  We consider the nonlinear Schroedinger equation on the one dimensional torus,
with a defocousing polynomial nonlinearity and study the dynamics corresponding
to initial data in a set of large measure with respect to the Gibbs measure. We
prove that along the corresponding solutions the modulus of the Fourier
coefficients is approximately constant for times of order
$\beta^{2+\varsigma}$, $\beta$ being the inverse of the temperature and
$\varsigma$ a positive number (we prove $\varsigma= 1/10$). The proof is
obtained by adapting to the context of Gibbs measure for PDEs some tools of
Hamiltonian perturbation theory.
