Liouville Energy Eigenvalue Problem

The mathematical dilemma of the direct determination of the reduced density matrix for an $N$-particle system in a stationary state is examined. An energy eigenvalue equation pertaining to the statistical Hamiltonian of a subsystem which determines the reduced density matrix is formulated. It is shown that the construction of this equation represents a classical, semi-quantum-mechanical attitude. The dilemma, in fact, arises from this attitude.