Raphael Ducatez

Analysis of the one dimensional inhomogeneous Jellium model with the Birkhoff-Hopf Theorem

We use the Hilbert distance on cones and the Birkhoff-Hopf Theorem to prove decay of correlation, analyticity of the free energy and a central limit theorem in the one dimensional Jellium model with non constant density charge background, both in the classical and quantum cases.

Introduction

The Jellium model describes a system of electrons interacting with each other in a continuous background of opposite charge. It is a very fundamental system in quantum chemistry and condensed matter physics [START_REF] Giuliani | Quantum Theory of the Electron Liquid[END_REF][START_REF] Parr | Density-Functional Theory of Atoms and Molecules[END_REF][START_REF] Brack | The physics of simple metal clusters: self-consistent Jellium model and semiclassical approaches[END_REF]. The model has been initially introduced by Wigner [START_REF] Wigner | On the interaction of electrons in metals[END_REF]. In (quasi-)one dimension it has then been rigorously studied when the background is uniform by Kunz [START_REF] Kunz | The one-dimensional classical electron gas[END_REF], Brascamp-Lieb [START_REF] Hj Brascamp | Some inequalities for Gaussian measures and the long-range order of the one-dimensional plasma[END_REF], Aizenman-Martin [START_REF] Aizenman | Structure of Gibbs states of one dimensional Coulomb systems[END_REF] and many others [START_REF] Blanc | The Crystallization Conjecture: A Review[END_REF][START_REF] Baxter | Statistical mechanics of a one-dimensional Coulomb system with a uniform charge background[END_REF][START_REF] Ph | On the statistical mechanics of one-dimensional Coulomb systems[END_REF][START_REF] Gruber | Equilibrium equations for classical systems with long range forces and application to the one dimensional Coulomb gas[END_REF][START_REF] Jansen | Symmetry breaking in Laughlin's state on a cylinder[END_REF][START_REF] Michael Aizenman | Symmetry Breaking in Quasi-1D Coulomb Systems[END_REF][START_REF] Jansen | Wigner crystallization in the quantum 1d jellium at all densities[END_REF][START_REF] Mathieu Lewin | Statistical Mechanics of the Uniform Electron Gas[END_REF]. This model is known to reveal a symmetry breaking called the Wigner crystal. One major difficulty is that the Coulomb potential is long range. In dimension one, the interaction is like -|x -y| and therefore the force between two particles does not depend on their mutual distance which simplifies a lot the problem.

In this paper we study the inhomogeneous Jellium model in which the background is not constant. The inhomogeneous case is very important for applications, at least in three dimensions [START_REF] Ch | Sum rules for inhomogeneous Coulomb systems[END_REF][START_REF] Hohenberg | Inhomogeneous electron gas[END_REF][START_REF]Theory of the Inhomogeneous Electron Gas[END_REF]. The Wigner crystal still appears for a periodic background, provided that the charge in one period is equal to the charge of the particles. Here we consider any background in one dimension and the system will not necessarily be crystallized.

One of the most important properties of the constant background model is that the classical partition function for N particles can be written in the form

Z N (β) = a, T (β) N b
where T (β) is a compact operator with positive kernel in some L 2 space, which depends smoothly on the inverse temperature β. By the Krein-Rutmann Theorem [START_REF] Amann | Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces[END_REF], T (β) has a unique largest eigenvalue λ(β) > 0 which is always non degenerate, hence is also a smooth function of β. As a consequence, the free energy per particle behaves as

f N (β) = - 1 N β log(Z N (β)) = - 1 β log(λ(β)) - 1 βN log( a, v v, b ) + O(κ N )
for large N , where v is the unique positive eigenvector associated with λ(β) and κ < 1. In fact, T (β) N /λ(β) N is close to the rank-one projection on v and this can also be used to prove the decay of correlations.

In the inhomogeneous case, the classical partition function takes the form

Z N (β) = a, 0≤i≤N -1 T i (β)b (1)
where the transitive operators T i (β) are no longer equal to each other. Our goal is to generalize the results proved in the homogeneous case to the inhomogeneous case. For this we will replace the spectral approach based on the Krein Rutmann theorem by the Birkhoff-Hopf theorem [START_REF] Birkhoff | Extensions of Jentzsch's theorem[END_REF][START_REF] Hopf | An inequality for positive linear integral operators[END_REF]. The main idea behind this method is to quantify how a product of many operators with positive kernels can be well approximated by a rank-one operator. A main tool is the so-called Hilbert distance on cones, a concept which will be discussed at length later on.

Using these tools we will prove the decay of correlations and the smoothness of the free energy in the inhomogeneous Jellium model. In the classical case, we can essentially handle any background, but in the quantum case we require it to be close to a constant. Our method is general and can be applied to other one dimensional inhomogeneous systems in statistical physics like the Ising model. It could also be useful for log gases [START_REF] Erdős | Universality for random matrices and log-gases[END_REF][START_REF] Peter | Log-gases and random matrices (LMS-34)[END_REF]. For this reason, we will present the theory in the abstract framework of cones on any Banach spaces, in a form which is well suited to the setting of statistical physics.

Our paper is organized as follows. We first describe in Section 2 the Jellium model and state our main results both for the classical and the quantum cases. We then introduce in Section 3.1 the framework required to state the Birkhoff-Hopf Theorem. Afterward, we suggest a new formulation of weak ergodicity using rank-one operators and prove it in Section 3.3. As it is shown in Section 3.4, the rank-one approximation implies that the k-particle marginals are well approximated by (independent) products of the 1-particle marginals. In Section 3.5 we prove the regularity of the abstract free energy. Finally we deal with the inhomogeneous Jellium model. Section 4.1 and Section 4.2 are dedicated to the proof of the classical and quantum cases, respectively. The main result of theses two sections is the construction of an appropriate cone such that the theorems of the previous sections can be applied.

Acknowledgement: I thank my PhD advisor Mathieu Lewin for proposing this problem, useful discussions and assiduous reading. I am also grateful to Lenaic Chizat who first mentioned me the Birkhoff-Hopf Theorem. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement MDFT No 725528).

The Jellium model

In this section, we present the Jellium model and state all our results. The proofs will be given in Section 4.1 and Section 4.2.

The classical Jellium model

Mathematical formalism

We consider N particles of negative charges q 1 , . . . , q N placed on a line -L < x 1 < x 2 < . . . < x N < L in an inhomogeneous fixed density of charge ρ ∈ L 1 ([-L; L]) such that ´L -L ρ(s)ds = -N i=1 q i . The one dimensional solution of u = 2δ 0 is u(x) = -|x|, which gives us the total energy of the system

E(x 1 , x 2 , . . . ,x N ) = - 1 2 ¨[-L,L] 2 ρ(y 1 )ρ(y 2 )|y 1 -y 2 |dy 1 dy 2 - 1 2
1≤i,j≤N

q i q j |x i -x j | + N i=1 q i ˆL -L ρ(y)|x i -y|dy.
The first term is the background-background interaction, the second term accounts for the electron-electron interaction and the third term for the backgroundelectron interaction.

Let us first calculate the state of minimum energy. For each particle i the position xi which minimizes the energy is such that ˆxi

-L ρ(y)dy = 1≤j<i q j + q i 2 .
It is the condition that for each particle there is the same amount of charge on its right side and on its left side, such that the particle is at equilibrium. In the homogeneous case, we have for any i, ρ|x i+1 -xi | = q. Therefore at T = 0, the electrons are located on q ρ Z (the Wigner cristal). But for a general background the lattice is not necessarily a solution.

We subtract the minimum of the energy and rewrite it as

E(x 1 , . . . , x N ) = E(x 1 , . . . , xN ) + 2 N i=1 q i ˆxi xi ρ(y)(y -x i )dy.
We denote by

U i (s) = -2q i ˆxi+s xi (y -xi -s)ρ(y)dy
the potential felt by the i th -particle around its stable position. We are interested in the canonical model at positive temperature. The position of the particles x i are now random and the probability of a set of positions

(x i ) i=1,...,N is proportional to e -βE(x1,••• ,x N ) (Gibbs measure).
The relevant physical properties of the system are obtained from the partition function given by

Z N (β) = e -βE(x1,. . . ,x N ) ˆ• • • ˆ-L<x1<x2<. . . <x N <L N i=1 e -2βqi ´xi xi ρ(y)(y-xi)dy dx i
and its free energy per particle

f N (β) = - 1 N β log(Z N (β)).
We also introduce the marginals ρ I (x i1 , x i2 , . . . , x i k ), for the probability of the positions of the k particles of the subset

I = {i 1 , i 2 , • • • , i k } ⊂ {1, • • • , N }.
More rigorously, it is the unique function such that for all test functions g ∈

L ∞ ([-L, L] k ), ˚g(x i1 , x i2 , • • • , x i k )ρ k (x i1 , • • • , x i k )dx i1 • • • dx i k = e -βE(x1,. . . ,x N ) Z N (β) ¨-L<x1<x2<. . . <x N <L g(x i1 , • • • , x i k )× × N k=1 e -2βqi ´xi xi ρ(y)(y-xi)dy dx i .
Our main interest is to know whether the particles are strongly correlated or not. This can be quantified by looking at the truncated correlation functions, which we introduce below.

Definition 1. (Cluster property) We say that

• the particles are independent if ρ {1,••• ,N } (x 1 , ..., x N ) = n i=1 ρ(x i ),
• the particles satisfy a cluster property if there exists

I ∪J = {1, • • • , n}, I ∩ J = ∅ such that ρ {1,••• ,N } (x 1 , ..., x n ) = ρ |I| ((x i ) i∈I )ρ |J| ((x j ) j∈J ).
In order to characterize the "clusters" we introduce the truncated marginal:

Definition 2. (Truncated marginal) The truncated marginals ρ T k are defined recursively as follows:

ρ T J (x j1 , ..., x j k ) = ρ J (x j1 , ..., x j k ) - I1∪I2∪...∪Ir=J r l=1 ρ T I l ((x i ) i∈I l ).
The truncated marginals appear to be the good indicator for clustering properties. Indeed we have the following proposition.

