Weak chaos and anomalous diffusion

The School of Mathematical Sciences of Queen Mary University of London invite applications for a PhD project commencing either in September 2016 (funded students) or at any point in the academic year (self-funded students).

Understanding anomalous dynamics developed into an extremely active area of research over the past few years. One access road to this highly interdisciplinary field are generalisations of simple random walks generating non-Markovian correlated dynamics and non- Gaussian processes. These processes often yield sub- or superdiffusion, where particles spread out slower or faster, respectively, in time than diffusion obtained from uncorrelated Markovian random walks. The theory has wide applications in many different fields of research such as, e.g., diffusion in biological systems, molecular diffusion in nanopores, and in socio-economics.

A challenging problem is to cross-link the stochastic theory of anomalous transport with the theory of chaotic dynamical systems. A simple, paradigmatic example is the intermittent Pomeau- Manneville map: It has a regime where it is not chaotic in the sense of having a positive Lyapunov exponent but nevertheless generates random-looking trajectories. The reason is that trajectories still diverge from each other but not exponentially. This model exhibits a non-trivial transition from normal diffusion to subdiffusion under parameter variation, which was understood by analytical approximations compared to results from computer simulations.

Conceptually the start of this project is analogous to a previous very successful PhD project analysing subdiffusion in the Pomeau-Manneville map. Now the more difficult problem of superdiffusion should be explored for which a specific transition from normal to superdiffusion was conjectured. This hypothesis should be checked by performing computer simulations before calculating analytical approximations by using stochastic theory. A more rigorous dynamical systems method to study the diffusive properties of this map may be developed by using a mathematical approach based on Lagrange-Chebychev polynomials.

There is the possibility to collaborate with O. Bandtlow on more rigorous mathematical aspects of dynamical systems theory, and with A.V. Chechkin, a world leading expert on advanced stochastic theory. This highly interdisciplinary project is right at the interface between stochastic theory, dynamical systems theory, statistical physics, and computer simulations.

This project will be supervised by Dr Rainer Klages and Dr Oscar Bandtlow. For full details, see the project abstract: http://www.maths.qmul.ac.uk/sites/default/files/phd%20projects%202015/DSSP/klages2.pdf

The application procedure is described on the School website. For further enquiries please contact Dr Rainer Klages ([Email Address Removed]) or Dr Oscar Bandtlow ([Email Address Removed]).

This project is eligible for several sources of full funding for the 2016/17 academic year, including support for 3.5 years’ study, additional funds for conference and research visits and funding for relevant IT needs. Applicants interested in the full funding will have to participate in a highly competitive selection process. The best candidates will be eligible to receive a prestigious Ian Macdonald Postgraduate Award of £1000, for which you will be considered alongside your application. The application deadline for full funding is January 31st 2016.

There is also 50% funding scheme available for students who are able to find the matching 50 % of the cost of their studies. Competition for these half-funded slots will be less intensive, and eligible students should mention their willingness to be considered for them in their application. The application deadline for 50 % funding is January 31st 2016.

This project can be also undertaken as a self-funded project, either through your own funds or through a body external to Queen Mary University of London. Self-funded applications are accepted year-round.