First mathematical ‘water machine’

Seven years ago the Australian mathematician Terence tao, awarded a Fields medal, proposed a new approach to solving the famous problem about the so-called Navier-Stokes equations, which describe the motion of the fluids.

The professor Eva miranda from the Polytechnic University of Catalonia (UPC) saw the post on Tao’s blog and it caught his attention, since at that time he was finishing a job with Daniel Peralta-Salas (ICMAT-CSIC) and Robert Cardona (BGSMath – UPC) on fluids in boundary spaces.

Now, these three authors, together with Francisco Presas (ICMAT-CSIC), for the first time have managed to build solutions for a fluid capable of simulating any Turing machine, motivated by the Tao approach. The result is published today in the journal Proceedings of the National Academy of Sciences (PNAS).

For the first time, solutions have been obtained for a fluid capable of simulating any Turing machine, an abstract construction capable of simulating any algorithm

A turing machine it is an abstract construction capable of simulating any algorithm. It receives, as input data, a sequence of zeros and ones and, after a number of steps, returns a result, also in the form of zeros and ones.

The fluid studied by the researchers can be considered as a ‘water machine‘: takes as input data a point in space, processes it –following the path of the fluid through that point– and offers as a result the next region to which the fluid has moved.

The result is an incompressible, non-viscosity fluid – the Navier-Stokes equations do consider viscosity – in dimension three. It is the first time that a water machine has been designed.

Unspeakable phenomena

One of the main consequences of the result is that it allows to prove that certain hydrodynamic phenomena are undecidable (problems without an algorithm leading to a correct yes or no answer). For example, if we send a message inside a bottle, we cannot guarantee that it reaches its recipient.

Something similar happened to the 29,000 rubber ducks that fell off a freighter during a storm and were lost in the ocean in 1992: no one could predict where they would appear. That is, there is no algorithm that allows us to ensure whether a fluid particle will pass through a certain region of space in finite time.

“Is inability to predict, which is different from that established by chaos theory, supposes a new manifestation of the turbulent behavior of fluids ”, the authors affirm.

The results show that certain hydrodynamic phenomena are undecidable, which is a new manifestation of the turbulent behavior of fluids

“In chaos theory, unpredictability is associated with the extreme sensitivity of the system with the initial conditions –the flapping of a butterfly can generate a tornado–, in this case it goes further: we prove that there can be no algorithm that solves the problem, It is not a limitation of our knowledge, but of the mathematical logic itself ”, highlight Peralta-Salas and Miranda, also ICREA Academia professor, member of the Center for Mathematical Research and the Paris Observatory.

This shows the complexity of the behavior of fluids, which appear in various fields, from weather forecasting to flow and waterfall dynamics.

Possible relationship to a millennium problem

Regarding its relationship with the Navier-Stokes problem, included in the Clay Foundation’s list of Millennium Problems, the researchers are cautious and assure that Tao’s proposal “is, for the moment, hypothetical,” they say.

The now reframed approach was proposed by mathematician Terence Tao as a strategy to try to solve one of the problems of the millennium, that of the Navier-Stokes equations.

Your idea is to use a water computer to force the fluid to accumulate more and more energy in smaller and smaller regions, until a singularity, that is, a point at which the energy becomes infinite. The existence or not of singularities in the equations is precisely the Navier-Stokes problem.

However, “at the moment we do not know how to do this for the Euler or Navier-Stokes equations,” say scientists who have discussed their results with Tao.

Combination of several mathematical areas

The Cardona, Miranda, Peralta-Salas y Presas water machine –the first that exists– is guided by the Euler equations but their solutions have no singularities. For its design, various geometry, topology and dynamic systems tools developed in the last 30 years have been key.

Specifically, symplectic and contact geometry and fluid dynamics are combined with the theory of computer science and mathematical logic. “It took us more than a year to understand how to connect the various cables of the demonstration,” the authors conclude.

Rights: Creative Commons.

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