GranCanonicalEnsemble

The grand canonical ensemble, often abbreviated as the GCE, is a statistical ensemble used to describe a system in thermal and chemical equilibrium with a reservoir that can exchange both energy and particles. The natural variables for the GCE are temperature T, volume V, and chemical potential μ. In this ensemble the particle number N can fluctuate, reflecting exchange of particles with the reservoir.

The grand partition function Ξ(T,V,μ) is defined by Ξ = sum_N Z_N(T,V) e^{β μ N} = sum_s e^{−β (E_s − μ N_s)}, where Z_N is the canonical partition function for fixed N and E_s − μ N_s is the enthalpy-like energy entering the Boltzmann weight. Equivalently, Ξ can be written as sum_s e^{−β (E_s − μ N_s)} over all microstates s. Thermodynamic averages follow from Ξ: ⟨N⟩ = (1/Ξ) sum_s N_s e^{−β (E_s − μ N_s)} and ⟨E⟩ = (1/Ξ) sum_s E_s e^{−β (E_s − μ N_s)}. These can be compactly expressed as ⟨N⟩ = (k_B T) ∂ ln Ξ / ∂ μ and ⟨E⟩ = −∂ ln Ξ / ∂ β + μ ⟨N⟩. The grand potential is Ω = −k_B T ln Ξ, and the pressure is P = −Ω/V.

Fluctuations in particle number obey Var(N) = ⟨N^2⟩ − ⟨N⟩^2 = k_B T ∂⟨N⟩/∂μ = k_B T ∂^2 ln Ξ / ∂ μ^2. The GCE is used for quantum systems with Fermi-Dirac or Bose-Einstein statistics, where μ fixes particle exchange; for photons, μ is typically zero.

Applications include open systems such as adsorption, quantum dots, and nanoscale or electrolyte contexts, and it underpins grand canonical Monte Carlo simulations. In the thermodynamic limit, the GCE often yields results equivalent to other ensembles.