O. Ajanki-and-f and . Huveneers, Rigorous Scaling Law for the Heat Current in Disordered Harmonic Chain, Communications in Mathematical Physics, vol.68, issue.3, pp.841-883, 2011.
DOI : 10.1007/s00220-010-1161-1

A. Anantharaman, R. Costaouec, C. Le-bris, F. Legoll, and A. F. Thomines, Introduction to numerical stochastic homogenization and the related computational challenges: some recent developments, in Multiscale modeling and analysis for materials simulation, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., World Sci. Publ, vol.22, pp.197-272

V. I. Arnold and A. Avez, Ergodic problems of classical mechanics, Translated from the French by, 1968.

G. Basile, C. Bernardin, and A. S. Olla, Thermal Conductivity for a Momentum Conservative Model, Communications in Mathematical Physics, vol.28, issue.1, pp.67-98, 2009.
DOI : 10.1007/s00220-008-0662-7

G. Basile and S. Olla, Energy Diffusion in Harmonic System with Conservative Noise, Journal of Statistical Physics, vol.8, issue.6, 2013.
DOI : 10.1007/s10955-013-0908-4
URL : https://hal.archives-ouvertes.fr/hal-00839712

G. Benabou, Homogenization of Ornstein-Uhlenbeck Process in Random Environment, Communications in Mathematical Physics, vol.233, issue.3, pp.699-714, 2006.
DOI : 10.1007/s00220-006-0046-9

A. Bensoussan, J. Lions, and A. G. Papanicolaou, Asymptotic analysis for periodic structures, 2011.

P. G. Bergmann and J. L. Lebowitz, New Approach to Nonequilibrium Processes, Physical Review, vol.99, issue.2, pp.99-578, 1955.
DOI : 10.1103/PhysRev.99.578

C. Bernardin, Hydrodynamics for a system of harmonic oscillators perturbed by a conservative noise, Stochastic Process, Appl, vol.117, pp.487-513, 2007.

C. Bernardin-and-p and . Gonçalves, Anomalous Fluctuations for a Perturbed Hamiltonian System with Exponential Interactions, Communications in Mathematical Physics, vol.126, issue.2, pp.291-332, 2014.
DOI : 10.1007/s00220-013-1860-5

C. Bernardin, P. Gonçalves, and A. M. , JARA, 3/4-fractional superdiffusion in a system of harmonic oscillators perturbed by a conservative noise, 2014.

C. Bernardin-and-f and . Huveneers, Small perturbation of a disordered harmonic chain by a noise and an anharmonic potential, Probab. Theory Related Fields, pp.301-331, 2013.

C. Bernardin, V. Kannan, J. L. Lebowitz, and A. J. Lukkarinen, Harmonic Systems with Bulk Noises, Journal of Statistical Physics, vol.8, issue.3, 2011.
DOI : 10.1007/s10955-011-0416-3
URL : https://hal.archives-ouvertes.fr/ensl-00635335

C. Bernardin-and-s and . Olla, Fourier???s Law for a Microscopic Model of Heat Conduction, Journal of Statistical Physics, vol.8, issue.n.1, pp.271-289, 2005.
DOI : 10.1007/s10955-005-7578-9

C. Bernardin-and-g and . Stoltz, Anomalous diffusion for a class of systems with two conserved quantities, Nonlinearity, vol.25, issue.4, pp.25-1099, 2012.
DOI : 10.1088/0951-7715/25/4/1099

L. Bertini, D. Gabrielli, and A. J. Lebowitz, Large Deviations for a Stochastic Model of Heat Flow, Journal of Statistical Physics, vol.1, issue.5-6, pp.843-885, 2005.
DOI : 10.1007/s10955-005-5527-2

F. Bonetto, J. L. Lebowitz, and A. J. Lukkarinen, Fourier's Law for a Harmonic Crystal with Self-Consistent Stochastic Reservoirs, Journal of Statistical Physics, vol.116, issue.1-4, pp.783-813, 2004.
DOI : 10.1023/B:JOSS.0000037232.14365.10

