Book contents
- Frontmatter
- Contents
- Preface
- Part one Basic concepts
- Part two Anomalous diffusion
- 5 Diffusion in the Sierpinski gasket
- 6 Diffusion in percolation clusters
- 7 Diffusion in loopless structures
- 8 Disordered transition rates
- 9 Biased anomalous diffusion
- 10 Excluded-volume interactions
- Part three Diffusion-limited reactions
- Part four Diffusion-limited coalescence: an exactly solvable model
- Appendix A The fractal dimension
- Appendix B The number of distinct sites visited by random walks
- Appendix C Exact enumeration
- Appendix D Long-range correlations
- References
- Index
6 - Diffusion in percolation clusters
Published online by Cambridge University Press: 19 January 2010
- Frontmatter
- Contents
- Preface
- Part one Basic concepts
- Part two Anomalous diffusion
- 5 Diffusion in the Sierpinski gasket
- 6 Diffusion in percolation clusters
- 7 Diffusion in loopless structures
- 8 Disordered transition rates
- 9 Biased anomalous diffusion
- 10 Excluded-volume interactions
- Part three Diffusion-limited reactions
- Part four Diffusion-limited coalescence: an exactly solvable model
- Appendix A The fractal dimension
- Appendix B The number of distinct sites visited by random walks
- Appendix C Exact enumeration
- Appendix D Long-range correlations
- References
- Index
Summary
In an insightful pioneering work de Gennes (1976a; 1976b) pondered the problem of a random walker in percolation clusters, which he described as “the ant in the labyrinth”. Similar ideas were presented at the time by Brandt (1975), Kopelman (1976), and Mitescu and Roussenq (1976). de Gennes' “ants” triggered intensive research on diffusion in disordered media. Here we describe the more important aspects of the subject. A brief account of percolation theory has been presented in Chapter 2.
The analogy with diffusion in fractals
As discussed in Chapter 2, percolation clusters may be regarded as random fractals. Below the critical threshold, p < pc the clusters are finite. The largest clusters have a typical size of the order of the (finite) correlation length ξ(p), and they possess a fractal dimension df = d − β/ν (Eq. (2.8)).
At criticality, p = pc, there emerges an infinite percolation cluster that may be described as a random fractal with the same dimension df = d − β/ν. The inception of the infinite cluster coincides with the divergence of the correlation length, ξ ∼ |p − pc|−ν. Along with the incipient infinite cluster there exist clusters of finite extent. The finite clusters may be regarded as fractals possessing the usual percolation dimension df. They are not different, practically, than the clusters that form below the percolation threshold, other than that there is no limit to their typical size – since the global correlation length diverges.
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- Publisher: Cambridge University PressPrint publication year: 2000
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