J Barés

M L Hattali

D Dalmas

D Bonamy

Fluctuations of global energy release and crackling in nominally brittle heterogeneous fracture

The temporal evolution of mechanical energy and spatially-averaged crack speed are both monitored in slowly fracturing artificial rocks. Both signals display an irregular burst-like dynamics, with power-law distributed fluctuations spanning a broad range of scales. Yet, the elastic power released at each time step is proportional to the global velocity all along the process, which enables defining a material-constant fracture energy. We characterize the intermittent dynamics by computing the burst statistics. This latter displays the scale-free features signature of crackling dynamics, in qualitative but not quantitative agreement with the depinning interface models derived for fracture problems. The possible sources of discrepancies are pointed out and discussed.

Predicting when and how solids break continues to pose significant fundamental challenges [1,2]. This problem is classically addressed within the framework of continuum mechanics, which links deterministically the degradation of a solid to the applied loading. Such an idealization, however, fails in several situations. In heterogeneous solids upon slowly increasing loading for instance, the fracturing processes are sometimes observed to be erratic, with random events of sudden energy release spanning a variety of scales. Such dynamics are e.g. revealed by the acoustic emission accompanying the failure of various materials [3][4][5][6] and, at much larger scale, by the seismic activity going along with earthquakes [7,8]; A generic observation in this field is the existence of scalefree statistics for the event energy [9].

These avalanche dynamics [10] have attracted much recent attention. They were originally thought to be inherent to quasi-brittle fracture, where the solid starts by accumulating diffuse damage through microfracturing events before collapsing when a macroscopic crack percolates throughout the microcrack cloud [START_REF] Van Mier | Concrete Fracture[END_REF]. Phenomenological models such as fiber bundle models (see [START_REF] Pradhan | [END_REF] for review) or random fuse models (see [2] for review) developed in this case reproduce qualitatively the avalanche dynamics with a minimal set of ingredients. More recently, it has been demonstrated [13] that a situation of nominally brittle fracture, involving the destabilization and propagation of a single crack, can also yield erratic dynamics. Within the linear elastic fracture mechanics (LEFM) framework, the in-plane motion of a crack front was mapped to the problem of a long-range (LR) elastic interface propagating within a two-dimensional (2D) random potential [14,15], so that the driving force selfadjusts around the depinning threshold [13]. This approach reproduces, in a simplified 2D configuration, the local and irregular avalanches evidenced in the space-time dynamics of an interfacial crack growing along a weak heterogeneous plane [16]. There exists theoretical arguments to extend this approach to the bulk fracture of real three-dimensional (3D) solids and crackling dynamics at the global (specimen) scale are anticipated [17,18]. Still, fracture experiments are crucially missing to demonstrate this point.

The study reported here aims at filling this gap. Fracture experiments in heterogeneous solids made of sintered polymer beads are found to display irregular burst-like dynamics at the global scale, with large, power-law distributed, fluctuations for the mean failure speed v(t) and overall mechanical energy E(t) stored in the specimen. Yet and despite their individual giant fluctuations, the ratio between v(t) and the power release dE(t)/dt remains constant and defines a continuum-level scale materialconstant fracture energy. The burst statistics displays the scale-free features predicted in elastic interface models. Still, the agreement remains qualitative only, and the scaling exponents are different from those predicted. The possible sources of discrepancies are discussed.

Experiments -The experiments were carried out on artificial rocks made of sintered polystyrene beads: i) a mold filled by monodisperse polystyrene beads was heated to T = 105 • C (90% of the temperature at glass transition) and compressed (pressure p = 4.2 MPa) between the two jaws of an Instron electromechanical machine while keeping T = 105 • C; ii) the mold was then unloaded and slowly cooled down to ambient temperature and the obtained sample was extracted from it. This sintering process provides heterogeneous solids with homogeneous microstructures, the length-scale of which is set by the bead diameter d. In all the experiments reported here, d = 500 µm. Large enough heterogeneity scales, indeed, is requested to observe global crackling at finite driving rate [18] and the fracture of sintered materials with smaller d (250 µm, 140 µm, 80 µm and 40 µm) were observed to display continuum-like dynamics.

In the so-obtained materials, stable cracks were driven by means of wedge splitting fracture tests (see Fig. 1A and Refs. [19,20] for details): Parallelepiped samples of size 140×125×15 mm in the x (propagation), y (loading) and z (sample thickness) were loaded in mode I by pushing a wedge at constant speed V wedge into a 25 × 25 mm cut out on one the two (y -z) edges. An initial seed crack (10 mm-long) was introduced with a razor blade in the middle of the cut. It prevents dynamic fracture and enables growing slow stable cracks. Two go-between steel blocks were placed between the wedge and the specimen to limit parasitic mechanical dissipation and ensure the damage and failure processes to be the sole dissipation source for mechanical energy in the system (see [START_REF] Barés | Failure in brittle heterogeneous materials: Intermittency, crackling and seismicity[END_REF] for details).

The wedge speed V wedge was varied from 16 nm/s to 1.6 µm/s. During each test, the force f (t) applied by the wedge was monitored in real time by a S-type Vishay cell force (acquisition rate of 50 kHz, accuracy of 1 N). As soon as the wedge starts to push on the specimen (time origin set at this onset), f increases. When f gets large enough (∼ 200 -300 N), the seed crack starts to propagate. This propagation was imaged at the specimen surface via a camera (USB2 uEye from IDS Imaging Development, space and time accuracy of 130 µm and 0.1 s, respectively), providing the instantaneous length c surf ace (t) of the crack edge at the surface. The instantaneous mechanical energy E(t) stored in the specimen is given by E(t) = 1 2 f (t)×V wedge ×t, and the instantaneous mean crack length c(t) (i.e. spatially averaged over specimen thickness) is obtained from the knowledge of the instantaneous specimen stiffness k(t) = f (t)/V wedge × t. Indeed, in a linear elastic material, the curve k vs. c is a continuous decreasing function set by the specimen geometry only, and independent of the other experimental parameters (e.g. V wedge , microstructure parameters...). We hence measured the curves k vs. c surf ace in each of our experiments, averaged them over all our experiments, and smoothed the result via a Tikhonov regularization. The so-obtained curve defines the curve k vs. c for our fracture geometry; it was checked this curve is identical to that obtained using 2D finite element calculations (software Castem 2007) on the exact experimental geometry, assuming plane stress conditions. This reference curve k(c) was used to infer c(t) = k -1 (f (t)/V wedge × t) from the signal f (t) [k -1 (x) denoting the inverse function of k(x)]. Time derivation of c(t) finally provides the instantaneous crack speed v(t).

Results -Figure 1(A) presents the time evolution of v(t) and E(t) in a typical fracture experiment. These profiles exhibit the intermittent features characteristic of crackling dynamics, with random violent bursts (resp. sudden drops) in v(t) (resp. in E(t)). The superposition of the two also reveals that the velocity bursts coincide with the energy drops. Beyond this occurrence coincidence, the fluctuation amplitude v(t) is proportional to the power P(t) = -dE/dt released at each moment t (fig. 1(B)). This proportionality was observed in all our experiments, irrespectively of the wedge speed. It betrays the characteristics of a nominally brittle fracture: Here Γ = 100 ± 10 J/m 2 .

We turn now to the statistical characterization of the crack dynamics. We analyzed the temporal evolution P(t) in preference to that of v(t) since the former is directly obtained from the experimental measurement of applied force f (t), while the latter calls for the addition of the k vs. c curve. The distribution of instantaneous power released is first analyzed. Note that, in experiments, an "instantaneous" quantity is actually averaged over a finite time scale δt, the value of which affects the fluctuation amplitude. The distributions of P(t), hence, have been computed for different values of δt. A Gaussian distribution (centered at zeros, standard deviation decreasing as 1/ √ δt) is observed at small scales (plain lines in Figs. 2A andB), and a power-law tail is observed at large scales (Fig. B). The Gaussian part at small P is observed throughout the whole experiments, even in the preliminary loading phases where cracks do not propagate. It results from the noise in the measurement of the force signal. This noise yields a δt-dependent res- Plain curves are Gaussian distributions with zero average and a variance σ(δt) prescribed so as the Gaussian curve fits the experimental data on the small scale plateau. C, main panel: Same graphs as in A after having withdrawn the noisedominated Gaussian part of each distribution. Empty and filled symbols correspond to V wedge = 16 nm/s and V wedge = 160 nm/s, respectively. The curves associated to V wedge = 160 nm/s have been shifted vertically for sake of clarity. Distributions involve 3094 events for V wedge = 16 nm/s and 5299 events for V wedge = 160 nm/s. Red and blue straight lines are power-law fits with exponents a small = 1.4 ± 0.15 (small scale regime) and a large = 2.5 ± 0.1 (large scale regime). Vertical dash line locates the crossover PC ≈ 0.34 mW. C, inset: maximum value Pmax observed for P(t) as a function of δt.

Red line shows the 1/ √ δt dependency.

olution limit P r below which the true fracture-induced fluctuations of P(t) cannot be deconvoluted from the Gaussian noise. Conversely, the probability that a fluctuation of size P(t) > P r is due to noise rapidly is insignificant. Keeping only the relevant part above P r , the distributions all collapse onto a single master curve (Fig. 2C) exhibiting two power-law scaling, a small-scale regime with a scaling exponent a small = 1.4 ± 0.15 and a large-scale regime with a large = 2.5 ± 0.1. The two scaling regimes, together with the value of the associated crossover (P C ≈ 0.34 mW), depend neither on δt, nor on the loading rate V wedge . Conversely, the maximal value P max (resp. v max ) of P(t) (resp. v(t)) decreases with δt, as 1/ √ δt (Inset in Fig. 2C), as expected for independent fluctuation peaks. Note that the large scale power-law exponent a large = 2.5±0.1 observed here at the global scale 3. A: Distribution of the avalanche size S, defined either as the energy released (black/bottom) or as the area swept (red/top) during the event. In the latter case, the area is normalized by d 2 where d = 500 µm is the grain diameter B: Scaling between normalized size S/ S and duration D/ D . In both panels, the various symbols correspond to various values for {δt, C} (specified in the inset of A) and V wedge (empty symbol for V wedge = 16 nm/s, filled ones for V wedge = 160 nm/s; the latter have been shifted vertically for sake of clarity). Analyzes involve 899 avalanches for V wedge = 16 nm/s and 473 events for V wedge = 160 nm/s. Straight lines are power-law fits P (S) ∝ S -τ and S ∝ D γ , with τ = 1.4 ± 0.1 (resp. τ = 1.1 ± 0.15) and γ = 1.38 ± 0.05 (resp. γ = 1.17 ± 0.05) for V wedge = 16 nm/s (resp. V wedge = 160 nm/s).

for fracture experiments in bulk solids is very close to that reported on the local velocity fluctuations in 2D situations of interfacial cracks, both experimentally [16,[START_REF] Tallakstad | [END_REF] and numerically [23]. This supports the conjecture that at large scales, brittle 3D fracture can be reduced to a 2D elastic interface problem [9,13,18]. Conversely, the small scale power-law regime with a small = 1.4 ± 0.1 observed here differs from that observed in the 2D interfacial configuration.

