Living organisms, ecosystems, and the Earth itself are incredibly complex systems that defy thermal equilibrium, as per the laws of physics. To describe such non-equilibrium systems, scientists have relied on dynamic density functional theory (DDFT) thus far.

However, a team of physicists from the University of Bayreuth has recently identified weaknesses in DDFT. In an article published in the Journal of Physics: Condensed Matter, they propose an alternative approach called power functional theory, which, when combined with artificial intelligence methods, offers more reliable descriptions and predictions of the dynamics of non-equilibrium systems over time.

Many-particle systems encompass atoms, electrons, molecules, and other minuscule particles that are invisible to the naked eye. A system is considered to be in thermal equilibrium when there is a balance in temperature, and no heat flow occurs. In such a state, the system’s properties remain unchanged unless influenced by external factors. Density functional theory has proven highly effective in studying equilibrium systems, providing accurate descriptions and predictions. However, to extend the theory to non-equilibrium systems, dynamic density functional theory (DDFT) was developed. DDFT aims to comprehend systems that can alter their states without the need for external influences, possessing their own inherent momentum. The insights and techniques derived from DDFT are particularly valuable in the study of models concerning living organisms or microscopic flows.

The researchers’ exploration of power functional theory introduces a promising avenue for advancing our understanding of non-equilibrium systems. By leveraging artificial intelligence methods, this approach holds the potential to enhance the accuracy and reliability of describing and predicting the dynamics of these intricate systems.

## The error potential of dynamic density functional theory

However, dynamic density functional theory (DDFT) faces limitations due to its auxiliary construction for describing non-equilibrium systems. It converts the continuous dynamics of these systems into a sequence of equilibrium states over time. This approach introduces a potential for errors, as highlighted by the recent study led by Prof. Dr. Matthias Schmidt and his team from the University of Bayreuth.

The researchers focused their investigation on a relatively simple example—the unidirectional flow of a gas, known as a “Lennard-Jones fluid” in physics. When interpreting this non-equilibrium system as a series of successive equilibrium states, DDFT overlooks a crucial aspect: the flow field associated with the system’s time-dependent dynamics. As a consequence, DDFT may yield inaccurate descriptions and predictions.

“We acknowledge that dynamic density functional theory can offer valuable insights and suggestions when applied to specific conditions in non-equilibrium systems. However, the issue we emphasize in our study, using fluid flow as an example, is the inability to ascertain with sufficient certainty whether those conditions are met in a given case. DDFT lacks the means to control and ensure the presence of the restricted framework conditions that would enable reliable calculations. Hence, it becomes even more crucial to develop alternative theoretical concepts for comprehending non-equilibrium systems,” explains Prof. Dr. Daniel de las Heras, the study’s lead author.

## Power functional theory proves to perform substantially better

Over the past decade, Prof. Dr. Matthias Schmidt’s research team has made substantial contributions to the development of power functional theory (PFT), a relatively young physical theory that has demonstrated remarkable success in studying many-particle systems. The goal of the physicists from Bayreuth is to achieve a level of precision and elegance in describing the dynamics of non-equilibrium systems, comparable to what classical density functional theory offers for equilibrium systems.

In their recent study, the team employs the example of fluid flow to showcase the significant superiority of power functional theory over dynamic density functional theory (DDFT) in understanding non-equilibrium systems. PFT allows for the direct description of system dynamics without the need for a detour through a chain of successive equilibrium states over time. The key to this achievement lies in the integration of artificial intelligence. Through machine learning, the time-dependent behavior of fluid flow can be captured, encompassing all relevant factors influencing the system’s inherent dynamics, including the flow field. Remarkably, the team has even achieved precise control over the flow of the Lennard-Jones fluid using PFT.

“Our investigation provides further evidence that power functional theory is an exceptionally promising concept for describing and explaining the dynamics of many-particle systems. In Bayreuth, our future focus will be on further developing this theory and applying it to non-equilibrium systems that possess a much higher degree of complexity compared to the fluid flow studied. This way, PFT has the potential to surpass dynamic density functional theory, effectively addressing its identified weaknesses. The original density functional theory, specifically tailored for equilibrium systems and proven to be valuable, is retained as an elegant special case within the broader framework of PFT,” states Prof. Dr. Matthias Schmidt, chair of theoretical physics II at the University of Bayreuth.

Source: Bayreuth University