Grand canonical diffusion-influenced reactions: a stochastic theory with applications to multiscale reaction-diffusion simulations

TL;DR

This paper introduces the grand canonical Smoluchowski master equation (GC-SME), a stochastic model that links probabilistic particle dynamics with concentration-based reaction models, enabling accurate multiscale reaction-diffusion simulations.

Contribution

It establishes a rigorous relationship between probabilistic and concentration-based models using GC-SME, providing a statistical mechanical interpretation and new multiscale numerical methods.

Findings

GC-SME recovers concentration-based approach in large number limit

Provides a statistical mechanical interpretation of concentration models

Enables accurate coupling of particle and bulk concentration simulations

Abstract

Smoluchowski-type models for diffusion-influenced reactions (A+B -> C) can be formulated within two frameworks: the probabilistic-based approach for a pair A, B of reacting particles and the concentration-based approach for systems in contact with a bath that generates a concentration gradient of B particles that interact with A. Although these two approaches are mathematically similar, it is not straightforward to establish a precise mathematical relationship between them. Determining this relationship is essential to derive particle-based numerical methods that are quantitatively consistent with bulk concentration dynamics. In this work, we determine the relationship between the two approaches by introducing the grand canonical Smoluchowski master equation (GC-SME), which consists of a continuous-time Markov chain that models an arbitrary number of B particles, each one of them following Smoluchowski's probabilistic dynamics. We show that the GC-SME recovers the concentration-based approach by taking either the hydrodynamic or the large copy number limit. In addition, we show that the GC-SME provides a clear statistical mechanical interpretation of the concentration-based approach and yields an emergent chemical potential for nonequilibrium spatially inhomogeneous reaction processes. We further exploit the GC-SME robust framework to accurately derive multiscale/hybrid numerical methods that couple particle-based reaction-diffusion simulations with bulk concentration descriptions, as described in detail through two computational implementations.