The Frenkel-Kontorova Model

Braun, O. M.; Kivshar, Yuri S.

doi:10.1007/978-3-662-10331-9

Cited by 657 publications

(410 citation statements)

References 4 publications

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“…Experimental observation of this kind of superlubric and anisotropic regime of motion has recently been reported [3,4]. The remarkable conclusion of frictionless sliding can be drawn, in particular, in the context of the Frenkel-Kontorova (FK) model (see [5] and references therein). Since however the physical contact between two solids is generally mediated by so-called "third bodies", the role of incommensurability has been recently extended [6] in the framework of a driven 1D confined model inspired by the tribological problem of two sliding interfaces with a thin solid lubricant layer in between.…”

Section: Introductionmentioning

confidence: 89%

Hysteresis from dynamically pinned sliding states

et al. 2007

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We report a surprising hysteretic behavior in the dynamics of a simple one-dimensional nonlinear model inspired by the tribological problem of two sliding surfaces with a thin solid lubricant layer in between. In particular, we consider the frictional dynamics of a harmonic chain confined between two rigid incommensurate substrates which slide with a fixed relative velocity. This system was previously found, by explicit solution of the equations of motion, to possess plateaus in parameter space exhibiting a remarkable quantization of the chain center-of-mass velocity (dynamic pinning) solely determined by the interface incommensurability. Starting now from this quantized sliding state, in the underdamped regime of motion and in analogy to what ordinarily happens for static friction, the dynamics exhibits a large hysteresis under the action of an additional external driving force F ext . A critical threshold value F c of the adiabatically applied force F ext is required in order to alter the robust dynamics of the plateau attractor. When the applied force is decreased and removed, the system can jump to intermediate sliding regimes (a sort of "dynamic" stick-slip motion) and eventually returns to the quantized sliding state at a much lower value of F ext . On the contrary no hysteretic behavior is observed as a function of the external driving velocity.

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Section: Introductionmentioning

confidence: 89%

Hysteresis from dynamically pinned sliding states

et al. 2007

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“…We can understand this phenomenology in terms of kinks (i.e., local compressions of the chain with substrate potential minima holding more than just one particle [11]), as described in [1]. Assume initially integer q.…”

Section: Results and Theorymentioning

confidence: 99%

Kink plateau dynamics in finite-size lubricant chains

et al. 2007

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We extend the study of velocity quantization phenomena recently found in the classical motion of an idealized 1D model solid lubricant -consisting of a harmonic chain interposed between two periodic sliding potentials [Phys. Rev. Lett. 97, 056101 (2006)]. This quantization is due to one slider rigidly dragging the commensurate lattice of kinks that the chain forms with the other slider. In this follow-up work we consider finite-size chains rather than infinite chains. The finite size (i) permits the development of robust velocity plateaus as a function of the lubricant stiffness, and (ii) allows an overall chain-length re-adjustment which spontaneously promotes single-particle periodic oscillations. These periodic oscillations replace the quasiperiodic motion produced by general incommensurate periods of the sliders and the lubricant in the infinite-size model. Possible consequences of these results for some real systems are discussed.

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“…[1][2][3] As a discrete model, the classical FK model is non-integrable and has been used as a generic tool to study many nonlinear effects such as chaos, kinks and breathers since 1970's. 4) Besides nonlinearity, another important feature of this model is the competition between two length scales: one is the average distance between the neighboring particles and the other is the length of the spacial period of the external potential. This competition can lead to quite interesting phenomena.…”

Section: Introductionmentioning

confidence: 99%

“…For K > K c , the particles are pinned and for K < K c , they are depinned and can slide along the external potential. 4) The above scenario will be changed once we go to the quantum regime. Intuitively, it should be expected that, for a classically pinned state, if the quantum fluctuation is high enough, it will get depinned and become a sliding one.…”

Section: Introductionmentioning

confidence: 99%

A Density-Matrix Renormalization Group Study of a One-Dimensional Incommensurate Quantum Frenkel–Kontorova Model

Wang

et al. 2014

J. Phys. Soc. Jpn.

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In this paper, the one-dimensional incommensurate quantum Frenkel-Kontorova model is investigate by a density-matrix renormalization group algorithm. Special attention is given to the entanglement and the ground state energy. The energy gap between the ground state and the first excited is also calculated. From all the numerical results, we have observed an obvious property changes from the pinned state to the sliding one as the quantum fluctuation is increased. But no expected quantum critical point can be justified by the present data.

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The Frenkel-Kontorova Model

Cited by 657 publications

References 4 publications

Hysteresis from dynamically pinned sliding states

Hysteresis from dynamically pinned sliding states

Kink plateau dynamics in finite-size lubricant chains

A Density-Matrix Renormalization Group Study of a One-Dimensional Incommensurate Quantum Frenkel–Kontorova Model

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