Scaling and crossover properties of a new solvable model of aggregation kinetics

In a previous paper, a new reaction kernel for the Smoluchowski equations of aggregation was solved exactly. This kernel, K(j, k) = 2 - q(j) - q(k), for 0 < q < 1 a real positive quantity, interpolates between two well understood exactly solved cases, namely that of K(j, k) = 2 and that of K(j, k) = j + k. This new model, however, shows a number of unexpected features, not found in either of the two limiting cases. It is shown that this model has a remarkable behaviour with respect to the commonly accepted scaling theory. On the one hand, it satisfies a rigorous form of the scaling hypothesis, but, on the other hand, it clearly violates some relations which are ordinarily assumed to follow from it. These issues are discussed, as well as the nature of the singular limit in which q is very close to one, for which our kernel becomes close to the sum kernel mentioned above. In particular, the form of the crossover between two kernels with different degrees of homogeneity can be discussed here in an exact way.