Derivation of Isothermal Quantum Fluid Equations with Fermi-Dirac and Bose-Einstein Statistics

By using the quantum maximum entropy principle we formally derive, from a underlying kinetic description, isothermal (hydrodynamic and diffusive) quantum fluid equations for particles with Fermi-Dirac and Bose-Einstein statistics. A semiclassical expansion of the quantum fluid equations, up to -terms, leads to classical fluid equations with statistics-dependent quantum corrections, including a modified Bohm potential. The Maxwell-Boltzmann limit and the zero temperature limit are eventually discussed.