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

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 $\frac{1}{\alpha }\mathbf{D}(\mathbf{x})$ and drift $\mathbf{b}(\mathbf{x})$, where $\alpha$ represents the size parameter'' of a system, we show that there are two distinctly different entropy balance equations. One reads ${\rm d} S^{(α)}/{\rm d} t = e^{(α)}_p + Q^{(α)}_{ex}$ for all $α$. However, the leading $α$-order, extensive’', terms of the entropy production rate $e_p^{(\alpha )}$ and heat exchange rate $Q_{ex}^{(\alpha )}$ are exactly cancelled. Therefore, in the asymptotic limit of $\alpha \to \mathrm{\infty }$, there is a second, local $\mathrm{d}S\mathrm{/}\mathrm{d}t=\mathrm{\nabla }\cdot \mathbf{b}(\mathbf{x}(t))+(\mathbf{D}:{\mathbf{\Sigma }}^{-1})(\mathbf{x}(t))$ on the order of $O(1)$, where $\frac{1}{\alpha }\mathbf{D}(\mathbf{x}(\mathbf{t}))$ represents the randomness generated in the dynamics usually represented by metric entropy, and $\frac{1}{\alpha }\mathbf{\Sigma }(\mathbf{x}(t))$ 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.