Ae-Lim Ahn

Eric Laporte
email: eric.laporte@univ-paris-est.fr

Jee-Sun Nam

Semantic Polarity of Adjectival Predicates in Online Reviews

Web users produce more and more documents expressing opinions. Because these have become important resources for customers and manufacturers, many have focused on them. Opinions are often expressed through adjectives with positive or negative semantic values. In extracting information from users' opinion in online reviews, exact recognition of the semantic polarity of adjectives is one of the most important requirements. Since adjectives have different semantic orientations according to contexts, it is not satisfying to extract opinion information without considering the semantic and lexical relations between the adjectives and the feature nouns appropriate to a given domain. In this paper, we present a classification of adjectives by polarity, and we analyze adjectives that are undetermined in the absence of contexts. Our research should be useful for accurately predicting semantic orientations of opinion sentences, and should be taken into account before relying on an automatic methods.

Introduction

With the high increase in the number of documents expressing opinions, Web opinion mining is becoming a challenging task. Users express their opinions about products on the web, and people share their opinions. These opinions become important resources for customers who want information about products and manufacturers who wish to improve their productivity. Therefore, the demand for automatic extraction of opinions from Web documents is increasing, and the research on classification of reviewers' opinion progresses steadily. Such studies are called Opinion Mining(OM), which covers a range of activities from retrieving opinion sentences in web documents to determining their meaning. Opinion mining in Korean web documents resorts to increasingly various approaches. However, serious linguistic analyses about opinion documents are still rare. In our opinion, in order to get efficient results from opinion mining, fundamental work on opinion sentences and construction of linguistic resources have to be performed in advance.

A sentence which contains one or more topic segments denoting product features, and one or more evaluative segments expressing opinions, is called an "opinion sentence" (Hu 2004). Such a sentence contains opinion words. The majority of opinion words are adjectival predicates. Therefore, to deduce the orientation of an opinion sentence, we examine semantic the polarity of adjectival predicates (i.e. the positive value [+] vs. the negative value [-]). However, some adjectives are context-dependent. That is, a given word may indicate different opinions depending on its domain, or even within one and the same domain, depending on product features. For example, "The battery life is long." expresses a positive opinion (+); and "It takes a long time to focus." a negative opinion (-), with the same opinion word "long" combined with distinct product features in the same domain (Ding and Liu 2007). Many OM researchers point out the ambiguity of adjectives, and emphasize the necessity of analysing them. Korean sentences are little different from English of French sentences: Korean adjectives may have context dependent polarity. We should consider semantic and lexical restrictions between adjectives and the co-occurring features (or topics) in one domain.

In addition, adjectives can mark serve as intensity markers for other opinion words. For example, in "I yenghwa-nun kwankayk-ul ppalatuli-nun hupiplyek-I kang-haysseyo."("This movie has strong attraction for audiences."), "kanghata"("strong") reinforces "hupiplyek"("attraction"), and itpresents a positive opinion. In other hands, in "Phoklyekseng-I kang-han yenghwa-tukunyo."( "It was a movie of strong violence."), "kanghata" does the same for "Phoklyekseng"("violence") which presents a negative opinion. As we can see, opinion mining should not be limited to simply counting "good" and "bad" words in a document. In this paper, we define the features that people consider when they evaluate the products and determine the polarity of adjectives depending on their features within various domains. Section 2 will survey previous research related to opinion mining. We classify adjectival predicates by semantic polarity in Section 3, and we present feature lists in each domain in Section 4. Then a case study on a specific adjective is reported in Section 5. Section 6 includes the conclusion and suggests future studies1 .

Related Work

Studies on the extraction of opinion sentences show that adjectives often express users' opinion.

Classification of opinions is conducted at document-level or sentence-level in recent studies. That is, the document can be interpreted as positive or negative as a whole, or each sentence is classified as positive or negative [START_REF] Yuk | Sacen kipan-ui hankuke sangpum lipyu uikyen cacil chuchul mich punlyu sisuthem (Dictionary-Based Opinion Features Extraction and Classification of Korean Product Reviews[END_REF]. Most studies adopt machine learning or fully automated approaches to assign a polarity to documents or sentences. One method for labeling positively and negatively oriented adjectives uses conjunctions. It is an unsupervised learning method for obtaining polarity of adjectives with accuracy over 90% (Hatzivassiloglou & Mckeown 1997). Extraction using seed words extends an initial set of words with predetermined orientation labels to construct a larger set of semantically oriented words [START_REF] Turney | Thumps up or thumbs down? Semantic orientation applied to unsupervised classification of reviews[END_REF]. . This method finds that some adjectives have neutral orientation, because it does not consider semantic ambiguity depending on domains. As an attempt to solve this limitation, correlations between the seed words and other adjectives are computed so as to enrich the sets of seed words with new domain-relevant opinion adjectives [START_REF] Harb | Web Opinion Mining: How to extract opinions from blogs?[END_REF]. This approach still produces much noise in certain types of text. Without linguistic analysis of each adjective, this problem is difficult indeed.

For Korean sentiment classification, several natural language processing techniques, including the use of a semantic dictionary, have been used. The semantic dictionary contains words used to express product features and customer's opinions: it was constructed semi-automatically [START_REF] Myeng | Pancatongulo kuchuktoyn uymi sacenul iyonghan hankuke sangphumphyeng punsuk sisutheym (A Korean Product Review Analysis System Using a Semi-Automatically Constructed Semantic Dictionary)[END_REF]. This dictionary was extended into a detailed description of opinion features, opinion words, independent opinion words, opinion phrases, and some negation expressions [START_REF] Yuk | Sacen kipan-ui hankuke sangpum lipyu uikyen cacil chuchul mich punlyu sisuthem (Dictionary-Based Opinion Features Extraction and Classification of Korean Product Reviews[END_REF].

In France, two partners of the DoXa project2 , Arisem and the LIGM laboratory, have undertaken the construction of language resources for a few domains, describing the following data: the vocabulary of opinions and sentiments [START_REF] Brizard | Méthodologie de création et d'enrichissement des ressources d'extraction des opinions et sentiments[END_REF][START_REF] Varga | Version de démonstration des ressources d'extraction des opinions et sentiments[END_REF]; the words and expressions used to refer to topics and subtopics in evaluative documents, such as "durée de vie de la batterie"("battery life"), these resources being structured as ontologies in the OWL-DL language; the markers used to intensify or attenuate the expression of opinions and sentiments, e.g. fort "strong".

In this paper, we describe a linguistic analysis of polarity of adjectives by considering various contexts to improve the accuracy of automatic systems. The goal of this study will be the construction of an "Opinion-Feature Dictionary" based on this analysis.

Classification of Adjectives by Semantic Polarity

Opinion classification is based on the meaning of words and their relations. As semantic orientation of opinion sentences is most affected by the polarity of the adjectives, study on polarity of adjectives is required. By analyzing a corpus built from Web documents, we can classify the observed adjectives We obtained the list of the most frequent adjectives in the five corpora, using the Korean lexical analyzer Geuljabi 3 . The frequent adjectival predicates are different depending on their domains (Table 1).

This means the features which people consider most significant for evaluation are different in each domain. For example, reviews about mobile phones contain adjectival predicates which express an evaluation of the functions or look of mobile phones, such as "ppaluta"("fast") and "mwukupta"("heavy"). In reviews about movies, emotional adjectives occur with high frequency, such as "caymiissta"("interesting") and "sulputa"("sad"). Consequentially, the choice of adjectives differs for each domain. This explains why adjectives can have different polarity depending on the features they express an opinion about.

We assigned a semantic polarity to adjectives extracted from the corpora, whose types are about 300.

We classified adjectival predicates into two types: adjectives of ABSOLUTE polarity and those of RELATIVE polarity.

A. Type 1: Adjectives of ABSOLUTE Polarity

A-1. Positively-Oriented Adjectives

Examples: cohta("good"), yeypputa("beautiful"), phyenhata("comfortable"), olhta("correct"), chinhata("friendly") etc.

A-2. Negatively-Oriented Adjectives

Examples: silhta("unpleasant"), simhata("harsh"), akkapta("wasteful"), telepta("dirty"), sikkulepta ("noisy") etc.

B. Type 2: Adjectives of RELATIVE Polarity

Examples: kapyepta("light"), mukepta("heavy"), kanghata("strong"), kiphta("deep"), khuta("big"), nophta("high") etc.

3 For more detailed information, see "www.sejong.or.kr" Adjectives of ABSOLUTE polarity do not depend on contexts. Their polarity is stable across domains and features. On the other hand, adjectives of RELATIVE polarity have context-dependent polarity. Their polarity is affected by several factors in their sentences.

Based on this classification, we obtain the frequency of each type in the five domains ( (1) a. "Lostey hotheyl cupyen-ey khun kenmul-i manh-supnita."

(" There are big buildings around the hotel.") b. "Hotheyl kyumo-ka khu-ko kunsaha-neyyo."

("The hotel is big and wonderful.")

Both sentences in (1) have "khuta" in the predicate. However, sentence (1a) expresses a fact; sentence (1b) expresses an opinion about hotel. This means that not every sentence with "khuta" is an opinion.

Some previous research uses an annotated corpus, where opinions and facts are tagged, but it is not possible to obtain reliable annotated corpora in sufficient quantity. In addition, determining whether adjectives express an opinion or a fact is not done in perfect way. Therefore, we require close observation of sentences classified by the polarity of their adjectives. Especially, sentences containing adjectives of RELATIVE polarity have to be analyzed closely. We examine them in Section 5.

Adjectives of Absolute Polarity

Adjectives of ABSOLUTE polarity are not affected by contexts and can be interpreted independently of domains. We determine their polarity to construct the Lexicon of ABSOLUTE Polarity. Frequent adjectives can be different depending on domains, but their semantic polarity does not change. Hence, we need not present information about domains(or features) in this lexicon.

Adjective

Construction of feature lists of each domain

We need to study the possible ways of expressing opinions for a given domain, and we can guess which opinions about products customers or manufacturers want to be aware of. One adjective can qualify a finite number of features within a given domain, so we can describe these relations between topic segments and adjectives. By determining the semantic polarity of adjectival predicates when they are applied to a given feature, we can also determine the semantic polarity of corresponding opinion sentences. Through the analysis of the corpora, we define topic categories which reviewers evaluate about products within each domain. We extract nouns from each corpus using Geuljabi, and classify feature nouns depending on topic categories.

Cosmetic Products

What evaluative subjects interest customers and manufactures regarding cosmetics? The reviewers evaluate components of cosmetic products such as color, scent and ingredients. Effects of cosmetic products are a major subject of such evaluations, which mean how effective a product is. Reviewers describe strong or weak points against physical symptoms. Price and design of products are also important considerations.

Domain Topic Category Feature Nouns

COSMETIC PRODUCT

Component

색상("color"), 향("scent"), 화학성분("chemicals"), 알코올("alcohol"), 양("quantity"), 비타민("vitamin"), 수분("moisture"), 촉감("touch") etc.

Effect

효과("effect"), 반응("reaction"), 기능("function"), 발림("application"), 흡수 ("absorption"), 지속력("resistance"), 차단력("protection") etc.

Physical Symptom

지성("oiliness"), 기미("freckles"), 여드름("pimple"), 상처("scar"), 각질 ("keratin"), 주름 ("winkle"), 손상("demage"), 자극("irritant") etc.

