Combinatorics of chemical reaction systems

TL;DR

This paper introduces a stochastic mechanics framework for chemical reaction systems that combines statistical physics and combinatorics, providing analytical solutions for various reaction types.

Contribution

It develops a unified approach to formulate and solve evolution equations for different generating functions in chemical reactions using combinatorial and orthogonal polynomial techniques.

Findings

Analytical solutions for all six elementary reaction types.

Connection between stochastic generators and probability distribution transformations.

Application of Sobolev-Jacobi polynomials to binary reactions.

Abstract

We propose a concise stochastic mechanics framework for chemical reaction systems that allows to formulate evolution equations for three general types of data: the probability generating functions, the exponential moment generating functions and the factorial moment generating functions. This formulation constitutes an intimate synergy between techniques of statistical physics and of combinatorics. We demonstrate how to analytically solve the evolution equations for all six elementary types of single-species chemical reactions by either combinatorial normal-ordering techniques, or, for the binary reactions, by means of Sobolev-Jacobi orthogonal polynomials. The former set of results in particular highlights the relationship between infinitesimal generators of stochastic evolution and parametric transformations of probability distributions.