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

We explore [1] 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 such a system, we evaluate the scalings with N of the terms describing the corrections to the Vlasov equation for the coarse-grained one particle phase space density. Considering an interaction with radial pair force \textit{F(r) $\sim$ 1/r}$^{a}$, regulated to a bounded behavior below a ``softening'' scale $l$, we find that there is an essential qualitative difference between the cases \textit{a\textless d} (i.e. the spatial dimension) and \textit{a\textgreater d}, i.e., depending on the the integrability at large distances of $F(r)$. For \textit{a\textless d} the corrections to the Vlasov dynamics for a given coarse-grained scale are essentially insensitive to the softening parameter $l$, while for \textit{a\textgreater d} the corrections are directly regulated by $l$, i.e. by the small scale properties of the interaction, in agreement with the Chandrasekhar approach [2]. This gives a simple physical criterion for a basic distinction between long-range (\textit{a\textless d}) and short range (\textit{a\textgreater d}) interactions, different from the thermodynamic one (\textit{a\textless d-1} or \textit{a\textgreater d-1}). This alternative classification, based purely on dynamical arguments, is relevant notably to understanding the conditions for the existence of so-called quasi-stationary states in long-range interacting systems. \\[4pt] [1] A. Gabrielli et al., \underline {arxiv.org/abs/1408.0999}, to appear in PRE (2014)\\[0pt] [2] A. Gabrielli et al., PRL, \textbf{115}, 210602 (2010)