Origin of universality in the onset of superdiffusion in L\'evy walks

TL;DR

This paper investigates the universal transition in superdiffusive behavior of Lévy walks at a critical parameter, revealing that finite velocity and fluctuation transitions in walk counts are key to understanding this phenomenon.

Contribution

It identifies the fundamental mechanisms behind the universality in the onset of superdiffusion in Lévy walks, focusing on velocity constraints and fluctuation transitions.

Findings

Universal transition at critical β=3/2 in Lévy walks

Finite velocity couples position and time in superdiffusion

Transition in fluctuations of the number of completed walks

Abstract

Superdiffusion arises when complicated, correlated and noisy motion at the microscopic scale conspires to yield peculiar dynamics at the macroscopic scale. It ubiquitously appears in a variety of scenarios, spanning a broad range of scientific disciplines. The approach of superdiffusive systems towards their long-time, asymptotic behavior was recently studied using the L\'evy walk of order $1<\beta<2$, revealing a universal transition at the critical $\beta_{c}=3/2$. Here, we investigate the origin of this transition and identify two crucial ingredients: a finite velocity which couples the walker's position to time and a corresponding transition in the fluctuations of the number of walks $n$ completed by the walker at time $t$.