Fakultät für Physik




Physics of Complex Systems – Übersicht


Infos for the Exam

  • We start at 10 am sharp, so be there a little bit earlier.
  • No notes or other material allowed!
  • Check out last years trial exam for a basic idea about the structure (pw: turing)
  • Focus on the concepts and not the mathematical details (its more important to know why you do a certain calculation more than to know how you do it).
  • Expect a large share of short questions (asking for concepts) and skteching (geometric objects in phase space and their meaning are key!).

Organization of the lecture:

  • If not stated otherwise, the lecture takes place on
    • Tuesday, 10:00 c.t. - 12:00 in room A348, Theresienstr. 37
    • Wednesday, 12:00 c.t. - 14:00 in room A 348, Theresienstr. 37

Credits: 4 SWS lecture + 2 SWS exercise class, 9 ECTS

  • Beginning: 17/10/2017
  • End: 07/02/2018
  • Final Exam: 23/02/2018 (10 am - 2 pm)

Physics of Complex Systems

How would one design a physical theory of systems like a living organism, financial markets and other economic systems, social networks on the world-wide web or the microbiome of mammals? Is this even possible given the “complexity” of these systems? Are there, despite the obvious difference between the constituents of these systems (cells, financial brokers, proteins, …), some common principles underlying the system-level behaviour of these systems? Does “complex” necessarily mean “too complicated” such that any endeavour to design a physical theory is bound to fail. The answer is obviously yes if one approaches these problems from the wrong angle, namely by starting say at an atomic description and trying to work one’s way up in length and time scales. In this lecture, I will argue that there is indeed a way to understand these complex systems in terms of well-founded physical theories if one is willing to think outside the box of what some people call “fundamental theories” and instead take the point of view of what is commonly “phenomenological theories”. The latter start from the striking and highly nontrivial observation that complex systems show “emergent behaviour”, i.e. these systems exhibit properties which are system-level features that are qualitatively different from the properties of the constituent building blocks and do not even exist at that level: Rather well-known examples are the thermodynamic concept such as temperature and entropy, and material properties like stiffness, viscosity and superfluidity. Sand ripples are seen below shallow wavy water and are formed whenever water oscillates over a bed of sand. Snowflakes exhibit a plethora of beautiful geometric patterns. On a more complex level, the cells of our body are able to perform a variety of biological functions like sensing, cell division and cell migration. Financial markets show some regularities and sometimes they crash. Without any orchestration or internal leaders, flocks of birds and schools of fish organise into beautiful spatio-temporal patterns. Once an egg is fertilised it evolves into a living being completely on its own. The list of beautiful and mind-boggling phenomena is endless! What is most amazing about these phenomena is that all of them are “self-organised”, i.e. spontaneously arises from local interactions between parts of an initially disordered system without any control by an external agent. Typically, these interactions are highly nonlinear which often leads to unexpected and sometimes counter-intuitive behaviour of complex systems.

It remains a huge scientific challenge to identify and to understand the fundamental and general principles of complex systems, and to put them on a quantitative and mathematical basis. This lecture will give you an introduction in the “way of thinking about complex systems”, new conceptual frameworks that show how these systems may be understood in a unified fashion, and also some of the essential mathematical and computational tools (see below).



The lecture will start with a general overview, the “big picture”. This will emphasise both the bewildering range of phenomena and the unifying point of view taken in analysing complex systems. It is intended to help you to fathom the overarching goals and unifying principles in a highly interdisciplinary field at the crossroads between physics, mathematics, information theory, biology, and many other disciplines.

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 stochastic systems: the role of noise and correlations in complex systems (networks, graph theory, non-equilibrium steady states and phase transitions, large deviations, generic scale invariance, self-assembly)

Recommended Literature

  • Nonlinear Dynamics and Chaos, Strogatz 
[for the first theme of the lecture]
  • Dynamical Systems in Neuroscience, Izhikevich [for the first theme of the lecture]