An open microscopic model of heat conduction: evolution and non-equilibrium stationary states
Tomasz Komorowski Stefano Olla Marielle Simon






Tomasz Komorowski: Institute of Mathematics, Polish Academy
  Of Sciences
Warsaw, Poland. 
komorow@hektor.umcs.lublin.pl


Stefano Olla: CNRS, CEREMADE

  Université Paris-Dauphine, PSL Research University

  75016 Paris, France 
  olla@ceremade.dauphine.fr


Marielle Simon: Inria, Univ. Lille, CNRS, UMR 8524, Laboratoire Paul Painlevé, F-59000 Lille, France
marielle.simon@inria.fr








  We consider a one-dimensional chain of coupled oscillators in contact at both ends
    with heat baths at different temperatures, and subject to an external force at one end.
    The Hamiltonian dynamics in the bulk is perturbed by random exchanges of the neighbouring
    momenta  such that  the energy is locally conserved.
 We prove that in the stationary state the energy and the volume stretch profiles,
      in large scale limit, converge to the solutions of a diffusive system with Dirichlet boundary conditions.
      As a consequence the macroscopic temperature stationary profile presents a maximum inside the
      chain higher than the thermostats temperatures,
      as well as the possibility of uphill diffusion (energy current against the temperature gradient). Finally, we are also able to derive the non-stationary macroscopic coupled diffusive equations followed by the energy and volume stretch profiles.


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  no715734) 

[2010]82C70, 60K35


An open microscopic model of heat conduction: evolution and non-equilibrium stationary states
    Marielle Simon
    December 30, 2023
=============================================================================================











§ 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 <cit.>) 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 <cit.>,   confirmed
by many numerical experiments on the NESS <cit.> 
and proved analytically for harmonic chains with
random exchanges of momenta that conserve energy,
  momentum and volume stretch, see <cit.>. 
  
  

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 <cit.> 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 <cit.> 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 <cit.> or <cit.> and references therein.
These numerical results have been confirmed in <cit.>, 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 τ̅_+ 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 <cit.>.

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 (<ref>) 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 <cit.>,
see also <cit.> for a similar model without boundary conditions.

In the present article we study the NESS of this dynamics. We prove, see 
Theorem <ref> 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 τ̅_+, 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 τ̅_+^2, i.e. the increase of the bulk temperature
is not a linear response term. In the case τ̅_+= 0, the NESS was studied in <cit.>,
where the temperature profile is  proved to be linear: more details are available in <cit.>.
 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. <cit.>).

The NESS for our model also provides a simple example of an uphill energy diffusion:
if the force τ̅_+ 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 <ref> 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 <cit.>,
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 <cit.>.



Let us add a comment on the proofs of our main results.
In proving Theorem  <ref> (on the asymptotics of energy and stretch profile)
we need to make an additional  hypothesis concerning
the strength >0 of the noisy part of the dynamics,
see (<ref>) and (<ref>). More precisely we suppose that =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 (<ref>)
of the microscopic energy profile of  the chain (the term ℋ_n^m).
We conjecture that this term vanishes in the stationary macroscopic limit,
making the conclusion of the theorem valid for any >0,
but are unable to prove this fact at the moment.
We do not need this hypothesis in our proof of Theorem  <ref> (on the uphill diffusion phenomenon).

In Appendix <ref> 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 <cit.>. 



The rest of the paper is organized as follows: in Section <ref> we define the microscopic model under investigation and give the expected macroscopic system of equations, showing the phenomenon of uphill diffusion. In Section <ref> 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 <ref> provides elements of the proofs and preliminary computations on the averages, while Section <ref> provides all the remaining technical lemmas, concerning the second order moments.







   
  

 




§ MICROSCOPIC DYNAMICS AND MACROSCOPIC BEHAVIOUR





 §.§ Open chain of oscillators



Let _n := {1,…,n}, _n:=_n ∪{0} and := [0,1]. 
The configuration space Ω_n:=^_n×^_n  consists of all sequences
(𝐪,𝐩):={q_x,p_x}_x ∈_n, where
p_x∈ stands for the momentum of the oscillator at site x, and
q_x∈ 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)
τ̅_+, 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∈_n. The configurations can then be described by
 
    (𝐫, 𝐩) = (r_1, …, r_n, p_0, …, p_n) ∈Ω_n.

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

    ℋ_n (𝐫, 𝐩) :=∑_x∈_n E_x+ p_0^2/2,

 with

    E_x:= p_x^2/2 + r_x^2/2,   x∈_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 dynamics are given in the bulk by 

    r_x(t)    = n^2 (p_x(t)- p_x-1(t))     t,       x∈_n 
       p_x(t)    = n^2 (r_x+1(t)- r_x(t))     t -  γ n^2  p_x(t)
           t 
          + n√(γ)(p_x-1(t)
           w_x-1,x(t)-p_x+1(t) w_x,x+1(t)),   
           x∈{1, …, n-1}

 and at the boundaries: 
    
    p_0(t)    = n^2   r_1(t)      t
         -  n^2/2(γ+γ̃)p_0(t)  t   - n √(γ)p_1(t)  w_0,1(t) + n√(γ̃T_-)w̃_0(t) 
       p_n(t)    = -n^2   r_n(t)     t  + n^2 τ̅_+(t)   t - n^2/2(γ +  γ̃)p_n(t)  t  
                             
                        + n √(γ)p_n-1(t) w_n-1,n(t)
                        + n√(γ̃T_+)w̃_n(t)

where w_x,x+1(t), x∈{0,…,n-1}, w̃_0(t) and w̃_n(t) are independent, standard one dimensional Wiener processes, and γ>0 (resp. γ̃>0) regulates the intensity of the random perturbation (resp. the Langevin thermostats).  See Figure <ref> for a representation of the chain.
Note that the purely Hamiltonian dynamics is perturbed
  by a stochastic noise which exchanges kinetic energy between the neighbouring atoms,
  and with the boundary thermostats.



We let 

    ( 𝐫_n(t),  𝐩_n(t)) := (r_1(t), …,
    r_n(t), p_0(t), …, p_n(t)),   t≥0
 be the
Ω_n–valued process whose dynamics is determined by the
equations (<ref>)–(<ref>).
Its generator is given by

    L:= n^2 (A + γ/2 S + γ̃/2S̃),

where

    A := ∑_x=1^n (p_x - p_x-1) ∂_r_x + ∑_x=1^n-1 (r_x+1 - r_x) ∂_p_x
      + r_1 ∂_p_0 + (τ̅_+(t) - r_n) ∂_p_n
         S f  := ∑_x=0^n-1_x ∘_x (f),

where _x is the momentum exchange operator defined as 

    _x:= p_x+1∂_p_x - p_x ∂_p_x+1,

and moreover the generator of the Langevin heat baths at the boundaries is given by

    S̃ :=  T_-∂_p_0^2 - p_0 ∂_p_0 + T_+∂_p_n^2 - p_n ∂_p_n.

From the microscopic energy conservation law
there exist microscopic energy currents j_x,x+1 which satisfy

    n^-2 L ℰ_x  = j_x-1,x - j_x,x+1 ,     for any  x ∈_n

and are given by 

    j_x,x+1 := - p_x r_x+1 + γ/2 (p_x^2 - p_x+1^2),    if  x ∈{0,...,n-1},
 while at the boundaries 
  
    j_-1,0 := γ̃/2(T_- - p_0^2 ),     
      j_n,n+1 := -γ̃/2(T_+ - p_n^2 ) - τ̅_+(t) p_n.





 §.§ Macroscopic equations


Suppose that r(t,u),  e(t,u), (t,u)∈_+×, 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 ∈_n r_x(t) δ_x/n(·) ,    and   1/n∑_x ∈_nℰ_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 <ref>. 
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 equations[See also
  Theorems 3.7 and 3.8 of <cit.> for a similar model which gives a similar coupled diffusive system
  for every value of γ.],
    
    ∂_t
            r(t,u)   =γ^-1 ∂^2_uur(t,u)
    ∂_t e (t,u)   =  1/2∂^2_uu{(γ^-1+γ) e(t,u) +
           1/2(γ^-1- γ)  r^2(t,u)},  
            (t,u)∈_+× ,

with the boundary conditions

    r(t,0) = 0
       ,          r(t,1) = τ̅_+(t), 
       e (t,0) = T_-,        e(t,1) = T_+ + (τ̅_+(t))^2/2

and with the initial condition

    r(0,u)= r_0(u),          
     e(0,u) = E_0(u).

In Appendix <ref> 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 (<ref>).



