= Solution
Let $f(\mathbf p)\,d^3p$ be particle number per unit volume in a <momentum> element, including the spin-state count in $f$. This is the local momentum part of a <phase-space distribution function>, normalized by the <number density> $n=\int f(\mathbf p)\,d^3p$. For an isotropic distribution with particle <energy> $E(p)=\sqrt{m^2c^4+p^2c^2}$, the <kinetic pressure of an isotropic gas> is the average momentum flux:
$$
P=\frac13\int \mathbf p\cdot\mathbf v\,f(\mathbf p)\,d^3p=\frac13\int\frac{p^2c^2}{E(p)}f(\mathbf p)\,d^3p.
$$
The <kinetic energy density>, explicitly excluding rest mass, is
$$
u_{\rm kin}=\int[E(p)-mc^2]f(\mathbf p)\,d^3p.
$$
For $p\ll mc$, $E-mc^2=p^2/(2m)$ and $\mathbf v=\mathbf p/m$, giving $P=2u_{\rm kin}/3$. For $p\gg mc$, $E\simeq pc$ and $v\simeq c$, giving $P\simeq u_{\rm kin}/3$. Thus
$$
\boxed{P=\frac23u_{\rm kin}\quad\text{nonrelativistically},\qquad P=\frac13u_{\rm kin}\quad\text{in the ultrarelativistic limit}.}
$$
Neither relation assumes a Maxwellian distribution. At intermediate momenta neither constant ratio is exact, and anisotropic distributions require a <pressure tensor> rather than this scalar <pressure>.
For a classical <Maxwell-Boltzmann distribution>, $f=n(2\pi mk_BT)^{-3/2}\exp[-p^2/(2mk_BT)]$. Three Gaussian component integrals give $\langle p^2\rangle=3mk_BT$, so the <ideal gas> has
$$
\boxed{P=nk_BT,\qquad u_{\rm kin}=\frac32nk_BT.}
$$
In a fully ionized mixture, sum over independent species to obtain $P_{\rm gas}=\rho k_BT/(\mu_{\rm mol}m_u)$, where the <mean molecular weight> counts ions and free <electrons>. This excludes partial-ionization and interaction corrections. A dilute classical ultrarelativistic gas still has $P=nk_BT$ but $u_{\rm kin}\simeq3nk_BT$; its momentum distribution is proportional to $e^{-pc/(k_BT)}$ instead of the nonrelativistic Gaussian.
For a thermal <photon> gas, the two polarization states and zero <photon chemical potential> give the <Planck photon distribution>
$$
f_\gamma(p)=\frac{2}{(2\pi\hbar)^3}\frac1{e^{pc/(k_BT)}-1}.
$$
Use $x=pc/(k_BT)$ and $\int_0^\infty x^3/(e^x-1)\,dx=\pi^4/15$. The <radiation constant> and <photon> <equation of state> are therefore
$$
\boxed{u_\gamma=a_{\rm rad}T^4,\qquad P_\gamma=\frac13a_{\rm rad}T^4,\qquad a_{\rm rad}=\frac{\pi^2k_B^4}{15\hbar^3c^3}.}
$$
Likewise $n_\gamma=2\zeta(3)(k_BT/\hbar c)^3/\pi^2$. Unlike a gas with a fixed particle number, <photons> do not have <pressure> proportional to baryonic <mass density>.
For fully degenerate <electrons>, the <Fermi-Dirac distribution> becomes a filled momentum sphere with two spin states. Counting them gives the <Fermi momentum>
$$
p_F=\hbar(3\pi^2n_e)^{1/3},\qquad n_e=\frac{\rho}{\mu_em_u}.
$$
Here $\mu_e$ is the <mean molecular weight per electron>. The <equation of state of a cold electron gas> follows by integrating momentum flux up to $p_F$:
$$
P_e=\frac1{3\pi^2\hbar^3}\int_0^{p_F}\frac{p^4c^2}{\sqrt{m_e^2c^4+p^2c^2}}\,dp.
$$
Its two limits are
$$
\boxed{P_e=\frac{\hbar^2(3\pi^2)^{2/3}}{5m_e}\left(\frac{\rho}{\mu_em_u}\right)^{5/3}\quad(p_F\ll m_ec),}
$$
$$
\boxed{P_e=\frac{\hbar c(3\pi^2)^{1/3}}4\left(\frac{\rho}{\mu_em_u}\right)^{4/3}\quad(p_F\gg m_ec).}
$$
The corresponding kinetic <energy> densities are $3n_e[p_F^2/(2m_e)]/5$ and $3n_ep_Fc/4$, respectively. Thus in the high-density relativistic limit the <electron> <pressure> scales as $\rho^{4/3}$ and is nearly independent of <temperature>. This is the ideal noninteracting, fixed-composition <electron> result, not a universal <equation of state> at nuclear densities where captures, interactions and the composition change.
The boundaries on a <stellar equation-of-state regime diagram> concern the <electron> component. Define the <electron relativistic density threshold> and <thermal electron relativistic threshold> by
$$
\rho_* =\frac{\mu_em_u}{3\pi^2}\left(\frac{m_ec}{\hbar}\right)^3\simeq9.74\times10^5\mu_e\;\mathrm{g\,cm^{-3}},\qquad T_* =\frac{m_ec^2}{k_B}\simeq5.93\times10^9\;\mathrm K.
$$
A degenerate <electron> gas is nonrelativistic well below $\rho_*$ and ultrarelativistic well above it: this is an approximately vertical division on a log-density plot. A nondegenerate <electron> gas instead becomes thermally relativistic near $T_*$: this is an approximately horizontal division. The regimes have broad crossovers rather than a discontinuity at either line.
