Laurent Bougrain

Matthieu Duvinage

Edouard Klein

Inverse reinforcement learning to control a robotic arm using a Brain-Computer Interface

teaching and research institutions in France or abroad, or from public or private research centers.

Inverse reinforcement learning to control a robotic arm using a Brain-Computer Interface Laurent Bougrain 1,2 , Matthieu Duvinage 3 , Research Fellow, FNRS, Edouard Klein 1,4 Abstract-The goal of this project is to use inverse reinforcement learning to better control a JACO robotic arm developed by Kinova in a Brain-Computer Interface (BCI). A self-paced BCI such as a motor imagery based-BCI allows the subject to give orders at any time to freely control a device. But using this paradigm, even after a long training, the accuracy of the classifier used to recognize the order is not 100%. While a lot of studies try to improve the accuracy using a preprocessing stage that improves the feature extraction, we work on a postprocessing solution. The classifier used to recognize the mental commands will provide as outputs a value for each command such as the posterior probability. But the executed action will not only depend on this information. A decision process will also take into account the position of the robotic arm and previous trajectories. More precisely, the decision process will be obtained applying an inverse reinforcement learning (IRL) on a subset of trajectories specified by an expert. At the end of the workshop, the convergence of the inverse reinforcement algorithm has not been achieved. Nevertheless, we developed a whole processing chain based on OpenViBE for controlling 2Dmovements and we present how to deal with this high dimensional time series problem with a lot of noise which is unusual for the IRL community.

Index Terms-Inverse reinforcement learning, Brain-Computer Interfaces, Motor imaginery, Robotic arm

I. INTRODUCTION

Brain-Computer interfaces (BCI) [START_REF] Wolpaw | Brain-computer interfaces for communication and control[END_REF] interpret brain activity to produce commands on a computer or other devices like a robotic arm (see figure 1). A BCI therefore allows its user, and especially a person with high mobility impairment, to interact with its environment only using its brain activity.

A major difficulty to properly interpret the mental command lies in the fact that brain activity is very variable even if a particular task is reproduced identically. Beyond the noise acquired by the recording system, background brain activity, concentration, fatigue or medication of the subject are the source of this variability. This variability makes it difficult for the classifier to recognize the different mental commands. Specific preprocessings such as common spatial pattern filter [START_REF] Lotte | Spatially regularized common spatial patterns for EEG classification[END_REF] are useful to help distinguish the mental command. However, this effort is not always sufficient. It therefore becomes necessary to explore new solutions to address this variability. Thus, it is now necessary to make decision systems able to deal with this variability. This is why some projects introduce a reinforcement learning in their BCI system such modifying the classifier [START_REF] Fruitet | Automatic motor task selection via a bandit algorithm for a brain-controlled button[END_REF]. We propose to use reinforcement learning in a broader context.

In this project we studied how a reinforcement learning can improve the control of a robotic arm. More precisely, the decision process will take into account a subset of trajectories specified by an expert and the position of the robotic arm in addition to the usual outputs of the mental commands classifier.

II. METHODS

The goal of this study is to present the possible improvement on command recognition obtained by a post-processing performed by an inverse reinforcement learning algorithm. In this section, we first present the almost standard processing chain we used to obtain four different commands using motor imagery. Then we present how inverse reinforcement learning can help to better identify the mental order provided by the user.

A. A BCI system based on motor imagery

For controlling a neuroprosthesis of the upper limb several options are available nowadays. Firstly, the neural activity in the arm/hand area of the motor cortex can be directly recorded and decoded using invasive [START_REF] Hochberg | Reach and grasp by people with tetraplegia using a neurally controlled robotic arm[END_REF] or noninvasive electrodes ( [START_REF] Bradberry | Reconstructing three-dimensional hand movements from noninvasive electroencephalographic signals[END_REF], [START_REF] Ofner | Decoding of hand movement velocities in three dimensions from the eeg during continuous movement of the arm[END_REF]). But it is also possible using noninvasive electrodes to exploit various physiological phenomena such as sensorimotor rhythms, event-related desynchronization/eventrelated synchronization, event-related potential or steady-state visual evoked potentials. In particular, motor imaginary [START_REF] Pfurtscheller | Current trends in graz brain-computer interface (bci) research[END_REF] can be used to control a 2D cursor ( [START_REF] Wolpaw | Control of a two-dimensional movement signal by a noninvasive brain-computer interface in humans[END_REF], [START_REF] Yuan | Relationship between speed and eeg activity during imagined and executed hand movements[END_REF]) or perform a 3D control [START_REF] Royer | EEG control of a virtual helicopter in 3-dimensional space using intelligent control strategies[END_REF]. They can even be combined in a hybrid BCI [START_REF] Horki | Combined motor imagery and ssvep based bci control of a 2 dof artificial upper limb[END_REF]. We selected motor imagery for several reasons: i) intending to produce a real movement is more natural for controlling a neuroprosthesis, ii) no additional device is needed to produce stimulations if used in a self-paced mode [START_REF] Townsend | Continuous eeg classification during motor imagery-simulation of an asynchronous bci[END_REF] iii) it has been already used successfully with healthy people [START_REF] Guger | How many people are able to operate an eeg-based brain-computer interface (bci)?[END_REF] and patients [START_REF] Ang | Clinical study of neurorehabilitation in stroke using eeg-based motor imagery brain-computer interface with robotic feedback[END_REF] and iv) it can be used for rehabilitation [START_REF] Kaiser | First steps towards a motor-imagery based stroke bci: New strategy to set up a classifier[END_REF]. Nevertheless, the number of commands is small (two or three usually); the information transfer rate is slow (1 action per 8s); and the accuracy is not very high (80 %).

We used motor imagery (MI) in a system-paced BCI. Having a self-paced BCI is not essential for this study and it is technically easy to switch from one mode to the other. We defined a standard processing chain for motor imagery based on the parameters used for the Graz paradigm. We want to identify four commands corresponding to four motor imageries: left hand, right hand, both hands and feet. These four MI will allow us to control a robotic hand in a 2D horizontal space using respectively left, right, forward and backward commands [START_REF] Naeem | Seperability of four-class motor imagery data using independent components analysis[END_REF].

We used a conventional montage for MI when applying a preprocessing based on common-spatial filters [START_REF] Wang | Classifying eeg-based motor imagery tasks by means of time?frequency synthesized spatial patterns[END_REF], [START_REF] Bashashati | A survey of signal processing algorithms in bci based on electrical brain signal[END_REF], [START_REF] Lotte | Spatially regularized common spatial patterns for EEG classification[END_REF]. Then, among various possible classifiers to detect the MI [START_REF] Lotte | A review of classification algorithms for eeg-based brain-computer interfaces[END_REF], we selected linear discriminant analysis for its stability. More details are presented in the following sections.

1) Signal acquisition: We used a TMSi Refa amplifier with 32 EEG channels. We only selected 13 electrodes : Fz, FC5, FC1, FC2, FC6, C3, Cz, C4, CP5, CP1, CP2, CP6, Pz (see Fig. 2) located according to a layout 10/10 on a WaveGuard 32 channel sintered Ag/AgCl. This system use a AFz ground and a common average. We used a sampling rate of 512 Hz. 2) Pre-processing: We used a 4th order Butterworth bandpass filter 8-30 Hz to only keep mu and beta bands.

Then we applied a Common Spatial Pattern (CSP). This filter takes into account the distribution of each class of a two-classes classification. The variance of the filtered signal is maximal for one class and minimal for the other class. Thus, we want to extremize using generalized eigen value decomposition:

J(w) = wX 1 X T 1 w T wX 2 X T 2 w T = wC 1 w T wC 2 w T
where X i is the multichannel EEG signals from class i, C i is the EEG spatial covariance matrix for class i and w is the spatial filter to optimize. We obtained features f = log(wCw T ).

3) Motor imagery paradigm: Figure 3 presents our timing for motor imagery. Each session contains 20 trials per class. After the presentation of the cue, we analysis the signal for 3,5 seconds. The features are extracted for a 1-s period. We use a sliding window of 100 ms to repeat the analysis and confirm the decision of the classifier using a vote.

4) Classifier: For discriminating four motor imageries, we combined one-versus-all linear discriminant classifiers (one per class). In case of ambiguity, the longest distance to the separation plane shows the winner class.

5) Device: By default, the JACO arm can be controlled using a joystick. An API by Kinova is available to read sensors and send commands of movement for a specific direction and a specific duration. This API provides a virtual joystick. This mode of operation does not make it possible to specify the final position of the arm. Thus, interacting with the JACO arm via the API necessitates the definition of elementary movements (right, left, forward, backward, up.). The VRPN protocol already implemented in OpenViBE is a natural candidate to control the arm. Thus, we used a VRPN client/server using predefined action IDs which can be interpreted by the JACO arm as virtual joystick commands but sent through our application. The recording features also supports the recording of VRPN clients' commands.

B. Inverse Reinforcement Learning

Inverse Reinforcement Learning (IRL) is the problem of eliciting a succinct description of a task from demonstrations by an expert [START_REF] Ng | Algorithms for inverse reinforcement learning[END_REF]. This succinct description of the task can then be used to train an agent in order to make it imitate the expert. More formally, IRL assumes that an expert is acting optimally in an Markov Decision Process (MDP) [START_REF] Puterman | Markov decision processes: Discrete stochastic dynamic programming[END_REF] and seeks the reward function for which this expert is optimal. As noted in the existing literature, this is an ill-posed problem in the Hadamard sense. However, recent advances [START_REF] Klein | Structured Classification for Inverse Reinforcement Learning[END_REF] in the domain may make solving the IRL problem on large or complex tasks feasible.

In our setting, we would like to use IRL to alleviate the problem of accuracy in order recognition from BCI signals. By using information about previously recognized commands and learning from human-labelled movement sequences, it should be feasible to gain a certain consistency in the overall arm movement. To put it in another way, after seeing a few examples of the arm moving in a direct manner from point A to point B, one is unlikely to admit a command that make the arm flail in seemingly random directions.

Using hand-labelled arm trajectories as expert demonstrations, we wish to extract a reward function that could be used to train an agent to recognize commands from BCI signals.

The main challenges behind this task are the difficulty of finding a suitable MDP setting for casting the problem, the high dimensionality of BCI signals, the sparsity of data for both reward function inference and its optimization by an agent once the expert's actions have been analyzed by the IRL algorithm.

One of the main assumption of IRL is that the expert is acting optimally in an MDP with respect to an unknown reward function. Our goal when choosing a MDP setting for our experiment is to try to make that assumption hold. In previous test for the algorithm we used, the expert was explicitly created from a reward function and an MDP. Although the reward function was unknown to the IRL algorithm, it existed. Sharing the same MDP as the expert is one of the basic assumption made by the analysis of our algorithm. In this setting, however, the so-called expert is an omniscient agent as the path the arm followed was fixed in advance and the operator only had to follow it. There may or may not exist a MDP describing the process. We tried more than one characterization of the problem, discovering various flaws, and understanding better and better the subtleties of the exercise as we went on. This is described in the next section.

BCI signals typically are high dimensional time series with a lot of noise. From a signal processing perspective they are quite a challenge. This is very unusual for the IRL community who is more used to toy problems (although impressive applications have been published [START_REF] Abbeel | Autonomous helicopter aerobatics through apprenticeship learning[END_REF]) where the dimension is low and the observation perfect. IRL can be applied to partially observable environments, although it is not the direction we wish to take here as it has its own set of challenges, mainly related to computation cost explosion. The high dimensionality problem has been circumvented by the use of SCIRL, a new IRL algorithm that among other advantages is quite fast to run. The low signal to noise ratio, however, is at the heart of our problem and the very reason for the existence of this project. It raised its lot of problems when trying to come up with a reasonable MDP setting.

The model for our system being unknown yet (although modeling the brain have been promised over and over by scifi authors, it is not yet within the reach of a one-month project) we had to rely on sampling to make things work. This means that we had to rely on expert demonstration only to retrieve a reward function and to optimize it. Reward inference from expert data only is one of the marketed features of SCIRL. Having access to samples drawn by a random policy is one of the many ways to run a Reinforcement Learning (RL) algorithm, and the most accessible to us. The high practical cost of generating samples with a BCI prevented us from getting even that in the allocated timeframe.

To wrap up, although IRL may alleviate the accuracy problem in BCI driven settings, the many challenges this approach implies are far outside the comfort zone of the community.

III. RESULTS

We installed OpenViBE, a user-friendly open-source tool for BCIs, on a windows XP system. This system supports both a JACO robotic arm driver and a Refa32 amplifier driver.

We built OpenViBE scenarii for i) signal acquisition ii) common spatial pattern filter training iii) classifier training and iv) offline use (see Fig. 4 and Fig. 5).

A. Standard motor imagery recognition

State-of-the-art similar results were obtained with imaginary and actual movements. The best combination strategy was the one-vs-all combined with a voting classifier. There were much less confusion and thus, better overall performance. It often happened that some classifier outputs had a very high confident level while the correct class was not represented. Confusion matrices were similar in both conditions (see Table I).

B. IRL

Let us disclose the end story immediately: not all challenges exposed earlier were overcomed. The most hacky topic in the whole ordeal clearly was the composition of the state and action space of the MDP. Encouraging results were obtained on a simulation built to validate an initial approach. Sadly, this failed to generalize to the real thing as the real noise was much higher than modelized. A second, more sound approach was built, in which the state space directly consists in the output of the spatial filters and the last decision taken by the agent. This parametrization did not show any deep flaw and would be our goto parametrization if we are given the opportunity to work on this problem again.

The high dimensionality of the MDP was not a problem for our IRL algorithm, which was indeed able to infer a reward only from a few expert demonstration (corresponding to less than an hour of work for the operator).

