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.