Kinetic theory of dense fluids

A kinetic theory of dense fluids is presented in this series of papers. The theory is based on a kinetic equation for subsystems which represents a subset of equations structurally invariant to the sizes of the subsystem that includes the Boltzmann equation as an element at the low density limit. There exists a H-function for the kinetic equation and the equilibrium solution is uniquely given by the canonical distribution functions for the subsystems comprising the entire system. The cluster expansion is discussed for the N-body collision operator appearing in the kinetic equation. The kinetic parts of transport coefficients are obtained by means of a moment method and their density expansions are formally obtained. The Chapman-Enskog method is discussed in the subsequent paper.     

Expansion formulae for time-correlation functions and transport coefficients: With an application to cyclotron resonance

New types of expansion formulae are obtained for time-correlation functions and kinetic coefficients. This is based on the previously developed damping theoretical expansion method combined with Mori's projection operator. A few simple applications are given for illustration and detailed calculations are done for the cyclotron resonance lineshape.     

A kinetic theory of dense fluids

A new kinetic equation is developed which incorporates the desirable features of the Enskog, the Rice-Allnatt, and the Prigogine-Nicolis-Misguich kinetic theories of dense fluids. Advantages of the present theory over the latter three theories are (1) it yields the correct local equilibrium hydrodynamic equations, (2) unlike the Rice-Allnatt theory, it determines the singlet and doublet distribution functions from the same equation, and (3) unlike the Prigogine-Nicolis-Misguich theory, it predicts the kinetic and kinetic-potential transport coefficients. The kinetic equation is solved by the Chapman-Enskog method and the coefficients of shear viscosity, bulk viscosity, thermal conductivity, and self-diffusion are obtained. The predicted bulk viscosity and thermal conductivity coefficients are singular at the critical point, while the shear viscosity and self-diffusion coefficients are not.     

Linear quantum Enskog equation. I. Homogeneous quantum fluids

The quantum-statistical generalization of the well-known classical, linear revised Enskog equation is derived for spatially uniform systems. This new quantum kinetic equation allows the study of equilibrium time correlation functions and their associated transport coefficients of normal quantum fluids where static correlations and degeneracy effects due to particle statistics (both are treated exactly) are important. Furthermore, we derive the quantum-statistical analog of the classical ring operator. These microscopic and systematic derivations are based on a recently developed superoperator formalism (including cluster expansion techniques) that, as a main feature, allows a clear distinction between static and dynamic correlations, which is crucial in the discussion of the Enskog approximation.     

A microscopic theory of nucleation. II

Using the nonequilibrium statistical operator obtained in the preceding paper of the authors [1], equations describing the kinetics of nucleation in a nonequilibrium medium are derived. A Fokker-Planck equation is found for embryo distribution functions in the number of particles, energy, momentum, and c.m. coordinates with additional random forces due to non equilibrium processes in the medium. Hydrodynamic equations are obtained for the medium with account of thermodynamic forces due to discontinuities of thermodynamic parameters at the interphase boundary. The symmetry of cross (interphase) kinetic coefficients is considered.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 44–52, May, 1978.     

Enskog-Landau kinetic equation for multicomponent mixture. Analytical calculation of transport coefficients

The Enskog-Landau kinetic equation is considered to describe non-equilibrium processes of a mixture of charged hard spheres. This equation has been obtained in our previous papers by means of the non-equilibrium statistical operator method. The normal solution of this kinetic equation found in the first approximation using the standard Chapman-Enskog method is given. On the basis of the found solution the flows and transport coefficients have been calculated. All transport coefficients for multicomponent mixture of spherical Coulomb particles are presented analytically for the first time. Numerical calculations of thermal conductivity and thermal diffusion coefficient are performed for some specific mixtures of noble gases of high density. We compare the calculations with those ones for point-like neutral and charged particles. Received 10 June 1999 and Received in final form 15 October 1999     

Diffusion Dynamics of Classical Systems Driven by an Oscillatory Force

We investigate the asymptotic behavior of solutions to a kinetic equation describing the evolution of particles subject to the sum of a fixed, confining, Hamiltonian, and a small time-oscillating perturbation. Additionally, the equation involves an interaction operator which projects the distribution function onto functions of the fixed Hamiltonian. The paper aims at providing a classical counterpart to the derivation of rate equations from the atomic Bloch equations. Here, the homogenization procedure leads to a diffusion equation in the energy variable. The presence of the interaction operator regularizes the limit process and leads to finite diffusion coefficients. AMS Subject classification: 74Q10, 35Q99, 35B25, 82C70     

