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

Abstract

We establish a commutator estimate for the Yukawa modulated energy associated with overdamped particle systems with repulsive interactions in every fixed dimension $d\ge 2$, with constants uniform for $0<\kappa \le {\kappa }_{\ast }$. The main difficulty is specific to the modified Helmholtz operator: truncating the Yukawa potential produces both a surface charge measure and a positive volume charge density, while the resulting smeared charge has total charge $1-{\kappa }^2{r}^2/(2d)+o({\kappa }^2{r}^2)$ as $\kappa r\downarrow 0$. We restore neutrality by rescaling the reference density and use a stress-energy identity whose interface term involves the arithmetic mean of the one-sided gradient traces. This yields an additive finite-$N$ correction of order $N^{-2/d}$ for $d\ge 3$ and $(1+\log N)/N$ for $d=2$, without an $N$-independent lower bound on interparticle distances. As consequences, we obtain quantitative weak--strong stability for the mean-field limit, control of the $k$-particle marginals for fixed $k$, and a time-integrated mean-square force discrepancy. For smooth compactly supported initial data, regular mean-field solutions exist on a time interval uniform in $0\le \kappa \le {\kappa }_{\ast }$. The same uniformity yields the Coulomb limit: the squared Wasserstein distance between the Yukawa and Coulomb mean-field solutions is $O({\kappa }^4)$ in $d\ge 3$ and $O({\kappa }^2)$ in $d=2$; in $d=3$ we also obtain a direct simultaneous mean-field/Coulomb estimate at the particle level.

59 pages