Quasistationarity in a long-range interacting model of particles moving on a sphere

TL;DR

This paper studies a long-range particle system on a sphere with Heisenberg interactions, revealing quasistationary states that persist for long times and depend on system size, with implications for understanding long-range dynamics.

Contribution

It analytically and numerically investigates quasistationary states in a long-range Heisenberg model, highlighting their stability and relaxation timescales in different regimes.

Findings

Long-lived non-magnetized quasistationary states exist in the model.

Relaxation time to equilibrium scales algebraically with system size N.

In unstable regimes, relaxation occurs rapidly, scaling as log N.

Abstract

We consider a long-range interacting system of $N$ particles moving on a spherical surface under an attractive Heisenberg-like interaction of infinite range, and evolving under deterministic Hamilton dynamics. The system may also be viewed as one of globally coupled Heisenberg spins. In equilibrium, the system has a continuous phase transition from a low-energy magnetized phase, in which the particles are clustered on the spherical surface, to a high-energy homogeneous phase. The dynamical behavior of the model is studied analytically by analyzing the Vlasov equation for the evolution of the single-particle distribution, and numerically by direct simulations. The model is found to exhibit long lived non-magnetized quasistationary states (QSSs) which in the thermodynamic limit are dynamically stable within an energy range where the equilibrium state is magnetized. For finite $N$, these states relax to equilibrium over a time that increases algebraically with $N$. In the dynamically unstable regime, non-magnetized states relax fast to equilibrium over a time that scales as $\log N$. These features are retained in presence of a global anisotropy in the magnetization.