  We develop a convergent reaction-drift-diffusion master equation (CRDDME) to
facilitate the study of reaction processes in which spatial transport is
influenced by drift due to one-body potential fields within general domain
geometries. The generalized CRDDME is obtained through two steps. We first
derive an unstructured grid jump process approximation for reversible
diffusions, enabling the simulation of drift-diffusion processes where the
drift arises due to a conservative field that biases particle motion.
Leveraging the Edge-Averaged Finite Element method, our approach preserves
detailed balance of drift-diffusion fluxes at equilibrium, and preserves an
equilibrium Gibbs-Boltzmann distribution for particles undergoing
drift-diffusion on the unstructured mesh. We next formulate a
spatially-continuous volume reactivity particle-based reaction-drift-diffusion
model for reversible reactions of the form $\textrm{A} + \textrm{B}
\leftrightarrow \textrm{C}$. A finite volume discretization is used to generate
jump process approximations to reaction terms in this model. The discretization
is developed to ensure the combined reaction-drift-diffusion jump process
approximation is consistent with detailed balance of reaction fluxes holding at
equilibrium, along with supporting a discrete version of the continuous
equilibrium state. The new CRDDME model represents a continuous-time
discrete-space jump process approximation to the underlying volume reactivity
model. We demonstrate the convergence and accuracy of the new CRDDME through a
number of numerical examples, and illustrate its use on an idealized model for
membrane protein receptor dynamics in T cell signaling.
