Alessia Colombo

Rachele Ossola

Mirko Magni

Dominique Roberto

Denis Jacquemin

Carlo Castellano

Francesco Demartin
email: francesco.demartin@unimi.it

Claudia Dragonetti
email: claudia.dragonetti@unimi.it

Intriguing C-H center dot center dot center dot Cu interactions in bis-(

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

We have synthesized and characterized a series of bis-(phenanthroline)Cu(I) complexes of interest as redox mediators for dye-sensitized solar cells. This study led to the discovery of intriguing anagostic interactions between the hydrogen atom and the copper center as evidenced by X-ray diffraction studies on single crystal. Remarkably, an anagostic interaction was found between a H atom of a methyl group and a copper site.

An important challenge of humanity is to replace fossil fuels with renewable energy sources to answer the growing worldwide energy demand, minimizing negative environmental and climate effects. Indeed, sunlight remains the cleanest, the most abundant and the cheapest energy source. 1 Dye-sensitized solar cells (DSSCs) 2-5 are a realistic solution for harnessing the sun energy and converting it into electrical energy. 6 The traditional redox mediator in DSSCs is the I -/I3 -couple as it presents ideal kinetic properties. However it has some disadvantages, e.g., I2 in equilibrium with I3 -is volatile, complicating long-term cell sealing, I3 -is darkly colored and limits the light harvesting efficiency of the dye, the couple presents large photovoltage loss due to the non-optimal matching with the dye redox potential, and I3 -/I -is corrosive and corrodes most metals. 7 Therefore, during the recent years, there has been numerous efforts to search for new electron transfer mediators, in particular based on transition metal complexes. 7 In this context, copper complexes are particularly appealing, as they stand as promising efficient and cheap electron mediators for solar cells. The use of copper complexes with a distorted tetrahedral geometry in which the structural change between the Cu(I) and Cu(II) complexes is minimized, provides a promising strategy to electron mediators for DSSCs, allowing a very fast electron transfer and a redox potential of the Cu(I)/Cu(II) couple high enough to increase open-circuit voltage (Voc) values. [9][10][11] In this framework, we have synthesized a series of different copper complexes (1-5) [10][11][12] (see Fig. 1, all with PF6 -counterion), with phenanthrolines that bear, in α position, a substituent with different steric hindrance, in order to clarify its contribution in achieving simultaneously: (i) an appropriate redox potential; (ii) a distorted tetrahedral geometry; and (iii) an effective shield of the metal core.

We have synthesized the proper phenanthroline ligand and the corresponding Cu(I) complex, with simple and fast reactions (Scheme 1). Ligands were prepared thanks to a reaction developed by Sauvage 13 as an effective and quick way to obtain 2-substituted and 2,9-disubstituted-1,10-phenanthrolines starting from commercially available 1,10-phenanthroline and organolithium reagents. 14 The reaction is a two-step addition-oxidation. The 2,9-disubstituted derivative is obtained using an excess of organolithium reagent in the first stage. In the second step re-aromatization is achieved treating the intermediate with activated manganese dioxide. In more details, copper complexes were prepared by reaction of the appropriate ligand with the stoichiometric amount of [Cu(CH3CN)4][PF6] (2:1) in the minimal amount of dichloromethane in order to favor the formation of the product that in some cases directly precipitates from the reaction environment in a pure form, avoiding the subsequent crystallization step. The starting white salt [Cu(CH3CN)4][PF6] was synthesized from Cu2O according to literature 15 (Scheme 1). The knowledge of metal complexes potentials is crucial to ascertain their applicability as redox mediators in DSSCs. Therefore the electrochemical properties of the copper coordination compounds were studied by cyclic voltammetry in acetonitrile, focusing on the electron transfer process centered on the metal core (see Fig. 2). In all complexes the anodic window showed one chemically reversible peak attributed to a Cu(I) to Cu(II) redox switching; a quite high electrochemical reversibility, i.e. a small potential difference between forward and backward peaks, indicative of a fast electron transfer occurs, is found in all the compounds indicating a limited rearrangement of the coordination sphere. The determination of the heterogeneous electron transfer rate constant, kheter, can be a valuable parameter to quantitatively assess the impact of the steric bulkiness of α-substituents on the self-exchange rate. 16 For the sake of comparison bulky tert-butyl chains in compound 3 induce a ca. 20 times faster heterogeneous electron transfer toward glassy carbon electrode surface with respect to the unsubstituted bis(1,10-phenanthroline)copper complex; kheter, being 9•10 -3 and 5•10 -4 cm s -1 respectively, with and without the bulky substituents, respectively. These values were estimated from charge transfer resistance, Rct, by electrochemical impedance spectroscopy according to the following relationship: 17

k heter = RT F 2 R ct C
where R is the gas constant, T the absolute temperature, F the Faraday constant and C the concentration of the copper complex (in mol cm -3 ). The Cu(I)/Cu(II) half-wave potentials, E1/2, are reported in Table 1. Electronic and steric effects of the substituents finely control the oxidation potentials. In agreement with a previous work, 10 complex 5 with 2,9disubtituted phenanthrolines exhibits E1/2 positively shifted with respect to the 2-substituted analogue (complex 2) due to an increased steric hindrance around the copper atom that progressively destabilizes the electrogenerated Cu(II) product that is forced to maintain a geometry similar to the preferred Cu(I) one. Remarkably, all 2-substituted complexes (3 being borderline) have redox potentials suitable for energetic application as electron mediators in DSSCs, being thermodynamically able to regenerate many common photosensitizers with oxidation potentials higher than 0.30-0.35 V vs Fc + |Fc. Electronic absorption spectra of the investigated copper complexes were recorded in acetonitrile (see Table 2 and ESI). They follow the pattern reported in literature, with a MLCT band between 426 and 470 nm, [START_REF] Kohler | [END_REF] associated with charge transfer from Cu(dπ) to phenanthroline (π*) orbitals. The extinction coefficients of the MLCT band decrease with increasing steric bulk at position 2, with compound 3 possessing the weakest visible absorption (2200 M -1 cm -1 ). To verify the influence of various substituents in α position of the phenanthrolines on the distorted tetrahedral geometry of the corresponding complexes, we have prepared crystals of complexes 2-4 suitable for X-ray characterization using a simple double layer procedure (slow diffusion in CH3CN/hexane). ‡ The structures are displayed in Figs. 3 to 6, for compounds 1 to 4. The structure of 5 has also been determined but is not detailed here because it is already known. 12a Unexpectedly, the synthesized complexes show a "peculiar" interaction between a hydrogen of a C-H bond (of the CH3 or of a CH of phenanthroline or of a tolyl ring ring) and the copper site, as shown in Table 3.

Complex

 (nm) 10 3  (M -1 cm -1 ) These interactions awoke our curiosity because in the literature there is some controversy regarding the agostic and anagostic interaction in copper complexes. 20 The strength, nonpolar nature, and low polarizability of C-H bonds make that they are generally considered as chemically inert. However, significant interactions between C-H bonds of -bound alkyl groups and metal atoms can occur. [START_REF] Kohler | [END_REF]21 The origin of the agostic interaction lies in the identification in the 1960s of hydridic H atoms in transition metal complexes by Trofimenko. 22 The term agostic was proposed in 1983 by Brookhart and Green in their seminal contribution 23 for a case in which a C-H group interacts with a transition metal with the formation of a two-electron three-center bond. In that case, the metal centre behaves like a Lewis acid. The term anagostic was coined in 1990 by Lippard and co-workers to distinguish sterically enforced M•••H-C contacts (M=Pd, Pt) in square-planar transition metal d 8 complexes from attractive agostic interactions. 24 In this case, the availability of an empty orbital is not required. Today the theme of agostic and anagostic interactions remains a hot topic, as evidenced, for example, in a recent paper on anagostic interactions under pressure. 25 It is evident that the assignment of an agostic or anagostic interaction requires caution. 26 In the case of an ephedrine copper complex, for example, an interaction initially described as agostic 27 was next disputed by other authors. 20 In the investigated complexes the Cu•••H lengths, superior to 2.3 Å, [START_REF] Brookhart | Proc. Nat. Acad. Sc[END_REF] preclude any agostic bonds; in fact as far as bond lengths are concerned agostic interactions are characterized by relatively short M-H distances (1.8-2.3 Å) whereas anagostic interactions are characterized by relatively longer M-H distances (2.3-2.9 Å).The shortest bonds between a H atom of a methyl group (complex 3) or of a CH in ortho position of tolyl group (complex 5) and copper can be consequently attributed to anagostic interactions. To our knowledge, this is the first time that an anagostic interaction is found between a H atom of a methyl group and a copper site, as happened for complex 3. DFT calculations were performed on all cations (see the ESI for technical details) to verify that the existence of "short" C-H•••Cu distances is not induced by crystal packing effects. We have found the following distances: for complex 1, 2.74 Å (between Cu and the H of the CH3) for complex 2, 2.60 Å (between Cu and H of CH3 of tolyl ring); for complex 3, 2.37 and 2.69 Å (between Cu and the H of the CH3 of tertbutyl); for complex 4, 2.90 and 3.20 Å (respectively between Cu and H of phenyl ring, and between Cu and H of phenantroline); and for complex 5, 2.36 and 2.41 Å (between Cu and the H of the tolyl ring). Concerning complex 4, different H atoms are involved in the shortest C-H•••Cu contact: in the X-ray structure the closest H atom belongs to the phen ligand whereas in DFT calculations the closest H atom belongs to the phenyl group: this difference is likely due to crystal packing effects that tune the dihedral angle between the phenyl ring and the phenantroline. As This journal is © The Royal Society of Chemistry 20xx

1
Please do not adjust margins Please do not adjust margins expected, the smallest distances obtained through DFT are in complexes 3 and 5 which is consistent with the solid-state results. This hints that the XRD finding should pertain in other environment than the crystals. The anagostic interactions evidenced in the present work protect the copper site and improve the stability of the structure by further minimizing the change of geometry between Cu (I) and Cu (II). This protection is of crucial importance for the design of efficient redox couples for DSSCs. Therefore our complexes, whose potential can be easily tuned by adequate substituents on the phenanthrolines, have a great potential as redox couple of an electron transfer mediator electrolyte in DSSC applications, along with a proper dye with adequate HOMO and LUMO levels, where the HOMO level lies under the potential of the redox couple. In this framework, we underline that the most popular redox couple, I -/I3 -, has several limitations (see Introduction), and, in particular it presents a too negative redox potential (-0.28 V vs Fc + |Fc in acetonitrile, converted from 0.35 V vs NHE) [START_REF] Rowley | [END_REF] that precludes its use in DSSCs relying on some of the most promising dyes that have a relatively high potential. Copper complexes, and specifically complex 3, are expected to be adequate redox mediators (along with the related Cu(II) complex) for DSSCs upon combination with organic dyes such as C218 30a , LEG4 30b or Y123. 30c It is known that copper complexes, which versatile coordination chemistry has significant influence over their optical and redox properties, 26 are of great interest as catalysts because the use of complexes deriving from earth-abundant elements is desirable from the standpoint of scalability and sustainability. Besides, agostic 31 or anagostic 32 interactions are often used to activate inert C-H bonds, facilitating new reactions. For example, recently the catalytic activity of bis(pyrazolyl)borate copper complexes toward carbenoide insertion into N-H bonds was reported. 31 In this system weak intramolecular C-H•••Cu interactions are of great importance and act as a switch which is turned "on" if interacting with the substrate and "off" if eliminating the product and regenerating the weak interaction. 31 Moreover the steric hindrance of groups in the positions 2,9 of 1,10-phenanthroline copper (I) complexes plays an important role in the regioselective addition of CBr4 to styrene, substituted phenyl rings leading to a better catalytic activity then methyl groups. 33 The complexes investigated in the present work are therefore of potential interest for such catalysis studies.

In summary we have fully characterized variously substituted 1,10phenanthroline copper(I) complexes. Their crystal structure, determined by single-crystal X-ray analysis, confirms a distorted tetrahedral geometry and reveals interesting anagostic interactions for complexes 3 and 5. Clearly the investigated copper complexes are excellent candidates as electron transfer mediators for DSSCs and are appealing for catalysis studies. Notes and references ‡ X-ray crystallographic data in CIF format for 2-4 have been deposited with the Cambridge Crystallographic Data Centre, CCDC 1564905-1564907.

8

 8

Fig. 1 .

 1 Fig.1. Chemical structures of the investigated Cu complexes.

Fig. 2

 2 Fig. 2 Cyclovoltammetric, CV, patterns of Cu-based complexes 1-5, in acetonitrile with 0.1 M TBAPF6 on glassy carbon electrode at 0.2 Vs -1 .

