Not yet reviewed by Pith; the record is open.

Hot matter with nucleons can be produced in the inner region of the neutrino-dominated accretion flow in gamma-ray bursts or during the proto-neutron star birth in successful supernova. The composition and equation of state of the matter depend on the dynamic $\beta$ equilibrium under various neutrino opacities. The strong interaction between nucleons may also play an important role. We plan to extend the previous studies by incorporating these two aspects in our model. The modification of the $\beta$-equilibrium condition from neutrino optically thin to thick has been modeled by an equilibrium factor $\chi$ ranging between the neutrino-freely-escaping case and the neutrino-trapped case. We employ the microscopic Brueckner-Hartree-Fock approach extended to the finite temperature regime to study the interacting nucleons. We show that the composition and chemical potentials of the hot nuclear matter for different densities and temperatures at each stage of $\beta$ equilibrium. We also compare our realistic equation of states with those of the free gas model. We find the neutrino opacity and the strong interaction between nucleons are important for the description and should be taken into account in model calculations.