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Mathematics of Complexity and Dynamical Systems pp 559–570Cite as

Fractals and Percolation

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Article Outline

Glossary

Definition of the Subject

Introduction

Percolation

Percolation Clusters as Fractals

Anomalous Transport on Percolation Clusters: Diffusion and Conductivity

Networks

Summary and Future Directions

Bibliography

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Abbreviations

Percolation:

In the traditional meaning, percolation concerns the movement and filtering of fluids through porous materials. In this chapter, percolation is the subject of physical and mathematical models of porous media that describe the formation of a long-range connectivity in random systems and phase transitions. The most common percolation model is a lattice, where each site is occupied randomly with a probability p or empty with probability \({1-p}\). At low p values, there is no connectivity between the edges of the lattice. Above some concentration \({p_\mathrm{c}}\), the percolation threshold , connectivity appears between the edges. Percolation represents a geometric critical phenomena where p is the analogue of temperature in thermal phase transitions .

Fractal:

A fractal is a structure which can be subdivided into parts, where the shape of each part is similar to that of the original structure. This property of fractals is called self‐similarity , and it was first recognized by G.C. Lichtenberg more than 200 years ago. Random fractals represent models for a large variety of structures in nature, among them porous media, colloids, aggregates, flashes, etc. The concepts of self‐similarity and fractal dimensions are used to characterize percolation clusters. Self‐similarity is strongly related to renormalization properties used in critical phenomena , in general, and in percolation phase transition properties.

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Author information

Authors and Affiliations

  1. Department of Physics, Bar–Ilan University, Ramat–Gan, Israel

    Yakov M. Strelniker & Shlomo Havlin

  2. Institut für Theoretische Physik III, Justus-Liebig‐Universität, Giessen, Germany

    Armin Bunde

Authors
  1. Yakov M. Strelniker
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  2. Shlomo Havlin
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  3. Armin Bunde
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Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Strelniker, Y.M., Havlin, S., Bunde, A. (2012). Fractals and Percolation. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_36

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