Laurent Boudin
email: laurent.boudin@upmc.fr

Julien Mathiaud
email: julien.mathiaud@cea.fr

ASYMPTOTIC BEHAVIOR OF A DIFFUSIVE SCHEME SOLVING THE INVISCID ONE-DIMENSIONAL PRESSURELESS GASES SYSTEM

Keywords: 2000 Mathematics Subject Classification. Primary: 65M12, 58F17; Secondary: 53C35 Pressureless gases, vanishing viscosity, diffusive scheme, regularity loss, asymptotic behavior

In this work, we discuss some numerical properties of the viscous numerical scheme introduced in [8] to solve the one-dimensional pressureless gases system, and study in particular, from a computational viewpoint, its asymptotic behavior when the viscosity parameter ε > 0 used in the scheme becomes smaller.

1. Introduction. In this work, we focus on the one-dimensional system describing a pressureless gas, which writes, for any T > 0,

∂ t ρ + ∂ x (ρu) = 0, (1)
∂ t q + ∂ x (qu) = 0, (2)
in (0, T ] × R, where ρ(t, x) ≥ 0 is the gas density and q(t, x) ∈ R is the momentum at time t ∈ [0, T ] and location x ∈ R. The gas velocity u(t, x) ∈ R must be somehow defined as a quotient of q by ρ, but there is trouble when ρ is 0. That is why one needs the notion of duality solutions [START_REF] Bouchut | Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness[END_REF], previously introduced for conservation laws [START_REF] Bouchut | One-dimensional transport equations with discontinuous coefficients[END_REF]. Clearly, ( 1) and ( 2) can be seen as conservation laws, for mass and momentum respectively. System (1)-( 2) is supplemented with initial conditions

ρ(0, •) = ρ in , q(0, •) = q in . (3)
This system arises from very various physical situations (cold plasmas [START_REF] Bouchut | On zero pressure gas dynamics[END_REF], astrophysics [START_REF] Ya | Gravitational instability: An approximate theory for large density perturbations[END_REF][START_REF] Brenier | A modified least action principle allowing mass concentrations for the early universe reconstruction problem[END_REF], traffic models [START_REF] Berthelin | A model for the formation and evolution of traffic jams[END_REF][START_REF] Vuyst | Nonlocal second order vehicular traffic flow models and Lagrange-remap finite volumes[END_REF]...) and has been mathematically studied in numerous articles, such as [START_REF] Rykov | Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics[END_REF][START_REF] Brenier | Sticky particles and scalar conservation laws[END_REF][START_REF] Poupaud | Measure solutions to the linear multi-dimensional transport equation with non-smooth coefficients[END_REF][START_REF] Bouchut | Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness[END_REF].

We here choose a periodic framework: we focus on [0, 1] and impose that mass density, velocity and momentum have the same values at both x = 0 and x = 1, so that the solutions are 1-periodic.

When ρ and u are smooth, and if ρ remains non zero, (2) can be modified into the standard Burgers equation thank to (1):

∂ t u + ∂ x u 2 2 = ∂ t u + u∂ x u = 0. ( 4
)
The mass density ρ then solves a basic transport equation, where u is given: it does not depend on ρ because of (4). But it is common knowledge that, in finite time, a mass concentration phenomenon can happen, for instance, when u does not increase. Consequently, the regularity properties of u and ρ are lost, and u does not solve (4) anymore.

From the numerical viewpoint, one can think about several ways to discretize (1)-( 3). Kinetic schemes [START_REF] Bouchut | On zero pressure gas dynamics[END_REF][START_REF] Bouchut | Numerical approximations of pressureless and isothermal gas dynamics[END_REF] or particle methods [START_REF] Chertock | A new sticky particle method for pressureless gas dynamics[END_REF] allow to use the kinetic framework underlying the pressureless gas dynamics. It is also natural to try numerical schemes related to hyperbolic conservation laws [START_REF] Gosse | Numerical approximations of one-dimensional linear conservation equations with discontinuous coefficients[END_REF] or relaxation schemes [START_REF] Berthon | A relaxation scheme for the approximation of the pressureless Euler equations[END_REF].

In [START_REF] Boudin | A numerical scheme for the one-dimensional pressureless gases system[END_REF], we prove that upwind schemes were not an option, since they failed to ensure the discrete one-sided Lipschitz (OSL) condition introduced [START_REF] Brenier | The discrete one-sided Lipschitz condition for convex scalar conservation laws[END_REF] for convex scalar conservation laws. Then, following the strategy of [START_REF] Boudin | A solution with bounded expansion rate to the model of viscous pressureless gases[END_REF] at the continuous level, we add an artificial viscosity. The new diffusive scheme we obtain is proven to be L ∞ -stable and consistent, hence converging towards the solution of the viscous pressureless gases system. In particular, it satisfies the discrete OSL condition. We here investigate, at the numerical level, the asymptotic behavior of the same numerical scheme when the artificial viscosity vanishes.

2. Diffusive numerical scheme. Let us recall the diffusive scheme we presented in [START_REF] Boudin | A numerical scheme for the one-dimensional pressureless gases system[END_REF]. Consider ∆t, ∆x > 0 such that N = T /∆t ∈ N and I = 1/∆x ∈ N, and set λ = ∆t/∆x. Denote ρ n i , q n i and u n i the approximate values of ρ, q and u at time n∆t ∈ [0, T ] and location (i + 1/2)∆x ∈ [0, 1], for 0 ≤ n ≤ N and 0 ≤ i < I. Of course, thanks to the periodicity property, ρ n i , q n i and u n i can be extended for any i ∈ Z. For the sake of readability, in the previous notations, we may drop the time iteration index n and replace n + 1 by a prime symbol " ′ ".

Note that the discrete OSL condition can be written as nλ(u n i+1 -u n i ) ≤ 1, for any i and n > 0.

Let us describe step by step the strategy for our scheme.

2.1. Periodic initial data. We choose arbitrary 1-periodic initial data ρ in ≥ 0, u in ≥ 0. Indeed, the viscous problem in [START_REF] Boudin | A solution with bounded expansion rate to the model of viscous pressureless gases[END_REF] deals with mass density and velocity, and not momentum.

2.2.

Regularizing initial data. We take a fixed ε > 0, small enough. We may have to regularize both u in and ρ in so that u in , ρ in ∈ C 1 (R; R * + ) satisfy the assumptions of Theorem 2 in [START_REF] Boudin | A numerical scheme for the one-dimensional pressureless gases system[END_REF], i.e.

ρ in (x) ≥ Cε 1/4 , u in (x) ≤ C, (u in ) ′ (x) ≤ C √ ε , ∀x ∈ [0, 1], (5)
where C ≥ 0 is a constant not depending on ε. Note that ρ in must lie in R * + , since the continuous viscous model involves a division by ρ.

The following quantities

U = max [0,1] u in > 0, V = min [0,1] u in > 0, A = max(0, max [0,1] (u in ) ′ ) ≥ 0, R = min [0,1]
ρ in > 0, may depend on ε. More precisely, they must satisfy properties inherited from (5), i.e.

R ≥ Cε 1/4 , V ≤ U ≤ C, A ≤ C √ ε , (6)
where C ≥ 0 does not depend on ε.

2.3.

Choosing the time and space steps. The steps ∆t and ∆x > 0 are then chosen such that

0 < ∆x ≤ 2V 1 + A , (7)
0 < ∆t ≤ min 1 4A + 1 , 1 4U ∆x, R 4ε(1 + AT ) ∆x 2 , (8)
where we set λ = ∆t/∆x and σ = ∆t/∆x 2 .

2.4. Writing the scheme. We eventually write the discretization of the viscous pressureless gases system as

u ′ i = u i -λ u i 2 2 - u i-1 2 2 + εσ ρ i (u i-1 + u i+1 -2u i ), (9)
ρ ′ i = (1 -λu ′ i )ρ i + λu ′ i-1 ρ i-1 . (10)
2.5. Properties of the scheme. In [START_REF] Boudin | A numerical scheme for the one-dimensional pressureless gases system[END_REF], we prove the following Theorem 2.1. We assume that (7)-( 8) hold. Then we have, for any i and n ≥ 0,

V ≤ u n i ≤ U, u n i -u n i-1 ≤ A∆x 1 + An∆t , ρ n i ≥ R 1 + An∆t ≥ R 1 + AT > 0.
Moreover, the discrete total mass is conserved. Finally, when ε > 0 is fixed, scheme

-( 10) is first-order consistent (in both t and x) with the viscous pressureless gases system, and is monotonic.

In other words, both discrete unknown functions satisfy maximum principles, and the discrete velocity satisfies the OSL condition. Assumptions [START_REF] Boudin | A numerical scheme for the one-dimensional pressureless gases system[END_REF] on ∆t are crucial to ensure the stability of the scheme. They are related to the standard assumptions to get stability for explicit schemes on transport or diffusion equations.

3. Numerical study. The viscosity parameter ε > 0 is chosen in the first place. We use several values of ε in the following computations: 10 , 10 -5 , 10 -6 , 5×10 -7 , 10 -7 and the smallest value ε 0 = 10 -6 /14, which will be considered as the reference situation. We take T = 1 s as the final time. Moreover, we focus on one case with (almost, see the discussion below) smooth initial data. It allows to point out all the problems arising the pressureless gases system, in particular the regularity loss phenomenon, already explained in Section 1.

We set, for any x ∈ [0, 1],

ρ in (x) = 1 + 1 2 cos(4πx), u in (x) = 2x(1 -x) + 1 2 .
We clearly have U = 1, V = 1/2, A = 2 and R = 1/2. When ε belongs to the set of values above, ( 6) is clearly satisfied. Nevertheless, u in is not C 1 (R; R * + ). Indeed, there is an issue at the bounds of [0, 1]. Therefore, we should need to regularize u in . Fortunately, it is not necessary: the regularization near 0 and 1 can happen on intervals whose length may be chosen smaller than 2∆x. We can choose it so that (u in ) ′ remains in [-2, 2], and consequently, A still equals 2.

