Stochastic approach to diffusion inside the chaotic layer of a resonance

Abstract

We model chaotic diffusion, in a symplectic 4D map by using the result of a theorem that was developed for stochastically perturbed integrable Hamiltonian systems. We explicitly consider a map defined by a free rotator (FR) coupled to a standard map (SM). We focus in the diffusion process in the action, $I$, of the FR, obtaining a semi--numerical method to compute the diffusion coefficient. We study two cases corresponding to a thick and a thin chaotic layer in the SM phase space and we discuss a related conjecture stated in the past. In the first case the numerically computed probability density function for the action $I$ is well interpolated by the solution of a Fokker-Planck (F-P) equation, whereas it presents a non--constant time delay respect to the concomitant F-P solution in the second case suggesting the presence of an anomalous diffusion time scale. The explicit calculation of a diffusion coefficient for a 4D symplectic map can be useful to understand the slow diffusion observed in Celestial Mechanics and Accelerator Physics.