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