麻豆传媒

A CASUAL observer might be disconcerted by Gene Stanley鈥檚 choice of reading
material. Stanley is a physicist. Yet on his desk, surrounded by a pile of
ruffled research papers on heart physiology, lies a text on the mathematics of
financial markets. Beneath it are books on the molecular structure of water and
the chemistry of the human stomach. And stacked on a nearby chair, ready to go
home for the evening, are volumes on computer simulation, ant behaviour and oil
exploration. Stanley seems to be working his way at random through Boston
University鈥檚 science library.

But this haphazard intellectual foraging isn鈥檛 a sign of incipient madness.
Nor is Stanley a compulsive and indiscriminate gatherer of facts. Far from it.
He is simply a man who has been seized by the power and beauty of an idea. And
in nearly every book he opens, he finds another place to apply it.

Recently, he has been grappling with the mystery of Alzheimer鈥檚 disease.
Before that it was the dynamics of the human heart and their relation to
breathing disorders, and before that the organisational structures of businesses
and their natural patterns of growth. An ambitious course of research, to say
the least. Yet in all these areas, Stanley has been gaining new insights into
problems that other researchers have been battling with for years.

The idea that has so gripped Stanley is the 鈥減rinciple of universality鈥�.
Outside physics it is virtually unknown. Yet in the past few years, Stanley and
a growing number of scientists have exploited the idea to forge crucial advances
in fields ranging from evolutionary theory to the science of earthquakes.

Universality is hard to pin down without resorting to mathematics, but you
don鈥檛 need to look very far to get a feel for it. The secret lies in the stuff
of the everyday world鈥攊n liquids, solids, gases and even in an ordinary
magnet.

An iron magnet will tug on a nail or a paper clip. But heat it up until it
glows and it will lose its magnetic power. Why? The answer has to do with
organisation. A piece of iron is made of millions of microscopic domains, each
about the size of a grain of sand, and each one a tiny magnet. Like all magnets,
these domains would like to be aligned with one another, and at room temperature
they are. It is their cooperation that makes the iron magnetic. But at high
temperatures鈥攁bove 770 掳C鈥攖hermal noise disrupts the domains鈥�
cooperation, and their magnets end up all askew. As a result, the iron becomes
nonmagnetic.

This change from one state of organisation to another is what physicists call
a phase transition. Similar changes take place when a puddle evaporates into
vapour, an ice cube melts in a gin and tonic or when a piece of tin, cooled to a
frigid 3 kelvin, becomes superconducting. In one state, the units鈥攂e they
magnets, molecules or electrons鈥攃ommunicate effectively to make the system
ordered, and in the other they don鈥檛. Physicists have been studying such things
for centuries. But there is more in a magnet than you might imagine.

In a piece of iron held at 770 掳C, the metal is on a knife edge between
its magnetic and nonmagnetic states. Some domains still manage to align with one
another, but they do so only in localised clumps that all point in different
directions
(see Diagram).
This clumping may look random, but it is not. It
shows an astonishing organisation. Cut out a small piece, magnify it, and you
regain an image that is indistinguishable from the one you started with. When
iron is in its 鈥渃ritical state鈥�, its domains somehow organise themselves into
what mathematicians call a self-similar pattern. Every bit of the image looks
just like the whole.FIG-21074601.jpg

This mysterious self-similarity is a defining characteristic of the critical
state. It crops up in all sorts of places鈥攊n water poised between its
liquid and vapour phases, in crystals under pressure as they start to collapse
from one molecular structure to another, in tin on the edge of becoming
superconducting and at critical points in just about any other system you can
think of. Under close inspection, the organisation of all these critical states
turns out to be not merely qualitatively alike, but precisely, exactly,
mathematically identical.

But how can this be? How can a bunch of mindless magnetic domains organise
themselves so thoroughly? And why do things that appear to have little in common
organise themselves according to the same rules? After all, the domains stay put
and interact through their magnetic fields while particles in a fluid move
around, and interact only when they hit one another. And in a superconductor or
superfluid, electrons interact according to the strange rules of quantum theory.
The component parts of these systems are about as different as they could be,
yet their critical states are utterly indistinguishable.

