In classical mechanics, one intuitive formulation of chaos/ergodicity (in the loose sense) is that most trajectories should fill up phase space densely over infinite time. A classic example of such a system is the Sinai billiard.
In classical field theory, the phase space becomes infinite dimensional, consisting of fields $\phi(x)$ and their conjugate momenta $\pi(x)$. It is hard for me to imagine how to formulate/prove that a field configuration can evolve over time to fill this infinite dimensional space. After some literature search, I could not find a paradigmatic solvable example (like the Sinai's billiard in classical chaos). Given that the chaotic real world (i.e. a world where statistical mechanics works) is described by field theories, an understanding of chaos restricted to classical mechanics seems incomplete.
Does anyone know of a systematic treatment of chaos and ergodicity in classical field theories? Are there solvable examples?