Mathematicians have finally succeeded in unifying the laws of physics that govern the motions of particles at different scales. Their efforts resolve a question set by mathematician David Hilbert in 1900 as part of an ambitious programme for all mathematicians of the 20th century – and could deepen our understanding of the complex behaviour of fluids in the atmosphere and oceans.
“This is a major result in my view. I thought it was completely beyond reach,” says at the University of Lyon in France.
Specifically, at the University of Michigan and his colleagues have demonstrated how to consistently and meaningfully stitch together physical laws at three different scales. First, there is the microscopic realm of single particles colliding with each other in accordance with Isaac Newton’s laws of motion. In the mesoscopic realm of larger objects, collections of such particles instead follow statistical laws pioneered by Ludwig Boltzmann. At the even larger scale macroscopic scale, where we reside, physicists turn to notoriously difficult mathematical tools such as the Navier-Stokes equation, which captures all the intricacies of how a fluid behaves.
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Over the years, physicists and mathematicians have established some links between the three frameworks, but until now they were never fully united. The quest to do so began in the 19th century, says Hani, after Boltzmann presented his statistical techniques and his contemporaries clamoured for a rigorous mathematical proof that they actually worked. This ultimately morphed into , which calls for a derivation of laws dictating the behaviour of fluids from the most basic, bare-bones mathematical axioms.
One reason this was so desirable is because some of these laws are reversible in time and some are not, says team member at the University of Chicago. For instance, Newton’s laws are not sensitive to the direction of time’s flow, which renders “before” and “after” interchangeable, while Boltzmann’s statistical equations suggest a way to demarcate the two. Deng says that his team’s work, which has been ongoing for more than half a decade, elucidates when and how this switch happens, eliminating the possibility of a time-related mathematical paradox.
A key ingredient of the team’s approach relies on recasting calculations in terms of diagrams originated by the physicist Richard Feynman, who used them for tackling problems in quantum field theory. Mathematicians have learned to use these diagrams to tackle difficult equations for particles that repeatedly interact with each other, as occurs in a fluid, but Hani says this can become overwhelming. Instead, the team found a way to reduce the number of diagrams they had to calculate exactly, which allowed them to build a clear mathematical path from Newton’s laws to the Navier-Stokes equation.
Texier says that while there is a long history of partial solutions to Hilbert’s sixth problem, the new work is a “real leap forward”, validating both the way Hilbert posed the problem and the intuition behind Boltzmann’s original work. In other words: the new proof reaffirms the way physicists have been thinking about fluids and gases for more than a century, while guaranteeing a firm mathematical foundation. But Hani says the team does not feel like their work closes the book on Hilbert’s quest.
“The importance of [Hilbert’s sixth] problem is really not just in terms of axiomatizing the laws of physics, but it’s also in terms of understanding the implications of these [mathematical] models. We know that they break down at some point. I think the modern motivation for Hilbert’s [sixth] problem should be in terms of understanding what happens when those models break,” he says.
Deng says that he is particularly interested in what happens at the smallest, most microscopic scale, when the more macroscopic fluids equations develop singularities, in other words when their solutions become meaningless. This can happen in a broad range of situations in oceanography and atmospheric science, but the researchers may now be able to get an exact picture because of their rigorous connection between the two scales.
For Texier, all the implications of the new work are not yet clear, simply because it is such a rich and complex piece of mathematics. “I think it’s going to take a lot of effort for the community to digest it,” he says.
arXiv
