Stéphane Brull

Jacques Schneider

On the ellipsoidal statistical model for polyatomic gases

. We also study the return to equilibrium of its solutions in the homogeneous case.

1 Introduction.

Classically the monatomic Boltzmann equation is used for describing rarefied gases. But in realistic cases polyatomic gases must be considered. In this case the density function depends on the velocity of particles but also on their internal energy (rotational, vibrational . . . ) and the density function writes f = f (t, x, v, I) with (t, x, v, I) ∈ R + × R 3 × R 3 × R + . The Boltzmann equation for this density is still

∂ ∂t + v • ∇ x f = Q(f, f ).
where Q(f, f ) is some quadratic collision term which can take different forms ( [START_REF]Kusker A model for rotational energy exchange in polyatomic gases[END_REF][START_REF]Pullin Kinetic models for polyatomic molecules with phenomenological energy exchange[END_REF][START_REF] Bourgat-L.Desvillettes-P.Le Tallec-B | Perthame Microreversible Collisions for Polyatomic Gases and Boltzmann's Theorem[END_REF][START_REF] Desvillettes | Sur un Modèle de type Borgnakke-Larsen Conduisant des lois d'Energie Non-linaires en Température pour les Gaz Parfaits Polyatomiques[END_REF]). But those models are very demanding in computer resources (see e.g [START_REF] Bahi | Contribution la simulation numrique des coulements en gaz rarfis[END_REF]). Therefore it is of interest to look for simplified models that preserve important properties of the former. The simplest one is the socalled BGK model introduced by Bhatnagar, Gross and Krook ( [START_REF] Bhatnagar-E | A model for collision processes in gases[END_REF]). It writes

R BGK (f ) = ρ (M -f ), (1.1)
where M = M(t, x, v, I) is the Maxwellian distribution defined by

M(v, I) = ρΛ δ (2πRT eq ) 3/2 (RT eq ) δ 2 exp - |v -u| 2 2RT eq - I 2 δ
RT eq (1.

2) with ρ(t, x) =

R 3 ×R + f dvdI, u(t, x) = 1 ρ R 3 ×R + v f dvdI, (1.3)
T eq = 2 (3 + δ)ρR R 3 ×R + ( 1 2 |v -u| 2 + I 2 δ ) f dvdI. (1.4)
δ is the total number of molecular internal degrees of freedom of the gas and R = k m , k being the Boltzmann constant and m the molecular mass. Finally

Λ -1 δ = R + e -I 2 δ dI.
Other important definitions are the total energy E E(t, x) =

R 3 ×R + ( 1 2 |v| 2 + I 2 δ )f dvdI, (1.5)
and the entropy functional H

H(f ) = R 3 ×R + f ln f dvdI. (1.6)
This model enjoys many important properties of the Boltzmann equation: mass, momentum and energy E are conserved and it is entropy dissipative. But it suffers a deficiency in the hydrodynamical limit at the Navier-Stokes level. Its Prandtl number P r is incorrect. Let us recall that P r is defined by

P r = γ γ -1 Rµ κ
where γ is the ratio of specific heats (γ = Cp Cv ) which is connected to δ by the relation γ = δ + 5 δ + 3 .

µ and κ are respectively the first viscosity and the thermal conductivity.

For most gases we have P r < 1 while P r = 1 for the BGK model. It is therefore of interest to look for relaxation models that do not have this drawback. Before proceeding further on let us recall an important fact proved by Toscani ([33]) and Collet ( [START_REF]Collet Extensive Lyapounov functionals for moment-preserving evolution equations[END_REF]). The definition of H is not merely the extension to polyatomic gases of the entropy functional for monatomic Boltzmann equation (see [START_REF] Cercignani | The Boltzmann equation and its applications[END_REF]) but it is the unique Lyapunov functional for the kinetic equation

∂ ∂t + v • ∇ x f = ρ (M -f ). (1.7)
This is a consequence of a variational principle. Let C f be defined by

C f = {g ≥ 0 / R 3 ×R + (1, v, 1 2 |v| 2 + I 2 δ )gdvdI = (ρ, ρu, E)}. (1.8)
Then there exists a unique (strictly) convex functional -namely Hsuch that M is the (unique) extremal point in C f (remark that conservation laws for the BGK model are equivalent to M ∈ C f ). This will explain the choice of entropy H below. Now the idea is to look for a "new" relaxation term of the form

R(f ) = λ(G -f ), (1.9)
where λ is a parameter and G is a function which are seek so as to obtain in the hydrodynamic limit the correct transport coefficients -the two viscosities µ and αµ and thermal conductivity κ-and thus the right Prandtl number. λ and G depend on macroscopic values of f . Moreover G is the extremal point in a set K f of the entropy functional. K f is included in C f to ensure conservation laws and it also contains constraints on other moments ( m i (v, I) gdvdI) i of g. More precisely

m i (v, I) R(f )dvdI = -λ i m i (v, I) f dvdI, ∀i
where (λ i ) i are strictly positive rates. Those constraints are chosen basing on physical considerations: in the hydrodynamic limit each moment m i (v, I) f dvdI becomes very small and measures in some sense the departure from local equilibrium. Thus (λ i ) i are connected to transport coefficients. So the whole procedure consists in 1. choosing the set of moments (m i (v, I)) i and thus defining K f , 2. finding the extremal point G of K f with respect to the entropy, 3. defining all parameters λ and (λ i ) i by computation of the hydrodynamic limit of (1.7) (with R(f ) instead of R BGK (f )) and comparison with the true Navier-Stokes equation so as to recover the correct transport coefficients.

Entropy minimization principles as well as the idea of using different BGKtype relaxation terms were first considered by Levermore ([20]) in the context of monatomic gases. But in his work the different models could only reach Prandtl numbers larger than 1. Then one of the author ( [START_REF] Schneider | Entropic approximation in kinetic Theory[END_REF]) has proposed to introduce a single relaxation term R(f ) that fits with the collision operator Q M ax (f, f ) for monatomic Maxwellian molecules in a weak sense: one adds in C f the constraints that moments of R(f ) coincide with those of Q M ax (f, f ) up to any chosen order (in this case the fluid limit is "correct").

Finally the approach we present here was proposed by the authors in the case of monatomic gases ( [START_REF] Brull | A new approach of the Ellipsoidal Statistical Model[END_REF]) who recovered the classical ellipsoidal statistical model by Holway ([17]). The paper is divided as follows. In section 2 we recall some physical backgrounds about polyatomic gases. Section 3 is devoted to finding additional moments constraints so as to define K f . In section 4 we address the problem of finding the extremal point G in K f with respect to the entropy functional H. Since this problem is subjected to the so-called realizability conditions (see Junk ([25,[START_REF] Junk | Maximum entropy for reduced moment problems[END_REF]) and one of the author ( [START_REF] Schneider | Entropic approximation in kinetic Theory[END_REF])) we prove that it has a unique solution under necessary and sufficient conditions on the relaxation rates (Theorem 4.1). Then we compute the hydrodynamical limit of the model and set the values of λ, (λ i ) i ) by comparison with the true Navier-Stokes system of equations. This allows to recover the proper transport coefficients and defines entirely (1.9). Moreover (1.9) turns out to be the ES-BGK relaxation model for polyatomic gases introduced in Andries&al ( [START_REF] Andries | Perthame Entropy condition for the ES BGK model of Boltzmann equation for mono and polyatomic gases[END_REF]). In the next section we study the return to equilibrium for the ES-BGK model (Theorem 5.1) in the homogeneous case. It is based on a sharp estimate of the relative entropy

H(f |M) = f ln( f M ) dv dI. (1.10)
Finally this result is illustrated numerically in the case of monatomic gases (section 6). The reason of this simplification is that we want to compare the L 1 -return to equilibrium (and also the relative entropy) between the ES-BGK model and the true Boltzmann equation. Those comparisons are not possible in the case of polyatomic gases since there exist only Monte-Carlo methods to solve such problem (except the method by Bahi [START_REF] Bahi | Contribution la simulation numrique des coulements en gaz rarfis[END_REF]). We also introduce in this section a new BGK model that allows to recover the proper viscosity for any kind of molecule interaction.

2 Physical backgrounds.

We have already defined the macroscopic moments ρ, u, E and T eq in (1.3,1.5). Recall firstly the conservation laws for the BGK model

R 3 ×R + (1, v, 1 2 |v| 2 + I 2 δ )MdvdI = R 3 ×R + (1, v, 1 2 |v| 2 + I 2 δ ) f dvdI. (2.1)
Remark that due to mass and momentum conservation the conservation of total energy is equivalent to that of the internal energy

R 3 ×R + ( 1 2 |c| 2 + I 2 δ ) MdvdI = R 3 ×R + ( 1 2 |c| 2 + I 2 δ ) f dvdI (2.2)
where c = vu. This definition is equivalent to that of the specific internal energy

e = 1 ρ E - 1 2 |u| 2 = 3 + δ 2 RT eq . (2.3)
e is naturally splitted into two parts e = e tr + e int with

e tr = 1 2ρ R 3 ×R + |c| 2 f dvdI, e int = 1 ρ R 3 ×R + I 2 δ f dvdI. (2.4)
Those energies correspond to the translational and internal energies of the particles. Temperatures are associated to them

T tr = 2 3R e tr , T int = 2 δR e int . (2.5)
Finally the opposite of the stress tensor Θ is defined by

Θ = R 3 ×R + c ⊗ cf dvdI. (2.6)
3 Setting of the problem.

The purpose of this paper is to construct a relaxation type term of the form

R(f ) = λ(G -f ),
where parameter λ and function G are to be fitted to the proper transport coefficients -the two viscosities µ and µα and thermal conductivity κ. This requires a particular definition of set of constraints K f which will define G (G ∈ K f ) through a variational principle. So let us determine K f . Remark firstly that macroscopic quantities introduced in the previous section are moments of f related to the polynomial space

P = span 1, v, v ⊗ v, I 2 δ . (3.1)
We will show in the next section that this polynomial space is well adapted to the variational problem defined below. Secondly conservation laws hold if and only if

G ∈ K f ⊂ C f . That is R 3 ×R + (1, v, 1 2 |v| 2 + I 2 δ ) g dvdI = R 3 ×R + (1, v, 1 2 |v| 2 + I 2 δ ) f dvdI. (3.2)
Again conservation of total energy can be replaced to that of specific energy using mass and momentum conservation

R 3 ×R + ( 1 2 |c| 2 + I 2 δ ) g dvdI = R 3 ×R + ( 1 2 |c| 2 + I 2 δ ) f dvdI. (3.3)
Then a mathematical and physical consideration is that f must tend to M in the homogeneous case. Since M is isotropic we set an additional constraint

R 3 ×R + c ⊗ c - 1 3 |c| 2 Id λ(g -f ) dvdI = -λ 1 R 3 ×R + c ⊗ c - 1 3 |c| 2 Id f dvdI, (3.4)
where λ 1 is a positive relaxation coefficient. Here Id is the identity matrix in R 3 . Finally following Borgnakke and Larsen ( [START_REF] Borgnakke-P | Statistical collision model for Monte-Carlo simulation of polyatomic gas mixtures[END_REF]) we take into account physical requirement that translational and internal energies tend to the specific internal energy e

R 3 ×R + 1 3 |c| 2 - 2 3 + δ ( |c| 2 2 + I 2 δ ) λ(g -f ) dvdI = -λ 2 R 3 ×R + 1 3 |c| 2 - 2 3 + δ ( |c| 2 2 + I 2 δ ) f dvdI = -λ 2 R 3 ×R + 1 3 |c| 2 f dvdI -ρRT eq , (3.5)
where λ 2 is a positive relaxation coefficient (here the last equation is obtained by using the definition of internal energy (2.3)). Now we are able to set our variational problem "Find G solution to the minimization problem

G = min g∈K f H(g) (3.6)
where K f is the set of functions g ≥ 0 such that (3.2),(3.4),(3.5) hold."

