Photon gas

In physics, a photon gas is a gas-like collection of photons, which has many of the same properties of a conventional gas like hydrogen or neon – including pressure, temperature, and entropy. The most common example of a photon gas in equilibrium is the black-body radiation.

Photons are part of a family of particles known as bosons, particles that follow Bose–Einstein statistics and with integer spin. A gas of bosons with only one type of particle is uniquely described by three state functions such as the temperature, volume, and the number of particles. However, for a black body, the energy distribution is established by the interaction of the photons with matter, usually the walls of the container, and the number of photons is not conserved. As a result, the chemical potential of the black-body photon gas is zero at thermodynamic equilibrium. The number of state variables needed to describe a black-body state is thus reduced from three to two (e.g. temperature and volume).

In a classical ideal gas with massive particles, the energy of the particles is distributed according to a Maxwell–Boltzmann distribution. This distribution is established as the particles collide with each other, exchanging energy (and momentum) in the process. In a photon gas, there will also be an equilibrium distribution, but photons do not collide with each other (except under very extreme conditions, see two-photon physics), so the equilibrium distribution must be established by other means. The most common way that an equilibrium distribution is established is by the interaction of the photons with matter. If the photons are absorbed and emitted by the walls of the system containing the photon gas, and the walls are at a particular temperature, then the equilibrium distribution for the photons will be a black-body distribution at that temperature.

A very important difference between a generic Bose gas (gas of massive bosons) and a photon gas with a black-body distribution is that the number of photons in the photon gas is not conserved. A photon can be created upon thermal excitation of an atom in the wall into an upper electronic state, followed by the emission of a photon when the atom falls back to a lower energetic state. This type of photon generation is called thermal emission. The reverse process can also take place, resulting in a photon being destroyed and removed from the gas. It can be shown that, as a result of such processes there is no constraint on the number of photons in the system, and the chemical potential of the photons must be zero for black-body radiation.

The thermodynamics of a black-body photon gas may be derived using quantum statistical mechanical arguments, with the radiation field being in equilibrium with the atoms in the wall. The derivation yields the spectral energy density u, which is the energy of the radiation field per unit volume per unit frequency interval, given by: $${\displaystyle u(\nu ,T)={\frac {8\pi h\nu ^{3}}{c^{3}}}~{\frac {1}{e^{\frac {h\nu }{kT}}-1}},}$$ where h is the Planck constant, c is the speed of light, ν is the frequency, k is the Boltzmann constant, and T is temperature.

Integrating over frequency and multiplying by the volume, V, gives the internal energy of a black-body photon gas: $${\displaystyle U=\left({\frac {8\pi ^{5}k^{4}}{15(hc)^{3}}}\right)VT^{4}.}$$

The derivation also yields the (expected) number of photons N: $${\displaystyle N=\left({\frac {16\pi k^{3}\zeta (3)}{(hc)^{3}}}\right)VT^{3},}$$ where ${\displaystyle \zeta (n)}$ is the Riemann zeta function. Note that for a particular temperature, the particle number N varies with the volume in a fixed manner, adjusting itself to have a constant density of photons.

If we note that the equation of state for an ultra-relativistic quantum gas (which inherently describes photons) is given by $${\displaystyle U=3PV,}$$ then we can combine the above formulas to produce an equation of state that looks much like that of an ideal gas: $${\displaystyle PV={\frac {\zeta (4)}{\zeta (3)}}NkT\approx 0.9\,NkT.}$$