Grand canonical ensemble
A system free to exchange both energy and particles with a reservoir, at fixed temperature and chemical potential.
Definition
The grand canonical ensemble describes a system that can exchange both energy and particles with a reservoir, so that temperature T and chemical potential μ are fixed while energy and particle number both fluctuate. The microstate weights carry an extra factor exp(μN/k_BT), and the relevant sum is the grand partition function Ξ = Σ exp(−(E_i − μN_i)/k_BT). The corresponding potential is the grand potential Φ = −k_BT ln Ξ.
It is the natural framework wherever particle number is not conserved or not controlled: quantum gases of bosons and fermions, adsorption of molecules onto a surface, electrons in a semiconductor, and chemical equilibrium. The Bose–Einstein and Fermi–Dirac distributions are most cleanly derived in this ensemble.
Like the canonical ensemble, it agrees with the microcanonical description for macroscopic systems, because the relative fluctuations in particle number also vanish as 1/√N. The chemical potential μ measures the free-energy cost of adding one particle and is the variable conjugate to N, just as temperature is conjugate to entropy.