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Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics

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Abstract

We prove that the spectral gap of the Kawasaki dynamics shrink at the rate of 1/L 2 for cubes of sizeL provided that some mixing conditions are satisfied. We also prove that the logarithmic Sobolev inequality for the Glauber dynamics in standard cubes holds uniformly in the size of the cube if the Dobrushin-Shlosman mixing condition holds for standard cubes.

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References

  • [AZ] Aizenman, M., Holley, R.: Rapid convergence to equilibrium of the stochastic Ising model in the Dobrushin-Shlosman regime. IMS Vol. in Math. & Appl., Berlin, Heidelberg, New-York: Springer 1987, pp. 1–11

    Google Scholar 

  • [CS] Carlen, E. Stroock, D.: An application of the Bakry-Emery criterion to infinite dimensional diffusions. In: Séminaire de Probabilités XX. Springer Lecture Notes in Mathematics1204, 1986, pp. 341–347

  • [DGS] Davies, E.B., Gross, L., Simon, B.: Hypercontractivity: A bibliographical review. In: Ideas and Methods of Mathematics and Physics. In Memoriam of Raphael Hoegh-Krohn. S. Albeverio, J.E. Fenstand, H. Holden, T. Lindstrom (eds.) Cambridge: Cambridge University Press, 1992

    Google Scholar 

  • [DP] De Masi A., Presutti, E.: Mathematical Methods for Hydrodynamic Limit. Lecture Notes in Math.1501, Berlin, Heidelberg, New York: Springer 1991

    Google Scholar 

  • [F] Funaki: private communication

  • [G] Gross, L.: Logarithmic Sobolev Inequalities. Am. J. Math.97, 1061–1083 (1976)

    Google Scholar 

  • [KLO] Kipnis, C., Landim, C., Olla, S.: Hydrodynamical limit for a nongradient system: The generalized symmetric exclusion process. Preprint

  • [MO1] Martinelli, F., Olivieri, E.: Approach to equilibrium of Glauber dynamics in the one phase region, I: The attractive case. Preprint

  • [MO2] Martinelli, F., Olivieri, E.: Finite volume mixing conditions for lattice spin systems and exponential approach to equilibrium of Glauber dynamics. Preprint

  • [O] Olivieri, E.: On a cluster expansion for lattice spin systems with a finite size condition for the convergence. J. Stat. Phys.50, 1179–1200 (1988)

    Google Scholar 

  • [OP] Olivieri, E., Picco, P.: Cluster expansion for ad-dimensional lattice system and finite volume factorization properties. J. Stat. Phys.59, 221–256 (1990)

    Google Scholar 

  • [Q] Quastel, J.: Diffusion of color in the simple exclusion process, Thesis, New York Univ. 1990

  • [R] Ruelle, D.: Statistical Mechanics: Rigorous Results. New York: Addison-Wesley 1989

    Google Scholar 

  • [S] Spohn, H.: Large Scale Dynamics of Interacting Particles. Berlin, Heidelberg, New York: Springer 1991

    Google Scholar 

  • [ST] Stroock, D.W.: Logarithmic Sobolev Inequalities for Gibbs States. MIT preprint

  • [SZ1] Stroock, D., Zegarlinski, B.: The logarithmic Sobolev inequality for lattice gases with continuous spins. J. Funct. Anal.104, 299–326 (1992)

    Google Scholar 

  • [SZ2] Stroock, D., Zegarlinski, B.: The equivalence of the logarithmic Sobolev inequality and the Dobrushin-Shlosman mixing condition. Commun. Math. Phys.144, 303–323 (1992)

    Google Scholar 

  • [SZ3] Stroock, D., Zegarlinski, B.: The Logarithmic Sobolev Inequality for Discrete Spin Systems on a Lattice. Commun. Math. Phys.149, 175–193 (1992)

    Google Scholar 

  • [V] Varadhan, S.R.S.: Nonlinear diffusion limit for a system with nearest neighbor interactions II. Proc. Taniguchi Symp., 1990 Kyoto

  • [Z1] Zegarlinski, B.: private communication

  • [Z2] Zegarlinski, B.: Dobrushin Uniqueness Theorem and Logarithmic Sobolev Inequalities. J. Funct. Anal.105, 77–111 (1992)

    Google Scholar 

  • [Z3] Zegarlinski, B.: Gibbsian Description and Description by Stochastic Dynamics in the Statistical Mechanics of Lattice Spin Systems with Finite Range Interactions. In: Proc. of IIIrd International Conference: Stochastic Processes, Physics and Geometry, Locarno, Switzerland, June 24–29, 1991

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, 48109-1092, Ann Arbor, MI, USA

    Sheng Lin Lu

  2. Courant Institute, New York University, 10012, New York, NY, USA

    Horng-Tzer Yau

Authors
  1. Sheng Lin Lu
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  2. Horng-Tzer Yau
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Communicated by J.L. Lebowitz

Research partially supported by U.S. National Science Foundation grant 9101196, Sloan Foundation Fellowship and David and Lucile Packard Foundation Fellowship

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Lu, S.L., Yau, HT. Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics. Commun.Math. Phys. 156, 399–433 (1993). https://doi.org/10.1007/BF02098489

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  • DOI: https://doi.org/10.1007/BF02098489

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