Modélisation, Simulation et Analyse Mathématique des Plasmas Quantiques et Classiques
Soutenue le 29 novembre 2010 devant le jury composé de:
Christophe Besse...........Examinateur......Université de Lille 1
Franck Boyer.............Examinateur........Université Paul Cézanne
Laurent Desvillettes.......Rapporteur........ ENS Cachan
Thierry Gallouet..........Rapporteur.....Université de Provence
Florence Hubert........Examinateur......Université de Provence
Ansgar Juengel...........Rapporteur.......Université de Wien
Anne Nouri.................Examinateur.....Université de Provence
Jean-Michel Roquejoffre.......Examinateur.......Université Paul Sabatier Toulouse
Yanick Sarazin..............Examinateur.......CEA Cadarache
And here you can find a version of my HDR: HDR.PDF
Summary of the HDR
The present dissertation is concerned with the mathematical modeling and analysis as well as the numerical simulation of the evolution of a system of $N$ charged particles (plasma), submitted to an electric or electromagnetic field. These systems come from different types of application, as for example the two treated in this memorandum, the turbulent transport in fusion plasmas or the electron/spin transport in nanoscale semiconductor devices.
These two physical domains are very modern and challenging, involving currently a lot of research effort. On the one hand, the development of new sources of energy, able to satisfy the requirements of safety, low environmental impact and illimited availability of resources, is indispensable today. The controlled thermonuclear fusion, based on an efficient magnetic Al confinement of very hot plasmas, is one of the most promising concepts. Classical plasmas, characterized by low densities and/or high temperatures are the core of this physical field.
On the other hand, semiconductors are the most essential components in today's electronic industry. The continued miniaturization of semiconductor devices permits to increase the performances such as power, speed and robustness, leading thus to significant technological advances. Quantum effects start to play an important role at this scale. Thus, quantum plasmas, characterized by high densities and/or low temperatures are the essential matter of this field.
A plasma is much more than a ionized gaz. Collective effects play an important role and the underlying physics is completely different from that of a neutral gaz. The dynamics of a plasma is very complex, the principal reason being the nonlinear and self-consistent nature of the coupled ``field-charged particle'' system. Indeed, the motion of the charged particles generates currents and magnetic fields, which, on their turn, affect the motion of the charged particles. Substantial effort is currently done to find good physical, mathematical and numerical approximations, in order to solve this highly nonlinear problem, with purpose the better understanding of the plasma dynamics.
The principal encountered mathematical and numerical difficulties in this research domain are related to the multi-scale nature of the treated physical phenomena and/or the high frequency regimes of the unknowns. All along this dissertation mathematical and numerical techniques are developed to cope with these multi-scale phenomena, as for example asymptotic studies to obtain macroscopic (gyro-kinetic) models, WKB approximations to filter out the oscillations, development of Asymptotic Preserving schemes based on micro-macro decompositions, and so on.
Keywords: Magnetically confined fusion plasmas, Kinetic equations, Fluid equations, Nonlinear equations, Instabilities, Turbulences, Asymptotic Preserving schemes, Anisotropies, Quantum ballistic transport models, Schroedinger equation, WKB approximation, Coherence length, Decoherence, Asymptotical analysis, Splitting methods, High oscillating regimes, Multi-scale methods