Diffusion in Fractal Media

Background

Fractals are sets with some form of self-similarity and are often regarded as good models for natural structures. A simple example is provided by certain soils which can show features over a range of length scales. However being able to calculate the dimension of a soil aggregate is purely a recognition of the fact that the soil is highly irregular and does not provide information about how the fractal nature of soil affects the processes that occur within it. In order to make good use of fractal models we need to develop the mathematics to describe the behaviour of partial differential equations, such as the heat and wave equation, in fractal media.

Over the past 10 years there has been developing mathematical interest in these questions, with an important approach being to construct stochastic processes on fractals. The connection between Brownian motion and heat flow allows us to give a probabilistic description of the solution to the heat equation. Early work in this area has concentrated on fractals which have exact self-similarity and showed that there is rather different behaviour of the diffusion than that observed in Euclidean space or on manifolds. Some of the phenomena observed are that the transition density of the diffusion is not Gaussian, that there can exist localised eigenfunctions of the Laplacian on a fractal, and that fractals can support stochastic processes which do not have equivalents in Euclidean space.

Recent Work

There has been a move to discuss general classes of fractals and the behaviour of diffusion in the class of deterministic finitely ramified fractals is quite well understood, though there are still fundamental mathematical problems to solve. One direction which has been partially explored, is how robust is the theory? If the fractal is perturbed, how close is the resulting diffusion to the original one, how close are the solutions to the heat equation?

In the general setting it is known that information about the fractal is only required down to a certain scale to estimate transition densities. This understanding is still quite crude and more sophisticated techniques could yield more information. In the case of sufficiently regular fractals a good description of the transition density can be provided using branching processes which leads to results on the large deviations for the diffusion. This area has yet to be fully exploited and deeper understanding will be useful for applications.

Other work has been to extend the theory to random fractals which provide a much more realistic picture of real fractals. So far only simple recursive fractals have been analysed and open questions are to develop a more general theory. It appears that randomness can have a positive affect in that certain properties of the fractal become simpler but in general, as the fractal can have very different spatial structure in different locations, it leads to more complications.

All of the above developments have been purely theoretical and there is a clear need to apply the theory to real problems. There are both mathematical modelling questions and statistical issues which are yet to be treated. An example is provided by the emission of nitrous oxide gas from soil. The gas is produced by anaerobic bacteria in the soil and hence can only occur in regions where there is no oxygen. This will typically be the case in wet soils or in sufficiently large soil aggregates where there are aerobic bacteria which remove all the oxygen. The analysis of the formation of anaerobic zones has just used classical diffusion theory, ignoring the disordered nature of the medium. How should we model the problem of determining the size of anaerobic zones and how can we estimate the parameters for the model applying the theoretical ideas on diffusions on fractals?

Possible Projects