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

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.

5 pages, 2 figures

References in corpus (9)

• Jamming at Zero Temperature and Zero Applied Stress: the Epitome of Disorder
• Marginal Stability in Structural, Spin and Electron Glasses
• Introduction to Random Matrices - Theory and Practice
• The Ioffe-Regel criterion and diffusion of vibrations in random lattices
• The Statistical Physics of Athermal Materials
• Exact theory of dense amorphous hard spheres in high dimension. I. The free energy
• Low-frequency vibrations of jammed packings in large spatial dimensions
• Jamming and replica symmetry breaking of weakly disordered crystals
• Random quench predicts universal properties of amorphous solids

Cited by in corpus (3)

• Complete mathematical theory of the jamming transition: A perspective
• Quasi-localized vibrational modes, Boson peak and sound attenuation in model mass-spring networks
• Two-level systems and harmonic excitations in a mean-field anharmonic quantum glass