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 (<ref>) and
(<ref>) we conclude that 

    ∂_t e_ mech(t,u)= ^-1(∂_uu^2
        e_ mech(t,u)-(∂_ur(t,u))^2),  
      (t,u)∈_+×

  with 
 
    e_ mech (t,0) = 0,     e_ mech(t,1) =
       (τ̅_+(t))^2/2 ,   e_ mech (0,0) = r_0^2(u)/2,   (t,u)∈_+×

and 

    ∂_t e_th (t,u)=
            1/2(γ^-1+γ)∂^2_uu e_th(t,u) +
            γ^-1 (∂_u r(t,u))^2,   
            (t,u)∈_+× ,
 with 

    e_th (t,0) = T_-,     e_th (t,1) = T_+,    t >0.


 



 §.§ Stationary non-equilibrium states



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

When τ̅_+=0 and T_- = T_+ = T, the system is in equilibrium
  and  the stationary probability distribution is given explicitly by the homogeneous Gibbs measure
  
    ν_T ( r, p):= g_T( r,  p)  p_0∏_x∈_n p_x  r_x,
 where
  
    g_T( r,  p) :=e^-p_0^2/2T/√(2π T)∏_x∈_ne^- E_x/T/2π T.

If τ̅_+ ≠ 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. (<ref>)) for the microscopic dynamics described by  the equations (<ref>)–(<ref>).
As a consequence
⟨ LF⟩_ ss = 0 
for any function F in the domain of the operator L, given by
(<ref>). Hereafter, we denote

    ⟨ F⟩_ ss := ∫_Ω_n F  μ_ ss .

The proof of the existence and uniqueness of a stationary state follows from the same argument as the one
used in <cit.> for τ̅_+=0. The fact that  in our case τ̅_+ does not vanish requires only minor modifications. In addition, one can show, see bound (A.1) in     <cit.>, that for a fixed n we have ⟨ H_n⟩_ ss<+∞ (cf.  (<ref>)).

The corresponding
stationary profiles, denoted respectively by r_ ss(u) and e_ th,ss(u),  will solve the stationary version of equations (<ref>) and (<ref>), i.e.:

    r_ss (u) = τ̅_+ u
 and 

    (γ^-1+γ)∂^2_uu e_th, ss(u) + 2γ^-1 τ̅_+^2 = 0,

with the boundary conditions 

    e_th, ss(0)= T_-,       e_th, ss(1) = T_+.
 
In other words

    e_th, ss(u) = τ̅_+^2/1+γ^2  u (1-u) + (T_+-T_-)u + T_-.

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

    ⟨ j_x,x+1⟩_ ss =: j̅_s,     for any 
     x∈{-1,…,n}.

The macroscopic stationary energy current is defined as the
limit of  nj̅_s, as n→+∞. It equals, see Theorem <ref> below,

    J_ ss=  - 1/2(γ^-1+γ)(T_+ - T_-) -τ̅_+^2/2γ.

Observe that  the energy
current can flow against the temperature gradient if T_- > T_+ and |τ̅_+| is large enough
(uphill diffusion). 
Assuming T_+ ≥ T_-
  the maximum stationary temperature e_th, ss^ max is reached at

    u_ max= (1/2+1+γ^2/τ̅_+^2(T_+-T_-)) ∧ 1

which implies that, if the condition 2(1+γ^2) (T_+ -T_-) ≤τ̅_+^2 is satisfied, then
 the maximum temperature of the chain is attained inside, since 
u_ max < 1 (see Figure <ref>), and
 it equals 

    e_th, ss^ max =  (T_+ - T_-)/2 + T_- +τ̅_+^2/4(1+γ^2) ≥ T_+.

Note that this does not depend on the sign of τ̅_+. 



This phenomenon was observed by dynamical numerical simulations in <cit.> for the
stationary states of the rotor model. It has attracted quite some
interest
from physicists, see <cit.> 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 <ref>.  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: 


 The following uniform convergence holds:

    sup_u ∈| ⟨ r_[nu]⟩_ ss - r_ ss(u) |  0,


where r_ ss(u):=τ̅_+ u. In particular, for any continuous test function G:→ R, 

    1/n∑_x∈_n G(x/n) ⟨ r_x⟩_ ss∫_ G(u) r_ ss(u)   u.



The averages under the stationary state ⟨ r_x⟩_ ss and
⟨ p_x⟩_ ss are computable explicitly, see Proposition
<ref>  in the next section. It turns out that
 ⟨ p_x⟩_ ss is constant for all  x∈_n and
 equals  p̅_s:= τ̅_+/(γ n +γ̃) (see (<ref>)).
 From (<ref>) we also have

    n(⟨ r_x+1⟩_ ss - ⟨ r_x⟩_ ss) = n γp̅_s  τ̅_+,     for   x∈{1,...,n-1}

    and 
    ⟨ r_1⟩_ ss⟶_n→∞ 0,     ⟨ r_n⟩_ ss τ̅_+.

 Finally,  (<ref>) 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.

 
    n j̅_s     - 1/2(γ^-1+γ)(T_+ - T_-) -τ̅_+^2/2γ.



Note that Theorems <ref> and <ref> 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_x^2⟩_ ss
       ℰ^th_x:= ℰ_x
      -ℰ^mech_x=1/2 p_x^2 + 1/2(r_x^2 -
      ⟨ r_x^2⟩_ ss),      x ∈_n .






Assume that γ=1. For any continuous test function G:→
R we have 

    1/n∑_x∈_n G(x/n)
      ℰ^mech_x     ∫_
      G(u) 1/2r_ ss^2(u)  u,
    1/n∑_x∈_n G(x/n) ⟨
      E_x ^th⟩_ ss    ∫_ G(u) e_
        th,ss(u)   u, 
    1/n∑_x∈_n G(x/n) ⟨
      E_x⟩_ ss    ∫_ G(u) ( e_
        th,ss(u) + 1/2 r_ ss^2(u))   u,

where 

    r_ ss(u)   =τ̅_+ u, 
     
    e_ th,ss(u)   =τ̅_+^2/2  u(1-u)+(T_+-T_-)u + T_+.
 


The remaining part of the paper deals with the  proofs of Theorems <ref> and <ref>.
 





§ THE STATIONARY STATE
 

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



 §.§ Elongation and momenta averages



The average stationary momenta are equal to

    ⟨ p_x⟩_ ss  = p̅_s := τ̅_+ /γ n + γ̃ ,     for any  x∈_n.

The average stationary elongations are equal to

    ⟨ r_x⟩_ ss =  p̅_s/2 (γ̃-γ + 2 γ x) = τ̅_+(2γ x+γ̃-γ)/2(γ n + γ̃)    for any   x ∈_n.


 
We start with some useful relations that hold for the stationary state:


  (1) since ⟨ Lr_x⟩_ ss = 0,
applying (<ref>), we conclude

    ⟨ p_x⟩_ ss = ⟨ p_x-1⟩_ ss = p̅_s,    for any  x ∈_n;


  (2) from ⟨ Lp_x⟩_ ss = 0 we get

    ⟨ r_x+1⟩_ ss - ⟨ r_x⟩_ ss = γp̅_s,     for any  x∈{2, …, n-2}
       ⟨ r_1⟩_ ss = 1/2(γ+γ̃) p̅_s,
       ⟨ r_n⟩_ ss  = - 1/2(γ+γ̃) p̅_s + τ̅_+.



These equations determine the average stationary momentum and
elongation as given in formulas (<ref>)
  and (<ref>).






 §.§ Elements of the proofs of Theorems <ref>
  and <ref>

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

 
We can write j̅_s as a discrete gradient, namely 

    j̅_s=∇ϕ(x):=ϕ(x+1)-ϕ(x),     x∈{1,…,n-1} ,

with 

    ϕ(x): = -1/2γ( ⟨ r_x^2⟩_ ss +
      ⟨ p_x-1 p_x⟩_ ss) - γ/4(
    ⟨ p_x^2 ⟩_ ss + ⟨ p_x-1^2
    ⟩_ ss) ,      x∈_n .




Thanks to (<ref>), the function ϕ(x) is  
harmonic, i.e.: 

    Δϕ(x):=ϕ(x+1)+ϕ(x-1)-2ϕ(x)=0,      for any  x ∈{2,…,n-1}.
 



By a direct calculation one can easily check that the energy currents j_x,x+1 (defined in (<ref>)) satisfy the following fluctuation-dissipation relation: 

    j_x,x+1 = n^-2 Lg_x -1/2γ∇(r_x^2 + p_x-1p_x) - γ/4∇(p_x-1^2 + p_x^2),
  
  for any x ∈{1,...,n-1},  with 
  
    g_x:=-1/4p_x^2 + 1/2γp_x(r_x+r_x+1).

Therefore,  (<ref>) is obtained by taking the average in (<ref>) with respect to the stationary state.



We can now sketch the proof of Theorem <ref>: straightforward computations, using the definition (<ref>) of ϕ, yield

    ⟨ E_x⟩_ ss= 1/2(⟨ p_x^2 ⟩_ ss + ⟨ r_x^2 ⟩_ ss)     = -2γ/1+γ^2ϕ(x) + γ^2/2(1+γ^2)(⟨ p_x^2 ⟩_ ss - ⟨ p_x-1^2 ⟩_ ss) 
          - 1/1+γ^2⟨ p_x p_x-1⟩_ ss+ 1-γ^2/2(1+γ^2)(⟨ p_x^2 ⟩_ ss - ⟨ r_x^2 ⟩_ ss).

