Level: MSci, MSc
Title: Chaotic diffusion in deterministic Langevin dynamics
Supervisor:

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

Style and difficulty:

  • If you are not sure what you are capable of doing research-wise, you may start this project like Deterministic chaos in the Bernoulli shift. Then either continue with the project above, or dive into more demanding aspects according to this project as explained below. Accordingly, the core of this project is based on simple and advanced textbooks, then moves on to research papers. Degree of difficulty can be tuned from simple to very demanding. Mostly analytical but may involve some easy computer work later on. For an excellent student, this should lead to some new research.

Contents:

  • If you want to be on the safe side, you may start like the project above by discussing basic chaos properties of the Bernoulli shift (Devaney's definition, Ljapunov exponents, ergodicity). Then explore the concept of Brownian motion by explaining what a Langevin equation is. Put this dynamics into context of stochastic theory by explaining what a Wiener process and an Ornstein-Uhlenbeck process is. This defines the textbook part of the project. On this basis, study research articles by my colleague Prof. Christian Beck and summarize the idea of a deterministic Langevin equation. This dynamical system can be used to generate chaotic diffusion: describe what this concept means. If you are very clever, you can now start to do research on this type of system: Calculate the model's diffusion coefficient analytically by a so-called Takagi function technique. If you can, compare your analytical findings to results from your own computer simulations.
Further Reading:
  • J.R. Dorfman, An Introduction to Chaos in Nonequilibrium Statistical Mechanics, Chapter 14. Cambridge Univ. Press, 1999.
  • R. Klages, Deterministic diffusion in one-dimensional chaotic dynamical systems, Chapter 5. Wissenschaft und Technik Verlag, Berlin, 1996. (Ph.D. thesis available on my homepage).
  • C. Beck, Dynamical systems of Langevin type, Physica A 233 419 (1996).
  • See also a previous precursor MSc thesis; the research part of this thesis should be continued in more depth.
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 (70%) and basic concepts of stochastic theory (10%) and statistical mechanics (20%).
Current Availability: Yes