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 rij like~1/rαij (α≥0), where α→∞ and α=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 α through the ratio α/d.

L. J.L. Cirto, A. Rodríguez, F. D. Nobre, C. Tsallis, Validity and failure of the Boltzmann weight, EPL 123(3) (2018)