Stability of jammed packings II: the transverse length scale

TL;DR

This paper investigates how large amorphous sphere packings respond to boundary perturbations, revealing a transverse length scale that diverges at the jamming transition and determines system stability.

Contribution

It introduces a detailed analysis of boundary response and vibrational modes, identifying a transverse length scale that governs stability near the jamming transition.

Findings

Boundary perturbations become irrelevant beyond a certain length scale.

The transverse length scale diverges at the jamming transition.

Stability is governed by the competition between plane waves and anomalous modes.

Abstract

As a function of packing fraction at zero temperature and applied stress, an amorphous packing of spheres exhibits a jamming transition where the system is sensitive to boundary conditions even in the thermodynamic limit. Upon further compression, the system should become insensitive to boundary conditions provided it is sufficiently large. Here we explore the linear response to a large class of boundary perturbations in 2 and 3 dimensions. We consider each finite packing with periodic-boundary conditions as the basis of an infinite square or cubic lattice and study properties of vibrational modes at arbitrary wave vector. We find that the stability of such modes be understood in terms of a competition between plane waves and the anomalous vibrational modes associated with the jamming transition; infinitesimal boundary perturbations become irrelevant for systems that are larger than a length scale that characterizes the transverse excitations. This previously identified length diverges at the jamming transition.