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A family of explicit minimizers for interaction energies

Ruiwen Shu

TL;DR

This work analyzes energy minimizers for nonlocal, power-law interaction kernels W({\bf x})=\frac{|{\bf x}|^a}{a}-\frac{|{\bf x}|^b}{b} in \mathbb{R}^d, focusing on LIC potentials where minimizers are unique up to translation. It develops a novel odd-dimensional construction by converting the Euler–Lagrange condition into a local PDE using successive Laplacians, enabling a radial power-series solution and an explicit determination of the minimizer via a determinant condition det M(R)=0, with special elementary forms in d=1 and d=3. For even dimensions, it introduces a dimension-reduction lemma that projects and rescales the higher-dimensional minimizer to obtain the d-dimensional minimizer, yielding explicit formulas such as ρ_2 in terms of ρ_3. Overall, the paper provides new explicit minimizers in low dimensions and a principled method to generate higher-dimensional counterparts through projection, enriching the understanding of nonlocal aggregation dynamics and steady states.

Abstract

In this paper we consider the minimizers of the interaction energies with the power-law interaction potentials $W({\bf x}) = \frac{|{\bf x}|^a}{a} - \frac{|{\bf x}|^b}{b}$ in $d$ dimensions. For odd $d$ with $(a,b)=(3,2-d)$ and even $d$ with $(a,b)=(3,1-d)$, we give the explicit formula for the unique energy minimizer up to translation. For the odd dimensions, the key observation is that successive Laplacian of the Euler-Lagrange condition gives a local partial differential equation for the minimizer. For the even dimensions $d$, the minimizer is given as the projection and rescaling of the previously constructed minimizer in dimension $d+1$ via a new lemma on dimension reduction.

A family of explicit minimizers for interaction energies

TL;DR

This work analyzes energy minimizers for nonlocal, power-law interaction kernels W({\bf x})=\frac{|{\bf x}|^a}{a}-\frac{|{\bf x}|^b}{b} in \mathbb{R}^d, focusing on LIC potentials where minimizers are unique up to translation. It develops a novel odd-dimensional construction by converting the Euler–Lagrange condition into a local PDE using successive Laplacians, enabling a radial power-series solution and an explicit determination of the minimizer via a determinant condition det M(R)=0, with special elementary forms in d=1 and d=3. For even dimensions, it introduces a dimension-reduction lemma that projects and rescales the higher-dimensional minimizer to obtain the d-dimensional minimizer, yielding explicit formulas such as ρ_2 in terms of ρ_3. Overall, the paper provides new explicit minimizers in low dimensions and a principled method to generate higher-dimensional counterparts through projection, enriching the understanding of nonlocal aggregation dynamics and steady states.

Abstract

In this paper we consider the minimizers of the interaction energies with the power-law interaction potentials in dimensions. For odd with and even with , we give the explicit formula for the unique energy minimizer up to translation. For the odd dimensions, the key observation is that successive Laplacian of the Euler-Lagrange condition gives a local partial differential equation for the minimizer. For the even dimensions , the minimizer is given as the projection and rescaling of the previously constructed minimizer in dimension via a new lemma on dimension reduction.
Paper Structure (8 sections, 6 theorems, 56 equations, 1 figure)

This paper contains 8 sections, 6 theorems, 56 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Qualitative results
  3. The cases of odd $d$ with $(a,b)=(3,2-d)$
  4. The case $d=1$
  5. The general case
  6. The case $d=3$ in elementary functions
  7. Connection between steady states in different dimensions
  8. The cases of even $d$ with $(a,b)=(3,1-d)$

Key Result

Proposition 2.1

Consider $E$ in E with $W$ given by Wab with $-d<b\le 2 \le a \le 4,\,b<a$. There exists a unique minimizer $\rho_\infty$ of $E$ up to translation. $\rho_\infty$ is compactly supported and radially symmetric. It is the only compactly supported probability measure satisfying the Euler-Lagrange condit for some $C_0\in\mathbb{R}$. If one further assumes $-d<b\le 2-d$, then $\rho_\infty$ is an $L^\inf

Figures (1)

  • Figure 1: The minimizers for $(d,a,b)$ being $(1,3,1)$, $(2,3,-1)$ and $(3,3,-1)$ (left, middle and right, respectively). The blue curves are the minimizers $\rho$ in the radial coordinate. The red curves are the generated potentials $W*\rho$, subtracted by the constant $C_0=\frac{1}{2}E[\rho]$.

Theorems & Definitions (13)

  • Proposition 2.1
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Lemma 4.1
  • proof
  • Remark 4.2
  • Lemma 5.1
  • proof
  • ...and 3 more