Tomasz Komorowski

Stefano Olla
email: olla@ceremade.dauphine.fr

Ceremade

BALLISTIC AND SUPERDIFFUSIVE SCALES IN MACROSCOPIC EVOLUTION OF A CHAIN OF OSCILLATORS

We consider a one dimensional infinite acoustic chain of harmonic oscillators whose dynamics is perturbed by a random exchange of velocities, such that the energy and momentum of the chain are conserved. Consequently, the evolution of the system has only three conserved quantities: volume, momentum and energy. We show the existence of two space-time scales on which the energy of the system evolves. On the hyperbolic scale (tǫ -1 , xǫ -1 ) the limits of the conserved quantities satisfy a Euler system of equations, while the thermal part of the energy macroscopic profile remains stationary. Thermal energy starts evolving at a longer time scale, corresponding to the superdiffusive scaling (tǫ -3/2 , xǫ -1 ) and follows a fractional heat equation. We also prove the diffusive scaling limit of the Riemann invariants -the so called normal modes, corresponding to the linear hyperbolic propagation.

Introduction

Consider a chain of coupled anharmonic oscillators in one dimension. Denote by q x and p x the position and momentum of the particle labeled by x ∈ Z. The interaction between particles x and x-1 is described by the potential energy V (r x ) of an anharmonic spring, where the quantity

r x := q x -q x-1 (1.1)
is called the inter-particle distance or volume strain. The (formal) Hamiltonian of the chain is given by where the energy of the oscillator x is defined by e x (q, p) := p 2

H(q, p) = x e x (q, p), (1.2)
x 2 + V (r x ).

(1.

3)

The respective Hamiltonian dynamics is given by the solution of the equations:

qx (t) = ∂H ∂p x (q, p) = p x , (1.4)
ṗx (t) = -∂H ∂q x (q, p), x ∈ Z.

There are three formally conserved quantities (also called balanced) for this dynamics The length and internal energy in equilibrium can be expressed as functions of λ by

r(τ, β) = r x ν λ = β -1 ∂ τ G(λ), p = β -1 ∂ p G(λ) (1.9) u(τ, β) = e x ν λ - p 2 2 = 1 2β + V (r x ) ν λ = -∂ β G(λ) + τ r.
Therefore the tension τ (r, u) and inverse temperature β(r, u) can be determined from the relations:

β = ∂ u S(u, r), τ β = -∂ r S(u, r), (1.10)
where S(u, r) is the thermodynamic entropy defined by the Legendre transform S(u, r) = inf τ,β

[-βτ r + βu + G(β, 0, τ )] .

(1.11)

It is expected that, after hyperbolic rescaling of space and time (tǫ -1 , xǫ -1 ), the empirical distribution of the balanced quantities w T x (t) := [r x (t), p x (t), e x (t)],

(1.12) defined for a smooth function J with compact support by the formula:

ǫ x J(ǫx)w x t ǫ
converges, as ǫ → 0, to the solution w T (t, y) = [r(t, y), p(t, y), e(t, y)] of the compressible Euler system of equations:

∂ t r = ∂ y p, ∂ t p = ∂ y τ (r, u), ∂ t e = ∂ y [pτ (r, u)] , (1.13)
where u(t, y) := e(t, y) -p 2 (t, y)/2 is the local internal energy. For a finite macroscopic volume, this limit has been proven using the relative entropy method, provided the microscopic dynamics is perturbed by a random exchange of velocities between particles, see [START_REF] Olla | Hydrodynamic Limit for a Hamiltonian System with Weak Noise[END_REF] and [START_REF] Braxmeier-Even | Hydrodynamic limit for a Hamiltonian system with boundary conditions and conservative noise[END_REF], in the regime when the system of Euler equations admits a smooth solution.

After a sufficiently long time, the solution of (1.13) should converge, in an appropriately weak sense, to some mechanical equilibrium described by: p(x) = p 0 , τ (r(x), u(x)) = τ 0 , (1.14) where p 0 and τ 0 are some constants.

To characterize all possible stationary solutions of (1.14) can possibly be a daunting task. Most likely they are generically very irregular. But it is quite obvious that if we start with smooth initial conditions that satisfy (1.14), the respective solutions of (1.13) remain stationary. Also by the same relative entropy method as used in [START_REF] Olla | Hydrodynamic Limit for a Hamiltonian System with Weak Noise[END_REF] and [START_REF] Braxmeier-Even | Hydrodynamic limit for a Hamiltonian system with boundary conditions and conservative noise[END_REF], it follows that starting with such initial profiles, the corresponding empirical distributions of the balanced quantities converge, in the hyperbolic time scale, to the same initial stationary solution, i.e. they do not evolve in time. On the other hand, we do know that the system will eventually converge to a global equilibrium, so this implies that there exists a longer time scale, on which these profiles (stationary at the hyperbolic scale) will evolve.

There is a strong argument, stemming from both a numerical evidence and quite convincing heuristics, suggesting the divergence of the Green-Kubo formula, defining the thermal diffusivity, for a generic unpinned one dimensional system, see [START_REF] Lepri | Thermal Conduction in classical low-dimensional lattices[END_REF] [START_REF] Lepri | Heat conduction in chains of nonlinear oscillators[END_REF]. Therefore, we expect that the aforementioned larger time scale (at which the evolution of these profiles is observed) is superdiffusive. Furthermore, an argument by H. Spohn [START_REF] Spohn | Nonlinear fluctuating hydrodynamics for anharmonic chains[END_REF], based on fluctuating hydrodynamics and mode coupling, also suggests a superdiffusive evolution of the heat mode fluctuation field, when the system is in equilibrium. Consequently, one can conjecture the following scenario: after the space-time rescaling (ǫ -α t, ǫ -1 x), the temperature T (t, y) = β -1 (t, y) evolves following some fractional (possibly non-linear) heat equation. The choice of the exponent α may depend on the particular values of the tension and of the interaction potential V (r).

In the present work we prove rigorously that this picture of two time scale evolution holds for the harmonic chain with a momentum exchange noise that conserves the total momentum and energy. We consider the quadratic Hamiltonian:

H(q, p) = x 1 2 p 2 x - 1 4
x,x ′ α x-x ′ (q xq x ′ ) 2 (1.15)

where harmonic coupling coefficients α x-x ′ , between atoms labelled by x and x ′ , are assumed to decay sufficiently fast. The detailed hypotheses made about the potential are contained in Section 2.1. Furthermore we perturb the Hamiltonian dynamics, given by (1.4), by a random exchange of momentum between the nearest neighbor atoms, in such a way that the total energy and momentum of the chain are conserved, see formula (2.7) below. We assume that the initial configuration is distributed according to a probability measure µ ǫ such that its total energy grows like ǫ -1 (i.e. macroscopically, it is of order 1). Additionally, we suppose that the family of measures (µ ǫ ) ǫ>0 possesses a macroscopic profile w T (y) = [r(y), p(y), e(y)], with r, p, e belonging to C ∞ 0 (R) -the space of all smooth and compactly supported functions. The above means that

lim ǫ→0+ ǫ x w x µǫ J(ǫx) = R w T (y)J(y)dy, (1.16)
for any test function J ∈ C ∞ 0 (R). We decompose the initial configuration into a phononic (low frequencies) and thermal (high frequences) terms, i.e. the initial configuration (r, p) is assumed to be of the form

r x = r ′ x + r ′′ x and p x = p ′ x + p ′′ x , x ∈ Z, (1.17)
where the variance of (r ′′ x , p ′′ x ) around the macroscopic profile (r(ǫx), p(ǫx)) x∈Z vanishes, with ǫ → 0+ (see (2.28) and (2.29) below). This obviously implies that the macroscopic profiles corresponding to (r ′

x , p ′ x ) equal to 0. Concerning the distribution of this configuration we suppose that its energy spectrum, defined in (2.23) below, is square integrable in the sense of condition (2.26). We call the respective energy profiles corresponding to (r ′

x , p ′ x ) and (r ′′ x , p ′′ x ) as the thermal and phononic (or mechanical) ones, see Definitions 2.2 and 2.3 below. In Section 9 we give several examples of families of initial configurations satisfying the above hypotheses. Among them are inhomogeneous Gibbs measures with the temperature, momentum and tension profiles changing on the macroscopic scale, see Section 9.2.3. These examples include also the families of measures corresponding to the local equilibrium states for the dynamics described by (1.2), see Section 9.2.5.

In our first result, see Theorem 3.1 and Corollary 3.2 below, we show that the empirical distribution of w x (t) converges, at the hyperbolic time-space scale, to a solution satisfy the linear version of (1.13) with τ (r, u) := τ 1 r, where the parameter τ 1 , called the speed of sound, is given by formula (2.5).

Notice that since we are working in infinite volume, relative entropy methods cannot be applied to obtain such hydrodynamic limit, as done in [START_REF] Braxmeier-Even | Hydrodynamic limit for a Hamiltonian system with boundary conditions and conservative noise[END_REF] and [START_REF] Olla | Hydrodynamic Limit for a Hamiltonian System with Weak Noise[END_REF]. Instead, we obtain the limit result, using only the techniques based on the L 2 bounds on the initial configurations. After a direct computation it becomes clear that the function T (y) := e(t, y) -e ph (t, y), (1.18) is stationary in time. The macroscopic phononic energy is given by

e ph (t, y) := p 2 (t, y) 2 + τ 1 r 2 (t, y), (1.19)
Also, the entropy in this case equals

S(u, r) = C log(u -τ 1 r 2 )
for some constant C > 0 depending on coefficients (α x ). In fact, as it turns out, see Theorem 3.1 below, T (y) and e ph (t, y), given by (1.18) and (1.19) above, are the respective limits of the thermal and phononic energy profiles.

Concerning the behavior of the thermal energy profile at the longer, superdiffusive time scale it has been shown in [START_REF] Jara | Superdiffusion of energy in a chain of harmonic oscillators with noise[END_REF] that after the spacetime rescaling (ǫ -3/2 t, ǫ -1 x), the thermal energy profile converges to the solution T (t, y) of a fractional heat equation

∂ t T (t, y) = -ĉ|∆ y | 3/4 T (t, y), with T (0, y) = T (y) (1.20)
and the coefficient ĉ given by formula (5.50) below. This result is generalized in the present paper to configurations of the form (1.17). Finally, in Theorem 3.4 we consider the empirical distributions of the microscopic estimators of the Riemann invariants of the linear wave equation system that describe the evolution of the macroscopic profiles of (r x (t), p x (t)) -the so called normal modes. We show that they evolve at the diffusive space-time scale (tǫ -2 , yǫ -1 ). A similar behavior is conjectured for some an-harmonic chains, e.g. those corresponding to the β-FPU potential at zero tension (cf. [START_REF] Spohn | Nonlinear fluctuating hydrodynamics for anharmonic chains[END_REF]).

Concerning the organization of the paper in Section 2 we rigorously introduce the basic notions that appear throughout the article. The main results are formulated in Section 3. In Sections 4 -8 we present their respective proofs. Finally in Section 9 we show examples of distributions of the initial data that are both of phononic and thermal types.

