For over two centuries, science has dreamt of a perfectly predictable universe. Formulated in 1814 by the French mathematician and astronomer Pierre-Simon Laplace, Laplace’s demon suggested that knowing the initial properties and governing laws of nature’s particles would make it possible to predict the future with absolute precision. However, mathematician Alan Turing dismantled this vision in computing with the halting problem, proving that no general method can determine, solely from initial data, whether a computer program will finish running or loop forever.

When fluids can ‘think’

Now, a study led by the Universitat Politècnica de Catalunya - BarcelonaTech (UPC) and the ICMAT of the Spanish National Research Council (CSIC), published in the scientific journal PNAS Nexus, has shown that undecidable trajectories can also emerge in systems like viscous fluids. In other words, just as in computing, there are scenarios where future particle behaviour cannot be predicted, even when the equations governing the system are perfectly known.

The findings are based on the Navier-Stokes equations, which describe fluid motion. The study demonstrates the existence of stationary solutions to these equations that can simulate a universal Turing machine—a computational model introduced by Alan Turing—meaning they can replicate any calculation.

To explain this phenomenon, Eva Miranda, a professor at the UPC’s School of Mathematics and Statistics (FME) and an ICREA Academia researcher, recalls an iconic event in oceanography: the rubber ducks that were lost in the Pacific Ocean in 1992 after accidentally falling from a cargo ship. “They were predicted to reach the coast of England by 2007, but many never arrived. It is as if they entered a ‘dead zone’ where we can no longer know if they will reach their final destination. We have found a possible explanation: there are trajectories that become unpredictable at a certain point. Mathematics simply loses track.”

By bridging three-dimensional cosymplectic geometry, mathematical analysis and computational theory, the study shows that when a particle crosses a harmonic velocity field, its trajectory can be so complex that tracking its path is equivalent to running computer calculations.

“Imagine a drop of ink falling onto a calm sea. The thin, winding thread it traces as it sinks corresponds to a sequence of zeros and ones—a computation tape like those used by computers. As the ink descends and the fluid transforms it into increasingly complex shapes, the water is essentially ‘computing’—solving problems, processing data and executing instructions. This means a fluid could perform any calculation a computer can: calculate the CPI, win a game of chess or crack an encrypted message,” Miranda explains.

Additionally, the researchers verified that determining whether a fluid particle will pass through a specific region in these simulation systems is equivalent to Turing’s halting problem. “Turing proved that no general method can tell us whether a program will stop, making it an undecidable problem. Therefore, knowing whether a particle will reach a specific space is also, generally, an undecidable problem,” the researcher states.

Eva Miranda’s team has shown that these stationary solutions can appear at all viscosity values. Consequently, this behaviour can occur both in low-viscosity fluids like water and air, and in highly viscous ones like volcanic lava.

The results reveal a new side to the Navier-Stokes equations, one of the Millennium Prize Problems set by the Clay Mathematics Institute: “Even in systems governed by well-known, deterministic laws, such as fluids under the Navier-Stokes equations, some questions remain unanswerable. Not because we lack information, but because the very nature of the problem prevents it,” Miranda concludes. This also suggests that fluids are not just a classical object of physics research, but a realm where the fundamental limits of knowledge and prediction manifest themselves.

A question that has challenged renowned scientists

In his book The Emperor’s New Mind, Nobel laureate physicist and mathematician Roger Penrose posed a riddle: are there fundamental limits to the computational capacity of physical systems? Can nature itself think? In 1991, computer scientist, mathematician and physicist Cris Moore sharpened this intuition by asking directly: could fluids compute?

Fields Medallist Terence Tao took up the gauntlet with an even bolder idea. If viscous fluids can simulate a Turing machine, researchers could theoretically design initial conditions that drive the fluid to an extreme state: a finite-time blow-up where the equations break down and singularities form. If successful, Tao would solve one of the seven Millennium Prize Problems, which carries a one-million-dollar reward. 

This very approach led the team headed by Eva Miranda and Daniel Peralta, a researcher at ICMAT, to embark on this mathematical venture —one that has managed to prove that nature not only can compute, but actually does.

Miranda notes that “Tao’s dream is stubborn. Current constructions, brilliant as they are, do not yet produce a blow-up. The door is open, but it has not yet been crossed. And this is where the story gets even more exciting: Google DeepMind, collaborating with mathematicians worldwide, has now set its sights on this very problem, recognising it as one of the most fascinating frontiers in contemporary mathematics. Artificial intelligence has also joined the fray; for now, however, the great enigma remains unsolved.”

The question Penrose tossed like a stone into a pond continues to create ripples. “From Barcelona, we keep pushing the boundaries because in mathematics, as in fluids, the movement never stops,” Eva Miranda concludes.

Further information

• Turing complete Navier–Stokes steady states via cosymplectic geometry. Paper published in PNAS Nexus
• Eva Miranda: “Research is an emotional rollercoaster”