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

TL;DR

This paper introduces a mean-field model for the density of states in jammed soft spheres, capturing key features of real packings and predicting localized modes and the low-frequency behavior.

Contribution

The paper develops a novel mean-field approach modeling contact networks as random graphs, accurately reproducing density of states features and low-frequency scaling in jammed systems.

Findings

Model reproduces main features of real packings' density of states.

Predicts localized modes near the lower band edge.

Shows nontrivial dependence of low-frequency exponent on coordination distribution.

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 $D(\omega)\sim\omega^\alpha$ for $\omega\to 0$ in the hyperstatic regime is studied. We find that the model predicts a nontrivial dependence of $\alpha$ on the details of the coordination distribution.