Open Access
June 2015 Tracer diffusion at low temperature in kinetically constrained models
Oriane Blondel
Ann. Appl. Probab. 25(3): 1079-1107 (June 2015). DOI: 10.1214/14-AAP1017

Abstract

We describe the motion of a tracer in an environment given by a kinetically constrained spin model (KCSM) at equilibrium. We check convergence of its trajectory properly rescaled to a Brownian motion and positivity of the diffusion coefficient D as soon as the spectral gap of the environment is positive (which coincides with the ergodicity region under general conditions). Then we study the asymptotic behavior of D when the density 1q of the environment goes to 1 in two classes of KCSM. For noncooperative models, the diffusion coefficient D scales like a power of q, with an exponent that we compute explicitly. In the case of the Fredrickson–Andersen one-spin facilitated model, this proves a prediction made in Jung, Garrahan and Chandler [Phys. Rev. E 69 (2004) 061205]. For the East model, instead we prove that the diffusion coefficient is comparable to the spectral gap, which goes to zero faster than any power of q. This result contradicts the prediction of physicists (Jung, Garrahan and Chandler [Phys. Rev. E 69 (2004) 061205; J. Chem. Phys. 123 (2005) 084509]), based on numerical simulations, that suggested Dgapξ with ξ<1.

Citation

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Oriane Blondel. "Tracer diffusion at low temperature in kinetically constrained models." Ann. Appl. Probab. 25 (3) 1079 - 1107, June 2015. https://doi.org/10.1214/14-AAP1017

Information

Published: June 2015
First available in Project Euclid: 23 March 2015

zbMATH: 1317.82054
MathSciNet: MR3325269
Digital Object Identifier: 10.1214/14-AAP1017

Subjects:
Primary: 82D30
Secondary: 60K37

Keywords: glassy systems , Kinetically constrained models , random environment , Tracer diffusion

Rights: Copyright © 2015 Institute of Mathematical Statistics

Vol.25 • No. 3 • June 2015
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