# Ergodic Hypothesis

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## Ergodic Hypothesis

in statistical mechanics, the assertion that the average values over time of the physical quantities that characterize a system are equal to the statistical average values of the quantities. The ergodic hypothesis is the basis of statistical mechanics. Physical systems for which the ergodic hypothesis is valid are called ergodic systems.

More precisely, in the classical statistical mechanics of equilibrium systems, the ergodic hypothesis is the assertion that the time averages of functions that depend on the coordinates and momenta of all the particles of a system in a phase space are equal to the statistical averages over a uniform distribution of phase points in a thin (in the limit, an infinitesimally thin) energy layer near a constant-energy surface. The time averages are taken over the trajectory of the system as represented by points in the phase space. A uniform distribution of phase points in a thin energy layer near a constant-energy surface is called a Gibbs microcanonical distribution.

In quantum statistical mechanics, the ergodic hypothesis is the assertion that all states in a thin energy layer are equally probable. Thus, the ergodic hypothesis is equivalent to the assertion that a closed system can be described by a Gibbs microcanonical distribution. This assertion is a fundamental postulate of equilibrium statistical mechanics, since a Gibbs canonical distribution and a Gibbs grand canonical distribution can be obtained on the basis of the microcanonical distribution (seeGIBBS DISTRIBUTION and MICROCANONICAL ENSEMBLE).

In a narrower sense, the ergodic hypothesis is the assertion made by L. Boltzmann in the 1870’s that, in the course of time, the phase trajectory of a closed system passes through every point of a constant-energy surface in phase space. In such form, the ergodic hypothesis is not valid because Hamilton’s canonical equations of motion uniquely define the tangent to a phase trajectory and do not allow the phase trajectory to intersect itself. Therefore, Boltzmann’s ergodic hypothesis was replaced by the quasi-ergodic hypothesis, which asserts that the phase trajectories of a closed system pass arbitrarily close to every point of a constant-energy surface.

Mathematical ergodic theory studies the conditions under which the time averages for dynamic systems are equal to the statistical averages. Ergodic theorems of this type were proved by the American scientists G. Birkhoff and J. von Neumann. According to the ergodic theorem of von Neumann, a system is ergodic when an energy surface cannot be divided into finite regions such that, if the initial phase point is located in one such region, the whole trajectory of the system remains entirely within that region. This property is called metric intransitivity.

To prove that real systems are ergodic is a very complicated and as yet unsolved problem.

### REFERENCES

Uhlenbeck, G., and J. Ford. Lektsii po statisticheskoi mekhanike. Moscow, 1965. Pages 126–30. (Translated from English.)
Khinchin, A. Ia. Matemalicheskie osnovaniia statisticheskoi mekhaniki. Moscow-Leningrad, 1943.
Ter Haar, D. “Osnovaniia statisticheskoi mekhaniki.” Uspekhi fizicheskikh nauk, 1956, vol. 59, issue 4 and vol. 60, issue 1. (Article translated from English.)
Arnold, V. I., and A. Avez. Ergodic Problems of Classical Mechanics. New York, 1968.

D. N. ZUBAREV

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