Equilibrium fluctuations for the disordered harmonic chain perturbed by an energy conserving noise

We investigate the macroscopic behavior of the disordered harmonic chain of oscillators, through energy diffusion. The Hamiltonian dynamics of the system is perturbed by a degenerate conservative noise. After rescaling space and time diffusively, we prove that energy fluctuations in equilibrium evolve according to a linear heat equation. The diffusion coefficient is obtained from the non-gradient Varadhan's approach, and is equivalently defined through the Green-Kubo formula. Since the perturbation is very degenerate and the symmetric part of the generator does not have a spectral gap, the standard non-gradient method is reviewed under new perspectives.


Introduction
In this paper we investigate diffusion problems in non homogeneous media for interacting particle systems. More precisely, we adress the problem of energy fluctuations for chains of oscillators with random defects. In the last fifty years, it has been recognized that introducing randomness in interacting particle systems has a drastic effect on the conduction properties of the material. Mathematically the only tractable model is the one dimensional system with harmonic interactions [1]. The aim of this paper is to study the diffusive behavior of disordered harmonic chains perturbed by an energy conserving noise. In some sense, the noise should simulate the non linearities effect, and the conductivity of the onedimensional chain should become finite and positive. We also expect that some homogenization effect occurs and that the conductivity does not depend on the statistics of the disorder in the thermodynamic limit.
The disorder effect has already been investigated for lattice gas dynamics, for example in [7,8,13,15]. These papers share one main feature: the models are non gradient due to the presence of the environment. Non gradient systems are usually solved by establishing a microscopic Fourier's laẘ UMPA, UMR-CNRS 5669, ENS de Lyon, 46 allée d'Italie, 69007 Lyonmarielle.simon@ens-lyon.fr up to a small fluctuating term, following the sophisticated method initially developed by Varadhan in [19], and generalized to non-reversible dynamics [10]. The previous works mainly deal with symmetric systems of particles that evolve according to an exclusion process in random environment: the particles are attempting jumps to nearest neighbor sites at rates which depend on both their position and the objective site, and the rates themselves come from a quenched random field. Different approaches are adopted to tackle this non gradient system: whereas the standard Varadhan's method is helpful only in dimension d ě 3 [7], the "long jump" variation developed by Quastel in [15] is valid in all dimensions.
The study of disordered chains of oscillators perturbed by a conservative noise has appeared more recently, see by instance [2,3,5]. In these papers, only the behavior of the thermal conductivity defined by the Green-Kubo formula is investigated. Here, the diffusion coefficient is defined through hydrodynamics.
In [18], we have obtained the diffusive scaling limit for a homogeneous chain of coupled harmonic oscillators perturbed by a noise, which randomly flips the sign of the velocities, so that the energy is conserved but not the momentum. We have derived a system of nonlinear hydrodynamic equations on the only two conserved quantities: the energy and the total length of the chain, thanks to the relative entropy method. One of the major ingredient is an exact fluctuation-dissipation equation (see for example [12]), which reproduces at the microscopic level the Fourier law up to a small fluctuating term.
Our first motivation was to investigate the same chain of harmonic oscillators, still perturbed by the velocity-flip noise, but now provided with i.i.d. random masses. This makes all previous computations pointless: in particular, the fluctuation-dissipation equations are not directly computable any more. As a consequence, the fluctuation-dissipation decomposition can only be approximated by a sequence of local functions, in the sense that the difference has a small space-time variance with respect to the dynamics in equilibrium. The main ingredients of the usual non gradient method are: first, a spectral gap for the symmetric part of the dynamics, and second, a sector condition for the total generator. The model has then special features that enforce the Varadhan's method to be considered with new perspectives. In particular, the symmetric part of the generator is poorly ergodic, and does not have a spectral gap when restricted to microcanonical manifolds. Moreover, due to the degeneracy of the noise, the asymmetric part of the generator is difficult to control by its symmetric part (in technical terms, the sector condition does not hold), with the only velocity-flip noise. Besides, let us remark that the energy current depends on the disorder, and has to be approximated by a fluctuation-dissipation equation which takes into account the fluctuations of the disorder itself.
Because of the high degeneracy of the velocity-flip noise, we add a second stochastic perturbation, that exchanges velocities (divided by the square root of mass) and positions at random independent Poissonian times, so that a kind of sector condition can be proved (see Proposition 5.7: we call it the weak sector condition). However, the spectral gap estimate and the usual sector condition still do not hold when adding the exchange noise. The harmonic chain has helpful features, in particular the generator of the dynamics preserves the degree of polynomials, and even a degenerate noise is sufficient to apply Varadhan's approach. The sector condition and the non gradient decomposition are only needed for a specific class of functions. The stochastic noise still does not have a spectral gap, but it does make no harm. Contrary to the standard Varadhan's approach, we do not need to prove any general result concerning the so-called closed forms (see [17] by instance). As far as we know, this is the first time that the non-gradient method is used successfully without the spectral gap estimate nor the usual sector condition.
Here, we study equilibrium macroscopic energy fluctuations. By instance, for the nonlinear ordered for some finite constant C ą 0. The equations of motions are given by The dynamics conserves the total energy To overcome the lack of ergodicity of deterministic chains, we add a stochastic perturbation to this new dynamics, so that the convergence of the energy fluctuations distribution holds (Theorem 3.1). The noise can be easily described: at independently distributed random Poissonian times, the quantity p x { ? M x and the interdistance r x are exchanged, or the momentum p x is flipped into´p x . This noise still conserves the total energy E, and is very degenerate.
Even if Theorem 3.1 could be proved for this harmonic chain, for pedagogical reasons we now focus on a simplified model (as in [4]), which has exactly the same features and involves less painful computations. From now on, we study the dynamics on the new configurations tη x u xP written as where m :" tm x u xP is the new disorder with the same characteristics as before. It is notationally convenient to change the variable η x into ω x :" ? m x η x , and the total energy reads Let us now introduce the corresponding stochastic energy conserving dynamics: the evolution is described by (1) between random exponential times, and at each ring one of the following interactions can happen: (i) Exchange noise: two nearest neighbors variables ω x and ω x`1 are exchanged, With these two perturbations, the dynamics conserves the total energy only: The other important conservation laws of the Hamiltonian part are destroyed by the stochastic noises. As a result, the following family tµ β u βą0 of grand-canonical Gibbs measures is invariant for the process: The index β stands for the inverse of the temperature. Notice that µ β does not depend on the disorder, and that the dynamics is not reversible with respect to the measure µ β . We define e β as the thermodynamical energy associated to β, namely the expectaction of ω 2 0 with respect to µ β , and χpβq " 2β´2 as the variance of ω 2 0 with respect to µ β . We consider the system starting with µ β and we denote by β the expectation for the stochastic dynamics starting with this invariant distribution. We prove a diffusive behavior for the energy: first, define the distributions-valued energy fluctuation field It is well-known that Y N converges in distribution as N ÝÑ 8 towards a centered Gaussian field Y, which satisfies for good test functions F, G. In this paper we prove that these energy fluctuations evolve diffusively in time (Theorem 3.1). More precisely, the following distribution converges in law as N ÝÑ 8 to the solution of the linear Stochastic Partial Differential Equation (SPDE) where D is the diffusion coefficient which has explicit expressions, and B is the standard normalized space-time white noise.
Let us now give the plan of the paper. We start with Section 2, which is devoted to introduce the model and all notations and definitions that are needed. The main point is to identify the diffusion term D (Section 5), by adapting the method introduced in [19]. In Section 4, we derive the Boltzmann-Gibbs principle. The convergence of the energy fluctuations field (in the sense of finite dimensional distributions) is proved in Section 3. Finally, Section 6 gives a precise description of the diffusion coefficient through several variational formulas. In Section 7, we present a second disordered model, where the interaction is described by a potential V. For this anharmonic chain, we need a very strong stochastic perturbation, which has a spectral gap, and satisfies the sector condition. In Appendices, technical points are detailed: in Appendix A, the space of square integrable functions w.r.t. the standard Gaussian law is studied through its orthonormal basis of Hermite polynomials. In Appendix B, the weak version of closed forms usual result is investigated. The sector condition is proved for a specific class of functions in Appendix C. Appendix D is devoted to prove the convergence of the Green-Kubo formula.
In Appendix E, the tightness for the energy fluctuation field is investigated.

