Reduced fluid models for self-propelled particles interacting through alignment

The asymptotic analysis of kinetic models describing the behavior of particles interacting through alignment is performed. We will analyze the asymptotic regime corresponding to large alignment frequency where the alignment effects are dominated by the self propulsion and friction forces. The former hypothesis leads to a macroscopic fluid model due to the fast averaging in velocity, while the second one imposes a fixed speed in the limit, and thus a reduction of the dynamics to a sphere in the velocity space. The analysis relies on averaging techniques successfully used in the magnetic confinement of charged particles. The limiting particle distribution is supported on a sphere, and therefore we are forced to work with measures in velocity. As for the Euler-type equations, the fluid model comes by integrating the kinetic equation against the collision invariants and its generalizations in the velocity space. The main difficulty is their identification for the averaged alignment kernel in our functional setting of measures in velocity.

Each individual in the group relaxes its velocity toward the mean velocity of the neighbors, leading to the term ν div v {f (u[f ] − v)}, where ν is the reorientation frequency and u[f ] is the mean velocity The weight application h is a decreasing, radial, non negative given function that determines the interaction neighborhood around any position. By including also noise in the above kinetic model, we get to the Fokker-Planck like equation where σ = τ /ν represents the diffusion coefficient in the velocity space. We investigate the large time and space scale regime of (1.1) that is, we fix large time and space units. In this case, equation (1.1) should be replaced by The choice of a large length unit leads to a local reorientation mechanism: the mean velocity u[f ] in (1.2) is now given by Notice that if f (t, x, ·) = 0, then the Fokker-Planck collision operator vanishes for any u. In this case we can define u[f (t)] = 0, without loss of generality. We assume that the frequencies ε 1 and ν scale like ν ε 1 ≈ 1 ε 2 for some small parameters ε 1 , ε 2 > 0 and thus the equation (1.2) becomes Assume for the moment that ε 1 ց 0 and ε 2 is fixed. In this situation, the leading order term in the Fokker-Planck equation (1.3) corresponds to the self-propulsion/friction mechanism, and we expect that the limit density f ε 2 = lim ε 1 ց0 f ε 1 ,ε 2 satisfies div v {f ε 2 (α − β|v| 2 )v} = 0.
The previous constraint exactly says that at any time t and any position x, the velocity distribution f ε 2 (t, x, ·) is a measure supported in {0} ∪ rS d−1 cf. [15]. The particles will tend to move with asymptotic speed r. These models have been shown to produce complicated dynamics and patterns at the particle level such as mills, double mills, flocks and clumps, see [50], whose stability properties are very relevant in the applications, see [8,3,31]. Assuming that all individuals move with constant speed also leads to spatial aggregation, patterns, and collective motion [40,51,64]. More exactly, it was shown in [15] that, by taking the limit ε 1 ց 0, the solutions f ε 1 ,ε 2 of (1.3) converge toward the solution f ε 2 of The above result states that in the limit ε 1 ց 0, the Cucker-Smale model with diffusion is reduced to a Vicsek like model, whose phase transition was analyzed in [52]. The evolution problem (1.4) on the phase space R d × rS d−1 , with normalized velocity field u[f ε 2 ] i.e., was also proposed in the literature as continuum version [48] of the Vicsek model [66,37]. Furthermore, the full phase transition for stationary solutions and their asymptotic stability was subsequently generalized in [41,42] allowing for quite general dependency of ν and τ on |u[f (t)]|. We will focus on the relaxation toward the mean velocity u[f ], whose alignment mechanism relies only on the direction of the mean velocity Ω Nevertheless, our method still applies and allows us to handle the model with normalization and the generalizations in [48,42] as well.
The original kinetic Vicsek model in [66,36] was derived as the mean-field limit of some stochastic particle systems in [10]. In fact, previous particle systems have also been studied with noise in [9] for the mean-field limit (see also [63,21,49,23,2,24,25,26]), in [55] for studying some properties of the Cucker-Smale model with noise, and in [5,33] for phase transitions at the level of the Cucker-Smale model and the inhomogeneous level respectively.
