{
  "id": "1306.6500",
  "version": "v4",
  "published": "2013-06-27T13:51:56.000Z",
  "updated": "2015-05-15T13:21:57.000Z",
  "title": "Tracer diffusion at low temperature in kinetically constrained models",
  "authors": [
    "Oriane Blondel"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/14-AAP1017 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Applied Probability 2015, Vol. 25, No. 3, 1079-1107",
  "doi": "10.1214/14-AAP1017",
  "categories": [
    "math.PR",
    "cond-mat.stat-mech",
    "math-ph",
    "math.MP"
  ],
  "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 $\\xi<1$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-02-28T02:18:27.000Z",
      "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 behaviour of $D$ when the density $1-q$ of the environment goes to 1 in two classes of KCSM. For non-cooperative 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 \\cite{junggarrahanchandler}. 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 proves that the trend found in numerical simulation results (\\cite{junggarrahanchandler}, \\cite{junggarrahanchandler2}), $D\\sim \\mathrm{gap}^\\xi$ with $\\xi<1$, cannot hold in the limit $q\\rightarrow 0$.",
      "comment": "24 pages, 4 figures; v2: presentation improved, minor corrections, added acknowledgement and remark - version to appear in Annals of Applied Probability; v3: modified end of abstract",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2015-05-15T13:21:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "kinetically constrained models",
      "tracer diffusion",
      "low temperature",
      "diffusion coefficient",
      "spectral gap"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.6500B"
    }
  }
}