Summary of Key Points
In 2025, the young mathematician Deng Yu (a professor at the University of Chicago), together with his collaborators Zaher Hani and Ma Xiao (both from the Young Scientists Program at the University of Science and Technology of China), formed the DHM team. They solved one of the most challenging parts of Hilbert's Sixth Problem: they rigorously derived the Boltzmann equation from microscopically reversible Newtonian mechanics, which describes macroscopic irreversible phenomena, and further connected it to fluid dynamics equations. This breakthrough filled a mathematical gap between classical mechanics and thermodynamics, explaining the microscopic origins of the "arrow of time" (such as a broken cup that cannot be restored to its original state). As a result, Deng Yu, along with Wang Hong, was awarded the Fields Medal that year—the highest honor in mathematics.
Detailed Explanation
Hilbert's Sixth Problem: The Centenary Question of “Mathematizing Physics”
In 1900, the mathematical genius David Hilbert proposed 23 directions for future research. The sixth problem focused on using mathematical axioms to make physical laws as rigorous as geometry (for example, Euclidean geometry has five axioms; physics should also have a clear mathematical foundation). The most difficult sub-problem was how to derive macroscopic thermodynamic laws (which are irreversible, such as the increase in entropy) from the microscopic motion of particles (which is reversible).
Classical Mechanics vs. Thermodynamics: Why Does Time Have a Direction?
- Microscopic Reversibility: The motion of individual molecules is bidirectional; reversing the trajectory of a molecule still conforms to Newton's laws (for example, a bouncing ball will rebound in the reverse direction as well).
- Macroscopic Irreversibility: All phenomena in life exhibit an “arrow of time”: a broken cup cannot be restored, hot water does not cool back on its own, and gas diffuses and does not return to its original state. This is due to the second law of thermodynamics: the disorder (entropy) in the universe always increases.
- Paradoxes:
- Loschmidt's Paradox: How can reversible microscopic particles lead to irreversible macroscopic phenomena?
- Poincaré Recurrence: Theoretically, a system could return to its initial state after a long enough time (for example, a broken cup could become whole again), but this has never been observed in reality. Boltzmann explained this as a matter of statistical probability, but there was a lack of rigorous mathematical proof.
The DHM Team's Breakthrough: Building a Mathematical Bridge from Microscopy to Macroscopy
Previous mathematicians (such as Lanford) had only proven that the Boltzmann equation held true for short periods of time, but Hilbert required a rigorous derivation for any length of time. The DHM team solved two key problems:
- Proving the Reasonableness of Molecular Chaos: A core assumption of the Boltzmann equation is that the speeds of colliding molecules are independent and do not affect each other. The DHM team used mathematical methods to rigorously verify that this assumption holds true over long periods.
- Connecting Microscopy and Macroscopy: They started with Newtonian mechanics (the motion of N particles) and derived the Boltzmann equation (which describes the statistical behavior of gas molecules), and then further derived fluid dynamics equations (such as Euler’s equations, which describe water and air flows). This established a complete mathematical link from “microscopic particles” to “macroscopic gases” to “fluids.”
The Significance of the Breakthrough:
- Mathematically: It filled a century-long gap, transforming the Boltzmann equation from an empirical assumption into a rigorous theory.
- Physically: It provided a stronger foundation for fields such as fluid dynamics and statistical mechanics (for example, weather forecasts and aircraft design now have a theoretical basis at the microscopic level).
- Philosophically: It finally provided a mathematical answer to the question of where the “arrow of time” comes from: it is not due to any special properties of microscopic particles, but rather the inevitable statistical behavior of a large number of particles leading to an increase in entropy, which in turn determines the direction of time.
Deng Yu: The “Atypical” Growth of a Genius Mathematician
Deng Yu is not the typical “bookworm”:
- Academic Path: Won the gold medal in the IMO (International Mathematical Olympiad) in 2006 → Studied at Peking University’s Department of Mathematics → Went to MIT → Became a professor at the University of Chicago.
- Academic Achievements: Has published papers in three top mathematical journals (The Annals of Mathematics, Advances in Mathematics, and the Journal of the American Mathematical Society), with recent publications in these prestigious journals.
- Broad Interests: His Zhihu profile states that he enjoys poetry, comics, Go, football, and all beautiful things, showcasing a multifaceted side of a mathematician.
Conclusion
The achievements of the DHM team are not only a major mathematical breakthrough but also a bridge connecting microscopic and macroscopic physics. They have shown us that what appears to be the irreversible arrow of time is actually an inevitable result of the statistical behavior of microscopic particles—a victory for both science and philosophy. Deng Yu’s award also demonstrates that mathematics is not just an abstract game of symbols but a powerful tool for understanding the essence of the world.