Approach to equilibrium for a particle interacting with a harmonic thermal bath
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915945232990208 |
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| author | Bonetto, Federico Maiocchi, Alberto Mario |
| author_facet | Bonetto, Federico Maiocchi, Alberto Mario |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_20003 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Approach to equilibrium for a particle interacting with a harmonic thermal bath Bonetto, Federico Maiocchi, Alberto Mario Statistical Mechanics Mathematical Physics 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. |
| title | Approach to equilibrium for a particle interacting with a harmonic thermal bath |
| topic | Statistical Mechanics Mathematical Physics |
| url | https://arxiv.org/abs/2510.20003 |