Linear and nonlinear diffusion and reaction-diffusion equations from discrete-velocity kinetic models

Abstract

Two-velocity kinetic models are used to derive, in the appropriate limit, the equations which govern the macroscopic density of fluid systems. Such equations are obtained from an asymptotic expansion in powers of a small parameter related to the microscopic mean free path. It is shown that the density of a fluid interacting with a non-equilibrium background satisfies a linear diffusion equation, and that the hierarchy of equations arising from the asymptotic expansion can be completely solved by a recursive scheme. For a system of interacting particles, a nonlinear diffusion equation is obtained and some of its solutions are analysed. Finally, the density of a system of particles undergoing chemical reactions is shown to satisfy a nonlinear equation which formally coincides with the reaction-diffusion equation proposed ad hoc at the macroscopic level.