Long-range interacting rotators: connection with the mean--field approximation

We analyze the equilibrium properties of a chain of ferromagnetically coupled rotators which interact through a force that decays as $r^{-\alpha}$ where r is the interparticle distance and $\alpha\ge 0$. Our model contains as particular cases the mean field limit ($\alpha=0$) and the first-neighbor model ($\alpha \to \infty$). By integrating the equations of motion we obtain the microcanonical time averages of both the magnetization and the kinetic energy. Concerning the long-range order, we detect three different regimes at low energies, depending on whether $\alpha$ belongs to the intervals $[0,1)$, $(1,2)$ or $(2,\infty)$. Moreover, for $0 \le \alpha < 1$, the microcanonical averages agree, after a simple scaling, with those obtained in the canonical ensemble for the mean-field XY model. This correspondence offers a mathematically tractable and computationally economic way of dealing with systems governed by slowly decaying long-range interactions.