Diène Ndiaye

Elisabeth Fanton

Sandrine Morlat-Therias

Loïc Vidal

Adams Tidjani

Jean-Luc Gardette
email: luc.gardette@univ-bpclermont.fr

Durability of wood polymer composites: 1. Influence of wood on the photochemical properties

Keywords: A polymer-matrix composites (PMCs), A wood, B Durability, D Infrared(IR) spectroscopy

The photooxidation of wood-based composites (WPC) using thermoplastics (PE, PP) as a continuous phase has been investigated. WPC are increasingly used in applications in which UV exposure has become a concern. The composites were produced by melt blending in a Brabender at 180°C. Characterization of the samples, with the aid of scanning electron microscopy supplemented by microscope photography, showed a good dispersion of wood in the polymeric material. Dispersion was improved by adding polypropylene grafted with maleic anhydride (PP-g-MA). The results clearly displayed that the presence of wood in polymer composites does not change the pathway of photooxidation of the host polymer.

However, wood absorbs in the UV-visible domain and can contribute to the degradation initiation through the creation of radicals. The outstanding outcome of this study is the significant improvement of the photostability of WPC made with PPgMA. This opens a door of better long term behaviour of this last formulation in the objective of outdoors applications.

Introduction

Composites made from blend of thermoplastic and natural fibres have attracted a number of researchers and manufacturers because of their great advantages [START_REF] Stark | Surface chemistry and mechanical property changes of wood-flour/High-density-polyethylene composites after accelerated weathering[END_REF]. Among them, one can quote a reduction of the production cost, a low density of the material, and good mechanical properties combining with renewability. So far, various thermoplastics, including polyethylene and polypropylene, have been used as host material for research and development, both on laboratory and industrial scales [START_REF] Sombatsompop | Influence of type and concentration of maleic anhydride grafted polypropylene and impact modifiers on mechanical properties of PP/wood sawdust composites[END_REF]. The enthusiasm in this field is explained by the number of potential applications. For example wood -polymer composites (WPC), mostly manufactured through extrusion and injection moulding processes are used in automotive (dashboards or screen-doors of the vehicles) and construction industry (interior floor coverings, profiled for doors and windows, ornamental panels, external shutters, pavements, garage or entrance doors, and so on). However, because they are constituted of organic materials, polymers and wood fibres, WPC are sensitive to UV light and thermal ageing. Then, in applications which demand routine exposure to solar radiations, WPC are likely to undergo degradation that limits their use.

The first key point for the production of acceptable WPC is the compatibility between wood and polymer host matrix. Wood is hydrophilic in nature (high surface tension), which lowers the compatibility with hydrophobic polymeric material (low surface tension) during composite preparation. Scanning the literature, one can find different surface treatments that have been experienced to improve wood/polymer adhesion in composites. Remind that the level of adhesion and/or the dispersion state of wood are the key points for the improvement of mechanical properties of the composites. Indeed, the wood particles which have high strength and modulus -with good adhesion and uniform dispersion-can impart better mechanical properties to the host polymer in order to obtain a composite with better properties than those of the unfilled polymer. Maleic anhydride is the usual choice as a 3 coupling agent [START_REF] Sombatsompop | Influence of type and concentration of maleic anhydride grafted polypropylene and impact modifiers on mechanical properties of PP/wood sawdust composites[END_REF]. Thus, the surface tension of wood fibres is reduced approaching that of the molten polymer. From this point, wetting and adhesion are improved via mechanisms such as diffusion and mechanical interlocking between the two entities.

WPC are produced starting from wood (up to 70 %), in the form of pellets, "flour" (fine particles) or fibres, mixed with a polymeric matrix: for instance polyethylene, polypropylene or polyvinylchloride. The unit is supplemented by many additives (antioxidants, pigments, fungicides, stabilizing anti-UV, etc...).

Wood is a composite material consisting mainly of three major polymers: cellulose, As mentioned above, WPC are very sensitive to UV-light degradation. It has been shown that lignins are keener to degradation among constituents of the wood [START_REF] Béarnais-Barbry | Université de Bordeaux 1: Etude de la photoréactivité de la méthoxy-p-quinone et de phenyl-p-benzoquinones en solution[END_REF][START_REF] Ruggiero | Photodegradation of sugar cane bagasse acidolysis lignins[END_REF][START_REF] George | Photodegradation and photostabilisation of wood -the state of the art[END_REF][START_REF] Mitsui | Low atmospheric temperature dependence on photodegradation of wood[END_REF][START_REF] Ayadi | Color stability of heat-treated wood during artificial weathering[END_REF][START_REF] Lundquist | Novel pulp fibre reinforced thermoplastic composites[END_REF]. The main degradation process of lignin identified by the authors is the demethoxylation of the guaiacyl or syringyl ring. Phenoxy radicals, which are the key intermediates, are formed through different pathways. Phenols present might be photo oxidized into quinines and muconic acids.

The explanation given is the abstraction of phenol hydrogen by excited α -carbonyl or singlet oxygen. Direct oxidation of phenols absorbing near UV light or homolysis of the β -O-4 bond in aryl glycerol-β -aryl ethers can then occur. Scission of chemical bonds can also take place leading to a production of radicals. This process is claimed to be responsible for yellowing of wood [START_REF] Mitsui | Low atmospheric temperature dependence on photodegradation of wood[END_REF].

The aim of this work is to develop photostable composite materials from polyolefin (polypropylene and polyethylene) and wood flour. A first series of specimen was obtained by blending wood flour and polymeric material. In a second series, polypropylene-grafted-maleic anhydride was added to the previous ingredients. The behaviour of these materials under UV light was studied. The evaluation of mechanical properties of the produced composites through several tests (thickness swelling, hardness and moisture content) was also carried out and will be reported in a forthcoming article.

Experimental

Materials

The wood flour particles of 425 microns (40-mesh) in size were kindly donated by American Wood fibbers (Schofield, WI) and are constituted predominantly with ponderosa pine, maple, oak, spruce, southern yellow pine, cedar. The wood was oven dried at 100°C for The isotactic polypropylene matrix (PP) has a density of 0.900 g/cm 3 and a melt flow index of 2.5g/10min. High density polyethylene (HDPE) used has a density 0.963g/cm 3 and a melt index of 0.72g/10min. Both polymers were provided by Solvay Co.

Polypropylene grafted with maleic anhydride (PP-g-MA) with an approximate maleic anhydride (MA) content of 3 wt% was purchased from Aldrich Chemical Company, Inc.

(Milwaukee, WI). All ingredients were used as received.

Compounding and processing

The composite materials were obtained by melt blending in a chamber of a Brabender mixer AEV 330 at 180°C for 10 min at 90 rpm. For the extraction of volatile and harmful gases, the hood was open. At first, polymer was introduced for 5 min at 170°C and the screw speed fixed at 40 rpm. In a second step the wood granulates and others compatibilizers were introduced. The system was mixed during 10 min at 180°C at 90 rpm. After the processing step, each sample was sandwiched during 1 min between two Teflon papers and pressed at 175°C under a pressure of 200 bars to obtain films with thicknesses between 100 to 200 microns. For each polymer matrix (PP, PE), four formulations were produced with various content of wood (table 1).

Table 1

Electron and optical microscopy

The state of dispersion of the wood inside the polymeric matrix was analyzed using optical microscopy on samples of 100 to 200 µm thick. Scanning Electron Microscopy (SEM) was used to obtain microphotographs of the fracture surfaces of the wood composites. These factures have been performed in liquid nitrogen to avoid any deformation. SEM has been performed using a FEI Quanta 400 microscope working at 30 kV.

The polymer surface was examined with LEICA optical microscope working in a transmission mode. Samples were thin enough that no special preparation of the samples was needed for their observations with the optical microscope.

UV-irradiation and chemical characterization

The UV absorption of each sample was measured using a UV-510 Perkin-Elmer spectrophotometer. Irradiation in conditions relevant to outdoor exposure (λ > 300 nm) in the presence of air was carried out in a SEPAP 12/24 unit working at a temperature of 60°C.

Details on this artificial accelerated photoageing device are given elsewhere [START_REF] Lemaire | Le vieillissement des polymères : Principes d'étude du photovieillissement[END_REF]. To follow the degradation of the UV-light irradiated specimen, the infrared spectra were recorded with a Nicolet 5SX FTIR-spectrometer, working with OMNIC software. Spectra were obtained using 32 scans and a 4 cm -1 resolution.

Experimental results

Characterization of the initial properties of the WPC samples

Morphology and dispersion of the composites

Figure 1 shows the SEM micrographs of the fracture surface of the WP composites made from PP, PE and PP with PPgMA.

Figure 1

Visual observation reveals a separation between particles and matrix in PP and PE composites probably because of the lack of interfacial adhesion. In composites in which a coupling agent is used, it looks smooth. This could result from the presence of PPgMA that could bind the two phases together due to the OH groups of the wood that react with maleic anhydride to form bonds [START_REF] Sombatsompop | Influence of type and concentration of maleic anhydride grafted polypropylene and impact modifiers on mechanical properties of PP/wood sawdust composites[END_REF] This provides sufficient adhesion between the components to resist cracks forming as the consequence of the Poisson effect. Note that in all cases, an absence of fibre pull-out, which indicates a weak interfacial adhesion [START_REF] Sombatsompop | Influence of type and concentration of maleic anhydride grafted polypropylene and impact modifiers on mechanical properties of PP/wood sawdust composites[END_REF][START_REF] Kandem | Properties of wood plastic composites made of recycled HDPE and wood flour from CCA-treated wood removed from service[END_REF][START_REF] Bengtsson | The effect of crosslinking on the properties of polyethylene/wood flour composites[END_REF]. The actual images of the wood dispersion acquired display a good dispersion of wood in the polymeric material whatever the host material (Fig 2). However, in PE and PP composites, the wood remains compact at opposite to PP/PPgMA in which the fibres are separated. This good dispersion could be attributed to the fact that some of the OH groups of wood react with maleic anhydride to form linkages and thereby improve the dispersion of wood in the composite [START_REF] Kandem | Properties of wood plastic composites made of recycled HDPE and wood flour from CCA-treated wood removed from service[END_REF].

UV-visible absorption

The WPC samples made with PP and different percentages (from 0 to 50%) of wood were characterized by their UV-visible spectrum before irradiation. Figure 3 shows the curves obtained for thin films (thickness around 100 microns) of WPC with 0%, 5%, 10%, 25%and 50% of wood (WPPC0 to 4). The UV-visible spectrum of a PP film shows the presence of two maxima at 230 nm and 280 nm which correspond to the processing phenolic antioxidant remaining in the processed polymer. In presence of wood, the UV-visible spectra of WPC reveal an increase of absorbance towards the long wavelengths in the visible part of the spectrum. The absorbance increases with the amount of wood, but the presence of the processing antioxidant is still observable even for the highest contents in wood.

Figure 3

Very similar observations can be made on the basis of the analysis of the UV-visible absorption spectra of the PE/wood composites.

Influence of UV-irradiation on the chemical structure of PP and PE wood polymer composites PP/wood composites (WPPC)

Infrared analysis

UV light irradiation of PP in air is known to produce dramatic modifications of the infrared spectra of the exposed samples. The modifications of the IR spectra of all the WPC films upon irradiation in the presence of oxygen are comparable to that of pristine polypropylene, and are mainly characterized by an increase of absorbance in the hydroxyl and carbonyl regions, as observed on figure 4, which shows the IR spectrum of the sample with 5 % of wood.

Figure 4

The photooxidation of polypropylene results in the formation of hydroxyl (mainly hydroperoxides and alcohols) and carbonyl groups (mainly ketones, esters and acids) easily detectable by infrared spectroscopy respectively in the 3200-3600 cm -1 and 1900-1500 cm -1 domains. The chemical structure of the main oxidation products can be considered as well established and the mechanisms by which oxidation occur is now fairly well understood [START_REF] Phillipart | Influence of light intensity on the photooxidation of polypropylene[END_REF].

In the carbonyl domain the photochemical oxidation of PP leads to the formation of a broad band with several maxima, peaking at 1713, 1735 and 1780 cm -1 . The photoproducts corresponding to these absorption maxima have been identified formerly [START_REF] Phillipart | Influence of the exposure parameters on the mechanism of photooxidation of polypropylene[END_REF][START_REF] Lacoste | Gamma-, photo-and thermally-initiated oxidation of isotactic polypropylene[END_REF], and it is agreed that the main absorption at 1713 cm -1 is that of the carboxylic acids in the dimer form.

