Kinetic theory of dense fluids. II. The generalized Chapman-Enskog solution

In the second paper of this series we solve the kinetic equation proposed in the previous paper by a method following the spirit of Chapman and Enskog (generalized Chapman-Enskog method). The zeroth-order solution to the kinetic equation leads to the Euler equations in hydrodynamics for real fluids, and the first-order solution to the Navier-Stokes equations for real fluids. General formulas for transport coefficients such as viscosity and heat-conductivity coefficients are obtained for dense fluids, which are given in terms of time-correlation functions of fluxes conjugate to the thermodynamic forces. The results have the same formal structures as the time-correlation functions in linear response theory except for the collision operator appearing in place of the Liouville operator in the evolution operator for the system.     

A convergent nonequilibrium statistical mechanical theory for dense gases. II. Transport coefficients to first order in the density

Using the two-body distribution function found earlier by the authors with the aid of new boundary conditions, the kinetic equation and the transport coefficients are obtained to zeroth and first order in the density. To zeroth order we recover the Boltzmann kinetic equation. To first order the resulting expressions differ from the ones obtained by Choh and Uhlenbeck, due to effects of the medium.3 Reference 2 will be referred to as I. Here we use the same notation as in I.     

The Modified Group Expansions for Construction of Solutions to the BBGKY Hierarchy

A solution to the BBGKY hierarchy for nonequilibrium distribution functions is obtained within modified boundary conditions. The boundary conditions take into account explicitly both the nonequilibrium one-particle distribution function as well as local conservation laws. As a result, modified group expansions are proposed. On the basis of these expansions, a generalized kinetic equation for hard spheres and a generalized Bogolubov–Lenard–Balescu kinetic equation for a dense electron gas are derived within the polarization approximation.     

Elementary theory of the equation of state and the pair distribution function for the hard sphere system

We present an intuitive method to calculate the pair distribution function and the equation of state for the hard sphere system by an elementary kinetic theory. The resulting equation of state not only reproduces exactly the second and the third virial coefficients, but also exhibits a remarkable agreement with molecular-dynamics results.     

A convergent nonequilibrium statistical mechanical theory for dense gases. III. Transport coefficients to second order in the density

The purpose of this paper is to obtain general expressions for the second-order terms of the transport coefficients of a dense gas. These expressions are obtained using the convergent kinetic theory proposed recently by Braun and Flores.     

Transport Coefficients in Some Stochastic Models of the Revised Enskog Equation

A stochastic model of the revised Enskog equation is considered. A choice of the smearing function suggested by the work of Leegwater is used to apply the model to the repulsive part of the Lennard-Jones potential and the inverse-power soft-sphere potential. The virial coefficients obtained from the equilibrium properties of the models are in excellent agreement with the known exact coefficients for these models. The transport coefficients for the repulsive Lennard-Jones (RLP) model are also computed and appear to be of comparable accuracy to the Enskog-theory coefficients applied directly to a hard-sphere system, although exact results for the RLP with which to make an extensive comparison are not yet available. The pressure and the transport coefficients obtained from the model (shear viscosity, thermal conductivity, and self-diffusion) are compared with the pressure and the corresponding transport coefficients predicted by the Enskog and square-well kinetic theories.     

Explicit and implicit finiteness corrections of virial coefficients

The equation of state of finite systems deviates from thermodynamic limit. The corresponding finiteness corrections of virial coefficients are studied. Most calculations are based on the canonical ensemble with periodic boundary conditions.Explicit andimplicit finiteness corrections occur. They are displayed up to the eighth virial coefficient. These results are applied to hard spheres in one, two, and three dimensions.     

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.     

Improved variational solutions of the Boltzmann–Ziman transport equation. Application to electronic transport in dense plasmas

We study inelastic electron-ion scattering contributions to the thermoelectronic transport coefficients in dense plasmas. They are deduced from improved variational solutions of the Boltzmann–Ziman equation explained in this work. A new expression for the thermopower is proposed which is evaluated in the case of the dense hydrogen plasma.     

Moment equations and the dynamic equilibrium condition for a relativistic electron beam propagating along an external magnetic field in dense and rarefied gas-plasma media

Equations of transfer of mass, momentum, and energy for a transverse segment of a paraxial relativistic electron beam propagating in dense and rarefied gas-plasma media along an external magnetic field are derived from the kinetic equation. A virial equation is obtained, and the dynamic equilibrium condition that generalizes the wellknown Bennet equation for the cases under study is found.     

