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29
From particle to kinetic and hydrodynamic descriptions of flocking
 Kinetic and Related Methods
"... Abstract. We discuss the CuckerSmale’s (CS) particle model for flocking, deriving precise conditions for flocking to occur when pairwise interactions are sufficiently strong long range. We then derive a Vlasovtype kinetic model for the CS particle model and prove it exhibits timeasymptotic floc ..."
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Cited by 28 (2 self)
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Abstract. We discuss the CuckerSmale’s (CS) particle model for flocking, deriving precise conditions for flocking to occur when pairwise interactions are sufficiently strong long range. We then derive a Vlasovtype kinetic model for the CS particle model and prove it exhibits timeasymptotic flocking behavior for arbitrary compactly supported initial data. Finally, we introduce a hydrodynamic description of flocking based on the CS Vlasovtype kinetic model and prove flocking behavior without closure of higher moments. 1. Introduction. Collective
Asymptotic Flocking Dynamics for the kinetic CuckerSmale model
, 2009
"... Abstract. In this paper, we analyse the asymptotic behavior of solutions of the continuous kinetic version of flocking by Cucker and Smale [16], which describes the collective behavior of an ensemble of organisms, animals or devices. This kinetic version introduced in [24] is here obtained starting ..."
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Cited by 19 (4 self)
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Abstract. In this paper, we analyse the asymptotic behavior of solutions of the continuous kinetic version of flocking by Cucker and Smale [16], which describes the collective behavior of an ensemble of organisms, animals or devices. This kinetic version introduced in [24] is here obtained starting from a Boltzmanntype equation. The largetime behavior of the distribution in phase space is subsequently studied by means of particle approximations and a stability property in distances between measures. A continuous analogue of the theorems of [16] is shown to hold for the solutions on the kinetic model. More precisely, the solutions will concentrate exponentially fast their velocity to their mean while in space they will converge towards a translational flocking solution.
DOUBLE MILLING IN SELFPROPELLED SWARMS FROM KINETIC THEORY
"... (Communicated by Tong Yang) Abstract. We present a kinetic theory for swarming systems of interacting, selfpropelled discrete particles. Starting from the Liouville equation for the manybody problem we derive a kinetic equation for the single particle probability distribution function and the rela ..."
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Cited by 14 (5 self)
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(Communicated by Tong Yang) Abstract. We present a kinetic theory for swarming systems of interacting, selfpropelled discrete particles. Starting from the Liouville equation for the manybody problem we derive a kinetic equation for the single particle probability distribution function and the related macroscopic hydrodynamic equations. General solutions include flocks of constant density and fixed velocity and other nontrivial morphologies such as compactly supported rotating mills. The kinetic theory approach leads us to the identification of macroscopic structures otherwise not recognized as solutions of the hydrodynamic equations, such as double mills of two superimposed flows. We find the conditions allowing for the existence of such solutions and compare to the case of single mills.
Particle, Kinetic, and Hydrodynamic Models of Swarming
"... Summary. We review the stateoftheart in the modelling of the aggregation and collective behavior of interacting agents of similar size and body type, typically called swarming. Starting with individualbased models based on “particle”like assumptions, we connect to hydrodynamic/macroscopic descr ..."
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Cited by 8 (3 self)
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Summary. We review the stateoftheart in the modelling of the aggregation and collective behavior of interacting agents of similar size and body type, typically called swarming. Starting with individualbased models based on “particle”like assumptions, we connect to hydrodynamic/macroscopic descriptions of collective motion via kinetic theory. We emphasize the role of the kinetic viewpoint in the modelling, in the derivation of continuum models and in the understanding of the complex behavior of the system. Key words: swarming, kinetic theory, particle models, hydrodynamic descriptions, mean fields, pattern formation 1
Asymptotic dynamics of attractiverepulsive swarms
, 2008
"... We classify and predict the asymptotic dynamics of a class of swarming models. The model consists of a conservation equation in one dimension describing the movement of a population density field. The velocity is found by convolving the density with a kernel describing attractiverepulsive social in ..."
