Hydrodynamic limit for a chain with thermal and mechanical boundary forces

We prove the hydrodynamic limit for a one dimensional harmonic chain with a random flip of the momentum sign. The system is open and subject to two thermostats at the boundaries and to an external tension at one of the endpoints. Under a diffusive scaling of space-time, we prove that the empirical profiles of the two locally conserved quantities, the volume stretch and the energy, converge to the solution of a non-linear diffusive system of conservative partial differential equations.


Introduction
The mathematical derivation of the macroscopic evolution of the conserved quantities of a physical system, from its microscopic dynamics, through a rescaling of space and time (so called hydrodynamic limit) has been the subject of much research in the last 40 years (cf. [9] and references within). Although heuristic assumptions like local equilibrium and linear response permit to formally derive the macroscopic equations [14], mathematical proofs are very difficult and most of the techniques used are based on relative entropy methods (cf. [9] and references within). Unfortunately, in the diffusive scaling when energy is one of the conserved quantities, relative entropy methods cannot be used: since one would need sharp bounds on higher energy moments, which cannot of time evolution of the Fourier-Wigner functions defined in Section 8.1. The last two terms in the right hand side of (8.9) correspond to the dissipation, due to the stochastic dynamics in the bulk. The remaining two terms describe the interaction between the fluctuation of the thermal and mechanical components of the kinetic energy at the boundary points and in the bulk of the system, respectively. In order to control these terms we need to bound the rate of damping of the mechanical energy, which is done in Lemma 5.4. These controls allow us to prove that the L 2 norm of the covariances of random fluctuations of momenta and stretches, at the given time, grows with the logarithm of the size of the system: this is the content of Proposition 8.1. This in turn enables us to show, using again the properties of the Fourier-Wigner function dynamics, the already mentioned equipartition property, which is stated in Proposition 4.6 and proved in Section 8.3.
The next ingredient that is important in the hydrodynamic limit argument is the linear bound, in the system size, for the relative entropy of the chain, with respect to both the thermal equilibrium and local equilibrium probability measures. We establish this bound, together with some of its consequences, in Section 7 (see Proposition 7.1). A crucial property that allows us to control the entropy production, coming from the action of the external force, is the estimate of the damping rate of the time average of the momentum expectation at the respective endpoint of the system obtained in Proposition 4.2.
As we have already mentioned the model we consider in the present work, with the random flip of the sign of momenta, is more difficult to handle than the random momentum exchange one investigated in [11], due to the fact that the energy is not transported by the stochastic part of the dynamics. We believe that the method used in the present paper can be also applied to that model. In addition, the assumption that the forcing acting at the boundary is constant in time, is only made here to simplify the already complicated arguments for the entropy bound of Section 7.1 and the momentum damping estimates formulated in Proposition 4.2 and Lemma 5.4. At the expense of increasing the volume of the calculations, with some additional effort, one could extend the results of the present paper to the case when the tension is a C 1 smooth function of time.
A proof of the Fourier law in the stationary state remains an open problem for the random flip model. We hope that in the future we will also be able to extend the results of the present paper to the more challenging case of the chain of anharmonic springs.
Finally, concerning the organization of the paper. The description of the model and basic notation is presented in Section 2. The formulation of the main result, together with the auxiliary facts needed to carry out the proof are done in Section 3. For a reader convenience we sketch the structure of the main argument in Section 4. The proof of the hydrodynamic limit is carried out in Section 5. It is contingent on a number of auxiliary results that are shown throughout the remainder of the paper. Namely, the estimates of the momentum and stretch averages are done in Section 6, the energy production bounds are obtained in Section 7, while Section 8 is devoted to showing the equipartition property. Finally, in the appendix sections we give the proofs of quite technical estimates used throughout Section 6. We suppose that the position and momentum of a harmonic oscillator at site x ∈ I n are denoted by (q x , p x ) ∈ R 2 . The interaction between two particles situated at x − 1, x ∈ I • n EJP 26 (2021), paper 19. is described by the quadratic potential energy