Fig. 3 Fig. 4 Fig. 5 Fig. 6

 3456 Fig.3 Structure of the cation in (1) (see ref. 11)

  C.D., A.C. and D.R deeply thank the bilateral project Italy-India "Cromofori a forma di Y coniugati al ferrocene come potenziali sensibilizzatori in celle DSSC in combinazione con innovativi mediatori redox" (Prot. nr. MAE0104617), "Con il contributo del Ministero degli Affari Esteri e della Cooperazione Internazionale, Direzione Generale per la Promozione del Sistema Paese". This work used computational resources from the CCIPL (Centre de Calcul Intensif des Pays de la Loire) installed in Nantes.

[Cu(phe) 2 ] + Scheme 1. Synthesis

  This journal is © The Royal Society of Chemistry 20xx of ligands and complexes.

	Please do not adjust margins
			1		AV22
	mol -1 V s ) 0.5 -0.5	1 unit	2 3		RO 14 MK 35 RO 20 MK 39
	3 dm				Serie1
	i/(cv 0.5 ) /(Acm -2		5 4		GC (0.071 cm 2 ); 0.2 Vs -1 0.1M sinossi mediatori 0.96 -1.1E-03M ACN TBAPF6
					positive feedback
					(85-105W)
	-1.0	-0.5	0.0	0.5	1.0
			E vs Fc + |Fc /V
	Please do not adjust margins

Table 1 .

 1 Oxidation half-wave potential, E1/2, in acetonitrile.

	Complex	1	2	3	4	5
	E1/2 vs Fc + |Fc /V	-0.02	0.09	0.34	0.10	0.50
	[Cu(1,10-phenanthroline)2][PF6] shows E1/2 = -0.38 V

Table 2 .

 2 Visible absorption spectra of 1-5 complexes in acetonitrile.

Table 3 .

 3 Shortest Cu•••H distances in Å in the synthesized complexes In the complex [Cu(1,10-phen)2][PF6] 19 the shortest Cu…H distance is 3.203 Å.

	Complex a	Shortest Cu…H distances (Å)	H belongs to:
	1 (ref 11)	3.18	CH3 of mesityl ring
	2	3.05	CH3 of tolyl ring
	3	2.49; 2.53	CH3 of tertbutyl
	4	3.11;(3.27)	CH of phenanthroline
			(CH3 of phenyl ring)
	5 (ref. 12a)	2.72; 2.86	CH of tolyl ring

a


L U Xu

Linjie Zhao

EQUILIBRIUM PERTURBATIONS FOR STOCHASTIC INTERACTING SYSTEMS

Keywords: Equilibrium perturbation, exclusion process, oscillator chain, hydrodynamic limit

We consider the equilibrium perturbations for two stochastic systems: the d-dimensional generalized exclusion process and the one-dimensional chain of anharmonic oscillators. We add a perturbation of order N -α to the equilibrium profile, and speed up the process by N 1+κ for parameters 0 < κ ≤ α. Under some additional constraints on κ and α, we show the perturbed quantities evolve according to the Burgers equation in the exclusion process, and to two decoupled Burgers equations in the anharmonic chain, both in the smooth regime.

Introduction

One of the main aims in the theory of hydrodynamic limit is to derive partial differential equations from microscopic systems. For asymmetric systems with one or several conservation laws, the conserved quantity/quantities usually evolve macroscopically according to the hyperbolic system under the Euler scaling (time accelerated by N and space divided by N ), cf. [START_REF] Rezakhanlou | Hydrodynamic limit for attractive particle systems on Z d[END_REF][START_REF] Olla | Hydrodynamical limit for a Hamiltonian system with weak noise[END_REF][START_REF] Kipnis | Scaling limits of interacting particle systems[END_REF] for example. To refine the hydrodynamic limit, when the system has only one conservation law, in the seminal paper [START_REF] Esposito | Diffusive limit of asymmetric simple exclusion[END_REF], Esposito, Marra and Yau perturb the asymmetric simple exclusion processes around the equilibrium point with order N -1 at the initial time in dimensions d ≥ 3, and show that the evolution of the perturbed quantity, namely the equilibrium perturbation, is governed by the viscous Burgers equation under the diffusive scaling (time accelerated by N 2 and space divided by N ). This is closely related to the understanding of Navier-Stokes corrections and is called the incompressible limit in the literature, cf. [START_REF] Kipnis | Scaling limits of interacting particle systems[END_REF]Section 7.7] for details. Later, such kind of result is extended by Seppäläinen [START_REF] Seppäläinen | Perturbation of the equilibrium for a totally asymmetric stick process in one dimension[END_REF] to dimension d = 1. He considers the one-dimensional Hammersley's model, adds a perturbation of order N -α to the equilibrium, and shows that the perturbation macroscopically obeys the invisid Burgers equation in the time scale N 1+α t if 0 < α < 1/2. The proof uses combinatorial properties of the Hammersley's model and coupling techniques, and the result holds even beyond the appearance of shocks. Independently, Tóth and Valkó [START_REF] Tóth | Between equilibrium fluctuations and eulerian scaling: perturbation of equilibrium for a class of deposition models[END_REF] obtain the same result for a large class of one-dimensional interacting particle systems, which is called deposition models in their paper, but only for 0 < α < 1/5. Tóth and Valkó employ Yau's relative entropy method [START_REF] Yau | Relative entropy and hydrodynamics of Ginzburg-Landau models[END_REF], thus the result holds only in the smooth regime of the solutions. Very recently, Jara, Landim and Tsunoda [START_REF] Jara | Derivation of viscous burgers equations from weakly asymmetric exclusion processes[END_REF] consider equilibrium perturbations in weakly asymmetric exclusion processes and derive viscous Burgers equation under the diffusive scaling. Equilibrium perturbations have also been considered for systems with two conservation laws. In [START_REF] Tóth | Perturbation of singular equilibria of hyperbolic two-component systems: a universal hydrodynamic limit[END_REF], Tóth and Valkó prove for a very rich class of systems that the small perturbations are universally driven by a two-by-two system. In [START_REF] Valkó | Hydrodynamic limit for perturbation of a hyperbolic equilibrium point in two-component systems[END_REF], Valkó shows that small perturbations around a hyperbolic equilibrium point evolve according to two decoupled Burgers equations. For system with three or more conservation laws, the problem remains generally open.

1.1. Nonlinear geometric optics for conservation laws. To illustrate the idea, for n ≥ 1, we start from an n-system of conservation laws

∂ t f (t, v) + m • ∇ v J f (t, v) = 0, f (0, •) = f ini , (1.1)
where

v ∈ R d , f ini ∈ C 1 (R d ; R n ), m ∈ R d , J : R n → R n is a smooth, nice function and m • ∇ v J f (v) := m 1 ∂ ∂v 1 + ... + m d ∂ ∂v d J f (v) ∈ R n .
To avoid the discussion on boundary terms, we adopt the periodic condition f (t, v + e i ) = f (t, v) for each i = 1, ..., d. It is equivalent to say that the space variable v ∈ T d := (R/Z) d . The stationary solutions to (1.1) are given by constant functions. We are interested in the behavior of the solution when the initial condition is slightly perturbed from the stationary. More precisely, fix a vector w * ∈ R n and consider the perturbed system

∂ t f ε (t, v) + m • ∇ v J f ε (t, v) = 0, f ε (0, •) = w * + εw ini ,
where ε > 0 and w ini ∈ C 1 (T d ; R n ). Suppose that f ε decomposes as

f ε (t, v) = w * + εw(t, v) + w ε (t, v), |w ε (t, v)| = o(ε).
By expanding J at w * , we obtain the linearized equation of the first order component:

∂w ∂t + A d i=1 m i ∂w ∂v i = 0, w(0, •) = w ini , (1.2)
where A = DJ(w * ) is the Jacobian matrix. Assume that (1.1) is strictly hyperbolic at w * : A has n distinct, non-zero eigenvalues λ 1 , ..., λ n , with the corresponding left (right) eigenvectors denoted by l 1 , ..., l n (r 1 , ..., r n ):

l j (λ j Id -A) = (λ j Id -A)r j = 0, l j r k = 1 {j=k} .

From (1.2), (d/dt)l j w = 0 along each characteristic lines (t, v + λ j tm). Hence,

w(t, v) = n j=1
l j w ini φ j (t, v) r j , φ j := v -λ j tm.

To observe non-trivial evolution along these lines, we investigate at longer time ε -1 t. Assume that the solution decomposes further to the second order as

f ε (t, x) = w * + ε n j=1 σ j (εt, φ j )r j + ε 2 n j=1 σj (εt, φ)r j + o(ε 2 ),
where φ = (φ 1 , . . . , φ n ) ∈ R d×n , σ j = σ j (s, u) and σj = σj (s, u 1 , . . . , u n ) are C 1 functions for j = 1, ..., n. Inserting the expansion into (1.1) and noting that Ar j = λ j r j , we see that the terms of order ε cancel autonomously, while the higher order terms read

ε 2 n j=1 ∂ s σ j (εt, φ j )r j + ε 2 j,j (λ j -λ j )m • ∇ u j σj (εt, φ)r j +ε 2 j,j σ j (εt, φ j ) m • ∇ u σ j (εt, φ j )H(r j , r j ) = o(ε 2 ).
Since l k r j = 1 {k=j} , the above holds true if and only if for each k = 1, ..., n,

∂ s σ k (s, u k ) + n j=1 (λ k -λ j )m • ∇ uj σk (s, u 1 , . . . , u n ) + j,j l k H(r j , r j )σ j (s, u j ) m • ∇ u σ j (s, u j ) = 0. (1.3)
We restrict our discussion to the non-resonant situation, in which the wave of each frequency λ k performs self-consistent time evolution governed by the Burgers equation

∂ s σ k + l k H(r k , r k ) m • ∇ u 2 -1 σ 2 k = 0, ∀ k = 1, . . . , n. (1.4)
Comparing (1.3) and (1.4), σk then has to solve the equation

n j=1 (λ j -λ k )m • ∇ uj σk (s, u 1 , . . . , u n ) = (j,j ) =(k,k) l k H(r j , r j )σ j (s, u j ) m • ∇ u σ j (s, u j ).
By superposition, we only need to construct σk as

σk (s, u 1 , . . . , u n ) := j =k l k H(r j , r j )σ 2 j (s, u j ) 2(λ j -λ k ) + j =j σk,j,j (s, u j , u j ),
where the functions σk,j,j = σk,j,j (s, u, u ) satisfies that

n j=1 (λ j -λ k )m • ∇ u j =j σk,j,j (s, u j , u j ) = j =j l k H(r j , r j )σ j (s, u j ) m • ∇ u σ j (s, u j ).
(1.5)

Hence, we formally obtain a sufficient condition for the non-resonant case:

(λ j -λ k )m • ∇ u + (λ j -λ k )m • ∇ u σk,j,j (s, u, u ) = l k H(r j , r j )σ j (s, u) m • ∇ u σ j (s, u ), (1.6)
is solvable for all k, j, j = 1, ..., n such that j = j . Notice that the formal calculation above apparently relies on the smoothness of σ k , so we need to assume that t < T shock , where T shock is the time when the shock first appears in the entropy solution to (1.4).

Observe that the non-resonant condition holds autonomously if n = 1. If n = 2, d = 1, without loss of generality we can set m = 1 and solve (1.6)

explicitly as σ1,1,2 = c 1 σ 1 (s, u)σ 2 (s, u ), σ1,2,1 = c 1 Σ 2 (s, u)∂ u σ 1 (s, u ), σ2,1,2 = c 2 Σ 1 (s, u)∂ u σ 2 (s, u ), σ2,2,1 = c 2 σ 2 (s, u)σ 1 (s, u ), where c 1 = -(λ 1 -λ 2 ) -1 l 1 H(r 1 , r 2 ), c 2 = (λ 1 -λ 2 ) -1 l 2 H(r 1 , r 2 ) and Σ 1 , Σ 2 are primitives of σ 1 , σ 2 , respectively, i.e., Σ 1 = • 0 σ 1 du, Σ 2 = • 0 σ 2 du.
To make Σ j well-defined on T, we shall assume in addition that T σ j (0, u)du = 0 for both j = 1, 2, or equivalently

T w ini (u)du = T σ 1 (0, u)r 1 + σ 2 (0, u)r 2 du = 0. (1.7)
The non-resonant expansion is not in general valid for the system with n ≥ 3. We refer the readers to [START_REF] Majda | Compressible fluid flow and systems of conseration laws in several space variables[END_REF][START_REF] Diperna | The validity of nonlinear geometric optics for weak solutions of conservation laws[END_REF] for rigorous justification and detailed discussion for the resonant case.

