A Mean Field Limit for the Hamiltonian Vlasov System

The derivation of effective equations for interacting many body systems has seen a lot of progress in the recent years. While dealing with classical systems, singular potentials are quite challenging (Hauray and Jabin in Annales scientifiques de l'École Normale Supérieure, , Lazarovici and Pickl in Arch Ration Mech Anal 225(3):1201–1231, ) comparably strong results are known to hold for quantum systems (Knowles and Pickl in Comm Math Phys 298:101–139, ). In this paper, we wish to show how techniques developed for the derivation of effective descriptions of quantum systems can be used for classical ones. While our future goal is to use these ideas to treat singularities in the interaction, the focus here is to present how quantum mechanical techniques can be used for a classical system and we restrict ourselves to regular two-body interaction potentials. In particular we compute a mean field limit for the Hamilton Vlasov system in the sense of (Fröhlich et al. in Comm Math Phys 288:1023–1058, ; Neiss in Arch Ration Mech Anal. 10.1007/s00205-018-1275-8) that arises from classical dynamics. The structure reveals strong analogy to the Bosonic quantum mechanical ensemble of the many-particle Schrödinger equation and the Hartree equation as its mean field limit (Pickl in arXiv:0808.1178v1, ).