The Dynamics of a Disordered Linear Chain

Freeman J. Dyson
Phys. Rev. 92, 1331 – Published 15 December 1953
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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.

  • Received 17 August 1953

DOI:https://doi.org/10.1103/PhysRev.92.1331

©1953 American Physical Society

Authors & Affiliations

Freeman J. Dyson

  • Institute for Advanced Study, Princeton, New Jersey

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Issue

Vol. 92, Iss. 6 — December 1953

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