Therefore, the microscopic energy profile can be decomposed as the sum of four terms: 

    H_n(G):= 1/n∑_x ∈_n
      G(x/n) ⟨ℰ_x⟩_ ss =  
      H_n^ϕ(G) +   H_n^∇(G) +   H_n^corr(G) +
       H_n^m(G),

where 

    H^ϕ_n(G)     := -2γ/1+γ^2 1/n∑_x ∈_n G(x/n) ϕ(x), 
     H^∇_n(G)     := γ^2/2(1+γ^2) 1/n∑_x∈_n G (x/n) ( ⟨ p_x^2 ⟩_ ss - ⟨ p_x-1^2 ⟩_ ss),
     H^corr_n(G)     :=-1/1+γ^2 1/n∑_x ∈_n G(x/n) ⟨ p_x-1p_x⟩_ ss, 
     H^m_n(G)     := 1-γ^2/2(1+γ^2) 1/n∑_x ∈_n G(x/n) (⟨ p_x^2 ⟩_ ss -⟨  r_x^2⟩_ ss) .

Note that, if γ=1, then ℋ_n^m≡ 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: 
 
For any continuous test function G:→, 

    ℋ_n^ϕ(G)    ∫_ G(u) ( τ̅_+^2/1+γ^2u + (T_+-T_-)u+T_- )  u, 
    ℋ_n^∇(G)     0 ,
    ℋ_n^corr(G)     0.


The complete proof of the proposition will be given in Section <ref>. Let us first comment
on the ideas used in the argument. The limit (<ref>)
will be concluded using the fact that ϕ is
harmonic,   (<ref>) is a consequence of the presence of a discrete gradient ⟨ p_x^2⟩_ ss - ⟨ p_x-1^2 ⟩_ ss inside the sum, and (<ref>) will be  shown thanks to the second order bounds, which are obtained in the next section. 


With the help of Proposition <ref>
the proof of Theorem
<ref> becomes straightforward. Assume that γ=1. From the decomposition (<ref>) and Proposition <ref>, we get

    1/n∑_x∈_n G(x/n) ⟨ E_x ⟩_ ss   ∫_ G(u) (  τ̅_+^2/2  u +  (T_+-T_-)u+T_- )  u 
       =∫_ G(u) ( τ̅_+^2/2 u(1-u)
      +(T_+-T_-)u+T_-+(τ̅_+ u)^2/2)   u.

Recalling (<ref>) and (<ref>) we conclude that the right
hand side equals

    ∫_ G(u) ( e_ th,ss(u)+r_ ss^2(u)/2)
       u.

 Thus   (<ref>) follows. From Theorem
<ref> we immediately conclude 
(<ref>). The convergence of the stationary microscopic
thermal energy profile in (<ref>) is an immediate consequence of these two statements. 





§ MOMENT BOUNDS UNDER THE STATIONARY STATE


In this section we present a complete proof of Proposition
  <ref> (see Section <ref>) and we show Theorem <ref>
  (see Proposition <ref>).
  



Before presenting the
proof, we need a few technical estimates on the entropy production (Section <ref>) and second order moments (Section <ref> and Section <ref>). 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 (<ref>). For a given T we will use
  
    ν_T ( r, p)= g_T( r,  p)  p_0∏_x∈_n p_x  r_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   <cit.>, 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  ν_T_+ .



 
Denote h:=g_T_-/g_T_+.The following formula holds 

    γ∑_x=0^n-1 D_x (f_s) + 
     γ̃T_-  (∂_p_0 (f_s/h))^2/(f_s/h)_T_-+ 
    γ̃T_+  (∂_p_n f_s)^2/f_s_T_+
      = 
      τ̅_+^2  /T_+(γ n+γ̃) + γ̃(1/T_+- 1/T_-) ( T_- - ⟨ p_0^2⟩_ ss),

where 

    D_x (f_s):=(_xf_s)^2/f_s_T_+.







Integration by parts yields

    gS̃f
    _T_+=T_+∂_p_nf ∂_p_ng
    _T_+ +T_-∂_p_0f ∂_p_0g
    _T_+
    +T_-(1/T_--1/T_+) g ∂_p_0f  _T_+

for any f,g∈ C^∞_0(_n), where  C^∞_0(_n) is the
space of compactly supported smooth functions. As 

    A(ℋ_n (𝐫, 𝐩))=τ̅_+p_n     _x(∑_y=0^np_y^2)=0,     x∈{0,…,n-1}

we conclude

    -  gAf  _T_+ =   f Ag _T_++τ̅_+ p_n f g _T_+
    ,
       
     -  g _x^2f _T_+ =_xf _xg _T_+ ,

for any x=0,…,n-1, and any f,g∈ C^∞_0(_n).
 We take the average of  -n^-2L (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∑_x=0^n-1 D_x(f_s) + 
        γ̃T_+ (∂_p_n f_s)^2/f_s_T_+ 
        - τ̅_+ ∂_p_n f_s _T_+ - γ̃⟨(T_- ∂_p_0^2 - p_0∂_p_0) log f_s ⟩_ ss.

From the definition h=g_T_-/g_T_+,  the last term can be rewritten in
the form: 

    - ⟨(T_- ∂_p_0^2 - p_0∂_p_0) log f_s ⟩_ ss    = - ∫f_s/h(T_- ∂_p_0^2 - p_0∂_p_0) ( log(f_s/h))   g_T_- dp_0∏_x=1^ndp_x dr_x 
          - ∫ f_s   (T_- ∂_p_0^2 - p_0∂_p_0)(log h)   g_T_+ dp_0∏_x=1^ndp_x dr_x 
        = T_- (∂_p_0 (f_s/h))^2/(f_s/h)_T_- + (1/T_-- 1/T_+) ( T_- - ⟨ p_0^2⟩_ ss).

Moreover, by integration by parts and (<ref>), we obtain

    ∂_p_n f_s _T_+ = T_+^-1  p_n f_s _T_+ = p̅_s/T_+  
    = τ̅_+ /T_+(γ n + γ̃)

and (<ref>) 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 <cit.>.







 §.§ 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 |j̅_s|, see
(<ref>)  and investigate the behaviour of ϕ(1)
and ϕ(n) as n→∞, see (<ref>). 

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


The following equality holds

    ⟨ p_0^2 ⟩_ ss + ⟨ p_n^2 ⟩_
         ss = T_+ + T_- + 2τ̅_+ p̅_s/γ̃    n≥ 1.

Moreover, there exists a constant
C=C(γ,γ̃,τ̅_+,T_+,T_-)>0, such that

    ⟨ r_1^2⟩_ ss  +  ⟨
       r_n^2⟩_ ss≤ C    n≥ 1.



By convention,  the constants appearing in the statements below depend only on the
parameters indicated in parentheses in the statement of the proposition.

 
The first identity (<ref>) is an easy consequence of
(<ref>) and (<ref>), which  yields

    j̅_s     =   γ̃/2(T_- - ⟨ p_0^2⟩_ ss),
    j̅_s    = - τ̅_+ p̅_s- γ̃/2(T_+- ⟨ p_n^2⟩_ ss).

 Identity (<ref>) is obtained by adding sideways the above
equalities.  
To show estimate (<ref>) note that

    n^-2L(p_0 r_1)    = (p_1-p_0)p_0+  r_1^2 - 1/2(γ̃+γ)
        p_0r_1 
    
         n^-2 L(p_n r_n)    = p_n(p_n - p_n-1) + (τ̅_+-r_n) r_n - 1/2(γ̃+γ)
        p_n r_n .

After taking the
average with respect to the stationary state from (<ref>) we conclude 

    ⟨ r_1^2⟩_ ss   = ⟨ p_0^2⟩_ ss-⟨ p_1p_0
                                 ⟩_ ss+1/2(γ̃+γ)
                            ⟨ p_0r_1⟩_ ss.
   
Recalling the definition of the current (<ref>) and
then invoking (<ref>), we get

    ⟨ r_1^2⟩_ ss   =⟨ p_0^2⟩_ ss-⟨ p_1p_0 ⟩_ ss-1/2(γ̃+
        γ)(⟨ j_0,1⟩_ ss + γ/2(⟨ p_1^2 ⟩_ ss - ⟨ p_0^2 ⟩_ ss) ) 
        =⟨ p_0^2⟩_ ss-⟨ p_1p_0 ⟩_ ss- γ̃/4 (γ̃+
        γ)( T_--⟨ p_0^2 ⟩_ ss)-γ/4(γ̃+γ) (⟨ p_1^2 ⟩_ ss - ⟨ p_0^2 ⟩_ ss)  .

