A Convergent Reaction-Drift-Diffusion Master Equation with Interaction Potentials
Abstract
We extend the convergent reaction-diffusion master equation (CRDME) framework on unstructured grids to include two-body interaction potentials between particles. Such interactions are essential for modeling systems in which particles experience electrostatic repulsion, volume exclusion, or van der Waals attraction. Our approach discretizes the weak form of the forward equation for a particle-based stochastic reaction- drift-diffusion model to obtain a new CRDME by combining two complementary techniques: an edge- averaged finite element method with mass-lumping quadrature for spatial transport terms that include potential interactions, and a mix of finite volume and mass-lumping quadrature approximations for reaction terms. This yields transition rates for a spatial jump process of particles hopping between dual mesh voxels and undergoing reactions. The resulting spatially discrete CRDME retains key physical properties of the continuum system in closed domains. It preserves the Gibbs-Boltzmann equilibrium distribution in the absence of reactions, and satisfies detailed balance of both drift-diffusion and reactive fluxes for systems with reversible reactions. Numerical comparisons with Brownian dynamics simulations validate our method, and we demonstrate empirical second-order convergence as the mesh is refined.
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BibTeX
@misc{heldman2026convergent,
title = {{A Convergent Reaction-Drift-Diffusion Master Equation with Interaction Potentials}},
author = {Max Heldman and Samuel A. Isaacson},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09546},
eprint = {2609.09546},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09546},
}
RIS
TY - UNPB
TI - A Convergent Reaction-Drift-Diffusion Master Equation with Interaction Potentials
AU - Max Heldman
AU - Samuel A. Isaacson
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09546
UR - https://arxiv.org/abs/2609.09546
AB - We extend the convergent reaction-diffusion master equation (CRDME) framework on unstructured grids to include two-body interaction potentials between particles. Such interactions are essential for modeling systems in which particles experience electrostatic repulsion, volume exclusion, or van der Waals attraction. Our approach discretizes the weak form of the forward equation for a particle-based stochastic reaction- drift-diffusion model to obtain a new CRDME by combining two complementary techniques: an edge- averaged finite element method with mass-lumping quadrature for spatial transport terms that include potential interactions, and a mix of finite volume and mass-lumping quadrature approximations for reaction terms. This yields transition rates for a spatial jump process of particles hopping between dual mesh voxels and undergoing reactions. The resulting spatially discrete CRDME retains key physical properties of the continuum system in closed domains. It preserves the Gibbs-Boltzmann equilibrium distribution in the absence of reactions, and satisfies detailed balance of both drift-diffusion and reactive fluxes for systems with reversible reactions. Numerical comparisons with Brownian dynamics simulations validate our method, and we demonstrate empirical second-order convergence as the mesh is refined.
ER -