Tristan Benoist

Vojkan Jaksic

Claude-Alain Pillet

Vojkan Jakšić

Energy statistics in open harmonic networks

come

Introduction

This note grew out of a set of remarks concerning energy fluctuations in thermal equilibrium for open harmonic Hamiltonian or stochastic networks made in [START_REF] Jakšić | Entropic fluctuations in thermally driven harmonic networks[END_REF][START_REF] Jakšić | Entropic Fluctuations in Gaussian Dynamical Systems[END_REF] (see Remark 5 in Section 3.2 of [START_REF] Jakšić | Entropic fluctuations in thermally driven harmonic networks[END_REF] and Remark 3.25 in Section 3.6 of [START_REF] Jakšić | Entropic Fluctuations in Gaussian Dynamical Systems[END_REF]) and our attempts to interpret the experimental results of [CINT] in this context. From a theoretical point of view, it turned out that for this very specific question one can go far beyond the large deviation results of [START_REF] Jakšić | Entropic fluctuations in thermally driven harmonic networks[END_REF][START_REF] Jakšić | Entropic Fluctuations in Gaussian Dynamical Systems[END_REF] and the above mentioned remarks: one can compute the statistics of energy fluctuations in steady state network in a closed form in terms of the variance-gamma distribution. Equally surprisingly, for some specific electronic RC-circuit this statistics and the resulting variance-gamma distribution were precisely measured in experiments of Ciliberto, Imparato, Naert, and Tanase [CINT], although the result was not explicitly reported (the focus of the experiment was the measurement of the non-equilibrium entropy production statistics). The purpose of this note is to comment on the theoretical derivation of the heat and work statistics in harmonic networks and RC-circuits, and in the latter case compare the theoretical prediction with experimental results of [CINT].

To describe more precisely the line of thought that has led to this note, consider a stationary stochastic process Z t , t ≥ 0, and set

Q t = Z t -Z 0 .
Obviously, Q t = 0, where • denotes expectation, and one may ask about order one fluctuations of 1 t Q t . A standard way to approach the respective large deviations problem is to study the limit q(α) = lim t→∞ 1 t log e αQt arXiv:1705.03970v1 [math-ph] 10 May 2017

for α in R. Of course, if Z 0 is almost surely bounded, then 1 t Q t has no order one fluctuations and q(α) = 0 for all α. However, in some physically important cases with unbounded Z 0 , one has q(α) = 0 for |α| ≤ α c and q(α) = ∞ for |α| > α c , where the critical α c is finite. This was precisely the case in the above mentioned remarks in [START_REF] Jakšić | Entropic fluctuations in thermally driven harmonic networks[END_REF][START_REF] Jakšić | Entropic Fluctuations in Gaussian Dynamical Systems[END_REF], where Z t -Z 0 was related to the energy flux across a finite part of harmonic Hamiltonian network in thermal equilibrium. If the Large Deviations Principle for 1 t Q t is controlled by the Legendre transform

I(θ) = α c |θ| of q(α), then for any open set O ⊂ R, lim t→∞ 1 t log Q t (tO) = -inf θ∈O I(θ), (1.1)
where Q t denotes the probability distribution of Q t . The rigorous justification of this step in [START_REF] Jakšić | Entropic fluctuations in thermally driven harmonic networks[END_REF][START_REF] Jakšić | Entropic Fluctuations in Gaussian Dynamical Systems[END_REF] required an application of a somewhat subtle extension of the Gärtner-Ellis theorem. The conclusion of the remarks is that a physically important quantity like energy variation per unit time in thermal equilibrium, which has expectation zero, may exhibit order one fluctuations due to the fact that the phase space space is unbounded.

The experimental results of [CINT] indicated that a similar phenomenon happens and can be precisely measured in a suitable electronic RC-circuit. In examining the connection we realized that the large deviation result (1.1) can be considerably refined and that one can express the large time statistics of the energy fluctuations in a finite part of an open harmonic network in closed form in terms of the variance-gamma distribution. That in turn reflected on the experimental findings of [CINT]. In their setting, the large time statistics of energy change turns into the large time statistics of the total power dissipated in the resistors and is also described by the variance-gamma distribution. This allowed us to compare the theoretical prediction of the variance-gamma distribution with the experimental results of [CINT]. The comparison is presented in Subsection 4.3. The theoretical prediction is nearly perfectly matched with experimental results.

Although the mathematical analysis presented in this note is elementary, the conclusions shed a new light on some important aspects of statistical mechanics of open system and can be viewed as a fine statistical characterization of the quantities entering the first law of thermodynamic for finite subsystems of extended harmonic systems. For example, we show that the total kinetic energy of a finite subsystem verifies a universal LDP (1.1) with α c = (k B T ) -1 . These points requires further study and we plan to return to them in the future.

The note is organized as follows. In Section 2 we review the derivation of the variance-gamma distribution and describe its basic properties. Hamiltonian harmonic networks are discussed in Section 3. Stochastic harmonic networks and electronic RC-circuits are discussed in Section 4.

2 Probabilistic setting 2.1 Variance-gamma distribution of Gaussian vectors

Let X and Y be independent R n -valued Gaussian vectors with zero mean and covariance M > 0, and let L > 0 be a matrix on R n . The probability distribution of the random variable

Q = X • LX -Y • LY
is called variance-gamma distribution of the pair (L, M ). We shall denote it by Q vg . Obviously,

Q vg (] -∞, s]) = x•N y≤s dN (x) dN (y)
where N denotes the standard (zero mean, unit covariance) Gaussian measure on R n and the matrix N is given by

N = 2L 1/2 M L 1/2 . (2.2)
In this subsection we shall compute the density of Q vg in terms of the modified Bessel functions of the second kind

K ν (z) = ∞ 0 e -z cosh t cosh(νt)dt, ν ∈ C, | arg z| < π 2 , (2.3)
and the function

R n x → κ(x) =   n j=1 λ 2 j x 2 j   1/2
, where 0 < λ 1 ≤ λ 2 • • • ≤ λ n denote the repeated eigenvalues of the matrix (2.2).

Proposition 2.1 The density of Q vg is given by

f vg (s) = |s| (n-1)/2 S n-1 K (n-1)/2 |s| κ( k) dσ( k) (2πκ( k)) (n+1)/2 , (2.4)
where σ denotes the Lebesgue measure on the unit sphere S n-1 in R n .

Remark 1. Although this result is well-known, due to the lack of a convenient reference we give a proof below.

Remark 2. It follows immediately from the definition (2.3) that K ν (x) > 0 for x > 0 and ν ∈ R. Hence (2.4) implies that f vg (s) > 0 for s = 0. For in depth discussion of Bessel functions we refer the reader to Watson's treatise [W]. In this note we shall make use of the following facts:

(1) The only singularity of K ν is a logarithmic branching point at z = 0, with the leading asymptotics

K ν (z) ∼ Γ(ν) 2 z 2 -ν , (Re ν > 0, z → 0) . Moreover, the function [0, ∞[ x → x ν K ν (x) is monotone decreasing. It follows that f vg is a monotone decreasing function of |s| satisfying 0 < f vg (s) ≤ f vg (0) = Γ n-1 2 4(2π) (n+1)/2 S n-1 dσ( k) κ( k) .
(2) For fixed ν the asymptotic expansion

K ν (z) ∼ π 2z 1/2 e -z ∞ k=0 a k (ν) z k , (|z| → ∞, | arg z| < 3π/2) (2.5)
holds with a 0 (ν) = 1 and

a k (ν) = (4ν 2 -1 2 )(4ν 2 -3 2 ) • • • (4ν 2 -(2k -1) 2 ) k! 8 k . (3) For half-integer ν = m + 1 2 , m ∈ N, one has K ν (z) = 2z π 1/2 k m (z). (2.6)
where k m denotes the modified spherical Bessel function of the second kind

k m (z) = π 2 (-z) m 1 z d dz m e -z
z .

Remark 3. If N = λI n , where I n denotes the identity matrix on R n and λ > 0, then the integrand in (2.4) is a constant and thus

f vg (s) = |S n-1 | K (n-1)/2 |s| λ |s| (n-1)/2 (2πλ) (n+1)/2 , (2.7)
where

|S n-1 | = 2π n/2 /Γ(n/2) is the area of S n-1 .
Remark 4. In the special case n = 2 and N = λI n it follows from (2.7) and (2.6) that

f vg (s) = e -|s|/λ 2λ . (2.8)
More generally, for n = 2, (2.4) reduces to

f vg (s) = 2π 0 e -|s|/κ(ϕ) 4πκ(ϕ) dϕ, with κ(ϕ) = λ 2 1 sin 2 (ϕ) + λ 2 2 cos 2 (ϕ). Elementary manipulations lead to f vg (s) = θ 2π π 0 1 -ε 2 1 + ε cos ϕ e -θ|s| √ 1+ε cos ϕ dϕ, (2.9)
where

ε = λ 2 2 -λ 2 1 λ 2 2 + λ 2 1 , θ = 1 2 λ -2 1 + λ -2 2 1/2 . Proof of Proposition 2.1. Setting U = N -1/2 L 1/2 (X -Y ) and V = N -1/2 L 1/2 (X + Y ) one has Q = U • N V .
Note that U and V are independent identically distributed Gaussian random vectors with zero mean and unit covariance. It follows that the characteristic function of Q vg can be written as

χ vg (α) = R 2n
e iαx•N y dN (x) dN (y).

