Bifurcation of global energy minimizers for a diffusion-aggregation model on sphere

TL;DR

This paper analyzes the bifurcation behavior of global energy minimizers for a diffusion-aggregation model on spheres, identifying thresholds and generalizing results to multiple sphere spaces, revealing how attraction strength influences spreading or aggregation.

Contribution

It introduces a threshold for attractive interactions that determines the global minimizers and extends the analysis to product spaces of spheres.

Findings

Identified a threshold value for attractive interactions affecting minimizer behavior.

Established the nature of bifurcation at the threshold.

Generalized results to spaces of multiple spheres, such as tori.

Abstract

We consider a free energy functional defined on probability densities on the unit sphere $\mathbb{S}^d$, and investigate its global minimizers. The energy consists of two components: an entropy and a nonlocal interaction energy, which favour spreading and aggregation behaviour, respectively. We find a threshold value for the size of the attractive interactions, and establish the global energy minimizers in each case. The bifurcation at this threshold value is investigated. We also generalize the results to spaces consisting of an arbitrary number of spheres (e.g., the flat torus $\mathbb{S}^1 \times \mathbb{S}^1$).