Using Young's inequality 

    | ⟨ p_1p_0 ⟩_ ss| ≤A/2⟨ p_1^2 ⟩_ ss + 1/2A⟨ p_0^2 ⟩_ ss,

with A = γ/2 (γ + γ̃), 
we get 
  
    ⟨ r_1^2 ⟩_ ss≤( 1/γ(γ+γ̃) + 1 + 1/4(γ+γ̃)^2 ) ⟨ p_0^2 ⟩_ ss.

From (<ref>) we conclude that ⟨ r_1^2 ⟩_ ss is bounded.


To estimate ⟨ r_n^2 ⟩_ ss, note that from (<ref>) we write

    ⟨ r_n^2⟩_ ss = ⟨ p_n^2⟩_ ss -  ⟨ p_n p_n-1⟩_ ss + τ̅_+ ⟨ r_n⟩_ ss -  1/2(γ̃+γ) ⟨ p_nr_n⟩_ ss.

We use again Young's inequality 

    | ⟨ p_np_n-1⟩_ ss| ≤A/2⟨ p_n^2
    ⟩_ ss + 1/2A⟨ p_n-1^2 ⟩_ ss ,
 
with A= 1/(2γ) and we get

    ⟨ r_n^2 ⟩_ ss≤(1+1/4γ) ⟨ p_n^2 ⟩_ ss +  γ⟨ p_n-1^2 ⟩_ ss + τ⟨ r_n ⟩_ ss - 1/2(γ̃+  γ) ⟨ p_n r_n⟩_ ss.

To replace ⟨ p_n-1^2 ⟩_ ss, note that 

    n^-2L(p_n^2) = 2 (τ̅_+ - r_n) p_n + γ (p_n-1^2 - p_n^2)
    +  γ̃(T_+-p_n^2).

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

    γ⟨ p_n-1^2 ⟩_ ss =2 ⟨ r_np_n⟩_ ss - 2 τ̅_+ ⟨ p_n ⟩_ ss +  (γ+γ̃) ⟨ p_n^2 ⟩_ ss -  γ̃T_+,

which, in (<ref>), gives

    ⟨ r_n^2 ⟩_ ss≤(1+1/4γ+γ + γ̃) ⟨ p_n^2 ⟩_ ss  +τ̅_+ (⟨ r_n ⟩_ ss -2⟨ p_n⟩_ ss)+ 1/2(4-γ̃- γ) ⟨ p_n r_n⟩_ ss -  γ̃T_+.

  Using again Young's inequality 
  
    | ⟨ p_nr_n⟩_ ss| ≤A/2⟨
        p_n^2 ⟩_ ss + 1/2A⟨ r_n^2 ⟩_ ss
 with
  A=1/2|4-γ-γ̃|, we finally arrive at 
   
    1/2⟨ r_n^2 ⟩_ ss≤(1+1/4γ+γ + γ̃+1/4(4-γ-γ̃)^2) ⟨ p_n^2 ⟩_ ss  +τ̅_+ (⟨ r_n ⟩_ ss -2⟨ p_n⟩_ ss) -  γ̃T_+.

We now invoke (<ref>), (<ref>) and (<ref>)
to conclude the bound on ⟨ r_n^2⟩_ ss, which combined with the
already obtained bound on ⟨ r_1^2 ⟩_ ss yields  (<ref>).



There exists C=C(γ,γ̃,τ̅_+,T_+,T_-)>0, such that 

    ⟨ p_1^2 ⟩_ ss + ⟨ p_n-1^2 ⟩_ ss≤ C    n≥ 1.






To bound  ⟨ p_n-1^2 ⟩_ ss we use formula (<ref>).
 From an elementary inequality |⟨ r_n p_n
 ⟩_ ss |≤1/2(⟨ p_n^2 ⟩_ ss + ⟨ r_n^2 ⟩_ ss)
 and Proposition <ref>, we easily conclude that
 ⟨ p_n-1^2⟩_ ss is bounded. 


The  bound for ⟨
 p_1^2 ⟩_ ss is obtained similarly. First, note that 

    n^-2 L(p_0^2) = 2 r_1 p_0 +  γ
      (p_1^2-p_0^2) + γ̃(T_–p_0^2).

 Taking the average with respect to the stationary state, using the
 inequality |⟨ r_1 p_0 ⟩_ ss |≤1/2(⟨ r_1^2 ⟩_ ss +
 ⟨ p_0^2⟩_ ss), and invoking Proposition
 <ref>, we conclude the desired bound on ⟨
 p_1^2 ⟩_ ss. Thus (<ref>) follows.


In the next proposition we provide a bound on the energy current
under the stationary state, which will be   further refined in Proposition <ref>.



There exists a constant C=C(γ,γ̃,τ̅_+,T_+,T_-)
>0, such that the stationary current satisfies

    |j̅_s| ⩽C/n    n≥ 1.






We sum the identity (<ref>) from x=1 to n-1 and apply
(<ref>) to express ϕ(n) and ϕ(1).
In this way we get 

    (n-1) j̅_s     = ϕ(n)-ϕ(1) 
        = ⟨ - p_n-1p_n + r_n^2/2γ - γ (p_n-1^2 +p_n^2)/4⟩_ ss +  ⟨p_1p_0 + r_1^2/2γ + γ (p_1^2 +p_0^2)/4⟩_ ss.

Therefore, (<ref>) follows from the elementary 
  inequalities
 
    |⟨ p_x-1p_x⟩_ ss |≤1/2( ⟨ p_x^2
    ⟩_ ss + ⟨ p_x-1^2 ⟩_ ss)

together with  the bounds obtained in Proposition <ref> and Corollary <ref>. 


Proposition <ref> permits to get a better estimate
on the entropy production. Namely, combining  (<ref>),
(<ref>) and (<ref>) we conclude the following.

There exists C=C(γ,γ̃,τ̅_+,T_+,T_-)>0, such that 

    γ∑_x=0^n-1 D_x (f_s) + 
     γ̃T_-  (∂_ p_0 (f_s/h))^2/(f_s/h)_T_- + 
    γ̃T_+  (∂_p_n f_s)^2/f_s_T_+≤C/n,     n≥1.




Thanks to Proposition <ref> we are now able to
estimate the covariances of  momenta and stretches at the  boundaries as follows: 

 
There exists C=C(γ,γ̃,τ̅_+,T_+,T_-)>0, such
that, at the left boundary point

    |⟨ p_0p_1⟩_ ss|+|⟨
      r_1p_1⟩_ ss|+|⟨ r_1p_0⟩_ ss| ⩽C/√(n),      n≥1,
 and at the
right boundary point
  
    |⟨ p_np_n-1⟩_ ss| +|⟨
    r_np_n⟩_ ss|+|⟨ r_np_n-1⟩_ ss| ⩽C/√(n),     n≥1.


 
Integration by parts yields

    ⟨ p_0 p_1⟩_ ss=-T_- p_1 (f_s/g_T-) ∂_p_0g_T-_T_+=T_- p_1 ∂_p_0 (f_s/h)_T_-.

We use the entropy production bound (<ref>) and estimate
(<ref>) on ⟨ p_1^2 ⟩_ ss, to estimate the right
hand side. As a result we get

    | ⟨ p_0 p_1⟩_ ss|= T_-| p_1 ∂_p_0 (f_s/h)_T_-| ≤ T_-⟨ p_1^2⟩_ ss^1/2 ( ∂_p_0 (f_s/h))^2/(f_s/h)_T_-^1/2≤C/√(n).

Similarly, 

    | ⟨ p_n p_n-1⟩_ ss|=T_+ | p_n-1∂_p_n f_s_T_+| ≤ T_+⟨ p_n-1^2⟩_ ss^1/2 ( ∂_p_n f_s)^2/f_s_T_+^1/2≤C/√(n).

Finally, note that, for any x ∈_n 

    n^-2L(r_x^2) = 2(p_x-p_x-1)r_x.

 Therefore,  upon averaging with respect to the NESS, we get

    ⟨ p_x r_x ⟩_ ss = ⟨ p_x-1 r_x ⟩_ ss,    x ∈_n.

In particular, applying (<ref>) 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_np_n
⟩_ ss|. This is done  using the entropy production bound
(<ref>) in the same manner as before, namely: 

    | ⟨ r_1 p_0⟩_ ss|= T_-| r_1 ∂_p_0( f_s/h)_T_-| ≤ T_-⟨ r_1^2⟩_ ss^1/2 ( ∂_p_0 (f_s/h))^2/(f_s/h)_T_-^1/2≤C/√(n),

from (<ref>) and (<ref>). We leave the last estimate for the reader. 


We now have all the ingredients necessary to prove moments convergences at the boundaries:


The following limits hold: at the left boundary point,  

    ⟨ p_x^2⟩_ ss     T_-     for  x∈{0,1}, 
    ⟨ r_1^2⟩_ ss     T_- ,
    ⟨ r_1r_2⟩ _s     0,

and at the right boundary point,
       
    ⟨ p_x^2⟩_ ss     T_+     for  x∈{n-1,n} ,
    ⟨ r_n^2⟩_ ss     T_+ + τ̅_+^2, 
    
      
      ⟨ r_n-1 r_n⟩ _s      τ̅_+^2.



