Particles that normally repel each other will suddenly 'collapse' and huddle together at the edges of their container if the repulsion strength crosses a specific tipping point.
March 31, 2026
Original Paper
Critical phase transitions in minimum-energy configurations for the exponential kernel family $e^{-|x-y|^q}$ on the unit interval
arXiv · 2603.28179
The Takeaway
Normally, things that push each other away spread out as evenly as possible. This study discovered a mathematical tipping point where increasing the repulsion force doesn't push the particles further apart, but instead causes the entire group to abandon the center and smash into the boundaries, creating a counter-intuitive new pattern of organization.
From the abstract
We study the optimal placement of $k$ ordered points on the unit interval for the bounded pair potential \[ K_q(d)=e^{-d^q}, \qquad q>0. \] The family interpolates between strongly cusp-like kernels for $0 1$. Our emphasis is on the transition from collision-free minimizers to endpoint-collapsed minimizers. We reformulate the problem in gap variables, record convexity, symmetry, and the Karush-Kuhn-Tucker conditions, and give a short proof that collisions are impossible for $0 1$ we identify cri