Price

가격("price"), 세일("sale") etc.

Design

케이스("case"), 튜브("tube"), 모양("shape"), 크기("size") etc.

Table 6. Features for evaluation in COSMETIC PRODUCT

Hotels

For HOTEL reviews, we define six topic categories. Reviewers evaluate hotel facilities and supplies. In addition, they describe how the staff provides services, how clean the hotel is and how good the food is.

They also give the value for the location, view from the room, and transportations.

Domain Topic Category

Feature Nouns

HOTEL

Facilities

호텔("hotel"), 걲물("building"), 로비("lobby"), 방("room"), 창("window"), 주차 장("parking lot"), 엘리베이터("elevator") etc.

Supplies in hotel

침대("bed"), 컴퓨터("computer"), 냉장고("refrigerator"), 욕조("bath"), 샴푸 ("shampoo"), 비누("soap"), 수걲("towel") etc.

Service

예약("reservation"), 체크아웃("check-out"), 체크인("check-in"), 앆내

("guidance"), 룸서비스("room service"), 서비스("service") etc.

Cleanliness

청소("cleaning"), 냄새("scent"), 먼지("dust"), 정돆("arrangement"), 관리("care") etc.

Food

조식("breakfast"), 점심("lunch"), 음식("food"), 메뉴("menu"), 레스토랑

("restaurant"), 맛("taste"), 음료("drinks"), 빵("bread") etc.

Surroundings

위치("location"), 전망("view"), 거리("distance"), 길("way"), 야외("outdoor"), 교 통("transportation") etc.

Table 7. Features for evaluation in HOTEL

Hospitals

Reviews of hospitals are increasingly numerous in specific fields such as Plastic Surgery and Dentistry, because results can be definitely different according to doctor's ability or experience. Facilities and services are considerable subjects as much as the doctor's ability nowadays.

Domain Topic Category

Feature Nouns HOSPITAL Facilities 병원("hospital"), 걲물("building"), 시설("facilities"), 인테리어("interior"), 대기실 ("waiting room") etc.

Ability and service of staffs

의사("doctor"), 갂호사("nurse"), 서비스("service"), 실력("ability"), 코디네이터 ("coordinator"), 상담가("consultant"), etc.

Symptom of body

통증("pain"), 멍("bruise"), 상처("wound"), 부작용("side effect"), 부기 ("swelling"), 주름("wrinkle") etc.

Result

효과("effect"), 회복("recover"), 결과("result"), 변화("change"), 이미지 ("image"), 모습("appearance"), 콤플렉스("complex") etc.

Price

가격("price"), 세일("sale") etc.

Time

대기시갂("waiting"), 회복시갂("recovery time"), 수술시갂("operation time") etc.

Table 8. Features for evaluation in PLASTIC SURGERY

Mobile Phones

The IT field covers products such as mobile phones, cameras, and PCs. Reviews of mobile phones are most numerous, because it is considered a necessity of life. When people choose their new phone, they consider the look, various functions, and other qualities of mobile phones. Generally, mobile phones are compared about various features.

Domain Topic Category Feature Nouns

MOBILE

PHONE

Part of mobile phone

카메라("camera"), 화면("screen"), 배터리("battery"), 케이스("case"), 스피커 ("speaker"), 버튼("button") etc.

Quality

속도("speed"), 시갂("duration"), 음질("sound"), 해상도("definition"), 움직임 ("movement"), 접속("connection"), etc.

Function 문자("text message"), 게임("game"), 전화("call"), 사짂("picture"), 벨소리("ring"), 사전("dictionary") etc.

Price 가격("price"), 세일("sale") etc.

Design

색깔("color"), 모양("shape"), 크기("size"), 디자인("design") etc.

Table 9. Features for evaluation in MOBILE PHONE

Movies

Reviews are not limited to the evaluation of material products, but include sentiments on performances such as movies, concerts and musicals. In this case, the reviewers express their emotions towards the performance. In addition to their sentiments on a whole movie, people tell their opinion about the actors' performance and the contents of the story.

Domain Topic Category

Feature Nouns

MOVIE

Character and

Director 감독("director"), 인물("character") 배우("actor"), 주인공("protagonist"), 스타("star"), 역할("role"), 작가("writer"), 제작자("producer") etc.

Story

이야기("story"), 줄거리("plot"), 시리즈("series"), 장면("scene"), 사걲("episode"), 결말("ending"), 갈등("trouble"), 구조("structure"), 주제("theme") etc.

Result

흥행("box-office"), 평가("evaluation"), 성공("success"), 인기

("popularity"), 반응("reaction"), 실패(failure") etc.

Elements of movie

음악("music"), 대사("line"), 출연("casting"), 표현("expression"), 구성("composition"), 묘사("description"), 배경("background"), 목소리 ("voice") etc.

Emotion

경험("experience"), 기억("memory"), 관심("interest"), 매력("attraction"), 만족("satisfaction"), 감동("impression"), 걱정("worry") etc.

Table 10. Features for evaluation in MOVIE

5. A case study on the Adjective of RELATIVE Polarity "Khuta" ("big")

In this section, we analyze opinion sentences with an adjective of RELATIVE polarity, and show restrictions between the adjective and the co-occurring features. We choose "khuta"("big"). It occurs in every domain at a high frequency because it is more ambiguous than other adjectives such as "ppaluta"("fast") and "twukkepta"("thick"). The same word "khuta" expresses size, or qualifies various types of magnitude: "Hwamyen-I nemu khu-yo."(" The screen is very large.") describes size of screen, "Peylsoli-ka nemu khup-nita."("The ring sound is loud.") sound volume, and "Caphan sayong-e khu-n cangcem-i isseyo"("There is a big advantage in using the keypad.") the importance of function. In the contrary, "ppaluta" expresses only speed, and "twukkepta" the distance between sides.

"Khuta" generally expresses a favorable opinion in the HOTEL domain, when it evaluates the size of the hotel as in (2a) below. It expresses an unfavorable opinion in the MOBILE PHONE domain, when it evaluates the size of mobile phone like (2b). Even with the same feature, its polarity may depend on the domain. In addition, in one domain, its polarity may depend on features. When "khuta" is applied to parts of a mobile phone, such as screen and buttons, it has a positive value as shown in (2c).

(2) a. "Lostey hotyel-un khu-ko wungcang-haysseyo."

(" The LOTTE hotel was big and magnificent.") b. "Aiphon-uy khuki-ka sayngkak-pota khu-n kes kath-ayo."

(" The Iphone is bigger than I expected.") c. "Hayntuphon pethun-i khe-se cal nullye-yo."

("A big button on a phone is easy to press.")

Consequently, prediction of polarity of adjectives cannot be achieved at the document or sentencelevel, but only by matching jointly the feature and the adjective. Thus, we should consider other contexts that affect the polarity of opinion sentences. This approach leads us to construct an "Opinion-Feature Dictionary". We show examples of an "Opinion-Feature Dictionary" of the MOBILE PHONE domain.

DOMAIN MOBILE PHONE CATEGORY

Part of mobile phone

Table11. Sample form Opinion-Features Dictionary of MOBILE PHONE domain

We mark positive polarity with + and negative polarity with -. The features which are not predicated by the adjectives are marked with "z"(zero). Feature noun lists can be extended with synonyms. For example, "kakyek"("price") has synonyms such as "piyoung", "wenga", "cengga" and "kumayk".

Conclusion

In extracting information about users' opinion from online reviews, exact recognition of the semantic polarity of the adjectives is one of the most important requirements. As adjectives have different semantic orientations according to their contexts, opinion information cannot be extracted satisfactorily without considering the semantic and lexical relations between adjectives and appropriate feature nouns.

Research about opinion mining with adjectival predicates is active, but linguistic properties of adjectives are underexploited. In this paper, we suggested a classification of adjectives by semantic polarity, and presented an approach to predicting the orientation of opinion sentences with adjectives. We emphasize the importance of describing adjectives of RELATIVE polarity depending on domains and contexts. The ultimate goal of our research will be to construct an Opinion-Features Dictionary classified by Domains, which will be necessary to extract accurately online users' opinions from Web documents.

In future work, we need to consider other cases of ambiguity of adjectives of RELATIVE polarity.

(3) a. "Nolahta"("yellow") in (3a) and "tungkulta"("round") in (3b) describe the color and shape of the

Figure 1 .

 1 Figure 1. The polarity of "Kanghata" ("strong") depending on features in the MOVIE domain.

  "Kheyisu-eyse yakkan nolanbich-i nayo." (" The case has a bit of yellow.") b. "Sol-i tungk-un phyen-ipnita." ("The brush is round.") c. "Hayntuphon-i thupakha-ki-potan tungk-un phyen-ieyyo." ("The mobile phone is round rather than rough.")

Table 1 . Five domains size for our corpora

 1

	Size	23,502 tokens	23,505 tokens	26,854 tokens	23,776 tokens	24,631 tokens
	Adjectival predicates	321 types	278 types	297 types	235 types	291 types

We classified adjectives using five domains represented by five corpora of online texts: movies [MOVIE], cosmetic products [COSMETIC PRODUCT], hospitals [HOSPITAL], mobile phones [MOBILE] and hotels [HOTEL]. These corpora consist of about 23,500 ~ 26,800 tokens each.

  Table 2). The average of the total number of occurrences of adjectives is 2,135 per domain: 884 occurrences of adjectives of ABSOLUTE polarity and 1,251 of adjectives of RELATIVE polarity.

		Cosmetic Product	Hotel	Hospital	Mobile Phone	Movie	Average
	Total frequency of adjectival predicates	2,151	2,262	2,174	1,901	2,186	2,135
	ABSOLUTE polarity	912	766	1,022	814	908	884(41.4%)
	RELATIVE polarity	1,239	1,496	1,152	1,087	1,278	1,251(58.6%)

Table 2 . The frequency of adjectives bearing ABSOLUTE or RELATIVE polarity in various corpora

 2

		HOTEL	MOBILE
	Total concordance	44 sentences	53 sentences
	Noise	9 sentences	12 sentences
	Total Opinion Sentence	35 sentences	41sentences

Adjectives of RELATIVE polarity account for more occurrences than adjectives of ABSOLUTE polarity. This shows that the current keyword-based approach in which adjectives bear fixed polarities have a serious drawback. If we perform keyword-based extraction method with adjectives of RELATIVE polarity, we cannot obtain reliable results. For instance, in the HOTEL and MOBILE corpora, when we extract sentences containing "khuta"("big"), we will get about 20% noise.

Table 3 . The result of sentence extraction by Khuta Table

 3 3 shows that keyword-based extraction allows only about 79% and 77% precision respectively. The Noise row shows the number of Non-opinion sentences, which express facts in a neutral way. Let us

compare (1a) to (1b).