 


From (<ref>) and Proposition
<ref> we get ⟨ p_0^2 ⟩_ ss→
T_-. Thanks to (<ref>)  and (<ref>) we deduce  ⟨ p_1^2 ⟩_ ss→
T_-, which in turn proves (<ref>). A similar argument proves
(<ref>).
Indeed, from (<ref>) and Proposition
<ref>,  we get
⟨ p_n^2 ⟩_ ss→ T_+ and from (<ref>)  and (<ref>),  we deduce
⟨ p_n-1^2 ⟩_ ss→ T_+.






  §.§.§ Proofs of (<ref>) and (<ref>)

The limit  (<ref>) follows directly from (<ref>) and
Proposition <ref>. From
(<ref>) we conclude that 
⟨ r_n ⟩_ ss→τ̅_+.
Using then (<ref>)  together with Proposition <ref> 
we conclude  (<ref>).



  §.§.§ Proofs of (<ref>)  and (<ref>)

Note that

    n^-2 L(r_1p_1)     = (p_1-p_0)p_1 + (r_2 - r_1)r_1 - γ r_1 p_1 
    
    n^-2 L(r_n p_n-1)    = (p_n-p_n-1)p_n-1 + (r_n-r_n-1)r_n -  γ r_np_n-1.

Taking the average with respect to the stationary state, and using
Proposition <ref> together with (<ref>) proved
above, we get 

    ⟨ r_1^2⟩_ ss - ⟨ r_1 r_2⟩ _s  T_-

and

    ⟨ r_n^2⟩_ ss - ⟨ r_n-1 r_n⟩ _s  T_+.

Using the already proved limits (<ref>)  and (<ref>) we conclude
(<ref>) and (<ref>). 



 
The following limits hold:

    ϕ(1)     -1/2(γ^-1+γ)T_-
    ϕ(n)      -1/2(γ^-1+γ)T_+ - τ̅_+^2/2.

In consequence, (<ref>) holds and Theorem <ref> is proved. 



Limits in (<ref>) and (<ref>) follow from formula (<ref>),
Proposition  <ref> and
the limits computed in Corollary <ref>. The
limit  (<ref>) is a consequence of (<ref>), (<ref>) and formula (<ref>).







 §.§ Energy bounds


We now provide bounds on the total energy under the stationary state:

 
 There exists C=C(γ,γ̃,τ̅_+,T_+,T_-)>0, such that 

    1/n∑_x=1^n ⟨ p_x^2 ⟩_ ss≤ C     and    1/n∑_x=1^n  ⟨ r_x^2
      ⟩_ ss≤ C ,   n≥1.



From the current decomposition given by (<ref>), we easily get that 

    ϕ(x) = (x-1) j̅_s + ϕ(1),     for any  x ∈_n.

Summing over x, this gives

    1/n∑_x=1^n ϕ(x) = 1/n∑_x=2^n (x-1) j̅_s + ϕ(1)  = n(n-1)/2n j̅_s +  ϕ(1).

Therefore, recalling (<ref>) and (<ref>),
we get

    1/n∑_x=1^n ϕ(x)   -1/4( γ^-1 + γ)(T_++T_-) - τ̅_+^2/4γ.

From (<ref>), we have 

    1/n∑_x=1^n ϕ(x)  = - 1/2γ n∑_x=1^n ⟨ r_x^2 ⟩_ ss - 1/2γ n∑_x=1^n ⟨ p_x p_x-1⟩_ ss - γ/2n∑_x=1^n ⟨ p_x^2 ⟩_ ss + γ/4n( ⟨ p_n^2 ⟩_ ss- ⟨ p_0^2 ⟩_ ss).

To compute the limit of the second sum in the right hand side of
(<ref>), we first write: 

    n^-2 L (p_x-1p_x)     = (r_x+1-r_x) p_x-1 + (r_x-r_x-1)p_x - 2γ p_xp_x-1,      x ∈{2,…,n-1}.

Thus,  taking the average with respect to the stationary state and
subsequently using (<ref>), we obtain 

    2γ⟨ p_xp_x-1⟩_ ss    = ⟨ p_x r_x⟩_ ss
    +⟨ p_x-1r_x+1⟩_ ss -  ⟨ p_xr_x-1⟩_ ss  -
                                          ⟨
                                          p_x-1r_x⟩_ ss
       =
    ⟨ p_x-1r_x+1⟩_ ss -  ⟨ p_xr_x-1⟩_ ss .

On the other hand

    n^-2 L (r_xr_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 (<ref>), we get

    0    = ⟨ p_x+1r_x⟩_ ss+⟨ p_xr_x+1⟩_ ss -  ⟨ p_xr_x⟩_ ss  -  ⟨ p_x-1r_x+1⟩_ ss
        =⟨ p_x+1r_x⟩_ ss+⟨ p_x+1r_x+1⟩_ ss -  ⟨ p_xr_x⟩_ ss  -  ⟨ p_x-1r_x+1⟩_ ss,

which yields

    ⟨ p_x+1r_x+1⟩_ ss -  ⟨ p_xr_x⟩_ ss  =  ⟨ p_x-1r_x+1⟩_ ss-⟨ p_x+1r_x⟩_ ss

for any x ∈{2,…,n-1}. Combining with (<ref>) we
get

    2γ⟨ p_xp_x-1⟩_ ss    = 
    ⟨ p_x+1r_x+1⟩_ ss -  ⟨ p_xr_x⟩_ ss+⟨ p_x+1r_x⟩_ ss-  ⟨ p_xr_x-1⟩_ ss ,

for any x ∈{2,…,n-1}. 
 Summing over x, one gets: 

    ∑_x=2^n-1⟨ p_xp_x-1⟩_ ss = 1/2γ(  ⟨ p_n r_n ⟩_ ss - ⟨ p_2r_2 ⟩_ ss + ⟨ p_n r_n-1⟩_ ss - ⟨ p_2 r_1 ⟩_ ss).

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
<ref>.
 We still need the bounds on the covariances ⟨ p_2r_2 ⟩_ ss,  ⟨ p_n r_n-1⟩_ ss and ⟨ p_2 r_1 ⟩_ ss. 
To deal with it write

    n^-2L(p_0p_1)     = (r_2-r_1)p_0 + r_1 p_1 - 1/2(3γ + γ̃) p_0p_1 
    
    n^-2L(r_1r_2)     = (p_1-p_0)r_2 + (p_2 -p_1) r_1 
    
    n^-2L(p_n-1p_n)     = (τ̅_+-r_n)p_n-1 + (r_n-r_n-1)p_n - 1/2(3γ + γ̃)  p_n-1p_n.

Taking averages  with respect to the stationary state and summing  (<ref>) and (<ref>)
sideways gives (using ⟨ p_2 r_2 ⟩_ ss = ⟨ p_1 r_2 ⟩_ ss from (<ref>))  

    ⟨ p_2 r_2 ⟩_ ss + ⟨ p_2 r_1 ⟩_ ss = ⟨ p_0 r_1 ⟩_ ss + 1/2(3γ + γ̃) ⟨ p_0 p_1 ⟩_ ss0,

 from Proposition <ref>. Moreover, (<ref>) gives (using ⟨ p_n r_n ⟩_ ss = ⟨ p_n-1 r_n ⟩_ ss)  
 
    ⟨ r_n-1p_n ⟩_ ss = τ̅_+ ⟨ p_n-1⟩_ ss - 1/2(3γ+γ̃) ⟨ p_n-1p_n⟩_ ss0,

 from (<ref>) and Proposition <ref>. Therefore,
 we have proved that (<ref>) vanishes as n→∞. In
 fact, due to the estimates obtained in Proposition <ref> we have even proved that there exists a constant C=C(γ,γ̃,τ̅_+,T_+,T_-)>0, such that
 
    |∑_x=1^n ⟨ p_x p_x-1⟩_ ss| ≤C/√(n) ,   n≥1.

From (<ref>) it follows that 

    1/2γ n∑_x=1^n ⟨ r_x^2 ⟩_ ss + γ/2n∑_x=1^n ⟨ p_x^2 ⟩_ ss    = - 1/2 γ n ∑_x=1^n ⟨ p_x p_x-1⟩_ ss - 1/ n∑_x=1^n ϕ(x) + γ/4 n(⟨ p_n^2⟩_ ss - ⟨ p_0^2 ⟩_ ss) 
       1/4( γ^-1 + γ)(T_++T_-)+ τ̅_+^2/4γ,

due to (<ref>) and (<ref>).
This in particular implies estimate (<ref>).



Thanks to the energy bounds, we  are finally able to prove one further
convergence, which will be essential in establishing  Proposition <ref>. 
  
