Entropy Production Beyond the Diffusion Limit: A Phase Transition in the Two-Allele Wright–Fisher Model

We study stationary entropy production as a measure of irreversibility in a two-allele Wright–Fisher population, comparing the finite-population chain with its diffusion, fixed-point Gaussian, and sequential weak-mutation reductions. Whole-generation sampling generates probability-current cycles among population compositions despite the acyclic mutation graph, making the finite chain generically irreversible, unlike its three reversible reductions. For a population of size N, with selection s=α/N and mutation μ=β/N, we establish a boundary-localization phase transition in entropy production at βc=1/2. Below it, entropy production is localized near fixation, where the minority allele has O(1) copies; above it, the polymorphic bulk dominates; at criticality, contributions are approximately uniform over logarithmic intervals of distance from fixation. In the N→∞ limit, the localization order parameter jumps from one to zero at βc, while the boundary width needed to capture a fixed fraction diverges as β→βc−, identifying a mixed-order transition. The entropy-production rate per unit diffusion time, NσN, diverges as N1−2β below criticality and as (αcothα/6)lnN at criticality, and converges to C(α,β) above it. For β>1, we also derive the first regular finite-N correction. The finite chain and its reversible diffusion share the leading drift and variance but differ in their third central moments. Since the diffusion is reversible for every β, the diffusion approximation conceals both finite-population irreversibility and its boundary-localization transition.