Macroscopic evolution of mechanical and thermal energy in a harmonic chain with random flip of velocities

. We consider an unpinned chain of harmonic oscillators with periodic boundary conditions, whose dynamics is perturbed by a random ﬂip of the sign of the velocities. The dynamics conserves the total volume (or elongation) and the total energy of the system. We prove that in a diﬀusive space-time scaling limit the proﬁles corresponding to the two conserved quantities con-verge to the solution of a diﬀusive system of diﬀerential equations. While the elongation follows a simple autonomous linear diﬀusive equation, the evolution of the energy depends on the gradient of the square of the elongation.


MACROSCOPIC EVOLUTION OF MECHANICAL AND THERMAL ENERGY IN A HARMONIC CHAIN WITH RANDOM FLIP OF VELOCITIES
TOMASZ KOMOROWSKI, STEFANO OLLA, AND MARIELLE SIMON Abstract. We consider an unpinned chain of harmonic oscillators with periodic boundary conditions, whose dynamics is perturbed by a random flip of the sign of the velocities. The dynamics conserves the total volume (or elongation) and the total energy of the system. We prove that in a diffusive space-time scaling limit the profiles corresponding to the two conserved quantities converge to the solution of a diffusive system of differential equations. While the elongation follows a simple autonomous linear diffusive equation, the evolution of the energy depends on the gradient of the square of the elongation. Harmonic chains with energy conserving random perturbations of the dynamics have recently received attention in the study of the macroscopic evolution of energy [1,2,5,8,10,12]. They provide models that have a non-trivial macroscopic behavior which can be explicitly computed. We consider here the dynamics of an unpinned chain where the velocities of particles can randomly change sign. This random mechanism is equivalent to the deterministic collisions with independent environment particles of infinite mass. Since the chain is unpinned, the relevant conserved quantities of the dynamics are the energy and the volume (or elongation).
Under a diffusive space-time scaling, we prove that the profile of elongation evolves independently of the energy and follows the linear diffusive equation Here u is the Lagrangian space coordinate of the system and γ ą 0 is the intensity of the random mechanism of collisions. The energy profile can be decomposed into the sum of mechanical and thermal energy ept, uq " e mech pt, uq`e thm pt, uq where the mechanical energy is given by e mech pt, uq " 1 2 rpt, uq 2 , while the thermal part e thm pt, uq, that coincides with the temperature profile, evolves following the non-linear equation: B t e thm pt, uq " 1 4γ B 2 uu e thm pt, uq`1 2γ pB u rpt, uqq 2 .
This is equivalent to the following conservation law for the total energy: B t ept, uq " 1 4γ B 2 uuˆe pt, uq`r pt, uq 2 2˙. (1. 3) The derivation of the macroscopic equations (1.1) and (1.2) from the microscopic dynamical system of particles, after a diffusive rescaling of space and time, is the goal of this paper. Concerning the distribution of the energy in the frequency modes: the mechanical energy e mech pt, uq is concentrated on the modes corresponding to the largest wavelength, while the thermal energy e thm pt, uq is distributed uniformly over all frequencies. Note that 1 2γ pB u rpt, uqq 2 is the rate of dissipation of the mechanical energy into thermal energy.
The presence of the non-linearity in the evolution of the energy makes the macroscopic limit non-trivial. Relative entropy methods (as introduced in [13]) identify correctly the limit equation (see [12]), but in order to make them rigorous one needs sharp bounds on higher moments than cannot be controlled by the relative entropy 1 . In this sense the proof in [12] is not complete.
We follow here a different approach based on Wigner distributions. The Wigner distributions permit to control the energy distribution over various frequency modes and provide a natural separation between mechanical and thermal energies. The initial positions and velocities of particles can be random, and the only condition we ask, besides to have definite mean asymptotic profiles of elongation and energy, is that the thermal energy spectrum has a square integrable density. In the macroscopic limit we prove that locally the thermal energy spectrum has a constant density equal to the local thermal energy (or temperature), i.e. that the system is, at macroscopic positive times, in local equilibrium, even though it is not at initial time. Also follows from our result that the mechanical energy is concentrated on the lowest modes. This is a stronger local equilibrium result than the one usually obtained with relative entropy techniques. The Wigner distribution approach had been successfully applied in different contexts for systems perturbed by noise with more conservation laws in [5,8]. Here we need a particular asymmetric version of the Wigner distribution, in order to deal with a finite size discrete microscopic system.
When the system is pinned, only energy is conserved and its macroscopic evolution is linear, and much easier to be obtained. In this case the thermalization and the correlation structure have been studied in [9,10].
When the chain of oscillators is anharmonic, still with velocity flip dynamics, the hydrodynamic limit is a difficult non-gradient problem, for the moment still open. In that case the macroscopic equations would be: where τ pr, eq is the thermodynamic equilibrium tension as function of the volume r and of the energy e, and β´1pr, eq is the corresponding temperature, while Dpr, eq is the thermal diffusivity defined by the usual Green-Kubo formula, as space-time variance of the energy current in the equilibrium infinite dynamics at average elongation r and energy e (see Section 4 for the definition of these quantities). The linear response and the existence of Dpr, eq have been proven in [3].

