On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids
For a consistent description of kinetic and hydrodynamic processes in dense gases and liquids the generalized non-Markovian equations for the nonequilibrium one-particle distribution function and potential part of the averaged enthalpy density are obtained. The inner structure of the generalized tra...
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| Опубліковано в: : | Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України
2010
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| Цитувати: | On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids / B.B. Markiv, I.P. Omelyan, M.V. Tokarchuk // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23005: 1-16. — Бібліогр.: 28 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859649831801192448 |
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| author | Markiv, B.B. Omelyan, I.P. Tokarchuk, M.V. |
| author_facet | Markiv, B.B. Omelyan, I.P. Tokarchuk, M.V. |
| citation_txt | On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids / B.B. Markiv, I.P. Omelyan, M.V. Tokarchuk // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23005: 1-16. — Бібліогр.: 28 назв. — англ. |
| collection | DSpace DC |
| container_title | Condensed Matter Physics |
| description | For a consistent description of kinetic and hydrodynamic processes in dense gases and liquids the generalized non-Markovian equations for the nonequilibrium one-particle distribution function and potential part of the averaged enthalpy density are obtained. The inner structure of the generalized transport kernels for these equations is established. It is shown that the collision integral of the kinetic equation has the Fokker-Planck form with the generalized friction coefficient in momentum space. It also contains contributions from the generalized diffusion coefficient and dissipative processes connected with the potential part of the enthalpy density.
Для узгодженого опису кінетичних і гідродинамічних процесів у густих газах і рідинах одержано узагальнені немарківські рівняння для нерівноважної одночастинкової функції розподілу та середнього значення густини потенціальної частини ентальпії. Розкрито внутрішню структуру узагальнених ядер переносу даних рівнянь. Показано, що інтеграл зіткнення кінетичного рівняння має структуру Фоккера - Планка з узагальненим коефіцієнтом тертя в імпульсному просторі. Він також містить вклади від узагальненого коефіцієнта дифузії в просторі імпульсів і дисипативних процесів, пов'язаних із густиною потенціальної частини ентальпії.
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| first_indexed | 2025-12-07T13:32:27Z |
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Condensed Matter Physics 2010, Vol. 13, No 2, 23005: 1–16
http://www.icmp.lviv.ua/journal
On the problem of a consistent description of kinetic
and hydrodynamic processes in dense gases and
liquids
B.B. Markiv, I.P. Omelyan, M.V. Tokarchuk
Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
Received March 19, 2010, in final April 8, 2010
For a consistent description of kinetic and hydrodynamic processes in dense gases and liquids the gener-
alized non-Markovian equations for the nonequilibrium one-particle distribution function and potential part of
the averaged enthalpy density are obtained. The inner structure of the generalized transport kernels for these
equations is established. It is shown that the collision integral of the kinetic equation has the Fokker-Planck
form with the generalized friction coefficient in momentum space. It also contains contributions from the gener-
alized diffusion coefficient and dissipative processes connected with the potential part of the enthalpy density.
Key words: kinetics, hydrodynamics, kinetic equations, memory functions, time correlation functions
PACS: 05.20.Dd, 05.60.+w, 52.25.Fi, 82.20.M
1. Introduction
A number of investigations [1–12] was devoted to the problem of constructing a consistent
description of kinetic and hydrodynamic processes in dense gases, liquids, and plasma. The paper
by D.N. Zubarev and V.G. Morozov [3] is a fundamental one in kinetic theory of dense gases and
liquids. In that paper the formulation of a new boundary condition to the BBGKY hierarchy for
nonequilibrium distribution functions which takes into account correlations connected with local
conservation laws was given. In a pair collisions approximation this modification of Bogolyubov’s
correlation weakening condition allows one to consistently derive the Enskog kinetic equations [3]
for neutral and one-component charged hard sphere systems. In the paper by Yu.L. Klimontovich [6]
the necessity and possibility of a consistent description of kinetic and hydrodynamic processes
in gases for values of the Knudsen number of order of unity were substantiated. Herewith, the
relation (τG
ph)min ∼ L2
min/D ∼ τph takes place, where τph denotes the physically infinitesimal time
interval during which one particle suffers a collision from a collection with Nph = nVph = N
V Vph
particles in small volume Vph with physically infinitesimal length scale lph, where D is the kinetic
coefficient, (τG
ph)min ∼ (lGph)
2
min/D (diffusive relation) is the minimal hydrodynamic time typical of
diffusive process, Lmin is the minimal length, on which the diffusive relation is still correct. The
latter is lesser than the free motion path l but larger with respect to the scale lph corresponding
to the kinetic description. Basing on these arguments, the generalized kinetic equation for the
description of nonequilibrium processes at arbitrary Knudsen numbers was proposed. In particular,
the collision integral of this equation contains the diffusion coefficient in velocity space. A regular
perturbation theory can be implemented for solving such an equation. Using the ideas of paper [3, 4]
the generalized transport equations for the hydrodynamic variables such as number of particles,
momentum and total energy densities were obtained in [7, 9] by means of nonequilibrium statistical
operator method. Such equations are consistent with the kinetic equation for the nonequilibrium
one-particle distribution function. They allow one to study the time correlation functions describing
the dynamics of a liquid system in states close to equilibrium. These results are equivalent to those
obtained in the investigations of [1, 2] based on the Mori projection operator method. An important
c© B.B. Markiv, I.P. Omelyan, M.V. Tokarchuk 23005-1
http://www.icmp.lviv.ua/journal
B.B. Markiv, I.P. Omelyan, M.V. Tokarchuk
common problem of these investigations consists in the calculation of the generalized transport
kernels (memory functions). The latter determine the collision integrals for various models [1, 13–
24] as well as the generalized transport coefficients such as viscosity and heat conductivity [1, 3, 4].
This problem is particularly significant in the region of short-ranged interactions. The reason is that
at small distances according to Klimontovich one obtains the changes not only for the spatially-
temporal characteristics of particles dynamics but also for momenta during the collisions of particles
inside the physical volume Vph at times τph. The importance of taking into account the kinetic
processes connected with irreversible collision processes at the scale of short-ranged interparticle
interactions was pointed out in [25], for instance. The short-wavelength collective modes in liquids
were investigated therein on the basis of a linearized kinetic equation of the revised Enskog theory
for the model of hard spheres.
In section 2 of this paper, a nonequilibrium statistical operator is derived within a consistent
description of kinetic and hydrodynamic processes for the system of classical interacting particles.
In section 3, on the basis of this operator, the set of transport equations for the nonequilibrium one-
particle distribution function and the potential part of the averaged enthalpy density is obtained in
the case of weakly nonequilibrium processes. In the fourth section, the inner structure of generalized
transport kernels of a consistent description of kinetic and hydrodynamic processes is analyzed. It
is shown that they can be expressed in terms of time correlation functions corresponding to the
basic set of dynamical variables, phase density of microscopic particle distribution n̂(x), potential
part of the enthalpy density ĥint
~k
(~r) and generalized transport kernels describing diffusive, viscous
and heat-conduction processes.
2. Nonequilibrium statistical operator for a consistent description of ki-
netics and hydrodynamics of system
In dense gases and liquids there is no small parameter and the characteristic time of interpar-
ticle correlations is comparable with those for the one-particle distribution function. This means
that during the particle collision process the many-particles correlations related to local mass,
momentum and energy conservation laws, underlying the hydrodynamic description of a system,
cannot be neglected. In this connection the local conservation laws impose some restrictions on the
kinetic processes. Their role is especially important at high densities, when the interaction between
a separate group of particles and other ones cannot be neglected. This indicates a close connection
between the kinetic and hydrodynamic processes in dense gases and liquids [3, 4]. Therefore, to
obtain the kinetic equations for these systems it is quite natural to choose the reduced description
of nonequilibrium states and in such a way to automatically take into account a proper dynam-
ics of conserved quantities. For this purpose we can include densities of hydrodynamic variables
together with f1(x; t) into the set of reduced-description parameters at the very beginning. The
phase functions
ρ̂(~r) = m
∫
d~pn̂1(x) = mn̂(~r), ~̂(~r) = m
∫
d~p~pn̂1(x),
ε̂(~r) =
∫
d~p
p2
2m
n̂1(x) +
1
2
∫
d~p
∫
d~p′
∫
d~r′Φ(|~r − ~r′|)n̂2(x, x′), (1)
correspond to the densities of hydrodynamic quantities ρ(~r; t), ~(~r; t) and ε(~r; t). Here
n̂1(x) =
N
∑
l=1
δ(x − xl) =
N
∑
l=1
δ(~p − ~pl)δ(~r − ~rl), n̂2(x, x′) =
N
∑
l6=j=1
δ(x − xl)δ(x
′ − xl), (2)
are the phase densities of the microscopic distribution of particles, xj = {~pj , ~rj} the phase variables
of j-particle, Φ(rlj) the interaction energy of two particles, ~pj the j-th particle momentum, m is
its mass, and rlj = |~rl − ~rj | denote the distance between a pair of interacting particles.
