Twist map, the extended Frenkel-Kontorova model and the devil's staircase
Exact results obtained on the discrete Frenkel Kontorova (FK) model and its extensions during the past few years are reviewed. 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. 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 is described. Necessary conditions for the trajectories to represent metastable configurations, which can be chaotic, are given. The existence of a finite Peierl Nabarro barrier for elementary discommensurations is connected with a property of non-integrability of the twist map. The existence of KAM tori corresponds to undefectible incommensurate ground-states and a theorem is given which asserts that when the phenon 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, allows 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 to the integrable limit varies. 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.
- Research Organization:
- Los Alamos National Lab., NM (USA)
- DOE Contract Number:
- W-7405-ENG-36
- OSTI ID:
- 6773274
- Report Number(s):
- LA-UR-82-2351; CONF-8205121-2; ON: DE82021778
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
75 CONDENSED MATTER PHYSICS
SUPERCONDUCTIVITY AND SUPERFLUIDITY
ANALYTICAL SOLUTION
CRYSTAL DEFECTS
CRYSTAL MODELS
CRYSTAL STRUCTURE
ENERGY LEVELS
EXCITED STATES
FRENKEL DEFECTS
GROUND STATES
MAPPING
MATHEMATICAL MODELS
MATHEMATICS
METASTABLE STATES
PHONONS
POINT DEFECTS
QUASI PARTICLES
STOCHASTIC PROCESSES
TOPOLOGY
TWO-DIMENSIONAL CALCULATIONS
VACANCIES