The stochastic dynamics

2.1. Hamiltonian dynamics with a noise. Concerning the interactions appearing in (1.15) we consider only the unpinned case, therefore we let

α 0 := - x =0 α x .
(2.1)

Define the Fourier transform of (α x ) x∈Z by α(k) :

= x α x exp {-2πikx} , k ∈ T, (2.2)
where T is the unit torus, that is the interval [-1/2, 1/2] with the identified endpoints. Then, from (2.1), we have α(0) = 0. Furthermore, we assume that: a1) coefficients (α x ) x∈Z are real valued, symmetric and there exists a constant C > 0 such that

|α x | ≤ Ce -|x|/C for all x ∈ Z. a2) stability: α(k) > 0, k = 0, a3) the chain is acoustic : α′′ (0) > 0,
The above conditions imply that α(k) = α(-k), k ∈ T. In addition, α ∈ C ∞ (T) and it can be written as

α(k) = 4τ 1 s 2 (k)ϕ 2 (s 2 (k)).
(2.

3)

The function ϕ : [0, +∞) → (0, +∞) is smooth and ϕ(0) = 1. For the abbreviation sake, we shall use the notation s(k) := sin(πk) and c(k) := cos(πk), k ∈ T.

(2.4)

The parameter τ 1 appearing in (2.3), called the sound speed, is defined by

τ 1 := α′′ (0) 8π 2 . (2.5)
Let also ϕ -:= inf ϕ and ϕ + := sup ϕ.

(2.6)

Since the Hamiltonian is invariant under the translations of the positions q x of the atoms, the latter are not well defined, and the configuration space of our system is given by ((r x , p x )) x∈Z , where r x should be thought as the distance between particle x and x -1. The evolution is given by a system of stochastic differential equations

dr x (t) = ∇ * p x (t) dt, (2.7
)

dp x (t) = -α * q x (t) - γ 2 (β * p(t)) x dt +γ 1/2 z=-1,0,1 (Y x+z p x (t))dw x+z (t), x ∈ Z.
with the parameter γ > 0 that determines the strength of the noise in the system. The vector fields (Y x ) x∈Z are given by

Y x := (p x -p x+1 )∂ p x-1 + (p x+1 -p x-1 )∂ px + (p x-1 -p x )∂ p x+1 . (2.8)
Here (w x (t), t ≥ 0) x∈Z are i.i.d. one dimensional, real valued, standard Brownian motions, that are non-anticipative over some filtered probability space (Ω, F , (F t ) , P). The symbol * denotes the convolution, but in particular, since α(0) = 0, we can make sense of (remember that q x are not really defined)

α * q x := x ′ α x-x ′ q x ′ ,x ,
where

q x ′ ,x :=            x ′ <x ′′ ≤x r x ′′ , if x ′ < x, 0, if x ′ = x, x<x ′′ ≤x ′ r x ′′ , if x < x ′ .
(2.9) Furthermore, β x = ∆β (0)

x and β

(1)

x := ∇ * β (0)

x , where

β (0) x =    -4, x = 0, -1, x = ±1, 0, if otherwise.
(2.10)

The lattice Laplacian of a given (g x ) x∈Z is defined as ∆g x := g x+1 + g x-1 -2g x . Let also ∇g x := g x+1 -g x and ∇ * g x := g x-1 -g x . A simple calculation shows that

β(k) = 8s 2 (k) 1 + 2c 2 (k) = 8s 2 (k) + 4s 2 (2k), (2.11) and β(1) (k) = 4ie -2iπk s(k) 1 + 2c 2 (k) .
(2.12)

The evolution equations are formulated on the Hilbert space ℓ 2 made of all real valued sequences ((r x , p x )) x∈Z that satisfy

x r 2 x + p 2 x < +∞. (2.13)
Concerning the initial data we assume that it is distributed according to a probability measure µ ǫ that satisfies sup ǫ∈(0,1]

ǫ x r 2 x + p 2
x µǫ < +∞.

(2.14)

The above means that the total macroscopic energy of the system is finite. Here • µǫ denotes the average with respect to µ ǫ .

Denote by E ǫ the expectation with respect to the product measure P ǫ = µ ǫ ⊗ P. The existence and uniqueness of a solution to (2.7) in ℓ 2 , with the aforementioned initial condition can be easily concluded from the standard Hilbert space theory of stochastic differential equations, see e.g. Chapter 6 of [START_REF] Da Prato | Stochastic Equations in Infinite Dimensions[END_REF].

2.2. Energy density functional. For a configuration that satisfies (2.13), the energy per atom functional can be defined by:

e x (r, p) := p 2 x 2 - 1 4 x ′ α x-x ′ q 2 x,x ′ .
(2.15)

Notice that x e x = H(q, p). We highlight here the fact that, although the total energy is non-negative, in light of the assumptions a1)-a3) made about the interaction potential, the energy per atom e x does not have definite sign. However we can prove, see Appendix A below, the following fact. We shall also assume that there exists a macroscopic profile for the family of measures (µ ǫ ) ǫ>0 , i.e. functions r(y), p(y) and e(y) belonging to C ∞ 0 (R) such that the respective w T (y) = [r(y), p(y), e(y)] satisfies (1.16).

Macroscopic profiles of temperature and phononic energy.

The macroscopic temperature profile corresponding to the family of laws (µ ǫ ) ǫ∈(0,1] is defined as a function T : R → [0, +∞) such that R T (y)J(y)dy = lim

ǫ→0+ ǫ x J(ǫx) (δ ǫ p x ) 2 µǫ - 1 4 x ′ α x-x ′ (δ ǫ q x,x ′ ) 2 µǫ (2.19) for any J ∈ C ∞ 0 (R)
, where δ ǫ p x := p x -p(ǫx) and

δ ǫ q x ′ ,x :=            x ′ <x ′′ ≤x δ ǫ r x ′′ , if x ′ < x, 0, if x ′ = x, x<x ′′ ≤x ′ δ ǫ r x ′′ , if x < x ′ , (2.20)
with δ ǫ r x := r x -r(ǫx). After a simple calculation one gets the following identity T (y) = e(y) -e ph (y), (2.21) with e ph (y) := 1 2 p 2 (y) + τ 1 r 2 (y) , y ∈ R, (2.22) the phononic (or mechanical) macroscopic energy profile.

2.4. Energy spectrum. The energy spectrum of the configuration distributed according to µ ǫ is defined as

w ǫ (k) := | p(k)| 2 µǫ + α(k) 4s 2 (k) |r(k)| 2 µǫ , (2.23)
where p(k) and r(k) are the Fourier transforms of (p x ) and (r x ), respectively. Using (2.3) we get

w ǫ (k) = | p(k)| 2 µǫ + τ 1 ϕ 2 (s 2 (k)) |r(k)| 2 µǫ , k ∈ T.
(2.24) Assumption (2.18) is equivalent with

K 0 = sup ǫ∈(0,1] ǫ T w ǫ (k)dk < +∞. (2.25)
Definition 2.2. The family of distributions (µ ǫ ) ǫ>0 is said to be of a thermal type if its energy spectrum w ǫ (k) satisfies

K 1 := sup ǫ∈(0,1] ǫ 2 T w 2 ǫ (k)dk < +∞. (2.26)
Remark. In Section 5.1 we show that (2.26) implies that the respective macroscopic profiles (r(y), p(y)) vanish. Therefore, we conclude that then the macroscopic phononic energy e ph (y) ≡ 0 (see (2.22)) and, as a result,

lim ǫ→0+ ǫ x J(ǫx) e x µǫ = R J(y)T (y)dy, J ∈ C ∞ 0 (R). ( 2

.27)

We stress that although r(y) ≡ 0 and p(y) ≡ 0, in case condition (2.26) holds, configuration (r, p) need not necessarily be centered (in the measure µ ǫ ). Condition (2.26) is related to the issue of variability of the initial data on the microscopic scale, as can be seen in examples presented in Section 9.

Definition 2.3. The family (µ ǫ ) is called to be of a phononic (or mechanical) type if lim ǫ→0+ ǫ x [r x -r(ǫx)] 2 µǫ = 0 (2.28)
and lim

ǫ→0+ ǫ x [p x -p(ǫx)] 2 µǫ = 0, (2.29)
where r(x) and p(x) are the macroscopic profiles of the strain and momentum, see (1.16).

Remark. Using (2.19) and (2.28) together with (2.29) it is straightforward to see that the temperature profile corresponding to a phononic type initial data vanishes, i.e. T (y) ≡ 0.

Statement of the main results

3.1. Hyperbolic scaling. Assume that the configuration (r, p) can be decomposed into two parts whose laws are respectively of the thermal and phononic types. More precisely, we assume that

r x = r ′ x + r ′′ x , p x = p ′ x + p ′′ x , x ∈ Z, (3.1)
where

K ′ 1 := sup ǫ∈(0,1] ǫ 2 T (w ′ ǫ ) 2 (k)dk < +∞, (3.2)
and

w ′ ǫ (k) is the energy spectrum of (r ′ , p ′ ), i.e. w ′ ǫ (k) = | p′ (k)| 2 µǫ + τ 1 ϕ 2 (s 2 (k)) |r ′ (k)| 2 µǫ , k ∈ T. (3.3)
with ( p′ (k), r′ (k)) the respective Fourier transforms of (p ′ , r ′ ). In addition, we assume that the configuration (r ′′ , p ′′ ) is of the phononic type in the sense of Definition 2.3. Suppose that (r ′ (t), p ′ (t)) and (r ′′ (t), p ′′ (t)) describe the evolution of the respective initial configurations (r ′

x , p ′ x ) and (r ′′ x , p ′′ x ) under the dynamics (2.7). Define the microscopic temperature and phononic energy density profiles by e th,x (t) := e x (r ′ (t), p ′ (t)) and e ph,x (t) := e x (r

′′ (t), p ′′ (t)) (3.4) Let v T (t, y) := (r(t, y), p(t, y)) (3.5)
be the solution of the linear wave equation

   ∂ t r(t, y) = ∂ y p(t, y), ∂ t p(t, y) = τ 1 ∂ y r(t, y), r(0, y) = r(y), p(0, y) = p(y). (3.6)
The macroscopic phononic energy at time t, see (2.22), is given by

e ph (t, y) = 1 2 p 2 (t, y) + τ 1 r 2 (t, y) . (3.7)
The components of w T (t, y) := [r(t, y), p(t, y), e ph (t, y)] evolve according to the system of linear Euler equations:

       ∂ t r(t, y) = ∂ y p(t, y), ∂ t p(t, y) = τ 1 ∂ y r(t, y), ∂ t e ph (t, y) = τ 1 ∂ y (r(t, y)p(t, y)) , w(0, y) = w(y). (3.8)
The following result is proven in Section 6.

