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

TL;DR

This paper investigates the equilibrium properties of a chain of ferromagnetically coupled rotators with interactions decaying as a power law, revealing different regimes of long-range order and connecting microcanonical and canonical ensemble results.

Contribution

It introduces a model interpolating between mean-field and short-range interactions and analyzes its equilibrium behavior, highlighting the connection between long-range decay and ensemble equivalence.

Findings

Identifies three regimes of long-range order depending on decay exponent 

Shows microcanonical averages match canonical results for <1 after scaling

Provides a computationally efficient approach for systems with long-range interactions

Abstract

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.