Bose-Einstein Distribution Formula
The Bose-Einstein distribution gives the average number of bosons occupying a quantum state at energy E.
Applies to photons, phonons, and integer-spin particles.
The Formula
The Bose-Einstein distribution gives the average number of particles occupying a quantum state at energy E for a system of bosons at thermal equilibrium. Bosons are particles with integer spin: photons (spin 1), phonons, helium-4 nuclei, and the Higgs boson. Unlike fermions (described by Fermi-Dirac statistics), bosons have no restriction on how many can occupy the same state — many can pile into a single quantum state.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| n̅(E) | Average number of particles in state at energy E | dimensionless |
| E | Energy of the quantum state | Joules or eV |
| μ | Chemical potential (for photons, μ = 0) | Joules or eV |
| k_B | Boltzmann constant = 1.381 × 10−23 J/K | J/K |
| T | Absolute temperature | Kelvin (K) |
Comparison of the three quantum distributions:
- Bose-Einstein: denominator has −1 (bosons, any occupation)
- Fermi-Dirac: denominator has +1 (fermions, maximum 1 per state)
- Maxwell-Boltzmann: denominator is e^(E/k_BT) (classical limit, high T or low density)
Example 1 — Planck Radiation Law
For photons in a blackbody cavity, μ = 0. Find the average number of photons in a mode at frequency f = 600 THz (visible light) at T = 5778 K (Sun surface).
E = hf = 6.626 × 10−34 × 6 × 10¹&sup4; = 3.976 × 10−19 J = 2.48 eV
k_BT = 1.381 × 10−23 × 5778 = 7.98 × 10−20 J = 0.498 eV
E/k_BT = 2.48/0.498 = 4.98
n̅ = 1/(e^4.98 − 1) = 1/(145.6 − 1) ≈ 0.0069 photons per mode — visible light modes are sparsely populated even at solar temperatures
Example 2 — Bose-Einstein Condensate
What happens when the chemical potential μ approaches E from below?
As μ → E: the denominator e^((E−μ)/k_BT) − 1 approaches 0
n̅(E) → ∞
This divergence signals Bose-Einstein condensation: a macroscopic number of bosons occupy the ground state. First observed by Eric Cornell and Carl Wieman in 1995 with rubidium-87 atoms cooled to 170 nanokelvin.
When to Use It
Use the Bose-Einstein distribution when:
- Deriving Planck's blackbody radiation law (photon gas with μ = 0)
- Calculating phonon contributions to heat capacity in solids (Debye model)
- Analyzing superfluidity in helium-4 and Bose-Einstein condensates
- Studying laser operation, where photons pile into a single mode
- Understanding the cosmic microwave background as a photon gas
The Bose-Einstein distribution has no upper limit on occupation number — in principle, all bosons can condense into the same quantum state. This is fundamentally different from fermions, which obey the Pauli exclusion principle and can never share a state.