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 1 D (x) and drift b (x), where represents the ``size parameter'' of a system, we show that there are two distinctly different entropy balance equations. One reads d S^ () / d t = e^ () ₚ + Q^ () ₄ₗ for all. However, the leading -order, ``extensive'', terms of the entropy production rate e^ () ₚ and heat exchange rate Q^ () ₄ₗ are exactly cancelled. Therefore, in the asymptotic limit of, there is a second, local d S/ d t = b (x (t) ) + (D: ^-1) (x (t) ) on the order of O (1), where 1 D (x (t) ) represents the randomness generated in the dynamics usually represented by metric entropy, and 1 (x (t) ) is the covariance matrix of the local Gaussian description at x (t), which is a solution to the ordinary differential equation ẋ= b (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 \`{a la} D. Ruelle. As a continuation of 17, mathematical details with sufficient care are given in four Appendices.