###### Inhaltsbereich

# T6: Self-organisation and pattern formation – Materials

## Self-organisation and pattern formation

How would one design a physical theory of the processes in a living cell, the collective dynamics of a group of interacting agents (cells, organisms), or other complex systems such as financial markets? Is this even possible given the "complexity" of these systems? Despite the obvious differences between the constituents of these systems (proteins, cells, financial brokers, ...), are there some common principles underlying the emergent behaviour of these systems at the system level? Does "complex" necessarily mean "too complicated", so that any attempt to design a physical theory is doomed to failure?

There is a way to understand these systems if you approach them with the right perspective and ask the right questions. The big challenge is to find ways to describe the tendency of these systems to "self-organize" or "self-assemble" into complex structures or to explain the emergence of spatio-temporal patterns. In this set of lectures, I will give an introduction to this area of complex systems and discuss new conceptual frameworks that show how these systems can be understood in a unified way, as well as some of the essential mathematical and computational tools.

The main themes of the lecture will be:

- Phase space analysis: a unifying geometric view of system dynamics (flows, fixed points, attractors, stability, excitability, synchronization, catastrophes, bifurcations, chaos)
- Pattern formation and self-organisation: a physical theory of spatially extended systems (phase separation, interface dynamics, reaction-diffusion systems, fronts and waves, neuronal networks, collective phenomena, active systems, hydrodynamic instabilities, turbulence)
- Non-equilibrium dynamics and self-assembly: broken detailed balance and its consequences (non-equilibrium steady states and phase transitions, generic scale invariance, self-assembly)

There are lecture notes for the first part of the lecture already available to download at ArXiv:2012.01797. This is based on a set of lectures that I gave at a Les Houches Summer School in 2018. In addition, the ideas of nonlinear dynamics can be applied to evolutionary game theory; see Physica A 389 (2010) 4265–4298.

## Lecture notes

- Preliminary lecture notes can be downloaded here(last updated on 30.06.2021).

## Lecture Slides

- Self-organisation, Introduction
- Well-mixed Systems
- Well-mixed Systems(extended version)
- Dynamical systems
- Hopf Bifurcation, Oscillators
- Diffusion Equation, notes
- Gradient Dynamics I
- Gradient Dynamics II
- Trigger waves I
- Trigger waves II
- Fisher waves
- Liquid-liquid phase separation
- Cahn-Hilliard equation
- Turing Patterns
- Turing instability and Swift-Hohenberg Equation
- Swift-Hohenberg Equation I
- Swift-Hohenberg Equation II
- Complex Ginzburg-Landau Eq.
- Mass Conserving Reaction-Diffusion Eq.

## Additional Reading Material

(Accessible via the LMU Library website)

- Steven Strogatz: Nonlinear Dynamics and Chaos, Westview Press, 2015,
- Eugene M. Izhikevich: Dynamical Systems in Neuroscience, MIT Press, 2010,
- James E. Ferrell Jr.: Bistability, Bifurcations, and Waddington's Epigenetic Landscape - [DOI],
- The geometry of soft materials: a primer - [DOI] - (Short introduction to differential

geometry). - Erwin Frey: Lecture notes: Advanced Statistical Physics, Stochastic Processes
- Lendert Gelens, Graham A. Anderson, and James E. Ferrell, Jr: Spatial trigger waves: positive feedback gets you a long way - [DOI]
- Jose Negrete Jr. and Andrew C. Oates: Towards a physical understanding of developmental patterning - [DOI]
- Oskar Hallatschek: The noisy edge of travelling waves (effects of demographic noise on front propagation in the Fisher equation) - [DOI]
- Jonathan H.P. Dawes: After 1952: The later development of Alan Turing's ideas on the mathematics of pattern formation - [DOI]
- Vladimir García-Morales and Katharina Krischer: The complex Ginzburg–Landau equation: an introduction - [DOI]
- Igor S. Aranson and Lorenz Kramer: The world of the complex Ginzburg-Landau equation (advanced review, way beyond the scope of the lecture) -[DOI]