Mean-field limit for particle systems with topological interactions

Abstract

Many interesting physical systems can be described at the microscopic level as particle dynamics and at the mesoscopic level with kinetic equations. In the wide field of two-body interactions, the link between these two regimes is mathematically well understood in the case of the mean-field limit, i.e. when the density of the particles diverges with their number N , the mean free path vanishes as 1{N and the interaction intensity scales with 1{N . In this limit, each particle feels the interaction with the others as a mean. A rigorous mathematical proof of this result can be done in the case of two-body interactions with sufficiently regular potentials. This classical achievement has been obtained independently by several authors in the ’70s (see [5, 14, 27]) and its explanation is particularly clear in the Dobrushin’s argument [14] where the result follows by noticing that the empirical measure associated with the particle system is a weak solution of the mean-field equation; the proof follows by showing the weak continuity, w.r.t the initial datum, of the weak solutions. Although the theory for regular pairwise interactions is sufficiently well understood, going beyond it considering singular potentials, is instead a harder task. This is the case of the three-dimensional VlasovPoisson equation, which is the most important equation of plasma physics and of galactic dynamics, based on the choice of the Coulomb or Newton potential, respectively. In this equation, the potential 1{r is singular at the origin and does not belong to any L space. Although the mean-field limit for the Vlasov-Poisson equation remains an open