= Photon gas
{wiki}
A <photon gas> is a collection of <photons> described statistically as radiation. Here the equilibrium model is homogeneous, isotropic <electromagnetic radiation> in vacuum at <temperature> $T>0$: each <photon> has <energy> $E=c|\mathbf p|=h\nu$, with <momentum> $\mathbf p$ and <frequency> $\nu$, and there are two independent transverse states of <electromagnetic polarization>. An absorbing cavity with walls at a common <temperature> provides a way to establish this <thermal equilibrium>. The bulk formulas assume cavity dimensions much larger than the typical <wavelength> $hc/(k_BT)$, so the <electromagnetic mode density in a box> can replace individual cavity modes by a continuum. https://davidtong.org/teaching/statistical-physics/statmechhtml/S3#S3.SS2[David Tong, Statistical Physics, §3.2: photon energies, polarizations and cavity equilibrium]{ref}
The crucial constraint is whether <photon emission> and <photon absorption>, or other number-changing processes, can equilibrate the <photon> population. When they can, <photon> number is not an independently conserved quantity and the equilibrium <photon chemical potential> is $\mu_\gamma=0$. For each polarization and <frequency>, the mean <occupation number> then follows the <Bose-Einstein distribution>:
$$
f(\nu)=\frac{1}{\exp[h\nu/(k_BT)]-1}.
$$
If number-changing processes are ineffective but number-conserving interactions redistribute <energy>, a constrained <Bose-Einstein distribution> can instead have $f=[\exp((h\nu-\mu_\gamma)/(k_BT))-1]^{-1}$ with <chemical potential> $\mu_\gamma\le0$ fixed by the number constraint. Thus <thermal equilibrium> with respect to <energy> exchange alone does not establish $\mu_\gamma=0$. The zero value can also persist in an already prepared <Planck photon distribution> during suitable subsequent evolution; continual <photon emission> is not required to preserve it. https://academic.oup.com/ptep/article/2014/6/06B107/1560917[Hiroyuki Tashiro, CMB spectral distortions and energy release in the early universe (2014), §3.1: kinetic equilibration without full photon-number equilibration]{ref}
Combining $f$ with the two-polarization <density of states>, $g_\nu/V=8\pi\nu^2/c^3$, gives the <Planck photon distribution> and the spectral <photon energy density>:
$$
\frac{dn_\gamma}{d\nu}=\frac{8\pi\nu^2}{c^3}f(\nu),\qquad
u_\nu\equiv\frac{du}{d\nu}=\frac{8\pi h\nu^3}{c^3}f(\nu)=\frac{4\pi}{c}B_\nu(T).
$$
Here $B_\nu$ is the <spectral radiance> in <Planck's law>; $u_\nu$ is stored <energy density> per unit <frequency>, not outgoing <power> per <area>. Their different units are connected by the full-sphere <solid angle> and the <speed of light> $c$. The <Boltzmann constant> is $k_B$, and $h$ is the <Planck constant>. https://farside.ph.utexas.edu/teaching/sm1/Thermalhtml/node103.html[Richard Fitzpatrick, Planck Radiation Law: two polarizations and radiation energy spectrum]{ref}
The equilibrium <equation of state> is particularly simple:
$$
u=a_RT^4,\qquad p=\frac{u}{3},\qquad
a_R=\frac{8\pi^5k_B^4}{15h^3c^3}=\frac{4\sigma}{c}
\simeq7.565733250\times10^{-16}\,\mathrm{J\,m^{-3}\,K^{-4}}.
$$
The <radiation constant> $a_R$ is the same constant called $a_{\rm rad}$ in <black-body radiation>; $\sigma$ is the <Stefan-Boltzmann constant>. An outline of the <energy density> calculation substitutes $x=h\nu/(k_BT)$ and uses $\int_0^\infty x^3/(e^x-1)\,dx=\pi^4/15$. The <radiation pressure> follows because the <angular average> of the squared directional component of <photon> <momentum> is $1/3$ for <isotropy>; the <thermodynamic derivation of blackbody radiation pressure> supplies an independent route. These formulas describe the radiation contribution, not the cavity walls. The isotropic massless relation $p=u/3$ survives a different isotropic spectrum, whereas the coefficient in $u=a_RT^4$ requires the zero-<chemical potential> <Planck photon distribution>.
