Finite temperature effective action and thermalization of perturbations

The effective action is computed for the \lphi--theory at finite temperature for small perturbations about a constant background field, using a generalized tadpole method. We find the complete effective action, including the real and imaginary parts, to all orders in derivatives and to order ${\cal O}(\lambda^2)$. We demonstrate that the high $T$ approximation, where only the zero Matsubara frequency is included, is incorrect for the imaginary part even though it is UV-finite. The solutions of the dispersion relations show that initial perturbations do not necessarily thermalize fast enough to be absent at the onset of phase transition.