Pattern development in cellular automata triggered by site-specific reactive processes: Dynamical aspects
Roberto A. Garza‐López, John J. Kozak · Physical Review A · 1989
In this paper we continue our study of the (discretized) evolution of regular and fractal patterns initiated by a site-specific reactive event. By formulating and solving (numerically) the stochastic master equation descriptive of the model introduced (for two classes of initial conditions and for two locations of the target site), we are able to demonstrate that symmetry-breaking instabilities which generate fractal patterns (here the Sierpinski gasket) propagate slower than those which generate Euclidean ones (here the triangular lattice). A connection is made between (two) characteristics of the system's evolution (the zero-mode relaxation time and an effective relaxation time descriptive of the overall decay of the initial state) and results calculated using the theory of finite Markov processes (the mean walk length of a coreactant diffusing on the underlying lattice). Using this relationship, the trends observed in our study are interpreted in light of recent theoretical work on the problem of diffusion on fractal lattices and in terms of the notion of a ``fractal'' valency, a concept that places stress on insights drawn from earlier analytic and numerical studies of random processes on finite planar lattice of integral dimension d=2. Finally, the possible relevance of our results to a specific problem in pattern formation and development, the generation of neural networks of the Purkinje type, is discussed.