  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.
