Tomasz Komorowski

Stefano Olla

Marielle Simon

An open

Keywords: 2010 Mathematics Subject Classification. 82C70, 60K35 Hydrodynamic limit, heat diffusion, non-equilibrium stationary states, uphill heat diffusion

Introduction

Non-equilibrium transport in one dimension presents itself to be an interesting phenomenon and in many models numerical simulations can be easily performed. Most of the attention has been focused on the study of the non-equilibrium stationary states (NESS), where the systems are subject to exterior heat baths at different temperatures and other external forces, so that the invariant measure is not the equilibrium Gibbs measure.

The most interesting models are those with various conserved quantities (energy, momentum, volume stretch...) whose transport is coupled. The densities of these quantities may evolve at different time scales, particularly when the spatial dimension of the system equals one. For example, in the Fermi-Pasta-Ulam (FPU) chain, volume stretch, mechanical energy and momentum all evolve in the hyperbolic time scale. Their evolution is governed by the Euler equations (see [START_REF] Even | Hydrodynamic Limit for an Hamiltonian System with Boundary Conditions and Conservative Noise[END_REF]) while the thermal energy is expected to evolve at a superdiffusive time scale, with an autonomous evolution described by a fractional heat equation. This has been predicted [START_REF] Spohn | Nonlinear Fluctuating Hydrodynamics for Anharmonic Chains[END_REF], confirmed by many numerical experiments on the NESS [START_REF] Lepri | Heat conduction in chains of nonlinear oscillators[END_REF][START_REF] Lepri | Thermal Conduction in classical low-dimensional lattices[END_REF] and proved analytically for harmonic chains with random exchanges of momenta that conserve energy, momentum and volume stretch, see [START_REF] Jara | Superdiffusion of Energy in a system of harmonic oscillators with noise[END_REF].

In contrast to the situation described above, the present paper deals with a system for which conserved quantities evolve macroscopically in the same diffusive time scale, and their macroscopic evolution is governed by a system of coupled diffusive equations. One example is given by the chain of coupled rotors, whose dynamics conserves the energy and the angular momentum. In [START_REF] Iacobucci | Negative thermal conductivity of chains of rotors with mechanical forcing[END_REF] the NESS of this chain is studied numerically, when Langevin thermostats are applied at both ends, while a constant force is applied to one end and the position of the rotor on the opposite side is kept fixed. While heat flows from the thermostats, work is performed by the torque, increasing the mechanical energy, which is then transformed into thermal energy by the dynamics of the rotors. The stationary temperature profiles observed numerically in [START_REF] Iacobucci | Negative thermal conductivity of chains of rotors with mechanical forcing[END_REF] present a maximum inside the chain higher than the temperature of both thermostats. Furthermore, a negative linear response for the energy flux has been observed for certain values of the external parameters. This phenomenon is referred to in the literature as an uphill diffusion, see [START_REF] Krishna | Uphill diffusion in multicomponent mixtures[END_REF] or [START_REF] Colangeli | Microscopic models for uphill diffusion[END_REF] and references therein. These numerical results have been confirmed in [START_REF] Iubini | Boundary induced instabilities in coupled oscillators[END_REF], as well as an instability of the system when thermostats are at zero temperature.

The present work aims at describing a similar phenomenon for the NESS, but for a different model. In particular, we are able to show rigorously that the maximum of the temperature profile occurs inside the system. According to our knowledge it is the first theoretical result that rigorously establishes the heating inside the system and uphill diffusion phenomena.

More specifically, we consider a chain of unpinned harmonic oscillators whose dynamics is perturbed by a random mechanism that conserves the energy and volume stretch: any two nearest neighbour particles exchange their momenta randomly in such a way that the total kinetic energy is conserved. Two Langevin thermostats are attached at the opposite ends of the chain and a constant force τ + acts on the last particle of the chain. This system has only two conserved quantities: total energy and volume. Since the random mechanism does not conserve the total momentum, the macroscopic behaviour of these two quantities is diffusive, and the non stationary hydrodynamic limit with periodic boundary conditions (no thermostats or exterior force present) has been proven in [START_REF] Komorowski | Macroscopic evolution of mechanical and thermal energy in a harmonic chain with random flip of velocities[END_REF].

The action of this constant force puts the system out of equilibrium, even when the temperatures of the thermostats are equal. As in the rotor chain described above, the exterior force performs positive work on the system, that increases the mechanical energy (concentrated on low frequency modes). The random mechanism, which consists in the kinetic energy exchange between neighbouring atoms, see definition (2.8) and the following explanations, transforms the mechanical energy into the thermal one (uniformly distributed in all frequencies, when the system is in a local equilibrium), which is eventually dissipated by the thermostats. This transfer of mechanical into thermal energy happens in the bulk of the system and is already completely predicted by the solution of the macroscopic diffusive system of equations obtained in the hydrodynamic limit [START_REF] Komorowski | Macroscopic evolution of mechanical and thermal energy in a harmonic chain with random flip of velocities[END_REF], see also [START_REF] Bernardin | Hydrodynamics for a system of harmonic oscillators perturbed by a conservative noise[END_REF] for a similar model without boundary conditions.

In the present article we study the NESS of this dynamics. We prove, see Theorem 3.3 below, that the energy and the volume stretch profiles converge to the stationary solution of the diffusive system, with the boundary conditions imposed by the thermostats and the external tension. It turns out that these stationary equations can be solved explicitly and the stretch profile is linear between 0 and τ + , while the thermal energy (temperature) profile is a concave parabola with the boundary conditions coinciding with the temperatures of the thermostats. The curvature of the parabola is proportional to τ 2 + , i.e. the increase of the bulk temperature is not a linear response term. In the case τ + = 0, the NESS was studied in [START_REF] Bernardin | Fourier Law for a Microscopic Model of Heat Conduction[END_REF], where the temperature profile is proved to be linear: more details are available in [START_REF] Bernardin | Heat conduction model: stationary non-equilibrium properties[END_REF][START_REF] Bernardin | Harmonic Systems with Bulk Noises[END_REF]. This heating inside the system phenomenon is similar to the ohmic loss, due to the diffusion of electricity in a resistive system (see e.g. [START_REF] Bradji | Discretization of the coupled heat and electrical diffusion problems by the finite element and the finite volume methods[END_REF]).

The NESS for our model also provides a simple example of an uphill energy diffusion: if the force τ + is large enough and applied on the side, where the coldest thermostat is acting, the sign of the energy current can be equal to the one of the temperature gradient, see Theorem 3.2 below. It is not surprising after understanding that this is regulated by a system of two diffusive coupled equations. On the other hand, the model does not work as a stationary refrigerator: i.e. a system where the heat on the coldest thermostat flows into it.

Our results suggest that there is a universal behaviour of the temperature profiles in the NESS when there are at least two conserved quantities. This should be tested on a system with three conserved quantities that evolves in the diffusive scale, such as e.g. a non-acoustic harmonic chain with a random exchange of momentum as considered in [START_REF] Komorowski | Diffusive propagation of energy in a non-acoustic chain[END_REF], where the non-stationary hydrodynamic limit is proven. An attempt to describe more generally the systems for which the phenomena of an uphill energy diffusion and heating inside the system occur is made in [START_REF] Olla | Role of conserved quantities in Fourier's law for diffusive mechanical systems[END_REF].

Let us add a comment on the proofs of our main results. In proving Theorem 3.3 (on the asymptotics of energy and stretch profile) we need to make an additional hypothesis concerning the strength γ > 0 of the noisy part of the dynamics, see (2.3) and (2.4). More precisely we suppose that γ = 1. This assumption is of purely technical nature and allows to discard the term corresponding to the equipartition of the mechanical and kinetic energy in the decomposition (4.8) of the microscopic energy profile of the chain (the term H m n ). We conjecture that this term vanishes in the stationary macroscopic limit, making the conclusion of the theorem valid for any γ > 0, but are unable to prove this fact at the moment. We do not need this hypothesis in our proof of Theorem 3.2 (on the uphill diffusion phenomenon).

In Appendix A we give a proof for the non-stationary macroscopic evolution of the energy and the volume stretch profiles in the diffusive space-time scaling. As for the NESS, the proof is rigorous only for γ = 1, for similar reasons. The corresponding result with periodic boundary conditions was contained in [START_REF] Bernardin | Hydrodynamics for a system of harmonic oscillators perturbed by a conservative noise[END_REF].

The rest of the paper is organized as follows: in Section 2 we define the microscopic model under investigation and give the expected macroscopic system of equations, showing the phenomenon of uphill diffusion. In Section 3 we state the main results of the paper, namely the convergence of the non-equilibrium stationary profiles of elongation, current and energy. In order to prove them, we need precise computations on the averages and second order moments taken with respect to the NESS. Section 4 provides elements of the proofs and preliminary computations on the averages, while Section 5 provides all the remaining technical lemmas, concerning the second order moments.

Microscopic dynamics and macroscopic behaviour

2.1. Open chain of oscillators. Let I n ∶= {1, . . . , n}, I n ∶= I n ∪ {0} and I ∶= [0, 1]. The configuration space Ω n ∶= R In × R In consists of all sequences (q, p) ∶= {q x , p x } x∈In , where p x ∈ R stands for the momentum of the oscillator at site x, and q x ∈ R represents its position. The interaction between two particles x and x + 1 is described by the quadratic potential energy V (q x -q x+1 ) ∶= 1 2 (q x -q x+1 ) 2 of a harmonic spring linking the particles. At the boundaries the system is connected to two Langevin heat baths at temperatures T -and T + . Furthermore, on the right boundary acts a force (tension) τ + , possibly slowly changing in time at a scale t/n 2 . Note that the system is unpinned, i.e. there is no external potential binding the particles. Consequently, the absolute positions q x do not have a precise meaning, and the dynamics only depends on the interparticle elongations r x ∶= q x -q x-1 , x ∈ I n . The configurations can then be described by (r, p) = (r 1 , . . . , r n , p 0 , . . . , p n ) ∈ Ω n .

(2.1)

The total energy of the system is defined by the Hamiltonian:

H n (r, p) ∶= ∑ x∈In E x + p 2 0 2 , ( 2.2)
with

E x ∶= p 2 x 2 + r 2 x 2 , x ∈ I n .
We investigate this system in the diffusive time scale (when the ratio of the microscopic vs macroscopic time is n 2 ), therefore the equations of the microscopic

q 0 q 1 q n q x-1 q x q x+1 r x T - T + τ + (t)
Figure 1. Oscillator chains with heat baths and one boundary force.

dynamics are given in the bulk by

dr x (t) = n 2 (p x (t) -p x-1 (t)) dt, x ∈ I n (2.3) dp x (t) = n 2 (r x+1 (t) -r x (t)) dt -γn 2 p x (t)dt + n √ γ(p x-1 (t)dw x-1,x (t) -p x+1 (t)dw x,x+1 (t)), x ∈ {1, . . . , n -1} (2.4)
and at the boundaries:

dp 0 (t) = n 2 r 1 (t)dt - n 2 2 (γ + γ)p 0 (t)dt -n √ γp 1 (t)
L ∶= n 2 (A + γ 2 S + γ 2 S), (2.8)
where

A ∶= n ∑ x=1 (p x -p x-1 )B rx + n-1 ∑ x=1 (r x+1 -r x )B px + r 1 B p 0 + (τ + (t) -r n ) B pn Sf ∶= n-1 ∑ x=0 X x ○ X x (f ),
where X x is the momentum exchange operator defined as

X x ∶= p x+1 B px -p x B p x+1 ,
and moreover the generator of the Langevin heat baths at the boundaries is given by

S ∶= T -B 2 p 0 -p 0 B p 0 + T + B 2 pn -p n B pn .
From the microscopic energy conservation law there exist microscopic energy currents j x,x+1 which satisfy

n -2 LE x = j x-1,x -j x,x+1 ,
for any x ∈ I n (2.9)

and are given by

j x,x+1 ∶= -p x r x+1 + γ 2 (p 2 x -p 2 x+1 ), if x ∈ {0, ..., n -1}, (2.10)
while at the boundaries

j -1,0 ∶= γ 2 (T --p 2 0 ) , j n,n+1 ∶= - γ 2 (T + -p 2 n ) -τ + (t)p n . ( 2.11)
2.2. Macroscopic equations. Suppose that r(t, u), e(t, u), (t, u) ∈ R + × I, are the macroscopic profiles of elongation and energy of the macroscopic system, obtained in the diffusive scaling limit. The profiles r(t, ⋅), e(t, ⋅) are the expected limits, as n gets large, of

1 n ∑ x∈In r x (t)δ x/n (⋅), and
1 n ∑ x∈In E x (t)δ x/n (⋅),
where δ u (⋅) is the delta Dirac function at point u. These convergences are expected to hold in the weak formulation sense: more details will be given in Appendix A. If both convergences do hold at time t = 0 to some given profiles r 0 (u) and E 0 (u), then we expect that they satisfy the following system of equations1 ,

B t r(t, u) = γ -1 B 2 uu r(t, u) (2.12) B t e(t, u)= 1 2 B 2 uu {(γ -1 + γ)e(t, u) + 1 2 (γ -1 -γ) r 2 (t, u)} , (t, u) ∈ R + × I, (2.13)
with the boundary conditions

r(t, 0) = 0, r(t, 1) = τ + (t), e(t, 0) = T -, e(t, 1) = T + + (τ + (t)) 2
2 and with the initial condition r(0, u) = r 0 (u), e(0, u) = E 0 (u). In Appendix A we will give the proof arguments for a derivation of these macroscopic equations, which are rigorous for γ = 1, and conditioned to a form of local equilibrium result for γ ≠ 1, stated in (A.46).