Open chain of oscillators
At the boundaries the system is connected to two Langevin heat baths at temperatures T 0 := T − and T n := T + . We also assume that a force (tension) of constant value τ + ∈ R is acting on the utmost right point x = n. Since the system is unpinned, the absolute positions q x do not have precise meaning, and the dynamics depends only on the interparticle stretch r x := q x − q x−1 for x = 1, . . . , n, and by convention throughout the paper we set r 0 := 0. The configurations are then described by (r, p) = (r 1 , . . . , r n , p 0 , . . . , p n ) ∈ Ω n := R n × R n+1 . (2.1) The total energy of the chain is defined by the Hamiltonian: H n (r, p) := x∈In E x (r, p), (2.2) where the microscopic energy density is given by Finally, we assume that for each x ∈ I n the momentum p x can be flipped, at a random exponential time with intensity γn 2 , to −p x , with γ > 0. Therefore, the microscopic dynamics of the process {(r(t), p(t))} t 0 describing the total chain is given in the bulk by dr x (t) = n 2 (p x (t) − p x−1 (t)) dt, x ∈ {1, . . . , n}, dp x (t) = n 2 (r x+1 (t) − r x (t)) dt − 2p x (t − ) dN x (γn 2 t), x ∈ I • n , (2.4) and at the boundaries by r 0 (t) ≡ 0, dp 0 (t) = n 2 r 1 (t) dt − 2p 0 (t − ) dN 0 (γn 2 t) − γn 2 p 0 dt + n 2 γT − dw 0 (t), (2.5) dp n (t) = −n 2 r n (t) dt + n 2 τ + dt − 2p n (t − ) dN n (γn 2 t) − γn 2 p n dt + n 2 γT + dw n (t), where w 0 (t) and w n (t) are independent standard Wiener processes and N x (t), x ∈ I n are independent of them i.i.d. Poisson processes of intensity 1. Besides, γ > 0 regulates the intensity of the Langevin thermostats. All processes are given over some probability space (Σ, F, P). The factor n 2 appearing in the temporal scaling comes from the fact that t, used in the equations above, is the macroscopic time, and the microscopic time scale is the diffusive one. We assume that the initial data is random, distributed according to the probability distribution µ n over Ω n . We denote by P n := µ n ⊗ P (resp. E n ) the product probability distribution over Ω n × Σ (resp. its expectation).
Equivalently, the generator of this dynamics is given by for any C 2 -class smooth function F . Here p x is the momentum configuration obtained from p with p x replaced by −p x . Finally, the generator of the Langevin heat bath at the boundary points equals: (2.9)

Notations
We collect here notations and conventions that we use throughout the paper.
• Given an integrable function G : I → C, its Fourier transform is defined by (2.10) If G ∈ L 2 (I), then the inverse Fourier transform reads as where the sum converges in the L 2 sense.
• Given a sequence {f x , x ∈ I n }, its Fourier transform is given by k ∈ I n := 0, 1 n+1 , . . . , n n+1 . (2.12) Reciprocally, for any f : I n → C, the inverse Fourier transform reads where we use the following short notation k∈ In for the averaged summation over frequencies k ∈ I n . The Parseval identity can be then expressed as follows k∈ In For a given function f we adopt the convention (2.16) According to our notation, given a configuration (r, p) = (r 1 , . . . , r n , p 0 , . . . , p n ) ∈ Ω n := R n × R n+1 EJP 26 (2021), paper 19.
we let, for any k ∈ I n , recalling the convention r 0 := 0. Since the configuration components are real valued, the corresponding Fourier transforms have the property: • For a function G : I → C, we define three discrete approximations: of the function itself, of its gradient and Laplacian, respectively by • Having two families of functions f i , g i : A → R, i ∈ I, where I, A are some sets we for any i ∈ I, a ∈ A.
If both f i g i , i ∈ I and g i f i , i ∈ I, then we shall write f i ≈ g i , i ∈ I.

