Electron Positron Capture Rates and the Steady State Equilibrium Condition for Electron-Positron Plasma with Nucleons

The reaction rates of the beta processes for all particles at arbitrary degeneracy are derived, and an {\it analytic} steady state equilibrium condition $\mu_n=\mu_p+2\mu_e$ which results from the equality of electron and positron capture rates in the hot electron-positron plasma with nucleons is also found, if the matter is transparent to neutrinos. This simple analytic formula is valid only if electrons are nondegenerate or mildly degenerate, which is generally satisfied in the hot electron-positron plasma. Therefore, it can be used to efficiently determine the steady state of the hot matter with plenty of positrons. Based on this analytic condition, given the baryon number density and the temperature, if the nucleons are nondegenerate, only one algebraic equation for determining the electron fraction is obtained, which shows the great advantage of the analytic equilibrium condition.