Ergodic hypothesis

Remarks

By Birkhoff's ergodic theorem, this holds exactly when the flow on $\Sigma_E$ is ergodic for $\mu_E$: every measurable invariant subset of $\Sigma_E$ has measure $0$ or $1$. Example: for $H(q, p) = \tfrac12 (p^2 + q^2)$ on $\R^2$ and $E > 0$, the energy surface is a circle, the flow is a rotation at constant speed, and $\mu_E$ is the normalised arc length. Every orbit runs once around the circle in time $2\pi$, so time averages equal the circle average and the hypothesis holds. Non-example: for two uncoupled oscillators, $H = \tfrac12(p_1^2 + q_1^2) + \tfrac12(p_2^2 + q_2^2)$, each energy $h_1 = \tfrac12(p_1^2 + q_1^2)$ is conserved. The set where $h_1 < E/2$ is then an invariant subset of $\Sigma_E$ of measure strictly between $0$ and $1$, so the hypothesis fails.