---
tag: AZRPHS478B
kind: concept
name: Ergodic transformation
msc: [37A25, 28D05]
specializes: [556WBDAHNP]
sources:
  - url: https://en.wikipedia.org/wiki/Ergodicity
---

## Summary

A [[measure-preserving transformation|556WBDAHNP]] of a probability space is ergodic if it has no invariant measurable set other than null sets and sets of full measure.
Ergodic maps cannot be split into two separate systems of positive measure.
By Birkhoff's ergodic theorem, for an ergodic map the time average of an integrable function along almost every orbit equals its space average.
This is the mathematical form of the ergodic hypothesis of statistical mechanics.

## Explanations

### L1 human

Stir a container of fluid again and again.
The stirring is ergodic if it mixes everything into one piece: no region of positive volume is trapped and kept apart from the rest.
A rotation of a circle by an irrational fraction of a turn is ergodic, since each orbit fills the circle evenly.
A rotation by half a turn is not ergodic: the union of two opposite quarter arcs is moved onto itself.