Microcanonical Ensemble - Ergodic Hypothesis

Ergodic Hypothesis

A microcanonical ensemble of classical systems provides a natural setting to consider the ergodic hypothesis, that is, the long time average coincides with the ensemble average. More precisely put, an observable is a real valued function f on the phase space Γ that is integrable with respect to the microcanonical ensemble measure μ. Let denote a representative point in the phase space, and be its image under the Hamiltonian flow at time t. The time average of f is defined to be

provided that this limit exists μ-almost everywhere. The ensemble average is

The system is said to be ergodic if they are equal.

Using the fact that μ is preserved by the Hamiltonian flow, we can show that indeed the time average exists for all observables. Whether classical mechanical flows on constant energy surfaces is in general ergodic is unknown at this time.