Abstract
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 ρ^S_Ψ=tr^B |Ψ Ψ| is approximately equal to the canonical density matrix ρcan=Z⁻¹can exp(-β H^S). Here, we discuss the analogous statement and related questions for the \emph{grand-canonical} density matrix ρgc=Z⁻¹gc exp(-β(H^S-μ₁ N₁^S-…-μ_rN_r^S)) with N_i^S the number operator for molecules of type i in the system S. This includes (i) the case of chemical reactions (which requires some novel considerations) and (ii) that of systems S defined by a spatial region which particles may enter or leave. It includes statements about how ρgc arises from the density matrix of the appropriate (generalized micro-canonical) Hilbert subspace Hgmc ⊂ H^S ⊗ H^B (defined by a micro-canonical interval of total energy and suitable particle number sectors) or from typical Ψ in Hgmc, as well as statements about the distribution of the (conditional) wave function ψ^S of S, which turns out to be a so-called GAP or Scrooge measure. 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. To this end (particularly for the chemical reactions), we also need to extend these considerations to the so-called generalized Gibbs ensembles, which apply to systems for which some macroscopic observables are conserved.