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

Article Outline


Definition of the Subject


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


This is a preview of subscription content, log in to check access.


Primary Literature

  1. 1.
    Alexander S, Orbach RD (1982) Density of states on fractals–fractons. J Phys Lett 43:L625–L631CrossRefGoogle Scholar
  2. 2.
    Bertacci D (2006) Asymptotic behavior of the simple random walk on the 2‑dimensional comb. Electron J Probab 45:1184–1203Google Scholar
  3. 3.
    Christou A, Stinchcombe RB (1986) Anomalous diffusion on regular and random models for diffusion‐limited aggregation. J Phys A Math Gen 19:2625–2636MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Condamin S, Bénichou O, Tejedor V, Voituriez R, Klafter J (2007) First‐passage times in complex scale‐invariant media. Nature 450:77–80Google Scholar
  5. 5.
    Coulhon T (2000) Random Walks and Geometry on Infinite Graphs. In: Ambrosio L, Cassano FS (eds) Lecture Notes on Analysis on Metric Spaces. Trento, CIMR, (1999) Scuola Normale Superiore di PisaGoogle Scholar
  6. 6.
    Durhuus B, Jonsson T, Wheather JF (2006) Random walks on combs. J Phys A Math Gen 39:1009–1037MATHCrossRefGoogle Scholar
  7. 7.
    Durhuus B, Jonsson T, Wheather JF (2007) The spectral dimension of generic trees. J Stat Phys 128:1237–1260MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Giona M, Roman HE (1992) Fractional diffusion equation on fractals – one‐dimensional case and asymptotic‐behavior. J Phys A Math Gen 25:2093–2105; Roman HE, Giona M, Fractional diffusion equation on fractals – 3‑dimensional case and scattering function, ibid., 2107–2117CrossRefGoogle Scholar
  9. 9.
    Grassberger P (1999) Conductivity exponent and backbone dimension in 2-d percolation. Physica A 262:251–263MathSciNetCrossRefGoogle Scholar
  10. 10.
    Havlin S, Ben-Avraham D (2002) Diffusion in disordered media. Adv Phys 51:187–292CrossRefGoogle Scholar
  11. 11.
    Klafter J, Sokolov IM (2005) Anomalous diffusion spreads its wings. Phys World 18:29–32Google Scholar
  12. 12.
    Klemm A, Metzler R, Kimmich R (2002) Diffusion on random‐site percolation clusters: Theory and NMR microscopy experiments with model objects. Phys Rev E 65:021112CrossRefGoogle Scholar
  13. 13.
    Metzler R, Klafter J (2000) The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 339:1–77MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Metzler R, Klafter J (2004) The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J Phys A Math Gen 37:R161–R208MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Metzler R, Glöckle WG, Nonnenmacher TF (1994) Fractional model equation for anomalous diffusion. Physica A 211:13–24Google Scholar
  16. 16.
    Nakayama T, Yakubo K, Orbach RL (1994) Dynamical properties of fractal networks: Scaling, numerical simulations, and physical realizations. Rev Mod Phys 66:381–443CrossRefGoogle Scholar
  17. 17.
    Özarslan E, Basser PJ, Shepherd TM, Thelwall PE, Vemuri BC, Blackband SJ (2006) Observation of anomalous diffusion in excised tissue by characterizing the diffusion‐time dependence of the MR signal. J Magn Res 183:315–323Google Scholar
  18. 18.
    O'Shaughnessy B, Procaccia I (1985) Analytical solutions for diffusion on fractal objects. Phys Rev Lett 54:455–458CrossRefGoogle Scholar
  19. 19.
    Schulzky C, Essex C, Davidson M, Franz A, Hoffmann KH (2000) The similarity group and anomalous diffusion equations. J Phys A Math Gen 33:5501–5511MATHCrossRefGoogle Scholar
  20. 20.
    Sokolov IM (1986) Dimensions and other geometrical critical exponents in the percolation theory. Usp Fizicheskikh Nauk 150:221–255 (1986) translated in: Sov. Phys. Usp. 29:924CrossRefGoogle Scholar
  21. 21.
    Sokolov IM, Klafter J (2005) From diffusion to anomalous diffusion: A century after Einstein's Brownian motion. Chaos 15:026103MathSciNetCrossRefGoogle Scholar
  22. 22.
    Sokolov IM, Mai J, Blumen A (1997) Paradoxical diffusion in chemical space for nearest‐neighbor walks over polymer chains. Phys Rev Lett 79:857–860CrossRefGoogle Scholar
  23. 23.
    Stauffer D (1979) Scaling theory of percolation clusters. Phys Rep 54:1–74CrossRefGoogle Scholar
  24. 24.
    Webman I (1984) Diffusion and trapping of excitations on fractals. Phys Rev Lett 52:220–223MathSciNetCrossRefGoogle Scholar

Additional Reading

  1. 25.
    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
  2. 26.
    Haus JW, Kehr K (1987) Diffusion in regular and disordered lattices. Phys Rep 150:263–406CrossRefGoogle Scholar
  3. 27.
    Bouchaud JP, Georges A (1990) Anomalous diffusion in disordered media – statistical mechanisms, models and physical applications. Phys Rep 195:127–293MathSciNetCrossRefGoogle Scholar
  4. 28.
    Stauffer D, Aharony A (2003) Introduction to Percolation Theory. Taylor & Fransis, LondonGoogle Scholar
  5. 29.
    Isichenko MB (1992) Percolation, statistical topography, and transport in random‐media. Rev Mod Phys 64:961–1043MathSciNetCrossRefGoogle Scholar
  6. 30.
    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