Remark that only two unknown parameters λ 1 λ and λ 2 λ appear in (3.4) and (3.5) which means that these two relations let us with an additional degree of freedom. Setting θ = λ 2 λ and using (3.3) constraint (3.5) can be formulated as follows

R 3 ×R + 1 3 |c| 2 gdvdI = (1 -θ) R 3 ×R + 1 3 |c| 2 f dvdI + θρRT eq .
Equation (3.4) can be transformed into

R 3 ×R + c ⊗ c g dvdI = 1 3 R 3 ×R + |c| 2 g dvdI + (1 - λ 1 λ ) R 3 ×R + (c ⊗ c - 1 3 |c| 2 Id)f dvdI.
In order to compare the present work with other works about the ellipsoidal statistical model for polyatomic gases we set

λ 1 λ = 1 -ν(1 -θ). (3.7)
and obtain

R 3 ×R + c ⊗ c g dvdI = (1 -θ) (1 -ν)ρRT tr Id + νΘ) + θρRT eq Id = ρT . (3.8)
This means that this tensor is defined entirely in terms of macroscopic quantities depending only on f . This will make easier the computation of G.

4 Construction of the relaxation model.

The aim of this section is to construct the relaxation model. It is performed in three steps.

1. Derive necessary and sufficient conditions on the values of θ and ν such that (3.6) possesses a solution. Give the shape of its solution G.

2. Define relaxation rates λ 1 ,λ 2 and λ such as to recover proper transport coefficients in the hydrodynamic limit up to Navier-Stokes level.

Comparison with the ellipsoidal statistical model for polyatomic gases ([2]

).

Solution to the minimization problem (3.6).

Our first concern is to solve minimization problem (3.6). Let us recall that such a minimization problem is subjected to the so-called realizability condition. For this problem we refer to the papers by Junk ([26]) and Schneider ( [START_REF] Schneider | Entropic approximation in kinetic Theory[END_REF]). A natural condition for realizability is K f = ∅.

Lemma 4.1. Let χ be a symmetric definite positive tensor of order two and consider the set K of functions g ≥ 0 s.t.

gdvdI = ρ, 1 ρ vgdvdI = u, δ 2Rρ |v -u| 2 gdvdI = T, ( |v -u| 2 2 + I 2 δ )gdvdI = E (4.1)
for some

(ρ, u, T, E) in R * + × R 3 × R * + × R * + with E > 2Rρ and
1 ρ R 3 c ⊗ c gdvdI = χ.
Then K is not empty.

Proof. The proof is given in appendix A.

It is clear that condition "χ is symmetric definite positive" in the above lemma imposes conditions on ν and θ in the minimization problem. As we are going to see those those conditions are also sufficient to ensure that the problem (3.6) possesses a solution for each density function f in the weighted

L 1 -space L 1 2 = {f / (1 + |v| 2 + I Theorem 4.1. Let f (f = 0) be a nonnegative function in L 1 2 , ν ∈ [-1 2 , 1[ and θ ∈ [0, 1]. Then minimization problem (3.6) admits a unique solution G defined by G(v, I) = ρΛ δ det(2πT )(RT eq ) δ 2 exp - 1 2 c, T -1 c - I 2 δ RT rel . (4.2)
where

T rel = θT eq + (1 -θ)T int . Conversely if K f = ∅ for all nonnegative functions f ∈ L 1 2 (f = 0) then ν ∈ [-1 2 , 1[, θ ∈ [0, 1
] and (3.6) admits a unique solution for each nonnegative function in L 1 2 .

Proof. (Theorem 4.1.)

Step 1: It is easy to see that

ν ∈ [- 1 2 , 1[ and θ ∈ [0, 1] ⇒ T ∈ S + 3 (R),
therefore K f = ∅ according to the above lemma. Let now m(v, I) be a generating vector basis of P and consider the set Λ defined with

Λ = {η ∈ R 11 / exp(η.m(v, I)) ∈ L 1 }.
Then λ = ∅ and λ ∩ ∂λ = ∅. This can be readily checked by writing any polynomial p(v, I) ∈ P as

p(v, I) = a + b.v + v T .A.v + cI 2 δ , with (a, b, A, c) ∈ R × R 3 × S 3 (R) × R. Then exp(p(v, I)) ∈ L 1 (R 3 × R if and only if -A ∈ S + 3 (R)
and c < 0. Then according to [START_REF] Junk | Maximum entropy for reduced moment problems[END_REF] there exists a unique solution to (3.6) which is of the form G(v) = exp(m(v, I)) with m(v, I) ∈ P. In this particular case the solution G can be computed explicitly using (3.2) and (3.8). Remark that this first step can also be obtained in a similar way using Theorem 2 in [START_REF] Schneider | Entropic approximation in kinetic Theory[END_REF].

Step 2: In the sequel we set ρ = 1 and R = 1 for the sake of clarity. Let f be a nonnegative function (f = 0) in L 1 2 and define T tr and T eq by (2.4), (2.3). Then

K f = ∅ implies (see (3.7)) 0 < R 3 ×R + 1 3 |c| 2 gdvdI = (1 -θ)T tr + θT eq . (4.3)
It is always possible to find a sequence of nonnegative functions f n ∈ L 1 2 such that

T n tr = 1 3 |c| 2 f n dv dI = 1 3 f n c → 0 as n → ∞ while T n eq = T eq = 2 3 + δ ( |c| 2 2 + I 2 δ )f n dvdI = 2 3 + δ ( 1 2 f n c + f n I )
is fixed. Then (4.3) writes

T n tr + θ(T eq -T n tr ) = 1 3 f n c + θ 2 3 + δ ( f n c 2 + f n I ) - 1 3 f n c > 0. (4.4) Suppose now that θ = 1 + ε > 1 for small ε then 2(1 + ε)f n I < (1 - εδ 3 )f n c
which is not possible for n big enough. Hence θ ≤ 1. Now let us consider constraint (3.8) and denote with (Θ i ) i=1...3 the eigenvalues of Θ and with (τ i ) i=1...3 the eigenvalues of T . On one side one has

Θ i ∈]0, 3T tr [, ∀i = 1 . . . 3
and on the other side τ i > 0, ∀ = 1 . . . 3. It is clear that Θ and T can be diagonalized in the same basis so that (3.8) leads to

(1 -θ) (1 -ν)T tr + νΘ i + θT eq > 0, ∀i = 1 . . . 3.
It is always possible to find a sequence of functions (f n ) such that T n tr = T tr > 0 is fixed while for a given index i one has Θ n i → 0. Hence by taking θ = 0 in the previous equation we get (1ν)T tr > 0 and thus 1 < ν. In the same way letting Θ n i tend to 3T tr we get (1 + 2ν)T tr ≥ 0 and so ν ≥ - 1 2 .

Remark 1. The maximal degree 2 (in v) of P ensures existence and uniqueness of the solution to minimization problem (3.6). A maximal degree of order 3 would remove this result. But P seems not well adapted to the requirement of finding right thermal conductivity κ since it involves moment of density function of degree 3. Nevertheless we will show in the next section that constraints related to P are sufficient to recover at the hydrodynamic level Navier-Stokes equations with proper transport coefficients.

Remark 2. This model is entropic as proved by Andries &al ( [START_REF] Andries | Perthame Entropy condition for the ES BGK model of Boltzmann equation for mono and polyatomic gases[END_REF]). This can also be obtained by generalizing the proof given in [START_REF] Brull | A new approach of the Ellipsoidal Statistical Model[END_REF].

4.2 Definition of λ 1 , λ 2 and λ.

In this part we determine the hydrodynamic limit of the equation

∂ ∂t + v • ∇ x f = λ ε (G -f ) (4.5)
so as to recover the Navier-Stokes transport with proper transport coefficients -the two viscosities α and αµ and the thermal conductivity κ. This will define relaxation coefficients λ 1 , λ 2 and λ. Here ε is the Knudsen number defined as the ratio between the mean free path and a typical length scale (see [START_REF] Cercignani | The Boltzmann equation and its applications[END_REF] for a discussion about this nondimensional number). Recall that the Navier-Stokes system of equations writes

∂ t   ρ ρu i E   + ∂ x j •   ρu j ρu i u j + P δ ij Eu j + P u j   = ε ∂ x j •   0 σ ij u i σ ij + κ ∂ x j T  
where the pressure is given by the polytropic law

P = ρRT = (γ -1)ρe, total energy by E = 1 2 ρ|u| 2 + ρe,
and viscosity tensor by

σ ij = µ (∂ x j u i + ∂ x i u j -αdiv(u)δ ij ).
The hydrodynamic limit of (4.5) can be obtained using the classical Chapman-Enskog expansion. This consists in letting ε → 0 and expanding f with respect to ε

f = f (0) + εf (1) + ε 2 f (2) + . . . (4.6)
Since G depends on f one also has

G = G (0) + εG (1) + ε 2 G (2) + . . . , (4.7)
where G (i) are functions of moments of (f (j) ) j=1...i . f (0) = G (0) is a Maxwellian distribution as a consequence of H-theorem (remark 1). Assuming that this function has the same first moments as f (t, x, v, I) then

f (0) = G (0) = M(t,
x, v, I) and the following orthogonality relations hold

1, v, 1 2 |c| 2 + I 2 δ f (1) dvdI = 1, v, 1 2 |c| 2 + I 2 δ G (1) dvdI = 0. (4.8)
Here equations on internal energies are obtained using the same argument as for equation (3.3). Then we are going to determine λ 1 , λ 2 and λ by using the set of constraints (3.2),(3.5),(3.4) and equation (4.5) at order 0 1) f (1) ). (4.9)

∂ ∂t + v • ∇ x M = λ(G (
Multiplying this equation with (1, v,

1 2 |v| 2 + I 2 δ
) and integrating it over v and I gives the Euler system of equations. This system is used to compute the left-hand side of (4.9) only in terms of space derivatives (see e.g [START_REF] Tallec | A hierarchy of hyperbolic models linking Boltzmann to Navier Stokes equations for polyatomic gases[END_REF])

∂ ∂t + v • ∇ x M = c c 2 + 2I 2 δ 2RT eq - 5 + δ 2 • ∇(ln T eq ) + 1 RT eq (c ⊗ c - 1 3 + δ (|c| 2 + 2I 2 δ ) Id : D(u) M (4.10)
where D(u) is defined with

D(u) = 1 2 (∇ x u + ∇ x u t ) - 1 3 div(u) Id.
Then

v i v j ∂ ∂t + v • ∇ x MdvdI = P ∂u i ∂x j + ∂u j ∂x i - 2 3 + δ div(u) δ ij . (4.

