Communications in Mathematical Physics

, Volume 354, Issue 2, pp 459–475 | Cite as

A Continuous Family of Equilibria in Ferromagnetic Media are Ground States

  • Xifeng SuEmail author
  • Rafael de la Llave


We show that a foliation of equilibria (a continuous family of equilibria whose graph covers all the configuration space) in ferromagnetic transitive models are ground states. The result we prove is very general, and it applies to models with long range and many-body interactions. As an application, we consider several models of networks of interacting particles including models of Frenkel–Kontorova type on \({\mathbb{Z}^d}\) and one-dimensional quasi-periodic media. The result above is an analogue of several results in the calculus of variations (fields of extremals) and in PDE’s. Since the models we consider are discrete and long range, new proofs need to be given. We also note that the main hypothesis of our result (the existence of foliations of equilibria) is the conclusion (using KAM theory) of several recent papers. Hence, we obtain that the KAM solutions recently established are minimizers when the interaction is ferromagnetic and transitive (these concepts are defined later).


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.School of Mathematical SciencesBeijing Normal UniversityBeijingPeople’s Republic of China
  2. 2.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA
  3. 3.JLU-GT Joint Institute for Theoretical ScienceJilin UniversityChangchunPeople’s Republic of China

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