On the mean-field limit for the Vlasov-Poisson system
Manuela Feistl-Held, Peter Pickl
TL;DR
The paper rigorously derives the Vlasov-Poisson equation from a 3D N-particle Coulomb system under an ultra-small cut-off $|q|>N^{- rac{5}{12}+\sigma}$, establishing propagation of chaos for typical initial data. It leverages a second-order Gronwall framework, a probabilistic partition of particles into good/bad/superbad groups, and LLN-type bounds across refined collision classes to bound the deviations between microscopic trajectories and the mean-field characteristics. A central technical achievement is proving that the microscopic dynamics stay within $N^{-\frac{1}{6}+\varepsilon}$ of the mean-field flow with high probability, for any fixed $T>0$, and that the mean-field flow converges to the VP solution as $N\to\infty$. This provides a robust probabilistic derivation of the VP equation from Coulomb interactions with a physically relevant cut-off, with explicit rates and high-probability estimates that improve on prior results and highlight the role of second-order dynamics in controlling singular interactions.
Abstract
We present a probabilistic proof of the mean-field limit and propagation of chaos of a classical N-particle system in three dimensions with Coulomb interaction force of the form $f^N(q)=\pm\frac{q}{|q|^3}$ and $N$-dependent cut-off at $|q|>N^{-\frac{5}{12}+σ}$ where $σ>0$ can be chosen arbitrarily small. This cut-off size is much smaller than the typical distance to the nearest neighbour. In particular, for typical initial data, we show convergence of the Newtonian trajectories to the characteristics of the Vlasov-Poisson system. The proof is based on a Gronwall estimate for the maximal distance between the exact microscopic dynamics and the approximate mean-field dynamics. Thus our result leads to a derivation of the Vlasov-Poisson equation from the microscopic $N$-particle dynamics with force term arbitrary close to the physically relevant Coulomb force.