Calibrated Configurations for Frenkel–Kontorova Type Models in Almost Periodic Environments

The Frenkel–Kontorova model describes how an infinite chain of atoms minimizes the total energy of the system when the energy takes into account the interaction of nearest neighbors as well as the interaction with an exterior environment. An almost periodic environment leads to consider a family of...

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Bibliographic Details
Published in:Annales Henri Poincaré 2017-09, Vol.18 (9), p.2905-2943
Main Authors: Garibaldi, Eduardo, Petite, Samuel, Thieullen, Philippe
Format: Article
Language:English
Subjects:
Calibration
Classical and Quantum Gravitation
Continuity (mathematics)
Dynamical Systems
Dynamical Systems and Ergodic Theory
Elementary Particles
Environment models
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Physics
Physics and Astronomy
Quantum Field Theory
Quantum Physics
Relativity Theory
Theoretical
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Summary:The Frenkel–Kontorova model describes how an infinite chain of atoms minimizes the total energy of the system when the energy takes into account the interaction of nearest neighbors as well as the interaction with an exterior environment. An almost periodic environment leads to consider a family of interaction energies which is stationary with respect to a minimal topological dynamical system. We focus, in this context, on the existence of calibrated configurations (a notion stronger than the standard minimizing condition). In any dimension and for any continuous superlinear interaction energies, we exhibit a set, called projected Mather set, formed of environments that admit calibrated configurations. In the one-dimensional setting, we then give sufficient conditions on the family of interaction energies that guarantee the existence of calibrated configurations for every environment. The main mathematical tools for this study are developed in the frameworks of discrete weak KAM theory, Aubry–Mather theory and spaces of Delone sets.
ISSN:1424-0637
1424-0661
DOI:10.1007/s00023-017-0589-7