Global and Local diffusion in the Standard Map

Abstract

We study the global and the local transport and diffusion in the case of the standard map, by calculating the diffusion exponent $μ$. In the global case we find that the mean diffusion exponent for the whole phase space is either $μ=1$, denoting normal diffusion or $μ=2$ denoting anomalous diffusion (and ballistic motion). The mean diffusion of the whole phase space is normal when no accelerator mode exist and it is anomalous (ballistic) when accelerator mode islands exist even if their area is tiny in the phase space. The local value of the diffusion exponent inside the normal islands of stability is $μ=0$, while inside the accelerator mode islands it is $μ=2$. The local value of the diffusion exponent in the chaotic region outside the islands of stability converges always to the value of 1. The time of convergence can be very long, depending on the distance from the accelerator mode islands and the value of the non linearity parameter $K$. For some values of $K$ the stickiness around the accelerator mode islands is maximum and initial conditions inside the sticky region can be dragged in a ballistic motion for extremely long times of the order of $10^7$ or more but they will finally end up in normal mode diffusion with $μ=1$. We study, in particular, cases with maximum stickiness and cases where normal and accelerator mode islands coexist. We find general analytical solutions of periodic orbits of accelerator type and we give evidence that they are much more numerous than the normal periodic orbits. Thus, we expect that in every small interval $ΔK$ of the non linearity parameter $K$ of the standard map there exist smaller intervals of accelerator mode islands. However, these smaller intervals are in general very small, so that in the majority of the values of $K$ the global diffusion is normal.