Deeper Understanding of the Physics of Complex Liquids

One method for quantitatively describing complex systems, such as liquids, is density functional theory. It allows for efficient capture of interactions between many microscopic particles. However, density functional theory has its limitations, and not all quantities necessary and useful for describing complex systems can be directly calculated with it. Dr. Florian Sammüller and doctoral researcher Johanna Müller from the Chair of Theoretical Physics II, led by Prof. Dr. Matthias Schmidt at the University of Bayreuth, have made significant contributions to practical applications and deeper understanding.

By generalising the concept of density functionals with a combination of theory, numerical methods, and machine learning, it is now possible to consider arbitrary quantities, such as the number of interconnected particles in gels. So-called hyper-density functionals enable the systematic study of physical mechanisms leading to structure formation. Thanks to new machine learning methods, hyper-density functionals are realised as neural networks on computers. The information required to train these networks is highly accurate, as it is derived from simulation work where each individual particle in the system can be represented

The statistical description of all particles in a complex system is accompanied by a fundamental gauge invariance: this describes the property that physical laws remain unchanged when certain mathematical transformations are applied to the quantities describing the system. Unexpectedly, the researchers from Bayreuth discovered a completely analogous mathematical structure for particles, including a previously unknown Lie algebra identified by Johanna Müller, which describes an antisymmetric linkage. In this case, the average of any possible measurement remains unchanged, regardless of certain shifts of all particles in space, where the magnitude and direction may vary at each point of measurement.

Dr. Florian Sammüller, who established neural density functional theory in the research, explains: “It is remarkable how well theoretical insights can be combined with numerical methods. On the one hand, there is density and hyper-density functional theory, which deals with the existence and mathematical properties of abstract functional mappings. On the other hand, many-particle simulations offer a more intuitive understanding of a liquid under investigation, as the movement of individual particles can be directly observed – similar to experiments using a microscope. Machine learning methods, especially neural networks, are perfectly suited as a link between these two approaches, as they can extract and represent the underlying functional relationships from numerical simulation data.”