BBGKY chain of kinetic equations, non-equilibrium statistical operator method and collective variable method in the statistical theory of non-equilibrium liquids

BBGKY chain of kinetic equations, non-equilibrium statistical operator method and collective variable method in the statistical theory of non-equilibrium liquids

A chain of kinetic equations for non-equilibrium one-particle, two-particle and s-particle distribution functions of particles which take into account nonlinear hydrodynamic fluctuations is proposed. The method of Zubarev non-equilibrium statistical operator with projection is used. Nonlinear hydr...

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Published in:Condensed Matter Physics
Date:2016
Main Authors: Yukhnovskii, I.R., Hlushak, P.A., Tokarchuk, M.V.
Format: Article
Language:English
Published: Інститут фізики конденсованих систем НАН України 2016
Online Access:https://nasplib.isofts.kiev.ua/handle/123456789/156560
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Cite this:BBGKY chain of kinetic equations, non-equilibrium statistical operator method and collective variable method in the statistical theory of non-equilibrium liquids / I.R. Yukhnovskii, P.A. Hlushak, M.V. Tokarchuk // Condensed Matter Physics. — 2016. — Т. 19, № 4. — С. 43705: 1–18. — Бібліогр.: 51 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
id nasplib_isofts_kiev_ua-123456789-156560
record_format dspace
spelling Yukhnovskii, I.R.
Hlushak, P.A.
Tokarchuk, M.V.
2019-06-18T16:35:44Z
2019-06-18T16:35:44Z
2016
BBGKY chain of kinetic equations, non-equilibrium statistical operator method and collective variable method in the statistical theory of non-equilibrium liquids / I.R. Yukhnovskii, P.A. Hlushak, M.V. Tokarchuk // Condensed Matter Physics. — 2016. — Т. 19, № 4. — С. 43705: 1–18. — Бібліогр.: 51 назв. — англ.
1607-324X
PACS: 74.40.Gh, 05.70.L, 64.70.F
DOI:10.5488/CMP.19.43705
arXiv:1612.07219
https://nasplib.isofts.kiev.ua/handle/123456789/156560
A chain of kinetic equations for non-equilibrium one-particle, two-particle and s-particle distribution functions of particles which take into account nonlinear hydrodynamic fluctuations is proposed. The method of Zubarev non-equilibrium statistical operator with projection is used. Nonlinear hydrodynamic fluctuations are described with non-equilibrium distribution function of collective variables that satisfies generalized Fokker-Planck equation. On the basis of the method of collective variables, a scheme of calculation of non-equilibrium structural distribution function of collective variables and their hydrodynamic speeds (above Gaussian approximation) contained in the generalized Fokker-Planck equation for the non-equilibrium distribution function of collective variables is proposed. Contributions of short- and long-range interactions between particles are separated, so that the short-range interactions (for example, the model of hard spheres) are described in the coordinate space, while the long-range interactions — in the space of collective variables. Short-ranged component is regarded as basic, and corresponds to the BBGKY chain of equations for the model of hard spheres.
Запропоновано ланцюжок кiнетичних рiвнянь для нерiвноважних одночастинкової, двочастинкової i sчастинкової функцiй розподiлу частинок з урахуванням нелiнiйних гiдродинамiчних флуктуацiй. Використовується метод нерiвноважного статистичного оператора Зубарєва з проектуванням. Нелiнiйнi гiдродинамiчнi флуктуацiї описуються нерiвноважною функцiєю розподiлу колективних змiнних, що задовольняє узагальнене рiвняння Фоккера-Планка. На основi методу колективних змiнних запропоновано спосiб розрахунку нерiвноважної структурної функцiї розподiлу колективних змiнних та їх гiдродинамiчних швидкостей (вище гаусового наближення), що мiстяться в узагальненому рiвняннi Фоккера-Планка для нерiвноважної функцiї розподiлу колективних змiнних. При цьому роздiленi вклади вiд короткодiючих i далекодiючих взаємодiй мiж частинками, що привело до того, що короткодiючi взаємодiї (наприклад, модель твердих сфер) описуються в координатному просторi, а далекодiючi — у просторi колективних змiнних. Короткодiюча складова розглядається як базисна, якiй вiдповiдає ланцюжок рiвнянь ББГКI для моделi твердих сфер.
en
Інститут фізики конденсованих систем НАН України
Condensed Matter Physics
BBGKY chain of kinetic equations, non-equilibrium statistical operator method and collective variable method in the statistical theory of non-equilibrium liquids
Ланцюжок кiнетичних рiвнянь ББГКI, метод нерiвноважного статистичного оператора та метод колективних змiнних в нерiвноважнiй статистичнiй теорiї рiдин
Article
published earlier
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
title BBGKY chain of kinetic equations, non-equilibrium statistical operator method and collective variable method in the statistical theory of non-equilibrium liquids
spellingShingle BBGKY chain of kinetic equations, non-equilibrium statistical operator method and collective variable method in the statistical theory of non-equilibrium liquids
Yukhnovskii, I.R.
Hlushak, P.A.
Tokarchuk, M.V.
title_short BBGKY chain of kinetic equations, non-equilibrium statistical operator method and collective variable method in the statistical theory of non-equilibrium liquids
title_full BBGKY chain of kinetic equations, non-equilibrium statistical operator method and collective variable method in the statistical theory of non-equilibrium liquids
title_fullStr BBGKY chain of kinetic equations, non-equilibrium statistical operator method and collective variable method in the statistical theory of non-equilibrium liquids
title_full_unstemmed BBGKY chain of kinetic equations, non-equilibrium statistical operator method and collective variable method in the statistical theory of non-equilibrium liquids
title_sort bbgky chain of kinetic equations, non-equilibrium statistical operator method and collective variable method in the statistical theory of non-equilibrium liquids
author Yukhnovskii, I.R.
Hlushak, P.A.
Tokarchuk, M.V.
author_facet Yukhnovskii, I.R.
Hlushak, P.A.
Tokarchuk, M.V.
publishDate 2016
language English
container_title Condensed Matter Physics
publisher Інститут фізики конденсованих систем НАН України
format Article
title_alt Ланцюжок кiнетичних рiвнянь ББГКI, метод нерiвноважного статистичного оператора та метод колективних змiнних в нерiвноважнiй статистичнiй теорiї рiдин
description A chain of kinetic equations for non-equilibrium one-particle, two-particle and s-particle distribution functions of particles which take into account nonlinear hydrodynamic fluctuations is proposed. The method of Zubarev non-equilibrium statistical operator with projection is used. Nonlinear hydrodynamic fluctuations are described with non-equilibrium distribution function of collective variables that satisfies generalized Fokker-Planck equation. On the basis of the method of collective variables, a scheme of calculation of non-equilibrium structural distribution function of collective variables and their hydrodynamic speeds (above Gaussian approximation) contained in the generalized Fokker-Planck equation for the non-equilibrium distribution function of collective variables is proposed. Contributions of short- and long-range interactions between particles are separated, so that the short-range interactions (for example, the model of hard spheres) are described in the coordinate space, while the long-range interactions — in the space of collective variables. Short-ranged component is regarded as basic, and corresponds to the BBGKY chain of equations for the model of hard spheres. Запропоновано ланцюжок кiнетичних рiвнянь для нерiвноважних одночастинкової, двочастинкової i sчастинкової функцiй розподiлу частинок з урахуванням нелiнiйних гiдродинамiчних флуктуацiй. Використовується метод нерiвноважного статистичного оператора Зубарєва з проектуванням. Нелiнiйнi гiдродинамiчнi флуктуацiї описуються нерiвноважною функцiєю розподiлу колективних змiнних, що задовольняє узагальнене рiвняння Фоккера-Планка. На основi методу колективних змiнних запропоновано спосiб розрахунку нерiвноважної структурної функцiї розподiлу колективних змiнних та їх гiдродинамiчних швидкостей (вище гаусового наближення), що мiстяться в узагальненому рiвняннi Фоккера-Планка для нерiвноважної функцiї розподiлу колективних змiнних. При цьому роздiленi вклади вiд короткодiючих i далекодiючих взаємодiй мiж частинками, що привело до того, що короткодiючi взаємодiї (наприклад, модель твердих сфер) описуються в координатному просторi, а далекодiючi — у просторi колективних змiнних. Короткодiюча складова розглядається як базисна, якiй вiдповiдає ланцюжок рiвнянь ББГКI для моделi твердих сфер.
issn 1607-324X
url https://nasplib.isofts.kiev.ua/handle/123456789/156560
citation_txt BBGKY chain of kinetic equations, non-equilibrium statistical operator method and collective variable method in the statistical theory of non-equilibrium liquids / I.R. Yukhnovskii, P.A. Hlushak, M.V. Tokarchuk // Condensed Matter Physics. — 2016. — Т. 19, № 4. — С. 43705: 1–18. — Бібліогр.: 51 назв. — англ.
