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Influence of the interaction range on the thermostatistics of a classical many-body system

We numerically study a one-dimensional system of $N$ classical localized planar rotators coupled through interactions which decay with distance as $1/r^α$ ($α\ge 0$). The approach is a first principle one (\textit{i.e.}, based on Newton's law), and yields the probability distribution of momenta. For $α$ large enough and $N\gg1$ we observe, for longstanding states, the Maxwellian distribution, landmark of Boltzmann-Gibbs thermostatistics. But, for $α$ small or comparable to unity, we observe instead robust fat-tailed distributions that are quite well fitted with $q$-Gaussians. These distributions extremize, under appropriate simple constraints, the nonadditive entropy $S_q$ upon which nonextensive statistical mechanics is based. The whole scenario appears to be consistent with nonergodicity and with the thesis of the $q$-generalized Central Limit Theorem.