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  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 ρˆ S  = trB || is approximately equal to the canonical density matrix ρˆcan = Z−1 can exp(−β ˆH S). Here, we discuss the analogous statement and related questions for the grand-canonical density matrix ρˆgc = Z−1 gc exp(−β( ˆH S−μ1 ˆN S 1 −. . .−μr ˆN S r )) with ˆN S i 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 ρˆgc 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  in Hgmc, as well as statements about the distribution of the (conditional) wave function ψS 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.