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 ${\mathrm{c}}_{\mathrm{k}}$(r)∼${\mathrm{k}}^{\mathrm{-}\mathrm{\tau }}$φ(${\mathrm{kr}}^{\mathrm{-}\mathrm{z}}$), 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 ${\mathrm{c}}_{\mathrm{k}}$(r)∼${\mathrm{k}}^{\mathrm{-}3/2}$φ(μ), with φ(μ)∼${\mathrm{\mu }}^{5/2}$ for μ=k/${\mathrm{r}}^2$→0, c(r)∼${\mathrm{r}}^{\mathrm{-}1}$, 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.