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.
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Last modified: April 11 2016 - 18:14:43