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. Recent experiments have demonstrated the hypothesis .

See also

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

References

Related reading

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