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

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 description of meaning of diffusion in biology and survey of derivation of kinetic equations by projective operator method, we pay an attention to the Lorentz gas with step potential. The gas is composed of $N$ particles: $N-1$ of which are immovable between $N-1$ 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.