Coagulation with a steady point monomer source

We investigate the phenomenon of coagulation with constant feed-in of monomers at a single point. For spatial dimension d>4, the steady-state cluster concentration, c(r), obeys Laplace's equation while for d<4, the steady-state concentration of clusters of mass k a distance r from the source scales as cₖ(r){~}k^-τ{φ}(kr^-z), with z=4-d, {τ}=(d-6)/(d-4) for d>2, and z=2, {τ}=1+d/2 for d<2. For the linear chain, we outline an exact solution for which cₖ(r){~}k^-3/2{φ}({μ}), with {φ}({μ}){~}μ5/2 for {μ}=k/r²{→}0, c(r){~}r^-1, and the number of clusters increases with time t as lnt. The effects of cluster drift and the presence of a sink are also considered.