Sadly, and this is the blocking point of the experiment so far, we were not able to train an agent on this reward. We need more data, specifically data sampled with a policy different from the expert's, in order to use the basic RL algorithm we tried to use [START_REF] Lagoudakis | Least-squares policy iteration[END_REF]. We were thus not able to assess the quality of the found reward, although the fast convergence of SCIRL let us hope that it was good. We hope to solve this problem by either using less data greedy algorithm [START_REF] Geist | Kalman Temporal Differences[END_REF] or brutally generating more data (cumbersome for the operator). Another solution would be to use spatial filters able to deal with a displacement of the BCI, in order to allow the use of data from different sessions.

IV. CONCLUSION

We developed a whole processing chain using OpenViBE for controlling a robotic arm. According to the literature, we designed OpenViBE's scenarii (acquisition, filtering, classification and on-line use) based on a classic motor imagery paradigm. We selected four motor imageries (left hand, right hand, both hands and feet). They are respectively associates with 2D-movement (left move, right move, forward, backward). We used common spatial filters and one-versus-therest (linear discriminant analysis) classifiers. Our goal was not to improve the paradigm parameters but study how inverse reinforcement learning can help to select the right movement according to the classifier outputs and stored trajectories. Our classifier accuracy corresponds to the state-of-the-art. Thus, it is possible to control the Jaco to press a button. But, up to now, the IRL algorithm is not converging so cannot help to perform the right movement. Nevertheless, a significant analysis of the difficulties to apply IRL on high dimensional and noisy problem and ways to overcome them has been done.

Hand-labelled expert's trajectories

Signals from the BCI are passed through the spatial filters and handlabelled with the action the expert was thinking about.

SCIRL

Reward function

The SCIRL algorithm outputs a reward function over the state space using only the expert's data.

LSPI

Policy

An agent must be trained on the reward to get a control policy. This can only be done if enough data is available. This was not the case here.

✗ Non expert data

A non trivial amount of data is needed in order to learn a policy from a reward function. Such an amount of data was not available. V. PERSPECTIVES A deeper study is necessary for understanding the nonconvergence of the IRL algorithm. If the IRL algorithm is robust enough, we will modify the processing chain to have a self-paced BCI. In the future, we also would like to use multiclass classifiers to avoid ambiguities due to the oneversus-therest approach. We would like to explore the tongue motor imagery to replace the both hands one. This choice avoids overlapped locations with the other motor imageries. We need to assess performance in offline and online conditions with a large population.

Fig. 1 .

 1 Fig. 1. The Brain-Computer Interface loop : from electroencephagraphic signals acquisition, feature extraction and classification to feedback. Our project will add a decision process based on an inverse reinforcement learning in the command identification module.

Fig. 2 .

 2 Fig. 2. Position of the selected electrodes for motor imagery of left hand, right hand, both hands and feet. The green electrode corresponds to the ground, the black ones are the main locations of our motor imageries and the red ones are useful for common spatial patterns.

Fig. 3 .

 3 Fig. 3. Timing used for the motor imagery paradigm.

Fig. 4 .

 4 Fig. 4. OpenViBE scenario designed for training the one-versus-all classifiers.

Fig. 5 .

 5 Fig. 5. OpenViBE scenario designed for on-line use. EEG signals are recorded, filtered, classified and one movement is sent to the robotic arm via the API.

Fig. 6 .

 6 Fig. 6. Visual explanation of the IRL pipeline.

TABLE I CONFUSION

 I MATRICE OBTAINED ON A TESTING SESSION.

	Correct classes	left hand right hand both hands feet	left hand 0.9 0.1 0.1 0.1	Predicted classes right hand t both hands 0 0.05 0.8 0.1 0.05 0.85 0.1 0	feet 0.05 0 0.05 0.8

Laurent Bougrain is an associate professor at the university of Lorraine (France). He is a member of the Inria team NeuroSys dedicated to computational neuroscience at LORIA/Inria Nancy grand Est. He has been working for more than a decade on time series analysis with a focus on experimental data obtained during neuroscientific experiments. In recent years, he has dedicated his research to Brain-Computer Interfaces (BCI). He is working on template-based classifier for single trial detection using multichannel denoising techniques. He is the winner of the international BCI competition IV of the challenge about predicting the finger flexion from ECoG in 2008. He is currently working on a project on reinforcement learning to control a robotic arm and a wheelchair from EEG. He also collaborates to the worldwide BCIsoftware OpenVibe (http://openvibe.inria.fr). E-mail: bougrain@loria.fr, http: //www.loria.fr/ ∼ bougrain.

Matthieu Divinage As a TIME student, Matthieu Duvinage holds an Electrical Engineering degree from the Facult Polytechnique of Mons (UMons, Belgium, 2009) and one degree from SUPELEC (France, 2009). He also holds a degree of fundamental and applied physics from Paris Sud XI Orsay (France, 2009) and a degree of management science from the School of Management at the University of Louvain (UCLouvain, 2011). His master thesis was performed at the Multitel research center (Mons, Belgium) and dealt with robust low complexity speech recognition using frame dropping based on voicing information and clustering techniques. He obtained an F.R.S-FNRS grant for pursuing a PhD thesis about the development of a lower limb prosthesis driven by a neural command in close partnership with the Universit Libre de Bruxelles (ULB).

Edouard Klein is a PhD student co-supervised by Yann Guermeur (ABC team, CNRS), Matthieu Geist (IMS research group, Suplec) and Olivier Pietquin (IMS research Group, Suplec and UMI 2958 (GeorgiaTech -CNRS)). The topic of his PhD is automatic feature selection in inverse reinforcement learning. He received the Electrical Engineering degree of PHELMA (Grenoble, France) in 2010. Since that, he has worked on reinforcement learning and learning from demonstration in the IMS research group of Suplec.


Laurent Navoret
email: laurent.navoret@math.unistra.fr

A two-species hydrodynamic model of particles interacting through self-alignment

Keywords: Individual based model, sheep behaviour, Vicsek model, asymptotic analysis, orientation interaction, hydrodynamic limit, collision invariants

 for the Vicsek model: we find out a regime where alignment in the whole population occurs. We obtain a new macroscopic model for the densities of each populations and the common mean direction of the particles. The treatment of the non-conservativity of the interactions requires a detail study of the linearised interaction operator.

Introduction

The modelling of flocking behaviour, like flock of birds or school of fish, has recently been the subject of a vast literature: one of the main issue tackled in these works is the emergence of collective movement from only local interactions between neighbouring animals, without leaders. To describe such dynamics, a first modeling approach is to consider a system of self-propelled particles, i.e. particles moving with a preferred speed, and to propose interaction rules: a first class of models have considered binary attraction-repulsion interactions [START_REF] Carrillo | Double milling in self-propelled swarms from kinetic theory[END_REF][START_REF] Chuang | State transitions and the continuum limit for a 2D interacting, self-propelled particle system[END_REF]. These models can be completed by alignment interactions [START_REF] Aoki | A simulation study on the schooling mechanism in fish[END_REF][START_REF] Couzin | Collective memory and spatial sorting in animal groups[END_REF]. Note that empirical studies on flock of starlings [START_REF] Ballerini | Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study[END_REF][START_REF] Cavagna | From empirical data to inter-individual interactions: unveiling the rules of collective animal behavior[END_REF] have also been carried out to characterise the interaction law. A vast literature also focuses only on alignment interactions dynamics (without attraction-repulsion interactions) and in particular on the Vicsek model [START_REF] Vicsek | Novel Type of Phase Transition in a System of Self-Driven Particles[END_REF]: each particles tends to align with the mean direction of its neighbours. In this paper, we propose a model where the particles also follow Vicsek alignment interactions but can be either moving or at rest. This model is motivated by the displacement dynamics of gregarious animals, like the movement of sheep herds. The population is thus split in two phases with two independent alignment dynamics: one phase made of particles with a prescribed non-zero speed and the other phase made of zero-speed particles. The passage of the particles from the moving to the non-moving state (and inversely) makes the two phases interact and so raises the question whether the large scale alignment in each phases and the alignment of the two phases themselves could occur.

To describe and to characterise the large scale dynamics of animal populations, we are interested in macroscopic models that provide the dynamics of macroscopic quantities like the density or the mean velocity inside the flock. Some of these models are phenomenological (e.g. [START_REF] Mogilner | A non-local model for a swarm[END_REF][START_REF] Topaz | A Nonlocal Continuum Model for Biological Aggregation[END_REF]) while others are obtained as the mean-field limits of individual based models [START_REF] Carrillo | Double milling in self-propelled swarms from kinetic theory[END_REF][START_REF] Degond | A macroscopic model for a system of swarming agents using curvature control[END_REF][START_REF] Chuang | State transitions and the continuum limit for a 2D interacting, self-propelled particle system[END_REF][START_REF] Carrillo | Self-propelled interacting particle systems with roosting force[END_REF]. Concerning the Vicsek model, several macroscopic (kinetic or continuum) models have been proposed [START_REF] Bertin | Boltzmann and hydrodynamic description for self-propelled particles[END_REF][START_REF] Czirók | Collective behavior of interacting self-propelled particles[END_REF] but Degond and Motsch [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF] first provided a mathematical derivation from the microscopic model. Such derivations raise interesting questions about propagation of chaos [START_REF] Bolley | Mean-field limit for the stochastic Vicsek model[END_REF][START_REF] Bolley | Stochastic Mean-Field Limit: Non-Lipschitz Forces & Swarming[END_REF], and enable to study the convergence rate to the long-time asymptotic flocking states [START_REF] Cucker | Emergent behavior in flocks[END_REF][START_REF] Ha | A simple proof of the Cucker-Smale flocking dynamics and mean-field limit[END_REF][START_REF] Ha | From particle to kinetic and hydrodynamic descriptions of flocking[END_REF][START_REF] Carrillo | Asymptotic flocking dynamics for the kinetic cucker-smale model[END_REF]. As the original Vicsek algorithm [START_REF] Vicsek | Novel Type of Phase Transition in a System of Self-Driven Particles[END_REF][START_REF] Grégoire | Onset of collective and cohesive motion[END_REF][START_REF] Aldana | Phase Transitions in Self-Driven Many-Particle Systems and Related Non-Equilibrium Models: A Network Approach[END_REF][START_REF] Grégoire | Onset of collective and cohesive motion[END_REF][START_REF] Frouvelle | Dynamics in a kinetic model of oriented particles with phase transition[END_REF], macroscopic models can also present phase transition phenomena [START_REF] Frouvelle | Dynamics in a kinetic model of oriented particles with phase transition[END_REF] from disordered to ordered configurations. Several variants of the Vicsek model have been studied: the noise can be implemented in different ways [START_REF] Chaté | Collective motion of self-propelled particles interacting without cohesion[END_REF], the noise and the alignment frequency can be made dependent on the local density, an angle of vision can be added [START_REF] Frouvelle | A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters[END_REF]. We also refer to [START_REF] Carrillo | Particle, kinetic, and hydrodynamic models of swarming[END_REF] for a review on swarming models. We mention that models have also been proposed for vehicular/pedestrian traffic [START_REF] Helbing | Traffic and related self-driven many-particle systems[END_REF][START_REF] Bellomo | On the modeling of traffic and crowds: A survey of models, speculations, and perspectives[END_REF] but they differ from the previous ones since each particle then pursue a goal.

As announced above, we study here a variant of the Vicsek model for the displacement of a system of particles which can either move or not. Indeed, this work is motivated by the behaviour of sheep in groups during grazing period: sheep alternate between motionless time when grazing and displacements to look for fresher grass. In our model, the (biological) state of each particles is then described by a speed variable which takes only two values. The change between the two states of the animal can be either spontaneous or triggered by fellows. Moreover, as biologists experimentally show it in [START_REF] Pillot | Moving together: incidental leaders and naïve followers[END_REF], any member of the group can initiate a movement: this is called distributed leadership. Therefore, in the model we consider, the change of speed is modelled by a Markovian jump process, whose jump rates are depending on the local alignment to the neighbouring particles: the more one motionless particle is aligned with moving particles, the more it is likely to interact and then to change its speed. Comparisons with experimental data should be required to confirm or improve such interacting law. For more information about herding, we refer to [START_REF] Pillot | Collective movements, initiation and stops: diversity of situations and law of parsimony[END_REF][START_REF] Pillot | Scalable Rules for Coherent Group Motion in a Gregarious Vertebrate[END_REF][START_REF] Couzin | Self-organization and collective behavior in vertebrates[END_REF]. Here after, we will study this model as a minimal model of synchronization between two phases with different characteristic speeds and exchanges of individuals between the two phases.

Based on this microscopic dynamics, a mean field kinetic equation with a discrete speed dynamics is considered: the Vicsek model is coupled with a Markovian jump process modelling the change of speed. This kinetic model is the starting point of our study. This model is then discrete in the speed variable and thus shares some analogies with the discrete kinetic models developed for traffic modeling in [START_REF] Delitala | Mathematical modeling of vehicular traffic: a discrete kinetic theory approach[END_REF][START_REF] Bellomo | On the modeling of traffic and crowds: A survey of models, speculations, and perspectives[END_REF]. The discreteness of the velocity modulus space is not a mathematical simplification (as it is the case for the Broadwell or Carleman models [START_REF] Gatignol | Théorie cinétique des gaz à répartition discrète de vitesses[END_REF]), but is here a modelling assumption: as in traffic dynamics, the distribution function in the speed variable is really discontinuous [START_REF] Pillot | Moving together: incidental leaders and naïve followers[END_REF]. However, we note that our model strongly differs from all these works since the velocity direction space is kept here as a continuum, i.e. the unit circle S 1 . Note that Markovian jump models have also been proposed for the population dynamics of cells, in particular to model proliferating or destructive interactions [START_REF] Bellouquid | Mathematical methods and tools of kinetic theory towards modelling complex biological systems[END_REF][START_REF] Bellomo | Multiscale biological tissue models and flux-limited chemotaxis for multicellular growing systems[END_REF]: unlike these models, the particles are here conserved in time.

