Universality in the Onset of Super-Diffusion in L\'evy Walks

TL;DR

This paper investigates how systems described by 1D Lévy walks transition towards superdiffusive behavior, revealing a universal critical point at β=3/2 that determines the nature of their approach to superdiffusion.

Contribution

It uncovers a universal transition in the approach to superdiffusion in Lévy walks, independent of model specifics, at the critical value β=3/2.

Findings

Transition at β=3/2 separates diffusive and superdiffusive corrections.

Above β_c, correction scales as |x| ~ t^{1/2} (diffusive).

Below β_c, correction remains superdiffusive, |x| ~ t^{1/(2β-1)}.

Abstract

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 L\'evy 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 $\lvert x \rvert \sim t^{1/2}$, describing simple diffusion. However, below $\beta_{c}$ it is instead found to remain superdiffusive, scaling as $\lvert x \rvert \sim t^{1/(2\beta-1)}$. This transition is shown to be independent of the precise model details and is thus argued to be universal.