Generator of the Markov process
We first describe the dynamics on the finite torus N :" t0, ..., Nu, meaning that boundary conditions are periodic. The configuration tω x u xP N evolves according to a dynamics which can be divided into two parts, a deterministic one and a stochastic one. The space of configurations of our system is given by Ω N " N . We recall that the disorder is an i.i.d. sequence m " tm x u xP which satisfies: for some finite constant C ą 0. The corresponding product and translation invariant measure on the space Ω D " rC´1, Cs is denoted by and its expectation is denoted by . For a fixed disorder field m " tm x u xP N , we consider the system of ODE's ?
? m x´1¸d t, t ě 0, x P N and we superpose to this deterministic dynamics a stochastic perturbation described as follows: to each atom x P N , and each bond tx, x`1u, x P N is associated an exponential clock of rate one, such that each clock is independent of each other. When the clock attached to x rings, ω x is flipped into´ω x , and when the clock attached to the bond tx, x`1u rings, the values ω x and ω x`1 are exchanged. This dynamics can be entirely defined by the generator of the Markov process tω x ptq ; x P N u tě0 , that is where, for all functions f : Here, the configuration ω x is the configuration obtained from ω by flipping the momentum of particle x: The configuration ω x,x`1 is obtained from ω by exchanging the momenta of particles x and x`1: We denote the total generator of the noise by S N :" γS flip N`λ S exch N , where γ, λ ą 0 are two positive parameters which regulate the respective strengths of noises.
One quantity is conserved: the total energy ř ω 2 x . The following translation invariant product Gibbs measures µ N β on Ω N are invariant for the process: In the following, the expectation of f with respect to µ N β is denoted by x f y β . The index β stands for the inverse temperature, namely xω 2 0 y β " 1{β. Let us highlight the fact that the Gibbs measures do not depend on the disorder m. This obvious remark will play further a crucial role. From the definition, our model is not reversible with respect to the measure µ N β . Precisely, A m N is an antisymmetric operator in L 2 pµ N β q, whereas S N is symmetric. We denote by Ω the space of configurations in the infinite line, that is Ω :" , and by µ β the product Gibbs measure on . Hereafter, for every β ą 0, we denote by ‹ β the probability measure on Ω DˆΩ defined by ‹ β :" b µ β . We notice that ‹ β is translation invariant and we write ‹ β for the corresponding expectation.

Energy current
Since the dynamics conserves the total energy, there exist instantaneous currents of energy j x,x`1 such that L m pω 2 x q " j x,x`1 pm, ωq´j x´1,x pm, ωq. The quantity j x,x`1 is the amount of energy between the particles x and x`1, and is equal to The energy conservation law can be read locally as where J x,x`1 ptq is the total energy current between x and x`1 up to time t. This can be written as where M x,x`1 ptq is a martingale which can be explicitely computed as Itô stochastic integral: where pN x,x`1 q xP are independent Poisson processes of intensity λ. We also write j x,x`1 " j A x,x`1j S x,x`1 where j A x,x`1 (resp. j S x,x`1 ) is the current associated to the antisymmetric (resp. symmetric) part of the generator: One can check that the current cannot be directly written as the gradient of a local function, neither by an exact fluctuation-dissipation equation (in other words, the current is not the sum of a gradient and a dissipative term of the form L m N pτ x hq, where h is a local function of the system configuration). This means that we are in the nongradient case. We also define the static compressibility that is equal to χpβq :" xω 4 0 y β´x ω 2 0 y 2 β " 2 β 2 .

Cylinder functions
For every x P and f a measurable function on Ω DˆΩ , we consider the translated function τ x f , which is the function on Ω DˆΩ defined by: τ x f pm, ωq :" f pτ x m, τ x ωq, where τ x m and τ x ω are the disorder and particle configurations translated by x P , respectively: If f is a measurable function on Ω DˆΩ , the support of f , denoted by Λ f , is the smallest subset of such that f pm, ωq depends only on tm x , ω x ; x P Λ f u and f is called a cylinder function if Λ f is finite.
For every cylinder function f : Ω DˆΩ ÝÑ , consider the formal sum which does not make sense but for which ∇ 0 pΓ f q :" Γ f pm, ω 0 q´Γ f pm, ωq, are well defined. Similarly, we define p∇ x f qpm, ωq :" f pm, ω x q´f pm, ωq, Let Λ Ť be a finite subset of , and denote by F Λ the σ-algebra generated by tm x , ω x ; x P Λu. For a fixed positive integer ℓ, we define Λ ℓ :" t´ℓ, ..., ℓu. If the box is centered at site x P , we denote it by Λ ℓ pxq :" t´ℓ`x, ..., ℓ`xu.
We denote by C the set of cylinder functions on Ω DˆΩ with compact support and null average with respect to µ β . We also introduce the set of quadratic cylinder functions on Ω DˆΩ , denoted by Q Ă C, and defined as follows: ϕ P Q if there exists a finite sequence pψ i, j pmqq i, jP of real cylinder functions on Ω D such that ϕpm, ωq " For ϕ P C, denote by s ϕ the smallest positive integer s such that Λ s contains the support of ϕ and then Λ ϕ " Λ s ϕ . Hereafter, we consider operators L m , A m and S acting on functions f P C as We also denote S x " γ∇ x`λ ∇ x,x`1 for x P . For Λ ℓ Ť defined as above, we denote by L m Λ ℓ , resp. S Λ ℓ , the restriction of the generator L m , resp. S, to the finite box Λ ℓ , assuming periodic boundary conditions. DEFINITION 2.1. Let C 0 (respectively Q 0 ) be the set of cylinder (respectively quadratic cylinder) functions ϕ on Ω DˆΩ such that there exists a finite subset Λ Ť , and cylinder functions tF x , G x u xPΛ , satisfying If ϕ belongs to Q 0 , we assume the cylinder functions F x , G x to be quadratic.
In the following, we will mostly deal with Q 0 . To conclude this section we introduce the quadratic form of the generator: for any x P and cylinder functions f , g P C, let us define and recall that The symmetric form D ℓ is called the Dirichlet form, and is well-defined on C. This is a random variable with respect to the disorder m.

