Localization behavior of vibrational modes in granular packings

TL;DR

This paper investigates vibrational mode localization in frictionless granular media, introducing a new method to analyze localization length across regimes, revealing quasi-localized modes near jamming and scaling relations consistent with Random Matrix Theory.

Contribution

A novel method for studying vibrational mode localization in granular packings, applicable across localized and extended regimes, and deriving new scaling relations.

Findings

Lowest frequency modes show quasi-localized resonances near jamming

Localization length scales with system size as L^{d/2} in extended regime

Data collapse indicates no signature of jamming proximity in higher frequency modes

Abstract

We study the localization of vibrational modes of frictionless granular media. We introduce a new method, motivated by earlier work on non-Hermitian quantum problems, which works well both in the localized regime where the localization length $\xi$ is much less than the linear size $L$ and in the regime $\xi$ grater or of order $L$ when modes are extended throughout our finite system. Our very lowest frequency modes show "quasi-localized" resonances away from the jamming point; the spatial extent of these regions increases as the jamming point is approached, as expected theoretically. Throughout the remaining frequency range, our data show no signature of the nearness of the jamming point and collapse well when properly rescaled with the system size. Using Random Matrix Theory we derive the scaling relation $\xi$ ~ $L^{d/2}$ for the regime $\xi$ >> $L$ in $d$ dimensions.