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The sphere expansion rate γ is defined as γ= 1 D dD dt, where D is the diameter of the sphere.
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In Ref., it was shown that jammed packings with a very narrow density distribution centered at any over a wide range ∈ [0.6,0.7408... ] with variable disorder can be generated in three dimensions. These results as well as those reported here support the view that there is no universal jamming point that is distinguishable based on the packing density alone and occurrence frequency. These results stress the importance of the "geometric- structure" approach to analyzing packing problems. In Ref., the SLP algorithm was employed to produce disordered jammed sphere packings in three dimensions with anomalously low densities. However, the algorithmic details of the SLP solution were not provided in that paper. Here we also use the SLP algorithm to generate disordered jammed sphere packings with anomalously low densities but using only initial conditions produced by the SLP algorithm. Additional details are provided in Sec. .
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In practice, the relative coordinates (rather than the global coordinates) are used in the algorithm due to the changing sphere positions with respect to the adaptive fundamental cell. Note that the terminology "global" coordinate was not explicitly used in Ref., even though the equations for the global positions of particle centers were explicitly given.
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S L P = 0.194, which indicate that the packings virtually have the same degree of disorder.
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