Scale-free cluster distributions from conserving merging-fragmentation processes

We propose a dynamical scheme for the combined processes of fragmentation and merging as a model system for cluster dynamics in nature and society displaying scale-invariant properties. The clusters merge and fragment with rates proportional to their sizes, conserving the total mass. The total number of clusters grows continuously but the full time-dependent distribution can be rescaled over at least 15 decades onto a universal curve which we derive analytically. This curve includes a scale-free solution with a scaling exponent of $-3/2$ for the cluster sizes