Time evolution of coarse-grained entropy in classical and quantum motions of strongly chaotic systems

We study relaxation of an ensemble of cat maps with initially localized phase-space distributions. Calculations of the coarse-grained entropy S-epsilon(t) for both classical and quantum motions are presented. It is shown that, within the relaxation period, both classical and quantum entropies increase with a nearly constant rate which can be identified as the largest Lyapunov exponent of the classical cat. After an empirical relaxation time, the time behavior for two entropies becomes different. While the classical entropy increases to the equilibrium entropy S-eqm and stays there, its quantum analogue fluctuates incessantly around a mean (S) over bar(epsilon) which is less than S-eqm. We regard the entropy difference Delta S = S-eqm - (S) over bar(epsilon) as a measure of nonergodicity of the quantum motion of strongly chaotic systems and investigate its dependence on the Planck constant h. For fixed initial phase-space distributions, numerical results suggest that there is a scaling law Delta S proportional to h(beta) with beta approximate to 0.72 in the semiclassical regime. (C) 1997 Published by Elsevier Science B.V.