Kim Christensen and Nicholas R. Moloney, Complexity and Criticality List of Content
Published October 2005 by Imperial College Press Summary:
The aim of this book is to introduce the concepts of critical phenomena
and explore the common ground between complexity and criticality.
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
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
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
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 via the book's associated
On this site, readers will also find animation codes to visualise the behaviour
of the models considered.