From expressions (1) we can see an important role of potential energy of interaction. Contrary
to ρ(~r; t) = 〈ρ̂(~r)〉t and ~(~r; t) = 〈~̂(~r)〉t, the nonequilibrium average ε(~r; t) = 〈ε̂(~r)〉t cannot
23005-2
On the problem of a consistent description
be expressed only via one-particle distribution function f1(x; t) = 〈n̂1(x)〉t. Indeed, as it follows
from (1), the potential part of energy εint(~r; t) = 〈ε̂int(~r)〉t is calculated by means of two-particle
distribution function f2(x, x′; t) = 〈n̂2(x, x′)〉t, where
ε̂int(~r) =
1
2
∫
d~p
∫
d~p′
∫
d~r′Φ(|~r − ~r′|)n̂2(x, x′) (3)
is the density of potential energy. Obviously, if f1(x; t) is chosen as one of a parameters of the
reduced description, then the average value of the interaction energy density can be taken as an
additional independent parameter. 〈. . .〉t =
∫
dΓN · · · %(xN ; t) denotes the nonequilibrium aver-
age calculated by means of nonequilibrium distribution function %(xN ; t). The latter satisfies the
Liouville equation
∂
∂t
%(xN ; t) + iLN%(xN ; t) = 0, (4)
where
iLN =
N
∑
l=1
~pl
m
· ∂
∂~rl
− 1
2
N
∑
l6=j=1
∂
∂~rl
Φ(rlj)
(
∂
∂~pl
− ∂
∂~pj
)
(5)
is the Liouville operator of the system. The function %(xN ; t) is symmetric with respect to inversion
of phase variables of any pair of particles (xl ↔ xj) and satisfies the normalization condition
∫
dΓN%(xN ; t) = 1, dΓ = (dx)N/N !, dx = d~pd~r.
In order to solve the Liouville equation (4) we will use the Zubarev nonequilibrium statistical
operator method [10, 11]. Within its framework we will be looking for solutions of the equation (4),
which are independent of the initial conditions. The solutions will depend on time only explicitly,
i.e. through the observable quantities selected for a reduced description of nonequilibrium states of
the system. When the basic parameters of a reduced description are chosen, the solution %(xN ; t)
can be presented using the nonequilibrium statistical operator method in the following general
form [10, 11]:
%(xN ; t) = %q(x
N ; t) −
t
∫
−∞
eε(t′−t)T (t, t′) (1 − Pq(t
′)) iLN%q(x
N ; t′)dt′, (6)
where T (t, t′) = exp+
{
−
∫ t
t′
(1 − Pq(t
′)) iLNdt′
}
is the evolution operator containing projection;
exp+ is the ordered exponential. Pq(t
′) is the generalized Kawasaki-Gunton projection operator
whose structure depends on the form of quasiequilibrium distribution function %q(x
N ; t). The latter
is determined according to Gibbs and corresponds to a maximum of the entropy of the system at
fixed parameters of the reduced description with taking into account the normalization condition. In
our case, the basic parameters are the nonequilibrium one-particle distribution function f1(x; t) and
the averaged density of the interaction energy of the system 〈ε̂int(~r)〉t. Then, the quasiequilibrium
distribution function %q(x
N ; t) providing a maximum for the entropy of the system at fixed averaged
values f1(x; t) and 〈ε̂int(~r)〉t can be obtained in a standard way. Such a function has the following
form [7, 9]:
%q(x
N ; t) = exp
{
−Φ(t) −
∫
d~rβ(~r; t)ε̂int(~r) −
∫
dxa(x; t)n̂1(x)
}
, (7)
where
Φ(t) = ln
∫
dΓN exp
{
−
∫
d~rβ(~r; t)ε̂int(~r) −
∫
dxa(x; t)n̂1(x)
}
(8)
23005-3
B.B. Markiv, I.P. Omelyan, M.V. Tokarchuk
has been determined from the normalization condition for distribution function
∫
dΓN%q(x
N ; t) =
1. β(~r; t) and a(x; t) are the Lagrange multipliers, which are defined from the self-consistency
conditions:
〈ε̂int(~r)〉t = 〈ε̂int(~r)〉tq , 〈n̂1(x)〉t = 〈n̂1(x)〉tq . (9)
In order to understand the physical sense of parameters β(~r; t) and a(x; t), let us transform the
function in the following way
%q(x
N ; t) = exp
{
−Φ(t) −
∫
d~rβ(~r; t)ε̂′(~r) −
∫
dxa′(x; t)n̂1(x)
}
, (10)
where
ε̂′(~r) = ε̂kin(~r) − ~v(~r; t) · ~̂(~r) +
m
2
v2(~r; t)n̂1(~r) + ε̂int(~r) (11)
is the total energy density in the reference frame moving together with a system element of the
mass velocity ~v(~r; t). Herewith, the parameter a′(x; t) is of the form
a′(x; t) = a(x; t) − β(~r; t)
{
p2
2m
− ~v(~r; t) · ~p +
m
2
v2(~r; t)
}
.
Consequently, the first self-consistency condition (9) transforms to
〈ε̂′(~r)〉t = 〈ε̂′(~r)〉tq . (12)
When the self-consistency conditions (9) and (12) are fulfilled, we obtain the relations
δΦ(t)
δβ(~r; t)
= −〈ε̂′(~r)〉t, δΦ(t)
δa′(x; t)
= −〈n̂1(x)〉tq = −〈n̂1(x)〉t = −f1(x; t). (13)
They mean that parameters β(~r; t) and a′(x; t) are conjugated to 〈ε̂′(~r)〉t and 〈n̂1(x)〉t, respectively.
Now let us determine the entropy of the system, taking into account the self-consistency conditions
(9) and (12):
S(t) = −〈ln %q(x
N ; t)〉tq = Φ(t) +
∫
d~rβ(~r; t)〈ε̂′(~r)〉t +
∫
dxa′(x; t)〈n̂1(x)〉t. (14)
Then, the thermodynamic relations can be obtained as
δS(t)
δ〈ε̂′(~r)〉t = β(~r; t),
δS(t)
δ〈n̂1(x)〉t = a′(x; t). (15)
From relations (15) we see that parameter β(~r; t) can be considered as an analogue of the local in-
verse temperature. Excluding a′(x; t) by means of self-consistency condition (9) in (10), one obtains
an explicit expression for the entropy of a nonequilibrium state when kinetics and hydrodynamics
are considered consistently [3, 7]:
S(t)=Φ(t) +
∫
d~rβ(~r; t)
{
〈ε̂(~r)〉t−~v(~r; t)〈~̂(~r)〉t +
m
2
v2(~r; t)〈n̂(~r)〉t
}
−
∫
dxf1(x; t) ln
f1(x; t)
u(~r; t)
, (16)
and
u(~r; t) =
∫
d~rN−1
(N − 1)!
exp
{
−Φ(t) −
∫
d~rβ(~r; t)ε̂′(~r)
}
N
∏
j=2
n(~rj ; t)
u(~rj ; t)
.
