Physical Models
In this chapter we discuss several examples of the physical systems where the FK model plays an important role. We demonstrate how to introduce a simplified model accounting for the basic features of the system’s dynamics, and also mention why in many cas
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In this chapter we discuss several examples of the physical systems where the FK model plays an important role. We demonstrate how to introduce a simplified model accounting for the basic features of the system's dynamics, and also mention why in many cases the standard FK model should be generalized to take into account other important features, such as nonsinusoidal external potential, anharmonic coupling between the particles, thermal effects, higher dimension effects, etc. However, the main purpose of this chapter is not only to list examples of different physical systems that can be analyzed effectively with the help of the FK model but also provide an important guideline for deriving low-dimensional models in other applications of nonlinear physics.
2.1 General Approach The basic approach for deriving the FK model is rather simple. First of all, from an original (usually rather complicated) discrete nonlinear system one should extract a low-dimensional sub-system and describe the remaining part as a substrate by introducing an effective potential to account for its action. The elements of the effective one-dimensional discrete array play the role of effective atoms in the FK model. In many cases, such elements correspond to real atoms, although they may model clusters of atoms, as in the case of the DNA-like chains, may correspond to spins in magnetic chains, or may even describe some complex objects such as point-like Josephson junctions in an array. In the model, the effective atoms interact with their neighbors, and the simplest interaction is a linear coupling between the nearest neighbors. In the systems described by the FK-type models, the effective substrate should have a crystalline structure. In the simplest case, when the structure corresponds to the Bravais elementary cell, i.e. it has one atom per an unit cell, one may keep only the first term in the Fourier expansion of the periodic potential and obtain a sinusoidal substrate potential. In the framework of this approach, the standard FK model can describe, more or less rigorously, a realistic physical system only when the one-dimensional sub-system and substrate are of a different origin, e.g. the FK atoms are light particles, while the substrate potential is composed by O. M. Braun et al., The Frenkel-Kontorova Model © Springer-Verlag Berlin Heidelberg 2004
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2 Physical Models
heavy particles which can be treated as being "frozen" so that their motion can be neglected. As a matter of fact, this corresponds to many physical systems. For example, this is the case of surface physics where the model atoms are adsorbed atoms and the substrate corresponds to a crystal surface, or hydrogen-bonded chains where atoms are light hydrogen atoms while the substrate potential is created by heavy oxygen atoms. However, even in other cases, such as in the dislocation theory where the atoms and substrate are of the same origin, the FK model remains a very useful approximation, and it allows to describe many nontrivial phenomena of the system dynamics, ofte
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