Superdiffusion of energy in a chain of harmonic oscillators with noise

We consider a one dimensional infinite chain of har- monic oscillators whose dynamics is perturbed by a stochastic term conserving energy and momentum. We prove that in the unpinned case the macroscopic evolution of the energy converges to a fractional diffusion. For a pinned system we prove that energy evolves diffusively, generalizing some of the results of [4].


Introduction
Superdiffusion of energy and the corresponding anomalous thermal conductivity have been observed numerically in the dynamics of unpinned FPU chains [15,16]. This is generally attributed to the small scattering rate for low modes, due to momentum conservation. When the interaction has a pinning potential, it is expected that the system undergoes a normal diffusion. More recently, the problem has been studied in models where the Hamiltonian dynamics is perturbed by stochastic terms that conserve energy and momentum, like random exchange of velocity between nearest neighbors particles [1,2]. In these models, the interaction is purely harmonic, and as a result, the Green-Kubo formula for thermal conductivity κ can be studied explicitly. It diverges for one and two dimensional lattices in case no pinning potential is present, while thermal conductivity stays finite for pinned systems or in dimension d ≥ 3. In the cases when the conductivity is finite it is proven in [4] that energy fluctuations in equilibrium evolve diffusively.
The main result of the present article concerns the nature of the superdiffusion in dimension 1, when the chain is unpinned. It has already been proven that in the weak noise limit (where the average number of stochastic collisions per unit time is kept finite, 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. as in the Grad limit) the Wigner distribution of the energy converges to an inhomogeneous phonon linear Boltzmann equation [5]. Since the corresponding scattering kernel R(k, k ) is positive, the resulting Boltzmann equation can be interpreted probabilistically as the evolution of the density for some Markov process: in this limit a phonon of mode k moves with the velocity given by the gradient of the dispersion relation ∇ω(k) and change mode with rate R(k, k ). Under a proper space-time rescaling, this process converges to a Lévy superdiffusion generated by the fractional Laplacian −| | 3/4 . This is proven in [3,10], using probabilistic techniques such as coupling and martingale convergence theorems. A completely analytic proof of the convergence, from a kinetic to a fractional diffusion equation, without the use of the probabilistic representation, has been proposed in [18]. All these results provide a two-step solution: first take a kinetic limit, then use a hydrodynamic rescaling of the kinetic equation. A kind of diagonal procedure is treated in [13]: using the probabilistic approach, one can push the time scale a little longer (matching suitably the size of a still small scattering rate) than in the kinetic limit case. As a result, it is possible to obtain the diffusive limit under the pinning potential and superdiffusive in the unpinned case.
In the present paper we prove a direct limit to the fractional superdiffusion, just by rescaling space and time, without the weak noise assumption. We also recover the diffusive limit results of [4] in the case of a finite diffusivity and study the cases of intermediate weaker noise limits. The rigorous formulations of our main results are listed in Sect. 3.
In a recent article [20], Herbert Spohn predicts the same fractional superdiffusive behavior for the heat mode in the β-FPU at zero pressure. This follows from an application of mode coupling approximation procedure to fluctuating hydrodynamic equations. Our present results concern a model, which has also three conserved quantities. They are in agreement with the predictions of [20], confirming that the harmonic stochastic model is a good approximation of some non-linear models, at least in the case of symmetric interactions.
The strategy of the proof is as follows: first, we formulate the result for the limit evolution of the Wigner distribution W (t) of the energy, when the initial data are in L 2 , see Theorems 5.1 and 5.2, proven in Sects. 10 and 11. These results concern the system with non-equilibrium initial data but of the finite total energy. The extension to homogeneous initial data (whose L 2 norm is infinite), in particular the equilibrium dynamics with Gibbs distributed initial data, is possible by a simple duality argument, see Sect. 12. Our results can be formulated in terms of a local energy functional, see Theorems 3.1 and 3.2 in the case of the L 2 integrable initial data, and Theorem 3.3 for the initial data in equilibrium, respectively. This is possible thanks to the asymptotic equivalence of the relevant energy functionals proven in Propositions 5.3 and 6.3 below.
Concerning the proof when the initial data have square summable realizations, which is the crucial part of our argument, we study first the time evolution of the Wigner distribution of the energy W (t), which represents the energy density in both the spatial variable and frequency modes (in fact, it is more convenient to work with the Fourier transform of W (t) in the spatial variable). As it has been already remarked in [5], the evolution of W (t) is not autonomous but involves another distribution Y (t), whose real and imaginary parts represent the difference between kinetic and potential energy and the energy current, respectively, see (9.13). The principal advantage of working with the pair (W (t), Y (t)) is that its evolution can be described by a system of ordinary differential equations. By performing the Laplace transform in the temporal domain, the system reduces further to an algebraic system of linear equations, see (10.5), and the problem of finding the asymptotics of the energy density for the chain of oscillators reduces to the question of asymptotics of solutions of the system. This is done in Sects. 10 and 11. First, we observe in Sect. 10 that due to the high number of random collisions in the time scales considered, both W (t) and Y (t) homogenize (unlike in the case of the kinetic limit considered in [5]), and their limits do not depend on the frequency mode variable. The homogenization is proven in Theorem 10.2. In addition, because of fast fluctuations, the time integral of Y (t) will disappear from the final equation, as in the case of the kinetic limit in [5]. The above implies that the phonon-Boltzmann equation gives a good approximation of the evolution of W (t), but the presence of the error term, which is of order o (1), as tends to 0, does not allow for a direct application of the probabilistic approach of [3,10]. Instead, we use a version of the analytic approach of [18], based on projections on the product components of the scattering kernel appearing in the homogenized dynamics, see (5.13) and (10.9) below. This is done in Sect. 11.
In our choice of the dynamics, a diffusive random exchange of momenta takes place between the three nearest neighbor particles in such a way that total kinetic energy and momentum are conserved in the process. However, our method can be applied to linear models with quite general stochastic scattering mechanisms, generating different scattering rates. The result does not depend on the particular type of stochastic perturbation, as long as it conserves the appropriate quantities. E.g., we could consider a model with a simple Poissonian exchange of the two nearest neighbor velocities described in Sect. 2.1.3 below. In fact this case is computationally less involved, due to a simpler structure of the respective scattering kernel.
Concerning the possible generalizations of our results to dimensions d ≥ 2, see [1] for the formulation of the model; we conjecture they can also be treated by the present method.
Although for the equilibrium fluctuations we prove only the convergence of the covariance function, our approach can be further developed to obtain the convergence in law for the equilibrium fluctuation field to the respective Ornstein-Uhlenbeck process. The question of the convergence in probability for the non-equilibrium case could possibly be more involved, as it requires the control of higher moments of the energy distribution.
A remark concerning the initial data of the system is also in order. We choose the initial probability distributions of the velocities and inter-particle distances whose energy spectrum satisfies condition (3.9). This condition implies that the initial data are macroscopically centered (see Sect. 3.2.1). While this choice is quite natural in the situation of a pinned chain, it requires some explanation in the unpinned case. In the latter situation, if we start with non-centered initial conditions, their respective macroscopic averages will evolve, at the hyperbolic space-time scale, following the linear wave equation. As a result, they will disperse to infinity, since we start with the data whose realization has a finite L 2 norm. This implies that at a larger superdiffusive time scale these averages will be null. Thus the initial condition for the macroscopic superdiffusive evolution is provided solely by the variance or the high oscillations of the initial data, i.e., the temperature profile. Summarizing, the initial energy profile can be decomposed into a temperature profile and a phonon energy profile. At the hyperbolic time scale the phonon energy profile will converge to 0 asymptotically, as time goes to infinity, while the temperature profile remains stationary. The latter starts evolving at the larger, superdiffusive time scale following a fractional heat equation. The details of this decomposition will be explained in a forthcoming article [11].
We mention here the article [6], where a result similar to ours is proven, by very different techniques, for a dynamics with two conserved quantities (energy and volume) in the case when the initial data is given by a Gibbs equilibrium measure.