Proposition 3. If ρ n (x 1 , ..., x n ) = ρ |I| ((x i ) i∈I )ρ |J| ((x j ) j∈J ) then for all I such that I ∩ I = ∅ and I ∩ J = ∅ then ρ T |I | ((x i ) i∈I ) = 0.
For the reader's convenience we have written the proof of Proposition 3 in Appendix A. We are now ready to state our main results.

Main results

In the classical case we make the following assumptions:

• (H1) There exist q, Q > 0 such that for all i, 0 < q ≤ q i ≤ Q .

• (H2) There exists 0 < m < M such that for all t ∈ [-L, L], m ≤ ρ(t) ≤ M . These assumptions (H1,H2) imply the following bounds for the potential:

U i (s) ≥ qms 2 and d ds U i (s) ≥ smq.
Our first result is to be understood as follows: If we consider particles which are far away from each other (meaning that there are a lot of other particles between them) then the marginal is exponentially close to the independent marginal. We also get the cluster property: if groups of particles are far from each others, then the marginal is exponentially close to the independent cluster marginal.

Theorem 4. For any β > 0, there exists κ < 1 such that for any

I ⊂ {0, • • • , N }, |I| = k, we have ρ I (x i1 , x i2 , . . . , x i k ) - i l ∈I ρ {i l } (x i l ) ≤ C k κ d
for some C k > 0, provided that between any two consecutive particle in I there are at least d others particles (in practice take d = inf |i l -i l+1 | -1). Also we have |ρ T I (x i1 , x i2 , . . . , x i k )| ≤ C k κ D when there exist two consecutive particles in I with at least D others particles between them (in practice take D = max |i l -i l+1 | -1).

Our next result concerns the regularity of the free energy, which is a fundamental property for one-dimensional systems in statistical physics.

Theorem 5. For any β 0 > 0, there exists ∆β > 0 such that the free energy is smooth on [β 0 -∆β, β 0 + ∆β] uniformly on N . More precisely we have

| d k dβ k f N | ≤ M k , with M k > 0 independent of N and for all β ∈ [β 0 -∆β, β 0 + ∆β].
In the proof we will show the following estimate on M k :

M k ≤ k!c k (2)
for some c > 0. From this bound we obtain the analyticity of the limiting free energy, when this limit exists.

Theorem 6. For any β 0 > 0, there exists ∆β > 0 such that if there exists f such that a

f N k → f on [β 0 -∆β, β 0 + ∆β] for a subsequence N k → ∞, then f is real analytic on [β 0 -∆β, β 0 + ∆β].
As a corollary, the system will not reveal any phase transition for β = ∞.

Corollary 7. If the charge background is periodic or if it is constructed randomly with an ergodic process, there exists a limiting function f such that f N → f (almost surely in the ergodic case) and f is real analytic on (0, ∞).

This generalizes the results of Kunz [START_REF] Kunz | The one-dimensional classical electron gas[END_REF].

The quantum model

Mathematical formalism

We now give our results for the quantum problem. In the classical case, we neglect the kinetic energy because in phase space momentum and position are independent for the Gibbs measure. This is no longer true in the quantum case and we have to consider the whole N -particle Hamiltonian

H = - 1 2 N i=1 ∂ 2 xi + E(x 1 , • • • , x N ).
For simplicity we choose Dirichlet boundary conditions at the two ends ∓L.

The quantum fermionic canonical function is

Z Q N (β) =
Tr(exp(-βH)) and the free energy is

f Q N (β) = - 1 βN log(Z Q N (β)).
We have the following Feynman-Kac formula [START_REF] Jansen | Wigner crystallization in the quantum 1d Jellium at all densities[END_REF] for the partition function

Z Q N (β). Proposition 8. (Feynman Kac formula) We have Z Q N (β) = ˆX µ x1x1 . . . µ x N x N (e -´β 0 U (γ1(t),. . . ,γ N (t))dt 1 (γ1,••• ,γ N )∈W N )dx 1 . . . dx N (3) and ρ(x; y) = 1 Z N µ x1y1 × µ x2,y2 × . . . × µ x N ,y N (e -´β 0 U (γ1(t),. . . ,γ N (t))dt )
where

X = {(x 1 , • • • , x N ) : -L < x 1 < • • • < x N < L}, W N = {(γ 1 , . . . , γ N )|∀t ∈ [0, β] : -L < γ 1 (t) < γ 2 (t) < . . . < γ N (t) < L}
is the Weyl chamber and µ x,y are the probability measures of a Brownian bridge from x to y of length β.

The random system we study in the quantum model is no longer the positions (x i ) i≤N but rather the paths (γ i ) i≤N . We define the extended marginals on the set of paths ρ Γ (γ 1 , ..., γ N ) and we are able to apply the theorems of Section 3.4 in this set up. However, for simplicity we will only states the results on the position marginals ρ k (x i1 , ..., x i k ) which satisfy, for any bounded function

g : [-L, L] N → R, ˆ• • • ˆρk (x i1 , • • • , x i k )g(x i1 , • • • , x i k )dx 1 • • • dx k = 1 Z Q N (β) ˆ-L<x1<•••<x N <L µ x1x1 . . . µ x N x N (e -´β 0 U (γ1(t),. . . ,γ N (t))dt 1 (γ1,••• ,γ N )∈W N ) g(x i1 , • • • , x i k )dx 1 . . . dx N

Main results

Unfortunately, in the quantum case we are only able to prove a result in a perturbation regime where ρ and the q i are almost constant. We therefore make the following assumptions :

• (HQ1) q(1 -) ≤ q i ≤ q(1 + ) for all i.

• (HQ2) ρ(1 -) ≤ ρ(t) ≤ ρ(1 + ) for all t.

Theorem 9. For any β > 0, under condition (HQ1-2) for > 0 small enough, there exists κ < 1, such that for all

I ⊂ {1, • • • , N }, |I| = k, |ρ I (x i1 , x i2 , . . . , x i k ) - i l ∈I ρ {i l } (x i l )| ≤ C k κ d
for some C k > 0, if between any two consecutive particle in I there are at least d others particles (in practice take d = inf |i l -i l+1 | -1).On the other hand,

|ρ T I (x i1 , x i2 , . . . , x i k )| ≤ C k κ D if
there exists two consecutive particles in I with at least D others particles between them (in practice take

D = max |i l -i l+1 | -1).
As in the classical case we obtain the regularity of the partition function f Q N . Theorem 10. For any β 0 > 0, there exists ∆β > 0 such that under condition (HQ1-2) with > 0 small enough, the free energy is C ∞ on [β 0 -∆β, β 0 + ∆β] and for all k we have

d k dβ k f Q N ≤ M k with M k independent of N .
Finally we can prove analyticity of the free energy with the same estimate as (2).

Theorem 11. For any β 0 > 0, there exists ∆β > 0 such that under condition (HQ1-2), for > 0 small enough, if f N k admits a limit f for a subsequence

N k → ∞, then f is real analytic on [β 0 -∆β, β 0 + ∆β].
Corollary 12. We make the same assumptions as in Theorem 9. If the charge background is periodic or if it is constructed randomly with an ergodic process, there exists a limiting function f such that f N → f (almost surely in the ergodic case) and f is real analytic.

General theory to apply the Birkhoff-Hopf theorem

The Birkhoff-Hopf theorem has been used for instance to study non linear integrable equations, weak ergodic theorems, or the so-called DAD problem [START_REF] Jonathan M Borwein | Entropy minimization, dad problems, and doubly stochastic kernels[END_REF].

We first introduce the notion of cone and the Hilbert distance. In this set up we can state the Birkhoff-Hopf theorem. Then we prove Theorem 4 and Theorem 5 with the extra assumption of strictly contracting operators.

Framework and Birkhoff-Hopf theorem.

We follow [START_REF] Simon | An elementary proof of the birkhoff-hopf theorem[END_REF] for the notation and we refer to this paper for a proof of the Birkhoff Hopf theorem (Theorem 18 bellow). Let E be a real linear Banach space.

Definition 13. (Abstract cone) C ⊂ E is called a cone if 1. C is convex, 2. λC ⊂ C for any λ ≥ 0, 3. C ∩ -C = {0} .
Using C we define a partial order on E Definition 14. (Partial order) For any x, y ∈ E, we write

x ≤ C y if y -x ∈ C.
For clarity we will use ≤ instead of ≤ C if there is no confusion about the cone.

Definition 15. If C is a cone, we define the dual cone C * by

C * := {f ∈ E * : ∀x ∈ C, (f, x) ≥ 0}.
The set C * is a cone if C -C is dense and in particular if C has nonempty interior. We say that x, y ∈ C are comparable and write x∼y if there exist α, β > 0 such that αx ≤ y ≤ βx. This defines an equivalence relation. We say that C is normal if there exists γ > 0 such that ∀x, y ∈ C, 0 ≤ x ≤ y ⇒ x ≤ γ y .

Definition 16. For any x, y ∈ C comparable, we define the Hilbert metric by

d C (x, y) = log β min (x, y) α max (x, y)
where

α max (x, y) = sup {α > 0 : αx ≤ y} and β min (x, y) = inf {β > 0 : y ≤ βx}
The Hilbert metric is a metric on the projective space of C.

We say that T :

E → E is order-preserving if x ≤ y ⇒ T (x) ≤ T (y).
If T is a linear operator (the only case we will consider here) this is equivalent to

T (C) ⊂ C.
Remark 17. If T is order-preserving then T is non-expanding for the Hilbert metric. Indeed αx ≤ y ≤ βx implies αT (x) ≤ T (y) ≤ βT (x).

