  Stochastic reaction-diffusion models are now a popular tool for studying
physical systems in which both the explicit diffusion of molecules and noise in
the chemical reaction process play important roles. The Smoluchowski
diffusion-limited reaction model (SDLR) is one of several that have been used
to study biological systems. Exact realizations of the underlying stochastic
process described by the SDLR model can be generated by the recently proposed
First-Passage Kinetic Monte Carlo (FPKMC) method. This exactness relies on
sampling analytical solutions to one and two-body diffusion equations in
simplified protective domains.
  In this work we extend the FPKMC to allow for drift arising from fixed,
background potentials. As the corresponding Fokker-Planck equations that
describe the motion of each molecule can no longer be solved analytically, we
develop a hybrid method that discretizes the protective domains. The
discretization is chosen so that the drift-diffusion of each molecule within
its protective domain is approximated by a continuous-time random walk on a
lattice. New lattices are defined dynamically as the protective domains are
updated, hence we will refer to our method as Dynamic Lattice FPKMC or
DL-FPKMC. We focus primarily on the one-dimensional case in this manuscript,
and demonstrate the numerical convergence and accuracy of our method in this
case for both smooth and discontinuous potentials. We also present applications
of our method, which illustrate the impact of drift on reaction kinetics.
