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 a deterministic dynamics exhibiting chaotic behaviors. By taking the former approach based on the general diffusion process with diffusion \(\alpha ^{-1}\varvec{D}(\textbf{x})\) and drift \(\textbf{b}(\textbf{x})\), where \(\alpha \) represents the “size parameter” of a system, we show that there are two distinctly different entropy balance equations. One reads \(\textrm{d}S^{(\alpha )}/\textrm{d}t = e^{(\alpha )}_p + Q^{(\alpha )}_{ex}\) for all \(\alpha \). Our key result addresses the asymptotic of the entropy production rate \(e^{(\alpha )}_p\) and heat exchange rate \(Q^{(\alpha )}_{ex}\) up to \(O(\tfrac{1}{\alpha })\)-corrections as system’s size \(\alpha \rightarrow \infty \). It yields in particular that the “extensive”, leading \(\alpha \)-order terms of \(e^{(\alpha )}_p\) and \(Q^{(\alpha )}_{ex}\) are exactly canceled out. Therefore in the asymptotic limit of \(\alpha \rightarrow \infty \), there is a second, local entropy balance equation \(\textrm{d}S/\textrm{d}t=\nabla \cdot \textbf{b}(\textbf{x}(t))+\left( \varvec{D}:\varvec{\varSigma }^{-1}\right) (\textbf{x}(t))\) on the order of O(1), where \(\alpha ^{-1}\varvec{D}(\textbf{x}(t))\) represents the randomness generated in the dynamics usually represented by metric entropy, \(\alpha ^{-1}\varvec{\varSigma }(\textbf{x}(t))\) is the covariance matrix of the local Gaussian description at \(\textbf{x}(t)\) that is a solution to the ordinary differential equation \(\dot{\textbf{x}}=\textbf{b}(\textbf{x})\) at time t, and \(\varvec{D}:\varvec{\varSigma }^{-1}\) is the Frobenius product of \(\varvec{D}\) and \(\varvec{\varSigma }^{-1}\). This latter equation is akin to the notions of volume-preserving conservative dynamics and entropy production in the deterministic dynamic approach to irreversible thermodynamics à la D. Ruelle [55]. Our study follows the rigorous approach and formalism of [28]; the mathematical details with sufficient care are given in the appendices.