  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.
