非平衡态热力学第一定律的对称唯一表述、热与功的统计定义、绝热定理和克劳修斯不等式的命运：一个微观的观点
Nonequilibrium Thermodynamics. Symmetric and Unique Formulation of the First Law, Statistical Definition of Heat and Work, Adiabatic Theorem and the Fate of the Clausius Inequality: A Microscopic View
摘要：
热和功在非平衡态热力学中的地位是相当混乱和不唯一的，即使在第一定律在交换热和功方面有很长的历史之后，目前仍有相互矛盾的解释，并且远未解决。此外，交换量缺乏一定的对称性。通过将传统的概念推广到包括它们的时间相关的不可逆分量，允许我们以对称形式dE（t）= dQ（t）-dW（t）来表示第一定律，其中dQ（t）和功dW（t）出现在平等的基础上并且具有对称性。我们证明了不可逆功转化为不可逆热。根据微观状态概率p_{i}（t）进行的统计分析唯一地将dW（t）确定为等熵的，而将dQ（t）确定为dE（t）的等距变化（见正文）;对于交换量，不会出现这种明显的分离。因此，我们的第一定律的新公式提供了巨大的优势，并导致了一个非常有用的非平衡态热力学公式，正如我们最近所展示的那样。我们证明了绝热过程不会改变p_{i}。无论系统偏离平衡多远，所有这些结果都是有效的。当系统处于内部平衡时，dQ（t）\equivT（t）dS（t）表示系统的瞬时温度T（t），这让人联想到平衡。我们证明了p_{i}（t）具有与平衡态非常不同的形式。第一定律和第二定律不再是独立的，所以我们只需要一个定律，这又让人想起平衡。传统的公式如克劳修斯不等式{ \oint } d_{e} Q（t）/ T_{0} <0等变为等式{ \oint }dQ（t）/T（t）\equiv 0等，这是一个从不可逆性角度看非常显著但出乎意料的结果。我们确定在两个简单的情况下，我们的方法的有用性的不可逆成分，在这里，传统的配方是没有用的。
英文摘要：
The status of heat and work in nonequilibrium thermodynamics is quite confusing and non-unique at present with conflicting interpretations even after a long history of the first law in terms of exchange heat and work, and is far from settled. Moreover, the exchange quantities lack certain symmetry. By generalizing the traditional concept to also include their time-dependent irreversible components allows us to express the first law in a symmetric form dE(t)= dQ(t)-dW(t) in which dQ(t) and work dW(t) appear on an equal footing and possess the symmetry. We prove that irreversible work turns into irreversible heat. Statistical analysis in terms of microstate probabilities p_{i}(t) uniquely identifies dW(t) as isentropic and dQ(t) as isometric (see text) change in dE(t); such a clear separation does not occur for exchange quantities. Hence, our new formulation of the first law provides tremendous advantages and results in an extremely useful formulation of non-equilibrium thermodynamics, as we have shown recently. We prove that an adiabatic process does not alter p_{i}. All these results remain valid no matter how far the system is out of equilibrium. When the system is in internal equilibrium, dQ(t)\equivT(t)dS(t) in terms of the instantaneous temperature T(t) of the system, which is reminiscent of equilibrium. We demonstrate that p_{i}(t) has a form very different from that in equilibrium. The first and second laws are no longer independent so that we need only one law, which is again reminiscent of equilibrium. The traditional formulas like the Clausius inequality {\oint}d_{e}Q(t)/T_{0}<0, etc. become equalities {\oint}dQ(t)/T(t)\equiv0, etc, a quite remarkable but unexpected result in view of irreversibility. We determine the irreversible components in two simple cases to show the usefulness of our approach; here, the traditional formulation is of no use.
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