Nonequilibrium Quantum Free Energy and Effective Temperature, Generating Functional and Influence Action
arXiv:2011.10468 · doi:10.1103/PhysRevD.103.065001
Abstract
A definition of nonequilibrium free energy $\mathcal{F}_{\textsc{s}}$ is proposed for dynamical Gaussian quantum open systems strongly coupled to a heat bath and a formal derivation is provided by way of the generating functional in terms of the coarse-grained effective action and the influence action. 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 $\mathcal{F}_{\textsc{s}}$, von Neumann entropy and internal energy $\mathcal{U}_{\textsc{s}}$ of the reduced system () 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 $\mathcal{F}_{\textsc{s}}(t)=\mathcal{U}_{\textsc{s}}(t)- T_{\textsc{eff}} (t)\,\mathcal{S}_{vN}(t)$ {\it 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 ultra-weak 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.
26 pages, 1 figure. No change in v2, added further comments
References in corpus (42)
- A nonequilibrium equality for free energy differences
- The Entropy Production Fluctuation Theorem and the Nonequilibrium Work Relation for Free Energy Differences
- Equilibrium free energy differences from nonequilibrium measurements: a master equation approach
- Open Quantum Systems. An Introduction
- Energy flow, partial equilibration and effective temperatures in systems with slow dynamics
- Fluctuation-Dissipation: Response Theory in Statistical Physics
- Second law and Landauer principle far from equilibrium
- Quantum optics in the phase space - A tutorial on Gaussian states
- The effective temperature
- Nonequilibrium entropy production for open quantum systems
- First and Second Law of Thermodynamics at strong coupling
- {\it Colloquium:} Statistical Mechanics and Thermodynamics at Strong Coupling: Quantum and Classical
- Off equilibrium dynamics and aging in unfrustrated systems
- Quantum Brownian Motion in a Bath of Parametric Oscillators: A model for system-field interactions
- Stochastic Theory of Relativistic Particles Moving in a Quantum Field: II. Scalar Abraham-Lorentz-Dirac-Langevin Equation, Radiation Reaction and Vacuum Fluctuations
- Strong Coupling Thermodynamics of Open Quantum Systems
- Acceleration Radiation for Orbiting Electrons
- Nonequilibrium linear response for Markov dynamics, I: jump processes and overdamped diffusions
- Off-equilibrium generalization of the fluctuation dissipation theorem for Ising spins and measurement of the linear response function
- Stochastic thermodynamics in the strong coupling regime: An unambiguous approach based on coarse-graining
- Thermodynamics of a subensemble of a canonical ensemble
- Open system trajectories specify fluctuating work but not heat
- Nonequilibrium Linear Response for Markov Dynamics, II: Inertial Dynamics
- Gaussian quantum operator representation for bosons
- Near-equilibrium measurements of nonequilibrium free energy
- Unruh and analogue Unruh temperatures for circular motion in 3+1 and 2+1 dimensions
- Equation of state and coarse grained free energy for matrix models
- Quantum thermodynamics from the nonequilibrium dynamics of open systems: energy, heat capacity and the third law
- Accelerated Electrons and the Unruh Effect
- Fluctuation-Dissipation relations far from Equilibrium
- Nonequilibrium Brownian motion beyond the effective temperature
- Hamiltonian of mean force for damped quantum systems
- Measurability of nonequilibrium thermodynamics in terms of the Hamiltonian of mean force
- Thermodynamics of quantum systems strongly coupled to a heat bath I. Operator thermodynamic functions and relations
- Fluctuation-Dissipation Relation from the Nonequilibrium Dynamics of a Nonlinear Open Quantum System
- Quantum free energy differences from non-equilibrium path integrals: II. Convergence properties for the harmonic oscillator
- Quantum free energy differences from non-equilibrium path integrals: I. Methods and numerical application
- Specific heat anomalies of small quantum systems subjected to finite baths
- Fluctuation-Dissipation and Correlation-Propagation Relations from the Nonequilibrium Dynamics of Detector-Quantum Field Systems
- Nonequilibrium Nonlinear Open Quantum Systems I. Functional Perturbative Analysis of a Weakly Anharmonic Oscillator
- Comment on "Measurability of nonequilibrium thermodynamics in terms of the Hamiltonian of mean force"
- Strong Coupling and non-Markovian Effects in the Statistical Notion of Temperature
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- Spectral Bounds on Entropy and Ergotropy via Statistical Effective Temperature in Classical Polarization and Quantum Thermal States
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- Quantum Thermodynamic Uncertainties in Nonequilibrium Systems from Robertson-Schrödinger Relations
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