  The $\lambda \phi^4$ model in a finite volume is studied in the infinite $N$
limit 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 shows 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, in the usual sense. 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 scheme.
