06-158 Nandor Simanyi

Conditional Proof of the Boltzmann-Sinai Ergodic Hypothesis (Assuming the Hyperbolicity of Typical Singular Orbits) (91K, AMS TeX) May 14, 06

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Abstract. We consider the system of $N$ ($\ge2$) elastically colliding hard balls of masses $m_1,\dots,m_N$ and radius $r$ on the flat unit torus $\Bbb T^\nu$, $\nu\ge2$. We prove the so called Boltzmann-Sinai Ergodic Hypothesis, i. e. the full hyperbolicity and ergodicity of such systems for every selection $(m_1,\dots,m_N;r)$ of the external geometric parameters, under the assumption that almost every singular trajectory is geometrically hyperbolic (sufficient), i. e. the so called Chernov-Sinai Ansatz holds true for the model. The present proof does not use at all the formerly developed, rather involved algebraic techniques, instead it employs exclusively dynamical methods and tools from geometric analysis.

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