We introduce the projective diameter

∆(T ) = sup {d C (T (x), T (y)) : x, y ∈ C, T (x) ∼ T (y)}
and the contracting ratio

κ(T ) = inf {c > 0 : ∀x, y d C (T (x), T (y)) ≤ cd C (x, y), T (x) ∼ T (y)}
Here is the main theorem we will use :

Theorem 18. (Birkhoff-Hopf [5, 22]) If T is order-preserving then κ(T ) = tanh ∆(T ) 4 .
The result has to be understood as follows: if the image of the cone of the order preserving operator is strictly inside the cone (∆(T ) < ∞), then the operator is strictly contracting (κ(T ) < 1) for the Hilbert metric.

Application to statistical physics

We now use the previous formalism to study the partition function and the marginals from statistical physics in the abstract framework of positive operators.

Definition 19. (Density function) Let u ∈ C * and v ∈ C, and let X, Y, (X i ) i be positive bounded operators. We define

• the partition function by

Z = (u, T N . . . T 0 v),
• the one-point density function by

ρ K1 (X) = 1 Z (u, T N . . . T K1+1 XT K1 . . . T 0 v),
• the pair correlation function by

ρ K2,K1 (Y, X) = 1 Z (u, T N . . . T K2+1 Y T K2 . . . T K1+1 XT K1 . . . T 0 v),
• the k-point correlation function by

ρ K k ,...,K2,K1 (X k , ..., X 1 ) = 1 Z (u, T N . . . T K k +1 X k T K k . . . T K1+1 X 1 T K1 . . . T 0 v).
The operators X, Y, (X i ) should be thought of as test functions acting on the position of the K th i particle. Remark 20. The simplest model that can be written in this formalism is the one-dimensional Ising model [START_REF] Pfeuty | The one-dimensional Ising model with a transverse field[END_REF]. All the results stated above for Jellium can be easily adapted to the inhomogeneous one-dimensional Ising model.

We also think of Markov processes on a finite or compact set, in which case T i is the transitive kernel from X i to X i+1 .

Decay of correlations

The following theorem states the exponential decay of the correlation functions.

Theorem 21. (Decay of correlations) Let (T i ) i=1,. . . ,N be positive operators such that ∆(T i (C)) ≤ M < ∞
for any i. Then there exist c > 0 which depends only on k, such that for min |K j+1 -K j | large enough, we have

1 -cκ minj |Kj+1-Kj | k i=1 ρ Ki (X i ) ≤ ρ K k ,...,K1 (X k , ..., X 1 ) ≤ 1 + cκ minj |Kj+1-Kj | k i=1 ρ Ki (X i ) with κ = tanh( M 4
). The decay of correlations is an important concept in statistical physics and it is ubiquitous in one-dimensional systems [START_REF] Ruelle | Statistical mechanics: Rigorous results[END_REF].

The proof of Theorem 21 is provided below in Section 3.4.

Regularity of the free energy

The second theorem states that the partition function depending on a parameter is smooth, if the transitive operators are smooth enough. In order to express the "regularity" of the operator in the framework of a cone and the Hilbert distance, we have to construct the following norm. The following result says that the distance is close to being a norm in the neighborhood of any point x 0 .

Proposition 22. Let x 0 ∈ C. For any > 0, there exists r > 0, a function f and a norm N defined on the projective space, such that d can be written as follows

d(x, y) = f (x, y)N (y -x)
for all x, y such that d(x, x 0 ) < r and d(y, x 0 ) < r, with |f (x, y) -1| < .

We can now state our second main result Theorem 23. Let C be a cone and let T i (β) be a family of smooth bounded operators for β in the neighborhood [β 0 -δ, β 0 + δ] of β 0 , which are contracting of parameter κ < 1, uniformly in β and i. For all i, we denote by N i the norms defined in Proposition 22 around

x i = i-1 j=0 T j (β)b.
Assume that the derivatives in β of the operator are uniformly bounded for theses norm, that is,

∃C , ∀i, d k T i dβ k Ni→Ni+1 ≤ C k for some constant C k independent of i and of β ∈ [β 0 -δ, β 0 + δ]. Then f N (β) = 1 N log a, N -1 i=0 T i (β)b
is uniformly smooth, meaning there is a constant C which depends only on κ and (M k ) such that :

d n f N (β) dβ n ≤ C(κ, (M k ) k≤n )
where

M k = sup i,β∈[β0-δ,β0-δ] d k Ti dβ k Ni→Ni+1 . Moreover if the following limit exists f (β) = lim k→∞ 1 N k log a, N k -1 i=0 T i (β)b
for a sequence N k → ∞, then it is smooth in a neighborhood of β 0 :

d n f (β) dβ n ≤ C(κ, (M k ) k≤n ).
If the positive operator appears to be uniformly analytic for the constructed norm then the free energy is analytic. More precisely we have the following theorem Theorem 24. With the same assumptions as in Theorem 23, if there exists r ≥ 0 such that

∂ n β T i Ni→Ni+1 n! ≤ r n
for all n, then f is real analytic around β with radius of convergence at least equal to (1 -κ)/r.

The two theorems of this section are proved later in Section 3.5.

Central Limit Theorem

We consider the particular case where the space is L 1 (Λ), with Λ a measurable set and the cone is

C = {f ∈ L 1 (Λ) : f ≥ 0}.
We construct the canonical random process y i as follows. Let A 1 , • • • , A N ⊂ Λ, and take as test functions 1 A1 , 1 A2 ..., 1 A N . Then we define

P(y 1 ∈ A 1 , y 2 ∈ A 2 , ..., y N ∈ A N ) = ρ 1,••• ,N (1 A1 , ..., 1 A N ) = 1 Z N i=1 T i 1 Ai .
(4) The decay of correlation in Theorem 21 is the mixing property of the process (y i ).

The Central Limit Theorem has been proved for a urge number of random processes like martingales [START_REF] Hall | Martingale limit theory and its application[END_REF], Markov processes [START_REF] Hervé | Vitesse de convergence dans le théorème limite central pour des chaînes de Markov fortement ergodiques[END_REF][START_REF] Guivarc | Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d[END_REF] or random products of matrices [START_REF] Le | Théoremes limites pour les produits de matrices aléatoires[END_REF]. One of the classical proofs of the central limit theorem uses the regularity of the Laplace transform, this is what we adapt here.

Theorem 25. (Central Limit) Let h i : Λ → R be such that E(exp(h i (y i ))) < ∞
where the mean is on the probability (4). Let T i (β) = e βhi(yi) T i . If the T i (β) satisfy the assumptions of Theorem 23, then we have

sup x∈R P 1 √ N (h i (y i ) -E(h i (y i ))) ≥ x -P N (0, σ 2 ≥ x) = O 1 √ N
where σ 2 is the second derivative of the free energy

σ 2 = d 2 dβ 2 1 N log a, N -1 i=0 T i (β)b .
This theorem is proved in Section 3.6.

Rank-one operator approximation

One of the first historical applications of the Birkhoff-Hopf Theorem was in population demography [START_REF] Cohen | Ergodic theorems in demography[END_REF]. An age structure diagram f evolves due to birth and death, with death and birth rates not constant in time and one can calculate its time evolution. It appears that even if f does not converge to an equilibrium, the long time evolution is independent of the initial age structure f (0). Namely this is a weak ergodicity property: if f 1 and f 2 are two solutions of the evolution with different initial data,

f 1 (t)/ f 1 (t) -f 2 (t)/ f 2 (t) → 0.
In this section, we formulate weak ergodicity in term of a rank-one operator approximation and we give a construction and an estimate of such an approximation in case where several contracting operators are composed one after another.

The cone of order preserving operators

We state here some simple results about the set of order preserving operators.

Lemma 26. Let C with C -C dense. The set of corresponding order-preserving operator is a cone.

We denote by P C this cone and only P if there is no confusion.

Proof. We check every point of the definition.

1. If A, B are order preserving operator then A + B is an order preserving operator. Indeed (A + B)(x) ∈ C for all x ∈ C.

2. The set of order preserving operator is invariant by product of strictly positive scalars. We have the following order on the set of operators :

B ≥ P A ⇔ (B -A)(C) ⊂ C
and the corresponding Hilbert distance

d P (A, B) = min log β α : αA ≤ B ≤ βA . Remark 28. If A ≤ P B and C ≤ P D then AC ≤ P BD. Indeed (B -A)C(C) ⊂ (B -A)(C) ⊂ C, and B(D -C)(C) ⊂ B(C) ⊂ C. Therefore BD ≥ P BC ≥ P AC.
Unfortunately that T is contracting does not imply that T : A → T A is contracting as well. One can take for example:

A = 1 2 2 1 , B = 2
Proof. Let α 1 , β 1 , α 2 , β 2 such that α 1 A ≤ C ≤ β 1 A, α 2 B ≤ D ≤ β 2 B and log β1 α1 -d P (A, C) ≤ , log β2 α2 -d P (B, D) ≤ . Then α 1 α 2 AC ≤ BD ≤ β 1 β 2 AC
and we have d P (AB, CD) ≤ log β1β2 α1α2 ≤ d P (A, C)+d P (B, D)+2 . Now we construct a rank-one operator L = z • l, with a vector z ∈ E, and a linear form l ∈ E * to approximate a contracting function T . It is natural to choose z ∈ T (C). We construct l in the following subsection.

Rank-one operator construction

We construct here the rank-one operator close to a contracting operator. Then there exists a linear form l ∈ C * such that, for any x ∈ C, a(x) ≤ l(x) ≤ b(x).

Proof. We check that b is a convex function,

b(tx + (1 -t)y) ≤ b(tx) + b((1 -t)y) = tb(x) + (1 -t)b(y),
and that a is a concave function,

a(tx + (1 -t)y) ≥ a(tx) + a((1 -t)y) = ta(x) + (1 -t)a(y).
Let us define two sets :

B = {(x, s) ∈ C × R : b(x) ≤ s} and A = {(x, s) ∈ C × R : a(x) > s}.
Then A ∩ B = ∅, A and B are convex. Because of the Hahn-Banach separation theorem, there exists l = 0 a linear form on E × R such that for all (x, s) ∈ B, l(x, s) ≥ 0 and all (x, s) ∈ A, l(x, s) ≤ 0. We have l(x, s) = l 1 (x) + αs with l 1 ∈ E * and α ∈ R.

