Discrete models of growth and dynamical percolation in chemistry
Simon J. Fraser · Journal of Computational Chemistry · 1987
Abstract Space‐time lattice (cellular automaton) models of pattern formation and growth are described. Suitable local rules for automation evolution represent the spreading of wave fronts of activity in an excitable medium. A random distribution of seeds produces expanding rings that fuse and are annihilated. The seeding density, pA, is used as a scaling parameter to give unique, reduced dynamics in an arbitrary dimension d. For d = 2, in this (continuum) picture, the rings fuse globally (percolate) at a critical instant, t̂c = 0.45. For the unscaled time evolution, dynamical percolation is examined in the pA × t plane. A swath of these percolating states is found. On the “explosion” boundary of this swath the percolation cluster just forms; on the “implosion” boundary it breaks up. Using a small‐sample method the fractal dimension of the critical (boundary) cluster is estimated to be 1.9 (±0.01). Also percolation for continuously emitting seeds, which produce “discs” of activity, is related to ring evolution.