Rate equation for cluster size distribution n_k(t):
dn_k/dt = ½ Σ K(i,j)nᵢnⱼ − n_k Σ K(k,j)nⱼ
where K(i,j) is the coagulation kernel. Exact solutions exist for special kernels.
Constant K=1: n_k(t) = 4/(t+2)² · (t/(t+2))^(k−1). Mean size ⟨k⟩ = 1 + t/2.
Additive K=i+j: sol via generating functions. Gelation occurs at finite time only for product kernel.
Product K=ij: gelation at t_c = 1 — a giant cluster forms suddenly (sol-gel transition). For t > t_c, mass is lost from the finite-cluster distribution.
The bottom panel shows the evolving cluster size distribution (log-log), which develops a power-law tail near gelation.