Vibrations and Diverging Length Scales Near the Unjamming Transition

Abstract

We numerically study the vibrations of jammed packings of particles interacting with finite-range, repulsive potentials at zero temperature. As the packing fraction $\ensuremath{\phi}$ is lowered towards the onset of unjamming at ${\ensuremath{\phi}}_{c}$, the density of vibrational states approaches a nonzero value in the limit of zero frequency. For $\ensuremath{\phi}>{\ensuremath{\phi}}_{c}$, there is a crossover frequency, ${\ensuremath{\omega}}^{*}$ below which the density of states drops towards zero. This crossover frequency obeys power-law scaling with $\ensuremath{\phi}\mathrm{\text{\ensuremath{-}}}{\ensuremath{\phi}}_{c}$. Characteristic length scales, determined from the dominant wave vector contributing to the eigenmode at ${\ensuremath{\omega}}^{*}$, diverge as power laws at the unjamming transition.