Table 4 . Example of ABSOLUTE polarity of adjectives 3.2. Adjectives of RELATIVE Polarity The

 4

		Polarity Adjective	Polarity Adjective	Polarity
	가능하다(possible)	+	밉다(hateful)	-	예민하다(sensitive)	-
	갂편하다(convenient)	+	부족하다(insufficient)	-	예쁘다(pretty)	+
	갑갑하다(stuffy)	-	부지런하다(diligent)	+	옳다(correct)	+
	걱정스럽다(worried)	-	불행하다(unhappy)	-	완벽하다(perfect)	+
	고맙다(grateful)	+	비싸다(expensive)	-	위험하다(dangerous)	-
	굉장하다(fantastic)	+	사랑스럽다(lovely)	+	유명하다(famous)	+
	귀엽다(cute)	+	사소하다(trivial)	-	이상하다(strange)	-
	깔끔하다(tidy)	+	상냥하다(affectionate)	+	잘생기다(handsome)	+
	나쁘다(bad)	-	새롭다(new)	+	잧미있다(interesting)	+
	낡다(used)	-	서투르다(unskilled)	-	젊다(youthful)	+
	Adjective	Polarity Adjective	Polarity Adjective	Polarity
	가깝다(near)	+/-	단순하다(simple)	+/-	작다(small)	+/-
	가볍다(light)	+/-	두껍다(thick)	+/-	잦다(frequent)	+/-
	강하다(strong)	+/-	딱딱하다(hard)	+/-	적다(little)	+/-
	길다(long)	+/-	많다(many)	+/-	좁다(narrow)	+/-
	깊다(deep)	+/-	멀다(far)	+/-	짧다(short)	+/-
	낮다(low)	+/-	무겁다(heavy)	+/-	느리다(slow)	+/-
	넓다(wide)	+/-	빠르다(quick)	+/-	어리다(young)	+/-
	높다(high)	+/-	얇다(thin)	+/-	약하다(weak)	+/-

meanings of adjectives of RELATIVE polarity are affected by co-occurring nouns, sentence patterns or contexts. Since we cannot determine their polarity out of context, further analysis of them and description through linguistic observation is required.

Table 5 . Example of RELATIVE polarity of adjectives

 5

This work was supported by the French Unique Interministerial Fund (FUI) and by French industrial cluster Cap Digital, through the funding of the DoXa project on automatic processing of opinion and sentiment.

DoXa is a French project on automatic processing of opinion and sentiment. https://www.projet-doxa.fr/

product, but they do not convey the reviewer's subjective opinion. These sentences are factual and neutral.

However, "tungkulta" in (3c) expresses a positive opinion. In this comparative sentence, it is contrasted with "thupakhata"("rough") which has a negative meaning. Therefore, when we predict the semantic orientation of opinion sentences, we also need an understanding of such sentence structures.


Pierre Degond
email: pierre.degond@math.univ-toulouse.fr

Amic Frouvelle
email: amic.frouvelle@math.univ-toulouse.fr

Jian-Guo Liu
email: jliu@phy.duke.edu

Macroscopic limits and phase transition in a system of self-propelled particles

Keywords: propelled particles, alignment interaction, Vicsek model, phase transition, hydrodynamic limit, diffusion limit, Chapman-Enskog expansion, non-hyperbolicity AMS subject classification: 35L60, 35K55, 35Q80, 82C05, 82C22, 82C70, 92D50

Macroscopic limits and phase transition in a system of self-propelled particles

Pierre Degond, Amic Frouvelle, Jian

Introduction

The context of this paper is that of [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF] and is concerned with a kinetic model for selfpropelled particles and its hydrodynamic or diffusion limits. The particles move with the same constant speed and their velocity directions (which belong to the sphere S) align to the local average orientation, up to the addition of some noise. This model has been proposed as a variant of the Vicsek particle model [START_REF] Vicsek | Novel type of phase transition in a system of self-driven particles[END_REF]. In this paper, we remove the normalization of the force intensity which was done in [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF]. This apparently minor modification leads to the appearance of phase transitions, which have been studied in the space-homogeneous setting in [START_REF] Frouvelle | Dynamics in a kinetic model of oriented particles with phase transition[END_REF]. In [START_REF] Frouvelle | Dynamics in a kinetic model of oriented particles with phase transition[END_REF], it is proved that the equilibrium distribution function changes type according to whether the density is below or above a certain threshold value. Below this value, the only equilibrium distribution is isotropic in velocity direction and is stable. Any initial distribution relaxes exponentially fast to this isotropic equilibrium state. By contrast, when the density is above the threshold, a second class of anisotropic equilibria formed by Von-Mises-Fischer distributions of arbitrary orientation appears. The isotropic equilibria become unstable and any initial distribution relaxes towards one of these anisotropic states with exponential speed of convergence. We would like to emphasize the connection of the presented alignment models to the the Doi-Onsager [START_REF] Doi | The theory of polymer dynamics[END_REF][START_REF] Onsager | The effects of shape on the interaction of colloidal particles[END_REF] and Maier-Saupe [START_REF] Maier | Eine einfache molekulare Theorie des nematischen kristallinflüssigen Zustandes[END_REF] models for phase transition in polymers. The occurrence of phase transitions makes a strong difference in the resulting macroscopic models as compared with the ones found in [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF][START_REF] Frouvelle | Dynamics in a kinetic model of oriented particles with phase transition[END_REF], where no such phase transitions were present.

In the present paper, we rely on this previous analysis to study the large-scale limit of the space-inhomogeneous system. In the regions where the density is below the threshold, the convection speed becomes zero and the large-scale dynamics becomes a nonlinear diffusion. On the other hand, in the region where the density is above the threshold, the large-scale dynamics is described by a similar hydrodynamic model for self-alignment interactions as derived in [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF][START_REF] Frouvelle | A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters[END_REF]. However, the modified normalization of the force gives rise to different convection speeds and the resulting model may lose its hyperbolicity in some regions of the state space.

The Vicsek model [START_REF] Vicsek | Novel type of phase transition in a system of self-driven particles[END_REF], among other phenomena, models the behaviour of individuals in animal groups such as fish schools, bird flocks, herds of mammalians, etc (see also [START_REF] Aldana | Phase transitions in self-driven many-particle systems and related non-equilibrium models: a network approach[END_REF][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][START_REF] Grégoire | Onset of collective and cohesive motion[END_REF]). This particle model (also called 'Individual-Based Model' or 'Agent-Based model') consists of a discrete stochastic system for the particle positions and velocities. A time-continuous version of the Vicsek model and its kinetic formulation have been proposed in [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF]. The rigorous derivation of this kinetic model has been performed in [START_REF] Bolley | Mean-field limit for the stochastic Vicsek model[END_REF].

Hydrodynamic models are more efficient than particle models for large numbers of particles, because they simply encode the different particles quantities into simple averages, such as the density or mean-velocity. We refer to [START_REF] Chuang | State transitions and the continuum limit for a 2D interacting, self-propelled particle system[END_REF][START_REF] D'orsogna | Self-propelled particles with soft-core interactions: patterns, stability and collapse[END_REF][START_REF] Mogilner | A non-local model for a swarm[END_REF][START_REF] Mogilner | Mutual interactions, potentials, and individual distance in a social aggregation[END_REF][START_REF] Toner | Flocks, herds, and schools: A quantitative theory of flocking[END_REF][START_REF] Topaz | Swarming patterns in a two-dimensional kinematic model for biological groups[END_REF][START_REF] Topaz | A nonlocal continuum model for biological aggregation[END_REF] for other models of self-propelled particle interactions. Rigorous derivations of hydrodynamic models from kinetic ones for self-propelled particles are scarce and [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF][START_REF] Frouvelle | A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters[END_REF] are among the first ones (see also some phenomenological derivations in [START_REF] Kulinskii | Hydrodynamic model for a system of self-propelling particles with conservative kinematic constraints[END_REF][START_REF] Ratushnaya | Collective behaviour of self propelling particles with kinematic constraints ; the relations between the discrete and the continuous description[END_REF][START_REF] Ratushnaya | Hydrodynamic model for the system of self propelling particles with conservative kinematic constraints; two dimensional stationary solutions[END_REF]). Similar models have also been found in relation to the so-called Persistent Turning Walker model of fish behavior [START_REF] Degond | Large scale dynamics of the Persistent Turning Walker model of fish behavior[END_REF][START_REF] Degond | A macroscopic model for a system of swarming agents using curvature control[END_REF]. Diffusive corrections have also been computed in [START_REF] Degond | Diffusion in a continuum model of self-propelled particles with alignment interaction[END_REF]. We refer to [START_REF] Carrillo | Double milling in self-propelled swarms from kinetic theory[END_REF][START_REF] Carrillo | Self-propelled interacting particle systems with roosting force[END_REF] for other macroscopic models of swarming particle systems derived from kinetic theory. In particular, we mention [START_REF] Frouvelle | A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters[END_REF] where a vision angle and the dependence of alignment frequency upon local density have been investigated.

The outline of this paper is as follows. In Section 2, we describe the Individual-Based Model (IBM), and its mean-field limit. In Section 3, we investigate the properties of the rescaled mean-field model. We prove that there are two possibilities for a local equilibrium, depending on the value of its density ρ. Section 4 is devoted to the derivation of the diffusion model when the density ρ is below the threshold. Finally, in Section 5, we derive the hydrodynamic model for self-alignment interactions in the region where ρ is above the threshold and study its hyperbolicity. A conclusion is drawn in section 6. Two appendices are added. In appendix 1, we calculate a Poincaré constant which provides us with a fine estimate of the rate of convergence to the equilibrium states. In appendix 2. some numerical computations of the coefficients of the model are given.

Particle system and mean-field limit

We consider N oriented particles in R n , described by their positions X 1 , . . . X N and their orientation vectors ω 1 , . . . , ω N belonging to S, the unit sphere of R n . We define the mean momentum J k of the neighbors of the particle k by

J k = 1 N N j=1 K(X j -X k )ω j .
In this paper, the observation kernel K will be supposed isotropic (depending only on the distance |X j -X k | between the particle and its neighbors), smooth and with compact support. Introducing a non-isotropic observation kernel, as in [START_REF] Frouvelle | A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters[END_REF] would lead to the same conclusion, with a slightly different convection speed for the orientation in the macroscopic model, but the computations are more complicated. Therefore we focus on an isotropic observation kernel for the sake of simplicity. The particles satisfy the following system of coupled stochastic differential equations (which must be understood in the Stratonovich sense), for k ∈ 1, N :

dX k = ω k dt (2.1) dω k = (Id -ω k ⊗ ω k )J k dt + √ 2d(Id -ω k ⊗ ω k ) • dB k t , (2.2)
The first equation expresses the fact that particles move at constant speed equal to unity, following their orientation ω k . The terms B k t stand for N independent standard Brownian motions on R n , and the projection term (Id -ω k ⊗ ω k ) (projection orthogonally to ω k ) constrains the norm of ω k to be 1. We have that (Id -

ω k ⊗ ω k )J k = ∇ ω (ω • J k )| ω=ω k ,
where ∇ ω is the tangential gradient on the sphere. So the second equation can be understood as a relaxation (with a rate proportional to the norm of J k ) towards a unit vector in the direction of J k , subjected to a Brownian motion on the sphere with intensity √ 2d. We refer to [START_REF] Hsu | Stochastic Analysis on Manifolds[END_REF] for more details on Brownian motions on Riemannian manifolds.

The interaction term (first term of (2.2)) is the sum of smooth binary interactions. This model is an intermediate between the Cucker-Smale model [START_REF] Cucker | Emergent behavior in flocks[END_REF], where there is no constraint on the velocity and no noise, and the time-continuous version of the Vicsek model proposed in [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF], where the velocity is constant and noise is added. Indeed, in [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF], J k is replaced by νΩ k , where Ω k = J k |J k | is the unit vector in the direction of J k and the relaxation frequency ν is a constant. Therefore, in [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF], the interaction term cannot be recast as a sum of binary interactions and has a singularity when J k is close to 0. The model presented here brings a modification consisting in letting ν depend (linearly) on the norm of the velocity J k . A related modification has previously been introduced in [START_REF] Frouvelle | A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters[END_REF], consisting in letting the relaxation parameter ν depend on a local density ρk , but the modification considered here brings newer phase transition phenomena.

