Tomasz Komorowski

Stefano Olla
email: olla@ceremade.dauphine.fr

Ceremade

DIFFUSIVE PROPAGATION OF ENERGY IN A NON-ACOUSTIC CHAIN

We consider a non acoustic chain of harmonic oscillators with the dynamics perturbed by a random local exchange of momentum, such that energy and momentum are conserved. The macroscopic limits of the energy density, momentum and the curvature (or bending) of the chain satisfy a system of evolution equations. We prove that, in a diffusive space-time scaling, the curvature and momentum evolve following a linear system that corresponds to a damped Euler-Bernoulli beam equation. The macroscopic energy density evolves following a non linear diffusive equation. In particular the energy transfer is diffusive in this dynamics. This provides a first rigorous example of a normal diffusion of energy in a one dimensional dynamics that conserves the momentum.

Introduction

Macroscopic transport in a low dimensional system, in particular the energy transport, has attracted attention in both the physics and mathematical physics literature in the latest decades. Anomalous energy transport has been observed numerically in Fermi-Pasta-Ulam (FPU) chains, with the diverging thermal conductivity [START_REF] Lepri | Heat conduction in chains of nonlinear oscillators[END_REF]. Generically this anomalous superdiffusive behavior is attributed to the momentum conservation properties of the dynamics [START_REF] Lepri | Thermal Conduction in classical low-dimensional lattices[END_REF]. Actually one dimensional FPU-type chains have potential energy depending on the interparticle distances (i.e. the gradients of the particles displacements), and have three main locally conserved quantities: volume stretch, momentum and energy. These conserved (or balanced) quantities have different macroscopic space-time scalings, corresponding to different type of initial non-equilibrium behaviour. A mechanical non-equilibrium initial profile due to the gradients of the tension induces a macroscopic ballistic evolution, at the hyperbolic space-time scale, governed by the Euler equations (cf. [START_REF] Even | Hydrodynamic Limit for an Hamiltonian System with Boundary Conditions and Conservative Noise[END_REF]). When the system approaches to, or is already at a mechanical equilibrium, the temperature profile will evolve at a superdiffusive time scale.

Recent heuristic calculations based on fluctuating hydrodynamics theory [START_REF] Spohn | Nonlinear fluctuating hydrodynamics for anharmonic chains[END_REF], connect the macroscopic space-time scale of the superdiffusion of the thermal (energy) mode to the diffusive or superdiffusive fluctuations of the other conserved quantities. It turns out that this superdiffusive behavior of the energy is governed by a fractional laplacian heat equation. This picture can be mathematically rigorously proven in the case of a harmonic chain perturbed by a local random exchange of momentum, see [START_REF] Komorowski | Ballistic and superdiffusive scales in macroscopic evolution of a chain of oscillators[END_REF][START_REF] Jara | Superdiffusion of energy in a chain of harmonic oscillators with noise[END_REF]. In particular, it has been shown in [START_REF] Komorowski | Ballistic and superdiffusive scales in macroscopic evolution of a chain of oscillators[END_REF], that in the models driven by the tension, there is a separation of the time evolution scales between the long modes (that evolve on a hyperbolic time scale) and the thermal short modes that evolve in a longer superdiffusive scale. In addition, from the explicit form of the macroscopic evolution appearing in these models, it is clear that this behavior is strongly dependent on a non-vanishing speed of sound. More specifically, when the speed of sound is null, there is no macroscopic evolution either at the hyperbolic or superdiffusive time scales. This suggests that the macroscopic evolution of the system should happen at a yet longer, possibly diffusive, time scale for all modes.

In the present article we investigate the harmonic chain model with the random exchange of momenta. The interaction potential depends only on the squares of the curvature (or bending) of the chain

k x := -∆q x := 2q x -q x-1 -q x+1 , x ∈ Z, (1.1)
where q x are the positions of the particles. This means that its hamiltonian is formally given by

H(k, p) = x e x (k, p), (1.2)
where the energy of the oscillator x is defined

e x (k, p) := p 2 x 2 + αk 2 x 2 . (1.3)
Here α is a positive parameter that indicates the strength of the springs. This corresponds to a special choice of attractive nearest neighbor springs and repulsive next nearest neighbor springs. It turns out that the respective speed of sound is null, even though the momentum is conserved by the dynamics. As the energy depends on the curvature and not on the volume, this system is tensionless, and the corresponding relevant conserved quantity, besides the energy and momentum, is the curvature and not the volume stretch.

Our first result, see Theorem 3.1 below. asserts that these three conserved quantities (curvature, momentum and energy) evolve together in the diffusive time scale. Curvature and momentum are governed macroscopically by the damped Euler-Bernoulli beam equations: ∂ t k(t, y) = -∆ y p(t, y), ∂ t p(t, y) = α∆ y k(t, y) + 3γp(t, y) , (1.4) where γ > 0 is the intensity of the random exchange of momentum.

Defining the mechanical macroscopic energy as e mech (t, y) = 1 2 p 2 (t, y) + αk 2 (t, y) (1.5) and its thermal counterpart (or temperature profile) as e th (t, y) = e(t, y) -e mech (t, y) (1.6) the evolution of the latter is given by

∂ t e th (t, y) = ( √ 3 -1)α 2 √ 3γ + 3γ
∆ y e th (t, y) + 3γ (∂ y p(t, y)) 2 , (

see Theorem 3.2. In particular, the thermal conductivity is finite and we have a normal diffusion in this system. Notice also that because of the viscosity term, a gradient of the macroscopic velocity profile induces a local increase of the temperature. This result puts in evidence two main differences between the present and the FPU-type models:

(i) the thermal conductivity is finite, even though the system is one dimensional and dynamics conserves the momentum. This suggests that the non-vanishing speed of sound is a necessary condition for the superdiffusion of the thermal energy, (ii) there is no separation of the time scales between low (mechanical) and high (thermal) energy modes: all the frequencies evolve macroscopically in the diffusive time scale. Furthermore there is a continuous transfer of energy from low modes to high modes, resulting in the rise of the temperature, due to the gradients of the momentum profile. These rigorous results on the harmonic non-acoustic chain lead us to conjecture that a similar behavior is expected for the deterministic non-linear hamiltonian dynamics corresponding to an interaction of the type V (k x ), i.e. the energy is a non linear function of the curvature of the chain.

About our proof of the hydrodynamic limit: this is a non-gradient dynamics (microscopic energy currents are not of the form of discrete space gradients of some functions). Therefore, we cannot use known techniques for such type of limits based on relative entropy methods (cf. e.g. [START_REF] Varadhan | Nonlinear diffusion limit for a system with nearest neighbor interactions II[END_REF], [START_REF] Yau | Relative entropy and hydrodynamics of GinzburgLandau models[END_REF]) for two reasons:

(i) lack of control of higher moments of the currents in terms of the relative entropy, (ii) degeneracy of the noise in the dynamics, as it acts only on the velocities.

Instead, we develop a method already used in [START_REF] Jara | Superdiffusion of energy in a chain of harmonic oscillators with noise[END_REF], based on Wigner distributions for the energy of the acoustic chain. Thanks to the energy conservation property of the dynamics we can easily conclude, see Section 5.4, that the Wigner distributions form a compact family of elements in a weak topology of an appropriate Banach space. Our main result concerning the identification of its limit is contained in Theorem 5.1 below. The spatial energy density is a marginal of the Wigner function. We would like to highlight the fact, that in addition to proving the hydrodynamic limit of the energy functional, we are also able to identify the distribution of the macroscopic energy in the frequency mode domain, see formula (5.30). In particular the thermal energy is uniformly distributed on all modes (which is a form of local equilibrium), while the macroscopic mechanical energy is concentrated on the macroscopic low modes, see (5.30).

To show Theorem 5.1 we investigate the limit of the Laplace transforms of the Wigner distributions introduced in Section 7. The main results, dealing with the asymptotics of the Laplace-Wigner distributions, are formulated in Theorems 7.1 -7.3. Having these results we are able to finish the identification of the limit of the Wigner distributions, thus ending the proof of Theorem 5.1. The proofs of the aforementioned Theorems 7.1 -7.3, which are rather technical, are presented in Sections 8 -10, respectively.

The dynamics

2.1. Non acoustic chain of harmonic oscillators. Since in the nonacoustic chain the potential energy depends only on the bendings, see (1.1), in order to describe the configuration of the infinite chain we only need to specify (k x ) x∈Z , and the configurations of our dynamics will be denoted by

((p x , k x )) x∈Z ∈ (R × R) Z .
In case when no noise is present the dynamics of the chain of oscillators can be written formally as a Hamiltonian system of differential

equations kx (t) = -∆∂ px H(p(t), k(t)) (2.1) ṗx (t) = ∆∂ qx H(p(t), k(t)), x ∈ Z.
where ∆f (x) = f (x + 1) + f (x -1) -2f (x). Let also ∇g x := g x+1 -g x and ∇ * g x := g x-1 -g x .

2.1.1. Continuous time noise. We add to the right hand side of (2.1) a local stochastic term that conserves both p 2

x-1 +p 2 x +p 2 x+1 and p x-1 +p x + p x+1 . The respective stochastic differential equations can be written as

dk x (t) = -∆p x (t) dt, (2.2
)

dp x (t) = α∆k x (t) - γ 2 β * p x (t) 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 indicates the strength of the noise in the system, and (Y x ) are vector fields 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 . (2.3)
Here (w x (t)) t≥0 , x ∈ Z are i.i.d. one dimensional, real valued, standard Brownian motions, over a probability space (Ω, F , P). Furthermore,

β x = ∆β (0)
x , where

β (0) x =    -4, x = 0, -1, x = ±1, 0, if otherwise.
As a result we obtain

β x =        6, x = 0, -2, x = ±1, -1, x = ±2, 0, if otherwise.
We can rewrite the system (2.2)

dk x (t) = -∆p x (t) dt, (2.4
)

dp x (t) = α∆k x (t) + γ 2 ∆(β (0) * p(t)) x dt +γ 1/2 z=-1,0,1 (Y x+z p x (t))dw x+z (t), x ∈ Z.
Remark. The particular choice of the random exchange in the above dynamics is not important. The result can be extended to any other random mechanism of moment exchange, as long as total energy and momentum are conserved. Most simple dynamics would be given by exchange of momentum between nearest neighbor atoms at independent exponential times.

Stationary Gibbs distributions

. Let λ = (β, p, τ ), with β -1 ≥ 0. The product measures dµ λ := x exp {-β (e x -pp x -τ k x ) -G(λ)} dk x dp x , G(λ) := 1 2 log 2πβ α + β(p 2 + ατ 2 ) 2 (2.5)
are stationary for the dynamics defined by (2.4). In this context τ is called the load of the chain, while as usual β -1 is the temperature and p is the average momentum. Notice that, when τ = 0, the above distribution is spatially translation invariant, only for the (k x , p x ) coordinates, but is is not translation invariant with respect to the position q x , or the stretch r x = q xq x-1 coordinates.

Initial data.

Concerning the initial data we assume that, given ǫ > 0, it is distributed according to a probability measure µ ǫ on the configuration of ((k x , p x )) x∈Z and satisfies sup ǫ∈(0,1] ǫ x e x µǫ < +∞.

