TITLE: Is there a mathematical and information theoretic explanation for this cube packing phenomenon?
QUESTION [58 upvotes]: I saw this unintuitive result on dice packing:

A jumble of thousands of cubic dice, agitated by an oscillating
rotation, can rapidly become completely ordered, a result that is hard
to produce with more conventional shaking.

The paper is not very rigorous.

Is there a mathematical explanation of this result?

Is such phenomenon studied?



Can the presence of centripetal force in oscillating rotation help explain the phenomenon?


Update: Why does entropy seemingly reduce? Is there  information theoretic explanation?



Is there a minimum and maximum volume below and beyond respectively of both the cylinder and the dice which the phenomenon fails?


How about radius of the cylinder having an effect and the mass of individual dice?

Note the complexity remains the same. One is simply explainable as random and the other has total order explainable succinctly. But crucially they go through an intermediate transition phase of very high complexity. But unlike the example in https://www.scottaaronson.com/blog/?p=762 the entropy reduces and s something non-intuitive is happening. I do not think there is any elementary explanation. The example in the link of mixing coffee produces high entropy coffee no matter how you stir (slow or fast with or without rotation but there might be a totally non-intuitive way to mix coffee and lower entropy).

REPLY [5 votes]: This should be seen as a comment, but it is too long for a comment:  
One aspect, that has not been addressed in the answers, is the influence of the container geometry. In the experiment, a cylindrical container has been subjected to two different kinds of motions: a coaxial rotation with alternating directions and tapping to the side of side of the cylinder, which can also be interpreted as a horizontal motion that keeps the axis upright.  
Assume for a moment, that the experiment were carried out without the influence of gravity and the cubes at least a distance of $\epsilon$ apart. then,  

Both kinds of motions would result in horizontal motions of the cube's centers of gravity.   
in the case of rotational motion the primary source of impact on the cubes would be friction between the cylinder and the cubes touching it, resulting in a pure rotation of those cubes around an axis through their center of gravity that is parallel to the one of the container. The "wave of motion" will proceed outside-in as the rotating cubes start to collide and the axes of rotation will start to deviate from the cylinder axis; there will also be forces on the cubes' centers that have a vertical component. All in all, the effects will be fairly moderate.  
in the case of tapping motion cubes will collide with cylinder and are roughly reflected like rays of light on a mirror, which means that there will be a caustic of the horizontical motion vectors or, put differently, the shockwaves will be bundled and result in a much higher motion energy acting on certain cubes. A secondary effect will be, that after a while cubes will be reflected from the opposite "side" of the cylinder and then collide with other cubes, that are still on their way to that side; the effect will be approximately the same as if two vertically aligned balls are dropped to the ground, namely that the reflection of the second cube will be accelerated towards the cylinder axis again.  

So, in a nutshell the difference can be explained by the different magnitude of shockwaves that are generated with the different kinds of motions; that can easily be checked by replacing the cubes with water and measuring the deviation of the surface from equilibrium.   
Further experimental investigations could check the effect of container geometry (elliptical or prismatic instead of cylindrical) or rotation aroung an axis off the center of symmetry.
But I am neither a physicist, nor do I have any background in statistical mechanics, so forgive me if my answer is nonsense.