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On the mean-field limit of Vlasov-Poisson-Fokker-Planck equations

Li Chen, Jinwook Jung, Peter Pickl, Zhenfu Wang

TL;DR

The paper proves a strong mean-field limit for a Newtonian $N$-body system with optional velocity white noise, deriving propagation of chaos in $L^1$ toward the VPFP equation (or VP in the noiseless case) via an $N$-dependent regularization of the Coulomb kernel. The main technique combines the relative-entropy method with trajectory-approximation results for the interacting system, using an intermediate McKean–Vlasov problem to bridge microscopic and mean-field dynamics. It furnishes explicit convergence rates in $N$, depending on the regularization scale and the noise strength, and shows how, under suitable regularity, one can obtain strong $L^1$-convergence of marginals to the corresponding mean-field solutions. The results unify and extend prior trajectory-based and entropy-based approaches, yielding quantitative propagation of chaos for VPFP and VP in three dimensions and clarifying the role of noise and kernel regularization in the convergence behavior. These findings provide rigorous justification for VP/VPFP as mean-field limits of large, interacting particle systems in kinetic theory contexts.

Abstract

The derivation of effective descriptions for interacting many-body systems is an important branch of applied mathematics. We prove a propagation of chaos result for a system of $N$ particles subject to Newtonian time evolution with or without additional white noise influencing the velocities of the particles. We assume that the particles interact according to a regularized Coulomb-interaction with a regularization parameter that vanishes in the $N\to\infty$ limit. The respective effective description is the so called Vlasov-Poisson-Fokker-Planck (VPFP), respectively the Vlasov-Poisson (VP) equation in the case of no or sub-dominant white noise. To obtain our result we combine the relative entropy method from \cite{jabinWang2016} with the control on the difference between the trajectories of the true and the effective description provided in \cite{HLP20} for the VPFP case respectively in \cite{LP} for the VP case. This allows us to prove strong convergence of the marginals, i.e. convergence in $L^1$.

On the mean-field limit of Vlasov-Poisson-Fokker-Planck equations

TL;DR

The paper proves a strong mean-field limit for a Newtonian -body system with optional velocity white noise, deriving propagation of chaos in toward the VPFP equation (or VP in the noiseless case) via an -dependent regularization of the Coulomb kernel. The main technique combines the relative-entropy method with trajectory-approximation results for the interacting system, using an intermediate McKean–Vlasov problem to bridge microscopic and mean-field dynamics. It furnishes explicit convergence rates in , depending on the regularization scale and the noise strength, and shows how, under suitable regularity, one can obtain strong -convergence of marginals to the corresponding mean-field solutions. The results unify and extend prior trajectory-based and entropy-based approaches, yielding quantitative propagation of chaos for VPFP and VP in three dimensions and clarifying the role of noise and kernel regularization in the convergence behavior. These findings provide rigorous justification for VP/VPFP as mean-field limits of large, interacting particle systems in kinetic theory contexts.

Abstract

The derivation of effective descriptions for interacting many-body systems is an important branch of applied mathematics. We prove a propagation of chaos result for a system of particles subject to Newtonian time evolution with or without additional white noise influencing the velocities of the particles. We assume that the particles interact according to a regularized Coulomb-interaction with a regularization parameter that vanishes in the limit. The respective effective description is the so called Vlasov-Poisson-Fokker-Planck (VPFP), respectively the Vlasov-Poisson (VP) equation in the case of no or sub-dominant white noise. To obtain our result we combine the relative entropy method from \cite{jabinWang2016} with the control on the difference between the trajectories of the true and the effective description provided in \cite{HLP20} for the VPFP case respectively in \cite{LP} for the VP case. This allows us to prove strong convergence of the marginals, i.e. convergence in .
Paper Structure (8 sections, 11 theorems, 120 equations)

This paper contains 8 sections, 11 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Propagation of chaos in $L^1$: Towards Vlasov-Poisson-Fokker-Planck
  3. The proof of Theorem \ref{['main_thm1']}
  4. Proof of Theorem \ref{['main_thm2']}
  5. Propagation of chaos in $L^1$: Towards Vlasov-Poisson
  6. Technical lemmas
  7. Proof of Theorem \ref{['cor_VP']}
  8. Remarks on $L^1$-convergence

Key Result

Theorem 1.1

Let $T>0$, $\delta \in (0,1/d)$ be given and assume that the initial data $f_0\geq 0$ satisfies that where $\mathcal{P}_2(\mathbb R^{2d})$ denotes the space of probability measures with finite 2nd moments. Let $f_t^N$ and $f_t$ be the joint distribution to the particle system micro_reg with the regularized interaction kN_LP and the (unique) weak solution to the limit PDE main_eq, respectively. As

Theorems & Definitions (20)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.2
  • Theorem 2.1
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.1
  • proof : Proof of Theorem \ref{['main_thm1']}
  • ...and 10 more