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Diffusivity in multiple scattering systems


Authors: Timothy Chumley, Renato Feres and Hong-Kun Zhang
Journal: Trans. Amer. Math. Soc. 368 (2016), 109-148
MSC (2010): Primary 60F05; Secondary 82B40
DOI: https://doi.org/10.1090/tran/6325
Published electronically: April 15, 2015
MathSciNet review: 3413858
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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).


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Additional Information

Timothy Chumley
Affiliation: Department of Mathematics, Washington University, Campus Box 1146, St. Louis, Missouri 63130

Renato Feres
Affiliation: Department of Mathematics, Washington University, Campus Box 1146, St. Louis, Missouri 63130

Hong-Kun Zhang
Affiliation: Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003

DOI: https://doi.org/10.1090/tran/6325
Received by editor(s): July 8, 2013
Received by editor(s) in revised form: October 20, 2013
Published electronically: April 15, 2015
Article copyright: © Copyright 2015 American Mathematical Society