Cédric Bernardin

François Huveneers
email: huveneers@ceremade.dauphine.fr

Joel L Lebowitz
email: lebowitz@math.rutgers.edu

Liverani Carlangelo

Stefano Olla
email: olla@ceremade.dauphine.fr

Carlangelo Liverani
email: liverani@mat.uniroma2.it

Ceremade

Green-Kubo formula for weakly coupled system with dynamical noise

Keywords: Thermal conductivity, Green-Kubo formula, coupling expansion, small noise

Introduction

Energy transport in nonequilibrium macroscopic systems is described phenomenologically by Fourier's law. This relates the energy flux J, at the position r in the system, to the temperature gradient at r, via J = -κ∇T . The computation of the thermal conductivity κ, which depends on the temperature and the constitution of the system, from the underlying microscopic dynamics is one of the central mathematical problems in nonequilibrium statistical mechanics (see [START_REF] Bonetto | Fourier's Law: a challenge to theorist[END_REF] [START_REF] Lepri | Thermal Conduction in classical low-dimensional lattices[END_REF] [START_REF] Dhar | Heat Transport in low-dimensional systems[END_REF] and references therein).

The Green-Kubo (GK) formula gives a linear response expression for the thermal conductivity. It is defined as the asymptotic space-time variance for the energy currents in an infinite system in equilibrium at temperature T = β -1 , evolving according to the appropriate dynamics. It is therefore always nonnegative. For purely Hamiltonian (or quantum) dynamics, there is no proof of convergence of the GK formula (and consequently no proof of Fourier law). One way to overcome this problem is to add a dash of randomness (noise) to the dynamics [START_REF] Bernardin | Transport Properties of a Chain of Anharmonic Oscillators with Random Flip of Velocities[END_REF].

In the present work we explore the resulting GK formula and start an investigation of what happens when the strength of the noise, ς, goes to zero.

Our basic setup is a chain of coupled systems described in Section 2. Each uncoupled system (to which we will refer as a cell) evolves according to Hamiltonian dynamics (like a billiard, a geodesic flow on a manifold of negative curvature, or an anharmonic oscillator...) perturbed by a dynamical energy preserving noise, with intensity ς. We will consider cases where the only conserved quantity for the dynamics with ς > 0, is the energy. The cells are coupled by a smooth nearest neighbor potential εV . We assume that the resulting infinite volume Gibbs measure has a convergent expansion in ε for small ε. We are interested in the behaviour of the resulting GK formula for κ(ε, ς) given explicitly by equation (3.1) below, for small ς and ε keeping the temperature β -1 and other parameters fixed.

We start in Section 3 by noting that for ς > 0, the GK formula is well defined and has a finite upper bound [START_REF] Bernardin | Transport Properties of a Chain of Anharmonic Oscillators with Random Flip of Velocities[END_REF]. We do not however have a strictly positive lower bound on κ(ε, ς) except in some special cases [START_REF] Bernardin | Transport Properties of a Chain of Anharmonic Oscillators with Random Flip of Velocities[END_REF]. We expect however that κ(ε, ς) > 0 whenever ε > 0, ς > 0, i.e. there are no (stable) heat insulators. The situation is different when we let ς → 0. In that case we have examples where κ(ε, ς) → 0 (disordered harmonic chains [START_REF] Bernardin | Small perturbation of a disordered harmonic chain by a noise and an anharmonic potential[END_REF]), and where κ(ε, ς) → ∞ (periodic harmonic systems).

To make progress in elucidating the properties of κ(ε, ς), when ς → 0, we carry out in Section 4 a purely formal expansion of κ(ε, ς) in powers of ε: κ(ε, ς) = n≥2 κ n (ς)ε n . This is formal for several reasons, among which is the fact that space-time correlations entering in the GK formula involve non-local functions and depend themselves on ε.

This, together with a similar formal expansion in Section 7, are the only non rigorous sections of the paper (apart from a technical assumption at the end of Section 5), yet they allow to identify the basic objects of interest.

In Section 5 we start the study of the objects loosely introduced in Section 4. More precisely we show that a formal operator is in fact well defined and is a Markov generator.

We then investigate in Section 6 the structure of the term κ 2 (ς), which we believe, but do not prove, coincides with the lim ε→0 κ(ε, ς)/ε 2 . We show in certain cases that κ 2 (ς) is finite and strictly positive for ς > 0 by proving that it is equal to the conductivity obtained from a weak coupling limit in which there is a rescaling of time as ε -2 t (cf. [START_REF] Liverani | Toward the Fourier law for a weakly interacting anharmonic crystal[END_REF][START_REF] Liverani | Macroscopic Energy diffusion after weak coupling[END_REF]). We argue further that lim ς→0 κ 2 (ς) exists and is closely related to the weak coupling macroscopic conductivity obtained for the purely Hamiltonian dynamics ς = 0 from the beginning. The latter is computed for a geodesic flow on a surface of negative curvature, and is strictly positive [START_REF] Dolgopyat | Energy transfer in a fast-slow Hamiltonian system[END_REF]. The rigorous study of the zero noise limit would require the extension to random perturbations of the theory developed for deterministic perturbations in [START_REF] Butterley | Smooth anosov flows: correlation spectra and stability[END_REF][START_REF] Butterley | Robustly invariant sets in fiber contracting bundle flows[END_REF]. A first step in such a direction has been achieved recently by S.Dyatlov and M.Zworski for a noise given by the full Laplace-Beltrami operator [32]. For the noise considered here a similar result should be possible by arguing as in the discrete time case [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF].

The identification of κ 2 (ς) with the weak coupling limit conductivity (suggested by H. Spohn [31], see also [START_REF] Huveneers | Energy fluctuations in simple conduction models[END_REF]) gives some hope that the higher order terms, can also be shown to be well defined and studied in the limit ς → 0. This could then lead (if nature and mathematics are kind) to a proof of the convergence and positivity of the GK formula for a Hamiltonian system.

We next, formally, show in Section 7 that we obtain the same κ 2 (ς) for the thermal conductivity of an open system: N coupled cells in which cell 1 and cell N are in contact with Langevin reservoirs at different temperatures, when we let N → ∞ and the two reservoir temperatures approach to β -1 .

Section 8 is devoted to a detailed study of κ 2 (ς) for three examples:

1) a chain of coupled pinned anharmonic oscillators;

2) a chain of rotors;

3) a harmonic chain with random (positive) pinnings.

In all cases we can prove that, generically, lim sup ς→0 κ 2 (ς) < +∞, as contrasted with the regular harmonic chain where κ 2 (ς) → ∞ when ς → 0 [1] [START_REF] Dhar | Heat conduction in disordered harmonic lattices with energy conserving noise[END_REF]. In case 1 we have no lower bound for this limit. In case 2 we expect but do not prove that κ 2 (ς) vanishes as ς goes to 0. In case 3 we prove that the limit of κ 2 (ς) is zero, as it is for the conductivity κ(ε, ς) of the harmonic chain with random pinning springs ( [START_REF] Bernardin | Small perturbation of a disordered harmonic chain by a noise and an anharmonic potential[END_REF]), when ς → 0. Phase mixing, due to lack of resonances between frequencies of different cells at different energies, is the relevant ingredient for the finiteness of κ 2 (ς) when ς → 0.

The System

2.1. Cell dynamics. We define first the dynamics of a single uncoupled cell. This will be given by a Hamiltonian dynamics generated by

H = p 2 /2 + W (q)
where the position q has values in some d-dimensional manifold, q ∈ M , and the momentum p belongs to the cotangent bundle of M , which can be locally identified with R d . We generally assume that W ≥ 0, and its minimum value is 0. In the case of the dynamics of a billiard, W = 0 and M ⊂ R 2 is the corresponding compact set of allowed positions with reflecting condition on the boundary. Another chaotic example is given by M a manifold with negative curvature and W = 0 (cf [START_REF] Dolgopyat | Energy transfer in a fast-slow Hamiltonian system[END_REF]). We will also consider cases where d = 1, that are completely integrable. Of course in the case W = 0, the manifold M will always be taken compact. If M is not compact, then we ask that W (q) goes to infinity with q fast enough. The Hamiltonian flow in a cell is perturbed by a noise that acts on the velocity, conserving the kinetic energy and thus the internal energy of each cell (as in [3][24]). Noises that exchange energy between different cells will not be considered here. Consequently the energy current will be due entirely to the deterministic interaction between the cells.

The time evolved configuration {q(t), p(t)} is given by a Markov process on the state space Ω = M × R d , generated by

L 0 = A 0 + ςS 0
where A 0 = p•∂ q -∇W (q)•∂ p is the Liouville operator associated to the Hamiltonian flow and S 0 is the generator of the stochastic perturbation, which acts only on the momentum p and is such that S 0 |p| 2 = 0. Here, | • | denotes the induced norm in R d and "•" the corresponding scalar product.

In dimension 1, we take an operator S that generates at random exponential times a flip on the sign of the velocity:

S 0 f (q, p) = 1 2 [f (q, -p) -f (q, p)]. (2.1)
More generally we can take in any dimension the random dynamics generated by1

S 0 f (q, p) = f |q, e -f (q, p) (2.2)
where at exponential times the momentum is renewed by choosing a new momentum with the uniform distribution •|q, e on the sphere {p ∈ R d ; |p|2 = 2(e -W (q))}. Alternatively for d ≥ 2, we can choose a continuous noise by just taking for S 0 the Laplacian on the sphere {p ∈ R d ; |p| 2 = 2(e -W (q))}, [START_REF] Liverani | Toward the Fourier law for a weakly interacting anharmonic crystal[END_REF].

In both cases S 0 has a spectral gap on the subspace of functions such that f | q, e = 0. Note that the measures • | q, e are all even in the momentum p.

2.2.

Interaction. Consider now the dynamics on Ω Z constituted by infinitely many processes {(q(t), p(t)} := {q x (t), p x (t)} x∈Z as above, but coupled by a nearest neighbor potential εV . The dynamics is then generated by 2

L ε = x∈Z ςS x + A x + ε∇V (q x -q x-1 ) • (∂ p x-1 -∂ px ) = L 0 + εG (2.3) where L 0 = A+ςS, A = x A x , S = x S x .
Here A x and S x act as A 0 and S 0 on the x-th component of Ω Z and like the identity on the other components. For simplicity, in general we assume that the interaction potential V is smooth and bounded together with its derivatives. Note however that in the special examples discussed in Section 8 we will consider also more general cases.

The energy of each cell, which is the sum of the internal energy and of the interaction energy, is defined by

e ε x = e x + ε 2 (V (q x+1 -q x ) + V (q x -q x-1 )) . (2.4)
To simplify notation we write e x for e 0 x = |px| 2 2 + W (q x ), the energy of the isolated system x. The dynamics generated by L 0 preserves all the energies e x . We denote by e := {e x ; x ∈ Z} the collection of the internal energies.

The dynamics generated by L ε conserves the total energy. The corresponding energy currents εj x,x+1 , defined by the local conservation law

L ε e ε x = ε (j x-1,x -j x,x+1
) are antisymmetric functions of the p's such that

j x,x+1 = - 1 2 (p x + p x+1 ) • ∇V (q x+1 -q x ). (2.5)
Let us denote by µ β,ε = • β,ε the canonical Gibbs measure at temperature β -1 > 0 defined by the Dobrushin-Lanford-Ruelle equations, which of course depends on the interaction εV . We shall assume in all the cases considered that µ β,ε is analytical in ε for sufficiently small ε (when applied to local bounded functions). Since we are considering here for simplicity only the one dimensional lattice with nearest neighbor interaction, it follows under great general conditions on V and W that the Gibbs state is unique and has spatial exponential decay of correlations for bounded local functions.

Also we assume that the equilibrium infinite dynamics is well defined, i.e. for a set of initial conditions which has probability measure one with respect to µ β,ε (this can be proven by standard techniques as in [START_REF] Fritz | Stationary states of random Hamiltonian systems[END_REF][START_REF] Lanford | Time evolution of infinite anharmonic systems[END_REF][START_REF] Marchioro | Existence of time evolution in νdimensional statistical mechanics[END_REF][START_REF] Olla | Equilibrium fluctuations for interacting Ornstein-Uhlenbeck particles[END_REF] and references therein). This defines a strongly continuous contracting semigroup of L 2 (µ β,ε ) with infinitesimal generator L ε for which the smooth local functions form a core.

We conclude this section by introducing some basic notation that will be used in the following sections. For any given bounded local functions f, g, define the semi-inner product

f, g β,ε = x∈Z [ τ x f, g β,ε -f β,ε g β,ε ]. (2.6)
Here τ x is the shift operator by x and •, • β,ε the scalar product in L 2 (µ β,ε ). The sum is finite in the case ε = 0, and converges for ε > 0 thanks to the exponential decay of the spatial correlations for local bounded functions, [START_REF] Georgii | Gibbs measures and phase transitions[END_REF]Proposition 8.34]. Remark that, since the velocities are always distributed independently, j x,x+1 , j x,x+1 β,ε < ∞.

Denote by

H ε = L 2 ( •, • β,ε ) the corresponding closure. Observe that if g is local, then g ∈ H ε is equivalent to g ∈ H 0 . In addition, if a local function g belongs to L 2 (µ β,ε ) then it belongs to H ε , even though in general L 2 (µ β,ε ) ⊂ H ε .
Note that the semigroup e tLε is a contraction semigroup on H ε as well, since, for each local functions f , e tLε f, e tLε f β,ε = lim

L→∞ 1 2L + 1 |x|,|y|≤L e tLε τ x f, e tLε τ y f β,ε ≤ f, f β,ε
where we have used the translation invariance of the semigroup. Also one can easily check that if f is a local smooth function, then

e tLε f -f, e tLε f -f β,ε ≤ t 2 L ε f, L ε f β,ε
from which it readily follows that L ε generates a strongly continuous semigroup on H ε as well.

