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Duality Relations for Non-Ohmic Composites, with Applications to Behavior near Percolation

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Abstract

Keller, Dykhne, and others have exploited duality to derive exact results for the effective behavior of two-dimensional Ohmic composites. This paper addresses similar issues in the non-Ohmic context. We focus primarily on three different types of nonlinearity: (a) the weakly nonlinear regime; (b) power-law behavior; and (c) dielectric breakdown. We first make the consequences of duality explicit in each setting. Then we draw conclusions concerning the critical exponents and scaling functions of “dual pairs” of random non-Ohmic composites near a percolation threshold. These results generalize, unify, and simplify relations previously derived for nonlinear resistor networks. We also discuss some self-dual nonlinear composites. Our treatment is elementary and self-contained; however, we also link it with the more abstract mathematical discussions of duality by Jikov and Kozlov.

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  1. Ohad Levy

    Present address: Levich Institute, CCNY–CUNY, New York, New York, 10031

Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, New York, New York, 10012

    Ohad Levy & Robert V. Kohn

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  1. Ohad Levy
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  2. Robert V. Kohn
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Levy, O., Kohn, R.V. Duality Relations for Non-Ohmic Composites, with Applications to Behavior near Percolation. Journal of Statistical Physics 90, 159–189 (1998). https://doi.org/10.1023/A:1023251701546

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