A probabilistic mean-field limit for the Vlasov–Poisson system for ions

Abstract

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 thermalised. Consequently, the Poisson coupling for the electrostatic potential contains an additional exponential non-linearity 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 thermalised 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), 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 thermalised 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 (2016), 1–16] and Lazarovici and Pickl [Arch. Ration. Mech. Anal. 225 (2017), 1201–1231] that is applicable to interaction forces defined through a non-linear 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.