11)

Here P = ρR T eq (4.12) since the Euler system of equations yields T eq = T . Now expanding relation (3.5) at order 1 (with G instead of g) and using (4.9) one finds that the trace of Θ (1) satisfies the relations

|c| 2 f (1) dvdI = - 1 λ 2 |c| 2 ( ∂ ∂t + v • ∇ x )MdvdI = - 1 λ 2 |v| 2 ∂ ∂t + v • ∇ x MdvdI.
Here last equation is obtained by using orthogonality relations (4.8) together with (4.9). Comparing this with (4.11) gives on one hand

|c| 2 f (1) dvdI = - P λ 2 2 3 + δ div(u).
But on the other hand the trace of -Θ (1) must be equal to that of stress tensor σ ij µ(2 -3α)div(u).

So that

λ 2 = P µ 2 3 + δ 1 2 -3α . ( 4
-c i c j f (1) dvdI = 1 λ 1 c i c j ( ∂ ∂t + v • ∇ x )M dvdI = - P λ 1 ∂u i ∂x j + ∂u j ∂x i i = j
must equate the traceless part of stress tensor σ ij so that

λ 1 = P µ . (4.14)
Finally notice that the first polynomials in the right hand side of equation (4.10) are nothing but the Hermite polynomials H 3 i (c, I) with respect to M. Hence they are orthogonal to P for the scalar product φ ψMdvdI. This makes easier the computation of

H 3 i (c, I) (G (1) -f (1) dvdI = - 1 λ H 3 i (c, I)( ∂ ∂t + v • ∇ x )M dvdI = - 1 λ P u i ∂u i ∂x j + ∂u j ∂x i - 2 3 + δ div(u) δ ij + 5 + δ 2λ P R∂ x i T
where we the last equation is deduced from (4.10). On the other hand comparing this expression with the Navier-Stokes term for thermal conductivity one obtains

λ = 5 + δ 2 P R κ . (4.15)
This last definition together with (4.13),(4.14) define entirely the relaxation model (1.9).

Comparison with the ellipsoidal statistical model

Let us recall (see (3.7)) that

ν = 1 1 -θ (1 - λ 1 λ ).
Therefore using (4.14)

λ = λ 1 1 -θν + ν = P µ(1 -ν + θν) so that R(f ) also writes R(f ) = P µ(1 -ν + θν) (G -f ) (4.16)
This turns out to be the ellipsoidal statistical model found in [START_REF] Andries | Perthame Entropy condition for the ES BGK model of Boltzmann equation for mono and polyatomic gases[END_REF] for polyatomic gases.

Remark 3. The Prandtl number of this model is

P r = 5 + δ 2 Rµ κ = λ λ 1 = 1 1 -ν + θν .
For example, in the situation of diatomic gases we have the experimental values P r = 5 7 and θ = 1 5 . Hence ν = -1 2 .

5 Return to equilibrium.

We consider in this part the behavior of the solution to the space homogeneous equation

∂ ∂t f = λ(G -f ) (5.1)
when t tends to infinity. We show that the relaxation rates λ 1 , λ 2 have still an influence on the asymptotic behavior of f . The idea is to derive a rate on the decay of relative entropy H(f |G) (1.10) together with a Gronwall lemma. The main result of this section is the following theorem.

Theorem 5.1. For λ 1 and λ 2 defined in (4.14) and (4.13) let us put λ 3 = min(λ 1 , λ 2 ). Then for all f solution to (5.1), we have

f -M L 1 R 3 ×R + (t) = O(e -λ 3 t ). (5.2)
where M has the same definition as usual (1.2) dropping out the space dependance.

Proof. By derivation of the relative entropy (1.10) it holds that

∂ ∂t f ln f G = ∂ ∂t f ln f G + ∂ ∂t f -f ∂ ∂t G G .
We have two cases: θ = 0 and 0 < θ ≤ 1. We choose to treat the second case which is more complicated. Let us first show that

∂ t G G ∞ = O(e -λ 3 t ). (5.3) Let (τ 1 , τ 2 , τ 3 ) (resp. Θ 1 , Θ 2 , Θ 3 ) be the eigenvalues of T (resp. Θ). Then ∂ ∂t G G has the expression ∂ ∂t G G = 2 δ ∂ ∂t T rel T rel + 1 2 ( τ ′ 1 τ 1 + τ ′ 2 τ 2 + τ ′ 3 τ 3 ) + 1 2 c, ∂ ∂t (T -1 )c -I 2 δ ∂ ∂t ( 1 T rel ) .
Using (3.5) we obtain after multiplication of (5.1) by ) and integration (T tr -T eq ) ′ = -λ 2 (T tr -T eq ).

(5.4)

So

T tr -T eq = O(e -λ 2 t ). (5.7)

The symmetric tensor Θ(0) can be diagonalized as Θ(0) = P t ∆(0)P where ∆(0) is the diagonal matrix whose terms are (θ 1 (0), θ 2 (0), θ 3 (0)).

From to the relations (3.8, 5.7), T (t) can be diagonalized in the same basis as Θ(t) and Θ(0). Then T (t) = P t D(t)P , where D is the diagonal matrix whose terms are (τ 1 , τ 2 , τ 3 ). Therefore

τ i = (1 -θ) (1 -ν)RT tr + νΘ i + θRT eq (5.8) and τ ′ i τ i = (1 -θ) (1 -ν)λ(T eq -T tr ) Id + λ 1 ν(T tr -Θ i ) (1 -θ) (1 -ν)T tr + νΘ i + θT eq .
As θ = 0 by putting λ 3 = min(λ 1 , λ 2 ) and by using (5.5, 5.7), it holds that

τ ′ i τ i = O(e -λ 3 t ). As ∂ ∂t ( 1
T 2 δ rel ) = 2 δ T ′ rel T 2 δ +1 rel and T ′ rel = (1 -θ)T ′ int , it holds that ∂ ∂t ( 1
T 2 δ rel ) = 2 3 (1 -θ)λ 2 T eq -T tr T 2 δ rel .
Hence from (5.5)

∂ ∂t ( 1
T 2 δ rel ) = O(e -λ 2 t ). Moreover c, ∂ ∂t (T -1 )c = c, ∂ ∂t (D -1 )c = - 3 i=1 τ ′ i τ 2 i |c i | 2 . Therefore ∂ ∂t G G = T ′ rel T rel (- 2 δ + I 2 δ T δ rel ) - 1 2 3 i=1 τ ′ i τ i (1 + |c i | 2 τ i )
and (5.3) follows. Now let us finish the proof of Theorem 5.1. By integrating (5.1) and by using mass conservation, it holds that

∂ ∂t R 3 f ln( f G )dv = λ R 3 (G -f ) ln( f G )dv - R 3 f ∂ ∂t G G dv. As G ln( f G ) ≤ 0, it holds that ∂ ∂t H(f |G) ≤ -λH(f |G) + h(t) with h(t) = - R 3 ×R + f (- 2 δ + I 2 δ T δ rel ) 2 3 (1 -θ)λ 2 T eq -T tr T rel ) - 1 2 3 i=1 (1 + |c i | 2 τ i (1 -θ) (1 -ν)λ(T eq -T tr ) + λ 1 ν(T tr -Θ i ) (1 -θ) (1 -ν)T tr + νΘ i + θT eq ) dvdI.
So the relative entropy H(f |G) satisfies the inequality

∂ ∂t H(f |G)(t) ≤ -λ H(f |G)(t) -H(f |G)(0) + h(t) -λH(f |G)(0).
According to Gronwall lemma it comes that

H(f |G)(t) ≤ h(t) -1 -tλH(f |G)(0) + t 0 h(s)ds - t 0 u 0 h(s)ds h(s)e -λs du.
Hence as H(f |G)(0) ≥ 0, we get for t big enough

H(f |G)(t) ≤ h(t) + t 0 h(s)ds - t 0 u 0 h (
s) ds e -λs du.

According to Lemma 5.3, h satisfies

h ∞ = O(e -λ 3 t ).
Therefore H(f |G) satisfies the estimate

H(f |G) = O(e -λ 3 t ). (5.9)
Therefore from Czsisar-Kullback ( [START_REF] Csiszár | I-divergence geometry of probability distributions and minimization problems Sanov property[END_REF]) inequality we get

2H(f |G) ≥ 1 ρ f -G 2 L 1 (t) (5.10)
(the factor ρ is added because of the normalization) and the estimate (5.9) gives

f -G 2 L 1 (t) = O(e -λ 3 t ). (5.11)
Moreover in order to estimate G-M L 1 (t), we consider the relative entropy

H(G|M ) = R 3 G ln G M dvdI.
By using the expressions of G and M , H(G|M ) has the expression

H(G|M ) = G ln( (RT eq ) 3 2 det(τ ) ) + |c| 2 2RT eq - 1 2 c, τ -1 c + I 2 δ R ( 1 T eq - 1 T rel ) dv.
(5.12)

The second term of (5.12) can be computed as follows

|c| 2 2RT eq - 1 2 c, τ -1 c = 3 i=1 c 2 i 2R 1 T eq - 1 τ i = 3 i=1 c 2 i 2R T eq τ i τ i -T eq = O(e -λ 3 t ).
According to the definition of T rel , T eq and of (5.5)

1 T eq - 1 T rel = O(e -λ 3 t ).
In order to estimate ln(

(RTeq ) 3 2 √ det(τ )
), we use that θ i (t) has the expression θ i (t) = Ae -λ 1 t + RT tr and T eq has the expression T eq = T tr + Be -λ 2 t . Hence we get ln( (RT eq )

3 2 det(τ ) ) = O(e -λ 3 t ) Therefore H(G|M ) = O(e -λ 3 t ).
Then from Czsisar-Kullback ( [START_REF] Csiszár | I-divergence geometry of probability distributions and minimization problems Sanov property[END_REF]) inequality we get

G -M L 1 = O(e -λ 3 t ). (5.13)
And the estimates (5.11, 5.13) lead to (5.2).

