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 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 [11]. Generically this anomalous superdiffusive behavior is attributed to the momentum conservation properties of the dynamics [10]. Actually one dimensional FPUtype chains have potential energy depending on the interparticle distances (that is 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 This paper has been partially supported by the ANR-15-CE40-0020-01 Grant LSD; T. Komorowski acknowledges the support of the Polish National Science Center Grant UMO-2012/07/B/ST1/03320. 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. [6]). When the system approaches, 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 [12], 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 [8,9]. In particular, it has been shown in [9] 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 where q x are the positions of the particles. This means that its hamiltonian is formally given by where the energy of the oscillator x is defined 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 then 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: the curvature, momentum and energy evolve together in the diffusive time scale. Denote by k(t, y), p(t, y) and e(t, y) the respective macroscopic limits of the fields corresponding to the aforementioned quantities. The evolution of the macroscopic curvature and momentum fields k(t, y) and p(t, y) is governed by the damped Euler-Bernoulli beam equations: ∂ t k(t, y) = − y p(t, y), ∂ t p(t, y) = y αk(t, y) + 3γ p(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 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 open up the question if a similar behavior could appear in some deterministic non-linear Hamiltonian dynamics corresponding to an interaction of the type V (K x ), where ∂ q x H is a non linear function of the curvature of the chain.
Concerning our proof of the hydrodynamic limit. Since the microscopic energy currents in our system are not on the form of discrete space gradients of some function, this is a non-gradient system. Nevertheless we cannot use known techniques based on relative entropy methods (cf. for example [13]) 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 [8], 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 the 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.

Non Acoustic Chain of Harmonic Oscillators
Since in the non-acoustic 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 In case when no noise is present, the dynamics of the chain of oscillators can be written formally as a Hamiltonian system of differential equationṡ . Let also ∇g x := g x+1 − g x and ∇ * g x := g x−1 − g x .

Energy-Momentum Conserving
Noise. Following [1,2] we perturb the Hamiltonian dynamics (2.1) by introducing the random momentum exchange between the neighboring sites in such a way that the total momentum and energy of the system are conserved. This is achieved by adding to the right hand side of (2.1) a local stochastic term that conserves both p 2 The respective stochastic differential equations can be written as with the parameter γ > 0 that indicates the strength of the noise in the system, and (Y x ) vector fields given by Here (w x (t)) t≥0 , x ∈ Z are i.i.d. one dimensional, real valued, standard Brownian motions, over a probability space ( , F, P). Furthermore, the field (Ř x ) is defined as follows: As a result we obtainŘ 0 = 3/2,Ř ±1 = −1/2,Ř ±2 = −1/4 andŘ x = 0, if |x| ≥ 3. In addition, for any two fields (a x ), (b x ) we define their convolution by letting (a * b) x := y∈Z a x−y b y . This definition makes sense for example in the case when one of the fields is of compact support (as for example (Ř x ) here). We can rewrite the system (2.2) 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 momentum 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, κ), with β −1 ≥ 0 andp, κ ∈ R. The product measures are stationary for the dynamics defined by (2.4). In this context κ is called the load of the chain, while as usual β −1 is the temperature andp is the average momentum. When β −1 = 0 the measure in (2.5) is, by a convention, the product of delta type measures, each concentrated at (κ,p).

