  We consider the large-N $\Phi^4$ theory with spontaneously broken symmetry at
finite temperature. We study, in the large-N limit, quantum states which are
characterized by a time dependent, spatially homogenous expectation value of
one of the field components, $\phi_N(t)$, and by quantum fluctuations of the
other $N-1$ components, that evolve in the background of the classical field.
Investigating such systems out of equilibrium has recently been shown to
display several interesting features. We extend here this type of
investigations to finite temperature systems. Essentially the novel features
observed at T=0 carry over to finite temperature. This is not unexpected, as
the main mechanisms that determine the late-time behavior remain the same. We
extend two empirical - presumably exact - relations for the late-time behavior
to finite temperature and use them to define the boundaries between the region
of different asymptotic regimes. This results in a phase diagram with the
temperature and the initial value of the classical field as parameters, the
phases being characterized by spontaneous symmetry breaking resp. symmetry
restoration. The time evolution is computed numerically and agrees very well
with the expectations.
