Anomalous Diffusion on Percolating Clusters

  • Amnon Aharony


Much of the recent renewed interest in percolation theory is related to the realization that percolation clusters are self-similar,1 and may thus be modeled by fractal structures.2 On a fractal structure, all the physical properties behave as powers of the relevant length scale, L. This behavior crosses over to a homogeneous one (i.e. independent of L, for appropriately defined quantities), on length scales larger than the percolation connectedness (or correlation) length, ξ∝|p−pc|−v. Assuming that ξ is the only important length in the problem, all other lengths should be measured in units of ξ, and thus depend on L only via the ratio L/ξ. This implies scaling. For example, above the percolation threshold (p≥pc) one has1
$$M\left( L \right) = {L^D}m\left( {L/\zeta } \right)$$
for the number of sites on the infinite incipient cluster within a volume of linear size L. The exponent D is the fractal dimensionality 2 of the cluster in the self-similar regime, and the scaling function m(x) behaves as a constant for x→0 and as m(x)∿xβ/v for x»1, so that M(L)∿LdP. Here, d is the Euclidean dimensionality of space, and P ∿ ξ−β/v ∿ (p−pc)β is the probability per site to belong to the infinite cluster. Thus, one identifies D=d−β/v.


Anomalous Diffusion Percolation Cluster Sierpinski Gasket Infinite Cluster Relevant Time Scale 
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  1. 1.
    e.g. A. Kapitulnik, A. Aharony, G. Deutscher and D. Stauffer, J. Phys. A16: L269 (1983).Google Scholar
  2. 2.
    B.B. Mandelbrot, The Fractal Geometry of Nature, ( Freeman, San Francisco, 1982 ).zbMATHGoogle Scholar
  3. 3.
    P.G. de Gennes, La Recherche 7: 919 (1976).Google Scholar
  4. 4.
    Y. Gefen, A. Aharony and S. Alexander, Phys. Rev. Lett. 50: 77 (1983).CrossRefGoogle Scholar
  5. 5.
    e.g. F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw Hill Kogakusha, Tokyo 1965 ), Chap. 15. 5.Google Scholar
  6. 6.
    H. Nakanishi, Y. Gefen and A. Aharony, unpublished.Google Scholar
  7. 7.
    J.P. Straley, Phys. Rev. B15: 5733 (1977).CrossRefGoogle Scholar
  8. 8.
    A.L. Efros and B.I. Shklovskii, Phys. Status Solidi (b). 76: 475 (1976)CrossRefGoogle Scholar
  9. D.J. Bergman and Y. Imry, Phys. Rev. Lett. 39: 1222 (1977).CrossRefGoogle Scholar
  10. 9.
    R.B. Laibowitz and Y. Gefen, Phys. Rev. Lett. 53: 380 (1984).CrossRefGoogle Scholar
  11. 10.
    P.G. de Gennes, J. de Phys. Lett. 200: 2197 (1979).Google Scholar
  12. 11.
    A. Bunde, D.C. Hong, I. Majid and H.E. Stanley, J. Phys. A18: L137 (1985).Google Scholar
  13. 12.
    J. Adler, A. Aharony and D. Stauffer, J. Phys. A18: L129 (1985).Google Scholar
  14. 13.
    M.J. Stephen, Phys. Rev. B17: 4444 (1978).CrossRefGoogle Scholar
  15. 14.
    J. Kertesz, J. Phys. A16: L471 (1983).Google Scholar
  16. 15.
    A. Coniglio and H.E. Stanley, Phys. Rev. Lett. 52: 1068 (1984).CrossRefGoogle Scholar
  17. 16.
    D. Stauffer, Phys. Rept. 54: 3 (1979)CrossRefGoogle Scholar
  18. D. Stauffer, Introduction to Percolation Theory ( Taylor and Frazer, London, 1985 ).zbMATHCrossRefGoogle Scholar
  19. 17.
    M. Sahimi, J. Phys. A17: L601 (1984).Google Scholar
  20. 18.
    A. Aharony and D. Stauffer, Phys. Rev. Lett. 52: 2368 (1984).CrossRefGoogle Scholar
  21. 19.
    S. Alexander and R. Orbach, J. de Phys. (Paris) Lett. 43: L625 (1982).CrossRefGoogle Scholar
  22. 20.
    R. Rammal and G. Toulouse, J. de Phys. (Paris) Lett. 44: L13 (1983).CrossRefGoogle Scholar
  23. 21.
    A. Aharony, S. Alexander, O. Entin-Wohlman and R. Orbach, Phys. Rev. B31: 2565 (1985).CrossRefGoogle Scholar
  24. 22.
    See also lectures by R. Orbach in these proceedings.Google Scholar
  25. 23.
    F. Leyvraz and H.E. Stanley, Phys. Rev. Lett. 51: 2048 (1983).CrossRefGoogle Scholar
  26. 24.
    S. Alexander, Ann. Israel Phys. Soc. 5: 149 (1983).Google Scholar
  27. 25.
    H.E. Stanley, I. Majid, A. Margolina and A. Bunde, Phys. Rev. Lett. 53: 1706 (1984).CrossRefGoogle Scholar
  28. 26.
    A.B. Harris and T.C. Lubensky, J. Phys. A17: L609 (1984).MathSciNetGoogle Scholar
  29. 27.
    A. Aharony, H.E. Stanley and A. Margolina, unpublished.Google Scholar
  30. 28.
    Y. Gefen, A. Aharony, B.B. Mandelbrot and S. Kirkpatrick, Phys. Rev. Lett. 47: 1771 (1981).MathSciNetCrossRefGoogle Scholar
  31. 29.
    S. Wilke, Y. Gefen, V. Ilkovic, A. Aharony and D. Stauffer, J. Phys. A17: 647 (1984)Google Scholar
  32. S. Havlin, Z. Djordjevic, I. Majid, H.E. Stanley and G.H. Weiss, Phys. Rev. Lett. 53: 178 (1984).CrossRefGoogle Scholar
  33. 30.
    S. Havlin, Phys. Rev. Lett. 53: 1705 (1984).CrossRefGoogle Scholar
  34. 31.
    D.J. Bergman and Y. Kantor, Phys. Rev. Lett. 53: 511 (1984)MathSciNetCrossRefGoogle Scholar
  35. Y. Kantor and I. Weburan, Phys. Rev. Lett. 52: 1891 (1984)CrossRefGoogle Scholar
  36. L. Benguigui, Phys. Rev. Lett. 53: 2028 (1984).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Amnon Aharony
    • 1
  1. 1.School of Physics and AstronomyTel-Aviv UniversityRamat AvivIsrael

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