Microscopic dynamics
2.1. Periodic chain of oscillators. In the following we denote by T n :" Z{nZ " t0, . . . , n´1u the discrete circle with n points, and, for any L ą 0, by TpLq thecontinuous circle of length L, and we set T :" Tp1q.
We consider a one-dimensional harmonic chain of n oscillators, all of mass 1, with periodic boundary conditions. The clearest way to describe this system is as a massive one dimensional discrete surface tϕ x P R, x P T n u. The element (or particle) x of the surface is at height ϕ x and has mass equal to 1. We call its velocity (that coincides with its momentum) p x P R. Each particle x is connected to the particles x´1 and x`1 by harmonic springs, so that n´1 and 0 are connected in the same way. The total energy of the system is given by the Hamiltonian: In addition to the Hamiltonian dynamics associated to the harmonic potentials, particles are subject to a random interaction with the environment: at independently distributed random Poissonian times, the momentum p x is flipped intó p x . The resulting equations of the motion are # dϕ x ptq " n 2 p x ptq dt, dp x ptq " n 2`ϕ x`1 ptq`ϕ x´1 ptq´2ϕ x ptq˘dt´2p x pt´q dN x pγn 2 tq, for any x P T n . Here tN x ptq ; t ě 0, x P T n u are n independent Poisson processes of intensity 1, and the constant γ is positive. We have already accelerated the time scale by n 2 , according to the diffusive scaling. Notice that the energy H n is conserved by this dynamics. There is another important conservation law that is given by the sum of the elongations of the springs, that we define as follows. We call r x " ϕ x´ϕx´1 the elongation of the spring between x and x´1, and since x P T n we have r 0 " ϕ 0´ϕn´1 . The equation of the dynamics in these coordinates are given by: This implies that the dynamics is completely defined giving the initial conditions tr x p0q, p x p0q, x P T n u. 4 The periodicity in the ϕ x variables would impose that ř n´1 x"0 r x p0q " 0. On the other hand the dynamics defined by (2.3) is well defined also if ř n´1 x"0 r x p0q ‰ 0 and has the conservation law ř n´1 x"0 r x ptq " ř n´1 x"0 r x p0q :" R n . Note that R n can also assume negative values. In this case we can picture the particles as n points q 0 , . . . , q n´1 P Tp|R n |q, the circle of length |R n |. These points can be defined as q x :" " ř x y"0 r y ‰ mod|Rn| , for x " 0, . . . , n´1. It follows that q n " q 0 . We will not use neither the q x coordinates nor the ϕ x coordinates, but we consider only the evolution defined by (2.3) with initial configurations ř n´1 x"0 r x p0q " R n P R.

