Mathematicians finally proved that melted ice will remain smooth

Put a piece of ice Cube into a glass of water. You might imagine the way it started to melt. You also know that no matter what shape it is, you will never see it melt into something like snowflakes, with sharp edges and tiny tips everywhere.

Mathematicians use equations to simulate this melting process. These equations worked well, but it took 130 years to prove that they fit the obvious facts about reality.in a Papers published in March, Alessio Figali with Joaquin Serra ETH Zurich and Xavier Rose-Orton Researchers at the University of Barcelona have determined that the equation is indeed intuitive. Snowflakes in the model may not be impossible, but they are extremely rare and completely fleeting.

“These results open up new perspectives in the field,” said Maria Colombo Swiss Federal Institute of Technology Lausanne. “I didn’t have such a deep and accurate understanding of this phenomenon before.”

The problem of how ice melts in water is called the Stefan problem, named after the physicist Josef Stefan. Pose It was in 1889. This is the most important example of the “free boundary” problem. Mathematicians consider how processes like thermal diffusion make the boundary move. In this case, the boundary is between ice and water.

For many years, mathematicians have been trying to understand these complex models of evolving boundaries. To make progress, this new work draws inspiration from previous studies of different types of physical systems: soap film. It is based on them to prove that along the ever-changing boundary between ice and water, sharp points such as sharp points or edges are rarely formed, and even if they form, they will disappear immediately.

These sharp points are called singularities, and it turns out that they are as short-lived in the free boundary of mathematics as in the physical world.

Melting hourglass

Consider again the ice cubes in a glass of water. These two substances are composed of the same water molecules, but the water is in two different phases: solid and liquid. There is a boundary at the intersection of the two. But when the heat of the water is transferred to the ice, the ice melts and the boundary moves. Eventually, the ice-and the borders that followed it-disappeared.

Intuition may tell us that this melting boundary always remains smooth. After all, when you take a piece of ice from a glass of water, you won’t be cut by the sharp edges. But with a little imagination, it is easy to imagine a scene with sharp points.

Take an hourglass-shaped ice cube and submerge it. As the ice cubes melted, the waist of the hourglass became thinner and thinner until the liquid continued to be eaten. At the moment this happens, the once smooth waist becomes two pointed tips or singularities.

“This is one of the problems that naturally exhibit singularities,” said Giuseppe Mingione University of Parma. “This is what physical reality tells you.”

However, reality also tells us that the singularity is controllable. We know that pointed tips shouldn’t last long, because warm water will melt them quickly. Maybe if you start with a huge block of ice made entirely of hourglasses, snowflakes may form. But it still won’t last more than an instant.

In 1889, Stefan conducted a mathematical study on this problem and spelled out two equations describing the melting of ice. One describes the diffusion of heat from warm water into cold ice, which causes the ice to shrink and at the same time cause the water area to expand. As the melting process progresses, the second equation tracks the changing interface between ice and water. (In fact​​, these equations can also describe the situation where the ice is too cold and the surrounding water freezes-but in the current work, the researchers ignore this possibility.)