A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit
Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 2, pp. 302-351.

Nous étudions un système sur réseau à variable de spin continue. Dans la première partie, nous établissons deux résultats abstraits : des conditions suffisantes pour une inégalité de Sobolev logarithmique avec constante indépendante de la dimension (Théorème 3), et des conditions suffisantes pour la convergence vers la limite hydrodynamique (Theorème 8). Dans la seconde partie, nous utilisons ces résultats abstraits pour traiter un exemple spécifique, à savoir la dynamique de Kawasaki avec un potentiel de type Ginzburg-Landau.

We consider the coarse-graining of a lattice system with continuous spin variable. In the first part, two abstract results are established: sufficient conditions for a logarithmic Sobolev inequality with constants independent of the dimension (Theorem 3) and sufficient conditions for convergence to the hydrodynamic limit (Theorem 8). In the second part, we use the abstract results to treat a specific example, namely the Kawasaki dynamics with Ginzburg-Landau-type potential.

DOI : 10.1214/07-AIHP200
Classification : 60K35, 60J25
Mots-clés : logarithmic Sobolev inequality, hydrodynamic limit, spin system, Kawasaki dynamics, canonical ensemble, Coarse-graining 
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     title = {A two-scale approach to logarithmic {Sobolev} inequalities and the hydrodynamic limit},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
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Grunewald, Natalie; Otto, Felix; Villani, Cédric; Westdickenberg, Maria G. A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 2, pp. 302-351. doi : 10.1214/07-AIHP200. http://www.numdam.org/articles/10.1214/07-AIHP200/

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