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 $\alpha \ge 0$. 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 $\alpha$ belongs to the intervals $[0,1)$, $(1,2)$, or $(2,\infty )$. For $0<\mathrm{}\alpha <1\mathrm{}$, the microcanonical averages agree, after a scaling, with those obtained in the canonical ensemble for the mean-field case $(\alpha =0\mathrm{})$. This correspondence offers a mathematically tractable way of dealing with systems governed by slowly decaying long-range interactions.