Approach to equilibrium for a particle interacting with a harmonic thermal bath
Federico Bonetto, Alberto Mario Maiocchi
TL;DR
The paper analyzes how a harmonic probe coupled to a large finite bath approaches equilibrium by studying two-time correlation functions. Using Laplace-transform techniques, it derives the infinite-bath limit C_α(s,t) and shows that resonance with the bath spectrum drives apparent thermalization at leading order, while non-resonant coupling yields negligible thermalization. Higher-order α corrections introduce oscillatory and power-law decays, indicating that the bath cannot be fully represented by a Markovian stochastic thermostat. A stochastic-thermostat model is discussed to illustrate the conditions under which the bath behaves like white noise, and the results are extended to two-time correlations, revealing distinct non-resonant and resonant regimes with different long-time behaviors. The analysis relies on an explicit spectral function f_+(λ) and residue calculations, providing a detailed picture of thermalization, memory effects, and the limitations of Markovian approximations in finite but large baths.
Abstract
We study the long time evolution of the position-position correlation function $C_{α,N}(s,t)$ for a harmonic oscillator (the {\it probe}) interacting via a coupling $α$ 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 $N$, $C_{α,N}(s,t)$ is very well approximated by its limit $C_α(s,t)$ as $N\to\infty$. 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 $α$, $C_α(s,t)$ 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 $\lim_{t\to\infty} C_α(t,t)=T_B/Ω^2$ while that $\lim_{τ\to\infty} C_α(τ,τ+t)$ exists and decays exponentially in $t$. Notwithstanding this, at higher order in $α$, $C_α(s,t)$ 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.