A new interaction potential for swarming models

TL;DR

This paper introduces Quasi-Morse interaction potentials for swarming models, enabling explicit macroscopic solutions that closely match particle system dynamics, advancing understanding of collective animal motion.

Contribution

The paper proposes a new class of interaction potentials called Quasi-Morse, allowing explicit solutions for flock and mill states in swarming models, with numerical coefficients.

Findings

Quasi-Morse potentials produce flock and mill states similar to natural observations.

Explicit macroscopic solutions match long-term particle system dynamics.

The approach simplifies analysis of collective motion phenomena.

Abstract

We consider a self-propelled particle system which has been used to describe certain types of collective motion of animals, such as fish schools and bird flocks. Interactions between particles are specified by means of a pairwise potential, repulsive at short ranges and attractive at longer ranges. The exponentially decaying Morse potential is a typical choice, and is known to reproduce certain types of collective motion observed in nature, particularly aligned flocks and rotating mills. We introduce a class of interaction potentials, that we call Quasi-Morse, for which flock and rotating mills states are also observed numerically, however in that case the corresponding macroscopic equations allow for explicit solutions in terms of special functions, with coefficients that can be obtained numerically without solving the particle evolution. We compare thus obtained solutions with long-time dynamics of the particle systems and find a close agreement for several types of flock and mill solutions.