We use molecular dynamics to investigate the glass transition occuring at large volume fraction, $\phi$, and low temperature, $T$, in assemblies of soft repulsive particles. We find that equilibrium dynamics in the ($\phi$, $T$) plane obey critical scaling in the proximity of a critical point at $T=0$ and $\phi=\phi_0$, which corresponds to the ideal glass transition of hard spheres. This glass critical point, `point $G$', is distinct from athermal jamming thresholds. A remarkable consequence of scaling behaviour is that the dynamics at fixed $\phi$ passes smoothly from that of a strong glass to that of a very fragile glass as $\varphi$ increases beyond $\varphi_0$. We explore correlations between fragility and various physical properties.

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(w) of vibrational modes of angular frequency w 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(w) toward a non-zero constant as w goes to 0, b) appearance of a low-frequency cutoff w*, and c) power-law increase of w* with compression. Differences between these two systems occur at lower frequency. The density of states of the modified system displays an abrupt plateau that appears at w*, 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 dz: dz > p^(1/2), 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.

Glasses have a large excess of low-frequency vibrational modes in comparison with crystalline solids. We show that such a feature is a necessary consequence of the geometry generic to a marginally connected solid. In particular, we analyze the density of states of a recently simulated system, comprised of weakly compressed spheres at zero temperature. We account for the observed a) constancy of the density of modes with frequency, b) appearance of a low-frequency cutoff, and c) power-law increase of this cutoff with compression. We predict a length scale below which the boundary conditions strongly affect the system.

By employing the adaptive network simulation method, we demonstrate that the ensemble-averaged stress caused by a local force for packings of frictionless rigid beads is concentrated along rays whose slope is consistent with unity: forces propagate along lines at 45 degrees to the horizontal or vertical. This slope is shown to be independent of polydispersity or the degree to which the system is sheared. Further confirmation of this result comes from fitting the components of the stress tensor to the null stress constitutive equation. The magnitude of the response is also shown to fall off with the -1/2 power of distance. We argue that our findings are a natural consequence of a system that preserves its volume under small perturbations.