Let us now choose ∆x = 20ε. Again, if ε belongs to our same set of values, (7) obviously holds. Then ∆t can be taken as the optimal possible value given by ( 8), i.e. ∆t = ε min (5, 50/(1 + 2T )) = 5ε.

We want to study the behavior of our numerical solution with respect to ε. In the following, since ̺ has a measure meaning when ε goes to zero, it is really more convenient to focus on the cumulative mass M , defined by

M (t, x) = x 0 ̺(t, y) dy.
We first draw the evolution of M and u with respect to x, at two different times, before and after approximately t = 0.5. Indeed, until t = 0.5, at the continuous level, u solves the Burgers equation, and there is no regularity loss. But, near time t = 0.5, a regularity loss happens because of mass concentration and subsequent vacuum appearance. The solutions are then displayed for various values of ε. More precisely, Fig. 1-2 respectively give the behavior of M and u at time t = 0.4, i.e. before the regularity loss. Of course, the curves are sharpened with smaller values of ε, but there is still no jump at all for either ρ or u. As the reader can see on Fig. 5, till the regularity loss, the numerical conservation of Q is quite satisfactorily ensured. The property is not recovered after the loss. This drawback of our scheme ( 9)-( 10) was already pointed out in [START_REF] Boudin | A numerical scheme for the one-dimensional pressureless gases system[END_REF]: we had to choose between the total momentum conservation (if we write a scheme for ρ and q) and the OSL condition (for which we wrote a scheme for ρ and u). The latter is in fact crucial to ensure that we select the right solution to the pressureless gases system. Nevertheless, we can at least state that the order of magnitude of Q is conserved.

Eventually, we focus on the behavior with respect to ε. We focus on the following quantities

E M (t) = max x∈[0,1] |M (t, x) -M 0 (t, x)|, (11)
E u (t) = max x∈[0,1] |u(t, x) -u 0 (t, x)|, (12)
for the values ε under study. Quantities indexed by 0 are of course the ones related to the reference value of ε, ε 0 . Quantities on coarser grids are projected on the (fine) grid of the reference value (ε 0 corresponds to 700,000 space cells) in order to compute the discretized versions of ( 11)-( 12). These results give us hints of what should be the asymptotic behavior of our solutions, in particular when we loose regularity. The convergence seems very slow in ε but it was expected since the smallest power of ε in the main hypotheses of the scheme is equal to one fourth, see Eqn. 6. That partially explains that the errors are quite large, even for quite small values of ε. In fact, we were not able to understand why ε = 10 -7 gives such a different behavior (in order of magnitude) compared to not so larger values of ε, whereas, from 10 -4 to 10 -6 , the results are close after the regularity loss. Perspectives. As expected, the crucial point in the asymptotic behavior of our scheme is strongly linked to the regularity loss phenomenon which occurs for the pressureless gases. We should be able to prove reasonable convergence estimates with respect to ε as long as the solutions remain smooth. Beyond the regularity loss, the situation remains quite unclear.

Figure 1 .Figure 2 .

 12 Figure 1. Behavior, at time 0.4, of M (a) on [0, 1], (b) focused around x = 0.2

Fig. 3 -Figure 3 .

 33 Fig.3-4 then present the typical profiles of both M and u after the regularity loss, here given at final time T = 1. The jump is now visible at x = 0.59 on both figures, and u of course still decreases at the jump abscissa.

Figure 4 .

 4 Figure 4. Behavior, at final time, of u (a) on [0, 1], (b) focused around x = 0.59Let us now study the evolution of the total momentumQ(t) =

Figure 5 .

 5 Figure 5. Checking the momentum for several values of ε

Fig. 6 -Figure 6 .

 66 Figure 6. Behavior of E M w.r.t. t for several values of ε

Figure 7 .

 7 Figure 7. Behavior of E u w.r.t. t for several values of ε 4.Perspectives. As expected, the crucial point in the asymptotic behavior of our scheme is strongly linked to the regularity loss phenomenon which occurs for the pressureless gases. We should be able to prove reasonable convergence estimates with respect to ε as long as the solutions remain smooth. Beyond the regularity loss, the situation remains quite unclear.

Acknowledgments. The authors want to thank Frédéric Lagoutière for the (ongoing) discussions they share about this topic.


Dr Diane S Lazard
email: diane.lazard@bjn.aphp.fr

Hyo-Jeong Lee

Eric Truy

Anne-Lise Giraud

Bilateral reorganization of posterior temporal cortices in post-lingual deafness and its relation to cochlear implant outcome

Keywords: disinhibition, fMRI, imagery, phonology, plasticity, sound memory, supramarginal gyrus

Post-lingual deafness induces a decline in the ability to process phonological sounds or evoke phonological representations. This decline is paralleled with abnormally high neural activity in the right posterior superior temporal gyrus/supramarginal gyrus (PSTG/SMG). As this neural plasticity negatively relates to cochlear implantation (CI) success, it appears important to understand its determinants. We addressed the neuro-functional mechanisms underlying this maladaptive phenomenon using behavioral and fMRI data acquired in ten normal-hearing subjects and ten post-lingual deaf candidates for CI. We compared two memory tasks where subjects had to evoke phonological (speech) and environmental sound representations from visually presented items. We observed dissociations in the dynamics of right versus left PSTG/SMG neural responses as a function of duration of deafness. Responses in the left PSTG/SMG to phonological processing and responses in the right PSTG/SMG to environmental sound imagery both declined. However, abnormally high neural activity was observed in response to phonological visual items in the right PSTG/SMG, i.e., contralateral to the zone where phonological activity decreased. In contrast, no such responses (overactivation) were observed in the left PSTG/SMG in response to environmental sounds. This asymmetry in functional adaptation to deafness, suggests that maladaptive reorganization of the right PSTG/SMG region is not due to balanced hemispheric interaction, but to a specific take-over of the right PSTG/SMG region by phonological processing, presumably because speech remains behaviorally more relevant to communication than the processing of environmental sounds. These results demonstrate that cognitive long-term alteration of auditory processing shapes functional cerebral reorganization.

INTRODUCTION

Cochlear implant (CI) is a hearing restoration technique used in congenital and acquired severe to profound sensorineural deafness, that relies on transforming acoustic information into electric input applied to the auditory nerve. In post-linguistically deaf adults, duration of auditory deprivation is a factor known to contribute to speech recognition ability after implantation [START_REF] Blamey | Factors affecting auditory performance of postlinguistically deaf adults using cochlear implants[END_REF]. Additional factors contribute to outcome variability, e.g. pre-implant residual hearing [START_REF] Rubinstein | Residual speech recognition and cochlear implant performance: effects of implantation criteria[END_REF]), but to date none of the identified clinical variables fully explains all of the variability [START_REF] Blamey | Factors affecting auditory performance of postlinguistically deaf adults using cochlear implants[END_REF][START_REF] Giraud | Predicting cochlear implant outcome from brain organisation in the deaf[END_REF][START_REF] Green | Predictors of audiological outcome following cochlear implantation in adults[END_REF]. Other factors may intervene, including the amount and trajectory of cerebral plasticity that may have taken place during the period of deafness [START_REF] Giraud | Predicting cochlear implant outcome from brain organisation in the deaf[END_REF]Lazard, et al. 2010b;[START_REF] Strelnikov | Does brain activity at rest reflect adaptive strategies? Evidence from speech processing after cochlear implantation[END_REF].

The right posterior superior temporal gyrus/supramarginal gyrus (PSTG/SMG) undergoes a profound reorganization during deafness that is manifest in a hypermetabolism at rest [START_REF] Giraud | Predicting cochlear implant outcome from brain organisation in the deaf[END_REF]Lee, et al. 2007a), and in abnormal activation levels during visual speech tasks (rhyming and lip-reading tasks) (Lazard, et al. 2010b;Lee, et al. 2007b). We previously argued that the involvement of this region in visually-driven speech processing may be maladaptive, as it predicts poor phonological scores in deaf subjects and poor auditory speech perception after receiving a CI (Lazard, et al. 2010b). The progressive engagement of the right PSTG/SMG following auditory deprivation duration is observed in conjunction with the disengagement of the left PSTG/SMG (Lee, et al. 2007b), a cortical region that normally underpins phonological processing [START_REF] Binder | Functional neuroimaging of language[END_REF]. The engagement of the right PSTG/SMG hence presumably follows from reduced neural activity in the left homologue reflecting phonological decline when auditory feedback is lacking [START_REF] Andersson | Cognitive correlates of visual speech understanding in hearing-impaired individuals[END_REF]Lazard, et al. 2010b;Lee, et al. 2007b;[START_REF] Schorr | Auditory-visual fusion in speech perception in children with cochlear implants[END_REF]. The decline in phonological skills in post-lingual deaf subjects [START_REF] Anderson | Recent results with the MED-EL COMBI 40+ cochlear implant and TEMPO+ behind-the-ear processor[END_REF]Lazard, et al. 2010b) could also arise from more general alterations of auditory memory (not exclusively phonological memory) due to the sparseness of auditory inputs [START_REF] Andersson | Cognitive correlates of visual speech understanding in hearing-impaired individuals[END_REF].

As the right PSTG/SMG is normally more involved in environmental sound processing [START_REF] Lewis | Human brain regions involved in recognizing environmental sounds[END_REF][START_REF] Toyomura | Neural correlates of auditory feedback control in human[END_REF]) than in phonological processing [START_REF] Hartwigsen | Phonological decisions require both the left and right supramarginal gyri[END_REF], we conducted a fMRI task in ten post-lingual deaf subjects and ten normal-hearing controls. Participants had to perform a mental imagery task in which they had to retrieve environmental sounds from pictures of noisy objects [START_REF] Halpern | When that tune runs through your head: a PET investigation of auditory imagery for familiar melodies[END_REF]. Seven of the deaf subjects had previously participated in another fMRI study, in which they performed a rhyming task on visual material (Lazard, et al. 2010b). We re-analyzed these data and compared the dynamics of functional interactions across the two PSTG/SMG regions during phonological and environmental sound processing using a cross-sectional approach. This approach involves observing deaf patients at one single time point, where they exhibit different durations of auditory deprivation and evolving hearing loss (i.e. from the beginning of hearing aid use). This represents an alternative to more demanding longitudinal studies, in which neuroplasticity is followed-up in each patient.