The scale-free statistics observed for the fluctuations P(t) (or equivalently for the fluctuations v(t)) is a first hint toward crackling dynamics. We adopt the standard procedure in the field, and identify the underlying avalanches with the bursts where P(t) is above a prescribed reference level P r = C P . Then, the avalanche duration D of each pulse is given by the interval between the two intersections of P(t) with P r , and the avalanche size S is defined as the energy released during the event, i.e. the integral of P(t) between the two intersection points. Note that, in conventional elastic interface formalism, the avalanche size S is expressed as the total area A swept by the front between two successive pinned configurations. The two definitions are equivalent since, in the nominally brittle situation experienced here, an event releasing an energy S creates fracture surfaces of area A = S/Γ. As expected for a crackling signal, S follows a power-law distribution P (S) ∝ S -τ over nearly three order of magnitude, with event areas up to ∼ 400 times the elementary one d 2 (Fig. 3:A). for V wedge = 16 nm/s and V wedge = 160 nm/s cannot be precluded. Also, the mean avalanche size goes as a power-law with D, S ∝ D γ (Fig. 3:B). The exponents τ and γ are independent of the prescribed values for δt and C. Conversely, they both decrease with the loading rate, from {τ = 1.4±0.1, γ = 1.38±0.05} at V wedge = 16 nm/s to {τ = 1.1±0.15, γ = 1.17±0.05} at V wedge = 160 nm/s.

To complete the dynamics characterization, we computed the average temporal avalanche shape. This observable, indeed, provides an accurate characterization of the considered crackling signal and has been measured in a variety of systems [24][25][26][27][28][29]. The standard procedure was adopted here: First, we identified all the pulses i of duration D i falling within a prescribed interval [D min , D max ]); and second, we averaged the shape P i (t)/P i max vs. t/D i over all the collected pulses i. Figure 4:A shows the resulting shape and its evolution with D. Two observations should be noted: i) The shape is parabolic for small D and flattens as D increases; ii) A small, but clear leftward asymmetry is observed as D increases: The bursts start faster than they stop. Quantitatively, the shape evolution (how fast it flattens and how large the asymmetry is) depends on V wedge .

A similar shape flattening was observed in Barkhausen experiments [27]. Therein, it was shown to result from the finite value of the demagnetizing factor k. Such a shape flattening is thus expected in the LR interface model for crack growth [13], where the unloading factor plays the same role as k [18]. Conversely, the numerical simulation of this LR interface model yielded symmetrical shapes [START_REF] Barés | Failure in brittle heterogeneous materials: Intermittency, crackling and seismicity[END_REF] (at finite driving rate) or slight rightward asymmetry [29] (at vanishing driving rate). In Barkhausen experiments, in contrast, a leftward asymmetry was observed, and attributed to the eddy currents, which provides a negative effective mass to the domain walls [24]. We conjecture that the viscoelastic nature of the polymer rocks fractured here acts in a similar way by providing a negative inertia to the crack front (i.e. the addition of a retardation term in the LR interface model of crack [13]).

Concluding discussion -The experiments reported here demonstrate that crackling at global scale can be observed in nominally brittle fractures (due to the propagation of a single crack) and is not restricted to quasibrittle (multi-fracturing) situation: Three main observations emerge: i) Despite their individual giant fluctuations, the ratio between spatially-averaged velocity and power release remains fairly constant and defines a continuum-level scale material-constant fracture energy; ii) The event size, defined either as the increase of crack length or as the energy release during the event, is powerlaw distributed, and scales as a power-law with the event duration; iii) the associated exponents depend on the crack loading rate.

These observations are in qualitative agreement with what is predicted by a recent model [13] identifying stable crack growth with a LR elastic interface driven in a random potential so that the driving force self-adjust around the depinning threshold. Still, the agreement is qualitative only: i) In the universality class of the LR depinning transition, the scaling exponents are predicted to be a small = 0.38 [START_REF] Dobrinevski | Dynamics of elastic interfaces in random media[END_REF], τ = 1.28 [9], and γ = 1.80 [9], significantly different from the experimental values measured here; ii) These predicted exponents are independent of the driving rate [30], contrary to what is observed here. This discrepancy is thought to result from the finite width of the fracture specimens; the LR elastic kernel in the interface model [13] arises from Rice's perturbative analysis of the elastic problem of a corrugated crack front embedded in a sample of infinite width [START_REF] Rice | [END_REF]. Conversely, it is interesting to note that the variations of τ and γ observed in our experiments are compatible with those expected in the mean-field ABBM model [START_REF] Alessandro | Montorsi[END_REF]: In the ABBM model, i) τ decreases from 3/2 to 1 as the driving rate increases; ii) γ exhibits two values: γ = 2 for short pulses, γ = 1 for large ones, with a crossover decreasing with driving rate [27]. In our experiments, the mean-field approximation may be relevant since the separation between the microstructure scale (bead size d = 500 µm) and the continuum-level scale (specimen width: 15 mm = 30d) is quite small. The simplicity of the ABBM model has allowed the derivation of exact analytical solutions for the avalanche distribution and shape [27,[34][35][36]. A very interesting future extension of this study is to accurately characterize how the avalanche statistics evolves with the driving rate and to confront those against the ABBM solutions. Work in this direction is under progress.

Support through ANR project MEPHYSTAR (ANR-09-SYSC-006-01) is gratefully acknowledged. Special

1 FIG. 2 .

 12 FIG.2. Distribution of instantaneous power release for different coarse-graining time δt plotted in logarithmic (panel A) and linear scales (panel B). Here, V wedge = 16 nm/s. Plain curves are Gaussian distributions with zero average and a variance σ(δt) prescribed so as the Gaussian curve fits the experimental data on the small scale plateau. C, main panel: Same graphs as in A after having withdrawn the noisedominated Gaussian part of each distribution. Empty and filled symbols correspond to V wedge = 16 nm/s and V wedge = 160 nm/s, respectively. The curves associated to V wedge = 160 nm/s have been shifted vertically for sake of clarity. Distributions involve 3094 events for V wedge = 16 nm/s and 5299 events for V wedge = 160 nm/s. Red and blue straight lines are power-law fits with exponents a small = 1.4 ± 0.15 (small scale regime) and a large = 2.5 ± 0.1 (large scale regime). Vertical dash line locates the crossover PC ≈ 0.34 mW. C, inset: maximum value Pmax observed for P(t) as a function of δt. Red line shows the 1/ √ δt dependency.

  FIG.3. A: Distribution of the avalanche size S, defined either as the energy released (black/bottom) or as the area swept (red/top) during the event. In the latter case, the area is normalized by d 2 where d = 500 µm is the grain diameter B: Scaling between normalized size S/ S and duration D/ D . In both panels, the various symbols correspond to various values for {δt, C} (specified in the inset of A) and V wedge (empty symbol for V wedge = 16 nm/s, filled ones for V wedge = 160 nm/s; the latter have been shifted vertically for sake of clarity). Analyzes involve 899 avalanches for V wedge = 16 nm/s and 473 events for V wedge = 160 nm/s. Straight lines are power-law fits P (S) ∝ S -τ and S ∝ D γ , with τ = 1.4 ± 0.1 (resp. τ = 1.1 ± 0.15) and γ = 1.38 ± 0.05 (resp. γ = 1.17 ± 0.05) for V wedge = 16 nm/s (resp. V wedge = 160 nm/s).

FIG. 4 .

 4 FIG.4. Temporal avalanche shape for different avalanche durations D (V wedge = 16 nm/s). Note the shape flattening and leftward asymmetry that develop as D increases.

  FIG.1. A: Sketch of the Wedge-Splitting test. B: Zoomed view of the crack tip speed v(t) (black) and stored mechanical energy E(t) (gray) as a function of time in a typical fracture experiment (Here, V wedge = 16 nm/s). C: Instantaneous power release P = -dE/dt as a function of v(t) for all t. The axes are linear in the main panel, and logarithmic in the inset. In both panels, straight lines indicate proportionality. The proportionality constant Γ = 100 ± 10 J/m 2 gives the material's fracture energy. In both panels B and C, the coarse-graining time is δt = 0.2 s

B C

P(t) = G(t) × v(t)

where G(t) = -dE/dc is the energy release rate. Now, for a stable crack slowly driven in a nominally brittle material, LEFM states that G(t) ∼ Γ where the fracture energy Γ is a material constant. In other words, a nominally brittle fracture compatible with LEFM assumptions yields P(t) = Γ × v(t) at all times t, irrespectively of the precise values of P(t) and v(t), as observed here. In this scenario, the proportionality constant in fig.

1

(B) gives Γ for the considered material:

thanks to Thierry Bernard for technical support.


Amosse Edouard
email: amosse.edouard@unice.fr

Nhan Le-Thanh
email: nhan.le-thanh@unice.fr

Introduction toSocialRadio Case Studies and Perspectives

Keywords: social networks, online communities, social radio, short messages. I

In this paper, we introduce a Social Radio platform designed to support contextual communication between communities. Radio communication was no longer used for information broadcast using different communication channels. The same concept is used in this work but for different purposes. We set up a microblogging community channel to report, comment or simply vote some events. In order to understand the concept we detail a scenario in the epidemic field.

