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Proof of the Ergodic Hypothesis for Typical Hard Ball Systems

Abstract.

We consider the system of \( N (\geq 2) \) hard balls with masses \( m_{1}, \ldots, m_{N} \) and radius r in the flat torus \( \mathbb{T}_{L}^{\nu} = \mathbb{R}^{\nu} / L \cdot \mathbb{Z}^{\nu} \) of size \( L, \nu \geq 3 \) . We prove the ergodicity (actually, the Bernoulli mixing property) of such systems for almost every selection \( (m_{1}, \ldots, m_{N}; L) \) of the outer geometric parameters. This theorem complements my earlier result that proved the same, almost sure ergodicity for the case \( \nu = 2 \) . The method of that proof was primarily dynamical-geometric, whereas the present approach is inherently algebraic.

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Correspondence to Nándor Simányi.

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Communicated by Eduard Zehnder

Submitted 17/10/02, accepted 01/12/03

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Simányi, N. Proof of the Ergodic Hypothesis for Typical Hard Ball Systems . Ann. Henri Poincaré 5, 203–233 (2004). https://doi.org/10.1007/s00023-004-0166-8

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  • DOI: https://doi.org/10.1007/s00023-004-0166-8

Keywords

  • Geometric Parameter
  • Early Result
  • Present Approach
  • Ball System
  • Ergodic Hypothesis