We then prove that α > 0. For any s 0 > 0 we have (0, s 0 ) ∈ B, (0, -s 0 ) ∈ A, αs 0 ≥ -αs 0 and as a conclusion α ≥ 0. If α = 0, because for any x ∈ C there exist s 1 and s 2 such that (x, s 1 ) ∈ B and (x, s 2 ) ∈ A, we have 0 ≥ l(x, s 2 ) = l 1 (x) = l(x, s 1 ) ≥ 0 and therefore l 1 (x) = 0. Let x 0 ∈ C, and V (x 0 ) ⊂ C a small ball with center x 0 and radius . Then for all y with y < , we have l 1 (y) = l 1 (y + x 0 ) = 0. As a conclusion l = l 1 = 0 which is absurd, so α = 0.

Let

l 0 = -l1 α . Since (x, b(x)) ∈ B, -l 0 (x) + b(x) ≥ 0 we have l 0 (x) ≤ b(x)
. Moreover for > 0, (x, a(x) -) ∈ A -l 0 (x) + a(x) -≤ 0 and therefore l 0 (x) ≥ a(x) . It is possible to check the hypothesis of Lemma 30. Indeed we have that

α 1 T (y 0 ) ≤ C T (x 1 ) ≤ C β 1 T (y 0 ), α 2 T (y 0 ) ≤ C T (x 2 ) ≤ C β 2 T (y 0 ) implies (α 1 + α 2 )T (y 0 ) ≤ C T (x 1 ) + T (x 2 ) ≤ C (β 1 + β 2 )T (y 0 )
and therefore a(x 1 + x 2 ) ≥ a(x 1 ) + a(x 2 ) and b(x 1 + x 2 ) ≤ b(x 1 ) + b(x 2 ). We also have a(λx) = λa(x) and b(λx) = λb(x) for all λ ≥ 0 and x ∈ C. We can then apply Lemma 30: there exists a linear form l with a(x) ≤ l(x) ≤ b(x). Moreover log b(x) a(x) ≤ ∆(T ) for all x ∈ C. We then have for all x l(x) a(x) ≤ e -∆(T ) and b(x) l(x) ≤ e ∆(T ) and therefore

e -∆(T ) T (x) ≤ C T (y 0 ) • l(x) ≤ C e ∆(T ) T (x).
As a conclusion d P (T (y 0 ) • l, T ) ≤ log(e 2∆(T ) ) ≤ 2∆(T ).

Corollary 32. Let (T i ) i=0,. . . ,N be positive operators. If

1. ∆(T 0 (C)) ≤ R < ∞, 2. T i i = 1, . . . , N are uniformly contracting of parameter κ < 1,
then there exists a linear form l, z 0 ∈ C, z 0 = 1 and l ∈ C * such that

d P (T N . . . T 0 ), z 0 • l) ≤ 2κ N R.
Proof. We have ∆(T N . . . T 0 ) ≤ κ N R and the result follows from the previous corollary.

Decay of correlation function

Here we prove Theorem 21. The idea is to replace the product of contracting operator between the k points of measure by a rank-one operator. We will do so for k = 2 and for k > 2 this will be exactly the same. More precisely we will prove the following Theorem 33. (Theorem 21 in the case k = 2) Let (T i ) i=1,. . . ,N be positive operators such that ∆(T i (C)) ≤ M < ∞ for any i, and let

K 1 , K 2 ∈ N be such that 1 ≤ K 1 ≤ K 2 ≤ N . Let u, v ∈ C,
and X, Y be two positive operators.Then:

e -8R(κ K 1 +κ K 2 -K 1 +κ N -K 2 ) ρ K1 (X)ρ K2 (Y ) ≤ ρ K2,K1 (Y, X) ≤ e 8R(κ K 1 +κ K 2 -K 1 +κ N -K 2 ) ρ K1 (X)ρ K2 (Y ).
One can use this theorem for K 1 ,K 2 -K 1 , N -K 2 large. In this case, the Taylor expansion of e x gives

|ρ K1 (X)ρ K2 (Y ) -ρ K1,K2 (X, Y )| ≤ 16R(κ K1 + κ K2-K1 + κ N -K2 ) X Y , which decays exponentially. Proof. Let us introduce L K10 = z K10 l K10 , L K2K1 = z K2K1 l K2K1 and L N K2 = z N K2 l N K2 which are rank-one operators such that      d P (T K1 . . . T 0 ), L K10 ) ≤ 2κ K1 R, d P (T K2 . . . T K1 ), L K2K1 ) ≤ 2κ K2-K1 R, d P (T N . . . T K2 ), L N K2 ) ≤ 2κ N -K2 R.
We then use Proposition 29, to obtain the inequality for the partition function,

d P (T N . . . T 0 , L N K2 L K2K1 L K10 ) ≤ 2R(κ N -K2 + κ K2-K1 + κ K1 ),
for the density function

d P (T N . . . T K1+1 XT K1 . . . T 0 , L N K2 L K2K1 XL K10 ) ≤ 2R(κ N -K2 +κ K2-K1 +κ K1 ) d P (T N . . . T K2+1 Y T K2 . . . T 0 , L N K2 Y L K2K1 L K10 ) ≤ 2R(κ N -K2 +κ K2-K1 +κ K1 ),
and the pair correlation function

d P (T N . . . T K2+1 Y T K2 . . . T K1+1 XT K1 . . . T 0 , L N K2 Y L K2K1 XL K10 ) ≤ 2R(κ N -K2 + κ K2-K1 + κ K1 ).

Moreover we have

(u, L N K2 L K2K1 L K10 v) • (u, L N K2 Y L K2K1 XL K10 v) = (u, z N K2 )(l N K2 z K2K1 )(l K2K1 z K10 )(l K10 v)(u, z N K2 )(l N K2 Y (z K2K1 )) (l K2K1 X(z K10 ))(l K10 v) = (u, z N K2 )(l N K2 z K2K1 )(l K2K1 X(z K10 ))(l K10 v)(u, z N K2 )(l N K2 Y (z K2K1 )) (l K2K1 z K10 )(l K10 v) = (u, L N K2 L K2K1 XL K10 v) • (u, L N K2 Y L K2K1 L K10 v)
and this allows us to conclude that

Z 2 ρ K1 (X)ρ K2 (Y ) ≤ (u, L N K2 L K2K1 XL K10 v) • (u, L N K2 Y L K2K1 L K10 v)e 4R(κ N -K 2 +κ K 2 -K 1 +κ K 1 ) ≤ (u, T N . . . T K2+1 XT K2 . . . T K1+1 XT K1 . . . T 0 v) • (u, T N . . . T 0 v) × e 8R(κ N -K 2 +κ K 2 -K 1 +κ K 1 ) ≤ Z 2 ρ K2,K1 (Y, X)e 8R(κ N -K 2 +κ K 2 -K 1 +κ K 1 ) .
Finally, we have

Z 2 ρ K1 (X)ρ K2 (Y ) ≥ Z 2 ρ K2,K1 (Y, X)e -8R(κ N -K 2 +κ K 2 -K 1 +κ K 1 ) .
The proof of the decay of the cluster correlation is the same. One should just replace X i by

X i = T i+l Y i,l-1 • • • T i+1 Y i,2 T i Y i,1
, which are positive operators.

Smoothness of the free energy

In this section, we prove Proposition 22 and Theorem 23.

Proof of Proposition 22

Proof. Let H a hyperplane such that E = Vect({x 0 }, H). The projective space is locally isomorph to H. Let B be the convex set containing all the s ∈ H for which there exist (α + , α -, β -, β + ) satisfying α + x 0 ≤ x 0 + s ≤ β + x 0 , with β +α + ≤ r, and α -x 0 ≤ x 0 -s ≤ β -x 0 , with β --α -≤ r, for some r small enough. This set is symmetric with respect to the transformation s → -s. Therefore, it is the ball of the norm s = r • inf(λ ∈ R, s λ ∈ B)). Let us check that this norm is close to the distance. Let s 1 , s 2 ∈ H be such that d(x 0 +s 1 , x 0 ) < r and d(x 0 + s 2 , x 0 ) < r. We have then α 1 x 0 ≤ x 0 +s 1 ≤ β 1 x 0 and α 2 x 0 ≤ x 0 +s 2 ≤ β 2 x 0 and because r is very small, we can write

