This repository contains the material of the course "theory of complex systems".
The course introduces students to the theory of complex systems with a main focus on critical phenomena. We will discuss the following topics: from microscopic to macroscopic description of complex systems, emergence, phase transitions, critical phenomena, universality, scale invariance, mean-field theory, renormalisation, and self-organized criticality. The course will focus mainly on equilibrium phase transitions, but will also touch upon phase transitions in dynamical systems.
We will work with simple models of complex systems, such as percolation and epidemic models, the Ising model, the linear voter model, Metropolis and Glauber dynamics, the TASEP model (a simple traffic model), the sandpile model, simple models of collective behavior, etc. Numerical simulations of the studied models will be briefly discussed in class, but not implemented. The course will be illustrated with examples of complex systems taken from various fields of science (e.g., physics, neuroscience, social sciences, animal collective behavior, earth science). Contemporary research directions will be explored through the review of recent papers, as well as with a few invited guest lecturers throughout the course.
The course and the assignments will be mainly theoretical (problems with analytical answers, open answers). Assignments can involve implementing simple numerical simulations. For this reason, it is recommended to have basic programming skills prior to taking this class.
Resources for the course include:
- Book Statistical Mechanics: Entropy, Order Parameters and Complexity, by J. Sethna (2020), available on the author’s personal website.
- Book Introduction to the Theory of Complex Systems, by S. Thurner, R. Hanel, and P. Klimek (2018)
- Book Complexity and Criticality, by K. Christensen and N. Moloney (2005)
- Website Complexity Explorables, which as a large collection of interactive simulations of complex systems.
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Chapter 0: Introduction
- Introduction to Complex Systems
- Introduction to Critical Phenomena
- Tutorial: Poisson Processes
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Chapter 1: From Microscopic to Macroscopic description of a Complex System
- Description of Complex Systems at Equilibrium
- Description of Complex systems Out-of-equilibrium and Markov Processes
- Tutorial: The Metropolis algorithm: from out-of-equilibrium to equilibrium description.
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Chapter 2: Examples of Critical Phenomena in Complex Systems at Equilibrium
- The Ising Model
- Percolation
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Chapter 3: Mean-field theory
- The Mean-Field approximation
- Landau Theory
- Tutorial: application to the TASEP model
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Chapter 4: Scale Invariance and Universality
- Emergent behaviors in random walks
- Introduction to Real-Space Renormalization
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Chapter 5: Examples of critical phenomena in Complex systems out-of-equilibrium
- Epidemic spreading on complex networks
- Voter model and opinion dynamics
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Chapter 6: Examples of collective behaviors
- Vicsek models and flocking behavior
- Noisy Kuramoto model
