Abstract
What physical concept can unite the flow of liquid through ground coffee, electrical conduction in a conductor–insulator mixture, target collapse on shooting, the birth of a continent, polymer gelification and the spread of epidemics or forest fires? One question is common to all these processes: how is “something” produced at large scales by contributions at small scales? In all these situations, a quantity (liquid, electric charges, target fracture, dry land, molecular crosslinks, disease or fire) may or may not propagate from one element to its neighbour. As in the previous chapters, we are interested in asymptotic properties resulting at large scale, that is to say in a system that is large compared with the size of the individual elements. The analogue of temperature T here is the inverse of the relative population density p of elements (pores, conducting regions, impacts, etc) which varies between 0 and 1 for the maximum population density. p ∼ 0 corresponds to a large amount of disorder for a dilute population and p ∼ 1 corresponds to a large amount of order established. In these situations we can ask ourselves the same questions as for changes of states of matter in thermal systems.
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Notes
- 1.
By the average connectivity length of a given cluster, we mean the average distance separating the cluster elements.
- 2.
That is to say it has classical geometric properties (non fractal), for instance the number of elements contained in a region of linear extension L of the infinite cluster varies as L d where d is the dimension of space (see also Sect. 3.1.1).
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Lesne, A., Laguës, M. (2012). The Percolation Transition. In: Scale Invariance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15123-1_5
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