INTRODUCTION

Since people moved an important part of their time on Internet [START_REF] Backstrom | Find Me If You Can: Improving Geographical Prediction with Social and Spatial proximity[END_REF], microblogs such as Twitter or social networks such Facebook, Google+ become more and more popular and can be considered as important tools for communication among people, sharing expertise and knowledge enrichment. It is important to note that people have different motivations in using these tools. Some are interested in sharing observations while others are concerned with consuming information. Peoples' needs depend on whether they are consumers or producers. While information are more and more available, finding the relevant information among all those contextdependant observations is an important issue for consumers and also a challenge for researchers or service providers. For example, a candidate can be interested in what people think of his candidacy for the next election.

Handlinglarge quantities of comments from all users on a given topic can be difficult or even impossible without appropriate tools. Such tools must be able to retrieve the common trends or the main idea among a set of observations.In this work we tackle this problem by introducing the social radio, a platform enabling users to extract the common trends on a specific subject (theme) by analysing reported observations within an online community.

The term radio is commonly used for information broadcasting using radio waves or frequencies. In this study, we use the same concept but for different purposes. The main parts of a radio are: frequency (its identity),information sources, a redaction service and an audience (people interested in this radio). Information sources can be employees (journalist, reporters), music producers (for a music radio) etc.

Activities on the web are quite similar toa radio becauseinformation is constantly shared between producers and consumers. The main differences between the web and a radio come from the organisation of the information.

Numerous online services provide multidisciplinary information at large scale to an open audience. Finding relevant information is an important challenge for consumers. Data analysis in online social networks or microblogging platform is largely explored since the past decade in many domains [START_REF] Passant | Microblogging: A Semantic and Distributed Approach[END_REF][START_REF] Le-Thanh | Introduction à l'Univers des BRIEFS[END_REF][START_REF] Da Silva | Online Communities Administration: Defining Tools for Different Profiles[END_REF][START_REF] Da Silva | What they want and what they get: A study of light-weight technologies for online communities[END_REF][START_REF] Naaman | Is it Really About Me? Message Content in Social Awareness Streams[END_REF][START_REF] Java | Why We Twitter: Understanding Microblogging Usage and Communities[END_REF][START_REF] Abel | Semantic Enrichment of Twitter Posts for UserProfile Construction on the Social Web[END_REF]. Lots of studieshave focused on motivation [START_REF] Java | Why We Twitter: Understanding Microblogging Usage and Communities[END_REF], member's activities and behaviours [START_REF] Stankovic | Directing Status Messages to their Audience in Online Communities[END_REF][START_REF] Schoberth | Online Communities: A Longitudinal Analysis of Communication Activities[END_REF], data filtering and enrichment [START_REF] Abel | Semantic Enrichment of Twitter Posts for UserProfile Construction on the Social Web[END_REF][START_REF] Passant | Meaning Of A Tag: A Collaborative Approach to Bridge the Gap Between Tagging and Linked Data[END_REF]. In [START_REF] Nagarajan | Spatio-Temporal-Thematic Analysis of Citizen Sensor Data: Challenges and Experiences[END_REF], the goal was to extract common thematic used in Twitter posts based on an n-dimensional analysis taking into accounttheme, space and time. The authors in [START_REF] Zhao | Temporal and Information Flow Based Event Detection From Social Text Streams[END_REF] proposed an approach for event detection from social text stream by exploring the content as well as temporal and social dimension. F. Abel has proposed in [START_REF] Abel | Semantic Enrichment of Twitter Posts for UserProfile Construction on the Social Web[END_REF] an approach to enrich status messages from Twitter based on external links referred in the messages.In [START_REF] Passant | Meaning Of A Tag: A Collaborative Approach to Bridge the Gap Between Tagging and Linked Data[END_REF], the authors used hash tags and linked data to enrich post from twitter.

In this paper we study the topology of the social radio, a platform for information synthesis within an online community. We analyse the main needs that can make the platform generic. By generic we mean, anyone can configure it according to his need with multiple information sources such as online social networks, microblogging, websites etc. The rest of this paper is organized as follows; Section 2 presents background about online communities and microblog messages;Section 3 introducesa generic platform for broadcasting information in online communities, which we call the social radio. We then detail a use case for the social radio in Section 4. Finally, we discuss future work and make conclusions in Section 5.

II. BACKGROUND

A. Online communities, structures and objectives

In an "online community", members do not necessarily meet and communicate with each other face-to-face. Online community can be built in two ways: implicitly from user behaviour (e.g. linking users who have the same hobbies or who took similar actions an a website) or explicitly (e.g. users link themselves to others: friends or co-workers).

According to [START_REF] Seufert | Design and Management of Online Learning Communities[END_REF], online communities can be broken down into overlapping classes as shown in Figure 1. Generally, members of online communities can be split into two categories [START_REF] Nonnecke | Non-public and public online community participation: Needs, attitudes and behavior[END_REF]:

-Active Participants (Posters): Users who participate actively in the community by posting messages -Inactive Participants (Consumers): Members who are interested in the information shared within the community.

As pointed out by J. Nielsen [START_REF] Nielsen | Participation Inequality: Encouraging More Users to Contribute[END_REF], user participation follows more or less a 90-9-1 rule. That is, 90% of users read/observe but don't contribute; 9% contribute from time to time; and most contributions come from just 1% of the users, who participate frequently. How information is used depends onwhether a participant is active or inactive.For instance, onan e-commerce platformusers (customers) comment or rate products independently of other users while buyerslurk for most rated products to buy. Users post information regarding their observations or personal opinions on fact they observe. While many users can report observations on the same fact, event or theme;how to present the relevant information to consumers is an important challenge.

B. Short messages Microblogging and social networks differ from blogsandgeneralweb sites becausethey enable anyone to sharetheir thoughts online to an open audience. This freedom of the crowd distributes a large amount of data on social platforms. Microblogginghas also been successful in corporate environments in facilitating informal communication, learning and knowledge exchange (almost real-time) and fast propagation of new information. However, data frommicroblogs is substantially differentfrom other that found in other information systems such as theenterprise web site.In particular, [START_REF] Passant | Microblogging: A Semantic and Distributed Approach[END_REF] itrepresentsrich social connections (between the information senders/authors and recipients/reviewers); and (2) the content is strongly context sensitive. That is, not only within the text piece content, the meaning of words is dependent from the context of the communication as well as relationships among social actors. In a communication process between humans, the interlocutors know the context.That's why they can understand the meaning even when a message is very short. As an example "This traffic jam bothers me"has a meaning for the recipientbecause he has the context (e.g. who the sender is, where the sender is, why he's there, his relationship with the sender). Inferring context is essentialwhen processing such messages.Hence, in our work, we willprocess messages based on both the thematic content and on the community in which the message was produced.

III. TYPOLOGY OF THE SOCIAL RADIO AND DEFINITION

This section defines terms and concept of the studied platform. In other to define the Social Radio for information broadcasting within online communities, we consider the following concepts according to member'sbehaviours and needs:

-A themedefines the context of shared information. -A communityis composed ofposters and consumers.

-Information processing mechanisms definehow to structure information based onconsumers'needs and information privacy.

Definition III.1 Informationtheme

The theme of information is what that defines the context 0of the information being shared. s the main part of a message, that what which defines the context of the information being shared. Because of the brevity of post in online social platforms, the theme is an important key for context grasping. Context inferring is widely explored in computer science [START_REF] Passant | Microblogging: A Semantic and Distributed Approach[END_REF][START_REF] Ehrlich | Microblogging Inside and Outside the Workplace[END_REF]and semantic web [START_REF] Abel | Semantic Enrichment of Twitter Posts for UserProfile Construction on the Social Web[END_REF]. Information context is the first step for information processing within online communities.

Definition III.1 Posters and Consumers

Sincemembers of a community have different information needs and behaviours; information processing it is important to we deliberately splitthe members who form a radio's community intoposters and consumers [START_REF] Nonnecke | Non-public and public online community participation: Needs, attitudes and behavior[END_REF]. Posters are those who post messages on the radio while consumers are those who consume information.This approach allows us to define how information should be presented to users according to their profiles.

Definition III.3 Information synthesis

When seeking for information, to begin with, users are interested in anoverview instead of details [START_REF] Nielsen | Participation Inequality: Encouraging More Users to Contribute[END_REF]. For example, the most important part of a meeting is the

A community can be at the intersection of one or more classes.

summary of points that have been discussed and decisions that have been taken. Details can come up if users ask for them. A radio social shall be able to summarize information in other to present the most relevant information to the user.

A. Types of radio

Consideringcommunity

structures, behaviour and expectations [START_REF] Nonnecke | Non-public and public online community participation: Needs, attitudes and behavior[END_REF], information is processingfrom two perspectives:posters (who post information) and consumers (who seek andconsume information). For posting and consuming there are three possibilities: public access, private access and hybrid access. A radio with public access for posting or consuming enable any users to access information on the radio; a radio with private access restrict access to some members of the radio; and radio with hybrid access manage access to information according to privacy and type of information. Considering the three possibilities for consuming and posting, the

B. Characteristics of the social radio

In a survey on Twitter in [START_REF] Java | Why We Twitter: Understanding Microblogging Usage and Communities[END_REF], Java et al. found that members' activity can group as:1) Daily Chatter (what a user is actually doing), 2) Conversations (direct exchange between members), 3) Sharing Information and 4) Reporting News. In [START_REF] Ehrlich | Microblogging Inside and Outside the Workplace[END_REF], Kate et al. extend those categories into:status, provide information, forward information, ask question direct post anddirect question. Although those studies are on the Twitter platform, the categories describe the main user activities within online communities. In the radio social, more than the categories defined in [START_REF] Java | Why We Twitter: Understanding Microblogging Usage and Communities[END_REF][START_REF] Ehrlich | Microblogging Inside and Outside the Workplace[END_REF], we attempt to capture main types of user's activities within online communities: reporting, evaluation, and opinion.In Table 2 In twitter, when reporting an event users use hash tag to highlight the most important word in the post, in most cases those keywords define the context of the post [START_REF] Passant | Microblogging: A Semantic and Distributed Approach[END_REF][START_REF] Abel | Semantic Enrichment of Twitter Posts for UserProfile Construction on the Social Web[END_REF].

