Thermalization process in bare and dressed coordinate approaches

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.