A property of a dynamical system with an invariant probability measure: the system is ergodic if, for almost every initial condition, time averages of integrable observables equal their ensemble (space) average with respect to that measure.

Principle

Indecomposability under the dynamics: the invariant measure cannot be written as a nontrivial convex combination of other invariant measures supported on disjoint invariant sets.

Demonstration

Example: rotation of the circle by an irrational angle is ergodic with respect to the uniform measure; time averages along a typical orbit sample the circle uniformly.

Misapplication

Equating ergodicity with fast decay of correlations; ergodicity guarantees equality of long-time averages and ensemble averages but does not quantify the rate at which averages converge.

Consequence

For ergodic systems, a long time series from a single typical trajectory suffices to estimate ensemble statistics, justifying single-trajectory experiments in stationary regimes.

Reversal

Non-ergodic dynamics decompose phase space into invariant components so that time averages depend on the initial component and differ from global ensemble averages.

Boundary

Defined for systems carrying an invariant probability measure; excludes nonstationary processes without invariant statistics or purely transient behavior.

Semantic Tension

Often confused with mixing: mixing is a stronger property implying loss of correlations and therefore ergodicity, but ergodicity alone does not imply decorrelation over finite times.

Synthesis

Ergodicity ensures that individual long-time observations represent ensemble expectations by requiring the dynamics to explore the invariant measure indecomposably.