Algebra for classical multiparticle complexes

Theoretical studies in biophysics, polymer physics, and other fields often focus on the behavior of large complexes of classically behaving molecules held together by site-specific interactions. Mathematical methods have yet to be developed, however, for handling the combinatorial explosion of complexes that often occur in these systems. We introduce a general mathematical language for describing the formation and thermodynamics of classical multiparticle complexes in terms of algebraically-defined assembly rules. At the heart of this formalism is a Fock space that supports the creation and annihilation of not only individual particles but also multiparticle interactions and binding site occupancies. The action of these rules on component particles occurs through a manifestation of Wick's theorem. To facilitate mathematical computations, we introduce a compact diagrammatic representation that makes the algebra visually intuitive. We demonstrate the formalism's utility in evaluating statistical properties of assembly in static and dynamic environments for a number of models in chemical kinetics and polymer physics, and highlight its efficacy as a basis for deterministic and stochastic analysis of infinitely-extendable systems with varying heterogeneous complexity.