Infinite chain of interacting harmonic oscillators.
2.1.1. Hamiltonian system. The dynamics of the chain of oscillators can be written formally as a Hamiltonian system of differential equationṡ The formal Hamiltonian is given by and we assume also (cf [5]) that: (a1) (α x ) x∈Z is real valued and there exists C > 0 such that |α x | ≤ Ce −|x|/C for all x ∈ Z, (a2)α(k) is also real valued andα(k) > 0 for k = 0 and in caseα(0) = 0 we havê α (0) > 0, (a3) to guarantee that the local energy functional, see (3.8) below, is non-negative we assume that α x ≤ 0, x = 0.
The p x component stands for the velocity (or momentum, as the mass of each particle is taken equal to 1) of the particle x. In the pinned case,α(0) > 0, the particle labelled with x feels a pinning harmonic potential centered at ax, where a ≥ 0 is an arbitrary equilibrium interparticle distance, so q x should be interpreted as the displacement of the position of the particle x from the point ax. Since the dynamics is linear, it does not depend on a, which assume is equal to 1.
In the unpinned case,α(0) = 0, the system is translation invariant, and only the interparticle distances are relevant for the dynamics. So in the unpinned case the variables q x are defined up to a common additive constant. Therefore, the relevant quantities are functionals of the relative distances between the particles. An important example is a wave function defined in Sect. 5.1. Its definition is unambiguous both in the pinned and unpinned cases. In the unpinned case, the total momentum and the energy of the chain are formally conserved, (besides the volume of course). Since we insist on preserving these properties, we choose a stochastic perturbation having the same conservation laws. This can be done either locally, via a time continuous stochastic exchange of momentum considered in this paper, or through a time discontinuous random exchange of momentum mechanism (see Sect. 2.1.3).

