Universality in the Onset of Superdiffusion in Levy Walks

Anomalous dynamics in which local perturbations spread faster than diffusion are ubiquitously observed in the long-time behavior of a wide variety of systems. Here, the manner by which such systems evolve towards their asymptotic superdiffusive behavior is explored using the 1D Levy walk of order 1 < beta < 2. The approach towards superdiffusion, as captured by the leading correction to the asymptotic behavior, is shown to remarkably undergo a transition as beta crosses the critical value beta(c) = 3/2. Above beta(c), this correction scales as vertical bar x vertical bar similar to t(1/)(2), describing simple diffusion. However, below beta(c) it is instead found to remain superdiffusive, scaling as vertical bar x vertical bar similar to t(1/)(2()(beta-1)). This transition is shown to be independent of the precise model details and is thus argued to be universal.