Stochastic model for the dynamics of a spin-oscillator coupled system

We construct a stochastic model for the dynamics of a one-dimensional system consisting of bilinearly coupled harmonic oscillators and spins. The spin dynamics is defined as a Glauber model where the spins are effectively coupled through their interaction with the oscillators. To maintain internal thermal equilibrium in the composite system, which does not exhibit Onsager symmetry, we introduce a phenomenological retarded friction in the oscillator equation of motion and relate it to the spin correlation function through a fluctuation-dissipation theorem. The oscillator susceptibility is derived and the behavior of its poles as functions of wavevector and temperature is studied. The results are compared to those obtained by other authors who have studied similar systems, using irreversible thermodynamics. In contrast to ours, these treatments do not give an explicit result for the wavevector dependence of the poles.