Abstract:
Many physical and biological systems consist of individuals with collective behaviour. In
reality, these individuals have a nite size, or at least prevent others from interacting
or becoming involved; hence, they exclude a volume in space. Volume exclusion can
be considered the simplest possible interaction within a population. It plays an important
role in determining transport properties from small-scale systems such as interior
cell motion or ion channels to large-scale systems such as pedestrian motion or animal
swarms. Individual-based models describe how interacting individuals give rise to collective
behaviour; however, these models become computationally intractable for large
systems. Our research aims to replace these expensive simulation models with continuum
population-level models based on partial di erential equations (PDEs). The di usion
of nite size hard-core interacting particles in one or two dimensions are examined systematically
using asymptotic methods. The result is a coupled system of PDEs for the
distribution functions of position, velocity and time. Due to size exclusion, the PDEs are
nonlinear in the transport term.