# Boltzmann and Fisher: The Role of Statistical Mechanical Theory in the Development of Mathematical Population Genetics

## By David R. Crawford

### Duke University / University of Pittsburgh

Abstract:

Statistical mechanical theory played a major role in Fisher’s contribution to the birth of mathematical population genetics. Fisher had a significant background in physics and he saw in Boltzmann’s work in statistical mechanics a counterpart to Darwin’s work in evolutionary theory. Fisher often drew comparisons between his models and the Kinetic Theory of Gases, and he likened the role of his Fundamental Theorem of Natural Selection in evolutionary theory to the role of the Second Law of Thermodynamics in statistical mechanics. In this paper I elaborate how Fisher employed conceptual tools from statistical mechanics in his population genetics models. I argue that Fisher did so without an adequate understanding of the formal roles or the empirical justifications of those concepts and models in physics. Three factors make the formal basis of Fisher’s theory in Boltzmann’s problematic: (a) the (in)sufficiency of basic model parameters for the systems studied (e.g., sample size, timescale, independence of units); (b) the empirical justification for idealizations (both idealized relations like Boltzmann’s Stosszahlansatz and idealized models like ensembles); and (c) the different roles of 1st- and 2nd-order relations and properties in the hierarchical models. I argue that the shortcomings of Fisher’s applications reveal important features of physical systems which make Boltzmann’s idealizations useful and they demonstrate how theorists can misapply statistical mechanical principles across disciplinary boundaries when they fail to consider these empirical constraints. I conclude with a discussion of how Fisher’s importation of conceptual tools from physics continues to affect contemporary discourse in evolutionary biology.