An ice drop Cubes in a glass of water. You can probably figure out a way to start melting it. You also know that no matter what shape it takes, you’ll never see it melt into something like a snowflake, with sharp edges and fine bristles.
Mathematicians model this melting process with equations. The equations work well, but it took 130 years to prove that they were consistent with the obvious information about reality. In a study posted in March, Alessio Figali and Joaquim Best of the Swiss Federal Institute of Technology Zurich and Xavier Ross-Otten of the University of Barcelona established that the equations really match the insights. Snowflakes in models may not be impossible, but they are extremely rare and completely transient.
“These results open up a whole new perspective on the field,” said Maria Colombo of the Swiss Federal Institute of Technology in Lausanne. “There has never been such a deep and precise understanding of this phenomenon before.”
The question of how ice melts in water is called the Stefan problem, named after the physicist Joseph Stefan, who posed for it in 1889. This is the most important example of a “free boundary” problem, where mathematicians consider how a process like expansion makes heat move a boundary. In this case, the boundary is between ice and water.
Over the years, mathematicians have tried to understand complex models of these evolutionary boundaries. To make progress, the new work draws inspiration from previous research on a different type of physical system: soap film. This builds on them proving that along the evolutionary boundaries between ice and water, sharp spots like coops or edges are rarely formed and even when they do they disappear immediately.
These sharp spots are called singularity, and as can be seen, they are as transient as the free boundaries of mathematics, as they are in the physical world.
Molten bell glasses
Again, consider an ice cube in a glass of water. Two substances are made up of the same water molecule, but water has two different stages: solid and liquid. A boundary exists where the two stages meet. But as the heat from the water is transferred to the ice, the ice melts and the boundaries move. Eventually, the ice – and with it the border – disappeared.
Insights can tell us that these melting borders are always smooth. After all, when you pull a piece of ice out of a glass of water you don’t cut yourself on the sharp edge. But with a little imagination, it is easy to imagine situations where sharp spots appear.
Take a piece of ice in the shape of an hourglass and dip it. As the ice melts, the hourglass waist becomes thinner and thinner until the liquid eats all the way. It happens at this moment, what was once a smooth waist becomes two point dots or singles.
“This is one of the problems that naturally demonstrates unity,” said Giuseppe Mingioni of the University of Parma. “That’s the physical reality that tells you.”
Yet reality tells us that loneliness is controlled. We know that wells should not be long lasting, because warm water melts them quickly. Perhaps if you start with a huge piece of ice that is made entirely of hour glass, a snowflake can form. But it still won’t last more than a moment.
In 1889 Stephen subjected the problem to mathematical experiments, spelling out two equations that describe the melting of ice. One describes the expansion of heat from warm water to cool ice, which compresses the ice as it expands over a region of water. A second equation tracks the variable interface between ice and water as the melting process progresses. (In fact, the equations can also describe situations where the ice is so cold that it turns the surrounding water into ice – but in current work, researchers have ruled out that possibility.)