Grand Canonical Ensemble — Chemical Potential & Fugacity

The grand canonical ensemble fixes temperature T and chemical potential μ, allowing particle number N to fluctuate. The grand partition function Ξ = Σ_N z^N Z_N (z=e^{βμ} fugacity) encodes all thermodynamics. The occupation statistics—Fermi-Dirac, Bose-Einstein, or Boltzmann—depend critically on μ relative to the single-particle energies.

Grand partition function:
Ξ = Σ_N z^N Z_N = e^{−βΩ}
z = e^{βμ} (fugacity)

Mean occupation:
FD: ⟨n_ε⟩ = 1/(e^{β(ε−μ)}+1)
BE: ⟨n_ε⟩ = 1/(e^{β(ε−μ)}−1)
MB: ⟨n_ε⟩ = e^{−β(ε−μ)}

Fugacity expansion (virial):
PV/NkT = Σ_l a_l z^l
a₁=1, a₂=−1/(2^(5/2)) (BE/FD)

BEC threshold:
z → 1 (μ → 0⁻) for bosons
Phase transition when n λ³ = 2.612