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Symmetry breaking in the periodic Thomas–Fermi–Dirac–von Weizsäcker model

Ricaud, Julien (2017), Symmetry breaking in the periodic Thomas–Fermi–Dirac–von Weizsäcker model. https://basepub.dauphine.fr/handle/123456789/17981

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RICAUD_TFDW(1).pdf (769.4Kb)
Type
Document de travail / Working paper
External document link
https://hal.archives-ouvertes.fr/hal-01487352
Date
2017
Series title
Cahier de recherche CEREMADE, Université Paris-Dauphine
Pages
41
Metadata
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Author(s)
Ricaud, Julien cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
We consider the Thomas--Fermi--Dirac--von~Weizsäcker model for a system composed of infinitely many nuclei placed on a periodic lattice and electrons with a periodic density. We prove that if the Dirac constant is small enough, the electrons have the same periodicity as the nuclei. On the other hand if the Dirac constant is large enough, the 2-periodic electronic minimizer is not 1-periodic, hence symmetry breaking occurs. We analyze in detail the behavior of the electrons when the Dirac constant tends to infinity and show that the electrons all concentrate around exactly one of the 8 nuclei of the unit cell of size 2, which is the explanation of the breaking of symmetry. Zooming at this point, the electronic density solves an effective nonlinear Schr\"odinger equation in the whole space with nonlinearity u7/3−u4/3. Our results rely on the analysis of this nonlinear equation, in particular on the uniqueness and non-degeneracy of positive solutions.
Subjects / Keywords
Thomas--Fermi--Dirac--von~Weizsäcker model; nonlinear equation

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