FOURIER’S LAW FOR A MICROSCOPIC MODEL OF HEAT CONDUCTION

We consider a chain of N harmonic oscillators perturbed by a conservative stochastic dynamics and coupled at the boundaries to two gaussian thermostat at different temperatures. The stochastic perturbation is given by a diffusion process that exchange momentum between nearest neighbor oscillators conserving the total kinetic energy. The resulting total dynamics is a degenerate hypoelliptic diffusion with a smooth stationary state. We prove for the stationary state, in the limit as N →∞, the Fourier’s law and the linear profile for the energy average.


Introduction
In insulating crystals heat is transported by lattice vibrations, and since the pioneering work of Debye, one-dimensional chains of anharmonic oscillators have been used as microscopic models for heat conduction (for a review cf. [13] and [5]). These chains are then connected at the extremities to two thermostats at different temperatures. Non-linear effects are extremely important in order to obtain finite conductivity. Enough strong non-linearity causes scattering between phonons and should imply a sufficiently fast decay of correlations for heat currents. In fact it is well known that harmonic chains, because of their infinitely many conserved quantities, have infinite conductivity and do not obey Fourier's law (cf. [15]). On the other hand a rigorous treatment of a nonlinear chain, even the proof of the existence of the conductivity coefficient, is out of reach of current mathematical techniques.
In the present paper we study a model of a chain of harmonic oscillators where the hamiltonian dynamic is perturbed by a random continuous exchange of kinetic energy between nearest neighbors oscillators. This random exchange conserves the total kinetic energy and destroy all other conservation laws. In this sense it simulates the long time effect of the non-linearities in the deterministic model. This random exchange on kinetic energy is realized by a diffusion on the circle of constant kinetic energy of the nearest neighbor oscillators. We expect the same macroscopic behavior and results if this diffusions are replaced by jump processes.
The interaction with the reservoirs are modeled by Ornstein-Uhlenbeck processes at the corresponding temperatures. It results that the total dynamics of system is a degenerate hypoelliptic diffusion on the phase space. By Hörmander theorem this process has a smooth stationary state. This stationary state is product and gaussian only if the temperatures of the thermostats are equal (equilibrium).
We prove that in the stationary state a Fourier's law is valid for the energy flow and that the total energy of the system is proportional to the size of it. Then we prove a linear profile for the energy. A corresponding law of large number (hydrodynamic limit) should be valid for this system, but at the moment we have not been able to prove this.
The macroscopic evolution of the dynamical fluctuation in equilibrium for the corresponding infinite model, have been proven in a companion paper (cf. [10]).
With similar motivations other stochastic models have been proposed before. In 1970, Bosterly, Rich and Visscher (cf. [3]) considered a chain of harmonic oscillators where each oscillator is also connected to an interior bath, modeled, like the boundary terms, by Ornstein-Uhlenbeck processes. The temperature of each bath is then chosen in a self-consistent way. The Fourier's law and the linear profile of temperature for this model in the steady state have been proven recently by Bonetto, Lebowitz and Lukkarinen (cf. [4]). There are two main difference between this model and ours. In the Bosterly, Rich and Visscher model, energy is not conserved by the bulk dynamics, event though the temperatures of the internal baths are regulated so that the average flow of energy between the oscillators and the internal baths is null. In our system the bulk dynamic conserves energy, and only the boundary reservoirs can change the total energy. The second difference is that the dynamic of the Bosterly, Rich and Visscher model is linear, and consequently the stationary state is fully gaussian. Fourier's law, linear profile of temperatures and other result can be then obtained by computing the limit of the 2-point correlations of the stationary state. The stochastic perturbation we consider is intrinsically non-linear and the stationary state is non-gaussian (except in the equilibrium case).
Another model has been introduced in 1982 by Kipnis, Marchioro and Presutti (cf. [11]) where the energy is microscopically conserved but the hamiltonian part of the dynamics is removed. The dynamics consist here only in a random exchange of energy between nearest-neighbor oscillators, given by properly defined jump processes. The striking duality properties of this process make it explicitly solvable, and in [11] Fourier's law and linear profile of temperature are proven. Recently a deterministic hamiltonian model has been proposed in [8] where, in a proper high temperature limit and under a chaoticity assumption, the model of Kipnis, Marchioro and Presutti can be recovered.
The main tool we use in our proof is a bound of the entropy production of the bulk dynamics. This tool has been successful in the analogous problem of Fick's law in some lattice dynamics (cf. [7], [12]).
One of the main difficulties in proving Fourier's law and hydrodynamic limit is to establish a fluctuation-dissipation relation, i.e. a decomposition of the current of the conserved quantity (here the energy) in a dissipative part (a spatial gradient) and a fluctuating part (a time derivative). Thanks to the stochastic perturbation one can write here an exact fluctuation-dissipation relation (cf. equation (27)). Then, in order to obtain the Fourier's law, we have to bound (uniformly in the size of the system) the second moment of the positions and velocity at the boundary. It results that we can bound the second moments of all the coordinates, that gives a bound of the expectation of the total energy proportional to the size of the system.

