Exact calculation to second order of the effective dielectric constant of a strongly nonlinear inhomogeneous composite

We study strongly nonlinear and inhomogeneous dielectrics that follow a power-law relation between the electric and the displacement fields, D=ε‖E${\Vert }^{\beta }$E. Considering a narrow distribution of local dielectric constants ε(r), we find the first correction to the potential field. Using this result we present an exact calculation of the effective dielectric constant of an isotropic system to second order in the fluctuations δε(r). Our result is independent of details of the local geometry and represents the nonlinear analog of the exact calculation in linear dielectrics.