In a conversation with Mateus we came across this issue: how to define the various chemical potentials that we have in our EoSes, particularly when the system might be out of beta equilibrium.

Chemical potentials are associated with conserved (or “almost conserved”, i.e. slowly equilibrating) quantities. So, for example, one should write the chemical potential of the electron in terms of its conserved charges which are electric charge Q (-1) and electron lepton number L_e (+1),

\mu_e = -\mu_Q + \mu_{Le} = \mu_{NQ} + \mu_{Le}

where NQ is negative charge (Note that \mu_{NQ}=-\mu_Q; we should not use the notation “\mu_Q” for the *negative* charge chemical potential!) and L_e is electron lepton number.

In neutrino-free matter the neutrinos carry lepton number away, so \mu_{Le}=0. But in neutrino-trapped matter \mu_{Le} will not be zero.

The nucleons carry baryon number (+1), electric charge (0 or +1), isospin (\pm\frac{1}{2}), and strangeness (0). So one would write

\mu_n = \mu_B - \frac{1}{2}\mu_I

\mu_p = \mu_B - \mu_{NQ} + \frac{1}{2}\mu_I

etc

When the system is out of cold beta equilibrium this means there is a nonzero \mu_I.

But \mu_e is *always* equal to \mu_{NQ} because the electron has charge Q=-1 (or NQ=+1).