Nonequilibrium Quantum Free Energy and Effective Temperature, Generating Functional and Influence Action

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 $\mathcal{S}_{vN}$ and internal energy $\mathcal{U}_{\textsc{s}}$ 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 $\mathcal{F}_{\textsc{s}}(t)=\mathcal{U}_{\textsc{s}}(t)- T_{\textsc{eff}} (t)\,\mathcal{S}_{vN}(t)$ {\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.