[论文解读] Stochastic Dynamical Structure (SDS) of Nonequilibrium Processes in the Absence of Detailed Balance. IV: Emerging of Stochastic Dynamical Equalities and Steady State Thermodynamics from Darwinian Dynamics

Ping Ao|ArXiv.org|Jun 1, 2008

Advanced Thermodynamics and Statistical Mechanics参考文献 137被引用 82

一句话总结

本文提出，达尔文动力学——最初用于生物进化——可作为非平衡统计力学与稳态热力学的根本统一框架，即使在不满足详细平衡的情况下亦成立。通过从随机达尔文动力学推导出系综、相对熵形式的第二定律，以及随机动力学等式（包括广义爱因斯坦关系和Jarzynski等式），本文表明热力学定律可自然地从演化型动力学中涌现，为统计力学提供了一种超越保守动力学的、逻辑上一致的新基础。

ABSTRACT

This is the fourth paper, the last one, on solution to the problem of absence of detailed balance in nonequilibrium processes. It is an approach based on another known universal dynamics: 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 enormously 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.

研究动机与目标

• 将达尔文动力学确立为非平衡统计力学与热力学的基础框架。
• 解决长期以来从动力学原理推导热力学第二定律而无需假设详细平衡的难题。
• 证明随机动力学等式（如Jarzynski等式与广义爱因斯坦关系）可自然地从达尔文动力学中导出。
• 将保守动力学与达尔文动力学统一为互补的、逻辑一致的物理与生物系统描述。
• 通过随机过程中可观测的物理量，提供可实验检验的预测。

提出的方法

• 从描述达尔文动力学的随机微分方程出发，正式推导Fokker-Planck方程。
• 利用相对熵动态证明热力学第二定律，无需依赖详细平衡。
• 应用Feynman-Kac公式推导随机动力学等式，包括Jarzynski等式。
• 引入广义爱因斯坦关系，关联非平衡系统中的扩散、漂移与外力。
• 从变异与选择的随机动力学中推导出系综分布及热力学量（温度、能量、熵）。
• 证明破坏详细平衡的动力学分量不贡献于相对熵的变化，从而保持第二定律成立。

实验结果

研究问题

• RQ1 在不假设详细平衡的前提下，热力学定律（如第二定律）能否从随机动力学理论中自然涌现？
• RQ2 如何从达尔文动力学中推导出系综分布与温度？
• RQ3 广义爱因斯坦关系在连接随机动力学与热力学中起什么作用？
• RQ4 能否在不依赖平衡假设的前提下，从达尔文动力学中推导出Jarzynski等式等随机动力学等式？
• RQ5 达尔文动力学是否为统计力学提供一个逻辑上一致且根本的起点，且与保守动力学互补？

主要发现

• 达尔文动力学内在的随机性意味着存在一种有效温度，从而导致Boltzmann-Gibbs系综分布。
• 通过相对熵动态证明了热力学第二定律，且无需详细平衡，该结论适用于任意大小的系统。
• 破坏详细平衡的动力学分量不贡献于相对熵的变化，从而保持第二定律成立。
• 广义爱因斯坦关系被成功推导并证明可实验检验，其中所有物理量在如半导体中电子扩散等系统中均可测量。
• Jarzynski等式及其他随机动力学等式自然地从该形式体系中涌现，为实验提供了直接可访问的途径。
• 达尔文动力学为统计力学与热力学提供了逻辑简洁且一致的基础，与保守动力学互补。