Caging and Transport in Simple Disordered Systems
Recent advances on the glass problem motivate reexamining classical models of caging and transport. In particular, seemingly incompatible percolation and mean-field caging descriptions on the localization transition call for better understanding both. In light of this fundamental inconsistency, we study the caging and transport of a series of simple disordered systems.
We first consider the dynamics of site percolation on hypercubic lattices. Using theory and simulations, we obtain that both caging and subdiffusion scale logarithmically for dimension d ≥ d_u, the upper critical dimension of percolation. The theoretical derivation on Bethe lattice and a random graph confirm that logarithmic scalings should persist in the limit d→∞. The computational validation evaluates directly the dynamical critical exponents below d_u as well as their logarithmic scaling above d_u. Our numerical results improve various earlier estimates and are fully consistent with our theoretical predictions.
Recent implementation of efficient simulation algorithms for high-dimensional systems also facilitates the study of dense packing lattices beyond the conventional hypercubic ones. Here, we consider the percolation problem on checkerboard D_d lattices and on E_8 relatives for d=6 to 9. Precise estimates for both site and bond percolation thresholds obtained from invasion percolation simulations are compared with dimensional series expansion based on lattice animal enumeration for D_d lattices. As expected, the bond percolation threshold rapidly approaches the Bethe lattice limit as d increases for these high-connectivity lattices. Corrections, however, exhibit clear yet unexplained trends.
The random Lorentz gas (RLG) is a minimal model for transport in disordered media. Despite the broad relevance of the model, theoretical grasp over its properties remains weak. Here, we first extend analytical expectations for asymptotic high-d bounds on the void percolation threshold, and then computationally evaluate both the threshold and its criticality in various d. A simple modification of the RLG is found to bring the mean-field-like caging down to d=3.
The RLG also provides a toy model of particle caging, which is known to be relevant for describing the discontinuous dynamical transition of glasses. Following the percolation studies, we consider its exact mean-field solution in the d→∞ limit and perform simulation in d=2...20. We find that for sufficiently high d the mean-field caging transition precedes and prevents the percolation transition, which only happens on timescales diverging with d. This perturbative correction is associated with the cage heterogeneity. We further show that activated processes related to rare cage escapes destroy the glass transition in finite dimensions, leading to a rich interplay between glassiness and percolation physics. This advance suggests that the RLG can be used as a toy model to develop a first-principle description of particle hopping in structural glasses.
While the cages in the RLG are formed by non-interacting obstacles, cage structure is important for the hopping process in three-dimensional glasses. As a final note and also a future direction, a study on the three-dimensional polydisperse hard spheres with modification, named as the Mari-Kurchan-Krzakala (MKK) model was proposed. This consideration provides a controllable way to interpolate between the mean-field and the real space glasses. These insights help chart a path toward a complete description of finite-dimensional glasses.
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