Scale-invariant regimes in one-dimensional models of growing and coalescing droplets

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Abstract
We consider several simplified models of breath figures in one dimension. For all these models, the combined effects of growth and of coalescence of droplets lead to a scale-invariant regime with a stable distribution of the distances between droplets. We show that at the mean-field level there exist one-parameter families of such stable distributions, each distribution being characterized by its decay at infinity. We explain how the mean-field theory can be improved by taking into account the effect of pair or higher correlations. For some models one can check that the pair and higher correlations are factorized, meaning that correlations are absent and that therefore the mean-field theory is exact. Finally, we show that a very simple model of domain growth related to spinodal decomposition, the one-dimensional Potts model in the limit of an infinite number of states, also possesses a one-parameter family of stable distributions analogous to what we obtained for breath figures.