Anomalous Diffusion Due to Accelerator Modes in the Standard Map

Abstract

Diffusion of chaotic orbits occurs in the widespread chaotic region of the standard map. Its conventional description by the diffusion constant breaks down if accelerator mode islands exist. We shall explore a new description of this anomalous diffusion due to the accelerator mode islands numerically and theoretically by use of Feller's theorem of recurrent events, and find that the probability distribution function P(v ; n) of the coarse-grained velocity vn ≡(Jn - J0)/n for action Jt at time t = 0, 1, 2, obeys a scaling law P(v ; n) = nδ\hatp(nδv) for large n with 1/2 > δ> 0, which has an inverse-power asymptotic form \hatp(x)∝|x|-(1+β) for |x| →∞ with 2 > β= 1/(1 - δ) > 1.