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We numerically produce fully amorphous assemblies of frictionless spheres inthree dimensions and study the jamming transition these packings undergo atlarge volume fractions. We specify four protocols yielding a critical value forthe jamming volume fraction which is sharply defined in the limit of largesystem size, but is different for each protocol. Thus, we directly establishthe existence of a continuous range of volume fraction where nonequilibriumjamming transitions occur. However, these jamming transitions share the samecritical behaviour. Our results suggest that, even in the absence of partialcrystalline ordering, a unique location of a random close packing does notexist, and that volume fraction alone is not sufficient to describe theproperties of jammed states.

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