Self-similarity in a cut-and-paste model of coarsening

We study certain redistribution processes for a resource distributed among a large number of resource holders (clusters). In the processes we consider, the smallest clusters are cut up into pieces according to a given statistical law, and the pieces are randomly redistributed among the remaining clusters. We derive an evolution equation for the cluster size distribution, show that self-similar solutions exist and characterize their structure. In a limiting case when the pieces are small, we show that in general solutions approach a singular self-similar form.