Coupling microscopic and macroscopic cellular automata

Cellular automata can model natural phenomena on many diierent levels of detail. Often, one speciic level of detail is appropriate for the problem under investigation. But in some cases a connection between the diierent levels of detail is necessary. One such case is the catalytic reaction on a surface. The homogeneous crystallographic surface can reasonably well be described by mesoscopic approaches, but in the presence of defects, a microscopic simulation (where each site of the crystal lattice is individually represented) is necessary. In this paper we present a framework for coupling diierent types of cellular automata to achieve an eecient simulation which is nevertheless detailed enough to resolve microscopic phenomena where necessary. Catalytic surface reactions Chemical reactions on a catalytic surface have often been modeled by Monte-Carlo methods or cellular automata. In fact, Monte-Caro methods can usually be converted into probabilistic cellular automata. Here we consider the surface reaction A + 1 2 B 2 ! 0 (1) which is understood to mean that some molecule B 2 is adsorbed at the surface, and splits into the molecular or atomic form B upon adsorption. A second molecule A is also adsorbed at the surface, and if A and B are neighbors on the surface, they can react to a molecule AB, which desorbs immediately in this model and does not participate in further reaction steps (therefore ! 0: reaction to \nothing"). This scheme is appropriate for the oxidation of carbon monoxide on platinum 4, 5]. A microscopic simulation of this reaction can be constructed fairly simply. Consider a regular lattice, the geometry of which reeects the crystallographic lattice present on the crystal face under consideration. On empty sites of this lattice, the species A can adsorb with probability p A. On two adjacent empty sites, a molecule of B 2 can adsorb and split into atoms B. If a site occupied by 1