Department Colloquium Summer 2024

Many processes in nature can be modeled on different scales; for
example with 'high resolution' using particles or 'zoomed out' using
partial differential equations (PDEs). One interest covered in this talk is
the derivation of PDEs describing the collective behaviour of (infinitely)
many particles: Can we derive the PDE? Can we machine learn the PDE from
particle data? It turns out that it is often important to understand the
structure of the equations; on the level of PDEs, a thermodynamic structure
based on energy and entropy is very helpful and will be described. The arrow
of time will play a central role: Small-scale models (of particle type) are
often can be reversible in time, while the large-scale description (of PDE type)
is irreversible. This seems at first puzzling, as the two models describe the
same system. We will discuss some mathematical approaches to understand this.