Percolation

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Fig. 1: In coffee percolation, soluble compounds leave the coffee grounds and join the water to form coffee. Insoluble compounds (and granulates) remain within the coffee filter. Manual coffee preperation.jpg
Fig. 1: In coffee percolation, soluble compounds leave the coffee grounds and join the water to form coffee. Insoluble compounds (and granulates) remain within the coffee filter.
Fig. 2: Percolation in a square lattice (Click to animate) Percolation.gif
Fig. 2: Percolation in a square lattice (Click to animate)

In physics, chemistry and materials science, percolation (from Latin percolare, "to filter" or "trickle through") refers to the movement and filtering of fluids through porous materials. It is described by Darcy's law. Broader applications have since been developed that cover connectivity of many systems modeled as lattices or graphs, analogous to connectivity of lattice components in the filtration problem that modulates capacity for percolation.

Contents

Background

During the last decades, percolation theory, the mathematical study of percolation, has brought new understanding and techniques to a broad range of topics in physics, materials science, complex networks, epidemiology, and other fields. For example, in geology, percolation refers to filtration of water through soil and permeable rocks. The water flows to recharge the groundwater in the water table and aquifers. In places where infiltration basins or septic drain fields are planned to dispose of substantial amounts of water, a percolation test is needed beforehand to determine whether the intended structure is likely to succeed or fail. In two dimensional square lattice percolation is defined as follows. A site is "occupied" with probability p or "empty" (in which case its edges are removed) with probability 1 – p; the corresponding problem is called site percolation, see Fig. 2.

Percolation typically exhibits universality. Statistical physics concepts such as scaling theory, renormalization, phase transition, critical phenomena and fractals are used to characterize percolation properties. Combinatorics is commonly employed to study percolation thresholds.

Due to the complexity involved in obtaining exact results from analytical models of percolation, computer simulations are typically used. The current fastest algorithm for percolation was published in 2000 by Mark Newman and Robert Ziff. [1]

Examples

See also

Related Research Articles

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The percolation threshold is a mathematical concept in percolation theory that describes the formation of long-range connectivity in random systems. Below the threshold a giant connected component does not exist; while above it, there exists a giant component of the order of system size. In engineering and coffee making, percolation represents the flow of fluids through porous media, but in the mathematics and physics worlds it generally refers to simplified lattice models of random systems or networks (graphs), and the nature of the connectivity in them. The percolation threshold is the critical value of the occupation probability p, or more generally a critical surface for a group of parameters p1, p2, ..., such that infinite connectivity (percolation) first occurs.

In the context of the physical and mathematical theory of percolation, a percolation transition is characterized by a set of universal critical exponents, which describe the fractal properties of the percolating medium at large scales and sufficiently close to the transition. The exponents are universal in the sense that they only depend on the type of percolation model and on the space dimension. They are expected to not depend on microscopic details such as the lattice structure, or whether site or bond percolation is considered. This article deals with the critical exponents of random percolation.

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Interdependent networks Subfield of network science

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The field of complex networks has emerged as an important area of science to generate novel insights into nature of complex systems The application of network theory to climate science is a young and emerging field. To identify and analyze patterns in global climate, scientists model climate data as complex networks.

Robustness, the ability to withstand failures and perturbations, is a critical attribute of many complex systems including complex networks.

References

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  2. Brunk, Nicholas E.; Twarock, Reidun (2021-07-23). "Percolation Theory Reveals Biophysical Properties of Virus-like Particles". ACS Nano. American Chemical Society (ACS). 15 (8): 12988–12995. doi: 10.1021/acsnano.1c01882 . ISSN   1936-0851. PMC   8397427 . PMID   34296852.
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  7. Grassberger, Peter (1983). "On the Critical Behavior of the General Epidemic Process and Dynamical Percolation". Mathematical Biosciences. 63 (2): 157–172. doi:10.1016/0025-5564(82)90036-0.
  8. Newman, M. E. J. (2002). "Spread of epidemic disease on networks". Physical Review E. 66 (1 Pt 2): 016128. arXiv: cond-mat/0205009 . Bibcode:2002PhRvE..66a6128N. doi:10.1103/PhysRevE.66.016128. PMID   12241447. S2CID   15291065.
  9. D. Li, B. Fu, Y. Wang, G. Lu, Y. Berezin, H.E. Stanley, S. Havlin (2015). "Percolation transition in dynamical traffic network with evolving critical bottlenecks". PNAS. 112 (3): 669–72. Bibcode:2015PNAS..112..669L. doi: 10.1073/pnas.1419185112 . PMC   4311803 . PMID   25552558.CS1 maint: uses authors parameter (link)
  10. Guanwen Zeng, Daqing Li, Shengmin Guo, Liang Gao, Ziyou Gao, HEugene Stanley, Shlomo Havlin (2019). "Switch between critical percolation modes in city traffic dynamics". Proceedings of the National Academy of Sciences. 116 (1): 23–28. Bibcode:2019PNAS..116...23Z. doi: 10.1073/pnas.1801545116 . PMC   6320510 . PMID   30591562.CS1 maint: uses authors parameter (link)
  11. X. Yuan, Y. Hu, H.E. Stanley, S. Havlin (2017). "Eradicating catastrophic collapse in interdependent networks via reinforced nodes". PNAS. 114 (13): 3311–3315. arXiv: 1605.04217 . Bibcode:2017PNAS..114.3311Y. doi: 10.1073/pnas.1621369114 . PMC   5380073 . PMID   28289204.CS1 maint: uses authors parameter (link)

Further reading