Ergodicity

In probability theory, an ergodic dynamical system is one that, broadly speaking, has the same behavior averaged over time as averaged over the space of all the system's states in its phase space. In physics the term implies that a system satisfies the ergodic hypothesis of thermodynamics. A random process is ergodic if its time average is the same as its average over the probability space, known in the field of thermodynamics as its ensemble average. The state of an ergodic process after a long time is nearly independent of its initial state. The term 'ergodic' was derived from the Greek words ἔργον (ergon: 'work') and ὁδός (hodos: 'path', 'way'). It was chosen by Ludwig Boltzmann while he was working on a problem in statistical mechanics. The branch of mathematics that studies ergodic systems is known as ergodic theory. Let ( X , Σ , μ ) {displaystyle (X,;Sigma ,;mu ,)} be a probability space, and T : X → X {displaystyle T:X o X} be a measure-preserving transformation. We say that T is ergodic with respect to μ {displaystyle mu } (or alternatively that μ {displaystyle mu } is ergodic with respect to T) if the following equivalent conditions hold: 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 {displaystyle mathbb {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. A discrete dynamical system ( X , T ) {displaystyle (X,T)} , where X {displaystyle X} is a topological space and T {displaystyle T} a continuous map, is said to be uniquely ergodic if there exists a unique T {displaystyle T} -invariant Borel probability measure on X {displaystyle X} . The invariant measure is then necessary ergodic for T {displaystyle T} (otherwise it could be decomposed as a barycenter of two invariant probability measures with disjoint support). In a Markov chain with a finite state space, a state i {displaystyle i} is said to be ergodic if it is aperiodic and positive-recurrent (a state is recurrent if there is a nonzero probability of exiting the state, and the probability of an eventual return to it is 1; if the former condition is not true, then the state is 'absorbing'). If all states in a Markov chain are ergodic, then the chain is said to be ergodic.