A new flocking model through body attitude coordination

We present a new model for multi-agent dynamics where each agent is described by its position and body attitude: agents travel at a constant speed in a given direction and their body can rotate around it adopting different configurations. In this manner, the body attitude is described by three orthonormal axes giving an element in $SO(3)$ (rotation matrix). Agents try to coordinate their body attitudes with the ones of their neighbours. In the present paper, we give the Individual Based Model (particle model) for this dynamics and derive its corresponding kinetic and macroscopic equations.

[1]  Peter Constantin,et al.  On the high intensity limit of interacting corpora , 2010 .

[2]  Pierre Degond,et al.  SELF-ORGANIZED HYDRODYNAMICS IN AN ANNULAR DOMAIN: MODAL ANALYSIS AND NONLINEAR EFFECTS , 2014, 1406.7852.

[3]  R. Illner,et al.  The mathematical theory of dilute gases , 1994 .

[4]  Christian A. Ringhofer,et al.  A Model for the Dynamics of large Queuing Networks and Supply Chains , 2006, SIAM J. Appl. Math..

[5]  Laplacian on Riemannian manifolds , 2010 .

[6]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[7]  Maximino Aldana,et al.  Phase Transitions in Self-Driven Many-Particle Systems and Related Non-Equilibrium Models: A Network Approach , 2003 .

[8]  Pierre Degond,et al.  Phase transition and diffusion among socially interacting self-propelled agents , 2012, 1207.1926.

[9]  Maher Moakher,et al.  To appear in: SIAM J. MATRIX ANAL. APPL. MEANS AND AVERAGING IN THE GROUP OF ROTATIONS∗ , 2002 .

[10]  Jian-Guo Liu,et al.  Macroscopic Limits and Phase Transition in a System of Self-propelled Particles , 2011, Journal of Nonlinear Science.

[11]  J. S. Rowlinson,et al.  PHASE TRANSITIONS , 2021, Topics in Statistical Mechanics.

[12]  Pierre Degond,et al.  Evolution of wealth in a non-conservative economy driven by local Nash equilibria , 2014, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[13]  Ning Jiang,et al.  Hydrodynamic Limits of the Kinetic Self-Organized Models , 2015, SIAM J. Math. Anal..

[14]  Yoshio Sone,et al.  Kinetic Theory and Fluid Dynamics , 2002 .

[15]  Tu,et al.  Long-Range Order in a Two-Dimensional Dynamical XY Model: How Birds Fly Together. , 1995, Physical review letters.

[16]  P. Degond.  Macroscopic limits of the Boltzmann equation: a review , 2004 .

[17]  Dirk Helbing,et al.  Dynamics of crowd disasters: an empirical study. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  A. Sznitman.  Topics in propagation of chaos , 1991 .

[19]  José A. Carrillo,et al.  Mean-field limit for the stochastic Vicsek model , 2011, Appl. Math. Lett..

[20]  Herbert Levine,et al.  Cooperative self-organization of microorganisms , 2000 .

[21]  Christian A. Ringhofer,et al.  Stochastic Dynamics of Long Supply Chains with Random Breakdowns , 2007, SIAM J. Appl. Math..

[22]  Du Q. Huynh,et al.  Metrics for 3D Rotations: Comparison and Analysis , 2009, Journal of Mathematical Imaging and Vision.

[23]  Peter Constantin,et al.  The Onsager equation for corpora , 2008, 0803.4326.

[24]  H. Chaté,et al.  Onset of collective and cohesive motion. , 2004, Physical review letters.

[25]  P. Degond,et al.  Macroscopic models of collective motion with repulsion , 2014, 1404.4886.

[26]  Amic Frouvelle,et al.  A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters , 2009, 0912.0594.

[27]  Pierre Degond,et al.  Continuum limit of self-driven particles with orientation interaction , 2007, 0710.0293.

[28]  Joseph J. Hale,et al.  From Disorder to Order in Marching Locusts , 2006, Science.

[29]  Pierre Degond,et al.  A multi-layer model for self-propelled disks interacting through alignment and volume exclusion , 2015 .

[30]  Pierre Degond,et al.  Phase Transitions, Hysteresis, and Hyperbolicity for Self-Organized Alignment Dynamics , 2013, 1304.2929.

[31]  Vicsek,et al.  Novel type of phase transition in a system of self-driven particles. , 1995, Physical review letters.

[32]  I. Couzin,et al.  Collective memory and spatial sorting in animal groups. , 2002, Journal of theoretical biology.

[33]  Thierry Mora,et al.  Flocking and Turning: a New Model for Self-organized Collective Motion , 2014, ArXiv.

[34]  Pierre Degond,et al.  HYDRODYNAMICS OF SELF-ALIGNMENT INTERACTIONS WITH PRECESSION AND DERIVATION OF THE LANDAU–LIFSCHITZ–GILBERT EQUATION , 2012 .

[35]  Axel Klar,et al.  Derivation of Continuum Traffic Flow Models from Microscopic Follow-the-Leader Models , 2002, SIAM J. Appl. Math..

[36]  A. Czirók,et al.  Collective Motion , 1999, physics/9902023.

[37]  Y. Gliklikh.  Stochastic Analysis on Manifolds , 2011 .

[38]  G. Parisi,et al.  Scale-free correlations in starling flocks , 2009, Proceedings of the National Academy of Sciences.

[39]  I. Holopainen.  Riemannian Geometry , 1927, Nature.

[40]  Giacomo Dimarco,et al.  HYDRODYNAMICS OF THE KURAMOTO-VICSEK MODEL OF ROTATING SELF-PROPELLED PARTICLES , 2013, 1306.3372.

[41]  D. Helbing.  Traffic and related self-driven many-particle systems , 2000, cond-mat/0012229.

[42]  Moon-Jin Kang,et al.  Global Well-posedness of the Spatially Homogeneous Kolmogorov–Vicsek Model as a Gradient Flow , 2015, 1509.02599.

[43]  P. Cochat,et al.  Et al , 2008, Archives de pediatrie : organe officiel de la Societe francaise de pediatrie.

[44]  Michel Droz,et al.  Hydrodynamic equations for self-propelled particles: microscopic derivation and stability analysis , 2009, 0907.4688.

[45]  H. Swinney,et al.  Collective motion and density fluctuations in bacterial colonies , 2010, Proceedings of the National Academy of Sciences.

[46]  Pierre Degond,et al.  A continuum model for nematic alignment of self-propelled particles , 2015, 1509.03124.