Aggregation with multiple conservation laws.

Irreversible aggregation processes underly many natural phenomena including, e. g., polymerization [1], gelation [2], island growth [3-4] and aerosols [5]. The classical rate theory of Smoluchowski describes the kinetics of such processes [6-9]. Recently, scaling [10-16] and exact [17-23] theoretical studies showed that spatial correlations play a crucial role in low dimensions. While the above examples are diffusive driven, there are physical situations such as formation of large-scale structure of the universe [24] and clustering in traffic flows [25], where the aggregates move ballistically. So far, theories of ballistic aggregation [26] have been restricted to scaling arguments [26-30]. In the ballistic aggregation process both the mass and the momentum are conserved. In polymerization processes involving copolymers, each monomer species mass is a conserved quantity. Hence, we study aggregation processes with multiple conservation laws. The simplest example for such a system is aggregation with k distinct species. Both the multivariate distribution and the single variable distributions are of interest. We present exact solutions to the time dependent and the steady state mean-field rate equations. Although they are straightforward generalizations to the well known solutions they exhibit interesting behaviors. An asymptotic analysis shows that fluctuations associated with a single conserved quantity are Gaussian in nature. As a result, an additional “diffusive” size scale emerges. We apply the above theory to ballistic aggregation as well as diffusive aggregation. In the case of ballistic aggregation, we use an approximate collision rate to obtain a solution to the Boltzmann equation in arbitrary dimension. While this approach agrees with the scaling arguments, it suggests that for a given mass, the momentum distribution is Gaussian. We compare these predictions with one and two dimensional simulations. Furthermore, we consider steady state properties of the aggregation process by introducing input of particles. For homogeneous input, a novel time scale describing the density relaxation is found. In the case of a localized input, clustering occurs only for d ≤ 2. Additionally, we apply the theory to two-species aggregation with diffusing particles. Using the density dependent reaction rate, we obtain the leading scaling behavior of the two relevant mass scales. This paper is organized as follows. In section II, we present exact solutions of the rate equation theory. We investigate time dependent as well as steady state properties of the process. We then apply the theory to ballistic aggregation with momentum conserving collisions (section III) and to diffusive driven aggregation (section IV). We conclude with a discussion and suggestions for further research in section V.