Mean-field model for the density of states of jammed soft spheres

Abstract

We propose a class of mean-field models for the isostatic transition of systems of soft spheres, in which the contact network is modeled as a random graph and each contact is associated to d degrees of freedom. We study such models in the hypostatic, isostatic, and hyperstatic regimes. The density of states is evaluated by both the cavity method and exact diagonalization of the dynamical matrix. We show that the model correctly reproduces the main features of the density of states of real packings and, moreover, it predicts the presence of localized modes near the lower band edge. Finally, the behavior of the density of states D(ω)∼ω^{α} for ω→0 in the hyperstatic regime is studied. We find that the model predicts a nontrivial dependence of α on the details of the coordination distribution.

Simple random matrix model for the vibrational spectrum of structural glasses

“…We see an ω 4 regime when bond strengths are uniform, but it is unclear what quantity would be analogous to a uniform bond weight in a d × d sub-block. Concurrent work by Benetti et al focused on d-dimensional Laplacian matrices where the magnitude of each bond is unity, but the geometry of the bond is randomly distributed, and these also generate scaling consistent with ω 4 at low frequencies [28]. To better understand the connections between these models and why both generate ω 4 scaling, one could study systems with random bond weights and ordered geometries, or have both be disordered.…”

Renormalization to localization without a small parameter

“…Due to quite general description, ERMs cover a considerable class of physical models and arise naturally in various systems, e.g., in the ones with non-crystalline structures like gases, liquids, amorphous materials, and glasses. Although such models sometimes arise in the systems with shortrange interactions, such as elastic networks [39], jammed soft spheres [40] or magnetic vortex plasma [41], more commonly ERMs are used to describe the long-range models. Indeed, longrange ERMs are applied to the analysis of the systems of particles with Coulomb interactions in two-dimensional irregular confinement [42], disordered classical Heisenberg magnets with uniform antiferromagnetic interactions [43], systems with dipole-dipole interactions such as dipolar gases [44], systems of ultracold Rydberg atoms [45] and so on.…”

Proof of a Conjecture on the Infinite Dimension Limit of a Unifying Model for Random Matrix Theory

“…However random block matrices were often analyzed by the cavity method [2], [3], [15], [16], familiar in statistical physics and this seems possible also in the present model.…”