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 is a concept from statistical mechanics and dynamical systems that relates to the long-term behavior of a dynamical system. It asserts that, under certain conditions, the time average of a physical quantity is equal to the ensemble average (or spatial average) over the state space of the system.
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