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^-α where r is the interparticle distance and α≥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 α belongs to the intervals $[0,1)$, $(1,2)$, or (2,∞). For 0<α<1, the microcanonical averages agree, after a scaling, with those obtained in the canonical ensemble for the mean-field case (α0ex0ex=0ex0ex0). This correspondence offers a mathematically tractable way of dealing with systems governed by slowly decaying long-range interactions.