J. Phys. I France
Volume 3, Numéro 11, November 1993
Page(s) 2161 - 2170
DOI: 10.1051/jp1:1993238
J. Phys. I France 3 (1993) 2161-2170

Decay of long-ranlye field fluctuations induced by random structures: a unified spectral approach

Didier Sornette

Laboratoire de Physique de la Matière Condensée, Université de Nice-Sophia Antipolis, B.P. 71, Parc Valrose, 06108 Nice Cedex 2, Fiance

(Received 14 June 1993, revised 9 August 1993, accepted 23 August 1993)

The problem of the determination of the powerlaw decay of the standard deviation $\sigma^2 \sim z^{-\alpha}$ of the fluctuations of the field generated by a random array of elements (multipoles, ensemble of dislocations, etc.) as a function of the distance z from the array is reduced to the determination of two quantities: 1) the spectral power of the disorder in the low k limit and 2) the structure of the Green function, as a function of wavenumber and distance, for a periodic array of the constituting elements. We thus recover straightforwardly all results known previously and derive new ones for more general constitutive elements. The general expression of the decay exponent is found to be a $\alpha = 3 - \beta + 2(b - c)$, where $\beta$ characterizes the self-affine structure of the disorder ( $\beta = 2$ : strong disorder and $\beta = 0$ : weak disorder) and the exponents b and c are the exponents of the algebraic powerlaw corrections, in wavenumber and distance respectively, to the dominating exponential decay of the Green function for a periodic array of the constituting elements. The proposed spectral method solves automatically the generally difficult problem of renormalization and screening in arbitrary random structures.

© Les Editions de Physique 1993

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