2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/99833We 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 $c=2-3\ln 2+\frac{27ζ(3)}{2π^2}$ 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.Number TheoryMathematical PhysicsDynamical SystemsPrimary: 11K60 Secondary: 11P21; 11J83; 37A60;37A60; 37D50; 82C05; 82C40text