Kinetic Theory of a Weakly Coupled Fluid

Abstract

Virtually all measurable properties of a classical fluid may be determined from the expectation value of the phase-space density operator f{rpt) =Z~B{r-r~(t)) 6(p -p (t)), and the phase-space density correlation function (f (rpt) f (r' p' t')) -(f (rp t)) (f(r' p' t')), a matrix with indices (rpt). Systematic procedures for approximating this matrix, unhindered by secular effects, are always based on approximations to its inverse. For a weakly coupled fluid, the inverse can be expanded in powers of~, the ratio of potential to kinetic energy.The leading term in this expansion gives rise to a Vlasov equation for the phase-space correlation function. The next term is the first that includes collisions, and results in relaxation towards equilibrium.This paper is concerned with the detailed study of the resulting fundamental nontrivial approximation.It is not Markovian and is perfectly reversible. Although the approximation is complicated, it is tractable analytically in various limits, and numerically for all wavelengths and frequencies. In this paper, only the behavior in certain limits is evaluated. Particular attention is directed toward its contractionsthe density correlation function, which is measured by inelastic neutron and light scattering, and the momentum correlation function. Calculation of the former at long wavelengths corroborates the Landau-Placzek expression for light scattering, and therefore demonstrates that the kinetic equation predicts hydrodynamic behavior at long times. Since the correlation function is correct to order X, it has, in contrast to a solution to the Boltzmann equation, the correct long-wavelength velocity of sound, c =-(dp/dmn), & 3 k&T/m. It also predicts different transport coefficients than those deduced from a Boltzman equation. These include a nonvanishing bulk viscosity. The transport coefficients reduce to those derived from the Boltzmann equation at low densities. Some aspects of the short-time behavior are also discussed.

Kinetic theory of self-diffusion in a moderately dense one-component plasma

“…The motion of the tagged particle does not conserve its momentum and energy; only the tagged-particle number is trivially conserved. The latter property leads with (9) to the conservation law 15) f dp tp '(kpp' ; z…”

A Consistent Description of Kinetics and Hydrodynamics of Systems of Interacting Particles by Means of the Nonequilibrium Statistical Operator Method

“…The third region of investigations on collective modes and time correlation functions in dense gases and liquids is connected with the kinetic theory [77,[107][108][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 description of a nonequilibrium state of the system.…”

“…Approximate calculations of memory functions (expansions on density, weak interaction) in the hydrodynamic limit were performed in papers by Mazenko [109][110][111][112][113][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][108][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.…”

“…Equation (3.61) was derived for the first time by using the Mori projection operators method in [107][108][109], when the microscopic phase densityn k (p) was a parameter of an abbreviated description. In this case the memory function ϕ ′ nn (k, p, p ′ ; z) has the following structure:…”

“…The kinetic equation (3.59) for the Fourier component of the nonequilibrium one-particle distribution function stimulates investigations of the dynamical structure factor S(k; ω), time correlation functions for transverse and longitudinal flows of the particles, diffusion coefficients, viscosities for dense gases and liquids [77,79,[107][108][109][110][111][112][113][114][115][116][117][118][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.…”