Define now e mech (t, u) ∶= 1 2 r 2 (t, u) and e th (t, u) ∶= e(t, u) -e mech (t, u) as respectively the mechanical and thermal components of the macroscopic energy. From (2.12) and (2.13) we conclude that

B t e mech (t, u) = γ -1 (B 2 uu e mech (t, u) -(B u r(t, u)) 2 ) , (t, u) ∈ R + × I with e mech (t, 0) = 0, e mech (t, 1) = (τ + (t)) 2 2 , e mech (0, 0) = r 2 0 (u) 2 , (t, u) ∈ R + × I and B t e th (t, u) = 1 2 (γ -1 + γ)B 2 uu e th (t, u) + γ -1 (B u r(t, u)) 2 , (t, u) ∈ R + × I, (2.14)
with e th (t, 0) = T -, e th (t, 1) = T + , t > 0.

Stationary non-equilibrium states.

From now on we assume τ + (t) ≡ τ + to be constant in time.

When τ + = 0 and T -= T + = T , the system is in equilibrium and the stationary probability distribution is given explicitly by the homogeneous Gibbs measure The proof of the existence and uniqueness of a stationary state follows from the same argument as the one used in [4, Appendix A] for τ + = 0. The fact that in our case τ + does not vanish requires only minor modifications. In addition, one can show, see bound (A.1) in [START_REF] Bernardin | Fourier Law for a Microscopic Model of Heat Conduction[END_REF], that for a fixed n we have ⟨H n ⟩ ss < +∞ (cf.

ν T (dr, dp) ∶= g T (
(2.2)).

The corresponding stationary profiles, denoted respectively by r ss (u) and e th,ss (u), will solve the stationary version of equations (2.12) and (2.14), i.e.:

r ss (u) = τ + u (2.16) and (γ -1 + γ)B 2 uu e th,ss (u) + 2γ -1 τ 2 + = 0, with the boundary conditions e th,ss (0) = T -, e th,ss (1) = T + .
In other words

e th,ss (u) = τ 2 + 1 + γ 2 u(1 -u) + (T + -T -)u + T -.
(2.17)

Taking the average with respect to the stationary state in (2.9), we get the stationary microscopic energy current

⟨j x,x+1 ⟩ ss =∶ j s , for any x ∈ {-1, . . . , n}.
(

The macroscopic stationary energy current is defined as the limit of nj s , as n → +∞. It equals, see Theorem 3.2 below,

J ss = - 1 2 (γ -1 + γ)(T + -T -) - τ 2 + 2γ .
Observe that the energy current can flow against the temperature gradient if T -> T + and |τ + | is large enough (uphill diffusion). Assuming T + ⩾ T -the maximum stationary temperature e max th,ss is reached at

u max = ( 1 2 + 1 + γ 2 τ 2 + (T + -T -)) ∧ 1 which implies that, if the condition 2(1 + γ 2 )(T + -T -) ⩽ τ 2
+ is satisfied, then the maximum temperature of the chain is attained inside, since u max < 1 (see Figure 2), and it equals e max th,ss =

(T + -T -) 2 + T -+ τ 2 + 4(1 + γ 2 ) ⩾ T + .
Note that this does not depend on the sign of τ + . This phenomenon was observed by dynamical numerical simulations in [START_REF] Iacobucci | Negative thermal conductivity of chains of rotors with mechanical forcing[END_REF] for the stationary states of the rotor model. It has attracted quite some interest from physicists, see [START_REF] Krishna | Uphill diffusion in multicomponent mixtures[END_REF] for a review. The present article is devoted to the proof of such a phenomenon, when γ = 1. This restriction is technical and will be further explained in Section 4.2. According to our knowledge it is the first rigorous proof of this fact in the existing literature.

Main results

Let us start with the following:

Theorem 3.1 (Stationary elongation profile). The following uniform convergence holds:

sup u∈I |⟨r [nu] ⟩ ss -r ss (u)| → n→∞ 0,
where r ss (u) ∶= τ + u. In particular, for any continuous test function

G ∶ I → R, 1 n ∑ x∈In G( x n )⟨r x ⟩ ss → n→∞ ∫ I G(u)r ss (u) du.
Proof. The averages under the stationary state ⟨r x ⟩ ss and ⟨p x ⟩ ss are computable explicitly, see Proposition 4.1 in the next section. It turns out that ⟨p x ⟩ ss is constant for all x ∈ I n and equals p s ∶= τ + /(γn + γ) (see (4.1)). From (4.2) we also have

n (⟨r x+1 ⟩ ss -⟨r x ⟩ ss ) = nγp s → n→∞ τ + , for x ∈ {1, ..., n -1} and ⟨r 1 ⟩ ss → n→∞ 0, ⟨r n ⟩ ss → n→∞ τ + .
Finally, (4.2) directly implies the conclusion of the theorem. □

Concerning the stationary energy flow and the validity of the Fourier law we show the following result on the macroscopic stationary energy current.

Theorem 3.2 (Stationary energy current and Fourier law)

.

nj s → n→∞ - 1 2 (γ -1 + γ)(T + -T -) - τ 2 + 2γ . (3.1)
Note that Theorems 3.1 and 3.2 are valid for any γ > 0. We now state our last main result about the stationary energy profile, which we are able to prove only for γ = 1. Before stating it we introduce the stationary microscopic mechanical and thermal energy per particle as follows

E mech x ∶= 1 2 ⟨r 2 x ⟩ ss E th x ∶= E x -E mech x = 1 2 p 2 x + 1 2 (r 2 x -⟨r 2 x ⟩ ss ), x ∈ I n .
Theorem 3.3 (Stationary energy profile). Assume that γ = 1. For any continuous test function G ∶ I → R we have

1 n ∑ x∈In G( x n )E mech x → n→∞ ∫ I G(u) 1 2 r 2 ss (u) du, (3.2)
1 n ∑ x∈In G( x n )⟨E th x ⟩ ss → n→∞ ∫ I G(u)e th,ss (u) du, (3.3)
1 n ∑ x∈In G( x n )⟨E x ⟩ ss → n→∞ ∫ I G(u)(e th,ss (u) + 1 2 r 2 ss (u)) du, (3.4)
where

r ss (u) = τ + u, e th,ss (u) = τ 2 + 2 u(1 -u) + (T + -T -)u + T + .
The remaining part of the paper deals with the proofs of Theorems 3.2 and 3.3.

The stationary state

Let us start with explicit computations for the average momenta and elongations with respect to the NESS.

⟨p x ⟩ ss = p s ∶= τ + γn + γ , for any x ∈ I n . (4.1)
The average stationary elongations are equal to

⟨r x ⟩ ss = p s 2 (γ -γ + 2γx) = τ + (2γx + γ -γ) 2(γn + γ) for any x ∈ I n . (4.2)
Proof. We start with some useful relations that hold for the stationary state:

(1) since ⟨Lr x ⟩ ss = 0, applying (2.8), we conclude

⟨p x ⟩ ss = ⟨p x-1 ⟩ ss = p s , for any x ∈ I n ;
(2) from ⟨Lp x ⟩ ss = 0 we get

⟨r x+1 ⟩ ss -⟨r x ⟩ ss = γp s , for any x ∈ {2, . . . , n -2} ⟨r 1 ⟩ ss = 1 2 (γ + γ)p s , ⟨r n ⟩ ss = - 1 2 (γ + γ)p s + τ + .
These equations determine the average stationary momentum and elongation as given in formulas (4. Proof of Proposition 4.2. By a direct calculation one can easily check that the energy currents j x,x+1 (defined in (2.10)) satisfy the following fluctuation-dissipation relation:

j s = ∇ϕ(x) ∶= ϕ(x + 1) -ϕ(x), x ∈ {1, . . . , n -1}, (4.3) with ϕ(x) ∶= - 1 2γ (⟨r 2 x ⟩ ss + ⟨p x-1 p x ⟩ ss ) - γ 4 (⟨p 2 x ⟩ ss + ⟨p 2 x-1 ⟩ ss ), x ∈ I n . ( 4
j x,x+1 = n -2 Lg x - 1 2γ ∇(r 2 x + p x-1 p x ) - γ 4 ∇(p 2 x-1 + p 2 x ), (4.6)
for any x ∈ {1, ..., n -1}, with

g x ∶= - 1 4 p 2 x + 1 2γ p x (r x + r x+1 ).
Therefore, (4.3) is obtained by taking the average in (4.6) with respect to the stationary state. □

We can now sketch the proof of Theorem 3.3: straightforward computations, using the definition (4.4) of ϕ, yield

⟨E x ⟩ ss = 1 2 (⟨p 2 x ⟩ ss + ⟨r 2 x ⟩ ss ) = - 2γ 1 + γ 2 ϕ(x) + γ 2 2(1 + γ 2 ) (⟨p 2 x ⟩ ss -⟨p 2 x-1 ⟩ ss ) - 1 1 + γ 2 ⟨p x p x-1 ⟩ ss + 1 -γ 2 2(1 + γ 2 ) (⟨p 2 x ⟩ ss -⟨r 2 x ⟩ ss ). (4.7)
Therefore, the microscopic energy profile can be decomposed as the sum of four terms:

H n (G) ∶= 1 n ∑ x∈In G( x n )⟨E x ⟩ ss = H ϕ n (G) + H ∇ n (G) + H corr n (G) + H m n (G), (4.8)
where

H ϕ n (G) ∶= - 2γ 1 + γ 2 1 n ∑ x∈In G( x n )ϕ(x), H ∇ n (G) ∶= γ 2 2(1 + γ 2 ) 1 n ∑ x∈In G( x n )(⟨p 2 x ⟩ ss -⟨p 2 x-1 ⟩ ss ), H corr n (G) ∶= - 1 1 + γ 2 1 n ∑ x∈In G( x n )⟨p x-1 p x ⟩ ss , H m n (G) ∶= 1 -γ 2 2(1 + γ 2 ) 1 n ∑ x∈In G( x n )(⟨p 2 x ⟩ ss -⟨r 2 x ⟩ ss ). Note that, if γ = 1, then H m n ≡ 0. If γ ≠ 1,
we conjecture that this last term vanishes as n → ∞, but we are not able to prove it at the moment. The limits of the other three terms will be obtained in the next section and are summarized in the following proposition:

Proposition 4.4. For any continuous test function G ∶ I → R, H ϕ n (G) → n→∞ ∫ I G(u)( τ 2 + 1 + γ 2 u + (T + -T -)u + T -) du, (4.9)
H ∇ n (G) → n→∞ 0, (4.10)
H corr n (G) → n→∞ 0. (4.11)
The complete proof of the proposition will be given in Section 5.4. Let us first comment on the ideas used in the argument. The limit (4.9) will be concluded using the fact that ϕ is harmonic, (4.10) is a consequence of the presence of a discrete gradient ⟨p 2

x ⟩ ss -⟨p 2

x-1 ⟩ ss inside the sum, and (4.11) will be shown thanks to the second order bounds, which are obtained in the next section.

Proof of Theorem 3.3. With the help of Proposition 4.4 the proof of Theorem 3.3 becomes straightforward. Assume that γ = 1. From the decomposition (4.7) and Proposition 4.4, we get

1 n ∑ x∈In G( x n )⟨E x ⟩ ss → n→∞ ∫ I G(u)( τ 2 + 2 u + (T + -T -)u + T -) du = ∫ I G(u)( τ 2 + 2 u(1 -u) + (T + -T -)u + T -+ (τ + u) 2 2 ) du.
Recalling (2.16) and (2.17) we conclude that the right hand side equals

∫ I G(u)(e th,ss (u) + r 2 ss (u) 2 ) du.
Thus (3.4) follows. From Theorem 3.1 we immediately conclude (3.2). The convergence of the stationary microscopic thermal energy profile in (3.3) is an immediate consequence of these two statements. □

Moment bounds under the stationary state

In this section we present a complete proof of Proposition 4.4 (see Section 5.4) and we show Theorem 3.2 (see Proposition 5.9). Before presenting the proof, we need a few technical estimates on the entropy production (Section 5.1) and second order moments (Section 5.2 and Section 5.3). In the whole section we do not assume γ = 1, since our results hold for any γ > 0.

