Billiards in a General Domain with Random Reflections

We study stochastic billiards on general tables: a particle moves according to its constant velocity inside some domain $$\fancyscript{D}\subset {\mathbb{R}}^d$$ until it hits the boundary and bounces randomly inside, according to some reflection law. We assume that the boundary of the domain is locally Lipschitz and almost everywhere continuously differentiable. The angle of the outgoing velocity with the inner normal vector has a specified, absolutely continuous density. We construct the discrete time and the continuous time processes recording the sequence of hitting points on the boundary and the pair location/velocity. We mainly focus on the case of bounded domains. Then, we prove exponential ergodicity of these two Markov processes, we study their invariant distribution and their normal (Gaussian) fluctuations. Of particular interest is the case of the cosine reflection law: the stationary distributions for the two processes are uniform in this case, the discrete time chain is reversible though the continuous time process is quasi-reversible. Also in this case, we give a natural construction of a chord “picked at random” in $$\fancyscript{D}$$ , and we study the angle of intersection of the process with a (d − 1)-dimensional manifold contained in $$\fancyscript{D}$$ .

[1]  J. R. Partington,et al.  The kinetic theory of gases : some modern aspects , 1934 .

[2]  H. Fédérer.  Geometric Measure Theory , 1969 .

[3]  D. Varberg.  Convex Functions , 1973 .

[4]  E. Jaynes.  The well-posed problem , 1973 .

[5]  C. Kipnis,et al.  Stationary states for a mechanical system with stochastic boundary conditions , 1985 .

[6]  H Robbins,et al.  Asymptotically minimax stochastic search strategies in the plane. , 1987, Proceedings of the National Academy of Sciences of the United States of America.

[7]  F. Morgan.  Geometric Measure Theory: A Beginner's Guide , 1988 .

[8]  Steven P. Lalley,et al.  Stochastic search in a convex region , 1988 .

[9]  C. Cercignani.  The Boltzmann equation and its applications , 1988 .

[10]  D. Stoyan,et al.  Stochastic Geometry and Its Applications , 1989 .

[11]  Konstantin Borovkov.  On a New Variant of the Monte Carlo Method , 1992 .

[12]  Richard L. Tweedie,et al.  Markov Chains and Stochastic Stability , 1993, Communications and Control Engineering Series.

[13]  Semi-Markov Process.  ON SIMULATION OF RANDOM VECTORS WITH GIVEN DENSITIES IN REGIONS AND ON THEIR BOUNDARIES , 1994 .

[14]  The Boltzmann-Grad limit for a one-dimensional Boltzmann equation in a stationary state , 1996 .

[15]  P. Spreij.  Probability and Measure , 1996 .

[16]  F. Golse,et al.  Diffusion approximation and hyperbolic automorphisms of the torus , 1997 .

[17]  A general framework for approximate sampling with an application to generating points on the boundary of bounded convex regions , 1998 .

[18]  H. Thorisson.  Coupling, stationarity, and regeneration , 2000 .

[19]  M. Coppens,et al.  Modeling of Diffusion in Zeolites , 2000 .

[20]  M. Coppens,et al.  Effects of surface roughness on self- and transport diffusion in porous media in the Knudsen regime. , 2001, Physical review letters.

[21]  Steven N. Evans,et al.  Stochastic billiards on general tables , 2001 .

[22]  Diffusion approximation of a Knudsen gas model: dependence of the diffusion constant upon the boundary condition , 2002 .

[23]  M. Coppens,et al.  Dynamic Monte-Carlo simulations of diffusion limited reactions in rough nanopores , 2003 .

[24]  Gregory S. Yablonsky,et al.  Knudsen's cosine law and random billiards , 2004 .

[25]  Lambert diffusion in porous media in the Knudsen regime: equivalence of self-diffusion and transport diffusion. , 2005 .

[26]  A. Bunde,et al.  Lambert diffusion in porous media in the Knudsen regime: equivalence of self-diffusion and transport diffusion. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  A. J. Dammers,et al.  Effects of heterogeneity on diffusion in nanopores—From inorganic materials to protein crystals and ion channels , 2006 .

[28]  G. Schütz,et al.  Amplification of molecular traffic control in catalytic grains with novel channel topology design. , 2005, The Journal of chemical physics.

[29]  E. Beckenbach.  CONVEX FUNCTIONS , 2007 .

[30]  R. Feres.  Random Walks Derived from Billiards , 2007 .