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Cited by 8 (0 self)
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We classify and predict the asymptotic dynamics of a class of swarming models. The model consists of a conservation equation in one dimension describing the movement of a population density field. The velocity is found by convolving the density with a kernel describing attractiverepulsive social interactions. The kernel’s first moment and its limiting behavior at the origin determine whether the population asymptotically spreads, contracts, or reaches steadystate. For the spreading case, the dynamics approach those of the porous medium equation. The widening, compactlysupported population has edges that behave like traveling waves whose speed, density and slope we calculate. For the contracting case, the dynamics of the cumulative density approach those of Burgers’ equation. We derive an analytical upper bound for the finite blowup time after which the solution forms one or more δfunctions.
Asymptotics of blowup solutions for the aggregation equation
, 2011
"... We consider the asymptotic behavior of radially symmetric solutions of the aggregation equation ut = ∇ · (u∇K ∗ u) in R n, for homogeneous potentials K = x  γ, γ> 0. For γ> 2, the aggregation happens in infinite time and exhibits a concentration of mass along a collapsing δring. We develop an a ..."
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Cited by 5 (2 self)
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We consider the asymptotic behavior of radially symmetric solutions of the aggregation equation ut = ∇ · (u∇K ∗ u) in R n, for homogeneous potentials K = x  γ, γ> 0. For γ> 2, the aggregation happens in infinite time and exhibits a concentration of mass along a collapsing δring. We develop an asymptotic theory for the approach to this singular solution. For γ < 2, the solution blows up in finite time and we present careful numerics of second type similarity solutions for all γ in this range, including additional asymptotic behavior in the limits γ → 0 + and γ → 2 −.
G.: A kinetic flocking model with diffusion
, 2009
"... We study the stability of the equilibrium states and the rate of convergence of solutions towards them for the continuous kinetic version of the CuckerSmale flocking in presence of diffusion whose strength depends on the density. This kinetic equation describes the collective behavior of an ensembl ..."
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Cited by 4 (1 self)
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We study the stability of the equilibrium states and the rate of convergence of solutions towards them for the continuous kinetic version of the CuckerSmale flocking in presence of diffusion whose strength depends on the density. This kinetic equation describes the collective behavior of an ensemble of organisms, animals or devices which are forced to adapt their velocities according to a certain rule implying a final configuration in which the ensemble flies at the mean velocity of the initial configuration. Our analysis takes advantage both from the fact that the global equilibrium is a Maxwellian distribution function, and, on the contrary to what happens in the CuckerSmale model [4], the interaction potential is an integrable function. Precise conditions which guarantee polynomial rates of convergence towards the
Phase transitions, hysteresis, and hyperbolicity for selforganized alignment dynamics. in preparation
"... We provide a complete and rigorous description of phase transitions for kinetic models of selfpropelled particles interacting through alignment. These models exhibit a competition between alignment and noise. Both the alignment frequency and noise intensity depend on a measure of the local alignmen ..."
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Cited by 3 (3 self)
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We provide a complete and rigorous description of phase transitions for kinetic models of selfpropelled particles interacting through alignment. These models exhibit a competition between alignment and noise. Both the alignment frequency and noise intensity depend on a measure of the local alignment. We show that, in the spatially homogeneous case, the phase transition features (number and nature of equilibria, stability, convergence rate, phase diagram, hysteresis) are totally encoded in how the ratio between the alignment and noise intensities depend on the local alignment. In the spatially inhomogeneous case, we derive the macroscopic models associated to the stable equilibria and classify their hyperbolicity according to the same function.
Emergence in Random Noisy Environments
, 909
"... We investigate the emergent behavior of four types of generic dynamical systems under random environmental perturbations. Sufficient conditions for nearlyemergence in various scenarios are presented. Recent fundamental works of F. Cucker and S. Smale on the construction and analysis of flocking mod ..."
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We investigate the emergent behavior of four types of generic dynamical systems under random environmental perturbations. Sufficient conditions for nearlyemergence in various scenarios are presented. Recent fundamental works of F. Cucker and S. Smale on the construction and analysis of flocking models directly inspired our present work. Keywords: emergence; CuckerSmale model; dynamical system; flocking; multiagent system; consensus problems; random noise; linguistic evolution.