Quantum Statistics Comparer — Atri Mishra

Distributions

Maxwell-Boltzmann

f(E) = A·e^(−E/k_BT)

Classical particles — distinguishable

Bose-Einstein

f(E) = 1 / (e^((E−μ)/k_BT) − 1)

Bosons — integer spin (photons, phonons)

Fermi-Dirac

f(E) = 1 / (e^((E−μ)/k_BT) + 1)

Fermions — half-integer spin (electrons)

Parameters

Physics Scenario

Temperature T (K) 300 K

Chemical Potential μ (k_BT units) −2.0

Fermi Energy E_F (eV) — for FD only

Show E_F line (Fermi-Dirac) Show k_BT thermal energy Dark background Fill area under curves

Quick Comparison

Property	MB	BE	FD
Particles	Classical	Bosons	Fermions
Spin	Any	Integer	Half-int
Pauli Excl.	No	No	Yes
f(E) at E=μ	—	∞	½
Example	Gas atoms	Photons	Electrons
f(E=E_F)	—	—	0.500