Complexity Group @
Niels Bohr Institute


Topics in Complex Systems - 2016
Time and place

Weeks 36-44, 2016, Aud D, NBI.
Tuesday : 11:15-12:00
Thursday : 10:15-12:00

Weeks 36-44, 2016.
Tuesday : 9:15-11:00, Aud D, NBI
Thursday : 13:15-15:00, Aud D, NBI

Topics in Complex Systems (7.5 ECTS) is one of the courses offered yearly by the Center for Models of Life of the Niels Bohr Institute for both Master and PhD students. The course focuses on a wide variety of complex phenomena ranging from phase transitions and critical phenomena, and econophysics. The common themes will be the self-organization (complex patterns and properties arising from simple rules) and universality , i.e. that broad classes of phenmena can be captured by the same simple model. Students could benefit from a background knowledge of dynamical systems, e.g., Dynamical Systems and Chaos, but it is not a necessity. Also students from economics could find the couse usefull.

We use the book J. M. Yeomans, "Statistical Mechanics of Phase Transitions" (Oxford Science Publications) and notes by Kim Sneppen.

This course is ideal for students who plan to write their thesis in complex systems and bio-complexity.

All lectures and problem sessions will be given in English and the final oral examination will also be held in English (or Danish).

The final examination will be based on the 10 questions/subjects from the list below; the questions should be prepared using the indicated materials. Students will draw one question and are expected to answer it within 10-15 minutes, followed by 10 minites questions in other subjects. The grade is set solely by the final examination, i.e., there are no midterm reports of any kind. Attendance to lectures and problem sessions is beneficial but not mandatory; it therefore does not influence the final grade. The students will be graded in the exam (page numbers refer to K. Sneppen's notes "Topics Course 2015", see below).

  1. Phase transitions, The Ising model and Mean field theory: scaling, critical exponents, correlation functions, Bogoliubov inequality, Landau expansion (Yeomans: Chaps. 1,2, 3, 4)
  2. The Monte Carlo method: Markov chain, Detailed balance and equilibrum distribution, solutions to F(x), Metropolis algorithm and heat bath, show your own Monte Carlo curves (on paper) (Monte Carlo Notes, Chap. 7)
  3. Diffusion and Burgers' equation: Diffusion equation, solution to Diff.eq., Burgers without viscosity, conservations laws, with viscosity, Galilean invariance, rescaling (Reynolds nr), Hopf-Cole transformation (Fractals p.1, Wiin-Nielsen, pp1-7)
  4. Percolation, critical point, Bethe lattice and scaling, directed percolation, dimension equation (Chap. 1)
  5. Networks:Basic concepts, Percolation and spreading of disease on networks, Motifs and topoligical characteristics. (Chap. 2)
  6. Networks: Dynamics on networks, dynamics of networks and possible models for emergence of scale-free networks. (Chap. 2)
  7. Agent based models: Schelling model, voter model, persistantly competing states, Threshold dynamics and Self organized criticality (Chap. 3)
  8. Interfaces:Transformation from Burgers to KPZ, scaling exponents, quenched noise, Sneppen model, directed percolation (Bohr, Chap. 7.1, 7.2, 7.3,7.4 (until 7.4.1), 7.5 (until 7.5.3)
  9. Game Theory: information games and bet-hedging (Chap. 4),
  10. Econophysics: time series, Hurst exponents, volatility, risk, inverse statistics, Fear factor model, collapse and exponential growth (Chap. 5)

  11. Question hour: Aud. D, Saturday 5 Nov 10.15-12.

    Oral Exam, Aud. C, NBI:

    Tuesday 8 Nov.:

    9.00: Benjamin Lazar
    9.30: Benjamin Halager
    10.00: Johann Kollath
    10.30: Stavros Kyzalas
    11.00: Luca Troise
    11.30: Felix Weidenmuller

    Wednesday 9 Nov.:
    9.00: Nanna Oesterlund
    9.30: Sidsel Winther
    10.00: Kasper Lund
    10.30: Allan Hansen
    11.00: Sigurd Carlsen
    11.30: Silas Boye
    12.00: Marius Simonsen
    14.00: Andreas Thomas Eilersen
    14.30: Kim Joensen

Reading materials and curriculum for the oral exam. All exercises are also part of the exam curriculum.

J. M. Yeomans, "Statistical Mechanics of Phase Transitions", Chap. 1, 2, 3, 4.1, 4.2, 7 (pdf, scanned copy).

Notes on Monte Carlo (pdf).

Diffusion: Notes on Fractals (pdf), p.1; A. Wiin-Nielsen, "Cascade Processes in one-dimensional Models". pp. 1-7 (and figures). (pdf). T. Bohr et al., "Dynamical Systems Approach to Turbulence", Chap. 7.1, 7.2, 7.3,7.4 (until 7.4.1), 7.5 (until 7.5.3). (pdf)

Kim Sneppen, "Topics Course 2015" (pdf).


Kim Sneppen (, course responsible
Mogens Hogh Jensen (