From the Individual-Based Model (2.1), (2.2), we derive a mean-field limit as the number of particles N tends to infinity. We define the empirical distribution f N by

f N (x, ω, t) = 1 N N i=1 δ (X i (t),ω i (t)) (x, ω),
where the Dirac distribution is defined by duality by δ (X,Ω) , ϕ R n ×S = ϕ(X, Ω) for any smooth function ϕ ∈ C(R n × S), the duality product •, • R n ×S extending the usual inner product of L 2 (R n ×S). For convenience, the integration measure is supposed of total mass equal to 1 on the sphere S, and we have f N , 1 R n ×S = 1. Denoting the convolution with respect to the space variable by * , and the duality product on the sphere by •, • S , we get

J k = K * f N (X k ), ω S .
If there is no noise (d = 0), it is easy to see that f N satisfies the following partial differential equation (in the sense of distributions):

∂ t f N + ω • ∇ x f N + ∇ ω • ((Id -ω ⊗ ω) Jf N f N ) = 0,
where ∇ ω • denotes the divergence operator on the unit sphere, and

Jf N (x, t) = (K * f N )(x), ω S .
When noise is present (d = 0), the empirical distribution f N tends to a probability density function f satisfying the following partial differential equation:

∂ t f + ω • ∇ x f + ∇ ω • ((Id -ω ⊗ ω) Jf f ) = d∆ ω f, (2.3) with Jf (x, t) = S (K * f )(x, ω, t) ω dω. (2.4)
This result has been shown in [START_REF] Bolley | Mean-field limit for the stochastic Vicsek model[END_REF], under the assumption that the kernel K is Lipschitz and bounded. Eqs. (2.3), (2.4) are the starting point of our study. We notice that there is a competition between the alignment and diffusion terms. The alignment term is quadratic while the diffusion term is linear. So we can expect that alignment wins over diffusion for high densities while at low densities, diffusion dominates. This is the source of the phase transition rigorously studied in the space-homogeneous setting in [START_REF] Frouvelle | Dynamics in a kinetic model of oriented particles with phase transition[END_REF]. In this reference, it is proven that there is a unique isotropic equilibrium at low density but beyond a certain density threshold, another family of non-isotropic equilibria in the form of Von-Mises-Fischer distributions arises. Above this threshold, the isotropic equilibria become unstable and the anisotropic ones become the stable ones. Therefore, we expect different large-scale limits according to whether the density is lower or larger than this threshold.

We now make some preliminary remarks and assumptions. We suppose that the kernel K is integrable, and that its total weight

K 0 = R n K(x)dx is positive. Writing f (x, ω, t) = f ( 1 d x, ω, 1 d t) and K(x) = 1 K 0 d n K( 1 d x),
we get that f satisfies (2.3) with d = 1 and K replaced by K in (2.4), and we have

R n K(x)dx = 1.
So without loss of generality, we can suppose that d = 1 and that K 0 = 1.

We are now ready to investigate the large-scale behavior of (2.3), (2.4) in space and time. The derivation of the macroscopic limit proceeds as in [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF], and follows closely the presentation of [START_REF] Frouvelle | A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters[END_REF], so we only give a summary, focusing on the points which are specific to the present model, in particular the distinction between the ordered and disordered phases.

3 The macroscopic limit

Hydrodynamic scaling

In order to observe the system at large scales, we perform a hydrodynamic scaling. We introduce a small parameter ε, and the change of variables x ′ = εx, t ′ = εt. We write

f ε (x ′ , ω, t ′ ) = f (x, ω, t), and K ε (x ′ ) = 1 ε n K(x). Then f ε satisfies ε(∂ t f ε + ω • ∇ x f ε ) = -∇ ω • ((Id -ω ⊗ ω) Jε f ε f ε ) + ∆ ω f ε , (3.1) with Jε f ε (x, t) = S (K ε * f ε )(x, ω, t) ωdω. (3.2)
The purpose of this paper is to derive a formal limit of this rescaled mean-field model when the parameter ε tends to 0. The first effect of this hydrodynamic scaling is that, up to order 1 in ε, the equation becomes local. Indeed, supposing that f ε does not present any pathological behavior as ε → 0, we get the following expansion:

Jε f ε (t, x) = J f ε (t, x) + O(ε 2 ), (3.3)
where the local flux J f is defined by

J f (x, t) = S f (x, ω, t) ω dω. (3.4)
The proof of this expansion is elementary and omitted here (see e.g. Appendix A.1 of [START_REF] Frouvelle | A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters[END_REF]). We also define the density ρ f associated to f by

ρ f (x, t) = S f (x, ω, t) dω. (3.5)
Hence, Eq. (3.1) becomes, after dropping the O(ε 2 ) term:

ε(∂ t f ε + ω • ∇ x f ε ) = Q(f ε ), (3.6)
with

Q(f ) = -∇ ω • ((Id -ω ⊗ ω)J f f ) + ∆ ω f. (3.7)
This paper is concerned with the formal limit ε → 0 of this problem. We remark that the collision operator Q acts on the ω variable only. The derivation of the macroscopic model relies on the properties of this operator. An obvious remark is that

ω∈S Q(f ) dω = 0 (3.8)
which expresses the local conservation of mass. The first step of the study consists in characterizing the equilibria, i.e. the functions f such that Q(f ) = 0. Indeed, when ε → 0, Q(f ε ) → 0 and the limit f = lim ε→0 f ε belongs to the set of equilibria. This characterization is the purpose of the next subsection.

Equilibria

For any unit vector Ω ∈ S, and κ 0, we define the so-called Von-Mises-Fisher distribution [START_REF] Watson | Distributions on the circle and sphere[END_REF] with concentration parameter κ and orientation Ω by

M κΩ (ω) = e κ ω•Ω S e κ υ•Ω dυ . (3.9)
We note that the denominator depends only on κ. M κΩ is a probability density on the sphere, and we will denote by • M κΩ the average over this probability measure. For functions γ depending only on ω • Ω, the average γ(ω • Ω) M κΩ does not depend on Ω and will be denoted by γ(cos θ) Mκ . Using spherical coordinates, this average is given by: γ(cos θ) Mκ = π 0 γ(cos θ) e κ cos θ sin n-2 θ dθ π 0 e κ cos θ sin n-2 θ dθ .

The flux of the Von-Mises-Fisher distribution is given by

J M κΩ = ω M κΩ = c(κ)Ω, (3.10)
where the so-called order parameter c(κ) is such that 0 c(κ) 1 and is defined by

c(κ) = cos θ Mκ = π 0 cos θ e κ cos θ sin n-2 θ dθ π 0 e κ cos θ sin n-2 θ dθ . (3.11)
c(κ) measures how the distribution M κΩ is concentrated about Ω. When c(κ) = 0, M κΩ is the uniform distribution M κΩ = 1, and when c(κ) → 1, we have M κΩ → δ Ω (ω).

We remark that the dependence of M κΩ upon κ and Ω only appears through the product κΩ. In this way, we can consider M J for any given vector J ∈ R n . We also note that

∇ ω (M J ) = (Id -ω ⊗ ω)J M J . Therefore Q(f ) = ∇ ω • M J f ∇ ω f M J f .
Using Green's formula, we have

S Q(f ) g M J f dω = - S ∇ ω f M J f • ∇ ω g M J f M J f dω, and S Q(f ) f M J f dω = - S ∇ ω f M J f 2 M J f dω 0. (3.12) Definition 3.1. A function f (ω) is said to be an equilibrium of Q if and only if Q(f ) = 0.
Let f be an equilibrium. Using (3.12), we deduce that f M J f is a constant. Therefore, f = ρ f M J f is of the form ρM κΩ with κ 0 and Ω ∈ S (we note that in the case |J f | = 0, then κ = 0 and we can take any Ω ∈ S because f is then just the uniform distribution). Using (3.10), we get

κΩ = J f = ρJ M κΩ = ρc(κ)Ω,
which leads to the following equation for κ (compatibility condition):

ρc(κ) = κ. (3.13)
The study of this condition and the classification of the equilibria can be found in [START_REF] Frouvelle | Dynamics in a kinetic model of oriented particles with phase transition[END_REF].

The key point is to notice that the function κ → c(κ) κ is decreasing and tends to 1 n as κ → 0. Therefore, there is no other solution than κ = 0 if ρ n. By contrast, if ρ > n, there is a unique strictly positive solution in addition to the trivial solution κ = 0. This leads to the following proposition. Proposition 3.2. (i) If ρ n, κ = 0 is the only solution to the compatibility relation (3.13). The only equilibria are the isotropic ones f = ρ, with arbitrary ρ 0.

(ii) If ρ > n, the compatibility relation (3.13) has exactly two roots: κ = 0 and a unique strictly positive root denoted by κ(ρ). The set of equilibria associated to the root κ = 0 consists of the isotropic equilibria f = ρ, with arbitrary ρ > n. The set of equilibria associated to the root κ(ρ) consist of the Von Mises-Fischer distributions ρM κ(ρ)Ω with arbitrary ρ > n and arbitrary Ω ∈ S and forms a manifold of dimension n.

The rate of convergence to the equilibria have been studied in [START_REF] Frouvelle | Dynamics in a kinetic model of oriented particles with phase transition[END_REF] in the spatially homogeneous setting. The results are recalled in the next section.

Rates of convergence to equilibrium in the spatially homogeneous setting

Denoting by g ε = f ε /ρ f ε the velocity probability distribution function, we can rewrite (3.6) under the following form (omitting the superscripts ε for the sake of clarity and neglecting the O(ε 2 ) term):

ε(∂ t (ρg) + ω • ∇ x (ρg)) = -(ρ) 2 ∇ ω • ((Id -ω ⊗ ω)J g g) + ρ∆ ω g.
In the spatially homogeneous setting, we let ∇ x (ρg) = 0 and get

ε∂ t (ρg) = -(ρ) 2 ∇ ω • ((Id -ω ⊗ ω)J g g) + ρ∆ ω g = Q(ρg). (3.14)
Integrating this equation with respect to ω and using (3.8), we find that ∂ t ρ = 0. Therefore, ρ is independent of t and can be cancelled out. The homogeneous equation (3.14) therefore takes the form:

ε∂ t g = -ρ ∇ ω • ((Id -ω ⊗ ω)J g g) + ∆ ω g. (3.15)
We now remind the definitions of global and asymptotic rate.

Definition 3.3. Let X be a Banach space with norm • and let f (t): R + → X be a function of t with values in X . We say that f (t) converges to f ∞ with global rate r if and only if there exists a constant C which only depends on f 0 , such that

f (t) -f ∞ Ce -rt . (3.16)
We say that f (t) converges to f ∞ with asymptotic rate r if and only if there exists a constant C depending on f 0 (but not only on f 0 ) such that (3.16) holds. Finally, we say that f (t) converges to f ∞ with asymptotic algebraic rate α if and only if there exists a constant C depending on f 0

f (t) -f ∞ C t α .
Now, concerning problem (3.15), we can state the following theorem:

Theorem 3.4. [START_REF] Frouvelle | Dynamics in a kinetic model of oriented particles with phase transition[END_REF] Suppose g 0 is a probability measure, belonging to H s (S). There exists a unique weak solution g to (3.15), with initial condition g(0) = g 0 . Furthermore, this solution is a classical one, is positive for all time t > 0, and belongs to C ∞ ((0, +∞) × S).

