Ergodic hypothesis

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The Ergodic hypothesis essentially states that an ensemble average (i.e. an instance of a Monte Carlo simulation) of an observable,  \langle O \rangle_\mu is equivalent to the time average, \overline{O}_T of an observable (i.e. molecular dynamics). i.e.

\lim_{T \rightarrow \infty} \overline{O}_T (\{q_0(t)\},\{p_0(t)\}) = \langle O \rangle_\mu.

A restatement of the ergodic hypothesis is to say that all allowed states are equally probable. This holds true if the metrical transitivity of general Hamiltonian systems holds true. Recent experiments have demonstrated the hypothesis [1].

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