Nuclear Equation of State (EoS) Overview

• Nuclear Equation of State (EoS) is a fundamental relationship that defines how pressure, energy density, and composition vary with density, temperature, and proton fraction.
• It integrates low-density virial expansion with high-density mean-field models to achieve a thermodynamically consistent framework across extreme astrophysical conditions.
• This comprehensive model is crucial for simulating core-collapse supernovae and neutron star mergers by accurately capturing nuclear composition and thermodynamic responses.

The nuclear equation of state (EoS) is a fundamental relationship describing how the thermodynamic properties of nuclear matter—such as pressure, energy density, free energy, and composition—change as functions of baryon density, temperature, and proton fraction. Constructing a realistic EoS requires accounting for the various constituents of nuclear matter (free nucleons, light and heavy nuclei) and their interactions, especially under extreme conditions relevant to core-collapse supernovae and neutron star mergers. A technically rigorous treatment over a broad range of densities and temperatures is essential for astrophysical simulations and for interpreting multi-messenger observational data.

1. Virial Expansion and Composition at Subnuclear Densities

The virial expansion provides an exact description of low-density nuclear matter where interactions are sufficiently weak that the system behaves as a dilute nonideal gas. In this approach, the grand partition function is expanded in fugacities ($z_a$) for all relevant species (neutrons, protons, alpha particles, and thousands of heavy nuclei), with second virial coefficients ($b_{ij}$) encoding the corrections from two-body elastic scattering:

$$\begin{aligned} \frac{\log \mathcal{Q}}{V} = \frac{P}{T} = & \frac{2}{\lambda_n^3} \left[ z_n + z_p + (z_n^2 + z_p^2) b_n + 2 z_n z_p b_{pn} \right] \ & + \frac{1}{\lambda_\alpha^3} \left[ z_\alpha + z_\alpha^2 b_\alpha + 2 z_\alpha(z_n + z_p) b_{\alpha n} \right] \ & + \sum_i \frac{1}{\lambda_i^3} z_i \Omega_i \,, \end{aligned}$$

where $\lambda_a = \sqrt{2\pi/(m_a T)}$ denotes the thermal wavelength. For heavy nuclei, the fugacity is determined by chemical equilibrium:

$$z_i = z_p^Z z_n^N \exp\left(\frac{E_i - E_i^{C}}{T}\right)$$

with $E_i$ the nuclear binding energy and $E_i^{C}$ the Coulomb correction, evaluated in the Wigner–Seitz approximation as

$$E_i^{C} = \frac{3}{5} \frac{Z_i^2 \alpha}{r_A} \left[ -\frac{3}{2}\left(\frac{r_A}{r_i}\right) + \frac{1}{2}\left(\frac{r_A}{r_i}\right)^3 \right].$$

The internal partition function of heavy nuclei, $\Omega_i$, includes discrete and continuum excited states, typically parameterized by a back-shifted Fermi gas model. This formalism enables tracking $\sim 9000$ nuclear species with $A\geq12$, critical for determining the matter composition, particularly under supernova and neutron star merger conditions.

At very low densities, the virial expansion reduces to nuclear statistical equilibrium (NSE). As density increases, additional terms in the expansion or matching to mean-field models become necessary to treat many-body correlations beyond dilute gas interactions.

2. Matching to Mean Field Models at Higher Density

At increasing baryon densities ($n_B \gtrsim 10^{-3}$ fm$^{-3}$), interactions between nucleons and nuclei become stronger, invalidating the low-density virial expansion. In this regime, the free energy from the virial approach is smoothly matched to relativistic mean field (RMF) calculations. The matching is typically performed using a thermodynamically consistent interpolation procedure, selecting the branch with the lower free energy for each state point, and ensuring the derivative conditions for the pressure and its continuity. The transition density is temperature- and composition-dependent but for $T = 1$ MeV, $Y_P = 0.3$ occurs around $n_B \sim 4 \times 10^{-3}$ fm$^{-3}$.