Continuous time noise.
We add to the right hand side of (2.1) a local stochastic term that conserves 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 with the parameter γ > 0 that determines the strength of the noise in the system, and (Y x ) are vector fields given by Here (w x (t)) t≥0 , x ∈ Z are i.i.d. one dimensional, real valued, standard Brownian motions, that are non-anticipative over some filtered probability space ( , F, The lattice Laplacian of (g x ) x∈Z is defined as g x := g x+1 + g x−1 − 2g x . Let also ∇g x := g x+1 − g x and ∇ * g x := g x−1 − g x . For a future reference we let β 1,x := ∇ * β (0) x . A simple calculation shows that where, for the abbreviation sake, we have written s(k) := sin(π k) and c(k) := cos(π k), k ∈ T. (2.9)

Random momentum exchange.
Another possible stochastic dynamics that conserves the volume, energy and momentum (in the unpinned case) can be obtained by a "jump" type mechanism of the momentum exchange. More precisely, let (N x,x+1 (t)) x∈Z be i.i.d. Poisson processes with intensity 3γ /2. The dynamics of the position component (q x (t)) x∈Z is the same as in (2.5), while the momentum (p x (t)) x∈Z is a càdlàg process given by dp

Remarks on hyperbolic scaling. Euler equations.
Consider now the unpinned casê α(0) = 0. For a configuration (p x (t), r x (t)) x∈Z we define the energy per atom: Thanks to condition (a3) we have e x (t) ≥ 0. Notice also that, since . Then x r x (t), when finite, represents the total length of the system when the equilibrium interparticle distance a = 0. The chain has three formally conserved (also called balanced) quantities Because the noise is added to the system, these are the 'only' conserved quantities. More precisely, the only stationary probability measures for the infinite dynamics (2.1), that are also translation invariant and have a finite density entropy property (see Definition 4.2.1 of [7]), are mixtures of the Gibbs measures x dr x dp x parametrized by the temperature T , momentump and tension τ , properly defined locally by the appropriate DLR equations on their conditional distributions (see Section 4 of [7]). It can be proven that after the hyperbolic space-time scaling, these conserved quantities evolve deterministically following the system of Euler equations: with the initial datā r (0, y) =r 0 (y),p(0, y) =p 0 (y),ē(0, y) =ē 0 (y) determined by the limits of quantities given by (3.2) at time t = 0. Here the parameter τ 1 , called the sound speed, is defined by (3.4) More precisely, consider the empirical distributions associated to the conserved quantities: u x (t) := (r x (t), p x (t), e x (t)). Then, with J -a smooth test function with compact support, and the convergence holds in probability for any t > 0, provided it holds for the initial distribution at t = 0. The functionsr 0 ,p 0 ,ē 0 are assumed to belong to C ∞ 0 (R)-the space of all smooth and compactly supported functions. The components ofū(t, y) := (r (t, y),p(t, y),ē(t, y)) satisfy (3.3). Note that system (3.3) decouples. Quantities (r (t, y),p(t, y)) satisfy the linear wave equation. Define the energy of the phonon modes as e ph (t, y) := τ 1r 2 (t, y) The residual energy component, called the local temperature profile, is given by The above definition leads to the decomposition of the energy profileē(t, y) into the temperature profile, that remains stationary under the hyperbolic scaling, and the phononic energyē ph (t, y) whose evolution is driven by the linear wave equation, see (3.3). Observe that, starting with compactly supported initial data, the phonon energy will disperse to infinity, as t → ∞, and the energy profile will converge (weakly) to the temperature profile. This is the reason why at any larger time scale, we have only to look at the evolution of the temperature profile.
In the case of a finite number of particles N = [ −1 ], with periodic or other boundary conditions, convergence in probability stated in (3.5) can be proven by using relative entropy methods, see [19] and [8]. In fact in the latter paper the limit has been shown in the non-linear case, in the smooth regime of the Euler equations. In the infinite volume, starting with the initial distribution μ on the space of configurations (r x , p x ) x∈Z satisfying sup with · μ denoting the expectation with respect to μ , the relative entropy method cannot be applied. The detailed analysis of the behavior of the energy component corresponding to the phononic modes, under the hyperbolic scaling is not the subject of the present paper and we shall deal with it in our future work. Our purpose here is to go beyond the hyperbolic time scale and understand the behavior of the energy component corresponding to the local temperature profile on the diffusive or (if necessary) superdiffusive space-time scale.

Behavior of the energy functional.
Our main results deal with the macroscopic behavior of the energy functional, for a given configuration (p(t), q(t)). The energy per site is defined as . (3.8) In this section we shall assume that condition (3.7) is satisfied. Denote by E the expectation with respect to the product measure P := μ ⊗ P.

