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

Communications in Mathematical Physics volume 269, pages425–471(2007)Cite this article

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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Affiliations

  1. Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL, 61801, USA

    Florin P. Boca & Alexandru Zaharescu

  2. Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, RO-014700, Bucharest, Romania

    Florin P. Boca & Alexandru Zaharescu

Authors
  1. Florin P. Boca
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  2. Alexandru Zaharescu
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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