# nLab ergodicity

Contents

## Surveys, textbooks and lecture notes

#### Representation theory

representation theory

geometric representation theory

## Geometric representation theory

#### Measure and probability theory

measure theory

probability theory

# Contents

## Idea

Ergodicity is an idea which originated from statistical physics, and which spread to probability theory, representation theory and dynamical systems, forming the basis of the entire mathematical field of ergodic theory.

An ergodic state can be interpreted as a situation where we have a strong form of equilibrium, where all the populated states of the system are connected and accessible to each other.

For example, one can imagine a drop of ink completely dissolved in a glass of water which one is stirring. It is at equilibrium (i.e. the ink is completely dissolved), and each molecule of ink can swap places with any other molecule of ink under stirring (i.e. the glass of water is not divided into mutually inaccessible regions, unlike for example two glasses).

Sometimes the term ergodic system denotes a system which is not yet at equilibrium, but such that if it transitions to equilibrium, it will reach an ergodic state.

## Main definitions

Let $X$ be a measurable space. Let $M$ be a monoid (for example a group) with an action on $X$ via measurable functions $m:X\to X$ or via Markov kernels $k_m:X\to X$.

An invariant probability measure $p$ on $X$ is called ergodic if and only if for each invariant set $A$,

$p(A) \;=\; 0 \qquad or \qquad p(A) \;=\; 1.$

Equivalently, $p$ is a zero-one measure on the invariant sigma-algebra.

A (measure-preserving) dynamical system is called ergodic if its stationary measure is ergodic.

Similarly, a (stationary) stochastic process, for example a stationary Markov chain, is called ergodic if its stationary measure is ergodic.

Sometimes, more generally, a dynamical system or a stochastic process are called ergodic if any of the following equivalent conditions hold:

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## References

category: probability

Last revised on July 18, 2024 at 09:42:27. See the history of this page for a list of all contributions to it.