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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 $$h(T)=-2 \ln \epsilon +c+o(1)$$ of the KS entropy of the billiard map in this model, as conjectured by P. Dahlqvist.

The Distribution of the Free Path Lengths in the Periodic Two-Dimensional Lorentz Gas in the Small-Scatterer Limit