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Journal ArticleDOI

The Dynamics of a Disordered Linear Chain

15 Dec 1953-Physical Review (American Physical Society)-Vol. 92, Iss: 6, pp 1331-1338
TL;DR: In this paper, the distribution function of the frequencies of normal modes of vibration of a disordered chain of one-dimensional harmonic oscillators is calculated analytically, in the limit when the chain becomes infinitely long.
Abstract: By a disordered chain we mean a chain of one-dimensional harmonic oscillators, each coupled to its nearest neighbors by harmonic forces, the inertia of each oscillator and the strength of each coupling being a random variable with a known statistical distribution law. A method is presented for calculating exactly the distribution-function of the frequencies of normal modes of vibration of such a chain, in the limit when the chain becomes infinitely long. For some special examples, in which the distribution law of the oscillator parameters is assumed to be of exponential form, the frequency spectra are calculated analytically. The theory applies equally well to a chain of masses connected by elastic springs and making mechanical vibrations, or to an electrical transmission line composed of alternating inductances and capacitances with random characteristics.
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TL;DR: In this article, the authors studied the distribution of eigenvalues for two sets of random Hermitian matrices and one set of random unitary matrices in the energy spectra of disordered systems.
Abstract: In this paper we study the distribution of eigenvalues for two sets of random Hermitian matrices and one set of random unitary matrices. The statement of the problem as well as its method of investigation go back originally to the work of Dyson [i] and I. M. Lifsic [2], [3] on the energy spectra of disordered systems, although in their probability character our sets are more similar to sets studied by Wigner [4]. Since the approaches to the sets we consider are the same, we present in detail only the most typical case. The corresponding results for the other two cases are presented without proof in the last section of the paper. §1. Statement of the problem and survey of results We shall consider as acting in iV-dimensiona l unitary space ///v, a selfadjoint operator BN (re) of the form

2,594 citations

Journal ArticleDOI
TL;DR: In this paper, the physics of Anderson transition between localized and metallic phases in disordered systems is reviewed, including both metal-insulator transitions and quantum-Hall-type transitions between phases with localized states.
Abstract: The physics of Anderson transitions between localized and metallic phases in disordered systems is reviewed The term ``Anderson transition'' is understood in a broad sense, including both metal-insulator transitions and quantum-Hall-type transitions between phases with localized states The emphasis is put on recent developments, which include: multifractality of critical wave functions, criticality in the power-law random banded matrix model, symmetry classification of disordered electronic systems, mechanisms of criticality in quasi-one-dimensional and two-dimensional systems and survey of corresponding critical theories, network models, and random Dirac Hamiltonians Analytical approaches are complemented by advanced numerical simulations

1,505 citations


Cites background from "The Dynamics of a Disordered Linear..."

  • ...The single-channel model with chiralclass disorder has been studied, in its various incarnations, in a large number of works, starting form the pioneering paper by Dyson (1953)....

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  • ...The DOS shows at criticality the Dyson singularity (Dyson, 1953; McKenzie, 1996; Titov et al., 2001), ρ(E) ∼ 1/|E ln3E| ....

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Journal ArticleDOI
TL;DR: A review of the methods for determining the behavior of solids whose properties vary randomly at the microscopic level, with principal attention to systems having composition variation on a well-defined structure (random "alloys") can be found in this paper.
Abstract: We review the methods which have been developed over the past several years to determine the behavior of solids whose properties vary randomly at the microscopic level, with principal attention to systems having composition variation on a well-defined structure (random "alloys"). We begin with a survey of the various elementary excitations and put the dynamics of electrons, phonons, magnons, and excitons into one common descriptive Hamiltonian; we then review the use of double-time thermodynamic Green's functions to determine the experimental properties of systems. Next we discuss these aspects of the problem which derive from the statistical specification of the microscopic parameters; we examine what information can and cannot be obtained from averaged Green's functions. The central portion of the review concerns methods for calculating the averaged Green's function to successively better approximation, including various self-consistent methods, and higher-order cluster effects. The last part of the review presents a comparison of theory with the experimental results of a variety of properties---optical, electronic, magnetic, and neutron scattering. An epilogue calls attention to the similarity between these problems and those of other fields where random material heterogeneity has played an essential role.

1,213 citations

Journal ArticleDOI
TL;DR: A tentative theory is proposed to combine various features of the problem which have been revealed by some of the different approaches to the theory of noninteracting electrons in a static disordered lattice.

1,084 citations

Journal ArticleDOI
TL;DR: In this article, scaling theories and numerical simulations are used to describe diffusion processes on percolation systems and fractals, and different types of disordered systems exhibiting anomalous diffusion are presented.
Abstract: Diffusion in disordered systems does not follow the classical laws which describe transport in ordered crystalline media, and this leads to many anomalous physical properties. Since the application of percolation theory, the main advances in the understanding of these processes have come from fractal theory. Scaling theories and numerical simulations are important tools to describe diffusion processes (random walks: the 'ant in the labyrinth') on percolation systems and fractals. Different types of disordered systems exhibiting anomalous diffusion are presented (the incipient infinite percolation cluster, diffusion-limited aggregation clusters, lattice animals, and random combs), and scaling theories as well as numerical simulations of greater sophistication are described. Also, diffusion in the presence of singular distributions of transition rates is discussed and related to anomalous diffusion on disordered structures.

859 citations