Mean-field limit for the stochastic Vicsek model

Bolley, François; Cañizo, José A.; Carrillo, José A.

doi:10.1016/j.aml.2011.09.011

Cited by 112 publications

(145 citation statements)

References 8 publications

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“…By contrast, the SOH macroscopic model is a rigorous coarse-graining of the Vicsek model [35]. Its derivation starts from a mean-field equation (proved equivalent to the Vicsek model in the large particle number limit [40]) for the probability distribution of the particle positions x and orientations ω at time t . At large spatio-temporal scales, the alignment intensity and noise are large, whereas the interaction radius R is small.…”

Section: Methodsmentioning

confidence: 99%

Symmetry-breaking phase transitions in highly concentrated semen

Creppy

Plouraboué

Praud

et al. 2016

J. R. Soc. Interface.

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New experimental evidence of self-motion of a confined active suspension is presented. Depositing fresh semen sample in an annular shaped microfluidic chip leads to a spontaneous vortex state of the fluid at sufficiently large sperm concentration. The rotation occurs unpredictably clockwise or counterclockwise and is robust and stable. Furthermore, for highly active and concentrated semen, richer dynamics can occur such as self-sustained or damped rotation oscillations. Experimental results obtained with systematic dilution provide a clear evidence of a phase transition towards collective motion associated with local alignment of spermatozoa akin to the Vicsek model. A macroscopic theory based on previously derived self-organized hydrodynamics models is adapted to this context and provides predictions consistent with the observed stationary motion.

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Section: Methodsmentioning

confidence: 99%

Symmetry-breaking phase transitions in highly concentrated semen

Creppy

Plouraboué

Praud

et al. 2016

J. R. Soc. Interface.

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“…The rigorous mean-field limit has been proven for the Vicsek model in [5]. A key difference between the Vicsek model and the system (24)-- (25) is the way we compute the average in (21).…”

Section: Mean-field Limitmentioning

confidence: 99%

“…Notice that, effectively, \partial rel q is a purely imaginary quaternion, since Re((\partialq)q \ast ) = q \cdot \partialq = 0 by (1), and it can be identified with a vector in \BbbR 3 (recall Remark 2.1). With this notation the SOHQ corresponds to \partial t \rho + \nabla x \cdot (c 1 e \bfone (\= q)\rho ) = 0, (5) \rho (\partial t \= q + c 2 (e \bfone (\= q) \cdot \nabla x )\= q) + c 3 [e \bfone (\= q) \times \nabla x \rho ] \= q + c 4 \rho [\nabla x,rel \= q e \bfone (\= q) + (\nabla x,rel \cdot \= q)e \bfone (\= q)] \= q = 0, (6) where e \bfone is a vector in \BbbR 3 and e \bfone (\= q) denotes the rotation of e \bfone by the quaternion \= q, that is,…”

Section: Preliminary On Quaternionsmentioning

confidence: 99%

Quaternions in Collective Dynamics

Degond¹,

Frouvelle²,

Merino-Aceituno³

et al. 2018

Multiscale Model. Simul.

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Abstract. We introduce a model of multiagent dynamics for self-organized motion; individuals travel at a constant speed while trying to adopt the averaged body attitude of their neighbors. The body attitudes are represented through unitary quaternions. We prove the correspondence with the model presented in [P. Degond, A. Frouvelle, and S. Merino-Aceituno, Math. Models Methods Appl. Sci., 27 (2017), pp. 1005--1049], where the body attitudes are represented by rotation matrices. Differently from this previous work, the individual-based model introduced here is based on nematic (rather than polar) alignment. From the individual-based model, the kinetic and macroscopic equations are derived. The benefit of this approach, in contrast to that of the previous one, is twofold: first, it allows for a better understanding of the macroscopic equations obtained and, second, these equations are prone to numerical studies, which is key for applications.

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“…As the number of particles n increases, one could estimate the convergence of the particle model (2.1) by means of Wasserstein metrics [38,39]. Moreover, we anticipate that the convergence analysis of our spectral method with respect to the number of spectral modes N can be performed by adapting existing work on conservative spectral schemes [25,35].…”

Section: Vortex Formationmentioning

confidence: 98%

Spectral method for a kinetic swarming model

Gamba

Haack

Motsch

2015

Journal of Computational Physics

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In this paper we present the first numerical method for a kinetic description of the Vicsek swarming model. The kinetic model poses a unique challenge, as there is a distribution dependent collision invariant to satisfy when computing the interaction term. We use a spectral representation linked with a discrete constrained optimization to compute these interactions. To test the numerical scheme we investigate the kinetic model at different scales and compare the solution with the microscopic and macroscopic descriptions of the Vicsek model. We observe that the kinetic model captures key features such as vortex formation and traveling waves.

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Mean-field limit for the stochastic Vicsek model

Cited by 112 publications

References 8 publications

Symmetry-breaking phase transitions in highly concentrated semen

Symmetry-breaking phase transitions in highly concentrated semen

Quaternions in Collective Dynamics

Spectral method for a kinetic swarming model

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