Rémi Peyre

Some ideas about quantitative convergence of collision models to their mean field limit

Some Ideas About Quantitative Convergence of Collision Models to Their Mean Field Limit

Rémi

Introduction

The Boltzmann equation was written down by L. Boltzmann [START_REF] Boltzmann | Weitere Studien über das W ärmegleichgenicht unter Gasmol äkuler[END_REF] in 1872, five years after Maxwell's seminal paper [START_REF] Maxwell | On the Dynamical Theory of Gases[END_REF], to describe the behaviour of a large number of gas molecules interacting by pairwise collisions. Proving rigorously the heuristic arguments of Boltzmann to get some convergence of the N -particle model to the continuous Boltzmann equation when N -→ ∞ is an extremely difficult challenge that mathematicians are still dealing with.

Here we are only going to handle the spatially homogeneous Boltzmann equation (also called mean field Boltzmann equation), in which one forgets the positions of the gas particles to concentrate only on the collision phenomenon. Then proving the convergence of the N -particle system to the continuous equation is a typical mean field limit problem-a particle model is said to be mean field when each particle interacts with comparable strength with all the other ones. Such a problem, which was first proposed by Kac [START_REF] Kac | Foundations of kinetic theory[END_REF], is far more tractable than the original one, and convergence results, mostly qualitative, have already been obtained for it (see § 6.4).

Here however we are interested in a quantitative and non-asymptotic version of these results. We would also like to set our results in an infinitedimensional setting, that is, to say that not only any reasonable functional of the particle model converges to the corresponding functional of the limit system, but moreover that all these functionals converge uniformly. The quantitative convergence we are going to prove will even have an N -1/2 speed, typical of the uniform central limit theory (see [START_REF] Dudley | Uniform central Limit Theorems[END_REF] about it).

Concerning concrete Boltzmann models, in the actual state of my work I am only able to use my results for Maxwellian systems, and moreover constants in convergence bounds deteriorate rapidly with time. However that does not seem to be a fundamental feature of my approach, and further improvements might overcome these issues.

Important Remark.

There are two sides in this work. The first one, whose climax is Theorem 3.3, is abstract: it consists in showing how Hilbert spaces can be used to prove a new powerful type of convergence results for collision models like Boltzmann's. That work is a priori likely to be applied to a wide range of situations, but for each of them checking the hypotheses of the abstract theorem is a different challenge. The second side, which is more physical, consists in studying one particular case of application of our formulas, namely the Boltzmann model looked at in the Ḣ-s space, for which we obtain precise numerical bounds (cf. § 6.3). Though the results got for that particular choice can be proved to be definitely limited in some way (cf. § 4.1), that may not be true any longer for a smarter choice of Hilbert space-which would however be more complicated to handle. So this article highlights a way of studying collision models, but remains at a simple level in the applications, hence the title "some ideas".

Here is some notation which will be used throughout this paper:

• The space R d is equipped with its Euclidean structure, whose norm is denoted by | • |.

• f : E -→ F being a measurable function and µ a measure on E, the image measure of µ by f on F will be denoted f #µ.

• δ x denotes a Dirac mass at x.

• S(R d ) is the Schwartz space on R d , i.e. the set of (complex-valued) C ∞ functions on R d which tend to 0 at infinity faster than any |x| -k , as well as all their derivatives.

• The Fourier transform of a function f ∈ S(R d ) is denoted by f , with the unitary convention f (ξ) = (2π) -d/2 R d f (x)e -iξ•x dx.

• The notation || • || will be used to denote Hilbert norms in functional spaces. If Q is a linear operator between two Hilbert spaces, its operator norm sup ||x|| 1 ||Qx|| will be denoted |||Q|||.

• x, y and z being three points of an affine Hilbert space with y, z = x, yxz denotes the angle between -→ xy and -→ xz, which is an element of [0, π].

• The identity matrix of size d is denoted I d .

1 The model

The microscopic model

Let us describe the particle model for the spatially homogeneous Boltzmann evolution. Such models have been first proposed by Kac [START_REF] Kac | Foundations of kinetic theory[END_REF] and later thoroughly studied by Sznitman [START_REF] Sznitman | Topics in propagation of chaos[END_REF], Spohn [START_REF] Spohn | Large scale dynamics of interacting particles[END_REF] and others. There are N identical particles indexed by 0, . . . , N -1, each particle i being characterized by its velocity v i ∈ R d . One imposes random collision times, so that the microscopic evolution is a Markov process. The way two particles with respective velocities v and w hit each other is described by some positive measure γ v,w on (R d ) 2 , N -1 dγ v,w (v , w ) being the collision rate from state (v, w) to state (v , w ).

In other words, the generator L of the Markov process is

Lf (v 0 , . . . , v N -1 ) = 1 2N 0 i,j<N (R d ) 2
-f (v 0 , . . . , v N -1 ) + f (. . . , v i , . . . , v j , . . .) dγ v i ,v j (v i , v j ). (1.1) We may add to this model some extra physical conditions. First, we will always suppose that the momentum and energy are conserved by collisions, and that the model is invariant by velocity translation or rotation, i.e. that for all v, w ∈ R d , for any (positive) isometry J of R d ;

γ v,w -a.e. v + w = v + w, (1.2) γ v,w -a.e. |w -v | = |w -v|, (1.3) γ Jv,Jw = (J, J)#γ v,w .
(1.4)

When conditions (1.2) to (1.4) are satisfied, the model is completely described by the family of measures γ u u∈(0,∞) on (0, π], where dγ u (θ) is the proportion, by unit of time, of particles with relative speed u which undergo a collision making them deviate by an angle θ in the collision referential.

Moreover, it is often assumed that the γ u have a scale invariance property, in the sense that there exists a real parameter g such that for any λ ∈ (0, +∞),

γ λu = λ g γ u .
(1.5)

For instance, the hard sphere model is scale-invariant with g = 1. Another very interesting particular case is when g = 0-then one says that the model is Maxwellian. In this article the concrete results obtained will actually concern Maxwellian models.

Before turning to the macroscopic model, let us make some remarks on the microscopic one: 1.1 Remark.

1. The N -1 factor in Equation (1.1) is essential to get the mean field limit: it morally says that the global collision rate of one particle is independent of the total number of particles.

2. Strictly speaking, generator (1.1) allows a particle to collide with itself, which is physically absurd. Yet because of the conservation law (1.3), the auto-collision term is actually zero, so there is no problem.

3. The γ v,w have to satisfy some integrability conditions for the Markov process to be well-defined. For instance, if conditions (1.2) to (1.5) are statisfied, then it suffices that for an arbitrarily chosen u ∈ (0, ∞),

π 0 θ d-1 dγ u (θ) is finite, cf. [21].

The macroscopic model

The macroscopic space-homogeneous Boltzmann equation [START_REF] Carleman | Sur la théorie de l'équation intégrodifférentielle de Boltzmann[END_REF] is obtained informally by letting N tend to infinity in the microscopic evolution. Then the particles' velocities are described by they empirical measure, which is a (nonatomic in general) probability measure µ t on R d . The evolution of that measure is deterministic and is governed by the equation:

D t µ = Q(µ t , µ t ), (1.6)
where Q is the Boltzmann collision kernel of the system, formally defined by:

Q(µ, ν) = 1 2 -δ v -δ w + δ v + δ w dγ v,w (v , w ) dµ(v)dν(w). (1.7)
Equation (1.6) is an ordinary differential equation in an infinite-dimensional space; that equation is non-linear because of the quadratic term Q(µ, µ). Unique existence of a solution to it has been thoroughly studied over the last decades [START_REF] Desvillettes | Boltzmann's kernel and the spatially homogeneous Boltzmann equation[END_REF][START_REF] Villani | A review of mathematical topics in collisional kinetic theory[END_REF]. For our theory to work, we will need to consider a setting where that unique existence is achieved in some convenient space-which is quite logical altogether. Later we will see concrete examples where (1.6) behaves well for our purpose.

Conservation laws, convergence to equilibrium

Because of the conservation laws (1.2) and (1.3), we get d + 1 invariant functions for the microscopic system: the first d are synthetised in the momentum P = N -1 i=0 v i , and the last one is the energy

K = 1 2 N -1 i=0 |v i | 2 .
In the macroscopic model, these invariants become p = v dµ(v) and k = 1 2 |v| 2 dµ(v). Moreover the fact that the macroscopic model derives from the description of an evolution of particles implies two extra properties for it: first positivity of Equation (1.6), which means that if µ 0 is a positive measure, then so are the µ t for t positive; secondly conservation of mass which gives the (d + 2)-nd invariant m = dµ(v) for the macroscopic equation.

Concerning equilibrium, if we impose some minimal non-degeneracy condition (see [START_REF] Villani | A review of mathematical topics in collisional kinetic theory[END_REF]), then it is a well-known beautiful result due to Boltzmann [START_REF] Boltzmann | Weitere Studien über das W ärmegleichgenicht unter Gasmol äkuler[END_REF] that Equation (1.6) is dissipative for positive measures and converges to an equilibrium measure µ eq depending only on p, k and m: for m = 1 and p = 0, it is

dµ eq (v) = d 4πk d/2 e -d|v| 2 /4k dv, (1.8)
and it has the invariance properties µ eq (p, k, 1) = τ p #µ eq (0, k -p 2 /2, 1), τ p being the translation by vector p, and µ eq (λp, λk, λm) = λµ eq (p, k, m). More recently a beautiful quantitative version of that convergence result has been proved by Carlen, Gabetta and Toscani [START_REF] Carlen | Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas[END_REF].

For the microscopic model, there is also a unique ergodic equilibrium measure for each value of P and K (N being fixed), which is merely the uniform measure on the (dN -d-1)-dimensional sphere ( * ) of (R d ) N made of the N -uples of vectors having these P and K. Note that for N particles with momentum N p and energy N k, the marginals of that measure tend to the continuous equilibrium measure µ eq (p, k, 1) when N -→ ∞.

Finally it is worth recalling that the microscopic process is reversible under its equilibrium measure, while on the contrary the macroscopic equation (1.6) exhibits a dissipative behaviour-a phenomenon which caused much trouble at Boltzmann's time, but has been well understood today.

Homogeneous Sobolev spaces 2.1 Why homogeneous Sobolev spaces?

To be able to speak of quantitative convergence, we will work in some Banach space. Which one will we take ? As we want to compare the empirical measure of our particle system to its limit evolution, a natural choice is to take some coupling distance between measures-say, the W 1 Wasserstein distance [24, § 7], defined for µ, ν two positive measures with the same mass by:

W 1 (µ, ν) = sup f 1-Lip. f d(ν -µ) , (2.1)
( * ) Possibly of radius 0.

where "f 1-Lip." means that the supremum is taken over all 1-Lipschitz functions on R d . However it turns out that it is hopeless to get an N -1/2 rate of convergence in such a space, because testing ν -µ against so much test functions makes the uniform central limit theory fail: see [8, § 6.4] for more details. We also give a more intuitive, completely different explanation of that fact in § A.

Thus the idea is to test ν -µ against a smaller space made of more regular functions. Sobolev spaces W s,p , s > 0, are such natural test spaces; then ν -µ will be seen as an element of the dual space W -s,p/(p-1) . For our theory we will have to work in a Hilbert space, so we take p = 2 and work in W -s,2 = H -s ; then we can take s fractional, which will turn out to be useful indeed. Yet since defining a norm for H -s spaces requires to choose some aribtrary length, which is physically annoying, we will rather consider homogeneous Ḣ-s spaces, which do have a canonical norm (plus other advantages). Note however, cf. Remark 0.1, that this choice is only one possibility-certainly particularly reasonable-among other ones, and that trickier choices might also be relevant.

