Comptes Rendus
Fourier and the science of today / Fourier et la science d'aujourd'hui
Role of conserved quantities in Fourier's law for diffusive mechanical systems
Comptes Rendus. Physique, Volume 20 (2019) no. 5, pp. 429-441.

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.

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.

Published online:
DOI: 10.1016/j.crhy.2019.08.001
Keywords: Diffusive transport, Linear response, Hydrodynamic limit, Non-equilibrium stationary states, Weak coupling limit
Mot clés : Transport diffusif, Réponse linéaire, Limite hydrodynamique, États stationnaires hors équilibre, Limite de couplage faible
Stefano Olla 1

1 Ceremade, UMR CNRS, Université Paris-Dauphine, PSL Research University, place du Maréchal-de-Lattre-de-Tassigny, 75016 Paris, France
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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/

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