arXiv subjects

Xuanrui Feng

Publications and source records attributed to Xuanrui Feng.

7 recordsLinked to original sources

arXiv2026

Modulated Gibbs Measure and Mean-Field Limit for 3D Vlasov--Poisson--Fokker--Planck Equations

We introduce a modulated Gibbs measure for the usual tensorized initial data for stochastic Newton's systems with singular repulsive interactions. Using the uniform-in-$N$ partition-function estimates of Wang--Zhao \cite{wang2026uniform}, we show that the modulated Gibbs measure and its tensorized reference are $O(N^{-1})$-close in normalized relative entropy. A stability argument then transfers propagation of chaos from modulated Gibbs measures to tensorized data and other initial laws at the same entropy scale. Combined with the weighted BBGKY estimates of Bresch--Jabin--Soler \cite{bresch2025new}, this yields the first cutoff-free mean-field limit and propagation of chaos for the three-dimensional Vlasov--Poisson--Fokker--Planck (VPFP) equation on an $N$-independent short-time interval.

math.AP↗

arXiv2026

Uniform-in-time relative entropy estimates for Kac's approximation of the Landau equation

We prove uniform-in-time quantitative relative entropy estimates for Kac's particle approximation of the three-dimensional spatially homogeneous Landau equation for Maxwellian molecules. Using the relative entropy method and logarithmic derivative estimates, we obtain a finite-time normalized entropy bound of order $N^{-1/2}$. The proof relies on explicit covariance identities and a new law-of-large-numbers estimate for the particle system using a new duality method. The uniform-in-time propagation of chaos is derived from an interpolation with an algebraic relaxation estimate toward equilibrium.

math.AP↗

arXiv2025

Kac's Program for the Landau Equation

We study the derivation of the spatially homogeneous Landau equation from the mean-field limit of a conservative $N$-particle system, obtained by passing to the grazing limit on Kac's walk in his program for the Boltzmann equation. Our result covers the full range of interaction potentials, including the physically important Coulomb case. This provides the first resolution of propagation of chaos for a many-particle system approximating the Landau equation with Coulomb interactions, and the first extension of Kac's program to the Landau equation in the soft potential regime. The convergence is established in weak, Wasserstein, and entropic senses, together with strong $L^1$ convergence. To handle the singularity of soft potentials, we extend the duality approach of Bresch-Duerinckx-Jabin \cite{bresch2024duality} and establish key functional inequalities, including an extended commutator estimate and a new second-order Fisher information estimate.

math.AP↗

arXiv2024

Propagation of Chaos for 2D Log Gas on the Whole Space

We derive the quantitative propagation of chaos in the sense of relative entropy for the first time for the 2D Log gas or the weakly interacting particle systems with 2D Coulomb interactions on the whole space. We resolve this problem by adapting the modulated free energy method in [BJW23] to the whole space setting and establishing the crucial logarithmic growth estimates for the mean-field Poisson-Nernst-Planck (PNP) equation of single component via the parabolic maximum principle.

math.AP↗

arXiv2024

Quantitative Propagation of Chaos for 2D Viscous Vortex Model with General Circulations on the Whole Space

We derive quantitative propagation of chaos in the sense of relative entropy for the 2D viscous vortex model with general circulations, approximating the vorticity formulation of the 2D Navier-Stokes equation on the whole Euclidean space. Our results work on the general setting that the vortices are positioned on the whole space $\R^2$ and that the circulations are allowed to be in different magnitudes and orientations, which can be adapted to general unconfined realistic fluids with vorticity that may change sign. We provide explicit convergence rates which are optimal in $N$ and optimal in $t$ among existing literature. The key technical tools, which are our major novelty, are the sharp logarithmic growth estimates and a new ODE hierarchy and iterated integral estimates.

math.AP↗

arXiv2024

Relative Entropy Method for Particle Approximation of the Landau Equation for Maxwellian Molecules

We derive the spatially homogeneous Landau equation for Maxwellian molecules from a natural stochastic interacting particle system. More precisely, we control the relative entropy between the joint law of the particle system and the tensorized law of the Landau equation. To obtain this, we establish as key tools the pointwise logarithmic gradient and Hessian estimates of the density function and also a new Law of Large Numbers result for the particle system. The logarithmic estimates are derived via the Bernstein method and the parabolic maximum principle, while the Law of Large Numbers result comes from crucial observations on the control of moments at the particle level.

math.AP↗

arXiv2023

Quantitative Propagation of Chaos for 2D Viscous Vortex Model on the Whole Space

We derive the quantitative estimates of propagation of chaos for the large interacting particle systems in terms of the relative entropy between the joint law of the particles and the tensorized law of the mean field PDE. We resolve this problem for the first time for the viscous vortex model that approximates 2D Navier-Stokes equation in the vorticity formulation on the whole space. We obtain as key tools the Li-Yau-type estimates and Hamilton-type heat kernel estimates for 2D Navier-Stokes in the whole space.

math.AP↗