Unilateral cortical lesions can induce further dysfunction in the same cortical hemisphere [START_REF] He | Breakdown of functional connectivity in frontoparietal networks underlies behavioral deficits in spatial neglect[END_REF][START_REF] Price | Dynamic diaschisis: anatomically remote and context-sensitive human brain lesions[END_REF] or elicit a compensation by recruiting homologous areas in the contralateral hemisphere [START_REF] Heiss | Differential capacity of left and right hemispheric areas for compensation of poststroke aphasia[END_REF][START_REF] Winhuisen | Role of the contralateral inferior frontal gyrus in recovery of language function in poststroke aphasia: a combined repetitive transcranial magnetic stimulation and positron emission tomography study[END_REF][START_REF] Winhuisen | The right inferior frontal gyrus and poststroke aphasia: a follow-up investigation[END_REF]), e.g., through disinhibition of transcallosal connections [START_REF] Thiel | Direct demonstration of transcallosal disinhibition in language networks[END_REF][START_REF] Warren | Anterior temporal lobe connectivity correlates with functional outcome after aphasic stroke[END_REF]). As most cognitive functions are asymmetrically implemented in the brain [START_REF] Formisano | Who" is saying "what"? Brainbased decoding of human voice and speech[END_REF][START_REF] Hickok | The cortical organization of speech processing[END_REF][START_REF] Hugdahl | Lateralization of cognitive processes in the brain[END_REF], the engagement of the contralateral cortex may be less efficient than the primary functional organization, and hence maladaptive [START_REF] Kell | How the brain repairs stuttering[END_REF][START_REF] Marsh | Recovery from aphasia following brain injury: the role of reorganization[END_REF][START_REF] Martin | Measurements of auditory-verbal STM span in aphasia: effects of item, task, and lexical impairment[END_REF][START_REF] Naeser | Improved picture naming in chronic aphasia after TMS to part of right Broca's area: an open-protocol study[END_REF][START_REF] Preibisch | Evidence for compensation for stuttering by the right frontal operculum[END_REF][START_REF] Van Oers | Contribution of the left and right inferior frontal gyrus in recovery from aphasia. A functional MRI study in stroke patients with preserved hemodynamic responsiveness[END_REF].

We assumed that the ability to mentally evoke stored sounds, whether phonological or environmental, i.e. auditory memory, declines during acquired deafness. We thus expected to observe a parallel decrease in neural responses to phonological processing in the left PSTG/SMG and to environmental sound in the right PSTG/SMG activation. If demonstrated, the latter loss of cortical function could play a facilitating role in the enhanced engagement of the right PSTG/SMG in phonological processing (Lazard, et al. 2010b). This asymmetry in brain plasticity could reflect the fact that maintaining environmental sound processing during deafness is less relevant than maintaining oral communication. Alternatively, however, if communication needs play no special role in neural plasticity, there should be no asymmetry in neural adaptation to the decline in the processing of phonological and environmental sounds. We therefore expected our results to distinguish between two possibilities: i) if right phonological maladaptive plasticity results from reciprocal hemispheric interaction, a decline in environmental sound processing should prompt equivalent overactivation in the left PSTG/SMG during the evocation of sounds, ii) if phonological maladaptive plasticity is facilitated by environmental sound decline and the stronger need to maintain efficient oral communication, only the right PSTG/SMG should display an over-activation. From a clinical perspective, validating the latter hypothesis would imply that it may be important to train environmental sound memory in profound hearing deficits because it prevents the reorganization of the right PSTG/SMG, a functional adaptation that is deleterious to the success of cochlear implantation.

METHODS

Subjects (Table I)

Twenty adults participated in this fMRI study, which was approved of by the local ethics committee (CPP, Sud-Est IV, Centre Léon Bérard, Lyon, France). Written informed consent was obtained from all participants. The samples were composed of ten post-lingually severe to profound deaf subjects (8 women and 2 men, mean age ± s.d = 53 ± 14.7 years), who were candidates for cochlear implantation, and ten age-matched normal-hearing controls (6 women and 4 men, mean age ± s.d = 41 ± 13.8 years). The hearing of subjects was assessed by audiometric testing. Normal hearing for control subjects was defined by pure tone thresholds ≤ 20 dB HL for frequencies from 500 to 4000 Hz. All subjects had normal or corrected-tonormal vision, no history of neurological pathology, and were right-handed according to the Edinburgh handedness inventory [START_REF] Oldfield | The assessment and analysis of handedness: the Edinburgh inventory[END_REF]. CI candidates 1 to 9 (Table 1) met the classical criteria for cochlear implantation, i.e. average of 0.5, 1, 2 kHz hearing threshold >90 dB, <30% sentence recognition score with best-fitted hearing aid (according to NIH Consensus Statement, 1995), and no word recognition using a list composed of threephoneme monosyllabic words presented at 60 dB (HL) (Lafon test, Lafon 1964) with bestfitted hearing aid. CI candidate number 10 was referred to cochlear implantation in agreement with extended indications for implantation (≤50% sentence recognition score with best-fitted hearing aids) [START_REF] Cullen | Cochlear implantation in patients with substantial residual hearing[END_REF]). The evolution of her deafness from mild to severe had been steep and occurred during childhood. It then stayed stable for many years until this study was performed. All but one CI candidate had progressive sensorineural hearing loss and wore hearing aids. Subject number 7 had had bilateral petrous bone fracture, yielding sudden deafness. Hearing loss was first severe and rapidly deteriorated within 6 months into profound deafness. No attempt was made to fit a hearing aid for this subject.

Table 1 provides the duration of auditory deprivation, i.e. the time elapsed since subjects could no longer communicate by hearing, even with the best-fitted hearing aids. Subject 10, was excluded from correlations between duration of auditory deprivation and imaging results because he presented some residual hearing. The Table also provides the duration of hearing loss, i.e. the time elapsed since subjective auditory acuity decreased, leading to the use of hearing aids. The use of hearing aids was bilateral in the case of bilateral progressive hearing loss (Subjects 2 to 6 and 8 to 10), or unilateral in the case of progressive hearing loss in one ear and sudden severe deafness in the other ear (idiopathic etiology 15 years before, Subject 1).

None of the ten CI candidates used sign language, and they all relied on speechreading and written language for communication. Deaf patients were either implanted with a Cochlear device (Melbourne, Australia) or with a MXM device (Vallauris, France). Word recognition scores were measured six months after implantation using the Lafon test (as for the preoperative assessment).

Experimental paradigm

fMRI data were acquired during mental imagery of sounds evoked by visually presented items. This experiment was performed by controls and CI candidates prior to surgery.

Acquired data were related to post-operative word recognition scores and clinical variables (auditory deprivation duration and hearing loss duration) in the latter group.

Subjects were asked to retrieve the sound produced by a visually presented object or animal (white and black pictures). Objects included musical instruments (piano, trumpet, etc), transportation means (car, train, plane, etc), and other everyday objects like a door and a clock.

Two control tasks consisted of either imagining the color of those items (only silent objects, e.g., shirt, snowman) or silently naming them. The aim of the color imagery task was to target specifically sound memory processing. The aim of the naming task was to eliminate subjects with semantic dysfunctions. Conditions were randomized across subjects. The visual stimuli comprised 80 pictures, all shown once. Subjects were requested to self-rate their ability to mentally evoke the sound or color of each item or to name them. They gave their response by button press (left button if possible, right button if not) after each presentation within the scanner. Answers and related reaction times were recorded. No objective control was provided in this experiment because the items used for the experiment were sampled from everyday life (piano, car, dog…) and could not be unknown by the participants [START_REF] Bunzeck | Scanning silence: mental imagery of complex sounds[END_REF]. The fMRI experiment included two runs of event-related design. Images were presented for 1.5 seconds and were followed by a fixation cross whose duration randomly varied from 1 to 7 seconds. A screen showing written instructions (3.5 seconds) preceded each image. All subjects performed a training session using Presentation software version 9.90 (Neurobehavioral system, Inc., Albany, CA, USA).

Cerebral activity of the left and right PSTG/SMG was compared in seven CI candidates who also performed a phonological (rhyming) task in a separate session. The phonological task consisted in comparing the ending of two visually presented common regular French words.

The material involved three possible phonological endings with different orthographic spellings, so that judgements could not be based on orthography, but had to be based solely on phonology. The results from this session are reported in a separate manuscript (Lazard, et al. 2010b). The stimuli were comprised of 80 words. The same scanner and the same methods of acquisition as in the imagery task were used. Subjects were requested to determine if words rhymed or not by a button press after each presentation of pairs of words. The seven subjects who performed this task are numbered 1 to 3, 6, and 8 to 10 in Table I D. S. Lazard

Imaging parameters for fMRI experiment

Gradient echo-planar fMRI data with blood oxygenation level dependent contrast were acquired with a 1.5T magnetic resonance scanner (Siemens Sonata, Medical Systems, Erlangen, Germany) with standard head coil to obtain volume series with 33 contiguous slices (voxel size 3.4 x 3.4 x 4 mm, no gap, repetition time 2.95 s, echo time 60 ms) covering the whole brain. Earplugs (mean sound attenuation 30 dB) and earmuffs (mean sound attenuation 20 dB) were provided both to controls and deaf subjects to equate as much as possible the experimental environments of both groups. We acquired 498 functional images in two runs per subject.