Definition and useful properties

Let us define properly the Ḣ-s spaces. (2.2)

Then those of the f ∈ S(R d ) for which ||f || Ḣ-s < ∞, equipped with the norm || • || Ḣ-s , constitute a pre-Hilbert space with scalar product

f, g Ḣ-s = R d f (ξ) g(ξ) |ξ| -2s dξ. (2.3)
The Hilbert space obtained by completing it is denoted Ḣ-s .

2.2 Remark. For a physicist, f : R d -→ C has some homogeneity: say, the elements in R d are measured in x (generally x is a unit of length, say meters) and the elements in C are measured in y (which will often be a density, say

kg • m -d ). Then ||f || Ḣ-s is measured in y • x s+d/2 (in our example, ||f || Ḣ-s would be measured in kg • m s-d/2 ). Equivalently, if µ is a measure on R d , the physical dimension of ||µ|| Ḣ-s is z • x s-d/2
, x being the physical dimension of the elements of R d and z the physical dimension of µ (which in our example would be kg).

As we told in § 2.1, bounding a function or a measure in Ḣ-s means bounding uniformly its integral against some class of regular functions: 2.3 Proposition. Define Ḣs in the same way as Ḣ-s . Then, for any f for which it makes sense:

||f || Ḣ-s = sup

||g|| Ḣs 1 R d f (x)g(x)dx . (2.4)
2.4 Lemma. For s ∈]0, d[, let φ s be the locally integrable function

φ s (x) = |x| -(d-s) , (2.5)
then one has for all f, g ∈ S(R d ):

f, g = c(s, d) 2 f * φ s , g * φ s L 2 (R d ) , (2.6
)

with c(s, d) = Γ (d -s)/2 (2π) d/2 Γ(s/2) , (2.7)
Γ(•) being Euler's Gamma function.

Proof. Use that the Fourier transform of |ξ| -s is (2π

) d/2 c(s, d)φ s (x), cf. [19, ex- ercise V-10].
2.5 Immediate proposition. Let J λ be a similarity of R d with dilation factor λ, then for any map f ∈ Ḣ-s ,

||f • J λ || Ḣ-s = λ s+d/2 ||f || Ḣ-s . (2.8)
Equivalenty, for any measure µ ∈ Ḣ-s ,

||J λ #µ|| Ḣ-s = λ s-d/2 ||µ|| Ḣ-s .
(2.9)

From now on, we will always write implicitly s = d/2 + r.

2.6 Proposition. Suppose d 2 ( †) and let µ be a compactly supported signed measure on R d with total mass 0, then for any r ∈ (0, 1), µ can be seen as an element of Ḣ-s .

Proof. Thanks to Lemma 2.4 we just need to prove that µ * φ s is a squareintegrable function. Suppose that µ is supported by the ball B(R) of radius R centered at 0 and splits into µ + -µ -with µ + and µ -positive measures each of total mass M . Then for ρ > 0, on

B(ρ) µ * φ s is equal to µ * (1 B(R+ρ) φ s ), so the L 2 norm of 1 B(ρ) (µ * φ s ) is bounded above by 2M • ||1 B(R+ρ) φ s || L 2 < ∞. Thus µ * φ s is locally L 2 . On the other hand, for |x| = ρ > R, |(µ * φ s )(x)| M 1 (ρ -R) d/2-r - 1 (ρ + R) d/2-r 2M R d/2 -r (r -R) d/2+1-
r , (2.10) so µ * φ s is L 2 at infinity, which finishes the proof. ( †) The proposition remains valid with d = 1, except that it must be demanded that r < 1/2.

Corollary. Still suppose d

2, then for r ∈ (0, 1), any signed measure with zero total mass, if it has an r-th polynomial momentum, can be seen as en element of Ḣ-s .

Proof. Let µ = µ + -µ -be such a measure with its Hahn decomposition, µ + and µ -each having total mass M . Then the integral Minkowski inequality gives

||µ|| Ḣ-s 1 M (R d ) 2 ||δ x -δ y || Ḣ-s dµ + (x)dµ -(y) = C r M • (R d ) 2 |x -y| r dµ + (x)dµ -(y) < ∞, (2.11)
C r being the Ḣ-s norm of any δ x -δ y for |x -y| = 1, which is finite by the previous proposition.

2.8 Remark. The Ḣ-s norm allows us to measure the distance between two (sufficiently integrable) probability measures, but speaking of the Ḣ-s norm of a single probability measure would be nonsense! Note also that, by Sobolev imbedding, one can bound above ||ν -µ|| Ḣ-s , for any two probability measures µ and ν, by (up to some explicit multiplicative constant)

W 1,r (µ, ν) = sup f dµ -f dν ; ∀x, y |f (x) -f (y)| |y -x| r .
(2.12)

3 Dynamic control

Abstract setting

Now let us study the evolution of our particle system along time. We first give our main result in an abstract setting to alleviate its proof; the reader more comfortable with physical settings may read Theorem 3.6 instead.

Let H be a Hilbert space, let A be an H-affine space and let ( X t ) t 0 be some jump Markov process on A with generator L. Fix o an arbitrary point of A and define

I : A -→ H x → -→ ox : (3.1)
since I is defined up to an additive constant, the operator (LI) :

A -→ H ( ‡)
does not depend on the choice of o and we can therefore define (X t ) t 0 as the ( ‡) Stricto sensu L acts on some space of real functions on A, say the space of continuous bounded functions C b (A, R). Yet we can straightforwardly extend L to the space C b (A, E) for any Banach space E by defining the operator

L (E) : C b (A, E) -→ C b (A, E) through: ∀φ ∈ E ∀f ∈ C b (A, E) φ, L (E) f = L φ, f . (3.2)
That is what we do here: I is a function from A to H, so LI actually denotes L (H) I.

deterministic process on A following the differential equation

D t X = LI (X t ). (3.3)
Our goal is to control the distance between X t and X t . Here what is important for us is to have a good control of large deviations for that distance. As Cramér's method cannot be applied directly because of the infinitedimensional setting, we introduce an exponential utility function U : H -→ R defined by:

U(x) = e ||x|| + e -||x|| . (3.4)
The following proposition gathers the properties of U we will use in our work:

3.1 Immediate proposition. 1. For all x ∈ H, U(x) e ||x|| ;
2. U(0) = 2;

3. For all x, h ∈ H, U(x + h) e ||h|| U(x);

4. U is of class C ∞( * ) ;
5. For all x ∈ H, ∇U(x) is positively colinear to x;

6. For all x ∈ H, |||∇ 2 U||| U(x).
Then one can state the theorem which will be our central tool. We first need some notation to alleviate our formulas:

Definition.

We denote e 1 (t) = (e t -1)/t, extended by e 1 (0) = 1, resp. e 2 (t) = (e t -1 -t)/t 2 , extended by e 2 (0) = 1/2. We also denote κ -the negative part of κ, i.e. κ -= max{-κ, 0}.

3.3 Theorem. Suppose that Equation (3.3) has a κ-contracting semigroup for some κ ∈ R, in the sense that for all x ∈ A, h ∈ H:

D x (LI) • h, h -κ||h|| 2 . (3.5)
Suppose moreover that the Markov process-which we recall to be a jump process-has the amplitude of all its jumps bounded above by some L < ∞, and satisfies:

∀x ∈ A L(|| • -x|| 2 )(x) V (3.6) for some V < ∞.
Then, denoting X 0 the (random) initial value of the Markov process and X 0 the (deterministic) initial value of the differential equation (3.3), one has for any T 0, for any λ > 0:

ln E U λ( X T -X T ) ln E U λe -κT ( X 0 -X 0 ) + λ 2 e 2 (λe 2κ -T L)e 1 (-2κT )V T. (3.7)
Proof. The principle of the proof is to show that some time-depending functional

F ( X t ) = e h(t) U λe κ(t-T ) ( X t -X t ) , (3.8)
for a well-chosen function h, is a supermartingale.

To make our computations completely rigorous, throughout the proof we will assume that the expected number of collisions per unit of time is uniformly bounded, that is, that there is some M < ∞ such that |(L1 A )(x)| M for all x ∈ A and any Borel subset A ⊂ A. Then the general result can be recovered by a standard truncation argument.

Let us fix some t ∈ [0, T ] and suppose ( X t ) t ∈[0,t] is known. Let δt be a small amount of time devised to tend to 0; O(δt n ) will denote any quantity bounded by some Cδt n when δt tends to 0, where C depends only on κ, V , M , λ, T , t and ||X t ||. With this notation, the law of X t+δt depends on ( X t ) t ∈[0,t] only through X t , and our goal is

to show that E[F ( X t+δt )] -F ( X t ), which is O(δt), is nonpositive-more precisely, we only need to prove that E[F ( X t+δt )] -F ( X t ) O(δt 2 ) ( †) . Set Y = X -X. Denote δ X = X t+δt -X t , resp. δX = X t+δt -X t , δ Y = Y t+δt -Y t , δF = F ( X t+δt ) -F ( X t ). The fundamental observation is that E δ X = LI ( X t ) δt + O(δt 2 ).
(3.9) Now, admitting temporarily that h will be of class C 2 , we write:

δF = h (t)F (t)δt (3.10) + e h(t) λe κ(t-T ) ∇U(λe κ(t-T ) Y t ) • LI( X t ) -LI(X t ) + κ Y t δt (3.11) + e h(t) U λe κ(t-T ) Y t+δt -U λe κ(t-T ) { Y t + [LI( X t ) -LI(X t )]δt} (3.12) + O(δt 2 ).
In that sum we first see that the term (3.11) is nonpositive: (3.5) indeed implies, for all x ∈ A, y ∈ H,

(LI)(x + y) -(LI)(x) + κy , y 0, (3.13)
which we apply here with x = X t and y = Y t , using that ∇U(λe

κ(t-T ) Y t ) is positively colinear to Y t (Proposition 3.1-5).
Now let us look at term (3.12). Because of (3.9), the expectation of the random variable

λe κ(t-T ) Y t+δt -Y t + [LI( X t ) -LI(X t )]δt (3.14)
is O(δt 2 ). We will use it thanks to the following ( †) Beware that "expr. O(δt n )" does not mean "expr. = O(δt n )" but actually "(expr.) + = O(δt n )".

3.4 Lemma. Let X ∈ H; let y be an H-valued random variable with zero mean. Then one has:

E[U(X + y)] U(X) 1 + E e 2 (||y||) ||y|| 2 . (3.15)
Proof of the lemma. Taylor's formula yields

U(X + y) = U(X) + ∇U(X) • y + 1 0 (1 -θ)∇ 2 U(X + θy)dθ • (y ⊗ y). (3.16)
In that sum the third term is bounded above by

||y|| 2 1 0 (1 -θ) U(X + θy)dθ (3.17)
by Proposition 3.1-6, which in turn is bounded by What does it give for us? Let E be the event "some collision occurs between t and t+δt". E is an event of probability O(δt), on E, the random variable (3.14) is O(1), and on c E it is O(δt). Hence, denoting temporarily * for that variable,

||y|| 2 U(X)
E || * || 2 e 2 (|| * ||) , up to some O(δt 2 ), is merely λ 2 e 2κ(t-T ) E ||δ Y || 2 e 2 (||λe κ(t-T ) δ Y ||) ,
which is bounded above by λ 2 e 2κ(t-T ) e 2 (λe 2κ -T L)V uniformly in t.

