15/07/2020

IMPA researchers discuss the accuracy of modeling.

Imagem das nuvens ilustra a instabilidade de Kelvin – Helmholtz. Abaixo, sistema retirado do estudo mostra dois futuros diferentes emergindo da mesma condição inicial.

Cecilia Manzoni

With the novel coronavirus, mathematical models gained visibility and became a subject of general interest, as they projected possible scenarios for the unfolding of the pandemic. But the disparity between these projections has raised a question in society that also worries scientists: should we always expect precise answers from mathematical models? According to chaos theory, the randomness present in some natural phenomena is due to the imprecision of the initial data. In an article published in July in the journal Communications Physics, from Nature , researchers from IMPA point to a new paradigm. Some phenomena of the classical world observed in nature may be intrinsically random.

Written by Simon Thalabard, a distinguished postdoctoral fellow at IMPA; Alexei Mailybaev, a researcher at the institute; and Jérémie Bec, from MINES ParisTech, the study “From the butterfly effect to spontaneous stochasticity in singular shear flows ” analyzed the Kelvin-Helmholtz instability. This phenomenon is found in nature when two flows have different velocities, creating a velocity jump at the boundary between them. “It’s a phenomenon of fluid dynamics and classical physics, which can be observed at the meeting of two rivers and also in cloud formation, creating visually beautiful structures,” explains Mailybaev.

Among the equations used to project these phenomena is the Navier-Stokes equation. “These are very old and challenging equations from a mathematical point of view because, even if the initial conditions are very good, they don't necessarily deliver a regular prediction of the future,” points out Thalabard. The impasse represented a “major question” for the mathematical community, says the postdoctoral researcher. “From the point of view of physical modeling, what to do with these equations? Discard them or reinterpret them?” he asks.

After studying the topic for over two years, the experts discovered that even in classical mechanics phenomena, an area where predictions are usually deterministic, with a cause and consequence for each effect, spontaneous stochastic characteristics can be found. Specifically addressing the Kelvin-Helmholtz instability, the group concluded that the Navier-Stokes equations are useful for studying the phenomenon, but should be used in projections in a probabilistic, not deterministic, way, as previously thought. "The projection will give a percentage of the chances of obtaining a result," explains the postdoctoral researcher.

“It’s a connection between two opposing worlds: the deterministic and the probabilistic. In the deterministic world, if you know the state of your system at the initial moment, it’s possible to predict the future exactly. The probabilistic world, on the other hand, is described by random quantities. From the beginning, you assume you don’t know everything. A phenomenon can generate several predictions, not just one,” Mailybaev adds.

For the IMPA duo, the article also brings a “philosophical” discovery. “Chaos theory suggests that the randomness we see in the world is due to the fact that we are 'too lazy' to make good measurements. Randomness would reflect our lack of ability to measure things. But perhaps the world around us is free from determinism, and this brings a new perspective to mathematical modeling in general,” Thalabard mentioned.

Mailybaev believes that the “philosophical” aspect of the study questions some concepts we have ingrained about science since school. “We carry this notion that science is linked to determinism, that the laws of nature are all determined, as is the future. Quantum mechanics has somewhat changed this perception, with intrinsic probabilities. But what we show with the article is that even the classical world can spontaneously contain a probabilistic outcome, with various possibilities.”

Check out the full article.