The main purpose of the paper is to investigate the large scale dynamics of this twophase kinetic model. Two time scales corresponding to the two types of interactions, are involved in this model: the time scale ε of the Vicsek interactions and the time scale δ of the speed changes. Here, we focus on the asymptotic regime ε ≪ δ ≪ 1, where the Vicsek interactions are more frequent than the speed changes and where both time scales are small compared to the macroscopic one: to this aim, we first consider the dynamics in the regime ε ≪ 1, δ = O(1) before investigating the δ → 0 limit. Following the methodology introduced in [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF], in the large scale Vicsek limit (ε → 0), we obtain two macroscopic Vicsek models, giving the dynamics of the densities and the mean directions for each phases (moving and non-moving phase), coupled through the speed change operator. The specificity of this model is that the macroscopic speed change operator do not conserve the global mean direction. Indeed, they are derived from the integration of the kinetic speed change operator against "generalised collisional invariants" [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF] which are specific for each Vicsek operators: they depend on the parameters of the Vicsek operator, which can be different for the moving and the non-moving phase.

The second step now consists in obtaining an averaged two-phase model, once the equilibria of the speed change dynamics are reached. The derivation of such "simplified" averaged models is a very challenging issue in two-phase fluid dynamics and has been the subject of a lot of works [START_REF] Ishii | Thermo-fluid dynamic theory of two-phase flow[END_REF]. To achieve our goal, we first find out the equilibria of the speed-change operator: thanks to a careful study of the speed change operator, we are able to show that the two phases are either locally aligned or locally in the opposite direction. Moreover, the densities of the two phases are linked through a non-linear balance equation. To find out the dynamics of these equilibria, we face once again the non-conservativity nature of the model: indeed, this property prevent us from having obvious "collisional invariants" that would result in balance simplifications. To overcome this difficulty, we perform an Hilbert expansion around the equilibria and we figure out the kernel and the image of the linearised exchange operator, which acts on the two-dimensional space of the mean directions: any non-zero element of the one-dimensional orthogonal space of its image then provide a "generalised collisional invariants". Supposing that the equilibria are reached, we thus find out the dynamical system satisfied by the total density and the common direction of the two-phases.

All this methodology enables to provide the dynamics of the two-phases, once they aligned. It provides a new non-conservative model for swarming population, whose several mathematical properties are yet left open. For instance, the stability of the equilibria and the hyperbolicity of the macroscopic model will be investigated in future works. In addition, this model raises some interesting questions about the modelling of alignment interactions in herds: what is the macroscopic behaviour in other regime of parameters (for instance if the Vicsek interactions are less frequent than the speed changes) ? We can ask also about the appearance of phase transitions in such discrete speed dynamics.

The outline of this article is as follows. In Section 2, we introduce the two-phase model at the particle and the kinetic level. In Section 3, we present the macroscopic regime we are interested in: we perform a hydrodynamic rescaling and uncouple the Vicsek and the speed change time scales. We then state the two main results. Sections 4 and 5 are devoted to the proofs of the derived hydrodynamic models: as explained above, we first focus on the large scale Vicsek dynamics (sect. 4) and then in the large scale speed change asymptotic (sec. 5). In the two steps, we figure out the equilibria and close the system using collisional invariants. Appendices A, B and C provide some detailed computations.

A two-speed Vicsek model

We present in this section an individual-based model and its mean-field kinetic version to describe alignment interactions in a system of self-propelled particles, that can move and stop.

The microscopic model

We consider N particles with positions X k ∈ R 2 and velocities The Vicsek dynamics within the subgroups. The interactions among particles of the same phase are given by the Vicsek model, as described in [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF]: a particle is supposed to have a mimetic behaviour with the neighbouring congeners being in the same state (moving or at rest). The dynamics of positions X k and orientations ω k are given by the following equations:

V k = cη k ω k ∈ R 2 for k ∈ {1, . . . , N}, where η k ∈ {0, 1} and ω k ∈ S 1 = {ω ∈ R 2 , |ω| = 1}
dX k dt = c η k ω k , dω k = (Id -ω k ⊗ ω k )(ν k ωk dt + 2d k dB t ), (1)
where Id denotes the identity matrix, w ⊗v denotes the tensor product of the two vectors w and v. The operator (Idω k ⊗ ω k ) is the projection operator onto the orthogonal plane to ω k : it ensures the norm of the direction ω k to be unity. Two dynamics are in competition: each particle tend to align with the mean direction ωk of their neighbours in the disc of radius R around them :

ωk = J k |J k | , J k = j, η j =η k , |X j -X k |≤R ω j , (2)
and noise is applied to the direction with a Brownian motion B t on R 2 . Note that in the definition of the local mean direction J k , only the particles of the same phase are taken into account. This two behaviours are quantified by the alignment intensities ν k and the noise intensities d k . These two parameters are uniform in each subgroup:

(ν k , d k ) = (ν 0 , d 0 ), if η k = 0, (ν 1 , d 1 ), if η k = 1.
At this level, the dynamics of the two phases are totally independent: the moving particles follows the Vicsek model with parameters (ν 1 , d 1 ) and the motionless particles1 follows a static Vicsek model with parameter (ν 0 , d 0 ). The static Vicsek model is similar to the Ising model, that provides the dynamics of spins distributed on a lattice.

The speed change: the Markov process η k . The particles can also change their speeds η k from the moving (η k = 1) to the motionless state (η k = 0) and from the motionless to the moving state: it results in a permanent exchange of the particles between the moving and the unmoving phases. This exchange between the two subgroups are described by the dynamics of η k : it is a time-continuous Markov process on the state space {0, 1}. The transition rates are given by:

g k = τ k     1 + α 1 N j, η j =η k , |X j -X k |≤R (1 + ω k • ω j ) 2     , (3)
where τ k is the intrinsic rate which can be can take different values for the two subgroups:

τ k = τ 0 , if η k = 0, τ 1 , if η k = 1.
The second term in the sum is of order ατ k and makes the rate depend on the local alignment with the members of the other phase : a particle at rest (resp. moving) is all the more likely to change its speed as it is locally aligned with its neighbouring moving congeners (resp. congeners at rest). It is aimed at describing the distributed leadership observed in herds [START_REF] Pillot | Moving together: incidental leaders and naïve followers[END_REF]: each particle can bring forth a speed change of its neighbours that are aligned with it.

The two Vicsek models are now coupled via the passage of the particles from one subgroup to the other. The dependence of the transition rates on the alignment of the two subgroups might bring forth the alignment of the whole population. The goal of this paper is to determine for which set of parameters, alignment in the whole system occurs.

The mean field kinetic model

We introduce the two distribution functions in phase space: f 0 (x, ω, t) for the particles at rest and f 1 (x, ω, t) for the moving particles. The mean field model we consider is the following:

∂ t f 0 = Q 0 (f 0 ) + E(f 0 , f 1 ), ( 4
) ∂ t f 1 + c ω • ∇ x f 1 = Q 1 (f 1 ) -E(f 0 , f 1 ), (5)
where ∇ x denotes the space gradient operator. The left-hand sides of these two equations are the transport operators of the particles with velocities 0 × ω and 1 × ω, while the right-hand sides model the velocity dynamics of the particles.

The operator Q 0 and Q 1 are the Vicsek operators:

Q 0 (f 0 ) = -∇ ω • (ν 0 F [f 0 ]f 0 ) + d 0 ∆ ω f 0 , (6) Q 1 (f 1 ) = -∇ ω • (ν 1 F [f 1 ]f 1 ) + d 1 ∆ ω f 1 , (7)
with ∇ ω • and ∆ ω are respectively the divergence and the Laplace operators2 on the circle S 1 . Note that the operators, Q 0 and Q 1 , differ from one to each other only in the couple of parameters (ν 0 , d 0 ) and (ν 1 , d 1 ). The operator F denotes the alignment forces in each group and is given by:

F [f ](x, ω, t) = (Id -ω ⊗ ω) ω[f ](x, t), (8) ω
[f ](x, t) = J [f ](x, t) |J [f ](x, t)| , J [f ](x, t) = y∈R 2 ,v∈S 1 K(|x -y|)vf (y, v, t)dydv, (9)
where K(|x|) is the interaction kernel equal to the indicator function of the disc of radius R. These operators was derived in [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF]: without Brownian motion (d 0 = d 1 = 0) nor the operator E(f 0 , f 1 ), equations ( 4)-( 5) are also satisfied by the empirical distribution functions of the particles following the Vicsek rules (1) and so the mean-field limit can be investigated: a rigorous study, carried out in [START_REF] Bolley | Mean-field limit for the stochastic Vicsek model[END_REF], justifies the resulting kinetic equations for the one-particle distribution functions. We refer also to [START_REF] Spohn | Large Scale Dynamics of Interacting Particles[END_REF] for theoretical developments on mean-field limits. Moreover, numerical simulations [START_REF] Motsch | Numerical simulations of a non-conservative hyperbolic system with geometric constraints describing swarming behavior[END_REF] provide a numerical validation of this kinetic description at least in some range of parameters.

The exchange term E(f 0 , f 1 ) in the right-hand sides of both ( 4) and ( 5) is given by:

E(f 0 , f 1 ) = -τ 0 G[f 1 ]f 0 + τ 1 G[f 0 ]f 1 , (10)
G[f ](x, ω, t) = 1 + α y∈R 2 ,v∈S 1 K(|x -y|) (1 + v • ω) 2 f (y, v, t)dydv. (11)
Without the Vicsek operators, equations ( 4)-( 5) formally represent the forward Kolmogorov equation of the time-inhomogeneous Markov process defined by the rates (3). We refer to [START_REF] Stroock | An introduction to Markov processes[END_REF] for more detail on this subject. System [START_REF] Bellomo | Multiscale biological tissue models and flux-limited chemotaxis for multicellular growing systems[END_REF][START_REF] Bellomo | From the mathematical kinetic, and stochastic game theory to modelling mutations, onset, progression and immune competition of cancer cells[END_REF][START_REF] Bellomo | On the modeling of traffic and crowds: A survey of models, speculations, and perspectives[END_REF][START_REF] Bellouquid | Mathematical methods and tools of kinetic theory towards modelling complex biological systems[END_REF][START_REF] Bertin | Boltzmann and hydrodynamic description for self-propelled particles[END_REF][START_REF] Bolley | Stochastic Mean-Field Limit: Non-Lipschitz Forces & Swarming[END_REF][START_REF] Bolley | Mean-field limit for the stochastic Vicsek model[END_REF][START_REF] Carrillo | Asymptotic flocking dynamics for the kinetic cucker-smale model[END_REF] is the starting point of our study of the large time and space scale dynamics of the microscopic model presented in section 2.1.

The macroscopic dynamics: rescaling and main results

In this section, we focus on the large scale dynamics of system [START_REF] Bellomo | Multiscale biological tissue models and flux-limited chemotaxis for multicellular growing systems[END_REF][START_REF] Bellomo | From the mathematical kinetic, and stochastic game theory to modelling mutations, onset, progression and immune competition of cancer cells[END_REF][START_REF] Bellomo | On the modeling of traffic and crowds: A survey of models, speculations, and perspectives[END_REF][START_REF] Bellouquid | Mathematical methods and tools of kinetic theory towards modelling complex biological systems[END_REF][START_REF] Bertin | Boltzmann and hydrodynamic description for self-propelled particles[END_REF][START_REF] Bolley | Stochastic Mean-Field Limit: Non-Lipschitz Forces & Swarming[END_REF][START_REF] Bolley | Mean-field limit for the stochastic Vicsek model[END_REF][START_REF] Carrillo | Asymptotic flocking dynamics for the kinetic cucker-smale model[END_REF]. We are specifically interested in regimes where the time scale of the two interactions present in the model, the internal Vicsek dynamics inside the subgroups and the exchange dynamics between the subgroups, are small compared with the time scale of observation. To investigate asymptotic regimes, we first begin by performing a time and space rescaling to obtain a dimensionless system. Let us write system (4-11) is rescaled variables. Let ν V the typical Vicsek interaction frequency,

ν 0 = ν V ν ′ 0 , ν 1 = ν V ν ′ 1 ,
and τ E the typical speed change frequency,

τ 0 = τ E τ ′ 0 , τ 1 = τ E τ ′ 1 .
The dimensionless diffusion coefficients are also given by:

d 0 = ν V d ′ 0 , d 1 = ν V d ′ 1 .
We then introduce the dimensionless time and space variables: t ′ = ν V t, x ′ = xν V /c. After dropping the tildes, system (4-11) becomes in the new variables:

∂ t f 0 = Q 0 (f 0 ) + 1 δ E(f 0 , f 1 ), (12)
∂ t f 1 + ω • ∇ x f 1 = Q 1 (f 1 ) - 1 δ E(f 0 , f 1 ), (13)
where δ = ν V /τ E quantifies the relative intensity of the two interactions and is supposed to be order O(1). All the coefficients of the operators Q 0 , Q 1 and E (i.e. ν 0 , ν 1 , d 0 , d 1 , τ 0 , τ 1 ) are now dimensionless and are supposed to be of order O(1).

