Finite-temperature electron equation of state ($P_e(\rho ,T)$)

Neglecting interactions and thermal pairs, the Fermi-Dirac distribution $f(p)=[\exp ((\epsilon (p)-{\mu }_{\mathrm{kin}})/(k_{B}T))+1{]}^{-1}$ with kinetic energy $\epsilon (p)=\sqrt{m_e^2{c}^4+p^2{c}^2}-m_e{c}^2$ determines the Electron contribution to a stellar equation of state. The kinetic chemical potential is fixed by the electron number density, not an independently chosen pressure fraction:The electron kinetic internal energy per volume is $({\pi }^2{\hslash }^3{)}^{-1}{\int }_0^{\infty }{p}^2\epsilon (p)f(p)dp$. In the nondegenerate limit $P_e=n_e{k}_{B}T$; at zero temperature the integrals become the equation of state of a cold electron gas. These are limits of one component and must not be added together as independent electron pressures. When electron-positron thermal pair abundance matters, both charge species must be included, with charge neutrality fixing their net density.