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

Abstract

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.