Table of Contents
International Journal of Statistical Mechanics
Volume 2013, Article ID 738345, 11 pages
Research Article

Thermalization of Lévy Flights: Path-Wise Picture in 2D

Institute of Physics, University of Opole, 45-052 Opole, Poland

Received 23 May 2013; Accepted 25 August 2013

Academic Editor: Antonina Pirrotta

Copyright © 2013 Mariusz Żaba and Piotr Garbaczewski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We analyze two-dimensional (2D) random systems driven by a symmetric Lévy stable noise which in the presence of confining potentials may asymptotically set down at Boltzmann-type thermal equilibria. In view of the Eliazar-Klafter no-go statement, such dynamical behavior is plainly incompatible with the standard Langevin modeling of Lévy flights. No explicit path-wise description has been so far devised for the thermally equilibrating random motion we address, and its formulation is the principal goal of the present work. To this end we prescribe a priori the target pdf ρ in the Boltzmann form ~exp[ ] and next select the Lévy noise (e.g., its Lévy measure) of interest. To reconstruct random paths of the underlying stochastic process we resort to numerical methods. We create a suitably modified version of the time honored Gillespie algorithm, originally invented in the chemical kinetics context. A statistical analysis of generated sample trajectories allows us to infer a surrogate pdf dynamics which sets down at a predefined target, in consistency with the associated kinetic (master) equation.