(i) If J g 0 = 0, the large time behavior of the solution is given by one of the three cases below:

-Case ρ < n: g converges exponentially fast to the uniform distribution, with global rate

r(ρ) = (n -1)(n -ρ) nε , (3.17)
in any H p norm.

-Case ρ > n: There exists Ω ∈ S such that g converges exponentially fast to M κ(ρ)Ω , with asymptotic rate greater than

r(ρ) = 1 ε [ρc(κ(ρ)) 2 + n -ρ]Λ κ(ρ) > 0,
in any H p norm, where Λ κ is the best constant for the following Poincaré inequality:

|∇g| 2 M κΩ Λ κ (g -g M κΩ ) 2 M κΩ , (3.18)
We have

r(ρ) ∼ 1 ε 2(n -1)( ρ n -1), when ρ → n. (3.19)
-Case ρ = n: then g converges to the uniform distribution in any H p norm, with algebraic asymptotic rate 1/2. More precisely, we have:

g -1 H p ≤ C ε t 1/2
.

(ii) If J g 0 = 0: Then, (3.15) reduces to the heat equation on the sphere. So g converges to the uniform distribution, exponentially fast, with global rate r = 2n ε in any H p norm. Remark 3.1. That g 0 is a probability measure implies that g 0 ∈ H s (S) for all s < -n-1 2 . However, the theorem holds for all s. So for s -n-1 2 , that g 0 ∈ H s (S) is not a mere consequence of being a probability measure and must added to the hypothesis. Now, we comment the results of this theorem. First, in the supercritical case (when ρ > n), the uniform distribution is an unstable equilibrium: for any perturbation g of the uniform distribution such that J g = 0, the associated solution converges to a given Von-Mises distribution, with a fixed concentration parameter κ(ρ) defined by the compatibility condition (3.13). Second, the rates of convergence to the equilibrium are exponential. In the supercritical case, these rates are only asymptotic ones, but we can prove a uniform bound on these rates for ρ in any compact interval. A more precise study of the behavior of these rates is left to future work. Therefore, when ε is small, the function f ε converges rapidly to a given equilibrium, provided that the rate satisfies r(ρ) → ∞ when ε → 0. In the case ρ < n, thanks to (3.17), this condition is equivalent to saying that ε = o(n -ρ). In the case ρ > n, thanks to (3.19), the condition ε = o(n -ρ) implies that r(ρ) → ∞ when ε → 0 uniformly in any bounded ρ interval of the form [n, A] with A < ∞. However, a uniform estimate from below of r(ρ) is lacking when ρ → ∞. But we can reasonably conjecture that away from a buffer region |ρ -n| = O(ε), the convergence to the equilibrium is exponentially fast. Some elements towards a uniform estimate of the rate r(ρ) are provided in Appendix 1. Furthermore, in Appendix 2, we compute Λ κ and then r(ρ) numerically. The results are depicted in Fig. 1 for dimensions 2, 3, and 4. We observe that for ρ > n, r(ρ) grows linearly with ρ, which supports our conjecture.

Therefore, in the general space-inhomogeneous case, we will assume that the formal limit of f ε as ε → 0 is given by a function f (x, ω, t) which has a different velocity profile according to the position of the local density ρ(x, t) with respect to the threshold value n. For this purpose, we define the disordered region R d and the ordered region R o as

R d = {(x, t) | n -ρ ε (x, t) ≫ ε, as ε → 0 }, (3.20) R o = {(x, t) | ρ ε (x, t) -n ≫ ε, as ε → 0 }. (3.21)
We assume that as ε → 0 we have

f ε (x, ω, t) → ρ(x, t), ∀(x, t) ∈ R d , (3.22) f ε (x, ω, t) → ρ(x, t) M κ(ρ)Ω(x,t) , ∀(x, t) ∈ R o , (3.23)
and that the convergence is as smooth as needed.

The goal is now to derive evolution equations for ρ(x, t) and Ω(x, t). This is the subject of the following two sections. We already note that, integrating (3.6) with respect to ω and using (3.8), we get the mass conservation equation

∂ t ρ ε + ∇ x • (J f ε ) = 0. (3.24)
4 Diffusion model in the disordered region

We derive the macroscopic model in the disordered region R d ⊂ R n , using (3.22). With (3.24) and the fact that J f ε → J f = 0, the mass conservation equation reduces to

∂ t ρ = 0.
To obtain more precise information, we look for the next order in ε, using a Chapman-Enskog method, similarly to the case of rarefied gas dynamics (see [START_REF] Degond | Macroscopic limits of the Boltzmann equation: a review[END_REF] for a review). We prove the following theorem:

Theorem 4.1. When ε tends to zero, the (formal) first order approximation to the solution of the rescaled mean-field system (3.6), (3.7) in the disordered region R d defined by (3.20) is given by

f ε (x, ω, t) = ρ ε (x, t) -ε n ω • ∇ x ρ ε (x, t) (n -1)(n -ρ ε (x, t)) , (4.1)
where the density ρ ε satisfies the following diffusion equation

∂ t ρ ε = ε n -1 ∇ x • 1 n -ρ ε ∇ x ρ ε . (4.2)
Proof. We let ρ ε = ρ f ε and write

f ε = ρ ε (x, t) + εf ε 1 (x, ω, t) with S f ε 1 dω = 0. Inserting this Ansatz into (3.4), we get J ε f ε = εJ f ε 1 (t, x
), and the model (3.6), (3.7) becomes:

∂ t ρ ε + ω • ∇ x ρ ε + ε(∂ t + ω • ∇ x )f ε 1 = -∇ ω ((Id -ω ⊗ ω)J f ε 1 ρ ε ) + ∆ ω f ε 1 -ε∇ ω ((Id -ω ⊗ ω)J f ε 1 ρ ε ). (4.3)
Additionally, (3.24) gives:

∂ t ρ ε + ε∇ x • (J f ε 1 ) = 0. (4.4)
In particular ∂ t ρ ε = O(ε). We need to compute f ε 1 to find the expression of the current. But, with this aim, we may retain only the terms of order 0 in (4.3). Since

∇ ω ((Id -ω ⊗ ω)A) = -(n -1)A • ω,
for any constant vector A ∈ R n , the equation for f ε 1 reads:

∆ ω f ε 1 = (∇ x ρ ε -(n -1)ρ ε J f ε 1 ) • ω + O(ε)
. This equation can be easily solved, since the right-hand side is a spherical harmonic of degree 1 (i.e. is of the form A • ω; we recall that ∆ ω (A • ω) = -(n -1)A • ω and that A • ω is of zero mean). Then:

f ε 1 = - 1 n -1 (∇ x ρ ε -(n -1)ρ ε J f ε 1 ) • ω + O(ε).
We immediately deduce, using that S ω ⊗ ωdω = 1 n Id:

J f ε 1 = -1 n(n -1) (∇ x ρ ε -(n -1)ρ ε J f ε 1 ) + O(ε),
which implies that

J f ε 1 = -1 (n -1)(n -ρ ε ) (∇ x ρ ε + O(ε)).
Inserting this equation into (4.4) leads to the diffusion model (4.2) and ends the proof.

Remark 4.1. The expression of f ε 1 , which is given by the O(ε) term of (4.1) confirms that the approximation is only valid when n -ρ ε ≫ ε. The diffusion coefficient is only positive in the disordered region and it blows up as ρ ε tends to n, showing that the Chapman-Enskog expansion loses its validity.

5 Hydrodynamic model in the ordered region

Derivation of the model

We now turn to the ordered region R o ⊂ R n defined by (3.21). The purpose of this section is to give a formal proof of the following: Theorem 5.1. When ε tends to zero, the (formal) limit to the solution f ε (x, ω, t) of the rescaled mean-field system (3.6), (3.7), in the ordered region R o ⊂ R n defined by (3.21), is given by

f (x, ω, t) = ρ(x, t) M κ(ρ(x,t))Ω(x,t) (ω), (5.1)
where the Von-Mises-Fischer distribution M κΩ is defined at (3.9), and the parameter κ is the unique positive solution to the compatibility condition (3.13). Moreover, the density ρ > n and the orientation Ω ∈ S satisfy the following system of first order partial differential equations:

∂ t ρ + ∇ x • (ρcΩ) = 0, (5.2) ρ(∂ t Ω + c(Ω • ∇ x )Ω) + λ(Id -Ω ⊗ Ω)∇ x ρ = 0, (5.3)
where the coefficient c = c(κ(ρ)) is defined at (3.11), the coefficient c = c(κ(ρ)) will be defined later on at (5.9), and the parameter λ = λ(ρ) is given by

λ = ρ -n -κ c κ(ρ -n -κc)
.

(5.4)

Proof: From now on, we will drop the dependence on ρ in the coefficients when no confusion is possible. With (3.23), f ε → f , where f is the stable local equilibrium (5.1).

We now derive the evolution equations (5.2), (5.3) for ρ and Ω.

We recall that the concentration parameter κ satisfies the compatibility condition (3.13) where the order parameter c is defined by (3.11) and that we have J f = ρcΩ. Therefore, eq. (3.24) in the limit ε → 0, reads

∂ t ρ + ∇ x • (ρcΩ) = 0.
To compute the evolution equation for Ω, the method proposed originally in [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF] consists in introducing the notion of generalized collisional invariant (GCI). This method has been then applied to [START_REF] Degond | A macroscopic model for a system of swarming agents using curvature control[END_REF][START_REF] Frouvelle | A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters[END_REF]. The first step is the definition and determination of the GCI's. We define the linear operator L κΩ associated to a concentration parameter κ and a direction Ω as follows:

L κΩ (f ) = ∆ ω f -κ∇ ω • ((Id -ω ⊗ ω)Ωf ) = ∇ ω • M κΩ ∇ ω f M κΩ , so that Q(f ) = L J f (f ).
We define the set C κΩ of GCI's associated to κ ∈ R and Ω ∈ S by:

C κΩ = ψ| ω∈S L κΩ (f ) ψ dω = 0, ∀f such that (Id -Ω ⊗ Ω)J f = 0 .
Hence, if ψ is a GCI associated to κ and Ω, we have:

ω∈S Q(f ) ψ dω = 0, ∀f such that J f = κΩ.
The determination of C κΩ closely follows [START_REF] Frouvelle | A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters[END_REF]. We define the space

V = {g | (n -2)(sin θ) n 2 -2 g ∈ L 2 (0, π), (sin θ) n 2 -1 g ∈ H 1 0 (0, π)}, (5.5)
and we denote by g κ the unique solution in V of the elliptic problem

L * κ g(θ) = sin θ, (5.6)
where L * κ g(θ) = -(sin θ) 2-n e -κ cos θ d dθ (sin θ) n-2 e κ cos θ dg dθ (θ) + n-2 sin 2 θ g(θ).