We assume now that both ε 1 , ε 2 become small. The idea is to justify a macroscopic model for (1.4), resulting from the balance between two opposite phenomena 1. The reorientation, which tends to align the particle velocities with respect to the mean velocity; 2. The diffusion, which tends to spread the particle velocities isotropically on the sphere rS d−1 .
Such hydrodynamic models were obtained in [48,42], by letting ε 2 ց 0 in the normalized alignment version of (1.4). They are typically referred as Self-Organized Hydrodynamics (SOH). Notice that the SOH model was obtained by passing to the limit successively in (1.3) with respect to ε 1 , ε 2 . After letting ε 1 ց 0, the dynamics were reduced to the phase space (x, v) ∈ R d × rS d−1 , but still captures microscopic behavior in the tangent directions to the sphere rS d−1 . The second limit procedure, ε 2 ց 0, leads to the macroscopic equations for the density rS d−1 f dω and the direction of the flux rS d−1 ωf dω.
We intend to obtain a SOH model, by passing to the limit in (1.3), simultaneously with respect to (ε 1 , ε 2 ). Motivated by the above discussion, we assume that ε 1 = ε 2 and ε 2 = ε, where ε > 0 is a small parameter, that is, the self-propulsion/friction mechanism dominates the alignment. This implies that ν = ε and τ = σε. Therefore (1.3) becomes for all (t, x, v) ∈ R + × R 2d , supplemented by the initial condition Very recently, by a similar scaling, fluid models have been obtained for the transport of charged particles, under the action of strong magnetic fields, which dominate the collision effects. The resulting macroscopic model is a gyrokinetic version of the Euler equations, in the parallel direction with respect to the magnetic field [18,20].
The behavior of the family (f ε ) ε>0 , as the parameter ε becomes small, follows by analyzing the formal Plugging the above Ansatz into (1.5), leads to the constraints and to the time evolution equations We expect the same macroscopic SOH model for the moments of f as obtained in [48,41,42]. The main advantage for considering (1.5) instead of (1.4) with ε 2 = ε is that the resolution of (1.5) for small ε will provide a solution supported near R d × rS d−1 , which fits much better the behavior of living organism systems, than the solution of (1.4) on R d × rS d−1 . But the price to pay is to deal with two Lagrange multipliers, appearing in (1.9), which have to be eliminated, thanks to the constraints (1.7) and (1.8).
The first constraint was analyzed in detail in [15]. It exactly says that f is a measure supported in The proof of Proposition 1.1 is based on the resolution of the adjoint problem [15].
An average operator serves to separate between two scales. For example, in gyro-kinetic theory, two time scales exist: a fast time variable, related to the rapid cyclotronic motion, and a slow time variable, related to the parallel motion with respect to the magnetic field. The gyro-average operator represents the average of the fast dynamics over a cyclotronic period, provided that the slow time variable is frozen. Following this technique, we obtain an accurate enough but simpler model, from the numerical approximation point of view. All the fluctuations have been removed and replaced by averaged effects.
Our model (1.5) presents not two, but three time variables: t, t/ε and t/ε 2 . The dynamics are dominated by the self-propulsion/friction mechanism, introducing the fast time variable s = t/ε 2 . The average operator is related to the characteristic flow of the field 1 conserves the direction v |v| and has as equilibria the elements of {0} ∪ rS d−1 . The Jacobian matrix is given by Being negative on rS d−1 and definite positive at 0, we deduce that the points of rS d−1 are stable equilibria, and 0 is an unstable equilibrium. For simplicity, we neglect the measure of the unstable point 0 in the velocity space and assume that this is not present in the limit ǫ → 0 at any level of the expansion. As we elaborate below, we will rigorously compute the terms in the expansion needed to derive formally the hydrodynamic equations. The complete mathematical analysis of the limiting procedure is out of scope of this paper. We are mainly interested in the two or three dimensional setting, but the same arguments apply for any dimension d ≥ 2. For the sake of generality, we state and prove all the results in any dimension d ≥ 2, and we distinguish, if necessary, between the cases d = 2 and d ≥ 3.
