Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review

The Ergodic Theorem in Quantum Statistical Mechanics

Martin J. Klein

Published Jul 1, 1952• 25 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.87.111

Related papers

Ergodic Theorem for Interacting Systems

H. Ekstein

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 $N$ 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 $V$.

Physical Review•Jul 15, 1957•11 citations

Pauli, an ergodic theorem and related matters

Peter T. Landsberg

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.

American Journal of Physics•Jan 13, 2005•5 citations

Ergodic Theorem in Quantum Mechanics

P. Bocchieri, A. Loinger

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.

Physical Review•Aug 1, 1958•36 citations

A NOTE ON THE ERGODIC THEOREM FOR QUANTIZED SYSTEMS

CHANG TSUNG-SUI, 中国科学院数学研究所

In a paper by Klein, it is pointed out that for the quantum mechanical ergodic theorem (time average = average over states) to be valid, all integrals of motion R must satisfy the condition ∑ρ"(α′ρ"|R|β′ρ")=const δα′β′ (1) where α′, β′, γ, … refer to the assembly under question and α", β",…, ρ" ,… refer to the surrounding with which our assembly is in equilibrium. In this short note, it is pointed out that the argument of Klein is doubtful at one point and a slightly different approach is given. In this approach, it is shown that (1) is actually sufficient, but only after introducing an additional assumption that the different states of the surroundings have equal probabilities, and that in general, a stronger condition such as (α′ρ″|R|β′θ″)=const δα′β′δρ″θ″ (2) is needed. Cases with integrals of motion is also considered. We consider first an isolated system, write the wave function as (α1α2>, where α1 represents a set of commuting integrals of motion and α2 a set of observables not containing integrals of motion and introduce the assumption that for any integral of motion R, the matrix 1α′2Rα1"α2"> representing it is of the form φ(α′1α1")δ(α′2α2"). (3) Under such conditions, we prove that if the initial state of our system is an eigenstate of α1 corresponding to the eigenvalue α10, then the average of an observable F over time is given by ∑α210α2|F|α10α2>/∑α210α2|1|α10α2>, a result which is clearly to be expected. Extension to systems interacting with external surroundings is easily made.

Acta Physica Sinica•Jan 1, 1958•0 citations

An Ergodic Theorem for Quantum Counting Processes

Burkhard Kümmerer, Hans Maassen

For a quantum-mechanical counting process we show ergodicity, under the condition that the underlying open quantum system approaches equilibrium in the time mean. This implies equality of time average and ensemble average for correlation functions of the detection current to all orders and with probability 1.

arXiv:quant-ph/0102134•Feb 27, 2001•2 citations

An ergodic theorem for quantum counting processes

B K mmerer, Hans Maassen

For a quantum-mechanical counting process we show ergodicity, under the condition that the underlying open quantum system approaches equilibrium in the time mean. This implies equality of time average and ensemble average for correlation functions of the detection current to all orders and with probability 1.

Journal of Physics A Mathematical and General•Feb 12, 2003•28 citations
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.
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.
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.
In a paper by Klein, it is pointed out that for the quantum mechanical ergodic theorem (time average = average over states) to be valid, all integrals of motion R must satisfy the condition ∑ρ"(α′ρ"|R|β′ρ")=const δα′β′ (1) where α′, β′, γ, … refer to the assembly under question and α", β",…, ρ" ,… refer to the surrounding with which our assembly is in equilibrium. In this short note, it is pointed out that the argument of Klein is doubtful at one point and a slightly different approach is given. In this approach, it is shown that (1) is actually sufficient, but only after introducing an additional assumption that the different states of the surroundings have equal probabilities, and that in general, a stronger condition such as (α′ρ″|R|β′θ″)=const δα′β′δρ″θ″ (2) is needed. Cases with integrals of motion is also considered. We consider first an isolated system, write the wave function as (α1α2>, where α1 represents a set of commuting integrals of motion and α2 a set of observables not containing integrals of motion and introduce the assumption that for any integral of motion R, the matrix 1α′2Rα1"α2"> representing it is of the form φ(α′1α1")δ(α′2α2"). (3) Under such conditions, we prove that if the initial state of our system is an eigenstate of α1 corresponding to the eigenvalue α10, then the average of an observable F over time is given by ∑α210α2|F|α10α2>/∑α210α2|1|α10α2>, a result which is clearly to be expected. Extension to systems interacting with external surroundings is easily made.