Vacuum bubble and fissure formation in collective motion with competing attractive and repulsive forces

TL;DR

This work analyzes the continuum limit of many interacting agents with competing short-range repulsion and long-range attraction on 2D periodic domains, focusing on a vertical bifurcation from the uniform density and the formation of vacuum bubbles and fissures. By employing periodized potentials on square and hexagonal lattices, the authors derive explicit vertical branches, universal near-onset scaling laws for vacuum regions, and diffusive corrections via center-manifold reductions. They perform extensive numerical continuation in both full PDE and reduced finite-rank settings, revealing a second vertical branch and a topology-change pathway from bubbles to clusters, with stability depending on lattice symmetry. Comparisons with finitely many-particle dynamics show qualitative agreement and reveal discretization-induced hysteresis, underscoring the relevance of the continuum predictions for finite particle systems and suggesting broad applicability to pattern formation in active matter. The results advance understanding of vacuum formation, pattern selection, and topology transitions in nonlocal-interaction models with lattice symmetry.

Abstract

We study the continuum limit of the motion of agents in the plane driven by competing short-range repulsion and long-range attractive forces. At a critical parameter value, we find destabilization of a trivial branch of uniformly distributed solutions and analyze bifurcating solutions. Curiously, the bifurcating branch is vertical, leading to a reversible, non-hysteretic phase transition. Near the bifurcation point, we demonstrate scaling laws for the size of vacuum regions, which can form fissures or bubbles. We also study the effect of small noise and the eventual topological transition from vacuum bubbles to isolated particle clusters.

Figures (15)

• Figure 1.1: Contour plots of potentials $V_\mathrm{per}$ from \ref{['e:Vexpl']} in the square case (left) and the hexagonal case (right) with maxima yellow, minima blue. Note that the square potential is odd, but the hexagonal potential is not: level sets near maxima and minima have square-like corrections to the leading-order round shape in the square potential; the corrections are triangular near minima and hexagonal near maxima in the hexagonal potential case.
• Figure 2.1: An illustration of vacuum bubbles (left) and fissures (right) in a square of width $2\pi$, comprising two (!) fundamental domains $\Omega$ of the square lattice. Blue regions represent areas of vacuum; $u$ is positive in the green regions $\Omega_0$. Note that there is one bubble and one fissure in each fundamental domain (boundaries of fundamental domain indicated by dashed lines); fissures have width $2\ell$ and total length $\sqrt{2}\pi$ in a fundamental domain, bubbles have radius $\ell$.
• Figure 2.2: Inner figures (blue and green): An illustration of the vacuum bubbles on the hexagonal lattice. Left, triangles; right, hexagons. Blue regions represent areas of vacuum. In the case of triangles, bubbles are initially approximately circular, as pictured here; the triangular corrections become more apparent as ${\widetilde{\mu}}$ increases. All bubbles have (inner) radius $\ell$. Outer figures (pink and blue): example boundaries of triangle and hexagon vacuum regions from numerics, for various $\widetilde{\mu}$ (size not to scale with inner diagram). Triangles: $\widetilde{\mu} =$ 1e-5, 5e-5, 1e-4, 2.5e-4, 5e-4, 1e-3, 2e-3. Hexagons: $\widetilde{\mu} =$ 5e-5, .001, .005, .01, .03, .07.
• Figure 3.1: Numerical agreement and comparison for the almost-vertical branches. Dots are numerical; curves are the $\mathcal{O}(\varepsilon)$ predictions from \ref{['e:mu_pred_sq']} and \ref{['e:mu1hex']}; square symmetry (left) and hexagonal symmetry (right) shown. Bubble branches are in green; fissure branches are in blue.
• Figure 3.2: Center manifold predictions vs. numerical values, for $\varepsilon = 10^{-4}$ (square symmetry, left) and $\varepsilon = 2.5 10^{-3}$ (hexagonal symmetry, right). The parameter is shifted relative to the bifurcation point at $\varepsilon=0$. Center manifold predictions are in lighter colors. Fissures are in blue; bubbles are in green and black. Black represents branches found to be stable in center manifold predictions and in numerical continuation, respectively.

Theorems & Definitions (13)

• Theorem 1: Vacuum region scaling --- square lattice
• Remark 2.1: Maximal Isotropy --- square lattice
• Theorem 2: Vacuum region scaling --- hexagonal lattice
• Remark 2.2: Vacuum area, hexagon versus triangle
• Remark 2.3: Polygonal shapes
• Remark 2.4: Maximal Isotropy --- hexagonal lattice
• Proposition 3.1: Diffusive corrections --- square bubbles
• Remark 3.2: Diffusive corrections --- square fissures
• Remark 3.3: Diffusive corrections --- no elliptical bubbles
• Proposition 3.4: Diffusive corrections --- triangles and hexagons