The hierarchy equations for the quantum thermal density matrices of an assembly of interacting particles in an external potential are derived in a compact form. This approach can complement and finally replace the phenomenological treatment of the thermodynamic properties of quantum gasses, liquids, and solids. We also derive a temperature Hartree-Fock approximation for fermions based on the symmetry properties of the exact hierarchy and finally we discuss the present approximation in relation to that of Kaplan and Argyres [T. A. Kaplan and P. N. Argyres, Ann. Phys. (N.Y.) 92, 1 (1975)].
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