Entropy production of the stationary state. Recall definition (2.15).

For a given T we will use

ν T (dr, dp) = g T (r, p)dp 0 ∏ x∈In dp x dr x ,
as a reference measure, and denote its respective expectation by ⟪⋅⟫ T .

Stationarity of µ ss under the microscopic dynamics implies that L ⋆ µ ss = 0 (in the sense of distributions). The operator L ⋆ is hypoelliptic, thus by [9, Theorem 1.1, p. 149], the measure µ ss has a smooth density f s with respect to ν T+ , i.e.

⟨F ⟩ ss = ⟪F f s ⟫ T+ = ∫ F f s dν T+ .
∑ x=0 D x (f s ) + γT -⟪ (B p 0 (f s /h)) 2 (f s /h) ⟫ T- + γT + ⟪ (B pn f s ) 2 f s ⟫ T+ = τ 2 + T + (γn + γ) + γ ( 1 T + - 1 T - ) (T --⟨p 2 0 ⟩ ss ), (5.1)
where

D x (f s ) ∶= ⟪ (X x f s ) 2 f s ⟫ T+ .
Proof. Integration by parts yields

⟪g Sf ⟫ T+ = T + ⟪B pn f B pn g⟫ T+ + T -⟪B p 0 f B p 0 g⟫ T+ + T -( 1
T - - 1 T + ) ⟪gB p 0 f ⟫ T+ for any f, g ∈ C ∞ 0 (Ω n ), where C ∞ 0 (Ω n ) is the space of compactly supported smooth functions. As A (H n (r, p)) = τ + p n and X x ( n ∑ y=0 p 2 y ) = 0, x ∈ {0, . . . , n -1} we conclude -⟪gAf ⟫ T+ = ⟪f Ag⟫ T+ + τ + ⟪p n f g⟫ T+ , -⟪gX 2 x f ⟫ T+ = ⟪X x f X x
g⟫ T+ , for any x = 0, . . . , n-1, and any f, g ∈ C ∞ 0 (Ω n ). We take the average of -n -2 L(log f s ) with respect to the stationary state µ ss . Taking into account the above identities we obtain

0 = -n -2 ⟨L log f s ⟩ ss = -n -2 ⟪f s L log f s ⟫ T+ = γ n-1 ∑ x=0 D x (f s ) + γT + ⟪ (B pn f s ) 2 f s ⟫ T+ -τ + ⟪B pn f s ⟫ T+ -γ ⟨(T -B 2 p 0 -p 0 B p 0 ) log f s ⟩ ss .
From the definition h = g T-/g T+ , the last term can be rewritten in the form:

-⟨(T -B 2 p 0 -p 0 B p 0 ) log f s ⟩ ss = -∫ f s h (T -B 2 p 0 -p 0 B p 0 )( log ( f s h )) g T-dp 0 n ∏ x=1 dp x dr x -∫ f s (T -B 2 p 0 -p 0 B p 0 )( log h) g T+ dp 0 n ∏ x=1 dp x dr x = T -⟪ (B p 0 (f s /h)) 2 (f s /h) ⟫ T- + ( 1 T - - 1 T + ) (T --⟨p 2 0 ⟩ ss ).
Moreover, by integration by parts and (4.1), we obtain

⟪B pn f s ⟫ T+ = T -1 + ⟪p n f s ⟫ T+ = p s T + = τ + T + (γn + γ)
and (5.1) follows. Since f s needs not be compactly supported the above calculation is somewhat formal. A rigorous argument (using variational principles) can be found in [5, Section 3]. □

Bounds on second moments.

In the present section we obtain some bounds on the covariance functions of momenta and positions, with respect to the stationary states. In particular, we estimate the magnitude of the average current |j s |, see (2.18) and investigate the behaviour of ϕ(1) and ϕ(n) as n → ∞, see (4.4).

Let us first state a rough estimate on the second moments at the boundaries, which is going to be refined further.

Proposition 5.2 (Second moments at the boundaries: Part I). The following equality holds

⟨p 2 0 ⟩ ss + ⟨p 2 n ⟩ ss = T + + T -+ 2τ + p s γ for all n ⩾ 1. (5.2)
Moreover, there exists a constant C = C(γ, γ, τ + , T + , T -) > 0, such that

⟨r 2 1 ⟩ ss + ⟨r 2 n ⟩ ss ⩽ C for all n ⩾ 1.
(5.3) Remark 5.3. By convention, the constants appearing in the statements below depend only on the parameters indicated in parentheses in the statement of the proposition.

Proof of Proposition 5.2. The first identity (5.2) is an easy consequence of (2.11) and (2.18), which yields

j s = γ 2 (T --⟨p 2 0 ⟩ ss ) , ( 5.4
)

j s = -τ + p s - γ 2 (T + -⟨p 2 n ⟩ ss ) .
(5.5)

Identity (5.
2) is obtained by adding sideways the above equalities. To show estimate (5.3) note that

n -2 L(p 0 r 1 ) = (p 1 -p 0 )p 0 + r 2 1 - 1 2 (γ + γ)p 0 r 1 (5.6) n -2 L(p n r n ) = p n (p n -p n-1 ) + (τ + -r n )r n - 1 2 (γ + γ)p n r n .
(5.7)

After taking the average with respect to the stationary state from (5.6) we conclude

⟨r 2 1 ⟩ ss = ⟨p 2 0 ⟩ ss -⟨p 1 p 0 ⟩ ss + 1 2 (γ + γ)⟨p 0 r 1 ⟩ ss . ( 5.8)
Recalling the definition of the current (2.10) and then invoking (5.4), we get

⟨r 2 1 ⟩ ss = ⟨p 2 0 ⟩ ss -⟨p 1 p 0 ⟩ ss - 1 2 (γ + γ)(⟨j 0,1 ⟩ ss + γ 2 (⟨p 2 1 ⟩ ss -⟨p 2 0 ⟩ ss )) = ⟨p 2 0 ⟩ ss -⟨p 1 p 0 ⟩ ss - γ 4 (γ + γ)(T --⟨p 2 0 ⟩ ss ) - γ 4 (γ + γ)(⟨p 2 1 ⟩ ss -⟨p 2 0 ⟩ ss ).
Using Young's inequality

|⟨p 1 p 0 ⟩ ss | ⩽ A 2 ⟨p 2 1 ⟩ ss + 1 2A ⟨p 2 0 ⟩ ss , with A = γ 2 (γ + γ), we get ⟨r 2 1 ⟩ ss ⩽ ( 1 γ(γ + γ) + 1 + 1 4 (γ + γ) 2 )⟨p 2 0 ⟩ ss .
(5.9)

From (5.2) we conclude that ⟨r 2 1 ⟩ ss is bounded. To estimate ⟨r 2 n ⟩ ss , note that from (5.7) we write

⟨r 2 n ⟩ ss = ⟨p 2 n ⟩ ss -⟨p n p n-1 ⟩ ss + τ + ⟨r n ⟩ ss - 1 2 (γ + γ)⟨p n r n ⟩ ss .
(5.10)

We use again Young's inequality

|⟨p n p n-1 ⟩ ss | ⩽ A 2 ⟨p 2 n ⟩ ss + 1 2A ⟨p 2 n-1 ⟩ ss ,
with A = 1/(2γ) and we get

⟨r 2 n ⟩ ss ⩽ (1 + 1 4γ )⟨p 2 n ⟩ ss + γ⟨p 2 n-1 ⟩ ss + τ ⟨r n ⟩ ss - 1 2 (γ + γ)⟨p n r n ⟩ ss . (5.11) To replace ⟨p 2 n-1 ⟩ ss , note that n -2 L(p 2 n ) = 2(τ + -r n )p n + γ(p 2 n-1 -p 2 n ) + γ(T + -p 2 n ).
(5.12)

Taking the average with respect to the stationary state, we obtain:

γ⟨p 2 n-1 ⟩ ss = 2⟨r n p n ⟩ ss -2τ + ⟨p n ⟩ ss + (γ + γ)⟨p 2 n ⟩ ss -γT + , ( 5.13)
which, in (5.11), gives

⟨r 2 n ⟩ ss ⩽ (1 + 1 4γ + γ + γ)⟨p 2 n ⟩ ss + τ + (⟨r n ⟩ ss -2⟨p n ⟩ ss ) + 1 2 (4 -γ -γ)⟨p n r n ⟩ ss -γT + .
Using again Young's inequality In the next proposition we provide a bound on the energy current under the stationary state, which will be further refined in Proposition 5.9. (5.17)

|⟨p n r n ⟩ ss | ⩽ A 2 ⟨p 2 n ⟩ ss + 1 2A ⟨r 2 n ⟩ ss with A = 1 2 |4 -γ -γ|, we finally arrive at 1 2 ⟨r 2 n ⟩ ss ⩽ (1 + 1 4γ + γ + γ + 1 4 (4 -γ -γ) 2 )⟨p 2 n ⟩ ss + τ + (⟨r n ⟩ ss -2⟨p n ⟩ ss ) -γT + . ( 5
Proof. We sum the identity (4.3) from x = 1 to n -1 and apply (4.4) to express ϕ(n) and ϕ [START_REF] Bernardin | Hydrodynamics for a system of harmonic oscillators perturbed by a conservative noise[END_REF]. In this way we get

(n -1)j s = ϕ(n) -ϕ(1) = ⟨ - p n-1 p n + r 2 n 2γ - γ(p 2 n-1 + p 2 n ) 4 ⟩ ss + ⟨ p 1 p 0 + r 2 1 2γ + γ(p 2 1 + p 2 0 ) 4 ⟩ ss . (5.18)
Therefore, (5.17) follows from the elementary inequalities

|⟨p x-1 p x ⟩ ss | ⩽ 1 2 (⟨p 2 x ⟩ ss + ⟨p 2
x-1 ⟩ ss ) for x = n and x = 1 together with the bounds obtained in Proposition 5.2 and Corollary 5.4. □ Proposition 5.5 permits to get a better estimate on the entropy production. Namely, combining (5.1), (5.4) and (5.17) we conclude the following.

Corollary 5.6. There exists

C = C(γ, γ, τ + , T + , T -) > 0, such that γ n-1 ∑ x=0 D x (f s ) + γT -⟪ (B p 0 (f s /h)) 2 (f s /h) ⟫ T- + γT + ⟪ (B pn f s ) 2 f s ⟫ T+ ⩽ C n , n ⩾ 1. (5.19)
Thanks to Proposition 5.5 we are now able to estimate the covariances of momenta and stretches at the boundaries as follows: Proposition 5.7 (Second moment at the boundaries: Part III). There exists C = C(γ, γ, τ + , T + , T -) > 0, such that, at the left boundary point

|⟨p 0 p 1 ⟩ ss | + |⟨r 1 p 1 ⟩ ss | + |⟨r 1 p 0 ⟩ ss | ⩽ C √ n , n ⩾ 1, (5.20)
and at the right boundary point

|⟨p n p n-1 ⟩ ss | + |⟨r n p n ⟩ ss | + |⟨r n p n-1 ⟩ ss | ⩽ C √ n , n ⩾ 1.
(5.21)

Proof. Integration by parts yields

⟨p 0 p 1 ⟩ ss = -T -⟪p 1 (f s /g T -)B p 0 g T -⟫ T+ = T -⟪p 1 B p 0 (f s /h)⟫ T-.
We use the entropy production bound (5.19) and estimate (5.15) on ⟨p 2 1 ⟩ ss , to estimate the right hand side. As a result we get

|⟨p 0 p 1 ⟩ ss | = T -|⟪p 1 B p 0 (f s /h)⟫ T-| ⩽ T -⟨p 2 1 ⟩ 1 2 ss ⟪ (B p 0 (f s /h)) 2 (f s /h) ⟫ 1 2 T- ⩽ C √ n .
Similarly,

|⟨p n p n-1 ⟩ ss | = T + |⟪p n-1 B pn f s ⟫ T+ | ⩽ T + ⟨p 2 n-1 ⟩ 1 2 ss ⟪ (B pn f s ) 2 f s ⟫ 1 2 T+ ⩽ C √ n .
Finally, note that, for any

x ∈ I n n -2 L(r 2 x ) = 2(p x -p x-1 )r x . (5.22)
Therefore, upon averaging with respect to the NESS, we get

⟨p x r x ⟩ ss = ⟨p x-1 r x ⟩ ss , x ∈ I n .
(5.23)

In particular, applying (5.23) for x = 1 and x = n, we remark that the only quantities we need to yet estimate are |⟨r 1 p 0 ⟩ ss | and |⟨r n p n ⟩ ss |. This is done using the entropy production bound (5.19) in the same manner as before, namely:

|⟨r 1 p 0 ⟩ ss | = T -|⟪r 1 B p 0 (f s /h)⟫ T-| ⩽ T -⟨r 2 1 ⟩ 1 2 ss ⟪ (B p 0 (f s /h)) 2 (f s /h) ⟫ 1 2 T- ⩽ C √ n , from (5.
3) and (5.1). We leave the last estimate for the reader. □

We now have all the ingredients necessary to prove moments convergences at the boundaries: Corollary 5.8 (Second moments at the boundaries: Part IV). The following limits hold: at the left boundary point, (5.29)

⟨p 2 x ⟩ ss → n→∞ T - for x ∈ {0, 1}, (5
Proofs of (5.24) and (5.27). From (5.4) and Proposition 5.5 we get ⟨p 2 0 ⟩ ss → T -. Thanks to (5.16) and (5.20) we deduce ⟨p 2 1 ⟩ ss → T -, which in turn proves (5.24). A similar argument proves (5.27). Indeed, from (5.5) and Proposition 5.5, we get ⟨p 2 n ⟩ ss → T + and from (5.13) and (5.21), we deduce ⟨p 2 n-1 ⟩ ss → T + . Proofs of (5.25) and (5.28). The limit (5.25) follows directly from (5.8) and Proposition 5.7. From (4.2) we conclude that ⟨r n ⟩ ss → τ + . Using then (5.10) together with Proposition 5.7 we conclude (5.28).