F. Bonetto, J. L. Lebowitz, J. Lukkarinen, and A. S. Olla, Heat Conduction and Entropy Production in??Anharmonic Crystals with Self-Consistent Stochastic Reservoirs, Journal of Statistical Physics, vol.233, issue.4, pp.1097-1119, 2009.
DOI : 10.1007/s10955-008-9657-1
URL : https://hal.archives-ouvertes.fr/hal-00318755

F. Bonetto, J. L. Lebowitz, and A. L. Rey-bellet, FOURIER'S LAW: A CHALLENGE TO THEORISTS, Mathematical physics, pp.128-150, 2000.
DOI : 10.1142/9781848160224_0008

A. A. Bourgeat and . Piatnitski, Approximations of effective coefficients in stochastic homogenization, Annales de l?Institut Henri Poincare (B) Probability and Statistics, vol.40, issue.2, pp.153-165, 2004.
DOI : 10.1016/j.anihpb.2003.07.003

T. Brox and H. Rost, Equilibrium Fluctuations of Stochastic Particle Systems: The Role of Conserved Quantities, The Annals of Probability, vol.12, issue.3, pp.742-759, 1984.
DOI : 10.1214/aop/1176993225

E. Cancès, F. Legoll, and A. G. Stoltz, Theoretical and numerical comparison of some sampling methods for molecular dynamics, M2AN Math. Model. Numer. Anal, pp.41-351, 2007.

A. Casher and J. L. Lebowitz, Heat Flow in Regular and Disordered Harmonic Chains, Journal of Mathematical Physics, vol.12, issue.8, pp.1701-1711, 1971.
DOI : 10.1063/1.1665794

C. W. Chang, D. Okawa, H. Garcia, A. Majumdar, and A. A. Zettl, Breakdown of Fourier???s Law in Nanotube Thermal Conductors, Physical Review Letters, vol.101, issue.7, 2008.
DOI : 10.1103/PhysRevLett.101.075903

D. Cioranescu-and-p and . Donato, An introduction to homogenization, of Oxford Lecture Series in Mathematics and its Applications, 1999.

D. Dawson-and-l and . Gorostiza, Generalized solutions of a class of nuclear-space-valued stochastic evolution equations, Applied Mathematics & Optimization, vol.12, issue.1, pp.241-263, 1990.
DOI : 10.1007/BF01447330

A. Dhar, Heat Conduction in the Disordered Harmonic Chain Revisited, Physical Review Letters, vol.86, issue.26, pp.5882-5885, 2001.
DOI : 10.1103/PhysRevLett.86.5882

A. Dhar, V. Kannan, and A. J. Lebowitz, Heat conduction in disordered harmonic lattices with energy-conserving noise, Physical Review E, vol.83, issue.2, p.83, 2011.
DOI : 10.1103/PhysRevE.83.021108

J. Eckmann-and-m and . Hairer, Non-Equilibrium Statistical Mechanics??of Strongly Anharmonic Chains of Oscillators, Communications in Mathematical Physics, vol.212, issue.1, pp.105-164, 2000.
DOI : 10.1007/s002200000216

A. Egloffe, A. Gloria, J. Mourrat, and A. T. Nguyen, Random walk in random environment, corrector equation and homogenized coefficients: from theory to numerics, back and forth, IMA Journal of Numerical Analysis, vol.35, issue.2, 2012.
DOI : 10.1093/imanum/dru010
URL : https://hal.archives-ouvertes.fr/hal-00749667

B. Engquist and P. E. Souganidis, Asymptotic and numerical homogenization, Acta Numerica, vol.17, pp.147-190, 2008.
DOI : 10.1017/S0962492906360011

N. S. Even and . Olla, Hydrodynamic limit for an hamiltonian system with boundary conditions and conservative noise, 2011.

A. Faggionato-and-f and . Martinelli, Hydrodynamic limit of a disordered lattice gas, Probab. Theory Related Fields, pp.535-608, 2003.

I. Fatt, The network model of porous media, Trans. AIME, vol.207, pp.144-181, 1956.

J. Fritz, T. Funaki, and A. J. Lebowitz, Stationary states of random Hamiltonian systems, Probab. Theory Related Fields, pp.211-236, 1994.