It is also convenient to define the following semi-inner product

f, g 1 = f, (-S)g β,ε .
Let H 1 ε be the associated Hilbert space. We also define the Hilbert space H -1 ε via the duality given by the H ε norm, that is

f 2 -1 = sup g { 2 f, g β,ε -g, g 1 }
where the supremum is taken over local bounded functions g. Define the subspace of the antisymmetric functions in the velocities

H a ε = {f ∈ H ε : f (q, -p) = -f (q, p)} . (2.7)
where

q = {q x ∈ M } x∈Z , p = {p x ∈ R d } x∈Z .
Similarly define the subspace of the symmetric functions in p as H s ε . Remark that H s ε ⊥ H a ε and H s ε ⊕ H a ε = H ε . Let us define P a ε and P s ε the corresponding orthogonal projections, whose definition, in fact, depends on ε only in a trivial way. Therefore we sometime omit the index ε to denote them.

Finally, given a σ-algebra F we will use

E β,ε (• | F), µ β,ε (• | F) or • | F β,
ε to designate the conditional expectation. Then, for any function f ∈ L 2 (µ β,ε ), we define

(Π ε f )(e) = µ β,ε (f |e), Q ε = 1 -Π ε .
Note that Π ε is a projector and is self-adjoint also when seen as an operator acting on H ε , hence it is a well defined operator both on L 2 (µ β,ε ) and H ε . We conclude this section with a useful Lemma. Lemma 2.1. There exists δ > 0 such that, for each smooth local func-

tion f , f, -Sf β,ε ≥ δ [ f -f | e, q β,ε ] 2 β,ε .
Proof. For any local function let ψ f = f -f | e, q β,ε , then

ψ 2 f β,ε = x f f | {e z , q z } z≤x β,ε -f | {e z , q z } z<x β,ε β,ε ≤ 2 x f | {e z , q z } z≤x β,ε 2
β,ε
.

Thus, remembering the spectral gap property for each S x , we get that

f, -Sf β,ε ≥ x f, (-S x )f β,ε ≥ x 2δ f | {e z , q z } z =x β,ε 2
β,ε ≥ δ ψ 2 f 2 β,ε .
If needed, the above result can be extended to a more general class of functions by density.

The Green-Kubo formula

The argument of Section 5 in [START_REF] Bernardin | Transport Properties of a Chain of Anharmonic Oscillators with Random Flip of Velocities[END_REF], that will be recalled and extended shortly, gives the convergence of the thermal conductivity defined by the Green-Kubo formula

3 κ(ε, ς) = β 2 ε 2 ∞ 0 x∈Z E β,ε (j x,x+1 (t)j 0,1 (0)) dt. (3.1)
Here E β,ε indicates the expectation of the infinite dynamics in equilibrium at temperature β -1 . The convergence of the integral in (3.1) is in fact defined as

lim ν→0 j 0,1 , (ν -L ε ) -1 j 0,1 β,ε . (3.2)
Moreover, since the derivatives of V are assumed to be uniformly bounded, there exists a constant C > 0 such that

sup ν>0 j 0,1 , (ν -L ε ) -1 j 0,1 β,ε ≤ C ς . (3.3)
The above facts follow from the next proposition since j 0,1 ∈ H a ε so that

j 0,1 , (ν -L ε ) -1 j 0,1 β,ε = j 0,1 , P a ε (ν -L ε ) -1 j 0,1 β,ε . Proposition 3.1.
Let δ > 0 be the constant appearing in the statement of Lemma 2.1. There exists ε 0 , C > 0 such that, for all ε ∈ [0, ε 0 ], there exists a bounded operator

T ε from H -1 ε to H 1 ε , with norm bounded by ς -1 δ -1 2 , such that T ε g = lim ν→0 P a ε (ν -L ε ) -1 g. (3.4)
Further, for each smooth local function h ∈ H 1 0 ∩H 0 such that P a 0 h = 0, we have

T 0 L 0 h = T * 0 L * 0 h = 0, (3.5)
where L * 0 = -A + ςS and T * 0 = lim ν→0 P a 0 (ν -L * 0 ) -1 are the adjoint operators of L 0 and T 0 in L 2 (µ β,0 ).

Proof. We follow a strategy put forward in [START_REF] Bernardin | Transport Properties of a Chain of Anharmonic Oscillators with Random Flip of Velocities[END_REF].

The first step is to show that S has a spectral gap on H a ε . Let f be a local function such that P s ε f = 0, note that this implies f | q, e β,ε = 0. Set F L = L -1/2 |y|≤L τ y f , and note that P s ε F L = 0, then by Lemma 2.1 we have δ F 2 L β,ε ≤ F L , (-S)F L β,ε Then since, for L → ∞, we have that

F 2 L β,ε → f, f β,ε , F L (-S)F L β,ε → f, (-S)f β,ε ,
the spectral gap property follows.

We then study νf ν -L ε f ν = g. Since A ε inverts the parity and S preserves it, for each function ϕ ∈ H ε we set ϕ + = P s ε ϕ and ϕ -= P a ε ϕ. We can then write

ν f + µ , f + ν β,ε -f + µ , Af - ν + ς f + µ , f + ν 1 = f µ , g + β,ε µ f - ν , f - µ β,ε -f - ν , Af + µ + ς f - µ , f - ν 1 = f ν , g - β,ε . (3.6)
Summing the above equations we have

ν f + µ , f + ν +µ f - ν , f - µ +ς f µ , f ν 1 = f µ , g + β,ε + f ν , g - β,ε (3.7) Putting µ = ν we get ν f 2 ν β,ε + ς f ν , f ν 1 ≤ f ν 1 g -1 .
Hence f ν is uniformly bounded in H 1 ε and by the spectral gap property so is f - ν in H ε . Moreover, νf ν converges strongly to 0 in H ε . We can then extract weakly convergent subsequences. Taking first the limit, in (3.7), ν → 0 and then µ → 0 along one such subsequences (converging to f * ) we have ς f * , f * 1 = f * , g β,ε . Taking again the limit along such subsequence, with µ = ν, we have then lim

ν→0 ν f 2 ν β,ε = 0. (3.8)
Next, taking the limit along different weakly convergent subsequences (let f * be the other limit) we have

ς f * , f * 1 = f * , g + β,ε + f * , g - β,ε
and, exchanging the role of the two sequences

2ς f * , f * 1 = f * , g β,ε + f * , g β,ε = ς f * , f * 1 + ς f * , f * 1 which implies f * = f * ,
that is all the subsequences have the same limit. Finally, arguing similarly to the above, for ν(

f ν -f µ ) + (ν -µ)f µ - L ε (f ν -f µ ) = 0
we obtain that the convergence takes place in the strong norm. We are left with the proof of (3.5). Note that

L 0 h = Ah + ςSh, then Sh ∈ H -1 0 while Ah ∈ H a 0 ⊂ H -1 0 .
We can thus apply the previous consideration to g = L 0 h and obtain, for each smooth local function

ϕ ∈ H 0 , ϕ, T 0 g β,0 = lim ν→0 P a 0 ϕ, (ν -L 0 ) -1 L 0 h β,0 = -P a 0 ϕ, h β,0 + lim ν→0 ν (ν -L * 0 ) -1 P a 0 ϕ, h β,0 .
Next, note that P a 0 ϕ ∈ H -1 0 and that all the above discussion applies verbatim to the operator L * 0 , thus (3.8) implies that the above limit is zero, hence the claim of the Proposition (the proof for the adjoint being the same).

Remark 3.2. Since if g ∈ H a ε , by the spectral gap property, δ g 2 -1 ≤ g 2
β,ε , it follows that T ε is a bounded operator on P a ε H ε . Also note that by essentially the same proof it is a bounded operator on P a ε L 2 . Remark 3.3. Note that the operator T 0 is a local operator in the sense that if g is a local function then T 0 g is also a local function.

Formal expansion of κ(ε, ς)

It follows from (3.3) that κ(ε, ς) is of order ε 2 , i.e. lim sup ε→0 ε -2 κ(ε, ς) = κ2 (ς) < +∞. (4.1)
We conjecture that the limit exists and it is given by κ 2 (ς), the lowest term in the formal expansion of κ(ε, ς) in powers of ε:

κ(ε, ς) = ∞ n=2 ε n κ n (ς). (4.2)
It turns out that, for calculating the terms in this expansion, it is convenient to choose ν = ε 2 λ in (3.3), for a λ > 0, and solve the resolvent equation

(λε 2 -L ε )u λ,ε = εj 0,1 (4.
3) for the unknown function u λ,ε . The reason for considering λ > 0 is to have well defined solutions also for the infinite system. The factor ε 2 is the natural scaling in view of the subsequent computations. In fact, in order to see an energy diffusion, we need to look at times of order ε -2 , as suggested by the weak coupling limit [START_REF] Liverani | Toward the Fourier law for a weakly interacting anharmonic crystal[END_REF] [START_REF] Dolgopyat | Energy transfer in a fast-slow Hamiltonian system[END_REF].

We have already remarked that P a ε u λ,ε converges in H ε , for any fixed ε > 0.

Let us formally assume that a solution of (4.3) is in the form

u λ,ε = n≥0 (v λ,n + w λ,n )ε n , (4.4)
where Πv λ,n = Qw λ,n = 0. Here Π = Π 0 and Q = Q 0 refer to the uncoupled measure µ β,0 . This should be considered as an ansatz, and the manipulations of the rest of this section are done without worrying about the existence and the local regularity of the functions involved.

Given the expression (4.4) we can, in principle, use it in (3.2) to write

β -2 κ(ε, ς) = ε 2 lim λ→0 j 0,1 , (λε 2 -L ε ) -1 j 0,1 β,ε = lim λ→0 n≥0 ε n+1 j 0,1 , v λ,n + w λ,n β,ε = n≥1 lim λ→0 ε n j 0,1 , v λ,n-1 β,ε (4.5)
where we have used the fact that j 0,1 , w λ,n β,ε = 0, by symmetry4 and we have, arbitrarily, exchanged the limit with the sum. Note that (4.5) is not of the type (4.2) since the terms in the expansion depend themselves on ε. To identify the coefficients κ n we would need to expand in ε also the expectations. The existence of such an expansion is not obvious, in spite of the assumption on the Gibbs measure, since we will see that the functions v λ,n are non local. Nevertheless, we conjecture that

β -2 κ(ε, ς) = ε 2 lim λ→0 j 0,1 , v λ,1 β,ε + o(ε 2 ) = ε 2 lim λ→0 j 0,1 , v λ,1 β,0 + o(ε 2 ). ( 4.6)
Recalling (4.1) this yields the formula

κ2 (ς) = κ 2 (ς) = β 2 lim λ→0 j 0,1 , v λ,1 β,0 . (4.7)
In fact, in Section 6 we will prove the second equality of (4.6) in the special case W = 0. Our next task is then to find explicit formulae for v λ,n , w λ,n . Observe that L 0 w λ,n = 0 for all n ≥ 0 and that ΠGP s = 0, since G, defined by (2.3), changes the symmetry. Accordingly

v λ,0 = 0 -L 0 v λ,1 -Gw λ,0 = j 0,1 λw λ,n-2 + λv λ,n-2 -L 0 v λ,n -Gv λ,n-1 -Gw λ,n-1 = 0, n ≥ 2. (4.8)
Let us consider first the last equation for n = 2. Note that ΠL 0 = 0, thus applying Π we have, together with the second of (4.8),

-L 0 v λ,1 = j 0,1 + Gw λ,0 λw λ,0 = ΠGv λ,1 . (4.9)
Equations (4.9) can be written as5

P a v λ,1 = P a (-L 0 ) -1 (j 0,1 + Gw λ,0 ) λw λ,0 = ΠGP a v λ,1 . (4.10)
We can then apply ΠG to obtain6

ΠGP a v λ,1 = ΠGP a (-L 0 ) -1 [j 0,1 + Gw λ,0 ] λw λ,0 = ΠGP a v λ,1 .
It is then natural to consider the operator

L = ΠGP a (-L 0 ) -1 GΠ. (4.11)
We will show in Proposition 5.1 below that the operator L is a generator of a Markov process so that (λ -L) -1 is well defined for λ > 0. Hence, we have

w λ,0 = (λ -L) -1 ΠGP a 0 (-L 0 ) -1 j 0,1 P a v λ,1 = P a (-L 0 ) -1 [j 0,1 + Gw λ,0 ] . (4.12)
We can now analyse the case n > 2. If we apply Π and Q at the last equation of (4.8) we obtain

λw λ,n-2 = ΠGv λ,n-1 L 0 v λ,n = λv λ,n-2 -QGv λ,n-1 -Gw λ,n-1 .
Arguing as before we have

w λ,n = (λ -L) -1 ΠG(-L 0 ) -1 [-λv λ,n-1 + QGv λ,n ] , n ≥ 1 v λ,n+1 = (-L 0 ) -1 [-λv λ,n-1 + Gw λ,n + QGv λ,n ] , n ≥ 1. (4.13)
In the following we will discuss explicitly only P a v λ,1 and w λ,0 , showing that they are well defined. The study of the higher order terms is harder due to the presence of the QGv λ,n which, on the one hand, depends on the symmetric part of v λ,n (on which we have poor bounds) and, on the other hand, has no obvious reason to be in H -1 (on which we know how to invert L 0 ). In order to make further progresses one must, at least, extend Proposition 3.1 to all the functions that have zero average with respect to each microcanonical measure. This, in principle, can be done in specific cases (e.g., see [START_REF] Liverani | Toward the Fourier law for a weakly interacting anharmonic crystal[END_REF]) but at the price of dealing with less conventional functional spaces, hence making the argument much more delicate.

