Clausius inequality beyond the weak-coupling limit: The quantum Brownian oscillator

Abstract

We consider a quantum linear oscillator coupled at an arbitrary strength to a bath at an arbitrary temperature. We find an exact closed expression for the oscillator density operator. This state is noncanonical but can be shown to be equivalent to that of an uncoupled linear oscillator at an effective temperature T*(eff) with an effective mass and an effective spring constant. We derive an effective Clausius inequality deltaQ*(eff)< or =T*(eff)dS , where deltaQ*(eff) is the heat exchanged between the effective (weakly coupled) oscillator and the bath, and S represents a thermal entropy of the effective oscillator, being identical to the von-Neumann entropy of the coupled oscillator. Using this inequality (for a cyclic process in terms of a variation of the coupling strength) we confirm the validity of the second law. For a fixed coupling strength this inequality can also be tested for a process in terms of a variation of either the oscillator mass or its spring constant. Then it is never violated. The properly defined Clausius inequality is thus more robust than assumed previously.

Stochastic and Macroscopic Thermodynamics of Strongly Coupled Systems

“…Others have argued that the validity of the second law is restored by accounting for the heat associated with coupling the system to its reservoir [35,73,74] or by introducing an effective equilibrium temperature [75]. Additional subtleties arise in the evaluation of specific heat for a quantum system strongly coupled to a bath of harmonic oscillators [76][77][78][79].…”

Nonequilibrium thermodynamics in the strong coupling and non-Markovian regime based on a reaction coordinate mapping

“…In particular, if one wishes to address the performance of a steadily working heat engine, the general results derived in [11][12][13][14][15][16][17][18][19][20][21] are not of great help because they either focus on integrated changes of thermodynamic values (e.g., the total heat exchanged in a finite time instead of the rate of heat exchange) and additionally rely on an initially decorrelated system-environment state [11][12][13] and/or coupling only to a single thermal reservoir [11,[13][14][15][16][17];or they remain very formal [18][19][20][21]. Furthermore, model-specific studies are either based on simple or exactly solvable models from the field of quantum transport [22][23][24][25] and quantum Brownian motion [26][27][28][29], or spin-boson models [30][31][32] often in combination with specific transformations applicable only to special Hamiltonians (polaron transformations) [31,[33][34][35][36]; or the investigations are restricted to numerical studies [37].The goal of this paper is to close the gap between the general results, which are often hard to apply in practice, and studies restricted to...…”

Quantum Thermodynamics: A Nonequilibrium Green’s Function Approach

“…However, in case of strong system-reservoir interactions, finding definitions for heat, work, entropy, and entropy production, which satisfy the basic laws of thermodynamics is an open problem. Each proposal has its own limitations [16][17][18][19][20][21][22][23], even at equilibrium [24][25][26][27][28][29][30]. Reversible transformations, for instance, are never explicitly characterized.…”