(2.10)

Performing the Gaussian integral over y and invoking the invariance of N under orthogonal transformations further yields

χ vg (α) = R n e -α 2 2 x•N 2 x dN (x) = R n e -α 2 κ(x) 2 /2 dN (x).
It follows that

f vg (s) = R e -iαs R n e -α 2 κ(x) 2 /2 dN (x) dα 2π .
Performing the Gaussian integral over α, we derive

f vg (s) = (2π) -1/2 R n e -s 2 /2κ(x) 2 dN (x) κ(x) ,
and passing to the spherical coordinates r = x and k = x/ x we obtain

f vg (s) = (2π) -(n+1)/2 S n-1 ∞ 0 e -1 2 (r 2 +r -2 s 2 κ( k) -2 ) r n-2 dr dσ( k) κ( k) .
The change of variable

r 2 = |s|e t /κ( k) gives ∞ 0 e -1 2 (r 2 +r -2 s 2 κ( k) -2 ) r n-2 dr = |s| κ( k) (n-1)/2 ∞ 0 e - |s| κ( k) cosh(t) cosh((n -1)t/2)dt,
and comparison with (2.3) yields the result. 2

Variance-gamma distribution of a Gaussian process

Consider a stationary R n -valued Gaussian stochastic process

X t = (X (1) t , . . . , X (n)
t ), t ≥ 0, with mean zero and covariance M > 0. Given a matrix L > 0 on R n we set

Q t = X t • LX t -X 0 • LX 0 , and denote by Q t the law of Q t . Setting again N = 2L 1/2 M L 1/2 and U t = N -1/2 L 1/2 (X t -X 0 ), V t = N -1/2 L 1/2 (X t + X 0 ), one has Q t = U t • N V t .
The Gaussian random vectors U t and V t are identically distributed, with zero mean, but they do not need to be independent. To deal with this point we introduce the n × n-matrix

∆(t) with entries ∆ ij (t) = X (i) t X (j) 0
and formulate

Assumption (A) lim t→∞ ∆(t) = 0.
Proposition 2.2 Suppose that (A) holds. Then Q t → Q vg weakly as t → ∞. Moreover, this convergence is accompanied with the following Large Deviations Principle: for any open set O ⊂ R,

lim t→∞ 1 t log Q t (tO) = -inf θ∈O I(θ), (2.11)
where the rate function is given by

I(θ) = |θ| λ n .
Proof. In terms of the R 2n -valued random variable Z t = (U t , V t ) one has

Q t = Z t • N Z t , N = 1 2 0 N N 0 .
Z t is Gaussian with mean zero and covariance

M t = I n -A(t) B(t) B T (t) I n + A(t) ,
where A(t) = A T (t) and B(t) = -B T (t) are the n × n matrices given by

A(t) = N -1/2 L 1/2 (∆(t) + ∆ T (t))L 1/2 N -1/2 , B(t) = N -1/2 L 1/2 (∆(t) -∆ T (t))L 1/2 N -1/2 . The characteristic function of Q t is χ t (α) = R 2n e iαz• N z e -1 2 z• Mtz dz det(2π M t ) 1/2 . Assumption (A) gives that lim t→∞ M t = I 2n , (2.12)
and comparing with (2.10) we deduce that for any α ∈ R,

lim t→∞ χ t (α) = χ vg (α).
The weak convergence Q t → Q vg then follows from Lévy's continuity theorem.

To prove the LDP, it suffices to consider the case O =]a, b[. If 0 ∈ O, then tO is an increasing function of t and since Q vg has a strictly positive density, the weak convergence

Q t → Q vg implies that Q t (tO) ≥ Q t (O) → Q vg (O) = c > 0 for 1 ≤ t → ∞. Hence, 0 ≥ 1 t log Q t (tO) ≥ 1 t log c → 0
and the result follows. So we may assume that 0 ∈ O. We consider the case 0 < a < b, the other case is similar.

From the convergence (2.12) we infer that for large enough t > 0 there exists 0 < δ t < 1/2 such that

lim t→∞ δ t = 0, (1 -δ t )I 2n ≤ M t ≤ (1 + δ t )I 2n .
(2.13)

Denoting by N ±δ the Gaussian measure on R 2n with zero mean and covariance (1 ± δ)I 2n and setting

Q ±δ (A) = z• N z∈A dN ±δ (z), it follows from (2.13) that 3 -n Q -δt ≤ Q t ≤ 3 n Q +δt
for large enough t > 0. Further noticing that

Q ±δt (tO) = Q vg (τ t O), τ t = t 1 ± δ t ,
and τ t /t → 1 as t → ∞ we conclude that it suffices to show that

lim τ →∞ 1 τ log Q vg (τ O) = - a λ n .
(2.14)

We proceed to prove (2.14).

Obviously,

Q vg (τ O) = τ b a f vg (τ s)ds,
and from the asymptotic expansion (2.5) we infer that there is a constant C > 0 such that for τ large enough, all s ∈ [a, b], and all k ∈ S n-1 ,

C -1 1 √ τ e - τ |s| κ( k) ≤ K (n-1)/2 τ |s| κ( k) ≤ C 1 √ τ e - τ |s| κ( k) .
(2.15)

Since max k∈S n-1 κ( k) = λ n , the formula (2.4) and the upper bound in (2.15) give, for large τ ,

Q vg (τ O) ≤ C √ τ S n-1 dσ( k) (2πκ( k)) (n+1)/2 b a s (n-1)/2 e -τ s/λn ds ≤ C S n-1 dσ( k) (2πκ( k)) (n+1)/2 (b -a)a (n-1)/2 √ τ e -τ a/λn
from which the upper bound

lim sup τ →∞ 1 τ log Q vg (τ O) ≤ - a λ n ,
follows. To prove the corresponding lower bound, fix 0 < < b -a and pick a neighborhood O of (0, 0, . . . , 1) in S n-1 such that

inf k∈O κ( k) ≥ λ n -.
Replacing integration over S n-1 with integration over O in (2.4), the lower bound in (2.15) gives

Q vg (τ O) ≥ C -1 √ τ O dσ( k) (2πκ( k)) (n+1)/2 b a s (n-1)/2 e -τ s/(λn-) ds ≥ C -1 √ τ O dσ( k) (2πκ( k)) (n+1)/2 a+ a s (n-1)/2 e -τ s/(λn-) ds ≥ C -1 O dσ( k) (2πκ( k)) (n+1)/2 (a + ) (n-1)/2 √ τ e -τ (a+ )/(λn-) ,
from which we infer lim inf

τ →∞ 1 τ log Q vg (τ O) ≥ - a + λ n - .
Taking ↓ 0 we derive the result. 2

3 Hamiltonian harmonic networks

Setup

A Hamiltonian harmonic network is specified by a quadruple (G, m, ω, C), where G is a countable set describing vertices of the network, m = (m x ) x∈G and ω = (ω x ) x∈G are elements of ∞ R (G) describing the masses and frequencies of the oscillators, and C is a non-negative bounded operator on2 R (G) describing the coupling between the oscillators. We denote by C xy = δ x , Cδ y the matrix elements of C w.r.t. the standard basis {δ x } x∈G of 2 R (G). We identify m and ω with the induced multiplication operators on 2 R (G) and assume that, as such, m and ω are strictly positive1 . The finite energy configurations of the network are described by vectors in the Hilbert space

K = 2 R (G) ⊕ 2 R (G) 2 R (G) ⊗ R 2
, where the pair (p, q) ∈ K describes the momenta and positions of the oscillators. The Hamiltonian of the network is

H(p, q) = x∈G p 2 x 2m x + m x ω 2 x q 2 x 2 + 1 2
x,y∈G C xy q x q y , which is the quadratic form associated to the linear map H : K → K defined by

H = 1 2 m -1 0 0 mω 2 + C .
H is a bounded strictly positive operator on K so that φ, ψ H = φ, Hψ .

defines an inner product on K. We denote by K H the Euclidean space K equipped with this inner product.

The Hamilton equation of motion writes φ = Lφ with

L = 0 -(mω 2 + C) m -1 0 .
Note that L is a bounded skew-symmetric operator on K H . It follows that the finite energy trajectories of the network are described by the action of the one parameter group of orthogonal transformations R t → e tL of K H .