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AT tokarchukmv bbgkychainofkineticequationsnonequilibriumstatisticaloperatormethodandcollectivevariablemethodinthestatisticaltheoryofnonequilibriumliquids
AT yukhnovskiiir lancûžokkinetičnihrivnânʹbbgkimetodnerivnovažnogostatističnogooperatoratametodkolektivnihzminnihvnerivnovažniistatističniiteoriíridin
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first_indexed 2025-11-25T23:28:47Z
last_indexed 2025-11-25T23:28:47Z
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fulltext Condensed Matter Physics, 2016, Vol. 19, No 4, 43705: 1–18 DOI: 10.5488/CMP.19.43705 http://www.icmp.lviv.ua/journal BBGKY chain of kinetic equations, non-equilibrium statistical operator method and collective variable method in the statistical theory of non-equilibrium liquids I.R. Yukhnovskii, P.A. Hlushak, M.V. Tokarchuk Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii St., 79011 Lviv, Ukraine Received July 6, 2016, in final form November 1, 2016 A chain of kinetic equations for non-equilibrium one-particle, two-particle and s-particle distribution functions of particles which take into account nonlinear hydrodynamic fluctuations is proposed. The method of Zubarev non-equilibrium statistical operator with projection is used. Nonlinear hydrodynamic fluctuations are described with non-equilibrium distribution function of collective variables that satisfies generalized Fokker-Planck equa- tion. On the basis of the method of collective variables, a scheme of calculation of non-equilibrium structural distribution function of collective variables and their hydrodynamic speeds (above Gaussian approximation) contained in the generalized Fokker-Planck equation for the non-equilibrium distribution function of collective variables is proposed. Contributions of short- and long-range interactions between particles are separated, so that the short-range interactions (for example, the model of hard spheres) are described in the coordinate space, while the long-range interactions — in the space of collective variables. Short-ranged component is re- garded as basic, and corresponds to the BBGKY chain of equations for the model of hard spheres. Key words: non-linear fluctuations, non-equilibrium statistical operator, distribution function, Fokker–Planck equation, simple fluid PACS: 74.40.Gh, 05.70.L, 64.70.F 1. Introduction A development of equilibrium and non-equilibrium statistical mechanics of classical and quantum systems, which began in 1970-ies following the work by Bogoliubov [1] and works by Born, Green [2], Kirkwood [3, 4], Yvon [5] and continues today, has led to a substantial progress in the theory of gases, liquids, plasma. Bogoliubov in the book [1], which turns 70 years, in a strict form formulated the idea of a hierarchy of relaxation times in a system of many interacting particles and a reduced number of parameters describing the evolution of the system. This idea played a major role in the development of modern methods of non-equilibrium theory to study the dynamics of macroscopic systems at kinetic and hydrodynamic stages, based on the fundamental principles of statistical mechanics. Important for the development of this direction were the works by Zubarev [6–8], Zwanzig [9–11], Robertson [12, 13], Kawasaki and Gunton [14], Peletminskii and Yatsenko [15], Zubarev and Kalashnikov [16]. The results of the research done in this field are detailed in books [17–23] and reviews [24, 25]. However, along with the important results of theories in statistical physics, as well as in other fields of modern physics, there are still many unsolved problems, especially in the theory of non-equilibrium processes. In dense gases and fluids in the field of phase transitions, the heterophase fluctuations play an im- portant role [26–34]. They always appear and disappear during the diffusion processes. In the field of © I.R. Yukhnovskii, P.A. Hlushak, M.V. Tokarchuk, 2016 43705-1 http://dx.doi.org/10.5488/CMP.19.43705 http://www.icmp.lviv.ua/journal I.R. Yukhnovskii, P.A. Hlushak, M.V. Tokarchuk phase transitions, the heterophase fluctuations are factors forming a new phase, in particular, forming the bubbles in a liquid or drops in a gas. The non-equilibrium gas–liquid phase transition is characterized by nonlinear hydrodynamic fluc- tuations of mass, momentum and particle energy, which describe a collective nature of the process and define the spatial and temporal behavior of the transport coefficients (viscosity, thermal conductivity), time correlation functions and a dynamic structure factor. At the same time, due to heterogeneity in col- lective dynamics of these fluctuations, liquid drops emerge in the gas phase (in case of transition from the gas phase to the liquid phase), or the gas bubbles emerge in the liquid phase (in case of transition from the liquid phase to the gas phase), the formation of which is of a kinetic nature described by a redistribu- tion of momentum and energy, i.e., when a certain group of particles in the system receives a significant decrease (in the case of drops), or increase (in the case of bubbles) of kinetic energy. The particles, that form bubbles or droplets, diffuse out of their phases in the liquid or in the gas and vice versa. They have different values of momentum, energy and pressure in different phases. All these features are related to the non-equilibrium one-, two-, s-particle distribution functions (which depend on the coordinate, mo- mentum and time) that satisfy the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) chain of equations. They are the real heterophase systems inwhich bubble embryoses, drops or small crystals are of a kinetic nature caused by nonlinear fluctuations, changes in temperature, pressure. In our case, the heterophase formations (containing afinite particle number in this or that phase) can be described by non-equilibrium distribution function fs (xs ; t). The kinetic processes within heterophase formations are described respec- tively by the kinetic equations [35], in which the right-hand side contains the summands that take into ac- count the mutual effect of kinetic and hydrodynamic processes. Obviously, such heterophase formations form and disappear (with finite lifetime), exchanging by both the particles and energy with the surround- ing particles in the background of nonlinear hydrodynamic fluctuations of densities of particle number, momentum, energy; the contribution of such fluctuations grows at phase transformations. These nonlin- ear fluctuations are described by the Fokker-Planck equation [35]. Here, in the processes of interaction between kinetic and hydrodynamic fluctuations with appropriate changes of temperature and pressure in heterophase formations due to spontaneous symmetry breaking there can be self-organizing processes of sorts of particle motion with group velocity fs (xs ; t) = fs (r1 − vt ,p1, . . . ,rs − vt ,ps), which leads to an automodel (quasi-soliton) spreading of heterophase formations in the respective system. Such processes require a detailed separate study due to the calculation difficulties