The Photon Gas
A photon gas is the thermodynamic system you get when you confine electromagnetic radiation inside a cavity — or, more honestly, when you look at any region of space filled with light and treat the photons as a gas of particles that obey Bose-Einstein statistics with zero chemical potential. This is black-body radiation. This is the Rayleigh-Jeans catastrophe that killed classical physics. This is where Planck was forced to bite the bullet and quantize.
What it is
Photons are massless bosons with two polarization states. Their energy dispersion is ε = ħω = ℏck, linear in momentum. Because photon number is not conserved — you can create or destroy them freely by exchanging energy with the cavity walls — the chemical potential vanishes: μ = 0. The occupation number for each mode is therefore the Planck distribution:
n̄(ω) = 1 / (e^(ℏω/k_BT) − 1)
This single formula contains all of thermal radiation. Integrate it over the density of states and you get the energy, pressure, entropy, specific heat — everything a thermodynamicist could ask for.
The density of states
In a volume V, the number of modes with frequencies between ω and ω + dω is:
g(ω)dω = V · ω² / (π²c³) dω
Multiply by the average energy ℏω·n̄(ω) and you get the spectral energy density u(ω, T). The integral over all frequencies gives the total energy:
U = (π²/15) · (V · (k_BT)⁴) / (ℏ³c³)
That T⁴ is the Stefan-Boltzmann law in disguise. U ∝ V T⁴. Pressure P = U/(3V). Entropy S ∝ V T³. These are the equations of state for a photon gas, and they are exact.
Why it matters
The photon gas was the first system where classical statistical mechanics produced infinities. The Rayleigh-Jeans law, u(ω) ∝ ω²k_BT, diverges at high frequency — the ultraviolet catastrophe. Planck's quantization cut off the high-frequency modes exponentially, and the integral converged. The UV catastrophe was not a mathematical error in integration; it was a physical signal that energy comes in quanta.
Beyond its historical role, the photon gas is still a working model. It describes:
- The cosmic microwave background, a near-perfect black body at 2.725 K.
- The interiors of stars, where radiation pressure contributes to supporting the star against gravity.
- Optical cavities and laser resonators, where the mode structure determines the thermodynamic and quantum properties of the field.
- Early-universe cosmology, where photons, neutrinos, and electrons were all in thermal equilibrium.
Key numbers
The energy density of a photon gas at temperature T is:
u = U/V = a · T⁴
where a = 4σ/c ≈ 7.56 × 10⁻¹⁶ J·m⁻³·K⁻⁴ is the radiation constant (σ is the Stefan-Boltzmann constant). At room temperature, T ≈ 300 K, the energy density is vanishingly small — that's why we don't notice black-body radiation in daily life. At T = 5800 K (the Sun's surface), u ≈ 6.3 × 10⁷ J/m³. The difference is dramatic.
The number density of photons is:
n = N/V ≈ 20.3 · (k_BT)³ / (π²ℏ³c³) ≈ 2.0 × 10⁷ · (T/K)³ m⁻³
At 300 K, there are roughly 5 × 10¹¹ photons per cubic meter. Not many — but at 5800 K, the density is 4 × 10¹⁷ m⁻³.
Pressure and work
A photon gas exerts pressure even at zero temperature — no, wait, at zero temperature there are no photons. But at any finite T, P = u/3. Compress a cavity adiabatically (keep entropy fixed) and the temperature rises. The adiabatic law is VT³ = constant, or equivalently T ∝ V⁻¹/³ ∝ L⁻¹ where L is a characteristic cavity size. Photons redshift when the cavity expands. This is the same physics that cools the cosmic microwave background as the universe expands — cosmological redshift applied to a thermal gas.
Summary
A photon gas is a thermodynamic system of massless bosons with μ = 0, governed by the Planck distribution. Its energy scales as T⁴, its pressure as T⁴/3, and its entropy as T³. It was the system that forced the birth of quantum mechanics. It still fills the universe.