{\alpha\gamma}(\vec{r}_{12})\cdot\frac{\partial}{\partial\vec{v}_{1}}g_{\alpha\gamma}(\vec{r}_{12}|n,\beta;t)f_{\alpha}(\vec{r}_{1},\vec{v}_{1};t)n_{\gamma}(\vec{r}_{2};t)\\\
+\frac{1}{m_{\alpha}}\sum_{\gamma}\int_{\sigma_{\gamma}}^{r_{\rm
eff}}d\vec{R}_{2}\,\frac{\partial}{\partial\vec{r}_{1}}\Phi^{\text{sh-
att}}_{\alpha
s}(\vec{r}_{1},\vec{R}_{2},)\cdot\frac{\partial}{\partial\vec{v}_{1}}g_{\alpha
s}(\vec{r}_{1},\vec{R}_{2}|n,\beta;t)f_{\alpha}(\vec{r}_{1},\vec{v}_{1};t)n_{s}(\vec{R}_{2};t).$

The last term is the integral of Landau–type collisions 37, 39

$I^{(1)}_{\alpha L}(x_{1};t)=\sum_{\gamma}\int_{\sigma_{\gamma}}^{r_{\rm
eff}}d\vec{r}_{2}\int d\vec{p}_{2}\,iL_{\alpha\gamma}^{\text{sh-
att}}(12)\int_{-\infty}^{0}d\tau\,e^{(\varepsilon+iL_{\alpha\gamma}^{0}(12)+iL_{\alpha\gamma}^{\text{sh-
att}}(12))\tau}\\\
\times\left(\frac{\partial}{\partial\tau}+iL_{\alpha\gamma}^{\text{sh-
att}}(12)\right)g_{\alpha\gamma}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t+\tau)f_{\alpha}(x_{1};t+\tau)f_{\gamma}(x_{2};t+\tau)\\\
+\sum_{\gamma}\int_{r_{\rm eff}}^{\infty}d\vec{r}_{2}\int
d\vec{p}_{2}\,iL_{\alpha\gamma}^{l}(12)\int_{-\infty}^{0}d\tau\,e^{(\varepsilon+iL_{\alpha\gamma}^{0}(12)+iL_{\alpha\gamma}^{l}(12))\tau}\\\
\times\left(\frac{\partial}{\partial\tau}+iL_{\alpha\gamma}^{l}(12)\right)g_{\alpha\gamma}(\vec{r}_{1},\vec{r}_{2}|n,\beta;t+\tau)f_{\alpha}(x_{1};t+\tau)f_{\gamma}(x_{2};t+\tau)\\\
+\int_{\sigma_{s}}^{r_{\rm eff}}d\vec{R}_{2}iL_{\alpha s}^{\text{sh-
att}}(12)\int_{-\infty}^{0}d\tau\,e^{(\varepsilon+iL_{\alpha
s}^{0}(12)+iL_{\alpha s}^{\text{sh-att}}(12))\tau}\\\
\times\left(\frac{\partial}{\partial\tau}+iL_{\alpha s}^{\text{sh-
att}}(12)\right)g_{\alpha
s}(\vec{r}_{1},\vec{R}_{2}|n,\beta;t+\tau)f_{\alpha}(x_{1};t+\tau)n_{s}(\vec{R}_{2};t+\tau).$

By solving 43 kinetic Enskog–Vlasov–Landau equations (10) for charged solid
spheres can be constructed as equations hydrodynamics and obtained analytical
expressions for mutual diffusion coefficients, thermodiffusion, viscosity, and
thermal conductivity through particle distribution functions and them the
nature of the interaction.

##  5 Conclusions

The kinetic approach is applied to the description of ion transfer processes
in the system ionic solution – a porous medium. The nonequilibrium state of
the system is described using a modified one chain of equations BBGKI 37, 38,
39, 41, 42 for partial nonequilibrium distribution functions of ions and
particles of the porous matrix. For this purpose, we used the approach
proposed in 37, 38, 39, 41, 42, 44, where the modified chain of BBGKI
equations is built taking into account the concept of the consistent
description of kinetics and hydrodynamics of nonequilibrium processes of
interacting particles in the Zubarev method of nonequilibrium statistical
operator. A generalized kinetic equation of the revised Enskog–Vlasov–Landau
theory for the nonequilibrium ion distribution function in the model of
charged solid spheres is obtained, taking into account attractive short-range
interactions for the ionic solution – porous medium system. Two cases of
construction of kinetic equations depending on the initial conditions of the
interaction of ionic solution particles and porous matrix are considered. In
the case of non-interacting subsystems (at the initial moment of time), the
modified boundary conditions of the BBGKI chain of equations do not include
quasi-equilibrium coordinate (pair and higher-order) functions describing
correlations between solution ions and porous matrix particles. In the case of
interacting subsystems (at the initial moment of time), modified boundary
conditions in the chain of equations BBGKI include vase-equilibrium coordinate
(pair and higher-order) functions describing the correlations between solution
ions and porous matrix particles. For this case, a generalized kinetic
equation of the revised Enskog–Vlasov–Landau theory for the nonequilibrium ion
distribution function in the model of charged solid spheres is obtained,
taking into account attractive short-range interactions for the ionic solution
– porous medium system. To calculate the paired quasi-equilibrium distribution
coordinate functions for ions and particles of the porous matrix, it is
proposed to use inhomogeneous, time-dependent Ornstein–Zernike equations for
the corresponding complete correlation functions
$h_{\alpha\gamma}(\vec{r},\vec{r}^{\prime}|n,\beta;t)$ ($h_{++}$, $h_{-+}$,
$h_{--}$), $h_{\alpha s}(\vec{r},\vec{R}_{s}|n,\beta;t)$ ($h_{+s}$, $h_{-s}$).

