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A Continuous Family of Equilibria in Ferromagnetic Media are Ground States

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Abstract

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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Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, No. 19, XinJieKouWai St., HaiDian District, Beijing, 100875, People’s Republic of China

    Xifeng Su

  2. School of Mathematics, Georgia Institute of Technology, 686 Cherry St., Atlanta, GA, 30332, USA

    Rafael de la Llave

  3. JLU-GT Joint Institute for Theoretical Science, Jilin University, Changchun, 130012, People’s Republic of China

    Rafael de la Llave

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  1. Xifeng Su
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  2. Rafael de la Llave
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Correspondence to Xifeng Su.

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Communicated by C. Liverani

Xifeng Su supported by National Natural Science Foundation of China (Grant No. 11301513) and “the Fundamental Research Funds for the Central Universities”. Rafael de la Llave has been partially supported by NSF Grant DMS-1500943. He also acknowledges the support of the Tang Aoqing visiting professorship in Jilin University and the hospitality of Beijing Normal University.

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Su, X., de la Llave, R. A Continuous Family of Equilibria in Ferromagnetic Media are Ground States. Commun. Math. Phys. 354, 459–475 (2017). https://doi.org/10.1007/s00220-017-2913-y

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