Transport on percolation clusters with power-law distributed bond strengths

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Journal Title
Journal ISSN
Volume Title
School of Science | A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
Date
2003
Major/Subject
Mcode
Degree programme
Language
en
Pages
056106/1-8
Series
Physical Review E, Volume 67, Issue 5
Abstract
The simplest transport problem, namely finding the maximum flow of current, or maxflow, is investigated on critical percolation clusters in two and three dimensions, using a combination of extremal statistics arguments and exact numerical computations, for power-law distributed bond strengths of the type P(σ)∼σ−α. Assuming that only cutting bonds determine the flow, the maxflow critical exponent v is found to be v(α)=(d−1)ν+1/(1−α). This prediction is confirmed with excellent accuracy using large-scale numerical simulation in two and three dimensions. However, in the region of anomalous bond capacity distributions (0<~α<~1) we demonstrate that, due to cluster-structure fluctuations, it is not the cutting bonds but the blobs that set the transport properties of the backbone. This “blob dominance” avoids a crossover to a regime where structural details, the distribution of the number of red or cutting bonds, would set the scaling. The restored scaling exponents, however, still follow the simplistic red bond estimate. This is argued to be due to the existence of a hierarchy of so-called minimum cut configurations, for which cutting bonds form the lowest level, and whose transport properties scale all in the same way. We point out the relevance of our findings to other scalar transport problems (i.e., conductivity).
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Keywords
transport properties, percolation clusters
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Citation
Alava, Mikko J. & Moukarzel, Cristian F. 2003. Transport on percolation clusters with power-law distributed bond strengths. Physical Review E. Volume 67, Issue 5. 056106/1-8. ISSN 1539-3755 (printed). DOI: 10.1103/physreve.67.056106.