Moment analysis of the cluster-size-distribution approach to scaling during coagulation

We study the temporal approach of a cluster size distribution to its asymptotic scaling form. By enforcing consistency between the distribution's zeroth moment derived from both the Smoluchowski equation and the scaling distribution ansatz, we find values for the scaling exponents w and z in terms of the scaling exponent {τ} and the kernel homogeneity {λ} which are not equivalent to their asymptotic, scaling forms. The predicted values do agree well, however, with intermediate time values found in simulations by Kang, Redner, Meakin, and Leyvraz [Phys Rev. A 33, 1171 (1986)]. By enforcing consistency between all moment orders, the asymptotic exponent values are found. These results imply the lowest-order moments approach their scaling values quickest.