The problem then is that proving ergodicity is extremely hard for anything that is not a Markov chain to begin with. IIRC ergodicity of hard spheres is not yet

The problem then is that proving ergodicity is extremely hard for anything that is not a Markov chain to begin with. IIRC ergodicity of hard spheres is not yet proved in full generality (n spheres, d dimensions), and that's basically the simplest physical system there is!