Theorem 3.1. Suppose that the initial configuration (r, p) satisfies the assumptions formulated in the foregoing. Then, (r ′ (tǫ -1 ), p ′ (tǫ -1 )) is of thermal type for all t ≥ 0 and

lim ǫ→0+ ǫ x +∞ 0 J(t, ǫx)E ǫ e th,x t ǫ dt = +∞ 0 dt R T (y)J(t, y)dy (3.9) for any J ∈ C ∞ 0 ([0, +∞) × R). Configurations (r ′′ (tǫ -1 ), p ′′ (tǫ -1
)) are of phononic type for all t ≥ 0 and

lim ǫ→0+ ǫ x E ǫ e ph,x t ǫ -e ph (t, ǫx) = 0. (3.10)
In addition,

lim ǫ→0+ ǫ x +∞ 0 J(t, ǫx)E ǫ e x t ǫ dt = +∞ 0 R
[e ph (t, y) + T (y)] J(t, y)dtdy,

(3.11) for any J ∈ C ∞ 0 ([0, +∞) × R).
Concerning the macroscopic evolution of the vector w x (t), see (1.12), the above result implies the following.

Corollary 3.2. Assume that lim ǫ→0+ ǫ x J(ǫx) w x (0) µǫ = R J(y)w(y) dy, J ∈ C ∞ 0 (R). (3.12) Then, for any J ∈ C ∞ 0 ([0, +∞) × R) we have lim ǫ→0+ ǫ x +∞ 0 J(t, ǫx)E ǫ w x t ǫ dt = +∞ 0 R J(t, y)w(t, y)dtdy,
(3.13) with the components of w(t, y) being the solutions of the linear Euler system (3.8).

3.2. Super-diffusive scaling. It follows from Theorem 3.1 that the phononic energy evolution takes place on the hyperbolic scale that is described by the linear Euler equation. In consequence, it gets dispersed to infinity on time scales longer than tǫ -1 . On the other hand, it has been shown, see Theorem 3.1 of [START_REF] Jara | Superdiffusion of energy in a chain of harmonic oscillators with noise[END_REF], that the respective temperature profile T (y), see (2.19), evolves on a superdiffusive scale (tǫ -3/2 , xǫ -1 ). Therefore, we have the following result.

Theorem 3.3. Suppose that the distribution of the initial configuration (r, p) satisfies condition (3.1). Then, for any test function J ∈ C ∞ 0 ([0, +∞) × R) we have:

lim ǫ→0+ ǫ x +∞ 0 J(t, ǫx)E ǫ e x t ǫ 3/2 dt = +∞ 0 R
T (t, y)J(t, y)dtdy.

(3.14) Here T (t, y) satisfies the fractional heat equation (5.49) with the initial condition T (0, y) = T (y), given by (2.27).

The proof of the above result is presented in Section 7.

Evolution of the normal modes at the diffusive time scale.

Finally we consider the Riemann invariants of the linear wave equation system (3.6). They are given by f (±) (t, y) := p(t, y) ± √ τ 1 r(t, y).

(3.15)

The quantities defined in (3.15) are constant along the characteristics of (3.6), which are given by the straight lines y ± √ τ 1 t = const. Therefore

f (±) (t, y) = f (±) (y ± √ τ 1 t), (3.16)
with f (±) (y) determined from the initial datum. This motivates the introduction of the microscopic normal modes given by

f (+) y (t) := p y (t) + √ τ 1 1 + 1 2 3γ √ τ 1 -1 ∇ * r y (t), (3.17) f (-) y (t) := p y (t) - √ τ 1 1 - 1 2 3γ √ τ 1 + 1 ∇ * r y (t),
that are the second order approximations (up to a diffusive scale) of f (±) (t, y). The particular form of f (±) (t) is determined by the fact that we are looking for quantities of the form p(t

)± √ τ 1 (1 ± c ± ∇ * ) r(t) that,
at the hyperbolic scale, approximate the Riemann invariants (3.15) and possess limits at the diffusive time scale. The latter requirement determines that

c ± = (1/2) 3γ/ √ τ 1 ± 1 .
Remark. The normal modes f (±) (t) capture the fluctuations around the Riemann invariants of the linear hyperbolic system (1.13). Their analogues, called normal modes φ ± (t), are considered in Appendix 1 b) of [START_REF] Spohn | Nonlinear fluctuating hydrodynamics for anharmonic chains[END_REF], in the context of anharmonic chains with FPU-potential, see (8.18) of ibid.

To state our result rigorously, assume that (µ ǫ ) ǫ -the initial laws of configurations (r, p) satisfy (1.16). Denote also the heat kernel

P t (y) := 1 √ 4πDt exp - y 2 4Dt , (3.18)
where D := 3γ. Let f

(d) ι (t, y) := P t * f (ι) (y), ι = ± and f (±) (y) := p(y) ± √ τ 1 r(y). (3.19)
With the above notation we can formulate the following result.

Theorem 3.4. Suppose that condition (1.16) is in force. Then, the phonon modes f

(±) y (t) satisfy lim ǫ→0+ ǫ x J(ǫx)E ǫ f (±) x t ǫ = R J(y)f (±) (t, y)dy. (3.20)
for any J ∈ C ∞ 0 (R). In addition, for any ι ∈ {-, +} lim

ǫ→0+ ǫ x J ǫx -ι √ τ 1 t ǫ E ǫ f (ι) x t ǫ 2 = R f (d) ι (t, y)J(y)dy, (3.21)
The proof of this theorem is contained in Section 8.

Remark. The results of the present paper are valid also for other stochastic dynamics obtained by a stochastic perturbation of the harmonic chain that conserves the volume, energy and momentum. For example we can take a harmonic chain perturbed by a noise of the "jump" type. More precisely, let (N x,x+1 (t)) x∈Z be i.i.d. Poisson processes with intensity 3γ/2. The dynamics of the strain component (r x (t)) x∈Z is the same as in (2.7), while the momentum (p x (t)) x∈Z is a càdlàg process given by

dp x (t) = -(α * q(t)) x dt (3.22) + [∇p x (t-)dN x,x+1 (t) + ∇ * p x (t-)dN x-1,x (t)] , x ∈ Z
where (α * q) x (t) is defined as in (2.9). One can show, with essentially the same arguments as the ones used in the present paper, that Theorems 3.1 -3.4 hold also for this dynamics.

Asymptotics of the phononic ensemble

In this section we assume that the laws (µ ǫ ) ǫ>0 of the initial configuration (r, p) is of the phononic type, i.e. they satisfy (2.28) and (2.29).

4.1. Approximation of the macroscopic phononic energy. The solution of (3.6) is given by

r(t, y) = r (y + √ τ 1 t) + r (y - √ τ 1 t) + 1 √ τ 1 [p (y + √ τ 1 t) -p (y - √ τ 1 t)] (4.1) p(t, y) = p (y + √ τ 1 t) + p (y - √ τ 1 t) + √ τ 1 [r (y + √ τ 1 t) -r (y - √ τ 1 t)] ,
where r(y), p(y) are the initial data in (3.6). Using the above formulas and (3.7) we can write e ph (t, y) in terms of the profiles r(y) and p(y).

We can easily infer the following. 3) The respective energy density is defined then as

Lemma 4.1. Suppose that r, p ∈ C ∞ 0 (R).
ē(ǫ) x (t) := e x (r ǫ (t), pǫ (t)) = p2 ǫ,x (t) 2 - 1 4 x ′ α x-x ′ q2 ǫ,x,x ′ (t), (4.4)
where qǫ,x,x ′ (t) is the respective mean of q x,x ′ (t) defined by (2.9

E * := sup t≥0,ǫ∈(0,1] ǫ -1 T (| rǫ (t, k)| 2 + | pǫ (t, k)| 2 )dk < +∞. (4.6)
From (2.7) we obtain

d dt vǫ (t, k) = 1 ǫ δ-1 A ǫ (k) vǫ (t, k), (4.7)
where vT

ǫ (t, k) = [ rǫ (t, k), pǫ (t, k)] (4.8)
and

A ǫ (k) :=   0 a -τ (ǫk)a * -b   , (4.9)
with a * -the complex conjugate of a and τ (k

) := τ 1 ϕ 2 (sin 2 (πk)), a := 1 ǫ (1 -exp {-2πiǫk}), b := γ ǫ |1 -exp {-2πiǫk} | 2 [2 + cos(2πǫk)]. (4.10)
Observe that for a given M > 0 we have lim

ǫ→0+ sup |k|≤M |A ǫ (k) -A 0 (k)| = 0 (4.11)
and

A 0 (k) :=   0 2πik 2πiτ 1 k 0   .
Since both eigenvalues of A 0 (k) are imaginary we have

D * := sup (t,k)∈R 2 exp {A 0 (k)t} < +∞. (4.12)
Here • is the matrix norm defined as the norm of the corresponding linear operator on a euclidean space. In fact, an analogue of (4.12) holds in the case of the linear dynamics governed by (4.7).

Lemma 4.2. We have

C * := sup (t,k)∈R 2 ,ǫ∈(0,1] exp {A ǫ (k)t} < +∞. (4.13)
Proof. The eigenvalues of A ǫ (k) equal

λ ± = -1 2 b ± b 2 -4τ (ǫk)|a| 2 .
Since both b and τ (ǫk) (see (4.10)) are real and non-negative it is clear that Reλ ± ≤ 0 and the conclusion of the lemma follows.

4.3. Asymptotics of the mean ensemble. Let v(0) ǫ (t, k) be the solution of the following equation

d dt v(0) ǫ (t, k) = 1 ǫ δ-1 A 0 (k) v(0) ǫ (t, k), v(0) ǫ (0, k) = vǫ (0, k). ( 4
lim ǫ→0+ M -M | vǫ (t, k) -v(0) ǫ (t, k)| 2 dk = 0, t ≥ 0. (4.17) Proof. Note that A ǫ (k) = A 0 (k) + ǫB ǫ (k) (see (4.9)), where b * (M) := sup ǫ∈(0,1] sup |k|≤M B ǫ (k) < +∞. (4.18)
From (4.7) we can write

d dt vǫ (t, k) = 1 ǫ δ-1 A 0 (k) vǫ (t, k) + ǫ 2-δ B ǫ (k) vǫ (t, k), (4.19)
Therefore, by Duhamel's formula, we conclude

vǫ (t, k) -v(0) ǫ (t, k) = ǫ 2-δ t 0 exp t -s ǫ δ-1 A 0 (k) B ǫ (k) vǫ (s, k)dk. ( 4
lim ǫ→0+ ǫ -1 T vǫ (t, ℓ) -v t ǫ δ-1 , ℓ 2 dℓ = 0 (4.21)
and, in consequence,

lim ǫ→0+ ǫ x vx t ǫ δ -v t ǫ δ-1 , ǫx 2 = 0. (4.22)
Proof. Formula (4.22) follows from (4.21) so we only focus on the proof of the latter. We have

vǫ (t, ℓ) = exp A ǫ (ℓ) t ǫ δ-1 vǫ (0, ℓ).
Therefore,from the above and (4.15) we can write lim sup

