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

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

2003-01-23

arxiv.org/abs/math/0301270v6

Mathematics

Number Theory

Scientific paper

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 $c=2-3\ln 2+\frac{27\zeta(3)}{2\pi^2}$ in the asymptotic formula $h(T)=-2\ln \eps+c+o(1)$ of the KS entropy of the billiard map in this model, as conjectured by P. Dahlqvist.

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