The Annals of Applied Probability

Tracer diffusion at low temperature in kinetically constrained models

Oriane Blondel

Full-text: Open access

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 $\xi<1$.

Article information

Source
Ann. Appl. Probab., Volume 25, Number 3 (2015), 1079-1107.

Dates
First available in Project Euclid: 23 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1427124124

Digital Object Identifier
doi:10.1214/14-AAP1017

Mathematical Reviews number (MathSciNet)
MR3325269

Zentralblatt MATH identifier
1317.82054

Subjects
Primary: 82D30: Random media, disordered materials (including liquid crystals and spin glasses)
Secondary: 60K37: Processes in random environments

Keywords
Tracer diffusion kinetically constrained models glassy systems random environment

Citation

Blondel, Oriane. Tracer diffusion at low temperature in kinetically constrained models. Ann. Appl. Probab. 25 (2015), no. 3, 1079--1107. doi:10.1214/14-AAP1017. https://projecteuclid.org/euclid.aoap/1427124124


Export citation

References

  • Aldous, D. and Diaconis, P. (2002). The asymmetric one-dimensional constrained Ising model: Rigorous results. J. Stat. Phys. 107 945–975.
  • Bertini, L. and Toninelli, C. (2004). Exclusion processes with degenerate rates: Convergence to equilibrium and tagged particle. J. Stat. Phys. 117 549–580.
  • Cancrini, N., Martinelli, F., Roberto, C. and Toninelli, C. (2008). Kinetically constrained spin models. Probab. Theory Related Fields 140 459–504.
  • Cancrini, N., Martinelli, F., Schonmann, R. and Toninelli, C. (2010). Facilitated oriented spin models: Some nonequilibrium results. J. Stat. Phys. 138 1109–1123.
  • Chang, I. and Sillescu, H. (1997). Heterogeneity at the glass transition: Transational and rotational self-diffusion. J. Phys. Chem. B 101 8794–8801.
  • Chleboun, P., Faggionato, A. and Martinelli, F. (2012). Time scale separation and dynamic heterogeneity in the low temperature East model. Available at arXiv:1212.2399.
  • Cicerone, M. T. and Ediger, M. D. (1996). Enhanced translation of probe molecules in supercooled o-terphenyl: Signature of spatially heterogeneous dynamics? J. Chem. Phys. 104 7210–7218.
  • De Masi, A., Ferrari, P. A., Goldstein, S. and Wick, W. D. (1989). An invariance principle for reversible Markov processes. Applications to random motions in random environments. J. Stat. Phys. 55 787–855.
  • Edmond, K. V., Elsesser, M. T., Hunter, G. L., Pine, D. J. and Weeks, E. R. (2012). Decoupling of rotational and translational diffusion in supercooled colloidal fluids. Proc. Natl. Acad. Sci. USA 109 17891–17896.
  • Jung, Y., Garrahan, J. P. and Chandler, D. (2004). Excitation lines and the breakdown of Stokes–Einstein relations in supercooled liquids. Phys. Rev. E (3) 69 061205.
  • Jung, Y., Garrahan, J. P. and Chandler, D. (2005). Dynamical exchanges in facilitated models of supercooled liquids. J. Chem. Phys. 123 084509.
  • Kipnis, C. and Varadhan, S. R. S. (1986). Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 1–19.
  • Komorowski, T., Landim, C. and Olla, S. (2012). Fluctuations in Markov Processes: Time Symmetry and Martingale Approximation. Grundlehren der Mathematischen Wissenschaften 345. Springer, Heidelberg.
  • Spohn, H. (1990). Tracer diffusion in lattice gases. J. Stat. Phys. 59 1227–1239.
  • Swallen, S. F., Bonvallet, P. A., McMahon, R. J. and Ediger, M. D. (2003). Self-diffusion of tris-naphthylbenzene near the glass transition temperature. Phys. Rev. Lett. 90 015901.