Trafic jam on a road:

The is any personal appreciation when reporting traffic jam, the user observes that this event exist right now in this place.

@BLTEDI:

"This #traffic bothers me". This means(somewhere in the world) a traffic jam exists. C. Discussion In the previous section, we have definedtypes and characteristics of information in online social services; in this section we discuss the needs for the social radio to be a generic service for information broadcasting within online communities. Consistent to previous research, the social radio shall take in consideration type of communities [START_REF] Da Silva | Online Communities Administration: Defining Tools for Different Profiles[END_REF] as information theme depends on the group in which it is sharing [START_REF] Schoberth | Online Communities: A Longitudinal Analysis of Communication Activities[END_REF]. On the other hand, information processing shall consider consumers need. As pointed out in [START_REF] Da Silva | What they want and what they get: A study of light-weight technologies for online communities[END_REF], in a community members have different expectations: 1) content presentation, 2)Interactive Realism, 3) privacy, 4) leadership and 5) decision making. Consequently, withina community information can be processed differently according to the consumers needs (e.g. service providers and customers in a E-commerce platform). According to E. José et al.,a way to reduce the gap between real and virtual communities is through a proper intervention by one or various person in charge of the community. We can adopt this management approach to allow users define the structure of a social radio.

Opinion: The post describes the user's appreciation on

Information theme can exploit semantic web technics [START_REF] Passant | Microblogging: A Semantic and Distributed Approach[END_REF][START_REF] Abel | Semantic Enrichment of Twitter Posts for UserProfile Construction on the Social Web[END_REF] to define characteristics of shared data on the radio. Indeed, web semantic and ontologies enable definition of concept and relation among concepts in an informationsharing environment. Defining a vocabulary will benefit for the radio in this it defines clearly the radio thematic and his relation to other thematic on the web as well as dependant concept and entities.

Information sources are the sources that provide information to the radio. A radio can define external as well as internal information sources according to the user needs. Internal sources can be dedicated applications that providedata directly to the radio (e.g. users with smart devices, sensors etc.). Existing social services platform expose their data in API such external application can use them. Based on its thematic a radio can connect to an external source and gather information shared within the community. Because of the brevity of information in microblogs and social networks, data retrieved from external sources must be enriched in other to grasp their context. Enrichment mechanism can rely on previous research in the field [START_REF] Passant | Microblogging: A Semantic and Distributed Approach[END_REF][START_REF] Abel | Semantic Enrichment of Twitter Posts for UserProfile Construction on the Social Web[END_REF][START_REF] Passant | Meaning Of A Tag: A Collaborative Approach to Bridge the Gap Between Tagging and Linked Data[END_REF].

The main part of the social radio is automatic processing of information according to user needs, types of information and characteristics. According to that a radio shall provide generic algorithms for information processing and enable a user to configure them according to his needs.

IV. RASE "RADIO SOCIAL FOR EPIDEMIOLOGY", A CASE STUDY

In this section, the focused use case is illustrated by an application developed with health workers in Biot to report epidemic symptom based on the social radio concept. The system is composed of two parts:an administrative part that enables management and configuration settings and the client part that enable users to interact the system. Interaction can be report new epidemic symptom or consume information that has been shared within the community. In the current version only health workers (e.g. doctors, nurses, pharmacists etc.)are allowed to postwhile other users can consume shared information. The Figure 2 presents a use case of the application.

In this application we are facing two constraints:a) time constraint since event reporting is not the primary part of health workers' activity and b) medical confidentialities.

a) Medical confidentialities

Patient information is not required for the system; posters report symptoms without any personal information regarding the patient.

b) Dealing with time constraint

In order to make the application less time consuming, interfaces shall be user friendly and intuitive. In this scenario, information theme is well known; members of the community exchange information on epidemic symptom and diseases. Unfortunately, existing medical ontologies did not applicable in our case because they are domain-related. Diseases or symptoms are treated differently from a medical domain to another. Pathology of diseases or symptoms are not relevantto our system, we do not need a complete ontology. For that, we have built anontology to that defines a shared vocabulary for the actors of the system. The ontology has two concepts:Epidemic and Symptoms and a set of properties as shown in Figure 4. Instances of the ontology are created using and administrative interface (Figure 5). A message is a quintuplet θ =< 𝑃, 𝑂, 𝑆, 𝐼, 𝐴 >. [START_REF] Le-Thanh | Introduction à l'Univers des BRIEFS[END_REF] a) P is the phenomenon (e.g. influenza) an instance of the ontology; b) O is the observer (e.g. a doctor); c) S is the location of the observation; d) I the time (instant) of the observation and e) A content of the observation (e.g. cough, headache) instance of the ontology related to the phenomenon. To reportan epidemic a user pick an instance of epidemic in the vocabulary and the system retrieves automatically symptoms related; the user chooses one or many symptoms. Space and timeare inherent to epidemiology [START_REF] Camara | Un système de veilleépidémiologique à base d'ontologie[END_REF]. Asking the user to provide those properties will overload the user actions. Hopefully, smart devices enable to retrieve user location, current time and information about the user.We consider a "broadcast delay" that consists in postponing the message over a short period while we enrich the message.

There exist two main approaches for spatial representation: a quantitative approach that represents spatial entities using numerical values (e.g. longitude, latitude, geometric form); and a qualitative approach that represents spatial entities using lexical term close to the human perception of spatial notions [START_REF] Renz | Qualitative spatial reasoning using constraint calculi[END_REF]. The main advantage of the qualitative approach on the quantitative approach is the ability to deal with incomplete information. Because the user location is not always available (e.g. GPS disabled) and the system does not require fine-grained location; spatial entities are represented qualitatively.

For example, when a user reports and epidemic case, we associate the nearest locality to the message he reports. Spatial settings are defined according to the Geonames ontology 1 in which geographic spatial entities are grouped in a subsumption relationship. We have restricted ourselves to five levels in the ontology as presented in table 3. To retrieve spatial entities, we have used the Google Reverse Geocoding API to map the user location with named spatial entities. Mapping between Google spatial entities and Geonames are resumed in Table 3.

Google parameters Geonames Properties

Name

Data processing

Because an epidemiologic symptom is reported in a region does not mean that an epidemic exists. To detect the emergence of an epidemic in a region, we have defined analgorithmthat process periodically information based on their spatiotemporal properties. At this stage, we cannot confirm automatically that an epidemic exist but instead weraise a notification when the incidence of an epidemic exceeds a given threshold. For each period, the algorithm computes the number of cases reported in each distinct region and determines a ratio according to the population. If for a region the incidence is greater than the threshold,the system raises an alert to notify the administrators.

Given T a time interval defined by𝑇 = [𝑡 𝑖 , 𝑡 𝑗 ].The set of observations in this interval is 𝛺 = {θ 1 , θ 2 , θ 3 , . . . . . . , θ 𝑛 }, where θ 𝑖 is a message.

Δ 𝑠 = 1 if θ 𝑖 ∈ 𝛺 𝑎𝑛𝑑 θ 𝑖 S = s 0 else 𝑖=𝑐𝑎𝑟𝑑 𝛺 𝑖=1
The incidence is a probabilistic value given by: 𝜅

𝑠 = Δ 𝑠 𝜂 𝑆
where 𝜂 𝑆 is the population of the region.

The dataset do not contain information about the population of a region. DBpedia is a crowd-source effort to make structured information publicly available on the Web. It links resources from Wikipedia with other datasets and enables asking complex queries using SPARQL, a query language for the semantic Web. From the previous SPARQL request we have the population of the region as:

𝜂 𝑆 = 𝑞(Where q is the response obtained from DBpedia)

Then the incidence for a region is:

𝜅 𝑠 = Δ 𝑠 𝜂 𝑆
According to the INSERM vector scale, we have associated a colour to each region regarding the position of the incidence in the scale. If the incidence is greater than the threshold we raise an alert to the system managers.

Conclusion

We have presented the RaSE platform an application built on the concept of the social radio. This application provided tools and interfaces that enable health workers to exchange information on epidemic symptoms in a distributed environment. Data reported by health workers are processed periodically according to their spatiotemporal properties and displayed on base map in other to make them available for consumers. In France, there exist the "Sentinels Network " that provide information about epidemic incidence on the territory. However in their approach "sentinels" using paper and phone calls to report information.Furthermore "sentinels" are dedicated users that journey over the country to report information. In our approach, data are real-time reported and can be reported by several health workers such as (doctors, pharmacists, nurses etc.). Moreover, "sentinel" can use the platform to report information to the "Sentinels Network".

V. CONCLUSION &PERSPECTIVES

In this paper we have introduced the concept of "Social Radio", a platform for information broadcasting in online communities.

Online communities' members have different expectations and we have found that information cannot process in the same way for all members in a community. The social radio aims to provide tools enabling members of online communities building their own "radio" according to their needs. We have defined the types and characteristics for a social to be generic. To prove the concept we have presented RaSE an application that enables health workers to report epidemiologic symptoms. Data reported on the platform are automatically processed and enrich before sharing them to consumers. The following points resumed the direction for future works:

Figure 1 :

 1 Figure 1: Different Online Communities Profiles (extracted from [17]).

Figure 2 :

 2 Figure 2: A use case's scenario of RASE Let S be a set of unique location in the dataset 𝑆 = {𝑠 1 , 𝑠 2 , . . . , 𝑠 𝑚 / 𝑚 ≤ 𝑛}. For each region in S the number of reported observations is determined by:

Table 1

 1 summarizes the different types of radio including appropriate examples.

