Ergodic hypothesis - Definition

In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.

Liouville's Theorem shows that, for classical systems, the local density of microstates following a particle path through phase space is constant (ie the total or convective time derivative is zero). Thus, if the microstates are uniformly distributed in phase space initially, this will remain true at later time.

Ergodic theory is a branch of mathematics which deals with dynamical systems which satisfy a version of this hypothesis, phrased in the language of measure theory.