Mathematics of Complexity and Dynamical Systems

2011 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Anomalous Diffusion on Fractal Networks

  • Igor M. Sokolov
Reference work entry
DOI: https://doi.org/10.1007/978-1-4614-1806-1_2

Article Outline

Glossary

Definition of the Subject

Introduction

Random Walks and Normal Diffusion

Anomalous Diffusion

Anomalous Diffusion on Fractal Structures

Percolation Clusters

Scaling of PDF and Diffusion Equationsl on Fractal Lattices

Further Directions

Bibliography

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Bibliography

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Additional Reading

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    The present article gave a brief overview of what is known about the diffusion on fractal networks, however this overview is far from covering all the facets of the problem. Thus, we only discussed unbiased diffusion (the effects of bias may be drastic due to e. g. stronger trapping in the dangling ends), and considered only the situations in which the waiting time at all nodes was the same (we did not discuss e. g. the continuous‐time random walks on fractal networks), as well as left out of attention many particular systems and applications. Several review articles can be recommended as a further reading, some of them already mentioned in the text. One of the best-known sources is [10] being a reprint of the text from the “sturm und drang” period of investigation of fractal geometries. A lot of useful information on random walks models in general an on walks on fractals is contained in the review by Haus and Kehr from approximately the same time. General discussion of the anomalous diffusion is contained in the work by Bouchaud and Georges. The classical review of the percolation theory is given in the book of Stauffer and Aharony. Some additional information on anomalous diffusion in percolation systems can be found in the review by Isichenko. A classical source on random walks in disordered systems is the book by Hughes.Google Scholar
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    Hughes BD (1995) Random Walks and random Environments. Oxford University Press, New YorkMATHGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Igor M. Sokolov
    • 1
  1. 1.Institute of PhysicsHumboldt-Universität zu BerlinBerlinGermany