Mark Buchanan
Ubiquity
The Science of History Why the World is Simpler than we Think
Weidenfels&Nicolson 2000



pg 37

FRACTALS – NON-EQUILIBRIUM PHYSICS – IRREVERSIBILITY – POWER LAWS

Keywords: 'power law' - scale-invariance or self-similarity - physics is naturally suited to producing fragments over a tremendous range of scales -

Determinism - the spirit of determinism implicit in Newton's ideas had fixed a debilitating stranglehold on the scientific imagination. To physicists grappling with the mysteries of the atomic world, Newtonian physics was mostly a hindrance, although ultimately a few scientists managed heroically to wrestle themselves free, and to create the new quantum theory. Along with the ideas of Niels Bohr, Werner Heisenberg and Paul Dirac, Schrödinger's creation liberated scientists once again, enabling them to see the world as if transformed. By the late 1920s, a world of confusing and inconsistent facts had again fallen naturally into place. More recently, a mathematician working for IBM kicked off a trans­formation of comparable scope - Benoit Mandelbrot - the study of fractals





Power laws
..In terms of energy, it turns out that the Gutenberg-Richter law boils down to one very simple rule: if earthquakes of type A release twice the energy of those of type B, then type A quakes happen four times less frequently. Double the energy, that is, and an earthquake becomes four times as rare - that's what the graph says. This simple pattern holds for quakes over a tremendous range of energies. Physicists refer to any relationship of this sort as a 'power law', and such laws have an importance out of all proportion to their rather simple appearance.

Frozen potatoes are like rocks - brittle, and ready to shatter under the force of a sharp impact. Throw one against a wall and you end up with a pile of shards of different sizes, some like golf balls, others as big as cherries, and others as small as either peas or grape seeds. What is the typical size? To find out, you might throw a thousand or so potatoes against the wall to generate a lot of pieces, and then proceed a la Gutenberg and Richter. First, separate the pieces into piles according to how much they weigh. Being careful, perhaps you could establish ten different piles of pieces, with the smallest weighing a gram or so. There would also be some smaller pieces, rather difficult to handle, which you could ignore. Now, plot on a graph the number of pieces in each pile versus the weight of those pieces.

With potato fragments standing in for earthquakes, you would find a featureless curve of the Gutenberg-Richter sort. There would be a huge number of tiny fragments the size of grape seeds, and the number of fragments would then fall off gradually as you considered larger sizes. If you worked carefully, in fact, you would find that the numbers of the larger fragments dwindle in an exceptionally regular way: every time you double the weight of the fragments you're looking at, their number will decrease by a factor of about six. This is the same kind of power law pattern found by Gutenberg and Richter, the only difference being that here the reduction in numbers that follows from each doubling in weight involves a factor of six rather than four.

But wait a minute - what about all those exceptionally tiny pieces that we ignored? Those are frozen potato fragments too, and really deserve to be included. Realising this, you might get a magnifying glass and sort out these pieces too into piles. In so doing, you would extend the distribution into the realm of tinier shards. What do things look like on this scale? Surprisingly, it turns out that the pieces down here again follow the very same law. The further down you go, the more pieces you see, with the numbers changing in a regular way: each time you reduce by two the weight of the pieces, their numbers increase by six. Three physicists from the University of Southern Denmark actually did this experiment in 1993, which is why I can report that the number in this power-law pattern is six rather than four. In the experiment, the fragments ranged in weight from 100gram chunks all the way down t tiny specks of only one‑thousandth of a gram, with this simple patter holding the whole way.

I haven't yet told you why this pattern is called a power law; we come to that shortly. First, what does it imply about our original question regarding the 'typical' or ‘normal' size of a fragment? Imagine you were a being who could change your size at will. With a snap of your fingers you could go instantly from being cherry‑sized to being as big as a pea or from that size down to the scale of an ant. This would be a great aid in inspecting the pile of fragments. Whatever scale you wanted to look at, you could simply make yourself that size, walk about and see what you find. Suppose you begin at the size of a pea. You look around for while, and get a feel for the landscape. You notice some pieces that are roughly pea‑sized and others that are somewhat larger or smaller. Perhaps then you decide to shrink yourself down to a size ten times smaller. You would be surprised to find that things at this smaller scale look much the same. When pea‑sized, you might have noticed that for every frag­ment that was about your weight, there were roughly six about half that weight. Shrinking yourself, you would discover exactly the same thing ‑ still about six fragments half your weight for every one your own. At all scales, the landscape would have precisely the same feel to it, and if you lost track of how many times you had shrunk yourself, you wouldn't be able to tell merely by looking around. This is what the power‑law pattern means. The way a frozen potato breaks may be extraordinarily complicated. In fact, the exact pattern of fracturing is different for every potato you hurl at the wall. And yet there is also something about the process that is astonishingly simple, since the resulting pile of shards always has a special property called scale-invariance or self‑similarity.

