Diffusivity in multiple scattering systems

Abstract:

We consider random flights of point particles inside $n$-dimensional channels of the form $\mathbb {R}^{k}\times \mathbb {B}^{n-k}$, where $\mathbb {B}^{n-k}$ is a ball of radius $r$ in dimension $n-k$. The sequence of particle velocities taken immediately after each collision with the boundary of the channel comprise a Markov chain whose transition probabilities operator $P$ is determined by a choice of (billiard-like) random mechanical model of the particle-surface interaction at the “microscopic” scale. Markov operators obtained in this way are natural, which means, in particular, that (1) the (at the surface) Maxwell-Boltzmann velocity distribution with a given surface temperature, when the surface model contains moving parts, or (2) the so-called Knudsen cosine law, when this model is purely geometric, is the stationary distribution of $P$.

Our central concern is the relationship between the surface scattering properties encoded in $P$ and the constant of diffusivity of a Brownian motion obtained by an appropriate limit of the random flight in the channel. We show by a suitable generalization of a central limit theorem of Kipnis and Varadhan how the diffusivity is expressed in terms of the spectrum of $P$ and compute, in the case of $2$-dimensional channels, the exact values of the diffusivity for a class of parametric microscopic surface models of the above geometric type (2).