Effective multipoles in random media

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.

Surface lattice Green's functions for high-entropy alloys

“…Our results show that the A-atom potential could be used for calculating the mean response. The per-site fluctuations could then be obtained from the local structure within regions of size ∼ λ c around the site of interest using perturbative approaches [53][54][55][56][57] or multipole expansions [58]. Since a local evaluation up to a cutoff distance scales linearly with the number of sites, this would give rise to a feasible computational scheme.…”

Higher-Order Linearization and Regularity in Nonlinear Homogenization

“…See [4,Theorem 3.8] for the full statement, which was first proved in the periodic setting by Avellaneda and Lin [8]. Subsequent versions of this result, which are based on the ideas of [7,8] in their more quantitative formulation given in [6], were proved in various works [17,15,2], with the full statement here given in [3,10]. In all of its various forms, higher regularity in stochastic homogenzations is based on the simple idea that solutions of the heterogeneous equation should be close to those of the homogenized equation, which should have much better regularity.…”

The Choice of Representative Volumes in the Approximation of Effective Properties of Random Materials

“…Higher-order approximation results in terms of homogenized problems have been derived in [19,20,21,54,67], relying on the concept of higher-order correctors which was first used in the stochastic homogenization context in [42] to establish Liouville principles of arbitrary order in the spirit of Avellaneda and Lin's result in periodic homogenization [11]. Further works in quantitative stochastic homogenization include the analysis of nondivergence form equations [7], a regularity theory up to the boundary [43], denerate elliptic equations [2,44], and the homogenization of parabolic equations [3,64].…”