Multipole Distributions and Hyper-Flux Fields
Vladimir Gol'dshtein, Reuven Segev
TL;DR
The paper develops a unified, distributional framework for multipoles of a general extensive property in continuum mechanics, extending beyond electrostatics to include hyper-fluxes for higher-order transport. It represents multipolar distributions by measures through the action $Q(φ)=\sum_{0≤k≤r}∫ φ_{,i_1...i_k} \, dq^{i_1...i_k}$, enabling computation of bound charges, bound dipoles, and bound quadrupoles, and it couples this with a force/power functional that arises under motion of the distribution. It introduces higher-order fluxes (hyperfluxes) and analyzes how moving polarization and edge densities contribute to surface and edge terms, integrating stresses and hyper-stresses into a coherent transport picture. Finally, it generalizes the construction to differentiable manifolds using jet extensions and de Rham currents, providing an invariant geometric formulation that subsumes the Euclidean-space expression as a local representation.
Abstract
We outline here a simple mathematical introduction to the notions of multipoles for a general extensive property $Π$ from the point of view of continuum mechanics. Classically, $Π$ is the electric charge, but the theory is not limited to electrostatics. The proposed framework allows a simple computation of the bound "charges" and bound multipoles of lower orders. In addition, if the property $Π$ has a potential function in the sense described below, a general expression for the mechanical force (power) functional acting on bodies containing the property is presented. Finally, using a similar viewpoint, we consider hyper-fluxes -- flux fields of tensorial order greater than one -- and show that moving multipoles (in particular, a moving dielectric) give rise to hyper-fluxes.