Pregelation Behaviour of Coagulation Processes with the Constant-Reaction-Number Kernel

We propose an irreversible binary coagulation model with a constant-reaction-number kernel, in which, among all the possible binary coagulation reactions, only p reactions are permitted to take place at every time. By means of the generalized rate equation we investigate the kinetic behaviour of the system with the reaction rate kernel K(i;j)=(ij)^ω (0≤ ω <1/2), at which an i-mer and a j-mer coagulate together to form a large one. It is found that for such a system there always exists a gelation transition at a finite time t_c, which is in contrast to the ordinary binary coagulation with the same rate kernel. Moreover, the pre-gelation behaviour of the cluster size distribution near the gelation point falls in a scaling regime and the typical cluster size grows as (t_c-t)^-1/(1-2ω). On the other hand, our model can also provide some predictions for the evolution of the cluster distribution in multicomponent complex networks.