  The dynamics of the true-vacuum bubbles nucleated during a first-order phase
transition is affected by the distribution functions of the particle species in
the plasma, driven out-of-equilibrium by the travelling domain wall. An
accurate modelling of this phenomenon is relevant for a quantitative
description of phase transitions in the early universe and for the
determination of the corresponding cosmic relics, such as, among the others,
the stochastic background of gravitational waves. We address this problem by
developing a new spectral method devised for a fast and reliable computation of
the collision integral in the Boltzmann equations. In a scalar singlet
extension of the Standard Model chosen as a benchmark scenario, we test our
algorithm, determining the bubble speed and profile, and we asses the impact of
the out-of-equilibrium dynamics.
