The Boltzmann-Grad limit of the periodic Lorentz gas

We study the dynamics of a point particle in a periodic array of spherical scatterers, and construct a stochastic process that governs the time evolution for random initial data in the limit of low scatterer density (Boltzmann-Grad limit). A generic path of the limiting process is a piecewise linear curve whose consecutive segments are generated by a Markov process with memory two.

[1]  F. Golse.  The periodic Lorentz gas in the Boltzmann-Grad limit , 2006 .

[2]  G. Gallavotti,et al.  Divergences and the Approach to Equilibrium in the Lorentz and the Wind-Tree Models , 1969 .

[3]  Jens Marklof,et al.  Kinetic transport in the two-dimensional periodic Lorentz gas , 2008 .

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

[5]  D. Szász,et al.  Limit Laws and Recurrence for the Planar Lorentz Process with Infinite Horizon , 2007 .

[6]  François Golse,et al.  On the distribution of free path lengths for the periodic Lorentz gas. II , 2000 .

[7]  On the Periodic Lorentz Gas and the Lorentz Kinetic Equation ( , 2007, math/0703812.

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

[9]  A. Zaharescu,et al.  The Distribution of the Free Path Lengths in the Periodic Two-Dimensional Lorentz Gas in the Small-Scatterer Limit , 2003, math/0301270.

[10]  D. Dolgopyat,et al.  Recurrence properties of planar Lorentz process , 2008 .

[11]  E. Davies,et al.  One-parameter semigroups , 1980 .

[12]  B. Dundas,et al.  DIFFERENTIAL TOPOLOGY , 2002 .

[13]  R. Newton.  Scattering theory of waves and particles , 1966 .

[14]  Matthew Nicol,et al.  Communications in Mathematical Physics Almost Sure Invariance Principle for Nonuniformly Hyperbolic Systems , 2022 .

[15]  François Golse,et al.  On the Distribution of Free Path Lengths¶for the Periodic Lorentz Gas , 1998 .

[16]  J. Marklof,et al.  The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems , 2007, 0706.4395.

[17]  Emanuele Caglioti,et al.  On the Distribution of Free Path Lengths for the Periodic Lorentz Gas III , 2000, math/0301300.

[18]  I. Tóth,et al.  Exponential Decay of Correlations in Multi-Dimensional Dispersing Billiards , 2008 .

[19]  Matthew Nicol,et al.  A vector-valued almost sure invariance principle for hyperbolic dynamical systems , 2006, math/0606535.

[20]  J. Marklof,et al.  The Periodic Lorentz Gas in The Boltzmann–Grad Limit: Asymptotic Estimates , 2010, 1008.3811.

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

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

[23]  N. Chernov.  Statistical properties of the periodic Lorentz gas. Multidimensional case , 1994 .

[24]  Per Dahlqvist.  The Lyapunov exponent in the Sinai billiard in the small scatterer limit , 1997 .

[25]  Emanuele Caglioti,et al.  The Boltzmann-Grad limit of the periodic Lorentz gas in two space dimensions , 2008 .