  The dynamics of bubbles nucleated during a first-order phase transition is
controlled by the non-equilibrium fluctuations generated by the traveling
domain wall. An accurate modelling of the out-of-equilibrium properties of the
plasma is necessary for the characterization of the phase transition relics,
such as the signal of gravitational waves. In this thesis we provide a solution
to the Boltzmann equation that describes the plasma fluctuations without
imposing any ansatz on the perturbation by devising a new spectral method that
leverages on the rotational invariance of the collision operator. This allows
for a robust and fast computation of the terminal velocity of the wall. We then
compare our results with the previous approaches in the literature finding
important quantitative and qualitative differences. We also employ our method
to determine the terminal velocity of the wall for two benchmark configurations
in the singlet extended Standard Model. We finally assess the impact of
different approximations and we find that the most important source of
uncertainty is given by the infrared modes of the electroweak gauge bosons.
