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Minimal Configurations for the Frenkel-Kontorova Model on a Quasicrystal

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Abstract

In this paper, we consider the Frenkel-Kontorova model of a one dimensional chain of atoms submitted to a potential. This potential splits into an interaction potential and a potential induced by an underlying substrate which is a quasicrystal. Under standard hypotheses, we show that every minimal configuration has a rotation number, that the rotation number varies continuously with the minimal configuration, and that every non negative real number is the rotation number of a minimal configuration. This generalizes well known results obtained by S. Aubry and P.Y. le Daeron in the case of a crystalline substrate.

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Authors and Affiliations

  1. Centro de Modelamiento Matemático, U.M.I. CNRS 2807, Universidad de Chile, Av. Blanco Encalada 2120, Santiago, Chile

    Jean-Marc Gambaudo

  2. Departamento de Ingeniería Matemática, Fac. Ciencias Físicas y Matemáticas, Universidad de Chile, Av. Blanco Encalada 2120 5to Piso, Santiago, Chile

    Pierre Guiraud

  3. Institut de Mathématiques de Bourgogne, U.M.R. CNRS 5584, Université de Bourgogne, U.F.R. des Sciences et Téchniques, B.P. 47870, 21078, Dijon Cedex, France

    Samuel Petite

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  1. Jean-Marc Gambaudo
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  2. Pierre Guiraud
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  3. Samuel Petite
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Correspondence to Jean-Marc Gambaudo.

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Communicated by G. Gallavotti

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Gambaudo, JM., Guiraud, P. & Petite, S. Minimal Configurations for the Frenkel-Kontorova Model on a Quasicrystal. Commun. Math. Phys. 265, 165–188 (2006). https://doi.org/10.1007/s00220-006-1531-x

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  • DOI: https://doi.org/10.1007/s00220-006-1531-x

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