α 1 = 1 + δα 1 , α 2 = 1 + δα 2 , β 1 = 1 + δβ 1 , β 2 = 1 + δβ 2 . At first order we have d(x 0 + s 1 , x 0 ) = δβ 1 -δα 1 + o(|δβ 1 |, |δα 1 |) and d(x 0 + s 2 , x 0 ) = δβ 2 -δα 2 + o(|δβ 2 |, |δα 2 |).
We now check that d(x 0 + s 1 , x 0 ) = (1 + O(r)) s 1 . First we have

d(x 0 + s 1 , x 0 ) ≥ (1 + O(r)) s 1 .
Indeed, for any λ ∈ R, we have λ(1

+ δα 1 )x 0 + (1 -λ)x 0 ≤ x 0 + λs ≤ λ(1 + δβ 1 )x 0 +(1-λ)x 0 and x 0 +λ(δα 1 )x 0 ≤ x 0 +λs ≤ x 0 +λδβ 1 x 0 . With λ = r δβ1-δα1 , we obtain x 0 + r δβ 1 -δα 1 (δα 1 )x 0 ≤ x 0 + r δβ 1 -δα 1 s ≤ x 0 + r δβ 1 -δα 1 δβ 1 x 0 Therefore s ≤ δβ 1 -δα 1 = (1 + O(r))d(x 0 , x 0 + s).
Then we claim that

d(x 0 + s 1 , x 0 ) ≤ (1 + O(r)) s 1 .
Indeed let λ be such that for any α, β αx

0 ≤ x 0 + s λ ≤ βx 0 ⇒ β -α ≥ r. Then for any α, β λαx 0 + (1 -λ)x 0 ≤ x 0 + s ≤ λβx 0 + (1 -λ)x 0 ⇒ β -α ≥ r. Therefore d(x 0 , x 0 + s) ≤ log 1 -λ + λβ 1 -λ + λα = log 1 + λδβ 1 + λδα = (λ(δβ -δα))(1 + O(r)) ≤ s (1 + O(r)).
We finally check that d(

x 0 + s 1 , x 0 + s 2 ) = (1 + O(r))d(x 0 + s 1 -s 3 , x 0 + s 2 -s 3 ) for any s 1 , s 2 , s 3 ∈ B. We have, α 3 x 0 ≤ x 0 +s 3 ≤ β 3 x 0 , and d(x 0 +s 1 , x 0 +s 2 ) = (δβ -δα)(1 + O(r)) with (1 + δα)(x 0 + s 1 ) ≤ x 0 + s 2 ≤ (1 + δβ)(x 0 + s 1 ). Then (1 + δα + O(r)δα)(x 0 + s 1 + s 3 ) ≤ (1 + δα)(x 0 + s 1 + s 3 ) -δαs 3 ≤ x 0 + s 2 + s 3 and x 0 + s 2 + s 3 ≤ (1 + δβ)(x 0 + s 1 + s 3 ) -δαs 3 ≤ (1 + δβ + O(r))(x 0 + s 1 + s 3 ). We conclude that d(x 0 + s 1 , x 0 + s 2 ) = s 2 -s 1 (1 + O(r)).
Let C be the cone of positive vectors in R n and let x 0 = (x 1 , • • • , x n ) and H = {s : i s i = 0}. Then in a neighborhood of x 0 , we have

αx 0 ≤ x 0 + s ≤ βx 0 with α = max α αx i ≤ x i + s i = 1 + min s i x i and β = min β βx i ≥ x i + s i = 1 + max s i x i . In addition d(x 0 , x 0 + s) = log 1 + max s i x i 1 + min s i x i ≈ max s i x -min s i x = max s i x i + max -s i x i .
Finally the constructed norm is then:

s 1 -s 2 = max s i 1 -s i 2 x i + max s i 2 -s i 2 x i .
In order to prove Theorem 23 we will need the following lemma.

Lemma 34. Let E n be Banach spaces with norms n . Consider the functions u n : R →E n iteratively defined by

u n+1 (s) = g n (s, u n (s)),
with g n (s, 0) = 0 which are assumed to be uniformly contracting, ∂ 2 g n n → n+1 ≤ κ with κ < 1. If the g n are uniformly C k then the u n are uniformly C k .

Proof. We prove by induction that there exists constant a C k such that for all n, d k ds k u n n ≤ C k . Computing the derivative gives

d k ds k u n+1 = ∂ 2 g n • d k ds k u n + Q g n , (∂ r 1 ∂ p 2 g n ), ( d i ds i u n ) i<k
where Q is a polynomial involving lower order derivatives of u n and the derivatives of g n . Because of the induction hypothesis, there exists C k-1 such that for all n and all i < k, d i ds i u n ≤ C k-1 . Therefore Q can be uniformly bounded by a constant Ck which depends only on sup n∈N ∂ r 1 ∂ p 2 g n and C k-1 . We have therefore

d k ds k u n+1 n+1 ≤ κ d k ds k u n n + Ck and we can then set C k = 1 1 -κ Ck . (5)
We can now conclude because if g is contracting for d then it is contracting for . and log(λ(β)) for β around 1. In the usual positive cone,

∆(T (β)(C)) = d C 1 , 1 = |2 log( )|.
The Birkhoff-Hopf theorem gives κ = tanh(log( )/2) ≈ 1-2 . Around the point 1 1 , the norm N is equal to the norm . ∞ (see example 3.5.1). The iterative formula ( 5) gives a constant behave like C k ≈ (2 ) (1-k) and this is what we get with the exact calculation of d dβ k [log(λ(β))]. Remark 36. If T is contracting for the distance d, then T is locally contracting for N .

We can now finish the proof of Theorem 23.

Proof. [Theorem 23]We denote

u n (β) = n-1 i=0 T i (β)b n-1 i=0 T i (β)
and we decompose the log of the product as

f N (β) = 1 N log a, N -1 i=0 T i (β)b = 1 N log a, u N (β) + 1 N log( T i (β)u i (β) ).
Because the T i are smooth we only have to make sure that the u i are smooth as well. This follows from the previous lemma. The function u i (β) is smooth for the constructed norm i . But because the cone is normal,

αx ≤ y ≤ βx ⇒ y - β + α 2 x ≤ β -α 2 x ,
we have then that x • x -y i ≥ x -y and we conclude because u n (β) = 1.

Proposition 37. Let g 1 , g 2 , • • • , g n • • • , be analytic functions such that for any k, g k (x) = i b k,n x n with |b k,n | ≤ r n and |b k,0 | ≤ 1. Let f 0 = g 0 and f k+1 = (1 + κg k+1 f k ). Then for any k, f k = c k,n x n
with c k,n ≤ d n where d n are the coefficient of the Taylor expansion of 1-rx (1-κ)-rx . In particular, if f n admits a limit f ∞ , then f ∞ is analytic.

Proof. We can assume b k,n = r n for any k, n. Indeed another configuration would give a smaller c k,n . We expand f k and have:

f k = k i=0 κ 1-rx i whose coefficients are then smaller than those of ∞ i=0 κ 1-rx i = 1-rx (1-κ)-rx .
Corollary 38. Let g n and u n be as in Proposition 34. Suppose that there exists r ≥ 0 such that

∂ i s g i!
≤ r i for all i, then u n are analytic with coefficients of its Taylor series bounded by that of 1-rx (1-κ)-rx . In particular if u n admits a limit then it is analytic with convergence radius 1-κ r . Proof. This follows from the fact that

u n (s) -u n (0) = g n (s, u n-1 (s) -u n-1 (0)) + g n (s, u n-1 (0)) -u n (0).

Proof of Theorem 25

We now prove the central limit theorem 25 from the regularity of the Laplace transform.

Proof. By Theorem 24 f (α) is smooth with ∂ α f | α=0 = γ √ N ≤ C √ N , ∂ 2 α [f - γ √ N α]| α=0 = σ 2 ≤ C and ∂ 3 α [f -γ √ N α] ≤ C √ N . Then (f (α)-γα) = 1+ (σα) 2 2 + O( 1 √ N ).
Therefore the Laplace transform is close to the one of a Gaussian and we can conclude with the usual Berry Essen inequality.

4 Proofs for the Jellium model

Proof for the classical Jellium model

We first write the partition function in the form of products of operators. Recall that U i (s) = -2q i ´xi+s xi (y -xi )ρ(y)dy with xi the equilibrium position of the particle i. We note δ = 1 2 min(|ã i -ãi+1 |). Definition 39. (Iterative operator) Let T i be the operator defined for any function f in L 1 or L ∞ by

T i f (x) = ˆ∞ s=x-xi+1+ xi e -βU (s) f (s)ds.
In particular, we can rewrite the partition function as

Z N (β) = e -βE(x1,. . . ,x N ) 1 x N <L-x N , N -1 i=0 T i 1 x1>-L-x1
We are then in the setting of Section 3.4.

Construction of a uniform invariant cone

We first notice that we cannot directly apply the Birkhoff-Hopf Theorem with the cone of positive functions C 0 . Indeed we have the Remark 40. For any

T i , ∆ C0 (T i ) = ∞. For example Supp[T i (1 [0,1] )] = (-∞; 1 + (x i+1 -xi )]
and Supp(T i (1 [START_REF] Aizenman | Structure of Gibbs states of one dimensional Coulomb systems[END_REF][START_REF] Amann | Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces[END_REF] )) = (-∞, 3 + (x i+1 -xi )] and then we have for any α > 0 and

1+(x i+1 -xi ) < t < 3+(x i+1 -xi ), T i (1 [0,1] )-αT i (1 [2,3] ) (t) < 0 so α min = 0.
The solution is to construct another cone. If we were restricted to a bounded interval, then the simplest solution would be to consider finite products of T i , instead of one by one. The kernel of n+k-1 i=n T i is strictly positive on {(x, y), y ≥ x-2nδ}, and therefore with n such that 2nδ > 2A, the kernel is strictly positive.

T i are then contracting for the cone {f ≥ 0}.

In our case, because of the multiplication by e -Ui(s) , we will be able to neglect the influence of f e -Ui(s) outside (-A, A). We choose A such that ˆ∞ A e -Ui(s) ds ≤ δ 2 e -Ui(A)

(for example, because of d ds U (s) ≥ qms, we can choose A = 2 δqm ). In Proposition 41 we will define a cone such that f can be slightly negative for {x : x ≥ A}. We also make it so that T i are contracting and not only k+n k T i . The price to pay is more restrictions. Intuitively, it is how T i f looks like for f ≥ 0.

Let us divide the interval [-A, A] in small intervals with

I k = [kδ/2, (k + 1)δ/2] with k ∈ Z and -2 A δ -1 = k min ≤ k ≤ k max = 2 A δ + 1. We suppose
f ∈ C ⇔                    ∀t ≥ A f (t) + I kmax (f ) ≥ 0, ∀t ≤ A f (t) ≥ 0, on -A ≤ t ≤ A f is decreasing , ∀t ≤ -A f (t) ≥ f (-A), ∀k ∈ [-2 A δ , 2 A δ ] I k-1 (f ) ≤ 1 k I k (f ), ∀t ≤ -A f (t) ≤ 1 I kmin (f ), satisfies that, for any i, T i is d C contracting.
This cone may seem a bit artificial, however it behaves nicely with respect to the iteration of T i . For the proof we need the following Lemma 42. Let y, x > 0, K linear and a, b, u, v ≥ 0 such that Kx ≥ ax + uy and Ky ≤ by + vx. If a > b or u > 0, then there exist > 0 such that if 1 x ≥ y then 1 Kx > Ky.

