Nonlinear susceptibilities of granular matter

We discuss the nonlinear behavior of a random composite material in which current density and electric field are related by J=σE+a‖E${\Vert }^2$E, with σ and a position dependent. To first order in the nonlinear coefficient a, the effective nonlinear conductivity of the composite material is shown to be expressible as $a_e$=〈a‖E${\Vert }^4$〉/$E_0^4$, where $E_0$ is the magnitude of the applied field, the angular brackets denote a volume average, and E is the electric field in the linear limit (a=0). To the same order, the coefficient $a_e$ is also shown to be related to the mean-square conductivity fluctuation in an analogous problem in which the composite is linear but the conductivity fluctuates: The connection is λ$a_e$=V(δ${\sigma }_{\mathrm{rms}{)}^2}$, where V is the volume, δ${\sigma }_{\mathrm{rms}}$ is the rms conductivity fluctuation, and λ is a constant with dimensions of energy. In the low-concentration regime (p≪1, where p is the concentration of nonlinear material), an expression for $a_e$ is derived which is exact to first order in p. The ratio $a_e$/${\sigma }_e^2$ (where ${\sigma }_e$ is the conductivity of the composite) is shown to diverge near the percolation threshold for both a metal-insulator composite and a normal-metal–perfect-conductor composite; the power law characterizing the divergence is estimated. The results are generalized to nonlinear dielectric response at finite frequencies.