Hydrodynamic limits: statements of the main results
In this section we state our main results, given below in Theorem 3.2, Theorem 3.3 and Theorem 3.5. Before that, we formulate our first assumption on the initial probability distribution of the configurations.
Suppose that T > 0. We first introduce ν T (dr, dp) which is defined as the product Gaussian measure on Ω n of zero average and variance T > 0 given by ν T (dr, dp) := e −E0/T √ 2πT dp 0 n x=1 e −Ex/T √ 2πT dp x dr x .
(3.1) Let µ n (t) be the probability law on Ω n of the configurations (r(t), p(t)) and let f n (t, r, p) be the density of the measure µ n (t) with respect to ν T . We define the linear interpolation between the inverse boundary temperatures T −1 − and T −1 Recall the definition of its discrete approximation: β x := β(x/n), x ∈ I n . Let ν be the corresponding inhomogeneous product measure with tension τ + : where the Gibbs potential is for β > 0, τ ∈ R. Consider the density f n (t) := f n (t) dν T d ν . H n (t) := Ωn f n (t) log f n (t)d ν. (3.6) In the entire paper we assume f n (0) ∈ C 2 (Ω n ) and H n (0) n, n 1. (3.7)

Empirical distributions of the averages
We are interested in the evolution of the microscopic profiles of stretch, momentum, and energy, which we now define. For any n 1, t 0 and x ∈ I n , let Moreover, we denote by r (n) (t, k), p (n) (t, k), with k ∈ I n , the Fourier transforms of the first two fields defined in (3.8). We shall make the following hypothesis: Assumption 3.1. We assume 1. an energy bound on the initial data: 2. a uniform bound satisfied by the spectrum of the initial averages: The assumption (3.10) prevents the average of the initial profiles to concentrate at any particular mode k, as n → ∞. This is a natural regularity condition on the macroscopic profiles of volume stretch and momenta.

Convergence of the average stretch and momentum
In order to state the convergence results for the profiles, we extend the definition (3.8) to profiles on I, as follows: for any u ∈ I and x ∈ I n let Let r(t, u) be the solution of the following partial differential equation with the boundary and initial conditions: for any (t, u) ∈ R + × I. To guarantee the existence and uniqueness of a C 2 -regular solution of the above problem we assume that r 0 ∈ C 2 (I) and r 0 (1) = τ + . (3.14) Let p 0 ∈ C(I) be an initial momentum profile. Our first result can be formulated as follows.
EJP 26 (2021), paper 19. Theorem 3.2 (Convergence of the stretch and momentum profiles). Assume that the initial distribution of the stretch and momentum weakly converges to r 0 (·), p 0 (·) introduced above, i.e. for any test function G ∈ C ∞ (I) we have Then, under Assumption 3.1, for any t > 0 the following holds: weakly in L 2 (I), where r(·) is the solution of (3.12)- (3.13). In addition, we have The proof of this theorem is given in Section 5.2. It is not difficult to prove (see Section 4 below) that, under the same assumptions as in Theorem 3.2, for each t > 0 the sequence of the squares of the mean stretches {[r (n) ] 2 (·)} n 1 -the mechanical energy density -is sequentially −weakly compact in L 1 ([0, t]; C(I)) . However, in order to characterize its convergence one needs substantial extra work, and this is why we state it as an additional important result.
The proof of this theorem is contained in Section 5.3.

Convergence of the energy density average
Our last result concerns the microscopic energy profile. To obtain the convergence of E (n) x (t) for t > 0, we add an assumption on the fluctuating part of the initial data distribution. For any x ∈ I n , let Similarly as before, let r (t, k) be the Fourier transforms of the fields defined in (3.19). We shall assume the following hypothesis on the covariance of the stretch and momentum fluctuations.  Assumption 3.4 constitutes a rather weak hypothesis about the spatial decorrelation for the initial distribution of the stretch and momentum fields. Under this assumption the convergence in probability for the local (weak) law of large numbers for the initial profiles is guaranteed. For example under the local Gibbs measures x (0) x∈Z decorrelates for any x = x and the first two conditions of (3.20) are trivially satisfied, with the bound provided by inf x β −1 x . The third one also holds, as then E x p (n) Let e(t, u) be the solution of the initial-boundary value problem for the inhomogeneus heat equation  for any (t, u) ∈ R + × I. Here r(t, u) is the solution of (3.12)- (3.13), and e 0 is non-negative.
Our principal result concerning the convergence of the energy functional is contained in the following theorem:  where e(·) is the solution of (3.21)-(3.22).
The proof of this theorem is presented in Section 5.4.

