Minimal Configurations for the Frenkel-Kontorova Model on a Quasicrystal

Gambaudo, Jean-Marc; Guiraud, Pierre; Petite, Samuel

doi:10.1007/s00220-006-1531-x

Cited by 14 publications

(29 citation statements)

References 11 publications

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“…Extensive research has been done in the last 30 years about the properties of ground states resulting from the minimization of potential H with F = 0, see [4,8,14,15,18,19,21,24] and the references therein. LetH…”

Section: Ground Statesmentioning

confidence: 99%

Existence and Modulation of Uniform Sliding States in Driven and Overdamped Particle Chains

Qin

2011

Commun. Math. Phys.

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In this paper we are mainly concerned with existence and modulation of uniform sliding states for particle chains with damping γ and external driving force F. If the on-site potential vanishes, then for each F > 0 there exist trivial uniform sliding states x n (t) = nω + νt + α for which the particles are uniformly spaced with spacing ω > 0, the sliding velocity of each particle is ν = F/γ , and the phase α is arbitrary. If the particle chain with convex interaction potential is placed in a periodic on-site potential, we show under some conditions the existence of modulated uniform sliding states of the formwhere the modulation function u is periodic and unique up to phase. The conditions are that the system is overdamped and the driving force F exceeds some critical value F d (ω) ≥ 0 depending on mean spacing ω. If F ∈ [0, F d (ω)], the system possesses a set of rotationally ordered equilibrium states for irrational ω, which can be described by a non-decreasing hull function, just as the case γ = F = 0, where Aubry-Mather theory applies to ground states. Meanwhile, we prove that F d (ω) = 0, which was argued physically much earlier, if the hull function of ground states with irrational rotation number ω for F = 0 is continuous.

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Section: Ground Statesmentioning

confidence: 99%

Existence and Modulation of Uniform Sliding States in Driven and Overdamped Particle Chains

Qin

2011

Commun. Math. Phys.

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“…Short-range potentials and X -equivariant functions have been studied in different contexts (see, e.g., [GGP06,Hof95,Kel03]). In terms of the physical model, the value of a short-range potential at a given point t depends on the neighborhood of t up to a given radius.…”

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confidence: 99%

“…By equation (7), the leaves coincide with the orbits of the action T and are 1-manifolds isometric to R (for more details, see, e.g., [BBG06,BG03,GGP06]). By equation (7), the leaves coincide with the orbits of the action T and are 1-manifolds isometric to R (for more details, see, e.g., [BBG06,BG03,GGP06]).…”

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confidence: 99%

Translation numbers for a class of maps on the dynamical systems arising from quasicrystals in the real line

Aliste-Prieto¹

2009

Ergod. Th. Dynam. Sys.

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In this paper, we study translation sets for non-decreasing maps of the real line with a pattern-equivariant displacement with respect to a quasicrystal. First, we establish a correspondence between these maps and self maps of the continuous hull associated with the quasicrystal that are homotopic to the identity and preserve orientation. Then, by using first-return times and induced maps, we provide a partial description for the translation set of the latter maps in the case where they have fixed points and obtain the existence of a unique translation number in the case where they do not have fixed points. Finally, we investigate the existence of a semiconjugacy from a fixed-point-free map homotopic to the identity on the hull to the translation given by its translation number. We concentrate on semiconjugacies that are also homotopic to the identity and, under a boundedness condition, we prove a generalization of Poincaré’s theorem, finding a sufficient condition for such a semiconjugacy to exist depending on the translation number of the given map.The author acknowledges support from a CONICYT doctoral fellowship and grants: ECOSCONICYT C03EC03, Nucleo Milenio P04-069-F, ANR Crystal Dyn and Basal-CMM

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“…[GGP06] shows that the quasi-periodic case can be considered as a dynamical system on a Cantor set ( Delone set).…”

Section: Introductionmentioning

confidence: 99%

“…In the mathematical literature, quasi-periodic Frenkel-Kontorova models have been considered in [GGP06,AP10], which use mainly topological methods to study the existence of orbits with rotation number. In the periodic case, the critical points of the energy, i.e.…”

Section: Introductionmentioning

confidence: 99%

KAM theory for quasi-periodic equilibria in 1D quasi-periodic media: II. Long-range interactions

Llave

2012

J. Phys. A: Math. Theor.

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Abstract. We consider Frenkel-Kontorova models corresponding to 1 dimensional quasicrystals.We present a KAM theory for quasi-periodic equilibria. The theorem presented has an a-posteriori format. We show that, given an approximate solution of the equilibrium equation, which satisfies some appropriate non-degeneracy conditions, then, there is a true solution nearby. This solution is locally unique.Such a-posteriori theorems can be used to validate numerical computations and also lead immediately to several consequences a) Existence to all orders of perturbative expansion and their convergence b) Bootstrap for regularity c) An efficient method to compute the breakdown of analyticity.Since the system does not admit an easy dynamical formulation, the method of proof is based on developing several identities. These identities also lead to very efficient algorithms.Quasi-periodic solutions, quasicrystals, hull functions,KAM theory [2000] 70K43, 52C23, 37A60, 37J40, 82B20

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Minimal Configurations for the Frenkel-Kontorova Model on a Quasicrystal

Cited by 14 publications

References 11 publications

Existence and Modulation of Uniform Sliding States in Driven and Overdamped Particle Chains

Existence and Modulation of Uniform Sliding States in Driven and Overdamped Particle Chains

Translation numbers for a class of maps on the dynamical systems arising from quasicrystals in the real line

KAM theory for quasi-periodic equilibria in 1D quasi-periodic media: II. Long-range interactions

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