Dynamics of infinite-range ballistic aggregation

We calculate the width of the growing interface of ballistic aggregation in the limit in which the range of the sticking interaction between the particles becomes infinite. We derive a scaling form for the width, a n d we compute the shortand long t ime exponents finding U = $ and CI =;. Furthermore, we find that the crossover exponent defining the argument of the scaling function is y = 4. We compare these exact results with computer simulations, finding excellent agreement. We also discuss the relation of' these results to those of ordinary finite-range ballistic aggregation. Finally, we present a simple expression for the density of all ballistic aggregation clusters, regardless of the range of the interaction, which agrees with known results and interpolates between the infiniteand finite-range cases. Because of its apparent simplicity and evident applicability to a variety of experimental situations, ballistic aggregation is one of the most commonly studied of the many models of non-equilibrium growth (Vold 1963, Sutherland 1966, Meakin 1985, Bensimon et a1 1984, Family and Vicsek 1985, Meakin et a1 1986). The model is believed to be applicable to several diflerent kinds of deposition processes including physical vapour deposition and molecular-beam epitaxy. In addition, the growth algorithms defining ballistic aggregation are extremely simple to state and to implement on a computer, and the resulting aggregates display many surprising and intriguing features. Unfortunately, despite its simplicity, attempts to directly analyse ballistic aggregation have not met with much success. Nevertheless, some analytical progress has been made by studying models related to ballistic aggregation, but which are easier to solve (Meakin et a1 1986, Kardar et a1 1986, Gelband and Strenski 1985). One such model is an extreme limit of ordinary ballistic aggregation in which the range of the interaction between the particles is taken to infinity. The static properties of this infinite-range ballistic aggregation ( IRBA'I have been studied recently by Gelband and Strenski (1985), who derived a number of exact results for the density and correlation functions of the aggregate. In this paper, we will discuss the growth dynamics of this model. In particular, we will compute exactly the exponents describing the longand short-time behaviour of the growing interface. To begin, let us first review the model. The simplest manifestation of ordinary ballistic aggregation is obtained by considering a two-dimensional square lattice. (The generalisation to higher dimensions will be obvious.) Place a line of L particles (the substrate) along the x axis. For simplicity, we will choose periodic boundary conditions in the x direction. At each time step, randomly choose a value of x and drop a particle in the y direction along the lattice line s = c until it sticks to the aggregate. The simplest sticking rules are that the particle dropped along the line x = c sticks at the lattice site x = c, y = max[h(c 11, h ( c ) + 1, h ( c + l ) ] , where h ( a ) is the largest value of y at which 0305-4470/87/ 186391 + 06$02.50