From friction scaling to an efficient method for estimating bubble wall velocity

We present a unified description of first-order cosmological phase transition dynamics that links the phenomenological friction model employed in hydrodynamic simulations to the microscopic treatment based on Boltzmann equations. We derive an approximate analytical expression for the chemical potential and demonstrate that the resulting friction parameter $\tilde{\eta}$ follows a simple power-law dependence on the transition strength ($\propto v_n^4/T_n^4$). Incorporating this scaling into a phenomenological framework accurately reproduces the terminal wall velocities obtained from the full microscopic analysis performed using \texttt{WallGo}. This approach offers an efficient method to quantify out-of-equilibrium contributions to friction and reliably estimate bubble-wall velocities.