Nonextensive scaling in a long-range Hamiltonian system

The nonextensivity of a classical long-range Hamiltonian system is discussed. The system is the so-called $\alpha$-XY model, a lattice of inertial rotators with an adjustable parameter $\alpha$ controlling the range of the interactions. This model has been explored in detail over the last years. For sufficiently long-range interactions, namely $\alpha<d$, where $d$ is the lattice dimension, it was shown to be nonextensive and to exhibit a second order phase transition. However, conclusions in apparent contradiction with the findings above have also been drawn. This picture reveals the fact that there are aspects of the model that remain poorly understood. Here we perform a thorough analysis, essaying an explanation for the origin of the apparent discrepancies.