Grand-Canonical Typicality

• URL: http://arxiv.org/abs/2601.03253v1
• Date: Tue, 06 Jan 2026 18:57:29 GMT
• Title: Grand-Canonical Typicality
• Authors: Cedric Igelspacher, Roderich Tumulka, Cornelia Vogel,
• Abstract summary: We study how the grand-canonical density matrix arises in macroscopic quantum systems.<n>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 $hatS_=mathrmtrB |ranglelangle |$ is approximately equal to the canonical density matrix $hat_mathrmcan=Z-1_mathrmcan exp(-hatH
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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