A collisional approach to the calculation of time correlation functions. Transport coefficients of gases

A method of successive approximations is proposed for the evaluation of the time-correlation functions such as those that give the thermal transport coefficients of gases. The method is based on a calculation of the changes in correlations of appropriate functions of the molecular velocity which are a result of collisions in the gas. The decaying rates of the correlations are expressed as integrals of the differential collision cross section. When the first approximation is introduced in the expressions for thermal transport coefficients, results are obtained for the coefficient of binary diffusion and the viscosity and thermal conductivity of single-component systems which are identical with those of the first Chapman-Enskog solutions of the Boltzmann and Enskog equations. For the coefficients of viscosity and thermal conductivity in multicomponent systems, it is shown that the first approximation leads to expressions of the form of the Sutherland and Wassiljewa relations, respectively.     

Kinetic theory in the weak-coupling approximation: I. Formal theory and application to classical open systems

A general formalism, where irreversible processes are related to singularities of the resolvent of the Liouville operator, is applied to classical open systems. For a system weakly coupled to a thermal reservoir, a kinetic equation is derived. It is shown that the method leads to equations defining a positivity-preserving semigroup with the Maxwell-Boltzmann distribution as a stationary solution and obeying an H-theory. It is pointed out that these properties are not always shared by irreversible equations obtained as asymptotic approximations of the so-called generalized master equation.     

Simple model for the calculation of the coefficient of self-diffusion in a liquid

The coefficient of self-diffusion in three-dimensional classical liquid is computed approximately from the hierarchy of kinetic equations for the time-correlation functions (TCF).     

Method of Green's functions in the theory of quantum kinetic equations. I

The solution of the dynamic equation is strongly degenerate for systems with an infinite number of degrees of freedom. A causality principle is stated, whereby the particular solution of the dynamic equation, which is at the same time a solution of the kinetic equation, can be selected. The principle is applied here to the multitime formalism of correlation functions. A basis is thus obtained for the method of Green's time-temperature functions in the theory of kinetic equations.The author thanks all those who have discussed the topics in this paper with him, particularly V. L. Bonch-Bruevich, D.A. Kirzhnits, V.M. Fain, E.S. Fradkin, and A. S. Shekhter.     

Diffusion and correlation in a coherent representation

Using a Fock-space formalism for the Master equation and introducing the density operator we present an unified method to derive kinetic equations for hopping processes with and without exclusion on a lattice. The corresponding Liouvillians are written in terms of Fermi or Bose operators, respectively. Although the Liouvillians are different the averaged particle numbers obey the same diffusion equation. Differences appear in the correlation functions only. The Master equation can be transformed into a differential equation in a coherent state representation. Using the algebraic properties of Grassmann numbers we are able to find the exact statonary solution for diffusion with exclusion. The conductivity can be derived in the bosonic and the fermionic case. The results are in accordance with those obtained with different other methods.     

Memory function for dressed particles

This paper is concerned with the calculation of the memory function and derivation of a kinetic equation for one-body phase space correlation functions. The theory uses a one-body additive projection operator and a division of the Liouville operator with an unperturbed part that describes dressed particles. Binary collisions are neglected, for the theory aims at describing the screening and backflow effects of a type contained in the plasma kinetic theory of Balescu and Lenard. We obtain an explicit kinetic equation which is an improvement of these theories for the plasma case, and involves the exact equilibrium pair and triplet distributions. The equation also describes systems with strong short-range forces and shows how the screening effects occur in this case as well. The unifying function is the direct correlation function. The theory is meant to provide understanding for a more complete theory of fluids where a proper account is given of close collisions.Work supported by National Science Foundation, Grant No. GH 35691.     

截断展开方法和广义变系数KdV方程新的精确类孤子解

利用特殊的截断展开方法求出了广义变系数KdV方程新的类孤子解.这种方法的基本思想是假定形式解具有截断展开形式,以致可把广义变系数KdV方程转化为一组待定函数的代数方程组,进而给出待定函数容易积分的常微分方程.利用例子证明了这种方法是十分有效的. 关键词： 截断展开法 变系数 KdV方程 孤波解     

