Mean field limits of particle-based stochastic reaction-drift-diffusion models

TL;DR

This paper derives and proves the mean field limit of particle-based stochastic reaction-drift-diffusion models, resulting in nonlinear PDEs that incorporate potential interactions, with numerical evidence supporting the convergence and impact of these interactions.

Contribution

It extends previous models by including drift and potential interactions, deriving new PDEs for the mean field limit, and demonstrating their properties through numerical simulations.

Findings

Mean field PDEs include nonlinear concentration-dependent coefficients.

Two-body repulsive potentials significantly affect reaction dynamics.

Numerical solutions converge to the mean field equations as population size increases.

Abstract

We consider particle-based stochastic reaction-drift-diffusion models where particles move via diffusion and drift induced by one- and two-body potential interactions. The dynamics of the particles are formulated as measure-valued stochastic processes (MVSPs), which describe the evolution of the singular, stochastic concentration fields of each chemical species. The mean field large population limit of such models is derived and proven, giving coarse-grained deterministic partial integro-differential equations (PIDEs) for the limiting deterministic concentration fields' dynamics. We generalize previous studies on the mean field limit of models involving only diffusive motion, with care to formulating the MVSP representation to ensure detailed balance of reversible reactions in the presence of potentials. Our work illustrates the more general set of PIDEs that arise in the mean field limit, demonstrating that the limiting macroscopic reactive interaction terms for reversible reactions obtain additional nonlinear concentration-dependent coefficients compared to the purely diffusive case. Numerical studies are presented which illustrate that two-body repulsive potential interactions can have a significant impact on the reaction dynamics, and also demonstrate the empirical numerical convergence of solutions to the PBSRDD model to the derived mean field PIDEs as the population size increases.