Granular crystal cluster

Minkowski structure metrics are well-suited to detect order in amorphous point patterns [1].

For example, in this simulation of a granular gas in a shallow cell, an ordered cluster forms:

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(Data by Thomas Schindler [2])

In projection, the the particles of the cluster form a hexagonal lattice. This kind of order is obvious in the q_6 order parameter, because all the normals of the Voronoi cells point into directions which are multiples of 60 degrees.

In a q_6 histogram, the growth of the cluster is evidenced by a growing peak at q_6=1:

The unordered background appears as the continuous part of this “spectrum”.

The shallow cell granular gas has a few more phases with different symmetries which can be analysed using other IMT order parameters such as q_4 [2].

[1] [doi] W. Mickel, S. C. Kapfer, G. E. Schröder-Turk, and K. Mecke, “Shortcomings of the bond orientational order parameters for the analysis of disordered particulate matter,” Journal of Chemical Physics, vol. 138, iss. 4, p. 44501, 2013.
[2] [doi] T. Schindler and S. C. Kapfer, “Nonequilibrium steady states, coexistence, and criticality in driven quasi-two-dimensional granular matter,” Phys. Rev. E, vol. 99, p. 22902, 2019.