Day: November 8, 2021

Drop an ice cube into a glass of water. You can probably picture the way it starts to melt. You also know that no matter what shape it takes, you’ll never see it melt into something like a snowflake, composed everywhere of sharp edges and fine cusps.

Mathematicians model this melting process with equations. The equations work well, but it has taken 130 years to prove that they conform to obvious facts about reality. In a paper posted in March, Alessio Figalli and Joaquim Serra of the Swiss Federal Institute of Technology Zurich and Xavier Ros-Oton of the University of Barcelona have established that the equations really do match intuition. Snowflakes in the model may not be impossible, but they are extremely rare and entirely fleeting.

“These results open a new perspective on the field,” said Maria Colombo of the Swiss Federal Institute of Technology Lausanne. “There was no such deep and precise understanding of this phenomenon previously.”

The question of how ice melts in water is called the Stefan problem, named after the physicist Josef Stefan, who posed it in 1889. It is the most important example of a “free boundary” problem, where mathematicians consider how a process like the diffusion of heat makes a boundary move. In this case, the boundary is between ice and water.

For many years, mathematicians have tried to understand the complicated 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. It builds on them to prove that along the evolving boundary between ice and water, sharp spots like cusps or edges rarely form, and even when they do they immediately disappear.

These sharp spots are called singularities, and, it turns out, they are as ephemeral in the free boundaries of mathematics as they are in the physical world.

Melting Hourglasses

Consider, again, an ice cube in a glass of water. The two substances are made of the same water molecules, but the water is in two different phases: solid and liquid. A boundary exists where the two phases meet. But as heat from the water transfers into the ice, the ice melts and the boundary moves. Eventually, the ice—and the boundary along with it—disappear.

Intuition might tell us that this melting boundary always remains smooth. After all, you do not cut yourself on sharp edges when you pull a piece of ice from a glass of water. But with a little imagination, it is easy to conceive of scenarios where sharp spots emerge.

Take a piece of ice in the shape of an hourglass and submerge it. As the ice melts, the waist of the hourglass becomes thinner and thinner until the liquid eats all the way through. At the moment this happens, what was once a smooth waist becomes two pointy cusps, or singularities.

“This is one of those problems that naturally exhibits singularities,” said Giuseppe Mingione of the University of Parma. “It’s the physical reality that tells you that.”

Yet reality also tells us that the singularities are controlled. We know that cusps should not last long, because the warm water should rapidly melt them down. Perhaps if you started with a huge ice block built entirely out of hourglasses, a snowflake might form. But it still wouldn’t last more than an instant.

In 1889 Stefan subjected the problem to mathematical scrutiny, spelling out two equations that describe melting ice. One describes the diffusion of heat from the warm water into the cool ice, which shrinks the ice while causing the region of water to expand. A second equation tracks the changing interface between ice and water as the melting process proceeds. (In fact, the equations can also describe the situation where the ice is so cold that it causes the surrounding water to freeze—but in the present work, the researchers ignore that possibility.)

Drop an ice cube into a glass of water. You can probably picture the way it starts to melt. You also know that no matter what shape it takes, you’ll never see it melt into something like a snowflake, composed everywhere of sharp edges and fine cusps.

Mathematicians model this melting process with equations. The equations work well, but it has taken 130 years to prove that they conform to obvious facts about reality. In a paper posted in March, Alessio Figalli and Joaquim Serra of the Swiss Federal Institute of Technology Zurich and Xavier Ros-Oton of the University of Barcelona have established that the equations really do match intuition. Snowflakes in the model may not be impossible, but they are extremely rare and entirely fleeting.

“These results open a new perspective on the field,” said Maria Colombo of the Swiss Federal Institute of Technology Lausanne. “There was no such deep and precise understanding of this phenomenon previously.”

The question of how ice melts in water is called the Stefan problem, named after the physicist Josef Stefan, who posed it in 1889. It is the most important example of a “free boundary” problem, where mathematicians consider how a process like the diffusion of heat makes a boundary move. In this case, the boundary is between ice and water.

