Vibrational spectrum of Granular packings with random matrices

Area of Science:

• Physics
• Statistical Mechanics
• Materials Science

Background:

• The vibrational spectrum of granular packings is a key indicator of the jamming transition.
• Random matrix theory has been proposed to approximate these vibrational spectra.

Purpose of the Study:

• To validate the use of random matrix theory for describing granular packing vibrations.
• To identify the specific random matrix ensemble that accurately models jammed granular matter.

Main Methods:

• Dynamical numerical simulations of frictionless bead packs near the jamming point.
• Analysis of the autocorrelation function of the density of states.
• Comparison with predictions from the Laguerre orthogonal ensemble and Gaussian orthogonal ensemble of random matrices.
• Development and analysis of a physically motivated random lattice model.

Main Results:

• The vibrational spectrum autocorrelation function from simulations closely matches the Laguerre orthogonal ensemble predictions.
• Significant disagreement was found with the Gaussian orthogonal ensemble predictions.
• The proposed random lattice model successfully reproduces key features of granular vibrational density of states and Laguerre ensemble correlations.

Conclusions:

• The Laguerre orthogonal ensemble correctly captures the universal statistical properties of jammed granular matter.
• The Gaussian orthogonal ensemble is excluded as a suitable model for these systems.
• The new random lattice model provides a physically grounded approach to studying granular vibrations and their statistical properties.