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

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+ 27 zeta(3)/2 pi(2) in the asymptotic formula h(T) = - 2 ln epsilon+ c+ o(1) of the KS entropy of the billiard map in this model, as conjectured by P. Dahlqvist.