Merging multidimensional equations of state of strongly interacting matter via a statistical mixture

TL;DR

This work introduces a thermodynamically consistent two-fluid mixing framework to merge distinct equations of state (EoSs) for strongly interacting matter by minimizing a grand potential density $ω(T,μ_B)$ with respect to an internal order-parameter $p$. The method replaces naive switching with an entropy- and interaction-driven mixing term, enabling crossover, a critical point with mean-field Ising universality, and a first-order line, while preserving convexity and stability. As a concrete application, the authors merge a van der Waals hadron-resonance-gas EoS with a holographic Einstein–Maxwell–Dilaton EoS, yielding a single global EoS that spans hadronic and deconfined matter up to $μ_B\sim 1$ GeV and $T\sim 600$ MeV, in good agreement with lattice QCD at $μ_B=0$ and with $T'$ expansions at finite density. The resulting EoS supports realistic hydrodynamic simulations and provides a robust, extensible framework for exploring the QCD phase diagram, including the possible observability of phase-transition signatures in heavy-ion collisions and neutron-star phenomena.

Abstract

We introduce a general method to merge multidimensional equations of state (EoSs) by combining them in a two-fluid equilibrium statistical mixture in the grand canonical ensemble. The merged grand potential density $ω$ is built directly from the input EoSs and the fluid fractions are fixed by minimizing $ω$ at fixed temperature $T$ and baryon chemical potential $μ_B$. Thermodynamic consistency and stability are guaranteed as all thermodynamic quantities are consistently derived from a single merged grand potential $ω(T,μ_B)$ with the correct convexity properties. Our method can accommodate a first-order phase transition and a critical endpoint with mean-field critical exponents. We use this method to merge a van der Waals Hadron-Resonance-Gas EoS with a holographic Einstein-Maxwell-Dilaton EoS that has a critical point and a first-order line. The result is a single EoS, spanning hadronic and deconfined matter over a broad range in $(T,μ_B)$, which can be readily used in heavy-ion hydrodynamic simulations. Our merging method can be generalized to consider a higher dimensional phase diagram (e.g., by considering more chemical potentials) and more than two input EoSs.

Figures (14)

• Figure 1: Schematic figure of the thermodynamic potential when $P_1=P_2$, as a function of the probability $p$, for three different scenarios: i) $b<2a$ (continuous black line), corresponding to a crossover; ii) $b=2a$ (dashed blue line), corresponding to the critical point; iii) $b>2a$ (dashed-dotted red line), corresponding to a first order phase transition.
• Figure 2: Mixing weight $\overline{p}$ of the EMD EoS as a function of temperature at different values of chemical potential (left) and as a function of chemical potential at different values of the temperature (right).
• Figure 3: Pressure as a function of the temperature for different values of chemical potential. At chemical potential $\mu_B=0$, we compare the results with LQCD data from Ref. Borsanyi:2013bia (green points). In all panels, the solid black line indicates the equation of state obtained by merging the QvdW-HRG EoS at low temperature (blue, dashed line) with the EMD EoS at high temperature (red, dotted line).
• Figure 4: Entropy density as a function of temperature for different values of chemical potential. At vanishing chemical potential $\mu_B=0$, we compare the results with LQCD data from Ref. Borsanyi:2013bia (green points). In all panels, the solid black line indicates the equation of state obtained by merging the QvdW-HRG EoS at low temperature (blue, dashed line) with the EMD EoS at high temperature (red, dotted line).
• Figure 5: Net baryon density as a function of temperature for different values of chemical potential. In all panels, the solid black line indicates the equation of state obtained by merging the QvdW-HRG EoS at low temperature (blue, dashed line) with the EMD EoS at high temperature (red, dotted line).