Approach to equilibrium for a particle interacting with a harmonic thermal bath

We study the long time evolution of the position-position correlation function $C_{\alpha,N}(s,t)$ for a harmonic oscillator (the {\it probe}) interacting via a coupling $\alpha$ with a large chain of $N$ coupled oscillators (the {\it heat bath}). At $t=0$ the probe and the bath are in equilibrium at temperature $T_P$ and $T_B$, respectively. We show that for times $t$ and $s$ of the order of $N$, $C_{\alpha,N}(s,t)$ is very well approximated by its limit $C_{\alpha}(s,t)$ as $N\to\infty$. We find that, if the frequency $\Omega$ of the probe is in the spectrum of the bath, the system appears to thermalize, at least at higher order in $\alpha$. This means that, at order 0 in $\alpha$, $C_\alpha(s,t)$ equals the correlation of a probe in contact with an ideal stochastic {\it thermostat}, that is forced by a white noise and subject to dissipation. In particular we find that $\lim_{t\to\infty} C_\alpha(t,t)=T_B/\Omega^2$ while that $\lim_{\tau\to\infty} C_\alpha(\tau,\tau+t)$ exists and decays exponentially in $t$. Notwithstanding this, at higher order in $\alpha$, $C_{\alpha}(s,t)$ 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.