	Posters	Consumers	Example
	Public	Public	Traffic event report:Anyone is
			allowed to report or consume
			information within the community
	Public	Private	Customer satisfaction survey:Anyone
			is allowed to post information while
			only employees of the service providers
			are allowed to consume reported
			information.
	Public	Hybrid	E-Commerce platform: Anyone is
			allowed to buy and comment on
			product they have bought, rating and
			comments are visible to other
			customers while information on a
			customer such as address or phone
			number is accessible only by employees
			of the company.
	Private	Public	An online blog: Only manager (s) can
			publish information on a blog while
			anyone is granted to consume
			published information.
	Private	Private	Instant Messaging: Only member can
			post information that can be consumed
			only by other members
	Private	Hybrid	Twitter: Only account owners can
			twitt; some information is available for
			public access while other is available
			for (some) followers.
	Hybrid	Public	TODO:
	Hybrid	Private	Air pollution survey: in an air
			pollution survey, although anyone can
			report observations but only authorized
			members can access the reported
			information
	Hybrid	Hybrid	Epidemic survey: This community
			allows anyone to report epidemic
			symptoms, but information reported by
			a health worker is more reliable than
			one reported by an unknown user. In
			the output only confirmed epidemics
			are reported with public access.
			Information such as unverifiedcases is
			granted to some members (e.g.
			doctors).

Tableau 1. Type of radio and examples.

  we summarize our codingactivitiessupported by examples.

	Category		Example
	Reporting: Users report news
	or event based on observation
	in their environment. Mostly
	such actions are objective;
	users report what they observe
	in the universe. There is no
	personal	appreciation	or
	interpretation of the fact that
	the user is being described.

User activities and examples. Examples are extracted from Twitter; usernames are removed for privacy;

				Descriptive Opinion:
			the	"#enjoying #holidays
	given subject.Activities such as	with	family	and
	evaluation,	notation	and	friends."
	comment fall into user's
	opinion.	An opinion is	Qualitative
	subjective and strongly depends	Opinion:@MATITA:
	on the user point of view. More	"#S5 like this phone"
	generally,	opinion	groups
	appreciation and evaluation of
	facts, services and events.
	Opinion can be qualitative (the
	set of value is finite; e.g. like vs.
	dislike) or descriptive (the set of
	value that a user can use is
	indefinite).
	Questioning: Posts that are	@BKFD:	"Anyone
	looking for feedback. The user	knows how to import
	looks for a response within the	data from a #RDBMS
	community.			to #RDF?"
	Tableau 2.

Mapping Google reverse geocoding attributes and Geonames ontology

	country	A	Country
	administrative_area_level_1	ADM1	Region
	administrative_area_level_2	ADM2	County
	administrative_area_level_3	ADM3	City
	locality	ADM4	Locality
	Tableau 3

  The following request to the DBpedia SPARQL endpoint retrieves the population of the region S. Note we assume S is Nice in the request.

	SELECT ?q{
	SERVICE <http://dbpedia.org/sparql> {
	<http://dbpedia.org/resource/Nice>
	<http://dbpedia.org/property/population> ?q. }}
	Request 1: Request to the DBpedia SPARQL endpoint retrieving the
	population of the city of Nice

http://www.w3.org/2005/Incubator/geo/XGR-geo-ont-20071023/

-Data enrichment: Short messages are widely used in online communities and can be from several sources,in future works we will define an approach for data enrichment and reconciliation. We believe that spatiotemporal properties are a good alternative for data enrichment. -Information processing: We have identified in the Section III.C that algorithm for data processing can be: 1) probabilistic, 2) statistical and 3) evidential.


Julien Barré

Raphaël Chétrite

Massimiliano Muratori

Fernando Peruani

Motility-induced phase separation of active particles in the presence of velocity alignment

Self-propelled particle (SPP) systems are intrinsically out of equilibrium systems, where each individual particle converts energy into work to move in a dissipative medium. When interacting through a velocity alignment mechanism, and with the medium acting as a momentum sink, even momentum is not conserved. In this scenario, a mapping into an equilibrium system seems unlikely. Here, we show that an entropy functional can be derived for SPPs with velocity alignment and density-dependent speed, at least in the (orientationally) disordered phase. This non-trivial result has important physical consequences. The study of the entropy functional reveals that the system can undergo phase separation before the orientational-order phase transition known to occur in SPP systems with velocity alignment. Moreover, we indicate that the spinodal line is a function of the alignment sensitivity and show that density fluctuations as well as the critical spatial diffusion, that leads to phase separation, dramatically increase as the orientational-order transition is approached.

Introduction

Examples of biological interacting self-propelled particle (SPP) systems include animal groups [START_REF] Cavagna | Scale-free correlations in starling flocks[END_REF][START_REF] Bhattacharya | Collective decision making in cohesive flocks[END_REF], insect swarms [START_REF] Buhl | From disorder to order in marching locusts[END_REF][START_REF] Romanczuk | Collective motion due to individual escape and pursuit response[END_REF], bacteria [START_REF] Zhang | Collective motion and density fluctuations in bacterial colonies[END_REF][START_REF] Peruani | Collective motion and nonequilibrium cluster formation in colonies of gliding bacteria[END_REF][START_REF] Starruss | Pattern-formation mechanisms in motility mutants of Myxococcus xanthus[END_REF], and, at the microcellular scale, microtubules driven by molecular motors [START_REF] Schaller | Polar patterns of driven filaments[END_REF]. Even though most examples of SPPs come from biology, there exist non-living SPP systems. There are several ways of fabricating artificial SPPs. The self-propulsion of such particles typically requires an asymmetry in the particle: two distinct friction coefficients [START_REF] Kudrolli | Swarming and swirling in self-propelled polar granular rods[END_REF][START_REF] Deseigne | Collective motion of vibrated polar disks[END_REF][START_REF] Weber | Long-range ordering of vibrated polar disks[END_REF], light absorption coefficients [START_REF] Jiang | Active motion of a janus particle by selfthermophoresis in a defocused laser beam[END_REF][START_REF] Golestanian | Collective behavior of thermally active colloids[END_REF][START_REF] Theurkauff | Dynamic clustering in active colloidal suspensions with chemical signaling[END_REF][START_REF] Palacci | Living crystals of light-activated colloidal surfers[END_REF], or catalytic properties [START_REF] Paxton | Catalytic nanomotors: autonomous movement of striped nanorods[END_REF][START_REF] Mano | Bioelectrochemical propulsion[END_REF][START_REF] Rückner | Chemically powered nanodimers[END_REF][START_REF] Howse | Self-motile colloidal particles: from directed propulsion to random walk[END_REF][START_REF] Golestanian | Propulsion of a molecular machine by asymmetric distribution of reaction products[END_REF] depending on whether energy injection is done through vibration, light emission, or chemical reaction, respectively. Interestingly, this asymmetry does not need to be an intrinsic particle property. Self-propelled Quincke rollers [START_REF] Bricard | Emergence of macroscopic directed motion in populations of motile colloids[END_REF] as well as actively moving drops [START_REF] Thutupalli | Swarming behavior of simple model squirmers[END_REF] are remarkable examples where the asymmetry results from a spontaneous symmetry breaking that sets the particle to move in a given direction.

At a theoretical level, we have learned in the recent years that the largescale properties of SPP systems depend on few microscopic details. The symmetry associated to the self-propulsion mechanism of the particles, which can be either polar [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] Peruani | A mean-field theory for self-propelled particles interacting by velocity alignment mechanisms[END_REF] or apolar [START_REF] Chaté | Simple model for active nematics: quasi-longrange order and giant fluctuations[END_REF], as well as the symmetry of the particleparticle interactions, that often occur via a velocity alignment mechanism, which can be either ferromagnetic [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] or nematic [START_REF] Chaté | Simple model for active nematics: quasi-longrange order and giant fluctuations[END_REF][START_REF] Peruani | A mean-field theory for self-propelled particles interacting by velocity alignment mechanisms[END_REF][START_REF] Peruani | Nonequilibirum clustering of self-propelled rods[END_REF], play a fundamental role in the resulting self-organized patterns. Equally important is the dimension of the space where particles move, whether this space is continuous (off-lattice) [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] Chaté | Simple model for active nematics: quasi-longrange order and giant fluctuations[END_REF][START_REF] Peruani | A mean-field theory for self-propelled particles interacting by velocity alignment mechanisms[END_REF] or discrete (on lattice) [START_REF] Bussemaker | Mean-field analysis of a dynamical phase transition in a cellular automaton model for collective motion[END_REF][START_REF] Csahók | Lattice-gas model for collective biological motion[END_REF][START_REF] O'loan | Alternating steady state in one-dimensional flocking[END_REF][START_REF] Raymond | Flocking regimes in a simple lattice model[END_REF], and whether particles move on a homogeneous or heterogeneous medium [START_REF] Chepizhko | Optimal noise maximizes collective motion in heterogeneous media[END_REF][START_REF] Chepizhko | Diffusion, subdiffusion, and trapping of active particles in heterogeneous media[END_REF][START_REF] Reichhardt | Active matter transport and jamming on disordered landscapes[END_REF][START_REF] Quint | Swarming in disordered environments[END_REF].

