A consistent description of kinetics and hydrodynamics of systems of interacting particles by means of the nonequilibrium statistical operator method

A statistical approach to a self-consistent description of kinetic and hydrodynamic processes in systems of interacting particles is formulated on the basis of the nonequilibrium statistical operator method by D.N.Zubarev. It is shown how to obtain the kinetic equation of the revised Enskog theory...

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Дата:	1998
Автори:	Tokarchuk, M.V., Omelyan, I.P., Kobryn, A.E.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут фізики конденсованих систем НАН України 1998
Назва видання:	Condensed Matter Physics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/119887
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Цитувати:	A consistent description of kinetics and hydrodynamics of systems of interacting particles by means of the nonequilibrium statistical operator method / M.V. Tokarchuk, I.P. Omelyan, A.E. Kobryn // Condensed Matter Physics. — 1998. — Т. 1, № 4(16). — С. 687-751. — Бібліогр.: 181 назв. — англ.

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spelling	irk-123456789-1198872017-06-11T03:03:05Z A consistent description of kinetics and hydrodynamics of systems of interacting particles by means of the nonequilibrium statistical operator method Tokarchuk, M.V. Omelyan, I.P. Kobryn, A.E. A statistical approach to a self-consistent description of kinetic and hydrodynamic processes in systems of interacting particles is formulated on the basis of the nonequilibrium statistical operator method by D.N.Zubarev. It is shown how to obtain the kinetic equation of the revised Enskog theory for a hard sphere model, the kinetic equations for multistep potentials of interaction and the Enskog-Landau kinetic equation for a system of charged hard spheres. The BBGKY hierarchy is analyzed on the basis of modiﬁed group expansions. Generalized transport equations are obtained in view of a self-consistent description of kinetics and hydrodynamics. Time correlation functions, spectra of collective excitations and generalized transport coefﬁcients are investigated in the case of weakly nonequilibrium systems of interacting particles. Представлено один із статистичних підходів узгодженого опису кінетичних та гідродинамічних процесів систем взаємодіючих частинок, що сформульований на основі методу нерівноважного статистичного оператора Д.М.Зубарєва. Показано, як із ланцюжка рівнянь ББГКІ з модифікованими граничними умовами отримуються кінетичне рівняння ревізованої теорії Енскога для моделі твердих сфер, кінетичне рівняння для багатосходинкового потенціалу та кінетичне рівняння Енскога-Ландау для моделі заряджених твердих сфер. Проаналізовано ланцюжки рівнянь ББГКІ на основі модифікованих групових розкладів. Отримано узагальнені рівняння переносу узгодженого опису кінетики та гідродинаміки. Для випадку слабо нерівноважних систем класичних взаємодіючих частинок при взаємному врахуванні кінетичних та гідродинамічних процесів досліджено часові кореляційні функції, спектр колективних збуджень та узагальнені коефіцієнти переносу. 1998 Article A consistent description of kinetics and hydrodynamics of systems of interacting particles by means of the nonequilibrium statistical operator method / M.V. Tokarchuk, I.P. Omelyan, A.E. Kobryn // Condensed Matter Physics. — 1998. — Т. 1, № 4(16). — С. 687-751. — Бібліогр.: 181 назв. — англ. 1607-324X DOI:10.5488/CMP.1.4.687 PACS: 05.20.Dd, 05.60.+w, 52.25.Fi, 71.45.G, 82.20.M http://dspace.nbuv.gov.ua/handle/123456789/119887 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	A statistical approach to a self-consistent description of kinetic and hydrodynamic processes in systems of interacting particles is formulated on the basis of the nonequilibrium statistical operator method by D.N.Zubarev. It is shown how to obtain the kinetic equation of the revised Enskog theory for a hard sphere model, the kinetic equations for multistep potentials of interaction and the Enskog-Landau kinetic equation for a system of charged hard spheres. The BBGKY hierarchy is analyzed on the basis of modiﬁed group expansions. Generalized transport equations are obtained in view of a self-consistent description of kinetics and hydrodynamics. Time correlation functions, spectra of collective excitations and generalized transport coefﬁcients are investigated in the case of weakly nonequilibrium systems of interacting particles.
format	Article
author	Tokarchuk, M.V. Omelyan, I.P. Kobryn, A.E.
spellingShingle	Tokarchuk, M.V. Omelyan, I.P. Kobryn, A.E. A consistent description of kinetics and hydrodynamics of systems of interacting particles by means of the nonequilibrium statistical operator method Condensed Matter Physics
author_facet	Tokarchuk, M.V. Omelyan, I.P. Kobryn, A.E.
author_sort	Tokarchuk, M.V.
title	A consistent description of kinetics and hydrodynamics of systems of interacting particles by means of the nonequilibrium statistical operator method
title_short	A consistent description of kinetics and hydrodynamics of systems of interacting particles by means of the nonequilibrium statistical operator method
title_full	A consistent description of kinetics and hydrodynamics of systems of interacting particles by means of the nonequilibrium statistical operator method
title_fullStr	A consistent description of kinetics and hydrodynamics of systems of interacting particles by means of the nonequilibrium statistical operator method
title_full_unstemmed	A consistent description of kinetics and hydrodynamics of systems of interacting particles by means of the nonequilibrium statistical operator method
title_sort	consistent description of kinetics and hydrodynamics of systems of interacting particles by means of the nonequilibrium statistical operator method
publisher	Інститут фізики конденсованих систем НАН України
publishDate	1998
url	http://dspace.nbuv.gov.ua/handle/123456789/119887
citation_txt	A consistent description of kinetics and hydrodynamics of systems of interacting particles by means of the nonequilibrium statistical operator method / M.V. Tokarchuk, I.P. Omelyan, A.E. Kobryn // Condensed Matter Physics. — 1998. — Т. 1, № 4(16). — С. 687-751. — Бібліогр.: 181 назв. — англ.
series	Condensed Matter Physics
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last_indexed	2025-07-08T16:51:19Z
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fulltext	Condensed Matter Physics, 1998, Vol. 1, No. 4(16), p. 687–751 A consistent description of kinetics and hydrodynamics of systems of interacting particles by means of the nonequilibrium statistical operator method M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine 1 Svientsitskii St., 290011 Lviv–11, Ukraine Received December 14, 1998 A statistical approach to a self-consistent description of kinetic and hydro- dynamic processes in systems of interacting particles is formulated on the basis of the nonequilibrium statistical operator method by D.N.Zubarev. It is shown how to obtain the kinetic equation of the revised Enskog theory for a hard sphere model, the kinetic equations for multistep potentials of in- teraction and the Enskog-Landau kinetic equation for a system of charged hard spheres. The BBGKY hierarchy is analyzed on the basis of modified group expansions. Generalized transport equations are obtained in view of a self-consistent description of kinetics and hydrodynamics. Time corre- lation functions, spectra of collective excitations and generalized transport coefficients are investigated in the case of weakly nonequilibrium systems of interacting particles. Key words: kinetics, hydrodynamics, kinetic equations, transport coefficients, (time) correlation functions PACS: 05.20.Dd, 05.60.+w, 52.25.Fi, 71.45.G, 82.20.M 1. Introduction The problem of a self-consistent description of fast and slow processes which are connected with both linear and non-linear fluctuations of observed quantities for various physical systems, such as dense gases, liquids, their mixtures and plasma, remains an actual problem in nonequilibrium statistical mechanics. An important task in this direction is the construction of kinetic equations for dense systems with taking into account collective effects (hydrodynamic contribution) into the collision integrals. On the other hand, there is a problem of the calculation of time c© M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn 687 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn correlation functions and collective excitations spectra in the range of intermedi- ate values of wavevectors and frequencies, since low- and high-frequencies regions are described sufficiently well within the frameworks of molecular hydrodynam- ics and kinetic equations, respectively. In the intermediate region, the kinetic and hydrodynamic processes are connected and must be considered simultaneously. A significant interest in these problems was exhibited by Prof. D.N.Zubarev in his investigations. One of the approaches to the unification of kinetics and hydrody- namics in the theory of transport phenomena for systems of interacting particles was proposed by Zubarev and Morozov in [1] and developed in further papers [2–12]. This statistical approach is based on a modification of the boundary condi- tions for the weakening of correlations to the BBGKY hierarchy for many-particle distribution functions and to the Liouville equation for a full nonequilibrium dis- tribution function (nonequilibrium statistical operator – NSO). The modification of the boundary conditions consists in the fact that both the one-particle nonequi- librium distribution function and the local conservation laws for mass, momentum and total energy are included into the parameters for an abbreviated description of a nonequilibrium state of the system. The concept of a self-consistent description of kinetics and hydrodynamics was applied to plasma in an electro-magnetic field [13–15], to quantum systems with taking into account coupled states [9,10,16], quantum Bose systems [17] and the nonequilibrium thermo field dynamics [18]. The questions as to the necessity and possibility of a unified description of kinetic and hydrodynamic processes were discussed in papers by Klimontovich [19–21]. In this paper we present some actual results obtained recently within the con- cept of a self-consistent description of kinetics and hydrodynamics for dense gases and simple liquids. In section 2 we present the results for kinetic equations of RET [3,7], GDRS [8] and Enskog-Landau theories [3,7], obtained in the polar- ization approximation for a system of hard spheres and a Lenard-Balescu kinetic equation [11,12] derived from the BBGKY hierarchy with modified boundary con- ditions. Generalized transfer equations, time correlation functions and transport coefficients are presented in section 3 on the basis of a self-consistent description of kinetics and hydrodynamics within the framework of the nonequilibrium statistical operator method. 2. Conception of a consistent description of kinetics and hy - drodynamics for dense gases and liquids 2.1. Overview Bogolubov [22] proposed a consequent approach to the construction of kinetic equations which is based on the chain of equations for s-particle distribution func- tions and a weakening correlation principle. There are a lot of theories [23–26] which differ in forms of presentation, but all these approaches use the same weak- ening correlation condition and are most effective when a small parameter (inter- action, density, etc.) is presented. 688 A consistent description of kinetics and hydrodynamics In the kinetic theory of classical gases we can point out two principal problems. The first problem is connected with the fact that collision integrals can depend nonanalytically on density. Because of this, even in the case of a small-density gas, in order to calculate corrections to transport coefficients it is necessary to perform a partial resummation of the BBGKY hierarchy [27,28]. The second problem is the construction of kinetic equations for the gases of high densities. In this case we cannot restrict ourselves by several terms of the expansion of collision integrals on densities and the analysis of the BBGKY hierarchy becomes very complicated (generally speaking, the density is not a small parameter here). That is why the construction of kinetic equations for dense gases and liquids with model inter- particle potentials of interaction has a great importance in the problem under investigation. The first theory in this direction is a semi-phenomenological standard Enskog theory (SET) [29,30] of dense gases. Ideas similar to those used at the derivation of the Boltzmann equation were also used in this approach. The Enskog equation plays an essential role in the kinetic theory, next to the Boltzmann one [30]. This equation was obtained at the modelling of molecules by hard spheres for dense gases. As a result, the collision integral was presented in an analytical form. It is obtained by means of a hard sphere model where collisions can be considered as momentary and by the fact that the multiparticle contact at the same time is reputed to be infinitely small. Density correction introduced by Enskog proved to be considerable, as far as transport, due to collisions in a dense system, is the main mechanism of transport. Each molecule is almost localized at one point of space by the surrounding neighbour molecules and, therefore, the flow transport is suppressed. Though this theory properly describes density dependence of the kinetic coefficients, at the same time the suppositions about the structure of the collision integral remain sufficiently rough and phenomenological. Despite approxi- mate assumptions on the collision integral in the kinetic equation for a one-particle distribution function of hard spheres, the Enskog theory very well describes a set of properties for real dense gases [25,31]. Davis, Rise and Sengers [32] proposed a kinetic DRS theory. Within the frame- work of this theory, the interparticle potential of an interaction is chosen in a square well form. An attractive part of the real potential is approximated here by a finite height wall. From the point of view of the statistical theory of nonequilibrium processes, SET and DRS contain two essential drawbacks. The first one is that their kinetic equations are not obtained within the framework of some consequential theoretical scheme and one does not know how to improve these theories. And the second one is that the H-theorem has not been proved. Nevertheless, not long ago a way of constructing the SET entropy functional was given [33]. In order to overcome these drawbacks the authors of [34] obtained a kinetic equation for a revised version of the Enskog theory (RET) using the diagram method. Résibois [35,36] proved the H-theorem for it. In 1985, a revised version of the DRS theory (RDRS) was proposed [37]. The kinetic equation of RDRS satisfies the H-theorem as well. A 689 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn generalized version of RDRS — GDRS was considered in [2,8], being based on the approaches of [1,3]. Here, in order to treat a more realistic model, a kinetic equation for dense classical systems is offered with an interparticle interaction potential in the shape of a multistep function, where its sequential derivation and the normal solution are obtained. In the paper by Stell et al. [38,39] the authors completed the construction of the kinetic variational theory (KVT) introducing a whole family of theories, namely, KVT I, KVT II and KVT III. The KVT III refers to the local energy constraint in maximizing entropy as distinguished from the global energy constraint (KVT II) or the hard-core constraint (KVT I). According to this classification, the kinetic equation of RDRS [37] is obtained by applying KVT III to the square well potential and KMFT [38] is derived from KVT I as its application to the potential with a smooth tail. In [39], the KVT III version of KMFT was proposed. In this case the quasiequilibrium binary distribution function (QBDF) of hard spheres is to be substituted by the full QBDF, which takes into account the full potential of the interaction. The main conclusion of the KVT I version of KMFT [39,40] is that the smooth part of the potential does not contribute explicitly to transport coefficients. This fact is caused by the approximations of the theory. When the full QBDF is used, as it is in the KVT III version of KMFT, transport coefficients are determined by the soft part of the potential. The case when the potential between the hard sphere and the attractive walls is considered as a smooth tail (instead of the constant potential in DRS) has also been investigated [41]. The kinetic equation was obtained for this potential applying a mean field approximation for a smooth tail. At this stage of investigations in the kinetic theory of dense gases and liquids, paper [1] by Zubarev and Morozov played a fundamental role. In this paper a new formulation of boundary conditions for the BBGKY hierarchy is presented. Such a formulation took into account correlations connected with local conservation laws. In the binary collision approximation, this modification of the weakening correla- tion condition by Bogolubov led to a kinetic equation for a system of hard spheres which is close in structure to the usual Enskog kinetic equation. Similar ideas were proposed independently by the authors of paper [37] at the derivation of a kinetic equation when an interaction between the particles is modelled by a square well potential. It is necessary to point out that a somewhat different modification of the Bogolubov approach [22] was presented in the papers by Rudyak [42–44]. This allows one to obtain an Enskog-like kinetic equation and extend it to the systems with a soft potential of interaction between the particles. The modification of weakening correlation conditions to a chain of equations by Bogolubov was developed in papers [2–12]. An important achievement of the given approach is the fact that the modified boundary condition gives the possibility [4,7,14] to derive consequently the kinetic equation of the RET theory [34] for the first time. On the basis of result [1], a kinetic equation was obtained [2,8] for systems with a multistep potential of interaction (in particular, the H-theorem was proved for this equation in paper [8]), whereas an Enskog-Landau kinetic equation 690 A consistent description of kinetics and hydrodynamics was derived [3,7] for a system of charged hard spheres. Normal solutions for the obtained kinetic equations were found using the Chapman-Enskog method. With the help of these solutions, the numerical calcu- lations of transport coefficients (bulk and shear viscosities, thermal conductivity) were carried out for Argon-like systems [2,45], ionized Argon [7] and mixtures [46]. It is important to stress that the RET kinetic equation, the kinetic equation for systems with multistep potentials of interaction and the Enskog-Landau kinetic equation for charged hard spheres were derived from the BBGKY hierarchy with a modified boundary condition in a binary collision approximation. Obviously, the presented equations have a restricted region of application. They cannot be ap- plied to the description of systems with significantly collective effects caused by Coulomb, dipole or other long-range forces of interaction between the particles. To describe the collective effects in systems with a long-range character of interaction it is necessary to consider higher-order approximations for collision integrals. An example of such an equation is the Lenard-Balescu kinetic equation [47–49] for Coulomb plasma. To analyze solutions to the BBGKY hierarchy with a modified boundary con- dition in higher-order approximations on interparticle correlations we have applied a concept of group expansions [11,12]. 