Return to Equilibrium for an Anharmonic Oscillator coupled to a Heat Bath

TL;DR

This paper proves that a particle modeled as a harmonic or anharmonic oscillator coupled to a heat bath returns to equilibrium exponentially fast, using explicit formulas and Dyson's expansion, with detailed spectral analysis.

Contribution

It introduces a novel analysis of return to equilibrium for anharmonic oscillators coupled to heat baths, extending previous harmonic oscillator results.

Findings

Exponential return to equilibrium for small coupling constants

Explicit formulas for Weyl operator evolution in harmonic case

Spectral analysis showing continuous spectrum and unique eigenvalue

Abstract

We study a $C^*$-dynamical system describing a particle coupled to an infinitely extended heat bath at positive temperature. For small coupling constant we prove return to equilibrium exponentially fast in time. The novelty in this context is to model the particle by a harmonic or anharmonic oscillator, respectively. The proof is based on explicit formulas for the time evolution of Weyl operators in the harmonic oscillator case. In the anharmonic oscillator case, a Dyson's expansion for the dynamics is essential. Moreover, we show in the harmonic oscillator case, that $\R$ is the absolute continuous spectrum of the Standard Liouvillean and that zero is a unique eigenvalue.