Effective Multipoles in Random media

TL;DR

This paper demonstrates that in three-dimensional random media, effective dipole and quadrupole moments can be inferred without detailed knowledge of the medium far from the source, extending classical multipole concepts.

Contribution

The authors develop a higher-order two-scale expansion framework that constructs isomorphisms between heterogeneous and homogeneous harmonic function spaces, enabling effective multipole analysis in random media.

Findings

Effective dipole and quadrupole moments can be inferred without full medium realization.

Higher-order correctors are crucial for constructing isomorphisms.

The approach extends classical multipole theory to stochastic homogenization contexts.

Abstract

In a homogeneous medium, the far-field generated by a localized source can be expanded in terms of multipoles; the coefficients are determined by the moments of the localized charge distribution. We show that this structure survives to some extent for a random medium in the sense of quantitative stochastic homogenization: In three space dimensions, the effective dipole and quadrupole - but not the octupole - can be inferred without knowing the realization of the random medium far away from the (overall neutral) source and the point of interest. Mathematically, this is achieved by using the two-scale expansion to higher order to construct isomorphisms between the hetero- and homogeneous versions of spaces of harmonic functions that grow at a certain rate, or decay at a certain rate away from the singularity (near the origin); these isomorphisms crucially respect the natural pairing between growing and decaying harmonic functions given by the second Green's formula. This not only yields effective multipoles (the quotient of the spaces of decaying functions) but also intrinsic moments (taken with respect to the elements of the spaces of growing functions). The construction of these rigid isomorphisms relies on a good (and dimension-dependent) control on the higher-order correctors and their flux potentials.