C Dombry
email: dombry@univ-lyon1.fr

N Guillotin-Plantard

The Curie-Weiss model with dynamical external field

Keywords: Curie-Weiss model, dynamic random walk, dynamical system, ergodic theory, diophantine approximations, limit theorems, large deviation principle, statistical physics AMS Subject classification: Primary: 60K35, secondary: 82B41, 82B44, 60G50

.

Introduction

The Curie-Weiss model is a well-known approximation to the Ising model (see [START_REF] Ellis | Large Deviations and Statistical Mechanics[END_REF]). Probabilistic limit theorems for the Curie-Weiss model have been proven by the following authors: Ellis and Newman [START_REF] Ellis | Limit theorems for sums of dependent random variables occuring in statistical mechanics[END_REF], Ellis, Newman and Rosen [START_REF] Ellis | Limit theorems for sums of dependent random variables occuring in statistical mechanics[END_REF]... The purpose of the present paper is to prove limit theorems for the Curie-Weiss model with random external field generated by a dynamical 1 system, namely weak law of large numbers and central limit theorems for the associated magnetization. Our main motivation is the understanding of the statistical properties of the following physical model: consider n particles (iron atoms for instance) distributed along a lattice Γ = {1, . . . , n}. The value of ±1 at a site represents the spin, or magnetic moment, of the particle at that site. The particles are placed in a magnetic field which is given in terms of a dynamical system S = (E, A, µ, T ), that is a probability space (E, A, µ), T a transformation of E and a function f defined on E with values in [0, 1]. Let β > 0 be the inverse temperature and J a coupling constant assumed strictly positive. Given a configuration σ = (σ i ) i=1...,n ∈ Ω n = {-1, +1} n and x ∈ E, we define the Hamiltonian,

H n,x (σ) = βJ 2n n i=1 σ i 2 + 1 2 n i=1 log f (T i x) 1 -f (T i x) σ i .
We denote by Q n,x the Gibbs measure on Ω n defined by

Q n,x (σ) = 1 Z n,x exp[H n,x (σ)]
where Z n,x is the normalizing constant, called partition function

Z n,x = σ∈Ωn exp[H n,x (σ)].
Equivalently, Q n,x is the probability measure defined on (Ω n , P(Ω n )) such that, for any A ∈ P(Ω n ),

Q n,x (A) = 1 Zn,x A exp[H n,x (σ)] dP n (σ)
where P n is the uniform distribution on Ω n (i.e. for any σ ∈ Ω n , P n (σ) = 1 2 n ) and Zn,x is the partition function Zn,x = Ωn exp[H n,x (σ)] dP n (σ).

For each configuration σ = (σ i ) i=1,...,n we define the associated magnetization (or total spin)

M n = n i=1 σ i .
Remark that when f ≡ 1/2 our model corresponds to the Curie-Weiss model without external field studied in [START_REF] Ellis | Large Deviations and Statistical Mechanics[END_REF][START_REF] Ellis | Limit theorems for sums of dependent random variables occuring in statistical mechanics[END_REF][START_REF] Ellis | Limit theorems for sums of dependent random variables occuring in statistical mechanics[END_REF]. Moreover it is worth remarking that any field (g(T i x)) i≥1 can be considered by choosing the function f as e g /(1+e g ). In particular, it includes the case where the field is given in terms of a sequence of independent and identically Bernoulli random variables taking the values -ε and ε with probability 1/2 considered in [START_REF] Külske | Metastates in disordered mean field models: random field and Hopfield models[END_REF] and [START_REF] Fontes | On the averaged dynamics of the random field Curie-Weiss model[END_REF]. We will be mostly interested in the special case when the dynamical system is the irrational rotation on the torus which corresponds to a quasiperiodic random field; we refer to [START_REF] Koukiou | Extension of the Pirogov-Sinai theory to a class of quasiperiodic interactions[END_REF] for a complete and precise discussion about the relevance of this model in the modelization of certain physical models. We are interested in studying the asymptotic behaviour of M n in the socalled thermodynamical limit n → +∞. In [START_REF] Ellis | Limit theorems for sums of dependent random variables occuring in statistical mechanics[END_REF] a physical interpretation of this limit behaviour is given in relation with stable states (mixed or pure) and metastable states of the underlying physical system. An illustrative example derived from thermodynamics is given, namely a detailed description of states as well as the phase transition in a gas-liquid system. At infinite temperature (i.e. β = 0) the probability measure Q n,x is equal to the product measure

n i=1 f (T i x)δ 1 + (1 -f (T i x))δ -1
This implies that the random variable M n is just a sum of independent random variables σ i taking the value 1 with probability f (T i x) and -1 with probability 1f (T i x). So, in this particular case, the sequence of random variables (M n ) n≥1 is a so-called dynamic Z-random walk (see Section 2). The dynamic Z-random walks were introduced by the second author in [START_REF] Guillotin | Asymptotics of a dynamic random walk in a random scenery I. A law of large numbers[END_REF], then generalized to dimension d > 1 in [START_REF] Guillotin-Plantard | Dynamic Z d -random walks in a random scenery: a strong law of large numbers[END_REF]. Theoretical results about dynamic random walks and their applications can be found in the recent book [START_REF] Guillotin-Plantard | Dynamic random walks: Theory and applications[END_REF]. We are mainly interested in limit theorems (i.e. strong law of large numbers, central limit theorem and large deviation principle) for dynamic Z-random walks. We recall some of them here under simplified assumptions: assume that E is a compact metric space, A the associated Borel σ-field, T a continuous transformation of E. If there exists an unique invariant measure µ i.e. (E, A, µ, T ) is uniquely ergodic and if f is continuous with integral equal to 1/2, then for every x ∈ E, (M n /n) n≥1 converges almost surely to 0 as n goes to infinity. Moreover, if we assume that a

= E 4f (1 -f ) dµ > 0 and that n i=1 [f (T i x) -1/2] = o( √ n) (1)
then the sequence (M n / √ n) n≥1 converges in distribution as n → +∞ to the Normal distribution N (0, a). Let us recall that an important feature of certain Ising model is the existence of a critical value β c of β: For 0 < β < β c , the spins are weakly correlated and the probabilistic limit theorems obtained at β = 0 are valid. For β > β c , the correlation between the spins is strongly positive and the limit results are completely different. The model is then said to present a phase transition at β = β c . We will prove for our model that the limit theorems obtained at β = 0 for the dynamic random walk are still valid for any β < β c under the same hypotheses. The critical value β c is shown to fluctuate between 1/J and 1/(Ja) according to the dynamical system and the function f we consider. For an explicit class of dynamical systems and functions f , we are able to prove that β c is equal to 1/(Ja) and that at β = β c , under suitable assumptions on f , there exists some γ ∈ (0, 1) so that as n → +∞, (M n /n γ ) n converges in distribution to an explicit non Gaussian random variable. For β > β c , the situation is not so well understood. Let us recall that for the Curie-Weiss model with zero external field (e.g. f ≡ 1/2), the sequence (M n /n) n converges in distribution to 1/2(δ m + δ -m ) where m > 0 (see for instance Theorem IV.4.1 in [START_REF] Ellis | Large Deviations and Statistical Mechanics[END_REF]). In Section 4 we show that if the function f is not identically equal to 1/2, when β > β c , the sequence (M n /n) n does not converge in distribution.

We conjecture that Theorem 1 in [START_REF] Külske | Metastates in disordered mean field models: random field and Hopfield models[END_REF] should be true for our model, under suitable assumptions; this work is in progress.

In [START_REF] Koukiou | Extension of the Pirogov-Sinai theory to a class of quasiperiodic interactions[END_REF], the authors extended the Pirogov-Sinai theory to a class of models with small quasiperiodic interactions as perturbations of the periodic ones. More precisely, the low temperature phase diagram for spin systems with periodic hamiltonians perturbated by quasiperiodic interactions is studied. Under diophantine conditions and derivability conditions on the interaction potentials they prove that the low temperature phase diagram is a homeomorphic deformation of the phase diagram at zero temperature. Our model in the case when the dynamical system S is an irrational rotation on the torus belongs to this class of hamiltonians perturbated by quasiperiodic ones. In Section 5 the same kind of conditions on the diophantine approximation of the irrational angle and on the smoothness of the function f will be needed in order to state the limit theorems.

The outline of the paper is as follows: In Section 2, we define the dynamic Z-random walk and recall some results which will be useful in the sequel. In Section 3, we state and prove our results under general assumptions. In Section 4, we apply results of Section 3 when the integral of f is equal to 1/2 and a > 0. In Section 5, we study the case when the dynamical system is given by an irrational rotation on the torus.

Dynamic Z-random walks

The dynamic random walks were introduced in [START_REF] Guillotin | Asymptotics of a dynamic random walk in a random scenery I. A law of large numbers[END_REF] and generalized to upper dimensions in [START_REF] Guillotin-Plantard | Dynamic Z d -random walks in a random scenery: a strong law of large numbers[END_REF]. We now recall some of the results obtained in dimension one. Let S = (E, A, µ, T ) be a dynamical system where (E, A, µ) is a probability space and T is a measure-preserving transformation defined on E. Let f be a measurable function defined on E with values in [0, 1]. For each x ∈ E, we denote by P x the distribution of the time-inhomogeneous random walk:

S 0 = 0, S n = n i=1 X i for n ≥ 1 with step distribution P x (X i = z) =    f (T i x) if z = 1 1 -f (T i x) if z = -1 0 otherwise. (2)
The expectation with respect to P x will be denoted by E x . It is worth remarking that if the function f is not constant, (S n ) n∈N is a non-homogeneous Markov chain. This Markov chain can be classified in the large class of random walks evolving in a random environment. In most of the papers (see for instance [START_REF] Greven | Large deviations for a random walk in random environment[END_REF], [START_REF] Comets | annealed and functional large deviations for one-dimensional random walk in random environment[END_REF],...), the environment field takes place in space but it can also take place in space and time (see [START_REF] Boldrighini | Almostsure central limit theorem for a Markov model of random walk in dynamical random environment[END_REF]). Following the formalism used in the study of these random walks, when x is fixed, the measure P x is called quenched and the measure averaged on values of x defined as P(.) = E P x (.) dµ(x) is called annealed. In the quenched case, the random variables X i , i ≥ 1 are independent, but not necessarily identically distributed. In the annealed case, the X ′ s defines a stationary sequence of dependent random variables whose the correlations are related to the ones of the underlying dynamical system. We refer to [START_REF] Guillotin-Plantard | Transient random walks on 2d-oriented lattices[END_REF] (Section 2.1) for a more precise discussion of these two cases. Let C 1/2 (S) denote the class of functions f ∈ L 1 (µ) satisfying the following condition: for every x ∈ E,

n i=1 f (T i x) - E f dµ = o √ n .
Let us assume that f ∈ C 1/2 (S); if E f dµ = 1 2 and a = E 4f (1f ) dµ > 0, then, for every x ∈ E, the sequence of random variables (S n / √ n) n≥1 converges in distribution to the Normal law N (0, a) (see [START_REF] Guillotin-Plantard | Distributed algorithms with dynamical random transitions[END_REF]). A strong law of large numbers for the dynamic Z-random walk can be obtained for µ-almost every point x ∈ E from Kolmogorov's theorem assuming that the function f is measurable (see Chapter 2 in [START_REF] Guillotin-Plantard | Dynamic random walks: Theory and applications[END_REF] for details). The limit is then given by 2 E(f |I) -

1
-inf x∈B o Λ * (x) ≤ lim inf n 1 v n log P(Y n ∈ B) ≤ lim sup n 1 v n log P(Y n ∈ B) ≤ -inf x∈ B Λ * (x).
Theorem 2.1.

