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