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Nonlinear diffusion limit of non-local interactions on a sphere

Mark A. Peletier, Anna Shalova

TL;DR

This work proves a nonlinear diffusion limit for a nonlocal aggregation equation on the sphere, showing that strongly localized repulsion coupled with fixed attraction yields a porous-medium-type diffusion in the limit. The authors leverage a gradient-flow framework and, crucially, a harmonic-analysis-based spectral decomposition of spherical convolutions to handle noncommutativity on the sphere; they also establish compactness via Wasserstein theory and heat-flow interchange. A key contribution is providing a concrete, verifiable condition for the existence of a convolution square root using spherical-harmonics, enabling rigorous convergence from the nonlocal model to the local diffusion limit. The results connect to transformer models by interpreting the local repulsion as a diffusion mechanism and discuss implications for the design and analysis of toy transformer dynamics on manifolds.

Abstract

We study an aggregation PDE with competing attractive and repulsive forces on a sphere of arbitrary dimension. In particular, we consider the limit of strongly localized repulsion with a constant attraction term. We prove convergence of solutions of such a system to solutions of the aggregation-diffusion equation with a porous-medium-type diffusion term. The proof combines variational techniques with elements of harmonic analysis on a sphere. In particular, we characterize the square root of the convolution operator in terms of the spherical harmonics, which allows us to overcome difficulties arising due to the convolution on a sphere being non-commutative. The study is motivated by the toy model of transformers introduced by Geshkovski et al. (2025); and we discuss the applicability of the results to this model.

Nonlinear diffusion limit of non-local interactions on a sphere

TL;DR

This work proves a nonlinear diffusion limit for a nonlocal aggregation equation on the sphere, showing that strongly localized repulsion coupled with fixed attraction yields a porous-medium-type diffusion in the limit. The authors leverage a gradient-flow framework and, crucially, a harmonic-analysis-based spectral decomposition of spherical convolutions to handle noncommutativity on the sphere; they also establish compactness via Wasserstein theory and heat-flow interchange. A key contribution is providing a concrete, verifiable condition for the existence of a convolution square root using spherical-harmonics, enabling rigorous convergence from the nonlocal model to the local diffusion limit. The results connect to transformer models by interpreting the local repulsion as a diffusion mechanism and discuss implications for the design and analysis of toy transformer dynamics on manifolds.

Abstract

We study an aggregation PDE with competing attractive and repulsive forces on a sphere of arbitrary dimension. In particular, we consider the limit of strongly localized repulsion with a constant attraction term. We prove convergence of solutions of such a system to solutions of the aggregation-diffusion equation with a porous-medium-type diffusion term. The proof combines variational techniques with elements of harmonic analysis on a sphere. In particular, we characterize the square root of the convolution operator in terms of the spherical harmonics, which allows us to overcome difficulties arising due to the convolution on a sphere being non-commutative. The study is motivated by the toy model of transformers introduced by Geshkovski et al. (2025); and we discuss the applicability of the results to this model.

Paper Structure

This paper contains 33 sections, 27 theorems, 177 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Aggregation equations with and without diffusion
  3. Heuristic explanation.
  4. Related work
  5. Non-linear diffusion limit of non-local interactions.
  6. Aggregation equations with nonlinear diffusion
  7. Aggregation PDEs on manifolds
  8. Main contributions
  9. Notation
  10. Properties of the interaction kernels
  11. Spherical harmonics
  12. Spectral properties of the convolution
  13. Admissible interaction kernels
  14. Estimates
  15. Solutions of PDEs on the sphere
  16. ...and 18 more sections

Key Result

Theorem 2.2

Let $Y_{l, k}$ be the spherical harmonics defined by eq:harmonics, then the set is an orthonormal basis of $L^2({\mathbb S}^{n-1})$. In particular for any $f \in L^2({\mathbb S}^{n-1})$ the following identity holds in the sense that $\lim_{n\to \infty} \|f - \sum_{l=1}^n \mathrm{proj}_l f\|_{L_2} =0$.

Figures (1)

  • Figure 1:

Theorems & Definitions (55)

  • Definition 2.1: Gegenbauer polynomials
  • Theorem 2.2: Fourier decomposition on ${\mathbb S}^{n-1}$ dai2013approximation
  • Proposition 2.3: Zonal harmonics dai2013approximation
  • Definition 2.4: Zonal kernels
  • Definition 2.5: Convolution on ${\mathbb S}^{n-1}$
  • Definition 2.6: Spherical harmonics decomposition
  • Lemma 2.7: dai2013approximation
  • Theorem 2.8: Convolution theorem on ${\mathbb S}^{n-1}$
  • Lemma 2.11: Bounds on the energy
  • proof
  • ...and 45 more