Some insights in superdiffusive transport

Abstract

In a wide range of systems, the relaxation in response to an initial pulse has been experimentally found to follow a nonlinear relationship for the mean squared displacement, of the kindwhere α may be greater or smaller than 1. Such phenomena have been described under the generic term of anomalous diffusion. "Lévy flights" stochastic processes lead to superdiffusive behavior ( 2 1 < < α ) and have been recently proposed to model -among the others -the subsurface contaminant spread in highly heterogeneous media under the effects of water flow. In this paper, within the Continuous-Time Random Walk (CTRW) approach to anomalous diffusion, we compare the analytical solution of the approximated Fractional Diffusion Equation (FDE) with the Monte Carlo one, obtained by simulating the superdiffusive behavior of an ensemble of particle in a medium. We show that the two are neatly different as the process approaches the standard diffusive behavior. We argue that this is due to a truncation in the Fourier space expansion introduced by the FDE approach. We propose a second order correction to this expansion and numerically solve the CTRW model under this hypothesis: the accuracy of the results thus obtained is validated through Monte Carlo simulation over all the superdiffusive range. The same kind of discrepancy is shown to occur also in the derivation of the fractional moments of the distribution: analogous corrections are proposed and validated through the Monte Carlo approach.

Fractional Laplacian in bounded domains

“…This approach, however, has some shortcomings: first, the convergence of the discretized matrix to the continuum operator largely deteriorates as α → 2, i.e. when approaching the regular Laplacian [26,36,37]. Secondly, it is strictly limited to the range α ∈ (0, 2], due to its probabilistic underpinnings.…”

Stretched-Gaussian asymptotics of the truncated Lévy flights for the diffusion in nonhomogeneous media

“…where µ ′ = µ + β > 2 to ensure the existence of the second moment. The Fourier transform is given by [36] λ…”

Carbon Nanotubes as Fluorescent Probes in Living Drosophila Embryos

“…The MSD is a measure for the area x 2 (τ ) an object passes during a time step τ . MSD analyses are commonly used in biology as a measure to categorize different types of motility based on the diffusive behavior of tracer particles in a medium [6,166,167]. In this work we use this widely accepted idea to obtain information about the dynamical behavior of fluorescent CNTs in the Drosophila embryos.…”

“…The anomaly parameter is the exponent describing the power law in (Eq. 6.8) and is commonly used to numerically distinguish sub-diffusive, diffusive and super-diffusive motions [167]. Molecular motors of the kinesin family usually display directed, i. e. super-diffusive motility [6].…”