ǫ→0+ ǫ -1 T vǫ (t, ℓ) -v t ǫ δ-1 , ℓ 2 dℓ ≤ 2 lim sup ǫ→0+ ǫ -1 T exp A ǫ (ℓ) t ǫ δ-1 vǫ (0, ℓ) -v(0, ℓ) 2 dℓ (4.23) +2 lim sup ǫ→0+ ǫ -1 T exp A ǫ (ℓ) t ǫ δ-1 v(0, ℓ) -exp A 0 (ℓ) t ǫ δ-1 v(0, ℓ) 2 dℓ.
Using Lemma 4.2 and (4.16) we conclude that the first term on the right hand side vanishes. To estimate the second term divide the domain of integration into the regions |ℓ| ≤ M and its complement. On the first region we use Lemma 4.3 to conclude that the respective limit vanishes. Therefore, we can write that, modulo multiplication by factor 2, the second term on the right hand side of (4.23) is equal to lim sup

ǫ→0+ M ≤|ℓ|≤1/(2ǫ) exp A ǫ (ℓ) t ǫ δ-1 v(0, ℓ) -exp A 0 (ℓ) t ǫ δ-1 v(0, ℓ) 2 dℓ ≤ 2 lim sup ǫ→0+ M ≤|ℓ|≤1/(2ǫ) exp A ǫ (ℓ) t ǫ δ-1 v(0, ℓ) 2 + exp A 0 (ℓ) t ǫ δ-1 v(0, ℓ) 2 dℓ ≤ 4 (C * + D * ) lim sup ǫ→0+ M ≤|ℓ|≤1/(2ǫ) |v(0, ℓ)| 2 dℓ, (4.24)
the last estimate following from Lemma 4.2 and estimate (4.12). We can adjust M > 0 to become sufficiently large so that the utmost right hand side of (4.24) can be arbitrarily small. The conclusion of the lemma therefore follows.

Suppose also that (r(t, y), p(t, y)) is the solution of the linear wave equation (3.6) and ēph (t, y) is the corresponding macroscopic phononic energy density profile, defined by (3.7). As a direct corollary from Lemma 4.4 we conclude the following. Concerning the behavior of energy functional ē(ǫ)

x (t) corresponding to the average phononic ensemble (see (4.4)) we get the following.

Corollary 4.6. For any t ≥ 0 and δ ∈ [1, 2) we have

lim ǫ→0+ ǫ x ē(ǫ) x t ǫ δ -e ph t ǫ δ-1 , ǫx = 0, (4.26)
where e ph (t, y) is given by (3.7). In addition, for any δ ∈ (1, 2) we have

lim ǫ→0+ ǫ x J(ǫx)ē (ǫ) x t ǫ δ = 0, J ∈ C ∞ 0 (R). (4.27)
Proof. In light of Lemma 4.4 only equality (4.27) requires a proof. Assume that δ ∈ (1, 2). Note that the left hand side of (4.27) equals lim ǫ→0+ [J (ǫ)

1 (t) + J (ǫ) 2 (t)],
where

J (ǫ) 1 (t) := ǫ 2 x p2 ǫ,x t ǫ δ J(ǫx) (4.28) = 1 2 1/(2ǫ) -1/(2ǫ) Jǫ (ℓ) dℓ 1/(2ǫ) -1/(2ǫ) pǫ t ǫ δ-1 , -ℓ -k pǫ t ǫ δ-1 , k dk and J (ǫ) 2 (t) := - ǫ 4 x,x ′ J(ǫx)α x-x ′ q2 ǫ,x t ǫ δ = - z α z 4 1/(2ǫ) -1/(2ǫ) Jǫ (ℓ) dℓ (4.29) × 1/(2ǫ) -1/(2ǫ) h z (-ǫ(k + ℓ))h z (ǫk) rǫ t ǫ δ-1 , -ℓ -k rǫ t ǫ δ-1 , k dk, where h z (k) := exp {-2πikz} -1 exp {-2πik} -1 , z ∈ Z, k ∈ T and Jǫ (ℓ) := m Ĵ ℓ + m ǫ .
Recall that vǫ (t, k) and v(0) ǫ (t, k) are given by (4.7) and (4.14) respectively. Using Lemma 4.4, we conclude that lim

ǫ→0+ [J (ǫ) 1 (t) - J (ǫ) 1 (t)] = 0 and lim ǫ→0+ [J (ǫ) 2 (t) - J (ǫ) 2 (t)] = 0,
where

J (ǫ) 1 (t) := R Ĵ(ℓ)dℓ R p t ǫ δ-1 , -ℓ -k p t ǫ δ-1 , k dk (4.30)
and

J (ǫ) 2 (t) := - z z 2 α z 4 R Ĵ(ℓ) R r t ǫ δ-1 , -ℓ -k r t ǫ δ-1 , k dℓdk. (4.31) Recall that p t ǫ δ-1 , k = exp A 0 (k) t ǫ δ-1 v(k), e 2 C d where e T 2 = [0, 1] and •, • C d is the scalar product on C d . Therefore J (ǫ) 1 (t) = R Ĵ(ℓ)dℓ R v(-k -l), exp A T 0 (-k -ℓ) t ǫ δ-1 e 2 C d v(k), exp A T 0 (k) t ǫ δ-1 e 2 C d dk.
By an elementary application of the Riemann-Lebesgue theorem we get lim ǫ→0+

J (ǫ) j (t) = 0, j = 1, 2.
This ends the proof of (4.27).

4.4. Evolution of the conserved quantities at the hyperbolic scale. Our goal in this section is to prove the following result. Since, for any J ∈ C ∞ 0 (R) we have

ǫ x J(ǫx)E ǫ e x r t ǫ δ , p t ǫ δ (4.44) = ǫ x J(ǫx)E ǫ e x δr ǫ t ǫ δ , δp ǫ t ǫ δ + ǫ x J(ǫx)ē (ǫ) x t ǫ δ .
The conclusion of the theorem follows directly from (4.43) and (4.27).

Evolution of a thermal ensemble

In this section we investigate the limit of dynamics of ensemble families whose initial laws (µ ǫ ) ǫ>0 satisfy condition (2.26). Then, as it turns out, the respective macroscopic profile of (r, p) is trivial, i.e. r(y) ≡ 0 and p(y) ≡ 0. In fact, we show that condition (2.26) persists in time and the temperature profile remains stationary at the hyperbolic time scale.

Wave function. Define the wave function of a given configuration as

ψ x := (ω * q) x + ip x = x ′ ωx-x ′ q 0,x ′ + ip x , x ∈ Z, (5.1)
where q 0,x ′ is given by (2.9) and (ω x ) are the Fourier coefficients of the dispersion relation defined as, see (2.2),

ω(k) := α1/2 (k) = 2 √ τ 1 |s(k)|ϕ(s 2 (k)). (5.2)
Using (5.2) we can rewrite (5.1) in the form

ψ x = ω(1) * r x + ip x , x ∈ Z, with (ω (1)
x ) are the Fourier coefficients of ω (1) (k) = -ie πik sign(k) √ τ 1 ϕ(s 2 (k)), k ∈ T.

(5.

3)

The Fourier transform ψ(k) of the wave function (ψ x ) can be written as ψ(k) = ω (1) (k)r(k) + i p(k).

(5.4) Therefore,

| ψ(k)| 2 µǫ = |ω (1) (k)r(k)| 2 + | p(k)| 2 µǫ + 2 Im r(k)ω (1) (k) p * (k) µǫ .
(5.5) On the other hand, since r(-k) = r * (k) and p(-k) = p * (k) we obtain

| ψ(-k)| 2 µǫ = |ω (1) (k)r(k)| 2 + | p(k)| 2 µǫ -2 Im r(k)ω (1) (k) p * (k) µǫ .
(5.6) From (5.5) and (5.6) we obtain where

ψ(y) = √ τ 1 H(r)(y) + ip(y)
and

H(f )(y) := - 1 π p.v. R f (y ′ )dy ′ y -y ′ is the Hilbert transform of a given function f ∈ C ∞ 0 (R).
The integral on the right hand side is understood in the sense of the principal value. In particular, ψ(y) ≡ 0 implies that both p(y) ≡ 0 and r(y) ≡ 0.

To show (5.10) observe that by the Plancherel identity we can write that the absolute value of the expression under the limit in (5.10) equals

ǫ T ψ(k) µǫ Ĵǫ (k)dk , (5.12)
where Ĵǫ (k) :=

x J(ǫx) exp {-2πikx} ≈ 1 ǫ Ĵ k ǫ
and Ĵ(k) is the Fourier transform of J(x). Expression in (5.12) is therefore estimated as follows

T ψ(k) µǫ Ĵ k ǫ dk ≤ T | ψ(k)| 2 2 µǫ dk 1/4 T Ĵ k ǫ 4/3 dk 3/4
, where the estimate follows by Hölder inequality. Using the change of variables k ′ := k/ǫ in the second integral on the right hand side we conclude that it is bounded by ǫK 1/4 1 Ĵ L 4/3 (R) for ǫ ∈ (0, 1], which proves (5.10).

Evolution of the energy functional at the hyperbolic time scale.

Define by w ǫ (t, k) the energy spectrum (2.23) corresponding to the configuration (r(t), p(t)). The following result asserts that there is no evolution of the macroscopic temperature profile at the hyperbolic scale. Moreover, if the initial energy distribution is of the thermal type it remains so at this time scale. (5.13) Moreover, at any time t > 0 the energy spectrum satisfies

K 1 (t) := sup ǫ∈(0,1] ǫ 2 T w 2 ǫ t ǫ , k dk < +∞ (5.14)
and

lim ǫ→0+ ǫ x E ǫ r t ǫ J(ǫx) = 0 and lim ǫ→0+ ǫ x E ǫ p t ǫ J(ǫx) = 0
(5.15) for any J ∈ C ∞ 0 (R). The proof of this result is given in Section 5.5.

Wigner transform. Define ψ (ǫ)

x (t) := ψ x (tǫ -1 ) and ψ(ǫ) (t, k) its Fourier transform (belonging to L 2 (T)). By W ǫ (t) we denote the (averaged) Wigner transform of ψ (ǫ) (t), see (5.9) of [START_REF] Jara | Superdiffusion of energy in a chain of harmonic oscillators with noise[END_REF], given by

W ǫ (t), J := R×T W ǫ (t, p, k) Ĵ * (p, k)dpdk, (5.16)
where W ǫ (t, p, k) the Fourier -Wigner transform of ψ(ǫ) (t) given by

W ǫ (t, p, k) := ǫ 2 E ǫ ψ(ǫ) * t, k - ǫp 2 ψ(ǫ) t, k + ǫp 2 (5.
(1 + y 2 ) n |∂ l y ∂ m k J(y, k)| < +∞.
Let A be the completion of S under the norm

J A := R sup k | Ĵ(p, k)|dp. (5.18)
In what follows we shall also consider the Fourier -Wigner antitransform of ψ(ǫ) (t) given by

Y ǫ (t, p, k) := ǫ 2 E ǫ ψ(ǫ) t, -k + ǫp 2 ψ(ǫ) t, k + ǫp 2 (5.19)
From the Cauchy-Schwartz inequality we get

| W ǫ (t), J | ≤ ǫ 2 J A E ǫ ψ(ǫ) (t) 2 L 2 (T) .
Thanks to energy conservation property of the dynamics and Cauchy-Schwartz inequality we get sup

ǫ∈(0,1] sup t≥0 ( Y ǫ (t) A ′ + W ǫ (t) A ′ ) ≤ 2K 0 , (5.20)
where K 0 is the constant appearing in condition (2.25). As a direct consequence of the above estimate we infer that the family (W ǫ (•)) ǫ∈(0,1] is * -weakly sequentially compact as ǫ → 0+ in any L ∞ ([0, T ]; A ′ ), where T > 0, i.e. for any ǫ n → 0+ one can choose a subsequence ǫ n ′ such that W ǫ n ′ (•) n ′ that is * -weakly converging. In fact, using hypothesis (2.26) one can prove that, see Proposition 9.1 of [START_REF] Jara | Superdiffusion of energy in a chain of harmonic oscillators with noise[END_REF], the following estimate holds.