The formation of carboxylic acids occurs with chain scissions, which are partly responsible for the formation of nano/microcracks at the surface of the polymer and lead to the degradation of the mechanical properties of the material. The chemical modifications of the polymer measured by the concentration of the oxidation photoproducts can be quantitatively related to the mechanical degradation of the material [START_REF] Fréchou | Influence des conditions d'injection sur le comportement photochimique de polypropylenes[END_REF].

Rates of photooxidation

Figure 5 shows the variation of the ratio 1713 cm -1 / 2722 cm -1 as a function of irradiation time respectively for PP and PP/wood films with different percentages of wood flour (WPPC 0 to 4). Because the content of polymer was different for the various samples as a result of the different contents of wood flour, the kinetic curves of oxidation were plotted as the ratio of the absorbance at 1713 cm -1 versus the absorbance at 2722 cm -1 ( (C-H) stretching vibration band characteristic of PP). Fig. 5 shows that in the case of pristine PP (WPPC0), no increase of absorbance is observed during the first 20 hours of exposure. This is also the case in the domain of hydroxyl not shown here. Oxidation can be detected only after this first period. This induction period results from the disappearance of the processing antioxidant, which is consumed as a sacrificial additive [START_REF] Morlat | Photooxidation of Polypropylene/Montmorillonite nanocomposites Part I : Influence of nanoclay and compatibilising agent[END_REF].

At opposite, it can be observed on the infrared spectra of all the samples containing wood (WPPC1 to 4) that the absorbance in the carbonyl domain increases within the first 20 hours of irradiation. A careful examination of the hydroxyl domain shows that no increase of absorbance occurs in this last domain for the same irradiation time (see for example Fig. 4 relative to WPPC2). This observation suggests that oxidation is not coming from the PP matrix. A second important point observed in Figure 5 is that, within these first 20 hours, the carbonyl absorbance in the case of PP/wood composites increases with the amount of wood in the samples. These observations indicate that the initial increase of the carbonyl absorbance is related to the oxidation of wood and not that of PP.

After the first period, oxidation of the polymer starts with a rate which is more important for the samples with wood than for the PP one. The rate is slightly increasing with the amount of wood in the composite but a plateau is reached for concentrations above 25% of wood. These results indicate that the presence of wood considerably shortens the durability of the composite.

UV-visible analysis

Figure 6 shows the modifications of the UV-visible spectrum of WPPC4 (PP/ (50 % wood) during photooxidation. This figure shows that the absorbance in the visible domain increases with irradiation time. No modifications of the absorption spectrum of pristine PP are expected in this domain of the visible absorption (above 300 nm) [START_REF] Morlat | Photooxidation of Polypropylene/Montmorillonite nanocomposites Part I : Influence of nanoclay and compatibilising agent[END_REF], even for longer irradiation times. This result suggests that the modifications of the UV-visible absorption can be unambiguously attributed to a photochemical transformation of wood in the polypropylene matrix. The reaction involved in this process has been largely reported in the literature [START_REF] Béarnais-Barbry | Université de Bordeaux 1: Etude de la photoréactivité de la méthoxy-p-quinone et de phenyl-p-benzoquinones en solution[END_REF][START_REF] Ruggiero | Photodegradation of sugar cane bagasse acidolysis lignins[END_REF][START_REF] Mitsui | Low atmospheric temperature dependence on photodegradation of wood[END_REF]. Considering the chemical composition of wood, lignin is the key structure in wood photodegradation as only lignin absorbs in the UV-visible spectral domain [START_REF] George | Photodegradation and photostabilisation of wood -the state of the art[END_REF][START_REF] Mitsui | Low atmospheric temperature dependence on photodegradation of wood[END_REF][START_REF] Ayadi | Color stability of heat-treated wood during artificial weathering[END_REF]. The mechanism of lignin photodegradation is complex, involving different pathways giving the formation of radical species. These free radicals react with oxygen to produce chromophoric groups as carbonyl and carboxyl groups, which also lead to chain cleavage and yellowing.

One of the main radicals is a long life guaiacoxy radical, which undergoes transformation into quinoid structures which are the origin of the yellowing of wood [START_REF] George | Photodegradation and photostabilisation of wood -the state of the art[END_REF][START_REF] Mitsui | Low atmospheric temperature dependence on photodegradation of wood[END_REF][START_REF] Ayadi | Color stability of heat-treated wood during artificial weathering[END_REF].

HDPE/wood composites

Rate of oxidation

Photooxidation of WPC films with HDPE as polymer matrix (samples WPEC 0 to 4) was followed by IR analysis. The same conclusions as for WPC with PP (samples WPPC 0 to 4) can be drawn concerning the reaction stoechiometry. The photooxidation of pure HDPE gives an induction time of 30 hours, and in presence of wood the oxidation of the polymer starts earlier (Fig. 7). Moreover, an increase of the rate of oxidation is observed by increasing the percentage of wood. These observations indicate that the presence of wood participates in the initiation of degradation but does not change the reaction stoechiometry as seen previously.

Figure 7 WPC made with PP/PPgMA/wood: Beside the study of composites of wood and PP or HDPE, the photooxidation of composites with wood, PP and a compatibilizing agent (PPgMA) was examined in order to characterize the influence of a compatibilizing agent (PPgMA) on the photodegradation of WPC. Figure 8 shows the modifications of the infrared spectrum of a sample made with 25% wood resulting from UV light irradiation in the presence of air (WPPG 1). The initial spectrum of a WPPG 1 film shows a broad complex absorption band in the carbonyl domain.

This band results of the presence of both wood (25%) and maleic anhydride (3%). During the UV irradiation, absorption bands develop in the carbonyl domain (Fig. 8). Previous studies of the photooxidation of PPgMA samples have shown that no specific photoproducts of maleic anhydride degradation were detected [START_REF] Morlat | Photooxidation of Polypropylene/Montmorillonite nanocomposites Part I : Influence of nanoclay and compatibilising agent[END_REF]. The modifications observed in Fig. 8 indicate then the formation of oxidation photoproducts that come only from wood and PP, and not from MA. As reported above, in the first 20 hours, the modifications reveal a rapid oxidation of wood. The same behaviour is observed in the case of the other samples not presented here (WPPG 2 to 4).

Figure 8

On the basis of the infrared spectra of the various samples, the rates of photooxidation of WPC containing increasing amount of PPgMA (WPPG1 to 4) can be compared. Figure 9 represents the kinetic curves of oxidation for these samples and compares them with the curve obtained in the case of the sample of PP/wood (25%) (WPPC3) extracted from fig. 5. Two sets of curves are given, one for the carbonyl domain (fig. 9a) and another one for the hydroxyl domain (fig. 9b). In both cases one can make the same remarks. At first it is clearly observed that the presence of MA as a compatibilizer has for effect to decrease the oxidation rate of PP in the WPC. All the curves presented in figure 9 are below the curve corresponding to the sample that has no compatibilizer. Moreover it can be clearly observed that increasing the amount of PPgMA increases the stability of the polymer. After 80 hours of irradiation, no oxidation can be observed in the case of the samples with PPgMA (WPPG 3 and 4), whereas in the case of WPPC 3, oxidation started after only 20 hours under irradiation. This stabilizing effect resulting from the presence of maleic anhydride has to be considered as rather important in terms of applications of WPC.

Conclusion

The wood polymer composites produced using PE and PP as the host polymer showed interesting features. The characterization of the samples with the help of electron microscopy supplemented with microscope photographs revealed an acceptable level of dispersion of the wood particle in the polymeric material. Addition of PPgMA in polypropylene composites appreciably improved this dispersion, which might be explained by the formation of linkages between the OH groups of wood and maleic anhydride.

The following of the photooxidation behaviour of the samples highlights that the presence of wood in polymer composites has no influence on the mechanism of photooxidation of the polymer (PP or PE). This means that the same photoproducts are formed as well in WPC and as the pristine polymer, in the same relative concentrations.

However, wood absorbs in the UV-visible region, as observed on the spectra of the WPC samples that present a higher absorbance in the UV domain (300-400 nm) than the polymer samples. Wood is not inert under UV-visible light and specific photoproducts coming from the oxidation of wood are detected in the first 20 hours of irradiation.

Under irradiation, lignins that are considered in the literature as the key structures involved in the degradation of wood generate phenoxy radicals, and phenols might be oxidized [START_REF] Béarnais-Barbry | Université de Bordeaux 1: Etude de la photoréactivité de la méthoxy-p-quinone et de phenyl-p-benzoquinones en solution[END_REF][START_REF] Ruggiero | Photodegradation of sugar cane bagasse acidolysis lignins[END_REF][START_REF] George | Photodegradation and photostabilisation of wood -the state of the art[END_REF][START_REF] Colom | Structural analysis of photodegraded wood by means of FTIR spectroscopy[END_REF]. The influence of wood in composites expresses itself in the durability of the WPC by the modification of the kinetics of photooxidation. The results show that wood has no marked effect on the oxidation induction time of the polymer, which corresponds to the disappearance of the residual processing antioxidant [START_REF] Morlat-Thérias | Photooxidation of Polypropylene/Montmorillonite nanocomposites Part II : Interactions with antioxidants[END_REF][START_REF] Marcovich | Thermal and mechanical characterization of linear lowdensity polyethylene/wood flour composites[END_REF]. However one can clearly observe that the presence of wood results in an increase of the oxidation rate of the polymeric matrix, after the induction period. This effect is more and more accentuated with increasing amounts of wood within the WPC, up to 25%. This can be explained by radicalar species produced by photooxidation of wood that are likely to initiate the oxidation of the polymer once the antioxidant has been consumed within the induction period. Wood acts then as an initiator on the photooxidation of the polymer matrix, PP or PE, in the WPC material. However, by increasing the percentage of wood, the amount of polymer in the composite is lower, and the global effect on the durability of the WPC material might be positive if one considers the level of degradation of the composite and of the pure polymer after the same ageing time.

In PP/PPgMA/wood composites the fibres are separated. This better dispersion has been attributed to the possible reaction of some of the OH groups of wood with maleic anhydride to form linkages and thereby improve the dispersion of wood in the composite.

Interestingly this results in an improvement of the photostability of WPC composites made with PPgMA and a better long-term behaviour in the objective of outdoors applications.

The next step of this study will be the analysis of the evolution of the used properties of the WPC material, which have only been characterized before ageing.

Figure captions

  Scheme1: Chemical structure of cellulose

Figure 2

 2 Figure 2

Figure 6

 6 Figure 6

Figure 9

 9 Figure 9

Figure 1 :

 1 Figure 1: SEM micrographs of WP composites with PE (left) PP (middle) and PP /PPgMA

Figure 2 :

 2 Figure 2: Photographs of the composite surfaces: PE/wood (left), PP/wood (middle) and

Figure 3 :

 3 Figure 3: UV-Visible spectra of film of WPC made with PP and wood: the curves go from

Figure 4 :

 4 Figure 4: IR spectra of a WPPC2 film (PP/wood (5%)) during photooxidation.

Figure 5 :

 5 Figure 5: Variations of the normalized carbonyl absorbance (∆(OD(1713 cm -1 )/OD(2722 cm -1 )) as a function of irradiation time for films of PP/wood during photooxidation at λ > 300nm, 60°C (WPPC0 to 4).

Figure 6 :

 6 Figure 6: UV-visible spectra of a WPPC4 film (e~100 µm) during photooxidation (λ > 300

Figure 7 :

 7 Figure 7: Variations of normalized absorbance (∆(OD (1712 cm -1 )/OD(720 cm -1 )) as a

Figure 8 :

 8 Figure8: Modifications of the infrared spectra of PP/PPgMA/wood (WPPG 1) under photooxidation at λ > 300nm, 60°C (differences of spectra in the carbonyl domain 10h-0h, 20h-0h,…80h-0h).

Figure 9 :Figure 1 :

 91 Figure 9: Variations of a) normalized carbonyl absorbance (∆(OD(1718 cm -1 )/OD(2722 cm -1 )) and b) normalized hydroxyl absorbance

Figure 2 :Figure 3 :Figure 4 :Figure 5 :

 2345 Figure 2: Photographs of the composite surfaces: PE/wood (left), PP/wood (middle) and PP/PPgMA/wood (right).

Figure 6 :Figure 7 :

 67 Figure 6: UV-visible spectra of a WPPC4 film (e~100 µm) during photooxidation (λ > 300 nm, 60°C)(the curves go from bottom to the top according to the irradiation time 0h, 30h, 60h and 80h)

Figure 8 :

 8 Figure 8: Modifications of the infrared spectra of PP/PPgMA/wood (WPPG 1) under photooxidation at λ > 300nm, 60°C (differences of spectra in the carbonyl domain 10h-0h, 20h-0h,…80h-0h)

Figure 9 :

 9 Figure 9: Variations of a) normalized carbonyl absorbance (∆(OD (1718 cm -1 )/OD(2722cm - 1 )) and b) normalized hydroxyl absorbance(∆(OD (3400 cm -1 )/OD(2722 cm -1 )) as a function of irradiation time for films of PP/PPgMA/wood (25% of wood) during photooxidation at λ > 300nm, 60°C ( WPPG1 to 4)and film of PP/wood (WPPC3).


