The steady equilibrium conditions for a mixed gas of neutrons, protons, electrons, positrons and radiation fields （abbreviated as npe^± gas） with or without external neutrino flux are investigated, and a general chemical potential equilibrium equation μn = μp ＋ Cμe is obtained to describe the steady equilibrium at high temperatures （T 〉 10^9 K）. An analytic fitting formula of coefficient C is presented for the sake of simplicity, when neutrinos and antineutrinos are transparent. It is a simple method to estimate the electron fraction for the steady equilibrium npe^± gas that adopts the corresponding equilibrium condition. As an example, we apply this method to the GRB accretion disk and confirm that the composition in the inner region is approximately in equilibrium when the accretion rate is low. For the case with external neutrino flux, we calculate the initial electron fraction of neutrino-driven wind from the proto-neutron star model M15-l1-r1. The results show that the improved equilibrium condition makes the electron fraction decrease significantly more than the case μn = μp ＋ μe when the time is less than 5s post bounce, which may be useful for r-process nucleosynthesis models.
By employing an improved simulation of the evolution of black holes （BHs） based on the merger tree of dark matter halos, we explore the relationship between the central BH mass Mbh and velocity dispersion σ＊ at high redshift z ≥ 6 and quantify the mini-QSO＇s （with BH mass M = 200 - 105M⊙） contribution to cosmic reionization. The simulation demonstrates how seed BHs migrate onto the MBH-σ＊ relation by merging with each other and accreting gas at z ≥ 6： 1. The correlation between BHs and their host halos increases as the BHs grow; 2. The slope, i.e. Ф = dlog（Mbh）/dlog（σ＊） in the relationship, is insensitive to the redshift at z 〉 6. In agreement with previous work, we find that mini-QSOs＇ ionizing capability to the Universe lies in the range - 25% - 50% if early miniquasars have extremely high duty cycles, i.e. P（z 〉 6） - 0.9 - 1.
