the photon gas

Seal an empty box, heat its walls to temperature T, and the cavity fills with electromagnetic radiation in thermal equilibrium — a glowing gas whose particles are photons continually absorbed and re-emitted by the walls. This is the photon gas, and treating blackbody radiation as a gas of quantum particles obeying Bose-Einstein statistics turns the mysterious spectrum of a hot oven into ordinary thermodynamics.

Photons are massless, spin-1 bosons with two polarization states, and crucially their number is NOT conserved — the walls create and destroy them freely — so the chemical potential is fixed at mu = 0. The mean number of photons in a mode of angular frequency omega is therefore n = 1/(exp(hbar omega/kT) - 1). Counting modes in a box (density of states proportional to omega^2) and multiplying by the energy hbar omega per photon and the occupation gives the spectral energy density, which integrates to the total energy density u = a T^4 with a = pi^2 k^4 / (15 hbar^3 c^3) — the Stefan-Boltzmann law. Because photons are relativistic, the equation of state is P = u/3: the radiation pressure is one-third of the energy density, not the (2/3)u of a nonrelativistic gas.

The photon gas describes the light inside a furnace, the surface of a star, and the cosmic microwave background — a nearly perfect blackbody at 2.725 K filling all of space. It is the cleanest example in physics of a Bose gas with mu = 0, and its spectrum is exactly Planck's radiation law. One honest idealization: 'blackbody' means a perfect absorber and emitter in full thermal equilibrium; real cavities and stars only approximate this, and radiation streaming freely through space is not strictly in equilibrium.