The operator L

In this section we rigorously identify the quadratic (Dirichlet) form associated to the operator L = ΠGT 0 GΠ defined in (4.11).

Let us denote by ρ β (de) the distribution of the internal energies e = {e x ; x ∈ Z} under the Gibbs measure µ β,0 . It can be written in the form

dρ β (e) = x∈Z Z -1 β exp(-βe x -U (e x ))de x (5.1)
for a suitable function U . We denote the formal sum x U (e x ) by U := U(e). We denote also, for a given value of the internal energy ẽx in the cell x, by ν x ẽx the microcanonical probability measure in the cell x, i.e. the uniform probability measure on the manifold

Σ ẽx := {(q x , p x ) ∈ Ω ; e x (q x , p x ) = ẽx }.
Note that it is not obvious that the operator is well defined in H ε since we do not know if T 0 maps the range of G in its domain, yet, by Proposition 3.1, it is well defined as an operator from smooth local functions to distributions.

We are going to identify L as the Markov generator of a Ginzburg-Landau dynamics

L GL = x e U (∂ e x+1 -∂ ex ) e -U γ 2 (e x , e x+1 )(∂ e x+1 -∂ ex ) , (5.2)
where γ 2 (e 0 , e 1 ) = Σe 0 ×Σe 1 j 0,1 T 0 j 0,1 dν 0 e 0 dν 1 e 1 .

(5.3)

We recall that T 0 is defined by (3.4) and that T 0 j 0,1 is a local function.

Since ∇V is uniformly bounded, j 0,1 ∈ L 2 (µ β,0 ) so that j 0,1 ∈ L 2 (ν 0 e 0 ⊗ ν 1 e 1 ) for almost every e 0 , e 1 . Recalling that T 0 is a bounded operator on P a L 2 (see Remark 3.2), we conclude that T 0 j 0,1 ∈ L 2 (µ β,0 ) and thus that T 0 j 0,1 ∈ L 2 (ν 0 e 0 ⊗ ν 1 e 1 ) for almost every e 0 , e 1 . Therefore, γ 2 (e 0 , e 1 ) is finite for almost every e 0 , e 1 and γ belong both to L 2 and H ε .

Formally the previous formula reads

γ 2 (e 0 , e 1 ) = ∞ 0 dt
Σe 0 ×Σe 1 j 0,1 (e tL 0 j 0,1 ) dν 0 e 0 dν 1 e 1 (5.4) where e tL 0 denotes the semigroup of the uncoupled dynamics generated by L 0 .

The operator L GL is well defined only if γ 2 has some regularity properties, that are actually proven in specific examples [START_REF] Liverani | Toward the Fourier law for a weakly interacting anharmonic crystal[END_REF] [START_REF] Dolgopyat | Energy transfer in a fast-slow Hamiltonian system[END_REF]. In the generality we consider in this work we can only prove that (5.3) is well defined and that the corresponding drift α(e x , e x+1 ) = e U (e) (∂ e x+1 -∂ ex ) e -U (e) γ(e x , e x+1 ) 2 .

(5.5) is a well defined distribution. So we show that the Dirichlet forms associated to L and L GL coincide. Then in the cases where γ 2 is proven smooth (5.2) is well defined and L = L GL . A simple computation shows that

g, (-L GL )f β,0 = γ 2 (e 0 , e 1 ) [(∂ e 1 -∂ e 0 )Γ f ] [(∂ e 1 -∂ e 0 )Γ g ] β,0 (5.6)
where for any function f of the energies, we denote by Γ f = x τ x f (intended as a formal sum).

Proposition 5.1. For each local smooth functions f, g of the energies only we have g, (-L)f β,0 = g, (-L GL )f β,0 .

(5.7) In addition, α(e x , e x+1 ) = ΠGT 0 j x,x+1 is a well defined distribution equal to (5.5).

Proof. Let us start by noting that, for each local smooth function f depending only on the energies,

Gf = - x ∇V (q x -q x-1 ) • (p x ∂ ex -p x-1 ∂ e x-1 )f = x j x-1,x (∂ ex -∂ e x-1 )f - 1 2 x [L 0 V (q x -q x-1 )](∂ ex + ∂ e x-1 )f = x j x-1,x (∂ ex -∂ e x-1 )f -L 0 ω = x j x-1,x (∂ ex -∂ e x-1 )f + L * 0 ω, (5.8)
where ω = 1 2

x V (q x -q x-1 )(∂ ex + ∂ e x-1 )f.

Since j 0,1 (∂ e 0 -∂ e 1 )f ∈ H -1 0 and ω ∈ H 0 ∩H 1 with P a ω = 0, Proposition 3.1 implies that T 0 Gf is in H a 0 and it is equal to

T 0 Gf = x [T 0 j x-1,x ](∂ ex -∂ e x-1 )f.
(5.9)

Next, note that the adjoint of G y in L 2 (µ β,0 ) is given by

G * y = -G y -β∇V (q y -q y-1 ) • (p y -p y-1 ) = -G y -βL 0 V (q y -q y-1 ) = -G y + βL * 0 V (q y -q y-1 ).
(5.10)

and that G * y g ∈ H a 0 . It follows that the adjoint of G = y G y in H 0 (that we still denote by G * ) is given by

-G + β y L * 0 V (q y -q y-1 ).
Thus, if f, g are smooth local functions of the energies only then we have g ,

Lf β,0 = G * g , T 0 Gf β,0 = -Gg , T 0 Gf β,0 + β y L * 0 V (q y -q y-1 )g , T 0 Gf β,0 = -Gg , T 0 Gf β,0
where we have used Proposition 3.1 again. By using (5.9), (5.8) and Proposition 3.1 one last time we get then

g , Lf β,0 = - x,y j y-1,y (∂ ey -∂ e y-1 )g , T 0 j x-1,x (∂ ex -∂ e x-1 )f β,0
where in the expressions above the sums are restricted to the finite support of the corresponding functions. This can be rewritten as

g , Lf β,0 = - x,y,z j y+z-1,y+z τ z (∂ ey -∂ e y-1 )g , T 0 j x-1,x (∂ ex -∂ e x-1 )f β,0
Note that T 0 j x-1,x is a local function depending only on p x-1 , q x-1 , p x , q x and that we have

j y-1,y T 0 j x-1,x |e β,0 = δ xy γ 2 (e x-1 , e x ),
and consequently we obtain that the Dirichlet form g , -Lf β,0 coincides with the RHS of (5.6). Equation (5.5) is proved by similar arguments.

We conclude this section by noting that the operator L GL is the generator of a Ginzburg-Landau dynamics which is reversible with respect to ρ β , for any β > 0. It is conservative in the energy x e x and the microscopic current corresponding to this conservation law is given by the righthand side of (5.5). The corresponding finite size dynamics appears in [START_REF] Liverani | Toward the Fourier law for a weakly interacting anharmonic crystal[END_REF][START_REF] Dolgopyat | Energy transfer in a fast-slow Hamiltonian system[END_REF] as the weak coupling limit of a finite number N (fixed) of cells weakly coupled by a potential V in the limit → 0 when time t is rescaled as tε -2 . Remark 5.2. It should be possible to apply existing arguments to prove that L GL , and hence L, is a closed operator also in H 0 and that the local smooth functions constitute a core of self-adjointness of L in H 0 (see [START_REF] Fritz | Gradient dynamics of infinite point systems[END_REF] and [START_REF] Spohn | Equilibrium Fluctuations for Interacting Brownian Particles[END_REF], where this statement is proven for the more difficult model of interacting brownian motions). A proof of this fact exceeds the scopes of the present article, thus in the following we will simply assume it.

The hydrodynamic limit of the Ginzburg-Landau dynamics is then given (in the diffusive scale tN 2 , N → +∞), by a heat equation with diffusion coefficient which coincides with κ 2 as given by (6.20) below ( [START_REF] Varadhan | Nonlinear diffusion limit for a system with nearest neighbor interactions II, Asymptotic problems in probability theory: stochastic models and diffusions on fractals[END_REF], [START_REF] Liverani | Macroscopic Energy diffusion after weak coupling[END_REF]).

6. The lowest order term κ 2 (ς)

In this section we restrict our study to the unpinned case W = 0, obtaining an explicit formula for κ 2 (ς). In particular we prove, for this case, the second equality of (4.6). Note however that if one accepts the formula (4.7) for κ 2 (ς), then the following arguments, with ε = 0, yield an explicit formula for κ 2 (ς) in quite some generality.

We assume also in the following that γ -1 0,1 ∈ L 2 (µ ε,β ), such an assumption is verified in all the examples discussed in this paper. Also we assume that γ 2 is smooth, as proven in the examples considered in [START_REF] Liverani | Toward the Fourier law for a weakly interacting anharmonic crystal[END_REF] and [START_REF] Dolgopyat | Energy transfer in a fast-slow Hamiltonian system[END_REF]. Thus we have L = L GL .

Observe that by the definition of ρ β (given by (5.1)), if f depends only on the energies, we have f β,0 = ρ β (f ). In the following we will denote by • β the integration with respect to ρ β . Moreover the semi-inner product •, • β corresponding to • β is defined as in (2.6).

To simplify notation, we denote γ(e x , e x+1 ) (resp. α(e x , e x+1 )) by γ x,x+1 (resp. α x,x+1 ) and w λ,0 by w λ . Define the operator D x,x+1 = γ x,x+1 (∂ e x+1 -∂ ex ), then the adjoint, with respect to ρ β , is given by

D * x,x+1 = -e -U (∂ e x+1 -∂ ex )e U γ x,x+1
and consequently L = x D * x,x+1 D x,x+1 . First note that for any λ > 0, we have by (4.12)

λw λ -Lw λ = α 0,1 = D * 0,1 γ 0,1 . (6.1)
This resolvent equation, which involves only functions of the energies, has a well defined solution w λ ∈ H 0 for any λ > 0.

For each smooth local functions f, g, relation (5.6) imply

f, -Lg β = γ 2 0,1 [(∂ e 1 -∂ e 0 )Γ f ] [(∂ e 1 -∂ e 0 )Γ g ] β = (D 0,1 Γ f )(D 0,1 Γ g ) β , f, α β = -γ 0,1 D 0,1 Γ f β . (6.2)
Since w λ is in the domain of L, by Remark 5.2 and equation (6.1) we have

λ w λ , w λ β + (D 0,1 Γ w λ ) 2 β = -γ 0,1 D 0,1 Γ w λ β (6.3)
thus by Schwarz inequality

λ w λ , w λ β + (D 0,1 Γ w λ ) 2 β ≤ γ 2 0,1 1/2 β (D 0,1 Γ w λ ) 2 1/2
β and this gives the bounds

λ w λ , w λ β ≤ γ 2 0,1 β , (D 0,1 Γ w λ ) 2 β ≤ γ 2 0,1 β . (6.4)
The standard Kipnis-Varadhan argument ( [START_REF] Kipnis | Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions[END_REF], [START_REF] Komorowski | Fluctuations in Markov processes. Time symmetry and martingale approximation[END_REF] chapter 1) then gives lim λ→0 λ w λ , w λ β = 0.

It also follows form the same argument ( [START_REF] Kipnis | Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions[END_REF], [START_REF] Komorowski | Fluctuations in Markov processes. Time symmetry and martingale approximation[END_REF] Chapter 1) that D 0,1 Γ w λ converges strongly in L 2 (ρ β ) to a limit that we denote with η and that satisfies the relation

η 2 β = -γ 0,1 η β .
Before continuing we need a small technical Lemma. Lemma 6.1. Consider the linear functional defined on smooth local functions g of the energies by ε (g) = j 0,1 , T 0 Gg β,ε .

(6.5)

If γ -1 ∈ L 2 (µ β,ε
), then ε can be continuously extended to the domain of L.

Proof. To start with, note that ε is well defined since Gg is local and in H a ε for ε > 0. Moreover, T 0 Gg is a local function. By using (5.9) we compute

j 0,1 , T 0 Gg β,ε = z j z,z+1 , T 0 Gg β,ε = z,x j z,z+1 [T 0 j x,x+1 ] (∂ e x+1 -∂ ex )g β,ε = y=-1,0,1 x j x+y,x+y+1 [T 0 j x,x+1 ] (∂ e x+1 -∂ ex )g β,ε = y=-1,0,1 j y,y+1 [T 0 j 0,1 ](∂ e 1 -∂ e 0 )Γ g β,ε = y=-1,0,1 j y,y+1 T 0 j 0,1 γ 0,1 D 0,1 Γ g β,ε
.

That can be bounded by y=-1,0,1

j y,y+1 T 0 j 0,1 |e 2 β,ε γ -2 0,1 1/2 β,ε (D 0,1 Γ g ) 2 1/2 β,ε .
Since we assumed here that there is no pinning potential, we have that energies are function of the only velocities and

(D 0,1 Γ g ) 2 β,ε = (D 0,1 Γ g ) 2
β,0 = g, (-L)g β,0 . Since we assumed that the smooth local function are a core for L (see Remark 5.2), it follows that ε can be extended to any non-local function of the energies g that belongs to the domain of L.

