the Bose-Einstein distribution
/ BOHZ INE-stine /
Bosons are gregarious. Unlike fermions, any number of them can crowd into the same quantum state, and in fact the presence of one boson in a state makes it more likely that another will join — this 'stimulated' bunching is what makes a laser possible. The Bose-Einstein distribution gives the precise rule for the average number of bosons occupying a state of energy epsilon at temperature T, and unlike the Fermi function it has no ceiling.
The mean occupation of a single-particle state of energy epsilon is n(epsilon) = 1 / (exp((epsilon - mu)/kT) - 1), with the crucial -1 in the denominator. As epsilon approaches mu the exponential approaches 1, the denominator approaches 0, and the occupation diverges — bosons pile without limit into the lowest available states. Consistency requires mu to be less than or equal to the lowest single-particle energy (otherwise the occupation would be negative). For particles whose number is not conserved — photons in a cavity, phonons in a solid — there is no constraint fixing N, so the chemical potential is pinned at mu = 0. The formula again comes from the grand canonical ensemble applied to one mode, but now summing over 0, 1, 2, 3, ... particles, a geometric series.
The Bose-Einstein distribution underlies blackbody radiation (the photon gas), the heat capacity of solids (the phonon gas), superfluidity, and Bose-Einstein condensation. As with Fermi-Dirac, it reduces to the classical Maxwell-Boltzmann distribution only in the dilute, high-temperature limit where the exponential is large and the -1 is negligible; there the bunching disappears and particles behave as independent classical objects. A common misconception to avoid: the divergence at epsilon = mu is not a pathology but the mathematical seed of condensation, the macroscopic occupation of the ground state.
For photons in a cavity mu = 0, so the mean number of photons in a mode of angular frequency omega is n = 1/(exp(hbar omega/kT) - 1). At high frequency (hbar omega >> kT) this is exponentially tiny — the exponential cutoff that tames the ultraviolet catastrophe and gives Planck's law its shape.
With mu = 0, the Bose-Einstein occupation of high-frequency modes is exponentially suppressed.
The whole difference from Fermi-Dirac is the sign of the 1 in the denominator (-1 for bosons, +1 for fermions); both reduce to Maxwell-Boltzmann when occupancies are much less than 1. For number-conserving bosons mu is temperature-dependent and negative; for photons and phonons it is exactly zero.