Proofs of (5.26) and (5.29). Note that

n -2 L(r 1 p 1 ) = (p 1 -p 0 )p 1 + (r 2 -r 1 )r 1 -γr 1 p 1 (5.30) n -2 L(r n p n-1 ) = (p n -p n-1 )p n-1 + (r n -r n-1 )r n -γr n p n-1 .
(5.31)

Taking the average with respect to the stationary state, and using Proposition 5.7 together with (5.24) proved above, we get

⟨r 2 1 ⟩ ss -⟨r 1 r 2 ⟩ s → n→∞ T - (5.32) and ⟨r 2 n ⟩ ss -⟨r n-1 r n ⟩ s → n→∞ T + .
(5.33)

Using the already proved limits (5.25) and (5.28) we conclude (5.26) and (5.29). □ Proposition 5.9 (The stationary current: Part II). The following limits hold:

ϕ(1) → n→∞ - 1 2 (γ -1 + γ)T - (5.34) ϕ(n) → n→∞ - 1 2 (γ -1 + γ)T + - τ 2 + 2 . ( 5
C = C(γ, γ, τ + , T + , T -) > 0, such that 1 n n ∑ x=1 ⟨p 2 x ⟩ ss ⩽ C and 1 n n ∑ x=1 ⟨r 2 x ⟩ ss ⩽ C, n ⩾ 1.
(5.36)

Proof. From the current decomposition given by ( 4.3), we easily get that ϕ(x) = (x -1)j s + ϕ( 1), for any x ∈ I n .

Summing over x, this gives

1 n n ∑ x=1 ϕ(x) = 1 n n ∑ x=2 (x -1)j s + ϕ(1) = n(n -1) 2n j s + ϕ(1).
Therefore, recalling (5.34) and (3.1), we get

1 n n ∑ x=1 ϕ(x) → n→∞ - 1 4 (γ -1 + γ)(T + + T -) - τ 2 + 4γ .
(5.37) From (4.4), we have

1 n n ∑ x=1 ϕ(x) = - 1 2γn n ∑ x=1 ⟨r 2 x ⟩ ss - 1 2γn n ∑ x=1 ⟨p x p x-1 ⟩ ss - γ 2n n ∑ x=1 ⟨p 2 x ⟩ ss + γ 4n (⟨p 2 n ⟩ ss -⟨p 2 0 ⟩ ss ).
(5.38) To compute the limit of the second sum in the right hand side of (5.38), we first write:

n -2 L(p x-1 p x ) = (r x+1 -r x )p x-1 + (r x -r x-1 )p x -2γp x p x-1 ,
x ∈ {2, . . . , n -1}.

(5.39) Thus, taking the average with respect to the stationary state and subsequently using (5.23), we obtain

2γ⟨p x p x-1 ⟩ ss = ⟨p x r x ⟩ ss + ⟨p x-1 r x+1 ⟩ ss -⟨p x r x-1 ⟩ ss -⟨p x-1 r x ⟩ ss = ⟨p x-1 r x+1 ⟩ ss -⟨p x r x-1 ⟩ ss .
(5.40)

On the other hand

n -2 L(r x r x+1 ) = (p x -p x-1 )r x+1 + (p x+1 -p x )r x , x ∈ {1, . . . , n -1}.
Hence, taking the average and using again (5.23), we get

0 = ⟨p x+1 r x ⟩ ss + ⟨p x r x+1 ⟩ ss -⟨p x r x ⟩ ss -⟨p x-1 r x+1 ⟩ ss = ⟨p x+1 r x ⟩ ss + ⟨p x+1 r x+1 ⟩ ss -⟨p x r x ⟩ ss -⟨p x-1 r x+1 ⟩ ss ,
which yields

⟨p x+1 r x+1 ⟩ ss -⟨p x r x ⟩ ss = ⟨p x-1 r x+1 ⟩ ss -⟨p x+1 r x ⟩ ss
for any x ∈ {2, . . . , n -1}. Combining with (5.40) we get

2γ⟨p x p x-1 ⟩ ss = ⟨p x+1 r x+1 ⟩ ss -⟨p x r x ⟩ ss + ⟨p x+1 r x ⟩ ss -⟨p x r x-1 ⟩ ss , (5.41)
for any x ∈ {2, . . . , n -1}. Summing over x, one gets:

n-1 ∑ x=2 ⟨p x p x-1 ⟩ ss = 1 2γ (⟨p n r n ⟩ ss -⟨p 2 r 2 ⟩ ss + ⟨p n r n-1 ⟩ ss -⟨p 2 r 1 ⟩ ss ).
(5.42)

To compute the limit as n → ∞, we need to estimate the covariances appearing in the right hand side. The covariance ⟨p n r n ⟩ ss can be estimated thanks to Proposition 5.7. We still need the bounds on the covariances ⟨p 2 r 2 ⟩ ss , ⟨p n r n-1 ⟩ ss and ⟨p 2 r 1 ⟩ ss . To deal with it write

n -2 L(p 0 p 1 ) = (r 2 -r 1 )p 0 + r 1 p 1 - 1 2 (3γ + γ)p 0 p 1 (5.43) n -2 L(r 1 r 2 ) = (p 1 -p 0 )r 2 + (p 2 -p 1 )r 1 (5.44) n -2 L(p n-1 p n ) = (τ + -r n )p n-1 + (r n -r n-1 )p n - 1 2 (3γ + γ)p n-1 p n .
(5.45)

Taking averages with respect to the stationary state and summing (5.43) and (5.44) sideways gives (using ⟨p 2 r 2 ⟩ ss = ⟨p 1 r 2 ⟩ ss from (5.23))

⟨p 2 r 2 ⟩ ss + ⟨p 2 r 1 ⟩ ss = ⟨p 0 r 1 ⟩ ss + 1 2 (3γ + γ)⟨p 0 p 1 ⟩ ss → n→∞ 0, (5.46)
from Proposition 5.7. Moreover, (5.45) gives (using

⟨p n r n ⟩ ss = ⟨p n-1 r n ⟩ ss ) ⟨r n-1 p n ⟩ ss = τ + ⟨p n-1 ⟩ ss - 1 2 (3γ + γ)⟨p n-1 p n ⟩ ss → n→∞ 0,
(5.47) from (4.1) and Proposition 5.7. Therefore, we have proved that (5.42) vanishes as n → ∞. In fact, due to the estimates obtained in Proposition 5.7 we have even proved that there exists a constant

C = C(γ, γ, τ + , T + , T -) > 0, such that | n ∑ x=1 ⟨p x p x-1 ⟩ ss | ⩽ C √ n , n ⩾ 1.
(5.48) From (5.38) it follows that

1 2γn n ∑ x=1 ⟨r 2 x ⟩ ss + γ 2n n ∑ x=1 ⟨p 2 x ⟩ ss = - 1 2γn n ∑ x=1 ⟨p x p x-1 ⟩ ss - 1 n n ∑ x=1 ϕ(x) + γ 4n (⟨p 2 n ⟩ ss -⟨p 2 0 ⟩ ss ) → n→∞ 1 4 (γ -1 + γ)(T + + T -) + τ 2 + 4γ ,
due to (5.37) and (5.48). This in particular implies estimate (5.36). □

Thanks to the energy bounds, we are finally able to prove one further convergence, which will be essential in establishing Proposition 4.4.

Proposition 5.11. For any continuous test function

G ∶ I → R we have 1 n n-1 ∑ x=1 G( x n )⟨p x p x+1 ⟩ ss → n→∞ 0.
(5.49)

Proof. Assume first that G ∈ C 1 (I).
For the brevity sake we denote G x ∶= G(x/n) for any x ∈ I n and ψ(x) = ⟨p x+1 r x+1 ⟩ ss + ⟨p x+1 r x ⟩ ss . Then (5.41) says that

⟨p x p x+1 ⟩ ss = 1 2γ (ψ(x + 1) -ψ(x)), for any x ∈ {1, . . . , n -2}.
Therefore, by an application of summation by parts formula, we get

1 n n-1 ∑ x=1 G x ⟨p x p x+1 ⟩ ss = 1 2γn 2 n-2 ∑ x=2 n(G x-1 -G x )ψ(x) (5.50)
+ 1 n ⟨p n p n-1 ⟩ ss G n-1 + 1 2γn (⟨p n r n ⟩ ss + ⟨p n r n-1 ⟩ ss ) (5.51) - 1 2γn
(⟨p 2 r 2 ⟩ ss + ⟨p 2 r 1 ⟩ ss )G 1 .

(5.52)

The boundary terms (5.51) and (5.52) vanish, as n → ∞, thanks to (5.21), (5.46) and (5.47). To deal with the sum in the right hand side of (5.50) note that, since

G ∈ C 1 (I), we have sup x∈{2,...,n-2} n|G x-1 -G x | ⩽ ∥G ′ ∥ ∞ . (5.53) Since |ψ(x)| ⩽ 2 (2⟨p 2 x+1 ⟩ ss + ⟨r 2 x ⟩ ss + ⟨r 2 x+1 ⟩ ss ) , x ∈ {1, . . . , n -1} we conclude that 1 2γn 2 | n-2 ∑ x=2 n(G x-1 -G x )ψ(x)| ⩽ C n { 1 n n ∑ x=1 (⟨p 2 x ⟩ ss + ⟨r 2 x ⟩ ss )} , ( 5.54)
which vanishes, as n → +∞, thanks to the energy bound (5.36). This proves (5.49) for any test function G ∈ C 1 (I). The result can be extended to all continuous functions by the standard density argument and the energy bound (5.36). □

Proof of Proposition 4.4.

We now have at our disposal all components needed to prove Proposition 4.4 and thus conclude the proof of Theorem 3.3. There are three convergences to prove:

Proof of (4.9). From Proposition 4.2 (in particular (4.5)), the function ϕ is linear and is completely determined from the values at the endpoints. More precisely,

ϕ(x) = ϕ(n) -ϕ(1) n -1 x + nϕ(1) -ϕ(n) n -1 , for any x ∈ I n .
Since, from Proposition 5.9, the values ϕ(1) and ϕ(n) are of order 1 as n → ∞, we see that

ϕ(x) ≃ (ϕ(n) -ϕ(1)) x n + ϕ(1), as n → ∞,
and therefore we easily obtain

1 n ∑ x∈In G( x n )ϕ(x) → n→∞ ∫ I G(u){ - τ 2 + 2γ u - 1 2 (γ -1 + γ)[(T + -T -)u + T -]} du.
which proves directly (4.9).

Proof of (4.10). Concerning H ∇ n (G) we use a summation by parts formula (with the notation G x = G(x/n)), which leads to:

H ∇ n (G) = γ 2 2(1 + γ 2 ) ( G n n ⟨p 2 n ⟩ ss - G 1 n ⟨p 2 0 ⟩ ss + 1 n 2 n-1 ∑ x=1 n(G x -G x+1 )⟨p 2 x ⟩ ss ).
The boundary terms in the right hand side vanish, as n → +∞, since ⟨p 2 n ⟩ ss and ⟨p 2 0 ⟩ ss are bounded, due to (5.2). To deal with the limit of the last sum in the right hand side, we can repeat the argument made in (5.53)-(5.54), which shows that the expression vanishes. Thus (4.10) holds.

Proof of (4.11). This is a consequence of (5.49).

Appendix A. Non-Stationary behaviour

In this section we explain how to derive (2.12) and (2.13): while the derivation of (2.12) is rigorous, in order to obtain (2.13) we need to assume a form of local equilibrium that allows for the local equipartition of kinetic and potential energy, see (A.46) below. In the stationary setting this term corresponds to H m n (G) in (4.8) and, similarly, does not appear in the case γ = 1. Unfortunately, quite analogously with the stationary situation, the relative entropy method does not allow us to treat the case γ / = 1. Throughout the present section we allow τ + (t) to be a C 1 function.

