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 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 {\em 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).

39 pages, 11 images

References in corpus (3)

• The cosine law at the atomic scale: Toward realistic simulations of Knudsen diffusion
• Random billiards with wall temperature and associated Markov chains
• Multiple scattering in random mechanical systems and diffusion approximation

Cited by in corpus (5)

• Diffusion and escape from polygonal channels: extreme values and geometric effects
• Slow Convergence in Generalized Central Limit Theorems
• Fermi acceleration in rotating drums
• Revisiting Maxwell-Smoluchowski theory: low surface roughness in straight channels
• Knudsen diffusivity in random billiards: spectrum, geometry, and computation