Quantum Energy-Transport and Drift-Diffusion models

We show that Quantum Energy-Transport and Quantum Drift-Diffusion models can be derived through diffusion limits of a collisional Wigner equation. The collision operator relaxes to an equilibrium defined through the entropy minimization principle. Both models are shown to be entropic and exhibit fluxes which are related with the state variables through spatially non-local relations. Thanks to an $\hbar$ expansion of these models, $\hbar^2$ perturbations of the Classical Energy-Transport and Drift-Diffusion models are found. In the Drift- Diffusion case, the quantum correction is the Bohm potential and the model is still entropic. In the Energy-Transport case however, the quantum correction is a rather complex expression and the model cannot be proven entropic.