The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems

Pages 1949-2033 from Volume 172 (2010), Issue 3 by Jens Marklof, Andreas Strömbergsson

Abstract

The periodic Lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers, and is one of the fundamental models for chaotic diffusion. In the present paper we investigate the Boltzmann-Grad limit, where the radius of each scatterer tends to zero, and prove the existence of a limiting distribution for the free path length. We also discuss related problems, such as the statistical distribution of directions of lattice points that are visible from a fixed position.

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Mathematical Subject Classification
Primary: , , , ,
DOI
http://doi.org/10.4007/annals.2010.172.1949
MR
2726104
Zbl
1211.82011
Milestones
Received: 29 June 2007
Revised: 14 January 2008
Accepted: 24 March 2008
Published online: 5 October 2010

Authors

Jens Marklof

School of Mathematics
University of Bristol
Bristol, BS8 1TW
United Kingdom

Andreas Strömbergsson

Department of Mathematics
Box 480
Uppsala University
SE-75106 Uppsala
Sweden