Journal of Nonlinear Science

, Volume 20, Issue 2, pp 153–218 | Cite as

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

  • Rafael de la LlaveEmail author
  • Enrico Valdinoci


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.

Ground states Quasiperiodic equilibria Long-range interaction Peierls–Nabarro barrier 

Mathematics Subject Classification (2000)

82C20 37J45 82C22 37A60 


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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Texas at AustinAustinUSA
  2. 2.Dipartimento di MatematicaUniversità di Roma Tor VergataRomeItaly

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