Method of reduced description in theory of long wave nonequilibrium fluctuations
The regular method for construction of kinetic equations of long-wave fluctuation theory is developed in a microscopic approach on the base of generalization of the kinetic Bogolyubov theory. The transition to the hydrodynamic theory of long-wave fluctuations is investigated in detail. The derived h...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Цитувати: | Method of reduced description in theory of long wave nonequilibrium fluctuations / S.V. Peletminskii, Yu.V. Slyusarenko // Вопросы атомной науки и техники. — 2001. — № 6. — С. 283-286. — Бібліогр.: 8 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859914060823265280 |
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| author | Peletminskii, S.V. Slyusarenko, Yu.V. |
| author_facet | Peletminskii, S.V. Slyusarenko, Yu.V. |
| citation_txt | Method of reduced description in theory of long wave nonequilibrium fluctuations / S.V. Peletminskii, Yu.V. Slyusarenko // Вопросы атомной науки и техники. — 2001. — № 6. — С. 283-286. — Бібліогр.: 8 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | The regular method for construction of kinetic equations of long-wave fluctuation theory is developed in a microscopic approach on the base of generalization of the kinetic Bogolyubov theory. The transition to the hydrodynamic theory of long-wave fluctuations is investigated in detail. The derived hydrodynamic equations describe a turbulent liquid state. The concept of nonequilibrium entropy for fluctuating systems is introduced. The H-theorem is proved.
|
| first_indexed | 2025-12-07T16:03:52Z |
| format | Article |
| fulltext |
METHOD OF REDUCED DESCRIPTION IN THEORY OF LONG WAVE
NONEQUILIBRIUM FLUCTUATIONS
S.V. Peletminskii, Yu.V. Slyusarenko
Institute for Theoretical Physics
National Science Center "Kharkov Institute of Physics and Technology", Kharkov, Ukraine
e-mails: mailto:spelet@kipt.kharkov.ua, slusarenko@kipt.kharkov.ua
The regular method for construction of kinetic equations of long-wave fluctuation theory is developed in a
microscopic approach on the base of generalization of the kinetic Bogolyubov theory. The transition to the
hydrodynamic theory of long-wave fluctuations is investigated in detail. The derived hydrodynamic equations
describe a turbulent liquid state. The concept of nonequilibrium entropy for fluctuating systems is introduced. The
H-theorem is proved.
PACS: 05.20.y, 05.20.Dd, 05.40.+j, 82.20.M
INTRODUCTION
It is well known [1-3] that for the evolution process
of a system at times t>>τr (a hydrodynamic evolution
stage; τr is the relaxation time) the states in which the
correlation radius of many-particles distribution
functions increases with time (long-wave fluctuations)
inevitably appear. In this connection it is of great
interest to construct a long fluctuation kinetic theory
which must underlie a long fluctuation hydrodynamic
theory just as the usual kinetic theory underlies
hydrodynamics.
A set of works (see, for example [4,5], etc.) deals
with the long fluctuation kinetic theory. In the works the
main attention was focused on the derivation of the
specific equations for a one-particle and a binary
correlation function in a certain approximation. In such
an approach the basic ideas of the kinetic Bogolyubov
theory [6] (the functional hypothesis; the boundary
conditions problem representing a formulation of the
functional hypothesis in zeroth-order of perturbation
theory; the principle of spatial weakening correlations)
used to be lost. Besides, the ideas did not seem to work
in the long fluctuation theory. One of the basic objects
of the present work is to show that the long fluctuation
kinetic theory not only agrees with the general ideas of
the kinetic Bogolyubov theory but needs the latter for its
specific applications.
1. METHOD OF REDUCED
DESCRIPTION OF LONG NON-
EQUILIBRIUM FLUCTUATIONS
In studying the kinetics of long-wave fluctuations it
is necessary to deal with systems having a large
correlation radius increasing with time so that the
assumption forming the basis for [6] about rapid decays
of correlations at |xi - xj|≥r0 (r0 is a particle interaction
radius) is not fulfilled. Therefore, for such systems the
formalism of [6] has to be modified in a certain way.
