Field distributions and effective-medium approximation for weakly nonlinear media

An effective-medium theory is proposed for random weakly nonlinear dielectric media. It is based on a Gaussian approximation for the probability distributions of the electric field in each component of a multiphase composite. These distributions are computed to linear order from a Bruggeman-like self-consistent formula. The resulting effective-medium formula for the nonlinear medium reduces to Bruggeman’s in the linear case. It is exact up to second order in a weak-disorder expansion, and close to the exact result in the dilute limit (in particular, it is exact for $d=1\mathrm{}$ and $d=\infty ).\mathrm{}$ In a high contrast situation, the noise exponents are $\kappa ={\kappa }^{\prime }=0\mathrm{}$ near the percolation threshold. Numerical results are provided for different weak nonlinearities. The use of the Bruggeman formula as a starting point for nonlinear homogeneization theories in dimensions $d>\mathrm{}2\mathrm{}$ is questioned on the basis of known exact bounds on the noise exponents.