Vibrations in jammed solids: Beyond linear response

We propose a `phase diagram’ for particulate systems that interact via purely repulsive contact forces, such as granular media and colloidal suspensions. We identify and characterize two distinct classes of behavior as a function of the input kinetic energy per degree of freedom $T_0$ and packing fraction deviation above and below jamming onset $\mathrm{\Delta }\varphi =\varphi -{\varphi }_J$ using numerical simulations of purely repulsive frictionless disks. Iso-coordinated solids (ICS) only occur above jamming for $\mathrm{\Delta }\varphi >\mathrm{\Delta }{\varphi }_c(T_0)$; they possess average coordination number equal to the isostatic value ($< z> = z_{\rm iso}$) required for mechanically stable packings. ICS display harmonic vibrational response, where the density of vibrational modes from the Fourier transform of the velocity autocorrelation function is a set of sharp peaks at eigenfrequencies ${\omega }_k^d$ of the dynamical matrix evaluated at $T_0=0$. Hypo-coordinated solids (HCS) occur both above and below jamming onset within the region defined by $\mathrm{\Delta }\varphi >\mathrm{\Delta }{\varphi }_{-}^{\ast }(T_0)$, $\Delta \phi < \Delta \phi^+(T_0)$,and$\Delta \phi > \Delta \phi{cb}(T_0)$. In this region, the network of interparticle contacts fluctuates with $< z> \approx z_{\rm iso}/2$, but cage-breaking particle rearrangements do not occur. The HCS vibrational response is nonharmonic, {\it i.e} the density of vibrational modes $D(\omega )$ is not a collection of sharp peaks at ${\omega }_k^d$, and its precise form depends on the measurement method. For $\Delta \phi > \Delta \phi_{cb}(T_0)$and$\Delta \phi < \Delta \phi^_{-}(T_0)$, the system behaves as a hard-particle liquid.