6 Numerical study.

In this section we want to illustrate theorem 5.1 by computing in the homogeneous case the L 1 -convergence to equilibrium for different types of molecular interactions and different models -namely the BGK, the ES-BGK and the true Boltzmann equation. Remark that it does not exists a single Boltzmann equation for a chosen type of molecular interaction (see e.g [START_REF]Kusker A model for rotational energy exchange in polyatomic gases[END_REF][START_REF]Pullin Kinetic models for polyatomic molecules with phenomenological energy exchange[END_REF][START_REF] Bourgat-L.Desvillettes-P.Le Tallec-B | Perthame Microreversible Collisions for Polyatomic Gases and Boltzmann's Theorem[END_REF][START_REF] Desvillettes | Sur un Modèle de type Borgnakke-Larsen Conduisant des lois d'Energie Non-linaires en Température pour les Gaz Parfaits Polyatomiques[END_REF]). Moreover solving such an equation requires to use Monte-Carlo method due to its prohibitive deterministic computational cost [START_REF] Bahi | Contribution la simulation numrique des coulements en gaz rarfis[END_REF]. For such method it is neither possible to estimate a L 1 -convergence nor to compute a relative entropy. For those reasons we shall consider only monatomic gases. Now let us recall in this case different modelings:

1. the BGK model (we divide it by the molecular mass m as for the Boltzmann collision operator)

R BGK = ρ m ( M(v) -f (t, v)) (6.1) with M M(v) = ρ (2πRT ) 3/2 exp - |v -u| 2 2RT . (6.2)
For the sake of simplicity we still denote M by M.

The ES-BGK model

R ES-BGK = ρRT µ(1 -ν) G -f (6.3)
where G is defined by (4.2) together with the relations (2.4, 2.3, 2.6, 3.8),

the Boltzmann collision operator

R BE = 1 m R 3 ×S 2 + (f ′ f ′ * -f f * )B(θ, g)dθdεdv * , (6.4)
with

v ′ = v -n(n • g), v ′ * = v * -n(n • g), g = v -v * .
n is the unit vector such that

g • n = cos(θ) |g|
and the integral over the half sphere means g • n > 0. Finally B(θ, g) is the cross section.

It is well known that the BGK operator was designed for Maxwellian molecules that is when B(θ, g) = B(θ). Here we propose to replace this operator with

RBGK = ρRT µ (M(v) -f (v)) (6.5)
This model will be discussed later on (see section 6.3). We choose to compute the relaxation time for the argon (R = 208) with two different initial data and two different cross sections. The initial data are 1. an anisotropic Gaussian function

f (0, v) = f 1 (v) = ρ (2πRT 0 ) 3 2 exp - v 2 x 2 + v 2 y + 2v 2 z 2RT 0
with ρ ≃ 10 -5 kg/m 3 and T 0 = 234K. Then

u = 1 ρ f 1 (v) vdv = 0 m/s, T = 1 3ρR |v -u| 2 f (v)dv = 273K.
2. the sum of two Maxwellian functions

f (0, v) = f 2 (v) = ρ 2 1 (2πRT 1 ) 3 2 exp(- (v x -400) 2 + v 2 y + v 2 z 2RT 1 ) + 1 (2πRT 2 ) 3 2 exp(- v 2 x + v 2 y + v 2 z 2RT 2 )
with T 1 = 30K and T 2 = 300K. Then

u x = 200m/s, u y = u z = 0m/s, T = 229.1K.
We choose to do all computations using discrete velocity models (DVM). This enables to compute the L 1 -convergence and even the relative entropy which is not possible with DSMC. For the BGK and the ES-BGK numerical method we just use a simple Euler scheme. Recall that one can compute the exact solution for the homogeneous Boltzmann equation and that implicit schemes were designed by L. Mieussens ([22,[START_REF] Mieussens | Discrete velocity models and implicit scheme for the BGK equation rarefied gas dynamic[END_REF]) for those models. The reasons for this are one that we require accuracy in time and two because DVM for the Boltzmann equation do not enable implicit scheme. In this case we refer to [START_REF] Goldstein-B | Investigation of the motion of discrete-velocity gases[END_REF][START_REF] Palczewski | A consistency result for a discrete-velocity model of the Boltzmann equation[END_REF] for the computation of the quadratic integral collision operator. Comparisons with other quadrature formula such as [START_REF] Rogier | A direct method for solving the Boltzmann equation[END_REF][START_REF] Michel | Approximation simultanée de réels par des nombres rationnels et noyau de collision de l'équation de Boltzmann[END_REF] give the same results. Now we are going to treat the case of Maxwellian molecules and Hard Spheres.

Maxwellian Molecules.

In this case the cross section in the Boltzmann collision operator depends only on θ: B(θ, g) = β(θ). We choose to take β(θ) = cos(θ) which is well suited for the quadrature formula ( [START_REF] Goldstein-B | Investigation of the motion of discrete-velocity gases[END_REF]). Then the viscosity can be easily computed ( [START_REF] Cercignani | The Boltzmann equation and its applications[END_REF]) and µ = 2kT . Comparing the different models it is easy to see that a factor ρ 2 m appears in (6.5, 6.3, 6.4). Hence we can define a rescaled time with t ′ = ρ m t. For the numerical calculations, we use 20 × 20 × 20 velocities with bounds [-1000, 1000] 3 (in m/s). Figures 1 and2 show the L 1 -convergence to equilibrium with initial data f 1 and f 2 . The main feature in both test cases is the exponential decay of f -M L 1 for each model (time in logarithmic scale). First of all this illustrates Theorem 5.1 for the ES-BGK model but it is also the result proved in [START_REF]Arkeryd Stability in L 1 for the spatially homogeneous equation[END_REF][START_REF] Wennberg | Stability and exponential convergence for the Boltzmann Equation[END_REF] for the homogeneous Boltzmann equation for Maxwellian molecules with cut-off (the velocity domain is bounded). Comparisons between the rates of relaxation of ||f -M|| L 1 and of the relative entropy H(f |M) -e.g with initial data f 1 , see figure 3-for the ES-BGK and Boltzmann collision operators illustrate perfectly the Csiszàr-Küllback inequality (5.10) and show that it is quite optimal. This is an important tool that is used in Theorem 5.1 for the ES-BGK model. As concerns the Boltzmann equation the same inequality is used but the essential result is the inequality for Maxwellian molecules. This result was proved by C. Villani ([34]). Again the difference between the slopes fits perfectly with the theory.

H(Q + (f, f )) ≤ H(f ) ( 6

Hard Sphere.

In this case the cross section B is equal to σ 2 |(vv * ) • n| where σ is the molecular diameter. In this context it is proved in [START_REF] Chapman-T | The Mathematical Theory of non-uniform gases[END_REF] that the viscosity µ is given as a first approximation by the expression

µ = 5 16σ 2 kmRT π 1 2 = 5m 16σ 2 RT π 1 2
.

Hence

ρRT µ = 5 16 ρ σ 2 m (RT π) 1 2 .
Recall that in the hard sphere case the Boltzmann equation is given by

R BE = σ 2 m R 3 ×S 2 |(v -v * ) • n|(f ′ f ′ * -f f * )dv * dn. (6.7)
where n belongs to the whole unit sphere. Hence one can define the rescaled time t ′ = ρσ 2 m t. Figures 4 and5 show the L 1 -convergence to equilibrium with initial data f 1 and f 2 in the hard-sphere case. Since there is still a cut-off, the result proved in [START_REF]Arkeryd Stability in L 1 for the spatially homogeneous equation[END_REF][START_REF] Wennberg | Stability and exponential convergence for the Boltzmann Equation[END_REF] remains valid and decay of f -M L 1 in time is still exponential.

Finally figure 6 shows the the rates of relaxation of ||f -M|| L 1 and of the

R BGK = 1 τ (M -f )
where τ is the relaxation rate. Then using the Chapman-Engskog expansion we remark that this relaxation rate appears only in the space derivatives and the viscosity is given by

µ = τ ρRT = kT.
This viscosity is not far from that obtained with the modified BGK model (6.5) for Maxwellian molecules: µ = 2kT . Remark however that the rate of the L 1 -convergence to equilibrium obtained with the second model fits better with the Boltzmann equation. But for other type of molecular interaction the classical BGK model is not well adapted to any kind of computations.

With the modified BGK model one recovers the true viscosity µ though this model suffers the same deficiency as the classical one: the Prandtl number P r = 5 2 Rµ κ is equal to 1 contrarily to physical case where P r ≃ 2 3 . Nonetheless the convergence slopes are approximatively of the same order as the ES-BGK model (up to the factor 1 (1-ν) = 2 3 ). Observe that the modified BGK model and the ES-BGK model are essentially constructed -beside classical properties such as conservation laws and H-theorem-so as to give the proper viscosity (and heat conduction coefficient in the second case) for small Knudsen number. This means that there convergence rates to equilibrium is connected with the behavior of the moments f A(V ) (respectively f B(V )). It is clear for the Boltzmann equation that the behavior of those moments is not the main factor of the convergence rate to equilibrium except when the solution is closed to equilibrium. In this case the analysis of the spectrum of the linearized operator L shows the importance of those moments ( [START_REF] Cercignani | The Boltzmann equation and its applications[END_REF]). But in the cases we have chosen to treat, all convergence slopes show an important correlation with all models and henceforth with the above moments.

χ being being symmetric definite and nonnegative such that χ = S 2 .

. 13 )

 13 Correspondingly using relation(3.4), the traceless part of -Θ(1)

(5. 5 )

 5 On the other side multiplying the equation (5.1) by c ⊗ c -1 3 |c| 2 Id and integrating on R 3 × R + we have according to (3.4)Θ -RT tr Id ′ = -λ 1 Θ -RT tr Id .(5.6)Therefore Θ(t) satisfies Θ -RT tr Id = O(e -λ 1 t ).

Figure 1 :

 1 Figure 1: L 1 -convergence to equilibrium.

Figure 2 :

 2 Figure 2: L 1 -convergence to equilibrium.

Figure 3 :

 3 Figure 3: L 1 -convergence and relative entropy.

Figure 5 :Figure 6 :

 56 Figure 5: L 1 -convergence to equilibrium.

R 3 2 δ 2 R 3 2 δR 3

 322323 ×R + |S c| 2 h(-α |c| 2 -βI ×R + |S c 1 | 2 h(|c| 2 -βI ×R + |c 1 | 2 h(-|c| 2 -βI 2 δ )dcdI R 3 ×R + h(-|c| 2 )dcdIthe fourth relation of (4.1) is obtained. Therefore C f = ∅.

  |M) with initial data f 1 . In this case the estimate (6.6) is no longer valid but the convergence rates still show the importance of the Csiszàr-Küllback inequality.Remark that if we use the classical BGK model (6.1) we do not recover the true viscosity. The reason for this is quite simple. Let us write the BGK

		1
									BGK
									ES-BGK
									HS
		0.1
	Return to equilibrium	0.01
		0.001
		0.0001
		0	0.001		0.002	0.003	0.004		0.005	0.006
						Time
		1
									BGK
									ES-BGK
									HS
		0.1
	Return to equilibrium	0.01
		0.001
		0.0001
		0	0.0005	0.001	0.0015	0.002	0.0025	0.003	0.0035	0.004
						Time
	Figure 4: L 1 -convergence to equilibrium.
	relative entropy H(f 6.3 Discussion.