Explicit contributions of the averaged values of hydrodynamic variables 〈ε̂(~r)〉t, 〈~̂(~r)〉t, and 〈n̂(~r)〉t
and the contribution of kinetics in the form analogous to the Boltzmann entropy (the last term)
23005-4
On the problem of a consistent description
are present in the expression (16). However, if we formally omit the contribution of potential
energy, it is possible to show that u(~r; t) = e and the expression (16) explicitly transforms into
the Boltzmann entropy. For the quasiequilibrium statistical operator (10) the Kawasaki-Gunton
projection operator Pq(t) has the following form:
Pq(t)%
′(xN ; t) =
{
%q(x
N ; t) −
∫
d~r
δ%q(x
N ; t)
δ〈ε̂′(~r)〉t 〈ε̂′(~r)〉t −
∫
dx
δ%q(x
N ; t)
δ〈n̂1(x)〉t 〈n̂1(x)〉t
}
∫
dΓN%′(xN ; t)
+
∫
d~r
δ%q(x
N ; t)
δ〈ε̂′(~r)〉t
∫
dΓN ε̂′(~r)%′(xN ; t) +
∫
d~r
δ%q(x
N ; t)
δ〈n̂1(x)〉t
∫
dΓN n̂1(x)%′(xN ; t).
(17)
The projection operator Pq(t) acts on distribution functions only and possesses the following prop-
erties: Pq(t)Pq(t
′) = Pq(t), Pq(t)%q(t
′) = %q(t
′), Pq(t)%(t′) = %q(t
′).
The solution (6) is exact and corresponds to the idea of a reduced description of kinetic and
hydrodynamic processes of nonequilibrium states of the system. In [9] by means of %(xN ; t) (6) a set
of the coupled equations for a nonequilibrium one-particle distribution function and the averaged
density of the total energy was obtained. This set is strongly nonlinear. It takes into account
complicated kinetic and hydrodynamic processes and can be used for the description of both
strongly and weakly nonequilibrium states. Now we will show that in the case of nonequilibrium
sates which are close to equilibrium these equations can be significantly simplified.
3. Equations for a consistent description of kinetics and hydrodynamics in
weakly nonequilibrium states
Studying the nonequilibrium states of that system which is close enough to equilibrium it is
quite natural to suppose that the averaged value of energy 〈ε̂(~r)〉t, the nonequilibrium distribution
function f1(x; t) as well as the parameters β(~r; t) and b(x; t) = a(x; t)−β(~r; t)p2/2m slightly differ
from their corresponding equilibrium values. Then, we can expand the quasiequilibrium distribution
function (10)
%q(x
N ; t) = exp
{
−Φ(t) −
∫
d~rβ(~r; t)ε̂(~r) −
∫
dxb(x; t)n̂1(x)
}
into a series with respect to deviations of parameters δβ(~r; t) = β(~r; t)−β and δb(x; t) = b(x; t)−βµ,
and restrict the consideration to the linear approximation:
%q(x
N ; t) = %0(x
N )
{
1 −
∫
d~rδβ(~r; t)ε̂(~r) −
∫
dxδb(x; t)n̂1(x)
}
. (18)
Here, β = 1/kBT , T is the equilibrium value of temperature, %0(x
N ) = Z−1e−β(H−µN) is the
equilibrium distribution function, Z =
∫
dΓNe−β(H−µN) is the grand partition function, µ is the
equilibrium value of chemical potential. The values of parameters δb(x; t) and δβ(~r; t) are deter-
mined from the self-consistency conditions:
〈ε̂(~r)〉t = 〈ε̂(~r)〉tq , 〈n̂1(x)〉t = 〈n̂1(x)〉tq . (19)
Then, using the Fourier transformation for spatial coordinates and self-consistency equations (19),
we write down the distribution function (18) in the form [9]:
%q(x
N ; t) = %0(x
N )
1 +
∑
~k
′
〈ĥint
~k
〉tΦ−1
hh (~k)ĥint
~k
+
∑
~k
′
∫
d~p
∫
d~p ′〈n̂~k(~p)〉tΦ−1
~k
(~p, ~p ′)n̂~k(~p ′)
, (20)
where
∑
~k
′
=
∑
~k(~k 6=0),
~k is the wave-vector.
n̂~k(~p) =
∫
d~re−i~k~rn̂1(~r, ~p) (21)
23005-5
B.B. Markiv, I.P. Omelyan, M.V. Tokarchuk
is the Fourier-components of microscopic phase density of particles number,
ĥint
~k
= ε̂int
~k
− 〈ε̂int
~k
n̂−~k〉0S
−1
2 (k)n̂~k (22)
the Fourier-components of the potential part of the enthalpy density, ε̂int
~k
= 1
2
∑N
l6=j=1 Φ(|~rlj |)e−i~k~rj
and n̂~k =
∑N
l=1 e−i~k~rl are the Fourier-components of the potential energy and particle num-
ber densities, respectively. Φ−1
hh (~k) is the function inverse to the equilibrium correlation function
Φhh(~k) = 〈ĥint
~k
ĥint
−~k
〉0, 〈. . .〉0 =
∫
dΓN . . . %0(x
N ), 〈n̂~k(~p)〉0 = 0(~k 6= 0). Φ−1
~k
(~p, ~p ′) is the function
inverse to Φ~k(~p, ~p ′) = 〈n̂~k(~p)n̂−~k(~p ′)〉0 = nδ(~p − ~p ′)f0(p
′) + n2f0(p)f0(p
′)h2(~k). It is equal to
Φ−1
~k
(~p, ~p ′) =
δ(~p − ~p ′)
nf0(p′)
− c2(k), (23)
where n = N/V , f0(p) = (β/2πm)3/2 exp(−βp2/2m) is the Maxwellian distribution and c2(k) is
the direct correlation function, related to correlation function h2(k): h2(k) = c2(k)[1 − nc2(k)]−1.
S2(k) = 〈n̂~kn̂−~k〉0 is the static structure factor. It is important to note that dynamic variables
ĥint
~k
and n̂~k(~p) in the distribution (20) are orthogonal in the sense that 〈ĥint
~k
n̂~k(~p)〉0 = 0. In the
approximation (20), the nonequilibrium distribution function %(xN ; t) (6) has the following form:
%(xN ; t) = %0(x
N )
1 +
∑
~k
′
∫
d~p
∫
d~p ′〈n̂~k(~p)〉tΦ−1
~k
(~p, ~p ′)n̂−~k(~p ′) +
∑
~k
′
〈ĥint
~k
〉tΦ−1
hh (~k)ĥint
−~k
−
∑
~k
′
∫
d~p
∫
d~p ′
t
∫
−∞
eε(t′−t)T0(t, t
′)In(−~k; ~p)Φ−1
~k
(~p, ~p ′)〈n̂~k(~p ′)〉t′dt′
−
∑
~k
′
t
∫
−∞
eε(t′−t)T0(t, t
′)I int
h (−~k)Φ−1
hh (~k)〈ĥint
~k
〉t′dt′
, (24)
where
In(~k; ~p) = (1 − P0)iLN n̂~k(~p) = (1 − P0) ˙̂n~k(~p), I int
h (~k) = (1 − P0)iLN ĥint
~k
= (1 − P0)
˙̂
hint
~k
(25)
are the generalized flows in linear approximation and T0(t, t
′) = exp[(t − t′)(1 − P0)iLN ] the
evolution operator with regard to projection P0. Operator P0 is the linear approximation of the
Mori projection operator and is constructed on the orthogonal dynamic variables n̂~k(~p), ĥint
~k
[9]:
P0Â~k =
∑
~k
′
〈Â~kĥint
−~k
〉0Φ−1
hh (~k)ĥint
~k
+
∑
~k
′
∫
d~p
∫
d~p ′〈Â~kn̂−~k(~p)〉0Φ−1
~k
(~p, ~p ′)n̂~k(~p ′). (26)
It possesses the properties P0P0 = P0, P0(1 − P0) = 0, P0n̂~k(~p) = n̂~k(~p), P0ĥ
int
~k
= ĥint
~k
.
