  We consider a quantum system composed of a spatially infinitely extended free
Bose gas with a condensate, interacting with a small system (quantum dot) which
can trap finitely many Bosons. Due to spontaneous symmetry breaking in the
presence of the condensate, the system has many equilibrium states for each
fixed temperature.
  We extend the notion of Return to Equilibrium to systems possessing a
multitude of equilibrium states and show in particular that a condensate
coupled to a quantum dot has the property of Return to Equilibrium in a weak
coupling sense: any local perturbation of an equilibrium state of the coupled
system, evolving under the interacting dynamics, converges in the long time
limit to an asymptotic state. The latter is, modulo an error term, an
equilibrium state which {\it depends} in an explicit way on the local
perturbation (an effect due to long-range correlations). The error term
vanishes in the small coupling limit.
  We deduce the stability result from properties of structure and regularity of
eigenvectors of the generator of the dynamics, called the Liouville operator.
Among our technical results is a Virial Theorem for Liouville type operators
which has new applications to systems with and without a condensate.