The exact <electron Fermi temperature>, subtracting <electron> rest <energy>, is
$$
\boxed{T_F=\frac{m_ec^2}{k_B}\left[\sqrt{1+(\rho/\rho_*)^{2/3}}-1\right].}
$$
For <electrons> with negligible thermal pairs, strong degeneracy requires $T\ll T_F$, whereas $T\gg T_F$ gives a <nondegenerate gas>. In the pair-rich regime the actual <electron> and <Positron> distributions must instead be treated with their <chemical potentials>, as discussed below. The two limiting degeneracy boundaries are
$$
T_F\simeq\frac{T_*}{2}(\rho/\rho_*)^{2/3}\simeq3.02\times10^5\left[\frac{\rho/(\mathrm{g\,cm^{-3}})}{\mu_e}\right]^{2/3}\mathrm K,
$$
$$
T_F\simeq T_*(\rho/\rho_*)^{1/3}\simeq5.98\times10^7\left[\frac{\rho/(\mathrm{g\,cm^{-3}})}{\mu_e}\right]^{1/3}\mathrm K.
$$
These slopes $2/3$ and $1/3$ explain the bent degeneracy boundary on the logarithmic graph. A thermal-wavelength test $n_e[2\pi\hbar^2/(m_ek_BT)]^{3/2}\sim2$ gives the same nonrelativistic density/<temperature> scaling, with an order-one definition of the crossover.
The <radiation-to-gas pressure boundary> in the nondegenerate fully ionized regime is
$$
P_\gamma=P_{\rm gas}\quad\Longleftrightarrow\quad T^3=\frac{3k_B\rho}{a_{\rm rad}\mu_{\rm mol}m_u}.
$$
Radiation dominates above this line. Once <electrons> are strongly degenerate, compare <radiation pressure> with $P_e(\rho)$ instead of continuing the ideal-gas comparison into that regime. The <radiation-to-degeneracy pressure boundary> is $T=[3P_e(\rho)/a_{\rm rad}]^{1/4}$, with logarithmic slopes $5/12$ and $1/3$ in the nonrelativistic and ultrarelativistic limits. Degeneracy of the <electrons> and dominance of their <pressure> are distinct criteria.
For the pair curve, distinguish baryonic net <electrons> from thermally created <electrons> and <Positrons>. <Chemical equilibrium> with <photons> requires opposite <electron>/<Positron> <chemical potentials> when the one-particle <energy> includes rest <energy>, as in the distributions below. At low <temperature> and low degeneracy the zero-chemical-potential density per charge species is
$$
n_0(T)=2\left(\frac{m_ek_BT}{2\pi\hbar^2}\right)^{3/2}e^{-m_ec^2/(k_BT)}.
$$
<Charge neutrality> gives $n_--n_+=n_b=\rho/(\mu_em_u)$, while the nondegenerate equilibrium product is $n_-n_+=n_0^2$. Hence
$$
\boxed{n_+=\frac{\sqrt{n_b^2+4n_0^2}-n_b}{2}.}
$$
Pairs become important when $n_0$ is comparable with or exceeds $n_b$, approximately the <electron-positron thermal pair abundance> curve $\rho_{\rm pair}(T)=\mu_em_un_0(T)$. Below its density at a fixed <temperature>, pairs dominate over the net charge <electrons>. The exponential makes the low-temperature portion steep in a log-log plot. For $\mu_e=2$, the full zero-potential integral gives approximate pair-marker densities $1.80\times10^3\,\mathrm{g\,cm^{-3}}$ at $10^9\,\mathrm K$ and $7.52\times10^5\,\mathrm{g\,cm^{-3}}$ at $3\times10^9\,\mathrm K$. At relativistic <temperature> use the full zero-chemical-potential <Fermi-Dirac distribution> instead; it gives $n_0\to[3\zeta(3)/(2\pi^2)](k_BT/\hbar c)^3$, so the high-temperature pair curve approaches slope three in log-density versus log-temperature. Strong net-electron degeneracy suppresses <Positrons> and requires the full chemical-potential-dependent distribution.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/iii/paper-317-eos-regimes.png]
{title=Approximate stellar equation-of-state regimes and thermal pair boundary}
The original diagram uses fully ionized helium, $\mu_e=2$ and $\mu_{\rm mol}=4/3$, only to set numerical locations. It plots the full $T_F$ curve and the zero-chemical-potential pair integral, rather than extending their asymptotes into the crossover. The pair boundary is deliberately an approximate abundance marker, not a phase transition; at high <temperature> the nonrelativistic ion approximation and fixed-composition model also have limits. For $T\gg T_*$ with abundant nondegenerate pairs, the combined <electron>/<Positron> <energy> density is $7a_{\rm rad}T^4/4$, so <photons> plus pairs have $u=11a_{\rm rad}T^4/4$ and $P=u/3$. Real stellar matter adds Coulomb effects, partial ionization, nuclear reactions and, at sufficiently high density, nuclear-matter physics beyond the ideal regime map.
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