Stable Distributions of Growing and Coalescing Droplets

We consider two simplified models of the formation of patterns emerging from the growth and coalescence of water droplets on a substrate (breath figures). The study is restricted here to the case of a one-dimensional substrate. In the first model we assume a monodisperse distribution of droplet sizes. In the second model, obtained as a mean-field approximation of the first one, the distribution of distances between neighbouring droplets obeys a Smoluchowski equation. We solve this equation analytically to obtain the coverage of the line (fraction of it covered by droplets) and the distribution of distances between droplets. We conclude by discussing the relevance of the random parking problem for breath figures.