Collective behavior of self-propelling particles with kinematic constraints: The relation between the discrete and the continuous description

Abstract

In two papers we proposed a continuum model for the dynamics of systems of self propelling particles with kinematic constraints on the velocities and discussed some of its properties. The model aims to be analogous to a discrete algorithm used in works by T. Vicsek et al. In this paper we derive the continuous hydrodynamic model from the discrete description. The similarities and differences between the resulting model and the hydrodynamic model postulated in our previous papers are discussed. The results clarify the assumptions used to obtain a continuous description.

Continuum Limit of Self-Driven Particles With Orientation Interaction

“…Now, we would like to take the limit ∆t → 0 and find a time-continuous dynamics. To do so, we first consider the deterministic algorithm (2.1), (2.5) and following [29], make some elementary remarks. First, because |ω…”

Large Scale Dynamics of the Persistent Turning Walker Model of Fish Behavior

“…[26,36,47,48]). Several attempts to derive continuum models from the CVA model are also reported in the literature [34,44,45]. In [21,22], a derivation of a continuum model from a kinetic version of the CVA model is proposed.…”

Macroscopic limit of self-driven particles with orientation interaction

“…So, to each of the right-hand sides χ = − sin φ sin θM Ω or χ = cos φ sin θM Ω which have zero average on the sphere, there exist solutions ψ 1 and ψ 2 respectively (unique up to constants) of problem (12). We single out unique solutions by requesting that ψ 1 and ψ 2 have zero average on the sphere: ψ k dω = 0, k = 1, 2.…”

“…[10,14,15]) but they are phenomenological and not based on a 'rigorous' derivation from individual interactions. Several attempts to derive continuum models from the CVA model are also reported in the literature [9,12,13], but some steps in the derivation would need further inspection and the mathematical 'qualities' of the resulting models have not been analysed yet. One can also refer to [6,11] for related models.…”

“…Plusieurs modèles de ce type existent [10,14,15] mais ils sont phénoménologiques et non basés sur un établissement rigoureux à partir des interactions individuelles. Plusieurs tentatives pour établir des modèles continus à partir du modèle CVA ont été proposées dans la littérature [9,12,13] mais certaines étapes de la dérivation demandent à être précisées et les qualités mathématiques des modèles obtenus n'ont pas encore été analysées. On pourra égale-ment se référer à [6,11] pour des modèles connexes.…”