For many years, mathematicians have tried to understand the complicated 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. It builds on them to prove that along the evolving boundary between ice and water, sharp spots like cusps or edges rarely form, and even when they do they immediately disappear.

These sharp spots are called singularities, and, it turns out, they are as ephemeral in the free boundaries of mathematics as they are in the physical world.

Melting Hourglasses

Consider, again, an ice cube in a glass of water. The two substances are made of the same water molecules, but the water is in two different phases: solid and liquid. A boundary exists where the two phases meet. But as heat from the water transfers into the ice, the ice melts and the boundary moves. Eventually, the ice—and the boundary along with it—disappear.

Intuition might tell us that this melting boundary always remains smooth. After all, you do not cut yourself on sharp edges when you pull a piece of ice from a glass of water. But with a little imagination, it is easy to conceive of scenarios where sharp spots emerge.

Take a piece of ice in the shape of an hourglass and submerge it. As the ice melts, the waist of the hourglass becomes thinner and thinner until the liquid eats all the way through. At the moment this happens, what was once a smooth waist becomes two pointy cusps, or singularities.

“This is one of those problems that naturally exhibits singularities,” said Giuseppe Mingione of the University of Parma. “It’s the physical reality that tells you that.”

Yet reality also tells us that the singularities are controlled. We know that cusps should not last long, because the warm water should rapidly melt them down. Perhaps if you started with a huge ice block built entirely out of hourglasses, a snowflake might form. But it still wouldn’t last more than an instant.

In 1889 Stefan subjected the problem to mathematical scrutiny, spelling out two equations that describe melting ice. One describes the diffusion of heat from the warm water into the cool ice, which shrinks the ice while causing the region of water to expand. A second equation tracks the changing interface between ice and water as the melting process proceeds. (In fact, the equations can also describe the situation where the ice is so cold that it causes the surrounding water to freeze—but in the present work, the researchers ignore that possibility.)

Noted tax-lover Elon Musk is the latest entrant in Texas’ Wild West energy market. The CEO of SpaceX and Tesla is now officially licensed to sell electricity to retail customers in the state through a company called Tesla Energy Ventures.

Musk filed paperwork to operate in the state over the summer, a move first noticed by Texas Monthly. Late last week, the state gave its blessing to Musk’s venture. Texas’s Public Utility Commission wrote in its finding that Tesla Energy Ventures is now licensed to “provide retail electric services throughout the area served by ERCOT,” which is basically all of Texas.

The full details of the new company are still a bit of a mystery. But the plan appears to be to target existing Tesla customers to start. Tesla sells solar panels, batteries, and cars that can act as batteries. Together, these components could essentially be a virtual power plant.

Tesla has experience on that front; the company has a virtual power plant up and running in the state of South Australia and a beta version that launched in California this summer. How fast a Texas version of this—or something else altogether with existing customers in the state—expands is unclear. Energy firm Wood Mackenzie wrote at the time it was announced that it expects “less than 5%” of the state’s existing 50,000 Powerwall owners to participate, which it said would “not have a meaningful impact on grid operations in California.”

Those numbers might not move the needle in the near term, but Musk has said he sees energy being a huge part of Tesla’s future. “I think long-term, Tesla Energy will be roughly the same size as Tesla Automotive,” he told investors last year, speaking of the company’s energy side that’s different from the Texas venture. “The energy business is collectively bigger than the automotive business.” And the company could fill a unique niche in Texas.

“Usually an electricity service provider that doesn’t have generation on the ground essentially acts as a middleman, they buy on the wholesale market and turn around and sell to customers,” Ted Kury, director of energy studies for the Public Utility Research Center (PURC) at the University of Florida, told UtilityDive at the time of the Tesla filing. “Tesla would have the ability to act as a buyer and seller simultaneously and have access to a lot of data. Regulators are going to have to think about what the implications might be.”

