Juan José Mendoza-Arenas Found a New Use for an Existing Mathematical Tool
A paper from the lab of Juan José Mendoza-Arenas illustrating an extraordinary example of mathematical efficiency was recently highlighted by the journal Physical Review Applied as an Editor’s Choice.
The work shows a method originally developed to analyze correlated quantum systems—later adapted to simulate classical fluids—can also be used to calculate the properties of an entirely different kind of quantum system: quantum turbulence. The method, based on tensor network theory, uses much less computing power than standard simulation techniques, paving the way to analyze bigger systems than ever before, said Mendoza-Arenas, assistant professor of mechanical engineering and materials science in the Swanson School of Engineering.
He also holds a secondary assistant professor position in the department of Physics and Astronomy in the Kenneth P. Dietrich School of Arts and Sciences.
The realm of the quantum is often described as unintuitive or just plain weird. In many ways, it is, but in a twist, when it comes to the theoretical analysis of turbulence in fluids, the quantum phenomenon is easier to understand than its classical counterpart. For this reason, quantum turbulence is sometimes seen as a potential intermediate step to better understanding the classical turbulence of our everyday experience.
“Classical turbulence is very, very difficult to understand,” Mendoza-Arenas said of the chaotic motion that develops when a fluid like water swirls and flows with currents and eddies. When some substances are cooled to very low temperatures, however, their properties change dramatically and they almost act like one giant quantum object. In some cases, these so-called quantum fluids, such as superfluid helium-4 or electron currents in superconductors, flow with no friction.
An important feature of the physics of turbulent motion is that the circulation—a measure of how much the fluid rotates, or swirls, around a closed loop—takes on only discrete values in a quantum fluid. In a classical flow, however, circulation can take on any continuous value. Because of these properties, the quantum scenario is simpler and widely viewed as a steppingstone to the classical case.
To better understand a turbulent system requires probing it at many different length scales. The complexity of its simulation grows exponentially as the number of resolved length scales increases (i.e., as the grid becomes finer). “At a certain point, the problem becomes so complex that our existing, standard methods for analysis just don't allow us to go any further,” Mendoza-Arenas said.
A few years ago, however, a team of researchers, including Nikita Gourianov, Peyman Givi, and Dieter Jaksch (colleagues of Mendoza-Arenas) borrowed an idea from an all-together different branch of physics, many-body quantum physics, where the mathematical objects needed to understand a system grows exponentially with the number of quantum particles.
Sound familiar?
“Some colleagues had the idea to bring the methodologies invented to deal with that issue, but to apply them to classical fluid dynamics,” Mendoza-Arenas said. “That to me is a very exciting transfer of knowledge. Something was invented in one kind of science and now it's been transferred to solve problems in a very different field.”
Crucially, the mathematical tools used to describe these quantum mechanical systems, called tensor networks, were adapted to be applied to classical fluids. This tailored approach meant the new calculations needed less computing power.
Mendoza-Arenas, who has studied quantum systems for many years, thought, why stop there? “Maybe we can do the same thing for quantum fluids.”
In the paper, Simulating quantum turbulence with matrix-product states (DOI: 10.1103/x5xg-whh1), led by PhD student Felipe Gómez-Lozada, Mendoza-Arenas’s team showed they could also use these pared down tensor networks to more efficiently compute the properties of complex turbulence in quantum fluids, too. The tensor network calculations can accurately reproduce the physical properties of these systems, using up to ten thousand times less computational memory than standard simulation methods.
These findings may prove useful in applications to many low-temperature systems, such as superfluid helium, mixtures in Bose-Einstein condensates, or even the recently achieved supersolid states. They may also lead to a better understanding of turbulence in classical fluids.
But for Mendoza-Arenas, the interest is more intrinsic. Yes, quantum turbulence may help deepen our understanding of classical turbulence, but, he said, “The quantum offers much more richness and more variety, which, to me, enhances the interest at a more fundamental level.”