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A Geometric Approach to Semi-Dispersing Billiards

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Hard Ball Systems and the Lorentz Gas

Part of the book series: Encyclopaedia of Mathematical Sciences ((EMS,volume 101))

Abstract

This section contains a survey of a few results obtained by a particular realization of V. Arnold’s old idea that hard ball models of statistical physics can be “considered as the limit case of geodesic flows on negatively curved manifolds (the curvature being concentrated on the collisions hypersurface)”. The approach is based on representing billiard trajectories as geodesics in appropriate spaces. These spaces are not even topological manifolds: they are lengths spaces of curvature bounded above in the sense of A. D. Alexandrov. Nevertheless, this method allows to transforms a certain type of problems about billiards into purely geometric statements; and a problem looking difficult in its billiard clothing sometimes turns into a relatively easy statement (by the modern standards of metric geometry). In particular, this approach helped to solve an old problem of whether the number of collisions in a hard ball model is bounded from above by a quantity depending only on the system (and thus uniform for all initial conditions).

We would like to express our sincere gratitude to M.Brin, N.Chernov, G.Galperin, M.Gromov, A.Katok and Ya.Pesin for very helpful comments and fruitful discussions.

partially supported by a Sloan Foundation Fellowship and NSF grant DMS-98-05175

partially supported by NSF DMS-99-71587

partially supported by NSF DMS-98-03092

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  1. D. Burago
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  2. S. Ferleger
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  3. A. Kononenko
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  1. Institute of Mathematics, Budapest University of Technology and Economics, 91, 1521, Budapest, Hungary

    D. Szász

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Burago, D., Ferleger, S., Kononenko, A. (2000). A Geometric Approach to Semi-Dispersing Billiards. In: Szász, D. (eds) Hard Ball Systems and the Lorentz Gas. Encyclopaedia of Mathematical Sciences, vol 101. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04062-1_2

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