Exact ground state of the Frenkel-Kontorova model with repeated parabolic potential. I. Basic results

Griffiths, Robert B.; Schellnhuber, Hans Joachim; Urbschat, H.

doi:10.1103/physrevb.56.8623

Cited by 4 publications

(3 citation statements)

References 28 publications

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“…Among the main results, one may cite the existence of a well-defined 'rotation number' for any minimizing configuration, the 'total ordering property' of the set of minimizing configurations with a fixed rotation number, and the existence of a unique 'hull function' up to a phase shift which parametrizes all minimizing configurations with a fixed rotation number. These main results have been obtained by several authors, first by Aubry and Le Daeron [3], Aubry et al [4], Griffiths et al [37], independently by Mather [48,51], more recently by Gomes and Oberman, Rorro and Falcone [20,33,34,53], Baesens and MacKay [6], and in a similar context of ergodic optimization by Bousch, Brémont, Jenkinson and Morris, Bressaud and Quas, Leplaideur [9,10,12,13,39,40,42]. The minimizing configurations previously described correspond to equilibrium fixed points of the steady-state Frenkel-Kontorova equation.…”

Section: Introductionsupporting

confidence: 54%

Minimizing orbits in the discrete Aubry–Mather model

Garibaldi

Thieullen

2011

Nonlinearity

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We consider a generalization of the Frenkel-Kontorova model in higher dimension leading to a new theory of configurations with minimal energy, as in Aubry's theory or in Mather's twist approach in the periodic case. We consider a one dimensional chain of particles and their minimizing configurations and we allow the state of each particle to possess many degrees of freedom. We assume that the Hamiltonian of the system satisfies some twist condition. The usual "total ordering" of minimizing configurations does not exist any more and new tools need to be developed. The main mathematical tool is to cast the study the minimizing configurations into the framework of discrete Lagrangian theory. We introduce forward and backward Lax-Oleinik problems and interpret their solutions as discrete viscosity solutions as in Hamilton-Jacobi methods. We give a fairly complete description of a particular class of minimizing configurations: the calibrated class. These configurations may be thought of as "ground states" obtained in the thermodynamic limit at temperature zero. We obtain, in particular, Mather's graph property or the non-crossing property of two calibrated configurations and the existence of a rotation number for most of the calibrated configurations.

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Section: Introductionsupporting

confidence: 54%

Minimizing orbits in the discrete Aubry–Mather model

Garibaldi

Thieullen

2011

Nonlinearity

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“…This model was first proposed by Griffiths et al 14 Several interesting new phenomena such as the nonrecurrent minimum energy ͑NRME͒ configuration in the incommensurate case, discontinuous cantorus-cantorus phase transitions ͑i.e., phase transitions in the gap structure͒, and independent orbits of gaps composing the complement of the CAM set ͑i.e., a gap structure with multiple discontinuity classes or holes 15 ͒ were found in the dϭ2 case. Recent work on this model 16,17 concentrated on acquiring ground state configurations through studying directional derivatives of the energy function, giving the average energy per atom, with respect to the elements in the phase parameter ͑defined later͒. However, as we shall see, for a given set of winding number and phase parameter, the depicted RO configurations may not be unique up to shift operations ͑defined later͒.…”

Section: ͑12͒mentioning

confidence: 99%

Exactly solved Frenkel-Kontorova model with multiple subwells

Lee

Tzeng

2002

Phys. Rev. B

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We exactly solve a class of Frenkel-Kontorova models with a periodic potential composed of piecewise convex parabolas having the same curvature. All rotationally ordered stable configurations can be depicted with appropriate phase parameters. The elements of a phase parameter are grouped into subcommensurate clusters. Phase transitions, manifested in the gap structure changes previously seen in numerical simulations, correspond to the splitting and merging of subcommensurate clusters with the appearance of incommensurate nonrecurrent rotationally ordered stable configurations. Through the notion of elementary phase shifts, all the possibilities for the existence of configurations degenerate with the ground state are scrutinized and the domains of stability in the phase diagram are characterized. At the boundaries of a domain of stability, nonrecurrent minimum energy configurations are degenerate with the ground state configurations and phase transitions occur.

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“…The description of FK models by an underlying one parameter continuum flow also constitutes a novel technique which should be applicable to the larger class of problems 2-4 mentioned above, as well as FK models with more complex external potentials 19,20 . The article is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

Frenkel-Kontorova models, pinned particle configurations, and Burgers shocks

Mungan

Yolcu

2010

Phys. Rev. B

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We analyze the relationship between the lowest energy configurations of an infinite harmonic chain of particles in a periodic potential and the evolution of characteristics in a periodically-forced inviscid Burgers equation. The shock discontinuities in the the Burgers evolution arise from thermodynamical considerations and play an important role as they separate out flows related to lowest energy configurations from those associated with higher energies. We study in detail the exactly solvable case of an external potential consisting of parabolic segments, and calculate analytically the lowest energy configurations, as well as excited states containing discommensurations.

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Exact ground state of the Frenkel-Kontorova model with repeated parabolic potential. I. Basic results

Cited by 4 publications

References 28 publications

Minimizing orbits in the discrete Aubry–Mather model

Minimizing orbits in the discrete Aubry–Mather model

Exactly solved Frenkel-Kontorova model with multiple subwells

Frenkel-Kontorova models, pinned particle configurations, and Burgers shocks

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