  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.

Statistical Mechanics