A Mexican hat dance: Clustering in spatially non-exclusive particle systems

Area of Science:

• Mathematical modeling
• Applied mathematics
• Complex systems

Background:

• The study of coupled particle dynamics and spontaneous organization is a fundamental problem.
• Mexican hat potentials feature local repulsion and distal attraction, establishing a preferred particle distance.

Purpose of the Study:

• To investigate the self-organization of particles interacting via a Mexican hat potential.
• To analyze the resulting structures when particles are confined by a background potential well.
• To explore the bifurcations and complex dynamics of these arrangements.

Main Methods:

• Simulation of particle systems with Mexican hat potentials.
• Analysis of particle configurations under confinement.
• Examination of bifurcations in high-dimensional arrangements.

Main Results:

• Particles spontaneously self-organize into "stacks" of varying sizes and positions.
• The Mexican hat potential drives the formation of preferred inter-particle distances.
• Complex, intricate structures emerge from the interplay of particle interactions and confinement.

Conclusions:

• The Mexican hat potential facilitates intricate self-organization in coupled particle systems.
• Bifurcation analysis provides insights into the diverse dynamical behaviors observed.
• This research offers a glimpse into a rich zoo of complex dynamics in mathematical modeling.