Smoluchowski Coagulation — Cluster Dynamics
Clusters of size i and j merge at rate K(i,j) to form size i+j. Depending on the kernel, clusters grow indefinitely or a gel (infinite cluster) forms in finite time — a sol-gel transition.
Total mass: — | Mean size: — | Largest: — | t: 0
Smoluchowski equation: ∂n_k/∂t = ½Σ_{i+j=k}K(i,j)n_i n_j − n_k Σ_j K(k,j)n_j + fragmentation. For K=ij (product kernel), gelation occurs at t_gel=1 — mass drains into a giant cluster (sol-gel transition). For K=i+j (sum kernel), the mean cluster size diverges as t→∞ but no finite-time gelation.