Equilibrium Combinatorial Self-Assembly via Generating Functions

Abstract

We develop a generating-function calculus for equilibrium combinatorial self-assembly. Starting from a bond-level specification of allowable interactions, we define a symmetry-weighted species generating function whose evaluation yields equilibrium concentrations, and an ensemble generating function (an exponential transform) that packages equilibrium probabilities of mixtures. We ground these objects in both deterministic (coagulation-fragmentation) and stochastic (master equation) dynamics, showing how detailed balance leads to the equilibrium expressions and how the exponential generating function arises as a partition function. We develop a formal power series calculus -- derivatives, integrals, exponentials, composition -- where each operation acquires a precise combinatorial interpretation. The paper is organized around two regimes: cycle-free assembly, where binding equations for the species generating function are nonlinear and the ensemble equation couples to the species generating function; and assembly with cycles, where a cycle-opening term enters the species equation and the exponential transform linearizes the ensemble equation into a closed linear PDE with operator-exponential solutions. Each regime is developed with a linear polymer worked example in which we compute equilibrium concentrations, extract canonical partition functions, and derive canonical factorial moments. A cross-linked polymer example -- combining heterotypic chain bonds with homotypic cross-links -- illustrates both regimes together, yielding a factorized canonical partition function and an explicit gelation surface.