Generalized Frenkel-Kontorova Model Research Articles

Rotation numbers and bounded deviations for quasi-periodic monotone recurrence relations

It is known that monotone recurrence relations are defined by finding equilibria of generalized Frenkel-Kontorova models. In this paper, we intend to study quasi-periodic monotone recurrence relations. By introducing a countable set consisting of integers with bounded distances and improving a method of Angenent used for studying periodic monotone recurrence relations, we derive a criterion for the existence of solutions with bounded deviations from the rigid rotation. With the help of the criterion, a range of parameters is provided to guarantee the quasi-periodic standard map has orbits with bounded deviations from the rigid rotation corresponding to each given rotation number.

A generalized Frenkel–Kontorova model for a propagating austenite–martensite phase boundary: revisited numerically

We explain the propagating austenite–martensite phase boundary by a Frenkel–Kontorova model for a chain of meshes along a ledge of the phase transitions. We demonstrate such steps for example chains of 16 and 47 meshes. We can represent a Langevin solution which describes possible cases of a consecutive excitation of a higher phase under a low external force.Graphic abstract

Mountain pass solutions to a generalized Frenkel-Kontorova model

Mountain pass solutions to a generalized Frenkel-Kontorova model

Multitransition solutions for a generalized Frenkel-Kontorova model

We study a generalized Frenkel-Kontorova model. Using minimal and Birkhoff solutions as building blocks, we construct a lot of homoclinic solutions and heteroclinic solutions for this generalized Frenkel-Kontorova model under gap conditions. These new solutions are not minimal and Birkhoff any more. We use constrained minimization method to prove our results.

Heteroclinic solutions for a generalized Frenkel-Kontorova model by minimization methods of Rabinowitz and Stredulinsky

We study heteroclinic solutions of a generalized Frenkel-Kontorova model. Using the methods of Rabinowitz and Stredulinsky, we prove that if the rotation vector of the configuration is rational and if there is an adjacent pair of periodic configurations, then there is a solution that is heteroclinic in one fixed direction and periodic in other directions. Furthermore, if the above heteroclinic solutions have an adjacent pair, then there is a solution that is heteroclinic in two directions and periodic in other directions. The procedure can be repeated to produce more complex solutions. Thus we obtain a variational construction for these minimal and Birkhoff solutions.

Equilibrium Configurations for Generalized Frenkel–Kontorova Models on Quasicrystals

I study classes of generalized Frenkel–Kontorova models whose potentials are given by almost-periodic functions which are closely related to aperiodic Delone sets of finite local complexity. Since such Delone sets serve as models for quasicrystals, this setup presents Frenkel–Kontorova models for the type of aperiodic crystals which have been discovered since Shechtman’s discovery of quasicrystals. Here I consider models with configurations $$u:{\mathbb {Z}}^r \rightarrow {\mathbb {R}}^d$$ , where d is the dimension of the quasicrystal, for any r and d. The almost-periodic functions used for potentials are called pattern-equivariant and I show that if the interactions of the model satisfies a mild $$C^2$$ requirement, and if the potential satisfies a mild non-degeneracy assumption, then there exist equilibrium configurations of any prescribed rotation rotation number/vector/plane. The assumptions are general enough to satisfy the classical Frenkel–Kontorova models and its multidimensional analoges. The proof uses the idea of the anti-integrable limit.

Buckling induced by a topological defect in a chain

The chain buckling induced by a topological defect (anti-kink) is investigated using a generalized Frenkel–Kontorova model. Instead of a single-peak appeared in the classical Euler instability, the buckling shape induced by the topological defect is a Sine-like pattern (two-peaks) that is antisymmetric regarding the point of the maximum stress. Evolution of the unstable buckling mode is clearly characterized by two time scales: one is the lifetime of the unstable phase, the other is the growth rate of the new phase. The phase transition curve of the buckling is a linear function of the chain length.