Superdiffusive behavior of the unpinned chain.
We assume first thatα(0) = 0, i.e. the pinning potential vanishes and the Hamiltonian dynamics conserves both the momentum and energy. Define, the energy spectrum of a configuration (p x , q x ) x∈Z as wherep(k) andq(k) are the Fourier transforms of (p x ) and (q x ), respectively (see Sect. 4 below), andα(k) is given by (2.3). Assumption (3.7) is equivalent with In what follows we shall suppose a stronger integrability condition on w (k). Namely, we assume that sup According to the remark made below formula (4.9) the above assumption implies that both lim Suppose that the initial distribution of energy satisfies the following assumptions: where W 0 ∈ L 1 (R) (it is obviously non-negative).
Theorem 3.1. Let δ = 3/2, then, under the conditions on the initial distribution stated in the foregoing, for any test function J ∈ C ∞ 0 ([0, +∞) × R) we have: where W (t, y) satisfies the fractional heat equation: with the initial condition W (0, y) = W 0 (y) and The proof of this result is a direct consequence of Theorem 5.2 and Proposition 5.3 formulated below. In fact, (as can be seen from the aforementioned results) it can be formulated in a more general way to cover also the case of a weaker noise, i.e. parameter γ can be replaced by s γ 0 , for some s ∈ [0, 1) and γ 0 > 0. Then, the result is still valid at the time scale corresponding to the exponent δ = (3 − s)/2. The limit W (t, y) is the same as in the case s = 0, covered by Theorem 3.1.

Diffusive behavior of the pinned chain.
Ifα(0) > 0 there is a pinning potential and the Hamiltonian dynamics does not conserve the momentum. Energy is the only relevant conserved quantity but it does not evolve at the hyperbolic space-time scale.
Since ω (k) ≈ k and R(k) ≈ k 2 , as k 1 (see (2.4) and (2.7)), we haveσ 2 < +∞ (it is infinite in the unpinned case, due to ω (k) ≈ sign k). As a result, the evolution is diffusive and we have the following: Then, under the assumptions made in the foregoing, for any J (t, y) as in Theorem 3.1 we have Hereĉ The above theorem follows directly from Theorem 5.1 and the already mentioned Proposition 5.3 formulated below.

Equilibrium fluctuations.
The results formulated in Sect. 3.2 hold under the condition of finite microscopic total energy (3.7). By a duality argument they can be applied to obtain the following macroscopic behavior of the fluctuations when the system starts in an equilibrium measure μ E 0 ,0,0 . For the fluctuations of the energy mode we assume that γ = γ 0 s for some γ 0 > 0 and s ∈ [0, 1). Consider the energy fluctuation field where e x (t) is given by (3.8) and δ is chosen as before, i.e. δ = (3 − s)/2 in the unpinned case, and δ = 2 − s in the pinned one. The covariance field is defined as The following theorem is a direct corollary from Theorems 6.1 and 6.2, and Proposition 6.3 formulated below.

Theorem 3.3. For any functions J
with the initial condition and A = −ĉ| y | 3/4 in the unpinned, or A = D y in the pinned case, respectively. Coefficientsĉ and D are the same as in Theorems 3.1 and 3.2, respectively.
Remark. We remark here that Theorems 3.1 through 3.3 hold also for the dynamics corresponding to the random momentum exchange model described by (2.10).

Some Basic Notation
The one dimensional torus T is the interval [−1/2, 1/2] with identified endpoints. Let 2 be the space of all complex valued sequences (ψ x ) x∈Z , equipped with the norm

Formula (2.3) determines also an isometric isomorphism between h m and H m (T)-the
For an arbitrary J : T → C, k ∈ T, p ∈ R and > 0 we define 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 sup y∈R, k∈T For J ∈ S we letĴ be its Fourier transform in the first variable, i.e.
We introduce the norm By A we denote the completions of S in the respective norm.

Averaged Wigner transform.
For a given ∈ (0, 1] we let ψ be a random element distributed on 2 according to a Borel probability measure μ . We assume that (cf (3.9)) where · μ is the expectation with respect to μ . Define and for any J ∈ A. From the Cauchy-Schwartz inequality we get Thanks to Jensen's inequality we conclude from (4.3) that Therefore sup Functional W (0) ∈ A is called the averaged Wigner transform of ψ. We refer to Y (0) as the averaged anti-Wigner transform. By Plancherel's identity we obtain for any J ∈ S. As a consequence of (4.7), both W and Y are * -weakly (sequentially) compact in A , as → 0+, i.e. for any sequence n → 0 we can choose a n ) n≥1 whose each component is * -weakly convergent in A , see Section 4.1 of [5].
One can show, see Theorem B4 of [17], that if (W (0) n ) n is * -weakly convergent then there exists a finite Borel measure W 0 (dy, dk) on R × T whose total mass does not exceed K 0 and such that Applied to functions J (y, k) = J (y) the Wigner distribution becomes: Indeed, by Plancherel's identity we can write that the absolute value of the expression under the limit equals , where the estimate follows by Hölder inequality. Using the change of variables k := k/ in the second integral on the right hand side we conclude that it is bounded by , which proves (4.10).