1. For µ-almost every x ∈ E, the sequence (S n /n) n satisfies a LDP with speed n and good rate function

Λ * (y) = sup λ∈R {< λ, y > -Λ(λ)} where Λ(λ) = E log e λ f + (1 -f )e -λ I ,
I being the σ-field generated by the fixed points of the transformation T .

2. Assume that E is a compact metric space, A the associated Borel σ-field and T a continuous transformation of E. If (E, A, µ, T ) is uniquely ergodic (i.e. there exists an unique invariant measure µ) and if f is continuous, then 1. holds for every x ∈ E. The rate function is then deterministic and equal to

Λ(λ) = E log e λ f (x) + (1 -f (x))e -λ dµ(x).
Let us mention that an annealed large deviations statement for (S n /n) n under the measure P can easily be proved using results of [START_REF] Dinwoodie | Large deviations for exchangeable random vectors[END_REF] (Remark that E is assumed to be compact). The proof of the above theorem can be found in [START_REF] Dombry | Data Structures with Dynamical Random Transitions[END_REF].

Limit theorems for the magnetization

In this section, E is assumed to be a compact metric space, A the associated Borel σ-field, µ a probability measure on (E, A), T a continuous measurepreserving transformation of E, f a continuous function from E to [0, 1] and x a fixed point of E. The system (E, A, µ, T ) is asumed to be uniquely ergodic. In the sequel, the sequence (S n ) n will denote the dynamical random walk introduced in Section 2 and (M n ) n the magnetization defined in the introduction.

Weak law of large numbers for the magnetization

For every n ≥ 1, we define the function

G n (s) = βJ 2 s 2 - 1 n log E x (exp(βJsS n )) = βJ 2 s 2 - 1 n n i=1 L(f (T i x), βJs)
where the function

L is defined on [0, 1] × R by L(φ, s) = log φ e s + (1 -φ) e -s .
We also define the function G by

G(s) = βJ 2 s 2 - E L(f (y), βJs) dµ(y).
2k i = min{j ≥ 1 | G (j) (m i ) = 0}, λ i = G (2k i ) (m i ).
Let us remark that g is nonpositive since G(0) = 0.

For every α ∈ [0, 1], we define C α (S) the class of µ-integrable functions h : E → R satisfying the following condition: for every point x ∈ E,

n k=1 h(T k x) - E h dµ = o n α .
Remark that since the dynamical system is uniquely ergodic, the class C 1 (S) always contains the set of continuous functions on E. Theorem 3.2.

1. Assume that for every i ∈ {1, . . . , r} and every j ∈ {1, . . . , 2k i }, the function

E → R y → ∂ j ∂s j L(f (y), βJm i )
belongs to the set C j/2k i (S).

Then, for every bounded continuous function h, the expectation of h(M n /n) under Q n,x is equivalent, as n goes to infinity, to

r i=1 b i,n h(m i ) r i=1 b i,n where b i,n = n -1/2k i e -nGn(m i ) λ -1/2k i i +∞ -∞ exp(-s 2k i /(2k i )!) ds.
In particular, if G achieves its minimum at a unique point m, then the distribution of M n /n under Q n,x converges to δ m the Dirac mass at m.

2.

The distribution of M n /n under Q n,x verifies a large deviation principle with speed n and good rate function

I β,x (z) = Λ * (z) - βJ 2 z 2 -inf z∈R {Λ * (z) - βJ 2 z 2 }
where Λ * is defined in Theorem 2.1.

Scaling limit for the magnetization

Theorem 3.3. Assume that G has a unique global minimum m of type 2k and strength λ and that for every j ∈ {1, . . . , 2k}, the function ∂ j ∂s j L(f (.), βJm) belongs to the set C j/2k (S). Then, the following convergence of measures holds:

M n -nm n 1-1/2k ⇒ Z(2k, λ)
where Z(2k, λ) is the probability measure with density function

C exp -λs 2k /(2k)! ,
where C is a normalizing constant and λ is defined by

λ = 1 λ -1 βJ -1 if k = 1 λ if k ≥ 2 .
Remark: Note that the case of a minimum of type 2 yields a central limit theorem: the fluctuations of M n /n around m are of order n -1/2 and Gaussian. When the type of the minimum is greater than 4, the limit distributions are non standard.

Technical lemmas

Lemma 3.1. Let Y be a random variable with distribution N (0, 1/(βJ)), independent of M n for every n ≥ 1. Then, given m and γ real, the probability density function of the random variable

Y n 1/2-γ + M n -nm n 1-γ is equal to exp (-nG n (m + sn -γ )) R exp (-nG n (m + sn -γ )) ds
.

Proof:

The probability density function of the random variable

n 1/2 Y + M n is given by 1 Z n βJ 2πn e -βJ(s-x) 2 /(2n) e βJx 2 /2n dρ n (x) with ρn = ρ 1 * . . . * ρ n where ρ i = f (T i x)δ 1 + (1 -f (T i x))δ -1 .
Y n 1/2-γ + Mn-nm n 1-γ is given by 1 Z n βJ 2π n 1/2-γ e -βJn(m+sn -γ ) 2 /2 E x (e βJ(m+sn -γ )Sn )
The lemma is then easily deduced.

The previous lemma suggest that the behaviour of the sequence of random variables M n and of the sequence of functions G n are linked together.

Lemma 3.2. The sequence of functions (G n ) n≥1 converges to G uniformly on compacta of R as n goes to infinity. Furthermore, for every k ≥ 1, the sequence of derivative functions (G

(k) n ) n≥1 converges to G (k)
uniformly on compacta of R as n goes to infinity.

Proof: Note that the function L is of class C ∞ on [0, 1] × R, so that for any s ∈ R, the function y → ∂ k ∂s k L(f (y), βJs) is continuous. For any s ∈ R and k ≥ 0, G (k) n (s)-G (k) (s) = (βJ) k 1 n n i=1 ∂ k ∂s k L(f (T i x), βJs) - E ∂ k ∂s k L(f (y), βJs)dµ(y)
The unique ergodicity hypothesis implies that this quantity converges to 0 as n goes to infinity.

We prove the uniform convergence with the following majoration of the difference

|G (k) n (s) -G (k) (s)| on the compact [-α; α]: |G (k) n (s) -G (k) (s)| ≤ |G (k) n (0) -G (k) (0)| + α -α |G (k+1) n (s) -G (k+1) (s)| ds.
The unique ergodicity hypothesis implies that for any s ∈ R, |G

(s) -G (k+1) (s)| converges to zero as n goes to infinity. Furthermore, the function

∂ k ∂s k L is bounded on the compact set [0, 1] × [-βJα, βJα], and hence the difference |G (k+1) n (s) -G (k+1) (s)| is uniformly bounded on [-α, α]. Finally, by dominated convergence theorem, |G (k) n (s)-G (k) (s)| ≤ |G (k) n (0)-G (k) (0)|+ α -α |G (k+1) n (s)-G (k+1) (s)| ds → 0,
and the convergence is uniform for s ∈ [-α, α].

Lemma 3.3. Let m be a global minimum of G of type 2k and strength λ. Suppose that for every j ∈ {1, . . . , 2k}, the function y → ∂ j ∂s j L(f (y), βJm) belongs to the class C j 2k (S). Then, for every s ∈ R,

lim n→+∞ n G n m + sn -1/2k -G n (m) = λ s 2k (2k)! . (3)
Furthermore, there exist δ > 0 and N ≥ 1 such that for every n ≥ N and s ∈ [-δn 1/2k ; δn 1/2k ],

n G n m + sn -1/2k -G n (m) ≥ λ 2 s 2k (2k)! - 2k-1 j=1 |s| j . (4)

Proof:

Let s ∈ R and u = sn -1/2k . Taylor's formula implies that

G n (m + u) -G n (m) = 2k j=1 G (j) n (m) j! u j + R n (u),
where the remainder R n has the integral form

R n (u) = u 2k+1 (2k)! 1 0 (1 -θ) 2k G (2k+1) n (m + θu) dθ.
The j-th derivative of G n at point m is equal to

G (j) n (m) = P j (m) - (βJ) j n n i=1 ∂ j ∂s j L(f (T i x), βJm),
where

P j (m) =    βJm if j = 1 βJ if j = 2 0 otherwise .
As n goes to infinity, this quantity converges to

G (j) (m) = P j (m) -(βJ) j E ∂ j ∂s j L(f (y), βJm) dµ(y).
The hypothesis that the function y → ∂ j ∂s j L(f (y), βJm) belongs to the class C j 2k (S) implies that for every j ∈ {1, . . . , 2k},

n|G (j) n (m) -G (j) (m)| = o n j 2k .
Since the point m is a global minimum of G of type 2k and strength λ, G (j) (m) = 0 for every j ∈ {1, . . . , 2k -1} and G (2k) (m) = λ > 0. This implies that as n goes to infinity, for every j ∈ {1, . . . , 2k -1},

G (j) n (m)n 1-j 2k → 0, (5)
and that for j = 2k,

G (2k) n (m) → λ. (6)
The integral remainder satisfies

nR n (sn -1/2k ) = s 2k+1 n -1/2k (2k)! 1 0 (1 -θ) 2k G (2k+1) n (m + θsn -1/2k ) dθ.
The fact that the functions G (2k+1) n are uniformly bounded on the compact set [m -|s|, m + |s|] implies that the integral is bounded as n goes to infinity, so that nR n (sn -1/2k ) → 0.

