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.Comment: 23 pages, 8 figure

Optimal lattice configurations for interacting spatially extended particles

“…We also note that systems with localizing and delocalizing interactions, although in a dynamic setting, are also relevant in biological models related to swarming and flocking, see e.g. [4,46,3,14] and the references therein. We show that for particles with radially symmetric mass distribution, the triangular lattice is still optimal if the mass of each particle is either sufficiently concentrated near its center, or if the mass distribution is described by a completely monotone function.…”

Swarming in bounded domains

“…Here, we note that other potentials have been considered such as the quasi-morse potential by Carrillo et al (2013Carrillo et al ( , 2014b or Log-Newtonian potential by Fetecau et al (2011).…”

Explicit equilibrium solutions for the aggregation equation with power-law potentials

“…However, getting conclusions about the qualitative properties of the continuum problem (2) turned out to be mathematically involved. The existence of explicit formulas for flock and mill patterns for particular potentials such as the Morse potentials, originally used in [25], was possible due to the particular properties of associated differential operators [36,6,21,17]. Let us point out that mill patterns are also characterized by a probability density or measure satisfying a similar equation to (2) giving the balance between attractive-repulsive and centrifugal forces of the form ∇K * ρ = −s 0 ∇ ln |x| in supp(ρ).…”