Compared to other formulations of the second law of thermodynamics, the Clausius statement that heat does not spontaneously flow from cold to hot concerns a system in nonequilibrium states, and in that respect is more ambitious but also more ambiguous. We discuss two scenarios when the Clausius statement in its plain form does not hold. First, for ergodic systems, the energy transfer may be consistent with the statement on a coarse-grained timescale, but be anomalously directed during time intervals shorter than the thermalization time. In particular, when an initially colder system is brought in contact to a hotter bath, the internal energy of the former increases with time in a long run but not monotonically. Second, the heat transfer may not respect the Clausius statement on any timescale in nonergodic systems due to the formation of localized vibrational modes. We illustrate the two scenarios with a familiar model of an isotope atom attached to a semi-infinite harmonic atomic chain. Technically, the discussion is based on a Langevin equation for the isotope, using the initial condition when the isotope and chain are initially prepared in uncorrelated canonical states under the constraint that the boundary atom between the isotope and chain is initially fixed and later released. In such setting, the noise in the Langevin equation is nonstationary, and the fluctuation-dissipation relation has a nonstandard form.

中文翻译：

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非克劳修斯传热：非平稳朗文方程的方法

与热力学第二定律的其他表述相比，克劳修斯（Clausius）的说法是，热量不会自冷自发地流向热，这关系到系统处于非平衡状态，在这方面更雄心勃勃，但也更加模棱两可。我们讨论了两种情况，当它们的普通形式的Clausius语句不成立时。首先，对于遍历系统，能量传递可能与粗粒度时间尺度上的陈述一致，但在比热化时间短的时间间隔内会异常定向。特别地，当使最初较冷的系统接触较热的浴时，从长远来看，前者的内部能量随时间增加，但不是单调的。第二，在非遍历系统中，由于局部振动模式的形成，热传递在任何时间尺度上都可能不遵守Clausius陈述。我们用与半无限谐波原子链相连的同位素原子的熟悉模型来说明这两种情况。从技术上讲，该讨论基于同位素的Langevin方程，使用初始条件，即当同位素和链之间的边界原子最初是固定的，然后释放时，最初以不相关的规范状态制备同位素和链。在这种情况下，Langevin方程中的噪声是非平稳的，并且波动-耗散关系具有非标准形式。从技术上讲，该讨论基于同位素的Langevin方程，使用初始条件，即当同位素和链之间的边界原子最初是固定的，然后释放时，最初以不相关的规范状态制备同位素和链。在这种情况下，Langevin方程中的噪声是非平稳的，并且波动-耗散关系具有非标准形式。从技术上讲，该讨论基于同位素的Langevin方程，使用初始条件，即当同位素和链之间的边界原子最初是固定的，然后释放时，最初以不相关的规范状态制备同位素和链。在这种情况下，Langevin方程中的噪声是非平稳的，并且波动-耗散关系具有非标准形式。