title: The free energy method and the Wright-Fisher model with 2 alleles.
abstract: We systematically investigate the Wright-Fisher model of population genetics with the free energy functional formalism of statistical mechanics and in the light of recent mathematical work on the connection between Fokker-Planck equations and free energy functionals. In statistical physics, entropy increases, or equivalently, free energy decreases, and the asymptotic state is given by a Gibbs-type distribution. This also works for the Wright-Fisher model when rewritten in divergence to identify the correct free energy functional. We not only recover the known results about the stationary distribution, that is, the asymptotic equilibrium state of the model, in the presence of positive mutation rates and possibly also selection, but can also provide detailed formulae for the rate of convergence towards that stationary distribution. In the present paper, the method is illustrated for the simplest case only, that of two alleles.
authors.affiliation: Max Planck Institute for Mathematics in the Sciences, Inselstr.22, 04103, Leipzig, Germany. trandat@mis.mpg.de.
authors.forename: Tat Dat
authors.initials: TD
authors.lastname: Tran
authors.affiliation: Max Planck Institute for Mathematics in the Sciences, Inselstr.22, 04103, Leipzig, Germany. julian.hofrichter@mis.mpg.de.
authors.forename: Julian
authors.initials: J
authors.lastname: Hofrichter
authors.affiliation: Max Planck Institute for Mathematics in the Sciences, Inselstr.22, 04103, Leipzig, Germany. jost@mis.mpg.de.|Santa Fe Institute for the Sciences of Complexity, Santa Fe, NM, 87501, USA. jost@mis.mpg.de.
authors.forename: Jürgen
authors.initials: J
authors.lastname: Jost
mesh_terms: D000483:Alleles; D000818:Animals; D005075:Biological Evolution; D003198:Computer Simulation; D019277:Entropy; D040961:Genetic Drift; D005828:Genetics, Population; D006801:Humans; D008957:Models, Genetic; D015233:Models, Statistical; D009154:Mutation / Q000235:genetics*
keywords: Entropy; Entropy production; Free energy functional; Mutations; Random genetic drift; Reversible distribution; Selection; Stationary distribution; Wright–Fisher model