Another aspect of key importance, and central to the present study, is whether there exists a coupling between the speed and the density of the SP particles. Notice that the importance lies on the existence of such coupling and not on the mere fact that the speed may fluctuate. In the context of SPPs with a velocity alignment, it has been shown first with a lattice model [START_REF] Peruani | Traffic jams, gliders, and bands in the quest of collective motion of self-propelled particles[END_REF] and later on with an off-lattice model [START_REF] Farrell | Pattern formation in self-propelled particles with densitydependente motiliy[END_REF] that such a coupling induces spontaneous phase separation and a zoology of complex patterns. The most evident physical mechanism that can introduce a coupling between speed and density is simple volume exclusion as showed with simple lattice models by adding exclusion process rules in the absence of particle-particle alignment in [START_REF] Thompson | Lattice models of nonequilibrium bacterial dynamics[END_REF] and with alignment in [START_REF] Peruani | Traffic jams, gliders, and bands in the quest of collective motion of self-propelled particles[END_REF], and with an off-lattice model of self-propelled disks interacting by a soft-core repulsion [START_REF] Fily | Athermal phase separation of self-propelled particles with no alignment[END_REF]. The observed non-equilibrium phase separation can be traced back to the non-equilibrium Motility Induced Phase Separation (MIPS) introduced in the context of interacting run-and-tumble particles by J. Tailleur and M.E. Cates in [START_REF] Tailleur | Statistical mechanics of interacting run-and-tumble bacteria[END_REF][START_REF] Cates | When are active Brownian particles and run-and-tumble particles equivalent? Consequences for motility-iniduced phase separation[END_REF]. In absence of an alignment mechanism, MIPS is a generic feature of active particles interacting by volume exclusion as shown in simulations with self-propelled disks [START_REF] Fily | Athermal phase separation of self-propelled particles with no alignment[END_REF][START_REF] Redner | Structure and dynamics of a phaseseparating active colloidal fluid[END_REF][START_REF] Fily | Freezing and phase separation of self-propelled disks[END_REF] and spheres [START_REF] Mognetti | Living clusters and crystals from low-density suspensions of active colloids[END_REF][START_REF] Wysocki | Cooperative motion of active brownian spheres in three-dimensional dense suspensions[END_REF], and argued theoretically in [START_REF] Thompson | Lattice models of nonequilibrium bacterial dynamics[END_REF][START_REF] Stenhammar | Continuum theory of phase separation kinectis for active brownian particles[END_REF][START_REF] Speck | Effective Cahn-Hilliard equation for phase separation of active Brownian particles[END_REF]. One exciting aspect of the MIPS, as first pointed out in [START_REF] Thompson | Lattice models of nonequilibrium bacterial dynamics[END_REF][START_REF] Tailleur | Statistical mechanics of interacting run-and-tumble bacteria[END_REF][START_REF] Cates | When are active Brownian particles and run-and-tumble particles equivalent? Consequences for motility-iniduced phase separation[END_REF], is the remarkable similarity with equilibrium phase-separation, which allows the mapping between these non-equilibrium active systems with the analogous equilibrium systems.

The goal of the present study is to look at MIPS in the context of SPPs with a velocity alignment mechanism. Specifically, we want to understand the role played by the alignment mechanism in the phase separation process. Let us recall that SPP systems with a velocity alignment mechanism exhibit a phase transition from a disordered to an ordered phase. In the disordered phase, the large-scale behavior of the particles is diffusive as occurs for SPPs without a velocity alignment. Thus, we may hope that a mapping to an equilibrium scenario, as the one performed in [START_REF] Tailleur | Statistical mechanics of interacting run-and-tumble bacteria[END_REF][START_REF] Cates | When are active Brownian particles and run-and-tumble particles equivalent? Consequences for motility-iniduced phase separation[END_REF], remains possible. We push for such an analogy as far as possible. To be exact, to the onset of the ordered phase.

Before starting, let us review briefly some of the most relevant theoretical results for (dried) SPP systems with velocity alignment and in the absence of density-dependent speed. The first hydrodynamical equations were derived based on symmetry arguments and contained all allowed terms by symmetry [START_REF] Toner | Long-range order in a two-dimensional dynamical XY model: how birds fly together[END_REF][START_REF] Toner | Flocks, herds, and schools: a quantitative theory of flocking[END_REF]. These initial studies provided a theoretical basis to understand the emergence of long-range order (LRO) in two dimensional systems with continuum symmetry as well as the presence of giant number fluctuations in the ordered phase. The drawback of these initial approaches is the impossibility of connecting the parameters of the hydrodynamic equations with those of the microscopic models. In [START_REF] Bertin | Hydrodynamic equations for self-propelled particles: microscopic derivation and stability analysis[END_REF], the macroscopic equations were derived from given microscopic equations. Such an approach revealed that the "parameters" of the hydrodynamic theory are in fact non-linear functions of the density. This has allowed to understand the emergence of macroscopic structures, such as bands, in this type of SPP systems [START_REF] Mishra | Fluctuations and pattern formation in self-propelled particles[END_REF][START_REF] Caussin | Emergent spatial structures in flocking models: a dynamical system insight[END_REF]. For a detailed review, we refer the reader to [START_REF] Marchetti | Hydrodynamics of soft active matter[END_REF]. Here, we just mention that macroscopic equations have been derived for ferromagnetic [START_REF] Toner | Long-range order in a two-dimensional dynamical XY model: how birds fly together[END_REF][START_REF] Toner | Flocks, herds, and schools: a quantitative theory of flocking[END_REF][START_REF] Bertin | Hydrodynamic equations for self-propelled particles: microscopic derivation and stability analysis[END_REF][START_REF] Marchetti | Hydrodynamics of soft active matter[END_REF] and nematic velocity alignment [START_REF] Peshkov | Nonlinear field equations for aligning self-propelled rods[END_REF] in the dilute approximation and close to the order-disorder transition, with the exception of [START_REF] Ihle | Kinetic theory of flocking: derivation of hydrodynamic equations[END_REF] and [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF]. Even though we have now a fairly good qualitative understanding of the hydrodynamics of SPP systems (in homogeneous media), many open questions and fundamental problems remain unsolved.

The paper is organized as follows: we start by introducing the microscopic model we are interested in. The following sections are devoted to the rather long computation which starts from the microscopic model and ends with the entropy functional describing the spatial density in the system. The main steps of the computation are outlined at the beginning of the third section. Finally, we draw some physical conclusions from the derived coarse-grained equations.

Model

We consider a system of N active particles, i = 1, . . . , N , moving in a twodimensional space. The position of the i-th particle is given by x i = (x i , y i ) and what we refer to as its active velocity (AV) by v(n i )u(θ i ), where u(θ i ) ≡ (cos(θ i ), sin(θ i )) defines the direction of the AV and v(n i ) ≡ v 0 ṽ(n i ) its norm, with v 0 a constant and ṽ(n i ) a function that depends on n i . The term n i refers to the local density around the i-th particle. More specifically, n i = N j=1 g(|x ix j |/R) where the function g(∆/R), with ∆ = |x ix j | defines the interaction range. Finally, we consider an over-damped dynamics for the evolution of the x i = (x i , y i ) and θ i such that the equations of motion of the i-th particle take the form:

ẋi = v(n i )u(θ i ) + 2D x σ i (t) (1) θi
= - γ n i N j=1 g(|x i -x j |/R) sin(θ i -θ j ) + 2D θ η i (t) . (2)
σ i (t) = (σ x i (t), σ y i (t)
) and η i (t) for i = 1, . . . , N are white, gaussian and uncorrelated noises with unit covariance. These noises represents a "bath" with very short time correlations or memory. The alignment sensitivity γ is not directly related to the fluctuation amplitude D θ through an Einstein relation. A suitable option for the function g is to take g(∆/R) = 1 for ∆/R ≤ 1 and 0 otherwise, definition by which R defines the interaction range. Notice that the model definition, in particular v(n i ), implies a mesoscale description; i.e., we assume that there is a microscopic physical mechanism that leads to v(n i ). Furthermore, we require v(x) to be a differentiable function, which is not necessary applicable to lattice models with strict exclusion rules [START_REF] Soto | Run-and-tumble dynamics in a crowded environment: persistent exclusion process for swimmers[END_REF].

It is convenient to write the equations of motion in adimensional form. Calling L the box size, we write x i = Lx i , y i = Lỹ i . We also rescale the time t = t/D θ , and adopt v 0 as the velocity scale. By using

σ i ( t/D θ ) = √ D θ σ i ( t) and η i ( t/D θ ) = √ D θ η i ( t)
, we arrive to:

dx i d t = ǫṽ(n i )u(θ i ) + ǫ 2 Dx σ i ( t) (3)
dθ i d t = - γ n i N j=1 g((x i -xj )/α) sin(θ i -θ j ) + √ 2 η i ( t) , (4)
where we have introduced the dimensionless parameters γ = γ/D θ , α = R/L, ε = v 0 /(LD θ ), and Dx = D x D θ /v 2 0 , and so n i = N j=1 g((x i -xj )/α). We notice that Dx = D x D θ /v 2 0 is the ratio between the passive (D x ) and active (v 2 0 /[2D θ ]) diffusion coefficient, which we consider to be of order ε 0 . From now on, we work with the adimensional equations and drop the ˜to simplify the notation. In the following, we will assume α → 0.

Main computation

As is clear from the scaling introduced through the parameter ε, we are interested in the situation where the dynamics over the angles θ i is fast with respect to the spatial dynamics. Furthermore, we want to study the density of particles and their local mean orientation over large length scales. Our ultimate goal is to obtain a static large deviation principle yielding an entropy functional, which describes the fluctuations of the spatial empirical density, along the lines of [START_REF] Tailleur | Statistical mechanics of interacting run-and-tumble bacteria[END_REF][START_REF] Cates | When are active Brownian particles and run-and-tumble particles equivalent? Consequences for motility-iniduced phase separation[END_REF], but accounting for the existence of a velocity alignment mechanism. To make the forthcoming computations easier to follow, we outline here the general scheme:

1. Write an effective equation for the phase space empirical density (that is in the variables x, y and θ), keeping the finite N fluctuations. This leads to a stochastic PDE with a noise term of order 1/ √ N . 2. Take advantage of the time-scale separation to obtain a closed effective dynamics for the empirical density of the slow spatial variables x, y. 3. Write a functional Fokker-Planck equation for the macroscopic density field ρ. 4. Look for the stationary probability density of the macroscopic field ρ, under the asymptotic form e N S[ρ] and solve for S at leading order in N .

Dynamical equation for the phase space empirical density

We assume that N is large, but finite, and denote by f d (x, θ, t) the empirical density of particles in the 3D space given but [x, y, θ]. The temporal evolution of f d (x, θ, t) is expressed in terms of the following stochastic partial differential equation:

∂f d ∂t = -ε∇. (v(ρ(x, t)))u(θ)f d (x, θ, t)) + γ ρ(x, t) ∂ ∂θ f d (x, θ, t) dθ ′ sin(θ -θ ′ )f d (x, θ ′ , t) + ∂ 2 f d ∂θ 2 + ε 2 D x ∇ 2 f d + 2 N ∂ ∂θ η(x, θ, t) f d +ε √ 2D x √ N ∇ • σ(x, θ, t) f d , (5)
where η(x, θ, t) and σ(x, θ, t) are gaussian noises, delta-correlated in time and space, and ρ(x, t) is the empirical spatial density defined by:

ρ(x, t) = dθf d (x, θ, t). (6)
This can be shown at a formal level by following [START_REF] Dean | Langevin equation for the density of a system of interacting Langevin processes[END_REF].

