We provide a stochastic mathematical representation for Clausius’ and Kelvin-Planck’s statements of the Second Law of Thermodynamics in terms of the entropy productions of a finite, compact driven Markov system and its lift . A surjective map is rigorously established through the lift when the state space is either a discrete graph or a continuous n -dimensional torus $${\mathbb {T}}^n$$ T n . The corresponding lifted processes have detailed balance thus a natural potential function but no stationary probability. We show that in the long-time limit the entropy production of the finite driven system precisely equals the potential energy decrease in the lifted system. This theorem provides a dynamic foundation for the two equivalent statements of Second Law of Thermodynamics, à la Kelvin’s and Clausius’. It suggests a modernized, combined statement: “A mesoscopic engine that works in completing irreversible internal cycles statistically has necessarily an external effect that lowering a weight accompanied by passing heat from a warmer to a colder body.”

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克劳修斯和开尔文关于第二定律和不可逆性陈述的数学表示

我们根据有限的紧凑驱动马尔可夫系统的熵产生及其升力，为克劳修斯和开尔文-普朗克关于热力学第二定律的陈述提供了随机数学表示。当状态空间是离散图或连续 n 维环面 $${\mathbb {T}}^n$$ T n 时，通过提升严格建立满射图。相应的提升过程有详细的平衡，因此是一个自然的潜在函数，但没有平稳概率。我们表明，在长时间限制下，有限驱动系统的熵产生精确地等于提升系统的势能减少。该定理为热力学第二定律的两个等价陈述提供了动态基础，即开尔文和克劳修斯。它提出了一种现代化的综合声明：