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

TL;DR

This paper demonstrates that at strong coupling in quantum thermodynamics, the traditional notion of equal temperature in equilibrium does not hold, proposing 'in equilibrium' as a more fundamental concept.

Contribution

It establishes that strong coupling leads to inequivalence of 'equal temperature' and 'in equilibrium', and introduces a generalized fluctuation-dissipation relation for such systems.

Findings

Effective temperature can differ from bath temperature at strong coupling.

Two systems can be in equilibrium but have different effective temperatures.

A generalized fluctuation-dissipation relation is valid at strong coupling.

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 $T^{*}$ 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 $T^*$ 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 \neq 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.