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

arXiv:1403.3542 · Physical Review E

Fluctuation theorems and the generalized Gibbs ensemble in integrable systems

James M. Hickey, Sam Genway

Published Aug 8, 2014• 14 citations
arXivHTMLPDF

Abstract

We derive fluctuation relations for a many-body quantum system prepared in a generalized Gibbs ensemble subject to a general nonequilibrium protocol. By considering isolated integrable systems, we find generalizations to the Tasaki-Crooks and Jarzynski relations. Our approach is illustrated by studying the one-dimensional quantum Ising model subject to a sudden change in the transverse field, where we find that the statistics of the work done and irreversible entropy show signatures of quantum criticality. We discuss these fluctuation relations in the context of thermalization.

DOI
10.1103/physreve.90.022107

Related papers

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

Generalized Gibbs ensemble in integrable lattice models

Lev Vidmar, Marcos Rigol

Abstract The generalized Gibbs ensemble (GGE) was introduced ten years ago to describe observables in isolated integrable quantum systems after equilibration. Since then, the GGE has been demonstrated to be a powerful tool to predict the outcome of the relaxation dynamics of few-body observables in a variety of integrable models, a process we call generalized thermalization. This review discusses several fundamental aspects of the GGE and generalized thermalization in integrable systems. In particular, we focus on questions such as: which observables equilibrate to the GGE predictions and who should play the role of the bath; what conserved quantities can be used to construct the GGE; what are the differences between generalized thermalization in noninteracting systems and in interacting systems mappable to noninteracting ones; why is it that the GGE works when traditional ensembles of statistical mechanics fail. Despite a lot of interest in these questions in recent years, no definite answers have been given. We review results for the XX model and for the transverse field Ising model. For the latter model, we also report original results and show that the GGE describes spin–spin correlations over the entire system. This makes apparent that there is no need to trace out a part of the system in real space for equilibration to occur and for the GGE to apply. In the past, a spectral decomposition of the weights of various statistical ensembles revealed that generalized eigenstate thermalization occurs in the XX model (hard-core bosons). Namely, eigenstates of the Hamiltonian with similar distributions of conserved quantities have similar expectation values of few-spin observables. Here we show that generalized eigenstate thermalization also occurs in the transverse field Ising model.

arXiv:1604.03990•Journal of Statistical Mechanics Theory and Experiment•Jun 1, 2016•735 citations

Fluctuation-dissipation relations and critical quenches in the transverse field Ising chain

Laura Foini, Leticia F. Cugliandolo, Andrea Gambassi

Dynamic correlation and response functions of classical and quantum systems in thermal equilibrium are connected by fluctuation-dissipation theorems, which allow an alternative definition of their (unique) temperature. Motivated by this fundamental property, we revisit the issue of thermalization of closed many-body quantum systems long after a sudden quench, focusing on the nonequilibrium dynamics of the Ising chain in a critical transverse field. We show the emergence of distinct observable-dependent effective temperatures, which rule out Gibbs thermalization in a strict sense but might still have a thermodynamic meaning.

arXiv:1107.5956•Physical Review B•Dec 15, 2011•107 citations

Generalised Quantum Fluctuation Relations

Jordi Mur-Petit, Armando Relaño, Rafael A. Molina +1 more

The non-equilibrium dynamics of quantum many-body systems is one of the most fascinating problems in physics. Open fundamental and practical questions range from how they relax to equilibrium, to how to extract useful work from them. Here, we derive a set of exact results that relate out-of-equilibrium fluctuations in the energy and other observables of a quantum system to its equilibrium properties for a very general family of initial conditions. These quantum fluctuation relations generalise the Jarzynski and Crooks relations to quantum systems with conserved quantities, and can be applied to protocols driving the system between integrable and chaotic regimes, or coupling it to different reservoirs. We illustrate our results with simulations of an integrable model subject to quenches realisable with current technology. Our findings will help guiding research on the interplay of quantum and thermal fluctuations in quantum simulation, and their exploitation in the design of new quantum devices.

•Nov 2, 2017•0 citations

Relations between dissipated work and Rényi divergences in the generalized Gibbs ensemble

Bo-Bo Wei

In this work, we show that the dissipation in a many-body system under an arbitrary nonequilibrium process is related to the R\'enyi divergences between two states along the forward and reversed dynamics under a very general family of initial conditions. This relation generalizes the links between dissipated work and R\'enyi divergences to quantum systems with conserved quantities whose equilibrium state is described by the generalized Gibbs ensemble. The relation is applicable for quantum systems with conserved quantities and can be applied to protocols driving the system between integrable and chaotic regimes. We demonstrate our ideas by considering the one-dimensional transverse quantum Ising model and the Jaynes-Cummings model which are driven out of equilibrium.