2.2. The Liouville equation and the BBGKY hierarchy with a mo dified boundary condition Let us consider a system of N identical classical particles which are enclosed in volume V , with the Hamiltonian: H = N∑ j=1 p2j 2m + 1 2 N∑ j=1 N∑ k=1 j 6=k Φ (\|rjk\|) , (2.1) where Φ (\|rjk\|) is the interaction energy between two particles j and k; \|rjk\|=\|rj − rk\| is the distance between a pair of interacting particles; pj is the momentum of jth particle and m denotes its mass. A nonequilibrium state of such a system is described by the N-particle nonequilibrium distribution function ̺ ( xN ; t ) = ̺ (x1, . . . , xN ; t) which satisfies the Liouville equation: ( ∂ ∂t + iLN ) ̺ ( xN , t ) = 0, (2.2) where i = √ −1, xj = {rj,pj} is a set of phase variables (coordinates and mo- menta), LN is the Liouville operator: LN = N∑ j=1 L(j) + 1 2 N∑ j=1 N∑ k=1 j 6=k L(j, k), (2.3) 691 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn where L(j) = −i pj m ∂ ∂rj , L(j, k) = i ∂Φ (\|rjk\|) ∂rjk ( ∂ ∂pj − ∂ ∂pk ) . The function ̺ ( xN ; t ) is symmetrical with respect to permutations xl ⇆ xj of phase variables for an arbitrary pair of particles. This function satisfies the normalization condition ∫ dΓN̺ ( xN ; t ) = 1, dΓN = (dx)N/N ! = drdp. (2.4) In order to solve the Liouville equation (2.2), it is necessary to introduce a boundary condition. It must be chosen in such a way that the solution to the Liouville equation will correspond to a physical state of the system under con- sideration. Using the nonequilibrium statistical operator method by D.N.Zubarev [9,10,50,51] (NSO method), we shall search for such solutions to the Liouville equa- tion depending on time explicitly via the values for some set of observable variables, which is sufficient for describing a nonequilibrium state without depending on the initial time t0. The solution to the Liouville equation, which satisfies the initial condition ̺ ( xN ; t ) \|t=t0 = ̺q ( xN ; t0 ) , has the form: ̺ ( xN ; t, t0 ) = e−iLN (t−t0)̺q ( xN ; t0 ) . (2.5) We shall consider such times t≫ t0 when the details of the initial state of the system can be neglected. Then, to avoid the dependence on t0, let us average (2.5) with respect to the initial times from t0 to t and perform the boundary transition (t− t0) → ∞. As a result, one obtains [9,10,49]: ̺ ( xN ; t ) = ε 0∫ −∞ dt′ eεt ′ eiLN t′̺q ( xN ; t+ t′ ) , t′ = t0 − t, (2.6) where ε tends to +0 after the thermodynamic limit transition. It can be shown by straightforward differentiations that solution (2.6) satisfies the Liouville equation with an infinitesimal source in the right-hand side: ( ∂ ∂t + iLN ) ̺ ( xN , t ) = −ε ( ̺ ( xN , t ) − ̺q ( xN , t ) ) . (2.7) The source breaks the symmetry of the Liouville equation with respect to time inversion and selects retarded solutions which correspond to an abbreviated de- scription of a nonequilibrium state of the system. The auxiliary function ̺q ( xN ; t ) denotes a quasiequilibrium distribution function that is defined from an extreme 692 A consistent description of kinetics and hydrodynamics condition for the informational entropy of the system, provided the normaliza- tion condition is preserved and the average values for variables of the abbreviated description are fixed. The choice of ̺q ( xN ; t ) depends mainly on a nonequilibrium state of the system under consideration. In the case of low density gases, where times of free motion are essentially larger than collision times, higher-order distribution functions of particles become dependent on time only via one-particle distribution functions [22,52,53]. It does mean that an abbreviated description of nonequilibrium states is available, and the total nonequilibrium distribution function depends on time via f1(x; t). In such a case, the quasiequilibrium distribution function ̺q ( xN ; t ) reads [52,53]: ̺q ( xN ; t ) = N∏ j=1 f1(xj ; t) e , (2.8) where e is the natural logarithm base. Then, the Liouville equation with a small source (2.7) in view of (2.8) corresponds to the abbreviated description of time evolution of the system on a kinetic stage, when only a one-particle distribution function is considered as a slow variable. However, there are always additional quantities which vary in time slowly because they are locally conserved. In the case of a one-component system, the mass density ρ(r; t), momentum j(r; t) and total energy E(r; t) belong to such quantities. At long times they satisfy the gen- eralized hydrodynamics equations. Generally speaking, the equation for f1(x; t) must be conjugated with these equations. For low density gases, such a conjuga- tion can be done, in principle, with an arbitrary precision in each order on density. In high density gases and liquids, when a small parameter is absent, the correla- tion times corresponding to hydrodynamic quantities become commensurable with the characteristic times of varying one-particle distribution functions. Therefore, in dense gases and liquids, the kinetics and hydrodynamics are closely connected and should be considered simultaneously. That is why many-particle correlations, related to the local conservation laws of mass, momentum and total energy, can- not be neglected [1,5,9,10]. The local conservation laws affect the kinetic processes due to an interaction of the selected particles group with other particles of the system. This interaction is especially important in the case of high densities, and it must be taken into consideration. Then, at the construction of kinetic equa- tions for high densities, it is necessary to choose the abbreviated description of a nonequilibrium system in the form to satisfy the true dynamics of conserved quan- tities automatically. To this end, the densities of hydrodynamic variables should be included together with the one-particle distribution function f1(x; t) into the initial set of parameters of the abbreviated description [1,5,9,10]. The next phase functions correspond to the densities of the hydrodynamic variables ρ(r; t), j(r; t) and E(r; t): 693 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn ρ̂(r) = ∫ dp n̂1(x)m, ĵ(r) = ∫ dp n̂1(x)p, Ê(r) = ∫ dp n̂1(x) p2 2m + 1 2 ∫ dr′ dp dp′ n̂2(x, x ′)Φ(\|r − r′\|), (2.9) where n̂1(x) and n̂2(x, x ′) are one- and two-particle microscopic phase densities by Klimontovich [24]: n̂1(x) = N∑ j=1 δ(x− xj) = N∑ j=1 δ(r − rj)δ(p− pj), (2.10) n̂2(x, x ′) = N∑ j 6=k=1 δ(x− xj)δ(x ′ − xk). (2.11) Relations (2.9) show a distinctive role of the potential interaction energy. Contrary to ρ(r; t) = 〈ρ̂(r)〉t and j(r; t) = 〈ĵ(r)〉t, the nonequilibrium values of the total energy E(r; t) = 〈Ê(r)〉t cannot be expressed via the one-particle distribution function f1(x; t) = 〈n̂1(x)〉t only, because in order to evaluate the potential part of Eint(r; t) = 〈Êint(r)〉t it is necessary to involve the two-particle distribution function f2(x, x ′; t) = 〈n̂2(x, x ′)〉t. Here Êint(r) = 1 2 ∫ dr′ dp dp′ n̂2(x, x ′)Φ(\|r − r′\|) (2.12) is the density of the potential energy of interaction. The next conclusion can be formulated as follows. If the one-particle distribution function f1(x; t) is chosen as a parameter of an abbreviated description, then the density of the interaction energy (2.12) can be considered as an additional independent parameter. One can find the quasiequilibrium distribution function ̺q ( xN ; t ) from the condition of extremum for a functional of the informational entropy Sinf(t) = − ∫ dΓN ̺ ( xN ; t ) ln ̺ ( xN ; t ) at fixed average values of 〈n̂1(x)〉t = f1(x; t), 〈Êint(r)〉t = Eint(r; t), including the normalization condition for ̺q ( xN ; t ) [5,9,10,49]: ∫ dΓN̺q ( xN ; t ) = ∫ dΓN̺ ( xN ; t ) = 1. This is equivalent to finding an unconditional extreme for the functional L(̺) = ∫ dΓN̺ ( xN ; t ){ 1− ln ̺ ( xN ; t ) − Φ(t)− ∫ drβ(r; t)Êint(r)− ∫ dx a(x; t)n̂1(x) } , where Φ(t), β(r; t), a(x; t) are the Lagrange multipliers. Taking the variation δ δ̺ L(̺), after some simple manipulations one obtains the quasiequilibrium distribution function ̺q ( xN ; t ) = exp { −Φ(t) − ∫ dr β(r; t)Êint(r)− ∫ dx a(x; t)n̂1(x) } , (2.13) Φ(t) = ln ∫ dΓN exp { − ∫ dr β(r; t)Êint(r)− ∫ dx a(x; t)n̂1(x) } .(2.14) 694 A consistent description of kinetics and hydrodynamics Here, Φ(t) is the Massieu-Planck functional. It is determined from the condition of normalization for the distribution ̺q ( xN ; t ) . Relation (2.13) was obtained for the first time in [1]. To determine the physical meaning of parameters β(r; t) and a(x; t) let us rewrite ̺q ( xN ; t ) (2.13) in the form: ̺q ( xN ; t ) = exp { −Φ(t) − ∫ dr β(r; t)Ê ′(r)− ∫ dx a′(x; t)n̂1(x) } , (2.15) Φ(t) = ln ∫ dΓN exp { − ∫ dr β(r; t)Ê ′(r)− ∫ dx a′(x; t)n̂1(x) } , where Ê ′(r) is the density of the total energy in a reference frame which moves together with a system element of the mass velocity V (r; t) [5] Ê ′(r) = Ê(r)− V (r; t)ĵ(r) + m 2 V 2(r; t)n̂(r). (2.16) Here n̂(r) = ∫ dp n̂1(x) is the density of the particles number. Parameters β(r; t) and a′(x; t) in (2.15) are determined from the conditions of self-consistency, namely, the equality of quasi-average values 〈n̂1(x)〉tq and 〈Ê ′(r)〉tq with their real averages 〈n̂1(x)〉t, 〈Ê ′(r)〉t: 〈n̂1(x)〉tq = 〈n̂1(x)〉t = f1(x; t), 〈Ê ′(r)〉tq = 〈Ê ′(r)〉t, here 〈. . .〉tq = ∫ dΓN . . . ̺q ( xN ; t ) . (2.17) In these transformations the parameters a′(x; t) and a(x; t) are connected by the relation a′(x; t) = a(x; t)− β(r; t) { p2 2m − V (r; t)p+ m 2 V 2(r; t) } . In the case, when the conditions (2.17) take place, one can obtain some relations taking into account self-consistency conditions and varying the modified Massieu- Planck functional Φ(t) after (2.15) with respect to parameters β(r; t) and a′(x; t) δΦ(t) δβ(r; t) = −〈Ê ′(r)〉tq = −〈Ê ′(r)〉t, δΦ(t) δa(x; t) = −〈n̂1(x)〉tq = −〈n̂1(x)〉t = −f1(x; t). (2.18) It means that parameter β(r; t) is conjugated to the average energy in an accompa- nying reference frame, and a′(x; t) is conjugated to the nonequilibrium one-particle distribution function f1(x; t). To determine the physical meaning of these param- eters let us define the entropy of a system taking into account the self-consistency conditions (2.17): S(t) = −〈ln ̺q ( xN ; t ) 〉tq = Φ(t) + ∫ dr β(r; t)〈Ê ′(r)〉t + ∫ dx a′(x; t)〈n̂1(x)〉t. (2.19) 695 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn Taking functional derivatives of S(t) (2.19) with respect to 〈Ê ′(r)〉t and 〈n̂1(x)〉t at fixed corresponding averaged values gives the following thermodynamic relations: δS(t) δ〈Ê ′(r)〉t = β(r; t), δS(t) δf1(x; t) = a′(x; t). (2.20) Hence, β(r; t) is an analogue of the inverse local temperature. Two limiting cases follow from the structure of expression (2.19) for entropy. At a′(x; t) = −β(r; t)µ(r; t), (2.19) transforms into an expression for entropy which corresponds to the hydrodynamic description of the nonequilibrium state of the system [5,9,10]: S(t) = Φ(t) + ∫ dr β(r; t) ( 〈Ê ′(r)〉t − µ(r; t)〈n̂(r)〉t ) , (2.21) with the following quasiequilibrium distribution function [5,9,10]: ̺q ( xN ; t ) = exp { −Φ(t)− ∫ dr β(r; t) ( Ê ′(r)− µ(r; t)n̂(r) )} , (2.22) Φ(t) = ln ∫ dΓN exp {∫ dr β(r; t) ( Ê ′(r)− µ(r; t)n̂(r) )} and the corresponding self-consistency conditions for the definition of thermody- namic parameters β(r; t), µ(r; t) (local value of the chemical potential): 〈n̂(r)〉tq = 〈n̂(r)〉t, 〈Ê ′(r)〉tq = 〈Ê ′(r)〉t. In the case when the contribution of the interaction energy between the par- ticles can be neglected (a dilute gas), the quasiequilibrium distribution function (2.15) has the form: ̺q ( xN ; t ) = exp { −Φ(t)− ∫ dr β(r; t)Ê ′ kin(r)− ∫ dx a′(x; t)n̂1(x) } , or, taking into account (2.16) and the relation between a′(x; t) and a(x; t), one obtains: ̺q ( xN ; t ) = exp { −Φ(t) − ∫ dx a(x; t)n̂1(x) } , (2.23) Φ(t) = ln ∫ dΓN exp { − ∫ dx a(x; t)n̂1(x) } , where Ê ′ kin(r) = Êkin(r)− V (r; t)ĵ(r) + m 2 V 2(r; t)n̂(r), Êkin(r) = ∫ dp p2 2m n̂1(x) is the density of the kinetic energy. Now, determining in (2.23) parameter a(x; t) with the help of the self-consistency condition 〈n̂(x)〉tq = 696 A consistent description of kinetics and hydrodynamics 〈n̂(x)〉t, one can show [5] that (2.23) for ̺q ( xN ; t ) transforms into distribution (2.8) when it is assumed that the only parameter of the abbreviated description for a nonequilibrium state of the system is a one-particle distribution function. As it is known [51,53], the quasiequilibrium distribution function (2.23) corresponds to the Boltzmann entropy of a dilute gas: SB(t) = − ∫ dx f1(x; t) ln f1(x; t) e . (2.24) In a general case, when kinetic and hydrodynamic processes are considered simultaneously, the quasiequilibrium distribution function (2.15) or (2.13) can be rewritten in a somewhat different form. This form is more convenient for the comparison with ̺q ( xN ; t ) (2.8), obtained in a usual way [22,52], when f1(x; t) is the only parameter of the abbreviated description. First of all, let us note that one can include parameter Φ(t) from (2.13) into parameter a(x; t) as a term which does not depend on x. Parameter a(x; t) in ̺q ( xN ; t ) can be excluded with the help of the self-consistency condition 〈n̂(x)〉tq = 〈n̂(x)〉t = f1(x; t). Reduction of ̺q ( xN ; t ) results in ̺q ( xN ; t ) = exp { −UN ( rN ; t )} N∏ j=1 f1(xj ; t) u(rj ; t) , (2.25) where functions u(rj ; t) are obtained from the relations u(rj ; t) = ∫ drN−1 (N − 1)! exp { −UN ( r, rN−1; t )} N∏ j=2 n(rj; t) u(rj ; t) , (2.26) UN ( rN ; t ) = UN (r1, . . . , rN ; t) = 1 2 N∑ j 6=k=1 Φ(\|rj − rk\|)β(rk; t), n(r; t) = 〈n̂(r)〉t = ∫ dp f1(x; t) is the nonequilibrium particles concentration. In expression (2.25), UN (r; t) and u(rj ; t) respectively, depend explicitly and implic- itly on n(r; t) and β(r; t) (or 〈Ê ′(r)〉t). To obtain the ordinary Bogolubov scheme [22,52], it is necessary to put UN(r; t) = 0 in (2.25) and (2.26). Then, one can define u = e, and (2.25) transforms into the quasiequilibrium distribution (2.8), as it should be. In a general case, u(r; t) is a functional of the nonequilibrium density of particles number n(r; t) and β(r; t), which is an analogue of the inverse local temperature. Nevertheless, one should handle this analogy with care, as far as definition (2.25) can describe states which are far from local equilibrium. In particular, f1(x; t) can differ considerably from the local Maxwellian distribution. The entropy expression (2.19) can be transformed according to the structure of the quasiequilibrium distribution function (2.25) S(t) = ∫ dr β(r; t)〈Êint(r)〉t − ∫ dx f1(x; t) ln f1(x; t) u(r; t) . (2.27) 697 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn Here the potential and kinetic parts are separated. In the case of low density gases, the influence of the potential energy can be neglected and u(r; t) = e. Then, expression (2.27) tends to the usual Boltzmann entropy. Thus, determining the quasiequilibrium distribution function ̺q ( xN ; t ) (2.25) and entropy S(t) (2.27) of the system, when the nonequilibrium one-particle dis- tribution function, as well as the average values of densities for the number of particles, momentum and energy are parameters of an abbreviated description of the nonequilibrium state which are locally conserved, the Liouville equation with the source (2.7) can be presented in the form [1,5]: ( ∂ ∂t + iLN ) ̺ ( xN ; t ) = −ε ( ̺ ( xN ; t ) − exp { −UN ( rN ; t )} N∏ j=1 f1(xj ; t) u(rj ; t) ) . (2.28) Further, on the basis of this equation, one obtains the BBGKY hierarchy for nonequilibrium distribution functions of classical particles with modified bounda- ry conditions which take into account the nonequilibriumnes of the one-particle distribution function, as well as the local conservation laws. To obtain the first equation of the hierarchy, let us integrate with respect to the variables xN−1 = {x2, . . . , xN} the both parts of (2.28). Taking into account (2.10), (2.11) and (2.17), one obtains: ( ∂ ∂t + iL(1) ) f1(x1; t) + ∫ dx2 iL(1, 2)f2(x1, x2; t) = 0. (2.29) Integrating now (2.28) over the variables xN−2 = {x3, . . . , xN}, after simple trans- formations one can obtain an equation for the two-particle distribution function f2(x1, x2; t) which differs from the corresponding equation in the Bogolubov hier- archy [1,3] by a source in the right-hand side: ( ∂ ∂t + iL2 ) f2(x1, x2; t) + ∫ dx3 ( iL(1, 3) + iL(2, 3) ) f3(x1, x2, x3; t) = −ε ( f2(x1, x2; t)− g2(r1, r2; t)f1(x1; t)f1(x2; t) ) . (2.30) In this equation, L2 = L(1) + L(2) + L(1, 2) is the Liouville two-particle oper- ator and g2(r1, r2; t) denotes the binary coordinate distribution function for the quasiequilibrium state (2.25); f2(x1, x2; t) is the nonequilibrium two-particle dis- tribution function: f2(x1, x2; t) = 〈n̂2(x1, x2; t)〉t = ∫ dΓN−2 ̺(x1, x2, x N−2; t). Similarly, integrating equation (2.28) over xN−s = {xs+1, . . . , xN} phase variables, one derives next equations of the chain with the corresponding sources which define boundary conditions for reduced distribution functions. For the s-particle nonequilibrium distribution function fs(x s; t) = 〈n̂s(x s)〉t = ∫ dΓN−s ̺(x1, . . . , xs, x N−s; t) 698 A consistent description of kinetics and hydrodynamics one has the following equation: ( ∂ ∂t + iLs ) fs(x s; t) + ∫ dxs+1 s∑ j=1 iL(j, s+ 1)fs+1(x s+1; t) = −ε ( fs(x s; t)− gs(rs; t) s∏ j=1 f1(xj ; t) ) , (2.31) where Ls = s∑ j=1 L(j) + 1 2 s∑ j 6=k=1 L(j, k), gs(r s; t) = fs(r s; t) / s∏ j=1 n(rj ; t), n̂s(r s) = s∑ j1=1 . . . s∑ js=1 j1 6=j2 6=...js s∑ k=1 δ(rk − r′ jk ), fs(r s; t) = 〈n̂s(r s)〉tq. (2.32) As usual, we shall assume that the principle of weakening spatial correlations is valid for the quasiequilibrium state. Then, in the thermodynamic limit, the coordinate distribution functions fs(r s; t), gs(r s; t) satisfy the boundary relations: lim (min \|rj−rk\|)→∞ fs(r1, . . . , rs; t) = s∏ j=1 n(rj ; t), (2.33) lim (min \|rj−rk\|)→∞ gs(r1, . . . , rs; t) = 1. (2.34) Therefore, taking into account “slow” hydrodynamical variables (the density of the interaction energy in the present case) leads to a modification of boundary conditions for the chain of equations (2.29)–(2.31) for nonequilibrium distribution functions. In order to reproduce the usual Bogolubov boundary conditions for the weakening of correlations [22], it is necessary to replace all gs(r s; t) by their limiting values (2.34). Such a replacing is valid in the case of small density, however, new boundary conditions can be more useful for dense gasses, since they automatically take into account spatial correlations connected with the interaction of a pair of particles with the rest of the particles of the system. It is obvious that the influence of such an interaction increases as the density rises. We note that the chain of equations (2.29)–(2.31) requires us to add equations for coordinate quasiequilibrium distribution functions which are functionals on the nonequilibrium density of the particles number n(r; t) and the inverse local tem- perature β(r; t). In particular, in [54] it is shown that the binary quasiequilibrium distribution function g2(r1, r2; t) is connected with the pair quasiequilibrium cor- relation function h2(r1, r2; t) = g2(r1, r2; t)−1 which satisfies the Ornstein-Zernike equation: h2(r1, r2; t) = c2(r1, r2; t) + ∫ dr3 c2(r1, r3; t)h2(r2, r3; t)n(r3; t), (2.35) 699 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn where c2(r1, r2; t) is a direct quasiequilibrium correlation function. Thus, the chain of equations (2.29)–(2.31) is distinguished from the usual Bo- golubov hierarchy by the existence of sources in the right-hand sides, beginning from the second equation, and it takes into account both one-particle and collec- tive hydrodynamical effects. It is important to investigate solutions to this chain of equations in the simplest approximations which lead to model kinetic equations for dense gases and simple liquids. We shall consider the most investigated binary collision approximation. 2.3. Binary collision approximation In this section, we consider as an example of the use of hierarchy (2.31), the simplest approximation of binary collisions which in the case of the Bogolubov ordinary conditions of correlations weakening leads to the Boltzmann equation for f1(x1; t). In equation (2.30) for f2(x1, x2; t) we omit the term with the three-particle distribution function, i.e. we take into account the influence of the “medium” on the evolution of the distinguished pair of particles only through the correlation corrections in the boundary condition. Then, we arrive at the equation: ( ∂ ∂t + iL2 + ε ) f2(x1, x2; t) = εg2(r1, r2; t)f1(x1; t)f1(x2; t). (2.36) The formal solution of (2.36) has the form: f2(x1, x2; t) = ε 0∫ −∞ dτ e(ε+iL2)τg2(r1, r2; t+ τ)f1(x1; t+ τ)f1(x2; t+ τ). (2.37) Following the Abel theorem [9,10,51,55,56], this solution can be written in the form: f2(x1, x2; t) = lim τ→−∞ eiL2τg2(r1, r2; t+ τ)f1(x1; t+ τ)f1(x2; t+ τ). (2.38) Substituting expression (2.38) into equation (2.29), one obtains a kinetic equation in the binary collision approximation ( ∂ ∂t + iL(1) ) f1(x1; t) = Icoll(x1; t), (2.39) where Icoll(x1; t) = ∫ dx2 iL(1, 2) lim τ→−∞ eiL2τg2(r1, r2; t+τ)f1(x1; t+τ)f1(x2; t+τ) (2.40) is a collision integral. It is necessary to point out that equation (2.35) for the pair quasiequilibrium correlation function must be added to kinetic equation (2.29). Equation (2.35) takes into account an essential part of the many-particles corre- lations. 