Initial Data
Concerning the initial data we assume that: (A1) given > 0, it is distributed according to a probability measure μ on the configuration of ((K x , p x )) x∈Z and satisfies Here · μ denotes the average with respect to μ . We denote also by E the expectation with respect to the product measure P = μ ⊗ P.
The existence and uniqueness of a solution to (2.2) in 2 , with the aforementioned initial condition can be easily concluded from the standard Hilbert space theory of stochastic differential equations, see for example Chapter 6 of [4]. We assume furthermore that (A2) the mean of the initial configuration varies on the macroscopic spatial scale: for some functions κ, p ∈ C ∞ 0 (R). Then, the respective Fourier transformŝ κ andp 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 > 0 for some r > 1. We have denoted byf the Fourier transform 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 The energy spectrum is defined as: wherep(k) andK(k) are the Fourier transforms of (p x ) and (K x ), respectively. Assumption (2.6) implies in particular that The announced property of the L r integrability of the energy spectrum can be formulated as follows: (A3) there exists r > 1 such that: Thanks to the hypothesis (2.7) we conclude that for any G ∈ C ∞ 0 (R) we have The quantities p(·), κ(·) are called the macroscopic velocity and curvature profiles. We assume furthermore that: (A4) the following limits exist Here j (·), p 2 (·), κ 2 (·) are some functions belonging to C ∞ 0 (R). As a consequence, we conclude that the limit also exists for any G ∈ C ∞ 0 (R). Here e(y) -the macroscopic energy profile -is given by Remark. An important example of initial distributions that satisfy the above conditions is provided by local Gibbs measures (see Section 9.2.5 of [9]). That is inhomogeneous product probability measures of the type Here the vector λ ) and the functions β −1 (·), p(·), κ(·) belong to C ∞ 0 (R). On the sites where β( x) −1 = 0, we let the corresponding exponential factor in (2.17) be a delta distribution concentrated at the point (κ( x), p( x)). In this case j (y) = κ(y) p(y), p 2 (y) = p 2 (y) + β −1 (y),
Remark. The above system is uniformly parabolic in the Petrovskii sense for any γ > 0, see for example p. 8 of [5]. The solution exists and is uniquely determined by the fact that its Fourier transform [p(t, η),κ(t, η)] satisfies the system of ordinary differential equations see pp. 44-47 in [5]. Our first result concerns the evolution of the macroscopic profiles of the velocity and curvature.
where p(t, y) and κ(t, y) is the solution of (3.1).
The proof of this result is fairly standard and we show it in Section 11. Define the macroscopic profile of the mechanical energy of the chain by with e  where e mech (t, y) is given by (3.4), while the thermal energy (temperature) e th (t, y) is the solution of the following Cauchy problem: (3.7) The diffusivity coefficient equals (see formula (9.21) below) Notice that the gradient of the macroscopic momentum p(t, y) appearing in (3.7) causes a local increase of the temperature. It is also straightforward to understand the appearance of this term in the aforementioned equation. Consider for simplicity the case α = 0. The dynamics is constituted then only by the random exchanges of the momentum. The conserved quantities that evolve macroscopically are the momentum p x and the kinetic energy p 2 x /2. The corresponding macroscopic equations are (3.9) These can be proven easily since the microscopic dynamics is of gradient type. It follows that the macroscopic equation for the temperature field, defined by e th = e − p 2 /2, is given by 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(π k) and c(k) := cos(π k), k ∈ T. (4.1) Given a set A and two functions f, g : 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 For J ∈ S we letĴ be its Fourier transform in the first variable, that iŝ For any M > 0 let A M be the completion of S in the norm We drop the subscript from the notation if M = +∞. Let A and A M be the respective topological dual spaces of A and A M . The space A M is made of equivalence classes of measurable functions, for which the pseudo-norm is finite.

The Wave Function
The wave function corresponding to the configuration ((p x , K x )) x∈Z is defined as Its Fourier transform is given bŷ The energy and its spectrum (2.10) can be written as Using the decomposition into the macroscopic profile and the fluctuation part, see (2.9), we can write 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 ,± , Y ,± that we often write together in the form of a vector and for any J ∈ A. Here W ,± (η, k) and Y ,± (η, k)-called the Fourier-Wigner functionsare given by For any J ∈ A we can write Using the Cauchy-Schwarz inequality and (2.11) we get By Plancherel's identity we obtain that for any J ∈ S where Using the decomposition of the wave function into its mean, following a macroscopic profile φ(·), and the fluctuation part {ψ ( ) x , x ∈ Z}, see (5.4), we can correspondingly decompose the vector of the Wigner functions. Namely, where the Fourier-Wigner function corresponding to these wave functions shall be denoted by and We let where, using the Poisson summation formula, we have defined 14) The formulas for Y ,± , J and Y ,± , J are constructed analogously using the respective Fourier-Wigner functions. Notice that for small the expression above of W ,± is well approximated by the more natural definition: As a consequence of assumption (2.14) we conclude that for functions J (y, k) = J (y): with j (·), κ 2 (·) and p 2 (·) given by (2.14). A simple calculation also shows that One can also easily check that

Evolution of the Wave Function
Adjusted to the macroscopic time, we can define the wave function corresponding to the configuration at time t/ 2 is the unique solution of the Itô stochastic differential equation, understood in the mild sense (see for example Theorem 7.4 of [4]) is the dispersion relation, and Here The process B(dt, dk) is a space-time Gaussian white noise, that is Since the total energy of the system is conserved in time, see Section 2 of [3], for each ∈ (0, 1] we have

Wigner Functions Corresponding to ψ ( ) (t)
Denote by the vector made of Wigner functions corresponding to the wave functions ψ ( ) (t). They can be defined by formulas (5.6) and (5.7), where the respective Fourier-Wigner functions W ,± (t, η, k) and Y ,± (t, η, k) are given by analogues of (5.8) in which the wave functions are substituted by ψ ( ) (t) and the average · μ is replaced by E , From (5.23) we conclude, thanks to (5.9), that 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 (·)) ∈(0,1] are * −weakly sequentially compact in L 1 ([0, +∞); A) * as → 0+, that is given a component of the above family, for example W ,+ (·), and any sequence n → 0+, one can choose a subsequence W 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 (3.7), while the mechanical one e mech (t, y) is defined by (3.4). The limit of the Wigner functions corresponding to the macroscopic profile wave function Our main result concerning the limit of the Wigner transform can be stated as follows.
Theorem 5.1. Suppose that the initial data satisfy the assumptions. Then, Analogously to formulas (5.10) and (5.11), we can write 32) (see (5.12)). Therefore, the conclusion of Theorem 3.2 is a direct consequence of Theorem 5.1.

Evolution of the Wigner Functions
Using (5.20) we can derive the equations describing the time evolution of the Wigner functions. In particular, one can conclude that for a fixed the components of (W (t)) t≥0 belong to C([0, +∞); A ). After a straightforward calculation (see Section 8 of [8] for details) we obtain that their Fourier transforms satisfy Here (cf (5.20)) where, with some abuse of notation cf (5.21), we have denoted the scattering kernel where e ± and f ± are the L 1 (T) normalized vectors given by Note also that (cf (5.21)) In addition, R (k) = 2π(s(2k) + s(4k)) (6.9) and R (k) = 4π 2 (4c 2 (2k) + c(2k) − 2). (6.10)

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 equationsD (6.12) Here In addition, we letD ( ) Remark. Note that R (0) > 0. Therefore, given M > 0 there exists 0 depending only on M > 0 and such that R (η, k) > 0 for all |η| ≤ M and k ∈ T.
The right hand sides of (6.11) and (6.12) are respectively equal R ( ) Here, for the abbreviation sake we have let can be rewritten in the matrix formD where andD is a 4 × 4 matrix that can be written in the block form where A , B , C are 2 × 2 matrices given by and B =D ( ) − I 2 , with I n denoting the n × n identity matrix. We have also denoted and W (η, k) is the column vector corresponding to the Fourier-Wigner transforms of the components of (5.5) and r T := r (1) , r (2) , r (2) ,− , r (1) ,− , (6.25) ,− (λ, η, k) := (r (2)

Invertibility of MatrixD
We prove that the matrixD 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.

Inverse ofD (λ, p, k)
Recall thatD (λ, p, k) is a 2 × 2 block matrix of the form (6.21). Since B is diagonal we have [A , B ] = [C , B ] = 0. A simple calculation shows that also (6.36) Therefore, provided that detD = 0. Note that Substituting into (6.37), using also (6.22) we conclude that the inverse matrixD −1 is a 2 × 2 block matrix of the formD −1 =δ −1 adj(D ) where the adjugate ofD equals where M , P and Q are 2 × 2 matrices given by For the abbreviation sake we denote by d j, , j = 1, . . . , 4 the vectors corresponding to the rows of the adjugate ofD 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 n → 0+ there exists a subsequence W 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 n by showing that for any M > 0 there exists λ 0 > 0 such that the vector w n (λ) made of Laplace transforms of the components of W 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 (t)), as → 0+ finishing in this way the proof of Theorem 5.1. From (6.20) we obtain w =D −1 R .
Unfortunately, the right hand side of the above system contains also terms that depend on the vector w , via the projections of its components onto the vectors e ± and f ± . To describe the behavior of w we need to determine first these projections. Using (6.20) the above system can be rewritten in the form where z T (λ, η, k) = z (1) , z (2) , z ,− , z ,− :=D −1 h , h is given by (6.25) and the 4 × 4 matrixẼ (λ, η, k) equals Multiplying both sides of (7.2) by e ι , ι ∈ {−, +} and then integrating over T we get a system of 8 equations Here w (ι) are column vectors obtained by a scalar multiplication of the entries of w (see (6.19)) by e ι . The same concerns ± are 4 × 4 matrices defined as follows: Note that vector v appearing on the right hand side of (7.6) still depends on the projections of w onto f ± , cf (6.25) and (6.16). It turns out however that the asymptotics of these projections, as → 0+, can be described by only one of them, for example w (−) . This is a conclusion of our next result. Denote by δw := w (+) − w (−) . We shall also use the following convention: for a given M > 0 the constants 0 , λ 0 > 0 are selected as in the statement of Proposition 6.1 so that δ (λ, η, k) δ (0) (λ, η, k) for all k ∈ T, |η| ≤ M and λ > λ 0 . In particular, then we have (7.1). The proof of the theorem is presented in Section 8.
To describe the limit of w (−) (λ, η) we can use the the system (7.6), which is "almost closed" with respect to the components of w (−) , that is 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 Here Hereê th (η) is the Fourier transform of e th (y) appearing in (3.5) andĉ is given by (3.8).
We can show, see Section 9 below for the proof, the following result. To obtain the asymptotics of w (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 (t) is sequentially pre-compact , as → 0+, in the * -weak topology of L 1 ([0, +∞), A) * . To identify its limiting points we consider w (λ, η, k) the vector of the Laplace-Fourier transforms of W (t). Given λ > 0 this family is sequentially pre-compact in the * -weak topology of A , as → 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 (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 ,+ (λ, η, k) (the limit of w ,− can then be trivially concluded) agrees for λ > λ 0 with the Laplace transform of W (t, dy, dk) appearing in (5.30).
Using the proposition we conclude that the third term appearing in the right hand side of the second equation of (7.19) equals On the other hand, the second term equals We can see therefore that e th (t, y) satisfies (3.7). Thus the conclusion of Theorem 5.1 follows.

Proof of Theorem 7.1
We start with the following result.
Since (see (6.16)) the right hand side of the second equation of system (7.6) can be written as where (d j, are the rows of the adjugate matrix toD , given by (6.38)) 14) We can write 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 ,1 | = O(1), as 1. A similar argument allows us to conclude that also |Z , j | = O(1), as 1 for j = 2, 3. Thus, the second equality in (8.7) follows as well.

Proof of (7.8)
The left hand side of the first equation of the system (7.6) can be rewritten in the following form Note that e − e + R 3 (see (6.6)). From Lemma 8.1 and the Lebesgue dominated convergence theorem we conclude that a After a direct computation we obtain Taking into account (6.28) we conclude that e −ẽ δ . In addition, Combining this with the second formula of (8.18) we obtain, by the Lebesgue dominated convergence theorem, Using the above together with bound (7.9) we conclude that expression (8.16) can be written as 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,−) , is of order of magnitude O(1), as 1. To see that we write where the terms U ,i , i = 1, 2, 3 are given by The fact that z (1,−) = O(1), as 1, can be argued in a similar way as it has been done in the case of z (2,−) , see (8.14) and (8.15) above.

Proof of (7.10)
From (6.20) we obtain Using the above and the already proved estimates (7.8), (7.9) and Lemma 8.1 we obtain lim Here f(k) ≡ 1.

Convergence of II
, III and IV Thanks to Lemma 8.1 we can write The remaining terms appearing in expressions II , III and IV can be estimated in the same manner allowing us to conclude that 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 We conclude in this way that all R I , R II , R III and R IV tend to 0 in the L 1 sense. Thus, (7.11) follows.

Determining w (−)
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 Here a where z (1,±) are the scalar products of z (1) by e ± (cf (7.3) and (7.7)).

Limit of V ,1
Using the decomposition of the Wigner functions of the initial data into the parts corresponding to the macroscopic profile and the fluctuations, see (5.13), we can write an analogous decomposition W = W + W , for the Laplace-Fourier transforms of the respective Wigner functions. It allows us to write V ,1 = V (1) ,1 + V (2) ,1 , where ,1 := Here are the solutions of the analogues of (9.18) in which the right hand side has been replaced by W and W , respectively.

Macroscopic Wigner Functions and Their Dynamics. From (3.1) we get
Therefore the Fourier transforms W φ (t) of the macroscopic Wigner functions (cf (7.14)) satisfy Taking the Laplace transforms of both sides of (9.25) we obtaiñ for any λ > 0 and (η, k) ∈ R 2 .

Limit of V
(1) ,1 The limit in question is a special case of the following result.
Proof. We only prove (9.40), as the argument for (9.41) is very similar. The left hand side of (9.40) for w ,+ can be rewritten in the form Denote by J j, , j = 1, 2, 3, 4 the respective terms arising after opening of the square bracket. Changing variables k := k/ we can write (cf (5.14)) In fact, thanks to the rapid decay of the macroscopic wave function φ, we can write By virtue of the Lebesgue dominated convergence theorem, we conclude that the limit in (9.42) equals Dealing similarly with the remaining terms J j, , j = 2, 3, 4 we conclude (9.40) for w ,+ . The cases of w ,− and y ,± can be handled similarly.
Using a similar argument we infer that for any λ > λ 0 , |η| ≤ M we have Thanks to (10.11) to compute the last limit we can use the Lebesgue dominated convergence and conclude, using (9.35) and (9.39), that the right hand side of the above equality coincides with the right hand side of (10.8).

Proof of (7.18)
We use the notation from Section 8.4 and carry out our analysis only for y ,+ , as the argument for y ,− is very similar. For any ϕ ∈ C(T) we have The analysis of the terms on the right hand side of (10.13) is very similar to the one done in Section 10.2. As a result we obtain lim →0+ T (|I | + |III | + |IV |)dk = 0.