Henrik Jeldtoft Jensen.
Self-Organized Criticality: emergent complex behaviour in physical and biological systems.
CUP. 1998
rating : 3.5 : worth reading
review : 13 October 2001
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?]
