Statistics and Quantum Maximum Entropy Principle

By using the reduced Wigner formalism we consider a kinetic theory for a quantum gas. We introduce a set of generalized kinetic fields and obtain a hierarchy of Quantum Hydrodynamic (QHD) equations for the corresponding macroscopic variables. To close the QHD system a maximum entropy principle is asserted, and to explicitly incorporate particles indistinguishability a proper quantum entropy is analyzed in terms of the reduced density matrix. This approach implies a quantum generalization of the corresponding Lagrange multipliers. Quantum contributions are expressed in powers of $\hbar^2.$