Diffusion approximation of a Knudsen gas model: dependence of the diffusion constant upon the boundary condition

Abstract

In this article we study a model for the dynamics of particles moving freely between two horizontal plates. Our goal is to characterize the diffusive behavior for such systems in the long‐time and large (horizontal) scale limits. Using homogenization techniques for PDEs, we obtain a diffusion equation as a limit of the original kinetic equation, appropriately scaled. This limiting diffusion can be understood intuitively by observing that in the limit of long times and vanishing vertical distance between the plates, for a given particle, the mean free path between two successive reflections at the plates also vanishes, and therefore the number of reflections grows unboundedly with the length of the time interval. The nature of a particular reflection law then efficiently randomizes the particle motion. More specifically, we study the dependence of the diffusion constant D_α on mixed boundary conditions: the case of specular reflection (based on the “Arnold cat map”) with an isotropic component (i.e., with small “accommodation” coefficient α). We find that the diffusion tensor D_α is positive definite for every α∈(0,1]. Furthermore, in the limit of vanishing isotropic component (α→0), we recover the result of Bardos, Golse and Colonna [Physica D, 104 (1997), 32–60].