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American Journal of Physics

Pauli, an ergodic theorem and related matters

Peter T. Landsberg

Published Jan 13, 2005• 5 citations
Publisher

Abstract

Time averages are important in physics and in statistical mechanics. An ergodic theorem is a way of justifying the replacement of time averages by averages in phase space. A derivation of this theorem from quantum mechanics was given by von Neumann in 1929. Pauli and Fierz found a shorter argument in 1937. They agreed that for ergodicity to hold there should be no energy degeneracy in the Hamiltonian. I trace the circumstances and consequences of a disproof of these arguments.

DOI
10.1119/1.1811622

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Time averages are important in physics and in statistical mechanics. An ergodic theorem is a way of justifying the replacement of time averages by averages in phase space. A derivation of this theorem from quantum mechanics was given by von Neumann in 1929. Pauli and Fierz found a shorter argument in 1937. They agreed that for ergodicity to hold there should be no energy degeneracy in the Hamiltonian. I trace the circumstances and consequences of a disproof of these arguments.
An ergodic theorem is a statement of the equality of the time averages of the physical properties of a system and the averages of these quantities obtained from the consideration of a Gibbsian ensemble of identical systems. It is shown that, in quantum statistical mechanics, a necessary and sufficient condition for the existence of an ergodic theorem can be derived for a system in weak energetic interaction with its surroundings. This necessary and sufficient condition for the equality of time and (microcanonical) ensemble averages is that all operators invariant in time shall reduce to constant multiples of the unit operator when applied to the system itself. It is shown that the conditions for ergodicity are formally analogous to those derived by von Neumann for classical statistical mechanics. No assumption of "molecular chaos" is made.
Landsberg and Farquhar have shown that the von Neumann-Fierz approach to the quantum ergodic problem does not give any criterion of ergodicity.In this paper it is shown that von Neumann's procedure of averaging over the macro-observers is sufficient by itself to "justify" at every time the use of the quantum microcanonical ensemble. It is then proved that the time behavior of the system is completely irrelevant to the demonstration of the von Neumann and von Neumann-Fierz ergodic theorems. It is concluded that the quantum ergodic problem must be attacked on entirely new lines.
von Neumann has proved a quantum-mechanical ergodic theorem which deals with a closed macroscopic system and macroscopic observables. Certain unsatisfactory features of this theorem are probably due to the overly-strong idealization of a completely closed system.Since the claims of statistical mechanics go beyond macroscopic systems, there is a need for a more general theorem. The present paper deals with a system of any number of degrees of freedom and with any observable, during interaction with a temperature bath. In order to avoid an assumption of randomness, the bath must be described explicitly and precisely by a time-independent Hamiltonian. The model of the bath is obtained by taking the Gibbs ensemble seriously, i.e., as a set of NNN identical systems interacting through a potential $\ensuremath{\lambda}V$.The following theorem is proved. The time average of the quantum-mechanical expectation value of any observable with respect to any initial state is equal to its statistical average, in the double limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\lambda}\ensuremath{\rightarrow}0$, for the overwhelming majority of all interaction potentials VVV.
We discuss the content and significance of John von Neumann’s quantum ergodic theorem (QET) of 1929, a strong result arising from the mere mathematical structure of quantum mechanics. The QET is a precise formulation of what we call normal typicality, i.e., the statement that, for typical large systems, every initial wave function ψ0 from an energy shell is “normal”: it evolves in such a way that |ψt〉〈ψt | is, for most t, macroscopically equivalent to the micro-canonical density matrix. The QET has been mostly forgotten after it was criticized as a dynamically vacuous statement in several papers in the 1950s. However, we point out that this criticism does not apply to the actual QET, a correct statement of which does not appear in these papers, but to a different (indeed weaker) statement. Furthermore, we formulate a stronger statement of normal typicality, based on the observation that the bound on the deviations from the average specified by von Neumann is unnecessarily coarse and a much tighter (and more relevant) bound actually follows from his proof.
We discuss the content and significance of John von Neumann’s quantum ergodic theorem (QET) of 1929, a strong result arising from the mere mathematical structure of quantum mechanics. The QET is a precise formulation of what we call normal typicality, i.e. the statement that, for typical large systems, every initial wave function ψ 0 from an energy shell is ‘normal’: it evolves in such a way that | ψ t 〉〈 ψ t | is, for most t , macroscopically equivalent to the micro-canonical density matrix. The QET has been mostly forgotten after it was criticized as a dynamically vacuous statement in several papers in the 1950s. However, we point out that this criticism does not apply to the actual QET, a correct statement of which does not appear in these papers, but to a different (indeed weaker) statement. Furthermore, we formulate a stronger statement of normal typicality, based on the observation that the bound on the deviations from the average specified by von Neumann is unnecessarily coarse and a much tighter (and more relevant) bound actually follows from his proof.
Abstract In incompletely specified situations the passage from quantum mechanics to statistical mechanics requires an averaging process. Typically, one has to average a scalar product of a fixed unit vector α and a unit vector β of random direction over a probability distribu­tion. A mathematically elementary class of such averaging processes is considered. It is used to define the concept of weak uniformity of the distribution of the end point of β over the unit sphere in a many-dimensional unitary space. In the case of strict uniformity, von Neumann’s method of averaging results. The distributions contain a parameter in terms of which a condition for probable ergodicity can be formulated. It expresses a restriction which bears most explicitly on the averaging process, and less explicitly on the Hamiltonian of the system. One finds that, while strict uniformity and large phase cells are together sufficient for probable ergodicity, each condition by itself can occur when ergodicity is not over­-whelmingly probable. The averaging processes over macro-observers, initial states and Hamiltonians occur as special cases.