COMPLEXITY & CRITICALITY (V1)

This book provides a challenging and stimulating introduction to the contemporary topics of complexity and criticality, and explores their common basis of scale invariance, a central unifying theme of the book. Criticality refers to the behaviour of extended systems at a phase transition where scale invariance prevails. The many constituent microscopic parts bringing about macroscopic phenomena that cannot be understood by considering a single part alone. The phenomenology of phase transitions is introduced by considering percolation, a simple model with a purely geometrical phase transition, thus enabling the reader to become intuitively familiar with concepts such as scale invariance and renormalisation. The Ising model is then introduced, which captures a thermodynamic phase transition from a disordered to an ordered system as the temperature is lowered in zero external field. By emphasising analogies between percolation and the Ising model, the reader's intuition of phase transiti

Percolation: Percolating Phase Transition; In One and Two Dimensions, and in the Bethe Lattice; Geometric Properties of Clusters; Scaling Ansatz, Scaling Functions and Scaling Relations; Universality; Real-Space Renormalisation Group; Ising Model: Review of Thermodynamics and Statistical Mechanics; Symmetry Breaking; Ferromagnetic Phase Transition; In One and Two Dimensions, and in the Mean-Field; Landau Theory of Continuous Phase Transitions; Scaling Ansatz, Scaling Functions and Scaling Relations; Universality; Real-Space Renormalisation Group; Self-Organised Criticality: BTW Model in One and Two Dimensions, and in the Mean-Field; A Rice Pile Experiment and the Oslo Model; Earthquakes and the OFC Model; Rainfall; Self-Organised Criticality as a Unifying Principle.