Homogeneous random fields on
Let (ξ y ) y∈Z be a sequence of i.i.d. complex Gaussian random variables such that Eξ 0 = 0 and E|ξ 0 | 2 = 1. Defineψ a Gaussian, random H −m (T)-valued element, where m > 1/2. Its covariance field equals for any J 1 , J 2 ∈ C ∞ (T). Then, is a complex Gaussian, stationary field. Function E(k) is called the spectral measure of the field (ψ x ) x∈Z . In the particular case when

The wave function and its evolution.
The wave function, adjusted to the macroscopic time, is defined as (see [5]) The Fourier transform of the wave function is given bŷ is an L 2 (T)-valued, adapted process that is the unique solution of the Itô stochastic differential equation, understood in the mild sense (see e.g. Theorem 7.4 of [9]) The process B(dt, dk) is a cylindrical Wiener noise on L 2 (T) given by where (w x ) are i.i.d. standard, 1-dimensional real Brownian motions.

Asymptotics of the Wigner transform.
In what follows we assume that condition (4.3) holds. Suppose also that s ∈ [0, 1) and that γ = γ 0 s . The noise in (2.5) is called weak (resp. strong) if s > 0 (resp. s = 0). Furthermore assume that for any J ∈ S such that J (y, k) ≡ J (y) we have where W 0 (·) belongs to L 1 (R) and is non-negative. Its Fourier transform shall be denoted by Since the total energy of the system x∈Z |ψ x (t)| 2 is conserved in time, see Section 2 of [5], for each ∈ (0, 1] we have (5.9) Here, as we recall, E is the expectation with respect to P = μ ⊗ P. From (5.8) we conclude, thanks to (4.7), that where K 0 is the constant appearing in condition (4.6). As a direct consequence of the above estimate we infer that the family (W (·)) ∈(0,1] is * -weakly sequentially compact in any Our main result states that, given s ∈ [0, 1), the exponent δ can be adjusted so that (W (·)) is * -weakly convergent, as → 0+, in any L ∞ ([0, T ]; A ), where T > 0. The cases of pinned (α(0) > 0) and unpinned chains (α(0) = 0) are considered in Sects. 5.2.1 and 5.2.2 respectively. Before presenting our results let us recall briefly the case of the kinetic limit treated in [5], see Theorem 5 in ibid., corresponding to s = 1, which is outside of the scope of our results. Then, taking δ = 1 the family W (·) is * -weakly convergent, as → 0+, to the unique weak solution of the linear kinetic equation The scattering operator L, acting on the k-variable, is defined by Here R(k) is given by (3.18) and Case of a pinning potential-diffusive transport of energy. Suppose that Since (5.17) together with the assumptionα (0) > 0 imply that From the above and (5.16) we infer thatσ 2 given by formula (3.17) is finite. where, andĉ is defined by (3.21), if s ∈ (0, 1) (weak noise), or by (3.20), if s = 0 (strong noise).

Case of a no pinning potential-3/2 fractional superdiffusion. Suppose that
Recall that in this case the dispersion relation satisfies (2.4). Therefore, the integral appearing on the right hand side of (3.17) becomes divergent. Define with W 0 ( p) given by (5.7) andĉ Our result can be formulated as follows. The proofs of the above two theorems are presented in Sect. 11.2.

Energy modes.
Thanks to condition e x (t) defined in (3.8) are non-negative. A simple calculation, using the definition of the Wigner transform, see (4.8), shows that Theorem 3.1 (resp. Theorem 3.2) is a consequence of (5.23), Theorem 5.2 (resp. Theorem 5.1) and the following result, proved in Sect. 13.1.
for δ as in the statement of Theorem 5.1 (resp. Theorem 5.2).

Fluctuations in Equilibrium
In this section we assume that the system is in equilibrium, i.e. that (ψ x ) x∈Z is a homogeneous, complex Gaussian random field whose covariance function is given by (4.15).
As we have already mentioned, its law x (t)) as the field given by the Fourier coefficients of the solution (ψ ( ) (t, k)) of the Eq. (5.4) whose initial data is distributed according to μ E 0 . It has been shown in [12], see We will also denote W (t; J ) := W (ψ ( ) (t); J ). From the time invariance of the law of ψ ( ) (t) and (4.15) we obtain Given J 1 , J 2 ∈ S define also the covariance field In the particular case when t = 0 we obtain Using the Parseval identity we conclude that Our result dealing with this situation can be formulated as follows.

6.2.2.
Case of a no pinning potential. The result in this case can be formulated as follows.
The proofs Theorems 6.1 and 6.2 are presented in Sect. 12.