700 A consistent description of kinetics and hydrodynamics 2.3.1. Kinetic equation of the revised Enskog theory In papers [1,3] it was shown how the collision integral (2.40) in the case of a system of hard spheres transforms into the collision integral of the RET kinetic equation [34]. For subsequent manipulations, it is convenient to represent the par- ticle interaction potential in the form: Φhs(\|rjk\|) = lim a→+∞ Φa(\|rjk\|), Φa(\|rjk\|) = { a, \|rjk\| < σ, 0, \|rjk\| > σ, (2.41) where σ is a hard sphere diameter. Since the potential Φhs(\|rjk\|) is strongly singular (in particular, the operator iL(1, 2) is not well defined), we shall operate with the potential Φa(\|rjk\|), when deriving the kinetic equation, and set a → +∞ in the final expressions. The limit τ → −∞ in the collision integral (2.40) is mathematically formal. At the physical level of description, this limit assumes that \|τ \| ≫ τ0, where τ0 > 0 is some characteristic time scale. Depending on the choice of τ0, we obtain different stages of the evolution of the system (kinetic, hydrodynamic). At the kinetic stage, τ0 is a characteristic interaction time for which the singular potential (2.41) is an arbitrarily small quantity (τ0 → +0). Therefore, the limit τ − τ0 → −∞ keeps its form even in the case τ → −0, provided τ0 is of the higher order of smallness relatively to τ : lim τ→−0 { lim τ0→+0 τ τ0 } = −∞. (2.42) In this case the collision integral (2.40) [1,3,54] for the hard spheres interparticle potential of interaction transforms into the following final expression: Ihscoll(x1; t) = σ2 ∫ dr̂12 dv2 θ(r̂12 · g)(r̂12 · g)× (2.43) { ghs2 (r1, r1 + σ+r̂12; t)f1(r1, v ′ 1; t)f1(r1 + σ+r̂12, v ′ 2; t)− ghs2 (r1, r1 − σ+r̂12; t)f1(r1, v1; t)f1(r1 − σ+r̂12, v2; t) } , where v′ 1 = v1 + r̂12(r̂12 · g), v′ 2 = v2 − r̂12(r̂12 · g). (2.44) The collision integral in the form (2.43) is identical to the collision integral of the RET theory first introduced by van Beijeren and Ernst [34] on the basis of a diagram method. As Résibois showed [35,36], the kinetic equation of the RET theory satisfies an H-theorem. We represent the collision integral (2.43) in a more compact form by using the two-particle quasi-Liouvillian (evolution pseudo-operator) T̂ (1, 2) [26]. Then, the kinetic equation takes the form: ( ∂ ∂t + v1 ∂ ∂r1 ) f1(x1; t) = ∫ dx2 T̂ (1, 2)g hs 2 (r1, r2; t)f1(x1; t)f1(x2; t), (2.45) 701 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn T̂ (1, 2) = σ2 ∫ dr̂12 θ(r̂12 · g)(r̂12 · g)× (2.46) { δ(r12 + σ+r̂12)B̂(r̂12)− δ(r12 − σ+r̂12) } , B̂(r̂12)Ψ(v1, v2) = Ψ(v′ 1, v ′ 2), (2.47) where Ψ is an arbitrary function of the velocities. Thus, on the basis of the considered approach we have derived the kinetic equation of the RET theory without additional phenomenological assumptions. We have shown that within the framework of the employed method this equation corresponds to the simplest pair-collision approximation without allowance for retardation in time. 2.3.2. Kinetic equation for a multistep potential of intera ction The collision integral (2.40) is still rather complicated to be written in an explicit form for an arbitrary potential of interaction. As it is now known [30,32], only in two particular cases, namely, for hard spheres and square-well potentials, this integral is reduced to an analytical form. In order to extend these previous results, we consider the interparticle potential in the form of a multistep function Φjk ≡ Φms jk = lim ε0→∞ Φε0 jk(rjk), (2.48) where Φε0 jk(rjk) =    ε0, rjk < σ0; εl, σl−1 < rjk < σl, l = 1, . . . , N∗; 0, σN∗ < rjk; (2.49) and N∗ is the total number of walls except for a hard sphere wall. Potential of a hard sphere contains strong singularity (operator L(j, k) is hard to define). That is why we shall deal with the potential Φε0 jk(rjk) and only in final expressions we shall put ε0 → ∞. Let us for convenience separate the systems of attractive and repulsive walls. Let n∗ be the number of repulsive walls at the distance σri ≡ σi and of the height △εri = εri − εri+1 > 0, where εri ≡ εi, i = 1, . . . , n∗; and m∗ – the number of attractive walls with the corresponding parameters σaj ≡ σj+n∗, △εaj = εaj+1 − εaj > 0, where εaj ≡ εj+n∗, j = 1, . . . , m∗; εrn∗+1 = εa1, ε a m∗+1 = 0, N∗ = n∗ +m∗. Thus, the parameters σ0 (hard sphere diameter), n∗, σri, △ε r i, m ∗, σaj and △εaj completely determine the geometry of the multistep potential (see figure 1 where a specific case n∗ = 1, m∗ = 3 is shown and the multistep function approximates some real smooth potential). The limit τ → −∞ in the collision integral (2.40) is a formal one in the mathe- matical sense. On the physical level of description this means that \|τ \| ≫ τ0, where τ0 > 0 is some characteristic interval of time. For different values of τ0 we can in- vestigate different stages of the evolution of the system. In the Boltzmann theory [57] \|τ \| ≫ τc, where τc is the time of binary interactions (collision time). On the other hand, \|τ \| is far less than the characteristic scale of time τm for hydrodynamic 702 A consistent description of kinetics and hydrodynamics e e e e s s s s s4 4 3 3 2 21 1 0 e0 Figure 1. The multistep potential at n∗ = 1 and m∗ = 3. variables. The above situation is possible because of the fact that for dilute gases the kinetic (τ ∼ τc) and hydrodynamic (τ ∼ τm) stages of the evolution are far from one another in time, i.e. τc ≪ τf ≪ τm, where τf is the characteristic time of free motion. The pattern is qualitatively different in dense gases and liquids with a realistic smooth potential of interaction. The kinetic and hydrodynamic stages appear to be closely connected. Besides, such quantities as the length of free motion and the time of interaction are not defined in a usual way, because all particles make influence on the dynamics of interaction for some chosen pair of particles. However, for special types of potentials such a classification of times remains valid even for high densities. For a multi-step potential the region Ω of binary in- teractions consists of the following subregions [σk− △r0, σk+ △r0] , k = 0, 1, . . . , N∗, where △r0 → +0 due to the singular nature of the potential under consideration. This potential has the finite range max {σk} = σN∗ > 0 of action. We can intro- duce the following set of specific time intervals: τ0 =△r0/g0 → +0 is the time of pair interactions on the walls, △τ = min{σk − σk−1} /g0 > 0 is the time of motion between the two nearest neighbouring walls and twhole = σN∗/g0 > 0 is the time of motion of the whole systems of walls for some pair of particles, where g0 is an average relative velocity of two particles. As the potential contains the horizontal parts, where the force of the interparticle interaction is equal to zero, it is also possible to introduce time τf as an average time of free motion of particles in the system. Changing the geometry of the potential and increasing the density, we can make this time arbitrarily small in order to match the relation τ0 ≪ τf ≪△τ < twhole < τm. (2.50) 703 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn The kinetic stage of the evolution corresponds to small times of order τ ∼ τf . Therefore, as far as the relation (2.50) is satisfied, the formal limit τ → −∞ should be considered as τ/τ0 → −∞ or merely as \|τ \| ≫ τ0. For smooth non- singular potentials the region of interaction has some non-zero size and τ0 is finite. For singular potentials the case τ0 → +0 is possible and the limit τ/τ0 → −∞ remains valid even for τ → −0 if only τ0 is a value of the higher infinitesimal order lim τ→−0 { lim τ0→+0 τ τ0 } → −∞. (2.51) Thus, the formal limit τ → −0 can be applied to the collision integral (2.40) in the sense of (2.50) and (2.51). As it is now well established, the limit τ → −0 leads to the kinetic equations of the RET [3,34,38] and RDRS [37] theories. As it was shown in [2,8], the collision integral (2.40) for potential (2.48) taking into account (2.50), (2.51), can be presented in a compact form: Ims coll(x1; t) = ∫ dx2 T̂12g2(r1, r2; t)f1(x1; t)f1(x2; t) (2.52) in terms of the two-particle pseudo-Liouville operator T̂12 = T̂hs + n∗∑ i=1 ∑ p=b,c,d T̂ p ri + m∗∑ j=1 ∑ p=b,c,d T̂ p aj , (2.53) which consists of the pseudo-Liouville operators for a hard sphere wall, for ith repulsive wall and for jth attractive wall, correspondingly: T̂hs = σ2 0 ∫ dr̂12(r̂12 · g)θ(r̂12 · g) { δ(r12 + σ+ 0 r̂12)B̂ a(r̂12)− δ(r12 − σ+ 0 r̂12) } , (2.54) T̂ p ri = σ2 ri ∫ dr̂12(r̂12 · g)θpri(r̂12 · g) { δ(r12 + σ1,p ri r̂12)B̂ p ri(r̂12)− δ(r12 − σ2,p ri r̂12) } , (2.55) T̂ p aj = σ2 aj ∫ dr̂12(r̂12 · g)θpaj(r̂12 · g) { δ(r12 + σ1,p aj r̂12)B̂ p aj(r̂12)− δ(r12 − σ2,p aj r̂12) } . (2.56) Here, θbri(r̂12 · g) = θ(−r̂12 · g), θbaj(r̂12 · g) = θ(r̂12 · g), θcri(r̂12 · g) = θ(r̂12 · g− αr i), θcaj(r̂12 · g) = θ(−r̂12 · g− αa j), θdri(r̂12 · g) = θ(r̂12 · g)θ(αr i − r̂12 · g), θdaj(r̂12 · g) = θ(−r̂12 · g)θ(αr i + r̂12 · g) (2.57) are the corresponding unit step functions, σ1,b ri = σ+ ri , σ2,b ri = σ− ri , σ1,b aj = σ− aj , σ2,b aj = σ+ aj , σ1,c ri = σ− ri , σ2,c ri = σ+ ri , σ1,c aj = σ+ aj , σ2,c aj = σ− aj , (2.58) σ1,d ri = σ+ ri , σ2,d ri = σ+ ri , σ1,d aj = σ− aj , σ2,d aj = σ− aj , 704 A consistent description of kinetics and hydrodynamics and B̂p are operators of the velocity displacement caused by an interaction of the p-type at each of the walls, namely, B̂a(r̂12)Ψ(v1, v2) = B̂d ri(r̂12)Ψ(v1, v2) = B̂d aj(r̂12)Ψ(v1, v2) = Ψ(v′ 1, v ′ 2), B̂b ri(r̂12)Ψ(v1, v2) = Ψ(v′′ r1, v ′′ r2), B̂b aj(r̂12)Ψ(v1, v2) = Ψ(v′′ a1, v ′′ a2), B̂c ri(r̂12)Ψ(v1, v2) = Ψ(v′′′ r1, v ′′′ r2), B̂c aj(r̂12)Ψ(v1, v2) = Ψ(v′′′ a1, v ′′′ a2). (2.59) Physically, the values σ1,p ri (σ1,p aj ) (2.58) correspond to the distances between the particles just after the interaction of the p-type on the ith repulsive (jth attractive) wall of the potential, whereas the values σ2,p ri (σ2,p aj ) correspond to the distances just before the collision. The H-theorem for the kinetic equation with a multistep potential of interac- tion (2.52) was proved in [8]. The normal solution by using the boundary conditions method [40,58] was found in [2,45]. On the basis of the solution found, numerical calculations of transport coefficients for liquid Argon along the liquid-vapour curve were performed. 2.3.3. Enskog-Landau kinetic equations for systems of char ged hard spheres We consider kinetic equation (2.39) in the binary collision approximation with collision integral (2.40), when the interaction potentials Φ(\|rjk\|) of classical par- ticles at short distances can be modelled by the hard-sphere potential Φhs(\|rjk\|) (2.41), and at large distances by a certain long-range smooth “tail” Φl(\|rjk\|), i.e. Φ(\|rjk\|) = Φhs(\|rjk\|) + Φl(\|rjk\|), (2.60) where Φl(\|rjk\|) = { 0, \|rjk\| < σ, Φl(\|rjk\|), \|rjk\| > σ. (2.61) We note that breaking the particle interaction potential (2.60) into short- and long-range parts is not unique and, therefore, there arises the problem of optimal separation. These problems have often been discussed in equilibrium statistical mechanics [59]. The similar ideas about the separation of the interaction potential at the derivation of kinetic equations were used by Rudyak [60,61]. With allowance for (2.3) and (2.41), the collision integral (2.40) for the inter- particle potential (2.60), (2.61) takes the form: Icoll(x1; t) = I1(x1; t) + I2(x1; t), (2.62) I1(x1; t) = − lim a→+∞ σ+∫ 0 dr12 r 2 12 ∫ dr̂12 ∫ dv2 iL a(1, 2) lim τ→−0 eiL a 2τ × g2(r1, r2; t+ τ)f1(x1; t+ τ)f1(x2; t+ τ), (2.63) 705 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn I2(x2; t) = − ∞∫ σ+ dr12 r 2 12 ∫ dr̂12 ∫ dv2 iL l(1, 2) lim τ→−∞ e i ( L (0) 2 +Ll(1,2) ) τ × g2(r1, r2; t+ τ)f1(x1; t+ τ)f1(x2; t+ τ), (2.64) where L (0) 2 = L(1) + L(2), Ll(1, 2) = i r̂12 m∗ ∂ ∂r12 Φl(\|r12\|) ( ∂ ∂v1 − ∂ ∂v2 ) (2.65) and we have used the idea of time separation of the interactions: instant collisions of hard spheres (τ → −0) and an extended process of interaction with the long- range potential (τ → −∞). In accordance with the results obtained in subsection 3.1, the first term in the right-hand side of (2.62) is identical to the collision integral of the hard-sphere system (the right-hand side of (2.45)): I1(x1; t) = ∫ dx2 T̂ (1, 2)g2(r1, r2; t)f1(x1; t)f1(x2; t) (2.66) with the only difference that here g2(r1, r2; t) is a quasiequilibrium binary distri- bution function of the particles in the system with a total interparticle interaction potential (2.60). In the case when the long-range part of the interaction potential is absent (Φl(r) = 0), the first part I1(x1; t) of the collision integral is identical to the collision integral of the RET theory; the second part I2(x1; t) is then identically equal to zero. In the second limiting case, when the density of the system is low (rarefied gases: n → 0, g2(r1, r2; t) → 1 and the hard-sphere part of the potential vanishes (σ → +0), the second part I2(x1; t) of the collision integral is identical to the collision integral of the Boltzmann equation [30], and at the same time I1(x1; t)→0. In a special case, when the long-range interaction is weak, we make an expan- sion for exp{(iL(0) 2 +iLl(1, 2))τ}, restricting ourselves to the term linear in iLl(1, 2). Then, the second part I2(x1; t) (2.64) reduces to the form: I2(x1; t) = I (0) 2 (x1; t) + I (1) 2 (x1; t), (2.67) I (0) 2 (x1; t) = − ∞∫ σ+ dr12 r 2 12 ∫ dr̂12 ∫ dv2 iL l(1, 2) lim τ→−∞ eiL (0) 2 τ × g2(r1, r2; t+ τ)f1(x1; t+ τ)f1(x2; t+ τ), (2.68) I (1) 2 (x1; t) = ∞∫ σ+ dr12 r 2 12 ∫ dr̂12 ∫ dv2 iL l(1, 2) lim τ→−∞ eiL (0) 2 τ × (2.69) τ∫ 0 dτ ′ e−iL (0) 2 τ ′ iLl(1, 2)eiL (0) 2 τ ′g2(r1, r2; t+ τ)f1(x1; t+ τ)f1(x2; t+ τ). The first term I (0) 2 (x1; t) is a generalization of the Vlasov mean field in KMFT [38], and the second I (1) 2 (x1; t) is a generalized Landau collision integral with allowance 706 A consistent description of kinetics and hydrodynamics for retardation in time in the approximation of the second order on interaction. Indeed, if we set formally g2(r; t) ≡ 1 (rarefied gases) and σ → 0, and completely ignore the spatial inhomogeneity of f1(x1; t) and the time retardation, then the second term I (1) 2 (x1; t) (2.69) is transformed into the ordinary Landau collision integral [62]. In the case when Φl(\|r12\|) is the Coulomb interaction potential, the complete collision integral (2.62), (2.66), (2.67) can be called the Enskog-Landau collision integral for a system of charged hard spheres which, in contrast to the ordinary Landau collision integral, does not diverge at short distances. However, at large distances we will observe a divergence, as usual. To eliminate this, it is necessary to take into account the effects of screening [24,63,64]. To avoid this problem sequentially we have to consider the kinetic equation with taking into account the dynamical screening effects [63]. But this way is impossible with the Enskog-Landau kinetic equation. The only thing we can do for further calculation is to change the upper integral limit to some finite value which could have a meaning of the statical screening value in our system (see the following subsections). To solve this problem we must consider dynamical screening effects. Using the boundary conditions method, a normal solution to the generalized Enskog-Landau kinetic equation was found in [65,66]. This solution coincides with that obtained in [3,7] for a stationary case. On the basis of normal solutions, numerical calculations of such transport coefficients as viscosity and thermal con- ductivity were performed for once-ionized Argon [7] and for mixtures of ionized inert gases [46]. 2.4. Modified group expansions for the construction of solut ions to the BBGKY hierarchy 2.4.1. Modified group expansions Recently, a kinetic equation of the revised Enskog theory for a dense system of hard spheres (2.45) and an Enskog-Landau kinetic equation for a dense system of charged hard spheres with collision integrals (2.66), (2.67) have been obtained from the BBGKY hierarchy in the binary collisions approximation [3,7]. It should be noted that this approximation does not correspond to the usual two-particle approximation inherent in the Boltzmann theory, because an essential part of the many-particle correlations is implicitly taken into account by the pair quasiequi- librium distribution function g2(r1, r2; t). To analyze solutions to the BBGKY hierarchy (2.31) [1,7] in higher approxi- mations on interparticle correlations, it is more convenient to use the concept of group expansions [67–69]. This was applied to the BBGKY hierarchy in previous investigations [26,28,52,67,69] using the boundary conditions which correspond to the weakening correlations principle by Bogolubov [22]. The same conception was envolved in papers by Zubarev and Novikov [52], where a diagram method for obtaining solutions to the BBGKY hierarchy was developed. To analyze the BBGKY hierarchy (2.31), we turn to the papers by Zubarev and Novikov [52] and earlier ones by Green [67] and Cohen [68,69], and pass from 707 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn the nonequilibrium distribution functions fs(x s; t) to the irreducible distribution ones Gs(x s; t), which can be introduced by the equalities presented in [52,67]. In our case, with some modifications we obtain: f1(x1; t) = G1(x1; t), (2.70) f2(x1, x2; t) = G2(x1, x2; t) + g2(r1, r2; t)G1(x1; t)G1(x2; t), f3(x1, x2, x3; t) = G3(x1, x2, x3; t) + ∑ P G2(x1, x2; t)G1(x3; t) + g3(r1, r2, r3; t)G1(x1; t)G1(x2; t)G1(x3; t), ... ... Here, the position-dependent quasiequilibrium distribution functions g2(r1, r2; t), g3(r1, r2, r3; t), gs(r s; t) are defined in [11,12]. The modification of group expan- sions (2.70) consists in the fact that a considerable part of space time correlations is accumulated in the quasiequilibrium functions gs(r s; t). If all gs(r s; t) = 1 for s = 2, 3, . . ., these group expansions coincide with those of papers [52,67–69]. As far as each line in (2.70) brings in new functions Gs(r s; t), s = 1, 2, 3, . . ., the corre- sponding equations can be solved with respect to irreducible distribution functions and we can write the following: G1(x1; t) = f1(x1; t), (2.71) G2(x1, x2; t) = f2(x1, x2; t)− g2(r1, r2; t)f1(x1; t)f1(x2; t), G3(x1, x2, x3; t) = f3(x1, x2, x3; t)− ∑ P f2(x1, x2; t)f1(x3; t)− h3(r1, r2, r3; t)f1(x1; t)f1(x2; t)f1(x3; t), ... .... In (2.70) and (2.71), the symbol ∑ P denotes the sum of all different permutations of coordinates for three and more particles h3(r1, r2, r3; t) = g3(r1, r2, r3; t)− g2(r1, r2; t)− g2(r1, r3; t)− g2(r2, r3; t) ≡ h′3(r1, r2, r3; t)− 2, (2.72) where h′3(r1, r2, r3; t) is a three-particle quasiequilibrium correlation function. Now let us write the BBGKY hierarchy (2.31) [1,3,7] for the irreducible distribution functions Gs(x s; t), namely, the first two equations, ( ∂ ∂t + iL(1) ) G1(x1; t) + (2.73) ∫ dx2 iL(1, 2)g2(r1, r2; t)G1(x1; t)G1(x2; t) + ∫ dx2 iL(1, 2)G2(x1, x2; t) = 0. Differentiating the relation for G2(x1, x2; t) in (2.71) with respect to time and using the second equation from the BBGKY hierarchy for the function f2(x1, x2; t), we 708 A consistent description of kinetics and hydrodynamics can get for the pair irreducible distribution function G2(x1, x2; t) an equation, which reads: ( ∂ ∂t + iL2 + ε ) G2(x1, x2; t) = − ( ∂ ∂t + iL2 ) g2(r1, r2; t)G1(x1; t)G1(x2; t)− ∫ dx3 { iL(1, 3) + iL(2, 3) } { G3(x1, x2, x3; t) + ∑ P G2(x1, x2; t)G1(x3; t) + g3(r1, r2, r3; t)G1(x1; t)G1(x2; t)G1(x3; t) } . (2.74) In a similar way, we can obtain other equations for the three-particle irreducible functionG3(x1, x2, x3; t) and the higherGs(x s; t) ones. One remembers now that the appearance of the quasiequilibrium distribution functions g2(r1, r2; t), g3(r1, r2, r3; t), gs(r s; t) in the hierarchy is closely connected with the fact that the boundary conditions for the solutions of the Liouville equation take into consider- ation both the nonequilibrium character of the one-particle distribution function and the local conservation laws, which corresponds to a consistent description of the kinetics and hydrodynamics of the system [1,3]. Since in the present paper we analyze two first equations, (2.73) and (2.74) only, we will not write the others. It is important to note that, if we put formally gs(r s; t) ≡ 1 for all s = 2, 3, . . . in (2.73) and (2.74), we come to the first two equations of the BBGKY hierarchy for the ir- reducible distribution functions G1(x1; t) and G2(x1, x2; t), which were obtained in paper [52] by D.N.Zubarev and M.Yu.Novikov. The first term in the right-hand side of (2.74) is a peculiarity of (2.73) and (2.74) equation system. This is a term with a time derivative of the pair quasiequilibrium distribution function g2(r1, r2; t). As it was shown in [1,3], the binary quasiequilibrium distribution function is a functional of the local values of temperature β(r; t) and mean particle density n(r; t). Thus, time derivatives of g2(r1, r2\|β(t), n(t)) will conform to β(r; t) and n(r; t). These quantities, in their turn, according to the self-consistency conditions [1,7], will be expressed via the average energy value 〈Ê ′(r)〉t in an accompanying reference frame and via 〈n̂(r)〉t, which constitute a basis of the hydrodynamical description of a nonequilibrium state of the system. Solving equation (2.74) for the irreducible quasiequilibrium two-particle distribution function G2(x1, x2; t) in the generalized polarization approximation and taking into account the first equation of the chain (2.73), lead to [11,12]: ( ∂ ∂t + iL(1) ) G1(x1; t) + ∫ dx2 iL(1, 2)g2(r1, r2; t)G1(x1; t)G1(x2; t) = ∫ dx2 t∫ −∞ dt′ eε(t ′−t)iL(1, 2)U(t, t′)× (2.75) ( ∂ ∂t′ + iL2 + L (x1, x2; t ′) ) g2(r1, r2; t ′)G1(x1; t ′)G1(x2; t ′), 709 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn where U(t, t′) = exp+ { − ∫ t t′ dt′′ [iL2 + L (x1, x2; t ′′)] } , L (x1, x2; t) = L (x1; t) + L (x2; t). (2.76) The operator L (x1; t) can be obtained by a variation of the Vlasov collision integral near the nonequilibrium distribution G1(x1; t), namely, δ (∫ dx3 iL(1, 3)G1(x3; t)G1(x1; t) ) = (2.77) ∫ dx3 iL(1, 3)G1(x3; t)δG1(x1; t) = L (x1; t)δG1(x1; t). This is a kinetic equation for a nonequilibrium one-particle distribution function with the non-Markovian collision integral in the generalized polarization approx- imation. It should be noted that the presence of the Vlasov operator L (x1, x2; t) in the collision integral (2.75) indicates taking into consideration collective effects. Analysis of the collision integral (2.75) in a general case is a rather complicated problem. But it is obvious that the collision integral in (2.75) or an expression for G2(x1, x2; t) with (2.74) may be much simplified for every physical model of a particle system or for each nonequilibrium state of the collision integral in (2.75). To show this, we shall consider two particular cases: a hard spheres model and Coulomb plasma. 