, Volume 177, Issue 2, pp 381-413
Date: 17 Feb 2009

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.

Dedicated to Yakov G. Sinai and Domokos Szász.
Research supported by the National Science Foundation, grants DMS-0457168, DMS-0800538.