How Nature Works by Per Bak, Copernicus/Springer-Verlag,
$27, ISBN 0 387 94791 4
IMAGINE a child steadily adding a stream of grains to a pile of sand. As long
as the pile is small, not much happens. But when the slope of the pile reaches a
critical value, it will grow no more. Adding sand just causes the surface grains
to slide off, leaving the slope essentially unchanged. Most of the sand
avalanches will be small, some will be medium-sized and, occasionally, there
will be landslides which will affect a large part of the pile. It is not
possible to predict whether the next grain of sand will cause a large or a small
avalanche. Build another sand pile, you will end up with the same slope, and a
system poised in a similarly delicate 鈥渃ritical鈥� state.
Per Bak鈥檚 original paper on self-organised criticality has become a citation
classic and began a landslide of its own. The sand pile paradigm is well on the
way to joining the butterfly effect in popularity. I do not quote politicians
lightly, but I must confess that I tend to agree with US Vice President Al Gore
who wrote that 鈥渢he sandpile theory鈥攕elf-organised criticality鈥攊s
irresistible as a metaphor鈥�.
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Bak used a computer model of the pile of sand to show how power laws can
arise. Roughly speaking, avalanches that are 10 times larger are 10 times less
frequent. More precisely, if we plot how many avalanches there are of each size
on a logarithmic scale, we obtain a straight line. As in so many phase
transitions, there are no natural length or time scales at the critical
state.
Such power laws arise in all kinds of fields. The statistics of earthquakes,
for instance, follow the Gutenberg-Richter law. Way back in the 1940s, George
Zipf had found similar laws concerning the size of cities, or the frequency of
words.
Our brains have a tendency to look for periodic patterns in space and time,
and to find such patterns even where there are none. But a lasting benefit of
the related concepts of power laws and fractals is that they help us to overcome
our fixation with periodicity and understand that there are other types of
meaningful regularities. Some scientists, however, are now prone to exaggerate
the prevalence of power laws. Besides, the literature is biased because it is
difficult to publish a paper stating that a phenomenon has been shown not to
obey a power law鈥攗nless you happen to be dealing with a sand pile.
Ironically, it has turned out that in real piles of sand, avalanches do not
follow a power law. This has been used to deride Bak鈥檚 model, rather unfairly, I
believe. The sand pile was a metaphor; the inertia of sand was not included in
the model.
What Bak and his co-workers really do is computer simulations of
patterns on grids which are updated regularly according to fixed local rules.
Depending on these transition rules, you can model all kinds of things: sand
piles, earthquake faults, pulsar glitches or solar flares. A fascinating array
of complex behaviour appears.
The best known example of such a model is certainly Conway鈥檚 Game of Life.
This is a cellular automaton where every square on a huge chequerboard is deemed
either 鈥渄ead鈥� or 鈥渁live鈥�. If it is dead, and three of its eight neighbours are
alive, the square becomes alive. If it is alive and fewer than two or more than
three of its neighbours are alive, then it dies. To everyone鈥檚 surprise Bak has
shown that the Game of Life is nearly critical: if, in a static pattern, you
alter the state of one square at random, you start off an avalanche of changes
whose size obeys a power law.
Bak and his co-workers have developed a knack for designing situations which
lead from amazingly simple local rules to a critical overall behaviour. Arrange
random numbers on a circle, for instance, and then replace the lowest number and
its two neighbouring numbers with some new random numbers. Repeating this, the
size of the lowest number on the circle increases on average, until it reaches a
threshold. A number above this threshold will change only if its neighbour has a
low value. It can then acquire a low value which will cause its neighbours to
change, and so on. This domino effect obeys a power law. So what? Well, listen
to Bak: 鈥淥ur simulations demonstrated that there is no contradiction between
Darwin鈥檚 theory and punctuated equilibria.鈥�
In case you are wondering what a circular array of random numbers has to do
with Darwin, you are urged to think of the numbers as the fitness of interacting
species. If one species goes extinct, other species may be doomed. A landslide
of extinctions is a mass extinction. Bak flatly states that 鈥渋t makes no sense
to distinguish between background extinctions happening all the time and major
ecological catastrophes鈥�.
This may well be, but the link with the computer model seems rather tenuous.
Ordinary biologists define fitness as the average reproductive success of an
individual. For Bak, the fitness is the ability of a species to survive鈥攁n
altogether different concept that is nowhere properly distinguished from the
fitness of Darwin, Ronald Fisher and Sewall Wright. The same high-handed
attitude towards biology pervades Bak鈥檚 whole book. It never mentions that many
alternative models exist for punctuated equilibriums. It systematically
confounds genetic code and genotype, and wrongly asserts that the logistic map
describes predator-prey interactions.
It may lead the guileless reader to believe that co-evolution has recently
been invented at the Santa Fe Institute and that an organism is just anything
whose parts are interconnected.
Before physicists took the matter in hand, 鈥渙nly a scattered handful of
oddballs were working on understanding life鈥�. An unexpected fit of
modesty鈥斺�漷his is not the last word [on macroevolution]鈥濃�攊s quickly
mastered: 鈥淧robably it is the first鈥�. But in truth a power law on species
extinction, while exciting, will be of no more help in 鈥渦nderstanding life鈥� than
Zipf鈥檚 law on the frequency of words helps us to understand grammar.
The pace of the book, which is never slow, quickens alarmingly towards the
end. Take the brain, for instance. Being complex, it must be critical, suggests
Bak. Neurons can excite neurons just as sand grains can fall against others and
start an avalanche. Economics? Well, a producer receiving an order may in turn
place some orders and start an avalanche. 鈥淓conomics is like sand, not water.鈥�
Traffic jams?
Actually, traffic jams seem to be an excellent example of self-organised
criticality鈥攅ven more convincing than the sand pile. And there is more
than a sand grain of truth in Bak鈥檚 message that purely local interactions can
trigger changes on a global scale. But just when you feel like conceding that
such avalanche effects have been shamefully neglected, Bak spoils it all by
brazenly concluding that 鈥渟elf-organised criticality is a law of nature for
which there is no dispensation鈥�.
I picture Bak as a kind of scientific musketeer: flamboyant, touchy, full of
swagger and ready to join every fray鈥攔eady to start it, in fact, anywhere,
anytime. His book is written with panache. The style is brisk, the content
stimulating. I recommend it as a bracing experience. But if you end up believing
that self-organised criticality explains How Nature Works鈥攚ell,
don鈥檛 blame me.