Semi-inner products and diffusion coefficient
For cylinder functions g, h P C, let ! g, h " β,‹ :" ÿ xP ‹ β rg τ x hs and ! g " β,‹‹ :" which are well defined because g and h belong to C and therefore all but a finite number of terms vanish. Notice that !¨,¨" β,‹ is an inner-product, since the following equality holds: Since ! f´τ x f , g " β,‹ " 0 for all x P , this scalar product is only semidefinite. In the next proposition we give explicit formulas for elements of C 0 .
Proof. The proof is straightforward. l DEFINITION 2.2. We define the diffusion coefficient Dpβq for β ą 0 as ) .
The first term in the sum is only due to the exchange noise, whereas the second one comes from the Hamiltonian part of the dynamics. Formally, this formula could be read as but the last term is ill-defined because j A 0,1 is not in the range of L m . More rigorously, we should define The last expression is now well-defined, and the problem is reduced to prove convergence as z ÝÑ 0. From Hille-Yosida Theorem (see Proposition 2.1 in [6] by instance) (5) is equal to the infinite volume Green-Kubo formula: In Section 6.2, we prove that (6) converges, by inspiring the argument from [2]. It follows that the diffusion coefficient can be equivalently defined in the two ways. Thanks to the Green-Kubo formula, one can easily see that Dpβq does not depend on β. We denote by Lpzq the second term of the right-hand side of (6), that is The function L is smooth on p0,`8q (see [16]). Let h z :" h z pm, ω; βq be the solution of the resolvent equation in L 2 p!¨,¨" β,‹ q: pz´L m qh z " j A 0,1 .
Then we have Observe that if ω is distributed according to µ β then β 1{2 ω is distributed according to µ 1 . Since h z pm, ω; 1q " h z pm, ω; βq and j A x,x`1 is an homogeneous function of degree two in ω, it follows that the diffusion coefficient does not depend on β.

Macroscopic fluctuations of energy
In this section we are interested in the fluctuations of the empirical energy. We prove that the limit fluctuation process is governed by a generalized Ornstein-Uhlenbeck process, whose covariances are given in terms of the diffusion coefficient. We adapt the non-gradient method introduced by Varadhan. In particular, we establish rigorously the variational formula that appears in the definition of the diffusion coefficient (Definition 2.2). Varadhan's approach is investigated in Sections 4, 5 and 6.

Energy fluctuation field
Recall that we denote by e β the thermodynamical energy associated to the inverse temperature β ą 0, namely e β " β´1. We define the energy empirical distribution π N t,m on the torus " r0, 1q as where δ u states for the Dirac measure. We denote by tωptqu tě0 the Markov process generated by N 2 L m N and by M 1 the set of probability measures on , endowed with the weak topology. The space of trajectories in M 1 , which are right-continuous and left-limited (i.e. the Skorokhod space) is denoted by D`r0, Ts, M 1˘. If the initial state of the dynamics is given by the equilibrium Gibbs measure µ N β , then π N t,m weakly converges towards the deterministic measure on , equal to te β duu. Our goal is to investigate the fluctuations of the empirical measure π N with respect to this limit. Let us fix the disorder m, and the inverse of temperature β ą 0. Consider the system under the equilibrium measure µ N β .

DEFINITION 3.1 (Empirical energy fluctuations).
We denote by Y N t,m the empirical energy fluctuation field defined as where H : ÝÑ is a smooth function.
We are going to prove that the distribution Y N t,m converges in law towards the solution of the linear SPDE: is a standard normalized space-time white noise, and D is the diffusion coefficient defined in Theorem 5.9. Observe that there is no dependence on the disorder m in the limit process. In other words, the latter is described by the stationary generalized Ornstein-Uhlenbeck process with zero mean and covariances given by for all t ě 0 and smooth functions H, G : ÝÑ . Here, H (resp. G) is the periodic extension to the real line of H (resp. G). We denote by Y N m the probability measure on Dpr0, Ts, M 1 q induced by the energy fluctuation field Y N t,m and the Markov process tωptqu tě0 generated by N 2 L m N , starting from the equilibrium probability measure µ N β . Let Y be the probability measure on the space Dpr0, Ts, M 1 q corresponding to the generalized Ornstein-Uhlenbeck process Y t . The main result of this section is the following.

Strategy of the proof
We follow the lines of Section 3 in [14]. The proof of Theorem 3.1 is divided into three steps. First, we need to show that the sequence tY N m u Ně1 is tight. This point follows a standard argument, given for instance in Section 11 of [9], and recalled in Appendix E for the sake of completeness. Then, we prove that the one-time marginal of any limit point Y ‹ of a convergent subsequence of tY N m u Ně1 is the law of a centered Gaussian field Y with covariances given by where H, G : ÝÑ are smooth functions. This statement comes from the central limit theorem for independent variables. Finally, we prove the main point in the next subsections: all limit points Y ‹ of convergent subsequences of tY N m u Ně1 solve the martingale problems below.

Martingale decompositions
Let us fix a smooth function H : ÝÑ . We rewrite Y N t,m pHq as Hereafter, ∇ N denotes the discrete gradient: Hˆx`1 N˙´Hˆx N˙ , and the discrete Laplacian ∆ N is defined in a similar way: To close the equation, we are going to replace the term involving the microscopic currents with a term involving Y N t,m . In other words, the most important part in the fluctuation field represented by is its projection over the conservation field Y N t,m (recall that the total energy is the unique conserved quantity of the system). The non-gradient approach consists in using the fluctuation-dissipation approximation of the current j x,x`1 given by Theorem 5.9 as D`ω 2 Here, pN x,x`1 q xP , pN x q xP are independent Poisson processes of intensity (respectively) λ and γ. The strategy of the proof is based on the two following results.

LEMMA 3.2. For every smooth function H : ÝÑ
, and every function f P Q,

THEOREM 3.3 (Boltzmann-Gibbs principle).
There exists a sequence of functions t f k u kP P Q such that (i) for every smooth function H : ÝÑ , (ii) and moreover As a result, the martingale M We have proved that the limit solves the martingale problems (7) and (8), which uniquely characterized the generalized Ornstein-Uhlenbeck process Y t . The proof of Lemma 3.2 is postponed to the end of this section. The proof of Theorem 3.3 is more challenging, and Sections 4, 5 and 6 are devoted to it.

