Renormalized coordinate approach to the thermalization process

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 elapsed time, in both approaches, the occupation number of the particle becomes independent of its initial value. The value of the occupation 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.

Figure 1

(Color online) Time behavior for ${\overline{n}}_0\left(t\right)$ given by Eq. (53) for $\left(t>1\right)$, $n_0=1$, $\overline{\omega }=1$, $\beta =2$, and $g=0.1$; arbitrary units are used.

Figure 2

(Color online) Time behavior for $n_0^{\prime }\left(t\right)$ given by Eq. (105) for $\left(t>1\right)$, $n_0=1$, $\overline{\omega }=1$, $\beta =2$, and $g=0.1$; arbitrary units are used.