The random allocation model as a framework for constrained optimization and entropy stability

Abstract

Phase transitions and constrained optimization are central to statistical mechanics, yet undergraduate instruction often relies on models that obscure their combinatorial origins or require advanced mathematical machinery. The random allocation model with a power-law weight w(m)=m−β offers a minimal, exactly solvable alternative. We derive the equilibrium occupation distribution using the most probable configuration method, explicitly tracking the emergence of the Lagrange multiplier as a conjugate variable to particle density. Analysis of the entropy density s(ρ) reveals a real-space condensation transition at a finite critical density ρc(β) for β>2. The transition is third-order in the Ehrenfest sense for 23, with the exponent β playing a role mathematically analogous to spatial dimensionality in the ideal Bose gas. The derivation isolates the statistical core of condensation, providing a transparent training ground for the saddle-point method without the overhead of kinetic energy or temperature. Furthermore, the model resolves a persistent pedagogical challenge by explicitly distinguishing the concavity of the extensive entropy S(M) from the convexity of the intensive entropy density s(ρ) in the condensed phase. Because the analysis requires only elementary combinatorics and Stirling’s approximation, the system serves as a tractable example of higher-order transitions that can be integrated directly into advanced undergraduate or early graduate statistical mechanics courses.