Duality relations for non-Ohmic composites, with applications to behavior near percolation

Ohad Levy, Robert V. Kohn

Research output: Contribution to journalArticle

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 sealing 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.

Original languageEnglish (US)
Pages (from-to)159-189
Number of pages31
JournalJournal of Statistical Physics
Volume90
Issue number1-2
DOIs
StatePublished - Jan 1998

Keywords

  • Duality
  • Effective properties
  • Nonlinear Composites
  • Percolation

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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