Universality in the onset of superdiffusion and anomalous transport in Lévy 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.

The manner by which such systems evolve towards their asymptotic superdiffusive behavior is explored using the 1D Lévy walk of order 1<β<2.
The approach towards superdiffusion, as captured by the leading correction to the asymptotic propagator, is shown to undergo a transition as β crosses the critical value 3/2. Above it, this correction scales as |x|~t^(1/2), describing simple diffusion. However, below the critical β it instead remains superdiffusive, scaling as |x|~t^(2β-1).

The transition is independent of the precise model details and is thus argued to be universal.

Such finite corrections typically play a crucial role in experimental and numerical studies of superdiffusive systems. This holds true both in equilibrium settings, as well as in nonequilibrium settings, where superdiffusion gives raise to anomalous transport.

References:

×        AM, “Universality in the onset of superdiffusion in Lévy walks” - PRL 124, 140601 (2020)

×        AM, “Lévy walks on finite intervals: a step beyond asymptotics” - PRE 100, 012106 (2019)

×        AM, J. Cividini, A. Kundu, D. Mukamel,  "Derivation of fluctuating hydrodynamics and crossover from diffusive to anomalous transport in a hard-particle gas" PRE 99, 012124 (2019)