Cluster growth kinetics and gelation transition
Smoluchowski's coagulation equation: dn_k/dt = ½Σ K(i,j)n_i n_j [i+j=k] − n_k Σ K(k,j)n_j. Constant kernel: n_k ~ t^{k−1}/t^{k+1} (no gelation). Sum kernel: n_k ~ exp(−k/⟨k⟩) (marginal). Product kernel K=ij: gelation at t_gel=1/N₀ — a runaway giant cluster forms in finite time.