arXiv:1711.03855•Physical Review A•Apr 30, 2018•6 citations

Fluctuation relations for heat exchange in the generalized Gibbs ensemble

Bo-Bo Wei

In this work, we investigate the heat exchange between two quantum systems whose initial equilibrium states are described by the generalized Gibbs ensemble. First, we generalize the fluctuation relations for heat exchange discovered by Jarzynski and Wójcik to quantum systems prepared in the equilibrium states described by the generalized Gibbs ensemble at various generalized temperatures. Secondly, we extend the connections between heat exchange and the Rényi divergences to quantum systems under generic initial conditions. These relations are applicable for quantum systems with conserved quantities and universally valid for quantum systems in the integrable and chaotic regimes.

arXiv:1711.07139•Frontiers of Physics•Sep 12, 2018•0 citations
We derive fluctuation relations for a many-body quantum system prepared in a generalized Gibbs ensemble subject to a general nonequilibrium protocol. By considering isolated integrable systems, we find generalizations to the Tasaki-Crooks and Jarzynski relations. Our approach is illustrated by studying the one-dimensional quantum Ising model subject to a sudden change in the transverse field, where we find that the statistics of the work done and irreversible entropy show signatures of quantum criticality. We discuss these fluctuation relations in the context of thermalization.
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.
Abstract The generalized Gibbs ensemble (GGE) was introduced ten years ago to describe observables in isolated integrable quantum systems after equilibration. Since then, the GGE has been demonstrated to be a powerful tool to predict the outcome of the relaxation dynamics of few-body observables in a variety of integrable models, a process we call generalized thermalization. This review discusses several fundamental aspects of the GGE and generalized thermalization in integrable systems. In particular, we focus on questions such as: which observables equilibrate to the GGE predictions and who should play the role of the bath; what conserved quantities can be used to construct the GGE; what are the differences between generalized thermalization in noninteracting systems and in interacting systems mappable to noninteracting ones; why is it that the GGE works when traditional ensembles of statistical mechanics fail. Despite a lot of interest in these questions in recent years, no definite answers have been given. We review results for the XX model and for the transverse field Ising model. For the latter model, we also report original results and show that the GGE describes spin–spin correlations over the entire system. This makes apparent that there is no need to trace out a part of the system in real space for equilibration to occur and for the GGE to apply. In the past, a spectral decomposition of the weights of various statistical ensembles revealed that generalized eigenstate thermalization occurs in the XX model (hard-core bosons). Namely, eigenstates of the Hamiltonian with similar distributions of conserved quantities have similar expectation values of few-spin observables. Here we show that generalized eigenstate thermalization also occurs in the transverse field Ising model.
Dynamic correlation and response functions of classical and quantum systems in thermal equilibrium are connected by fluctuation-dissipation theorems, which allow an alternative definition of their (unique) temperature. Motivated by this fundamental property, we revisit the issue of thermalization of closed many-body quantum systems long after a sudden quench, focusing on the nonequilibrium dynamics of the Ising chain in a critical transverse field. We show the emergence of distinct observable-dependent effective temperatures, which rule out Gibbs thermalization in a strict sense but might still have a thermodynamic meaning.
The non-equilibrium dynamics of quantum many-body systems is one of the most fascinating problems in physics. Open fundamental and practical questions range from how they relax to equilibrium, to how to extract useful work from them. Here, we derive a set of exact results that relate out-of-equilibrium fluctuations in the energy and other observables of a quantum system to its equilibrium properties for a very general family of initial conditions. These quantum fluctuation relations generalise the Jarzynski and Crooks relations to quantum systems with conserved quantities, and can be applied to protocols driving the system between integrable and chaotic regimes, or coupling it to different reservoirs. We illustrate our results with simulations of an integrable model subject to quenches realisable with current technology. Our findings will help guiding research on the interplay of quantum and thermal fluctuations in quantum simulation, and their exploitation in the design of new quantum devices.
In this work, we show that the dissipation in a many-body system under an arbitrary nonequilibrium process is related to the R\'enyi divergences between two states along the forward and reversed dynamics under a very general family of initial conditions. This relation generalizes the links between dissipated work and R\'enyi divergences to quantum systems with conserved quantities whose equilibrium state is described by the generalized Gibbs ensemble. The relation is applicable for quantum systems with conserved quantities and can be applied to protocols driving the system between integrable and chaotic regimes. We demonstrate our ideas by considering the one-dimensional transverse quantum Ising model and the Jaynes-Cummings model which are driven out of equilibrium.
In this work, we investigate the heat exchange between two quantum systems whose initial equilibrium states are described by the generalized Gibbs ensemble. First, we generalize the fluctuation relations for heat exchange discovered by Jarzynski and Wójcik to quantum systems prepared in the equilibrium states described by the generalized Gibbs ensemble at various generalized temperatures. Secondly, we extend the connections between heat exchange and the Rényi divergences to quantum systems under generic initial conditions. These relations are applicable for quantum systems with conserved quantities and universally valid for quantum systems in the integrable and chaotic regimes.