Dynamical systems, statistical physics, wireless networks.

A more complete description of activities may be found at my home page, https://people.maths.bris.ac.uk/~macpd/

PhD projects

I am happy to supervise well motivated students who want to work on the above general areas. Two areas of particular interest are:

Deterministic diffusion A particle moves according to a deterministic law in an unbounded domain, starting from a random initial condition.  How does the displacement depend on time?  A good class of such systems to visualise are mathematical billiards: The particle (“billiard ball”) moves in straight lines except for reflections from obstacles. Different shapes lead to dynamical properties from regular to chaotic and including many finer distinctions. The obstacle locations can be deterministic or random.  Depending on these factors the diffusion can be anything from normal (Brownian motion) to highly anomalous.  See my paper "Diffusion in the Lorentz gas," available from the above website.

Spatial networks What happens if you place points randomly in space, link them with a probability that depends on pairwise distance, and study the resulting network? How does this depend on the shape of the domain, the density of points, generalisations such as angular dependence, obstacles, time-dependent locations? These questions are interesting from both a theoretical and practical point of view.  One of the main applications is the design of wireless ad-hoc networks, of interest to Toshiba research labs in Bristol.  But there are many other kinds of networks with spatial structure that could be considered.  See my paper "Random geometric graphs with general connection functions", available from the above website.

For more ideas, please look at other recent publications available from the above website. They are research articles, hence rather technical; please read only to get the general idea, and then contact me to discuss your interests.