On the Relaxation of the Hard—Sphere Rayleigh and Lorentz Gas

Abstract

As part of a study of the relaxation of nonequilibrium systems, the (translational) relaxation of a hardsphere Rayleigh and Lorentz gas is investigated. From a detailed analysis of the collision dynamics an exact expression is derived for the kernel A (x | x′) of the collision integral which gives the probability per unit time for a change of the reduced kinetic energy from x′ to x during a binary collision between a subsystem and a heat bath particle. A master equation, i.e., a linearized Boltzmann equation, incorporating this kernel is then formulated to represent the time variation of the distribution function of the subsystem particles. Making use of the special property of this kernel that it is a strongly peaked function around x—x′=0 for both the Rayleigh and Lorentz gas, a technique is developed for transforming this integral master equation into differential Fokker—Planck equations consistent in the order of the expansion parameter λ, the ratio of the mass of the heat bath particles to the subsystem particles. The Fokker—Planck equation for the Rayleigh gas is solved analytically and explicit solutions are presented for the relaxation of initial Maxwell and initial δ-function distributions of the energy. It is shown that an initial Maxwell distribution of the energy (or speed) of the subsystem particles relaxes to the final equilibrium Maxwell distribution via a continuous sequence of Maxwell distributions. The mean energy of the subsystem particles is shown to relax exponentially to its equilibrium value independent of the form of the initial distribution. For the hard-sphere Lorentz gas the Fokker—Planck equation is not susceptible of an analytical solution. Machine solutions are presented for various initial distributions which show that the Maxwell distribution is not preserved in the relaxation of the hard sphere Lorentz gas. Finally, a brief discussion is given of the relation between the hard-sphere model (r—∞) considered here and the more general model of the Rayleigh and Lorentz gas with a r—8 repulsive central force law.