2.2.
Generator and invariant measures. The generator of the stochastic dynamics (rptq :" tr x ptqu xPTn , pptq :" tp x ptqu xPTn ), is given by where the Liouville operator A n is formally given by while, for f : Ω n Ñ R, where p x is the configuration that is obtained from p by reversing the sign of the velocity at site x, namely: pp x q y " p y if y ‰ x and pp x q x "´p x . The two conserved quantities H n " ř xPTn E x and R n " ř xPTn r x , are determined by the initial data (eventually random), and typically they should be proportional to n: H n " ne, R n " nr, with e P R`the average energy per particle, and r P R the average spring elongation. Consequently the system has a two parameters family of stationary measures given by the canonical Gibbs distributions µ n τ,β pdr, dpq " ź xPTn exp`´βpE x´τ r x q´G τ,β˘d r x dp x , As usual, the parameters β´1 ą 0 and τ P R are called respectively temperature and tension. Observe that the function gives the average equilibrium length in function of the tension τ , and is the corresponding thermodynamic internal energy function. Note that the energy Epτ, βq is composed by a thermal energy β´1 and a mechanical energy τ 2 2 . 2.3. Hydrodynamic limits. Let µ n pdr, dpq be an initial Borel probability distribution on Ω n . We denote by P n the law of the process tprptq, pptqq ; t ě 0u starting from the measure µ n and generating by L n , and by E n its corresponding expectation. We are given initial continuous profiles of tension tτ 0 puq ; u P Tu and of temperature tβ´1 0 puq ą 0 ; u P Tu. The thermodynamic relations (2.4) and (2.5) give the corresponding initial profiles of elongation and energy as r 0 puq :" τ 0 puq and e 0 puq :" The initial distributions µ n are assumed to satisfy the following mean convergence statements: for any test function G that belongs to the set C 8 pTq of smooth functions on the torus. We expect the same convergence to happen at the macroscopic time t: where the macroscopic evolution for the volume and energy profiles follows the system of equations: t, uq, pt, uq P R`ˆT, with the initial condition rp0, uq " r 0 puq, ep0, uq " e 0 puq.
The solutions ept,¨q, rpt,¨q of (2.10) are smooth when t ą 0 (the system of partial differential equations is parabolic). Note that the evolution of rpt, uq is autonomous of ept, uq. The precise assumptions that are needed for the convergence (2.9) are stated in Theorems 3.7 and 3.8 below.

Notations.
3.1.1. Discrete Fourier transform. Let us denote by p f the Fourier transform of a finite sequence tf x u xPTn of numbers in C, defined as follows: Reciprocally, for any f : p T n Ñ C, we denote by q f x ( xPTn its inverse Fourier transform given by The Parseval identity reads If tf x u xPTn and tg x u xPTn are two sequences indexed by the discrete torus, their convolution is given by

Continuous Fourier transform. Let
CpTq be the the space of continuous, complex valued functions on T. For any function G P CpTq, let F G : Z Ñ C denote its Fourier transform given as follows: Similar identities to (3.2) and (3.3) can easily be written: for instance, we shall repeatedly use the following Note that when G is smooth the Fourier coefficients satisfy If J : TˆT Ñ C is defined on a two-dimensional torus, we still denote by F Jpη, vq, pη, vq P ZˆT, its Fourier transform with respect to the first variable. We equip the set C 8 pTˆTq of smooth (with respect to the first variable) functions with the norm Let A 0 be the completion of C 8 pTˆTq in this norm and pA 1 0 , }¨} 1 0 q its dual space. 7

A fundamental example.
In what follows, we often consider the discrete Fourier transform associated to a function G P CpTq, and to avoid any confusion we introduce a new notation: let F n G : p T n Ñ C be the discrete Fourier transform of the finite sequence tGp x n qu xPTn defined similarly to (3.1) as In particular, we have the Parseval identity ÿ for any η P Z.