Putting all things together, we get

E[δF ] h (t) + λ 2 e 2κ(t-T ) e 2 (λe 2κ -T L)V F (t)δt + O(δt 2 ), (3.19)
which will be O(δt 2 ) provided

h (t) -λ 2 e 2κ(t-T ) e 2 (λe 2κ -T L)V. (3.20)
To achieve that optimally with h(T ) = 0, we choose

h(t) = λ 2 e 2 λe 2κ -T L e 1 2κ(t -T ) V (T -t), (3.21)
which is of class C 2 indeed. Formula (3.7) then follows by the supermartingale property.

3.5 Remark. Strictly speaking our proof only shows that F ( X t ) is a local supermartingale. But this local supermartingale is nonnegative, so it is actually a global supermartingale (see [18, § IV-1.5]). Let N ∈ N . The stochastic process ( X t ) t 0 on A will be the empirical measure µ N t of our microscopic process v 0 (t), . . . , v N -1 (t) on (R d ) N . It is a Markov process indeed; let us denote L N its generator. Regardless of N , one has as expected:

Application to Boltzmann's model

∀µ ∈ A L N I (µ) = Q(µ, µ), (3.22)
so the deterministic process (X t ) t 0 on A will be our macroscopic process following the Boltzmann equation (1.6). Finally, for ) . So Theorem 3.3 becomes:

µ ∈ A, ν ∈ H, D µ (LI) • ν = 2Q(µ, ν) ( ‡
3.6 Theorem. Let H be a Hilbert space of measures and A the corresponding affine space containing probability measures (or a subset of it, cf. Remark 3.7).

Consider our microscopic and macroscopic models for some N ∈ N , with certain initial conditions ( * ) (v 0 , . . . , v N -1 ), resp. µ 0 .

Suppose that there exists some constants κ ∈ R, L < ∞, V < ∞ such that:

1. For all µ ∈ A, ν ∈ H, Q(µ, ν), ν - κ 2 ||ν|| 2 ; (3.23)
2. For all µ ∈ A,

E ||µ t+ -µ t || 2 µ t = µ V ; (3.24)
3. The effect of collisions for the microscopic model in A is always bounded by L, i.e. one has almost surely

∀t 0 µ N t+ -µ N t L. (3.25)
Then for any T 0, for any λ > 0,

ln E U λ( µ T -µ T ) ln E U λe -κT ( µ 0 -µ 0 ) + λ 2 e 2 (λe 2κ -T L)e 1 (-2κT )V T. (3.26)
3.7 Remark. Theorem 3.6 remains valid, with the same proof, if we replace A by any subset A ⊂ A such that almost surely ∀t 0 X N t , X t ∈ A. An important ( ‡) Stricto sensu Q(µ, •) is an affine operator from A to H, not a linear operator on H: in fact here Q(µ, ν) denotes Q(µ, ν), Q(µ, •) being the linear part of Q(µ, •). Identifying notations is relevant because Q, like Q, is formally defined by (1.7).

( * ) The initial condition for the stochastic process can be random.

example of it is that, when A contains nonpositive measures, one can always take for A the subset of the true probability measures of A (which subset is stable because of positivity and conservation of mass for the evolutions, cf. § 1.3), for which the properties of positive measures can be used.

Constants for the Sobolev setting

From now on, when dealing with Boltzmann models we work in the space Ḣ-s (R d ) for some r ∈ (0, 1). We denote by C r the Ḣ-s norm of any δ x -δ y for |x -y| = 1, which is some finite explicit function of d and r.

To apply Theorem 3.6, we have to compute the values of L, V , κ and

E U λe -κT ( X 0 -X 0 )
. Here let us just look at the first two quantities-the last two ones will be the objects of separate sections.

Recall that K denotes the energy of the N -particle system, which is conserved along the stochastic evolution-note by the way that up to translating the origin of R d , we can replace K by the internal energy

K = K - |P | 2 2N . (3.27)
Then at any time no particle has speed greater than √ 2K, so the effect of a collision between two particles on the empirical measure can be no more than 2 • (8K) r/2 C r N -1 , which yields an admissible value for L.

3.8 Remark. To get the bound L 2 • (8K) r/2 C r N -1 we have used that the relative speed between two particles is at most 2 √ 2K, and that the effect of a collision with relative speed u is at most 2u r N -1 . Actually one can do slightly better: the relative speed between two particles is at most 2 √ K and the effect of a collision with relative speed u is at most 2 √ 2 1-r -1 C r u r /N (corresponding to the deviation angle θ = π/2), so we could have taken

L = 2 1+r √ 2 1-r -1 C r K r/2 N -1 . (3.28)
It is that bound that we will use in the sequel.

Anyway remember that, since K is going to be of order of magnitude O(N ), one has L = O(N r/2-1 ) when N -→ ∞. Now let us compute V : V is defined by (3.24), which is bounded above by

2C r 2 N -1 (R d ) 2 |w -v| 2r dµ(v)dµ(w) Jensen 2C r 2 N -1 (R d ) 2 |w -v| 2 dµ(v)dµ(w) r = 2 1+2r C r 2 N -1 K N r 2 1+2r C r 2 K r N -(1+r) ; (3.29)
taking into account Remark 3.8, we could even take

V = (2 1-r -1)2 1+2r C r 2 K r N -(1+r) . (3.30)
Anyway remember that V = O(N -1 ) when N -→ ∞.

Comments on the results

All the computations in this subsection are heuristic, so we will drop lower order terms without wondering when we can do so. C 1 , C 2 , . . . will denote constants depending only on κ, V , L and T , whose exact expression does not interest us.

In the right-hand side of Formula (3.26) there are two terms: the first one, ln E[U(λe -κT ( X 0 -X 0 ))], merely expresses the difference between the experimental initial condition and its continuous limit. There is obviously no surprise in getting such a term, whose study is deferred to § 5: for the time being just notice the presence of the factor e -κT in front of X 0 -X 0 , which means that the effect of initial fluctuations will be quite large if κ < 0, and conversely quite small if κ > 0.

The actual dynamic effect in (3.26) lies in the term λ 2 e 2 (λe 2κ -T L)e 1 (-2κT ) V T . Let us study it in the case of our Boltzmann model, according to § 3.2. We have noticed that, when N becomes large, one has L = O(N r/2-1 ), resp. V = O(N -1 ). So let us write L N r/2-1 , resp. V ωN -1 . Then the dynamic term of (3.26) becomes λ 2 e 2 (λe 2κ -T L)e 1 (-2κT )V T λ 2 N -1 e 2 (λe 2κ -T N r/2-1 )e 1 (-2κT )ωT.

(3.31)

The λ 2 N -1 factor hints that the good order of magnitude for λ will be λ = O(N 1/2 ). So write λ = yN 1/2 ; then (3.31) becomes λ 2 e 2 (λe 2κ -T L)e 1 (-2κT )V T e 2 (ye 2κ -T N (r-1)/2 )e 1 (-2κT )ωy 2 T.

(3.32)

In our case (r -1)/2 < 0 so, if N is sufficiently large, ye 2κ -T N (r-1)/2 is very close to zero and the e 2 ( * ) term is very close to e 2 (0) = 1/2, finally giving

λ 2 e 2 (λe 2κ -T L)e 1 (-2κT )V T 1 2 e 1 (-2κT )ωy 2 T. (3.33)
For a fixed T , (3.33) shows that the dynamic term in Formula (3.26) is approximately C 1 y 2 . Moreover, as we will see in § 5, the static term

ln E[U(λe -κT ( X 0 -X 0 ))] is approximately C 2 y 2 + C 3 .
In the end, one gets

ln E U yN 1/2 ( X t -X t ) C 4 y 2 + C 3 , (3.34)
hence by Markov's inequality and Proposition 3.1-1, for all x > 0,

P yN 1/2 || X T -X T || x e C 4 y 2 +C 3 -x . (3.35)
Optimizing Formula (3.35) for fixed x/y ratio, one finally finds:

∀ε 0 P || X T -X T || ε exp C 3 -C 5 N ε 2 .
(3.36) So Theorem 3.6 gives a Gaussian control for the fluctuations between X T and X T for any fixed value of T -provided the existence of some contractivity constant κ, which for H = Ḣ-s will be proved for the Maxwellian case in § 4. Moreover the order of magnitude of the fluctuations we get is N -1/2 , the typical deviation size in central limit theorems. So we may say that the bounds we have got are a kind of explicit dynamic central limit bound for the Boltzmann model.

3.9 Remark. Actually the approximations we made to get (3.34) are sensible only if y is not too large, otherwise λe 2κ -T N r/2-1 1 and then the e 2 ( * ) term in (3.31) cannot be considered as close to 1/2. It follows that our computations are valid only for λ N 1-r/2 / , i.e. for y N (1-r)/2 / . Tracking that constraint throughout our reasoning, it finally turns out that (3.36) is only valid for ε T ωN -r/2 / . So our Gaussian control does not hold up to large deviations but only till some intermediate deviations ( †) . Fortunately (3.36) tells us that the probability of such intermediate deviations is bounded above by something like e -C 6 N 1-r , which goes very fast to 0 anyway. Moreover, even for ε ωT N -r/2 / one can still use (3.35) with y = N (1-r)/2 / and k yN 1/2 ε, which gives an exponential control of the tail of the law of || X T -X T || applicable to large deviations.

The behaviour of Formula (3.33) as T becomes large depends on the sign of κ: ( ‡) • If κ < 0 (the worst case), then the e 1 (-2κT ) factor becomes exponentially large as soon as T 1/|κ|. Thus the dynamic control given by Theorem 3.6 is relevant only for moderate values of T corresponding to durations for which each particle makes only a couple of collisions. Moreover, as we noticed in the beginning of that subsection, in that case the term due to the control of initial fluctuations will become huge as T increases. Note however that qualitatively we get a Gaussian control for any fixed T , only the constants in that control becoming bad.

• If κ = 0 the dynamic term of (3.26) increases proportionally to T , so our bound remains good even for moderately large values of T , but ultimately becomes uninteresting.

( †) Note however that our control (3.28) on L was very coarse: in real situations indeed the maximal relative speed between two particles is ∼ √ ln N with very large probability (think about the Maxwell distribution), so in most cases L ∼ (ln N ) r/2 N -1 , and then we can study deviation orders N -η with η arbitrarily close to 0. However such a study needs a control on the probability that L becomes large, in other words a control on the probability of appearance of an abnormally hot particle, which would require another article. ( ‡) In § 4.1 we will see that for H = Ḣ-s , κ is actually always negative. Our the discussion is relevant nevertheless, because it remains valid for other applications of abstract Theorem 3.3, therefore highlighting the interest of choosing a Hilbert space better than Ḣ-s .

• If κ > 0 (the best case) then T e 1 (-2κT ) -→ 1/2κ when T -→ ∞, so the right-hand side of (3.26) remains bounded uniformly in T , implying that the N -particle model approximates well its continuous limit for any time ( * ) . Note that κ > 0 is tantamount to having an exponential convergence of (1.6) to equilibrium in A, so in that case our bound rather looks like a result of convergence to "equilibrium" for the empirical measure µ N t .

