Exact asymptotic solution of an aggregation model with a bell-shaped distribution.

Abstract

We present in a detailed manner the scaling theory of irreversible aggregation characterized by the set of reaction rates K(k,l)=1/k+1/l. In this case, it is possible to determine the behavior of large-size aggregates in the limit of large times in a way that allows a highly detailed analysis of the behavior of the system. This is the so-called scaling limit, in which the cluster size distribution collapses to a function of the ratio of the cluster size to a time-dependent typical size. The results confirm the far more general results of earlier work concerning a general scaling theory for so-called reaction rates of Type III, which are characterized by the property that aggregates of very different sizes react faster than comparable aggregates of similar sizes. For these, the cluster size distribution decays rapidly to zero both for sizes much larger and much smaller than the typical size, and is thus often described as being "bell-shaped". For clusters much larger than the typical size, however, an unexpected subleading correction is discovered. Finally, several results going beyond the scope of the scaling limit are obtained: in particular the behavior of concentrations for fixed cluster size in the large-time limit and the large-size behavior for clusters at a fixed time. The latter again shows subleading deviations from the expected behavior.