A.1. Preliminaries. In the present section we establish non-stationary asymptotics corresponding to Corollary 5.8. They will be useful in proving the hydrodynamic limit in Section A.2. Since ν T+ is not stationary, except for the corresponding equilibrium boundary conditions, the relative entropy H n (t) defined as

H n (t) ∶= ∫ f n (t) log f n (t)dν T+
is not strictly decreasing in time, where hereafter f n (t) is the density of the Ω nvalued random variable (r n (t), p n (t)) (recall (2.7)), with respect to ν T+ . However, the effect of the boundary condition can be controlled and one can obtain a linear in n bound at any time t, i.e. (see the proof of Proposition 4.1 in [START_REF] Letizia | Non-equilibrium isothermal transformations in a temperature gradient from a microscopic dynamics, with V. Letizia[END_REF] for the details of the argument)

H n (t) ⩽ C(t)n, n ⩾ 1, t ⩾ 0.
(A.1) Both here and throughout the remainder of the paper C(t) shall denote a generic constant, always independent of n and locally bounded in t.

Furthermore, one obtains the bounds on the Dirichlet form controlling the entropy production, similar to (5.19),

∫ t 0 ds [γ n-1 ∑ x=0 D x (f n (s)) + γT -⟪ (B p 0 (f n (s)/h)) 2 (f n (s)/h) ⟫ T- + γT + ⟪ (B pn f n (s)) 2 f n (s) ⟫ T+ ] ⩽ C(t) n , (A.2)
where h = g T-/g T+ and n ⩾ 1, t ⩾ 0, see Proposition 4.1 and Appendix D of [START_REF] Letizia | Non-equilibrium isothermal transformations in a temperature gradient from a microscopic dynamics, with V. Letizia[END_REF] for the proof. Below we list some consequences of the above bounds on the entropy and Dirichlet forms.

Lemma A.1. The following equalities hold: Thus, by the Cauchy-Schwarz inequality

lim n→∞ E[ ∫ t 0 (p 2 n (s) -T + ) ds] = 0, lim n→∞ E[ ∫ t 0 (p 2 0 (s) -T -) ds] = 0 (A.
|E[ ∫ t 0 (p 2 n (s) -T + ) ds]| ⩽ T + ( ∫ t 0 ds ∫ p 2 n f n (s)dν T+ ) 1/2 ( ∫ t 0 ds ∫ (B pn f n (s)) 2 f n (s) dν T+ ) 1/2
|E [ ∫ t 0 (p 2 n (s) -T + ) ds]| ⩽ C, n ⩾ 1.
Hence, in particular we obtain

∫ t 0 ds ∫ p 2 n f n (s)dν T+ ⩽ C. (A.7)
Using this estimate in (A.5) together with (A.2) we conclude that for any t ⩾ 0 there exists C > 0 for which

|E[ ∫ t 0 (p 2 n (s) -T + ) ds]| ⩽ C √ n , n ⩾ 1. (A.8)
Hence the first equality of (A.3) follows. The proof of the second equality of (A.3) is similar.

Concerning (A.4): from the energy conservation it follows that

n -2 LE n = j n-1,n + γ 2 (T + -p 2 n ) + τ + (t)p n .
To deal with the term τ + (t)p n , note that

| ∫ t 0 τ + (s)E[p n (s)]ds| ⩽ ∥τ + ∥ ∞ ∫ t 0 | ∫ p n f n (s)dν T+ | ds = ∥τ + ∥ ∞ T + ∫ t 0 | ∫ B pn f n (s)dν T+ | ds ⩽ ∥τ + ∥ ∞ T + ∫ t 0 ( ∫ f n (s)dν T+ ) 1/2 ( ∫ (B pn f n (s)) 2 f n (s) dν T+ ) 1/2 ds ⩽ ∥τ + ∥ ∞ T + C √ n → n→∞ 0, (A.9)
by virtue of (A.2). The first equality of (A.4) is then a direct consequence of (A.3). The same argument works for j 0,1 . □

Using the energy conservation it follows immediately:

Corollary A. and at the right boundary point:

E [ ∫ t 0 p n-1 (s)r n (s)ds] → n→∞ 0 (A.17) E [ ∫ t 0 (p 2 n-1 (s) -p 2 n (s)) ds] → n→∞ 0, (A.18) E [ ∫ t 0 (r n-1 (s)r n (s) -τ 2 + (s)) ds] → n→∞ 0. (A.19)
Proof. Since n -2 Lr 2 n = 2r n p n -2r n p n-1 , using (A.6) we conclude that (A.17) holds, provided that we can prove

E [ ∫ t 0 r n (s)p n (s)ds] → n→∞ 0. (A.20)
The latter is a consequence of the following estimate, cf. (A.2),

|E [ ∫ t 0 r n (s)p n (s)ds]| = | ∫ t 0 ds ∫ r n B pn f n (s)dν T+ | ⩽ ( ∫ t 0 ds ∫ r 2 n f n (s)dν T+ ) 1/2 ( ∫ t 0 ds ∫ (B pn f n (s)) 2 f n (s) dν T+ ) 1/2 ⩽ ( ∫ t 0 ds ∫ r 2 n f n (s)dν T+ ) 1/2 C √ n → n→∞ 0.
The proof of (A.14) is similar.

To show (A.18) note that (see (5.12))

γ ∫ t 0 E [p 2 n-1 (s) -p 2 n (s)] ds = 2 ∫ t 0 τ + (s)E[p n (s)]ds -2 ∫ t 0 E[r n (s)p n (s)]ds + γ ∫ t 0 E [T + -p 2 n (s)] ds - 1 n 2 E [p 2 n (t) -p 2 n (0)] .
The second, third and fourth terms in the right hand side vanish due to (A.20), Lemma A.1 and (A.6), respectively. The first term has been already treated in (A.9). An analogous argument, starting from (5.16) allows us to prove (A.15).

Besides, we have

E [ ∫ t 0 p n-1 (s)p n (s)ds] = E [ ∫ t 0 p n-1 (s)(p n (s) -T + )ds] + T + E [ ∫ t 0 p n-1 (s)ds] → n→∞ 0. (A.21)
The above convergence is proved as follows: the first term in the right hand side can be estimated by the Cauchy-Schwarz inequality. Then we can use the bound

E [ ∫ t 0 p 2 n-1 (s)ds] ⩽ C, n ⩾ 1
(it follows from the already proved (A.18)) and Lemma A.1 to prove that it vanishes, as n → +∞. To show that the second term vanishes we can use estimates analogous to (A.9). The argument for (A.14) follows essentially the same lines. By (5.7) we can now write

E [ ∫ t 0 (r 2 n (s) -T + -τ 2 + (s)) ds] = ∫ t 0 τ + (s)E[r n (s) -τ + (s)]ds + ∫ t 0 E[p 2 n (s) -T + ]ds -∫ t 0 E[p n (s)p n-1 (s)]ds - 1 2 (γ + γ) ∫ t 0 E[p n (s)r n (s)]ds - 1 n 2 E[p n (t)r n (t) -p n (0)r n (0)]. (A.22)
By the previous results, the last four terms in the right hand side vanish, as n → +∞. Concerning the first term we use

n -2 Lp n (s) = τ + (s) -r n (s) - 1 2 (γ + γ)p n (s) (A.23) to conclude that ∫ t 0 τ + (s)E[r n (s) -τ + (s)]ds = 1 2 (γ + γ) ∫ t 0 τ + (s)E[p n (s)]ds - 1 n 2 ∫ t 0 τ + (s) d ds E[p n (s)]ds. (A.24)
The first term vanishes, as n → +∞, by (A.9). By integration by parts the second term equals

1 n 2 ∫ t 0 τ ′ + (s)E[p n (s)]ds - 1 n 2 (τ + (t)E[p n (t)] -τ + (0)E[p n ( 0
)]) which also vanishes, thanks to (A.9) and (A.6). Therefore,

E [ ∫ t 0 (r 2 n (s) -T + -τ 2 + (s)) ds] → 0.
To see (A. [START_REF] Lepri | Heat conduction in chains of nonlinear oscillators[END_REF]) it suffices to show that

E [ ∫ t 0 (r 2 n (s) -r n (s)r n-1 (s) -T + ) ds] → 0.
For that purpose we invoke (5.31), which permits to write

E [ ∫ t 0 ((r n-1 (s) -r n (s))r n (s) + T + )ds] = E [ ∫ t 0 (p 2 n (s) -p 2 n-1 (s))ds] + E [ ∫ t 0 p n (s)p n-1 (s)ds] + E [ ∫ t 0 (T + -p 2 n (s))ds] -γE [ ∫ t 0 r n (s)p n-1 (s)ds] - 1 n 2 E[r n (t)p n-1 (t) -r n (0)p n-1 (0)].
Each term in the right hand side of the above equality vanishes, as n → +∞, by virtue of the already proved estimates. With a similar procedure we obtain (A.16). □ A.2. Hydrodynamic limit. Let us now turn to equation (2.12), which can be formulated in a weak form as:

∫ 1 0 du G(u)(r(t, u) -r(0, u)) = 1 γ ∫ t 0 ds ∫ 1 0 du G ′′ (u)r(s, u) - 1 γ G ′ (1) ∫ t 0 τ + (s)ds, (A.25) for any test function G ∈ C 2 0,1 (I) -the class of C 2 functions on [0, 1] such that G(0) = G(1) = 0.
Existence and uniqueness of such weak solutions in an appropriate space of integrable functions are standard. By the microscopic evolution equations (2.3) we have (recall that G

x = G(x/n)) E [ 1 n n ∑ x=1 G x (r x (t) -r x (0))] = E [ ∫ t 0 ds n n ∑ x=1 G x (p x (s) -p x-1 (s))] = E [ ∫ t 0 ds {- n-1 ∑ x=1 (∇ n G) x p x (s) -nG 1 p 0 (s)}] , (A.26) where (∇ n G) x ∶= n(G x+1 -G x ). Using (2.4) we can write (A.26) as E [-∫ t 0 ds { n-1 ∑ x=1 1 γ (∇ n G) x (r x+1 (s) -r x (s)) + 1 2(γ + γ) nG 1 r 1 (s)}] (A.27) + E [ 1 γn 2 n-1 ∑ x=1 (∇ n G) x (p x (t) -p x (0)) + 1 2(γ + γ)n 2 nG 1 (p 0 (t) -p 0 (0))] . (A.28) Since G is smooth, nG 1 → G ′ (0) and (∇ n G) x → G ′ (x),
as n → +∞. Using this and (A.6), one shows that the expression (A.28) converges to 0. The only significant term is therefore the first one (A.27). Summing by parts, using the notation

(∆ n G) x ∶= n 2 (G x+1 + G x-1 -2G x )
and recalling that G(0) = 0, it can be rewritten as

E [ ∫ t 0 1 γ { 1 n n-1 ∑ x=2 (∆ n G) x r x (s) -(∇ n G) n-1 r n (s)} ds] -( 1 2(γ + γ) (∇ n G) 0 - 1 γ (∇ n G) 1 ) E [ ∫ t 0 r 1 (s)ds] . (A.29)
It is easy to see, using (2.5), that

lim n→+∞ E [ ∫ t 0 r 1 (s)ds] = γ + γ 2 lim n→+∞ ∫ t 0 ds ∫ p 1 f n (s)dν T+ = γ + γ 2T + lim n→+∞ ∫ t 0 ds ∫ B p 1 f n (s)dν T+ = 0,
by (A.2). Using (A.23) we obtain also

lim n→+∞ E [ ∫ t 0 r n (s)ds] = ∫ t 0 τ + (s)ds.
Therefore, we can rewrite (A.29) as

∫ t 0 ds { 1 γ ∫ 1 0 G ′′ (u)r (n) (s, u)du -G ′ (1)τ + (s)} + o n (t), (A.30) where r (n) (t, u) = E[r x (t)] for u ∈ [ x n ; x+1 n ), n ⩾ 1 (A.
∥r (n) (s, ⋅)∥ L 2 (I) =∶ R < +∞. (A.32)
The above means that for each s ∈ [0, t] the sequence {r (n) (s, ⋅)} n⩾1 is contained in B R -the closed ball of radius R > 0 in L 2 (I), centered at 0. The ball is compact in L 2 w (I) -the space of square integrable functions on I equipped with the weak L 2 topology. The topology restricted to B R is metrizable. From the above argument it follows in particular that for each t > 0 the sequence {r (n) (⋅)} is equicontinuous in C ([0, t], B R ). Thus, according to the Arzela theorem, see e.g. [13, p. 234], it is sequentially pre-compact in the space C ([0, t], L 2 w (I)) for any t > 0. Consequently, any limiting point of the sequence satisfies (A.25).