of kinetic and hy- drodynamic transport kernels in equations of transport. In this connection we would like to draw atten- tion to the Klimontovich article [36], in which to a certain extent there is realized a consistent description of kinetics and hydrodynamics (diffusion processes are taken into account) for gas-liquid phase transi- tion. Themain difficulty is that the kinetics and hydrodynamics of these processes are strongly interrelated and should be considered simultaneously. In articles [37–39] there is proposed a consistent description of kinetic and hydrodynamic processes in dense gases and liquids on the basis of Zubarev non-equilibrium statistical operator [19, 20]. This article is an extension or generalization of article [35]. Here, we want to emphasize certain as- pects. In [35], the main parameter for the description of kinetic processes was one-particle non-equilib- rium distribution function that corresponds to the gas description. In order to get a consistent description of kinetic and hydrodynamic processes (which is important for dense gases and liquids [37–39]) it is nec- essary to include the density of average potential energy as an important parameter. Having included this parameter, we can identify explicit contributions connected with short- and long-range interactions be- tween particles. For dilute gases, this parameter can be neglected, while for dense gases and liquids one makes an important contribution compared to kinetic energy. In this contribution, we will develop an approach [35] for a consistent description of kinetic and hydrodynamic processes that are characterized by non-linear fluctuations taking into account short-range and long-range interactions between particles. The approach is important for the description of non-equilibrium gas–liquid phase transition. In Section 2, we will obtain the non-equilibrium statistical operator for non-equilibrium state of the system when the parameters of a reduced description are represented by a non-equilibrium one-particle distribution function, the density of non-equilibrium average value of potential energy of particle in- teraction and a distribution function of non-equilibrium nonlinear hydrodynamic variables. Using this operator, we construct kinetic equations for non-equilibrium one-, two-, s-particle distribution functions 43705-2 BBGKY chain of kinetic equations, non-equilibrium statistical operator method which take into account nonlinear hydrodynamic fluctuations, for which the non-equilibrium distribu- tion function satisfies a generalized Fokker–Planck equation. In Section 3, we will consider a scheme for calculation of the structural distribution function of hydro- dynamic collective variables and their hydrodynamic velocities (in approximation higher than Gaussian), which enter the generalized Fokker–Planck equation for the non-equilibrium distribution function of hy- drodynamic collective variables. We separate the contributions from short-range and long-range inter- actions between particles, which will be described in the coordinate space and in the space of collective variables, respectively. Moreover, the short-range component will be considered in a simplified manner, which in our case will correspond to the BBGKY chain of equations for the model of hard spheres [40]. 2. Non-equilibrium distribution function and BBGKY chain of kinetic eq- uations in the Zubarev non-equilibrium statistical operator method For a consistent description of kinetic and hydrodynamic fluctuations in a classical one-component fluid, it is necessary to select the description parameters for one-particle and collective processes. For these parameters, we choose the non-equilibrium one-particle distribution function f1(x; t) = 〈n̂1(x)〉t , the density of non-equilibrium average value of potential energy of particle interaction H int(r, t) = 〈Ĥ int(r)〉t and distribution function of hydrodynamic variables f (a; t) = 〈δ(â−a)〉t . Here, the phase func- tion n̂1(x) = N ∑ j=1 δ(x − x j ) = N ∑ j=1 δ(r−r j )δ(p−p j ) (2.1) is the microscopic particle number density. x j = (r j ,p j ) is the set of phase variables (coordinates and momenta), N is the total number of particles in a volume V . Ĥ int(r)= 1 2 N ∑ j,l=1 Φ(|rl j |)δ(r−r j ) (2.2) is the microscopic density of the potential energy of particle interaction. Amicroscopic phase distribution of hydrodynamic variables is given by f̂ (a) = δ(â −a) = 3 ∏ l=1 ∏ k δ(âlk −alk), (2.3) where â1k = n̂k , â2k = Ĵk , â3k = ε̂k are the Fourier components of the densities of particle number, mo- mentum and energy: n̂k = N ∑ j=1 e−ikr j , Ĵk = N ∑ j=1 p j e−ikr j , ε̂k = N ∑ j=1 [ p2 j 2m + 1 2 N ∑ l, j=1 Φ(|rl j |) ] e−ikr j , (2.4) and amk = (nk,Jk,εk) are the corresponding collective variables. Φ(|rl j |) =Φ(|rl −r j |) is the pair interac- tion potential between particles, which we assume can be represented as the sum of short-range interac- tion Φ sh(|rl j |) and long-range interaction Φ long(|rl j |) potentials: Φ(|rl j |) =Φ sh(|rl j |)+Φ long(|rl j |). The introduction of 〈Ĥ int(r)〉t as a parameter is important step, as for liquids and dense gases it gives a larger contribution than the kinetic part of energy connected with 〈n̂1(x)〉t . In addition this parameter describes the collective dynamics of short- and long-range interactions: 〈Ĥ int(r)〉t = 〈Ĥ sh(r)〉t + 1 2V 2 ∑ q,k ν(k)eiqr ( 〈n̂q+kn̂−k〉 t −〈n̂q〉 t ) , 〈n̂q+kn̂−k〉 t = F (q,k; t) is the non-equilibrium scattering function which connected with non-equilibrium dynamic factor of system, ν(k) = ∫ dr e−ikr Φ long(|r|) is the Fourier component of long-range potential of 43705-3 I.R. Yukhnovskii, P.A. Hlushak, M.V. Tokarchuk particle interactions. The average values 〈n̂1(x)〉t , 〈Ĥ int(r)〉t and 〈δ(â − a)〉t are calculated by means of the non-equilibrium N -particle distribution function ̺(xN ; t), that satisfies the Liouville equation. In line with the idea of reduced description of non-equilibrium states this function is the functional ̺ ( xN ; t ) = ̺ ( . . . , f1(x; t),〈Ĥ int(r)〉t , f (a; t), . . . ) . (2.5) In order to find a non-equilibrium distribution function ̺(xN ; t) we use Zubarev’s method [19, 20, 41], in which a general solution of Liouville equation taking into account a projection procedure can be pre- sented in the form: ̺ ( xN ; t ) = ̺rel ( xN ; t ) − t ∫ −∞ dt ′eǫ(t ′−t )T (t , t ′) [ 1−Prel(t ′) ] iLN̺rel ( xN ; t ′ ) , (2.6) where ǫ→+0 after thermodynamic limiting transition. The source selects the retarded solutions of Liou- ville equationwith operator iLN . T (t , t ′)= exp+{− ∫t t ′ dt ′[1−Prel(t ′)]iLN } is the generalized time evolution operator taking into account Kawasaki-Gunton projection Prel(t ′). The structure of Prel(t ′) depends on the relevant distribution function ̺rel(xN ; t), which in method by Zubarev is determined from extremum of the information entropy at simultaneous conservation of normalization condition ∫ dΓN̺rel ( xN ; t ) = 1, dΓN = (dx)N h3N N ! = (dx1, . . . ,dxN ) h3N N ! , dx = drdp, (2.7) and the fact that the parameters of the reduced description, f1(x; t) and f (a; t) are fixed. Then, a relevant distribution function can be written as follows: ̺rel ( xN ; t ) = exp [ −Φ(t)− ∫ dxγ(x; t)n̂1(x)− ∫ drβ(r; t)Ĥ int(r)− ∫ daF (a; t) f̂ (a) ] , (2.8) where da is the integration over collective variables: da = ∏ k dnkdjkdεk , dnk = dRenk dImnk , dεk = dReεk dImεk , djk = d jx,k d jy,k d jz,k , d jx,k = dRe jx,k dIm jx,k , . . . . The Massieu-Planck functional Φ(t) is determined from the normalization condition for the relevant dis- tribution function Φ(t)= ln ∫ dΓN exp [ − ∫ dxγ(x; t)n̂1(x)− ∫ drβ(r; t)Ĥ int(r)− ∫ daF (a; t) f̂ (a) ] . The functions γ(x; t), β(r; t) and F (a; t) are the Lagrange multipliers and are determined from the self-consistent conditions: f1(x; t) = 〈n̂1(x)〉t = 〈n̂1(x)〉t rel , 〈Ĥ int(r)〉t = 〈Ĥ int(r)〉t rel , f (a; t) = 〈δ(â −a)〉t = 〈δ(â−a)〉t rel , (2.9) where 〈. . .〉t = ∫ dΓN . . .