δ )f ∈ L 1 }.

APPENDIX A: Proof of lemma 4.1.

For any function h ∈ L 2 1 consider the function g α,β defined with

Then one easily see that g α,β has the same mass and momentum as f . Next, in order to get the relation


Emmanuel Duclos
email: duclos@esia.univ-savoie.fr

Maurice Pillet

AN OPTIMAL CONTROL CHART FOR NON-NORMAL PROCESSES

Keywords: Least-squares estimation, Non-Gaussian processes, Optimal estimation, Process control, Statistics

In this paper we present the application of a control chart for non-normal processes. This chart, which looks like an XS control chart is built with a leastsquares L-estimator, which can replace the arithmetic mean and standard deviation usually calculated for Shewhart charts. This estimator has the property to provide a minimum variance estimation of the process position and scattering. This, disregarding data distribution. We focused our attention on « multi-generators » processes, like screw-machines or multi-die holder for injection molding, these processes have the property to generate non-normally distributed pieces.

INTRODUCTION

When piloting a process with S.P.C. control charts ( X R / ) we usually assume that the mean is normally distributed. In spite of it robustness, this assumption is far from always being satisfied. In fact, either the X distribution is close to a normal distribution or the sample is large enough to satisfy the assumption of normality of the mean. However, large samples are prohibited for economic reasons. Disregarding this assumption can cause problems when the hypothesis of normality is not valid. First of all, statistical error, type I and II, correspond no longer with those defined for a normal distribution when control limits are placed at  3 from the target [START_REF] Shilling | Effect of Non-Normality on the Control Limits of X Charts[END_REF]. The second point concerns precision of estimations. Indeed, we can show that the mean is not the optimal estimator in term of variance when the population is non-normal. It means that we can find an estimator without bias which can provide better performances than the average. So in order to solve these problems, we propose in this article the use of a control chart (L chart) build with a minimum variance estimator whose performances have been compared to those of the average in term of variance and distribution shape. We will study this estimator in the case of data incoming from a « Multi -generator » process.

MULTI-GENERATOR PROCESSES

Consider a process which consists of several elementary machines. Among these processes, we can quote for example injection presses. Each criterion produced by an elementary machine is distributed among an elementary characteristic which can be normal or non-normal according to the criterion studied. Then, the global distribution or population is said to be a « mix of probability distributions » which is unlikely to give à normal population. To illustrate this article we have chosen the following mix of distributions (Figure 1). Assuming that each elementary distribution has the same probability (/   16 ) , the population is defined by :

fX f X f X f X pN N N () () (). . . ()       12 6
Even when a Multi-generator process is under control, observations are non-normally distributed.

Because of its physical characteristics, it is sometimes impossible to make it normal or economically unsuitable.

PRINCIPAL OF AN L-ESTIMATOR

If a population is normally distributed, we can prove that the variance of the arithmetic mean reaches the Frechet limit. So, the arithmetic mean is said to be the optimal estimator in terms of variance. Unfortunately, this is not the case when the population is non-normally distributed. Therefore, we have been interested in an estimator based on order statistics known as L-estimators, which have the capability to take into account the population shape, through their coefficients [START_REF] David | Order Statistics[END_REF].

The L-estimator proposed by E.H. Lloyd (1952), based on a Least-Squares algorithm, requires no hypothesis on the population shape whereas the method of maximum likelihood supposes a normal parent.

L-Estimators and L-Statistics

Consider a sample (x 1 , x 2 , ..., x n ) of n independent observations sampled at a time k.

x (1) , x (2) , ...., x (n) are ordered observations of Xk such as xx x n

() () () 12    .
x (n) is called n ordered statistics of the ordered vector Xk.

The linear combination of Xk's ordered statistics with a vector Cj of real coefficient is an L-statistic or L-estimator.

The estimation of  parameter at a time k is given by :

 ()  k t ki i i n CX C x      1 (1)
The average is a particular case of L-estimator since all coefficients are equal :

  C n in i   1 1;
. The range is another particular L-estimator for a scale parameter :

C 1 = -1, C n = 1 and C i =0   ii n
1, The problem is then, to calculate the L-estimator's coefficients in order to obtain the expected performances.

The Least Square L-estimator

Choice of  p and  p parameters. The population's parameters of location and scale which can be estimated by the least-squares L-estimator are not necessarily the mean and standard deviation of the population. However, we show that they are the most appropriate parameters, if we want to maximise the process capability indice. [START_REF] Pillet | SPC -Generalisation of capability index Cpm[END_REF]. The purpose of S.P.C. is to improve production quality and therefore to minimise production cost. Taguchi defined the loss function (2) which represents the cost of non-quality :

    LK X    2 2 Target (2) Cpm IT X   6 22  () Target (3)
Where K is a constant, X is the average of the population and  its standard deviation. We notice that minimising Taguchi's loss function is equivalent to maximising the capability indice Cpm (3). So to maximise the Cpm indice we have to minimise the population standard deviation and the quadratique error between sample mean and the process target. In consequence, we will consider X and  respectively as localisation  p and scale  p parameters of the population.

Generalised Least-Squares. In this paragraph we describe the construction of the least-squares Lestimator. We also expose some basic results concerning the Generalised Least Square theory.

Suppose the following multiple linear model X Wb u

  where :

 X is the variable of interest (sample n observations of the process).  u is a random vector modelling noise on Y (common causes).

 b is the vector of parameters that we expect to estimate (localisation and scale of the process).  W is a matrix including non random variables. The following conditions are supposed to be satisfied :

Eu e tVu I n () ()    0 2 
Estimating parameter b by the least-squares consists of minimising the u random variable influence. The Ordinary Least Square estimator is defined by :

   bW W W Y tt  1
Application on an ordered sample Assume (x (1) ,x (2) ,...,x (n) ) are ordered observations of a vector X k where xx x n

() () () 12     U (r) is a standardized variable :   U x r rp p () ()    
and moments of the order statistic U (r) are :

      EU V a rU C o vU U rr rr r r sr s () () () () ,    
Lets resume now the problem exposed by Lloyd. The moment of order statistics X(r) can be written under vectorial form :

 

EXr e pp () .

  

where  is a vector of a r and e is a unit vector

Xe a                                  x xn n () () 11 1 1    
The preceding equation can be written :

  EX A   where                          and n A 1 1 1  The variance-covariance matrix of X is :  2 
where  is an (nxn) matrix of  rs elements. [START_REF] Aitken | On least squares and linear combinations of observations[END_REF] proved that such a problem could be solved by applying a least-squares algorithm on ordered statistics. Since Var u Var X () ( )   2  , the model is general. The solution of this model derives from the ordinary least-squares. In fact, it is noticeable that  is a positive matrix, so a M(n,n) regular matrix exists such as:

M M t    1 A Generalised Least Square Estimation of  is defined as (4) :                      AA AX tt  1 1 1 (4)
Variance of both estimators of location and scale is (5) :

        Var AA Var ee AA p t t p t t  det  det            2 1 1 2 1 1     (5)
This theoretical description of the Least-Squares Lestimator brings to light, that this estimator can't be used without knowing moments of the ordered observations. This can cause some problems of application. The next paragraph will present an application of this estimator for Statistical Control of a multi-generator process.

L-CHARTS PERFORMANCES

Calculation of the variance-covariance matrix.

To be able to appreciate improvement brought by the L-estimator compared to the arithmetic mean, we achieved computer simulations. In order to determine the L-estimator coefficients, it is necessary to calculate the matrix of variances-covariances of standardized ordered statistics. These calculations were made with random data generated according to a known distribution. Each coefficient of the matrix is calculated with the relation ( 6)

      ij i i j j m m xx x x     1 1 () () ( ) ( ) (6)
Where m is the number of samples.

Coefficients of variance-covariance matrix can also be calculated from the theoretical expression :

      Cov X X x y f x
(,) ! () ! ( ) ! ( ) ! () () () () () ()           11 1 1 1
This expression necessitates powerful means of calculation and even more, if the sample size is large and the distribution is non-normal. Further more, most practical applications deal with unknown populations, so that, this calculus can't be achieved.

Although theoretical calculus was made to reach a great precision, computer simulations were preferred because of their similarity with the practical situation.

Since processes are not necessarily well known and operating conditions are always evolving, building a model of the population's distribution does not seem realistic.

Computer Simulations

Since a control chart is a set of two estimations : Punctual estimation and confidence interval estimation, we studied the L-estimator performances in terms of variance and shape. Since the method is non parametric, performances of the L-estimator are different according to the distribution of the population.

The more different from a normal law the distribution is, the more efficient the L-estimator. In consequence we studied a case where the population is significantly non-normal (figure 1) to appreciate the L-estimator efficiency compared to the mean.

Variance and bias of estimations. Results in table 2, show that the variance of the L-estimator of location is always much lower than the average. The relative efficiency of the L-estimator compared to the mean is (8) :



Var   4.29 2.45 1.52 1.01 0.72 0.55 0.44 0.36

 

Var X 5.32 3.99 3.19 2.66 2.28 2.00 1.77 1.60 eff 0.80 0.62 0.48 0.38 0.32 0.28 0.25 0.23

    eff Var Var X    (8)
To provide the same performances in term of variance as the L-estimator when n=4, the average requires a sample size n=7. The benefits of using an L-estimator can be either the sample size reduction, or the improvement of commandability when sample size is maintained and the variance of the estimation is reduced.

In opposition, the L-estimator for the scale parameter brings no improvement compared to the standard deviation, in terms of variance. However, it gives a non biased estimation of the population dispersion even when the population is non-normal (Figure 3, Table 2). Whereas the empiric standard deviation Sn, is a biased estimator when sample size is small. In fact, the coefficients c4 usually used to correct this bias is unsuitable when the population is non-normal.

A relative bias for estimators of a scale parameter can be defined as ( 9) :

  b pp       (9)
Of course these values are specific to this example, but similar results can be found for other significantly non-normal populations.

Symmetry of distributions. The distribution shape was studied to determine how to apply confidence interval tests for control charts. The setting of limits on X control charts are based on the assumption of normality, justified by the central limit theorem.