Proposition 5.2. For any M > 0 there exists C 1 > 0 such that

W ǫ (t, p, •) 2 L 2 (T) + Y ǫ (t, p, •) 2 L 2 (T) (5.21) ≤ W ǫ (0, p, •) 2 L 2 (T) + Y ǫ (0, p, •) 2 L 2 (T) e C 1 ǫt , ∀ ǫ ∈ (0, 1], |p| ≤ M, t ≥ 0.
The right hand side of (5.21) remains bounded for ǫ ∈ (0, 1] thanks to (5.8).

In fact asymptotically, as ǫ → 0+, the function |ψ

(ǫ)
x (t) | 2 coincides with the energy density e x (tǫ -1 ), which can be concluded from the following result, see Proposition 5.3 of [START_REF] Jara | Superdiffusion of energy in a chain of harmonic oscillators with noise[END_REF].

Proposition 5.3. Suppose that condition (5.8) holds. Then,

lim ǫ→0+ ǫ x J(ǫx)E ǫ e x t ǫ - 1 2 ψ (ǫ) x (t) 2 = 0, t ≥ 0, J ∈ C ∞ 0 (R).
(5.22)

From the above result we conclude that for a real valued function J(y, k) ≡ J(y) we can write where

lim ǫ→0+ W ǫ (t), J = lim ǫ→0+ ǫ 2 x J(ǫx)|ψ (ǫ) x (t) | 2
U ǫ,+ (t, p, k) := 1 2 Y ǫ (t, p, k) + Y * ǫ (t, -p, k) , U ǫ,-(t, p, k) := 1 2i Y ǫ (t, p, k) -Y * ǫ (t, -p, k) .
Thanks to Proposition 5.2 the Laplace transform is defined for any λ > 0 and for any M > 0, compact interval I ⊂ (0, +∞). In addition, we have

C I := sup ǫ∈(0,1] sup λ∈I,|p|≤M   wǫ,ι (λ, p) L 2 (T) + ι∈{-,+} ūǫ,ι (λ, p) L 2 (T)   < +∞.
ǫ 2 T E ǫ | ψ(ǫ) (t, k)| 2 2 dk < +∞.
(5.29) Condition (5.14) then follows directly from (5.7). In addition equalities (5.15) can be inferred from (5.29) and (5.9). Concerning the proof of the convergence of the energy functional in (5.13), recall that we already know that (W ǫ (•)) is * -weakly sequentially compact in (L 1 ([0, +∞), A))

* . Therefore for any ǫ n → 0, as n → +∞, we can choose a subsequence, denoted in the same way, such that * -weakly converging to some W ∈ (L 1 ([0, +∞), A))

* . In light of (5.20) we have

sup t≥0 W ǫ (t) A ′ ≤ K 0 ,
(5.30)

with K 0 the same as in (2.25). Therefore, the respective Laplace-Fourier transform wǫ (λ, p, k) can be defined for any λ > 0. To identify W it suffices to identify the * -weak limit in A ′ of the Laplace transforms wǫn (λ), as n → +∞. Thanks to Theorem 5.4, any limit w(λ, p, k) ≡ w(λ, p) obtained in this way will be constant in k. In fact

w(λ, p) = lim n→+∞ w (±) ǫn (λ, p).
We claim that in fact for any J ∈ C 0 (R) w(λ), J = R J(y)T (y)dy, (5.31) which implies that in fact W ∈ L ∞ ([0, +∞), A ′ ) and

W (t, y, k) ≡ T (y), t ≥ 0.
In light of Proposition 5.3 this allows us to claim (5.13). The only thing yet to be shown is therefore (5.31).

Proof of (5.31). Letting W (0) ǫ,± = W ǫ,± (0) and U (0) ǫ,± = U ǫ,± (0) we obtain that for any λ > 0 (see (10.5) of [START_REF] Jara | Superdiffusion of energy in a chain of harmonic oscillators with noise[END_REF])

λ wǫ,+ -W (0)
ǫ,+ = -iδ ǫ ω wǫ,+ -iγR ′ pū ǫ,-+ γ ǫ L ( wǫ,+ -ūǫ,+ ) + ǫr (1) ǫ , (5.32)

λū ǫ,+ -U (0) ǫ,+ = 2ω ǫ ūǫ,-+ γ ǫ L ūǫ,+ - 1 2 ( wǫ,+ + wǫ,-) + ǫr (2) ǫ , λū ǫ,--U (0) ǫ,-= - 2ω ǫ ūǫ,+ - 2γ ǫ Rū ǫ,-- iγR ′ p 2 ( wǫ,--wǫ,+ ) + ǫr (3) ǫ λ wǫ,--W (0) ǫ,-= iδ ǫ ω wǫ,-+ iγR ′ pū + γ ǫ L ( wǫ,--ūǫ,+ ) + ǫr (4) ǫ ,
where

δ ǫ ω(k, p) := 1 ǫ ω k + ǫp 2 -ω k - ǫp 2 , (5.33) ω(k, p
) := 1 2 ω k + p 2 + ω k - p 2 , Lw(k) := 2 T R(k, k ′ )w(k ′ )dk ′ -2R(k)w(k), w ∈ L 1 (T), and
R(k, k ′ ) := 3 4 ι∈{-,+} e ι (k)e -ι (k ′ ), (5.34)
R(k) := T R(k, k ′ )dk ′ ,
with e ± (k) given by (5.26). The remainder terms satisfy sup

ǫ∈(0,1] sup λ∈I,|p|≤M 4 j=1 r(j) ǫ (λ, p) L 2 (T) < +∞ (5.35)
for any M > 0, compact interval I ⊂ (0, +∞).

From the first equation of the system (5.32) we get

D (ǫ) wǫ = ǫ W (0) ǫ + 3 2 γ ι∈{-,+} e ι w (-ι) ǫ + q ǫ , (5.36)
where

D (ǫ) := ǫλ + 2γR + iǫδ ǫ ω, (5.37)
and q ǫ := 3 i=1 q (i) ǫ , with q (1) ǫ := ǫ 2 r(1) ǫ , q (2) ǫ := -γLū ǫ,+ , q (3) ǫ := -iǫγR ′ pū ǫ,-.

(5.38)

Computing wǫ from (5.36) and then multiplying scalarly both sides of the resulting equation by γe ι , ι ∈ {-, +} we get the following system

γw (ι) ǫ T 1 - 3γe -e + 2D (ǫ) dk - 3γ 2 2 w (-ι) ǫ T e 2 ι D (ǫ) dk = γǫ T e ι W (0) ǫ D (ǫ) dk + γ T e ι q ǫ D (ǫ)
dk, ι ∈ {-, +}. Adding sideways the above equations corresponding to both values of ι and then dividing both sides of the resulting equation by ǫ we obtain

a (ǫ) w w (+) ǫ -a (ǫ) + w (+) ǫ -w (-) ǫ = 4γ 3 T R W (0) ǫ D (ǫ) dk + 4γ 3ǫ T
Rq ǫ D (ǫ) dk, (5.39) where

a (ǫ) w (λ, p) := 4γ 3 T λ + iδ ǫ ω D (ǫ) R dk (5.40) a (ǫ)
+ (λ, p) := γ T λ + iδ ǫ ω D (ǫ) e + dk. Equality (5.31) is then a consequence of the following. Proposition 5.5. For any J ∈ S such that J(y, k) ≡ J(y) and λ > λ 0 we have

lim ǫ→0+ R×T 2γR D (ǫ) W (0) ǫ Ĵ dpdk - R×T T Ĵdpdk = 0, (5.41)
In addition, for any M > 0 and a compact interval I ⊂ (0, +∞)

lim ǫ→0+ sup λ∈I,|p|≤M a (ǫ) + (λ, p) w (+) ǫ (λ, p) -w (-) ǫ (λ, p) = 0, (5.42)

Rq

(3) ǫ

D (ǫ) dk ≤ 4γ 2 |p| R ′ ∞ 3 T R|ū ǫ,-| |D (ǫ) | dk ≤ C T |ū ǫ,-|dk
and (5.45) for j = 3 follows from (5.28).

In the case j = 2 note that T q

(2)

ǫ dk = 0, therefore 4γ 3ǫ T Rq (2) ǫ D (ǫ) dk = 2 3 T (λ + iδ ǫ ω)q (2) ǫ D (ǫ) dk.
Since k → δ ǫ ω(k, p) is odd for each p we get that the latter integral equals

I ǫ := - 2γ 3 T λ(ǫλ + 2γR) + ǫ(δ ǫ ω) 2 |D (ǫ) | 2 Lū ǫ,+ dk. (5.46) = - γ 2 ι∈{-,+} u -ι ǫ,+ T e ι |D (ǫ) | 2 λ(ǫλ + 2γR ǫ ) + ǫ(δ ǫ ω) 2 dk + 4γ 3 T Rū ǫ,+ |D (ǫ) | 2 λ(ǫλ + 2γR ǫ ) + ǫ(δ ǫ ω) 2 dk, with u (ι ′ )
ǫ,ι (λ, p) := ūǫ,ι (λ), e ι ′ L 2 (T) , ι, ι ′ ∈ {-, +}. We conclude therefore that there exists a constant C > 0 such that

|I ǫ | ≤ Cγǫ ι∈{-,+} |u -ι ǫ,+ | T e ι (δ ǫ ω) 2 |D (ǫ) | 2 dk + γǫ T R|ū ǫ,+ |(δ ǫ ω) 2 |D (ǫ) | 2 dk. (5.47)
Denote the terms appearing on the right hand side by I

ǫ , respectively. Since |D (ǫ) | 2 ≥ 2ǫλγR we conclude that there exists C > 0 such that

I (1) ǫ ≤ C ι∈{-,+} |u ι ǫ,+ | and I (2) ǫ ≤ C ūǫ,+ L 1 (T) .
Therefore (5.45) for j = 2 follows directly from (5.28). 5.6. The limit of energy functional at the superdiffusive time scale. We have shown in Theorem 5.1 that the evolution of the temperature profile takes place on a scale longer than the hyperbolic one.

