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Stochastic dynamics
Stochastic dynamics is the study of dynamical processes that occur on sufficiently large time scales such that the fast microscopic degrees of freedom can be effectively described by stochastic noise. The paradigmatic case is the movement of a Brownian particle (e.g. a plastic bead of micrometer dimensions) in a fluid (e.g. water). Then the particle trajectory performs a random walk effected by the random forces exerted by the molecules of the fluid. The mathematical tool required to describe this situation is a stochastic differential equations, also known as the Langevin equation. Alternatively one can use the Fokker-Planck equation, which is a partial differential equation for the probability density p(x,t) of the particle to be at position x at time t. For stochastic processes with jumps, the appropriate equation is the master equation.
Stochastic dynamics has many applications, e.g. in physics, chemistry, biology and economics. In this course, we will provide an introduction into the fundamentals of this field, in particular to the three fundamental types of equations. Applications will be chosen from the fields of biophysics, finance and materials science. If time permits, we will also discuss the path integral approach to stochastic processes as well as recent developments in non-equilibrium physics and stochastic thermodynamics. The basic material for this course is covered well by the book by Honerkamp (see below).
The course is designed for physics students in advanced bachelor and beginning master semesters (students from other disciplines are also welcome). It will be given in English. A basic understanding of physics and differential equations is sufficient to attend. A background in statistical physics is helpful, but not required. The course takes place every Monday from 2.15 - 3.45 pm in lecture hall HS2 in INF 308. Every second week on Wednesday afternoons the solutions to the exercises will be discussed in a tutorial. If you attend the course and solve more than 60 percent of the exercises, you earn 4 credit points. We recommend to complement this course by the one on non-linear dynamics (Wednesday 9.15 - 10.45 am at kHS Philosophenweg 12, tutorial in the complementary weeks).
Material for the course
- Introduction Oct 17 2016
- Stochastic Processes in Finance Jan 9 2017
- Current version of the script (Jan 23)
Exercises
Solutions are to be handed in at the lecture one week after assignment. You can work in groups of two if you want to.- Assignment No. 1 (Oct 24)
- Assignment No. 2 (Nov 7)
- Assignment No. 3 (Nov 21)
- Assignment No. 4 (Dec 5)
- Assignment No. 5 (Dec 19)
Literature
- J. Honerkamp, Stochastische Dynamische Systeme, VCH 1990
- W. Paul and J. Baschnagel, Stochastic Processes: From Physics to Finance, Springer 1999
- R. Zwanzig, Nonequilibrium Statistical Mechanics, Oxford University Press 2001
- U. Seifert, Stochastic thermodynamics, fluctuation theorems, and molecular machines, Rep. Prog. Phys. 75, 126001, 2012
- C.W. Gardiner, Handbook of stochastic methods, Springer 2004
- N.G. van Kampen, Stochastic Processes in Physics and Chemistry, Elsevier 1992
- W. Horsthemke und R. Lefever, Noise-induced transitions. Theory and Applications in Physics, Chemistry, and Biology, Springer 1984
- H. Risken, The Fokker-Planck Equation, Springer 1996
- H. C. Berg, Random Walks in Biology, Princeton University Press 1993
- P. Nelson, Biological Physics, Freeman 2003
- R. Phillips and coworkers, Physical Biology of the Cell, 2nd edition Garland Sci. 2012
