Anomalous diffusion and the correspondence principle

We study the quantum behavior of the standard map in the so-called accelerator state, which, within the theoretical framework of classical physics, would result in anomalous diffusion. In agreement with the behavior of the systems, which classically exhibit full chaos and are consequently characterized by positive Lyapunov coefficients λ, quantum uncertainty increases very quickly and leads to the breakdown of the overwhelming majority of the classical trajectories in a time ${\mathit{t}}_{\mathrm{\chi }}$=(1/λ)ln(1/ħ). In the case of normal diffusion, the diffusion process is unaffected by this rapid transition from classical to quantum physics. However, in the case of anomalous diffusion, we find the existence of a new breakdown process, corresponding to a statistical departure of quantum from classical dynamics. We argue that this new kind of breakdown, which does not have anything to do with the well known phenomenon of localization, takes place on a time scale ${\mathit{t}}_{\mathit{B}}$=(1/${\lambda }_{\mathit{f}}$)ln(1/ħ) larger than ${\mathit{t}}_{\mathrm{\chi }}$ and with Lyapunov coefficient ${\lambda }_{\mathit{f}}$ determined essentially by the stochastic trajectories moving on the border between the stochastic sea and the accelerator islands. If our arguments are confirmed, they would lead to the possibility of observing the breakdown of the correspondence principle in the statistical sense in times compatible with experimental observation.