  The classical models for irreversible diffusion-influenced reactions can be
derived by introducing absorbing boundary conditions to over-damped continuous
Brownian motion (BM) theory. As there is a clear corresponding stochastic
process, the mathematical description takes both Kolmogorov forward equation
for the evolution of the probability distribution function and the stochastic
sample trajectories. This dual description is a fundamental characteristic of
stochastic processes and allows simple particle based simulations to accurately
match the expected statistical behavior. However, in the traditional theory
using the back-reaction boundary condition to model reversible reactions with
geminate recombinations, several subtleties arise: it is unclear what the
underlying stochastic process is, which causes complications in producing
accurate simulations; and it is non-trivial how to perform an appropriate
discretization for numerical computations. In this work, we derive a discrete
stochastic model that recovers the classical models and their boundary
conditions in the continuous limit. In the case of reversible reactions, we
recover the back-reaction boundary condition, unifying the back-reaction
approach with those of current simulation packages. Furthermore, all the
complications encountered in the continuous models become trivial in the
discrete model. Our formulation brings to attention the question: With
computations in mind, can we develop a discrete reaction kinetics model that is
more fundamental than its continuous counterpart?
