Grand-Canonical Typicality
Cedric Igelspacher, Roderich Tumulka, Cornelia Vogel
TL;DR
The paper establishes a quantum foundation for the grand-canonical ensemble by introducing a generalized micro-canonical subspace $\mathscr{H}_\mathrm{gmc}$ defined via conserved macroscopic observables $\hat{Q}_k$ that commute and are extensive. It proves that tracing out the bath yields a generalized Gibbs density matrix $\hat{\rho}_\mathrm{gG}$ and that, for typical pure states in $\mathscr{H}_\mathrm{gmc}$, the subsystem S is described by the grand-canonical density matrix $\hat{\rho}_\mathrm{gc}$, with the conditional wave function distributed according to GAP measures. The results extend to generalized Gibbs ensembles and clarify chemical equilibrium with explicit expressions for equilibrium populations via derivatives of the grand partition function. The analysis combines canonical typicality, the PSW theorem on typicality, GAP measures, and ETH/MITE-based approach-to-equilibrium arguments, providing a robust quantum justification for both density-matrix and wave-function descriptions of grand-canonical thermalization in macroscopic systems and chemical reactions.
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 $\hatρ^S_Ψ=\mathrm{tr}^B |Ψ\rangle\langle Ψ|$ is approximately equal to the canonical density matrix $\hatρ_\mathrm{can}=Z^{-1}_\mathrm{can} \exp(-β\hat{H}^S)$. Here, we discuss the analogous statement and related questions for the \emph{grand-canonical} density matrix $\hatρ_\mathrm{gc}=Z^{-1}_\mathrm{gc} \exp(-β(\hat{H}^S-μ_1 \hat{N}_{1}^S-\ldots-μ_r\hat{N}_{r}^S))$ 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 and (ii) that of systems $S$ defined by a spatial region which particles may enter or leave. It includes the statements (a) that the density matrix of the appropriate (generalized micro-canonical) Hilbert subspace $H_\mathrm{gmc} \subset H^S \otimes H^B$ (defined by a micro-canonical interval of total energy and suitable particle number sectors), after tracing out $B$, yields $\hatρ_\mathrm{gc}$; (b) that typical $Ψ$ from $H_\mathrm{gmc}$ have reduced density matrix $\hatρ^S_Ψ$ close to $\hatρ_\mathrm{gc}$; and (c) that the conditional wave function $ψ^S$ of $S$ has probability distribution $\mathrm{GAP}_{\hatρ_\mathrm{gc}}$ if a typical orthonormal basis of $H^B$ is used. 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. We also extend these considerations to the so-called generalized Gibbs ensembles, which apply to systems for which some macroscopic observables are conserved.