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arXiv:cond-mat/0306306 · Physica A Statistical Mechanics and its Applications

Hierarchy Bloch equations for the reduced statistical density operators in canonical and grand canonical ensembles

Liqiang Wei, Chia‐Chung Sun

Published Dec 2, 2003• 4 citations
arXivHTMLPDF
DOI
10.1016/j.physa.2003.10.072

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We deal with many-electron systems having a noninteger number of electrons, which cannot be described properly by means of pure states or by canonical statistical ensemble states. The study of the one-electron reduced density matrix for these systems raises the problem of its representability in statistical ensembles of grand canonical type. We derive the necessary and sufficient conditions for the representability of the one-electron reduced density matrix of grand-canonical statistical ensembles.
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 NNN particles. The pair distribution function g(2)(r)g^{(2)}(r)g(2)(r) of fermions clearly exhibits antibunching with g(2)(0)=0g^{(2)}(0)=0g(2)(0)=0 due to the Pauli exclusion principle at all temperatures, whereas that of normal bosons shows bunching with g(2)(0)≈2g^{(2)}(0)\approx 2g(2)(0)≈2, corresponding to the Hanbury Brown--Twiss effect. For bosons below the Bose--Einstein condensation temperature T0T_0T0​, an off-diagonal long-range order develops in the one-particle density matrix to reach g(1)(r)=1g^{(1)}(r)=1g(1)(r)=1 at T=0T=0T=0, and the pair correlation starts to decrease towards g(2)(r)≈1g^{(2)}(r)\approx 1g(2)(r)≈1 at T=0T=0T=0. The results for N→∞N\rightarrow \inftyN→∞ are seen to converge to those of the grand canonical ensemble obtained by assuming the average ⟨ψ^(r)⟩\langle\hatψ({\bf r})\rangle⟨ψ^​(r)⟩ of the field operator ψ^(r)\hatψ({\bf r})ψ^​(r) below T0T_0T0​. This fact justifies the introduction of the "anomalous" average ⟨ψ^(r)⟩≠0\langle\hatψ({\bf r})\rangle\neq 0⟨ψ^​(r)⟩=0 below T0T_0T0​ in the grand canonical ensemble as a mathematical means of removing unphysical particle-number fluctuations to reproduce the canonical results in the thermodynamic limit.