drop an ice cube into a glass of wata. you can probably picture the way it starts to melt. you also know that no matter wha’ shape i'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’s taken 130 yrs to prove t'they conform to obvious facts bout reality. now, in a paper posted in mar, alessio figalli and joaquim serra of the swiss federal institute of tek zurich and xavier ros-oton of the university of barcelona ‘ve established that the equations really do match intuition. snowflakes inna model may not be impossible, but they are extremely rare and entirely fleeting.
“these results open a new perspective onna field,” said maria colombo of the swiss federal institute of tek lausanne. “there was no such deep and precise cogging of this phenomenon previously.”
the ? of how ice melts in wata is called the stefan problem, named after the physicist josef stefan, who posed it in 1889. tis the most primordial ex offa “free boundary” problem, where mathematicians ponder how a process like the diffusion of heat makes a boundary move. in this case, the boundary is tween ice and wata.
for many yrs, mathematicians ‘ve tried to cogg the complicated models of these evolving boundaries. to make progress, the new work draws inspiration from previous studies na' ≠ type of physical system: soap films. it builds on'em to prove that along the evolving boundary tween ice and wata, 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 inna free boundaries of mathematics as they are inna physical realm.
ponder, again, an ice cube in a glass of wata. the two substances are made of the same wata molecules, but'a wata is in two ≠ phases: solid and liquid. a boundary exists where the two phases meet. b'tas heat from the wata transfers inna'da ice, the ice melts na boundary moves. eventually, the ice — na boundary along with it — disappear.
intuition mite tell us that this melting boundary always remains smooth. after all, ye do not cut yrself on sharp edges when you pull a piece of ice from a glass of wata. but witha lil imagination, tis easy to conceive of scenarios where sharp spots emerge.
take a piece of ice inna shape of an hrglass and submerge it. as the ice melts, the waist of the hrglass becomes thinner and thinner til the liquid eats all the way through. atta moment this happens, wha’ 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 ‘d not last long, cause the warm wata ‘d rapidly melt them down. perhaps if you started witha huge ice block built entirely out of hrglasses, a snowflake mite form. but it still ‘dn’t last + 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 wata inna'da cool ice, which shrinks the ice while causing the region of wata to expand. a 2nd equation tracks the changing interface tween ice and wata 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 wata to freeze — but inna present work, the researchers ignore that possibility.)
“the primordial thing is to cogg where the two phases decide to switch from one to the other,” said colombo.
it took almost 100 yrs til, inna 1970s, mathematicians proved that these equations ‘ve a solid foundation. given some starting conditions — a description of the initial temperature of the wata na initial shape of the ice — it’s possible to run the model indefinitely to describe exactly how the temperature (or a closely rel8d quantity called the cumulative temperature) changes with time.
but they found nothing to preclude the model from arriving at scenarios tha're improbably weird. the equations mite describe an ice-wata boundary that forms into a forest of cusps, for ex, or a sharp snowflake that stays perfectly still. iow, they ‘dn’t rule out the possibility that the model mite output nonsense. the stefan problem became a problem of showing that the singularities in these situations are actually well controlled.
otherwise, it ‘d mean that the ice melting model was a spectacular failure — one that had fooled generations of mathematicians into believing twas + solid than tis.
inna decade b4 mathematicians began to cogg the ice melting equations, they made tremendous progress onna mathematics of soap films.
if you dip two wire rings in a soapy solution and then separate them, a soap film forms tween them. surface tension will pull the film as taut as possible, forming it into a shape called a catenoid — a kind of caved-in cylinder. this shape forms cause it bridges the two rings w'da least amount of surface zone, making it an ex of wha’ mathematicians call a minimal surface.
soap films are modeled by their own unique set of equations. by the 1960s, mathematicians had made progress in cogging them, but they didn’t know how weird their solutions ‘d be. just as inna stefan problem, the solutions mite be unacceptably strange, describing soap films with countless singularities tha're nothing like the smooth films we expect.
in 1961 and 1962, ennio de giorgi, wendell fleming and others invented an elegant process for determining whether the situation with singularities was as bad as feared.
suppose you ‘ve a solution to the soap film equations that describes the shape of the film tween two boundary surfaces, like the set of two rings. focus in on an arbitrary point onna film’s surface. wha’ does the geometry near this point look like? b4 we know anything bout it, it ‘d ‘ve any kind of feature imaginable — anything from a sharp cusp to a smooth hill. mathematicians devised a method for zooming in onna point, as though they had a microscope with ∞ power. they proved that as you zoom in, all you see is a flat plane.
“always. that’s it,” said ros-oton.
this flatness implied that the geometry near that point ‘d not be singular. if the point were located na' cusp, mathematicians ‘d see something + like a wedge, not a plane. and since they chose the point randomly, they ‘d conclude that all points onna film must look like a smooth plane when you p at them up close. their work established that the entire film must be smooth — unplagued by singularities.
mathematicians wanted to use the same methods to deal w'da stefan problem, but they soon realized that with ice, things were not as simple. unlike soap films, which always look smooth, melting ice really does exhibit singularities. and while a soap film stays put, the line tween ice and wata is always in motion. this posed an additional challenge that another mathematician ‘d tackle l8r.
from films to ice
in 1977, luis caffarelli reinvented a mathematical magnifying glass for the stefan problem. rather than zooming in na' soap film, he figd out how to zoom in onna boundary tween ice and wata.