In the following works, solutions for the Enskog–Vlasov–Landau kinetic
equations for nonequilibrium distribution functions will be found by the
boundary conditions method and the corresponding equations of hydrodynamics
with generalized ion transfer coefficients in the ionic solution-porous matrix
system will be obtained.

## References

  * 1 Sahimi M. Non-linear and non-local transport processes in heterogeneous media: from long-range correlated percolation to fracture and materials breakdown. Physics Reports. 306 (4–6), 213–395 (1998).
  * 2 Hatano Y., Hatano N. Dispersive transport of ions in column experiments: An explanation of long-tailed profiles. Water Resources Research. 34 (5), 1027–1033 (1998).
  * 3 Gelb L. D., Gubbins K. E., Radhakrishnan R., Sliwinska-Bartkowiak M. Phase separation in confined systems. Reports on Progress in Physics. 62 (12), 1573–1659 (1999).
  * 4 Advances in Lithium-Ion Batteries. Eds.: W. A. van Schalkwijk, B. Scrosati. Springer New York, NY (2002).
  * 5 Wagemaker M. Structure and Dynamics of Lithium in Anatase TiO2. Delft Univer. Press, Netherland (2002).
  * 6 Berkowitz B., Klafter J., Metzler R., Scher H. Physical pictures of transport in heterogeneous media: Advection-dispersion, random-walk, and fractional derivative formulations. Water Resources Research. 38 (10), 1191 (2002).
  * 7 Berkowitz B., Cortis A., Dentz M., Scher H. Modeling non-Fickian transport in geological formations as a continuous time random walk. Reviews of Geophysics. 44 (2), RG2003 (2006).
  * 8 Smith J. J., Zharov I. Ion Transport in Sulfonated Nanoporous Colloidal Films. Langmuir. 24 (6), 2650–2654 (2008).
  * 9 Neuman S. P., Tartakovsky D. M. Perspective on theories of non-Fickian transport in heterogeneous media. Advances in Water Resources. 32 (5), 670–680 (2009).
  * 10 Rotenberg B., Pagonabarrag I., Frenkel D. Coarse-grained simulations of charge, current and flow in heterogeneous media. Faraday Discussions. 144, 223–243 (2010).
  * 11 Yang C., Nakayama A. A synthesis of tortuosity and dispersion in effective thermal conductivity of porous media. International Journal of Heat and Mass Transfer. 53 (15–16), 3222–3230 (2010).
  * 12 Maffeo C., Bhattacharya S., Yoo J., Wells D., Aksimentiev A. Modeling and Simulation of Ion Channels. Chemical Reviews. 112 (12), 6250–6284 (2012).
  * 13 Bijeljic B., Raeini A., Mostaghimi P., Blunt M. J. Predictions of non-Fickian solute transport in different classes of porous media using direct simulation on pore-scale images. Physical Review E. 87 (1), 013011 (2013).
  * 14 Grygorchak I. I., Kostrobij P. P., Stasiuk I. V., Tokarchuk M. V., Velychko O. V., Ivashchyshyn F. O., Markovych B. M. Physical processes and their microscopic models in periodic inorganic / organic clathrates. Lviv, Rastr-7 (2015), (in Ukrainian).
  * 15 Lithium batteries. Eds.: G.-A. Nazri, G. Pistoia. USA, Springer (2009).
  * 16 Tyukhova A., Dentz M., Kinzelbach W., Willmann M. Mechanisms of Anomalous Dispersion in Flow Through Heterogeneous Porous Media. Physical Review Fluids. 1 (7), 074002 (2016).
  * 17 Comolli A., Dentz M. Anomalous dispersion in correlated porous media: a coupled continuous time random walk approach. The European Physical Journal B. 90, 166 (2017).
  * 18 Waisbord N., Stoop N., Walkama D. M., Dunkel J., Guasto J. S. Anomalous percolation flow transition of yield stress fluids in porous media. Physical Review Fluids. 4 (6), 063303 (2019).
  * 19 Zhao T., Qing L., Long T., Xu X., Zhao S., Lu X. Dynamical coupling of ion adsorption with fluid flow in nanopores. AIChE J. 67, e17266 (2021).
  * 20 Bisquert J., Compte A. Theory of the electrochemical impedance of anomalous diffusion. Journal of Electroanalytical Chemistry. 499 (1), 112–120 (2001).