For any continuous test function G:→ we have

    1/n∑_x=1^n-1G(x/n)⟨ p_x p_x+1⟩_ ss 0.

 



Assume first that G∈ C^1(). For the brevity sake 
we denote G_x:=G( x/ n) for any x ∈_n and  ψ(x) = ⟨ p_x+1 r_x+1⟩_ ss + ⟨ p_x+1 r_x⟩_ ss. Then (<ref>)  says that 

    ⟨ p_xp_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∑_x=1^n-1 G_x ⟨ p_x p_x+1⟩_ ss    = 1/2γ n^2∑_x=2^n-2 n (G_x-1-G_x) ψ(x) 
          + 1/n⟨ p_np_n-1⟩_ ss G_n-1 + 1/2γ n( ⟨ p_n r_n ⟩_ ss + ⟨ p_n r_n-1⟩_ ss) 
          - 1/2γ n(⟨ p_2 r_2 ⟩_ ss + ⟨ p_2 r_1⟩_ ss) G_1.

The boundary terms (<ref>) and (<ref>) vanish,
as n →∞, thanks to (<ref>), (<ref>) and
(<ref>). To deal with the sum in the right hand side of (<ref>) 
note that, since G∈ C^1(), we have

    sup_x∈{2,…,n-2}n |G_x-1-G_x|≤G'_∞.

Since 

    |ψ(x)|≤ 2(2⟨ p^2_x+1⟩_ ss+⟨ r_x^2
      ⟩_ ss+⟨ r^2_x+1⟩_ ss),       x∈{1,…,n-1}

we conclude that 

    1/2γ n^2|∑_x=2^n-2 n (G_x-1-G_x)
      ψ(x)|≤C/n{1/n∑_x=1^n ( ⟨ p_x^2 ⟩_ ss+ ⟨ r_x^2 ⟩_ ss)},

which vanishes, as n→+∞, thanks to the energy bound (<ref>). This proves
(<ref>) for any test function G∈ C^1(). The result can
be extended to all continuous functions by the standard density argument and
the energy bound (<ref>).





 §.§ Proof of Proposition <ref>
 
We now have at our disposal all components needed to prove Proposition <ref> and thus conclude the proof of Theorem <ref>.  There are three convergences to prove: 



  §.§.§ Proof of (<ref>)

From Proposition <ref> (in particular
(<ref>)),  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 + n ϕ(1)-ϕ(n)/n-1,     for any  x ∈_n.

Since, from Proposition <ref>, 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 ∈_n G(x/n) ϕ(x) ∫_   G(u) { -τ̅_+^2/2γ  u - 1/2(γ^-1+γ)[(T_+-T_-)u+ T_-] }  u.

which proves directly (<ref>).



  §.§.§ Proof of (<ref>)

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_n^2 ⟩_ ss - G_1/n⟨ p_0^2 ⟩_ ss + 1/n^2∑_x=1^n-1 n(G_x-G_x+1) ⟨ p_x^2 ⟩_ ss).
 
The boundary terms in the right hand side vanish, as n→+∞,
since ⟨ p_n^2⟩_ ss and ⟨ p_0^2 ⟩_ ss are
bounded, due to (<ref>). To deal with the limit of the last sum in the right
hand side, we can repeat the argument made in (<ref>)-(<ref>),
which shows that the expression vanishes. Thus 
(<ref>) holds. 



  §.§.§ Proof of (<ref>)
 
This is a consequence of  (<ref>).














§ NON-STATIONARY BEHAVIOUR





In this section we explain how to derive (<ref>)
  and (<ref>): while the derivation of (<ref>)
is rigorous, in order to obtain (<ref>) we need to assume a form of local equilibrium that
allows for the local equipartition of  kinetic and potential energy,
see (<ref>) below. In the stationary setting this term corresponds to H_n^m(G) in (<ref>) and, similarly, does not appear in the
case =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 τ̅_+(t) to be a C^1 function.





 §.§ Preliminaries







 In the present section we establish non-stationary asymptotics
corresponding to Corollary <ref>. They will be
useful in proving the hydrodynamic limit in Section <ref>.
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) ν_T_+
 is not strictly decreasing in time, where hereafter f_n(t) is
the density of the _n–valued  random variable
(𝐫_n(t),𝐩_n(t)) (recall (<ref>)), 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 <cit.> for the details of the argument)

    H_n(t) ≤  C(t) n,   n≥ 1, t≥0.

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 (<ref>),

    ∫_0^t  s [ γ∑_x=0^n-1 D_x (f_n(s)) + 
     γ̃T_-  (∂_p_0 (f_n(s)/h))^2/(f_n(s)/h)_T_-+ 
    γ̃T_+  (∂_p_n f_n(s))^2/f_n(s)_T_+] 
    ≤C(t)/n,

where h = g_T_-/g_T_+  and n≥1, t≥0,  see
  Proposition 4.1 and Appendix D of <cit.> for the
  proof. 









Below we list some consequences of the
  above bounds on the entropy and Dirichlet forms.

 The following equalities hold:
  
    lim_n→∞𝔼[   ∫_0^t  (p_n^2(s) - T_+) s]  = 0,    lim_n→∞𝔼[   ∫_0^t  (p_0^2(s) - T_-) s]  = 0

and
 
    lim_n→∞𝔼[   ∫_0^t  j_n-1,n(s)  s]  = 0,    lim_n→∞𝔼[   ∫_0^t  j_0,1(s)  s]  = 0.





  Note that, 
  
    𝔼[   ∫_0^t  (p_n^2(s) - T_+) s]      =
          ∫_0^t   s ∫(p_n^2- T_+) f_n(s) ν_T_+
        =
          T_+  ∫_0^t   s ∫ p_n∂_p_n f_n(s) ν_T_+ .

Thus, by the Cauchy-Schwarz inequality

    |𝔼[   ∫_0^t  (p_n^2(s) - T_+) s]|
     
    ≤ T_+ ( ∫_0^t   s ∫ p_n^2 f_n(s) ν_T_+)^1/2( ∫_0^t   s ∫( ∂_p_n f_n(s) )^2/f_n(s)ν_T_+)^1/2.

  By the entropy inequality, see e.g. 
  <cit.>, we can write (recall (<ref>))

    ∫ℋ_n (𝐫, 𝐩)  f_n(s, 𝐫, 𝐩) ν_T_+ (𝐫, 𝐩) 
    ≤1/{log[∫exp{ℋ_n (𝐫, 𝐩) }ν_T_+ (𝐫, 𝐩)]+H_n(s)}

for any >0.
 By (<ref>), for any t>0 and a sufficiently small >0, there exists C>0
 such that
  
    sup_s∈[0,t]∫ℋ_n (𝐫, 𝐩)  f_n(s, 𝐫, 𝐩) ν_T_+ (𝐫, 𝐩) ≤  Cn,   n≥ 1.

  Consequently, by (<ref>) and (<ref>), there exists C>0
  such that 
  
    sup_s∈[0,t]|𝔼[  ∫_0^t  (p_n^2(s) - T_+)
       s]| ≤ C,   n≥1.

 Hence, in particular we obtain
  
    ∫_0^t  s ∫ p_n^2 f_n(s) ν_T_+≤ C.

  Using this estimate in  (<ref>) together with  (<ref>)
  we
  conclude that for any t≥0 there exists C>0 for which
  
    |𝔼[   ∫_0^t  (p_n^2(s) - T_+) s]|
      ≤C/√(n),   n≥1.

 Hence the first equality of (<ref>) follows.
 The proof of the second equality of  (<ref>) is similar.



Concerning (<ref>): from the energy
 conservation it follows that
 
    n^-2 L ℰ_n = j_n-1,n+γ̃/2  (T_+ - p_n^2) + τ̅_+(t) p_n.

To deal with the  term τ̅_+(t) p_n, note that

    |∫_0^t τ̅_+ (s)  𝔼[p_n(s)] 
       s|    ≤τ̅_+_∞∫_0^t |∫ p_n f_n(s)
       ν_T_+| s 
        =τ̅_+_∞/T_+∫_0^t|∫∂_p_n f_n(s) ν_T_+| s
     
       ≤τ̅_+_∞/T_+∫_0^t (∫  f_n(s)
       dν_T_+)^1/2( ∫( ∂_p_n
           f_n(s) )^2/f_n(s)ν_T_+)^1/2 s 
       ≤τ̅_+_∞/T_+C/√(n)⟶_n→∞ 0,

by virtue of (<ref>).

 The first equality of (<ref>) is then a direct consequence of
 (<ref>). The same argument works for j_0,1. 


Using the energy conservation it follows immediately:

The currents, defined in (<ref>) and
  (<ref>),  satisfy
  
    lim_n→∞𝔼[   ∫_0^t  j_x,x+1(s)
           s]  = 0,    x=-1,…,n,   t≥0.





Concerning the potential energy at the boundary
  points we have the following bound.