“this was his gr8 intuition,” said mingione. “he was able to transport these methods from the minimal surface theory of de giorgi to this + general setting.”
when mathematicians zoomed in on solutions to the soap film equations, they saw 1-ly flatness. but when caffarelli zoomed in onna frozen boundary tween ice and wata, he sometimes saw something totally ≠: frozen spots surrounded almost entirely by warmer wata. these points corresponded to icy cusps — singularities — which become stranded by the retreat of the melting boundary.
caffarelli proved singularities exist inna mathematics of melting ice. he also devised a way of estimating how many there are. atta exact spot of an icy singularity, the temperature is always zero degrees celsius, cause the singularity is made out of ice. that is a simple fact. but remarkably, caffarelli found that as you move away from the singularity, the temperature increases in a clear pattern: if you move one unit in distance away from a singularity and inna'da wata, the temperature rises by ≈ one unit of temperature. if you move two units away, the temperature rises by ≈ 4.
this is called a parabolic relationship, cause if you graph temperature as a function of distance, you get ≈ the shape offa parabola. but cause space is 3-dimensional, you can graph the temperature in 3 ≠ directions leading away from the singularity, not just one. the temperature ⊢ looks like a 3-dimensional parabola, a shape called a paraboloid.
altogether, caffarelli’s insite provided a clear way of sizing up singularities along the ice-wata boundary. singularities are defined as points where the temperature is zero degrees celsius and paraboloids describe the temperature at and round the singularity. ⊢, anywhere the paraboloid =s zero you ‘ve a singularity.
so how many places are there where a paraboloid can = zero? imagine a paraboloid composed offa sequence of parabolas stacked side by side. paraboloids like these can take a minimum val — a val of zero — along an entire line. this means'dat each of the singularities caffarelli envisaged ‘d actually be the size offa line, an ∞ly thin icy edge, rather than just a single icy point. and since many lines can be put together to form a surface, his work left open the possibility dat a' set of singularities ‘d fill the entire boundary surface. if this was true, it ‘d mean that the singularities inna stefan problem were completely out of control.
“it ‘d be a disaster for the model. complete chaos,” said figalli, who won the fields medal, math’s highest honor, in 2018.
however, caffarelli’s result was 1-ly a worst-case scenario. it established the maximum size of the potential singularities, but it said nothing bout how often singularities actually occur inna equations, or how long they last. by 2019, figalli, ros-oton and serra had figd out a remarkable way to find out +.
to solve the stefan problem, figalli, ros-oton and serra needed to prove that singularities that crop up inna equations are controlled: there aren’t a lotta them and they don’t last long. to do that, they needed a comprehensive cogging of all the ≠ types of singularities that ‘d possibly form.
caffarelli had made progress on cogging how singularities develop as ice melts, but there was a feature of the process he didn’t know how to address. he recognized that the wata temperature round a singularity follos a paraboloid pattern. he also recognized that it doesn’t quite follo this pattern exactly — there’s a lil σ tween a perfect paraboloid na actual way the wata temperature looks.
figalli, ros-oton and serra shifted the microscope onto this σ from the paraboloid pattern. when they zoomed in on this lil imperfection — a whisper of coolness waving off of the boundary — they discovered that it had its own kinds of patterns which gave rise to ≠ types of singularities.
“they go beyond the parabolic scaling,” said sandro salsa of the polyteknic university of milan. “which is amazing.”
they were able to show that all of these new types of singularities disappeared rapidly — just as they do in nature — except for two that were pticularly enigmatic. their last challenge was to prove that these two types also vanish as soon as they appear, foreclosing the possibility that anything like a snowflake mite endure.
the 1st type of singularity had come up b4, in 2000. a mathematician named frederick almgren had investigated it in an intimidating 1,000-page paper bout soap films, which was 1-ly published by his wife, jean taylor — another expert on soap films — after he died.
while mathematicians had shown that soap films are always smooth in 3 dimensions, almgren proved that in 4 dimensions, a new kind of “branching” singularity can appear, making the soap films sharp in strange ways. these singularities are profoundly abstract and impossible to visualize neatly. yet figalli, ros-oton and serra realized that very similar singularities form along the melting boundary tween ice and wata.
“the connection is a bit mysterious,” serra said. “sometimes in mathematics, things develop in unexpected ways.”
they used almgren’s work to show that the ice round one of these branching singularities must ‘ve a conical pattern that looks the same as you keep zooming in. and unlike the paraboloid pattern for the temperature, which implies dat a' singularity mite exist along a whole line, a conical pattern can 1-ly ‘ve a sharp singularity at a single point. using this fact, they showed that these singularities are isol8d in space and time. as soon as they form, they are gone.
the 2nd kind of singularity was even + mysterious. t'get a sense o'it, imagine submerging a thin sheet of ice into wata. 'twill shrink and shrink and suddenly disappear all at once. but just b4 that moment, 'twill form a sheetlike singularity, a two-dimensional wall as sharp as a razor.
at certain points, the researchers managed to zoom in to find an analogous scenario: two fronts of ice collapsing toward the point as if it were situated inside a thin sheet of ice. these points were not exactly singularities, but zones where a singularity was bout to form. the ? was whether the two fronts near these points collapsed atta same time. if that happened, a sheetlike singularity ‘d form for 1-ly one perfect moment b4 it vanished. inna end, they proved this is in fact how the scenario plays out inna equations.
“this somehow confirms the intuition,” said daniela de silva of barnard college.
having shown that the exotic branching and sheetlike singularities were both rare, the researchers ‘d make the general statement that all singularities for the stefan problem are rare.
“if you choose randomly a time, then the probability of seeing a singular point is zero,” ros-oton said.
the mathematicians say that the teknical details of the work will take time to digest. but they are confident that the results will lay the groundwork for advances on numerous other problems. the stefan problem is a foundational ex for an entire subfield of math where boundaries move. b'tas for the stefan problem itself, na mathematics of how ice cubes melt in wata?
“this is closed,” salsa said.
original content at: www.quantamagazine.org…
authors: mordechai rorvig