Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Kinetically constrained lattice gases: Tagged particle diffusion

O. Blondel and C. Toninelli

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Kinetically constrained lattice gases (KCLG) are interacting particle systems on the integer lattice $\mathbb{Z}^{d}$ with hard core exclusion and Kawasaki type dynamics. Their peculiarity is that jumps are allowed only if the configuration satisfies a constraint which asks for enough empty sites in a certain local neighborhood. KCLG have been introduced and extensively studied in physics literature as models of glassy dynamics. We focus on the most studied class of KCLG, the Kob Andersen (KA) models. We analyze the behavior of a tracer (i.e. a tagged particle) at equilibrium. We prove that for all dimensions $d\geq2$ and for any equilibrium particle density, under diffusive rescaling the motion of the tracer converges to a $d$-dimensional Brownian motion with non-degenerate diffusion matrix. Therefore we disprove the occurrence of a diffusive/non diffusive transition which had been conjectured in physics literature. Our technique is flexible enough and can be extended to analyse the tracer behavior for other choices of constraints.

Résumé

Les gaz réticulaires avec contrainte cinétique (KCLG) sont des systèmes de particules en interaction sur le réseau $\mathbb{Z}^{d}$ avec au plus une particule par site et une dynamique de type Kawasaki. Leur particularité est que les sauts ne sont autorisés que si la configuration satisfait une contrainte exigeant suffisamment de sites vides dans un certain voisinage local. Les KCLG ont été introduits et massivement étudiés dans la littérature physique comme modèles pour les systèmes vitreux. Nous nous concentrons sur l’une des classes de KCLG les plus étudiées : les modèles de Kob–Andersen (KA). Nous analysons le comportement d’un traceur (c.-à-d. une particule marquée) à l’équilibre. Pour toute dimension $d\geq2$ et toute densité de particules, nous montrons qu’à l’échelle diffusive la trajectoire du traceur converge vers un mouvement brownien $d$-dimensionnel avec matrice de diffusion non dégénérée. Par conséquent nous démontrons la non-existence d’une transition (qui avait été conjecturée dans la littérature physique) entre un régime diffusif et un régime non diffusif. Notre technique est assez flexible et peut être étendue pour analyser le comportement du traceur sous d’autres choix de contraintes.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 4 (2018), 2335-2348.

Dates
Received: 26 January 2017
Revised: 23 October 2017
Accepted: 7 November 2017
First available in Project Euclid: 18 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1539849801

Digital Object Identifier
doi:10.1214/17-AIHP873

Mathematical Reviews number (MathSciNet)
MR3865675

Zentralblatt MATH identifier
06996567

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J27: Continuous-time Markov processes on discrete state spaces

Keywords
Kawasaki dynamics Tagged particle Kinetically constrained models

Citation

Blondel, O.; Toninelli, C. Kinetically constrained lattice gases: Tagged particle diffusion. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 4, 2335--2348. doi:10.1214/17-AIHP873. https://projecteuclid.org/euclid.aihp/1539849801


Export citation

References

  • [1] L. Bertini and C. Toninelli. Exclusion processes with degenerate rates: Convergence to equilibrium and tagged particle. J. Statist. Phys. 117 (3–4) (2004) 549–580.
  • [2] O. Blondel. Tracer diffusion at low temperature in kinetically constrained models. Ann. Appl. Probab. 25 (3) (2015) 1079–1107.
  • [3] N. Cancrini, F. Martinelli, C. Roberto and C. Toninelli. Kinetically constrained spin models. Probab. Theory Related Fields 140 (3–4) (2008) 459–504.
  • [4] N. Cancrini, F. Martinelli, C. Roberto and C. Toninelli. Kinetically constrained lattice gases. Comm. Math. Phys. 297 (2) (2010) 299–344.
  • [5] J. T. Chayes and L. Chayes. Bulk transport properties and exponent inequalities for random resistor and flow networks. Comm. Math. Phys. 105 (1) (1986) 133–152.
  • [6] A. De Masi, P. A. Ferrari, S. Goldstein and W. D. Wick. An invariance principle for reversible Markov processes. Applications to random motions in random environments. J. Stat. Phys. 55 (3–4) (1989) 787–855.
  • [7] S. Franz, R. Mulet and G. Parisi. Kob–Andersen model: A nonstandard mechanism for the glassy transition. Phys. Rev. E 65, 021506.
  • [8] J. P. Garrahan, P. Sollich and C. Toninelli. Kinetically constrained models. In Dynamical Heterogeneities in Glasses, Colloids, and Granular Media, L. Berthier, G. Biroli, J.-P. Bouchaud, L. Cipelletti and W. van Saarloos (Eds). Oxford Univ. Press, London, 2011.
  • [9] P. Gonçalves, C. Landim and C. Toninelli. Hydrodynamic limit for a particle system with degenerate rates. Ann. Inst. Henri Poincaré Probab. Stat. 45 (4) (2009) 887–909.
  • [10] H. Kesten. Percolation Theory for Mathematicians. Progress in Probability and Statistics 2, iv+423 pp. Birkhäuser, Boston, MA, 1982.
  • [11] W. Kob and H. C. Andersen. Kinetic lattice-gas model of cage effects in high-density liquids and a test of mode-coupling theory of the ideal-glass transition. Phys. Rev. E (3) 48 (1993) 4359–4363.
  • [12] J. Kurchan, L. Peliti and M. Sellitto. Aging in lattice-gas models with constrained dynamics. Europhys. Lett. 39 (4) (1997) 365–370.
  • [13] E. Marinari and E. Pitard. Spatial correlations in the relaxation of the Kob–Andersen model. Europhys. Lett. 69 (2005) 35–241.
  • [14] Y. Nagahata. Lower bound estimate of the spectral gap for simple exclusion process with degenerate rates. Electron. J. Probab. 17 (2012) 92.
  • [15] J. Quastel. Diffusion of color in the simple exclusion process. Comm. Pure Appl. Math. 45 (6) (1992) 623–679.
  • [16] F. Ritort and P. Sollich. Glassy dynamics of kinetically constrained models. Adv. Phys. 52 (2003) 219–342.
  • [17] H. Spohn. Tracer diffusion in lattice gases. J. Stat. Phys. 59 (5–6) (1990) 1227–1239.
  • [18] H. Spohn. Large Scale Dynamics of Interacting Particles. Springer, Berlin, 1991.
  • [19] C. Toninelli and G. Biroli. Dynamical arrest, tracer diffusion and kinetically constrained lattice gases. J. Stat. Phys. 117 (1–2) (2004) 27–54.
  • [20] C. Toninelli, G. Biroli and D. S. Fisher. Cooperative behavior of kinetically constrained lattice gas models of glassy dynamics. J. Stat. Phys. 120 (1–2) (2005) 167–238.
  • [21] H.-T. Yau. Logarithmic Sobolev inequality for generalized simple exclusion processes. Probab. Theory Related Fields 109 (4) (1997) 507–538.