Scientists Can't Quite Believe What A Rock And Soap Have In Common

If you've ever dropped a glass or a plate onto a hard floor, you know the struggle of painstakingly finding all the pieces and putting them safely in the trash. One thing you may have noticed, but given little thought to, is that there are tons of small pieces, but only a handful of large pieces. This mass distribution of fragments has been observed by scientists for decades, but recently, one researcher in France has found an equation that accurately describes it, and it applies to everything from rocks to soap bubbles.

The man behind this research is Emmanuel Villermaux, a French physicist who publishes prodigiously on fragmentation and mixing. The truly unique thing about his latest work (published in Physical Review Letters) is that its predictions work regardless of the material being studied. Villermaux isn't concerned with the how or why of fragmentation here, just the outcome, and except for a few cases, the outcomes are largely the same. 

There are a couple of principles underlying Villermaux's discovery. The first (and possibly the most crucial) is that fragmentations occur with the most randomness possible. In other words, fragmentations maximize disorder, or entropy. Villermaux limits this disorder based on his previous work, and what emerges is a formula that very accurately predicts the size distribution of many different objects, from spaghetti to sugar cubes.

When most people think of distributions (if they think of it at all), they think of a Gaussian or normal distribution. If you've heard of the bell curve, then you already know what we're talking about, but if not, think of the results of a standardized test like the SAT. Most people will score around a 1050, give or take a couple hundred, whereas a much smaller minority will score above 1400 or below 600. This distribution of scores is called "normal."

This distribution isn't what's observed for breaking objects. What's long been observed, and what Emmanuel Villermaux has put a formula to, is a power law distribution of fragment size. Normal distributions may be better known, but power law distributions are arguably just as common and can be observed in wealth distributions and scales that measure earthquake magnitude. As for breaking objects, this generally means that larger fragments are less likely, though slightly different power law distributions govern the predictions, varying for 1D (spaghetti), 2D (plates), and 3D (rocks) fragmentations. In other words, the exponent seen in each power law is directly related to the number of dimensions that a given object occupies.

This work is important for two reasons. First, it manages to mathematically describe phenomena that have been previously observed, providing a solid foundation for future research. Second, it has real world implications. Knowing the size of a landslide, you can now predict the maximum size of the debris and how much of it there will be, allowing for better emergency response planning.