Energy fluctuations.
Applying the Wigner fluctuating field to a function J (y) constant in k we obtain the fluctuation field Denote the empirical fluctuation of energy field by and the respective second mixed moment by Our next result shows the fields defined by (6.8) and (6.9) are asymptotically equal. Its proof is presented in Sect. 13.2.
As a result the conclusions of Theorems 6.1 and 6.2 hold for C (e) (t; J 1 , J 2 ) substituted in place of C (t; J 1 , J 2 ), which in turn implies Theorem 3.3.

Outline of the Proofs of Theorems 5.1 and 5.2
This section is intended to outline the proof of Theorem 5.2 (Theorem 5.1 follows from a similar consideration). First, in Sect. 8, we describe the evolution of the Wigner transform W (t, y, k) of the wave function ψ ( ) x (t) introduced in Sect. 5. In fact, for our purposes it is more convenient to deal with its Fourier transform in the spatial domain, given by (8.1). It satisfies the following equation Here, γ = γ 0 s for some s ∈ [0, 1), with δ = (3−s)/2 and O( ) is some expression that becomes negligible, as → 0+. Here U ,+ (t, p, k) represents the difference between the kinetic and potential energy, while U ,− (t, p, k) is related to the energy current (the product of the momentum and inter-particle distance). They are highly oscillatory and their averages (in time and in k) turn out to vanish in the limit as → 0+.
To simplify the presentation we assume also here that the scattering kernel equals R(k, k ) = R(k)R(k ), where R(k) = 2 sin 2 (π k), which is actually the case for the random momentum exchange model described in Sect. 2.1.3. The scattering operator (see (5.12)) is then of the form Since the wave function at time t = 0 is L 2 bounded, see (4.6), this bound persists in time, due to the energy conservation. In turn this implies the bound on the norm of (W (·)) in L ∞ ([0, +∞); A ), see (5.10). In consequence this family is compact in the * -weak topology in L ∞ ([0, T ]; A ) for any T > 0. Our goal is to identify its limit as the function W (t) appearing in the statement of Theorem 5.2. To do so we modify the argument put forward in [18]. To further simplify our presentation we drop the negligible term appearing on the right hand side of (7.1). Performing the Laplace transform on both sides of (7.1) and using the formula (7.2) for the scattering operator L we obtain that (dropping the arguments (λ, p, k) to abbreviate the notation) Herew (λ, p, k) andū ,± (λ, p, k) are the Laplace transforms of W (t, p, k) and U ,± (t, p, k), respectively, see (10.5). After some simple computations we get Performing the scalar product of both sides of the equation against 2γ R/ δ we conclude where W ( ) 0 is the Fourier-Wigner transform of the initial condition and Here O is the expression that arises from the scalar multiplication of the right hand side of (7.4). It is quite simple to show that the second term on the left hand side of (7.5) tends to W 0 ( p), given by (5.7). Our main effort goes into proving that the right hand side of (7.5) vanishes as → 0+ and that a ( ) w → λ + C| p| 3/2 for an appropriate C > 0, as → 0+, when ω(0) = 0 (we have a ( ) w → λ + C p 2 in the unpinned case). The first fact is a consequence of the aforementioned oscillatory behavior ofū ,± , while the convergence of a ( ) w follows from detailed calculations, see Proposition 11.1. This allows us to conclude that w(λ, p), R L 2 (T) -the limit of w (λ, p), R L 2 (T) , as → 0+, satisfies Since in the macroscopic time the number of random collisions grows as s−δ 1 (recall that is proportional to γ / δ ∼ s−δ ) the limit w(λ, p, k) of energy densityw (λ, p, k) for a fixed p, as → 0+, should become independent of the k-variable, therefore, since T R(k)dk = 1, we ought to have w(λ, p, k) ≡ w(λ, p) = w(λ, p), R L 2 (T) .
This homogenization result is proved in Theorem 10.2 and allows us to conclude (7.7). By virtue of (7.6) and the uniqueness property of the Laplace transform we infer that W (t), appearing in the statement of Theorem 5.2, satisfies (5.21). The "true" argument is a bit more involved, due to the fact that the scattering kernel R(k, k ) corresponding to the noise considered in this paper is not of a product type, see (5.13), which complicates the actual calculations.

