Diffusivity in multiple scattering systems

Abstract

We consider random flights of point particles inside n-dimensional channels of the form R k ×B n−k , where B n−k is a ball of radius r in dimension n−k. The particle velocities immediately after each collision with the boundary of the channel comprise a Markov chain with a transition probabilities operator P that is determined by a choice of (billiard-like) random mechanical model of the particle-surface interaction at the "microscopic" scale. Our central concern is the relationship between the 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. Markov operators obtained in this way are natural (definition below), 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 . 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).

Diffusion and escape from polygonal channels: extreme values and geometric effects

“…The boundary is required to be piecewise smooth, except in a countable number of points. Multiple extensions of this simple scenario have been considered, including interacting particles [26,28], timedependent boundaries [29], thermostated dynamics [30,31], stochastic boundaries [32], and boundaries with holes [33].…”

Diffusion and escape from polygonal channels: extreme values and geometric effects

“…The boundary is required to be piecewise smooth, except in a countable number of points. Multiple extensions of this simple scenario have been considered, including interacting particles [26,28], timedependent boundaries [29], thermostated dynamics [30,31], stochastic boundaries [32], and boundary with holes [33].…”

Fermi acceleration in rotating drums

“…The relationship between rough billiard boundaries and random reflections (including Lambertian reflections; equivalently Knudsen law) was studied in [3,10,11,12,17,18,19,20,29,30,31].…”