Bose-Einstein Distribution

Equivalent forms

n(\nu) = \frac{2h\nu^3/c^2}{e^{h\nu/(k_B T)} - 1}

T_c = \frac{2\pi\hbar^2}{m k_B}\left(\frac{n}{\zeta(3/2)}\right)^{2/3}

N_0 = N\left[1-\left(\frac{T}{T_c}\right)^3\right]

The −1 (vs +1 in FD) leads to condensation; the planck_radiation_law is this formula applied to photons with μ = 0.

Symbol	Name	SI unit	Dimension	Range
$⟨n_i⟩$	Mean occupation numberoutput Average number of bosons in state i	dimensionless	1	0 – 100
$εᵢ$	State energy Energy of the single-particle state	eV	ML²T⁻²	0.01 – 5
$μ$	Chemical potential Must be ≤ ε₀ (ground state energy) to prevent divergence	eV	ML²T⁻²	-2 – 0
$T$	Temperature Absolute temperature	K	Θ	10 – 10000

Unit systems

SI:: $k_B = 1.380649e-23\,\mathrm{J/K}$
eV:: $k_B T \approx 0.02585\,\mathrm{eV}$ at room temperature
natural:: $\hbar = k_B = 1$

Where it holds

Applies to bosons (integer spin) in thermal equilibrium. Requires indistinguishability and quantum occupation statistics.

Discovery

Satyendra Nath Bose & Albert Einstein · 1924

Bose derived Planck's law by treating photons as indistinguishable quanta. Einstein generalised it to massive particles and predicted condensation — confirmed experimentally in rubidium atoms in 1995 (Nobel Prize 2001).

Try this

How does a laser produce perfectly coherent light?

Photons are bosons — unlimited numbers can occupy the same state. At low temperature, bosons condense into the ground state en masse (BEC), giving lasers, superfluids, and superconductors their macroscopic quantum coherence.

Research status: stable

Common misconceptions

BEC is NOT simply 'cooling gas'; it requires indistinguishable bosons with phase coherence. Cooper pairs in BCS theory are spin-0 composite bosons, not elementary bosons.

Derivation

Grand canonical:

\Xi _i = \Sigma _{n=0}^{\infty } e^{-n\beta (\varepsilon _i - \mu )} = 1/(1 - e^{-\beta (\varepsilon _i - \mu )})

.

Mean ⟨n_i⟩

= -\partial \ln \Xi _i/\partial (\beta \varepsilon _i) = 1/(e^{\beta (\varepsilon _i-\mu )} - 1)

.

Limiting cases

\varepsilon - \mu

≫ k_B T⟶ ⟨n⟩

\approx e^{-\beta (\varepsilon -\mu )}

→ Maxwell-Boltzmann (classical limit)

\mu \to \varepsilon _{0} (BEC

onset)⟶ Ground state occupation

N_{0} \to N

; macroscopic quantum state forms

\mu = 0

, photons⟶ Planck radiation law; photon number not conserved

Fermi-Dirac Distribution→Planck Radiation Law→Canonical Partition Function→