With the purpose of formulation of modified
functional hypothesis we introduce smoothed s-particle
distribution functions fS(x1,...,xS;t) which are obtained
from usual many-particle distribution functions
fS(x1,...,xS;t) in going to the asymptotic domain |xi - xj|≥
r0,
( ) ( ) ≡ →
> >−
t;x,...,xft;x,...,xf sSr||sS
ji
11
0xx
( )t;x,...,xPf sS 1≡ (1.1)
(here xi≡(xi,pi) is the coordinate of the phase point of the
i-th particle, i = 1,2,...,S; P is a symbol of the smoothing
operation).
We explain the concept of smoothing operation in
detail. If the initial many-particle distribution functions
have been smooth (on the scale of r0) functions of xi (i =
1,2,...,S), then on account of the temporal evolution the
functions fS(x1,...,xS;t) will have at |xi - xj|<r0, (i,j = 1,…
,S) a complex irregular character displaying irregular
properties on scale r0 of the potential energy of the S-
particle interaction. The trend of this irregular
dependence can be shown by way of example of the
function f(x) (x stands for |xi-xj|) having two spatial
scales of variation, f(x) = f(x/r0,x/L) (r0 is the
characteristic microscopic scale of the variation of f(x)
on small distances, L>>r0 is a characteristic macroscopic
scale of the variation on large distances). It is the
function which one deals with solving the BBGYK
equation chain (). Then the smoothing operation of the
function f(x) is defined by the formula
( ) ( )
∞≡=
L
x,fxPfxf , 12 =P (1.2)
According to the reduced description method of
Bogolyubov [6] we will consider that a system state is
completely described by smoothed many-particle
distribution functions at times t>>τ0 (τ0 ≈ r0/v is the
characterization time; v is the average particle velocity).
It means that the exact many-particle distribution
functions fS will be dependent on time and initial many-
particle distribution functions only by smoothed many-
particle distribution functions at t>>τ0,
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 283-286. 27
mailto:slusarenko@kipt.kharkov.ua
mailto:spelet@kipt.kharkov.ua
( ) →
τ> > 0
1 tSS t;x,...,xf
( ) ( )( ).,...tf,tf;x,...,xf SSt 211
0
→
τ> > (1.3)
Thus although the distribution functions fS(x1,...,xS;t)
do depend, generally speaking, on the initial many-
particle distribution functions fS(x1,...,xS;0), at times well
in excess of τ0 the dependence is simplified and
contained only functions f1(t), f2(t), ..., the functionals of
which become the quantities of fS.
In this sense the functionals (1.3) are universal and
independent of the pattern of initial conditions for many-
particle distribution functions.
The considered functionals (1.3) in accordance with
(1.1) have to satisfy the following relationship:
( ) ( )( ) →
> >− 0211 r||SS
ji
,...tf,tf;x,...,xf
xx
( ) ( )( ) ≡ →
> >−
,...tf,tf;x,...,xPf SSr|| ji 211
0xx
( )t;x,...,xf SS 1≡ (1.4)
We will study system dynamics having proceeded
from the BBGYK equation chain for the many-particle
distribution functions
10
1 SSS,SSSS
S fRfi
t
f
Λ+Λ=Λ+Λ−=
∂
∂
+ (1.5)
Here operators ΛS and RS are defined by formulae
{ } ∑
≤≤ ∂
∂
≡=Λ−
Si i
Si
SSSS ,
f
m
f,Hfi
1
00
x
p (1.6)
{ } ≡=Λ− SSSS f,Vfi 1
( )
,
V
jiSji i
ji
∂
∂−
∂
∂
∂
−∂
≡ ∑
≤<≤ ppx
xx
1
≡
= ∫ ∑ +
≤≤
+++ 1
1
111 S
Si
S,iSSS f,VdxfR
( )
∑ ∫
+≤≤
++
∂
∂
∂
−∂
≡
11
11
Si i
S
i
Si fV
px
xx
the operator RS transforms a function of a phase space of
S+1 particles in one of a phase space of s particle),
where HS
0=∑1≤i≤SH(xl), V=∑1≤i<j≤SV(xl-xj), H(xl)=pl
2/2m is
the free particle Hamiltonian and Vij=V(xl-xj) is the
Hamiltonian of pair interaction between particles (the
symbol {,} denotes Poisson's brackets).
