grandkanonischen

Grandkanonischen is a term used in physics to refer to the grand canonical ensemble, a statistical framework for describing systems that are in thermal and chemical equilibrium with a reservoir. In this ensemble the system can exchange both energy and particles with the surroundings, allowing the particle number to fluctuate.

The central quantity is the grand partition function Ξ(β, μ, V) = ∑N e^{β μ N} ZN(β, V), where β = 1/(kB T), μ is the chemical potential, V the volume, and ZN is the canonical partition function at fixed N. The probability of observing a microstate with energy E and particle number N is proportional to e^{-β (E − μ N)}. The corresponding density operator is ρ ∝ e^{-β (H − μ N)}, with Ξ normalizing the state: Ξ = Tr e^{-β (H − μ N)}.

Thermodynamic relations follow from Ξ. The grand potential is Ω = −kB T ln Ξ, and ⟨N⟩ = (1/β) ∂ ln Ξ / ∂ μ, ⟨E⟩ = −∂ ln Ξ / ∂ β + μ ⟨N⟩. Fluctuations in particle number are given by Var(N) = ∂⟨N⟩/∂(β μ) = ∂^2 ln Ξ / ∂(β μ)^2. In the appropriate limits, the grand canonical ensemble reduces to the canonical ensemble (fixed N) when particle exchange is suppressed or μ is set to a value effectively imposing a fixed N.

Applications of the grandkanonischen framework appear in quantum gases (Fermi/Bose systems), condensed matter physics, and lattice simulations, where control of particle exchange is essential. In German-language literature, the term is commonly rendered as grandkanonisches Ensemble.