1.2. Equilibrium perturbation. The goal of the present paper is to derive the non-resonant system (1.4) of Burgers equations as a decent scaling limit for some stochastic interacting system. Formally speaking, for a scaling parameter N ∈ N + , we study a particle system ζ(t) = {ζ x (t)}, where the microscopic position x belongs to the periodic lattice T d N := Z d /(N Z d ). Suppose that the system has n conserved quantities g = (g 1 , . . . , g n ) such that, under the Euler space-time rescaling (N t, [N v]) their empirical densities evolve with the macroscopic conservative system (hydrodynamic limit):

lim N →∞ 1 N d x∈T d N g ζ x (N t) δ x N (dv) ⇒ f (t, v)dv, N → ∞,
where f is the solution to (1.1). To do the perturbation, fix α > 0,

w * ∈ R n , w ini ∈ C 1 (T d ; R n ) and
start the dynamics from some initial distribution such that 1

N d x∈T d N g ζ x (0) δ x N (dv) ⇒ w * + N -α w ini (v) dv, N → ∞.
Let {σ k ; k = 1, . . . , n} be the smooth solutions to (1.4). In the non-resonant situation, the arguments in the previous part suggests the formal asymptotic formula

1 N d x∈T d N g ζ x (N t) δ x N (dv) ≈   w * + N -α n j=1 σ j (N -α t, v -λ j tm)r j   dv.
Choosing κ ≤ α and using the variables s = N -κ t, u = v -λ j tm, one obtains that

1 N d-α x∈T d N l k g ζ x (N 1+κ s) -w * δ x N -N κ λ k sm (du) ⇒ σ (α,κ) k (s, u)du, (1.8)
as N → ∞ for each k = 1, ..., n, where

σ (α,κ) k (s, u) = σ k (s, u), if κ = α, σ ini k (u) := l k w ini (u), if κ < α.
Remark 1.1 (Fluctuation). The convergence in (1.8) is available only if α < d/2. To see that, let us assume that the equilibrium states of the dynamics are given by the family of canonical Gibbs measures ⊗ x ν w (dζ x ), where w is the corresponding equilibrium value of the conserved vector g. Starting the dynamics from equilibrium initial state with w = w * . If g possesses finite variance, then the central limit theorem for i.i.d. random variables yields that for each time s > 0, 1

N d/2 x∈T d N l k g ζ(s) -w * δ x N (du)
converges weakly, as N → ∞, to some Gaussian random field. The macroscopic time evolutions of these fields are called the equilibrium fluctuations. For the equilibrium fluctuations for the models studied in this paper, see e.g. [START_REF] Chang | Equilibrium fluctuations of asymmetric simple exclusion processes in dimension d ≥ 3[END_REF][START_REF] Gonçalves | Central limit theorem for a tagged particle in asymmetric simple exclusion[END_REF][START_REF] Olla | Equilibrium fluctuation for an anharmonic chain with boundary conditions in the Euler scaling limit[END_REF].

In this paper, we rigorously prove (1.8) for two microscopic models: (i) the generalized exclusion in any dimension, where the total number of particles is the only conservation law (n = 1) and (ii) the nonlinear Hamiltonian dynamics in one dimension, disturbed by noises that preserve only the total momentum and volume stretch (n = 2). 1.2.1. Generalized exclusion process. In Section 2 and 3, we study the d-dimensional generalized exclusion process with at most K > 0 particles per site. By [START_REF] Rezakhanlou | Hydrodynamic limit for attractive particle systems on Z d[END_REF][START_REF] Kipnis | Scaling limits of interacting particle systems[END_REF], the density of the particles under the Euler scaling evolves according to the hyperbolic equation

∂ t + m • ∇J( ) = 0, (0, •) = ini , (1.9)
where J( ) = (K -) is the macroscopic flux at density , and m = 0 is the mean of the underlying transition kernel. Fix * ∈ (0, K). We start the process from a perturbation of the constant profile ini

N (u) := * + N -α ρ ini (u), (1.10)
where ρ ini ∈ C ∞ (T d ). We speed up the process by N 1+κ , κ ≤ α, and choose the reference measure associated to the smooth profile * + N -α ρ(N κ-α t, u -N κ λtm),

where λ := J ( * ) = K -2 * and ρ is the classical solution to the Burgers equation

∂ s ρ(s, u) -m • ∇ ρ 2 (s, u) = 0, ρ(0, u) = ρ ini (u), (1.11)
up to the first shock appears. Then, in Theorem 2.1, we show that under some restrictions on κ and α, the relative entropy is of order o(N d-2α ). In particular, we partially extend the results in [START_REF] Esposito | Diffusive limit of asymmetric simple exclusion[END_REF][START_REF] Tóth | Between equilibrium fluctuations and eulerian scaling: perturbation of equilibrium for a class of deposition models[END_REF] to 0 < α < 1/3 if d = 1, and to 0 < α < 1 if d ≥ 2. As a result, in Theorem 2.3, we show the perturbed quantity evolves according to (1.11).

The proof relies on the refined relative entropy method recently introduced by Jara and Menezes [START_REF] Jara | Non-equilibrium fluctuations of interacting particle systems[END_REF][START_REF] Jara | Non-equilibrium fluctuations for a reaction-diffusion model via relative entropy[END_REF], also used in many other contexts, e.g., [START_REF] Funaki | Motion by mean curvature from glauber-kawasaki dynamics[END_REF][START_REF] Jara | The stochastic heat equation as the limit of a stirring dynamics perturbed by a voter model[END_REF][START_REF] Jara | Derivation of viscous burgers equations from weakly asymmetric exclusion processes[END_REF]. Compared to the technique in [START_REF] Tóth | Between equilibrium fluctuations and eulerian scaling: perturbation of equilibrium for a class of deposition models[END_REF], our proof does not involve the spectral gap estimate or logarithmic Sobolev inequality for the underlying microscopic dynamics, which is known as a hard problem for general interacting particle systems. 1.2.2. Anharmonic chain of oscillators. In Section 4 and 5, we study the one-dimensional chain of (unpinned) anharmonic oscillators with conservative noise. The noise is modeled by Langevin thermostats acting at each position and fixing the temperature to T = β -1 . The only two conserved quantities of the dynamics are the momentum p and the length r, hence it corresponds to the case n = 2, d = 1. The hydrodynamic equation is given by the following p-system

∂ t p = ∂ v τ (r), ∂ t r = ∂ v p, (p, r)(0, •) = (p, r) ini ,
where τ = τ β is the equilibrium tension. It is proved in [START_REF] Braxmeier-Even | Hydrodynamic limit for an Hamiltonian system with boundary conditions and conservative noise[END_REF] together with the energy conservation law in the smooth regime and in [START_REF] Marchesani | Hydrodynamic limit for an anharmonic chain under boundary tension[END_REF] after the appearance of shocks.

Define J(p, r) := (-τ (r), -p) and fix (p * , r * ) ∈ R 2 such that τ (r) = 0. Denote by A = DJ(p * , r * ) and note that Aυ -= τ (r * )υ -, Aυ + = -τ (r * )υ + for υ ± = (± τ (r * ), 1) . Similarly to the exclusion process, we start the dynamics from a perturbed profile

(p * , r * ) + N -α σ ini -υ -+ N -α σ ini + υ + , where σ ini ± ∈ C 1 (T)
. We also speed up the time to be N 1+κ t for some κ ≤ α. Choose the reference measure associated to the slowly varying parameters

(p * , r * ) + N -α j=± σ j N κ-α t, v + jN κ τ (r * )t υ j ,
where (σ -, σ + ) is the smooth solution to

∂ s σ -(s, u) + τ (r * ) 4 τ (r * ) ∂ u σ 2 -(s, u) = 0, σ -(0, u) = σ ini -(u), ∂ s σ + (s, u) - τ (r * ) 4 τ (r * ) ∂ u σ 2 + (s, u) = 0, σ + (0, u) = σ ini + (u).
(1.12)

We prove in Theorem 4.1 that in the smooth regime, the relative entropy grows with the order o(N 1-2α ). As a consequence, we prove in Theorem 4.2 that the macroscopic perturbation is governed by the decoupled system (1.12). The proof of the relative entropy estimate is based on the equivalence of ensembles for inhomogeneous Gibbs states and the uniform gradient estimate for the Poisson equation associated to the generator of the stochastic dynamics, cf. [START_REF] Xu | Hyperbolic scaling limit of non-equilibrium fluctuations for a weakly anharmonic chain[END_REF]Section 8 & 9]. It is worth pointing out that, unlike the exclusion, the equilibrium perturbation for Hamiltonian dynamics has not been investigated before, mainly due to the technical difficulties in obtaining fine estimate for the relative entropy.

Asymmetric generalized exclusion process

In this and the next sections, we consider the asymmetric generalized exclusion process. Fix a positive integer K > 0, which denotes the maximum number of particles at each site. The state space of the generalized exclusion process is

Ω d N := {0, 1, . . . , K} T d N . For a configuration η ∈ Ω d N , η x denotes the number of particles at site x ∈ T d N .
Denote by {e i } 1≤i≤d the canonical basis of T d N and let e i = -e i-d for i = d + 1, ..., 2d. Given the jump rates {p i ≥ 0; 1 ≤ i ≤ 2d}, consider the generator L N which is given for any function f :

Ω d N → R by L N f (η) = 2d i=1 x∈T d N c x,i (η) f (η x,x+ei ) -f (η) ,
where the jump rate is given by c x,i (η) := p i η x (K -η x+ei ) and η x,y is the configuration obtained from η after a particle jumps from x to y,

(η x,y ) z =        η x -1, z = x, η y + 1, z = y, η z , otherwise. Assume that p i + p i+d > 0 for each 1 ≤ i ≤ d. Denote m = (m i ) 1≤i≤d ∈ R d where m i = p i -p i+d .
Also assume that m is a nonzero vector, hence L N is asymmetric. The generator L N has a family of product invariant measures indexed by the particle density. For ∈ [0, K], let ν 1 be the binomial measure B(K, K -1 ):

ν 1 (k) := K k K k 1 - K K-k , ∀ k = 0, . . . , K. Denote by ν N the product measure on Ω d N such that ν N (η x ) = ν 1 (η x ) for each x ∈ T d N . It is not hard to check that the family {ν N ; 0 ≤ ≤ K} is invariant under L N . Observe that the average number of particles per site under ν N is = E ν 1 [η x ].
Let µ N,0 be associated to the profile ini N in (1.10) with some α > 0. For 0 < κ ≤ α, denote by {η = η(t); t ≥ 0} the Markov process generated by N 1+κ L N starting from µ N,0 . Define

N (t, u) := * + N -α ρ(N κ-α t, u -N κ λtm), (2.1)
where ρ(t, u) is the smooth solution to (1.11). Note that for N sufficiently large, N ∈ (0, K). Denote by µ N,t the distribution of η(t) and by ν N,t the slowly varying product measure

ν N,t (dη) = x∈T d N ν 1 N,x (dη x ), N,x := N t, x N . (2.2) Let f N,t (η) := ν N,t (η) -1 µ N,t (
η) be the Radon-Nikodym derivative. For a probability measure µ on Ω d N and some µ-density f , i.e. f ≥ 0 and Ω d N f dµ = 1, define the relative entropy as

H(f ; µ) := Ω d N f log f dµ = η∈Ω d N f (η) log f (η)µ(η).
To shorten the notations, denote

H N (t) := H(f N,t ; ν N,t ), ∀ t ∈ [0, T ].
It is well known that the the entropy solution to the Burgers equation (1.11) may develop shocks in a finite time interval even if the initial density is smooth, cf. [START_REF] Málek | Weak and Measure-valued Solutions to Evolutionary PDEs[END_REF] for example. Since our proof depends on the relative entropy method, which requires the reference density profile to be smooth, throughout the article, we assume the initial density profile ρ ini is smooth, and when κ = α, fix a time horizon T > 0 such that the solution to the Burgers equation (1.11) is smooth during the time interval [0, T ].

The followings are the main results of this part.

Theorem 2.1. Suppose that H N (0) = o(N d-2α ). (i) If d = 1, α ∈ (0, 1/2), then for any t ∈ [0, T ], H N (t) = o(N 1-2α ) for κ ∈ (0, α] ∩ (0, 1 -2α); ii) If d ≥ 2, α > 0, then for any t ∈ [0, T ], H N (t) = o(N d-2α ) for κ ∈ (0, α] ∩ (0, 1). Remark 2.2. It is believed that in dimension d = 1
, the above result extends to κ ∈ (0, α] even beyond the appearance of shocks, cf. [START_REF] Tóth | Between equilibrium fluctuations and eulerian scaling: perturbation of equilibrium for a class of deposition models[END_REF].

As a direct consequence of the above theorem, we have the following law of large numbers for the perturbed quantities.

Theorem 2.3. Suppose the assumptions in Theorem 2.1 hold and further that α < d/2. Then, for any ϕ ∈ C(T d ) and any ε > 0, lim

N →∞ µ N,t 1 N d-α x∈T d N η x - * ϕ x N -N κ λtm - T d ρ (α,κ) (t, u)ϕ(u)du > ε = 0,
with the function ρ (α,κ) defined as

ρ (α,κ) (t, u) = ρ ini (u), κ < α, ρ(t, u), κ = α,
where ρ(t, u) is the entropy solution to (1.11).

