  We study the detailed out of equilibrium time evolution of a homogeneous
Bose-Einstein condensate.We consider a nonrelativistic quantum theory for a
self-interacting complex scalar field, immersed in a thermal bath, as an
effective microscopic model for the description of the Bose-Einstein
condensate. This approach yields the following main results:(i) the interaction
between fluctuations proves to be crucial in the mechanism of instability
generation; (ii) there are essentially two regimes in the $k$-space, with a
crossover for $k^2/2m \sim 2\lambda |\phi_0|^2$, where, in our notation,
$\lambda$ is the coupling constant and $|\phi_0|^2$ is the condensate density;
(iii) a set of coupled equations that determines completely the nonequilibrium
dynamics of the condensate density as a function of the temperature and of the
total density of the gas.
