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

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 $\omega$ is built directly from the input EoSs and the fluid fractions are fixed by minimizing $\omega$ at fixed temperature $T$ and baryon chemical potential $\mu_B$. Thermodynamic consistency and stability are guaranteed as all thermodynamic quantities are consistently derived from a single merged grand potential $\omega(T,\mu_B)$ 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 $(T,\mu_B)$, 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.