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The evolution of entropy is derived with respect to dynamical systems. For astochastic system, its relative entropy $D$ evolves in accordance with thesecond law of thermodynamics; its absolute entropy $H$ may also be so, providedthat the stochastic perturbation is additive and the flow of the vector fieldis nondivergent. For a deterministic system, $dH/dt$ is equal to themathematical expectation of the divergence of the flow (a result obtainedbefore), and, remarkably, $dD/dt = 0$. That is to say, relative entropy isalways conserved. So, for a nonlinear system, though the trajectories of thestate variables, say $\ve x$, may appear chaotic in the phase space, say$\Omega$, those of the density function $\rho(\ve x)$ in the new ``phasespace'' $L^1(\Omega)$ are not; the corresponding Lyapunov exponent is alwayszero. This result is expected to have important implications for the ensemblepredictions in many applied fields, and may help to analyze chaotic data sets.

Remarkable evolutionary laws of absolute and relative entropies with dynamical systems