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Smoluchowski-type models for diffusion-influenced reactions (A+B -> C) can beformulated within two frameworks: the probabilistic-based approach for a pairA, B of reacting particles and the concentration-based approach for systems incontact with a bath that generates a concentration gradient of B particles thatinteract with A. Although these two approaches are mathematically similar, itis not straightforward to establish a precise mathematical relationship betweenthem. Determining this relationship is essential to derive particle-basednumerical methods that are quantitatively consistent with bulk concentrationdynamics. In this work, we determine the relationship between the twoapproaches by introducing the grand canonical Smoluchowski master equation(GC-SME), which consists of a continuous-time Markov chain that models anarbitrary number of B particles, each one of them following Smoluchowski'sprobabilistic dynamics. We show that the GC-SME recovers theconcentration-based approach by taking either the hydrodynamic or the largecopy number limit. In addition, we show that the GC-SME provides a clearstatistical mechanical interpretation of the concentration-based approach andyields an emergent chemical potential for nonequilibrium spatiallyinhomogeneous reaction processes. We further exploit the GC-SME robustframework to accurately derive multiscale/hybrid numerical methods that coupleparticle-based reaction-diffusion simulations with bulk concentrationdescriptions, as described in detail through two computational implementations.

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