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arXiv:2011.10468 · Physical review. D/Physical review. D.

Nonequilibrium quantum free energy and effective temperature, generating functional, and influence action

Jen-Tsung Hsiang, B. L. Hu

Published Mar 4, 2021• 16 citations
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Abstract

A definition of nonequilibrium free energy ${\mathcal{F}}_{\mathrm{s}}$ is proposed for dynamical Gaussian quantum open systems strongly coupled to a heat bath and the formal relation with the generating functional, the coarse-grained effective action and the influence action is indicated. For Gaussian open quantum systems exemplified by the quantum Brownian motion model studied here, a time-varying effective temperature can be introduced in a natural way, and, with it, the nonequilibrium free energy ${\mathcal{F}}_{\mathrm{s}}$, von Neumann entropy ${\mathcal{S}}_{vN}$ and internal energy ${\mathcal{U}}_{\mathrm{s}}$ of the reduced system ($S$) can be defined accordingly. In contrast to the nonequilibrium free energy found in the literature which references the bath temperature, the nonequilibrium thermodynamic functions we find here obey the familiar relation ${\mathcal{F}}_{\mathrm{s}}(t)={\mathcal{U}}_{\mathrm{s}}(t)\ensuremath{-}{T}_{\mathrm{EFF}}(t){\mathcal{S}}_{vN}(t)$ at any and all moments of time in the system's fully nonequilibrium evolution history. After the system equilibrates they coincide, in the weak coupling limit, with their counterparts in conventional equilibrium thermodynamics. Since the effective temperature captures both the state of the system and its interaction with the bath, upon the system's equilibration, it approaches a value slightly higher than the initial bath temperature. Notably, it remains nonzero for a zero-temperature bath, signaling the existence of system-bath entanglement. Reasonably, at high bath temperatures and under ultraweak couplings, it becomes indistinguishable from the bath temperature. The nonequilibrium thermodynamic functions and relations discovered here for dynamical Gaussian quantum systems should open up useful pathways toward establishing meaningful theories of nonequilibrium quantum thermodynamics.