Proof. If a > b, then for small enough, b + v < a and we have Ky ≤ by

+ vx ≤ ( b +v)x < a x ≤ 1 Kx. If u > 0, then we have 1 Kx-Ky ≥ ( a -v)x-(b-u )y > 0 for small enough.
We can now carry on the proof of Proposition 41.

Proof. We construct the k recursively. Because f is decreasing and e -Ui are uniformly integrable, there exists u such that ´∞ -α f e -Ui(s) ds ≤ uI kmin (f ). We then have

sup T i f = ˆR e -Ui(s) f (s)ds ≤ sup f • ˆ-A -∞
e -Ui(s) ds + uI kmin (f ).

Moreover I kmin (T f ) ≥ δe -Ui(-A) I kmin (T f ) and then

´-A -∞ e -Ui(s) ds < δe -Ui(-A) . By Lemma 42 there exists such that for any i and t , T i f (t) < 1 I kmin (T i f ).

Suppose we have constructed every k up to k = k 0 , and let us construct k0+1 . Because of the induction hypothesis there exist b k0 such that sup t f (t) ≤ b k0 I k0 (f ) so there exists b k0 such that

I k0 (T i f ) ≤ b k0 I k0 (f ). Moreover ∀a ∈ I k0+1 , T i f (a) = ˆ∞ a-xi+1-xi f (s)e -Ui(s) ds ≥ ˆIk 0 f (s)e -max s∈I k 0 Ui(s) ds ≥ e -max s∈I k 0 Ui(s) I k0 (f ).
So thanks to Lemma 42, there exists k0+1 such that if for all k ≤ k 0 , I k (f ) ≤

1 k+1 I k+1 (f ) then for all k ≤ k 0 + 1, I k (T i f ) < 1 k+1 I k0 (T i f ).
We also have for any a ≤ A

T i f (a) = ˆ∞ a-xi+1-xi f (s)e -U (s) ds ≥ ˆA A-δ f (s)e -Ui(s) ds - ˆ∞ A e -Ui(s) ds • I kmax (f ) ≥ e -Ui(A) (1 - δ 2 )I kmax (f ).
In particular T i f (a) ≥ 0. Moreover for any a > A, we have

T i f (a) ≥ - ˆ∞ A e -Ui(s) ds • I kmax (f ) ≥ - δ 2 e -Ui(A) I kmax (f ) ≥ - 1 1 -δ 2 I kmax (Kf ). Because f ≥ 0 on (-∞, A], T f is decreasing on ]-∞, A+δ].
To conclude, it will be enough to compare T i f with T i g for f, g ∈ C and I kmax (f ) = I kmax (g) = 1. Because all the inequalities become strict, there exists such that for any f ∈ C with

I kmax (f ) = 1, if g L ∞ ≤ then T i (f -g) ∈ C. Moreover for any g ∈ C with I kmax (g) = 1, g L ∞ ≤ 1 k . So T i (f - k g) ∈ C
. And this concludes the proof because then ∆ ≤ 2 log( k ).

Remark 43. If we denote by H k the assertion "I k-1 (f ) ≤ 1 k I k (f )", we have actually proved that if f satisfies all the condition of C except (H i ) i=r,. . . ,kmax , then T i f satisfies all the condition of C except (H i ) i=r+1,. . . ,kmax . This implies that if f ≥ 0 and supp(f ) ⊂ [-A, A], then k+n i=k T i f ∈ C is as wanted.

Decay of correlation.

Here we prove Theorem 4.

We can carry on with the construction of conditions like

I k-1 (f ) ≤ 1 k I k (f ) after k max in Proposition 41. We denote by C m this more specified cone replacing ∀k ∈ [-2 A δ , 2 A δ ]I k-1 (f ) ≤ 1 k I k (f ) by ∀k ∈ [-2 A δ , 2 A δ + m]I k-1 (f ) ≤ 1 k I k (f ). We have then the Proposition 44. If g ∈ C then k+n i=k T i g ∈ C i . Proof. By Proposition 41, k+n i=k T i g ∈ C.
Therefore it is enough to prove the remaining conditions. Let g i ∈ C i . As previously,

T i g(a) ≥ - δ 2 e -Ui(A+δi) I kmax (f ) ≥ - 1 1 -δ 2 I kmax+i (T f ).
As a consequence we have that for f ≥ 0 and supp(f

) ∈ (-∞, A + nδ] then f k+n i=k T i g ≥ 0 for all g ∈ C. Remark 45. If f ≥ 0 and supp(f ) ∈ [A, ∞) then T i f ∈ C.Therefore for f , supp(f ) ∈ [a -
T i f ≤ e -n(| a δ | 2 -c) 1, f 1, k+n i=k T i 1 .

By iteration supp(

i T f ) ∈ [a -n max(ã i -ãi+1 ), ∞). Therefore T i f ∞ ≤ 1, f e -mini(Ui(a-k(max(ãi-ãi+1))))
= 1, f e -γmin (a-k(max(ãi-ãi+1)) 2 .

Proof. There exists λ > 0 such that 1,

k+n-1 i=k

T i 1 ≥ λ n , and we set c = log λ. The result follows.

We are now ready to prove the decay of the correlation functions. Recall that for i 1 , i 2 , • • • , i k , we have the k-th marginal defined by

ρ k (x i1 , x i2 , . . . , x i k ) = 1 Z N (β) e -βE(x1,. . . ,x N ) ˆ• • • ˆ-L<x1<x2<. . . <x N <L e -2βqi ´xi xi ρ(y)(y-xi)dy i =i1,i2,...,i N dx i .
Note that there exists r such that for ρ k (x i1 , x i2 , . . . , x i k ) ≤ e -r max |x| 3 .

Corollary 47. There exists κ < 1 and C k > 0 such that n) be an approximation of the Dirac δ 0 . We evaluate

|ρ k (x i1 , x i2 , . . . , x i k ) - k l=1 ρ 1 (x i l )| ≤ C k κ inf |i l -i l+1 | . Proof. Let x i1 , • • • , x i k ∈ R k and let δ (
˘[ρ k (y i1 , y i2 , . . . , y i k ) - ρ 1 (y i1 )] δ (n) (y i k -x i k )dy i k = ˘[ρ k (y i1 , y i2 , . . . , y i k ) - ρ 1 (y i1 )] δ (n) xi k (y i k )dy i k
which, in our formalism, is equal to

1 Z N (β) (u, T N . . . T K k +1 δ (n) x k T K k . . . T K1+1 δ (n) 1 T K1 . . . T 0 v) - 1 Z (u, T N . . . T K1+1 δ (n) x k T K1 . . . T 0 v).
To begin with, assume

-A < x 1 , • • • , x k < A. Because of Remark 45, for any i ∈ [1, k], T K k +m . . . T K k +1 δ (n) x k T K k is a positive operator. Changing C k , we can suppose inf(|i l -i l+1 |) > m. Therefore, denoting X l = T K l +m . . . T K l +1 δ (n)
x l T K l , we can apply Theorem 33 and we obtain :

1 Z N (β) (u, T N . . . T K k +1 δ (n) x k T K k . . . T K1+1 δ (n) 1 T K1 . . . T 0 v) - 1 Z (u, T N . . . T K1+1 δ (n) x k T K1 . . . T 0 v) ≤ C k κ inf(i l -i l+1 ) ,
where

C k = 2k(2k + 1)Rκ -m if inf(i l -i l+1 ) is larger that a constant c. Suppose that there exist |x i | > inf(|i l -i l+1 |δ) then ρ k (x i1 , x i2 , . . . , x i k ) ≤ e -r( δ) 2 inf l |i l -i l+1 | 2 and we are done. If for all i, |x i | ≤ inf(|i l -i l+1 |δ), then as previously T K k +m . . . T K k +1 δ (n) x k T K k is positive for m = inf |i l -i l+1 |, hence | 1 Z N (β) (u, T N . . . T K k +1 δ (n) x k T K k . . . T K1+1 δ (n) 1 T K1 . . . T 0 v) - 1 Z (u, T N . . . T K1+1 δ (n) x k T K1 . . . T 0 v)| ≤ C k κ (1-) inf(i l -i l+1 )
with C k = 2k(2k + 1)R. We can conclude replacing κ by κ (1-) .

N (∂ n β T i (β)u 0 ) < ∞.
To simplify the calculation, we introduce an approximating norm.We define

a 1 (s) = sup t≥A s(t) + I kmax (s) ν , a 2 (s) = sup t≤A s(t) ν , a 3 (s) = sup -A≤t≤A s (t) ν , a 4 (s) = sup t≤-A s(t) -s(-A) ν , a 5 (s) = sup k 1 k I k (s) -I k-1 (s) ν , a 6 (s) = sup t≤-A s(t) -1 I kmin (s) ν ,
and we set A ν (s) = max(a 1 (s), a 2 (s), a 3 (s), a 4 (s), a 5 (s), a 6 (s)). Finally we define s ν = max(A(-s), A(s)).

Proposition 49. For u 0 in T i (C), there exists ν > 0 such that . i+1 ≤ . ν .

