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.
Mots clés : logarithmic Sobolev inequality, hydrodynamic limit, spin system, Kawasaki dynamics, canonical ensemble, Coarse-graining
@article{AIHPB_2009__45_2_302_0, author = {Grunewald, Natalie and Otto, Felix and Villani, C\'edric and Westdickenberg, Maria G.}, title = {A two-scale approach to logarithmic {Sobolev} inequalities and the hydrodynamic limit}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {302--351}, publisher = {Gauthier-Villars}, volume = {45}, number = {2}, year = {2009}, doi = {10.1214/07-AIHP200}, mrnumber = {2521405}, zbl = {1179.60068}, language = {en}, url = {http://www.numdam.org/articles/10.1214/07-AIHP200/} }
TY - JOUR AU - Grunewald, Natalie AU - Otto, Felix AU - Villani, Cédric AU - Westdickenberg, Maria G. TI - A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2009 SP - 302 EP - 351 VL - 45 IS - 2 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/07-AIHP200/ DO - 10.1214/07-AIHP200 LA - en ID - AIHPB_2009__45_2_302_0 ER -
%0 Journal Article %A Grunewald, Natalie %A Otto, Felix %A Villani, Cédric %A Westdickenberg, Maria G. %T A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit %J Annales de l'I.H.P. Probabilités et statistiques %D 2009 %P 302-351 %V 45 %N 2 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/07-AIHP200/ %R 10.1214/07-AIHP200 %G en %F AIHPB_2009__45_2_302_0
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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