Vibrational density of states of jammed packing: mean-field theory

TL;DR

This paper develops a mean-field approach to approximate the vibrational density of states in amorphous solids by mapping the Hessian matrix to a random matrix, showing good agreement with simulations especially in higher dimensions.

Contribution

It introduces a novel method to relate the Hessian of amorphous solids to a random matrix, improving understanding of vibrational properties near the jamming transition.

Findings

Good agreement with numerical simulations in 3D at low pressure

Better agreement in higher dimensions, supporting mean-field predictions

Approximation becomes exact as spatial dimension increases

Abstract

Several mean-field theories predict that Hessian matrices of amorphous solids can be written by using the random matrix in the limit of the large spatial dimensions $d\to\infty$. Motivated by these results, we here propose a way to map a Hessian of the amorphous solid to a random matrix. This is possible by determining the coefficients of a random matrix so that the trace of the random matrix coincides with the Hessian of the original system. We compare our result with that of previous numerical simulations of harmonic spheres in several spatial dimensions $d=3$, $5$, and $9$. For small pressure $p\ll 1$ (near jamming), we find a good agreement even in $d=3$, and obtain better agreements in larger $d$, suggesting that the approximation indeed becomes exact in the limit of large spatial dimensions.