Averaging step -Time scale separation

We now take Fourier components of f d of increasing order, stopping the expansion as soon as possible. This is a standard strategy, see e.g. [START_REF] Bertin | Hydrodynamic equations for self-propelled particles: microscopic derivation and stability analysis[END_REF]. Notice that finite N fluctuations are taken into account. We use the following notations: P = (P x , P y ), with

P(x, t) = dθu(θ)f d (x, y, θ, t) . (7)
Integrating Eq. ( 5) respectively over dθ, cos θdθ and sin θdθ, we obtain:

∂ρ ∂t = -ε∇.(vP) + ε 2 D x ∇ 2 ρ + ε √ 2D x √ N ∇ • (ξ(x, y, t)) (8)
∂P ∂t = - 1 2 ε∇ (vρ) + γ 2 -1 P + ε 2 D x ∇ 2 P + 2 N η(x, y, t) + O( ε √ N ) (9)
where the noises are defined by

η x (x, y, t) = dθ sin θ f d η(x, y, θ, t) (10)
η y (x, y, t) = -dθ cos θ f d η(x, y, θ, t) (11)
ξ(x, y, t) = dθ f d σ(x, y, θ, t) .

By construction, the noises are gaussian, delta-correlated in time and space. Furthermore

η x (x, y, t)η x (x ′ , y ′ , t ′ ) ≃ δ(x -x ′ )δ(y -y ′ )δ(t -t ′ ) 1 2 ρ(x, y, t) (13)
η y (x, y, t)η y (x ′ , y ′ , t ′ ) ≃ δ(x -x ′ )δ(y -y ′ )δ(t -t ′ ) 1 2 ρ(x, y, t) (14)
η x (x, y, t)η y (x ′ , y ′ , t ′ ) ≃ 0 (15) ξ(x, y, t)ξ(x ′ , y ′ , t ′ ) = δ(x -x ′ )δ(y -y ′ )δ(t -t ′ )ρ(x, y, t) . (16)
Consistently with our approximation, we have dropped in the noise correlation all Fourier coefficients beyond the first. Notice that in Eq. ( 9) we have neglected higher order Fourier coefficients since we are interested in characterizing the system dynamics in the disordered phase -i.e. when collective motion is not observed. This implies that our approximation is only valid below the onset of collective motion. Moreover, without higher Fourier components, Eq. ( 9) predicts that |P| grows unboundedly for γ > 2. To obtain a system of equations that is physically well-behaved in the ordered phase, we have to go at least one component further in Fourier. Such extra Fourier component is connected to the nematic order, while P to polar order. To understand the disordered phase, which is our objective here, we insist that it is enough to develop up to polar order. Furthermore, we stress that Eq. ( 9) is consistent with a small ε expansion. If γ smaller than 2 and not too close to 2, P very quickly reaches its stationary value and remains small: particle motion is locally disordered, since the interaction promoting alignment is not strong enough to create a local orientational order. In this regime, we can take the l.h.s. of Eq. ( 9) to be 0 in order to determine the stationary value of P. Neglecting terms of order ε 2 and ε/ √ N , we obtain:

P = ε -1 2(1 -γ 2 ) ∇(v(ρ)ρ) + 2 N 1 1 -γ 2 η ( 17
)
These computations are formal, and could in general lead to incorrect results: one should in particular be cautious about the meaning of the noise term, which is multiplicative. However, since we will eventually take a small noise limit (large N ), this formal approach will correctly yield the leading order in N . We insert Eq.( 17) into ( 8) and look for the long-time behavior of ρ by introducing a new time-scale t = ε 2 t:

∂ρ ∂t = 1 2 ∇ • v 1 -γ 2 ∇[v(ρ)ρ] + D x ∇ 2 ρ + √ 2D x √ N ∇ • (ξ(r, t)) + 2 N ∇ • v 1 -γ 2 η , (18)
where again we have dropped the ˜and replace η by -η. Notice that due to the involved change of time-scale, all ǫ's have disappeared of the final equation, and both noise terms give a contribution. The expansion in powers of ε is formally consistent, in the sense that further Fourier components would contribute terms which are formally of higher order. This means that one can hope that Eq. ( 18) is in some sense exact in the limit N → ∞ , ε → 0. Eq. ( 18) can be expressed in a more compact notation in the following way:

∂ρ ∂t = U [ρ](x) + 1 √ N ν(x, t) (19)
where

U [ρ](x) = 1 2 ∇ • v(ρ) 1 -γ 2 ∇[v(ρ)ρ] + D x ∇ 2 ρ (20) and ν(x, y, t)ν(x ′ , y ′ , t ′ ) = D[ρ](x, x ′ )δ(t -t ′ ) with D[ρ](x, x ′ ) = ∂ x ∂ x ′ [b[ρ](x)δ(x -x ′ )] + ∂ y ∂ y ′ [b[ρ](x)δ(x -x ′ )] (21)
where

b[ρ] = 2D x ρ + ρv 2 (ρ) (1 -γ 2 ) 2
We have combined here the two independent gaussian noise terms into a single one.

Functional Fokker-Planck equation

From Eq. ( 19), one can write a functional Fokker-Planck equation (see for details [START_REF] Gardiner | Handbook of stochastic methods[END_REF]) for the probability distribution of the density field µ t [ρ]:

∂µ t ∂t = -dx δ δρ(x) (U [ρ](x)µ t ) + 1 2N dx δ δρ(x) dx ′ D[ρ](x, x ′ ) δ δρ(x ′ ) µ t (22)
Note that we have here assumed an interpretation of the noise ν corresponding to Ito's convention. This has no consequence at leading order in N . We look for a stationary solution taking the asymptotic form

µ [ρ] ∼ e N S[ρ] (23)
and compute S at leading order in N . The drift U and the noise correlation D depend on ρ. However, we see that the relevant terms at leading order in N are obtained when the functional derivatives with respect to ρ act on µ t rather than on U or D. This leads to the following equation for S:

U [ρ](x) δS δρ(x) dx = 1 2 dx ∂ x δS δρ(x) dx ′ ∂ x ′ (b[ρ](x ′ )δ(x -x ′ )) δS δρ(x ′ ) + dx ∂ y δS δρ(x) dx ′ ∂ y ′ (b[ρ](x ′ )δ(x -x ′ )) δS δρ(x ′ ) = - 1 2 dx∇ • b[ρ](x)∇ δS δρ(x) δS δρ(x) (24)
By comparison and using the expression [START_REF] Golestanian | Propulsion of a molecular machine by asymmetric distribution of reaction products[END_REF] for U , one sees that a sufficient condition to find S is to solve the equation

1 2 b[ρ]∇ δS δρ(x) = - 1 2 v(ρ) 1 -γ 2 ∇(ρv(ρ)) -D x ∇ρ (25)
Formally, it can be shown that this expression represents the equilibrium condition for µ [ρ], that is the condition for a zero-flux solution of the functional Fokker-Planck equation, Eq. ( 22). Equivalently, this corresponds to the reversibility of the dynamics given by Eq. ( 19) with respect to the density µ[ρ].

We look for a solution S, which is a local functional of ρ, and of the form:

S[ρ] = dxs(ρ(x)) (26)
where the entropy density s is a real function to be determined. One finds

s"(ρ) = - v 2 (ρ) + ρv(ρ)v ′ (ρ) 1 -γ 2 b[ρ] + 2D x b[ρ] (27)
When the integration of Eq. ( 27) is possible, the function s can be explicitly retrieved. This is the main result of this article. In the absence of alignment, i.e. γ = 0, Eq. ( 27) leads to the same results derived in [START_REF] Cates | When are active Brownian particles and run-and-tumble particles equivalent? Consequences for motility-iniduced phase separation[END_REF]. For instance, it is straightforward to see that if in addition we make D x = 0, the spinodal line is given by the condition dv/dρ = -v/ρ as explained in [START_REF] Tailleur | Statistical mechanics of interacting run-and-tumble bacteria[END_REF][START_REF] Cates | When are active Brownian particles and run-and-tumble particles equivalent? Consequences for motility-iniduced phase separation[END_REF]. Notice that the procedure followed to arrive to the free energy, consisted in deriving a stochastic equation for the empirical density, and using Ito's calculus to obtain an expression for the density field µ, in turn connected to the free energy S. We stress that the described procedure is fundamentally different from recent approaches [START_REF] Speck | Effective Cahn-Hilliard equation for phase separation of active Brownian particles[END_REF] used to describe phase separation in non-aligning active particles, where a free energy is obtained by deriving first a non-fluctuating BBGKY hierarchy of equations, performing a perturbation expansion, and making a direct analogy between the derived equation for the density field at second leading order and the Cahn-Hilliard equation, whose free energy is well known. Our derivation, on the contrary, contains finite N fluctuations, going beyond mean-field, and allows us to obtain directly a free-energy-like functional form, which is not necessary a Cahn-Hilliard free energy with a local cubic term as in [START_REF] Speck | Effective Cahn-Hilliard equation for phase separation of active Brownian particles[END_REF]. In the following section, we discuss the physical meaning of equations here derived.

