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

TL;DR

This paper investigates the long-term behavior of a harmonic oscillator coupled to a large chain of oscillators, revealing conditions under which the system appears to thermalize and highlighting the limitations of modeling the bath as a stochastic thermostat.

Contribution

It provides a detailed analysis of the correlation function dynamics, showing how thermalization depends on the oscillator's frequency relative to the bath spectrum and the order of coupling.

Findings

Correlation function approximates its limit for large N and times of order N.

Thermalization occurs when the probe's frequency is in the bath spectrum at higher order in coupling.

When far from the bath spectrum, no thermalization is observed.

Abstract

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.