Sketches of proof and Equipartition of energy
In this section we present some essential intermediate results which will be used to prove the convergence theorems, and which are consequences of the various assumptions made. We have decided to expose them in an independent section in order to emphasize the main steps of the proofs, and to highlight the role of our hypotheses.

The boundary terms
An important feature of our model is the presence of τ + = 0. A significant part of the work consists in estimating boundary terms. We first state in this section the crucial bounds that we are able to get, under Assumption 3.1, and which concern the extremity points x = 0 and x = n. One of the most important result is the following:    This result is proved in Section 6.1. Another consequence of Assumption 3.1 is the following one-point estimate, which uses the previous result (4.1), but allows us to get a sharper bound:  This proposition is proved in Section 6.2.

Remark 4.3.
In fact, in the whole paper, only the second estimate in (4.3) will be used. However, in its proof, the first estimate comes freely.

Estimates in the bulk
Provided with a good control on the boundaries, one can then obtain several estimates in the bulk of the chain. Two of them are used several times in the argument, and can be proved independently of each other. The first one is   The proof of Proposition 4.4 is given in Section 6.3 below, and makes use of Proposition 4.2. Here we formulate some of its immediate consequences: • thanks to (4.4) we conclude that for each t > 0 the sequence of the averages {r (n) (t)} n 1 is bounded in L 2 (I), thus it is weakly compact. Therefore, to prove Theorem 3.2 one needs to identify the limit in (3.16), which is carried out in Section 5.2, • the second equality (3.17) of Theorem 3.2 simply follows from (4.5), • finally, the estimate (4.4) implies in particular that  Therefore, we conclude that, for each t > 0 the sequence {[r (n) ] 2 (·)} n 1 is sequentially −weakly compact in L 1 ([0, t]; C(I)) , as claimed. This is the first step to prove Theorem 3.3.
The second important estimate focuses on the microscopic energy averages and is formulated as follows:  This estimate is proved in Section 7.1, using a bound on the entropy production, given in Proposition 7.3 below. Thanks to Proposition 4.5 the sequence {E (n) (·)} n 1 is sequentially −weakly compact in (L 1 ([0, t]; C 2 0 (I))) for each t > 0. Therefore, to prove Theorem 3.5, one needs to identify the limit. This identification requires the extra Assumption 3.4.

Consequence of Assumption 3.4
The proof of Theorem 3.5 is based on a mechanical and thermal energy equipartition result stated as follows: The proof of this result is presented in Section 8 (cf. conclusion in Section 8.3), and uses some of the results above, namely Proposition 4.1 and Proposition 4.5.

Proofs of the hydrodynamic limit theorems
In the present section we show Theorems 3.2, 3.3 and 3.5 announced in Section 3. The proof of the latter is contingent on several intermediate results: • first of all, to prove the three results we need specific boundary estimates which will be all stated in Section 5.1 (see Lemma 5.1), and which are byproducts of Proposition 4.2, Proposition 4.4 and Proposition 4.5; • the proof of Theorem 3.3 requires moreover Lemma 5.2, which is based on a detailed analysis of the average dynamics (r (n) x , p (n) x ) x∈In (that will be carried out in Section 6); • finally, to show Theorem 3.5 we need: first, a uniform L 2 bound on the averages of momentum, see Lemma 5.4 below. The latter will be proved in Section 6.4, as a consequence of Proposition 4.1; second, the equipartition result for the fluctuation of the potential and kinetic energy of the chain, which has already been stated in Section 4, see Proposition 4.6.

