A study on the evolution of dynamic systems, funded by the European Union, has unexpectedly led to a better understanding of the chaos in the quantum world.

In mathematics, the theory of dynamical systems studies the evolution of systems that change over time. Using the so-called evolution rule in this setting, researchers should be able to describe the future state of a phenomenon based on its current state. A wide range of scientific fields, beyond mathematics alone, could benefit from these predictions, such as physics, biology, chemistry, engineering, economics or medicine.

The EU-funded LDMRD project (Large Deviations and Measurement of Rigidity in Dynamics) was therefore initiated to develop new tools adapted to this theory of dynamical systems, while exploring possible applications for problems related to mathematical physics, geometry or arithmetic. One of the main objectives of the project was to continue the study of “rigidity of measurement”, a parameter that would facilitate precision measurements when more than one dynamic system behaves very differently over time.

Within dynamic systems theory, the so-called “evolution” rule is largely deterministic, which means that for a specific period of time only, a future state follows the current state. Yet some are more chaotic, as random external events can concretely influence the future of some dynamic systems. It is for this reason that the theory of dynamical systems has proved particularly applicable to quantum mechanics. As the LDMRD project coordinator, Dr. Tuomas Sahlsten, explains: “This theory is particularly useful for studying “chaotic “systems in the sense that two systems can evolve on radically different trajectories even if they start from different positions close to each other.

The tools needed to explore the rigidity of dynamic systems (based on the work of Bernard Host in the 1990s) far exceeded those currently available, forcing the team to learn from neighboring disciplines. They have borrowed from number theory, representation theory, quantum chaos and other disciplines.

As a by-product of this process, researchers have identified the applicability of this project to quantum mechanics, which has led them to formulate a new theorem of thermodynamic quantum chaos combining ideas from large networks. Large networks — such as Big Data, social networks, convolutional neural networks of Artificial Intelligence models — which, although commonly used today, do not yet have adequate theories to explain their operation. One of the approaches generally adopted is spectral theory, and the LDMRD team has succeeded in generating new ideas about the theoretical spectral properties of large-scale surfaces, which serve as analogies to these large-scale networks. .

As Dr. Sahlsten puts it: “By far the most extraordinary and exciting discovery of the project concerns this connection with quantum mechanics. It is sometimes difficult to predict, before you begin research, what discoveries will emerge when you actually start working on the subject.”

By deepening their knowledge of quantum chaos, the researchers also make a major contribution to understanding the temporal evolution of quantum states of chaotic dynamical systems over a long period of time. Since the project’s research focuses on pure mathematics, the immediate impact of the project will be felt most directly by a scientific community dedicated to quantum mechanics. Indirectly though, as this community plays a major role in modern scientific advances and technologies, be it meteorological/climatic studies or computer engineering, the progress of this project will also address some of the major social challenges.

In the shorter term, as Dr. Sahlsten explains: “The next research should continue in this direction and study the questions on quantum chaos that we have opened thanks to this major breakthrough achieved in collaboration with our colleague Etienne Le Masson. Our work has opened a new line of research that could lead to major breakthroughs and allow us to understand the theory of large networks “.