  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.
