Friction forces on phase transition fronts

TL;DR

The paper tackles friction forces on cosmological first-order phase-transition fronts, focusing on ultra-relativistic (UR) walls and the possibility of runaway propagation. It develops a decomposition of the UR total force into a driving term and an ultra-relativistic friction coefficient $\eta_{\text{UR}}$, and introduces a two-parameter phenomenological model with $\eta_{\text{NR}}$ and $\eta_{\text{UR}}$ (plus a saturation mechanism controlled by a parameter $\lambda_z$) to interpolate between non-relativistic and UR limits. Using the bag equation of state, it derives analytical expressions for the wall velocity and runaway conditions, showing that stationary and runaway solutions can coexist in a parameter range and detailing how supercooling and latent heat influence the outcomes. The study provides practical criteria for when walls run away or settle into stationary detonations/deflagrations, thereby informing predictions of gravitational wave signals and other cosmological relics from phase transitions. The results establish a concrete, two-parameter friction framework that remains consistent with known NR and UR limits and bridges to more realistic models of phase-transition dynamics.$

Abstract

In cosmological first-order phase transitions, the microscopic interaction of the phase transition fronts with non-equilibrium plasma particles manifests itself macroscopically as friction forces. In general, it is a nontrivial problem to compute these forces, and only two limits have been studied, namely, that of very slow walls and, more recently, ultra-relativistic walls which run away. In this paper we consider ultra-relativistic velocities and show that stationary solutions still exist when the parameters allow the existence of runaway walls. Hence, we discuss the necessary and sufficient conditions for the fronts to actually run away. We also propose a phenomenological model for the friction, which interpolates between the non-relativistic and ultra-relativistic values. Thus, the friction depends on two friction coefficients which can be calculated for specific models. We then study the velocity of phase transition fronts as a function of the friction parameters, the thermodynamic parameters, and the amount of supercooling.

Figures (7)

• Figure 1: A friction force of the form $v_w/\sqrt{1-(1-\lambda_z^2)v_w^2}$. From top to bottom the curves correspond to $\lambda=0,0.1,0.2,0.5,1,1.5$ and $3$. Red lines indicate the cases $\lambda=0$ and $\lambda=1$.
• Figure 2: Regions in the $(T_n,\eta_{\mathrm{UR}})$-plane where runaway and detonation solutions can exist. The runaway necessary condition (\ref{['necessbag']}) is fulfilled below the blue lines. The sufficient condition (\ref{['sufibag']}) is fulfilled below the red lines.
• Figure 3: The wall velocity for the bag model with $\alpha_c=4.45\times 10^{-3}$ and $T_n=0.89T_c$. Detonations are in blue, "traditional" deflagrations are in red, and Jouguet deflagrations are in black. The right panels show only the stable solutions.
• Figure 4: The wall velocity as a function of $\eta_{\mathrm{UR}}$, for the same bag parameters of Fig. \ref{['figvaretanr']} and several values of $\eta_{\mathrm{NR}}$. The vertical lines indicate the values of $\eta_{\mathrm{suf}}$ and $\eta_{\mathrm{nec}}$.
• Figure 5: The wall velocity as a function of $\alpha_n/\alpha_c=(T_c/T_n)^4$, for $\alpha_c=4.45\times 10^{-3}$, $\eta_{\mathrm{NR}}/L=0.1$, and several values of $\lambda_z$.