Solutions of stationary McKean-Vlasov equation on a high-dimensional sphere and other Riemannian manifolds

TL;DR

This paper investigates stationary solutions of the McKean-Vlasov equation on high-dimensional spheres and Riemannian manifolds, extending energetic problem formulations and analyzing bifurcations and phase transitions.

Contribution

It extends the energetic formulation to manifold settings, characterizes critical points, and analyzes bifurcations and phase transitions on spheres and other manifolds.

Findings

Characterization of critical points of free energy on manifolds

Identification of bifurcation branches around uniform states

Conditions for discontinuous phase transitions

Abstract

We study stationary solutions of McKean-Vlasov equation on a high-dimensional sphere and other compact Riemannian manifolds. We extend the equivalence of the energetic problem formulation to the manifold setting and characterize critical points of the corresponding free energy functional. On a sphere, we employ the properties of spherical convolution to study the bifurcation branches around the uniform state. We also give a sufficient condition for an existence of a discontinuous transition point in terms of the interaction kernel and compare it to the Euclidean setting. We illustrate our results on a range of system, including the particle system arising from the transformer models and the Onsager model of liquid crystals.