2.4.2. Hard spheres model in the polarization approximatio n In this subsection we shall investigate kinetic processes for a hard spheres model in the approximations which are higher than a binary collisions one. We take into account the character of the model parameters and the results of the previous section of this article and papers [26,49]. This investigation is convenient to carry out on the basis of the equation chain (2.73), (2.74) at the formal substitution of a potential part of the Liouville operator iL(1, 2) by the Enskog collision operator T̂ (1, 2) [26,70]. In this case, equations (2.73) and (2.74) have the form: ( ∂ ∂t + iL(1) ) G1(x1; t) + (2.78) ∫ dx2 T̂ (1, 2)g2(r1, r2; t)G1(x1; t)G1(x2; t) + ∫ dx2 T̂ (1, 2)G2(x1, x2; t) = 0, ( ∂ ∂t + iL0 2 + T̂ (1, 2) + ε ) G2(x1, x2; t) = (2.79) − ( ∂ ∂t + iL0 2 + T̂ (1, 2) ) g2(r1, r2; t)G1(x1; t)G1(x2; t) − ∫ dx3 { T̂ (1, 3) + T̂ (2, 3) }{ G3(x1, x2, x3; t) + ∑ P G2(x1, x2; t)G1(x3; t) + g3(r1, r2, r3; t)G1(x1; t)G1(x2; t)G1(x3; t) } . 710 A consistent description of kinetics and hydrodynamics Further, we will consider the same approximations concerning equation (2.79), in which G3(x1, x2, x3; t) and h3(r1, r2, r3; t) are neglected. Then, if we introduce similarly to (2.77) the Boltzmann-Enskog collision operator C(x1; t) (L (x1; t) → C(x1; t)), equation (2.79) could be rewritten in the next form: δ ∫ dx3 T̂ (1, 3)G1(x1; t)G1(x3; t) = C(x1; t)δG1(x1; t), (2.80) ( ∂ ∂t + iL0 2 + T̂ (1, 2) + C(x1, x2; t) + ε ) G2(x1, x2; t) = (2.81) − ( ∂ ∂t + iL0 2 + T̂ (1, 2) + C(x1, x2; t) ) g2(r1, r2; t)G1(x1; t)G1(x2; t), Hence it appears that the formal solution to G2(x1, x2; t) reads: G2(x1, x2; t) = − 0∫ −∞ dt′ eε(t ′−t)Uhs(t, t ′)× (2.82) { ∂ ∂t′ + iL0 2 + T̂ (1, 2) + C(x1, x2; t ′) } g2(r1, r2; t ′)G1(x1; t ′)G1(x2; t ′), where Uhs(t, t ′) is an evolution operator for the system of hard spheres: Uhs(t, t ′) = exp+ { − ∫ t t′ dt′′ [ iL0 2 + T̂ (1, 2) + C(x1, x2; t ′′) ]} , (2.83) C(x1, x2; t) = C(x1; t) + C(x2; t). Now let us put (2.82) into the first equation (2.78). Then, the resulting equation takes the form [11,12]: ( ∂ ∂t + iL(1) ) G1(x1; t) = ∫ dx2 T̂ (1, 2)g2(r1, r2; t)G1(x1; t)G1(x2; t)− (2.84) ∫ dx2 T̂ (1, 2) 0∫ −∞ dt′ eε(t ′−t)Uhs(t, t ′) { ∂ ∂t′ + iL0 2 + T̂ (1, 2) + C(x1, x2; t ′) } × g2(r1, r2; t ′)G1(x1; t ′)G1(x2; t ′). This equation can be called a generalized kinetic equation for the nonequilibrium one-particle distribution function of hard spheres with a non-Markovian collision integral in the generalized polarization approximation. The first term in the right- hand side of this equation is the collision integral from the revised Enskog theory [1,7,34,54]. Neglecting time retardation effects and assuming that the operator C(x1, x2; t) does not depend on time when G1(x1; t) = f0(p) = n ( m 2πkT )3/2 exp { − p2 2mkT } 711 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn is a local equilibriumMaxwell distribution function, the next term can be rewritten in a simplified form: IR(x1; t) = − t∫ −∞ dt′ eε(t ′−t)R0(x1; t, t ′)G1(x1; t ′)− t∫ −∞ dt′ eε(t ′−t)R1(x1; t, t ′)G1(x1; t ′), (2.85) where R0(x1; t, t ′) = ∫ dx2 T̂ (1, 2) exp { (t′ − t) [ iL0 2 + T̂ (1, 2) + C(x1, x2) ]} × [ iL0 2 + C(x1, x2) ] g2(r1, r2; t ′)G1(x2; t ′), (2.86) R1(x1; t, t ′) = ∫ dx2 T̂ (1, 2) exp { (t′ − t) [ iL0 2 + T̂ (1, 2) + C(x1, x2) ]} × T̂ (1, 2)g2(r1, r2; t ′)G1(x2; t ′), (2.87) R1(x1; t, t ′) is a generalized ring operator. The kinetic equation (2.84) together with (2.86) and (2.87) is the generalization of the kinetic equation for a system of hard spheres which was obtained by Bogolubov in [49,70]. It coincides with the case, when the quasiequilibrium pair distribution function of hard spheres is set formally to be a unity. 2.4.3. Coulomb plasma in the polarization approximation Here we shall study an electron gas, which is contained in a homogeneous pos- itively charged equilibrating background. This background can be created, for ex- ample, by hard motionless ions. Then, electrons interact according to the Coulomb law: Φ(\|r12\|) = e2 \|r1 − r2\| = e2 \|r12\| , the Fourier transform of which exists in the form of a real function Φ(\|k\|): e2 r12 = ∫ dk (2π)3 Φ(\|k\|)eik·r12 , Φ(\|k\|) = Φ(k) = 4πe2 k2 , (2.88) here k is a wavevector, e is an electron charge. Let us consider equation chain (2.73), (2.74), when G3(x1, x2, x3; t) = 0, g3(r1, r2, r3; t) = 0 in the homogeneous case, when G1(x1; t) = G1(p1; t) and pair distribution functions depend on \|r12\|. Following the Bogolubov method [22], we shall assume that the one-particle dis- tribution function G1(p1; t) is calculated in the “zeroth” order on the interaction constant q, pair distribution functions G2(r12,p1,p2; t) and g2(r12; t) in the first or- der on q, and G3(x1, x2, x3; t), g3(r1, r2, r3; t) ∼ q2, where q = e2 rd Θ, rd = √ Θ/4πe2n is the Debye radius, n = N/V , Θ = kBT , kB is the Boltzmann constant, T is ther- modynamic temperature. Therefore, to obtain an equation for G2(r12,p1,p2; t) in the first approximation on the interaction constant q without time retardment ef- fects, it is necessary to retain all integral terms, but omit all the others. In this case, 712 A consistent description of kinetics and hydrodynamics using the Fourier transform with respect to spatial coordinates for a homogeneous Coulomb electron gas, the set of equations (2.73), (2.74) yields: ∂ ∂t G1(p1; t) = − ∂ ∂p1 ∫ dk dp2 iΦ(\|k\|)g2(k; t)G1(p1; t)G1(p2; t) − ∂ ∂p1 ∫ dk dp2 iΦ(\|k\|)G2(k,p1,p2; t), or ∂ ∂t G1(p1; t) = ∂ ∂p1 G1(p1; t) ∫ dk kΦ(\|k\|)ℑm g2(k; t) + ∂ ∂p1 ∫ dk kΦ(\|k\|)ℑmG2(k,p1; t) (2.89) and an equation for G2(k,p1,p2; t): ( ∂ ∂t + ik p12 m + ε ) G2(k,p1,p2; t) = (2.90) ikΦ(\|k\|) { ∂ ∂p1 G1(p1; t) ∫ dp3 G2(k,p2,p3; t)− ∂ ∂p2 G1(p2; t) ∫ dp3 G2(k,p1,p3; t) } + ikΦ(\|k\|) { ∂ ∂p1 G1(p1; t)g2(−k; t)G1(p2; t)− ∂ ∂p2 G1(p2; t)g2(k; t)G1(p1; t) } , ε → +0, and G2(k,p1; t) = ∫ dp2 G2(k,p1,p2; t); ℑm g2(k; t), ℑmG2(x1, x2; t) are imaginary parts of the corresponding distribution functions. The following properties should be noted: G2(−k,p1,p2; t) = G∗ 2(k,p1,p2; t), g2(−k; t) = g∗2(k; t), where ∗ denotes a complex conjugation. The solution to (2.90), neglecting time retardment effects, reads: G2(k,p1,p2; t) = (2.91) kΦ(\|k\|) k · p12 m − i0 { ∂ ∂p1 G1(p1; t)G2(−k,p2; t)− ∂ ∂p2 G1(p2; t)G2(k,p1; t) } + kΦ(\|k\|) k · p12 m − i0 { ∂ ∂p1 G1(p1; t)g2(−k; t)G1(p2; t)− ∂ ∂p2 G1(p2; t)g2(k; t)G1(p1; t) } . It should also be noted that equation (2.89) contains an imaginary part of the irreducible pair nonequilibrium distribution function, to be integrated with respect to momentum of the second particle. Now one integrates equation (2.91) over all the values of momentum p2 and defines in such a way some function G2(k,p1; t):  1 + ∫ dp2 kΦ(\|k\|) k · p12 m − i0 ∂ ∂p2 G1(p2; t)  G2(k,p1; t) = (2.92) 713 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn ∂ ∂p1 G1(p1; t) ∫ dp2 kΦ(\|k\|) k · p12 m − i0 G2(−k,p2; t) + ∫ dp2 kΦ(\|k\|) k · p12 m − i0 { ∂ ∂p1 G1(p1; t)g2(−k; t)G1(p2; t)− ∂ ∂p2 G1(p2; t)g2(k; t)G1(p1; t) } . Further, we should exclude from (2.92) the term with G2(−k,p2; t). To do this, we follow Lenard [47,49] and integrate equation (2.92) over the momentum component p1⊥, which is perpendicular to wavevector k. The resulting expression then reads: [1 + Φ(\|k\|)χ(k, p1; t)]G2(k, p1; t) = ∂ ∂p1 G1(p1; t) ∫ dp2 kΦ(\|k\|) k · p12 m − i0 G2(−k,p2; t) + ∫ dp2 kΦ(\|k\|) k · p12 m − i0 { ∂ ∂p1 G1(p1; t)g2(−k; t)G1(p2; t)− ∂ ∂p2 G1(p2; t)g2(k; t)G1(p1; t) } ,. (2.93) Here the following conventional designations have been introduced: χ(k, p1; t) = ∫ dp2 k k · p12 m − i0 ∂ ∂p2 G1(p2; t), p1 = p1 · k k , G1(p1; t) = ∫ dp1⊥ G1(p1; t), p2 = p2 · k k , G2(k, p1; t) = ∫ dp1⊥ G2(k,p1; t), k = \|k\|. (2.94) Now we multiply the both equations, (2.92) and (2.93), by ∂ ∂p1 G1(p1; t) and by ∂ ∂p1 G1(p1; t), respectively, and subtract them: ( 1 + Φ(\|k\|)χ(k, p1; t) )[ G2(k,p1; t) ∂ ∂p1 G1(p1; t)−G2(k, p1; t) ∂ ∂p1 G1(p1; t) ] = Φ(\|k\|)χ(k, p1; t)g2(k; t) [ G1(p1; t) ∂ ∂p1 G1(p1; t)−G1(p1; t) ∂ ∂p1 G1(p1; t) ] . (2.95) If we extract the imaginary part of this equation, one can find the unknown quan- tity ℑmG2(k,p1; t), provided ℑmG2(k, p1; t) = 0 [11,12]: ∂ ∂p1 G1(p1; t)ℑmG2(k,p1; t) = Φ(\|k\|)ℑm [χ(k, p1; t)g2(k; t)] \|1 + Φ(\|k\|)χ(k, p1; t)\|2 × [ G1(p1; t) ∂ ∂p1 G1(p1; t)−G1(p1; t) ∂ ∂p1 G1(p1; t) ] . (2.96) Since ℑmχ(k, p1; t)=-π ∂ ∂p1 G1(p1; t) [64], putting an expression for ℑmG2(k,p1; t) into equation (2.89) gives the generalized Bogolubov-Lenard-Balescu kinetic equa- tion for an electron gas in an equilibrating background: ∂ ∂t G1(p1; t) = ∂ ∂p1 G1(p1; t) ∫ dk kΦ(\|k\|)ℑm g2(k; t) + (2.97) ∂ ∂p1 ∫ dp2 Q(p1,p2; t) [ ∂ ∂p1 − ∂ ∂p2 ] G1(p1; t)G1(p2; t), 714 A consistent description of kinetics and hydrodynamics where Q(p1,p2; t) is a second rank tensor Q(p1,p2; t) = −π ∫ dk \|Φ(\|k\|)\|2 k · k \|1 + Φ(\|k\|)χ(k, p1; t)\|2 ℑm g2(k; t)δ(k · (p1 − p2)), (2.98) which coincides with Q(p1,p2) [64] at ℑm g2(k; t) = 1. In this case, the ki- netic equation (2.97) transforms into the well-known Lenard-Balescu equation [47–49,64]. Evidently, the generalized Bogolubov-Lenard-Balescu kinetic equation (2.97) claims the description of a dense electron gas, since in both the general- ized mean field and the generalized Bogolubov-Lenard-Balescu collision integrals, many-particle correlations are treated by the imaginary part of g2(k; t). Neverthe- less, the problem of divergence in the collision integral of equation (2.97) at small distances (k → ∞) still remains. There are papers where the divergence of colli- sion integrals is avoided with the help of a special choice of the differential cross section (quantum systems [71]), or via a combination of simpler collision integrals (classical systems [24]). These generalizations for collision integrals are attractive by their simplicity and helpful for ideal plasma. But, contrary to the obtained by us Bogolubov-Lenard-Balescu kinetic equation, they do not work for nonideal plasma. In accordance with the proposed structure of the collision integral Itotal = IBoltzmann − ILandau + ILenard−Balescu the influence of particles interaction on plasma energy will be defined by the cor- relation function g2(r). Its asymptotic is lim r→∞  exp { − eaeb rkBT } − 1 ︸︷︷︸ Boltzmann + eaeb rkBT︸︷︷︸ Landau − eaeb rkBT e−r/rD ︸︷︷︸ Lenard−Balescu   = 1 r2 (!), where r D denotes the Debye radius. In other combinations one arrives at false expressions for thermodynamic functions [24]. Dynamical screening, which appears in the generalized Bogolubov-Lenard-Balescu collision integral obtained by us, is free of these discrepancies. Generally speaking, the problem of divergency could be solved within the framework of a charged hard spheres model, combining the results of this section and the preciding one. But this step is an intricate and complicated problem and needs a separate consideration. Evidently, an investigation of the obtained kinetic equation is important in view of its solutions and studying transport coefficients and time correlation functions for model systems. In view of the dense systems study, where the consideration of spatial in- terparticle correlations is important, the BBGKY hierarchy (2.73), (2.74) with the modified boundary conditions and group expansions has quite a good per- spective. The kinetic equation (2.84)–(2.87) is a generalization of the Bogolubov one [49,70] for a system of hard spheres. M.Ernst and J.Dorfman [72] investi- gated collective modes in an inhomogeneous gas and showed that the solution of 715 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn a dispersion equation for hydrodynamic modes leads to a nonanalytic frequency dependence on a wavevector. This is connected with the fact that the ring oper- ator for inhomogeneous systems at small wavenumbers has a term proportional to √ k. Similar investigations of collective modes and time correlation functions in the hydrodynamic region were carried out by Bogolubov [70]. Nevertheless, it is necessary to carry out analogous investigations of hydrodynamic collective modes and time correlation functions on the basis of kinetic equation (2.84), taking into account (2.85)–(2.87), where some part of space correlations is considered in the pair quasiequilibrium function g2(r1, r2; t). Obviously, these results may appear to be good for very dense gases, which could be described by a hard spheres model. An important factor is that in the kinetic equation (2.84)–(2.87), as well as in the generalized Bogolubov-Lenard-Balescu one, collective effects are taken into account both via the Vlasov mean field and the binary quasiequilibrium correla- tion function which is a functional of nonequilibrium values of temperature and a chemical potential. Transferring the obtained results to quantum systems is not obvious. Such a procedure is rather complicated and needs additional investigations. Nevertheless, some steps in this way have been already done by Morozov and Röpke [9,10,16]. 3. Generalized transport equations and time correlation functions 3.1. Overview One of the main problems of the nonequilibrium statistical mechanics of liq- uids is an investigation of collective excitations, time correlation functions and transport coefficients because, using these quantities, we can compare the corre- sponding theoretical results with the data on light and neutron scattering as well as with the results obtained by the method of molecular dynamics (MD) [73–79]. At present, one can distinguish three regions in these investigations. The first one is the usual hydrodynamics in a linear approximation when the time and spatial evolution of small excitations in liquids can be described in terms of hydrodynamic modes (in terms of eigenvalues and eigenvectors of the linear hydrodynamics equations) [26,31,77,80–82]. Such hydrodynamic modes are the heat mode zH(k) = −DTk 2, two sound modes z±(k) = ±ick − Γk2 and two vis- cosity modes zν1,2 = −νk2 (see, for example, [81]). In our notations k denotes a wavenumber which describes spatial dispersion of excitations, DT = λ/ncp is a coefficient of thermal diffusion, ν = η/mn is the kinematical viscosity, Γ = 2 3 ν + 1 3 κ/mn + 1 2 (cp/cV − 1)DT is the constant of sound decrement. λ, κ and η are the thermal conductivity, bulk and shear viscosity coefficients defined by the Green-Kubo formulas, n is the average density, m is the mass of a separate par- ticle, cp and cV are specific heats at constant pressure and volume, c is the sound velocity. In this region, where \|k\|−1 is much larger than atomic sizes of liquid σ, and ωτ ≪ 1, where ω is frequency and τ is the characteristic correlation time, 716 A consistent description of kinetics and hydrodynamics the dynamical structure factor S(k;ω) is well described by the Landau-Plachek formula [26,81,84] in terms of Brillouin lines which corresponds to heat and sound modes. With increasing wavevector values k, the precision of the hydrodynamic description decreases, since correlations at short times and small spatial distances which are inherent in neutron scattering in liquids are not described by linear hydrodynamic equations. Extensions of the usual hydrodynamics were performed on the basis of modern methods of nonequilibrium statistical mechanics by Zwanzig and Mori [85–87] with the use of the method of projection operators, by Sergeev and Tishchenko [88,89] using the NSO method [50,51] and by Tserkovnikov with the use of the method of Green functions [90–92]. Transport equations were obtained for the average values of the densities of mass, momentum and energy. These equations generalize the hydrodynamic ones. In these equations, thermodynamic quantities cp, cV depend on wavevector k, whereas the transport coefficients λ, κ and η depend on k, as well as on frequency ω. Moreover, new generalized transport coefficients appear in the generalized hydrodynamic equations. These coefficients describe dynamical correlations between the thermal and viscous motions which vanish in the limit k → 0 and ω → 0. This is the second region, namely, the region of the generalized hydrodynamics [26,77,83,87,59,89–94]. Papers by Götze, Lücke, Bosse and others [95–101] are of special interest. In these papers, an investigation of spectra for fluctuations of densities for the number of particles, their longitudinal and transverse fluxes for small as well as for intermediate values of k and ω is performed. For Argon [73,76] and Rubidium [74–76], a good coincidence with the experimental data was obtained. It is necessary to point out papers by Yulmetyev and Shurygin [102–106] where the dynamical structure factor S(k;ω) for Rubidium [103–105] and Argon [102,106] was investigated by the projection operator method taking into account non-Markovian effects. The results obtained in these papers are valid in the low-frequency approximation. The third region of investigations on collective modes and time correlation func- tions in dense gases and liquids is connected with the kinetic theory [77,107–109] on the basis of the method of projection operators by Mori and its generalizations [77,79,87]. In this approach, the nonequilibrium one-particle distribution function in phase space of coordinates and momenta is a variable for an abbreviated de- scription of a nonequilibrium state of the system. At the same time, collective effects arise explicitly in memory functions. Approximate calculations of mem- ory functions (expansions on density, weak interaction) in the hydrodynamic limit were performed in papers by Mazenko [109–114], Forster and Martin [108], Forster [115] and others [100,116,117]. John and Forster unified paper [118] and proposed a formalism similar to that of the generalized hydrodynamics [107–109,114,115]. This formalism is distinguished by the fact that the nonequilibrium one-particle distribution function in phase space of coordinates and momenta, together with the density of total energy are included into a set of variables of an abbreviated description. The results of this theory for the dynamical structure factor S(k;ω) of Argon agree well with the experimental data and MD calculations in the re- 717 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn gions of small and intermediate values of k and ω. The same formalism was used later in papers by Sjodin and Sjolander [119] in which the dynamical structure factor S(k;ω) for Rubidium was investigated. The authors of this paper touched a number of interesting questions about the influence of one-particle motion and temperature fluctuations on S(k;ω) in the regions of intermediate and great val- ues of wavenumber k, as well as about the existence of short-length collective modes [75]. Such questions and a number of others arise in discussing the results of neutron scattering with ℓ ∼ σ, (\|k\| = 2π/ℓ), where the kinetic theory of liquids is developed insufficiently. From the experimental results of neutron scattering in Rubidium [75] it follows that there are sharp side peaks for S(k;ω) at points ω = ±ck, where c is an adiabatic velocity of sound. Such side peaks could be identified as Brillouin ones. However, the Landau-Plachek formula is not valid in this region, where ℓ ∼ σ. During the last period, the question of the existence of short-length collective modes, beginning from the papers by de Schepper and Cohen [120,121], is inten- sively discussed [122–135]. The proposed in [120,121] approach is based on the kinetic equation of the revised Enskog theory (RET) [1,3,34], which was derived by us sequentially in subsection 2.3.1. The linearized kinetic equation of RET for hard spheres, which was used in [120,121], was obtained in papers by Mazenko and others as well [112,113,117]. Although the kinetic theory considered by de Schep- per and Cohen for short-length collective modes is related to a model system of hard spheres, it has explained many questions which are significant for understand- ing collective modes in the region of large k. First of all, it proposed a concept of extending hydrodynamic modes into the region of large wavevectors [121,122]. Solving the problem on eigenvalues for the generalized Enskog operator (taking into account a mean field) for hard spheres has shown [122,132] the existence of five generalized (extended) hydrodynamic and kinetic modes. The generalized hydro- dynamic modes are a generalization of hydrodynamic modes to the region of large k and tend to zero at k → 0. The kinetic modes at k → 0 take positive nonzero values. The dynamical structure factor S(k;ω) is presented in this approach in the form of a sum of spectral terms which correspond to the Lorentzian form of lines. Secondly, to confirm the theoretical predictions of papers [120–135], experiments on neutron scattering in liquid Argon [123], Neon [125], as well as MD calculations for a system of hard spheres [126] and for a liquid with the Lennard-Jones-like interparticle potential of interaction [128] were made. In [123] short-length sound modes and their decrement in the region ℓ ∼ σ for a liquid Argon were investigated. The obtained results [131] for a nonanalytical dependence of the sound dispersion on k were compared with the experimental data and the results of the theory of coupled modes [136]. Eigenvalues for the heat mode zH(k), obtained on the basis of the generalized Enskog equation, were investigated in [130]. The calculations performed agree quantitatively with the MD data for hard spheres [125] and with the data on neutron scattering for Argon [123,124,137] and Crypton [138]. Interesting results of the investigation of the dynamical structure factor S(k;ω) and time correlation functions of flow particle and enthalpy densities, as well as 718 A consistent description of kinetics and hydrodynamics collective modes for a liquid with the Lennard-Jones potential of interaction were obtained in [139]. There a system of equations for time correlation functions of den- sities for the number of particles, momentum and enthalpy, the generalized stress tensor and the flow of energy obtained by the Mori projection operator method was used. The basis of this system are equations of a generalized hydrodynamic description obtained for the first time in [93] for simple liquid and ionic systems with the use of the NSO method [140]. Solving the system of equations for time correlation functions in the Markovian approximation on eigenvalues in the hydro- dynamic limit k → 0, ω → 0 showed [139] that apart from the eigenvalues which correspond to pure hydrodynamic modes, in particular, to heat zH(k) and two sound modes z±(k), there exist two eigenvalues that correspond to kinetic modes. They differ from zero in the limit k → 0. The analogous results were obtained in our paper [141]. Further, such an approach was developed for a Lennard-Jones fluid in a number of works [142–154]. The concept of generalized collective modes was developed [142,143] on the basis of equations of extended hydrodynamics and the relevant equilibrium static quantities which normally are obtained within the method of molecular dynamics (MD). This approach gives the possibility to calculate generalized-mode spectra of a Lennard-Jones fluid using nine- and four- mode descriptions for longitudinal [142,147] and transverse [142,146] fluctuations, respectively. The investigation of generalized transport coefficients dependent on wavevector and frequency was carried out for the first time in [148]. Further, this approach was developed for mixtures [154–156] and polar liquids, in particular, for Stockmayer models and TIP4P water [157–165]. Problems of the description of kinetic and hydrodynamic fluctuations, taking into account correlations in a simple fluid, were discussed in papers by Peletminskii, Sokolovskii and Slusarenko [166,167]. They used a functional hypothesis and the method of an abbreviated description [168]. On the basis of general equations of fluctuational hydrodynamics obtained in [166], we carried out investigations of liquid hydrodynamics near the equilibrium without taking into account far correlations and spatial dispersion of transport coefficients [169,170]. It is necessary to point out a series of papers by Balabanyan [171–175] devoted to the investigation of classical time correlation and Poisson Green functions and spectra of collective modes on the basis of the Boltzmann kinetic equation by the method of moments for a system of hard spheres and Maxwellian molecules. The problem of the construction of interpolation formulas for density-density and flow-flow correlation functions which are valid for a wide range of frequen- cies and wavevectors was considered in papers by Tserkovnikov [91,92,176,177]. Here a method of Green functions was applied to the molecular hydrodynamics of quantum Bose-systems. These questions were presented in detail in [176,177] for a weakly nonideal Bose-gas. The obtained interpolation formulas for the Green functions of transverse components of flow density [176], as well as of fluctuations of the number of particles and energy [177], appear to be valid in the hydrody- namic region and in the region of great frequencies and small wavelengths. These 719 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn results are obviously valid in the classical case as well. From the presented survey, one can draw the following conclusions: firstly, for a more detailed study of collective modes, generalized transport coefficients and time correlation functions in liquids and dense gases, it is necessary to have a theory in which kinetic and hydrodynamic processes are considered simultaneously; secondly, at the present time, a consistent theory for the description of light and neutron scattering in the whole range of wavevector k and frequency ω has not been formulated yet. In our papers [1,3,5,7] we proposed an approach in which the kinetics and hydrodynamics of transport processes in dense gases and liquids are considered as coupled. As a result, we obtained a system of coupled generalized equations for the nonequilibrium one-particle distribution function and the mean density of the total energy. It can be applied to the description of nonequilibrium states of the system of particles which are as close as they are far from the equilibrium state. In the next subsections of this section we shall present this approach in comparison with other theories and investigate time correlation functions, generalized transport coefficients and collective modes in comparison with the molecular hydrodynamics [77] and its extended versions [139,141–152]. 3.2. The generalized kinetic equation for the nonequilibri um one-particle distribution function with taking into account transport e quations for the mean density of energy According to the formulation of a modified boundary condition for the Li- ouville equation (2.4) with choosing parameters of an abbreviated description 〈n̂1(x)〉t = f1(x; t) and 〈Ê(r)〉t (the definition of the quasiequilibrium distribution function is presented by equation (2.12)), the dependence of the the quasiequilib- rium distribution function on time is defined by variation in time of the average values 〈n̂1(x)〉t and 〈Ê(r)〉t: ̺ ( xN ; t ) = ̺ ( . . . , {〈n̂1(x)〉t, 〈Ê(r)〉t} . . . ) . (3.1) Time evolution of the parameters of an abbreviated description f1(x; t), 〈Ê(r)〉t is described by transport equations. For obtaining these equations it is more conve- nient to use the method of projection operators, which was widely used by Robert- son [178,179] and modified by Kawasaki and Gunton [180]. We shall reformulate the NSO method in order to take into account projection operators. Let us write the Liouville equation with source (2.4) in the form: ( ∂ ∂t + iLN + ε ) △̺ ( xN ; t ) = − ( ∂ ∂t + iLN ) ̺q ( xN ; t ) , (3.2) introducing △̺ ( xN ; t ) = ̺ ( xN ; t ) − ̺q ( xN ; t ) . We shall use the quasiequilibrium distribution function ̺q ( xN ; t ) defined as follows: ̺q ( xN ; t ) = exp { −Φ(t)− ∫ dr β(r; t)Ê(r)− ∫ dx b(x; t)n̂1(x) } , (3.3) 720 A consistent description of kinetics and hydrodynamics which is found from (2.12), taking into account (2.13), by redefinition of the pa- rameter conjugated to 〈n̂1(x)〉t: b(x; t) = a(x; t)− β(r; t) p2 2m . (3.4) Parameters β(r; t) and b(x; t) in (3.3) are defined from the corresponding self- consistency conditions: 〈Ê(r)〉t = 〈Ê(r)〉tq, 〈n̂1(x)〉t = 〈n̂1(x)〉tq. (3.5) Taking into account the structure of the quasiequilibrium distribution function (3.3), the time derivative ∂ ∂t of this function in the right-hand side of equation (3.2) can be presented as ∂ ∂t ̺q ( xN ; t ) = −Pq(t)iLN̺ ( xN ; t ) , (3.6) where Pq(t) is the Kawasaki-Gunton projection operator, defined as follows: Pq(t)̺ ′ = (3.7){ ̺q ( xN ; t ) − ∫ dr ∂̺q ( xN ; t ) ∂〈Ê(r)〉t 〈Ê(r)〉t − ∫ dx ∂̺q ( xN ; t ) ∂〈n̂1(x)〉t 〈n̂1(x)〉t }∫ dΓN ̺′ + ∫ dr ∂̺q ( xN ; t ) ∂〈Ê(r)〉t ∫ dΓN Ê(r)̺′ + ∫ dx ∂̺q ( xN ; t ) ∂〈n̂1(x)〉t ∫ dΓN n̂1(x)̺ ′. This operator acts only on distribution functions and has the properties: Pq(t)̺(x N ; t′) = ̺q(x N ; t), Pq(t)̺q(x N ; t′) = ̺q(x N ; t), Pq(t)Pq(t ′) = Pq(t). Taking into account (3.6), the Liouville equation (2.4) transforms into the form ( ∂ ∂t − ( 1− Pq(t) ) iLN + ε ) △̺ ( xN ; t ) = − ( 1− Pq(t) ) iLN̺q ( xN ; t ) , (3.8) the formal solution to which is △̺ ( xN ; t ) = − t∫ −∞ dt′ eε(t ′−t)T (t, t′) ( 1− Pq(t ′) ) iLN̺q ( xN ; t′ ) , (3.9) and one finds an expression for the nonequilibrium distribution function ̺ ( xN ; t ) = ̺q ( xN ; t ) − t∫ −∞ dt′ eε(t ′−t)T (t, t′) ( 1− Pq(t ′) ) iLN̺q ( xN ; t′ ) , (3.10) 721 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn where T (t, t′) = exp+ { − ∫ t t′ dt′′ ( 1− Pq(t ′′) ) iLN } (3.11) is a generalized evolution operator with taking account the Kawasaki-Gunton pro- jection operator, exp+ is an ordering exponent. Solution (3.10) is exact. It corre- sponds to the idea of an abbreviated description of a nonequilibrium state of the system (3.1). Acting by the operators (1 − Pq(t ′)) and iLN on ̺q(x N ; t′) in the right-hand side of expression (3.10), we write an expression for the nonequilibrium distribution function in the explicit form: ̺ ( xN ; t ) = ̺q ( xN ; t ) + ∫ dr t∫ −∞ dt′ eε(t ′−t)T (t, t′)IE(r, t ′)β(r; t′)̺q ( xN ; t′ ) − ∫ dx t∫ −∞ dt′ eε(t ′−t)T (t, t′)In(x, t ′)b(x; t′)̺q ( xN ; t′ ) , (3.12) where IE(r, t ′) = ( 1− P(t′) ) ˙̂E(r), (3.13) In(x, t ′) = ( 1− P(t′) ) ˙̂n1(x) (3.14) are generalized flows, ˙̂E(r) = iLN Ê(r), ˙̂n1(x) = iLN n̂1(x). P(t′) is a generalized Mori projection operator which acts on dynamical variables Â(r) and has the following structure: P(t)Â(r′) = 〈Â(r′)〉tq + (3.15) ∫ dr ∂〈Â(r′)〉tq ∂〈Ê(r)〉t ( Ê(r)− 〈Ê(r)〉t ) + ∫ dx ∂〈Â(r′)〉tq ∂〈n̂1(x)〉t ( n̂1(x)− 〈n̂1(x)〉t ) . The projection operator P(t) has the following properties: P(t)Ê(r) = Ê(r), P(t)(1− P(t)) = 0, P(t)n̂1(x) = n̂1(x), P(t)P(t′) = P(t). (3.16) With the help of the solution to the Liouville equation (3.12), one obtains a sys- tem of coupled equations for the nonequilibrium one-particle distribution function f1(x; t) and the average density of energy 〈Ê(r)〉t. To do it explicitly, it is necessary to calculate ∂ ∂t 〈n̂1(x)〉t = ∂ ∂t f1(x; t) = 〈 ˙̂n1(x)〉t, ∂ ∂t 〈Ê(r)〉t = 〈 ˙̂E(r)〉t. 722 A consistent description of kinetics and hydrodynamics To this end let us use the equalities 〈( 1− P(t) ) ˙̂n1(x) 〉t = 〈 ˙̂n1(x)〉t − 〈 ˙̂n1(x)〉tq, 〈( 1− P(t) ) ˙̂E(r) 〉t = 〈 ˙̂E(r)〉t − 〈 ˙̂E(r)〉tq and perform averaging in the right-hand side with the nonequilibrium distribution function (3.12). Then we obtain, calculating 〈 ˙̂n1(x)〉tq, 〈 ˙̂E(r)〉tq [5]: ∂ ∂t f1(x; t) + p m ∂ ∂r f1(x; t) = ∫ dx′ ∂ ∂r Φ(\|r − r′\|) ∂ ∂p g2(r, r ′; t)f1(x; t)f1(x ′; t) + ∫ dx′ t∫ −∞ dt′ eε(t ′−t)ϕnn(x, x ′; t, t′)b(x′; t′) + ∫ dr′ t∫ −∞ dt′ eε(t ′−t)ϕnE(x, r ′; t, t′)β(r′; t′), (3.17) − ∂ ∂t 〈Ê(r)〉t = 1 2 ∫ dr′ dp dp′ p m ∂ ∂r Φ(\|r − r′\|)g2(r, r′; t)f1(x; t)f1(x ′; t) (3.18) + ∫ dr′ dp dp′ p m ∂ ∂r [ p2 2m + 1 2 Φ(\|r − r′\|) ] g2(r, r ′; t)f1(x; t)f1(x ′; t) − ∫ dx′ t∫ −∞ dt′ eε(t ′−t)ϕEn(r, x ′; t, t′)b(x′; t′) − ∫ dr′ t∫ −∞ dt′ eε(t ′−t)ϕEE(r, r ′; t, t′)β(r′; t′), where g2(r, r ′; t) is a binary quasiequilibrium distribution function (2.32), and ϕnn(x, x ′; t, t′) = ∫ dΓN In(x; t)T (t, t ′)In(x ′; t′)̺q(x N ; t′), (3.19) ϕnE(x, r ′; t, t′) = ∫ dΓN In(x; t)T (t, t ′)IE(r ′; t′)̺q(x N ; t′), (3.20) ϕEn(r, x ′; t, t′) = ∫ dΓN IE(r; t)T (t, t ′)In(x ′; t′)̺q(x N ; t′), (3.21) ϕEE(r, r ′; t, t′) = ∫ dΓN IE(r; t)T (t, t ′)IE(r ′; t′)̺q(x N ; t′) (3.22) are generalized transport kernels which describe the kinetic and hydrodynamic dissipative processes in a system. We obtained generalized transport equations for a nonequilibrium one-particle distribution function and the average energy den- sity, which describe both strong and weak nonequilibrium processes. From these 723 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn equations one can derive interesting limiting cases. First of all, if two last terms in the right-hand side of equation (3.17) are not taken into account (these terms de- scribe non-Markovian dissipative processes), then one obtains a generalized kinetic equation in the mean field approximation [5,38]: ∂ ∂t f1(x; t) + p m ∂ ∂r f1(x; t) = ∫ dx′ ∂ ∂r Φ(\|r − r′\|) ∂ ∂p g2(r, r ′; t)f1(x; t)f1(x ′; t). (3.23) From equation (3.23) at g2(r, r ′; t) = 1 we have a usual Vlasov kinetic equation for a nonequilibrium one-particle distribution function. The kinetic equation (3.23) for f1(x; t) must be complemented by an equation for the average energy density (3.18), without taking into account two last terms which describe non-Markovian dissipative processes. It is important to remember here that the binary quasiequi- librium distribution function g2(r, r ′; t) is a functional of the local values of inversed temperature β(r; t) and average density n(r; t): g2(r, r ′; t) = g2(r, r ′\|β(t), n(t)). If spatial correlations connected with an interaction between a separated group of particles and the media are neglected (that is valid in the case of small densities), i.e. formally putting for ̺q ( xN ; t ) in (2.22) the density of interaction energy Ê int(r) to be equal to zero (UN(r N ; t) = 0), then (2.22) transforms into a quasiequilibrium distribution (2.5) that corresponds to the usual boundary conditions of the weak- ening of correlations by Bogolubov [22] for the solution of the Liouville equation. In this case, from the system of equations (3.17), (3.18), we have a generalized kinetic equation for the nonequilibrium one-particle distribution function which was obtained earlier by Zubarev and Novikov [52] ∂ ∂t f1(x; t) + p m ∂ ∂r f1(x; t) = ∫ dx′ ∂ ∂r Φ(\|r − r′\|) ∂ ∂p f1(x; t)f1(x ′; t) + ∫ dx′ t∫ −∞ dt′ eε(t ′−t)ϕnn(x, x ′; t, t′)a(x′; t′). (3.24) However, in this equation, contrary to the kinetic equation for f1(x; t) of paper [52], we used a projection procedure for the exclusion of time derivatives for parameters a(x; t) which are determined from the self-consistency condition (3.5) and in our case, according to (2.12), (2.22), (2.23) and u = e, they are equal to e−a(x;t) = f1(x; t)/e. In the kinetic equation (3.24), the transport kernel has the following structure: ϕnn(x, x ′; t, t′) = ∫ dΓN ( 1− P ′(t) ) ˙̂n1(x)T (t, t ′) ( 1− P ′(t′) ) ˙̂n1(x ′) N∏ j=1 f1(xj; t) e , (3.25) where the projection operator P ′(t) is defined as P ′(t)Â(x′) = 〈Â(x′)〉tq + ∫ dx ∂〈Â(x′)〉tq ∂〈n̂1(x)〉t ( n̂1(x)− 〈n̂1(x)〉t ) , (3.26) 724 A consistent description of kinetics and hydrodynamics and the average values 〈Â(x′)〉tq are calculated with the help of the quasiequilibrium distribution function (2.5). The kernel ϕnn(x, x ′; t, t′) describes kinetic processes in the system. A coupled system of equations for the nonequilibrium one-particle distribution function and the density of the total energy (3.17), (3.18) is strongly nonlinear. It takes into account complicated kinetic and hydrodynamic processes and such a system can be used for describing both strongly and weakly nonequilibrium states. In the next subsection we apply equations (3.17), (3.18) to the investigation of nonequilibrium states of the system, which are close to equilibrium. In this case the equations are simplified significantly. 3.3. Kinetics and hydrodynamics of nonequilibrium state ne ar equilibrium Let us assume that the average energy density 〈Ê(r)〉t, the nonequilibrium one- particle distribution function f1(x; t) and parameters β(r; t), b(x; t) deviate slightly from their equilibrium values. Then, being restricted to the linear approximation [5], the quasiequilibrium distribution function ̺q ( xN ; t ) (3.3) can be expanded over deviations of the parameters β(r; t), b(x; t) from their equilibrium values: ̺q ( xN ; t ) = ̺0(x N) [ 1− ∫ dr δβ(r; t)Ê(r)− ∫ dx δb(x; t)n̂1(x) ] , (3.27) where ̺0(x N) = Z−1e−β(H−µN) is an equilibrium distribution of particles, Z =∫ dΓN e−β(H−µN) is a grand partition function, β = 1/kBT is an equilibrium value for inverse temperature, kB is the Boltzmann constant, δβ(r; t) = β(r; t) − β, δb(x; t) = b(x; t)+βµ, µ is an equilibrium value of the chemical potential. In formula (3.27) it is convenient to transform the dynamical variables Ê(r), n̂1(x) = n̂1(r,p) and parameters δβ(r; t), δb(x; t) into Fourier-components. Then we obtain: ̺q ( xN ; t ) = ̺0(x N) [ 1− ∑ k ′ δβ−k(t)Êk − ∑ k ′ ∫ dp δb−k(p; t)n̂k(p) ] , (3.28) where ∑′ k = ∑ k(k 6=0) and Êk = ∫ dr e−ik·rÊ(r), δβ−k(t) = ∫ dr eik·rδβ(r; t), n̂k(p) = ∫ dr e−ik·rn̂1(r,p), δb−k(p; t) = ∫ dr eik·rδb(x; t). Using the self-consistency conditions (3.5), let us consequently exclude parameters δβ−k(t), δb−k(p; t) in equation (3.28). From 〈n̂k(p)〉t = 〈n̂k(p)〉tq one finds: ∫ dp′ Φk(p,p ′)δb−k(p ′; t) = −〈n̂k(p)〉t − 〈n̂k(p)Ê−k〉0δβk(t), (3.29) where 〈. . .〉0 denotes averaging over the equilibrium distribution, namely 〈. . .〉0 =∫ dΓN . . . ̺0(x N), 〈n̂k(p)〉0 = 0, (k 6= 0); Φk(p,p ′) = 〈n̂k(p)n̂−k(p ′)〉0 = nδ(p− p′)f0(p ′) + n2f0(p)f0(p ′)h2(k) (3.30) 725 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn is an equilibrium correlation function, n = N/V , f0(p) = (β/2πm)3/2e−βp2/2m is a Maxwell distribution, h2(k) is a Fourier-image of the function g2(R)− 1: h2(k) = ∫ dR e−ik·R ( g2(R)− 1 ) , (3.31) g2(R) = g2(\|r − r′\|) is a pair equilibrium distribution function. Now let us define the function Φ−1 k (p,p′) inversed to the function Φk(p,p ′) by means of the relation ∫ dp′ Φ−1 k (p′′,p′)Φk(p ′,p) = δ(p′′ − p). (3.32) Taking into account (3.30), from (3.32) we find Φ−1 k (p′′,p′) in an explicit form Φ−1 k (p′′,p′) = δ(p′′ − p′) nf0(p′′) − c2(k), (3.33) where c2(k) denotes a direct correlation function which is connected with the correlation function h2(k) as: h2(k) = c2(k)[1 − nc2(k)] −1. Let us multiply the left-hand side of (3.29) by the function Φ−1 k (p′′,p′) and integrate it with respect to p′. Taking into account (3.32), we find: δb−k(p; t) = − ∫ dp′ Φ−1 k (p,p′)〈n̂k(p ′)〉t − δβk(t) ∫ dp′ Φ−1 k (p,p′)〈n̂k(p ′)Ê−k〉0. Substituting the obtained value of the parameter δb−k(p; t) in (3.28) we find after some transformations: ̺q ( xN ; t ) = ̺0(x N) [ 1− ∑ k ′ δβ−k(t)ĥ int k + ∑ k ′ ∫ dp dp′ 〈n̂k(p ′)〉tΦ−1 k (p′,p)n̂k(p) ] , (3.34) where ĥint k = Êk − ∫ dp dp′ 〈Êkn̂−k(p ′)〉0Φ−1 k (p′,p)n̂k(p) = Ê int k − 〈Ê int k n̂−k〉0S−1 2 (k)n̂k, (3.35) Ê int k = 1 2 N∑ l 6=j=1 Φ(\|rlj\|)e−ik·rl , n̂k = N∑ l=1 e−ik·rl (3.36) are the Fourier-components of densities for the interaction energy and the number of particles, respectively. Further, it is more convenient, instead of the dynamical variable of energy Êk, to use the variable ĥint k (3.35) which is orthogonal to n̂k(p) by means of the equality: 〈ĥint k n̂−k〉0 = 0. (3.37) From the structure of the dynamical variable ĥint k (3.35) it can be seen that it cor- responds to a potential part of the Fourier-component of the generalized enthalpy ĥk, which is introduced in molecular hydrodynamics [5,77,79]: ĥk = Êk − 〈Êkn̂−k〉0n̂k = ĥkin k + ĥint k , (3.38) 726 A consistent description of kinetics and hydrodynamics where ĥkin k = Êkin k − 〈Êkin k n̂−k〉0S−1 2 (k)n̂k (3.39) is a kinetic part of the generalized enthalpy, Êkin k = N∑ l=1 p2l 2m e−k·rl is the Fourier-component of the kinetic energy density. S2 = 〈n̂kn̂−k〉0 is a static structure factor of the system. Taking into account the orthogonality of the dynam- ical variables ĥint k and n̂k (3.37), from the self-consistency condition 〈ĥint k 〉t = 〈ĥint k 〉tq that is equivalent to 〈Êk〉t = 〈Êk〉tq one defines the parameter δβk(t): δβk(t) = −〈ĥint k 〉tΦ−1 hh (k), (3.40) where Φhh(k) = 〈ĥint k ĥint−k 〉0. Finally, let us substitute (3.40) into (3.34). As a result, we obtain a quasiequilibrium distribution function in the linear approximation: ̺q ( xN ; t ) = ̺0(x N ) [ 1 + ∑ k ′ 〈ĥint k 〉tΦ−1 hh (k)ĥ int k + ∑ k ′ ∫ dp dp′ 〈n̂k(p ′)〉tΦ−1 k (p′,p)n̂k(p) ] . (3.41) Taking into account (3.41), the nonequilibrium distribution function ̺(xN ; t) (3.10) or (3.12) has the following form in this approximation [5]: ̺ ( xN ; t ) = ̺0(x N)× (3.42) [ 1 + ∑ k ′ 〈ĥint k 〉tΦ−1 hh (k)ĥ int k + ∑ k ′ ∫ dp dp′ 〈n̂k(p ′)〉tΦ−1 k (p′,p)n̂k(p)− ∑ k ′ t∫ −∞ dt′ eε(t ′−t)〈ĥint k 〉t′Φ−1 hh (k)T0(t, t ′)I inth (−k)− ∑ k ′ ∫ dp dp′ t∫ −∞ dt′ eε(t ′−t)〈n̂k(p ′)〉t′Φ−1 k (p′,p)T0(t, t ′)In(−k,p) ] , where In(k,p) = (1− P0) ˙̂nk(p), (3.43) I inth (k) = (1− P0) ˙̂ hint k (3.44) are generalized fluxes in the linear approximation, ˙̂nk(p) = iLN n̂k(p), ˙̂ hint k = iLN ĥ int k , T0(t, t ′) = e(t−t′)(1−P0)iLN is a time evolution operator with the projec- tion operator P0 which is a linear approximation of the generalized Mori projec- tion operator P(t) (3.15), with taking into account orthogonalization of variables 727 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn {Êk, n̂k(p)} → {ĥint k , n̂k(p)}. According to the structure (3.41), P0 acts on the dynamical variables Âk P0Âk ′ = ∑ k 〈Âk ′ ĥint−k 〉0Φ−1 hh (k)ĥ int k + ∑ k ∫ dp dp′ 〈Âk ′ n̂−k(p ′)〉0Φ−1 k (p′,p)n̂k(p). (3.45) For kinetics and hydrodynamics of nonequilibrium processes which are close to an equilibrium state, the generalized transport equations (3.17), (3.18) in the linear approximation for the nonequilibrium distribution function ̺ ( xN ; t ) (3.42) transform into a transport equation for fk(p; t) = 〈n̂k(p)〉t, hintk (t) = 〈ĥint k 〉t ∂ ∂t fk(p; t)+ ik · p m fk(p; t) = − ik · p m nf0(p)c2(k) ∫ dp′ fk(p ′; t)+iΩnh(k,p)h int k (t)− ∫ dp′ t∫ −∞ dt′ eε(t ′−t)ϕnn(k,p,p ′; t, t′)fk(p ′; t′)− t∫ −∞ dt′ eε(t ′−t)ϕnh(k,p; t, t ′)hint k (t′), (3.46) ∂ ∂t hint k (t) = ∫ dp iΩhn(k,p)fk(p; t)− (3.47) ∫ dp t∫ −∞ dt′ eε(t ′−t)ϕhn(k,p; t, t ′)fk(p; t ′)− t∫ −∞ dt′ eε(t ′−t)ϕhh(k; t, t ′)hint k (t′), where iΩnh(k,p), iΩhn(k,p) are normalized static correlation functions: iΩnh(k,p ) = 〈 ˙̂nk(p)ĥ int −k 〉0Φ−1 hh (k), (3.48) iΩhn(k,p) = ∫ dp′ 〈 ˙̂h int k n̂−k(p ′)〉0Φ−1 k (p′,p) (3.49) and ϕnn(k,p,p ′; t, t′) = ∫ dp′′ 〈In(k,p)T0(t, t′)In(−k,p′′)〉0Φ−1 k (p′′,p′), (3.50) ϕhn(k,p; t, t ′) = ∫ dp′ 〈I inth (k)T0(t, t ′)In(−k,p′)〉0Φ−1 k (p′,p), (3.51) ϕnh(k,p; t, t ′) = 〈In(k,p)T0(t, t′)I inth (−k)〉0Φ−1 hh (k), (3.52) ϕhh(k; t, t ′) = 〈I inth (k)T0(t, t ′)I inth (−k)〉0Φ−1 hh (k) (3.53) are generalized transport kernels (memory functions) which describe kinetic and hydrodynamic processes. The system of transport equations (3.46), (3.47) is closed. Eliminating hint k (t), it is possible to obtain a closed kinetic equation for a Fourier component of the nonequilibrium one-particle distribution function. For this pur- pose one uses a Laplace transform with respect to time, assuming that at t > 0 the quantities fk(p; t = 0), hint k (t = 0) are known A(z) = i ∫ ∞ 0 dt eiztA(t), z = ω + iε, ε→ +0. (3.54) 728 A consistent description of kinetics and hydrodynamics Then, equations (6.46) and (6.47) are presented in the form: zfk(p; z) + ik · p m fk(p; z) = − ik · p m nf0(p)c2(k) ∫ dp′ fk(p ′; z) + (3.55) Σnh(k,p; z)h int k (z)− ∫ dp′ ϕnn(k,p,p ′; z)fk(p ′; z) + fk(p; t = 0), zhint k (z) = ∫ dp′ Σhn(k,p ′; z)fk(p ′; z)− ϕhh(k; z)h int k (z) + hint k (t = 0), (3.56) where Σnh(k,p; z) = iΩnh(k,p)− ϕnh(k,p; z), (3.57) Σhn(k,p; z) = iΩhn(k,p)− ϕhn(k,p; z). (3.58) Let us solve equation (3.56) with respect to hint k (z) and substitute the result into (3.55). Then one obtains a closed kinetic equation for fk(p; z) (at h int k (t = 0) = 0): zfk(p; z) + ik · p m fk(p; z) = − ik · p m nf0(p)c2(k) ∫ dp′ fk(p ′; z)− ∫ dp′ Dnn(k,p,p ′; z)fk(p ′; z) + fk(p; t = 0), (3.59) where Dnn(k,p,p ′; z) = ϕnn(k,p,p ′; z)− Σnh(k,p; z) 1 z + ϕhh(k; z) Σhn(k,p ′; z) (3.60) is a generalized transport kernel of kinetic processes, which is renormalized tak- ing into account the processes of transport of the potential energy of interaction between the particles. If we put formally in Dnn(k,p,p ′; z) that ĥint−k = 0 (such equality is valid when the contribution of the average potential energy of interac- tion is much smaller than that of the average kinetic energy), then from (3.59) one obtains a kinetic equation for fk(p; z), zfk(p; z) + ik · p m fk(p; z) = − ik · p m nf0(p)c2(k) ∫ dp′ fk(p ′; z)− ∫ dp′ ϕ′ nn(k,p,p ′; z)fk(p ′; z) + fk(p; t = 0). (3.61) Equation (3.61) was derived for the first time by using the Mori projection opera- tors method in [107–109], when the microscopic phase density n̂k(p) was a param- eter of an abbreviated description. In this case the memory function ϕ′ nn(k,p,p ′; z) has the following structure: ϕ′ nn(k,p,p ′; t, t′) = ∫ dp′′ 〈I0n(k,p)T ′ 0(t, t ′)I0n(−k,p′′)〉0Φ−1 k (p′′,p′), (3.62) where I0n(k,p) = (1− P ′ 0) ˙̂nk(p) (3.63) 729 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn is a generalized flow, P ′ 0 is the Mori projection operator introduced in [107–109]: P ′ 0Âk ′ = ∑ k ∫ dp dp′ 〈Âk ′n̂−k(p ′)〉0Φ−1 k (p′,p)n̂k(p), (3.64) T ′ 0(t, t ′) = e(t−t′)(1−P′ 0)iLN is the corresponding time evolution operator. The kinetic equation (3.59) for the Fourier component of the nonequilibrium one-particle distri- bution function stimulates investigations of the dynamical structure factor S(k;ω), time correlation functions for transverse and longitudinal flows of the particles, dif- fusion coefficients, viscosities for dense gases and liquids [77,79,107–119]. In papers by Mazenko, there was performed a derivation of the linearized Boltzmann-Enskog equation by expanding memory functions ϕ′ nn(k,p,p ′; t, t′) on density. In the case of weak interactions a Fokker-Planck equation was obtained. Mazenko [114] and Forster [115,136] calculated the generalized viscosity coefficient with the help of kinetic equation (3.59). Self-diffusion of a one-component plasma [181], as well as spectra of mass and charge fluctuations in ionic solutions were investigated on the basis of this equation. In actual calculations, certain approximations for the memory functions ϕ′ nn(k,p,p ′; t, t′) were used. However, the main drawback of the kinetic equation (3.59) lies in its non-self-consistency with the conservation law of the total energy for dense gases and liquids when the contribution of the inter- action energy between the particles to thermodynamic quantities and transport coefficients becomes important. This fact was also pointed out earlier in the paper by John and Forster [118], where they performed an investigation of the dynamical structure factor S(k;ω) in the intermediate region of wavevector k and frequency values ω for a simple liquid on the basis of the set of dynamical variables of an abbreviated description n̂k(p) and Êk. In the next subsection, on the basis of a system of transport equations for Fourier components of the nonequilibrium one-particle distribution function and the potential part of enthalpy (3.46), (3.47), we shall obtain equations for time cor- relation functions. We shall also investigate the spectrum of collective excitations and the structure of generalized transport coefficients. 3.4. Time correlation functions, collective modes and gene ralized transport coefficients With the help of combined equations (3.46), (3.47) one obtains a system for time correlation functions: Φnn(k,p,p ′; t) = ∫ dp′′ 〈n̂k(p; t)n̂−k(p ′′; 0)〉0Φ−1 k (p′′,p′), (3.65) Φhn(k,p; t) = ∫ dp′ 〈ĥint k (t)n̂−k(p ′; 0)〉0Φ−1 k (p′,p), (3.66) Φnh(k,p; t) = 〈n̂k(p; t)ĥ int −k (0)〉0Φ−1 hh (k), (3.67) Φhh(k; t) = 〈ĥint k (t)ĥint−k (0)〉0Φ−1 hh (k), (3.68) where n̂k(p; t) = e−iLN tn̂k(p; 0), ĥ int k (t) = e−iLN tĥint k (0). 730 A consistent description of kinetics and hydrodynamics One uses the Fourier transform with respect to time 〈a〉ω = ∫ ∞ −∞ dt eiωt〈a〉t. Then we write the system of equations (3.46), (3.47) in the form: −iω〈n̂k(p)〉ω = ∫ dp′ Σnn(k,p,p ′;ω + iε)〈n̂k(p ′)〉ω +Σnh(k,p;ω + iε)〈ĥint k 〉ω,(3.69) −iω〈ĥint k 〉ω = ∫ dp′ Σhn(k,p ′;ω + iε)〈n̂k(p ′)〉ω − ϕhh(k;ω + iε)〈ĥint k 〉ω, (3.70) where Σnn(k,p,p ′;ω + iε) = iΩnn(k,p,p ′)− ϕnn(k,p,p ′;ω + iε), (3.71) Σnh(k,p;ω + iε) = iΩnh(k,p)− ϕnh(k,p;ω + iε), (3.72) Σhn(k,p;ω + iε) = iΩhn(k,p)− ϕhn(k,p;ω + iε). (3.73) It is more convenient to present the system of equations (3.69), (3.70) in a matrix form: −iω〈ãk〉ω = Σ̃(k;ω + iε)〈ãk〉ω, (3.74) where ãk = col(n̂k(p), ĥ int k ) is a vector-column and Σ̃(k;ω + iε) = [ ∫ dp′ Σnn(k,p,p ′;ω + iε) Σnh(k,p;ω + iε) ∫ dp′ Σhn(k,p ′;ω + iε) −ϕhh(k;ω + iε) ] , (3.75) Σ̃(k;ω + iε) = ∫ ∞ 0 dt ei(ω+iε)tΣ̃(k; t). Now, one uses the solution to the Liouville equation in approximation (3.41) with- out introducing the projection operator Pq(t): ̺ ( xN ; t ) = ̺q ( xN ; t ) − t∫ −∞ dt′ eε(t ′−t)eiLN (t′−t) ( ∂ ∂t ′ + iLN ) ̺q ( xN ; t′ ) . Then, from the self-consistency conditions 〈ãk〉t = 〈ãk〉tq one obtains a system of equations which connects the average values 〈n̂k(p)〉ω and 〈ĥint k 〉ω with spectral functions of time correlation functions: iω [ ∫ dp′ Φnn(k,p,p ′;ω + iε) Φint nh(k,p;ω + iε) ∫ dp′ Φint hn(k,p ′;ω + iε) −Φint hh(k;ω + iε) ] × [ 〈n̂k(p)〉ω 〈ĥint k 〉ω ] = [ ∫ dp′ Φnn(k,p,p ′) 0 0 −Φhh(k) ] − i(ω + iε) [ ∫ dp′ Φnn(k,p,p ′;ω + iε) Φint nh(k,p;ω + iε) ∫ dp′ Φint hn(k,p ′;ω + iε) −Φint hh(k;ω + iε) ] × [ 〈n̂k(p)〉ω 〈ĥint k 〉ω ] . 731 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn Such a system can be presented in the next compact form: iωΦ̃(k;ω + iε)〈ãk〉ω = [ Φ̃(k)− i(ω + iε)Φ̃(k;ω + iε) ] 〈ãk〉ω. (3.76) Let us multiply equation (3.74) by the matrix Φ̃(k;ω+iε) and compare the result with equation (3.76). So we find zΦ̃(k; z) = Σ̃(k; z)Φ̃(k; z)− Φ̃(k), z = ω + iε, (3.77) Φ̃(k; z) = i ∫ ∞ 0 dt eiztΦ̃(k; t), ε → 0 or in an explicit form: zΦnn(k,p,p ′; z) = ∫ dp′′ Σnn(k,p,p ′′; z)Φnn(k,p ′′,p′; z) + (3.78) Σnh(k,p; z)Φ int hn(k,p ′; z)− Φnn(k,p,p ′), zΦint nh(k,p; z) = ∫ dp′′ Σnn(k,p,p ′′; z)Φint nh(k,p ′′; z) + Σnh(k,p; z)Φ int hh(k; z), (3.79) zΦint hn(k,p ′; z) = ∫ dp′′ Σhn(k,p ′′; z)Φint nn(k,p ′′,p′; z)− ϕhh(k; z)Φ int hn(k,p ′; z),(3.80) zΦint hh(k; z) = ∫ dp′′ Σhn(k,p ′′; z)Φint nh(k,p ′′; z)− ϕhh(k; z)Φ int hh(k; z)− Φhh(k), (3.81) where the condition Φhn(k,p ′) = Φnh(k,p) = 0 is taken into account. Therefore, we obtained a system of equations for normalized correlation func- tions (3.65)–(3.68) for the description of nonequilibrium states of the system when kinetic and hydrodynamic processes are considered simultaneously. A similar sys- tem of equations for time correlation functions in the method of projection opera- tors was obtained in the paper by John and Forster [118]. In this paper the param- eters of an abbreviated description were orthogonal dynamical variables n̂k(p) and Êk. Solutions of such an equation system in the hydrodynamic limit were found using the projection procedure for the first five moments of the nonequilibrium one-particle distribution function. In view of this, the dynamical structure factor S(k;ω) for intermediate values of k and ω was investigated using a parametric approximation of memory functions for liquid Argon. However, in this paper the question concerning collective modes and generalized transport coefficients, when kinetics and hydrodynamics are connected between themselves, is not discussed. In order to solve the system of equations (3.78)–(3.81) we also apply the pro- jection procedure [108]. Let us introduce the dimensionless momentum ξ = k mv0 , v20 = (mβ)−1. Then the system of equations (3.77) can be rewritten in the matrix form: zΦ̃(k; ξ, ξ′; z)− Σ̃(k; ξ, ξ′′; z)Φ̃(k; ξ′′, ξ′; z) = −Φ̃(k; ξ, ξ′), (3.82) where it is clear that the integration must be performed with respect to the re- peating indices ξ′′. Further, let us introduce the scalar product of two functions, 732 A consistent description of kinetics and hydrodynamics φ(ξ) and ψ(ξ), as 〈φ\|ψ〉 = ∫ dξ φ∗(ξ)f0(ξ)ψ(ξ). (3.83) Then, the matrix element for some “operator” M can be determined as 〈φ\|M \|ψ〉 = ∫ dξ dξ′ φ∗(ξ)M(ξ, ξ′)f0(ξ ′)ψ(ξ′). (3.84) Let φ(ξ) = {φµ(ξ)} be the orthogonalized basis of functions with the weight f0(ξ), so that the following condition is satisfied: 〈φν \|φµ〉 = δνµ, ∑ ν \|φν〉〈φν \| = 1, (3.85) where φµ(ξ) = φlmn(ξ) = (l!m!n!)−1/2H̄l(ξx)H̄m(ξy)H̄n(ξz), (3.86) H̄l(ξ) = 2−l/2Hl(ξ/2), Hl(ξ) is a Hermite polynomial. Then, each function in the system of equations (3.82), which depends on momentum variables ξ, ξ ′, can be expanded over functions φµ(ξ) in the series: Φ̃(k; ξ, ξ′; z) = ∑ ν,µ φ∗ ν(ξ)Φ̃νµ(k; z)φµ(ξ ′)f0(ξ ′), (3.87) Σ̃(k; ξ, ξ′; z) = ∑ ν,µ φ∗ ν(ξ)Σ̃νµ(k; z)φµ(ξ ′)f0(ξ ′), (3.88) where Φ̃νµ(k; z) = 〈φν \|Φ̃(k; ξ, ξ′; z)\|φµ〉 = ∫ dξ dξ′ φ∗ ν(ξ)f0(ξ)Φ̃(k; ξ, ξ ′; z)φµ(ξ ′), (3.89) Σ̃νµ(k; z) = 〈φν \|Σ̃(k; ξ, ξ′; z)\|φµ〉 = ∫ dξ dξ′ φ∗ ν(ξ)f0(ξ)Σ̃(k; ξ, ξ ′; z)φµ(ξ ′). (3.90) Let us substitute expansions (3.87)–(3.90) into equation (3.82). As a result, one obtains: zΦ̃νµ(k; z)− ∑ γ Σ̃νγ(k; z)Φ̃γµ(k; z) = −Φ̃νµ(k). (3.91) In actual calculations, a finite number of functions from the set φν(ξ) is used. Taking into account this fact, let us introduce the projection operator P which projects arbitrary functions ψ(ξ) onto a finite set of functions φµ(ξ): P = n∑ ν=1 \|φν〉〈φν \| = 1−Q, P 〈ψ\| = n∑ ν=1 〈ψ\|φν〉〈φν\|. (3.92) Here n denotes a finite number of functions. Then, from (3.91) we obtain a system of equations for a finite set of functions φµ(ξ), n∑ γ=1 [ zδ̄νγ − iΩ̃νγ(k) + D̃νγ(k; z) ] Φ̃γµ(k; z) = −Φ̃νµ(k), (3.93) 733 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn where D̃νµ(k; z) = 〈φν\|ϕ̃(k; z) + Σ̃(k; z)Q [ z −QΣ̃(k; z)Q ]−1 QΣ̃(k; z)\|φµ〉 (3.94) are generalized hydrodynamic transport kernels and iΩ̃νµ(k) = 〈φν\|iΩ̃(k)\|φµ〉 (3.95) is a frequency matrix. Note that matrices iΩ̃(k) and ϕ̃(k; z) are defined according to (3.48), (3.49) and (3.50)–(3.53). Let us find solutions to the system of equations (3.93) in the hydrodynamic region when a set of functions φµ(ξ) present five moments of a one-particle distri- bution function: φ1(ξ) = 1, φ2(ξ) = ξz, φ3(ξ) = 1√ 6 (ξ2− 3), φ4(ξ) = ξx, φ5(ξ) = ξy. (3.96) Then, the following relations are fulfilled: 〈1\|n̂k(ξ) = ∫ dξ n̂k(ξ) = n̂k, 〈ξγ\|n̂k(ξ) = ∫ dξ n̂k(ξ) ξγ = p̂γ k , (3.97) 〈6−1/2(ξ2 − 3)\|n̂k(ξ) = ∫ dξ n̂k(ξ) 6 −1/2(ξ2 − 3) = Êkin k − 3n̂kβ −1 = ĥkin k , for the Fourier components of densities for the number of particles, momentum and the kinetic part of generalized enthalpy. Besides that, the microscopic conservation laws for densities of the number of particles and momentum can be written in the form: 〈1\| ˙̂nk(ξ) = −ikγ p̂ γ k m−1, 〈ξα\| ˙̂nk(ξ) = −ikγ ↔̂ T k γα, (3.98) where ↔̂ T k γα is a Fourier component of the stress-tensor. If we choose the direction of wavevector k along oz-axis, then φν(ξ), ν = 1, 2, 3 will correspond to longitudinal modes, whereas φν(ξ) at ν = 4, 5 will be related to transverse modes. From the system of equations (3.93), at ν = 4, 5, φ4(ξ) = ξx, φ5(ξ) = ξy, one obtains an equation for the Fourier component of the time correlation function connected with the transverse component of the momentum density Φ44(k; z). From this equation one finds: Φ44(k; z) = Φ⊥ pp(k; z) = − 1 z +D⊥ pp(k; z) , (3.99) 734 A consistent description of kinetics and hydrodynamics where Φ⊥ pp(k; z) = 〈ξx\|Φnn(k, ξ, ξ ′; z)\|ξ′x〉, (3.100) D⊥ pp(k; z) = D⊥(kin) pp (k; z) +D⊥(int) pp (k; z), (3.101) D⊥(kin) pp (k; z) = 〈ξx\|ϕnn(k, ξ, ξ ′; z)\|ξ′x〉, (3.102) D⊥(int) pp (k; z) = 〈ξx\| [ Σ̃(k, ξ, ξ′; z)Q [ z −QΣ̃(k, ξ, ξ′; z)Q ]−1 QΣ̃(k, ξ, ξ′; z) ] nn \|ξ′x〉, (3.103) D⊥ pp(k; z) = ik2η(k; z)(mn)−1, (3.104) where η(k; z) denotes a generalized coefficient of shear viscosity. Such a coefficient consists of two main contributions. The first one is D ⊥(kin) pp (k; z), whereas the sec- ond contribution D ⊥(int) pp (k; z) describes a relation of kinetic and hydrodynamic processes. If the last term is neglected, which formally corresponds to ĥint k = 0, then one obtains an expression for the generalized coefficient of shear viscosity η(k; z), obtained earlier by the authors of [114,115] when solving the equation zΦnn(k,p,p ′; z)− ∫ dp′′ iΩnn(k,p,p ′′)Φnn(k,p ′′,p′; z) + (3.105) ∫ dp′′ ϕ′ nn(k,p,p ′′; z)Φnn(k,p ′′,p′; z) = −Φnn(k,p,p ′), by using the method of projections [108] in the hydrodynamic region. Equation (3.105) corresponds to kinetic equation (3.61) for the Fourier component of the nonequilibrium one-particle distribution function. In ϕ′ nn(k,p,p ′; z) the projection is performed on a space of dynamics of the “slow” variable n̂k(p) connected with the microscopic conservation laws for the particle-number density and momentum (3.98). In our approach in D⊥ pp(k; z), the projection is carried out on an extended space of dynamics of the “slow” variables n̂k(p) and ĥ int k . Moreover, ĥint k is a hydro- dynamic variable, whereas n̂k(p) is a kinetic one and only its average value with five moments (3.96) can be related to hydrodynamic quantities. If we put ν = 1, 2, 3, φ1(ξ) = 1, φ2(ξ) = ξz, φ3(ξ)− 6−1/2(ξ2 − 3) in the system of equation (3.93), then we obtain: zΦna(k; z)− iΩnp(k)Φpa(k; z) = −Φna(k), (3.106) zΦpa(k; z)− iΩpn(k)Φna(k; z) +D\|\| pp(k; z)Φpa(k; z) − (3.107) Σphkin(k; z)Φhkina(k; z)− Σphint(k; z)Φhinta(k; z) = −Φpa(k), zΦhkina(k; z)− Σhkinp(k; z)Φpa(k; z) + (3.108) Dhkinhkin(k; z)Φhkina(k; z) +Dhkinhint(k; z)Φhinta(k; z) = −Φhkina(k), zΦhinta(k; z)− Σhintp(k; z)Φpa(k; z) + (3.109) Dhinthkin(k; z)Φhkina(k; z) +Dhinthint(k; z)Φhinta(k; z) = −Φhinta(k), 735 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn where a = {n̂k, p̂k , ĥkin k , ĥint k } and Σphkin(k; z) = iΩphkin(k) − Dphkin(k; z), Σphint(k; z) = iΩphint(k) − Dphint(k; z), Σhkinp(k; z) = iΩhkinp(k) − Dhkinp(k; z), Σhintp(k; z) = iΩhintp(k) − Dhintp(k; z), (3.110) iΩab(k) and Dab(k; z) are determined according to (3.95), (3.94). From the system of equations (3.106)–(3.109) one can define the Fourier components of the particle number density correlation functions Φnn(k; z) = Φ11(k; z) = 〈1\|Φnn(k, ξ, ξ ′; z)\|1′〉, , (3.111) as well as of the longitudinal component of the momentum density Φ\|\| pp(k; z) = Φ22(k; z) = 〈ξz\|Φnn(k, ξ, ξ ′; z)\|ξ′z〉, (3.112) for the kinetic part of generalized enthalpy Φhkinhkin(k; z) = Φ33(k; z) = 〈6−1/2(ξ2− 3)\|Φnn(k, ξ, ξ ′; z)\|6−1/2((ξ′)2− 3)〉 (3.113) as well as for the potential part of generalized enthalpy Φhinthint(k; z) and cross cor- relation functions, especially Φhinthkin(k; z), Φnhkin(k; z), Φnhint(k; z), Φphkin(k; z), Φphint(k; z). It is important to point out that the system of equations (3.106)– (3.109) corresponds to the system of equations for Fourier components of the av- erage values of densities for the number of particles 〈n̂k〉z, longitudinal momentum 〈p̂k〉z, kinetic 〈ĥkin k 〉z and potential 〈ĥint k 〉z parts of generalized enthalpy: z〈n̂k〉z − iΩnp(k)〈p̂k〉z = −〈n̂k(t = 0)〉, (3.114) z〈p̂ k 〉z − iΩpn(k)〈n̂k〉z + (3.115) D\|\| pp(k; z)〈p̂k 〉z − Σphkin(k; z)〈ĥkin k 〉z − Σphint(k; z)〈ĥint k 〉z = −〈p̂ k (t = 0)〉, z〈ĥkin k 〉z − Σhkinp(k; z)〈p̂k〉z + (3.116) Dhkinhkin(k; z)〈ĥkin k 〉z +Dhkinhint(k; z)〈ĥint k 〉z = −〈ĥkin k (t = 0)〉, z〈ĥint k 〉z − Σhintp(k; z)〈p̂k〉z + (3.117) Dhinthkin(k; z)〈ĥkin k 〉z +Dhinthint(k; z)〈ĥint k 〉z = −〈ĥint k (t = 0)〉. This system of equations is similar in construction to the equations of molecu- lar hydrodynamics [77]. The difference consists in the fact that instead of the equations for the Fourier component of the mean enthalpy density 〈ĥk〉t which is introduced in molecular hydrodynamics, there are two connected equations for the Fourier components of mean values of the kinetic and potential parts of en- thalpy density. Moreover, instead of the generalized thermal conductivity which appears in molecular hydrodynamics, the dissipation of energy flows is described in equations (3.114)–(3.117) by a set of generalized transport kernels Dhkinhkin(k; z), 736 A consistent description of kinetics and hydrodynamics Dhkinhint(k; z), Dhinthkin(k; z), Dhinthint(k; z). Obviously, transport kernels give more detailed information on the dissipation of energy flows in the system because they describe the time evolution of dynamical correlations between the kinetic and po- tential flows of enthalpy density. Solving the system of equation (3.106)–(3.109) at a = n, one obtains an ex- pression for the correlation function “density-density” Φnn(k; z) Φnn(k; z) = −S2(k) [ z − iΩnp(k)iΩpn(k) z + D̄ \|\| pp(k; z) ]−1 , (3.118) where D̄\|\| pp(k; z) = D\|\| pp(k; z)− Σ̄phkin(k; z) [ z + D̄hkinhkin(k; z) ]−1 Σ̄hkinp(k; z) − Σphint(k; z) [z + Dhinthint(k; z)]−1Σhintp(k; z), (3.119) Σ̄phkin(k; z) = Σphkin(k; z)− (3.120) Σphint(k; z) [z +Dhinthint(k; z)]−1Dhinthkin(k; z), Σ̄hkinp(k; z) = Σhkinp(k; z)− (3.121) Dhkinhint(k; z) [z +Dhinthint(k; z)]−1Σhintp(k; z), D̄hkinhkin(k; z) = Dhkinhkin(k; z)− (3.122) Dhkinhint(k; z) [z +Dhinthint(k; z)]−1Dhinthkin(k; z). In expressions (3.119)–(3.122) we can observe an interesting renormalization of the functions Σab and Dab via the generalized transport kernels for fluctuations of flows of the potential part of enthalpy density. It is important to point out that in the mode-coupling theory developed by Götze [95,96], an expression for Φnn(k; z) has the same form as in (3.118). However, D̄ \|\| pp(k; z) is connected only with the generalized shear viscosity η\|\|(k; z), since the densities of the number of particles n̂k and momentum p̂ k are included in the set of variables of an ab- breviated description. In our case D̄ \|\| pp(k; z) takes into account both thermal and viscous dynamical correlation processes. Excluding from (3.118) the imaginary part Φ \|\| nn(k;ω) of the correlation function Φnn(k; z), one obtains an expression for the dynamical structure factor S(k;ω) in which contributions of transport ker- nels corresponding to the kinetic and potential parts of the enthalpy density ĥk are separated. It is evident that the main contribution of the transport kernel Dhinthint(k; z) to the S(k;ω) for liquids was in the hydrodynamical region (the re- gion of small values of wavevector k and frequency