Zeroth Law in Quantum Thermodynamics at Strong Coupling: `in Equilibrium', not `Equal Temperature'

Abstract

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 \cite{NEqFE} for Gaussian quantum systems shows that an effective temperature $T^{\ast }$ 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 $T^{\ast }$ is slightly higher than the bath temperature $T_{\textsc{b}}$, with the deviation depending on the coupling. A second object coupled with a different strength with an identical bath at temperature $T_{\textsc{b}}$ will not have the same equilibrated temperature as the first object. Thus $ET\ne EQ$ for strong coupling thermodynamics. We then investigate the conditions for dynamical equilibration for two objects 1 and 2 strongly coupled with a common bath $B$, 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.

arXiv comment22 pages, 2 figures

References in corpus (25)

• Quantum entanglement
• Statistical mechanics and dynamics of solvable models with long-range interactions
• Quantum optics in the phase space - A tutorial on Gaussian states
• {\it Colloquium:} Statistical Mechanics and Thermodynamics at Strong Coupling: Quantum and Classical
• Exact analytical solutions to the master equation of quantum Brownian motion for a general environment
• Backreaction and Unruh effect: New insights from exact solutions of uniformly accelerated detectors
• Temporal and Spatial Dependence of Quantum Entanglement from a Field Theory Perspective
• Gaussian quantum operator representation for bosons
• Quantum thermodynamics from the nonequilibrium dynamics of open systems: energy, heat capacity and the third law
• Hamiltonian of mean force for damped quantum systems
• Nonequilibrium Steady State in Open Quantum Systems: Influence Action, Stochastic Equation and Power Balance
• Gravity and Nonequilibrium Thermodynamics of Classical Matter
• Thermodynamics of quantum systems strongly coupled to a heat bath I. Operator thermodynamic functions and relations
• Influence on electron coherence from quantum electromagnetic fields in the presence of conducting plates
• Distance and Coupling Dependence of Entanglement in the Presence of a Quantum Field
• Fluctuation-Dissipation Relation from the Nonequilibrium Dynamics of a Nonlinear Open Quantum System
• Entanglement, recoherence and information flow in an accelerated detector - quantum field system: Implications for black hole information issue
• The Clausius inequality beyond the weak coupling limit: The quantum Brownian oscillator revisited
• Fluctuation-Dissipation and Correlation-Propagation Relations from the Nonequilibrium Dynamics of Detector-Quantum Field Systems
• Atom-Field Interaction: From Vacuum Fluctuations to Quantum Radiation and Quantum Dissipation / Radiation Reaction
• Nonequilibrium Quantum Free Energy and Effective Temperature, Generating Functional and Influence Action
• Fluctuation-Dissipation and Correlation-Propagation Relations in $(1+3)$D Moving Detector-Quantum Field Systems
• Fluctuation-Dissipation Relation for Open Quantum Systems in Nonequilibrium Steady State
• Strong Coupling and non-Markovian Effects in the Statistical Notion of Temperature
• The zeroth law of thermodynamics is redundant

Cited by in corpus (2)

• Equilibrium Temperature for Black Holes with Nonextensive Entropy
• Strong Coupling Thermodynamics and Stochastic Thermodynamics from the Unifying Perspective of Time-Scale Separation