Physics Overview
Our EoS describes the relationship between energy density and pressure in a large three-dimensional (\(\mathrm{nB}\), \(\mathrm{Ye}\), \(\mathrm{T}\)) space,
There are several physical regimes each constrained by different observables and theoretical approaches.
First the zero temperature nuclear matter at nuclear saturation density is closely connected to nuclear masses, charge radii, giant resonances, and other laboratory observables. Global fits to experimental data have been performed with Skyrme and covariant mean-field models.
Secondly, cold neutron matter below nuclear saturation density, is difficult to probe experimentally but is well-constrained by theoretical calculations based on semi-phenomenological nuclear forces or microscopic chiral effective field theory-based interactions.
The third regime, strongly-interacting high-temperature matter, is best described by interactions and many-body approaches similar to those applied to cold neutron matter near saturation density.
The fourth regime, low-density and high-temperature matter that is nearly non-degenerate, is best described by the virial expansion. The equation of state in this regime is determined from nucleon-nucleon scattering phase shifts.
Finally, neutron-rich matter at densities above twice saturation density is most strongly constrained by observations of neutron star masses and radii, particularly the observation of neutron stars with \(M \simeq 2M_\odot\).
In this work, we construct a phenomenological free energy density that is consistent with observational and theoretical constraints in the five aforementioned physical regimes. This is in contrast to works which attempt to describe matter over the entire density and temperature range with a single detailed model of the nucleon-nucleon interaction.
Our second advance is in the treatment of uncertainties. The most relevant parameters which describe the uncertainties in different density and temperature regimes are not clearly related. In this work, through the construction of a phenomenological model one can vary uncertainties in different regimes independently, without spoiling agreement elsewhere.
The module produces an EoS table in a 3 dimensional grid of \(\mathrm{nB}\), \(\mathrm{Ye}\) and \(\mathrm{T}\) and stores various physical and thermodynamics properties in the data files (detailed documentation here).
Rest mass contribution
The Helmholtz free energy of homogeneous nucleonic matter is denoted \(f_{\mathrm{Hom}}(n_n^{\prime},n_p^{\prime},T)\) in Du et al. (2022) where the primes refer to the local nucleonic densities in the gaseous or low-density phase. The rest mass energy density corresponding to this part of the full free energy density is then \(m_n n_n^{\prime} + m_p n_p^{\prime}\). The rest mass energy density is also omitted from the free energy density of nuclei, referred to in Du et al. (2022) as \(\sum_i f_i\). The rest mass energy density associated with the nuclear contribution is then \(\sum_i N_i n_i m_n + \sum_i Z_i n_i m_p\). Multiplying \(f_{\mathrm{Hom}}\) by \(\xi\) and then combining these two contributions to the free energy, we find that the total rest mass energy density (which is not included in the published tables) is:
then by Eq. 2 in Du et al. (2022) this is equal to
Dividing this by \(n_B\) gives the contribution which has
been subtracted from Fint as described in Table Format.
Variable Transformations
It is useful to be able to convert derivative operators between the various sets of composition variables. In the relations below, we omit the “bars” and simply write \(n_n,n_p\) for \(\bar{n}_n,\bar{n}_p\). In other words, all of the nucleon densities below are presumed to include nucleons both inside and outside of nuclei.
Converting between \((n_n,n_p)\) and \((n_B,n_e)\)
Since \(n_p=n_e\) and \(n_n=n_B-n_e\),
For second derivatives
Converting between \((n_n,n_p)\) and \((n_B,Y_e)\)
Since \(n_p=n_B Y_e\) and \(n_n=n_B(1-Y_e)\),
The inverse transformation is:
This transformation is used in stability() in eos_nuclei.cpp.
There is a Maxwell relation:
which implies
or
thus
This equality is also used in stability() in eos_nuclei.cpp.
Converting between \((n_n,n_p)\) and \((n_B,n_e)\) with muons
When muons are included, the expressions change, since \(n_p = n_e + n_{\mu}(n_e)\) and \(n_n = n_B - n_e - n_{\mu}(n_e)\),
where
For second derivatives
Chemical Potentials
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)
To simplify the discussion we use the following notation:
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
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. \(X_n
\equiv n_n/n_B \neq \bar{n}_n/n_B\).
Speed of sound in a multicomponent system
Using \(\varepsilon\) for energy density \(S\) for entropy, \(s\) for entropy density, and \(\tilde{s}\) for entropy per baryon, the speed of sound is
The energy density in the denominator must include the rest mass contribution to the energy density. In infinite matter, it is useful to rewrite this derivative in terms of fixed volume rather than fixed number.
The second derivative is
and first derivative is
Putting these two results together gives
To re-express this in terms of derivatives of the free energy (which again must include the rest mass contribution),
For the sum over \(k\), all densities are constant except for \(n_i\), thus
To compute this we need
which requires
Finally, we get
and
Note that, when applying this expression, one must be consistent about the free energy which one differentiates and the densities and chemical potentials which are used. See Chemical Potentials for more information regarding this issue.
WIP
First, we define the symmetry energy to include a zero temperature contribution which combines the QMC EOS near saturation density, the neutron star fit at higher densities, and the Skyrme interaction for isospin-symmetric matter
Defining the isospin asymmetry :math: delta = 1-2x_p, we can combine this with the model described in Du et al. 2019 to obtain the free energy density of degenerate matter
Finally, we ensure that the total nucleonic free energy gives the result from the virial expansion at high temperatures using
When we need to include the electrons, positrons, and photons, we define the free energy density
Using this formalism, the chemical potentials and entropy can be computed directly (eq. 28-32 in Du et al. 2019).
We enforce causality at high densities.