The distribution of the free path lengths in the periodic two-dimensional Lorentz gas in the small-scatterer limit. (English) Zbl 1143.37002

Summary: 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.


37A25 Ergodicity, mixing, rates of mixing
11B57 Farey sequences; the sequences \(1^k, 2^k, \dots\)
37A60 Dynamical aspects of statistical mechanics
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
82C40 Kinetic theory of gases in time-dependent statistical mechanics
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