1998*Self-Organized Criticality*.*Stochastic Dynamics of Complex Systems*. 2013, with Paolo Sibani

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?]

Dynamical evolution over long time scales is a prominent feature
of all the systems we intuitively think of as complex – for example,
ecosystems, the brain or the economy. In physics, the term ageing
is used for this type of slow change, occurring over time scales much
longer than the patience, or indeed the lifetime, of the observer.
The main focus of this book is on the stochastic processes which
cause ageing, and the surprising fact that the ageing dynamics of
systems which are very different at the microscopic level can be
treated in similar ways.

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.