It is easy to obtain an equation of motion for the
smoothed distribution functions fS from the equation
chain for many-particle distribution functions. For this
purpose< going in (1.5) to the asymptotic region |xi-xj|
>>r0 (i,j = 1,...,S) and allowing for
( ) ,V
||ji
ji
0 →−
∞→− xx
xx
we have accordingly (1.2)
,Lf
t SSSS LL ≡+=
∂
∂ 0
,SSSSSS fPRL,fi 1
00
+=Λ−=L (1.7)
(we took into account that P{VS,fS}=0).
A further problem will be to find a solution of the
equation chain (1.5) in the form fS=fS(x1,...,xS;f1,f2,...) not
researching the initial evolution stage. To obtain a
unique solution of the problem we formulate for the
functionals (1.3) boundary conditions which in accord
with (1.4), (1.6) take the form
( ) ( ) ( ) ( ) ( ) ,...)tfS,tfS;x,...,x(fS SSS 2
0
21
0
11
0 τ−τ−τ
( ).t;x,...,xf SS 1 →
∞→τ (1.8)
Here SS
0(τ)=exp(iτΛS
0).
To obtain the equation of motion for the parameters
fS(t) of reduced description it is necessary (see (1.7),
(1.8)) to find an apparent form of the functionals
fS(x1,...,xS;f1,f2,...) in a certain approximation, i.e. to
solve Eq. (1.5) with allowance for (1.3), (1.8). In such
cases the usual ways of calculation are iterations over a
weak interaction (for an arbitrary particle density) or a
low density of particles (for an arbitrary interaction
between particles provided that the interaction does not
lead to the production of bound states of particles).
2. GENERAL KINETIC EQUATION
OF LONG FLUCTUATION THEORY
Introduce the generating functional of smoothed
many-particle distribution functions
( ) += 1fu;F (2.1)
( ) ( ) ( ).x,...,xfxu...xudx...dx
!S SSS
SS 1
1
11
1∑ ∫ ∫
∞
=
+
A functional G(u;g) connected with the functional F(u;f)
by the relationship
( ) ( )]gu,Gexpgu,F [= (2.2)
is the generating functional of smoothed correlation
functions gS(x1,...,xS),
( ) =g;uG (2.3)
( ) ( ) ( ) ,x,...,xgxu...xudx...dx
!S SSS
SS 1
1
11
1∑ ∫ ∫
∞
=
=
where g1 ≡ f. Represent the generating functional G(u;g)
in the form
( ) ( ) ( ) ( )∫ += ,g;uxfxdxug;uG G (2.4)
where G(u;g) is the generating functional of the proper
correlation function
( ) =g;uG (2.5)
( ) ( ) ( ).x,...,xgxu...xudx...dx
!S SSS
SS 1
2
11
1∑ ∫ ∫
∞
=
=
The kinetic equation for the one-particle distribution
function f(x) and generating functional G(u;g) is to be
produced in the form
( ) ( ) ( ) ,f;xLg;
f
expxf
mt
xf
δ
δ=
∂
∂+
∂
∂ G
x
p
(2.6a)
( ) ( ) ( )
( )∫ =
δ
δ
∂
∂+
∂
∂
xu
g;u
m
xdxu
t
g;u GG
x
p
( )
δ
δ−
−
δ
δ+= g;
f
expg;ug;
f
uexp GGG
( ) ( ) ,f;xLxdxu∫× (2.6b)
28
where G(δ/δf;g) is generating functional G(u;g) in which
an operation of functional differentiation over f(x) is
substituted instead of a functional argument u(x). It is
the equations (2.6) which are general equations of the
theory of long nonequilibrium fluctuations. As is known,
a usual kinetic equation for the one-particle distribution
function takes the form
( ) ( ) ( ).f;xLxf
mt
xf =
∂
∂+
∂
∂
x
p
(2.7)
We see that the dynamics of long nonequilibrium
fluctuations is determined by the functional L(x;f) -
collision integral of Bogolyubov's kinetic theory. It
agrees with Onsager's principle according to which
macroscopic (long) fluctuations evolve in time with laws
of macroscopic physics; such a law of macroscopic
physics in case of kinetic theory is Eq. (2.7).
