Propagation of chaos for first-order mean-field systems with non-attractive moderately singular interaction

Abstract

We consider particle systems that evolve by inertialess binary interaction through general non-attractive kernels of singularity $|x{|}^{-\alpha }$ with $\alpha <d-1$. We prove a quantitative mean-field limit in terms of Wasserstein distances under certain conditions on the initial configuration while maintaining control of the particle configuration in the form of the minimal distance and certain singular sums of the particle distances. As a corollary, we show propagation of chaos for $\alpha <\frac{d-1}{2}$ for $d⩾3$ and $\alpha <\frac{1}{3}=\frac{2d-3}{3}$ for $d=2$. This extends the results of Hauray [Math. Models Methods Appl. Sci. 19 (2009), 1357–1384], which yield propagation of chaos for $\alpha <\frac{d-2}{2}$ without an assumption on the sign of the interaction. The main novel ingredient is that due to the non-attraction property it is enough to control the distance to the next-to-nearest neighbour particle.