The evolutionary dynamics first conceived by Darwin and Wallace, referring to as Darwinian dynamics in the present paper, has been found to be universally valid in biology. The statistical mechanics and thermodynamics, while enormous successful in physics, have been in an awkward situation of wanting a consistent dynamical understanding. Here we present from a formal point of view an exploration of the connection between thermodynamics and Darwinian dynamics and a few related topics. We first show that the stochasticity in Darwinian dynamics implies the existence temperature, hence the canonical distribution of Boltzmann-Gibbs type. In term of relative entropy the Second Law of thermodynamics is dynamically demonstrated without detailed balance condition, and is valid regardless of size of the system. In particular, the dynamical component responsible for breaking detailed balance condition does not contribute to the change of the relative entropy. Two types of stochastic dynamical equalities of current interest are explicitly discussed in the present approach: One is based on Feynman-Kac formula and another is a generalization of Einstein relation. Both are directly accessible to experimental tests. Our demonstration indicates that Darwinian dynamics represents logically a simple and straightforward starting point for statistical mechanics and thermodynamics and is complementary to and consistent with conservative dynamics that dominates the physical sciences. Present exploration suggests the existence of a unified stochastic dynamical framework both near and far from equilibrium.

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从达尔文动力学中出现随机动力学方程和稳态热力学

达尔文和华莱士首先构想的进化动力学，在本文中称为达尔文动力学，已被发现在生物学中普遍适用。统计力学和热力学虽然在物理学中取得了巨大的成功，但一直处于需要一致的动力学理解的尴尬境地。在这里，我们从正式的角度介绍了对热力学和达尔文动力学之间联系的探索以及一些相关主题。我们首先证明达尔文动力学中的随机性意味着存在温度，因此是 Boltzmann-Gibbs 型的正则分布。就相对熵而言，热力学第二定律是在没有详细平衡条件的情况下动态证明的，并且无论系统大小如何都有效。特别是，负责打破详细平衡条件的动力分量对相对熵的变化没有贡献。本方法明确讨论了当前感兴趣的两种随机动态等式：一种基于 Feynman-Kac 公式，另一种是爱因斯坦关系的推广。两者都可以直接进行实验测试。我们的论证表明，达尔文动力学在逻辑上代表了统计力学和热力学的一个简单而直接的起点，并且与主导物理科学的保守动力学相辅相成并与之一致。目前的探索表明存在一个统一的随机动力学框架，无论是接近平衡还是远离平衡。