Ground States and Critical Points for Aubry–Mather Theory in Statistical Mechanics
TL;DR: A generalization of the celebrated criterion of existence of invariant circles if and only iff the Peierls–Nabarro barrier vanishes is provided.
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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01 Apr 1995
418 citations
TL;DR: In this paper, an Aubry-Mather-type theory for solutions of non-linear, possibly degenerate, elliptic PDEs and other pseudo-differential operators is discussed.
Abstract: We discuss an Aubry–Mather-type theory for solutions of non-linear, possibly degenerate, elliptic PDEs and other pseudo-differential operators. We show that for certain PDEs and ΨDEs with periodic coefficients and a variational structure it is possible to find quasi-periodic solutions for all frequencies. This results also hold under a generalized definition of periodicity that makes it possible to consider problems in covers of several manifolds, including manifolds with non-commutative fundamental groups. An abstract result will be provided, from which an Aubry–Mather-type theory for concrete models will be derived.
53 citations
Cites methods from "Ground States and Critical Points f..."
...See, however, [5,15,69,83,16,9,28,27] and [65] for related results....
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...Extensions of this method to difference equations on lattices and on graphs were considered in [44,17,28,26,25]....
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TL;DR: In this article, the authors consider a multidimensional model of the Frenkel-Kontorova type with non-nearest-neighbour interactions and show that there are quasiperiodic ground states which enjoy further geometric properties.
Abstract: We consider a multidimensional model of the Frenkel–Kontorova type but we allow non-nearest-neighbour interactions, which satisfy some weak version of ferromagnetism.For every possible frequency vector, we show that there are quasiperiodic ground states which enjoy further geometric properties. The ground states we produce are either bigger or smaller than their integer translates. They are at a bounded distance from the plane wave with the given frequency.The comparison property above implies that the ground states and the translations are organized into laminations. If these leave a gap, we show that there are critical points inside the gap which also satisfy the comparison properties.In particular, given any frequency, we show that either there is a continuous parameter of ground states or there is a ground state and another critical point which is not a ground state. This is a higher dimensional analogue of the criterion of the non-existence of invariant circles if and only if there is a positive Peierls–Nabarro barrier.All the above results are higher dimensional extensions of similar results in Aubry–Mather theory.
32 citations
TL;DR: In this paper, it was shown that every Aubry-mather set can be interpolated by a continuous gradient-flow invariant family, the so-called ghost circle.
29 citations
TL;DR: In this paper, the authors consider models of interacting particles situated in the points of a discrete set, where the state of each particle is determined by a real variable, and the particles are interacting with each other and are interested in ground states and other critical points of the energy.
Abstract: We consider models of interacting particles situated in the points of a discrete set Λ. The state of each particle is determined by a real variable. The particles are interacting with each other and we are interested in ground states and other critical points of the energy (metastable states).
25 citations
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Book•
01 Jan 1941
TL;DR: In this paper, the authors present a theory for linear PDEs: Sobolev spaces Second-order elliptic equations Linear evolution equations, Hamilton-Jacobi equations and systems of conservation laws.
Abstract: Introduction Part I: Representation formulas for solutions: Four important linear partial differential equations Nonlinear first-order PDE Other ways to represent solutions Part II: Theory for linear partial differential equations: Sobolev spaces Second-order elliptic equations Linear evolution equations Part III: Theory for nonlinear partial differential equations: The calculus of variations Nonvariational techniques Hamilton-Jacobi equations Systems of conservation laws Appendices Bibliography Index.
25,734 citations
4,135 citations
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01 Jan 1995TL;DR: In this article, Katok and Mendoza introduced the concept of asymptotic invariants for low-dimensional dynamical systems and their application in local hyperbolic theory.
Abstract: Part I. Examples and Fundamental Concepts Introduction 1. First examples 2. Equivalence, classification, and invariants 3. Principle classes of asymptotic invariants 4. Statistical behavior of the orbits and introduction to ergodic theory 5. Smooth invariant measures and more examples Part II. Local Analysis and Orbit Growth 6. Local hyperbolic theory and its applications 7. Transversality and genericity 8. Orbit growth arising from topology 9. Variational aspects of dynamics Part III. Low-Dimensional Phenomena 10. Introduction: What is low dimensional dynamics 11. Homeomorphisms of the circle 12. Circle diffeomorphisms 13. Twist maps 14. Flows on surfaces and related dynamical systems 15. Continuous maps of the interval 16. Smooth maps of the interval Part IV. Hyperbolic Dynamical Systems 17. Survey of examples 18. Topological properties of hyperbolic sets 19. Metric structure of hyperbolic sets 20. Equilibrium states and smooth invariant measures Part V. Sopplement and Appendix 21. Dynamical systems with nonuniformly hyperbolic behavior Anatole Katok and Leonardo Mendoza.
3,962 citations