Olivier Finkel
email: finkel@logique.jussieu.fr

Some Problems in Automata Theory Which Depend on the Models of Set Theory

Keywords: Automata and formal languages, logic in computer science, computational complexity, infinite words, ω-languages, 1-counter automaton, 2-tape automaton, cardinality problems, decision problems, analytical

We prove that some fairly basic questions on automata reading infinite words depend on the models of the axiomatic system ZFC. It is known that there are only three possibilities for the cardinality of the complement of an ω-language L(A) accepted by a Büchi 1-counter automaton A. We prove the following surprising result: there exists a 1-counter Büchi automaton A such that the cardinality of the complement L(A) -of the ω-language L(A) is not determined by ZFC:

(1). There is a model

. We prove a very similar result for the complement of an infinitary rational relation accepted by a 2-tape Büchi automaton B. As a corollary, this proves that the Continuum Hypothesis may be not satisfied for complements of 1-counter ω-languages and for complements of infinitary rational relations accepted by 2-tape Büchi automata. We infer from the proof of the above results that basic decision problems about 1-counter ωlanguages or infinitary rational relations are actually located at the third level of the analytical hierarchy. In particular, the problem to determine whether the complement of a 1-counter ωlanguage (respectively, infinitary rational relation

. This is rather surprising if compared to the fact that it is decidable whether an infinitary rational relation is countable (respectively, uncountable).

Introduction

In Computer Science one usually considers either finite computations or infinite ones. The infinite computations have length ω, which is the first infinite ordinal. The theory of automata reading infinite words, which is closely related to infinite games, is now a rich theory which is used for the specification and verification of non-terminating systems, see [START_REF] Grädel | Automata, Logics, and Infinite Games: A Guide to Current Research [outcome of a Dagstuhl seminar[END_REF][START_REF] Perrin | Infinite words, automata, semigroups, logic and games[END_REF].

Connections between Automata Theory and Set Theory have arosen in the study of monadic theories of well orders. For example, Gurevich, Magidor and Shelah proved in [START_REF] Gurevich | The monadic theory of ω 2[END_REF] that the monadic theory of ω 2 , where ω 2 is the second uncountable cardinal, may have different complexities depending on the actual model of ZFC (the commonly accepted axiomatic framework for Set Theory in which all usual mathematics can be developped), and the monadic theory of ω 2 is in turn closely related to the emptiness problem for automata reading transfinite words of length ω 2 . Another example is given by [START_REF] Neeman | Finite state automata and monadic definability of singular cardinals[END_REF], in which Neeman considered some automata reading much longer transfinite words to study the monadic theory of some larger uncountable cardinal.

However, the cardinal ω 2 is very large with respect to ω, and therefore the connections between Automata Theory and Set Theory seemed very far from the practical aspects of Computer Science. Indeed one usually thinks that the finite or infinite computations appearing in Computer Science are "well defined" in the axiomatic framework of mathematics, and thus one could be tempted to consider that a property on automata is either true or false and that one has not to take care of the different models of Set Theory (except perhaps for the Continuum Hypothesis CH which is known to be independent from ZFC).

In [START_REF] Finkel | The complexity of infinite computations in models of set theory[END_REF] we have recently proved a surprising result: the topological complexity of an ωlanguage accepted by a 1-counter Büchi automaton, or of an infinitary rational relation accepted by a 2-tape Büchi automaton, is not determined by the axiomatic system ZFC. In particular, there is a 1-counter Büchi automaton A (respectively, a 2-tape Büchi automaton B) and two models V 1 and V 2 of ZFC such that the ω-language L(A) (respectively, the infinitary rational relation

L(B)) is Borel in V 1 but not in V 2 .
We prove in this paper other surprising results, showing that some basic questions on automata reading infinite words actually depend on the models of ZFC. In particular, we prove the following result: there exists a 1-counter Büchi automaton A such that the cardinality of the complement L(A) -of the ω-language L(A) is not determined by ZFC. Indeed it holds that:

(1). There is a model

V 1 of ZFC in which L(A) -is countable. (2). There is a model V 2 of ZFC in which L(A) -has cardinal 2 ℵ 0 . (3). There is a model V 3 of ZFC in which L(A) -has cardinal ℵ 1 with ℵ 0 < ℵ 1 < 2 ℵ 0 .
Notice that there are only these three possibilities for the cardinality of the complement of an ω-language accepted by a Büchi 1-counter automaton A because the ω-language L(A) is an analytic set and thus L(A) -is a coanalytic set, see [START_REF] Jech | Set theory[END_REF]page 488].

We prove a very similar result for the complement of an infinitary rational relation accepted by a 2-tape Büchi automaton B. As a corollary, this proves that the Continuum Hypothesis may be not satisfied for complements of 1-counter ω-languages and for complements of infinitary rational relations accepted by 2-tape Büchi automata.

In the proof of these results, we consider the largest thin (i.e., without perfect subset) effective coanalytic subset of the Cantor space 2 ω , whose existence was proven by Kechris in [START_REF] Kechris | The theory of countable analytical sets[END_REF] and independently by Guaspari and Sacks. An important property of C 1 is that its cardinal depends on the models of set theory. We use this fact and some constructions from recent papers [START_REF] Finkel | Borel ranks and Wadge degrees of omega context free languages[END_REF][START_REF] Finkel | On the accepting power of two-tape Büchi automata[END_REF] to infer our new results about 1-counter or 2-tape Büchi automata.

Combining the proof of the above results with Shoenfield's Absoluteness Theorem we get that basic decision problems about 1-counter ω-languages or infinitary rational relations are actually located at the third level of the analytical hierarchy. In particular, the problem to determine whether the complement of a 1-counter ω-language (respectively, infinitary rational relation

) is countable is in Σ 1 3 \ (Π 1 2 ∪ Σ 1 2 )
. This is rather surprising if compared to the fact that it is decidable whether an infinitary rational relation is countable (respectively, uncountable). As a by-product of these results we get a (partial) answer to a question of Castro and Cucker about ω-languages of Turing machines.

The paper is organized as follows. We recall the notion of counter automata in Section 2. We expose some results of Set Theory in Section 3, and we prove our main results in Section 4. Concluding remarks are given in Section 5.

Notice that the reader who is not familiar with the notion of ordinal in set theory may skip part of Section 3 and just read Theorems 3.3 and 3.5 in this section. The rest of the paper relies mainly on the set-theoretical results stated in Theorem 3.5.

Counter Automata

We assume the reader to be familiar with the theory of formal (ω-)languages [START_REF] Thomas | Automata on infinite objects[END_REF][START_REF] Staiger | ω-languages[END_REF]. We recall the usual notations of formal language theory.

If Σ is a finite alphabet, a non-empty finite word over Σ is any sequence x = a 1 . . . a k , where a i ∈ Σ for i = 1, . . . , k , and k is an integer ≥ 1. The length of x is k, denoted by |x|. The empty word has no letter and is denoted by λ; its length is 0. Σ ⋆ is the set of finite words (including the empty word) over Σ.

The first infinite ordinal is ω. An ω-word over Σ is an ω -sequence a 1 . . . a n . . ., where for all integers i ≥ 1, a i ∈ Σ. When σ = a 1 . . . a n . . . is an ω-word over Σ, we write σ

(n) = a n , σ[n] = σ(1)σ(2) . . . σ(n) for all n ≥ 1 and σ[0] = λ.
The usual concatenation product of two finite words u and v is denoted u.v (and sometimes just uv). This product is extended to the product of a finite word u and an ω-word v: the infinite word u.v is then the ω-word such that:

(u.v)(k) = u(k) if k ≤ |u| , and (u.v)(k) = v(k -|u|) if k > |u|.
The set of ω-words over the alphabet Σ is denoted by Σ ω . An ω-language V over an alphabet Σ is a subset of Σ ω , and its complement (in

Σ ω ) is Σ ω -V , denoted V -.
We now recall the definition of k-counter Büchi automata which will be useful in the sequel. Let k be an integer ≥ 1. A k-counter machine has k counters, each of which containing a non-negative integer. The machine can test whether the content of a given counter is zero or not. And transitions depend on the letter read by the machine, the current state of the finite control, and the tests about the values of the counters. Notice that in this model some λ-transitions are allowed. During these transitions the reading head of the machine does not move to the right, i.e. the machine does not read any more letter.

Formally a k-counter machine is a 4-tuple M=(K, Σ, ∆, q 0 ), where K is a finite set of states, Σ is a finite input alphabet, q 0 ∈ K is the initial state, and

∆ ⊆ K × (Σ ∪ {λ}) × {0, 1} k × K × {0, 1, -1} k is the transition relation. The k-counter machine M is said to be real time iff: ∆ ⊆ K × Σ × {0, 1} k × K × {0, 1, -1} k , i.e.
iff there are no λ-transitions.

If the machine M is in state q and c i ∈ N is the content of the i th counter C i then the configuration (or global state) of M is the (k + 1)-tuple (q, c 1 , . . . , c k ).

For a ∈ Σ ∪ {λ}, q, q ′ ∈ K and (c 1 , . . . , c k ) ∈ N k such that c j = 0 for j ∈ E ⊆ {1, . . . , k} and c j > 0 for j / ∈ E, if (q, a, i 1 , . . . , i k , q ′ , j 1 , . . . , j k ) ∈ ∆ where i j = 0 for j ∈ E and i j = 1 for j / ∈ E, then we write: a : (q, c 1 , . . . , c k ) → M (q ′ , c 1 + j 1 , . . . , c k + j k ). Thus the transition relation must obviously satisfy: if (q, a, i 1 , . . . , i k , q ′ , j 1 , . . . , j k ) ∈ ∆ and i m = 0 for some m ∈ {1, . . . , k} then j m = 0 or j m = 1 (but j m may not be equal to -1).

Let σ = a 1 a 2 . . . a n . . . be an ω-word over Σ. An ω-sequence of configurations r = (q i , c

i 1 , . . . c i k ) i≥1 is called a run of M on σ, starting in configuration (p, c 1 , . . . , c k ), iff: (1) (q 1 , c 1 1 , . . . c 1 k ) = (p, c 1 , . . . , c k ) (2) for each i ≥ 1, there exists b i ∈ Σ∪{λ} such that b i : (q i , c i 1 , . . . c i k ) → M (q i+1 , c i+1 1 , . . . c i+1 k ) and such that either a 1 a 2 . . . a n . . . = b 1 b 2 . . . b n . . . or b 1 b 2 . . . b n . . . is a finite word, prefix (i.e. initial segment) of a 1 a 2 . . . a n . . .
The run r is said to be complete when a 1 a 2 . . . a n . . . = b 1 b 2 . . . b n . . . For every such run r, In(r) is the set of all states entered infinitely often during r.

A complete run r of M on σ, starting in configuration (q 0 , 0, . . . , 0), will be simply called "a run of M on σ". Definition 2.1 A Büchi k-counter automaton is a 5-tuple M=(K, Σ, ∆, q 0 , F ), where M ′ =(K, Σ, ∆, q 0 ) is a k-counter machine and F ⊆ K is the set of accepting states. The ω-language accepted by M is:

L(M)= {σ ∈ Σ ω | there exists a run r of M on σ such that In(r) ∩ F = ∅}
The class of ω-languages accepted by Büchi k-counter automata is denoted BCL(k) ω . The class of ω-languages accepted by real time Büchi k-counter automata will be denoted r-BCL(k) ω . The class BCL(1) ω is a strict subclass of the class CFL ω of context free ω-languages accepted by Büchi pushdown automata.

We recall now the definition of classes of the arithmetical hierarchy of ω-languages, see [START_REF] Staiger | ω-languages[END_REF]. Let X be a finite alphabet. An ω-language L ⊆ X ω belongs to the class Σ n if and only if there exists a recursive relation

R L ⊆ (N) n-1 × X ⋆ such that: L = {σ ∈ X ω | ∃a 1 . . . Q n a n (a 1 , . . . , a n-1 , σ[a n + 1]) ∈ R L },
where Q i is one of the quantifiers ∀ or ∃ (not necessarily in an alternating order). An ω-language L ⊆ X ω belongs to the class Π n if and only if its complement X ω -L belongs to the class Σ n . The class Σ 1 1 is the class of effective analytic sets which are obtained by projection of arithmetical sets. An ω-language L ⊆ X ω belongs to the class Σ 1 1 if and only if there exists a recursive relation

R L ⊆ N × {0, 1} ⋆ × X ⋆ such that: L = {σ ∈ X ω | ∃τ (τ ∈ {0, 1} ω ∧ ∀n∃m((n, τ [m], σ[m]) ∈ R L ))}.
Then an ω-language L ⊆ X ω is in the class Σ 1 1 iff it is the projection of an ω-language over the alphabet X × {0, 1} which is in the class Π 2 . The class Π 1 1 of effective co-analytic sets is simply the class of complements of effective analytic sets.

