Anomalous transmission and drifts in one-dimensional Lévy structures

Abstract

We study the transmission of random walkers through a finite-size inhomogeneous material with a quenched, long-range correlated distribution of scatterers. We focus on a finite one-dimensional structure where walkers undergo random collisions with a subset of sites distributed on deterministic (Cantor-like) or random positions, with Lévy spaced distances. Using scaling arguments, we consider stationary and time-dependent transmission and we provide predictions on the scaling behavior of particle current as a function of the sample size. We show that, even in absence of bias, for each single realization a non-zero drift can be present, due to the intrinsic asymmetry of each specific arrangement of the scattering sites. For finite systems, this average drift is particulary important for characterizing the transmission properties of individual samples. The predictions are tested against the numerical solution of the associated master equation. A comparison of different boundary conditions is given.

Infinite densities for Lévy walks

“…Surprisingly, in many cases the continuous spectrum qν(q) exhibits a bi-linear scaling (see details below). Examples for this piecewise linear behavior of qν(q) include nonlinear dynamical systems [2,[6][7][8][9], stochastic models with quenched and annealed disorder, in particular, the Lévy walk [16][17][18][19][20][21] and sand pile models [22]. Recent experiments on the active transport of polymers in the cell [15], theoretical investigation of the momentum [23] and the spatial [24] spreading of cold atoms in optical lattices and flows in porous media [25] further confirmed the generality of strong anomalous diffusion of the bi-linear type.…”

Large Fluctuations for Spatial Diffusion of Cold Atoms

“…The ICD was previously found, for example, for different models of Lévy walks [15,39]. Since dual scaling of the moments and fat tailed distributions are very common, we speculate that ICDs will describe a large class of systems, e.g., Lévy glasses [40], fluctuating surfaces [41], motion of tracer particles in the cell [42] and diffusion on lipid bi-layers [43]. To identify the ICDs in these diverse systems requires further work.…”

Strong anomalous diffusion in two-state process with Lévy walk and Brownian motion

“…Similarly, for the scaling analyses when s ≪ k, Eqs. (20) and (22) are also valid if we replace cos(πα + /2) by | cos(πα + /2)|.…”

“…Compared with the infinite density I(z), the different asymptotic forms of p(x, t) on the central part within different scaling regimes are both normalized, since taking k = 0 both yield p(0, s) ≃ 1/s in Eqs. (20) and (22). However, the high order (bigger than α + ) moments will diverge if we use the asymptotic forms of p(x, t) in the central part, since the large-x behavior in Eq.…”

“…Otherwise, one can find a nonlinear function ν(q) of q for the motions with multiple modes; this phenomenon is named as strong anomalous diffusion [9]. There have been vast systems exhibiting strong anomalous diffusion, such as the nonlinear dynamical systems [9,[12][13][14][15], the annealed or quenched Lévy walk [16][17][18][19][20], sand pile models [21][22][23], the active transport of polymeric particles in living cells [24], and the spreading of cold atoms in optical lattices [25][26][27]. The mechanisms of the strong anomalous diffusion for Lévy walk are studied in detail in Refs.…”