Hydrodynamic rescaling

We here perform a hydrodynamic rescaling to look at the large time and space scale dynamics. The hydrodynamic rescaling consists in introducing macroscopic variables in space and time: x = εx, t = εt, with ε ≪ 1. In the new variables, the distribution function of the zero speed particles f ε 0 (x, ω, t) = f 0 (x, ω, t) and of the moving particles

f ε 1 (x, ω, t) = f 1 (x, ω, t
) satisfy the following system (dropping the tildes):

ε(∂ t f ε 0 ) = Q ε 0 (f ε 0 ) + 1 δ E ε (f ε 0 , f ε 1 ), (14)
ε(∂ t f ε 1 + ω • ∇ x f ε 1 ) = Q ε 1 (f ε 1 ) - 1 δ E ε (f ε 0 , f ε 1 ). ( 15
)
The Vicsek operators Q ε 0 and Q ε 1 and the exchange operator E ε are given by:

Q ε 0 (f ε 0 ) = -∇ ω • (ν 0 F ε [f ε 0 ]f ε 0 ) + d 0 ∆ ω f ε 0 , (16)
Q ε 1 (f ε 1 ) = -∇ ω • (ν 1 F ε [f ε 1 ]f ε 1 ) + d 1 ∆ ω f ε 1 , (17)
E ε (f ε 0 , f ε 1 ) = -τ 0 G ε [f ε 1 ]f ε 0 + τ 1 G ε [f ε 0 ]f ε 1 , (18)
where F ε is the rescaled interaction forces given by:

F ε [f ε ](x, ω, t) = (Id -ω ⊗ ω) ωε [f ε ](x, t), ωε [f ε ](x, t) = J ε [f ε ](x, t) |J ε [f ε ](x, t)| , J ε [f ε ](x, t) = 1 ε 2 y∈R 2 ,v∈S 1 K x -y ε vf ε (y, v, t)dydv, (19)
and G ε is the rescaled coupling coefficient:

G ε [f ε ](x, ω, t) = 1 + α ε 2 y∈R 2 ,v∈S 1 K x -y ε (1 + v • ω) 2 f ε (y, v, t)dydv. (20)
Let us make an expansion with respect to ε of these interaction terms. To this aim, let us introduce the macroscopic quantities related to a distribution function f (x, ω, t), that is the density ρ[f ](x, t), the momentum j[f ](x, t) and the mean direction

Ω[f ](x, t): ρ[f ](x, t) = v∈S 1 f (x, v, t)dv, j[f ](x, t) = v∈S 1 vf (x, v, t)dv, Ω[f ](x, t) = j[f ](x, t) |j[f ](x, t)| .
The following lemma provides the expansion of ωε and G ε with respect to ε.

Lemma 1. We have the following expansions:

ωε (x, t) = Ω[f ε ](x, t) + O(ε 2 ), (21)
G ε [f ε ](x, ω, t) = 1 + α ρ[f ε ](x, t) + ω • j[f ε ](x, t) 2 + O(ε 2 ), ( 22
)
where

ρ[f ε ], j[f ε ], Ω[f ε
] are the density, the momentum and the mean direction related to f ε .

The proof of Lemma 1 is omitted. Lemma 1 enables to write system (14-20) as following:

∂ t f ε 0 = 1 ε Q 0 (f ε 0 ) + 1 εδ E(f ε 0 , f ε 1 ) + O (ε) , (23)
∂ t f ε 1 + ω • ∇ x f ε 1 = 1 ε Q 1 (f ε 1 ) - 1 εδ E(f ε 0 , f ε 1 ) + O (ε) , (24)
where Q 0 , Q 1 and E are the operators of order O(1) in the expansions of Q ε 0 , Q ε 1 and E ε . They are given by:

Q 0,1 (f ε 0,1 ) = -∇ ω • (ν 0,1 F [f ε 0,1 ]f ε 0,1 ) + d 0,1 ∆ ω f ε 0,1 , (25)
E(f ε 0 , f ε 1 ) = -τ 0 g[f ε 1 ]f ε 0 + τ 1 g[f ε 0 ]f ε 1 , (26)
where the expressions of F and g result from lemma 1:

F [f ](x, ω, t) = (Id -ω ⊗ ω)Ω[f ](x, t), (27)
g[f ](x, ω, t) = 1 + α ρ[f ](x, t) + ω • j[f ](x, t) 2 . ( 28
)
Note that equation ( 25) contains terms indexed by "0, 1" : in all the following, an equation whose all the quantities f a,b are indexed by "a, b" represents the set of two equations3 , the equation with the left indexes "a" and the equation with the right indexes "b".

The macroscopic regime: main results

We would like now to investigate the hydrodynamic limit ε → 0 in system [START_REF] Degond | A macroscopic model for a system of swarming agents using curvature control[END_REF][START_REF] Degond | Congestion in a macroscopic model of self-driven particles modeling gregariousness[END_REF][START_REF] Delitala | Mathematical modeling of vehicular traffic: a discrete kinetic theory approach[END_REF][START_REF] Frouvelle | A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters[END_REF][START_REF] Frouvelle | Dynamics in a kinetic model of oriented particles with phase transition[END_REF][START_REF] Gatignol | Théorie cinétique des gaz à répartition discrète de vitesses[END_REF]. It corresponds to a regime where the Vicsek interaction and the speed change time scales are both small compared to the observation time scale and of order O(ε). However, the different mathematical natures of the Vicsek operators, Q 0 and Q 1 , and of the exchange operator E prevent us from achieving this goal in one unique step.

Instead of supposing that the Vicsek and exchange time scales are both of order O(ε), we begin by supposing that only the Vicsek time scale is so, while the exchange time scale remains of order O(1) at large scale:

δ ′ = εδ = O(1).
We here uncouple the time scales of the two interactions and first link the macroscopic time scales to the Vicsek interactions only. In the following, we will provide the asymptotic dynamics of system (where we omit the prime):

∂ t f ε 0 = 1 ε Q 0 (f ε 0 ) + 1 δ E(f ε 0 , f ε 1 ) + O (ε) , (29)
∂ t f ε 1 + ω • ∇ x f ε 1 = 1 ε Q 1 (f ε 1 ) - 1 δ E(f ε 0 , f ε 1 ) + O (ε) , (30)
in the limit:

ε → 0, δ = O(1). (31)
In a second step, we will take the limit:

δ → 0, (32)
It consists in considering that the time scales of the exchange interactions are also small compared to the time scale of the observation. Note that instead of first taking a small Vicsek interaction time scale and then a small exchange time scale, we would make it in the opposite order. Such a study would be the subject of future works.

In the two following theorems, the two limits [START_REF] Ha | From particle to kinetic and hydrodynamic descriptions of flocking[END_REF] and [START_REF] Helbing | Traffic and related self-driven many-particle systems[END_REF] in system (29-30) provides the macroscopic dynamics in the regime ε ≪ δ ≪ 1. Before stating the results, let us introduce the Von-Mises velocity distributions M λ,Ω :

M λ,Ω (ω) = C λ exp ω • Ω λ , ω∈S 1 M λ,Ω (ω)dω = 1. ( 33
)
where λ stands for the temperature of the distribution and C λ are a re-normalisation constant to ensure the total mass to be unity. The following theorem states the hydrodynamic system satisfied by the densities and the mean directions of the two phases in the regime (31):

Theorem 2. [Limit ε → 0, δ = O(1)].
1. The limits of the distribution functions as ε goes to 0 are given by:

f ε 0 (x, ω, t) → ρ 0 (x, t)M λ 0 ,Ω 0 (x,t) (ω), f ε 1 (x, ω, t) → ρ 1 (x, t)M λ 1 ,Ω 1 (x,t) (ω). with λ 0,1 = d 0,1 /ν 0,1 .
2. The densities ρ 0,1 (x, t) and the mean directions Ω 0,1 (x, t) satisfy the system:

∂ t ρ 0 = 1 δ R(ρ 0 , Ω 0 , ρ 1 , Ω 1 ), (34)
∂ t ρ 1 + ∇ x • (c 1 ρ 1 Ω 1 ) = - 1 δ R(ρ 0 , Ω 0 , ρ 1 , Ω 1 ), ( 35
)
ρ 0 ∂ t Ω 0 = λ 0 (Id -Ω 0 ⊗ Ω 0 ) -∇ x ρ 0 + 1 δ β 0 S 0 (ρ 0 , Ω 0 , ρ 1 , Ω 1 ) , (36)
ρ 1 ∂ t Ω 1 + γ 1 ρ 1 (Ω 1 • ∇ x )Ω 1 = λ 1 (Id -Ω 1 ⊗ Ω 1 ) -∇ x ρ 1 + 1 δ β 1 S 1 (ρ 0 , Ω 0 , ρ 1 , Ω 1 ) , ( 37
)
where c 1 , γ 1 , β 0,1 are constants defined in (44) and (59). The operators R, S 0 and S 1 comes from the exchange dynamics and are defined in (53) and (57-58).

Before developing the proof of this theorem in section 4, let us point out the main features of system [START_REF] Mogilner | A non-local model for a swarm[END_REF][START_REF] Motsch | Numerical simulations of a non-conservative hyperbolic system with geometric constraints describing swarming behavior[END_REF][START_REF] Pillot | Collective movements, initiation and stops: diversity of situations and law of parsimony[END_REF][START_REF] Pillot | Scalable Rules for Coherent Group Motion in a Gregarious Vertebrate[END_REF]. First, as the original macroscopic Vicsek model [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF], system (34-37) is non-conservative due to the macroscopic geometric constraints |Ω 0,1 | = 1 guaranteed by the projection operators (Id -Ω 0,1 ⊗ Ω 0,1 ). But, the above two-speed model is all the more non-conservative since the presence of the exchange operators R, S 0,1 , that model the large scale change of speed of the particles. Note also that, unlike the mass exchange, the momentum exchange operators are not the same for the two subgroups (S 0 = S 1 ) and thus do not compensate. Then, we take the limit δ → 0 in system [START_REF] Mogilner | A non-local model for a swarm[END_REF][START_REF] Motsch | Numerical simulations of a non-conservative hyperbolic system with geometric constraints describing swarming behavior[END_REF][START_REF] Pillot | Collective movements, initiation and stops: diversity of situations and law of parsimony[END_REF][START_REF] Pillot | Scalable Rules for Coherent Group Motion in a Gregarious Vertebrate[END_REF] to obtain the final result:

Theorem 3. [Limit δ → 0].
1. The set of equilibria of the density exchange operator R is given by:

(ρ 0 , ρ 1 ) ∈ (R + ) 2 , ρ 0 = f Φ (ρ 1 ) ,
where f Φ , defined in (62), is a non-linear function depending on Ω 0 and Ω 1 through the local alignment Φ (54).

2. The set of equilibria of the mean direction exchange operators (Id -Ω 0 ⊗ Ω 0 )S 0 and (Id -Ω 1 ⊗ Ω 1 )S 1 is the following: [START_REF] Mogilner | A non-local model for a swarm[END_REF][START_REF] Motsch | Numerical simulations of a non-conservative hyperbolic system with geometric constraints describing swarming behavior[END_REF][START_REF] Pillot | Collective movements, initiation and stops: diversity of situations and law of parsimony[END_REF][START_REF] Pillot | Scalable Rules for Coherent Group Motion in a Gregarious Vertebrate[END_REF] yields the following closed system for the total density ρ = ρ 0 + ρ 1 and the common mean direction

(Ω 0 , Ω 1 ) ∈ (S 1 ) 2 , Ω 0 = Ω 1 or Ω 0 = -Ω 1 .

Once the set of equilibria

Ω 0 = Ω 1 (resp. Ω 0 = -Ω 1 ) is reached, system
Ω = Ω 0 = Ω 1 (resp. Ω = Ω 1 = -Ω 0 ): ∂ t ρ + ∇ x • (c 1 ρ 1 [ρ]Ω) = 0, ( 38
) M + (ρ)∂ t Ω + γ 1 N + (ρ) (Ω • ∇ x )Ω = -P + (ρ)(Id -Ω ⊗ Ω) ∇ x ρ, (39)
(resp. M -(ρ)∂ t Ω + γ 1 N -(ρ) (Ω • ∇ x )Ω = -P -(ρ)(Id -Ω ⊗ Ω) ∇ x ρ, (40)
where The proof of this theorem is developed in section 5. This theorem provides the macroscopic dynamics of the density of particles ρ and the momentum Ω once the two subgroups of particles are aligned. Note that the derivation of system (38-39) is a priori not obvious since the difference between the momentum exchange operators (S 0 = S 1 ) prevents us from directly cancelling them when summing the two momentum equations [START_REF] Pillot | Collective movements, initiation and stops: diversity of situations and law of parsimony[END_REF][START_REF] Pillot | Scalable Rules for Coherent Group Motion in a Gregarious Vertebrate[END_REF]. We overcome this difficulty by carefully analysing the exchange operators and their linearisations.

ρ 1 [ρ] = (Id + f Φ ) -1 (ρ).
4

The large scale dynamics of the Vicsek interactions: limit ε → 0

This part is devoted to the main steps of the proof of theorem 2, while technical details are put in appendix A. The derivation of the macroscopic equations (34-37) from the kinetic system (29-30) is similar to the one of the macroscopic Vicsek model [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF] except that there are extra macroscopic exchange terms. These macroscopic exchange terms depend on the collisional invariants of the Vicsek operator.