In fact the right time-space scaling is given by (tǫ -3/2 , xǫ -1 ) as it has been shown in Theorem 3.1 of [START_REF] Jara | Superdiffusion of energy in a chain of harmonic oscillators with noise[END_REF].

Theorem 5.6. Suppose that the distribution of the initial configuration (r, p) satisfies condition (2.26). Then, for any test function J ∈ C ∞ 0 ([0, +∞) × R) we have:

lim ǫ→0+ ǫ x +∞ 0 J(t, ǫx)E ǫ e x t ǫ 3/2 dt = +∞ 0 R
T (t, y)J(t, y)dtdy.

(5.48) Here T (t, y) satisfies the fractional heat equation:

∂ t T (t, y) = -ĉ|∆ y | 3/4 T (t, y) (5.49)
with the initial condition T (0, y) = T (y), given by (2.27) and

ĉ := [α ′′ (0)] 3/4 2 9/4 (3γ) 1/2 .
(5.50)

6. Proof of Theorem 3.1 and Corollary 3.2

Since the dynamics is linear, the solutions of (2.7) are of the form

r x (t) = r ′ x (t) + r ′′ x (t) and p x (t) = p ′ x (t) + p ′′ x (t), t ≥ 0, x ∈ Z, where (r ′ (t), p ′ (t)
) and (r ′′ (t), p ′′ (t)) are the solutions corresponding to the initial configurations (r ′

x , p ′ x ) and (r ′′ x , p ′′ x ) under the dynamics (2.7), see Section 3.1. The convergence of the r and p components of the vector w(t) from Corollary 3.2 is therefore a direct consequence of the conclusions of Theorems 5.1 and 4.7

The statements concerning the asymptotics of the thermal and phononic components of the energy functional contained in (3.9) and (3.10) also follow from the aforementioned theorems. To prove (3.11), finishing in this way also the proof of Corollary 3.2, note that

e x t ǫ = e th,x t ǫ +e ph,x t ǫ +2e x r ′ t ǫ , p ′ t ǫ ; r ′′ t ǫ , p ′′ t ǫ (6.1)
where e th,x (t) and e ph,x (t) are defined in (3.4). Given two configurations (r (j) , p (j) ), j = 1, 2 the "mixed" energy functional is defined as e x (r (1) , p (1) ; r (2) , p (2) ) := 1 2 p (1) x p (2) x -

1 4 x ′ α x-x ′ q (1)
x,x ′ q

(2)

x,x ′ . (6.2)

Here q (j)

x,x ′ are computed from (2.9) for the respective configurations r (j) , j = 1, 2. Formula (3.11) is a simple consequence of the following. The first factor on the right hand side stays bounded, due to the conservation of energy property of the dynamics, while the second one vanishes thanks to (4.38).

We can conclude therefore that lim

ǫ→0+ ǫ x E ǫ e x r ′ t ǫ , p ′ t ǫ ; δ ǫ r ′′ t ǫ , δ ǫ p ′′ t ǫ = 0, (6.4)
for any t ≥ 0, Equality (6.3) would follow, provided we can show that lim ǫ→0+ ǫ

x J(ǫx)E ǫ e x r ′ t ǫ , p ′ t ǫ ; r ǫ (t), p ǫ (t) = 0, (

for any J ∈ C ∞ 0 (R) and t ≥ 0, where (r ǫ (t), p ǫ (t)) := (r(t, ǫx), p(t, ǫx)) x∈Z .

The latter however is a direct consequence of (5.15).

Proof of Theorem 3.3

Using the notation from Section 6 we can write the analogue of (6.1) at the time scale t/ǫ 3/2 . Thanks to (4.40) for any J ∈ C ∞ 0 (R) we have lim ǫ→0+ x J(ǫx)E ǫ e ph,x t ǫ 3/2 = 0, (

The respective time-space weak limit of e th,x t/ǫ 3/2 can be evaluated using Theorem 3.3. Finally, to finish the proof of Theorem 3.3 we need the following analogue of Lemma 6.1.

ǫ x J(ǫx)E ǫ e x r ′ t ǫ 3/2 , p ′ t ǫ 3/2 ; r ′′ t ǫ 3/2 , p ′′ t ǫ 3/2 ≤ ǫ J ∞ x E ǫ e x r ′ t ǫ 3/2 , p ′ t ǫ 3/2 1/2 × ǫ x J(ǫx)E ǫ e x r ′′ t ǫ 3/2 , p ′′ t ǫ 3/2 1/2
The first factor on the right hand side stays bounded, due to the conservation of energy property of the dynamics, while the second one vanishes thanks to (7.1). This ends the proof of the lemma.

Proof of Theorem 3.4

From the definition of the normal modes, see (3.17), we conclude that

d f(±) (t, k) = ± (1 -e -2iπk ) √ τ 1 f(±) (t, k) (8.1)
+ D ± (1 -e 2iπk ) 2 f(±) (t, k) + O(s 3 (k)) dt + dM t (k),
where

dM t (k) := 2iγ 1/2 T r(k, k ′ ) p(t, k -k ′ )B(dt, dk ′ ), (8.2)
with B(dt, dk) Gaussian white noise in (t, k), satisfying

B * (dt, dk) = B(dt, -dk), E[B(dt, dk)B * (dt ′ , dk ′ )] = δ(t -t ′ )δ(k -k ′ )
and D ± are given by

D ± := 3γ ± √ τ 1 2 . (8.3) Let f(±) ǫ (t, k) := ǫE ǫ f(±) t ǫ , ǫk , k ∈ ǫ -1 T. It satisfies d dt f(±) ǫ (t, k) = ± 1 ǫ (1 -e -2iǫπk ) f(±) ǫ (0, k) + O(ǫ). (8.4)
An elementary stability theory for solutions of ordinary differential equations guarantees that for any T, M > 0 we have f(±)

ǫ (t, k) = e ±2iπkt (1 + O(ǫ)) f(±) ǫ (t, k) , |k| ≤ M, |t| ≤ T. (8.5)
After a straightforward calculation we obtain that the left hand side of (3.20) equals

lim ǫ→0+ R Ĵ(p) f(±) ǫ (t, -p) dp = lim ǫ→0+ R Ĵ(p)e ∓2iπptf(±) ǫ (0, -p) dp,
which tends to the expression on the right hand side of (3.20), as

ǫ → 0+ for any J ∈ C ∞ 0 (R). Let v(ι) ǫ (t, k) := f(ι) ǫ t ǫ , k exp -ι(1 -e -2iπǫk ) √ τ 1 t ǫ 2 , ι ∈ {-, +}
for k ∈ ǫ -1 T. For any T, M > 0 and ι ∈ {-, +} we have

d v(ι) ǫ dt (t, k) = D ι ǫ 2 (1 -e 2iπǫk ) 2 v(ι) ǫ (t, k) + O(ǫ), |k| ≤ M, t ∈ [0, T ]. (8.6)
Using again the elementary stability theory of ordinary differential equations we conclude that lim

ǫ→0+ sup t∈[0,T ],|k|≤M v(ι) ǫ (t, k) -f(ι) ǫ (0, k) e -4π 2 tDι|k| 2 = 0, ι ∈ {-, +}.
For any J ∈ C ∞ 0 (R) we can write that the expression under the limit on the left hand side of (3.21) equals

lim ǫ→0+ R Ĵ(p) exp -ι2πip √ τ 1 t ǫ f(ι) ǫ t ǫ , -p dp = lim ǫ→0+ R Ĵ(p) exp ι √ τ 1 1 -e -2πiǫp -2πiǫp t ǫ 2 v(ι) ǫ (t, -p) dp = lim ǫ→0+ R Ĵ(p) exp 2ι √ τ 1 π 2 p 2 t + O(ǫ) v(ι) ǫ (t, -p) dp
and the latest limit equals R Ĵ(p) f (ι) (-p)e -4π 2 tD|p| 2 dp, with f (ι) (p) the Fourier transform of f (ι) (y), given by (3.19), which ends the proof of (3.21).

Examples

In the final section we give examples of the initial data that are either of thermal or phononic types introduced in Definitions 2.2 and 2.3. The examples are formulated in terms of the wave function. 9.1. Non-random initial data. Suppose that φ(x) is a function that belongs to C ∞ 0 (R). Let a ≥ 0 and let µ ǫ be δ-type measures on ℓ 2 concentrated at ψ (ǫ) x := ǫ (a-1)/2 φ(ǫ a x), x ∈ Z. (9.1)

We have

ψǫ (k) = ǫ (a-1)/2 R φ(p) x∈Z exp {2πix(ǫ a p -k)} dp.
Using Poisson summation formula (see e.g. p. 566 of [START_REF] Lax | Functional Analysis[END_REF])

x∈Z exp {ibxξ} = 2π |b| x∈Z δ 0 ξ - 2π b x , ξ ∈ R, (9.2)
(understood in the distribution sense) that holds for any b = 0, and the fact that φ(k) is rapidly decaying we conclude

ψǫ (k) ≈ 1 ǫ (a+1)/2
φ k ǫ a . 9.1.1. Case a = 1 -macroscopic initial data. Note that then lim sup

ǫ→0+ T dk ǫ | ψ(k)| 2 µǫ = R | φ(p)| 2 dp < +∞, (9.3)
where φ(p) is the Fourier transform of φ(y). The data is of phononic type, in the sense of Definition 2.3, with the macroscopic profile given by φ(y). On the other hand, condition (2.26) fails, as can be seen from the following computation:

T dk ǫ | ψ(k)| 2 µǫ 2 = ǫ 2 T | ψǫ (k)| 4 dk ≈ 1 ǫ 2 1/2 -1/2 φ k ǫ 4 dk ≈ 1 ǫ R | φ (p) | 4 dp.
9.1.2. Case a ∈ (0, 1) -oscillating (but not too fast) data. In this case one can easily verify that condition (2.25) holds, but again condition (2.26) fails. However, the rate of the blow-up is slower than in the case of the macroscopic data. Indeed,

T dk ǫ | ψ(k)| 2 µǫ 2 = ǫ 2 T | ψǫ (k)| 4 dk ≈ 1 ǫ a 1/(2ǫ a ) -1/(2ǫ a ) | φ (k) | 4 dk ≈ 1 ǫ a R | φ (p) | 4 dp.
On the other hand, it is easy to verify that the macroscopic profile for the initial data vanishes but

K 0 = lim ǫ→0+ ǫ x |ψ (ǫ) x | 2 µǫ = R | φ (k) | 2 dk.
Hence the family (µ ǫ ) ǫ>0 is neither of phononic, nor thermal type. 9.1.3. Case a = 0 -microscopically oscillatory data. We have

ψǫ (k) = ǫ -1/2 x∈Z φ(x) exp {-2πixk} = ǫ -1/2 φ(k),
where, the periodized Fourier transform of φ(y) is given by φ

(k) = x∈Z φ(x + k). (9.4)
In this case condition (2.26) holds, since

T dk ǫ | ψ(k)| 2 µǫ 2 = T | φ (k) | 4 dk.
The data is of thermal type. 9.2. Random initial data. 9.2.1. Modified stationary field. Assume that (η x ) x∈Z is a zero mean, random stationary field such that E|η 0 | 2 < +∞. We suppose that its covariance can be written as

r x = E (η x η * 0 ) = T exp {2πikx} R(k)dk, x ∈ Z, (9.5)
where R ∈ C(T) is non-negative. Given a ≥ 0 and φ(x) ∈ C ∞ 0 (R) define the wave function as ψ (ǫ) x := ǫ (a-1)/2 φ(ǫ a x)η x , x ∈ Z. (9.6) One can easily check that condtion (2.25) holds. We show that both micro-and macroscopically varying initial data satisfy condition (2.26).