In the sequel we shall make use of the following dynamical assumption:

Assumption (B)

Internal and external work, heat

We fix a finite subset G 0 ⊂ G and define the total/kinetic/potential energy of the corresponding subnetwork by

H G 0 (p, q) = x∈G 0 p 2 x 2m x + m x ω 2 x q 2 x 2 + 1 2
x,y∈G 0

C xy q x q y , K G 0 (p) = x∈G 0 p 2 x 2m x , V G 0 (q) = x∈G 0 m x ω 2 x q 2 x 2 + 1 2
x,y∈G 0

C xy q x q y .
To avoid trivialities, we shall assume that there exists x ∈ G 0 and y ∈ G c 0 such that C xy = 0. The force acting on a vertex x ∈ G 0 decomposes into an internal component

F int x (q) = -m x ω 2 x q x - y∈G 0 C xy q y
and an external one

F ext x (q) = - y∈G c 0 C xy q y .
The work done by the external forces over the time interval [0, t] is equal to the change in internal energy of the subnetwork

W ext (t) = x∈G 0 t 0 F ext x (q(s)) qx (s)ds = H G 0 (p(t), q(t)) -H G 0 (p(0), q(0)).
(3.17)

Adding to it the work done by the internal forces we get the total work done on the subnetwork which equals the change in its kinetic energy:

W ext (t) + W int (t) = x∈G 0 t 0 F int x (q(s)) + F ext x (q(s)) qx (s)ds = K G 0 (p(t)) -K G 0 (p(0)).
The change in potential energy of the subnetwork is V G 0 (q(t)) -V G 0 (q(0)) and it is the opposite of the work done by the internal forces,

-W int (t) = - x∈G 0 t 0 F int x (q(s)) qx (s)ds = V G 0 (q(t)) -V G 0 (q(0)).
Thinking of the subnetwork as an open system connected to a reservoir, we can interpret Eq. (3.17) as a statement of the first law of thermodynamics: the increase of the internal energy of the subnetwork equals the amount of heat ∆Q t extracted from the reservoir.

The equilibrium case

In this subsection we assume that the network is in thermal equilibrium at temperature T > 0. This means that X t = (p(t), q(t)) is a stationary Gaussian process over the Hilbert space K with mean zero and covariance

M = k B T m 0 0 (mω 2 + C) -1 , (3.18)
where k B is the Boltzmann constant.

Since p x (t)p y = e tL (δ x ⊕ 0), M (δ y ⊕ 0) ,

Assumption (B) implies (recall (3.16)) that lim t→∞ p x (t)p y = 0.
In the same way one shows that lim t→∞ p x (t)q y = lim t→∞ q x (t)p y = lim t→∞ q x (t)q y = 0, and so the process (X t ) t≥0 satisfies Assumption (A). Applying Proposition 2.2 to the quadratic form K G 0 we infer that the increase in kinetic energy of the subnetwork converges in law, as t → ∞, to the variance-gamma distribution of the pair (M, K G 0 ). One easily shows that

N = 2K 1/2 G 0 M K 1/2 G 0 = k B T I |G 0 | ,
and so, by Remark 3, the limiting law has density (2.7) with λ = k B T and n = |G 0 |. The LDP that accompanies this convergence holds with the rate function

I(θ) = |θ| k B T .
Applying Proposition 2.2 to H G 0 we deduce in the same way that the heat ∆Q t absorbed by the subnetwork over the time interval [0, t] converges in law, as t → ∞, to the variance-gamma distribution of the pair (M, H G 0 ). In this case

N = 2H 1/2 G 0 M H 1/2 G 0 = k B T I |G 0 | 0 0 V 1/2 G 0 V -1 V 1/2 G 0
, where V = mω 2 + C and V G 0 denote its restriction to G 0 with Dirichlet boundary conditions on the complement G c 0 . The density of the limiting law is given by (2.4) and, in this case, depends on the details of the interaction. It is interesting to notice that, by Schur's complement formula,

V 1/2 G 0 V -1 V 1/2 G 0 = (I |G 0 | -V -1/2 G 0 CV -1 G c 0 CV -1/2 G 0 ) -1 > I |G 0 |
so that the LDP for heat holds with rate function

I(θ) = |θ| (1 -ϑ)k B T
,

where ϑ = V -1/2 G 0 CV -1 G c 0 CV -1/2 G 0 > 0.

The non-equilibrium case

Let G = a G a
be a finite partition of G and

K = a K a
the induced decomposition of Hilbert space K. We restrict C to 2 R (G a ) by setting δ x , C a δ y = C xy if x, y ∈ G a and zero otherwise. This leads to (non-interacting) subnetworks (G a , m a , ω a , C a ). Suppose that the a-th subnetwork is in thermal equilibrium at temperature T a , let M a be the corresponding covariance and set

M = a M a .
Suppose also that the full network was initially described by a Gaussian random field over K with mean zero and covariance M . The initial probability measure P evolves under the interacting dynamics determined by L and at time t, P t is a Gaussian probability measure with mean zero and covariance

M t = e tL M e tL * . Set L = a L a ,
where L a denotes the generator of the dynamics of the a-th subnetwork. Assuming that Ca C a is a trace class operator and that Condition (B) holds, a standard trace-class scattering argument [LS] gives that the strong limits + . The Gaussian measure P + over K with mean zero and covariance M + is the non-equilibrium steady state of the network associated to the initial state determined by M . Obviously, P t → P + weakly as t → ∞.

W + = s -lim t→∞ e -t L e tL
One can now repeat the analysis of the previous section for the stationary Gaussian process X t = (p(t), q(t)) with mean zero and covariance M + . Both the change in kinetic and total energy of a finite subnetwork converge in law to a variance-gamma distribution as t → ∞. However, in this case there is no simple expression for the λ j 's. These eigenvalues depend on detailed dynamical properties of the model encoded in the scattering matrix W + . Even in the simplest example of a harmonic crystal analyzed in [JPS2, Section 3.2], not much is known about the λ j 's outside perturbative regimes.

Z t = e tA Z 0 + t 0 e (t-s)A Bdw s .
We shall assume that A is stable, that is, that all its eigenvalues have strictly negative real part, and set

M = ∞ 0 e sA BB * e sA * ds. (4.20)
Note that M is non-negative and satisfies the Lyapunov equation

AM + M A * + BB * = 0.
The pair (A, B) is called controllable if the smallest A-invariant subspace that contains Ran B is equal to R n . If (A, B) is controllable, then M > 0 and the centered Gaussian measure P on R n with covariance M is the unique stationary measure for the process (Z t ) t≥0 . Moreover

∆ ij (t) = Z (i) t Z (j) 0 = (e tA M ) ij ,
satisfies Assumption (A).

We now turn to two examples of the above framework: stochastic harmonic networks and stochastic RC-circuit.

Stochastic harmonic networks

We start with a Hamiltonian harmonic network (G, m , ω, C) where the vertex set G is assumed to be finite. To each x ∈ G we associate two parameters describing a thermal reservoir coupled to the oscillator at x: the temperature T x > 0 and the relaxation rate γ x ≥ 0 (γ x = 0 meaning: no reservoir attached to x). The external force acting on the oscillator at x has the usual Langevin form

F x (p, q) = (2γ x k B T x ) 1/2 ẇx -γ x p x ,
where the ẇx are independent standard white noises. The equation of motion of stochastic network is

m x qx = p x , ṗx = -m x ω 2 x q x - y∈G C xy q y + F x (p, q).
With Z t = (p(t), q(t)) and obvious notation it can be cast to the form (4.19) with The matrix A is stable and we shall assume that the pair (A, B) is controllable. Distributing Z 0 according to the centered Gaussian measure with covariance M given by (4.20) yields a stationary Gaussian process satisfying Assumption (A). Note that in the equilibrium case, T x = T for all x ∈ G, M is given by (3.18) which is now a finite matrix.

A = -γ -(mω 2 + C) m -1 0 , B = (2γk B T ) 1/2 0 0 0 . T 2 R 2 u 2 T 1 R 1 u 1 C 2 C 1 C V 1 V 2
The variations of the total/kinetic energy of the network can be analyzed in the same way as in Subsections 3.3 and 3.4. In particular, in both cases one has the convergence in law to an appropriate variance-gamma distribution. The advantage of the stochastic setting over the Hamiltonian setting described in Section 3 is that the covariances are finite-dimensional matrices and thus accessible to numerical computations. In some important examples they can be explicitly computed. In the next subsection we analyze one such example.

Stochastic RC-circuit

The RC-circuit subject of the experimental investigations of [CINT] is described in Figure 1. We set

V = V 1 V 2 , C = C + C 2 -C -C C + C 1 , R = R 1 0 0 R 2 , T = 2k B T 1 0 0 T 2 .
The circuit analysis gives that the voltages satisfy the SDE

dV t = AV t dt + Bdw t ,
where A = -C -1 R -1 and B = C -1 (T R -1 ) 1/2 . One easily verifies that the matrix A is stable and that the pair (A, B) is controllable. The stochastic term is due to Nyquist's thermal noise in the resistors. We denote by Q 1 (t) and Q 2 (t) the powers dissipated in the resistances R 1 and R 2 at time t. They satisfy, in Stratonovich sense,

dQ 1 (t) = V 1 (t)((C 1 + C)dV 1 (t) -CdV 2 (t)), dQ 2 (t) = V 2 (t)(-dV 1 (t) + (C 2 + C)dV 2 (t)).
We are interested in the total dissipated power defined by

Q t = Q 1 (t) + Q 2 (t),
where we set Q 0 = 0. By direct integration,

Q t = 1 2 (V t • CV t -V 0 • CV 0 ) .
The eigenvalues of

N = ∞ 0
C 1/2 e sA BB * e sA * C 1/2 ds can be computed by diagonalizing A and evaluating the integral. The eigenvalues are given by

λ ± = k B 2 T 1 + T 2 ± |T 1 -T 2 | 1 -Λ 2 ,
where

Λ = √ R 1 R 2 C 1 2 (R 1 + R 2 )C + 1 2 (R 1 C 1 + R 2 C 2 ) ∈]0, 1[.
Recalling Remark 4 after Proposition 2.1, we can now complete the theoretical analysis of the total dissipated power. Figure 2: Empirical and theoretical distribution of Q t with the experimental parameters of [CINT]: R 1 = R 2 = 10 8 Ω, C = 10 -10 F , C 1 = 6.8×10 -10 F and C 2 = 4.2×10 -10 F . The heat units are in multiples of k B T 2 . For the empirical distribution t = 0.2 s, the theoretical distribution is the limiting t → ∞ variance-gamma distribution3 . Both at equilibrium (left pane, T 1 = T 2 = 296K) and out of equilibrium (right pane, T 1 = 88K, T 2 = 296K), the theoretical prediction (black dashed lines) matches almost perfectly the empirical distribution computed from the experimental data (grey dots). Note that we have taken the theoretical prediction as a reference for the normalization of the empirical distribution. The discrepancy between the theoretical and empirical distribution for |Q t | ≥ 10k B T 2 is due to the lack of experimental data in this energy range.

