Kolmogorov-Sinai Entropy Rate versus Physical Entropy

We elucidate the connection between the Kolmogorov-Sinai entropy rate $\kappa$ and the time evolution of the physical or statistical entropy $S$. For a large family of chaotic conservative dynamical systems including the simplest ones, the evolution of $S\left(t\right)$ for far-from-equilibrium processes includes a stage during which $S$ is a simple linear function of time whose slope is $\kappa$. We present numerical confirmation of this connection for a number of chaotic symplectic maps, ranging from the simplest two-dimensional ones to a four-dimensional and strongly nonlinear map.