For a ∈ (0, 1] (the oscillatory case) we have

ψ(ǫ) (k) ≈ 1 ǫ (a+1)/2 T φ k -ℓ ǫ a η(dℓ), (9.7)
where η(dℓ) is the stochastic spectral measure corresponding to (η x ) x∈Z . Then, thanks to (9.7) we get

T dk ǫE| ψ(ǫ) (k)| 2 2 ≈ ǫ -2a T dk T φ k -ℓ ǫ a 2 R(ℓ)dℓ 2 (9.8) = ǫ -2a T dk T 2 φ k -ℓ ǫ a 2 φ k -ℓ ′ ǫ a 2 R(ℓ) R(ℓ ′ )dℓdℓ ′ .
Changing variables l′ := ℓ ′ /ǫ a and k := k/ǫ a we get that the last expression equals

T R(ℓ)dℓ 1/(2ǫ a ) -1/(2ǫ a ) 1/(2ǫ a ) -1/(2ǫ a ) φ k - ℓ ǫ a 2 φ (k -ℓ ′ ) 2 R(ǫ a ℓ ′ )dkdℓ ′ , which, as ǫ → 0+, tends to R(0) T R(ℓ)dℓ R φ (k) 2 dk 2 .
Thus, condition (2.26) is clearly satisfied by this family of fields. In the case a = 0 (microscopically varying initial data) the condition is also valid, as then

ψ(ǫ) (k) = ǫ -1/2 T φ(k -ℓ)η(dℓ),
with φ(k) given by (9.4). As a result

T dk ǫE| ψ(ǫ) (k)| 2 2 = T dk T φ (k -ℓ) 2 R(ℓ)dℓ 2 < +∞.
9.2.2. Locally stationary initial data. Assume that (r x,ǫ , p x,ǫ ) x∈Z , ǫ ∈ (0, 1] is a family locally stationary random fields over a probability space (Ω, F , P). By the above we mean the fields that satisfy the following: 1) they are square integrable for each ǫ and there exist C ∞ 0 functions r, p : R → R, called the mean profiles satisfying r ′′

x,ǫ := Er x,ǫ = r(ǫx), p ′′ x,ǫ := Ep (ǫ) x = p(ǫx),

2) the covariance matrix of the field

r ′ x,ǫ := r x,ǫ -r ′′ x,ǫ , p ′ x,ǫ := p x,ǫ -p ′′ x,ǫ , is given by E[r ′ x,ǫ r ′ x+x ′ ,ǫ ] = C 11 (ǫx, x ′ ), E[p ′ x,ǫ p ′ x+x ′ ,ǫ ] = C 22 (ǫx, x ′ ), E[r ′ x,ǫ p ′ x+x ′ ,ǫ ] = C 12 (ǫx, x ′ ), E[p ′ x,ǫ r ′ x+x ′ ,ǫ ] = C 21 (ǫx, x ′ ), x, x ′ ∈ Z, where C ij : R × Z → R, i, j = 1, 2 are functions that satisfy C * := 2 i=1 x ′ R C ii (x, x ′ )dx 2 < +∞. (9.9)
The energy spectrum of the field (r ′ x,ǫ , p ′ x,ǫ ) x∈Z , cf (3.3), equals

w ′ ǫ (k) = x C 22 (ǫx, k) + α(k) 4s 2 (k) x C 11 (ǫx, k) where Ĉij (ǫx, k) = x ′ C ij (ǫx, x ′ ) exp {-2πikx ′ } , (x, k) ∈ Z × T, i, j = 1, 2.
We can write

ǫ 2 T (w ′ ǫ ) 2 dk ≤ Cǫ 2 2 j=1 x 1 ,x 2 T Ĉjj (ǫx 1 , k) Ĉ * jj (ǫx 2 , k)dk, ǫ ∈ (0, 1],
for some constant C > 0. By an applications of the Plancherel identity we conclude that the right hand side approximates CC * , as ǫ ≪ 1.

Thus condition (3.1) holds in this case. 9.2.3. Local Gibbs measures. Another important example of random initial data is furnished by the local Gibbs measure. Given the profiles of temperature β(ǫx) -1 , momentum p(ǫx), and tension τ (ǫx), where β -1 , p, τ ∈ C ∞ 0 (R) and β -1 ≥ 0 we define a product measure analogous to (1.6), in which the constant profiles are replaced by slowly varying functions. These measures are formally written as dν λ,ǫ := x exp -β(ǫx) e x -p(ǫx)p x -τ (ǫx)r x -G(λ(ǫx)) dr x dp x , (9.10) where λ(x) = (β(x), p(x), τ (x)) and G(•) is an appropriate Gibbs potential that normalizes the respective measure. In order to make the above "definition" rigorous one would have to consider the Gibbs measure in question as solutions of the respective DLR equations. We shall omit that issue by dealing only with local Gibbs measures, i.e. the case when β(y) -1 is compactly supported so the relevant measure is defined on a finite dimensional space. On the sites where β(ǫx) -1 = 0, we let the corresponding exponential factor in (9.10) be a delta distribution concentrated at the point (0, 0). The corresponding profile of volume stretch r(ǫx) and temperature β(ǫx) -1 are given by analogues of relations (1.9) with r and u appearing there replaced by the respective slowly varying functions.

The natural decomposition in thermal and mechanical initial conditions is now given by Let us assume that there exists c * > 0, for which c * r 2 ≤ V (r). Then, 0 ≤ C 11 (ǫx, x ′ ) ≤ 1 c * δ x ′ ,0 R V (r)e -β(ǫx)V (r) dr R e -β(ǫx)V (r) dr = 1 c * δ x ′ ,0 ũ(ǫx) -1 2 β -1 (ǫx) .

Here ũ(ǫx) := u(0, β(ǫx)) and u(τ, β) is the internal energy function defined in (1.9), see e.g. (2.1.9) of [START_REF] Bernardin | Thermodynamics and non-equilibrium macroscopic dynamics of chains of anharmonic oscillators notes[END_REF]. Condition (9.9) is satisfied, once we assume that R β -1 (y)dy < +∞ and R ũ(y)dy < +∞. 9.2.5. Gaussian, locally Gibbs measures. We assume that supp β -1 = [-K, K] and that β -1 (y) > 0 for |y| < K. In addition, we suppose that the sequence (α x ), besides satisfying conditions a1)-a3) and (2.1), is compactly supported, i.e. there exists a positive integer ℓ such that α x = 0 for |x| > ℓ and α x < 0 for 0 < |x| ≤ ℓ (obviously in light of (2.1) we have α 0 > 0). In this case, the formal expression (9.11) is a probability measure on ℓ 2 (Z) that equals (we omit writing primes) ν λ,ǫ := µ ǫ,K ⊗ δ K c , (9.13)

where δ K c is the Borel probability measure on the space of sequences (r x ) |x|≥ǫ -1 K , concentrated on the sequence r x ≡ 0, |x| ≥ K/ǫ. Measure µ ǫ,K is Gaussian on the Euclidean space corresponding to finite sequences (ρ x ) |x|<ǫ -1 K whose characteristic functional equals exp

   - 1 2 |x|,|x ′ |<ǫ -1 K σ x,x ′ ρ x ρ x ′    ,
where Σ := [σ x,y ] is the inverse of the symmetric matrix S := [S x,y ] corresponding to the quadratic form

|x|,|x ′ |<ǫ -1 K S x,x ′ r x r x ′ = - 1 4 |x|,|x ′ |<ǫ -1 K β(ǫx)α x-x ′ q 2 x,x ′ ,
where q x,x ′ is defined in (2.9). We claim that there exists c * > 0 independent of ǫ and such that

|x|,|x ′ |<ǫ -1 K S x,x ′ r x r x ′ ≥ c * |x|<ǫ -1 K r 2 x , ∀ (r x ) |x|<ǫ -1 K . (9.14)
According to our assumptions there exists β * > 0 such that β(y) ≥ β * for all y ∈ R. We can write therefore inf

   |x|,|x ′ |<ǫ -1 K S x,x ′ r x r x ′ : |x|<ǫ -1 K r 2 x = 1    (9.15) ≥ β * 4 inf    - |x|,|x ′ |<ǫ -1 K α x-x ′ q 2 x,x ′ : |x|<ǫ -1 K r 2 x = 1    ≥ β * 4 inf - x,x ′ α x-x ′ q 2 x,x ′ : x r 2 x = 1 .
One can easily verify the following identity (cf (2.3)) A simple calculation, using the fact that x α x = 0, shows that x (α * q) x q x = -1 2 x,y α x-y (q xq y ) 2 (A.1) for any (q x ) such that r := ∇ * q is square summable. Using (A.1), the Plancherel identity and (2.3) we can write Here r(k), p(k), q(k) are the Fourier transforms of r x , p x , qx respectively. In light of the assumptions made about ϕ(•) it is clear that the utmost right hand side is equivalent to x (r 2 x + p 2 x ), so (2.16) follows. Obviously only the lower bound in (2.17) requires a proof. Note that

- 1 2 x,x ′ α x-x ′ q 2 x,x ′ =
x |e x | ≤ 1 2 x p 2 x + x,x ′ |α x-x ′ |q 2 x,x ′ . (A.3)
Here q x,x ′ is given by (2.9). By Cauchy-Schwartz inequality for x ≥ x ′ we have q 2

x,x ′ ≤ (x -x ′ )

x ′ <x ′′ ≤x r 2

x ′′ and an analogous inequality holds also for x < x ′ . Therefore Denote the first and the second term by I and II respectively. We have

I = z>0 z|α z | x ′′ r 2 x ′′ x ′′ +z>x≥x ′′ 1 ≤ z>0 z 2 |α z | x ′′ r 2 x ′′ .
On the other hand where C := max 1/2, z>0 z 2 |α z | . This combined with the already shown estimate (2.16) ends the proof of the lower bound in (2.17).

II = z>0 z|α z | x ′′

Date:

  December 23, 2015. File: ko-euler-scaling-revision-submitted.tex. This work has been partially supported by the European Advanced Grant Macroscopic Laws and Dynamical Systems (MALADY) (ERC AdG 246953), and by the CAPES and CNPq program Science Without Borders. Work of T.K. has been partially supported by the Polish National Science Centre grant UMO-2012/07/B/SR1/03320.