Treatment of boundary terms
First of all, the conservation of the energy gives the following microscopic identity: x ∈ I o n , are the microscopic currents. At the boundaries we have One can see that boundaries play an important role. Before proving the hydrodynamic limit results, one needs to understand very precisely how boundary variables behave. This is why we start with collecting here all the estimates that are essential in the following argument. Their proofs require quite some work, and for the sake of clarity this will be postponed to Section 7.3.
Lemma 5.1 (Boundary estimates). The following holds: for any t 0 (iv) (Boundary temperatures, part I) for any n 1 and at the right boundary point Provided with all the previous results which have been stated (but not proved yet), we are ready to prove Theorem 3.2 and 3.5. Before that, in order to make the presentation unequivocal, we present in Figure 1 a diagram with the previous statements, and the sections where they will be proved into parentheses.  : An arrow from A to B means that A is used to prove B, but is not necessarily a direct implication.

Proof of Theorem 3.2
Recall the diffusive equation (3.12), which can be formulated in a weak form as: for any test function G ∈ C 2 0 (I As usual, the symbol o n (1) denotes an expression that tends to zero with n → +∞. The dynamics of the averages (r(t), p(t)) is easy to deduce from the evolution equations (see also (6.2) where it is detailed). We can therefore rewrite the right hand side of (5.15) as EJP 26 (2021), paper 19.
Since G is smooth we have lim n→+∞ sup x∈In (∇ n G) x − G (x) = 0. Using this and Proposition 4.4 one shows that the second expression in (5.16) converges to 0, leaving, as the only possible significant, the first term. Summing by parts and recalling that G(0) = 0, it can be rewritten as Therefore, we need to understand the macroscopic behavior of the boundary strech variables, which is done thanks to Lemma 5.1: from (5.7) we conclude that the second term tends to zero, as n → +∞. Using again (5.7) but for the right boundary we infer that (5.17) can be written as The above means, in particular that the sequence is its -weakly limiting point. Any limiting point of the sequence satisfies (5.14), which shows that has to be unique and as a result {r (n) (·)} n 1 is -weakly convergent to r ∈ L ∞ ([0, t * ]; L 2 (I)), the solution to (3.12)-(3.13).

Proof of Theorem 3.3
The following estimate shall be crucial in our subsequent argument.
The proof of the lemma uses Proposition 4.2 and is postponed to Section 6.5.
Define r (n) int : [0, +∞) × I → R as the function obtained by the piecewise linear interpolation between the nodal points (x/(n + 1), r x ), x = 0, . . . , n + 1. Here we let r n+1 := r n . As a consequence of Lemma 5.2 above we obtain the following.
Proof. It is easy to see that From the proof of Theorem 3.2 given in Section 5.2 we know that the sequence t 0 r (n) int (s, u)ds weakly converges in L 2 (I) to t 0 r(s, u)ds. From (5.21) and the compactness of Sobolev embedding into C(I) in dimension 1 we conclude (5.22).
Thanks to (5.21) we know that for any t * > 0 we have The above implies that the sequence {[r ; C(I)) . One can choose a subsequence, that for convenience sake we denote by the same symbol, which is −weakly convergent in any L 1 ([0, t * ]; C(I)) , t * > 0. We prove now that for any G where r(·) is the solution of (3.12)-(3.13). By a density argument it suffices only to consider functions of the form x−1 (s) dsdt.

Proof of Theorem 3.5
Concerning equation (3.21)-(3.22), its weak formulation is as follows: for any test Given a non-negative initial data e 0 ∈ L 1 (I) and the macroscopic stretch r(·, ·) (determined via (5.14)) one can easily show that the respective weak formulation of the boundary value problem for a linear heat equation, resulting from (5.30), admits a unique measure valued solution.
Recall that the averaged energy density function E (n) (t, u) has been defined in (3.11).
It is easy to see, thanks to Proposition 4.5, that for any t * > and the initial condition E(0, u) = 0. Here r(t, u) is the solution of (3.12).
Concerning the limit identification for {E (n) } n 1 we write and, by passing to the limit n → ∞, we get that the left hand side converges to Hence, any −weak limiting point e ∈ (L 1 ([0, t * ]; C 2 0 (I))) of the sequence {E (n) } n 1 is given by e(t, u) = ∂ t E(t, u), which in turn satisfies (5.30) and Theorem 3.5 would then follow. Therefore one is left with proving (5.32).
By a direct calculation we conclude the following fluctuation-dissipation relation for the microscopic currents: (5.37) Using the notation g x (t) := g x (r(t), p(t)) (and similarly for other local functions), this allows us to write II n,1 = 4 j=1 II n,1,j , where We have  From Lemma 5.1-(5.10) we conclude that the second term tends to zero, with n → +∞.
By integration by parts the first term equals which tends to zero, thanks to Proposition 4.5. Summarizing, we have shown that E n (∆ n G) x (s)p x (s)p x−1 (s) ds .