DOI
10.1103/physrevd.103.065001

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A definition of nonequilibrium free energy Fs{\mathcal{F}}_{\mathrm{s}}Fs​ is proposed for dynamical Gaussian quantum open systems strongly coupled to a heat bath and the formal relation with the generating functional, the coarse-grained effective action and the influence action is indicated. For Gaussian open quantum systems exemplified by the quantum Brownian motion model studied here, a time-varying effective temperature can be introduced in a natural way, and, with it, the nonequilibrium free energy Fs{\mathcal{F}}_{\mathrm{s}}Fs​, von Neumann entropy SvN{\mathcal{S}}_{vN}SvN​ and internal energy Us{\mathcal{U}}_{\mathrm{s}}Us​ of the reduced system (SSS) can be defined accordingly. In contrast to the nonequilibrium free energy found in the literature which references the bath temperature, the nonequilibrium thermodynamic functions we find here obey the familiar relation Fs(t)=Us(t)\ensuremath−TEFF(t)SvN(t){\mathcal{F}}_{\mathrm{s}}(t)={\mathcal{U}}_{\mathrm{s}}(t)\ensuremath{-}{T}_{\mathrm{EFF}}(t){\mathcal{S}}_{vN}(t)Fs​(t)=Us​(t)\ensuremath−TEFF​(t)SvN​(t) at any and all moments of time in the system's fully nonequilibrium evolution history. After the system equilibrates they coincide, in the weak coupling limit, with their counterparts in conventional equilibrium thermodynamics. Since the effective temperature captures both the state of the system and its interaction with the bath, upon the system's equilibration, it approaches a value slightly higher than the initial bath temperature. Notably, it remains nonzero for a zero-temperature bath, signaling the existence of system-bath entanglement. Reasonably, at high bath temperatures and under ultraweak couplings, it becomes indistinguishable from the bath temperature. The nonequilibrium thermodynamic functions and relations discovered here for dynamical Gaussian quantum systems should open up useful pathways toward establishing meaningful theories of nonequilibrium quantum thermodynamics.
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A central open question in quantum thermodynamics is the identification of thermodynamic entropy away from equilibrium. Although thermodynamic entropy coincides with the von Neumann entropy for a Gibbs state at equilibrium, the thermodynamic role of quantum coherence generated during nonequilibrium evolution remains unclear. Here, we show that von Neumann entropy does not, in general, represent the thermodynamic entropy governing heat exchange in coherent nonequilibrium quantum systems. By integrating the resource theory of quantum coherence with thermodynamic principles, we develop a general framework for nonequilibrium quantum thermodynamics. We identify thermodynamic entropy with the energy entropy derived from energy-measurement statistics and introduce a dynamical temperature extending the equilibrium concept to nonequilibrium states. Consequently, we obtain an exact entropy-balance relation separating the entropy flux associated with heat exchange from the coherence-loss contribution to entropy production, establishing an operational connection between quantum coherence, heat flow, and total entropy change. Unlike approaches relying on equilibrium reference states or stationary Gibbs ensembles, our framework uses instantaneous dynamical quantities and remains valid throughout transient nonequilibrium evolution. It provides consistent dynamical definitions of temperature, work, heat, and free energy. Applying the formalism to an exactly solvable open quantum system, we show that the first and second laws hold throughout transient evolution and that equilibrium thermodynamics emerges dynamically in the weak-coupling long-time limit. Our results establish a unified framework for coherent nonequilibrium quantum thermodynamics and suggest opportunities for controlling heat dissipation through coherence engineering in quantum thermal technologies.
In a series of papers, we intend to take the perspective of open quantum systems and examine from their nonequilibrium dynamics the conditions when the physical quantities, their relations, and the laws of thermodynamics become well defined and viable for quantum many-body systems. We first describe how an open-system nonequilibrium dynamics (ONEq) approach is different from the closed combined system + environment in a global thermal state (CGTs) setup. Only after the open system equilibrates will it be amenable to conventional thermodynamics descriptions, thus quantum thermodynamics (QTD) comes at the end rather than assumed in the beginning. The linkage between the two comes from the reduced density matrix of ONEq in that stage having the same form as that of the system in the CGTs. We see the open-system approach having the advantage of dealing with nonequilibrium processes as many experiments in the near future will call for. Because it spells out the conditions of QTD's existence, it can also aid us in addressing the basic issues in quantum thermodynamics from first principles in a systematic way. We then study one broad class of open quantum systems where the full nonequilibrium dynamics can be solved exactly, that of the quantum Brownian motion of N strongly coupled harmonic oscillators, interacting strongly with a scalar-field environment. In this paper, we focus on the internal energy, heat capacity, and the third law. We show for this class of physical models, amongst other findings, the extensive property of the internal energy, the positivity of the heat capacity, and the validity of the third law from the perspective of the behavior of the heat capacity toward zero temperature. These conclusions obtained from exact solutions and quantitative analysis clearly disprove claims of negative specific heat in such systems and dispel allegations that in such systems the validity of the third law of thermodynamics relies on quantum entanglement. They are conceptually and factually unrelated issues. Entropy and entanglement will be the main theme of our second paper on this subject matter.
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.
We build a framework for the thermodynamics of macroscopic quantum systems. In contrast with approaches requiring access to the full density matrix, our framework relies on a coarse-grained description, based on measurement statistics of a few observables. When these observables commute, the outcomes define classical macrostates whose entropy is quantified by observational entropy, accounting for uncertainty about both the macrostate and the microstate within it. We extend this notion to non-commuting observables forming a subalgebra of the operator space, and use Jaynes' principle to define an algebra-dependent entropy interpolating between von Neumann and observational entropies. Given initial and final measurement sets, connected by internal and/or environment-induced dynamics, we derive a second law for the coarse-grained dynamics. Unlike formulations based on von Neumann entropy, our inequality captures irreversibility from both non-unitary environment-induced dynamics and internal equilibration. It takes the usual form of a positive entropy production when the system is initially at internal equilibrium, while correction terms capture nonequilibrium resources ignored by the coarse-graining. We also derive fluctuation theorems for coarse-grained thermodynamic quantities. Along a quasi-static path of measurement schemes, we identify quantum macroscopic notions of work and heat fulfilling the first and second laws, including an additional work contribution from manipulating the algebra to which the system is confined, through external constraints or quantum measurement backaction. Finally, we apply our framework to examples illustrating the impact of varying the coarse-graining scheme. Our approach unifies macroscopic and stochastic thermodynamics in a genuinely quantum framework, laying the basis for a versatile, experimentally friendly toolbox to analyze complex quantum dynamics.
The zeroth law of thermodynamics involves a transitivity relation (pairwise between three objects) expressed either in terms of ``equal temperature'' (ET), or ``in equilibrium'' (EQ) conditions. In conventional thermodynamics conditional on vanishingly weak system-bath coupling these two conditions are commonly regarded as equivalent. In this work we show that for thermodynamics at strong coupling they are inequivalent: namely, two systems can be in equilibrium and yet have different effective temperatures. A recent result [J.-T. Hsiang and B. L. Hu, Phys. Rev. D 103, 065001 (2021) for Gaussian quantum systems shows that an effective temperature T∗{T}^{*}T∗ can be defined at all times during a system's nonequilibrium evolution, but because of the inclusion of interaction energy, after equilibration the system's T∗{T}^{*}T∗ is slightly higher than the bath temperature TB{T}_{\mathrm{B}}TB​, with the deviation depending on the coupling. A second object coupled with a different strength with an identical bath at temperature TB{T}_{\mathrm{B}}TB​ will not have the same equilibrated temperature as the first object. Thus $ET\ensuremath{\ne}EQ$ for strong coupling thermodynamics. We then investigate the conditions for dynamical equilibration for two objects 1 and 2 strongly coupled with a common bath BBB, each with a different equilibrated effective temperature. We show this is possible, and prove the existence of a generalized fluctuation-dissipation relation under this configuration. This affirms that in equilibrium is a valid and perhaps more fundamental notion which the zeroth law for quantum thermodynamics at strong coupling should be based on. Only when the system-bath coupling becomes vanishingly weak that ``temperature'' appearing in thermodynamic relations becomes universally defined and makes better physical sense.