Comment on ‘Out-of-equilibrium Frenkel–Kontorova model’ (Imparato A 2021 J. Stat. Mech. 013214)

We explain the ‘phases’ of a Frenkel–Kontorova chain of atoms in a different way to the commented article. We reject the decision of states of the chain into commensurate and incommensurate states introduced by Aubry.

Nano-friction phenomenon of Frenkel-Kontorova model under Gaussian colored noise

The nano-friction phenomenon in a one-dimensional Frenkel-Kontorova model under Gaussian colored noise is investigated by using the molecular dynamic simulation method. The role of colored noise is analyzed through the inclusion of a stochastic force via a Langevin molecular dynamics method. Via the stochastic Runge-Kutta algorithm, the relationship between different parameter values of the Gaussian colored noise (the noise intensity and the correlation time) and the nano-friction phenomena such as hysteresis, the maximum static friction force is separately studied here. Similar results are obtained from the two geometrically opposed ideal cases: incommensurate and commensurate interfaces. It was found that the noise strongly influences the hysteresis and maximum static friction force and with an appropriate external driving force, the introduction of noise can accelerate the motion of the system, making the atoms escape from the substrate potential well more easily. Interestingly, suitable correlation time and noise intensity give rise to super-lubricity. It is noteworthy that the difference between the two circumstances lies in the fact that the effect of the noise is much stronger on triggering the motion of the FK model for the commensurate interface than that for the Incommensurate interface.

Description of zero field steps on the potential energy surface of a Frenkel-Kontorova model for annular Josephson junction arrays

We explain the emergence of zero field steps (ZFS) in a Frenkel-Kontorova (FK) model for a 1D annular chain being a model for an annular Josephson junction array. We demonstrate such steps for a case with a chain of 10 phase differences. We necessarily need the periodic boundary conditions. We propose a mechanism for the jump from M fluxons to $$M+1$$ M + 1 in the chain. Graphic abstract

Description of Shapiro steps on the potential energy surface of a Frenkel–Kontorova model, Part II: free boundaries of the chain

We explain Shapiro steps in a Frenkel–Kontorova (FK) model for a 1D chain of particles with free boundaries. The action of an external alternating force for the oscillating structure of the chain is important here. The different ’floors’ of the potential energy surface (PES) of this model play an important role. They are regions of kinks, double kinks, and so on. We will find out that the preferable movements are the sliding of kinks or antikinks through the chain. The more kinks / antikinks are included the higher is the ’floor’ through the PES. We find the Shapiro steps moving and oscillating anywhere between the floors. They start with a single jump over the highest SP in the global valley through the PES, like in part I of this series. They finish with complicated oscillations in the PES, for excitations directly over the critical depinning force. We use an FK model with free boundary conditions. In contrast to other results in the past, for this model, we obtain Shapiro steps in an unexpected, inverse sequence. We demonstrate Shapiro steps for a case with soft ’springs’ between an 8-particle FK chain. Graphic abstract

Description of Shapiro steps on the potential energy surface of a Frenkel–Kontorova model Part I: The chain in a variable box

We explain the vibrations of a Frenkel–Kontorova (FK) model under Shapiro steps by the action of an external alternating force. We demonstrate Shapiro steps for a case with soft ‘springs’ between an 8-particles FK chain. Shapiro steps start with a single jump over the highest $$\hbox {SP}_4$$ SP 4 in the global valley through the PES. They finish with doubled, and again doubled oscillations. We study in this part I a traditional FK model with periodic boundary conditions.

An example for nonexistence of minimal foliations

For the high-dimensional Frenkel–Kontorova model on lattices, we have concluded that there are heteroclinic connections between neighboring Birkhoff minimizers which are more periodic. This conclusion is based on the existence of neighboring elements, i.e., the existence of gaps. By adding a large enough oscillation to the local potential, I prove that all minimal foliations can be destroyed into minimal laminations, and hence there always exist gaps.