The random walk's guide to anomalous diffusion: A fractional dynamics approach

Ralf Metzler, Joseph Klafter

Research output: Contribution to journalReview articlepeer-review

Abstract

Fractional kinetic equations of the diffusion, diffusion-advection, and Fokker-Planck type are presented as a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns. These fractional equations are derived asymptotically from basic random walk models, and from a generalised master equation. Several physical consequences are discussed which are relevant to dynamical processes in complex systems. Methods of solution are introduced and for some special cases exact solutions are calculated. This report demonstrates that fractional equations have come of age as a complementary tool in the description of anomalous transport processes.

Original languageEnglish
Pages (from-to)1-77
Number of pages77
JournalPhysics Report
Volume339
Issue number1
DOIs
StatePublished - Dec 2000

Keywords

  • 02.50.Ey
  • 05.40.-a
  • 05.40.Fb
  • Anomalous diffusion
  • Anomalous relaxation
  • Dynamics in complex systems
  • Fractional Fokker-Planck equation
  • Fractional diffusion equation
  • Mittag-Leffler relaxation

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