Vectorial crystallization problems and collective behavior

TL;DR

This paper introduces a class of vectorial crystallization problems modeling anisotropic molecules and collective animal behavior, analyzing configurations with orientation-dependent interactions and predicting various formation patterns.

Contribution

It develops a variational framework for vectorial crystallization, incorporating angle-dependent interactions and predicting emergent collective structures.

Findings

Ground states include row and diamond formations.

Angular dependence influences the emergence of specific patterns.

The model applies to biological and material systems with anisotropic interactions.

Abstract

We propose and analyze a class of vectorial crystallization problems, with applications to crystallization of anisotropic molecules and collective behavior such as birds flocking and fish schooling. We focus on two-dimensional systems of "oriented" particles: Admissible configurations are represented by vectorial empirical measures with density in $\mathcal S^1$. We endow such configurations with a graph structure, where the bonds represent the "convenient" interactions between particles, and the proposed variational principle consists in maximizing their number. The class of bonds is determined by hard sphere type pairwise potentials, depending both on the distance between the particles and on the angles between the segment joining two particles and their orientations, through threshold criteria. Different ground states emerge by tuning the angular dependence in the potential, mimicking ducklings swimming in a row formation and predicting as well, for some specific values of the angular parameter, the so-called {\it diamond formation} in fish schooling.