(5.7)

Then defining h κ by g κ (θ) = h κ (cos θ) sin θ, we get

C κΩ = {h κ (ω • Ω)A • ω + C | C ∈ R, A ∈ R n , with A • Ω = 0 }.
C κΩ is a vector space of dimension n, since A is a vector with n-1 independent components. The next step consists in multiplying (3.6) by a GCI associated to κ ε and Ω ε such that J f ε = κ ε Ω ε , and to integrate it with respect to ω. For any vector

A ∈ R n , with A•Ω ε = 0, we get ω∈S Q(f ε )h κ ε (ω • Ω ε ) A • ω dω = 0.
So, the vector

X ε = 1 ε ω∈S Q(f ε )h κ ε (ω • Ω ε ) ω dω, is parallel to Ω ε , or equivalently (Id -Ω ε ⊗ Ω ε ) X ε = 0. Using (3.
6), we get:

X ε = ω∈S (∂ t f ε + ω • ∇ x f ε ) h κ ε (ω • Ω ε ) ω dω.
In the limit ε → 0, we get (Id -Ω ⊗ Ω) X = 0, (

where

X = ω∈S (∂ t (ρM κΩ ) + ω • ∇ x (ρM κΩ )) h κ (ω • Ω) ω dω .
Finally it has been proved in [START_REF] Frouvelle | A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters[END_REF] that (5. Therefore we get

c dρ dκ = κ ρ dρ dκ = 1 -ρ dc dκ = 1 -ρ (1 -(n -1) c κ -c 2 ) = n -ρ + κc, (5.11)
and finally

λ = 1 κ + c -c n -ρ + κc = n -ρ + κ c κ(n -ρ + κc) ,
which ends the proof of Theorem 5.1.

The next part is devoted to the study of the properties of the model (5.2)-( 5.3) in the ordered region.

Hyperbolicity of the hydrodynamic model in the ordered region

We first investigate the hyperbolicity of the hydrodynamic model (5.2)-(5.3). We recall some definitions. Let

∂ t U + n i=1 A i (U)∂ x i U = 0, (5.12)
be a first order system where x ∈ R n , t 0, U = (U 1 , . . . U m ) is a m-dimensional vector and (A i (U)) i=1,...,n are n m × m-dimensional matrices. Let U 0 ∈ R m . The constant and uniform state U(x, t) = U 0 is a particular solution of (5.12). The linearization of (5.12) about this constant and uniform state leads to the following linearized system:

∂ t u + n i=1 A i (U 0 )∂ x i u = 0. (5.13)
We look for solutions of (5.13) in the form of plane waves u(x, t) = ū e i(k•x-ωt) , with k ∈ R n and ω ∈ C. Such solutions exist if and only if ω/|k| is an eigenvalue of the matrix A(k/|k|) and ū is the related eigenvector, where for a direction ξ ∈ S, the matrix A(ξ) is defined by

A(ξ) = n i=1 A i (U)ξ i . (5.14)
The problem (5.12) is said to be hyperbolic about U 0 , if only purely propagative plane waves with real ω can exist or equivalently, if A(ξ) has real eigenvalues for any ξ. We also must rule out polynomially increasing in time solutions which could exist if the matrix would not be diagonalizable. This leads to the following definitions:

Definition 5.2. (i) Let U 0 ∈ R m . System (5.12
) is hyperbolic about U 0 if and only if for all directions ξ ∈ S, the matrix A(ξ) is diagonalizable with real eigenvalues.

(ii) System (5.12) is hyperbolic, if and only if it is hyperbolic about any state U 0 in the domain of definition of the matrices A i (U).

The linearization of system (5.2)-( 5.3) about a stationary uniform state (ρ 0 , Ω 0 ) is obtained by inserting the following expansion

ρ = ρ 0 + δr + o(δ), (5.15)
Ω = Ω 0 + δW + o(δ), (5.16)
with δ ≪ 1 a small parameter and r = r(x, t), W = W (x, t), the first order perturbations of ρ and Ω. Given that |Ω| = |Ω 0 | = 1, we have W • Ω 0 = 0. Inserting (5.15), (5.16) into (5.2)-( 5.3) leads to the following linearized system:

∂ t r + γ 0 (Ω 0 • ∇ x )r + ρ 0 c 0 (∇ x • W ) = 0, (5.17)
∂ t W + c 0 (Ω 0 • ∇ x )W + λ 0 ρ 0 (Id -Ω 0 ⊗ Ω 0 )∇ x r = 0, (5.18)
W • Ω 0 = 0, (5.19) with γ(ρ) = c + ρ dc dρ ,
and γ 0 = γ(ρ 0 ), c 0 = c(ρ 0 ), c0 = c(ρ 0 ) and λ 0 = λ(ρ 0 ). Next, we show that system (5.17)-(5. [START_REF] Frouvelle | Dynamics in a kinetic model of oriented particles with phase transition[END_REF]) is invariant under rotations. This will allow us to choose one arbitrary direction ξ in the definition (5.14) instead of checking all possible directions. For this purpose, let R be a rotation matrix of R n , i.e. R is a n×n matrix such that R T = R -1 , where the exponent T denotes transposition. We introduce the change of variables x = Rx ′ and define new unknowns

r(x) = r ′ (x ′ ), W (x) = RW ′ (x ′ ), Ω 0 = RΩ ′ 0 .
We note the following identities

Ω ′ 0 • W ′ (x ′ ) = Ω 0 • W (x) = 0, ∇ x r(x) = R ∇ x ′ r ′ (x ′ ), ∇ x W (x) = R ∇ x ′ W ′ (x ′ ) R T , (∇ x • W )(x) = (∇ x ′ • W ′ )(x ′ ), (Ω 0 • ∇ x )W (x) = (∇ x W (x)) T Ω 0 = R (∇ x ′ W (x ′ )) T Ω ′ 0 = R (Ω ′ 0 • ∇ x ′ )W ′ (x ′ ), (Ω 0 • ∇ x )r(x) = (Ω ′ 0 • ∇ x ′ )r ′ (x ′ ).
With these identities, it is easy to show that (r ′ , W ′ ) satisfies system (5.17)- (5.19) with Ω 0 replaced by Ω ′ 0 . The rotational invariance of (5.17)- (5.19) shows that, in order to check the hyperbolicity, it is enough to choose any particular direction ξ. Let us call this arbitrary direction z, with unit vector in this direction denoted by e z . To check the hyperbolicity of waves propagating in the z direction it is sufficient to look at the system where all unknowns only depend only on the space coordinate z and on the time t. Denoting by θ the angle between the z direction and Ω, we can write:

Ω = cos θ e z + sin θ v, θ ∈ [0, π], v ∈ S n-2 ,
where S n-2 is the sphere of dimension n -2 collecting all unit vectors orthogonal to e z . With these hypotheses, system (5.2)-(5.3) is written.

∂ t ρ + ∂ z (ρc(ρ) cos θ) = 0. (5.20) ρ[∂ t (cos θ) + c(ρ) cos θ ∂ z (cos θ)] + λ sin 2 θ ∂ z ρ = 0.
(5.21)

∂ t v + c(ρ) cos θ ∂ z v = 0, with |v| = 1 and e z • v = 0.
(5.22)

In the special case of dimension n = 2, the system reduces to (5.20)-(5.21), with θ ∈ (-π, π) and Ω = cos θ e z + sin θ v 0 , where v 0 is one of the two unit vectors orthogonal to e z . The hyperbolicity of this system depends on the sign of λ. Proposition 5.5 below shows that λ < 0 in the two limits ρ → n and ρ → ∞. Additionally, the numerical computation of λ, displayed in Fig. 2, provides evidence that λ < 0 for all values of ρ, at least in dimensions n = 2, 3, and 4. Therefore, we assume that λ < 0.

(5.23)

We first check the local hyperbolicity criterion:

Proposition 5.3. We assume (5.23). Then, system (5.20)- (5.22) is hyperbolic about (ρ, θ, v) if and only if

| tan θ| < tan θ c := | c -c n-ρ+κc | 2 √ -λc . ( 5

.24)

Proof: We apply [START_REF] Frouvelle | A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters[END_REF] and find that the hyperbolicity criterion is written:

| tan θ| < | c -d dρ (ρc)| 2 √ -λc .
Using the compatibility condition (3.13) and (5.11), eq. ( 5.24) follows.

As for global hyperbolicity, we have Proposition 5.4. We assume (5.23). Then, system (5.20)-(5.22) is not hyperbolic.

Proof: It has been proved in [START_REF] Frouvelle | A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters[END_REF] that system (5.2)-( 5.3) is hyperbolic if and only if λ > 0.

As we assume (5.23), it follows that the system is not hyperbolic.

We now provide asymptotic expansions of the coefficients which show that, at least when ρ → n or ρ → ∞, we have λ < 0. Proposition 5.5. We have the following expansions:

(i) When ρ → n: c = √ n+2 n √ ρ -n + O(ρ -n), c = 2n-1 2n √ n+2 √ ρ -n + O(ρ -n), λ = -1 4 √ n+2 1 √ ρ -n + O(1), θ c = π 2 - 2 √ n+2 √ n √ ρ -n + O(ρ -n).
(ii) When ρ → ∞:

c = 1 -n-1 2 ρ -1 + (n-1)(n+1) 8 ρ -2 + O(ρ -3 ), c = 1 -n+1 2 ρ -1 -(n+1)(3n+1) 24 ρ -2 + O(ρ -3 ), λ = -n+1 6 ρ -2 + O(ρ -3 ), θ c = arctan( √ n+1 √ 6 4
) + O(ρ -1 ).

Proof: Using the compatibility condition (3.13), the expression (5.4) depends only on κ, c, and c. With the asymptotic expansion of c and c as κ → 0 and κ → ∞ given in [START_REF] Frouvelle | A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters[END_REF], we can get an expansion for λ. We have

c = 1 n κ -1 n 2 (n+2) κ 3 + O(κ 5 ) as κ → 0, 1 -n-1 2κ + (n-1)(n-3) 8κ 2 + O(κ -3 ) as κ → ∞, c = 2n-1 2n(n+2) κ + O(κ 2 ) as κ → 0, 1 -n+1 2κ + (n+1)(3n-7) 24κ 2 + O(κ -3 ) as κ → ∞.
We first compute an expansion of ρ = κ c . We get

ρ = n + 1 n+2 κ 2 + O(κ 4 ) as κ → 0, κ + n-1 2 + (n-1)(n+1) 8κ + O(κ -2 ) as κ → ∞. (5.25)
Using the definition (5.4), we then get

λ = -1 4κ + O(1) as κ → 0 -n+1 6κ 2 + O(κ -3 ) as κ → ∞.
We can also expand the threshold angle θ c in terms of κ. We get

θ c = π 2 - 2 (n+2) √ n κ + O(κ 2 )
as κ → 0, arctan(

√ n+1 √ 6 4
) + O(κ -1 ) as κ → ∞.

We can now reverse the expansion (5.25) to get an expansion of κ (and then of the other coefficients) in terms of the density ρ. We get

κ = √ n + 2 √ ρ -n + O(ρ -n) as ρ → n, ρ -n-1 2 -(n-1)(n+1) 8ρ + O(ρ -2 ) as ρ → ∞.
Inserting this expansion into the previous ones, we finally deduce the expressions stated in proposition 5.5.

