Lévy flights in external force fields: Langevin and fractional Fokker-Planck equations and their solutions

Sune Jespersen, Ralf Metzler, and Hans C. Fogedby
Phys. Rev. E 59, 2736 – Published 1 March 1999
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Abstract

We consider Lévy flights subject to external force fields. This anomalous transport process is described by two approaches, a Langevin equation with Lévy noise and the corresponding generalized Fokker-Planck equation containing a fractional derivative in space. The cases of free flights, constant force, and linear Hookean force are analyzed in detail, and we corroborate our findings with results from numerical simulations. We discuss the non-Gibbsian character of the stationary solution for the case of the Hookean force, i.e., the deviation from Boltzmann equilibrium for long times. The possible connection to Tsallis’s q statistics is studied.

  • Received 14 October 1998

DOI:https://doi.org/10.1103/PhysRevE.59.2736

©1999 American Physical Society

Authors & Affiliations

Sune Jespersen*

  • Institute of Physics and Astronomy, University of Århus, 8000 Århus C, Denmark

Ralf Metzler

  • School of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel

Hans C. Fogedby

  • Institute of Physics and Astronomy, University of Århus, 8000 Århus C, Denmark
  • NORDITA, Blegdamsvej 17, 2100 København Ø, Denmark

  • *Electronic address: sune@ifa.au.dk
  • Electronic address: metzler@post.tau.ac.il
  • Electronic address: fogedby@ifa.au.dk

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Vol. 59, Iss. 3 — March 1999

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