Recall that a Büchi Turing machine is just a Turing machine working on infinite inputs with a Büchi-like acceptance condition, and that the class of ω-languages accepted by Büchi Turing machines is the class Σ 1 1 of effective analytic sets [START_REF] Cohen | ω-computations on Turing machines[END_REF][START_REF] Staiger | ω-languages[END_REF]. On the oher hand, one can construct, using a classical construction (see for instance [START_REF] Hopcroft | Introduction to automata theory, languages, and computation[END_REF]), from a Büchi Turing machine T , a 2-counter Büchi automaton A accepting the same ω-language. Thus one can state the following proposition.

Proposition 2.2 An ω-language L ⊆ X ω is in the class Σ 1 1 iff it is accepted by a non deterministic Büchi Turing machine, hence iff it is in the class BCL(2) ω .

Some Results of Set Theory

We recall that the reader who is not familiar with the notion of ordinal in set theory may skip part of this section: the main results in this section, which will be used later in this paper, are stated in Theorems 3.3 and 3.5.

We now recall some basic notions of set theory which will be useful in the sequel, and which are exposed in any textbook on set theory, like [START_REF] Jech | Set theory[END_REF]. The usual axiomatic system ZFC is Zermelo-Fraenkel system ZF plus the axiom of choice AC. The axioms of ZFC express some natural facts that we consider to hold in the universe of sets. For instance a natural fact is that two sets x and y are equal iff they have the same elements. This is expressed by the Axiom of Extensionality:

∀x∀y [ x = y ↔ ∀z(z ∈ x ↔ z ∈ y) ].
Another natural axiom is the Pairing Axiom which states that for all sets x and y there exists a set z = {x, y} whose elements are x and y:

∀x∀y [ ∃z(∀w(w ∈ z ↔ (w = x ∨ w = y)))]
Similarly the Powerset Axiom states the existence of the set of subsets of a set x. Notice that these axioms are first-order sentences in the usual logical language of set theory whose only non logical symbol is the membership binary relation symbol ∈. We refer the reader to any textbook on set theory for an exposition of the other axioms of ZFC.

A model (V, ∈) of an arbitrary set of axioms A is a collection V of sets, equipped with the membership relation ∈, where "x ∈ y" means that the set x is an element of the set y, which satisfies the axioms of A. We often say " the model V" instead of " the model (V, ∈)".

We say that two sets A and B have same cardinality iff there is a bijection from A onto B and we denote this by A ≈ B. The relation ≈ is an equivalence relation. Using the axiom of choice AC, one can prove that any set A can be well-ordered so there is an ordinal γ such that A ≈ γ. In set theory the cardinal of the set A is then formally defined as the smallest such ordinal γ.

The infinite cardinals are usually denoted by ℵ 0 , ℵ 1 , ℵ 2 , . . . , ℵ α , . . . The cardinal ℵ α is also denoted by ω α , when it is considered as an ordinal. The first infinite ordinal is ω and it is the smallest ordinal which is countably infinite so ℵ 0 = ω (which could be written ω 0 ). There are many larger countable ordinals, such as ω 2 , ω 3 , . . . , ω ω , . . . ω ω ω , . . . The first uncountable ordinal is ω 1 , and formally ℵ 1 = ω 1 . In the same way ω 2 is the first ordinal of cardinality greater than ℵ 1 , and so on.

The continuum hypothesis CH says that the first uncountable cardinal ℵ 1 is equal to 2 ℵ 0 which is the cardinal of the continuum. Gödel and Cohen have proved that the continuum hypothesis CH is independent from the axiomatic system ZFC, i.e., that there are models of ZFC + CH and also models of ZFC + ¬ CH, where ¬ CH denotes the negation of the continuum hypothesis, [START_REF] Jech | Set theory[END_REF].

Let ON be the class of all ordinals. Recall that an ordinal α is said to be a successor ordinal iff there exists an ordinal β such that α = β + 1; otherwise the ordinal α is said to be a limit ordinal and in this case α = sup{β ∈ ON | β < α}.

The class L of constructible sets in a model V of ZF is defined by L = α∈ON L(α), where the sets L(α) are constructed by induction as follows:

(1). L(0) = ∅ (2). L(α) = β<α L(β), for α a limit ordinal, and

(3). L(α+1) is the set of subsets of L(α) which are definable from a finite number of elements of L(α) by a first-order formula relativized to L(α).

If V is a model of ZF and L is the class of constructible sets of V, then the class L is a model of ZFC + CH. Notice that the axiom (V=L), which means "every set is constructible", is consistent with ZFC because L is a model of ZFC + V=L.

Consider now a model V of ZFC and the class of its constructible sets L ⊆ V which is another model of ZFC. It is known that the ordinals of L are also the ordinals of V, but the cardinals in V may be different from the cardinals in L.

In particular, the first uncountable cardinal in L is denoted ℵ L 1 , and it is in fact an ordinal of V which is denoted ω L 1 . It is well-known that in general this ordinal satisfies the inequality ω L 1 ≤ ω 1 . In a model V of the axiomatic system ZFC + V=L the equality ω L 1 = ω 1 holds, but in some other models of ZFC the inequality may be strict and then ω L 1 < ω 1 : notice that in this case ω L 1 < ω 1 holds because there is actually a bijection from ω onto ω L 1 in V (so ω L 1 is countable in V) but no such bijection exists in the inner model L (so ω L 1 is uncountable in L). The construction of such a model is presented in [Jec02, page 202]: one can start from a model V of ZFC + V=L and construct by forcing a generic extension V[G] in which ω V 1 is collapsed to ω; in this extension the inequality ω L 1 < ω 1 holds.

We assume the reader to be familiar with basic notions of topology which may be found in [Mos80, LT94, Sta97, PP04]. There is a natural metric on the set Σ ω of infinite words over a finite alphabet Σ containing at least two letters which is called the prefix metric and is defined as follows. For u, v ∈ Σ ω and u = v let δ(u, v) = 2 -l pref(u,v) where l pref(u,v) is the first integer n such that the (n + 1) st letter of u is different from the (n + 1) st letter of v. This metric induces on Σ ω the usual Cantor topology in which the open subsets of Σ ω are of the form

W.Σ ω , for W ⊆ Σ ⋆ . A set L ⊆ Σ ω is a closed set iff its complement Σ ω -L is an open set. Definition 3.1 Let P ⊆ Σ ω ,
where Σ is a finite alphabet having at least two letters. The set P is said to be a perfect subset of Σ ω if and only if :

(1) P is a non-empty closed set, and (2) for every x ∈ P and every open set U containing x there is an element y ∈ P ∩ U such that x = y. So a perfect subset of Σ ω is a non-empty closed set which has no isolated points. It is well known that a perfect subset of Σ ω has cardinality 2 ℵ 0 , i.e. the cardinality of the continuum, see [Mos80, page 66]. Definition 3.2 A set X ⊆ Σ ω is said to be thin iff it contains no perfect subset.

The following result was proved by Kechris [START_REF] Kechris | The theory of countable analytical sets[END_REF] and independently by Guaspari and Sacks.

Theorem 3.3 (see [Mos80] page 247) (ZFC) Let Σ be a finite alphabet having at least two letters. There exists a thin

Π 1 1 -set C 1 (Σ ω ) ⊆ Σ ω which contains every thin, Π 1 1 -subset of Σ ω . It is called the largest thin Π 1 1 -set in Σ ω .
An important fact is that the cardinality of the largest thin Π 1 1 -set in Σ ω depends on the model of ZFC. The following result was proved by Kechris, and independently by Guaspari and Sacks, see [Kan97, page 171].

Theorem 3.4 (ZFC) The cardinal of the largest thin

Π 1 1 -set in Σ ω is equal to the cardinal of ω L 1 .
This means that in a given model V of ZFC the cardinal of the largest thin Π 1 1 -set in Σ ω is equal to the cardinal in V of ω L 1 , the ordinal which plays the role of the cardinal ℵ 1 in the inner model L of constructible sets of V.

We can now state the following theorem which will be useful in the sequel. It follows from Theorem 3.4 and from some constructions of models of set theory due to Cohen (for (a)), Levy (for (b)) and Cohen (for (c)), see [START_REF] Jech | Set theory[END_REF].

Theorem 3.5 (a) There is a model

V 1 of ZFC in which the largest thin Π 1 1 -set in Σ ω has cardinal ℵ 1 with ℵ 1 = 2 ℵ 0 . (b) There is a model V 2 of ZFC in which the largest thin Π 1 1 -set in Σ ω has cardinal ℵ 0 , i.e.

is countable.

(c) There is a model V 3 of ZFC in which the largest thin

Π 1 1 -set in Σ ω has cardinal ℵ 1 with ℵ 0 < ℵ 1 < 2 ℵ 0 .
In particular, all models of (ZFC + V=L) satisfy (a). The models of ZFC satisfying (b) are the models of (ZFC + ω L 1 < ω 1 ).

Cardinality problems for ω-languages

Theorem 4.1 There exists a real-time 1-counter Büchi automaton A such that the cardinality of the complement L(A) -of the ω-language L(A) is not determined by the axiomatic system ZFC:

(1). There is a model

V 1 of ZFC in which L(A) -is countable. (2). There is a model V 2 of ZFC in which L(A) -has cardinal 2 ℵ 0 . (3). There is a model V 3 of ZFC in which L(A) -has cardinal ℵ 1 with ℵ 0 < ℵ 1 < 2 ℵ 0 .
Proof. From now on we set Σ = {0, 1} and we shall denote by C 1 the largest thin

Π 1 1 -set in {0, 1} ω = 2 ω . This set C 1 is a Π 1 1 -set defined by a Π 1 1 -formula φ, given by Moschovakis in [Mos80, page 248]. Thus its complement C - 1 = 2 ω -C 1 is a Σ 1 1 -set defined by the Σ 1 1 -formula ψ = ¬φ. By Proposition 2.2, the ω-language C -
1 is accepted by a Büchi Turing machine M and by a 2-counter Büchi automaton A 1 which can be effectively constructed.

We are now going to use some constructions which were used in a previous paper [START_REF] Finkel | Borel ranks and Wadge degrees of omega context free languages[END_REF] to study topological properties of context-free ω-languages, and which will be useful in the sequel.

Let E be a new letter not in Σ, S be an integer ≥ 1, and θ S : Σ ω → (Σ ∪ {E}) ω be the function defined, for all x ∈ Σ ω , by: θ S (x) = x(1).E S .x(2).E S 2 .x(3).E S 3 .x(4) . . . x(n).E S n .x(n + 1).E S n+1 . . . We proved in [START_REF] Finkel | Borel ranks and Wadge degrees of omega context free languages[END_REF] that if L ⊆ Σ ω is an ω-language in the class BCL(2) ω and k = cardinal(Σ)+2, S = (3k) 3 , then one can effectively construct from a Büchi 2-counter automaton A 1 accepting L a real time Büchi 8-counter automaton A 2 such that L(A 2 ) = θ S (L).

On the other hand, it is easy to see that θ S (Σ ω ) -= (Σ ∪ {E}) ωθ S (Σ ω ) is accepted by a real time Büchi 1-counter automaton. The class r-BCL(8) ω ⊇ r-BCL(1) ω is closed under finite union in an effective way, so θ S (L) ∪ θ S (Σ ω ) -is accepted by a real time Büchi 8-counter automaton A 3 which can be effectively constructed from A 2 .

In [START_REF] Finkel | Borel ranks and Wadge degrees of omega context free languages[END_REF] we used also another coding which we now recall. Let K = 2 × 3 × 5 × 7 × 11 × 13 × 17 × 19 = 9699690 be the product of the eight first prime numbers. Let Γ be a finite alphabet; here we shall set Γ = Σ ∪ {E}. An ω-word x ∈ Γ ω is coded by the ω-word

h K (x) = A.C K .x(1).B.C K 2 .A.C K 2 .x(2).B.C K 3 .A.C K 3 .x(3).B . . . B.C K n .A.C K n .x(n).B . . .
over the alphabet Γ 1 = Γ ∪ {A, B, C}, where A, B, C are new letters not in Γ. We proved in [START_REF] Finkel | Borel ranks and Wadge degrees of omega context free languages[END_REF] that, from a real time Büchi 8-counter automaton A 3 accepting L(A 3 ) ⊆ Γ ω , one can effectively construct a Büchi 1-counter automaton A 4 accepting the ω-language h K (L(A 3 ))∪h K (Γ ω ) -.