̺(xN ; t) and 〈. . .〉t rel = ∫ dΓN . . .̺rel(xN ; t). To find the explicit form of non-equi- librium distribution function ̺(xN ; t), we exclude the factor F (a; t) in a relevant distribution function and thereafter, by means of self-consistent conditions (2.9), we have ̺rel ( xN ; t ) = ̺ kin-hyd rel ( xN ; t ) f (a; t) W (a; t) ∣ ∣ ∣ ∣ a=â . (2.10) Here, W (a; t) = ∫ dΓN exp [ −Φkin-hyd(t)− ∫ dxγ(x; t)n̂1(x)− ∫ drβ(r; t)Ĥ int(r) ] f̂ (a) = ∫ dΓN̺ kin-hyd rel ( xN ; t ) f̂ (a) (2.11) 43705-4 BBGKY chain of kinetic equations, non-equilibrium statistical operator method is the structure distribution function of hydrodynamic variables, which could be also considered as a Jacobian for transition from f̂ (a) into space of collective variables nk, Jk, εk averaged with the “kinetic” relevant distribution function ̺ kin-hyd rel ( xN ; t ) = exp [ −Φkin-hyd(t)− ∫ dxγ(x; t)n̂1(x)− ∫ drβ(r; t)Ĥ int(r) ] , Φ kin-hyd(t) = ln ∫ dΓN exp [ − ∫ dxγ(x; t)n̂1(x)− ∫ drβ(r; t)Ĥ int(r) ] . (2.12) Here, the entropy S(t)=− 〈 ln̺rel ( xN ; t )〉t rel =Φ(t)+ ∫ dxγ(x; t)〈n̂1(x)〉t + ∫ drβ(r; t)〈Ĥ int(r)〉t + ∫ da f (a; t) ln f (a; t) W (a; t) . (2.13) corresponds to the relevant distribution (2.10). In combination with the self-consistent conditions (2.9), it can be considered as entropy of non-equilibrium state. In accordance with (2.6), in order to obtain the explicit form of non-equilibrium distribution function, it is necessary to disclose the action of Liouville operator on ̺rel(xN ; t) and the action of the Kawasaki-Gunton projection operator, which in our case has the following structure according to (2.10): Prel(t)̺′ = ̺rel ( xN ; t ) ∫ dΓN̺′+ ∫ dx ∂̺rel ( xN ; t ) ∂〈n̂1(x)〉t [ ∫ dΓN n̂1(x)̺′−〈n̂1(x)〉t ∫ dΓN̺′ ] + ∫ dr ∂̺rel ( xN ; t ) ∂〈Ĥ int(r)〉t [ ∫ dΓN Ĥ int(r)̺′−〈Ĥ int(r)〉t ∫ dΓN̺′ ] + ∫ da ∂̺rel ( xN ; t ) ∂ [ f (a;t ) W (a;t ) ] 1 W (a; t) [ ∫ dΓN f̂ (a)̺′− f (a; t) ∫ dΓN̺′ ] + ∫ dx ∫ da ∂̺rel ( xN ; t ) ∂ [ f (a;t ) W (a;t ) ] f (a; t) W (a; t) ∂ lnW (a; t) ∂〈n̂1(x)〉t [ ∫ dΓN n̂1(x)̺′−〈n̂1(x)〉t ∫ dΓN̺′ ] + ∫ dr ∫ da ∂̺rel ( xN ; t ) ∂ [ f (a;t ) W (a;t ) ] f (a; t) W (a; t) ∂ lnW (a; t) ∂〈Ĥ int(r)〉t [ ∫ dΓN Ĥ int(r)̺′ −〈Ĥ int(r)〉t ∫ dΓN̺′ ] . (2.14) Next, we consider the action of Liouville operator on a relevant distribution function (2.10): iLN̺rel ( xN ; t ) =− ∫ dxγ(x; t) ˙̂n1(x)̺rel ( xN ; t ) − ∫ drβ(r; t) ˙̂H int(r)̺rel ( xN ; t ) + [ iLN f (a; t) W (a; t) ∣ ∣ ∣ ∣ a=â ] ̺ kin-hyd rel ( xN ; t ) , (2.15) where ˙̂n1(x) = iLN n̂1(x), ˙̂H int(r)= iLN Ĥ int(r). Having used thereafter the relation iLN f̂ (a) = iLN f̂ (nk,Jk,εk) = ∑ k [ ∂ ∂nk f̂(a) ˙̂nk + ∂ ∂Jk f̂(a)˙̂Jk + ∂ ∂εk f̂(a)˙̂εk ] , where ˙̂nk = iLN n̂k , ˙̂Jk = iLN Ĵk , ˙̂εk = iLN ε̂k , the last expression in (2.15) can be rewritten in the following form: [ iLN f (a; t) W (a; t) ∣ ∣ ∣ ∣ a=â ] ̺ kin-hyd rel ( xN ; t ) = ∫ da ∑ k { ˙̂nkW (a; t) [ ∂ ∂nk f (a; t) W (a; t) ] + ˙̂JkW (a; t) [ ∂ ∂Jk f (a; t) W (a; t) ] + ˙̂εkW (a; t) [ ∂ ∂εk f (a; t) W (a; t) ]} ̺L ( xN ; t ) . (2.16) 43705-5 I.R. Yukhnovskii, P.A. Hlushak, M.V. Tokarchuk Here, we introduced a new relevant distribution function ̺L(xN , a; t) with the microscopic distribution of large-scale collective variables ̺L ( xN , a; t ) = ̺ kin-hyd rel ( xN ; t ) f̂ (a) W (a; t) . (2.17) This relevant distribution function is connected with ̺rel(xN ; t) by the relation ̺rel ( xN ; t ) = ∫ da f (a; t)̺L ( xN , a; t ) (2.18) and is obviously normalized to unity ∫ dΓN̺L ( xN ; t ) = 1. (2.19) Using the relation (2.17), the average values with a relevant distribution are conveniently represented in the following form: 〈. . .〉t rel = ∫ da〈. . .〉t L f (a; t), 〈. . .〉t L = ∫ dΓN . . .̺L(xN ; t). (2.20) Now, in accordance with (2.16) and (2.17) we can rewrite the action of the Liouville operator on ̺rel(xN ; t) as follows: iLN̺rel ( xN ; t ) =− ∫ da ∫ dxγ(x; t) ˙̂n1(x)̺L ( xN , a; t ) f (a; t)− ∫ da ∫ drβ(r; t) ˙̂H int(r)̺L ( xN , a; t ) f (a; t) + ∫ da ∑ k [ ˙̂nkW (a; t) ∂ ∂nk f (a; t) W (a; t) +W (a; t)˙̂Jk · ∂ ∂Jk f (a; t) W (a; t) + ˙̂εkW (a; t) ∂ ∂εk f (a; t) W (a; t) ] ̺L ( xN , a; t ) . (2.21) Substituting this expression into (2.6), one obtains, for non-equilibrium distribution function, the follow- ing result: ̺ ( xN ; t ) = ∫ da f (a; t)̺L ( xN , a; t ) + ∫ da ∫ dr t ∫ −∞ dt ′eǫ(t ′−t )Trel(t , t ′) [ 1−Prel(t ′) ] ˙̂H int(r)̺L ( xN ; t ) f (a; t ′)β(r; t ′) − ∫ da ∫ dx t ∫ −∞ dt ′eǫ(t ′−t )Trel(t , t ′) [ 1−Prel(t ′) ] ˙̂n1(x)̺L ( xN , a; t ′ ) f (a; t ′)γ(x; t ′) − ∫ da ∑ k t ∫ −∞ dt ′eǫ(t ′−t )Trel(t , t ′) [ 1−Prel(t ′) ] [ ˙̂nkW (a; t ′) ∂ ∂nk f (a; t ′) W (a; t ′) +W (a; t ′)˙̂Jk · ∂ ∂Jk f (a; t ′) W (a; t ′) + ˙̂εkW (a; t ′) ∂ ∂εk f (a; t ′) W (a; t ′) ] ̺L ( xN , a; t ′ ) (2.22) and the corresponding generalized transport equations: ( ∂ ∂t + p m · ∂ ∂r ) f1(x; t)− ∫ dx′ ∂ ∂r Φ(|r−r′|) · ( ∂ ∂p − ∂ ∂p′ ) g2(x, x′; t) =− ∫ dr′ ∫ da t ∫ −∞ dt ′eǫ(t ′−t )φnH (x,r′, a; t , t ′) f (a; t ′)β(r′; t ′) − ∫ dx′ ∫ da t ∫ −∞ dt ′eǫ(t ′−t )φnn(x, x′, a; t , t ′) f (a; t ′)γ(x′; t ′) − ∑ k ∫ da t ∫ −∞ dt ′eǫ(t ′−t ) [ φn j (x,k, a; t , t ′) · ∂ ∂Jk +φnε(x,k, a; t , t ′) ∂ ∂εk ] f (a; t ′) W (a; t ′) , (2.23) 43705-6 BBGKY chain of kinetic equations, non-equilibrium statistical operator method ∂ ∂t 〈Ĥ int(r)〉t = 〈 ˙̂H int(r)〉t rel− ∫ dr′ ∫ da t ∫ −∞ dt ′eǫ(t ′−t )φH H (r,r′, a; t , t ′) f (a; t ′)β(r′; t ′) − ∫ dx′ ∫ da t ∫ −∞ dt ′eǫ(t ′−t )φHn (r, x′, a; t , t ′) f (a; t ′)γ(x′; t ′) − ∑ k ∫ da t ∫ −∞ dt ′eǫ(t ′−t ) [ φH j (r,k, a; t , t ′) · ∂ ∂Jk +φHε(r,k, a; t , t ′) ∂ ∂εk ] f (a; t ′) W (a; t ′) , (2.24) ∂ ∂t f (a; t)= ∑ k [ ∂ ∂nk vn(a; t)+ ∂ ∂Jk ·v j (a; t)+ ∂ ∂εk vε(a; t) ] f (a; t) = ∑ k ∂ ∂Jk · ∫ dr′ ∫ da′ t ∫ −∞ dt ′eǫ(t ′−t )φ j H (r′,k, a, a′; t , t ′) f (a; t ′)β(r′; t ′) − ∑ k ∂ ∂εk ∫ dr′ ∫ da′ t ∫ −∞ dt ′eǫ(t ′−t )φεH (r′,k, a, a′; t , t ′) f (a; t ′)β(r′; t ′) + ∑ k ∂ ∂Jk · ∫ dx′ ∫ da′ t ∫ −∞ dt ′eǫ(t ′−t )φ j n(x′,k, a, a′; t , t ′) f (a; t ′)γ(x′; t ′) − ∑ k ∂ ∂εk ∫ dx′ ∫ da′ t ∫ −∞ dt ′eǫ(t ′−t )φεn(x′,k, a, a′; t , t ′) f (a; t ′)γ(x′; t ′) + ∑ k,q ∫ da′ t ∫ −∞ dt ′eǫ(t ′−t ) ∂ ∂Jk ·φ j j (k,q, a, a′; t , t ′) · ∂ ∂Jq f (a; t ′) W (a; t ′) + ∑ k,q ∫ da′ t ∫ −∞ dt ′eǫ(t ′−t ) ∂ ∂εk φεε(k,q, a, a′; t , t ′) ∂ ∂εq f (a; t ′) W (a; t ′) + ∑ k,q ∫ da′ t ∫ −∞ dt ′eǫ(t ′−t ) [ ∂ ∂Jk ·φ jε(k,q, a, a′; t , t ′) ∂ ∂εq + ∂ ∂εk φε j (k,q, a, a′; t , t ′) · ∂ ∂Jq ] f (a; t ′) W (a; t ′) . (2.25) The generalized transport equations (2.23), (2.24) include a relevant two-particle distribution function of particles g2(x, x′; t) : g2(x, x′; t) = ∫ dΓN−2̺rel ( xN ; t ) = 〈n̂1(x)n̂1(x′)〉t rel = 〈Ĝ(x, x′)〉t rel = ∫ dagL 2 (x, x′; a; t) f (a; t), (2.26) where Ĝ(x, x′) = n̂1(x)n̂1(x′), and gL 2 (x, x′; a; t) = ∫ dΓN−2̺L(xN ; a; t) is the two-particle relevant distribu- tion function of large-scale collective variables. The generalized transport kernels φαβ(t , t ′) = 〈Iα(t)Trel(t , t ′)Iβ(t ′)〉t ′ L , α,β= {n, H , j,ε}, (2.27) that describe non-Markovian kinetic and hydrodynamic processes, are non-equilibrium correlation func- tions of generalized fluxes Iα, Iβ: În (x; t) = [1−P (t)] ˙̂n1(x), ÎH (r; t) = [1−P (t)] ˙̂H int(r), (2.28) Îj(k; t) = [1−P (t)]˙̂Jk , Îε(k;t) = [1−P(t)]˙̂εk . (2.29) Here, P (t) is the generalized Mori operator related to Kawasaki-Gunton projection operator Prel(t) by the following relation Prel(t)a(x)̺rel ( xN ; t ) = ̺rel ( xN ; t ) P (t)a(x). 