Finite Temperature and Composition

Hot compact-object matter cannot be specified by a cold barotrope. Pressure and energy depend on temperature or entropy, electron fraction, nuclear composition, and possibly trapped-neutrino content. Collapse and merger trajectories cross regimes containing nuclei, nucleons, leptons, photons, and new particles. A usable equation-of-state table must define those degrees of freedom, derive fields consistently from one free energy, recover cold and analytic thermal limits, and interpolate without inventing instability or acausality.

Required background. Hilbert Stress–Energy from Metric Variation supplies total energy density and pressure.

Helpful background. The Einstein Equation supplies collapse and merger gravity; Relativistic Matter and Compact Objects gives the dense-matter setting; and Computational Relativity supplies table-driven evolution.

State variables and thermodynamic derivatives

A common table uses baryon density, temperature, and net electron fraction,

$$(n_B,T,Y_e), \qquad Y_e=\frac{n_{e^-}-n_{e^+}}{n_B}.$$

Let $a(n_B,T,Y_e)$ be free energy per baryon. Then

$$p=n_B^2\left(\frac{\partial a}{\partial n_B}\right)_{T,Y_e}, \qquad s_B=-\left(\frac{\partial a}{\partial T}\right)_{n_B,Y_e},$$ $$e=a+Ts_B, \qquad \varepsilon=n_B e.$$

Here $a$ and $e$ must include the same declared reference rest energy. For charge-neutral neutron–proton–electron matter without muons, changing $Y_e=Y_p$ at fixed $n_B$ converts a neutron to a proton and an electron. Consequently,

$$\left(\frac{\partial a}{\partial Y_e}\right)_{n_B,T} =\mu_p+\mu_e-\mu_n, \qquad de=\frac{p}{n_B^2}\,dn_B+T\,ds_B +(\mu_p+\mu_e-\mu_n)\,dY_e.$$

This shows why a composition column is dynamically necessary: weak reactions change energy as well as particle fractions. With additional charged species, the independent fractions and their conjugate chemical combinations must be redefined rather than reusing this three-species formula.

The evolution may use $(n_B,s_B,Y_e)$ instead, requiring an inversion for $T$. At fixed $(n_B,Y_e)$, positive heat capacity makes $s_B(T)$ increase and the inversion locally unique. A nonmonotone branch is a warning about instability or interpolation; coexistence can require tracking phase fractions explicitly.

The table must say whether rest masses, electrons and positrons, photons, muons, and trapped neutrinos are included. Adding an external lepton or photon module to a table that already contains it double counts pressure and energy. Oertel and collaborators describe equilibrium requirements and common table contents in Oertel et al. 2017, §II.B and §V.D.

Composition from nuclei to uniform matter

At subnuclear density and moderate temperature, nuclear statistical equilibrium distributes baryons among free nucleons and an ensemble of nuclei. The free energy includes nuclear binding, excitations, excluded volume or other dissolution physics, Coulomb corrections, and translational entropy. A single representative nucleus can approximate bulk thermodynamics but may misrepresent weak rates and composition-sensitive transport.

As density or temperature rises, nuclei dissolve into nonuniform pasta-like configurations and then uniform nucleonic matter. The transition treatment differs across models; a discontinuity in composition is not necessarily a first-order bulk thermodynamic transition. At still higher density, hyperons, meson condensates, or quarks may appear, each requiring chemical equilibrium and interaction parameters.

Weak reactions determine whether $Y_e$ is evolved, in beta equilibrium, or constrained by a trapped lepton fraction. In transparent cold matter, $\mu_{\nu_e}=0$ and beta equilibrium fixes $Y_e$. During collapse after neutrino trapping, or in opaque parts of a hot merger remnant, $Y_L=Y_e+Y_{\nu_e}$ is approximately advected on times shorter than diffusion; $Y_{\nu_e}=(n_{\nu_e}-n_{\bar\nu_e})/n_B$ is a net fraction. Before trapping, electron capture and neutrino escape instead reduce $Y_L$. The same nuclear model can therefore yield different pressures along different composition constraints.

