Kinetic theory of the standard map in the localized weak-stochasticity regime

Abstract

The well-known Chirikov–Taylor standard map is studied using the methods of non-equilibrium statistical mechanics. It appears possible to simplify the master equation whenever the stochasticity parameter is small and the initial density profile is sharply localized in phase space: this defines the ‘localized weak-stochasticity’ (LWS) regime. The resulting equation of evolution of the density profile involves a convolution in time. Its most conspicuous feature is the absence of a ‘short’ intrinsic time scale: the memory time is infinite. The master equation can therefore not be Markovianized as in usual kinetic theory (or as in the diffusive regime of the standard map). Moreover, the initial value of the fluctuations influences the rate of change of the density profile at all times. Quite unexpectedly, the contributions of the two terms of the master equation to the density profile are of the same order of magnitude. As a result, they practically cancel each other. It is this curious feature that explains the suppression of the motion by the islands and the KAM barriers that is well known from the study of individual standard map orbits. The LWS approximation is valid for a limited time (which is very long when the initial localization is very sharp).