Single-file mobility of water-like fluid in a generalized Frenkel-Kontorova model.

In this work, we used a generalized Frenkel-Kontorova model to study the mobility of water molecules inside carbon nanotubes with small radius at low temperatures. Our simulations show that the mobility of confined water decreases monotonically increasing the amplitude of the substrate potential at fixed commensurations. On the other hand, the mobility of the water molecules shows a non-monotonic behavior when varying the commensuration. This result indicates that the mobility of the confined fluid presents different behavior regimes depending on the amplitude of the water-nanotube interaction. In order to qualitatively understand these results, we study analytically the driven Frenkel-Kontorova model at finite temperatures. This analysis allows us to obtain the curves of the mobility versus commensurations, at fixed substrate potentials. Such curves show the existence of three regimes of mobility behavior as a function of the commensuration ratio. Additionally, our study indicates a nontrivial and strong dependence of the mobility with a quantity that can be interpreted as an effective amplitude of the substrate potential, depending on the bare amplitude of the substrate potential, the commensuration ratio, and temperature.

Secondary invariants of Birkhoff minimizers and heteroclinic orbits

For the generalized Frenkel–Kontorova model on high-dimensional lattices, we study the classification of Birkhoff minimizers with the same rotation vector which is rationally dependent. We introduce secondary invariants to distinguish Birkhoff minimizers with the same rotation vector and show that the collection of Birkhoff minimizers with the same secondary invariants is totally ordered. We also construct heteroclinic connections between neighboring Birkhoff minimizers which are more periodic.

Many Denjoy minimal sets for monotone recurrence relations

We extend Mather's work (1985 Comment. Math. Helv. 60 508–57) to high-dimensional cylinder maps defined by monotone recurrence relations, e.g. the generalized Frenkel–Kontorova model with finite range interactions. We construct uncountably many Denjoy minimal sets provided that the Birkhoff minimizers with some irrational rotation number ω do not form a foliation.

Lubricated friction in Frenkel–Kontorova model between incommensurate surfaces

We study the superlubricity in the generalized Frenkel–Kontorova model. Each particle of the top layer is driven by external dc driving force Fext. When the ratios of the three inherent length scales of the system are chosen to be the Golden Mean (a/b=233/144,c/a=144/89), we find that there exists a critical interparticle interaction strength above which the static friction force Fsu of the top layer is zero. And Fsu could easily flow into the low friction with existing the lubricant layer.

Monotone gradient dynamics and the location of stationary (p,q)-configurations

By exploiting the monotone property of the gradient dynamics of the Frenkel–Kontorova model, we are able to locate the ordered and unordered stationary states in the space of (p, q)-configurations, as well as forbidden regions for such states. Moreover, we will show that some generalized Frenkel–Kontorova models (which are associated with multiharmonic standard maps) can have ordered (p, q)-configurations that are neither action minimizing nor mini-maximizing, and we give their location with respect to the set of (p, q)-minimizers and mini-maximizers.

The Aubry-Mather theorem for driven generalized elastic chains

We consider uniformly (DC) or periodically (AC) driven generalizedinfinite elastic chains (a generalized Frenkel-Kontorova model) withgradient dynamics. We first show that the union of supports of allspace-time invariant measures, denoted by $\mathcal{A}$, projectsinjectively to a dynamical system on a 2-dimensional cylinder. We also proveexistence of space-time ergodic measures supported on a set of rotationalyordered configurations with an arbitrary (rational or irrational) rotationnumber. This shows that the Aubry-Mather structure of ground states persistsif an arbitrary AC or DC force is applied. The set $\mathcal{A}$ attractsalmost surely (in probability) configurations with bounded spacing. In theDC case, $\mathcal{A}$ consists entirely of equilibria and uniformly slidingsolutions. The key tool is a new weak Lyapunov function on the space oftranslationally invariant probability measures on the state space, whichcounts intersections.

