Hydrodynamic model for the system of self propelling particles with conservative kinematic constraints; two dimensional stationary solutions

Abstract

In a first paper we proposed a continuum model for the dynamics of systems of self propelling particles with kinematic constraints on the velocities. The model aims to be analogous to a discrete algorithm used in works by T. Vicsek et al. Phys. Rev. Lett. 75 (1995) 1226. In this paper we prove that the only types of the stationary planar solutions in the model are either of translational or axial symmetry of the flow. Within the proposed model we differentiate between finite and infinite flocking behavior by the finiteness of the kinetic energy functional. © 2005 Elsevier B.V. All rights reserved

Continuum Limit of Self-Driven Particles With Orientation Interaction

“…[25,34,35]). Several attemps to derive continuum models from the CVA model are also reported in the literature [23,29,30], but the derivation and the mathematical 'qualities' of the resulting models have not been fully analyzed yet. One can also refer to [16,26] for related models.…”

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

“…Before comparing this expression to the one we used in Refs. [21,22] (19), is satisfied for each density distribution which varies only in directions normal to the flow direction. It follows that the continuous analog of the CVA model has stationary linear and vortical solutions.…”

Macroscopic limit of self-driven particles with orientation interaction

“…[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.…”

“…Ils ne sont pas connus explicitement mais le Lemme 3.2 permet de montrer que l'espace des invariants collisionnels généralisés (pour une direction de flux donnée) est un espace de dimension 3 (Proposition 3.1). Ils dépendent en fait d'une même fonction scalaire g(μ) de la variable μ ∈ (−1, 1) solution du problème elliptique (13). Une fois ces invariants collisionnels déterminés, le modèle macroscopique s'obtient en intégrant le modèle cinétique (1) contre ces invariants collisionnels et en faisant tendre ε vers 0.…”