Proof of Lemma 3.2
In this paragraph we give a proof of Lemma 3.2. We define As before, we can rewrite , .
-´∇ Nˆx N˙Γ f‚ pm, sqd Therefore, On the one hand, This last quantity is of order 1{N 2 , because f is a local function of zero average, and H is smooth. On the other hand, let us define Then, the expectation of the second term of (12) is equal to Again, since f is local and H is smooth, this quantity is of order 1{N 2 . Indeed, in the expression ∇ x,x`1 pY x q, there is a sum over z P , but in which only terms with |z´x| ď 2 remain. The same holds for the third term of (12).

Central limit theorem variances at equilibrium
In this section we are going to identify the diffusion coefficient D that appears in (9). Roughly speaking, D can be viewed as the asymptotic component of the energy current j x,x`1 in the direction of the gradient ω 2 x`1´ω 2 x , and makes the expression below vanish: Here, the infimum is taken over all smooth local functions f . Let us try to give in the following subsection an intuition for the origin of Theorem 4.5.

An insight through additive functionals of Markov processes
Consider a continuous time Markov process tY s u sě0 on a complete and separable metric space E, which has an invariant and ergodic measure π. We denote by x¨y π the inner product in L 2 pπq and by L the infinitesimal generator of the process. The adjoint of L in L 2 pπq is denoted by L ‹ . Fix a function V : E ÝÑ in L 2 pπq such that xVy π " 0. Theorem 2.7 in [11] gives conditions on V which guarantee a central limit theorem for VpY s qds and shows that the limiting variance equals Let the generator L be decomposed as L " S`A, where S " pL`L ‹ q{2 and A " pL´L ‹ q{2 denote, respectively, the symmetric and antisymmetric parts of L. Let H 1 be the completion of L 2 pπq with respect to the semi-norm }¨} 1 defined as: Let H´1 be the dual space of H 1 with respect to L 2 pπq, in other words, the Hilbert space generated by local functions and the norm }¨}´1 defined by where the supremum is carried over all local functions g. Formally, } f }´1 can also be thought as Notice the difference with the variance σ 2 pV, πq which formally reads Hereafter, B s represents the symmetric part of the operator B. We can write, at least formally, that where A ‹ stands for the adjoint of A. We have therefore that " p´Lq´1 ‰ s ď p´Sq´1. The following result is a rigorous estimate of the time variance in terms of the H´1 norm, which is proved in [11], Lemma 2.4.

LEMMA 4.1. Given T ą 0 and a mean zero function
If we compare the previous left-hand side to the Boltzmann-Gibbs principle (11), the next step should be to take V as and then take the limit as N goes to 8. In the right-hand side of (13) we will obtain a variance that depends on N, and the main task will be to show that this variance converges: this is studied in more details in what follows. Precisely, we prove that the limit of the variance results in a semi-norm, which is denoted by~¨~β and defined in (20). We are going to see that (20) involves a variational formula, which formally reads~f~2 The final step consists in minimizing this semi-norm on a well-chosen subspace in order to get the Boltzmann-Gibbs principle, through orthogonal projections in Hilbert spaces. The hard point is that¨~β only depends on the symmetric part of the generator S, and the latter is really degenerate, since it does not have a spectral gap.
In Subsection 4.2, we investigate the variance x f , p´Sq´1 f y β , and prove its well-posedness for every function f in C 0 . In Subsection 4.3, we relate the previous limiting variance (taking the limit as N goes to infinity) to the suitable semi-norm. Subsection 4.4 is devoted to prove the Boltzmann-Gibbs principle inspired from Lemma 4.1. Then, in Section 5 we investigate the Hilbert space generated by the semi-norm, and prove some decompositions into direct sums. Finally, Section 6 focuses on the diffusion coefficient and its different expressions.

Microcanonical measures and integration by part 4.2.1 Decomposition on microcanonical measures
The thermodynamic ensemble which is naturally associated with a Hamiltonian dynamics is the microcanonical ensemble, which describes the system at fixed energy. It is possible to devise a probability measure on the configurations ω P Ω N with constant energy β´1 ą 0 such that the measure is stationary with respect to the Hamiltonian flow. The corresponding probability measure denoted by µ mc N,β is the normalized uniform probability measure on the sphere , .
-. Now, for β´1 ą 0 fixed, we disintegrate the microcanonical measure µ mc N,β on S N,β . Let G be the group generated by the following matrices: the permutation matrices P σ , defined for any permutation σ of t1, ..., Nu by 0 otherwise, and the sign matrices S k , defined for k P t1, ..., Nu by The group G acts on S N,β . For x P S N,β , we denote by G N,β x the orbit of x " px 1 , ..., x N q under the action of G. More precisely, and each orbit is finite, with cardinality 2 NpN`1q{2 . This group action defines a projection Then, the disintegration of µ mc N,β with respect to π writes as follows: for all test functions It is not difficult to see that µ N,β,x p¨q :" µ mc N,β p¨|xq is the uniform measure on the orbit G N,β x , since its support is invariant under a subgroup of rotations (of the total sphere). Let us denote by x¨y N,E,x the corresponding expectation. We obtain that, for all test functions f : S N,β ÝÑ , To conclude, let us fix the energy β´1 ą 0, take x in the microcanonical sphere S N,β , and look at the dynamics generated by L N restricted on the orbit G N,β x . Then, observe that the kernel of S N in the Hilbert space L 2 pµ N,β,x q has dimension 1. As a result, the range of S N in L 2 pµ N,β,x q has codimension 1, and is equal to the mean-zero functions.
In the sequel, we give some properties of the two spaces C 0 and Q 0 : for instance, the energy current is among the elements of Q 0 (Proposition 4.2). We also prove an integration by parts formula for the functions of C 0 (Proposition 4.3). Moreover, the following elements belong to Q 0 :

Properties of
Proof. The first statement is straightforward as a consequence of the definition. Besides, paq is directly obtain from the following identities: for x P , and k ě 1, Then, if f P Q, it is easy to see that (14) and (15) are sufficient to prove pbq. For instance, l Conversely, let us now consider a function ϕ P C 0 . From the previous subsection together with Proposition 4.2, we can write the cylinder function ϕ as ϕ " p´S Λ ϕ qp´S Λ ϕ q´1ϕ for some mean-zero function p´S Λ ϕ q´1ϕ, measurable with respect to the variables m x , ω x ; x P Λ ϕ ( . The reversibility of the measure µ ℓ,β,x implies that the following decomposition holds in L 2 pµ ℓ,β,x q: The following proposition is a direct consequence of these comments.
for all rectangles Λ ℓ that contain Λ ϕ , for all β ą 0, and for all functions g P L 2 pµ ℓ,β,x q. For all y P , where Cpϕ, β, xq is equal to By reintegration of the desintegrated measure µ ℓ,β,x the same result (16) may be restated with microcanonical measures µ ℓ,β,x replaced by µ β and ‹ β and (17) becomes: for some constants C 1 , C 2 which can be written in terms of variances and do not depend on β.
Proof. Inequalities (17) and (18) follow from Cauchy-Schwarz inequality applied to (16). Let us notice that the last inequality (19) uses the translation invariance of the measure ‹ β . l ) .

