Level: MSci, MSc
Title: Anomalous diffusion in weakly chaotic dynamical systems
Supervisor:

Dr Rainer Klages
Research Area: Dynamical Systems and Statistical Physics
Description:

Style and difficulty:

  • This is a challenging project that goes mostly beyond textbooks. It is more based on scientific reviews and research papers. It can be performed purely analytically or in combination with computer work. Although the main outcomes of this project are known, it poses a very challenging task that connects directly with active research. Hence, this is an ideal project for someone thinking of doing a PhD later on. If successful, it could immediately be expanded into a PhD project.

Contents:

  • Anomalous diffusion defines a very active field of current research. What does it mean to say that a system exhibits anomalous diffusion? Outline the basic idea. Show that anomalous diffusion can be generated by weakly chaotic dynamical systems: Introduce the so-called Pomeau-Manneville map and qualitatively discuss its intermittent dynamics. Explain the concept of weak chaos and argue that this map is weakly chaotic. By using this map, construct a deterministic model that exhibits subdiffusion. Calculate the anomalous diffusion coefficient of this model by continuous time random walk theory by explaining what this theory is about. This assumes familiarity with Fourier-Laplace transforms and the like. Explain the idea of fractional derivatives and derive a fractional diffusion equation for this model.
Further Reading:
  • R. Metzler, J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339 1 (2000).
  • R. Klages, Weak chaos, infinite ergodic theory, and anomalous dynamics, in From Hamiltonian Chaos to Complex Systems, Edited by X. Leoncini and M. Leonetti, Springer, Berlin, July 2013, 3–42 [article as pdf file http://www.maths.qmul.ac.uk/~klages/papers/klages_wchaos.pdf].
  • R. Klages, Microscopic Chaos, Fractals and Transport in Nonequilibrium Statistical Mechanics, monograph, Advanced Series in Nonlinear Dynamics Vol. 24, World Scientific, Singapore, 2007.
  • N. Korabel, R. Klages, A.V. Chechkin, I.M. Sokolov, V.Yu. Gonchar, Fractal properties of anomalous diffusion in intermittent maps, Phys. Rev. E 75 036213 (2007) [article as pdf file http://www.maths.qmul.ac.uk/~klages/papers/PhysRevE_75_036213.pdf].
Key Modules:

MTH6107 Chaos and Fractals

MTH744U/MTH744P Dynamical Systems

MTH743U/MTH743P Complex Systems
Other Information:

This project consists of a mix between dynamical systems theory (20%) and basic concepts of stochastic theory (70%) and statistical mechanics (10%). Hence you should have interest and some working knowledge of this theory as provided by at least one of the modules above.
Current Availability: Yes