Around 1970, Leo Kadanoff of the University of Chicago hit on the secret. In
all these systems, he pointed out, the interactions between domains or molecules
extend only over a short range, from one unit to its closest neighbours. Yet in
the critical state, these interactions manage to link up with one another and
propagate order all the way across the system. Kadanoff suggested that it is
this peculiar 鈥渓inking up鈥� that is crucial, rather than the details of the
interactions themselves.

In other words, at the critical point there is a kind of universal
organisation in which details the particular system get obliterated. Molecules,
atoms, magnets or electron spins鈥攊t simply doesn鈥檛 matter what is
interacting. 鈥淚n the critical state,鈥� says Stanley, 鈥渙rder spreads across the
system much as diseases spread through orchards. What matters is not so much the
kind of tree, but the connectivity between trees.鈥�

There are a few things, of course, that do matter, such as the number of
physical dimensions of the system. The connectivity between units arranged in a
line differs from that of units in a plane, or scattered through space. Also
important are what physicists refer to as the basic symmetry properties of the
units. Many molecules are roughly spherical, so that in a collection of them the
order propagates equally in all directions. But magnetic domains in iron are
more like tiny rods. When they begin to align with one another, order propagates
in one way along the direction of the rods, and in another way within the plane
perpendicular to those rods.

Class acts

But these few essential properties don鈥檛 spoil the show. They simply mean
that instead of a single kind of universal critical organisation, there are a
handful corresponding to different dimensions and symmetries. In the language of
physics, each one is a 鈥渦niversality class鈥�: the set of systems that share the
same few essential properties. And all systems from a particular universality
class, no matter how different they may be, behave identically at the critical
point.

Kadanoff鈥檚 idea kindled a mathematical brushfire that has been spreading
through the sciences ever since. Universality gives us a new understanding of
how apparently very different things can act the same way. And the beauty is
that those things don鈥檛 have to have a physical existence. Mathematical things
count too. So if you want to model something such as a magnet or a fluid near
its critical point, you don鈥檛 have to worry about accurately representing how
every component interacts with its neighbours. Any model, however abstract or
ridiculous, will do, so long as it belongs to the right universality class.

It was the breadth of the possible uses of universality that hooked Stanley
30 years ago when he was a graduate student studying mathematical models for
magnets. 鈥淚t struck me,鈥� he says, 鈥渢hat the kinds of models I was using might
apply also to a huge number of other cooperative systems.鈥� Physicists, after
all, aren鈥檛 the only ones who deal with such things. The behaviour of an
economy, a company or an ecosystem arises from the interactions between the
individuals that make it up. Cooperative systems are everywhere, be they flocks
of birds or colonies of bacteria. And according to universality, the precise
nature of the elements that make up those systems and how they interact are
often irrelevant. 鈥淯niversality gives us confidence,鈥� says Stanley, 鈥渢hat we
really can model and understand complex systems like this.鈥�

But before this seems too miraculous, remember that it only works if a system
really is in a critical state. So for the past 25 years Stanley has been rooting
out the traces of critical organisation in everything he can find.
Mathematically, the signature of a critical state is what physicists call a
power law. In the magnetic critical state, for example, small clumps are more
numerous than large ones. But the number of clumps (N) of a particular
size (s) turns out to be inversely proportional to s raised to
some power (P). That is
狈(蝉)鈭漵^-P. Along with this
mathematical form come a number of properties. It is a sure sign that the system
has an element of self-similarity in its organisation.

Stanley has found power laws like this in the structure of DNA, in the
dynamics of the human heart and lungs, and even in systems in which the units
have 鈥渇ree will鈥�. In 1995, for example, he and a team of researchers including
Michael Salinger of Boston University鈥檚 School of Management made a thorough
study of the growth statistics of manufacturing firms. Looking at every company
traded publicly in the US between 1975 and 1991, they found that smaller firms
grow faster but less predictably than larger ones. That is, while large firms
tend to grow at more or less the same slow pace, year after year, small firms
spring up quickly but face an uncertain fate. A tiny company might double or
triple in size in one year, and go bankrupt the next.

The finding that growth rates diminish with size overturns an aspect of
traditional economic theory which held that growth is determined mostly by the
type of technology used for production. Obviously, this should vary greatly from
one business to another: turning out potato crisps just isn鈥檛 the same as
putting together jumbo jets. But the group鈥檚 results show a universal pattern of
growth that holds for firms of all types, whatever business they are in. What a
company produces doesn鈥檛 matter nearly so much as its size.

