Kinetic equation for a gas with attractive forces as a functional equation

Abstract

Diffusion problems studied in the time scale comparable with time of particles collision lead to kinetic equations which for step-wise potentials are functional equations in the velocity space. After a survey of derivation of kinetic equations by projective operator method, an attention is paid to the Lorentz gas with step potential. The gas is composed of N particles: N - 1 of which are immovable; between those N - 1 immovable particles - scatterers, particle number 1 is moving, and we describe its movement by means of one-particle distribution function satisfying a kinetic equation. Solutions of the kinetic equation for some simple potentials are given. We derive also a kinetic equation for one-dimensional Lorentz gas, which is a functional equation.