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Conditional proof of the Boltzmann-Sinai ergodic hypothesis

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Abstract

We consider the system of N (≥ 2) elastically colliding hard balls of masses m 1,…,m N and radius r on the flat unit torus \(\mathbb{T}^{\nu}\) , ν≥2. We prove the so called Boltzmann-Sinai Ergodic Hypothesis, i.e. the full hyperbolicity and ergodicity of such systems for every selection (m 1,…,m N ;r) of the external parameters, provided that almost every singular orbit is geometrically hyperbolic (sufficient), i.e. the so called Chernov-Sinai Ansatz is true. The present proof does not use the formerly developed, rather involved algebraic techniques, instead it employs exclusively dynamical methods and tools from geometric analysis.

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Authors and Affiliations

  1. Department of Mathematics, University of Alabama at Birmingham, Campbell Hall, Birmingham, AL, 35294, USA

    Nándor Simányi

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  1. Nándor Simányi
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Correspondence to Nándor Simányi.

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Dedicated to Yakov G. Sinai and Domokos Szász.

Research supported by the National Science Foundation, grants DMS-0457168, DMS-0800538.

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Simányi, N. Conditional proof of the Boltzmann-Sinai ergodic hypothesis. Invent. math. 177, 381–413 (2009). https://doi.org/10.1007/s00222-009-0182-x

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