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Propagation of chaos for multi-species moderately interacting particle systems up to Newtonian singularity

José Antonio Carrillo, Shuchen Guo, Alexandra Holzinger

TL;DR

This work delivers a rigorous, quantitative propagation-of-chaos result for multi-species moderately interacting particle systems with Coulomb/Newtonian singularities, deriving the limiting aggregation-diffusion PDEs without additional particle-level cutoffs. The authors integrate a two-step limiting strategy: a mean-field/relative-entropy transition from the particle system to an intermediate regularised PDE, followed by a PDE-error estimate to the singular limiting system, aided by a stopping-time argument to prove convergence in probability. The main contributions include an algebraic-in-$N$ strong $L^1$ convergence rate, a coupled relative-entropy/$L^2$-distance framework between intermediate and limiting PDEs, and a global well-posedness theory for the multi-species aggregation-diffusion system under small initial data. The results bridge microscopic stochastic dynamics with singular cross-diffusion PDEs applicable to physics, biology, and social dynamics, and lay groundwork for subsequent fluctuation analysis and higher-regularity improvements.

Abstract

We derive a class of multi-species aggregation-diffusion systems from stochastic interacting particle systems via relative entropy method with quantitative bounds. We show an algebraic $L^1$-convergence result using moderately interacting particle systems approximating attractive/repulsive singular potentials up to Newtonian/Coulomb singularities without additional cut-off on the particle level. The first step is to make use of the relative entropy between the joint distribution of the particle system and an approximated limiting aggregation-diffusion system. A crucial argument in the proof is to show convergence in probability by a stopping time argument. The second step is to obtain a quantitative convergence rate to the limiting aggregation-diffusion system from the approximated PDE system. This is shown by evaluating a combination of relative entropy and $L^2$-distance.

Propagation of chaos for multi-species moderately interacting particle systems up to Newtonian singularity

TL;DR

This work delivers a rigorous, quantitative propagation-of-chaos result for multi-species moderately interacting particle systems with Coulomb/Newtonian singularities, deriving the limiting aggregation-diffusion PDEs without additional particle-level cutoffs. The authors integrate a two-step limiting strategy: a mean-field/relative-entropy transition from the particle system to an intermediate regularised PDE, followed by a PDE-error estimate to the singular limiting system, aided by a stopping-time argument to prove convergence in probability. The main contributions include an algebraic-in- strong convergence rate, a coupled relative-entropy/-distance framework between intermediate and limiting PDEs, and a global well-posedness theory for the multi-species aggregation-diffusion system under small initial data. The results bridge microscopic stochastic dynamics with singular cross-diffusion PDEs applicable to physics, biology, and social dynamics, and lay groundwork for subsequent fluctuation analysis and higher-regularity improvements.

Abstract

We derive a class of multi-species aggregation-diffusion systems from stochastic interacting particle systems via relative entropy method with quantitative bounds. We show an algebraic -convergence result using moderately interacting particle systems approximating attractive/repulsive singular potentials up to Newtonian/Coulomb singularities without additional cut-off on the particle level. The first step is to make use of the relative entropy between the joint distribution of the particle system and an approximated limiting aggregation-diffusion system. A crucial argument in the proof is to show convergence in probability by a stopping time argument. The second step is to obtain a quantitative convergence rate to the limiting aggregation-diffusion system from the approximated PDE system. This is shown by evaluating a combination of relative entropy and -distance.
Paper Structure (12 sections, 12 theorems, 225 equations)

This paper contains 12 sections, 12 theorems, 225 equations.

Table of Contents

  1. Introduction
  2. Strategy of the proofs
  3. Proof of convergence in probability
  4. Distance between particle system and intermediate PDEs
  5. Distance between intermediate PDEs and the limiting system
  6. Global well-posedness of aggregation-diffusion system
  7. Global-in-time existence of solutions of (\ref{['aggregation-diffusion']})
  8. Uniqueness of the solution of (\ref{['aggregation-diffusion']})
  9. Appendix
  10. Proof of Lemma \ref{['subadditivity']}
  11. Proof of Lemma \ref{['LLN']}
  12. Compactness argument

Key Result

Theorem 1.2

Under assumptions (H1)-(H7), let $\bar{f}=(\bar{f}_1,\ldots,\bar{f}_n)$ be the solution of aggregation-diffusion system aggregation-diffusion satisfying $\bar{f}_\alpha\in L^\infty(0,T;L^1\cap L^\infty(\mathbb{R}^d))\cap L^2(0,T;H^1(\mathbb{R}^d))$ for $\alpha=1,\ldots,n$. And let $\widetilde{f}_{\v We further assume that the parameter $\varepsilon$ has algebraic connection with $N$ as $\varepsilo

Theorems & Definitions (30)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 2.1: Relative entropy
  • Lemma 2.2
  • Proposition 2.3: Mean-field estimate
  • Proposition 2.4: PDE error estimate
  • Theorem 2.5: Global well-posedness of \ref{['aggregation-diffusion']}
  • Proposition 3.1
  • Lemma 3.2: Law of large numbers
  • ...and 20 more