Phase transitions in a mechanical system coupled to Glauber spins

TL;DR

This paper studies a harmonic oscillator coupled to a chain of Glauber spins, revealing a second order phase transition at a critical temperature where the oscillator's equilibrium position shifts from zero to non-zero, with analytical and numerical analysis.

Contribution

It introduces a model coupling a harmonic oscillator with Glauber spins, analyzing phase transitions and oscillator dynamics under this interaction, including effective potential and nonlinear friction effects.

Findings

A second order phase transition at critical temperature T_c.

Oscillator's stable position shifts from zero to non-zero below T_c.

Analytical results agree with numerical simulations.

Abstract

A harmonic oscillator linearly coupled with a linear chain of Ising spins is investigated. The $N$ spins in the chain interact with their nearest neighbours with a coupling constant proportional to the oscillator position and to $N^{-1/2}$, are in contact with a thermal bath at temperature $T$, and evolve under Glauber dynamics. The oscillator position is a stochastic process due to the oscillator-spin interaction which produces drastic changes in the equilibrium behaviour and the dynamics of the oscillator. Firstly, there is a second order phase transition at a critical temperature $T_c$ whose order parameter is the oscillator stable rest position: this position is zero above $T_c$ and different from zero below $T_c$. This transition appears because the oscillator moves in an effective potential equal to the harmonic 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 oscillator dynamical behaviour is described by an effective equation containing a nonlinear friction term that drives the oscillator towards the stable equilibrium state of the effective potential. The analytical results are compared with numerical simulation throughout the paper.