  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.
