LSI for Kawasaki Dynamics with Weak Interaction

Article

Abstract

We consider a large lattice system of unbounded continuous spins that are governed by a Ginzburg-Landau type potential and a weak quadratic interaction. We derive the logarithmic Sobolev inequality (LSI) for Kawasaki dynamics uniform in the boundary data. The scaling of the LSI constant is optimal in the system size and our argument is independent of the geometric structure of the system. The proof consists of an application of the two-scale approach of Grunewald, Otto, Westdickenberg & Villani. Several ideas are needed to solve new technical difficulties due to the interaction. Let us mention the application of a new covariance estimate, a conditioning technique, and a generalization of the local Cramér theorem.

References

  1. 1.
    Bakry, D., Émery, M.: Diffusions hypercontractives. Sem. Probab. XIX, Lecture Notes in Math, 1123. Berlin-Heidelberg-New York: Springer-Verlag, 1985, pp. 177–206Google Scholar
  2. 2.
    Bodineau T., Helffer B.: The log-Sobolev inequality for unbounded spin systems. J. Funct. Anal. 166(1), 168–178 (1999)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bodineau, T., Helffer, B.: Correlations, spectral gap and log-Sobolev inequalities for unbounded spins systems. In: Differential equations and mathematical physics (Birmingham, AL, 1999), Volume 16 of AMS/IP Stud. Adv. Math., Providence, RI: Amer. Math. Soc., 2000, pp. 51–66Google Scholar
  4. 4.
    Cancrini N., Martinelli F., Roberto C.: The logarithmic Sobolev constant of Kawasaki dynamics under a mixing condition revisited. Ann. Inst. H. Poincaré Probab. Statist. 38(4), 385–436 (2002)MathSciNetADSMATHCrossRefGoogle Scholar
  5. 5.
    Caputo P.: Uniform Poincaré inequalities for unbounded conservative spin systems: the non-interacting case. Stochastic Process. Appl. 106(2), 223–244 (2003)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Chafaï D.: Glauber versus Kawasaki for spectral gap and logarithmic Sobolev inequalities of some unbounded conservative spin systems. Markov Process. Rel. Fields 9(3), 341–362 (2003)MATHGoogle Scholar
  7. 7.
    Feller, W.: An introduction to probability theory and its applications. Vol II. 2nd ed. Wiley Series in Probability and Mathematical Statistics. New York: John Wiley and Sons, Inc., 1971Google Scholar
  8. 8.
    Gross L.: Logarithmic Sobolev inequalities. Amer. J. Math. 97, 1061–1083 (1975)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Grunewald N., Otto F., Villani C., Westdickenberg M.: A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit. Ann. Inst. H. Poincaré Probab. Statist. 45(2), 302–351 (2009)MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10.
    Guionnet, A., Zegarlinski, B.: Lectures on logarithmic Sobolev inequalities. In: Séminaire de Probabilités, XXXVI, Volume 1801 of Lecture Notes in Math, Berlin: Springer, 2003, pp. 1–134Google Scholar
  11. 11.
    Guo M., Papanicolau G., Varadhan S.: Nonlinear diffusion limit for a system with nearest neighbor interactions. Commun. Math. Phys. 118, 31–59 (1988)ADSMATHCrossRefGoogle Scholar
  12. 12.
    Helffer B.: Remarks on decay of correlations and Witten Laplacians. III. Application to logarithmic Sobolev inequalities. Ann. Inst. H. Poincaré Probab. Statist. 35(4), 483–508 (1999)MathSciNetADSMATHCrossRefGoogle Scholar
  13. 13.
    Holley R., Stroock D.: Logarithmic Sobolev inequalities and stochastic Ising models. J. Stat. Phys. 46, 1159–1194 (1987)MathSciNetADSMATHCrossRefGoogle Scholar
  14. 14.
