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arXiv:2601.03253 · Journal of Statistical Physics

Grand-Canonical Typicality

Cedric Igelspacher, Roderich Tumulka, Cornelia Vogel

quant-phSubmitted Jan 6, 2026• Announced Jan 7, 2026• 0 citations• Revised Jan 24, 2026 (v2)
v1 · Jan 6, 2026v2 · Jan 24, 2026v3 · May 21, 2026
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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.

Journal reference
Journal of Statistical Physics 193: 120 (2026)
DOI
10.1007/s10955-026-03656-5
Comments
43 pages LaTeX, no figures; v2 minor improvements and additions

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For a quantum system in a macroscopically large volume V, prepared in a pure state and subject to maximally noisy or ergodic unitary dynamics, the reduced density matrix of any sub-system v≪V is almost surely totally mixed. We show that the fluctuations around this limiting value, evaluated according to the invariant measure of these unitary flows, are captured by the Gaussian unitary ensemble (GUE) of random matrix theory. An extension of this statement, applicable when the unitary transformations conserve the energy but are maximally noisy or ergodic on any energy shell, allows to decipher the fluctuations around canonical typicality. According to typicality, if the large system is prepared in a generic pure state in a given energy shell, the reduced density matrix of the sub-system is almost surely the canonical Gibbs state of that sub-system. We show that the fluctuations around the Gibbs state are encoded in a deformation of the GUE whose covariance is specified by the Gibbs state. Contact with the eigenstate thermalization hypothesis is discussed.
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