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



pg 68

Keywords: ubiquitous patterns of change and organisation - fractals and power laws are at work in all these settings, very possibly because the critical state underlies their dynamics - critical state – organisation – heat: the temperature of anything measures how much disorganised energy there is in it –

phase transition - critical point - At some intermediate temperature, the forces of order and chaos must fight to a stalemate. This is known as the critical point, and in iron, it occurs at 770 degrees Celsius - not only that the critical state and the rise of factions appear in every phase transition, but that the precise mathematical character of this state depends in almost no way on the details of the things involved - basic geometrical issue of how easy it is for an ordering influence at one point to bring similar order to another nearby. This is not physics, but geometry - critical state universality - 'universality classes' - Things that live in critical states tend to show similar kinds of organisation, and this organisation arises not from specific details of those systems and the elements that make them up, but from the far deeper skeleton of basic geometry and logic behind these details. The critical form wells up in things regardless of what they are. So when something is recognised to be in a critical state, its essential character can be understood even by ignoring most of the details -



World Out of Balance

Ultimately, this book is about ubiquitous patterns of change and organisation that run through our world at all levels. I have begun with earthquakes and discussed them at some length only to illustrate a way of thinking that we will soon meet in other settings. When it comes to earthquakes, disastrous episodes of financial collapse, revolutions or catastrophic wars, we all quite understandably long to identify the causes that make these things happen, so that we may avoid them in the future. We shall soon find that fractals and power laws are at work in all these settings, very possibly because the critical state underlies their dynamics. As a result, our human longing for explanation may be terribly misplaced, and doomed always to go unsatisfied. If our world is at all times tuned to be on the edge of sudden, radical change, then these and other upheavals may all be strictly unavoidable and unforeseeable, even just moments before they strike. . .

pg 70
…We might try to wade ever more deeply into the details of specific earthquakes, hoping to tease out which really matter and which don't. But the wealth of detail is nearly infinite, and, fortunately, there is another way to answer the objections about the wilful neglect of reality. I have so far been writing as if the critical state, which seems to lie behind the remarkable behaviour of both the sand‑pile game and earth­quakes alike, was discovered by Bak, Tang and Weisenfeld in 1987. This is hardly the truth. They recognised the critical state in their game, and drew conclusions from their discovery. But the science of the critical state has roots going back several centuries. These roots dig deeply into the physics of seemingly mundane things ‑such as the workings of clunky iron magnets and the molecular details of how liquid water turns into a vapour when heated.


pg 72
In the year 1600, an English physician and scientist named William Gilbert published a monumental treatise entitled De Magnete, then the most comprehensive study ever compiled on the properties of ordinary iron magnets. Gilbert's magnets could pick up nails and tug on swords and horseshoes, and would attract or repel one another depending on how they were oriented. In his book, he also reported one other less obvious detail. Gilbert had put one of his magnets inside an iron­monger's furnace, where it grew hot and glowed orange, and, to his surprise, lost its ability to attract nails. The extreme heat seemed to nullify a magnet's power.

The all‑devouring jaws, one would think, would have swallowed up any mysteries about clunky iron magnets many years ago. Not true. The first decent explanation for Gilbert's observation only appeared in 1907, more than 300 years later. It took another forty years for physicists to learn why this first theory was actually quite erroneous, and another thirty to replace it with a better one. All in all, science took nearly four centuries to consume the iron magnet. Yet in unravelling the mystery, physicists learned a profound lesson: the world is simpler than it seems. And that when it comes to understanding some things, the details most certainly do not matter.

Coming to Order

Every atom in a chunk of iron is itself a tiny magnet, and can point in any direction: up, down, left, right, etc. So you might imagine the inside of a piece of iron as an army of arrows. Physicists knew even a century ago that whether a piece of iron is magnetic or not has something to do with the organisation of this army. The iron might be sitting on a table at room temperature, or be piping hot in a furnace. The crucial question is: where do all the arrows point?

Being what they are, these atomic magnets would like to line up with one another, and, if left to themselves, they would do so, falling quickly into formation like any well‑disciplined army. But the arrows have a disrupting enemy to contend with: heat. The temperature of anything measures how much disorganised energy there is in it; in warm air, the molecules fly about more violently than they do in cold. In solid iron, the atoms do not fly about, but quiver about fixed positions, the vibrations becoming ever more violent as the iron gets hotter. So while the magnetic forces between the iron atoms try to line them up, heat treats them to a storm of disrupting abuse. There is a war between the forces of order and of chaos, and its outcome determines how a magnet behaves outwardly.

If the iron sits on a table at room temperature, the jostling of the atomic magnets is fairly weak, and they succeed in lining up. The strength of each atomic magnet is extremely tiny, of course, but the number in even a small chunk of iron is well over 1024

(that is, 1,000,000,000,000,000,000,000,000). Working together, this army amounts to something, and the iron can pick up nails. If the iron sits glowing in a furnace, on the other hand, then an annihilating storm of noise will overwhelm the forces of order. A snapshot now would show an army in disarray. In this case, the effects of all the tiny magnets cancel out, and the iron cannot pick up nails.

This is just one example of what physicists refer to as a phase transition. When ice melts in a gin and tonic, or when a puddle evaporates and is lost to the air, these too are phase transitions: each being the trans­formation of a substance from one form or 'phase' to another. In every case, there is a change in the internal workings of the stuff, as its atoms or molecules organise themselves differently. In the case of the magnet, then, the story seems rather simple: when cold, the forces of order win out; when hot, the battle goes the other way and chaos rules.

But this isn't a complete story, for it leaves out one very juicy detail. At some intermediate temperature, the forces of order and chaos must fight to a stalemate. This is known as the critical point, and in iron, it occurs at 770 degrees Celsius. What happens to the army of arrows at this point?

