A trio of mathematicians, Yu Deng of the University of Chicago and Zaher Hani and Xiao Ma of the University of Michigan, have reportedly made a significant breakthrough in mathematics, providing a rigorous derivation that explains the one-way flow of time. Their work, detailed in a recent preprint paper on arXiv.org, addresses a crucial aspect of David Hilbert's Sixth Problem, posed over 125 years ago, by mathematically linking the chaotic behavior of individual gas particles to the predictable, fluid dynamics described by equations like Navier-Stokes. This achievement offers a foundational understanding of why phenomena like aging and entropy increase are irreversible.
Hilbert's Sixth Problem challenged mathematicians to place physics on a firm axiomatic foundation, specifically seeking to derive macroscopic fluid mechanics from microscopic particle behavior. For decades, scientists had struggled to fully connect Newton's reversible laws of motion, which govern individual particles, to Boltzmann's kinetic theory and subsequently to the Navier-Stokes equations that describe fluid flow. The primary hurdle was rigorously demonstrating how the statistical behavior of countless particles leads to the continuous, one-way processes observed in fluids over extended periods.
Deng, Hani, and Ma's paper, titled "Hilbert's sixth problem: derivation of fluid equations via Boltzmann's kinetic theory," presents a comprehensive proof. They successfully demonstrated how the dynamics of hard sphere particle systems, under specific conditions, rigorously give rise to the Boltzmann equation, and then to the fundamental partial differential equations of fluid mechanics. This intricate derivation bridges the gap between the microscopic, mesoscopic, and macroscopic descriptions of physical systems, completing a long-sought chain of logic.
The profound implication of this mathematical proof lies in its connection to the arrow of time. As stated in the tweet by Mario Nawfal, "Newton’s laws say particles can run forwards or backwards in time. But the Boltzmann and Navier-Stokes equations? Pure one-way traffic." The mathematicians' work provides a rigorous mathematical backing for this irreversibility, showing how entropy, a measure of disorder, inherently increases in isolated systems. This explains why ink disperses in water but never spontaneously recoalesces, or why biological systems age without de-aging.
While the paper is currently a preprint undergoing peer review, its potential impact on theoretical physics and applied sciences is significant. This mathematical grounding could enhance the accuracy and reliability of complex simulations, such as those used in weather forecasting and engine design. The resolution of this long-standing challenge offers a deeper, unified understanding of fundamental physical laws and the inherent directionality of time itself.