nLab ergodicity

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:

• There is a unique invariant probability measure;
• There is a unique ergodic measure;
• All invariant measures are ergodic, and there exists at least one.

Equivalent descriptions

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Examples

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In terms of category theory

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Properties

• The ergodic decomposition theorem says that, in some cases, every invariant probability measure is a convex mixture of ergodic ones.
• The ergodic theorem? says that, in some cases, empirical means of random variables on an ergodic system tend to expectations (and on more general stationary systems, they tend to conditional expectations w.r.t. the invariant sigma-algebra).

Related entries

• probability theory, categorical probability
• invariant measure, invariant set
• ergodic decomposition theorem, ergodic theorem?
• de Finetti's theorem, zero-one law
• Markov process?, Markov kernel, Markov category
• shifts, Bernoulli process

References

• Wikipedia, Ergodic process

• Terence Tao, What’s new? Lecture 9: Ergodicity, blog entry.

• Sean Moss, Paolo Perrone, A category-theoretic proof of the ergodic decomposition theorem, Ergodic Theory and Dynamical Systems, 2023. (arXiv:2207.07353)

• Noé Ensarguet, Paolo Perrone, Categorical probability spaces, ergodic decompositions, and transitions to equilibrium, arXiv:2310.04267

category: probability