Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Kinetically constrained lattice gases: Tagged particle diffusion

O. Blondel and C. Toninelli

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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.


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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Kawasaki dynamics Tagged particle Kinetically constrained models


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.

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