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) similar to f(theta e), the force distribution of such pairs and phi(c) the packing fraction at which pressure diverges, we predict that (i) the density of states has a low-frequency peak at a scale.*, rising up to it as D(omega)similar to omega(2+a), and decaying above omega* as D(omega)similar to omega(-a) where a=(1-theta(e))/(3+theta(e)) and omega is the frequency, (ii) shear modulus and mean-squared displacement are inversely proportional with <delta R-2>similar to 1/mu similar to(phi(c)-phi)(k), where kappa= 2-2=(3+theta(e)), and (iii) continuum elasticity breaks down on a scale l(c) similar to 1/root delta z similar to(phi(c)-phi)(-b), where b=(1+theta(e))/(6+2 theta(e)) and partial derivative z = z - 2d, where z is the coordination and d the spatial dimension. We numerically test (i) and provide data supporting that theta(e) approximate to 0.41 in our bidisperse system, independently of system preparation in two and three dimensions, leading to kappa approximate to 1.41, a approximate to 0.17, and b approximate to 0.21. Our results for the mean-square displacement are consistent with a recent exact replica computation for d =infinity, whereas some observations differ, as rationalized by the present approach.