[Rôle des quantités conservées dans la loi de Fourier pour les systèmes mécaniques diffusifs]
Le transport d'énergie peut être influencé par la présence d'autres quantités conservées. Nous considérons ici des systèmes diffusifs dans lesquels l'énergie et les autres quantités conservées évoluent macroscopiquement à la même échelle diffusive spatio-temporelle. Dans ces situations, la loi de Fourier dépend aussi du gradient des autres quantités conservées. La chaîne du rotor est un exemple classique de ces systèmes, où l'énergie et le moment angulaire sont conservés. Nous passons en revue ici quelques résultats mathématiques récents sur le transport diffusif de l'énergie et d'autres quantités conservées, en particulier relatifs à des systèmes dans lesquels la dynamique hamiltonienne est perturbée par des termes stochastiques conservateurs. La dynamique stochastique permet de définir les coefficients de transport (conductivité thermique) et, dans certains cas, de prouver l'équilibre local et la réponse linéaire nécessaire pour obtenir les équations diffusives qui régissent l'évolution macroscopique des quantités conservées. Les profils de température et les autres profils des quantités conservées dans les états stationnaires hors équilibre peuvent alors être compris à partir du comportement diffusif non stationnaire. Nous passons également en revue certains résultats et problèmes ouverts concernant l'approche en deux étapes (par couplage faible ou limites cinétiques) de l'équation de la chaleur à partir de modèles mécaniques dans lesquels seule l'énergie est conservée.
Energy transport can be influenced by the presence of other conserved quantities. We consider here diffusive systems where energy and the other conserved quantities evolve macroscopically on the same diffusive space–time scale. In these situations, the Fourier law depends also on the gradient of the other conserved quantities. The rotor chain is a classical example of such systems, where energy and angular momentum are conserved. We review here some recent mathematical results about the diffusive transport of energy and other conserved quantities, in particular for systems where the bulk Hamiltonian dynamics is perturbed by conservative stochastic terms. The presence of the stochastic dynamics allows us to define the transport coefficients (thermal conductivity) and in some cases to prove the local equilibrium and the linear response argument necessary to obtain the diffusive equations governing the macroscopic evolution of the conserved quantities. Temperature profiles and other conserved quantities profiles in the non-equilibrium stationary states can be then understood from the non-stationary diffusive behavior. We also review some results and open problems on the two step approach (by weak coupling or kinetic limits) to the heat equation, starting from mechanical models with only energy conserved.
Mot clés : Transport diffusif, Réponse linéaire, Limite hydrodynamique, États stationnaires hors équilibre, Limite de couplage faible
Stefano Olla 1
@article{CRPHYS_2019__20_5_429_0,
author = {Stefano Olla},
title = {Role of conserved quantities in {Fourier's} law for diffusive mechanical systems},
journal = {Comptes Rendus. Physique},
pages = {429--441},
publisher = {Elsevier},
volume = {20},
number = {5},
year = {2019},
doi = {10.1016/j.crhy.2019.08.001},
language = {en},
}
Stefano Olla. Role of conserved quantities in Fourier's law for diffusive mechanical systems. Comptes Rendus. Physique, Volume 20 (2019) no. 5, pp. 429-441. doi : 10.1016/j.crhy.2019.08.001. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2019.08.001/
[1] Stationary states of random Hamiltonian systems, Probab. Theory Relat. Fields, Volume 99 (1994) no. 2, pp. 211-236
