A Path Integral Approach to Effective Non-Linear Medium

TL;DR

This paper introduces a novel path integral method to compute the effective properties of non-linear disordered media, enabling perturbation analysis for various non-linearity strengths using many-body techniques.

Contribution

It presents a new approach using path integrals to analyze effective properties of non-linear disordered systems, extending perturbation methods to second order.

Findings

Perturbation expansion of effective constants to second order in disorder.

Method applicable to both strong and weak non-linearities.

Results align with previous studies and are extendable to other non-linear disordered systems.

Abstract

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 both cases of strong and weak non-linearities. Our results are in agreement with previous ones, and could be easily extended to other types of non-linear problems in disordered systems.