Remark 2.4. If K = 1, the above results could be interpreted in terms of second class particles. The dynamic is defined as follows: there are two kinds of particles, called first and second class particles, in the system. On top of the exclusion rule, the first class particles have priorities to jump over the second class ones. Precisely speaking, if a first class particle jumps to a site occupied by a second class one, then the jump is performed and the two particles exchange their positions; while if a second class particle jumps to a site occupied by a first class one, then the jump is suppressed. At the initial time, independently at each site x, put one first class particle with probability * , and one second class particle with probability N -α ρ ini (x/N ). Then, at the macroscopic time t, the density profile of the second class particles, along the characteristic line of the PDE (1.9), is described by the function ρ α,κ . Indeed, denote by η 1 x (t) (resp. η 2 x (t)) the number of the first (resp. the second) class particles at site x at time t, then η x (t) = η 1

x (t) + η 2 x (t). Since the process of the first class particle is in equilibrium, the distribution of η 1 (t) is given by ν N * for any time t ≥ 0. Therefore, for any ϕ ∈ C(T d ),

Var 1 N d-α x∈T d N (η 1 x (t) - * )ϕ x-λtN 1+κ m N = O N 2α-d ,
which converges to zero as N → ∞ since α < d/2. As a result, for any ε > 0, lim

N →∞ µ N,t 1 N d-α x∈T d N (η 1 x - * )ϕ x-λtN 1+κ m N > ε = 0.
Together with Corollary 2.3, we have

lim N →∞ µ N,t 1 N d-α x∈T d N η 2 x ϕ x-λtN 1+κ m N - T d ρ α,κ (t, u)ϕ(u) du > ε = 0.

Relative entropy for the generalized exclusion

In this section, we prove Theorems 2.1 and 2.3. Recall the profile N defined in (2.1). To make notations simple, in the following calculations, we denote

x = N,x (t) := N t, x N , ∀ x ∈ T d N .
For a probability measure µ on Ω d N and a µ-density f , define the Dirichlet form

D N (f ; µ) := 2d i=1 x∈T d N η∈Ω d N c x,i (η) f (η x,x+ei ) -f (η) 2 µ(η). (3.1)
We claim that for any ν N,t -density f , there exists δ 0 = δ 0 ( * ) such that

D N (f ; ν N,t ) ≥ δ 0 2d i=1 x∈T d N η∈Ω d N η x (K -η x+ei ) f (η x,x+ei ) -f (η) 2 ν N,t (η). (3.2)
Indeed, the claim holds obviously if p i > 0 for all 1 ≤ i ≤ 2d. Suppose that p i = 0 for some 1 ≤ i ≤ d, then p i+d = 0 due to our assumption. Recalling that e i+d = -e i ,

η η x (K -η x-ei ) f (η x,x-ei ) -f (η) 2 ν N,t (η) = η (η x + 1)(K -η x-ei + 1) f (η) -f (η x-ei,x ) 2 ν N,t (η x-ei,x ), for each x ∈ T d N . Observe from the definition of ν N,t in (2.2) that ν N,t (η x-ei,x ) ν N,t (η) = θ x θ x-ei η x-ei (K -η x ) (K -η x-ei + 1)(η x + 1)
,

where θ x := x K -x . Since | x - * | ≤ N -α ||ρ|| ∞ and * ∈ (0, K)
, hence there exists some C = C( * ) > 0, such that C -1 < θ x < C for all x and sufficiently large N . Therefore,

η η x (K -η x-ei ) f (η x,x-ei ) -f (η) 2 ν N,t (η) ≥ C η η x-ei (K -η x ) f (η) -f (η x-ei,x ) 2 ν N,t (η),
which is enough to conclude the claim. Recall that f N,t = µ N,t /ν N,t and let ψ N,t := (ν N * ) -1 ν N,t be the derivative of ν N,t with respect to the stationary measure ν N * . By Yau's relative entropy inequality (see Lemma A.1),

d dt H N (t) ≤ -N 1+κ D N (f N,t ; ν N,t ) + η∈Ω d N N 1+κ L * N,t 1(η) - d dt log ψ N,t (η) µ N,t (η),
where L * N,t is the adjoint of L N with respect to ν N,t . The main idea is to write the right-hand side of the above inequality as CH N (t) + o(N d-2α ) for some finite constant C independent of N , and the result then follows from Grönwall's inequality.

Since

c x,i (η) is the jump rate from η ∈ Ω d N to η x,x+ei , Lemma A.1 yields that L * N,t 1(η) = 2d i=1 x∈T d N ν N,t (η x+ei,x ) ν N,t (η) c x,i (η x+ei,x ) -c x,i (η) = 2d i=1 x∈T d N p i x (K -x+ei ) x+ei (K -x ) η x+ei (K -η x ) -η x (K -η x+ei ) = 2d i=1 x∈T d N p i x (K -x+ei ) η x+ei (K -η x ) x+ei (K -x ) - η x (K -η x+ei ) x (K -x+ei )
.

For any configuration η ∈ Ω d N and any site x ∈ T d N , denote

ω x = ω x (η) := η x -x x (K -x ) . (3.3)
It could be checked directly that for x = y,

η x (K -η y ) x (K -y ) - η y (K -η x ) y (K -x ) = K ω x -ω y + ( x -y )ω x ω y . (3.4)
Recall that e i+d = -e i for 1 ≤ i ≤ d, and (3.4) permits us to write L * N,t 1(η) as

L * N,t 1(η) = d i=1 x∈T d N b N (x, x + e i ) ω x+ei -ω x + ( x+ei -x )ω x ω x+ei ,
where for 1

≤ i ≤ d, b N (x, x + e i ) := K p i x (K -x+ei ) -p i+d x+ei (K -x ) .
Now we calculate d dt log ψ N,t (η). Since ν N,t and ν N * are both product measures,

log ψ N,t (η) = x∈T d N η x log θ x * -K log 1 + θ x 1 + * .
The time derivative then reads

d dt log ψ N,t (η) = x∈T d N η x ∂ t θ x θ x -K ∂ t θ x 1 + θ x = x∈T d N K η x ∂ t x x (K -x ) - ∂ t x K -x = x∈T d N Kω x ∂ t x .
To sum up, we have shown that

d dt H N (t) ≤ -N 1+κ D N (f N,t ; ν N,t ) + R N,t + E N,t , (3.5)
where

R N,t = η∈Ω d N d i=1 x∈T d N a N i,x ω x ω x+ei µ N,t (η), (3.6)
a N i,x = N 1+κ b N (x, x + e i )( x+ei -x ), (3.7)
and

E N,t = η∈Ω d N d i=1 x∈T d N N 1+κ b N (x, x + e i )(ω x+ei -ω x ) - x∈T d N Kω x ∂ t x µ N,t (η).
3.1. Proofs of Theorems 2.1 and Corollary 2.3. In this subsection, we prove Theorems 2.1 and 2.3. We first deal with the error term E N,t .

Lemma 3.1. There exists a constant C independent of N such that

E N,t ≤ H N (t) + CN d+2κ-2α-2 .
Proof. For a sequence {a

x ; x ∈ T d N } and 1 ≤ i ≤ d, let ∇ i a x := a x+ei -a x and ∇ * i a x := a x-ei -a x .
Using the summation by parts formula,

E N,t = η∈Ω d N µ N,t (η) x∈T d N ω x N 1+κ d i=1 ∇ * i b N (x, x + e i ) -K∂ t x .
By the definition of b N and Taylor's expansion,

∇ * i b N (x, x + e i ) = N -1-2α Km i ∂ ui ρ 2 N κ-α t, x N -N κ λtm -N -1-α Kλm i ∂ ui ρ N κ-α t, x N -N κ λtm + N,i (x),
where N,i (x) = O(N -2-α ). Meanwhile, by the definition of N in (2.1),

∂ t x = N κ-2α ∂ s -N κ-α λ(m • ∇ u ) ρ N κ-α t, x N -N κ λtm .
Since ρ solves the Burgers equation (1.11), we can rewrite the E N,t as

E N,t = N 1+κ η∈Ω d N d i=1 x∈T d N ω x N,i (x) µ N,t (η).
It is easy to see |E N,t | ≤ CN d-1+κ-α . To get a better bound, by entropy inequality,

E N,t ≤ H N (t) + log η∈Ω d N exp N 1+κ d i=1 x∈T d N ω x N,i (x) ν N,t (η) = H N (t) + x∈T d N log η∈Ω d N exp N 1+κ ω x d i=1 N,i (x) ν N,t (η) . (3.8)
Note that ω x is bounded and has zero mean with respect to the measure ν N,t . Using the basic inequality

e a ≤ 1 + a + (1/2)a 2 e |a| , log(1 + a) ≤ a,
we have

E N,t ≤ H N (t) + O(N d-2+2κ-2α ). (3.9)
This concludes the proof of the lemma.

The following result bounds the term R N,t , whose proof is postponed to Subsection 3.2.

Proposition 3.2. For ≥ 1, let

g d ( ) =        , d = 1, log , d = 2, 1, d ≥ 3.
(3.10)

Then, there exists a constant C independent of N such that

R N,t ≤ N 1+κ D N (f N,t ; ν N,t ) + C N κ-α + N κ-2α-1 d g d ( ) H N (t) + N d d + C H N (t) + N d+2κ-4α-2 d g d ( ) .
Now, we are ready to prove Theorem 2.1.

Proof of Theorem 2.1. In Proposition 3.2, take

= (N ) =        N (1+2α-κ)/2 , d = 1, N (1+2α-κ)/2 / √ log N , d = 2, N (1+2α-κ)/d , d ≥ 3.
Therefore, in dimension d = 1, if κ < 1 -2α and κ ≤ α, then

d dt H N (t) ≤ C(1 + N κ-α )H N (t) + C N (1-2α+κ)/2 + N κ-2α + N 2κ-2α-1 ≤ CH N (t) + o(N 1-2α ).
In dimension d = 2, if κ < 1 and κ ≤ α, then

d dt H N (t) ≤ C(1 + N κ-α )H N (t) + C N 1-2α+κ log N + N 2κ-2α ≤ CH N (t) + o(N 2-2α ).
In dimensions d ≥ 3, if κ < 1 and κ ≤ α, then

d dt H N (t) ≤ C(1 + N κ-α )H N (t) + C N d-2α+κ-1 + N d+2κ-2α-2 ≤ CH N (t) + o(N d-2α ).
We conclude the proof by using Grönwall's inequality.

The proof of Theorem 2.3 is a direct application of the entropy inequality, cf. [12, Corollary 6.1.3] for example. We sketch it below for completeness.

Proof of Theorem 2.3. It suffices to prove lim

N →∞ η∈Ω d N 1 N d-α x∈T d N (η x -E ν N,t [η x ])ϕ x-λtN 1+κ m N µ N,t (η) = 0.
By the entropy inequality, for any γ > 0, the integral above is bounded by

H N (t) γN d-2α + 1 γN d-2α log η∈Ω d N exp γN -α x∈T d N (η x -E ν N,t [η x ])ϕ x-λtN 1+κ m N ν N,t (η) .
By Theorem 2.1, the first term above converges to zero as N → ∞. Since e |c| ≤ e c + e -c , log(a + b) ≤ log 2 + max{log a, log b} for any c and any positive reals a, b, and since α < d/2, we could remove the absolute value inside the exponential in the second term above, and rewrite it as

1 γN d-2α x∈T d N log η∈Ω d N exp γN -α (η x -E ν N,t [η x ])ϕ x-λtN 1+κ m N ν N,t (η) .
Using the basic inequality e a ≤ 1 + a + (1/2)a 2 e |a| and log(1 + a) ≤ a, there exists a finite constant C independent of N such that the above formula is bounded by

C γN d-2α × N d × (γN -α ) 2 = Cγ.
Since γ could be taken arbitrarily small, the proof is completed.

3.2. Proof of Proposition 3.2. It remains to prove Proposition 3.2. The first step is to properly decompose the term R N,t defined in Eq. (3.6). For ≥ 1, let p (•) be the uniform measure on Λ d = {0, 1, . . . , -1} d , i.e. p (x) = -d if x ∈ Λ d and = 0 otherwise. Let q = p * p be the convolution of p with itself, q (z) =

y∈T d N p (y)p (z -y), z ∈ T d N .
For x ∈ T d N and ≥ 1, the spatial average of ω x in the box Λ d 2 -1 is defined as

ω x = z∈T d N ω x+z q (z). Define R N,t = η∈Ω d N d i=1 x∈T d N a N i,x ω x ω x+ei µ N,t (η). (3.11)
Using the definition of flows introduced in Subsection A.2 and summation by parts formula,

R N,t -R N,t = η∈Ω d N d i=1 x,z∈T d N a N i,x ω x ω x+ei+z (δ 0 (z) -q (z))µ N,t (η) = η∈Ω d N d i,j=1 x,z∈T d N a N i,x ω x ω x+ei+z (φ (z, e j ) -φ (z -e j , e j ))µ N,t (η) = η∈Ω d N d i,j=1 x,z∈T d N a N i,x ω x (ω x+ei+z -ω x+ei+ej +z )φ (z, e j )µ N,t (η).
Make the change of variables x → x -z -e i , and put

h j,x = d i=1 z∈T d N a N i,x-z-ei ω x-z-ei φ (z, e j ), (3.12)
then we have

R N,t -R N,t = η∈Ω d N d j=1 x∈T d N h j,x (ω x -ω x+ej ) µ N,t (η).
To further decompose the term on the right-hand side of the last line, we introduce the following integration by parts formula.