The second of Eqs. (2.6) admits the solution G=0.
With that, the equations becomes (2.7). However, the
solution G=0 corresponds to the very special initial
conditions gS|t=0=0 (S=2,3...). If at the initial moment the
correlation functions g2(x1,x2)=f(x1)f(x2)ξ (x1,x2),..., are
small in comparison with f(x1)f(x2) (|ξ (x1,x2)|<<1), then
the one-particle distribution function f(x) exponentially
decays up to times τ0=τrln[1/|ξ (x1,x2)|] (τr=l/v is
relaxation time, l is a mean free path) according to the
standard kinetic equation (2.7).The general kinetic
equations (2.6) are decisive at t≥τ0.
We note that the functional hypothesis in the form
(1.3)was decisive for the construction of the long
fluctuation theory. However, for the construction of the
usual kinetic theory, in which only the one-particle
distribution function is a parameter of the reduced
description, the functional hypothesis is formulated as
( ) ( )( ) ,t,'xf;x,...,xft;x,...,xf SStSS 11
0
→
τ> > (2.8)
where fS(x1,...,xS;f) are universal functionals of one-
particle distribution functions. In our opinion the
functional hypothesis (1.3) describes a general situation
corresponding to arbitrary initial conditions while the
hypothesis in the form (2.8) is not valid in the general
case and corresponds to very special initial conditions
which are described by the universal functionals
fS(x1,...,xS;f) where an arbitrary initial one-particle
distribution function f(x;0) figures as f(x):
( ) ( )( ).;'xf;x,...,xf;x,...,xf SSSS 00 11 =
It should be noted as well that stationary solutions of
Eqs. (2.6) for a statistical equilibrium state take the form
0320 ff,...,,S,g S === .
It means there are no long fluctuations in the statistical
equilibrium state. Short-wave fluctuations in the
equilibrium state are determined by the functionals
fS(x1,...,xS;f) in which the Maxwell distribution f0 has to
be substituted for a functional argument f(x). As is
shown in [8], many-particle distribution functions
obtained by such way completely coincide with the
many-particle distribution Gibbs' functions.
3. GENERAL HYDRODYNAMIC EQUATION
OF LONG FLUCTUATION THEORY
Smoothed hydrodynamic averages of products of the
additive motion integral,
( ) =ζ αα t;,..., S... S
xx11 (3.1)
( ) ( ) ( )∫ ∫ αα ζζ= ,t;x,...,xf...d...d SSSS S 111 1
pppp
will be denoted by ζa(t) (here ζα(p) (α=0,i,4;i=1,2,3) are
additive integral of motion; ζ0(p)=p2/2m is an energy, ζ
i(p)=pi is a momentum, ζ4(p)=m is a particle mass). It is
easy to see that the generating functional F(v; ζa) of the
smoothed hydrodynamic averages ζα1...αS(x1,...,xS) is
connected with the generating functional F(u;f) of the
smoothed many-particle distribution function fS by the
formula
( ) ( ) ( ) ( ) ==ζ
αα ζ= pvxua f;uF;vF (3.2)
( ) ( ) ×+ ∑ ∫ ∫
∞
=
αα
1
11 1
11
S
SS S
v...vd...d
!S
xxxx
( )t;,..., S... S
xx11 ααζ×
(it has extended the summation over repeated index
"α"). The functional G(v;ξa) connected with the
functional F(v; ζa) by the relationship
( ) ( )[ ]av;Gexp ξ=ς a;vF (3.3)
is the generating functional of the hydrodynamic
correlation functions ξα1...αS(x1,...,xS),
( ) ( ) ( ) ×=ξ ∑ ∫ ∫
∞
=
ααα
