Ergodic hypothesis
The Ergodic hypothesis (Ref 1 and 2) essentially states that an ensemble average (i.e. Monte Carlo) of an observable, 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 \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
- George D. Birkhoff, "Proof of the Ergodic Theorem", PNAS 17 pp. 656-660 (1931)
- Adrian Patrascioiu "The Ergodic-Hypothesis, A Complicated Problem in Mathematics and Physics", Los Alamos Science, 15 pp. 263- (1987)