Commutator Estimates Uniform in the Screening Parameter and Mean-Field Limits for Yukawa Interactions

Abstract

We study quantitative mean-field limits for classical particles with Yukawa (screened Coulomb) interactions in every fixed dimension $d\ge2$, uniformly as the screening parameter $\kappa$ tends to zero. Our main result is a first-order commutator estimate in the natural Yukawa modulated energy, with an additive error of order $N^{-2/d}$ for $d\ge3$ and $(1+\log N)/N$ for $d=2$. It requires only a bounded reference density and a Lipschitz transport field, with no uniform lower bound on interparticle distances and no negative power of $\kappa$. The modified Helmholtz operator $-\Delta+\kappa^2$ creates the main new difficulty. Truncating the potential to a constant inside each truncation ball produces a surface charge and a positive volume charge whose total mass is strictly less than one. We keep the reference density unchanged and control this loss of mass through an exact Green function representation and renormalized energy identities. A stress-energy identity with interface terms and averaging over the truncation radii then give the uniform commutator estimate. Combined with the modulated energy dissipation identity and a normalized quadratic transport cost, this estimate yields weak--strong stability, propagation of chaos, and time-integrated control of the mean-square difference between empirical and mean-field forces. We also prove a quantitative Yukawa-to-Coulomb limit. For smooth product data, $N\to\infty$ and $\kappa\downarrow0$ may be taken simultaneously with no relation between their rates; in dimension three, a direct comparison at the particle level also holds for general symmetric initial laws with finite initial error.