Approach to Equilibrium for a Particle Interacting with a Harmonic Thermal Bath

We study the long time evolution of the position–position correlation function Cα,N(s,t) for a harmonic oscillator (the probe) interacting via a coupling α with a large chain of N coupled oscillators (the heat bath). At t=0 the probe and the bath are in equilibrium at temperature TP and TB, respectively. We show that for times t and s of the order of N, Cα,N(s,t) is very well approximated by its limit Cα(s,t) as N→∞. We find that, if the frequency Ω of the probe is in the spectrum of the bath, the system appears to thermalize, at least at higher order in α. This means that, at order 0 in α, Cα(s,t) equals the correlation of a probe in contact with an ideal stochastic thermostat, that is forced by a white noise and subject to dissipation. In particular we find that limt→∞Cα(t,t)=TB/Ω2 while that limτ→∞Cα(τ,τ+t) exists and decays exponentially in t. Notwithstanding this, at higher order in α, Cα(s,t) contains terms that oscillate or vanish as a power law in |t-s|. That is, even when the bath is very large, it cannot be thought of as a stochastic thermostat. When the frequency of the bath is far from the spectrum of the bath, no thermalization is observed.