标题： 介观与宏观熵平衡方程在随机动力学及其确定性极限中 显示英文标题

标题： Mesoscopic and Macroscopic Entropy Balance Equations in a Stochastic Dynamics and Its Deterministic Limit

摘要： 熵、其产生及其在动力系统中的变化可以从完全随机的动力学描述中理解，也可以从表现出混沌行为的确定性动力学中理解。 通过基于具有扩散$\tfrac{1}{\alpha}{\bf D}(\bf x)$和漂移$\bf b(\bf x)$的一般扩散过程的前一种方法，其中$\alpha$表示系统的“大小参数”，我们表明存在两个截然不同的熵平衡方程。 其中一个为${\rm d} S^{(\alpha)}/{\rm d} t = e^{(\alpha)}_p + Q^{(\alpha)}_{ex}$对所有$\alpha$。 然而，熵产生率 $e^{(\alpha)}_p$ 和热交换率 $Q^{(\alpha)}_{ex}$ 的主导 $\alpha$ 阶，“广泛”的项恰好被抵消。 Therefore, in the asymptotic limit of $\alpha\to\infty$, there is a second, local ${\rm d} S/{\rm d} t = \nabla\cdot{\bf b}({\bf x}(t))+\left({\bf D}:{\bf \Sigma}^{-1}\right)({\bf x}(t))$ on the order of $O(1)$, where $\tfrac{1}{\alpha}{\bf D}(\bf x(t))$ represents the randomness generated in the dynamics usually represented by metric entropy, and $\tfrac{1}{\alpha}{\bf \Sigma}({\bf x}(t))$ is the covariance matrix of the local Gaussian description at ${\bf x}(t)$, which is a solution to the ordinary differential equation $\dot{\bf x}={\bf b}(\bf x)$ at time $t$. 这个后一个方程类似于非平衡热力学确定性动力学方法中的体积保持保守动力学和熵产生概念 {\it 按照} D. Ruelle。 作为[17]的延续，四个附录中给出了足够细致的数学细节。 显示英文摘要

摘要： Entropy, its production, and its change in a dynamical system can be understood from either a fully stochastic dynamic description or from a deterministic dynamics exhibiting chaotic behavior. By taking the former approach based on the general diffusion process with diffusion $\tfrac{1}{\alpha}{\bf D}(\bf x)$ and drift $\bf b(\bf x)$, where $\alpha$ represents the ``size parameter'' of a system, we show that there are two distinctly different entropy balance equations. One reads ${\rm d} S^{(\alpha)}/{\rm d} t = e^{(\alpha)}_p + Q^{(\alpha)}_{ex}$ for all $\alpha$. However, the leading $\alpha$-order, ``extensive'', terms of the entropy production rate $e^{(\alpha)}_p$ and heat exchange rate $Q^{(\alpha)}_{ex}$ are exactly cancelled. Therefore, in the asymptotic limit of $\alpha\to\infty$, there is a second, local ${\rm d} S/{\rm d} t = \nabla\cdot{\bf b}({\bf x}(t))+\left({\bf D}:{\bf \Sigma}^{-1}\right)({\bf x}(t))$ on the order of $O(1)$, where $\tfrac{1}{\alpha}{\bf D}(\bf x(t))$ represents the randomness generated in the dynamics usually represented by metric entropy, and $\tfrac{1}{\alpha}{\bf \Sigma}({\bf x}(t))$ is the covariance matrix of the local Gaussian description at ${\bf x}(t)$, which is a solution to the ordinary differential equation $\dot{\bf x}={\bf b}(\bf x)$ at time $t$. This latter equation is akin to the notions of volume-preserving conservative dynamics and entropy production in the deterministic dynamic approach to nonequilibrium thermodynamics {\it \`{a} la} D. Ruelle. As a continuation of [17], mathematical details with sufficient care are given in four Appendices.