Evolution of the Wigner Transform
For a given > 0 letψ ( ) (t) be a solution of (5.4) with the initial conditionψ distributed according to a probability measure μ on L 2 (T). The Fourier transform of the Wigner transform ofψ ( ) (t) is given by where, as we recall, E is the average with respect to the initial condition and the realization of the noise. To close the equations governing the dynamics of W (t, p, k) we shall also need the following functions We shall also write W ,+ = W and Y ,+ = Y . Define (cf (5.5)) With the above definition we can write (see (3.18) for definition R(·)) where δ ω(k, p) andR(k, p) are defined in (4.1) and is the Fourier transform of the momentum. Since the latter is real valued, its Fourier transform is complex even. The last term appearing on the right hand side of (8.4) can be replaced by and In addition, After straightforward calculations (cf (8.4)-(8.7)) we conclude that Then, (8.10) From (8.7) and (8.10) we conclude that for any fixed p ∈ R the evolution ( W (t), is governed by a closed system of four linear equations with a generator that is a bounded operator in (L r (T)) 4 for any r ∈ [1, +∞]. In particular, under the assumption that the initial distribution of the wave functions satisfies (4.3) the components of ( W (t), Y (t), Y ,− (t), W ,− (t)) belong to C([0, +∞); A ).
and C 1 is as in (9.18). The "remainder" termr (i) (λ, p, k), that is the Laplace transforms t, p, k), has the following property: for any M > 0 and compact interval I ⊂ (λ 0 , +∞) for i = 1, 2, 3, 4 and ∈ (0, 1], | p| ≤ M, λ ∈ I . Therefore, from Proposition 9.1 we conclude that (10.4) Taking the Laplace transform of the both sides of equations of the system (9.13) we obtain (2) , where L := L+(1/2)( p) 2 (δ 2 L) and L, δ 2 L are given by (5.12) and (9.6), respectively. Performing the real parts of the scalar products in L 2 (T) of the respective equations of the above system with (1/2)w ,± (λ, p, k),ū ,± (λ, p, k) and adding them sideways we get Given M > 0 and I ⊂ (λ 0 , +∞) compact, we havē In fact it is possible to get a more precise result.  D (w (λ, p)) We postpone the proof of the above Proposition till Sect. 14.2 and use it first to show a homogenization result formulated below. Define By virtue of (10.8) we have The last estimate follows from (see (5.13)) This together with (10.4) imply (10.10). The case ι = + can be argued similarly. The proof of (10.11) is a consequence of (10.4) and (10.8).

Identification of the Limit of the Wigner Transform
Recall that (W (·)) is * -weakly sequentially compact in L ∞ ([0, T ], A ) for any T > 0. Therefore for any n → 0, as n → +∞, we can choose a subsequence, denoted in the same way, such that it * -weakly converges to some W (·) ∈ L ∞ ([0, T ], A ). In light of (5.10) we have sup with K 0 the same as in (4.6). Therefore, we can define its Laplace-Fourier transform w(λ, p, k) for any λ > 0. Thanks to Theorem 10.2, any limit w(λ, p, k) obtained this way will be constant in k.

Proof of Proposition 11.1.
Proof of (11.10). It suffices only to prove that, for any J as in the statement of (11.10) Then equality (11.10) is a consequence of (5.6). Note that 2γ R/D ( ) is bounded and convergent to 1, as → 0+. Using Cauchy-Schwarz inequality we can estimate the expression under the limit in (11.16) by The first integral tends to 0, as → 0+, by virtue of the Lebesgue dominated convergence theorem, while the second one remains bounded thanks to condition (4.3). Thus (11.10) follows.

The limit of I
(11.20) In the integral appearing in I Define Then, Note that, according to (11.20), From (3.18) (and (2.7)) we conclude that bothR(k) andR (k) converge uniformly to 6π 2 k 2 when k ∈ I , | p| ≤ M. Likewise, δ ω(k, p) converges uniformly to 1 when k ∈ I and λ ∈ I , | p| ≤ M. Since in additioñ and the convergence is uniform in λ ∈ I and | p| ≤ M. Using the calculus of residua one can show that Thus, The limit of I ( ) 3 . Then, as → 0+, uniformly in λ ∈ I and | p| ≤ M (recall that ρ 2 ∈ (0, 2 − δ − s)). It ends the proof of (11.19), thus finishing the proof of (11.14).
Proof of (11.11). It is a simple consequence of the following.
Proof. Similarly as in (11.17) we get a ( ) Term |a ( ) +,1 | is bounded, due to the fact that γ e + ( δ λ + 2γ R ) |D ( ) | 2 . To bound the term a ( ) +,2 in the pinned case we use the fact that then (δ ω) 2 R . In the case ω(0) = 0 we use the bound e + R. Then, the conclusion of the lemma follows from (11.19).
Proof of (11.13) and (11.15). Denote The equalities in question follow easily from our next result.
Next, (11.26) Denote the terms appearing on the utmost right hand of (11.26) by I , II and III , respectively. Since T L f dk = 0 for any f ∈ L 1 (T) we have III = 0. In addition, (see (9.3)) Here u , and, using the estimate |D | ≥ δ + γ R (k), we get This leads to estimate (recall that δ = 2 − s) In the unpinned case,ω(k, p) ≈ R (k), therefore from (10.8) we get (11.31) and, using again |D | ≥ δ + γ R (k), we obtain We have shown therefore that in both cases lim →0+ I = 0. Concerning term II note that, thanks to the fact that k → δ ω(k, p) is odd and k → Lū ,+ (λ, p, k) is even we have We conclude therefore that Denote the terms appearing on the right hand side by II (1) and II (2) , respectively.
11.3. The dual dynamics. The equations (8.7) and (8.10) describing the dynamics of the column vector W ε (t, p, k) given by can be written in the form where L is some matrix operator. We now define the dual dynamics that runs on test functions. Suppose that is the solution of the system dual to (11.35), i.e.
. (11.36) with given initial conditions that are the Fourier transforms of some functions belonging to S. The adjoint matrix L * is given explicitly by The operators L * p , L ± p * and R * p are the adjoints of L p , L ± p and R * p (see (8.6) and (8.8)) with respect to the Lebesgue measure on T. Given M > 0 we introduce the norm 12. Proofs of Theorems 6.1 and 6.2

