Smoluchowski Coagulation & Gelation
Clusters aggregate via multiplicative kernel K(i,j) = ij. A spanning gel forms in finite time tgel = 1 as mass drains from the sol into the infinite cluster.
dn_k/dt = (1/2)Σᵢ₊ⱼ₌ₖ K(i,j)nᵢnⱼ − nₖ Σⱼ K(k,j)nⱼ
Multiplicative kernel K(i,j)=ij. Exact solution: n_k(t) = e^{−2t}(1−e^{−t})^{k−1}/k for t<1.
At gelation t=1: power law n_k ~ k^{−5/2}. Gold dashed line shows this slope.