Concerning equation (2.13) with the respective boundary condition, its weak formulation is as follows: for any G ∈ L 1 ([0, +∞); C It is easy to see, thanks to (A.6), that (E n (t)) n⩾1 is bounded in the dual to the separable Banach space L 1 ([0, +∞); C 2 0,1 (I)). Thus it is ⋆-weakly compact. In what follows we identify its limit e(t) by showing that it satisfies (A.33). To achieve this goal we are going to use the microscopic energy currents given in (2.10).

Consider now a smooth test function G ∈ C ∞ 0 ([0, +∞) × I) such that G(s, 0) = G(s, 1) ≡ 0, s ⩾ 0. Then, from (2.9), we get Using this relation we can rewrite the last term (A.34) as

- 1 n E [ n ∑ x=0 G x (0)E x (0)] = - 1 n E [ n-1 ∑ x=1 G x (0)E x (0)] = - 1 n ∫ t 0 ds E [ n-1 ∑ x=0 B s G x (s)E x (s)]
∫ t 0 ds E[ 1 n n-2 ∑ x=2 (∆ n G) x (s)V x (s)-(∇ n G) n-2 (s)V n-1 (s)+(∇ n G) 1 (s)V 1 (s)] (A.36)
plus expressions involving the average fluctuating term

∫ t 0 dsE[ 1 n 2 n-2 ∑ x=1 (∇ n G) x (s)Lg x (s)] = E [ 1 n 2 n-2 ∑ x=1 ((∇ n G) x (t)g x (t) -(∇ n G) x (0)g x (0))] -∫ t 0 dsE [ 1 n 2 n-2 ∑ x=1 (∇ n B s G) x (s)g x (s)]
which turns out to be small, as n → +∞, thanks to the energy bound (A.6). By Lemmas A.1 and A.4 we have: On the other hand, using (A.35), the term (A.38) equals Finally, for γ ≠ 1 we expect that the following term to vanish as n → +∞:

lim n→∞ E [ ∫ t 0 ds (∇ n G) 1 (s)V 1 (s)] = -∫ t 0 dsB u G(s, 0) 1 2 (γ
∫ t 0 ds E [ 1 n n-2 ∑ x=2 (∆ n G) x (p 2
x (s) -(r x (s) -E[r x (s)])

2 )] , (A.46) as can be guessed by local equilibrium considerations. Unfortunately in order to prove the last limit one needs some higher moment bounds that are not available from relative entropy considerations. One prospective work could be to proceed in an analogous way as in the periodic case [START_REF] Komorowski | Macroscopic evolution of mechanical and thermal energy in a harmonic chain with random flip of velocities[END_REF], by studying the evolution of the Wigner distribution of the thermal energy in Fourier coordinates.

Figure 2 .

 2 Figure 2. Temperature profile when T + -T -< 2τ 2 + .

4. 1 .Proposition 4 . 1 .

 141 Elongation and momenta averages. The average stationary momenta are equal to

. 4 ) 4 . 3 .

 443 Remark Thanks to (4.3), the function ϕ(x) is harmonic, i.e.: ∆ϕ(x) ∶= ϕ(x + 1) + ϕ(x -1) -2ϕ(x) = 0, for any x ∈ {2, . . . , n -1}. (4.5)

Proposition 5 . 1 (

 51 Entropy production). Denote h ∶= g T-/g T+ .The following formula holds γ n-1

Proposition 5 . 5 (

 55 The stationary current: Part I). There exists a constant C = C(γ, γ, τ + , T + , T -) > 0, such that the stationary current satisfies|j s | ⩽ C n for all n ⩾ 1.

2 n

 2 (s) -T + ) ds] = ∫ t 0 ds ∫ (p 2 n -T + ) f n (s)dν T+ = T + ∫ t 0 ds ∫ p n B pn f n (s)dν T+ .

(

  B s G(s, u) + 1 2 (γ -1 + γ)B 2 u G(s, u)) e(s, u) du s, u)r 2 (s, u) du -∫ +∞ 0 (B u G(s, 1) [(γ -1 + γ)T + + 1 4 (γ -1 -γ) τ + (s) 2 ] -B u G(s, 0)T -) ds. (A.33)Given a non-negative initial data e(0, ⋅) ∈ L 1 (I) and the macroscopic stretch r(⋅) (determined via (A.25)) one can easily show that the respective weak formulation of the boundary value problem for a linear heat equation, resulting from (A.33), admits a unique measured value solution.The averaged thermal energy function is defined asE n (t, u) ∶= E[E x (t)], u ∈ [ x n , x+1n ), x = 0, . . . , n -1.

  G) x (s) j x,x+1 (s) -nG n-1 (s)j n-1,n (s) + nG 1 (s)j 0,1 (s)]. (A.34)Here G x (s) ∶= G(s, x/n) and we use a likewise notation for∇ n G x (s), ∆ n G x (s).Concerning (A.34): by (A.4), the expectation of its last two terms are negligible. In order to treat the first term of (A.34), we use the fluctuation-dissipation relation (4.6), i.e. j x,x+1 = n -2 Lg x + ∇V x ,

- 1 +2γ τ 2 +

 12 γ) T -, (A.37) which takes care of the left boundary condition. Concerning the right one we haveE [ ∫ t 0 ds (∇ n G) n-2 (s)V n-1 (s)] = -E [ ∫ t 0 ds (∇ n G) n-2 (s)∇V n-1 (s)] (A.38) + E [ ∫ t 0 ds (∇ n G) n-2 (s)V n (s)] . (A.39)Using again the results of Lemmas A.1 and A.4 we conclude that the limit of the second term (A.39) equals-∫ t 0 dsB u G(s, 1) (1 2(γ -1 + γ) T + + 1 (s)) .

r, p)dp 0 ∏ x∈In dp x dr x , where g T (r, p) ∶=

	e -p 2 0 /2T √ 2πT	∏ x∈In	e -Ex/T 2πT	.	(2.15)
	If τ + ≠ 0, or T -≠ T + , the stationary measure exists and is unique, but it is not
	given explicitly. More precisely, we know that there exists a unique stationary
	probability distribution µ ss on Ω n (cf. (2.1)) for the microscopic dynamics de-
	scribed by the equations (2.3)-(2.6). As a consequence ⟨LF ⟩ ss = 0 for any function
	F in the domain of the operator L, given by (2.8). Hereafter, we denote
	⟨F ⟩ ss ∶= ∫

Ωn

F dµ ss .

  One of the main characteristics of this model is the existence of an explicit fluctuation-dissipation relation, which permits to write the stationary current j s as a discrete gradient of some local function, as given in the following:

	1) and (4.2).	□
	4.2. Elements of the proofs of Theorems 3.2 and 3.3.

Proposition 4.2 (Decomposition of the stationary current). We can write j s as a discrete gradient, namely

2 .

 2 The currents, defined in (2.10) and (2.11), satisfyThe term in the left hand side vanish, as n → +∞, due to estimate (A.6). We can repeat then the same arguments as we have used to obtain(5.14) and conclude that there exists C > 0 such that

	Lemma A.3. There exists a constant C < ∞ such that
						E [ ∫ 0	t	(r 2 1 (s) + r 2 n (s)) ds] ⩽ C, n ⩾ 1.	(A.11)
	Proof. Using (5.7) we get
	n -2 E[p n (t)r n (t) -p n (0)r n (0)]	(A.12)
	= ∫	0	t	E [p n (s)(p n (s) -p n-1 (s)) + (τ + (s) -r n (s))r n (s) -	1 2	(γ + γ)p n (s)r n (s)] ds.
				E [ ∫	0	t	r 2 n (s)ds] ⩽ C {E [ ∫ 0	t	p 2 n (s)ds] + 1} , n ⩾ 1.	(A.13)
	Estimate (A.7) can be used to obtain the desired bound for E [ ∫ 0 r 2 t n (s) ds]. An analogous estimate on E [ ∫ t 1 (s) ds] follows from the same argument, using (5.6) 0 r 2 and the second equality in (A.3) instead. □
	Lemma A.4. The following convergences hold: at the left boundary point:
								E [ ∫	0	t	p 0 (s)r 1 (s)ds]	→ n→∞	0	(A.14)
							E [ ∫	0	t	(p 2 1 (s) -p 2 0 (s)) ds]	→ n→∞	0	(A.15)
								E [ ∫	0	t	r 1 (s)r 2 (s) ds]	→ n→∞	0	(A.16)
				lim n→∞	E[ ∫	0	t	j x,x+1 (s)ds] = 0, x = -1, . . . , n, t ⩾ 0.	(A.10)
	Concerning the potential energy at the boundary points we have the following
	bound.

  31)and lim n→+∞ sup s∈[0,t] o n (s) = 0. Thanks to (A.6) we know that there exists R > 0 such that sup

n⩾1

sup s∈

[0,t]

  G) n-2 (s)E[j n-1,n (s)]. (A.40)From (A.10) we conclude that the second term vanishes, with n → +∞. By integration by parts the first term equals1 n 2 E[(∇ n G) n-2 (t)g n-1 (t) -(∇ n G) n-2 (0)g n-1 (0)] B s G) n-2 (s)g n-1 (s)] , (A.41)which vanishes, thanks to (A.6). Summarizing we have shown that Now, for the bulk, it follows from (5.22) and (5.39) thatn -2 Lh x = ∇W x -2γp x p x-1 , x = 2, . . . , n -1, ∶= r x-1 p x .Therefore by (A.2) and an argument similar to the one used above we conclude that

	t
	-ds (∇ n G) n-2 (s)V n-1 (s)] 1 n 2 E [ ∫ = -∫ t 0 dsB u G(s, 1) [ t 0 h x ∶= p x p x-1 -r 2 x 2 , W x lim n→∞ ∫ t 0 ds E [ 1 n n-2 ∑ V x = -E x + 1 4 (p 2 x -p 2 x-1 ) -1 2 p x p x-1 1 2 (γ -1 + γ) T + + so that it is easy to see that (A.36) is equivalent to n→+∞ E [ ∫ 0 with ds (∇ n lim -∫ t 0 ds E [ n-2 1 n ∑	1 2γ	τ 2 + (s)] . (A.42) (A.44)

1 n 2 E [ ∫ t 0 ds (∇ n G) n-2 (s)Lg n-1 (s)] + ∫ t 0 ds (∇ n x=2 (∆ n G) x (s)p x (s)p x-1 (s)] = 0.

(A.43)

In the case γ = 1 we can rewrite

x=2 (∆ n G) x (s)E x (s)] + o n (

1), (A.45)

closing the energy conservation equation and concluding the proof.

See also Theorems 3.7 and 3.8 of[START_REF] Komorowski | Macroscopic evolution of mechanical and thermal energy in a harmonic chain with random flip of velocities[END_REF] for a similar model which gives a similar coupled diffusive system for every value of γ.

This work was partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund, and by the ANR-15-CE40-0020-01 grant LSD. T.K. acknowledges the support of the National Science Centre: NCN grant 2016/23/B/ST1/00492. M. S. thanks Labex CEMPI (ANR-11-LABX-0007-01) and the project EDNHS ANR-14-CE25-0011 of the French National Research Agency (ANR) for their support. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovative programme (grant agreement n o 715734).

1


Dr Stéphanie Halbert

Dr Jimmy Lauberteaux

Dr Charlie Blons

Renata Marcia De Figueiredo

Dr Christophe Crévisy

Dr Olivier Baslé

Jean-Marc Campagne

Marc Mauduit
email: marc.mauduit@ensc-rennes.fr

Pr. Dr Hélène Gérard
email: helene.gerard@sorbonne-universite.fr

Jean-Marc Campagne

Pr. Dr Marcia De Figueiredo

J.-M Campagne

Catalytically Active Species in Copper/DiPPAM-Catalyzed 1,6-Asymmetric Conjugate Addition of Dialkylzinc to Dienones A Computational Overview

Keywords: C-C bond, DFT, catalysis, Cu/Zn heterobimetallic active species, enantioselectivity

published or not. The documents may come

Introduction

Copper-catalyzed Asymmetric Conjugate Addition (ACA) to electro-deficient substrates is an efficient method to form C-C bonds. [START_REF]Copper-Catalyzed Asymmetric Synthesis[END_REF][START_REF] Woodward | Compr. Org. Synth[END_REF] The 1,6-addition to ,,,-dienone is especially challenging, as a complete regiocontrol is difficult to reach because of the presence of numerous active sites. However, 1,6-ACA has been described with a range of electrophiles, nucleophiles, metal-based catalysts, and chiral ligands families. [START_REF] Hénon | [END_REF][4][5][6][7][8][9] Mauduit et al. have developed an efficient chiral tridentate P,N,O ligand, named DiPPAM (DiPhenylPhosphinoAzoMethinylate salt L), [10] which has proved its efficiency for copper-catalyzed 1,6-ACA of dialkylzinc reagents to cyclic [5,7] and more recently to acyclic [11][12][13] dienones (Scheme 1). While good to excellent regioselectivity and enantioselectivity could be achieved with this catalytic system, attempts to characterize intermediates were unsuccessful and the active catalytic species has remained unidentified. Scheme 1. Cu(OTf)2/DiPPAM L efficiently catalyzed 1,6-ACA of dialkylzinc reagents to cyclic and acyclic ,,,-dienones.

