Coagulation-fragmentation kinetics and gelation transition
The Smoluchowski coagulation equation dc_k/dt = ½Σᵢ₊ⱼ₌ₖ K(i,j)cᵢcⱼ − cₖΣⱼ K(k,j)cⱼ governs cluster size distribution. The product kernel K(i,j)=ij leads to gelation at finite time t_gel = 1/M₀ (mass diverges — a sol-gel transition). The constant kernel gives exponential distributions; the sum kernel gives log-normal. Scaling solutions c_k ~ k^(-τ)f(k/s(t)) reveal universal exponents τ=5/2 (product) and τ=3/2 (sum).