43705-7 I.R. Yukhnovskii, P.A. Hlushak, M.V. Tokarchuk It should be emphasized that in (2.26) the averages are calculated with a distribution function ̺L(xN , a; t) (2.20), so that the transport kernels are functions of collective variables ak. In equation (2.25), the func- tions (called hydrodynamic velocities) vn,k(a; t), v j ,k(a; t), vε,k(a; t) represent the fluxes in the space of collective variables and are defined as: vn(a; t) = ∫ dΓN ˙̂nk̺L ( xN , a; t ) = 〈 ˙̂nk〉 t L, v j (a; t)= ∫ dΓN ˙̂Jk̺L ( xN , a; t ) = 〈˙̂Jk〉 t L, vε(a; t) = ∫ dΓN ˙̂εk̺L ( xN , a; t ) = 〈 ˙̂εk〉 t L. (2.30) The presented system of transport equations provides a consistent description of kinetic and hydrody- namic processes of classical fluids which take into account long-living fluctuations. The system of transport equations (2.23)–(2.25) is not closed due to Lagrange parameters γ(x; t), β(r′; t ′), which are determined from the corresponding self-consistent conditions. To describe the kinetics of heterophase fluctuations, as discussed in the Introduction, in addition to the equation for non-equilibrium one-particle distribution function it is important to consider the equa- tions for higher non-equilibrium distribution functions of groups of interacting particles. The kinetics of these phases is described by kinetic equations for higher non-equilibrium distribution functions of finite number of particles in heterophase formations. Therefore, to describe these heterophase kinetic processes we must supplement this system of transport equations with the kinetic equation f2(x, x′; t), and hence for fs (x1 . . . xs ; t), s < N : ∂ ∂t f2(x, x′; t)+ iL2 f2(x, x′; t)− ∫ dx′′[iL(13)+ iL(23)] f3 (x, x′, x′′; t) = iL2∆ f2(x, x′; t)− ∫ d x′′[iL(13)+ iL(23)]∆ f3 (x, x′, x′′; t) + ∫ dx′′ ∫ da t ∫ −∞ dt ′eǫ(t ′−t )φGn (x, x′, x′′, a; t , t ′) f (a; t ′)γ(x′′; t ′) − ∫ dr′′ ∫ da t ∫ −∞ dt ′eǫ(t ′−t )φG H (x, x′,r′′, a; t , t ′) f (a; t ′)β(r′′; t ′) − ∑ k ∫ da t ∫ −∞ dt ′eǫ(t ′−t ) [ φG j (x, x′,k, a; t , t ′) · ∂ ∂Jk + φGε(x, x′,k, a; t , t ′) ∂ ∂εk ] f (a; t ′) W (a; t ′) , (2.31) ∂ ∂t fs (xs ; t)+ iLs fs (xs ; t)− ∑ j 1 s! ∫ dxs+1iL( j , s +1) fs+1(xs , xs+1; t) = iLs∆ fs (xs ; t)− ∑ j 1 s! ∫ d xs+1iL( j , s +1)∆ fs+1(xs , xs+1; t) + ∫ dx′′ ∫ da t ∫ −∞ dt ′eǫ(t ′−t )φGs n (xs , x′′, a; t , t ′) f (a; t ′)γ(x′′; t ′) − ∫ dr′′ ∫ da t ∫ −∞ dt ′eǫ(t ′−t )φGs H (xs ,r′′, a; t , t ′) f (a; t ′)β(r′′; t ′) − ∑ k ∫ da t ∫ −∞ dt ′eǫ(t ′−t ) [ φGs j (xs ,k, a; t , t ′) · ∂ ∂Jk +φGsε(xs ,k, a; t , t ′) ∂ ∂εk ] f (a; t ′) W (a; t ′) , (2.32) where ∆ f2(x, x′; t) = f2(x, x′; t)− g2(x, x′; t), ∆ fs (xs ; t) = fs (xs ; t)− gs (xs ; t). 43705-8 BBGKY chain of kinetic equations, non-equilibrium statistical operator method In equation (2.31), the two-particle Liouville operator iL2 = iL0(x)+ i L0(x′)+ iL(x, x′) was introduced. It contains a one-particle operator iL0(x) = p m · ∂ ∂r , x = {r,p}, as well as a potential part iL(x, x′) = ∂ ∂r Φ(|r−r′|) · ( ∂ ∂p − ∂ ∂p′ ) . Accordingly, in equation (2.32), iLs is the s-particle Liouville operator, gs(x1 . . . xs ; t) = 〈Ĝs(x1 . . . xs )〉t rel = ∫ dagL s (x1 . . . xs ; a; t) f (a; t), where gL s (x1 . . . xs ; a; t) = ∫ dΓN Ĝs(x1 . . . xs )̺L ( xN ; a; t ) is the s-particle relevant distribution function of large-scale variables and Ĝs (xs ) = n̂1(x1) . . . n̂1(xs ). Thus, we obtained a system of equations for non-equilibrium one-, two-, s-particle distribution func- tions which take into account nonlinear hydrodynamic fluctuations. Now, we discuss the equation (2.25) that is of Fokker-Planck type for a non-equilibrium distribution function of collective variables which take into account the kinetic processes. The transport kernel in this equation φnn(x, x′; t , t ′) describes a dissipation of kinetic processes, while the kernels φn j (x,k, a; t , t ′), φnε(x,k, a; t , t ′), φ j n(x,k, a; t , t ′), φεn(x,k, a; t , t ′) describe a dissipation of correlations between kinetic and hydrodynamic processes. To uncover amore detailed structure of transport kernelsφnn(x; x′, a; t , t ′), φGn (x; x′, x′′, a; t , t ′) we consider the action of Liouville operator on n̂1(x) and Ĝ(x, x′): iLN n̂1(x) =− ∂ ∂r · 1 m ĵ(r,p)+ ∂ ∂p · F̂(r,p), (2.33) iLN Ĝ(x, x′) =− ∂ ∂r · 1 m ĵ(r,p)n̂1(x′)− n̂1(x) ∂ ∂r′ · 1 m ĵ(r′,p′)+ ∂ ∂p · F̂(r,p)n̂1(x′)+ n̂1(x) ∂ ∂p′ · F̂(r′,p′), (2.34) where ĵ(r,p) = N ∑ j=1 p jδ(r−r j )δ(p−p j ) (2.35) is the microscopic density of the momentum vector in coordinate-momentum space, F̂(r,p) = ∑ l, j ∂ ∂r j Φ(|r j −rl |)δ(r−r j )δ(p−p j ) (2.36) is the microscopic density of force vector in coordinate-momentum space. Taking into account the equa- tions (2.33)–(2.36) for the kinetic transport kernels, we obtain: φnn(x; x′, a; t , t ′) =− [ ∂ ∂r ·D j j (x, x′, a; t , t ′) · ∂ ∂r′ − ∂ ∂p ·DF j (x, x′, a; t , t ′) · ∂ ∂r′ − ∂ ∂r ·D j F (x, x′, a; t , t ′) · ∂ ∂p′ + ∂ ∂p ·DF F (x, x′, a; t , t ′) · ∂ ∂p′ ] , (2.37) where D j j (x, x′, a; t , t ′)= ∫ dΓN ĵ(x)T (t , t ′)[1−P (t ′)]ĵ(x′)ρL ( xN ; t ′ ) , DF F (x, x′, a; t , t ′) = ∫ dΓN F̂(x)T (t , t ′)[1−P (t ′)]F̂(x′)ρL ( xN ; t ′ ) 43705-9 I.R. Yukhnovskii, P.A. Hlushak, M.V. Tokarchuk are the generalized diffusion and the particle friction coefficients in the coordinate-momentum space. Moreover, ∫ dp ∫ dp′D j j (x, x′; t , t ′) = D j j (r,r′; t , t ′), ∫ dp ∫ dp′DF F (x, x′; t , t ′) = DF F (r,r′; t , t ′) are the generalized coefficients of diffusion and friction. Similarly, we obtain the expression for the trans- port kernel φGn (x; x′, x′′; t , t ′): φGn (x; x′, x′′, a; t , t ′) =− [ ∂ ∂r ·D j j n(x, x′, x′′, a; t , t ′) · ∂ ∂r′ + ∂ ∂r ·D j n j (x, x′, x′′, a; t , t ′) · ∂ ∂r′′ − ∂ ∂p ·DF j n(x, x′, x′′, a; t , t ′) · ∂ ∂r′ − ∂ ∂p ·DF n j (x, x′, x′′, a; t , t ′) · ∂ ∂r′′ − ∂ ∂r ·D j F n(x, x′, x′′, a; t , t ′) · ∂ ∂p′ − ∂ ∂r ·D j nF (x, x′, x′′, a; t , t ′) · ∂ ∂p′′ + ∂ ∂p ·DF F n(x, x′, x′′, a; t , t ′) · ∂ ∂p′ + ∂ ∂p ·DF nF (x, x′, x′′, a; t , t ′) · ∂ ∂p′′ ] . (2.38) It is remarkable that the expression ∫ dx′ ∫ da t ∫ −∞ dt ′eǫ(t ′−t )φnn(x, x′, a; t , t ′) f (a; t ′)γ(x′; t ′) in equation (2.23) with (2.37) is the generalized collision integral of Fokker-Planck type in the coordinate- momentum space. That is, taking into account (2.26) and (2.37), the kinetic equation (2.23) can be written as follows: ( ∂ ∂t + p m ∂ ∂r ) f1(x; t)− ∫ dx′ ∫ da ∂ ∂r Φ(|r−r′|) ( ∂ ∂p − ∂ ∂p′ ) g l 2(x, x′, a; t) f (a; t) =− ∫ dr′ ∫ da t ∫ −∞ dt ′eǫ(t ′−t )φnH (x,r′, a; t , t ′) f (a; t ′)β(r′; t ′) − ∫ dx′ ∫ da t ∫ −∞ dt ′eǫ(t ′−t ) ∂ ∂r ·D j j (x, x′, a; t , t ′) · ∂ ∂r′ γ(x′; t ′) f (a; t ′) + ∫ dx′ ∫ da t ∫ −∞ dt ′eǫ(t ′−t ) [ ∂ ∂p ·DF j (x, x′, a; t , t ′) · ∂ ∂r′ + ∂ ∂r ·D j F (x, x′, a; t , t ′) · ∂ ∂p′ − ∂ ∂p ·DF F (x, x′, a; t , t ′) · ∂ ∂p′ ] γ(x′; t ′) f (a; t ′) − ∑ k ∫ da t ∫ −∞ dt ′eǫ(t ′−t ) [ φn j (x,k, a; t , t ′) · ∂ ∂Jk +φnε(x,k, a; t , t ′) ∂ ∂εk ] f (a; t ′) W (a; t ′) . (2.39) In the equation (2.25), the quantities φ j j (k,q, a, a′; t , t ′), φ jε(k,q, a, a′; t , t ′), φε j (k,q, a, a′; t , t ′), φεε(k,q, a, a′; t , t ′) correspond to the dissipative processes connected with the correlations between vis- cous and heat hydrodynamic processes. Note that the equations (2.31), (2.32) have the structure similar to the equation (2.39) but with more complex many-particle transport kernels. Thus, we obtained a sys- tem of equations for non-equilibrium one-, two-, s-particle distribution functions on collision integral of Fokker-Planck type which take into account nonlinear hydrodynamic fluctuations. The set of equations (2.23), (2.25), (2.31), (2.32) allow for two limiting cases. The first, if the description of non-equilibrium processes does not take into account nonlinear hydrodynamic fluctuations, we will obtain the system of equations for a consistent description of kinetics and hydrodynamics obtained in 43705-10 BBGKY chain of kinetic equations, non-equilibrium statistical operator method papers [39, 42, 43]. These systems can be written as follows: ( ∂ ∂t + p m · ∂ ∂r ) f1(x; t)− ∫ dx′ ∂ ∂r Φ(|r−r′|) · ( ∂ ∂p − ∂ ∂p′ ) g2(x, x′; t) =− ∫ dr′ t ∫ −∞ dt ′eǫ(t ′−t )φnH (x,r′; t , t ′)β(r′; t ′)− ∫ dx′ t ∫ −∞ dt ′eǫ(t ′−t )φnn(x, x′; t , t ′)γ(x′; t ′), (2.40) ∂ ∂t 〈Ĥ int(r)〉t = 〈 ˙̂H int(r)〉t rel− ∫ dr′ t ∫ −∞ dt ′eǫ(t ′−t )φH H (r,r′; t , t ′)β(r′; t ′) − ∫ dx′ t ∫ −∞ dt ′eǫ(t ′−t )φHn(r, x′; t , t ′)γ(x′; t ′), (2.41) where the transport kernels φnn(x, x′; t , t ′), φnH (x,r′; t , t ′), φHn(r, x′; t , t ′), φH H (r,r′; t , t ′) are given by expression (2.27). An averaging in these kernels is performed using the relevant statistical operator ̺ kin-hyd rel (xN ; t). If in equations (2.40), (2.41) we do not consider the contribution from the potential en- ergy of interaction, which is true for a dilute gas, then we shall obtain a generalized kinetic equation for a non-equilibrium one-particle distribution function [44]: ( ∂ ∂t + p m · ∂ ∂r ) f1(x; t)− ∫ dx′ ∂ ∂r Φ(|r−r′|) ( ∂ ∂p − ∂ ∂p′ ) g2(x, x′; t) = ∫ dx′ t ∫ −∞ dt ′eǫ(t ′−t )φnn(x, x′; t , t ′)γ(x′; t ′). (2.42) The second, if we do not take into account the kinetic processes, then we shall obtain generalized (non-Markov) Fokker–Planck equation for non-equilibrium distribution function of collective variables, which can be obtained using the method of Zwanzig projection operators or using the method of Zubarev non-equilibrium statistical operator [45]: ∂ ∂t f (a; t)= ∑ k [ ∂ ∂nk vn(a; t)+ ∂ ∂Jk ·v j (a; t)+ ∂ ∂εk vε(a; t) ] f (a; t) = ∑ k,q ∫ da′ t ∫ −∞ dt ′eǫ(t ′−t ) ∂ ∂Jk ·φ j j (k,q, a, a′; t , t ′) · ∂ ∂Jq f (a; t ′) W (a; t ′) + ∑ k,q ∫ da′ t ∫ −∞ dt ′eǫ(t ′−t ) ∂ ∂εk φεε(k,q, a, a′; t , t ′) ∂ ∂εq f (a; t ′) W (a; t ′) + ∑ k,q ∫ da′ t ∫ −∞ dt ′eǫ(t ′−t ) × [ ∂ ∂Jk ·φ jε(k,q, a, a′; t , t ′) ∂ ∂εq + ∂ ∂εk φε j (k,q, a, a′; t , t ′) · ∂ ∂Jq ] f (a; t ′) W (a; t ′) . (2.43) One of the main problems for the analysis of transport equations (2.23)–(2.25) and transport kernels are the calculations of the structure functions W (a; t) of collective variables and of hydrodynamic veloc- ities vn(a; t), v j (a; t), vε(a; t). We consider one of these ways in the next section. 3. Calculation of structure function W (a; t ) and hydrodynamical veloci- ties vα(a; t ) by the collective variable method In the Kawasaki theory [46] of non-linear fluctuations, the structure function is approximated by a Gaussian dependence on collective variables. In this case, as it can be seen, the hydrodynamic velocities 43705-11 I.R. Yukhnovskii, P.A. Hlushak, M.V. Tokarchuk vl ,k(a; t), l = n, j ,ε are the linear function of a. Another approach for the calculation of hydrodynamical velocities vl ,k(a; t) was proposed on the basis of local thermodynamics [45]. The resulting expressions are obviously valid at low frequencies and for small values of the wave vector, when the conditions of the local thermodynamics are valid. Structure function W (a; t) and hydrodynamical velocities vl ,k(a; t) in a case of study of hydrodynamic fluctuations were calculated in [47, 48] using the method of collective variables [49]. The basic idea of this approach is that the structure function W (a; t) and hydrodynamic velocities vl ,k(a; t) can be calculated in approximations higher than Gaussian. Next, we use the method of collective variables [35, 47–49] to calculate the structure function W (a; t) and hydrodynamic velocities vl ,k(a; t). Further, the collective variable method is used to calculate the structure function W (a; t) and hydrodynamic velocities vl ,k(a; t). We considered earlier the case when the interactions between parti- cles on small distances were described by short-range potentialΦsh(|rl j |), in particular, by the potential of hard spheres. At long distances, the interactions between particles are described by the long-range poten- tial Φlong(|rl j |). Accordingly, we define the short- and long-acting parts of the interaction of the Liouville operator: iLN = iL0 N + T̂N + iL long N , where iL0 N is the Liouville operator of N non-interacting particles, T̂N is the scattering operator of hard sphere system and iL long N is the potential part of Liouville operator with long-range interaction between particles. First, we calculate the structure function W (a; t) for collective variables. To do this, we use the inte- gral representation for δ-functions: f̂ (a) = ∫ dωexp [ − iπ ∑ l ,k ωl ,k(âl ,k −al ,k) ] , l = n, j,ε. (3.1) Next, using a cumulant expansion [35, 48] for W (a; t), one obtains: W (a; t) = ∫ dΓN̺ kin-hyd rel ( xN ; t ) f̂ (a) = ∫ dωexp [ − iπ ∑ l ,k ωl ,k āl ,k − 1 2V 2 ∑ q ∑ k β−q(t)ν(k)(nq+kn−k −nq) − π2 2 ∑ l1,l2 ∑ k1 ,k2 M l1,l2 2 (k1,k2; t)ωl1,k1 ωl2 ,k2 ] exp [ ∑ nÊ3 Dn(ω; t) ] , (3.2) where āl ,k = al ,k −〈âl ,k〉 t kin-sh , dω= ∏ l ,k dωr l ,kdωs l ,k , ωl ,k =ωr l ,k − iωs l ,k , ωl ,−k =ω∗ l ,k , Dn(ω; t) = (−iπ)n n! ∑ l1 ,...,ln ∑ k1,...,kn M l1,...,ln n (k1, . . . ,kn ; t)ωl1 ,k1 . . .ωln ,kn , (3.3) M l1,...,ln n (k1, . . . ,kn ; t) = 〈âl1,k1 . . . âln ,kn 〉 t ,c kin-sh (3.4) are the non-equilibrium cumulant averages in approximations of the n-order, which are calculated using distribution ̺kin-sh rel (xN ; t) for hard sphere model: ̺kin-sh rel ( xN ; t ) = exp [ −Φ kin-sh(t)− ∫ drβ(r; t)Ĥ sh(r)− ∫ dxγ(x; t)n̂1(x) ] . (3.5) In (3.4) superscript “c” means the cumulant averages. It should be noted that in (3.2) we shared the con- tributions from short- and long-range interactions. The short-range interactions are take into account in a relevant distribution (3.5) (which can be considered as a base distribution) and long-range interactions are presented through collective variables ∫ drβ(r; t)Ĥ long(r) = 1 2V 2 ∑ q ∑ k β−q(t)ν(k) ( nq+kn−k −nq ) . 