On the other hand, mechanistic investigations on conjugate addition were previously undertaken in the case lithium organocuprate as nucleophiles, resorting on kinetic [14] NMR spectroscopy [15][16][17][18] studies. The rate-determining step is the C-C bond formation as described by investigations on kinetic isotope effects and activation parameters. In addition, the formation of a π-complex between copper and C=C bond was predicted by NMR studies. [17,19,20] These data are completed by computational studies. [21][22][23][24][START_REF] Mori | Mod. Organocopper Chem[END_REF][START_REF] Yoshikai | [END_REF] The mechanism comprises different steps: the formation of the lithium organocuprate, the coordination of the carbonyl compound as a -complex, an oxidative addition to form a Cu(III) intermediate (Figure 1, a), followed by a reductive elimination corresponding to the C-C bond formation. For Cu-catalyzed ACA, the nature of the catalytically active species becomes a major issue. Whereas a Cu(II) salt is generally used as the precatalyst, a Cu(I) complex is considered to be the active species, [START_REF]Copper-Catalyzed Asymmetric Synthesis[END_REF] in line with NMR and EPR evidences supporting the reduction of Cu(II) to Cu(I). [27] In this context, a halide-bridged bimetallic monocopper is proposed for bidentate phosphine ligand (Josiphos) when resorting to a Grignard reagent (EtMgBr) as an alkyl source (Figure 1,b). [28] A similar intermediate was more recently described within the framework of the first direct experimental detection of transmetalation intermediates in copper-catalyzed addition reaction using the monodentate phosphoramidite ligand and diethylzinc as nucleophile. [29] Nevertheless, another bimetallic intermediate was detected in the latter study, this time exhibiting two Cu centers and three phosphine ligands. Two Cu centers are also present in the [Zn(μ-X)Cu(μ-X)CuR] trimetallic structure proposed as a catalytically active species (Figure 1,c) from a computational point of view by Calhorda et al. In this case, intermetallic bridge X can be a variety of suitable ligand, like an acetate (or thiophene carboxylate, triflate, ...). [30] Figure 1. Proposed intermediates in conjugate addition reaction with lithium organocuprate cluster (a), with halide-bridged bimetallic monocopper (b) and with trimetallic specie (c) from computational studies.

Having these features in mind, our group has recently proposed a mechanism for Cu-catalyzed ACA of dialkylzinc to dienones using anionic and chiral ligands as a bridge between the Cu and Zn centers. [31,32] This model was found to be applicable to a variety of ligands (Figure 2, a), as both DiPPAM L or hydroxylalkyl N-Heterocyclic Carbene ligands (NHC), and to different electrophiles (cyclic and acyclic dienones with various substituents). The relevance of this catalytic active species was supported by the good agreement with selectivity between experimental observations and calculations. Indeed, it was used to rationalize the effect of the nature of the substituent at C=O position in substrate on the regioselectivity towards 1,4-or 1,6conjugate addition, [31] evidencing the possibility for an electronrich substituent (methyl-imidazole for instance) to establish an interaction with the electro-deficient copper. The same model also allowed to reproduce and analyze the enantioselectivity of the ACA, which was proposed to originate from a stabilizing stacking network in the -complex. [32] Nevertheless, no experimental or computational evidences were available to support the nature of the postulated intermediates. In this paper, we aim at determining the origin of the catalytic activity through comparison of various reaction mechanisms. In a first part, the origin of the regio-and stereo-control is determined for ACA of ZnEt2 on acyclic ,,,-dienone, a more challenging than the previously examined ones due to its increased versatility (Figure 2). In a second part, alternative catalytically active species, resulting in alternative catalytic effects, are examined. It allows us altogether to confirm our mechanistic hypothesis and to evaluate the potential (or risk) of emergence of competitive pathways under different reaction conditions.

Results and Discussion

Regio-and enantio-selectivity in acyclic ketones.

The copper-catalyzed asymmetric addition of ZnEt2 to (3E,5E)trideca-3,5-dien-2-one (1) had previously been experimentally investigated using L-DiPPAM L (Table 1). [11] Two dienones were considered which differ by the groups at terminal position (R 1 ) and at C=O function (R 2 ). When the reaction was performed at room temperature with 1a, very good regioselectivity (98:2) with good yield (70%) was obtained in favor of the 1,6-product with a large enantiomeric excess (92%). When the positions of the methyl and phenyl groups were exchanged (1b), very different results were obtained as both yield (30%), regioselectivity (30:70) and enantioselectivity (53% ee) were found to be moderate. We thus undertook a systematic study of the reaction path in 1a to better determine the origin of the excellent selectivities.

Table 1. Cu/DiPPAM-catalyzed enantioselective 1,6-addition of ZnEt2 to linear dienones. [11] Dienone R 1 /R

Accepted Manuscript

Similarly to our previous studies as well as in the study reported by Nakamura et al., [33] the C-C bond formation step takes place in a bimetallic species (Figure 3 and Table S1, for detailed geometrical parameters in SI), in which DiPPAM ligand is the only bridge between Cu and Zn centers. As Zn(II) is coordinated to the carboxylate arm and an alkyl chain, whereas Cu(I) is bonded to the phosphine site and another alkyl chain, this heterobimetallic species is formally composed of two neutral metal centers. Bonding of the electrophile (1a) at the Zn(II) center through one of the carbonyl lone pair is referred to I0. An additional binding is observed as the C=C bond of 1a comes into interaction through a π-bond with the Cu(I) center, resulting in intermediate I4 when 1,4-conjugate addition is examined, and I6 in the case of 1,6-conjugate addition. I6 and I0 intermediates are in equilibrium as they are isoergonic, with an exchange between the two structures exhibiting a Gibbs free energy equal to 13 kcal.mol -1 which proved to be lower than the C-C bond formation step. In contrast, I4 is found to be significantly destabilized (rG = + 7.2 kcal.mol -1 ) and its formation is disfavored (TS4 higher than C-C bond formation in TSC-C6).

The C-C bond formation profile (in red for 1,4-and in blue for 1,6-conjugate addition in Figure 3) clearly suggests that the most favorable pathway is the 1,6-conjugate addition from both thermodynamic (as the enolate product P6 is more stable than P4 by 7.2 kcal.mol -1 ) and kinetic (as TSC-C6 is lower than TSC-C4 by more than 10 kcal.mol -1 ) points of view. It is in line with previous theoretical studies for addition of Me group at the terminal C=C bond of penta-2,4-dienal in a cuprate aggregate model. [34] This can be associated to the loss of conjugation during the 1,4-addition, which is already well advanced in TSC-C4, as highlighted by the evolution of the Cα=Cβ bond length along the 1,4-addition pathway (1.36 Å in I0, lengthened to 1.42 Å through formation of the π-adduct in I4, quasi-single with 1.47 Å in TSC-C4 compared to 1.53 Å for the Cα-Cβ single bond in the product P4). These conjugation effects fully justify the strong difference observed between 1a to 1b. In 1b, in the case of 1,6addition, the C-C bond formation fully breaks the conjugation of the phenyl group with the  system whereas, in the 1,4-addition, conjugation of the terminal C=C bond with Ph is kept.

Conjugation is thus proposed to be the factor tuning regioselectivity between 1a and 1b, in parallel to the already observed ligand effects (reported for BINAP) [11] or to the additional substrate-catalyst interaction (reported for Acyl-Nmethylimidazole). [31] This is confirmed by computations on a simple model including only electronic effects in 1a and 1b (see Tables S2-S4 in SI).

Accepted Manuscript

FULL PAPER

We next investigate the origin of the enantioselectivity of the 1,6addition to dienone 1a using the previously reported results showing that it is possible to predict the experimentally formed enantiomer from the stability of adduct I6. [35] Such an approach proved efficient in the case of cyclic dienone in agreement with the VCD and X-ray data, [32] provided an adapted computational level (dispersion correction was added to DFT calculations, see Computational Details) and an adapted docking procedure are used. Indeed, it appeared necessary to envision not only the two potential docking sides of the C=C double bond, but also all possible conformations relative to the C(O)-C and Cβ-Cγ bonds, as the most stable structure of the free ketone is not necessarily the one preferred in the bimetallic π-complex. In the case of acyclic ketones, a similar study was thus undertaken, which required investigation of a larger number of conformers as both C(O)-Cα and Cβ-Cγ single bonds can afford both s-cis and s-trans conformations (Figure 4). Eight structures of the π-complex I6 were computed, corresponding to the two docking sides for the four possible conformations of the substrate, introduced in Figure 4 and noted CT, TT, TC and CC. These structures are expected to be in fast equilibrium, considering the high C-C bond formation barriers compared to the coordination/decoordination of the ketone to the bimetallic complex reported above. The CT conformer is the most stable for the free dienone (by 1.3 kcal.mol -1 compared to the TT analogue). This preference is conserved in I6 intermediates. The most stable arrangement calculated is affiliated with the formation of the S-enantiomer (structure (a) in Figure 4) and this strong (> 4 kcal.mol -1 ) preference is conserved when adding entropic contributions (See SI, Figure S1). An NCI analysis, given in Figure 5, allows determination of the origin of this preference thanks to the observation of two characteristic non-bonding isosurfaces. This conformer seems to be stabilized by CH- interaction between the C-H bond of tert-butyl group and a phenyl group of diphenylphosphino moiety. A second - interaction is observed between C=C bond of substrate and imine group of DiPPAM ligand. This network of CH- interactions (in SI, Figure S2) is similar to that obtained in the intermediate with cyclic dienone. [32] As conclusion, the enantioselectivity of the addition is controlled by two main non-covalent interactions, one ensuring the folding of the DiPPAM ligand (CH-π interaction) and one associated with the docking of the ketone and related to the interaction of its unsaturated π system with the C=N bond of DiPPAM.

An experimental derivatization was initiated to confirm the enantioselectivity obtained from DFT calculations. In order to access to the absolute configuration of 2a obtained from Cu/L-DiPPAM catalyzed 1,6-ACA to 1a, the former was converted to the known ester 4 [36] following a slightly modified described procedure (Scheme 2 and Experimental Section). [37] Scheme 2. Experimental process to determine the absolute configuration of 2a.

Accepted Manuscript

FULL PAPER

The literature reported an S absolute configuration with a positive rotation ([α]D 20 = +10.0 (c 1.0, CHCl3)). [36] In our case, compound 4 obtained from 2a have the same sign, confirming S is the major enantiomer for 2a. The evaluated value for [α]D 21 = +6.0 (c 1.0, CHCl3) is consistent with 85% ee obtained for 2a. Thus, a (S) configuration can be assigned to compound 2a, in accordance to the calculated data.

Alternative mechanisms.

The mechanism proposed above, based on the similarity with previous studies, [21][22][23][24][START_REF] Mori | Mod. Organocopper Chem[END_REF][START_REF] Yoshikai | [END_REF] nicely confronts to experimental selectivity data (regio-and enantio-selectivity). It nevertheless relies on a catalytically active species (bimetallic Cu-Zn complex of DiPPAM) which could not be characterized through experimental analysis. We thus decided to undertake theoretical mechanistic investigations of the 1,6-addition pathway to understand the electronic factors governing the formation and activation with this catalytic intermediate. This interpretative part of the theoretical study was carried out using models compounds including all electronic effects but minimizing steric ones: ZnMe2 is reacted in the presence of a copper (I) complex and a model of DiPPAM ligand (M), for which the tert-butyl group as well as the phenyl group at the phosphine are replaced by hydrogen atoms. This small model provides similar result to the full one in the case of the addition to 1a (Figure S3 and Table S5). Three alternative mechanisms are envisioned for the 1,6conjugate addition. The first one (going through intermediate J, Figure 6) is deduced from direct interaction of the ZnMe2 with a preformed Cu(M) complex without methyl transfer from Zn to Cu: Cu is thus bonded to the phosphine chain of the DiPPAM whereas ZnMe2 interacts with the carboxylate arm. The second one (K in Figure 6) is similar to I0 except for the exchange of the Cu and Zn binding sites. The last one is obtained from I6 through decoordination of the ZnMe + center from the carboxylate arm of the DiPPAM to form a separated ion pair (L in Figure 6). Detailed studies of the energetics and structural features associated to these three mechanisms are next reported. (Figure 7, complete data in SI, Tables S5-S6). In order to better characterize the docking of the dienone in the intermediates given in Figure 5, the associated Cu-Zn bimetallic complexes have also been optimized (Figure 8).