[2] Coupled transport in rotor models, New J. Phys., Volume 18 (2016) https://iopscience.iop.org/article/10.1088/1367-2630/18/8/083023/pdf
[3] Transport properties of a chain of anharmonic oscillators with random flip of velocities, J. Stat. Phys., Volume 145 (2011), pp. 1224-1255
[4] Fluctuating hydrodynamics for a chain of nonlinearly coupled rotators, 2014 | arXiv
[5] Role of conserved quantities in normal heat transport in one dimension, 2014 (arXiv preprint) | arXiv
[6] Nonlinear fluctuating hydrodynamics for the classical XXZ spin chain, 2019 | arXiv
[7] Nonlinear diffusion limit for a system with nearest neighbor interactions. II, Sanda/Kyoto, 1990 (K.D. Elworthy; N. Ikeda, eds.) (Pitman Res. Notes Math. Ser.), Volume vol. 283, Longman Scientific & Technical, Harlow, UK (1993), pp. 75-128
[8] Scaling Limits of Interacting Particle Systems, Springer-Verlag, Berlin, 1999
[9] Diffusion of colors in the simple exclusion process, Commun. Pure Appl. Math., Volume 45 (1992) no. 6, pp. 623-679
[10] Macroscopic energy diffusion for a chain of anharmonic oscillators, Probab. Theory Relat. Fields, Volume 157 (2013), pp. 721-775 | DOI
[11] Hydrodynamics for a system of harmonic oscillators perturbed by a conservative noise, Stoch. Process. Appl., Volume 117 (2007), pp. 487-513
[12] Macroscopic evolution of mechanical and thermal energy in a harmonic chain with random flip of velocities, Kinet. Relat. Models, Volume 11 (2018) no. 3, pp. 615-645
[13] Microscopic derivation of an adiabatic thermodynamic transformation, Braz. J. Probab. Stat., Volume 29 (2015) no. 2, pp. 540-564 | DOI
[14] S. Olla, M. Simon, in preparation.
[15] Diffusive propagation of energy in a non-acoustic chain, Arch. Ration. Mech. Appl., Volume 223 (2017) no. 1, pp. 95-139
[16] A. Iacobucci, S. Olla, G. Stoltz, Stationary non-equilibrium states in rotors models, in preparation.
[17] On the relaxation rate of short chains of rotors interacting with Langevin thermostats, Electron. Commun. Probab., Volume 22 (2017) no. 8 | DOI
[18] Non-equilibrium steady state and subgeometric ergodicity for a chain of three coupled rotors, Nonlinearity, Volume 28 (2015) no. 7, pp. 2397-2421
[19] Uphill diffusion in multicomponent mixtures, Chem. Soc. Rev., Volume 44 (2015), pp. 2812-2836
[20] Negative thermal conductivity of chains of rotors with mechanical forcing, Phys. Rev. E, Volume 84 (2011)
[21] Microscopic models for uphill diffusion, J. Phys. A, Volume 50 (2017)
[22] An open microscopic model of heat conduction: evolution and non-equilibrium stationary states, 2019 (accepted for publication in Communications in Mathematical Sciences) | arXiv
[23] Energy transfer in a fast–slow Hamiltonian system, Commun. Math. Phys., Volume 308 (2011), pp. 201-225
[24] Toward the Fourier law for a weakly interacting anharmonic crystal, J. Am. Math. Soc., Volume 25 (2012) no. 2, pp. 555-583
[25] Green-Kubo formula for weakly coupled systems with noise, Commun. Math. Phys., Volume 334 (2015) no. 3, pp. 1377-1412 | DOI
[26] Diffusive scaling in energy Ginzburg–Landau dynamics, 2015 | arXiv
[27] Ergodic systems of n balls in a billiard table, Commun. Math. Phys., Volume 146 (1992) no. 2, pp. 357-396 | DOI
[28] Heat conduction and Fourier's law by consecutive local mixing and thermalization, Phys. Rev. Lett., Volume 101 (2008) no. 2 | DOI
[29] Spectral gap for stochastic energy exchange model with non-uniformly positive rate function, Ann. Probab., Volume 43 (2015) no. 4, pp. 1663-1711 | DOI
[30] On the limiting Markov process of energy exchanges in a rarely interacting ball-piston gas, J. Stat. Phys., Volume 166 (2017) no. 3–4, pp. 903-925
[31] Transport Fourier's law for a microscopic model of heat conduction, J. Stat. Phys., Volume 121 (2005), pp. 271-289
Cité par Sources :
Commentaires - Politique