  * 21 Sibatov R. T., Uchaikin V. V. Fractional differential approach to dispersive transport in semiconductors. Physics-Uspekhi. 52 (10), 1019–1043 (2009).
  * 22 Sibatov R. T. Drobno-differencialnaja teorija anomalnoj kinetiki nositelej zarjada v neuporjadochennyh poluprovodnikovyh sistemah. Thesis for the Degree of Doctor of Sciences in Physics and Mathematics. Uljanovsk (2012), (in Russian).
  * 23 Khamzin A. A., Popov I. I., Nigmatullin R. R. Correction of the power law of ac conductivity in ion-conducting materials due to the electrode polarization effect. Physical Review E. 89 (3), 032303 (2014).
  * 24 Ferguson T. R., Bazant M. Z. Nonequilidrium Thermodynamics of Porous Electrodes. Journal of The Electrochemical Society. 159 (12), A1967–A1985 (2012).
  * 25 Xie Y., Li J., Yuan C. Mathematical modeling of the electrochemical impedance spectroscopy in lithium ion battery cycling. Electrochimica Acta. 127, 266–275 (2014).
  * 26 Sibatov R. T., Uchaikin V. V. Fractional differential approach to the description of dispersive transfer in semiconductors. Physics-Uspekhi. 52 (10), 1019–1043 (2009).
  * 27 Kostrobij P. P., Markovych B. M., Chernomorets Yu. I., Tokarchuk R. M., Tokarchuk M. V. Statistical description of electro-diffusion processes of ions intercalation in “electrolyte–electrode” system. Mathematical Modeling and Computing. 1 (2), 178–194 (2014).
  * 28 Kostrobij P., Markovych B., Viznovych O., Tokarchuk M. Generalized electrodiffusion equation with fractality of space-time. Mathematical Modeling and Computing. 3 (2), 163–172 (2017).
  * 29 Grygorchak I. I., Ivashchyshyn F. O., Tokarchuk M. V., Pokladok N. T., Viznovych O. V. Modification of properties of GaSe $\beta$-cyclodexterin Clathrat by synthesis in superposed electric and light-wave fields. Journal of Applied Physics. 121, 185501 (2017).
  * 30 Kostrobij P. P., Ivashchyshyn F. O., Markovych B. M., Tokarchuk M. V. Microscopic theory of the influence of dipole superparamagnetics (type $\langle\beta\mathrm{-CD}\langle\mathrm{FeSO}_{4}\rangle\rangle$) on current flow in semiconductor layered structures (type GaSe, InSe). Mathematical Modeling and Computing. 8 (1), 89–105 (2021).
  * 31 Kostrobij P., Grygorchak I., Ivashchyshyn I., Markovych B., Viznovych O., Tokarchuk M. Generalized electrodiffusion equation with fractality of space time: experiment and theory. The Journal of Physical Chemistry A. 122 (16), 4099–4110 (2018).
  * 32 Jardat M., Hribar-Lee B., Vlachy V. Self-diffusion of ions in charged nanoporous media. Soft Matter. 8, 954–964 (2012).
  * 33 Jardat M., Hribar-Lee B., Dahirel V., Vlachy V. Self-diffusion and activity coefficients of ions in charged disordered media. The Journal of Chemical Physics. 137 (11), 114507 (2012).
  * 34 Omelyan I. P., Zhelem R. I., Sovyak E. M., Tokarchuk M. V. Calculation of distribution functions and diffusion coefficients for ions in the system “initial electrolyte solution – membrane”. Condensed Matter Physics. 2 (1), 53–62 (1999).
  * 35 Yukhnovskii I. R., Zhelem R. I., Tokarchuk M. V. Physical processes in the fuel containing masses interacting with aqueous solutions in the “Shelter” object. Inhomogeneous diffusion of ions UO${}_{2}^{2+}$, Cs+ in the system “glassy nuclear magma – water”. Condensed Matter Physics. 2 (2), 351–360 (1999).
  * 36 Yukhnovskii I. R., Omelyan I. P., Zhelem R. I., Tokarchuk M. V. Statistical theory for diffusion of radionuclides in ground and subterranean water. Radiation Physics and Chemistry. 59 (4), 361–375 (2000).
  * 37 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), 412–424 (1991).
  * 38 Tokarchuk M. V., Omelyan I. P. Model kinetic equations for dense gases and liquids. Ukrainian Journal of Physics. 35 (8), 1255–1261 (1990), (in Ukrainian).
  * 39 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). Kharkov, Kharkov Physico Technical Institute. 3 (24), 60–65 (1992), (in Russian).