    Kipnis, C., Landim, C.: Scaling limits of interacting particle systems. Grundlehren der Mathematischen Wissenschaften. 320. Berlin: Springer., 1999Google Scholar
  15. 15.
    Kosygina E.: The behavior of the specific entropy in the hydrodynamic scaling limit. Ann. Probab. 29(3), 1086–1110 (2001)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Landim C., Panizo G., Yau H.T.: Spectral gap and logarithmic Sobolev inequality for unbounded conservative spin systems. Ann. Inst. H. Poincaré Probab. Statist. 38(5), 739–777 (2002)MathSciNetADSMATHCrossRefGoogle Scholar
  17. 17.
    M. Ledoux. Logarithmic Sobolev inequalities for unbounded spin systems revisted. Sem. Probab. XXXV, Lecture Notes in Math. 1755. Berlin-Heidelberg-New York: Springer-Verlag, 2001, pp. 167–194Google Scholar
  18. 18.
    Lu S.L., Lu S.L., Lu S.L.: Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics. Commun. Math. Phys. 156(2), 399–433 (1993)ADSMATHCrossRefGoogle Scholar
  19. 19.
    Menz, G.: Equilibrium dynamics of continuous unbounded spin systems. Dissertation, University of Bonn (2011). urn:nbn:de:hbz:5N-25331Google Scholar
  20. 20.
    Menz, G., Otto, F.: A new covariance estimate. In preparation, 2011Google Scholar
  21. 21.
    Menz, G., Otto, F.: Uniform logarithmic sobolev inequalities for conservative spin systems with super-quadratic single-site potential. MPI-MIS preprint, Leipzig, 2011Google Scholar
  22. 22.
    Otto F., Reznikoff M.: A new criterion for the logarithmic Sobolev inequality and two applications. J. Funct. Anal. 243(1), 121–157 (2007)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Otto F., Villani C.: Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173, 361–400 (2000)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Royer, G.: Une initiation aux inégalités de Sobolev logarithmiques. Cours Spéc., Soc. Math. France, 1999Google Scholar
  25. 25.
    Shiryaev, A.N.: Probability. New York: Springer-Verlag, 2nd edition, 1996Google Scholar
  26. 26.
    Stroock D., Zegarlinski B.: The equivalence of the logarithmic Sobolev inequality and the Dobrushin-Shlosman mixing condition. Commun. Math. Phys. 144, 303–323 (1992)MathSciNetADSMATHCrossRefGoogle Scholar
  27. 27.
    Stroock D., Zegarlinski B.: The logarithmic Sobolev inequality for discrete spin systems on the lattice. Commun. Math. Phys. 149, 175–193 (1992)MathSciNetADSMATHCrossRefGoogle Scholar
  28. 28.
    Stroock D., Zegarlinski B.: On the ergodic properties of glauber dynamics. J. Stat. Phys. 81, 1007–1019 (1995)MathSciNetADSMATHCrossRefGoogle Scholar
  29. 29.
    Yau H.-T.: Relative entropy and hydrodynamics of Ginzburg-Landau models. Lett. Math. Phys. 22, 63–80 (1991)MathSciNetADSMATHCrossRefGoogle Scholar
  30. 30.
    Yau H.-T.: Logarithmic Sobolev inequality for lattice gases with mixing conditions. Commun. Math. Phys. 181(2), 367–408 (1996)ADSMATHCrossRefGoogle Scholar
  31. 31.
    Yoshida N.: Application of log-Sobolev inequality to the stochastic dynamics of unbounded spin systems on the lattice. J. Funct. Anal. 173, 74–102 (2000)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Yoshida N.: The equivalence of the log-Sobolev inequality and a mixing condition for unbounded spin systems on the lattice. Ann. Inst. H. Poincaré Probab. Statist. 37(2), 223–243 (2001)ADSMATHCrossRefGoogle Scholar
  33. 33.
    Zegarlinski B.: The strong decay to equilibrium for the stochastic dynamics of unbounded spin systems on a lattice. Commun. Math. Phys. 175, 401–432 (1996)MathSciNetADSMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Max Planck Institute for Mathematics in the SciencesLeipzigGermany

Personalised recommendations