Estimates of J n,2
After a direct calculation, it follows from (2.6) that with h x := 1 2γ Substituting into the expression for J n,2 we conclude that J n,2 = K n,1 + K n,2 , where K n,1 and K n,2 correspond to W x+1 − W x and n −2 Lh x , respectively. Using the summation by parts to deal with K n,1 , performing time integration in the case of K n,2 and subsequently invoking the energy bound from Proposition 4.5, we conclude that lim n→+∞ J n,2 = 0.
Summarizing, the results announced above, allow us to conclude that where lim n→∞ sup s∈[0,t * ] |o n (s)| = 0 for a fixed t * > 0. Given t > 0 we can take, as test function, G(s, u) := H(t, u), for any s ∈ [0, t], with an arbitrary compactly supported H ∈ C([0, +∞); C 2 0 (I)). Integrating over t ∈ [0, t * ] we obtain that E n (t), cf.  All the proofs are based on a refined analysis of the system of equations satisfied by the averages of momenta and stretches.
To simplify the notation, in the present section we omit writing the superscript n by the averages p  x (t) defined in (3.11). Their dynamics is given by the following system of ordinary differential equations and at the boundaries: r 0 (t) ≡ 0, d dt p 0 (t) = n 2 r 1 (t) − n 2 (2γ + γ)p 0 (t), (6.3) d dt p n (t) = −n 2 r n (t) + n 2 τ + (t) − n 2 (2γ + γ)p n (t).
The resolution of these equations will allow us to get several crucial estimates. For that purpose, we first rewrite the system in terms of Fourier transforms, and we will then take its Laplace transform. We define r(t, k) = E n r(t, k) , p(t, k) = E n p(t, k) .

Proof of Proposition 4.1
For any x ∈ I n and λ ∈ C such that Re(λ) > 0 we define the Laplace transforms: Performing the Laplace transform on both sides of (6.5) we obtain the following system   We fix t * > 0 and consider τ + (t) := τ + 1 [0,t * ] (t). Then, for λ ∈ C, Re(λ) > 0, By the Plancherel Theorem we have   A similar argument, using (6.23) and (6.24), shows that P d n,j n −2 , for j = 1, 2. As a result we conclude (4.1).
As a result, we obtain I n,3 (b − a) p /n. Estimates for I n,j , j = 1, 2 are similar. Hence, for any t > 0 we can find p > 0 such that b a p 0 (s) + p n (s) ds (b − a) p n , n 1, (6.45) and this together with (6.40) implies (4.3).

Proof of Proposition 4.4
Let Multiplying the equations (6.1), (6.2) by r x (t) and p x (t), respectively and (6.3), (6.4) by p 0 (t) and p n (t) and summing up we get Hence, by virtue of (4.3), we conclude that for any t * > 0    Therefore (4.5) follows upon an application of the Gronwall inequality.

Proof of Lemma 5.4
To prove Lemma 5.4, we need an estimate for which can be done using the explicit formula (6.7). Note first that we have the following inequalities for λ ± (k): Therefore, in order to estimate the members which appear in the right hand side of (6.7), we introduce, for = 0, 1, 2, EJP 26 (2021), paper 19.
Since the left hand side of (6.51) is obviously bounded it suffices to show that k∈ In which follows easily from the fact that sin 2 (πk) ∼ k 2 , as |k| 1.
From (6.7) we get that for any t * > 0 sup x∈In |p x (t)|1 [0,t * ] (t) I n (t) + II n (t) + III n (t), We also have q (n) 1 In order to estimate |p diff 0,n |, recall Proposition 4.1: using (4.1) and the Young inequality for convolution we obtain log(n + 1) n 2 .
Thus, the conclusion of Lemma 5.4 follows.

