Framework for phase transitions between the Maxwell and Gibbs constructions at finite temperature

TL;DR

The paper generalizes a thermodynamically consistent framework for hadron-quark first-order transitions to finite temperature and out-of-$\beta$ equilibrium by introducing a continuous charge-neutrality parameter $\eta$ that interpolates between Maxwell and Gibbs constructions. It provides a finite-$T$ EOS built from hadronic and quark sectors, includes antiparticles, and computes thermodynamic quantities (e.g., $C_V$, $C_P$, $c_{ad}^2$, and the thermal index $\Gamma$) across a range of $n_B$, $T$, and $Y_e$, showing that the mixed phase is not generally at constant pressure when $\beta$-equilibrium is not enforced. The results reveal that higher $T$ and lower $Y_e$ push the transition to lower densities, and that the pressure in the mixed phase and the thermal response depend sensitively on $\eta$ because multiple globally conserved charges govern equilibrium. The work demonstrates that constant-$\Gamma$ extrapolations can be ill-defined in the mixed phase and provides a thermodynamically consistent route for EOS tables applicable to CCSNe and BNSMs, with clear paths for extensions to include additional degrees of freedom and neutrino physics.

Abstract

The characteristics of the hadron-to-quark first-order phase transition differ depending on whether charge neutrality is locally or globally fulfilled. In $β$-equilibrated matter, these two possibilities correspond to the Maxwell and Gibbs constructions. Recently, we presented a new framework in which a continuously-varying parameter allows one to describe a first-order phase transition in intermediate scenarios to the two extremes of fully local and fully global charge neutrality. In this work, we extend the previous framework to finite temperatures and out-of-$β$ equilibrium conditions, making it available for simulations of core-collapse supernovae and binary neutron star mergers. We investigate its impact on key thermodynamic quantities across a range of baryon densities, temperatures, and electron fractions. We find that when matter is not in $β$-equilibrium, the pressure in the mixed phase is not constant even for the case of fully-local charge neutrality. Moreover, we compute the thermal index using three different approaches, demonstrating that the finite-temperature extension of an equation of state using a constant thermal index can be ill-defined when applied to the mixed phase.

Figures (12)

• Figure 1: The volume fraction of nucleons $\chi$ vs. baryon density $n_{\rm B}$ in units of nuclear saturation density $n_0$ for the indicated values of the local-to-total lepton ratio $\eta$, with net electron fraction $Y_e=0.1$ and temperature $T=10$ MeV [panel (a)], $Y_e=0.1$ and $T=50$ MeV [panel (b)], $Y_e=0.4$ and $T=10$ MeV [panel (c)], and $Y_e=0.4$ and $T=50$ MeV [panel (d)].
• Figure 2: Phase diagram in the $T-n_{\rm B}$ plane for the indicated values of the local-to-total lepton ratio $\eta$. For each case, phase boundaries($T$ vs. $n_{\rm B}$) are shown as dashed(solid) lines at a fixed specific entropy $S=s/n_{\rm B}$ (at either $S=1$ or $S=2$) and fixed $Y_e$ (either $Y_e=0.1$ or $Y_e=0.4$) as stated in the plot titles. $Y_e=0.25$ cases [panels (c) and (d)] are reported here as well since Ref. Kuroda:2021eiv obtains it as a typical central value after the core bounce in a CCSN simulation of massive progenitors.
• Figure 3: Pressure $P$ vs. energy density $\varepsilon$ for the indicated values of the local-to-total lepton ratio $\eta$, net electron fraction $Y_e$, and temperature $T$. Different from the $\beta$-equilibrium case, the pressure is not constant in the mixed phase even in the $\eta=1$ (Maxwell) case.
• Figure 4: Physical particle fractions $Y_i^*$ vs. baryon density $n_{\rm B}$ at net electron fraction $Y_e=0.1$, for the indicated values of the local-to-total lepton ratio $\eta$ and temperature $T$. Contributions from electrons and positrons are reported separately, while in the other cases, the net values are presented.
• Figure 5: Same as Fig. \ref{['Fig:Y01']} but at net electron fraction $Y_e=0.4$.