Law of 'maximal randomness' explains how broken objects shatter in the most annoying way possible

A dropped vase, a crushed sugar cube and an exploding bubble all have something in common: They break apart in similar ways, a new mathematical equation reveals.

A French scientist recently discovered the mathematical equation, which describes the size distribution of fragments that form when something shatters. The equation applies to a variety of materials, including solids, liquids and gas bubbles, according to a new study, published Nov. 26 in the journal Physical Review Letters.

Though cracks spread through an object in often unpredictable ways, research has shown that the size distribution of the resulting fragments seems to be consistent, no matter what they're made of — you can always expect a certain ratio of larger fragments to smaller ones. Scientists suspected that this consistency pointed to something universal about the process of fragmenting.

Rather than focusing on how fragments form, Emmanuel Villermaux, a physicist at Aix-Marseille University in France, studied the fragments themselves. In the new study, Villermaux argued that fragmenting objects follow the principle of "maximal randomness." This principle suggests that the most likely fragmentation pattern is the messiest one — the one that maximizes entropy, or disorder.

But that randomness has to obey certain limits. To account for this, Villermaux introduced a conservation law that he and his colleagues discovered in 2015. This law adds physical constraints on the density of fragments in space when an object shatters.

By combining the two principles, Villermaux derived a mathematical equation that describes the pattern of fragment sizes from a shattered object. He then validated the equation by comparing the equation's predictions to years' worth of fragmentation data collected on various objects, including glass, spaghetti, liquid droplets, gas bubbles, plastic fragments in the ocean, and even flakes from early stone tools. All matched the predicted size distribution.

Villermaux also tested the equation by dropping heavy objects onto sugar cubes and observing how they fragmented. "That was a summer project with my daughters," Villermaux told New Scientist. "I did this a long time ago when my children were still young and then came back to the data, because they were illustrating my point well."

However, the newly discovered law doesn't always apply: It doesn't apply in situations with no randomness, such as a smooth stream of liquid breaking into droplets of equal size; and it doesn't cover conditions where the fragments interact with each other, such as in certain plastics.

Ferenc Kun, a physicist at the University of Debrecen in Hungary, told New Scientist that understanding fragmentation could help scientists determine how energy is spent on shattering ore in industrial mining or how to prepare for rockfalls.

Future work could involve determining the smallest possible size a fragment could have, Villermaux told New Scientist.

It's also possible that the shapes of different fragments could follow a similar relationship, Kun wrote in an accompanying viewpoint article.