Table 2 Relative bias of the L-estimator of dispersion and standard deviation

n 3 4 5 6 7 8 b   -9.2 % -5.2% -3.4% -2.4% -1.7% -1.4%
b Sn 0.1% 0.03% 0.06% 0.03% 0.03% 0.01% The theorem essentially states, that under general conditions, the distribution of sample means approaches normality. Nevertheless [START_REF] Burr | The effect of Non-Normality on Constants for X and R Charts[END_REF] showed that for significantly non-normal distributions, sample mean was far from being normal. Yourstone (1992) proposed recently modified control limits to keep risks close to 0.27% in the case of skewed population. Figure 4 underlines the fact that when the sample size is small, normality is far from being satisfied. Coefficients of Kurtosis approaches 3 by inferior value as n increases. It means that the distribution has heavier tails than the normal law.

On the contrary we notice that the distribution of the L-estimator doesn't converge to a normal law (Figure 5), but a narrower one. Coefficients of Kurtosis are actually around 6. Since coefficients of the Means-Squares L-estimator are non zero for extreme values, this estimator doesn't reject outliers. As a result distribution tails are as heavy as the average ones, hence control limits are kept at 3 n to minimise type I false alarms.

EXAMPLE OF APPLICATION

The L-chart

The interest for such an estimator is evident for an industrial application since the sample size can be reduced without loss of efficiency compared to the mean. The application of this chart requires a preliminary run to calculate the variance-covariance matrix. However this run isn't restricting as it was established by simulation that 40 samples were sufficient to calculate  precisely. Such a preliminary run is then short enough for an industrial application. This procedure is equivalent to using a preliminary control chart to determinate the mean and the standard deviation of the process.

When using standard control charts, the process which is under control, is always supposed to be stationary. Statistical Process Control aim is to keep the process under control, which means evolution of the process position or dispersion. In consequence, each time an out of control state is detected, the model (matrix  and vector ) has to be computed again to take into account the evolution of the process.

L-chart for Start-up processes

In order to reduce to a minimum the period of reference and then control the process with the only values  p and  p fixed by the user's knowledge, we propose a start-up procedure for the L-chart by exponentially weighted moving average (EWMA) of the L-estimator's coefficients.

Coefficients of the estimator (C EWMA ) evolve from an initial value (C initial ) to the optimal coefficients (C L- estimator ) by using relationship (10).

C k k C k k C EWMA L Estimator initial      max max () 1 (10)
Where k is the number of a samples and k max is the number of samples required to calculate the matrix of variance-covariance with precision. The coefficient k max should be larger than 40.

Coefficients of the vector C initial depend on the parameter to estimate. In order to estimate the parameter of location  p , coefficient of C initial are equal to the mean (1/n). As a result, the estimator will provide the mean of samples at the beginning of the run. On the contrary, as k increases, more weight is given to the coefficients of the L-estimator C L- estimator . which are more and more reliable. In order to estimate the parameter of scale  p , the vector C initial defined as : C initial (1)= -1/d2, C initial (1)= 1/d2 and C initial (i)= 0   ii n 1, gives an estimation of population's standard deviation with R/d2. Thanks to this step, variance of estimation for the location parameter, decreases from  p n 2 to  Le s t i m a t o r  2 . 6. CONCLUSION Through this short example, we have shown that it is interesting to replace the traditional estimators of Statistical Process Control (the average and the standard deviation), by the Least-Squares Lestimator. This estimator has the advantage of providing non biased estimations with minimum variance. These two characteristics are essential to minimise the cost of non quality. In fact, it is shown that Taguchi's loss function is minimised when observations are centred on the target and their variance is minimal. In addition, the application of this L-estimator is not limited to Shewhart control charts. It can easily be extended to CUSUM charts. On the other hand, the use of the Least-Squares Lestimator is facilitated by its systematic method of calculation whatever the distribution of the population. In order to make this method transparent for a manufacturer, we have proposed a procedure to launch production without any preliminary run. Finally, for the construction of L charts, we have systematically placed control limits to Target  3 n . This specification allows us to reduce  risks (as compared to the average), however it seems unsatisfactory because, there are unquantifiable risks on both sides of the distribution. L-estimators are non non-normally distributed in spite of the fact that they are asymptotically normally distributed. We are presently working on a systematic method of calculation for control limits that would make L charts more efficient in term of Average Run Length (A.R.L.).

Fig. 1 .

 1 Fig.1. Non-normal population resulting of a mix of elementary distributions.

Fig. 2 .

 2 Fig. 2. Relative efficiency of the L-estimator compared to the arithmetic mean

Fig. 3 .

 3 Fig. 3. Relative bias of the L-estimator of dispersion.

  Fig. 8 L-chart of location compared to an X chart.

Table 1

 1

			Variance of both location estimators and
	relative efficiency of the L-estimator compared to the
				arithmetic mean
	n	3	4	5	6	7	8	9	10


Peter Wirtz

Governance and missionary effectiveness of the Company of Jesus: lessons from an extended theory of governance

Keywords: Governance, cognitive lever; behavioral lever, jesuits

From its very beginning, the governance of the Company of Jesus featured a certain number of specificities : vow of obedience to the pope, strong authority of the order's general, as well as the ignatian spirit which is transmitted through the spiritual exercises to all generations of jesuits. This article shows that the young Company's governance system played a decisive role for the order's missionary effectiveness. This is due, in great part, to its action as a cognitive and behavioral lever.

Introduction

The Company of Jesus, also called the Jesuit Order, was born of a vow made at Montmartre in 1534 by Ignatius of Loyola and his companions, and was officially recognized with the promulgation of the Papal bull Regimini militantis ecclesiae in 1540 in Rome by Pope Paul III. This bull contained the formula instituti, which was a first sketch of the order's structures, following the deliberations of the founding companions of 1539. The Institute thus prefigures the company's Constitutions, which was the central document framing the order's governance, and whose final written form was the result of an accumulation of several years' experience in concrete experimentation of missionary activity in the Ignatian spirit. Thus it was not a document which had been imagined in advance, but it was a description, carefully documented and prepared, about practices whose usefulness and effectiveness had been recognized and tried by Ignatius and his companions [START_REF] Bertrand | Un corps pour l'esprit : Essai sur l'expérience communautaire selon les Constitutions de la Compagnie de Jésus[END_REF] 2 .

What fundamentally distinguishes Jesuit governance from those of other religious orders is the explicit vow of obedience to the Pope, as well as the very strong central authority exercised by the Superior General, who is elected for life. Moreover, the principle of obedience revolves around a spirituality focusing on each individual's progress in a personal approach of the divine mysteries and the work of discernment as faced with choices to be made by each one in his present walk of life. Jesuit governance thus combines a strong governance marked by obedience to the central authority of the Church and the different hierarchical echelons of the order (Superior General, Provincials, Superiors of Houses, Colleges, and Universities) with a great inner freedom, each one working, "for the greater glory of God" 3 .

In the midst of a rapidly-changing world, at the dawn of modernity (16 th Century), the Company of Jesus quickly proved to have a formidable effectiveness in its apostolic and missionary activity, as witnessed by its rapid expansion throughout the world (India, China, Japan, and Latin America) [START_REF] Hours | Histoire des ordres religieux, PUF : Que sais-je Inauen E. and Frey B[END_REF], as well as the success of its schools and other institutions for formation, to the point of having a quasi-monopoly over colleges in Catholic Europe at the time (Calvez, 2001, p. 208). When the constitutions were approved in 1558, the order was already a globalized organization, with missions in East Asia and Latin America. Very quickly, and in addition to his spiritual role, the Superior General thus became the director of a truly "multinational" enterprise. The study of the genesis of its constitutions leads us to believe that the Company's system of governance, as experienced during its first two decades, formed a fundamental base of support for its rapid global expansion as well as for the management of this international missionary opus. The example given by the apostolic and missionary effectiveness as displayed by the Company of Jesus therefore interests management studies, and most especially research on governance. The latter in fact searches for possible explanations for the effectiveness of governance mechanisms in realizing organizational goals. However, the dominant approach of governance research deals mostly with big listed corporations and tries to measure their effectiveness through a measure of their impact on the creation of shareholder value. Due to this fact, this field is not suited to studying the impact of the governance of non-profit organizations, such as religious orders, on the accomplishment of their mission. Recently, researchers in economics and management have started to explore the governance of religious orders (Inauen and Frey, 2008;[START_REF] Inauen | Back to the Future: A Monastic Perspective on Corporate Governance[END_REF][START_REF] Rost | The Corporate Governance of Benedictine Abbeys: What can Stock Corporations Learn from Monasteries?[END_REF][START_REF] Wirtz | Le secret d'une organisation qui dure : la gouvernance cognitive des Dominicains[END_REF]. Notably, [START_REF] Wirtz | Le secret d'une organisation qui dure : la gouvernance cognitive des Dominicains[END_REF][START_REF] Wirtz | Au-delà des théories : leçons de la gouvernance des Dominicains[END_REF] studied the impact of the Dominican order's governance on the apostolic activity proper to it4 , and consider that in order to find answers to this kind of investigation, it is necessary to adopt an extended conceptual framework of governance, such as that proposed by [START_REF] Charreaux | A la recherche du lien perdu entre caractéristiques des dirigeants et performance de la firme: Gouvernance et latitude managériale[END_REF].

Two central aspects in Charreaux's work allow for an understanding of the logic and effectiveness in the governance of organizations as complex and specific as the religious orders, which are non-profit, but which have particular ends: the analysis of governance in a systemic approach, as well as an analysis of its mode of action according to three types of lever (disciplinary, mental or cognitive, and behavioral). For the Company of Jesus and its missionary action, we may believe that mental or cognitive levers (Ignatian spirituality, especially in what it means for discernment and intellectual formation) and behavioral levers (the principle of obedience as a means for resolving uncertainty) have historically played an especially important role (it is the central hypothesis of this article, as represented in Figure 1). The article aims at expanding upon this intuition. To do this, we shall firstly (1) summarize the expanded model of governance as proposed by [START_REF] Charreaux | A la recherche du lien perdu entre caractéristiques des dirigeants et performance de la firme: Gouvernance et latitude managériale[END_REF] which will serve as our interpretive framework. Then (2), we shall describe the specificities of the Company of Jesus' governance system, before analyzing

(3) the levers of its effectiveness. Which interpretive framework should be used to analyze jesuit governance?

In Management Science, the dominant approach towards analyzing systems of governance is strongly marked by agency theory (see [START_REF] Daily | Corporate Governance : Decades of Dialogue and Data[END_REF], which focuses the analysis on the management of conflicts of interest (so called "agency conflicts")

and measures the effectiveness of governance systems in their capacity to minimize agency costs and thus guarantee an appropriate return on investment for financial investors [START_REF] Shleifer | A Survey of Corporate Governance[END_REF]. Most empirical studies carried out according to this approach study listed companies and often focus on particular governance mechanisms (i.e. the board of directors) studied in an isolated fashion, rather than approaching the governance system as a complex and dynamic whole.

