The twist map, the extended Frenkel-Kontorova model and the devil's staircase

Abstract

This paper reviews exact results which we obtained on the discrete Frenkel-Kontorova (FK) model and its extensions, during the past few years. These models are associated with area preserving twist maps of the cylinder (or a part of it) onto itself. The theorems obtained for the FK model thus yields new theorems for the twist maps. We describe the exact structure of the ground-states which are either commensurate or incommensurate and assert the existence of elementary discommensurations under certain necessary and sufficient conditions. Necessary conditions for the trajectories to represent metastable configurations, which can be chaotic, are given. The existence of a finite Peierls-Nabarro barrier for elementary discommensurations is connected with a property of non-integrability of the twist map. We next prove that the existence of KAM tori corresponds to “undefectible” incommensurate ground-states and give a theorem which asserts that when the phonon spectrum of an incommensurate ground-state exhibits a finite gap, then the corresponding trajectory is dense on a Cantor set with zero measure length. These theorems, when applied to the initial FK model, allow one to prove the existence of the transition by “breaking of analyticity” for the incommensurate structures when the parameter which describes the discrepancy of the model from the integrable limit varies. These theorems also allow one to obtain a series of rigorous upper bounds for the stochasticity threshold of the standard map which for the fifth order approximation already approaches within 25% the value which is numerically known. Finally, we describe a theorem proving the existence of a devil's staircase for the variation curve of the atomic mean distance versus a chemical potential, for certain properties of the twist map which are generally satisfied.