Ergodic hypothesis

The Ergodic hypothesis essentially states that an ensemble average (i.e. an instance of a Monte Carlo simulation) of an observable, $\langle O\rangle _{\mu }$ is equivalent to the time average, ${\overline {O}}_{T}$ of an observable (i.e. molecular dynamics). i.e.

$\lim _{T\rightarrow \infty }{\overline {O}}_{T}(\{q_{0}(t)\},\{p_{0}(t)\})=\langle O\rangle _{\mu }.$

A restatement of the ergodic hypothesis is to say that all allowed states are equally probable. This holds true if the metrical transitivity of general Hamiltonian systems holds true.

See also

• Fermi-Pasta-Ulam experiment
• Markov chain
• Mixing systems

References

1. George D. Birkhoff, "Proof of the Ergodic Theorem", PNAS 17 pp. 656-660 (1931)
2. J. V. Neumann "Proof of the Quasi-ergodic Hypothesis", PNAS 18 pp. 70-82 (1932)
3. J. V. Neumann "Physical Applications of the Ergodic Hypothesis", PNAS 18 pp. 263-266 (1932)
4. G. D. Birkhoff and B. O. Koopman "Recent Contributions to the Ergodic Theory", PNAS 18 pp. 279-282 (1932)
5. 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))
6. Ya G Sinai "Dynamical systems with elastic reflections", Russian Mathematical Surveys 25 pp. 137-189 (1970)
7. Adrian Patrascioiu "The Ergodic-Hypothesis, A Complicated Problem in Mathematics and Physics", Los Alamos Science, 15 pp. 263- (1987)
8. Jan von Plato "Boltzmann's ergodic hypothesis", Archive for History of Exact Sciences 42 pp. 71-89 (1991)
9. Domokos Ssász "Boltzmann's Ergodic Hypothesis, a Conjecture for Centuries?", Studia Scientiarum Mathematicarum Hungarica 31 pp. 299-322 (1996) (reprint)