Quantum mechanical inspired factorization of the molecule pair propagator in theories of diffusion-influenced reactions

TL;DR

This paper introduces a quantum mechanics-inspired factorization of the molecule pair propagator in diffusion-influenced reactions, enabling easier computation and broader applicability without solving complex PDEs.

Contribution

It presents a general factorization method for the propagator that simplifies calculations and extends to various models, aiding stochastic simulation algorithms.

Findings

Allows calculation of the full propagator without solving PDEs

Applicable to classical and alternative diffusion models

Facilitates the development of more efficient stochastic algorithms

Abstract

Building on mathematical similarities between quantum mechanics and theories of diffusion-influenced reactions, we discuss how the propagator of a reacting molecule pair can be represented as a product of three factors in the Laplace domain. This representation offers several advantages. First, the full propagator can be calculated without ever having to solve the corresponding partial differential equation or path integral. Second, the representation is quite general and capable of capturing not only the classical Smoluchowski-Collins-Kimball model, but also alternative theories, as is here exemplified by the case of a delta- and step-function potential in one and two dimensions, respectively. Third, the three factors correspond to physical quantities that feature prominently in stochastic spatially-resolved simulation algorithms and hence the interpretation of current and the design of future algorithms may benefit. Finally, the representation may serve as a suitable starting point for numerical approximations that could be employed to enhance the efficiency of stochastic simulations.