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

Natalie Grunewald; Felix Otto; Cédric Villani; Maria G. Westdickenberg

Annales de l'I.H.P. Probabilités et statistiques (2009)

  • Volume: 45, Issue: 2, page 302-351
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Grunewald, Natalie, et al. "A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit." Annales de l'I.H.P. Probabilités et statistiques 45.2 (2009): 302-351. <http://eudml.org/doc/78025>.

@article{Grunewald2009,
abstract = {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.},
author = {Grunewald, Natalie, Otto, Felix, Villani, Cédric, Westdickenberg, Maria G.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {logarithmic Sobolev inequality; hydrodynamic limit; spin system; Kawasaki dynamics; canonical ensemble; Coarse-graining; Spin system; Hawasaki dynamics; canonical ensembles; coarse0graining},
language = {eng},
number = {2},
pages = {302-351},
publisher = {Gauthier-Villars},
title = {A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit},
url = {http://eudml.org/doc/78025},
volume = {45},
year = {2009},
}

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
PB - Gauthier-Villars
VL - 45
IS - 2
SP - 302
EP - 351
AB - 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.
LA - eng
KW - logarithmic Sobolev inequality; hydrodynamic limit; spin system; Kawasaki dynamics; canonical ensemble; Coarse-graining; Spin system; Hawasaki dynamics; canonical ensembles; coarse0graining
UR - http://eudml.org/doc/78025
ER -

References

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  1. [1] D. Bakry and M. Émery. Diffusions hypercontractives. In Sem. Probab. XIX. Lecture Notes in Mathematics 1123 177–206. Springer, Berlin, 1985. Zbl0561.60080MR889476
  2. [2] G. Blower and F. Bolley. Concentration of measure on product spaces with applications to Markov processes. Studia Math. 175 (2006) 47–72. Zbl1101.60009MR2261699
  3. [3] Th. Bodineau and B. Helffer. On log Sobolev inequalities for unbounded spin systems. J. Funct. Anal. 166 (1999) 168–178. Zbl0972.82035MR1704666
  4. [4] D. Chafaï. Glauber versus Kawasaki for spectral gap and logarithmic Sobolev inequalities of some unbounded conservative spin systems. Markov Process. Related Fields 9 (2001) 341–362. Zbl1040.60081MR2028218
  5. [5] P. Caputo. Uniform Poincaré inequalities for unbounded conservative spin systems: The non-interacting case. Stochastic Process. Appl. 106 (2003) 223–244. Zbl1075.60581MR1989628
  6. [6] A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications, 2nd edn. Appl. Math. 38. Springer, New York, 1998. Zbl0896.60013MR1619036
  7. [7] W. Feller. An Introduction to Probability Theory and Its Applications, Vol. 2. Wiley, New York, 1971. Zbl0219.60003MR270403
  8. [8] L. Gross. Logarithmic Sobolev inequalities. Amer. J. Math. 97 (1975) 1061–1083. Zbl0318.46049MR420249
  9. [9] A. Guionnet and B. Zegarlinski. Lectures on logarithmic Sobolev inequalities. In Sminaire de Probabilits, XXXVI. Lecture Notes in Mathematics 1801 1–134. Springer, Berlin, 2003. Zbl1125.60111MR1971582
  10. [10] M. Z. Guo, G. C. Papanicolaou and S. R. S. Varadhan. Nonlinear diffusion limit for a system with nearest neighbor interactions. Commun. Math. Phys. 118 (1988) 31–59. Zbl0652.60107MR954674
  11. [11] R. Holley and D. Stroock. Logarithmic Sobolev inequalities and stochastic Ising models. J. Statist. Phys. 46 (1987) 1159–1194. Zbl0682.60109MR893137
  12. [12] C. Kipnis and C. Landim. Scaling Limits of Interacting Particle Systems. Springer, Berlin, 1999. Zbl0927.60002MR1707314
  13. [13] E. Kosygina. The behavior of the specific entropy in the hydrodynamic scaling limit. Ann. Probab. 29 (2001) 1086–1110. Zbl1018.60096MR1872737
  14. [14] C. Landim, G. Panizo and H. T. Yau. Spectral gap and logarithmic Sobolev inequality for unbounded conservative spin systems. Ann. Inst. H. Poincaré Probab. Statist. 38 (2002) 739–777. Zbl1022.60087MR1931585
  15. [15] M. Ledoux. The Concentration of Measure Phenomenon, Math. Surveys and Monographs 89. Amer. Math. Soc., Providence, RI, 2001. Zbl0995.60002MR1849347
  16. [16] M. Ledoux. Logarithmic Sobolev inequalities for unbounded spin systems revisted. In Sem. Probab. XXXV. Lecture Notes in Mathematics 1755 167–194. Springer, Berlin. 2001. Zbl0979.60096MR1837286
  17. [17] S. L. Lu. Hydrodynamic scaling with deterministic initial configurations. Ann. Probab. 23 (1995) 1831–1852. Zbl0860.60086MR1379170
  18. [18] S. L. Lu and H. T. Yau. Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics. Commun. Math. Phys. 156 (1993) 399–433. Zbl0779.60078MR1233852
  19. [19] F. Otto and M. G. Reznikoff. A new criterion for the logarithmic Sobolev inequality and two applications. J. Funct. Anal. 243 (2007) 121–157. Zbl1109.60013MR2291434
  20. [20] F. Otto and C. Villani. Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173 (2000) 361–400. Zbl0985.58019MR1760620
  21. [21] G. Royer. Une Initiation aux Inégalités de Sobolev Logarithmiques. Cours Spécialisés. Soc. Math. de France, Paris, 1999. Zbl0927.60006MR1704288
  22. [22] G. Toscani and C. Villani. On the trend to equilibrium for some dissipative systems with slowly increasing a-priori bounds. J. Stat. Phys. 98 (2000) 1279–1309. Zbl1034.82032MR1751701
  23. [23] C. Villani. Topics in Optimal Transportation. Graduate Texts in Mathematics 58. Amer. Math. Soc., Providence, RI, 2003. Zbl1106.90001MR1964483
  24. [24] H.-T. Yau. Relative entropy and hydrodynamics of Ginzburg–Landau models. Lett. Math. Phys. 22 (1991) 63–80. Zbl0725.60120MR1121850
  25. [25] H.-T. Yau. Logarithmic Sobolev inequality for lattice gases with mixing conditions. Commun. Math. Phys. 181 (1996) 367–408. Zbl0864.60079MR1414837

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