1
11 1
1
S
SS S
v...vd...d
!S
;vG xxxx
( )t;,..., S... S
xx11 ααξ× , (3.4)
which will be denoted by ξa(t) (ξα(x)≡ζα(x)=∫dpζα
(p)f1(x) are densities of additive motion integrals). The
generating functional G(u;g) of the proper correlation
function ξα1...αS(x1,...,xS;t), S≥2 similarly to (2.5) have the
form
( ) ( ) ( ) ×=ξ ∑ ∫ ∫
∞
=
ααα
2
11 1
1
S
SS S
v...vd...d
!S
;v xxxxG
( )t;,..., S... S
xx11 ααξ× , (3.5)
To obtain the closed equation of motion for the
generating functional F(v; ζa) of hydrodynamic averages
(or equations of motions for the densities of additive
motion integrals ζα(x,t) and generating functional G(u;g)
of the proper correlation function ξα1...αS(x1,...,xS;t)) it is
necessary to find a solution of Eqs.(2.6) in a
hydrodynamic approximation. With this purpose we
used the functional hypothesis
( ) ( )( ) ,t;x,...,xft;x,...,xf aSStSS r
ζ →
τ> > 11
or
( )( ) ( )( )( ),tf;uFtf;uF at r
ζ →
τ> > (3.6)
which has a simple physical sense: according to the
method of reduced description it is considered that at a
moment t>>τr (τr is a relaxation time) for the
hydrodynamic stage of evolution a system state is
completely described by densities ζα(x,t) of additive
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2000, №2.
Серия: Ядерно-физические исследования (36), с. 3-6.
motion integrals and smoothed correlation functions of
hydrodynamic parameters ξα1...αS(x1,...,xS;t)) at the same
moment.
For the hydrodynamic stage of evolution of the
system the specific sizes of spatial inhomogeneities
(over all spatial coordinates of correlation functions) are
large in comparison with a mean free particle path,
which develops the theory of perturbations over spatial
gradients of reduced description parameters.
Perturbation theory is developed to solve equations
for kinetics of long-wave fluctuations. This theory is
analogous to the Chapman-Enscog procedure which is
used to derive equations of usual (nonfluctuating)
hydrodynamics proceeding from usual kinetic equation.
As a result the following general equations for
fluctuation hydrodynamics can be derived
( ) ( ) ,;T;exp
t
t,
a ζ
ξ
δ ζ
δ=
∂
ζ∂
α
α x
x
G (3.7)
( ) ( )
−
ξ−
ξ
δ ζ
δ+=
∂
ξ∂
αα ;v;vexp
;v a GG
G
t
( ) ( )∫ ζ
ξ
δ ζ
δ− αα ,;Tvd;exp a xxxG (3.8)
where
( ) ( )( ) ( )( ) ( )
k
ik
ik
k
xxx
T
∂
ζ∂
ζη
∂
∂+
∂
ζζ∂
−≡ζ γ
α γ
α
α
x
x
x
x
(3.8)
In the expression (3.8) the current densities of
hydrodynamic parameters ζαk are determined by the
formula
( )( ) ( ) ( ) ,xf
m
p
d k
k ∫ αα ζ=ζζ 0ppx
where f0(x) is local-equilibrium Maxwell distribution
and the quantity ηαγ;ik(ζ(x)) determines dissipative
kinetic coefficients (see in connection with this, for
example, [7]).
We emphasize that at the fluctuation hydrodynamic
stage of evolution the system dynamics is determined by
the unique quantity Tα(x;t) (see (3.8)) describing the
usual (without fluctuations) hydrodynamics of a viscous
liquid
( ) ( ) ,;T
t
t,
ζ=
∂
ζ∂
α
α x
x
(3.9)
as well as that at fluctuation kinetic stage of evolution
the dynamics of long-wave fluctuations has been
determined by the unique quantity L(x;f) which is a
functional of Bogolyubov's theory (see (2.6), (2.7)).
