plasmaEnergy_eq

In the grand-canonical ensemble at zero chemical potential, a relativistic fluid is fixed by a single potential: the pressure $P(T)$ (equivalently grand potential density $\Omega=-P$). Entropy density is $s=dP/dT$ and energy density is the Legendre transform $\rho=T\cdot s-P$. This module uses that structure to discharge the Euler and Gibbs–Duhem identities that EntropyConservationFRW previously took as named hypotheses for comoving entropy conservation.

The concrete energy density is the occupation-number integral $\rho=(g_B/(2\pi^2))T^4\int_0^\infty t^3/(e^t-1),dt+(g_F/(2\pi^2))T^4\int_0^\infty t^3/(e^t+1),dt$. Upstream, FermionWeightIntegral proves those integrals equal $\pi^4/15$ (Bose–Einstein) and $7\pi^4/120$ (Fermi–Dirac) via Mellin transforms. The companion pressure closed form is $P=(\pi^2/90)(g_B+(7/8)g_F)T^4$.

Unfold the occupation-integral definition of plasma energy. Rewrite the two improper integrals by the Bose–Einstein theorem $\int t^3/(e^t-1),dt=\pi^4/15$ and the Fermi–Dirac theorem $\int t^3/(e^t+1),dt=7\pi^4/120$. After field_simp clears the nonzero factor $\pi$, a ring reduction yields the prefactors: $g_B/(2\pi^2)\cdot(\pi^4/15)=g_B\pi^2/30$ and $g_F/(2\pi^2)\cdot(7\pi^4/120)=(7/8)g_F\pi^2/30$.

This is the energy half of the statistical-mechanical plasma realization in GrandPotential. Downstream, plasma_energyOf uses it for Legendre consistency: the abstract energy $T\cdot s-P$ equals the independently computed occupation integral ("two separate ensemble computations meet"). plasma_eos derives the radiation equation of state $p=\rho/3$ purely from the grand-canonical integrals, with no EOS input. PhaseSpaceReduction.phaseSpaceEnergy_closed_form reuses the same identity to match three-dimensional phase-space energy density to $(\pi^2/30)(g_B+(7/8)g_F)T^4$. Together with the pressure closed form, it collapses the Euler/Gibbs–Duhem hypotheses of comoving entropy conservation into one structural statement: the coupled sector's pressure is a differentiable potential.