In view of its own structure, the nonequilibrium distribution function (24) is a functional of
the reduced-description parameters 〈n̂~k(~p)〉t, 〈ĥint
~k
〉t, dynamic variables n̂~k(~p), ĥint
~k
along with their
generalized flows (25). Using %(xN ; t) (24) for the parameters of reduced description f~k(~p; t) =
〈n̂~k(~p)〉t, hint
~k
(t) = 〈ĥint
~k
〉t we can obtain the following set of equations [9]:
∂
∂t
f~k(~p; t) +
i~k · ~p
m
f~k(~p; t) = − i~k · ~p
m
nf0(p)c2(k)
∫
d~p ′f~k(~p ′; t) + iΩnh(~k; ~p)hint
~k
(t)
−
∫
d~p ′
t
∫
−∞
eε(t′−t)ϕnn(~k; ~p, ~p ′; t, t′)f~k(~p ′; t′)dt′ −
t
∫
−∞
eε(t′−t)ϕnh(~k; ~p; t, t′)hint
~k
(t′)dt′, (27)
23005-6
On the problem of a consistent description
∂
∂t
hint
~k
(t) =
∫
d~p ′iΩhn(~k; ~p ′)f~k(~p ′; t) −
∫
d~p ′
t
∫
−∞
eε(t′−t)ϕhn(~k; ~p ′; t, t′)f~k(~p ′; t′)dt′
−
t
∫
−∞
eε(t′−t)ϕhh(~k; t, t′)hint
~k
(t′)dt′. (28)
Here, iΩnh(~k; ~p), iΩhn(~k; ~p) are the normalized static correlation functions
iΩnh(~k; ~p) = 〈 ˙̂n~k(~p)ĥint
−~k
〉0Φ−1
hh (~k), iΩhn(~k; ~p) =
∫
d~p ′〈 ˙̂hint
~k
n̂−~k(~p ′)〉0Φ−1
~k
(~p ′, ~p) (29)
and
ϕnn(~k; ~p, ~p ′; t, t′) =
∫
d~p ′′〈In(~k; ~p)T0(t, t
′)In(−~k; ~p ′′)〉0Φ−1
~k
(~p ′′, ~p ′),
ϕhn(~k; ~p; t, t′) =
∫
d~p ′〈I int
h (~k)T0(t, t
′)In(−~k; ~p ′)〉0Φ−1
~k
(~p ′, ~p),
ϕnh(~k; ~p; t, t′) = 〈In(~k; ~p)T0(t, t
′)I int
h (−~k)〉0Φ−1
hh (~k),
ϕhh(~k; ~p; t, t′) = 〈I int
h (~k)T0(t, t
′)I int
h (−~k)〉0Φ−1
hh (~k) (30)
are the generalized transport kernels (memory functions) describing kinetic and hydrodynamic
processes. The set of equations (27) and (28) is closed with respect to the parameters of reduced
description f~k(~p; t), hint
~k
(t). If in this set of equations one formally puts ĥint
~k
= 0, then we obtain
the kinetic equation for f~k(~p; t):
∂
∂t
f~k(~p; t) +
i~k · ~p
m
f~k(~p; t) = − i~k · ~p
m
nf0(p)c2(k)
∫
d~p ′f~k(~p ′; t)
−
∫
d~p ′
t
∫
−∞
eε(t′−t)ϕ′
nn(~k; ~p, ~p ′; t, t′)f~k(~p ′; t′)dt′. (31)
This is true when the contribution from the potential energy is considerably smaller than the
averaged kinetic energy (e.g. in the case of gases or weakly coupled liquids). The equation (31) was
obtained for the first time by means of the Mori projection operators method in [13–15]. Therein,
the basic parameter of the reduced description was a nonequilibrium one-particle distribution
function f~k(~p; t) which corresponds to the microscopic phase density n̂~k(~p) (the Klimontovich
function). In this case the memory function ϕ′
nn(~k; ~p, ~p ′; t, t′) has the following structure:
ϕ′
nn(~k; ~p, ~p ′; t, t′) =
∫
d~p ′′〈I0
n(~k, ~p)T ′
0(t, t
′)I0
n(−~k, ~p ′′)〉0Φ−1
~k
(~p ′′, ~p ′), (32)
where I0
n(~k, ~p) = (1−P ′
0)
˙̂n~k(~p) is the generalized flow, P ′
0 is the Mori operator, introduced in [13–15]
P ′
0Â~k′
=
∑
~k
′
∫
d~p
∫
d~p ′〈Â~k′
n̂−~k′
(~p ′)〉0Φ−1
~k
(~p ′, ~p)n̂~k(~p), (33)
and T ′
0(t, t
′) is the corresponding evolution operator with regard to projection. Based on the ki-
netic equation (31), the investigations of the dynamic structure factor, transverse and longitudinal
current time correlation functions, diffusion and viscosity coefficients for dense gases and liquids
where carried out [1, 2, 13–24, 26]. In particular, in Mazenko’s papers the linearized Boltzmann-
Enskog equation was obtained by means of expansion of the memory functions ϕ′
nn(~k; ~p, ~p ′; t, t′)
in density. For the case of weak coupling, the equation of Fokker-Planck type was obtained. How-
ever, the main drawback of kinetic equation (31) consists in its inconsistency with the total energy
conservation law, especially for dense gases and liquids, when the contribution of potential energy
23005-7
B.B. Markiv, I.P. Omelyan, M.V. Tokarchuk
into thermodynamic functions and transport coefficients is determinant. In [1] this drawback was
studied in detail. Therein, the investigations of the dynamic structure factor at intermediate values
of wave-vector ~k and frequency ω for simple liquids were carried out by using the Mori projec-
tion operators method for the reduced-description parameters n̂~k(~p), ε̂~k. Contrary to the transport
equations presented in [1], our set of equations (27) and (28) is constructed on the orthogonal
dynamic variables n̂~k(~p), ĥint
~k
. Therefore, “kinetic” and “hydrodynamic” contributions are sepa-
rated and correlation between them is described by the generalized memory functions (30). It is
important to reveal their inner structure.
4. Memory functions of a consistent description of kinetic and hydrody-
namic processes
In order to study the structure of the memory functions (30) let us look at the form of the
corresponding generalized flows (25) on which the memory functions are built. In particular, let us
take into consideration the fact that
˙̂n~k(~p) = iLN n̂~k(~p) = − i~k
m
· ~̂~k(~p) +
∂
∂~p
· ~F~k(~p), (34)
where
~̂~k(~p) =
N
∑
j=1
~pje
−i~k·~rj δ(~p − ~pj) (35)
is the momentum density in the (~k, ~p) space and
~F~k(~p) =
1
2
∑
j 6=l
∂
∂~r j
· Φ|~rj−~rl|δ(~p − ~pj)e
−i~k~rj . (36)
The action of the Mori projection operator P0 on ˙̂n~k(~p) can be presented as
P0
˙̂n~k(~p) = − β
m
~p · ~ΦFh(~k)ĥint
~k
f0(p) − i~k
m
· ~pn̂~k(~p), (37)
where
~ΦFh(~k) = 〈~F~kĥint
~k
〉0Φ−1
hh (~k). (38)
Taking into account (23) we write down the memory function ϕnn(~k; ~p, ~p ′; t, t′) in the following
form
ϕnn(~k; ~p, ~p ′; t, t′) =
∫
d~p ′′
〈
In(~k; ~p)T0(t, t
′)In(−~k; ~p ′′)
〉
0
{
δ(~p ′′ − ~p ′)
nf0(p′)
− c2(k)
}
=
〈
In(~k; ~p)T0(t, t
′)In(−~k; ~p ′)
〉
0
1
nf0(p′)
−
∫
d~p ′′
〈
In(~k; ~p)T0(t, t
′)In(−~k; ~p ′′)
〉
0
c2(k). (39)
The second term in the right-hand side of (39) is equal to zero, because
∫
d~pIn(~k; ~p) = 0. (40)
The transport kernel ϕnn(~k; ~p, ~p ′; t, t′) enters the kinetic equation (27) as the term
∫
ϕnn(~k; ~p, ~p ′; t, t′)f~k(~p ′; t)d~p ′. Taking into account (39) and (40) we can write the last one in
23005-8
On the problem of a consistent description
the form
∫
d~p ′ϕnn(~k; ~p, ~p ′; t, t′)f~k(~p ′; t) =
∫
d~p ′
{
ϕ̄(~k; ~p, ~p ′; t, t′) − ∂
∂~p
ϕFF (~k; ~p, ~p ′; t, t′)
×
(
β
mnf0(p′)
~p ′ − ∂
∂~p ′
)
}
f~k(~p ′; t′) − ϕ̄(2)
n (~k; ~p; t, t′) · 〈~̂~k〉
t, (41)
where the second term has the structure of a generalized Fokker-Planck operator containing the
generalized friction coefficient ϕFF (~k; ~p, ~p ′; t, t′) in the spatially-impulse space.
