Ensemble inequivalence and absence of quasi-stationary states in long-range random networks

Ensemble inequivalence has been previously displayed only for long-range interacting systems with non-extensive energy. In order to perform the thermodynamic limit, such systems require an unphysical, so-called, Kac rescaling of the coupling constant. We here study models defined on long-range random networks, which avoid such a rescaling. The proposed models have an extensive energy, which is however non-additive. For such long-range random networks, pairs of sites are coupled with a probability decaying with the distance $r$ as $1/r^\delta$. In one dimension and with $0 \leq \delta <1$, surface energy scales linearly with the network size, while for $\delta >1$ it is $O(1)$. By performing numerical simulations, we show that a negative specific heat region is present in the microcanonical ensemble of a Blume-Capel model, in correspondence with a first-order phase transition in the canonical one. This proves that ensemble inequivalence is a consequence of $non$-$additivity$ rather than $non$-$extensivity$. Moreover, since a mean-field coupling is absent in such networks, relaxation to equilibrium takes place on an intensive time scale and quasi-stationary states are absent.