Scaling properties of a structure intermediate between quasiperiodic and random

Journal Article · Wed Jun 01 04:00:00 EDT 1988 · J. Stat. Phys.; (United States)

DOI:https://doi.org/10.1007/BF01014899· OSTI ID:6356825

Aubry, S; Godreche, C; Luck, J M

We consider a one-dimensional structure obtained by stringing two types of beads (short and long bonds) on a line according to a quasiperiodic rule. This model exhibits a new kind of order, intermediate between quasiperiodic and random, with a singular continuous Fourier transform (i.e., neither Dirac peaks nor a smooth structure factor). By means of an exact renormalization transformation acting on the two-parameter family of circle maps that defines the model, we study in quantitative way the locally scaling properties of its Fourier spectrum. We show that it exhibits power-law singularities around a dense set of wavevectors q, with a local exponent ..gamma..(q) varying continuously with the ratio of both bond lengths. Our construction also sheds some new light on the interplay between three characteristic properties of deterministic structures, namely: (1) a bounded fluctuation of the atomic positions with respect to their average lattice; (2) a quasiperiodic Fourier transform, i.e., made of Dirac peaks; and (3) for sequences generated by a substitution, the number-theoretic properties of the eigenvalue spectrum of the substitution.

• Research Organization: Univ. of California, Santa Barbara (USA)
• OSTI ID: 6356825
• Journal Information: J. Stat. Phys.; (United States), Journal Name: J. Stat. Phys.; (United States) Vol. 51:5-6; ISSN JSTPB
• Country of Publication: United States
• Language: English

Related Subjects

657002* -- Theoretical & Mathematical Physics-- Classical & Quantum Mechanics
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ATOMS
CONVERGENCE
CRYSTAL LATTICES
CRYSTAL MODELS
CRYSTAL STRUCTURE
DISTANCE
EIGENVALUES
FLUCTUATIONS
FOURIER TRANSFORMATION
FRACTALS
INTEGRAL TRANSFORMATIONS
INTERATOMIC DISTANCES
INVARIANCE PRINCIPLES
MAPPING
MATHEMATICAL MODELS
MATRICES
MECHANICS
ONE-DIMENSIONAL CALCULATIONS
ORDER PARAMETERS
RECURSION RELATIONS
RENORMALIZATION
SCALING LAWS
SINGULARITY
SPECTRA
STATISTICAL MECHANICS
TOPOLOGICAL MAPPING
TRANSFORMATIONS
VARIATIONS