Notice that, using (4.10), Proposition 3.1 and Lemma 6.1

j 0,1 , v λ,1 β,ε = j 0,1 , P a v λ,1 β,ε = j 0,1 , T 0 j 0,1 β,ε + j 0,1 , T 0 Gw λ β,ε . (6.6)
Thus, by the first line of (4.6), κ 2 (ς) is given by

β -2 κ 2 (ς) = lim ε→0 lim λ→0 { j 0,1 , T 0 j 0,1 β,ε + j 0,1 , T 0 Gw λ β,ε } . (6.7)
Next, we will use the above formula for κ 2 (ς) for a rigorous study of the lowest order term. The first step consists to make sense, for ε > 0 and λ > 0 fixed, of the two terms involved.

To compute the limit of the second term of the RHS of (6.7) it is convenient to use the following Lemma. Observe first that if g is a smooth local function depending only on the energies then j 0,1 g is an antisymmetric function and T 0 (j 0,1 g) = (T 0 j 0,1 ) g. Lemma 6.2. There exists a constant C, ε 0 > 0 such that for any 0 ≤ ε < ε 0 and for each function g of the energies {e x } we have

| j -1,0 , T 0 j 0,1 g β,ε | ≤ C ε g 2 β,0 . (6.8)
and

j 0,1 , T 0 j 0,1 g β,ε -γ 2 0,1 g β,ε ≤ C ε g 2 β,0 . (6.9)
Proof. Note that the adjoint of L 0 with respect to • β,ε is given by

L 0 f = L * 0 f -εβ x (p x+1 -p x )∇V (q x+1 -q x )f,
where L * 0 is the adjoint with respect to • β,0 . Notice that the second term of the above expression contains a formal infinite sum, i.e. L 0 f is a distribution well defined against any local function.

Then, observing that

p -1 • ∇V (q 0 -q -1 ), T 0 j 0,1 g β,ε = 0 we have j -1,0 , T 0 j 0,1 g β,ε = 1 2 (p 0 -p -1 ) • ∇V (q 0 -q -1 ), T 0 j 0,1 g β,ε = 1 2 L * 0 V (q 0 -q -1 ), T 0 j 0,1 g β,ε = 1 2 L 0 V (q 0 -q -1 ), T 0 j 0,1 g β,ε + 1 2 εβ x (p x+1 -p x )
• ∇V (q x+1 -q x )V (q 0 -q -1 ), T 0 j 0,1 g β,ε .

(6.10)

In the above equation, the term L 0 V (q 0 -q -1 ), T 0 j 0,1 g β,ε is well defined as well as the second one because V (q 0 -q 1 ) is a local function and

x (p x+1 -p x ) • ∇V (q x+1 -q x )V (q 0 -q -1 ), T 0 j 0,1 g β,ε is a finite sum, in fact equal to x=-1,0,1 [T 0 j 0,1 ](p x+1 -p x )|q, e β,ε • ∇V (q x+1 -q x )V (q 0 -q -1 )g β,ε (6.11) since [T 0 j 0,1 ](p x+1 -p x )|q, e β,ε = 0 if x = -1, 0, 1.
The first term of the RHS of the last equality in (6.10) is equal to L 0 V (q 0 -q -1 ), T 0 j 0,1 g β,ε = V (q 0 -q -1 ), L 0 (T 0 j 0,1 g) β,ε = V (q 0 -q -1 ) , [L 0 (T 0 j 0,1 )] g) β,ε = f , [L 0 (T 0 j 0,1 )] β,ε (6.12) where f = gV (q 0 -q -1 ) is a symmetric function of the velocities. We claim now that f , [L 0 (T 0 j 0,1 )] β,ε = 0. (6.13) Indeed, let ν > 0 and write

j 0,1 = (ν -L 0 )P a ((ν -L 0 ) -1 j 0,1 + (ν -L 0 )P s ((ν -L 0 ) -1 j 0,1
and take the scalar product of both sides with f to get

0 = (ν -L 0 )P a (ν -L 0 ) -1 j 0,1 , f β,ε + (ν -L 0 )P s (ν -L 0 ) -1 j 0,1 , f β,ε .
The first term of the RHS goes to f , [-L 0 (T 0 j 0,1 )] β,ε as ν → 0 and the second term is equal to

ν P s (ν -L 0 ) -1 j 0,1 , f β,ε -L 0 P s (ν -L 0 ) -1 j 0,1 , f β,ε .
It is easy to see that the second term of the previous expression is equal to 0. This is because L 0 = A + ςS, A maps a symmetric function of the p x 's into an antisymmetric function of the p x 's, S is symmetric w.r.t. µ β,ε and Sf = 0.

Thus, to prove the claim (6.13) we are reduced to show that lim

ν→0 ν P s (ν -L 0 ) -1 j 0,1 , f β,ε = lim ν→0 ν (ν -L 0 ) -1 j 0,1 , f β,ε = 0.
(6.14) By Schwarz inequality it is sufficient to bound the L 2 (µ β,ε ) norm of the local function u ν = (ν -L 0 ) -1 j 0,1 . By definition we have

νu ν -L 0 u ν = j 0,1 . Since j 0,1 = -1 2 Sj 0,1 a classical argument ([20] Chapter 1) shows that u ν , u ν β,0 ≤ Cν -1 (6.15)
where C is a constant independent of ν. Observe that the support of the local function u ν is fixed independently of ν so that there exist K, ε 0 > 0 such that for any 0 < ε < ε 0 , we have

u ν , u ν β,ε ≤ K u ν , u ν β,0 ≤ CKν -1 . (6.16)
This concludes the proof of (6.13) and it remains only to show that there exists C, ε 0 > 0 independent of g such that (6.11) is bounded by

C ε g 2 β,0 , ε < ε 0 . (6.17)
This follows from Schwarz inequality. Therefore (6.8) is proved. Proof of (6.9) follows a similar line.

Applying the above lemma with g ≡ 1 to (6.7), it follows j 0,1 , T 0 j 0,1 β,ε = γ 2 0,1 β,0 + O(ε). We are left with the second term in (6.7). Using again Lemma 6.2 we have

y j 0,1 , T 0 j y-1,y (∂ ey -∂ e y-1 )w λ β,ε = x,y j x-1,x , T 0 j y-1,y (∂ ey -∂ e y-1 )w λ β,ε = x j x-1,x , T 0 j x-1,x (∂ ex -∂ e x-1 )w λ β,ε + O(ε) = j 0,1 , T 0 j 0,1 (∂ e 1 -∂ e 0 )Γ w λ β,ε + O(ε) = γ 2 0,1 (∂ e 1 -∂ e 0 )Γ w λ β,0 + O(ε) = γ 0,1 D 0,1 Γ w λ β,0 + O(ε)
In the second equality we used the fact that the sum over y can be reduced to the sum over y = x -1, x, x + 1 since the other terms are 0. Observe that he remainder terms O(ε) are uniform in λ.

Therefore, we have that

lim λ→0 j 0,1 , v λ,1 β,ε = γ 2 0,1 β,0 + lim λ→0 γ 0,1 D 0,1 Γ w λ β,0 + O(ε) = γ 2 0,1 β,0 -η 2 β,0 + O(ε) = γ 2 0,1 β,0 + η 2 β,0 + 2 γ 0,1 η β,0 + O(ε) = (γ 0,1 + η) 2 β,0 + O(ε). (6.18)
We conclude that

κ 2 (ς) = β 2 (γ 0,1 + η) 2 β,0 ≥ 0. (6.19)
It follows from the above calculation that (recalling the notations introduced at the beginning of this section)

β -2 κ 2 (ς) = γ 2 0,1 β -α 0,1 , (-L) -1 α 0,1 β . (6.20)
The right hand side of (6.20) is exactly the macroscopic diffusion of the energy in the autonomous stochastic dynamics describing the evolution of e, obtained in the weak coupling limit [START_REF] Dolgopyat | Energy transfer in a fast-slow Hamiltonian system[END_REF][START_REF] Liverani | Toward the Fourier law for a weakly interacting anharmonic crystal[END_REF][START_REF] Liverani | Macroscopic Energy diffusion after weak coupling[END_REF]. Thus even if (6.20) is obtained from a formal expansion it is a mathematically well defined object and we expect it coincides with lim ε→0 ε -2 κ(ε, ς).

So that we have proved the following proposition. Proposition 6.3. Assume that the pinning potential W = 0 and that V, ∇V are uniformly bounded. Then we have

lim ε→0 lim λ→0 { j 0,1 , T 0 j 0,1 β,ε + j 0,1 , T 0 Gw λ β,ε } (6.21)
exists and is equal to

γ 2 0,1 β -α 0,1 , (-L) -1 α 0,1 β .
Remark: Lower bounds on κ 2 (ς). Notice that γ -1 0,1 η β = 0, so

1 = γ -1 0,1 (γ 0,1 + η) β ≤ γ -2 0,1 1/2 β (γ 0,1 + η) 2 1/2 β
In particular using the last line of (6.19)

γ -2 0,1 -1 β ≤ β -2 κ 2 (ς) ≤ γ 2 0,1 β (6.22)
In [START_REF] Liverani | Toward the Fourier law for a weakly interacting anharmonic crystal[END_REF], a single particle Hamiltonian of the form H = p 2 /2 + W (q) is considered in dimension d = 2. It is shown there under suitable assumptions on the potentials V and W that the bound γ 2 (e 0 , e 1 ) ≥ c -(ς)e 0 e 1 holds, for small energies and c -(ς) > 0 for ς > 0. It follows that the lower bound (6.22) is strictly positive as soon as ς > 0 for that system. We conjecture that this holds in general for ς > 0 and we prove it for the examples of Section 8.

When the Hamiltonian part of the cell dynamics is given by a geodesic flow on a manifold of negative curvature, the lower bound in (6.22) is strictly positive even without the noise (ς = 0), in dimension d ≥ 3 ([12]).

The non-equilibrium stationary state

Instead of studying the energy flux via the GK, an alternative, more direct, approach is possible: one can consider the stationary state in a finite open system with Langevin thermostats at the boundaries having temperatures T and T + δT respectively, [START_REF] Bernardin | Transport Properties of a Chain of Anharmonic Oscillators with Random Flip of Velocities[END_REF]. To simplify the study we assume that d = 1. The generator of the dynamics is then

L ε,N,δT = N x=1 (A x + ςS x ) + εG + B 1,T +δT + B N,T
where B 1,T +δT , B N,T are the generators of the corresponding Langevin dynamics at the boundaries:

B x,T = T 2 ∂ 2 px -p x ∂ px and G = N x=2 ∇V (q x -q x-1 )(∂ p x-1 -∂ px ) .
Our goal is to compute the thermal conductivity of the stationary state, e.g. the stationary current divided by the temperature gradient δT /N :

κ N,T,ε = lim δT →0 N δT ε j 0,1 N,δT,ε , (7.1)
where < • > N,δT,ε is the expectation with respect to the stationary measure. To this end we are going to expand the stationary measure in ε and δT .

Let us reiterate once more that the following is only formal. Indeed, to simplify the presentation, we do not insist on issues that have already been treated more carefully in the previous sections. As a preliminary step, we use as a reference measure the inhomogeneous Gibbs distribution with linear profile of inverse temperature {β x } x=1,...,N , interpolating between the two inverse temperatures by setting β x+1 -β x ∼ -δT N T 2 . We will call E the expectation with respect to such a measure, that is

E(f ) = Z -1 e -N x=1 βxe ε x f (q, p)dqdp, (7.2)
where as before e ε x = 1 2 p 2 x + W (q x ) + 1 2 ε[V (q x -q x-1 ) + V (q x+1 -q x )], for x = 2, . . . , N -1 and e ε 1 = 1 2 p 2 1 + W (q x ) + 1 2 εV (q 2 -q 1 ), e ε N = 1 2 p 2 N + W (q x ) + 1 2 εV (q N -q N -1 ). 7 To keep consistency with previous notations, we will use e x to designate e 0

x , the internal energy of the isolated cell.

The corresponding adjoint operator is

L * ε,N,δT = N x=1 (-A x +ςS x )-εG+ε N -1 x=1 (β x+1 -β x )j x,x+1 +B 1,T +δT +B N,T .
We assume that there exists a unique stationary probability distribution with smooth density. The existence and uniqueness of such a probability measure still remains an open problem for most of the dynamics that appear in this work, though for some models, proofs can be found in [START_REF] Bernardin | Transport Properties of a Chain of Anharmonic Oscillators with Random Flip of Velocities[END_REF] (see also [START_REF] Rey-Bellet | Open Classical System[END_REF]). For certain choices of the local dynamics L 0 and interaction V , the smoothness of the density follows by applying results of [START_REF] Eckmann | Non-Equilibrium Statistical Mechanics of Strongly Anharmonic Chains of Oscillators[END_REF], [START_REF] Carmona | Existence and uniqueness of an invariant measure for a chain of oscillators in contact with two heat baths[END_REF].

Let f ε,N,δT be the density of this stationary measure with respect to this inhomogeneous Gibbs measure, i.e. the solution of L * ε,N,δT f ε,N,δT = 0, f ε,N,δT ≥ 0. It is observed that the energy of the particles at the boundary sites (x = 1, N ) is not conserved due to the action of the reservoirs. So, if e = {e 2 , . . . , e N -1 }, it is convenient to define the projector 8Πf (e 2 , . . . , e N -1 ) = E(f | e) .