The energy flow of discrete extended gradient systems

We study the energy flow of spatially discrete, extended gradient systems (infinite lattices), allowing the total energy to be infinite and considering formally gradient dynamics. We show that in spatial dimensions 1,2, the flow is for almost all times arbitrarily close to the set of equilibria, and in dimensions ⩾3, the size of the set with non-equilibrium dynamics for a positive density of times is two dimensions less than the space dimension. The theory applies to first- and second-order dynamics of elastic chains in a periodic or polynomial potential, chains with interactions beyond the nearest neighbour, deterministic dynamics of spin glasses, the discrete complex Ginzburg–Landau equation, and others. In particular, we apply the theory to show the existence of coarsening dynamics for a class of generalized Frenkel–Kontorova models in bistable potential.

Total destruction of invariant tori for the generalized Frenkel–Kontorova model

We consider generalized Frenkel–Kontorova models on higher dimensional lattices. We show that the invariant tori which are parameterized by continuous hull functions can be destroyed by small perturbations in the Cr topology with r < 1.

Coverage-Dependent Collective Diffusion of a Dense Pb Wetting Layer on Si(111)

The dynamics of a dense Pb wetting layer on the Si(111) surface is studied in the framework of a generalized Frenkel-Kontorova model. Instead of the typical diffusion by random hopping processes, a liquidlike collective motion of the Pb atoms within the dense wetting layer is revealed to give rise to ultrafast kinetics of the wetting layer even at low temperatures. A kinetic Monte Carlo simulation including this collective spreading mechanism of the dense wetting layer quantitatively reproduces the experimental observations.

Ground-State Transition in a Two-Dimensional Frenkel—Kontorova Model

The ground state of a generalized Frenkel—Kontorova model with a transversal degree of freedom is studied. When the coupling strength, K, and the frequency of a single-atom vibration in the transversal direction, ω0y, are increased, the ground state of the model undergoes a transition from a two-dimensional configuration to a one-dimensional one. This transition can manifest in different ways. Furthermore, we find that the prerequisite of a two-dimensional ground state is θ ≠ 1/q.

Goal-oriented adaptive mesh refinement for the quasicontinuum approximation of a Frenkel-Kontorova model

The quasicontinuum approximation [E.B. Tadmor, M. Ortiz, R. Phillips, Quasicontinuum analysis of defects in solids, Philos. Mag. A 73(6) (1996) 1529–1563] is a method to reduce the atomistic degrees of freedom of a crystalline solid by piecewise linear interpolation from representative atoms that are nodes for a finite element triangulation. In regions of the crystal with a highly nonuniform deformation such as around defects, every atom must be a representative atom to obtain sufficient accuracy, but the mesh can be coarsened away from such regions to remove atomistic degrees of freedom while retaining sufficient accuracy. We present an error estimator and a related adaptive mesh refinement algorithm for the quasicontinuum approximation of a generalized Frenkel-Kontorova model that enables a quantity of interest to be efficiently computed to a predetermined accuracy.

Ground states and critical points for generalized Frenkel–Kontorova models in

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.

Plastic depinning in artificial vortex channels: Competition between bulk and boundary nucleation

We study the depinning transition of a driven chain-like system in the presence of frustration and quenched disorder. The analysis is motivated by recent transport experiments on artificial vortex-flow channels in superconducting thin films. We start with a London description of the vortices and then map the problem onto a generalized Frenkel-Kontorova model and its continuous equivalent, the sine-Gordon model. In the absence of disorder, frustration reduces the depinning threshold in the commensurate phase, which nearly vanishes in the incommensurate regime. Depinning of the driven frustrated chain occurs via unstable configurations that are localized at the boundaries of the sample and evolve into topological defects which move freely through the entire sample. In the presence of disorder, topological defects can also be generated in the bulk. Further, disorder leads to pinning of topological defects. We find that weak disorder effectively reduces the depinning threshold in the commensurate phase, but increases the threshold in the incommensurate phase.