  The $\lambda \phi^4$ model in a finite volume is studied in the infinite $N$
limit and within a non-gaussian Hartree-Fock approximation both at equilibrium
and out of equilibrium, with particular attention to certain fundamental
features of the broken symmetry phase. The numerical solution of the dynamical
evolution equations show that the zero-mode quantum fluctuations cannot grow
macroscopically large starting from microscopic initial conditions. Thus we
conclude that there is no evidence for a dynamical Bose-Einstein condensation.
On the other hand, out of equilibrium the long-wavelength fluctuations do scale
with the linear size of the system, signalling dynamical infrared properties
quite different from the equilibrium ones characteristic of the same
approximation schemes. This result suggests the cause, and the possible remedy,
of some unlikely features of the application to out--of--equilibrium dynamics
of the standard HF factorization scheme, which coincides with the gaussian
restriction of our Hartree--Fock approximation.
