Stochastic dynamics

Description

Stochastic dynamics is the study of dynamical processes of macroscopic objects 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. A very recent development are diffusion models for generative machine learning, like DALL-E or Midjourney. 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 soft matter physics, biophysics, finance and machine learning.

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. A script is available from earlier versions of this course.

Literature

• J. Honerkamp, Stochastische Dynamische Systeme, VCH 1990
• W. Paul and J. Baschnagel, Stochastic Processes: From Physics to Finance, Springer 1999
• C.W. Gardiner, Handbook of stochastic methods, Springer 2004
• N.G. van Kampen, Stochastic Processes in Physics and Chemistry, Elsevier 1992
• H. Risken, The Fokker-Planck Equation, Springer 1996