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 $F(r) \sim 1/r^\gamma$ at large scales, and regulated to a bounded behaviour below a "softening" scale $\varepsilon$, we find that there is an essential qualitative difference between the cases $\gamma < d$ and $\gamma > d$, i.e., depending on 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 $\varepsilon$, while for $\gamma > d$ the amplitude of these terms is directly regulated by $\varepsilon$, and thus by the small scale properties of the interaction. This corresponds to a simple physical criterion for a basic distinction between long-range ($\gamma \leq d$) and short range ($\gamma > d$) interactions, different to the canonical one ($\gamma \leq d +1$ 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 quasi-stationary states in long-range interacting systems.