Hence the formula

n G n m + sn -1/2k -G n (m) = 2k j=1 G (j)
n (m) j! n 1-j/2k s j + nR n (sn -1/2k ), yields the limit λs 2k /(2k)! as n goes to infinity and this proves equation (3). Let us prove (4), from ( 5) and ( 6), there exists N ≥ 1 such that for every n ≥ N , and every j ∈ {1, . . . , 2k -1},

G (j) n (m)n 1-j/2k j! ≤ 1
and, for j = 2k, G (2k) n (m) ≥ 3λ/4. There also exists δ > 0 such that for every n ≥ 1, and every s ∈ [-δn 1/2k ; δn 1/2k ],

sn -1/2k 1 0 (1 -θ) 2k G (2k+1) n (m + θsn -1/2k ) dθ ≤ λ/4,
which implies that for every n ≥ 1 and every s ∈ [-δn 1/2k ; δn 1/2k ],

nR n (sn -1/2k ) ≥ - λ 4
s 2k (2k)! .
(There exists M such that |G

(s)| ≤ M for every n ≥ 1 and every s ∈ [m -1, m + 1], then choose δ as the minimum of (2k + 1)λ/(4M ) and 1 ). This implies that for every n ≥ N and every s ∈ [-δn 1/2k ; δn 1/2k ], This implies that for any s such that |s| ≥ 3, G n (s) ≥ 3βJ 2 > g (since G(0) = 0, g is nonpositive). From Lemma 3.2, on the compact set W = V ∩ {s; |s| ≤ 3}, the sequence of functions G n converges uniformly to G. Hence, the sequence inf W G n (s) converges to inf W G(s) > g. Let

n G n m + sn -1/2k -G n (m) ≥ λ 2 s 2k (
ε = min( 1 2 (inf W G(s) -g), 3βJ 2 -g) > 0.
Then, for large n, G n (s) ≥ g + ε on the set V . Hence, for large n, e ng V exp (-nG n (s)) ds ≤ e ng e -(n-1)(g+ε) R e -Gn(s) ds.

The inequality

e -Gn(s) ≤ exp - βJ 2 s 2 + βJ|s| implies that e ng V exp (-nG n (s)) ds ≤ e g e -(n-1)ε R exp - βJ 2 s 2 + βJ|s| ds = O(e -nε ).

Proof of Theorem 3.1

For every y ∈ E fixed, the function s → L(f (y), s) is real analytic. Moreover, if s ∈ [-α, α], then for every y ∈ E, |L(f (y), s)| ≤ α. This implies that the functions s → E L(f (y), s) dµ(y) and G are real analytic.

We now prove that G has a finite number of global minima. The function G goes to infinity as s goes to infinity, since for every s ∈ R,

G(s) ≥ βJ 2 s 2 -βJ|s|.
This implies that the continuous function G is bounded below and has a global minimum. Furthermore, the set where G achieves its mimimum is bounded. Since G is a non constant analytic function, the set where its first derivative G (1) vanishes is discrete. The set where G is minimum is thus discrete. Being also bounded, it must be a finite set. Remark: It is worth remarking that every minimum point of G is of finite type. Indeed, if G has a minimum m of infinite order, then G (j) (m) = 0 for every j ≥ 1. Since G is analytic, this implies that G is constant. It contradicts the fact that G tends to infinity when s goes to infinity.

Proof of Theorem 3.2

A consequence of Lemma 3.1 is that the density of Sn n + Y √ n is equal to exp(-nG n (s)) exp(-nG n (s)) ds .

Note that the contribution of the Gaussian random variable Y √ n vanishes in the limit n → +∞. We thus have to prove that for any bounded continuous function h,

exp(-nG n (s))h(s) ds exp(-nG n (s)) ds = r i=1 b i,n h(m i ) r i=1 b i,n + o(1)
For every i ∈ {1, . . . , r}, m i is a minimum of G of type 2k i and strength λ i .

For every minimum m i , we apply Lemma 3.3. Then, for every i ∈ {1, . . . , r}, there exist δ i > 0 and N i ≥ 1 such that (4) holds. Let N be the maximum of (N i ) i=1,...,r and let δ be such that δ ≤ min i=1,...,r δ i and the sets ]m i -δ, m i +δ[ be disjoint. Let V be the closed set

V = R \ r i=1 ]m i -δ, m i + δ[.
m i +δ m i -δ exp(-nG n (s))h(s) ds = n -1/2k i e -nGn(m i ) δn 1/2k i -δn 1/2k i exp -n(G n (m i + sn -1/2k i ) -G n (m i )) h(m i + sn -1/2k i ) ds = n -1/2k i e -nGn(m i ) +∞ -∞ exp(-λ i s 2k i /(2k i )!)h(m i ) ds + o(1)
where the last equality is obtained by combining (3), ( 4) and dominated convergence. Hence, the integral is equivalent as n goes to infinity to

b i,n h(m i )
where

b i,n = n -1/2k i e -nGn(m i ) λ -1/2k i i +∞ -∞ exp(-s 2k i /(2k i )!) ds.
This yields the asymptotic

exp(-nG n (s))h(s) ds exp(-nG n (s)) ds = 1≤i≤r b i,n h(m i ) 1≤i≤r b i,n + o(1).
To prove the large deviations property, we use Laplace method. From Theorem 2.1, the distribution of S n /n under P x satisfies a large deviation principle with speed n and good rate function Λ ⋆ . The distribution Q n,x is absolutely continuous with respect to P x with density

dQ n,x dP x = 1 Zn,x exp n βJ(S n /n) 2 2 .
Since S n /n takes its values in [0, 1] and that the function z → βJz 2 /2 is continuous and bounded on [0, 1], it follows from Varadhan integral lemma (see [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF]) that the distribution of M n /n under Q n,x satifies a large deviation principle with speed n and good rate function

I β,x (z) = Λ * (z) - βJ 2 z 2 -inf z∈R {Λ * (z) - βJ 2 z 2 }.

Proof of Theorem 3.3

From Lemma 3.1, the probability density function of

M n -nm n 1-1/2k + Y n 1/2-1/2k
is given by exp(-nG n (m + sn -1/2k )) exp(-nG n (m + sn -1/2k )) ds .

The theorem is a consequence of the following result: for any bounded continuous function h, exp(-nG n (m + sn -1/2k ))h(s) ds exp(-nG n (m + sn -1/2k )) ds → exp(-λs 2k /(2k)!)h(s) ds exp(-λs 2k /(2k)!) ds [START_REF] Ellis | Large Deviations and Statistical Mechanics[END_REF] Let δ given by Lemma 3.3 and let V = R\ ]mδ, m + δ[. Lemma 3.4 yields the following estimation

|s|≥δn 1/2k exp(-nG n (m + sn -1/2k ))h(s) ds = n 1/2k V exp(-nG n (s))h((s -m)n 1/2k ) ds = O n 1/2k e -nε-ng (8)
From Lemma 3.3 and dominated convergence,

|s|<δn 1/2k exp(-nG n (m + sn -1/2k ))h(s) ds = e -nGn(m) |s|<δn 1/2k exp -n(G n (m + sn -1/2k ) -G n (m)) h(s) ds = e -nGn(m) R exp(-λs 2k /(2k)!)h(s) ds + o(1) (9)
By combining equations ( 8) and ( 9) we obtain [START_REF] Ellis | Large Deviations and Statistical Mechanics[END_REF].

The phase transition

In this section, we keep the hypotheses of the previous section and we focus on the case where E f dµ = 1 2 and a = E 4f (1f ) dµ > 0. We prove that a phenomenon of phase transition occurs: at high temperature (i.e. β small), the system has the same behaviour as at infinite temperature (i.e. β = 0). We prove that for β < β c , the magnetization vanishes in the thermodynamical limit (i.e. M n /n converges in distribution to zero) and we also give a caracterization of the critical inverse temperature β c . The study of the system at lower temperature (β > β c ) is quite difficult since the study of the minima of G strongly depends on the dynamical system S and on the function f . We will give some general conditions under which the minima of G are well known, and also study some examples.

The critical temperature

In this section the parameter β is not fixed and we write G β instead of G to enhance this dependency. We recall that G β (s) = βJ 2 s 2 -Λ(βJs), where

Λ(y) = E log f (x)e y + (1 -f (x))e -y dµ(x).
It follows from the change of variables u = βJs that β c is also defined by

β c = sup β > 0 ∀u ∈ R , Λ(u) ≤ u 2 2βJ .
Equivalently, the critical temperature is defined by the more explicit relation

1 2β c J = sup u∈R ⋆ Λ(u) u 2 .
Proposition 4.1. The critical inverse temperature β c satisfies

1 J ≤ β c ≤ 1 Ja .
Furthermore, if β < β c , then G β has a unique minimum at point 0.

Proof :

We use the fact that Λ(u) ∼ a 2 u 2 as u → 0. Hence, u → Λ(u) u 2 is continuous on R ⋆ , tends to a/2 as u → 0 and to 0 as u → ±∞. This implies that

a 2 ≤ sup u∈R ⋆ Λ(u) u 2
and then β c ≤ 1 Ja . Note that the equality β c = 1 Ja holds if and only if u → Λ(u) u 2 reaches its maximum at point 0. The inequality β c ≥ 1 J follows from the fact that the second derivative of Λ is bounded above by 1 which implies that Λ(u) ≤ 1 2 u 2 . The relation

G β s βJ = G βc s β c J + 1 2βJ - 1 2β c J s 2
implies that if β < β c , the function G β has a unique minimum at point 0 equal to 0.

Theorem 4.1. Let us assume that f ∈ C 1/2 (S). Then, for every β < β c , the distribution of M n under Q n,x satisfies a law of large numbers:

1 n M n ⇒ δ 0 , as n → ∞
and a central limit theorem:

1 √ n M n ⇒ N (0, σ 2 ) , as n → ∞ with σ 2 = a/(1 -βJa).
Proof:

The law of large numbers is an application of Theorem 3.2 and the central limit theorem is an application of Theorem 3.3. We verify that the assumptions of these theorems are satisfied. From Proposition 4.1, the hypothesis E f dµ = 1 2 and β < β c implies that the function G β has a unique minimum at point m = 0 of type 2k = 2 and strength λ = G

(2)

β (0) = βJ(1 -βJa) > 0. Since the function y → ∂ ∂s L(f (y), 0) is equal to βJ(2f -1), the hypothesis f ∈ C 1/2 (S) implies that y → ∂ ∂s L(f (y), 0) belongs to C 1/2 (S). Moreover, the function y → ∂ 2 ∂s 2 L(f (y), 0) is equal to (βJ) 2 (4f (1 -f )) which belongs to C 1 (S) since f is continuous.

General study for a specific class of systems

We consider here a class of systems for which we can study the minima of the function G β for every β > 0. We suppose that the system satisfies the following hypothesis (H) The function Λ is even and its derivative is concave on (0, +∞). Before introducing our main results, we exhibit some cases where the hypothesis (H) is satisfied. Note that (H) is an assumption on the function

Λ : u → E log f (x)e u + (1 -f (x))e -u dµ(x)
that does not really depend on the dynamical system S = (E, A, µ, T ) but only on the image distribution of µ under the application f , that we denote by µ f that is the measure on [0, 1] such that for every Borel set A,

µ f (A) = µ(f -1 (A)).
Proposition 4.2. The hypothesis (H) is satisfied in the following cases:

1. the measure µ f is equal to 1 [0,1] (x) dx.

the measure

µ f is equal to 1 2 (δ λ + δ 1-λ ), with 1 2 - √ 3 6 ≤ λ ≤ 1 2 , (and if 0 ≤ λ < 1 2 - √ 3
6 , the hypothesis (H) is not satisfied.)

