They started by imagining a very fine grid superimposed on their fluid. They then computed how long particles spent in each square of the grid, on average. In some squares, the fluid acted like a rushing river: Particles tended to sweep straight across the square, spending only a brief period of time there. In other squares, small eddies might push particles around, slowing them down.

The problem was that the numbers the mathematicians calculated might vastly differ from square to square — revealing precisely the kind of small-scale disorder that usually prevented mathematicians from using homogenization.

Armstrong, Bou-Rabee and Kuusi needed to find a way around that.

Ordering Disorder

The mathematicians hoped to show that at slightly larger scales than the one their grid had captured, the fluid’s behavior would be a bit less noisy and disordered. If they could do that, they’d be able to use typical homogenization techniques to understand what was happening at the largest scale.

But other mathematicians thought that even if they succeeded in analyzing those intermediate small scales, the fluid would only look noisier. Before things got smoother, eddies would first merge and interact in even more complicated ways. Trying to show otherwise would be a fool’s errand.

The team decided to try anyway. They started by drawing a slightly coarser grid, in which each square encompassed several squares from the original one. Smaller eddies that had lived in separate squares of the original grid might now get grouped together, changing the average amount of time a particle spent in the new square. Or more complicated behaviors might emerge.

The team once again computed how long particles stayed in each square and how much the numbers associated with adjacent squares might differ. This took painstaking effort: They had to keep track of how the fluid’s behavior in each square would change, and how it would change a particle’s probable motion. They then showed that in this coarser grid, adjacent numbers tended to differ by smaller amounts.

They did this for coarser and coarser grids, until they showed that at a larger — though still relatively small — scale, the fluid looked nice enough for them to use typical homogenization. “You have to do this procedure, which by itself was totally new, infinitely many times,” Vicol said. “The fact that they were able to do this was, from a math perspective, really insane.” It required more than 300 pages of calculation and proof, and took the mathematicians nearly two years.

“It was a very intense experience,” Bou-Rabee said. “There were many Saturday mornings where we’d wake up at 6 a.m. and go to the office to work, and then repeat the next day.”

But once they were able to apply the usual set of homogenization techniques, they had enough information about the fluid at large scales to know that two solid particles dropped into it would spread according to the equation for diffusion. The trio then evaluated the rate of that diffusion and found that it was precisely what the physicists had conjectured decades earlier.

They’d proved the superdiffusion conjecture.

A Long View

The result, which the mathematicians divided into two different papers, provides the first rigorous mathematical understanding of a peculiarity of turbulent fluids: the way they spread particles around with breathtaking efficiency. It’s the first proof of the kind of phenomenon that Richardson observed a century ago in the distribution of balloon enthusiasts across Europe. “You don’t get these sort of definitive results that often,” Quastel said. “I’m pretty impressed — lots of people are pretty impressed.”

Armstrong, for his part, sees the work as a vindication of his ambitions for homogenization. “Nobody expected us to get out of our lane anytime soon,” he said. “So the idea that we would come and then start solving problems in other domains using those methods, there was no sign of that.”

Antti Kupiainen, a mathematician at the University of Helsinki, agreed. “I think even more important is they have a new method, a new way to approach these problems,” he said. In real-life turbulence — which the simplified fluid from the conjecture only modeled in the roughest sense — the scales interact in stronger and more complex ways, leading to more extreme superdiffusive behavior. Perhaps Armstrong, Bou-Rabee and Kuusi’s technique could help researchers chip away at related questions for more realistic models of turbulence, as well as other problems.

Renormalization, after all, is used throughout physics to make sense of systems that exhibit different behaviors at different scales. Armstrong hopes that his techniques can be adapted to prove statements in some of those contexts as well — including particle physics, the area of study where renormalization was first developed.

“I feel like there are so many open possibilities at the moment,” Kuusi said. “I think that this is the last time this will happen to me in my life, and right now I’m going to enjoy the ride.”