Lemma 3.3 (Integration by parts formula). Let h : Ω d

N → R be such that h does not depend on the values of η x or η z for some x, z ∈ T d N . Then, for any ν N,t -density f ,

η∈Ω d N h(η) ω z -ω x f (η)ν N,t (η) = 1 K η∈Ω d N h(η)s N x,z (η) f (η x,z ) -f (η) ν N,t (η) -( z -x ) η∈Ω d N h(η)f (η)ω z ω x ν N,t (η),
where

s N x,z (η) = η x (K -η z ) x (K -z )
.

Proof. For K = 1, the result is proved in [10, Lemma E.1], and we extend it to general K. Since h does not depend on the values of η x and η z , using the change of variables η → η z,x , we may rewrite the second term on the right-hand side of the above equation as

1 K η∈Ω d N h(η) s N x,z (η z,x )ν N,t (η z,x ) ν N,t (η) -s N x,z (η) f (η)ν N,t (η). Direct calculations show that s N x,z (η z,x )ν N,t (η z,x ) ν N,t (η) = η z (K -η x ) z (K -x ) = s N z,x (η).
We conclude the proof by Eq. (3.4).

Since φ is supported in Λ d 2 -1 , the value of h j,x does not depend on those of η x and η x+ej for 1 ≤ j ≤ d. Recall µ N,t (η) = f N,t (η)ν N,t (η). By Lemma 3.3, we have

R N,t -R N,t = 1 K d j=1 x∈T d N η∈Ω d N h j,x s N x+ej ,x f N,t (η x+ej ,x ) -f N,t (η) ν N,t (η) - d j=1 x∈T d N ( x -x+ej ) η∈Ω d N h j,x ω x ω x+ej f N,t ν N,t (η). (3.13)
By Cauchy-Schwarz inequality, for any γ > 0, the first term on the right hand side of the last equation is bounded by

γ 2 d j=1 x∈T d N η∈Ω d N s N x+ej ,x f N,t (η x+ej ,x ) -f N,t (η) 2 ν N,t (η) + 1 2K 2 γ d j=1 x∈T d N η∈Ω d N (h j,x ) 2 s N x+ej ,x f N,t (η x+ej ,x ) + f N,t (η) 2 ν N,t (η). (3.14)
Since * ∈ (0, K), for N large enough, there exists a constant C = C( * , K) > 0 such that S N x,z ≤ C for any x, z ∈ T d N . Then, by (3.2), the first term in (3.14) is bounded by C 1 γD N (f N,t ; ν N,t ) for some

C 1 = C 1 ( * , K, δ 0 ). Since for any η ∈ Ω d N and any x, z ∈ T d N , ν N,t (η x,z ) ν N,t (η) = η x (K -η z ) (K + 1 -η z )(η z + 1)) z (K -x ) x (K -z ) ≤ C := C( * , K),
by Cauchy-Schwarz inequality and exchange of variables η → η x,x+ej , the second term in (3.14) is bounded from above by C γ

d j=1 x∈T d N η∈Ω d N (h j,x ) 2 f N,t (η)ν N,t (η).
Take γ = N 1+κ /C 1 , together with (3.13), then we have shown that

R N,t ≤ N 1+κ D N (f N,t ; ν N,t ) + R N,t + C N 1+κ d j=1 x∈T d N η∈Ω d N (h j,x ) 2 f N,t (η) ν N,t (η) + d j=1 x∈T d N ( x+ej -x ) η∈Ω d N h j,x ω x ω x+ej f N,t (η)ν N,t (η) (3.15)
for some C = C( * , K, δ 0 ).

Next, we shall deal with the last three terms on the right-hand side of the above decomposition (3.15) respectively. We first deal with the term R N,t defined in (3.11). Observe that

d i=1 x∈T d N a N i,x ω x ω x+ei = d i=1 x,y,z∈T d N a N i,x ω x ω x+ei+z p (y)p (z -y) = d i=1 x∈T d N y∈T d N a N i,x-y ω x-y p (y) z∈T d N ω x+ei+z p (z) .
In the last identity, we reindex x by x -y and z by z + y. By entropy inequality, for any γ > 0,

R N,t ≤ 1 γ H N (t) + log η∈Ω d N exp γ d i=1 x∈T d N y∈T d N a N i,x-y ω x-y p (y) z∈T d N ω x+ei+z p (z) ν N,t (η) . (3.16)
Note that for large enough, the two random variables

y∈T d N a N i,x -y ω x -y p (y) z∈T d N ω x +ei+z p (z) , y∈T d N a N i,x -y ω x -y p (y) z∈T d N ω x +ei+z p (z) are independent with respect to ν N,t if ||x -x || ∞ > 3 . Denote x = z (mod ) if x i = z i + k i for some k i ∈ Z and for all 1 ≤ i ≤ d. Note that for any x ∈ T d N , there exists a point z ∈ Λ d
3 -1 such that x = z (mod 3 ), and for fixed z, if x , x = z (mod 3 ) and x = x , then ||x -x || ∞ ≥ 3 . By Hölder's inequality and independence, the second term in (3.16) is bounded by

1 d(3 ) d d i=1 z∈Λ d log η∈Ω d N exp γd(3 ) d x:x=z (mod 3 ) y∈T d N a N i,x-y ω x-y p (y) z∈T d N ω x+ei+z p (z) ν N,t (η) ≤ 1 d(3 ) d d i=1 x∈T d N log η∈Ω d N exp γd(3 ) d y∈T d N a N i,x-y ω x-y p (y) z∈T d N ω x+ei+z p (z) ν N,t (η) . (3.17)
We claim that the for each x ∈ T d N , the random variable

y∈T d N a N i,x-y ω x-y p (y)
is sub-Gaussian of order C 2 N 2κ-2α -d with respect to the measure ν N,t for some constant C 2 = C 2 ( * , K, ∂ ui ρ). Indeed, first note that by the definition of a N i,x in (3.7), there exists a constant

C = C( * , K) such that |a N i,x | ≤ C||∂ ui ρ|| ∞ N κ-α . Since ω x ≤ C( * , K)
for N large enough, and has mean zero with respect to ν N,t , by Lemma A.5, the random variable a N i,x ω x is sub-Gaussian of order CN 2κ-2α for some constant C = C( * , K, ∂ ui ρ) with respect to ν N,t . Therefore, for any θ ∈ R,

log η∈Ω d N exp θ y∈T d N a N i,x-y ω x-y p (y) ν N,t (η) = y∈T d N log η∈Ω d N exp θa N i,x-y ω x-y p (y) ν N,t (η) ≤ 1 2 θ 2 y∈T d N C 2 N 2κ-2α p (y) 2 = C 2 2 θ 2 N 2κ-2α -d .
Similarly, one could prove that z∈T d N ω x+ei+z p (z) is sub-Gaussian of order C 2 -d . By Corollary A.4, taking γ = (4C 2 d3 d ) -1 N α-κ , we bound the term in (3.17) by N d log 3/(3 ) d . Therefore, there exists a constant C independent of N such that

R N,t ≤ C N α-κ H N (t) + N d d . (3.18)
The third term in (3.15) is treated in the same way as above. Using Lemma A.2, there exists a constant C = C(ρ * , K) such that the random variable h j,x defined in (3.12) is sub-Gaussian of order

CN 2κ-2α z∈T d N φ (z, e j ) 2 ≤ C 3 N 2κ-2α g d ( )
with respect to the measure ν N,t for some constant C 3 = C 3 ( * , K, C 0 ). Also note that the two random variables h j,x and h j,x are independent under the measure ν N,t if ||x -x || ∞ > 3 . By entropy inequality and Hölder's inequality, for any γ > 0,

1 N 1+κ η∈Ω d N d j=1 x∈T d N (h j,x ) 2 f N,t (η) ν N,t (η) ≤ 1 γN 1+κ H N (t) + log η∈Ω d N exp γ d j=1 x∈T d N (h j,x ) 2 ν N,t (η) ≤ 1 γN 1+κ H N (t) + 1 d(3 ) d d j=1 x∈T d N log η∈Ω d N exp γd(3 ) d (h j,x ) 2 ν N,t (η) . (3.19) Take γ = 4C 3 d3 d N 2κ-2α d g d ( )
-1 , then by Lemma A.3, there exists a finite constant C independent of N such that the third term in (3.15) is bounded by

C d g d ( ) N 1+2α-κ H N (t) + N d d . (3.20)
Now we deal with the last term in (3.15). First note that

| x+ej -x | ≤ N -1-α ||∂ uj ρ|| ∞ .
By entropy inequality and Hölder's inequality, for any γ > 0, we may bound the last term in (3.15) by

1 γN 1+α H N (t) + log η∈Ω d N exp γ d j=1 x∈T d N N 1+α ( x+ej -x )h j,x ω x ω x+ej ν N,t (η) ≤ 1 γN 1+α H N (t) + 1 d(3 ) d d j=1 x∈T d N log η∈Ω d N exp γd(3 ) d N 1+α ( x+ej -x )h j,x ω x ω x+ej ν N,t (η) . (3.21) Since |ω x ω x+ej | ≤ C := C( * , K
), and we have already shown that h j,x is sub-Gaussian of order C 3 N 2κ-2α g d ( ), then, for any θ ∈ R,

log η∈Ω d N exp{θh k,x ω x ω x+ej }ν N,t (η) ≤ log η∈Ω d N exp{C|θh k,x |}ν N,t (η) ≤ log η∈Ω d N exp{Cθh j,x } + exp{-Cθh j,x } ν N,t (η) ≤ log 2 + Cθ 2 N 2κ-2α g d ( ) (3.22)
for some constant C = C( * , K, C 3 ). Therefore, there exists some constant C independent of N , such that the term on the right hand side of (3.21) is bounded by

C γN 1+α H N (t) + N d d × 2d g d ( )γ 2 N 2κ-2α + N d d .
Take γ = N -1-α , then the last line is bounded by

C H N (t) + d g d ( )N d+2κ-4α-2 + N d d . (3.23)
We conclude the proof of Proposition 3.2 by using (3.15), (3.18), (3.20) and (3.23).

1-d chain of anharmonic oscillators

In this and the next sections, we consider a chain of N coupled oscillators in one-dimensional lattice space. All particles have identical mass 1. The momentum and position of the particle x = 1, ..., N are denoted by p x ∈ R and q x ∈ R, respectively. The interaction between two particles x -1 and x is determined by an anharmonic spring with the potential energy V (q x -q x-1 ), where V is some nice function. The total energy is given by the Hamiltonian

H N (η) := x∈T N p 2 x 2 + V (q x -q x-1 ), ∀ η ∈ Ω N .
The corresponding Hamiltonian dynamics then reads

ṗx = -∂ qx H N , qx = ∂p x H N , ∀ x ∈ T N .
Assume that V ∈ C 2 (R; R) and some constant c > 0, such that

c -1 ≤ V (r) ≤ c, ∀ r ∈ R.
Define r x := q x -q x-1 to be the inter-particle distance and require the periodic boundary condition:

(p N +1 , r N +1 ) = (p 1 , r 1 )
. The configuration space is then Ω N = (R 2 ) T N , with its elements denoted by

η = {η x = (p x , r x ); x ∈ T N }.
Observe that the total momentum, the total volume and the Hamiltonian are conserved. Under a generic assumption of local equilibrium, Euler equations can be formally obtained as the evolution of these quantities. However, to prove it for the purely deterministic system turns out to be a difficult task. Proper stochastic noise helps us solve this problem. Suppose that at each bond (x, x + 1), the deterministic system is contact with a thermal bath at fixed temperature. More precisely, fix some inverse temperature β > 0 and define

Y x := ∂ ∂r x+1 - ∂ ∂r x , Y * x = β V (r x+1 ) -V (r x ) -Y x .
For some deterministic parameter γ N > 0 that regulates the strength of the heat bath, consider the operator L N given by

L N := A N + γ N S N , S N := - 1 2 x∈T N Y * x Y x , (4.1)
where A N is the Liouville operator given by

A N := x∈T N (p x -p x-1 ) ∂ ∂r x + V (r x+1 ) -V (r x ) ∂ ∂p x .
The Markov process generated by L N is equivalently expressed by the following system of stochastic differential equations: for each x ∈ T N ,

         dp x (t) = V (r x+1 ) -V (r x ) dt, dr x (t) = (p x+1 -p x )dt + βγ N 2 V (r x+1 ) + V (r x-1 ) -2V (r x ) dt + √ γ N dB x-1 t -dB x t , where {B x • ;
x ≥ 1} is an infinite system of independent, standard Brownian motions. Notice that the total momentum x p x and the total length x r x are the only conserved quantities of the microscopic dynamics. The conservation law of energy is no longer preserved by S N . 4.1. Stationary states. The stationary states of L N are given by the family of canonical Gibbs measures indexed by the global momentum p ∈ R and tension τ ∈ R:

ν N p,τ (dp dr) = x∈T N β 2π exp - β(p x -p) 2 2 dp x ⊗ π τ (dr x ),
where the probability measure π τ is defined as

π τ (dr) := 1 Z(τ ) e -β(V (r)-τ r) dr, Z(τ ) := R e -β(V (r)-τ r) dr.
Observe that the dependence on β is omitted, since it is fixed hereafter. It is easy to see that A N , S N are respectively anti-symmetric and symmetric with respect to the Gibbs states. Moreover, for all smooth functions f , g on Ω N ,

Ω N f S N g dν N p,τ = - 1 2 Ω N x∈T N Y x f Y x g dν N p,τ .
Define the Gibbs potential G = G(τ ) for τ ∈ R and the free energy F = F (r) for r ∈ R by the following Legendre transform

G(τ ) := 1 β log Z(τ ), F (r) := sup τ ∈R τ r -G(τ ) .
The average length r = r(τ ) and equilibrium tension τ = τ (r) are then given by the convex conjugate variables r(τ ) := E πτ [r] = G (τ ), τ (r) := F (r).