The equilibria of the Vicsek operators

We suppose that δ = O(1) and we want to take the limit ε → 0 in system ( 29)- [START_REF] Ha | A simple proof of the Cucker-Smale flocking dynamics and mean-field limit[END_REF]. Therefore, assuming that the distribution function f ε 0 and f ε 1 converge to limits denoted by f 0 and f 1 as ε → 0, these limits satisfy the equilibria :

Q 0 (f 0 ) = 0 and Q 1 (f 1 ) = 0.
According to lemma 4.2 in [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF], the kernels of the Vicsek operators Q 0,1 are two-dimensional manifolds: there exists ρ 0,1 (x, t) ∈ R and Ω 0,1 (x, t) ∈ S 1 such that distributions f 0 and f 1 equals

f 0 (x, ω, t) = ρ 0 (x, t)M λ 0 ,Ω 0 (x,t) (ω), f 1 (x, ω, t) = ρ 1 (x, t)M λ 1 ,Ω 1 (x,t) (ω), (41)
where M λ,Ω denotes the Von Mises distribution defined in eq. [START_REF] Ishii | Thermo-fluid dynamic theory of two-phase flow[END_REF]. We actually note that ρ 0,1 (, x, t) and Ω 0,1 (x, t) are the density and the mean direction of the equilibria distribution function:

ρ[f 0,1 ](x, t) = ρ 0,1 (x, t), (42)
j[f 0,1 ](x, t) = c 0,1 ρ 0,1 (x, t)Ω 0,1 (x, t), Ω[f 0,1 ](x, t) = Ω 0,1 (x, t), (43)
where c 0,1 are constants defined as:

c 0,1 = cos θ M λ 0,1 . (44)
For any function s(cos θ), the brackets s(cos θ) M λ will denote the average of s(ω • Ω) with respect to M λ,Ω :

s(cos θ) M λ = 2π θ=0 s(cos θ)C λ e cos θ λ dθ. (45)
Note that this definition is independent of Ω.

The generalised collisional invariants

We would like now to obtain the dynamics of the macroscopic quantities, ρ 0,1 (x, t) and Ω 0,1 (x, t). With this aim, the usual method is to integrate equations (29-30) against collisional invariants, which are velocity functions I(ω) belonging to the orthogonal of the image of Q 0,1 . A condition to recover the dynamics of the equilibria is that the dimension of the vector space of collisional invariants equals the dimension of the vector space of local equilibria, which here is 2. It is not the case in the Vicsek dynamics since the only known collisional invariant is mass: I 1 (ω) = 1. In [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF], this difficulty is overcome by considering generalised collisional invariants I(ω), that are collisional invariants valid only for the subset of the distribution functions with a prescribed mean direction:

ω∈S 1 Q 0,1 (f )(x, ω, t)I(ω) dω = 0, ∀f such that Ω[f ] = Ω.
In dimension 2, it has been shown in [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF] that the vector space of the generalised collisional invariants for Q 0 (resp. Q 1 ) is spanned by I 1 and a function (I 2 ) 0 (resp. (I 2 ) 1 ), which is the unique solution with zero average of the elliptic equation:

∂ θ e cos θ/λ 0,1 ∂ θ (I 2 ) 0,1 = sin θe cos θ/λ 0,1 , (46)
where we identified the functions on S 1 and the 2π-periodic functions of R, using polar coordinates in the basis (Ω, Ω ⊥ ). Their explicit expressions are:

(I 2 ) 0,1 (θ) = λ 0,1 π θ 0 e -cos ϕ/λ 0,1 dϕ π 0 e -cos ϕ/λ 0,1 dϕ -θ . (47)
We would like here to stress that these second generalized invariants (I 2 ) 0,1 depend on the parameter λ 0,1 = d 0,1 /ν 0,1 and then are a priori different for the two Vicsek operators: this would not be the case if the model conserved momentum. To simplify the following computations, let us introduce the function h 0,1 (cos θ) = (I 2 ) 0,1 (θ)/ sin θ. We thus have:

(I 2 ) 0,1 (ω) = h 0,1 (ω • Ω[f ]) Ω[f ] ⊥ • ω . (48)

The macroscopic system

As announced in the previous section, to obtain the dynamics of the densities ρ 0,1 (x, t) and the mean directions Ω 0,1 (x, t) introduced in eq. ( 41), we now integrate system ( 29)-( 30) with respect to the velocity variable ω, after having multiplied it by the collisional invariants I 1 and I 2 .

Mass equations. Integrating equations ( 29)-(30) (multiplied by the collisional invariant I 1 (ω) = 1) with respect to ω, we easily obtain:

∂ t ρ[f ε 0 ] = 1 δ ω∈S 1 E(f ε 0 , f ε 1 )dω, (49)
∂ t ρ[f ε 1 ] + ∇ x • j[f ε 1 ] = - 1 δ ω∈S 1 E(f ε 0 , f ε 1 )dω. ( 50
)
So, in the limit ε → 0, using the relations (42-43) , system (49)-(50) results in the following mass equations:

∂ t ρ 0 = 1 δ R(ρ 0 , Ω 0 , ρ 1 , Ω 1 ), (51)
∂ t ρ 1 + ∇ x • (c 1 ρ 1 Ω 1 ) = - 1 δ R(ρ 0 , Ω 0 , ρ 1 , Ω 1 ), (52)
where the macroscopic exchange operator R is derived from the microscopic one E (defined in ( 26)-( 28)) and easy computations provides the following expression:

R(ρ 0 , Ω 0 , ρ 1 , Ω 1 ) = ω∈S 1 E(ρ 0 M λ 0 ,Ω 0 , ρ 1 M λ 1 ,Ω 1 )dω = τ 1 ρ 1 -τ 0 ρ 0 + α (τ 1 -τ 0 ) ρ 0 ρ 1 Φ, ( 53
)
and Φ is the macroscopic alignment of the two populations:

Φ = 1 + c 0 c 1 (Ω 0 • Ω 1 ) 2 . ( 54
)
System (51-52-53) exactly corresponds to equations (34-35) of theorem 2.

Momentum equations.

Let us now multiply system (29-30) by the second generalised collisional invariants (I 2 ) 0,1 (ω) and then integrate with respect to ω. After some computations reported in appendix A and similar to those made in [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF], we obtain in the limit ε → 0:

λ -1 0 (sin θ) 2 h 0 M Ω 0 ρ 0 ∂ t Ω 0 = (Id -Ω 0 ⊗ Ω 0 ) -(sin θ) 2 h 0 M Ω 0 ∇ x ρ 0 + 1 δ S 0 (ρ 0 , Ω 0 , ρ 1 , Ω 1 ) , (55)
λ -1 1 (sin θ) 2 h 1 M Ω 1 ρ 1 ∂ t Ω 1 + λ -1 1 (sin θ) 2 cos θh 1 M Ω 1 ρ 1 (Ω 1 • ∇ x )Ω 1 = (Id -Ω 1 ⊗ Ω 1 ) -(sin θ) 2 h 1 M Ω 1 ∇ x ρ 1 + 1 δ S 1 (ρ 0 , Ω 0 , ρ 1 , Ω 1 ) , (56)
where S 0 and S 1 are the terms coming from the exchange operator:

S 0 (ρ 0 , Ω 0 , ρ 1 , Ω 1 ) = ω∈S 1 E(ρ 0 M λ 0 ,Ω 0 , ρ 1 M λ 1 ,Ω 1 )h 0 (ω • Ω 0 )ωdω, (57)
S 1 (ρ 0 , Ω 0 , ρ 1 , Ω 1 ) = ω∈S 1 -E(ρ 0 M λ 0 ,Ω 0 , ρ 1 M λ 1 ,Ω 1 )h 1 (ω • Ω 1 )ωdω. ( 58
)
where the exchange operator E, defined in ( 26)-( 28), can be expressed as function of the macroscopic quantities:

E(ρ 0 M λ 0 ,Ω 0 , ρ 1 M λ 1 ,Ω 1 ) = -τ 0 g 1 (ρ 1 , Ω 1 )ρ 0 M λ 0 ,Ω 0 + τ 1 g 0 (ρ 0 , Ω 0 )ρ 1 M λ 1 ,Ω 1 ,
with:

g 0,1 (ρ 0,1 , Ω 0,1 )(x, ω, t) = g[ρ 0,1 M λ 0,1 ,Ω 0,1 ](x, ω, t) = 1 + αρ 0,1 (x, t) 1 + c 0,1 (ω • Ω 0,1 (x, t)) 2 .
Note that the index of the functions g 0,1 refers to the parameters λ 0,1 and then to the constant c 0,1 appearing when computing the flux j[ρ 0,1 M λ 0,1 ,Ω 0,1 ] of the local equilibrium (in ( 28)).

Unlike the exchange terms in the mass equations (51-52), the operators S 0 and S 1 are not opposite because of the presence of h 0 and h 1 : according to their definitions (47)-(48), h 0 and h 1 are different as soon as λ 0 = λ 1 . However, the operators S 0 and S 1 are still symmetric in subscripts: when changing subscripts 0 to 1 and 1 to 0 in S 0 , we obtain S 1 .

After introducing the following constants:

γ 1 = (sin θ) 2 cos θh 1 M Ω 1 (sin θ) 2 h 1 M Ω 1 , β 0,1 = (sin θ) 2 h 0,1 M Ω 0,1 -1 , (59)
system ( 55)-( 56) can be also written as follows:

ρ 0 ∂ t Ω 0 = λ 0 (Id -Ω 0 ⊗ Ω 0 ) -∇ x ρ 0 + 1 δ β 0 S 0 (ρ 0 , Ω 0 , ρ 1 , Ω 1 ) , (60)
ρ 1 ∂ t Ω 1 + γ 1 ρ 1 (Ω 1 • ∇ x )Ω 1 = λ 1 (Id -Ω 1 ⊗ Ω 1 ) -∇ x ρ 1 + 1 δ β 1 S 1 (ρ 0 , Ω 0 , ρ 1 , Ω 1 ) , (61)
which are the mean direction equations ( 36)- [START_REF] Pillot | Scalable Rules for Coherent Group Motion in a Gregarious Vertebrate[END_REF] given in theorem 2.

5 Large scale speed change dynamics: limit δ → 0

In this section, we provide the proof of theorem 3: we first work out the equilibria of the exchange operators, R, S 0 and S 1 , appearing in the right-hand-side of system [START_REF] Mogilner | A non-local model for a swarm[END_REF][START_REF] Motsch | Numerical simulations of a non-conservative hyperbolic system with geometric constraints describing swarming behavior[END_REF][START_REF] Pillot | Collective movements, initiation and stops: diversity of situations and law of parsimony[END_REF][START_REF] Pillot | Scalable Rules for Coherent Group Motion in a Gregarious Vertebrate[END_REF]. Note that they are non linear operators acting on the macroscopic densities and mean directions and note also that this is the projected momentum operators (Id -Ω 0 ⊗ Ω 0 )S 0 and (Id -Ω 1 ⊗ Ω 1 )S 1 that are acting in the momentum equations. However, we will prove that the only equilibria of the momentum exchange operators, S 0 and S 1 , are given by Ω 0 = Ω 1 or Ω 0 = -Ω 1 . We then obtained the macroscopic system [START_REF] Pillot | Moving together: incidental leaders and naïve followers[END_REF][START_REF] Spohn | Large Scale Dynamics of Interacting Particles[END_REF] by employing an Hilbert expansion with respect to δ and thanks to a "discrete" version of the generalised collisional invariants methodology, we close the system.

Equilibria for the densities

Let us first explicitly re-introduce the dependency of the macroscopic variables with respect to δ: the solutions of system (34-37) with δ > 0 are denoted ρ δ 0,1 (x, t) and Ω δ 0,1 (x, t). Supposing that ρ δ 0,1 and Ω δ 0,1 converge to ρ 0,1 and Ω 0,1 as δ goes to 0 and taking this limit in mass equations ( 34)-( 35), we formally obtain:

R(ρ 0 , Ω 0 , ρ 1 , Ω 1 ) = 0,
From the definition of R (eq. ( 53)), we easily obtain the balance equation for the densities

ρ 0 = f Φ (ρ 1 ), f Φ (ρ 1 ) = ρ 1 τ 1 τ 0 + α τ 1 τ 0 -1 Φρ 1 -1 , (62)
which can be equivalently expressed in terms of the total density ρ = ρ 0 + ρ 1 :

ρ = k Φ (ρ 1 ), k Φ (ρ 1 ) = ρ 1 + f Φ (ρ 1 ). ( 63
)
Note that in the general case, these relations are not explicit since the macroscopic alignment parameter Φ (defined in (54)) depends on the directions Ω 0 and Ω 1 , whose evolutions are non-linearly related to the densities through mass and momentum equations ( 36)-( 37). This provides the first part of theorem 3.

Let us now make some remarks on this relation. Simple computations shows that the function f Φ (and then function k Φ ) is increasing. The following proposition gives the domain where both ρ 0 and ρ 1 are non negative. Proposition 4. (Conditions for positivity) If α > 0, τ 0 = τ 1 and supposing that ρ 0 and ρ 1 are non-negative, then the following results hold :

1. if τ 1 /τ 0 > 1, then ρ 0 is bounded: ρ 0 1 αΦ (τ 1 /τ 0 -1) 1 αΦ min (τ 1 /τ 0 -1) , 2. if τ 1 /τ 0 < 1, then ρ 1 is bounded: ρ 1 1 αΦ (τ 0 /τ 1 -1) 1 αΦ min (τ 0 /τ 1 -1)
,

where Φ min = (1 -c 0 c 1 )/2 is the minimal value of the macroscopic alignment parameter.

The proof of this proposition is easy and omitted. This proposition shows that the dependency of the exchange rates on the local alignment Φ implies that the population with the higher interaction frequency is bounded in time.