(2.6)

Here • µǫ denotes the average with respect to µ ǫ . We denote also by E ǫ the expectation with respect to the product measure P ǫ = µ ǫ ⊗ P.

The existence and uniqueness of a solution to (2.2) 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].

We assume furthermore that the mean of the initial configuration varies on the macroscopic spatial scale:

k x µǫ = κ(ǫx), p x µǫ = p(ǫx), x ∈ Z (2.7)
for some functions κ, p ∈ C ∞ 0 (R). Their Fourier transforms κ and p belong to the Schwartz class S(R). As for the fluctuations around the mean we assume that their energy spectrum is uniformly L r integrable with respect to ǫ > 0 for some r > 1. We have denoted by

f (k) := x f x exp {-2πikx} , k ∈ T.
(2.8)

the Fourier trasform of a given sequence f x , x ∈ Z. Here T is the unit torus, understood as the interval [-1/2, 1/2] with the identified endpoints. Let kx := k xk x µǫ , and px := p xp x µǫ , x ∈ Z.

(2.9)

The energy spectrum is defined as:

E ǫ (k) := 1 2 | p(k)| 2 µǫ + α | k(k)| 2 µǫ , k ∈ T, (2.10)
where p(k) and k(k) are the Fourier transforms of ( px ) and ( kx ), respectively. Assumption (2.6) implies in particular that

K 0 = sup ǫ∈(0,1] ǫ T E ǫ (k)dk < +∞. (2.11)
The announced property of the L r integrability of the energy spectrum means that there exists r > 1 such that: The quantities p(•), κ(•) are called the macroscopic velocity and curvature profiles. We assume furthermore that the following limits exist

K 1 := sup ǫ∈(0,1] ǫ r T E r ǫ (k)dk < +∞. ( 2
lim ǫ→0+ ǫ x G(ǫx) p 2 x µǫ = R G(y)p 2 (y)dy, (2.14) lim ǫ→0+ ǫ x G(ǫx) k 2 x µǫ = R G(y)κ 2 (y)dy, lim ǫ→0+ ǫ x G(ǫx) k x p x µǫ = R G(y)j(y)dy, for any G ∈ C ∞ 0 (R). Here j(•), p 2 (•), κ 2 (•) are some functions belonging to C ∞ 0 (R).
As a consequence, we conclude that the limit lim (2.16)

Remark. An important example of initial distributions that satisfy the above conditions is provided by local Gibbs measures, i.e. inhomogeneous product probability measures of the type

x∈Z exp {-β x (e x -p x p x -τ x k x ) -G(λ x )} dk x dp x , (2.17)
Here the vector λ x = (β x , p x , τ x ) is given by λ x := λ(ǫx), where λ(x) := (β(x), p(x), τ (x)) and the functions

β -1 (•), p(•), τ (•) belong to C ∞ 0 (R).
The deterministic field G(λ x ), x ∈ Z, called the Gibbs potential is given by an analogue of the second equality of (2.5).

In this case j(y) = p(y)τ (y), p 2 (y) = p(y) + β -1 (y), and κ 2 (y) = κ(y) + β -1 (y). For a proof of this fact see [START_REF] Komorowski | Ballistic and superdiffusive scales in macroscopic evolution of a chain of oscillators[END_REF].

Formulation of the main results

Suppose that p(t, y), κ(t, y) satisfy the following Cauchy problem ∂ t κ(t, y) = -∆ y p(t, y), (3.1) ∂ t p(t, y) = α∆ y κ(t, y) + 3γ∆ y p(t, y), p(0, y) = p(y), κ(0, y) = κ(y), with p(•), κ(•) given by (2.13).

Our first result concerns the evolution of the macroscopic profiles of the velocity and curvature. Theorem 3.1. Under the assumptions spelled about in the foregoing for any G ∈ C ∞ 0 (R) and t ≥ 0 we have where e mech (t, y) is given by (3.3), while the thermal energy (temperature) e th (t, y) is the solution of the following Cauchy problem:

lim ǫ→0+ ǫ x G(ǫx)E ǫ p x t ǫ 2 = R G(y)p(t, y)dy, (3.2)
lim ǫ→0+ ǫ x G(ǫx)E ǫ k x t ǫ 2 = R G(y)κ(t,
∂ t e th (t, y) = ĉ∂ 2 y e th (t, y) + 3γ(∂ y p) 2 (t, y), e th (0, y) = e th (y).

(3.6)

The diffusivity coefficient equals

ĉ := ( √ 3 -1)α 2γ √ 3 + 3γ. ( 3
The interaction α affects the thermal diffusivity, but does not influence the nonlinearity appearing in the evolution of the temperature profile.

Some basic notation

To abbreviate our notation we write s(k) := sin(πk) and c(k

) := cos(πk), k ∈ T. (4.1)
Let ℓ 2 be the space of all complex valued sequences (f x ) x∈Z , equipped with the norm f 2

ℓ 2 := x |f x | 2 .
Obviously f belongs to L 2 (T)the space of all complex valued functions equipped with the norm

f L 2 (T) := f , f 1/2 L 2 (T) , where f , ĝ L 2 (T) := T f (k)ĝ * (k)dk, f , ĝ ∈ L 2 (T).
Given a set A and two functions f, g :

A → R + we say that f (x) ≈ g(x), x ∈ A if there exists C > 1 such that f (x) C ≤ g(x) ≤ Cf (x), ∀ x ∈ A.
We write g(x) f (x), when only the upper bound on g is satisfied.

Denote by S the set of functions J : R × T → C that are of C ∞ class and such that for any integers l, m, n we have sup y∈R, k∈T

(1 + y 2 ) n |∂ l y ∂ m k J(y, k)| < +∞.
For J ∈ S we let Ĵ be its Fourier transform in the first variable, i.e.

Ĵ(η, k) := R e -2πiyη J(y, k)dy, (η, k) ∈ R × T.
For any M > 0 let A M be the completion of S in the norm

J A M := B M dη T | Ĵ(η, k)|dk . ( 4
ψ x := √ αk x + ip x , x ∈ Z. (5.1)
Its Fourier transform is given by

ψ(k) = √ α k (k) + i p (k) , k ∈ T. (5.2)
The energy and its spectrum (2.10) can be written as

e x = 1 2 |ψ x | 2 , x ∈ Z and E ǫ (k) = 1 2 | ψ(k)| 2 µǫ , k ∈ T. (5.3)
Using the decomposition into the macroscopic profile and the fluctuation part, see (2.9), we can write

ψ x = φ(ǫx) + ψ(ǫ) x , x ∈ Z, (5.4)
where φ(y) := √ ακ(y) + ip(y) and ψ(ǫ)

x := √ α kx + i px are the wave functions corresponding to the macroscopic profile and the fluctuation part, respectively.

Wigner functions.

By the Wigner functions corresponding to the wave function field (ψ x ) x∈Z we understand four tempered distributions W ǫ,± , Y ǫ,± that we often write together in the form of a vector

W T ǫ := [W ǫ,+ , Y ǫ,+ , Y ǫ,-, W ǫ,-], (5.5)
where

W ǫ,± , J := R×T W ǫ,± (η, k) Ĵ * (η, k)dηdk, (5.6)
and

Y ǫ,± , J := R×T Y ǫ,± (η, k) Ĵ * (η, k)dηdk (5.7)
for any J ∈ A. Here W ǫ,± (η, k) and Y ǫ,± (η, k) -called the Fourier-Wigner functions -are given by

W ǫ,+ (η, k) := ǫ 2 ψ * k - ǫη 2 ψ k + ǫη 2 µǫ , Y ǫ,+ (η, k) := ǫ 2 ψ -k + ǫη 2 ψ k + ǫη 2 µǫ , (5.8)
Y ǫ,-(η, k) := Y * ǫ,+ (-η, k), W ǫ,-(η, k) := W ǫ,+ (η, -k). For any J ∈ A we can write | W ǫ,+ , J | ≤ ǫ 2 J A sup η T ψ(ǫ) k - ǫη 2 ψ(ǫ) k + ǫη 2 µǫ dk.
Using the Cauchy-Schwartz inequality and (2.11) we get sup ǫ>0 ι=±

( Y ǫ,ι A ′ + W ǫ,ι A ′ ) ≤ 4K 0 .
(5.9)

A simple calculation shows that for any function J(y, k) ≡ J(y)

W ǫ,± , J = ǫ x e x µǫ J * (±ǫx) (5.10) and Y ǫ,± , J = ǫ x l x ± i √ αj x µǫ J * (ǫx), (5.11)
where

l x := 1 2 αk 2 x -p 2 x , j x := k x p x , x ∈ Z.
Using the decomposition of the wave function into its mean, following a macroscopic profile φ(•), and the fluctuation part { ψ(ǫ)

x , x ∈ Z}, see (5.4), we can correspondingly decompose the vector of the Wigner functions. Namely

W ǫ = W ǫ + W ǫ , (5.12
) where the Fourier-Wigner function corresponding to these wave functions shall be denoted by

W T ǫ := [W ǫ,+ , Y ǫ,+ , Y ǫ,-, W ǫ,-]. and W T ǫ := [ W ǫ,+ , Y ǫ,+ , Y ǫ,-, W ǫ,-]. We let W ǫ,± , J := R×T W ǫ,± (η, k)J * (η, k)dηdk, W ǫ,± , J := R×T W ǫ,± (η, k)J * (η, k)dηdk,
where, using the Poisson summation formula, we have defined

W ǫ,± (η, k) = 1 2ǫ x,x ′ φ * ±k + x ǫ - η 2 φ ±k + x ′ ǫ + η 2 , W ǫ,± (η, k) = ǫ 2 ψ(ǫ) * ±k - ǫη 2 ψ(ǫ) ±k + ǫη 2 µǫ . (5.13)
The formulas for Y ǫ,± , J and Y ǫ,± , J are constructed analogously using the respective Fourier-Wigner functions. Notice that for small ǫ the expression above of W ǫ,± is well approximated by the more natural definition:

W ǫ,± (η, k) ∼ ǫ 2 φ * ±k ǫ - η 2 φ ±k ǫ + η 2 .
As a consequence of assumption (2.14) we conclude that for functions J(y, k) = J(y):

lim ǫ→0+ W ǫ,± , J = R e(±y)J * (y)dy, lim ǫ→0+ Y ǫ,± , J = R l(y) ± i √ αj(y) J * (y)dy and l(y) := 1 2 (ακ 2 (y) -p 2 (y)) ,
with j(•), κ 2 (•) and p 2 (•) given by (2.14). A simple calculation also shows that

lim ǫ→0+ W ǫ,± (J) = W ± (J) := 1 2 R 2 φ * h - η 2 φ h + η 2 Ĵ * (η, 0)dηdh = 1 2 R 2 |φ * (y) | 2 e -2πiyη Ĵ * (η, 0)dηdy (5.14)
Thus,

W ± (dy, dk) = 1 2 |φ(y)| 2 δ 0 (dk)dy. (5.15)
One can also easily check that

Y + (dy, dk) = 1 2 φ 2 (y)δ 0 (dk)dy and Y -(dy, dk) = 1 2 [φ * (y)] 2 δ 0 (dk)dy.
(5.16) We denote the respective vector W

T := [W + , Y + , Y -, W + ].

Evolution of the wave function.

Adjusted to the macroscopic time, we can define the wave function corresponding to the configuration at time t/ǫ 2

ψ (ǫ) x (t) := √ αk x ǫ -2 t + ip x ǫ -2 t , x ∈ Z, (5.17)
where (p

x (t), k x (t)) x∈Z satisfies (2.2). Its Fourier transform ψ(ǫ) (t, k) = √ α k t ǫ 2 , k + i p t ǫ 2 , k , k ∈ T, (5.18)
is the unique solution of the Itô stochastic differential equation, understood in the mild sense (see e.g. Theorem 7.4 of [START_REF] Da Prato | Stochastic Equations in Infinite Dimensions[END_REF])

d ψ(ǫ) (t, k) = -iω(k) ǫ 2 ψ(ǫ) (t, k) - γR(k) ǫ 2 ψ(ǫ) (t, k) -( ψ(ǫ) ) * (t, -k) dt (5.19) + iγ 1/2 ǫ T r(k, k ′ ) ψ(ǫ) (t, k -k ′ ) -( ψ(ǫ) ) * (t, k ′ -k) B(dt, dk ′ ), where ψ(ǫ) (0) ∈ L 2 (T), ω(k) := 2 √ αs 2 (k), k ∈ T (5.20) is a dispersion relation, R(k) := 2s 2 (k) 1 + 2c 2 (k) = 2s 2 (k) + 4s 2 (2k).
(5.21) and

r(k, k ′ ) := 4s(k)s(k -k ′ )s(2k -k ′ ), k, k ′ ∈ T.
(5.22)

A simple calculation shows that β(k) = 4R(k).

The process B(dt, dk) is a space-time Gaussian white noise, i.e.

E [B(dt, dk)B * (ds, dk ′ )] = δ(t -s)δ(k -k ′ )dtdsdkdk ′ .
Since the total energy of the system is conserved in time, see Section 2 of [START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF], for each ǫ ∈ (0, 1] we have

ψ(ǫ) (t) L 2 (T) = ψ(ǫ) L 2 (T) , t ≥ 0, P ǫ a.s.
(5.23) 5.4. Wigner functions corresponding to ψ (ǫ) (t). Denote by

W T ǫ (t) := [W ǫ,+ (t), Y ǫ,+ (t), Y ǫ,-(t), W ǫ,- (t)
] the vector made of Wigner functions corresponding to the wave functions ψ (ǫ) (t). They can be defined by formulas (5.6) and (5.7), where the respective Fourier-Wigner functions W ǫ,± (t, η, k) and Y ǫ,± (t, η, k) are given by analogues of (5.8) in which the wave functions are substituted by ψ (ǫ) (t) and the average • µǫ is replaced by E ǫ . From (5.23) we conclude, thanks to (5.9), that sup

t≥0 ι=± sup ǫ∈(0,1] W ǫ,ι (t) A ′ + sup ǫ∈(0,1] Y ǫ,ι (t) A ′ ≤ 4K 0 , (5.24)
where K 0 is the constant appearing in condition (2.11). As a direct consequence of the above estimate we infer that the components of (W ǫ (•)) ǫ∈(0,1] are * -weakly sequentially compact in (L 1 ([0, +∞); A)) * as ǫ → 0+, i.e. given a component of the above family, e.g. W ǫ,+ (•), and any sequence ǫ n → 0+ one can choose a subsequence W ǫ n ′ ,+ (•) converging * -weakly.

To characterize the limit we recall that the thermal energy density e th (t, y) is given by the solution of the Cauchy problem (3.6), while the mechanical one e mech (t, y) is defined by (3.3). The limit of the Wigner functions corresponding to the macroscopic profile wave function

φ(t, y) := √ ακ(t, y) + ip(t, y), (t, y) ∈ [0, +∞) × R. (5.25) equals W T (t) := [W + (t), Y + (t), Y -(t), W + (t)], (5.26)
where (5.28) Our main result concerning the limit of the Wigner transform can be stated as follows.

W ± (t,
Theorem 5.1. Suppose that the initial data satisfy the assumptions Then, (W ǫ (t)) t≥0 converge, as ǫ → 0+, * -weakly over

(L 1 ([0, +∞), A)) * to W T (t) = [W (t), Y + (t), Y -(t), W (t)], t ≥ 0, (5.29)
where W (t) is a measure on R × T given by W (t, dy, dk) := e th (t, y)dydk + e mech (t, y)dyδ(dk).

(5.30)

Analogously to formulas (5.10) and (5.11) we can write

ǫ x E ǫ e x t ǫ 2 J * (±ǫx) = W ǫ,± (t), J (5.31)
and

ǫ x E ǫ l x t ǫ 2 ± i √ αj x t ǫ 2 J * (ǫx) = Y ǫ,± (t), J , J ∈ S(R).
(5.32) Therefore, the conclusion of Theorem 3.2 is a direct consequence of Theorem 5.1.

Evolution of the Wigner functions

Using (5.19) we can derive the equations describing the time evolution of the Wigner functions. In particular, one can conclude that for a fixed ǫ the components of (W ǫ (t)) t≥0 belong to C([0, +∞); A ′ ). After a straightforward calculation (see Section 8 of [START_REF] Jara | Superdiffusion of energy in a chain of harmonic oscillators with noise[END_REF] for details) we obtain that their Fourier transforms satisfy

∂ t W ǫ,+ (t) = - i ǫ δ ǫ ω W ǫ,+ (t) + γ ǫ 2 L ǫη W ǫ,+ (t) - γ 2ǫ 2 ι=± L + ιǫη Y ǫ,-ι (t), (6.1)
and

∂ t Y ǫ,+ (t) = - 2i ǫ 2 ω Y ǫ,+ (t) + γ ǫ 2 L ǫη Y ǫ,+ (t) (6.2)
+ γ ǫ 2 R ǫη ( Y ǫ,--Y ǫ,+ )(t) - γ 2ǫ 2 ι=± L + ιǫη W ǫ,-ι (t).
Here (cf (5.20))

δ ǫ ω := 1 ǫ ω k + ǫη 2 -ω k - ǫη 2 = 2 √ αs(ǫη)s(2k), ω := 1 2 ω k + ǫη 2 + ω k - ǫη 2 = 4 √ α s 2 (k) c 2 ǫη 2 + c 2 (k) s 2 ǫη 2 , L η f (k) := 2R η f (k) -2f (k) T R(k, k ′ , η)dk ′ , (6.3)
L ± η f (k) := 2R η f (k) -2R k ± η 2 f (k), R η f (k) := T R(k, k ′ , η)f (k ′ )dk ′ ,
where the scattering kernel (cf (5.22))

R(k, k ′ , η) := 1 2 ι=±1 r k - η 2 , k -ιk ′ r k + η 2 , k -ιk ′ , k, k ′ ∈ T. (6.4)
A direct calculation yields

R(k, k ′ , η) = R(k, k ′ ) -s 2 η 2 R 1 (k, k ′ ) + s 4 η 2 R 2 (k, k ′ ; η). (6.5) Here R(k, k ′ ) := R(k, k ′ , 0) = 3 4 e -⊗ e + + e + ⊗ e -(k, k ′ ), R 1 (k, k ′ ) := 16f + ⊗ f + + f + ⊗ e -+ e -⊗ f + + 3f -⊗ e + + 3e + ⊗ f -(k, k ′ ), R 2 (k, k ′ ; η) = 16 f + (k) + f + (k ′ ) + 4 4f + ⊗ f + + f + ⊗ f -+ f -⊗ f + (k, k ′ ) -32s 2 η 2 f + (k) + f + (k ′ ) + 2f(k) + 64s 4 η 2 f(k),
where e ± and f ± are the L 1 (T) normalized vectors given by

e + (k) := 8 3 s 4 (k), e -(k) := 2s 2 (2k) (6.6)
and f ≡ 1,

f + (k) := 2s 2 (k), f -(k) := 2c 2 (k), k ∈ T. (6.7)
Note also that (cf (5.21))

R(k) = T R(k, k ′ )dk ′ = 3 4
ι∈{-,+} e ι (k). (6.8)

In addition R ′ (k) = 2π(s(2k) + s(4k)) (6.9) and R ′′ (k) = 4π 2 (4c 2 (2k) + c(2k) -2). (6.10) 6.1. System of equations for the Laplace-Fourier transform of the Wigner functions. Taking the Laplace transform of both sides of (6.1) and (6.2), we get the following equations

D(ǫ) 1 w ǫ,+ + D(ǫ) + y ǫ,+ + D(ǫ) -y ǫ,-= R (ǫ) 1 (6.11) and D(ǫ) + w ǫ,+ + D(ǫ) 2 y ǫ,+ + D(ǫ) -w ǫ,-= R (ǫ)
2 . (6.12) Here

w ǫ,± (λ, η, k) := +∞ 0 e -λt W ǫ,± (t, η, k)dt, y ǫ,± (λ, η, k) := +∞ 0 e -λt Y ǫ,± (t, η, k)dt.
In addition, we let

D(ǫ) 1 (λ, η, k) := ǫ 2 λ + 2γR ǫ + iǫδ ǫ ω (6.13) D(ǫ) 2 (λ, η, k) := ǫ 2 λ + 2γR ǫ + 2iω, D(ǫ) ± (λ, η, k) := -γR ǫ ± γǫ 2 R ′ η,
where

R ǫ := R + (ǫη) 2 8 R ′′ . (6.14)
The right hand sides of (6.11) and (6.12) are respectively equal

R (ǫ) 1 := g ǫ + ǫ 2 W ǫ,+ (η, k) - γ(πǫη) 2 2 f ǫ + ǫ 3 r (1) ǫ , (6.15)
R (ǫ) 2 := -g ǫ + ǫ 2 Y ǫ,+ (η, k) + γ(πǫη) 2 2 f ǫ + ǫ 3 r (2)
ǫ , where

g ǫ := 3 2 γ ι=± e ι v ǫ , e -ι L 2 (T) , (6.16
)

f ǫ := f + v ǫ , 16f + + e -L 2 (T) + e -v ǫ , f + L 2 (T) + 3f -v ǫ , e + L 2 (T) + 3e + v ǫ , f -L 2 (T) .
Here, for the abbreviation sake we have let

v ǫ (λ, η, k) := w ǫ,+ (λ, η, k) - 1 2 y ǫ,o (λ, η, k), (6.17)
where

y ǫ,o (λ, η, k) := y ǫ,+ (λ, η, k) + y ǫ,-(λ, η, k).
In addition, the remainder terms r

(i) ǫ , i = 1, 2 satisfy lim sup ǫ→0+ sup λ≥λ 0 r (i) ǫ (λ) A ′ M < +∞, i = 1, 2, λ 0 , M > 0. ( 6
.18)

A closed system of equations on

w T ǫ (λ, η, k) := [w ǫ,+ , y ǫ,+ , y ǫ,-, w ǫ,-] (6.19)
can be rewritten in the matrix form Dǫ

w ǫ = R ǫ , (6.20)
where

R T ǫ := R (ǫ) 1 , R (ǫ) 2 , R (ǫ) 2,-, R (ǫ) 1,-
and Dǫ is a 4 × 4 matrix that can be written in the block form

Dǫ = A ǫ B ǫ B ǫ C ǫ , (6.21)
where A ǫ , B ǫ , C ǫ are 2 × 2 matrices given by

A ǫ := D(ǫ) 1 D(ǫ) + D(ǫ) + D(ǫ) 2 , C ǫ :=   D(ǫ) 2 * D(ǫ) + D(ǫ) + D(ǫ) 1 *   (6.22)
and B ǫ = D(ǫ) -I 2 , with I n denoting the n × n identity matrix. We have also denoted

R (ǫ) 1,-(λ, η, k) := R (ǫ) 1 (λ, η, -k), R (ǫ) 2,-(λ, η, k) := R (ǫ) 2 (λ, -η, k) * . Let w (ι)
ǫ (λ, η) be the column vectors obtained by scalar multiplication of each component of w ǫ (λ, η, k) by e ι . Note that

R ǫ = 3γ 2 ι=± e ι F w (-ι) ǫ (λ, η) + ǫ 2 h ǫ , (6.23)
where the matrix F = (1/2)e T ⊗ e, vector e

T := [1, -1, -1, 1], h ǫ (λ, η, k) = W ǫ (η, k) - γ(πη) 2 2 f ǫ e + r ǫ (λ, η, k) (6.24)
and W ǫ (η, k) is the column vector corresponding to the Fourier-Wigner transforms of the components of (7.3), and

r T ǫ := r (1) ǫ , r (2) ǫ , r (2)
ǫ,-, r

ǫ,-, (6.25)

r (1) ǫ,-(λ, η, k) := r (1) ǫ (λ, η, -k), r (2)
ǫ,-(λ, η, k) := (r (2) ǫ (λ, -η, k)) * . Recall that a ⊗ b = [a i b j ], if a = [a 1 , . . . , a n ] and b = [b 1 , . . . , b m ].
6.2. Invertibility of matrix Dǫ . We prove that the matrix Dǫ appearing in (6.20) is invertible, thus the vector of the Laplace-Fourier transforms of Wigner functions is uniquely determined by the system. It turns out to be true, provided that λ is sufficiently large.

Let us denote δǫ (λ, η, k) := det Dǫ (λ, η, k). Since matrices B ǫ and C ǫ commute we have (see p. 56 of [START_REF] Gantmakher | The theory of matrices[END_REF])

δǫ = det(A ǫ C ǫ -B 2 ǫ ) = D(ǫ) 1 D(ǫ) 2 * + [ D(ǫ) + ] 2 -[ D(ǫ) -] 2 2 -4[ D(ǫ) + ] 2 Re D(ǫ) 1 Re D(ǫ) 2 .

TOMASZ KOMOROWSKI AND STEFANO OLLA

After a direct calculation we get δǫ = ǫ 8 λ 4 + 8ǫ 6 λ 3 γR ǫ + 4ǫ 4 λ 2 5(γR

ǫ ) 2 + ω2 + ǫδ ǫ ω 2 2 - γǫR ′ η 2 2 +4ǫ 2 λ(γR ǫ ) 4(γR ǫ ) 2 + 4ω 2 + (ǫδ ǫ ω) 2 -(γǫR ′ η) 2 (6.26) +4ǫ 2 (γR ǫ δ ǫ ω) 2 -2(γR ǫ )δ ǫ ω ωγR ′ η + (ωδ ǫ ω) 2 + 16(γR ǫ ω) 2 . Define δ(0) ǫ := (ǫ 2 λ + R ǫ ) 4 . (6.27) Proposition 6.1. For any M > 0 there exist ǫ 0 (M), λ 0 (M) > 0 such that δǫ (λ, η, k) ≈ δ(0) ǫ (λ, η, k), k ∈ T, |η| ≤ M, λ > λ 0 , ǫ ∈ (0, ǫ 0 ]. (6.28) In particular, we have δǫ (λ, η, k) > 0, k ∈ T, |η| ≤ M, λ > λ 0 , ǫ ∈ (0, ǫ 0 ]. (6.29)
Proof. Using (6.10) we conclude that for any M > 0 there is ǫ 0 > 0 such that

R ǫ ≈ R(k) + (ǫη) 2 , k ∈ T, |η| ≤ M, ǫ ∈ (0, ǫ 0 ]. (6.30)
Comparing the second formula from (6.3) with (6.30) we get

ω ≈ R ǫ , k ∈ T, |η| ≤ M, ǫ ∈ (0, ǫ 0 ]. (6.31)
From (6.9), the first formula of (6.3) and (6.31) we get also

|δ ǫ ωR ′ | R ǫ , |η| ≤ M, k ∈ T. (6.32) Therefore 8(ǫγ) 2 R ǫ |δ ǫ ω ωR ′ η| ǫ 2 R 3 ǫ , |η| ≤ M, k ∈ T, ǫ ∈ (0, ǫ 0 ]
. Choosing λ 0 sufficiently large we can guarantee also that

ǫ 2 λ(γR ǫ ) 3 ≥ 8(ǫγ) 2 R ǫ |δ ǫ ω ωR ′ η| (6.33) for |η| ≤ M, k ∈ T, λ > λ 0 , ǫ ∈ (0, ǫ 0 ].
In a similar fashion we can argue that

ǫ 4 λ 2 (γR ǫ ) 2 ≥ 4ǫ 4 λ(γR ǫ ) (δ ǫ ω) 2 -γ R′ η 2 (6.34) and ǫ 6 λ 3 γR ǫ ≥ ǫ 6 λ 2 (δ ǫ ω) 2 -(γR ′ η) 2 (6.35)
for |η| ≤ M, k ∈ T, λ > λ 0 , ǫ ∈ (0, ǫ 0 ]. From estimates (6.33)-( 6.35) we conclude that

δǫ ǫ 8 λ 4 + ǫ 6 λ 3 R ǫ + ǫ 4 λ 2 R 2 ǫ + ǫ 2 λR 3 ǫ + R 4 ǫ .
Therefore, cf (6.27), we get δ(0) ǫ δǫ . The reverse estimate is a simple consequence of the first two formulas from (6.3) and (6.32). 6.3. Inverse of Dǫ (λ, p, k). Recall that Dǫ (λ, p, k) is a 2 × 2 block matrix of the form (6.21). Since B ǫ is diagonal we have [A ǫ , B ǫ ] = [C ǫ , B ǫ ] = 0. A simple calculation shows that also

A ǫ C ǫ = C ǫ A ǫ . (6.36) Therefore, D-1 ǫ = (C ǫ A ǫ -B 2 ǫ ) -1 0 0 (C ǫ A -B 2 ǫ ) -1 C ǫ -B ǫ -B ǫ A ǫ , (6.37)
provided that det Dǫ = 0. Note that

(C ǫ A-B 2 ǫ ) -1 = δ-1 ǫ     D(ǫ) 1 * D(ǫ) 2 + [ D(ǫ) + ] 2 -[ D(ǫ) -] 2 -2 D(ǫ) + Re D(ǫ) 2 -2 D(ǫ) + Re D(ǫ) 1 D(ǫ) 1 D(ǫ) 2 * + [ D(ǫ) + ] 2 -[ D(ǫ) -] 2     .
Substituting into (6.37), using also (6.22) we conclude that the inverse matrix D-1 ǫ is a 2×2 block matrix of the form D-1 ǫ = δ-1 ǫ adj( Dǫ ) where the adjugate of Dǫ equals

adj( Dǫ ) = P ǫ Q ǫ Q ǫ M ǫ , (6.38)
where M ǫ , P ǫ and Q ǫ are 2 × 2 matrices given by

P ǫ := d(ǫ) 1 d(ǫ) - d(ǫ) - d(ǫ) 2 , Q ǫ := ( d(ǫ) + ) * d(ǫ) o d(ǫ) o d(ǫ) + , M ǫ := ( d(ǫ) 2 ) * ( d(ǫ) -) * ( d(ǫ) -) * ( d(ǫ) 1 ) * .
Here

d(ǫ) 1 := | D(ǫ) 2 | 2 D(ǫ) 1 * - D(ǫ) + 2 + D(ǫ) - 2 Re D(ǫ) 2 -i D(ǫ) + 2 - D(ǫ) - 2 Im D(ǫ) 2 , d(ǫ) 2 := D(ǫ) 1 2 D(ǫ) 2 * - D(ǫ) + 2 + D(ǫ) - 2 Re D(ǫ) 1 -i D(ǫ) + 2 - D(ǫ) - 2 Im D(ǫ) 1 , (6.39) d(ǫ) -:= D(ǫ) + D(ǫ) + 2 - D(ǫ) - 2 - D(ǫ) + D(ǫ) 1 D(ǫ) 2 * , d(ǫ) + := - D(ǫ) - D(ǫ) 1 D(ǫ) 2 * + D(ǫ) + 2 - D(ǫ) - 2 , d(ǫ) o := 2 D(ǫ) + D(ǫ) -Re D(ǫ) 2 .
For the abbreviation sake we denote by d j,ǫ , j = 1, . . . , 4 the vectors corresponding to the rows of the adjugate of Dǫ given by (6.38). Combining the above with (6.13) and (6.26) we get.

Proof of Theorem 5.1

As we have already mentioned for any sequence ǫ n → 0+ there exists a subsequence W ǫ n ′ (t) that convergences * -weakly to some W ∈ (L 1 ([0, +∞); A))

* . We prove that the element W does not depend on the choice of the sequence ǫ n ′ by showing that for any M > 0 there exists λ 0 > 0 such that the vector w ǫ n ′ (λ) made of Laplace transforms of the components of W ǫ n ′ (t) converges * -weakly over A ′ M for any λ > λ 0 . In fact one can describe the respective limit as the Laplace transform of the vector W(t) appearing in the statement of Theorem 5.1. This identifies the limit of (W ǫ (t)), as ǫ → 0+ finishing in this way the proof of Theorem 5.1.

From (6.20) we obtain

w ǫ = D-1 ǫ R ǫ . (7.1)
Unfortunately, the right hand side of the above system contains also terms that depend on the vector w ǫ , via the projections of its components onto the vectors e ± and f ± . To describe the behavior of w ǫ we need to determine first these projections. Using (6.20) the above system can be rewritten in the form

1 ǫ 2 w ǫ - 3γ 2 ι=± e ι Ẽǫ w (-ι) ǫ = z ǫ , (7.2)
where

z T ǫ (λ, η, k) = z (1) ǫ , z (2) ǫ , z (2)
ǫ,-, z

ǫ,-:= D-1 ǫ h ǫ , (1)
h ǫ is given by (6.25) and the 4 × 4 matrix Ẽǫ (λ, η, k) equals

Ẽǫ := 1 2 e T ⊗ ∆ ǫ , (7.4) with ∆
T ǫ := [ ∆1,ǫ , ∆2,ǫ , ∆ * 2,ǫ , ∆ * 1,ǫ ] and ∆1,ǫ := d(ǫ) 1 + d(ǫ) o - d(ǫ) --( d(ǫ) + ) * , ∆2,ǫ := d(ǫ) -+ d(ǫ) + - d(ǫ) 2 -d(ǫ) o . (7.5)
Multiplying both sides of (7.2) by e ι , ι ∈ {-, +} and then integrating over T we get a system of 8 equations

G ǫ u ǫ = v ǫ , (7.6)
where

u ǫ (λ, η) := w (-) ǫ w (+) ǫ , v ǫ (λ, η) := z (-) ǫ z (+) ǫ .
Here w

(ι)
ǫ are column vectors obtained by a scalar multiplication of the entries of w ǫ (see (6.19)) by e ι . The same concerns

z (ι) ǫ T (λ, η, k) = z (1,ι) ǫ , z (2,ι) ǫ , z (2,ι) ǫ,-, z (1,ι) ǫ,- . (7.7) Matrix G ǫ (λ, η) is a 2 × 2 block matrix of the form G ǫ =   A (ǫ) o A (ǫ) - A (ǫ) + A (ǫ) o   ,
where

A (ǫ) o , A (ǫ)
± are 4 × 4 matrices defined as follows:

A (ǫ) ι := - 3γ 2ǫ 2 T e 2 ι δǫ Ẽǫ dk, ι ∈ {-, +}, A (ǫ) o := ǫ -2 I - 3γ 2 T
e -e + δǫ Ẽǫ dk .

Note that vector v ǫ appearing on the right hand side of (7.6) still depends on the projections of w ǫ onto f ± , cf (6.25) and (6.16). It turns out however that the asymptotics of these projections, as ǫ → 0+, can be described by only one of them, e.g. w

(-)

ǫ . This is a conclusion of our next result. Denote by δw ǫ := w

(+) ǫ -w (-)
ǫ . We shall also use the following convention: for a given M > 0 the constants ǫ 0 , λ 0 > 0 are selected as in the statement of Proposition 6.1 so that δǫ (λ, η, k) ≈ δ(0) ǫ (λ, η, k) for all k ∈ T, |η| ≤ M and λ > λ 0 . In particular, then we have (7.1).

Theorem 7.1. For any M > 0 and λ > λ 0 we have The proof of the theorem is presented in Section 8.

|δw ǫ (λ, η)| ǫ 2 (7.
To describe the limit of w (-) ǫ (λ, η) we can use the the system (7.6), which is "almost closed" with respect to the components of w (-) ǫ , i.e. it is closed modulo some corrections that in light of Theorem 7.1 are of lower order of magnitude.

Let us first introduce some additional notation. Given the wave function φ(t, y) we define the vector of the Laplace-Fourier transforms of the respective macroscopic Wigner functions w T φ (λ, η, h) = [w φ,+ , y φ,+ , y φ,-, w φ,-], (7.12) where w φ,± (λ, η, h) := +∞ 0 e -λt W φ,+ (t, η, h)dt, (7.13)

y φ,± (λ, η, h) := +∞ 0 e -λt Y φ,+ (t, η, h)dt, λ > 0, (η, h) ∈ R 2 .
Here

W φ,+ (t, η, h) := 1 2 φ * t, h - η 2 φ t, h + η 2 , (7.14) Y φ,+ (t, η, h) := 1 2 φ t, -h + η 2 φ t, h + η 2 , Y φ,-(t, η, k) := Y φ,+ * (t, -η, h), W φ,-(t, η, h) := W φ,+ (t, η, -h).
Define wT φ (λ, η) := w φ , y φ,+ , y φ,-, w φ , where Here êth (η) is the Fourier transform of e th (y) appearing in (3.4) and ĉ is given by (3.7).

w φ (λ, η) := R w φ,± (λ, η, h)dh, y φ,± (λ, η) := R y φ,± (λ, η, h)dh.
We can show, see Section 9 below for the proof, the following result.

Theorem 7.2. For any M > 0 and

J ∈ S(R) such that supp Ĵ ⊂ [-M, M] we have R w (-) (λ, η) Ĵ * (η)dη = lim ǫ→0+ R w (-) ǫ (λ, η) Ĵ * (η)dη
for all λ > λ 0 .

To obtain the asymptotics of w ǫ (λ, η, k) we use (7.1), which allows us to describe the Fourier-Laplace transforms of the Wigner functions in terms of their projections onto e ± and f ± . We obtain then the following result. The proof of this result is contained in Section 10.

The end of the proof of Theorem 5.1. Thanks to (5.24) we know that W ǫ (t) is sequentially pre-compact , as ǫ → 0+, in the * -weak topology of (L 1 ([0, +∞), A)) * . To identify its limiting points we consider w ǫ (λ, η, k) the vector of the Laplace-Fourier transforms of W ǫ (t). Given λ > 0 this family is sequentially pre-compact in the * -weak topology opf A ′ , as ǫ → 0+. Thanks to Theorems 7.2 and 7.3 we conclude that given M > 0 one can choose λ 0 as in the statement of Proposition 6.1, such that the the components of w ǫ (λ, η, k) converge * -weakly over A ′ M to the Laplace-Fourier transforms of the respective functions appearing in the claim of Theorem 5.1 for any λ > λ 0 . To finish the proof we only need to verify that w(λ, dη, dk) -the limit of w ǫ,+ (λ, η, k) (the limit of w ǫ,-can then be trivially concluded) agrees for λ > λ 0 with the Laplace transform of W (t, dy, dk) appearing in (5.30).

According to Theorem 7.3 the limit in question is the Fourier-Laplace transform of the measure-valued function W ′ (t, dy, dk) = e ′ th (t, y)dydk + e mech (t, y)dyδ 0 (dk), where, according to (7.16), we have e ′ th (0, y) = e th (y), (7.19)

∂ t e ′ th (t, y) = ĉ∂ 2 y e ′ th (t, y) + 3γ 4 ∂ 2 y wφ (t, y) • e + 12γπ 2 R h 2 W φ (t, y, h) • e dh.
Here wφ (t, y) is defined in (7.15),

W T φ (t, y, k) := [W φ,+ (t, y, k), Y φ,+ (t, y, k), Y φ,-(t, y, k), W φ,-(t, y, k)] and W φ,± (t, y, k) = R e 2πiyη W φ,± (t, η, k)dη, Y φ,± (t, y, k) = R e 2πiyη Y φ,± (t, η, k)dη,
with W φ,± and Y φ,± given by (7.14). An elementary calculation yields the following.

A similar consideration leads to the estimate

| D(ǫ) j | R ǫ + λǫ 2 , k ∈ T, |η| ≤ M, ǫ ∈ (0, ǫ 0 ], λ > λ 0 (8.4)
for any j ∈ {1, 2, +, -}. In particular we can conclude from (6.39) that

j | d(ǫ) j | (R ǫ + λǫ 2 ) 3 ≈ 3 j=0 R 3-j ǫ (λǫ 2 ) j . (8.5)
Thanks to (6.28) we infer that

R| d(ǫ) j | (R ǫ + λǫ 2 ) 4 δǫ , j ∈ {1, 2, o, +, -}.
8.1. Proof of (7.9). We show (7.9) for (ι, ι ′ ) = (-, +). The cases of other values of (ι, ι ′ ) can be handled in the same way. We use the second equation of the system (7.6). Estimate in question follows, provided we can show that the left hand side of the equation can be written in the form

ǫ -2 y (-) ǫ (λ, η) + Tǫ (λ, η) = z (2,-) ǫ (λ, η), (8.6)
where Tǫ (λ, η) = O(1), (8.7)

z (2,-) ǫ (λ, η) = O(1), as ǫ ≪ 1.
We can write Tǫ (λ, η)

= -2b (ǫ) o v (-) ǫ -2b (ǫ) -v (+) ǫ , (8.8)
where v

(±) ǫ (λ, η) := v ǫ (λ, η, •
), e ± (see (6.17

± := - 3γ 4ǫ 2 T e 2 ± ∆2,ǫ dk δǫ .
Substituting from (6.13) into (7.5) we find

ǫ -2 ∆2,ǫ = δ ǫ ω [2η ωR ′ -2(γR ǫ )δ ǫ ω] -4λ(γR ǫ ) (γR ǫ ) + ǫ 2 λ + ǫ 2 λ (δ ǫ ω) 2 -(R ′ η) 2 + ǫ 4 λ 3 (8.10) +i 4λω(γR ǫ ) + ǫ 2 λδ ǫ ωR ′ η + 2ǫ 2 ω(δ ǫ ω) 2 + 2ǫ 2 λ 2 ω .
Therefore, cf (6.27), we conclude that

ǫ -2 | ∆2,ǫ | e -e + + e 2 -+ e 2 + ǫ -2 | ∆2,ǫ |R 2 δ(0) ǫ . (8.11)
Thus,

|b (ǫ) o | + |b (ǫ) -| + |b (ǫ) + | 1 (8.12)
and the first equality of (8.7) follows.

Since (see (6.16))

f * ǫ (λ, -η, k) = f ǫ (λ, η, -k) = f ǫ (λ, η, k
) the right hand side of the second equation of system (7.6) can be written as

z (2,-) ǫ = Z ǫ,1 + Z ǫ,2 + Z ǫ,3 , (8.13)
where (d j,ǫ are the rows of the adjugate matrix to Dǫ , given by (6.38))

Z ǫ,1 := T d 2,ǫ • W ǫ e -dk δǫ , (8.14)
Z ǫ,2 := - γ(πη) 2 2 T ∆2,ǫ e -f ǫ δǫ dk, Z ǫ,3 := ǫ T d 2,ǫ • r ǫ e -dk δǫ .
We can write

|Z ǫ,1 | ≤ j sup k e -| d(ǫ) j | | δǫ | ι=± W ǫ,ι A ′ + Y ǫ,ι A ′ . (8.15)
Using Lemma 8.1 we conclude that for any M > 0 there exists λ 0 such that for any λ > λ 0 we have |Z ǫ,1 | = O(1), as ǫ ≪ 1. A similar argument allows us to conclude that also |Z ǫ,j | = O(1), as ǫ ≪ 1 for j = 2, 3. Thus, the second equality in (8.7) follows as well.

8.2. Proof of (7.8). The left hand side of the first equation of the system (7.6) can be rewritten in the following form

a (ǫ) w,-w (-) ǫ -a (ǫ) - δw ǫ ǫ 2 + a (ǫ) o y (-) ǫ,o ǫ 2 + a (ǫ) - y (+) ǫ,o
w,± := ǫ -2 1 -2 T (γR)e ± ∆1,ǫ δǫ dk , (8.17)
a (ǫ) o := 3γ 4 T e -e + ∆1,ǫ δǫ dk, a (ǫ)
± := 3γ 4 T e 2 ± ∆1,ǫ δǫ dk.
Note that e -e + R 3 (see (6.6)). From Lemma 8.1 and the Lebesgue dominated convergence theorem we conclude that a

ω ωR ′ + 2η 2 γR ′′ (γR ǫ )ω 2 + 4 (γR ǫ δ ǫ ω) 2 -2(γR ǫ )δ ǫ ω ω R′ η + (ωδ ǫ ω) 2 +4λ(γR ǫ ) 4(γR ǫ ) 2 + 4ω 2 + (ǫδ ǫ ω) 2 -ǫ R′ η 2 -4(γR)λ 2(γR ǫ ) 2 + 2ω 2 -2 ǫR ′ η 2 2 +4ǫ 2    λ 2   5(γR ǫ ) 2 + ω2 + (ǫδ ǫ ω) 2 - ǫ R′ η 2 2   -2γRλ 2 (γR ǫ )    -2(γR)λ 3 ǫ 4 + 8ǫ 4 λ 3 γR ǫ + ǫ 6 λ 4 .
Taking into account (6.28) we conclude that e -ẽǫ δǫ . In addition,

ẽǫ = 2η 2 γR ′′ (γR ǫ )ω 2 + 8λ(γR ǫ )[(γR ǫ ) 2 + ω2 ] + o(1), as ǫ ≪ 1.
Combining this with the second formula of (8.18) we obtain, by the Lebesgue dominated convergence theorem,

ā := lim ǫ→0+ γa (ǫ) w,-< +∞. (8.19)
Using the above together with bound (7.9) we conclude that expression (8.16) can be written as

ā γ [1 + o(1)]w (-) ǫ -ā-[1 + o(1)] δw ǫ ǫ 2 + O(1), as ǫ ≪ 1. (8.20)
Then bound (7.8) would follow, provided we can show that the right hand side of the first equation of the system (7.6), given by z

(1,-) ǫ
, is of order of magnitude O(1), as ǫ ≪ 1. To see that we write

z (1,-) ǫ = U ǫ,1 + U ǫ,2 + U ǫ,3 , (8.21)
where the terms U ǫ,i , i = 1, 2, 3 are given by

U ǫ,1 = T d 1,ǫ • W ǫ e -dk δǫ , (8.22)
U ǫ,2 := - γ(πη) 2 2 T ∆1,ǫ e -f ǫ δǫ dk, U ǫ,3 := ǫ T d 1,ǫ • r ǫ e -dk δǫ .
The fact that z

(1,-) ǫ

= O(1), as ǫ ≪ 1, can be argued in a similar way as it has been done in the case of z (2,-) ǫ , see (8.14) and (8.15) above. 8.3. Proof of (7.10). From (6.20) we obtain

w ǫ,+ (λ, η, k) = I ǫ + II ǫ + III ǫ + IV ǫ ,
where

I ǫ := 3γ 2 δǫ ∆(ǫ) 1 ι∈{-,+} v ǫ , e ι L 2 (T) e -ι , (8.23
) Using the above and the already proved estimates (7.8), (7.9) and Lemma 8.1 we obtain

II ǫ := - γ(πǫη) 2 2 δǫ ∆(ǫ) 1 f + v ǫ , 16f + + e -L 2 (T) + e -v ǫ , f + L 2 (T) +3f -v ǫ , e + L 2 (T) + 3e + v ǫ , f -L 2 (T) , (8.24)
III ǫ := ǫ 2 δ-1 ǫ d ǫ,1 • W ǫ , IV ǫ := ǫ 3 δ-1 ǫ d ǫ,1 • r ǫ .
lim ǫ→0+ I ǫ -w (-) ǫ f L 1 (T) = 0. ( 8
ǫ 2 T | δ-1 ǫ W ǫ d(ǫ) 1 |Rdk ǫ 2 W ǫ A ′ → 0, as ǫ → 0 + .
The remaining terms appearing in expressions II ǫ , III ǫ and IV ǫ can be estimated in the same manner allowing us to conclude that

lim ǫ→0+ T (|II ǫ | + |III ǫ | + |IV ǫ |) Rdk = 0.
8.4. Proof of (7.11). From (6.20) we obtain

y ǫ (λ, η, k) = I ǫ + II ǫ + III ǫ + IV ǫ ,
where

I ǫ := 3γ 2 δǫ ∆(ǫ) 2 ι∈{-,+} v ǫ , e ι L 2 (T) e -ι , II ǫ := - γ(πǫη) 2 2 δǫ ∆(ǫ) 2 f + v ǫ , 16f + + e -L 2 (T) + e -v ǫ , f + L 2 (T) +3f -v ǫ , e + L 2 (T) + 3e + v ǫ , f -L 2 (T) , III ǫ := ǫ 2 δ-1 ǫ d ǫ,2 • W ǫ , IV ǫ := ǫ 3 δ-1 ǫ d ǫ,2 • r ǫ .
The analysis of the above terms is very similar to what has been done in the precious section. Using (8.10) we conclude that for any λ > λ 0

∆(ǫ) 2 (λ, η, k) δǫ (λ, η, k) ǫ 2 , k ∈ T, |η| ≤ M. (8.27)
We conclude in this way that all RI ǫ , RII ǫ , RIII ǫ and RIV ǫ tend to 0 in the L 1 sense. Thus, (7.11) follows. ǫ . Since functions e ± (k) are both even the fourth and eighth equation of the system (7.6) coincide with the first and the fifth ones respectively.

Adding the first and fifth equations of the system (7.6) we get

γa (ǫ) w w (-) ǫ - ι∈{-,+} a (ǫ) y,-ι ŷ(ι) ǫ,o = γz (1,o) ǫ + γa (ǫ) w,-δw ǫ (9.1) Here a (ǫ) w,ι , a (ǫ)
ι , ι ∈ {o, -, +} are given by (8.17) and ŷ(±)

ǫ,o := ǫ -2 y (±) ǫ,o .
In addition

a (ǫ) w := a (ǫ) w,-+ a (ǫ) w,+ = 4 3ǫ 2 T γR 1 -2(γR) ∆1,ǫ δǫ dk, a (ǫ)
y,± := a

(ǫ) ± + a (ǫ) o := T e ± ∆1,ǫ γRdk δǫ and z (1,o) ǫ := z (1,-) ǫ + z (1,+) ǫ , (9.2)
where z

(1,±) ǫ are the scalar products of z

ǫ by e ± (cf (7.3) and (7.7)). The second and third equations of (7.6) read (cf (8.9))

γb (ǫ) o w (-) ǫ + γb (ǫ) -w (+) ǫ + ŷ(-) ǫ,+ = γz (2,-) ǫ,+ -ǫ 2 b (ǫ) o ŷ(-) ǫ,o + b (ǫ) - ŷ(+) ǫ,o , (9.3)
γ(b (ǫ) o ) * w (-) ǫ + γ(b (ǫ) -) * w (+) ǫ + ŷ(-) ǫ,+ = γz (2,-) ǫ,--ǫ 2 (b (ǫ) o ) * ŷ(-) ǫ,o + (b (ǫ) -) * ŷ(+) ǫ,o .
Adding sideways these equations we get

2γw (-) ǫ Re b (ǫ) o + 2γw (+) ǫ Re b (ǫ) -+ ŷ(-) ǫ,o = γz (2,-) ǫ,o + r (-) ǫ . (9.4)
Here z

(2,±) ǫ,o

:= z (2,±) ǫ + z (2,±) ǫ,- and
r (-) ǫ := -2ǫ 2 ŷ(-) ǫ,o Re b (ǫ) o + ŷ(+) ǫ,o Re b (ǫ) - , (9.5)
The sixth and seventh equations of (7.6) yield 2γw (-) ǫ Re b

(ǫ) + + 2γw (+) ǫ Re b (ǫ) o + ŷ(+) ǫ,o = γz (2,+) ǫ,o + r (+) ǫ , (9.6)
and

r (+) ǫ := -2ǫ 2 ŷ(-) ǫ,o Re b (ǫ) + + ŷ(+) ǫ,o Re b (ǫ) o . (9.7)
Summarizing, we have obtained the following system

γa (ǫ) w w (-) ǫ - ι∈{-,+} a (ǫ) y,-ι ŷ(ι) ǫ,o = γz (1,o) ǫ + γa (ǫ) w,-δw ǫ , 2γw (-) ǫ Re b (ǫ) o + 2γw (+) ǫ Re b (ǫ) -+ ŷ(-) ǫ,o = γz (2,-) ǫ,o + r (-) ǫ , (9.8)
2γw (-) ǫ Re b

(ǫ) + + 2γw (+) ǫ Re b (ǫ) o + ŷ(+) ǫ,o = γz (2,+) ǫ,o + r (+) ǫ .
Using Theorem 7.1 we conclude that given M > 0 and λ > λ 0 the family (w

(-) ǫ , ŷ(-) ǫ,o , ŷ(+) ǫ,o ) remains bounded in L ∞ [-M, M],
as ǫ → 0+. It is therefore * -weakly sequentially compact in this space. Denote by (w (-) , ŷ(-) o , ŷ(+) o ) (9.9) its * -weak limit. Thanks to (8.19), (8.12) and the results of Theorem

= 4γ(R -R ǫ )ηδ ǫ ω ωR ′ + 2γη 2 R ′′ (k)(γR ǫ )ω 2 +4 γ(R -R ǫ )(γR ǫ )(δ ǫ ω) 2 + (ωδ ǫ ω) 2 -4(γR)λ 2ω 2 -2 ǫR ′ η 2 2 +4λ(γR ǫ ) 4γ(R -R ǫ )(γR ǫ ) + 4ω 2 + (ǫδ ǫ ω) 2 -(ǫR ′ η) 2 (9.14) +4ǫ 2 λγ 2 R(δ ǫ ω) 2 + λ 2 5γ(R -R ǫ )(γR ǫ ) + ω2 + (ǫδ ǫ ω) 2 + 8ǫ 4 λ 3 γR ǫ + ǫ 6 λ 4 .
Using Lemma 8.1 and the Lebesgue dominated convergence theorem we obtain

lim ǫ→0+ 4γ 3 T R fǫ δǫ dk = 2λ 3 + η 2 3γ T (ω ′ ) 2 R dk. (9.15)
Using the above formula and substituting

z (1,o) ǫ - z (2,o) ǫ,o 2 = z (-) ǫ + z (+) ǫ • e
(cf (7.3) and (7.7)) we can rewrite (9.12) in the form

2λ 3 + η 2 3γ T (ω ′ ) 2 R dk w (-) = lim ǫ→0+ 3 j=1 V ǫ,j , (9.16)
where

V ǫ,1 = 2γ 3 T w (0) ǫ (λ, η, k) • e Rdk, (9.17)
V ǫ,2 = - 2γ(πη) 2 3 T ( ∆1,ǫ -∆2,ǫ ) Rf ǫ δǫ dk, V ǫ,3 := γǫ T ∆1,ǫ Rr (1) ǫ δǫ dk + γǫ 2 T (2 d(ǫ) -- d(ǫ) 2 -d(ǫ) o ) Rr (2) ǫ δǫ dk + γǫ 2 T [2( d(ǫ) + ) * -d(ǫ) o -( d(ǫ) 2 ) * ] Rr (2) ǫ,- δǫ dk.
Here (w (0) ǫ ) T (λ, η, k) := [w