Contractivity of the collision kernel 4.1 Limitations due to our settings

In this section we are going to look for computing constant κ in (3.5). Unfortunately it turns out that, for the choices we have made, our results are unavoidably limited, as we quickly explain in this foreword. Let me stress however that all the issues encoutered may be solved by working in a trickier space than the plain Ḣ-s (cf. Remark 0.1).

First κ can only be negative, which is the worst case (see page [START_REF] Maxwell | On the Dynamical Theory of Gases[END_REF]. Why that? Well, if κ were positive, as we said previously it would imply convergence of Equation (1.6) to a unique equilibrium for all probability measures. Yet there are several different equilibrium probability measures for the Boltzmann evolution (see Formula (1.8) and below), whose differences lie in Ḣ-s , which is a contradiction. So κ is nonpositive. Then we could prove, using that the model is nondegenerate, that κ cannot be zero and thus is negative. To have a chance to get positive values of κ, Ḣ-s should be replaced by a Hilbert space containing only signed measures η such that η(dx), xη(dx), |x| 2 η(dx) = 0-but which one?

Secondly, the only chance for κ to be finite is the case of Maxwellian models (remember definition below (1.5)): this is due to a bad scale invariance property for non-Maxwellian models, cf. Remark 4.4. Though the Maxwellian case is often a useful first step for theoretists, the physical models encountered in real life do not have any reason for being so! To have a chance to get results for non-Maxwellian models, Ḣ-s should be replaced by some non-homogeneous space-but non-homogeneous spaces are often less tractable than homogeneous spaces and more difficult to interpret physically.

Principle to the computation of κ

To check Hypothesis (3.23), according to Remark 3.7, we can consider our Markov processes restricted to the set of probability measures, and then by positive linearity it suffices to prove (3.23) when µ is a Dirac mass: 4.1 Immediate proposition. If, for one arbitrary (then for all) v ∈ R d , the linear operator Q(δ v , •) : Ḣ-s -→ Ḣ-s satisfies the "contractivity" property

∀f ∈ Ḣ-s Q(δ v , f ), f - κ 2 ||f || 2 , ( 4
: Ḣ-s -→ L 2 f → f * φ s . (4.2)
Then Q(δ v , •) : Ḣ-s -→ Ḣ-s satisfies property (4.1) if and only if

( * φ s ) • Q(δ v , •) • ( * φ s ) -1 : L 2 -→ L 2 (4.3)
satisfies the same property in the space L 2 (R d ).

Proof. It follows directly from the isomorphism formula (2.6).

4.4 Remark. Now we can understand why κ cannot be finite for a non-Maxwellian model: suppose the model satisfies (1.5) with g = 0, and for λ ∈ (0, +∞) denote by I λ the homothety transforming v into λv, then you get

Q δ 0 , I λ #µ = λ g I λ #Q δ 0 , µ , (4.4)
so if Q(δ 0 , •) were κ-contracting for a κ < 0 it would also be λ g κ-contracting for all λ, thus 0-contracting, which is impossible. 4.6 Remark. Qθ (δ v ) respresents the post-collisional distribution of velocity of a particle at initial velocity v which has collided with a particle at initial velocity 0, undergoing an angular deviation θ in the collision referential, the precise direction of that deviation being random.

Effective computation

Proof. Let us give first a neat proof working when d is even. Call R θ the set of

the rotations R of R d satisfying v 0(Rv) = θ/2 for all v ∈ R d . If d is even, R θ is
non-empty and has some canonical probability measure π θ equipping it. Then we notice that

Qθ (µ) = R θ [cos(θ/2)R]#µ dπ θ (R). (4.6)
Because of the rotational invariance of φ s , for any R ∈ R θ ,

( * φ s ) • R# = R# • ( * φ s ). (4.7)
Similarly, the scale invariance of φ s makes that for any λ ∈ (0, ∞),

( * φ s ) • I λ # = λ s • I λ # • ( * φ s ). (4.8)
The result then follows by applying Formulas (4.7) and (4.8) to the integral (4.6).

When d is odd unfortunately I have nothing better than a calculationwhich by the way also works for d even. Choose an arbitrary v > 0, we will prove that ( Qθ δ v ) * φ s = Qθ (δ v * φ s ), where v also denotes the point (v, 0, . . . , 0) ∈ R d . Since these two functions are invariant by any rotation around v, we will locate a point in R d merely by its first coordinate z and its distance ρ to the z axis; we will also denote Z = z 2 + ρ 2 its distance to 0. In the following calculations S denotes the unit sphere in R d-1 , equipped with its Lebesgue probability measure σ, and ρ also denotes the point (ρ, 0, . . . , 0) ∈ R d-1 ; points of S are denoted y = (y 0 , y 1 ) with y 0 ∈ R, y 1 ∈ R d-2 . Treating ( Qθ δ v ) * φ s as a function, we find:

( Qθ δ v ) * φ s (z, ρ) = S cos(θ/2)v -z 2 + sin θ y 0 v/2 -ρ 2 + (sin θ) 2 |y 1 | 2 v 2 /4 -(d-s)/2 dσ(y 0 , y 1 ) = S Z 2 + cos(θ/2) 2 v 2 -2 cos(θ/2) 2 vz -sin θ vρy 0 -(d-s)/2
dσ(y 0 , y 1 ). (4.9)

For Qθ (δ v * φ s ) it is more complicated since that case needs computing a expression of type Qθ f , f being a function. Usually that kind of computation raises no difficulty, but here the operator Qθ has some singularity which makes it less tractable: in Qθ f , the "mass" (in the measure sense) received by the point (z, 0, . . . , 0) comes only from a (d-2)-dimensional sphere in R d -more precisely the sphere of points (z, ρ), ρ ∈ R d-1 , with |ρ| = tan(θ/2)z. That regularity problem can be overcome by an approximation technique, yielding:

Qθ f (z, 0, . . . , 0) = 1 cos(θ/2) d S f z, [tan(θ/2)z]y dσ(y) (4.10)
-that formula also allowing to compute Qθ f at points not located on the z axis by rotational invariance.

So

Qθ (δ v * φ s ) (z, ρ) = cos(θ/2) -d • S z -tan(θ/2)ρy 0 -v 2 + ρ + tan(θ/2)zy 0 2 + tan(θ/2) 2 Z 2 |y 1 | 2 -(d-s)/2 dσ(y 0 , y 1 ) = cos(θ/2) -d S 1 + tan(θ/2) 2 Z 2 -2v z -tan(θ/2)ρy 0 + v 2 -(d-s)/2 dσ(y 0 , y 1 ) = cos(θ/2) -s ( Qθ δ v ) * φ s (z, ρ). (4.11) 4.7 Corollary. Let Q θ = Qθ + Qπ-θ -Q0 -Qπ . Then, for any f ∈ Ḣ-s , Q θ f, f cos(θ/2) r + sin(θ/2) r -1 • ||f || 2 . (4.12)
Proof. Observe first that Q0 is the identity and that Qπ = 0, so it suffices to prove that the operator norm of Qθ in Ḣ-s is bounded above by cos(θ/2) r . By isomorphism Formula (2.6), that is also the norm of ( * φ s )

• Qθ • ( * φ s ) -1 in L 2 , which is cos(θ/2) s
Qθ by Lemma 4.5. So we just have to bound the norm of Qθ , regarded as an operator in L 2 , by cos(θ/2) -d/2 . Now we note that one can write

Qθ f = I cos(θ/2) # Q θ f, ( 4
Q θ ||| L 2 1, thus ||| Qθ ||| L 2 cos(θ/2) -d/2
, quod erat demonstrandum. Now we are ready to state the main result of this section: 4.8 Theorem. In a Maxwellian model, calling γ the common value of all the measures γ u , the collision kernel Q, when restricted to the probability measures, satisfies hypothesis (3.5) with

κ = π 0 1 -cos(θ/2) r -sin(θ/2) r dγ(θ). (4.14)
Proof. Note that

Q(δ 0 , •) = 1 2 π 0 Q θ dγ(θ) (4.15)
and apply all the previous work of this section (Lemmas 4.1, 4.3, 4.5 and 4.7).

4.9 Example. The "Kac" model ( ‡) is the case where the measure γ v,w always has total mass 1 and is uniform on the sphere supporting it, i.e. it is the Maxwellian model with

dγ(θ) = Γ(d -1) 2 d-2 Γ (d -1)/2 2 (sin θ) d-2 dθ. (4.16)
By Theorem 4.8, for that model one has -∞ < κ < 0 for any r ∈ (0, 1).

4.10 Example. The model of Maxwellian potential corresponds to particles having a repulsive force with a radially symmetric potential decreasing like ρ -(2d-2) as the distance ρ between two particles increases. For that model dγ ∼ θ -3/2 dθ when θ -→ 0 for any d (thus the measure γ is not finite, however it remains possible to define both the N -particle and the limit models, cf. Remark 1.1-3), so by Theorem 4.8 one also has -∞ < κ < 0 for any r ∈ (0, 1).

Initial value

In Formula (3.26) given by Theorem 3.6, as we saw, besides the dynamic term there is a term due to the fluctuations of the initial empirical measure. In this section we control these fluctuations in the case of i.i.d. initial particles.

Let µ be a probability measure on R d and let r ∈ (0, 1). We assume that µ has an r-th exponential moment, i.e. that there exists some a > 0 such that R d e a|v| r dv < ∞.

(5.1)

In the sequel we suppose a fixed.

If v is a random variable of R d with law µ, then δ v -µ is a random variable in Ḣ-s , whose law will be denoted D µ : D µ is a centered probability measure on Ḣ-s . I claim that D µ has an exponential moment with parameter a, i.e.

Ḣ-s

e a||ν|| dD µ (ν) < ∞ :

To prove it it suffices to note that

||δ v -µ|| ||δ v -δ 0 || + ||δ v 0 -µ|| = C r |v -v 0 | r + ||δ v 0 -µ||, (5.3)
whose a-parameter exponential is integrable because of (5.1).

So the law D µ has a finite exponential moment, hence a fortiori a finite variance. Let us denote it by σ 2 :

σ 2 = Ḣ-s ||ν|| 2 dD µ (ν).
(5.4) ( ‡) Actually this is not exactly the Kac model of [START_REF] Kac | Foundations of kinetic theory[END_REF], but the spirit is the same.

Now we have all the definitions at hand to state the main result of this section:

5.1 Theorem. Let v 0 , . . . , v N -1 be N i.i.d. random variables on R d with law µ, and denote µ N = N -1 N -1 i=0 δ v i their empirical measure. Then there exists an explicit constant A(µ), which is easy to bound, such that for all λ aN :

E U λ( µ N -µ) 2 exp λ 2 σ 2 2N + λ 3 A(µ) N 2 a 3 . (5.5)
Before proving Theorem 5.1, let us further examine Formula (5.5): the term in the exponential remains bounded when N -→ ∞ if λ increases as N 1/2 , like in (3.31). Thus, writing λ = yN 1/2 like in (3.34):

E U yN 1/2 ( µ N -µ) 2 exp σ 2 y 2 2 + A(µ)y 3 a 3 N -1/2 . (5.6)
Though we will not use it in the sequel, note the following 5.2 Corollary. For S σ 2 , for all x 0, for all N N 0 • • = x 2 /a 2 S 2 :

P || µ N -µ|| xN -1/2 exp - x 2 2S + ln 2 + A(µ)N 0 1/2 N -1/2 .
(5.7)

5.3 Remark. (5.7) works as soon as N x 2 /a 2 S 2 , i.e. as soon as x aSN 1/2 , so that estimate is valid up to the large deviations setting.