Data analyses

Answers and reaction times (expressed as means ± standard deviation) were compared across groups using non-parametric Mann-Whitney-Wilcoxon tests.

The fMRI data were analyzed using SPM5 (Statistical Parametric Mapping, Centre for Neuroimaging, London, UK, http: //www.fil.ion.ucl.ac.uk/spm) in a Matlab 7.1 (Mathworks, Natick, MA, USA) environment and displayed using MRIcron software (www.sph.sc.edu/comd/rorden/mricron). We performed standard preprocessing (realignment, unwarping, normalization and spatial smoothing with an 8-mm full width at half-maximum Gaussian kernel), and calculated contrast images versus baseline in each single subject for each task (imagine sound, imagine color, and naming). We performed group analyses of each contrast (one sample t-tests), plus contrasts comparing each condition with every other. Group differences (simple group effects and group-by-task interactions) between CI candidates and controls in the sound imagery task (minus color) were also explored using ANOVA modeling groups and conditions, and two sample t-tests. In group-by-task interactions, results were masked by the simple main effect of sound imagery in each sample (inclusive mask). Onesample t-tests analyses were reported using the false discovery rate correction (FDR) thresholded at p<0.05. Uncorrected statistics was used for the two-sample t-tests and interactions at p<0.001. Correlations with behavioral data (i.e., hearing loss/deafness duration, post-CI word recognition) using non-parametric Spearman tests (p<0.05) were also tested.

For the CI candidates group, we entered contrast images into a regression analysis using post-CI word recognition scores, hearing loss duration, and auditory deprivation duration. Because we only used this whole brain analyses to confirm post-hoc correlations, we used a p=0.01 threshold, which is acceptable when exploring a-priori defined regions (see Table III).

Values of the peak voxel for the right and left PSTG/SMG in the two tasks (sound imagery and phonological processing) were extracted for the CI candidates (n=10 for the imagery task and n=7 in common for the phonology task) and the respective controls. Values were compared across groups using non-parametric Mann-Whitney-Wilcoxon tests, and tested for correlations with deafness duration (i.e. either hearing loss or auditory deprivation durations) and post-CI word recognition scores.

RESULTS

Behavioral assessment and control tasks

In the sound imagery task, the mean self-stimations of sound imagery were 79% (± 23) and 84% (± 16), and the delays to imagine the sounds corresponding to the images were 1771 ms (±1096) and 1673 ms (±546), for CI candidates and controls, respectively. These behavioral data did not statistically differ across groups (Fig. 1, A andB). Although self-estimated performance did not differ across groups, we tested for a possible decline in sound imagery as a function of deafness duration. There was no significant correlation for scores or reaction times (Fig. 1 C andD).

Regarding the control tasks during scanning, we did not observe a group difference for the control task "imagine color". Crucially, there was also no group difference in the naming task that we used to assure that CI candidates did not have any semantic deficit. This control task was entered in the statistical model but not used in specific contrasts, as it was not central to our hypotheses.

Sound imagery in normal hearing and post-lingually deaf subjects

In normal hearing subjects, sound imagery relative to baseline (Fig. 2, blue blobs and Table II) activated the bilateral inferior frontal gyri, the bilateral PSTG/SMG, the bilateral cerebellum, the left anterior superior temporal gyrus (STG), the left superior frontal gyrus, and the left inferior parietal/angular gyrus. Sound imagery relative to color imagery (Fig. 2, purple blobs, Table II) activated the left inferior and superior frontal gyrus, the left inferior parietal lobule, and the right PSTG/SMG. Many of these regions were not significantly activated in deaf patients (Fig. 2, yellow blobs and Table III), who performed the task recruiting only the left inferior and superior frontal gyri and the bilateral cerebellum. Notably, they recruited none of the brain regions usually involved in environmental sound acoustic analysis and representation (i.e. the left parietal lobule and the right posterior temporal cortex) [START_REF] Engel | Different categories of living and non-living sound-sources activate distinct cortical networks[END_REF][START_REF] Kraut | Neuroanatomic organization of sound memory in humans[END_REF][START_REF] Lewis | Human brain regions involved in recognizing environmental sounds[END_REF]. No activation during sound relative to color imagery in the CI candidate group was observed. We detected a task-by-group interaction. Control subjects showed stronger neural activation than patients (sound>color; p<0.001, uncorrected, Fig. 2, pink blobs and Table III) in the left inferior frontal gyrus and the right PSTG/SMG.

Comparing processing of sound imagery and phonology in CI candidates and normalhearing controls

As 7 of the 10 CI candidates also performed a phonological rhyming task (Lazard, et al. 2010b) in a separate session, we could compare responses in the left and right PSTG/SMG in each task. Patterns of activation in normal-hearing subjects and in CI candidates, depicted in Fig. 3A andB, were inverted in the two groups. In normal-hearing controls during the sound imagery task, the region that was most strongly activated was the right PSTG/SMG (Fig. 3B, light purple, p<0.01 relative to the activation of the left PSTG/SMG, 4A light purple), and during the phonological task, the left PSTG/SMG, even though slightly activated, was significantly more activated than the right PSTG/SMG (p<0.01; Fig. 3 A&B, light green).

Conversely, the CI candidates over-activated both right and left PSTG/SMG during the phonological task (significant group difference in the right PSTG/SMG, p<0.01, and in the left PSTG/SMG, p=0.02, Fig. 3 A&B, dark green, and stars), but CI candidates did not activate these regions during the imagery task (significant group differences in the right and left PSTG/SMG, p<0.01, Fig. 3 A&B, dark purple, and stars).

Activity in PSTG/SMG and deafness durations (auditory deprivation and hearing loss durations)

We assessed how activity was related to deafness duration and post-implant speech perception scores in bilateral PSTG/SMG. Despite low individual activation levels in these regions in the CI candidate group for the sound imagery task (not significantly different from baseline), correlations indicated that the response inter-subject variability was meaningfully related to CI outcome (Fig. 3F). The levels of activation of the right PSTG/SMG during sound imagery and the left PSTG/SMG during phonological processing, were both negatively correlated with the duration of deafness (with the duration of hearing loss for the right PSTG/SMG shown in Fig. 3D and supplemental Figure 1, and with the duration of auditory deprivation for the left PSTG/SMG shown in Fig. 3C and supplemental figure 1). Regarding contralateral recruitment, different behaviors were observed between the right and left PSTG/SMG. The involvement of the left PSTG/SMG in sound imagery tended to decrease with the duration of auditory deprivation (Fig. 3C and Supplemental figure 1), while the over-activation in phonological processing of the right PSTG/SMG increased with hearing loss duration (Fig. 3D and Supplemental figure 1).

Activity in PSTG/SMG and cochlear implant outcome

Phonological processing by the left PSTG/SMG correlated positively with post-implant speech perception measured at 6 months (Fig. 3E), as well as the level of activation in the right PSTG/SMG during environmental sound (Fig. 3F). Conversely, enhanced phonological responses by the right PSTG/SMG negatively correlated with post-CI scores (Fig. 3F). Note that those deaf subjects who did not over-activate the left PSTG/SMG during the phonological task were those for whom the right PSTG/SMG was recruited most strongly for the same task (Fig. 3E&F).

Whole brain regressions with hearing loss duration and CI-outcome

We confirmed our results using independent whole brain regression analyses. The activation of the right PSTG/SMG during sound imagery (Fig. 4 and Table III) correlated positively with post-CI speech perception, and negatively with hearing loss duration. These results confirm that although the right PSTG/SMG progressively disengages from environmental sound processing, its activation is associated with good speech perception scores. The negative correlation with post-CI scores showed that subjects, who reorganized the right PSTG/SMG to process phonology, performed environmental sound imagery by recruiting alternative regions, namely the right middle superior temporal gyrus (Fig. 4, red blob and Table III).

DISCUSSION

Even though patients did not report difficulties with sound imagery, bilateral PSTG/SMG activation profiles denote profound functional reorganization compared to controls. When retrieving the sound of everyday objects, hearing subjects involved a large part of the network that has previously been related to the acoustic processing of environmental sounds, including the bilateral inferior prefrontal gyrus/insula, the left anterior temporal cortex [START_REF] Leff | The cortical dynamics of intelligible speech[END_REF][START_REF] Scott | Identification of a pathway for intelligible speech in the left temporal lobe[END_REF] and the right posterior temporal lobe [START_REF] Beauchamp | Integration of auditory and visual information about objects in superior temporal sulcus[END_REF][START_REF] Binder | Function of the left planum temporale in auditory and linguistic processing[END_REF][START_REF] Lewis | Human brain regions involved in recognizing environmental sounds[END_REF]. They additionally recruited the inferior parietal lobule/angular gyrus previously linked to long-term memory retrieval [START_REF] Kraut | Neuroanatomic organization of sound memory in humans[END_REF][START_REF] Shannon | Functional-anatomic correlates of memory retrieval that suggest nontraditional processing roles for multiple distinct regions within posterior parietal cortex[END_REF] and the bilateral supramarginal gyrus, where speech sensorymotor transformation is carried out [START_REF] Hickok | Area Spt in the human planum temporale supports sensory-motor integration for speech processing[END_REF]. These data confirmed that the right PSTG/SMG region is normally involved in environmental sound processing. They also suggest that mental evocation of sounds, e.g. animal sounds, relies to some extent on soundto-articulation mapping (presumably inner onomatopoeia). This refers to the process by which a sound is mapped onto the motor routines that lead to its production. This process typically involves an interaction between the supramarginal gyrus/SPT area and the left inferior prefrontal gyrus. The left inferior frontal gyrus is a multimodal area involved in hierarchical language processing [START_REF] Sahin | Sequential processing of lexical, grammatical, and phonological information within Broca's area[END_REF]) and in language production planning [START_REF] Hickok | The cortical organization of speech processing[END_REF][START_REF] Turkeltaub | Localization of sublexical speech perception components[END_REF] and the supramarginal gyrus belongs to the articulatory sensory-motor integration circuit [START_REF] Hickok | Area Spt in the human planum temporale supports sensory-motor integration for speech processing[END_REF]).

