Nonequilibrium dynamics of a fast oscillator coupled to Glauber spins

Abstract

A fast harmonic oscillator is linearly coupled with a system of Ising spins that are in contactwith a thermal bath, and evolve under a slow Glauber dynamics at dimensionless temperatureθ. The spins have a coupling constant proportional to the oscillator position.The oscillator–spin interaction produces a second order phase transition atθ = 1 with the oscillator position as its order parameter: the equilibrium position is zero forθ > 1 and nonzerofor θ < 1. For θ < 1, the dynamics of this system is quite different from relaxation to equilibrium. Formost initial conditions, the oscillator position performs modulated oscillationsabout one of the stable equilibrium positions with a long relaxation time. Forrandom initial conditions and a sufficiently large spin system, the unstable zeroposition of the oscillator is stabilized after a relaxation time proportional toθ. If the spin system is smaller, the situation is the same until the oscillator position is closeto zero, then it crosses over to a neighborhood of a stable equilibrium position about whichit keeps oscillating for an exponentially long relaxation time. These results of stochasticsimulations are predicted by modulation equations obtained from a multiple scale analysisof macroscopic equations.