4.2. Equilibrium perturbation. As illustrated in Section 1, we fix (p * , r * ) ∈ R 2 and consider the distribution µ N,0 associated to the profile (p ini N , r ini N ) given by

p ini N r ini N := p * r * + N -α j=± σ ini j υ j , υ ± := ± τ (r * ) 1 ,
where α > 0 and σ ini ± ∈ C ∞ (T). In addition, we require that (cf. (1.7))

T σ -(u)du = T σ + (u)du = 0.
For 0 < κ ≤ α, denote by {η(t); t ≥ 0} the Markov process generated by N 1+κ L N and the initial distribution µ N,0 . As usual, we use the notation µ N,t for the distribution of η(t) on Ω N . For (t, u) ∈ [0, T ] × T, define (p, r) = (p, r)(t, u) by

p N r N := p * r * + N -α j=± σ j N κ-α t, u + jN κ τ (r * )t υ j ,
where σ -, σ + solve the decoupled system of Burgers equations (1.12). Denote by ν N,t the slowly varying product measure

ν N,t (dη) = x∈T d N ν 1 p N x ,τ N x (dη x ), p N x , τ N x := p N , τ (r N ) t, x N .
Let f N,t be the Radon-Nikodym derivative dµ N,t /dν N,t and recall the relative entropy

H N (t) = H(f N,t ; ν N,t ) := Ω N f N,t log f N,t dν N,t . (4.2)
Recall that our argument relies on the smoothness of σ ± , hence we require that t ∈ [0, T ] for (i) any T < T shock , the first time when shock appears in the entropy solution to (1.12) if κ = α and (ii) any T > 0 if κ < α. The first result is stated below, cf. the case d = 1 in Theorem 2.1.

Theorem 4.1. Suppose that H N (0) = o(N 1-2α ) and α ∈ (0, 1/2). If κ ∈ (0, α] ∩ (0, (1 -2α)/3) and

N 5κ+4α-1 γ N N 1-κ , then H N (t) = o(N 1-2α ) for any t ∈ [0, T ].
lim N →∞ µ N,t 1 N 1-α x∈T N r x -r * 2 - p x -p * 2 τ (r * ) ϕ x N -N κ τ (r * )t - T N σ (α,κ) - (t, u)ϕ(u)du > ε = 0, lim N →∞ µ N,t 1 N 1-α x∈T N r x -r * 2 + p x -p * 2 τ (r * ) ϕ x N + N κ τ (r * )t - T N σ (α,κ) + (t, u)ϕ(u)du > ε = 0,
for any t ∈ [0, T ], ϕ ∈ C(T) and ε > 0, where

σ (α,κ) ± (t, u) := σ ± (t, u), if 0 < α < 1 5 , κ = α, σ ini ± (u), if 0 < α < 1 2 , 0 < κ < min α, 1-2α 3 .

Relative entropy for the oscillator chain

We have seen in (1.6) that, for a system of 2 conservation laws, the non-resonant system of perturbations (1.12) requires proper second order correction terms. Hence, we choose the modified profile (p N , rN )(t, u) given by

p N r N + N -2α j=± σj N κ-α t, u -N κ τ (r * )t, u + N κ τ (r * )t υ j ,
where for (t, u -, u

+ ) ∈ [0, T ] × R 2 , σ-(t, u -, u + ) := - τ (r * ) 8τ (r * ) σ 2 + (t, u + ) - τ (r * ) 4τ (r * ) ∂ u σ -(t, u -)Σ + (t, u + ) + σ -(t, u -)σ + (t, u + ) , σ+ (t, u -, u + ) := - τ (r * ) 8τ (r * ) σ 2 -(t, u -) - τ (r * ) 4τ (r * ) σ -(t, u -)σ + (t, u + ) + Σ -(t, u -)∂ u σ + (t, u + ) . (5.1)
Here Σ ± =

• 0 σ ± (t, u)du are the primitive functions of σ ± . Let νN,t be the product measure on Ω N associated to the profile (p, r)(t, •). Recall that µ N,t is the distribution of the dynamics η(t). Denote by fN,t Radon-Nikodym derivative of µ N,t with respect to νN,t and by H N (t) the corresponding relative entropy. Also recall the relative entropy H N (t) in (4.2). The following lemma is straightforward.

Lemma 5.1. For any t ≥ 0, H N (t) = o(N 1-2α ) if and only if H N (t) = o(N 1-2α ).
Proof. Assume first that H N (t) = o(N 1-2α ). By the definition of relative entropy,

H N (t) -H N (t) = Ω N g N,t dµ N,t , g N,t := log dν N,t dν N,t .
Direct calculations show that g N,t (η) is equal to

β x∈T N (p N x -p N x )(p x -pN x ) + (τ N x -τ N x )(r x -rN x ) -β x∈T N (p N x -p N x ) 2 2 + G(τ N x ) -G(τ N x ) -rN x (τ N x -τ N x )
for each η = (p x , r x ) x∈T N ∈ Ω N . The second line is bounded from above by O(N 1-4α ). Meanwhile, by the entropy inequality, the integral of the first line is bounded from above by

H N (t) + x∈T N log Ω N exp β pN x -p N x τ N x -τ N x • p x -pN x r x -rN x dν N,t .
Observe that the last term is bounded by O(N 1-4α ) and hence

H N (t) ≤ 2 H N (t) + CN 1-4α = o(N 1-2α
). The inverse assertion follows similarly.

Let ψ N,t be the density function of νN,t with respect to some fixed reference measure ν N p,τ . Without loss of generality, we can choose (p, τ ) = (0, 0). Define the Dirichlet form (cf. (3.1))

D N fN,t ; ν N,t := Ω N x∈T N Y x fN,t 2 dν N,t .

Standard manipulation gives

d dt H N (t) = Ω N fN,t N 1+κ L N log fN,t - d dt log ψ N,t dν N,t .
Recalling the definition of L N in (4.1), we have

fN,t L N log fN,t = L N fN,t - γ N 2 x∈T N f -1 N,t Y x fN,t 2 = L N fN,t -2γ N x∈T N Y x fN,t 2 .
Integrate it with respect to νN,t and notice that

Ω N A N fN,t dν N,t = - Ω N fN,t A N ψ N,t dν N,t ψ N,t = - Ω N fN,t A N log ψ N,t dν N,t , Ω N S N fN,t dν N,t = - 1 2 x∈T N Ω N Y x fN,t • Y x ψ N,t dν N,t ψ N,t ≤ D N fN,t ; νN,t + 1 4 Ω N fN,t x∈T N Y x log ψ N,t 2 dν N,t ,
where the last estimate follows from Cauchy-Schwarz inequality. Hence,

d dt H N (t) ≤ -γ N N 1+κ D fN,t ; νN,t + Ω N fN,t J N,t dν N,t + γ N N 1+κ 4 Ω N fN,t x∈T N Y x log ψ N,t 2 dν N,t , where J N,t := (-N 1+κ A N -d/dt) log ψ N,t .
With the choice (p, τ ) = (0, 0), log ψ N,t = log(dν N,t /ν N 0,0 ) reads

log ψ N,t = β x∈T N pN x p x - (p N x ) 2 2 + τ N x r x -G τ N x + G(0) .
Elementary computation then shows that

x∈T N Y x log ψ N,t 2 = β 2 x∈T N τ N x+1 -τ N x 2 , -A N log ψ N,t = β x∈T N τ N x+1 -τ N x pN x -pN x-1 • p x -pN x V (r x ) -τ N x , - d dt log ψ N,t = -β x∈T N d dt pN x rN x • p x -pN x τ (r N x )(r x -rN x )
.

Therefore, we finally obtain the following inequality, cf. (3.5):

d dt H N (t) ≤ -γ N N 1+κ D fN,t ; νN,t + β R N,t + E N,t , (5.2)
where

R N,t := Ω N fN,t x∈T N dr N x dt V (r x ) -τ N x -τ (r N x )(r x -rN x ) dν N,t ; E N,t := Ω N fN,t x∈T N γ N N 1+κ β 4 ∇τ N x 2 + N x • p x -pN x V (r x ) -τ N x dν N,t .
Here we use the abbreviation Proposition 5.2. There exists a constant C independent of N , such that

∇f x = f x+1 -f x , ∇ * f x = f x-1 -f x
E N,t ≤ H N (t) + C γ N N κ-2α + N -1+2κ-2α + N 1+2κ-6α . Proof. Recall that the profile (p N , rN ) is explicitly given by pN rN (t, u) = p * r * + j=± σ j (s, u j ) N α + σj (s, u -, u + ) N 2α υ j ,
where σ ± = σ ± (s, u), σ± = σ(s, u 1 , u 2 ) are smooth functions given respectively by (1.12) and (5.1), s = N κ-α t, υ -= (-τ (r * ), 1) , υ + = ( τ (r * ), 1) and

u -= u -N κ τ (r * )t, u + = u + N κ τ (r * )t.
For the first term in E N,t , since ∇τ N

x ≤ N -1 sup x |∂ x τ (r(t, x))| ≤ CN -1-α , γ N N 1+κ β 2 4 Ω N fN,t x∈T N ∇τ N x 2 dν N,t ≤ Cγ N N κ-2α .
We focus on the second term in E N,t . By Taylor's expansion,

∇τ N x ∇p N x-1 = 1 N ∂ ∂u τ (r N ) pN t, x N + O N -2-α .
Expanding the function (p, r) → (τ (r), p) at (p * , r * ) up to the second order,

∇τ N x ∇p N x-1 = A j=± ∂ u σ j (s, u j ) N 1+α + (∂ u1 + ∂ u2 )σ j (s, u -, u + ) N 1+2α υ j + ∂ u [(σ -(s, u -) + σ + (s, u + )) 2 ] 2N 1+2α b + O N -2-α + N -1-3α ,
where A = 0 τ (r * ) 1 0

and b = (τ (r * ), 0) . Meanwhile,

d dt pN x rN x = j=± ∂ s σ j (s, u j ) N -κ+2α + j τ (r * )∂ u σ j (s, u j ) N -κ+α υ j + τ (r * ) N -κ+2α j=± (-∂ u1 + ∂ u2 )σ j (s, u -, u + )υ j + O N κ-3α . Noticing that Aυ ± = ± τ (r * )υ ± for j = ±, N x = 1 N -κ+2α -∂ s σ -(s, u -) -2 τ (r * )∂ u2 σ-(s, u -, u + ) υ - + -∂ s σ + (s, u + ) + 2 τ (r * )∂ u1 σ+ (s, u -, u + ) υ + + ∂ u [(σ -(s, u -) + σ + (s, u + )) 2 ] 2 b + O N -1+κ-α + N κ-3α .
(5.3)

We show that the terms in the first bracket vanishes. From (5.1),

∂ u2 σ-(s, u -, u + ) = - τ (r * ) 8τ (r * ) ∂ u σ 2 + (t, u + ) + 2σ -(s, u -)σ + (s, u + ) , ∂ u1 σ+ (s, u -, u + ) = - τ (r * ) 8τ (r * ) ∂ u σ 2 -(t, u + ) + 2σ -(s, u -)σ + (s, u + ) .
Therefore, with the identity τ (r * )(υ --υ + ) = -2 τ (r * )b,

-2 τ (r * )∂ u2 σ-(s, u -, u + )υ -+ 2 τ (r * )∂ u1 σ+ (s, u -, u + )υ + = τ (r * ) 4 τ (r * ) ∂ u σ 2 + (s, u + )υ --σ 2 -(s, u -)υ + -∂ u σ -(s, u -)σ + (s, u + ) b.
The bracket in (5.3) is then equal to

-∂ s σ -(s, u -)υ -- τ (r * ) 4 τ (r * ) υ + - 1 2 b ∂ u σ 2 -(s, u -) -∂ s σ + (s, u + )υ + + τ (r * ) 4 τ (r * ) υ -+ 1 2 b ∂ u σ 2 + (s, u + ).
Observe from (1.12) that it is identically zero, hence N x ≤ C N -1+κ-α + N κ-3α . The conclusion then follows from the relative entropy inequality.