Proof of Theorem 5.1. The principle of the proof is exactly the same as for Theorem 3.3, except that here time will be discrete.

Let v 0 , . . . , v N -1 be N i.i.d. random variables with law µ and set

M i = i-1 j=0 N -1 (δ v i -µ), then ( M i ) i is a
Markov chain and a martingale, and M N has the same law as µ N . So it suffices to prove that for all 0 i < N ,

E U(λ M i+1 ) M i exp λ 2 σ 2 2N + λ 3 A(µ) a 3 N 2 U(λ M i ). (5.8)
To get (5.8), thanks to Lemma 3.4 it suffices to prove that

e λN -1 ||ν|| -N -1 λN -1 ||ν|| dD µ (ν) exp λ 2 σ 2 2N 2 + λ 3 A(µ) a 3 N 3 . (5.9)
We set A(µ) = e a||ν|| -a||ν|| -1 dD µ (ν).

(5.10)

The function e 2 (t) = (e t -1 -t)/t 2 is convex on R + , so

∀t 0 ∀θ ∈ [0, 1] e θt -θt -1 1 2 (1 -θ)θ 2 t 2 + θ 3 (e t -t -1). (5.11)
Consequently

e N -1 λ||ν|| -N -1 λ||ν|| -1 dD µ (ν) 1 -λaN -1 λ 2 σ 2 2N 2 + λ 3 A(µ) a 3 N 3 , (5.12)
whence (5.9).

Discussion

Examples of synthetic results

Until now in this article I have just given separate results, mainly Theorem 3.6, Formulas (3.28) and (3.30), and Theorems 4.8 and 5.1. Obviously all these results are to be put together to get synthetic results on the convergence of N -particle dynamic models to their mean field limit; yet I did not do it in the previous sections.

There are several reasons why I have postponed the presentation of such synthetic results to the last section. The most obvious one is that these global results would have been quite unreadable if put in the beginning of the article. More important, the different "bricks" of results given within the core of the paper are open to improvements different for each, some of which may work for some cases but not for others, so that there may be no ideal general result.

Let us however give some examples of formulas got by piling our theorems together-proofs will not be given since they really consist in plain gluing game:

= 2 1+r √ 2 1-r -1C r k r/2 1 , (6.2) ω = (2 1-r -1)2 1+2r C r 2 k r 1 , (6.3)
σ 2 = R d ||δ v -µ 0 || 2 Ḣ-s dµ 0 (v). (6.4)
Let N 2; let v 0 0 , . . . , v 0 N -1 be N i.i.d. random variables with law µ 0 and let µ N 0 be their empirical measure; denote

K N = 1 2 N -1 i=0 |v 0 i | 2 .
Let µ N t be the empirical measure at time t of the Markov process with generator (1.1) for the "Kac" model (4.16) and initial condition (v 0 0 , . . . , v 0 N -1 ). Let (µ t ) t 0 be the deterministic evolution (1.6) for the same model with initial value µ 0 .

Then for any a > 0, there is a (easily bounded) constant A(a, µ) such that, for any T > 0, as soon as λ ae -|κ|T N :

ln E 1 b K N N k 1 U λ( µ N t -µ t ) ln 2 + e 2|κ|T λ 2 σ 2 2N + e 2|κ|T λ 3 A(a, µ) N 2 a 3 + λ 2
ωT N e 1 (2|κ|T )e 2 λe 2|κ|T N r/2-1 . (6.5) ( * ) Warning, κ is negative.

Corollary.

For the same model, for any y 0:

lim

N -→∞ ln E 1 b K N N k 1 U yN 1/2 (
• • = (2 1-r -1)2 1+2r C r 2 k r .

Optimality

Theorem 6.1 essentially gives a convergence to the continuous limit at rate N -1/2 with Gaussian control. Qualitatively it is the best result one could hope for, because it is the same way of convergence as for central limit theorems. Quantitatively however, is the parameter in the Gaussian bound optimal?

Here we will look at what happens for Theorem 5.1 (Theorem 3.3 exhibits the same behaviour, but it is harder to see). Through Corollary 5.2, Theorem 5.1 gives some Gaussian bound in an infinite-dimensional frame. Yet its proof, whose main ingredient is the use of the utility function U, would work as well in a finite-dimensional setting. So let us imagine that we replace Ḣ-s by R d and D µ by the centered normal law with variance I d , denoted by N ; then σ 2 becomes R d |x| 2 dN (x) = d. In that case µ N turns into a random variable X N on R d which is centered normal with variance N -1 I d , and we get:

P(|X N | xN -1/2 ) exp - x 2 2d + ln 2 + AN 0 1/2 N -1/2 (6.8)
for some A and N 0 not depending on N , so making N -→ ∞:

N (|X| x) 2e -x 2 /2d , (6.9)
whereas the exact result is

N (|X| x) = 2 1-d/2 Γ(d/2) ∞ x y d-1 e -y 2 /2 dy ≈ x-→∞ e -x 2 /2 , (6.10)
where "≈" means "having equivalent logarithms".

So for d > 1 the parameter in the Gaussian bound is underestimeted by a factor d. Why that? Well, the proof of Theorem 5.1 uses the bound on the curvature of the utility function U given by Proposition 3.1-6. But as soon as x is reasonably large, the Hessian of U at x is much more curved in one direction than in all the other ones, so that Formula (3.15) in Lemma 3.4 becomes strongly suboptimal since the factor ||y|| 2 in it should morally be replaced by the sole component of the variance of y in the direction along which ∇ 2 U(x) is most curved. So the techniques involving U are poor as soon as the dimension in which the random phenomena occur becomes large. For our particle models we work in Ḣ-s , whose dimension is. . . infinite! Does that mean that our results are "infinitely bad"? Actually not, because each increment of the martingale M (see the proof of Theorem 5.1) is determined by the value of one v i , so the law of this increment may be seen as a probability on R d . More precisely, the support of

D µ is isometric to R d equipped with the distance | • -• | r , whose
Hausdorff dimension is d/r, so that the "effective" dimension of Ḣ-s in our theorem is d/r.

As a consequence we had better not choose r too close to 0. On the other hand, the bigger s is, the more regular the test functions in the definition of ||•|| Ḣ-s (see Proposition 2.3) are, so the less small-scale details ||•|| Ḣ-s catches ( †) . So it should be advised to take medium values of r, e.g. r = 1/2.

A numerical computation

One important side of our work is that it gives non-asymptotic results. The idea behind it is that, to understand Boltzmann's evolution, we will not actually look at N -→ ∞, but rather take some fixed large N and say that the behaviour of the N -particle system for that N is very close to the limit evolution with very large probability. In particular, think about the case of numerical simulation: we cannot afford dealing with 10 24 particles on our computers! Here I will compute numerical values for the following case: the collision kernel is the one of "Kac" model for d = 3 and we take µ 0 = 1 2 (δ -1 + δ 1 ). Physically speaking, it means that we crash together two same-sized sets of frozen particles with relative speed 2. Then the collisions between particles of differents sets will tend to scatter the distribution of velocities of the particles, which will morally converge to the law (1.8) with k = 1/2 in a few units of time-this is the behaviour of Boltzmann's equation (1.6) indeed. The question is, which N shall we choose to be almost certain that the evolution of the particle system will be fairly close to (1.6)? Say we take r = 1/2 and we want to have || µ N T -µ T || Ḣ-s greater than ε = 10 -2 with probability less than q = 10 -1 for T = 3. As in our case K N N k almost surely, we take k We choose arbitrarily a = 1; then (5.10) gives A(a, µ) 0.0213. We have to take λ | ln q|/ε, so let us put λ = 500. For N = 8 • 10 5 we find by (6.5): So with a discrete system of 8 • 10 5 particles one will much probably find a quite good approximation of Boltzmann mean field limit by running the particle system 3 units of time. Now simulating 8 • 10 5 particles is easy for today's computers, which shows that our bounds can actually be useful in practice. However there is little doubt that the true speed of convergence is much faster than what our computations suggest.

1 = k = 1/2.
ln E U λ( µ N t -µ t ) < 2.692, ( 6
6.5 Remark. Here we have bypassed the problem of the 1 b K N N k 1 factor in (6.5) by a specific argument. How can we do for it in the general case? Well, merely note that, as soon as one wants to have a result in terms of probability, they will just have to add P( K N > N k 1 ) to the probability they get forgetting the indicator. But the event { K N > N k 1 } is a large deviations event, so as soon as µ 0 has some square-exponential moment its probability will decrease exponentially with N and thus cause no problem actually.

Comparison to older results

The usual method to tackle mean field limit problems relies on the concept of propagation of chaos devised by Kac [START_REF] Kac | Foundations of kinetic theory[END_REF]. Briefly speaking, propagation of chaos consists in studying the law of the N -particle assembly (v 0 , . . . , v N -1 ) through its finite-dimensional marginals, i.e. in studying the laws of the (v 0 , . . . , v k-1 )'s for k finite. One says that there is "chaos" when these finitedimensional laws tend to product laws u ⊗k when N -→ ∞, and the goal is to prove that, if there is chaos at time 0, then this chaos "propagates" for all t.

Propagation of chaos has been proved by Sznitman [START_REF] Sznitman | Équations de type de Boltzmann, spatialement homogènes[END_REF] for spatially homogeneous Boltzmann models, and more recently by Graham and Méléard [START_REF] Graham | Stochastic particle approximations for generalized Boltzmann models and convergence estimates[END_REF] for the more general Povzner equation. Actually, proving propagation of chaos is the same as proving the convergence of empirical distributions of the Nparticle system to some deterministic distribution, but propagation of chaos emphasizes the individual behaviour of each particle, which is described by the nonlinear particle [START_REF] Sznitman | Topics in propagation of chaos[END_REF]. On the other hand, the finite-dimensional setting of that method makes that the quantitative results got thanks to it (for instance in [START_REF] Fournier | Monte-Carlo approximations and fluctuations for 2D Boltzmann equations without cutoff[END_REF][START_REF] Malrieu | Logarithmic Sobolev inequalities for some nonlinear PDE's[END_REF]) do not translate very well when one tries to control the difference between µ N and µ in some metric space.

My paper was motivated by the reading of [START_REF] Bolley | Quantitative concentration inequalities for empirical measures on non-compact spaces[END_REF], in which Bolley, Guillin and Villani tackle some mean field limit problems in a quantitative way by working with W 1 Wasserstein distances for the empirical measures. They get an explicit control on the large deviations of the difference between the empirical measure of the N -particle system and its theoretical limit for positive times. Yet there are two annoying shortcomings in their work:

• First, it seems to be limited to McKean-Vlasov models, that is, systems where the interactions between particles are due to forces rather than collisions. The proofs of [START_REF] Bolley | Quantitative concentration inequalities for empirical measures on non-compact spaces[END_REF] indeed fundamentally relie on a coupling technique (popularized by Sznitman [START_REF] Sznitman | Équations de type de Boltzmann, spatialement homogènes[END_REF]), in which one defines a coupling between the real assembly of particles and a virtual assembly of N independent nonlinear particles. Such a technique has little relevance when one deals with collisions, because these events imply two particles at the same moment each time they occur, so there is no natural way of coupling with independent particles.