Indeed, Musk will now be able to sell power to the people in Texas at a time when the state’s grid and its reliability have come under scrutiny after deadly blackouts. The main driver of those blackouts was, despite Republican claims otherwise, natural gas infrastructure freezing up and failing to meet demand. A Tesla virtual power plant could help keep the lights on. At least some Tesla customers actually did rely on their cars to keep warm.

Say what you will about the billionaire jostling his way into major markets that include cars, tunnels with cars, energy, the internet, and space ( is there anything else left?), but the Texas grid is hardly the paragon of public goods. “Tire fire” would be a generous descriptor after watching the grid go down in February and almost went down this summer (twice). Meanwhile, the folks that oversee Texas oil and gas that played a key role in the grid’s February collapse have largely evaded scrutiny or oversight. I could literally write a whole other story about how poorly run the grid is. Now, I’m not saying a meme-loving billionaire will singlehandedly fix Texas energy (he won’t) or that there aren’t better solutions out there (there are). But it’s not like the immediate alternative is much better.

Noted tax-lover Elon Musk is the latest entrant in Texas’ Wild West energy market. The CEO of SpaceX and Tesla is now officially licensed to sell electricity to retail customers in the state through a company called Tesla Energy Ventures.

Musk filed paperwork to operate in the state over the summer, a move first noticed by Texas Monthly. Late last week, the state gave its blessing to Musk’s venture. Texas’s Public Utility Commission wrote in its finding that Tesla Energy Ventures is now licensed to “provide retail electric services throughout the area served by ERCOT,” which is basically all of Texas.

The full details of the new company are still a bit of a mystery. But the plan appears to be to target existing Tesla customers to start. Tesla sells solar panels, batteries, and cars that can act as batteries. Together, these components could essentially be a virtual power plant.

Tesla has experience on that front; the company has a virtual power plant up and running in the state of South Australia and a beta version that launched in California this summer. How fast a Texas version of this—or something else altogether with existing customers in the state—expands is unclear. Energy firm Wood Mackenzie wrote at the time it was announced that it expects “less than 5%” of the state’s existing 50,000 Powerwall owners to participate, which it said would “not have a meaningful impact on grid operations in California.”

Those numbers might not move the needle in the near term, but Musk has said he sees energy being a huge part of Tesla’s future. “I think long-term, Tesla Energy will be roughly the same size as Tesla Automotive,” he told investors last year, speaking of the company’s energy side that’s different from the Texas venture. “The energy business is collectively bigger than the automotive business.” And the company could fill a unique niche in Texas.

“Usually an electricity service provider that doesn’t have generation on the ground essentially acts as a middleman, they buy on the wholesale market and turn around and sell to customers,” Ted Kury, director of energy studies for the Public Utility Research Center (PURC) at the University of Florida, told UtilityDive at the time of the Tesla filing. “Tesla would have the ability to act as a buyer and seller simultaneously and have access to a lot of data. Regulators are going to have to think about what the implications might be.”

Indeed, Musk will now be able to sell power to the people in Texas at a time when the state’s grid and its reliability have come under scrutiny after deadly blackouts. The main driver of those blackouts was, despite Republican claims otherwise, natural gas infrastructure freezing up and failing to meet demand. A Tesla virtual power plant could help keep the lights on. At least some Tesla customers actually did rely on their cars to keep warm.

Say what you will about the billionaire jostling his way into major markets that include cars, tunnels with cars, energy, the internet, and space ( is there anything else left?), but the Texas grid is hardly the paragon of public goods. “Tire fire” would be a generous descriptor after watching the grid go down in February and almost went down this summer (twice). Meanwhile, the folks that oversee Texas oil and gas that played a key role in the grid’s February collapse have largely evaded scrutiny or oversight. I could literally write a whole other story about how poorly run the grid is. Now, I’m not saying a meme-loving billionaire will singlehandedly fix Texas energy (he won’t) or that there aren’t better solutions out there (there are). But it’s not like the immediate alternative is much better.