3. the measure µ f has its support included into

[ 1 2 - √ 3 6 , 1 2 + √ 3
6 ] and satisfies the symmetry condition:

µ f = µ 1-f .
Proof: 1.: In this case, the function Λ is equal to

Λ(u) = 1 0 log xe u + (1 -x)e -u dx = u tanh(u) -1 if u = 0 0 if u = 0.
It is an even function. In order to prove the concavity of Λ ′ , we compute the third derivative Λ (3) for u ∈ (0, +∞) :

Λ (3) (u) = 2(1 -tanh(u) 2 ) tanh(u) 4 (3 tanh(u) -3u + u tanh(u) 2 )
which is negative on (0, +∞), so Λ satisfies the hypothesis (H).

2.: When

µ f = 1 2 (δ λ + δ 1-λ ), the function Λ is equal to Λ(u) = 1 2 log(λ e u + (1 -λ) e -u ) + log((1 -λ) e u + λ e -u ) = log(cosh(u)) + 1 2 log 1 -(2λ -1) 2 tanh(u) 2 ,
which is an even function and its third derivative is given for u ∈ (0, +∞) by

Λ (3) (u) = 2t(1 -t 2 )(1 -g 2 ) (1 -g 2 t 2 ) 3 g 2 (g 2 -3)t 2 + (3g 2 -1)
with t = tanh(u) ∈ (0, 1) and g = 2λ -1 ∈ (-1, 1). The fraction is nonnegative and for every λ (3) is negative on (0, +∞) and assumption (H) is satisfied.

∈ [ 1 2 - √ 3 6 , 1 2 ], the bracket is nonpositive since g 2 (g 2 -3)t 2 + (3g 2 -1) ≤ (3g 2 -1) ≤ 0. Thus, Λ

3.:

The condition that f and 1f have the same distribution under µ implies that the function Λ is even : for every u ∈ R,

Λ(u) = E log f (x)e u + (1 -f (x))e -u dµ(x) = E log (1 -f (x))e u + (1 -(1 -f (x)))e -u dµ(x) = Λ(-u).
Writing Λ(u) = Λ(u)+Λ(-u) 2 yields the following expression :

Λ(u) = log(cosh(u)) + 1 2 E log 1 -(2f (x) -1) 2 tanh(u) 2 dµ(x).
We compute the third derivative using this last formula. This yields

Λ (3) (u) = 2t(1-t 2 ) E 1 -g(x) 2 (1 -g(x) 2 t 2 ) 3 g(x) 2 (g(x) 2 -3)t 2 + (3g(x) 2 -1) dµ(x) with t = tanh(u) ∈ (-1, 1) and g = 2f -1. When the measure µ f has its support included into [ 1 2 - √ 3 6 , 1 2 + √ 3 6 ]
, then (3g 2 -1) is µ-almost everywhere nonpositive and hence Λ (3) (u) and -u are of the same sign, so assumption (H) is satisfied.

We are now ready to give a complete description of the minima of the function G β in function of the inverse temperature β and to show a phase transition at the critical inverse temperature β = β c . Theorem 4.2. Let us assume that hypothesis (H) is satisfied. Then, the following results hold:

1. the critical inverse temperature β c is equal to 1/(Ja).

2. for β < β c , G β admits 0 as unique minimum of type 2 and strength βJ(1 -βJa).

3. for β = β c , G β admits 0 as unique minimum of type ≥ 4. Proof:

1.: We prove that for every u ∈ R,

Λ(u) ≤ a 2 u 2 . ( 10
)
This implies that

1 2β c J = sup u∈R ⋆ Λ(u) u 2 ≤ a 2
and hence that β c ≥ 1 Ja . From Proposition 4.1, the equality is proved. We now prove inequality (10) using hypothesis (H). Since Λ ′ is concave on (0, +∞), its derivative Λ (2) is a nonincreasing function. Hence, for every u > 0, Λ (2) (u) ≤ Λ (2) (0) = a. The function Λ ′ vanishes at point 0 and has a derivative bounded above by a: this implies that Λ ′ (u) ≤ au. Integrating one more time yields Λ(u) ≤ a 2 u 2 for every u > 0. Since the function Λ is even, the same inequality holds for u < 0 and this proves inequality [START_REF] Feller | Introduction to Probability Theory and its Applications[END_REF]. 2.: This result is contained in Proposition 4.1.

3.: From assumption (H) the function G ′

β (s) equal to βJ(s -Λ ′ (βJs)) is odd and convex on (0, +∞). For

β = β c , G ′ β (0) = G (2)
β (0) = 0, so from convexity, G ′ β is nonnegative on (0, +∞). The point 0 is the only point where G ′ β Otherwise, if there exists some u > 0 such that G ′ β (u) = 0, the convex function would be identically zero on [0, u] , and being real analytic, it would be identically zero on R, which is not the case. Thus zero is the only point where G We now prove that the type of m is equal to 2: by the mean value theorem there exists m 0 ∈ (0, βJm) such that Λ (2)

(m 0 ) = Λ ′ (βJm) βJm = 1 βJ .
The real analytic function Λ (2) is not constant on [0, +∞[ (since Λ (2) (0) = a > 0 and lim s→+∞ Λ (2) (s) = 0). Then, since Λ (3) is nonpositive on (0, +∞), the function Λ (2) is strictly decreasing on (0, +∞). It follows that Λ (2) (m 0 ) > Λ (2) (βJm) and that

G

(2)

β (βJm) = (βJ) 2 1 βJ -Λ (2) (βJm) = (βJ) 2 λ (2) (m 0 ) -Λ (2) (βJm) > 0.
This completes the proof of the theorem.

As a consequence, the asymptotic behaviour of M n under Q n,x for β ≥ β c is deduced for the systems satisfying assumption (H). Recall that Theorem 4.1 treats the case β < β c for general systems.

A particular case: the rotation on the torus

In this section we treat the particular case of the irrational rotation on the torus which corresponds to a quasiperiodic random field already mentioned in [START_REF] Koukiou | Extension of the Pirogov-Sinai theory to a class of quasiperiodic interactions[END_REF]. It is one of the dynamical systems S in ergodic theory for which we are able to provide a large subclass of C α (S). In the first section we first give a precise description of this subclass in terms of the diophantine properties of the irrational angle. In the second one we apply results of Section 4 to this particular dynamical system when the function f is the identity function.

Some Results on Diophantine Approximations

Let us consider the dynamical system (T r , B(T r ), λ, T α ) where λ is the Lebesgue measure on the torus T r and T α is the irrational rotation over T r defined by x → x + α mod 1. It is well known that under these conditions this dynamical system is ergodic and for every f ∈ L 1 (λ), for almost every x ∈ T r ,

M n = 1 n n l=1 f (T l α x) - T r f (t)dt → n → ∞ 0
When f is with bounded variation, this result holds for every x ∈ T r and it is possible to determine the speed of convergence of the sequence M n to 0 in terms of arithmetic properties of the irrational vector α. When r = 1, for all irrational badly approximated by rationals, Denjoy-Koksma's inequality gives us a majorization of M n uniformly in x for n large enough. But when r ≥ 2, Denjoy-Koksma's inequality does not hold (see Yoccoz [START_REF] Yoccoz | Sur la disparition de la propriété de Denjoy-Koksma en dimension 2[END_REF]) and the method of low discrepancy sequences has to be used.

Case of one-dimensional torus

Let α be an irrational. We call a rational p q with p, q relatively prime such that |α -p q | < 1 q 2 , a rational approximation of α. When α has the continued fraction expansion α = [α] + [a 1 , . . . , a n , . . .], the n-th principal convergent of α is pn qn where, ∀n ≥ 2,

p n = a n p n-1 + p n-2
q n = a n q n-1 + q n-2 ;

the recurrence is given by defining the values of p 0 , p 1 and q 0 , q 1 .

Denjoy-Koksma's inequality Let f : R → [0, 1] be a function with bounded variation V (f ) and p q a rational approximation of α. Then, for every

x ∈ T 1 , | q l=1 f (T l α x) -q T 1 f (t)dt| ≤ V (f ).
Proposition 5.1. Let f be a function with bounded variation V (f ). For every irrational α such that the inequality a m < m 1+ǫ , where ǫ > 0, is satisfied eventually for all m,

sup x∈T 1 n l=1 f (T l α x) - T 1 f (t)dt = O(log 2+ǫ n).

Proof:

The sequence of integers (q i ) i≥1 being strictly increasing, for a given n ≥ 1, there exists m n ≥ 0 such that q mn ≤ n < q mn+1 .

By Euclidean division, we have n = b mn q mn +n mn-1 with 0 ≤ n mn-1 < q mn . We can use the usual relations

q 0 = 1, q 1 = a 1 q n = a n q n-1 + q n-2 , n ≥ 2. ( 11
)
We obtain that (a mn+1 + 1)q mn > q mn+1 > n and so b mn ≤ a mn+1 . If m n > 0, we may write n mn-1 = b mn-1 q mn-1 + n mn-2 with 0 ≤ n mn-2 < q mn-1 .

Again, we find b mn-1 ≤ a mn . Continuing in this manner, we arrive at a representation for n of the form n = mn i=0 b i q i with 0 ≤ b i ≤ a i+1 for 0 ≤ i ≤ m n and b mn ≥ 1. Using Denjoy-Koksma's inequality, we get

| n l=1 f (T l α x) -n T 1 f (x)dx| ≤ V (f ) mn i=0 i ≤ V (f ) mn i=0 a i+1 .
By hypothesis, there exists m 0 ≥ 1 such that,

a m < m 1+ǫ , ∀m ≥ m 0 .
Let n be such that m n > m 0 . Thus,

| n l=1 f (T l α x) -n T 1 f (t)dt| ≤ V (f )( m 0 -1 i=0 a i+1 + (m n + 1) 2+ǫ ).
We need to know the asymptotic behavior of m n . When α is the golden ratio, a n = 1, ∀n ≥ 1 and the relation [START_REF] Fontes | On the averaged dynamics of the random field Curie-Weiss model[END_REF] implies that q n ∼ 1 √ 5 α n+1 . Let α ′ be another irrational; its partial quotients a ′ n satisfy necessarily a ′ n ≥ 1. Using the relation [START_REF] Fontes | On the averaged dynamics of the random field Curie-Weiss model[END_REF], we see that q ′ n ≥ q n , ∀n ≥ 1. Therefore, m n = O(log n) and the proposition is proved.

Generalization to r-dimensional torus

We recall some definitions and well known results from the method of low discrepancy sequences in dimension r ≥ 1.