Mechanism through J.

The docking of the carbonyl compound to Cu(M) forms intermediate noted IJ in Figure 7, b. As expected, no interaction of Zn center with C=C bond could be obtained. [38] In this case, Cu plays the role of Lewis acid while Zn acts as a Me donor. The sole coordination energy of the C=O bond to copper in IJ can be evaluated relative to CJ to -6.6 kcal.mol -1 . In contrast, substrate binding to I6 combines a coordination of the C=O to Zn and of the C=C to copper, leading to an overall interaction energy computed with respect to C0 of -11.1 kcal.mol - 1 . It is found to be the quasi-additive contribution of the sole coordination of the C=O to Zn in I0 (ΔE = -6.6 kcal mol -1 with respect to C0) and of the binding of the double bond to Cu (-3.9 kcal mol -1 for binding of ethylene to (Me)Cu(M)). [39] Intermediate I6 allows thus both a good coordination of the C=O to the metal center and a significant coordination of the  system to Cu. Let us nevertheless notice that the strong binding of the carbonyl compound in I6 is associated with a large entropic loss compared to IJ, for which more degrees of freedom of the carbonyl chain are kept. As a consequence, similar Gibbs energies with respect to separated reactants were found for I6 and IJ (7.2 kcal.mol -1 ). Discrimination between these two mechanisms thus originated from kinetic factors. The activation barrier in the non-transmetalated mechanism is equal to 27.2 kcal.mol -1 relative to IJ, so that TSJ is significantly higher than TSC-C6 (in Figure 7, a, associated with 17.5 kcal.mol -1 barrier). TSJ is found to be a little more reactant-like, as evidenced from the shorter C=C bond (1.403 Å in TSJ in comparison with 1.466 in TSC-C6), in line with the strongest exothermicity of the reaction and the Hammond postulate. This structural feature is in fact associated with absence of activation of the C=C bond in IJ (1.345 Å, which is similar to the free electrophile) as expected in absence of interaction between the metal center and the double bond. We can thus propose that this mechanism, in which IJ is a non transmetalated species, is not effective because i) copper cannot act as an efficient Lewis acid and ii) there is no activation of the C=C bond in IJ. 7, c compared to 17.5 kcal.mol -1 calculated above). Increasing the cuprate character and Lewis acidity favor kinetically the step of C-C bond forming (TSK). Nevertheless, the metal bonding environments are less favorable because IK is found to be 5.9 kcal.mol -1 above I6. It can be associated to the lowest stability of the bimetallic complex, as CK is found to be significantly higher than C0 (+5.8 kcal.mol -1 ). As a consequence, the decrease of the activation barrier between IK and TSK is lost by the destabilization of the adduct IK, already evidenced in the bimetallic catalyst CK.

Accepted Manuscript

Mechanism through L. The activation at stake in reaction path K relies on the synergetic double activation of the carbonyl compound by a Lewis acidic cation and a strong back donation of the electron rich copper center toward the C=C double bond. [START_REF] Yoshikai | [END_REF] To obtain the best of this effect, an "extreme" mechanism was thus searched using a ZnR + entity as an activating cation and docking the C=C double bond to the [Cu(M)R] -cuprate. The resulting docking intermediate is noted IL in Figure 7,d. Intermediate IL has a tetracoordinated geometry, in which the copper center is bonded to the phosphine and carboxylate groups of the ligand (M), to a methyl, and to the C terminal atom of substrate according to a short Cu-C distance of 2.017 Å. The square planar geometry of Cu and the pyramidalization of C in IL confirms a strong back-donation of copper toward the dienone and thus a Cu(III) nature of the intermediate. The efficiency of this electron transfer in activating the electrophile is confirmed by the small activation barrier for C-

Accepted Manuscript

FULL PAPER

C bond formation (6.4 kcal.mol -1 ), which is found to be 9.3 kcal.mol -1 lower than the smallest found up to now, in the case of mechanism K. Nevertheless, intermediate IL is very high in energy relative to I6 (+31.3 kcal.mol -1 ), as could be expected from the decoordination of the Zn center from the DiPPAM ligand. Considering that this is associated to a very strong Lewis acidity of the ZnMe + entity and that the reaction takes place experimentally in ethereal solvent, an explicitly solvated model must be used to counterbalance this decoordination. Two molecules of dimethylether, as a model of solvent, namely MeTHF, are added to the ZnMe + moiety and decrease the energies of IL, TSL and PL by 10 kcal.mol -1 with respect to their analogues in gas phase (See SI, Table S7). Addition of single point implicit solvation to the latest model does not change the conclusions and mechanism through L remains higher than in our proposed mechanism (See SI, Table S7).

As a conclusion, we have examined four bimetallic mechanisms, using either Zn (mechanism J) or Cu as a nucleophilic center.

The absence of C=C activation in the former increases significantly its activation barrier and disfavors this pathway. The three nucleophilic-Cu mechanisms differ by an increasing cuprate character, which is associated with lowering of activation barrier. Nevertheless, associated intermediates are at the same time destabilized. It thus appears that the preferred reaction pathway results from an equilibrium between the stability of the adduct and the activation of the substrate.

Conclusions

In this paper, the Cu-Zn bimetallic pathway and the resulting selectivity (regio-and enantioselectivity) of the Cu-catalyzed conjugate addition reaction of dialkylzinc to acyclic dienones are investigated by DFT calculations. The 1,6-addition is the kinetically and thermodynamically favored step, unless the conjugation of the ----dienone chain is altered by a terminal phenyl group. Furthermore, the retained bimetallic model allows the theoretical prediction of the absolute configuration, which was confirmed experimentally by converting a 1,6-adduct formed in the Cu/L-DiPPAM catalyzed ACA into a known chiral product. As this bimetallic mechanism seems general and proved compatible with different ligands (DiPPAM and NHC ligands) and various substrate types (cyclic and acyclic with different substituents), different competing pathways are considered to understand the factors that govern the mechanism. This confirms not only that the copper has to be a C=C bond activator and an alkyl donor while zinc plays the role of a Lewis acid which stabilizes the enolate during the C-C bond formation step but also that the C=C bond activation lowers that activation barrier of the C-C bond formation step. However, developing a strong Lewis acidity toward Zn and electronically enriching Cu appears energetically costly. As a consequence, the catalytic pathway appears as a balance between the stable neutral bimetallic complex and the activating ion-pair formation.

Experimental Section

Experimental Details. Compound 4 was prepared according to a slightly modified described procedure. [37] Compound 2a (164.5 mg, 0.815 mmol, 85% ee) was dissolved along with K2CO3 (56 mg, 0.407 mmol) in a mixture of t-BuOH (40 mL) and distilled water (9 mL). A solution of K2CO3 (56 mg, 0.407 mmol), NaIO4 (732.1 mg, 3.423 mmol) and KMnO4 (32.2 mg, 0.204 mmol) in distilled water (31 mL) was added and the mixture was stirred at rt for 16 h. The reaction media was acidified to pH = 2 by adding HCl (1N) and then stirred another 6 h. Afterwards, Na2S2O5 was added until disappearance of the reddish color. The mixture was extracted with Et2O (100 mL) and the aqueous layer was removed. The organic layer was extracted with a 1M K2CO3 aqueous solution (100 mL) and after separation, the pH of the aqueous layer was lowered to 1 by a carefully addition of concentrated HCl (strong gas release). Finally, the aqueous layer was extracted with Et2O (3x100 mL) and the combined organic layers were dried over MgSO4, filtered and concentrated under reduced pressure (20°C, 520 mbar). The crude mixture, containing both carboxylic acids formed in this step, was directly used in the next step. The phenyldiazomethane in toluene solution was obtained according to Biffis' procedure. [40] To the mixture of the two carboxylic acids previously obtained was added 13 mL (2 equiv) of the phenyldiazomethane solution and the red mixture was stirred at rt for 16 h (persistence of the reddish color after this time). The mixture was concentrated under reduced pressure and purified by silica gel chromatography using a deposit solid (pentane/EtOAc: 100/0 to 80/20 Snap ultra 10g, Biotage puriflash). Many purifications were performed in order to obtain a fraction of 4 having an acceptable purity (25 mg; 16% yield over 2 steps, slightly yellow oil).

[]D 21 = +6.0 (c 1.0, CHCl3).

Computational Details. The calculations were carried out with the Gaussian09 package. [41] All geometries and frequencies were obtained using the Density Functional Theory (DFT) with the hybrid B3PW91 functional. [42,43] This method is believed to be appropriate despite the bimetallic character of the studied complexes since both Cu(I) and Zn(II) centers are d 10 closed-shell and are never in direct interaction. The copper, zinc metals and phosphorous were represented with the quasirelativistic SDD ECP [44,45] completed by the associated basis-set. [46,47] A 6-31G(d,p) basis set was used for all other atoms (H, C, N, O, P). [48,49] Geometry optimizations were performed without any constraint and the nature of the extrema (minima and transition state) were verified by analytical calculations of frequencies. In addition, the connections between the transition state and minima were verified by carrying out a small displacement along the reaction coordinate in each direction and optimizing geometry starting from these structures. The Gibbs free energies were calculated assuming an ideal gas, unscaled harmonic frequencies, and the rigid rotor approximation in the standard conditions (P = 1 atm and T = 298 K). Dispersion effects were examined for the complete system with the hybrid B3PW91-D3 corrected functional. [50] A single point evaluation was carried out on all the stationary points of the 1,4 and 1,6 reaction pathways (see S.I. Table S8) and confirmed that dispersion effects don't alter our conclusions. For the enantioselectivity aspects, geometry optimizations were directly carried out including dispersion. To illustrate weak interactions, Non Covalent interactions isosurfaces were obtained with NCIPLOT code. [51,52] For alternative mechanisms part, copper and zinc metals were represented with the quasi-relativistic SDD ECP [44] completed by the associated basis-set. [46] A 6-31++G(d,p) basis set was used for all other atoms (H, C, N, O, P). [48,49] In addition, Density-based Solvation Model (SMD) [53] was used as it is considered as a good method to calculate the free energy of solvation Gsolv for neutral and ionic molecules in solvent (mean unsigned errors of 1 kcal.mol -1 for neutral molecules). SMD Single point calculations (THF is used) are given in the solvent molecule (Me2O instead of MeTHF) as a supermolecule, was used as the second method of solvation model.

Copper/DiPPAM-catalyzed 1,6-Asymmetric Conjugate Addition (ACA) of dialkylzinc to acyclic dienones pathway is demonstrated to originate from a balance between the activating potential of the two Cu and Zn centers and the stability of the bimetallic complex. The retained mechanism is supported by the experimentally observed regio-and enantioselectivities.

Figure 2 .

 2 Figure 2. DFT study on 1,6 vs 1,4 selectivity of Cu-ACA of dialkylzincs to ,,,-unsaturated Michael acceptors: (a) previously reported and (b) this work.

Figure 3 .

 3 Figure 3. DFT calculations on the regioselectivity of the copper-catalyzed addition of diethylzinc to α,β,γ,δ-unsaturated dienone 1a (R 1 =Me ; R 2 =Ph): Gibbs free energy (in kcal.mol -1 ) profile of the C-C bond formation steps relative to separated Cu(L-DiPPAM), ZnEt2 and 1a.

Figure 4 .

 4 Figure 4. Eight possible structures computed for I6 with L = L-(S)-DiPPAM and acyclic dienone 1a in the various conformations of the C-C single bonds. Structures on the left yield the S enantiomer and those on the right lead to the R. Energies (in kcal mol -1 ) are given with respect to structure (a).

Figure 5 .

 5 Figure 5. NCI snapshot of the most stable structures computed for I6 with L = L-(S)-DiPPAM and acyclic dienone 1a (structure (a) in Figure 4)

Figure 6 .

 6 Figure 6. Proposed alternative intermediates. [Zn] refers to Zn(II) coordinated to explicit solvent molecules (taken as OMe2).

Figure 8 .

 8 Figure 8. Optimized intermediates without docking of dienone (energies are given relative to the separated reactants Cu(M) + ZnMe2)

Figure 7 .

 7 Figure 7. Gibbs free energy profile (in kcal mol -1 ) for the three alternative mechanisms envisioned: proposed intermediate (a), non transmetalated species noted IJ (b), lateral Cu noted IK (c) and separated ion pair noted IL (d) (ΔG are computed with respect to separated reactants 1c, ZnMe2 and Cu(M)).

  Stéphanie Halbert, Jimmy Lauberteaux, Charlie Blons, Renata Marcia de Figueiredo, Christophe Crévisy, Olivier Baslé, Jean-Marc Campagne, Marc Mauduit and Hélène Gérard Page No. -Page No.

Acknowledgements

This work was supported by the Agence Nationale de la Recherche (SCATE grant no. ANR 12-BS07-0009-01). The authors thank Pr. A. Alexakis for fruitul discussions.

Accepted Manuscript