ω), whereas Dhkinhkin(k; z) will contribute to the kinetic region (orders of interatomic distances, small correlation times). In the region of intermediate values of wavevector k and frequency ω, it is necessary to take into account all the transport kernels Σphkin(k; z), Σphint(k; z), Dhkinhkin(k; z), Dhinthkin(k; z), Dhinthint(k; z). Since it is impossible to perform ex- act calculations of the described above functions, it is necessary in each separate 737 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn region to accept models corresponding to the physical processes. Obviously, it is necessary to provide modelling on the level of generalized transport kernels ϕnn(k,p,p ′; t, t′), ϕnh(k,p; t, t ′), ϕhn(k,p ′; t, t′), ϕhh(k; t, t ′) (3.50)–(3.53). In par- ticular, we can accept for ϕnn(k,p,p ′; t, t′) the Boltzmann-Enskog model for hard spheres or the Fokker-Planck model for the case of weak correlation, as in the papers by Mazenko [111,114,135]. Such approximations can be applicable to ki- netic processes at interatomic distances (region of large values of wavevector k) and small correlation times (region of large frequencies ω) of interatomic collisions. Besides, it is necessary to take into account information about interactions of par- ticles at short distances. In the transport kernels ϕnn(k; ξ, ξ ′; t, t′), ϕnh(k; ξ; t, t ′), ϕhn(k; ξ ′; t, t′)ϕhh(k; t, t ′) it is important to distinguish processes connected with a type of particles interaction at short and long distances, analogously to the case of Enskog-Landau kinetic equation for a model of charged hard spheres. The problems of modelling for these kernels are complicated by the absence of a small parameter in terms of which the perturbation theory might be developed. The transport kernel ϕ′ nn(k,p,p ′; t, t′) (3.62) and its moments on space[1, pl, p2 2m ] were analyzed in the limit z → ∞ and the hydrodynamic limit k → 0, z → 0 in the paper by Forster [115]. The modelling problems of transport kernels for inter- mediate values of k and ω are reflected in the details of the description of spectra for collective excitations, as well as in the dynamical structure factor. In the limit k → 0, ω → 0, the cross correlations between the kinetic and potential flows of energy and shear flows become not so important and the system of equations (3.118)–(3.122) gives a spectrum of the collective modes of molecular hydrodynam- ics [77,79]. For intermediate values of k and ω, the spectrum of collective modes can be found from the condition   z iΩnp(k) 0 0 iΩpn(k) z +D \|\| pp(k; z) Σphkin(k; z) Σphint(k; z) 0 Σhkinp(k; z) z +Dhkinhkin(k; z) Dhkinhint(k; z) 0 Σhintp(k; z) Dhinthkin(k; z) z +Dhinthint(k; z)   = 0, (3.123) in which contributions of the kinetic and potential parts of generalized enthalpy are separated. This will be reflected in the structure of a heat mode at concrete model calculations of the wavevector- and frequency-dependent transport kernels Dhkinhkin(k; z), Dhkinhint(k; z), Dhinthkin(k; z), Dhinthint(k; z) depending on k and ω. The system of equations (3.93) for time correlation functions allows an extended description of collective modes in liquids, which includes both hydrodynamic and kinetic processes. Including on the basis of functions φν(ξ) (3.96) some additional functions, ψl Q(ξ) = 1 5 (ξ2 − 5)ξl, ψlj Π(ξ) = √ 2 2 (ξlξj − 1 3 ξ2δlj), (3.124) which corresponds to a 13-moments approximation of Grad, one obtains a system of equations for time correlation functions of an extended set of hydrodynamic variables {n̂k, p̂k , ĥkin k , ↔̂ Πk, Q̂k , ĥint k }: zΦ̃H(k; z)− iΩ̃H(k)Φ̃H(k; z) + ϕ̃H(k; z)Φ̃H(k; z) = −Φ̃H(k), (3.125) 738 A consistent description of kinetics and hydrodynamics where Φ̃H(k; z) =   Φnn Φnp Φnhkin ΦnΠ ΦnQ Φnhint Φpn Φpp Φphkin ΦpΠ ΦpQ Φphint Φhkinn Φhkinp Φhkinhkin ΦhkinΠ ΦhkinQ Φhkinhint ΦΠn ΦΠp ΦΠhkin ΦΠΠ ΦΠQ ΦΠhint ΦQn ΦQp ΦQhkin ΦQΠ ΦQQ ΦQhint Φhintn Φhintp Φhinthkin ΦhintΠ ΦhintQ Φhinthint   (k;z) (3.126) is a matrix of Laplace images of the time correlation functions, ↔̂ Πk= ∫ dξϕΠ(ξ)n̂k(ξ), Q̂ k = ∫ dξ ϕQ(ξ)n̂k(ξ), iΩ̃H(k) =   0 iΩnp 0 0 0 0 iΩpn 0 iΩphkin iΩpΠ 0 iΩphint 0 iΩhkinp 0 0 iΩhkinQ 0 0 iΩΠp 0 0 iΩΠQ 0 0 0 iΩQhkin iΩQΠ 0 iΩQhint 0 iΩhintp 0 0 iΩhintQ 0   (k) (3.127) is an extended hydrodynamic frequency matrix, ϕ̃H(k; z) =   0 0 0 0 0 0 0 DH pp DH phkin DH pΠ DH pQ DH phint 0 Dhkinp Dhkinhkin DhkinΠ DhkinQ Dhkinhint 0 DΠp DΠhkin DΠΠ DΠQ DΠhint 0 DQp DQhkin DQΠ DQQ DQhint 0 Dhintp Dhinthkin DhintΠ DhintQ Dhinthint   (k;z) (3.128) is a matrix of generalized memory functions, elements of which are transport ker- nels (3.94) projected on the basis of functions φν(ξ) (3.96), (3.124). For such a description, the spectrum of generalized collective modes of the system is deter- mined for intermediate k and ω by the relation det ∣∣∣zĨ − iΩ̃H(k) + ϕ̃H(k) ∣∣∣ = 0 which takes into account kinetic and hydrodynamic processes. In the hydrody- namic limit k → 0, ω → 0, when the contribution of cross dissipative correla- tions between the kinetic and hydrodynamic processes practically vanishes, the system of equations for the time correlation function (3.125), after some trans- formations, can be reduced to a system of equations for time corelation functions of densities for the number of particles n̂k, momentum p̂k, total enthalpy ĥk, the generalized stress tensor π̂↔k = (1 − PH)iLN p̂k and the generalized enthalpy flow q̂k = (1−PH)iLN ĥk, where PH is the Mori operator constructed on the dynamical variables {n̂k, p̂k, ĥk}. For such a system of equations, the spectrum of collective excitations is determined from [139,141]: ∣∣∣∣∣∣∣∣∣∣ z iΩnp 0 0 0 iΩpn z iΩph iΩpπ 0 0 iΩhp z 0 0 0 iΩπp 0 z + ϕππ iΩπQ + ϕπQ 0 0 0 iΩQπ + ϕQπ ϕQQ ∣∣∣∣∣∣∣∣∣∣ (k;z) = 0. (3.129) 739 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn In the hydrodynamic limit this gives: the heat mode zH(k) = DTk 2 +O(k4), (3.130) two complex conjugated sound modes z±(k) = ±iωS(k) + zS(k), (3.131) where ωS(k) = ck+o(k3) is a frequency of sound propagation, zS(k) = Γk2+O(k4) is a frequency of sound damping with the damping coefficient Γ; two nonvanishing in the limit k → 0 kinetic modes zπ(k) = ϕππ(0) +O(k2), zQ(k) = ϕQQ(0) +O(k2). (3.132) Here DT denotes a thermal diffusion coefficient DT = v2TQ γϕQQ(0) = λ mncp , v2TQ = mΦQQ − h2 ncV , γ = cp/cV cp and cV are, correspondingly, thermodynamic values of specific heats at the constant pressure and volume, λ is a thermal conductivity coefficient, h denotes a thermodynamic value of enthalpy, c = γ/mnS(k = 0) denotes an adiabatic sound velocity. In (3.131) Γ = 1 2 (γ − 1)DT + 1 2 η\|\| (3.133) is a sound damping coefficient with η\|\| = v2pπ ϕππ(0) = ( 4 3 η + κ ) /nm, v2pπ = mS(0)Φππ(0)− γ mnS(0) , where η and κ are shear and bulk viscosities. This spectrum coincides with the results of papers [139,141,142]. However, at fixed values of k and ω, the transport kernels ϕππ, ϕπQ, ϕQπ, ϕQQ are expressed via the generalized transport kernels Dνµ(k; z) of matrix (3.128), i.e. via ϕnn(k,p,p ′; t, t′), ϕnh(k,p; t, t ′), ϕhn(k,p ′; t, t′), ϕhh(k; t, t ′) (3.50)–(3.53), according to the definition Dνµ(k; z) (3.94). Here, it is important to point out that passing from the system of transport equations of a self-consistent description of kinetics and hydrodynamics to the equations of gen- eralized hydrodynamics, we can connect the generalized transport kernels (3.50)– (3.53) with the hydrodynamic transport kernels Dνµ(k; z) in (3.123) or (3.128). Therefore, we gain the aim of the present section, namely, to connect transport kernels of a self-consistent description of kinetics and hydrodynamics with hy- drodynamic transport kernels for time correlation functions for densities of the number of particles n̂k, momentum p̂k, enthalpy ĥk, which can be calculated by using the MD method [142–152,159], as well as by experiments on light scatter- ing for different real systems. Evidently, the most important problem is model calculations of transport kernels (3.50)–(3.53). Such calculations must take into 740 A consistent description of kinetics and hydrodynamics account a type of interaction of particles at short and long distances, as well as their structural distribution. From this point of view, of significant interest is the modelling of transport kernels (3.50)–(3.53) for a system of charged hard spheres, since for a system of hard spheres a transport kernel of the type ϕnn(k;p,p ′; t, t′) (3.50) was computed via the Enskog collision integral [120,121]. 4. Conclusion We have presented one of the approaches for a self-consistent description of ki- netic and hydrodynamic processes in systems of interacting particles, formulated by the NSO method. It is based on a modification of the boundary condition to the Liouville equation, which takes into account a nonequilibrity of the one-particle distribution function, as well as the local conservation laws which constitute a ba- sis for the hydrodynamic description. Using such a description, generalized kinetic equations for dense gases and liquids can be derived. The result obtained can be ex- tended to quantum systems of interacting particles [16–18]. In particular, it can be applied to nuclear matter and chemically reacting systems, where kinetic processes play an important role together with hydrodynamic ones. The method applied can be extended to the investigations of nonlinear and hydrodynamic fluctuations in polar, ionic and magnetic liquids and electrolyte solutions. There is a significant interest in investigations of collective excitations, time correlation functions and transport coefficients on the basis of equations (3.93), (3.94), (3.123), (3.128) at the presence of transport kernels ϕ(k; z) which describe an interference between kinetic and hydrodynamic processes. In particular, the transport kernel ϕ(k,p,p′; t) can take into account processes of interaction of par- ticles at short and long distances (at Coulomb and dipole interactions of particles). To do this, we can apply an approach of collective modes [142–152]. References 1. Zubarev D.N., Morozov V.G. Formulation of boundary conditions for the Bogolubov hierarchy with allowance for local conservation laws. // Teor. Mat. Fiz., 1984, vol. 60, No. 2, p. 270-279 (in Russian)1. 2. Zubarev D.N., Morozov V.G. Omelyan I.P., Tokarchuk M.V. Unification of kinetics and hydrodynamics in theory of transport phenomena. In: Collection of scientific works of Institute for Theoretical and Applied Mechanics of Siberian Branch of USSR Academy of Sciences. Models of mechanics of continuous media. Novosibirsk, 1989, p. 34-51 (in Russian). 3. Zubarev D.N., Morozov V.G., Omelyan I.P., Tokarchuk M.V. On kinetic equations for dense gases and liquids. // Teor. Mat. Fiz., 1991, vol. 87, No. 1, p. 113-129 (in Russian). 1Publications in that journal are usually translated into English and published by the Plenum Publishing Corporation in the USA annually. If we know where they appear in a translated version, we shall give this information next to the data of the original. 741 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn 4. Zubarev D.N., Morozov V.G., Omelyan I.P., Tokarchuk M.V. The Enskog-Landau ki- netic equation for charged hard spheres. In: Problems of atomic science and technique. Series: Nuclear physics investigations (theory and experiment). Kharkov, Kharkov Physico Technical Institute, 1992, vol. 3(24), p. 60-65 (in Russian). 5. Zubarev D.N., Morozov V.G., Omelyan I.P., Tokarchuk M.V. The unification of ki- netic and hydrodynamic approaches in the theory of dense gases and liquids. // Teor. Mat. Fiz., 1993, vol. 96, No. 3, p. 325-350 (in Russian). 6. Morozov V.G., Kobryn A.E., Tokarchuk M.V. Modified kinetic theory with consid- eration for slow hydrodynamical processes. // Cond. Matt. Phys., 1994, vol. 4, p. 117-127. 7. Kobryn A.E., Morozov V.G., Omelyan I.P., Tokarchuk M.V. Enskog-Landau kinetic equation. Calculation of the transport coefficients for charged hard spheres. // Phys- ica A, 1996, vol. 230, No. 1&2, p. 189-201. 8. Omelyan I.P., Tokarchuk M.V. Kinetic equation for liquids with a multistep potential of interaction. H-theorem. // Physica A, 1996, vol. 234, No. 1&2, p. 89-107. 9. Zubarev D., Morozov V., Röpke G. Statistical mechanics of nonequilibrium processes. Vol. 1, Basic concepts, kinetic theory. Berlin, Academie Verlag, 1996. 10. Zubarev D., Morozov V., Röpke G. Statistical mechanics of nonequilibrium processes. Vol. 2, Relaxation and hydrodynamical processes. Berlin, Academie Verlag, 1997. 11. Tokarchuk M.V., Omelyan I.P., Kobryn A.E. On the kinetic theory of classical inter- acting particles by means of nonequilibrium statistical operator method. // J. Phys. Studies, 1997, vol. 1, No. 4, p. 490-512 (in Ukrainian). 12. Kobryn A.E., Omelyan I.P., Tokarchuk M.V. The modified group expansions for constructions of solutions to the BBGKY hierarchy. // J. Stat. Phys., 1998, vol. 92, No. 5/6 p. 973-994. 13. Tokarchuk M.V. On the statistical theory of a nonequilibrium plasma in its electro- magnetic self-field. // Teor. Mat. Fiz., 1993, vol. 97, No. 1, p. 27-43 (in Russian). En- glish transl. appears In: Teoreticheskaya i Matematicheskaya Fizika, 1994, by Plenum Publishing Corporation, p. 1126-1136. 14. Tokarchuk M.V., Ignatyuk V.V. Consistent kinetic and hydrodynamic description of plasma in self-electromagnetic field. I. Transport equations. // Ukr. Fiz. Zhurn., 1997, vol. 42, No. 2, p. 153-160 (in Ukrainian). 15. Tokarchuk M.V., Ignatyuk V.V. Consistent kinetic and hydrodynamic description of plasma in self-electromagnetic field. II. Time correlation functions and collective modes. // Ukr. Fiz. Zhurn., 1997, vol. 42, No. 2, p. 161-167 (in Ukrainian). 16. Morozov V., Röpke G. Quantum kinetic equation for nonequilibrium dense systems. // Physica A, 1995, vol. 221, p. 511-538. 17. Vakarchuk I.O., Glushak P.A., Tokarchuk M.V. A consistent description of kinetics and hydrodynamics of quantum Bose-systems I. Transport equations. // Ukr. Fiz. Zhurn., 1997, vol. 42, No. 9. p. 1150-1158 (in Ukrainian). 18. Tokarchuk M.V., Arimitsu T., Kobryn A.E. Thermo field hydrodynamic and kinetic equations of dense quantum nuclear systems. // Cond. Matt. Phys., 1998, vol. 1, No. 3(15), p. 605-642. 19. Klimontovich Yu.L. On the necessity and possibility of the united description of kinetic and hydrodynamic processes. // Teor. Mat. Fiz., 1992, vol. 92, No. 2, p. 312-330 (in Russian). 20. Klimontovich Yu.L. The united description of kinetic and hydrodynamic processes 742 A consistent description of kinetics and hydrodynamics in gases and plasmas. // Phys. Lett. A, 1992, vol. 170, p. 434. 21. Klimontovich Yu.L. From the Hamiltonian mechanics to a continuous media. Dissi- pative structures. Criteria of self-organization. // Cond. Matt. Phys., 1994, vol. 4, p. 89-116. 22. Bogolubov N.N. Problems of a dynamical theory in statistical physics. In: Studies in statistical mechanics, vol. 1 (eds. J. de Boer and G.E.Uhlenbeck). Amsterdam, North-Holland, 1962. 23. Liboff R.L. Introduction to the Theory of Kinetic Equations. New York, John Willey and Sons, 1969. 24. Klimontovich Yu.L. Kinetic Theory of Nonideal Gases and Nonideal Plasmas. Oxford, Pergamon Press, 1982. 25. Ferziger J.H., Kaper H.G. Mathematical Theory of Transport Processes in Gases. Amsterdam, North-Holland, 1972. 26. Résibois P., de Leener M. Classical Kinetic Theory of Fluids. New York, John Willey and Sons, 1977. 27. Kawasaki K., Oppenheim J. Logarithmic term in the density expansion of transport coefficients. // Phys. Rev., 1965, vol. 139, No. 6, p. 1763-1768. 28. Weistock J. Nonanalyticity of transport coefficients and the complete density ex- pansion of momentum correlation function. // Phys. Rev., 1971, vol. 140, No. 2, p. 460-465. 29. Enskog D. Kinetische Theorie der Warmeleitung Reibung und Selbstdiffusion in gevissen verdichtetten Gasen und Flussigkeiten. // Svenska Vetenskapsaked. Handl., 1922, vol. 63, p. 4. 30. Chapman S., Cowling T.G., The Mathematical Theory of Nonuniform Gases. Cam- bridge, Cambridge University Press, 1952. 31. Hanley H., MacCarthy R., Cohen F. Analysis of the transport coefficients for simple dense fluids: application of the modified Enskog theory. // Physica, 1972, vol. 60, No. 2, p. 322-356. 32. Davis H.T., Rise S.A., Sengers J.V. On the kinetic theory of dense fluids. The fluid of rigid spheres with a square well attraction. // J. Chem. Phys., 1961, vol. 35, No. 6, p. 2210-2233. 33. Grmela M., Rosen R., Garcia-Colin L.S. Compatibility of the Enskog theory with hydrodynamics. // J. Chem. Phys., 1981, vol. 75, No. 11, p. 5474-5484. 34. Ernst M.H., van Beijeren H. The modified Enskog equation. // Physica, 1973, vol. 68, No. 3, p. 437-456. 35. Résibois P. H-theorem for the (modified) nonlinear Enskog equation. // Phys. Rev. Lett., 1978, vol. 40, No. 22, p. 1409-1411. 36. Résibois P. H-theorem for the (modified) nonlinear Enskog equation. // J. Stat. Phys., 1978, vol. 19, No. 6, p. 593-603. 37. Karkheck J., van Beijeren H., de Schepper I.M. Kinetic theory and H-theorem for a dense square well fluid. // Phys. Rev. A, 1985, vol. 32, No. 4, p. 2517-2520. 38. Karkheck J., Stell G. Kinetic mean-field theories. // J. Chem. Phys., 1981, vol. 75, No. 3, p. 1475-1487. 39. Stell G., Karkheck J., van Beijeren H. Kinetic mean-field theories: rezultz of energy constraint in maximizing entropy. // J. Chem. Phys., 1983, vol. 79, No. 6, p. 3166- 3167. 40. Tokarchuk M.V., Omelyan I.P. Normal solution of Enskog-Vlasov kinetic equation 743 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn using boundary conditions method. In: Proceedings, Contributed papers of confer- ence “Modern problems of statistical physics”, vol. 1. Lviv, February 3-5, 1987, p. 245-252 (in Russian). 41. Leegwater J.A., van Beijeren H., Michels J.P.J. Linear kinetic theory of the square well fluid. // J. Phys: Cond. Matt., 1989, vol. 1, p. 237-255. 42. Rudyak V.Ya. On the theory of kinetic equations for dense gas. // Jurn. Tehn. Fiz.2, 1984, vol. 54, No. 7, p. 1246-1252 (in Russian). 43. Rudyak V.Ya. About derivation of Enskog-like kinetic equation for dense gas. // Teplofiz. Vys. Temp.3, 1985, vol. 23, No. 2, p. 268-272 (in Russian). 44. Rudyak V.Ya. Statistical Theory of Dissipative Processes in Gases and Liquids. Novosibirsk, Nauka, 1987 (in Russian). 45. Tokarchuk M.V., Omelyan I.P. Model kinetic equations for dense gases and liquids. // Ukr. Fiz. Zhurn., 1990, vol. 36, No. 8, p. 1255-1261 (in Ukrainian). 46. Kobryn A.E., Omelyan I.P., Tokarchuk M.V. Normal solutions and transport coeffi- cients to the Enskog-Landau kinetic equation for a two-component system of charged hard spheres. The Chapman-Enskog method. // Physica A (to be published). 47. Lenard A. On Bogolubov’s kinetic equation for a spatially homogeneous plasma. // Ann. Phys., 1960, vol. 3, p. 390-401. 48. Balescu R. Irreversible processes in ionized gases. // Phys. Fluids., 1960, vol. 3, p. 52-71. 49. Shelest A.V. Bogolubov’s Method in Dynamical Theory of Kinetic Equations. Moscow, Nauka, 1990 (in Russian). 50. Zubarev D.N. Nonequilibrium Statistical Thermodynamics. New York, Consultant Bureau, 1974. 51. Zubarev D.N. Modern methods of statistical theory of nonequilibrium processes. In: Reviews of science and technology. Modern problems of mathematics. Moscow, VINITI, 1980, vol. 15, p. 131-220 (in Russian). 52. Zubarev D.N., Novikov M.Yu. Generalized formulation of the boundary condition for the Liouville equation and for BBGKY hierarchy. // Teor. Mat. Fiz. 1972, vol. 13, No. 3, p. 406-420 (in Russian). 53. Zubarev D.N., Novikov M.Yu. Die Boltzmangleichung und mögliche Wege zur En- twicklung dynamischer Methoden in der kinetische Theorie. // Fortschritt für Physik, 1973, vol. 21, No. 12, p. 703-734. 54. Zubarev D.N., Morozov V.G., Omelyan I.P., Tokarchuk M.V. Derivation of kinetic equations for the system of hard spheres using nonequilibrium statistical operator method. / Preprint ITP-90-11R, Kiev 1990 (in Russian). 55. Viner N. Fourier Integral and Some its Applications. Moscow, Fizmatgiz, 1963 (in Russian). 56. Ditkin V.A., Kuznetsov P.I. Reference Book on Operating Calculus. Moscow-Lenin- grad, Gostehizdat, 1951 (in Russian). 57. Boltzmann L. In: Lectures on Gas Theory. Barkeley, University of California, 1964. 58. Zubarev D.N., Khonkin A.D. The method for construction of normal solutions for kinetic equations with the help of boundary conditions. // Teor. Mat. Fiz., 1972, vol. 11, No. 3, p. 403-412 (in Russian). 2Engl. transl. reads “Journal for Technical Physics” 3Engl. transl. reads “Heat Physics of High Temperatures” 744 A consistent description of kinetics and hydrodynamics 59. Yukhnovskii I.R., Holovko M.F. Statistical Theory of Classical Equilibrium Systems. Kiev, Naukova Dumka, 1980 (in Russian). 60. Rudyak V.Ya., Yanenko N.N. On taking into account intermolecular attractive forces at the derivation of kinetic equations. // Teor. Mat. Fiz., 1985, vol. 64, No. 2, p. 277- 286 (in Russian). 61. Rudyak V.Ya. Kinetic equations of nonideal gas with real interaction potentials. // Jurn. Tehn. Fiz., 1987, vol. 57, No. 8, p. 1466-1475 (in Russian). 