Ergodicity

In probability theory, an ergodic dynamical system is one that, broadly speaking, has the same behavior averaged over time as averaged over the space of all the system's states in its phase space. In physics the term is used to imply that a system satisfies the ergodic hypothesis of thermodynamics.

A random process is ergodic if its time average is the same as its average over the probability space, known in the field of thermodynamics as its ensemble average. In an ergodic process, the state of the process after a long time is nearly independent of its initial state.

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

Let ${\displaystyle (X,\;\Sigma ,\;\mu \,)}$ be a probability space, and ${\displaystyle T:X\to X}$ be a measure-preserving transformation. We say that T is ergodic with respect to ${\displaystyle \mu }$ (or alternatively that ${\displaystyle \mu }$ is ergodic with respect to T) if one of the following equivalent statements is true:

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