Andrea Gabrielli — Università di Roma "La Sapienza" # 
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 \(F(r)\sim1/ra\), regulated to a 
bounded behavior below a "softening" scale \(l\), we find that there is an essential qualitative difference between 
the cases \( a < d \) (i.e. the spatial dimension) and \(a > d\) , i.e., depending on the the integrability at large 
distances  of \(F(r)\). For \( a < d \) the corrections to the Vlasov dynamics for a given coarse-grained scale 
are essentially  insensitive to the softening parameter \(l\), while for \(a>d\) 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 (\( a < d \)) and short 
range (\(a>d\)) interactions, different from the thermodynamic one (\( a < d-1 \) or \( a > d-1 \)). 
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. <br/><br/>

[1] A. Gabrielli et al., Phys. Rev. E, 90, 062910 (2014).<br/>
[2] A. Gabrielli et al., Phys. Rev. Lett., 115, 210602 (2010).<br/>