  
  There exists a  constant C<∞ such that
  
    𝔼[ ∫_0^t (r_1^2(s) + r_n^2(s))   s] ≤ C,   n≥1.



 Using (<ref>) we get

    n^-2[p_n(t) r_n(t)-p_n(0) r_n(0)]
       = ∫_0^t [p_n(s)(p_n(s) - p_n-1(s)) + (τ̅_+(s)-r_n(s)) r_n(s) - 1/2(γ̃+γ)
        p_n(s) r_n(s) ] s.

The term in the left hand side vanish, as n→+∞, due to
estimate (<ref>).
 We can repeat then
  the same arguments as we have used to obtain 
  (<ref>) and conclude that there exists C>0 such that
  
    𝔼[ ∫_0^t r_n^2(s) s]  ≤   C
      {𝔼[ ∫_0^t p_n^2(s) s] +1},  
      n≥ 1.
 
Estimate (<ref>) can be used to obtain the desired bound for 𝔼[ ∫_0^t r_n^2(s)   s]. An analogous  estimate on 𝔼[ ∫_0^t r_1^2(s)   s] follows from the same argument, using (<ref>)
    and the second equality in (<ref>) instead.



  The following convergences hold: at the left boundary point:
  
    𝔼[ ∫_0^t p_0(s) r_1(s)  s]    
                                                                                      0 
    𝔼[ ∫_0^t  (p_1^2(s) - p_0^2(s)) s]    
                                                                                      0 
    𝔼[  ∫_0^t r_1(s) r_2(s)  s]       0

and at the right boundary point:

    𝔼[ ∫_0^t p_n-1(s) r_n(s)  s]    
                                                                                      0 
    𝔼[ ∫_0^t  (p_n-1^2(s) - p_n^2(s)) s]      0 ,
    𝔼[  ∫_0^t  ( r_n-1(s) r_n(s) - τ̅_+^2(s)) s]     0 .



Since
  n^-2Lr_n^2 = 2 r_n p_n - 2r_n p_n-1,
 using
  (<ref>) we conclude that (<ref>) holds, provided
  that we can prove

    𝔼[ ∫_0^t r_n(s) p_n(s)  s]    
                                                                                      0.

 The latter is a consequence of the following estimate, cf. (<ref>), 
  
    |𝔼[ ∫_0^t r_n(s) p_n(s)  s ]|
              = |∫_0^t  s ∫ r_n ∂_p_n f_n(s)
            ν_T_+| 
       ≤(∫_0^t  s ∫ r_n^2 f_n(s) ν_T_+)^1/2( ∫_0^t   s ∫( ∂_p_n f_n(s) )^2/f_n(s)ν_T_+)^1/2
       ≤(∫_0^t  s ∫ r_n^2 f_n(s) ν_T_+)^1/2C/√(n) 0.

  The proof of (<ref>) is similar.


  To show (<ref>)
note that (see (<ref>)) 

    γ∫_0^t  [p_n-1^2 (s)- p_n^2(s)]  s =       2
      ∫_0^t τ̅_+ (s) [ p_n (s)]   s - 2
      ∫_0^t  [ r_n(s) p_n(s) ]  s
       
    +  γ̃∫_0^t  [T_+-p_n^2(s) ]  s -1/n^2[p_n^2(t)-p_n^2(0)] .

 The second, third and fourth terms in the right hand side vanish due
 to 
  (<ref>), Lemma <ref> and (<ref>), respectively.
The first term has been already treated in (<ref>). An analogous argument, starting from
(<ref>) allows us to prove (<ref>).








  Besides, we have
  
    𝔼[ ∫_0^t p_n-1(s) p_n(s)
            s] 
     =𝔼[ ∫_0^t p_n-1(s) (p_n(s)-T_+)
            s]+T_+𝔼[ ∫_0^t p_n-1(s)  s]
           0.

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[ ∫_0^t p_n-1^2(s)  s] ≤ C,    n≥1
 (it follows from the already proved (<ref>)) and Lemma
<ref> to prove that it vanishes, as n→+∞.
To show that the second term vanishes we can use estimates analogous
to (<ref>). The argument for (<ref>) follows essentially
the same lines.

  By (<ref>) we can now write
  
    𝔼[  ∫_0^t  (r_n^2(s) - T_+ -
         τ̅_+^2(s)) s] =     ∫_0^tτ̅_+ (s)
         𝔼[r_n(s) -τ̅_+ (s)] s 
         
         +   ∫_0^t𝔼[  p_n^2 (s)-
         T_+  ]  s 
    - ∫_0^t𝔼[  p_n (s)p_n-1(s)
         ]  s 
        - 1/2 (γ̃+ γ) ∫_0^t𝔼[  p_n (s)r_n(s) ]  s 
        - 1/n^2𝔼[   p_n(t) r_n(t)-p_n(0) r_n(0)].

 By the previous results, the last four terms in the right hand side
 vanish, as n→+∞. Concerning the first term we use

    n^-2Lp_n(s)=τ̅_+ (s)-r_n(s) -1/2 (γ̃+ γ)p_n(s)

to conclude that

    ∫_0^tτ̅_+ (s)
         𝔼[r_n(s) -τ̅_+ (s)] s=       1/2
         (γ̃+ γ)∫_0^tτ̅_+ (s)𝔼[p_n (s)]
         s  
        - 1/n^2∫_0^tτ̅_+ (s) / s𝔼[ p_n(s)] s.

The first term vanishes, as n→+∞, by  (<ref>). By
integration by parts  the second term equals

    1/n^2∫_0^tτ̅_+' (s) 𝔼[ p_n(s)] s-1/n^2(τ̅_+ (t) 𝔼[ p_n(t)]-τ̅_+ (0) 𝔼[
        p_n(0)])

which also vanishes, thanks to (<ref>) and (<ref>). Therefore,
  
    𝔼[  ∫_0^t  (r_n^2(s) - T_+ - τ̅_+^2(s)) s] ⟶ 0.

To see  (<ref>) 
it suffices to show that

    𝔼[  ∫_0^t  (r_n^2(s) - r_n(s) r_n-1(s)-T_+) s] ⟶ 0.

For that purpose we invoke (<ref>), which permits to write

    𝔼[ ∫_0^t   ((r_n-1(s)-r_n(s))r_n(s)+T_+)  s]
    =𝔼[ ∫_0^t   (p_n^2(s)-p_n-1^2(s))
      s] 
           
    +𝔼[ ∫_0^t   p_n(s)p_n-1(s) 
      s]  +𝔼[ ∫_0^t   (T_+-p_n^2(s))
      s]
           
    -
    γ𝔼[ ∫_0^t r_n(s)p_n-1(s) s] -1/n^2[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 (<ref>). 




 §.§  Hydrodynamic limit




Let us now turn to equation (<ref>), which can be formulated in a weak form as:

    ∫_0^1  u  G(u) (r(t,u) - r(0,u))  
     = 1/γ∫_0^t  s
      ∫_0^1     u   G”(u) r(s,u)  -  1/γ  G'(1) ∫_0^t τ̅_+(s)  s
      ,

for any  test function G∈ C^2_0,1() – 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
(<ref>) we have  (recall that G_x=G(x/n)) 

    [  1/n∑_x=1^n 
        G_x(r_x(t) -  r_x(0)) ]    
        = [ ∫_0^t  s  n ∑_x=1^n 
       G_x (p_x(s) -  p_x-1(s)) ]
        = [  ∫_0^t  s { -∑_x=1^n-1 
        (∇_n G)_x p_x(s) - n
        G_1 p_0(s)}],

where (∇_n G)_x:= n(G_x+1 - G_x). 
Using 
(<ref>) we can write (<ref>) as

    [ -∫_0^t   s  {∑_x=1^n-11/γ (∇_n
          G)_x(r_x+1(s) - r_x(s)) +1/2(γ+ γ̃) n
          G_1 r_1(s) }]  
       +[ 1/γ n^2∑_x=1^n-1 (∇_n
          G)_x(p_x(t) - p_x(0)) + 1/2(γ +γ̃)n^2   
        n G_1(p_0(t) - p_0(0)) ].

Since G is smooth, nG_1 → G'(0) and (∇_nG)_x→ G'(x), as n→ +∞.
Using this and  (<ref>), one shows that the expression
(<ref>) converges to 0.
The only significant term  is therefore the first one (<ref>). 
Summing by parts, using the notation 

    (Δ_n G)_x:=
    n^2(G_x+1+G_x-1-2G_x)
 and recalling  that G(0)=0, it can be rewritten as

    [ ∫_0^t  1/γ{1/n∑_x=2^n-1
          (Δ_n G)_x  r_x(s) - (∇_n G)_n-1  r_n(s) } s]
    
          - (1/2(γ+ γ̃)  
          (∇_n G)_0 -1/γ  (∇_n G)_1 )
       [∫_0^t r_1(s)  s].