Corollary 5.3 (Harmonic chain). When the oscillators perform harmonic interaction, i.e., V (r) ∝ r 2 , R N,t is identically zero. Hence, Theorems 4.1 and 4.2 hold autonomously. First, one can replace φ x by its block average. The error is bounded by Since κ ≤ α < 1/2, N -1+2κ-2α + N 1+2κ-6α = o(N 1-2α ), so that

R N,
d dt H N (t) ≤ C H N (t) + C γ N N κ-1 + γ -1 N N -1+5κ+4α 1 5 + o(1) N 1-2α .
The estimate then follows from the choice of γ N and Grönwall's inequality.

Theorem 4.2 follows from the exactly same argument as we used in the proof of Theorem 2.3, hence we omit the proof here.

Theorem 4 .Theorem 4 . 2 .

 442 1 is proved in Section 5. With Theorem 4.1 and exactly the same argument used in the proof of Theorem 2.3, we obtain the equilibrium perturbation. Under the assumption of Theorem 4.1,

5 . 1 .

 51 Proofs of Theorem 4.1 and 4.2. Theorem 4.1 follows from the inequality (5.2), Propositions 5.2 and 5.4 below.

Proposition 5 . 4 . 1 - 1 y=0φ

 5411 There exists a constant C independent of N , such that R N,t ≤ γ N N 1+κ β D N fN,t ; νN,t + C H N (t) + γ We follow the proof of [24, Lemma 3.1]. To shorten the notation, leta N x := ∂ t rN t, x N , φ x := V (r x ) -τ (r N x ) -τ (r N x ) r x -rN x .Observe thata N x ≤ N -α j=± jN κ ∂ u σ j |υ j | + O N κ-2α ≤ CN κ-α , a N x -a N y ≤ C|x -y|N -1+κ-α .Fix a mesoscopic scale = (N ) N and define φ x := x+y , φ x := E νN,t φ x -1 y=0 r x+y .

2 3 ) 5 N N 1+α 5 and

 355 and δ sufficiently small but fixed, logΩ N exp δ(N α-κ a N x ) φ x dν N,t ≤ C. Since N κ-α ≤ O(1), N,t ≤ C H N (t) + -1 N 1+κ-α . (5.6)Finally, the proof is concluded by choosing (N ) = γ 1 adding up the estimates (5.4), (5.5) and(5.6).Proof of Theorem 4.1. By (5.2), Proposition 5.2 and 5.4,

  d dt H N (t) ≤ C H N (t) + γ N N κ-2α + N -1+2κ-2α + N 1+2κ-6α + γ

  t -Then, for each x ∈ T N , Summing up for x ∈ T N and applying Cauchy-Schwarz inequality, The first term in the righthand side gives β -1 γ N N 1+κ D N ( fN,t ; νN,t ). To estimate the second term, note that since V is bounded, we can apply the gradient estimate for Poisson equation (see[START_REF] Xu | Hyperbolic scaling limit of non-equilibrium fluctuations for a weakly anharmonic chain[END_REF] Proposition 9.1], also cf.[START_REF] Wu | Gradient estimates of poisson equations on Riemannian manifolds and applications[END_REF] Theorem 1.1]) to obtain a constant C independent of x or , such that Since φx is a function of r x + . . . + r x+ -1 , Y x+y φ x ≡ 0. Also,Y x+y φ x = 1 V (r x+y+1 ) -V (r x+y ) -τ (r N x+y+1 ) + τ (r N x+y ) ≤ C .for any δ > 0. Observe that the extra factor in the last term above is because that φ x is independent of φ y for any |x -y| ≥ , see, e.g.,[START_REF] Jara | Non-equilibrium fluctuations for a reaction-diffusion model via relative entropy[END_REF] Lemma F.12] and[START_REF] Xu | Hyperbolic scaling limit of non-equilibrium fluctuations for a weakly anharmonic chain[END_REF] Lemma D.3]. Recall that N α-κ a Nx is bounded. To treat the exponential moment in above, we apply the equivalence of inhomogeneous ensembles[START_REF] Xu | Hyperbolic scaling limit of non-equilibrium fluctuations for a weakly anharmonic chain[END_REF] Proposition 8.3] to obtain that for = o(N

	Direct computation then shows that
											-2
							(a N x ) 2		fN,t Y x+y ψ x	2 dν N,t ≤ C	(a N x ) 2 3 ≤ C 3 N 1+2κ-2α .
			Ω N x∈T N					y=0				x∈T N
	Therefore, we obtain the estimate
			Ω N	fN,t	x∈T N		a N x φ x -φ	x dν N,t ≤	γ N N 1+κ β	D N fN,t ; νN,t + C β	4 N κ-2α γ N	.	(5.5)
	For the space variance of φ	x , relative entropy inequality reads
			Ω N	fN,t	x∈T N	a N x	φ	x dν N,t ≤	N κ-α δ	H N (t) +	1	x	log	Ω N	e δN α-κ a N x	φ	x dν N,t ,
								fN,t	a N x φ x dν N,t =	fN,t	φ x	-1	a N x -a N x-y dν N,t
							Ω N				x∈T N				Ω N	x∈T N	y=0
													-2
					y=0 x dν N,t = Y Ω N fN,t a N x φ x -φ	Ω N	a N x	-2
	Ω N	fN,t	x∈T N	a N x φ x -φ	x dν N,t ≤	γ N N 1+κ β( -1) Ω N x∈T N	-2 y=0	(Y x+y fN,t ) 2 4 fN,t	dν N,t
																+	β( -1) γ N N 1+κ	Ω N x∈T N	(a N x ) 2	-2 y=0	fN,t Y x+y ψ x	2 dν N,t .
							-2
										Y x+y ψ x	2 ≤ C 4	sup
							y=0

≤ H N (t) + C 2 N -1+2κ-2α . (5.4) Next, let ψ x = ψ x (r x , . . . , r x+ -1 ) solve the Poisson equation * t,x+y Y x+y ψ x = φ x -φ x ,

where for each x, Y * t,x is the adjoint operator of Y x with respect to νN,t :

Y * t,x := β V (r x+1 ) -V (r x ) -τ N x+1 + τ N x -Y x .

y=0 Y x+y fN,t Y x+y ψ x dν N,t . (rx,...,r x+ -1 ) -2 y=0 Y x+y [φ x -φ x ] 2 .

Acknowledgments: This project is supported by the ANR grant MICMOV (ANR-19-CE40-0012) of the French National Research Agency (ANR).

Appendix A. General tools

In this appendix, we state some model independent tools that is used through the paper.

A.1. Relative entropy inequality. In this subsection, we introduce a version of Yau's relative entropy inequality. Let {X t } t≥0 be a continuous-time Markov chain on a finite state space S, whose infinitesimal generator is defined as

Above, r(x, y) ≥ 0 is the rate at which the chain jumps from x to y, and f : S → R is any function. Define the carré du champ operator associated to L as

Denote by µ t the distribution of the process at time t with initial measure µ 0 . Let {ν t } t≥0 and ν be a family of measures in S such that ν t is differentiable in time t, and ν t (x) > 0, ν(x) > 0 for any x ∈ S and any t ≥ 0. Denote by f t (respectively ψ t ) the Radon-Nikodym derivative of µ t (respectively ν t ) with respect to ν t (respectively ν),

Define the relative entropy H(t) as

with the convention 0 log 0 = 0.

Lemma A.1 (Yau's relative entropy inequality). For any t ≥ 0,

where 1 is the constant function identical to one, and L * t is the adjoint of L with respect to L 2 (ν t ), which acting on any function g : S → R is given by

We refer the readers to [10, Lemma A.1] for proof of the above lemma. Compared to the classical Yau's relative entropy inequality (cf. [START_REF] Kipnis | Scaling limits of interacting particle systems[END_REF]Lemma 6.1.4]), an extra term Γ √ f t dν t is subtracted in the above version.

A.2. Flow lemma. In this subsection, we state a flow lemma introduced by Jara and Menezes [START_REF] Jara | Non-equilibrium fluctuations of interacting particle systems[END_REF][START_REF] Jara | Non-equilibrium fluctuations for a reaction-diffusion model via relative entropy[END_REF]. For two measures p and q on Z d , we say a function φ :

The support of the flow φ is defined as the set of points {z, z + e i } such that φ(z, z + e i ) = 0. Using the summation by parts formula, for any function f : Z d → R and any flow φ connects p to q,

For ≥ 1, let p (•) be the uniform measure on Λ d = {0, 1, . . . , -1} d , i.e. p (x) = -d if x ∈ Λ d and = 0 otherwise. Let q = p * p be the convolution of p with itself,

In the sequel, we shall use q to define the spatial average of a random variable over a large box instead of the usual p . Note that the support of q is contained in Λ d 2 -1 . Let δ 0 (•) be the Dirac measure concentrated at the origin. For ≥ 1, denote

The following lemma states that we could construct a flow, which connects δ 0 to q , such that the cost is at most of order g d ( ). We refer the readers to [START_REF] Jara | Non-equilibrium fluctuations for a reaction-diffusion model via relative entropy[END_REF][START_REF] Jara | Non-equilibrium fluctuations of interacting particle systems[END_REF] for its proof.

Lemma A.2 (Flow lemma).

There exists a finite constant C 0 such that for any ≥ 1, there exists a flow φ connecting δ 0 to q with support in

A.3. Concentration inequalities. In this subsection, we focus on properties of sub-Gaussian random variables. We say a real-valued random variable X is sub-Gaussian of order

The following lemma controls the expectation of the exponential of X 2 , and an elementary proof could be found in [START_REF] Jara | Non-equilibrium fluctuations of interacting particle systems[END_REF]Proposition F.7]. Below we shall present a different proof.

] ≤ 3.

The constant 3 above is not optimal and we only need it to be a constant. By Cauchy-Schwarz inequality, it is easy to have the following result.

Corollary A.4. Let X i be sub-Gaussian of order σ 2 i for i = 1, 2. Then, for any γ ≤ (4σ

Proof of Lemma A.3. Let N be a standard normal distribution independent of X, and denote by • the expectation with respect to N . Then, for any θ ∈ R,

Therefore,

Since X is sub-Gaussian of order σ 2 , the last formula is bounded by

If γ ≤ (4σ 2 ) -1 , it is easy to see the last formula is bounded by 3. This concludes the proof.

The following lemma states that centered bounded random variables are sub-Gaussian. Inria Lille-Nord Europe, Lille 59000, France.

Email address: lu.xu@inria.fr Inria Lille-Nord Europe, Lille 59000, France.

Email address: linjie.zhao@inria.fr


Patrick Tounian
email: p.tounian@aphp.fr

préparations infantiles est une erreur ? Why was it a mistake not to make the addition of arachidonic

acid to infant formula mandatory ?

2

Un apport adéquat en acides gras essentiels (acides linoléique (LA) et α-linolénique (ALA)) et polyinsaturés à longue chaine (AGPILC) qui en sont dérivés (acides arachidonique (ARA), eicosapentaénoïque (EPA) et docosahexaénoïque (DHA)) est capital pour assurer le développement, notamment neurologique, des nourrissons. Le lait de mère contient les acides gras essentiels, mais aussi de l'ARA et du DHA avec un rapport ARA/DHA supérieur à 1. En revanche, actuellement en France et dans beaucoup de pays européens, toutes les préparations infantiles sont enrichies en LA et ALA, mais la majorité d'entre elles ne contiennent ni ARA, ni DHA [START_REF] Tounian | Alimentation de l'enfant de 0 à 3 ans[END_REF].

L'agence européenne de sécurité des aliments (EFSA) recommande pourtant un apport minimum de 100 mg/j de DHA de 0 à 24 mois et de 140 mg/j d'ARA de 0 à 6 mois depuis 2013 [START_REF] Efsa Panel On Dietetic Products | Scientific Opinion on nutrient requirements and dietary intakes of infants and young children in the European Union[END_REF]. Compte tenu de ces éléments, la délégation de la Commission Européenne qui régit la composition des préparations infantiles pour nourrissons (1 er âge) et de suite (2 e âge) a imposé un ajout obligatoire de DHA dans les préparations infantiles fabriquées à partir du 22 février 2020, mais elle a étonnamment rendu facultative l'addition d'ARA [START_REF]and information requirements for infant formula and follow-on formula and as regards requirements on information relating to infant and young child feeding[END_REF].

Comme de nombreux experts en nutrition pédiatrique [START_REF] Koletzko | Should formula for infants provide arachidonic acid along with DHA? A position paper of the European Academy of Paediatrics and the Child Health Foundation[END_REF], nous considérons que l'ajout simultané d'ARA et de DHA dans les préparations infantiles est indispensable pour assurer un développement optimal des nourrissons. L'objet de cet article est de fournir les arguments qui étayent cette position.

