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 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 | ρ ^ Ψ S = tr B | Ψ ⟩ ⟨ Ψ | is approximately equal to the canonical density matrix ρ ̂can=Z⁻¹canexp (-β ĤS) ρ ^ can = Z can - 1 exp ( - β H ^ S ) . Here, we discuss the analogous statement and related questions for the grand-canonical density matrix ρ ̂gc=Z⁻¹gcexp (-β (ĤS-μ ₁ N̂₁S-… -μ ᵣN̂ᵣS)) ρ ^ gc = Z gc - 1 exp ( - β ( H ^ S - μ 1 N ^ 1 S - … - μ r N ^ r S ) ) with N̂ᵢS 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 $${