When ρ ∼ n, since |λ| = -λ is large compared to ρ, which is large compared to ρ c, the behavior of the orientation equation (5.3) can be compared to the behavior of

∂ t Ω = |λ| ρ (Id -Ω ⊗ Ω)∇ x ρ,
which relaxes Ω to the unit vector ∇ x ρ/|∇ x ρ|, with rate

λ ρ |∇ x ρ| ∼ 1 4n √ n+2 √ ρ-n |∇ x ρ|.
This actually makes sense only if the rate of convergence to the equilibrium

1 ε r(ρ) ∼ 2n-1 nε (ρ -n) in the neighborhood of n is large compared to this relaxation rate. This requires ε ≪ (ρ -n) 3 2 |∇ x ρ|.
In this case the leading behavior of the system is given by

∂ t ρ + ∇ x • ρc |∇ x ρ| ∇ x ρ = 0,
which is an ill-posed problem, being some kind of nonlinear backwards heat equation. To stabilize this system, a possibility is to derive a first order diffusive correction to model (5.2), (5.3) using a Chapman-Enskog expansion. Such a correction has been derived in [START_REF] Degond | Diffusion in a continuum model of self-propelled particles with alignment interaction[END_REF] for the model of [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF], but leads to complicated terms. Another possibility is to add some contribution of the non-locality of the interaction in the spirit of [START_REF] Degond | Hydrodynamic models of self-organized dynamics: derivation and existence theory[END_REF].

When ρ is large, c and c are close to 1, and λ is small. In the intermediate regime, numerical computations (see Appendix 2) show that there is a significant difference between c and c. This means that the information about velocity orientation travels slower than the fluid. Fig. 3 displays c and c as functions of ρ, in dimension 2 and 3.

Finally, when ρ → ∞, the critical angle θ c tends to a positive value arctan(

√ n+1 √ 6 4
). Numerically, we see that θ c is always larger than this limit value. Then, system (5.20)-(5.22) is hyperbolic in the region where the angle θ between Ω and the direction of propagation is less than this limit value, independently of the density ρ. Fig. 5.2 summarizes the different the types of macroscopic limits of the system in dimension 2, when the density ρ, and the angle θ between Ω and the propagation direction vary. The behavior of the system at the crossings either between the hyperbolic and non-hyperbolic regions or between the ordered and disordered regions, remains an open problem. We note that non-hyperbolicity problems appear in other areas such as the motion a an elastic string on a plane [START_REF] Pego | Instabilities in Glimm's scheme for two systems of mixed type[END_REF].

Conclusion

In this paper, we have derived a macroscopic model for particles undergoing self-alignment interactions with phase transitions. This model is derived from a time-continuous version of the Vicsek model. We have identified two regimes. In the disordered regime, the macroscopic model is given by a nonlinear diffusion equation depending on the small parameter ε describing the ratios of the microscopic to macroscopic length scales. In the ordered regime, the model is given by a hydrodynamic model for self-alignment interaction which is not hyperbolic. Many problems remain open. Among others, a first one is to determine the evolution of the boundary between the ordered and disordered regions and to understand how the models in the two regions are connected across this boundary. The . second one is to understand how to cope with the non-hyperbolicity of the model in the ordered region and possibly modify it by adding small diffusive corrections. Numerical simulations of the particle model are in progress to understand the behavior of the model in the two regimes.

Then one of the following three possibilities is true:

(i) Λ κ is the smallest eigenvalue of the Sturm-Liouville problem

L * κ (g) = λg, (6.4)
for g ∈ C 2 ([0, π]) with Neumann boundary conditions (g ′ (0) = g ′ (π) = 0) and such that π 0 (sin θ) n-2 e κ cos θ g(θ)dθ = 0, and the eigenspace of L * κΩ associated to the eigenvalue Λ κ is of dimension 1, spanned by ω → h 0 κ (ω • Ω), where the function θ → h 0 (cos θ) is smooth, positive for 0 θ < θ 0 and negative for θ 0 < θ π.

(ii) Λ κ is the smallest eigenvalue of the Sturm-Liouville problem

L * κ (g) = L * κ (g) + n-2 sin 2 θ g(θ) = λg, (6.5)
for g ∈ C 2 ([0, π]) with Dirichlet boundary conditions (g(0) = g(π) = 0), and the eigenspace of L * κΩ associated to Λ κ is of dimension n-1, consisting in the functions of the form ψ A (ω) = h 1 κ (ω • Ω)A • ω for any vector A ∈ R n such that Ω • A = 0, with θ → h 1 κ (cos θ) a smooth positive function for 0 < θ < π.

(iii) The two above Sturm-Liouville problems have the same smallest eigenvalue Λ κ , and the eigenspace of L * κΩ associated to Λ κ is of dimension n, spanned by the two types of function of the above cases.

Proof. First of all, we have

|∇ ω ψ| 2 M κΩ (min M κΩ ) S |∇ ω ψ| 2 (min M κΩ ) (n -1) S ψ - S ψ 2 , (6.6)
and

(ψ -ψ M κΩ ) 2 M κΩ (ψ - S ψ) 2 M κΩ (max M κΩ ) S ψ - S ψ 2 . (6.7)
The second inequality of (6.6) follows from the Poincaré inequality on the sphere:

S ψ - S ψ 2 1 n -1 S |∇ ω ψ| 2 .
The first inequality of (6.7) follows from the fact that

(ψ - S ψ) 2 M κΩ -(ψ -ψ M κΩ ) 2 M κΩ = S ψ - S ψM κΩ 2 0.
Eqs. (6.6) and (6.7) lead to the Poincaré inequality (6.1) with

Λ κ (n -1) min M κΩ max M κΩ = (n -1) e 2κ .
We use the inner product (ϕ, ψ) → ϕψ M κΩ , adapted to M κΩ . We denote by L2 κ (S) (resp. Ḣ1 κ (S)) the functions ψ ∈ L 2 (S) (resp. in H 1 (S)) such that ψ M κΩ = 0. The operator L * κΩ given by (6.2) is self-adjoint since ∇ ω ψ • ∇ ω ϕ M κΩ = ψL * κΩ ϕ M κΩ . It is then easy to see, using Lax-Milgram theorem, that if ϕ belongs to L2 κ (S) then there is a unique solution ψ ∈ Ḣ1 κ (S) to the equation L * κΩ ψ = ϕ. The so-obtained inverse operator is then compact and self-adjoint. By the spectral theorem, we get a basis of eigenfunctions, in the Hilbert space L2 κ (S), which are also eigenfunctions of L * κΩ . If we denote Λ -1 κ the largest eigenvalue of the inverse of L * κΩ , then it is easy to see that Λ κ is the best constant for the following Poincaré inequality, in the space Ḣ1 κ (S):

|∇ ω ψ| 2 M κΩ Λ κ ψ 2 M κΩ Λ κ (ψ -ψ M κΩ ) 2
M κΩ . Since the constants trivially satisfy this inequality, this shows that Λ κ is the best constant for the Poincaré inequality (6.1) in H 1 (S).

The goal is now to reduce the computation of the eigenvalues to simpler problems, using separation of variables. We write ω = cos θ Ω + sin θ v, where v belongs to the unit sphere, orthogonal to Ω. We identify Ω with the last element of an orthogonal basis of R n , and we write v ∈ S n-2 .

By spherical harmonic decomposition in an adapted basis (see for example [START_REF] Frouvelle | Dynamics in a kinetic model of oriented particles with phase transition[END_REF], appendix A), we have a unique decomposition of the form

ψ(ω) = k,m g k m (θ)Z k m (v), (6.8)
where (Z k m (v)) k∈ 1,km is a given orthonormal basis of the spherical harmonics of degree m on S n-2 , for m ∈ N, with k m = n+m-2 n-2

-n+m-4 n-2
. If ψ is continuous, g k m is given by

g k m (θ) = S n-2
ψ(cos θ Ω + sin θ v)Z k m (v)dv. (6.9)

We now show that the decomposition (6.8) remains stable under the action of the operator L κΩ , so that its spectral decomposition can be performed independently for each term of the decomposition. First, we examine the case of dimension n 3. Let ψ(ω) = g(θ)Z(v). We have

∇ ω ψ(ω) = g ′ (θ)e θ Z(v) + g(θ) sin θ ∇ v Z(v),
where the unit vector e θ is given by

e θ = ∇ ω θ = - 1 sin θ (Id -ω ⊗ ω)Ω.
We take functions

ψ(ω) = g k m (θ)Z k m (v) and ϕ(ω) = k,m f k m (θ)Z k m (v).
Since the spherical harmonics are orthonormal, and are eigenfunctions of ∆ v for the eigenvalues -m(m + n -3), we get:

∇ ω ψ • ∇ ω ϕ M κΩ = π 0 [f k m ′ (θ)g k m ′ (θ) + m(m+n-3) sin 2 θ f k m (θ)g k m (θ)
](sin θ) n-2 e κ cos θ dθ. (6.10) Suppose m 1. Then, it is easy to see that the function ψ belongs to Ḣ1 κ (S) if and only if (sin θ) n 2 -1 g ′ ∈ L 2 (0, π) and (sin θ) n 2 -2 g ′ ∈ L 2 (0, π). This condition is equivalent to the fact that g ∈ V , where V is defined by (5.5), and which we denote by V m κ for convenience:

V m κ = {g | (sin θ) n 2 -2 g ∈ L 2 (0, π), (sin θ) n 2 -1 g ∈ H 1 0 (0, π)}.
Suppose now that m = 0. Then Z k m is a constant, and the condition ψ ∈ Ḣ1 κ (S) is equivalent to the first condition only: (sin θ) n 2 -1 g ′ ∈ L 2 (0, π), under the constraint that π 0 (sin θ) n-2 e κ cos θ g(θ)dθ = 0. We will denote this space by V 0 κ :

V 0 κ = {g | (sin θ) n 2 -1 g ′ ∈ L 2 (0, π),
π 0 (sin θ) n-2 e κ cos θ g(θ)dθ = 0}. Formula (6.10) then suggests to define the operator

L * κ,m : V m κ → (V m κ ) * by π 0 f (θ) L * κ,m g(θ) (sin θ) n-2 e κ cos θ dθ = π 0 [f ′ g ′ + m(m+n-3) sin 2 θ
f g] (sin θ) n-2 e κ cos θ dθ.

(6.11) From (6.10), it follows that, if we decompose ψ

(ω) = k,m g k m (θ)Z k m (v), then L * κΩ ψ(ω) = k,m L * κ,m g k m (θ) Z k m (v),
showing that L * κΩ is block diagonal on each of these spaces V m κ (tensorized by the spherical harmonics of degree m on S n-2 ). So we can perform the spectral decomposition of L * We notice that

λ κ,m = inf π 0 f (θ) L * κ,m f (θ) (sin θ) n-2 e κ cos θ dθ f ∈ V m κ , π 0 f 2 (θ) (sin θ) n-2 e κ cos θ dθ = 1 ,
but since all the V m κ are the same for m 1, and since

π 0 1 sin 2 θ f 2 (sin θ) n-2 e κ cos θ dθ π 0 f 2 (sin θ) n-2 e κ cos θ dθ, we get λ κ,m+1 λ κ,m + (m + 1)(m + n -2) -m(m + n -3) = λ κ,m + 2m + n -2.
Finally, Λ κ is the minimum between λ κ,0 and λ κ,1 . The eigenfunctions for the operator L * κΩ being smooth, this is also true for the operators L * κ,m , by formula (6.9). So we can transform the definitions (6.11) by integration by parts.

