Fluctuations around the mean-field limit for attractive Riesz potentials in the moderate regime

TL;DR

This paper establishes a central limit theorem for fluctuations of moderately interacting particles with attractive Riesz potentials, showing they become Gaussian in the large particle limit, extending previous results to attractive regimes.

Contribution

It introduces a novel approach allowing for attractive potentials in the moderate regime while proving Gaussian fluctuations, using mean-square convergence and systematic term separation.

Findings

Fluctuations are asymptotically Gaussian.

Applicable to singular attractive or repulsive potentials of sub-Coulomb type.

Provides quantitative convergence rates for empirical measures.

Abstract

A central limit theorem is shown for moderately interacting particles in the whole space. The interaction potential approximates singular attractive or repulsive potentials of sub-Coulomb type. It is proved that the fluctuations become asymptotically Gaussians in the limit of infinitely many particles. The methodology is inspired by the classical work of Oelschl\"ager on fluctuations for the porous-medium equation. The novelty in this work is that we can allow for attractive potentials in the moderate regime and still obtain asymptotic Gaussian fluctuations. The key element of the proof is the mean-square convergence in expectation for smoothed empirical measures associated to moderately interacting $N$-particle systems with rate $N^{-1/2-\varepsilon}$ for some $\varepsilon>0$. To allow for attractive potentials, the proof uses a quantitative mean-field convergence in probability with any algebraic rate and a law-of-large-numbers estimate as well as a systematic separation of the terms to be estimated in a mean-field part and a law-of-large-numbers part.