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| The '''Ergodic hypothesis''' essentially states that an ensemble average (i.e. an instance of a [[Monte Carlo]] simulation) 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.'' | | 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.'' |
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| :<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> |
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| A restatement of the ergodic hypothesis is to say that all allowed states are equally probable. This holds true if | | A restatement of the ergodic hypothesis is to say that all allowed states are equally probable. |
| the ''metrical transitivity'' of general [[Hamiltonian]] systems holds true. Recent experiments have demonstrated the hypothesis <ref>[http://dx.doi.org/10.1002/anie.201105388 Florian Feil, Sergej Naumov, Jens Michaelis, Rustem Valiullin, Dirk Enke, Jörg Kärger, and Christoph Bräuchle "Single-Particle and Ensemble Diffusivities—Test of Ergodicity", Angewandte Chemie International Edition Early View (2011)]</ref>.
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| ==See also== | | ==See also== |
| *[[Fermi-Pasta-Ulam experiment]]
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| *[[Markov chain]] | | *[[Markov chain]] |
| *[[Mixing systems]]
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| ==References== | | ==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)] |
| ;Related reading
| | #[http://www.pnas.org/cgi/reprint/18/1/70 J. V. Neumann "Proof of the Quasi-ergodic Hypothesis", PNAS '''18''' pp. 70-82 (1932)] |
| *[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/18/3/263 J. V. Neumann "Physical Applications of the Ergodic Hypothesis", PNAS '''18''' pp. 263-266 (1932)] |
| *[http://www.pnas.org/cgi/reprint/18/1/70 J. V. Neumann "Proof of the Quasi-ergodic Hypothesis", PNAS '''18''' pp. 70-82 (1932)]
| | #[http://www.pnas.org/cgi/reprint/18/3/279 G. D. Birkhoff and B. O. Koopman "Recent Contributions to the Ergodic Theory", PNAS '''18''' pp. 279-282 (1932)] |
| *[http://www.pnas.org/cgi/reprint/18/3/263 J. V. Neumann "Physical Applications of the Ergodic Hypothesis", PNAS '''18''' pp. 263-266 (1932)]
| | #[http://library.lanl.gov/cgi-bin/getfile?15-18.pdf Adrian Patrascioiu "The Ergodic-Hypothesis, A Complicated Problem in Mathematics and Physics", Los Alamos Science, '''15''' pp. 263- (1987)] |
| *[http://www.pnas.org/cgi/reprint/18/3/279 G. D. Birkhoff and B. O. Koopman "Recent Contributions to the Ergodic Theory", PNAS '''18''' pp. 279-282 (1932)]
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| *Ya. G. Sinai "On the Foundation of the Ergodic Hypothesis for a Dynamical System of Statistical Mechanics", Doklady Akademii Nauk SSSR '''153''' pp. 1261–1264 (1963) (English version: Soviet Math. Doklady '''4''' pp. 1818-1822 (1963))
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| *[http://dx.doi.org/10.1070/RM1970v025n02ABEH003794 Ya G Sinai "Dynamical systems with elastic reflections", Russian Mathematical Surveys '''25''' pp. 137-189 (1970)]
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| *[http://library.lanl.gov/cgi-bin/getfile?15-18.pdf Adrian Patrascioiu "The Ergodic-Hypothesis, A Complicated Problem in Mathematics and Physics", Los Alamos Science, '''15''' pp. 263- (1987)]
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| *[http://dx.doi.org/10.1007/BF00384333 Jan von Plato "Boltzmann's ergodic hypothesis", Archive for History of Exact Sciences '''42''' pp. 71-89 (1991)]
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| * Domokos Ssász "Boltzmann's Ergodic Hypothesis, a Conjecture for Centuries?", Studia Scientiarum Mathematicarum Hungarica '''31''' pp. 299-322 (1996) [http://iml.univ-mrs.fr/~lafont/rencontres/esi098.pdf (reprint)]
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| [[category: Computer simulation techniques]] | | [[category: Computer simulation techniques]] |