The model
Atoms are labeled by x ∈ {1, . . . , N − 1}. Atom 1 and N − 1 are in contact with two separate heat reservoirs at two different temperatures T l and T r . The interaction between the reservoirs is modeled by two Ornstein-Uhlenbeck processes at the corresponding temperatures. The moments of the atoms are denoted by p 1 , . . . , p N −1 and the positions by q 1 , . . . , q N −1 . The distances between the positions are denoted by r 1 , . . . , r N −2 , where r x = q x+1 − q x . The hamiltonian of the system that represents the total energy inside the system is given by (1) The dynamics is described by the following system of stochastic differential equations: where γ > 0 is a parameter that regulates the strength of the random exchange of momenta between the nearest neighbor particles. Observe that by translating r x in r x −ρ one has the same equations for the new coordinate but with ρ = 0. So we set ρ = 0 without any loss of generality.
The generator writes as One can check easily that the Lie algebra generated by these vector fields and the hamiltonian part of L N has full rank at every point of the state space R N −1 × R N −2 . By Hörmander theorem it follows that this operator is hypoelliptic (cf. thm 22.2.1 in [9]), so the stationary measure has a smooth density. We denote with < · > the expectation with respect to the stationary measure. The existence of a unique stationary measure can be proved similarly as in [16] or in [6].
Energy is conserved by the bulk part of the dynamics and we have Consequently j x,x+1 is called instantaneous current of energy. Because of stationarity, for any x = 1, N − 1 we have The following theorems are the main results of this paper.
Theorem 1. For any γ > 0 Furthermore there exists a constant C depending only on γ, T l and T r such that where T (q) = T l + (T r − T l )q is the linear profile interpolating T l and T r .

Entropy production
Denote by g Tr (p 1 , r 1 , . . . , p N −2 , r N −2 , p N −1 ) the density of the product on gaussians with mean 0 and variance T r . We denote by f N the density of the stationary measure with respect to g Tr . By hypoellipticity this density is smooth.
By stationarity we have where L l = (T l ∂ 2 p 1 − p 1 ∂ p 1 ). Define h = g T l /g Tr , then we can rewrite the last term as So by (29) we have the following bound In section 4, we prove that this last expression is bounded by CN −1 for some constant C (cf. (29) and (6)). This relation also gives us the right sign for the energy current, i.e. if T l < T r we have < j x,x+1 >< 0.

Some bounds
From (6) and (7) we have Since this last is equal to < j 0,1 >, using (15), we obtain Then by Schwarz inequality there exists a constant C, depending only on γ, such that (18) < p 2 2 >≤ C < r 2 1 > + < p 2 1 > Analogous computation for the index x = N − 2 gives Observe now that and by use of (17) (22) and by Schwarz inequality, for any α > 0 choosing properly α one obtains a constant C depending only on γ, such that (24) < r 2 1 >≤ C < p 2 1 > and an analogous bound is obtained for < r 2 N −2 >. Putting all together we have obtained the following lemma: Lemma 1. There exists a constant C depending only on γ and linearly on T l and T r such that (25) The bulk dynamics is only apparently non-gradient since defining where the discrete gradient ∇ of a discrete function w is defined by (∇w)(x) = w(x + 1) − w(x). Using again (7) we have and by (25) we obtain that there exists a constant C depending only on T l , T r and γ such that

Fourier's law
Proof. Let us prove the case x = 1, for x = N − 2 the proof is similar. By (13), (29) and (25) < r 1 p 2 >=< r 1 p 1 >= r 1 p 1 (f N /h)g T l dp dr = T l r 1 ∂ p 1 (f N /h)g T l dp dr The proof for < p 1 p 2 > is similar. Now by (29) for x = 1 we have Then by (21) i.e. the law of Fourier.