Suppose we are given a function f (x) = f (x (1) , . . . , x (r) ) with r ≥ 1. By a partition P of [0, 1] r , we mean a set of r finite sequences η

(j) 0 , η (j) 1 , . . . , η (j) m j (j = 1, . . . , r), with 0 = η (j) 0 ≤ η (j) 1 ≤ . . . ≤ η (j)
m j = 1 for j = 1, . . . , r. In connection with such a partition, we define, for j = 1, . . . , r an operator ∆ j by ∆ j f (x (1) , . . . , x (j-1) , η (j) i , x (j+1) , . . . , x (r) ) = f (x (1) , . . . , x (j-1) , η (j) i+1 , x (j+1) , . . . , x (r) )f (x (1) , . . . , x (j-1) , η (j) i , x (j+1) , . . . , x (r) ), for 0 ≤ i < m j . Definition 5.1.

1. For a function f on [0, 1] r , we set

V (r) (f ) = sup P m 1 -1 i 1 =0 . . . mr-1 ir=0 |∆ 1,...,r f (η (1)
i 1 , . . . , η (r) ir )|,
where the supremum is extended over all partitions P of [0, 1] r . If V (r) (f ) is finite, then f is said to be of bounded variation on [0, 1] r in the sense of Vitali.

2. For 1 ≤ p ≤ r and 1 ≤ i 1 < i 2 < . . . < i p ≤ r, we denote by V (p) (f ; i 1 , . . . , i p ) the p-dimensional variation in the sense of Vitali of the restriction of f to

E r i 1 ...ip = {(t 1 , . . . , t r ) ∈ [0, 1] r ; t j = 1 whenever j is none of the i r , 1 ≤ r ≤ p}.
If all the variations V (p) (f ; i 1 , . . . , i p ) are finite, the function f is said to be of bounded variation on [0, 1] r in the sense of Hardy and Krause.

Let x 1 , . . . , x n be a finite sequence of points in [0, 1] r with x l = (x l 1 , . . . , x lr ) for 1 ≤ l ≤ n. We introduce the function For a real number t, let t denote its distance to the nearest integer, namely,

R n (
t = inf n∈Z | t -n | = inf({t}, 1 -{t})
where {t} is the fractional part of t. Definition 5.3. For a real number η, a r-tuple α = (α 1 , . . . , α r ) of irrationals is said to be of type η if η is the infimum of all numbers σ for which there exists a positive constant c = c(σ; α 1 , . . . , α r ) such that r σ (h) < h, α > ≥ c holds for all h = 0 in Z r , where r(h) = r i=1 max(1, |h i |) and < •, • > denotes the standard inner product in R r .

The type η of α is also equal to sup{γ : inf h∈(Z r ) * r γ (h) < h, α > = 0}.

We always have η ≥ 1 (see [START_REF] Osgood | Diophantine approximation and its applications[END_REF]). Now we give a result (see [START_REF] Kuipers | Uniform distribution of sequences[END_REF]) which yields the asymptotic behavior of the discrepancy of the sequence w = (x 1 + lα 1 , . . . , x r + lα r ), l = 1, 2, . . . as a function of the mutual irrationality of the components of α. Proposition 5.2. Let α = (α 1 , . . . , α r ) be an irrational vector. Suppose there exists η ≥ 1 and c > 0 such that r η (h) < h, α > ≥ c for all h = 0 in Z r . Then, for every x ∈ [0, 1] r , the discrepancy of the sequence w = (x 1 +lα 1 , . . . , x r +lα r ), l = 1, 2, . . . The proof is based on the Erdös-Turán-Koksma's theorem: For h ∈ Z r , define p(h) = max 1≤j≤r |h j |. Let x 1 , . . . , x n be a finite sequence of points in R r . Then, for any positive integer m, we have Proof: Let η ′ be such that η ≤ η ′ < 1 + 1 r . There exists c > 0 such that

D * n ≤ C r   1 m + 0≤p(h)≤m 1 r(h)
r η ′ (h) < h, α > ≥ c
holds for all h = 0 in Z r . Suppose we are given a p-tuple α p = (α i 1 , . . . , α ip ), 1 ≤ p ≤ r, of α, then r η ′ (h) < h, α p > ≥ c holds for all h = 0 in Z p , 1 ≤ p ≤ r. Thus, every p-tuple, 1 ≤ p ≤ r, is of type δ such that 1 ≤ δ ≤ η and (α i 1 , . . . , α ip ) is an irrational vector. For every p, 1 ≤ p ≤ r, we define w i 1 ...ip by the projection of w on E r i 1 ...ip . From the previous proposition, we have for every p, 1 ≤ p ≤ r,

nD * n (w i 1 ...ip ) = O(log p+1 n) if δ = 1 nD * n (w i 1 ...ip ) = O(n 1- 1 ((δ-1)p+1) log n) if 1 < δ ≤ η.
Now, ∀p = 1, . . . , r,

0 ≤ 1 - 1 (δ -1)p + 1 ≤ 1 - 1 (η -1)r + 1 ≤ 1.
Therefore, using Hlawka-Zaremba's theorem, we obtain Proposition 5.3.

Theorem 3 . 1 .Definition 3 . 1 .

 3131 The function G is real analytic, and the set where G achieves its minimum is non-empty and finite. We will denote by g = min{G(s); s ∈ R} the minimum of G and by m 1 , • • • , m r the points where G is minimal. Let us define the type 2k i and the strength λ i > 0 of the minimum m i by

Lemma 3 . 4 .

 34 Let V be any closed subset of R containing no global minima of G. There exists ε > 0 such that e ng V exp (-nG n (s)) ds = O e -nε . Proof: Since |S n | ≤ n, the expectation E x (exp(βJsS n )) is bounded above by exp(nβJ|s|) and the function G n satisfies for every s ∈ R G n (s) ≥ βJ 2 s 2 -βJ|s|.

  nG n (s))h(s) ds = O e -nε-ng . Use a change of variables and Lemma 3.3 to estimate the contribution of the set ]m iδ, m i + δ[,

Definition 4 . 1 .

 41 Let β c be the critical inverse temperature defined by the relationβ c = sup {β > 0 |∀s ∈ R , G β (s) ≥ 0} .

4 .

 4 for β > β c , G β admits two global minima at points ±m (with m > 0), of same type equal to 2 and same strength. The point m called spontaneous magnetization is the unique positive solution of the equation m = E tanh(βJm) + (2f -1) 1 + (2f -1) tanh(βJm) dµ.

′β

  vanishes, and the function G β has a unique global minimum at point 0. 4.: For β > β c , under assumption (H), the function G ′ β is odd and convex on (0, +∞), moreover, G ′ β (0) = 0, G (2) β (0) = βJ(1 -βJa) < 0 and lim s→+∞ G ′ β (s) = +∞. Hence, there exists an unique real m ∈ (0, +∞) such that G ′ β (m) = 0, thus the function G ′ β vanishes only at points -m, 0 and m. The function G β reaches its global minimum at points -m and m, and has a local maximum at 0. Since Λ is even, the minima m and -m have same type and same strength.

Theorem 4 . 3 .

 43 Assume that assumption (H) is satisfied.1. When β = β c , denote by 2k ≥ 4 and λ the type and the strength of m = 0 the minimum of G β . Assume that for every j ∈ {1, . . . , 2k}, the function f j belongs to C j/2k (S). Then,M n n ⇒ δ 0 and M n n 1-1/2k ⇒ Zwhere Z is the probability measure with density function C exp -λs 2k /(2k)! , C being the normalizing constant.2. When β > β c , assume that the functions ∂ ∂s L(f (.), ±βJm) belong to the set C 1/2 (S). Then, for every bounded continuous function h, the expectation of h(M n /n) under Q n,x is equivalent, as n goes to infinity, tob m,n h(m) + b -m,n h(-m) b m,n + b -m,n where b m,n = n j=1 f (T j x) e βJm + (1f (T j x)) e -βJm .Remark: A straightforward computation givesG (4) β (0) = 2(βJ) 4 (3I 4 -4I 2 + 1)whereI 2 = E (2f-1) 2 dµ and I 4 = E (2f -1) 4 dµ. So, when β = β c , the type of 0 is equal to 4 if and only if 3I 4 -4I 2 + 1 > 0. It is always verified when the support of f is strictly included into [

1 (

 1 satisfies D * n (w) = O(n -1 log r+1 n) for η = 1 and D * n (w) = O(n -(η-1)r+1) log n) for η > 1.

  only depends on the dimension r. This theorem combined with the results of[START_REF] Kuipers | Uniform distribution of sequences[END_REF] (p.131) gives us the result.

Theorem 5 . 1 (1≤i 1 1 O(n 1 - 1 (

 511111 Hlawka, Zaremba). Let f be of bounded variation on [0, 1] r in the sense of Hardy and Krause, and let ω be a finite sequence of points x 1 , . . . , x n in [0, 1] r . Then, we have <i 2 <...<ip≤rV (p) (f ; i 1 , . . . , i p )D * n (ω i 1 ...ip ),whereD * n (ω i 1 ...ip ) is the discrepancy in E r i 1 ...ip of the sequence ω i 1 ...ip obtained by projecting ω onto E r i 1 ...ip .Proposition 5.3. Let f be a function with bounded variation in the sense of Hardy and Krause, and α an irrational vector of type η, thensup x∈T r n l=1 f (T l α x) -T r f (t)dt = O(log r+1 n) if η = (η-1)r+1) log n) if η > 1.

  t 1 , . . . , t r ) = A(t 1 , . . . , t r ; n) n t 1 . . . t r for (t 1 , . . . , t r ) ∈ [0, 1] r , where A(t 1 , . . . , t r ; n) denotes the number of elements x l , 1 ≤ l ≤ n, for which x l i < t i for 1 ≤ i ≤ r. Definition 5.2. The discrepancy D * n of the sequence x 1 , . . . , x n in [0, 1] r is defined to be D * n = sup (t 1 ,...,tr)∈[0,1] r |R n (t 1 , . . . , t r )|.

A particular example: f (x) = x

Consider the irrational rotation on the one-dimensional torus with angle of type η and we choose f (x) = x. Clearly, the integral of f is equal to 1/2 and a = 2/3. We apply Theorems 4.2 and 4.3 for this particular example.

Theorem 5.2. The following results hold:

1. The critical inverse temperature β c is equal to 3/(2J).

When


Nicolas Lopez Ferber

Doan Pham

• Q Falcoz

• A Meffre

N Tessier-Doyen

• A Nzihou

• V Goetz

Ceramics from Municipal Waste Incinerator Bottom Ash and Wasted Clay for Sensible Heat Storage at High Temperature

Keywords: Thermal energy storage material, Waste-based ceramic, Sintering, Municipal waste incinerator bottom ashes

Although work has been done to understand the sintering behavior

Introduction

Municipal waste incineration is a common technique for household wastes (usually high carbon content) treatment. It consists in the combustion of these wastes at high temperature (900-1200 °C), therefore reducing their volume by up to 90%, decreasing land use compared to direct household wastes landfilling [START_REF] Crillesen | International Solid Waste Association: Management of bottom ash from WTE plants-an overview of management options and treatment methods[END_REF]. Most of the incinerators recover the heat generated to produce industrial steam or electricity. Considered as renewable energy, it represents around 1.7 TWh e in 2015 in France (comparable to biogas, or solid biomass) [START_REF] Cavaud | Chiffres clés des énergies renouvelables-Edition 2016[END_REF].