ϕ̄(~k; ~p, ~p ′; t, t′) =
~k
m
· ϕ(~k; ~p, ~p ′; t, t′) ·
~k
m
− i~k
m
· ϕF (~k; ~p, ~p ′; t, t′) · ∂
∂~p ′
+
∂
∂~p
· ϕF(~k; ~p, ~p ′; t, t′) · i~k
m
+ ϕ̄(1)
nn(~k; ~p, ~p ′; t, t′), (42)
ϕFF (~k; ~p, ~p ′; t, t′) = 〈~F~k(~p)T0(t, t
′)~F−~k(~p ′)〉0, ϕ(~k; ~p, ~p ′; t, t′) = 〈~̂~k(~p)T0(t, t
′)~̂−~k(~p ′)〉0, (43)
ϕF (~k; ~p, ~p ′; t, t′) = 〈~̂~k(~p)T0(t, t
′)~F−~k(~p ′)〉0, ϕF(~k; ~p, ~p ′; t, t′) = 〈~F~k(~p)T0(t, t
′)~̂−~k(~p ′)〉0,
moreover,
∫
d~p
∫
d~p ′ϕ(~k; ~p, ~p ′; t, t′) = D(~k; t, t′), (44)
is the generalized coefficient of diffusion of particles and ϕ(~k; ~p, ~p ′; t, t′) is the generalized diffusion
coefficient in momentum space.
∫
d~p
∫
d~p ′ϕFF (~k; ~p, ~p ′; t, t′) = ξ(~k; t, t′) (45)
is the generalized friction coefficient and ϕFF (~k; ~p, ~p ′; t, t′) is the generalized friction coefficient in
momentum space.
ϕ̄(1)
nn(~k; ~p, ~p ′; t, t′) =
β
m
~p · ~ΦFh(~k)f0(p)ϕh(~k; ~p ′; t, t′) · i~k
m
−
~k
m
· ~pϕn(~k; ~p, ~p ′; t, t′) ·
~k
m
β
m
~p · ~ΦFh(~k)f0(p)ϕhF (~k; ~p ′; t, t′) · ∂
∂~p ′
+
i~k
m
· ~pϕnF (~k; ~p, ~p ′; t, t′) · ∂
∂~p ′
−
~k
m
· ϕn(~k; ~p, ~p ′; t, t′)
~k
m
· ~p ′ − ∂
∂~p
· ~pϕFn(~k; ~p, ~p ′; t, t′)
i~k
m
· ~p ′
− β
m
~p · ~ΦFh(~k)f0(p)ϕhn(~k; ~p ′; t, t′)
i~k
m
· ~p ′ +
~k
m
· ~pϕ(0)
nn(~k; ~p ′; t, t′)
~k
m
· ~p ′, (46)
and contains the following time correlation functions
ϕ
(0)
nn(~k; ~p, ~p ′; t, t′) = 〈n̂~k(~p)T0(t, t
′)n̂−~k(~p ′)〉0, ϕhn(~k; ~p ′; t, t′) = 〈ĥint
~k
T0(t, t
′)n̂−~k(~p ′)〉0,
ϕh(~k; ~p ′; t, t′) = 〈ĥint
~k
T0(t, t
′)~̂−~k(~p ′)〉0, ϕhF (~k; ~p ′; t, t′) = 〈ĥint
~k
T0(t, t
′) ~̂F−~k(~p ′)〉0,
ϕn(~k; ~p, ~p ′; t, t′) = 〈n̂~k(~p)T0(t, t
′)~̂−~k(~p ′)〉0, ϕnF (~k; ~p, ~p ′; t, t′) = 〈n̂~k(~p)T0(t, t
′) ~̂F−~k(~p ′)〉0,
ϕFn(~k; ~p, ~p ′; t, t′) = 〈 ~̂F~k(~p)T0(t, t
′)n̂−~k(~p ′)〉0, ϕnh(~k; ~p ′; t, t′) = 〈n̂~k(~p ′)T0(t, t
′)ĥint
−~k
〉0, (47)
constructed on the basic set of dynamic variables n̂~k(~p), ĥint
~k
along with the Fourier-components of
the momentum density ~̂~k(~p) and the force ~F~k(~p) in momentum space. Moreover, n̂~k(~p), ~̂~k(~p) and
23005-9
B.B. Markiv, I.P. Omelyan, M.V. Tokarchuk
~F~k(~p) are connected by the equation of motion (34). In this respect we assume T0(t, t
′) = eiLN(t′−t).
Then (47) represents the time correlation functions whose evolution does not contain projection.
The time correlation functions (47) enter ϕ
(2)
n (~k; ~p; t, t′) as well:
ϕ(2)
n (~k; ~p; t, t′) =
i~k
m
· ~pϕnh(~k; ~p; t, t′)
β
mn
~ΦFh(~k) +
β
m
~p · ~ΦFh(~k)f0(p)ϕhh(~k; t, t′)
β
mn
~ΦFh(~k)
− i~k
m
· ϕh(~k; ~p; t, t′)
β
mn
~ΦFh(~k) +
∂
∂~p
· ϕFh(~k; ~p; t, t′)
β
mn
~ΦFh(~k), (48)
where
ϕh(~k; ~p; t, t′) = 〈~̂~k(~p)T0(t, t
′)ĥint
−~k
〉0 , ϕFh(~k; ~p; t, t′) = 〈~F~k(~p)T0(t, t
′)ĥint
−~k
〉0 ,
ϕhh(~k; t, t′) = 〈ĥint
~k
T0(t, t
′)ĥint
−~k
〉0 (49)
the time correlation functions of the Fourier-components of densities of the potential part of en-
thalpy, momentum ~̂~k(~p) and force ~F~k(~p) in impulse space.
In the kinetic equation (27) the transport kernel ϕnh(~k; ~p; t, t′) describes dynamic correlations
between the kinetic and hydrodynamic processes. Performing the action of operators (1−P0) and
iLN as well as taking into account (37) and
P0
˙̂
hint
~k
= ~Φ′
hF (~k) · ~̂~k , ~Φ′
hF (~k) = 〈ĥint
~k
~F−~k〉0
β
mn
, (50)
the kernel ϕnh(~k; ~p; t, t′) can be presented as
ϕnh(~k; ~p; t, t′) = − i~k
m
· ϕ̄ḣ(~k; ~p; t, t′) +
∂
∂~p
· ϕ̄Fḣ(~k; ~p; t, t′) +
β
m
~p · ~ΦFh(~k)f0(p)ϕ̄hḣ(~k; t, t′)
+
i~k
m
· ~pϕ̄nḣ(~k; ~p; t, t′) +
i~k
m
· ϕ(~k; ~p; t, t′) · ~Φ′
hF (~k) − ∂
∂~p
· ϕF(~k; ~p; t, t′)~Φ′
hF (~F )
− β
m
~p · ~ΦFh(~k)f0(p)ϕh(~k; t, t′) · ~Φ′
hF (~k) − i~k
m
· ~pϕn(~k; ~p; t, t′)~Φ′
hF (~k), (51)
where
ϕ̄ḣ(~k; ~p; t, t′) = 〈~̂~k(~p)T0(t, t
′)
˙̂
hint
−~k
〉0Φ−1
hh (~k), ϕ̄Fḣ(~k; ~p; t, t′) = 〈~F~k(~p)T0(t, t
′)
˙̂
hint
−~k
〉0Φ−1
hh (~k),
ϕ̄hḣ(~k; t, t′) = 〈ĥint
~k
(~p)T0(t, t
′)
˙̂
hint
−~k
〉0Φ−1
hh (~k),
ϕ̄nḣ(~k; t, t′) = −〈 ˙̂n~k(~p)T0(t, t
′)ĥint
−~k
〉0Φ−1
hh (~k) =
i~k
m
· ϕ̄h(~k; ~p; t, t′) − ∂
∂~p
· ϕ̄Fh(~k; ~p; t, t′), (52)
ϕ̄h(~k; ~p; t, t′) = ϕh(~k; ~p; t, t′)Φ−1
hh (~k), ϕ̄Fh(~k; ~p; t, t′) = ϕFh(~k; ~p; t, t′)Φ−1
hh (~k) (53)
are the normalized time correlation functions. The correlation functions ϕ(~k; ~p; t, t′), ϕF(~k; ~p; t, t′),
ϕh(~k; ~p; t, t′), and ϕn(~k; ~p; t, t′) have the structure similar to (47). From the structure of trans-
port kernels (41), (42), (46), (48) and (50) in the kinetic equation (27) for the nonequilibrium
one-particle distribution function one can see that the contributions of hydrodynamic processes
are described, besides ĥint
~k
, by the moments
∫
d~pf~k(~p; t) = n~k(t) = 〈n̂~k〉t,
∫
d~p~pf~k(~p; t) = 〈~̂~k〉t. As
in the case of equation (27), let us find the inner structure of transport kernels in equation (28)