  M + = W + M W *

Figure 1 :

 1 Figure 1: The RC-circuit of [CINT]: The resistances R 1 and R 2 are kept at temperatures T 1 and T 2 and are coupled via capacitance C. C 1 and C 2 are the inner circuit capacitances. u 1 and u 2 denote the fluctuating Johnson-Nyquist voltage. The voltages V 1 and V 2 are amplified by two low noise amplifiers.

  In this section we shall consider R n -valued Gaussian stochastic processes (Z t ) t≥0 that arise as solutions of SDE's of the formdZ t = AZ t +Bdw t , (4.19) where A and B are n × n and n × m matrices and w(t) a standard R m -valued Wiener process. The solution of the Cauchy problem (4.19) with initial condition Z 0 is

	4 Stochastic networks
	4.1 Preliminaries

The operator A acting on a Hilbert space is strictly positive whenever φ, Aφ ≥ λ φ

for all φ and some λ > 0.

Since the largest eigenvalue of A is smaller than -12 and exp(-.2 × 12) ≤ .091, the experimental data is comparable to the limit t → ∞.

Acknowledgments. We are grateful to Sergio Ciliberto and David Stephens for useful discussions. We also wish to thank Sergio Ciliberto for making available to us the raw data of the experiment reported in [CINT] which allowed one of us (T.B.) to analyze the data and plot the Figure 2 in Section 4.3. The research of T.B. was partly supported by ANR-11-LABX-0040-CIMI within the program ANR-11-IDEX-0002-02 and by ANR contract ANR-14-CE25-0003-0. The research of V.J. was partly supported by NSERC. The work of C.-A.P. was partly funded by Excellence Initiative of Aix-Marseille University -A*MIDEX, a French "Investissements d'Avenir" programme.

Thermal equilibrium case T 1 = T 2 = T . Then λ -= λ + = k B T and it follows from Proposition 2.2 and Eq. (2.8) that Q t converges in law as t → ∞ to the probability distribution with density

and Proposition 2.2 and Eq. (2.9) yield that Q t converges in law as t → ∞ to the probability distribution with density

where

.

Figure 2 summarizes the comparison of these theoretical predictions with the experimental data from [CINT].


Louisiane Verger

Olivier Dargaud

Mathieu Chassé

Nicolas Trcera

Gwenaëlle Rousse

Laurent Cormier

Synthesis, Properties and Uses of Chromium-Based Pigments from the Manufacture de Sèvres

Keywords: Pigments, chromium, color, Rietveld refinements, porcelain

Chromium (Cr) is at the origin of a wide variety of coloration (green, pink and brown) in porcelain glazes from the French Manufacture de Sèvres. This element was introduced for the first time at the factory in 1804, just a few years after its discovery by Louis-Nicolas Vauquelin. Pigments for glazes were developed at the laboratory of the factory, leading to a palette of 76 references. While the synthesis protocol and the nature of the precursors of these pigments are kept in laboratory notebooks, the products have never been characterized. In this work, the pigments composed of Cr were characterized by X-ray diffraction (XRD), UVvisible spectroscopy and X-ray absorption near edge structure spectroscopy (XANES). The analyses reveal the presence of common crystalline phases, and open new perspectives for the synthesis of pigments containing chromium at the Manufacture de Sèvres.

Introduction

Chromium was first discovered and isolated by Louis-Nicolas Vauquelin in 1797, from a crocoite (PbCrO4) sample from Siberia (N. L. Vauquelin 1797;Nicolas Louis Vauquelin 1797;[START_REF] Spiesser | Nicolas Louis Vauquelin -La découverte de deux nouveaux éléments : le chrome (1797) et le glucinium (béryllium 1798)[END_REF]. The name chromium, as proposed by Haüy (Vauquelin 1797) a prominent mineralogist who previously described the minerals emerald and beryl, originates from the word χρῶµα (khrôma) meaning color in greek referring to its ability to impart various coloration to minerals, such as green for eskolaite Cr2O3 and red for lead chromate PbCrO4. Soon after this discovery, by analyzing ruby, spinel and emerald, Vauquelin has evidenced that a low content of chromium was responsible for the red and green colors of these three minerals (N. L. Vauquelin 1797;Vauquelin 1798a;Vauquelin 1798b;[START_REF] Spiesser | Découverte du chrome et du béryllium dans l'émeraude par Louis-Nicolas Vauquelin, 200ème anniversaire[END_REF]. Since then, numerous different colors have been attributed to this element due to its various oxidation states and chemical environments. Based on these discoveries, Vauquelin has already foreseen in his report to the « Journal de la Société des Pharmaciens de Paris » in 1797 (N. L. Vauquelin 1797), that the use of chromium oxides will provide "solid and nice colors to Painting and Enameller's art".

Chromium discovery paved the way for its rapid use in porcelain manufacturing. At that time in France, one of the most innovative institutions was the Manufacture Impériale de Sèvres; a manufacture producing since 1740 fine quality porcelain, mastering all stages of production and with a special attention to colors. In the early 19 th century, under the vigorous impulsion of Alexandre Brongniart, chemist and director of the manufacture, chromium was used as pigment for deep green paints and background glazes. Brongniart might have known chromium possibilities thanks to his closeness with Haüy in Paris and his work at the Mines de Paris at the very same moment, as shown by its publication a few months earlier in the same journal as Vauquelin [START_REF] Bouillon-Lagrange | Note lithologique -Sur la colline de Champigny près de Paris[END_REF]. The development of chromium green colors appeared relatively simple since the color is obtained with the direct addition of the precipitated oxide in a lead flux to produce a glaze [START_REF] Salvétat | Spectroscopic study of chromium, iron, OH, fluid and mineral inclusions in uvarovite and fuchsite[END_REF][START_REF] D'albis | Traité de la Porcelaine de Sèvres[END_REF]. The first trials of chromium based glazes at the manufacture date back to 1802 [START_REF] Brongniart | Sur les couleurs obtenues des oxydes métalliques, et fixées par la fusion sur les différents corps vitreux[END_REF][START_REF] Préaud | The S{è}vres porcelain manufactory: Alexandre Brongniart and the triumph of art and industry[END_REF]. In 1806, the new green glazes obtained with chromium were used in production and subsequently replaced copper based green glazes [START_REF] Brongniart | Préparation chimique des couleurs vitrifiables et des métaux[END_REF]. During the first half of the 19 th century, many chromates were also identified as potentially useful pigments: lead chromate (red), barium chromate (yellow-green) [START_REF] Brongniart | Préparation chimique des couleurs vitrifiables et des métaux[END_REF], potassium chromate (yellow) [START_REF] Salvétat | Spectroscopic study of chromium, iron, OH, fluid and mineral inclusions in uvarovite and fuchsite[END_REF], mercury chromate (red/violet and green) (Godon-Saint-Memin 1804) and iron chromate [START_REF] Brongniart | Préparation chimique des couleurs vitrifiables et des métaux[END_REF].

In 1845, chromium green under-glaze is eventually introduced as a genuine marker of Sèvres porcelains with a mark stamped for each product [START_REF] Préaud | The S{è}vres porcelain manufactory: Alexandre Brongniart and the triumph of art and industry[END_REF] (Figure 1 (a)).

