Out-of-equilibrium dynamics of \phi^4 QFT in finite volume

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.