Ergodic (adjective)

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For other uses, see ergodic (disambiguation).

In mathematics and physics, the adjective ergodic is used to imply that a system satisfies the ergodic hypothesis of thermodynamics or that it is a system studied in ergodic theory.

A more rigorous definition may be given as follows:

Let $(X,Σ,μ)$ be a probability space, and $T : X \to X$ be a measure-preserving transformation, i.e.

$\mu \left( T^{-1} (E) \right) = \mu (E)$ for all $E \in \Sigma,$

so $μ$ is an invariant measure under $T$ . We call $T$ an ergodic transformation (with respect to $μ$ ) and call $μ$ an ergodic measure (with respect to $T$ ) if, whenever $T (E) = E$ for some $E \in \Sigma$ , then