2024/03/01 by José A. Cañizo, Cañizo, José A., Alejandro Ramos-Lora +1 · 1 citation
Biochemistry, Genetics and Molecular Biology · Engineering · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Insect and Arachnid Ecology and Behavior #Mathematical Physics (math-ph) #Numerical Analysis (math.NA) #Opinion Dynamics and Social Influence #Slime Mold and Myxomycetes Research
paper · pdf · doi:10.48550/arxiv.2403.00594
openalex publication_date 2024/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.