• Secondly, the results of [START_REF] Bolley | Quantitative concentration inequalities for empirical measures on non-compact spaces[END_REF] are good for large deviations, but the control they give for medium deviations is far too poor to get, as we would wish, some N -1/2 convergence rate. As we tell in § A, that is actually an intrinsic shortcoming of W 1 distances.

After writing my article I discovered some other papers sharing certain features with mine:

• The microscopic model exposed in this paper is an example of Bird's direct simulation Monte-Carlo method [1], whose convergence for the Boltzmann equation was proved in [START_REF] Wagner | A convergence proof for Bird's direct simulation Monte Carlo method for the Boltzmann equation[END_REF][START_REF] Pulvirenti | Convergence of particle schemes for the Boltzmann equation[END_REF], with explicit estimates on L 1 distance of the marginals.

• The first having looked at the empirical distribution of the particles in Hilbert spaces to bypass the coupling problems were Fernandez and Méléard [START_REF] Fernandez | A Hilbertian approach for fluctuations on the McKean-Vlasov model[END_REF], who analysed the fluctuations of the particle distribution when N -→ ∞ in a spirit close to uniform central limit theory [START_REF] Dudley | Uniform central Limit Theorems[END_REF].

• Aldéric Joulin pointed out to me that my convergence theorems could be interpreted as results of concentration of measure for a Markov process with positive curvature, according to the geometric notions for Markov chains introduced by Ollivier in [START_REF] Ollivier | Ricci curvature of Markov chains on metric spaces[END_REF]. Hypotheses 1, 2 and 3 of Theorem 3.6 indeed correspond resp. to the hypotheses about the discrete Ricci curvature κ, the coarse diffusion constant σ(x) and the granularity σ ∞ in Theorem 33 of [START_REF] Ollivier | Ricci curvature of Markov chains on metric spaces[END_REF].

Uniform in time bounds

The results we have given work for some fixed T , i.e. they control || µ N T -µ T ||. It may be more natural to control sup t∈[0,T ] || µ N t -µ t ||, i.e. to say that the system is always close to the Boltzmann mean field limit between times 0 and T , as [START_REF] Bolley | Quantitative concentration inequalities for empirical measures on non-compact spaces[END_REF] does for McKean-Vlasov models. We do not do it here, but note that, as we have used martingale techniques, getting results valid for all t ∈ [0, T ] could be easily achieved from the previous work by using stopping times. Actually for κ 0 it would turn out that uniform in time results are not much different from fixed time results (which is quite logical because then the control on || µ N t -µ t || is worst for t = T ). For κ > 0 yet, when T is large the maximum of the difference between µ N t and µ t is much less well controlled than its terminal value, as we already noticed in Footnote ( * ) at page 16.

A Why Wasserstein distances cannot yield

N -1/2 convergence

This appendix aims at explaining quickly why, in quite general situations, the W 1 distance cannot yield a N -1/2 rate of convergence for the empirical distribution of an assembly of N particles to its continuous limit. As it is not the main matter of this article, I will remain at a heuristic level.

My explanation relies on the transportation interpretation of W 1 distances. Recall that a coupling between two finite measures of same mass µ and ν on respective spaces X and Y is a measure π on X × Y whose marginals are resp. µ and ν, i.e. s.t. π(A × Y ) = µ(A) for all measurable A ⊂ X, resp. π(X × B) = ν(B). π is also called a transportation plan because it describes a way to transport a mass distributed according to µ into the mass distribution ν. The set of couplings between µ and ν is always non-void; we denote it Π(µ, ν).

When X = Y = R d , for π ∈ Π(µ, ν) we define the transportation cost

I[π] = (R d ) 2
|y -x| dπ(x, y), (A.1) which represents the total effort you have to put in to transform µ into ν following the plan π. Then the optimal transportation cost is merely

W 1 (µ, ν) = inf π∈Π(µ,ν) c(π). (A.2)
It is a deep result due to Kantorovitch [START_REF] Kantorovitch | On the translocation of masses[END_REF] that Definitions (2.1) and (A.2) actually coincide. For more details on all that, see [24, § 1]. Now, consider a N -particle system whose empirical measure µ N is expected to converge to some density measure µ, say the Lebesgue measure on [0, 1] d , and look at the Wasserstein distance W 1 (µ, µ N ). Let π be a coupling between µ N and µ. Write µ N = N -1 N i=1 δ v i , and call A i ⊂ R d the image of N -1 δ v i by the transportation plan π. A i has Lebesgue measure N -1 , so its observable diameter, which by isoperimetry is minimal when A i is a ball, is at least ∼ N -1/d , therefore A i |v -v i |dµ(v) N -1 • N -1/d . Thus the total transportation cost between µ and µ N is

I[π] = i A i |v -v i | dµ(v)
N -1/d , and since that is true for any transportation plan, in the end W 1 (µ, µ N ) N -1/d . But that is always true, however cleverly you might choose the v i 's (in other words, the phenomenon we describe is not due to fluctuations but to discretization), so for d > 2 ( ‡) it is hopeless getting an N -1/2 convergence rate of µ N to µ for the Wasserstein distance.

2. 1 2 .

 12 Definition. Let s ∈ R, and for f ∈ S(R d ), set ||f || Ḣ-s = R d | f (ξ)| 2 |ξ| -2s dξ 1/

1 0( 1

 11 -θ)e θ||y|| dθ = e 2 (||y||) ||y|| 2 . (3.18) by Proposition 3.1-3. Taking expectation gives the result since the second term in sum (3.16) has zero mean by assumption.

4. 5

 5 Lemma. Let θ ∈ [0, π]; define the linear operator Qθ on measures on R d , such that Qθ (δ v ) is the uniform probability measure on the (d -2)-dimensional sphere ( †) of velocities v such that |v -v/2| = |v|/2 and v v 2 v = θ. Then ( * φ s ) • Qθ = cos(θ/2) s Qθ • ( * φ s ). (4.5) ( †) That sphere degenerates into a point if θ ∈ {0, π}.

. 13 ) 2 .

 132 where Q θ is the kernel of the Markov chain on R d which sends x uniformly to the (d -2)-dimensional sphere of points y such that |y| = |x| and x0y = θ/But that Markov chain has the Lebesgue measure on R d as reversible equilibrium measure, so |||

6. 1 2 |v| 2 2 π 0 sin(θ/ 2 )

 122202 Theorem. Let d 2, r ∈ (0, 1). Let µ 0 be a probability measure on R d with finite r-exponential moments for all r < 1. Up to translating the origin of R d we can suppose that p • • = R d vdµ 0 (v) = 0; then let k = 1 dµ 0 (v). Choose some k 1 > k and defineκ = 1 -Γ(d -1)2 d-3 Γ (d -1)/2 r sin(θ) d-2 dθ ( * ) , (6.1)

Translation of Theorem 3.3

  Let us see what Theorem 3.3 gives for the Boltzmann model. For the time being, according to Remark 0.1 we do not precise what H and A are: all you have to know is that H is some Hilbert space of measures and A the corresponding affine space, which is assumed to contain all the probability measures having enough moments.

  It is not hard to see that conversely the best κ possible in (3.23) is exactly the best κ possible in (4.1). We do not prove it as it is not essential, but it will be implicitly used in Remark 4.4.

	.1)
	then the restriction of Q to probability measures satisfies Hypothesis (3.23).
	4.2 Remark.

4.3 Lemma. Recall definition (2.5) of φ s . Note ( * φ s ) the convolution operator * φ s

  As (6.7) is true for any value of k 1 , we can make k 1 approach k in it, which allows to replace ω by ω 0

		µ N t -µ t )	ln 2+e 2|κ|T σ 2 y 2 2	+e 1 (2|κ|T )	ωT y 2 2	. (6.6)
	6.3 Corollary. Still for the same model, for any x 0:
	lim N -→∞	P || µ N t -µ t || xN -1/2	2 exp	-x 2 2 e 2|κ|T σ 2 + e 1 (2|κ|T )ωT	.	(6.7)
	6.4 Remark.

  Then one computes the following numerical values, which are all rounded above:

	-κ	0.600;	(6.11)
		0.432;	(6.12)
	ω	0.0933;	(6.13)
	σ 2	0.0398.	(6.14)

( * ) To prove it, note that U(x) = f (||x|| 2 ), where f = 2 cosh( √ •) is (the restriction to [0, +∞)of) an analytic function on R.

( * ) Beware: it does not mean that one random particle system has large probability to stay always close to the continuous limit-which is trivially false by ergodicity-but that at any given time, most of the particle systems will be close to the limit.

( †) Remember however that homogeneous Sobolev spaces have no inclusion relations.


Moussa Kafal

Andrea Cozza

Lionel Pichon

An Efficient Technique Based on DORT Method to Locate Multiple Soft Faults in Wiring Networks

Decomposition of the time reversal operator (DORT) recently adopted to wiring fault detection and localization presented effectual results when dealing with a single soft fault along with complex network configurations. On the other hand, it failed in handling the task of locating multiple faults within even simple ones. In this article, we propose an enhanced version of the standard DORT technique (EDORT) which is based on a complementary procedure enabling the accurate and selective localization of multiple soft faults in various wiring systems.

Introduction

The emergence of electrical employments geared the indispensable use of electrical cables in nearly all modern systems. They became the backbone structures that are widely utilized to distribute power and communication signals throughout nuclear power plants, industrial machinery, human facilities and transportation systems which made their reliable and safe operation a pivotal issue. The increasing complexity of modern systems was accompanied with a huge increase in the lengths of cables. From few hundreds of meters in vehicles few decades ago to about 4 km in modern ones, wires lengths increased to several hundred km in civil and military airplanes. Moreover, estimates show that about 5000 km are present in power plants which skyrockets to 40000 km of cables in railway infrastructures of large countries [START_REF] Auzanneau | Wire troubleshooting and diagnosis: Review and perspectives[END_REF]. All these examples, show that the massive use of wiring networks in nowadays systems became crucial, especially that some of these cables are integrated in control and safety operations of many of these systems. Consequently, the searching phase for the detection and repair of a wiring fault counting on the human capabilities intervention is both expensive, time consuming and non-efficient.

Besides, this was accompanied with an inability of detection in many of the cases especially that most of the wires are embedded in the body structure of the systems where as an example, studies reported that 70 % of checked vehicles were returned fault free although they were defected. Thus, proper and efficient wire diagnosis tool is strongly recommended to prevent the drastic consequences that may occur [START_REF] Auzanneau | Wire troubleshooting and diagnosis: Review and perspectives[END_REF].

