Swarming by curvature control in arbitrary dimension

Abstract

We consider an interacting particle system proposed in the literature to model fish behavior. In this model, the agents move at constant speed and control the curvature of their trajectory (i.e. the time-derivative of their velocity) so as to align their velocity with that of their neighbors, up to some noise. We provide a novel $n$-dimensional formulation of this model for any $n \geq 3$ and derive its mean-field kinetic formulation using bundle geometry concepts. The target of the paper is the derivation of a fluid model in the hydrodynamic limit. We show that this fluid model is the "self-organized hydrodynamic" (SOH) model already found in earlier work pertaining to the Vicsek model. The derivation is based on the introduction of appropriate "generalized collision invariants" (GCI). The action of the $n$-dimensional orthogonal group is used to reduce the expression of the GCI to a set of two functions satisfying a system of equations which is solved by means of a variational formulation. This leads to explicit formulas for the coefficients of the SOH model in terms of those of the original interacting particle system.