The very same year, Brongniart called on a young scientist, Jacques-Joseph Ebelmen to be deputy director of the factory. When Brongniart died in 1847, Ebelmen succeeded Brongniart as the head of the factory. Besides his work to improve porcelain production, Ebelmen conducted research on the synthesis of minerals. Drawing its inspiration from solution chemistry, Ebelmen developed the crystalline growth method by the use of fluxes: precursors are mixed with a flux, such as boric acid or borax, and brought at high temperature in platinum foils. The flux plays the role of solvent at high temperature. Using this method, Ebelmen succeeded to synthesize some minerals for the first time, such as spinels [START_REF] Ebelmen | Sur une nouvelle méthode pour obtenir des combinaisons cristallisées par la voie sèche, et sur ses applications à la reproduction des espèces minérales[END_REF]. This crystalline growth method using fluxes initiated by Ebelmen is now commonly used to synthesize various compounds, including spinels [START_REF] Halenius | Structural relaxation around Cr$^{3+}$ and the red-green color change in the spinel (sensu stricto)-magnesiochromite (MgAl$_2$O$_4$-MgCr$_2$O$_4$) and gahnite-zincochromite (ZnAl$_2$O$_4$-ZnCr$_2$O$_4$) solid-solution series[END_REF][START_REF] Bosi | Blue spinel crystals in the MgAl$_2$O$_4$-CoAl$_2$O$_4$ series: Part II. Cation ordering over short-range and long-range scales[END_REF][START_REF] D'ippolito | Blue spinel crystals in the MgAl$_2$O$_4$-CoAl$_2$O$_4$ series: Part I. Flux growth and chemical characterization[END_REF][START_REF] Andreozzi | Compositional dependence of cation distribution in some synthetic (Mg,Zn)(Al,Fe$^{3+}$)O$_4$ spinels[END_REF], garnets [START_REF] Andrut | The crystal chemistry of birefringent natural uvarovites. Part III. Application of the superposition model of crystal fields with a characterization of synthetic cubic uvarovite[END_REF][START_REF] Lowell | Synthesis of Uvarovite Using a Sodium-Potassium-Borate Flux[END_REF] or layered oxides [START_REF] Hobbie | Zum Aufbau von NaRhO2[END_REF][START_REF] Yubuta | High-resolution electron microscopy for incommensurate structure of K x RhO 2 crystal[END_REF]). After the sudden death of Ebelmen (1852), Alphonse Louis Salvétat, in charge of chemical works at the Manufacture de Sèvres until 1880, benefited from Ebelmen's experience and continued his work. The size of crystals obtained by Ebelmen or Salvétat was small, they were only observable under an optical microscope. However, since pigments are ground to be used in porcelain decoration, this new synthesis method was perfectly suited to obtain new pigments composed of chromium.

Brongniart and his successors are at the origin of series of new pigments composed of chromium oxides: research conducted at the laboratory leads to a palette of pigments which was finalized in the middle of the 20 th century. Chromium is the most used coloring element in the palette at Sèvres: among the 138 pigments regularly prepared, 76 are composed of chromium oxides [START_REF] D'albis | Traité de la Porcelaine de Sèvres[END_REF] and used in a wide range of temperature from 900°C to 1400°C. Pigments are regularly synthesized (Fig. 1 (b-c)) and mixed in different uncolored frits. The mixture is painted on porcelains and fired at high or low temperature. The uncolored frit vitrifies during firing and its role is to fix the color on the porcelain. Glazes are obtained, i.e. glassy decorative layers on the porcelain surface. Since 1882, laboratory notebooks have been kept to follow the production (Figure 1 (d)).

While the synthesis protocol of pigments is known (D'Albis 2003), the crystalline phases obtained after the thermal treatment have never been characterized and chromium speciation remained unknown. Such knowledge is essential to address (i) environmental issues associated with the toxicity of Cr 6+ , (ii) technical issues in order to better understand the history of synthesis of the pigments and (iii) scientific issues surrounding the origin of the colors of pigments. Furthermore, such analysis is mandatory to improve the coloring properties of the pigments from Sèvres and to update their synthesis. In this work, we characterize exhaustively the pigments composed of chromium from the Manufacture de Sèvres. By doing so, we have determined the crystalline nature of these pigments, allowing to propose a classification based on the Cr-bearing phases, and we have evidenced the chromium speciation, allowing to improve the synthesis processes and to understand the origin of the coloration.

Materials and methods

The pigments characterized in this work were taken from the collection of pigments of the Sèvres manufacture. The collection is composed of 138 references, indexed from 10001 to 10138, all available and regularly synthesized between a few grams to ten kilos. Only the 76 composed of chromium were characterized. Most of them are prepared by a classic ceramic route: metallic oxides are mixed, ground and then calcined at 1280 °C or 1400 °C in porcelain crucibles. The detailed historic synthesis of Sèvres pigments can be found in d'Albis (D'Albis 2003).

X-ray powder diffraction (XRD) data were collected using a PANalytical X'Pert PRO diffractometer with Cu Kα radiations (λCu,Kα1 = 1.54056 Å, λCu,Kα2 = 1.54439 Å) in the 10°-130° 2θ range, with 2θ steps of 0.008° and a counting time of 25 s per step. Rietveld refinements [START_REF] Rietveld | A profile refinement method for nuclear and magnetic structures[END_REF] were carried on some pigments out using the FullProf Suite program [START_REF] Rodríguez-Carvajal | Recent advances in magnetic structure determination by neutron powder diffraction[END_REF]. This method is based on the least squares refinement of structural model against the experimental XRD power pattern, and allows to obtain meaningful structural information (lattice parameters, atomic positions, bond lengths…) as well as phase quantification. The following parameters are refined: zero-shift, scale factor, background coefficients, lattice parameters, peak shape, atomic positions, atom occupancies, isotropic thermal parameter, microstructural parameters.

Diffuse reflectance spectra were recorded using a Perkin Elmer Lambda 1050 spectrometer equipped with an integrating sphere, between 4000-36,000 cm -1 . The pigments were finely ground and the powdered samples were deposited on a support. A perfectly flat tablet of BaSO4 was used as a white reference material. Diffuse reflectance spectra of a powder are considered as representative of the absorption spectra but not completely equivalent due to particle size, refractive index and sample absorption coefficient (Lindberg and Snyder, Am. Mineral. 57(1972)485-493). A Kubelka-Munk conversion applied to a diffuse reflectance spectrum enables to obtain the absorption spectrum [START_REF] Kubelka | Ein Beitrag zur Optik der Farbanstriche[END_REF].

X-ray absorption near edge structure (XANES) spectra at the Cr K-edge were collected on the LUCIA beamline at the SOLEIL synchrotron facility in Saint-Aubin (France). The synchrotron operated with a storage ring current of 450 mA and an energy of 2.75 GeV. The layout of the LUCIA beamline is described in [START_REF] Vantelon | The LUCIA beamline at SOLEIL[END_REF]. XANES spectra were collected using a Si(311) double crystal monochromator. The monochromator was calibrated against the maximum intensity of the first inflection of the first derivative of a metallic Cr standard (5989 eV). XANES measurements were performed in the energy range 5940-6200 eV. The energy steps were fixed to 2, 0.05, 0.2, 0.5 and 1 eV for energy ranges of [5940 -5983] eV, [5983 -5999] eV, [5999 to 6040] eV, [6040 -6120] eV and [6120 -6200] eV, respectively. Spectra were collected under vacuum at room temperature (around 10 -2 mbar) in X-ray fluorescence mode using a 4-elements Silicon Drift Diode detector. Samples were prepared in order to obtain pellets from powdered materials and were mounted on carbon tapes. Two to four spectra were recorded for each sample with a counting time of about 1s per step. Multiple Cr K-edge XANES spectra recorded for each sample were merged to obtain average spectra before background subtraction and normalization using the ATHENA software (Ravel & Newville 2005).

Results and discussion

Classification of chromium-based pigments

The color of each of the 76 pigments containing Cr of the Sèvres factory has been quantified using the chromatic parameters x and y in the system defined in 1976 by the International Commission on Illumination (abbreviated CIE). These parameters have been calculated from Kubelka-Munk converted diffuse reflectance spectra using the equation given by Wyszecki and Stiles [START_REF] Wyszecki | Color Science: Concepts and Methods, Quantitative Data and Formulae[END_REF] (Figure 2). The pigments were also characterized by Xray diffraction. Based on the nature of the crystalline phases determined by X-ray diffraction, five different groups of pigments were identified.

The simplest chromium pigment: eskolaite (Cr2O3)

Eskolaite (Cr2O3) is the first chromium oxide used as a pigment at the Manufacture de Sèvres [START_REF] Brongniart | Préparation chimique des couleurs vitrifiables et des métaux[END_REF][START_REF] Ducrot | Second Empire et IIIème République, de l'audace à la jubilation -Sèvres, une histoire céramique[END_REF], giving a vibrant deep green color. Mixed with an uncolored frit, the pigment can be used directly to decorate porcelain artifacts. Thanks to its high stability in silicate melts, eskolaite was devised in decoration fired at high temperature, up to 1380°C. Eskolaite, mixed with calcium carbonate, quartz, kaolin or porcelain paste, is the precursor of four pigments (10035, 10065, 10085 and 10100). The resulting pigments after the thermal treatment consist of crystalline phases (Cr2O3 and SiO2) mixed with a glass formed by the partial vitrification of quartz, kaolin or of porcelain paste. The purpose of this process was certainly to lighten the green color of eskolaite. Four pigments using uranium or beryllium precursors are no longer used and, consequently, were not investigated. : uvarovite (Ca3Cr2(SiO4)3) A series of pigment is historically obtained from potassium dichromate (K2Cr2O7), calcium carbonate (CaCO3) and quartz (SiO2). Uvarovite (Ca3Cr2(SiO4)3), also known as Victorian green [START_REF] Colomban | The Destructive/Non-Destructive Identification of Enameled Pottery, Glass Artifacts and Associated Pigments-A Brief Overview[END_REF], belongs to the garnet group and is the only chromium-bearing phase identified in these pigments by XRD after calcination. Trivalent chromium is localized in octahedral sites in this mineral [START_REF] Salvétat | Spectroscopic study of chromium, iron, OH, fluid and mineral inclusions in uvarovite and fuchsite[END_REF]). Pigments composed of uvarovite are characterized by a green color, lighter than the one due to eskolaite (Figure 2).

