Dimensional study of the dynamical arrest in a random Lorentz gas

Yuliang Jin and Patrick Charbonneau

Phys. Rev. E 91, 042313 – Published 27 April 2015

Abstract

The random Lorentz gas (RLG) is a minimal model for transport in heterogeneous media. Upon increasing the obstacle density, it exhibits a growing subdiffusive transport regime and then a dynamical arrest. Here, we study the dimensional dependence of the dynamical arrest, which can be mapped onto the void percolation transition for Poisson-distributed point obstacles. We numerically determine the arrest in dimensions $d = 2 – 6$ . Comparison of the results with standard mode-coupling theory reveals that the dynamical theory prediction grows increasingly worse with $d$ . In an effort to clarify the origin of this discrepancy, we relate the dynamical arrest in the RLG to the dynamic glass transition of the infinite-range Mari-Kurchan-model glass former. Through a mixed static and dynamical analysis, we then extract an improved dimensional scaling form as well as a geometrical upper bound for the arrest. The results suggest that understanding the asymptotic behavior of the random Lorentz gas may be key to surmounting fundamental difficulties with the mode-coupling theory of glasses.

Received 1 September 2014
Revised 9 March 2015

DOI:https://doi.org/10.1103/PhysRevE.91.042313

Authors & Affiliations

Yuliang Jin^1,2,3,* and Patrick Charbonneau^1,4

¹Department of Chemistry, Duke University, Durham, North Carolina 27708, USA
²Dipartimento di Fisica, Sapienza Università di Roma and INFN, Sezione di Roma I, IPFC-CNR, Piazzale Aldo Moro 2, I-00185 Roma, Italy
³LPT, École Normale Supérieure, UMR 8549 CNRS, 24 Rue Lhomond, 75005 Paris, France
⁴Department of Physics, Duke University, Durham, North Carolina 27708, USA

^*Corresponding author: yuliang.jin@lpt.ens.fr

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Vol. 91, Iss. 4 — April 2015

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Images

Figure 1
Determination of $η_{p}$ in $d = 4 – 6$ (a)–(c) from fitting Eq. (2) (lines) to the large-system-size results [14]. (d) Critical scaling $Δ η_{p} \sim N^{- 1 / d ν}$ [from Eqs. (1) and (2)] in $d = 2 – 6$ (top to bottom). In $d = 2 – 3$ $ν$ is extracted by fitting Eq. (1) to the numerical results (lines), but in $d = 4 – 6$ large finite-size effects prevent reliable fitting, so the lattice percolation values, $ν = 0.68, 0.57$ , and 0.50, respectively, are used instead (lines) [15].
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Figure 2
The critical nonergodicity parameter $f_{p} (q)$ in large dimensions. As $d$ increases, $f_{p} (q)$ converges to a step function $f_{p} (q) = θ (q_{0} - q)$ , with the cutoff wave vector scaling as $q_{0} σ ≅ 0.15 d^{3 / 2}$ , which is fully consistent with the HS result [16].
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Figure 3
Numerical void percolation threshold $φ_{p}$ (circles) compared with the numerical MCT prediction (squares), the analytical MCT prediction under the generalized hydrodynamic approximation ${\bar{φ}}_{p}^{MCT} = d 2^{- d}$ (black solid line), and our estimated asymptotic high- $d$ scaling $φ_{p}^{\infty} ≃ 3.7 d 2^{- d}$ (dashed line).
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Figure 4
Dimensional behavior of the numerical MCT critical packing fraction $φ_{p}^{MCT}$ (squares). The blue solid line is the asymptotic large- $d$ scaling $φ_{p}^{MCT} = 0.22 d^{2} 2^{- d}$ . The black solid line is the MCT result with the generalized hydrodynamic approximation, ${\bar{φ}}_{p}^{MCT} = d 2^{- d}$ .
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Figure 5
Schematics of (a) the cage network suggested by the finite-dimensional analysis of the MK model [25], where balls represent cages and tubes the hopping channels. For the MK model (b), the central green circle is the caged particle, blue circles are its neighbors, and the arrow represents a possible hopping path; for void percolation (c), the central green point is the pointlike tracer and blue circles are scatterers; for the RLG (d), the tracer and scatterer sizes are exchanged. The MK model (b) is mapped onto void percolation (c) by rescaling the diameter of neighbors $σ \to 2 σ$ , and replacing the central particle by a point. A similar mapping can also be done from the MK model (b) to the RLG (d).
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Figure 6
Schematic plots of the accessible space for the tracer in the RRLG at different densities (left panels) and corresponding MSD behavior (right panels, logarithmic scale, $φ$ increasing from top to bottom) in dimensions (a) $d < d_{c}$ , (b) $d > d_{c}$ , and (c) $d = \infty$ . The MSD is expected to exhibit different regimes: (1) at short times, before any collision occurs, ballistic motions gives $δ r^{2} (t) \equiv 〈 {[r (t) - r (0)]}^{2} 〉 \sim t^{2}$ ; (2) in the delocalized phase, the tracer is diffusive at very long times, $δ r^{2} (t) \sim t$ ; (3) close to the percolation transition, the fractal nature of the void space gives a subdiffusive scaling, $δ r^{2} (t) \sim t^{2 / z}$ with $z > 2$ ; (4) by contrast to the RLG, the rattling contribution from scatterers should give rise to a caging plateau on time scales shorter than hopping; (5) in the localized phase the MSD reaches a second plateau that corresponds to the typical size of the cage network. At $φ_{d}$ , the typical cage size (the first plateau) is equal to $A_{d}$ (red dashed lines) in the MK model. Across $d_{c}, φ_{d}$ and $φ_{p}^{RRLG}$ switch order. In $d = \infty, φ_{p} = φ_{p}^{RRLG}$ , and the two plateaus merge into a single one. We expect the MK model to have an almost identical behavior to that of the RRLG at and above $φ_{d}$ , although with shrinking cages with $φ$ .
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