Mathematicians finally prove that thaw remains smooth

drop the snow cube in a glass of water. You can probably imagine the way the melting starts. You also know that no matter what shape it takes, you’ll never see it melt into something like a snowflake, made up all around of sharp edges and tiny crevices.

Mathematicians model this dissolution process with equations. The equations work well, but it took 130 years to prove that they agree with the obvious facts about reality. in a Paper published in MarchAnd Alessio Figali And Joaquim Serra From the Swiss Federal Institute of Technology Zurich and Xavier Ross Otton from the University of Barcelona that the equations do indeed match intuition. Snowflakes in the form may not be impossible, but they are extremely rare and completely transient.

“These findings open up a new perspective in the field,” he said. Maria Colombo From the Swiss Federal Institute of Technology Lausanne. “There has never been such a deep and accurate understanding of this phenomenon before.”

The question of how ice melts in water is called the Stefan problem, which is named after the physicist Joseph Stefan, who subtracted in 1889. It is the most important example of a “free boundary” problem, where mathematicians think about how a process like heat diffusion moves the boundary. In this case, the boundary is between ice and water.

For many years, mathematicians have tried to understand the complex models of these evolving boundaries. To make progress, the new work draws inspiration from previous studies on a different type of physical system: soap films. He relies on them to prove that along the evolving boundary between ice and water, sharp spots such as balconies or ledges rarely form, and even when they do they disappear instantly.

These sharp points are called singularities, and they turn out to be as ephemeral in the free limits of mathematics as they are in the physical world.

melting hourglass

Consider, again, an ice cube in a glass of water. The two substances consist of the same water molecules, but water is in two different phases: solid and liquid. There are boundaries where the two phases meet. But as heat transfers from water to ice, the ice melts and the boundaries move. Eventually, the ice – and the boundaries with it – disappear.

Intuition might tell us that these melting boundaries always remain smooth. After all, do not cut yourself from the sharp edges when you pull a piece of ice from a glass of water. But with a little imagination, it’s easy to visualize scenarios in which sharp spots appear.

Take a piece of ice in the shape of an hourglass and submerge it. As the ice melts, the hourglass’s waist becomes thinner and thinner so the liquid will feed along the way. The moment this happens, what was once a smooth waist becomes two pointed balconies, or slits.

“This is one of those problems that naturally present singularities,” he said. Giuseppe Mingion from the University of Parma. “It’s the physical truth that tells you that.”

Joseph Stefan formulated a pair of equations representing melting ice.

Vienna University Archives Originator: R. Fenzl Signatur: 135,726

However, reality also tells us that singularities are controlled. We know balconies shouldn’t last long, because warm water should dissolve them quickly. Perhaps if you start with a huge ice block built entirely of hourglasses, a snowflake may form. But it still won’t last more than a moment.

In 1889, Stefan subjected the problem to mathematical scrutiny, and demonstrated two equations describing melting ice. One describes the spread of heat from warm water to cold ice, which causes the ice to shrink while causing the area of ​​water to expand. The second equation tracks the changing interface between ice and water as the melting process continues. (Indeed, the equations can also describe a situation in which ice is so cold that it causes the surrounding water to freeze—but in the present work, researchers ignore this possibility.)

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