For a study of the levers of effectiveness in the governance of a religious order such as the Company of Jesus, this dominant approach towards governance seems unsuitable for at least three reasons. First of all, the goal of the Company such as it is stated in its motto (ad maiorem Dei gloriam5 ), and in the light of which its effectiveness should be judged, does not translate into a return on financial investment6 .

Secondly, agency theory analyzes the economic efficiency of governance systems only in terms of savings on agency costs, whose lever is essentially of a disciplinary nature.

That is to say that agency costs are minimized thanks to a set of incentive and control mechanisms whose principal role would be to minimize costs linked to agency conflicts among different stakeholders. Such a purely disciplinary approach to governance systems implicitly assumes that an enterprise's set of "value-creating" projects already exists such as could be found in a restaurant's menu [START_REF] Wirtz | Meilleures pratiques' de gouvernance et création de valeur : une appréciation critique des codes de bonne conduite[END_REF]. Said otherwise, (good and bad) projects represent a closed whole. However, such a purely disciplinary approach towards governance assumes the existence of a world closed to possibilities, which is in contradiction with the world contemporary to Ignatius of Loyola which witnessed an opening (at once intellectual and geographical) without precedent. The theological corollary to this opening is the idea of magis (more), so dear to the Jesuits, who always sought to further advance the glory of God, be it by novel means. This implies that, in the Ignatian world, marked as it was by sometimes radical discoveries and innovations, all possibilities were open 7 and able to move forwards thanks to the intelligent work of discernment by one and many. As soon as it is admitted that the scale of measurement for organizational effectiveness is the magis, this effectiveness cannot be assessed by a simple economy of agency costs, but must include the vectors of an authentic creation of value (spiritual in the case of a religious order). Working on the levers of an authentic creation in an open and uncertain world, beyond a simple economy therefore requires an extension of the analysis of governance to include cognitive and behavioral dimensions [START_REF] Charreaux | Gouvernance des entreprises : nouvelles perspectives[END_REF] 8 .

Thirdly, the Company of Jesus was meant to form an organic whole, a true "body", whose Superior General was the "head" (see especially the 9 th Part of the Constitutions, and more especially n° 719). However, as indicated above, many studies taking the dominant approach towards governance analyze certain mechanisms in a relatively isolated fashion. If the community of Jesuits really lives as an organic body, a systemic approach towards governance should thus be used, one in which the different actors and mechanisms of the system interact in a complex and dynamic way.

Charrreaux (1994) proposes such a systemic approach. In fact, he defines the governance system of an organization in a broader fashion as, "the set of mechanisms which have as an effect to delimit the powers and influence the decisions of top managers, in other words, 'governing' their conduct and defining their discretionary space." (p. 421-422). Thus, according to Charreaux, the governance of an organization is a system, made up of various mechanisms, which interact with the top executive and which give him a more-or-less sizeable discretionary space. The governance system and the manager's discretionary space are thus two sides of the same coin, namely the organization of authority at the highest level of an organization. Let us note that, given the vow of obedience and his election for life, the discretionary space of the Superior General of the Company is potentially very large. We shall revisit that point.

Different governance mechanisms can a priori be classified according to two criteria (Charreaux, 1997, p. 427), namely their degree of intentionality (the way a mechanism functions can be intentional, such as in the case of legislation and law, or spontaneous, such as in the case of cultural values), and their degree of specificity in relation to the organization (thus the general assembly of a public corporation is specific to itself, but the corporate law of a country is a non-specific mechanism). The intersection of these two criteria allows Charreaux to propose a typology, which is useful for describing the composition and characteristics of the governance system of a concrete organization, and which is proper to it. We have attempted to describe the configuration of the Jesuit system of governance in the second section of this article (table 1). Charreaux's typology, beyond allowing for an orderly classification of a list of mechanisms for a given organization, emphasizes the fundamentally systemic nature of governance, for it highlights: (1) the multiplicity of mechanisms susceptible of interacting with one another in a dynamic way; (2) their different possible modes of functioning, with the dynamic effects potentially induced by and for the system9 , as well as;

(3) the social nesting of any system of governance, even the most specific, in cultural, political, economic, and social logics10 , which are not specific to the system of governance, but with which it interacts in a constant manner.

Beyond its representation as a complex and dynamic system, [START_REF] Charreaux | A la recherche du lien perdu entre caractéristiques des dirigeants et performance de la firme: Gouvernance et latitude managériale[END_REF] proposes a model of governance, which is extended to include the cognitive and behavioral dimensions. Thus he goes further than agency theory does, for it only sees in governance a simple disciplinary lever. However the, "mechanisms governing top managers' conduct," are equally susceptible of acting like cognitive levers in the construction of future projects as well as acting like behavioral levers, allowing for the solving of problems faced in situations where standard rationality is inoperative. execution [START_REF] Fama | Separation of Ownership and Control[END_REF]. The initiation of strategic projects and their implementation would thus be the top manager's exclusive responsibility, the governance system merely playing the role of a control mechanism. Such an approach ignores the genesis of strategic projects and the fact that, in the reality of certain organizations, governance mechanisms are susceptible of offering support to the top executive in the very conception of his strategic projects. It is moreover the case for the Company of Jesus, to the extent that the vow of obedience made to the Sovereign

Pontiff for example, confides to the latter the choice of destinations for missions.

However, if the system of governance provides support to the top manager in the formulation (the initiation) of strategic projects, that is to say in their very conception, it functions as a truly cognitive and behavioral lever. This means that it potentially helps the director to better reflect upon, and put into practice his strategy by improving it through a better understanding of the possibilities. To understand the action of governance as a cognitive lever, reference can be made to resource-based theory [START_REF] Penrose | The Theory of the Growth of the Firm[END_REF]Wenerfeld, 1984). It allows for an understanding of the dynamics present in the construction of the cognitive resources of an organization and the interaction of their construction with strategic development. For its part, the behavioral lever makes reference to another theoretical field, partially intersecting the reflections on the cognitive approach, but which puts an accent on a certain number of psychological and behavioral biases which have been identified by studies of individuals placed in decision-making situations (i.e. [START_REF] Kahneman | Prospect Theory : An Analysis of Decision Under Risk[END_REF].

This literature shows, for example, that individuals naturally have a tendency to turn towards simple heuristics when placed before complex decisions. These are mental shortcuts, used to make a decision. Heuristics are thus a means for individuals whose rationality is limited, even procedural, to face a complex world. Their limited rationality means that they do not have complete knowledge of the parameters allowing them to make an optimal decision [START_REF] Simon | From Substantive to Procedural Rationality[END_REF].

The system of governance of the Company of Jesus

This section has the goal of representing the major features of the Company of Jesus'

governance system, such as it emerged during the first decades of the order. This [START_REF] Bertrand | La politique de S. Ignace de Loyola[END_REF]. 12 In fact, the cognitive approach considers educational systems as cognitive governance mechanisms.

Specific mechanisms

Non-specific mechanisms

Intentional mechanisms

-Institute -Constitutions of the Company -General Congregations ("To the authority of the Superior General over the Company is counterposed the authority of the Company over the Superior General", [START_REF] Bertrand | Un corps pour l'esprit : Essai sur l'expérience communautaire selon les Constitutions de la Compagnie de Jésus[END_REF] The different concrete governance mechanisms (institute, constitutions of the company, general congregations, vows, Ignatian spirituality, process of incorporation, Canon

Law…) together and in their interaction with each other define the discretionary space of the Superior General, who directs the Company. This discretionary space is intentionally very large. The Constitutions render the principle and reasons explicit in these terms: "Since experience, the practice of government, the knowledge of each in particular, and the authority exercized over all are of great value to perform this duty, the General should be elected for life and not for a determined time. To the other benefits shall also be added this one: the Company will have less to suffer from the difficulty and time spent in General Congregations, being nearly always fairly busy with matters of great importance for the glory of God." (719) And further on: "The authority of the Superior will be greater if he cannot be changed than if he had been elected for one or several years, vis-à-vis people from the outside, because he will be better known by all, and for the same reason, vis-à-vis those of the Company. To the contrary, the fact of knowing that he will leave his role one day, and will be equal or inferior to others, and also the fact that he will have little experience in the role could diminish his authority." (721). The great authority of the Superior General, starting with his large discretionary space, is thus anchored in the constitutions, but already the principle dates back to the deliberations of the first companions in 1539, who then decided to institutionalize a vow of obedience to one among them.

Beyond non-specific limitations, such as those imposed by Canon Law, and relating to the general political and regulatory environment, the Superior General's discretionary space, however, knew two specific and important limitations. The first is that of obedience to the Pope which is institutionalized in a very explicit vow (specific to the Jesuits). We shall see later on that it was instituted to play an important role as a behavioral lever, at the service of the Order's missionary service.

The second important limitation is the authority of the whole community in answer to that of its Superior General. This authority of the community hinges on the congregations (provincial and general)14 . In fact, the constitutions describe a profoundly dynamic conception of the relationships between actors in the governance system (and more especially between the "body" and the "head" of the company Congregations for their part interact with the general Father in an unceasing back-andforth where the initiative moves from one (Chapter 2,3,6) to the other (4,5,7). The 3x3 plan in Part IX expresses this same game admirably. Thus, and this is the most important point, a certain richness of concrete relationships is linked to the solidity of the two poles among which they are tied." (Bertrand, 1974, p. 188). That it is the community which circumscribes the discretionary space, large as it is, of the Superior General through the governance mechanism of the congregation is clearly expressed in the following quote: "…the community (finding its expression through the means of congregations) is the supreme body, which the companions scattered throughout the universe have found the strength to represent by delegates. It absolutely precedes the Superior General, when it concerns his election (…); it accompanies him as an equal towards an equal, when it concerns important questions other than the election (Chapter 7 of Part VIII of the Constitutions)." (Bertrand, 1974, p. 190).

This way of organizing the field of interaction among actors in the governance system, where the very strong authority of the Superior General interacts with an equally strong authority from the community (Bertrand speaks of "the solidity of two poles"), acts as a real mental lever in the Order at the service of the magis, as we shall see further on.

Thus, faced with a strong top executive, an equally strong system of governance is susceptible of acquiring an enabling force. "Governing the conduct" of a top manager (in reference to Charreaux's definition, 1997) is thus not necessarily a restriction in the director's latitude over an organization, but is potentially a source of support.

Cognitive and behavioral governance at the service of the mission

After the preceding brief presentation of the pillars of Jesuit governance, let us now see according to which modalities the principal governance mechanisms influenced the missionary and apostolic success of the young Company. We shall see, in this context, that they above all acted as behavioral and cognitive levers to rapidly spread the Company's missionary activity across the world and assure its success.