Consider now the mapping φ K : (Γ ∪ {A, B, C}) ω → (Γ ∪ {A, B, C, F }) ω which is simply defined by: for all x ∈ (Γ ∪ {A, B, C}) ω ,

φ K (x) = F K-1 .x(1).F K-1 .x(2) . . . F K-1 .x(n).F K-1 .x(n + 1).F K-1 . . . Then the ω-language φ K (L(A 4 )) = φ K (h K (L(A 3 ))∪h K (Γ ω ) -
) is accepted by a real time Büchi 1-counter automaton A 5 which can be effectively constructed from the Büchi 8-counter automaton A 4 , [START_REF] Finkel | Borel ranks and Wadge degrees of omega context free languages[END_REF].

On the other hand, it is easy to see that the ω-language (Γ ∪ {A, B, C, F }) ωφ K ((Γ ∪ {A, B, C}) ω ) is ω-regular and to construct a (1-counter) Büchi automaton accepting it. Then one can effectively construct from A 5 a real time Büchi 1-counter automaton A 6 accepting the ω-language φ K (h

K (L(A 3 ))∪h K (Γ ω ) -) ∪ φ K ((Γ ∪ {A, B, C}) ω ) -.
To sum up: we have obtained, from a Büchi Turing machine M accepting the ω-language C - 1 ⊆ Σ ω = 2 ω , a 2-counter Büchi automaton A 1 accepting the same ω-language, a real time Büchi 8-counter automaton A 3 accepting the ω-language L(A 3 ) = θ S (C - 1 )∪θ S (Σ ω ) -, a Büchi 1counter automaton A 4 accepting the ω-language h K (L(A 3 ))∪h K (Γ ω ) -, and a real time Büchi 1counter automaton A 6 accepting the ω-language φ K (h

K (L(A 3 ))∪h K (Γ ω ) -)∪φ K ((Γ∪{A, B, C}) ω ) -.
From now on we shall denote simply A 6 by A.

Therefore we have successively the following equalities:

L(A 1 ) = C - 1 , L(A 1 ) -= C 1 , L(A 3 ) -= θ S (C 1 ), L(A 4 ) -= h K (L(A 3 ) -) = h K (θ S (C 1 )), L(A 6 ) -= φ K (h K (L(A 3 ) -)) = φ K (h K (θ S (C 1 ))).
This implies easily that the ω-languages L(A 1 ) -, L(A 3 ) -, L(A 4 ) -, and L(A 6 ) -= L(A) - all have the same cardinality as the set C 1 , because each of the maps θ S , h K and φ K is injective.

Thus we can infer the result stated in the theorem from the above Theorem 3.5.

The following corollary follows directly from Item (3) of Theorem 4.1.

Corollary 4.2 It is consistent with ZFC that the Continuum Hypothesis is not satisfied for complements of 1-counter ω-languages, (hence also for complements of context-free ω-languages).

Remark 4.3 This can be compared with the fact that the Continuum Hypothesis is satisfied for regular languages of infinite trees (which are closed under complementation), proved by Niwinski in [START_REF] Niwinski | On the cardinality of sets of infinite trees recognizable by finite automata[END_REF]. Notice that this may seem amazing because from a topological point of view one can find regular tree languages which are more complex than context-free ω-languages, as there are regular tree languages in the class ∆ 1 2 \ Σ 1 1 ∩ Π 1 1 while context-free ω-languages are all analytic, i.e. Σ 1 1 -sets.

Recall that a real-time 1-counter Büchi automaton C has a finite description to which can be associated, in an effective way, a unique natural number called the index of C. From now on, we shall denote, as in [START_REF] Finkel | Highly undecidable problems for infinite computations[END_REF], by C z the real time Büchi 1-counter automaton of index z (reading words over Ω = {0, 1, A, B, C, E, F }).

We can now use the proofs of Theorem 3.5 and 4.1 to prove that some natural cardinality problems are actually located at the third level of the analytical hierarchy. The notions of analytical hierarchy on subsets of N and of classes of this hierarchy may be found for instance in [START_REF] Castro | Nondeterministic ω-computations and the analytical hierarchy[END_REF] or in the textbook [START_REF] Rogers | Theory of Recursive Functions and Effective Computability[END_REF].

Theorem 4.4 (1). {z ∈ N | L(C z ) -is finite } is Π 1 2 -complete. (2). {z ∈ N | L(C z ) -is countable } is in Σ 1 3 \ (Π 1 2 ∪ Σ 1 2 ). (3). {z ∈ N | L(C z ) -is uncountable} is in Π 1 3 \ (Π 1 2 ∪ Σ 1 2 ).
Proof. Item (1) was proved in [START_REF] Finkel | Highly undecidable problems for infinite computations[END_REF], and item (3) follows directly from item (2). We now prove item (2). We first show that {z ∈ N | L(C z ) -is countable } is in the class Σ 1 3 . Notice first that, using a recursive bijection b : (N ⋆ ) 2 → N ⋆ , we can consider an infinite word over a finite alphabet Ω as a countably infinite family of infinite words over the same alphabet by considering, for any ω-word σ ∈ Ω ω , the family of ω-words (σ i ) i≥1 such that for each i ≥ 1 the ω-word σ i ∈ Ω ω is defined by σ i (j) = σ(b(i, j)) for each j ≥ 1.

We can now express "L(C z ) -is countable " by the formula:

∃σ ∈ Ω ω ∀x ∈ Ω ω [(x ∈ L(C z )) or (∃i ∈ N x = σ i )]
This is a Σ 1 3 -formula because "(x ∈ L(C z ))", and hence also "[(x ∈ L(C z )) or (∃i ∈ N x = σ i )]", are expressed by Σ 1 1 -formulas. We can now prove that {z ∈ N | L(C z ) -is countable } is neither in the class Σ 1 2 nor in the class Π 1 2 , by using Shoenfield's Absoluteness Theorem from Set Theory. Let A be the real-time 1-counter Büchi automaton cited in Theorem 4.1 and let z 0 be its index so that A = C z 0 . Assume that V is a model of (ZFC + ω L 1 < ω 1 ). In the model V, the integer z 0 belongs to the set {z ∈ N | L(C z ) -is countable }, while in the inner model L ⊆ V, the language L(C z 0 ) -has the cardinality of the continuum: thus in L the integer z 0 does not belong to the set {z ∈ N | L(C z ) -is countable }. On the other hand, Shoenfield's Absoluteness Theorem implies that every Σ 1 2 -set (respectively, Π 1 2 -set) is absolute for all inner models of (ZFC), see [Jec02, page 490]. In particular, if the set {z ∈ N | L(C z ) -is countable } was a Σ 1 2 -set or a Π 1 2 -set then it could not be a different subset of N in the models V and L considered above. Therefore, the set

{z ∈ N | L(C z ) -is countable } is neither a Σ 1 2 -set nor a Π 1 2 -set.
Remark 4.5 Using an easy coding we can obtain a similar result for 1-counter automata reading words over Σ, where Σ is any finite alphabet having at least two letters.

Notice that the same proof gives a partial answer to a question of Castro and Cucker. They stated in [START_REF] Castro | Nondeterministic ω-computations and the analytical hierarchy[END_REF] that the problem to determine whether the complement of the ω-language accepted by a given Turing machine is countable (respectively, uncountable) is in the class Σ 1 3 (respectively, Π 1 3 ), and asked for the exact complexity of these decision problems.

Theorem 4.6

The problem to determine whether the complement of the ω-language accepted by a given Turing machine is countable (respectively, uncountable) is in the class

Σ 1 3 \ (Π 1 2 ∪ Σ 1 2 ) (respectively, Π 1 3 \ (Π 1 2 ∪ Σ 1 2 )).
We now consider acceptance of binary relations over infinite words by 2-tape Büchi automata, firstly considered by Gire and Nivat in [START_REF] Gire | Relations rationnelles infinitaires[END_REF]. A 2-tape automaton is an automaton having two tapes and two reading heads, one for each tape, which can move asynchronously, and a finite control as in the case of a (1-tape) automaton. The automaton reads a pair of (infinite) words (u, v) where u is on the first tape and v is on the second tape, so that a 2-tape Büchi automaton B accepts an infinitary rational relation L(B) ⊆ Σ ω 1 × Σ ω 2 , where Σ 1 and Σ 2 are two finite alphabets. Notice that L(B) ⊆ Σ ω 1 × Σ ω 2 may be seen as an ω-language over the product alphabet Σ 1 × Σ 2 .

We shall use a coding used in a previous paper [START_REF] Finkel | On the accepting power of two-tape Büchi automata[END_REF] on the topological complexity of infinitary rational relations. We first recall a coding of an ω-word over the finite alphabet Ω = {0, 1, A, B, C, E, F } by an ω-word over the alphabet Ω ′ = Ω ∪ {D}, where D is an additionnal letter not in Ω. For x ∈ Ω ω the ω-word h(x) is defined by : h(x) = D.0.x(1).D.0 2 .x(2).D.0 3 .x(3).D . . . D.0 n .x(n).D.0 n+1 .x(n + 1).D . . . It is easy to see that the mapping h from Ω ω into (Ω∪{D}) ω is injective. Let now α be the ω-word over the alphabet Ω ′ which is simply defined by: We can now prove our second main result.

Theorem 4.8 There exists a 2-tape Büchi automaton B such that the cardinality of the complement of the infinitary rational relation L(B) is not determined by ZFC. Indeed it holds that:

(1). There is a model V 1 of ZFC in which L(B) -is countable.

(2). There is a model V 2 of ZFC in which L(B) -has cardinal 2 ℵ 0 .

(3). There is a model V 3 of ZFC in which L(B) -has cardinal ℵ 1 with ℵ 0 < ℵ 1 < 2 ℵ 0 .

Proof. Let A be the real time 1-counter Büchi automaton constructed in the proof of Theorem 4.1, and B be the 2-tape Büchi automaton which can be constructed from A by the above Proposition 4.7. Letting L = L(A), the complement of the infinitary rational relation R = L(B) is equal to [(Ω ∪ {D}) ω -L] × {α} = h(L -) × {α}. Thus the cardinality of R -= L(B) -is equal to the cardinality of the ω-language h(L -), so that the result follows from Theorem 4.1.

As in the case of ω-languages of 1-counter automata, we can now state the following result, where T z is the 2-tape Büchi automaton of index z reading words over Ω ′ × Ω ′ . Theorem 4.9

(1).

{z ∈ N | L(T z ) -is finite } is Π 1 2 -complete. (2). {z ∈ N | L(T z ) -is countable } is in Σ 1 3 \ (Π 1 2 ∪ Σ 1 2 ). (3). {z ∈ N | L(T z ) -is uncountable} is in Π 1 3 \ (Π 1 2 ∪ Σ 1 2 ).
Proof. Item (1) was proved in [START_REF] Finkel | Highly undecidable problems for infinite computations[END_REF]. Items (2) and (3) are proved similarly to the case of ω-languages of 1-counter automata, using Shoenfield's Absoluteness Theorem.

On the other hand we have the following result.

Proposition 4.10 It is decidable whether an infinitary rational relation R ⊆ Σ ω 1 × Σ ω 2 , accepted by a given 2-tape Büchi automaton B, is countable (respectively, uncountable).

α

  = D.0.D.0 2 .D.0 3 .D.0 4 .D . . . D.0 n .D.0 n+1 .D . . .The following result was proved in[START_REF] Finkel | On the accepting power of two-tape Büchi automata[END_REF].Proposition 4.7 ([Fin06b]) Let L ⊆ Ω ω be in r-BCL(1) ω and L = h(L) ∪ (h(Ω ω )) -. Then R = L × {α} (Ω ′ ) ω × ((Ω ′ ) ω -{α})is an infinitary rational relation. Moreover one can effectively construct from a real time 1-counter Büchi automaton A accepting L a 2-tape Büchi automaton B accepting the infinitary relation R.

