DUALITY RELATIONS FOR NONOHMIC COMPOSITES, WITH APPLICATIONS TO BEHAVIOR NEAR PERCOLATION

Keller, Dykhne, and others have exploited duality to derive exact resu lts for the effective behavior of two-dimensional Ohmic composites. Th is paper addresses similar issues in the non-Ohmic context. We focus p rimarily on three different types of nonlinearity: (a) the weakly nonl inear 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 fun ctions of ''dual pairs'' of random non-Ohmic composites near a percola tion threshold. These results generalize, unify, and simplify relation s previously derived for nonlinear resistor networks. We also discuss some self-dual nonlinear composites. Our treatment is elementary and s elf-contained; however, we also link it with the more abstract mathema tical discussions of duality by Jikov and Kozlov.