Proof of the Ergodic Hypothesis for Typical Hard Ball Systems

We consider the system of $N$ ($\ge 2$) hard balls with masses $m_1,\mathrm{.}\mathrm{.}\mathrm{.},m_N$ and radius $r$ in the flat torus ${\mathbb{T}}_L^{\nu }={\mathbb{R}}^{\nu }\mathrm{/}L\cdot {\mathbb{Z}}^{\nu }$ of size $L$, $\nu \ge 3$. We prove the ergodicity (actually, the Bernoulli mixing property) of such systems for almost every selection $(m_1,\mathrm{.}\mathrm{.}\mathrm{.},m_N;L)$ 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.