This result can be restated with
Proof. This follows from the decomposition of every function in L 2 pµ β q over the Hermite polynomials basis: see Proposition A.3 in Section A. l

Limiting variance and semi-norm
In this subsection, we obtain a variational formula for the variance where ϕ P Q 0 and ℓ ϕ " ℓ´s ϕ´1 . We first introduce a semi-norm on Q 0 . For any cylinder function ϕ in Q 0 , let us definẽ As we previously noticed, this formula can be restated as Since ϕ belongs to Q 0 , the results of Subsection 4.2 are valid, namely: the first term in the right-hand side of (22) Here, ℓ ϕ stands for ℓ´s ϕ´1 so that the support of τ x ϕ is included in Λ ℓ for every x P Λ ℓ ϕ .
The proof is done in two steps that we separate as two different lemmas for the sake of clarity. In the first lemma, we bound the variance of a cylinder function ϕ P Q 0 , with respect to the canonical measure µ β , by the semi-norm~ϕ~2 β . In the second step, a lower bound for the variance can be easily deduced from the variational formula which expresses the variance as a supremum.
Proof. We follow the proof given in [14], Lemma 4.3 and we assume first that ϕ " ∇ 0 pFq`∇ 0,1 pGq, for two quadratic cylinder functions F, G. Then, the general case follows by linearity. We write the variational formula Since ϕ is quadratic, we can restrict the supremum in the class of quadratic functions h that are localized in Λ ℓ (the proof of that statement is detailed in Proposition A.3). It turns out that we can also restrict the supremum to functions h such that " D ℓ pµ β ; hq ı ď Cℓ. This follows from the fact that the first term can be bounded as follows (according Proposition 4.3 in addition to the convexity of the Dirichlet form): Recall that C ϕ is a constant that depends on ϕ. Next, we want to replace the sums over Λ ℓ ϕ with the same sums over Λ ℓ (recall that ℓ ϕ " ℓ´s ϕ´1 ď ℓ). For that purpose, we denote First of all, from Cauchy-Schwarz inequality, we have Then, we also can write as beforěˇˇˇˇˇˇ These last two inequalities give the upper bound Let us choose a sequence th ℓ u satisfying " D ℓ pµ β ; h ℓ q ı ď Cℓ. Then, the sequence tζ ℓ 0 ph ℓ q, ζ ℓ 1 ph ℓ qu is uniformly bounded in L 2 p ‹ β q, and this implies the existence of a weakly convergent subsequence. We denote by pζ 0 , ζ 1 q a weak limit and assume that the sequence tζ ℓ 0 ph ℓ q, ζ ℓ 1 ph ℓ qu weakly converges to pζ 0 , ζ 1 q. The conclusion is now based on the weak version of closed forms result that we prove in Appendix B, Theorem B.1: the pair pζ 0 , ζ 1 q can be written in L 2 p ‹ β q as with g P Q and a P . We have obtained that The inequality above is a consequence of the following fact: the L 2 -norm may only decrease along weakly convergent subsequences. The result follows, after recalling (21). l

LEMMA 4.7. Under the assumptions of Theorem 4.5,
Proof. We define, for f P Q, Spτ y f q.
The following limits hold: We only prove (24), the other relations can be obtained in a similar way. As previously, we assume for the sake of simplicity that ϕ " ∇ 0 pFq`∇ 0,1 pGq. We recall the elementary identity Therefore, The last limit comes from Proposition 2.1 and the fact that ℓ ϕ " ℓ´s ϕ´1 . Then, we obtain from the variational formula written with h " p´S Λ ℓ q´1paJ ℓ`H f ℓ q: The result follows after taking the supremum on f P Q, and recalling (21). l

Proof of Theorem 3.3
In this paragraph, we prove Theorem 3.3 by using the central limit theorem variances given in Theorem 4.5. First, we show how to relate (10) to such variances.
The previous result is proved for example in [11] (Section 2, Lemma 2.4). We are going to use this bound for functions of type ř x Gpx{Nqτ x ϕ, where ϕ belongs to Q 0 . The main result of this subsection is the following. THEOREM 4.9. Let ϕ P Q 0 , and G a smooth function on . Then, Proof. From Proposition 4.8, the left-hand side of (29) is bounded by that can be written with the variational formula as -.
Since ϕ P Q 0 , from Proposition A.3 we can restrict the supremum over f P Q. Proposition 4.3 gives β and by Cauchy-Schwarz inequality, The supremum on f can be explicitely computed, and gives the final bound We are now going to show that the constant on the right-hand side is proportional to~ϕ~2 β . For that purpose, we average on microscopic boxes: for k ! N, we denote and we want to substitute The error term that appears is estimated by From (30), the expression above is bounded by Ck{N, and then vanishes as N ÝÑ 8. We are reduced to estimate C sup By the same argument, this is bounded by The supremum on f can be explicitely computed, and gives the final bound Taking the limit as N ÝÑ 8 and then k ÝÑ 8, we obtain (29) from the central limit theorem for variances at equilibrium (Theorem 4.5). l We apply Theorem 4.9 to I 1,N t,m, f pHq, and we get lim sup NÝÑ8 C sup

Hilbert space and projections
We now focus on the semi-norm~¨~β that was introduced in the previous section by (20). We can easily define from~¨~β a semi-inner product on C 0 through polarization. Denote by N the kernel of the semi-norm~¨~β on C 0 . Then, the completion of Q 0 | N denoted by H β is a Hilbert space. Let us explain how the well-known Varadhan's approach is modified. Usually, the Hilbert space on which orthogonal projections are performed is the completion of C 0 | N , in other words it involves all local functions. Then, the standard procedure aims at proving that each element of that Hilbert space can be approximated by a sequence of functions in the range of the generator plus an additional term which is proportional to the current. A crucial step for obtaining this decomposition consists in: first, controlling the antisymmetric part of the generator by the symmetric one for every cylinder function, and second, proving a strong result on germs of closed forms (see Appendix B). These two key points are not satisfied in our model, but they can be proved when restricted to quadratic functions. It turns out that these weak versions are sufficient, since we are looking for a fluctuation-dissipation approximation that involves quadratic functions only.
In Subsection 5.1, we show that H β is the completion of SQ| N`t j S 0,1 u. In other words, all elements of H β can be approximated by a j S 0,1`S g for some a P and g P Q. This is not irrelevant since the symmetric part of the generator preserves the degree of polynomial functions. Moreover, the sum of the two subspaces t j S 0,1 u and SQ| N is orthogonal, and we denote it by Nevertheless, this decomposition is not satisfactory, because we want the fluctuating term to be on the form L m p f k q, and not Sp f k q. In order to make this replacement, we need to prove the weak sector condition, that gives a control of~A m g~β by~S g~β , when g is a quadratic function. The argument is explained is Subsection 5.2 and 5.3, and the weak sector condition is proved in Appendix C. The only trouble is that this new decomposition is not orthogonal any more, so that we can not express the diffusion coefficient as a variational formula, like (36). This problem is solved in Section 6.