Also let B = B 1,T + B N,T and k = 1 N T 2 N -1 x=1 j x,x+1 .
We expand the stationary measure as follows

f ε,N,δT = 1 + δT w 0 + n≥1 (v n + w n ) ε n + O((δT ) 2 ) (7.3)
where Πw n = w n and Πv n = 0. Next, it is convenient to set

L B = N x=1 (A x + ςS x ) + B = L 0,N,0 . Note that 9 L * B = N x=1 (-A x + ςS x ) + B = L * 0,N,0 and that L * B Π = ΠL * B = 0. Since L * ε,N,δT 1 = kδT + O((δT ) 2
), if we compute at the first order in δT we have

-εk -εGw 0 + n≥1 ε n {L * B v n -εGv n -εGw n } = 0.
From the above it follows

L * B v 1 = k + Gw 0 L * B v n+1 = Gw n + Gv n for n > 0. (7.4)
We proceed similarly to Section 4, starting from n = 1. Since, again, ΠGΠ = 0, applying Π to the second equation in (7.4), when n = 1, we

obtain 10 v 1 = (L * B ) -1 [k + Gw 0 ] ΠGv 1 = 0. (7.5)
We can then multiply the first equation by ΠG and define the operator

L B = ΠG(L * B ) -1 GΠ. (7.6)
This readily implies that

w 0 = L -1 B ΠG(-L * B ) -1 k v 1 = (L * B ) -1 [k + Gw 0 ] , (7.7)
solve (7.5). Next, we consider n > 1. Applying Π to the second of the (7.4) we have that v n must satisfy ΠGv n = 0 for all n ∈ N (recall that ΠGΠ = ΠL * B = 0). As we have seen, this is indeed the case for n = 1. Assume it and try for n > 1. Then, we can apply (L * B ) -1 and obtain

v n+1 = (L * B ) -1 [Gw n + Gv n ] . Multiplying for ΠG yields, for n ≥ 1, w n = L -1 B ΠG(-L * B ) -1 Gv n v n+1 = (L * B ) -1 [Gw n + Gv n ] . (7.8)
Note that

ΠGv n+1 = ΠG(L * B ) -1 [Gw n + Gv n ] = L B w n + ΠG(L * B ) -1 Gv n = 0 as needed.
Next, we want to compute how L B acts on the space of function {f : Πf = f }.

Gf = N x=2 ∇V (q x -q x-1 )(p x ∂ ex -p x-1 ∂ e x-1 )f = N x=2 j x-1,x (∂ ex -∂ e x-1 )f - 1 2 N x=2 [L * B V (q x -q x-1 )](∂ ex + ∂ e x-1 )f.
Thus, given two function of the energies f (e 2 , . . . , e N -1 ) and g(e 2 , . . . , e N -1 ), we have 11

E β (gL B f ) = E β (gΠG(L * B ) -1 GΠf ) = N x=2 E β (gG(L * B ) -1 j x-1,x (∂ ex -∂ e x-1
)f ), (7.9) 10 To simplify the presentation we assume that L -1 B is well defined. For a more rigorous argument it suffices to use the analogous of Proposition 3.1.

11 By E β we mean the measure (7.2) with δT = 0.

where we have used the antisymmetry in p of the measure. Also, taking the adjoint with respect to E β yields

G * = -G + β N x=2 ∇V (q x -q x-1 )(p x -p x-1 ) = -G + βL B V. (7.10)
Inserting the above in (7.9) and using again the antisymmetry in p we have

-E β (gL B f ) = 1 4 N x,y=2 E β (j y,y-1 (∂ ey -∂ e y-1 )g • (-L * B ) -1 j x,x-1 (∂ ex -∂ e x-1 )f .
Finally, we have

γ(e x-1 , e x ) 2 δ xy = E β (j y-1,y (-L * B ) -1 j x-1,x | e) x = 2, N γ(e 1 , e 2 ) 2 δ y,2 = E β (j y-1,y (-L * B ) -1 j 1,2 | e) γ(e N -1 , e N ) 2 δ y,N = E β (j y-1,y (-L * B ) -1 j N -1,N | e) Thus -E β (gL B f ) = 1 4 N x=2 E β (γ 2 (e x-1 , e x )(∂ ex -∂ e x-1 )g • (∂ ex -∂ e x-1 )f ) = 1 4 N -1 x=3 E β (γ 2 (e x-1 , e x )(∂ ex -∂ e x-1 )g • (∂ ex -∂ e x-1 )f ) + 1 4 E β (γ 2 (e 1 , e 2 )∂ e 2 g • ∂ e 2 f ) + 1 4 E β (γ 2 (e N -1 , e N )∂ e N -1 g • ∂ e N -1 f )
which shows that L B is the operator that one would expect in [START_REF] Liverani | Toward the Fourier law for a weakly interacting anharmonic crystal[END_REF][START_REF] Dolgopyat | Energy transfer in a fast-slow Hamiltonian system[END_REF] when adding the appropriate boundary terms. Also note that, in analogy with Proposition 5.1 and (6.1),

α(e x , e x+1 ) = ΠG(-L * B ) -1 j x,x+1 = ΠG * (-L B ) -1 j x,x+1 = D * x,x+1 γ(e x , e x+1 ). (7.11)
We can, at last, compute the current:

E(f ε,N,δT j 0,1 ) = δT E({w 0 + ε(v 1 + w 1 )}j 0,1 ) + O(ε 2 δT + (δT ) 2 ).
Thus, setting

j 0,N = lim δT →0 1 δT E(f ε,N,δT j 0,1 )
we have formally12

j 0,N = εE β (v 1 j 0,1 ) + O(ε 2 ) = ε N T 2 E β (γ(e 0 , e 1 ) 2 ) + εE β (k • (L * B ) -1 Gw 0 ) + O(ε 2 ) = εβ 2 N E β (γ(e 0 , e 1 ) 2 ) + εβ 2 N N -1 x=1 E β (α(e x-1 , e x )[(-L B ) -1 α(e 0 , e 1 )]) + O N (ε 2 )
Therefore, formally, the limit

lim ε→0 lim N →∞ lim δT →0 1 ε 2 κ N,T,ε = lim ε→0 lim N →∞ N ε j 0,N ,
with κ N,T,ε given by (7.1), yields the formula for κ 2 (ς) in agreement with the Green-Kubo formula expansion of Section 6.

8. Behavior of κ 2 (ς) in the limit ς → 0 for some model systems

We now study the behavior of κ 2 (ς) in the deterministic limit ς → 0. This limit is singular, since the operator L = ΠG(-L 0 ) -1 GΠ formally vanishes at ς = 0 for the whole class of systems considered in this work: both operators (-L 0 ) -1 and G exchange symmetric and antisymmetric functions under the operation p → -p, while Π annihilates antisymmetric functions. It is therefore important to analyse some particular cases in more detail. When the dynamics of individual cells is chaotic, the operator (-L 0 ) -1 can be defined at ς = 0 only on appropriate spaces of distributions, [START_REF] Liverani | On contact Anosov flows[END_REF], in which the operator induced by the map p → -p is unbounded. Thus the above formal argument does not hold and the operator L does not trivially vanishes for ς = 0, [START_REF] Dolgopyat | Energy transfer in a fast-slow Hamiltonian system[END_REF].

Another way to bypass the above problems is by looking at integrable isolated dynamics for which fairly explicit computations can be carried out. Here we consider three such examples. In all these cases, the uncoupled cells are one-dimensional and the stochasticity is the random velocity flip with rate ς -1 .

1. Anharmonic oscillators. It is a common belief, based on extensive numerical simulation, that the transport of energy in anharmonic onedimensional pinned chains is diffusive [START_REF] Lepri | Thermal Conduction in classical low-dimensional lattices[END_REF][9] (see also [START_REF] Gaspard | Heat conduction and Fourier's law by consecutive local mixing and thermalization[END_REF] for physical approaches passing through a weak coupling limit). However, to our knowledge, there are no rigorous mathematical arguments supporting this. We show here that lim sup ς→0 κ 2 (ς) < ∞ for one-dimensional oscillators with rather generic pinning potentials W and interaction V .

We consider the Hamiltonian (8.1) below which allows for an explicit description. The fact that as ς → 0, κ 2 (ς) does not diverge results from averaging oscillations in the uncoupled cells, and not from decay of correlations as it would be the case for a chaotic dynamics. The control of the time integrated current-current correlations in the limit ς → 0 is possible if resonances between near atoms occur with small probability in the Gibbs state. This condition is violated if the pinning W is harmonic, but is otherwise typically satisfied.

2. Disordered oscillators and rotors. We next consider in more details two examples of chains of one dimensional systems that display a similar structure: the disordered harmonic chain and the rotor model. In each case, the atoms are one-dimensional systems, so that, when both noise and coupling are removed, the full dynamics becomes again integrable. Moreover, then, neighboring particles typically oscillate at different frequencies. For these two examples, we are able to give explicit formulas for the weak coupling operator L (see Proposition 8.3 and Proposition 8.5).

In the absence of noise (ς = 0), the disordered chain is well known to be a perfect insulator: κ = 0 [START_REF] Bernardin | Small perturbation of a disordered harmonic chain by a noise and an anharmonic potential[END_REF], while it is conjectured that the conductivity of the rotor chain is finite and positive [START_REF] Lepri | Thermal Conduction in classical low-dimensional lattices[END_REF], but decays faster than any power law in ε as ε → 0 [START_REF] De Roeck | Asymptotic localization of energy in nondisordered oscillator chains[END_REF]. Thus in these two cases it is expected that the conductivity of the deterministic system κ(ε, ς) has no expansion in powers of ε. What we are actually able to prove is that for the disordered harmonic chain lim ς→0 κ 2 (ς) = 0. We also show in Subsection 8.3 that for the rotor chain lim sup ς→0 κ 2 (ς) < +∞, extending the conclusions of Proposition 8.1 to this case. 8.1. Upper bound on the conductivity for pinned anharmonic oscillators. Let

H(q, p) = x p 2 x 2 + W (q x ) + εV (q x+1 -q x ) = x H 0 (q x , p x ) + εV (q x+1 -q x ) (8.1)
with (q x , p x ) ∈ R 2 . The potential W is assumed to be smooth, strictly convex, except possibly at the origin, and symmetric. The potential V is also taken smooth, symmetric, bounded below, and of polynomial growth, always satisfying the requirement that µ β,ε is analytic in ε for small ε. To make things simple and concrete, we will actually focus on W given by

W (q) = |q| r r , r > 2. (8.2)
Proposition 8.1. Let W be given by (8.2) for some r > 2. Then, with the assumptions on V given after (8.1), lim sup κ 2 (ς) < +∞.

Proof. Because of its length, the proof as well as the needed introductory material are postponed to Appendix A.

Remark. In Proposition 8.1, we have limited ourselves to a case leading to rather clean computations. A closer look at the proof in appendix A shows that our hypotheses are too restrictive: what is important is that the map ω(I), giving the frequency of oscillation as a function of the action, can be inverted. The main advantage of taking the W given by (8.2), is that this can be done explicitly.

It then arises as a natural question whether the proof could be further generalized to cases where ω(I) is invertible everywhere but on a finite or countable number of points. This would for example be the case if we consider the pinning potential W (q) = q 2 + a cos(q) for some small enough constant a > 0. This is unfortunatly not the case, as some logarithmic divergence in ς shows up in the limit ς → 0, if one just tries to mimic the proof of Proposition 8.1. Unless the system posseses some hidden symmetry, this in fact means that γ 2 (e 0 , e 1 ) β,0 diverges logarithmically in the deterministic limit. This however does not necessarily imply that κ 2 itself will diverge in this limit, as the term η 2 β,0 in (6.19) can compensate this divergence. This is in fact what is expected to happen. 8.2. The disordered harmonic chain. The hamiltonian part of the generator is now given by

A = x p x ∂ qx -ω 2 x q x ∂ px , G = x (q x-1 -2q x + q x+1 )∂ px , (8.3)
where ω 2 x are random, independent and identically distributed squared frequencies, that satisfy the bound c -1 ≤ ω 2

x ≤ c, for some constant c > 0. The internal energy is given by e x = p 2

x /2 + ω 2 x q 2 x /2, while for ε ≥ 0 the energy flux εj x,x+1 between two adjacent oscillators is given by

ε j x,x+1 = -ε p x + p x+1 2 (q x+1 -q x ).
Lemma 8.2. Let x, y ∈ Z. A solution ψ x,y to the equation -L 0 ψ x,y = q x p y is given by ψ x,y = 4ς ω 2 x q x p y -ω 2 y q y p x + (ω 2 x -ω 2 y )p x p y + (ω 2

x -ω 2 y -8ς 2 )ω 2 y q x q y ∆(x, y)

with ∆ x,y = 8ς 2 (ω 2 x + ω 2 y ) + (ω 2 x -ω 2 y ) 2 . (8.4)
Proof. This follows by a direct computation.

This lemma allows us to give an explicit form of the operator L = ΠG(L 0 ) -1 GΠ. We know that L is the generator of a Ginzburg-Landau dynamics.

