Conditions for phase equilibrium in supernovae, protoneutron, and neutron stars

We investigate 1st order phase transitions in a general way, if not the single particle numbers of the system but only some particular charges like e.g. baryon number are conserved. In addition to globally conserved charges, we analyze the implications of locally conserved charge fractions, like e.g. local electric charge neutrality or locally fixed proton or lepton fractions. The conditions for phase equilibrium are derived and it is shown, that the properties of a phase transformation do not depend on the locally conserved fractions but only on the number of globally conserved charges. Finally, the general formalism is applied to the liquid-gas phase transition of nuclear matter and the hadron-quark phase transition for typical astrophysical environments like in supernovae, protoneutron, or neutron stars. We demonstrate that the Maxwell construction known from cold-deleptonized neutron star matter with two locally charge neutral phases requires modifications and further assumptions concerning the applicability for hot lepton-rich matter. All relevant combinations of local and global conservation laws are analyzed, and the physical meaningful cases are identified. Several new kinds of mixed phases are presented, as e.g. a locally charge neutral mixed phase in protoneutron stars which will disappear during the cooling and deleptonization of the protoneutron star.