3.2.
Assumptions on initial data. Without losing too much of generality, one can put natural assumptions on the initial probability measure µ n pdr, dpq. The first assumption concerns the mean of the initial configurations, and is sufficient in order to derive the first of the hydrodynamic equations (2.10) : ‚ The initial total energy can be random but with uniformly bounded expectation: ‚ We assume that there exist continuous initial profiles r 0 : T Ñ R and e 0 : T Ñ p0,`8q such that E n rp x p0qs " 0, E n rr x p0qs " r 0´x n¯f or any x P T n (3.10) and for any G P C 8 pTq 1 n Identity (3.10), in particular, implies the mean convergence of the initial elongation: for any G P C 8 pTq.
Remark 3.2. By energy conservation (3.9) implies that Remark 3.3. Conditions in (3.10) are assumed in order to simplify the proof, but they can be easily relaxed.
Next assumption is important to obtain the macroscopic equation for the energy in (2.10). It concerns the energy spectrum of fluctuations around the means at initial time. Define the initial thermal energy spectrum u n p0, kq, k P p T n , as follows: let p rp0, kq and p pp0, kq denote respectively the Fourier transforms of the initial random configurations tr x p0qu xPTn and tp x p0qu xPTn , and let (3.14) Due to the Parseval identity (3.3) we have Assumption 3.4. (Square integrable initial thermal energy spectrum) This technical assumption can be seen as a way to ensure that the thermal energy does not concentrate on one (or very few) wavelength(s).
Remark 3.5. Assumptions 3.1 and 3.4 are satisfied if the measures µ n are given by local Gibbs measures (non homogeneous product), corresponding to the given initial profiles of tension and temperature tτ 0 puq, β´1 0 puq ; u P Tu, defined as follows: with r 0 puq " τ 0 puq and e 0 puq " β´1 0 puq`r 2 0 puq 2 , see [7, Sections 9.2.3-9.2.5]. Note that our assumptions are much more general, as we do not assume any specific condition on the correlation structure of µ n . In particular microcanonical versions of (3.16), where total energy and total volumes are conditioned at fixed values ne and nr, are included by our assumptions.
Remark 3.6. We will see that macroscopically, our assumptions state that the initial energy has a mechanical part, related to τ 0 p¨q, that concentrates on the longest wavelength (i.e. around k " 0), see in Section 5 equation (5.33) for the precise meaning. For what concerns the thermal energy, (3.15) states that it has a square integrable density w.r.t. k.

3.3.
Formulation of mean convergence. In this section we state two theorems dealing with the mean convergence of the two conserved quantities, namely the elongation and energy. The first one (Theorem 3.7) is proved straightforwardly in Section 3.4 below. The second one is more involved, and is the main subject of the present paper.
Theorem 3.7 (Mean convergence of the elongation profile). Assume that tµ n u nPN is a sequence of probability measures on Ω n such that (3.10) is satisfied, with r 0 P CpTq. Let rpt, uq be the solution defined on R`ˆT of the linear diffusive equation: Then, for any G P C 8 pTq and t P R`, Theorem 3.8 (Mean convergence of the empirical profile of energy). Let tµ n u nPN be a sequence of probability measures on Ω n such that Assumptions 3.1 and 3.4 are satisfied. Then, for any smooth function G : R`ˆT Ñ R compactly supported with respect to the time variable t P R`, we have where ept, uq " e mech pt, uq`e thm pt, uq, with ‚ the mechanical energy, given by e mech pt, uq :" 1 2 prpt, uqq 2 and the function rpt, uq being the solution of (3.17), ‚ the thermal energy e thm pt, uq, defined as the solution of The proof of Theorem 3.8 is the aim of Sections 5 -7.
This new way of seeing the macroscopic equations is more convenient, as it naturally arises from the proof. More precisely, using (3.17) we conclude that the mechanical energy e mech pt, uq satisfies the equation B t e mech pt, uq " 1 2γ´B 2 uu e mech pt, uq´`B u rpt, uq˘21 0 and the macroscopic energy density function satisfies uu`e pt, uq`e mech pt, uq˘, ep0, uq " e 0 puq.
Remark 3.10. We actually prove a stronger result that includes a local equilibrium statement, see Theorem 7.5 below.
3.4. Proof of the hydrodynamic limit for the elongation. Here we give a simple proof of Theorem 3.7. From the evolution equations (2.3) we have the following identities: Substituting from the second equation into the first one we conclude that where ∇ n G´x n¯" n´G´x`1 n¯´G´x n¯∆ n G´x n¯" n´∇ n G´x n¯´∇ n G´x´1 n¯¯. By energy conservation and Assumption 3.1 it is easy to see thaťˇˇˇ1 Let us definer pnq pt, uq :" E n " r x ptq ‰ , for any u P " x n , x`1 n˘, n ě 1.