for the average value of the potential part of enthalpy. In particular, taking into account (50), for
ϕhh(~k; t, t′) we obtain:
ϕhh(~k; t, t′) = ϕ̄
(0)
ḣḣ
(~k; t, t′) − ~Φ′
hF (~k) · ϕ̄(0)
ḣ
(~k; t, t′)
− ϕ̄ḣ(
~k; t, t′) · ~Φ′
hF (~k) + ~Φ′
hF (~k) · ϕ(0)
(~k; t, t′) · ~ΦFh(~k). (54)
23005-10
On the problem of a consistent description
ϕ̄
(0)
ḣḣ
(~k; t, t′) = 〈 ˙̂hint
~k
T0(t, t
′)
˙̂
hint
−~k
〉0Φ−1
hh (~k), ϕ̄
(0)
ḣ
(~k; t, t′) = 〈~̂int
~k
T0(t, t
′)
˙̂
hint
~k
〉0Φ−1
hh (~k), (55)
ϕ̄ḣ(
~k; t, t′) = Φ−1
hh (~k)〈 ˙̂hint
~k
T0(t, t
′)~̂~k〉0 , ϕ(0)
(~k; t, t′) = 〈~̂~kT0(t, t
′)~̂~k〉0 = D(~k; t, t′) (56)
is the generalized diffusion coefficient (44). Taking into account (37), (51) the transport kernel
ϕhn(~k; ~p; t, t′) entering (28) as
∫
d~p ′ϕhn(~k; ~p ′; t, t′)f~k(~p ′; t′) can be presented in the following way
∫
d~p ′ϕhn(~k; ~p ′; t, t′)f~k(~p ′; t′) =
i~k
m
·
∫
d~p ′Whn(~k; ~p ′; t, t′)
1
nf0(p′)
f~k(~p ′; t)
−
∫
d~p ′WhF (~k; ~p ′; t, t′)
1
nf0(p′)
·
(
β
~p ′
mn
− ∂
∂~p ′
)
f~k(~p ′; t′) + Whj(~k; t, t′)
β
mn
· 〈~̂~k〉
t′ , (57)
where
Whn(~k; ~p ′; t, t′) = ϕḣ(
~k; ~p ′; t, t′) − ϕḣn(~k; ~p ′; t, t′) · ~p ′
− ~Φ′
hF (~k)ϕ(~k; ~p ′; t, t′) + ~Φ′
hF (~k)ϕn(~k; ~p ′; t, t′) · ~p ′,
WhF (~k; ~p ′; t, t′) = ϕḣF (~k; ~p ′; t, t′) + ~Φ′
hF (~k)ϕF (~k; ~p ′; t, t′),
Wh(~k; t, t′) = ϕḣh(~k; t, t′)~Φ′
hF (~k) − ~Φ′
hF (~k)ϕh(~k; t, t′)~Φ′
hF (~k)
are the transport kernels, formed by the time correlation functions of the type of (52), (53) and
(55). Taking into account the structure of the memory functions (41), (51), (54) and (57), we
present the set of equations (27), (28) in the form
∂
∂t
f~k(~p; t) +
i~k
m
· ~pf~k(~p; t) = − i~k
m
· ~pnf0(p)c2(~k)
∫
d~p ′f~k(~p ′; t) + iΩnh(~k; ~p)hint
~k
(t)
−
∫
d~p ′
t
∫
−∞
eε(t′−t)
{
ϕ(~k; ~p, ~p ′; t, t′)
− ∂
∂~p
· ϕFF (~k; ~p, ~p ′; t, t′) ·
(
β~p ′
mnf0(p′)
− ∂
∂~p ′
)
}
f~k(~p ′; t′)dt′
+
t
∫
−∞
eε(t′−t)ϕ(2)
n (~k; ~p, ; t, t′) · ~~k(t)dt′ +
t
∫
−∞
eε(t′−t)
{
i~k
m
· Wnḣ(~k; ~p; t, t′) − ∂
∂~p
· WFh(~k; ~p; t, t′)
− β
m
f0(p)~p · WFḣ(~k; ~p; t, t′)
}
hint
~k
(t′)dt′, (58)
∂
∂t
hint
~k
(t) =
∫
d~p ′iΩhn(~k; ~p ′)f~k(~p ′; t) −
t
∫
−∞
eε(t′−t)
{
ϕ̄
(0)
ḣḣ
(~k; t, t′) − ~Φ′
hF (~k) · ϕ̄(0)
h (~k; t, t′)
− ϕ̄h(~k; t, t′) · ~Φ′
hF (~k) + ~Φ′
hF (~k) · ϕ̄(0)
(~k; t, t′) · ~ΦFh(~k)
}
hint
~k
(t′)dt′
− i~k
m
·
∫
d~p ′
t
∫
−∞
eε(t′−t)Whn(~k; ~p ′; t, t′)
1
nf0(p′)
f~k(~p ′; t′)dt′
+
∫
d~p ′
t
∫
−∞
eε(t′−t)WhF (~k; ~p ′; t, t′)
1
nf0(p′)
·
(
β
mn
~p ′ − ∂
∂~p ′
)
f~k(~p ′; t′)dt′
−
t
∫
−∞
eε(t′−t)Wh(~k; t, t′)
β
mn
· ~~k(t′)dt′, (59)
23005-11
B.B. Markiv, I.P. Omelyan, M.V. Tokarchuk
where in the first equation the transport kernels have the following structure:
Wnḣ(~k; ~p; t, t′) = ϕ̄ḣ(~k; ~p; t, t′) − ~p · ϕ̄nḣ(~k; ~p; t, t′)
− ϕ(~k; ~p; t, t′) · ~Φ′
hF (~k) + ~p · ϕn(~k; ~p; t, t′) · ~Φ′
hF (~k),
WFh(~k; ~p; t, t′) = ϕ̄Fh(~k; ~p; t, t′) − ϕF(~k; ~p; t, t′) · ~Φ′
hF (~k),
WFḣ(~k; ~p; t, t′) = ~ΦFh(~k) · ϕhḣ(~k; t, t′) − ~ΦFh(~k) · ϕh(~k; t, t′) · ~Φ′
hF (~k).
The transport equations (58), (59) are functionally connected concerning the basic parameters of
reduced description f~k(~p; t), hint
~k
(t). However, the equations contain the average values of densities
of particles number 〈n̂~k〉t and momentum 〈~̂~k〉t, which, generally speaking, are the hydrodynamic
variables. Integrating the equation (58) over momentum, we obtain the equation for n~k(t)
∂
∂t
n~k(t) +
i~k
m
~~k(t) = 0, (60)
that represents the conservation law for an average value of number of particles. The transport
equations (58), (59) for f~k(~p; t) and hint
~k
(t) are written in {~k, ~p, t} space. According to (42), (47),
(48), f~k(~p; t) and hint
~k
(t) are determined in terms of the time correlation functions of the basic set
n̂~k(~p), ĥint
~k
(47), (48). This means that the system of equation (58), (59) should be complemented
by the equations for the appropriate time correlation functions of dynamic variables n̂~k(~p), ~̂~k(~p),
~F~k(~p) and ĥint
~k
.
In [9], on the basis of the equations (58) and (59) there has been obtained the following set of
transport equations for the time correlation functions:
Φnn(~k; ~p, ~p ′; t) = 〈n̂~k(~p; t)n̂−~k(~p ′; 0)〉0 , Φint
hn(~k; ~p ′; t) = 〈ĥint
~k
(t)n̂−~k(~p ′; 0)〉0 ,
Φint
nh(~k; ~p; t) = 〈n̂~k(~p; t)ĥint
−~k
(0)〉0 , Φint,int
hh (~k; t) = 〈ĥint
~k
(t)ĥint
−~k
(0)〉0 . (61)
Using the Laplace transform A(z) = i
∫∞
0 dteiztA(t), let us represent it in a matrix form,
zΦ̃(~k; z) = Σ̃(~k; z)Φ̃(~k; z) − Φ̃(~k), (62)
where n̂~k(~p; t) = e−iLN tn̂~k(~p; 0), ĥint
~k
(t) = e−iLNtĥint
~k
(0), and Φ̃(~k; z) is the matrix whose elements
are the Laplace transforms of the time correlation functions (61).