The main solid byproduct of incineration is Municipal Waste Incineration Bottom Ash (MWIBA), representing around 25% of the total input weight, the others being fly ash, air pollution control residues and dusts [START_REF]International Solid Waste Association: Waste-to-Energy Stateof-the-Art-Report[END_REF]. This is a quite complex material, with composition varying in time and space, as revealed by the discrepancies in composition of MWIBA samples in the literature (Table 1).

The bottom ash is typically classified as "non-dangerous waste" and landfilled, although some countries extensively use it as a filler for road sub-base construction. For example, around 80% of the 3 Mt of bottom ashes generated yearly in France are used as road sub-base filler after several treatment steps involving milling, sieving, ferrous metals removal (magnetic separation) and non-ferrous metals removal (Eddy currents), and ageing (stabilization of leaching properties through calcium oxide carbonation) [START_REF]Acceptabilité environnementale de matériaux alternatifs en technique routière-Les mâchefers d'incinération de déchets non dangereux (MIDND)[END_REF].

Even though most of the bottom ash is successfully reused in France, this is not the case among other European countries, so most of the 12 Mt produced annually is landfilled [START_REF] Crillesen | International Solid Waste Association: Management of bottom ash from WTE plants-an overview of management options and treatment methods[END_REF]. Being mostly a mixture of mineral compounds, namely alumina, hematite, silica and alkali oxides in the form of a heterogeneous gravel, it can be compatible with ceramic materials production [START_REF] Bethanis | Properties and microstructure of sintered incinerator bottom ash[END_REF][START_REF] Rambaldi | The recycling of MSWI bottom ash in silicate based ceramic[END_REF], which can represent a new valorization path, instead of road subbase filler production, providing new feedstock for ceramic industry and saving resources and environmental impacts related to clay and sand extraction. Another interesting application calls for high temperature storage systems, using ceramics as sensible thermal storage material in thermocline packed bed or structured bed systems [START_REF] Lopez Ferber | Flexibility and robustness of a high-temperature air/ceramic thermocline heat storage pilot[END_REF]. Some authors worked on such waste-based ceramics, using molten asbestos containing wastes [START_REF] Py | Recycled material for sensible heat based thermal energy storage to be used in concentrated solar thermal power plants[END_REF], metallurgical slags [START_REF] Dejean | Valorisation de laitiers sidérurgiques comme matériaux de stockage thermique pour procédés énergétiques durables[END_REF] or coal fly ash [START_REF] Kere | Applicability of thermal energy storage recycled ceramics to high temperature and compressed air operating conditions[END_REF], mostly using the petrurgic method implying casting and controlled cooling from molten state, a technique that is not considered as easily transferable to industry and mass-production. Sintering alumino-silicate industrial sub products appears to be preferable, due to the higher production capabilities of the sintering production pathway [22].

Research projects have been carried out regarding the sintering behavior of MWIBA, but being focused on sintering or crystallization behavior rather than production considerations; the processes and materials they proposed might not be suited for mass-production at reasonable cost (micronic wet-milling [START_REF] Bethanis | Properties and microstructure of sintered incinerator bottom ash[END_REF], compression for uniaxial shaping, casting at molten state or sinter-crystallization from a glass [START_REF] Schabbach | Integrated approach to establish the sinter-crystallization ability of glasses from secondary raw material[END_REF], valuable additives like high-grade kaolin [START_REF] Taurino | New fired bricks based on municipal solid waste incinerator bottom ash[END_REF] and so on).

In this work another process has been developed for the production of a 100% waste-based ceramic, using production protocols as simple and widely available as possible This process is based on the conventional extrusion technique, used in the ceramic building bricks industry. The ceramics obtained should exhibit properties suitable for sensible heat thermal storage up to 1000 °C, under air. Hence, the desired properties of the resulting material are as follows:

• Maximal volumetric thermal capacity (ρ × Cp), implying maximizing densification during sintering. This is especially important for thermal storage applications, since it would increase thermal storage capacity per unit volume. • Regarding thermocline packed bed air/ceramic storage systems, a high thermal conductivity is beneficial for storage behavior, ensuring that the Biot number is significantly lower than 1, but this could also be achieved by decreasing the size of the particles, at the cost of increased pressure drop. • Satisfying mechanical strength, measured as compressive strength (> 20 MPa). A high compressive strength for a thermal storage material will allow increasing the height of the packed bed without damaging the pieces. • Thermal stability over a large range of temperatures that can be encountered in thermal energy storage (up to 800-1000 °C),under air • Satisfying thermomechanical behavior, measured as the Coefficient of Thermal Expansion (CTE). This CTE should be as low as possible, or at least present predictable variations with temperature, to avoid increasing mechanical stress during use.

The strategy chosen focused on the simplicity of the production process, because the amount of materials required for sensible heat storage technology can be important, in both the sectors of industrial energetic efficiency and concentrated solar power techniques [24], implying that the developed material should be compatible with mass-production at low cost.

Materials and Methods

Materials Processing

Sampling 450 kg of an aged MWIBA have been collected in the maturation platform of a commercial incinerator in southern France using coning and quartering method as defined by the UIPAC [START_REF]IUPAC, Compendium of Chemical Terminology-Gold Book[END_REF], after post-treatment process (magnetic removal of ferrous and non-ferrous metals, ageing). The bottom ashes complied with the requirements for homologation as "MWIBA gravel 0-40 mm" as defined in the Setra guide for MWIBA management [START_REF]Acceptabilité environnementale de matériaux alternatifs en technique routière-Les mâchefers d'incinération de déchets non dangereux (MIDND)[END_REF]. Like most MWIBA, it consists in a mixture of ashes, sand, glassy material, metallic and ceramic fragments and residual unburnt organic compounds [START_REF] Becquart | Monotonic aspects of the mechanical behaviour of bottom ash from municipal solid waste incineration and its potential use for road construction[END_REF].

Milling

The particle size distribution (PSD) of as-collected MWIBA was not compatible with shaping and firing protocols, so the MWIBA had to be milled. The goal being to assess the feasibility of MWIBA ceramics production in building bricks and tile industry, we disregarded extensive wet milling. These techniques, although relevant to produce thin and homogeneous powders, seemed too complicated and not easily transferable to industrial settings, so an easier method has been favored. MWIBA has been dried at 105 °C for 1 day, before being milled with a tungsten jaw mill. MWIBA has been milled with three passes inside the jaw mill, the jaw being enclosed between each pass, from a 10 mm gap to a 1 mm gap. The powder obtained has then be sieved at 500 µm. The passing fraction has then been used for the trials. Milling several hundred grams with this technique requires approximately 45 s, and can be upscaled reasonably easily due to the great availability of jaw mills in the mineral industries.

Shaping

Shaping the green bodies before firing being an important step for any applications, adding a binding agent to the MWIBA was considered, in order to give the mixture high enough plasticity to undergo extrusion. Thus, a wasted clay was used as binder. This clay is a by-product of a quartz producing company located in southern France, with a generation rate of 100 t/day. It is a fine powder (d 50 = 16 µm) of an illitic clay, obtained by flocculation/decantation of a washing mud. For these trials, we used an extrusion device located in the CTMNC technology transfer facility, which is a scaled-down model of an industrial extrusion device (single screw, two stages, equipped with a void pump), with a conical end-piece with an opening of 20 mm wide, to produce 40 mm long cylinders. The added clay represents 20, 30 or 40% of the paste dry weight. Hence, this paper will mostly focus on the influence of the clay to bottom ash ratio in the formulation and firing temperature on the resulting properties of the sintered samples.

Firing

After being dried at 105 for 6 h, green bodies have been fired in electrical furnaces, with a heating rate of 5 °C/min from room temperature to 500 °C, then 2 °C/min from 500 °C to the maximum temperature, with isothermal firing duration varying from 1 to 4 h depending of the samples. The furnaces are not equipped with an active cooling system, implying cooling durations of about 10 h. The different stages of the process are illustrated by Fig. 1.

It is worth noting that, although formulations with 20 and 30 wt% clay required plasticizer (Zschimmer & Schwartz, Zusoplast C39) to form a plastic paste suitable for extrusion, 40 wt% clay formulation only required water (Table 2).

Raw Material Characterization

Chemical composition of the raw materials has been characterized with ICP-AES. Samples have been milled and sieved at 40 µm. The clay and the MWIBA have been dissolved into a mix of nitric and fluorhydric acid into a micro-wave oven (Anton Paar) under heat. The samples have been heated up to 240 °C and 60 bars in 10 min, with 40 min dwell time. After total dissolution, the sample have been characterized using a Perkin-Elmer Optima 8300 ICP device.

PSD of the powders has been measured using a Malvern Mastersizer 3000 granulometer, on carefully dried powders, for a particle size range of 0-1000 µm.

Loss On Ignition has been measured for the raw powders by comparing the mass of a sample of powder, before and after being heated up to 1000 °C for 2 h. This characteristic of the powders also had been confirmed with TG-DSC (Labsys Evo from Setaram, with 3 L/h air, and a heating rate of 5 °C/min from room temperature to 1100 °C).

The mineral phases present in the raw materials have been identified with X-Ray Diffraction, using a Philips PANalytical X'pert Pro MPD device (copper emitter, K α = 1543 Å), from 2θ = 10° to 2θ = 80°.

Characterization of the Sintered Samples

The goal being to assess the potential of these ceramics for thermal storage at high temperature, the main properties investigated are as follows:

• Volumic mass, expressed in kg/m 3 or in relative density, has been measured with the Archimede method, comparing dry and immersed sample apparent weight. • Thermal capacity, expressed in J/(g K), has been measured by a modular DSC (Q200 from TA), with a 50 mL/ min nitrogen flow, at room temperature and 390 °C, 390 °C being close to the average temperature of use. Four samples have been also characterized by NETZSCH Applications Laboratory, using a DSC 404 F1 Pegasus differential scanning calorimeter from room temperature to 1000 °C, using a 10 K/min heating ramp, under air. • Thermal diffusivity has been measured with a Laser Flash Analysis device from Netzsch (LFA 457) from room temperature to 1000 °C, with a heating ramp of 20°/min. The knowledge of the variations of specific heat with temperature allowed calculating the thermal conductivity of these samples. • Thermal stability in the temperature range of 0-1000 °C is of highest importance. The behavior of the sintered samples under thermal cycling has been investigated with DTA and TMA (Setsys device from Setaram). The coefficient of thermal expansion should also been as low as possible, or at least predictable so to avoid generating strains inside a bed of these materials. • Compressive strength of the ceramics will set the limit to the system's size. Therefore, high compressive strength can be beneficial to this material. It has been measured by submitting samples to an uniaxial force, using a hydraulic press and increasing the load until sample's rupture, on small cylindrical samples.

