超越弱耦合极限的克劳修斯不等式：量子布朗振子的再认识
The Clausius inequality beyond the weak coupling limit: The quantum Brownian oscillator revisited
摘要：
我们考虑一个量子线性振子以任意强度耦合到一个任意温度的水浴中。我们找到了振子密度算符的精确封闭表达式。这种状态是非正则的，但可以被证明是等效的，在一个有效的温度T_{eff}的有效质量和有效的弹性常数的非耦合线性振荡器。我们推导出一个有效的克劳修斯不等式Δ Q_{eff} =< T_{eff} dS，其中Δ Q_{eff}是有效（弱耦合）振子和浴之间交换的热，S代表有效振子的热熵，与耦合振子的冯诺伊曼熵相同。使用这个不等式（对于耦合强度变化的循环过程），我们确认了第二定律的有效性。对于固定的耦合强度，这个不等式也可以用振子质量或其弹性常数的变化来检验。它永远不会被侵犯。因此，适当定义的克劳修斯不等式比以前假设的更鲁棒。
英文摘要：
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.
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