Kinetics of migration-driven aggregation processes

Finite-size effects in addition and chipping processes

We investigate analytically and numerically a system of clusters evolving via collisions with clusters of minimal mass (monomers). Each collision either leads to the addition of the monomer to the cluster or the chipping of a monomer from the cluster, and emerging behaviors depend on which of the two processes is more probable. If addition prevails, monomers disappear in a time that scales as lnN with the total mass N≫1, and the system reaches a jammed state. When chipping prevails, the system remains in a quasistationary state for a time that scales exponentially with N, but eventually, a giant fluctuation leads to the disappearance of monomers. In the marginal case, monomers disappear in a time that scales linearly with N, and the final supercluster state is a peculiar jammed state; i.e., it is not extensive.

Role of zero clusters in exchange-driven growth with and without input

The exchange-driven growth model describes the mean-field kinetics of a population of composite particles (clusters) subject to pairwise exchange interactions. Exchange in this context means that upon interaction of two clusters, one loses a constituent unit (monomer) and the other gains this unit. Two variants of the exchange-driven growth model appear in applications. They differ in whether clusters of zero size are considered active or passive. In the active case, clusters of size zero can acquire a monomer from clusters of positive size. In the passive case they cannot, meaning that clusters reaching size zero are effectively removed from the system. We show that the large-time behavior is very different for the two variants of the model. We first consider an isolated system. In the passive case, the cluster size distribution tends towards a self-similar evolution and the typical cluster size grows as a power of time. In the active case, we identify a broad class of kernels for which the the cluster size distribution tends to a nontrivial time-independent equilibrium in which the typical cluster size is finite. We next consider a nonisolated system in which monomers are input at a constant rate. In the passive case, the cluster size distribution again attains a self-similar profile in which the typical cluster size grows as a power of time. In the active case, a surprising new behavior is found: the cluster size distribution asymptotes to the same equilibrium profile found in the isolated case but with an amplitude that increases linearly with time.

Reversible clustering under the influence of a periodically modulated binding rate

We study cluster-cluster aggregation and fragmentation with a periodically modulated binding rate. Using the Smoluchowski mean-field equations, we derive exact solutions for the cluster size distribution for two time courses of the binding rate: (i) harmonic modulations and (ii) on-off switching of the binding rate with dwell times tau(b) and tau(f). In both cases, the asymptotic cluster size distribution is oscillatory in time. The formation of small-sized clusters is enhanced compared to a constant binding rate below a critical cluster size s(c). There is another critical cluster size s(p) close to which the size distribution becomes quasistationary. We calculate analytically the dependence of the critical points s(c) and s(p) on the relevant system parameters for on-off switching of the binding rate. Our results are relevant whenever clustering can be externally controlled.

Competition between the catalyzed birth and death in the exchange-driven growth

We propose a three-species ( A , B , and C ) exchange-driven aggregate growth model with competition between catalyzed birth and catalyzed death. In the system, exchange-driven aggregation occurs between any two aggregates of the same species with the size-dependent rate kernel Kn(k,j)=Knkj (n=1,2,3) , and, meanwhile, monomer birth and death of species A occur under the catalysis of species B and C with the catalyzed birth and catalyzed death rate kernels I(k,j)=Ikjv and J(k,j)=Jkjv , respectively. The kinetic behavior is investigated by means of the mean-field rate equation approach. The form of the aggregate size distribution ak(t) of species A is found to depend crucially on the competition between species- B -catalyzed birth of species A and species- C -catalyzed death of species A , as well as the exchange-driven growth. The results show that (i) when exchange-driven aggregation dominates the process, ak(t) satisfies the conventional scaling form; (ii) when catalyzed birth dominates the process, ak(t) takes the conventional or generalized scaling form; and (iii) when catalyzed death dominates the process, the aggregate size distribution of species A evolves only according to some modified scaling forms.

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.

Mutually catalyzed birth of population and assets in exchange-driven growth

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.

Exchange-driven growth

We study a class of growth processes in which clusters evolve via exchange of particles. We show that depending on the rate of exchange there are three possibilities: (I) Growth-clusters grow indefinitely, (II) gelation-all mass is transformed into an infinite gel in a finite time, and (III) instant gelation. In regimes I and II, the cluster size distribution attains a self-similar form. The large size tail of the scaling distribution is Phi(x) approximately exp(-x(2-nu)), where nu is a homogeneity degree of the rate of exchange. At the borderline case nu=2, the distribution exhibits a generic algebraic tail, Phi(x) approximately x(-5). In regime III, the gel nucleates immediately and consumes the entire system. For finite systems, the gelation time vanishes logarithmically, T approximately [lnN](-(nu-2)), in the large system size limit N--> infinity. The theory is applied to coarsening in the infinite range Ising-Kawasaki model and in electrostatically driven granular layers.

Kinetics of a migration-driven aggregation process with birth and death

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<or=2 case. This model may provide a more natural description for diverse aggregation processes such as the evolution of the distribution of city population and individual wealth.