On the mean free path for a periodic array of spherical obstacles

We prove theorems pertaining to periodic arrays of spherical, obstacles which show how the macroscopic limit of the mean free path depends on the scaling of the size of the obstacles. We treat separately the cases where the obstacles are totally and partially absorbing, and we also distinguish between two-dimensional arrays, where our results are optimal, and higher dimensional arrays, where they are not. The cubically symmetric arrays to which these results apply do not have finite horizon.

[1]  H. Dumas.  Ergodization rates for linear flow on the torus , 1991 .

[2]  H. Spohn.  The Lorentz process converges to a random flight process , 1978 .

[3]  Giovanni Gallavotti,et al.  Rigorous Theory Of The Boltzmann Equation In The Lorentz Gas , 1972 .

[4]  F. Golse.  Transport dans les milieux composites fortement contrastés : le modèle du billard , 1994 .

[5]  R. Illner,et al.  Global validity of the Boltzmann equation for two- and three-dimensional rare gas in vacuum: Erratum and improved result , 1989 .

[6]  Leonid A. Bunimovich,et al.  Statistical properties of lorentz gas with periodic configuration of scatterers , 1981 .

[7]  Leonid A. Bunimovich,et al.  Statistical properties of two-dimensional hyperbolic billiards , 1991 .

[8]  L. Bunimovich,et al.  Markov Partitions for dispersed billiards , 1980 .

[9]  Global validity of the Boltzmann equation for a three-dimensional rare gas in vacuum , 1986 .

[10]  L. Bunimovich,et al.  Markov partitions for two-dimensional hyperbolic billiards , 1990 .

[11]  Halim Doss.  Sur une Resolution Stochastique de l'Equation de Schrödinger à Coefficients Analytiques , 1980 .

[12]  L. Bunimovich,et al.  On the Boltzmann equation for the Lorentz gas , 1983 .

[13]  P. Bleher.  Statistical properties of two-dimensional periodic Lorentz gas with infinite horizon , 1992 .

[14]  R. Dautray,et al.  Théorèmes de trace Lp pour des espaces de fonctions de la neutronique , 1984 .

[15]  L. Schwartz.  Théorie des distributions , 1966 .

[16]  O. Lanford.  Time evolution of large classical systems , 1975 .

[17]  R. Illner,et al.  Global validity of the Boltzmann equation for a two-dimensional rare gas in vacuum , 1986 .

[18]  Alan Baker,et al.  DIOPHANTINE APPROXIMATION (Lecture Notes in Mathematics, 785) , 1981 .

[19]  Vladimir Igorevich Arnold,et al.  Geometrical Methods in the Theory of Ordinary Differential Equations , 1983 .