Proof of Lemma 5.2
To prove Lemma 5.2 we need to get an expression for Using (6.1) and then summing by parts we get d dt Computing p x (t) from (6.2) we rewrite the utmost right hand side as − n 2 2γ(n + 1) Therefore, after integration by parts in the temporal variable, we get Using (4.4) from Proposition 4.4 we conclude that the first and third expressions in the right hand side stay bounded, as n 1. Summing by parts we conclude that the second expression equals The expression stays bounded, due to Proposition 4.4-(4.5).
Multiplying (6.3) by p 0 (t) and integrating we get   This ends the proof of (5.20).

Entropy production
This section is mainly devoted to proving Proposition 4.5, which will be concluded in Section 7.2. For that purpose, we will obtain an entropy production bound stated in Proposition 7.1 below, in Section 7.1. Its proof uses Proposition 4.2. This new result, together with the estimates given in Proposition 4.2 and Proposition 4.4, will also allow us to conclude the proof of Lemma 5.1 (which give all boundary estimates), in Section 7.3.

Entropy production bound
To prove Proposition 4.5, we need to show that sup s∈[0,t] x∈In E n E x (s) grows at most linearly in n. We first relate this quantity to the entropy production, as follows: recall that f n (t) is the density of the distribution µ n (t) of (r(t), p(t)) with respect to ν T , see (3.1). We denote the expectation with respect to ν T by · T . Given a density F ∈ L 2 (ν T ) we define the relative entropy H n,T [F ] of the measure dµ := F dν T , with respect to ν T by H n,T [F ] := F log F T = Ωn F log F dν T .   Then, by virtue of the entropy inequality, see e.g. [9, p. 338] (and also (7.18) below), and from our assumption (3.7) on the initial condition, we conclude that: for any α > 0 we can find C α > 0 such that H n,T (s) n, n 1. In order to prove Proposition 7.1 (which will be achieved in Section 7.1.3), we first introduce another relative entropy which takes into account the boundary temperatures fixed at T − and T + and explains how relate them to each other.

Relative entropy of an inhomogeneous product measure
Recall the definition of the non-homogeneous product measure ν given in (3.3) and of the density f n (t) given in (3.5). The relative entropies H n (t) (defined in (3.6)) and H n,T (t) (defined in (7.1)) are related by the following formula.
In addition, for any t * > 0 H n,T (t) H n (t) + n, n 1, t ∈ [0, t * ]. Proof. Formula (7.5) can be obtained by a direct calculation. To prove the bound (7.6) note first that one can choose a sufficiently small α > 0 so that Thus (7.6) follows from (7.5).

Estimate of H n (t)
Next step consists in estimating H n (t) by computing its derivative. Using the regularity theory for solutions of stochastic differential equations and Duhamel formula, see e.g. Section 8 of [3], we can argue that f n (t, r, p) is twice continuously differentiable in (r, p) and once in t, provided that f n (0) ∈ C 2 (Ω n ), which is the case, due to (3.7). Using the dynamics (2.4)-(2.5) we therefore obtain: E n j x,x+1 (t) + n 2 T −1 + τ + p (n) n (t) − n 2 D( f n (t)), (7.8) where j x,x+1 (t) := j x,x+1 (r(t), p(t)), with j x,x+1 given in (5.2), and the operator D is defined for any F 0 such that F log F ∈ L 1 ( ν) and ( It is standard to show, using the inequality a log(b/a) 2 √ a( √ b − √ a) for any a, b > 0, that: for any positive, measurable function F on Ω n , and any x ∈ I n , The main result of this section is the following: EJP 26 (2021), paper 19. H n (s) H n (0) + n, n 1.
Proof. From (7.8) we get H n (t) = H n (0) + I n + II n + III n , (7.14) where where the last inequality follows from (7.9) and (7.11). We now estimate I n , II n .

