Electronic Journal of Probability

Entropy decay for interacting systems via the Bochner-Bakry-Émery approach

Paolo Dai Pra and Gustavo Posta

Full-text: Open access


We obtain estimates on the exponential rate of decay of the relative entropy from equilibrium for Markov processes with a non-local infinitesimal generator. We adapt some of the ideas coming from the Bakry-Emery approach to this setting. In particular, we obtain volume- independent lower bounds for the Glauber dynamics of interacting point particles and for various classes of hardcore models.

Article information

Electron. J. Probab., Volume 18 (2013), paper no. 52, 21 pp.

Accepted: 4 May 2013
First available in Project Euclid: 4 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 39B62: Functional inequalities, including subadditivity, convexity, etc. [See also 26A51, 26B25, 26Dxx]
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Entropy decay functional inequalities

This work is licensed under a Creative Commons Attribution 3.0 License.


Dai Pra, Paolo; Posta, Gustavo. Entropy decay for interacting systems via the Bochner-Bakry-Émery approach. Electron. J. Probab. 18 (2013), paper no. 52, 21 pp. doi:10.1214/EJP.v18-2041. https://projecteuclid.org/euclid.ejp/1465064277

Export citation


  • Bakry, D.; Émery, Michel. Diffusions hypercontractives. (French) [Hypercontractive diffusions] Séminaire de probabilités, XIX, 1983/84, 177–206, Lecture Notes in Math., 1123, Springer, Berlin, 1985.
  • Bertini, Lorenzo; Cancrini, Nicoletta; Cesi, Filippo. The spectral gap for a Glauber-type dynamics in a continuous gas. Ann. Inst. H. Poincaré Probab. Statist. 38 (2002), no. 1, 91–108.
  • Boudou, Anne-Severine; Caputo, Pietro; Dai Pra, Paolo; Posta, Gustavo. Spectral gap estimates for interacting particle systems via a Bochner-type identity. J. Funct. Anal. 232 (2006), no. 1, 222–258.
  • Caputo, Pietro; Dai Pra, Paolo; Posta, Gustavo. Convex entropy decay via the Bochner-Bakry-Emery approach. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009), no. 3, 734–753.
  • Caputo, Pietro; Posta, Gustavo. Entropy dissipation estimates in a zero-range dynamics. Probab. Theory Related Fields 139 (2007), no. 1-2, 65–87.
  • M. Disertori, A. Giuliani; The nematic phase of a system of long hard rods. Comm. Math. Phys. (to appear). ARXIV1112.5564v2.
  • Diaconis, P.; Saloff-Coste, L. Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 (1996), no. 3, 695–750.
  • Erbar, Matthias; Maas, Jan. Ricci curvature of finite Markov chains via convexity of the entropy. Arch. Ration. Mech. Anal. 206 (2012), no. 3, 997–1038.
  • Grunewald, Natalie; Otto, Felix; Villani, Cédric; Westdickenberg, Maria G. A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009), no. 2, 302–351.
  • Kelly, F. P. Loss networks. Ann. Appl. Probab. 1 (1991), no. 3, 319–378.
  • Y. Kondratiev, K. Tobias, N. Ohlerich. Spectral gap for Glauber type dynamics for a special class of potentials. ARXIV1103.5079v1
  • Ledoux, M. Logarithmic Sobolev inequalities for unbounded spin systems revisited. Séminaire de Probabilités, XXXV, 167–194, Lecture Notes in Math., 1755, Springer, Berlin, 2001.
  • Levin, David A.; Peres, Yuval; Wilmer, Elizabeth L. Markov chains and mixing times. With a chapter by James G. Propp and David B. Wilson. American Mathematical Society, Providence, RI, 2009. xviii+371 pp. ISBN: 978-0-8218-4739-8
  • Li, Xiang-Dong. Perelman's entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry-Emery Ricci curvature. Math. Ann. 353 (2012), no. 2, 403–437.
  • Luby, Michael; Vigoda, Eric. Fast convergence of the Glauber dynamics for sampling independent sets. Statistical physics methods in discrete probability, combinatorics, and theoretical computer science (Princeton, NJ, 1997). Random Structures Algorithms 15 (1999), no. 3-4, 229–241.
  • Ma, Yutao; Wang, Ran; Wu, Liming. Transportation-information inequalities for continuum Gibbs measures. Electron. Commun. Probab. 16 (2011), 600–613.
  • Maas, Jan. Gradient flows of the entropy for finite Markov chains. J. Funct. Anal. 261 (2011), no. 8, 2250–2292.
  • Vigoda, Eric. A note on the Glauber dynamics for sampling independent sets. Electron. J. Combin. 8 (2001), no. 1, Research Paper 8, 8 pp. (electronic).
  • Wu, Liming. A new modified logarithmic Sobolev inequality for Poisson point processes and several applications. Probab. Theory Related Fields 118 (2000), no. 3, 427–438.
  • Wu, Liming. Estimate of spectral gap for continuous gas. Ann. Inst. H. Poincaré Probab. Statist. 40 (2004), no. 4, 387–409.