Noninteracting classical spins coupled to a heat bath of one-dimensional classical harmonic oscillators: Exact bath variable average

Abstract

A system of noninteracting classical spins coupled to the modes of a chain of one-dimensional classical harmonic oscillators via −S z Σc k x k was investigated to see whether the spin-bath coupling could relax the spins toward equilibrium. We considered two different cases for the initial conditions on the classical harmonic oscillator variables when we took the harmonic oscillator bath variable averages for the total spin components. In the first case, the harmonic oscillators were initially in equilibrium while they did not recognize the presence of the spin, whereas in the second case, the spins were in equilibrium and did recognize the presence of the spin. For the first case, the bath variable averages of the x- and the y-components of the total spin showed that the effective angular velocity transiently slowed down after an initial increase; then, it recovered its initial angular velocity continually. In the second case, the effective angular velocity was fixed. For both cases, the z-component of the total spin vector remained constant. If the x- and the y-components of the single spins were randomly distributed, we would get equilibrium values. For the z-components of single spins, unless they are at equilibrium from the beginning, they do not attain equilibrium.