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Simulations of Cellular Automata

HARMEN J. BUSSEMAKERa

Department of Physics, University of Maryland, College Park, Maryland 20742

Fluid-type lattice gas automata (LGA) have provided a fruitful method for studyingspatially extended systems of interacting particles. LGA are discrete dynamical sys-tems in which the space variable is restricted to a lattice and the time variable is in-teger-valued. In addition, the velocity states are discrete and finite in number. Thedynamics of LGA is specified by free motion for each particle between lattice pointson each time step, followed by “local collision rules” specifying transitions to a newset of velocity states depending on the particles at each node. Fluid-type LGA arefurther restricted to have collision rules respecting mass and momentum conserva-tion, and a lattice structure consistent with macroscopic fluid dynamics at largespace and time scales. A primary advantage of such models is that they are ideallysuited to the architecture of modern computers and the consequent efficient simula-tion of the dynamics using massively parallel computer algorithms. Originally mo-tivated as an alternative means to study nonlinear fluid dynamics (e.g., turbulence),LGA have proved to be fruitful testing grounds for the methods of nonequilibriumstatistical mechanics. In this presentation we describe a recent study of long-rangecorrelations and pattern formation using LGA. The adaptation of kinetic theorymethods to LGA allow more detailed comparisons with computer simulations thanare possible for real fluids.

LGA with strictly local collision rules that violate detailed balance have a non-Gibbsian equilibrium state, and we have studied equal time spatial correlations inthis state. The existence of local conservation laws for mass and momentum leads tothe existence of long-range algebraic decay of the pair correlation function. The pri-mary theoretical analysis is based on a ring kinetic theory going beyond the Boltz-mann equation for the occupation numbers to describe equal time correlations aswell [1], [2]. The kinetic equations have been solved numerically to determine boththe equilibrium occupation numbers and the correlation functions for an initial statewhich approaches the asymptotic equilibrium state. In this way it is possible to studyboth the equilibrium correlations and the approach to equilibrium. One important re-sult is the confirmation of the mode coupling mechanism for long-range correlationsas well as long time algebraic approach to equilibrium [3]. Two specific LGA with-out detailed balance collision rules were considered, a system of interacting randomwalkers on a square lattice with Fermi exclusion rules, and a fluid-type LGA definedon a triangular lattice. The agreement between computer simulation of the micrody-namics and the results of kinestic theory are excellent.

aPresent address: The Rockefeller University, Box 25, 1230 York Avenue, New York, NewYork 10021.

27BUSSEMAKER: SIMULATIONS OF CELLULAR AUTOMATA

REFERENCES

1. ERNST, M.H. & H.J. BUSSEMAKER. 1995. J. Stat. Phys. 81: 515–528.2. BUSSEMAKER, H.J. & M.H. ERNST. 1996. Phys. Rev. E. 53: 5837–5851.3. BRITO, R., H.J. BUSSEMAKER, M.H. ERNST & J. MATSUI. 1995. Phys. Rev. E. 52:

2657–2667.