Ergodic hypothesis: Difference between revisions
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The Ergodic hypothesis (Ref 1 and 2) essentially states that an ensemble average (MC) of an observable, <math> | The Ergodic hypothesis (Ref 1 and 2) essentially states that an ensemble average (MC) of an observable, <math> \langle O \rangle_\mu</math> is equivalent to the time average, <math>\overline{O}_T</math> of an observable (MD). ''i.e.'' | ||
:<math>\lim_{T \rightarrow \infty} \overline{O}_T (\{q_0(t)\},\{p_0(t)\}) = | :<math>\lim_{T \rightarrow \infty} \overline{O}_T (\{q_0(t)\},\{p_0(t)\}) = \langle O \rangle_\mu.</math> | ||
A restatement of the ergodic hypothesis is to say that all allowed states are equally probable. | A restatement of the ergodic hypothesis is to say that all allowed states are equally probable. | ||
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==References== | ==References== | ||
#[http://www.pnas.org/cgi/reprint/17/12/656 George D. Birkhoff, "Proof of the Ergodic Theorem", PNAS '''17''' pp. 656-660 (1931) ] | #[http://www.pnas.org/cgi/reprint/17/12/656 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) | ||
Revision as of 11:27, 23 February 2007
The Ergodic hypothesis (Ref 1 and 2) essentially states that an ensemble average (MC) of an observable, is equivalent to the time average, of an observable (MD). i.e.
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)