The lambda phi(4) model in a finite volume is studied in the infinite N Lim
it both at equilibrium and out of equilibrium, with particular attention to
certain fundamental features of the broken symmetry phase. The numerical s
olution of the dynamical evolution equations shows that the zero-mode quant
um fluctuations cannot grow macroscopically large starting from microscopic
initial conditions. Thus we conclude that there is no evidence for a dynam
ical Bose-Einstein condensation, in the usual sense. On the other hand, out
of equilibrium the long-wavelength fluctuations do scale with the linear s
ize of the system, signaling dynamical infrared properties quite different
from the equilibrium ones characteristic of the same approximation scheme.