Quantum hydrodynamic models from a maximum entropy principle
Massimo Trovato, Lino Reggiani · Journal of Physics A Mathematical and Theoretical · 2010
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 ℏ 2 . Quantum contributions are expressed in powers of ℏ 2 while classical results are recovered in the limit ℏ → 0.