In the modelling of diffusion of particles it is useful to get a model for the collective motion of the particles from the individual particle dynamics. One assumption that is often made is that the particles are of negligible size. If this is the case then particle collisions can also be ignored and one can also assume that particles move independently. This makes it easy to obtain the PDE for the marginal position probability density of a single particle. The most common example of this is the heat equation which arises from Brownian motion of particles.

In this project we aim to find PDEs for such dynamics in the case that individual particles exclude space available to other particles. We use a new asymptotic analysis technique to obtain PDEs for the collective dynamics. The PDEs are usually nonlinear. The validity of the PDEs is verified using simulations of the particle dynamics.

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**Excluded volume diffusion**## Department of Mathematics

# Excluded volume diffusion

The modelling of diffusion of particles where the particle size is taken into account leads to nonlinear PDEs for the collective dynamics.

### The researchers at The University of Auckland are:

- Community for Understanding and Learning in the Mathematical Sciences (CULMS)
- Centre for Mathematical Social Science (CMSS)
- Department of Computer Science
- Department of Engineering Science
- Department of Physics
- Department of Statistics
- Auckland Bioengineering Institute
- New Zealand Journal of Mathematics

**Programmes, Centres and Partners**

- Community for Understanding and Learning in the Mathematical Sciences (CULMS)
- Centre for Mathematical Social Science (CMSS)
- Department of Computer Science
- Department of Engineering Science
- Department of Physics
- Department of Statistics
- Auckland Bioengineering Institute
- New Zealand Journal of Mathematics

**Programmes, Centres and Partners**