Another green pigment

The first synthesis of uvarovite at Sèvres is proven by a mention in the laboratory notebooks dating back to 1885. Furthermore, this kind of pigment is only used on one porcelain type (new hard-paste porcelain), the use of which started in 1882.

From green to pink along the solid solution Cr2O3-Al2O3

Twelve pigments are obtained from eskolaite (Cr2O3) and corundum (Al2O3). The main phase identified by XRD in those pigments belongs to the solid solution Al2-xCrxO3, which color varies with chromium content from green to pink [START_REF] Gaudry | From the green color of eskolaite to the red color of ruby: an X-ray absorption spectroscopy study[END_REF][START_REF] Gaudry | Structural and electronic relaxations around substitutional Cr$^{3+}$ and Fe$^{3+}$ ions in corundum[END_REF].

Chromium(III) is in octahedral sites in the solid solution and it was previously shown by [START_REF] Gaudry | From the green color of eskolaite to the red color of ruby: an X-ray absorption spectroscopy study[END_REF]) that the modification of color can be ascribed to both the crystal field and the covalency of the Cr-O bonds, resulting from the broadening of the 3d band by Cr-Cr coupling.

The first synthesis of corundum doped with chromium were conducted by Ebelmen in the kilns of the Manufacture de Sèvres and presented to "the Academy" in 1847 [START_REF] Ebelmen | Sur une nouvelle méthode pour obtenir des combinaisons cristallisées par la voie sèche, et sur ses applications à la reproduction des espèces minérales[END_REF]).

Ebelmen obtained it by mixing alumina with borax, adding chromium oxide (1% of the alumina mass), and calcining it on a platinum foil in a porcelain kiln. By this process, he obtained small transparent red crystals, embedded in an amorphous part. He isolated the crystals by the addition of hydrochloric acid at 70 °C or 80 °C. Ebelmen's works were continued by Salvétat to design new colored porcelain glazes. In the laboratory notebooks, numerous synthesis using alumina and potassium dichromate are reported. The pink color obtained is called "rose Salvétat" or "rose S.V.T".

Tin-based pink pigments CaSnSiO5 and SnO2 doped with Cr

One pigment, called at Sèvres Pink or Pinck, is used to color porcelain glazes in a particular shade of pink [START_REF] Faurel | Pink pigment optimization by resonance Raman spectroscopy[END_REF][START_REF] Colomban | Differentiation of antique ceramics from the Raman spectra of their coloured glazes and paintings[END_REF]. The main phase identified by XRD in this pigment (pigment 10081) is Cr-doped malayaite (CaSnSiO5). The name Pink was kept since the introduction of this pigment in Sèvres from a British sample. At the beginning of the 19 th century, the British industry had developed a pink pigment to color under-glazes on earthenware. The synthesis being secret, Brongniart asked Malaguti, chemist at the Manufacture de Sèvres in 1838, to find the composition of this pigment. Malaguti succeeded to obtain the chemical composition and to identify the elements necessary to retrieve the pink color, as well as unneeded constituents which were added on purpose to mask the real pigment elements (Gay-Lussac & Arago 1836; Bouillon-Lagrange et al. 1836). He identified the key role of calcium, tin, silica and chromium. His researches also led him to synthesize a new lilac pigment, by simply calcining tin oxide and chromium oxide, leading to Cr-doped cassiterite (SnO2). Malayaite and cassiterite were detected by XRD in one and five pigments, respectively, giving pink to lilac colors (Figure 2). Chromium speciation in the malayaite and cassiterite structures is less straightforward than in the corundum structure. Authors agree on the presence of Cr 4+ in both structures, mainly substituting for Sn 4+ in octahedral symmetry [START_REF] Lopez-Navarrete | Oxidation state and localization of chromium ions in Crdoped cassiterite and Cr-doped malayaite[END_REF][START_REF] Cruciani | Malayaite ceramic pigments: A combined optical spectroscopy and neutron/X-ray diffraction study[END_REF][START_REF] Doménech | Electrochemical Detection of High Oxidation States of Chromium(IV andV) in Chromium-Doped Cassiterite and Tin-Sphene Ceramic Pigmenting Systems[END_REF][START_REF] Heyns | Evidence for the existence of Cr(IV) in chromium-doped malayaite Cr$^{4+}$:CaSnOSiO$_4$: a resonance Raman Study[END_REF]. In the case of malayaite, a very small amount of Cr 4+ in tetrahedral symmetry in substitution for Si 4+ was also detected by optical spectroscopy [START_REF] Cruciani | Malayaite ceramic pigments: A combined optical spectroscopy and neutron/X-ray diffraction study[END_REF][START_REF] Lopez-Navarrete | Oxidation state and localization of chromium ions in Crdoped cassiterite and Cr-doped malayaite[END_REF]. Furthermore, the presence of Cr 3+ substituting for Sn 4+ with the formation of oxygen vacancies to accommodate the charge was evidenced by optical spectroscopy and X-ray and neutron diffraction [START_REF] Cruciani | Malayaite ceramic pigments: A combined optical spectroscopy and neutron/X-ray diffraction study[END_REF]). The distribution of chromium between tetrahedral and octahedral sites and its valence variability are certainly the reasons why the color of pigment composed of Cr-doped malayaite is known to be hardly reproducible, and strongly dependent on synthesis conditions such as thermal treatment, used fluxes or precursors [START_REF] Faurel | Pink pigment optimization by resonance Raman spectroscopy[END_REF][START_REF] Harisanov | Influence of crystallinity on chromatic parameters of enamels coloured with malayaite pink pigments[END_REF][START_REF] Cordoncillo | Influence of some mineralizers in the synthesis of sphene-pink pigments[END_REF].

A wide color palette obtained with spinels

Spinels were synthesized for the first time at the Manufacture de Sèvres by Ebelmen, around the middle of 19 th century. In 1847, he reported the synthesis of Cr-doped red spinel (Cr 3+ :MgAl2O4), chromite (FeCr2O4) and magnesiochromite (MgCr2O4) in the wood-kiln for porcelain at Sèvres. Ebelmen used also kilns from a manufacturer of buttons, Jean-Félix Bapterosses, which can maintain a constant temperature. With those kilns he succeeded to obtain colorless gahnite (ZnAl2O4) and Cr-doped gahnite. In the laboratory notebooks from 1882, syntheses of spinels are frequent, representing more than 50% of the laboratory's production between 1882 and 1904.

The addition of various transition metals such as chromium or cobalt in the spinel structure leads to a wide range of colors [START_REF] Biagioni | The systematics of the spinel-type minerals: An overview[END_REF]. Spinels are indeed identified by XRD in a majority of pigments (59.2 %). Transition elements combined with chromium at the Sèvres laboratory are cobalt, iron and manganese. When chromium is used in low concentration in gahnite (ZnAl2O4) or spinel (MgAl2O4), pink colors are obtained [START_REF] Verger | Spectroscopic properties of Cr$^{3+}$ in the spinel solid solution ZnAl$_{2-x}$Cr$_{x}$O$_4[END_REF][START_REF] Bosi | Zn-O tetrahedral bond length variations in normal spinel oxides[END_REF][START_REF] Ikeda | Optical spectra of synthetic spinels in the system MgAl$_2$O$_4$-MgCr$_2$O$_4[END_REF]). The addition of cobalt enables to explore colors from green-blue to blue [START_REF] D'ippolito | Blue spinel crystals in the MgAl$_2$O$_4$-CoAl$_2$O$_4$ series: Part I. Flux growth and chemical characterization[END_REF]. The use of iron and manganese leads to pinkbrown to black shades (Figure 2).

Principle of use of chromium oxides deduced from physical and chemical properties

In this part, the most relevant observations and results obtained from the study of the series of pigments composed of chromium are reported, in order to distinguish common properties to the pigments and to open new perspectives for the synthesis of pigments.

Chromium speciation

Among the 76 pigments regularly synthesized at the laboratory, 18 were historically obtained from potassium dichromate (K2Cr2O7), a potential source for Cr 6+ . The use of Cr 6+ species is limited nowadays because of their high toxicity [START_REF] Wetterhahn | Mechanism of chromium(VI) carcinogenesis[END_REF]). Chromium (VI) is classified among carcinogenic agents, with the main effects on the respiratory system. Therefore, syntheses using K2Cr2O7 as a precursor were abandoned at Sèvres laboratory.

However, these pigments were devised to give unique colors (light green, pink and lilac), and their interruption results in a considerable loss in porcelain decorations.

