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

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.

• The paper introduces a unified merging method for multidimensional equations of state by minimizing the grand potential with respect to a mixing parameter.
• It demonstrates the method by combining a quantum van der Waals HRG model with a holographic Einstein-Maxwell-Dilaton EoS, achieving seamless interpolation between hadronic and deconfined phases.
• The approach yields a globally valid EoS that captures crossover, critical phenomena, and first-order transitions while ensuring thermodynamic stability and analytic tractability.

Merging Multidimensional Equations of State for Strongly Interacting Matter via a Statistical Mixture

Introduction and Motivation

The phase diagram of QCD spans wide-ranging temperatures ($T$) and baryon chemical potentials ($\mu_B$), encompassing regimes relevant to relativistic heavy-ion collisions and neutron star mergers. Modeling the equation of state (EoS) across its entire domain remains a persistent challenge because no single theoretical framework robustly covers both the hadronic (confined) and deconfined quark-gluon plasma (QGP) phases. Lattice QCD (LQCD) methods are restricted by the $\mu_B$ sign problem, and effective models such as the hadron resonance gas (HRG) and holographic approaches are limited in the regions of applicability and physical content.

This paper introduces a general and thermodynamically consistent procedure for merging distinct multidimensional EoSs. The merging leverages a statistical mixture in the grand canonical ensemble, minimizing the global grand potential density $\omega$ over an internal mixing parameter $p$. This formulation admits a seamless interpolation, supports critical phenomena (critical point and first-order transition), and systematically derives all thermodynamic observables with correct convexity properties. The formalism is demonstrated by merging a quantum van der Waals HRG EoS (QvdW-HRG) with a holographic Einstein-Maxwell-Dilaton (EMD) EoS, resulting in a single EoS suitable for hydrodynamic applications that spans both domains without introducing unphysical artifacts.

Formulation of the Merging Method

The commonly used weighted-average approach to EoS merging introduces artifacts such as spurious oscillations and can destabilize thermodynamic observables due to the nontrivial derivatives of switching functions. This paper instead replaces the deterministic switching function $S(T,\mu_B)$ with an internal variable $p \in [0,1]$, analogous to an order parameter. The merged grand potential density is constructed:

$$\omega = -p\,P_1 - (1-p)\,P_2 + a\,[p\,\ln p + (1-p)\,\ln(1-p)] + p(1-p) b$$

where $P_{1,2}$ are pressures from the two input EoSs, $a$ controls mixing entropy, and $b$ models fluid interactions. The equilibrium mixing weight $\overline{p}$ is obtained by minimizing $\omega$ at fixed $(T, \mu_B)$, subject to stability $\partial^2 \omega / \partial p^2 \geq 0$.

Thermodynamic quantities (pressure, entropy, baryon density, susceptibilities) are derived by differentiating the minimized $\omega$. Notably, the mixing entropy term renders the transition smooth for $b < 2a$ (crossover), introduces a critical point for $b=2a$, and produces a first-order transition for $b>2a$. The model's critical behavior is rigorously shown to be in the mean-field Ising universality class.

Figure 1: Schematic thermodynamic potential as a function of mixing probability $p$: crossover ($b<2a$), critical point ($b=2a$), and first-order phase transition ($b>2a$).

Models for the QCD Equation of State

Quantum van der Waals Hadron Resonance Gas (QvdW-HRG)

The QvdW-HRG model extends the ideal HRG description by introducing excluded-volume and mean-field baryon-baryon interactions. This supports the nuclear liquid-gas phase transition and yields LQCD-consistent observables at small $\mu_B$ and around the crossover region. The meson sector receives a repulsive term to ensure suppression beyond the hadronic regime. The model parameters are fixed to reproduce nuclear saturation properties.

Holographic Einstein-Maxwell-Dilaton (EMD) Model

The EMD model is a bottom-up holographic construction in $AdS_5$ with an explicit non-conformal breaking via a dilaton field. The baryon sector is sourced by a $U(1)$ gauge field and tuned to reproduce LQCD susceptibilities and entropy at $\mu_B=0$. The EMD framework predicts a critical point and first-order line at intermediate $(T, \mu_B)$, aligning well with LQCD data where available.

