Thermalization process in bare and dressed coordinate approaches

TL;DR

This paper investigates the thermalization process of a harmonic oscillator particle coupled to a field environment, using bare and dressed coordinates to analyze the time evolution of occupation numbers and demonstrate Markovian thermalization.

Contribution

It introduces the use of dressed coordinates to study thermalization without renormalization, providing a finite and direct analysis of the particle's approach to equilibrium.

Findings

Dressed coordinates eliminate the need for renormalization.

Occupation number becomes independent of initial conditions at large times.

The process is Markovian, indicating thermalization with the environment.

Abstract

We consider a particle in the approximation of a harmonic oscillator, coupled linearly to a field modeling an environment. The field is described by an infinite set of harmonic oscillators, and the system (particle--field) is considered in a cavity at thermal equilibrium. We employ the notions of bare and dressed coordinates to study the time evolution of the occupation number. With dressed coordinates no renormalization procedure is required, leading directly to a finite result. In particular, for a large time, the occupation number of the particle becomes independent of its initial value. So we have a Markovian process, describing the particle thermalization with the environment.