Entropy inequalities for unbounded spin systems

Entropy inequalities for unbounded spin systems

Paolo Dai Pra, Anna Maria Paganoni, and Gustavo Posta

Source: Ann. Probab. Volume 30, Number 4 (2002), 1959-1976.

Abstract

We consider nonconservative, reversible spin systems, with unbounded discrete spins. We show that for a class of these dynamics in a high temperature regime, the relative entropy with respect to the equilibrium distribution decays exponentially in time, although the logarithmic-Sobolev inequality fails. To this end we prove a weaker modification of the logarithmic-Sobolev inequality.

First Page: Show

Primary Subjects: 60K35, 82C22

Keywords: Spin systems; entropy; logarithmic-Sobolev inequality; spectral gap

Full-text: Open access

PDF File (162 KB)

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aop/1039548378
Digital Object Identifier: doi:10.1214/aop/1039548378
Mathematical Reviews number (MathSciNet): MR1944012
Zentralblatt MATH identifier: 1013.60076

back to Table of Contents

References

[1] AN, C., BLACHRE, S., CHAFAÏ, D., FOUGRES, P., GENTIL, I., MALRIEU, F., ROBERTO, C.

Mathematical Reviews (MathSciNet): MR1980261

Digital Object Identifier: doi:10.4064/aa109-3-5

and SCHEFFER, G. (2000). Sur les ingalits de Sobolev logarithmiques. Panor. Sy nth. 10.

[2] BERTINI, L., CANCRINI, N. and CESI, F. (2000). The spectral gap for a Glauber-ty pe dy namics in a continuous gas. Ann. Inst. H. Poincaré Probab. Statist. To appear.

Mathematical Reviews (MathSciNet): MR1899231

Zentralblatt MATH: 0994.82054

Digital Object Identifier: doi:10.1016/S0246-0203(01)01085-8

[3] BOBKOV, S. G. and LEDOUX, M. (1998). On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal. 156 347-365.

Mathematical Reviews (MathSciNet): MR99e:60051

Zentralblatt MATH: 0920.60002

Digital Object Identifier: doi:10.1006/jfan.1997.3187

[4] CESI, F. (2001). Quasi-factorization of the entropy and logarithmic Sobolev inequalities for Gibbs random fields. Probab. Theory Related Fields 120 569-584.

Mathematical Reviews (MathSciNet): MR2003f:82003

Digital Object Identifier: doi:10.1007/PL00008792

[5] DIACONIS, P. and SALOFF-COSTE, L. (1996). Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 695-750.

Mathematical Reviews (MathSciNet): MR97k:60176

Zentralblatt MATH: 0867.60043

Digital Object Identifier: doi:10.1214/aoap/1034968224

Project Euclid: euclid.aoap/1034968224

[6] DOBRUSHIN, R. L. and SHLOSMAN, S. B. (1985). Constructive criterion for the uniqueness of Gibbs field. In Statistical physics and Dy namical Sy stems 347-370. Birkhäuser, Boston.

Mathematical Reviews (MathSciNet): MR87d:82006

[7] GROSS, L. (1993). Logarithmic Sobolev inequalities and contractivity properties of semigroups. Dirichlet Forms. Lecture Notes in Math. 1563 54-88. Springer, Berlin.

Mathematical Reviews (MathSciNet): MR1292277

Digital Object Identifier: doi:10.1007/BFb0074091

[8] LAWLER, G. F. and SOKAL, A. D. (1988). Bounds on the L2 spectrum for Markov chains and Markov processes: A generalization of Cheeger's inequality. Trans. Amer. Math. Soc. 309 557-580.

Mathematical Reviews (MathSciNet): MR89h:60105

Digital Object Identifier: doi:10.2307/2000925

JSTOR: links.jstor.org

[9] MARTINELLI, F. (1999). Lectures on Glauber dy namics for discrete spin models. Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1717 93-191. Springer, Berlin.

Mathematical Reviews (MathSciNet): MR1746301

[10] MARTINELLI, F. and OLIVIERI, E. (1994). Approach to equilibrium of Glauber dy namics in the one phase region. II. The general case. Comm. Math. Phy s. 161 487-514.

Mathematical Reviews (MathSciNet): MR1269388

Digital Object Identifier: doi:10.1007/BF02101930

Project Euclid: euclid.cmp/1104270007

[11] MICLO, L. (1999). An example of application of discrete Hardy inequalities. Markov Process. Related Fields 5 319-330.

Mathematical Reviews (MathSciNet): MR2000h:60081

[12] MICLO, L. and ROBERTO, C. (2001). Modified logarithmic Sobolev inequality and Hardy ty pe inequalities. Preprint.

[13] STROOCK, D. W. and ZEGARLINSKI, B. (1992). The logarithmic Sobolev inequality for discrete spin sy stems on a lattice. Comm. Math. Phy s. 149 175-193.

Mathematical Reviews (MathSciNet): MR93j:82013

Digital Object Identifier: doi:10.1007/BF02096629

Project Euclid: euclid.cmp/1104251144

[14] STROOCK, D. W. and ZEGARLINSKI, B. (1992). The logarithmic Sobolev inequality for continuous spin sy stems on a lattice. J. Funct. Anal. 104 299-326.

Mathematical Reviews (MathSciNet): MR93f:82015

Zentralblatt MATH: 0794.46025

Digital Object Identifier: doi:10.1016/0022-1236(92)90003-2

previous :: next