# Density Matrices and Two-Particle Correlations

## Abstract

In this chapter, we first introduce two new concepts named *density matrix* and *reduced density matrices* as (5.1) and (5.3). The reduced density matrices are closely connected with the *n*-particle correlations in equilibrium (*n* = 1, 2, ⋯ ). Next, we give a proof of the *Bloch–De Dominicis theorem*, i.e., a thermodynamic extension of *Wick’s theorem*, which enables us to express the *n*-particle correlations of ideal gases in terms of one-particle correlations as (5.11). Finally, the theorem is applied to obtain the two-particle correlations of homogeneous ideal Bose and Fermi gases in three dimensions. The results are summarized in Fig. 5.1 below. It clearly shows that there exists a special quantum-mechanical correlation between each pair of identical particles due to the permutation symmetry, which is completely different in nature between Bose and Fermi gases.

## Keywords

Density Matrix Pair Distribution Function Microcanonical Ensemble Reduce Density Matrice Thermodynamic Extension## References

- 1.P.W. Anderson, Rev. Mod. Phys.
**38**, 298 (1966)CrossRefADSGoogle Scholar - 2.C. Bloch, C. De Donimicis, Nucl. Phys.
**7**, 459 (1958)CrossRefMATHGoogle Scholar - 3.M. Gaudin, Nucl. Phys.
**15**, 89 (1960)MathSciNetCrossRefMATHGoogle Scholar - 4.J.R. Johnston, Am. J. Phys.
**38**, 516 (1970)CrossRefADSGoogle Scholar - 5.O. Penrose, Philos. Mag.
**42**, 1373 (1951)CrossRefMATHGoogle Scholar - 6.O. Penrose, L. Onsager, Phys. Rev.
**104**, 576 (1956)CrossRefADSMATHGoogle Scholar - 7.G.C. Wick, Phys. Rev.
**80**, 268 (1950)MathSciNetCrossRefADSMATHGoogle Scholar - 8.C.N. Yang, Rev. Mod. Phys.
**34**, 694 (1962)CrossRefADSGoogle Scholar