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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{O}_T} of an observable (i.e. molecular dynamics). i.e.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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)