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Langevin description of superdiffusive Lévy processes

S. Eule1, V. Zaburdaev2, R. Friedrich3, and T. Geisel1,4

  • 1Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany
  • 2School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts, USA
  • 3Institute for Theoretical Physics, University of Münster, Münster, Germany
  • 4Department of Physics, University of Göttingen, Göttingen, Germany

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Phys. Rev. E 86, 041134 – Published 19 October, 2012

DOI: https://doi.org/10.1103/PhysRevE.86.041134

Abstract

The description of diffusion processes is possible in different frameworks such as random walks or Fokker-Planck or Langevin equations. Whereas for classical diffusion the equivalence of these methods is well established, in the case of anomalous diffusion it often remains an open problem. In this paper we aim to bring three approaches describing anomalous superdiffusive behavior to a common footing. While each method clearly has its advantages it is crucial to understand how those methods relate and complement each other. In particular, by using the method of subordination, we show how the Langevin equation can describe anomalous diffusion exhibited by Lévy-walk-type models and further show the equivalence of the random walk models and the generalized Kramers-Fokker-Planck equation. As a result a synergetic and complementary description of anomalous diffusion is obtained which provides a much more flexible tool for applications in real-world systems.

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Outline

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Phys. Rev. E 86, 041134– Published 19 October, 2012
  • Received 7 November 2011

©2012 American Physical Society

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