Evolution of the random Wigner transform.
To describe the evolution of the fluctuating Wigner transform W (t; J ), see (6.1), we shall also need the following quantities We Computing the time differential as in Sect. 8, we obtain , are some square integrable, continuous trajectory martingales. Summarizing, if the test functions J w,± and J y,± are such that their respective Fourier transforms in the x variableĴ w,± andĴ y,± belong to C ∞ c (R × T), then, using (12.2) and (12.3), we obtain where L * is given by (11.37). Suppose that J ( ) (t) is the solution of the Eq. (11.36). From part (ii) of Proposition 11.4 we conclude that provided that M > 0 is such that J ( ) (0, p, k) ≡ 0 for all | p| ≥ M, k ∈ T. Combining (12.4) with (11.36) we obtain Suppose that the initial data satisfies the hypothesis of part (ii) of Proposition 11.4 and thatĴ is compactly supported. According to (6.3) the right hand side of (12.6) equals The last equality holds, thanks to the Poisson summation formula, see [14], formula (50) on p. 566. Since the supports ofĴ w,+ (t) andĴ are both compact in p for a sufficiently small we can write that the right hand side equals Using (11.40) we conclude that for any compactly supported φ ∈ L 1 [0, +∞) we have withĉ given by (3.21) when δ < 2, or (3.20) when δ = 2 in the case of a pinning potential, or withĉ given by (5.22) in the unpinned case. Generalization to arbitrary J w,+ , J ∈ S and φ ∈ L 1 [0, +∞) is standard and can be done via an approximation, due to the fact that process W (t; J ) t≥0 is stationary.
Remark. Observe that the proof does not really use time stationarity of the initial distribution, in fact it follows that for any initial homogeneous distribution with energy density given by some E(k) such that T E(k)dk = 2E 0 , we have the same result. On the other hand, we do use the stationarity in order to prove the equivalence of the energy distribution (6.11), see Sect. 13.

Equivalence of Energy Functionals
]ω x−y dp.

Lemma 13.2. For any J
Proof. Using (13.2) we can write φ x in the Fourier transform coordinates as Note that F(k, −k) = 0. Moreover, according to Lemma 13.1 it is bounded. Observe that under the condition (13.3), function ω(k)q(k) is square integrable on T (althougĥ q(k) need not be so). Furthermore, By the Schwarz inequality and symmetry in k and k + p, we obtain Since Z ( p) is bounded, the result follows upon an application of the Lebesgue dominated convergence theorem.

13.2.
Proof of Proposition 6.3. Obviously, stationarity implies that the limit in (6.11) does not depend on t therefore it suffices to prove (6.11) for t = 0. For that purpose it is enough to show that 13.2.1. The case of an unpinned chain. We assume thatα(0) = 0, therefore φ (3) x = 0. Then, the field (q x ) is Gaussian given by whereŵ(dk) is a complex even, Gaussian white noise in L 2 (T), i.e.
Observe that F(−k, k) = 0. Summing first over x and then over x we obtain that the utmost right hand side of (13.15) equals n∈Z RĴ ( p)Ĵ n − p T F(−k − p, k)dk dp. (13.16) Therefore (13.11) (thus also the conclusion of the proposition) is a consequence of the Lebesgue dominated convergence theorem and Lemma 13.1.
14.2. Proof of Proposition 10.1. From the third equation of (10.5) Denote the terms appearing on the right hand side by J j , j = 1, 2, 3, 4. Thanks to (10.7) we have J 1 δ−s . Also, (since λ ≥ λ 0 ) From here we get that J j δ−s , j = 2, 3. Finally, which also yields J 4 δ−s that finally leads to an estimate |ū ,+ | 2 dk δ−s . (14.17) To obtain the estimate of D (w (λ, p)) it suffices to prove that D ū ,+ (λ, p) δ−s , which follows, provided we can show that is bounded from below by γ −2 , in the unpinned case (cf (14.8)) and by γ −2 R −1 in the pinned one we can bound the integral over the region by δ−s , due to (14.17). Hence, (14.18) follows.