Cluster distributions in physics and genetic diversity

An underlying similarity in the mathematical structure between a problem in the physical sciences and a problem in the biological sciences is presented. An isomorphism is established between cluster distributions in physics and issues related to genetic diversity in biology as given by the Ewens sampling theory [Theor. Population Bio. 3, 87 (1972)]. Allelic or genetic diversity, as measured by the number and frequency of different alleles (gene types), has a correspondence with the size distribution of clusters in physics. The rate of mutation in genetics is shown to have its parallel in Richardson’s thermionic emission rate in physics. Using methods from combinatorial analysis and from the symmetric group ${\mathit{S}}_{\mathit{n}}$, simple formal connections between these two areas are developed. The logarithmic series of Fisher, Corbet, and Williams [J. Animal Ecol. 12, 42 (1943)] for species abundance appears in the solution developed as does a scale-invariant hyperbolic function. Maximum entropy methods are discussed. Even though the underlying dynamical processes are quite different in these two areas, new insights and the possibility that methods developed in one area may be used with advantage in the other area may follow from this correspondence.