By Emma Johnston, Queensland University of Technology
The motion of a particle suspended in a fluid, a malaria infected mosquito zipping around a village or the price of fluctuating stock are all processes that can be modelled using random walks. In the words of British statistician, Karl Pearson, a random walk can be described as:
“A man starts from the point O and walks l yards in a straight line; then he turns through any angle whatever and walks another l yards in a second straight line. He repeats this process n times.”
In standard models of diffusion, it is assumed that the path undertaken by these agents will have a mean distance between changes of direction or collisions and a mean time taken to perform a step. However, a number of processes can be described by anomalous diffusion; situations in which these assumptions do not hold. The motion of objects might be better described by steps taking very different times and that are not independent.
These processes may be faster or slower than standard diffusion. For example, as a result of trapping, the travel times of contaminants in groundwater are much longer than expected from standard diffusion. Stagnant regions, we can imagine eddies in rivers, correspond to long times between steps. By using an anomalous diffusion model, researchers found they could more accurately predict how long pollutants from environmental accidents will remain before they are flushed out to sea. An example of super diffusion is the flight paths of albatrosses. They for long distances in a single direction before finding land, their flight paths can be modelled by random walks with steps sizes that can vary significantly.
What both these examples have in common is that the system is dominated by the largest steps or the longest periods of time with no motion. The system’s ‘memory’ about rare events is not erased. Since there is dependency at all in points in time, this can make anomalous diffusion equations difficult to numerically solve.
For my summer research project, we investigated the potential in including anomalous diffusion in the modelling of the growth of brain tumours called gliomas. This type of diffusion has already been observed in the diffusion of water molecules in the brain, and we hope to investigate this new framework further in the future.
Emma Johnston was one of the recipients of a 2015/16 AMSI Vacation Research Scholarship.