Nonequilibrium dynamics of a fast oscillator coupled to Glauber spins

TL;DR

This paper studies the nonequilibrium behavior of a fast oscillator coupled to Glauber spins, revealing a phase transition at a critical temperature and complex oscillatory dynamics influenced by initial conditions and system size.

Contribution

It introduces a coupled oscillator-spin model exhibiting a second order phase transition and analyzes its complex dynamics using multiple scale analysis and stochastic simulations.

Findings

A phase transition at temperature θ=1 with the oscillator position as order parameter.

Oscillatory dynamics with long relaxation times near stable equilibrium positions.

Stabilization of the zero position for large spin systems under certain conditions.

Abstract

A fast harmonic oscillator is linearly coupled with a system of Ising spins that are in contact with a thermal bath, and evolve under a slow Glauber dynamics at dimensionless temperature $\theta$. The spins have a coupling constant proportional to the oscillator position. The oscillator-spin interaction produces a second order phase transition at $\theta=1$ with the oscillator position as its order parameter: the equilibrium position is zero for $\theta>1$ and non-zero for $\theta< 1$. For $\theta<1$, the dynamics of this system is quite different from relaxation to equilibrium. For most initial conditions, the oscillator position performs modulated oscillations about one of the stable equilibrium positions with a long relaxation time. For random initial conditions and a sufficiently large spin system, the unstable zero position of the oscillator is stabilized after a relaxation time proportional to $\theta$. If the spin system is smaller, the situation is the same until the oscillator position is close to zero, then it crosses over to a neighborhood of a stable equilibrium position about which keeps oscillating for an exponentially long relaxation time. These results of stochastic simulations are predicted by modulation equations obtained from a multiple scale analysis of macroscopic equations.