Ergodic hypothesis

Summary

The ergodic hypothesis is an assumption of statistical mechanics about the long-run behaviour of a mechanical system. It says that the system spends time in each region of its energy surface in proportion to the volume of that region. In other words, time averages of an observable along a trajectory equal its average over the energy surface. It justifies replacing hard time averages by phase-space averages, which are the averages that equilibrium statistical mechanics computes. In mathematical terms it says that the Hamiltonian flow restricted to an energy surface is ergodic for the Liouville measure.