Discrete minimizers of the interaction energy in collective behavior: a brief numerical and analytic review

TL;DR

The paper surveys discrete minimizers of the N-particle interaction energy with repulsive-attractive power-law potentials, emphasizing numerical computation in two dimensions via basin-hopping with the GMIN software. It reviews existence theory, crystallization phenomena, and the connection to continuous mean-field limits, summarizing rigorous results in 1D, notable progress in 2D (including lattice optimality results) and the open status in 3D and higher. It highlights the basin-hopping strategy as an unbiased global search tool for challenging minimization landscapes and discusses the relevance to aggregation dynamics and pattern formation. The work also outlines open questions on regime boundaries and scaling limits that bridge discrete minimizers and continuum evolutions, guiding both theory and computation.

Abstract

We consider minimizers of the N-particle interaction potential energy and briefly review numerical methods used to calculate them. We consider simple pair potentials which are repulsive at short distances and attractive at long distances, focusing on examples which are sums of two powers. The range of powers we look at includes the well-known case of the Lennard-Jones potential, but we are also interested in less singular potentials which are relevant in collective behavior models. We report on results using the software GMIN developed by Wales and collaborators for problems in chemistry. For all cases, this algorithm gives good candidates for the minimizers for relatively low values of the particle number N. This is well-known for potentials similar to Lennard-Jones, but not for the range which is of interest in collective behavior. Standard minimization procedures have been used in the literature in this range, but they are likely to yield stationary states which are not minimizers. We illustrate numerically some properties of the minimizers in 2D, such as lattice structure, Wulff shapes, and the continuous large-N limit for locally integrable (that is, less singular) potentials.

Figures (7)

• Figure 1: Minimizer for the Lennard-Jones potential with 91 particles.
• Figure 2: Our best numerical candidate for a minimizer of the potential \ref{['eq:V-powers']} with $a=2$, $b=0.5$ and $10^4$ particles. (We thank David J. Wales for providing this result)
• Figure 3: Numerical candidate for minimizer of the potential \ref{['eq:V-powers']} with $a=2$, $b=-1$ and $10^4$ particles. (We thank David J. Wales for providing this result.) This corresponds to the "strongly repulsive" regime considered in Carrillo_2016. In this regime it is conjectured that a sequence of minimizers $(X_N)_{N \geq 2}$ must converge to a measure $\nu$ which must have a smooth density with respect to Lebesgue measure. In fact, the most likely candidate measure is given in A_Carrillo_2017 and is of the form $\nu(x) = C (R^2 - |x|^2)_+^{1 - \frac{b+d}{2}},$with $(\cdot)_+$ denoting the positive part, with suitable constants $C, R > 0$ which ensure in particular that the total mass is $1$. In the case corresponding to this figure we have $d=2$, $b=-1$, so the density of $\nu$ vanishes at the boundary. By contrast, in the case of Figure \ref{['fig:a2b05-10000']} the exponent $1 - (b+d)/2$ becomes negative and the density of $\nu$ should diverge at the boundary.
• Figure 4: Minimizer for the power-law potential \ref{['eq:V-powers']} with $a=2$ and several values of $b$, always with $10^3$ particles.
• Figure 5: Zoomed-in view of the last 4 minimizers in Figure \ref{['fig:a2']}.