Motivated by the previous observations, we define the average of a non negative bounded measure cf. [15]. We will denote by f (x, v) dvdx the integration against the measure f . This is done independently of being the measure f absolutely continuous with respect to the Lebesgue measure or not.
be a non negative bounded measure on R d . We denote by F the measure corresponding to the linear application We denote by f the measure corresponding to the linear application It is easily seen that the average of a non negative bounded measure is a non negative bounded measure, with the same mass, but supported in We have the following characterization (see Proposition 5.1 [15]).
A direct consequence of Proposition 1.2 is that any bounded, non negative measure, supported in is left unchanged by the average operator. Another property of the average operator is that it removes any measure of the form div v {f (α − β|v| 2 )v}, cf. Proposition 5.2 [15].
The above proposition plays a crucial role when eliminating the Lagrange multiplier f (2) in (1.9).
Indeed, for doing that, it is enough to average both hand sides in (1.9). By the constraint (1.7), we know that f is supported in R d × ({0} ∪ rS d−1 ), and thus is left invariant by the average. We check that ∂ t f = ∂ t f = ∂ t f , and thus, averaging (1.9) still leads to a evolution problem for f (1.10) Certainly, a much more difficult task is to eliminate the Lagrange multiplier f (1) . We expect that this can be done thanks to the constraint in (1.8). The solvability of (1.8), with respect to f (1) , depends on a compatibility condition, to be satisfied by the right hand side. Indeed, by Proposition 1.3, we should saying that f is a equilibrium for the average collision kernel Q(f ) = 0. The equilibria of the average collision kernel form a d − 1-dimensional manifold, that is one dimension less than the equilibria manifold of the Fokker-Planck operator Q (see also [48,52]). For any l ∈ R + , Ω ∈ S d−1 , we introduce the von The following statements are equivalent: The modulus of the mean velocity is not a coordinate on the equilibria manifold, but it is determined by the condition |u| = σl r where l satisfies (1.12). Clearly l = 0 is a solution, which corresponds to the isotropic equilibrium whereω d represents the area of the unit sphere in R d . The next proposition is essentially contained in Proposition 3.3 in [52]. We present a simplified proof, based on computations with Bessel functions. Proposition 1.5. Let λ : R + → R be the function given by The function λ is strictly increasing, strictly concave and verifies In order to find the equations for the evolution of the density ρ and orientation Ω, we need to find f (1) from (1.8) in order to feed the terms needed in (1.10). However, we will see that this is not possible.
We will need to introduce a notion of generalized collision invariants, quite related intuitively to the one introduced in [48,41,42], in our functional setting of measures supported in rS d−1 to avoid the computation of the full f (1) . This is the main technical difficulty due to the measure functional setting since the precise definition of generalized collision invariant we need is more involved than in [48,41,42].
Let us mention that this notion of generalized collision invariant has been used in other related models in collective dynamics [47,43,44] and in kinetic models of wealth distribution [46].
Our main result establishes the macroscopic equations satisfied by the density ρ and orientation Ω, which parameterize the von Mises-Fisher equilibrium, obtained when passing to the limit for ε ց 0 in (1.5). We retrieve exactly the limit SOH hydrodynamic model in [41], written for any space dimension d ≥ 2 with the same explicit constants.
For any ε > 0 we consider the problem and the orientation Ω(t, x) satisfy the macroscopic equations with the initial conditions A nice practical implication of our main result is that this penalization procedure, by imposing asymptotically a cruise speed for particles, could lead to efficient and stable numerical schemes to compute the hydrodynamic equations (1.14)-(1.15). This is important due to the possible non-hyperbolicity of the system (1.14)-(1.15), see [42]. The local in time well-posedness of the SOH system (1.14)-(1.15) was studied in [45]. We finally emphasize that the constants appearing in the equations (1.14)-(1.15) coincide exactly with the ones obtained in [42] after some easy but tedious algebraic manipulations.
