Upper critical dimension in irreversible aggregation

It is shown that the mean-field approach to irreversible aggregation (Smoluchowski’s coagulation equation) leads to incorrect predictions at large times for dimensions d≤$d_c$, where the upper critical dimension $d_c$ is model dependent, nd may be arbitrarily large. The value of $d_c$ is calculated for a large class of aggregation models. Estimates for the time $t_D$, at which the Smoluchowski approach breaks down , are given. The behavior of the system for t≳$t_D$ is discussed.