On thermopower in hopping transport

Certain general aspects of hopping transport in the context of thermopower are reinvestigated. Onsager symmetry of the macroscopic kinetic coefficients is derived from detailed balance of the microscopic coefficients of the kinetic equation by an expansion of the kinetic equation around local equilibrium in an external potential and temperature gradient. The resulting expression for the thermopower differs from expressions proposed in the literature. The difference however, seems to be small.     

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.     

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     

稠密气体中的输运现象和方位势氩的粘滞率和热导率

建议用推广的恩斯柯克方程(14),讨论稠密气体中的输运现象。指出空间关联函数Y(r)可以从状态方程的维里展开求得。文中给出了计算方位势稠密气体的粘滞率和热导率的公式以及计算关联函数的方法。用得到的结果,计算了温度为173K和348K的氩的粘滞率和温度为183K和348K的氩的热导率。在高密度区,计算结果与实验测定的符合程度,较之现有理论,有明显的改进。 关键词：     

Quantum kinetic equation for nonequilibrium dense systems

Using the density matrix method in the form developed by Zubarev, equations of motion for nonequilibrium quantum systems with continuous short range interactions are derived which describe kinetic and hydrodynamic processes in a consistent way. The T-matrix as well as the two-particle density matrix determining the nonequilibrium collision integral are obtained in the ladder approximation including the Hartree-Fock corrections and the Pauli blocking for intermediate states. It is shown that in this approximation the total energy is conserved. The developed approach to the kinetic theory of dense quantum systems is able to reproduce the virial corrections consistent with the generalized Beth-Uhlenbeck approximation in equilibrium. The contribution of many-particle correlations to the drift term in the quantum kinetic equation for dense systems is discussed.     

Analytical representation of the higher virial coefficients of binary mixtures of additive hard spheres

Approximate expressions for the fourth and fifth virial coefficients of binary hard-sphere fluid mixtures are derived. The procedure used to obtain these expressions is based on that previously proposed by Wheatley [J. chem. Phys., 111, 5455 (1999)], but slightly modified. Wheatley's procedure starts from a prescribed general analytical form of the virial coefficients, from which the particular expression for each virial coefficient is obtained by imposing to the general form a number of limiting conditions. Here, we propose an alternative general expression of the virial coefficients and derive one more condition. This condition is satisfied when the fourth and fifth virial coefficients are expressed in the form we propose, but not when they are expressed in Wheatley's form. The agreement of the proposed analytical expressions with exact numerical data is excellent. The procedure can be extended to higher virial coefficients, although the lack of exact numerical data prevents any comparison.     

Kinetic derivation of the second virial coefficient for a quantum gas

A kinetic equation has been proposed earlier for an inhomogeneous quantum dilute gas obeying Boltzmann statistics. We show here that it leads to the correct equilibrium properties of the gas by computing the second virial coefficient as it can be deduced from the trace of the pressure tensor and by checking the result with the value deduced from various approaches.     

Kinetic equation for a dense soliton gas

We propose a general method to derive kinetic equations for dense soliton gases in physical systems described by integrable nonlinear wave equations. The kinetic equation describes evolution of the spectral distribution function of solitons due to soliton-soliton collisions. Owing to complete integrability of the soliton equations, only pairwise soliton interactions contribute to the solution, and the evolution reduces to a transport of the eigenvalues of the associated spectral problem with the corresponding soliton velocities modified by the collisions. The proposed general procedure of the derivation of the kinetic equation is illustrated by the examples of the Korteweg-de Vries and nonlinear Schr?dinger (NLS) equations. As a simple physical example, we construct an explicit solution for the case of interaction of two cold NLS soliton gases.     

Kinetic approach to the envelope equation for a relativistic electron beam propagating through a scattering gas-plasma medium in the presence of the reverse plasma current with an arbitrary radial profile

A kinetic approach is applied to derive the transport equations, the virial equation, the dynamic-equilibrium equation, and the envelope equation for an axially symmetric paraxial relativistic electron beam propagating through a scattering gas-plasma medium in the presence of a reverse plasma current with a radial density profile that is generally different from the beam density profile. The equations obtained include additional terms that account for this difference.     

An Inverse Problem of Determining Coefficients in a One-Dimensional Radiative Transport Equation

We consider an inverse problem of determining unknown coefficients for a one-dimensional analogue of radiative transport equation. We show that some combination of the unknown coefficients can be uniquely determined by giving pulse-like inputs at the boundary and observing the corresponding outputs. Our result can be applied for determination of absorption and scattering properties of an optically turbid medium if the radiative transport equation is appropriate for describing the propagation of light in the medium.