Φ̃(~k) =
(
Φnn(~k; ~p, ~p ′) 0
0 Φint,int
hh (~k)
)
, Σ̃(~k; z) =
(
∫
d~p ′′Σnn(~k; ~p, ~p ′′; z) Σnh(~k; ~p; z)
∫
d~p ′′Σhn(~k; ~p ′′; z) −ϕhh(~k; z)
)
, (63)
where
Σnn(~k; ~p, ~p ′′; z) = iΩnn(~k; ~p, ~p ′′) − ϕnn(~k; ~p, ~p ′′; z), (64)
Σnh(~k; ~p; z) = iΩnh(~k; ~p) − ϕnh(~k; ~p; z), Σhn(~k; ~p ′′; z) = iΩhn(~k; ~p ′′) − ϕhn(~k; ~p ′′; z).
Taking into account the structure of memory functions (41), (51), (54) and (57), we present
the set of equations (62) in the explicit form:
zΦnn(~k; ~p, ~p ′; z) +
i~k · ~p
m
Φnn(~k; ~p, ~p ′; z) =
− i~k · ~p
m
nf0(p)c2(k)Φnn(~k; ~p, ~p ′; z) − β
m
~p · ~ΦFh(~k)f0(p)Φint
hn(~k; ~p ′; z)
−
∫
d~p ′′
{
ϕ̃(~k; ~p, ~p ′′; z) − ∂
∂~p
ϕFF (~k; ~p, ~p ′′; z)
(
β~p ′′
mnf0(p′′)
− ∂
∂~p ′′
)
}
Φnn(~k; ~p ′′, ~p ′; z)
− ϕ̃(2)
n (~k; ~p; z)Φn(~k; ~p ′; z) − ϕnh(~k; ~p; z)Φint
hn(~k; ~p ′; z) − Φnn(~k; ~p, ~p ′), (65)
23005-12
On the problem of a consistent description
zΦint
nh(~k; ~p; z) +
i~k · ~p
m
Φint
nh(~k; ~p; z) =
− i~k · ~p
m
nf0(p)c2(k)Φint
nh(~k; ~p; z) − β
m
~p · ~ΦFh(~k)f0(p)Φint,int
hh (~k; z)
−
∫
d~p ′′
{
ϕ̃(~k; ~p, ~p ′′; z) − ∂
∂~p
ϕFF (~k; ~p, ~p ′′; z)
(
β~p ′′
mnf0(p′′)
− ∂
∂~p ′′
)
}
Φint
nh(~k; ~p ′′; z)
− ϕ̃(2)
n (~k; ~p; z)Φint
h (~k; z)− ϕnh(~k; ~p; z)Φint,int
hh (~k; z), (66)
zΦint
hn(~k; ~p ′; z) = −
∫
d~p ′′
{ i~k
m
· Whn(~k; ~p ′′; z)
1
nf0(p′′)
− WhF (~k; ~p ′′; z)
1
nf0(p′′)
(
β~p ′′
mn
− ∂
∂~p ′′
)
}
Φnn(~k; ~p ′′, ~p ′; z)
− Wh(~k; z)
β
mn
Φn(~k; ~p ′; z) − ϕhh(~k; z)Φint
hn(~k; ~p ′; z), (67)
zΦint,int
hh (~k; z) = −
∫
d~p ′′
{ i~k
m
· Whn(~k; ~p ′′; z)
1
nf0(p′′)
− WhF (~k; ~p ′′; z)
1
nf0(p′′)
(
β~p ′′
mn
− ∂
∂~p ′′
)
}
Φint
nh(~k; ~p ′; z)
− Wh(~k; z)
β
mn
Φint
h (~k; z) − ϕhh(~k; z)Φint,int
hh (~k; z) − Φint,int
hh (~k). (68)
The extended set of equations (58), (59) and (65)–(68) at fixed values of normalized correla-
tion functions (29) and transport kernels (55) describing diffusive, viscous and heat-conduction
processes (ϕ
(0)
ḣḣ
is the potential part of the generalized heat-conductivity coefficient) can be solved,
in principle, in {~k, ~p, t} space with the help of numerical methods. Obviously, such solutions could
provide interesting and important information, in particular, on the behavior of the time corre-
lation function 〈n̂~k(~p; t)n̂~k(~p ′; 0)〉0, on momentum. By integrating over momentum variables, this
function can be connected with the time correlation function 〈n̂~k(t)n̂~k(0)〉0 and, thus, with the
dynamic structure factor of the system S(~k; ω).
Projecting the set of equations (58), (59) onto the first moments of the nonequilibrium one-
particle distribution function Ψ1(~p) = 1, Ψα(~p) =
√
2pα/2kBT , Ψε(~p) =
√
2/3(p2/2mkBT − 3/2)
(α = x, y, z) one can obtain the set of equations [9] for the averaged values of densities of particles
number n~k(t), momentum ~~k(t), kinetic hkin
~k
(t) and potential hint
~k
(t) parts of enthalpy. Similarly,
projecting the set of equations (65)–(68) onto the same moments [9], we obtain a set of equations
of the type of (62) for the appropriate time correlation functions
Φ̃(~k; z) =
Φnn Φn Φkin
nh Φint
nh
Φn Φ Φkin
h Φint
h
Φkin
hn Φkin
h Φkin,kin
hh Φkin,int
hh
Φint
hn Φint
h Φint,kin
hh Φint,int
hh
(~k;z)
(69)
with the matrix of memory kernels Σ̃G(~k; z).
Σ̃G(~k; z) = iΩ̃G(~k) + Π̃(~k; z), (70)
where
iΩ̃(~k) =
0 iΩn 0 0
iΩn 0 iΩkin
h iΩint
h
0 iΩkin
h 0 0
0 iΩint
h 0 0
(~k)
(71)
23005-13
B.B. Markiv, I.P. Omelyan, M.V. Tokarchuk
is the frequency matrix, ĥkin
~k
= ε̂kin
~k
− 〈ε̂kin
~k
n̂−~k〉0〈n̂~kn̂−~k〉
−1
0 n̂~k are the Fourier-components of the
kinetic part of enthalpy density.
Π̃(~k; z) =
0 0 0 0
0 Π Πkin
h Πint
h
0 Πkin
h Πkin,kin
hh Πkin,int
hh
0 Πint
h Πint,kin
hh Πint,int
hh
(~k;z)
(72)
is the matrix of transport kernels. Its elements have the following structure:
Πµν(~k; z) = 〈Ψµ|ϕ̃(~k; z) + Σ̃(~k; z)Q[zĨQΣ̃(~k; z)Q]−1QΣ̃(~k; z)|Ψν〉, (73)
where Q = 1 − P . P is the projection operator constructed on the eigenfunctions |Ψα(~p)〉 of the
nonequilibrium one-particle function and P〈Ψ| =
∑n
ν=1〈Ψ|Ψν〉〈Ψν |. 〈Ψ|Ψν〉 =
∫
d~ξΨ(~ξ)f0(ξ)Ψν(~ξ),
while Ψν(ξ) satisfy the conditions 〈Ψµ|Ψν〉 = δµν ,
∑
ν |Ψν〉〈Ψν | = 1. As we can see from the struc-
ture of elements of the matrixes iΩ̃G(~k) and Π̃( ~k; z), the contributions of kinetic and potential parts
of enthalpy are separated. However, all the transport kernels of Π̃( ~k; z) are determined in terms
of the time correlation functions (47), (49) and the transport kernels ϕ̄
(0)
ḣḣ
(~k; t, t′), ϕ̄
(0)
ḣ
(~k; t, t′),
ϕ̄
(0)
ḣ
(~k; t, t′) and D(~k; t, t′) (55). Herewith, a question arises regarding the study of time corre-
lation functions (69) and collective modes for liquids using the generalized collective modes ap-
proach [27, 28].
