Three Statistics

Consider an energy level ε. The occupation number ⟨n(ε)⟩ as a function of energy depends on what kind of particle:

Maxwell-Boltzmann (distinguishable, classical limit):
⟨n⟩ = e^(−(ε − μ)/kT) — exponential decay.

Bose-Einstein (identical bosons; symmetric wavefunction):
⟨n⟩ = 1 / (e^((ε − μ)/kT) − 1) — diverges as ε → μ⁺. Below the critical temperature for a finite-density gas, μ → 0 and a macroscopic fraction condenses into the ground state. This is a Bose-Einstein condensate.

Fermi-Dirac (identical fermions; antisymmetric wavefunction):
⟨n⟩ = 1 / (e^((ε − μ)/kT) + 1) — bounded in [0, 1]. Pauli exclusion. At T → 0, occupation is a step at the Fermi energy μ_F. The "Fermi sea" of cold electrons.

The three differ only by the ±1 in the denominator, but physically:
• Photons in a black body — Bose-Einstein, μ = 0 — gives Planck's law and explains everything from the CMB temperature to the color of red-hot iron.
• Electrons in a metal — Fermi-Dirac — gives the band gap, electrical conductivity, and why solids are stiff.
• Atoms in dilute gases — Maxwell-Boltzmann — gives the ideal gas law.

Where the statistics merge: when e^((ε − μ)/kT) ≫ 1 for all relevant ε — i.e., at high temperature or low density — both BE and FD become approximately MB. The "classical limit" of quantum statistics. Below this regime, statistics matter and quantum effects dominate.

Spin-statistics theorem (Pauli 1940): in relativistic quantum field theory, integer-spin particles MUST be bosons, half-integer MUST be fermions. This is not a postulate but a theorem — provable from Lorentz invariance, causality, and unitarity. The reason matter is solid and stars don't collapse is fundamentally relativistic.