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

Max Heldman, Samuel A. Isaacson, Qianhan Liu, Konstantinos Spiliopoulos

TL;DR

The work develops a rigorous mean-field theory for particle-based stochastic reaction-drift-diffusion models where particles diffuse and drift under one- and two-body potentials. Using measure-valued stochastic processes and Stroock–Varadhan martingale methods, it proves a large-population limit in which the stochastic particle dynamics converge to a deterministic system of partial integro-differential equations for the molar concentration fields, with reaction terms incorporating detailed balance through acceptance probabilities. A generalized Fröhner-Noé construction is employed to ensure detailed balance, yielding nonlinear concentration-dependent coefficients in the macroscopic PDEs. Numerical simulations, including CRDME discretizations and Fourier-based mean-field solvers, demonstrate empirical convergence of the particle model to the mean-field PIDEs as the system size grows and reveal the significant impact of potential interactions on reaction dynamics. The framework unifies drift-diffusion, inter-particle interactions, and chemistry into a rigorous mean-field limit, with applications to reversible reactions and complex multispecies networks.

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.

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

TL;DR

The work develops a rigorous mean-field theory for particle-based stochastic reaction-drift-diffusion models where particles diffuse and drift under one- and two-body potentials. Using measure-valued stochastic processes and Stroock–Varadhan martingale methods, it proves a large-population limit in which the stochastic particle dynamics converge to a deterministic system of partial integro-differential equations for the molar concentration fields, with reaction terms incorporating detailed balance through acceptance probabilities. A generalized Fröhner-Noé construction is employed to ensure detailed balance, yielding nonlinear concentration-dependent coefficients in the macroscopic PDEs. Numerical simulations, including CRDME discretizations and Fourier-based mean-field solvers, demonstrate empirical convergence of the particle model to the mean-field PIDEs as the system size grows and reveal the significant impact of potential interactions on reaction dynamics. The framework unifies drift-diffusion, inter-particle interactions, and chemistry into a rigorous mean-field limit, with applications to reversible reactions and complex multispecies networks.

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.
Paper Structure (26 sections, 7 theorems, 167 equations, 1 figure)

This paper contains 26 sections, 7 theorems, 167 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notations and preliminary definitions
  3. Generator and process level description
  4. Process level description.
  5. Assumptions for the Mean Field Limit and Main Results
  6. Assumptions for the Mean Field Limit.
  7. Assumptions on the molar concentration fields
  8. Assumptions on potentials
  9. Assumptions on reaction functions, placement densities and acceptance probabilities
  10. Mean field limit and examples.
  11. A specific example: Fröhner-Noé Model
  12. Acceptance probability.
  13. Examples.
  14. Simulations
  15. Description of model problem
  16. ...and 11 more sections

Key Result

Theorem 5.1

(Mean field large population limit) Given Assumptions L:uniformboundofmeasures-convpi, the sequence of measure-valued processes $\{\bm{\mu}_{t}^{\zeta}\}_{t \in[0, T]} \in \mathbb{D}_{\otimes_{j=1}^{J} M_{F}(\mathbb{R}^{d})}([0, T])$ is relatively compact in $\mathbb{D}_{\otimes_{j=1}^{J} M_{F}(\mat

Figures (1)

  • Figure 1: Comparison of the CRDME and MFM solutions. The CRDME data are obtained by averaging the results of 280,000 simulations.

Theorems & Definitions (33)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 4.1
  • Remark 4.1
  • Definition 4.2
  • ...and 23 more