Rotators with long-range interactions: 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 greater than or equal to 0. By integrating the equations of motion we obtain the microcanonical time averages of both the magnetization and the kinetic energy. We detect three different regimes depending on whether fu belongs to the intervals [0, 1), (1.2), or (2, infinity). For 0 < alpha < 1, the microcanonical averages agree, after a scaling, with those obtained in the canonical ensemble for the mean-field case (alpha = 0). This correspondence offers a mathematically tractable way of dealing with systems governed by slowly decaying long-range interactions.