Pigments composed of uvarovite (3.1.2), corundum (3.1.3), malayaite or cassiterite (3.1.4) doped with chromium were obtained from K2Cr2O7. One sample of each historical type was characterized by XANES spectroscopy at the Cr-K edge, respectively 10098 (uvarovite), 10059 (corundum), 10081 (malayaite) and 10095 (cassiterite). Pre-edge features are due to the s-d electronic transitions and is particularly sensitive to the presence of Cr 6+ down to a few ppm [START_REF] Peterson | Differential redox and sorption of Cr (III/VI) on natural silicate and oxide minerals: EXAFS and XANES results[END_REF][START_REF] Zachara | Chromium speciation and mobility in a high level nuclear waste vadose zone plume[END_REF][START_REF] Villain | XANES Determination of Chromium Oxidation States in Glasses: Comparison With Optical Absorption Spectroscopy[END_REF]. The absence of a symmetry center in the tetrahedral site of Cr 6+ induces a strong absorption in the pre-edge region, as illustrated by the spectra of K2Cr2O7 in Figure 3 (a). Spectra of pigments 10098 and 10059 exhibit features typical of Cr 3+ in uvarovite and corundum [START_REF] Farges | Chromium speciation in oxide-type compounds: application to minerals, gems, aqueous solutions and silicate glasses[END_REF][START_REF] Gaudry | Structural and electronic relaxations around substitutional Cr$^{3+}$ and Fe$^{3+}$ ions in corundum[END_REF]. The pre-edge area of pigment 10081 is characterized by two features, at different energies from the one of Cr 3+ in octahedral sites in Cr2O3. These two features are still not assigned, because of the complexity of Cr speciation in malayaite, and the poor solubility of Cr in malayaite which gives a bad signal to noise ratio [START_REF] Lopez-Navarrete | Oxidation state and localization of chromium ions in Crdoped cassiterite and Cr-doped malayaite[END_REF][START_REF] Cruciani | Malayaite ceramic pigments: A combined optical spectroscopy and neutron/X-ray diffraction study[END_REF]).

The pre-edge XANES spectra did not reveal any detectable Cr 6+ in these pigments. Due to the sensitivity of the method [START_REF] Peterson | Differential redox and sorption of Cr (III/VI) on natural silicate and oxide minerals: EXAFS and XANES results[END_REF][START_REF] Zachara | Chromium speciation and mobility in a high level nuclear waste vadose zone plume[END_REF][START_REF] Villain | XANES Determination of Chromium Oxidation States in Glasses: Comparison With Optical Absorption Spectroscopy[END_REF], we can conclude that the Cr 6+ precursor is reduced during the synthesis in Cr 3+ and Cr 4+ . The use of K2Cr2O7 as a starting material is thus not mandatory. The synthesis of pigments 10081 and 10098 have recently been modified at the laboratory of Sèvres by using Cr2O3 as a precursor, based on the work of Andreola et al. [START_REF] Andreola | Synthesis of chromium containing pigments from chromium galvanic sludges[END_REF].

The spectrum of the historical pigment 10095 (cassiterite) presents a strong absorption band in the pre-edge region, characteristic of the presence of Cr 6+ . Historically, to prepare the pigment after the calcination of the precursors, the product was grounded in water and the pigment was isolated by decantation. Former synthesis used K2Cr2O7 as a precursor and the water exhibited a yellow coloration, indicative of Cr 6+ in solution [START_REF] De Ruiter | Selective optical recognition and quantification of parts per million levels of Cr$^{6+}$ in aqueous and organic media by immobilized polypyridyl complexes on glass[END_REF]. For the purpose of the study, pigment 10095 was washed with water and removed by decantation. This process was repeated until the yellow coloration of the solution disappeared. Historical pigment 10095 before and after this process was characterized by UV-visible spectroscopy (Figure 3 (b)). The absorption band at 27500 cm -1 (due to ligand-metal charge transfer between O 2-and Cr 6+ ) disappears when the pigment is washed. Such observation indicates that Cr 6+ species in pigment 10095 are highly soluble, and easily removed with water. The final concentration of Cr 6+ in pigment 10095 was therefore uncertain in historical pigment preparation and was strongly dependent upon the washing step.

Our results indicate that no Cr 6+ is needed for the final coloration, pleading for a substitution of chromates by less hazardous chromium species in the synthesis protocol.

Thermal treatment applied to the pigment

We have observed in several pigments from the spinel group that homogeneity in the pigment is not achieved after the thermal treatment, since a single spinel phase was not obtained. As an example, Figure 4 shows the XRD diagram of the pigment 10075, belonging to the spinel group. This pigment is obtained from Cr2O3, Fe2O3, ZnO and Al(OH)3 after a heat treatment at 1280 °C, the furnace being switched off as soon as the temperature is reached (no subsequent plateau). The heat treatment procedure was developed during the 19 th century: pigments were calcined in wood-fired kilns, along with porcelain artifacts. The heat ramp was controlled to be slow enough to fire the porcelain and, once the desired firing temperature was reached, the kiln was turned off. It would have been particularly challenging at that time to maintain a constant temperature with such kilns (Figure 4 (b)). After this thermal processing, several spinel phases are detected, each exhibiting distinct peak widths in the XRD patterns.

However, when an annealing treatment at 1400 °C during 20 h is performed, the pattern of a single-phase with sharp diffraction peaks is obtained, corresponding to a spinel phase of homogeneous composition (Al,Fe,Zn)(Al,Cr,Fe)2O4. This shows that the firing treatment established in the 19 th century does not allow a complete reaction during the synthesis of the pigment.

This protocol is still in use at the Manufacture de Sèvres. However, as we have shown for the pigment 10075, such heat treatment leads to multi-phase pigments. The proportion of the different phases is likely to vary according to the milling conditions, the quantity of pigment synthesized, and the kiln local properties and performances. The use of a plateau during the thermal treatment in modern furnaces will enable the formation of a single phase and improve the reproducibility between different productions of the same pigment. Figuier [START_REF] Figuier | Les merveilles de l'industrie ou description des principales industries modernes[END_REF].

Use of solid solutions

Various types of solid solutions were detected in pigments from Sèvres, such as Al2-xCrxO3 (corundum structure) or MAl2-xCrxO4 (M = Zn, Mg, Co) (spinel structure). However, the pigments do not cover the whole range of chromium concentrations, i.e. x = 0 to 2. As a typical example, the end-members CoCr2O4 and CoAl2O4 were not detected by XRD for pigments mainly composed of the solid solution CoAl2-xCrxO4. Along this solid solution, spinel colors can vary from blue (x = 0) to greenish blue (x = 2) (Figure 5 Blue vertical tick marks indicate Bragg positions in space group Fd! " m for CoAl2-xCrxO4 and R! " c for Al2-xCrxO3. The peak indexed with a star corresponds to a quartz (SiO2) impurity.

Color attenuation

Pigments synthesized at the Manufacture de Sèvres are never composed of a single crystalline phase. Quartz (SiO2) was detected by XRD in almost every pigment (e.g., 10118 Figure 5 (c)), though it is not introduced as a precursor. This is probably due to a contamination from the porcelain crucible in which precursors are calcined. The impact of this contamination is limited because pigments are grounded with colorless frit composed of quartz in order to obtain the glaze.

As shown in the previous part (3.2.3), precursors are usually not introduced in stoichiometric proportion in the synthesis protocol of spinels MAl2-xCrxO4 (M=Zn, Mg, Co). For instance, the composition of pigment 10118 is the most distant from the stoechiometric line and is mainly composed of Al2O3, with a minority of CoAl1.2Cr0.8O4 (Figure 5(c)). Due to the presence of colorless Al2O3, the blue color is muted in pigment 10118 compared to pure CoAl1.2Cr0.8O4. Similarly, malayaite and uvarovite are prepared with an Al2O3 excess. The addition of an excess of colorless precursors such as Al2O3 softens the saturation of the color and enables the modification of the shade of the expected pure phase.

The objective at Sèvres is not to obtain pigments composed of pure crystalline phases, but rather to get pigments with the most interesting and attractive colors, with different shades.

Far from being an imperfection, the multiphasic aspect of Sèvres's pigments is a characteristic of the production of the Manufacture de Sèvres and, likely, it was purposely designed to obtain a vast and diverse palette of colors. This is also a signature of the process and the history of the production of pigments.

Conclusion

The synthesis of chromium-based pigments at the laboratory of the Manufacture de Sèvres is linked to the discovery of this element at the end of the 18 th century and the research conducted to synthesize minerals during the 19 th century. This work puts in perspective the different type of pigments still synthesized nowadays with the history of the manufacture.

Chromium is used as a precursor to obtain green and pink colors, and together with other transition metals, green-blue, brown and black colors. Despite the variety of colors in the pigments synthesized at the laboratory, six crystallized phases are evidenced, allowing the differentiation of five main groups: (1) eskolaite (Cr2O3), ( 2) uvarovite (Ca3Cr2(SiO4)3), (3) solid solution eskolaite-corundum (Al2-xCrxO3), (4) Cr-doped cassiterite (SnO2) and malayaite (CaSnSiO5), and (5) spinels. This works rationalizes the link between the crystallized phases identified in the pigment and their color on a wide set of pigments. The absence of Cr 6+ in the phases identified opens new perspectives for the synthesis of pigments avoiding environmentally hazardous Cr 6+ as a precursor. Eventually, this study highlights different approaches to set up a new palette of pigments composed of chromium, simplified and more complete: (i) synthesis of a unique crystalline phase, (ii) controlled variation of Cr occupancy over the whole solid solution in order to cover all possible shades, (iii) monitored dilution of every phase in colorless amorphous matrices (quartz, alumina for instance) in order to lighten the color.