Proof. Let R ⊆ Σ ω 1 × Σ ω 2 be an infinitary rational relation accepted by a 2-tape Büchi automaton B. It is known that Dom(R) = {u ∈ Σ ω 1 | ∃v ∈ Σ ω 2 (u, v) ∈ R} and Im(R) = {v ∈ Σ ω 2 | ∃u ∈ Σ ω 1 (u, v) ∈ R} are regular ω-languages and that one can find Büchi automata A and A ′ accepting Dom(R) and Im(R), [START_REF] Gire | Relations rationnelles infinitaires[END_REF]. On the other hand Lindner and Staiger have proved that one can compute the cardinal of a given regular ω-language L(A) (see [START_REF] Kuske | First-order and counting theories of omega-automatic structures[END_REF] where Kuske and Lohrey proved that this problem is actually in the class PSPACE). But it is easy to see that the infinitary rational relation R is countable if and only if the two ω-languages Dom(R) and Im(R) are countable, thus one can decide whether the infinitary rational relation R is countable (respectively, uncountable).

Remark 4.11

The results given by Items ( 2) and (3) of Theorem 4.9 and Proposition 4.10 are rather surprising: they show that there is a remarkable gap between the complexity of the same decision problems for infinitary rational relations and for their complements, as there is a big space between the class ∆ 0 1 of computable sets and the class

Concluding remarks

We have proved that amazingly some basic cardinality questions on automata reading infinite words depend on the models of the axiomatic system ZFC.

In [START_REF] Finkel | The complexity of infinite computations in models of set theory[END_REF] we have proved that the topological complexity of an ω-language accepted by a 1-counter Büchi automaton, or of an infinitary rational relation accepted by a 2-tape Büchi automaton, is not determined by ZFC.

In [START_REF] Finkel | Decision problems for recognizable languages of infinite pictures[END_REF], we study some cardinality questions for Büchi-recognizable languages of infinite pictures and prove results which are similar to those we have obtained in this paper for 1-counter ω-languages and for infinitary rational relations.

The next step in this research project would be to determine which properties of automata actually depend on the models of ZFC, and to achieve a more complete investigation of these properties.

Annexe

Proof of Thorem 3.5.

(a). In the model L, the cardinal of the largest thin Π 1 1 -set in Σ ω is equal to the cardinal of ω 1 . Moreover the continuum hypothesis is satisfied thus

Its cardinal is ℵ 0 , and therefore this is also the cardinal in V of the largest thin Π 1 1 -set in Σ ω .

(c). It suffices to show that there is a model V 3 of ZFC in which ω L 1 = ω 1 and ℵ 1 < 2 ℵ 0 . Such a model can be constructed by Cohen's forcing: start from a model V of ZFC + V=L (in which ω L 1 = ω 1 ) and construct by forcing a generic extension V[G] in which are added ℵ 2 (or even more) "Cohen's reals", which are in fact ℵ 2 subsets of ω. Notice that the cardinals are preserved under this extension (see [Jec02, page 219]), and that the constructible sets of V[G] are also the constructible sets of V, thus in the new model V[G] of ZFC we still have ω L 1 = ω 1 , but now ℵ 1 < 2 ℵ 0 .


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

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

Hydrodynamics of self-alignment interactions with precession and derivation of the Landau-Lifschitz-Gilbert equation

Keywords: Self-propelled particles, alignment dynamics, precession, hydrodynamic limit, hyperbolicity, diffusion correction, weakly non-local interaction, Landau-Lifschitz-Gilbert, spin dynamics AMS Subject classification: 35L60, 35K55, 35Q80, 82C05, 82C22, 82C70, 92D50

Hydrodynamics of self-alignment interactions with precession and derivation of the Landau-Lifschitz-Gilbert equation

Introduction

The setting of this paper is the same as in [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF] and considers a kinetic model of selfpropelled particles in three dimensions and its hydrodynamic limit. The particles are supposed to move with the same constant speed and their velocity orientation (which are vectors of the two-dimensional sphere S 2 ) tend to align to the local average orientation, with some noise. This model has been proposed as a kinetic formulation 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 investigate how the addition of a precession force around the local average orientation modifies the resulting kinetic and hydrodynamic models.

The derivation of the hydrodynamic model proceeds like in [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF] and relies on both the determination of local equilibria in the form of Von-Mises-Fischer (VMF) distributions and on the new concept of 'Generalized Collision Invariants' (GCI) which allows for a non-conservative closure. In the present paper, we show that the VMF equilibria are still equilibrium states of the system with precession and we extend the derivation of the GCI's to this system. As in [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF], the resulting hydrodynamics (later on referred to as hydrodynamic model for self-alignment with precession) consists of a conservation law for the density and a non-conservative equation for the velocity orientation. By contrast to [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF], the velocity orientation equation contains an additional term which accounts for the influence of the precession force and which results in the loss of hyperbolicity of the model as soon as the precession force is non-zero. This loss of hyperbolicity occurs when waves propagate in the same direction as that of the average velocity direction.

In order to stabilize the model, we introduce a weak non-locality in the alignment interaction: the local average which determines the alignment direction is now taken over a certain spatial extension around the particle. If the spatial extension of this local average is scaled like the square root of the alignment mean free path ε, then, the alignment direction possesses a small correction of order ε to the local average direction. This correction involves the Laplacian of the velocity orientation and results in diffusion terms in the macroscopic equation. Due to the precession force, this diffusion term has two contribution: the first one corresponds to the heat flow for harmonic maps ; the second one is in the form of a Landau-Lifschitz-Gilbert term. In the particular case of zero speed, the resulting model reduces to the phenomenological Landau-Lifschitz-Gilbert equations.

Therefore the present theory provides a kinetic formulation of classical micromagnetization models and spin dynamics [START_REF] Brown | Micromagnetics[END_REF].

In the non-local alignment interaction, we also include a small (of order ε) repulsive force which, in the macroscopic model, gives riseto both an additional pressure force and a pressure precession term.

The Vicsek model [START_REF] Vicsek | Novel type of phase transition in a system of self-driven particles[END_REF] has been proposed as a model for the interaction of individuals among animal societies 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]). It is a particle model (or 'Individual-Based Model' or 'Agent-Based model') which consists in an Ordinary Differential Equation system for the particle positions and velocities. As the number of particles increases, its computational cost increases more than linearly and becomes prohibitive for large particle numbers. The Vicsek model is set up in a time-discrete framework. A time-continuous version of the Vicsek model has been proposed in [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF] and a kinetic model has been deduced from it on formal grounds. The rigorous derivation of this kinetic model has been performed in [START_REF] Bolley | Mean-field limit for the stochastic Vicsek model[END_REF]. In the present paper, we will directly start from the kinetic level.

For the modeling of systems consisting of a large number of individuals, it is more efficient to use hydrodynamic models, which describe the system by macroscopic averages. Many such models exist [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] 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] but few of them are rigorously derived from particle or kinetic dynamics. The first rigorous derivation of the hydrodynamic limit of the Vicsek model is done in [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF] (earlier phenomenological derivations can be found 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] but do not lead to the same model). The resulting hydrodynamic model has also been found in the hydrodynamics of the so-called Persistent Turning Walker model [START_REF] Degond | A Macroscopic Model for a System of Swarming Agents Using Curvature Control[END_REF] of fish behavior, where particles interact through curvature control. Diffusive corrections to the hydrodynamic model have also been computed in [START_REF] Degond | Diffusion in a continuum model of self-propelled particles with alignment interaction[END_REF] through a Chapman-Enskog expansion up to the second order. Other macroscopic models of swarming particle systems derived from kinetic theory can be found e.g. in [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]. In particular, variants of the kinetic model of [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF] and its hydrodynamics have been proposed. In [START_REF] Frouvelle | A continuous model for alignment of self-propelled particles with anisotropy and density-dependent parameters[END_REF], the influence of a vision angle and dependency of the alignement frequency upon the local density has been investigated. In [START_REF] Frouvelle | Dynamics, in a kinetic model of oriented particles with phase transition[END_REF][START_REF] Degond | Macroscopic limits and phase transition in a system of self-propelled particles[END_REF], the modification of the normalization constant of the alignment force results in phase transitions from disordered to ordered equilibria as the density increases and reaches a threshold. In the spatially homogeneous case, these phase transition models are analogs of 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. This makes a strong difference in the resulting macroscopic models as compared with the ones we are presenting here.

The organization of the paper is as follows. In section 2, we set up the kinetic model and discuss the equilibria and GCI's. In section 3, the hydrodynamic limit is established. The hyperbolicity of the resulting model is investigated in section 4. Then, the weakly non-local model is derived in section 5. Finally, a conclusion is drawn at section 6.

2 Setting of the problem and preliminaries

The kinetic model

The starting point is the following kinetic model for a distribution function f (x, v, t), where x ∈ R 3 , v ∈ S 2 , t ≥ 0:

f ε t + cv • ∇ x f ε = 1 ε ∇ v • [-(P v ⊥ Ω f ε )f ε + d∇ v f ε + α(Ω f ε × v)f ε ] (2.1) := 1 ε Q(f ε ) := 1 ε (Q 0 (f ε ) + R(f ε )),
where

Q 0 (f ) = ∇ v • [-(P v ⊥ Ω f )f + d∇ v f ] , R(f ) = α∇ v • [(Ω f × v)f ] .
Q 0 (f ) is the operator studied in [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF]. R(f ) is the precession operator about an axis which is that of the average particle velocity. c ≥ 0 is the speed of the self-propelled particles. We let P v ⊥ = Id -v ⊗ v be the projection operator onto the plane normal to v and

ρ f = S 2 f (v) dv, Ω f = S 2 f (v) v dv | S 2 f (v) v dv| ,
the density and mean velocity direction associated to f . We define the equilibria (Von-Mises-Fischer distribution):

F Ω = exp(βv • Ω) S 2 exp(βv • Ω) dv ,
with β = 1/d. We note the compatibility relation:

ρ ρF Ω = ρ, Ω ρF Ω = Ω.
We define

M f = ρ f F Ω f ,
the equilibrium distribution associated to f .

Equilibria

In this section, we investigate the solutions of the equation Q(f ) = 0, which are the so-called local equilibria of the kinetic model.

Proposition 2.1 We have:

Q(f ) = 0 ⇐⇒ ∃ρ ≥ 0, ∃Ω ∈ S 2 , such that f = ρF Ω .
Proof: We note the following properties:

S 2 Q 0 (f ) f M f dv = -d S 2 ∇ v f M f 2 M f dv ≤ 0, S 2 R(f ) f M f dv = -α S 2 (Ω f × v) • ∇ v f M f -(P v ⊥ Ω f ) f M f f (v) dv = 0.
The first line is proved in [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF]. To prove the second line, we remark that

∇ v M f = βP v ⊥ ω M f , (Ω f × v) • (P v ⊥ Ω f ) = 0
and that the first term can be written as

- α 2 2 S 2 ∇ v (Ω f × v) f 2 M f dv = 0. Then, if Q(f ) = 0, we have 0 = S 2 Q(f ) f M f dv = -d S 2 ∇ v f M f 2 M f dv.
This implies that f M f is a constant, and that therefore, f is of the form ρF Ω for a given ρ ≥ 0, Ω ∈ S 2 . Conversely, by the compatibility relation, it is obvious that f = ρF Ω satisfies Q(f ) = 0.

Remark 2.1 We comment on the connection between this model and rod like polymers. In the case of rod-like polymers, the interaction operator is written

∇ v • [(-∇ v φ + w × v)f + d∇ v f ],
where φ(v) is a potential and w is the precession direction. In general, finding equilibria for polymer models is difficult [START_REF] Otto | Continuity of velocity gradients in suspensions of rod-like molecules[END_REF]. Here, we deal with the special case where φ(v) = -v •w which makes this computation possible. In general, the computation is possible if the level curves of φ are invariant under precession rotation.

Generalized Collisions Invariants

We recall the definition of a Generalized Collision invariant (GCI) (see [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF]). First, we define the collision operator

Q(f, Ω) = ∇ v • [-(P v ⊥ Ω)f + d∇ v f + α(Ω × v)f ] ,
where now Ω ∈ S 2 is not necessarily equal to Ω f . For given Ω, Q(•, Ω) is a linear operator and we will also consider its formal adjoint Q * (•, Ω). Then, we have the Definition 2.2 For given Ω ∈ S 2 , a function ψ Ω (v) is said to be a GCI associated to Ω iff the following property holds:

S 2 Q(f, Ω) ψ Ω dv = 0, ∀f such that Ω f = ±Ω.
We note that the condition Ω f = ±Ω is a linear condition on f which can equivalently be written A • S 2 f (v) v dv = 0, for all vectors A such that A • Ω = 0. On the other hand, we have

S 2 Q(f, Ω) ψ Ω dv = S 2 Q * (ψ Ω , Ω) f dv. Therefore, that ψ Ω is a GCI is equivalent to ∃A ∈ R 3 such that A • Ω = 0 and Q * (ψ Ω , Ω) = A • v, or (P v ⊥ Ω) • ∇ v ψ Ω + d∆ v ψ Ω -α(Ω × v) • ∇ v ψ Ω = A • v.
In spherical coordinates (with polar axis Ω, latitude θ ∈ [0, π] and longitude ϕ ∈ [0, 2π]), this equation is written (dropping the index Ω):

-sin θ ∂ θ ψ + d 1 sin θ ∂ θ (sin θ ∂ θ ψ) + 1 sin 2 θ ∂ 2 ϕ ψ -α sin θ ∂ ϕ ψ = = A 1 sin θ cos ϕ + A 2 sin θ sin ϕ, (2.2)
where A = (A 1 , A 2 , 0) are the coordinates of A, in a ortho-normal coordinate basis (e 1 , e 2 , Ω).

