Long-time behavior of the cluster size distribution in joint coagulation processes

Kinetics of migration-driven aggregation processes on scale-free networks

We propose a solvable model for the migration-driven aggregate growth on completely connected scale-free networks. A reversible migration system is considered with the produce rate kernel K(k;l|i;j) approximately k(u)i(upsilon)(lj)(nu) or the generalized kernel K(k;l|i;j) approximately (k(upsilon)i(omega)+k(omega)i(upsilon)(lj)(nu), at which an i-mer aggregate locating on the node with j links gains one monomer from a k-mer aggregate locating on the node with l links. It is found that the evolution behavior of the system depends crucially on the details of the rate kernel. In some cases, the aggregate size distribution approaches a scaling form and the typical size S(t,l) of the aggregates locating on the nodes with l links grows infinitely with time; while in other cases, a gelation transition may emerge in the system at a finite critical time. We also introduce a simplified model, in which the aggregates independently gain or lose one monomer at the rate I(1)(k;l)=I(2)(k;l) proportional to k(omega)l(nu), and find the similar results. Most intriguingly, these models exhibit that the evolution behavior of the total distribution of the aggregates with the same size is drastically different from that for the corresponding system in normal space. We test our analytical results with the population data of all counties in the U.S. during the past century and find good agreement between the theoretical predictions and the realistic data.