Thermodynamic limit, quasi-stationary states and the range of pair interactions

``Quasi-stationary'' states are approximately time-independent out of equilibrium states which have been observed in a variety of systems of particles interacting by long-range interactions. We investigate here the conditions of their occurrence for a generic pair interaction $V(r \rightarrow \infty) \sim 1/r^\gamma$ with $\gamma > 0$, in $d>1$ dimensions. We generalize analytic calculations known for gravity in $d=3$ to determine the scaling parametric dependences of their relaxation rates due to two body collisions, and report extensive numerical simulations testing their validity. Our results lead to the conclusion that, for $\gamma < d-1$, the existence of quasi-stationary states is ensured by the large distance behavior of the interaction alone, while for $\gamma > d-1$ it is conditioned on the short distance properties of the interaction, requiring the presence of a sufficiently large soft-core in the interaction potential.