On the driven Frenkel-Kontorova model: I. Uniform sliding states and dynamical domains of different particle densities

TL;DR

This paper investigates the complex dynamical behavior of a harmonic chain in a periodic potential, revealing uniform and nonuniform sliding states, pattern formation, and proposing experimental observations with Josephson junctions.

Contribution

It introduces a detailed analysis of sliding states and domain patterns in the driven Frenkel-Kontorova model at zero temperature, including equations for density and velocity dynamics.

Findings

Identification of uniform and nonuniform sliding states.

Derivation of equations for particle density and velocity.

Proposal of experiments with Josephson junctions.

Abstract

The dynamical behavior of a harmonic chain in a spatially periodic potential (Frenkel-Kontorova model, discrete sine-Gordon equation) under the influence of an external force and a velocity proportional damping is investigated. We do this at zero temperature for long chains in a regime where inertia and damping as well as the nearest-neighbor interaction and the potential are of the same order. There are two types of regular sliding states: Uniform sliding states, which are periodic solutions where all particles perform the same motion shifted in time, and nonuniform sliding states, which are quasi-periodic solutions where the system forms patterns of domains of different uniform sliding states. We discuss the properties of this kind of pattern formation and derive equations of motion for the slowly varying average particle density and velocity. To observe these dynamical domains we suggest experiments with a discrete ring of at least fifty Josephson junctions.