11
Thanks to the energy conservation we know that there exists R ą 0 such that sup ně1 sup tPr0,T s }r pnq pt,¨q} L 2 pTq ": R ă`8. (3.24) The above means that for each t P r0, T s the sequence r pnq pt,¨q ( ně1 is contained inB R -the closed ball of radius R ą 0 in L 2 pTq, centered at 0. The ball is compact in L 2 w pTq -the space of square integrable functions on the torus T equipped with the weak L 2 topology. The topology restricted toB R is metrizable, with the respective metric given e.g. by dpf, gq :"`8 ÿ n"1 where tφ n u is a countable and dense subset of L 2 pTq that can be chosen of elements of C 8 pTq. From (3.22) and (3.23) we conclude in particular that for each T ą 0 the sequence r pnq p¨q ( is equicontinuous in C`r0, T s,B R˘. Thus, according to the Arzela Theorem, see e.g. [6, p. 234], it is sequentially pre-compact in the space C pr0, T s, L 2 w pTqq for any T ą 0. Consequently, any limiting point of the sequence satisfies the partial differential equation (3.17) in a weak sense in the class of L 2 pTq functions. Uniqueness of the weak solution of the heat equation gives the convergence claimed in (3.19) and the identification of the limit as the strong solution of (3.17).
Concerning ( where ∇nGp x n q " n`Gp x n q´Gp x´1 n q˘. Using again energy conservation and the Cauchy-Schwarz inequality, it is easy to see that the right hand side of the above vanishes as n Ñ 8. Remark 3.11. Note that we have not used the fact that the initial average of the velocities vanishes. Additionally, by standard methods it is possible to obtain the convergence of elongation and momentum empirical distributions in probability (see (3.18 ) and (3.19)), but we shall not pursuit this here.

Conjecture for anharmonic interaction and thermodynamic considerations
Our results concern only harmonic interactions, but we can state the expected macroscopic behavior for the anharmonic case. Consider a non-quadratic potential V prq, of class C 1 and growing fast enough to`8 as |r| Ñ 8. The dynamics is now defined by # dr x ptq " n 2`p x ptq´p x´1 ptq˘dt dp x ptq " n 2`V 1 pr x`1 ptqq´V 1 pr x ptqq˘dt´2p x pt´q dN x pγn 2 tq, x P T n . (4.1) The stationary measures are given by the canonical Gibbs distributions dµ n τ,β " ź xPTn e´β pEx´τ rxq´G τ,β dr x dp x , τ P R, β ą 0, where we denote the energy that we attribute to the particle x, and Thermodynamic entropy Spr, eq is defined as Then we obtain the inverse temperature and tension as functions of the volume r and internal energy u: β´1pr, eq " B e Spr, eq, τ pr, eq "´β´1pr, eqB r Spr, eq (4.5) The macroscopic profiles of elongation rpt, uq and energy ept, uq will satisfy the equations (4.6) Here the diffusivity Dpr, eq ą 0 is defined by a Green-Kubo formula for the infinite dynamics in equilibrium at the given values pr, eq. The precise definition and the proof of the convergence of Green-Kubo formula for this dynamics can be found in [3]. A straightforward calculation gives the expected increase of thermodynamic entropy:

Time-dependent Wigner distributions
Before exposing the strategy of the proof of Theorem 3.8, let us start by introducing our main tool: the Wigner distributions associated to the dynamics. 13