Equilibria for the mean directions

Secondly, as δ goes to 0 in the mean direction equations ( 36)-( 37), the limit ρ 0,1 and Ω 0,1 of ρ δ 0,1 and Ω δ 0,1 satisfies:

(Id -Ω 0 ⊗ Ω 0 )S 0 (ρ 0 , Ω 0 , ρ 1 , Ω 1 ) = 0, (Id -Ω 1 ⊗ Ω 1 )S 1 (ρ 0 , Ω 0 , ρ 1 , Ω 1 ) = 0. ( 64
)
The most simple solutions of (64) are Ω 0 = Ω 1 or Ω 0 = -Ω 1 , i.e. when the two populations are in the same or in the opposite direction. Indeed, for instance in the case Ω 0 = Ω 1 , considering the polar coordinates θ in the basis (Ω 0 , Ω ⊥ 0 ), we can easily checked from (57) that the momentum exchange operator writes as follows:

(Id -Ω 0 ⊗ Ω 0 )S 0 (ρ 0 , Ω 0 , ρ 1 , Ω 0 ) = 2π θ=0 f 0 (cos θ) sin θdθ Ω ⊥ 0 , f 0 (cos θ) = [-τ 0 (1 + α 2 ρ 1 (1 + c 1 cos θ))ρ 0 exp(cos θ/λ 0 ) + τ 1 (1 + α 2 ρ 0 (1 + c 0 cos θ))ρ 1 exp(cos θ/λ 1 )]h 0 (cos θ),
and thus vanishes by even argument. Let us now explicit the expression of (Id-Ω 0 ⊗Ω 0 )S 0 in the general case:

(Id -Ω 0 ⊗ Ω 0 )S 0 (ρ 0 , Ω 0 , ρ 1 , Ω 0 ) = -(Id -Ω 0 ⊗ Ω 0 )(1 + α ρ 1 2 )ρ 0 τ 0 ω∈S 1 M λ 0 ,Ω 0 h 0 (ω • Ω 0 )ωdω -(Id -Ω 0 ⊗ Ω 0 )α c 1 ρ 1 2 ρ 0 τ 0 ω∈S 1 (ω ⊗ ω)M λ 0 ,Ω 0 h 0 (ω • Ω 0 )dωΩ 1 + (Id -Ω 0 ⊗ Ω 0 )(1 + α ρ 0 2 )ρ 1 τ 1 ω∈S 1 M λ 1 ,Ω 1 h 0 (ω • Ω 0 )ωdω + (Id -Ω 0 ⊗ Ω 0 )α c 0 ρ 0 2 ρ 1 τ 1 ω∈S 1 (ω ⊗ ω)M λ 1 ,Ω 1 h 0 (ω • Ω 0 )dωΩ 0 .
By even arguments, the first term vanishes and we can easily check that the integral in the second term can be written as follows:

ω∈S 1 (ω ⊗ ω)M λ 0 ,Ω 0 h 0 (ω • Ω 0 )dω = (cos θ) 2 h 0 M λ 0 Ω 0 ⊗ Ω 0 + (sin θ) 2 h 0 M λ 0 (Id -Ω 0 ⊗ Ω 0 ).
where the bracket notation refers to (45). We deduce the following simplified expression for the projection of S 0 :

(Id -Ω 0 ⊗ Ω 0 )S 0 (ρ 0 , Ω 0 , ρ 1 , Ω 0 ) = -α c 1 ρ 1 2 ρ 0 τ 0 (sin θ) 2 h 0 M λ 0 (Id -Ω 0 ⊗ Ω 0 )Ω 1 + (1 + α ρ 0 2 )ρ 1 τ 1 ω∈S 1 M λ 1 ,Ω 1 h 0 (ω • Ω 0 )(ω • Ω ⊥ 0 )dω Ω ⊥ 0 (65) + α c 0 ρ 0 2 ρ 1 τ 1 (Id -Ω 0 ⊗ Ω 0 ) ω∈S 1 (ω ⊗ ω)M λ 1 ,Ω 1 h 0 (ω • Ω 0 )dω Ω 0 ,
where all the terms collinear to Ω 0 was removed. An equivalent expression for (Id -Ω 1 ⊗ Ω 1 )S 1 can be obtained. Note that the involved integrals include the products of functions of the variable (ω • Ω 0 ) times functions of the variable (ω • Ω 1 ). Therefore, equilibria different from Ω 0 = ±Ω 1 might exist. However, the following proposition states that they are the only two possible solutions and thus completes the first statement of theorem 3.

Proposition 5. The only solutions to equations ( 64) are given by the set:

(ρ 0 , Ω 0 , ρ 1 , Ω 1 ) ∈ (R + × S 1 ) 2 , Ω 0 = Ω 1 or Ω 0 = -Ω 1 .
The proof of this proposition can be found in appendix B. This proposition proves that the only two equilibria are those where the two sub-population are locally directed in the same (Ω 0 = Ω 1 ) or in the opposite (Ω 0 = -Ω 1 ) direction. However, the question of the stability of the equilibria remains open and will be addressed in a future work.

and (Dg 0,1 ) and (DρM λ 0,1 ,Ω ) denote the linearisation of the operators g 0,1 and (ρ, Ω) → ρM λ 0,1 ,Ω and are given by:

(Dg 0,1 ) (ρ,Ω) (ρ, Ω) = α 2 ρ + c 0,1 (ρΩ + ρ Ω) • ω , (DρM λ 0,1 ,Ω ) (ρ,Ω) (ρ, Ω) = ρ + ρλ -1 0,1 (ω • Ω) M λ 0,1 ,Ω .
The expression of (DS 1 ) is just obtained by changing the index 0 into 1 and 1 into 0.

The proof of this lemma is elementary and is omitted. Let us note that the expansion of the projected momentum exchange operator is as follows:

(Id -Ω δ 0 ⊗ Ω δ 0 )S 0 (ρ δ 0 , Ω δ 0 , ρ δ 1 , Ω δ 1 ) = (Id -Ω 0 ⊗ Ω 0 )S 0 (ρ 0 , Ω 0 , ρ 1 , Ω 1 ) + δ (Id -Ω 0 ⊗ Ω 0 ) (DS 0,1 ) (ρ 0 ,Ω 0 ,ρ 1 ,Ω 1 ) (ρ 0 , Ω0 , ρ1 , Ω1 ) -δ ( Ω0 • S 0 (ρ 0 , Ω 0 , ρ 1 , Ω 1 ))Ω 0 -δ(Ω 0 • S 0 (ρ 0 , Ω 0 , ρ 1 , Ω 1 )) Ω0 + O(δ 2 ). ( 68
)
Hilbert expansion. We then insert expansions (66)-(67) into equations ( 34)-( 35)-( 36)- [START_REF] Pillot | Scalable Rules for Coherent Group Motion in a Gregarious Vertebrate[END_REF].

• At the leading order O(δ -1 ), we obviously obtain that (ρ 0 , Ω 0 , ρ 1 , Ω 1 ) are equilibria of the exchange operators:

R(ρ 0 , Ω 0 , ρ 1 , Ω 1 ) = 0, (69)
(Id -Ω 0 ⊗ Ω 0 )S 0 (ρ 0 , Ω 0 , ρ 1 , Ω 1 ) = 0, (70)
(Id -Ω 1 ⊗ Ω 1 )S 1 (ρ 0 , Ω 0 , ρ 1 , Ω 1 ) = 0. (71)
Consequently, according to previous sections 5.1 and 5.2, the following relations hold:

ρ = k Φ (ρ 1 ), Ω 0 = Ω 1 , with Φ = (1 + c 0 c 1 )/2, or ρ = k Φ (ρ 1 ), Ω 0 = -Ω 1 , with Φ = (1 -c 0 c 1 )/2.
where ρ = ρ 0 + ρ 1 is the total density and the function k Φ is defined by (63). Therefore, we would like to stress that the very simple form of these equilibria is essentially due to the fact that both mean directions equilibria and density equilibria are characterised by the alignment of the mean directions.

• At order O(1), system (34)-( 35)-( 36)-(37) becomes:

∂ t ρ 0 = (DR) (ρ 0 ,Ω 0 ,ρ 1 ,Ω 1 ) (ρ 0 , Ω0 , ρ1 , Ω1 ), (72)
∂ t ρ 1 + ∇ x • (c 1 ρ 1 Ω 1 ) = -(DR) (ρ 0 ,Ω 0 ,ρ 1 ,Ω 1 ) (ρ 0 , Ω0 , ρ1 , Ω1 ), (73)
ρ 0 ∂ t Ω 0 + λ 0 (Id -Ω 0 ⊗ Ω 0 )∇ x ρ 0 = λ 0 β 0 (Id -Ω 0 ⊗ Ω 0 ) (DS 0 ) (ρ 0 ,Ω 0 ,ρ 1 ,Ω 1 ) (ρ 0 , Ω0 , ρ1 , Ω1 ) -λ 0 β 0 (Ω 0 • S 0 (ρ 0 , Ω 0 , ρ 1 , Ω 1 )) Ω0 , (74)
ρ 1 ∂ t Ω 1 + γ 1 ρ 1 (Ω 1 • ∇ x )Ω 1 + λ 1 (Id -Ω 1 ⊗ Ω 1 )∇ x ρ 1 = λ 1 β 1 (Id -Ω 1 ⊗ Ω 1 ) (DS 1 ) (ρ 0 ,Ω 0 ,ρ 1 ,Ω 1 ) (ρ 0 , Ω0 , ρ1 , Ω1 ) -λ 1 β 1 (Ω 1 • S 1 (ρ 0 , Ω 0 , ρ 1 , Ω 1 )) Ω1 , (75)
where DR, DS 0 , DS 1 are the linearised exchange operators, whose expressions are given in lemma 6. Note that the terms ( Ω0 • S 0 )Ω 0 (resp. ( Ω1 • S 1 )Ω 1 ) in (68) do not appear in equation (74) (resp. (75)): indeed, according to (70) (resp. (71)), S 0 is parallel to Ω 0 (resp. S 1 is parallel to Ω 1 ).

Closure. System (72)-( 73)-( 74)-( 75) is not closed: it depends on the dynamics of the first order correction terms (ρ 0 , ρ1 , Ω0 , Ω1 ) that would be provided by the equations at order O(δ). However, by expressing compatibility conditions, we are able to close the system. We consider the case

Ω 0 = Ω 1 .
Actually, adding the two density equations, we get the following closed equation:

∂ t (ρ 0 + ρ 1 ) + ∇ x • (c 1 ρ 1 Ω 1 ) = 0,
where the exchange terms have been cancelled. For equations ( 74)-( 75), supposing that ρ0 and ρ1 are zero, the operator at the righthand side of equations ( 74)-(75) reduces to a linear operator in the two-dimensional space vect(Ω ⊥ 0 ) × vect(Ω ⊥ 0 ) acting on Ω0 , Ω1 ∈ vect(Ω ⊥ 0 ), where vect(Ω ⊥ 0 ) denotes the line spanned by Ω ⊥ 0 . Its Null-Space is not reduced to {0} since it is not the case for the original operator. Thus its image is one-dimensional (the operator is neither bijective nor zero) and a closed equation can be obtained just by expressing that the left-hand side of equations ( 74)-( 75) have to belong to the one-dimensional image of the linearised operator. Actually, an explicit expression of this linear operator can be obtained. Proposition 7. The following identities hold:

∀ρ 0 , ρ 1 , ρ0 , ρ1 ∈ R + , ∀ Ω 0 = ±Ω 1 ∈ S 1 , ∀ Ω0 , Ω1 ∈ vect(Ω ⊥ 0 ) = vect(Ω ⊥ 1 ), λ 0 β 0 (Id -Ω 0 ⊗ Ω 0 ) (DS 0 ) (ρ 0 ,Ω 0 ,ρ 1 ,Ω 1 ) (ρ 0 , Ω0 , ρ1 , Ω1 ) -λ 0 β 0 (Ω 0 • S 0 (ρ 0 , Ω 0 , ρ 1 , Ω 1 )) Ω0 = A + 0 (ρ 0 , ρ 1 )( Ω1 -Ω0 ) if Ω 0 = Ω 1 , A - 0 (ρ 0 , ρ 1 )( Ω1 + Ω0 ) if Ω 0 = -Ω 1 , λ 1 β 1 (Id -Ω 1 ⊗ Ω 1 ) (DS 1 ) (ρ 0 ,Ω 0 ,ρ 1 ,Ω 1 ) (ρ 0 , Ω0 , ρ1 , Ω1 ) -λ 1 β 1 (Ω 1 • S 1 (ρ 0 , Ω 0 , ρ 1 , Ω 1 )) Ω1 = A + 1 (ρ 0 , ρ 1 )( Ω0 -Ω1 ) if Ω 0 = Ω 1 , A - 1 (ρ 0 , ρ 1 )( Ω1 + Ω0 ) if Ω 0 = -Ω 1 ,
where A + 0 and A - 0 are two bilinear functions with respect to ρ 0 and ρ 1 and whose expressions are given by:

A + 0 (ρ 0 , ρ 1 ) = λ 0 β 0 τ 1 (1 + α ρ 0 2 )ρ 1 λ -1 1 sin 2 θh 0 M λ 1 + τ 1 α c 0 ρ 0 2 ρ 1 λ -1 1 sin 2 θ cos θh 0 M λ 1 -τ 0 α c 1 ρ 1 2 ρ 0 sin 2 θh 0 M λ 0 , A - 0 (ρ 0 , ρ 1 ) = λ 0 β 0 τ 1 (1 + α ρ 0 2 )ρ 1 λ -1 1 sin 2 θh - 0 M λ 1 -τ 1 α c 0 ρ 0 2 ρ 1 λ -1 1 sin 2 θ cos θh - 0 M λ 1 -τ 0 α c 1 ρ 1 2 ρ 0 sin 2 θh 0 M λ 0 ,
and A + 1 and A - 1 with the same expression by changing 1 to 0 and 0 to 1, and where, for any function h, h -: x → h(-x) denotes its symmetric.

The proof of this proposition is developed in appendix C: it relies on the simplification coming from the equilibria relations Ω 0 = ±Ω 1 . Multiplying equation (74) by A + 0 (resp. A - 0 ) and equation (75) by A + 0 (resp. A - 0 ) and then adding them (resp. subtracting them), it is an easy matter to derive the following closed system when equilibrium Ω 0 = Ω 1 (resp. Ω 0 = -Ω 1 ) is reached: Proposition 8. In the case where Ω 0 = Ω 1 (resp. Ω 0 = -Ω 1 ), system (72)-( 73)-( 74)-(75) yields the following closed system for the total density ρ = ρ 0 + ρ 1 and the common mean direction Ω = Ω 0 = Ω 1 (resp. Ω = Ω 1 = -Ω 0 ):

∂ t ρ + ∇ x • (c 1 ρ 1 [ρ]Ω) = 0, ( 76
) M + (ρ)∂ t Ω + γ 1 N + (ρ) (Ω • ∇ x )Ω = -P + (ρ)(Id -Ω ⊗ Ω) ∇ x ρ, (77)
[resp.