The landscape of rubble looks the same at all scales, as if every part were a tiny image of the whole. Another way to put it is that in the pile of shards, there is no one scale that is 'preferred' over another. This is a very special situation indeed. Chickens never lay eggs as large as a basketball or as small as a dust mite: the design of a chicken gives it an in‑built bias towards producing eggs falling on a bell curve around a familiar, typical, normal size. But in the fracturing process that produces frozen‑potato shards, there is no bias: the physics is naturally suited to producing fragments over a tremendous range of scales (although there are, of course, limits: you never find fragments bigger than whole potatoes, or smaller than single atoms).

So the power‑law pattern says that there is no such thing as a normal or typical fragment. This is the important point. It is worth mentioning one slightly technical point, however, since there is a very simple way to find out when this power‑law pattern is at work. In algebra, a power law is any curve for which the height changes in proportion to the horizontal distance raised to some power; that is, multiplied by itself a certain number of times. For example, the equation - height = (distance)2 - represents a curve that bends upwards ever more steeply. This is a power law with power equal to 2. In the case of earthquakes, if we think in terms of energy rather than magnitude, the Gutenberg‑Richter curve says that the number of earthquakes having energy equal to some value E is inversely proportional to E raised to the power 2, or E2. This power law captures the simple pattern we've been talking about: each time you double the energy, the earthquakes being considered become four (that is, two squared) times as rare.

The way to find whether a power‑law pattern is at work is simply to make a graph of the distribution of whatever it is you are interested in, and then see if the curve has this algebraic power‑law form. If it does, then you are dealing with something for which the words 'normal', 'typical', 'abnormal' and 'exceptional' do not apply. Power laws always have this implication, whatever the subject being considered. And this brings us to the most important conclusion that follows from Gutenberg's and Richter's observation.

Scale invariance in the pile of potato shards implies that a large shard is simply a scaled‑up version of a small one. Shards of all sizes result from a fracture process that works the same way at all scales. The Gutenberg­-Richter law implies the same thing for earthquakes, and the process in the earth's crust that generates them. Because earthquakes are distributed in energy according to a power law, that distribution is scale invariant. There is nothing whatsoever about the large quakes to suggest they have an origin different from that of the small quakes. In the absence of any other reason for suggesting that the really large quakes are special, the paradoxical implication is that what triggers small and large quakes is precisely the same. From this perspective, it does not make sense to look for special explanations for the largest earthquakes. They are no more special or unusual than the tiny shudders constantly rippling beneath our feet.

It is absolutely crucial to realise that we cannot draw this conclusion from any mathematical form other than a power law. But a power law demands it. In view of the Gutenberg‑Richter power law, it is extremely unlikely that the project of mega‑quake prediction is possible. Indeed, the entire project of earthquake prediction may be fundamentally mis­guided, and practically impossible. This is not to say that a science of earthquakes is impossible. In Chapter 5, we shall see how the Gutenberg­ Richter law can be explained, and the fascinating new direction in which earthquake science is now heading. And we shall explore in some detail the precise meaning of the odd truth that 'great earthquakes happen for no reason at all'.

Before turning to these matters, however, we should step back for a moment to gain some perspective. In the early 1980s, scientists in all fields simply were not aware of the deep significance of the power law. Since then a quiet revolution has been overturning traditional perspectives in literally hundreds of scientific areas.

The intellectual historian Isaiah Berlin once described the history of thought and culture as 'a changing pattern of great liberating ideas which inevitably turn into suffocating straitjackets'. Any idea, no matter how beautiful, unprecedented, powerful and flexible, ultimately runs up against limits. In London, in the spring of 1686, Isaac Newton presented the first book of his Principia to the Royal Society. The manuscript contained his insights concerning the law of gravitation and the general laws of motion, and provided the map for more than two centuries of clear scientific sailing.

