What is the Bose-Einstein distribution?

It is the statistical distribution f(E) = 1/(exp((E−μ)/k_B T)−1) describing the average number of identical bosons in a quantum state of energy E at temperature T with chemical potential μ.

Why must μ be less than E?

If μ ≥ E, the denominator becomes zero or negative, which is unphysical. For photons and phonons μ = 0 by convention; for other bosons μ < E_min.

How does the Bose-Einstein distribution differ from Fermi-Dirac?

The Fermi-Dirac distribution uses (exp((E−μ)/k_B T)+1) with a "+" sign, capping f(E) at 1. Bose-Einstein uses "−1", allowing f(E) > 1 — bosons can pile into the same state.

What happens to f(E) at very low temperature?

As T → 0 and μ → E_min, states near the ground level fill macroscopically — this is Bose-Einstein condensation. States with E >> μ become nearly empty.

What units should I use?

Use eV for energy and chemical potential. The calculator operates directly in eV using k_B = 8.617×10⁻⁵ eV/K. Temperature is in kelvin.

Bose-Einstein Distribution Calculator

Calculate the average occupation number f(E) for a boson energy state given energy, chemical potential, and temperature.

Distribution Parameters

Energy E (eV)

Energy of the quantum state

Chemical Potential μ (eV)

Must be < E (use 0 for photons/phonons)

Temperature T (K)

Absolute temperature (kelvin)

Presets

Result

— average occupation number

f(E) = 1 / ( exp((E − μ) / k_B T) − 1 )

Exponent (E−μ)/k_BT

—

Thermal energy k_BT

—

exp((E−μ)/k_BT)

—

Regime

—

f(E) vs Energy

μ = 0 eV, T = 300 K

f(E) vs Temperature

Summary

The Bose-Einstein distribution gives the average number of bosons occupying a single quantum state at energy E, temperature T, and chemical potential μ. Unlike fermions, bosons have no Pauli exclusion limit, so f(E) can exceed 1.

How it works

Enter the energy E of the quantum state in electron-volts (eV).
Enter the chemical potential μ (must be less than E for a valid result).
Enter the temperature T in kelvin.
The calculator evaluates f(E) = 1 / (exp((E − μ) / k_B T) − 1).
Results update instantly; the plot shows f(E) across a range of energies.

Use cases

Determine the photon occupation number at a given frequency and temperature.
Analyze phonon distributions in solid-state physics.
Study Bose-Einstein condensation near the critical temperature.
Compare boson vs. fermion statistics in quantum optics courses.

Frequently Asked Questions

Last updated: 2026-06-19 · Reviewed by Nham Vu