Temperature dependent Bogoliubov approximation in the classical fields approach to weakly interacting Bose gas

A classical fields approximation to the finite temperature microcanonical thermodynamics of weakly interacting Bose gas is applied to the idealized case of atoms confined in a box with periodic boundary conditions. We analyze in some detail the microcanonical temperature in the model. We also analyze the spectral properties of classical amplitudes of the plane waves -- the eigenmodes of the time averaged one--particle density matrix. Looking at the zero momentum component -- the order parameter of the condensate, we obtain the nonperturbative results for the chemical potential. Analogous analysis of the other modes yields nonperturbative temperature dependent Bogoliubov frequencies and their damping rates. Damping rates are linear functions of momenta in the phonon range and show more complex behavior for the particle sector. Where available, we make comparison with the analytic estimates of these quantities.