Ergodicity

In mathematics, the term ergodic is used to describe a dynamical system which, broadly speaking, has the same behavior averaged over time as averaged over the space of all the system's states (phase space). In physics the term is used to imply that a system satisfies the ergodic hypothesis of thermodynamics.

In statistics, the term describes a random process for which the time average of one sequence of events is the same as the ensemble average. In other words, for a Markov chain, as one increases the steps, there exists a positive probability measure at step that is independent of the probability distribution at initial step 0 (Feller, 1971, p. 271).

Etymology

The term "ergodic" was derived from the Greek words έργον (ergon: "work") and οδός (odos: "path" or "way"). It was chosen by Boltzmann while he was working on a problem in statistical mechanics.

Formal definition

Let be a probability space, and be a measure-preserving transformation. We say that T is ergodic with respect to (or alternatively that is ergodic with respect to T) if one of the following equivalent statements is true:

Ergodic (disambiguation)

Aside from its generic use as the generic adjective ergodic, ergodic may relate to: