Analytical and computational challenges arise in a remarkable variety of contexts, and some of the favourite hard analytical questions from classical kinetic theory arise over and over again. In this talk I will present three examples:

1. Coal dust in an incinerator, modelled as an inelastic rarefied particle system in a diffusive background. The steady boundary value problem associated with the corresponding Boltzmann-Fokker-Planck equation is very challenging from an analytical point of view; a potential solution is likely to require combinations of variational techniques for the Fokker-Planck case, pioneered by Baouendi and Grisvard, and structural information about the inelastic Boltzmann collision term as investigated by Gamba, Panferov and Villani.

2. Fokker-Planck equations modeling the evolution of the average orientation of fiber pieces moving in Stokes flow. This is a problem relevant in the industrial problem of plastic moulding, where the inclusion of glass or steel ”fibers” in liquid plastic is used to change the elastic properties of the finished product. The equations driving the orientation of the individual fiber are a system of ODEs known as Jeffery’s equation (derived by Jeffery in 1922), and we are going to discuss the Fokker-Planck equation driving the distribution of many such orientations on the surface of the unit sphere.

3. Coupled systems of Fokker-Planck equations as models of multilane traffic flow. It will be shown how reasonable assumptions on lane change probabilities, reaction times and braking behaviour give rise to bifurcated fundamental diagrams, traffic synchronization, and stop-and-go waves.