Universality in the Onset of Superdiffusion in Lévy Walks

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 Levy walk of order 1 < β < 2. The approach towards superdiffusion, as captured by the leading correction to the asymptotic behavior, is shown to remarkably undergo a transition as β crosses the critical value βc = 3/2. Above βc, this correction scales as |x| ∼ t 1/2 , describing simple diffusion. However, below βc it is instead found to remain superdiffusive, scaling as |x| ∼ t 1/(2β−1) . This transition is shown to be independent of the precise model details and is thus argued to be universal.

Origin of universality in the onset of superdiffusion in Lévy walks

“…(10) is consistent with the known Lévy walk propa-gatorP LW (k, s) =ψ (s−ivk)+ψ (s+ivk) 2−φ(s−ivk)−φ(s+ivk) [20], which is immediately recovered when summingP n (k, s) over n. For large t and small |k|,P LW (k, s) was shown in Ref. [22] to asymptotically approachP LW (q, t ) ∼ = e −tI(q) , where φ(s ± ivk), and ultimately yields the transition in the onset of superdiffusion in Lévy walks. While uncovering the origin of this transition, we shall see that the coupling between the walk's position and time is, in fact, a simple, natural, and intuitive mechanism which serves to intertwine the walk's position with the corresponding transition in the walk-number fluctuations.…”

“…This approach, captured by the leading correction to P 0 (x, t ) as t → ∞, was shown to transition at β c = 3/2 between a diffusive scaling |x| ∝ t 1/2 for β > β c and a superdiffusive scaling |x| ∝ t 1/(2β−1) for β < β c . This transition [23] is regarded universal as it was shown to be insensitive to φ(τ )'s short-time behavior, depending only on its heavy tail ∝τ −1−β [22]. Indeed, recent results concerning anomalous transport in a class of 1D systems [24] modeled by a Lévy walk of order β = 5/3 [3,8,25] are consistent with the diffusive correction predicted in Lévy walks for β > β c [22].…”

“…This transition [23] is regarded universal as it was shown to be insensitive to φ(τ )'s short-time behavior, depending only on its heavy tail ∝τ −1−β [22]. Indeed, recent results concerning anomalous transport in a class of 1D systems [24] modeled by a Lévy walk of order β = 5/3 [3,8,25] are consistent with the diffusive correction predicted in Lévy walks for β > β c [22]. This raises the exciting possibility that Lévy walks may remarkably remain a valid description of superdiffusive phenomena, even beyond the asymptotic limit.…”

“…In this Rapid Communication, we investigate the mechanism responsible for the universal transition observed in the onset of superdiffusion in Lévy walks [22]. We find it to be twofold, consisting of the finite speed v which couples the walker's position to time and a corresponding transition at β c = 3/2 in the fluctuations n 2 t ≡ n 2 t − n t 2 of the number of walks n completed by the walker at time t. At large t these become n 2 t ∼ = n 2 t 0 + δn 2 t with the asymptotic fluctuations n 2 t 0 = κ 0 t 3−β interpolating between a ballistic scaling for β = 1 and a diffusive scaling for β = 2.…”

“…This study shows that this robust and accessible observable can be used to precisely predict which systems are expected to exhibit a transition in the onset of superdiffusion, be they experimental or numerical. However, besides its theoretical value in uncovering the mechanism responsible for the transition in the onset of superdiffusion [22], the transition in n 2 t can itself be used as a tool for precisely determining the value of β. This collateral contribution is important since only a few such instruments are currently known, in spite of the well-known and often devastating difficulties posed by finite-time corrections in both experimental and numerical studies of superdiffusive phenomena [3,[34][35][36][37][38][39][40][41].…”