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 $\Psi$ 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 $\hat{\rho}^S_\Psi=\mathrm{tr}^B |\Psi\rangle\langle \Psi|$ is approximately equal to the canonical density matrix $\hat{\rho}_\mathrm{can}=Z^{-1}_\mathrm{can} \exp(-\beta \hat{H}^S)$. Here, we discuss the analogous statement and related questions for the \emph{grand-canonical} density matrix $\hat{\rho}_\mathrm{gc}=Z^{-1}_\mathrm{gc} \exp(-\beta(\hat{H}^S-\mu_1 \hat{N}_{1}^S-\ldots-\mu_r\hat{N}_{r}^S))$ 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 (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 $\hat{\rho}_\mathrm{gc}$ arises from the density matrix of the appropriate (generalized micro-canonical) Hilbert subspace $\mathscr{H}_\mathrm{gmc} \subset \mathscr{H}^S \otimes \mathscr{H}^B$ (defined by a micro-canonical interval of total energy and suitable particle number sectors) or from typical $\Psi$ in $\mathscr{H}_\mathrm{gmc}$, as well as statements about the distribution of the (conditional) wave function $\psi^S$ 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.