Synthèse et fonctions de l'ARA et du DHA

C'est à partir des deux acides gras essentiels, qui proviennent obligatoirement de l'alimentation car les êtres humains ne sont pas capables de les produire, que sont synthétisés l'ARA et le DHA. Le LA est converti en ARA (série oméga 6) et l'ALA en EPA et DHA (série oméga 3) (Figure 1). Le taux de conversion de l'ALA en EPA est de 8 à 12% alors qu'il n'est que d'environ 1% pour le DHA [START_REF] Efsa Panel On Dietetic Products | Scientific opinion on dietary reference values for fats, including saturated fatty acids, polyunsaturated fatty acids, monounsaturated fatty acids, trans fatty acids, and cholesterol[END_REF]. Ce processus consiste à allonger la chaine carbonée par les élongases et ajouter des doubles liaisons par les désaturases. Il est important de noter que les enzymes permettant la conversion des acides gras essentiels en AGPILC sont les mêmes pour les séries oméga 3 et 6, les substrats entrent donc en compétition selon leur concentrations respectives.

Le DHA est principalement présent dans le cerveau, où il participe surtout au fonctionnement du cortex préfrontal impliqué notamment dans la mémoire, et au niveau des photorécepteurs de la rétine [START_REF] Hadley | The essentiality of arachidonic acid in infant development[END_REF]. Le contenu en EPA du cortex et de la rétine est en revanche 100 fois inférieur à celui en DHA [START_REF] Calder | Docosahexaenoic acid[END_REF]. L'ARA est plus ubiquitaire, il est aussi présent en grande quantité dans le cerveau, mais on le retrouve également dans le coeur, les muscles, l'endothélium vasculaire, les lymphocytes T, les surrénales, les reins, le foie et le placenta [START_REF] Hadley | The essentiality of arachidonic acid in infant development[END_REF]. L'ARA est aussi le précurseur des éicosanoïdes qui interviennent en particulier dans le développement de la tolérance immunitaire chez le nourrisson [START_REF] Hadley | The essentiality of arachidonic acid in infant development[END_REF]. groupes [START_REF] Makrides | Fatty acid composition of brain, retina, and erythrocytes in breast-and formula-fed infants[END_REF]. On notera que le cerveau a une activité spécifique de désaturation et d'élongation de LA en ARA qui explique probablement ces résultats [START_REF] Hadley | The essentiality of arachidonic acid in infant development[END_REF].

L'ajout d'ARA dans les préparations infantiles n'améliore pas la fonction visuelle. Les nourrissons alimentés avec une formule enrichie en DHA pendant les 4 premiers mois de vie ont une fonction visuelle à l'âge de 4 ans identique à celle des enfants allaités et supérieure à celle des nourrissons recevant une formule non enrichie [START_REF] Birch | Visual acuity and cognitive outcomes at 4 years of age in a double-blind, randomized trial of long-chain polyunsaturated fatty acid-supplemented infant formula[END_REF]. Dans cette même étude, l'utilisation d'un lait enrichi en ARA et en DHA dans un rapport proche de celui du lait de mère n'apporte pas de bénéfice par rapport à la formule enrichie uniquement en DHA [START_REF] Birch | Visual acuity and cognitive outcomes at 4 years of age in a double-blind, randomized trial of long-chain polyunsaturated fatty acid-supplemented infant formula[END_REF].

Pourquoi l'ajout d'ARA dans les préparations infantiles est-elle nécessaire ?

Les arguments soutenant la nécessité d'ajouter de l'ARA en plus du DHA dans les préparations infantiles sont bien plus nombreux et solides que ceux qui justifient la seule addition de DHA.

Le lait de mère contient de l'ARA et du DHA. L'objectif des industriels de l'alimentation infantile est de fabriquer des formules dont la composition s'approche le plus possible de celle du lait de mère. La concentration d'ARA dans le lait maternel est presque toujours supérieure à celle de DHA avec un rapport qui peut aller jusqu'à 2/1 [START_REF] Fu | An updated review of worldwide levels of docosahexaenoic and arachidonic acid in human breast milk by region[END_REF]. Le contenu en ARA du lait maternel ne dépend pas de l'alimentation de la mère et est donc assez constant dans le temps, alors que la concentration en DHA est variable car elle se modifie selon les apports en produits de la mer, principale source de DHA [START_REF] Fu | An updated review of worldwide levels of docosahexaenoic and arachidonic acid in human breast milk by region[END_REF]. Depuis de longues années, les industriels rivalisent entre eux pour enrichir leurs formules avec des molécules présentes dans le lait de mère (oligosaccharides, lactoferrine), avec des effets bénéfiques qui demandent encore à être confirmés [START_REF] Tounian | Enrichissement des formules infantiles : réels effets bénéfiques ou simple marketing ?[END_REF]. Dans ce contexte, ne pas ajouter d'ARA serait incompréhensible. Dans la mesure où le lait de mère est le gold standard pour ajuster la composition des formules infantiles, l'argumentation en faveur de l'ajout obligatoire d'ARA pourrait s'arrêter là.

La synthèse d'ARA est insuffisante pour assurer ses fonctions autres que neurologiques. Si, comme nous l'avons vu précédemment, la synthèse locale d'ARA permet d'assurer les besoins cérébraux indépendamment des apports alimentaires en ARA, les nombreuses autres fonctions biologiques assurées par l'ARA sont altérées en cas d'apports exogènes insuffisants.

En effet, le contenu érythrocytaire en ARA décroit de 40% au cours des premiers jours de vie chez les nouveau-nés alimentés avec un lait non enrichi en ARA [START_REF] Hadley | The essentiality of arachidonic acid in infant development[END_REF] et seules les formules enrichies en ARA permettent de maintenir des taux circulants d'ARA identiques à ceux des enfants allaités [START_REF] Richard | DHA and ARA addition to infant formula: current status and future research directions[END_REF]. Le contenu érythrocytaire en ARA et les taux circulants d'ARA sont des bons marqueurs du statut extra-cérébral en ARA [START_REF] Hadley | The essentiality of arachidonic acid in infant development[END_REF]. De plus, seuls les nourrissons alimentés avec un lait enrichi en ARA ont des marqueurs plasmatiques des fonctions immunes proches de ceux de leurs congénères allaités, alors que ce n'est pas le cas lorsque la formule qu'ils reçoivent ne contient pas d'ARA [START_REF] Calder | Is there a role for fatty acids in early life programming of the immune system?[END_REF]. Cela pourrait expliquer la moindre incidence des infections chez les enfants ayant été alimentés avec un lait enrichi en ARA et DHA, comparés à ceux qui buvaient une formule non enrichie [START_REF] Lapillonne | Infants fed formula with added long chain polyunsaturated fatty acids have reduced incidence of respiratory illnesses and diarrhea during the first year of life[END_REF]. Donc, si le contenu cérébral en ARA [START_REF] Makrides | Fatty acid composition of brain, retina, and erythrocytes in breast-and formula-fed infants[END_REF] et le développement de la fonction visuelle [START_REF] Birch | Visual acuity and cognitive outcomes at 4 years of age in a double-blind, randomized trial of long-chain polyunsaturated fatty acid-supplemented infant formula[END_REF] semblent ne pas dépendre des apports alimentaires d'ARA grâce à une synthèse locale suffisante [START_REF] Hadley | The essentiality of arachidonic acid in infant development[END_REF], les fonctions biologiques périphériques auxquelles l'ARA participe nécessitent un apport alimentaire pour être correctement assurées.

Des variants génétiques fréquents des désaturases peuvent diminuer la synthèse d'ARA. Les Δ5-et Δ6-désaturases sont des enzymes-clés dans la synthèse des AGPILC (Figure 1). Trente pourcent de la population générale possèdent des variants génétiques des gènes codant pour ces enzymes qui rendent ces dernières moins efficaces, réduisant ainsi la synthèse d'ARA à partir de LA [START_REF] Lattka | Genetic Variations in Polyunsaturated Fatty Acid Metabolism -Implications for Child Health?[END_REF]. En comparant le quotient intellectuel à l'âge de 8 ans d'enfants ayant été allaités ou nourris avec une formule non enrichie en AGPILC, Steer et al. ont montré que la différence de quotient intellectuel en faveur des enfants allaités était bien plus importante avec les homozygotes pour l'un de ces variants génétiques qu'elle ne l'était avec ceux qui en étaient dépourvus, témoignant d'une synthèse insuffisante d'ARA et de DHA chez les homozygotes [START_REF] Steer | FADS2 polymorphisms modify the effect of breastfeeding on child IQ[END_REF]. Un autre travail a révélé que la composition membranaire en AGPILC des nourrissons alimentés avec un lait enrichi en ARA et en DHA à des concentrations moindres que celles du lait de mère, était comparable à celle des enfants allaités uniquement chez ceux qui n'étaient pas porteurs de ces variants génétiques [START_REF] Lorenzo | The effect of an infant formula supplemented with aa and dha on fatty acid levels of infants with different FADS genotypes: The COGNIS study[END_REF], suggérant ainsi une synthèse insuffisante d'ARA et de DHA chez les porteurs de ces variants. Donc, ne pas mettre d'ARA dans les formules infantiles risque de compromettre le statut en ARA et des fonctions biologiques qu'il assure chez les nombreux nourrissons possédant les variants génétiques fréquents des gènes des désaturases qui en réduisent l'activité.

Le rapport ARA/DHA dans les formules infantiles influence le développement neurologique.

Les proportions respectives d'ARA et de DHA dans les formules infantiles doivent également être prises en considération. En effet, l'étude DIAMOND a comparé différents paramètres de la fonction intellectuelle à l'âge de 5-6 ans entre 3 groupes d'enfants ayant été nourris au cours de leurs 4 premiers mois de vie par 3 formules à contenu croissant en DHA (0,32, 0,64 et 0,96% des acides gras totaux) alors que la concentration d'ARA était fixe (0,64% des acides gras totaux). Tous les paramètres intellectuels testés sauf un étaient significativement moins bons dans le groupe avec 0,96% de DHA comparé aux deux autres [START_REF] Colombo | Long-term effects of LCPUFA supplementation on childhood cognitive outcomes[END_REF]. Ces résultats suggèrent que le rapport ARA/DHA dans les formules infantiles doit être égal ou supérieur à 1 pour assurer une utilisation optimale de ces AGPILC par le cerveau.

Conclusion

  Pourquoi l'ajout d'ARA dans les préparations infantiles n'a pas été imposé ? L'EFSA s'est probablement basée sur deux arguments pour ne pas rendre obligatoire l'ajout d'ARA dans les préparations infantiles : la synthèse suffisante d'ARA par le cerveau et l'absence d'effet sur la fonction visuelle des préparations infantiles enrichies en ARA. Le DHA contenu dans le cerveau provient principalement de l'alimentation alors que l'ARA cérébral résulte avant tout d'une synthèse locale. En effet, une étude portant sur des nourrissons décédés de mort subite a montré que le contenu cortical en DHA était plus important chez les enfants allaités que chez ceux alimentés avec une formule non enrichie en DHA et en ARA, alors que l'accumulation d'ARA dans le cerveau était identique dans les 2

LFigure 1 .

 1 Figure 1. Synthèse des acides gras polyinsaturés à longue chaine à partir des acides gras

risque de compromettre le développement neurologique et immunologique d'un enfant en n'ajoutant pas d'ARA dans les formules infantiles serait probablement imprudent. Nous espérons qu'à l'écoute des experts en nutrition pédiatrique, la Commission Européenne modifiera la réglementation dans ce sens.

Points essentiels • L'ajout d'acide docosahexaénoïque (DHA) dans les formules infantiles est obligatoire • L'ajout d'acide arachidonique (ARA) dans les formules infantiles est facultatif • Un enrichissement des formules infantiles en DHA sans ARA est délétère • Seules les formules contenant du DHA et de l'ARA devront être prescrites

Abstract

Arachidonic (ARA) and docosahexaenoic (DHA) acids are polyunsaturated fatty acids (PUFA) necessary for optimal infant neurological development, and immunologic one also for ARA. European Commission mandated DHA addition to infant formulas elaborated from February 2020, while ARA addition remained optional. The aim of our paper is to show that both PUFA should absolutely be added in infant formulas. Arguments supporting this position are as follows. 1) The ARA concentration in breast milk is higher than DHA one. 2) Without any diet intake, ARA synthesis may not be sufficient in regard to its wide-range biological non-neurologic functions. 3) Variants in genes coding for desaturases present in 30% of individuals alter LCPUFA status and compromise infant health status, in the absence of ARA intake. 4) A ARA/DHA ratio lower than 1 in formulas alters neurologic development. For all these reasons, ARA and not only DHA should be added to infant formulas. Key words : arachidonic acid, docosahexaenoic acid, neurologic development, infant formula.