Anomalous diffusion induced by external force in the standard map

The diffusion of chaotic orbits in a widespread chaotic sea is normal with a variance proportional to the time t even when islands of tori exist as long as there are no accelerator-mode islands. If an external driving force is applied, however, the diffusion becomes anomalous with a variance t(zeta), zeta > 1. Indeed, if one uses the coordinate system moving with the mean velocity of chaotic orbits due to the driving force, the chaotic orbits become identical to the Levy flight due to the intermittent sticking to the islands of tori. This is shown numerically using the standard map with an external force, i.e., The Josephson map. The probability distribution function of the coarse-grained velocity is determined explicitly and turns out to obey an anomalous scaling law characterized by the exponent zeta.