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 ${\hat{\rho }}_{\Psi }^S={\text{tr}}^B|\Psi ⟩⟨\Psi |$ is approximately equal to the canonical density matrix ${\hat{\rho }}_{\text{can}}=Z_{\text{can}}^{-1}exp(-\beta {\hat{H}}^S)$ . Here, we discuss the analogous statement and related questions for the grand-canonical density matrix ${\hat{\rho }}_{\text{gc}}=Z_{\text{gc}}^{-1}exp(-\beta ({\hat{H}}^S-{\mu }_1{\hat{N}}_1^S-\dots -{\mu }_r{\hat{N}}_r^S))$ 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 ${\hat{\rho }}_{\text{gc}}$ arises from the density matrix of the appropriate (generalized micro-canonical) Hilbert subspace $H_{\text{gmc}}\subset H^S\otimes H^B$ (defined by a micro-canonical interval of total energy and suitable particle number sectors) or from typical $\Psi$ in $H_{\text{gmc}}$ , 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.