Published October 2005 by Imperial College Press

The word `complexity' takes on a variety of meanings depending on the context, and its official definition is continuously being revised. This is because the study of complexity is in its infancy and is a rapidly developing field at the forefront of many areas of science including mathematics, physics, geophysics, economics and biology, to name just a few. Institutes and departments have been formed, conferences and workshops organised, books and countless articles written, all in the name of complexity. And yet, nobody agrees on a clear and concise theoretical formalism with which to study complexity. The danger is therefore that complexity research may become unstructured or even misleading. For our purposes, complexity refers to the repeated application of simple rules in systems with many degrees of freedom that gives rise to emergent behaviour not encoded in the rules themselves.

The word `criticality', on the other hand, is well defined among statistical physicists. Criticality refers to the behaviour of extended systems at a phase transition where observables are scale free, that is, no characteristic scales exist for these observables. At a phase transition, the many constituent microscopic `parts' give rise to macroscopic phenomena that cannot be understood by considering the laws obeyed by a single part alone. Criticality is therefore a cooperative feature emerging from the repeated application of the microscopic laws of a system of interacting `parts'. The phenomenology of phase transitions is well developed and there exists a sound theoretical formalism for its description.

The book is divided into three chapters. In the first two chapters, we carefully introduce the reader to the concepts of critical phenomena using percolation and the Ising model as paradigmatic examples of isolated equilibrium systems. These systems undergo a phase transition only if an external agent finely tunes certain external parameters to particular values. The underlying theoretical formalism of criticality is carefully explained through the concept of scale invariance, a central unifying theme of the book.

However, there are many examples in Nature of complexity, that is, the spontaneous emergence of criticality in slowly-driven non-equilibrium systems: earthquakes in seismic systems, avalanches in granular media and rainfall in the atmosphere. Key models of self-organised criticality illustrate how such systems may naturally evolve into a stationary state displaying scale invariance, and analogies are drawn between complexity and criticality.

Although mathematical methods have been developed to describe complexity and criticality, it is our experience that these methods are unfamiliar to scientists outside the field. Therefore, throughout the book we emphasise the mathematical quantitative techniques available. Our hope is that this book will help students and researchers to treat complexity and criticality more quantitatively.

The book is based on the lecture notes developed for the Statistical Mechanics course. The target audiences are undergraduate and graduate students and researches in various fields. The book will be self-contained and accessible to readers not familiar with the concepts of complexity and criticality. The text can form the basis for advanced undergraduate or graduate courses, and serve as an introductory reference for researches in various fields. The book includes a generous number of figures, and has an associated website containing solutions to exercises and animations of the models considered. Each chapter is accompanied by exercises, full solutions to which can be obtained by contacting the authors. On www.complexityandcriticality.com, readers will find animation codes to visualise the behaviour of the models considered and (a few) corrections to the book.