Cellular automata for reactive systems

Two classes of cellular automata for reaction-diiusion systems are developed. These are the Reactive Lattice Gas Automata and the Moving Average Cellular Automata. For both classes explicit procedures for the construction of the automata are given. The construction can be based on given reaction mechanisms or on partial diierential equations. The cellular automata methods are used to investigate several nonlinear reaction-diiusion systems. The reactive lattice gas automata possess intrinsic uctuations that closely reeect uctuations in real systems. These uctuations are investigated in a bistable system and the correlations compared with the theoretical predictions from a Landau approach. The innuence of the spatial dimension and the aspect ratio on the correlations is considered and it is demonstrated how geometrical restrictions increase the uctuations and thereby increase the rate of transition between stable states. For investigations of macroscopic phenomena the moving average cellular automata are more eecient. They are used to simulate several models: { The complex Ginzburg-Landau equation is used to demonstrate oscillatory spiral waves. { The FitzHugh-Nagumo model is investigated as an example for excitable media. Excitable waves in circular and spiral form are shown, and it is demonstrated how the curvature of the wave front innuences the speed of propagation. Spiral waves can be initiated either through special initial conditions or through intrinsic uctuations. There are many analogies to the fronts in bistable media modeled with reactive lattice gas automata. { Simulations of another reaction model (the Lengyel-Epstein model of the CIMA reaction) show Turing patterns. Here the emphasis is on the possible coexistence of diierent types of spatial or spatio-temporal patterns. The cellular automata methods are extended to include advection by a ow eld, and two methods are developed for generating the ow eld. One is the lattice Boltzmann method for simulating the Navier-Stokes equations, and the other is a cellular automaton method for generating a stochastic velocity eld with characteristics similar to those observed in homogeneous turbulent ows. It is demonstrated how the nonlaminar or turbulent ow aaects the reaction-diiusion system by mixing and also by increasing the eeective diiusion coeecient. This work demonstrates that cellular automata are a powerful tool for modeling reaction-diiusion systems, including the eeects of uctuations. Acknowledgements Many thanks to everybody who has helped me along during these years, be it through advice, encouragement, comments, proofreading, or simply by being there. I would like to thank especially Professor J.-P. Boon for inviting me …