Note: Effect of localization on mean-field density of state near jamming

TL;DR

This paper investigates how localized vibrational modes influence the density of states near the jamming transition, providing a theoretical framework that matches numerical results and explains the observed scaling behavior.

Contribution

The study introduces a theoretical approach that captures the effect of localized modes on the density of states near jamming, including the pressure-dependent $\,\,\,\,D(\,\,\,\, ext{ extomega}) ext{ extasciicircum}4$ scaling.

Findings

Theoretical model reproduces numerical results near jamming.

Successfully explains the $\, ext{ extomega}^4$ scaling of $D( ext{ extomega})$.

Captures pressure dependence of the scaling pre-factor.

Abstract

We discuss the effects of the localized modes on the density of state $D(\omega)$ by introducing the probability distribution function of the proximity to the marginal stability. Our theoretical treatment reproduces the numerical results in finite dimensions near the jamming point., in particular, successfully captures the novel $D(\omega)\sim \omega^4$ scaling including its pressure dependence of the pre-factor.