Charged particles solve mathematical problems

Joakim Cronvall will receive his doctoral degree in mathematics from Lund University in 2025. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position with Professor Aron Wennman, KU Leuven, Belgium.

In mathematical physics, physical phenomena are studied using mathematical models. The aim is to explain and predict physical behaviour using more fundamental principles. One often-studied model comprises a collection of randomly scattered charged particles that repel each other pairwise, while confined by an external electric field that pushes them together. In physics, this type of cloud of charged particles is called a Coulomb gas. In mathematics, these models are used in various ways, for example as eigenvalues of a random matrix.  

Because the number of particles is very large, studying the behaviour of individual particles is impossible. Instead, the model provides a probabilistic description of the particle cloud. The behaviours that are typical of large systems of particles can thus be discovered by studying the model. 

Some properties of a Coulomb gas, such as the shape of the particle cloud and the density of particles, depend on the external electric field, while other properties appear to be largely independent of the electric field – they are universal. For example, if you focus close to a point inside the particle cloud, the system will behave the same no matter the shape of the electric field.

The aim of the research project is to find new properties of the Coulomb gas and either show that they are universal or describe how they can be linked to the electric field. This provides new tools for both studying the model itself and for opening a path for other mathematical problems related to Coulomb gas.