Proposition 2.3

The set C Ω of the GCI's associated to Ω is a vector space of dimension 3 spanned by {1, ψ (1) , ψ (2) }, where ψ (k) is associated to A = e k , for k = 1, 2. Furthermore,

ψ (k) (θ, ϕ) = ψ (k) 1 (θ) cos ϕ + ψ (k) 2 (θ) sin ϕ, k = 1, 2,
and ψ

(2)

1 = -ψ (1) 2 , ψ (2) 2 = ψ (1)
1 . Finally, (ψ

(1) 1 , ψ (1)
2 ) satisfies the following elliptic system:

-sin θ ∂ θ ψ (1) 1 + d 1 sin θ ∂ θ (sin θ ∂ θ ψ (1) 1 ) - 1 sin 2 θ ψ (1) 1 -α sin θ ψ (1) 2 = sin θ, (2.3)
-sin θ ∂ θ ψ (1) 2 + d 1 sin θ ∂ θ (sin θ ∂ θ ψ (1)
2 ) -

1 sin 2 θ ψ (1) 2 + α sin θ ψ (1) 1 = 0. (2.4) (ψ (2) 1 , ψ (2)
2 ) satisfies the same system with right hand sides 0 and sin θ for the first and second equation respectively. The elliptic system has a unique zero-average solution, which defines ψ (k) uniquely.

Proof: The reduction of (2.2) to (2.3), (2.4) follows from using Fourier transform in the ϕ variable. To solve this system, we construct the complex-valued function ψ(k

) (θ) = ψ (k) 1 (θ) + iψ (k)
2 (θ). ψ(1) satisfies the complex-valued elliptic problem:

-sin θ ∂ θ ψ(1) + d 1 sin θ ∂ θ (sin θ ∂ θ ψ(1) ) - 1 sin 2 θ ψ(1) + iα sin θ ψ(1) = sin θ,
whereas ψ(2) satisfies the same equation with right-hand side i sin θ. Since the imaginary part of the operator is skew-adjoint, it does not modify the energy identity. Therefore, the same theory as in [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF] applies and shows the existence and uniqueness of the solution in the space of zero average functions. The remaining part of the statement follows easily.

Hydrodynamic limit

The following theorem establishes the limit ε → 0.

Theorem 3.1 We suppose that f ε → f 0 as smoothly as needed. Then, we have

f 0 = ρF Ω , (3.1)
where ρ = ρ(x, t) and Ω = Ω(x, t) satisfy the following system

∂ t ρ + cc 1 ∇ x (ρΩ) = 0, (3.2) ρ(∂ t Ω + cc 2 cos δ (Ω • ∇ x )Ω + cc 2 sin δ Ω × ((Ω • ∇ x )Ω)) + cd P Ω ⊥ ∇ x ρ = 0. ( 3.3)
The coefficient c 1 is defined by

c 1 = S 2 F Ω (v) (v • Ω) dv = S 2 exp(βv • Ω) (v • Ω) dv S 2 exp(βv • Ω) dv . (3.4)
The coefficients c 2 and δ are defined as follows. First, define a k and b k for k = 1, 2 by

a k = 1 2 θ∈[0,π] F Ω (cos θ) ψ (1)
k (θ) sin 2 θ dθ, k = 1, 2, (3.5)
b k = 1 2 θ∈[0,π] F Ω (cos θ) ψ (1)
k (θ) cos θ sin 2 θ dθ, k = 1, 2, (3.6)
where ψ Remark 3.2 We note the following identities:

(Ω • ∇ x )Ω = (∇ x × Ω) × Ω, Ω × ((∇ x × Ω) × Ω) = P Ω ⊥ (∇ x × Ω).
Remark 3.3 The reason for keeping the dependence upon the speed c explicit is that we will later investigate the case c = 0.

Proof. Letting ε → 0 in (2.1) and using proposition 2.1, we get that Q(f ε ) → 0 = Q(f 0 ), and that eq. (3.1) is satisfied. Now, we look for the equations satisfied by (ρ, Ω) and for this purpose, we use the GCI.

According to Proposition 2.3, 1 is a classical collision invariant. Therefore, multiplying (2.1) by 1 and letting ε → 0, we get the mass conservation equation (3.2). Now, we successively multiply (2.1) by the GCI's ψ (k) , k = 1, 2 associated to the polar vector Ω f ε . From the fact that ψ (k) ∈ C Ω f ε , we get that

S 2 Q(f ε )ψ (k) dv = S 2 Q(f ε , Ω f ε )ψ (k) dv = 0, k = 1, 2,
thanks to definition 2.2. We deduce that

S 2 T (f ε )ψ (k) dv = 0, k = 1, 2,
where T f = ∂ t f + cv • ∇ x f is the transport operator. Letting ε → 0, we deduce that

T (k) := S 2 T (ρF Ω )ψ (k) dv = 0, k = 1, 2.
(3.8)

We note that

ψ (1) = ψ 1 (θ) cos ϕ + ψ 2 (θ) sin ϕ, (3.9) ψ (2) = -ψ 2 (θ) cos ϕ + ψ 1 (θ) sin ϕ, (3.10)
where we define ψ 1 = ψ

(1)

1 and ψ 2 = ψ (1)
2 for simplicity. A simple computation [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF] shows that

T (ρF Ω ) = F Ω {∂ t ρ + c(v • ∇ x )ρ + βρ(v • (∂ t + cv • ∇ x )Ω)} . We decompose v = v ⊥ + v , v ⊥ = P Ω ⊥ v, v = (v • Ω)Ω.
In spherical coordinate system, we have v ⊥ = (sin θ cos ϕ, sin θ sin ϕ, 0) T , v = (0, 0, cos θ).

Therefore, v ⊥ is an odd degree trigonometric polynomial of ϕ, while v is an even degree such polynomial. We decompose T (ρF Ω ) into even and odd degree trigonometric polynomials in ϕ using the above defined decomposition of v. We get:

T (ρF Ω ) = T e + T o , (3.11) T o = F Ω {cv ⊥ • ∇ x ρ + βρv ⊥ • (∂ t + c(v • Ω)Ω • ∇ x )Ω} ,
where we have used that v • ((v ⊥ • ∇ x )Ω) = 0 because |Ω| = 1. We can write

T o = F Ω v ⊥ • D(v • Ω) = F Ω sin θ (cos ϕ D 1 (cos θ) + sin ϕ D 2 (cos θ)), with D(v • Ω) = c∇ x ρ + βρ(∂ t + c(v • Ω)Ω • ∇ x )Ω, (3.12)
and D = (D 1 , D 2 , D 3 ) are the coordinates of D in the cartesian coordinate basis associated to the definitions of the angles (θ, ϕ). Now, inserting (3.11), (3.12) into (3.8), and noting that the odd degree polynomial in ϕ vanish away upon integration with respect to ϕ ∈ [0, 2π], we get,

T (1) = θ∈[0,π],ϕ∈[0,2π] F Ω (cos θ) (cos ϕ D 1 (cos θ) + sin ϕ D 2 (cos θ)) (ψ 1 (θ) cos ϕ + ψ 2 (θ) sin ϕ) sin 2 θ dθ dϕ, T (2) = θ∈[0,π],ϕ∈[0,2π] F Ω (cos θ) (cos ϕ D 1 (cos θ) + sin ϕ D 2 (cos θ)) (-ψ 2 (θ) cos ϕ + ψ 1 (θ) sin ϕ) sin 2 θ dθ dϕ.
Performing the ϕ integration leads to:

T (1) = 1 2 θ∈[0,π] F Ω (cos θ) (D 1 (cos θ)ψ 1 (θ) + D 2 (cos θ)ψ 2 (θ)) sin 2 θ dθ, (3.13)
T (2) = 1 2 θ∈[0,π] F Ω (cos θ) (-D 1 (cos θ)ψ 2 (θ) + D 2 (cos θ)ψ 1 (θ)) sin 2 θ dθ. (3.14)
Now, introducing the matrices

[a] = a 1 a 2 -a 2 a 1 , [b] = b 1 b 2 -b 2 b 1 ,
and T (2) 0

[A] =   a 1 a 2 0 -a 2 a 1 0 0 0 0   , [B] =   b 1 b 2 0 -b 2 b 1 0 0 0 0   ,
  = [A]P Ω ⊥ (c∇ x ρ + βρ∂ t Ω) + βρc[B]P Ω ⊥ ((Ω • ∇ x )Ω) = 0. (3.15)
The matrix [a] being a similitude matrix, it is invertible and we can introduce the matrix

[c 2 ] = [a] -1 [b] = c 21 c 22 -c 22 c 21 ,
and,

[C 2 ] =   c 21 c 22 0 -c 22 c 21 0 0 0 0   .
Multiplying the first two lines of the vector equation (3.15) by d[a] -1 , we get:

ρ(∂ t Ω + c[C 2 ](Ω • ∇ x )Ω) + cdP Ω ⊥ ∇ x ρ = 0. (3.16)
This is the momentum equation.

The equation can be transformed using the fact that [b] and consequently [c 2 ] are similitude matrices. Therefore, we can write

[c 2 ] = c 2 R δ , (3.17)
where c 2 and δ are given by (3.7) and

R δ = cos δ sin δ -sin δ cos δ ,
is the rotation matrix of angle δ. Then, (3.16) can be equivalently written according to (3.3), which ends the proof.

Hyperbolicity of the hydrodynamic model

In this section, we investigate the hyperbolicity of the hydrodynamic model for selfalignment interactions with precession. For this purpose, we use the spherical coordinates associated to a fixed cartesian basis. In this basis, denoting by θ ∈ [0, π] the latitude and ϕ ∈ [0, 2π] the longitude, we have Ω = (sin θ cos ϕ, sin θ sin ϕ, cos θ) T , and we let Ω θ and Ω ϕ be the derivatives of Ω with respect to θ and ϕ. We note that

|Ω θ | = 1, |Ω ϕ | = sin θ.
We will use the formulas

∇ x • Ω = Ω θ • ∇ x θ + Ω ϕ • ∇ x ϕ, P Ω ⊥ a = (Ω θ • a)Ω θ + (Ω ϕ • a) sin 2 θ Ω ϕ , (Ω • ∇ x )Ω = ((Ω • ∇ x )θ) Ω θ + ((Ω • ∇ x )ϕ) Ω ϕ , Ω × (Ω • ∇ x )Ω = ((Ω • ∇ x )θ) sin θ Ω ϕ -sin θ((Ω • ∇ x )ϕ) Ω θ , Ω t = Ω θ θ t + Ω ϕ ϕ t ,
where a is an arbitrary vector.

As in [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF], we use the time rescaling t ′ = cc 1 t and introduce

a = c 2 c 1 , λ 2 = d c 1 , ρ = λ lnρ.
With this change of variables and unknowns, system (3.2), (3.3) is written (dropping the primes on t):

ρt + Ω • ∇ x ρ + λ∇ x • Ω = 0, Ω t + a (cos δ (Ω • ∇ x )Ω + sin δ Ω × ((Ω • ∇ x )Ω)) + λ P Ω ⊥ ∇ x ρ = 0, or, ρt + Ω • ∇ x ρ + λ(Ω θ • ∇ x θ + Ω ϕ • ∇ x ϕ) = 0, (4.1)
θ t + a (cos δ (Ω • ∇ x )θ -sin δ sin θ (Ω • ∇ x )ϕ) + λ Ω θ • ∇ x ρ = 0, (4.2)
ϕ t + a (cos δ (Ω • ∇ x )ϕ + sin δ sin θ (Ω • ∇ x )θ) + λ Ω ϕ • ∇ x ρ sin 2 θ = 0. (4.3)
In passing, we note a mistake in formula (4.70) of [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF] where the last term should be divided by sin 2 θ. This mistake does not affect the eigenvalues of the system which are correct. Introducing

U =   ρ θ ϕ   ,
this system is written

U t + A(U)U x + B(U)U y + C(U)U z = 0,
in cartesian coordinates x = (x, y, z). Assuming translation invariance along the z direction, the system is reduced to

U t + C(U)U z = 0, with C(U) =   cos θ -λ sin θ 0 -λ sin θ a cos δ cos θ -a sin δ sin θ cos θ 0 a sin δ cos θ sin θ a cos δ cos θ   .
The system is hyperbolic, if and only if the eigenvalues of C(U) are real for all values of U. The characteristic polynomial of C(U) is given by

P (X) = X 3 -(1 + 2a cos δ) cos θ X 2 + (a (a + 2 cos δ) cos 2 θ -λ 2 sin 2 θ) X + +a (λ 2 cos δ sin 2 θ -a cos 2 θ) cos θ.
We note that in the case sin δ = 0, we recover the results of [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF] where all eigenvalues are real. In the present sin δ = 0 case, the roots of P (X) cannot be analytically computed. However, they are analytically computable in the two cases θ = 0 and θ = π/2.