Decomposition according to the symmetric part
We begin this subsection with a table of calculus, very useful in the sequel.
Proof. The first two identities are direct consequences of Theorem 4.5 and of Equality (27). The last two ones follow directly. l COROLLARY 5.2. For all a P and g P Q, In particular, the variational formula for~h~β , h P Q 0 , writes Proof. We divide the proof into two steps.
(a) The space is well generated -The inclusion SQ| N`t j S 0,1 u Ă H β is obvious. Moreover, from the variational formula (31) we know that: if h P H β satisfies ! h, j S 0,1 " β " 0 and ! h, S g " β " 0 for all g P Q, then~h~β " 0.
(b) The sum is orthogonal -This follows directly from the previous proposition and from the fact that: ! j S 0,1 , Sh " β " 0 for all h P Q. l

Replacement of S with L
In this subsection, we prove identities which mix the antisymmetric and the symmetric part of the generator, which will be used to get the weak sector condition (Proposition 5.7).
Proof. This easily follows from the first identity of Proposition 5.1 and from the invariance by translations of the measure ‹ β : l LEMMA 5.5. For all g P Q, ! S g, j A 0,1 " β "´! A m g, j S 0,1 " β .
Proof. By the first identity of Proposition 5.1, l Then, these two lemmas together with the second identity of Proposition 5.1 imply the following: We are now in position to state the main result of this subsection. Proof. The proof is technical because made of explicit computations for quadratic functions. For that reason, we report it to Appendix C. l

Decomposition of the Hilbert space
We now deduce from the previous two subsections the expected decomposition of H β .
PROPOSITION 5.8. We denote by L m Q the space tL m g ; g P Qu. Then, Proof. We first prove that H β can be written as the sum of the two subspaces. Then, we show that the sum is direct.
(a) The space is well generated -The inclusion L m Q| N`t j S 0,1 u Ă H β follows from Proposition 4.2. To prove the converse inclusion, let h P H β so that ! h, j S 0,1 " β " 0 and ! h, L m g " β " 0 for all g P Q. From Corollary 5.3, h can be written as h " lim kÝÑ8 S g k for some sequence tg k u P Q. More precisely, since ! S g k , A m g k " β " 0 by Lemma 5.4, Moreover, we also have by assumption that ! h, S g k " β " 0 for all k, and from Proposition 5.7, sup kP ~L m g k~β ď pCpβq`1q sup kP ~S g k~β ": C h pβq is finite. Therefore, The sum is direct -Let tg k u P Q be a sequence such that, for some a P , By a similar argument, where the last equality comes from the fact that ! j S 0,1 , S g k " β " 0 for all k. On the other hand, by Proposition 5.7,~L m g k~β ď pCpβq`1q~S g k~2 β . Then, a " 0. This concludes the proof. l Recall that j S 0,1 pm, ωq " λpω 2 1´ω 2 0 q. We have obtained the following result.
This concludes the first statement of Theorem 3.3. We prove the second statement (11) in Proposition 6.5 in Section 6.

On the diffusion coefficient
The main goal of this section is to express the diffusion coefficient in several variational formulas. We also prove the second statement of Theorem 3.3. First, recall Definition 2.2, which can be written as From Theorem 5.9, there exists a unique number D such that We are going to obtain a more explicit formula for that D, and relate it to (36), by following the argument in [14]. In Subsection 6.1, we first rewrite the decomposition of the Hilbert space given in Proposition 5.8, by replacing j S 0,1 with j 0,1 . This new statement is based on Corollary 5.6, which gives an orthogonality relation. The second step is to find an other orthogonal decomposition (see (37) below), which will enable us to prove the variational formula (36) for D. In Subsection 6.2, we study the convergence of the Green-Kubo formula given in (6), and then, in the last subsection, we investigate its behavior when the intensity of the exchange noise vanishes.

LEMMA 6.1. The following decompositions hold
Proof. We only sketch the proof of the first decomposition, since it is done in [14]. Let us recall from Proposition 5.8 that L m Q has a complementary subspace in H β which is one-dimensional. Therefore, it is sufficient to prove that H β is generated by L m Q and the total current. Let h P H β such that ! h, j 0,1 " β " 0 and ! h, L m g " β " 0 for all g P Q. By Corollary 5.3, h can be written as h " lim kÝÑ8 S g k`a j S 0,1 for some sequence tg k u P Q, and a P , and from Corollary 5.6, h~2 β " lim kÝÑ8 ! a j S 0,1`S g k , a j 0,1`L m g k " β .
Moreover, from Proposition 5.7, The same arguments apply to the second decomposition. l We define bounded linear operators T, T ‹ : H β ÝÑ H β as From the following identitỹ we can easily see that T ‹ is the adjoint operator of T and we also have the relations In particular, and there exists a unique number Q such that We are going to show that D " λQ.
Proof. The first identity follows from the fact that The second identity is obtained from the following statement l After an easy computation, we can also prove that ! Tg, g " β "! Tg, Tg " β for all g P H β . Since j S 0,1´T j S 0,1 is orthogonal to T j S 0,1 , we have: By the fact we obtain the variational formula for~T j S 0,1~β : Proof. With a similar argument (in the proof of the previous proposition), we have which concludes the proof. l THEOREM 6.4.
Proof. By the definition, j 0,1´D j S 0,1 {λ P L m Q and therefore So, D " λQ, and the variational formula for D can be deduced from the one for Q. l REMARK 6.1. We can rewrite the variational formula for D as: We use the fact that in (43), we can restrict the infimum on functions f satisfying ! j A 0,1´A m f , j S 0,1 " β " 0. Let us notice that (44) and (45) recover the variational formula (36).  Then, the result follows from D " λQ " χpβq (46) l

Convergence of Green-Kubo formula
Linear response theory predicts that the diffusion coefficient is given by the homogenized Green-Kubo formula, defined asκ where !¨" β,‹ is the inner product defined by (4). The Laplace transform is defined and is smooth on p0,`8q, and can be rewritten: The forthcoming theorem is proved in Appendix D, by considering the resolvent equation exists, and is finite.
Let us recall the link between the variational formula in Definition 2.2 and the Green-Kubo formula (see the end of Subsection 2.4). Since (47) converges as z goes to 0, it follows that D "D.

