  The dynamics and thermostatistics of a classical inertial XY model,
characterized by long-range interactions, are investigated on $d$-dimensional
lattices ($d=1,2,$ and 3), through molecular dynamics. The interactions between
rotators decay with the distance $r_{ij}$ like~$1/r_{ij}^{\alpha}$ ($\alpha
\geq 0$), where $\alpha\to\infty$ and $\alpha=0$ respectively correspond to the
nearest-neighbor and infinite-range interactions. We verify that the momenta
probability distributions are Maxwellians in the short-range regime, whereas
$q$-Gaussians emerge in the long-range regime. Moreover, in this latter regime,
the individual energy probability distributions are characterized by long
tails, corresponding to $q$-exponential functions. The present investigation
strongly indicates that, in the long-range regime, central properties fall out
of the scope of Boltzmann-Gibbs statistical mechanics, depending on $d$ and
$\alpha$ through the ratio $\alpha/d$.