Thermodynamic Results of the Merged EoS

The merging procedure yields an EoS that, at low $T$, recovers the QvdW-HRG, while at high $T$ and/or $\mu_B$ smoothly transitions to the EMD EoS. The equilibrium mixing weight $\overline{p}(T, \mu_B)$ sharply delineates the phase transition loci:

Figure 2: Mixing weight $\overline{p}$ of EMD as a function of $T$ and $\mu_B$, showing discontinuity at the first-order line and smooth crossover at low $\mu_B$.

Pressure, entropy density, baryon density, and energy density are traced across various $\mu_B$ values. Observables continuously interpolate between input EoSs in the crossover region and exhibit discontinuities characteristic of first-order transitions above the critical $\mu_B$.

Figure 3: Pressure versus temperature at different $\mu_B$, compared to LQCD at $\mu_B=0$. The merged EoS tracks the maximum input pressure and features a kink at the first-order line.

Figure 4: Entropy density versus $T$ for several $\mu_B$ values, illustrating crossover and phase transition across the merged EoS.

Figure 5: Net baryon density as a function of $T$ for increasing $\mu_B$, showing discontinuities at first-order transition.

Figure 6: Energy density as a function of $T$, compared against LQCD data at $\mu_B=0$.

Second-order susceptibilities and speed of sound squared reveal critical phenomena:

Figure 7: Net baryon number susceptibility versus $T$, displaying diverging behavior at the critical point in $\mu_B$.

Figure 8: Speed of sound squared as a function of $T$, vanishing at the critical endpoint.

Specific heat $C_V$ mirrors these features:

Figure 9: Specific heat at constant volume versus $T$ highlights discontinuous jumps and critical enhancement.

Phase Structure and Lattice QCD Validation

The merged EoS delivers a comprehensive QCD phase diagram, reproducing both crossover and first-order transition lines. The critical point location and associated phase boundaries are found to align with predictions from the standalone EMD model.

Figure 10: Phase diagram in the $T$--$\mu_B$ plane for the merged EoS and EMD model: crossover, first-order line, and critical endpoint.

Direct comparison to LQCD results via $T'$ expansion shows robust agreement across $T$ and $\mu_B/T$ for both pressure and conserved charge densities.

Figure 11: Quantitative comparison between merged EoS and LQCD $T'$-expansion for pressure, entropy, baryon density, and energy density at varying $\mu_B/T$.

Three-dimensional visualizations elucidate the merged EoS's structure:

Figure 12: Surface plots of pressure, energy density, entropy density, net baryon density, speed of sound squared, baryon susceptibility, and specific heat over $(T, \mu_B)$.

Isentropic trajectories reveal the response of the system near phase boundaries and the critical point:

Figure 13: Isentropic trajectories in the $T-\mu_B$ plane; pronounced bending near the critical endpoint.

Heat map visualizations of susceptibility highlight sharp features at both liquid-gas and QCD critical points:

Figure 14: Heat map of normalized second-order baryon susceptibility $\chi_2/(T^2 + \mu_B^2)$, demonstrating divergence at both critical points.

Implications, Extensions, and Future Directions

The merging framework maintains full thermodynamic consistency and avoids instabilities common in other switching-based interpolation schemes. Numeric robustness is improved by analytic expressions for all derivatives, critical for downstream modeling (hydrodynamics, fluctuation analyses).

The method admits extension to include more than two EoSs and higher-dimensional phase diagrams incorporating additional conserved charges. Parameter uncertainty quantification (Bayesian inference) and application to isospin-asymmetric matter are practical next steps. Embedding the merged EoS into dynamical simulations (hydrodynamics and transport) will advance predictive capability for low energy heavy-ion collisions and neutron star phenomena.

The EoS and merging code are openly available, facilitating widespread adoption and further comparative studies across candidate EoSs.

Conclusion

This work establishes a robust formalism for merging multidimensional EoSs for strongly interacting matter. By modeling the system as an equilibrium statistical mixture and minimizing the grand potential, the resulting framework integrates hadronic and deconfined phases in a single, globally valid EoS, supports critical phenomena, and maintains thermodynamic stability. Quantitative agreement with LQCD is achieved throughout the accessible phase diagram, and features such as critical points and first-order transitions emerge naturally, enabling realistic modeling of QCD matter for a broad spectrum of applications. Extension, validation, and further dynamical application of this methodology represent fertile ground for future research.

Reference: "Merging multidimensional equations of state of strongly interacting matter via a statistical mixture" (2601.07987)