Bose-Einstein Distribution Calculator
Average number of bosons in a quantum state n = 1/(exp((E-mu)/kT) - 1).
Compare to Fermi-Dirac and Maxwell-Boltzmann across energy, temperature, mu.
The Bose-Einstein distribution tells you, on average, how many bosons sit in a quantum state of energy E when the system is at temperature T and chemical potential μ.
n̄(E) = 1 / (exp((E − μ) / k_B T) − 1)
Bosons are particles with integer spin: photons, phonons (lattice vibrations), helium-4 nuclei, the Higgs. Unlike fermions, there’s no Pauli rule — any number of bosons can pile into the same state. That “−1” in the denominator (vs “+1” for Fermi-Dirac) is the whole reason lasers and Bose-Einstein condensates exist.
Three regimes the formula lives in:
When (E − μ) ≫ k_B T, the exponential is huge and n̄ ≈ exp(−(E − μ) / k_B T). That is Maxwell-Boltzmann, the classical limit. High-energy states are empty, and quantum statistics don’t matter.
When (E − μ) ≈ k_B T, you’re in the genuinely quantum regime. Bose-Einstein gives noticeably more occupation than Maxwell-Boltzmann (about 58% more at E − μ = k_B T). This regime is where blackbody radiation lives. It’s why Planck’s law differs from the Rayleigh-Jeans classical prediction.
When (E − μ) → 0⁺, the denominator collapses toward zero and n̄ diverges. That’s the signature of Bose-Einstein condensation. As you cool a dilute boson gas, μ climbs toward the ground-state energy from below, and at the critical temperature a macroscopic number of atoms suddenly pour into the single lowest state. Cornell, Wieman, and Ketterle won the 2001 Nobel Prize for observing this directly in rubidium-87 cooled to 170 nanokelvin.
For photons, μ = 0. Photons can be created and destroyed freely, which forces zero chemical potential. Set μ = 0 in this calculator and the result is exactly the number of photons per mode in a thermal cavity at temperature T — the Planck radiation law in disguise.
Units to know. k_B is the Boltzmann constant. In SI, k_B = 1.381 × 10⁻²³ J/K. In electron-volts (much friendlier for atomic-scale problems), k_B = 8.617 × 10⁻⁵ eV/K. This calculator does the unit conversion for you. Pick eV or Joules for the energy and just enter the number. T is always in Kelvin (room temperature is 293 K, the Sun’s surface is 5778 K, the cosmic microwave background is 2.725 K).