(0) ǫ,+ , y (0)
ǫ,+ , y

ǫ,-, w

(0) ǫ,-] (9.18)
is the solution of the system Dǫ w

(0) ǫ (λ, η, k) = W ǫ (η, k), (9.19)
where W ǫ (η, k) is the column vector of Fourier-Wigner functions corresponding to the initial data, see (5.5). In addition, f ǫ is given by (6.16) respectively, and r

ǫ,1 + V

ǫ,1 , where

V (1) ǫ,1 := 2γ 3 T w (0) ǫ (λ, η, k) • e Rdk, V (2)
ǫ,1 := 2γ 3 T w (0) ǫ (λ, η, k) • e Rdk.
Here

w (0) ǫ = [w (0) ǫ,+ , y (0)
ǫ,+ , y

ǫ,-, w

(0) ǫ,-], w (0) ǫ = [ w (0) ǫ,+ , y (0)
ǫ,+ , y

ǫ,-, w

(0) ǫ,-]
are the solutions of the analogues of (9.19) in which the right hand side has been replaced by W ǫ and W ǫ , respectively. 9.2.1. Macroscopic Wigner functions and their dynamics. From (3.1) we get

∂ t φ(t, k) = -iτ 2 (πk) 2 φ(t, k) -6γπ 2 k 2 φ(t, k) -φ * (t, -k) . (9.23)
Therefore the Fourier transforms W φ (t) of the macroscopic Wigner functions (cf (7.14)) satisfy

∂ t W φ,+ = -2π 2 iτ 2 kη + 6γ k 2 + η 2 2 W φ,+ + 6γπ 2 ι ′ ∈{-,+} k -ι ′ η 2 2 Y φ,ι ′ , (9.24)
∂ t Y φ,+ = -2π 2 (iιτ 2 + 6γ) k 2 + η 2 2 Y φ,+ + 6γπ 2 ι ′ =±1 k -ι ′ η 2 2 W φ,ι ′ .
Taking the Laplace transforms of both sides of (9.24) we obtain D1 w φ,+ + D+ y φ,+ + Dy φ,-= W φ,+ (0, η, k) (9.25) and D+ w φ,+ + D2 y φ,+ + Dw φ,-= Y φ,+ (0, η, k), (9.26) where for each λ > 0 and (η, k) ∈ R 2 . The above, combined with (6.28), implies that for any M > 0 and λ 0 as in Proposition 6.1 we have

D1 := λ + 2π 2 6γ k 2 + η 2 2 + iτ 2 kη , (9.27)
D2 := λ + 2π 2 k 2 + η 2 2 (6γ + iτ 2 ), D± := -6γπ 2 k ∓ η 2 2 . ( 9
δ(λ, η, k) > 0, |η| < M, k ∈ R, λ > λ 0 . ( 9
( D) = P Q Q M , (9.36)
where M, P and Q are 2 × 2 matrices given by

P := d1 d- d-d2 , Q := ( d+ ) * do do d+ , M := ( d2 ) * ( d-) * ( d-) * ( d1 ) * . Here d1 := | D2 | 2 D * 1 -( D2 + + D2 -)Re D2 -i( D2 + -D2 -)Im D2 , d2 := | D1 | 2 D * 2 -( D2 + + D2 -)Re D1 -i( D2 + -D2 -)Im D1 , d-:= D+ ( D2 + -D2 -) -D+ D * 1 Re D2 + i D * 1 D+ Im D2 = D+ ( D2 + -D2 -) -D+ D * 2 Re D1 + i D * 2 D+ Im D1 , d+ := -D-( D1 D * 2 + D2 + -D2 -), do := 2 D+ D-Re D2 . ( 9
ǫ,1 . The limit in question is a special case of the following result. Proposition 9.1. Suppose that ϕ ∈ C 2 (T) is such that ϕ(0) = ϕ ′ (0) = 0. Then, for any M > 0 we have

lim ǫ→0+ T w (0) ǫ,± (λ, η, k)ϕ(k)dk = 1 2 ϕ ′′ (0) R k 2 w φ,± (λ, η, k)dk (9.39) and lim ǫ→0+ T y (0) ǫ,± (λ, η, k)ϕ(k)dk = 1 2 ϕ ′′ (0) R k 2 y φ,± (λ, η, k)dk, |η| ≤ M, λ > λ 0 .
(9.40) Here w φ,± and y φ,± are given by (7.13).

Proof. We only prove (9.39), as the argument for (9.40) is very similar. The left hand side of (9.39) for w (0) ǫ,+ can be rewritten in the form

T δ-1 ǫ d(ǫ) 1 W ǫ,+ + d(ǫ) -Y ǫ,+ + ( d(ǫ) + ) * Y ǫ,-+ d(ǫ) o W ǫ,-ϕ(k)dk.
Denote by J j,ǫ , j = 1, 2, 3, 4 the respective terms arising after opening of the square bracket. Changing variables k := k/ǫ we can write (cf (5.13))

J 1,ǫ = 1 2 x,x ′ 1/(2ǫ) -1/(2ǫ) ( δ-1 ǫ d(ǫ) 1 )(λ, η, ǫk) φ * k + x ǫ - η 2 φ k + x ′ ǫ + η 2 ϕ(ǫk)dk.
Thanks to Lemma 8.1 we conclude that there exist λ 0 , ǫ 0 > 0 such that for any λ > λ 0 we have sup

ǫ∈(0,ǫ 0 ] sup k,|η|≤M R(ǫk) d(ǫ) ι (λ, η, ǫk) δǫ (λ, η, ǫk) < +∞.
In addition, we have sup

ǫ∈(0,1] sup |k|≤1/(2ǫ) (ǫ|k|) 2 R(ǫk) < +∞. Therefore, sup ǫ∈(0,ǫ 0 ] sup k,|η|≤M |ϕ(ǫk)| d(ǫ) ι (λ, η, ǫk) δǫ (λ, η, ǫk) < +∞.
In fact, thanks to the rapid decay of the macroscopic wave function φ, we can write

lim ǫ→0+ J 1,ǫ = 1 4 lim ǫ→0+ 1/(2ǫ) -1/(2ǫ) ( δ-1 ǫ d(ǫ) 1 )(λ, η, ǫk) φ * k - η 2 φ k + η 2 ǫ 2 (ϕ ′′ (0)k 2 +o(1))dk.
(9.41) By virtue of the Lebesgue dominated convergence theorem, we conclude that the limit in (9.41) equals ϕ ′′ (0) 4

R ( δ-1 d1 )(λ, η, k)k 2 φ * k - η 2 φ k + η 2 dk.
Dealing similarly with the remaining terms J j,ǫ , j = 2, 3, 4 we conclude (9.39) for w

ǫ,± can be handled similarly.

Since R(0) = R ′ (0) = 0 and R ′′ (0) = 12π 2 (cf (6.9) and (6.10)), by a direct application of Proposition 9.1, we obtain

lim ǫ→0+ V (1) ǫ,1 = 8γπ 2 R k 2 w φ (λ, η, k) • e dk (9.42) for all |η| ≤ M and λ > λ 0 . 9.2.3. Limit of V (2) ǫ,1 . For any J ∈ S(R) such that Ĵ is supported in [-M, M] we can write R V (2) ǫ,1 (λ, η) Ĵ * (η)dη (9.43) = 2γ 3 R T Ĵ * { W ǫ,+ ∆1,ǫ + Y ǫ,+ ∆2,ǫ + Y ǫ,- ∆ * 2,ǫ + W ǫ,- ∆ * 1,ǫ } Rdηdk δǫ .
By virtue of Lemma 8.1 we can use the Lebesgue dominated convergence theorem to enter with the limit, as ǫ → 0+, under the integral. Combining (8.25) and (8.27) we conclude that

lim ǫ→0+ R V (2) ǫ,1 Ĵ * (η)dη = 2 3 lim ǫ→0+ R T Ĵ * (η) W ǫ,+ (η, k)dηdk = lim ǫ→0+ ǫ 3 x | ψ(ǫ) x | 2 J(ǫx) = 2
3 R e th (y)J * (y)dy. (9.44)

The penultimate equality follows from (5.10).

10. Proof of Theorem 7.3 10.1. Proof of (7.17). Recall that dj (λ, η, k), j ∈ {1, 2, o, -, +} and det D(λ, η, k) are given by (9.37) and (9.33) respectively. We recall also ∆j (λ, q, k), j = 1, 2 are defined by a modification of formulas (7.5) where the coefficients d(ǫ) ι have been replaced by the corresponding dι . Given ϕ ∈ C(T) we can write

T w ǫ,+ (λ, η, k)ϕ(k)dk = T (I ǫ + II ǫ + III ǫ + IV ǫ )ϕ(k)dk.
Here I ǫ , II ǫ , III ǫ , IV ǫ are given by (8.23). By virtue of (8.26) we conclude that Proof. Note that, according to Proposition 6.1, for each M > 0 we can choose λ 0 , ǫ 0 > 0 such that for λ > λ 0 The computation of the limit, as ǫ → 0+, of each of the four expressions J (ǫ) j , j = 1, . . . , 4 that arise in the right hand side after opening of the bracket is almost identical so we explain only how to deal with the first one. Using (5.13) we can write that J Thanks to (10.11) to compute the last limit we can use the Lebesgue dominated convergence and conclude, using (9.34) and (9.38), that the right hand side of the above equality coincides with the right hand side of (10.8).

T |IV ǫ (λ, p, k)|dk ≤ ǫ 3 4 j=1 r (j) ǫ A ′ sup k δ-1 ǫ j | d(ǫ) j | (
10.1.2. Proof of (10.7). Using condition (2.12) we conclude that for some r > 1

  .12) Thanks to the hypothesis (2.7) we conclude that for any G ∈ C ∞ 0 ) k x µǫ = R G(y)κ(y)dy.

2 p 2

 22 any G ∈ C ∞ 0 (R). Here e(y) -the macroscopic energy profile -is given by e(y) = 1 (y) + ακ 2 (y) .

. 2 ) 5 . 1 .

 251 Here B M := [η : |η| < M]. We drop the subsrcipt from the notation if M = +∞. Let A ′ and A ′ M be the respective topological dual spaces of A and A M . Wigner function and its evolution 5.The wave function. The wave function corresponding to the configuration ((p x , k x )) x∈Z is defined as

Proposition 6 . 2 .

 62 For any M, λ > 0 we have d(ǫ) 1 = 4(γR) 3 + 8γRω 2 + o(1), d(ǫ) 2 = 4(γR) 3 -8i(γR) 2 ω + o(1), d(ǫ) ± = 4(γR) 3 -4i(γR) 2 ω + o(1), (6.40) d(ǫ) o = 4(γR) 3 + o(1), δǫ = 16(γRω) 2 + o(1), as ǫ ≪ 1, uniformly in |η| ≤ M for any k ∈ T.

  ′ (λ, η)| ǫ 2 , ǫ ∈ (0, ǫ 0 ], |η| ≤ M, ι, ι ′ ∈ {-, +}. (7.9) Moreover, for any |η| ≤ M and λ > λ 0 we have lim ǫ→0+ T w ǫ,+ (λ, η, k) -w (-) ǫ (λ, η) R(k)dk = 0 (7.10) and lim ǫ→0+ T |y ǫ,± (λ, η, k)| R(k)dk = 0. (7.11)

( 7 . 2 Rh 2

 722 15) Define w (-) (λ, η) by the formula λ + ĉ(2πη) 2 w (-) (λ, η) = -3γ(πη)2 wφ (λ, η) w φ (λ, η, h) • e dh + êth (η), (λ, η) ∈ (0, +∞) × R.

