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

Abstract

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 $\frac{1}{\hat{I}}\pm \mathbf{D}(\mathbf{x})$ and drift $\mathbf{b}(\mathbf{x})$, where $\hat{I}\pm$ represents the ``size parameter'' of a system, we show that there are two distinctly different entropy balance equations. One reads $\mathrm{d}{S}^{(\hat{I}\pm )}/\mathrm{d}t=e_p^{(\hat{I}\pm )}+Q_{ex}^{(\hat{I}\pm )}$ for all $\hat{I}\pm$. However, the leading $\hat{I}\pm$-order, ``extensive'', terms of the entropy production rate $e_p^{(\hat{I}\pm )}$ and heat exchange rate $Q_{ex}^{(\hat{I}\pm )}$ are exactly cancelled. Therefore, in the asymptotic limit of $\hat{I}\pm \to \infty$, there is a second, local $\mathrm{d}S/\mathrm{d}t=\nabla \cdot \mathbf{b}(\mathbf{x}(t))+\left(\mathbf{D}:{\hat{\mathbf{I}}\pounds }^{-1}\right)(\mathbf{x}(t))$ on the order of $O(1)$, where $\frac{1}{\hat{I}}\pm \mathbf{D}(\mathbf{x}(\mathbf{t}))$ represents the randomness generated in the dynamics usually represented by metric entropy, and $\frac{1}{\hat{I}}\pm \hat{\mathbf{I}}\pounds (\mathbf{x}(t))$ is the covariance matrix of the local Gaussian description at $\mathbf{x}(t)$, which is a solution to the ordinary differential equation $\dot{\mathbf{x}}=\mathbf{b}(\mathbf{x})$ 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 à la} D. Ruelle. As a continuation of [17], mathematical details with sufficient care are given in four Appendices.

24 pages