In contrast with normal hearing subjects, deaf subjects only involved the inferior prefrontal/insula/anterior temporal region and did not activate the bilateral PSTG/SMG during the imagery task. Assuming that their self-report they could retrieve the sound of familiar objects and animals was accurate, they presumably invoked abstract, semantic-level representations, as suggested by the activation of the left insula/ventral prefrontal, known to participate in sound retrieval and categorization [START_REF] Doehrmann | Semantics and the multisensory brain: how meaning modulates processes of audio-visual integration[END_REF]. These data may indicate that a detailed acoustic representation of sounds, at least for the sounds presented during the task, was disrupted.

Activation profiles of the right and left PSTG/SMG in CI candidates were inverted compared to normal-hearing controls. Because phonology may be i) more relevant to oral communication and speech-reading and ii) more strongly reinforced by visual cues, we hypothesized that the over-activation of the right PSTG/SMG during the phonological task (Lazard, et al. 2010b) was part of a global reorganization involving the usual function of this region, i.e. environmental sound processing. The right activation of the PSTG/SMG observed during sound imagery in controls was an expected finding, as the task was designed to recruit it [START_REF] Beauchamp | Integration of auditory and visual information about objects in superior temporal sulcus[END_REF][START_REF] Binder | Function of the left planum temporale in auditory and linguistic processing[END_REF][START_REF] Bunzeck | Scanning silence: mental imagery of complex sounds[END_REF][START_REF] Lewis | Human brain regions involved in recognizing environmental sounds[END_REF]). In contrast, low activation levels in the left PSTG/SMG during the phonological task were presumably due to the fact that this task was too easy for normal-hearing subjects. We cannot exclude the possibility, however, that this lack of sensitivity was due to the small number of subjects in the samples, or that lateralization may be different in the case of more difficult tasks [START_REF] Hartwigsen | Phonological decisions require both the left and right supramarginal gyri[END_REF]. That controls activated this region during sound imagery denotes that they may perform automatic sound-to-articulation mapping, e.g. transforming sounds into onomatopoeia when possible. Deaf subjects did not activate the right and left PSTG/SMG during the imagery task. This suggests that their ability to evoke non-speech sounds through acoustic representations may be altered. The global over-activation of the left PSTG/SMG in deaf subjects relative to controls during the phonological task may reflect difficulties in processing the task.

The goal of this study was to understand the interactions across bilateral PSTG/SMG in acquired deafness and their possible implication in CI outcome. The main issue was whether the declining activation of the right PSTG/SMG in environmental sounds memory following deafness onset could prompt or facilitate its take-over in phonological processing, or whether its implication in phonological processing merely reflects hemispheric reciprocal interaction, presumably disinhibition [START_REF] Thiel | Direct demonstration of transcallosal disinhibition in language networks[END_REF][START_REF] Warren | Anterior temporal lobe connectivity correlates with functional outcome after aphasic stroke[END_REF].

The neural response level during phonological processing in the left PSTG/SMG declined with the duration of auditory deprivation in deaf subjects. However, that this decline was not correlated with the duration of hearing loss may reflect the possibility that phonological processing is maintained as long as the meaning of speech inputs is available without resorting to lip-reading, and starts to decline when speech becomes unintelligible. In contrast, the engagement of the right PSTG/SMG in environmental sound memory was negatively correlated with the duration of hearing loss, but not with the duration of auditory deprivation. This negative correlation with the duration of hearing loss may be explained by an earlier beginning of the process of decline of right environmental sound processing than the beginning of the process of decline of left phonological processing in the time course of the auditory handicap.

The dynamics of right and left PSTG/SMG activation in phonological processing and sound imagery was even more drastically different. In the course of deafness, we observed a growing phonological implication of the right PSTG/SMG, while the implication of the left PSTG/SMG in sound imagery decreased. Transcallosal disinhibition is an adaptive behavior that is frequently reported after left strokes [START_REF] Marsh | Recovery from aphasia following brain injury: the role of reorganization[END_REF]; if contralateral compensation resulted from bilateral interhemispheric disinhibition following the declining efficiency of both PSTG/SMG, we should not observe this asymmetry, but instead a bilateral positive correlation with deafness (either with the duration of hearing loss or of auditory deprivation). We therefore conclude that the remarkable involvement of the right PSTG/SMG in phonological processing resulted from specific cognitive factors, i.e. the need to maintain phonological processing for visual communication, such as in speechreading (Lazard, et al. 2010b;[START_REF] Mortensen | Restored speech comprehension linked to activity in left inferior prefrontal and right temporal cortices in postlingual deafness[END_REF][START_REF] Suh | Speech experience shapes the speechreading network and subsequent deafness facilitates it[END_REF]. Importantly, both growing phonological responses and declining responses to sound imagery in the right PSTG/SMG (Fig. 34) correlated with hearing loss duration, meaning that these two functional reorganizations happened jointly and early in the evolution of auditory loss.

Central to our interpretation is the hypothesis that, during deafness, it is more important to maintain phonological representations than environmental sound representations, because oral and written communication depends on the former. Post-lingually deaf individuals usually rely on lip-reading to perceive speech, and this process has been shown to involve the left PSTG/SMG (Lee, et al. 2007b). Although normal-hearing subjects use the left PSTG/SMG not to lip-read, but only to process auditory speech sounds, deafness prompts an immediate take-over of this region (Lee, et al. 2007b). This indicates that the routing of visual speech information to auditory areas preexists even though it is not predominantly used under normal hearing conditions. The involvement of the left PSTG/SMG in lip-reading presumably results from the patients resorting to phonological representations that have previously been associated with specific phonological oro-facial configurations. While this cross-modal mechanism does not prevent the decline in phonological processing, it presumably delays it until auditory cues are too distorted to be utilizable. Yet, the involvement of the left PSTG/SMG in visual speech may progressively enhance the visual specificity of this primarily auditory region [START_REF] Champoux | Visual stimuli can impair auditory processing in cochlear implant users[END_REF]). Thus, although the left PSTG/SMG progressively looses its specificity to phonology, its steady involvement in speechreading helps maintain a physiological level of activity. This may not be the case for the right PSTG/SMG as environmental sounds, music and prosody processing are not as systematically visually reinforced as phonological sounds. Therefore, the asymmetry in the visual take-over of the PSTG/SMG could contribute to the right asymmetric take-over by phonology. Accordingly, we previously showed that the right PSTG/SMG is more slowly involved by the visual system during speechreading than its left homologue (Lee, et al. 2007b). The right PSTG/SMG might be available to phonological processing because it is under-stimulated in deaf people's everyday life in its usual function (environmental sound processing), and not immediately taken-over by the visual system.

Overall, present and previous data converge to suggest that the involvement of the right PSTG/SMG in phonological processing and speechreading results from an asymmetry in the use of the PSTG region during deafness, possibly arising from a difference in the propensity to be visually recycled.

The present data also show that the involvement of the right PSTG/SMG in environmental sound memory is positively correlated with CI speech perception outcome. As long as the right PSTG/SMG is still involved in environmental sound processing, it does not seem to be available to contralateral reorganization. Supporting this hypothesis, we observed a dissociation between the level of activation of the right PSTG/SMG by sound imagery vs. phonological processing in both correlations with hearing loss duration (Fig. 3D) and speech perception after CI (Fig. 3F). Although marginally significant (interaction p=0.09), this dissociation is visible in individual subjects in Fig. 3F: the individuals with strong responses during sound imagery in the right PSTG/SMG had low responses during the rhyming task, and vice versa. This dissociation shows that the implication of the right PSTG/SMG in phonological processing is presumably maladaptive because it is normally involved in nonspeech complex acoustic processing [START_REF] Thierry | Hemispheric dissociation in access to the human semantic system[END_REF][START_REF] Toyomura | Neural correlates of auditory feedback control in human[END_REF]). The positive correlation with post-CI scores in the sound imagery task may thus be explained by the fact that sustained use of the right PSTG/SMG protects against deleterious contralateral phonological take-over.

Designing behavioral tests that detect these reorganizations before cochlear implantation is appealing. It seems possible to elaborate dedicated phonological tests based on written material that could identify CI candidates who have abandoned phonological strategies to communicate. Testing sound memory with purely visual material seems more arduous and presumably less directly relevant, since CI candidates did not report any difficulty in evoking environmental sounds. Yet, these results illustrate that communication needs play an important role in the cortical adaptation to function loss [START_REF] Dye | Attentional enhancements and deficits in deaf populations: an integrative review[END_REF][START_REF] Strelnikov | Does brain activity at rest reflect adaptive strategies? Evidence from speech processing after cochlear implantation[END_REF]. They further suggest that it is beneficial to maintain normal functional organization during deafness, not only with respect to phonological processing, but more generally to sound memory.

FIGURE LEGENDS

Figure 1 :

 1 Figure 1: Behavioral results during fMRI imagery tasks A. Performance, B. Reaction times.

Figure 2 :

 2 Figure 2: Sound imagery in normal hearing and post-lingually deaf subjects.

Figure 3 :

 3 Figure 3: Comparing residual processing of environmental sound and phonology in CI

Figure 4 :

 4 Figure 4: Right hemisphere and surface rendering for whole brain correlation with hearing

Funding: This work was supported by grants from Neurelec, MXM, Vallauris, France, and l'Association des Anciens Internes des Hôpitaux de Paris, without conflict of interest.

D. S. Lazard


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

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

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

A note on phase transitions for the Smoluchowski equation with dipolar potential

In this note, we study the phase transitions arising in a modified Smoluchowski equation on the sphere with dipolar potential. This equation models the competition between alignment and diffusion, and the modification consists in taking the strength of alignment and the intensity of the diffusion as functions of the order parameter.

