Colloquium

Our work is inspired by a remarkable study of the size distribution of schools of fish in the mid-ocean, that was performed by fisheries scientist H. S. Niwa. A natural model for rates of merging and splitting of such schools or clusters takes the form of coagulation-fragmentation equations, which resemble Boltzmann's equation from the kinetic theory of gases. In our case the equations lack an analog of Boltzmann's `H-theorem', but we are anyway able to rather completely describe the large-time dynamics and equilibrium distributions. This helps to explain the highly non-Gaussian statistics seen by Niwa and others. Our analysis relies on recent advances in complex function theory for Bernstein and Pick functions (aka Herglotz functions). Further mathematical consequences relate to the combinatorics of Fuss-Catalan numbers and to infinitely divisible sequences in probability.