Ergodic hypothesis

The Ergodic hypothesis (Ref 1 and 2) essentially states that an ensemble average (i.e. Monte Carlo) 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.

References

1. George D. Birkhoff, "Proof of the Ergodic Theorem", PNAS 17 pp. 656-660 (1931)
2. Adrian Patrascioiu "The Ergodic-Hypothesis, A Complicated Problem in Mathematics and Physics", Los Alamos Science, 15 pp. 263- (1987)