Transport properties of Lévy walks: An analysis in terms of multistate processes

¹ Dipartimento di Matematica, Università di Bologna - Piazza di Porta S. Donato 5, I-40126 Bologna, Italy
² Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles C. P. 231, Campus Plaine, B-1050 Brussels, Belgium
³ Istituto Nazionale di Fisica Nucleare, Sezione di Bologna - Via Irnerio 46, I-40126 Bologna, Italy
⁴ Departamento de Física, Facultad de Ciencias, Universidad Nacional Autónoma de México Ciudad Universitaria, 04510 México D.F., Mexico

^(a) thomas.gilbert@ulb.ac.be

Received: 5 August 2014
Accepted: 10 November 2014

Abstract

Continuous time random walks combining diffusive and ballistic regimes are introduced to describe a class of Lévy walks on lattices. By including exponentially distributed waiting times separating the successive jump events of a walker, we are led to a description of such Lévy walks in terms of multistate processes whose time-evolution is shown to obey a set of coupled delay differential equations. Using simple arguments, we obtain asymptotic solutions to these equations and rederive the scaling laws for the mean squared displacement of such processes. Our calculation includes the computation of all relevant transport coefficients in terms of the parameters of the models.