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.

翻译：我们发展了一个收敛的反应-漂移-扩散主方程（CRDDME），以促进对在一般区域几何内受单体力场漂移影响的空间输运反应过程的研究。广义CRDDME通过两步构建。首先，我们推导了可逆扩散的非结构化网格跳跃过程近似，从而模拟由保守场偏置粒子运动引发的漂移-扩散过程。通过利用边平均有限元方法，我们的方法保持了平衡态下漂移-扩散通量的细致平衡，并保留了非结构化网格上经历漂移-扩散的粒子的平衡吉布斯-玻尔兹曼分布。其次，我们为形如$\textrm{A} + \textrm{B} \leftrightarrow \textrm{C}$的可逆反应建立了一个空间连续的体积反应性粒子基反应-漂移-扩散模型。采用有限体积离散化对该模型中的反应项生成跳跃过程近似。该离散化确保组合的反应-漂移-扩散跳跃过程近似与平衡态下反应通量的细致平衡一致，同时支持连续平衡态的离散版本。新的CRDDME模型是对底层体积反应性模型的连续时间离散空间跳跃过程近似。通过多个数值算例展示了新CRDDME的收敛性和准确性，并将其应用于T细胞信号传导中膜蛋白受体动力学的理想化模型。