We characterize the stable and unstable equilibrium states. For stable equilibria, we provide the exponential rate of convergence. We detail special cases, giving rise to second order and first order phase transitions, respectively. We study the hysteresis diagram, and provide numerical illustrations of this phenomena.

Introduction

In this short note, we study the following modified Smoluchowski equation (also called Fokker-Planck equation), for an orientation distribution f (ω, t) defined for a time t 0 and a direction ω ∈ S (the unit sphere S of R n ) as follows:

∂ t f = -ν(|J f |)∇ ω • (f ∇ ω (ω • Ω f )) + τ (|J f |)∆ ω f =: Q(f ), (1)
Ω f = J f |J f | , J f (t) = υ∈S υ f (υ, t) dυ. ( 2
)
where ∆ ω , ∇ ω •, and ∇ ω are the Laplace-Beltrami, divergence, and gradient operators on the sphere. The vector J f ∈ R n is the first moment associated to f (the measure on the sphere is the uniform measure such that S dυ = 1), and Ω f ∈ S represents the mean direction of the distribution f . The first term of the right-hand side of ( 1) is an alignment term towards Ω f , and the function ν represents the strength of this alignment. The function τ is the intensity of the diffusion on the sphere.

When τ is a constant and ν(|J f |) = |J f |, we can recover the standard Smoluchowski equation on the sphere, with dipolar potential [START_REF] Frouvelle | Dynamics in a kinetic model of oriented particles with phase transition[END_REF]. Indeed, the dipolar potential is given by K(ω, ω) = -ω • ω, and the equation can be recast as:

∂ t f = ∇ ω • f ∇ ω S K(ω • ω)f (ω, t)dω + τ ∆ ω f. (3)
Other classical kernels [START_REF] Doi | The Theory of Polymer Dynamics[END_REF][START_REF] Maier | Eine einfache molekulare Theorie des nematischen kristallinflüssigen Zustandes[END_REF][START_REF] Onsager | The effects of shape on the interaction of colloidal particles[END_REF] for the study of semi-dilute and concentrated suspensions of polymers are the so-called Maier-Saupe potential K(ω, ω) = -(ω• ω) 2 , or the original Onsager potential K(ω, ω) = |ω × ω|. In these cases, there are a lot of studies regarding the phase transition phenomenon for equilibrium states [START_REF] Chen | On the stationary solutions of the 2D Doi-Onsager model[END_REF][START_REF] Constantin | Asymptotic states of a Smoluchowski equation[END_REF][START_REF] Fatkullin | Critical points of the Onsager functional on a sphere[END_REF][START_REF] Fatkullin | A note on the Onsager model of nematic phase transitions[END_REF][START_REF] Liu | Axial symmetry and classification of stationary solutions of Doi-Onsager equation on the sphere with Maier-Saupe potential[END_REF][START_REF] Lucia | Exact multiplicity of nematic states for an Onsager model[END_REF][START_REF] Wang | A unified view on the rotational symmetry of equilibria of nematic polymers, dipolar nematic polymers and polymers in higher dimensional space[END_REF][START_REF] Wang | Multiple branches of ordered states of polymer ensembles with the Onsager excluded volume potential[END_REF][START_REF] Zhang | Stable dynamic states at the nematic liquid crystals in weak shear flow[END_REF][START_REF] Zhou | A new proof on axisymmetric equilibria of a three-dimensional Smoluchowski equation[END_REF][START_REF] Zhou | Characterization of stable kinetic equilibria of rigid, dipolar rod ensembles for coupled dipole-dipole and Maier-Saupe potentials[END_REF], and in particular a hysteresis phenomenon occurs for the Maier-Saupe potential in dimension 3. The case of the dipolar potential has also been studied precisely [START_REF] Frouvelle | Dynamics in a kinetic model of oriented particles with phase transition[END_REF], with an analysis of the rates of convergence to the equilibrium as time goes to infinity. In this case, there exists a so-called continuous phase transition for a critical threshold τ c : when τ < τ c the solution converges exponentially fast to a non-isotropic equilibrium; when τ > τ c it converges exponentially fast to the uniform distribution. At the critical value τ = τ c , the solution converges to the uniform distribution at a rate t -1/2 .

Here we study the modifications arising when ν and τ depend on |J f |, which can be motivated by some biological modeling (see [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF] and references therein). In that case, we cannot use τ as a bifurcation parameter anymore. Instead, we will use the initial mass ρ (a conserved quantity) as the key parameter to study the phase transition. We will assume that: Hypothesis 1.1.

(i) The functions ν and τ are C 1 , with ν(0) = 0 and τ > 0.

(ii) The function |J| → h(|J|) = ν(|J|)
τ (|J|) is an increasing function. We denote by σ its inverse, i.e.

κ = h(|J|) ⇔ |J| = σ(κ).
The first part of this hypothesis implies that we do not have any singularity of Q in [START_REF] Aldana | On the emergence of collective order in swarming systems: A recent debate[END_REF] as

|J f | → 0: if |J f | = 0, we simply have Q(f ) = τ (0)∆ ω f
. The second part is made for the sake of simplicity. It leaves enough flexibility to to reveal key behaviors in terms of phase transitions. It would be easy to remove it at the price of an increased technicality. Additionally, it means that when f is more concentrated in the direction of Ω f , the relative strength of the alignment force compared to diffusion is increased as well. This can be biologically motivated by the existence of some social reinforcement mechanism.

We will see that we can observe a wealth of phenomena, including hysteresis. The purpose of this note is to summarize the analytical results, as well as some numerical simulations which illustrate this phenomena. All the proofs are detailed in [START_REF] Degond | Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics[END_REF], where (1) arises as the spatially homogeneous version of a space-dependent kinetic equation, obtained as the mean-field limit of a self-propelled particle system interacting through alignment. This spatially homogeneous study is crucial to determine the macroscopic behavior of this space-dependent kinetic equation.

General study

Existence and uniqueness

We first state results about existence, uniqueness, positivity and regularity of the solutions of (1). Under hypothesis 1.1, we have the following Theorem 1. Given an initial finite non-negative measure f 0 in H s (S), there exists a unique weak solution f of (1) such that f (0) = f 0 . This solution is global in time.

Moreover, f ∈ C ∞ ((0, +∞) × S), with f (t, ω) > 0 for all positive t, and we have the following instantaneous regularity and uniform boundedness estimates (for m ∈ N, the constant C being independent of f ), for all t > 0:

f (t) 2 H s+m C 1 + 1 t m f 0 2 H s .
For later usage, we define Φ(|J|) as an anti-derivative of h: dΦ d|J| = h(|J|). In this case, the dynamics of (1) corresponds to the gradient flow of the following free energy functional:

F (f ) = S f ln f dω -Φ(|J f |).
Indeed, if we define the dissipation term D(f ) by

D(f ) = τ (|J f |) S f |∇ ω (ln f -h(|J f |) ω • Ω f )| 2 dω,
we get the following conservation relation:

d dt F (f ) = -D(f ) 0. ( 4
)

Equilibria

We now define the von Mises distribution which provides the general shape of the non-isotropic equilibria of Q.

Definition 2.1. The von Mises distribution of orientation Ω ∈ S and concentration parameter κ 0 is given by:

M κΩ (ω) = e κ ω•Ω S e κ υ•Ω dυ . ( 5
)
The order parameter c(κ) is defined by the relation

J M κΩ = c(κ)Ω,
and has expression:

c(κ) = π 0 cos θ e κ cos θ sin n-2 θ dθ π 0 e κ cos θ sin n-2 θ dθ . ( 6
)
The concentration parameter c(κ) defines a one-to-one correspondence κ ∈ [0, ∞) → c(κ) ∈ [0, 1). The case κ = c(κ) = 0 corresponds to the uniform distribution, while when κ is large (or c(κ) is close to 1), the von Mises distribution is closed to a Dirac delta mass at the point Ω. Some comments are necessary about the interval of definition of σ. First note that, under hypothesis 1.1, h is defined from [0, +∞), with values in an interval [0, κ max ), where we may have κ max = +∞. So σ is an increasing function from [0, κ max ) onto R + . Moreover, for later usage, we can define τ 0 = τ (0) > 0, and ρ c = lim

|J|→0 n|J| h(|J|) = nτ 0 ν ′ (0) (7)
where ρ c > 0 may be equal to +∞, and where we recall that n denotes the dimension. The equilibria are given by the following proposition:

Proposition 2.1. The following statements are equivalent:

(i) f ∈ C 2 (S) and Q(f ) = 0. (ii) f ∈ C 1 (S) and D(f ) = 0.
(iii) There exists ρ 0 and Ω ∈ S such that f = ρM κΩ , where κ 0 satisfies the compatibility equation:

σ(κ) = ρc(κ). (8)
Let us first remark that the uniform distribution, corresponding to κ = 0 is always an equilibrium. Indeed, we have c(0) = σ(0) = 0 and (8) is satisfied. However, Proposition 2.1 does not provide any information about the number of the non-isotropic equilibria. Indeed, equation ( 8) can be recast into:

c(κ) σ(κ) = 1 ρ , (9)
which is valid as long as σ = 0. We know that σ is an increasing unbounded function from its interval of definition [0, κ max ) onto [0, +∞), and thanks to hypothesis 1.1 and to [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF], we know that σ(κ) ∼ ρc n κ as κ → 0 (if ρ c < +∞). So since c(κ) ∼ 1 n κ as κ → 0 (see for instance [START_REF] Frouvelle | Dynamics in a kinetic model of oriented particles with phase transition[END_REF]), we have the two following results (also valid in the case ρ c = +∞):

c(κ) σ(κ) → 1 ρ c as κ → 0 and c(κ) σ(κ) → 0 as κ → κ max . (10)
We deduce that this function reaches its maximum, and we define

ρ * = min κ∈R + σ(κ) c(κ) . ( 11
)
For ρ < ρ * , the only solution to the compatibility condition is κ = 0, and the only equilibrium is the uniform distribution f = ρ. Except from these facts, we have no further direct information of this function κ → c(κ)/σ(κ), since c and σ are both increasing. Figure 1 depicts some examples of the possible shapes of the function κ → c(κ)/σ(κ). We see that depending on the value of ρ, the number of families of non-isotropic equilibria, given by the number of positive solutions of the equation ( 9), can be zero, one, two or even more. We now turn to the study of the stability of these equilibria, through the study of the rates of convergence.

