Ever since Per Bak and his sandpiles, there has been excitement about self-organising critical systems. Now people are seeing potentially SOC systems everywhere -- not only sandpile avalanches, but earthquakes, solar flares, lattice gases, extinction events, forest fire percolation, .... The subject area is still rather hand-wavey, and Jensen's intent is to bring some rigour and coherence to it.
He starts out with an overview of what a SOC system is, and some candidate examples. The "critical" part of SOC comes from the lack of scale in both time and space, which leads to power laws: both 1/f temporal fluctuations and spatial fractals. The "self organising" part comes from the fact that no external tuning is needed to move the system to this critical point. (Unlike, for example, phase transitions, which are critical systems, but have to be held at the right temperature, so are not self organising.) Hence the existence of power laws is a necessary, but not sufficient, property of SOC systems.
SOC systems have a driving force timescale very much longer than the relaxation force timescale. Some kind of pressure slowly builds up on the slower timescale, until it is big enough to overcome a threshold, leading to "cascades" of relaxation on the faster timescale. Ironically, although simulated sandpiles were the original inspiration for the subject area, real sandpiles don't seem to be SOC systems. Sand is too dense, and once an avalanche starts, inertia overcomes friction, and the avalanche doesn't stop until the system is totally relaxed. Rice piles are better (as long as you choose the grain shape carefully).
Jensen next discusses simulating SOC systems, and getting statistical results from the simulations. There are some problems simulating them. Edge effects can be important. For example, the usual simulator's trick of using periodic boundary conditions can lead to a periodic time behaviour. And they can take a very long time to converge. So if the system does have a scale factor, but it is very much larger than can be effectively simulated, the system might appear to be SOC.
Finally, he discussed analytical results, including statistical mechanical mean field theory, exact results using Abelian groups, and renormalisation group calculations.
This is a slim book, less than 150 pages, so the broad coverage requires quite dense exposition in places. But this is leavened by a readable style, and clear indications of which parts are precise, and which are still somewhat hand-wavey. [Standard complaint, however: why do some authors feel it is sufficient to provide bibliographies that omit the titles of the papers?]
The first part of this book provides the necessary mathematical and computational tools and the second part helps the reader develop the intuition needed to deal with these systems. The content of some of the first few chapters has been covered in several other books, but the emphasis and selection of the topics reflect both the authors’ interests and the overall theme of the book. The second part contains an introduction to the scientific literature and deals in some detail with the description of the complex phenomena of a physical and biological nature, for example, disordered magnetic materials, superconductors and glasses, models of co-evolution in ecosystems and even of ant behaviour. These heterogeneous topics are all dealt with in detail using similar analytical techniques.
This book emphasizes the unity of complex dynamics and provides the tools needed to treat a large number of complex systems of current interest. The ideas and the approach to complex dynamics it presents have not appeared in book form elsewhere.