We note one more important feature of the system (27)–(28). Let us suppose particles interact
through the potential presented as
Φ(|~rij |) = Φhs(|~rij |) + Φl(|~rij |), (74)
where Φhs(|~rij |) is the hard sphere interaction potential, and Φl(|~rij |) is the long-range part of
the potential. Taking into account the features of the hard sphere model dynamics [4] and in-
vestigations [20, 23, 25], one can separate Enskog-Boltzmann collision integral from the function
ϕnn(~k; ~p, ~p ′; t, t′). Then the equation (27) can be written in the form
∂
∂t
f~k(~p; t) +
i~k · ~p
m
f~k(~p; t) = − i~k · ~p
m
nf0(~p)(c2(k) − g2(σ)c0
2(k))
∫
d~p ′f~k(~p ′; t)
− ng2(σ)σ2
∫
dΩσ
∫
d~p1
(~p − ~p1) · ~̂σ
m
Θ−(~̂σ · (~p − ~p1))
×
[
f0(p
∗
1)f~k(~p; t) − f0(p1)f~k(~p∗; t) + ei~k·~̂σσf0(p
∗
1)f~k(~p∗1; t) − ei~k·~̂σσf0(p)f~k(~p1; t)
]
+ iΩnh(~k; ~p)hint
~k
(t) −
∫
d~p ′
t
∫
−∞
dt′eε(t−t′)ϕl
nn(~k; ~p, ~p ′; t, t′)f~k(~p ′; t′)
−
t
∫
−∞
dt′eε(t−t′)ϕnh(~k; ~p; t, t′)hint
~k
(t′), (75)
where c0
2(
~k) is the low-density limit of the direct correlation function. σ is the hard-sphere diameter
and the pair distribution function g2(σ). The step function Θ−(x) is unity for x < 0 and vanishes
otherwise. dΩσ is the differential solid angle, ~̂σ is unity vector. The pre- and postcollision momenta
of the colliding hard spheres are denoted as (~p, ~p1) and (~p∗, ~p∗1), respectively. ϕl
nn(~k; ~p, ~p ′; t, t′) is
the part of transport kernel, related to long-range interaction potential Φl(|~rij |). That, is presented
equation contains the Enskog-Boltzmann collision integral describing the short-time dynamics of
the hard sphere model. And the collective effects connected with the long-range interactions be-
tween particles are described by the functions iΩnh(~k; ~p), ϕl
nn(~k; ~p, ~p ′; t, t′), ϕnh(~k; t, t′) and the
equation for hint
~k
(t). Since the collective modes for the Enskog-Boltzmann model are well stud-
ied [25], the investigation of time correlation functions and collective modes for the system of
particles interacting through the potential (74) turns out to be of great interest.
23005-14
On the problem of a consistent description
5. Conclusions
In this paper in order to consistently describe the kinetic and hydrodynamic processes in dense
gases and liquids, the generalized non-Markovian equations for a nonequilibrium one-particle dis-
tribution function and the averaged value of the potential part of the enthalpy density are obtained
using the nonequilibrium statistical operator method. The inner structure of generalized transport
kernels (memory functions) for these equations was analyzed in detail. It is shown that they are ex-
pressed in terms of time correlation functions related to the basic set of dynamical variables n̂~k(~p),
ĥint
~k
(47), (49) along with the transport kernels (55) including the generalized diffusion coeffici-
ent. Herewith, the collision integral of the kinetic equation (58) for f~k(~p; t) has the Fokker-Planck
structure with the generalized friction coefficient in momentum space. It also contains contribu-
tions from a generalized diffusion coefficient in momentum space as well as contributions from
the dissipative hydrodynamic processes connected with the potential part of the enthalpy density.
On the basis of these transport equations, a set of equations for the time correlation functions
corresponding to the basic set n̂~k(~p), ĥint
~k
is obtained. Projecting it onto the first three moments of
a nonequilibrium one-particle distribution function, a system of equations is obtained for the time
correlation functions of hydrodynamic variables n̂~k, ~̂~k, ĥkin
~k
and ĥint
~k
with separate contributions of
the kinetic and potential parts of enthalpy density. Such a system in the Markovian approximation
can present an interest from the point of view of investigating the dynamic structure factor, time
correlation functions Φkin,kin
hh (~k; t), Φkin,int
hh (~k; t), Φint,int
hh (~k; t) as well as collective modes for liquids
within the generalized collective modes approach [27, 28].
References
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Concepts, Kinetic Theory. Alademie Verlag, Berlin, 1996.
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23005-15
B.B. Markiv, I.P. Omelyan, M.V. Tokarchuk
До проблеми узгодженого опису кiнетичних та
гiдродинамiчних процесiв у густих газах та рiдинах
Б.Б. Маркiв, I.П. Омелян, М.В. Токарчук
Iнститут фiзики конденсованих систем НАН України,
79011 Львiв, вул. Свєнцiцького, 1
Для узгодженого опису кiнетичних та гiдродинамiчних процесiв у густих газах та рiдинах отримано
узагальненi немаркiвськi рiвняння для нерiвноважної одночастинкової функцiї розподiлу та серед-
нього значення густини потенцiальної частини ентальпiї. Розкрито внутрiшню структуру узагальне-
них ядер переносу даних рiвнянь. Показано, що iнтеграл зiткнення кiнетичного рiвняння має струк-
туру Фоккера-Планка з узагальненим коефiцiєнтом тертя в iмпульсному просторi. Вiн також мiстить
вклади вiд узагальненого коефiцiєнта дифузiї в просторi iмпульсiв i дисипативних процесiв, пов’яза-
них iз густиною потенцiальної частини ентальпiї.
Ключовi слова: кiнетика, гiдродинамiка, кiнетичнi рiвняння, функцiї пам’ятi, часовi кореляцiйнi
функцiї
23005-16
Introduction
Nonequilibrium statistical operator for a consistent description of kinetics and hydrodynamics of system
Equations for a consistent description of kinetics and hydrodynamics in weakly nonequilibrium states
Memory functions of a consistent description of kinetic and hydrodynamic processes
Conclusions
|
| id | nasplib_isofts_kiev_ua-123456789-32091 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T13:32:27Z |
| publishDate | 2010 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Markiv, B.B. Omelyan, I.P. Tokarchuk, M.V. 2012-04-08T15:39:03Z 2012-04-08T15:39:03Z 2010 On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids / B.B. Markiv, I.P. Omelyan, M.V. Tokarchuk // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23005: 1-16. — Бібліогр.: 28 назв. — англ. 1607-324X PACS: 05.20.Dd, 05.60.+w, 52.25.Fi, 82.20.M https://nasplib.isofts.kiev.ua/handle/123456789/32091 For a consistent description of kinetic and hydrodynamic processes in dense gases and liquids the generalized non-Markovian equations for the nonequilibrium one-particle distribution function and potential part of the averaged enthalpy density are obtained. The inner structure of the generalized transport kernels for these equations is established. It is shown that the collision integral of the kinetic equation has the Fokker-Planck form with the generalized friction coefficient in momentum space. It also contains contributions from the generalized diffusion coefficient and dissipative processes connected with the potential part of the enthalpy density. Для узгодженого опису кінетичних і гідродинамічних процесів у густих газах і рідинах одержано узагальнені немарківські рівняння для нерівноважної одночастинкової функції розподілу та середнього значення густини потенціальної частини ентальпії. Розкрито внутрішню структуру узагальнених ядер переносу даних рівнянь. Показано, що інтеграл зіткнення кінетичного рівняння має структуру Фоккера - Планка з узагальненим коефіцієнтом тертя в імпульсному просторі. Він також містить вклади від узагальненого коефіцієнта дифузії в просторі імпульсів і дисипативних процесів, пов'язаних із густиною потенціальної частини ентальпії. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids До проблеми узгодженого опису кінетичних та гідродинамічних процесів у густих газах та рідинах Article published earlier |
| spellingShingle | On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids Markiv, B.B. Omelyan, I.P. Tokarchuk, M.V. |
| title | On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids |
| title_alt | До проблеми узгодженого опису кінетичних та гідродинамічних процесів у густих газах та рідинах |
| title_full | On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids |
| title_fullStr | On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids |
| title_full_unstemmed | On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids |
| title_short | On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids |
| title_sort | on the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/32091 |
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