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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 $1-q$ 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 $D\sim\operatorname{gap}^{\xi}$ with $\xi<1$.

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

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.25 • No. 3 • June 2015
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