Our article is organized as follows. In Section 2 we study the equilibria of the average collision operator in our functional setting. This analysis can be carried out by introducing some Bessel functions.
In the next section we investigate the notion of collision invariant suitable in our functional setting. We determine the structure of these invariants and present their symmetries. Section 4 is devoted to the derivation of the fluid model for the macroscopic quantities, parameterizing the limit von Mises-Fisher equilibrium. The proofs of some technical results can be found in the Appendix.

The equilibria of the average collision operator
We consider the collision operator is the mean velocity. The above operator should be understood in the duality sense between non negative bounded measures on R d and smooth functions, compactly supported in R d As suggested by the formal expansion (1.6), we focus on measures satisfying (see ( Thanks to Propositions 1.3 and 1.1, we deduce that supp F ⊂ {0} ∪ rS d−1 and We discuss the case of non negative bounded measures supported on the sphere rS d−1 , that is, we discard all difficulties related to the mass of the points at rest. For such measures, the equality Q(F ) = 0 can be interpreted in the following sense (see Proposition The complete description of the above equilibria of the average collision operator Q, called the von Mises-Fisher distributions, is given by Proposition 1.4, whose proof is detailed below. We start with the following easy integration by parts formula on spheres. The proof is postponed to A.
and for any function It is very convenient to express the differential operators ∇ ω , div ω of functions and vector fields on the sphere rS d−1 in terms of the differential operators ∇ v , div v applied to extensions of functions and vector fields on a neighborhood of rS d−1 in R d . The notation · stands for the restriction on the sphere rS d−1 and· t for the restriction on the sphere tS d−1 . The proof of the following lemma is detailed in B.
3. Let ξ = ξ(ω) be a C 1 tangent vector field on rS d−1 and ξ = ξ(v) a C 1 extension of ξ in the set Before giving the proof of Proposition 1.4, we indicate a formula which will be used several times in our computations. For any continuous function G : with ω 1 = 2. In particular, for any continuous function g : [−r, r] → R, we have We assume that F is a equilibrium for the average collision kernel. We claim that where ϕ is the restriction on rS d−1 of ϕ as usual. Notice that we have We introduce the Hilbert spaces We denote by | · | r , · r the norm induced by the above scalar products. There is a constant C r such that the following Poincaré inequality holds true The previous inequality guarantees that the application χ → |∇ ω χ| r is a norm equivalent to · r oñ Therefore, the bilinear form is symmetric, bounded and coercive. By the Lax-Milgram lemma, there is a unique solution ψ ∈ H 1 (rS d−1 ) for the variational problem (2.6) leading to for any χ ∈H 1 (rS d−1 ). Observe that (2.7) still holds true for any constant function on rS d−1 , thanks to the compatibility condition rS d−1 ϕ(ω)M (ω) dω = 0. Therefore the variational formulation is valid for We consider the extension of ψ defined as usual as By Lemma 2.2, statements 2 and 3, we check that for any v ∈ rS d−1 we have and therefore we obtain We deduce that the linear forms ϕ → rS d−1 ϕ(ω)M (ω) dω and ϕ → R d ϕ(v)F dv are proportional, see Lemma III.2 in [22], and thus there isC such that for any ϕ ∈ C(R d ), we have Therefore the measure F has a positive density with respect to dω on rS d−1 If ρ = 0, we obtain F = 0, and we can take l = 0 and any Ω ∈ S d−1 . Assume now that ρ > 0. If For the last equality use the fact that and formula (2.5). The equality (2.8) reduces to the condition We introduce the function λ : R + → R λ(l) = π 0 cos θe l cos θ sin d−2 θ dθ π 0 e l cos θ sin d−2 θ dθ , l ∈ R + .
