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 (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 $\hatρ_\mathrm{gc}$ 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 $Ψ$ in $\mathscr{H}_\mathrm{gmc}$ , as well as statements about the distribution of the (conditional) wave function $ψ^S$ 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.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 (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 $\hatρ_\mathrm{gc}$ 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 $Ψ$ in $\mathscr{H}_\mathrm{gmc}$, as well as statements about the distribution of the (conditional) wave function $ψ^S$ 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.

quant-pharXiv:2601.03253Journal of Statistical PhysicsJan 7, 20260 citations

Quantum Correlations of Ideal Bose and Fermi Gases in the Canonical Ensemble

Kazumasa Tsutsui, Takafumi Kita

We derive an expression for the reduced density matrices of ideal Bose and Fermi gases in the canonical ensemble, which corresponds to the Bloch--De Dominicis (or Wick's) theorem in the grand canonical ensemble for normal-ordered products of operators. Using this expression, we study one- and two-body correlations of homogeneous ideal gases with $N$ particles. The pair distribution function $g^{(2)}(r)$ of fermions clearly exhibits antibunching with $g^{(2)}(0)=0$ due to the Pauli exclusion principle at all temperatures, whereas that of normal bosons shows bunching with $g^{(2)}(0)\approx 2$ , corresponding to the Hanbury Brown--Twiss effect. For bosons below the Bose--Einstein condensation temperature $T_0$ , an off-diagonal long-range order develops in the one-particle density matrix to reach $g^{(1)}(r)=1$ at $T=0$ , and the pair correlation starts to decrease towards $g^{(2)}(r)\approx 1$ at $T=0$ . The results for $N\rightarrow \infty$ are seen to converge to those of the grand canonical ensemble obtained by assuming the average $\langle\hatψ({\bf r})\rangle$ of the field operator $\hatψ({\bf r})$ below $T_0$ . This fact justifies the introduction of the "anomalous" average $\langle\hatψ({\bf r})\rangle\neq 0$ below $T_0$ in the grand canonical ensemble as a mathematical means of removing unphysical particle-number fluctuations to reproduce the canonical results in the thermodynamic limit.We derive an expression for the reduced density matrices of ideal Bose and Fermi gases in the canonical ensemble, which corresponds to the Bloch--De Dominicis (or Wick's) theorem in the grand canonical ensemble for normal-ordered products of operators. Using this expression, we study one- and two-body correlations of homogeneous ideal gases with $N$ particles. The pair distribution function $g^{(2)}(r)$ of fermions clearly exhibits antibunching with $g^{(2)}(0)=0$ due to the Pauli exclusion principle at all temperatures, whereas that of normal bosons shows bunching with $g^{(2)}(0)\approx 2$, corresponding to the Hanbury Brown--Twiss effect. For bosons below the Bose--Einstein condensation temperature $T_0$, an off-diagonal long-range order develops in the one-particle density matrix to reach $g^{(1)}(r)=1$ at $T=0$, and the pair correlation starts to decrease towards $g^{(2)}(r)\approx 1$ at $T=0$. The results for $N\rightarrow \infty$ are seen to converge to those of the grand canonical ensemble obtained by assuming the average $\langle\hatψ({\bf r})\rangle$ of the field operator $\hatψ({\bf r})$ below $T_0$. This fact justifies the introduction of the "anomalous" average $\langle\hatψ({\bf r})\rangle\neq 0$ below $T_0$ in the grand canonical ensemble as a mathematical means of removing unphysical particle-number fluctuations to reproduce the canonical results in the thermodynamic limit.

arXiv:1610.03617Journal of the Physical Society of JapanOct 12, 20164 citations

Theory of Density Matrices

K. Husimi, Yoshihiro Kitano, Toshiyuki Nishiyama

Fortschritte der PhysikJan 1, 195827 citations

Some Recent Advances in Density Matrix Theory

R. McWeeny

Reviews of Modern PhysicsApr 1, 19601,460 citations

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 (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

\hatρ_\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

Ψ

\mathscr{H}_\mathrm{gmc}

, as well as statements about the distribution of the (conditional) wave function

ψ^S

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.

N

particles. The pair distribution function

g^{(2)}(r)

of fermions clearly exhibits antibunching with

g^{(2)}(0)=0

due to the Pauli exclusion principle at all temperatures, whereas that of normal bosons shows bunching with

g^{(2)}(0)\approx 2

, corresponding to the Hanbury Brown--Twiss effect. For bosons below the Bose--Einstein condensation temperature

T_0

, an off-diagonal long-range order develops in the one-particle density matrix to reach

g^{(1)}(r)=1

T=0

, and the pair correlation starts to decrease towards

g^{(2)}(r)\approx 1

T=0

. The results for

N\rightarrow \infty

are seen to converge to those of the grand canonical ensemble obtained by assuming the average

\langle\hatψ({\bf r})\rangle

of the field operator

\hatψ({\bf r})

below

T_0

. This fact justifies the introduction of the "anomalous" average

\langle\hatψ({\bf r})\rangle\neq 0

below

T_0

in the grand canonical ensemble as a mathematical means of removing unphysical particle-number fluctuations to reproduce the canonical results in the thermodynamic limit.