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

Ergodic Theorem for Interacting Systems

H. Ekstein

Published Jul 15, 1957• 11 citations
Publisher

Abstract

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 $V$.

DOI
10.1103/physrev.107.333

Related papers

The Ergodic Theorem in Quantum Statistical Mechanics

Martin J. Klein

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.

Physical Review•Jul 1, 1952•25 citations

Ergodic Foundation of Quantum Statistical Mechanics

P. Bocchieri, A. Loinger

Relations, which are ergodic in a generalized sense, are deduced from the laws of quantum theory without any hypothesis of a priori probabilities. These relations are satisfied for the overwhelming majority of the initial state-vectors of any isolated system with very many degrees of freedom.By means of these relations, it is proved that a system---when weakly coupled with another system having a comparatively very large number of degrees of freedom---does not deviate significantly from the canonical distribution at the overwhelming majority of the time instants.

Physical Review•May 15, 1959•74 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

Quantum statistical ergodic and H -theorems for incompletely specified systems

P. T. Landsberg

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.

Proceedings of the Royal Society of London A Mathematical and Physical Sciences•Jun 13, 1961•5 citations

Nonequilibrium Quantum Systems: Fluctuations and Interactions

Yiğit Subaşı

We explore some aspects of nonequilibrium statistical mechanics of classical and quantum systems. Two chapters are devoted to fluctuation theorems which were originally derived for classical systems. The main challenge in formulating them in quantum mechanics is the fact that fundamental quantities of interest, like work, are defined via the classical concept of a phase space trajectory. We utilize the decoherent histories conceptual framework, in which classical trajectories emerge in quantum mechanics as a result of coarse graining, and provide a first-principles analysis of the nonequilibrium work relation of Jarzynski and Crooks's fluctuation theorem for a quantum system interacting with a general environment based on the quantum Brownian motion (QBM) model. We indicate a parameter range at low temperatures where the theorems might fail in their original form. Fluctuation theorems of Jarzynski and Crooks for systems obeying classical Hamiltonian dynamics are derived under the assumption that the initial conditions are sampled from a canonical ensemble, even though the equilibrium state of an isolated system is typically associated with the microcanonical ensemble. We address this issue through an exact analysis of the classical Brownian motion model. We argue that a stronger form of ensemble equivalence than usually discussed in equilibrium statistical mechanics is required for these theorems to hold in the infinite environment limit irrespective of the ensemble used, and proceed to prove it for this model. An exact expression for the probability distribution of work is obtained for finite environments. Intuitively one expects a system to relax to an equilibrium state when brought into contact with a thermal environment. Yet it is important to have rigorous results that provide conditions for equilibration and characterize the equilibrium state. We consider the dynamics of open quantum systems using the Langevin and master equations and rigorously show that under fairly general conditions quantum systems interacting with a heat bath relax to the equilibrium state defined as the reduced thermal state of the system plus environment, even in the strong coupling regime. Our proof is valid to second-order in interaction strength for general systems and exact for the linear QBM model, for which we also show the equivalence of multi-time correlations. In the final chapter we give a sampling of our investigations into macroscopic quantum phenomena. We work out in detail a specific example of how and under what conditions the center of mass (CoM) coordinate of a macroscopic object emerges as the relevant degree of freedom. Interaction patterns are studied in terms of the couplings they induce between the CoM and relative coordinates of two macroscopic objects. We discuss the implications of these interaction patterns on macroscopic entanglement.

•Jan 1, 2013•0 citations
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 \ensuremathλV\ensuremath{\lambda}V\ensuremathλV.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→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞, \ensuremathλ\ensuremath→0\ensuremath{\lambda}\ensuremath{\rightarrow}0\ensuremathλ\ensuremath→0, for the overwhelming majority of all interaction potentials VVV.
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.
Relations, which are ergodic in a generalized sense, are deduced from the laws of quantum theory without any hypothesis of a priori probabilities. These relations are satisfied for the overwhelming majority of the initial state-vectors of any isolated system with very many degrees of freedom.By means of these relations, it is proved that a system---when weakly coupled with another system having a comparatively very large number of degrees of freedom---does not deviate significantly from the canonical distribution at the overwhelming majority of the time instants.
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.
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.
We explore some aspects of nonequilibrium statistical mechanics of classical and quantum systems. Two chapters are devoted to fluctuation theorems which were originally derived for classical systems. The main challenge in formulating them in quantum mechanics is the fact that fundamental quantities of interest, like work, are defined via the classical concept of a phase space trajectory. We utilize the decoherent histories conceptual framework, in which classical trajectories emerge in quantum mechanics as a result of coarse graining, and provide a first-principles analysis of the nonequilibrium work relation of Jarzynski and Crooks's fluctuation theorem for a quantum system interacting with a general environment based on the quantum Brownian motion (QBM) model. We indicate a parameter range at low temperatures where the theorems might fail in their original form. Fluctuation theorems of Jarzynski and Crooks for systems obeying classical Hamiltonian dynamics are derived under the assumption that the initial conditions are sampled from a canonical ensemble, even though the equilibrium state of an isolated system is typically associated with the microcanonical ensemble. We address this issue through an exact analysis of the classical Brownian motion model. We argue that a stronger form of ensemble equivalence than usually discussed in equilibrium statistical mechanics is required for these theorems to hold in the infinite environment limit irrespective of the ensemble used, and proceed to prove it for this model. An exact expression for the probability distribution of work is obtained for finite environments. Intuitively one expects a system to relax to an equilibrium state when brought into contact with a thermal environment. Yet it is important to have rigorous results that provide conditions for equilibration and characterize the equilibrium state. We consider the dynamics of open quantum systems using the Langevin and master equations and rigorously show that under fairly general conditions quantum systems interacting with a heat bath relax to the equilibrium state defined as the reduced thermal state of the system plus environment, even in the strong coupling regime. Our proof is valid to second-order in interaction strength for general systems and exact for the linear QBM model, for which we also show the equivalence of multi-time correlations. In the final chapter we give a sampling of our investigations into macroscopic quantum phenomena. We work out in detail a specific example of how and under what conditions the center of mass (CoM) coordinate of a macroscopic object emerges as the relevant degree of freedom. Interaction patterns are studied in terms of the couplings they induce between the CoM and relative coordinates of two macroscopic objects. We discuss the implications of these interaction patterns on macroscopic entanglement.