# Research Fields

Complex systems are out-of-equilibrium systems like lasers, hydrodynamic flows and chemical reactions that are often composed of many microscopic parts that interact nonlinearly. Hence, they can spontaneously form spatial, spatio-temporal, and functional structures that can not be understood as a superposition of the behavior of the parts. The resulting formation of spatio-temporal patterns is often seen as self-organization process

The scientific aim of the working group on Self-Organization and Complexity is to explore universal properties of non-equilibrium systems with theoretical and numerical

methods. Of considerable interest are methods of nonlinear dynamics like bifurcation theory, chaos theory combined with methods of statistical physics and the theory of stochastic processes.

### Current areas of research:

- Analysis of complex systems
- Turbulent fields and thermal convection
- Spatio-temporal pattern formation in self-assembling systems and its control
- Interface dynamics for complex liquids & soft and active matter
- Dynamics of phase transitions and growth processes

### Thermal Convection and Turbulence:

Thermal convection is the flow of a fluid that is driven by temperature differences. Almost all convection-powered flows that occur in nature (e.g. atmosphere, oceans, plate tectonics, ...) are turbulent. These flows are especially hard to handle due to their erratic, chaotic behavior, e.g., a space- and time-resolved prediction of the temperature fields is nearly impossible. Instead, we aim at a combined statistical and numerical description that characterizes the system through statistical quantities, for which analytical expressions can be derived from first principles. Next, we numerically solve the basic equations with 10^{7}-10^{8} degrees of freedom on supercomputers consisting of hundreds of CPUs and analyze the obtained data with our statistical methods. This combined effort leads to an understanding of the statistical properties of turbulent convection.

### Complex Liquids & Soft and Active Matter

In many cases the dynamics of complex liquids and soft matter is interface-dominated, i.e., controlled by capillarity and/or wettability. Examples include (driven) droplets on homo- and heterogeneous substrates, (active) liquid crystals and colloidal suspensions, self-propelled droplets, and multicomponent multilayer. An important objective is to understand the structure-forming interaction of the various interdependent advective and diffusive transport processes and phase transitions. The inclusion of chemical reactions and chemo-mechanical coupling naturally leads to questions related to cell locomotion, tissue growth and morphogenesis, and the motion of micro swimmers.

### Pattern Formation in Self-Assembling Systems

Experimental techniques like Langmuir--Blodgett transfer where a surfactant layer is transferred from a bath onto a plate, dip-coating or vapor deposition are widely used to create coatings of a precise thickness and/or structure. Using the self-assembly of microscopic building blocks into macroscopic structures facilitates the production of coatings with highly regular patterns over large areas. One aim is to understand how the basic properties of each system lead to the formation of particular functional patterns, as this allows one to develop ways to control the patterning process, e.g., by means of prestructured substrates or external fields.

### Control of Spatio-Temporal Patterns

Control and engineering of complex systems is a central issue of applied nonlinear science. Feedback loops represent an important concept to control spatial-temporal patterns by using the internal dynamics of the system. One example is provided by time-delayed feedback control that is modeled and analyzed through delay differential equations. The goal is to provide an analytical treatment of such systems using tools of bifurcation theory and nonlinear dynamics and to apply them to a range of systems from nonlinear optics to neurophysics.

### Analysis of Complex System:

The temporal evolution of complex systems on the macroscopic level can be described employing order parameters, while the much faster microscopic degrees of freedom can be treated as fluctuations. As a consequence, the order parameters evolve in time as stochastic processes. In many cases the equations describing the stochastic process can not be rigorously derived from the basic equations of the complex system and one tries to estimate the stochastic equations from measured data. Our goal is to improve these estimation methods and to apply them to a range of systems from physics to biology and medicine.