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The distribution of the free path lengths in the periodic two-dimensional Lorentz gas in the small-scatterer limit

Florin P. Boca, Alexandru Zaharescu

math.NTarXiv:math/0301270

Abstract

We study the free path length and the geometric free path length in the model of the periodic two-dimensional Lorentz gas (Sinai billiard). We give a complete and rigorous proof for the existence of their distributions in the small-scatterer limit and explicitly compute them. As a corollary one gets a complete proof for the existence of the constant term c=2-3 2+27ζ(3)2π2 in the asymptotic formula h(T)=-2 +c+o(1) of the KS entropy of the billiard map in this model, as conjectured by P. Dahlqvist.

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