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).
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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