Phase transitions in a mechanical system coupled to Glauber spins

Abstract

A harmonic oscillator linearly coupled with a linear chain of Ising spins is investigated. TheN spins in the chain interact with their nearest neighbours with acoupling constant proportional to the oscillator position and toN − 1/2, are in contact with a thermal bath at temperatureT, and evolve under Glauber dynamics. The oscillator position is a stochastic process due to theoscillator–spin interaction which produces drastic changes in the equilibrium behaviour and thedynamics of the oscillator. Firstly, there is a second order phase transition at a critical temperatureTc whose order parameter is the oscillator stable rest position: this position is zero aboveTc and differentfrom zero below Tc. This transition appears because the oscillator moves in an effective potential equal to theharmonic term plus the free energy of the spin system at fixed oscillator position. Secondly,assuming fast spin relaxation (compared to the oscillator natural period), the oscillatordynamical behaviour is described by an effective equation containing a nonlinear frictionterm that drives the oscillator towards the stable equilibrium state of the effectivepotential. The analytical results are compared with numerical simulation throughout thepaper.