Obedience to the Pope is a governance mechanism which, in the emerging Company, acted as a behavioral lever. In fact, obedience to the Roman Pontiff, far from being motivated by an irenic representation of his role, in fact acted as a heuristic technique for solving problems of indeterminacy in making choices in a world whose frontiers were expanding and which would thus become only more complex. Obedience to the Sovereign Pontiff supplied an external answer to the question of "where" the apostolic mission should be carried out. In their seventh part, the Constitutions of the Company affirm this fact, recalling the deliberations of 1539: "The intention of the forth vow made to the Sovereign Pontiff was not aiming at any place in particular, but that those making this vow be spread over different parts of the world. In fact, those who first came together in this Company came from many different provinces and kingdoms, and it did not seem clear to them to which country of faithful or infidels they should go. In order not to err along the way of the Lord, they made this promise or this vow so that the Sovereign Pontiff spread them out for the greater glory of God and accordingly to their intention to travel the world, and so that, if they did not find the desired spiritual fruit in one place, they could go from one to another, seeking the greater glory of God and greater support for souls." (n° 605)

The following commentary by Bertrand (1974, p. 139) presents the behavioral role of these vows of obedience to the Pope very well, acting as a heuristic device allowing for the solving of the dilemma of decision-making in an indeterminate framework. "The first most important thing for the first companions was the search for an 'other' variable to free them from the indetermination in which the very composition of their group placed them. More and more, so that their universal desire for service could take flesh without disappearing, they needed this voice of authority external to the group, even if it was criticized at the time by many, namely the voice of the Pope. The originality of the process, which was in a way scandalously prosaic, was a process of faith and, was to take the Pope as a means 'to better succeed'. Thinking that the Pope could be useful was a fundamentally realistic principle from which the Company was born. Some may be scandalized that the Holy Father could thus be reduced to a means and that such a reduction could be attributed to Saint Ignatius. The texts are there. Moreover, those who would be scandalized would no doubt have reflected less than Ignatius did on the definition of a means. Finally who, below the servus servorum Dei, could be scandalized to serve something, I mean to say, to serve men who unconditionally trust him, not for him, but for the single end which is one: the greater glory of God and the more universal salvation of souls?"

Once the companions were spread across the world, Jesuit spirituality and education acted as cognitive levers for the accomplishment of the apostolic mission. This spirituality allowed the necessary inner freedom to imagine the precise means for an effective mission according to the very particular and ever changing circumstances in which the companions sent on mission found themselves. Obedient submission to the Pope concerning the place they were sent to moreover had the effect of concentrating the companion's cognitive resources solely on the accomplishment of the mission in the precise place where they had been placed. Historically, the rapid successes of the missions were for a large part attributable to inculturation, which proceeded precisely from the Ignatian spirit and which encouraged the best ways of announcing the Gospel to be found, according to the particular context and circumstances. "In China and India, (…) the Jesuits defined a new approach to missions prefiguring that which we have since called 'inculturation'. It advocated a mastery of local languages and a deep knowledge of cultures, the development of a Christian body of literature in these languages, the adoption of local customs and rites compatible with Christianity, and the formation of a local clergy." (Hours, 2012, p. 113) This missionary approach by the Jesuits in fact proved to be particularly effective, to the point that a "new missiology" was gradually adopted as the dominant approach from the 19 th century on, borrowed from the founding intuition of the Jesuits (see Hours, 2012, p. 119-121). The development of a "new approach to missions" represented a real act of cognitive creation (in the sense of the generation of new knowledge and skills), and it is likely that a specifically Ignatian spirituality was an important lever for making this creation possible.

Let us note at this point that, later on (in 1645), Canon Law acted, not as a cognitive lever, but as a means to impose a strict discipline by forbidding the Chinese Rite as developed by the Jesuits. In fact, the Propaganda and the Holy Office proclaimed their condemnation of the Chinese Rite after the affair was carried before the Holy See by the Dominicans, who did not work according to the same methods. Thereafter, it is probable that missionary activity lost its effectiveness, just as the new missiology eventually returned to the inculturation principle for missions during the second, very strong missionary wave of the 19 th and 20 th centuries (Hours, 2012, p. 120).

One of the important pillars of Jesuit governance, as described previously, is the community's authority, as expressed through general congregations. It faces the equally strong authority of the Superior General in a game of dynamic interaction. The interaction between these two "strong poles" of governance that are the "head" and "body" of the Company act as a cognitive lever for initiatives at the scale of the Order.

Thus, Bertrand (1974, p. 190) can declare, "it is clear that the community seems like a possible place of ferment for initiatives, worries, and projects (…). It is a happy ferment where the spirit of the Company awakens. Saint Ignatius, less through a gift of prophecy than through a true knowledge of man, was certain that this ferment would occur." Yet all the energy of this "ferment" must not be dispersed, which explains the rare holdings of general congregations comparatively to other Orders. Once the great projects and the man leading them are determined (during the Superior's election), each one need only concentrate on his mission, whence he has been sent, and to place all his energy there, ad maiorem Dei gloriam."

Conclusion

The Jesuit Order was born more than four centuries ago. It experienced rapid growth, meeting with important missionary and apostolic successes very early on in a rapidly expanding world. The Company of Jesus' system of governance was characterized from its beginnings by a certain number of features, including the vow of obedience to the Pope, the Superior General's great authority, as well as the Ignatian spirit which is transmitted through the Spiritual Exercises to every generation of Jesuits. In the present article, we have demonstrated that the particular system of governance of the young Company played a decisive role in the missionary effectiveness of the Order, especially thanks to its action as a cognitive and behavioural lever. In an extension of this study, from the point of view of studies on governance systems, it would be interesting to develop a comparison with other religious orders, as well as studying issues of competition among them 15 . More generally speaking, given their long-time survival over the centuries, the study of the governance systems of religious orders, by comparison, may allow for important insights into the requirements of effective governance of modern organizations. Such a comparative analysis is yet to be developed.

Figure 1 .

 1 Figure 1. Jesuit governance and effectiveness of the international expansion of its mission

Figure 2 .

 2 Figure 2. Charreaux's extended model of governance(2008, p. 1861)

  ): "Part IX (of the Constitutions) (…) is (…) shared equally. Three chapters concern the general Father and his relationships with the Company, and three other chapters, the relationships of the Company towards the general Father. (…) All (in Part VIII and IX of the Constitution) is always organized in a way to move constantly between one pole and the other; the union of hearts embracing all game-plays in the representation of the whole during provincial and general congregations (Chapter 1 versus Chapters 2 to 7 in Part VIII).

  representation (table 1) is mainly based on two documents: The Constitutions of the Company in their official version (dating to said version B approved in 1558), and whose French version is found on the internet site of the Province de France; and Father Dominique Bertrand's work 11 , who published a very fine study of the genesis and structure of the Constitutions in 1974, as well as the spirit animating them.

	11 Dominique Bertrand is a Jesuit Father and holder of a State Doctorate. His thesis was a very
	methodical analysis of the vast Ignatian body of correspondence (see

Let us note beforehand that according to Father Bertrand, the spirit of St Ignatius' Spiritual Exercises is also fundamentally the same spirit animating the Constitutions of the Company, which by definition possess a social dimension, whereas the Exercises concern a mostly personal experience. The relation between the individual and the social dimensions of ignatian spirituality is thus complex but very real, for all of those who, through a long process of incorporation as prescribed by the Constitutions and including several important steps (general exam, noviciate, studies…), become companions do the Exercises. Since they are an integrative part of the Jesuit socialization process, the Exercises can equally be considered a governance mechanism 12 . To characterize it in a few words, let us say that the Ignatian spirit combines a sustained and progressive exercise in personal discernment with a spirit of obedience. The Ignatian spirit was born between a certain tension among a very direct,

Table 1 .

 1 The Company of Jesus' system of governance 13 The Chinese Jesuit Rite was forbidden following proceedings by other Orders who had maintained the traditional Latin Rite in their missionary activity (see below).

			-Canon Law (Papal Bulls…)
			-State (in France, the relationship between
			the monarchy and the Order have sometimes
			been conflictual)
		, p.
		76)
		-Vows (obedience to the Pope, to the
		Superior General)
		-Process of incorporation
		(education/socialisation)
	Spontaneous	-Ignatian spirit/spirituality (Spiritual	-Cultures of the missionary lands
	mechanisms	Exercises)	-Ambiant intellectual environment
			-Competition among religious
			congregations 13

The Jesuit Father Dominique Bertrand retraces, in a meticulous work (see especially Chapter 2), the genesis and structure of this important document, which had at least three different versions and which stabilized in 1558, about two years after the death of Ignatius and nearly two decades after the foundation of the order. By that time, the Company was already present with Provinces in India and Brazil, among other places. It is also to be noted that Jesuits were at the origin of the foundation of the city of Sao Paulo.

The Dominicans share with the Jesuits an intense intellectual productivity, as well as an apostolic mission. Yet, the governance systems of the two orders are radically different. In fact, Dominican governance is characterized by a very pronounced democratic functioning, and contrarily to the Jesuits, Dominican brothers spend a lot of time in regular and formalized deliberation processes, and this at every level (convent, province, order)[START_REF] Wirtz | Au-delà des théories : leçons de la gouvernance des Dominicains[END_REF].

Concretely, work for "the greater glory of God" is done by an intense apostolic and missionary activity throughout the world.

Especially given that the companions take a vow of poverty.

It is even potentially unlimited, which creates a problem of indeterminacy of choice, where the specific system of governance can be an effective remedy, as we shall demonstrate further down.

The fact of extending the analysis to include a cognitive and behavioral dimension does not in any way imply that the disciplinary dimension is absent from Jesuit governance. The Company always maintained numerous and complex relationships with ecclesial and temporal power, and there were frequent conflicts between the Papacy and the Jesuits over the course of history. The experiment of "reductions" for example, allowing native villages in Paraguay to live in autarchy, clashed with the economic interests of colonial powers, who finally obtained the expulsion of the Jesuits.

Spontaneous mechanisms, notably, through their indeterminate nature, are susceptible a priori to causing new things to emerge (and potentially supporting the magis, if we wish to link it with the Jesuit mission's problematic.) Ignatian spirituality, which puts a strong emphasis on the personal discernment of each person, sets the example.

The Company of Jesus was born in a particular socio-cultural environment and Ignatius, its first Superior General, interacted in a regular way with different authorities and political, cultural, and spiritual institutions of his era, to which his very rich correspondence attests (seeBernard, 1985).

Let us note that, contrary to other Orders who hold chapters at regular intervals, the Jesuits did not wish such a regularity, General Congregations being called in only two cases: for the election of a Superior General, and for important matters concerning the Company.

We thank Gerard Charreaux for this remark. personal approach to the Divine mysteries and an obedient submission to legitimate authority (Pope, Superior General). "This tension no doubt characterizes that which Ignatian spirituality authorizes: believing in the immediacy of the experience of God, and in the fruitfulness of long meditations to inscribe it into society and history." (See, "Petite Introduction à la Spiritualité Ignatienne.", http://www.jesuites.com/spiritualite/intro04.htm )