Computation of Power Law Equilibrium Measures on Balls of Arbitrary Dimension

Abstract

We present a numerical approach for computing attractive-repulsive power law equilibrium measures in arbitrary dimension. We prove new recurrence relationships for radial Jacobi polynomials on $d$-dimensional ball domains, providing a substantial generalization of the work based on recurrence relationships of Riesz potentials on arbitrary dimensional balls. Among the attractive features of the numerical method are good efficiency due to recursively generated banded and approximately banded Riesz potential operators and computational complexity independent of the dimension $d$, in stark contrast to the widely used particle swarm simulation approaches for these problems which scale catastrophically with the dimension. We present several numerical experiments to showcase the accuracy and applicability of the method and discuss how our method compares with alternative numerical approaches and conjectured analytical solutions which exist for certain special cases. Finally, we discuss how our method can be used to explore the analytically poorly understood gap formation boundary to spherical shell support.

arXiv comment40 pages, 14 figures

References in corpus (12)

• Ten equivalent definitions of the fractional Laplace operator
• Dimensionality of Local Minimizers of the Interaction Energy
• Nonlocal porous medium equation: Barenblatt profiles and other weak solutions
• Existence of Compactly Supported Global Minimisers for the Interaction Energy
• Tensor calculus in polar coordinates using Jacobi polynomials
• Explicit Flock Solutions for Quasi-Morse potentials
• Zernike expansions of derivatives and Laplacians of the Zernike circle polynomials
• Conquering the pre-computation in two-dimensional harmonic polynomial transforms
• Computing Equilibrium Measures with Power Law Kernels
• On the solution of a Riesz equilibrium problem and integral identities for special functions
• Recurrence relations for orthogonal polynomials on a triangle
• Generalized Zernike polynomials: Operational formulae and generating functions

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