Energy bound
We claim now there exists a constant C > 0 independent of N such that By (5) and (27), we have Here, (∆w)(x) = w(x+1)+w(x−1)−2w(x) is the usual discrete Laplacian of the function w(x). By (25) and the maximum principle, it follows that there exists a constant C independent of N such that Hence, by Schwarz inequality and (39), we get (41) In fact, by (25), we can extend this inequality for x ∈ {1, . . . , N − 2} (with a slight modification of the constant C): Using the trivial inequality √ z ≤ z/4 + 1 valid for any z, we obtain We prove here Theorem 2 for γ = 1. In this case we have φ(x) =< r 2 By (30) and (33), lim N →∞ φ(1) = 2T l and lim N →∞ φ(N − 2) = 2T r . This gives the boundary conditions for the Laplace equation (38). It follows that for any q ∈ [0, 1] Then, in order to prove (10), we are left to prove Since we have the uniform bound on the energy (44), we can assume that G has a compact support in (0, 1) (meaning we can forget the boundary terms in the following discussion). By equation (29), we have Remark that we have forget the boundary terms in the discrete integration by parts since G vanishes at the boundaries. Since G is continuously diffrentiable and because of (44), we have hence We recall now the following trivial computations (valid in the bulk) Using the second equation of (49) and the equation (48), we have: Hence, by the first equation of (49), (48) and (50), we have just to prove In the same way, the last equation of (49) says that It follows by (52) that (54) the last equality is the consequance of the discrete integration by parts and the differentiability of G. We have hence proved (51) and we are done.

Open problems and other models
We have proven for our stochastic model the Fourier law for any value of the coupling γ and the linear profile of the energy for the case γ = 1. The essential tool used has been a bound on the entropy production. This bound on the entropy production together with a uniform bound on < p 2+δ x > will provide a proof of the linear temperature profile and of local equilibrium for any value of γ. Unfortunately we have not been able to prove yet such uniform bound of the higher moments of the velocities, but we conjecture that it is certainly satisfied.
A generalization in more dimension looks like a difficult problem, since the decomposition of the current in a gradient plus a time derivative (cf. equation (27)) will be much more complex, involving non-local functions.
The proof we have exposed in the present paper can be adapted for some modification of the model. For example one can add a pinning given by on site harmonic potential, adding to the hamiltonian a term N −1 x=1 ν 2 q 2 x /2. Or adding stochastic reservoirs like in the model of Bolsterli-Rich-Visscher (cf. [3] and [4]) with self consistent temperatures, i.e. we can add to the generator a term where the temperatures T x are imposed to be equal to < p 2 x >. In this case we find that the self-consistent profile T x is asymptotically linear and the Fourier law is given by (55) lim N →∞ N < j x,x+1 >= 1 2(γ + λ) + γ 2 (T l − T r ).
which, in the limit as γ → 0 is in agreement with the results of Bonetto-Lebowitz-Lukkarinen (cf. [4]). The proof of (55) is very close to the one exposed in sections 3, 4, 5. In fact one has the decomposition of the current in the form ∇φ x + L N h x with the function h x given by (56) h x = 1 2(γ + λ) p x+1 (r x + r x+1 ) + 1 4 p 2 x+1 , x = 1, . . . , N − 3 Observe that this works also in the case γ = 0 if λ > 0. In this last model one can also prove local equilibrium by proper use of Log-Sobolev inequalities and the entropy production bound, similarly as done in [14]. In the case γ = 0 and in presence of pinning, local equilibrium is proved in [4].
One can also consider different stochastic perturbations that conserve energy and also momentum (as proposed in [2] and [8]). We prove in [2] that also these momentum conserving models have finite conductivity, i.e. the average energy current decrease like 1/N .