A Probabilistic Mean-Field Limit for the Vlasov-Poisson System for Ions

The Vlasov-Poisson system for ions is a kinetic equation for dilute, unmagnetised plasma. It describes the evolution of the ions in a plasma under the assumption that the electrons are thermalized. Consequently, the Poisson coupling for the electrostatic potential contains an additional exponential nonlinearity not present in the electron Vlasov-Poisson system. The system can be formally derived through a mean-field limit from a microscopic system of ions interacting with a thermalized electron distribution. However, it is an open problem to justify this limit rigorously for ions modelled as point charges. Existing results on the derivation of the three-dimensional ionic Vlasov-Poisson system, obtained by the author and Iacobelli [J. Math. Pures Appl. 135 (2020), pp. 199-255], require a truncation of the singularity in the Coulomb interaction at spatial scales of order $N^{-\beta}$ with $\beta<1/15$, which is more restrictive than the available results for the electron Vlasov-Poisson system. In this article, we prove that the Vlasov-Poisson system for ions can be derived from a microscopic system of ions and thermalized electrons with interaction truncated at scale $N^{-\beta}$ with $\beta<1/3$. We develop a generalisation of the probabilistic approach to mean-field limits developed in the works of Boers and Pickl [J. Stat. Phys. 164(1) (2016), pp. 1-16] and Lazarovici and Pickl [Arch. Ration. Mech. Anal. 225(3) (2017), pp. 1201-1231] that is applicable to interaction forces defined through a nonlinear coupling with the particle density. The proof is based on a quantitative uniform law of large numbers for convolutions between empirical measures of independent, identically distributed random variables and locally Lipschitz functions.