Indeed, if g 0 is an eigenfunction (in V 0 κ ) associated to L * κ,0 and an eigenvalue λ, then g 0 is smooth and satisfies the Sturm-Liouville eigenvalue problem L * κ g 0 (θ) = -(sin θ) 2-n e -κ cos θ ((sin θ) n-2 e κ cos θ g ′ (θ)) ′ = λg 0 (θ).

Conversely, a smooth function with the condition π 0 (sin θ) n-2 e κ cos θ g(θ)dθ = 0 belongs to V 0 κ . Actually, in dimension n 3, we do not need to impose the Neumann boundary conditions: they appear naturally, since we have

L * κ g 0 = -e -κ cos θ (e κ cos θ g ′ 0 ) ′ -n-2 tan θ g ′ 0 = λg 0 .
Therefore by continuity at θ = 0 and π, g ′ 0 (0) = g ′ 0 (π) = 0. Then, using classical Sturm-Liouville oscillation theory (see [START_REF] Weidmann | Spectral theory of ordinary differential operators[END_REF] for example), we find that the first eigenspace of L * κ is of dimension 1, spanned by a function g κ,0 (θ), which is positive for 0 θ < θ 0 and negative for θ 0 < θ π.

Similarly, if g 1 is an eigenfunction (in V 1 κ ) associated to L * κ,1 and an eigenvalue λ,then g 1 is smooth, with g 1 (0) = g 1 (π) = 0 and satisfies the Sturm-Liouville eigenvalue problem

L * κ,1 g 1 (θ) = L * κ g 1 (θ) + n-2 sin 2 θ g 1 (θ) = λg 1 (θ).
And conversely, if a function satisfies Dirichlet boundary conditions while being in C 2 ([0, π]), it belongs to V 1 κ . Once again, if n 3, we do not need to impose the Dirichlet boundary conditions in the C 2 ([0, π]) framework, since we have

L * κ g 1 = -e -κ cos θ (e κ cos θ g ′ 1 ) ′ -n-2 tan θ g ′ 1 + n-2 sin 2 θ g 1 = λg 1 .
So, by continuity at θ = 0 and π, g 1 (0) = g 1 (π) = 0, and then a first order expansion shows that continuity holds whatever the values of g ′ 0 (θ) at the endpoints are. Again, using classical Sturm-Liouville theory, we find that the first eigenspace of L * κ is of dimension 1, spanned by a function g κ,1 (θ), which keeps the same sign on (0, π).

The case λ κ,0 < λ κ,1 corresponds to case (i) of the proposition. Since a spherical harmonic of degree 0 on the sphere S n-2 is a constant, introducing h 0 κ such that h 0 κ (cos θ) = g κ,0 (θ) allows us to state that the eigenspace of L * κΩ associated to the lowest eigenvalue is spanned by ω → h 0 κ (ω • Ω). The case λ κ,0 > λ κ,1 corresponds to case (ii) of the proposition. The spherical harmonics of degree 1 on the sphere S n-2 are the functions of the form v → A • v, with A • Ω = 0. Introducing h 0 κ such that h 0 κ (cos θ) sin θ = g κ,0 (θ) allows us to state that the eigenspace of L * κΩ associated to the lowest eigenvalue is of dimension n-1, consisting of the functions of the form ω → h 1 κ (ω • Ω) A • ω, with A any vector in R n such that A • Ω = 0. Finally, the case λ κ,0 = λ κ,1 corresponds to case (iii) of the proposition and this ends the proof in the case of dimension n 3.

We now examine the special case of dimension n = 2. We identify H 1 (S) with the 2πperiodic functions in H 1 loc (R). So, Λ κ is the smallest eigenvalue of the periodic Sturm-Liouville problem L * κ (g) = L * κ (g) = -e -κ cos θ (e κ cos θ g ′ ) ′ = λg, for functions g such that π -π e κ cos θ g(θ)dθ = 0. Here the decomposition corresponding to (6.8) is the even-odd decomposition (there are only two spherical harmonics on S 0 : the constant function of degree 0 and the odd function of degree 1). The odd part g o of g can be identified with a function of H 1 0 (0, π), and it is easy to see that the odd part of L * κ (g) is L * κ (g o ), and similarly for the even part g e . So, we can perform the spectral decomposition of L * κ separately on the spaces of even and odd functions. Actually, if g is a solution of the Sturm-Liouville periodic problem, the function g(θ) = e -κ cos θ ∂ θ g(π -θ) is another solution with the same eigenvalue. Furthermore, if g is odd, then g is even and conversely. So the eigenvalues are the same for the odd and even spaces problems. Therefore, in dimension n = 2, proposition 6.1 can be refined and we can state that case (iii) is the only possibility: the eigenspace of L * κΩ associated to Λ κ is of dimension 2, spanned by an odd function g o κ , positive on (0, π), and an even function g e κ = g o κ , positive for 0 < θ < θ 0 and negative for θ 0 < θ < π. The proof of Proposition 6.1 is complete.

We can now state a conjecture, which refines proposition 6.1, if true, and which is based on numerical experiments. (ii) The function κ → Λ κ is increasing.

We also observe numerically that λ 1 ∼ κ when kappa is large. Some investigations are in progress to prove the monotonicity of the eigenvalue with respect to κ, based on formal expansions similar to those used in Section 5 of [START_REF] Frouvelle | A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters[END_REF]. Remark 6.1. At the end of the proof of Proposition 6.1, we have seen that in dimension n = 2 only statement (iii) is true. The proof uses a transformation of the solution of an eigenvalue problem into the solution of another eigenvalue problem. We can try to find a similar transformation in dimensions n 3: if f satisfies L κ,0 f = λf (with Neumann boundary conditions) then f = e -κ cos θ ∂ θ f (π -θ) (with Dirichlet boundary conditions) satisfies π 0 f L 1 f (sin θ) n-2 e κ cos θ dθ = λ π 0 f 2 (sin θ) n-2 e κ cos θ dθ -κ(n -2) π 0 cos θ f 2 (sin θ) n-2 e κ cos θ dθ, so if we can prove that π 0 cos θ f 2 (sin θ) n-2 e κ cos θ dθ > 0, we can deduce that λ 0 > λ 1 . So far we have been unable to prove this estimate.

Actually, λ κ,1 is the smallest eigenvalue associated to problem (6.5), with g ∈ V . So, considering once again the function f such that f (θ) = (sin θ) n 2 -1 g(θ), the vector AF , with A defined by (6.1), gives a second order approximation of (sin θ) n 2 -1 L * κ g(θ) = λf (θ) at the points θ i . So we can take the smallest eigenvalue of A as an approximation of λ κ,1 .

We now look for an approximation of λ κ,0 . Let g be a solution of the Sturm-Liouville problem (6.4) with Neumann boundary conditions. We introduce G = (g i+ 1

N -1 2 κ             , (6.2)
So we can take the smallest positive eigenvalue of B as an approximation of λ κ,0 (excluding the constant functions). The computations of Fig. 1 have been performed with N = 300 points.

Figure 1 :

 1 Figure 1: Rates of convergence to equilibria in dimensions 2, 3, and 4, as functions of the density ρ.

8 ) 10 )cos θ e κ cos θ sin n-2 θ dθ π 0 e κ cos θ sin n-2 θ dθ = π 0 cos 2 θ e κ cos θ sin n-2 θ dθ π 0 e κ cos θ sin n-2 θ dθ - π 0 cos θ e κ cos θ sin n-2 θ dθ π 0 e κ cos θ sin n-2 θ dθ 2 = 1 - π 0 sin 2 θ e κ cos θ sin n-2 θ dθ π 0 e κ cos θ sin n-2 θ dθ -c 2 = 1 -

 8102121 is equivalent to (5.3) with c = cos θ Mκ = π 0 cos θh κ (cos θ)e κ cos θ sin n θ dθ π 0 h κ (cos θ)e κ cos θ sin n θ dθ , We can now compute a simpler expression of λ. We differentiate the compatibility condition (3.13) with respect to κ, and we get (n -1) c κ -c 2 .

Figure 2 :

 2 Figure 2: Coefficient λ in dimensions 2, 3, and 4.

Figure 3 :

 3 Figure 3: The velocities c and c in dimension 2 (left) and 3 (right).

Figure 4 :

 4 Figure 4: Types of macroscopic limits in dimension 2. Around the threshold value ρ = 2, none of the diffusion or hydrodynamic limit is valid. The study of this transition is still open.

  κΩ by means of the spectral decomposition of each of the L * κ,m . It is indeed easy to prove, using Lax-Milgram theorem, that the operators L * κ,m have self-adjoint compact inverses for the dot product (f, g) = π 0 f g(sin θ) n-2 e κ cos θ dθ. Therefore the eigenfunctions and eigenvalues of L * κΩ correspond to those of the operators L * κ,m , for all m ∈ N. If we denote by λ κ,m the smallest eigenvalue of L * κ,m , we finally get Λ κ = min{λ κ,m , m ∈ N}.

Conjecture 6 . 1 .

 61 (i) When κ > 0 and n 3, only statement (ii) of Proposition 6.1 is true.

The authors wish to acknowledge the hospitality of Mathematical Sciences Center and Mathematics Department of Tsinghua University where this research was completed. The research of J.-G. L. was partially supported by NSF grant DMS 10-11738.

Appendix 1. Poincaré constant

In this appendix, we prove the following: Proposition 6.1. We have the following Poincaré inequality, for ψ ∈ H 1 (S):

The best constant Λ κ in this inequality is the smallest positive eigenvalue of the operator

We define the linear operator L * κ by L * κ (g)(θ) = -(sin θ) 2-n e -κ cos θ ((sin θ) n-2 e κ cos θ g ′ (θ)) ′ . (6.3)

Appendix 2. Numerical computations of the coefficients

We adopt a finite difference approach to compute the function g κ associated to the GCI's and defined by (5.6). We consider the function f κ such that f κ (θ) = (sin θ) n 2 -1 g κ (θ). In particular, since g κ ∈ V defined by (5.5), f κ belongs to H 1 0 (0, π). Since g κ satisfies (5.6), f κ satisfies

We discretize the interval (0, π) with N + 1 points θ i = 1 N iπ, and denote by f i κ an approximation of f κ at these points. Since

We define e i κ = e κ cos θ i . A second order approximation of (e κ cos θ f ′ κ ) ′ at θ i is then given by

Introducing

the vector F = (f i κ ) i∈ 1,N -1 is the solution of the linear system AF = S, where the vector S is (sin n 2 θ i ) i∈ 1,N -1 , and the tridiagonal matrix A is defined by

We use the trapezoidal method to perform the integrations in the definitions (3.11) and (5.9) of c and c. The other coefficients ρ, λ and θ c are then directly computed from c and c. The numerical results provided in Figures 345.2 have been obtained for N = 3000. We now detail how we obtain an approximation of the Poincaré constant Λ κ . By Appendix 1, Λ κ is the minimum between λ κ,1 and λ κ,0 , which are the smallest eigenvalue of two Sturm-Liouville problems. Several algorithms exist to compute eigenvalues of singular Sturm-Liouville problems (which is the case here whenever n 3) with a good precision [START_REF] Bailey | Computing eigenvalues of singular Sturm-Liouville problems[END_REF]. However, we use a simpler method based on finite differences.