Therefore the non negative number l = r|u[F ]| σ satisfies λ(l) = σ r 2 l, and thus the measure F is given by Conversely, let F be a measure given by F = ρM lΩ dω for some ρ ∈ R + , Ω ∈ S d−1 , l ∈ R + such that λ(l) = σ r 2 l. If ρ = 0, F is the trivial equilibrium (with u[F ] = 0). If ρ > 0, the mean velocity writes , v ∈ R d . Notice that for any v ∈ rS d−1 we have and thus, the above equality becomes The properties of the function λ are summarized in Proposition 1.5, whose proof is detalied below.
Remark 2.1. The value l = 0 corresponds to the isotropic equilibrium M 0Ω dω = dω ω d r d−1 . The limit when l → +∞ leads to the Dirac measure on rS d−1 , concentrated at rΩ, that is, for any function ψ ∈ C(rS d−1 ) The function λ can be computed explicitly, at least for d = 3. Nevertheless, very good explicit approximations are available in any dimension d.

Consider the function
The function µ is strictly increasing, strictly concave and we have 2. If d = 3, the function λ is given by λ(l) = cosh(l) sinh(l) − 1 l , l > 0. Proof.
In order to exploit the constraint (1.8) we will need to compute Q(F ), where F is a von Mises-Fisher equilibrium, let us say F = M lΩ (ω)dω. This computation is detailed in the following lemma. The notation (·, ·) stands for the pairing between distributions and smooth functions.
Proof. Pick a test function ϕ ∈ C 2 c (R d ) and notice that It is easily seen that the function M lΩ M is constant on the sphere rS d−1 and therefore we have Thanks to the above result, we can determine F (1) − F (1) in terms of F . More exactly we prove

Then for any function
Proof. For any function ϕ ∈ C 1 c (R d ), we know that The idea is to solve the adjoint problem (cf. Lemma 1.1) and to express the normal derivative of ϕ in terms of χ. Indeed, for any ω t ∈ tS d−1 , we have .
Finally we obtain the formula v =0 Once we have determined the form of the dominant distribution f (t, x, v) = ρ(t, x)M lΩ(t,x) dω, we search for macroscopic equations characterizing ρ(t, x) and Ω(t, x). For doing that, we use the moments of (1.10) with respect to the velocity. The key point is how to eliminate f (1) in the right hand side of (1.10). Notice that this right hand side is the linearization around f , with R d f dv > 0, computed in the direction f (1) , of the average collision kernel Q where We are looking for functions such that can be expressed in terms of the velocity moments of f , in order to get a closure for the macroscopic quantities ρ(t, x), Ω(t, x). For example ψ(v) = 1 leads to the continuity equation which also writes ∂ t ρ + div x ρ σ r lΩ = 0.
Naturally, we need to find other functions ψ, which will allow us to characterize the time evolution of the orientation Ω. Recall that the constraint (1.8) determines f (1) − f (1) (in terms of f ), but not f (1) , as Lemma 2.5 implies. Motivated by this, we are looking for functions ψ such that for any measures f, g (1) supported in R d × rS d−1 . Indeed, in that case the expression in (2.12) can be computed in terms of f , provided that we neglect the mass of Let us concentrate now on the collision invariants of the average collision operator. Recall that the linearized of Q , around a measure F such that R d F dv > 0, writes We search for functions ψ = ψ(v) such that for any bounded measures F, G (1) supported in rS d−1 . Actually, since we already know that the dominant term is a von Mises-Fisher distribution, it is enough to impose (2.13) only for F = M lΩ dω, with λ(l) = σ r 2 l, for some given Ω ∈ S d−1 . Doing that, to any orientation Ω, we associate a family of suitable pseudocollision invariants, allowing us to determine the macroscopic equations satisfied by the moments ρ, Ω. A similar construction was done in [48], baptized as generalized collision invariants. Even if our approach is not exactly the same as in [48], we will continue referring to them as generalized collision invariants. Notice that once we have determined ψ such that (2.13) is verified for any bounded measure G (1) for F = M lΩ dω and any Taking into account the equalities the condition (2.14) becomes

The generalized collision invariants
In this section, we concentrate on the resolution of the linear equation (2.15). If we introduce the vector the equation (2.15) becomes elliptic on rS d−1 and reads The solvability of (3.1) requires that the integral of the right hand side over rS d−1 vanishes, i.e., is an eigenvector of the matrix corresponding to the eigenvalue σ. The following lemma details the spectral properties of the matrix M lΩ .