(i) Estimates of I n
Recall the fluctuation-dissipation relation (5.36) and recall also the notation g x (t) := g x (r(t), p(t)) (and similarly for other local functions). We write |I n | I n,1 + I n,2 + I n,3 , where I n,1 := 1 n T −1  To deal with I n,1 we invoke the entropy inequality: for any α > 0 we have To deal with I n,2 , which involves boundary terms, we shall need some auxiliary estimates.

Boundary estimates: proof of Lemma 5.1
The entropy production bound from Proposition 7.3 is also crucial in order to get information on the behavior of boundary quantities. We prove here all the estimates of Lemma 5.1.
Proof of Lemma 5.1, estimate (5.5). We start with the right boundary point x = n. The proof for x = 1 is similar. Using the definition (3.3) we write      Since the entropy H n,T (t) and the form D T (f n (s)) are both non-negative (from a similar argument as in (7.11)), and the right hand side of (7.42) grows at most linearly in n (from Proposition 4.2) we conclude n . (7.43) and (5.13) follows.

Energy balance identity and equipartition
The main result which is left to be proved is Proposition 4.6, which describes an equipartition phenomenon between the mechanical and thermal energies. To prove that result, we will use the Fourier-Wigner distributions which permit to control the energy profiles over various frequency modes, and have been successfully used in previous works. The major difficulty here is the presence of boundary terms, which need to be controlled. In Section 8.1 we introduce definitions and write down the evolution equation satisfied by wave functions. In Section 8.2 we obtain an energy balance identity (Proposition 8.1). The proof of Proposition 4.6 is presented in Section 8.3.

The wave and Wigner functions
In the present section we restore the superscript n when referring to the mean and fluctuation of the stretch and momentum. We define the fluctuating wave function as x ∈ I n , t 0, (8.1) and its Fourier transform, The wave function extends to a periodic function on 1 n+1 Z, by letting ψ for any k ∈ I n . In particular ψ is well defined for any η ∈ Z. Then for EJP 26 (2021), paper 19.
Here δ x,y is the usual Kronecker delta function, which equals 1 if x = y and 0 otherwise. The process N (t, k) is a semi-martingale whose mean and covariation can be computed from the relations d N (t, k) = γn 2 (n + 1)δ k,0 dt, and d N (t, k), d N (t, k ) = γn 2 (n + 1)tδ k,−k dt.

Energy balance for the fluctuating Wigner functions
After straightforward computations one gets, for ι = ±: We are interested in the time evolution of the following quantity: One can verify that E n (t) = 1 2(n + 1) EJP 26 (2021), paper 19.
Given s ∈ (0, 1) we let I n,s := k ∈ I n : 0 k (n + 1) −s and I s n := I n \ I n,s .
We write η∈In k∈ In t 0 V n (s, η, k) G (s, η, k)ds = O n,s + O s n , (8.17) where terms O n,s and O s n correspond to the summation in k over I n,s and I s n , respectively, and s ∈ (0, 1) is to be determined later on. Denoting G 1 := − G/(σ n s) and G 2 := δ n s G/(σ n s), and using (8.16) we write O s n = I n + II n + III n + IV n + V n ,  R n (t)∂ t G 1 (t) n 3s/2−3 log(n + 1), n 1.
Estimates of IV n and V n The argument in both cases is the same, so we only consider IV n . We write IV n = IV n,1 + IV n,2 , where IV n,1 := − 1 n + 1 η∈In k∈ I s n t 0 E n ( p n (s) − p 0 (s)) p s, k + η n+1 G 1 (s, η, k)ds, IV n,2 := − 1 n + 1 η∈In k∈ I s n t 0 E n ( p n (s) − p 0 (s)) p (s, k) G 1 (s, η, k)ds. As a result, invoking (4.7), the Cauchy-Schwarz inequality and Plancherel identity, we can find a constant C > 0, independent of n, and such that |IV n,2 | 1 n + 1 x∈In t 0 E n | p n (s)| + | p 0 (s)| . The proof of the fact that lim n→+∞ IV n,1 follows the same lines as the argument presented above.

Conclusion
For s ∈ (0, 1 3 ) we have proved that both O n,s and O s n tend to zero as n → ∞, and therefore we conclude (8.15). This ends the proof of Proposition 4.6.