Mentioning the increasing importance of superfluids in connection with superconducting material, Bose-Einstein condensates and quantum computing, the author notices that modeling quantum fluids by the nonlinear Schrödinger equation involves mainly dissipative terms, whereas the allowance of quantum diffusion is not complete. Thus, the aim of the paper consists in the derivation of macroscopic models for diffusive quantum systems from Wigner-Boltzmann equations by moment methods. Three model classes are derived which depend on the behavior of the collision operator. (I) The quantum drift-diffusion equation, applied in spinorial systems and consisting of a nonlinear parabolic fourth-order equation for the particle density supplemented with the Poisson equation for the electrostatic potential, is deduced from the Wigner equation using a mass conservation BGK collision operator and diffusive scaling. Performing the asymptotic limit for the Planck constant results in a simpler semiclassical drift-diffusion model with the Poisson equation used in semiconductor device modeling. (II) Viscous quantum hydrodynamic equations are derived from the Wigner-Fokker-Planck equation. The hydrodynamic equations consist of equations for the energy densities, current and particle density, and are coupled with the Poisson equation. They are compared with the quantum hydrodynamic equations derived from the Schrödinger equation and from the Wigner equation. The quantum hydrodynamic model can be reduced to the Euler equations in the semiclassical limit. (III) Applying the Chapman-Enskog expansion to the Wigner-BGK model results in the quantum Navier-Stokes equations. It is shown that the quantum Navier-Stokes equations are consistent with the viscous quantum hydrodynamic models introducing a so-called osmotic velocity.
Additionally, analytical properties of the equations, such as existence, long-time behavior of the solutions, stationary equations, transient equations and asymptotic limits, are exhibited
The results are frequently presented in form of theorems and proofs containing the main ideas. For details, it is often referred to a comprehensive bibliography.
The presented approach to derive quantum fluid models is alternative to Madelung equations which describe only the ballistic transport.