For a pulsation faster than weak reactions, use a frozen-composition adiabatic derivative. For a secular equilibrium calculation, minimize the free energy over the reacting composition first. Differentiating these two paths generally gives different sound speeds. The explicit fixed-composition chain rule is derived in Thermodynamic Consistency.

Analytic and asymptotic checks

The $T\to0$ limit at fixed density and composition should converge to the declared cold equation of state, with entropy tending to zero for a nondegenerate ground state. The derivative must be checked from both positive-temperature tables and the cold slice; joining inconsistent releases creates a pressure or energy kink.

In a dilute nonrelativistic ideal-gas regime,

$$p\to n_{\rm part}T, \qquad e_{\rm th}\to\frac32\frac{n_{\rm part}}{n_B}T$$

for monatomic nondegenerate particles, with composition-dependent particle number. For equilibrium photons,

$$p_\gamma=\frac13\varepsilon_\gamma, \qquad \varepsilon_\gamma=\frac{\pi^2}{15}T^4.$$

A degenerate Fermi component has thermal corrections proportional to $T^2$ at leading order rather than ideal-gas $T$. These limits catch unit, degeneracy, rest-mass, and particle-counting mistakes.

Thermodynamic identities provide additional checks: first-law residuals, Maxwell mixed derivatives, positive heat capacity, stable susceptibility matrices, and $0\leq c_s^2\leq1$ along the actual adiabatic composition constraint. Smooth table values do not ensure smooth derivatives.

Table benchmark and rerun boundary

The continuum inputs are a versioned free-energy model or release on explicit $(n_B,T,Y_e)$ axes, particle inventory, masses, unit system, phase construction, valid domain, and interpolation prescription. The file checksum and any errata are part of the input. CompOSE Core Team 2022, Reference Manual v3.01, §§3.3, 3.5–3.6, 4.2.2–4.2.3, and Appendix A.1 (PDF) specifies the fields and interpolation conventions; the title-page version differs from the older version embedded in the URL.

Record the actual interpreter, library versions, precision, code revision, table identifier, download URL, checksum, unit conversion, axis ordering, interpolation method, and derivative tolerances. A potential-based interpolator reduces inconsistency among fields but does not itself guarantee stable second derivatives. The following is a proposed production-table protocol; this page does not claim that these numerical table checks were run.

Construct analytic ideal-gas, photon, and degenerate-Fermi reference tables on nested $24^3$, $48^3$, and $96^3$ grids over common physical domains. Recover $p$, $s_B$, $e$, chemical derivatives, and sound speed, then report $L_1$ and $L_\infty$ errors. Refine one axis at a time and compare linear, monotone cubic, and potential-consistent interpolation. Test $T\to0$ using decreasing temperature at fixed $(n_B,Y_e)$, and compare the result with the independent cold slice.

For a production table, sample every cell corner and interior points for positivity, monotonicity where physically required, convexity, Maxwell residuals, and causal sound speed. State how out-of-range queries are handled; silent extrapolation is unacceptable. Halve root-finding and derivative tolerances independently from grid refinement to separate solver error from interpolation uncertainty.

One failure mode is a nonmonotone entropy–temperature relation that makes primitive recovery select the wrong branch. Others include negative heat capacity, pressure jumps at a crust or phase join, composition fractions that do not sum to the declared baryon and charge totals, duplicated electron or photon terms, superluminal interpolated derivatives, and spurious oscillations near nuclear dissolution. Floors can hide these failures while changing the energy budget.

Exercises

Why can a table with smooth $p(n_B,T,Y_e)$ and $\varepsilon(n_B,T,Y_e)$ still violate thermodynamics, and what construction reduces that risk?

Solution

Smoothness of separately interpolated columns does not ensure that both are derivatives of one potential, so first-law and Maxwell relations can fail between nodes. Interpolate a declared free energy with a scheme designed for its derivatives, then derive pressure, entropy, energy, and chemical potentials consistently. Independent analytic limits and derivative residuals must still be checked.

References

• CompOSE Core Team. CompOSE Reference Manual, Version 3.01, 1 April 2022. Official PDF.
• Oertel, Micaela, Matthias Hempel, Thomas Klähn, and Stefan Typel. “Equations of State for Supernovae and Compact Stars.” Reviews of Modern Physics 89, 015007 (2017). DOI.