For a computed example with stipulated $T=2.725\,\mathrm K$, the <photon energy density> is $u\simeq4.17174\times10^{-14}\,\mathrm{J\,m^{-3}}$ and the <radiation pressure> is $p\simeq1.39058\times10^{-14}\,\mathrm{Pa}$. The <photon number density> in the <International System of Units> is
$$
n_\gamma=16\pi\zeta(3)\left(\frac{k_BT}{hc}\right)^3
\simeq4.10501\times10^8\,\mathrm{m^{-3}},\qquad
\frac{u}{n_\gamma}=\frac{\pi^4}{30\zeta(3)}k_BT\simeq2.70118\,k_BT.
$$
Here $\zeta$ is the <Riemann zeta function>. The mean <photon> <energy> differs from the <energy> at the peak of a spectral curve: counting <photons> and weighting them by <energy> emphasize different <frequencies>. The calculations use the exact <International System of Units> values of the <Planck constant>, <Boltzmann constant> and <speed of light>. https://www.nist.gov/si-redefinition/meet-constants[NIST, Meet the Constants: exact SI defining constants]{ref}
\Image[/photon-gas-entropy-spectral-weights.png]
{title=Analytic <photon gas> spectra: counting <photons> and weighting their <energies>}
{description=Original equilibrium model, not measured data. The left panel evaluates $u_\nu$ at stipulated $T=2.725\,\mathrm K$ per gigahertz. The right panel uses $x=h\nu/(k_BT)$ and plots the normalized spectral fractions $F_n=x^2/[2\zeta(3)(e^x-1)]$ and $F_u=15x^3/[\pi^4(e^x-1)]$, each with total <integral> one on $0<x<\infty$. Their different shapes show why a photon-counting spectrum and an <energy density> spectrum have different peaks.}
A motivating observation is the <cosmic microwave background> spectrum measured by the Far Infrared Absolute Spectrophotometer (FIRAS) aboard COBE. Its apparatus used a polarizing <Michelson interferometer>, with one input viewing a roughly seven-degree sky patch and the other an internal <blackbody>. Scanning mirrors changed the <optical path length> difference; heat-sensitive detectors recorded the varying <power>, and <Fourier transform spectroscopy> recovered the spectral difference. Operation outside the <terrestrial atmosphere> removed atmospheric emission and absorption, while shielding and cooling the optics in a helium cryostat suppressed the instrument's own radiation. https://lambda.gsfc.nasa.gov/data/cobe/firas/doc/FES4_APP_B.pdf[Mather, Fixsen and Shafer, Design for the COBE Far Infrared Absolute Spectrophotometer, apparatus and systematic-error controls]{ref}
Absolute calibration came from moving an external <blackbody> into the sky aperture, filling the same beam used for the sky. This was an iron-loaded epoxy absorber shaped as a re-entrant cone. Repeated reflections made it a close approximation to a <blackbody>, while thermometers supplied its <temperature>. Mather and colleagues explicitly evaluated reflections, <temperature> gradients, radiation leaking through the material and leakage around its edge. These are limits on how faithfully the calibrator realizes <thermal equilibrium> radiation, rather than details that can be eliminated merely by calling it black. https://arxiv.org/abs/astro-ph/9810373[Mather et al., Calibrator Design for COBE FIRAS (1999): absorber geometry and calibration error budget]{ref}
During sky observations the internal <blackbody> <temperature> was adjusted to nearly null the spectral difference, reducing gain and drift errors. Fixsen and colleagues' 1996 analysis used calibrated spectra over <wavenumbers> $2$ to $21\,\mathrm{cm^{-1}}$, about $60$ to $630\,\mathrm{GHz}$. They separated the sky's spatially uniform component, its dipole and Galactic emission, and varied sky cuts and foreground templates as controls. The resulting <cosmic microwave background> spectrum agreed with a single <Planck function>, with reported RMS deviations below 50 parts per million of its peak brightness. This is a residual statistic over the analyzed spectrum, not a pointwise relative error bound at every <frequency>. https://arxiv.org/abs/astro-ph/9605054[Fixsen et al., The Cosmic Microwave Background Spectrum from the Full COBE/FIRAS Data Set (1996), §§2, 5 and 5.3]{ref}
The observation motivates using a <photon gas> as a physical model rather than merely a formal counting exercise. Its interpretation has a clear boundary: agreement with <Planck's law> constrains departures from an equilibrium spectral shape within the calibrated band and foreground assumptions; it does not directly measure <radiation pressure> or <entropy density of a photon gas>. Nor does it establish that today's <cosmic microwave background> is continually re-equilibrating with matter. A previously thermal spectrum can retain its <Planck function> shape as <expansion of the universe> lowers every <photon> <frequency> in the same proportion. The usefulness of the model is that one measured spectral <temperature>, together with its equilibrium assumptions, predicts <energy density>, <radiation pressure>, <photon number density> and <entropy density of a photon gas> consistently.
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