Finite-N corrections to Vlasov dynamics and the range of pair interactions

We explore the conditions on a pair interaction for the validity of the Vlasov equation to describe the dynamics of an interacting N-particle system in the large N limit. Using a coarse graining in phase space of the exact Klimontovich equation for the N-particle system, we evaluate, neglecting correlations of density fluctuations, the scalings with N of the terms describing the corrections to the Vlasov equation for the coarse-grained one-particle phase space density. Considering a generic interaction with radial pair force F(r), with F(r) similar to 1/r(gamma) at large scales, and regulated to a bounded behavior below a "softening" scale epsilon, we find that there is an essential qualitative difference between the cases gamma < d and gamma > d, i.e., depending on the the integrability at large distances of the pair force. In the former case, the corrections to the Vlasov dynamics for a given coarse-grained scale are essentially insensitive to the softening parameter epsilon, while for gamma > d the amplitude of these terms is directly regulated by e, and thus by the small scale properties of the interaction. This corresponds to a simple physical criterion for a basic distinction between long-range (gamma <= d) and short-range (gamma > d) interactions, different from the canonical one (gamma <= d + 1 or gamma > d + 1) based on thermodynamic analysis. This alternative classification, based on purely dynamical considerations, is relevant notably to understanding the conditions for the existence of so-called quasistationary states in long-range interacting systems.