  The phenomenon of spontaneous symmetry breaking is investigated in the
dynamic thermalization of a degenerate quantum system. A three-level system
interacting with a heat bath is carefully studied to this end. It is shown that
the three-level system with degenerate ground states might have different
behaviors depending on the details of the interaction with the heat bath when
the temperature approaches zero. If we introduce an external field to break the
degeneracy of the ground states and let it approach zero after letting the
temperature approach zero, then two possibilities will arise: the steady state
is a definite one of the degenerate states independent of the initial state, or
the steady state is dependent on the initial state in a complicated way. The
first possibility corresponds to a spontaneous symmetry breaking of the system
and the second one implies that the heat bath could not totally erase the
initial information in certain cases.
