Thermal Equilibrium from the Hu-Paz-Zhang Master Equation

TL;DR

This paper analyzes the exact master equation for a harmonic oscillator coupled to a heat bath, showing that its late-time solution reaches thermal equilibrium and revealing the effective frequency shift due to bath interaction.

Contribution

It simplifies the Hu-Paz-Zhang master equation in the weak-coupling, late-time limit and demonstrates the unique thermal equilibrium solution.

Findings

The late-time solution is the canonical ensemble at the bath temperature.

The oscillator's frequency is effectively reduced by the bath interaction.

The simplified master equation confirms thermal equilibrium as the steady state.

Abstract

The exact master equation for a harmonic oscillator coupled to a heat bath, derived recently by Hu, Paz and Zhang, is simplified by taking the weak-coupling, late-time limit. The unique time-independent solution to this simplified master equation is the canonical ensemble at the temperature of the bath. The frequency of the oscillator is effectively lowered by the interaction with the bath.