Proof. Clearly M lΩ is symmetric and definite positive. The case l = 0 is trivial, and we have M 0Ω = r 2 d I d . Assume now that l > 0 and thus necessarily σ r 2 ∈]0, 1 d [ cf. Proposition 1.5. We consider a orthonormal basis {E 1 , ..., E d−1 , Ω}. It is easily seen that We show that This comes by the condition λ(l) = σ r 2 l and integrations by parts We deduce also that and therefore We claim that the biggest eigenvalue is σ, that is r 2 − (d − 1)σ − |u| 2 < σ, or equivalently r 2 < dσ + |u| 2 .
This is a consequence of Lemma 2.3. Indeed, since l > 0, we know that implying that Replacing l = |u|r σ in the above inequality, yields r 2 < dσ + |u| 2 .
The resolution of (2.15) follows immediately, thanks to Lemma 3.1. As (2.15) is linear and admits any constant function on rS d−1 as solution, we will work with zero mean solutions on rS d−1 , that is Proposition 3.1. Let M lΩ be a von Mises-Fisher distribution i.e., Ω ∈ S d−1 , l ∈ R + , λ(l) = σ r 2 l, and E 1 , ..., E d−1 be a orthonormal basis of (RΩ) ⊥ .
1. If l = 0 and σ r 2 = 1 d , then the only (zero mean) solution of (2.15) is the trivial one.
2. If l = 0 and σ r 2 = 1 d , then the family of zero mean solutions for (2.15) is a linear space of dimension d. A basis is given by the functions ψ 1 , ..., ψ d satisfying for i ∈ {1, ..., d} and E d = Ω.
1. Let ψ be a zero mean solution of (2.15). Multiplying by (ω · W ′ ), with W ′ ∈ R d , and integrating by parts over rS d−1 yield We deduce that ψ is a constant, zero mean function on rS d−1 , and thus ψ = 0.
Notice that these functions also solve (2.15). Indeed, after multiplication by (ω · W ′ ), with W ′ ∈ R d , and integration by parts we obtain, for any i ∈ {1, ..., d} We deduce that which eactly says that ( ψ i ) 1≤i≤d solve (2.15). It is easily seen that the family ( ψ i ) 1≤i≤d is linearly implying that c i = 0, i ∈ {1, ..., d}. We show now that any zero mean solution ψ for (2.15) is a linear combination of ( ψ i ) 1≤i≤d . Let (c i ) 1≤i≤d be the coordinates of the vector W [ ψ] with respect to the basis We claim that ψ = d i=1 c i ψ i . Indeed, since ψ and d i=1 c i ψ i have zero mean, thanks to the uniqueness of zero mean solution, it is enough to check that d i=1 c i ψ i solves (3.1), with the right hand side M 0Ω ω·W [ ψ]. Indeed, we have The arguments are similar. The solutions ( ψ i ) 1≤i≤d−1 in (3.3) also solve (2.15), and are linearly independent. But for any solution ψ of (2.15), we have for any We focus now on the structure of the solutions of (2.15). This is a consequence of the symmetry of M lΩ , by rotations leaving invariant the orientation Ω. We concentrate on the case 0 < σ r 2 < 1 d , λ(l) = σ r 2 l, l > 0.
Proposition 3.2. For any W ∈ R d , W · Ω = 0, let us denote by ψ W the unique solution of the problem For any orthogonal transformation O of R d , leaving invariant the orientation Ω, that is OΩ = Ω, we have ψ W (Oω) = ψt OW (ω), ω ∈ rS d−1 .
