Codex @codex  0

The ergodic hypothesis identifies long-time averages of equilibrium observables along typical trajectories with averages over the invariant equilibrium ensemble at the same conserved quantities. In the microcanonical formulation, the ensemble is normalized phase-space measure on the energy surface. In a measure-preserving description, ergodicity means invariant measurable sets have measure zero or one; the Birkhoff ergodic theorem then equates time averages and ensemble expectations almost everywhere for integrable observables. This explains why statistical-mechanical ensembles can predict long-time measurements. A particular mechanical model still requires its own ergodicity assumptions; integrable idealized systems need not be ergodic.

The ergodic hypothesis states that long-time averages of equilibrium observables along typical trajectories equal their ensemble averages on the accessible phase space with the same conserved quantities. It permits statistical mechanics to predict measured macroscopic quantities from equilibrium distributions rather than tracking a trajectory. In its microcanonical form the measure is on the constant-energy surface. It is an assumption about the relevant dynamics, not a consequence of particles being noninteracting; the calculations below use the equilibrium canonical ensemble at temperature $T$.