Vanishing exchange noise
With the same ideas of the previous subsection, it can be easily shown that the homogenized Green-Kubo formula also converges if the strength λ of the exchange noise vanishes. The aim of this paragraph is to study the limit of (48) as λ goes to 0. First, we turn (47) into a new definition that highlights the dependence on λ ą 0. For that purpose we introduce new notations: we denote S 0 " γS flip , S λ " S 0`λ S exch , and then Proof. The proof is divided into two steps. For the sake of readability, we erase the notation m in J 0 pmq, and keep in mind its dependence on the disorder. We also write L 2 ‹ for L 2 p!¨" β,‹ q.
Step 1 -Convergence of the diffusion coefficient. Let us denote by h z,0 and h z,λ the two solutions of the following resolvent equations in L 2 ‹ : pz´L m 0 qh z,0 " J 0 , pz´L m,‹ λ qh z,λ " J 0 . We look at the following difference, for λ, z ą 0 fixed,ˇˇ!
To complete the proof, we are reduced to show that λˇˇ! h z,λ , S exch ph z,0 q " β,‹ˇv anishes when we first let z ÝÑ 0 and then λ ÝÑ 0. For that purpose, we need more precise information on the two solutions h z,λ and h z,0 . Since the generator L m λ (resp. L m 0 ) conserves the degree of homogeneous polynomial functions, we know that the solution of the resolvent equation h z,λ (resp. h z,0 ) has to be homogeneous polynomial of degree two, precisely: where φ z,λ pm,¨,¨q : 2 ÝÑ is a square integrable symmetric function. Every degree two function h can be written as h " h "`h‰ , where h " belongs to the subspace Q " generated by tω 2 x , x P u and h ‰ belongs to the subspace Q ‰ generated by tω x ω y , x ‰ yu. These two subspaces of L 2 ‹ have the following properties: (i) Q " and Q ‰ are orthogonal in L 2 ‹ . (ii) Q " and Q ‰ are stable by S exch .
(iii) If h P Q " , then for all g P L 2 ‹ , ! S exch phq, g " β,‹ " 0. As a result, the two solutions h z,λ and h z,0 write and from the previous remarks we get according to the Cauchy-Schwarz inequality for the scalar product !¨, p´S exch q¨" β,‹ . We treat separately the two terms into the two lemmas below. We prove that the first term is bounded by C{ ? λ, and the second one is uniformly bounded for λ, z ą 0. Here we state the two lemmas: LEMMA 6.8. There exists a constant C ą 0 such that, for all z, λ ą 0, LEMMA 6.9. There exists a constant C ą 0 such that, for all z ą 0, From these statements we deduce λˇˇ! h z,λ , S exch ph z,0 q " β,‹ˇď C 0 ? λ where C 0 does not depend on λ, z ą 0, and Theorem 6.7 follows.
Step 2 -Proofs of the two lemmas. We begin with the proof of Lemma 6.8. We recall the resolvent We multiply (52) by h z,λ and integrate with respect to !¨" β,‹ , in order to get The right-hand side rewrites as p2γq´1 ! p´S 0 qpJ 0 q, h z,0 " β,‹ .
We now turn to Lemma 6.9. We prove a general result, precisely: there exists a constant C ą 0 such that, for all g P Q ‰ , ! g, p´S exch qg " β,‹ ď C ! g, g " β,‹ .
This fact is proved through explicit computations. Let us write g P Q ‰ in the form A straightforward computation gives that In the last inequality, we use the fact that the measure on the disorder is translation invariant and that pa´bq 2 ď 2pa 2`b2 q for all a, b P . Besides, one can also check that thanks to the translation invariance of . The bound (53) follows directly, with C " 4. To prove Lemma 6.9, it remains to show that ! h ‰ z,0 , h ‰ z,0 " β,‹ is uniformly bounded in z. We recall the resolvent equation in L 2 ‹ : zh z,0´p S 0`A m qh z,0 " J 0 .
Notice that, with the decomposition (51), we can write S 0 ph z,0 q "´2γh ‰ z,0 . We multiply (54) by h z,0 and integrate with respect to !¨" β,‹ , in order to get As previously, Cauchy-Schwarz inequality for the scalar product !¨, p´S 0 q¨" β,‹ on the right-hand side gives

The anharmonic chain perturbed by a diffusive noise
In this last main section we say a few words about the anharmonic chain, meaning that the interaction between atoms are non linear, and given by a potential V. As in [14], we assume that the function V : ÝÑ `s atisfies the following properties: (i) Vp¨q is a smooth symmetric function, (ii) there exist δ´and δ`such that 0 ă δ´ď V 2 p¨q ď δ`ă`8, Using the same notations as in the introduction, the configuration tp x , r x u now evolve according to We define π x :" p x { ? M x , and the dynamics on tπ x , r x u rewrites: The total energy is conserved. The flip and exchange noises have poor ergodicity properties, and can be used for harmonic chains only. For the anharmonic case, we introduce a stronger stochastic perturbation. Now, the total generator of the dynamics writes L m " A m`γ S, where where Y x, y " π x B r y´V 1 pr y qB π x , and X x " Y x,x . For this anharmonic case, the two needed ingredients can be proved directly from [14]. First, notice that the symmetric part of the generator does not depend on the disorder and is exactly the same as in [14]: the proof of the spectral gap is done in Section 12 of this paper. The sector condition can also be proved by inspiring from [14]. After taking into account the random environment and its fluctuation, the same argument of Lemma 8.2, Section 8 can be applied. Indeed, it is mainly based on the fact that the antisymmetric part of the generator can be written in terms of the symmetric one.

A.1 Hermite polynomials on
Let χ be the set of positive integer-valued functions ξ : ÝÑ , such that ξ x vanish for all but a finite number of x P . The length of ξ, denoted by |ξ|, is defined as For ξ P χ, we define the polynomial function on Ω where th n u nP are the normalized Hermite polynomials w.r.t. the centered one-dimensional Gaussian law with variance β´1. The sequence tH ξ u ξPχ forms an orthonormal basis of the Hilbert space L 2 pµ β q, where µ β is the infinite product Gibbs measure defined by (2). As a result, every function f P L 2 pµ β q can be decomposed in the form f pωq " ÿ ξPχ FpξqH ξ pωq.
Moreover, we can compute the scalar product x f , gy β for f " ř ξ FpξqH ξ and g " DEFINITION A.1. We denote by χ n Ă χ the subset sequences of length n, i.e. χ n :" ξ P χ ; |ξ| " n ( . A function f P L 2 pµ β q is of degree n if its decomposition f " ÿ ξPχ FpξqH ξ satisfies: Fpξq " 0 for all ξ R χ n .
It is not hard to check the following proposition: If a local function f P L 2 pµ β q is written in the form f " ř ξPχ FpξqH ξ , then where S is the operator acting on functions F : χ ÝÑ as Here, ξ x, y is obtained from ξ by exchanging ξ x and ξ y .
From this result we deduce: COROLLARY A.2. For any f " ř ξPχ FpξqH ξ P L 2 pµ β q, we have

A.2 Dirichlet forms and weakly sequences of quadratic functions
In this subsection we focus on the set of quadratic functions in L 2 pµ β q, namely degree two functions, that we denote by Q. We first restrict a variational formula to this class of functions, and then we study sequences of functions that weakly converge in L 2 . PROPOSITION A.3. If f P L 2 pµ β q is quadratic in the sense above, then the following variational formula can be restricted over quadratic functions g.
Proof. This fact follows after decomposing g as ř ξPχ GpξqH ξ . Then, it is an easy consequence of Corollary A.2 and of the orthogonality of Hermite polynomials. l PROPOSITION A. 4. Let t f n u n be a sequence of quadratic functions in L 2 pµ β q. Suppose that t f n u weakly converges to f P L 2 pµ β q. Then, f is quadratic.
Proof. For all n P , and ξ R χ 2 , the scalar product @ f n , H ξ D β vanishes (by definition). From weak convergence, we know that as n goes to infinity, for all ξ P χ. This implies:

B A weak version of closed forms results
In that section we prove a theorem that should be thought as a kind of closed forms results, as they are stated in [17] or in [9] (Section A.3.4). We give the link between Theorem B.1 below and closed forms at the end of this paragraph.