Case θ = π/2. Then, P (X) = X(X -λ)(X + λ) has real roots X = 0 and X = ±λ. These roots are the same as in the δ = 0 case [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF].

Case θ = 0. Then, P (X) = (X -1)((X -a cos δ) 2 + a 2 sin 2 δ) has one real root X = 1 and two complex conjugate roots X = ae ±iδ . Therefore, as soon as sin δ = 0, the system loses its hyperbolicity near θ = 0. Due to the loss of hyperbolicity of the hydrodynamic model, we look for diffusive corrections by introducing a nonlocal evaluation of the alignment direction. This task is performed in the next section.

Taking into account non-locality in the interaction

Now, we suppose that the Ω f vector is replaced by a non-local evaluation denoted by Ωη f . So, we introduce the following kinetic model:

f t + cv • ∇ x f = 1 ε ∇ v • -(P v ⊥ Ωη f )f + d∇ v f + α( Ωη f × v)f , (5.1)
where

Ωη f = j η f + η 2 r η f |j η f + η 2 r η f | , j η f = (x ′ ,v ′ )∈R 3 ×S 2 K |x ′ -x| η v ′ f (x ′ , v ′ , t) dv ′ dx ′ , r η f = -∇ x (x ′ ,v ′ )∈R 3 ×S 2 Φ |x ′ -x| η f (x ′ , v ′ , t) dv ′ dx ′ ,
is a force term. The first term (given by j η f ) expresses the alignment interaction (like in the Vicsek dynamics) while the second term (given by r η f ) expresses the repulsion interaction. We denote:

x ′ ∈R n K(|ξ|) dξ = k 0 , 1 2n x ′ ∈R n K(|ξ|) |ξ| 2 dξ = k, x ′ ∈R n Φ(|ξ|) dξ = φ.
We can always assume that k 0 = 1. Repulsive interaction here means that we assume φ ≥ 0. Defining u f by

ρ f u f = v ′ ∈S 2 f (v ′ ) v ′ dv ′ ,
we have the following Taylor expansion of v η f :

Ω η f = Ω f + η 2 1 ρ f |u f | ℓ f + o(η 2 ), ℓ f := P Ω ⊥ f (k∆(ρ f u f ) -φ∇ x ρ f ).
Inserting this expression into the kinetic equation (2.1), we get

f t + cv • ∇ x f = 1 ε ∇ v • [-(P v ⊥ Ω f )f + d∇ v f + α(Ω f × v)f ] + η 2 ε 1 ρ f |u f | ∇ v • [-(P v ⊥ ℓ f )f + α(ℓ f × v)f ] + o( η 2 ε ).
We now let η 2 ε = 1 and we let ε → 0. Dropping terms of order o(1) or smaller, this leads to the following problem:

f ε t + cv • ∇ x f ε = 1 ε ∇ v • [-(P v ⊥ Ω f ε )f ε + d∇ v f ε + α(Ω f ε × v)f ε ] + 1 ρ f ε |u f ε | ∇ v • [-(P v ⊥ ℓ f ε )f ε + α(ℓ f ε × v)f ε ] (5.2) = 1 ε Q(f ε ) -L(f ε ), with Lf = - 1 ρ f |u f | ∇ v • [-(P v ⊥ ℓ f )f + α(ℓ f × v)f ] .
The following theorem establishes the formal ε → 0 limit.

Theorem 5.1 We suppose that f ε → f 0 as smoothly as needed. Then, we have

f 0 = ρF Ω , (5.3)
where ρ = ρ(x, t) and Ω = Ω(x, t) satisfy the following system:

∂ t ρ + cc 1 ∇ x (ρΩ) = 0, (5.4) ρ{∂ t Ω + cc 2 cos δ (Ω • ∇ x )Ω + cc 2 sin δ Ω × ((Ω • ∇ x )Ω)} + cdP Ω ⊥ ∇ x ρ + + 1 c 1 {-(2d + c 2 cos δ)P Ω ⊥ (kc 1 ∆(ρΩ) -φ∇ x ρ)+ + (c 2 sin δ -α)(Ω × (kc 1 ∆(ρΩ) -φ∇ x ρ))} = 0. (5.5)
and the coefficients c 1 , c 2 and δ are the same as in Theroem 3.1. We restrict ourselves to the case 2d + c 2 cos δ ≥ 0 which is the condition for the system to be stable.

Remark 5.1 A special case is c = 0, ρ = 1, φ = 0. This leads to

∂ t Ω + k(2d + c 2 cos δ) Ω × (Ω × ∆Ω) + k(c 2 sin δ -α)(Ω × ∆Ω) = 0,
which is the classical Landau-Lifschitz-Gilbert equation, provided that 2d + c 2 cos δ ≥ 0.

Proof. Since Lf is a divergence in velocity space, the mass conservation equation is unaffected by this additional term compared to section 3. The additional contribution of Lf to the momentum equation is a term of the form:

L (k) := S 2 L(ρF Ω )ψ (k) dv, k = 1, 2, (5.6)
at the left hand side of (3.15). We have

L(ρF Ω ) = - 1 c 1 ∇ v • [-(P v ⊥ ℓ)F Ω + α(ℓ × v)F Ω ] ,
where ℓ := ℓ ρF Ω := P Ω ⊥ (kc 1 ∆(ρΩ) -φ∇ x ρ).

Therefore, using Green's formula, we get:

L (k) = 1 c 1 S 2 [-(P v ⊥ ℓ) + α(ℓ × v)] • ∇ v ψ (k) F Ω dv = 1 c 1 S 2 [-ℓ + α(ℓ × v)] • ∇ v ψ (k) F Ω dv = 0 = 1 c 1 S 2 -∇ v ψ (k) + α(v × ∇ v ψ (k) ) F Ω dv • ℓ.
Now, we use the formulas:

S 2 ∇ v g dv = 2 S 2 vg dv, S 2 (∇ v g)h dv = 2 S 2 vgh dv - S 2 (∇ v h)g dv, S 2 (v × ∇ v g)h dv = - S 2 (v × ∇ v h)g dv,
for any pair of scalar functions g, h on S 2 . We get:

L (k) = 1 c 1 S 2 [-2vF Ω + ∇ v F Ω -α(v × ∇ v F Ω )] ψ (k) dv • ℓ. Now, we note that ∇ v F Ω = β(P v ⊥ Ω)F Ω , which leads to L (k) = 1 c 1 S 2 [-2v + β(P v ⊥ Ω) -αβ(v × Ω)] ψ (k) F Ω dv • ℓ, (we used that v × Ω = v × (P v ⊥ Ω)). We use the decomposition v = v ⊥ + v . Since ℓ⊥Ω and P Ω ⊥ P v ⊥ Ω = -(v • Ω)v ⊥ ,
we can write:

L (k) = 1 c 1 S 2 [-(2 + β(v • Ω))v ⊥ -αβ(v ⊥ × Ω)] ψ (k) F Ω dv • ℓ.
Introducing the coordinates ℓ = (ℓ 1 , ℓ 2 , 0) in the cartesian basis associated to Ω, and decomposing v ⊥ accordingly, we get:

L (k) = 1 c 1 S 2 [-(2 + β cos θ)(cos ϕℓ 1 + sin ϕℓ 2 ) -αβ(sin ϕℓ 1 -cos ϕℓ 2 )] ψ (k) F Ω sin θ dv = 1 c 1 S 2 [(-(2 + β cos θ)ℓ 1 + αβℓ 2 ) cos ϕ+ +(-(2 + β cos θ)ℓ 2 -αβℓ 1 ) sin ϕ] ψ (k) F Ω sin θ dv.
Finally, using (3.9), (3.10), we get:

L (1) = 1 c 1 S 2 [(-(2 + β cos θ)ℓ 1 + αβℓ 2 ) cos ϕ + (-(2 + β cos θ)ℓ 2 -αβℓ 1 ) sin ϕ] (ψ 1 (θ) cos ϕ + ψ 2 (θ) sin ϕ) F Ω sin θ dv, L (2) = 1 c 1 S 2 [(-(2 + β cos θ)ℓ 1 + αβℓ 2 ) cos ϕ + (-(2 + β cos θ)ℓ 2 -αβℓ 1 ) sin ϕ] (-ψ 2 (θ) cos ϕ + ψ 1 (θ) sin ϕ) F Ω sin θ dv,
and performing the integration with respect to ϕ ∈ [0, 2π], we are led to F Ω sin 2 θ dθ.

L (1) = 1 2c 1 π 0 [(-(2 + β cos θ)ℓ 1 + αβℓ 2 )ψ 1 (θ) + (-(2 + β cos θ)ℓ 2 -αβℓ 1 )ψ 2 (θ)] F Ω sin 2 θ dθ, L (2)
In vector notations, this is written:

  L (1) L (2) 0   = 1 c 1 {-(2[A] + β[B])ℓ + αβ[A](ℓ × Ω)} .
Finally, collecting all terms leads to the following equation: + α((kc 1 ∆(ρΩ) -φ∇ x ρ) × Ω)} = 0, Using (3.17), we can also write this equation in the form (5.5), which ends the proof of the theorem.

Conclusion

In this paper, we have derived the hydrodynamic limit of a kinetic model of self-propelled particles with alignment interaction and with precession about the alignment direction.

We have shown that the resulting system consists of a conservative equation for the local density and a non-conservative equation for the orientation. We have observed that this system may lose its hyperbolicity. Then, we have provided an extension of the model including diffusion terms under the assumption of weakly non-local interaction. In the particular case of zero self-propelling speed, we have noted that the resulting model is nothing but the classical Landau-Lifschitz-Gilbert equation. Therefore the present theory provides one of the very few (if not the only) kinetic justification of the phenomenological Landau-Lifschitz-Gilbert equation. Future works in this direction will include the sensitivity analysis of the hydrodynamic model in terms of the modeling parameters, the accounting for phase transitions, the influence of a vision angle, and the inclusion of the particle interactions with a surrounding fluid in the perspective of modeling active particles suspensions. Another research direction consists in investigating the mathematical structure of these models, prove well-posedness and investigate its qualitative properties.

( 1 ). 7 ) 3 . 1

 1731 k are the GCI's found in proposition 2.3. Then, introducing (ρ a , θ a ), (ρ b , θ b ) the polar coordinates of the vectors (a 1 , a 2 ) and (b 1 , b 2 ), i.e. a 1 + ia 2 = ρ a e iθa , b 1 + ib 2 = ρ b e iθ b , c 2 and δ are defined byc 2 = ρ b ρ a , δ = θ b -θ a . (3Remark Model (3.2), (3.3) will be referred to as the hydrodynamic model for selfalignment interactions with precession.

with a k

  and b k (k = 1, 2) defined at (3.5), (3.6) and inserting (3.12) into (3.13),[START_REF] Bolley | Mean-field limit for the stochastic Vicsek model[END_REF]

2 +

 2 β cos θ)ℓ 1 + αβℓ 2 )ψ 2 (θ) + (-(2 + β cos θ)ℓ 2 -αβℓ 1 )ψ 1 (θ)]

[

  A]P Ω ⊥ (c∇ x ρ + βρ∂ t Ω) + βρc[B]P Ω ⊥ ((Ω • ∇ x )A] + β[B])ℓ + αβ[A](ℓ × Ω)} = 0.Now, multiplying the first two lines of this vector equation by β[a] -1 , we getρ(∂ t Ω + c[C 2 ](Ω • ∇ x )Ω) + cdP Ω ⊥ ∇ x ρ + 1 c 1 {-(2dId + [C 2 ])ℓ + α(ℓ × Ω)} = 0,or, expliciting the expression of ℓ:ρ(∂ t Ω + c[C 2 ](Ω • ∇ x )Ω) + cdP Ω ⊥ ∇ x ρ + 1 c 1 {-(2dId + [C2])P Ω ⊥ (kc 1 ∆(ρΩ) -φ∇ x ρ)+

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