A new interaction potential for swarming models

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.

23 pages, 8 figures

References in corpus (8)

• Novel type of phase transition in a system of self-driven particles
• Interaction Ruling Animal Collective Behaviour Depends on Topological rather than Metric Distance: Evidence from a Field Study
• Onset of collective and cohesive motion
• Self-organization in systems of self-propelled particles
• A well-posedness theory in measures for some kinetic models of collective motion
• State Transitions and the Continuum Limit for a 2D Interacting, Self-Propelled Particle System
• Continuum limit of self-driven particles with orientation interaction
• Self-organised complex aerial displays of thousands of starlings: a model

Cited by in corpus (23)

• A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure
• The derivation of Swarming models: Mean-Field Limit and Wasserstein distances
• Existence of Compactly Supported Global Minimisers for the Interaction Energy
• Nonlinear stability of flock solutions in second-order swarming models
• Instantaneous control of interacting particle systems in the mean-field limit
• Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms
• Explicit Flock Solutions for Quasi-Morse potentials
• Convergence of a Fully Discrete and Energy-Dissipating Finite-Volume Scheme for Aggregation-Diffusion Equations
• Online Flocking Control of UAVs with Mean-Field Approximation
• On gradient flow and entropy solutions for nonlocal transport equations with nonlinear mobility
• Optimal lattice configurations for interacting spatially extended particles
• Adhesion and volume constraints via nonlocal interactions lead to cell sorting
• How cells stay together; a mechanism for maintenance of a robust cluster explored by local and nonlocal continuum models
• Controlling a self-organizing system of individuals guided by a few external agents -- particle description and mean-field limit
• Single to Double Mill Small Noise Transition via Semi-Lagrangian Finite Volume Methods
• Well-posedness of a coupled system of Skorohod-like stochastic differential equations
• Interpretable machine learning models: a physics-based view
• Comparing first order microscopic and macroscopic crowd models for an increasing number of massive agents
• Carathéodory Theory and A Priori Estimates for Continuity Inclusions in the Space of Probability Measures
• Zoology of a non-local cross-diffusion model for two species
• Explicit Equilibrium Solutions For the Aggregation Equation with Power-Law Potentials
• Space mapping-based Receding Horizon Control for Stochastic Interacting Particle Systems: dogs herding sheep
• Confinement for Repulsive-Attractive Kernels