T. Funaki, K. Uchiyama, and A. H. Yau, Hydrodynamic Limit for Lattice Gas Reversible under Bernoulli Measures, Nonlinear stochastic PDEs, pp.1-40, 1994.
DOI : 10.1007/978-1-4613-8468-7_1

A. Gloria, S. Neukamm, and A. F. Otto, An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations, ESAIM: Mathematical Modelling and Numerical Analysis, vol.48, issue.2, pp.325-346, 2014.
DOI : 10.1051/m2an/2013110
URL : https://hal.archives-ouvertes.fr/hal-00863488

M. Jara and C. Landim, Quenched non-equilibrium central limit theorem for a tagged particle in the exclusion process with bond disorder, Annales de l'Institut Henri Poincare (B) Probability and Statistics, vol.44, issue.2, pp.341-361, 2008.
DOI : 10.1214/07-AIHP112

V. V. Jikov, S. M. Kozlov, and A. O. Oleinik, Homogenization of differential operators and integral functionals, 1994.
DOI : 10.1007/978-3-642-84659-5

C. Kipnis and C. Landim, Scaling limits of interacting particle systems, of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences, 1999.
DOI : 10.1007/978-3-662-03752-2

C. Kipnis and S. R. Varadhan, Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions, Communications in Mathematical Physics, vol.28, issue.1, pp.1-19, 1986.
DOI : 10.1007/BF01210789

K. Komoriya, Hydrodynamic limit for asymmetric mean zero exclusion processes with speed change, Annales de l'Institut Henri Poincare (B) Probability and Statistics, vol.34, issue.6, pp.767-797, 1998.
DOI : 10.1016/S0246-0203(99)80003-X

T. Komorowski, C. Landim, and A. S. Olla, Time symmetry and martingale approximation, Fluctuations in Markov processes of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences
URL : https://hal.archives-ouvertes.fr/hal-00722537

S. M. Kozlov, Geometric aspects of averaging, Uspekhi Mat, Nauk, vol.44, pp.79-120, 1989.

R. Künnemann, The diffusion limit for reversible jump processes onZ d with ergodic random bond conductivities, Communications in Mathematical Physics, vol.80, issue.1, pp.27-68, 1983.
DOI : 10.1007/BF01209386

C. Landim, S. Olla, and A. H. Yau, Some properties of the diffusion coefficient for asymmetric simple exclusion processes, The Annals of Probability, vol.24, issue.4, pp.1779-1808, 1996.
DOI : 10.1214/aop/1041903206

C. Landim, M. Sued, and A. G. Valle, Hydrodynamic Limit of Asymmetric Exclusion Processes Under Diffusive Scaling in d???3, Communications in Mathematical Physics, vol.22, issue.2, pp.215-247, 2004.
DOI : 10.1007/s00220-004-1076-9

O. E. Lanford, I. , J. L. Lebowitz, and A. E. Lieb, Time evolution of infinite anharmonic systems, Journal of Statistical Physics, vol.50, issue.6, pp.453-461, 1977.
DOI : 10.1007/BF01152283

F. and L. Maître, Sur les groupes pleins préservant une mesure de probabilité, 2014.

J. L. Lebowitz, Stationary Nonequilibrium Gibbsian Ensembles, Physical Review, vol.114, issue.5, pp.1192-1202, 1959.
DOI : 10.1103/PhysRev.114.1192

J. L. Lebowitz, E. Lieb, and A. Z. Rieder, Properties of harmonic crystal in a stationary nonequilibrium state, J. Math. Phys, pp.1073-1078, 1967.

J. L. Lebowitz-and-p and . Résibois, Microscopic Theory of Brownian Motion in an Oscillating Field; Connection with Macroscopic Theory, Physical Review, vol.139, issue.4A, pp.139-1101, 1965.
DOI : 10.1103/PhysRev.139.A1101

F. Legoll, W. Minvielle, A. Obliger, and A. M. Simon, A parameter identification problem in stochastic homogenization, ESAIM: Proceedings and Surveys, vol.48, 2014.
DOI : 10.1051/proc/201448008
URL : https://hal.archives-ouvertes.fr/hal-00942730

B. Leimkuhler, C. Matthews, and A. G. Stoltz, The computation of averages from equilibrium and non-equilibrium langevin molecular dynamics, 2013.

