A path integral approach to effective non-linear medium

In this article, we propose a new method to compute the effective properties of non-linear disordered media. We use the fact that the effective constants can be defined through the minimum of an energy functional. We express this minimum in terms of a path integral allowing us to use many-body techniques. We obtain the perturbation expansion of the effective constants to second order in disorder, for any kind of non-linearity. We apply our method to the case of strong non-linearities (i.e. D = epsilon(E-2)E-kappa/2, where epsilon is fluctuating from point to point), and to the case of weak non-linearity (i.e. D = epsilon E + chi(E-2)E where epsilon and chi fluctuate from point to point). Our results are in agreement with previous ones: and could be easily extended to other types of non-linear problems in disordered systems.