Living organisms, ecosystems, and planet Earth are physically examples of extraordinarily large and complex systems that are not in thermal equilibrium. To physically describe non-equilibrium systems, dynamical density functional theory has been used up to now.

However, this theory has weaknesses, as physicists from the University of Bayreuth have now shown in a paper published in the *Physics Journal: Condensed Matter*. Power functional theory proves to work substantially better in combination with artificial intelligence methods, allowing for more reliable descriptions and predictions of the dynamics of non-equilibrium systems over time.

Many-particle systems are all kinds of systems composed of atoms, electrons, molecules, and other particles invisible to the eye. They are in thermal equilibrium when the temperature is balanced and no heat flow occurs. A system in thermal equilibrium changes state only when the external conditions change. Density functional theory is tailor-made for the study of such systems.

For more than half a century, it has proven its unlimited value in chemistry and materials science. Based on a powerful classical variant of this theory, equilibrium system states can be described and predicted with high accuracy. Dynamic density functional theory (DDFT) extends the scope of this theory to non-equilibrium systems. This implies the physical understanding of systems whose states are not fixed by their external boundary conditions.

These systems have a momentum of their own: they have the ability to change their states without external influences affecting them. The results and application methods of DDFT are therefore of great interest, for example, for studying models for living organisms or microscopic flows.

## The error potential of dynamic density functional theory

However, DDFT uses an auxiliary construction to make non-equilibrium systems accessible to physical description. It translates the continuous dynamics of these systems into a time sequence of equilibrium states. This results in a potential for errors that should not be underestimated, as the Bayreuth team led by Prof. Dr. Matthias Schmidt shows in the new study.

The investigations focused on a relatively simple example, the unidirectional flow of a gas known in physics as the “Lennard-Jones fluid”. If this non-equilibrium system is interpreted as a chain of successive equilibrium states, one aspect involved in the time-dependent dynamics of the system, namely the flow field, is neglected. As a result, DDFT may provide inaccurate descriptions and forecasts.

“We do not deny that dynamic density functional theory can provide valuable insights and suggestions when applied to non-equilibrium systems under certain conditions. The problem, however, and we want to draw attention to this in our study using fluid flow as example, is that it is not possible to determine with sufficient certainty whether these conditions are met in a particular case. The DDFT does not provide any control over whether narrow framework conditions are provided under which it allows for reliable calculations. This makes it all the more useful to develop alternative theoretical concepts for understanding non-equilibrium systems,” says Prof. Dr. Daniel de las Heras, first author of the study.

## The power functional theory proves to work substantially better

For ten years, the research group of Prof. Dr. Matthias Schmidt has made a significant contribution to the development of a still young physical theory, which has so far proved to be very successful in the physical study of many-particle systems: the power functional theory (PFT). The Bayreuth physicists pursue the goal of being able to describe the dynamics of non-equilibrium systems with the same precision and elegance with which the classical density functional theory allows the analysis of equilibrium systems.

In their new study, they now use the example of a fluid flow to show that power functional theory is significantly superior to DDFT when it comes to understanding non-equilibrium systems. PFT allows to describe the dynamics of these systems without having to deviate through a chain of successive equilibrium states in time. The deciding factor here is the use of artificial intelligence. Machine learning opens up the time-dependent behavior of fluid flow by including all factors relevant to the intrinsic dynamics of the system, including the flow field. In this way, the team was even able to control the flow of the Lennard-Jones fluid with great precision.

“Our investigation provides further evidence that power function theory is a very promising concept that can be used to describe and explain the dynamics of many-particle systems. At Bayreuth, we intend to elaborate this theory further in the coming years, applying it to non-equilibrium systems that have a much higher degree of complexity than the fluid flow we have studied. In this way, PFT will be able to replace dynamical density functional theory, whose systemic weaknesses it avoids according to our findings so far The original density functional theory, which is adapted to equilibrium systems and has proven its worth, is thought to be an elegant special case of PFT,” says Prof. Dr. Matthias Schmidt, who is chair of Theoretical Physics II at the University of Bayreuth.

**More information:**

Daniel de las Heras et al, Perspective: how to overcome dynamic density functional theory, *Physics Journal: Condensed Matter* (2023). DOI: 10.1088/1361-648X/accb33

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