Coagulation processes with a phase transition

Abstract Smoluchowski's coagulation equation with a collection kernel K(x, y) ∼ (xy)ω with 1 2 describes a gelation transition (formation of an infinite cluster after a finite time tc (gel point)). For general ω and t > tc the size distribution is c(x, t) ∼ x−τ for x → ∞ with τ = ω + 3 2 . For ω = 1, we determine c(x, t) and the time dependent sol mass M(t) for arbitrary initial distribution in pre- and post-gel stage, where c(x, t) ∼ x − 5 2 exp (−x/x c ) for large x and t c(x, t) ∼ (− M ) 1 2 x − 5 2 for large xt and t > tc. Here xc is a critical cluster size diverging as (t - tc)−2 as t ↑ tc. For initial distributions such that c(x, 0) ∼ xp-2 as x → 0, we find M(t) ∼ t−p/(p+1) as t → ∞. New explicit post-gel solutions are obtained for initial gamma distributions, c(x, 0) ∼ xp-2e−px (p > 0) in the form of a power series (convergent for all t), and reducing for p = ∞ to the solution for monodisperse initial conditions. For p = 1, the solution is found in closed form.