Kinetic Models for Granular Flow

The generalization of the Boltzmann and Enskog kinetic equations to allow inelastic collisions provides a basis for studies of granular media at a fundamental level. For elastic collisions the significant technical challenges presented in solving these equations have been circumvented by the use of corresponding model kinetic equations. The objective here is to discuss the formulation of model kinetic equations for the case of inelastic collisions. To illustrate the qualitative changes resulting from inelastic collisions the dynamics of a heavy particle in a gas of much lighter particles is considered first. The Boltzmann–Lorentz equation is reduced to a Fokker–Planck equation and its exact solution is obtained. Qualitative differences from the elastic case arise primarily from the cooling of the surrounding gas. The excitations, or physical spectrum, are no longer determined simply from the Fokker–Planck operator, but rather from a related operator incorporating the cooling effects. Nevertheless, it is shown that a diffusion mode dominates for long times just as in the elastic case. From the spectral analysis of the Fokker–Planck equation an associated kinetic model is obtained. In appropriate dimensionless variables it has the same form as the BGK kinetic model for elastic collisions, known to be an accurate representation of the Fokker–Planck equation. On the basis of these considerations, a kinetic model for the Boltzmann equation is derived. The exact solution for states near the homogeneous cooling state is obtained and the transport properties are discussed, including the relaxation toward hydrodynamics. As a second application of this model, it is shown that the exact solution for uniform shear flow arbitrarily far from equilibrium can be obtained from the corresponding known solution for elastic collisions. Finally, the kinetic model for the dense fluid Enskog equation is described.     

Semiclassical approximation for the twodimensional Fisher–Kolmogorov–Petrovskii– Piskunov equation with nonlocal nonlinearity in polar coordinates

The two-dimensional Kolmogorov–Petrovskii–Piskunov–Fisher equation with nonlocal nonlinearity and axially symmetric coefficients in polar coordinates is considered. The method of separation of variables in polar coordinates and the nonlinear superposition principle proposed by the authors are used to construct the asymptotic solution of a Cauchy problem in a special class of smooth functions. The functions of this class arbitrarily depend on the angular variable and are semiclassically concentrated in the radial variable. The angular dependence of the function has been exactly taken into account in the solution. For the radial equation, the formalism of semiclassical asymptotics has been developed for the class of functions which singularly depend on an asymptotic small parameter, whose part is played by the diffusion coefficient. A dynamic system of Einstein–Ehrenfest equations (a system of equations in mean and central moments) has been derived. The evolution operator for the class of functions under consideration has been constructed in explicit form.     

Large Time Dynamics of a Classical System Subject to a Fast Varying Force

We investigate the asymptotic behavior of solutions to a kinetic equation describing the evolution of particles subject to the sum of a fixed, confining, Hamiltonian, and a small, time-oscillating, perturbation. The equation also involves an interaction operator which acts as a relaxation in the energy variable. This paper aims at providing a classical counterpart to the derivation of rate equations from the atomic Bloch equations. In the present classical setting, the homogenization procedure leads to a diffusion equation in the energy variable, rather than a rate equation, and the presence of the relaxation operator regularizes the limit process, leading to finite diffusion coefficients. The key assumption is that the time-oscillatory perturbation should have well-defined long time averages: our procedure includes general “ergodic” behaviors, amongst which periodic, or quasi-periodic potentials only are a particular case.     

Transport Theory and Collective Modes. I. The Case of Moderately Dense Gases

The complex spectral representation of the Liouville operator introduced by Prigogine and others is applied to moderately dense gases interacting through hard-core potentials in arbitrary d-dimensional spaces. Kinetic equations near equilibrium are constructed in each subspace as introduced in the spectral decomposition for collective, renormalized reduced distribution functions. Our renormalization is a nonequilibrium effect, as the renormalization effect disappears at equilibrium. It is remarkable that our renormalized functions strictly obey well-defined Markovian kinetic equations for all d, even though the ordinary distribution functions obey nonMarkovian equations with memory effects. One can now define transport coefficients associated to the collective modes for all dimensional systems including d = 2. Our formulation hence provides a microscopic meaning of the macroscopic transport theory. Moreover, this gives an answer to the long-standing question whether or not transport equations exist in two-dimensional systems. The non-Markovian effects for the ordinary distribution function, such as the long-time tails for arbitrary n-mode coupling, are estimated by superposition of the Markovian evolutions of the dressed distribution functions.     

Comments on the Grad procedure for the Fokker-Planck equation

If the kinetic equation of a macroscopic system is expanded with respect to the velocity in terms of orthogonal functions, e.g., in terms of Hermite functions, one obtains an infinite hierarchy of equations for the expansion coefficients. Grad's method consists in truncating this hierarchy and investigating the remaining finite system. In this paper we set up conditions under which this procedure is rigorously justified in case of the Fokker-Planck equation.