Diffusivity in multiple scattering systems

Author:

Chumley Timothy,Feres Renato,Zhang Hong-Kun

Abstract

We consider random flights of point particles inside n n -dimensional channels of the form R k × B n k \mathbb {R}^{k}\times \mathbb {B}^{n-k} , where B n k \mathbb {B}^{n-k} is a ball of radius r r in dimension n k 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 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 P .

Our central concern is the relationship between the surface scattering properties encoded in P 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 P and compute, in the case of 2 2 -dimensional channels, the exact values of the diffusivity for a class of parametric microscopic surface models of the above geometric type (2).

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Cited by 6 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Revisiting Maxwell-Smoluchowski theory: Low surface roughness in straight channels;Chemical Engineering Science;2024-02

2. Fermi acceleration in rotating drums;Journal of Mathematical Physics;2022-06-01

3. Diffusion and escape from polygonal channels: extreme values and geometric effects;Journal of Statistical Mechanics: Theory and Experiment;2021-07-01

4. Knudsen Diffusivity in Random Billiards: Spectrum, Geometry, and Computation;SIAM Journal on Applied Dynamical Systems;2021-01

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