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Modélisation Mathématique et Simulation Numérique de Systèmes Fluides Quantiques

Abstract : The PhD thesis is concerned with the study of a new class of quantum transport models: the quantum fluid models derived from the entropy principle. These models have been derived in two articles published in 2003 and 2005 by Degond, Méhats and Ringhofer in the Journal of Statistical Physics, by adapting to the quantum framework the moment method developed by Levermore in the classical framework. This method consists in taking the moments of the Quantum Liouville equation and closing this system by a local equilibrium (or quantum Maxwellian) defined as the minimizer of a quantum entropy with constraints on some physical quantities such as the mass, current, and energy. The main interest of such macroscopic models is their low cost in terms of numerical implementation compared to microscopic models such as the Schrödinger equation or the Wigner equation. Moreover, such models take implicitly into account collisions which are much more difficult to handle with quantum microscopic models. The goal of this thesis is thus to propose numerical methods to implement these models and to test them on some physical devices.
We have started in chapter I by proposing a discretization for the most simple of these models which is the Quantum Drift-Diffusion model on a closed domain. We have then decided in chapter II and III to apply this model to electron transport in semiconductors by choosing as open device the resonant tunneling diode. We have then studied in chapter IV the Isothermal Quantum Euler model, before considering in chapter V the study of non isothermal models such as the Quantum Hydrodynamic and the Quantum Energy Transport models. Finally, chapter VI is concerned with a slightly different problem which is the implementation of an asymptotically stable scheme in the semiclassical limit for the fluid formulation of the Schrödinger equation: the Madelung system.
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Contributor : Samy Gallego <>
Submitted on : Friday, January 25, 2008 - 6:28:07 PM
Last modification on : Friday, January 10, 2020 - 9:08:06 PM
Long-term archiving on: : Thursday, April 29, 2010 - 7:48:26 PM

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  • HAL Id : tel-00218256, version 1

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Samy Gallego. Modélisation Mathématique et Simulation Numérique de Systèmes Fluides Quantiques. Mathématiques [math]. Université Paul Sabatier - Toulouse III, 2007. Français. ⟨tel-00218256⟩

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