To assess the potential of optimization of these materials, they also have been characterized in terms of water absorption, related to open porosity. Samples exhibiting higher values of water absorption might be indicating lower sintering quality. XRD allowed also to identify the main mineral phases inside the sintered bodies, as well as to quantify their respective proportions, using a D8 Advance Bruker XRD device (5° < 2θ < 80°), coupled with a LynxEye device and using a Rietveld protocol.

Results

Raw Material Characterization

Composition and Particle Size Distribution

Both the clay and MWIBA are mostly composed of silica. The clay has higher proportion of alumina and potassium oxides, whereas the MWIBA contains more sodium oxides (Table 3). These compounds are known for their relevance as fluxing agents, and so might facilitate sintering.

The composition is on part with other MWIBA studied in the scientific literature, although this MWIBA in particular is richer in silicon oxide than most of the samples studied in the literature. Its "lower than usual" content in alumina could be explained by differences of in waste-sorting policies.

The clay was not milled, and exhibit a Gaussian-like particle size distribution, centered on its d 50 of 16 µm. The milled MWIBA exhibits a completely different reparation of particle size, with an over-representation of coarse particles (> 100 µm), mixed with smaller particles. The d 50 of the milled MWIBA is 166 µm, superior of that's of the clay by an order of magnitude (Fig. 2).

This situation, where one of the powder is much coarser and with a sharper distribution and the other one is smaller with a broader PSD is favorable to maximal mixture quality and compacity of the green bodies, the smaller particles being able to fill the porosity between the coarser ones. A difference of particle size of around 1:10 is usually described as relevant for increasing green bodies compacity [START_REF] Rahaman | Ceramic Processing[END_REF]. The two powders comply with this requirement (Fig. 2), increasing the maximal compacity and homogeneity achievable for the green bodies.

Raw Materials Mineralogy

Both clay and MWIBA powders have been characterized in terms of mineral composition. The clay exhibits typical illitic clay phases, such as illite and kaolinite, associated with quartz and K-feldspath, (presumably microcline). The MWIBA powder exhibits typical quartz and calcite phases, associated with wollastonite and pyrite (Fig. 3). Pyrite is not usually reported in the literature as being present in bottom ashes, the iron being mostly present as oxide (hematite).

Raw Materials Thermal Behavior

MWIBA have been milled in a tungsten carbide ring mill and sieved at 50 µm to obtain a fine powder suitable for DSC-TG analysis. This sample exhibits a thermal behavior similar to those studied in literature (Fig. 4), namely:

• An endothermic phase between room temperature and around 250 °C, associated with mass loss, which is most often attributed to the vaporization of residual, bound and hydration water. • An exothermic phase between 300 and 500 °C, associated with mass loss. These peaks are presumably due to the combustion of unburnt organic matter and the release of occluded gases [START_REF] Bethanis | Properties and microstructure of sintered incinerator bottom ash[END_REF]. • A sharp endothermic peak around 700 °C, associated with a sharp weight loss. This is typically associated with calcite decarbonation (CaCO 3 ⟶ CaO + CO 2 ). The usual temperature for calcite decarbonation is around 900 °C, but this phenomenon appears to occur at lower temperature when calcite is found in MWIBA [18, 28, 29] • Above 1000 °C, the thermal behavior becomes chaotic due to simultaneous melting and crystallization of the different phases as well as specific heat variations. At high temperature, some weight losses can be attributed to alkali metal sulfates dissociation [START_REF] Bethanis | Properties and microstructure of sintered incinerator bottom ash[END_REF].

The weight loss is higher than noticed during sintering tests, which is presumably due to concentration of volatile organic matter into the small-sized particles.

The clay exhibits very common thermal behaviour, with an endothermic peak around 100-150 °C associated with bounded water removal, followed by a sharp weight loss between 450 and 600 °C associated to kaolinite deshydroxylation and a 980 °C exothermic peak related to structural transformation of illite to mullite. This clay starts vitrifying at 1250 °C, a temperature where it reaches its maximum shrinkage (7%) (information provided by supplier).

Sintered Samples Properties

All the samples will be designated as follows: XXXX-YY-Z-H, with XXXX the firing temperature (°C), YY the clay content (% dry weight), Z the dwell time (h) and H the use of plasticizer additive (0 if no plasticizer is used, 1 if it is). Due to the considerable number of samples produced, only the ones considered interesting (due to their superior properties or interesting to compare with others) have been characterized further than volumic mass (Table 4).

Volumic Mass and Water Absorption

Fired density of samples have been measured with Archimede's method. Water absorption has been measured repeatedly on several samples by comparing dry and wet mass. Both are illustrated on Fig. 5. It appears that the fired density variations between samples were minor, all of them exhibiting values between 1.75 and 1.91 g/cm 3 . Depending on the composition, the maximum density was obtained at different temperatures. For example, with 20 wt% clay, the highest density was achieved for T firing = 1115 °C. There was no increase in density between samples fired at 1115 °C and those fired at 1120 °C. This can be explained by the creation of black bodies, related to partial melting of the material, leading to closed porosity. Some authors mentioned also that the decomposition of sulfates compounds can occur in this temperature range, the production of gases limiting density. Water absorption varies widely between samples, the lowest values being reached by samples 1115-20-4-1, 1115-30-4-1 and 1120-40-4-1. Low water absorption is somehow related to higher densities, but the difference is quite low. Water absorption could be related to the sintering process, open and interconnected porosities being the first closed during firing [START_REF]Techniques de l'ingénieur, Frittage: aspects physico-chimiques Partie 1: frittage en phase solide[END_REF], before overfiring and melting lead to new open pores formation, as water absorption is linked to open porosity and pore size distribution [START_REF] Kato | Water absorption and retention of porous ceramics fabricated by waste resources[END_REF].

Thermal Properties

The volumetric thermal capacity is the product of volumic mass and specific heat. As previously mentioned, all samples exhibit relatively close volumetric mass, but the quite different thermal capacity per unit mass leads to significant differences in terms of volumetric thermal capacity, whether taken at room temperature or 390 °C. Similar to the compressive strength, the samples containing 30 wt% clay and fired at 1115 °C exhibited superior properties by a significant margin (Fig. 6). Thermal conductivity was in the typical range for ceramic-based materials, the highest being 1 W/(m K) for the 1115-30 sample, at room temperature.

Mechanical/Thermomechanical Properties

For samples with the lower amount of clay, compressive strength increased from 10 to 28 MPa when firing temperature increased from 1100 to 1115 °C. However, at higher temperature, the high melting tendency of the bottom ash led to a decrease in compressive strength. For the highest proportion of clay, compressive strength stayed stable between 1100 and 1115 °C, before slightly increasing at 1120 °C. With 30% clay, compressive strength increased vastly in this temperature range, the increase being especially fast between 1115 and 1120 °C firing temperatures (Fig. 7). The samples containing 30 and 40 wt% clay and fired at 1120 °C exhibited compressive strength superior to the 40 MPa threshold, considered in the literature as a minimum requirement for heavy-duty construction materials [START_REF] Appendino | Production of glass-ceramic bodies from the bottom ashes of municipal solid waste incinerators[END_REF][START_REF] Vichaphund | Properties of ceramic produced from clay and MSW incineration bottom ash mixtures[END_REF], although variability is especially high with the 40% clay formulation. The samples' CTE has been measured using a TMA device, for all mentioned samples except 1115-40, which broke during the preparations for analysis.

The coefficient of thermal expansion (CTE) of samples is relatively low, and is lower for higher firing temperatures of the samples and for higher clay content. The CTE in the temperature range of 25-1000 °C is between 0.7 and 0.85%, which are comparable to alumina (Fig. 8). No significant hysteresis was detected between the heating and the cooling phase, further confirming the stability of the samples. The smooth profile of these curves also indicate the absence or weakness of thermally-induced mineral transformations that might have led to anomalies in CTE (the quartz-point being an example of this type of phenomenon).

Thermal Stability

In a given air/ceramic thermocline storage system, TES material will be subjected to thermal cycling from ambient up to 1000 °C, under air atmosphere. It should therefore be thermally stable between room temperature and 1000 °C, regarding both mineralogy and thermal expansion aspects. Selected samples (1120-40 and 1100-40) have been thermally cycled 3 times from 30 to 1000 °C under air, with 10 °C/min heating ramp and 30 min isotherm in a TG-ATD device (TMA Setsys, from Setaram, TG-ATD Labsys from Setaram). Analysis revealed that the samples exhibited endothermic phenomena around 900 °C, whether it occurs during the heating or the cooling phase (Fig. 9). These are not related to the melting of the material, and are not linked to weight losses. Therefore, it is more realistic to attribute them to reversible mineralogical transformations or specific heat variations. Anyway, those behaviors are highly reproducible, and did not influence the coefficient of thermal expansion in this temperature range. Thus, the material can be considered stable enough in this temperature range.

Mineralogy

The XRD conducted on the 9 samples revealed the presence of the same mineral phases, although their relative proportions changed depending on formulation and firing temperature. There is a low-level noise, as well as much noise for angles less than 20°, indicating glassy/amorphous phases. Moreover, the sheer number of low-intensity peaks indicates a complex mineralogical matrix (Fig. 10).

The semi-quantitative analysis of relative proportion of phases has been conducted in an attempt to link particular macro-properties of the samples to the relative proportions of different phases. 5).

First, it is noticeable that the proportions of Augite and Clinoenstatite were quite stable for all samples, notwithstanding firing temperature or clay content. The low concentrations of Magnesium in both raw materials can be considered as a limit for these phases' development. Wollastonite and Albite proportions varied in wider proportions, with overall increasing presence of Albite for highest clay content in mix, and decreasing presence of Wollastonite for highest clay content. The fact that the melting temperature of Albite is given at around 1115 °C [START_REF] Anovitz | Dry melting of high albite[END_REF] implies that some viscous Albite could react with other phases, or did not completely crystallize, leading to under-estimation on XRD quantification. It is possible that

Discussion

Comparison to a Reference Material: Alumina

Alumina can be considered as a reference for thermal storage applications, due to its high density, high specific heat as well as its high thermal conductivity [START_REF] Auerkari | Mechanical and physical properties of engineering alumina ceramics[END_REF]. Hence, we compared the specific heat and thermal capacity of alumina and several samples of MWIBA ceramics (Fig. 11). The impact of the supposed mineral transformation on the value of specific heat is very noticeable, and occurs for all samples, although it is less important for the 1120-30 sample. This variation of the specific heat around 900 °C, combined with overall lower specific heat and severely lower density lead to a significant gap in terms of volumetric specific heat between MWIBA ceramics and alumina.

