dn_k/dt = ½Σ_{i+j=k} K(i,j)n_i n_j − n_k Σ_j K(k,j)n_j
The
Smoluchowski equation governs the kinetics of cluster aggregation. The kernel K(i,j) determines how fast clusters of size i and j merge. For the constant kernel, mean cluster size grows linearly in time. The sum kernel K~(i+j) gives exponential growth. The product kernel K~ij undergoes
gelation — a phase transition where a single infinite cluster forms at finite time t_gel=1. Real systems: aerosol physics, colloidal aggregation, cloud droplet growth, polymerization, and galaxy formation.