Equilibrium in a Reaction Network of Assemblies

Abstract

We study a mean-field reaction network whose species are assemblies built from identical atoms by reversible coagulation and fragmentation. Each assembly is an ordered binary tree, so the number of species of a given length grows combinatorially, as the Catalan numbers. The model nonetheless admits an explicit equilibrium and tractable stochastic dynamics. A finite volume $V$ sets a crossover length $l_c \sim \ln V$ that splits the equilibrium into two sectors. Below $l_c$ each assembly occurs in many copies and the rank-frequency distribution is Zipf-like; above $l_c$ individual species are rare and fluctuation-dominated. The statistical weight of the rare sector decays slowly with volume, controlling the finite-size scaling of diversity, Shannon entropy, and other assembly-weighted observables. The equilibrium also admits a transparent grand-canonical description in terms of a bond energy and an atomic chemical potential. Together these results make the model a controlled neutral baseline against which selection and driving in richer assembly networks can be measured.