Eur. Phys. J. B 78, 207-216 (2010)
https://doi.org/10.1140/epjb/e2010-10422-4

Continuous Markovian model for Lévy random walks with superdiffusive and superballistic regimes

¹ University of Aizu, Tsuruga, Ikki-machi, Aizu-Wakamatsu City, 965-8580, Fukushima, Japan
² Institut für Physikalische Chemie, Westfälische Wilhelms Universität Münster, Corrensstr. 30, 48149 Münster, Germany
³ Center of Nonlinear Science CeNoS, Westfälische Wilhelms Universität Münster, 48149 Münster, Germany
⁴ Institut für Theoretische Physik, Westfälische Wilhelms Universität Münster, Wilhelm-Klemm. 9, 48149 Münster, Germany
⁵ Moscow Technical University of Radioengineering, Electronics, and Automation, Vernadsky pr., 78, 117454 Moscow, Russia
⁶ A.M. Prokhorov General Physics Institute, Russian Academy of Sciences, Vavilov Str. 38, Moscow, 119991, Russia

Corresponding authors: ^a ialub@fpl.gpi.ru - ^b andheuer@uni-muenster.de - ^c fiddir@uni-muenster.de

Received: 26 May 2010
Revised: 10 October 2010
Published online: 12 November 2010

Abstract

We consider a previously devised model describing Lévy random walks [I. Lubashevsky, R. Friedrich, A. Heuer, Phys. Rev. E 79, 011110 (2009); I. Lubashevsky, R. Friedrich, A. Heuer, Phys. Rev. E 80, 031148 (2009)]. It is demonstrated numerically that the given model describes Lévy random walks with superdiffusive, ballistic, as well as superballistic dynamics. Previously only the superdiffusive regime has been analyzed. In this model the walker velocity is governed by a nonlinear Langevin equation. Analyzing the crossover from small to large time scales we find the time scales on which the velocity correlations decay and the walker motion essentially exhibits Lévy statistics. Our analysis is based on the analysis of the geometric means of walker displacements and allows us to tackle probability density functions with power-law tails and, correspondingly, divergent moments.