Approach to Thermal Equilibrium in the Caldeira-Leggett Model

TL;DR

This paper analytically demonstrates how the Caldeira-Leggett model approaches thermal equilibrium in the high temperature and weak coupling limit, for both free particles and harmonic oscillators, using different master equations.

Contribution

It provides explicit calculations showing the asymptotic thermalization in the Caldeira-Leggett model, including generalization to higher dimensions and comparison of master equations.

Findings

Exact thermal equilibrium for free particles.

Approximate thermal equilibrium for harmonic oscillators.

Thermalization independent of initial conditions.

Abstract

We provide an explicit analytical calculation that shows the asymptotic approach of the one dimensional Caldeira-Leggett model to thermal equilibrium in the high temperature and weak coupling limit. We investigate a free particle and a harmonic oscillator system, using both the Lindblad and the non-Lindblad type master equations for each case, and show that thermal equilibrium is reached exactly for the free particle and aproximately for the harmonic oscillator, irrespective of the initial preparation of the system. We also generalize our calculation to higher dimensions.