M -(ρ)∂ t Ω + γ 1 N -(ρ) (Ω • ∇ x )Ω = -P -(ρ)(Id -Ω ⊗ Ω) ∇ x ρ, ( 78
)
where functions M ± , N ± , P ± are given by:

M ± (ρ) = A ± 1 [ρ]ρ 0 [ρ] + A ± 0 [ρ]ρ 1 [ρ], N ± (ρ) = ρ 1 [ρ]A ± 0 [ρ], P ± (ρ) = λ 0 A ± 1 [ρ]ρ ′ 0 [ρ] + λ 1 A ± 0 [ρ]ρ ′ 1 [ρ],
where A ± 0 , A ± 1 , ρ 0 and ρ 1 are here considered as functions of ρ:

ρ 1 [ρ] = k -1 Φ (ρ), ρ 0 [ρ] = ρ -k -1 Φ (ρ), A ± 0,1 [ρ] = A ± 0,1 (ρ 0 [ρ], ρ 1 [ρ])
with k Φ given by ( 63) and A ± 0 and A ± 1 given in proposition ??.

System (76)-(77) provides the dynamics of the two populations at equilibrium. The study of its mathematical properties will be the subject of future work.

Remark: The derivation of this closed system can be interpreted with a "collisional invariant" viewpoint. To obtain the density equation, we use that the exchange interactions preserve mass and thus the vector (R, -R) as well as its linearised operator ( R, -R) satisfies:

∀ρ 0 , ρ 1 ∈ R + , ∀Ω 0 , Ω 1 ∈ S 1 , 1 1 • R -R = 0.
Since momentum is not conserved, such a relation is lacking for the momentum exchange operator (Id -Ω 0 ⊗ Ω 0 )S 0 , (Id -Ω 1 ⊗ Ω 1 )S 1 . However, the previous results show that the linearised operator around an equilibria have such a collisional invariant (equal to (A 0 , A 1 )) and that it is sufficient to conclude.

Conclusion

In this article, we have studied the dynamics at large time and space scale of a two-phase Vicsek model, in which particles speed can take only the two values 0 or 1. To this aim, we proceed in two steps. First, using generalised collision invariants [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF], we provides a two-phase Vicsek continuum model, where the densities and the directions dynamics are coupled through a speed change operator. Due to the non-conservativity of the model, the mean direction is not conserved. However, once figuring out the equilibria of the speed change operator, the averaged two-phase model can be obtained by integrating the dynamics against a vector orthogonal to the image of the linearised speed change operator. This leads to a new non-conservative model where coefficients non-linearly depend on the local mean density. Future works would consist into investigating the mathematical properties of this new model (stability of equilibria, hyperbolicity). As for the Vicsek model [START_REF] Motsch | Numerical simulations of a non-conservative hyperbolic system with geometric constraints describing swarming behavior[END_REF], appropriate numerical schemes would be also devised. Another point concerns the macroscopic dynamics of the two-phase model when speed changes are more frequent (or as frequent as) than Vicsek interactions: the question whether flocking occurs in that asymptotic regimes is still unknown.

The model considered in this article is designed to analyze alignment interactions during grazing period of mammal herds. However, the model is minimalist and several improvements should be investigated. First, long range attraction and short range repulsion interactions could be added to ensure cohesion of the group and to model congestion [START_REF] Carrillo | Particle, kinetic, and hydrodynamic models of swarming[END_REF][START_REF] Degond | Congestion in a macroscopic model of self-driven particles modeling gregariousness[END_REF]. Secondly, as in any living system, the assumption of homogeneous interactions among the animals is not satisfactory: animals present different behaviours, which can be modelled by internal variables such as the degree of attention or the degree of hunger of each individual. Such heterogeneous behaviour has been already considered for cells population [START_REF] Bellomo | From the mathematical kinetic, and stochastic game theory to modelling mutations, onset, progression and immune competition of cancer cells[END_REF][START_REF] Bellouquid | Mathematical methods and tools of kinetic theory towards modelling complex biological systems[END_REF]. Finally, we could also make the model depending on environment variables such as the local resource (the level of grass) or/and the topography. All these modeling refinements may affect the global herd dynamics.

where h 1 and M λ 1 Ω 1 are functions of (ω • Ω 1 ) and M λ 0 ,Ω 0 depends only on (ω • Ω 0 ). The derivative of M λ 1 ,Ω 1 with respect to Ω 1 acting on a tangent vector dΩ to the circle is given by: ∂M

λ 1 ,Ω 1 ∂Ω 1 (dΩ) = λ -1 1 (ω • Ω 1 )(ω • dΩ)M λ 1 ,Ω 1 .
Thus, we have:

X = ω∈S 1 ∂ t ρ 1 + ω • ∇ x ρ 1 + λ -1 1 ρ 1 (ω • ∂ t Ω 1 + (ω ⊗ ω) : ∇ x Ω 1 ) M Ω 1 h 1 ωdω - 1 δ S 1 ,
where S 1 is given by (58). The symbol ':' denotes the contracted product of two tensors (if A = (A i,j ) i,j=1,2 and B = (B i,j ) i,j=1,2 are two tensors then A : B = i,j=1,2 A i,j B i,j ) and ∇ x Ω is the gradient tensor of the vector Ω: (∇ x Ω) i,j = ∂ x i Ω j . The four first terms in this formula, denoted X 1 to X 4 , are computed using the polar coordinate θ related to the Cartesian basis (Ω 1 , Ω ⊥ 1 ): in this basis h 1 and M λ 1 ,Ω 1 are even functions depending on cos θ.

By writing ω = (ω

• Ω 1 )Ω 1 + (ω • Ω ⊥ 1 )Ω ⊥ 1
, we decompose X 1 as follows:

X 1 = ω∈S 1 ∂ t ρ 1 M λ 1 ,Ω 1 h 1 ωdω = ∂ t ρ 1 π θ=-π h 1 M λ 1 ,Ω 1 cos θdθ Ω 1 + ∂ t ρ 1 π θ=-π h 1 M λ 1 ,Ω 1 sin θdθ Ω ⊥ 1 ,
The second term is zero (the integration of an odd function on a symmetric interval) and thus we obtain:

(Ω ⊥ 1 ⊗ Ω ⊥ 1 )X 1 = 0. (79)
A similar computation shows that:

X 2 = ω∈S 1 [(ω ⊗ ω)∇ x ρ 1 ] M Ω 1 h 1 dω = π θ=-π h 1 M λ 1 ,Ω 1 cos 2 θdθ (Ω 1 ⊗ Ω 1 )∇ x ρ 1 + π θ=-π h 1 M λ 1 ,Ω 1 cos θ sin θdθ [(Ω 1 ⊗ Ω ⊥ 1 ) + (Ω ⊥ 1 ⊗ Ω 1 )]∇ x ρ 1 + π θ=-π h 1 M λ 1 ,Ω 1 sin 2 θdθ (Ω ⊥ 1 ⊗ Ω ⊥ 1 )∇ x ρ 1 .
The first term is parallel to Ω 1 and the second term is zero. Therefore, we obtain:

(Ω ⊥ 1 ⊗ Ω ⊥ 1 )X 2 = sin 2 θh 1 M λ 1 (Ω ⊥ 1 ⊗ Ω ⊥ 1 )∇ x ρ 1 . (80)
With similar computations, we obtain the following results for X 3 :

X 3 = λ -1 1 ρ 1 ω∈S 1 [(ω ⊗ ω) • ∂ t Ω 1 ] M λ 1 ,Ω 1 h 1 dω, (Ω ⊥ 1 ⊗ Ω ⊥ 1 )X 3 = λ -1 1 ρ 1 sin 2 θh 1 M λ 1 (Ω ⊥ 1 ⊗ Ω ⊥ 1 )∂ t Ω 1 . (81)
For X 4 , we have:

X 4 = λ -1 1 ρ 1 ω∈S 1 [(ω ⊗ ω) : ∇ x Ω 1 ] M λ 1 ,Ω 1 h 1 ωdω = λ -1 1 ρ 1 π θ=-π cos 2 θ(Ω 1 ⊗ Ω 1 ) + sin θ cos θ(Ω 1 ⊗ Ω ⊥ 1 ) + sin θ cos θ(Ω ⊥ 1 ⊗ Ω 1 ) + sin 2 θ(Ω ⊥ 1 ⊗ Ω ⊥ 1 ) : ∇ x Ω 1 M λ 1 ,Ω 1 h 1 cos θdθ Ω 1 + λ -1 1 ρ 1 π θ=-π cos 2 θ(Ω 1 ⊗ Ω 1 ) + sin θ cos θ(Ω 1 ⊗ Ω ⊥ 1 ) + sin θ cos θ(Ω ⊥ 1 ⊗ Ω 1 ) + sin 2 θ(Ω ⊥ 1 ⊗ Ω ⊥ 1 ) : ∇ x Ω 1 M λ 1 ,Ω 1 h 1 sin θdθ Ω ⊥ .
The odd function of θ vanishes and thus we are left with:

X 4 = λ -1 1 ρ 1 π θ=-π cos 2 θ(Ω 1 ⊗ Ω 1 ) + sin 2 θ(Ω ⊥ 1 ⊗ Ω ⊥ 1 ) : ∇ x Ω 1 M λ 1 ,Ω 1 h 1 cos θdθ Ω 1 +λ -1 1 ρ 1 π θ=-π sin θ cos θ (Ω 1 ⊗ Ω ⊥ 1 ) + (Ω ⊥ 1 ⊗ Ω 1 ) : ∇ x Ω M λ 1 ,Ω 1 h 1 sin θdθ Ω ⊥ 1 .
Besides, we have for all Ω ∈ S 1 :

(

Ω ⊗ Ω ⊥ ) : ∇ x Ω = Ω i Ω ⊥ j ∂ x i Ω j = ((Ω • ∇ x )Ω) • Ω ⊥ , and (Ω ⊥ ⊗ Ω) : ∇ x Ω = Ω ⊥ i Ω j ∂ x i Ω j = 1 2 Ω ⊥ • ∇|Ω| 2 = 0.
So, we finally have

(Ω ⊥ 1 ⊗ Ω ⊥ 1 )X 4 = λ -1 1 ρ sin 2 θ cos θh 1 M λ 1 (Ω ⊥ 1 ⊗ Ω ⊥ 1 )((Ω 1 • ∇ x )Ω 1 ). (82)
Equations ( 79)-( 80)-( 81)-(82) results in the momentum equations (56).

B Proof of proposition 5 (Momenta balance)

Proof. We show here that if the equilibria condition holds:

(Id -Ω 0 ⊗ Ω 0 )S 0 (ρ 0 , Ω 0 , ρ 1 , Ω 0 ) = 0 then we have Ω 0 = ±Ω 1 .
Let φ the angle between Ω 0 and Ω 1 . We have the following computations.

1. We can integrate expressions between 0 and π instead of -π and π:

ω∈S 1 M λ 1 ,Ω 1 h 0 (ω • Ω 0 )(ω • Ω ⊥ 0 )dω = = C 1 π θ=0 sin θ(e 1 λ 1 cos(θ-φ) -e 1 λ 1 cos(θ+φ) )h 0 (cos θ)dθ = 2C 1 π θ=0 sin θe 1 λ 1 cos θ cos φ sinh 1 λ 1 sin θ sin φ h 0 (cos θ)dθ,
where C 1 is the normalisation constant of the θ → e cos θ/λ 1 .

2. We simplify the last terms of the equilibrium equation ( 65):

(Id -Ω 0 ⊗ Ω 0 ) ω∈S 1 (ω ⊗ ω)M λ 1 ,Ω 1 h 0 (ω • Ω 0 )dωΩ 0 = = (Id -Ω 0 ⊗ Ω 0 ) π θ=-π (cos θ) 2 C 1 e 1 λ 1 cos(θ-φ) h 0 (cos θ)dθ (Ω 0 ⊗ Ω 0 )Ω 0 + π θ=-π sin θ cos θC 1 e 1 λ 1 cos(θ-φ) h 0 (cos θ)dθ (Ω ⊥ 0 ⊗ Ω 0 ) + (Ω 0 ⊗ Ω ⊥ 0 ) Ω 0 + π θ=-π (sin θ) 2 C 1 e 1 λ 1 cos(θ-φ) h 0 (cos θ)dθ (Ω ⊥ 0 ⊗ Ω ⊥ 0 )Ω 0 = π θ=-π sin θ cos θC 1 e 1 λ 1 cos(θ-φ) h 0 (cos θ)dθ Ω ⊥ 0 = 2C 1 π θ=0 sin θ cos θe 1 λ 1 cos θ cos φ sinh 1 λ 1 sin θ sin φ h 0 (cos θ)dθ Ω ⊥ 0 .
So, we obtain from (65):

2C 1 π θ=0 ρ 1 τ 1 + αρ 0 ρ 1 2τ 1 (1 + c 0 cos θ) sin θe 1 λ 1 cos θ cos φ sinh 1 λ 1 sin θ sin φ h 0 (cos θ)dθ - αc 1 2τ 0 β 0 ρ 0 ρ 1 sin φ = 0.
Since (I 2 ) 0 (θ) = h 0 (cos θ) sin θ is non positive on (0, π) (application of the maximum principle for the elliptic equations (46)), the two terms of the last equations have the same sign (the sign of sin φ) and thus the two equal zero. Therefore, the angle φ equals 0 or π, which implies

Ω 0 = Ω 1 or Ω 0 = -Ω 1 .
C Appendix: Proof of proposition 7 (The linearised exchange operator)

Proof. We here provide a detailed proof of the computation of the exchange term:

(Id -Ω 0 ⊗ Ω 0 ) (DS 0 ) (ρ 0 ,Ω 0 ,ρ 1 ,Ω 1 ) (ρ 0 , Ω0 , ρ1 , Ω1 ) -(Ω 0 • S 0 (ρ 0 , Ω 0 , ρ 1 , Ω 1 )) Ω0

once the directions are at equilibria, i.e. Ω 0 = Ω 1 or Ω 0 = -Ω 1 .