In conclusion let us note the following. In case of
usual (without fluctuations) hydrodynamics an entropy
density s(x) is determined by the hydrodynamic
parameters ζα(x,t) (s(x)≡s(ζ(x,t))) and satisfy the
equation
( ) ( ) ( ) ,;I
x
;s
t
,s
k
k ζ+
∂
ζ∂
−=
∂
ζ∂ x
xx
(3.10)
where sk(x;ζ) is an entropy current density and I(x;ζ) is
an entropy production, and what is more I(x;ζ)≥0 since
the H-theorem is true in usual hydrodynamics (see for
instance [7]). It is shown [8], that at the fluctuation
hydrodynamic stage of evolution equation for entropy
has the form
( ) ( ) ( ) ,;I
x
;s
t
,s
a
k
aka ξ+
∂
ξ∂
−=
∂
ξ∂
x
xx
(3.11)
where
( ) ( ) ,,s;exp,s aa ζ
ξ
δ ζ
δ=ξ xx G (3.12)
( ) ( ) ,,s;exp,s kaak ζ
ξ
δ ζ
δ=ξ xx G
( ) ( ) ,,I;exp,I aa ζ
ξ
δ ζ
δ=ξ xx G
and what is more by force of positivity of I(x;ζ) the
entropy production I(x;ξ) is positive too, I(x;ξ)≥0, at
given fluctuations. Similarly the H-theorem is to be
proved for fluctuation kinetics.
ACKNOWLEDGMENT
The authors acknowledge INTAS for the partial
financial support (Project № 00-00577).
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30
METHOD OF REDUCED DESCRIPTION IN THEORY OF LONG WAVE NONEQUILIBRIUM FLUCTUATIONS
S.V. Peletminskii, Yu.V. Slyusarenko
INTRODUCTION
1. METHOD OF REDUCED DESCRIPTION OF LONG NON-EQUILIBRIUM FLUCTUATIONS
2. GENERAL KINETIC EQUATION
OF LONG FLUCTUATION THEORY
|
| id | nasplib_isofts_kiev_ua-123456789-80033 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T16:03:52Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Peletminskii, S.V. Slyusarenko, Yu.V. 2015-04-09T16:05:08Z 2015-04-09T16:05:08Z 2001 Method of reduced description in theory of long wave nonequilibrium fluctuations / S.V. Peletminskii, Yu.V. Slyusarenko // Вопросы атомной науки и техники. — 2001. — № 6. — С. 283-286. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 05.20.y, 05.20.Dd, 05.40.+j, 82.20.M https://nasplib.isofts.kiev.ua/handle/123456789/80033 The regular method for construction of kinetic equations of long-wave fluctuation theory is developed in a microscopic approach on the base of generalization of the kinetic Bogolyubov theory. The transition to the hydrodynamic theory of long-wave fluctuations is investigated in detail. The derived hydrodynamic equations describe a turbulent liquid state. The concept of nonequilibrium entropy for fluctuating systems is introduced. The H-theorem is proved. The authors acknowledge INTAS for the partial financial support (Project № 00-00577). en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Kinetic theory Method of reduced description in theory of long wave nonequilibrium fluctuations Метод сокращенного описания в теории длинноволновых неравновесных флуктуаций Article published earlier |
| spellingShingle | Method of reduced description in theory of long wave nonequilibrium fluctuations Peletminskii, S.V. Slyusarenko, Yu.V. Kinetic theory |
| title | Method of reduced description in theory of long wave nonequilibrium fluctuations |
| title_alt | Метод сокращенного описания в теории длинноволновых неравновесных флуктуаций |
| title_full | Method of reduced description in theory of long wave nonequilibrium fluctuations |
| title_fullStr | Method of reduced description in theory of long wave nonequilibrium fluctuations |
| title_full_unstemmed | Method of reduced description in theory of long wave nonequilibrium fluctuations |
| title_short | Method of reduced description in theory of long wave nonequilibrium fluctuations |
| title_sort | method of reduced description in theory of long wave nonequilibrium fluctuations |
| topic | Kinetic theory |
| topic_facet | Kinetic theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/80033 |
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