Figure 1 :

 1 Figure 1: (a) Actual marker of Sèvres porcelains, the green color is obtain with chromium oxide Cr2O3; (b-c) the storehouse of pigments in the laboratory of Sèvres manufacture; (d) snapshot of a laboratory notebook of 1893, describing the synthesis of a pigment.

Figure 2 :

 2 Figure 2: Chromaticity diagram (y as a function of x) with the coordinates of the 76 pigments from the Manufacture de Sèvres containing Cr. The coordinates are calculated from the corresponding diffuse reflectance spectra. The percentage indicates the proportion of each type of pigments. Pigments obtained from uranium and beryllium precursors are not represented.

Figure 3 :

 3 Figure 3: Analysis of pigments using potassium dichromate as a precursor. (a) XANES spectra at the Cr K-edge of pigments 10098, 10059, 10081, 10095 and a reference of potassium dichromate. (b) Diffuse reflectance spectra of pigment 10095, and pigment 10095 after washing.

Figure 4 :

 4 Figure 4: (a) XRD pattern of pigment 10075 before (green) and after (black) the annealing treatment at 1400 °C during 20h. Grey vertical tick marks indicate Bragg positions of spinel (Al,Fe,Zn)(Al,Cr,Fe)2O4. (b) Illustration of a wood-kiln in the Sèvres factory from Les

  Figure 5(a) illustrates the distribution of the nominal molar compositions of equivalent oxide

Figure 5 :

 5 Figure 5: (a) Ternary diagram (mol%) CoO, Cr2O3, Al2O3. The blue dots correspond to the precursors molar percentage used to synthesize the pigments. The orange line corresponds to composition along the solid solution CoAl2-xCrxO4. The two pictures illustrate the color of spinel phases CoCr2O4 and CoAl2O4. (b-c) Rietveld refinements for pigments 10094 and 10118. The green dots are the observed intensities; the black line is the calculated intensity.

Figure 1 :

 1 Figure 1: (left) Remission function F calculated from the di↵use reflectance spectra of pigment 10100 and (right) XRD diagram of pigment 10100 with the main phases identified: Cr 2 O 3 , SiO 2 quartz and SiO 2 cristobalite.

Figure 2 :

 2 Figure 2: (left) Remission function F calculated from the di↵use reflectance spectra of pigment 10098 and (right) XRD diagram of pigment 10098 with the main phases identified: Ca 3 Cr 2 (SiO 4 ) 3 and CaF 2 .

Figure 3 :

 3 Figure 3: (left) Remission function F calculated from the di↵use reflectance spectrum of pigment 10046 and pigment 10059 and (right) XRD diagram of pigment 10046 with the main phases identified: Al 0.1 Cr 1.9 O 3 and Al 2 O 3 .

Figure 4 :

 4 Figure 4: (left) Remission function F calculated from the di↵use reflectance spectra of pigment 10081 and (right) XRD diagram of pigment 10081 with the main phases identified: CaSnSiO 5 , SiO 2 quartz and SnO 2 .

Figure 5 :

 5 Figure 5: (left) Remission function F calculated from the di↵use reflectance spectra of pigment 10095 and (right) XRD diagram of pigment 10095 with the main phases identified: CaSnO 3 , SnO 2 and SiO 2 cristobalite.

Figure 6 :

 6 Figure 6: (left) Remission function F calculated from the di↵use reflectance spectra of pigment 10045. The high absorption in the near-UV region is due to the presence of ZnO, a semi conductor compound with a gap energy located at 26600 cm 1 .(right) XRD diagram of pigment 10045 with the main phases identified: ZnAl 1 .59Cr 0 .41O 4 and ZnO.

Table 1 :

 1 Summary of the di↵erent type of pigments composed of chromium and synthesised at the Manufacture de Sèvres. Pigments 10028, 10029, 10030 and 10064 are not included (contain beryllium or uranium).

	Example	10100	10098		10046		10059		10081		10095	10045	10024	10048				10014
	Number of pigments Reference of pigments	4 10035, 10065, 10085 ¤ , 10100	5 10096, 10097, 10098, 10099, 10120	5 10005, 10027, 10046, 10123 ¶ , 10124 ¶		7 10003 § , 10018, 10019, 10020 10059, 10061, 10134	1 10081			5 10079, 10086, 10087, 10088, 10095	4 10015, 10016, 10045, 10058	2 10023, 10024	8 10048, 10050, 10071, 10094, 10116, 10118, 10119, 10122	10007 # , 10009, 10010, 10012 # , 10014,	13 10022, 10041, 10049, 10051, 10067,	10121, 10130, 10135	10004, 10008, 10025, 10026, 10039,	18 10040, 10043, 10044, 10052, 10054, 10074, 10075, 10077, 10078, 10084,	10091, 10101, 10132	2 SiO 4 is detected as main phase
	Color	Green	Light green	Green			Pink	Pink			Purple-pink	Pink	Pink	Greenish blue to blue	Pink, greenish blue,	blue	Dark green, brown,	black
	Chromium valency	Cr 3+	Cr 3+	Cr 3+			Cr 3+	Cr 3+ , Cr 4+			Cr 3+ , Cr 4+	Cr 3+	Cr 3+	Cr 3+		Cr 3+		Cr 3+
	Main phase composed of chromium	Cr 2 O 3 Eskolaite	Ca 3 Cr 2 (SiO 4 ) 3 Uvarovite	Al 2-x Cr x O 3 (x > 1.9)	Solid solution	eskolaite-corundum	Al 2-x Cr x O 3 (x < 0.2)	CaSnSiO 5 : Cr	Sn and Cr-based	pigment	SnO 2 : Cr	ZnAl 2 O 4 : Cr	MgAl 2 O 4 : Cr	CoAl 2-x Cr x O 4		(Mg,Al,Co,Zn)(Al,Cr,Co) 2 O 4		4 2 O (Mg,Al,Mn,Fe,Co,Zn)(Al,Cr,Co,Fe)	¤ calcium chromium oxide CaCr 2 O 4 is detected	¶ a spinel phase is detected as minority phase

10075

Spinel § a spinel phase and eskolaite are detected as minority phases # willemite Zn

Table 2 :

 2 Chromaticity parameters L*, a*, b* of the pigments calculated from the di↵use reflectance spectra.

Table 3 :

 3 Rietveld refinement on pigment 10094: crystallographic data and atomic positions of the two phases CoAl 1.4 Cr 0.6 O 4 and Al 1.9 Cr 0.1 O 3 . Biso, atomic positions and occupancy are not fitted for the phase Al 1.9 Cr 0.1 O 3 because of the low intensity of the characteristic peaks of the phase. Pigment 10094 2 =2.434 CoAl 1.4 Cr 0.6 O 4 Fd3m R Bragg = 11.2 % a = 8.17661(5) Å V = 546.664(6) Å3 Cr 0.1 O 3 R3c R Bragg = 41.2 % a = 4.77631(16) Å c = 13.0377(7) Å V = 257.583(18) Å3

	Atom Wycko↵ position Occupancy x	y		z	Biso (	Å2 )
	Co	8a	1	0.125	0.125	0.125	1.01
	Al	16d	0.7	0.5	0.5		0.5	0.49
	Cr	16d	0.3	0.5	0.5		0.5	0.49
	O	32e	1	0.26402(12) 0.26402(12) 0.26402(12) 0.86
	Al 1.9 Atom Wycko↵ position Occupancy x	y	z	Biso (	Å2 )
	Al	12c		0.95	0.5	0.5 0.35217 1
	Cr	12c		0.05	0.5	0.5 0.35217 1
	O	18e		1	0.69365 0	0.25	1

Table 4 :

 4 Rietveld refinement on pigment 10118: crystallographic data and atomic positions of the two phases CoAl 1.2 Cr 0.8 O 4 and Al 2 O 3 . Pigment 10118 2 =7.190 CoAl 1.2 Cr 0.8 O 4 Fd3m R Bragg = 8.88 % a = 8.20253(10) Å V = 551.878(12) Å3 R3c R Bragg = 9.06 % a = 4.75930(2) Å c = 12.99156(6) Å V = 254.846(2) Å3

	Atom Wycko↵ position Occupancy x	y	z	Biso (	Å2 )
	Co	8a	1	0.125	0.125	0.125	1.05
	Al	16d	0.6	0.5	0.5	0.5	1.41
	Cr	16d	0.4	0.5	0.5	0.5	1.41
	O	32e	1	0.2622(5) 0.2622(5) 0.2622(5) 1.00
	Al 2 O 3 Atom Wycko↵ position Occupancy x	y	z	Biso (	Å2 )
	Al	12c	1	0.5	0.5 0.35224(3) 0.56
	O	18e	1	0.69136(15) 0	0.25	0.35

Acknowledgements

This work was supported by the Réseau Francilien sur les oxydes fonctionnels (DIM Oxymore) and the Région Ile-de-France. L. Verger acknowledges C. Bouttaz and D.

Ratajczak for their help at the laboratory of Sèvres.

Supporting Information