0 0 κ max 1 ρ c = 1 ρ * 1 ρ c = 1 ρ * 1 ρ c 1 ρ c 1 ρ * 1 ρ * κ c(κ) σ(κ)

Rates of convergence to equilibrium

The main tool to prove convergence of the solution to a steady state is LaSalle's principle, that we recall here (the proof follows exactly the lines of [START_REF] Frouvelle | Dynamics in a kinetic model of oriented particles with phase transition[END_REF]). By the conservation relation ( 4), we know that the free energy F is decreasing in time (and bounded from below since |J| is bounded). LaSalle's principle states that the limiting value of F corresponds to an ω-limit set of equilibria: Proposition 2.2. LaSalle's invariance principle.

Let f 0 be a positive measure on the sphere S. We denote by F ∞ the limit of F (f (t)) as t → ∞, where f is the solution to the modified Smoluchowski equation (1) with initial condition f 0 .

Then the set

E ∞ = {f ∈ C ∞ (S) s.t. D(f ) = 0 and F (f ) = F ∞ } is not empty. Furthermore f (t)
converges in any H s norm to this set of equilibria (in the following sense):

lim t→∞ inf g∈E∞ f (t) -g H s = 0.
Since we know the types of equilibria, we can refine this principle to adapt it to our problem: Proposition 2.3. Let f 0 be a positive measure on the sphere S, with initial mass ρ.

If no open interval is included in the set {κ, ρc(κ) = σ(κ)}, then there exists a solution κ ∞ to the compatibility solution [START_REF] Doi | The Theory of Polymer Dynamics[END_REF] such that we have:

lim t→∞ |J f (t)| = ρc(κ ∞ ) ∀s ∈ R, lim t→∞ f (t) -ρM κ∞Ω f (t) H s = 0.
This last proposition helps us to characterize the ω-limit set by studying the single compatibility equation [START_REF] Doi | The Theory of Polymer Dynamics[END_REF].

When κ = 0 is the unique solution, then this gives us that f converges to the uniform distribution. Otherwise, two cases are possible, either κ ∞ = 0, and f converges to the uniform distribution, or κ ∞ = 0, and the only unknown behavior is the one of Ω f (t) . If we are able to prove that it converges to Ω ∞ ∈ S, then f converges to a fixed non-isotropic steady-state ρM κ∞Ω∞ .

However, Proposition 2.3 does not give information about quantitative rates of convergence of |J f | to ρc(κ ∞ ), and of f (t) -ρM κ∞Ω f (t) H s to 0, as t → ∞. So we now turn to the study of the behavior of the difference between the solution f and a target equilibrium ρM κ∞Ω f (t) .

This study consists in two types of expansion. If we expand the solution around the uniform equilibrium, some simple energy estimates give us exponential convergence when ρ < ρ c . But when we expand the solution around a non-isotropic equilibrium ρM κ∞Ω f (t) , we see that the condition of stability is related to the monotonicity of the function κ → c(κ)/σ(κ). Hence we can see directly on the graph of this function (see examples on Figure 1) both the number of family of equilibria and their stability: if the function is decreasing, the family is stable. By contrast it is unstable when the function is increasing. When the difference between f and ρM κ∞Ω f (t) converges exponentially fast to 0 (on the stable branch), we are able to control the displacement of Ω f (t), which gives convergence to Ω ∞ ∈ S. We then have convergence of f to a given equilibriumρM κ∞ Ω∞ .

All these results are summarized in the following two theorems. In what follows, we say that a constant is a universal constant when it does not depend on the initial condition f 0 (that is to say, it depends only on ρ, n and the coefficients of the equation ν and τ , and on the exponent s of the Sobolev space H s in which the result is stated).

Theorem 2. We have the following instability and exponential stability results around the uniform equilibrium:

• Suppose that ρ < ρ c . We define

λ = (n -1)τ 0 (1 - ρ ρ c ) > 0. ( 12
)
There exists a universal constant C, such that if f 0 -ρ H s < λ C , then for all t 0, we have

f (t) -ρ H s f 0 -ρ H s 1 -C λ f 0 -ρ H s e -λt .
• If ρ > ρ c , and if J f 0 = 0, then we cannot have κ ∞ = 0 in Proposition 2.3: the solution cannot converge to the uniform equilibrium.

To study the stability around a non-isotropic equilibrium, we fix ρ, and we denote by κ a positive solution to the compatibility equation (we will not write the dependence of c and σ on κ when there is no possible confusion). We denote by F κ the value of F (ρM κΩ ) (independent of Ω ∈ S).

Theorem 3. We have the following instability and exponential stability results when starting close to a non-isotropic equilibrium:

• Suppose ( σ c ) ′ (κ) > 0.
For all s > n-1 2 , there exist universal constants δ > 0 and C > 0, such that for any initial condition f 0 satisfying f 0 -ρM κΩ H s < δ for some Ω ∈ S, there exists

Ω ∞ ∈ S such that f -ρM κΩ∞ H s C f 0 -ρM κΩ f 0 H s e -λt ,
where the rate is given by

λ = cτ (σ) σ ′ Λ κ ( σ c ) ′ . ( 13
)
The constant Λ κ is the best constant for the following weighted Poincaré inequality (see the appendix of [START_REF] Degond | Macroscopic limits and phase transition in a system of self-propelled particles[END_REF] for more details on this constant, which does not depend on Ω):

S |∇ ω g| 2 M κΩ dω Λ κ S g 2 M κΩ dω - S g M κΩ dω 2 . ( 14
)
• Suppose ( σ c ) ′ (κ) < 0. Then any equilibrium of the form ρM κΩ is unstable, in the following sense: in any neighborhood of ρM κΩ , there exists an initial condition f 0 such that F (f 0 ) < F κ . Consequently, in that case, we cannot have κ ∞ = κ in Proposition 2.3.

Second order phase transition

Let us now focus on the case where we always have ( σ c ) ′ > 0 for all κ > 0 (see for example the lowest two curves of Figure 1). In this case, the compatibility equation ( 9) has a unique positive solution for ρ > ρ c . With the results of the previous subsection about stability and rates of convergence, we obtain the behavior of the solution for any initial condition f 0 with initial mass ρ.

• If ρ < ρ c , then the solution converges exponentially fast towards the uniform distribution f ∞ = ρ.

• If ρ = ρ c , the solution converges to the uniform distribution.

• If ρ > ρ c and J f 0 = 0, then there Ω ∞ such that f converges exponentially fast to the von Mises distribution f ∞ = ρM κΩ∞ , where κ > 0 is the unique positive solution to the equation ρc(κ) = σ(κ).

The special case where J f 0 = 0 leads to the heat equation ∂ t f = τ 0 ∆ ω f . Its solution converges exponentially fast to the uniform distribution, but this solution is not stable under small perturbation of the initial condition. Let us remark that for some particular choice of the coefficients, as in [START_REF] Frouvelle | Dynamics in a kinetic model of oriented particles with phase transition[END_REF], it is also possible to get an algebraic rate of convergence in the second case ρ = ρ c . For example when σ(κ) = κ, we have f -ρ Ct -1 2 for t sufficiently large. So we can describe the phase transition phenomena by studying the order parameter of the asymptotic equilibrium c = |J f∞ | ρ , as a function of the initial density ρ. We have c(ρ) = 0 if ρ ρ c , and c is a positive continuous increasing function for ρ > ρ c . In the common situation where c σ = 1 ρc -aκ

1 β + o(κ 1
β
) when κ → 0, it is easy to see, since c(κ) ∼ 1 n κ when κ → 0, that we have

c(ρ) ∼ a(ρ -ρ c ) β , as ρ > → ρ c . (15)
Since c σ is Lipschitz, we always have β 1. So the first derivative of c is discontinuous at ρ = ρ c . This is the case of a second order phase transition (also called continuous phase transition). The critical exponent β can take arbitrary values in (0, 1], as can be seen by taking

h(|J|) such that σ(κ) = c(κ)(1 + κ 1 β ).
In general, we have the following practical criterion, which ensures a second order phase transition.

Lemma 1. If h(|J|)

|J| is a non-increasing function of |J|, then we have ( σ c ) ′ > 0 for all κ > 0. In this case, the critical exponent β in [START_REF] Onsager | The effects of shape on the interaction of colloidal particles[END_REF], if it exists, can only take values in [ 1 2 , 1].

Hysteresis

Typical example

We now turn to a specific example, where all the features presented in the stability study can be seen. We focus on the case where ν(|J|) = |J|, as in [START_REF] Frouvelle | Dynamics in a kinetic model of oriented particles with phase transition[END_REF], but we now take τ (|J|) = 1/(1 + |J|). From the modeling point of view, this occurs in the Vicsek model with vectorial noise (also called extrinsic noise) [START_REF] Aldana | On the emergence of collective order in swarming systems: A recent debate[END_REF][START_REF] Chaté | Collective motion of selfpropelled particles interacting without cohesion[END_REF]. In this case, we have h(|J|) = |J| + |J| 2 , so the assumptions of Lemma 1 are not fulfilled, and the function σ is given by σ(κ) = 1 2 ( √ 1 + 4κ -1). Expanding c σ when κ is large or κ is close to 0, we get

c σ = 1 n + 1 n κ + O(κ 2 ) as κ → 0, 1 √ κ + O(κ -1 ) as κ → ∞.
Consequently, there exist more than one family of non-isotropic equilibria when ρ is close to ρ c = n (and ρ > ρ c ).

