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The distribution of forces affects vibrational properties in hard sphere glasses

We study theoretically and numerically the elastic properties of hard sphere glasses, and provide a real-space description of their mechanical stability. In contrast to repulsive particles at zero-temperature, we argue that the presence of certain pairs of particles interacting with a small force $f$ soften elastic properties. This softening affects the exponents characterizing elasticity at high pressure, leading to experimentally testable predictions. Denoting $P(f)\sim f^{θ_e}$ the force distribution of such pairs and $ϕ_c$ the packing fraction at which pressure diverges, we predict that (i) the density of states has a low-frequency peak at a scale $ω^*$, rising up to it as $D(ω) \sim ω^{2+a}$, and decaying above $ω^*$ as $D(ω)\sim ω^{-a}$ where $a=(1-θ_e)/(3+θ_e)$ and $ω$ is the frequency, (ii) shear modulus and mean-squared displacement are inversely proportional with $\langle δR^2\rangle\sim1/μ\sim (ϕ_c-ϕ)^κ $ where $κ=2-2/(3+θ_e)$, and (iii) continuum elasticity breaks down on a scale $\ell_c \sim1/\sqrt{δz}\sim (ϕ_c-ϕ)^{-b}$ where $b=(1+θ_e)/(6+2θ_e)$ and $δz=z-2d$, where $z$ is the coordination and $d$ the spatial dimension. We numerically test (i) and provide data supporting that $θ_e\approx 0.41$ in our bi-disperse system, independently of system preparation in two and three dimensions, leading to $κ\approx1.41$, $a \approx 0.17$, and $b\approx 0.21$. Our results for the mean-square displacement are consistent with a recent exact replica computation for $d=\infty$, whereas some observations differ, as rationalized by the present approach.