5.1.
Wave function for the system of oscillators. Let p ppt, kq and p rpt, kq, for k P p T n , denote the Fourier transforms of, respectively, the momentum and elongation components of the microscopic configurations tp x ptqu xPTn and tr x ptqu xPTn , as in (3.1). Since they are real valued we have, for any k P p T n , p p ‹ pt, kq " ÿ xPTn e 2πikx p x ptq " p ppt,´kq and likewise p r ‹ pt, kq " p rpt,´kq. (5.1) The wave function associated to the dynamics is defined as Its Fourier transform equals p ψpt, kq :" p rpt, kq`ip ppt, kq, k P p T n .
With these definitions we have |ψ x | 2 " 2E x and the initial thermal energy spectrum, defined in (3.14), satisfies u n p0, kq " r u n p0, kq`Im Here Cov n pX, Y q :" E n rXY ‹ s´E n rXs E n rY ‹ s is the covariance of complex random variables X and Y , and r u n p0, kq :" 1 2n E n "ˇˇp ψp0, kq´E n r p ψp0, kqsˇˇ2 ı Using Cauchy-Schwarz inequality we conclude easily that 1 2 r u n p0, kq ď u n p0, kq ď 2r u n p0, kq, k P p T n .
Observe that we have p N ‹ pt, kq " p N pt,´kq. In addition, its mean and covariance equal respectively where δ x,y is the usual Kronecker delta function, which equals 1 if x " y and 0 otherwise. The conservation of energy, and Parseval's identity, imply together that: for all t ě 0. where the Wigner function Wǹ ptq is given for any pk, ηq P p T nˆZ by Wǹ pt, η, kq :" Here, we use the mapping Z Q η Þ Ñ η n P p T n , and F Gpη, vq denotes the Fourier transform with respect to the first variable.
Remark 5.1. Note that this definition of the Wigner function is not the standard symmetric one. Indeed, since the setting here is discrete, it turns out that (5.7) is the convenient way to identify the Fourier modes, otherwise we would have worked with ill-defined quantities, for instance η 2n , which are not always integers. The main interest of the Wigner distribution is that mean convergence of the empirical energy profile (3.20) can be restated in terms of convergence of Wigner functions (more precisely, their Laplace transforms, see Theorem 7.5 below), thanks to the following identity: if Gpu, vq " Gpuq does not depend on the second variable v P T, then Indeed, from (5.6) we obtain then @ Wǹ ptq, 1´x qk e 2πix η n pF Gq ‹ pηq.
(5.9) Performing the summation over k we conclude that the right hand side equals 1 2n where 1 Z is the indicator function of the set of integers. Since 0 ď x, x 1 ď n´1 and |ψ x ptq| 2 " 2E x ptq we conclude that the above expression equals 1 n for any t 0 and η P Z. This remark will be useful in what follows.
Thanks to identity (5.8), we have reduced the proof of Theorem 3.8 -and more precisely the proof of convergence (3.20) -to the investigation of the Wigner sequence tWǹ p¨q, Yǹ p¨q, Yń p¨q, Wń p¨qu n . The next sections are split as follows: (1) we first prove that this last sequence is pre-compact (and therefore admits a limit point) in Section 5.3.1; (2) then, we characterize that limit point in several steps: (a) we write a decomposition of the Wigner distribution into its mechanical and thermal parts in Section 5.3.2; (b) the convergence of the mechanical part is achieved in Section 5.3.3 and Section 5.3.4; (c) to solve the thermal part, we need to take its Laplace transform in Section 6.1, and then to study the dynamics it follows in Section 6.2 and Section 6.3; (d) the convergence statements for this Laplace transform are given in Section 7, the main results being Proposition 7.1 and Proposition 7.4; (e) finally, we go back to the convergence of the Wigner distributions in Theorem 7.5, and then to the conclusion of the proof of Theorem 3.8 in Section 7.4.

5.3.
Properties of the Wigner distributions.

5.3.1.
Weak convergence. From (3.13) we have that, for any n 1 and G P C 8 pTˆTq,ˇ@ where the norm }G} 0 is defined by (3.7). Hence, for the corresponding dual norm, we have the bound which implies weak convergence. Note that condition (3.11) ensures that, if Gpu, vq " Gpuq at the initial time t " 0, then we have

5.3.2.
Decomposition into the mechanical and fluctuating part. We decompose the wave function into its mean w.r.t. E n and its fluctuating part, as follows: Notice that for the initial data we have ψ x p0q " r 0`x n˘. It need not be true for t ą 0. The Fourier transform of the sequences tψ x ptqu and t r ψ with initial condition p ψp0, kq " pF n r 0 qpkq.

