Random walks derived from billiards. (English) Zbl 1145.37009

Feres, Renato

Hasselblatt, Boris (ed.), Dynamics, ergodic theory and geometry. Dedicated to Anatole Katok. Based on the workshop on recent progress in dynamics, Berkeley, CA, USA, from late September to early October, 2004. Cambridge: Cambridge University Press (ISBN 978-0-521-87541-7/hbk). Mathematical Sciences Research Institute Publications 54, 179-222 (2007).

The motivation of this paper is the interaction between a gas at very low temperature in a cylinder, with the inner surface of the latter, which is not perfectly flat as a result of its molecular structure. The gas-surface interaction can be thought of as the reflection of molecules on a random structure, therefore the terms of random billiards. It is shown that the problem can be reduced to a Markov chain of which the main properties are exhibited: reversibility and self-adjointness, ergodicity, relation between the geometry of the billiard and the spectrum of the Markov operator, diffusion limit.
For the entire collection see [Zbl 1128.37001].

Reviewer: Guy Jumarie (Montréal)

MSC:

37A50	Dynamical systems and their relations with probability theory and stochastic processes
82B05	Classical equilibrium statistical mechanics (general)
37D50	Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
37H10	Generation, random and stochastic difference and differential equations
60G50	Sums of independent random variables; random walks

Keywords:

random billiards; Markov chain; gas kinetics