Equations governing fluids can be difficult to understand
Vladimir Velyanovsky / Alami
In 1900, the mathematician David Hilbert presented to his colleagues a list of problems that, in his opinion, reflected both the current state of mathematics and its future. This year, 125 years later, Zaher Hani at the University of Michigan and his colleagues solved one of Hilbert's problems – and combined several laws of physics in the process.
Hilbert was a proponent of deriving all the laws of physics from mathematical axioms—statements that mathematicians consider to be fundamental truths. The sixth task on his list was to derive, from such axioms, the laws of physics governing the behavior of liquids.
Before 2025, physicists actually had three different ways to describe liquids, depending on their scale. Different rules governed the microscopic scale of individual particles, the mesoscopic world populated by aggregates of particles, and the macroscopic world populated by full-fledged liquids, such as water flowing in a sink. Researchers have made progress in finding connections between them, but the three were never put together until Hani and his colleagues figured out how.
The researchers' breakthrough came in part because they learned to use a diagram-based technique that physicist Richard Feynman had developed for a seemingly completely different field: quantum field theory. And it was hard work: their work, published earlier this year, marked the high point in a fifty-year project.
“We heard about the result from many people, especially from industry leaders who tested the work very carefully,” says Honey. The work, which was available as a preprint, is now being prepared for publication in a prestigious mathematics journal, he said.
In addition to being a remarkable mathematical achievement, the team's work also offers the chance to improve our understanding of the complex behavior of fluids in the atmosphere and oceans. Honey says they are now also working on the quantum version of the problem, where microscale mathematics allows a lot to be done. weirder and richer particle behavior.
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