Existence of Quasi-stationary states at the Long Range threshold

In this paper the lifetime of quasi-stationary states (QSS) in the $\alpha-$HMF model are investigated at the long range threshold ($\alpha=1$). It is found that QSS exist and have a diverging lifetime $\tau(N)$ with system size which scales as $\mbox{\ensuremath{\tau}(N)\ensuremath{\sim}}\log N$, which contrast to the exhibited power law for $\alpha<1$ and the observed finite lifetime for $\alpha>1$. Another feature of the long range nature of the system beyond the threshold ($\alpha>1$) namely a phase transition is displayed for $\alpha=1.5$. The definition of a long range system is as well discussed.