The function κ → c(κ) σ(κ) can be computed numerically. The results are displayed in Figure 2 in dimensions n = 2 and n = 3.

1 ρ * κ * κ * κ 1 ρ * c(κ) σ(κ)

We observe the following features:

• There exists a unique critical point κ * for the function c σ , corresponding to its global maximum 1 ρ * (in dimension 2, we obtain numerically ρ * ≈ 1.3726 and κ * ≈ 1.2619, in dimension 3 we get ρ * ≈ 1.8602 and κ * ≈ 1.9014).

• The function c

σ is strictly increasing in [0, κ * ) and strictly decreasing on (κ * , ∞). From these properties, it follows that the solution associated to an initial condition f 0 with mass ρ can exhibit different types of behavior, depending on the three following regimes for ρ.

• If ρ < ρ * , the solution converges exponentially fast to the uniform equilibrium f ∞ = ρ.

• If ρ * < ρ < n, there are two families of stable solutions: either the uniform equilibrium f = ρ or the von Mises distributions of the form ρM κΩ , for Ω ∈ S where κ is the unique solution with κ > κ * of the compatibility equation ( 8). If f 0 is sufficiently close to one of these equilibria, there is exponential convergence to an equilibrium of the same family.

The von Mises distributions of the other family (corresponding to solution of ( 8) such that 0 < κ < κ * ) are unstable in the sense given in Theorem 3.

• If ρ > n and J f 0 = 0, then there exists Ω ∞ ∈ S such that f converges exponentially fast to the von Mises distribution ρM κΩ∞ , where κ is the unique positive solution to the compatibility equation ρc(κ) = σ(κ).

At the critical point ρ = ρ * , the uniform equilibrium is stable (and for any initial condition sufficiently close to it, the solution converges exponentially fast to it), but the stability of the family of von Mises distribution {ρ * M κ * Ω , Ω ∈ S} is unknown.. At the critical point ρ = n, the family of von Mises distribution {nM κcΩ , Ω ∈ S} is stable, where κ c is the unique positive solution of [START_REF] Doi | The Theory of Polymer Dynamics[END_REF]. For any initial condition sufficiently close to nM κcΩ for some Ω ∈ S, there exists Ω ∞ such that the solution converges exponentially fast to nM κcΩ∞ . However, in this case, the stability of the uniform distribution f = n is unknown.

As previously, in the special case J f 0 = 0, the equation reduces to the heat equation and the solution converges to the uniform equilibrium.

Since c(κ) is an increasing function of κ, we can invert this relation κ → c(κ) into c → κ(c) and express the density ρ = σ(κ(c)) c as a function of c. The result is depicted in Figure 3 for dimension 2 or 3. With this picture, we recover the phase diagram in a conventional way: the possible order parameters c for the different equilibria are given as functions of ρ. The dashed lines corresponds to branches of equilibria which are unstable.

We can also obtain the corresponding diagrams for the free energy and the rates of convergences. For this particular example, the free energies F (ρ) and F κ (we recall that they correspond respectively to the free energy of the uniform distribution and of a von Mises distribution ρM κΩ for a positive solution κ of the compatibility equation ( 8), including both stable and unstable branches) are given by

F (ρ) = ρ ln ρ, F κ = ρ ln ρ + ρ ln M κΩ M - 1 2 σ 2 - 1 3 σ 3 = ρ ln ρ -ρ ln e κ cos θ dω - 1 6 (κ -σ) + 2 3 σκ.
The plots of these functions are depicted in dimensions 2 and 3 are depicted on the left part of Figure 4. Since the functions are very close in the figure for some range of interest, we depict the difference F κ -F (ρ) in a more appropriate scale, in the right part of Figure 4. The dashed lines correspond to unstable branches of equilibria. We observe that the free energy of the unstable non-isotropic equilibria (in dashed line) is always above that of the uniform distribution. There exist ρ 1 ∈ (ρ * , ρ c ) and a corresponding solution κ 1 of the compatibility solution (8) (with κ 1 > κ * , corresponding to a stable family of non-isotropic equilibria) such that F κ 1 = F (ρ F ). If ρ < ρ 1 , the global minimizer of the free energy is the uniform distribution, while if ρ > ρ 1 , then the global minimum is reached for the family of stable von Mises equilibria. The physical relevance of this value is not clear though, as we will see in the numerical illustration of the next subsection.

The rates of convergence to the stable equilibria, following Theorems 2 and 3, are given by

λ 0 = (n -1)(1 - ρ n ), for ρ < ρ c = n, λ κ = 1 1 + σ Λ κ (1 -( 1 c -c - n -1 κ )σ(1 + 2σ)), for ρ > ρ * ,
where λ 0 is the rate of convergence to the uniform distribution ρ, and λ κ is the rate of convergence to the stable family of von Mises distributions ρM κΩ , where κ is the unique solution of the compatibility condition [START_REF] Doi | The Theory of Polymer Dynamics[END_REF] such that κ > κ * . Details for the numerical computation of the Poincaré constant Λ κ are given in the appendix of [START_REF] Degond | Macroscopic limits and phase transition in a system of self-propelled particles[END_REF].

The computations in dimensions 2 and 3 are depicted in Figure 5.

Numerical illustrations of the hysteresis phenomenon

In order to highlight the role of the density ρ as the key parameter for this phase transition, we introduce the probability density function f = f ρ and we get

∂ t f = τ (ρ|J f |)∆ ω f -ν(ρ|J f |)∇ ω • ( f ∇ ω (Ω f • ω)). ( 16
)
When ρ is constant, this equation is equivalent to [START_REF] Aldana | On the emergence of collective order in swarming systems: A recent debate[END_REF]. We now consider ρ as a parameter varying slowly with time (compared to the time scale of the convergence to equilibrium, see Figure 5). If this parameter starts from a value ρ < ρ * , and increases slowly, the only stable distribution is initially the uniform distribution f = 1, and it remains stable. So we expect that the solution stays close to it, until ρ reaches the critical value ρ c . For ρ > ρ c , the only stable equilibria are the von Mises distributions, and the solution converges to one of these equilibria. The order parameter defined as c( f ) = |J f |, then jumps from 0 to c c = c(κ c ). If then the density ρ is further decreased slowly, the solution stays close to a von Mises distribution, and the order parameter slowly decreases, until ρ reaches ρ * back. For ρ < ρ * , the only stable equilibrium is the uniform distribution, and the concentration parameter jumps from c * = c(κ * ) to 0. This is a hysteresis phenomenon: the concentration parameter describes an oriented loop called hysteresis loop.

Let us now present some numerical simulations of the system (16) in dimension n = 2. We start with a initial condition which is a small perturbation of the uniform distribution, and we take ρ = 1.75 -0.75 cos( π T t), with T = 500. We use a standard central finite different scheme (with 100 discretization points), implicit in time (with a time step of 0.01). The only problem with this approach is that the solution converges so strongly to the uniform distribution for ρ < ρ c , so after passing ρ c , the linear rate of explosion for J f is given by e ( ρ ρc -1)t , and is very slow when ρ is close to ρ c . So since J f is initially very small when passing the threshold ρ = ρ c we would have to wait extremely long in order to see the convergence to the stable von Mises distribution. To overcome this problem, we adding a threshold ε and strengthen |J f | when f -1 ∞ ε, by

f f + max(0, ε -f -1 ∞ ) Ω f • ω.
We note that this transformation that we still have f -1 ∞ ε if it was the case before applying the transformation. Figure 6 depicts the result of a numerical simulation with a threshold ε = 0.02. We clearly see this hysteresis cycle, which agrees very well with the theoretical diagram. The jumps at ρ = ρ * and ρ = ρ c are closer to the theoretical jumps when T is very large. We were not able to see any numerical significance of the value ρ 1 (for which uniform and non-isotropic distributions have the same free energy) in all these numerical simulations. In particular, ρ 1 is close to ρ * (see Figure 4), so in most of the cases where both uniform and non-isotropic distribution are stable, the uniform distribution is not the global minimizer of the free energy, but in practical, metastability is very strong, and the solution still converges to the uniform distribution.

Conclusion

In this note, we have given the summary of strong results on the stability and instability of the equilibrium states of the modified Smoluchowski equation [START_REF] Aldana | On the emergence of collective order in swarming systems: A recent debate[END_REF]. This allows to have a precise description of the dynamics of the solution when time goes to infinity: it converges exponentially fast to a fixed equilibrium, with explicit formulas for the rates of convergence. We have also exhibited a specific example in which we observe a first order phase transition with a hysteresis loop (in contrast with the second order phase transition of the original Smoluchowski equation with dipolar potential [START_REF] Frouvelle | Dynamics in a kinetic model of oriented particles with phase transition[END_REF]). The details of the proofs will be found in a longer paper [START_REF] Degond | Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics[END_REF] as well as numerical comparisons between the particle and kinetic models to confirm that the hysteresis is really intrinsic to the system and not simply an artifact of the kinetic modeling. 16), with time varying ρ, in dimension 2. The red curve is the theoretical curve, the blue one corresponds to the simulation.

Figure 1 :

 1 Figure 1: The green, blue, red and purple curves correspond to various possible profiles for the function κ → c(κ) σ(κ) .

Figure 2 :

 2 Figure 2: The function κ → c(κ) σ(κ) , in dimensions 2 and 3.

Figure 3 :ρ * ρ 1 ρ * ρ 1 n=2Figure 4 :

 3114 Figure 3: Phase diagram of the model with hysteresis, in dimensions 2 and 3.

Figure 5 :

 5 Figure 5: Rates of convergence to both types of stable equilibria, as functions of the density ρ, in dimensions 2 and 3.

Figure 6 :

 6 Figure 6: Hysteresis loop for the concentration parameter c in a numerical simulation of the kinetic equation (16), with time varying ρ, in dimension 2. The red curve is the theoretical curve, the blue one corresponds to the simulation.