These findings also seemed to reveal something profound about the way
companies are organised. Any large firm contains within itself many smaller
units, each of which is organised much like the whole. To find out more about
how these smaller units are tied together, the researchers dreamt up a simple
hierarchical model for a firm. Capturing the way humans in a company behave in
mathematics is impossible, of course. But according to the principle of
universality, a firm that shows self-similarity must be in a critical state. And
we can understand the behaviour of the company as a whole without bothering with
details of how its constituent units鈥攊ts employees鈥攊nteract.

Model company

In the model, employees at each level of management make decisions that
directly influence employees one level below. But these subordinates don鈥檛
always do what they are told. Accordingly, the model contains a 鈥渟ubservience
parameter鈥� representing the probability that an employee will obey an
instruction received from above. A subservience of 0 signals anarchy鈥攖he
firm dissolves into an uncoordinated group of independent agents. On the other
hand, a subservience equal to 1 signifies a company staffed by automatons.
鈥淣either of these cases fits the facts,鈥� says Stanley. These model firms don鈥檛
grow like real ones.

But when the parameter lies between 0.7 and 0.9, the model behaves just like
the real thing. Small firms grow more irregularly than large ones because in the
smaller firm capricious acts of single individuals or groups within the company
show through more strongly. According to Stanley, companies grow as they do
because they are naturally hierarchical, and information flows imperfectly from
above to below. And employees, it seems, muck up or ignore about one out of
every four or five commands that come down from their bosses鈥攊t鈥檚 a
universal principle.

Just as Stanley鈥檚 work on companies has raised a few eyebrows in corporate
circles, other researchers鈥� work on earthquakes is sending tremors through the
world of seismologists. Earthquakes follow the famous Gutenberg-Richter law,
which says that over a given period the number of earthquakes that release
energy E is, on average, inversely proportional to E taken to
some power. This power law means that the system that generates quakes鈥攖he
Earth鈥檚 crust鈥攊s organised in a critical state. Accordingly, the precise
details of how stress builds up in the crust may not be important to
understanding what triggers an earthquake.

This has led Jean Carlson, James Langer and Brian Shaw of the University of
California at Santa Barbara to study a simple model of the crust
(see Diagram)
that deliberately ignores geological details. It consists of a chain of
blocks resting on a floor and connected to each other by springs. The blocks
represent the land on one side of a fault line. The floor is the other side of
the fault. The springs represent the rock鈥檚 elastic nature: when deformed it
stores energy that can later be released. Friction between the two
sides鈥攂etween the blocks and floor鈥攌eeps them in place, even when
the springs are slightly stretched or compressed.FIG-21074603.jpg

Faithful to a fault

In the Earth, the slow motion of continental plates drags tracts of land past
one another. This comes into the model through elastic bands that slowly drag
all the blocks to one side. When the bands stretch so far that the force on a
block is enough to overcome friction, it slips. That movement then changes the
forces acting on neighbouring blocks, and may cause them to slip too. So the
entire chain of blocks can suddenly shift in a 鈥渜uake鈥� that travels down the
fault as a wave of compression or extension in the springs and bands.

The model captures only the most rudimentary aspects of the geology of real
faults. Yet it faithfully mimics those elements of the dynamics of the Earth鈥檚
crust that are relevant to earthquakes. There are many interacting pieces.
Forces drive them out of equilibrium. And the system can store and release
elastic energy. Despite its crude character, the model鈥檚 behaviour follows the
Gutenberg-Richter law almost perfectly.

鈥淯niversality is a physicist鈥檚 dream,鈥� says Per Bak of the Niels Bohr
Institute in Copenhagen, another physicist fascinated by its power. Two months
ago in Nature, vol 388, p 764) along with physicists Ricard Sole and
Susanna Manrubia of the Polytechnic University of Catalonia in Barcelona, and
geologist Michael Benton of Bristol University, Bak reported an analysis of
major extinction events as they appear in the fossil record. Their results
reveal that the dynamics of extinctions have the same self-similar critical
structure as earthquakes. Massive extinctions happen far less frequently than
tiny ones. But a graph showing the frequency of extinctions versus size closely
follows a power law.

