Classical-field method for time dependent Bose-Einstein condensed gases

We propose a method to study the time evolution of Bose-Einstein condensed gases perturbed from an initial thermal equilibrium, based on the Wigner representation of the N-body density operator. We show how to generate a collection of random classical fields sampling the initial Wigner distribution in the number conserving Bogoliubov approximation. The fields are then evolved with the time dependent Gross-Pitaevskii equation, We illustrate the method with the damping of a collective excitation of a one-dimensional Bose gas.