After this course Master students of natural science and life science are able to do basic calculations using the most prominent toy models in biology, chemistry and physics, and transform complex problems into the simple mathematical rules that serve as input in these models. Moreover, students are able to critically read and understand the modern scientific literature on modeling complex behavior in the language of toy models in general.
Global course description. ‘Toy models’ are models that use as input (very) simple rules, and as ‘output’ are able to describe a wide variety of (complex) behavior. In this course, some of the most successful toy models will be treated. These models are able to put complex behavior into perspective in terms of generic underlying rules, and have led (and are still leading) to a deeper understanding in biology, chemistry and physics. Besides that, successful toy models have strong predictive power, and often have significant impact beyond the disciplinary boundaries for which they were originally designed.|
Aim of the course: introduce Master students of natural science and life science to some prominent toy models, and provide them with the mathematical and statistical mechanical tools and background that are necessary to ‘play’ with these toys.
Detailed course description
- Introduction to the Ising model and its different macroscopic (stationary) solutions or phases in 1,2,3 dimensions, properties of critical points, scale invariance and renormalization group. Tools: Boltzmann weight, partition function, thermodynamics, macroscopic order parameters, and mean-field theory. (Henk Stoof)
- The ‘random walk’ and applications in diffusion, polymer statistics and rare events. Tools: basic statistics (Willem Kegel)
- Random adsorption models. Fundamentals, MWC theory of allosteric interactions, simple genetic repression and activation. Tools: grand ensemble theory; undetermined multiplier method of Lagrange. (Willem Kegel)
- Topics in differential equations, bifurcations and tipping points, relaxation oscillations (van der Pol equation). Competition between species (Lotka-Volterra), replicator dynamics and evolutionary stable strategies. Tools: qualitative theory of ordinary differential equations, Liapunov theorem and the Poincaré-Bendixon theorem. (Sjoerd Verduyn Lunel)
- Topics in discrete dynamical systems and cellular automata going from individual dynamics to macroscopic behavior in biological models and artificial life. Tools: attractors, bifurcations, Liapunov exponents, and simulations. (Sjoerd Verduyn Lunel)