Swarming by curvature control in arbitrary dimension

TL;DR

The paper extends curvature-control swarming models to arbitrary dimensions by formulating PCC in a bundle geometry and deriving its mean-field KCC description. It proves (formally) that the hydrodynamic limit of the KCC model yields the Self-Organized Hydrodynamics (SOH) system, linking micro- and macro-scale behavior through generalized collision invariants. A key innovation is exploiting O(n) group actions to reduce the vector GCI to a two-scalar reduced GCI pair via variational formulations, enabling explicit expressions for SOH coefficients. The work lays groundwork for rigorous justifications, non-normalized extensions, and potential generalizations to richer interaction types and higher-dimensional swarming phenomena.

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.

Figures (3)

• Figure 1: Examples of single-particle trajectories in space in 2D (blue curve) and vector ${\mathcal{J}}$ (red arrow): (a) large alignment frequency $\nu > 4 c_0 \lambda |{\mathcal{J}}|$; (b) small alignment frequency $\nu < 4 c_0 \lambda |{\mathcal{J}}|$. Note that the Vicsek case corresponds to the overdamped limit of \ref{['eq:pk_single_noise_2D']} and that the Vicsek trajectories would thus resemble case (a).
• Figure 2: Diagram of the sets and maps involved in the geometric framework. Arrows with a triple head mean surjections and arrows with an anchor are for injections.
• Figure 3: The diffeomorphism ${\mathbf A}_\mathbf{u}$ depicted in dimension $n=3$. The velocity $v$ is changed into the pair $(\theta,w)$ where $\theta$ is the angle between $\mathbf{u}$ and $v$ and where $w$ belongs to the unit circle ${\mathbb S}^1$ in the plane orthogonal to $\mathbf{u}$ (diffeomorphism ${\mathbf V}_\mathbf{u}$ introduced in Section \ref{['subsubsec:hydro_chg_var']}). The tangent vector $\kappa$ to ${\mathbb S}^2$ at $v$ is changed into the pair $(\kappa_\parallel, \kappa_T)$ where $\kappa_\parallel$ is the projection of $\kappa$ on $e_\theta$, the director of the line normal to $v$ in the plane $\mathrm{Span}(\mathbf{u},v)$, and $\kappa_T$ is a tangent vector to ${\mathbb S}^1$ at $w$.

Theorems & Definitions (58)

• Lemma 3.1: Divergences of a field which does not depend on $\kappa$
• Lemma 3.2: Gradient of a function which only depends on $v$
• Lemma 3.3: Gradient of $\langle \kappa, \kappa \rangle_v$
• Theorem 3.4: Kinetic model for the single particle PCC model
• Remark 3.1
• Theorem 3.5: Kinetic model for the many-particle PCC model (non-normalized case)
• Conjecture 3.1: Kinetic model for the many-particle PCC model (normalized case)
• Lemma 4.1
• Lemma 4.2: Negativity of $Q$
• Lemma 4.3: Properties of the order parameter