Physical discussion and final remarks

Let us review the results we obtained. The equations of motion (1) and (2) were our starting point. We required v(x) to be a differentiable function, which implies that our derivation is, in principle, not adequate to describe sharp interfaces as the ones observed in lattice models with strict exclusion rules [START_REF] Soto | Run-and-tumble dynamics in a crowded environment: persistent exclusion process for swimmers[END_REF]. Under these assumptions, we derived an equation for the empirical density, Eq. ( 5), following [START_REF] Dean | Langevin equation for the density of a system of interacting Langevin processes[END_REF]. This equation becomes an exact description only in the limit of infinite N and infinite densities. Thus, for finite but large N , Eq. ( 5) should provide a good description for the particle density of a system whose microscopic dynamics is given by Eqs. ( 1) and (2). Our goal has been to derive an entropy-like functional for the particle density. In order to do that, we made an expansion in Fourier of Eq. ( 5), given by Eqs. ( 8) and ( 9), up to polar order P, and made use of the fast relaxation of P with respect of the that the transition between the (orientationally) disordered and ordered phase is controlled by the ratio γ = γ/D θ . In the disordered phase, the red dashed curve represents the spinodal line that diverges (according to our approximation) as the onset of orientational order is approached. As we come close to the disorder-order transition density fluctuations also diverge (see text). Above the red curve there is only one homogeneous phase, while below the system phase separates, i.e., the homogeneous phase is no longer stable. It has been shown that the ordered phase can exhibit a zoo of patterns, see [START_REF] Peruani | Traffic jams, gliders, and bands in the quest of collective motion of self-propelled particles[END_REF][START_REF] Farrell | Pattern formation in self-propelled particles with densitydependente motiliy[END_REF]. In the sketch we have assumed that v(ρ) = exp(-λρ), with λ large enough to allow MIPS. Notice that in absence of alignment interactions, D sp

x is a constant, whose value corresponds to γ = γ/D θ = 0 in the figure. We remind the reader that D sp

x is a dimensionless parameter, which means that for non-aligning particles the spinodal is given by DxD θ /v 2 0 = D sp x (γ = 0).

temporal evolution of ρ, to express P as function of ρ and its gradients. Such expansion up to polar order, as well as the separation of time-scales between the temporal evolution of P and ρ, are exclusively valid in the disordered phase. From Eq. ( 9), we can easily see that the onset of local (orientational) order occurs for γ/2 -1 > 0, that is, when the angular diffusion D θ is such that D θ < γ/2. This means that the entropy functional, given by Eqs. ( 26) and ( 27), is valid for D θ > γ/2, see solid vertical line in Fig. 1.

To understand the physical meaning of the derived equations, let us adopt a concrete functional form for v(ρ), e.g. v(ρ) = exp(-λρ). Notice that the qualitative features discussed below do not depend on the precise functional form for v(ρ), provided it is decreasing. Phase separation occurs below the binodal line defined by the double tangent construction on the s(ρ) curve. 28). While below each curve D SP x (ρ), the system is motility-induced phase separated (MIPS), above D SP x (ρ) the system remains homogeneous (between the binodal and spinodal lines, the homogeneous phase is only metastable). This phase diagram is the counterpart of the classical gas-liquid phase diagram Tρ. The critical point, defined by (ρc, D crit x ), see black dots and text, marks the value of Dx above which there is no more phase transition. In the figure we have assumed that v(ρ) = exp(-λρ). The curves in (a) and (b) correspond to λ = 1.

Easier to detect and compute is the spinodal line, below which an homogeneous phase cannot be stable. The spinodal can be found solving s ′′ (ρ) = 0, using expression Eq. ( 27). One finds

e -2λρ (λρ -1) 2 - γ = D sp x (ρ) (28)
For ρ, λ such that λρ > 1, Eq. ( 28) is represented by the red dashed line in Fig. 1. Above this line, the homogeneous solution is stable; below it, spinodal decomposition (Motility-Induced Phase Separation) occurs. Notice that this line, which corresponds to a dynamical instability, can also be obtained by linearizing Eq. ( 19), without noise, around a spatially homogeneous solution.

Eq. ( 27) also gives access to the metastable regions around the spinodal line.

Figure 1 shows that the critical spatial diffusion D sp x below which the homogeneous solution is unstable strongly depends on γ = γ/D θ : the divergence of D sp

x in our approximation is connected to the term in 1/(1 -γ/2) in Eq. ( 20). This high sensitivity of the spinodal line to the alignment strength, i.e. γ, is consistent with results obtained in simulations [START_REF] Peruani | Traffic jams, gliders, and bands in the quest of collective motion of self-propelled particles[END_REF]. As expected, the entropy s(ρ) is also affected by γ as shown in Fig. 2(a).

Thus, our results indicate that MIPS as described in [START_REF] Tailleur | Statistical mechanics of interacting run-and-tumble bacteria[END_REF][START_REF] Cates | When are active Brownian particles and run-and-tumble particles equivalent? Consequences for motility-iniduced phase separation[END_REF] also occurs in the presence of alignment interactions in the disordered phase, with γ affecting the spinodal (as well as binodal) line as shown in Fig. 2(b). Notice that Eq. ( 28) allows us to draw the phase diagram D x -ρ, which is the counterpart of the classical gas-liquid phase diagram T -ρ. From this expression we can obtain the so-called critical point (ρ c , D crit x ), with ρ c = 3/(2λ) and D crit x ≃ 0.0249/(2-γ) for the chosen functional form of v(ρ). For D x > D crit x (γ), Eq. ( 28) has no solution, which implies that in this case, the homogeneous phase is stable for any density and value of λ. In short, MIPS has disappeared above D crit x (γ). Hence, the critical value of the spatial diffusion needed to destroy the MIPS strongly increases when the alignment interactions increase. Moreover, in the current approximation, it diverges as order-disorder transition is approached, which suggests that there is no critical point in the ordered phase.

Finally, we stress that at this level of approximation, D x (i.e., the original, dimensional, spatial diffusion constant) does not affect the orientational order transition point (more precisely the instability of the homogeneous disordered phase), while γ and D θ play a role on both, the disordered and ordered phase. Now, let us turn to the analysis of density fluctuations related to the noise term present in Eq. ( 19). From Eq. ( 27), we learn that in the homogeneous phase, an alignment interaction γ > 0 makes s ′′ smaller in absolute value, and thus it increases the density fluctuations. Notice that density fluctuations diverge for s ′′ → 0. This occurs on the spinodal line, as well as in the MIPS phase, as we approach the instability of the (homogeneous) disordered phase, i.e. γ → 2D θ . It is worth noticing that for D x = 0, γ has no influence on the phase diagram shown in Fig. 1, while it still has on the density fluctuations. A note of caution is in order here. According to the proposed approach, D sp x , D crit

x , as well as density fluctuations diverge as the disorder-order transition is approached. As explained above, our approximation is not valid in the limit of γ → 2D θ . While we can be sure that D sp x , D crit x and density fluctuation increase as we approach the onset of collective motion, we cannot ensure that the system behavior at the disorder-order transition or in its vicinity is as predicted by the present approach.

In summary, our calculations indicate that phase separation can occur in the disorder phase with the alignment strength -more specifically with the "distance" to the instability of the (homogeneous) disordered phase -controlling the position of the spinodal line involved in the MIPS as well as the size of density fluctuations. In short, we have generalized the approach of [START_REF] Tailleur | Statistical mechanics of interacting run-and-tumble bacteria[END_REF][START_REF] Cates | When are active Brownian particles and run-and-tumble particles equivalent? Consequences for motility-iniduced phase separation[END_REF] in order to account for the presence of a velocity alignment mechanism. While the nature of the described phase separation remains a MIPS as observed in non-aligning systems [START_REF] Fily | Athermal phase separation of self-propelled particles with no alignment[END_REF][START_REF] Redner | Structure and dynamics of a phaseseparating active colloidal fluid[END_REF][START_REF] Fily | Freezing and phase separation of self-propelled disks[END_REF][START_REF] Mognetti | Living clusters and crystals from low-density suspensions of active colloids[END_REF][START_REF] Wysocki | Cooperative motion of active brownian spheres in three-dimensional dense suspensions[END_REF][START_REF] Stenhammar | Continuum theory of phase separation kinectis for active brownian particles[END_REF][START_REF] Speck | Effective Cahn-Hilliard equation for phase separation of active Brownian particles[END_REF], this does not exclude that in the orientationally ordered phase, phase separation can be of a different nature as suggested in [START_REF] Peruani | A kinetic model and scaling properties of non-equilibrium clustering of self-propelled particles[END_REF].

acknowledgements We acknowledge enlightening discussions with O. Dauchot, P. Degond, and J. Tailleur and financial support from the PEPS-PTI "Anomalous fluctuations in the collective motion of self-propelled particles". The suggestion by J. Tailleur of adding a white noise to Eq.( 1) has proved to be very fruitful for the present study. FP thanks the Kavli Institute for Theoretical Physics (University of California, Santa Barbara) and the organizers of the bioacter14 program for hospitality and financial support.

Fig. 1

 1 Fig. 1 Sketch of phase diagram: D SP x vs. γ = γ/D θ at fixed density. The figure indicatesthat the transition between the (orientationally) disordered and ordered phase is controlled by the ratio γ = γ/D θ . In the disordered phase, the red dashed curve represents the spinodal line that diverges (according to our approximation) as the onset of orientational order is approached. As we come close to the disorder-order transition density fluctuations also diverge (see text). Above the red curve there is only one homogeneous phase, while below the system phase separates, i.e., the homogeneous phase is no longer stable. It has been shown that the ordered phase can exhibit a zoo of patterns, see[START_REF] Peruani | Traffic jams, gliders, and bands in the quest of collective motion of self-propelled particles[END_REF][START_REF] Farrell | Pattern formation in self-propelled particles with densitydependente motiliy[END_REF]. In the sketch we have assumed that v(ρ) = exp(-λρ), with λ large enough to allow MIPS. Notice that in absence of alignment interactions, D spx is a constant, whose value corresponds to γ = γ/D θ = 0 in the figure. We remind the reader that D spx is a dimensionless parameter, which means that for non-aligning particles the spinodal is given by DxD θ /v 2 0 = D sp x (γ = 0).

Fig. 2

 2 Fig. 2 Phase separation in the disordered phase is controlled by the interplay between γ = γ/D θ and Dx. (a) Entropy s(ρ), derived from Eq. (27), for various values of γ = γ/D θ . (b) Phase diagram Dx vs. ρ for various values of γ = γ/D θ , see Eq. (28). While below each curve D SPx (ρ), the system is motility-induced phase separated (MIPS), above D SP x (ρ) the system remains homogeneous (between the binodal and spinodal lines, the homogeneous phase is only metastable). This phase diagram is the counterpart of the classical gas-liquid phase diagram Tρ. The critical point, defined by (ρc, D crit