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Ergodicity

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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 is used to imply that a system satisfies the ergodic hypothesis of thermodynamics.

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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. In an ergodic process, the state of the process after a long time is nearly independent of its initial state.

The term "ergodic" was derived from the Greek words έργον (ergon: "work") and οδός (odos: "path" or "way"). It was chosen by Ludwig Boltzmann while he was working on a problem in statistical mechanics.

Formal definition

Let ( X , Σ , μ ) be a probability space, and T : X → 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 statements is true:

  • for every E ∈ Σ with T − 1 ( E ) = E either μ ( E ) = 0 or μ ( E ) = 1 .
  • for every E ∈ Σ with μ ( T − 1 ( E ) △ E ) = 0 we have μ ( E ) = 0 or μ ( E ) = 1 (where △ denotes the symmetric difference).
  • for every E ∈ Σ with positive measure we have μ ( ⋃ n = 1 ∞ T − n E ) = 1 .
  • for every two sets E and H of positive measure, there exists an n > 0 such that μ ( ( T − n E ) ∩ H ) > 0 .
  • Every measurable function f : X → R with f ∘ T = f is almost surely constant.

    • 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

    μ ( T t ( A ) △ 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.

    Unique ergodicity

    A discrete dynamical system ( X , T ) , where X is a topological space and T a continuous map, is said to be uniquely ergodic if there exists a unique f -invariant Borel probability measure on X . The invariant measure is then necessary ergodic for T (otherwise it could be decomposed as a barycenter of two invariant probability measures with disjoint support).

    Markov chains

    In a Markov chain, a state 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 the state is "absorbing"). If all states in a Markov chain are ergodic, then the chain is said to be ergodic.

    Markov's theorem: a Markov chain is ergodic if there is a positive probability to pass from any state to any other state in one step.

    There is a simple test for ergodicity using eigenvalues of the transition matrix (which are always less or equal to one in absolute value). One is always an eigenvalue. If all other eigenvalues are less than one in absolute value then the chain is ergodic. This follows from the spectral decomposition of a non-symmetrical matrix.

    Examples in electronics

    Ergodicity means the ensemble average equals the time average. Each resistor has thermal noise associated with it and it depends on the temperature. Take N resistors (N should be very large) and plot the voltage across those resistors for a long period. For each resistor you will have a waveform. Calculate the average value of that waveform. This gives you the time average. You should also note that you have N waveforms as we have N resistors. These N plots are known as an ensemble. Now take a particular instant of time in all those plots and find the average value of the voltage. That gives you the ensemble average for each plot. If both ensemble average and time average are the same then it is ergodic.

    Ergodic decomposition

    Conceptually, ergodicity of a dynamical system is a certain irreducibility property, akin to the notions of irreducibility in the theory of Markov chains, 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.

    References

    Ergodicity Wikipedia
    (Text) CC BY-SA


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