Ergodicity

For other uses, see Ergodic (disambiguation).

In mathematics, the term ergodic is used to describe a dynamical system which, broadly speaking, has the same behavior averaged over time as averaged over space. In physics the term is used to imply that a system satisfies the ergodic hypothesis of thermodynamics.

Etymology

The word ergodic is derived from the Greek words έργον and οδός, work and path. This was chosen by Boltzmann while working on a problem in statistical mechanics.

Formal definition

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

• for every $E \in \Sigma$ with $T^{-1}(E)=E\,$ either $\mu(E)=0\,$ or $\mu(E)=1\,$.
• for every $E \in \Sigma$ with $\mu(T^{-1}(E)\bigtriangleup E)=0$ either $\mu(E)=0\,$ or $\mu(E)=1\,$ (where $\bigtriangleup$ denotes the symmetric difference).
• for every $E \in \Sigma$ with positive measure we have $\mu(\cup_{n=1}^\infty T^{-n}E) = 1$.
• for every two sets E and H of positive measure, there exists an n > 0 such that $\mu(T^{-n}E\cap H)>0$.

Measurable flows

These definitions have natural analogues for the case of measurable flows and, more generally, measure-preserving semigroup actions. Let {T^t} be a measurable flow on (X, Σ, μ). An element A of Σ is invariant mod 0 under {T^t} if

$\mu(T^{t}(A)\bigtriangleup A)=0$

for each t ∈ R. Measurable sets invariant mod 0 under a flow or a semigroup action form the invariant subalgebra of Σ, and the corresponding measure-preserving dynamical system is ergodic if the invariant subalgebra is the trivial σ-algebra consisting of the sets of measure 0 and their complements in X.

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.

Ergodic decomposition

Conceptually, ergodicity of a dynamical system is a certain irreducibility property, akin to the notions of irreducible representation in algebra and prime number in arithmetic. A general measure-preserving transformation or flow on a Lebesgue space admits a canonical decomposition into its ergodic components, each of which is ergodic.

See also

• Ergodic theory
• Mixing (mathematics)

References

• Walters, Peter (1982), An Introduction to Ergodic Theory, Springer, ISBN 0387951520 
• Brin, Michael; Garrett, Stuck (2002), Introduction to Dynamical Systems, Cambridge University Press, ISBN 0521808413 

External links

• Outline of Ergodic Theory, by Steven Arthur Kalikow