A continuous family of equilibria in ferromagnetic media are ground states

TL;DR

This paper proves that a continuous family of equilibria in ferromagnetic media are ground states, extending the understanding of minimal energy configurations to models with long-range and many-body interactions.

Contribution

It establishes that foliations of equilibria in ferromagnetic models are ground states, generalizing previous results to discrete, long-range, and many-body systems.

Findings

Foliations of equilibria are ground states in ferromagnetic models.

The result applies to models with long-range and many-body interactions.

KAM theory solutions are shown to be minimizers in this context.

Abstract

We show that a foliation of equilibria (a continuous family of equilibria whose graph covers all the configuration space) in ferromagnetic models are ground states. The result we prove is very general, and it applies to models with long range interactions and many body. 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 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).