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Ground States and Critical Points for Aubry–Mather Theory in Statistical Mechanics

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Abstract

We consider several models of networks of interacting particles and prove the existence of quasi-periodic equilibrium solutions. We assume (1) that the network and the interaction among particles are invariant under a group that satisfies some mild assumptions; (2) that the state of each particle is given by a real number; (3) that the interaction is invariant by adding an integer to the state of all the particles at the same time; (4) that the interaction is ferromagnetic and coercive (it favors local alignment and penalizes large local oscillations); and (5) some technical assumptions on the regularity speed of decay of the interaction with the distance. Note that the assumptions are mainly qualitative, so that they cover many of the models proposed in the literature. We conclude (A) that there are minimizing (ground states) quasi-periodic solutions of every frequency and that they satisfy several geometric properties; (B) if the minimizing solutions do not cover all possible values at a point, there is another equilibrium point which is not a ground state. These results generalize basic results of Aubry–Mather theory (take the network and the group to be ℤ). In particular, we provide with a generalization of the celebrated criterion of existence of invariant circles if and only iff the Peierls–Nabarro barrier vanishes.

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Authors and Affiliations

  1. Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, TX, 78712-0257, USA

    Rafael de la Llave

  2. Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, 00133, Rome, Italy

    Enrico Valdinoci

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  1. Rafael de la Llave
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  2. Enrico Valdinoci
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Correspondence to Rafael de la Llave.

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Communicated by A. Delshams.

The work of RdlL was supported by NSF grants. The work of EV was supported by GNAMPA Equazioni nonlineari su varietà: proprietà qualitative e classificazione delle soluzioni and MIUR Variational Methods and Nonlinear Differential Equations.

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de la Llave, R., Valdinoci, E. Ground States and Critical Points for Aubry–Mather Theory in Statistical Mechanics. J Nonlinear Sci 20, 153–218 (2010). https://doi.org/10.1007/s00332-009-9055-0

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