What does it mean for something to be neither organised nor disorganised, but somehow perched on the delicate boundary between the two? The answers to these questions are rather more elusive.

pg 81
In 1965, physicists were staring into the face of an almost unbelievable possibility: not only that the critical state and the rise of factions appear in every phase transition, but that the precise mathematical character of this state depends in almost no way on the details of the things involved. This idea remained a tantalising and half­formed possibility, until in 1970 a young physicist from the University of Chicago put it on more solid ground. If few details seemed to matter, Leo Kadanoff managed to put his finger on those few that really do.

At the critical point, pockets of organisation are just about to break out at any place at any moment, and are continually breaking out, as factions grow and then disappear. How large do the factions grow? How quickly do they dissolve? These questions are all down to the basic geometrical issue of how easy it is for an ordering influence at one point to bring similar order to another nearby. This is not physics, but geometry. In an ordinary magnet in three dimensions, any atomic magnet can reach out to influence its neighbours in three independent directions. In flatland, on the other hand, one of these possible directions has been yanked out of existence.

In studying the critical numbers that pop up in the critical states for different phase transitions, Kadanoff found that the basic physical dimension of the thing in question, of the very space in which it lives, is one of the factors that matters. He also found that only one other detail seems to matter, this being the general shape of the individual elements. In a gas of xenon, for example, each atom is like a tiny billiard ball. It can move around, but it can't point. In a magnet, the atoms are like arrows, and can 'do' more since they can potentially point in lots of directions. When the individual elements have more options, you can imagine that it is harder for order to propagate from one place to another. Sure enough, this detail also affects the precise form of the self­similarity in the critical state.

Incredibly, however, Kadanoff found that nothing else whatsoever seemed to matter. So forget the atomic masses and the electrical charges of the particles involved. Forget whether those particles are atoms of oxygen, nitrogen, krypton, nickel or iron. Forget even whether they are made of single atoms or are more complicated molecules made of several or even a hundred atoms. Forget everything, in fact, about the kinds of particles and how strongly or weakly they interact with one another. None of these details affects the organisation of the critical state even a tiny bit. Physicists refer to this considerable miracle as critical state universality, and it has now been supported by thousands of experiments and computer simulations.

In the critical state, the forces of order and chaos battle to an uneasy balance, neither ever fully winning or losing. And the character of the battle, and the perpetually shifting and changing strife to which it leads, is the same regardless of almost every last detail of the things involved. The physical dimension of the thing in question matters, as does the basic shape of its elements ‑ points, arrows and so on. But nothing else matters.

So let's take a tiny but informative step into the abstract, and imagine the conceptual world of all conceivable substances. This world will divide up naturally into nations. There will be the nation of 'arrow‑like things living in three dimensions', and another of 'point‑like things living in just one dimension', and so on. Physicists refer to these nations as 'universality classes'. The miracle of universality is that any two substances, real or imaginary, that fall into the same class will necessarily have exactly the same critical state organisation, regardless of how utterly dissimilar they may otherwise seem to be.

Critical Thinking

We have taken this brief detour in order to meet the critical state and its peculiar properties. And we are now ready to confront its deepest and most profound implication. For in critical state universality, nature has given scientists an amazing gift.

Since all physical systems fall into universality classes, if you succeed in understanding the critical state in any system from one class, you have immediately understood all systems in that class. But look: the crudest sorts of toy models, even Onsager's, also fall into these very same universality classes. So to understand any real physical system at its critical point, you may as well forget all the real, messy details about that system, and focus instead on the simplest mathematical game belonging to the same universality class. It can be crude, even ridiculously so. It can break the laws of physics, and ignore virtually every detail of the real system, and yet you have a guarantee that it will have exactly the same critical behaviour as the real physical thing, so long as those two crucial dimensions are correct. Even hideously crude models can work exactly like the real thing.

And this brings us back to those objections about the earthquake game. The block and spring model we looked at in Chapter 5 bears hardly any relation to the earth's real crust. No property of even a single real rock enters the model, nor does it respect the real world truth that earthquakes happen on networks of faults, rather than on single faults. To ask again a question we asked then: how can intentional falsification of what is known to be true physics possibly lead to any valuable insight into real earthquakes? If the toy model gives the Gutenberg‑Richter law, is this anything more than meaningless coincidence?

Now we have achieved a rather more sophisticated perspective. For something in a critical state, it is possible to understand its essential organisation while neglecting almost every single detail, as long as we do not neglect a few really crucial details. And in view of the Gutenberg­-Richter power law, and the self‑similar clustering of earthquakes in time, the earth's crust appears to be in a critical state, having no inherent or typical scale in either time or space. This undermines objections over lack of detail. It is indeed possible that the essential workings of the crust really can be understood in terms of incredibly crude models, such as the block and spring model of Burridge and Knopoff, or its modern cousins developed by Bak and Tang and by Olami, Feder and Chris­tensen.

So we have arrived at what we might call an attitude of critical thinking. Things that live in critical states tend to show similar kinds of organisation, and this organisation arises not from specific details of those systems and the elements that make them up, but from the far deeper skeleton of basic geometry and logic behind these details. The critical form wells up in things regardless of what they are. So when something is recognised to be in a critical state, its essential character can be understood even by ignoring most of the details.

We shall soon see that many other things such as economies, ecological communities, even the workings of science itself, may share aspects of this organisation as well. Leaving phase transitions behind and moving on to these systems, we may predict that since critical states come in only a few distinct kinds, the kinds of organisation that can exist in the world are actually quite limited. Things that seem on the surface very different may actually be deeply similar in their organisation.






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