Whatever the application is, wires are subject to unwanted modifications and breakdowns that can be classified into two main categories: hard faults (open or short circuits) or soft faults, typically featured by minor alterations as insulation defects, cracks, or frays which can lead to drastic consequences if not early rectified. It is worthy to note that our interest in soft faults shoved off from the studies that were conducted on cables and showed that 30% to 50% of detected wiring faults are considered to be soft which might be far dangerous than hard ones due to their detection difficulty [START_REF] Auzanneau | Wire troubleshooting and diagnosis: Review and perspectives[END_REF]. Several techniques have been lately innovated for the task of cable testing, but the widely used ones were the reflectometry based methods whose aim is to detect the presence of an impedance discontinuity by submitting a testing signal and monitoring the reflected one [START_REF] Furse | Down to the wire [aircraft wiring[END_REF]. These high frequency methods, include two main families: Time Domain Reflectometry (TDR) and Frequency Domain Reflectometry (FRD); they differ in the form of the injected wave, or in the post processing techniques used to extract information from the reflected wave. For example, in the TDR family, standard TDR injects a step or a pulse into the wire network to be tested, and relies on the analysis of the reflected wave in the temporal domain in order to extract the fault's signature [START_REF] Furse | A critical comparison of reflectometry methods for location of wiring faults[END_REF]. On the other hand, in FDR techniques stepped frequency sine waves are injected into the network under test (NUT), and determines the location of the fault depending on the parameter that is to be measured (frequency, phase, or stationary wave amplitude measurements) [START_REF] Furse | A critical comparison of reflectometry methods for location of wiring faults[END_REF]. In fact, these reflectometry methods showed promising results when dealing with hard faults but failed when addressing soft ones depicted by weak reflectivities [START_REF] Furse | A critical comparison of reflectometry methods for location of wiring faults[END_REF]. Moreover, they started flopping when complex networks evolved due to the appearance of junction related echoes that increased the complexity of analyzing the reflected signal. Differential DORT, an acoustically originating technique [START_REF] Fink | Self focusing in inhomogeneous media with time reversal acoustic mirrors[END_REF] was adopted to guided wave propagation on transmission lines [START_REF] Abboud | A noniterative method for locating soft faults in complex wire networks[END_REF] and succeeded in detecting and locating a single soft fault within an NUT. Besides, it has showed a great feasibility when addressing complex wiring configurations composed from several branches and junctions. On the other hand, it started failing when handling the mission of detecting multiple faults in different NUTs. Within this context, the aim of the article is to recall the main constraints faced by the standard DORT as presented in [START_REF] Kafal | Locating multiple soft faults in wire networks using an alternative dort implementation[END_REF] and extend the applicability of this technique to the case of multiple soft faults in transmission lines. The second section will elaborate the general concepts of DORT method along with presenting the main restrictions it faced when addressing multiple faults. This will be followed by presenting the new enhanced version of the standard DORT explaining the different steps it is composed of and interpreting the procedure of locating the multiple faults. After that, the third section will validate the new approach numerically and experimentally on different complexity NUTs with considering the presence of two different soft faults embedded in the network. Finally, the last section will draw a conclusion and a summary of this paper.

DORT METHOD

General Concept

Soft fault localization in complex NUT's is to a great extent similar to the problem of target location used in radar communication where the target's positioning becomes tough when its signature is relatively weak compared to other scatterers present in the medium [START_REF] Fink | Time-reversed acoustics[END_REF]. It was proven that by using the time reversal (TR) properties, the reflected echo from a target can be maximized while using the same amount of energy of the excitation signal thus leading to focal spots on the target's location after a series of several iterations. DORT method, derived from the iterative TR isolates and classifies different scattering spots without the need for iterations, which is achieved by the use of the time reversal operator (TRO) [8] [9]. This TRO contains valuable information of the scattering centers in the media encapsulated in its eigenspace structure where propagating the corresponding signals conducts a focusing on the desired scatterer thus providing separate images of the scatterers in a media. Taking advantage of the properties of DORT method adopted in a standard version in guided wave propagation, it became possible to synthesize signals intended to focus on the position of an eventual fault [START_REF] Abboud | A noniterative method for locating soft faults in complex wire networks[END_REF]. It is worthy to note that, all subsequent quantities are function of frequency thus the frequency variable will be dropped in all of what follows.

In order to approach the benefits of the standard DORT technique (SDORT) in locating soft faults, we shall introduce first S, the baselined scattering matrix of the NUT given by:

𝑺 = 𝑺 𝒇 -𝑺 𝒉 ( 1)
where 𝑺 𝒇 is the scattering matrix measured by observing the testing ports of the NUT containing the soft fault whereas 𝑺 𝒉 being the scattering matrix measured in the non-faulty (healthy) version of the same NUT. This operation allows removing the spurious echoes generated by impedance discontinuities like junctions, leaving only those echoes initially generated by the interaction between the testing signals and the faults. The resulting scattering matrix 𝑺 will thus only contain data related to the signals scattered by the fault, which will be used to synthesize the testing signals aiming to focus on the fault's position [START_REF] Abboud | A noniterative method for locating soft faults in complex wire networks[END_REF]. This is accomplished by creating the TRO 𝑲 = 𝑺𝑺 † , given that † refers to the Hermitian transpose, followed by an eigendecomposition process which produces eigenvalues with their corresponding eigenvectors. As a matter of fact, the number of non-zero eigenvalues hints at the number of faults present in the NUT. Whereas, the eigenvectors form the fundamentals of the testing signals which once propagated in the network will form focal spots indicating the position of the fault. Besides, theoretically DORT method as presented in open space target location techniques shows that it is possible to focus on any of the scatterers in the medium simply by propagating the eigenvectors corresponding to its eigenvalue as long as the scatterers are ideally resolved [START_REF] Prada | Eigenmodes of the time reversal operator: A solution to selective focusing in multiple-target media[END_REF].

This property no longer holds in guided wave propagation along transmission lines, while waves attenuate as a factor of 1 𝑟 ⁄ in free space [START_REF] Ott | Electromagnetic compatibility engineering[END_REF], guided waves experience negligible attenuation. On the contrary, a new mechanism of energy exchange appears where waves cannot focus solely on a fault without interacting with the others. As a matter of this fact, ideal resolve can no longer be satisfied preventing the selective focusing on each fault separately by using its eigenvalue components. Besides, as we are dealing with soft faults, SDORT faced an impasse whenever a considerable high severity difference is present between the faults in a system. In other words, a relatively weak soft fault in the NUT with respect to the other stronger one in the configuration will have a masked signature thus reducing the chances of localizing it. In addition to that, it has stood still when the complexity of the network increases that is manifested by the appearance of junction related echoes which are able to mask the eventual position of a weak soft fault.

Eventually, these restrictions are well detailed in [START_REF] Kafal | Locating multiple soft faults in wire networks using an alternative dort implementation[END_REF] in a clear manner.

Enhanced DORT Method

While SDORT does not perform well in locating multiple faults in an NUT, it gives accurate and effective results when dealing with a single fault even in complex network configurations. This fact motivated us to maximize the profit we can gain from the DORT concept in this field which resulted in an enhanced version (EDORT) of this method. The EDORT adopts the main steps followed by the standard method to locate a single fault in the NUT, and proceeds to a complementary procedure enabling the localization of other faults present in a configuration and thus bypassing the restrictions encountered by the SDORT.

This new method commences by measuring the global baselined scattering matrix of the multiple faulty NUT 𝑺 𝑮 𝟏 , which contains information of the signals scattered by all the faults in the system.

Applying the SDORT steps using this matrix will lead to several focusings on some of the faults' positions while others will not appear depending on the configuration's complexity and on the fault's severities. As a matter of what has been presented in [START_REF] Kafal | Locating multiple soft faults in wire networks using an alternative dort implementation[END_REF], increasing the system's complexity may lead to masking the signature of the very weak soft faults. Besides, as the intensity of the soft fault becomes relatively low with respect to the others, its detection becomes very hard and sometimes impossible [START_REF] Kafal | Locating multiple soft faults in wire networks using an alternative dort implementation[END_REF]. On the other hand, locating the position of the strongest soft fault in the NUT is always possible in the first step which is usually recognized by a dominant focal spot.

Thus, after localizing the first strongest fault we will move to a complementary similar process allowing locating other faults present in the wiring network. As a matter of what has been presented in [START_REF] Kafal | Locating multiple soft faults in wire networks using an alternative dort implementation[END_REF], the global scattering matrix 𝑺 𝑮 𝟏 containing all the faults' signatures, can be divided into multiple scattering matrices each containing one of the fault's response that is

𝑺 𝑮 𝟏 = ∑ 𝛼 𝑖 𝑺 𝒅 𝒊 𝑁 𝑓 𝑖-1 ( 2)
where 𝑁 𝑓 being the number of faults in the NUT, 𝛼 𝑖 the weighting factor which will be later presented, while 𝑺 𝒅 𝒊 designating the scattering matrix intending to focus on each soft fault separately [START_REF] Kafal | Locating multiple soft faults in wire networks using an alternative dort implementation[END_REF]. Thus, we will become capable of providing selective focusing on each fault separately by monitoring each 𝛼 𝑖 𝑺 𝒅 𝒊 corresponding to one of the faults. So, after localizing the strongest soft fault present in the NUT, a cloned network similar in structure to the original one will be created, but with a dummy soft fault on the position localized. The corresponding scattering matrix 𝑺 𝒅 𝟏 will embrace the response of the strongest fault solely. After that, a weighted version 𝛼 1 𝑺 𝒅 𝟏 will be subtracted from 𝑺 𝑮 𝟏 leading to new scattering matrix containing the remaining faults' signatures. The SDORT steps will be then applied in order to locate the new strongest fault in the NUT, after which the procedure described before will be followed to localize all faults in the configuration.

For the sake of clarity, the steps required in the EDORT process are summarized in what follows:

1) measure the global scattering matrix 𝑺 𝑮 𝟏 of the (eventually) multiple faulty NUT;

2) apply the SDORT method on 𝑺 𝑮 𝒊 , in order to estimate the position of the strongest soft fault;

3) include a dummy soft fault on the guessed position and compute its scattering matrix 𝑺 𝒅 𝒊 ;

4) compute the only significant eigenvalue of 𝑺 𝒅 𝒊 and its eigenvector 𝜇 𝑖 and 𝜔 𝑖 respectively;

RESULTS

In order to better illustrate the benefits introduced by EDORT over its standard version, a single junction network configuration will be analyzed and its results will be presented in this section.

Analyzed Configuration & Standards

In order to support the EDORT, some rules and standards have been adopted in our study in order to organize the process of obtaining results and their display. If we consider the general network structure, illustrated in Fig. 1, we first begin by assigning a number to each of the extremities where each designates a testing port. We will consider that the one numbered by ( 1), be the origin, and distances will be measured according to this origin, and consequently all graphs will plotted accordingly. Concerning the path designation, each path will be assigned number that is the part of the network between the origin testing port and any of the other ports. As an example, in Fig. 1, the third path is the part of the network linking the origin to the testing port numbered (3). All results obtained are treated by frequency domain simulation on Matlab, using the transmission line theory (TLT), where the transmission lines are considered to be uniform lossless lines with matched-ends in the simulation model created. This simulation model will be used for postprocessing the experimental data collected. Moreover, in order to display the results for determining the actual position of the fault, a time-space (ZT diagram) tool is used which allows the observing the voltage propagation in the network both in space and time.

Experimental results

Experimental tests were conducted, considering the NUT single junction structure depicted in Fig. 1 and implemented using standard 50 Ω coaxial cables as transmission lines as shown in Fig. 2.

Besides, 30 cm semi-rigid coaxial cables were used for the sake of creating the soft faults which is manifested by a crushing of a small portion of it whose severity depends on the length crushed as reproduced in Fig. 3. The stronger soft fault is produced by crushing the line for a distance of 2 cm whereas the weaker is designated by a 1 cm crushing. The ends of the cables were the testing ports for the purpose of simplicity, where three ends of the NUT were connected to a Rohde & Schwarz ZVB8 vector network analyzer (VNA) capable of covering a frequency range from 300 KHz to 8 GHz. Measurements of the scattering matrices was performed over a total bandwidth of 2 GHz and a sampling frequency of 20 MHz After calibrating the VNA using the calibration kit provided by the manufacturer studying the network consisted of two steps: sections.