This phenomenon should be investigated further, as there is no mention in the scientific littérature of what might cause it.

Benefits and Detriments

These MWIBA ceramics are easy to produce, and do not require rare additives or material. Although the firing temperature is higher than usual for the building bricks industry, it is still in the range of tile firing. Therefore, industrialization at reasonable cost might be feasible. The lower density of these ceramics will limit their relevance as thermal Fig. 11 Specific heat and thermal volumetric capacity of alumina, compared to several samples storage media, so further work has to be done to increase this characteristic, while still limiting the use of more complex additives and techniques. The production costs, as well as the environmental impacts have not been fully quantified to this date, but one can reasonably assume the production cost would be lower than for tiles, due to the low (or even negative) selling price of MWIBA on the market, with transformation processes being similar to those of the building bricks and tiles industries. These ceramics will also have the additional benefit of using byproducts instead of virgin ones, limiting the environmental impacts linked to extraction and purification, especially compared with alumina, for which extraction/ purification process is more complex and requires significant amounts of soda (Bayer process).

The optimal firing temperature has been determined to sit in the range of 1115-1125 °C, and all samples exhibit traces of melting. This temperature is high enough for the material to partially soften or melt, implying viscous flow sintering. Although this might have been beneficial to help filling the porosity of the material, it is a complex phenomenon that also could generate other problems, especially regarding shrinkage and sample deformation.

Shrinkage has not been properly investigated in this work, but it has been noticed that the cylinders tend to creep during firing, with a flattening of their round section, and an overall decrease of their diameter of around 10%. Further work will need to focus on how to limit shrinkage, to ensure minimal deformation of pieces during firing.

Potential for Optimization and Needs for Development

Although these ceramics are already suitable for thermal energy storage applications, increasing the density of the sintered pieces is critical.

One possible way could be to decrease the particle size of the MWIBA, to increase grains' surface area prior to sintering. Most of the works reported in the literature managed to reach higher densities of the sintered samples (around 2.3-2.5 g/cm 3 or even higher), while firing at lower temperature. But this approach will come into conflict with the goal of limiting production costs and complexity.

Another way might be to mill the MWIBA in order to reach a Funk-Dinger particle size distribution [START_REF] Funk | Predictive Process Control of Crowded Particulate Suspensions-Applied to Ceramic Manufacturing[END_REF][START_REF] Fennis | Using particle packing technology for sustainable concrete mixture design[END_REF][START_REF] Shi | A review on mixture design methods for self-compacting concrete[END_REF][START_REF] Zheng | Improved equation of the continuous particle size distribution for dense packing[END_REF]. This distribution would allow to increase green bodies compacity, thus might increase sintered density, as well as reduce shrinkage.

It would be also necessary to investigate the presumed phase transformation occurring around 900 °C, to confirm these materials are suitable for use as thermal storage media up to this temperature range.

Potential for Other Fields

Due to their mechanical properties, these ceramics could be interesting for other applications, especially in the building sector (bricks and tiles), where their moderate density, along with more than average compressive strength, could be relevant. The fact that these ceramics can be produced out of industrial byproducts with conventional methods in the building industry calls for more work on this material, focused on the industrialization potential and industrial relevance of these materials.

However, these applications will require more work regarding the decrease of shrinkage ratio during firing, to ensure minimal deformation of the material. Frost resistance, as well as leaching behavior of the ceramics, should be investigated before concluding about the relevance of this material for building purposes.

Conclusions and Perspectives

The feasibility of producing cylindrical pellets through extrusion of a plastic paste based on MWIBA and wasted clay is validated, using gross milling of the MWIBA and as-received clay powder. The viscous flow sintering ends up to high enough compressive strength. The samples did not show any significant modification of their behavior when cycled from 25 to 1000 °C. Thus, they can potentially be used as materials for sensible heat storage at high temperature.

The fired density of the samples is limited by high porosity, whether is it open or closed. The proximity of firing temperature and melting temperature end up to the formation of liquid phases, which could exit the sample during sintering and/or expand during firing process, forming wide macroscopic pores (black bodies). Increasing the dwell duration during firing might increase density, if the remaining porosity is due to insufficient mobility of the viscous phase.

These ceramics have been designed to be relatively easy to mass-produce. However, their low density limits their volumetric thermal capacity. This could be addressed by decreasing the particle size of the MWIBA, or mixing batches of different particle size to approach a Funk particle distribution, susceptible to increase the green body compacity and reduce shrinkage [START_REF] Kalpakli | Effects of particle size distribution on the refractory properties and corrosion mechanism of ultra-low cement castables[END_REF].

The relatively low thermal capacity of these ceramics can be a barrier for using as thermocline filler material, for prolonged standby period. However, they can still be effective to use the thermocline storage as regenerative devices, in which thermal capacity is of secondary concern, beneath the capabilities in term of heat transfer, for which samples with higher open porosity could be an advantage. These ceramics, although exhibiting moderate properties regarding their potential as thermal energy storage media, exhibit desired traits for the building sector, especially regarding compressive strength, coupled with moderate density and limited thermal conductivity. To investigate their potential further in this field, it would be required to evaluate their leaching behavior, as well as improving the sintering process.

Fig. 1

 1 Fig. 1 Stages of the production process. From left to right: dry powders before mixing, paste formation, extrusion, extruded green body, fired batch and individual piece, fired cylinders prepared for analysis

Fig. 2 a

 2 Fig. 2 a Particle size distribution of milled MWIBA (open circle) and clay (filled circle), b cumulative passing fractions of milled MWIBA (open circle) and clay (filled circle)

Fig. 3 aFig. 4

 34 Fig. 3 a XRD diffractogram of raw clay. filled circle Illite, hollow circle Quartz, filled diamond Kaolinite, open diamond K-Feldspath, b XRD diffractogram of raw MWIBA. Filled circle: quartz, filled triangle: calcite, filled square: Wollastonite, filled square: pyrite

Table 4 1 Fig. 5

 415 Fig. 5 Relative density (dots) and water absorption (squares), for each formulation (20, 30, 40% clay), as a function of firing temperature

Fig. 6 aFig. 7

 67 Fig. 6 a Volumetric thermal capacity and b thermal conductivity as a function of firing temperature, for 20% (blue coloured circle), 30% (red coloured circle) and 40% (black coloured circle) clay, at room temperature (squares) and 390 °C (dots). (Color figure online)

Fig. 8

 8 Fig. 8 Coefficient of thermal expansion as a function of temperature (heating and cooling) fired at 1100, 1115 and 1120 °C

  Five main mineral phases have been detected, identified and quantified, namely Wollastonite (CaSiO 3 , Crystallography Open Database Number 96-900-5779), Quartz (SiO 2 /Si 4 O 8 , COD Numbers 01-078-1852/96-900-6307), Albite (NaAlSi 3 O 8 , COD Number 96-900-1633), Augite [(Si, Al) 2 O 6 , COD Number 96-120-0007] and Clinoenstatite (Mg 2 Si 2 O 6 , COD Number 96-900-2714) (Table

Fig. 9

 9 Fig. 9 Differential thermal analysis graphics for 3 cycles from 25 to 1000 °C: a 1100-40 sample and b 1120-40

Table 1

 1 Composition of several MWIBA samples studied in the literature

	References	Vu et al. [13] Bethanis et al. [1] Barbieri et al. [6]	Schabbach et al. [5]	Aloisi et al. [14] Appendino et al. [15]	Bourtsa-las et al. [3]	Barbieri et al. [16]
	Analysis	ICP AES	ICP AES	XRF	ICP	XRF	AAS	XRD	ICP
	Compound (% dry weight)
	SiO 2 CaO	31.1 10.0	40.1-42.1 18.4-20.0	46.7 26.3	30.3-35.1 23.1-26.1	40.9 12.2	42.5-40.9 16.4-12.2	30.4 26.8	45.1 18.8
	Al 2 O 3 Na 2 O MgO	2.57 2.10 0.28	9.92-14.2 2.60-2.75 1.79-1.90	6.86 4.62 2.22	13.0-13.6 1.94-2.44 2.83-3.73	13.6 10.9 3.1	17.6-13.6 12.9-10.9 2.3-3.1	11.5 3.32 2.64	9.73 4.64 2.20
	Fe 2 O 3 TiO 2 P 2 O 5 SO 3 K 2 O CrO x MnO	4.36 0.83 2.07 0.60 0.86 0.90 -	6.20-6.60 1.00-1.14 1.34-1.44 -0.86-0.92 -0.08	4.69 0.77 -2.18 0.89 --	10.0-11.5 1.07-1.71 1.96-1.90 -0.94-1.45 -0.18-0.17	5.7 1.2 --1.8 --	5.7-5.7 2.5-1.2 --1.8-1.8 -0.1-0.1	13.1 1.39 2.18 3.05 1.33 -0.31	3.84 0.93 --1,29 --
	LOI (%drwt) 5.21	-	11	-	-	-	-	-

Table 2

 2 Formulation parameters of plastic paste

	Formulation	Clay [%dry-weight]	Plasticizer [%dry-weight]	Water [% total weight]
	20%ms-1	20	0.29	38
	30%ms-1	30	0.18	40
	40%ms-0	40	0.00	28
	40%ms-1	40	0.72

Table 3

 3 Chemical composition of clay and MWIBA (ICP-AES)

	Compound [wt%]	Clay	MWIBA
	Al 2 O 3 SiO 2 CaO	27.1 53.7 2.3	6.2 58.7 5.3
	Fe 2 O 3 K 2 O MgO	5.9 5.2 -	6.0 1.8 2.1
	Na 2 O SO x LOI (1000 °C) [wt%]	0.8 -4.9	7.7 1.0 5.2
	d 50 [µm]	16	166 (milled)

Table 5

 5 Relative proportions of the different phases, for 9 samples

	Sample ID Wol-lastonite (%)	Quartz (%)	Albite (%) Clinoen-(%) statite	Augite (%)
	1100-20-4-1	38	15	19	12	17
	1115-20-4-1	21	9	39	13	13
	1120-20-1-1	29	15	25	15	16
	1100-30-4-1	20	14	37	14	15%
	1115-30-4-1	19	15	35	14	18
	1120-30-1-1	29	13	26	16	16
	1100-40-4-0	18	17	37	14	13
	1115-40-4-0	19	17	33	14	17
	1120-40-4-0	17	10	40	15	18

Acknowledgements

This work was supported by the Programme "Investissements d'avenir" (Investment for the Future) of the Agence Nationale de la Recherche (National Agency for Research) of the French State under Award Number ANR-10-LABX-22-01-SOLSTICE. The author would also like to thank Eco-Tech Ceram for their scientific, material and financial support and the laboratory SPCTS CNRS UMR 7315 for their support.