It is easy to see,  using (<ref>), that 

    lim_n→+∞[∫_0^t r_1(s)  s]     =
     +/2lim_n→+∞∫_0^t   s∫
     p_1f_n(s) ν_T_+
       
    =
     +/2T_+lim_n→+∞∫_0^t   s∫∂_p_1f_n(s) ν_T_+ =0,

by (<ref>). Using (<ref>) we obtain also

    lim_n→+∞[∫_0^t r_n(s)  s]=∫_0^t
     τ̅_+(s)  s.

Therefore, we can rewrite (<ref>) as

    ∫_0^t  s{1/γ∫_0^1G”(u) r^(n)(s,u) u- G'(1) τ̅_+(s)
    }
    +o_n(t),

 where    

    r̅^(n)(t,u)=[r_x(t)]   u∈[  x n; x+1 n ),
        n≥1
 and lim_n→+∞sup_s∈[0,t]o_n(s)=0.
Thanks to (<ref>) we know that
there exists R>0 such that

    sup_n≥1sup_s∈[0,t]r̅^(n)(s,·)_L^2()=:R<+∞.

The above means that   for each s∈[0,t] the sequence {r̅^(n)(s,·)}_n≥1 is contained in B̅_R – the closed ball of radius R>0 in L^2(), centered at 0.
 The ball  is compact in L^2_w() –
the space of square integrable functions on  equipped
with the weak L^2 topology.  The  topology restricted to B̅_R is metrizable.
From the above argument it follows in particular that  for each t>0 the sequence  {r̅^(n)(·)} is equicontinuous 
 in C([0,t],B̅_R). Thus, according to the Arzela theorem, see e.g. <cit.>, it is
sequentially pre-compact in the space C([0,t],L^2_w()) for any t>0. Consequently, any limiting point of the
sequence satisfies  
(<ref>).  



 Concerning equation (<ref>) with the respective boundary
  condition, its weak
  formulation is as follows:
for any 
G∈ L^1([0,+∞);C_0,1^2()) we have

    0=     ∫_0^1 G(0,u) e(0,u)  u 
         +
     ∫_0^+∞ s ∫_0^1 (∂_sG(s,u) + 1/2 (γ^-1+γ)
        ∂_u^2G(s,u) )e(s,u)   u 
        + 1/4(γ^-1 - γ) ∫_0^+∞ s ∫_0^1 ∂_u^2 G(s,u)
       r^2(s,u)   u 
        -∫_0^+∞(∂_uG(s,1) [(γ^-1+γ) T_+
           + 1/4(γ^-1 -γ) τ̅_+(s)^2]
         - ∂_uG(s,0) T_-)  s.

Given a non-negative initial data e(0,·)∈ L^1() and
the macroscopic stretch r(·) (determined via (<ref>)) one can easily
    show that the respective weak formulation of the boundary value
    problem for a linear heat equation, resulting from (<ref>),
    admits a unique measured value solution.
 
The  averaged thermal energy  function is defined as 

    ℰ̅_n(t,u):=
    [ℰ_x(t)],   u∈[xn,x+1n),   x=0,…,n-1.

It is easy to see, thanks to (<ref>), that (ℰ̅_n(t))_n≥1
is  bounded in the dual to the separable Banach
space  L^1([0,+∞);C_0,1^2()).
Thus it is ⋆-weakly compact. In what follows we identify its limit e(t)
by showing  that it satisfies (<ref>).
To achieve this goal we are going to use the microscopic energy currents given in (<ref>). 

 Consider now a smooth test function G∈ C_0^∞([0,+∞)×)  such that G(s,0) = G(s,1) ≡ 0, s≥0.
Then,  from (<ref>), we get 

    - 1/n𝔼[∑_x=0^n  G_x (0) ℰ_x(0) ] = 
            -       1/n𝔼[ ∑_x=1^n-1 G_x (0)ℰ_x(0) ] 
    =- 1/n∫_0^t  s 𝔼[∑_x=0^n-1∂_sG_x (s) ℰ_x(s) ]
        + ∫_0^t  s 𝔼[∑_x=1^n-2 (∇_n G)_x(s)  j_x,x+1(s) - nG_n-1(s) j_n-1,n(s)
         + nG_1(s) j_0,1(s) ] .

Here G_x(s):=G(s,x/n) and we use a likewise notation for
∇_nG_x(s), Δ_nG_x(s).


Concerning (<ref>):  by (<ref>), the expectation of its last two terms  are negligible.
In order to treat the first term of (<ref>),
we use the fluctuation-dissipation relation  (<ref>), i.e.

    j_x,x+1 = n^-2 L g_x + ∇ V_x,

 with
  
    V_x= -1/2 γ  r_x^2 - γ/4(p_x^2 + p_x-1^2) - 1/2 γ p_xp_x-1 .

Using this relation we can rewrite the last term (<ref>) as

    ∫_0^t  s 𝔼[ 1/n∑_x=2^n-2
                                                      (Δ_n G)_x (s)V_x(s) 
                                                     -  (∇_n G)_n-2 (s)V_n-1(s) 
                                                     +  (∇_n G)_1(s)  V_1(s)]

plus expressions involving the average fluctuating term

    ∫_0^t  s  𝔼[ 1/n^2∑_x=1^n-2 (∇_n G)_x(s)     L g_x(s)] 
    
    =     𝔼[ 1/n^2∑_x=1^n-2((∇_n
      G)_x(t) g_x(t) -(∇_n G)_x(0) g_x(0))]
       
    -∫_0^t  s 𝔼[ 1/n^2∑_x=1^n-2 (∇_n ∂_sG)_x(s) g_x(s)]

which turns out to be small, as n→+∞, thanks to the energy bound (<ref>).

By Lemmas <ref> and <ref> we have:

    lim_n→∞𝔼[ ∫_0^t  s  (∇_n G)_1(s) V_1(s) ] =
        -  ∫_0^t   s ∂_u G(s,0)1/2(γ^-1+γ) T_- ,

which takes care of the left boundary condition. Concerning the right
one we have

    𝔼[ ∫_0^t  s   (∇_n G)_n-2 (s)
         V_n-1(s) ] =     
    - 𝔼[ ∫_0^t  s   (∇_n G)_n-2 (s) ∇ V_n-1(s)
     ] 
        +  𝔼[ ∫_0^t  s   (∇_n G)_n-2 (s)
         V_n(s) ].

Using again the results of 
Lemmas <ref> and <ref> we conclude that the limit of
the second term  (<ref>) equals

    - ∫_0^t   s ∂_u G(s,1) (1/2(γ^-1+γ) T_+
         + 1/2γτ̅_+^2(s)).

On the other hand, using (<ref>), the term (<ref>)
equals

    1/n^2𝔼[ ∫_0^t  s   (∇_n
         G)_n-2 (s) L g_n-1(s)
     ]
       +  ∫_0^t  s   (∇_n G)_n-2 (s)
      𝔼[j_n-1,n(s)].

From (<ref>) we conclude that the second term vanishes, with
n→+∞. By integration by parts the first term equals

    1/n^2𝔼[ (∇_n
      G)_n-2 (t) g_n-1(t) -(∇_n
      G)_n-2 (0) g_n-1(0)
     ]  
          -1/n^2𝔼[ ∫_0^t  s   (∇_n
         ∂_s G)_n-2 (s) g_n-1(s)
     ],

which vanishes, thanks to (<ref>). Summarizing we have shown that

    lim_n→+∞𝔼[ ∫_0^t  s   (∇_n G)_n-2 (s)
         V_n-1(s) ] 
      =
       - ∫_0^t   s ∂_u G(s,1) [1/2(γ^-1+γ) T_+
         + 1/2γτ̅_+^2(s)].

Now, for the bulk, it follows from (<ref>) and
  (<ref>) that

    n^-2 L h_x = ∇ W_x-2 p_x p_x-1,   x=2,…,n-1,

with

    h_x:=p_xp_x-1-r_x^2/2,     W_x:=r_x-1 p_x.

Therefore by
(<ref>) and an argument similar to the one used above we conclude that

    lim_n→∞∫_0^t  s 𝔼[1/n∑_x=2^n-2
                                                      (Δ_n G)_x (s) p_x(s)p_x-1(s) ]= 0.

In the case γ = 1 we can rewrite

    V_x= - ℰ_x  + 1/4(p_x^2 - p_x-1^2) - 1/2  p_xp_x-1

so that it is easy to see that (<ref>) is equivalent to

    -∫_0^t  s 𝔼[1/n∑_x=2^n-2  (Δ_n G)_x (s)  ℰ_x (s) ]
      + o_n(1),

closing the energy conservation equation and concluding the proof.



Finally, for γ≠ 1
we expect that the following term to vanish as n→ +∞:

    ∫_0^t  s  𝔼[1/n∑_x=2^n-2 (Δ_n G)_x  (p_x^2(s)
        - (r_x(s)-[ r_x(s)])^2)],

  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 <cit.>,
by studying the evolution of the Wigner distribution of the thermal energy in Fourier coordinates.













amsalpha
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