The world of mathematics cautions when the biggest living mathematicians reveal their vision for research in the next century. That was exactly what happened in 1900 at the International Conference of Mathematicians at the Sorbonne University in Paris. Legendary mathematician David Hilbert presented 10 unresolved questions as an ambitious guide post of the 20th century. He later expanded the list to include 23 issues, and could not overstate the impact on mathematical thinking over the past 125 years.
Hilbert’s sixth problem was one of the most advanced problems. He sought to “axiomize” physics or to determine the minimum number of mathematical assumptions behind all of its theories. Although it was interpreted roughly, it is not clear that mathematical physicists can know whether they have solved this problem. However, Hilbert mentioned some specific sub-goals, and the researchers then improved his vision to specific steps into that solution.
In March, University of Chicago mathematician Yu Deng and University of Michigan Zaher Hani, Xaher Hani and Xiao Ma submitted a new paper to Preprint Server arxiv.org, who claimed they had cracked one of these goals. If their work endures scrutiny, it could mark a major advancement in grounding the physics of mathematics and open the door to similar breakthroughs in other areas of physics.
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In the paper, researchers suggest that they understand how to integrate three physical theories explaining fluid movement. These theories manage a variety of engineering applications, from aircraft design to weather forecasting, but have previously been based on unproven assumptions. This breakthrough does not change theories themselves, but it mathematically justifies them and strengthens the confidence that equations work in our thinking.
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Each theory differs in how much it zooms in to the flowing liquid or gas. At the microscopic level, the liquid is made up of particles – a small Villiad ball sways and sometimes collides – and Newton’s Law of Motion works well to explain the trajectory.
However, zooming out to consider the collective behavior of a huge number of particles, so-called mesoscopic levels, is no longer convenient to model each individually. In 1872, Austrian theoretical physicist Ludwig Boltzmann addressed this when he developed what became known as the Boltzmann equation. Instead of tracking the behavior of all particles, the equation considers the possible behavior of typical particles. This statistical perspective smoothes out over low-level detail and supports a high-level trend. This equation allows physicists to calculate how to evolve the amounts of fluid momentum and thermal conductivity without painstakingly causing all microscopic collisions.
Zoom out further and you will find yourself in a macroscopic world. Here, liquids are considered to be a single continuous material rather than a collection of discrete particles. In this level of analysis, different equations (Euler and Navier equations) explain exactly how fluids move and how physical properties interact without relying on particles.
The three levels of analysis each explain the same underlying reality: how fluid flows. In principle, each theory must be built on the theory below the hierarchy. The Euler and Naviestokes equations at the macroscopic level must be followed logically from the Boltzmann equations at the mesoscopic level. This is the kind of “axiomization” that Hilbert requested in his sixth issue, and he explicitly referenced Boltzmann’s work on gas in the article in question. The complete theory of physics expects to follow mathematical rules that explain phenomena from the microscopic level to the macroscopic level. If scientists can’t fill that gap, it may suggest a misunderstanding of our existing theory.
Integrating three perspectives on fluid dynamics has posed a stubborn challenge to the field, but Deng, Hani and MA may have just done it. Their achievements are based on gradual progress over decades. However, all previous advancements came with some kind of asterisk. For example, the involved derivatives only worked on short timescales, vacuums, or other simplification conditions.
The new evidence consists of three steps. We derive macroscopic theory from mesoscopic theory. We will derive the neutral theory from microscopic theory. And we sew them together in a single derivation of macroscopic laws, far from microscopic laws.
The first step was previously understood, and even Hilbert himself contributed to it. On the other hand, deriving the neutral mirror from the microscope was much more mathematically challenging. Remember that mesoscopic settings are about the collective behavior of a huge number of particles. Thus, Deng, Hani, and Ma saw what happens to Newton’s equation as the number of individual colliding particles increases infinitely and their size shrinks to zero. They have demonstrated that stretching Newton’s equations to these extremes would converge the statistical behavior of the system, or the behavior of “typical” particles in a fluid, into the solution of the Boltzmann equation. This step forms the bridge by deriving mesoscopic mathematics from the extreme behavior of microscopic mathematics.
The main hurdle for this step was about the length of time the equation was modelling. Although there was already known how to derive the Boltzmann equation from Newton’s Law on a very short timescale, Hilbert’s program is not sufficient as actual liquids can flow for any time. Longer timescales become more complicated. More collisions may occur, and the entire history of particle interactions may withstand current behavior. The authors have overcome this by carefully explaining how much particle history affects its current influence and by leveraging new mathematical methods to assert that the cumulative effects of previous conflicts remain small.
In previous work to derive the Euler and Naviestokes equations from the Boltzmann equation, bonding their long-timescale breakthroughs integrates three theories of fluid mechanics. This finding converges mathematically into one ultimate theory that explains one reality, and therefore justifies taking different perspectives on fluids based on what is most useful in the context. Assuming the evidence is correct, it breaks new ground with Hilbert’s program. I hope that with such a fresh approach, the dam will explode into Hilbert’s challenges and more physics flow downstream.
This article was first published in Scientific American. ©ScientificAmerican.com. Unauthorized reproduction is prohibited. Follow Tiktok and Instagram, X and Facebook.
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