Stochastic approach to diffusion inside the chaotic layer of a resonance

We model chaotic diffusion in a symplectic four-dimensional (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 on the diffusion process in the action I of the FR, obtaining a seminumerical 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 (FP) equation, whereas it presents a nonconstant time shift with respect to the concomitant FP 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.