Disordered One-Dimensional Crystals

Schmidt, Helmut

doi:10.1103/physrev.105.425

Cited by 294 publications

(117 citation statements)

References 6 publications

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“…24 The model of δ-impurities with random positions was introduced and studied by Schmidt [188] but often refered as the Frisch & Lloyd model [99]. Compared to the model for white noise potential, which is characterized by one parameter, the Frisch & Lloyd model is characterized by two parameters : the strength of the δ-potential, and their density.…”

Section: Supersymmetric Random Hamiltonianmentioning

confidence: 99%

Functionals of Brownian motion, localization and metric graphs

Comtet¹,

Desbois²,

Texier³

2005

J. Phys. A: Math. Gen.

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We review several results related to the problem of a quantum particle in a random environment.In an introductory part, we recall how several functionals of the Brownian motion arise in the study of electronic transport in weakly disordered metals (weak localization).Two aspects of the physics of the one-dimensional strong localization are reviewed : some properties of the scattering by a random potential (time delay distribution) and a study of the spectrum of a random potential on a bounded domain (the extreme value statistics of the eigenvalues).Then we mention several results concerning the diffusion on graphs, and more generally the spectral properties of the Schrödinger operator on graphs. The interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated.Finally, we consider a two-dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices. We recall the connection between spectral properties of this model and winding functionals of the planar Brownian motion.

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Section: Supersymmetric Random Hamiltonianmentioning

confidence: 99%

Functionals of Brownian motion, localization and metric graphs

Comtet¹,

Desbois²,

Texier³

2005

J. Phys. A: Math. Gen.

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“…In Sect. 2, following Schmidt [18], we show that smoothness of k in E is connected to smoothness in E of the invariant measure on PR(1) (the projective line) associated to the "transfer matrix" for h ω u = Eu. We will discuss the reason why the attempt to analyze this measure directly appears to fail, and forces us to convolutions on SL(2, R).…”

Section: Theorem 11 Iff Has Compact Support and Fel\ For Some α > 0mentioning

confidence: 99%

Harmonic analysis on SL(2,R) and smoothness of the density of states in the one-dimensional Anderson model

Simon

Taylor

1985

Commun.Math. Phys.

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“…Onedimensional systems provide a simple framework to study the basic effects of disorder without consideration for the geometric complexity of higher dimensions, thus excluding the scattering of signals to other directions. Analysis of the spectrum and density of eigenstates was the subject of many early studies in disordered one-dimensional systems [4,8,33,42]. In the context of quantum mechanical particles, Anderson [1] noted the localization of the wavefunctions in the presence of sufficiently strong random potentials.…”

Section: Introductionmentioning

confidence: 99%

Frequency filtering in disordered granular chains

Lawney¹,

Luding²

2014

Acta Mech

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The study of disorder-induced frequency filtering is presented for one-dimensional systems composed of random, pre-stressed masses interacting through both linear and nonlinear (Hertzian) repulsive forces. An ensemble of such systems is driven at a specified frequency, and the spectral content of the propagated disturbance is examined as a function of distance from the source. It is shown that the transmitted signal contains only low-frequency components, and the attenuation is dependent on the magnitude of disorder, the input frequency, and the contact model. It is found that increased disorder leads to a narrower bandwidth of transmitted frequencies at a given distance from the source and that lower input frequencies exhibit less sensitivity to the arrangement of the masses. Comparison of the nonlinear and linear contact models reveals qualitatively similar filtering behavior; however, it is observed that the nonlinear chain produces transmission spectrums with a greater density at the lowest frequencies. In addition, it is shown that random masses sampled from normal, uniform, and binary distributions produce quantitatively indistinguishable filtering behavior, suggesting that knowledge of only the distribution's first two moments is sufficient to characterize the bulk signal transmission behavior. Finally, we examine the wave number evolution of random chains constrained to move between fixed end-particles and present a transfer matrix theory in wave number space, and an argument for the observed filtering based on the spatial localization of the higher-frequency normal modes.

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Disordered One-Dimensional Crystals

Cited by 294 publications

References 6 publications

Functionals of Brownian motion, localization and metric graphs

Functionals of Brownian motion, localization and metric graphs

Harmonic analysis on SL(2,R) and smoothness of the density of states in the one-dimensional Anderson model

Frequency filtering in disordered granular chains

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