2. Measurement of 𝑺 𝑓 (𝜔)of the faulty system, where the unaltered cable samples were replaced by the faulty ones.

Having measured the two transfer matrices of the considered NUT, the detailed EDORT steps are applied followed by post processing using the Matlab simulation model. The ZT diagram of Fig.

Figure 1

 1 Figure 1 Layout of the NUT considered for the experimental validation where two different severity soft faults were created by partial crushing of 30-cm semi-rigid coaxial cables as presented in Fig. 2 (fault's severity depends on the radius of the circle and their positions are measured with respect to the reference port (1)).

4

  shows a focal spot pointing the location of the first strongest soft fault in the NUT after the first iteration of the EDORT. Consequently, the second iteration was enough to clearly localize the weaker soft fault as designated in the ZT diagram of Fig. 5. The experimental results presented, allowed the localization of the two soft faults in the single junction network of Fig. 1 which imply the feasibility and efficiency of the proposed method in locating multiple faults within wiring networks which shall be extended to include more complex NUT's.

Figure 2

 2 Figure 2 Implementation of the NUT described in Fig. 1 and connected to a VNA for experimental tests.

Figure 3

 3 Figure 3 The two soft faults considered in the test whose severity depends on the length of crushing of the 30-cm semi-rigid cables.

Figure 4

 4 Figure 4 The voltage propagation along the second path of the network of Fig. 1 after the first EDORT iteration.

Figure 5

 5 Figure 5 The ZT diagram corresponding to the NUT of Fig. 2 after the second iteration of the EDORT showing clearly the position of the second soft fault in the configuration.


Edouard Cardenas Cabada

Nacer Hamzaoui

Quentin Leclere

Jérôme Antoni

Acoustic imaging applied to fault detection in rotating machine

Fault detection in rotating machines is generally based on vibration signal analysis. However, the sound radiated by a structure and its vibration are closely linked. We can therefore imagine that acoustic measurements could be useful for diagnostic improvement. In this paper, a fault diagnostic method based on acoustic imaging is proposed. Beamforming is used to describe the acoustic field generated by an operating machine. Usually, the source strength is mapped in order to identify the radiating areas. In this paper, fault detection features are plotted instead. The spectral kurtosis is mapped as a function of space and frequency to separate and localise impulsive sources.

Introduction

Acoustic imaging is most likely used as a noise source identification method than a diagnostic tool. Although some studies have already been done [START_REF] Coutable | Bearing fault detection based on Near-fiel Acoustic Holography[END_REF][START_REF] Choi | Near field impulsive source localization in a noisy environment[END_REF], fault detection based on acoustic imaging requires more attention. Actually, surveillance and diagnostic of rotating machines usually belongs to vibration analysis domain. One of the reason is that only a few accelerometers are needed to achieve the diagnosis whereas acoustic imaging requires an expensive antenna with a lot of microphones. Another reason is due to the scientific heritage which provides a wide range of signal processing tools for fault detection based on vibration analysis. However, acoustic imaging can be seen as a complement to vibration-based diagnostic. Signal processing methods used in the vibration domain can also be applied to acoustic signals. Moreover, acoustic imaging will provide spatial information of the fault allowing to localise it, which is not always possible in vibration analysis. Another advantage of acoustic measurement is that it does not require a contact with the part to be analysed.

The aim of this paper is to highlight the relevance of acoustic imaging as a fault detection tool. The spatial localisation aspect brought by acoustic imaging can be combined with standard fault diagnostic tools. The spectral kurtosis allows to separate impulsive components buried into a signal. Knowing that different faults in rotating machines produce a series of repetitive short transient forces justifies the use of kurtosis to detect them.

Beamforming [START_REF] Billingsley | The Acoustic Telescope[END_REF][START_REF] Christensen | Beamforming[END_REF] is first used to give displays of time varying source distribution on a machine surface. Then the spectral kurtosis [START_REF] Antoni | The spectral kurtosis: a useful tool for characterising non-stationary signals[END_REF][START_REF] Antoni | The spectral kurtosis: application to the vibratory surveillance and diagnostics of rotating machines[END_REF] can be applied to this generated signals in order to obtain an "impulsive" source distribution as a function of space and frequency. The efficiency of this approach for separating impulsive noise sources will be demonstrated.

Beamforming

Beamforming can be considered as one of the oldest acoustic imaging methods. This source location technique was inspired by telecomunication methods and is often refered to as "acoustic telescope" [START_REF] Billingsley | The Acoustic Telescope[END_REF]. In spite of its age, beamforming is still used in the industrial field [START_REF] Christensen | Beamforming[END_REF] because of its many advantages. Indeed, this array-based measurement technique can be used from short to long distances, requires a poor calculation time and is less sensitive to noise compared to other methods. However, beamforming has a low spatial resolution at low frequencies.

The principle of source location lays on the numerical focusing of the microphone array toward a direction (or point). The amplitude of planar (or spherical) waves coming from this direction (or point) is estimated.

Chosing a set of different directions will allow to map the source strength distribution over a surface. This calculation can either be processed in time or frequency domain depending on the results needed. In this paper, only the time approach is considered. The algorithm used is equivalent to a "delay and sum" beamformer. Each signal is delayed relatively to the distance separating the microphone and the source under consideration. Figure 1 shows a typical 2D setup for beamforming processing. A linear array made out with I microphones is placed in parallel with a radiating line. The distance d between the antenna and the identification line is taken so as to consider spherical waves. Thus, the linearly distributed sources q 1 to q J are monopole radiators. The pressure measured by microphone i can be written as follow

p i (t) = J ∑ j=1 q j (t -∆ i j ) r i j (1)
∆ i j = r i j c (2)
r i j being the distance from microphone i to source j, c the spherical waves speed, and ∆ i j the delay. Equation (1) can be written for each microphone, leading to a set of equations to be inverted in order to identify q j . The least squares solution gives

q j (t) = w j I ∑ i=1 p i (t + ∆ i j ) r i j (3)
w j = I ∑ i=1 1 r i j 2 -1 2 (4)
w j is a weighting factor which takes into account the mean distance between each microphone and the source. Thanks to equation (3) it is possible to identify all the sources by computing an iteration over the whole surface.

The results can be mapped as function of time and space.

The spectral kurtosis

It has been shown that the spectral kurtosis is a relevant tool for fault detection [START_REF] Antoni | The spectral kurtosis: a useful tool for characterising non-stationary signals[END_REF][START_REF] Antoni | The spectral kurtosis: application to the vibratory surveillance and diagnostics of rotating machines[END_REF]. Rotating machine components such as bearings or gears generate impulsive vibration when they fail. However, it is not an easy task to enhance these signatures because of the noise generated by the other parts of the machine.

The kurtosis is a statistical feature that measures the impulsiveness of a signal. It only gives a scalar result when applied to a time-varying signal, which isn't sufficient to conclude about a failure on a machine. The spectral kurtosis (SK) is an optimized version of the kurtosis. It makes possible to identify the frequency band in which the kurtosis is important. Therefore, calculations are made in the frequency domain. SK computing can be interpreted as follow : a filter bank decomposition of the signal will feed a spectral kurtosis algorithm. Such a decomposition can be processed by the short-Time Fourier Transform (STFT). Let us consider a signal Y (n), the sampled version of Y (t). A definition of the STFT is given by

Y w (kP, f ) = +∞ ∑ n=-∞ Y (n)w(n -kP)e -j2πn f (5)
w(n) is the N w length analysis window and P a temporal step. Now, the 2nth order spectral moment of Y w (kP, f ) has to be introduced

Ŝ2nY ( f ) = |Y w (kP, f )| 2n k (6)
• k being the time-averaged operator over the portion k. A definition of the SK estimated with STFT is given in reference [START_REF] Antoni | The spectral kurtosis: a useful tool for characterising non-stationary signals[END_REF] KY

( f ) = Ŝ4Y ( f ) Ŝ2Y ( f ) -2, f = 0 (7)
Notice that the spectral kurtosis results lay on one parameter : the window length N w introduced in equation ( 5). It will directly influence the spectral resolution of SK. It has been shown [START_REF] Antoni | The spectral kurtosis: application to the vibratory surveillance and diagnostics of rotating machines[END_REF] that too small values of N w lead to a strong bias whereas a too fine spectral resolution will make the SK decrease to zero. Thus, a compromise has to be found. Equation ( 7) can be used to design an optimal filter filtering out the mechanical signature of faults when applied to the original signal. Here, a different approach is proposed. The combination between acoustic imaging and SK will allow to visualise a new kind of mapping : the impulsiveness as a function of space and frequency.

Combination of beamforming and SK

In the previous sections, we've been through some theoretical knowledge and definitions. In this section we will get to the root of the problem which is to apply acoustic imaging to fault detection. As it has been said previously, we will use the signal generated by the beamformer to feed a SK algorithm.

Equation (3) provides a set of time varying signals q j (t). Lets apply equation ( 7) to these signals

Kq j ( f ) = Ŝ4q j ( f ) Ŝ2q j ( f ) -2 (8)
It is noteworthy to say that beamforming results which are function of time and space are turned into functions of frequency and space through equation (8) as it is illustrated in figure 2.

Figure 2: Illustration of the mapping of spectral kurtosis as a function of space and frequency.

In the next section, this approach will be tested on real signals.

Application to fault detection

The experimental application concerns a machine which can be considered as a linear distribution of point sources. A linear microphone array is placed in parallel with the radiating line. The antenna is made out with 30 microphones regularly distributed with a 2 cm step. A 15 seconds measurement is processed while the machine is operating. It could be useful to map the source strength as a function of frequency and space in order to visualize radiation features of the mechanism. The output of the beamformer in the frequency domain allows us to obtain such results which are presented in figure 4. At first sight, we can hardly identify sources that can be related to a fault. Moreover, the energy radiated is more important at low frequencies which is the domain where beamforming has a bad spatial resolution. Hence, this visualization is not optimal for fault detection in our case. As previously said, the spectral kurtosis makes possible to find the location of an impulsive excitation. Figure 5 shows the spectral kurtosis layout as a function of frequency and space. This mapping confirms the effectiveness of the method to separate impulsive sources. While figure 4 reveals a wide radiation zone, figure 5 highlights a few well-defined sources. Actually, we can spot two main areas where the kurtosis is high. The first one is located at x = -0.2 and the second and main one at x = -0.158. Forthcoming analysis should concentrate on this positions in order to identify precisely what kind of failure it is, which is not the point of this paper.

Conclusion

In this paper, a new method for fault detection, and more precisely fault location, was proposed. In the first section, the theory of beamforming and spectral kurtosis were introduced. Then, the relevance of combining beamforming and the spectral kurtosis was demonstrated. Since spectral kurtosis is a usefull tool for identifying the frequency bands in which the impulsiveness of the signal is important, adding the spatial dimension to it will make possible to localise impulsive sources. The method proposed was tested on a real case and showed its effectiveness.

Figure 1 :

 1 Figure 1: Beamforming setup focusing on q j

Figure 3 :

 3 Figure 3: Study setup.

Figure 4 :

 4 Figure 4: Beamforming output as a function of frequency and space

Figure 5 :

 5 Figure 5: Spectral kurtosis applied to beamforming output

  at x = -0.158 filtered beween 4900 and 5100 Hz Source Strength q(t) time[s]

Figure 6 :

 6 Figure 6: Filtered beamforming signal.