Mathematicians finally prove that melting ice is kept smooth

Put some ice put the bucket into a glass of water. You can probably imagine the way it starts to melt. You also know what shape it takes, that you will never see it melt into something like a snowflake, made up of sharp edges and fine cocoons everywhere.

Mathematicians model this fusion process with equations. The equation works well, but it took 130 years to prove that it coincides with the obvious facts about reality. In one paper published in March, Alessio Figalli and Joaquim Serra Zurich Swiss Federal Institute of Technology and Xavier Ros-Oton The University of Barcelona has established that the equations really match intuition. Snowflakes in the model may not be impossible, but they are very rare and completely transient.

“These results open up a whole new perspective on the field,” he said Maria Colombo Swiss Federal Institute of Technology in Lausanne. “Before there was no deep and precise understanding of this phenomenon.”

The question of how ice melts in water is called the Stefan problem, in honor of physicist Josef Stefan. raised 1889an. It is the most important example of the “free limit” problem, where mathematicians study how a process such as heat propagation forms a limit. In this case, it is the boundary between ice and water.

For many years, mathematicians have tried to understand the intricate patterns of these evolving boundaries. To move forward, the new work is inspired by previous research on another type of physical system: soap films. It is based on these that sharp points such as cocoons or ridges seldom form on the boundary between the evolution of ice and water, and disappear even immediately when they do.

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

Melting Watches

Consider, again, an ice cube in a glass of water. The two substances are made up of the same molecules of water, but water is in two different phases: solid and liquid. There is a limit to where the two phases meet. But as the heat of the water is transferred to the ice, the ice melts and the boundary moves. Eventually, the ice — and with it the boundary — disappears.

Intuition might say that this melting limit remains smooth. After all, you don’t have to cut your head off the sharp edges when you pull a piece of ice out of a glass of water. But with a little imagination, it’s easy to think of scenarios where sharp points emerge.

Take a piece of ice clock-shaped ice and immerse it. As the ice melts, the hourglass waist becomes thinner and thinner until it completely eats the liquid. At the moment this happens, what was once a smooth waist becomes two couscous points or peculiarities.

“This is one of those problems that shows peculiarities in itself,” he said Giuseppe Mingione University of Parma. “It’s physical reality that tells you that.”

Yet reality also tells us that singularities are controlled. We know that spits don’t have to last long because warm water needs to melt quickly. Perhaps if you started with a large block of ice built entirely of hourglass, it could create a snowflake. But it still shouldn’t last more than an instant.

In 1889 Stefan submitted the problem to mathematical analysis, specifying two equations that describe ice melting. One describes the spread of heat from warm water to cold ice, which shrinks the ice as it expands the water region. A second equation follows the variable interface between ice and water as the melting process progresses. (In fact, the equations may also describe a state where the ice is so cold, where it causes the surrounding water to freeze, but in this current work, the researchers rule out that possibility).