Appendix D: Percolation Theory

Percolation theory investigates whether a system which consists of elementary or microscopic states that are defined at the nodes of a given lattice is macroscopically connected or not (Binder 1984; Stauffer 1991; Stauffer and Aharony 1992; Sahimi 1994; Stauffer and Aharony 1995).In other words, percolation theory examines whether a given ensemble with microscopic states A, JB,... can be penetrated along a certain path defined by neighboring sites of identical state.This requires that from the start to the end of the path each site with, say, state A has a neighboring site within the same state.A trivial solution emerges if all sites are in the same state.Thus, if the rules that determine the respective state of the microscopic sites at a given time are put aside (e.g.spin up or spin down), percolation theory addresses a purely geometrical problem.In this view percolation models provide information about the topology of sites that share a common property.The underlying spatial lattices can be regular or irregular, e.g.Voronoi tessellations are admissible.Figure 6.10 showed some typical lattices that are frequently used in Monte Carlo, cellular automaton, and percolation simulations.Obvious macroscopic tasks of percolation theory are the prediction of forest fire propagation and the estimation of the connectivity of oil resources.In materials research, percolation is of considerable relevance in simulating current paths, microplastic behavior, diffusion, fracture mechanics, or properties of porous media (Stauffer and Aharony 1992; Kuo and Gupta 1995).Although percolation simulations are often used for predicting structure evolution at the microscopic level, they are not intrinsically calibrated and can therefore be applied to arbitrary regimes of space and time.