- Three mathematicians proved superdiffusion in turbulent fluids for the first time.
- The 300-page proof used enhanced homogenization to tame chaotic small-scale behaviour.
- A 2026 follow-up confirmed power-law predictions from the 1980s.
Scott Armstrong did not set out to solve superdiffusion, one of the oldest puzzles in fluid dynamics. A decade ago, the mathematician at NYU's Courant Institute barely knew what turbulence was.
His specialty, a technique called homogenization, seemed safely confined to its own lane. Then he saw a way to aim it at something far larger.
A Balloon Race That Revealed a Mystery
In 1906, over 200,000 Parisians gathered to watch gas balloons launched from the Tuileries Garden. Winds scattered them across Europe.
The British physicist Lewis Fry Richardson, analysing where the balloons landed, noticed something odd. They had dispersed with startling efficiency, far faster than ordinary diffusion could explain.
Richardson would later describe the pattern in a couplet: "Big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls and so on to viscosity."
The observation pointed to a property now called superdiffusion, where particles in turbulent fluids spread apart at anomalously fast rates.
What is superdiffusion?
In ordinary diffusion, particles spread gradually, like a drop of ink in still water. Superdiffusion is faster: turbulent eddies at multiple scales amplify the scattering. Two leaves dropped in a turbulent stream will separate much more quickly than calm-water physics predicts.
Physicists conjectured for decades that superdiffusion was real. Nobody could prove it mathematically.
Three Mathematicians and a 300-Page Proof
Armstrong, together with postdoctoral researcher Ahmed Bou-Rabee at NYU and Tuomo Kuusi of the University of Helsinki, spent nearly two years building their proof. The approach rested on homogenization, a method for showing how small-scale noise averages into simple behaviour at larger scales.
Key figure
300+ pages
Length of the mathematical proof confirming superdiffusion in turbulent fluids
The team superimposed progressively coarser grids on their fluid model. They calculated how long particles took to cross each grid square.
At fine scales, chaos dominated, much like the unexpected turbulence found in hypersonic simulations. But as the grid coarsened, adjacent values converged.
The crucial insight was iterative. Each pass through the grid tamed a little more disorder, until standard homogenization could take over.
Vlad Vicol, a mathematician at NYU's Courant Institute who was not involved in the work, put the difficulty plainly. "You have to do this procedure infinitely many times," he said. "The fact that they were able to do this was really insane."
You have to do this procedure infinitely many times. The fact that they were able to do this was really insane.
Vlad Vicol, NYU Courant Institute
The result confirmed what physicists had conjectured for decades. Particles in their model spread at precisely the enhanced rate predicted.
Why a Simplified Model Still Matters
The proof applies to a simplified representation of turbulence, not to the full Navier-Stokes equations that govern real fluids. That distinction matters.
Yet the team's achievement opens a path that did not exist before.
Jeremy Quastel, a mathematician at the University of Toronto, offered a measured assessment. "You don't get these sort of definitive results that often," he said.
In a follow-up paper published in January 2026, the same trio extended their framework to genuinely superdiffusive regimes. They confirmed predictions made by physicists Bouchaud and Georges in the 1980s about power-law growth rates.
The new result connects directly to Richardson's 4/3 law: two particles in turbulent fluid separate with squared distance scaling as the cube of time.
Armstrong sees the enhanced homogenization technique as portable. Problems in physics that involve reconciling behaviour at different scales, possibly including questions in particle physics, may prove accessible to the same approach.
The Ride of a Career
Kuusi, reflecting on the achievement, was candid. "I think that this is the last time this will happen to me in my life," he said, "and right now I'm going to enjoy the ride."
Armstrong's ambitions for homogenization had long struck colleagues as optimistic. "Nobody expected us to get out of our lane anytime soon," he recalled.
The lane turned out to be wider than anyone thought.
Sources
- Primary Research: New 'Superdiffusion' Proof Probes the Mysterious Math of Turbulence (Quanta Magazine, 2025)
- Additional Context:
- Power-law superdiffusion in self-similar incompressible random flows (Armstrong, Bou-Rabee, Kuusi, 2026)
- Superdiffusive central limit theorem for a Brownian particle in a critically-correlated incompressible random drift (arXiv, 2024)
