Asymptotic FixedSpeed Reduced Dynamics for Kinetic Equations in Swarming
Abstract
We perform an asymptotic analysis of general particle systems arising in collective behavior in the limit of large selfpropulsion and friction forces. These asymptotics impose a fixed speed in the limit, and thus a reduction of the dynamics to a sphere in the velocity variables. The limit models are obtained by averaging with respect to the fast dynamics. We can include all typical effects in the applications: shortrange repulsion, longrange attraction, and alignment. For instance, we can rigorously show that the CuckerSmale model is reduced to the Vicsek model without noise in this asymptotic limit. Finally, a formal expansion based on the reduced dynamics allows us to treat the case of diffusion. This technique follows closely the gyroaverage method used when studying the magnetic confinement of charged particles. The main new mathematical difficulty is to deal with measure solutions in this expansion procedure.
Keywords:
Vlasovlike equations, Measure solutions, Swarming, CuckerSmale model, Vicsek model, LaplaceBeltrami operator.
AMS classification:
92D50, 82C40, 92C10.
1 Introduction
This paper is devoted to continuum models for the dynamics of systems involving living organisms such as flocks of birds, school of fish, swarms of insects, myxobacteria… The individuals of these groups are able to organize in the absence of a leader, even when starting from disordered configurations [37]. Several minimal models describing such selforganizing phenomenon have been derived [38, 28, 19]. Most of these models include three basic effects: shortrange repulsion, longrange attraction, and reorientation or alignment, in various ways, see [33] and particular applications to birds [32] and fish [1, 2].
We first focus on populations of individuals driven by selfpropelling forces and pairwise attractive and repulsive interaction [34, 25]. We consider selfpropelled particles with Rayleigh friction [17, 16, 11, 14], leading to the Vlasov equation in dimensions:
(1) 
where represents the particle density in the phase space at any time , stands for the acceleration
and is the pairwise interaction potential modelling the repelling and attractive effects. Here, the propulsion and friction forces coefficients , are scaled in such a way that for particles will tend to move with asymptotic speed . These models have been shown to produce complicated dynamics and patterns such as mills, double mills, flocks and clumps, see [25]. Assuming that all individuals move with constant speed also leads to spatial aggregation, patterns, and collective motion [21, 26].
Another source of models arises from introducing alignment at the modelling stage. A popular choice in the last years to include this effect is the CuckerSmale reorientation procedure [20]. Each individual in the group adjust their relative velocity by averaging with all the others. This velocity averaging is weighted in such a way that closer individuals in space have more influence than further ones. The continuum kinetic version of them leads to Vlasovlike models of the form (1) in which the acceleration is of the form
where stands for the convolution, abusing a bit on the notation, with the nonnegative interaction kernel . In the original CuckerSmale work, the interaction is modelled by , with the weight function being a decreasing radial nonnegative function. We refer to the extensive literature in this model for further details [31, 29, 12, 13, 35].
In this work, we will consider the Vlasov equation (1) where the acceleration includes the three basic effects discussed above, and then takes the form:
(2) 
We will assume that the interaction potential , bounded continuous with bounded continuous derivatives up to second order, and with and nonnegative. Under these assumptions the model (1)(2) can be rigorously derived as a meanfield limit [36, 9, 24, 10, 3] from the particle systems introduced in [25, 20].
We will first study in detail the linear problem, assuming that the acceleration is a given globalintime bounded smooth field. We investigate the regime , that is the case when the propulsion and friction forces dominate the potential interaction between particles. At least formally we have
(3) 
where
(4) 
(5) 
up to first order. Therefore, to characterize the zeroth order term in the expansion we need naturally to work with solutions whose support lies on the sphere of radius denoted by with . In turn, we need to work with measure solutions to (4) which makes natural to set as functional space the set of nonnegative bounded Radon measures on denoted by . We will be looking at solutions to (1) which are typically continuous curves in the space with a suitable notion of continuity to be discussed later on. We will denote by the integration against the measure solution of (1) at time . For the sake of clarity, this is done independently of being the measure absolutely continuous with respect to Lebesgue or not, i.e., having a density or not.
Proposition 1.1
Assume that . Then is a solution to (4) if and only if .
The condition (4) appears as a constraint, satisfied at any time . The time evolution of the dominant term in the Ansatz (3) will come by eliminating the multiplier in (5), provided that verifies the constraint (4). In other words we are allowed to use those test functions which remove the contribution of the term i.e.,
Therefore we need to investigate the invariants of the field . The admissible test functions are mainly those depending on and . The characteristic flow associated to
will play a crucial role in our study. It will be analyzed in detail in Section 3. Notice that the elements of are the equilibria of . It is easily seen that the jacobian of this field
is negative on , saying that are stable equilibria. The point is unstable, . When the solutions concentrate on , leading to a limit curve of measures even if were smooth solutions. We can characterize the limit curve as solution of certain PDE whenever our initial measure does not charge the unstable point .
Theorem 1.1
Assume that , , . Then converges weakly in towards the solution of the problem
(6) 
(7) 
with initial data defined by
for all .
In the rest, we will refer to as the projected measure on the sphere of radius corresponding to . Let us point out that the previous result can be equivalently written in spherical coordinates by saying that is the measure solution to the evolution equation on given by
These results for the linear problem, when is given, can be generalized to the nonlinear counterparts where is given by (2). The main result of this work is (see Section 2 for the definition of ):
Theorem 1.2
Assume that , with nonnegative, , with . Then for all , the sequence converges in towards the measure solution on of the problem
(8) 
with initial data . Moreover, if the initial data is already compactly supported on , then the convergence holds in .
Let us mention that the evolution problem (8) on was also proposed in the literature as the continuum version [23] of the Vicsek model [38, 18] without diffusion for the particular choice and with some local averaging kernel. The original model in [38, 18] also includes noise at the particle level and was derived as the mean filed limit of some stochastic particle systems in [4]. In fact, previous particle systems have also been studied with noise in [3] for the meanfield limit, in [30] for studying some properties of the CuckerSmale model with noise, and in [22, 27] for analyzing the phase transition in the Vicsek model.
In the case of noise, getting accurate control on the particle paths of the solutions is a complicated issue and thus, we are not able to show the corresponding rigorous results to Theorems 1.1 and 1.2. Nevertheless, we will present a simplified formalism, which allows us to handle more complicated problems to formally get the expected limit equations. This approach was borrowed from the framework of the magnetic confinement, where leading order charged particle densities have to be computed after smoothing out the fluctuations which correspond to the fast motion of particles around the magnetic lines [5, 6, 7, 8]. We apply this method to the following (linear or nonlinear) problem
(9) 
with initial data where the acceleration and . By applying the projection operator to (9), we will show that the limiting equation for the evolution of on is given by
(10) 
where is the LaplaceBeltrami operator on .
Our paper is organized as follows. In Section 2 we investigate the stability of the characteristic flows associated to the perturbed fields . The first limit result for the linear problem (cf. Theorem 1.1) is derived rigorously in Section 3. Section 4 is devoted to the proof of the main Theorem 1.2. The new formalism to deal with the treatment of diffusion models is presented in Section 5. The computations to show that these models correspond to the Vicsek models, written in spherical coordinates, are presented in the Appendix A.
2 Measure solutions
2.1 Preliminaries on mass transportation metrics and notations
We recall some notations and result about mass transportation distances that we will use in the sequel. For more details the reader can refer to [39, 15].
We denote by the space of probability measures on with finite first moment. We introduce the socalled MongeKantorovichRubinstein distance in defined by
where denotes the set of Lipschitz functions on and the Lipschitz constant of a function . Denoting by the set of transference plans between the measures and , i.e., probability measures in the product space with first and second marginals and respectively
then we have
by Kantorovich duality. endowed with this distance is a complete metric space. Its properties are summarized below, see[39].
Proposition 2.1
The following properties of the distance hold:

Optimal transference plan: The infimum in the definition of the distance is achieved. Any joint probability measure satisfying:
is called an optimal transference plan and it is generically non unique for the distance.

Convergence of measures: Given and in , the following two assertions are equivalent:

tends to as goes to infinity.

tends to weakly as measures as goes to infinity and

Let us point out that if the sequence of measures is supported on a common compact set, then the convergence in sense is equivalent to standard weak convergence for bounded Radon measures.
Finally, let us remark that all the models considered in this paper preserve the total mass. After normalization we can consider only solutions with total mass and therefore use the MongeKantorovichRubinstein distance in . From now on we assume that the initial conditions has total mass .
2.2 Estimates on Characteristics
In this section we investigate the linear Vlasov problem
(11) 
(12) 
where and .
Definition 2.1
We introduce the characteristics of the field
We will prove that are well defined for any . Indeed, on any interval on which is well defined we get a bound
implying that the characteristics are global in positive time. For that we write
(13) 
and then, we get the differential inequality
for all , so that
Once constructed the characteristics, it is easily seen how to obtain a measure solution for the Vlasov problem (11)(12). It reduces to push forward the initial measure along the characteristics, see [10] for instance.
Proposition 2.2
Proof. The arguments are straightforward and are left to the reader. We only justify that meaning that for any the application
Taking into account that are locally bounded (in time, position, velocity) it is easily seen that for any compact set there is a constant such that
Our conclusion follows easily using the uniform continuity of and that . Notice also that the equality (14) holds true for any bounded continuous function .
We intend to study the behavior of when becomes small. This will require a more detailed analysis of the characteristic flows . The behavior of these characteristics depends on the roots of functions like , with , .
Proposition 2.3
Assume that and . Then the equation has two zeros on , denoted , satisfying
and
where .
Proof. It is easily seen that the function increases on and decreases on with change of sign on and . We can prove that are monotone with respect to . Take and observe that . In particular we have
implying , since is strictly increasing on . Similarly we have
and thus , since is strictly decreasing on . Passing to the limit in it follows easily that
Moreover we can write
and
saying that
The case can be treated is a similar way and we obtain
Proposition 2.4
Assume that and . Then the equation has one zero on , denoted , satisfying
Using the sign of the function we obtain the following bound for the kinetic energy.
Proposition 2.5
Proof. We know that
By comparison with the solutions of the autonomous differential equation associated to the righthand side, we easily deduce that
for any . This yields the following bound for the kinetic energy
The object of the next result is to establish the stability of around . We will show that the characteristics starting at points with velocities inside an annulus of length proportional to around the sphere get trapped there for all positive times for small .
Proposition 2.6
Assume that and that . Then, for any we have
Proof. As in previous proof, we know that
By comparison with the constant solution to the autonomous differential equation associated to the righthand side, we get that . Assume now that there is such that and we are done if we find a contradiction. Since , we can assume that by time continuity. Take now a minimum point of on . Obviously since
By estimating from below in (13) and using that is a minimum point of on , we obtain
But the function has negative sign on . Since we know that , it remains that
which contradicts the assumption .
Let us see now what happens when the initial velocity is outside . In particular we prove that if initially , then remains away from . We actually show that the characteristics starting away from zero speed but inside the sphere will increase their speed with respect to its initial value while those starting with a speed outside the sphere will decrease their speed with respect to its initial value, all for sufficiently small .
Proposition 2.7
Consider such that .
1. Assume that . Then for any we have
2. Assume that . Then for any we have
Proof. 1. Notice that if for some , then we deduce by Proposition 2.6 that for any and thus . It remains to establish our statement for intervals such that