= Beta equilibrium
<Beta equilibrium> is a condition in which weak reactions interconverting <neutrons> and <protons> produce no net change of composition. For matter where <neutrinos> are thermally trapped, the reaction $n+\nu_e\rightleftharpoons p+e^-$ gives the <chemical equilibrium> condition
$$
\mu_n+\mu_{\nu_e}=\mu_p+\mu_e,
$$
where each $\mu$ is the corresponding <chemical potential>. In cold neutrino-transparent <neutron star> matter, one usually takes $\mu_{\nu_e}=0$, giving $\mu_n=\mu_p+\mu_e$. The escaping neutrinos do not form an equilibrated thermal bath in that approximation.<Electron fraction constraints based on nuclear statistical equilibrium with beta equilibrium by A. Arcones and others>[Section 2, thermal versus dynamical beta equilibrium]{ref}
<Charge neutrality> in matter containing only neutrons, protons and electrons additionally requires $n_p=n_e$. Given a total baryon density, these relations and an <equation of state> determine the composition. As an illustrative chemical-potential check, $\mu_n=1000$ MeV and $\mu_p=900$ MeV require $\mu_e=100$ MeV in the transparent approximation. At finite temperature, balancing emission and capture rates is the more general kinetic condition; nuclear statistical equilibrium under strong reactions alone does not impose beta equilibrium.
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