Force distribution affects vibrational properties in hard-sphere glasses

We theoretically and numerically study 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)∼f^θe, 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)∼ω^2+a, and decaying above ω* as D(ω)∼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〈δR²〉∼1/μ∼(ϕ_c−ϕ)^κ, where κ=2−2/(3+θ_e), and (iii) continuum elasticity breaks down on a scale ℓ_c∼1/√δz∼(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≈0.41  in our bidisperse system, independently of system preparation in two and three dimensions, leading to κ≈1.41, a≈0.17, and b≈0.21. Our results for the mean-square displacement are consistent with a recent exact replica computation for d=∞, whereas some observations differ, as rationalized by the present approach.