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We present microscopic Molecular Dynamics simulations including the efficientEwald sum procedure to study warm and dense stellar plasmas consisting offinite-size ion charges immerse in a relativistic neutralizing electron gas.For densities typical of Supernova matter and crust in a proto-neutron star, weselect a representative single ion composition and obtain the virializedequation of state (vEoS). We scrutinize the finite-size and screeningcorrections to the Coulomb potential appearing in the virial coefficients $B_2,B_3$ and $B_4$ as a function of temperature. In addition, we study the thermalheat capacity at constant volume, $C_V$, and the generalized Mayer's relationi.e. the difference $C_P-C_V$ with $C_P$ being the heat capacity at constantpressure, obtaining clear features signaling the onset of the liquid-gas phasetransition. Our findings show that microscopic simulations reproduce thediscontinuity in $C_V$, whose value lies between that of idealized gas andcrystallized configurations. We study the pressure isotherms marking theboundary of the metastable region before the gaseous transition takes place.The resulting vEoS displays a behaviour where effective virial coefficientsinclude extra density dependence showing a generalized density-temperatureform. As an application we parametrize pressure as a function of density andtemperature under the form of an artificial neural network showing thepotential of machine learning for future regression analysis in more refinedmulticomponent approaches. This is of interest to size the importance of thesecorrections in the liquid-gas phase transition in warm and dense plasma phasescontributing to the cooling behaviour of early Supernova phases andproto-neutron stars.

Virialized equation of state for warm and dense stellar plasmas in proto-neutron stars and Supernova matter