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Problèmes mathématiques liés à la modélisation des solides à différentes échelles

Abstract : We investigate in this thesis various mathematical problems in solid-state modelling, at the atomic scale as well as the macroscopic one. The models we use here are highly simplified (compared to standard quantum models), but exhibit acceptable qualitative behaviours. In addition, their simplicity allows one to go further in their rigourous mathematical analysis.

The first part (chapters 2, 3, 4) is devoted to the study of the origin of crystalline symmetry. This problem may be stated as follows: do the models we study here account for the fact that at zero temperature, matter is ordered? or, equivalently, does the ground state of N identical atoms converge to some periodic structure as N goes to infinity? This kind of problem is related to the problem of thermodynamic limit, which is studied here in some particular cases.

In a second part, we study the case when matter does not show any order: if the system is periodic, the definition of its energy for the models we use here is possible through the thermodynamic limit process. We study it in the case when the system is not periodic, and give general hypotheses allowing to carry out the corresponding study.

In chapters 6 and 7, we study the possible link between these microscopic models and macroscopic solid-state theories, for mechanical features in the first case, and for electrostatic ones in the second. Finally, the last chapter presents numerical technics in use in the modelling of electronic structure in solids.
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Contributor : Xavier Blanc Connect in order to contact the contributor
Submitted on : Wednesday, February 27, 2002 - 4:42:16 PM
Last modification on : Tuesday, April 24, 2018 - 1:36:15 PM
Long-term archiving on: : Tuesday, September 11, 2012 - 5:00:23 PM


  • HAL Id : tel-00001163, version 1



Xavier Blanc. Problèmes mathématiques liés à la modélisation des solides à différentes échelles. Mathématiques [math]. Ecole des Ponts ParisTech, 2001. Français. ⟨tel-00001163⟩



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