Samy Gallego

An entropic quantum drift-diffusion model for electron transport in resonant tunneling diodes

We present an entropic quantum drift-diffusion model (eQDD) and show how it can be derived on a bounded domain as the diffusive approximation of the Quantum Liouville equation with a quantum BGK operator. Some links between this model and other existing models are exhibited, especially with the density gradient (DG) model and the Schrodinger-Poisson drift-diffusion model (SPDD). Then a finite difference scheme is proposed to discretize the eQDD model coupled to the Poisson equation and we show how this scheme can be slightly modified to discretize the other models. Numerical results show that the properties listed for the eQDD model are checked, as well as the model captures important features concerning the modeling of a resonant tunneling diode. To finish, some comparisons between the models stated above are realized.

Entropic Discretization of a Quantum Drift-Diffusion Model

This paper is devoted to the discretization and numerical simulation of a new quantum drift-diffusion model that was recently derived. In a first step, we introduce an implicit semi-discretization in time which possesses some interesting properties: this system is well-posed, it preserves the positivity of the density, the total charge is conserved, and it is entropic (a free energy is dissipated). Then, after a discretization of the space variable, we define a numerical scheme which has the same properties and is equivalent to a convex minimization problem. These results are illustrated by some numerical simulations.

Isothermal Quantum Hydrodynamics: Derivation, Asymptotic Analysis, and Simulation

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, th...

Quantum hydrodynamic and diffusion models derived from the entropy principle

In these notes, we review the recent theory of quantum hydrodynamic and diffusion models derived from the entropy minimization principle. These models are obtained by taking the moments of a collisional Wigner equation and closing the resulting system of equations by a quantum equilibrium. Such an equilibrium is defined as a minimizer of the quantum entropy subject to local constraints of given moments. We provide a framework to develop this minimization approach and successively apply it to quantum hydrodynamic models and quantum diffusion models. The results of numerical simulations show that these models capture well the various features of quantum transport.

Quantum Diffusion Models Derived from the Entropy Principle

In this chapter, we review the recent theory of quantum diffusion models derived from the entropy minimization principle. These models are obtained by taking the moments of a collisional Wigner equation and closing the resulting system of equations by a quantum equilibrium. Such an equilibrium is defined as a minimizer of the quantum entropy subject to local constraints of given moments. We provide a framework to develop this minimization approach. The results of numerical simulations show that these models capture well the various features of quantum transport.

On a New Isothermal Quantum Euler Model: Derivation, Asymptotic Analysis and Simulation

In the first part of this article, we derive a New Isothermal Quantum Euler model. Starting from the quantum Liouville equation, the system of moments is closed by a density operator which minimizes the quantum free energy. Several simplifications of the model are then 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 two situations: at the semiclassical limit and at the zero-temperature limit. The remarkable fact is that in each case we recover a known model: respectively the isothermal Euler system and the Madelung equations. Finally, we give in the third part some preliminary numerical simulations.