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In physics, chemistry and materials science, percolation concerns also the movement and filtering of fluids through porous materials. Examples include the movement of solvents through filter paper (chromatography) and the movement of petroleum through fractured rock. Electrical analogs include the flow of electricity through random resistor networks. During the last three decades, percolation theory, an extensive mathematical model of percolation, has brought new understanding and techniques to a broad range of topics in physics, materials science as well as geography.
Percolation typically exhibits universality. Combinatorics is commonly employed to study percolation thresholds.
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Applications of percolation theory
Among the applications are the study of percolation of petroleum and natural gas through semi-porous rock; here the theory has helped predict and improve the productivity of natural gas and oil wells.
In two dimensions, the percolation of water through a thin tissue (such as toilet paper) has the same mathematical underpinnings as the flow of electricity through two-dimensional random networks of resistors. In chemistry, chromatography can be understood with similar models.
The propagation of a tear or rip in a sheet of paper, in a sheet of metal, or even the formation of a crack in ceramic bears broad mathematical resemblance to the flow of electricity through a random network of electrical fuses. Above a certain critical point, the electrical flow will cause a fuse to pop, possibly leading to a cascade of failures, resembling the propagation of a crack or tear. The study of percolation helps indicate how the flow of electricity will redistribute itself in the fuse network, thus modeling which fuses are most likely to pop next, and how fast they will pop, and what direction the crack may curve in.
Examples can be found not only in physical phenomena, but also in biological and ecological ones (evolution), and also in economic and social ones (see diffusion of innovation).
Percolation can be considered to be a branch of the study of dynamical systems or statistical mechanics. In particular, percolation networks exhibit a phase change around a critical threshold.
See also
- Conductance (graph)
- Self-organization
- Self-organized criticality
- Percolation threshold
- Groundwater recharge
References
- Muhammad Sahimi. Applications of Percolation Theory. Taylor & Francis, 1994. ISBN 0-7484-0075-3 (cloth), ISBN 0-7484-0076-1 (paper)
- Geoffrey Grimmett. Percolation (2. ed). Springer Verlag, 1999.
External links
- Visual simulation of bond percolation. This application shows 38x38 bond percolation square lattice. The percolation threshold is reached when the slider position corresponds to p = 0.5
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