Towards deterministic equations for Lévy walks: The fractional material derivative

Abstract

Lévy walks are random processes with an underlying spatiotemporal coupling. This coupling penalizes long jumps, and therefore Lévy walks give a proper stochastic description for a particle's motion with broad jump length distribution. We derive a generalized dynamical formulation for Lévy walks in which the fractional equivalent of the material derivative occurs. Our approach will be useful for the dynamical formulation of Lévy walks in an external force field or in phase space for which the description in terms of the continuous time random walk or its corresponding generalized master equation are less well suited.

Lévy flights with power-law absorption

“…Lévy flights can be effectively approached via the socalled fractional calculus which provides a generalization of classical diffusion equations using fractional derivatives, especially fractional Laplacian operators (see also [29] and Appendix A). More generally, we mention that the description of superdiffusion can be accomplished in terms of fractional material derivatives (see [30,31]) which, as far as a dimensional analysis is carried out, give the same scalings as fractional derivatives.…”

Method of calculating densities for isotropic ballistic Lévy walks

“…We also provide explicit formulas for the densities of other coupled CTRWs -the so-called undershooting and overshooting Lévy walks (also known as wait-first and jump-first Lévy walks [14]). Similarly, these densities are given by elementary functions for odd dimensions d. Moreover these PDFs solve certain differential equations [22] with the fractional material derivative [23,24].…”

Discretized fractional substantial calculus

“…Lévy walk gives another proper dynamical description for the superdiffusion (roughly speaking, now the particle has finite physical speed), and the PDFs of waiting time and jump length are spatiotemporal coupling [13]. Friedrich and his co-workers discuss the CTRW model with position-velocity coupling PDF [5].…”

“…Carmi and Barkai use the CTRW model with functional of path and position coupling PDF [1]. Based on the CTRW models with coupling PDFs, they all derive the deterministic equations; and mathematically an important operator, fractional substantial derivative, is introduced [1,2,5,13,14].…”

“…The time fractional Fokker-Planck equation can well characterize the subdiffusion, and the space fractional Fokker-Planck equation can depict the Lévy flight. The Lévy flight has a diverging mean squared displacement, and can just be applied to rather exotic physical processes [13].…”