Proposition 8.3. Let L = ΠG(L 0 ) -1 GΠ. Then ρ β (e) = x βe -βex , (8.5
)

γ 2 (e x , e x+1 ) = 4ς ∆ x,x+1 e x e x+1 , (8.6)
α(e x , e x+1 ) = 8ς ∆ x,x+1

(e x -e x+1 ). (

Proof. To obtain the expression for the invariant measure, let us take an f that depends only on e x = (p 2 x + ω 2 x q 2 x )/2, and let us compute

f β,0 = Z x (β) -1 R 2 f p 2 x + ω 2 x q 2
x 2 e -β(p 2

x +ω 2 x q 2

x )/2 dq x dp x ∼ ∞ 0 f (e) e -βe de from which the expression for ρ β follows 13 . Next we have that

γ 2 (e x , e x+1 ) = Π j x,x+1 (-L 0 ) -1 j x,x+1 = 1 4 Π q x+1 p x -q x p x+1 + q x+1 p x+1 -q x p x ψ x+1,x -ψ x,x+1 + ψ x+1,x+1 -ψ x,x = 2ς ∆ x,x+1 Π ω 2 x+1 q 2 x+1 p 2 x + ω 2 x q 2 x p 2 x+1 13
Here and in the sequel, we use a ∼ b to say that there exist two postive constants

C 1 , C 2 such that a ≤ C 1 b and b ≤ C 2 a.
where we have used the fact that odd powers of q x , p x , q x+1 , p x+1 are annihilated by the projection Π. Using then polar coordinates

ω x q x √ 2 = √ e x cos θ x , p x √ 2 = √ e x sin θ x , it is computed that both Π(p 2 x ) = Π(ω 2 x q 2 x ) = 1 2π 2π 0 2e sin 2 θ x dθ x = e x .
This yields the announced expression for γ 2 (e x , e x+1 ).

The current α(e x , e x+1 ) follows using (5.5).

Corollary 8.4. For ς > 0, we have that a.s. in ω

κ 2 (ς) = 8ςβ 2 ∆ 0,1 (ς) * > 0
where • * represents the average with respect to the realizations of the disorder. In particular, a.s. in ω,

lim ς→0 κ 2 (ς) = 0.
Proof. The proof is given in Appendix B.

8.3. The rotor chain. The Hamiltonian part of the dynamics is given by

A = x p x ∂ qx , G =
x [sin(q x-1 -q x ) -sin(q x -q x+1 )] ∂ px , (8.8) with q x ∈ R/2πZ. The individual energy for the uncoupled dynamics (ε = 0) is e x = p 2

x /2. If ε > 0, there is a flux of energy which is given by εj x,x+1 where Proof. The proof is given in Appendix C.

j x,x+1 = - 1 2 (p x + p x+1 ) sin(q x+1 -q x ).
It is seen from the above expressions that as noted earlier the generator L formally vanishes as ς → 0. However, for ς small but positive, the coefficient γ 2 (e x , e x+1 ) can become of order 1/ς in case a resonance occurs, such that |e x+1 -e x | ≤ ς. We have unfortunately not been able to decide whether, despite of this phenomenon, the value of κ 2 (ς) still vanishes as ς → 0, as suggested by the results in [START_REF] De Roeck | Asymptotic localization of energy in nondisordered oscillator chains[END_REF].

We have however a result analogous to that of Proposition 8.1:

Proposition 8.6. For any ς > 0, κ 2 (ς) is strictly positive and

lim sup ς→0 κ 2 (ς) < +∞.
Proof. By (6.22) and the explicit form of γ we have that

β -2 κ 2 (ς) ≥ γ -2
(e 0 , e 1 ) -1 β,0 ≥ cς for a positive constant c independent of ς. By (6.22) it holds also that β -2 κ 2 ≤ γ 2 (e 0 , e 1 ) β,0 .

The function γ 2 (e 0 , e 1 ) β,0 has the behavior γ 2 (e 0 , e 1 ) β,0 ∼ R 2 + ς e 0 e 1 4ς 2 (e 0 + e 1 ) + (e 1 -e 0 ) 2 e -β(e 0 +e 1 ) de 0 de 1

√ e 0 e 1 ∼ R 2 ς x 2 y 2 8ς 2 (x 2 + y 2 ) + (y 2 -x 2 ) 2 e -β(x 2 +y 2 )/2 dxdy ∼ ∞ 0 dr 2π 0 dθ ς r 3 cos 2 θ sin 2 θ 8ς 2 + r 2 (cos 2 θ -sin 2 θ) 2 e -βr 2 /2 .
In the limit ς → 0, only the values of θ such that cos 2 θ -sin 2 θ ∼ 0 contribute (θ ∼ ±π/4 and θ ∼ ±3π/4), so that, by a Taylor expansion, To study the system at hand, it is convenient to pass to action-angle variables. Let I : R + → R + be defined by

I(E) = 1 2π A(E) dqdp with A(E) = {(q, p) ∈ R 2 : H 0 (q, p) ≤ E}.
Our assumptions on W ensure that I (E) = dI/dE (E) > 0 for any E > 0. Given E ≥ 0, we also set q * (E) = max{q ∈ R : H 0 (q, p) = E for some p ∈ R}.

Then we define the action-angle variables by

I x = I(q x , p x ) = I(H 0 (q x , p x )), θ x = θ(q x , p x ) = -sgn(p x ) I (H 0 (q x , p x )) q * (H 0 (qx,px)) qx dq 2(H 0 (q x , p x ) -W (q ))
.

It is checked that (I x , θ x ) ∈ R + × T with T = R/(2πZ). The potential W is such that this change of variable is invertible, except at origin. We denote by Q and P the inverse maps:

q x = Q(I x , θ x ), p x = P (I x , θ x ).
The change of variables (q x , p x ) ↔ (I x , θ x ) is known to be a canonical change of variables. Let H 0 : R + → R + be the inverse function of I: H 0 • I(E) = E for any E ∈ R + . In the action-angle variables, the Hamiltonian (8.1) reads

H(I, θ) = x H 0 (I x ) + V Q(I x+1 , θ x+1 ) -Q(I x , θ x ) . Defining ω(I x ) = H 0 (I x ) = dH 0 /dI x , Hamilton equations read İx = - ∂ ∂θ x V Q(I x+1 , θ x+1 ) -Q(I x , θ x ) , θx = ω x + ∂ ∂I x V Q(I x+1 , θ x+1 ) -Q(I x , θ x ) .
The current, given by (2.5), has the form

j x,x+1 = - 1 2 P (I x , θ x ) + P (I x+1 , θ x+1 ) V Q(I x+1 , θ x+1 ) -Q(I x , θ x ) (A.1) with V (x) = dV /dx.
Since we are in dimension d = 1, the noise written in the action-angle coordinates is given by

Sf (I, θ) = x f (I, θ x ) -f (I, θ) , (A.2)
with θ x is obtained from θ by changing θ x to -θ x (-θ x is the inverse of θ x for the addition on T). The symmetry of the potential W implies

P (I x , -θ x ) = -P (I x , θ x ) and Q(I x , -θ x ) = Q(I x , θ x ).
This implies that the noise S, as defined by (A.2), preserves the total energy, and that the relation

Sj x,x+1 = -4j x,x+1
holds.

A.1. The special case W given by (8.2). Let us now assume that W (q) = |q| r /r, i.e.

H 0 (q, p) = p 2 2 + |q| r r , r > 2.
(A.

3)

The following scaling relation are readily checked: A.2. Poisson equation for the uncoupled dynamics. In this subsection, we consider functions on R 2 + × T 2 , that depend on two actions (I 0 , I 1 ) and two angles (θ 0 , θ 1 ). The actions play the role of a parameter, and, for clarity, will be dropped from several notations. A function f ∈ C ∞ (T 2 ) is expanded in Fourier series as

f (I 0 , I 1 , θ 0 , θ 1 ) = (k 0 ,k 1 )∈Z f (I 0 , I 1 , k 0 , k 1 )e i(k 0 θ 0 +k 1 θ 1 ) with f (I 0 , I 1 , k 0 , k 1 ) = 1 (2π) 2 [-π,π] 2 f (I 0 , I 1 , θ 0 , θ 1 )e -i(k 0 θ 0 +k 1 θ 1 ) dθ 0 dθ 1 .
It is seen that the current satisfies ĵ0,1 (I 0 , I 1 , 0, 0) = 0 for all (I 0 , I 1 ) ∈ R 2 + . We introduce the notations

η(k 0 , k 1 ) = i k 0 ω(I 0 ) + k 1 ω(I 1 ) -2ς D(k 0 , k 1 ) = η(k 0 , k 1 )η(-k 0 , -k 1 ) - 16ς 4 η(-k 0 , k 1 )η(k 0 , -k 1 ) . Lemma A.1. Let f be a function on R 2 + × T 2 such that f (I 0 , I 1 , •, •
) is smooth and satisfies f (I 0 , I 1 , 0, 0) = 0, for any (I 0 , I 1 ) ∈ R 2 + . Writing f (k 0 , k 1 ) for f (I 0 , I 1 , k 0 , k 1 ), we define

g(I 0 , I 1 , k 0 , k 1 ) = f (k 0 , k 1 ) -ς f (-k 0 , k 1 ) η(-k 0 , k 1 ) + f (k 0 , -k 1 ) η(k 0 , -k 1 ) + ς 2 η(-k 0 , -k 1 ) 1 η(-k 0 , k 1 ) + 1 η(k 0 , -k 1 ) f (-k 0 , -k 1 ) -f (k 0 , k 1 ) . (A.7)
A solution u to the equation -L 0 u = f is given, in the Fourier variables, by

û(I 0 , I 1 , 0, 0) = 0, û(I 0 , I 1 , k 0 , k 1 ) = - η(-k 0 , -k 1 ) D(k 0 , k 1 ) g(I 0 , I 1 , k 0 , k 1 ) for (k x , k y ) = (0, 0).
Proof. In the Fourier variables, the equation

-L 0 u = f reads η(k 0 , k 1 )û(k 0 , k 1 ) + ς û(-k 0 , k 1 ) + ς û(k 0 , -k 1 ) = -f (k 0 , k 1 )
where we have written û(k 0 , k 1 ) for û(I 0 , I 1 , k 0 , k 1 ). The result is then checked by means of a direct computation.

Remarks. 1. All other solutions are obtained by taking for û(I 0 , I 1 , 0, 0) an arbitrary function of the actions I 0 , I 1 . This choice is irrelevant for the sequel. 2. Since |ς/η(k 0 , k 1 )| ≤ 1 for all (k 0 , k 1 ) ∈ Z 2 , we have the bound

|g(I 0 , I 1 , k 0 , k 1 )| ≤ 5 max{| f (k 0 , ±k 1 )|, | f (-k 0 , ±k 1 )|}.
3. For ς = 0, the solution simply becomes

û(I 0 , I 1 , k 0 , k 1 ) = i f (k 0 , k 1 ) k 0 ω(I 0 ) + k 1 ω(I 1 ) for (k x , k y ) = (0, 0). with h ω 0 , ω 1 , k 0 , k 1 ) = h c(r)ω r+2 r-2 0 , c(r)ω r+2 r-2 1 , k 0 , k 1 , c(r) > 0, ρ β (ω 0 , ω 1 ) = e -c (r)β(ω 2r r-2 0 +ω 2r r-2 1
) , c (r) > 0.

To proceed, we need some more technical informations on the function h(ω 0 , ω 1 , k 0 , k 1 ). The potential W is not strictly convex at the origin, implying that ω(I) vanishes as I → 0. For this reason, we need a relatively detailed knowledge on h(ω 0 , ω 1 , k 0 , k 1 ) for (ω 0 , ω 1 ) near the origin, in a order to exclude any divergence at small frequencies.

Using the general expression (A.1) for the current j 0,1 , the specific expression (A.6) for Q(I, θ) and P (I, θ), the definition (A.7) of g, and the definition (A.8) of h, we conclude that h is of the form h(I 0 , I 1 , k 0 , k 1 ) = I h1,1 (ω 0 , ω 1 , k 0 , k 1 ), (A.12)

where hi,j satisfies the following bounds: there exists a < +∞ and, for any b > 0, there exists a constant C b < +∞, such that hi,j (ω 0 , ω 1 , k 0 , k 1 ) ≤ C b (|ω 0 | + |ω 1 | + 1) a (|k 0 | + |k 1 | + 1) b , (i, j) = (0, 0), (0, 1), (1, 1). (A.13) Moreover, by symmetry, we have P (I, 0) = 0 for all I > 0, with P (I, 0) defined by (A.5). It follows that h0,0 (ω 0 , ω 1 , 0, k 1 ) = h0,1 (ω 0 , ω 1 , 0, k 1 ) = h0,1 (ω 0 , ω 1 , k 0 , 0) = h1,1 (ω 0 , ω 1 , k 0 , 0) = 0.

(A.14)

We now move back to the evaluation of (A.11). We distinguish three cases, according to the values of k 0 and k 1 ; resonances appear in case 3. The sum over (k 0 , k 1 ) ∈ Z 2 /{0, 0} can then be controlled thanks to the decay in (A.13) with b large enough.

Case 1: k 0 k 1 = 0. Let us, as an example, consider the case k 0 = 0, k 1 = 0. The integral (A.11) has a possible divergence only for k 1 → 0.

We have η(-k 0 , -k 1 ) D(k 0 , k 1 ) ≤ Cω -2

1
for ω 0 , ω 1 ≤ 1.

