Jamming transitions in amorphous packings of frictionless spheres occur over a continuous range of volume fractions

We numerically produce fully amorphous assemblies of frictionless spheres in three dimensions and study the jamming transition these packings undergo at large volume fractions. We specify four protocols yielding a critical value for the jamming volume fraction which is sharply defined in the limit of large system size, but is different for each protocol. Thus, we directly establish the existence of a continuous range of volume fraction where nonequilibrium jamming transitions occur. However, these jamming transitions share the same critical behaviour. Our results suggest that, even in the absence of partial crystalline ordering, a unique location of a random close packing does not exist, and that volume fraction alone is not sufficient to describe the properties of jammed states.