This power law agrees with a toy evolutionary model that Bak and Kim Sneppen
invented a few years ago when they were at the Brookhaven National Laboratory in
New York. They took the bare bones of Darwin鈥檚 theory and used them to see how
extinctions might occur by evolutionary dynamics alone鈥攚ithout any
external impositions, such as hypothetical asteroid impacts, to cause them. In
their model, species evolve and can go extinct, and they interact with one
another, as do species in the interconnecting web of real life.

Imagine a hundred broken matchsticks arranged along a line
(see Diagram).
Each matchstick represents a species, and its length鈥攁 random
number between 0 and 1鈥攎easures that species鈥� fitness. This mock ecosystem
evolves in steps. First, the species with the lowest fitness goes extinct. That
is, the shortest matchstick is discarded and replaced by a new one with length
again chosen randomly between 0 and 1. In addition, the two nearest neighbours
of that species also go extinct and are similarly replaced. This crudely mimics
the interaction between species in the real world, in which one extinction may
trigger others.FIG-21074604.jpg

To run evolution forwards, simply repeat these steps over and over. And
despite its simplicity, this model has a strikingly complex behaviour. After
many iterations, the ecosystem organises into a critical state, where the
pattern of fitnesses is self-similar. At each step, at least three species go
extinct. But over a series of cycles, the interaction between species makes it
possible for extinctions to travel across the system in a chain reaction. Run on
a computer, the model鈥檚 extinctions are distributed by size鈥攖he number of
species involved in a chain event鈥攊n much the same way as in real
evolution.

As a consequence, scientists may need to rethink their ideas about what
caused the great extinctions, like the one that wiped out the dinosaurs.
鈥淚t seems to be a widespread assumption,鈥� says Bak, 鈥渢hat some cataclysmic
impact must have been responsible for the mass extinctions.鈥� Those suggested so
far include drastic changes in weather, volcanic eruptions and, most popularly,
impacts from extraterrestrial objects. But the Bak-Sneppen model, combined with
the fossil data, suggests that maybe extinctions happen for no special reason at
all. 鈥淎 huge extinction doesn鈥檛 necessarily imply a corresponding catastrophic
cause,鈥� says Bak. The internal dynamics of Darwinian evolution seem to be enough
to cause the huge extinctions seen sporadically in the fossil record.

So from earthquakes to evolution, the notion of universality lies behind
theories that are adding an extra dimension to our understanding of the world.
But the consequences of the idea may ultimately be far more profound. For
hundreds of years science has proceeded on the notion that things can always be
understood鈥攁nd can only be understood鈥攂y breaking them down into
smaller pieces, and by coming to know those pieces completely. Systems in
critical states鈥攁nd they seem to be plentiful鈥攆lout this principle.
Important aspects of their behaviour have little to do with the detailed
properties of their component parts. Organisation in a magnet, a company or an
ecosystem isn鈥檛 down to the particles, people or organisms that make it up.

This doesn鈥檛 mean that geophysicists and evolutionary biologists need to quit
their jobs. Not all systems are in a critical state鈥攁nd details usually do
matter. But when something shows a critical organisation, scientists have in
universality a powerful tool for understanding that is utterly unlike anything
they have had before.

This new way of thinking is infiltrating some unexpected areas, with
important consequences. Stanley is now applying the arguments of universality to
the study of Alzheimer鈥檚 disease. Other scientists have used the ideas to
understand the behaviour of neutron stars, landscape formation and traffic jams.
A good bit of what goes on in all these complex things can be explained by
astonishingly simple mathematical models, all bolstered by a single idea.
鈥淲ithout universality,鈥� says James Crutchfield, a physicist at the University of
California at Berkeley, 鈥渆ach and every complex system would be a discipline to
itself, and the very enterprise of science would be doomed from the start.鈥� With
it, there seems to be a basis for a true 鈥渢heory of collectives鈥� of all
sorts.

Perhaps one day we may come to see knowledge itself as a system in a critical
state, with changes coming in avalanches, their sizes following a power law.
Universality may be one of the biggest avalanches of all. Who knows? It鈥檚 an
idea intriguing enough to send the indefatigable Stanley back to the library for
more books.