By integrating the virial and RMF domains into a unified EoS table, one captures the correct microphysics from the dilute virial limit through to uniform matter at and above nuclear saturation density. This is essential for realistic modeling of the full range of astrophysical environments.

3. Range of Applicability and Astrophysical Context

The EoS constructed in this framework is tabulated over a large parameter space:

Parameter	Range
Temperature ($T$)	0.158 to 15.8 MeV
Baryon density ($n_B$)	$10^{-8}$ to 0.1 fm$^{-3}$
Proton fraction ($Y_P$)	0.05 to 0.56

This broad range is dictated by the conditions encountered in core-collapse supernovae and neutron star mergers. Near the supernova neutrinosphere (at densities $\sim10^{11}$ g/cm$^3$), or during nucleosynthesis in mergers, the detailed nuclear composition—particularly the distribution of light elements and heavy nuclei—affects neutrino opacities and reaction rates, fundamentally influencing observable neutrino and gravitational wave signals.

Including detailed nuclear composition stands in contrast to single-average-nucleus treatments found in prior EoS tables (e.g., Lattimer–Swesty, Shen et al.), which may underestimate the impact of composition on weak interaction processes and thermodynamic response.

4. Thermodynamic Quantities and Calculation

The formalism provides a direct pathway to construct all relevant thermodynamic observables from the partition function and its derivatives:

• Free energy per nucleon: $F/A = f / n_B$
• Pressure: $$P_{\mathrm{th}} = n_B^2 \left(\frac{\partial (F/A)}{\partial n_B}\right)_{T, Y_P}$$
• Entropy and energy densities: obtained via standard thermodynamic relations.

This guarantees that the EoS tables are thermodynamically consistent and suitable for numerical hydrodynamics and neutrino transport codes. The inclusion of light particles, heavy nuclei, and Coulomb corrections results in a more accurate free energy landscape and properly captures the coexistence and dissolution of various nuclear species as functions of density and temperature.

5. Impact for Supernova and Merger Simulations

Realistic microphysical modeling of the nuclear EoS is essential for predictive simulations of stellar collapse and merger events:

• Neutrino interaction rates and the location of the neutrinospheres are set by the composition and thermodynamics, influencing the subsequent neutrino spectra.
• The onset of heavy element synthesis (r-process nucleosynthesis) depends on the precise abundance of neutron-rich nuclei and the equation of state.
• The thermodynamic response (adiabatic index, speed of sound) derived from the detailed EoS affects shock propagation, proto-neutron star evolution, and gravitational wave emission.

The ability to smoothly transition from the low-density virial regime to high-density mean-field matter ensures robustness across all relevant domains.

6. Central Mathematical Formulas

The key relationships underpinning the EoS construction include:

• Thermal wavelength: $\lambda_a = \sqrt{2\pi/(m_a T)}$
• Virial expansion of the partition function (see above)
• Chemical equilibrium: $$\mu_i = Z\mu_p + N\mu_n\,\Rightarrow\,z_i = \exp[(\mu_i + E_i - E_i^C)/T]$$
• Free energy and pressure (as above)

These expressions systematically relate microscopic particle interactions, binding energies, and Coulomb effects to macroscopic thermodynamic variables over a broad range of conditions.

7. Significance and Future Directions

The virial expansion EoS framework, matched to RMF models at higher densities, provides an internally consistent and computationally robust tool for simulating astrophysical phenomena involving nuclear matter far from equilibrium. Its use ensures a genuine accounting for state-dependent nuclear composition, thermal effects, and the necessary microphysics for interpreting high-energy transient signals.

Future developments involve matching low-density virial EoS tables to high-density RMF results in a unified dataset suitable for modern multi-dimensional supernova and merger simulations. Such efforts are essential for progress in constraining the nuclear EoS from multi-messenger observations and laboratory experiments (Shen et al., 2010).