Quantum hydrodynamic models from a maximum entropy principle

Abstract

We use a density matrix formalism to derive a kinetic theory for a quantum gas. Generalized kinetic fields are introduced and, employing the Wigner function, a certain hierarchy of quantum hydrodynamic (QHD) equations for the corresponding macroscopic variables is obtained. We assert a maximum entropy principle to obtain closure of the QHD system. For the explicit incorporation of statistics a proper quantum entropy is analyzed in terms of the reduced density matrix. The determination of the reduced Wigner function for equilibrium and non-equilibrium conditions is found to become possible only by assuming that the Lagrange multipliers can be expanded in powers of ℏ2. Quantum contributions are expressed in powers of ℏ2 while classical results are recovered in the limit ℏ → 0.