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 ($N\rightarrow \infty$) for all pressures $p$ (or compressions $Δϕ$), and for all $N$ near jamming onset $p\rightarrow 0$. We perform simulations on 2D frictionless bidisperse mechanically stable disk packings over a range of packing fractions $Δϕ= ϕ-ϕ_J$ above jamming onset $ϕ_J$. We apply perturbations with amplitude $δ$ 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 $δ> δ_c$, which we calculate analytically, a single contact breaks and fluctuations abruptly spread to all harmonic modes. As $δ$ increases further all discrete harmonic modes disappear into a continuous frequency band. We find that $<δ_c >\sim Δϕ/N^λ$, where $1 > λ> 0.5$, and thus jammed particulate systems are inherently nonharmonic with no linear vibrational response regime as $N\rightarrow \infty$ over the full range of $Δϕ$, and as $Δϕ\rightarrow 0$ at any $N$.

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