Equation of state for $\beta$-stable hot nuclear matter

We provide an equation of state for hot nuclear matter in $\beta$-equilibrium by applying a momentum-dependent effective interaction. We focus on the study of the equation of state of high-density and high-temperature nuclear matter, containing leptons (electrons and muons) under the chemical equilibrium condition in which neutrinos have left the system. The conditions of charge neutrality and equilibrium under $\beta$-decay process lead first to the evaluation of proton and lepton fractions and afterwards of internal energy, free energy, pressure and in total to the equation of state of hot nuclear matter. Thermal effects on the properties and equation of state of nuclear matter are assesed and analyzed in the framework of the proposed effective interaction model. Special attention is dedicated to the study of the contribution of the components of $\beta$-stable nuclear matter to the entropy per particle, a quantity of great interest for the study of structure and collapse of supernova.