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Fractals and Percolation

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

Percolation theory is useful for characterizing many disordered systems. Percolation is a pure random process of choosing sites to be randomlyoccupied or empty with certain probabilities.However, the topology obtained in such processes has a rich structure related to fractals.The structural properties of percolation clusters have become clearer thanks to the development of fractal geometry since the 1980s.

Introduction

Percolation represents the simplest model of a phase transition [1,8,13,14,26,27,30,48,49,61,64,65,68]. Assume a regular lattice (grid) where each site (or bond) is occupied with probability p or empty with probability \( { 1-p } \). At a critical threshold,\( { p_\mathrm{c} }\), a long-range connectivity first appears: \( { p_\mathrm{c} } \) iscalled the percolation threshold (see Fig. 1). Occupied and emptysites (or bonds) may stand for very different physical properties. For example, occupied sites may representelectrical conductors, empty...

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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. (2009). Fractals and Percolation. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_227

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