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

Abstract

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.

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Correspondence to Florin P. Boca.

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In memory of Walter Philipp

Communicated by P. Sarnak

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Boca, F.P., Zaharescu, A. The Distribution of the Free Path Lengths in the Periodic Two-Dimensional Lorentz Gas in the Small-Scatterer Limit. Commun. Math. Phys. 269, 425–471 (2007). https://doi.org/10.1007/s00220-006-0137-7

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Keywords

  • Asymptotic Formula
  • Diophantine Approximation
  • Free Path Length
  • Vertical Slit
  • Liouville Measure