Asymptotic Flocking Dynamics for the Kinetic Cucker–Smale Model

Abstract

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 Boltzmann-type equation. The large-time 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.

Collective Behavior of Animals: Swarming and Complex Patterns

“…On the other hand, in the regime g > 1/2, flocking can be expected under certain conditions on the initial configuration but there are counterexamples to the generic flocking [16]. We refer to [16,17,11] for further discussion about this model and its qualitative properties, but we do present now a short proof of the flocking behavior when g ≤ 1/2.…”

“…After several improvements, it has been shown in [16,17,22,21,11] that the asymptotic behavior of the system…”

“…our argument below can be carried out rigorously. We will show the proof for values of g < 1/2; improvements of this can be made following the steps in [11].…”

“…The case p = 2 needs a separate treatment to show that R v (t) is exponentially decreasing. These details can be found in [11] by the interested reader, here we will only report the details for the case p ≠ 2. Coming again to (2.5), we notice that W (t) is an increasing function of time and for 2 1 < c it diverges as t  ∞.…”

“…In fact, as usually done in statistical physics, there is a middle ground in modeling between particle and hydrodynamic descriptions given by the mesoscopic kinetic equations describing the probability of finding particles in phase space. Kinetic models of swarming has recently been proposed [22,10,11] and the connection between the above IBMs and the continuum models via kinetic theory has been tackled very recently in [21,10,9]. The interest of the kinetic theory models is to give a rigorous tool to connect IBMs and hydrodynamic descriptions as well as to interpret certain patterns as solutions of a given model, as in the case of the double mills [10].…”