Jammed particulate systems are inherently nonharmonic

Jammed particulate systems, such as granular media, colloids, and foams, interact via one-sided forces that are nonzero only when particles overlap. We find that systems with one-sided repulsive interactions possess no linear response regime in the large system limit ($N\to \mathrm{\infty }$) for all pressures $p$ (or compressions $\mathrm{\Delta }\varphi$), and for all $N$ near jamming onset $p\to 0$. We perform simulations on 2D frictionless bidisperse mechanically stable disk packings over a range of packing fractions $\Delta \phi = \phi-\phi_J$abovejammingonset$\phi_J$. We apply perturbations with amplitude $\delta$ to the packings along each eigen-direction from the dynamical matrix and determine whether the response of the system evolving at constant energy remains in the original eigenmode of the perturbation. For $\delta >{\delta }_c$, which we calculate analytically, a single contact breaks and fluctuations abruptly spread to all harmonic modes. As $\delta$ increases further all discrete harmonic modes disappear into a continuous frequency band. We find that $<{\delta }_c>\sim \mathrm{\Delta }\varphi \mathrm{/}{N}^{\lambda }$, where $1 > \lambda > 0.5$, and thus jammed particulate systems are inherently nonharmonic with no linear vibrational response regime as $N\to \mathrm{\infty }$ over the full range of $\mathrm{\Delta }\varphi$, and as $\mathrm{\Delta }\varphi \to 0$ at any $N$.