A Quantum Maximum Entropy Principle for closing the Moments of generalized Wigner Function.

By introducing a quantum entropy functional of the reduced density matrix, we report a rigorous scheme to develop quantum hydrodynamic models. The principle of quantum maximum entropy permits to solve the closure problem for a quantum hydrodynamic set of balance equations corresponding to an arbitrary number of moments of the reduced Wigner function. With this approach quantum contributions are obtained only by assuming that the Lagrange multipliers can be expanded in powers of \hbar^2. In particular, (i) the results available from literature in the framework of a quantum Boltzmann gas and a degenerate quantum Fermi gas are recovered as particular case; (ii) we generalize these results by incorporating explicitly the statistics for the quantum Fermi and Bose gases; (iii) we prove that the quantum maximum entropy principle keeps full validity in the classical limit, when \hbar -> 0. The principle of quantum maximum entropy is thus proposed as fundamental principle of quantum statistical mechanics