Ground States and Critical Points for Aubry–Mather Theory in Statistical Mechanics
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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.
Mathematics Subject Classification (2000)82C20 37J45 82C22 37A60
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- Bangert, V.: Mather sets for twist maps and geodesics on tori. In: Dynamics Reported. Dynam. Report. Ser. Dynam. Systems Appl., vol. 1, pp. 1–56. Wiley, Chichester (1988) Google Scholar
- Braun, O.M., Kivshar, Y.S.: Concepts, methods, and applications. In: The Frenkel–Kontorova Model. Texts and Monographs in Physics. Springer, Berlin (2004) Google Scholar
- Golé, C.: Global variational techniques. In: Symplectic Twist Maps. Advanced Series in Nonlinear Dynamics, vol. 18. World Scientific, River Edge (2001) Google Scholar
- Ruelle, D.: Rigorous results. In: Statistical Mechanics. World Scientific, River Edge (1999). Reprint of the 1989 edition Google Scholar
- Serre, J.-P.: Arbres, Malgames, SL 2. Société Mathématique de France, Paris (1977). Avec un sommaire anglais, Rédigé avec la collaboration de Hyman Bass, Astérisque, No. 46 Google Scholar
- Valdinoci, E.: Plane-like minimizers in periodic media: jet flows and Ginzburg–Landau. Ph.D. thesis, University of Texas at Austin (2001). MP_ARC # 01-356 Google Scholar
- Vallet, F.: Thermodynamique unidimensionelle, et structures bidimensionelles de quelques modèles pour des systèmes incommensurables. Ph.D. thesis, Université Pierre-et-Marie-Curie Paris VI (1986) Google Scholar