Proof. Let u 0 ∈ T i (C) We calculate the norm of Proposition 22. Let s be such that Definition 50. For any f we define

αu 0 ≤ u 0 + s ≤ βu 0 with α = max α                    ∀t ≥ A s(t) + I kmax (s) ≥ (α -1)[u 0 (t) + I kmax (t)], ∀t ≤ A s(t) ≥ (α -1)u 0 (t), on -A ≤ t ≤ A s (t) ≤ (α -1)u 0 (t), ∀t ≤ -A s(t) -s(-A) ≥ (α -1)[u 0 (t) -u 0 (-A)], ∀k ∈ [-2 A δ , 2 A δ ] I k-1 (s) -1 k I k (s) ≤ (1 -α)[I k-1 (u 0 ) -1 k I k (u 0 )], ∀t ≤ -A s(t) -1 I kmin (f ) ≤ (1 -α)[u 0 (t) -1 I kmin (u 0 )].
T i f (γ) = ˆE 1 ∀t,γ(t)<η(t)+δi f (η)ν i (dη),
where ν i = 1 c ´R ν i,xx dx with Radon Nikodym density dνi,xx dµxx (γ) = e -´β 0 Ui(γ(t))dt . We recall the result concerning the homogeneous case.

Theorem 51. T i is a compact operator on L 1 ((1 + x 2 ) -1 dx) with a unique largest eigenvalue λ M > 0.

For the reader's convenience, we have written again the proof.

Proof. T i is a compact operator. Indeed let u ∈ L 1 ((1 + x 2 ) -1 dx), then T u is bounded, with finite variation. T is Hilbert-Smicht: ¨E×E e -2 ´U (γ(t))dt 1 ∀t,γ(t)≤η(t)+δi dν(γ)dν(η) < ∞.
We have to check that ν i are bounded measures

P(|B t | > y) ≤ 2e -|y| 2 . Then ¨R×Γ e -´Ui(γ)dt dµ xx (γ)dx ≤ ˆR[µ xx (inf t γ(t) < x 2 )] + e -βUi( x 2 ) dx ≤ ˆR c[e -( x 2 ) 2 + e -βU ( x 2 ) ]dx < ∞
The largest eigenvalue is unique because the operator is irreducible and we can apply the Krein Rutmann Theorem.

Theorem 52. Let T be a bounded real operator whose spectral radius ρ(T ) is a non degenerate eigenvalue with eigenvector u 0 . Assume in addition that T = T -ρ(T )u 0 u * 0 has a spectral radius ρ(T ) < ρ(T ). Then there exists a cone such that the operator is contracting.

Proof. We can suppose that the largest eigenvalue is 1 and let u 0 be its eigenvector. We construct

C = positive linear combinations of ∪ n T n B(u 0 , (1 -n ))
with n a strictly decreasing sequence. Because there exists N such that (T -

u 0 u * 0 ) n is contracting for n ≥ N , C = positive linear combinations of ∪ n≤N1 T n B(u 0 , (1 -n )) .
Indeed, let x ∈ B(u 0 , (1 -N1+1 )), x = u 0 + y + su 0 with u * 0 y = 0 and y ≤ c and s ≤ c . There exists

N 1 such that (T -u 0 u * 0 ) N1+1 ≤ (1-0) 2c . Then T N1+1 (x) = (u 0 u * 0 )x + (T -u 0 u * 0 ) N1+1 (x) = (1 + s)u 0 + (T -u 0 u * 0 ) N1+1 (y) ∈ B((1 + s)u 0 , (1 + s) (1 -0 )), because (T -u 0 u * 0 ) N1+1 (y) ≤ c(1-0) 2c ≤ (1 + s) (1 -0 ). Let x ∈ C, with u * 0 x = 1. Then x = N1 i=0 a i x i , a i ≥ 0 with x i ∈ T i (B(u 0 , (1 -i ))
). Let us construct α and β such that αu 0 ≤ T (x) ≤ βu 0 .

First because B(u 0 , (1-0 )) ⊂ C, we choose β ≤ T (x) (1-0 ) , and we immediatly have u 0 -1 β T (x) ≥ 0. Second for all i, T (x i ) ∈ T i+1 (B(u 0 , (1i ))) and therefore T ( (1-i+1)

(1-i) x i ) ∈ T i+1 (B( (1-i+1)

(1-i) u 0 , (1 -i+1 ))). We have then

T ( (1 -i+1 ) (1 -i ) x i ) -(1 - (1 -i+1 ) (1 -i ) )u 0 ∈ T i+1 (B(u 0 , (1 -i+1 )))
and also

T ( (1 -i+1 ) (1 -i ) x i ) ≥ (1 - (1 -i+1 ) (1 -i ) )u 0 .
So with M = max (1-i+1) (1-i) and m = min (1-i+1)

(1-i) we have M T (x) ≥ m( a i )u 0 and we can conclude that m a i M u 0 ≤ T (x).

Such a construction is stable under small compact perturbations.

Proposition 53. There exists δ 0 > 0 such that for δT compact operator with δT ≤ δ 0 , ∆ C ((T + δT

)(C)) < 2∆ C (T (C))
where C is the cone constructed in Theorem 52 In particular T + δT is a positive contracting operator for d C .

Proof. We rewrite the proof of Theorem 52. We keep u 0 because (T +δT )(u 0 ) ∈ B(u 0 , ( (1 -0 ))). First, it is enough to change β ≤ (T +δT )(x)

(1-0 )

and we have u 0 -1 β (T + δT )(x) ≥ 0. Second, we also have

(T + δT )( (1 -i+1 ) (1 -i ) x i ) -δT ( (1 -i+1 ) (1 -i ) x i ) ≥ (T + δT )( (1 -i+1 ) (1 -i ) x i ) -u 0 δT ( (1-i+1) (1-i) x i ) (1 -0 ) ≥ [(1 - (1 -i+1 ) (1 -i ) ) - δ 0 (1 -0 )
]u 0 and we can finish the proof as previously for δ 0 small enough.

Remark 54. Actually we have that T (x) ∈ C for all x ∈ C. Indeed, as previously

m a i M u 0 ≤ T (x)
and then for any y ≤ (1-0) M m a i we have T (x) + y ≥ m ai M u 0 + y ≥ 0.

The construction of the cone is simple enough that we can calculate the norm of Theorem 22. Because of the previous remark it is equivalent to the space norm.

Proposition 55. There exists c > 0 such that for any y in the projected space, c y ≤ y N ≤ 1 c y

where N is the norm constructed in Proposition 22 for the cone C in a neighborhood of T (x).

Proof. Because of the previous remark, T (x) ∈ C. Therefore there exists r > 0 such that B(T (x), r) ⊂ C. For any y we have T (x) 1 -s y r ≤ T (x) + sy ≤ T (x) 1 + s y r and we obtain d C (T (x) + sy, T (x)) ≤ 2 s y r + o( s y r ). For the other direction, C ⊂ cone from B(u 0 , 1 2 ) (for small). In addition T (x) + y / ∈ C for y = 1 and then αT (x) ≤ T (x) + sy implies α ≤ (1 -s) and so y N ≤ y .

We can now finish the proof of Theorem 10

Proof. [Theorem 10] T (β) is C ∞ for the norm and thanks to the previous proposition also for any norm constructed | N around T (β)(x). We can then apply Theorem 23. The analyticity follows as well: if T (β) is analytic with coefficient bounded by r n for the norm then the coefficients are bounded by cr n for the norm N .

We focus now on the decay of correlation and the proof of Theorem 9. Recall that we want to prove that there exists κ < 1 such that |ρ k (x i1 , x i2 , . . . , x i k ) -

ρ 1 (x i1 )| ≤ C k κ min |i l -i l+1 |
Proof. [Theorem 9] There exists 0 > 0 such that for f with f L 1 ≤ 0 , T i+1 (1+ f )T i are positive operator for the cone C. Indeed, let u ∈ C, then T i (u) ∈ L ∞ (Γ), therefore f T i (u) ∈ L 1 with f T i (u) L 1 ≤ cδ 0 u , and finally T i+1 f T i (u) L ∞ ≤ c δ u . Because T (C) is compact on the projected space and thanks to Remark 54, there exists r > 0 such that for all i, u ∈ C, B(T i+1 T i (u), r u ) ⊂ C. As a conclusion T i+1 (1 + f )T i (u) = T i+1 T i (u) + T i+1 f T i (u) ∈ C. We can now apply Theorem 33 with the same notation. We find The rest follows from a induction on k and this concludes the proof of Theorem 9.

e -2(k+1)R( k j=0 κ K j+1 -K j ) k i=1 ρ Ki (T li+1 (1 + f i )T li ) ≤ ρ K k ,.

A Proof of Proposition 3

If I = {i 1 , i 2 }, then x i1 and x i2 are independent, ρ 2 (x i1 , x i2 ) = ρ(x i1 )ρ(x i2 ) and then ρ T (x i1 , x i2 ) = 0. For larger a I , we have

3 .Example 27 .

 327 Let A ∈ P ∩ -P, then (f, Ax) = 0 for all x ∈ C and all f ∈ C * . Therefore(f 1 -f 2 , A(x 1 -x 2 )) = 0 for all x 1 , x 2 ∈ C and all f 1 , f 2 ∈ C * . Therefore A = 0 since C * -C *and C -C are dense. One can think of C the positive vectors in R n and the set of matrices M n (R) with positive coefficients.

  d P (A, B) = log( 2 (1/2) ) = log(4) and d P (T A, T B) = log(4) as well. Lemma 29. Let A, B, C, D be increasing operators. Then d P (AB, CD) ≤ d P (A, C) + d P (B, D).

Lemma 30 .

 30 Let a : C → R + and b : C → R + be such that 1. there exist M 1 , M 2 < ∞, for all x, a(x) ≤ M 1 x and b(x) ≤ M 2 x , 2. for all λ ≥ 0 and x ∈ C, a(λx) = λa(x) and b(λx) = λb(x), 3. for all x ∈ C a(x) ≤ b(x), 4. for all x, y ∈ C a(x + y) ≥ a(x) + a(y) and b(x + y) ≤ b(x) + b(y).

Corollary 31 .