Thanks to (A.14), only the term in h1,1 survives in (A.12), and we conclude that the integrand behaves as as ω 1 → 0, so that there is in fact no singularity. Case 2: k 0 k 1 > 0. The only possible divergence of the integral (A.11) is at the origin. We have the bounds η(-k 0 , -k 1 ) D(k 0 , k 1 ) ≤ Cω -2 0 , Cω -1 0 ω -1 1 , Cω -2

1
for ω 0 , ω 1 ≤ 1, allowing to check, as in the previous case, that there is no singularity. Case 3: k 0 k 1 < 0. The integrand now becomes truly singular (resonances). Let us assume, for example, that k 0 > 0 and k 1 < 0. We split the integral (A.11) as As in the cases treated previously, it is seen that there is no singularity. Moreover, the integration domain is of size ς 2 , so that the integral is of order 1 at most.

We move to the second integral. We find it convenient to change again variables. With We observe that, in the domain of integration x ≥ ς:

4ς 2 x 2 x 2 + 4ς 2 ≥ 4 5 ς 2 .
Therefore, the integral converges to a finite value as ς → 0. where ω = (ω x ) x∈Z , and where {e x (t)} is the time evolved energy generated by the Ginzburg-Landau dynamics L with the coefficients γ 2 and α computed above, starting with the equilibrium distribution at temperature β -1 . Computing the time derivative, we have ∂ t S(x, t, ω) = 8ς∆ -1 x+1,x,ω [S(x + 1, t, ω) -S(x, t, ω)] -8ς∆ -1

x,x-1,ω [S(x, t, ω) -S(x -1, t, ω)] with ∆ x,y,ω defined by (8.4). Thus S(x, t, ω) = E 0,ω (δ x (X(t))), the transition probability of a 1-dimensional random walk on random bonds X(t) (so called bond diffusion). It is well known and easy to compute the asymptotic variance of this bond diffusion, it is given by the harmonic average of the bonds variables ( [START_REF] Komorowski | Fluctuations in Markov processes. Time symmetry and martingale approximation[END_REF] We start by the following lemma.

Lemma C.1. Let x, y ∈ Z. A solution ψ x,y to the equation -L 0 ψ x,y = sin(q x -q y )p x (C.1) is given by ψ x,y = ∆ -1

x,y [4ς 2 + (e x -e y )]e x + 1 2 (e x -e y )p x p y cos(q x -q y ) + ∆ -1

x,y {2ς(e y p x + e x p y )} sin(q x -q y ) (C. Proof. We compute Aψ x,y = 2ς∆ -1 x,y (e y p x + e x p y )(p x -p y ) cos(q x -q y ) -∆ -1

x,y 4ς 2 + (e x -e y ) e x + 1 2 (e x -e y )p x p y (p x -p y ) sin(q x -q y ) and Sψ x,y = -4ς∆ -1

x,y (e y p x +e x p y ) sin(q x -q y )-2∆ -1

x,y (e x -e y )p x p y cos(q x -q y ). Remembering that p 2

x = 2e x and p 2 y = 2e y , the terms in cos(q x -q y ) cancel in (A + ςS)ψ x,y , so that [A + ςS]ψ x,y = ∆ -1

x,y θ x,y sin(q x -q y ) with θ x,y = 4ς 2 + (e x -e y ) e x + 1 2 (e x -e y )p x p y (p x -p y ) -4ς 2 (e y p x + e x p y ) = -p x ∆ x,y . This proves the claim.

Proof of Proposition 8.5. The Gibbs measure at inverse temperature β is readily computed. For a function f depending only on the uncoupled energy e x = p 2

x /2, it holds that

f β,0 = β 2π R f (p 2 x /2)e -βp 2 x /2 dp x = β 4π ∞ 0
f (e)e -βe de √ e from which (8.9) follows.

Proposition 8 . 5 .γ 2

 852 For this system ρ(e) = x e -(U (ex)+βex) β/π with U (e x ) (e x , e x+1 ) = 2ς e x e x+1 ∆(e x , e x+1 ) , (8.10) α(e x , e x+1 ) = ς(e x -e x+1 ) ∆ 2 (e x , e x+1 ) ∆(e x , e x+1 ) + 8e x e x+1 (8.11) with ∆(e x , e x+1 ) = 4ς 2 (e x + e x+1 ) + (e x+1 -e x ) 2 .

γ 2 (

 2 e 0 , e 1 ) β,0 ∼ /r) 2 + u 2 du dr ∼ 1 as ς → 0. This proves the claim. Appendix A. Proof of Proposition 8.1

H

  0 (I) = H 0 (1)•I 2r/(r+2) and ω(I) = ω(1)•I (r-2)/(r+2) . (A.4) Moreover, writing Q(I, θ) = k∈Z Q(I, k)e ikθ , P (I, θ) = k∈Z P (I, k)e ikθ , (A.5) we obtain Q(I, θ) = I 2/(r+2) k∈Z Q(1, k)e ikθ , P (I, θ) = I r/(r+2) k∈Z P (1, k)e ikθ . (A.6) Because Q(1, θ) and P (1, θ) are smooth, the Fourier coefficients Q(1, k), P (1, k), with k ∈ Z, have good decay property as |k| → ∞.

1 hI 2r r+2 1 h 1 , 1 (

 1111 0 (I 0 , I 1 , k 0 , k 1 ) + I 0,1 (I 0 , I 1 , k 0 , k 1 )+ I 0 , I 1 , k 0 , k 1 ), so that in turn h takes the form h(ω 0 , ω 1 , k 0 , k 1 ) = ω 2r/(r-2) 0 h0,0 (ω 0 , ω 1 , k 0 , k 1 ) ω 0 , ω 1 , k 0 , k 1 ) + ω 2r/(r-2) 1

R 2 +

 2 (. . . ) = k 0 ω 0 +|k 1 |ω 1 <ς (. . . ) + k 0 ω 0 +|k 1 |ω 1 ≥ς (. . . ). (A.15)For the first integral, we are satisfied by the rough boundη(-k 0 , -k 1 ) D(k 0 , k 1 ) ≤ C ς 2 (k 0 ω 0 + |k 1 |ω 1 ) 2 .

xy 2 + 4ς 2 x 2 x 2

 222 = k 0 ω 0 + |k 1 |ω 1 , y = k 0 ω 0 -|k 1 |ω 1 ,the second integral in the right hand side of (A.15) becomesk 0 ω 0 +|k 1 |ω 1 ≥ς (. . . ) +4ς 2 φ(x, y, k 0 , k 1 ) ρβ (x, y, k 0 , k 1 ) with φ(x, y, k 0 , k 1 ) = h x + y 2k 0 , x -y 2|k 1 | , k 0 , k 1 (x + y)(x -y) 4k 0 |k 1 | 4/(r-2)ρβ (x, y, k 0 , k 1 ) = ρ β x + y 2k 0 , x -y 2|k 1 | .

Appendix B .

 . Proof ofCorollary 8.4 Consider the quenched space-time correlations of the energy:S(x, t, ω) = e x (t)e 0 (0) ρ β -β -2

  ω. By the Green-Kubo formula for the diffusivity for L, this is yieldsκ 2 (ς) = 8ςβ 2 ∆ 0,1 (ς) * → 0 as ς → 0, (B.2)which gives the claims.Appendix C. Proof of Proposition 8.5

  2) with ∆ x,y := ∆(e x , e y ) = 4ς 2 (e x + e y ) + (e y -e x ) 2 . (C.3)

The notation • | F stands for the conditional expectation with respect to the σ-algebra F. In particular • | e is the expectation with respect to the Liouville (microcanonical) measure on the energy surface e = H(q, p).

Note that, in general, we should write V (q x , q x-1 ) as q might not belong to a vector space. We avoid it to simplify notation, see[START_REF] Dolgopyat | Energy transfer in a fast-slow Hamiltonian system[END_REF] for details.

Here and in the following we work in units in which the Boltzmann constant equals one.

We will see that w λ,0 ∈ H s ε , hence j 0,1 w λ,0 is an integrale function antisymmetric in p. For the higher order terms the fact that the expectation is well defined is a conjecture.

The inverse of -L 0 should be understood as the limit of (ν -L 0 ) -1 , when ν → 0, in some appropriate topology.

Note however that there is no obvious reason why the functions should be in the domain of G. Thus the objects can only be interpreted as distributions. For the moment we do not worry about this issue since we are just doing formal computations.

Since we will compute a correction of order one, the correction to the local energies does not really matter.

Note that this projector is different from the one used in Section 4.

Here the adjoint is taken with respect to all the measures E(• | e).

Note that here, since we are dealing with finite systems, there is no problems in expanding the Gibbs measure in δT and ε.

Tom Spencer and Horng-Tzer Yau for the hospitality at the IAS where this work was completed, and all of us thank the BIRS-Banff. The research of SO and CL was founded in part by the European Advanced Grant Macroscopic Laws and Dynamical Systems (MALADY) (ERC AdG 246953). The research of CB was supported in part by the French Ministry of Education through the grant ANR-10-BLAN 0108 (SHEPI). The research of JLL was supported in part by NSF grant DMR1104500. Franc ¸ois Huveneers, CEREMADE,

A.3. Proof of Proposition 8.1. By (6.22) we have β -2 κ 2 (ς) ≤ γ 2 (e 0 , e 1 ) β,0 = j 0,1 (-L 0 ) -1 j 0,1 β,0 , with • β,0 the uncoupled Gibbs state. Writing u = (-L 0 ) -1 j 0,1 we have thus

with g as defined in (A.7), we obtain by Lemma A.1,

h(I 0 , I 1 , k 0 , k 1 ) e -β H 0 (I 0 )+H 0 (I 1 ) dI 0 dI 1 .

(A.9)

In this expression,

(A.10) We now come to the crux of the argument, and start using the specific form of H 0 . In view of (A.10), it looks desirable to change integration variables in (A.9) from (I 0 , I 1 ) to (ω 0 , ω 1 ) = (ω(I 0 ), ω(I 1 )). The anharmonicity of W , specifically expressed in this case by relation (A.4), makes this possible, giving

Next, γ 2 (e x , e x+1 ) is computed by means of Lemma C.1:

The terms in cos(q x -q x+1 ) in ψ x+1,x and ψ x,x+1 will vanish due to the projection Π, so that we are left with

4ς(e x+1 p x + e x p x+1 ) sin(q x+1 -q x ) ∆ x,x+1 .

Since 1 (2π) 2 [0,2π] 2 sin 2 (x -y) dxdy = 1/2, and since the projection of expressions containing uneven powers of p x or p x+1 vanishes, we obtain (8.10).

The current α(e x , e x+1 ) can be computed in two possible ways: directly by the definition α(e x , x x+1 ) = ΠG(-L 0 ) -1 j x,x+1 , or by means of the expression α(e x , e x+1 ) = e U (ex)+U (e x+1 ) ∂ e x+1 -∂ ex e -(U (ex)+U (e x+1 )) γ 2 (e x , e x+1 ) with U(x) = 1 2 log x. Both computations lead to (8.11).


Frédérique Laforest

Nicolas Le Sommer

Stéphane Frénot

François De Corbière

Yves Mahéo

Pascale Launay

Christophe Gravier

Julien Subercaze

Damien Reimert

Etienne Brodu

Nicolas Le Sommer

Idris Daikh

Nicolas Phelippeau

Xavier Adam

Frédéric Guidec

Stéphane Grumbach

C3PO: a Spontaneous and Ephemeral Social Networking Framework for a collaborative Creation and Publishing of Multimedia Contents

Keywords: Social Networking, Opportunistic Computing, Collaborative Edition

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

Introduction

Online Social Networks (OSNs), such as Facebook, Google+ or Twitter, allow people or institutions to communicate on themselves. They are also designed to let people make long-term contact with members of their family, with friends or with people that share the same interests or goals by a simple invitation mechanism or a system of followers. People can easily share data and communicate with the persons they are in touch using comments, content rating, chat, videos, etc. Although these OSNs are accessed every day by billions of people from their mobile devices, these social networks have several drawbacks. First, they rely on a centralized architecture. All personal data are stored on the servers of the OSN operators and are often exploited for commercial purposes, thus entailing data ownership issues. Therefore, many people are reluctant to use these OSNs. A permanent Internet connectivity is furthermore needed to access these servers. Such an approach excludes that services can be provided by mobile devices themselves. Information exchange is done following links between users in the network, limiting the communication to "known" people. Finally, OSNs do not provide efficient tools to permit people to produce multimedia contents in a cooperative manner.

In this paper, we present a new type of social networks, so called Spontaneous and Ephemeral Social Networks (SESNs). SESNs rely on a peer-to-peer distributed architecture formed spontaneously by mobile devices carried by people and by devices deployed for a particular occasion to support such networks. In these networks, devices opportunistically communicate with each others thanks to wireless interfaces such as Wi-Fi or Bluetooth. Due to their spontaneous and ephemeral nature, SESNs are suited to produce multimedia reports on sport or cultural events in a collaborative way. SESNs are expected to favor the emergence of a participatory micro-journalism. To do so, people are linked together for the duration of the event, and each member of a SESN is expected to offer a set of resources and services (e.g., data sharing service, bib recognition service) to the other members. Members are provided with tools exploiting these data and services, and allowing them to collaborate on the production of multimedia reports. These data and reports are replicated on remote servers in order to ensure their lasting and their large diffusion after Preprint submitted to Elsevier October 16, 2014 the end of the event. The multimedia contents produced by the SESN members are exchanged opportunistically and can be recommended by these members in order to ease the filtering and the retrieving of relevant contents. A SESN also provides mechanisms to incite users to share their own productions.