R

  the total volume, momentum and energy of the chain. The corresponding equilibrium Gibbs measures ν λ are parametrized by λ = [β, p, τ ], with the respective components equal to the inverse temperature, velocity and tension. They are product measures given explicitly by formulasdν λ = x exp -β e x -pp x -τ r x -G(λ) dr x dp x ,exp {-β (V (r) -τ r)} dr.(1.8)

2 . 4 . 4 .

 244 .20) The lemma then follows from (4.6), (4.18) and Lemma 4.Lemma For any t ≥ 0 and δ ∈ [1, 2) we have

Corollary 4 . 5 .

 45 For any t ≥ 0 we have lim

2 T

 2 | ψ(k)|2 2 µǫ dk < +∞.(5.8) Remark. Note that the macroscopic profile (r(y), p(y)) corresponding to a given configuration (r, p) satisfying (2.26) necessarily vanishes, i.e. lim ǫ→0+ ǫ x r x µǫ J(ǫx) = 0 and lim ǫ→0+ ǫ x p x µǫ J(ǫx) = 0 (5.9) for any J ∈ C ∞ 0 (R). To show (5.9) it suffices only to prove that lim ǫ→0+ ǫ x ψ x µǫ J(ǫx) = 0. (5.10) Indeed, from conditions (2.28) and (2.29), we obtain lim ǫ→0+ ǫ x ψ x µǫ J(ǫx) = R ψ(y)J(y)dy, (5.11)

Theorem 5 . 1 .

 51 For any J ∈ C ∞ 0 ([0, +∞) × R) we have lim y)T (y)dtdy.

  17

  )and J belongs to S-the set of functions on R × T that are of C ∞ class and such that for any integers l, m, n we have sup y∈R, k∈T

4 .

 4 Homogenization of the Wigner transform in the mode frequency domain. It turns out that in the limit, as ǫ → 0+, the Wigner transform W ǫ (t, x, k) becomes independent of the k variable for any t > 0. To avoid boundary layer considerations at the initial time t = 0 we formulate this property for the Laplace-Fourier transform of the respective Wigner function. More precisely, let wǫ,± (λ, p, k) := +∞ 0 e -λt W ǫ (t, p, ±k)dt,(5.24)ūǫ,± (λ, p, k) := +∞ 0 e -λt U ǫ,± (t, p, k)dt,

Theorem 5 . 4 . 5 .

 545 ǫ (λ, p) := wǫ,+ (λ), e ± L 2 (T) = wǫ,-(λ), e ± L 2 (T) , where e + (k) := 8 3 s 4 (k), e -(k) := 2s 2 (2k).(5.26)The following result holds, see Theorem 10.2 of[START_REF] Jara | Superdiffusion of energy in a chain of harmonic oscillators with noise[END_REF]. Suppose that the initial laws satisfy (2.26). Then, for any M > 0 and a compact interval I ⊂ (λ 0 , +∞) we have lim ǫ→0+ sup λ∈I,|p|≤M T wǫ,+ (λ, p, k) -w (±) ǫ (λ, p) dk = 0Proof of Theorem 5.1. Applying (5.21) at p = 0 we conclude that sup ǫ∈(0,1]

D

  Proof. Equality (5.41) follows from (2.27) and (5.23). Formulas (5.42) and (5.43) can be easily substantiated using the Lebesgue dominated convergence theorem. To argue (5.44) it suffices only to prove that lim (ǫ) dk = 0, j = 2, 3.

9 . 2 . 4 .

 924 r x = r(ǫx) + r ′ x , p x = p(ǫx) + p ′ where (r ′ x , p ′ x ) x are distributed by dν ′ λ,ǫ := x exp {-β(ǫx)e ′ x -G(β(ǫx), 0, 0)} dr ′ x dp ′ x (9.11)and e ′ x is the energy at site x of the configuration (r ′ x , p ′ x ) x∈Z . In this case we haveC 22 (ǫx, x ′ ) = δ x ′ ,0 (p ′ x ) 2 ν ′ λ,ǫ = δ x ′ ,0 β -1 (ǫx). (9.12) Nearest neighbor interactions. Consider first the nearest neighbor case, i.e. when e ′ x = V (r ′ x ). Then,C 11 (ǫx, x ′ ) = δ x ′ ,0 (r ′ x ) 2 ν ′ λ,ǫ = δ x ′ ,0R r 2 e -β(ǫx)V (r) dr R e -β(ǫx)V (r) dr .

τ 1 Tϕ(s 2

 12 (k))|r(k)| 2 dk, therefore the utmost left hand side (9.15) can be estimated from below byc * := τ 1 β * 2 inf ϕ.Hence (9.14) holds with c * as defined above. We can write|r(k)| 2 ν λ,ǫ = |x|,|x ′ |<ǫ -1 K σ x,x ′ e x (k)e * x ′ (k),with e x (k) := exp {-2πixk}. Thanks to (9.14) we obtain|r(k)| 2 ν λ,ǫ ≤ 1 c * |x|≤ǫ -1 K |e x (k)| 2 = 2(K + 1) c * ǫ , ∀ k ∈ T. (9.16)Thanks to (9.12) we conclude that there exists C > 0 such that| p(k)| 2 ν λ,ǫ = x p 2 x ν λ,ǫ = x β -1 (ǫx) ≤ C ǫ , ∀ ǫ ∈ (0, 1]. (9.17) Combining (9.16) with (9.17) we conclude condition (2.26). Appendix A. Proof of Proposition 2.1

x

  e x (r, p) = T | p(k)| 2 2 + α(k)|q(k)| 2 dk = T | p(k)| 2 2 + τ 1 ϕ 2 (s 2 (k))|r(k)| 2 dk. (A.2)

x

  ,x ′ |α x-x ′ |q 2 x,x ′ ≤ x≥x ′′ >x ′ |x -x ′ ||α x-x ′ |r 2 x ′′ + x ′ ≥x ′′ >x |x -x ′ ||α x-x ′ |r 2 x ′′ .Substituting z := x -x ′ in the first summation and z := x ′ -x in the second we get that the right hand side equalsz>0 z|α z | x≥x ′′ >x-z

  Theorem 4.7. Suppose that (2.28) and (2.29) are in force. Then, Proof. Equality (4.33) is a consequence of (4.32). We only need to substantiate the latter. Let (δr ǫ (t), δp ǫ (t)) := (δr ǫ,x (t), δp ǫ,x (t)) x∈Z , where Evolution of energy density at the superdiffusive scale. In this section we show that the energy density disperses to infinity at time scale t/ǫ δ , where δ ∈ (1, 2).

	δr ǫ,x (t) := r ǫ,x (t) -rǫ,x (t), δp ǫ,x (t) := p ǫ,x (t) -pǫ,x (t), with rǫ,x (t) and pǫ,x (t) given by (4.3). We have (see (4.4)) 4.5. Theorem 4.8. Suppose that (2.28) and (2.29) are in force, and δ ∈ (4.34) (1, 2). Then,
	E (4.35) By conservation of the total energy we obtain lim ǫ→0+ ǫ x J(ǫx)E ǫ e x t ǫ δ = 0, (4.40)
	ǫ for any J ∈ C ∞ x e x (r, p) µǫ = ǫ 0 (R), t ≥ 0. x Proof. In analogy to (4.36) we write E ǫ e x r	t ǫ	, p	t ǫ
	= ǫ	ǫ	x	x	E ǫ e x δr ǫ e x (r, p) µǫ = ǫ t ǫ , δp ǫ x	t ǫ E ǫ e x r + ǫ	x ǫ δ , p ē(ǫ) x t	t ǫ t ǫ δ
	≥ ǫ	x ≥ ǫ	E ǫ e x δr ǫ E ǫ e x δr ǫ t ǫ x	, δp ǫ t ǫ δ , δp ǫ t ǫ	t ε δ + ǫ	(4.41)	(4.36)
		+ǫ	x	ēph	t ǫ δ-1 , ǫx -ǫ	x	ē(ǫ) x	t ǫ δ -ēph	t ǫ δ-1 , ǫx .
	Therefore taking the limit, as ǫ → 0+, in (4.41) we obtain (using (4.2) and (4.26))
				R	e ph (0, y)dy ≥ lim sup ǫ→0+	ǫ	x	E ǫ e x δr ǫ	t ǫ δ , δp ǫ	t ε δ
		+ lim sup ǫ→0+	ǫ→0+ R e ph	ǫ	x ǫ δ-1 , y dy. E ǫ e x δr ǫ t	t ǫ	, δp ǫ	t ǫ	+	(4.42)
	From (4.37) we infer therefore that By (4.37) we conclude
					lim sup ǫ→0+ lim sup ǫ→0+ ǫ	ǫ	x	x	E ǫ e x δr ǫ E ǫ e x δr ǫ	t ǫ t ǫ δ , δp ǫ , δp ǫ	t ǫ t ε δ	= 0. = 0.	(4.38) (4.43)
	lim ǫ→0+ From (2.16) applied to the configuration (δr ǫ (tǫ -1 ) , δp ǫ (tǫ -1 )) we ǫ E ǫ r x t ǫ -r(t, ǫx) 2 = 0, conclude that there exists c * > 0 such that x (4.32) lim ǫ→0+ ǫ x E ǫ p x t ǫ -p(t, ǫx) 2 c -ǫ E ǫ δr ǫ,x t ǫ 2 2 t + δp ǫ,x ǫ x = 0 and ≤ ǫ x E ǫ e x δr ǫ t ǫ , δp ǫ t ǫ . (4.39)
	lim ǫ→0+ Equalities (4.32) then follow straightforwardly from (4.39) and Corol-ǫ lary 4.5.

x E ǫ e x t ǫ -e ph (t, ǫx) = 0, t ≥ 0. (4.33) ǫ e x (r ǫ (t), p ǫ (t)) = E ǫ e x (δr ǫ (t), δp ǫ (t)) + ē(ǫ) x (t) . x ēph (t, ǫx) -ǫ x ē(ǫ) x t ǫ -ēph (t, ǫx) Letting ǫ → 0+ and using assumptions (2.28) and (2.29) we conclude that the limit of the energy functional appearing on the the utmost left hand side equals R e ph (0, y)dy ≡ R e ph (t, y)dy, t ∈ R, (4.37) with e ph (t, y) given by (3.7). Using the above together with (4.26) we conclude that R e ph (0, y)dy ≥ lim sup R e ph (t, y)dy.

  Lemma 7.1. Under the assumptions of Theorem 3.3 we have

	lim ǫ→0+	ǫ	x	J(ǫx)E ǫ e x r ′	t ǫ 3/2 , p ′	t ǫ 3/2 ; r ′′	t ǫ 3/2 , p ′′	t ǫ 3/2	= 0
									(7.2)
	for any J ∈ C ∞ 0 (R)
	Proof. Using Cauchy-Schwartz inequality we can estimate