  * 40 Alvarez F. X., Jou D., Sellitto A. Pore-size dependence of the thermal conductivity of porous silicon: A phonon hydrodynamic approach. Applied Physics Letters. 97 (3), 033103 (2010).
  * 41 Kobryn O. E., Omelyan I. P., Tokarchuk M. V. The modified group expansions for constructions of solutions to the BBGKY hierarchy. Journal of Statistical Physics. 92 (5–6), 973–994 (1998).
  * 42 Tokarchuk M. V., Kobryn O. E., Omelyan I. P. Consistent description of kinetics and hydrodynamics of systems of interacting particles by means of nonequilibrium operator method. Condensed Matter Physics. 1 (4), 687–751 (1998).
  * 43 Kobryn A. E., Morozov V. G., Omelyan I. P., Tokarchuk M. V. Enskog–Landau kinetic equation. Calculation of the transport coefficient for charged hard spheres. Physica A: Statistical Mechanics and its Applications. 230 (1–2), 189–201 (1996).
  * 44 Tokarchuk M. V. To the kinetic theory of dense gases and liquids. Calculation of quasi-equilibrium particle distribution functions by the method of collective variables. Mathematical Modeling and Computing. 9 (2), 440–458 (2022).
  * 45 Polewczak J., Stell G. Transport Coefficients in Some Stochastic Models of the Revised Enskog Equation. Journal of Statistical Physics. 109, 569–590 (2002).
  * 46 Polewczak J. Hard-sphere kinetic models for inert and reactive mixtures. Journal of Physics: Condensed Matter. 28 (41), 414022 (2016).
  * 47 Karkheck J., Stell G. Maximization of entropy, kinetic equations, and irreversible thermodynamics. Physical Review A. 25 (6), 3302–3327 (1982).
  * 48 Yukhnovskii I. R., Holovko M. F. Statistical Theory of Classical Equilibrium Systems. Naukova Dumka, Kiev (1980), (in Russian).
  * 49 Zubarev D. N. Statistical thermodynammics of turbulent transport processes. Theoretical and Mathematical Physics. 53 (1), 1004–1014 (1982).
  * 50 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. Ukrainian Journal of Physics. 41 (5-6), 596–604 (1996), (in Ukrainian).
  * 51 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: Statistical Mechanics and its Applications. 443, 231–245 (2016).
  * 52 Yukhnovskii I. R., Hlushak P. A., Tokarchuk M. V. BBGKY chain of kinetic equations, non-equilibrium statistical operator method and collective variable method in the statistical theory of non-equilibrium liquids. Condensed Matter Physics. 19 (4), 43705 (2016).
  * 53 Ramshaw J. D. Time-dependent direct correlashion function. Physical Review A. 24 (3), 1567–1570 (1981).
  * 54 Eu B. C. Dynamic Ornstein–Zernike Equation. In: Transport Coefficients of Fluids. Chemical physics. Vol. 82, 221–240. Springer, Berlin, Heidelberg (2006).
  * 55 Gan H. H., Eu B. C. Theory of the nonequilibrium structure of dense simple fluids: Effects of shearing. Physical Review A. 45 (6), 3670–3686 (1992).
  * 56 Brader J. M., Schmidt M. Nonequilibrium Ornstein-Zernike relation for Brownian many-body dynamics. The Journal of Chemical Physics. 139 (10), 104108 (2013).
  * 57 Holovko M. F., Kalyuzhnyi Yu. V. On the effects of association in the statistical theory of ionic systems. Analytic solution of the PY-MSA version of the Wertheim theory. Molecular Physics. 73 (5), 1145–1157 (1991).
  * 58 Kalyuzhnyi Y. V., Vlachy V., Holovko M. F., Stell G. Multidensity integral equation theory for highly asymmetric electrolyte solutions. The Journal of Chemical Physics. 102 (14), 5770–5780 (1995).
  * 59 Holovko M. F., Sovyak E. M. On taking account of interactions in the statistical theory of electrolyte solutions. Condensed Matter Physics. 6, 49–78 (1995).
  * 60 Holovko M. F., Patsahan T. M., Shmotolokha V. I. What is liquid in random porous media: the Barker-Henderson perturbation theory. Condensed Matter Physics. 18 (1), 13607 (2015).