Chemical potentials

Here, we explain the ambiguities associated with the definition of the chemical potentials and how they relate to the code.

Denote the number density of neutrons in the low-density phase \(n_n\), the number density of protons in the low-density phase \(n_p\), the number density of electrons \(n_e\), and the number density of nuclei as \(\{n_i\}\). Denote the total number of neutrons and protons in both phases as \(\bar{n}_n\) and \(\bar{n}_p\). Using these definitions, one can write the free energy for hot and dense matter in (at least) four different ways, \(f_1(\bar{n}_n,\bar{n}_p,T)\), \(f_2(\bar{n}_n,\bar{n}_p,n_e,T)\), \(f_3(n_n,n_p,\{n_i\},T)\), \(f_4(n_n,n_p,\{n_i\},n_e,T)\). In the first form, the Saha equations have been solved to determine \(\{n_i\}\) and charge neutrality has been used to determine \(n_e\). In the second form, the Saha equations have been solved but charge neutrality has not been used. In the third form, the Saha equations have not been solved but charge neutrality has been used. The electron contribution to the free energy is included in all four free energies, but in the case of \(f_1\) and \(f_3\), the electron density is not independent of the other densities. For these four free energies, there are four corresponding proton chemical potentials, \(\partial f_1/\partial \bar{n}_p\), \(\partial f_2/\partial \bar{n}_p\), \(\partial f_3/\partial n_p\), and \(\partial f_4/\partial n_p\). None of these four proton chemical potentials are the same. This documentation attempts to explain how this complication relates to the code. In Du et al. (2022), we use a confusing notation because we do not clearly distinguish \(\bar{n}_n\) and \(n_n\). The function \(f_1\) is most directly related to the tables which are generated and one can simply identify \(\bar{n}_n=n_B(1-Y_e)\) and \(\bar{n}_p=n_B Y_e\).

The comparison between \(f_1\) and \(f_2\) is the simplest (now being a bit more careful about what is held constant)

\[\left(\frac{\partial f_1}{\partial \bar{n}_n}\right)_{\bar{n}_p,T} = \left(\frac{\partial f_2}{\partial \bar{n}_n}\right)_{\bar{n}_p,n_e,T} \quad \mathrm{and} \quad \left(\frac{\partial f_1}{\partial \bar{n}_p}\right)_{\bar{n}_n,T} = \left(\frac{\partial f_2}{\partial \bar{n}_p}\right)_{\bar{n}_n,n_e,T} + \left(\frac{\partial f_2}{\partial n_e}\right)_{\bar{n}_n,\bar{n}_p,T}\]

To simplify the discussion we use the following notation:

\[\mu_{p,i} \equiv \left( \frac{\partial f_i}{\partial \bar{n}_p} \right)\]

where all of the other densities are held constant, including either \(n_n\) or \(\bar{n}_n\) as appropriate. Thus \(f_1\) and \(f_2\) imply two thermodynamic identies

\[\begin{split}\varepsilon_1 &=& - P_1 + T s_1 + \bar{n}_n \mu_{n,1} + \bar{n}_p \mu_{p,1} \nonumber \\ \varepsilon_2 &=& - P_2 + T s_2 + \bar{n}_n \mu_{n,2} + \bar{n}_p \mu_{p,2} + n_e \mu_e\end{split}\]

When \(n_e=\bar{n}_p\), we have \(P_1=P_2\), \(\varepsilon_1=\varepsilon_2\), and \(s_1=s_2\). In the EOS literature, it has become standard to store \(\mu_{n,2}\) and refer to it as the “neutron chemical potential” and refer to \(\mu_{p,2}\) as the “proton chemical potential” even though charge neutrality has been assumed so the electron density is not independent. The tables generated at this website use the same notation.

The distinction between \(\mu_{n,1}\) and \(\mu_{n,3}\) is more complicated, see Eq. 36 of Du et al. (2022).

The neutron fraction Xn stored in the table refers only to neutrons outside of nuclei, i.e. \(X_n \equiv n_n/n_B \neq \bar{n}_n/n_B\).