S. Lepri, R. Livi, and A. A. Politi, Thermal conduction in classical low-dimensional lattices, Physics Reports, vol.377, issue.1, pp.1-80, 2003.
DOI : 10.1016/S0370-1573(02)00558-6

C. Liverani-and-s and . Olla, Toward the Fourier law for a weakly interacting anharmonic crystal, Journal of the American Mathematical Society, vol.25, issue.2
DOI : 10.1090/S0894-0347-2011-00724-8

G. H. Lunsford and . Ford, On the Stability of Periodic Orbits for Nonlinear Oscillator Systems in Regions Exhibiting Stochastic Behavior, Journal of Mathematical Physics, vol.13, issue.5, pp.700-705, 1972.
DOI : 10.1063/1.1666037

H. Mori, Transport, Collective Motion, and Brownian Motion, Progress of Theoretical Physics, vol.33, issue.3, p.423, 1964.
DOI : 10.1143/PTP.33.423

M. Mourragui-and-e and . Orlandi, Lattice Gas Model in Random Medium and Open Boundaries: Hydrodynamic and Relaxation to the Steady State, Journal of Statistical Physics, vol.16, issue.11, pp.685-714, 2009.
DOI : 10.1007/s10955-009-9796-z

S. Olla-and-m and . Sasada, Macroscopic energy diffusion for a chain of anharmonic oscillators, 2013.

S. Olla, S. R. Varadhan, and A. H. Yau, Hydrodynamical limit for a Hamiltonian system with weak noise, Communications in Mathematical Physics, vol.129, issue.3, pp.523-560, 1993.
DOI : 10.1007/BF02096727

J. Quastel, Bulk diffusion in a system with site disorder, The Annals of Probability, vol.34, issue.5, pp.1990-2036, 2006.
DOI : 10.1214/009117906000000322

M. Reed-and-b and . Simon, Methods of modern mathematical physics. I. Functional analysis, 1972.

H. Roos, On the problem of defining local thermodynamic equilibrium, p.eprint, 1997.

J. Roux, S. Rodts, and A. G. Stoltz, Introduction à la physique statistique et la physique quantique, 2009.

M. Sasada, Hydrodynamic limit for exclusion processes with velocity, Markov Process, pp.391-428, 2011.

M. Simon, Hydrodynamic limit for the velocity-flip model, Stochastic Processes and their Applications, pp.3623-3662, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01104050

H. Spohn, Large scale dynamics of interacting particles, 1991.
DOI : 10.1007/978-3-642-84371-6

D. Szász, Boltzmann's ergodic hypothesis, a conjecture for centuries?, in Hard ball systems and the Lorentz gas, Encyclopaedia Math. Sci, vol.101, pp.421-448, 2000.

D. W. Tang, Z. L. Wang, X. H. Zheng, W. G. Zhang, and A. Y. Zhu, Length-dependent thermal conductivity of single-wall carbon nanotubes: prediction and measurements, Nanotechnology, p.18, 2007.

C. Tremoulet, Hydrodynamic limit for interacting Ornstein-Uhlenbeck particles, Stochastic Process, Appl, vol.102, pp.139-158, 2002.
DOI : 10.1016/s0304-4149(02)00174-6
URL : http://doi.org/10.1016/s0304-4149(02)00174-6

S. R. Varadhan, Nonlinear diffusion limit for a system with nearest neighbor interactions. II, in Asymptotic problems in probability theory: stochastic models and diffusions on fractals, Pitman Res. Notes Math. Ser., Longman Sci. Tech, vol.283, pp.75-128, 1990.

H. T. Yau, Relative entropy and hydrodynamics of Ginzburg-Landau models, Letters in Mathematical Physics, vol.77, issue.1, pp.63-80, 1991.
DOI : 10.1007/BF00400379

R. J. Zwanzig, Memory Effects in Irreversible Thermodynamics, Physical Review, vol.124, issue.4, p.963, 1961.
DOI : 10.1103/PhysRev.124.983