 31 There exists a rank one operator L T = z • l with z ∈ C and l ∈ C * such that d P (T, L T ) ≤ 2∆(T ). Proof. Let z = T (y 0 ) ∈ T (C) and define a and b as follows: for any x ∈ C a(x) = def max {α : αT (y 0 ) ≤ C T (x)} and b(x) = def min {β : T (x) ≤ C βT (y 0 )}.

Example 35 . 1 ,

 351 Consider T (β) = β with largest eigenvalue λ(β) = (β + 1) + (β -1) 2 + 4 2 2

Aδ∈

  N to simplify the notation. Proposition 41. There exist ( k ) -2 A δ ≤k≤2 A δ such that the cone C defined by

  dl, a -dl] dl ≤ δ and A ≤ a ≤ A + nδ then T n+k+1 F k+n i=k T i is order preserving for the cone C. Proposition 46. For f ≥ 0 with supp(f ) ∈ [a-dl, a-dl] dl ≤ δ and a ≥ A+nδ,

4. 1 . 3

 13 Smoothness of the free energy for the classical Jellium model. Now we use Theorem 23 to prove the smoothness of the free energy. We first have to check its hypothesis. This is the aim of the following propositionProposition 48. Let u 0 ∈ T i (C). Then [∂ n β T i (β)] Ni→Ni+1 < ∞ for all n,where N i and N i+1 are the norm constructed in Proposition 22 around u 0 and T i (β)u 0 respectively, and

4. 2

 2 Proof for the quantum Jellium model 4.2.1 Decay of correlations, smoothness of the free energy.

ρ

  ..,K1 (T l k +1 (1 + f k )T l k , ..., T l1 (1 + f 1 )T l1 ) and ρ K k ,...,K1 (T l k +1 (1 + f k )T l k , ..., T l1 (1 + f 1 )T l1 ) ≤ e 2(k+1)R( k j=0 κ K j+1 -K j ) k i=1 Ki (T li+1 (1 + f i )T li ).

  |I l | ((x i ) i∈I l ) = I1∪I2∪...∪Ir={1,...,n} I l ⊂I or I l ⊂J. r l=1 ρ T |I l | ((x i ) i∈I l ) |I l | ((x i ) i∈I l ) |I l | ((x i ) i∈I l )   = ρ |I ∩I| ((x i ) i∈I∩I )ρ |I ∩J| ((x i ) i∈J∩I )and thereforeρ T |I | ((x i ) i∈I ) = ρ |I ∩I| ((x i ) i∈I∩I )ρ |I ∩J| ((x i ) i∈J∩I ) -I1∪I2∪...∪Ir=I , r l=1 ρ T |I l | ((x i ) i∈I l ) = 0.

Because u 0 ∈ T i (C), there exists 0 > 0 such that all the functions depending on u 0 on the right side of the equation ([u 0 (t) + I kmax (t)], u 0 (t), • • • ) can be bounded by 0 . We obtain

and we then have α ≤ α. We have constructed then ν with ν = 0 .

We finish the proof of Proposition 48.

Proof. We can now calculate ∂ n β T i (β) i→ ν . Let w with w i ≤ 1, In particular, there exists r > 0 such that d(u 0 , u 0 + r.w) ≤ 2r. Therefore

for all t < A. Hence |I kmax (r.w)| ≤ 2r.I kmax (u 0 ) and for all t > A:

U i (y) n e -βUi(y) w(y)dy and there exist c 1 , c 2 and c 3 such that

is a uniformly bounded operator for i → ν and then for i → i+1 .We can now conclude the proof of Theorem 5. Moreover,

We can then apply Theorem 24 which ends the proof of analyticity of the free energy of the classical Jellium model.


A.-L Perrier

Emmanuel Chereul

Denis Grenier

Fabrice Taborik

Mariam Abdallah

Thomas Chuzel

Stéphane Martin

Luc Magnier

Jean-Christophe Christophe Goebel

Xavier Pesesse

Anne-Laure Perrier

Sandra Pietri

Hugues Contamin

Olivier Beuf

Morphological assessment of non-human primate models of osteoarthritis: Comparisons of HR-MRI

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Many models of induced osteoarthritis (OA) are described in the literature, most of them in small animals. The limitations of these models, in adequately mimicking a condition as complex as human OA, need to be fully considered. Indeed, these models are rather ineffective in evaluating new treatments and especially in transposing observations to human disease considering the large genetic, phenotypic and pathophysiological heterogeneity between human and rodent. OA that closely resembles the human condition occurs naturally in primate. The histological lesions of OA in human and monkey knee joints include articular cartilage fibrillation and clefting. These lesions coincide with an increase in the thickness of the subchondral bone plate in both species and it has been suggested that this thickening may precede cartilage breakdown. These similarities make that these animals are useful to model the human disease. Non-invasive techniques to measure and qualify in-vivo the cartilage thickness in animal model of OA has been developed (1) and have greatly improve existing strategy to assess OA treatments and cartilage repair strategies. Established reproducibility confirmed that 3D HR-MRI could directly assess the cartilage thickness on guinea pigs (2) and recent instrumental developments demonstrated that volume quantification in the different compartments of the cartilage can be achieve on rat models of OA (3)(4). Nonetheless, spatial resolution is limited compared to CT scanner. The aim of this work was (i) to develop a dedicated protocol for knee joint examination of cynomolgus primates at 1.5T and with µ-CT arthroscanner (µCTA); (ii) to compare morphological parameters assessed based on MRI and µCTA acquisitions on a group of 10 old primates with spontaneous OA.

Material and Methods

The MRI experiments were performed on a 1.5T Siemens Sonata system. A pair of two-channel array coil was built on a thermoformable plastic support with about 32 mm outer diameter. Each element consists in a rectangular loop (30 x 35 mm 2 internal dimensions with 5 mm width and 35µm thickness copper track) etched on a flexible 508µm thick substrate (Rogers TM RT/Duroid 5880). The decoupling between the two channels was achieved with optimal coil overlapping to minimize coupling between the two elements (minimize |S 12 | parameter). Each dual-channels array coil was interfaced with a flex interface from Siemens and a coil configuration file was created to drive the interface in array mode. The designed phased array coil were compared to the Siemens 40mm diameter small loop coil. Experimental characterization (signal uniformity and SNR) was performed on cylindrical phantoms filled with 1,25g/L of NiSO 4 with 5g/L of NaCl solution. The ethical guidelines for experimental investigations with animals were followed, and the experimental protocol was approved by the Animal Ethics Committee of our institution. Ten female primates between 12 and 18 years old (mean 13.8±1.8) were examined. The primates were placed in supine position with both dual array coils was placed on top of patella to encompass the whole knee joint. A minimum distance of 100 mm between both knees was keep to insure at least 20dB decoupling between internal coil elements located at medial sides. HR-MRI was performed in the sagittal plane using a 3D water excitation FLASH sequence with 25° flip angle, 27 ms TR, 11.7 ms TE, 70 Hz/Pixel receiver bandwidth. A total of 120 partitions (220 µm thick) were acquired with a FOV of 50 x 50 mm 2 and an acquisition matrix size of 448 x 381 leading to an in-plane pixel of 112 x 131 µm 2 . The scan time was 20 min. µCTAs were performed on a GE Locus µ-CT at standard voltage and amperage parameters with an isotropic resolution of 90µm. The scan time was 15 min. For each animal, both knees were sequentially scanned. 3D thicknesses of the tibial plateau cartilage layers were assessed both on lateral and medial sides of the knee by using the same image processing protocol for each kind of acquisitions (MRI and µCTA). This protocol consisted in a double segmentation procedure: a first rough and manually handled contour segmentation to isolate the cartilage regions of interest (ROI) and avoid any divergence of the second region automatic global segmentation procedure which accurately extracts the morphology of both medial and lateral cartilage ROIs. Parameters of the second segmentation procedure were adapted for MRI or µCTA acquisitions. Inside the cartilage ROIs, the quantification of cartilage thicknesses was performed following the method previously described (5). Morphological results were compared in terms of distribution of the 3D thickness parameters obtained for each cartilage compartment. This analysis gives access to as many parameters as voxels describing the cartilage tissue. The spread of the thickness distribution with maximum-values at a given thickness characterizes the cartilage compartment.

Results

The SNR gain for the shaped two-channel array coil was 4.4 compared to the 40 mm diameter surface coil with 16% signal uniformity in the region of interest. In vivo images acquired with the array coil associated with the HR-MRI protocol nicely depicted the cartilage (Fig. 1). Such acquisitions were suitable to apply the segmentation procedure leading to articular cartilage volumes and thickness distributions. Simultaneous HR-MRI of both joints could not be reliably performed due to the lack of reproducibility in the magnetic field uniformity with two knees separated with air. In the examined group of old female primates, a coherent description with both imaging modalities was observed with superimpose 3D thickness distributions measured on the same animals. The presence of spontaneous OA was established (narrower thickness distribution) among the animals. For instance, the mean maximum thicknesses in the tibial plateaus without OA lesions were found to be 1.53 mm for the lateral compartment and 1.98 mm for the medial compartment. Furthermore, mean thickness distributions were found different in the lateral and medial compartments with the most common values determined at 0.81 mm and 1.89 mm respectively.

Conclusion

The developed two-channel phased array coil improved the SNR and has an acceptable signal uniformity allowing a relatively straightforward segmentation process and quantification of cartilage morphology (thickness, volume). Both imaging approaches gave similar normalized cartilage thickness distributions on the same animals. These coherent results showed that both imaging modalities are valuable to measure cartilage morphology (volume and thickness) and could be chosen depending on additional information requested e.g. indirect cartilage structure (T2, T1rho…) for MRI or subchondral bone density for µCTA.

Fig. 1 :

 1 Fig. 1: HR-MR image (lateral side) acquired on the right knee of a 13 years old primate female.

Acknowledgements

This work is funded by OSEO by grant E! 5671 from the Eurostars European program.