43705-12 BBGKY chain of kinetic equations, non-equilibrium statistical operator method We present the structure function W (a; t) for further calculations in the following form: W (a; t) = ∫ dωexp [ − iπ ∑ l ,k ωl ,k āl ,k − 1 2V 2 ∑ q ∑ k β−q(t)ν(k) ( nq+kn−k −nq ) − π2 2 ∑ l1,l2 ∑ k1,k2 M l1,l2 2 (k1,k2; t)ωl1,k1 ωl2,k2 ]( 1+B + 1 2! B2 + 1 3! B3 + . . .+ 1 n! Bn + . . . ) , (3.6) where B = ∑ nÊ3 Dn (ω; t). If in series of exponent (3.6), namely, exp [ ∑ nÊ3 Dn (ω; t) ] , one retains only the first term equal to unity, one will obtain the Gaussian approximation for W (a; t): W G(a; t)= ∫ dωexp [ iπ ∑ l ,k ωl ,kāl ,k − 1 2V 2 ∑ q ∑ k β−q(t)ν(k) ( nq+kn−k −nq ) − π2 2 ∑ l1,l2 ∑ k1,k2 M l1,l2 2 (k1,k2; t)ωl1,k1 ωl2,k2 ] , (3.7) whereM l1,l2 2 (k1,k2; t) are the matrix elements of non-equilibrium correlation functions: M2(k1,k2; t) = ∣ ∣ ∣ ∣ ∣ ∣ ∣ 〈n̂n̂〉c kin-sh 〈n̂Ĵ〉c kin-sh 〈n̂ε̂〉c kin-sh 〈Ĵn̂〉c kin-sh 〈ĴĴ〉c kin-sh 〈Ĵε̂〉c kin-sh 〈ε̂n̂〉c kin-sh 〈ε̂Ĵ〉c kin-sh 〈ε̂ε̂〉c kin-sh ∣ ∣ ∣ ∣ ∣ ∣ ∣ k1,k2 , (3.8) and, for example, the non-equilibrium cumulant average 〈n̂kn̂−k〉 t ,c kin-sh = 〈n̂kn̂−k〉 t kin-sh−〈n̂k〉 t kin-sh〈n̂−k〉 t kin-sh . (3.9) For integrating over dω in (3.7), we should transform the quadratic form in an exponential expression into a diagonal form with respect to ωl ,k . To this end, it is necessary to find the eigenvalues of the matrix (3.8) by solving the equation det ∣ ∣M̃2(k1,k2; t)− Ẽ (k; t) ∣ ∣= 0, (3.10) Ẽ (k; t) is the diagonal matrix. Further, the expression (3.7) can be written as follows: W G(a; t) = ∫ dω̃ detW̃ exp [ − iπ ∑ l ,k ãlkω̃lk − 1 2V 2 ∑ q ∑ k β−q(t)ν(k) ( nq+kn−k −nq ) − π2 2 ∑ l ∑ k El (k; t)ω̃lkω̃l ,−k ] , (3.11) where new variables ãlk , ω̃lk are connected with the old variables by the ratio: ãnk = ∑ l ālkωln , ωlk = 3 ∑ m=1 ωlmω̃mk , and ωlm are matrix elements: W̃ = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ω11 . . . ω15 ... . . . ... ω51 . . . ω55 ∣ ∣ ∣ ∣ ∣ ∣ ∣ (k;t ) . Integrand in (3.11) is a quadratic function ω̃nk and after integrating over dωnk we will obtain the follow- ing structure function in Gaussian approximation W G(a; t): W G(a; t) = exp [ − 1 2 ∑ l ,k E−1 l (k; t)ãlkãl ,−k − 1 2V 2 ∑ q ∑ k β−q(t)ν(k) ( nq+kn−k −nq ) ] ×exp [ − 1 2 ∑ k ln π det Ẽ(k; t) ] exp [ ∑ k ln detW̃ (k; t) ] , (3.12) 43705-13 I.R. Yukhnovskii, P.A. Hlushak, M.V. Tokarchuk or through variables ālk: W G(a; t) = Z (t) exp [ − 1 2 ∑ l ,k Ēl (k; t)ālkāl ,−k − 1 2V 2 ∑ q ∑ k β−q(t)ν(k) ( nq+kn−k −nq ) ] , (3.13) where Ēl (k; t) = ∑ l ′ ωl l ′E −1 l ′ (k; t)ωl ′l , Z (t)= exp [ − 1 2 ∑ k ln π det Ẽ(k; t) ] exp [ ∑ k ln detW̃ (k; t) ] . The structure function W G(a; t) provides a possibility to calculate (3.2) in higher approximations over Gaussian moments [35, 48]: W (a; t) =W G(a; t)exp [ ∑ nÊ3 〈D̃n(a; t)〉G ] , (3.14) where one presents 〈D̃n(a; t)〉G approximately as: 〈D̃3(a; t)〉G = 〈D̄3(a; t)〉G , 〈D̃4(a; t)〉G = 〈D̄4(a; t)〉G , 〈D̃6(a; t)〉G = 〈D̄6(a; t)〉G− 1 2 〈D̄3(a; t)〉2 G , 〈D̃8(a; t)〉G = 〈D̄8(a; t)〉G−〈D̄3(a; t)〉G〈D̄5(a; t)〉G− 1 2 〈D̄4(a; t)〉2 G , 〈D̃n(a; t)〉G = 1 W G(a; t) ∑ l1 ,...,ln ∑ k1,...,kn M̄ l1,...,ln n (k1, . . . ,kn ; t) 1 (iπ)n δn δāl1 ,k1 . . .δāln ,kn W G(a; t). 〈D̃n(a; t)〉G are the renormalized non-equilibrium cumulant averages of the order n for the variables ālk . In expression (3.14), the summands are with only even degrees over a since all odd Gaussian moments vanish. The method of calculation of the structure function W (a; t) can be used for approximate calculations of hydrodynamic velocities vl ,k(a; t). We present a general formula of velocities consistent with (2.30) in the following form: vl ,k(a; t) = ∫ dΓN ˙̂al ,k̺ kin-hyd rel ( xN ; t ) f̂ (a) (3.15) and introduce the function W (a,λ; t): W (a,λ; t) = ∫ dΓN exp ( − iπ ∑ l ,k λl ,k ˙̂al ,k ) ̺ kin-hyd rel ( xN ; t ) f̂ (a), (3.16) so that vl ,k(a; t) = ∂ ∂(−iπλl ,k) lnW (a,λ; t) ∣ ∣ ∣ λl ,k=0 . (3.17) We calculate the function W (a,λ; t) using the preliminary results of the calculation of the structural function W (a; t), and rewrite W (a,λ; t) as: W (a,λ; t) = ∫ dΓN ∫ dωexp ( − iπ ∑ l ,k λl ,k ˙̂al ,k ) exp [ − iπ ∑ l ,k ωl ,k ( âl ,k −al ,k ) ] ̺ kin-hyd rel ( xN ; t ) . (3.18) Now, we carry out an averaging in (3.18) using the following cumulant expansion: W (a,λ; t) = ∫ dωexp { − iπ ∑ l ,k ωl ,k āl ,k − 1 2V 2 ∑ q ∑ k β−q(t)ν(k) ( nq+kn−k −nq ) + ∑ nÊ1 [ Dn(ω; t)+Dn(λ; t)+Dn (ω,λ; t) ] } , (3.19) 43705-14 BBGKY chain of kinetic equations, non-equilibrium statistical operator method where Dn(ω; t) = (−iπ)n n! ∑ l1 ,...,ln ∑ k1,...,kn M l1,...,ln n (k1, . . . ,kn ; t)ωl1 ,k1 . . .ωln ,kn , Dn(λ; t) = (−iπ)n n! ∑ l1 ,...,ln ∑ k1,...,kn M (1)l1,...,ln n (k1, . . . ,kn ; t)λl1 ,k1 . . .λln ,kn , Dn (ω,λ; t) = (−iπ)n n! ∑ l1 ,...,ln ∑ k1,...,kn M (2)l1,...,ln n (k1, . . . ,kn ; t)ωl1,k1 . . .ωln−1 ,kn−1 . . .λln ,kn , with the cumulants of the following structure: M l1,...,ln n (k1, . . . ,kn ; t) = 〈âl1,k1 , . . . , âln ,kn 〉 t ,c kin-sh , M (1)l1,...,ln n (k1, . . . ,kn ; t) = 〈 ˙̂al1 ,k1 , . . . , ˙̂aln ,kn 〉 t ,c kin-sh , M (2)l1,...,ln n (k1, . . . ,kn ; t) = n [ (n− j )+ ( j −n+1)δl1 ,...,ln−1 ] ×〈âl1,k1 , . . . , âln− j ,kn− j , . . . , ˙̂aln− j+1 ,kn− j+1 , . . . , ˙̂aln ,kn 〉 t ,c kin-sh . First, we consider a Gaussian approximation for W (a,λ; t), namely in the exponent of an integrand we leave only the summands with n = 2 and linear over λl ,k: W G(a,λ; t) = ∫ dωexp [ iπ ∑ l ,k ωl ,k āl ,k − iπ ∑ l ,k 〈 ˙̂al ,k〉 t ,c kin-hyd λl ,k − 1 2V 2 ∑ q ∑ k β−q(t)ν(k) ( nq+kn−k −nq ) − π2 2 ∑ l1,l2 ∑ k1,k2 M l1,l2 2 (k1,k2; t)ωl1,k1 ωl2,k2 − π2 2 ∑ l1 ,l2 ∑ k1,k2 M (2)l1 ,l2 2 (k1,k2; t)ωl1,k1 λl2 ,k2 ] . (3.20) Then, transforming this expression in the exponent to a diagonal quadratic form over variables ωl ,k , similarly to W (a; t), after integrating with respect to the new variables ω̄l ,k , one obtains: W G(a,λ; t) = ∫ dωexp [ − iπ ∑ l ,k 〈 ˙̂al ,k〉 t kinλl ,k − 1 2V 2 ∑ q ∑ k β−q(t)ν(k) ( nq+kn−k −nq ) − π2 2 ∑ l ,k E−1 l (k; t)bl ,kbl ,−k − 1 2 ∑ k lnπdet Ẽ(k; t)+ ∑ k ln detW̃ (k; t) ] , (3.21) where bl ,k = ∑ j ωl j [ ā j ,k + iπ 2 ∑ j ′ M (2) j , j ′ 2 (k; t)λ j ′,k ] , and ωl j ,M (2) j , j ′ 2 (k; t) and El (k; t) are not dependent on λl ,k . Here, the cumulants M (2) j , j ′ 2 (k; t) have the following structure: M (2) j , j ′ 2 (k; t) = 〈 ˙̂a j ,kâ j ′,−k〉 t kin-sh−〈 ˙̂a j ,k〉 t kin〈â j ′,−k〉 t kin-sh. (3.22) Now, we calculate the hydrodynamic velocities vl ,k(a; t) in a Gaussian approximation according to the formula vl ,k(a; t) = ∂ ∂(−iπλl ,k) lnW G(a,λ; t) ∣ ∣ ∣ λl ,k=0 = 〈 ˙̂a j ,k〉 t kin− 1 2 ∑ j , j ′ E−1 l (k; t)ω j lω j ′lM (2) j ′,l 2 (k; t)āl ,k . (3.23) Specifically, we consider the particular case when one can divide the longitudinal and transverse fluctu- ations for collective variables. That is, we choose the direction of the wave vector k along the axis of Oz. Thus, one obtains: W G(a; t)= ∫ dωexp [ iπ ∑ l ,k ωl ,kāl ,k − 1 2V 2 ∑ q ∑ k β−q(t)ν(k) ( nq+kn−k −nq ) − π2 2 3 ∑ l1,l2=1 ∑ k1,k2 M ∥,l1 ,l2 2 (k1,k2; t)ωl1,k1 ωl2,k2 − π2 2 4 ∑ l1,l2=1 ∑ k1,k2 M ∥,⊥,l1 ,l2 2 (k1,k2; t)ωl1 ,k1 ωl2 ,k2 ] , (3.24) 43705-15 I.R. Yukhnovskii, P.A. Hlushak, M.V. Tokarchuk whereM ∥,l1,l2 2 (k1,k2; t) are the matrix elements of the non-equilibrium correlation functions of longitu- dinal fluctuations M ∥ 2(k1,k2; t) = ∣ ∣ ∣ ∣ ∣ ∣ ∣ 〈n̂n̂〉c kin-sh 〈n̂Ĵ∥〉c kin-sh 〈n̂ε̂〉c kin-sh 〈Ĵ∥n̂〉c kin-sh 〈Ĵ∥Ĵ∥〉c kin-sh 〈Ĵ∥ε̂〉c kin-sh 〈ε̂n̂〉c kin-sh 〈ε̂Ĵ∥〉c kin-sh 〈ε̂ε̂〉c kin-sh ∣ ∣ ∣ ∣ ∣ ∣ ∣ k1,k2 , (3.25) M ⊥l1,l2 2 (k1,k2; t) are the matrix elements of the non-equilibrium correlation functions of transverse and transverse-longitudinal fluctuations M ∥,⊥ 2 (k1,k2; t) = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ 0 〈n̂Ĵ⊥x 〉 c kin-sh 〈n̂Ĵ⊥y 〉 c kin-sh 0 〈Ĵ⊥x n̂〉c kin-sh 〈Ĵ⊥x Ĵ⊥x 〉 c kin-sh 〈Ĵ⊥x Ĵ⊥y 〉 c kin-sh 〈Ĵ⊥x ε̂〉 c kin-sh 〈Ĵ⊥y n̂〉c kin-sh 〈Ĵ⊥y Ĵ⊥x 〉 c kin-sh 〈Ĵ⊥y Ĵ⊥y 〉 c kin-sh 〈Ĵ⊥y ε̂〉 c kin-sh 0 〈ε̂Ĵ⊥x 〉 c kin-sh 〈ε̂Ĵ⊥y 〉 c kin-sh 0 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ k1,k2 . (3.26) In this case, the hydrodynamic velocities in the Gaussian approximation are as follows: v ∥G nk (a; t) = 〈 ˙̂nk〉 t kin-sh+E−1 1 (k; t) ( ω11n̄k +ω21J̄ ∥ k +ω31ε̄k ) Ωn(k; t), (3.27) v ∥G Jk (a; t) = 〈˙̂J ∥ k 〉t kin-sh+E−1 2 (k; t) ( ω12n̄k +ω22J̄ ∥ k +ω32ε̄k ) ΩJ (k; t), v ∥G εk (a; t) = 〈 ˙̂εk〉 t kin-sh+E−1 3 (k; t) ( ω13n̄k +ω23J̄ ∥ k +ω33ε̄k ) Ωε(k; t), where Ωn (k; t) =ω11〈n̂k ˙̂n−k〉 t ,c kin-sh +ω21〈Ĵ ∥ k ˙̂n−k〉 t ,c kin-sh +ω31〈ε̂k ˙̂n−k〉 t ,c kin-sh , (3.28) ΩJ (k; t) =ω12〈n̂k ˙̂J ∥ −k 〉 t ,c kin-sh +ω22〈Ĵ ∥ k ˙̂J ∥ −k 〉 t ,c kin-sh +ω32〈ε̂k ˙̂J ∥ −k 〉 t ,c kin-sh , Ωε(k; t) =ω13〈n̂k ˙̂ε−k〉 t ,c kin-sh +ω23〈Ĵ ∥ k ˙̂ε−k〉 t ,c kin-sh +ω33〈ε̂k ˙̂ε−k〉 t ,c kin-sh , and ωl j are the elements of matrix W̃ (k; t). As one can see the hydrodynamic velocities (3.27) in the Gaussian approximation for W G(a,λ; t) are linear functions of the collective variables nk , Jk and εk. It is remarkable that if the kinetic processes are not taken into account, i.e., ̺kin-sh rel (xN ; t) = 1, then 〈. . .〉t kin-sh →〈. . .〉0 is an average over a microscopic ensemble W (a); in this case, the expressions (3.27) for hydrodynamic velocities transform into the re- sults of the previous work [48], in which the nonlinear hydrodynamic fluctuations in simple fluids were investigated. 4. Conclusions Using the method of Zubarev non-equilibrium statistical operator, we have obtained amodified chain of BBGKY kinetic equations that take into account non-equilibrium hydrodynamic fluctuations for a sys- tem of interacting particles. At the same time, the non-equilibrium distribution function that describes the hydrodynamic fluctuations, satisfies a generalized Fokker-Planck equation. We divide the contribu- tions from short-range and long-range interactions between particles, and describe the short-range inter- actions (hard sphere model) in the coordinate space, while the long-range interactions are described in the space of collective variables. Moreover, the short-range component will be considered as a basis with distribution ̺kin-sh rel (xN ; t), which corresponds to the BBGKY chain of equations for the model of hard spheres [40]. The used method of collective variables [35, 47, 49] has made it possible to calculate the structural distribution function of hydrodynamic collective variables and their hydrodynamic velocities in approxi- mations higher than the Gaussian one. In particular, the hydrodynamic velocities above the Gaussian ap- proximation that follow from equation (3.19) and (3.27) will be proportional to āl ,k āl ′,k , and the transport kernels in the Fokker-Planck equation will be of fourth-order correlation functions over variables âl ,k . It is significant that the Fokker-Planck equation in a Gauss approximation for W̃ G(k; t), vG l ,k (a; t) leads to 43705-16 BBGKY chain of kinetic equations, non-equilibrium statistical operator method transport equations for 〈âl ,k〉 t and the structure of these equations is the same as in molecular hydrody- namics [50, 51], although the averaging is performed using ̺L(xN , a; t) = ̺ kin-hyd rel (xN ; t) f̂ (a)/W G(a; t). The proposed approach makes it possible to go beyond the Gaussian approximation for W̃ (k; t), vl ,k(a; t) and for transport kernels in the Fokker-Planck equation. As a result, we obtain a nonlinear system of equations for the 〈âl ,k〉 t . It is remarkable that the kinetic equation (2.39) contains generalized integrals of Fokker-Planck type with generalized coefficients of diffusion and friction in the phase space (r,p, t), where the limit of change of |r| is restricted by value |k|−1 hydr that corresponds to collective non- linear hydrodynamic fluctuations. 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Ланцюжок кiнетичних рiвнянь ББГКI, метод нерiвноважного статистичного оператора та метод колективних змiнних в нерiвноважнiй статистичнiй теорiї рiдин I.Р. Юхновський, П.А. Глушак, М.В. Токарчук Iнститут фiзики конденсованих систем НАН України, вул. Свєнцiцького, 1, 79011 Львiв, Україна Запропоновано ланцюжок кiнетичних рiвнянь для нерiвноважних одночастинкової, двочастинкової i s- частинкової функцiй розподiлу частинок з урахуванням нелiнiйних гiдродинамiчних флуктуацiй. Викори- стовується метод нерiвноважного статистичного оператора Зубарєва з проектуванням. Нелiнiйнi гiдроди- намiчнi флуктуацiї описуються нерiвноважною функцiєю розподiлу колективних змiнних, що задоволь- няє узагальнене рiвняння Фоккера-Планка. На основi методу колективних змiнних запропоновано спо- сiб розрахунку нерiвноважної структурної функцiї розподiлу колективних змiнних та їх гiдродинамiчних швидкостей (вище гаусового наближення), що мiстяться в узагальненому рiвняннi Фоккера-Планка для нерiвноважної функцiї розподiлу колективних змiнних. При цьому роздiленi вклади вiд короткодiючих i далекодiючих взаємодiй мiж частинками, що привело до того, що короткодiючi взаємодiї (наприклад, модель твердих сфер) описуються в координатному просторi, а далекодiючi — у просторi колективних змiнних. Короткодiюча складова розглядається як базисна, якiй вiдповiдає ланцюжок рiвнянь ББГКI для моделi твердих сфер. Ключовi слова: нелiнiйнi флуктуацiї, нерiвноважний статистичний оператор, функцiя розподiлу, рiвняння Фоккера-Планка, проста рiдина 43705-18 http://dx.doi.org/10.1063/1.1648300 http://dx.doi.org/10.1016/j.physa.2015.09.059 http://dx.doi.org/10.1007/BF02575494 http://dx.doi.org/10.1007/BF01018982 http://dx.doi.org/10.1007/BF01016582 http://dx.doi.org/10.1007/BF01019063 http://dx.doi.org/10.1023/A:1023044610690 http://dx.doi.org/10.5488/CMP.1.4.687 http://dx.doi.org/10.1007/s10955-014-0980-4 http://dx.doi.org/10.1016/0031-8914(62)90009-5 http://dx.doi.org/10.1016/0378-4371(83)90062-6 http://dx.doi.org/10.1007/BF01014797 http://dx.doi.org/10.1080/00268979500100181 Introduction Non-equilibrium distribution function and BBGKY chain of kinetic equations in the Zubarev non-equilibrium statistical operator method Calculation of structure function W(a;t) and hydrodynamical velocities v(a;t) by the collective variable method Conclusions

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