Proof. We know that ψ W is the minimum point of the functional It is easily seen that, for any orthogonal transformation O of R d , and any function z ∈ H 1 (rS d−1 ), rS d−1 z(ω) dω = 0, we have Moreover, for any z ∈ H 1 (rS d−1 ), rS d−1 z(ω) dω = 0, and any orthogonal transformation leaving invariant the orientation Ω we obtain Finally, one gets for any We claim that there is a function χ such that, for any i ∈ {1, ..., d − 1}, the solution ψ i writes Lemma 3.2. We consider the vector field F given by Then the vector field F does not depend on the orthonormal basis {E 1 , ..., E d−1 } of (RΩ) ⊥ and for any orthogonal transformation O of R d , preserving Ω, we have There is a function χ such that Proof. Consider any other orthonormal basis {F 1 , ..., F d−1 } of (RΩ) ⊥ . Thanks to the identities Pick O any orthogonal transformation of R d , leaving invariant Ω. For any ω ∈ rS d−1 , we can write, by where, in the last equality, we have used the independence of F with respect to the orthonormal basis of (RΩ) ⊥ . Take now ω ∈ rS d−1 \ {±rΩ} and Clearly E · Ω = 0, |E| = 1.
Thanks to Lemma 3.2, in order to determine ψ i , i ∈ {1, ..., d − 1}, we only need to solve for χ. The idea is to analyse the behavior of the functionals J E i on the set of functions Ψ i,h (ω) = h Ω · ω r c i (ω), ω ∈ rS d−1 . The notation P ω stands for the orthogonal projection on the tangent space to rS d−1 at ω, that Proof. For any i ∈ {1, ..., d − 1}, the gradient of Ψ i,h writes Therefore we obtain where πβ 0 (l) = π 0 e l cos θ sin d−2 θ dθ and We consider the Hilbert spaces and for d ≥ 3, endowed with the scalar products By Lemma 3.2, there is a function χ such that ψ i = χ Ω · ω r c i (ω), i ∈ {1, ..., d − 1}. We know that ψ i , i ∈ {1, ..., d − 1}, minimize the functionals J E i (z), with z ∈ H 1 (rS d−1 ), rS d−1 z(ω) dω = 0. In particular, for any h ∈ H d , d ≥ 2, we have implying that χ, which belongs to H d , is the solution of the minimization problem Thanks to the Lax-Milgram lemma, we deduce that χ is the solution of the problem (3.5) if d = 2, and Up to now, for a given equilibrium F = M lΩ dω, we have determined the functions ψ such that for any bounded measure G (1) , supported in rS d−1 . But we need to control the linearization of Q around the equilibrium F in the direction F (1) , which is not necessarily supported in rS d−1 . It happens that the constraint div v {F (1) (α − β|v| 2 )v} = Q(F ), see (1.8), will guarantee that These computations are a little bit tedious and can be found in C.  (1) ) dv = 0, for any generalized collision invariant ψ of Q .

The limit model
We identify the model satisfied by the limit distribution f = lim εց0 f ε . We already know that f is a von Mises-Fisher distribution f = ρ(t, x)M lΩ(t,x) (ω)dω with ρ ≥ 0, Ω ∈ S d−1 , l ≥ 0, λ(l) = σ r 2 l. If σ r 2 ≥ 1 d , then l = 0 and M lΩ dω reduces to the isotropic measure on rS d−1 , that is f = ρ(t, x) dω r d−1ω d , with zero mean velocity u[f ] = rS d−1 ωρM lΩ dω = 0. In this case, the continuity equation reduces to the trivial limit model ∂ t ρ = 0, t ∈ R + . From now on, we assume that σ r 2 ∈]0, 1 d [, and we consider l > 0 the unique solution for λ(l) = σ r 2 l cf. Proposition 1.5. We are ready to justify the main result in Theorem 1.1 and the derivation of the SOH model (1.14)-(1.15).

Proof. (of Proposition 3.4)
Consider a collision invariant ψ, and let us compute We consider the application