Target audience: M.Sc. Computational Sciences/Mathematik/Physik

Requirements: some advanced course on either statistical physics or stochastic processes

The liquid state comprises a large class of materials ranging from simple fluids (argon, methane) and molecular fluids (water) to soft matter systems such as polymer solutions (ketchup), colloidal suspensions (wall paint), and heterogeneous media (cell cytoplasm). The basic transport mode in liquids is that of diffusion due to thermal fluctuations, but already the simplest liquids exhibit a non-trivial dynamic response well beyond standard Brownian motion. From the early days of the field, computer simulations have played a central role in identifying complex dynamics and testing the approximations of their theoretical descriptions. On the other hand, theory imposes constraints on the analysis of experimental or simulation data.

The course is at the interface of probability theory and statistical mechanics. Noting that fluids constitute high-dimensional stochastic processes, I will give an introduction to the principles of liquid state theory, and we will derive the mathematical structure of the relevant correlation functions. The second part makes contact to recent research and gives an overview on selected topics. The exercises are split into a theoretical part, discussed in biweekly lessons, and a practical part in form of a small simulation project conducted during a block session (2 days) after the lecture phase.

Keywords:

• Brownian motion, diffusion, and stochastic processes in fluids
• harmonic analysis of correlation functions
• Zwanzig-Mori projection operator formalism
• mode-coupling approximations, long-time tails
• critical dynamics and transport anomalies

• Hansen and McDonald: Theory of simple liquids (Academic Press, 2006).
• Höfling and Franosch, Anomalous transport in the crowded world of biological cells, Rep. Prog. Phys. 76, 046602 (2013).

Further literature will be given during the course.