- Ergodicity
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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.
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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.[1]
Formal definition
Let be a probability space, and 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: [2]
- for every with either or .
- for every with either or (where denotes the symmetric difference).
- for every with positive measure we have .
- for every two sets E and H of positive measure, there exists an n > 0 such that .
Measurable flows
These definitions have natural analogues for the case of measurable flows and, more generally, measure-preserving semigroup actions. Let {Tt} be a measurable flow on (X, Σ, μ). An element A of Σ is invariant mod 0 under {Tt} if
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
Notes
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
Categories:- Ergodic theory
- Probability theory
- Stochastic processes
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