62. Lifshiz E.M., Pitaevskii L.P. Physical Kinetics. Oxford, Pergamon Press, 1981. 63. Balescu R. Statistical Mechanics of Charged Particles. New York, Interscience, 1963. 64. Ecker G. Theory of Fully Ionized Plasmas. New York, London, Academic Press, 1972. 65. Kobryn A.E., Omelyan I.P., Tokarchuk M.V. Normal solution to the Enskog-Landau kinetic equation using boundary conditions method. // Phys. Lett. A, 1996, vol. 223, No. 1&2, p. 37-44. 66. Kobryn A.E., Omelyan I.P., Tokarchuk M.V. Nonstationary solution to the Enskog- Landau kinetic equation using boundary conditions method. // Cond. Matt. Phys., 1996, No. 8, p. 75-98. 67. Green H.S. The Molecular Theory of Gases. Amsterdam, North Holland, 1952. 68. Cohen E.G.D. Cluster expansion and hierarchy. // Physica, 1962, vol. 28, No. 10, p. 1045-1059. 69. Cohen E.G.D. On the kinetic theory of dense gases. // J. Math. Phys., 1963, vol. 4, No. 2, p. 183-189. 70. Bogolubov N.N. On the stochastic processes in the dynamical systems. / Preprint JINR-E17-10514, Dubna, 1977 (in English). Bogolubov N.N. On the stochastic pro- cesses in the dynamical systems. // Fizika Elementarnyh Chastic i Atomnogo Yadra (Particles & Nuclei), 1978, vol. 9, No. 4, p. 501-579 (in Russian). 71. Gould H.A., DeWitt H.E. Convergent kinetic equation for a classical plasma. // Phys. Rev., 1967, vol. 155, No. 1, p. 68-74. 72. Ernst M.H., Dorfman J.R. Nonanalitic dispersion relations in classical fluids I. The hard sphere gas. // Physica, 1972, vol. 61, p. 157-181. 73. Scold K., Rowe J.M., Ostrowski G., Randolph R.D. Coherent and incoherent scat- tering laws of liquid Argon. // Phys. Rev. A, vol. 6, No. 3, p. 1107-1131. 74. Copley J.R.D., Rowe J.M. Density fluctuation in liquid Rubidium. I. Neutron scat- tering measurements. // Phys. Rev. A, 1974, vol. 9, No. 4, p. 1656-1666. 75. Copley J.G.D., Lovesey S.W. The dynamic properties of monoatomic liquids. // Rept. Progr. Phys., 1975, vol. 38, No. 4, p. 461-563. 76. Dynamics of Solids and Liquids by Neurton Scattering (eds. S.W.Lovesey and T.Schringer). Berlin, Springer Verlag, 1977. 77. Boon J.P., Yip S. Molecular Hydrodynamics. New York, 1980. 78. Hansen J.P., McDonald I.R. Theory of Simple Liquids, 2nd edn. London, Academic Press, 1986. 79. Balucani U., Zoppi M. Dynamics of the Liquid State. Oxford, Clarendon Press, 1994. 80. Kadanoff L.P., Martin P.C. Hydrodynamic equations and correlation functions. // Ann. Phys., 1963, vol. 24, No. 1, p. 419-469. 81. Mountain R.D. Spectral distribution of scattered ligth in a simple fluid. // Rev. Mod. Phys., 1966, vol. 38, No. 1, p. 205-214. 82. Résibois P. On linearized hydrodynamic modes in statistical physics. // J. Stat. Phys., 1970, vol. 2, No. 1, p. 21-51. 745 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn 83. Forster D. Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Func- tions. London, Amsterdam, W.A.Benjamin, Inc., 1975. 84. March N.H., Tosi M.P. Atomic Dynamics in Liquids. Macmillan Press, 1976. 85. Zwanzig R. Ensemble method in the theory of irreversibility. // J. Chem. Phys., 1960, vol. 33, No. 5, p. 1338-1341. 86. Zwanzig R. Memory effects in irreversible thermodynamics. // Phys. Rev., 1961, vol. 124, No. 4, p. 983-992. 87. Mori H. Transport, collective motion and Brownian motion. // Progr. Theor. Phys., 1965, vol. 33, No. 3, p. 423-455. 88. Sergeev M.V. Generalized transport equations in the theory of irreversible processes. // Teor. Mat. Fiz., 1974, vol. 21, No. 3, p. 402-414 (in Russian). 89. Tishchenko S.V. Construction of generalized hydrodynamics using the nonequilib- rium statistical operator method. // Teor. Mat. Fiz., 1976, vol. 26, No. 1, p. 96-110 (in Russian). 90. Tserkovnikov Yu.A. On a method of solution of infinite systems of equations for temperature Green functions. // Teor. Mat. Fiz., 1981, vol. 49, No. 2, p. 219-233 (in Russian). 91. Tserkovnikov Yu.A. Method of two-time Green functions in molecular hydrodynam- ics. // Theor. Mat. Fiz., 1985, vol. 63, No. 3, p. 440-457 (in Russian). 92. Zubarev D.N., Tserkovnikov Yu.A. Method of two-time temperature Green functions in equilibrium and nonequilibrium statistical mechanics. In: Collection of scientific works of Mathematical Institute of USSR Academy of Sciences. Moscow, Nauka, 1986, vol. 175, No. 3, p. 134-177 (in Russian). 93. Akcasu A.Z., Daniels E. Fluctuation analysis in simple fluids. // Phys. Rev. A, 1970, vol. 2, No. 3, p. 962-975. 94. Desai R.C., Kapral R. Translational hydrodynamics and ligth scattering from molec- ular fluids. // Phys. Rev. A, 1972, vol. 6, No. 6, p. 2377-2390. 95. Götze W., Lücke M. Dynamical current correlation functions of simple classical liq- uids for intermediate wave number. // Phys. Rev. A, 1975, vol. 11, No. 6, p. 2173- 2190. 96. Bosse J., Götze W., Lücke M. Mode-coupling theory of simple classical liquids. // Phys. Rev. A, 1978, vol. 17, No. 1, p. 434-445. 97. Bosse J., Götze W., Lücke M. Current fluctuation spetra of liquid Argon near its triple point. // Phys. Rev. A, 1978, vol. 17, No. 1, p. 447-454. 98. Bosse J., Götze W., Lücke M. Current-fluctuation spectra of liquid Rubidium. // Phys. Rev. A, 1978, vol. 18, No. 3, p. 1176-1182. 99. Bosse J., Götze W., Zippelius A. Velocity-autocorrelation spectrum of simple classical liquids. // Phys. Rev. A, 1978, vol. 18, No. 3, p. 1214-1221. 100. Zippelius A., Götze W. Kinetic theory for the coherent scattering function of classical liquids. // Phys. Rev. A, 1978, vol. 17, No. 1, p. 414-423. 101. Götze W. Aspects of Structural Glass Transitions. In: Liquids, Freezing and Glass Transition (eds. J.-P.Hansen, D.Levesque and J.Zinn-Justin). Amsterdam, North- Holland, 1991. 102. Shurygin V.Yu, Yulmetyev R.M. Influence of non-Markovian effects in thermal mo- tion of particles on intensity of non-coherent scattering of slow neutrons in liquids. // Teor. Mat. Fiz., 1990, vol. 83, No. 2, p. 222-235 (in Russian). 103. Shurygin V.Yu, Yulmetyev R.M. Calculation of the dynamic structure factor for 746 A consistent description of kinetics and hydrodynamics liquids by the reduced description technique. // Zhurn. Exp. Teor. Fiz., 1989, vol. 96, No. 3(9), p. 938-947 (in Russian). 104. Shurygin V.Yu, Yulmetyev R.M. Space dispersion of structure relaxation in simple liquids. // Zhurn. Exp. Teor. Fiz., 1991, vol. 99, No. 1, p. 144-154 (in Russian). 105. Shurygin V.Yu, Yulmetyev R.M. On spectre of non-Markovianity of relaxation pro- cesses in liquids. // Zhurn. Exp. Teor. Fiz., 1992, vol. 102, No. 3(9), p. 852-862 (in Russian). 106. Shurygin V.Yu, Yulmetyev R.M. Role of asymmetric time correlation functions in kinetics of relaxation processes in liquids. // Ukr. Fiz. Zhurn., 1990, vol. 35, No. 5, p. 693-695 (in Russian). 107. Akcasu A.Z., Duderstadt J.J. Derivation of kinetic equation from the generalized Langevin equation. // Phys. Rev., 1969, vol. 188, No. 1, p. 479-486. 108. Forster D., Martin P.C. Kinetic theory of a weakly coupled fluid. // Phys. Rev. A, 1970, vol. 2, No. 4, p. 1575-1590. 109. Mazenko G.F. Microscopic method for calculating memory functions in transport theory. // Phys. Rev. A, 1971, vol. 3, No. 6, p. 2121-2137. 110. Mazenko G.F. Properties of the low-density memory function. // Phys. Rev. A, 1972, vol. 5, No. 6, p. 2545-2556. 111. Mazenko G.F., Tomas Y.S., Sidney Yip. Thermal fluctuation of hard sphere gas. // Phys. Rev. A, 1972, vol. 6, No. 5, p. 1981-1995. 112. Mazenko G.F. Fully renormalized kinetic theory I. Self-diffusion. // Phys. Rev. A, 1973, vol. 7, No. 1, p. 209-222. 113. Mazenko G.F. Fully renormalized kinetic theory II. Low-density. // Phys. Rev. A, 1973, vol. 7, No. 1, p. 222-232. 114. Mazenko G.F. Fully renormalized kinetic theory III. Density flutuation. // Phys. Rev. A, 1974, vol. 9, Np 1. p. 360-387. 115. Forster D. Properties of the kinetic memory function in classical fluids. // Phys. Rev. A, 1974, vol. 9, No. 2, p. 943-956. 116. Boley C.D., Desai R.C. Kinetic theory of a dense gas: triple-collision memory func- tion. // Phys. Rev. A, 1973, vol. 7, No. 5, p. 1700-1709. 117. Furtado P.M., Mazenko G.F., Sidney Yip. Hard-sphere kinetic theory analisis of classical simple liquid. // Phys. Rev. A, 1975, vol. 12, No. 4, p. 1653-1661. 118. John M.S., Forster D. A kinetic theory of classical simple liquids. // Phys. Rev. A, 1975, vol. 12, No. 1, p. 254-266. 119. Sjodin S., Sjolander A. Kinetic model for classical liquids. // Phys. Rev. A, 1978, vol. 18, No. 4, p. 1723-1736. 120. de Schepper I.M., Cohen E.G.D. Collective modes in fluids and neutron scattering. // Phys. Rev. A, 1980, vol. 22, No. 1, p. 287-289. 121. de Schepper I.M., Cohen E.G.D. Very-short wavelength collective modes in fluids. // J. Stat. Phys., 1982, vol. 27, No. 2, p. 223-281. 122. Cohen E.G.D., de Schepper I.M., Zuilhof M.J. Kinetic theory of the eigenmodes of classical fluids and neutron scattering. // Physica B, 1984, vol. 127, p. 282-291. 123. de Schepper I.M., Verkerk R., van Well A.A., de Graaf L.A. Short-wavelength sound modes in liquid Argon. // Phys. Rev. Lett., 1983, vol. 50, No. 13, p. 974-977. 124. van Well A.A., de Graaf L.A. Density fluctuations in liquid Argon II. Coherent dynamic structure factor at large wave numbers. // Phys. Rev. A, 1985, vol. 32, No. 4, p. 2384-2395. 747 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn 125. van Well A.A., de Graaf L.A. Density fluctuations in liquid Neon studied by neutron scattering. // Phys. Rev. A, 1985, vol. 32, No. 4, p. 2396-2412. 126. Alley W.E., Alder B.J. Generalized transport coefficient for hard spheres. // Phys. Rev. A, 1983, vol. 27, No. 6, p. 3158-3173. 127. Alley W.E., Alder B.J., Sidney Yip. The neutron scattering function for hard spheres. // Phys. Rev. A, 1983, vol. 27, No. 6, p. 3177-3186. 128. de Schepper I.M., van Rijs J.C., van Well A.A., Verkerk P., de Graaf L.A., Bruin C. Microscopic sound waves in dense Lennard-Jones fluids. // Phys. Rev. A, 1984, vol. 29, No. 3, p. 1602-1605. 129. Sidney Yip., Alley W.E., Alder B.J. Evaluation of time correlation function from a generalized Enskog equation. // J. Stat. Phys., 1982, vol. 27, No. 1, p. 201-217. 130. de Schepper I.M., Cohen E.G.D., Zuilhof M.J. The width of neutron spectra and heat mode of fluids. // Phys. Lett. A, 1984, vol. 101, No. 8, p. 399-404. 131. de Schepper I.M., Verkerk P., van Well A.A., de Graaf L.A. Nonanalytic dispersion relations in liquid Argon. // Phys. Lett. A, 1984, vol. 104, No. 1, p. 29-32. 132. Bruin C., Mechels J.P.J., van Rijs J.C., de Graaf L.A., de Schepper I.M. Extended hydrodynamic modes in a dence hard sphere fluid. // Phys. Lett. A, 1985, vol. 110, No. 1, p. 40-43. 133. Kamgar-Parsi B., Cohen E.G.D., de Schepper I.M. Dynamical processes in hard sphere fluids. // Phys. Rev. A, 1987, vol. 35, No. 11, p. 4781-4795. 134. Cohen E.G.D., de Schepper I.M. Effective eigenmode description of dynamic pro- cesses in dense classical fluid mixtures. // Nuovo Cimento D, 1990, vol. 12, No. 4/5, p. 521–542. 135. de Schepper I.M., Cohen E.G.D., Kamgar-Parsi B. On the calculation of equilibrium time correlation functions in hard sphere fluids. // J. Stat. Phys., 1989, vol. 54, No. 1/2, p. 273-313. 136. Ernst M.H., Dorfman J.R. Nonanalitic dispersion relations for classical fluids II. General fluid. // J. Stat. Phys., 1975, vol. 12, No. 4, p. 311-359. 137. van Well A.A., Verkerk P., de Graaf L.A., Suck J.-B., Copley J.R.D. Density fluc- tuation in liquid Argon: coherent dynamic structure factor along the 120 K isoterm obtained by neutron scattering. // Phys. Rev. A, 1985, vol. 31, No. 5, p. 3391-3414. 138. Egelstaff P.A., Glase W., Litchensky D., Schneider E., Suck J.-B. Two-body time correlations in dense Krypton gas. // Phys. Rev. A, 1983, vol. 27, No. 2, p. 1106- 1115. 139. de Schepper I.M., Cohen E.G.D., Bruin C., van Rijs J.C., Montfrooij W., de Gra- af L.A. Hydrodynamic time correlation functions for a Lennard-Jones fluid. // Phys. Rev. A, 1988, vol. 38, No. 1, p. 271-287. 140. Zubarev D.N., Tokarchuk M.V. Nonequilibrium statistical hydrodynamics of ionic systems. // Teor. Mat. Fiz., 1987, vol. 70, No. 2, p. 234-254 (in Russian). 141. Mryglod I.M., Tokarchuk M.V. On statistical hydrodynamics of simple liquds. In: Problems of atomic science and technique. Series: Nuclear physics investigations (the- ory and experiment). Kharkov, Kharkov Physico Technical Institute, 1992, vol. 3(24), p. 134-139 (in Russian). 142. Mryglod I.M., Omelyan I.P. Generalized collective modes of a Lennard-Jones fluid. High-mode approximation. // Cond. Matt. Phys., 1994, vol. 4, p. 128-160. 143. Mryglod I.M., Hachkevych A.M. On nonequilibrium statistical theory of a fluid: linear relaxation theories with different sets of dynamic variables. // Cond. Matt. Phys., 748 A consistent description of kinetics and hydrodynamics 1995, vol. 5, p. 105-123. 144. Mryglod I.M., Omelyan I.P., Tokarchuk M.V. Generalized collective modes for the Lennard-Jones fluid. // Mol. Phys., 1995, vol. 84, No. 2, p. 235-259. 145. Mryglod I.M., Omelyan I.P. Generalized collective modes for a Lennard-Jones fluid in higher mode approximations. // Phys. Lett. A, 1995, vol. 205, No. 4, p. 401-406. 146. Mryglod I.M., Omelyan I.P. Generalized mode approach: I. Transverse time corre- lation functions and generalized shear viscosity of a Lennard-Jones fluid. // Mol. Phys., 1997, vol. 90, No. 1, p. 91-99. 147. Mryglod I.M., Omelyan I.P. Generalized mode approach: II. Longitudinal time cor- relation functions of a Lennard-Jones fluid. // Mol. Phys., 1997, vol. 91, No. 6, p. 1005-1015. 148. Mryglod I.M., Omelyan I.P. Generalized mode approach: III. Generalized transport coefficients of a Lennard-Jones fluid. // Mol. Phys., 1997, vol. 92, No. 5, p. 913-927. 149. Mryglod I.M. Generalized hydrodynamics of liquids. II. Time correlation functions using the formalism of generalized collective modes. // Ukr. Fiz. Zhurn., 1998, vol. 43, No. 2, p. 252-256 (in Ukrainian). 150. Mryglod I.M., Hachkevych A.M. Simple iteration schema for calculation of memory function: a “shoulder” problem for generalized shear viscosity. // Ukr. Fiz. Zhurn., 1999 (in press). 151. Mryglod I.M. Generalized statistical hydrodynamics of fluids: approach of generalized collective modes. // Cond. Matt. Phys., 1998, vol. 1, No. 4(16), p. 753-796. 152. Mryglod I.M. Mode-coupling behaviour of the generalized hydrodynamic modes for a Lennard-Jones fluid. // J. Phys. Studies, 1998 (submitted). 153. Mryglod I.M., Ignatyuk V.V. Generalized hydrodynamics of binary mixtures. // J. Phys. Studies, 1997, vol. 1, No. 2, p. 181-190 (in Ukrainian). 154. Bryk T.M., Mryglod I.M., Kahl G. Generalized collective modes in a binary He0.65- Ne0.35 mixture. // Phys. Rev. E, 1997, vol. 56, No. 3, p. 2903-2915. 155. Mryglod I.M. Generalized hydrodynamics of multicomponent fluids. // Cond. Matt. Phys., 1997, vol. 10, p. 115-135. 156. Bryk T.M., Mryglod I.M. Spectra of transverse excitations in liquid glass-forming metallic alloy Mg70Zn30: temperature dependence. // Cond. Matt. Phys., 1999 (in press). 157. Omelyan I.P. Wavevector- and frequency-dependent dielectric constant of the Stock- mayer fluid. // Mol. Phys., 1996, vol. 87, No. 6, p. 1273-1283. 158. Omelyan I.P. Temperature behaviour of the frequency-dependent dielectric constant for a Stockmayer fluid. // Phys. Lett. A, 1996, vol. 216, No. 1/5, p. 211-216. 159. Omelyan I.P. Wavevector dependent dielectric constant of the MCY water model. // Phys. Lett. A, 1996, vol. 220, No. 1/3, p. 167-177. 160. Omelyan I.P. On the reaction field for interaction site models of polar systems. // Phys. Lett. A, 1996, vol. 223, No. 4, p. 295-302. 161. Omelyan I.P. Generalized collective mode approach in the dielectric theory of dipolar systems. // Physica A, 1997, vol. 247, No. 1/4, p. 121-139. 162. Omelyan I.P., Zhelem R.I., Tokarchuk M.V. Generalized hydrodynamics of polar liquids in external inhomogeneous electric field. Method of nonequilibrium statistical operator. // Ukr. Fiz. Zhurn., 1997, vol. 42, No. 6, p. 684-692 (in Ukrainian). 163. Omelyan I.P. Longitudinal wavevector- and frequency-dependent dielectric constant of the TIP4P water model. // Mol. Phys., 1998, vol. 93, No. 1, p. 123-135. 749 M.V.Tokarchuk, I.P.Omelyan, A.E.Kobryn 164. Omelyan I.P., Mryglod I.M., Tokarchuk M.V. Generalized dipolar modes of a Stock- mayer fluid in high-order approximations. // Phys. Rev. E, 1998, vol. 57, No. 6, p. 6667-6676. 165. Omelyan I.P., Mryglod I.M., Tokarchuk M.V. Dielectric relaxation in dipolar fluid. Generalized mode approach. // Cond. Matt. Phys., 1998, vol. 1, No. 1(13), p. 179-200. 166. Peletminskii S.V., Sokolovskii A.I. General equations of fluctuation hydrodynamics. // Ukr. Fiz. Zhurn., 1992, vol. 37, No. 10, p. 1521-1528 (in Russian). 167. Peletminskii S.V., Slusarenko Yu.V. Kinetic and hydrodynamic theory of long-wave nonequilibrium fluctuations. In: Abstracts and contributed papers of International conference “Physics in Ukraine”. Kiev, 22-27 June, 1993. Vol.: Statistical physics and phase transitions. Kiev, Institute for Theoretical Physics, 1993, p. 103-106. 168. Akhiezer A., Peletminskii S. Methods of Statistical Physics. Oxford, Pergamon, 1981. 169. Sokolovskii A.I. Equations of hydrodynamics near equilibrium at absence of long- range correlations. // Ukr. Fiz. Zhurn., 1993, vol. 37, No. 10, p. 1528-1536 (in Rus- sian). 170. Peletminskii S.V., Sokolovskii A.I. Space dispersion of transport coefficients of a liquid near equilibrium. // Ukr. Fiz. Zhurn., 1992, vol. 37, No. 11, p. 1702-1711 (in Russian). 171. Balabanyan G.O. Construction of hydrodynamical asymptotics of classical equilib- rium correlation Green functions by means of the Boltzmann kinetic equation. // Teor. Mat. Fiz., 1990, vol. 82, No. 1, p. 117-132 (in Russian). 172. Balabanyan G.O. Classical equilibrium generalized hydrodynamic correlation Green functions. I. // Teor. Mat. Fiz., 1990, vol. 82, No. 3, p. 450-465 (in Russian). 173. Balabanyan G.O. Classical equilibrium generalized hydrodynamic correlation Green functions. II. Transversal autocorrelation Green function. // Teor. Mat. Fiz., 1990, vol. 83, No. 2, p. 311-319 (in Russian). 174. Balabanyan G.O. Classical equilibrium generalized hydrodynamic correlation Green functions. III. “Density-density” correlation Green function. // Teor. Mat. Fiz., 1990, vol. 85, No. 1, p. 102-114 (in Russian). 175. Balabanyan G.O. Construction of equations for classical equilibrium correlation Green functions by means of kinetic equations. // Teor. Mat. Fiz., 1991, vol. 88, No. 2, p. 225-245 (in Russian). 176. Tserkovnikov Yu.A. Molecular hydrodynamics of weakly nonideal nondegenerate Bose-gas. I. Green functions of transverse components of density of particle current. // Teor. Mat. Fiz., 1990, vol. 85, No. 1, s. 124-149 (in Russian). 177. Tserkovnikov Yu.A. Molecular hydrodynamics of weakly nonideal nondegenerate Bose-gas. II. Green functions of longitudinal fluctuations of particles and energy densities. // Teor. Mat. Fiz., 1990, vol. 85, No. 2, p. 258-287 (in Russian). 178. Robertson B. Equations of motion in nonequilibrium statistical mechanics. // Phys. Rev., 1966, vol. 144, No. 1, p. 151-161. 179. Robertson B. Equations of motion in nonequilibrium statistical mechanics II. Energy transport. // Phys. Rev., 1967, vol. 160, No. 1, p. 175-183. 180. Kawasaki K., Gunton J.D. Theory of nonlinear shear viscosity and normal stress effects. // Phys. Rev. A, 1973, vol. 8, No. 4, p. 2048-2064. 181. Gould H., Mazenko G.F. Microscopic theory of the self-diffusion in a classical one- component plasma. // Phys. Rev. A, 1977, vol. 15, No. 3, p. 1274-1287. 750 A consistent description of kinetics and hydrodynamics Узгоджений опис кінетики та гідродинаміки систем взаємодіючих частинок методом нерівноважного статистичного оператора М.В.Токарчук, І.П.Омелян, О.Є.Кобрин Інститут фізики конденсованих систем НАН України, 290011, Львів–11, вул. Свєнціцького, 1 Отримано 14 грудня 1998 р. Представлено один із статистичних підходів узгодженого опису кіне- тичних та гідродинамічних процесів систем взаємодіючих частинок, що сформульований на основі методу нерівноважного статистично- го оператора Д.М.Зубарєва. Показано, як із ланцюжка рівнянь ББГКІ з модифікованими граничними умовами отримуються кінетичне рів- няння ревізованої теорії Енскога для моделі твердих сфер, кінетич- не рівняння для багатосходинкового потенціалу та кінетичне рівнян- ня Енскога-Ландау для моделі заряджених твердих сфер. Проаналі- зовано ланцюжки рівнянь ББГКІ на основі модифікованих групових розкладів. Отримано узагальнені рівняння переносу узгодженого опису кінетики та гідродинаміки. Для випадку слабо нерівноважних систем класичних взаємодіючих частинок при взаємному врахуван- ні кінетичних та гідродинамічних процесів досліджено часові кореля- ційні функції, спектр колективних збуджень та узагальнені коефіцієн- ти переносу. Ключові слова: кінетика, гідродинаміка, кінетичні рівняння, коефіцієнти переносу, (часові) кореляційні функції PACS: 05.20.Dd, 05.60.+w, 52.25.Fi, 71.45.G, 82.20.M 751 752

A consistent description of kinetics and hydrodynamics of systems of interacting particles by means of the nonequilibrium statistical operator method

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