Isothermal quantum hydrodynamics: Derivation, asymptotic analysis, and simulation

International audience; This article is devoted to the reformulation of an isothermal version of the quantum hydrodynamic model derived by Degond and Ringhofer in [J. Statist. Phys., 112 (2003), pp. 587-628] (which will be referred to as the quantum Euler system). We write the model under a simpler (differential) form. The derivation is based on an appropriate use of commutators. Starting from the quantum Liouville equation, the system of moments is closed by a density operator which minimizes the quantum free energy. Some properties of the model are then exhibited, and most of them rely on a gauge invariance property of the system. Several simplifications of the model are also written for the special case of irrotational flows. The second part of the paper is devoted to a formal analysis of the asymptotic behavior of the quantum Euler system in three situations: at the semiclassical limit, at the zero-temperature limit, and at a diffusive limit. The remarkable fact is that in each case we recover a known model: respectively, the isothermal Euler system, the Madelung equations, and the entropic quantum drift-diffusion model. Finally, we give in the third part some preliminary numerical simulations.