The next sections of this paper are organized as follows. Section 2 presents a practical scenario of usage of SESNs. Section 3 presents the main characteristics and challenges of SESNs. Related work is discussed in Section 4. Section 5 concludes this paper and describes our plan for future works.

Spontaneous and Ephemeral Social Networks: A Practical Example of Usage

In this section, we describe by example how a SESN can help at making reports on sport event like a marathon, and what such a social network can bring to people. For the sake of illustration, let us consider a marathoner, a spectator and a reporter, named respectively Mike, Lea and Thierry.

Mike is 20 years old. He runs with his smartphone to follow his performances (speed, calorie expenditure, etc.) with a dedicated application. He launched a service that transmits his position regularly. Using the SESN, he can hear voiced information on the position of friends or opponents. Sometimes, he publishes photos and videos taken with his GoPro camera in the SESN. At the end of the race, Mike writes small messages on the SESN (e.g., difficulties, sensations, health state). He is informed of the arrival of friends and usual opponents. He reads content posted by them in the SESN. He also gets automatically all the contents that concern him (official time and ranking, photos and videos where he appears, etc.). When the event is finished, the SESN itself is closed, but most of the produced contents have been transfered to the organizer's web site. In the evening at home, he goes to this web site, reading or downloading photos, videos, organizers' comments or journalists' articles issued in the SESN.

Lea often drives her best friend Mike to marathons, but she does not run herself. As a spectator, she subscribes to the marathon SESN. She stands at the last km of the race. Other spectators provide photos, videos and messages at their relative location so she can follow Mike's race. She is also interested in general information on the marathon, posted by the organizer, journalists or other contributors. She does not hesitate to tag information with "like" when she has appreciated some posts. When competitors arrive at her location, she publishes in return as much information as she can about them. Lea publishes her photos with restricted rights, indicating that their commercial use require her explicit permission. The automatic bib recognition on photos service offered by the organizer is an appreciated service, even if she sometimes has to help the system by entering the number by hand. The tipping system in the SESN fills her virtual purse each time she publishes information. When other participants put "like" tags on the content she has provided, she gets additional tips. Her virtual purse allows her to get some goodies at the marathon organizer's desk. Still following her friend's race, Lea receives a warning when Mike arrives in her vicinity. She will not miss him as he passes.

Thierry is a journalist for the local newspaper, he quite always covers sport events in the local area. Entering the dedicated SESN, he has an inside view of the event happenings and relies on the participants posts to make his own opinion on the marathon. When organizers have accredited official reporters, Thierry uses the SESN to access their writings online. Thierry shares most of the photos taken with his professional camera within the SESN. Authoring rights can be put on his photos if he wants to get paid for photo reuse. Sometimes, Thierry does not have time to go to all sports events of the week-end. In the evening, he can go to the organizers' web sites to gather information.

Other profiles are interested in SESNs. Events organizers are often the initiators of SESNs for their events. Subcontractors of the event organizer are also involved, e.g. companies specialized in races timing, accredited official reporters, sponsors or communication staff provide content in the SESN. Computer scientists interested in data analysis can also provide dedicated services, like the automatic identification of the bib of participants on photos, or the automatic construction of personal journals for participants etc.

Spontaneous and Ephemeral Social Networks: Definitions and Design Proposal

In this section, we first define the main notions and concepts considered in SESNs. Then, we present the general architecture of the C3PO framework, and we detail the main features of this one.

Definitions

• A SESN is a social network that is limited in time and space and dedicated to a single event. A SESN can be created by a user of a mobile device that runs a C3PO-based application. The lifetime of a SESN is defined at its creation time. In most situations, the creation and the lifetime definition of a SESN will be achieved by organizer of an event. For example, a SESN is built for the 2015 Paris Marathon few hours before the beginning of the race, and is destroyed few hours after its end. A SESN is formed spontaneously by the devices carried by people who attend the event and by specific devices deployed to support such a network. A SESN is intended to be geographically limited to the area where the event takes place.

• Members of a SESN are users of mobile devices that attend the event related to this SESN. They provide resources and services in order to support the collaborative production of multimedia contents to make reports on this event. If a user has not received an announce from the creator of the SESN after a given time, he will no longer be considered as member of a SESN.

• Resources and services are offered by the members of a SESN in order to support the collaborative creation of multimedia contents. These services can be content sharing services, editorial services, ranking services ("like" mechanism) and processing services (e.g., face recognition, bib recognition). Members provide computation, communication, storage and energy resources.

• Valuable and relevant contents are multimedia contents (e.g., photos, videos, comments or reports) produced by the members of a SESN and that have been highly recommended by these ones. In order to ensure their lasting and their large diffusion, these contents are expected to be replicated on an centralized infrastructure (organizer's Web site).

General architecture of the C3PO framework

The general architecture of the C3PO framework, whose a description is given in Figure 2, relies on the latest advances in mobile applications. It is composed of 4 main modules, called respectively NetMan, OppCom, OppServ and SocialNet. NetMan is responsible of the network management and of device discovery. The OppCom module implements the "store, carry and forward" principle, and provides opportunistic communications between the devices of a SESN. OppServ makes it possible to discover and to invoke the services offered in a SESN. Finally, the SocialNet module provides most of the mechanisms we have in traditional online social networks, such as data sharing, comment publishing, etc. SocialNet is implemented as a set of services that can be discovered and invoked using the OppServ module. SocialNet also provides specific services that enable the collaborative production of multimedia contents by members of a SESN. This framework is currently under development. We have also designed a proof-of-concept Android-based application that uses this framework. Each member of a SESN supports and contributes to this social network by providing a set of resources and services. These services are discovered and invoked using opportunistic networking techniques. In C3PO, each device is responsible for discovering the services offered in the SESN, and for maintaining their own list of services. The reactive discovery is usually achieved when service providers broadcast service advertisements, while the proactive discovery occurs when service requesters broadcast services requests. Unicast and anycast service invocations are implemented in order to reduce the number of service requests disseminated in the network, and to improve the response time. Indeed, only one request is sent to invoke several providers at the same time.

Opportunistic Computing

In SESNs, services are provided using opportunistic communication to cope with connectivity disruptions due to the mobility of users and to the short communication range of the wireless interfaces. Thanks to these communications, the localization of members of a SESN is implicitly limited to the geographical area covered by the radio wireless interfaces of this collection of devices. When the event is finished, the members vanish and the network is destroyed automatically.

Content exchange

Content exchange must be done in a fast and easy way. Exchanged contents are mainly small messages that look like tweets. They reference photos, short videos, texts. This rough material can be enhanced by some processing services, with comments given by participants, etc.

Content exchange is based on a push principle : the content creator sends his message to the SESN. Each message can be thrown to all the SESN participants or to specific targets within the SESN. It is highly similar to the Twitter publishing mechanism. We do not use permanent followers, since a user may use a new id for each spontaneous and ephemeral group he belongs to. As soon as a user enters a group he gets all messages posted to this SESN, in a way similar to IRC. When you leave the SESN you do not get any remaining messages. Yet, some messages, or referenced material may be ranked by users. These messages are regularly reposted on the SESN. Thus, when a user enters a SESN he gets both live messages and top messages that have been highly ranked by participants. It is a way to get top-stories among more casual information.

Content recommendation

Content recommendation is used to organize published content on participants' user interface. The recommendation technique is a hybrid between content recommendation and social recommendation. As users of the SESN are any people in the vicinity, no relationship between people exists at the start of the SESN. Also, no user profile is available. The only way to characterize users is to learn from their activity in the SESN. Our content recommendation system thus will base its algorithm on three elements: activity traces, content popularity, and tags. Activity traces are built by observing the user service consumption; measures like duration of consultation or number of consultations of a specific message are used. Content popularity is built on exchanging information on the most viewed contents within the SESN network. Comments and tags are put by users when they have particularly appreciated a message. Tags are similar to the popular Facebook "like" feature. These tags and comments are used both in the content popularity measure and in the user's profile setting up.

Incentive mechanism

So that SESNs can be adopted and used by a large number of people, several features must be taken into account. The success of such social networks mainly depends on quality of the applications dedicated to these networks, but also on the quality of the information and the services offered in these networks [START_REF] Delone | The DeLone and McLean Model of Information Systems Success: A Ten-Year Update[END_REF]. Members' motivation to use SESNs is intrinsically dependent of this quality. Information quality is a multidimensional concept that depends of the context of use [START_REF] Wang | Beyond accuracy: what data quality means to data consumers[END_REF]. Each SESN is evaluated by its members on its ability to provide information that is fit for use [START_REF] Kane | Codification and collaboration: Information quality in social media[END_REF]. Relevance, timeliness and quantity of published contents in the SESN are defined as the three main dimensions for the evaluation of a SESN. Moreover, motivations of SESN members can be intrinsic and/or extrinsic [START_REF] Geiger | Crowdsourcing information systems -definition, typology and design[END_REF]. Intrinsic motivations refer to enjoyment-based and community-based motivations. Extrinsic motivations refer to recognition mechanisms (payoffs, social motivations). In order to favor the adoption of SESN by people, incentive mechanisms are envisioned (publication of the most popular contents and their owners, invitations to meet an athlete, tickets for next events, etc.). A content assessment system will be built, in which organizers and/or journalists will be able to choose among a set of criteria in order to nominate the most valuable contents and members in the SESN.

Related Work

Nowadays, it exists a plethora of social networks (see the non exhaustive list on Wikipedia [START_REF] Wikipedia | List of Online Social Networks[END_REF]). Most of them rely on a centralized architecture and require connectivity to the Internet. Over the last years, decentralized social networks have been studied and developed by researchers and non-academic open source projects. Diaspora [START_REF]Diaspora Web Site[END_REF] is a popular open source decentralized online social network based on multiple small independent servers. Several academic research works have addressed privacy issues in decentralized online social networks, such as LifeSocial [START_REF] Graffi | A distributed platform for multimedia communities[END_REF] Safebook [START_REF] Cutillo | Safebook: A Privacy-preserving Online Social Network Leveraging on Real-life Trust[END_REF], PeerSoN [START_REF] Buchegger | Peerson: P2p social networking: Early experiences and insights[END_REF], MyNet [START_REF] Kalofonos | Mynet: A platform for secure p2p personal and social networking services[END_REF] and Sodensson [START_REF] Baumgart | Towards secure user-centric networking: Service-oriented and decentralized social networks[END_REF], but none of them have considered social networks formed spontaneously by a set of mobile devices.

Data sharing in opportunistic networks have been studied in projects Haggle1 , Sarah2 , PodNet3 , SocialNet4 , Crowd5 and Scampi6 . Haggle [START_REF] Scott | Haggle: a Networking Architecture Designed Around Mobile Users[END_REF] takes advantage of contact opportunities and of device mobility in order to follow the "store, carry and forward" principle in the communication between devices. Haggle, PodNet and Crowd implements a publish/subscribe model, where users can express their interests via keywords and receive content items accordingly. PodNet extends the OSNs to opportunistic networks [START_REF] Distl | Extending the reach of online social networks to opportunistic networks with podnet[END_REF]. In Crowd [START_REF] Belblidia | Part-whole dissemination of large multimedia contents in opportunistic networks[END_REF], multimedia contents are exchanged via an online Web portal through Wi-Fi hotspots. SocialNet mainly aims at detecting the social links between people to improve the opportunistic forwarding of contents. Social links rely on community memberships, history of contacts, recurrent mobility patterns and/or user interests.

In C3PO, people implicitly express their common interests by physically attending sport or cultural events, and by being members of the SESNs dedicated to these events. Due to the ephemeral nature of the SESNs, the exploitation of information such as the history of contacts or the recurrent mobility patterns is not relevant. Projects Sarah and Scampi [START_REF] Conti | From Opportunistic Networks to Opportunistic Computing[END_REF] both have defined a middleware platform to support communication and service provision in opportunistic networks. They have also implemented Android applications based on these middleware platforms [START_REF] Mahéo | Beyond Opportunistic Networking Protocols: a Disruption-Tolerant Application Suite for Disconnected MANETs[END_REF]. Sarah allows to discover neighbors, manage a list of contacts, exchange contents in a secure manner. Nevertheless, none of these projects provides incentive tools to support a collaborative production of rich multimedia contents during ephemeral events.

Conclusion

In this paper, we have defined the concept of Spontaneous and Ephemeral Social Networks. We have shown their main difference with the traditional social networks, including peer-to-peer based architecture, opportunistic communication, event-related location and time limits, services dedicated to collaborative reporting so as to support micro-journalism from the crowd. The corresponding social networks are created by chance and vanish as people leave the event location. Created contents are persisted on external systems. Scientific locks concern the opportunistic network support for services, the recommendation of contents to members and the definition of an incentive mechanism that boosts the collaborative reporting.

A prototype dedicated to the reporting and the promotion of sport events is currently under development. The objective is to experiment this prototype in real situations, and to have a feedback in order to improve it, and to show how SESNs can contribute to support the micro journalism of the future.

Figure 1 :

 1 Figure 1: Illustration of a SESN dedicated to sport event reporting

Figure 2 :

 2 Figure 2: Architecture of the C3PO framework.

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Acknowledgements

This work is done in project C3PO. This project is supported by the French ANR (Agence Nationale de la Recherche) under contract ANR-13-CORD-0005. ❤tt♣✿✴✴✇✇✇✳❝✸♣♦✲❛♥r✳❢r✴