Theorem 7 . 3 .

 73 For any M > 0 we havelim ǫ→0+ T w ǫ,+ (λ, η, k)ϕ(k)dk -w (-) ǫ (λ, η) T ϕ(k)dk -w φ (λ, η)ϕ(0) = 0 (7.17) and lim ǫ→0+ T y ǫ,± (λ, η, k)ϕ(k)dk = y φ,± (λ, η)ϕ(0),(7.18)for all |η| ≤ M, λ > λ 0 and ϕ ∈ C(T).

  order O(1), as ǫ ≪ 1. Using (7.5) together with formulas (6.40) we infer that ∆1,ǫ = 8γR ǫ ω2 + o(1), as ǫ ≪ 14(γR)ηδ ǫ

8. 3 . 1 .

 31 Convergence of I ǫ . Note that, (see (6.26)) for any k =

9 . 2 9. 1 .

 921 Proof of Theorem 7.Determining w (-)

  8.1, equalities (6.40) and the Lebesgue dominated convergence theorem we conclude that lim ǫ→0+ a (ǫ) y,ι = -1 2 , ι ∈ {-, +}. (9.11) Subtracting sideways from the first equation of (9.8) the sum of the remaining two and taking into account (9.10) and (9.11) we obtain lim . Moreover, a direct calculation shows that γ a (ǫ) w -2γ ι=± Re b (ǫ)

ǫ 9 . 2 .

 92 , i = 1, 2 satisfy (6.18). Thanks to Lemma 8.1 we conclude, upon an application of the Lebesgue dominated convergence theorem, that lim ǫ→0+ V ǫ,3 = 0. (9.20) Using Theorems 7.1 and 7.3 and the definition of f ǫ (see (6.16)) we get lim ǫ→0+ V ǫ,2 = -8γ(πη) 2 w (-) -2γ(πη) 2 vφ , (9.21) with (cf (7.15)) vφ (λ, η) := wφ (λ, η) -1 2 (ȳ φ,-(λ, η) + ȳφ,+ (λ, η)) . (9.22) Limit of V ǫ,1 . Using the decomposition of the Wigner functions of the initial data into the parts corresponding to the macroscopic profile and the fluctuations, see (5.12), we can write an analogous decomposition W ǫ = W ǫ + W ǫ , for the Laplace-Fourier transforms of the respective Wigner functions. It allows us to write V ǫ,1 = V

ǫ - 2 D

 2 (ǫ) j (λ, q, ǫk) = Dj (λ, η, k), j ∈ {1, 2, -, +}(9.29) for any λ > 0 and (η, k) ∈ R 2 .The closed system of linear algebraic equations for the components of the Laplace-Fourier transforms w φ (cf (7.12)) takes the form Dw φ = W φ (0and B = D-I 2 (cf (6.21) and (6.22)). It can be checked by a direct inspection that[A, B] = [B, C] = [A, C] = 0. Therefore, δ(λ, η, k) := det D(λ, η, k) = det(AC -B 2 ) (9.33) = det D(λ, q, k) = | D1 D * 2 + D2 + -D2 -| 2 -4 D2 + Re D1 Re D2 .Thanks to (9.29) we conclude that lim ǫ→0+ ǫ -8 δǫ (λ, η, ǫk) = δ(λ, η, k) (9.34)

  .35) The matrix D(λ, η, k) is then invertible and, cf (6.38), D-1 = δ-1 adj( D). The adjugate of D equals adj

  . The cases of w (0) ǫ,-and y

Lemma 10 . 1 .

 101 dk, |η| ≤ M, λ > λ 0 . For any M > 0 lim ǫ→0+ T |IV ǫ |dk = 0, |η| ≤ M, λ > λ 0 . (10.1)

1 ǫ.ǫ 2 5 )

 125 |η| ≤ M and ǫ ∈ (0, ǫ 0 ]. Thanks to(8.5) we conclude thatǫ 3 δ(0) ǫ (λǫ 2 ) j . (10.3)Invoking the definition of δ(0) ǫ , see (6.27), we can bound the right hand side of (10.3) by ǫ 3 (R ǫ + λǫ 2 ) -The conclusion of the lemma follows then directly from the above estimate and (6.18).Using a similar argument we infer that for any λ > λ 0 , |η| ≤ M we havesup k |III ǫ | ǫ 2 supk ǫ ∈ (0, ǫ 0 ], thanks to (8.5) and (6.27). On the other hand, due to Proposition 6.2, for any λ > λ 0 , k = 0 and |η| ≤ M we have lim ǫ→0+ By virtue of the Lebesgue dominated convergence theorem we conclude therefore that limǫ→0+ T III ǫ ϕ(k)dk = 0.Finally, we haveT II ǫ ϕ(k)dk = ǫ 2 T δ-1 ǫ { W ǫ,+ d(ǫ) 1 + Y ǫ,+ d(ǫ) -+ Y ǫ,-( d(ǫ) + ) * + W ǫ,- d(ǫ) o }ϕ(k)dk. (10.6)

1 WJ 8 ) 1 {ǫJ- 1 /

 1811 ǫ,+ (η, k)ϕ(k)dk.In what follows we show that lim ǫ→0+We repeat the above argument to compute the limits of the remaining termsJ W φ,+ (0) d1 + Y φ,+ (0) d-+ Y φ,-(0)( d+ ) * + W φ,-(0) do }dk = ϕ(0) R w φ,+ (λ, η, k)dk. 10.1.1. Proof of(10.8). After the change of variables k ′ := k/ǫ we can writeJ -6 d(ǫ)1 (λ, η, ǫk) ǫ -8 δǫ (λ, η, ǫk) Using the argument from the proof of Lemma 10.1 we conclude that for any M > 0 and λ > λ 0 , where λ 0 , ǫ 0 > 0 are as in the statement of Proposition 6.1,ǫ -6 d(ǫ) 1 (λ, η, ǫk) ǫ -8 δǫ (λ, η, ǫk) 1,(10.11) for all k ∈ R, |η| ≤ M and ǫ ∈ (0, ǫ 0 ]. Due to the decay of the wave function φ we conclude that lim ǫ→0+ (2ǫ) ǫ -6 d(ǫ) 1 (λ, η, ǫk) ǫ -8 δǫ (λ, η, ǫk) φ * k -η 2 φ k + η 2 ϕ(ǫk)dk.

| 2 T| 2 . 1 {

 221 W ǫ,+ (η, k)| r dk < +∞.(10.12) Combining the above with estimate (10.4) together with the limit (10.5) we conclude that for any λ > λ 0 and |η| ≤ Mlim ǫ→0+ ǫ W ǫ,+ (η, k)ϕ(k)| d(ǫ) 1 (λ, η, k) δǫ (λ, η, k) dk = 0This obviously implies (10.7).10.Proof of (7.18). We use the notation from Section 8.4 and carry out our analysis only for y ǫ,+ , as the argument for y ǫ,-is very similar.For any ϕ ∈ C(T) we have T y ǫ (λ, q, k)ϕdk = T I ǫ ϕdk + T II ǫ ϕdk + T III ǫ ϕdk + T IV ǫ ϕdk.(10.13) The analysis of the terms on the right hand side of (10.13) is very similar to the one done in Section 10.1.2. As a result we obtainlim ǫ→0+ T (|I ǫ | + |III ǫ | + |IV ǫ |)dk = 0. W φ,+ d-+ Y φ,+ d2 + Y φ,-(do ) * + W φ,-d-}dk = ϕ(0) R y φ,+ (λ, η, k)dk and (7.18) follows.

(11. 2 ) 2 =

 22 After a straightforward calculation we obtain, using (2.13) that for anyG ∈ C ∞ 0 τ 2 π 2 k 2 and R(ǫk) ǫ 2 = 6π 2 k 2 ,uniformly on compact intervals, an elementary stability theory for solutions of ordinary differential equations guarantees that for any T, M > 0 we haveψǫ (t, k) = (1 + o(1)) ψ(0) ǫ (t, k),(11.4)uniformly on |k| ≤ M, |t| ≤ T , as ǫ ≪ 1, where ψ(0) ǫ (0, k) satisfies (11.1) with the initial condition ψ(0) ǫ (0, k) := ψǫ (0, k). Equation (11.1) can be solved explicitly. Taking into account (11.3) we obtain, upon letting ǫ → 0+, thatlim ǫ→0+ R Ĝ(k) ψ(0) ǫ (t, -k) dk = R Ĝ(k) ψ (t, -k) dk,(11.5)where ψ (t, k) satisfies (11.1) with the initial ψ(0, k) := φ(0, k). Therefore ψ(t, k) = φ(t, k) and, in conclusion,lim ǫ→0+ ǫ x G(ǫx)E ǫ ψ (ǫ) x (t) = limǫ→0+ R Ĝ(k) ψǫ (t, -k) dk = R Ĝ(k) φ (t, -k) dk = R G(y)φ (t, y) dy.

  Convergence of II ǫ , III ǫ and IV ǫ . Thanks to Lemma 8.1 we can write

	.26)
	Here f(k) ≡ 1.
	8.3.2.

  11. Proof of Theorem3.1 = -iτ 2 (πk) 2 φ(t, k) -6γπ 2 k 2 φ(t, k) -( φ) * (t, -k) . (11.1) Let ψǫ (t, k) := ǫE ǫ ψ(ǫ) (t, ǫk).

	Suppose that κ(t, y) and p(t, y) satisfy (3.1). Then, φ(t, k) -the
	Fourier transform of
			φ(t, y) :=	τ 2 4	κ(t, y) + ip(t, y)
	satisfies
	d dt φ(t, k) From (5.18) we obtain
	d dt	ψǫ (t, k) =	-iω(ǫk) ǫ 2	ψǫ (t, k) -	γR(ǫk) ǫ 2

ψǫ (t, k) -( ψǫ ) * (t, -k) .

This paper has been partially supported by the European Advanced Grant Macroscopic Laws and Dynamical Systems (MALADY) (ERC AdG 246953), T. K. acknowledges the support of the Polish National Science Center grant UMO-2012/07/B/SR1/03320.

 Proposition 7.4. Suppose that φ(t, y) is given by (5.25). Then,

Using the proposition we conclude that the third term appearing in the right hand side of the second equation of (7.19) equals We can see therefore that e ′ th (t, y) satisfies (3.6). Thus the conclusion of Theorem 5.1 follows.

Proof of Theorem 7.1

We start with the following result. Lemma 8.1. For any M > 0 and ǫ 0 , λ 0 as in Proposition 6.1 we have

The summation extends over j ∈ {1, 2, o, -, +}.

Proof. From the definition of D(ǫ) 1 , see (6.13), we obtain

for k ∈ T, |η| ≤ M, ǫ ∈ (0, ǫ 0 ], λ > λ 0 . Using the first formula of (6.3) we conclude that then .3)