The Planck Distribution
Thermal photons do not distribute themselves evenly. They obey the Planck distribution, describing how many photons occupy each mode in an enclosure at equilibrium. The formula is exact and universal.
The spectral energy density per unit frequency interval is:
u(ν, T) = (8πhν³ / c³) · 1 / (e^(hν / kT) − 1)
where h is Planck's constant, c is the speed of light, k is Boltzmann's constant, ν is frequency, and T is absolute temperature. The factor 8πhν³/c³ counts electromagnetic standing-wave modes per unit volume. The second factor is the quantum ingredient: the average occupation number of a mode with energy spacing hν at temperature T.
This occupation number is the Bose-Einstein distribution for particles with zero chemical potential. Photons are bosons and do not conserve number — the cavity walls create and destroy them freely. The distribution counts how nature distributes indistinguishable quanta among distinguishable modes.
At low frequencies (hν ≪ kT), the exponential expands as e^x ≈ 1 + x, reducing the distribution to the Rayleigh-Jeans result: u ≈ 8πν²kT/c³. Energy is shared equally — equipartition. Every mode gets kT regardless of frequency. This is the classical limit, wrong at high frequency.
At high frequencies (hν ≫ kT), the exponential dominates, giving the Wien tail: u falls off as e^(−hν/kT). A mode requires quantum hν to excite, but thermal fluctuations of order kT cannot supply it. The mode remains empty.
Integrating over all frequencies yields the Stefan-Boltzmann law. Total energy density is (8π⁵k⁴ / 15h³c³) · T⁴. The dimensionless coefficient ζ(4) = π⁴/90 enters naturally.
The peak obeys Wien's displacement law. Differentiating yields ν_max / T ≈ 5.88 × 10¹⁰ Hz/K. In wavelength space, λ_max · T = b ≈ 2.898 × 10⁻³ m·K.
Planck's constant, h ≈ 6.626 × 10⁻³⁴ J·s, sets the fundamental quantum grain. In units where ħ = 1, the distribution is a dimensionless function of ν/T alone. The same function describes stellar glow, microwave oven radiation, and thermal noise in resistors.