Asymptotics of Ă
Wǹ . Throughout the remainder of the paper we shall use the following notation: given a function f : p T n Ñ C we denote its k-average by " f p¨q The initial fluctuating Wigner function is related, as n Ñ`8, to the initial thermal energy e thm p0, uq " e 0 puq´r 2 0 puq{2 as follows: for any η P Z.
The last convergence follows from Assumption 3.1 and from an explicit computation that yields: In addition, Assumption 3.4 on the initial spectrum (see (3.15)) implies that Proof. We prove the proposition under the assumption that r P C 8 pTq. The general case can be obtained by an approximation of a continuous initial profile by a sequence of smooth ones. Using the dominated convergence theorem we conclude that the expression on the left hand side of (5.31) equals ř ηPZ lim nÑ`8 f n pηq, with f n pηq :" " Wǹ pη,¨qpF Gq ‹ pη,¨q ı n , η P Z. This can be written as Using Fourier representation (see (3.5)), we can write f n pηq as Due to smoothness of rp¨q, its Fourier coefficients decay rapidly. Therefore for a fixed η P Z and for any ρ P p0, 1q we have that the limit lim nÑ`8 f n pηq equals Summing over x, x 1 we conclude that lim nÑ`8 f n pηq equals Taking into account the fact that m P T n and the magnitude of ξ, ξ 1 is negligible when compared with n we conclude that the terms under the summation on the right hand side are non zero only if m`ξ 1 " 0 and m`ξ`η " 0, or m`ξ 1 " n and m`ξ`η " n. Therefore, One of the main point of the proof of our theorem is to show that this convergence holds for any macroscopic time t ą 0, i.e. that for any compactly supported G P C 8 pR`ˆT 2 q we have Wǹ pt, η,¨qpF Gq ‹ pt, η,¨q This would amount to showing that the Wigner distribution Wǹ ptq associated to ψ x ptq is asymptotically equivalent to the one corresponding to the macroscopic profile r pt,¨q via (5.29). This fact is not a consequence of Theorem 3.7, that implies only a weak convergence of ψ x ptq to r pt,¨q. We will prove (5.34) in Proposition 7.1 below, showing the convergence of the corresponding Laplace transforms (see (7.2)).
6. Strategy of the proof and explicit resolutions 6.1. Laplace transform of Wigner functions. Since our subsequent argument is based on an application of the Laplace transform of the Wigner functions, we give some explicit formulas for the object that can be written in case of our model. For any bounded complex-valued, Borel measurable function R`Q t Þ Ñ f t we define the Laplace operator L as: Given the solution r t " rpt,¨q of (3.17), we define the Laplace transform of the Wigner distribution (5.29) associated to a macroscopic profile, as follows: for any pλ, η, ξq P R`ˆZ 2 , wpr¨; η, ξqpλq " L`W pr¨; η, ξq˘pλq " The following formulas are easily deduced, by a direct calculation, from (3.17), and are left to the reader: Finally, we define wǹ the Laplace transform of Wǹ as the tempered distribution given for any G P C 8 pTˆTq and λ ą 0 by 3) where wǹ is the Laplace transform of Wǹ as follows: wǹ pλ, η, kq :" In a similar fashion we can also define wń pλq and yn pλq the Laplace transforms of Wń ptq and Yn ptq, respectively, and their counterparts wń , and yn .
As before, we shall often drop the variables pλ, η, kq from the notations. We are now ready to take the Laplace transform of both sides of (6.5): we obtain a linear system that can be written for any pλ, η, kq in the form pM n w n qpλ, η, kq " v 0 n pη, kq`γn 2 I n pλ, ηq e, (6.7) where the 2ˆ2 block matrix M n :" M n pλ, η, kq is defined as follows: where, given a positive integer N, Id N denotes the NˆN identity matrix, and A n , B n are 2ˆ2 matrices: Here and below, # a n :" λ`inpδ n sq`2γn 2 , b n :" λ`in 2 pσ n sq`2γn 2 , # γn :" γ˘sin`2π`k`η n˘˘, γ˘:" γ˘sinp2πkq. (6.9) An elementary observation yields the following symmetry properties γn p´η,´kq " γn pη, kq, γ˘p´η,´kq " γ¯pη, kq, (6.10) By the linearity of the Laplace transform we can write w n " w n`r w n , (6.11) where w n is the Laplace transform of pWǹ ptq, Yǹ ptq, Yń ptq, Wń ptqq, and r w n is the Laplace transform of p Ă Wǹ ptq, r Yǹ ptq, r Yń ptq, Ă Wń ptqq.
Performing the Laplace transform of both sides of (6.6) we conclude that w n solves the equation M n w n " v 0 n " Wǹ 1. (6.12) Using (6.7) we conclude that r w n solves M n r w n " r v 0 n`γ n 2 I n e. (6.13) Following (6.11) we also write I n " I n`r I n , where I n pλ, ηq :" e¨"w n pλ, η,¨q ı n and r I n pλ, ηq :" e¨" r w n pλ, η,¨q ı n .
In Section 8.1, we show that the matrix M n is invertible, therefore we can solve and rewrite (6.12) and (6.13) as: w n " Wǹ M´1 n 1, (6.14) r w n " M´1 n r v 0 n`γ n 2 I n M´1 n e. (6.15) In Section 7 we study the contribution of the terms appearing in the right hand sides of both (6.14) and (6.15) that reflect upon the evolution of the mechanical and fluctuating components of the energy functional.

