We study the kinetic behavior of the growth of aggregates driven by reversible migration between any two aggregates. For the simple system with the migration rate kernel K(k;j)=K(')(k;j) proportional, variant kj(upsilon) at which the monomers migrate between the aggregates of size k and those of size j, we find that for the upsilon< or =2 case the evolution of the system always obeys a scaling law. Moreover, the typical aggregate size grows as exp(2IA(0)t) in the case of upsilon=2 and as t(1/(2-upsilon)) in the case of -1
We propose an exchange-driven aggregation growth model of population and assets with mutually catalyzed birth to study the interaction between the population and assets in their exchange-driven processes. In this model, monomer (or equivalently, individual) exchange occurs between any pair of aggregates of the same species (population or assets). The rate kernels of the exchanges of population and assets are K(k,l) = Kkl and L(k,l) = Lkl , respectively, at which one monomer migrates from an aggregate of size k to another of size l. Meanwhile, an aggregate of one species can yield a new monomer by the catalysis of an arbitrary aggregate of the other species. The rate kernel of asset-catalyzed population birth is I(k,l) = Iklmu [and that of population-catalyzed asset birth is J(k,l) = Jklnu], at which an aggregate of size k gains a monomer birth when it meets a catalyst aggregate of size l . The kinetic behaviors of the population and asset aggregates are solved based on the rate equations. The evolution of the aggregate size distributions of population and assets is found to fall into one of three categories for different parameters mu and nu: (i) population (asset) aggregates evolve according to the conventional scaling form in the case of mu < or = 0 (nu < or = 0), (ii) population (asset) aggregates evolve according to a modified scaling form in the case of nu = 0 and mu > 0 (mu = 0 and nu > 0 ), and (iii) both population and asset aggregates undergo gelation transitions at a finite time in the case of mu = nu > 0.
We propose an irreversible aggregation model driven by migration and birth-death processes with the symmetric migration rate kernel K(k;j)=K'(k;j)=Ikj(upsilon), and the birth rate J(1)k and death rate J(2)k proportional to the aggregate's size k. Based on the mean-field theory, we investigate the evolution behavior of the system through developing the scaling theory. The total mass M1 is reserved in the J(1)=J(2) case and increases exponentially with time in the J1>J2 case. In these cases, the long-time asymptotic behavior of the aggregate size distribution a(k)(t) always obeys the scaling law for the upsilon
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