A mathematical problem with 125 years of age is finally resolved by a team of scientists from the University of Michigan and other research institutions. This is the sixth problem entitled “Mathematical treatment of the axioms of physics”, an ambitious challenge raised by the mathematician David Hilbert in 1900.
Hilbert wondered how it was possible to derive certain laws of physics, particularly those related to fluid behavior to different space scales. In an article published in El Portal Arxivthe authors claim to have successfully unified these laws, which could have deep implications in the field of fluid dynamics and improve our understanding of atmosphere behavior and oceans.
Hilbert’s mathematical problem
The issue raised by Hilbert is that there are three different treatments of the movement of fluids. The first is the microscopic: with the laws of Newton’s movement, you can follow, at least in principle, the trajectories and behavior of each of the particles that make up the fluid. The second is the mesoscopic, provided by the Boltzmann equation (⟨ecin⟩ = 23 kBT), which describes “statistically” the behavior of the fluid.
Finally, there is the macroscopic, provided by the fluidodynamic equations of Euler and Navier-Stokes, which describe the behavior of the fluid “as a whole.” The second part of Hilbert’s sixth problem is precisely the derivation and unification of these three treatments from the first axioms, in an organic and harmonious theory. And this is what Yu Deng, Zaher Hani and Xiao Ma, the three authors of the work, claim to have achieved, deriving the macroscopic equations of the fluids from Newton’s laws and through Boltzmann’s kinetic theory.
One of the main difficulties in solving this problem is related to time, a subtle and complex entity whose understanding and treatment has persecuted all physicists. As explained by the main author Yu Deng, some of the laws in question are symmetrical with respect to the investment of time, in the sense that they “work” whether time flows forward and backwards, that is, they do not identify a preferred direction for the flow of time. On the other hand, our daily experience tells us otherwise: we know that time always flows in one direction, as well as the laws of Boltzmann and the laws of thermodynamics, which indicate a precise direction for the flow of time, from the past to the future, or, to be more technical, of a minor state entropy to a state of greatest entropy. The three specialists managed to understand how, where, when and why this switch is activated that identifies a privileged direction for the flow of time, thus exceeding the paradox of the “temporal symmetry”.
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Simplification and resolution
Until now, several “partial solutions” had been proposed to the sixth Hilbert problem, but the newly published approach leads to a more general formulation. The three authors found a way to simplify the treatment of particle movement equations that interact repeatedly, and thus, reducing the complexity of the problem, they managed to make all the pieces of the mosaic fit.
But despite their success, they refer that: “The solution to Hilbert’s problems is not yet complete.” The importance of the sixth problem lies not only in the axiomatization of the laws of physics, but also in the understanding of the implications of these mathematical models. “We know that the models, in a certain point and a specific spatial scale, stop working. I think that the modern formulation of Hilbert’s sixth problem should be formulated in terms of ‘understanding what happens when this happens,” describes Deng. Scientists are especially interested in understanding what happens at even more microscopic scales, when fluid equations lead to calls singularities, that is, mathematical solutions that make no sense from the point of view of physics. It is not just an academic eagerness; In many scenarios of the fields of oceanography and atmospheric science, singularities are often found, and freshly published work can help understand how to address this issue. However, as the authors admit, this could take a long time.
Article originally published in Wired Italy. Adapted by Alondra Flores.
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