Mathematicians Solve Key Hilbert Sixth Problem: From Newtonian Particles to Boltzmann and Navier-Stokes

In this biweekly Quanta Magazine audio edition, a story unfolds about Hilbert’s sixth problem and a modern mathematical breakthrough. Three mathematicians, Yu Deng, Zaher Hani, and Zhao Ma, prove the harder microscopic-to-mesoscopic step for a dilute gas, showing that Newton’s laws governing individual particles can yield Boltzmann’s statistical description, which in turn leads to the Navier-Stokes equations for fluid flow. The episode traces how earlier partial results, including Lanford’s short-time proof, left a gap of recursion and recollisions, and how the researchers’ wave-inspired techniques were adapted to particles, enabling a complete microscopic-to-mesoscopic-to-macroscopic chain. It also delves into the time-arrow paradox: reversible microscopic dynamics versus irreversible macroscopic behavior.

Introduction: Hilbert’s Sixth Problem and a Modern Answer

This podcast episode from Quanta Magazine presents a sweeping view of a century-old aspiration: to axiomatize physics with the same mathematical rigor used in geometry. David Hilbert’s Sixth Problem challenged mathematicians to derive the laws of physics from first principles, in a manner parallel to axiomatizing geometry. The focus of the narrative is a trio of mathematicians—Yu Deng, Zaher Hani, and Zhao Ma—who finally made decisive progress on the most difficult link in the chain: deriving the microscopic Newtonian dynamics of individual particles (hard-sphere gas) from Boltzmann’s mesoscopic description, thereby bridging to the macroscopic Navier–Stokes equations. The story also engages with the conceptual stakes: even though the fundamental laws are time-reversible, the emergent behavior of gases exhibits irreversibility, prompting a deeper look at the arrow of time as a physical and mathematical phenomenon.

"the way the sixth problem is actually stated, it's never going to be solved." - Dave Levermore, University of Maryland

Background: From Microscopic Motion to Macroscopic Laws

To understand the breakthrough, the transcript reviews the three-tier modeling of gases. At the microscopic level, gas particles behave like billiard balls obeying Newton’s laws. Zooming out to the mesoscopic scale, Boltzmann’s equation predicts how particle numbers distribute across locations and speeds. At the macroscopic level, Navier–Stokes equations describe fluid density and velocity fields. The problem Hilbert posed was to demonstrate that these layers are compatible, i.e., that Newton’s laws imply Boltzmann’s equation and that Boltzmann’s description gives rise to Navier–Stokes. Mathematicians had shown partial progress on the second step, but the first step remained elusive for 125 years. Lanford’s theorem in 1975 provided a breakthrough for an extremely short time window, but could not guarantee the absence of recollisions for longer times. The story then introduces the key players and the prior landscape, including Boltzmann’s insight that recollisions must be rare yet mathematically proving this had remained out of reach for decades.

"what Boltzmann couldn't do was prove theorems about this because there wasn't structure or tools to do it at the time." - Sergio Simolina, Sapienza University

The Breakthrough: Adapting Wave Techniques to Particles

The core of the episode centers on how Deng, Hani, and Ma adapted methods from wave-like systems to a particle gas. Their strategy begins with the infinite space setting, where particles disperse and collisions eventually fade, providing a simplification a path forward could exploit. They first catalogued all possible collision patterns, pruning those with high recollision rates, which left a finite, but still vast, set of patterns to analyze. The real obstacle was that these patterns involve enormous numbers of particles and intricate indirect interactions, making a direct probabilistic tally seem intractable. The researchers drew on Deng and Hani’s earlier work on waves, which showed that complicated interactions could be decomposed into simpler components while preserving accuracy in estimating probabilities. They then systematically tackled progressively harder cases, extending their analysis from short to longer times and from free space to more realistic confined settings. The collaborative process included intensive Zoom sessions late at night as the team refined their slicing of complex collision structures into tractable estimates.

"It's as much an art as a science." - Yu Deng

By spring 2024, the team believed they had covered all essential scenarios. Their result demonstrated that recollisions are exceedingly rare in an infinite-space gas, enabling Boltzmann’s description to be derived from Newton’s laws within that setting. Princeton mathematician Alexandre Ianescu, Deng and Ma’s doctoral advisor, described the work as outstanding and among the most significant advances in years. The narrative then explains how they extended the infinite-space results to the boxed gas setting, completing the chain that Hilbert envisioned: microscopic dynamics → mesoscopic Boltzmann equation → macroscopic Navier–Stokes equations. In March 2025, they posted a paper formally combining their results with the known connections between Boltzmann and Navier–Stokes, achieving a holistic, rigorous framework for a realistic gas model.

"time is reversible at the microscopic scale but irreversible at the mesoscopic and macroscopic scales" - Simonella (as discussed in the article)

Implications: Time’s Arrow and the Power of Rigorous Proofs

The transcript emphasizes the philosophical and scientific payoff: although individual particle collisions are time-reversible, the aggregate behavior at larger scales exhibits irreversibility. The Boltzmann equation and Navier–Stokes equations are not time-reversible, so running time backward yields nonsensical results. The breakthrough thus not only completes a mathematical program but also provides a rigorous framework for understanding why a gas behaves the way it does across scales, shedding light on the emergence of the arrow of time. Looking forward, researchers hope to apply these techniques to more complicated particle systems, including non-spherical shapes and more intricate interactions, to see how robust the approach is in broader physical contexts. Physicists like Gregory Folkovich of the Weizmann Institute of Science describe how such rigorous proofs wake up physics by clarifying when particular models are appropriate and why some predictions hold across regimes.

"these sorts of rigorous proofs can help physicists understand why a gas behaves a certain way at various scales" - Gregory Folkovich

Looking Ahead: Expanding the Framework

The episode closes by noting that the methods developed to tame the combinatorial explosion of potential collision patterns could adapt to more realistic gases, including those with diverse particle shapes or more complicated interactions. The overarching achievement is not simply a specific theorem, but a paradigm—an approach to connecting the microscopic, mesoscopic, and macroscopic pictures with mathematical rigor. It opens doors to deeper explorations of kinetic theory, statistical mechanics, and the foundations of fluid dynamics, while highlighting the collaboration, patience, and creative problem-solving that drive major advances in mathematics and physics.

"time doesn’t rejuvenate as you go forward in time" - Simonella (in the context of time’s arrow)