Maths Project Repository Database

Dr Rainer Klages

Style and difficulty:

• This is a new, more advanced project that is more closely related to research.

Contents:

• This project requires some familiarity with stochastic processes, though not on a deep mathematical level, and the motivation to generally learn more about stochastic dynamics. You should start by explaining the concepts of a Langevin equation, anomalous diffusion, and so-called fluctuation relations. The first topic is about 100 years old, the second and third ones became very active fields of research over the past two decades. All of this material can be extracted from (advanced) textbooks and reviews, see [1,2] below. Then explain the idea of generalized Langevin dynamics by possibly touching upon fractional derivatives.
• After this introductory part, try to roughly understand what has been done in a recent research paper of mine [3] below (see also [4], to some extent). Recalculate as much as possible what has been stated in Section 4 therein.
• Hopefully there is some time left to now do some research in terms of new calculations. They may be performed by considering a slightly different version of the generalized Langevin equation studied in [3] below, checking for an anomalous fluctuation relation by using the very same kind of methods you have familiarized yourself with in the task before.

This is a nice project in this area of research that possibly could even be continued as a PhD project.

1. M. Toda, R. Kubo, N. Saito, Statistical Physics 1. Springer, Berlin, 1992.
2. R. Klages, A.V. Chechkin, P. Dieterich, Anomalous fluctuation relations, in Nonequilibrium Statistical Physics of Small Systems, Edited by R. Klages, W. Just, C.Jarzynski, Wiley-VCH, Weinheim, February 2013. 259–282; ISBN 978-3-527-41094-1 [preprint as pdf-file http://www.maths.qmul.ac.uk/~klages/papers/klages_chapter_rev.pdf].
3. A.V. Chechkin, F. Lenz, R. Klages, Normal and anomalous fluctuation relations for Gaussian stochastic dynamics, J. Stat. Mech. L11001/1-13 (2012) (Letter) [link to journal http://iopscience.iop.org/1742-5468/2012/11/L11001|article as pdf-file http://www.maths.qmul.ac.uk/~klages/papers/afr2_jstat.pdf].
4. A.V. Chechkin, R. Klages, Fluctuation relations for anomalous dynamics, J. Stat. Mech. L03002/1-11 (2009) (Letter) [link to journal http://www.iop.org/EJ/abstract/1742-5468/2009/03/L03002|article as pdf-file http://www.maths.qmul.ac.uk/~klages/papers/afr_jstat.pdf].

This project can be done purely analytically by using concepts of stochastic theory. Some familiarity with Fourier-Laplace transform techniques would be helpful but is not absolutely necessary. You should have interest and some working knowledge in stochastic processes, which relates to material covered by some of our UG and PG modules.