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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Diffusivity in multiple scattering systems
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by Timothy Chumley, Renato Feres and Hong-Kun Zhang PDF
Trans. Amer. Math. Soc. 368 (2016), 109-148 Request permission

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
  • ORCID: 0000-0003-2393-831X
  • Renato Feres
  • Affiliation: Department of Mathematics, Washington University, Campus Box 1146, St. Louis, Missouri 63130
  • MR Author ID: 262178
  • Hong-Kun Zhang
  • Affiliation: Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003
  • MR Author ID: 626279
  • Received by editor(s): July 8, 2013
  • Received by editor(s) in revised form: October 20, 2013
  • Published electronically: April 15, 2015
  • © Copyright 2015 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 368 (2016), 109-148
  • MSC (2010): Primary 60F05; Secondary 82B40
  • DOI: https://doi.org/10.1090/tran/6325
  • MathSciNet review: 3413858