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Analytic solution for kinetic equilibrium of beta-processes in nucleonic plasma with relativistic pairs

The analytic solution is obtained describing kinetic equilibrium of the $β$-processes in the nucleonic plasma with relativistic pairs. The nucleons $(n,p)$ are supposed to be non-relativistic and non-degenerate, while the electrons and positrons are ultra-relativistic due to high temperature $(T>6\cdot 10^9$K), or high density $(ρ>μ10^6$g/cm$^3$), or both, where $μ$ is a number of nucleons per one electron. The consideration is simplified because of the analytic connection of the density with the electron chemical potential in the ultra-relativistic plasma, and Gauss representation of Fermi functions. Electron chemical potential and number of nucleons per one initial electron are calculated as functions of $ρ$ and $T$.