B.1 Decomposition of quadratic functions
For the sake of clarity, we erase the dependence on the disorder m, and consider that the functions are defined on Ω, and square integrable w.r.t. the Gibbs measure µ β . We explain how to restate the same result for functions defined on Ω DˆΩ in Remark B.1. THEOREM B.1. Let t f n u nP a sequence of quadratic functions in L 2 pµ β q. Let us define g n :" ∇ 0´Γ f n¯a nd h n :" ∇ 0,1´Γ f n¯.
If tg n u, respectively th n u, weakly converges in L 2 pµ β q towards g, respectively h, then there exist a P and f P Q such that hpωq " apω 2 0´ω 2 1 q`∇ 0,1 pΓ f qpωq.
where ψ 1 , ψ 2 : 2 ÝÑ are square integrable symmetric functions. We are now going to give a list or equalities, being satisfied by the pair of sequences. Let us be more precise. We define, for a pair pf 1 , f 2 q of two L 2 pµ β q functions, the following identities, stated in L 2 pµ β q sense: It is straightforward to check that, for all n P , the pair pg n , h n q satisfies identities (R1-R3). Easily, one can show that the latter always take place after passing to the weak limit in L 2 pµ β q. Precisely, the weak limit pg, hq of tg n , h n u also satisfy (R1-R3). This follows from the following easy lemma (which is a consequence of the translation invariance of µ β ): LEMMA B.2. If tg n u n weakly converges in L 2 pµ β q towards g, then, for all x P , g n pω x q ( n weakly converges towards gpω x q, g n pω x,x`1 q ( n weakly converges towards gpω x,x`1 q. Notice that all equalities (R1-R3) turn into identities for ψ 1 and ψ 2 , defined in (60) and (61). Namely, ψ 1 and ψ 2 have to satisfy

(R2)
$ & % ψ 2 px, yq " 0 if x R t0, 1u and y R t0, 1u, The first two identities imply that g writes on the form and h rewrites as whereas the final equality makes a connection between g and h. In view of (58) and (59), we are going to need the following straightforward lemma: • the price to flip ω x when the configuration is ω should be equal to´f 1 x pω x q : this is (R1), • the price to exchange ω x and ω x`1 when the configuration is ω should also be equal to´f 2 x pω x,x`1 q : this is (R2).
In the context of interacting particle systems, closed forms are expected to give the same price for any 2-step path with equal end points. In our setting, the last equality (R3) can be translated into: "The quantity at site x is flipped, and then exchanged with the quantity at site x`1. Equally, the quantities at site x and x`1 are exchanged first, and then the quantity at site x`1 is flipped." There are three other such paths, that we do not need in our result: • two quantities are exchanged at sites x, x`1, and also independently at sites y, y`1, with tx, x`1u X t y, y`1u " H, • two quantities are flipped independently at sites x and y, with x ‰ y, • the quantity at site x is flipped, and then the quantities at sites y and y`1 are exchanged, for y R tx, x`1u, and the converse is also possible.
Recall that we have defined Ω :" . We denote by B the space of real-valued functions B :" t f : Ω ÝÑ u.
We are now interested in the space of forms, which are defined as pf 1 x , f 2 x q xP where f 1 x P B, and f 2 x P B, for every x P . To each function F : Ω ÝÑ is associated a form: x , f 2 x q xP is an exact form if there exists a continuous function F : Ω ÝÑ such that @ x P , @ ω P Ω, # f 1 x pωq " Fpω x q´Fpωq, f 2 x pωq " Fpω x,x`1 q´Fpωq.
Easily, one can prove that all exact forms are closed forms. We now present two examples of closed forms that play a central role.
EXAMPLE B.1. We denote by a " pa 1 , a 1 q the closed form defined by # a 1 x pωq " 0, a 2 x pωq " ω 2 x´ω 2 x`1 , for all x P and configurations ω P Ω. This closed form corresponds to the formal function Fpωq " ř x xω 2 x , but this is not an exact form.
EXAMPLE B.2. Let h be a cylinder function. Let us recall that we denote by Γ h the formal sum ř x τ x h, and define u h " pu 1 h , u 2 h q as # pu 1 h q x pωq " Γ h pω x q´Γ h pωq, pu 2 h q x pωq " Γ h pω x,x`1 q´Γ h pωq, for all x P , and configurations ω P Ω. Though ř x τ x h is a formal sum, these two equalities are well defined. Let us notice that u h is a closed form that is not exact, unless h is constant.
These two examples show that closed forms on Ω are not always exact forms. Let us introduce the notion of a germ of a closed form. Examples B.1 and B.2 provide two types of germs of closed forms. Consider the cylinder function Apωq " p0, ω 2 0´ω 2 1 q. The collection pτ x Aq xP is the closed form a of Example B.1. For a cylinder function h, the collection p∇ x Γ h , ∇ x,x`1 Γ h q xP obtained through translations of the cylinder function p∇ 0 Γ h , ∇ 0,1 Γ h q is the closed from of Example B.2. For a pair of L 2 p ‹ β q-functions f " p f 1 , f 2 q, we called it a germ of closed form if f " pτ x f q xP satisfies all of conditions as a closed form in L 2 p ‹ β q-sense. Usually, Theorem B.1 is replaced with a similar result that concerns every germ of closed form in L 2 p ‹ β q: see Theorem 5.1 in [17] or Theorem A.3.4.14 in [9].

C Proof of the weak sector condition
In this section we prove Proposition 5.7 that we recall here for the sake of clarity.
(ii) There exists a positive constant Cpβq such that, for all g P Q, A m g~β ď Cpβq~S g~β .
As a result, the variational formula (31) for~A m g~2 β gives: The result is proved. l

D Convergence of Green-Kubo formulas
In this section we prove Theorem 6.6, that we recall here for the sake of clarity: THEOREM D.1. The following limit lim zÝÑ0 zą0 ! j A 0,1 , pz´L m q´1 j A 0,1 " β,‹ exists, is positive and finite.
Proof. We recall that the completion of the space of square integrable local functions w.r.t. !¨" β,‹ is denoted by L 2 ‹ , and we also have defined the quantity Lpzq :" β 2 2 ż`8 0 e´z t ! j A 0,1 ptq, j A 0,1 p0q " β,‹ dt, which is well-defined on p0,`8q. We recall that h z :" h z pm, ω; βq is the solution of the resolvent THEOREM E.1. For almost all realization of the disorder m P Ω D , the sequence tY N m u Ně1 is tight in Dpr0, Ts, M 1 q.
Let us remind the decomposition of Y N t,m given in (9):