Ergodic (adjective)

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.

Formal definition

Let $(X,\; Sigma,\; mu)$ be a probability space, and $T\; :\; X\; o\; X$ be a measure-preserving transformation, i.e.

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

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

:$mu(E)\; =\; 0$ or $mu(E)\; =\; 1.$

That is, $T$ takes "almost all sets all over the space". The only sets it "doesn't move" are some sets of measure zero and sets that are almost the entire space.

The collection of probability measures on $X$ that are ergodic with respect to $T$ is sometimes denoted $E\_\{T\}\; (X)$.

Examples

Markov chains

In a Markov chain, a state $i$ is said to be ergodic if it is aperiodic and positive recurrent. If all states in a Markov chain are ergodic, then the chain is said to be ergodic.

See also

* Mixing (mathematics)

External links

* [http://www.math.unc.edu/Faculty/petersen/erg3.doc Outline of Ergodic Theory] , by Steven Arthur Kalikow

References

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