Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit

Natalie Grunewald, Felix Otto, Cédric Villani, and Maria G. Westdickenberg

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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.


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.

Article information

Ann. Inst. H. Poincaré Probab. Statist. Volume 45, Number 2 (2009), 302-351.

First available in Project Euclid: 29 April 2009

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J25: Continuous-time Markov processes on general state spaces

Logarithmic Sobolev inequality Hydrodynamic limit Spin system Kawasaki dynamics Canonical ensemble Coarse-graining


Grunewald, Natalie; Otto, Felix; Villani, Cédric; Westdickenberg, Maria G. A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit. Ann. Inst. H. Poincaré Probab. Statist. 45 (2009), no. 2, 302--351. doi:10.1214/07-AIHP200.

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