Computing Equilibrium Measures with Power Law Kernels

TL;DR

This paper presents a fast, stable numerical method for computing equilibrium measures with power law kernels, enabling analysis of existence, uniqueness, and support structure of solutions.

Contribution

It introduces a novel approach using banded operators on ultraspherical bases to efficiently solve equilibrium measure problems with power law kernels.

Findings

Method achieves high computational efficiency and stability.

Successfully compares with analytical solutions and particle simulations.

Explores conditions for existence, uniqueness, and support gaps in equilibrium measures.

Abstract

We introduce a method to numerically compute equilibrium measures for problems with attractive-repulsive power law kernels of the form $K(x-y) = \frac{|x-y|^\alpha}{\alpha}-\frac{|x-y|^\beta}{\beta}$ using recursively generated banded and approximately banded operators acting on expansions in ultraspherical polynomial bases. The proposed method reduces what is naively a difficult to approach optimization problem over a measure space to a straightforward optimization problem over one or two variables fixing the support of the equilibrium measure. The structure and rapid convergence properties of the obtained operators results in high computational efficiency in the individual optimization steps. We discuss stability and convergence of the method under a Tikhonov regularization and use an implementation to showcase comparisons with analytically known solutions as well as discrete particle simulations. Finally, we numerically explore open questions with respect to existence and uniqueness of equilibrium measures as well as gap forming behaviour in parameter ranges of interest for power law kernels, where the support of the equilibrium measure splits into two intervals.