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.
Categories: quant-ph cond-mat.stat-mech