From particle to kinetic and hydrodynamic descriptions of flocking

By Seung‐Yeal Ha, Eitan Tadmor. 2008

678 citations recorded by the source corpus.

We discuss the Cucker-Smale's (C-S) particle model for flocking, deriving precise conditions for flocking to occur when pairwise interactions are sufficiently strong long range. We then derive a Vlasov-type kinetic model for the C-S particle model and prove it exhibits time-asymptotic flocking behavior for arbitrary compactly supported initial data. Finally, we introduce a hydrodynamic description of flocking based on the C-S Vlasov-type kinetic model and prove flocking behavior without closure of higher moments.

Subfields: Computer Networks and Communications, Condensed Matter Physics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/44d08faad3d91fa26887ba2ed5849bc5cedb521e

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Citations in the corpus

Emergent dynamics of the Cucker-Smale flocking model and its variants (2016, methodology): “Preliminaries In this section, we briefly discuss hierarchical C-S models from microscopic scale to mesoscopic and macroscopic scales following the presentation in [42].”
A simple proof of the Cucker-Smale flocking dynamics and mean-field limit (2009, methodology): “Comparison with Cucker-Smale’s results.”
Numerical methods for kinetic equations* (2014, background): “2005), supply chains (Armbruster, Degond and Ringhofer 2007), flows on networks (Herty and Ringhofer 2011), flocking dynamics (Ha and Tadmor 2008) and many other.”
Avoiding Collisions in Flocks (2010, result): “[ 12 ] X. Feng, K. A. Loparo, Y. Ji, and H. J. Chizeck, “Stochastic stability”
Fokker–Planck equations in the modeling of socio-economic phenomena (2017, background): “[28, 82, 84] In the biological context, a consistent part of the recent research in population and behavioral biology and ecology has been looking for the emergent behavior of bird flocks, fish schools or bacteria aggregations.[26, 27, 38, 39, 40, 60, 61, 62] Other important examples of emergent beh”
Heterophilious Dynamics Enhances Consensus (2013, methodology): “Many models of self-organized dynamics are driven by examples from biology: these include aggregation of bacteria and amoeba [6, 51, 59, 74, 98], dynamics of insects [14, 33], schools of fish [1, 67, 111], flocking of birds [4, 26, 25, 24, 36, 37, 61, 93, 103, 107], and related models in ecology [58”
Mean Field Limit for Stochastic Particle Systems (2017, background): “One can see [17, 18, 40] and the references therein for a more detailed discussion of flocking or swarming models in the literature.”
A Kinetic Flocking Model with Diffusion (2010, methodology): “Equation (1.1) differs from the Ha-Tadmor model (1.4) in two essential features.”
A New Model for Self-organized Dynamics and Its Flocking Behavior (2011, methodology): “The flocking behavior of the C-S model derived in [15] was based on the 2-based arguments outlined in (1.”
Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system (2009, methodology): “In the mean-field limit (N →∞), the dynamics of the macroscopic variables is governed by the pure translation motion as in the noiseless case [2, 3, 8, 9].”
Asymptotic Flocking Dynamics for the Kinetic Cucker-Smale Model (2010, methodology): “The same equation has been derived and analyzed by Ha and Tadmor [24] as the mean-field limit of the discrete and finite dimensional model (2.1) by Cucker and Smale.”
Flocking estimates for the Cucker–Smale model with time lag and hierarchical leadership (2017, background): “[15, 17, 27, 37, 40, 46, 50, 52]), stochastic terms ([18, 23, 25]), pedestrian crowds (see [16, 31]), infinite-dimensional kinetic models (see [1, 4, 9, 12, 22, 28, 49]), topological models ([5, 29]), control models (see [6, 10, 11, 43, 51]).”
Emergent Behavior of a Cucker-Smale Type Particle Model With Nonlinear Velocity Couplings (2010, background): “Later Cucker and Smale’s flocking estimate was further simplified and improved to cover general communication weights and hierarchy structures in [9]–[12].”