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

TL;DR

This paper derives an exact quantum state for a coupled oscillator at arbitrary strength and temperature, introduces an effective Clausius inequality, and confirms the second law's validity beyond weak coupling assumptions.

Contribution

It provides a closed-form solution for the oscillator's density operator at arbitrary coupling and temperature, extending the Clausius inequality beyond weak coupling regimes.

Findings

The effective temperature and parameters can describe the coupled oscillator state.

The effective Clausius inequality holds for cyclic processes and parameter variations.

The second law remains valid beyond weak coupling, confirming the inequality's robustness.

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 non-canonical 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 delta Q_{eff} =< T_{eff} dS, where delta Q_{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.