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 converts to joules internally using k_B = 8.617×10⁻⁵ eV/K. Temperature is in kelvin.

Bose-Einstein Distribution Calculator

Name: Bose-Einstein Distribution Calculator
Availability: InStock
Author: Nham Vu

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

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

How it works

Enter the energy E of the quantum state in electron-volts (eV) or joules.
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: 2026-07-01 · Reviewed by Nham Vu