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/α D( x) and drift b( x), where α represents the ``size parameter'' of a system, we show that there are two distinctly different entropy balance equations. One reads d S(α)/ d t = e(α)ₚ + Q(α)ₑₓ for all α. However, the leading α-order, ``extensive'', terms of the entropy production rate e(α)ₚ and heat exchange rate Q(α)ₑₓ are exactly cancelled. Therefore, in the asymptotic limit of α→∞, there is a second, local d S/ d t = ∇· b( x(t))+( D: Σ⁻¹)( x(t)) on the order of $O(1)$, where 1/α D( x(t)) represents the randomness generated in the dynamics usually represented by metric entropy, and 1/α Σ( 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.