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Anomalous Diffusion on Percolating Clusters

  • Amnon Aharony

Abstract

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
$$M\left( L \right) = {L^D}m\left( {L/\zeta } \right)$$
(1)
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.

Keywords

Anomalous Diffusion Percolation Cluster Sierpinski Gasket Infinite Cluster Relevant Time Scale 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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