Unraveling the Secrets of Time Through Math

It’s crazy to think that something as abstract as a math problem from over a century ago could help us understand something as fundamental as time itself. But apparently, that's exactly what's happened. A few mathematicians just managed to crack a major piece of a puzzle laid out by David Hilbert back in 1900. He basically wanted to prove that the laws of physics, which describe the real world, are built on a solid foundation of mathematical logic. It was a massive, almost impossibly ambitious goal, but he started by focusing on a seemingly simple question about gases. You know, like the air we breathe.

What he wanted to know was this: Can we show that the equations physicists use to describe individual gas molecules—think of them as tiny billiard balls bouncing around—can be derived from the same mathematical principles that govern the gas as a whole, when it acts like one big, continuous substance? Physicists always assumed these different ways of looking at the same thing were connected, but no one had been able to prove it with the kind of airtight logic Hilbert was looking for. Now, after 125 years, three mathematicians—Yu Deng, Zaher Hani, and Xiao Ma—have finally managed to do just that. Their work is a huge step forward for Hilbert’s grand vision, but it also sheds light on something even more profound: why time only moves forward.


The View from Different Scales



Imagine looking at a gas. You could see it in a few different ways, right? On the smallest, most microscopic level, you have all the individual molecules zipping around, crashing into each other. Isaac Newton’s laws of motion are the tools you'd use to describe this. It’s a lot to keep track of, obviously.

Now, if you zoom out a bit—to what researchers are calling the “mesoscopic” scale—you can’t see every single particle anymore. Instead, you'd use a different model, the Boltzmann equation. This doesn't track individual molecules, but instead gives you a statistical picture of what's probably happening—where most of the molecules are, how fast they’re likely moving, and so on. It's the kind of math you’d use to figure out how air flows around something, like a space shuttle.

Zoom out even further, to the macroscopic scale, and the gas just looks like one big, continuous fluid. You can't even tell it’s made of particles at all. To describe this, you'd use the Navier-Stokes equations, which tell you about things like the gas’s density and overall flow.

Physicists have always treated these three models as different lenses for the same reality. But to really satisfy Hilbert's challenge, mathematicians had to prove that the whole chain of logic holds up: that Newton's microscopic laws lead to the Boltzmann equation, which in turn leads to the Navier-Stokes equations. While they'd had some success with the second step, that first one—connecting the chaotic world of individual particles to the statistical world of the Boltzmann equation—was the real sticking point. Until now, that is.

The Ghost of Recollisions











The big hurdle was proving a crucial assumption that the physicist Ludwig Boltzmann had to make. He showed that Newton’s laws could lead to his equation, but only if the particles in the gas weren't constantly recolliding. In other words, a pair of particles shouldn't hit each other, bounce off, and then come right back to hit each other again, over and over.

The problem, however, is that there are an infinite number of ways particles can collide and recollide, creating this massive, mind-boggling web of interactions. It's an utter nightmare to prove that these scenarios are as rare as Boltzmann needed them to be. A mathematician named Oscar Lanford got close back in 1975, but his proof only worked for an incredibly short amount of time—shorter than the blink of an eye. Beyond that, his math just fell apart, and no one could figure out how to extend his work for decades.

It's pretty amazing, then, that two mathematicians who weren't even working on particle systems—Deng and Hani—found a way in. They usually studied waves, and something in their work on that topic gave them a new way to think about particles. They were later joined by a graduate student, Xiao Ma, and together they figured out how to use their wave-based techniques to tackle the particle problem.


A Breakthrough from Constant Collaboration



The process wasn't easy. The team started with a simpler scenario, a gas in an infinite space where the particles would eventually disperse. This was a bit of a shortcut, but it let them develop their approach. They had to figure out how to break down the incredibly complex patterns of particle collisions into simpler, more manageable pieces. This was as much an art as it was a science. They’d often get stuck, spending late nights on Zoom calls trying to untangle their thinking. It took them months of trial and error, slowly building their proof case by case, from simple collisions to more and more complicated ones.

Eventually, they did it. In a paper posted online in the summer of 2024, they showed that recollisions are indeed incredibly rare, confirming that Boltzmann's statistical description of the gas can be rigorously derived from Newton’s laws for individual particles in this infinite space. Other mathematicians were floored. As one of their doctoral advisors, Alexandru Ionescu, put it, these were “some of the most significant advances in many, many years.”

With the hardest part of the puzzle solved, it didn’t take long for them to apply their findings to the more complex case of a gas trapped in a box—the very scenario needed to finally solve Hilbert’s sixth problem. The logical chain was complete.








The Arrow of Time



This work didn’t just solve a 125-year-old math problem; it also provided a rigorous mathematical explanation for one of physics's most enduring mysteries: the arrow of time.

At the microscopic level of individual particles, time is reversible. Newton’s equations don't care if you run them forward or backward. The past and future are interchangeable. But on our level, the macroscopic one, time only moves in one direction. We get older, not younger. A drop of ink spreads out in a glass of water, it doesn't spontaneously gather back into a perfect droplet. The equations that describe this macroscopic world, like the Boltzmann and Navier-Stokes equations, are not time-reversible.

This has always been a paradox. How could a time-irreversible reality emerge from a time-reversible foundation? Boltzmann's intuition was that while it’s technically possible for a gas to spontaneously contract, it’s so astronomically unlikely that it never happens. Lanford's proof confirmed this for an instant of time, but now Deng, Hani, and Ma have confirmed it for more realistic, longer-lasting situations.

It's a beautiful example of how deep mathematical proofs can not only provide answers but also force us to see the world differently. As a physicist named Gregory Falkovich said, “What mathematicians do to physicists is they wake us up.” Now that the groundwork has been laid, it’ll be interesting to see what other long-standing paradoxes they might tackle next.




Open Your Mind !!!

Source: Wired