Coaglation-fragmentation equations are simple, nonlocal models for evolution of the size distribution of clusters, appearing widely in science and technology. But few general analytical results characterize their dynamics. Solutions can exhibit self-similar growth, singular mass transport, and weak or slow approach to equilibrium. I will review some recent results in this vein, discussing: the cutoff phenomenon (as in card shuffling) for Becker-Doering equilibration dynamics; equilibrium and spreading profiles in a data-driven model of fish school size; and an individual-based jump-process description of group-size dynamics. A special role is played by Bernstein transforms and complex function theory for Pick or Herglotz functions.
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