A Mexican hat dance: clustering in Ricker-potential particle systems

TL;DR

This paper analyzes a one-dimensional ensemble of particles interacting via a modified Ricker (Mexican hat) potential under a quadratic confinement, uncovering spontaneous stacking and a rich bifurcation structure as confinement tightens. It combines analytical calculations (Lambert W solutions for two-particle equilibria, Jacobian stability analysis) with numerical continuation to map equilibria across n from small to large, revealing that major bifurcations scale with $w_0=\frac{s}{k}\sqrt{\frac{k-1}{n}}$ and persist in the large-$n$ limit, while non-origin equilibria become increasingly subtle. The study uncovers extensive multistability and hysteresis in stack configurations, including origin-fractionation bands and transfer branches between stacks, offering insights into soft-core interactions and potential connections to neuronal models and confined systems. The results highlight a coherent large-$n$ limiting picture where dominant bifurcations occur at fixed multiples of $w_0$, while the microstructure of equilibria becomes densely packed, motivating future work on variant confinements and higher-dimensional domains.

Abstract

The dynamics and spontaneous organization of coupled particles is a classic problem in modeling and applied mathematics. Here we examine the behavior of particles coupled by the Ricker potential, exhibiting finite local repulsion transitioning to distal attraction, leading to an energy-minimizing ``preferred distance''. When compressed by a background potential well of varying severity, these particles exhibit intricate self-organization into ``stacks" with varying sizes and positions. We examine bifurcations of these high-dimensional arrangements, yielding tantalizing glimpses into a rich dynamical zoo of behavior.

Figures (19)

• Figure 1: Ricker wavelet. Interaction potential $U(x)$ for a particle described by the Ricker wavelet potential (Eq. \ref{['eq:hat']}) with parameters $s=1$ and $k=2$.
• Figure 2: Three particles at equilibrium. Example of three particles at an equilibrium. The vertical solid blue lines show particle positions, the dotted blue curves show the particles' potentials, and the dashed maroon parabola is the background potential well. The solid black curve is the total potential, and we can see the derivative is zero at each particle's position, indicating this arrangement is at equilibrium. This arrangement is stable: since a particle does not influences itself, each particle effectively "sees" the global potential minus its own contribution, which makes each particle's position in this arrangement a "trough" from its own point of view.
• Figure 3: Equilibrium diagram. Apparently stable equilibrium positions for $512$ particles confined in a quadratic potential well. Color indicates particle abundance. The horizontal $w$-axis tick marks are placed at approximate bifurcation points, and persist on other diagrams of this type for comparison's sake. The horizontal scale is set by $w_0= \frac{s}{k}\sqrt{\frac{k-1}{n}}$, which is the critical $w$ value where the fully-stacked origin state becomes unstable (see section \ref{['sec:general-n_stability']}). See section \ref{['sec:methods']} for additional simulation details.
• Figure 4: Equilibrium diagram, $n=512$, symmetry enforced.Left:A symmetry-enforced version of Fig. \ref{['fig:512_particles']}. Besides being "cleaner," however, we also notice the apparent bifurcation points change slightly due to the minimum of two particles in a stack. Right: A 3D view of the data in the left panel, with stack-size information encoded in the vertical axis as well as color. This emphasizes the continuous shift in population fractionation and the structure of the major bifurcations.
• Figure 5: Symmetry-enforced system, $n$ convergence. Equilibrium diagrams for increasing population sizes $n$. The diagrams appear to converge in a visual sense to the same few "major" bifurcations, which occur at nearly the same multiples of the critical parameter value $w_0$ (given in Eq. \ref{['eq:w0_general']}). This motivates us to understand and characterize this large-$n$ generic behavior.