But by 1900, the spirit of determinism implicit in Newton's ideas had fixed a debilitating stranglehold on the scientific imagination. To physicists grappling with the mysteries of the atomic world, New­tonian physics was mostly a hindrance, although ultimately a few sci­entists managed heroically to wrestle themselves free, and to create the new quantum theory. 'Don't ask where it came from,' the American physicist Richard Feynman once said of the famous wave equation invented by Erwin Schrödinger, 'it came out of Schrödinger's head.' Along with the ideas of Niels Bohr, Werner Heisenberg and Paul Dirac, Schrödinger's creation liberated scientists once again, enabling them to see the world as if transformed. By the late 1920s, a world of confusing and inconsistent facts had again fallen naturally into place.

More recently, a mathematician working for IBM kicked off a trans­formation of comparable scope. In 1963, Benoit Mandelbrot was study­ing the patterns of ups and downs in the price of cotton on the Chicago mercantile exchange. Cotton prices rise and fall irregularly, and a record of prices over a few months looks like the silhouette of a wildly undu­lating mountain landscape. Even so, Mandelbrot thought he could spy a hidden order within these fluctuations. He noticed that the record showed price fluctuations on all time scales: there were rapid ups and downs occurring daily, or even from hour to hour or minute to minute, and slower, more gradual trends extending over weeks or months. This alone was not terribly surprising or illuminating. But Mandelbrot also discovered that if he took a small section of the record, over just one day, for example, and stretched it out so that it became as long as the entire record, it looked much like the whole. The rapid fluctuations, that is, seemed to be just like the longer‑term fluctuations, only stuffed into a shorter interval.

A less perceptive scientist might simply have recognised this as an oddity, and moved on to something else. Mandelbrot didn't. In looking at fluctuations in the prices of other commodities, such as gold or wheat, he found the same pattern. And he found it again in the way the values of stocks and bonds went up and down: each small fragment of the record seemed to be like a rough copy of the whole.

In the early 1970s, Mandelbrot altered the focus of his attention. Leaving the tumult of the markets far behind, he plunged instead into nature, and into the details of branching networks of rivers and streams. It is hard to imagine anything further removed from finance, and yet here too Mandelbrot stumbled over the same pattern. He noticed that in an aerial snapshot of the network of waterways that ultimately feed into the Mississippi, for example, any small portion of the photo, when magnified, looks much like the entire picture. Over the next decade, Mandelbrot spent countless hours in libraries and in conversation with scientists in an immense range of fields, following the intellectual trail that led away from his curious finding about cotton prices. He came upon similar patterns in the irregular shapes of mountain landscapes, in the wispy forms of clouds, in the jagged edges of a broken piece of glass and the rough surface of a shattered brick, and in the natural, erratic shapes of coastlines, trees, and so on.

For centuries, these irregular shapes had seemed to defy all scientific description; indeed; they seemed to lie largely beyond the grasp of mathematical science and in defiance of the very precepts of ordinary geometry. In a landmark book published in 1983,4 however, Mandelbrot invented a new kind of geometry, and so pulled the blinkers off scientists' eyes. Once you know how to look at these things, he pointed out, they often turn out to be very simple. The key lies in the notion that we met in our pile of potato fragments: self-similarity.

Scientists have now discovered self‑similarity in things ranging from the craters that scar the moon's surface to the plankton floating in the oceans, even in the way the human heart beats. If you think that your heart idles like the engine of a well‑tuned automobile, think again. A few years ago, cardiologist Ary Goldberger of Harvard Medical School and physicists from nearby Boston University recorded a volunteer's heartbeat for a whole day, and then subjected the data to intense math­ematical scrutiny. First they computed the time between each successive pair of heartbeats, arriving at a string of about one hundred thousand numbers. The larger numbers in each string signified moments when the heart went more slowly and left more time between beats; the smaller numbers marked times when the heart went faster.

Looking at those strings of numbers, Goldberger and his colleagues noticed something very much like the pattern Mandelbrot had seen in financial price records. There were fluctuations up and down that took place over hours, others over minutes, and still others over seconds (Figure 3). The human heart, it seems, is almost never content to do the same thing, but is constantly altering its rate. Goldberger and his colleagues found that there is a degree of self‑similarity at work here too. If you only snip out a fragment and blow it up, you find that the changes going on over seconds look much like those taking place more gradually over minutes or hours. There is a strange order lurking behind this irregularity, even if it is very far from the kind of order science tradi­tionally deals in.

There is, of course, a name for this kind of order. Mandelbrot invented the name himself, and so kicked off one of the most important scientific movements of recent years: the study of fractals.







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