  We consider a particle in the harmonic approximation coupled linearly to an
environment. modeled by an infinite set of harmonic oscillators. The system
(particle--environment) is considered in a cavity at thermal equilibrium. We
employ the recently introduced notion of renormalized coordinates to
investigate the time evolution of the particle occupation number. For
comparison we first present this study in bare coordinates. For a long ellapsed
time, in both approaches, the occupation number of the particle becomes
independent of its initial value. The value of ocupation number of the particle
is the physically expected one at the given temperature. So we have a Markovian
process, describing the particle thermalization with the environment. With
renormalized coordinates no renormalization procedure is required, leading
directly to a finite result.