Proof of the hydrodynamic behavior of the energy
In this section we conclude the proof of Theorem 3.8, up to technical lemmas that are proved in Section 8. 7.1. Mechanical energy w n . We start with the recollection of the results concerning the mechanical energy. The Laplace transform w n is autonomous from the thermal part and satisfies (6.14). Let us introduce, for any λ ą 0 and η P Z, the mechanical Laplace-Wigner function Wm ech pλ, ηq :" ÿ ξPZ W pr 0 ; η, ξq 2π 2 γ rξ 2`p ξ`ηq 2 s`λ .
Given M P N we denote by P M the subspace of C 8 pTˆTq consisting of all trigonometric polynomials that are finite linear combinations of e 2πiηu e 2πiξv , with η P t´M, . . . , Mu, ξ P Z and u, v P T.
The following lemma finalizes the identification of the limit for the Fourier transform of the thermal energy: The proofs of Lemma 7.2 and Proposition 7.4 go very much along the lines of the arguments presented in Section 8 and we will not present the details here. They are basically consequences of the following limit lim nÑ8 n 2 e 1¨M´1 n pλ, η, kq e ( " 1 2γ , which is proved in Section 8.2. 7.3. Asymptotics of r w n and w n . With a little more work one can prove the following local equilibrium result, which is an easy consequence of Proposition 7.1, Corollary 7.3 and Proposition 7.4 (recall also (6.15)). Wm ech pλ, ηq pF Gq ‹ pη, 0q, (7.11) We will not give the details for the proof of this last theorem, since the argument is very similar to Proposition 7.1.

7.4.
End of the proof of Theorem 3.8. The proof of convergence (3.20) has been reduced to the investigation of the Wigner distributions. Recall that from the uniform bound (5.18), we know that the sequence of all Wigner distributions tWǹ p¨q, Yǹ p¨q, Yń p¨q, Wń p¨qu n is sequentially pre-compact with respect to the ‹weak topology in the dual space of L 1 pR`, A 0 q. More precisely, one can choose a subsequence n m such that any of the components above, say for instance Wǹ m p¨q, ‹-weakly converges in the dual space of L 1 pR`, A 0 q to some W`p¨q.
To characterize its limit, we consider wǹ m pλq obtained by taking the Laplace transforms of the respective Wǹ m p¨q. For any λ ą 0, it converges ‹-weakly, as n m Ñ`8, in A 1 0 to some w`pλq that is the Laplace transform of W`p¨q. The latter is defined as xw`pλq, Gy :" Given a trigonometric polynomial G P C 8 pTˆTq we conclude, thanks to Theorem 7.5, that for any λ ą λ M , @ w`pλq, G D " ż R`ˆT 2 e´λ t ept, uqGpu, vq dt du dv, (7.12) where ept, uq is defined as in Theorem 3.8 and M P N is such that F Gpη, vq " 0 for all |η| ą M.
Due to the uniqueness of the Laplace transform (that can be argued by analytic continuation), this proves that in fact equality (7.12) holds for all λ ą 0. By a density argument it can be then extended to all G P A 0 and shows that W`pt, u, vq " ept, uq, for any pt, u, vq P R`ˆT 2 . This ends the proof of (3.20), and thus Theorem 3.8.

Proofs of the technical results stated in Section 7
In what follows we shall adopt the following notation: we say that the sequence C n pλ, η, kq ĺ 1 if for any given integer M P N, there exist λ M ą 0 and n M P N such that sup ! C n pλ, η, kq ; λ ą λ M , η P t´M, ..., Mu, n ą n M , k P p T n ) ă`8.
Proof. The block entries of the matrix M n defined in (6.9) satisfy the commutation relation rA n , B n s " A n B n´Bn A n " » -0´2γ´n 2 Rera n´bn ś 2γ`n 2 Rerb n´an s 0 fi fl " 0.