Ergodic hypothesis: Difference between revisions
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The Ergodic hypothesis (Ref 1 and 2) essentially states that an ensemble average ( | The Ergodic hypothesis (Ref 1 and 2) essentially states that an ensemble average (i.e. [[Monte Carlo]]) of an observable, <math> \langle O \rangle_\mu</math> is equivalent to the time average, <math>\overline{O}_T</math> of an observable (i.e. [[molecular dynamics]]). ''i.e.'' | ||
:<math>\lim_{T \rightarrow \infty} \overline{O}_T (\{q_0(t)\},\{p_0(t)\}) = \langle O \rangle_\mu.</math> | :<math>\lim_{T \rightarrow \infty} \overline{O}_T (\{q_0(t)\},\{p_0(t)\}) = \langle O \rangle_\mu.</math> | ||
Revision as of 14:09, 5 June 2007
The Ergodic hypothesis (Ref 1 and 2) essentially states that an ensemble average (i.e. Monte Carlo) of an observable, 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)