Cluster size distribution in irreversible aggregation at large times

It is assumed that the size distribution c k (t) satisfies Smoluchowski's coagulation equation with rate coefficients K(i, j), behaving as K(i, j) approximately i mu j nu (i<<j), and find the following: in gelling and nongelling systems of class I ( mu >0) the general solution c k (t) approaches for t to infinity the exact solution Cb k /t(k=1, 2, . . .), where the b k 's are independent of the initial conditions c k (0), and can be determined from a recursion relation. In class II systems ( mu =0), c k (t)/c 1 (t) to b k (t to infinity , k=1, 2, . . .), but the b k 's depend on c k (0). Only in the scaling limit (k to infinity , s(t) to infinity with k/s(t)=finite; s(t) is the mean cluster size) does c k (t) approach a form independent of the initial distribution. Class III, where c k (t)/c 1 (t) to infinity (t to infinity , k=2, 3, . . .), has not been considered here.