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Glasses have a large excess of low-frequency vibrational modes in comparison with most crystalline solids. We show that such a feature is a necessary consequence of the weak connectivity of the solid, and that the frequency of modes in excess is very sensitive to the pressure. We analyze, in particular, two systems whose density D(omega) of vibrational modes of angular frequency omega display scaling behaviors with the packing fraction: (i) simulations of jammed packings of particles interacting through finite-range, purely repulsive potentials, comprised of weakly compressed spheres at zero temperature and (ii) a system with the same network of contacts, but where the force between any particles in contact (and therefore the total pressure) is set to zero. We account in the two cases for the observed (a) convergence of D(omega) toward a nonzero constant as omega-->0, (b) appearance of a low-frequency cutoff omega*, and (c) power-law increase of omega* with compression. Differences between these two systems occur at a lower frequency. The density of states of the modified system displays an abrupt plateau that appears at omega*, below which we expect the system to behave as a normal, continuous, elastic body. In the unmodified system, the pressure lowers the frequency of the modes in excess. The requirement of stability despite the destabilizing effect of pressure yields a lower bound on the number of extra contact per particle deltaz:deltaz> or =p1/2, which generalizes the Maxwell criterion for rigidity when pressure is present. This scaling behavior is observed in the simulations. We finally discuss how the cooling procedure can affect the microscopic structure and the density of normal modes.
Wyart et al. (Wed,) studied this question.