Grand-Canonical Typicality

We study how the grand-canonical density matrix arises in macroscopic quantum systems. "Canonical typicality" is the known statement that for a typical wave function Ψ from a micro-canonical energy shell of a quantum system S weakly coupled to a large but finite quantum system B, the reduced density matrix hatρ^S_Ψ=tr^B |Ψranglelangle Ψ| is approximately equal to the canonical density matrix hatρ_can=Z^-1_can exp\(-βhat{H}^S\). Here, we discuss the analogous statement and related questions for the grand-canonical density matrix hatρ_gc=Z^-1_gc exp\(-β(hat{H}^S-μ₁ hat{N}₁^S-ldots-μ_rhat{N}_r^S\)) with hat{N}_i^S the number operator for molecules of type i in the system S. This includes (i) the case of chemical reactions and (ii) that of systems S defined by a spatial region which particles may enter or leave. It includes the statements (a) that the density matrix of the appropriate (generalized micro-canonical) Hilbert subspace H_gmc subset H^S otimes H^B (defined by a micro-canonical interval of total energy and suitable particle number sectors), after tracing out B, yields hatρ_gc; (b) that typical Ψ from H_gmc have reduced density matrix hatρ^S_Ψ close to hatρ_gc; and (c) that the conditional wave function ψ^S of S has probability distribution GAP_hatρgc if a typical orthonormal basis of H^B is used. That is, we discuss the foundation and justification of both the density matrix and the distribution of the wave function in the grand-canonical case. We also extend these considerations to the so-called generalized Gibbs ensembles, which apply to systems for which some macroscopic observables are conserved.