Let us begin by developing the expressions of the two components, X 0 and Y 0 , of the linearised exchange operator DS 0 introduced in lemma 6:

X 0 = α ρ0 2 ρ 1 ω∈S 1 ωM λ 1 ,Ω 1 h 0 dω + ω∈S 1 (ω ⊗ ω)M λ 1 ,Ω 1 h 0 dω (α c 0 ρ0 2 ρ 1 Ω 0 + α c 0 ρ 0 2 ρ 1 Ω0 ) + 1 + α ρ 0 2 ρ1 ω∈S 1 ωM λ 1 ,Ω 1 h 0 dω + ω∈S 1 (ω ⊗ ω)M λ 1 ,Ω 1 h 0 dω α c 0 ρ 0 2 ρ1 Ω 0 + (1 + α ρ 0 2 )λ -1 1 ρ 1 Ω1 + ω∈S 1 (ω • Ω1 )(ω ⊗ ω)M λ 1 ,Ω 1 h 0 dω α c 0 ρ 0 2 λ -1 1 ρ 1 Ω 0 + (1 + α ρ 0 2 )ρ 1 ω∈S 1 (ω ⊗ ω)M λ 1 ,Ω 1 h ′ 0 dω Ω0 + α c 0 ρ 0 2 ρ 1 ω∈S 1 (ω • Ω0 )(ω ⊗ ω)M λ 1 ,Ω 1 h ′ 0 dω Ω 0 , Y 0 = α ρ1 2 ρ 0 ω∈S 1 ωM λ 0 ,Ω 0 h 0 dω + ω∈S 1 (ω ⊗ ω)M λ 0 ,Ω 0 h 0 dω (α c 1 ρ1 2 ρ 0 Ω 1 + α c 1 ρ 1 2 ρ 0 Ω1 ) + 1 + α ρ 1 2 ρ0
ω∈S 1 ωM λ 0 ,Ω 0 h 0 dω

+ ω∈S 1 (ω ⊗ ω)M λ 0 ,Ω 0 h 0 dω α c 1 ρ 1 2 ρ0 Ω 1 + (1 + α ρ 1 2 )λ -1 0 ρ 0 Ω0 + ω∈S 1 (ω • Ω0 )(ω ⊗ ω)M λ 0 ,Ω 0 h 0 dω α c 1 ρ 1 2 λ -1 0 ρ 0 Ω 1 + (1 + α ρ 1 2 )ρ 0 ω∈S 1 (ω ⊗ ω)M λ 0 ,Ω 0 h ′ 0 dω Ω0 + α c 1 ρ 1 2 ρ 0 ω∈S 1 (ω • Ω0 )(ω ⊗ ω)M λ 0 ,Ω 0 h ′ 0 dω Ω 1 ,
where h 0 , h ′ 0 (resp. h 1 , h ′ 1 ) implicitly depend on (ω•Ω 0 ) (resp. (ω•Ω 1 )). These expressions can be simplified when computed at equilibria. Indeed, the following lemma gives some useful identities. Lemma 9. We have the following identities:

• ω∈S 1 ωM λ,Ω hdω = cos θh M λ Ω, ω∈S 1 ωM λ,-Ω hdω = -cos θh - M λ Ω, • ω∈S 1 (ω ⊗ ω)M λ,Ω hdω = cos 2 θh M λ Ω ⊗ Ω + sin 2 θh M λ Ω ⊥ ⊗ Ω ⊥ , ω∈S 1 (ω ⊗ ω)M λ,-Ω hdω = cos 2 θh - M λ Ω ⊗ Ω + sin 2 θh - M λ Ω ⊥ ⊗ Ω ⊥ , • ω∈S 1 (ω • Ω)(ω ⊗ ω)M λ,Ω hdω = ( Ω • Ω ⊥ ) sin 2 θ cos θh M λ Ω ⊗ Ω ⊥ + Ω ⊥ ⊗ Ω , ω∈S 1 (ω • Ω)(ω ⊗ ω)M λ,-Ω hdω = -( Ω • Ω ⊥ ) sin 2 θ cos θh - M λ Ω ⊗ Ω ⊥ + Ω ⊥ ⊗ Ω .
where h is a function depending on ω • Ω and h -: ω • Ω → h(-ω • Ω) is the symmetric function of h.

Therefore, at equilibria Ω 0 = Ω 1 , the two components of the linearised exchange operator, X 0 and Y 0 , simplified into:

X 0 = ρ0 α 2 ρ 1 cos θh 0 M λ 1 + αc 0 2 ρ 1 cos 2 θh 0 M λ 1 Ω 0 + ρ1 (1 + α ρ 0 2 ) cos θh 0 M λ 1 + α c 0 ρ 0 2 cos 2 θh 0 M λ 1 Ω 0 + α c 0 ρ 0 2 ρ 1 sin 2 θh 0 M λ 1 + (1 + α ρ 0 2 )ρ 1 sin 2 θh ′ 0 M λ 1
+(1 + α ρ 1 2 )ρ 0 sin 2 θh ′ 0 M λ 0 + α c 1 ρ 1 2 ρ 0 sin 2 θ cos θh ′ 0 M λ 0 Ω0 + α c 1 ρ 1 2 ρ 0 sin 2 θh 0 M λ 0 Ω1 .
Similarly, at equilibria Ω 0 = Ω 1 , the momentum exchange operator S 0 can be expressed as follows:

S 0 (ρ 0 , Ω 0 , ρ 1 , Ω 1 ) = τ 1 (1 + α ρ 0 2 )ρ 1 cos θh 0 M λ 1 + α c 0 ρ 0 2 ρ 1 cos 2 θh 0 M λ 1 Ω 0 -τ 0 (1 + α ρ 1 2 )ρ 0 cos θh 0 M λ 0 + α c 1 ρ 1 2 ρ 0 cos 2 θh 0 M λ 0 Ω 0 .

Assuming Ω 0 = Ω 1 , we thus obtain:

(Id -Ω 0 ⊗ Ω 0 ) (DS 0 ) (ρ 0 ,Ω 0 ,ρ 1 ,Ω 1 ) (ρ 0 , Ω0 , ρ1 , Ω1 ) -(Ω 0 • S 0 (ρ 0 , Ω 0 , ρ 1 , Ω 1 )) Ω0

= τ 1 (1 + α ρ 0 2 )ρ 1 λ -1 1 sin 2 θh 0 M λ 1 Ω1 + sin 2 θh ′ 0 M λ 1 Ω0 -cos θh 0 M λ 1 Ω0

+ τ 1 α c 0 ρ 0 2 ρ 1 sin 2 θh 0 M λ 1 Ω0 + sin 2 θ cos θh ′ 0 M λ 1 Ω0 +λ -1 1 sin 2 θ cos θh 0 M λ 1 Ω1 -cos 2 θh 0 M λ 1 Ω0

-τ 0 (1 + α ρ 1 2 )ρ 0 λ -1 0 sin 2 θh 0 M λ 0 Ω0 + sin 2 θh ′ 0 M λ 0 Ω0 -cos θh 0 M λ 0 Ω0

-τ 0 α c 1 ρ 1 2 ρ 0 sin 2 θh 0 M λ 0 Ω1 + sin 2 θ cos θh ′ 0 M λ 0 Ω0 +λ -1 0 sin 2 θ cos θh 0 M λ 0 Ω0 -cos 2 θh 0 M λ 0 Ω0 , = A 0 (ρ 0 , ρ 1 )( Ω1 -Ω0 ), with

A 0 (ρ 0 , ρ 1 ) = τ 1 (1 + α ρ 0 2 )ρ 1 λ -1 1 sin 2 θh 0 M λ 1 + τ 1 α c 0 ρ 0 2 ρ 1 λ -1 1 sin 2 θ cos θh 0 M λ 1 -τ 0 α c 1 ρ 1 2 ρ 0 sin 2 θh 0 M λ 0 ,
where we used the identities:

λ -1 0 sin 2 θh 0 M λ 0 = cos θh 0 M λ 0 -sin 2 θh ′ 0 M λ 0 , λ -1 0 sin 2 θ cos θh 0 M λ 0 = cos 2 θh 0 M λ 0 -sin 2 θh 0 M λ 0 -sin 2 θ cos θh ′ 0 M λ 0 .

Similar computations shows that at equilibria Ω 0 = -Ω 1 , (Id -Ω 0 ⊗ Ω 0 ) (DS 0 ) (ρ 0 ,Ω 0 ,ρ 1 ,Ω 1 ) (ρ 0 , Ω0 , ρ1 , Ω1 ) -(Ω 0 • S 0 (ρ 0 , Ω 0 , ρ 1 , Ω 1 )) Ω0 = B 0 (ρ 0 , ρ 1 )( Ω1 + Ω0 ), with

B 0 (ρ 0 , ρ 1 ) = τ 1 (1 + α ρ 0 2 )ρ 1 λ -1 1 sin 2 θh - 0 M λ 1 -τ 1 α c 0 ρ 0 2 ρ 1 λ -1 1 sin 2 θ cos θh - 0 M λ 1 -τ 0 α c 1 ρ 1 2 ρ 0 sin 2 θh 0 M λ 0 .

  denote respectively the velocity moduli and the velocity directions. The magnitude of the velocities can take only two values 0 or c > 0, and then the particles are separated into two subgroups: the subgroup {k, |V k | = 0} made of the particles at rest and the subgroup {k, |V k | = c} of the moving particles.

  The functions M ± (ρ) and N ± (ρ) and P ± (ρ) are trilinear with respect to ρ 1 [ρ] and ρ 0 [ρ] = ρ -ρ 1[ρ]. Their explicit expressions are given in proposition 8.

+α ρ 0 c 0 2 ρ 1 sin 2 θ cos θh ′ 0 M λ 1 Ω0ρ 1 sin 2 θh 0 M λ 1 + α c 0 ρ 0 2 ρ 1 λ -1 1 sin 2 θ

 12 cos θh 0 M λ 1 Ω1 , Y 0 =ρ 0 (1 + α ρ 1 2 ) cos θh 0 M λ 0 + α c 1 ρ 1 2 cos 2 θh 0 M λ 0 Ω 0 + ρ1 α 2 ρ 0 cos θh 0 M λ 0 + αc 1 2 ρ 0 cos 2 θh 0 M λ 0 Ω 0 + (1 + α ρ 1 2 )λ -1 0 ρ 0 sin 2 θh 0 M λ 0 + α c 1 ρ 1 2 ρ 0 λ -1 0 sin 2 θ cos θh 0 M λ 0

Note that particles at rest are not moving in space but their directions are changing in time.

If θ is the polar coordinate associated to an orthonormal basis (e 1 , e 2 ) of R 2 , then the divergence of a scalar function f (ω) is given by ∂ θ f and the divergence of a tangent vector field A = A θ e θ , where e θ = (-sin θ, cos θ) is the local polar basis, is given by ∂ 2 θ 2 A θ .

For instance, f 0,1 = k 1,0 means f 0 = k 1 and f 1 = k 0 .

Acknowledgements The author would like to express his gratitude to Pierre Degond and Francesco Ginelli for their fruitful suggestions. He wishes also to thank Richard Bon, Hugues Chaté, Marie-Hélène Pillot and Guy Théraulaz for stimulating discussions. This work has been supported by the "Agence Nationale de la Recherche", in the frame of the contract PANURGE 07-BLAN-0208.

Acknowledgment The author would like to express his gratitude to Pierre Degond and Francesco Ginelli for their fruitful suggestions. He wishes also to thank Richard Bon, Hugues Chaté, Marie-Hélène Pillot and Guy Théraulaz for stimulating discussions. This work has been supported by the "Agence Nationale de la Recherche", in the frame of the contract PANURGE 07-BLAN-0208.

The large scale dynamics

The results of the two previous sub-sections suggest that at equilibria, the dynamics of the two sub-populations could be described by the density of the whole population ρ = ρ 0 +ρ 1 and the common direction Ω = Ω 0 = ±Ω 1 . To recover these dynamics, let us consider the following expansions with respect to δ:

Let us note that Ω δ 0 and Ω 0 are of norm 1 and therefore Ω0 is orthogonal to Ω 0 (resp. Ω1 is orthogonal to Ω 1 ). Thus, we have the following expansions of the von-Mises distributions:

Let us also provide the expansions of the exchange operators R, S 0 and S 1 with respect to δ:

where DR, DS 0 , and DS 1 denote the linearised operators, whose expressions are provided by the following lemma.

Lemma 6. The linearised exchange operators are given by:

where X 0 and Y 0 stand for:

A Appendix: Derivation of the momentum equations

For the sake of completeness, we present here the derivation of the evolution equation of Ω 1 , given by (56). The derivation of equation ( 55) is similar.

We multiply system (30) by the generalised collisional invariant:

introduced in section 4.2 (eq. ( 48)) and we integrate it with respect to ω. In the limit ε → 0, we have Ω[f ε 1 ] → Ω 1 and (I 2 ) ε 1 → h 1 (ω • Ω 1 ) Ω ⊥ 1 • ω . Therefore, we obtain: