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

TL;DR

This paper derives and compares mesoscopic and macroscopic entropy balance equations in stochastic and deterministic dynamical systems, revealing how entropy behavior transitions in the limit of large system size.

Contribution

It introduces two distinct entropy balance equations for stochastic dynamics and shows their convergence to a deterministic form as system size increases.

Findings

Two different entropy balance equations are derived for stochastic systems.

In the large system limit, the equations converge to a deterministic entropy rate.

The deterministic limit relates to volume-preserving dynamics and entropy production.

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 $\tfrac{1}{\alpha}{\bf D}(\bf x)$ 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.