Interpolation inequalities of Gagliardo-Nirenberg type and compactness results for self-adjoint trace-class operators with finite kinetic energy are established. Applying these results to the minimization of various free energy functionals, we determine for instance stationary states of the Hartree problem with temperature corresponding to various statistics. Key-words. Compact self-adjoint operators – Trace-class operators – mixed states – occupation numbers – Lieb-Thirring inequality – Gagliardo-Nirenberg inequality – logarithmic Sobolev inequality – optimal constants – orthonormal and sub-orthonormal systems – Schrödinger operator – asymptotic distribution of eigenvalues – free energy – embeddings – compactness results AMS MSC (2000). Primary: 81Q10, 82B10; Secondary: 26D15, 35J10, 47B34 1.