Scaling in Aggregation with Breakup Simulations and Mean-Field Theory

An extended scaling description of the cluster size distribution $N_s\mathrm{}(t,k)\mathrm{}$ in time-dependent coagulation-fragmentation processes ($k$ is a small breakup rate constant) is presented as $N_s\mathrm{}(t,k)\mathrm{}\sim s^{-2\mathrm{}}h(s{k}^y,t{k}^x)$) and its validity is confirmed by computer simulations. This scaling form includes the scaling description of irreversible and steady-state processes as limiting cases for both short and long times compared to a crossover time $\tau \mathrm{}\left(k\right)\mathrm{}\sim k^{-x}$. A mean-field theory is analyzed and its limits of validity are explored.