Mathematical Impressions: Spontaneous Stratification

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spontaneous_stratification

It is an almost magical experience to simply pour a granular mixture and watch it separate into its components with no effort on your part. This spontaneous form of pattern formation is a natural example of what mathematicians call “symmetry breaking” and is very simple to demonstrate in your own kitchen. Scientists only partially understand the dynamics of the process, so it poses interesting open problems in applied mathematics. Mainly, though, it is just fun to observe.

The separation and striation phenomenon demonstrated in this video was discovered and studied by Hernan A. Makse, who described it in a series of papers, including these:

Hernan A. Makse, Shlomo Havlin, Peter R. King, and H. Eugene Stanley. “Spontaneous Stratification in Granular Mixtures.” Nature 386 (March 1997): 379–81, http://arxiv.org/pdf/cond-mat/9809432v1.

Pierre Cizeau, Hernan A. Makse, and H. Eugene Stanley. “Mechanisms of granular spontaneous stratification and segregation in two-dimensional silos.” Physical Review E 59 (April 1999).

 

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  • The term “spontaneous symmetry breaking” comes from physics, not mathematics.

  • Another similar effect is when you drop sugar cubes into a box, shake it, and it will take less space than the original random arrangement. I wonder how the thermodynamical entropy works for these systems. It seems the “more organized” state is the one with the higher entropy.

  • This is a demonstration of something called kinetic sieving. It is an extremely well known concept in geology (sedimentology specifically) and it has been studied through observation and experimentation for decades. In fact, in my introductory geology classes I use a similar demonstration, just with different grain sizes of sand. In my sedimentology course I have students make these flows in three dimensions. So this video, while interesting, and a great visual, is clearly unaware of a huge body of scholarship in another discipline. The effects of grain size, shape (angular vs rounded), and density have all been explored experimentally, and the results published in peer reviewed journals. The last one I read included a mathematical description of the effects of density on the process. So there is much more information out there about this subject for the interested reader!

  • We must all be grateful to the anonymous “Anonymous Commentator” for pointing out that “The term ‘spontaneous symmetry breaking’ comes from physics, not mathematics.”

    Heaven forfend anyone should think that mathematics and physics have anything in common!

    How good it is, and how blessed we are, that there are not merely typological minds like Commentator’s but also entire bureaucracies dedicated to keeping these distinctions firm and pure.

    -dlj.

  • The voiceover asks “what other simple phenomena have not been studied and mathematically analysed”.

    I’ve a paper in press at the moment that’s a case in point: the “spontaneous” knotting of cable, flex, string etc as a result of random jostling. I use the theory of self-avoiding random walks to show how the process works, and also to make a prediction as to how to reduce this annoying everyday phenomenon. The paper uses data from 12,000 jumblings of string by UK schoolchildren to confirm this prediction.

  • Robert, that string jumbling experiment sounds like a lot of fun! Would make a good video too!

    George – another great video! It looks like the mathematical model works that if a grain can rest completely on another grain below it stays, if not it rolls. Is that right? Would be great to have a simulation like this to play with on the Web…

  • Isn’t this just the brazil nut effect, as seen in boxes of muesli the world over? Probably just a cute name for the kinetic sieving mentioned above. The large particles rise to the top simply because the smaller ones fall through the gaps between them.
    Is this really spontaneous symettry breaking? Since the system is not in thermodynamic equilibrium (thermal noise is not a significant force here).
    (Sir) Sam Edwards had a series of papers on entropy in granular systems – good luck if you try to understand them.

  • It runs out to not be the Brazil Nut effect, which is traditionally attributed to convection rolls at the boundaries of the container. In the case of avalanche stratification, it appears to be associated with shear dilution of the avalanching surface, which enables percolation of the smaller particles to the lower layers, The key seem to be the “dead zone” of mixed phase material that always forms in the lower corner.
    http://groups.physics.umn.edu/disorder/sand/

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