Effective-medium approximation for lattice random walks with long-range jumps

Felix Thiel and Igor M. Sokolov
Phys. Rev. E 94, 012135 – Published 22 July 2016

Abstract

We consider the random walk on a lattice with random transition rates and arbitrarily long-range jumps. We employ Bruggeman's effective-medium approximation (EMA) to find the disorder-averaged (coarse-grained) dynamics. The EMA procedure replaces the disordered system with a cleverly guessed reference system in a self-consistent manner. We give necessary conditions on the reference system and discuss possible physical mechanisms of anomalous diffusion. In the case of a power-law scaling between transition rates and distance, lattice variants of Lévy-flights emerge as the effective medium, and the problem is solved analytically, bearing the effective anomalous diffusivity. Finally, we discuss several example distributions and demonstrate very good agreement with numerical simulations.

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  • Received 22 April 2016

DOI:https://doi.org/10.1103/PhysRevE.94.012135

©2016 American Physical Society

Physics Subject Headings (PhySH)

Statistical Physics & Thermodynamics

Authors & Affiliations

Felix Thiel* and Igor M. Sokolov

  • Institut für Physik, Humboldt-Universität zu Berlin, Newtonstrasse 15, 12489 Berlin, Germany

  • *thiel@posteo.de
  • igor.sokolov@physik.hu-berlin.de

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Issue

Vol. 94, Iss. 1 — July 2016

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