Syllabus

The aim of the statistical mechanics course is to provide a challenging and stimulating introduction to a selection of topics within modern physics. The course carefully relates the techniques and methods from phase transitions in simple disordered systems to concepts widely used in the study of complexity in Nature.

The lecture course is devoted to the study of systems consisting of many microscopic agents bringing about macroscopic phenomena that cannot be understood by considering a single agent alone. There are many examples in Nature of such cooperative phenomena or emergence of complex behaviour: fractals, phase transitions, earthquakes in seismic systems, avalanches in granular media, rainfall in the atmosphere, or metaphorical avalanches like mass extinctions in biology or stock market crashes in economics.

These phenomena are addressed in the framework of simple models.

Percolation theory is the simplest model displaying a phase transition. The analytic solutions to 1d and mean-field percolation are presented. While percolation cannot be solved exactly for intermediate dimensions, the model enables the students to become familiar with important concepts such as fractals, scaling, and renormalisation group theory in a very intuitive way.

The Ising model further develops the students' intuition of emergent cooperative phenomena by explicitly introducing a dynamic interaction between agents locally.

The students are then naturally prepared for the discussion of non-equilibrium systems where the constraint on having to tune an external parameter to obtain cooperative phenomena is relaxed. The lecturer invites the students to speculate whether self-organisation in non-equilibrium systems might be a unifying concept for dispirate fields such as statistical mechanics, geophysics, atmospheric physics, economics, and biology.

Seminal computer models, designed to capture the essential ingredients of the underlying physics in the simplest possible way, are presented. Furthermore, the statistical mechanics course is accompanied by exercises and various visual interactive simulations available on-line for all the models considered to allow the students to experience for themselves the behaviour of the models, in the spirit "seeing is believing".

Keywords for the SM lecture course: 26 lectures

Introduction: Themes and aims, cooperative phenomena, and emergence of complexity. [1 lecture]

Percolation: Percolation theory. Definition of percolation problem, cluster number density, cluster structure, critical phenomena near p_c, power laws, fractal dimension, correlation function, self-similar fractals, scaling theory, finite-size scaling, renormalisation group theory, fixed points, and universality. [10 lectures]

Ising model: Ising model. Definition of Ising model, phase transition, order parameter, mean-field theory, Landau theory of (2nd order) phase transitions, scaling theory, renormalisation group theory, fixed points, and universality. [10 lectures]

Self-organised criticality: Self-organisation, complexity, and self-organised criticality. Introduction to self-organisation in general and self-organised criticality in particular in nonequilibrium systems, data, and experiments. Models of granular media (sandpile model), earthquakes, economics, and biological evolution. Mean-field descriptions, universality, open questions. [5 lectures]